LMFDB - Embedded newform 338.5.d.g.239.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [338,5,Mod(99,338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("338.99"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	338.d (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,16,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$34.9390475223$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 494x^{6} + 90747x^{4} - 7343258x^{2} + 221087161 $ x^8 - 494x^6 + 90747x^4 - 7343258*x^2 + 221087161
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 13^{2} $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			239.3
Root			$12.0941 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	338.239
Dual form			338.5.d.g.99.3

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.00000 - 2.00000i) q^{2} +9.49600 q^{3} -8.00000i q^{4} +(19.2341 - 19.2341i) q^{5} +(18.9920 - 18.9920i) q^{6} +(13.9718 + 13.9718i) q^{7} +(-16.0000 - 16.0000i) q^{8} +9.17400 q^{9} -76.9364i q^{10} +(-23.2740 - 23.2740i) q^{11} -75.9680i q^{12} +55.8871 q^{14} +(182.647 - 182.647i) q^{15} -64.0000 q^{16} -214.962i q^{17} +(18.3480 - 18.3480i) q^{18} +(298.476 - 298.476i) q^{19} +(-153.873 - 153.873i) q^{20} +(132.676 + 132.676i) q^{21} -93.0960 q^{22} -962.105i q^{23} +(-151.936 - 151.936i) q^{24} -114.902i q^{25} -682.060 q^{27} +(111.774 - 111.774i) q^{28} +1544.79 q^{29} -730.588i q^{30} +(-750.040 + 750.040i) q^{31} +(-128.000 + 128.000i) q^{32} +(-221.010 - 221.010i) q^{33} +(-429.924 - 429.924i) q^{34} +537.469 q^{35} -73.3920i q^{36} +(498.332 + 498.332i) q^{37} -1193.90i q^{38} -615.491 q^{40} +(-401.607 + 401.607i) q^{41} +530.704 q^{42} -163.685i q^{43} +(-186.192 + 186.192i) q^{44} +(176.454 - 176.454i) q^{45} +(-1924.21 - 1924.21i) q^{46} +(1733.01 + 1733.01i) q^{47} -607.744 q^{48} -2010.58i q^{49} +(-229.804 - 229.804i) q^{50} -2041.28i q^{51} -4534.85 q^{53} +(-1364.12 + 1364.12i) q^{54} -895.309 q^{55} -447.097i q^{56} +(2834.33 - 2834.33i) q^{57} +(3089.57 - 3089.57i) q^{58} +(551.077 + 551.077i) q^{59} +(-1461.18 - 1461.18i) q^{60} +3948.46 q^{61} +3000.16i q^{62} +(128.177 + 128.177i) q^{63} +512.000i q^{64} -884.039 q^{66} +(-4874.72 + 4874.72i) q^{67} -1719.70 q^{68} -9136.15i q^{69} +(1074.94 - 1074.94i) q^{70} +(3902.47 - 3902.47i) q^{71} +(-146.784 - 146.784i) q^{72} +(2496.58 + 2496.58i) q^{73} +1993.33 q^{74} -1091.11i q^{75} +(-2387.81 - 2387.81i) q^{76} -650.358i q^{77} +6710.05 q^{79} +(-1230.98 + 1230.98i) q^{80} -7219.93 q^{81} +1606.43i q^{82} +(-1361.30 + 1361.30i) q^{83} +(1061.41 - 1061.41i) q^{84} +(-4134.61 - 4134.61i) q^{85} +(-327.370 - 327.370i) q^{86} +14669.3 q^{87} +744.768i q^{88} +(4416.84 + 4416.84i) q^{89} -705.815i q^{90} -7696.84 q^{92} +(-7122.38 + 7122.38i) q^{93} +6932.03 q^{94} -11481.8i q^{95} +(-1215.49 + 1215.49i) q^{96} +(-3419.23 + 3419.23i) q^{97} +(-4021.16 - 4021.16i) q^{98} +(-213.516 - 213.516i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{2} + 18 q^{5} - 10 q^{7} - 128 q^{8} + 396 q^{9} + 378 q^{11} - 40 q^{14} - 600 q^{15} - 512 q^{16} + 792 q^{18} + 1190 q^{19} - 144 q^{20} + 678 q^{21} + 1512 q^{22} - 1404 q^{27} - 80 q^{28}+ \cdots + 55296 q^{99}+O(q^{100}) $ 8 * q + 16 * q^2 + 18 * q^5 - 10 * q^7 - 128 * q^8 + 396 * q^9 + 378 * q^11 - 40 * q^14 - 600 * q^15 - 512 * q^16 + 792 * q^18 + 1190 * q^19 - 144 * q^20 + 678 * q^21 + 1512 * q^22 - 1404 * q^27 - 80 * q^28 + 1464 * q^29 - 368 * q^31 - 1024 * q^32 + 1206 * q^33 + 312 * q^34 + 684 * q^35 + 5296 * q^37 - 576 * q^40 - 696 * q^41 + 2712 * q^42 + 3024 * q^44 - 834 * q^45 + 792 * q^46 + 10896 * q^47 + 1456 * q^50 - 4728 * q^53 - 2808 * q^54 - 13260 * q^55 - 18990 * q^57 + 2928 * q^58 - 1194 * q^59 + 4800 * q^60 + 26340 * q^61 - 5652 * q^63 + 4824 * q^66 - 8314 * q^67 + 1248 * q^68 + 1368 * q^70 + 16830 * q^71 - 6336 * q^72 + 7586 * q^73 + 21184 * q^74 - 9520 * q^76 + 37824 * q^79 - 1152 * q^80 - 34176 * q^81 - 21864 * q^83 + 5424 * q^84 - 2226 * q^85 + 2592 * q^86 + 79416 * q^87 - 12558 * q^89 + 3168 * q^92 - 55416 * q^93 + 43584 * q^94 - 31166 * q^97 - 33728 * q^98 + 55296 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$.

$n$	$171$
$\chi(n)$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.00000	−	2.00000i	0.500000	−	0.500000i
$2$	2.00000	−	2.00000i	0.500000	−	0.500000i
$3$	9.49600			1.05511			0.527556	−	0.849521i	$-0.323109\pi$
$3$	9.49600			1.05511			0.527556	+	0.849521i	$0.323109\pi$
$4$		−	8.00000i		−	0.500000i
$5$	19.2341	−	19.2341i	0.769364	−	0.769364i	−0.208630	−	0.977995i	$-0.566901\pi$
$5$	19.2341	−	19.2341i	0.769364	−	0.769364i	0.977995	+	0.208630i	$0.0669005\pi$
$6$	18.9920	−	18.9920i	0.527556	−	0.527556i
$7$	13.9718	+	13.9718i	0.285138	+	0.285138i	0.835154	−	0.550016i	$-0.185378\pi$
$7$	13.9718	+	13.9718i	0.285138	+	0.285138i	−0.550016	+	0.835154i	$0.685378\pi$
$8$	−16.0000	−	16.0000i	−0.250000	−	0.250000i
$9$	9.17400			0.113259
$10$		−	76.9364i		−	0.769364i
$11$	−23.2740	−	23.2740i	−0.192347	−	0.192347i	0.604362	−	0.796709i	$-0.293428\pi$
$11$	−23.2740	−	23.2740i	−0.192347	−	0.192347i	−0.796709	+	0.604362i	$0.793428\pi$
$12$		−	75.9680i		−	0.527556i
$13$		0			0
$13$		0			0
$14$	55.8871			0.285138
$15$	182.647	−	182.647i	0.811765	−	0.811765i
$16$	−64.0000			−0.250000
$17$		−	214.962i		−	0.743814i	−0.928270	−	0.371907i	$-0.878704\pi$
$17$		−	214.962i		−	0.743814i	0.928270	−	0.371907i	$-0.121296\pi$
$18$	18.3480	−	18.3480i	0.0566296	−	0.0566296i
$19$	298.476	−	298.476i	0.826803	−	0.826803i	−0.160270	−	0.987073i	$-0.551237\pi$
$19$	298.476	−	298.476i	0.826803	−	0.826803i	0.987073	+	0.160270i	$0.0512365\pi$
$20$	−153.873	−	153.873i	−0.384682	−	0.384682i
$21$	132.676	+	132.676i	0.300853	+	0.300853i
$22$	−93.0960			−0.192347
$23$		−	962.105i		−	1.81872i	−0.416005	−	0.909362i	$-0.636570\pi$
$23$		−	962.105i		−	1.81872i	0.416005	−	0.909362i	$-0.363430\pi$
$24$	−151.936	−	151.936i	−0.263778	−	0.263778i
$25$		−	114.902i		−	0.183843i
$26$		0			0
$27$	−682.060			−0.935610
$28$	111.774	−	111.774i	0.142569	−	0.142569i
$29$	1544.79			1.83684			0.918422	−	0.395601i	$-0.129464\pi$
$29$	1544.79			1.83684			0.918422	+	0.395601i	$0.129464\pi$
$30$		−	730.588i		−	0.811765i
$31$	−750.040	+	750.040i	−0.780479	+	0.780479i	−0.979912	−	0.199432i	$-0.936090\pi$
$31$	−750.040	+	750.040i	−0.780479	+	0.780479i	0.199432	+	0.979912i	$0.436090\pi$
$32$	−128.000	+	128.000i	−0.125000	+	0.125000i
$33$	−221.010	−	221.010i	−0.202947	−	0.202947i
$34$	−429.924	−	429.924i	−0.371907	−	0.371907i
$35$	537.469			0.438750
$36$		−	73.3920i		−	0.0566296i
$37$	498.332	+	498.332i	0.364012	+	0.364012i	0.865288	−	0.501276i	$-0.167136\pi$
$37$	498.332	+	498.332i	0.364012	+	0.364012i	−0.501276	+	0.865288i	$0.667136\pi$
$38$		−	1193.90i		−	0.826803i
$39$		0			0
$40$	−615.491			−0.384682
$41$	−401.607	+	401.607i	−0.238910	+	0.238910i	−0.816398	−	0.577489i	$-0.804033\pi$
$41$	−401.607	+	401.607i	−0.238910	+	0.238910i	0.577489	+	0.816398i	$0.304033\pi$
$42$	530.704			0.300853
$43$		−	163.685i		−	0.0885264i	−0.999020	−	0.0442632i	$-0.985906\pi$
$43$		−	163.685i		−	0.0885264i	0.999020	−	0.0442632i	$-0.0140940\pi$
$44$	−186.192	+	186.192i	−0.0961735	+	0.0961735i
$45$	176.454	−	176.454i	0.0871376	−	0.0871376i
$46$	−1924.21	−	1924.21i	−0.909362	−	0.909362i
$47$	1733.01	+	1733.01i	0.784521	+	0.784521i	0.980590	−	0.196069i	$-0.0628177\pi$
$47$	1733.01	+	1733.01i	0.784521	+	0.784521i	−0.196069	+	0.980590i	$0.562818\pi$
$48$	−607.744			−0.263778
$49$		−	2010.58i		−	0.837392i
$50$	−229.804	−	229.804i	−0.0919214	−	0.0919214i
$51$		−	2041.28i		−	0.784806i
$52$		0			0
$53$	−4534.85			−1.61440			−0.807200	−	0.590278i	$-0.799018\pi$
$53$	−4534.85			−1.61440			−0.807200	+	0.590278i	$0.799018\pi$
$54$	−1364.12	+	1364.12i	−0.467805	+	0.467805i
$55$	−895.309			−0.295970
$56$		−	447.097i		−	0.142569i
$57$	2834.33	−	2834.33i	0.872369	−	0.872369i
$58$	3089.57	−	3089.57i	0.918422	−	0.918422i
$59$	551.077	+	551.077i	0.158310	+	0.158310i	0.781817	−	0.623507i	$-0.214293\pi$
$59$	551.077	+	551.077i	0.158310	+	0.158310i	−0.623507	+	0.781817i	$0.714293\pi$
$60$	−1461.18	−	1461.18i	−0.405882	−	0.405882i
$61$	3948.46			1.06113			0.530564	−	0.847645i	$-0.321980\pi$
$61$	3948.46			1.06113			0.530564	+	0.847645i	$0.321980\pi$
$62$			3000.16i			0.780479i
$63$	128.177	+	128.177i	0.0322946	+	0.0322946i
$64$			512.000i			0.125000i
$65$		0			0
$66$	−884.039			−0.202947
$67$	−4874.72	+	4874.72i	−1.08592	+	1.08592i	−0.0899815	+	0.995943i	$0.528681\pi$
$67$	−4874.72	+	4874.72i	−1.08592	+	1.08592i	−0.995943	+	0.0899815i	$0.971319\pi$
$68$	−1719.70			−0.371907
$69$		−	9136.15i		−	1.91896i
$70$	1074.94	−	1074.94i	0.219375	−	0.219375i
$71$	3902.47	−	3902.47i	0.774145	−	0.774145i	−0.204683	−	0.978828i	$-0.565616\pi$
$71$	3902.47	−	3902.47i	0.774145	−	0.774145i	0.978828	+	0.204683i	$0.0656163\pi$
$72$	−146.784	−	146.784i	−0.0283148	−	0.0283148i
$73$	2496.58	+	2496.58i	0.468489	+	0.468489i	0.901425	−	0.432936i	$-0.142522\pi$
$73$	2496.58	+	2496.58i	0.468489	+	0.468489i	−0.432936	+	0.901425i	$0.642522\pi$
$74$	1993.33			0.364012
$75$		−	1091.11i		−	0.193975i
$76$	−2387.81	−	2387.81i	−0.413402	−	0.413402i
$77$		−	650.358i		−	0.109691i
$78$		0			0
$79$	6710.05			1.07516			0.537578	−	0.843214i	$-0.319339\pi$
$79$	6710.05			1.07516			0.537578	+	0.843214i	$0.319339\pi$
$80$	−1230.98	+	1230.98i	−0.192341	+	0.192341i
$81$	−7219.93			−1.10043
$82$			1606.43i			0.238910i
$83$	−1361.30	+	1361.30i	−0.197605	+	0.197605i	−0.798972	−	0.601368i	$-0.794623\pi$
$83$	−1361.30	+	1361.30i	−0.197605	+	0.197605i	0.601368	+	0.798972i	$0.294623\pi$
$84$	1061.41	−	1061.41i	0.150426	−	0.150426i
$85$	−4134.61	−	4134.61i	−0.572264	−	0.572264i
$86$	−327.370	−	327.370i	−0.0442632	−	0.0442632i
$87$	14669.3			1.93808
$88$			744.768i			0.0961735i
$89$	4416.84	+	4416.84i	0.557611	+	0.557611i	0.928627	−	0.371016i	$-0.120990\pi$
$89$	4416.84	+	4416.84i	0.557611	+	0.557611i	−0.371016	+	0.928627i	$0.620990\pi$
$90$		−	705.815i		−	0.0871376i
$91$		0			0
$92$	−7696.84			−0.909362
$93$	−7122.38	+	7122.38i	−0.823492	+	0.823492i
$94$	6932.03			0.784521
$95$		−	11481.8i		−	1.27223i
$96$	−1215.49	+	1215.49i	−0.131889	+	0.131889i
$97$	−3419.23	+	3419.23i	−0.363400	+	0.363400i	−0.865063	−	0.501663i	$-0.832722\pi$
$97$	−3419.23	+	3419.23i	−0.363400	+	0.363400i	0.501663	+	0.865063i	$0.332722\pi$
$98$	−4021.16	−	4021.16i	−0.418696	−	0.418696i
$99$	−213.516	−	213.516i	−0.0217851	−	0.0217851i
$100$	−919.214			−0.0919214
$101$		−	8429.89i		−	0.826379i	−0.910645	−	0.413190i	$-0.864415\pi$
$101$		−	8429.89i		−	0.826379i	0.910645	−	0.413190i	$-0.135585\pi$
$102$	−4082.56	−	4082.56i	−0.392403	−	0.392403i
$103$			2398.45i			0.226076i	0.993591	+	0.113038i	$0.0360583\pi$
$103$			2398.45i			0.226076i	−0.993591	+	0.113038i	$0.963942\pi$
$104$		0			0
$105$	5103.81			0.462930
$106$	−9069.70	+	9069.70i	−0.807200	+	0.807200i
$107$	9412.21			0.822099			0.411049	−	0.911613i	$-0.365162\pi$
$107$	9412.21			0.822099			0.411049	+	0.911613i	$0.365162\pi$
$108$			5456.48i			0.467805i
$109$	−14993.7	+	14993.7i	−1.26199	+	1.26199i	−0.311860	+	0.950128i	$0.600952\pi$
$109$	−14993.7	+	14993.7i	−1.26199	+	1.26199i	−0.950128	+	0.311860i	$0.899048\pi$
$110$	−1790.62	+	1790.62i	−0.147985	+	0.147985i
$111$	4732.16	+	4732.16i	0.384073	+	0.384073i
$112$	−894.194	−	894.194i	−0.0712846	−	0.0712846i
$113$	−17564.9			−1.37559			−0.687795	−	0.725905i	$-0.741421\pi$
$113$	−17564.9			−1.37559			−0.687795	+	0.725905i	$0.741421\pi$
$114$		−	11337.3i		−	0.872369i
$115$	−18505.2	−	18505.2i	−1.39926	−	1.39926i
$116$		−	12358.3i		−	0.918422i
$117$		0			0
$118$	2204.31			0.158310
$119$	3003.40	−	3003.40i	0.212090	−	0.212090i
$120$	−5844.71			−0.405882
$121$		−	13557.6i		−	0.926005i
$122$	7896.92	−	7896.92i	0.530564	−	0.530564i
$123$	−3813.66	+	3813.66i	−0.252076	+	0.252076i
$124$	6000.32	+	6000.32i	0.390240	+	0.390240i
$125$	9811.28	+	9811.28i	0.627922	+	0.627922i
$126$	512.708			0.0322946
$127$			16236.5i			1.00667i	0.864092	+	0.503334i	$0.167893\pi$
$127$			16236.5i			1.00667i	−0.864092	+	0.503334i	$0.832107\pi$
$128$	1024.00	+	1024.00i	0.0625000	+	0.0625000i
$129$		−	1554.35i		−	0.0934051i
$130$		0			0
$131$	9149.90			0.533180			0.266590	−	0.963810i	$-0.414103\pi$
$131$	9149.90			0.533180			0.266590	+	0.963810i	$0.414103\pi$
$132$	−1768.08	+	1768.08i	−0.101474	+	0.101474i
$133$	8340.48			0.471506
$134$			19498.9i			1.08592i
$135$	−13118.8	+	13118.8i	−0.719825	+	0.719825i
$136$	−3439.39	+	3439.39i	−0.185953	+	0.185953i
$137$	−8330.28	−	8330.28i	−0.443832	−	0.443832i	0.449466	−	0.893298i	$-0.351614\pi$
$137$	−8330.28	−	8330.28i	−0.443832	−	0.443832i	−0.893298	+	0.449466i	$0.851614\pi$
$138$	−18272.3	−	18272.3i	−0.959478	−	0.959478i
$139$	−2026.81			−0.104902			−0.0524509	−	0.998624i	$-0.516703\pi$
$139$	−2026.81			−0.104902			−0.0524509	+	0.998624i	$0.516703\pi$
$140$		−	4299.75i		−	0.219375i
$141$	16456.6	+	16456.6i	0.827757	+	0.827757i
$142$		−	15609.9i		−	0.774145i
$143$		0			0
$144$	−587.136			−0.0283148
$145$	29712.6	−	29712.6i	1.41320	−	1.41320i
$146$	9986.30			0.468489
$147$		−	19092.5i		−	0.883542i
$148$	3986.66	−	3986.66i	0.182006	−	0.182006i
$149$	16993.1	−	16993.1i	0.765419	−	0.765419i	−0.211877	−	0.977296i	$-0.567958\pi$
$149$	16993.1	−	16993.1i	0.765419	−	0.765419i	0.977296	+	0.211877i	$0.0679577\pi$
$150$	−2182.21	−	2182.21i	−0.0969873	−	0.0969873i
$151$	1109.89	+	1109.89i	0.0486774	+	0.0486774i	0.731026	−	0.682349i	$-0.239042\pi$
$151$	1109.89	+	1109.89i	0.0486774	+	0.0486774i	−0.682349	+	0.731026i	$0.739042\pi$
$152$	−9551.23			−0.413402
$153$		−	1972.06i		−	0.0842438i
$154$	−1300.72	−	1300.72i	−0.0548455	−	0.0548455i
$155$			28852.7i			1.20095i
$156$		0			0
$157$	−17982.1			−0.729525			−0.364762	−	0.931101i	$-0.618850\pi$
$157$	−17982.1			−0.729525			−0.364762	+	0.931101i	$0.618850\pi$
$158$	13420.1	−	13420.1i	0.537578	−	0.537578i
$159$	−43062.9			−1.70337
$160$			4923.93i			0.192341i
$161$	13442.3	−	13442.3i	0.518588	−	0.518588i
$162$	−14439.9	+	14439.9i	−0.550216	+	0.550216i
$163$	32929.1	+	32929.1i	1.23938	+	1.23938i	0.960254	+	0.279126i	$0.0900449\pi$
$163$	32929.1	+	32929.1i	1.23938	+	1.23938i	0.279126	+	0.960254i	$0.409955\pi$
$164$	3212.86	+	3212.86i	0.119455	+	0.119455i
$165$	−8501.85			−0.312281
$166$			5445.20i			0.197605i
$167$	12360.2	+	12360.2i	0.443194	+	0.443194i	0.893084	−	0.449890i	$-0.148537\pi$
$167$	12360.2	+	12360.2i	0.443194	+	0.443194i	−0.449890	+	0.893084i	$0.648537\pi$
$168$		−	4245.63i		−	0.150426i
$169$		0			0
$170$	−16538.4			−0.572264
$171$	2738.22	−	2738.22i	0.0936431	−	0.0936431i
$172$	−1309.48			−0.0442632
$173$			32312.5i			1.07964i	0.841781	+	0.539820i	$0.181508\pi$
$173$			32312.5i			1.07964i	−0.841781	+	0.539820i	$0.818492\pi$
$174$	29338.6	−	29338.6i	0.969038	−	0.969038i
$175$	1605.38	−	1605.38i	0.0524206	−	0.0524206i
$176$	1489.54	+	1489.54i	0.0480868	+	0.0480868i
$177$	5233.02	+	5233.02i	0.167034	+	0.167034i
$178$	17667.4			0.557611
$179$			13337.3i			0.416258i	0.978101	+	0.208129i	$0.0667374\pi$
$179$			13337.3i			0.416258i	−0.978101	+	0.208129i	$0.933263\pi$
$180$	−1411.63	−	1411.63i	−0.0435688	−	0.0435688i
$181$			46226.0i			1.41101i	0.708706	+	0.705504i	$0.249279\pi$
$181$			46226.0i			1.41101i	−0.708706	+	0.705504i	$0.750721\pi$
$182$		0			0
$183$	37494.6			1.11961
$184$	−15393.7	+	15393.7i	−0.454681	+	0.454681i
$185$	19170.0			0.560116
$186$			28489.5i			0.823492i
$187$	−5003.03	+	5003.03i	−0.143070	+	0.143070i
$188$	13864.1	−	13864.1i	0.392260	−	0.392260i
$189$	−9529.58	−	9529.58i	−0.266778	−	0.266778i
$190$	−22963.7	−	22963.7i	−0.636113	−	0.636113i
$191$	11917.0			0.326664			0.163332	−	0.986571i	$-0.447776\pi$
$191$	11917.0			0.326664			0.163332	+	0.986571i	$0.447776\pi$
$192$			4861.95i			0.131889i
$193$	29776.3	+	29776.3i	0.799385	+	0.799385i	0.982999	−	0.183614i	$-0.0587795\pi$
$193$	29776.3	+	29776.3i	0.799385	+	0.799385i	−0.183614	+	0.982999i	$0.558779\pi$
$194$			13676.9i			0.363400i
$195$		0			0
$196$	−16084.6			−0.418696
$197$	39055.8	−	39055.8i	1.00636	−	1.00636i	0.00637973	−	0.999980i	$-0.497969\pi$
$197$	39055.8	−	39055.8i	1.00636	−	1.00636i	0.999980	−	0.00637973i	$-0.00203074\pi$
$198$	−854.062			−0.0217851
$199$			67507.8i			1.70470i	0.522971	+	0.852350i	$0.324823\pi$
$199$			67507.8i			1.70470i	−0.522971	+	0.852350i	$0.675177\pi$
$200$	−1838.43	+	1838.43i	−0.0459607	+	0.0459607i
$201$	−46290.3	+	46290.3i	−1.14577	+	1.14577i
$202$	−16859.8	−	16859.8i	−0.413190	−	0.413190i
$203$	21583.4	+	21583.4i	0.523755	+	0.523755i
$204$	−16330.2			−0.392403
$205$			15449.1i			0.367617i
$206$	4796.89	+	4796.89i	0.113038	+	0.113038i
$207$		−	8826.35i		−	0.205987i
$208$		0			0
$209$	−13893.5			−0.318066
$210$	10207.6	−	10207.6i	0.231465	−	0.231465i
$211$	−4367.34			−0.0980961			−0.0490481	−	0.998796i	$-0.515619\pi$
$211$	−4367.34			−0.0980961			−0.0490481	+	0.998796i	$0.515619\pi$
$212$			36278.8i			0.807200i
$213$	37057.8	−	37057.8i	0.816809	−	0.816809i
$214$	18824.4	−	18824.4i	0.411049	−	0.411049i
$215$	−3148.34	−	3148.34i	−0.0681090	−	0.0681090i
$216$	10913.0	+	10913.0i	0.233902	+	0.233902i
$217$	−20958.8			−0.445089
$218$			59974.7i			1.26199i
$219$	23707.5	+	23707.5i	0.494307	+	0.494307i
$220$			7162.47i			0.147985i
$221$		0			0
$222$	18928.7			0.384073
$223$	16161.8	−	16161.8i	0.324998	−	0.324998i	−0.525683	−	0.850681i	$-0.676190\pi$
$223$	16161.8	−	16161.8i	0.324998	−	0.324998i	0.850681	+	0.525683i	$0.176190\pi$
$224$	−3576.77			−0.0712846
$225$		−	1054.11i		−	0.0208219i
$226$	−35129.8	+	35129.8i	−0.687795	+	0.687795i
$227$	−31231.7	+	31231.7i	−0.606099	+	0.606099i	−0.941924	−	0.335825i	$-0.890985\pi$
$227$	−31231.7	+	31231.7i	−0.606099	+	0.606099i	0.335825	+	0.941924i	$0.390985\pi$
$228$	−22674.6	−	22674.6i	−0.436185	−	0.436185i
$229$	10025.3	+	10025.3i	0.191172	+	0.191172i	0.796202	−	0.605030i	$-0.206839\pi$
$229$	10025.3	+	10025.3i	0.191172	+	0.191172i	−0.605030	+	0.796202i	$0.706839\pi$
$230$	−74020.9			−1.39926
$231$		−	6175.80i		−	0.115736i
$232$	−24716.6	−	24716.6i	−0.459211	−	0.459211i
$233$		−	3033.83i		−	0.0558829i	−0.999610	−	0.0279415i	$-0.991105\pi$
$233$		−	3033.83i		−	0.0558829i	0.999610	−	0.0279415i	$-0.00889520\pi$
$234$		0			0
$235$	66665.7			1.20716
$236$	4408.61	−	4408.61i	0.0791549	−	0.0791549i
$237$	63718.6			1.13441
$238$		−	12013.6i		−	0.212090i
$239$	27446.8	−	27446.8i	0.480503	−	0.480503i	−0.424789	−	0.905292i	$-0.639652\pi$
$239$	27446.8	−	27446.8i	0.480503	−	0.480503i	0.905292	+	0.424789i	$0.139652\pi$
$240$	−11689.4	+	11689.4i	−0.202941	+	0.202941i
$241$	40981.9	+	40981.9i	0.705600	+	0.705600i	0.965607	−	0.260007i	$-0.0837249\pi$
$241$	40981.9	+	40981.9i	0.705600	+	0.705600i	−0.260007	+	0.965607i	$0.583725\pi$
$242$	−27115.3	−	27115.3i	−0.463003	−	0.463003i
$243$	−13313.6			−0.225468
$244$		−	31587.7i		−	0.530564i
$245$	−38671.7	−	38671.7i	−0.644260	−	0.644260i
$246$			15254.6i			0.252076i
$247$		0			0
$248$	24001.3			0.390240
$249$	−12926.9	+	12926.9i	−0.208495	+	0.208495i
$250$	39245.1			0.627922
$251$		−	55670.2i		−	0.883639i	−0.897104	−	0.441820i	$-0.854333\pi$
$251$		−	55670.2i		−	0.883639i	0.897104	−	0.441820i	$-0.145667\pi$
$252$	1025.42	−	1025.42i	0.0161473	−	0.0161473i
$253$	−22392.0	+	22392.0i	−0.349826	+	0.349826i
$254$	32473.1	+	32473.1i	0.503334	+	0.503334i
$255$	−39262.2	−	39262.2i	−0.603802	−	0.603802i
$256$	4096.00			0.0625000
$257$		−	19950.7i		−	0.302059i	−0.988529	−	0.151030i	$-0.951741\pi$
$257$		−	19950.7i		−	0.302059i	0.988529	−	0.151030i	$-0.0482589\pi$
$258$	−3108.71	−	3108.71i	−0.0467026	−	0.0467026i
$259$			13925.2i			0.207588i
$260$		0			0
$261$	14171.9			0.208040
$262$	18299.8	−	18299.8i	0.266590	−	0.266590i
$263$	−118460.			−1.71261			−0.856305	−	0.516470i	$-0.827246\pi$
$263$	−118460.			−1.71261			−0.856305	+	0.516470i	$0.827246\pi$
$264$			7072.31i			0.101474i
$265$	−87223.8	+	87223.8i	−1.24206	+	1.24206i
$266$	16681.0	−	16681.0i	0.235753	−	0.235753i
$267$	41942.3	+	41942.3i	0.588342	+	0.588342i
$268$	38997.7	+	38997.7i	0.542962	+	0.542962i
$269$	−12860.0			−0.177720			−0.0888600	−	0.996044i	$-0.528322\pi$
$269$	−12860.0			−0.177720			−0.0888600	+	0.996044i	$0.528322\pi$
$270$			52475.2i			0.719825i
$271$	−11279.4	−	11279.4i	−0.153585	−	0.153585i	0.626132	−	0.779717i	$-0.284637\pi$
$271$	−11279.4	−	11279.4i	−0.153585	−	0.153585i	−0.779717	+	0.626132i	$0.784637\pi$
$272$			13757.6i			0.185953i
$273$		0			0
$274$	−33321.1			−0.443832
$275$	−2674.22	+	2674.22i	−0.0353616	+	0.0353616i
$276$	−73089.2			−0.959478
$277$			60933.3i			0.794136i	0.917789	+	0.397068i	$0.129972\pi$
$277$			60933.3i			0.794136i	−0.917789	+	0.397068i	$0.870028\pi$
$278$	−4053.62	+	4053.62i	−0.0524509	+	0.0524509i
$279$	−6880.87	+	6880.87i	−0.0883965	+	0.0883965i
$280$	−8599.51	−	8599.51i	−0.109688	−	0.109688i
$281$	30774.2	+	30774.2i	0.389739	+	0.389739i	0.874594	−	0.484855i	$-0.161128\pi$
$281$	30774.2	+	30774.2i	0.389739	+	0.389739i	−0.484855	+	0.874594i	$0.661128\pi$
$282$	65826.5			0.827757
$283$		−	110848.i		−	1.38407i	−0.721866	−	0.692033i	$-0.756715\pi$
$283$		−	110848.i		−	1.38407i	0.721866	−	0.692033i	$-0.243285\pi$
$284$	−31219.7	−	31219.7i	−0.387073	−	0.387073i
$285$		−	109032.i		−	1.34234i
$286$		0			0
$287$	−11222.3			−0.136245
$288$	−1174.27	+	1174.27i	−0.0141574	+	0.0141574i
$289$	37312.3			0.446741
$290$		−	118850.i		−	1.41320i
$291$	−32469.0	+	32469.0i	−0.383428	+	0.383428i
$292$	19972.6	−	19972.6i	0.234244	−	0.234244i
$293$	23920.9	+	23920.9i	0.278639	+	0.278639i	0.832566	−	0.553926i	$-0.186871\pi$
$293$	23920.9	+	23920.9i	0.278639	+	0.278639i	−0.553926	+	0.832566i	$0.686871\pi$
$294$	−38184.9	−	38184.9i	−0.441771	−	0.441771i
$295$	21198.9			0.243596
$296$		−	15946.6i		−	0.182006i
$297$	15874.2	+	15874.2i	0.179962	+	0.179962i
$298$		−	67972.3i		−	0.765419i
$299$		0			0
$300$	−8728.86			−0.0969873
$301$	2286.97	−	2286.97i	0.0252423	−	0.0252423i
$302$	4439.57			0.0486774
$303$		−	80050.3i		−	0.871922i
$304$	−19102.5	+	19102.5i	−0.206701	+	0.206701i
$305$	75945.1	−	75945.1i	0.816395	−	0.816395i
$306$	−3944.13	−	3944.13i	−0.0421219	−	0.0421219i
$307$	−98055.5	−	98055.5i	−1.04039	−	1.04039i	−0.999149	−	0.0412382i	$-0.986870\pi$
$307$	−98055.5	−	98055.5i	−1.04039	−	1.04039i	−0.0412382	−	0.999149i	$-0.513130\pi$
$308$	−5202.86			−0.0548455
$309$			22775.6i			0.238536i
$310$	57705.4	+	57705.4i	0.600473	+	0.600473i
$311$			68074.1i			0.703819i	0.936034	+	0.351910i	$0.114468\pi$
$311$			68074.1i			0.703819i	−0.936034	+	0.351910i	$0.885532\pi$
$312$		0			0
$313$	−55684.0			−0.568383			−0.284192	−	0.958767i	$-0.591725\pi$
$313$	−55684.0			−0.568383			−0.284192	+	0.958767i	$0.591725\pi$
$314$	−35964.1	+	35964.1i	−0.364762	+	0.364762i
$315$	4930.74			0.0496926
$316$		−	53680.4i		−	0.537578i
$317$	23137.2	−	23137.2i	0.230246	−	0.230246i	−0.582549	−	0.812795i	$-0.697945\pi$
$317$	23137.2	−	23137.2i	0.230246	−	0.230246i	0.812795	+	0.582549i	$0.197945\pi$
$318$	−86125.8	+	86125.8i	−0.851685	+	0.851685i
$319$	−35953.3	−	35953.3i	−0.353312	−	0.353312i
$320$	9847.86	+	9847.86i	0.0961705	+	0.0961705i
$321$	89378.4			0.867406
$322$		−	53769.3i		−	0.518588i
$323$	−64161.0	−	64161.0i	−0.614988	−	0.614988i
$324$			57759.5i			0.550216i
$325$		0			0
$326$	131716.			1.23938
$327$	−142380.	+	142380.i	−1.33154	+	1.33154i
$328$	12851.4			0.119455
$329$			48426.4i			0.447394i
$330$	−17003.7	+	17003.7i	−0.156141	+	0.156141i
$331$	54201.8	−	54201.8i	0.494718	−	0.494718i	−0.415071	−	0.909789i	$-0.636243\pi$
$331$	54201.8	−	54201.8i	0.494718	−	0.494718i	0.909789	+	0.415071i	$0.136243\pi$
$332$	10890.4	+	10890.4i	0.0988024	+	0.0988024i
$333$	4571.70	+	4571.70i	0.0412277	+	0.0412277i
$334$	49441.0			0.443194
$335$			187522.i			1.67094i
$336$	−8491.26	−	8491.26i	−0.0752131	−	0.0752131i
$337$			15606.4i			0.137418i	0.997637	+	0.0687090i	$0.0218880\pi$
$337$			15606.4i			0.137418i	−0.997637	+	0.0687090i	$0.978112\pi$
$338$		0			0
$339$	−166796.			−1.45140
$340$	−33076.8	+	33076.8i	−0.286132	+	0.286132i
$341$	34912.9			0.300246
$342$		−	10952.9i		−	0.0936431i
$343$	61637.6	−	61637.6i	0.523911	−	0.523911i
$344$	−2618.96	+	2618.96i	−0.0221316	+	0.0221316i
$345$	−175726.	−	175726.i	−1.47638	−	1.47638i
$346$	64625.0	+	64625.0i	0.539820	+	0.539820i
$347$	7141.58			0.0593110			0.0296555	−	0.999560i	$-0.490559\pi$
$347$	7141.58			0.0593110			0.0296555	+	0.999560i	$0.490559\pi$
$348$		−	117354.i		−	0.969038i
$349$	−127134.	−	127134.i	−1.04378	−	1.04378i	−0.998997	−	0.0447840i	$-0.985740\pi$
$349$	−127134.	−	127134.i	−1.04378	−	1.04378i	−0.0447840	−	0.998997i	$-0.514260\pi$
$350$		−	6421.53i		−	0.0524206i
$351$		0			0
$352$	5958.14			0.0480868
$353$	−148458.	+	148458.i	−1.19139	+	1.19139i	−0.214714	+	0.976677i	$0.568882\pi$
$353$	−148458.	+	148458.i	−1.19139	+	1.19139i	−0.976677	+	0.214714i	$0.931118\pi$
$354$	20932.1			0.167034
$355$		−	150121.i		−	1.19120i
$356$	35334.7	−	35334.7i	0.278806	−	0.278806i
$357$	28520.3	−	28520.3i	0.223778	−	0.223778i
$358$	26674.6	+	26674.6i	0.208129	+	0.208129i
$359$	111995.	+	111995.i	0.868976	+	0.868976i	0.992359	−	0.123383i	$-0.0393744\pi$
$359$	111995.	+	111995.i	0.868976	+	0.868976i	−0.123383	+	0.992359i	$0.539374\pi$
$360$	−5646.52			−0.0435688
$361$		−	47854.8i		−	0.367207i
$362$	92452.1	+	92452.1i	0.705504	+	0.705504i
$363$		−	128743.i		−	0.977038i
$364$		0			0
$365$	96038.8			0.720877
$366$	74989.2	−	74989.2i	0.559804	−	0.559804i
$367$	58977.6			0.437880			0.218940	−	0.975738i	$-0.429740\pi$
$367$	58977.6			0.437880			0.218940	+	0.975738i	$0.429740\pi$
$368$			61574.7i			0.454681i
$369$	−3684.34	+	3684.34i	−0.0270587	+	0.0270587i
$370$	38339.9	−	38339.9i	0.280058	−	0.280058i
$371$	−63359.9	−	63359.9i	−0.460327	−	0.460327i
$372$	56979.1	+	56979.1i	0.411746	+	0.411746i
$373$	−55999.7			−0.402502			−0.201251	−	0.979540i	$-0.564501\pi$
$373$	−55999.7			−0.402502			−0.201251	+	0.979540i	$0.564501\pi$
$374$			20012.1i			0.143070i
$375$	93167.9	+	93167.9i	0.662528	+	0.662528i
$376$		−	55456.2i		−	0.392260i
$377$		0			0
$378$	−38118.3			−0.266778
$379$	74798.6	−	74798.6i	0.520733	−	0.520733i	−0.397060	−	0.917793i	$-0.629969\pi$
$379$	74798.6	−	74798.6i	0.520733	−	0.520733i	0.917793	+	0.397060i	$0.129969\pi$
$380$	−91854.7			−0.636113
$381$			154182.i			1.06215i
$382$	23834.0	−	23834.0i	0.163332	−	0.163332i
$383$	139190.	−	139190.i	0.948879	−	0.948879i	−0.0498768	−	0.998755i	$-0.515883\pi$
$383$	139190.	−	139190.i	0.948879	−	0.948879i	0.998755	+	0.0498768i	$0.0158829\pi$
$384$	9723.90	+	9723.90i	0.0659444	+	0.0659444i
$385$	−12509.1	−	12509.1i	−0.0843923	−	0.0843923i
$386$	119105.			0.799385
$387$		−	1501.65i		−	0.0100264i
$388$	27353.9	+	27353.9i	0.181700	+	0.181700i
$389$		−	178816.i		−	1.18170i	−0.806780	−	0.590852i	$-0.798792\pi$
$389$		−	178816.i		−	1.18170i	0.806780	−	0.590852i	$-0.201208\pi$
$390$		0			0
$391$	−206816.			−1.35279
$392$	−32169.3	+	32169.3i	−0.209348	+	0.209348i
$393$	86887.5			0.562564
$394$		−	156223.i		−	1.00636i
$395$	129062.	−	129062.i	0.827186	−	0.827186i
$396$	−1708.12	+	1708.12i	−0.0108925	+	0.0108925i
$397$	−179758.	−	179758.i	−1.14053	−	1.14053i	−0.988353	−	0.152178i	$-0.951371\pi$
$397$	−179758.	−	179758.i	−1.14053	−	1.14053i	−0.152178	−	0.988353i	$-0.548629\pi$
$398$	135016.	+	135016.i	0.852350	+	0.852350i
$399$	79201.2			0.497492
$400$			7353.71i			0.0459607i
$401$	−31188.8	−	31188.8i	−0.193959	−	0.193959i	0.603445	−	0.797404i	$-0.293794\pi$
$401$	−31188.8	−	31188.8i	−0.193959	−	0.193959i	−0.797404	+	0.603445i	$0.793794\pi$
$402$			185161.i			1.14577i
$403$		0			0
$404$	−67439.1			−0.413190
$405$	−138869.	+	138869.i	−0.846633	+	0.846633i
$406$	86333.6			0.523755
$407$		−	23196.4i		−	0.140033i
$408$	−32660.5	+	32660.5i	−0.196202	+	0.196202i
$409$	−85034.0	+	85034.0i	−0.508330	+	0.508330i	−0.914014	−	0.405684i	$-0.867034\pi$
$409$	−85034.0	+	85034.0i	−0.508330	+	0.508330i	0.405684	+	0.914014i	$0.367034\pi$
$410$	30898.2	+	30898.2i	0.183808	+	0.183808i
$411$	−79104.3	−	79104.3i	−0.468292	−	0.468292i
$412$	19187.6			0.113038
$413$			15399.0i			0.0902804i
$414$	−17652.7	−	17652.7i	−0.102994	−	0.102994i
$415$			52366.8i			0.304060i
$416$		0			0
$417$	−19246.6			−0.110683
$418$	−27786.9	+	27786.9i	−0.159033	+	0.159033i
$419$	−161655.			−0.920789			−0.460394	−	0.887714i	$-0.652292\pi$
$419$	−161655.			−0.920789			−0.460394	+	0.887714i	$0.652292\pi$
$420$		−	40830.5i		−	0.231465i
$421$	−80144.5	+	80144.5i	−0.452178	+	0.452178i	−0.896077	−	0.443899i	$-0.853595\pi$
$421$	−80144.5	+	80144.5i	−0.452178	+	0.452178i	0.443899	+	0.896077i	$0.353595\pi$
$422$	−8734.68	+	8734.68i	−0.0490481	+	0.0490481i
$423$	15898.6	+	15898.6i	0.0888543	+	0.0888543i
$424$	72557.6	+	72557.6i	0.403600	+	0.403600i
$425$	−24699.5			−0.136745
$426$		−	148231.i		−	0.816809i
$427$	55167.0	+	55167.0i	0.302568	+	0.302568i
$428$		−	75297.7i		−	0.411049i
$429$		0			0
$430$	−12593.4			−0.0681090
$431$	61470.4	−	61470.4i	0.330911	−	0.330911i	−0.522021	−	0.852932i	$-0.674822\pi$
$431$	61470.4	−	61470.4i	0.330911	−	0.330911i	0.852932	+	0.522021i	$0.174822\pi$
$432$	43651.8			0.233902
$433$		−	215352.i		−	1.14861i	−0.818641	−	0.574305i	$-0.805272\pi$
$433$		−	215352.i		−	1.14861i	0.818641	−	0.574305i	$-0.194728\pi$
$434$	−41917.6	+	41917.6i	−0.222544	+	0.222544i
$435$	282151.	−	282151.i	1.49109	−	1.49109i
$436$	119949.	+	119949.i	0.630994	+	0.630994i
$437$	−287165.	−	287165.i	−1.50373	−	1.50373i
$438$	94829.9			0.494307
$439$		−	294420.i		−	1.52770i	−0.645394	−	0.763850i	$-0.723307\pi$
$439$		−	294420.i		−	1.52770i	0.645394	−	0.763850i	$-0.276693\pi$
$440$	14324.9	+	14324.9i	0.0739925	+	0.0739925i
$441$		−	18445.1i		−	0.0948424i
$442$		0			0
$443$	−75284.4			−0.383617			−0.191808	−	0.981432i	$-0.561435\pi$
$443$	−75284.4			−0.383617			−0.191808	+	0.981432i	$0.561435\pi$
$444$	37857.3	−	37857.3i	0.192037	−	0.192037i
$445$	169908.			0.858012
$446$		−	64647.3i		−	0.324998i
$447$	161366.	−	161366.i	0.807602	−	0.807602i
$448$	−7153.55	+	7153.55i	−0.0356423	+	0.0356423i
$449$	99431.0	+	99431.0i	0.493207	+	0.493207i	0.909315	−	0.416108i	$-0.136606\pi$
$449$	99431.0	+	99431.0i	0.493207	+	0.493207i	−0.416108	+	0.909315i	$0.636606\pi$
$450$	−2108.22	−	2108.22i	−0.0104110	−	0.0104110i
$451$	18694.0			0.0919071
$452$			140519.i			0.687795i
$453$	10539.5	+	10539.5i	0.0513600	+	0.0513600i
$454$			124927.i			0.606099i
$455$		0			0
$456$	−90698.5			−0.436185
$457$	137724.	−	137724.i	0.659441	−	0.659441i	−0.295807	−	0.955248i	$-0.595589\pi$
$457$	137724.	−	137724.i	0.659441	−	0.659441i	0.955248	+	0.295807i	$0.0955885\pi$
$458$	40101.0			0.191172
$459$			146617.i			0.695920i
$460$	−148042.	+	148042.i	−0.699631	+	0.699631i
$461$	−231925.	+	231925.i	−1.09130	+	1.09130i	−0.0959131	+	0.995390i	$0.530577\pi$
$461$	−231925.	+	231925.i	−1.09130	+	1.09130i	−0.995390	+	0.0959131i	$0.969423\pi$
$462$	−12351.6	−	12351.6i	−0.0578681	−	0.0578681i
$463$	−195600.	−	195600.i	−0.912447	−	0.912447i	0.0840177	−	0.996464i	$-0.473225\pi$
$463$	−195600.	−	195600.i	−0.912447	−	0.912447i	−0.996464	+	0.0840177i	$0.973225\pi$
$464$	−98866.3			−0.459211
$465$			273985.i			1.26713i
$466$	−6067.66	−	6067.66i	−0.0279415	−	0.0279415i
$467$		−	213346.i		−	0.978252i	−0.872213	−	0.489126i	$-0.837316\pi$
$467$		−	213346.i		−	0.978252i	0.872213	−	0.489126i	$-0.162684\pi$
$468$		0			0
$469$	−136217.			−0.619278
$470$	133331.	−	133331.i	0.603582	−	0.603582i
$471$	−170758.			−0.769730
$472$		−	17634.5i		−	0.0791549i
$473$	−3809.61	+	3809.61i	−0.0170278	+	0.0170278i
$474$	127437.	−	127437.i	0.567204	−	0.567204i
$475$	−34295.4	−	34295.4i	−0.152002	−	0.152002i
$476$	−24027.2	−	24027.2i	−0.106045	−	0.106045i
$477$	−41602.7			−0.182846
$478$		−	109787.i		−	0.480503i
$479$	−102057.	−	102057.i	−0.444805	−	0.444805i	0.448818	−	0.893623i	$-0.351845\pi$
$479$	−102057.	−	102057.i	−0.444805	−	0.444805i	−0.893623	+	0.448818i	$0.851845\pi$
$480$			46757.6i			0.202941i
$481$		0			0
$482$	163928.			0.705600
$483$	127648.	−	127648.i	0.547168	−	0.547168i
$484$	−108461.			−0.463003
$485$			131532.i			0.559174i
$486$	−26627.3	+	26627.3i	−0.112734	+	0.112734i
$487$	−57040.7	+	57040.7i	−0.240507	+	0.240507i	−0.817060	−	0.576553i	$-0.804397\pi$
$487$	−57040.7	+	57040.7i	−0.240507	+	0.240507i	0.576553	+	0.817060i	$0.304397\pi$
$488$	−63175.4	−	63175.4i	−0.265282	−	0.265282i
$489$	312695.	+	312695.i	1.30768	+	1.30768i
$490$	−154687.			−0.644260
$491$		−	326366.i		−	1.35376i	−0.736093	−	0.676880i	$-0.763331\pi$
$491$		−	326366.i		−	1.35376i	0.736093	−	0.676880i	$-0.236669\pi$
$492$	30509.3	+	30509.3i	0.126038	+	0.126038i
$493$		−	332071.i		−	1.36627i
$494$		0			0
$495$	−8213.56			−0.0335213
$496$	48002.6	−	48002.6i	0.195120	−	0.195120i
$497$	109049.			0.441477
$498$			51707.6i			0.208495i
$499$	−321163.	+	321163.i	−1.28981	+	1.28981i	−0.354902	+	0.934904i	$0.615486\pi$
$499$	−321163.	+	321163.i	−1.28981	+	1.28981i	−0.934904	+	0.354902i	$0.884514\pi$
$500$	78490.3	−	78490.3i	0.313961	−	0.313961i
$501$	117373.	+	117373.i	0.467619	+	0.467619i
$502$	−111340.	−	111340.i	−0.441820	−	0.441820i
$503$	77129.2			0.304848			0.152424	−	0.988315i	$-0.451292\pi$
$503$	77129.2			0.304848			0.152424	+	0.988315i	$0.451292\pi$
$504$		−	4101.67i		−	0.0161473i
$505$	−162141.	−	162141.i	−0.635787	−	0.635787i
$506$			89568.1i			0.349826i
$507$		0			0
$508$	129892.			0.503334
$509$	−258772.	+	258772.i	−0.998807	+	0.998807i	−0.999999	−	0.00119244i	$-0.999620\pi$
$509$	−258772.	+	258772.i	−0.998807	+	0.998807i	0.00119244	+	0.999999i	$0.499620\pi$
$510$	−157049.			−0.603802
$511$			69763.2i			0.267168i
$512$	8192.00	−	8192.00i	0.0312500	−	0.0312500i
$513$	−203578.	+	203578.i	−0.773565	+	0.773565i
$514$	−39901.4	−	39901.4i	−0.151030	−	0.151030i
$515$	46131.9	+	46131.9i	0.173935	+	0.173935i
$516$	−12434.8			−0.0467026
$517$		−	80668.0i		−	0.301801i
$518$	27850.4	+	27850.4i	0.103794	+	0.103794i
$519$			306840.i			1.13914i
$520$		0			0
$521$	384986.			1.41830			0.709152	−	0.705055i	$-0.249078\pi$
$521$	384986.			1.41830			0.709152	+	0.705055i	$0.249078\pi$
$522$	28343.7	−	28343.7i	0.104020	−	0.104020i
$523$	101902.			0.372546			0.186273	−	0.982498i	$-0.440359\pi$
$523$	101902.			0.372546			0.186273	+	0.982498i	$0.440359\pi$
$524$		−	73199.2i		−	0.266590i
$525$	15244.7	−	15244.7i	0.0553096	−	0.0553096i
$526$	−236919.	+	236919.i	−0.856305	+	0.856305i
$527$	161230.	+	161230.i	0.580531	+	0.580531i
$528$	14144.6	+	14144.6i	0.0507369	+	0.0507369i
$529$	−645805.			−2.30776
$530$			348895.i			1.24206i
$531$	5055.58	+	5055.58i	0.0179301	+	0.0179301i
$532$		−	66723.8i		−	0.235753i
$533$		0			0
$534$	167769.			0.588342
$535$	181035.	−	181035.i	0.632494	−	0.632494i
$536$	155991.			0.542962
$537$			126651.i			0.439198i
$538$	−25720.0	+	25720.0i	−0.0888600	+	0.0888600i
$539$	−46794.2	+	46794.2i	−0.161070	+	0.161070i
$540$	104950.	+	104950.i	0.359912	+	0.359912i
$541$	52761.7	+	52761.7i	0.180270	+	0.180270i	0.791474	−	0.611203i	$-0.209314\pi$
$541$	52761.7	+	52761.7i	0.180270	+	0.180270i	−0.611203	+	0.791474i	$0.709314\pi$
$542$	−45117.8			−0.153585
$543$			438962.i			1.48877i
$544$	27515.2	+	27515.2i	0.0929767	+	0.0929767i
$545$			576780.i			1.94186i
$546$		0			0
$547$	414681.			1.38592			0.692961	−	0.720975i	$-0.256306\pi$
$547$	414681.			1.38592			0.692961	+	0.720975i	$0.256306\pi$
$548$	−66642.2	+	66642.2i	−0.221916	+	0.221916i
$549$	36223.2			0.120183
$550$			10696.9i			0.0353616i
$551$	461082.	−	461082.i	1.51871	−	1.51871i
$552$	−146178.	+	146178.i	−0.479739	+	0.479739i
$553$	93751.3	+	93751.3i	0.306568	+	0.306568i
$554$	121867.	+	121867.i	0.397068	+	0.397068i
$555$	182038.			0.590984
$556$			16214.5i			0.0524509i
$557$	164211.	+	164211.i	0.529287	+	0.529287i	0.920360	−	0.391073i	$-0.127896\pi$
$557$	164211.	+	164211.i	0.529287	+	0.529287i	−0.391073	+	0.920360i	$0.627896\pi$
$558$			27523.5i			0.0883965i
$559$		0			0
$560$	−34398.0			−0.109688
$561$	−47508.7	+	47508.7i	−0.150955	+	0.150955i
$562$	123097.			0.389739
$563$			479081.i			1.51144i	0.654893	+	0.755721i	$0.272714\pi$
$563$			479081.i			1.51144i	−0.654893	+	0.755721i	$0.727286\pi$
$564$	131653.	−	131653.i	0.413878	−	0.413878i
$565$	−337845.	+	337845.i	−1.05833	+	1.05833i
$566$	−221697.	−	221697.i	−0.692033	−	0.692033i
$567$	−100875.	−	100875.i	−0.313775	−	0.313775i
$568$	−124879.			−0.387073
$569$			131129.i			0.405019i	0.979280	+	0.202510i	$0.0649097\pi$
$569$			131129.i			0.405019i	−0.979280	+	0.202510i	$0.935090\pi$
$570$	−218063.	−	218063.i	−0.671170	−	0.671170i
$571$		−	131069.i		−	0.402003i	−0.979591	−	0.201002i	$-0.935580\pi$
$571$		−	131069.i		−	0.402003i	0.979591	−	0.201002i	$-0.0644196\pi$
$572$		0			0
$573$	113164.			0.344667
$574$	−22444.6	+	22444.6i	−0.0681223	+	0.0681223i
$575$	−110548.			−0.334359
$576$			4697.09i			0.0141574i
$577$	−265150.	+	265150.i	−0.796416	+	0.796416i	−0.982528	−	0.186112i	$-0.940411\pi$
$577$	−265150.	+	265150.i	−0.796416	+	0.796416i	0.186112	+	0.982528i	$0.440411\pi$
$578$	74624.5	−	74624.5i	0.223371	−	0.223371i
$579$	282756.	+	282756.i	0.843440	+	0.843440i
$580$	−237701.	−	237701.i	−0.706601	−	0.706601i
$581$	−38039.6			−0.112689
$582$			129876.i			0.383428i
$583$	105544.	+	105544.i	0.310525	+	0.310525i
$584$		−	79890.4i		−	0.234244i
$585$		0			0
$586$	95683.6			0.278639
$587$	396373.	−	396373.i	1.15034	−	1.15034i	0.163860	−	0.986484i	$-0.447606\pi$
$587$	396373.	−	396373.i	1.15034	−	1.15034i	0.986484	−	0.163860i	$-0.0523945\pi$
$588$	−152740.			−0.441771
$589$			447738.i			1.29061i
$590$	42397.9	−	42397.9i	0.121798	−	0.121798i
$591$	370874.	−	370874.i	1.06182	−	1.06182i
$592$	−31893.3	−	31893.3i	−0.0910030	−	0.0910030i
$593$	284953.	+	284953.i	0.810335	+	0.810335i	0.984684	−	0.174349i	$-0.0557821\pi$
$593$	284953.	+	284953.i	0.810335	+	0.810335i	−0.174349	+	0.984684i	$0.555782\pi$
$594$	63497.0			0.179962
$595$		−	115536.i		−	0.326349i
$596$	−135945.	−	135945.i	−0.382710	−	0.382710i
$597$			641054.i			1.79865i
$598$		0			0
$599$	−311888.			−0.869251			−0.434625	−	0.900611i	$-0.643119\pi$
$599$	−311888.			−0.869251			−0.434625	+	0.900611i	$0.643119\pi$
$600$	−17457.7	+	17457.7i	−0.0484936	+	0.0484936i
$601$	−348543.			−0.964957			−0.482478	−	0.875908i	$-0.660263\pi$
$601$	−348543.			−0.964957			−0.482478	+	0.875908i	$0.660263\pi$
$602$		−	9147.89i		−	0.0252423i
$603$	−44720.7	+	44720.7i	−0.122991	+	0.122991i
$604$	8879.14	−	8879.14i	0.0243387	−	0.0243387i
$605$	−260769.	−	260769.i	−0.712435	−	0.712435i
$606$	−160101.	−	160101.i	−0.435961	−	0.435961i
$607$	375731.			1.01976			0.509882	−	0.860244i	$-0.329689\pi$
$607$	375731.			1.01976			0.509882	+	0.860244i	$0.329689\pi$
$608$			76409.8i			0.206701i
$609$	204956.	+	204956.i	0.552619	+	0.552619i
$610$		−	303780.i		−	0.816395i
$611$		0			0
$612$	−15776.5			−0.0421219
$613$	−281017.	+	281017.i	−0.747845	+	0.747845i	−0.974074	−	0.226229i	$-0.927360\pi$
$613$	−281017.	+	281017.i	−0.747845	+	0.747845i	0.226229	+	0.974074i	$0.427360\pi$
$614$	−392222.			−1.04039
$615$			146705.i			0.387877i
$616$	−10405.7	+	10405.7i	−0.0274228	+	0.0274228i
$617$	−387417.	+	387417.i	−1.01767	+	1.01767i	−0.0178316	+	0.999841i	$0.505676\pi$
$617$	−387417.	+	387417.i	−1.01767	+	1.01767i	−0.999841	+	0.0178316i	$0.994324\pi$
$618$	45551.3	+	45551.3i	0.119268	+	0.119268i
$619$	−104044.	−	104044.i	−0.271540	−	0.271540i	0.558180	−	0.829720i	$-0.311500\pi$
$619$	−104044.	−	104044.i	−0.271540	−	0.271540i	−0.829720	+	0.558180i	$0.811500\pi$
$620$	230822.			0.600473
$621$			656213.i			1.70162i
$622$	136148.	+	136148.i	0.351910	+	0.351910i
$623$			123422.i			0.317993i
$624$		0			0
$625$	449236.			1.15004
$626$	−111368.	+	111368.i	−0.284192	+	0.284192i
$627$	−131932.			−0.335595
$628$			143856.i			0.364762i
$629$	107123.	−	107123.i	0.270757	−	0.270757i
$630$	9861.49	−	9861.49i	0.0248463	−	0.0248463i
$631$	278260.	+	278260.i	0.698863	+	0.698863i	0.964165	−	0.265302i	$-0.0854718\pi$
$631$	278260.	+	278260.i	0.698863	+	0.698863i	−0.265302	+	0.964165i	$0.585472\pi$
$632$	−107361.	−	107361.i	−0.268789	−	0.268789i
$633$	−41472.2			−0.103502
$634$		−	92548.8i		−	0.230246i
$635$	312295.	+	312295.i	0.774494	+	0.774494i
$636$			344503.i			0.851685i
$637$		0			0
$638$	−143813.			−0.353312
$639$	35801.2	−	35801.2i	0.0876791	−	0.0876791i
$640$	39391.5			0.0961705
$641$			97107.8i			0.236340i	0.992993	+	0.118170i	$0.0377028\pi$
$641$			97107.8i			0.236340i	−0.992993	+	0.118170i	$0.962297\pi$
$642$	178757.	−	178757.i	0.433703	−	0.433703i
$643$	174688.	−	174688.i	0.422513	−	0.422513i	−0.463555	−	0.886068i	$-0.653426\pi$
$643$	174688.	−	174688.i	0.422513	−	0.422513i	0.886068	+	0.463555i	$0.153426\pi$
$644$	−107539.	−	107539.i	−0.259294	−	0.259294i
$645$	−29896.6	−	29896.6i	−0.0718626	−	0.0718626i
$646$	−256644.			−0.614988
$647$			110830.i			0.264759i	0.991199	+	0.132379i	$0.0422617\pi$
$647$			110830.i			0.264759i	−0.991199	+	0.132379i	$0.957738\pi$
$648$	115519.	+	115519.i	0.275108	+	0.275108i
$649$		−	25651.5i		−	0.0609009i
$650$		0			0
$651$	−199025.			−0.469618
$652$	263433.	−	263433.i	0.619690	−	0.619690i
$653$	−173437.			−0.406739			−0.203370	−	0.979102i	$-0.565189\pi$
$653$	−173437.			−0.406739			−0.203370	+	0.979102i	$0.565189\pi$
$654$			569520.i			1.33154i
$655$	175990.	−	175990.i	0.410210	−	0.410210i
$656$	25702.8	−	25702.8i	0.0597274	−	0.0597274i
$657$	22903.6	+	22903.6i	0.0530607	+	0.0530607i
$658$	96852.7	+	96852.7i	0.223697	+	0.223697i
$659$	98603.7			0.227051			0.113525	−	0.993535i	$-0.463786\pi$
$659$	98603.7			0.227051			0.113525	+	0.993535i	$0.463786\pi$
$660$			68014.8i			0.156141i
$661$	−556659.	−	556659.i	−1.27405	−	1.27405i	−0.943944	−	0.330107i	$-0.892915\pi$
$661$	−556659.	−	556659.i	−1.27405	−	1.27405i	−0.330107	−	0.943944i	$-0.607085\pi$
$662$		−	216807.i		−	0.494718i
$663$		0			0
$664$	43561.6			0.0988024
$665$	160422.	−	160422.i	0.362760	−	0.362760i
$666$	18286.8			0.0412277
$667$		−	1.48625e6i		−	3.34071i
$668$	98882.0	−	98882.0i	0.221597	−	0.221597i
$669$	153473.	−	153473.i	0.342909	−	0.342909i
$670$	375043.	+	375043.i	0.835472	+	0.835472i
$671$	−91896.4	−	91896.4i	−0.204105	−	0.204105i
$672$	−33965.0			−0.0752131
$673$		−	491994.i		−	1.08625i	−0.839652	−	0.543125i	$-0.817241\pi$
$673$		−	491994.i		−	1.08625i	0.839652	−	0.543125i	$-0.182759\pi$
$674$	31212.9	+	31212.9i	0.0687090	+	0.0687090i
$675$			78369.9i			0.172005i
$676$		0			0
$677$	−556471.			−1.21413			−0.607065	−	0.794652i	$-0.707653\pi$
$677$	−556471.			−1.21413			−0.607065	+	0.794652i	$0.707653\pi$
$678$	−333593.	+	333593.i	−0.725700	+	0.725700i
$679$	−95545.5			−0.207239
$680$			132307.i			0.286132i
$681$	−296576.	+	296576.i	−0.639502	+	0.639502i
$682$	69825.7	−	69825.7i	0.150123	−	0.150123i
$683$	331221.	+	331221.i	0.710029	+	0.710029i	0.966541	−	0.256512i	$-0.0825732\pi$
$683$	331221.	+	331221.i	0.710029	+	0.710029i	−0.256512	+	0.966541i	$0.582573\pi$
$684$	−21905.7	−	21905.7i	−0.0468216	−	0.0468216i
$685$	−320451.			−0.682937
$686$		−	246550.i		−	0.523911i
$687$	95199.9	+	95199.9i	0.201708	+	0.201708i
$688$			10475.9i			0.0221316i
$689$		0			0
$690$	−702903.			−1.47638
$691$	433883.	−	433883.i	0.908691	−	0.908691i	−0.0874759	−	0.996167i	$-0.527880\pi$
$691$	433883.	−	433883.i	0.908691	−	0.908691i	0.996167	+	0.0874759i	$0.0278801\pi$
$692$	258500.			0.539820
$693$		−	5966.38i		−	0.0124235i
$694$	14283.2	−	14283.2i	0.0296555	−	0.0296555i
$695$	−38983.8	+	38983.8i	−0.0807077	+	0.0807077i
$696$	−234709.	−	234709.i	−0.484519	−	0.484519i
$697$	86330.3	+	86330.3i	0.177704	+	0.177704i
$698$	−508534.			−1.04378
$699$		−	28809.2i		−	0.0589627i
$700$	−12843.1	−	12843.1i	−0.0262103	−	0.0262103i
$701$		−	445654.i		−	0.906905i	−0.891280	−	0.453453i	$-0.850192\pi$
$701$		−	445654.i		−	0.906905i	0.891280	−	0.453453i	$-0.149808\pi$
$702$		0			0
$703$	297480.			0.601933
$704$	11916.3	−	11916.3i	0.0240434	−	0.0240434i
$705$	633057.			1.27369
$706$			593832.i			1.19139i
$707$	117781.	−	117781.i	0.235632	−	0.235632i
$708$	41864.2	−	41864.2i	0.0835172	−	0.0835172i
$709$	−351952.	−	351952.i	−0.700150	−	0.700150i	0.264293	−	0.964443i	$-0.414862\pi$
$709$	−351952.	−	351952.i	−0.700150	−	0.700150i	−0.964443	+	0.264293i	$0.914862\pi$
$710$	−300242.	−	300242.i	−0.595600	−	0.595600i
$711$	61558.0			0.121771
$712$		−	141339.i		−	0.278806i
$713$	721618.	+	721618.i	1.41948	+	1.41948i
$714$		−	114081.i		−	0.223778i
$715$		0			0
$716$	106699.			0.208129
$717$	260635.	−	260635.i	0.506984	−	0.506984i
$718$	447978.			0.868976
$719$		−	61435.6i		−	0.118840i	−0.998233	−	0.0594200i	$-0.981075\pi$
$719$		−	61435.6i		−	0.118840i	0.998233	−	0.0594200i	$-0.0189251\pi$
$720$	−11293.0	+	11293.0i	−0.0217844	+	0.0217844i
$721$	−33510.5	+	33510.5i	−0.0644631	+	0.0644631i
$722$	−95709.6	−	95709.6i	−0.183603	−	0.183603i
$723$	389164.	+	389164.i	0.744486	+	0.744486i
$724$	369808.			0.705504
$725$		−	177499.i		−	0.337691i
$726$	−257487.	−	257487.i	−0.488519	−	0.488519i
$727$		−	121549.i		−	0.229975i	−0.993367	−	0.114988i	$-0.963317\pi$
$727$		−	121549.i		−	0.229975i	0.993367	−	0.114988i	$-0.0366828\pi$
$728$		0			0
$729$	458388.			0.862538
$730$	192078.	−	192078.i	0.360438	−	0.360438i
$731$	−35186.1			−0.0658471
$732$		−	299957.i		−	0.559804i
$733$	−450529.	+	450529.i	−0.838522	+	0.838522i	−0.988664	−	0.150142i	$-0.952027\pi$
$733$	−450529.	+	450529.i	−0.838522	+	0.838522i	0.150142	+	0.988664i	$0.452027\pi$
$734$	117955.	−	117955.i	0.218940	−	0.218940i
$735$	−367226.	−	367226.i	−0.679766	−	0.679766i
$736$	123149.	+	123149.i	0.227341	+	0.227341i
$737$	226908.			0.417749
$738$			14737.4i			0.0270587i
$739$	−478183.	−	478183.i	−0.875600	−	0.875600i	0.117476	−	0.993076i	$-0.462520\pi$
$739$	−478183.	−	478183.i	−0.875600	−	0.875600i	−0.993076	+	0.117476i	$0.962520\pi$
$740$		−	153360.i		−	0.280058i
$741$		0			0
$742$	−253440.			−0.460327
$743$	47112.0	−	47112.0i	0.0853402	−	0.0853402i	−0.663148	−	0.748488i	$-0.730780\pi$
$743$	47112.0	−	47112.0i	0.0853402	−	0.0853402i	0.748488	+	0.663148i	$0.230780\pi$
$744$	227916.			0.411746
$745$		−	653693.i		−	1.17777i
$746$	−111999.	+	111999.i	−0.201251	+	0.201251i
$747$	−12488.6	+	12488.6i	−0.0223806	+	0.0223806i
$748$	40024.2	+	40024.2i	0.0715352	+	0.0715352i
$749$	131505.	+	131505.i	0.234412	+	0.234412i
$750$	372672.			0.662528
$751$		−	856766.i		−	1.51909i	−0.650457	−	0.759543i	$-0.725423\pi$
$751$		−	856766.i		−	1.51909i	0.650457	−	0.759543i	$-0.274577\pi$
$752$	−110912.	−	110912.i	−0.196130	−	0.196130i
$753$		−	528644.i		−	0.932338i
$754$		0			0
$755$	42695.6			0.0749012
$756$	−76236.7	+	76236.7i	−0.133389	+	0.133389i
$757$	970577.			1.69371			0.846853	−	0.531826i	$-0.178494\pi$
$757$	970577.			1.69371			0.846853	+	0.531826i	$0.178494\pi$
$758$		−	299194.i		−	0.520733i
$759$	−212635.	+	212635.i	−0.369106	+	0.369106i
$760$	−183709.	+	183709.i	−0.318056	+	0.318056i
$761$	−290858.	−	290858.i	−0.502241	−	0.502241i	0.409893	−	0.912134i	$-0.365566\pi$
$761$	−290858.	−	290858.i	−0.502241	−	0.502241i	−0.912134	+	0.409893i	$0.865566\pi$
$762$	308364.	+	308364.i	0.531073	+	0.531073i
$763$	−418977.			−0.719682
$764$		−	95336.2i		−	0.163332i
$765$	−37930.9	−	37930.9i	−0.0648142	−	0.0648142i
$766$		−	556760.i		−	0.948879i
$767$		0			0
$768$	38895.6			0.0659444
$769$	−106901.	+	106901.i	−0.180771	+	0.180771i	−0.791692	−	0.610921i	$-0.790799\pi$
$769$	−106901.	+	106901.i	−0.180771	+	0.180771i	0.610921	+	0.791692i	$0.290799\pi$
$770$	−50036.2			−0.0843923
$771$		−	189452.i		−	0.318706i
$772$	238210.	−	238210.i	0.399692	−	0.399692i
$773$	189040.	−	189040.i	0.316370	−	0.316370i	−0.531001	−	0.847371i	$-0.678184\pi$
$773$	189040.	−	189040.i	0.316370	−	0.316370i	0.847371	+	0.531001i	$0.178184\pi$
$774$	−3003.30	−	3003.30i	−0.00501322	−	0.00501322i
$775$	86181.0	+	86181.0i	0.143485	+	0.143485i
$776$	109415.			0.181700
$777$			132233.i			0.219028i
$778$	−357633.	−	357633.i	−0.590852	−	0.590852i
$779$			239740.i			0.395062i
$780$		0			0
$781$	−181652.			−0.297809
$782$	−413632.	+	413632.i	−0.676396	+	0.676396i
$783$	−1.05364e6			−1.71857
$784$			128677.i			0.209348i
$785$	−345869.	+	345869.i	−0.561270	+	0.561270i
$786$	173775.	−	173775.i	0.281282	−	0.281282i
$787$	−324921.	−	324921.i	−0.524600	−	0.524600i	0.394357	−	0.918957i	$-0.370967\pi$
$787$	−324921.	−	324921.i	−0.524600	−	0.524600i	−0.918957	+	0.394357i	$0.870967\pi$
$788$	−312446.	−	312446.i	−0.503180	−	0.503180i
$789$	−1.12489e6			−1.80699
$790$		−	516247.i		−	0.827186i
$791$	−245413.	−	245413.i	−0.392234	−	0.392234i
$792$			6832.50i			0.0108925i
$793$		0			0
$794$	−719032.			−1.14053
$795$	−828277.	+	828277.i	−1.31051	+	1.31051i
$796$	540063.			0.852350
$797$		−	254628.i		−	0.400857i	−0.979708	−	0.200429i	$-0.935767\pi$
$797$		−	254628.i		−	0.400857i	0.979708	−	0.200429i	$-0.0642335\pi$
$798$	158402.	−	158402.i	0.248746	−	0.248746i
$799$	372531.	−	372531.i	0.583537	−	0.583537i
$800$	14707.4	+	14707.4i	0.0229804	+	0.0229804i
$801$	40520.1	+	40520.1i	0.0631546	+	0.0631546i
$802$	−124755.			−0.193959
$803$		−	116211.i		−	0.180225i
$804$	370322.	+	370322.i	0.572886	+	0.572886i
$805$		−	517102.i		−	0.797966i
$806$		0			0
$807$	−122119.			−0.187514
$808$	−134878.	+	134878.i	−0.206595	+	0.206595i
$809$	149000.			0.227661			0.113830	−	0.993500i	$-0.463688\pi$
$809$	149000.			0.227661			0.113830	+	0.993500i	$0.463688\pi$
$810$			555476.i			0.846633i
$811$	−129192.	+	129192.i	−0.196424	+	0.196424i	−0.798465	−	0.602041i	$-0.794354\pi$
$811$	−129192.	+	129192.i	−0.196424	+	0.196424i	0.602041	+	0.798465i	$0.294354\pi$
$812$	172667.	−	172667.i	0.261877	−	0.261877i
$813$	−107110.	−	107110.i	−0.162049	−	0.162049i
$814$	−46392.7	−	46392.7i	−0.0700166	−	0.0700166i
$815$	1.26672e6			1.90707
$816$			130642.i			0.196202i
$817$	−48856.1	−	48856.1i	−0.0731939	−	0.0731939i
$818$			340136.i			0.508330i
$819$		0			0
$820$	123593.			0.183808
$821$	−337584.	+	337584.i	−0.500836	+	0.500836i	−0.911698	−	0.410862i	$-0.865228\pi$
$821$	−337584.	+	337584.i	−0.500836	+	0.500836i	0.410862	+	0.911698i	$0.365228\pi$
$822$	−316417.			−0.468292
$823$		−	1.05893e6i		−	1.56339i	−0.623660	−	0.781695i	$-0.714355\pi$
$823$		−	1.05893e6i		−	1.56339i	0.623660	−	0.781695i	$-0.285645\pi$
$824$	38375.1	−	38375.1i	0.0565191	−	0.0565191i
$825$	−25394.4	+	25394.4i	−0.0373104	+	0.0373104i
$826$	30798.1	+	30798.1i	0.0451402	+	0.0451402i
$827$	−114562.	−	114562.i	−0.167506	−	0.167506i	0.618376	−	0.785882i	$-0.287791\pi$
$827$	−114562.	−	114562.i	−0.167506	−	0.167506i	−0.785882	+	0.618376i	$0.787791\pi$
$828$	−70610.8			−0.102994
$829$		−	418099.i		−	0.608373i	−0.952612	−	0.304187i	$-0.901615\pi$
$829$		−	418099.i		−	0.608373i	0.952612	−	0.304187i	$-0.0983846\pi$
$830$	104734.	+	104734.i	0.152030	+	0.152030i
$831$			578622.i			0.837902i
$832$		0			0
$833$	−432198.			−0.622864
$834$	−38493.1	+	38493.1i	−0.0553415	+	0.0553415i
$835$	475477.			0.681956
$836$			111148.i			0.159033i
$837$	511572.	−	511572.i	0.730224	−	0.730224i
$838$	−323309.	+	323309.i	−0.460394	+	0.460394i
$839$	−92890.6	−	92890.6i	−0.131962	−	0.131962i	0.638041	−	0.770003i	$-0.279745\pi$
$839$	−92890.6	−	92890.6i	−0.131962	−	0.131962i	−0.770003	+	0.638041i	$0.779745\pi$
$840$	−81660.9	−	81660.9i	−0.115733	−	0.115733i
$841$	1.67908e6			2.37400
$842$			320578.i			0.452178i
$843$	292232.	+	292232.i	0.411218	+	0.411218i
$844$			34938.7i			0.0490481i
$845$		0			0
$846$	63594.4			0.0888543
$847$	189424.	−	189424.i	0.264040	−	0.264040i
$848$	290230.			0.403600
$849$		−	1.05262e6i		−	1.46034i
$850$	−49399.1	+	49399.1i	−0.0683724	+	0.0683724i
$851$	479448.	−	479448.i	0.662037	−	0.662037i
$852$	−296463.	−	296463.i	−0.408405	−	0.408405i
$853$	73320.6	+	73320.6i	0.100769	+	0.100769i	0.755694	−	0.654925i	$-0.227300\pi$
$853$	73320.6	+	73320.6i	0.100769	+	0.100769i	−0.654925	+	0.755694i	$0.727300\pi$
$854$	220668.			0.302568
$855$		−	105334.i		−	0.144091i
$856$	−150595.	−	150595.i	−0.205525	−	0.205525i
$857$			973943.i			1.32609i	0.748581	+	0.663043i	$0.230735\pi$
$857$			973943.i			1.32609i	−0.748581	+	0.663043i	$0.769265\pi$
$858$		0			0
$859$	−220355.			−0.298632			−0.149316	−	0.988790i	$-0.547707\pi$
$859$	−220355.			−0.298632			−0.149316	+	0.988790i	$0.547707\pi$
$860$	−25186.7	+	25186.7i	−0.0340545	+	0.0340545i
$861$	−106567.			−0.143753
$862$		−	245881.i		−	0.330911i
$863$	−224496.	+	224496.i	−0.301430	+	0.301430i	−0.841573	−	0.540143i	$-0.818370\pi$
$863$	−224496.	+	224496.i	−0.301430	+	0.301430i	0.540143	+	0.841573i	$0.318370\pi$
$864$	87303.6	−	87303.6i	0.116951	−	0.116951i
$865$	621502.	+	621502.i	0.830636	+	0.830636i
$866$	−430703.	−	430703.i	−0.574305	−	0.574305i
$867$	354317.			0.471361
$868$			167670.i			0.222544i
$869$	−156170.	−	156170.i	−0.206803	−	0.206803i
$870$		−	1.12860e6i		−	1.49109i
$871$		0			0
$872$	479798.			0.630994
$873$	−31368.0	+	31368.0i	−0.0411584	+	0.0411584i
$874$	−1.14866e6			−1.50373
$875$			274162.i			0.358089i
$876$	189660.	−	189660.i	0.247154	−	0.247154i
$877$	285275.	−	285275.i	0.370906	−	0.370906i	−0.496901	−	0.867807i	$-0.665529\pi$
$877$	285275.	−	285275.i	0.370906	−	0.370906i	0.867807	+	0.496901i	$0.165529\pi$
$878$	−588840.	−	588840.i	−0.763850	−	0.763850i
$879$	227153.	+	227153.i	0.293995	+	0.293995i
$880$	57299.8			0.0739925
$881$			391811.i			0.504807i	0.967622	+	0.252403i	$0.0812210\pi$
$881$			391811.i			0.504807i	−0.967622	+	0.252403i	$0.918779\pi$
$882$	−36890.1	−	36890.1i	−0.0474212	−	0.0474212i
$883$		−	579331.i		−	0.743029i	−0.928427	−	0.371514i	$-0.878839\pi$
$883$		−	579331.i		−	0.743029i	0.928427	−	0.371514i	$-0.121161\pi$
$884$		0			0
$885$	201305.			0.257021
$886$	−150569.	+	150569.i	−0.191808	+	0.191808i
$887$	−1.29354e6			−1.64412			−0.822059	−	0.569402i	$-0.807175\pi$
$887$	−1.29354e6			−1.64412			−0.822059	+	0.569402i	$0.807175\pi$
$888$		−	151429.i		−	0.192037i
$889$	−226853.	+	226853.i	−0.287040	+	0.287040i
$890$	339816.	−	339816.i	0.429006	−	0.429006i
$891$	168037.	+	168037.i	0.211665	+	0.211665i
$892$	−129295.	−	129295.i	−0.162499	−	0.162499i
$893$	1.03452e6			1.29729
$894$		−	645465.i		−	0.807602i
$895$	256531.	+	256531.i	0.320254	+	0.320254i
$896$			28614.2i			0.0356423i
$897$		0			0
$898$	397724.			0.493207
$899$	−1.15865e6	+	1.15865e6i	−1.43362	+	1.43362i
$900$	−8432.87			−0.0104110
$901$			974821.i			1.20081i
$902$	37388.0	−	37388.0i	0.0459535	−	0.0459535i
$903$	21717.1	−	21717.1i	0.0266334	−	0.0266334i
$904$	281039.	+	281039.i	0.343898	+	0.343898i
$905$	889117.	+	889117.i	1.08558	+	1.08558i
$906$	42158.2			0.0513600
$907$			1.33236e6i			1.61959i	0.586711	+	0.809797i	$0.300423\pi$
$907$			1.33236e6i			1.61959i	−0.586711	+	0.809797i	$0.699577\pi$
$908$	249853.	+	249853.i	0.303049	+	0.303049i
$909$		−	77335.9i		−	0.0935951i
$910$		0			0
$911$	224303.			0.270271			0.135135	−	0.990827i	$-0.456853\pi$
$911$	224303.			0.270271			0.135135	+	0.990827i	$0.456853\pi$
$912$	−181397.	+	181397.i	−0.218092	+	0.218092i
$913$	63365.8			0.0760174
$914$		−	550894.i		−	0.659441i
$915$	721175.	−	721175.i	0.861387	−	0.861387i
$916$	80202.1	−	80202.1i	0.0955861	−	0.0955861i
$917$	127840.	+	127840.i	0.152030	+	0.152030i
$918$	293234.	+	293234.i	0.347960	+	0.347960i
$919$	−1.16194e6			−1.37579			−0.687896	−	0.725809i	$-0.741466\pi$
$919$	−1.16194e6			−1.37579			−0.687896	+	0.725809i	$0.741466\pi$
$920$			592168.i			0.699631i
$921$	−931135.	−	931135.i	−1.09772	−	1.09772i
$922$			927699.i			1.09130i
$923$		0			0
$924$	−49406.4			−0.0578681
$925$	57259.3	−	57259.3i	0.0669210	−	0.0669210i
$926$	−782401.			−0.912447
$927$			22003.3i			0.0256053i
$928$	−197733.	+	197733.i	−0.229606	+	0.229606i
$929$	−687630.	+	687630.i	−0.796752	+	0.796752i	−0.982582	−	0.185830i	$-0.940503\pi$
$929$	−687630.	+	687630.i	−0.796752	+	0.796752i	0.185830	+	0.982582i	$0.440503\pi$
$930$	547971.	+	547971.i	0.633565	+	0.633565i
$931$	−600109.	−	600109.i	−0.692359	−	0.692359i
$932$	−24270.6			−0.0279415
$933$			646431.i			0.742607i
$934$	−426692.	−	426692.i	−0.489126	−	0.489126i
$935$			192458.i			0.220146i
$936$		0			0
$937$	−237066.			−0.270016			−0.135008	−	0.990845i	$-0.543106\pi$
$937$	−237066.			−0.270016			−0.135008	+	0.990845i	$0.543106\pi$
$938$	−272434.	+	272434.i	−0.309639	+	0.309639i
$939$	−528775.			−0.599708
$940$		−	533325.i		−	0.603582i
$941$	488611.	−	488611.i	0.551803	−	0.551803i	−0.375158	−	0.926961i	$-0.622411\pi$
$941$	488611.	−	488611.i	0.551803	−	0.551803i	0.926961	+	0.375158i	$0.122411\pi$
$942$	−341515.	+	341515.i	−0.384865	+	0.384865i
$943$	386388.	+	386388.i	0.434511	+	0.434511i
$944$	−35268.9	−	35268.9i	−0.0395775	−	0.0395775i
$945$	−366586.			−0.410499
$946$			15238.4i			0.0170278i
$947$	1.12367e6	+	1.12367e6i	1.25297	+	1.25297i	0.954383	+	0.298584i	$0.0965144\pi$
$947$	1.12367e6	+	1.12367e6i	1.25297	+	1.25297i	0.298584	+	0.954383i	$0.403486\pi$
$948$		−	509749.i		−	0.567204i
$949$		0			0
$950$	−137182.			−0.152002
$951$	219711.	−	219711.i	0.242935	−	0.242935i
$952$	−96108.9			−0.106045
$953$			541880.i			0.596647i	0.954465	+	0.298323i	$0.0964274\pi$
$953$			541880.i			0.596647i	−0.954465	+	0.298323i	$0.903573\pi$
$954$	−83205.4	+	83205.4i	−0.0914229	+	0.0914229i
$955$	229213.	−	229213.i	0.251324	−	0.251324i
$956$	−219575.	−	219575.i	−0.240252	−	0.240252i
$957$	−341413.	−	341413.i	−0.372783	−	0.372783i
$958$	−408226.			−0.444805
$959$		−	232778.i		−	0.253107i
$960$	93515.3	+	93515.3i	0.101471	+	0.101471i
$961$		−	201600.i		−	0.218295i
$962$		0			0
$963$	86347.6			0.0931103
$964$	327856.	−	327856.i	0.352800	−	0.352800i
$965$	1.14544e6			1.23004
$966$		−	510593.i		−	0.547168i
$967$	−906789.	+	906789.i	−0.969735	+	0.969735i	−0.999555	−	0.0298202i	$-0.990507\pi$
$967$	−906789.	+	906789.i	−0.969735	+	0.969735i	0.0298202	+	0.999555i	$0.490507\pi$
$968$	−216922.	+	216922.i	−0.231501	+	0.231501i
$969$	−609273.	−	609273.i	−0.648880	−	0.648880i
$970$	263064.	+	263064.i	0.279587	+	0.279587i
$971$	−113950.			−0.120858			−0.0604291	−	0.998172i	$-0.519247\pi$
$971$	−113950.			−0.120858			−0.0604291	+	0.998172i	$0.519247\pi$
$972$			106509.i			0.112734i
$973$	−28318.1	−	28318.1i	−0.0299115	−	0.0299115i
$974$			228163.i			0.240507i
$975$		0			0
$976$	−252701.			−0.265282
$977$	−312440.	+	312440.i	−0.327324	+	0.327324i	−0.851568	−	0.524244i	$-0.824348\pi$
$977$	−312440.	+	312440.i	−0.327324	+	0.327324i	0.524244	+	0.851568i	$0.324348\pi$
$978$	1.25078e6			1.30768
$979$		−	205595.i		−	0.214510i
$980$	−309374.	+	309374.i	−0.322130	+	0.322130i
$981$	−137552.	+	137552.i	−0.142932	+	0.142932i
$982$	−652732.	−	652732.i	−0.676880	−	0.676880i
$983$	−810.786	−	810.786i	−0.000839072	−	0.000839072i	0.706687	−	0.707526i	$-0.250189\pi$
$983$	−810.786	−	810.786i	−0.000839072	−	0.000839072i	−0.707526	+	0.706687i	$0.750189\pi$
$984$	122037.			0.126038
$985$		−	1.50241e6i		−	1.54851i
$986$	−664141.	−	664141.i	−0.683135	−	0.683135i
$987$			459857.i			0.472050i
$988$		0			0
$989$	−157482.			−0.161005
$990$	−16427.1	+	16427.1i	−0.0167607	+	0.0167607i
$991$	274971.			0.279988			0.139994	−	0.990152i	$-0.455292\pi$
$991$	274971.			0.279988			0.139994	+	0.990152i	$0.455292\pi$
$992$		−	192010.i		−	0.195120i
$993$	514700.	−	514700.i	0.521982	−	0.521982i
$994$	218098.	−	218098.i	0.220738	−	0.220738i
$995$	1.29845e6	+	1.29845e6i	1.31154	+	1.31154i
$996$	103415.	+	103415.i	0.104248	+	0.104248i
$997$	−1.53846e6			−1.54773			−0.773864	−	0.633352i	$-0.781679\pi$
$997$	−1.53846e6			−1.54773			−0.773864	+	0.633352i	$0.781679\pi$
$998$			1.28465e6i			1.28981i
$999$	−339892.	−	339892.i	−0.340573	−	0.340573i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.5.d.g.239.3		8
13.5	odd	4		338.5.d.f.99.3		8
13.6	odd	12		26.5.f.b.11.1	✓	8
13.8	odd	4	inner	338.5.d.g.99.3		8
13.10	even	6		26.5.f.b.19.1	yes	8
13.12	even	2		338.5.d.f.239.3		8
39.23	odd	6		234.5.bb.a.19.2		8
39.32	even	12		234.5.bb.a.37.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.5.f.b.11.1	✓	8	13.6	odd	12
26.5.f.b.19.1	yes	8	13.10	even	6
234.5.bb.a.19.2		8	39.23	odd	6
234.5.bb.a.37.2		8	39.32	even	12
338.5.d.f.99.3		8	13.5	odd	4
338.5.d.f.239.3		8	13.12	even	2
338.5.d.g.99.3		8	13.8	odd	4	inner
338.5.d.g.239.3		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	338.d (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 2.00000i) q^{2} +9.49600 q^{3} -8.00000i q^{4} +(19.2341 - 19.2341i) q^{5} +(18.9920 - 18.9920i) q^{6} +(13.9718 + 13.9718i) q^{7} +(-16.0000 - 16.0000i) q^{8} +9.17400 q^{9} -76.9364i q^{10} +(-23.2740 - 23.2740i) q^{11} -75.9680i q^{12} +55.8871 q^{14} +(182.647 - 182.647i) q^{15} -64.0000 q^{16} -214.962i q^{17} +(18.3480 - 18.3480i) q^{18} +(298.476 - 298.476i) q^{19} +(-153.873 - 153.873i) q^{20} +(132.676 + 132.676i) q^{21} -93.0960 q^{22} -962.105i q^{23} +(-151.936 - 151.936i) q^{24} -114.902i q^{25} -682.060 q^{27} +(111.774 - 111.774i) q^{28} +1544.79 q^{29} -730.588i q^{30} +(-750.040 + 750.040i) q^{31} +(-128.000 + 128.000i) q^{32} +(-221.010 - 221.010i) q^{33} +(-429.924 - 429.924i) q^{34} +537.469 q^{35} -73.3920i q^{36} +(498.332 + 498.332i) q^{37} -1193.90i q^{38} -615.491 q^{40} +(-401.607 + 401.607i) q^{41} +530.704 q^{42} -163.685i q^{43} +(-186.192 + 186.192i) q^{44} +(176.454 - 176.454i) q^{45} +(-1924.21 - 1924.21i) q^{46} +(1733.01 + 1733.01i) q^{47} -607.744 q^{48} -2010.58i q^{49} +(-229.804 - 229.804i) q^{50} -2041.28i q^{51} -4534.85 q^{53} +(-1364.12 + 1364.12i) q^{54} -895.309 q^{55} -447.097i q^{56} +(2834.33 - 2834.33i) q^{57} +(3089.57 - 3089.57i) q^{58} +(551.077 + 551.077i) q^{59} +(-1461.18 - 1461.18i) q^{60} +3948.46 q^{61} +3000.16i q^{62} +(128.177 + 128.177i) q^{63} +512.000i q^{64} -884.039 q^{66} +(-4874.72 + 4874.72i) q^{67} -1719.70 q^{68} -9136.15i q^{69} +(1074.94 - 1074.94i) q^{70} +(3902.47 - 3902.47i) q^{71} +(-146.784 - 146.784i) q^{72} +(2496.58 + 2496.58i) q^{73} +1993.33 q^{74} -1091.11i q^{75} +(-2387.81 - 2387.81i) q^{76} -650.358i q^{77} +6710.05 q^{79} +(-1230.98 + 1230.98i) q^{80} -7219.93 q^{81} +1606.43i q^{82} +(-1361.30 + 1361.30i) q^{83} +(1061.41 - 1061.41i) q^{84} +(-4134.61 - 4134.61i) q^{85} +(-327.370 - 327.370i) q^{86} +14669.3 q^{87} +744.768i q^{88} +(4416.84 + 4416.84i) q^{89} -705.815i q^{90} -7696.84 q^{92} +(-7122.38 + 7122.38i) q^{93} +6932.03 q^{94} -11481.8i q^{95} +(-1215.49 + 1215.49i) q^{96} +(-3419.23 + 3419.23i) q^{97} +(-4021.16 - 4021.16i) q^{98} +(-213.516 - 213.516i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{2} + 18 q^{5} - 10 q^{7} - 128 q^{8} + 396 q^{9} + 378 q^{11} - 40 q^{14} - 600 q^{15} - 512 q^{16} + 792 q^{18} + 1190 q^{19} - 144 q^{20} + 678 q^{21} + 1512 q^{22} - 1404 q^{27} - 80 q^{28}+ \cdots + 55296 q^{99}+O(q^{100}) \) 8 * q + 16 * q^2 + 18 * q^5 - 10 * q^7 - 128 * q^8 + 396 * q^9 + 378 * q^11 - 40 * q^14 - 600 * q^15 - 512 * q^16 + 792 * q^18 + 1190 * q^19 - 144 * q^20 + 678 * q^21 + 1512 * q^22 - 1404 * q^27 - 80 * q^28 + 1464 * q^29 - 368 * q^31 - 1024 * q^32 + 1206 * q^33 + 312 * q^34 + 684 * q^35 + 5296 * q^37 - 576 * q^40 - 696 * q^41 + 2712 * q^42 + 3024 * q^44 - 834 * q^45 + 792 * q^46 + 10896 * q^47 + 1456 * q^50 - 4728 * q^53 - 2808 * q^54 - 13260 * q^55 - 18990 * q^57 + 2928 * q^58 - 1194 * q^59 + 4800 * q^60 + 26340 * q^61 - 5652 * q^63 + 4824 * q^66 - 8314 * q^67 + 1248 * q^68 + 1368 * q^70 + 16830 * q^71 - 6336 * q^72 + 7586 * q^73 + 21184 * q^74 - 9520 * q^76 + 37824 * q^79 - 1152 * q^80 - 34176 * q^81 - 21864 * q^83 + 5424 * q^84 - 2226 * q^85 + 2592 * q^86 + 79416 * q^87 - 12558 * q^89 + 3168 * q^92 - 55416 * q^93 + 43584 * q^94 - 31166 * q^97 - 33728 * q^98 + 55296 * q^99

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