LMFDB - Embedded newform 338.5.d.f.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.f.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	−0.977995	−	0.208630i	$-0.933099\pi$
$5$	−19.2341	+	19.2341i	−0.769364	+	0.769364i	0.208630	+	0.977995i	$0.433099\pi$
$6$	−18.9920	+	18.9920i	−0.527556	+	0.527556i
$7$	−13.9718	−	13.9718i	−0.285138	−	0.285138i	0.550016	−	0.835154i	$-0.314622\pi$
$7$	−13.9718	−	13.9718i	−0.285138	−	0.285138i	−0.835154	+	0.550016i	$0.814622\pi$
$8$	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	0.796709	−	0.604362i	$-0.206572\pi$
$11$	23.2740	+	23.2740i	0.192347	+	0.192347i	−0.604362	+	0.796709i	$0.706572\pi$
$12$		−	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	−0.987073	−	0.160270i	$-0.948763\pi$
$19$	−298.476	+	298.476i	−0.826803	+	0.826803i	0.160270	+	0.987073i	$0.448763\pi$
$20$	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	−0.199432	−	0.979912i	$-0.563910\pi$
$31$	750.040	−	750.040i	0.780479	−	0.780479i	0.979912	+	0.199432i	$0.0639099\pi$
$32$	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	0.501276	−	0.865288i	$-0.332864\pi$
$37$	−498.332	−	498.332i	−0.364012	−	0.364012i	−0.865288	+	0.501276i	$0.832864\pi$
$38$		−	1193.90i		−	0.826803i
$39$		0			0
$40$	−615.491			−0.384682
$41$	401.607	−	401.607i	0.238910	−	0.238910i	−0.577489	−	0.816398i	$-0.695967\pi$
$41$	401.607	−	401.607i	0.238910	−	0.238910i	0.816398	+	0.577489i	$0.195967\pi$
$42$	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	0.196069	−	0.980590i	$-0.437182\pi$
$47$	−1733.01	−	1733.01i	−0.784521	−	0.784521i	−0.980590	+	0.196069i	$0.937182\pi$
$48$	−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	0.623507	−	0.781817i	$-0.285707\pi$
$59$	−551.077	−	551.077i	−0.158310	−	0.158310i	−0.781817	+	0.623507i	$0.785707\pi$
$60$	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	$-0.471319\pi$
$67$	4874.72	−	4874.72i	1.08592	−	1.08592i	0.995943	−	0.0899815i	$-0.0286808\pi$
$68$	−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	−0.978828	−	0.204683i	$-0.934384\pi$
$71$	−3902.47	+	3902.47i	−0.774145	+	0.774145i	0.204683	+	0.978828i	$0.434384\pi$
$72$	146.784	+	146.784i	0.0283148	+	0.0283148i
$73$	−2496.58	−	2496.58i	−0.468489	−	0.468489i	0.432936	−	0.901425i	$-0.357478\pi$
$73$	−2496.58	−	2496.58i	−0.468489	−	0.468489i	−0.901425	+	0.432936i	$0.857478\pi$
$74$	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	−0.601368	−	0.798972i	$-0.705377\pi$
$83$	1361.30	−	1361.30i	0.197605	−	0.197605i	0.798972	+	0.601368i	$0.205377\pi$
$84$	−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	0.371016	−	0.928627i	$-0.379010\pi$
$89$	−4416.84	−	4416.84i	−0.557611	−	0.557611i	−0.928627	+	0.371016i	$0.879010\pi$
$90$		−	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	−0.501663	−	0.865063i	$-0.667278\pi$
$97$	3419.23	−	3419.23i	0.363400	−	0.363400i	0.865063	+	0.501663i	$0.167278\pi$
$98$	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	$-0.399048\pi$
$109$	14993.7	−	14993.7i	1.26199	−	1.26199i	0.950128	−	0.311860i	$-0.100952\pi$
$110$	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	0.893298	−	0.449466i	$-0.148386\pi$
$137$	8330.28	+	8330.28i	0.443832	+	0.443832i	−0.449466	+	0.893298i	$0.648386\pi$
$138$	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	−0.977296	−	0.211877i	$-0.932042\pi$
$149$	−16993.1	+	16993.1i	−0.765419	+	0.765419i	0.211877	+	0.977296i	$0.432042\pi$
$150$	2182.21	+	2182.21i	0.0969873	+	0.0969873i
$151$	−1109.89	−	1109.89i	−0.0486774	−	0.0486774i	0.682349	−	0.731026i	$-0.260958\pi$
$151$	−1109.89	−	1109.89i	−0.0486774	−	0.0486774i	−0.731026	+	0.682349i	$0.760958\pi$
$152$	−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	$-0.909955\pi$
$163$	−32929.1	−	32929.1i	−1.23938	−	1.23938i	−0.279126	−	0.960254i	$-0.590045\pi$
$164$	−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	0.449890	−	0.893084i	$-0.351463\pi$
$167$	−12360.2	−	12360.2i	−0.443194	−	0.443194i	−0.893084	+	0.449890i	$0.851463\pi$
$168$		−	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	0.183614	−	0.982999i	$-0.441221\pi$
$193$	−29776.3	−	29776.3i	−0.799385	−	0.799385i	−0.982999	+	0.183614i	$0.941221\pi$
$194$			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	$0.502031\pi$
$197$	−39055.8	+	39055.8i	−1.00636	+	1.00636i	−0.999980	+	0.00637973i	$0.997969\pi$
$198$	−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	−0.850681	−	0.525683i	$-0.823810\pi$
$223$	−16161.8	+	16161.8i	−0.324998	+	0.324998i	0.525683	+	0.850681i	$0.323810\pi$
$224$	−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	−0.335825	−	0.941924i	$-0.609015\pi$
$227$	31231.7	−	31231.7i	0.606099	−	0.606099i	0.941924	+	0.335825i	$0.109015\pi$
$228$	22674.6	+	22674.6i	0.436185	+	0.436185i
$229$	−10025.3	−	10025.3i	−0.191172	−	0.191172i	0.605030	−	0.796202i	$-0.293161\pi$
$229$	−10025.3	−	10025.3i	−0.191172	−	0.191172i	−0.796202	+	0.605030i	$0.793161\pi$
$230$	−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	−0.905292	−	0.424789i	$-0.860348\pi$
$239$	−27446.8	+	27446.8i	−0.480503	+	0.480503i	0.424789	+	0.905292i	$0.360348\pi$
$240$	11689.4	−	11689.4i	0.202941	−	0.202941i
$241$	−40981.9	−	40981.9i	−0.705600	−	0.705600i	0.260007	−	0.965607i	$-0.416275\pi$
$241$	−40981.9	−	40981.9i	−0.705600	−	0.705600i	−0.965607	+	0.260007i	$0.916275\pi$
$242$	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	0.779717	−	0.626132i	$-0.215363\pi$
$271$	11279.4	+	11279.4i	0.153585	+	0.153585i	−0.626132	+	0.779717i	$0.715363\pi$
$272$			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	0.484855	−	0.874594i	$-0.338872\pi$
$281$	−30774.2	−	30774.2i	−0.389739	−	0.389739i	−0.874594	+	0.484855i	$0.838872\pi$
$282$	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	0.553926	−	0.832566i	$-0.313129\pi$
$293$	−23920.9	−	23920.9i	−0.278639	−	0.278639i	−0.832566	+	0.553926i	$0.813129\pi$
$294$	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	$0.0131302\pi$
$307$	98055.5	+	98055.5i	1.04039	+	1.04039i	0.0412382	+	0.999149i	$0.486870\pi$
$308$	−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	−0.812795	−	0.582549i	$-0.802055\pi$
$317$	−23137.2	+	23137.2i	−0.230246	+	0.230246i	0.582549	+	0.812795i	$0.302055\pi$
$318$	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	−0.909789	−	0.415071i	$-0.863757\pi$
$331$	−54201.8	+	54201.8i	−0.494718	+	0.494718i	0.415071	+	0.909789i	$0.363757\pi$
$332$	−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	$0.0142599\pi$
$349$	127134.	+	127134.i	1.04378	+	1.04378i	0.0447840	+	0.998997i	$0.485740\pi$
$350$		−	6421.53i		−	0.0524206i
$351$		0			0
$352$	5958.14			0.0480868
$353$	148458.	−	148458.i	1.19139	−	1.19139i	0.214714	−	0.976677i	$-0.431118\pi$
$353$	148458.	−	148458.i	1.19139	−	1.19139i	0.976677	−	0.214714i	$-0.0688819\pi$
$354$	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	0.123383	−	0.992359i	$-0.460626\pi$
$359$	−111995.	−	111995.i	−0.868976	−	0.868976i	−0.992359	+	0.123383i	$0.960626\pi$
$360$	−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	−0.917793	−	0.397060i	$-0.870031\pi$
$379$	−74798.6	+	74798.6i	−0.520733	+	0.520733i	0.397060	+	0.917793i	$0.370031\pi$
$380$	−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	−0.998755	−	0.0498768i	$-0.984117\pi$
$383$	−139190.	+	139190.i	−0.948879	+	0.948879i	0.0498768	+	0.998755i	$0.484117\pi$
$384$	−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	$0.0486288\pi$
$397$	179758.	+	179758.i	1.14053	+	1.14053i	0.152178	+	0.988353i	$0.451371\pi$
$398$	−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	0.797404	−	0.603445i	$-0.206206\pi$
$401$	31188.8	+	31188.8i	0.193959	+	0.193959i	−0.603445	+	0.797404i	$0.706206\pi$
$402$			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	−0.405684	−	0.914014i	$-0.632966\pi$
$409$	85034.0	−	85034.0i	0.508330	−	0.508330i	0.914014	+	0.405684i	$0.132966\pi$
$410$	−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	−0.443899	−	0.896077i	$-0.646405\pi$
$421$	80144.5	−	80144.5i	0.452178	−	0.452178i	0.896077	+	0.443899i	$0.146405\pi$
$422$	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	−0.852932	−	0.522021i	$-0.825178\pi$
$431$	−61470.4	+	61470.4i	−0.330911	+	0.330911i	0.522021	+	0.852932i	$0.325178\pi$
$432$	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	0.416108	−	0.909315i	$-0.363394\pi$
$449$	−99431.0	−	99431.0i	−0.493207	−	0.493207i	−0.909315	+	0.416108i	$0.863394\pi$
$450$	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	−0.955248	−	0.295807i	$-0.904411\pi$
$457$	−137724.	+	137724.i	−0.659441	+	0.659441i	0.295807	+	0.955248i	$0.404411\pi$
$458$	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	$-0.469423\pi$
$461$	231925.	−	231925.i	1.09130	−	1.09130i	0.995390	−	0.0959131i	$-0.0305771\pi$
$462$	12351.6	+	12351.6i	0.0578681	+	0.0578681i
$463$	195600.	+	195600.i	0.912447	+	0.912447i	0.996464	−	0.0840177i	$-0.0267752\pi$
$463$	195600.	+	195600.i	0.912447	+	0.912447i	−0.0840177	+	0.996464i	$0.526775\pi$
$464$	−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	0.893623	−	0.448818i	$-0.148155\pi$
$479$	102057.	+	102057.i	0.444805	+	0.444805i	−0.448818	+	0.893623i	$0.648155\pi$
$480$			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	−0.576553	−	0.817060i	$-0.695603\pi$
$487$	57040.7	−	57040.7i	0.240507	−	0.240507i	0.817060	+	0.576553i	$0.195603\pi$
$488$	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	$-0.384514\pi$
$499$	321163.	−	321163.i	1.28981	−	1.28981i	0.934904	−	0.354902i	$-0.115486\pi$
$500$	−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	−0.00119244	−	0.999999i	$-0.500380\pi$
$509$	258772.	−	258772.i	0.998807	−	0.998807i	0.999999	+	0.00119244i	$0.000379567\pi$
$510$	−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	0.611203	−	0.791474i	$-0.290686\pi$
$541$	−52761.7	−	52761.7i	−0.180270	−	0.180270i	−0.791474	+	0.611203i	$0.790686\pi$
$542$	−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	0.391073	−	0.920360i	$-0.372104\pi$
$557$	−164211.	−	164211.i	−0.529287	−	0.529287i	−0.920360	+	0.391073i	$0.872104\pi$
$558$			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	−0.186112	−	0.982528i	$-0.559589\pi$
$577$	265150.	−	265150.i	0.796416	−	0.796416i	0.982528	+	0.186112i	$0.0595888\pi$
$578$	−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	$0.552394\pi$
$587$	−396373.	+	396373.i	−1.15034	+	1.15034i	−0.986484	+	0.163860i	$0.947606\pi$
$588$	−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	0.174349	−	0.984684i	$-0.444218\pi$
$593$	−284953.	−	284953.i	−0.810335	−	0.810335i	−0.984684	+	0.174349i	$0.944218\pi$
$594$	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	−0.226229	−	0.974074i	$-0.572640\pi$
$613$	281017.	−	281017.i	0.747845	−	0.747845i	0.974074	+	0.226229i	$0.0726398\pi$
$614$	−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	$-0.494324\pi$
$617$	387417.	−	387417.i	1.01767	−	1.01767i	0.999841	−	0.0178316i	$-0.00567629\pi$
$618$	−45551.3	−	45551.3i	−0.119268	−	0.119268i
$619$	104044.	+	104044.i	0.271540	+	0.271540i	0.829720	−	0.558180i	$-0.188500\pi$
$619$	104044.	+	104044.i	0.271540	+	0.271540i	−0.558180	+	0.829720i	$0.688500\pi$
$620$	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	0.265302	−	0.964165i	$-0.414528\pi$
$631$	−278260.	−	278260.i	−0.698863	−	0.698863i	−0.964165	+	0.265302i	$0.914528\pi$
$632$	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	−0.886068	−	0.463555i	$-0.846574\pi$
$643$	−174688.	+	174688.i	−0.422513	+	0.422513i	0.463555	+	0.886068i	$0.346574\pi$
$644$	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	$0.107085\pi$
$661$	556659.	+	556659.i	1.27405	+	1.27405i	0.330107	+	0.943944i	$0.392915\pi$
$662$		−	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	0.256512	−	0.966541i	$-0.417427\pi$
$683$	−331221.	−	331221.i	−0.710029	−	0.710029i	−0.966541	+	0.256512i	$0.917427\pi$
$684$	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	−0.996167	−	0.0874759i	$-0.972120\pi$
$691$	−433883.	+	433883.i	−0.908691	+	0.908691i	0.0874759	+	0.996167i	$0.472120\pi$
$692$	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	0.964443	−	0.264293i	$-0.0851385\pi$
$709$	351952.	+	351952.i	0.700150	+	0.700150i	−0.264293	+	0.964443i	$0.585138\pi$
$710$	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	−0.150142	−	0.988664i	$-0.547973\pi$
$733$	450529.	−	450529.i	0.838522	−	0.838522i	0.988664	+	0.150142i	$0.0479732\pi$
$734$	−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	0.993076	−	0.117476i	$-0.0374803\pi$
$739$	478183.	+	478183.i	0.875600	+	0.875600i	−0.117476	+	0.993076i	$0.537480\pi$
$740$		−	153360.i		−	0.280058i
$741$		0			0
$742$	−253440.			−0.460327
$743$	−47112.0	+	47112.0i	−0.0853402	+	0.0853402i	−0.748488	−	0.663148i	$-0.769220\pi$
$743$	−47112.0	+	47112.0i	−0.0853402	+	0.0853402i	0.663148	+	0.748488i	$0.269220\pi$
$744$	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	0.912134	−	0.409893i	$-0.134434\pi$
$761$	290858.	+	290858.i	0.502241	+	0.502241i	−0.409893	+	0.912134i	$0.634434\pi$
$762$	−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	−0.610921	−	0.791692i	$-0.709201\pi$
$769$	106901.	−	106901.i	0.180771	−	0.180771i	0.791692	+	0.610921i	$0.209201\pi$
$770$	−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	−0.847371	−	0.531001i	$-0.821816\pi$
$773$	−189040.	+	189040.i	−0.316370	+	0.316370i	0.531001	+	0.847371i	$0.321816\pi$
$774$	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	0.918957	−	0.394357i	$-0.129033\pi$
$787$	324921.	+	324921.i	0.524600	+	0.524600i	−0.394357	+	0.918957i	$0.629033\pi$
$788$	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	−0.602041	−	0.798465i	$-0.705646\pi$
$811$	129192.	−	129192.i	0.196424	−	0.196424i	0.798465	+	0.602041i	$0.205646\pi$
$812$	−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	−0.410862	−	0.911698i	$-0.634772\pi$
$821$	337584.	−	337584.i	0.500836	−	0.500836i	0.911698	+	0.410862i	$0.134772\pi$
$822$	−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	0.785882	−	0.618376i	$-0.212209\pi$
$827$	114562.	+	114562.i	0.167506	+	0.167506i	−0.618376	+	0.785882i	$0.712209\pi$
$828$	−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	0.770003	−	0.638041i	$-0.220255\pi$
$839$	92890.6	+	92890.6i	0.131962	+	0.131962i	−0.638041	+	0.770003i	$0.720255\pi$
$840$	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	0.654925	−	0.755694i	$-0.272700\pi$
$853$	−73320.6	−	73320.6i	−0.100769	−	0.100769i	−0.755694	+	0.654925i	$0.772700\pi$
$854$	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	−0.540143	−	0.841573i	$-0.681630\pi$
$863$	224496.	−	224496.i	0.301430	−	0.301430i	0.841573	+	0.540143i	$0.181630\pi$
$864$	−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	−0.867807	−	0.496901i	$-0.834471\pi$
$877$	−285275.	+	285275.i	−0.370906	+	0.370906i	0.496901	+	0.867807i	$0.334471\pi$
$878$	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	−0.185830	−	0.982582i	$-0.559497\pi$
$929$	687630.	−	687630.i	0.796752	−	0.796752i	0.982582	+	0.185830i	$0.0594973\pi$
$930$	−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	−0.926961	−	0.375158i	$-0.877589\pi$
$941$	−488611.	+	488611.i	−0.551803	+	0.551803i	0.375158	+	0.926961i	$0.377589\pi$
$942$	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	$-0.903486\pi$
$947$	−1.12367e6	−	1.12367e6i	−1.25297	−	1.25297i	−0.298584	−	0.954383i	$-0.596514\pi$
$948$		−	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	−0.0298202	−	0.999555i	$-0.509493\pi$
$967$	906789.	−	906789.i	0.969735	−	0.969735i	0.999555	+	0.0298202i	$0.00949348\pi$
$968$	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	−0.524244	−	0.851568i	$-0.675652\pi$
$977$	312440.	−	312440.i	0.327324	−	0.327324i	0.851568	+	0.524244i	$0.175652\pi$
$978$	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	0.707526	−	0.706687i	$-0.249811\pi$
$983$	810.786	+	810.786i	0.000839072	+	0.000839072i	−0.706687	+	0.707526i	$0.749811\pi$
$984$	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.f.239.3		8
13.3	even	3		26.5.f.b.19.1	yes	8
13.5	odd	4		338.5.d.g.99.3		8
13.7	odd	12		26.5.f.b.11.1	✓	8
13.8	odd	4	inner	338.5.d.f.99.3		8
13.12	even	2		338.5.d.g.239.3		8
39.20	even	12		234.5.bb.a.37.2		8
39.29	odd	6		234.5.bb.a.19.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.5.f.b.11.1	✓	8	13.7	odd	12
26.5.f.b.19.1	yes	8	13.3	even	3
234.5.bb.a.19.2		8	39.29	odd	6
234.5.bb.a.37.2		8	39.20	even	12
338.5.d.f.99.3		8	13.8	odd	4	inner
338.5.d.f.239.3		8	1.1	even	1	trivial
338.5.d.g.99.3		8	13.5	odd	4
338.5.d.g.239.3		8	13.12	even	2

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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