LMFDB - Embedded newform 338.5.d.h.239.1

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)

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);

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")

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 = [8,16,0,0,30] 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:	8.0.120336834816.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 46x^{5} + 445x^{4} + 68x^{3} + 32x^{2} + 1136x + 20164 $ x^8 - 2x^7 + 2x^6 + 46x^5 + 445x^4 + 68x^3 + 32x^2 + 1136*x + 20164
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			239.1
Root			$-2.16680 - 2.16680i$ of defining polynomial
Character	$\chi$	$=$	338.239
Dual form			338.5.d.h.99.1

$q$-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} -11.9855 q^{3} -8.00000i q^{4} +(27.8195 - 27.8195i) q^{5} +(-23.9709 + 23.9709i) q^{6} +(-49.1528 - 49.1528i) q^{7} +(-16.0000 - 16.0000i) q^{8} +62.6514 q^{9} -111.278i q^{10} +(-59.0953 - 59.0953i) q^{11} +95.8837i q^{12} -196.611 q^{14} +(-333.429 + 333.429i) q^{15} -64.0000 q^{16} -416.844i q^{17} +(125.303 - 125.303i) q^{18} +(3.74806 - 3.74806i) q^{19} +(-222.556 - 222.556i) q^{20} +(589.119 + 589.119i) q^{21} -236.381 q^{22} -462.074i q^{23} +(191.767 + 191.767i) q^{24} -922.846i q^{25} +219.917 q^{27} +(-393.222 + 393.222i) q^{28} +918.874 q^{29} +1333.72i q^{30} +(67.6743 - 67.6743i) q^{31} +(-128.000 + 128.000i) q^{32} +(708.285 + 708.285i) q^{33} +(-833.689 - 833.689i) q^{34} -2734.81 q^{35} -501.211i q^{36} +(154.404 + 154.404i) q^{37} -14.9922i q^{38} -890.223 q^{40} +(-2081.41 + 2081.41i) q^{41} +2356.48 q^{42} +1407.69i q^{43} +(-472.762 + 472.762i) q^{44} +(1742.93 - 1742.93i) q^{45} +(-924.149 - 924.149i) q^{46} +(-1373.25 - 1373.25i) q^{47} +767.070 q^{48} +2430.99i q^{49} +(-1845.69 - 1845.69i) q^{50} +4996.07i q^{51} +2519.79 q^{53} +(439.834 - 439.834i) q^{54} -3288.00 q^{55} +1572.89i q^{56} +(-44.9222 + 44.9222i) q^{57} +(1837.75 - 1837.75i) q^{58} +(387.572 + 387.572i) q^{59} +(2667.43 + 2667.43i) q^{60} +2705.62 q^{61} -270.697i q^{62} +(-3079.49 - 3079.49i) q^{63} +512.000i q^{64} +2833.14 q^{66} +(-2425.83 + 2425.83i) q^{67} -3334.75 q^{68} +5538.18i q^{69} +(-5469.62 + 5469.62i) q^{70} +(-4973.62 + 4973.62i) q^{71} +(-1002.42 - 1002.42i) q^{72} +(1208.41 + 1208.41i) q^{73} +617.616 q^{74} +11060.7i q^{75} +(-29.9845 - 29.9845i) q^{76} +5809.40i q^{77} -3300.68 q^{79} +(-1780.45 + 1780.45i) q^{80} -7710.57 q^{81} +8325.63i q^{82} +(-1803.48 + 1803.48i) q^{83} +(4712.95 - 4712.95i) q^{84} +(-11596.4 - 11596.4i) q^{85} +(2815.39 + 2815.39i) q^{86} -11013.1 q^{87} +1891.05i q^{88} +(2917.53 + 2917.53i) q^{89} -6971.71i q^{90} -3696.59 q^{92} +(-811.109 + 811.109i) q^{93} -5493.02 q^{94} -208.538i q^{95} +(1534.14 - 1534.14i) q^{96} +(10446.8 - 10446.8i) q^{97} +(4861.98 + 4861.98i) q^{98} +(-3702.40 - 3702.40i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{2} + 30 q^{5} - 86 q^{7} - 128 q^{8} - 180 q^{9} - 474 q^{11} - 344 q^{14} - 1128 q^{15} - 512 q^{16} - 360 q^{18} - 518 q^{19} - 240 q^{20} + 1914 q^{21} - 1896 q^{22} - 2268 q^{27} - 688 q^{28}+ \cdots + 9504 q^{99}+O(q^{100}) $ 8 * q + 16 * q^2 + 30 * q^5 - 86 * q^7 - 128 * q^8 - 180 * q^9 - 474 * q^11 - 344 * q^14 - 1128 * q^15 - 512 * q^16 - 360 * q^18 - 518 * q^19 - 240 * q^20 + 1914 * q^21 - 1896 * q^22 - 2268 * q^27 - 688 * q^28 + 504 * q^29 - 3376 * q^31 - 1024 * q^32 - 246 * q^33 + 1080 * q^34 - 12948 * q^35 - 5056 * q^37 - 960 * q^40 - 4392 * q^41 + 7656 * q^42 - 3792 * q^44 + 2898 * q^45 + 24 * q^46 + 4848 * q^47 - 3152 * q^50 - 6840 * q^53 - 4536 * q^54 - 588 * q^55 - 4434 * q^57 + 1008 * q^58 - 3510 * q^59 + 9024 * q^60 - 17340 * q^61 - 3276 * q^63 - 984 * q^66 - 7334 * q^67 + 4320 * q^68 - 25896 * q^70 - 7998 * q^71 + 2880 * q^72 + 19294 * q^73 - 20224 * q^74 + 4144 * q^76 - 11328 * q^79 - 1920 * q^80 - 12096 * q^81 + 10344 * q^83 + 15312 * q^84 - 20670 * q^85 + 13536 * q^86 - 36744 * q^87 + 1998 * q^89 + 96 * q^92 - 12744 * q^93 + 19392 * q^94 + 13694 * q^97 + 24640 * q^98 + 9504 * 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
$3$	−11.9855			−1.33172			−0.665859	−	0.746077i	$-0.731935\pi$
$3$	−11.9855			−1.33172			−0.665859	+	0.746077i	$0.731935\pi$
$4$		−	8.00000i		−	0.500000i
$5$	27.8195	−	27.8195i	1.11278	−	1.11278i	0.120006	−	0.992773i	$-0.461709\pi$
$5$	27.8195	−	27.8195i	1.11278	−	1.11278i	0.992773	−	0.120006i	$-0.0382913\pi$
$6$	−23.9709	+	23.9709i	−0.665859	+	0.665859i
$7$	−49.1528	−	49.1528i	−1.00312	−	1.00312i	−0.999995	−	0.00312281i	$-0.999006\pi$
$7$	−49.1528	−	49.1528i	−1.00312	−	1.00312i	−0.00312281	−	0.999995i	$-0.500994\pi$
$8$	−16.0000	−	16.0000i	−0.250000	−	0.250000i
$9$	62.6514			0.773474
$10$		−	111.278i		−	1.11278i
$11$	−59.0953	−	59.0953i	−0.488391	−	0.488391i	0.419407	−	0.907798i	$-0.362238\pi$
$11$	−59.0953	−	59.0953i	−0.488391	−	0.488391i	−0.907798	+	0.419407i	$0.862238\pi$
$12$			95.8837i			0.665859i
$13$		0			0
$13$		0			0
$14$	−196.611			−1.00312
$15$	−333.429	+	333.429i	−1.48191	+	1.48191i
$16$	−64.0000			−0.250000
$17$		−	416.844i		−	1.44237i	−0.692744	−	0.721184i	$-0.743598\pi$
$17$		−	416.844i		−	1.44237i	0.692744	−	0.721184i	$-0.256402\pi$
$18$	125.303	−	125.303i	0.386737	−	0.386737i
$19$	3.74806	−	3.74806i	0.0103824	−	0.0103824i	−0.701897	−	0.712279i	$-0.747663\pi$
$19$	3.74806	−	3.74806i	0.0103824	−	0.0103824i	0.712279	+	0.701897i	$0.247663\pi$
$20$	−222.556	−	222.556i	−0.556389	−	0.556389i
$21$	589.119	+	589.119i	1.33587	+	1.33587i
$22$	−236.381			−0.488391
$23$		−	462.074i		−	0.873486i	−0.899586	−	0.436743i	$-0.856132\pi$
$23$		−	462.074i		−	0.873486i	0.899586	−	0.436743i	$-0.143868\pi$
$24$	191.767	+	191.767i	0.332930	+	0.332930i
$25$		−	922.846i		−	1.47655i
$26$		0			0
$27$	219.917			0.301669
$28$	−393.222	+	393.222i	−0.501559	+	0.501559i
$29$	918.874			1.09260			0.546298	−	0.837591i	$-0.316037\pi$
$29$	918.874			1.09260			0.546298	+	0.837591i	$0.316037\pi$
$30$			1333.72i			1.48191i
$31$	67.6743	−	67.6743i	0.0704208	−	0.0704208i	−0.671019	−	0.741440i	$-0.734143\pi$
$31$	67.6743	−	67.6743i	0.0704208	−	0.0704208i	0.741440	+	0.671019i	$0.234143\pi$
$32$	−128.000	+	128.000i	−0.125000	+	0.125000i
$33$	708.285	+	708.285i	0.650399	+	0.650399i
$34$	−833.689	−	833.689i	−0.721184	−	0.721184i
$35$	−2734.81			−2.23250
$36$		−	501.211i		−	0.386737i
$37$	154.404	+	154.404i	0.112786	+	0.112786i	0.761247	−	0.648461i	$-0.224587\pi$
$37$	154.404	+	154.404i	0.112786	+	0.112786i	−0.648461	+	0.761247i	$0.724587\pi$
$38$		−	14.9922i		−	0.0103824i
$39$		0			0
$40$	−890.223			−0.556389
$41$	−2081.41	+	2081.41i	−1.23820	+	1.23820i	−0.277458	+	0.960738i	$0.589492\pi$
$41$	−2081.41	+	2081.41i	−1.23820	+	1.23820i	−0.960738	+	0.277458i	$0.910508\pi$
$42$	2356.48			1.33587
$43$			1407.69i			0.761327i	0.924714	+	0.380663i	$0.124304\pi$
$43$			1407.69i			0.761327i	−0.924714	+	0.380663i	$0.875696\pi$
$44$	−472.762	+	472.762i	−0.244195	+	0.244195i
$45$	1742.93	−	1742.93i	0.860705	−	0.860705i
$46$	−924.149	−	924.149i	−0.436743	−	0.436743i
$47$	−1373.25	−	1373.25i	−0.621664	−	0.621664i	0.324293	−	0.945957i	$-0.394874\pi$
$47$	−1373.25	−	1373.25i	−0.621664	−	0.621664i	−0.945957	+	0.324293i	$0.894874\pi$
$48$	767.070			0.332930
$49$			2430.99i			1.01249i
$50$	−1845.69	−	1845.69i	−0.738277	−	0.738277i
$51$			4996.07i			1.92083i
$52$		0			0
$53$	2519.79			0.897040			0.448520	−	0.893773i	$-0.351951\pi$
$53$	2519.79			0.897040			0.448520	+	0.893773i	$0.351951\pi$
$54$	439.834	−	439.834i	0.150835	−	0.150835i
$55$	−3288.00			−1.08694
$56$			1572.89i			0.501559i
$57$	−44.9222	+	44.9222i	−0.0138265	+	0.0138265i
$58$	1837.75	−	1837.75i	0.546298	−	0.546298i
$59$	387.572	+	387.572i	0.111339	+	0.111339i	0.760582	−	0.649242i	$-0.224914\pi$
$59$	387.572	+	387.572i	0.111339	+	0.111339i	−0.649242	+	0.760582i	$0.724914\pi$
$60$	2667.43	+	2667.43i	0.740954	+	0.740954i
$61$	2705.62			0.727123			0.363562	−	0.931570i	$-0.381561\pi$
$61$	2705.62			0.727123			0.363562	+	0.931570i	$0.381561\pi$
$62$		−	270.697i		−	0.0704208i
$63$	−3079.49	−	3079.49i	−0.775885	−	0.775885i
$64$			512.000i			0.125000i
$65$		0			0
$66$	2833.14			0.650399
$67$	−2425.83	+	2425.83i	−0.540394	+	0.540394i	−0.923645	−	0.383250i	$-0.874805\pi$
$67$	−2425.83	+	2425.83i	−0.540394	+	0.540394i	0.383250	+	0.923645i	$0.374805\pi$
$68$	−3334.75			−0.721184
$69$			5538.18i			1.16324i
$70$	−5469.62	+	5469.62i	−1.11625	+	1.11625i
$71$	−4973.62	+	4973.62i	−0.986633	+	0.986633i	−0.999912	−	0.0132789i	$-0.995773\pi$
$71$	−4973.62	+	4973.62i	−0.986633	+	0.986633i	0.0132789	+	0.999912i	$0.495773\pi$
$72$	−1002.42	−	1002.42i	−0.193368	−	0.193368i
$73$	1208.41	+	1208.41i	0.226762	+	0.226762i	0.811338	−	0.584577i	$-0.198739\pi$
$73$	1208.41	+	1208.41i	0.226762	+	0.226762i	−0.584577	+	0.811338i	$0.698739\pi$
$74$	617.616			0.112786
$75$			11060.7i			1.96635i
$76$	−29.9845	−	29.9845i	−0.00519122	−	0.00519122i
$77$			5809.40i			0.979827i
$78$		0			0
$79$	−3300.68			−0.528870			−0.264435	−	0.964403i	$-0.585186\pi$
$79$	−3300.68			−0.528870			−0.264435	+	0.964403i	$0.585186\pi$
$80$	−1780.45	+	1780.45i	−0.278195	+	0.278195i
$81$	−7710.57			−1.17521
$82$			8325.63i			1.23820i
$83$	−1803.48	+	1803.48i	−0.261791	+	0.261791i	−0.825781	−	0.563991i	$-0.809265\pi$
$83$	−1803.48	+	1803.48i	−0.261791	+	0.261791i	0.563991	+	0.825781i	$0.309265\pi$
$84$	4712.95	−	4712.95i	0.667935	−	0.667935i
$85$	−11596.4	−	11596.4i	−1.60504	−	1.60504i
$86$	2815.39	+	2815.39i	0.380663	+	0.380663i
$87$	−11013.1			−1.45503
$88$			1891.05i			0.244195i
$89$	2917.53	+	2917.53i	0.368329	+	0.368329i	0.866867	−	0.498539i	$-0.166130\pi$
$89$	2917.53	+	2917.53i	0.368329	+	0.368329i	−0.498539	+	0.866867i	$0.666130\pi$
$90$		−	6971.71i		−	0.860705i
$91$		0			0
$92$	−3696.59			−0.436743
$93$	−811.109	+	811.109i	−0.0937806	+	0.0937806i
$94$	−5493.02			−0.621664
$95$		−	208.538i		−	0.0231067i
$96$	1534.14	−	1534.14i	0.166465	−	0.166465i
$97$	10446.8	−	10446.8i	1.11030	−	1.11030i	0.117188	−	0.993110i	$-0.462612\pi$
$97$	10446.8	−	10446.8i	1.11030	−	1.11030i	0.993110	−	0.117188i	$-0.0373879\pi$
$98$	4861.98	+	4861.98i	0.506246	+	0.506246i
$99$	−3702.40	−	3702.40i	−0.377758	−	0.377758i
$100$	−7382.77			−0.738277
$101$			1164.32i			0.114138i	0.998370	+	0.0570688i	$0.0181754\pi$
$101$			1164.32i			0.114138i	−0.998370	+	0.0570688i	$0.981825\pi$
$102$	9992.15	+	9992.15i	0.960414	+	0.960414i
$103$			13334.2i			1.25688i	0.777860	+	0.628438i	$0.216305\pi$
$103$			13334.2i			1.25688i	−0.777860	+	0.628438i	$0.783695\pi$
$104$		0			0
$105$	32778.0			2.97306
$106$	5039.57	−	5039.57i	0.448520	−	0.448520i
$107$	16300.3			1.42374			0.711868	−	0.702314i	$-0.247850\pi$
$107$	16300.3			1.42374			0.711868	+	0.702314i	$0.247850\pi$
$108$		−	1759.33i		−	0.150835i
$109$	14940.7	−	14940.7i	1.25753	−	1.25753i	0.305259	−	0.952269i	$-0.401257\pi$
$109$	14940.7	−	14940.7i	1.25753	−	1.25753i	0.952269	−	0.305259i	$-0.0987430\pi$
$110$	−6576.00	+	6576.00i	−0.543471	+	0.543471i
$111$	−1850.60	−	1850.60i	−0.150199	−	0.150199i
$112$	3145.78	+	3145.78i	0.250779	+	0.250779i
$113$	2940.09			0.230252			0.115126	−	0.993351i	$-0.463273\pi$
$113$	2940.09			0.230252			0.115126	+	0.993351i	$0.463273\pi$
$114$			179.689i			0.0138265i
$115$	−12854.7	−	12854.7i	−0.971997	−	0.971997i
$116$		−	7350.99i		−	0.546298i
$117$		0			0
$118$	1550.29			0.111339
$119$	−20489.1	+	20489.1i	−1.44687	+	1.44687i
$120$	10669.7			0.740954
$121$		−	7656.49i		−	0.522949i
$122$	5411.25	−	5411.25i	0.363562	−	0.363562i
$123$	24946.6	−	24946.6i	1.64893	−	1.64893i
$124$	−541.395	−	541.395i	−0.0352104	−	0.0352104i
$125$	−8285.92	−	8285.92i	−0.530299	−	0.530299i
$126$	−12318.0			−0.775885
$127$		−	29566.9i		−	1.83315i	−0.399863	−	0.916575i	$-0.630942\pi$
$127$		−	29566.9i		−	1.83315i	0.399863	−	0.916575i	$-0.369058\pi$
$128$	1024.00	+	1024.00i	0.0625000	+	0.0625000i
$129$		−	16871.9i		−	1.01387i
$130$		0			0
$131$	−16053.2			−0.935446			−0.467723	−	0.883875i	$-0.654925\pi$
$131$	−16053.2			−0.935446			−0.467723	+	0.883875i	$0.654925\pi$
$132$	5666.28	−	5666.28i	0.325200	−	0.325200i
$133$	−368.455			−0.0208296
$134$			9703.32i			0.540394i
$135$	6117.97	−	6117.97i	0.335691	−	0.335691i
$136$	−6669.51	+	6669.51i	−0.360592	+	0.360592i
$137$	−7714.53	−	7714.53i	−0.411025	−	0.411025i	0.471070	−	0.882096i	$-0.343868\pi$
$137$	−7714.53	−	7714.53i	−0.411025	−	0.411025i	−0.882096	+	0.471070i	$0.843868\pi$
$138$	11076.4	+	11076.4i	0.581619	+	0.581619i
$139$	13585.1			0.703129			0.351564	−	0.936164i	$-0.385650\pi$
$139$	13585.1			0.703129			0.351564	+	0.936164i	$0.385650\pi$
$140$			21878.5i			1.11625i
$141$	16459.1	+	16459.1i	0.827881	+	0.827881i
$142$			19894.5i			0.986633i
$143$		0			0
$144$	−4009.69			−0.193368
$145$	25562.6	−	25562.6i	1.21582	−	1.21582i
$146$	4833.65			0.226762
$147$		−	29136.6i		−	1.34835i
$148$	1235.23	−	1235.23i	0.0563930	−	0.0563930i
$149$	−18912.7	+	18912.7i	−0.851886	+	0.851886i	−0.990365	−	0.138480i	$-0.955778\pi$
$149$	−18912.7	+	18912.7i	−0.851886	+	0.851886i	0.138480	+	0.990365i	$0.455778\pi$
$150$	22121.5	+	22121.5i	0.983177	+	0.983177i
$151$	−10548.8	−	10548.8i	−0.462645	−	0.462645i	0.436877	−	0.899521i	$-0.356085\pi$
$151$	−10548.8	−	10548.8i	−0.462645	−	0.462645i	−0.899521	+	0.436877i	$0.856085\pi$
$152$	−119.938			−0.00519122
$153$		−	26115.9i		−	1.11563i
$154$	11618.8	+	11618.8i	0.489914	+	0.489914i
$155$		−	3765.33i		−	0.156725i
$156$		0			0
$157$	4252.07			0.172505			0.0862524	−	0.996273i	$-0.472511\pi$
$157$	4252.07			0.172505			0.0862524	+	0.996273i	$0.472511\pi$
$158$	−6601.36	+	6601.36i	−0.264435	+	0.264435i
$159$	−30200.8			−1.19460
$160$			7121.78i			0.278195i
$161$	−22712.2	+	22712.2i	−0.876210	+	0.876210i
$162$	−15421.1	+	15421.1i	−0.587606	+	0.587606i
$163$	9633.41	+	9633.41i	0.362581	+	0.362581i	0.864762	−	0.502181i	$-0.167469\pi$
$163$	9633.41	+	9633.41i	0.362581	+	0.362581i	−0.502181	+	0.864762i	$0.667469\pi$
$164$	16651.3	+	16651.3i	0.619098	+	0.619098i
$165$	39408.2			1.44750
$166$			7213.90i			0.261791i
$167$	−27778.0	−	27778.0i	−0.996021	−	0.996021i	0.00397084	−	0.999992i	$-0.498736\pi$
$167$	−27778.0	−	27778.0i	−0.996021	−	0.996021i	−0.999992	+	0.00397084i	$0.998736\pi$
$168$		−	18851.8i		−	0.667935i
$169$		0			0
$170$	−46385.6			−1.60504
$171$	234.821	−	234.821i	0.00803054	−	0.00803054i
$172$	11261.5			0.380663
$173$		−	6.52133i		−	0.000217893i	−1.00000		0.000108947i	$-0.999965\pi$
$173$		−	6.52133i		−	0.000217893i	1.00000		0.000108947i	$-3.46788e-5\pi$
$174$	−22026.3	+	22026.3i	−0.727515	+	0.727515i
$175$	−45360.4	+	45360.4i	−1.48116	+	1.48116i
$176$	3782.10	+	3782.10i	0.122098	+	0.122098i
$177$	−4645.23	−	4645.23i	−0.148272	−	0.148272i
$178$	11670.1			0.368329
$179$		−	25343.4i		−	0.790969i	−0.918473	−	0.395485i	$-0.870577\pi$
$179$		−	25343.4i		−	0.790969i	0.918473	−	0.395485i	$-0.129423\pi$
$180$	−13943.4	−	13943.4i	−0.430353	−	0.430353i
$181$			13510.0i			0.412380i	0.978512	+	0.206190i	$0.0661065\pi$
$181$			13510.0i			0.412380i	−0.978512	+	0.206190i	$0.933894\pi$
$182$		0			0
$183$	−32428.2			−0.968323
$184$	−7393.19	+	7393.19i	−0.218372	+	0.218372i
$185$	8590.88			0.251012
$186$			3244.43i			0.0937806i
$187$	−24633.5	+	24633.5i	−0.704439	+	0.704439i
$188$	−10986.0	+	10986.0i	−0.310832	+	0.310832i
$189$	−10809.5	−	10809.5i	−0.302610	−	0.302610i
$190$	−417.076	−	417.076i	−0.0115534	−	0.0115534i
$191$	−12999.2			−0.356329			−0.178165	−	0.984001i	$-0.557016\pi$
$191$	−12999.2			−0.356329			−0.178165	+	0.984001i	$0.557016\pi$
$192$		−	6136.56i		−	0.166465i
$193$	−12126.5	−	12126.5i	−0.325552	−	0.325552i	0.525340	−	0.850892i	$-0.323938\pi$
$193$	−12126.5	−	12126.5i	−0.325552	−	0.325552i	−0.850892	+	0.525340i	$0.823938\pi$
$194$		−	41787.2i		−	1.11030i
$195$		0			0
$196$	19447.9			0.506246
$197$	349.848	−	349.848i	0.00901462	−	0.00901462i	−0.702585	−	0.711600i	$-0.747971\pi$
$197$	349.848	−	349.848i	0.00901462	−	0.00901462i	0.711600	+	0.702585i	$0.247971\pi$
$198$	−14809.6			−0.377758
$199$		−	56143.5i		−	1.41773i	−0.705344	−	0.708865i	$-0.749207\pi$
$199$		−	56143.5i		−	1.41773i	0.705344	−	0.708865i	$-0.250793\pi$
$200$	−14765.5	+	14765.5i	−0.369138	+	0.369138i
$201$	29074.7	−	29074.7i	0.719653	−	0.719653i
$202$	2328.63	+	2328.63i	0.0570688	+	0.0570688i
$203$	−45165.2	−	45165.2i	−1.09600	−	1.09600i
$204$	39968.6			0.960414
$205$			115807.i			2.75568i
$206$	26668.4	+	26668.4i	0.628438	+	0.628438i
$207$		−	28949.6i		−	0.675619i
$208$		0			0
$209$	−442.985			−0.0101414
$210$	65555.9	−	65555.9i	1.48653	−	1.48653i
$211$	10570.7			0.237431			0.118716	−	0.992928i	$-0.462122\pi$
$211$	10570.7			0.237431			0.118716	+	0.992928i	$0.462122\pi$
$212$		−	20158.3i		−	0.448520i
$213$	59611.1	−	59611.1i	1.31392	−	1.31392i
$214$	32600.7	−	32600.7i	0.711868	−	0.711868i
$215$	39161.3	+	39161.3i	0.847189	+	0.847189i
$216$	−3518.67	−	3518.67i	−0.0754173	−	0.0754173i
$217$	−6652.76			−0.141281
$218$		−	59762.8i		−	1.25753i
$219$	−14483.4	−	14483.4i	−0.301983	−	0.301983i
$220$			26304.0i			0.543471i
$221$		0			0
$222$	−7402.42			−0.150199
$223$	27271.4	−	27271.4i	0.548401	−	0.548401i	−0.377577	−	0.925978i	$-0.623243\pi$
$223$	27271.4	−	27271.4i	0.548401	−	0.548401i	0.925978	+	0.377577i	$0.123243\pi$
$224$	12583.1			0.250779
$225$		−	57817.6i		−	1.14208i
$226$	5880.18	−	5880.18i	0.115126	−	0.115126i
$227$	−42485.7	+	42485.7i	−0.824500	+	0.824500i	−0.986750	−	0.162250i	$-0.948125\pi$
$227$	−42485.7	+	42485.7i	−0.824500	+	0.824500i	0.162250	+	0.986750i	$0.448125\pi$
$228$	359.378	+	359.378i	0.00691324	+	0.00691324i
$229$	−28796.6	−	28796.6i	−0.549124	−	0.549124i	0.377063	−	0.926188i	$-0.376934\pi$
$229$	−28796.6	−	28796.6i	−0.549124	−	0.549124i	−0.926188	+	0.377063i	$0.876934\pi$
$230$	−51418.6			−0.971997
$231$		−	69628.3i		−	1.30485i
$232$	−14702.0	−	14702.0i	−0.273149	−	0.273149i
$233$			54456.8i			1.00309i	0.865131	+	0.501546i	$0.167235\pi$
$233$			54456.8i			1.00309i	−0.865131	+	0.501546i	$0.832765\pi$
$234$		0			0
$235$	−76406.4			−1.38355
$236$	3100.57	−	3100.57i	0.0556696	−	0.0556696i
$237$	39560.2			0.704307
$238$			81956.2i			1.44687i
$239$	32089.5	−	32089.5i	0.561782	−	0.561782i	−0.368031	−	0.929813i	$-0.619968\pi$
$239$	32089.5	−	32089.5i	0.561782	−	0.561782i	0.929813	+	0.368031i	$0.119968\pi$
$240$	21339.5	−	21339.5i	0.370477	−	0.370477i
$241$	69395.6	+	69395.6i	1.19481	+	1.19481i	0.975701	+	0.219105i	$0.0703138\pi$
$241$	69395.6	+	69395.6i	1.19481	+	1.19481i	0.219105	+	0.975701i	$0.429686\pi$
$242$	−15313.0	−	15313.0i	−0.261474	−	0.261474i
$243$	74601.5			1.26338
$244$		−	21645.0i		−	0.363562i
$245$	67628.9	+	67628.9i	1.12668	+	1.12668i
$246$		−	99786.5i		−	1.64893i
$247$		0			0
$248$	−2165.58			−0.0352104
$249$	21615.5	−	21615.5i	0.348631	−	0.348631i
$250$	−33143.7			−0.530299
$251$			29730.6i			0.471906i	0.971764	+	0.235953i	$0.0758212\pi$
$251$			29730.6i			0.471906i	−0.971764	+	0.235953i	$0.924179\pi$
$252$	−24635.9	+	24635.9i	−0.387943	+	0.387943i
$253$	−27306.4	+	27306.4i	−0.426603	+	0.426603i
$254$	−59133.8	−	59133.8i	−0.916575	−	0.916575i
$255$	138988.	+	138988.i	2.13746	+	2.13746i
$256$	4096.00			0.0625000
$257$			12836.7i			0.194352i	0.995267	+	0.0971759i	$0.0309809\pi$
$257$			12836.7i			0.194352i	−0.995267	+	0.0971759i	$0.969019\pi$
$258$	−33743.7	−	33743.7i	−0.506937	−	0.506937i
$259$		−	15178.8i		−	0.226275i
$260$		0			0
$261$	57568.7			0.845095
$262$	−32106.4	+	32106.4i	−0.467723	+	0.467723i
$263$	59850.0			0.865272			0.432636	−	0.901569i	$-0.357584\pi$
$263$	59850.0			0.865272			0.432636	+	0.901569i	$0.357584\pi$
$264$		−	22665.1i		−	0.325200i
$265$	70099.1	−	70099.1i	0.998207	−	0.998207i
$266$	−736.910	+	736.910i	−0.0104148	+	0.0104148i
$267$	−34968.0	−	34968.0i	−0.490510	−	0.490510i
$268$	19406.6	+	19406.6i	0.270197	+	0.270197i
$269$	−80657.7			−1.11466			−0.557329	−	0.830292i	$-0.688174\pi$
$269$	−80657.7			−1.11466			−0.557329	+	0.830292i	$0.688174\pi$
$270$		−	24471.9i		−	0.335691i
$271$	54675.2	+	54675.2i	0.744478	+	0.744478i	0.973436	−	0.228958i	$-0.0735318\pi$
$271$	54675.2	+	54675.2i	0.744478	+	0.744478i	−0.228958	+	0.973436i	$0.573532\pi$
$272$			26678.0i			0.360592i
$273$		0			0
$274$	−30858.1			−0.411025
$275$	−54535.9	+	54535.9i	−0.721135	+	0.721135i
$276$	44305.4			0.581619
$277$		−	54949.1i		−	0.716145i	−0.933694	−	0.358073i	$-0.883434\pi$
$277$		−	54949.1i		−	0.716145i	0.933694	−	0.358073i	$-0.116566\pi$
$278$	27170.3	−	27170.3i	0.351564	−	0.351564i
$279$	4239.89	−	4239.89i	0.0544686	−	0.0544686i
$280$	43756.9	+	43756.9i	0.558124	+	0.558124i
$281$	−24920.8	−	24920.8i	−0.315609	−	0.315609i	0.531469	−	0.847078i	$-0.321640\pi$
$281$	−24920.8	−	24920.8i	−0.315609	−	0.315609i	−0.847078	+	0.531469i	$0.821640\pi$
$282$	65836.4			0.827881
$283$			40742.6i			0.508716i	0.967110	+	0.254358i	$0.0818642\pi$
$283$			40742.6i			0.508716i	−0.967110	+	0.254358i	$0.918136\pi$
$284$	39788.9	+	39788.9i	0.493316	+	0.493316i
$285$			2499.43i			0.0307716i
$286$		0			0
$287$	204614.			2.48411
$288$	−8019.38	+	8019.38i	−0.0966842	+	0.0966842i
$289$	−90238.2			−1.08043
$290$		−	102250.i		−	1.21582i
$291$	−125210.	+	125210.i	−1.47860	+	1.47860i
$292$	9667.31	−	9667.31i	0.113381	−	0.113381i
$293$	−86357.9	−	86357.9i	−1.00593	−	1.00593i	−0.999982	−	0.00594591i	$-0.998107\pi$
$293$	−86357.9	−	86357.9i	−1.00593	−	1.00593i	−0.00594591	−	0.999982i	$-0.501893\pi$
$294$	−58273.1	−	58273.1i	−0.674177	−	0.674177i
$295$	21564.1			0.247792
$296$		−	4940.93i		−	0.0563930i
$297$	−12996.1	−	12996.1i	−0.147333	−	0.147333i
$298$			75650.9i			0.851886i
$299$		0			0
$300$	88485.9			0.983177
$301$	69192.0	−	69192.0i	0.763701	−	0.763701i
$302$	−42195.1			−0.462645
$303$		−	13954.9i		−	0.151999i
$304$	−239.876	+	239.876i	−0.00259561	+	0.00259561i
$305$	75269.1	−	75269.1i	0.809127	−	0.809127i
$306$	−52231.7	−	52231.7i	−0.557817	−	0.557817i
$307$	−16827.4	−	16827.4i	−0.178542	−	0.178542i	0.612178	−	0.790720i	$-0.290294\pi$
$307$	−16827.4	−	16827.4i	−0.178542	−	0.178542i	−0.790720	+	0.612178i	$0.790294\pi$
$308$	46475.2			0.489914
$309$		−	159817.i		−	1.67380i
$310$	−7530.66	−	7530.66i	−0.0783627	−	0.0783627i
$311$		−	91661.6i		−	0.947691i	−0.880608	−	0.473845i	$-0.842866\pi$
$311$		−	91661.6i		−	0.947691i	0.880608	−	0.473845i	$-0.157134\pi$
$312$		0			0
$313$	−97693.4			−0.997186			−0.498593	−	0.866836i	$-0.666150\pi$
$313$	−97693.4			−0.997186			−0.498593	+	0.866836i	$0.666150\pi$
$314$	8504.14	−	8504.14i	0.0862524	−	0.0862524i
$315$	−171340.			−1.72678
$316$			26405.4i			0.264435i
$317$	13766.4	−	13766.4i	0.136994	−	0.136994i	−0.635284	−	0.772278i	$-0.719117\pi$
$317$	13766.4	−	13766.4i	0.136994	−	0.136994i	0.772278	+	0.635284i	$0.219117\pi$
$318$	−60401.6	+	60401.6i	−0.597302	+	0.597302i
$319$	−54301.1	−	54301.1i	−0.533614	−	0.533614i
$320$	14243.6	+	14243.6i	0.139097	+	0.139097i
$321$	−195367.			−1.89601
$322$			90848.9i			0.876210i
$323$	−1562.36	−	1562.36i	−0.0149753	−	0.0149753i
$324$			61684.5i			0.587606i
$325$		0			0
$326$	38533.7			0.362581
$327$	−179071.	+	179071.i	−1.67467	+	1.67467i
$328$	66605.0			0.619098
$329$			134999.i			1.24720i
$330$	78816.4	−	78816.4i	0.723750	−	0.723750i
$331$	−37790.7	+	37790.7i	−0.344928	+	0.344928i	−0.858216	−	0.513288i	$-0.828427\pi$
$331$	−37790.7	+	37790.7i	−0.344928	+	0.344928i	0.513288	+	0.858216i	$0.328427\pi$
$332$	14427.8	+	14427.8i	0.130895	+	0.130895i
$333$	9673.63	+	9673.63i	0.0872370	+	0.0872370i
$334$	−111112.			−0.996021
$335$			134971.i			1.20268i
$336$	−37703.6	−	37703.6i	−0.333968	−	0.333968i
$337$			13090.4i			0.115264i	0.998338	+	0.0576319i	$0.0183550\pi$
$337$			13090.4i			0.115264i	−0.998338	+	0.0576319i	$0.981645\pi$
$338$		0			0
$339$	−35238.4			−0.306631
$340$	−92771.1	+	92771.1i	−0.802518	+	0.802518i
$341$	−7998.47			−0.0687857
$342$		−	939.284i		−	0.00803054i
$343$	1474.16	−	1474.16i	0.0125301	−	0.0125301i
$344$	22523.1	−	22523.1i	0.190332	−	0.190332i
$345$	154069.	+	154069.i	1.29443	+	1.29443i
$346$	−13.0427	−	13.0427i	−0.000108947	−	0.000108947i
$347$	61798.8			0.513241			0.256620	−	0.966512i	$-0.417391\pi$
$347$	61798.8			0.513241			0.256620	+	0.966512i	$0.417391\pi$
$348$			88105.0i			0.727515i
$349$	−131328.	−	131328.i	−1.07822	−	1.07822i	−0.996670	−	0.0815455i	$-0.974014\pi$
$349$	−131328.	−	131328.i	−1.07822	−	1.07822i	−0.0815455	−	0.996670i	$-0.525986\pi$
$350$			181442.i			1.48116i
$351$		0			0
$352$	15128.4			0.122098
$353$	137041.	−	137041.i	1.09977	−	1.09977i	0.105333	−	0.994437i	$-0.466409\pi$
$353$	137041.	−	137041.i	1.09977	−	1.09977i	0.994437	−	0.105333i	$-0.0335910\pi$
$354$	−18580.9			−0.148272
$355$			276727.i			2.19581i
$356$	23340.2	−	23340.2i	0.184164	−	0.184164i
$357$	245571.	−	245571.i	1.92682	−	1.92682i
$358$	−50686.9	−	50686.9i	−0.395485	−	0.395485i
$359$	−24773.9	−	24773.9i	−0.192223	−	0.192223i	0.604433	−	0.796656i	$-0.293400\pi$
$359$	−24773.9	−	24773.9i	−0.192223	−	0.192223i	−0.796656	+	0.604433i	$0.793400\pi$
$360$	−55773.7			−0.430353
$361$			130293.i			0.999784i
$362$	27020.0	+	27020.0i	0.206190	+	0.206190i
$363$			91766.6i			0.696420i
$364$		0			0
$365$	67234.9			0.504671
$366$	−64856.3	+	64856.3i	−0.484162	+	0.484162i
$367$	168676.			1.25234			0.626169	−	0.779687i	$-0.284622\pi$
$367$	168676.			1.25234			0.626169	+	0.779687i	$0.284622\pi$
$368$			29572.8i			0.218372i
$369$	−130403.	+	130403.i	−0.957712	+	0.957712i
$370$	17181.8	−	17181.8i	0.125506	−	0.125506i
$371$	−123854.	−	123854.i	−0.899837	−	0.899837i
$372$	6488.87	+	6488.87i	0.0468903	+	0.0468903i
$373$	104703.			0.752559			0.376279	−	0.926506i	$-0.377203\pi$
$373$	104703.			0.752559			0.376279	+	0.926506i	$0.377203\pi$
$374$			98534.2i			0.704439i
$375$	99310.6	+	99310.6i	0.706209	+	0.706209i
$376$			43944.2i			0.310832i
$377$		0			0
$378$	−43238.1			−0.302610
$379$	125500.	−	125500.i	0.873709	−	0.873709i	−0.119165	−	0.992874i	$-0.538022\pi$
$379$	125500.	−	125500.i	0.873709	−	0.873709i	0.992874	+	0.119165i	$0.0380219\pi$
$380$	−1668.30			−0.0115534
$381$			354373.i			2.44124i
$382$	−25998.5	+	25998.5i	−0.178165	+	0.178165i
$383$	20232.1	−	20232.1i	0.137925	−	0.137925i	−0.634773	−	0.772698i	$-0.718907\pi$
$383$	20232.1	−	20232.1i	0.137925	−	0.137925i	0.772698	+	0.634773i	$0.218907\pi$
$384$	−12273.1	−	12273.1i	−0.0832324	−	0.0832324i
$385$	161614.	+	161614.i	1.09033	+	1.09033i
$386$	−48506.0			−0.325552
$387$			88193.9i			0.588866i
$388$	−83574.3	−	83574.3i	−0.555149	−	0.555149i
$389$		−	139258.i		−	0.920283i	−0.887846	−	0.460141i	$-0.847799\pi$
$389$		−	139258.i		−	0.920283i	0.887846	−	0.460141i	$-0.152201\pi$
$390$		0			0
$391$	−192613.			−1.25989
$392$	38895.9	−	38895.9i	0.253123	−	0.253123i
$393$	192405.			1.24575
$394$		−	1399.39i		−	0.00901462i
$395$	−91823.2	+	91823.2i	−0.588516	+	0.588516i
$396$	−29619.2	+	29619.2i	−0.188879	+	0.188879i
$397$	25795.6	+	25795.6i	0.163668	+	0.163668i	0.784190	−	0.620521i	$-0.213079\pi$
$397$	25795.6	+	25795.6i	0.163668	+	0.163668i	−0.620521	+	0.784190i	$0.713079\pi$
$398$	−112287.	−	112287.i	−0.708865	−	0.708865i
$399$	4416.10			0.0277392
$400$			59062.1i			0.369138i
$401$	−94955.8	−	94955.8i	−0.590517	−	0.590517i	0.347254	−	0.937771i	$-0.387114\pi$
$401$	−94955.8	−	94955.8i	−0.590517	−	0.590517i	−0.937771	+	0.347254i	$0.887114\pi$
$402$		−	116299.i		−	0.719653i
$403$		0			0
$404$	9314.54			0.0570688
$405$	−214504.	+	214504.i	−1.30775	+	1.30775i
$406$	−180661.			−1.09600
$407$		−	18249.1i		−	0.110167i
$408$	79937.2	−	79937.2i	0.480207	−	0.480207i
$409$	8116.40	−	8116.40i	0.0485195	−	0.0485195i	−0.682431	−	0.730950i	$-0.739077\pi$
$409$	8116.40	−	8116.40i	0.0485195	−	0.0485195i	0.730950	+	0.682431i	$0.239077\pi$
$410$	231615.	+	231615.i	1.37784	+	1.37784i
$411$	92462.3	+	92462.3i	0.547370	+	0.547370i
$412$	106674.			0.628438
$413$		−	38100.4i		−	0.223373i
$414$	−57899.2	−	57899.2i	−0.337809	−	0.337809i
$415$			100343.i			0.582630i
$416$		0			0
$417$	−162824.			−0.936369
$418$	−885.971	+	885.971i	−0.00507069	+	0.00507069i
$419$	−223820.			−1.27488			−0.637441	−	0.770499i	$-0.720007\pi$
$419$	−223820.			−1.27488			−0.637441	+	0.770499i	$0.720007\pi$
$420$		−	262224.i		−	1.48653i
$421$	−231747.	+	231747.i	−1.30753	+	1.30753i	−0.384333	+	0.923195i	$0.625568\pi$
$421$	−231747.	+	231747.i	−1.30753	+	1.30753i	−0.923195	+	0.384333i	$0.874432\pi$
$422$	21141.3	−	21141.3i	0.118716	−	0.118716i
$423$	−86036.3	−	86036.3i	−0.480840	−	0.480840i
$424$	−40316.6	−	40316.6i	−0.224260	−	0.224260i
$425$	−384683.			−2.12973
$426$		−	238444.i		−	1.31392i
$427$	−132989.	−	132989.i	−0.729390	−	0.729390i
$428$		−	130403.i		−	0.711868i
$429$		0			0
$430$	156645.			0.847189
$431$	−205616.	+	205616.i	−1.10689	+	1.10689i	−0.113330	+	0.993557i	$0.536152\pi$
$431$	−205616.	+	205616.i	−1.10689	+	1.10689i	−0.993557	+	0.113330i	$0.963848\pi$
$432$	−14074.7			−0.0754173
$433$			222253.i			1.18542i	0.805416	+	0.592710i	$0.201942\pi$
$433$			222253.i			1.18542i	−0.805416	+	0.592710i	$0.798058\pi$
$434$	−13305.5	+	13305.5i	−0.0706403	+	0.0706403i
$435$	−306379.	+	306379.i	−1.61913	+	1.61913i
$436$	−119526.	−	119526.i	−0.628764	−	0.628764i
$437$	−1731.88	−	1731.88i	−0.00906891	−	0.00906891i
$438$	−57933.6			−0.301983
$439$		−	303638.i		−	1.57553i	−0.615975	−	0.787765i	$-0.711238\pi$
$439$		−	303638.i		−	1.57553i	0.615975	−	0.787765i	$-0.288762\pi$
$440$	52608.0	+	52608.0i	0.271736	+	0.271736i
$441$			152305.i			0.783135i
$442$		0			0
$443$	103827.			0.529060			0.264530	−	0.964377i	$-0.414783\pi$
$443$	103827.			0.529060			0.264530	+	0.964377i	$0.414783\pi$
$444$	−14804.8	+	14804.8i	−0.0750996	+	0.0750996i
$445$	162328.			0.819736
$446$		−	109086.i		−	0.548401i
$447$	226678.	−	226678.i	1.13447	−	1.13447i
$448$	25166.2	−	25166.2i	0.125390	−	0.125390i
$449$	−11231.3	−	11231.3i	−0.0557104	−	0.0557104i	0.678703	−	0.734413i	$-0.262542\pi$
$449$	−11231.3	−	11231.3i	−0.0557104	−	0.0557104i	−0.734413	+	0.678703i	$0.762542\pi$
$450$	−115635.	−	115635.i	−0.571038	−	0.571038i
$451$	246003.			1.20945
$452$		−	23520.7i		−	0.115126i
$453$	126432.	+	126432.i	0.616113	+	0.616113i
$454$			169943.i			0.824500i
$455$		0			0
$456$	1437.51			0.00691324
$457$	−162517.	+	162517.i	−0.778155	+	0.778155i	−0.979517	−	0.201362i	$-0.935463\pi$
$457$	−162517.	+	162517.i	−0.778155	+	0.778155i	0.201362	+	0.979517i	$0.435463\pi$
$458$	−115187.			−0.549124
$459$		−	91671.1i		−	0.435118i
$460$	−102837.	+	102837.i	−0.485999	+	0.485999i
$461$	−37275.2	+	37275.2i	−0.175396	+	0.175396i	−0.789345	−	0.613950i	$-0.789580\pi$
$461$	−37275.2	+	37275.2i	−0.175396	+	0.175396i	0.613950	+	0.789345i	$0.289580\pi$
$462$	−139257.	−	139257.i	−0.652427	−	0.652427i
$463$	−137439.	−	137439.i	−0.641134	−	0.641134i	0.309700	−	0.950834i	$-0.399771\pi$
$463$	−137439.	−	137439.i	−0.641134	−	0.641134i	−0.950834	+	0.309700i	$0.899771\pi$
$464$	−58807.9			−0.273149
$465$			45129.2i			0.208714i
$466$	108914.	+	108914.i	0.501546	+	0.501546i
$467$			49120.0i			0.225229i	0.993639	+	0.112615i	$0.0359225\pi$
$467$			49120.0i			0.225229i	−0.993639	+	0.112615i	$0.964077\pi$
$468$		0			0
$469$	238473.			1.08416
$470$	−152813.	+	152813.i	−0.691774	+	0.691774i
$471$	−50963.0			−0.229728
$472$		−	12402.3i		−	0.0556696i
$473$	83188.1	−	83188.1i	0.371825	−	0.371825i
$474$	79120.4	−	79120.4i	0.352153	−	0.352153i
$475$	−3458.88	−	3458.88i	−0.0153302	−	0.0153302i
$476$	163912.	+	163912.i	0.723433	+	0.723433i
$477$	157868.			0.693837
$478$		−	128358.i		−	0.561782i
$479$	262401.	+	262401.i	1.14365	+	1.14365i	0.987777	+	0.155875i	$0.0498197\pi$
$479$	262401.	+	262401.i	1.14365	+	1.14365i	0.155875	+	0.987777i	$0.450180\pi$
$480$		−	85357.9i		−	0.370477i
$481$		0			0
$482$	277582.			1.19481
$483$	272217.	−	272217.i	1.16686	−	1.16686i
$484$	−61251.9			−0.261474
$485$		−	581248.i		−	2.47103i
$486$	149203.	−	149203.i	0.631691	−	0.631691i
$487$	−52753.7	+	52753.7i	−0.222431	+	0.222431i	−0.809521	−	0.587090i	$-0.800273\pi$
$487$	−52753.7	+	52753.7i	−0.222431	+	0.222431i	0.587090	+	0.809521i	$0.300273\pi$
$488$	−43290.0	−	43290.0i	−0.181781	−	0.181781i
$489$	−115461.	−	115461.i	−0.482856	−	0.482856i
$490$	270516.			1.12668
$491$			17315.8i			0.0718258i	0.999355	+	0.0359129i	$0.0114339\pi$
$491$			17315.8i			0.0718258i	−0.999355	+	0.0359129i	$0.988566\pi$
$492$	−199573.	−	199573.i	−0.824464	−	0.824464i
$493$		−	383027.i		−	1.57593i
$494$		0			0
$495$	−205998.			−0.840721
$496$	−4331.16	+	4331.16i	−0.0176052	+	0.0176052i
$497$	488934.			1.97942
$498$		−	86462.0i		−	0.348631i
$499$	−76303.8	+	76303.8i	−0.306440	+	0.306440i	−0.843527	−	0.537087i	$-0.819525\pi$
$499$	−76303.8	+	76303.8i	−0.306440	+	0.306440i	0.537087	+	0.843527i	$0.319525\pi$
$500$	−66287.4	+	66287.4i	−0.265149	+	0.265149i
$501$	332933.	+	332933.i	1.32642	+	1.32642i
$502$	59461.1	+	59461.1i	0.235953	+	0.235953i
$503$	−332406.			−1.31381			−0.656906	−	0.753973i	$-0.728135\pi$
$503$	−332406.			−1.31381			−0.656906	+	0.753973i	$0.728135\pi$
$504$			98543.6i			0.387943i
$505$	32390.7	+	32390.7i	0.127010	+	0.127010i
$506$			109226.i			0.426603i
$507$		0			0
$508$	−236535.			−0.916575
$509$	103718.	−	103718.i	0.400330	−	0.400330i	−0.478020	−	0.878349i	$-0.658645\pi$
$509$	103718.	−	103718.i	0.400330	−	0.400330i	0.878349	+	0.478020i	$0.158645\pi$
$510$	555952.			2.13746
$511$		−	118794.i		−	0.454938i
$512$	8192.00	−	8192.00i	0.0312500	−	0.0312500i
$513$	824.261	−	824.261i	0.00313206	−	0.00313206i
$514$	25673.5	+	25673.5i	0.0971759	+	0.0971759i
$515$	370950.	+	370950.i	1.39862	+	1.39862i
$516$	−134975.			−0.506937
$517$			162306.i			0.607230i
$518$	−30357.6	−	30357.6i	−0.113138	−	0.113138i
$519$			78.1612i			0.000290173i
$520$		0			0
$521$	−510268.			−1.87985			−0.939923	−	0.341385i	$-0.889104\pi$
$521$	−510268.			−1.87985			−0.939923	+	0.341385i	$0.889104\pi$
$522$	115137.	−	115137.i	0.422547	−	0.422547i
$523$	−531190.			−1.94199			−0.970993	−	0.239108i	$-0.923145\pi$
$523$	−531190.			−1.94199			−0.970993	+	0.239108i	$0.923145\pi$
$524$			128425.i			0.467723i
$525$	543666.	−	543666.i	1.97248	−	1.97248i
$526$	119700.	−	119700.i	0.432636	−	0.432636i
$527$	−28209.7	−	28209.7i	−0.101573	−	0.101573i
$528$	−45330.2	−	45330.2i	−0.162600	−	0.162600i
$529$	66328.4			0.237022
$530$		−	280396.i		−	0.998207i
$531$	24281.9	+	24281.9i	0.0861179	+	0.0861179i
$532$			2947.64i			0.0104148i
$533$		0			0
$534$	−139872.			−0.490510
$535$	453467.	−	453467.i	1.58430	−	1.58430i
$536$	77626.6			0.270197
$537$			303753.i			1.05335i
$538$	−161315.	+	161315.i	−0.557329	+	0.557329i
$539$	143660.	−	143660.i	0.494491	−	0.494491i
$540$	−48943.8	−	48943.8i	−0.167846	−	0.167846i
$541$	−266982.	−	266982.i	−0.912194	−	0.912194i	0.0842501	−	0.996445i	$-0.473151\pi$
$541$	−266982.	−	266982.i	−0.912194	−	0.912194i	−0.996445	+	0.0842501i	$0.973151\pi$
$542$	218701.			0.744478
$543$		−	161923.i		−	0.549174i
$544$	53356.1	+	53356.1i	0.180296	+	0.180296i
$545$		−	831284.i		−	2.79870i
$546$		0			0
$547$	412153.			1.37748			0.688738	−	0.725010i	$-0.258165\pi$
$547$	412153.			1.37748			0.688738	+	0.725010i	$0.258165\pi$
$548$	−61716.3	+	61716.3i	−0.205513	+	0.205513i
$549$	169511.			0.562411
$550$			218143.i			0.721135i
$551$	3443.99	−	3443.99i	0.0113438	−	0.0113438i
$552$	88610.8	−	88610.8i	0.290809	−	0.290809i
$553$	162238.	+	162238.i	0.530519	+	0.530519i
$554$	−109898.	−	109898.i	−0.358073	−	0.358073i
$555$	−102966.			−0.334277
$556$		−	108681.i		−	0.351564i
$557$	147814.	+	147814.i	0.476438	+	0.476438i	0.903990	−	0.427553i	$-0.140624\pi$
$557$	147814.	+	147814.i	0.476438	+	0.476438i	−0.427553	+	0.903990i	$0.640624\pi$
$558$		−	16959.6i		−	0.0544686i
$559$		0			0
$560$	175028.			0.558124
$561$	295244.	−	295244.i	0.938115	−	0.938115i
$562$	−99683.1			−0.315609
$563$		−	213487.i		−	0.673527i	−0.941589	−	0.336763i	$-0.890668\pi$
$563$		−	213487.i		−	0.673527i	0.941589	−	0.336763i	$-0.109332\pi$
$564$	131673.	−	131673.i	0.413940	−	0.413940i
$565$	81791.8	−	81791.8i	0.256220	−	0.256220i
$566$	81485.2	+	81485.2i	0.254358	+	0.254358i
$567$	378996.	+	378996.i	1.17888	+	1.17888i
$568$	159156.			0.493316
$569$		−	430345.i		−	1.32921i	−0.747196	−	0.664603i	$-0.768601\pi$
$569$		−	430345.i		−	1.32921i	0.747196	−	0.664603i	$-0.231399\pi$
$570$	4998.85	+	4998.85i	0.0153858	+	0.0153858i
$571$			413840.i			1.26929i	0.772805	+	0.634644i	$0.218853\pi$
$571$			413840.i			1.26929i	−0.772805	+	0.634644i	$0.781147\pi$
$572$		0			0
$573$	155802.			0.474530
$574$	409228.	−	409228.i	1.24206	−	1.24206i
$575$	−426423.			−1.28975
$576$			32077.5i			0.0966842i
$577$	−349122.	+	349122.i	−1.04864	+	1.04864i	−0.0498841	+	0.998755i	$0.515885\pi$
$577$	−349122.	+	349122.i	−1.04864	+	1.04864i	−0.998755	+	0.0498841i	$0.984115\pi$
$578$	−180476.	+	180476.i	−0.540213	+	0.540213i
$579$	145342.	+	145342.i	0.433544	+	0.433544i
$580$	−204501.	−	204501.i	−0.607909	−	0.607909i
$581$	177292.			0.525214
$582$			500838.i			1.47860i
$583$	−148908.	−	148908.i	−0.438106	−	0.438106i
$584$		−	38669.2i		−	0.113381i
$585$		0			0
$586$	−345432.			−1.00593
$587$	5609.16	−	5609.16i	0.0162788	−	0.0162788i	−0.698921	−	0.715199i	$-0.746336\pi$
$587$	5609.16	−	5609.16i	0.0162788	−	0.0162788i	0.715199	+	0.698921i	$0.246336\pi$
$588$	−233092.			−0.674177
$589$		−	507.295i		−	0.00146228i
$590$	43128.2	−	43128.2i	0.123896	−	0.123896i
$591$	−4193.09	+	4193.09i	−0.0120049	+	0.0120049i
$592$	−9881.86	−	9881.86i	−0.0281965	−	0.0281965i
$593$	−135707.	−	135707.i	−0.385917	−	0.385917i	0.487312	−	0.873228i	$-0.337978\pi$
$593$	−135707.	−	135707.i	−0.385917	−	0.385917i	−0.873228	+	0.487312i	$0.837978\pi$
$594$	−51984.2			−0.147333
$595$			1.13999e6i			3.22008i
$596$	151302.	+	151302.i	0.425943	+	0.425943i
$597$			672907.i			1.88802i
$598$		0			0
$599$	43856.1			0.122230			0.0611149	−	0.998131i	$-0.480534\pi$
$599$	43856.1			0.122230			0.0611149	+	0.998131i	$0.480534\pi$
$600$	176972.	−	176972.i	0.491588	−	0.491588i
$601$	−239364.			−0.662690			−0.331345	−	0.943510i	$-0.607502\pi$
$601$	−239364.			−0.662690			−0.331345	+	0.943510i	$0.607502\pi$
$602$		−	276768.i		−	0.763701i
$603$	−151982.	+	151982.i	−0.417981	+	0.417981i
$604$	−84390.1	+	84390.1i	−0.231322	+	0.231322i
$605$	−213000.	−	213000.i	−0.581926	−	0.581926i
$606$	−27909.8	−	27909.8i	−0.0759995	−	0.0759995i
$607$	713286.			1.93591			0.967957	−	0.251116i	$-0.0807975\pi$
$607$	713286.			1.93591			0.967957	+	0.251116i	$0.0807975\pi$
$608$			959.503i			0.00259561i
$609$	541326.	+	541326.i	1.45957	+	1.45957i
$610$		−	301076.i		−	0.809127i
$611$		0			0
$612$	−208927.			−0.557817
$613$	−25856.4	+	25856.4i	−0.0688093	+	0.0688093i	−0.740674	−	0.671865i	$-0.765494\pi$
$613$	−25856.4	+	25856.4i	−0.0688093	+	0.0688093i	0.671865	+	0.740674i	$0.265494\pi$
$614$	−67309.7			−0.178542
$615$		−	1.38800e6i		−	3.66978i
$616$	92950.3	−	92950.3i	0.244957	−	0.244957i
$617$	−344111.	+	344111.i	−0.903917	+	0.903917i	−0.995772	−	0.0918557i	$-0.970720\pi$
$617$	−344111.	+	344111.i	−0.903917	+	0.903917i	0.0918557	+	0.995772i	$0.470720\pi$
$618$	−319633.	−	319633.i	−0.836902	−	0.836902i
$619$	−112057.	−	112057.i	−0.292453	−	0.292453i	0.545596	−	0.838049i	$-0.316303\pi$
$619$	−112057.	−	112057.i	−0.292453	−	0.292453i	−0.838049	+	0.545596i	$0.816303\pi$
$620$	−30122.6			−0.0783627
$621$		−	101618.i		−	0.263504i
$622$	−183323.	−	183323.i	−0.473845	−	0.473845i
$623$		−	286809.i		−	0.738954i
$624$		0			0
$625$	115759.			0.296343
$626$	−195387.	+	195387.i	−0.498593	+	0.498593i
$627$	5309.38			0.0135055
$628$		−	34016.6i		−	0.0862524i
$629$	64362.5	−	64362.5i	0.162679	−	0.162679i
$630$	−342679.	+	342679.i	−0.863389	+	0.863389i
$631$	59131.7	+	59131.7i	0.148512	+	0.148512i	0.777453	−	0.628941i	$-0.216511\pi$
$631$	59131.7	+	59131.7i	0.148512	+	0.148512i	−0.628941	+	0.777453i	$0.716511\pi$
$632$	52810.9	+	52810.9i	0.132218	+	0.132218i
$633$	−126694.			−0.316191
$634$		−	55065.5i		−	0.136994i
$635$	−822535.	−	822535.i	−2.03989	−	2.03989i
$636$			241606.i			0.597302i
$637$		0			0
$638$	−217204.			−0.533614
$639$	−311604.	+	311604.i	−0.763135	+	0.763135i
$640$	56974.3			0.139097
$641$			195783.i			0.476496i	0.971204	+	0.238248i	$0.0765731\pi$
$641$			195783.i			0.476496i	−0.971204	+	0.238248i	$0.923427\pi$
$642$	−390734.	+	390734.i	−0.948007	+	0.948007i
$643$	−241575.	+	241575.i	−0.584293	+	0.584293i	−0.936080	−	0.351787i	$-0.885574\pi$
$643$	−241575.	+	241575.i	−0.584293	+	0.584293i	0.351787	+	0.936080i	$0.385574\pi$
$644$	181698.	+	181698.i	0.438105	+	0.438105i
$645$	−469366.	−	469366.i	−1.12822	−	1.12822i
$646$	−6249.43			−0.0149753
$647$		−	778145.i		−	1.85888i	−0.368970	−	0.929441i	$-0.620289\pi$
$647$		−	778145.i		−	1.85888i	0.368970	−	0.929441i	$-0.379711\pi$
$648$	123369.	+	123369.i	0.293803	+	0.293803i
$649$		−	45807.3i		−	0.108754i
$650$		0			0
$651$	79736.5			0.188146
$652$	77067.3	−	77067.3i	0.181290	−	0.181290i
$653$	542695.			1.27271			0.636355	−	0.771396i	$-0.280441\pi$
$653$	542695.			1.27271			0.636355	+	0.771396i	$0.280441\pi$
$654$			716285.i			1.67467i
$655$	−446591.	+	446591.i	−1.04094	+	1.04094i
$656$	133210.	−	133210.i	0.309549	−	0.309549i
$657$	75708.8	+	75708.8i	0.175394	+	0.175394i
$658$	269997.	+	269997.i	0.623602	+	0.623602i
$659$	776835.			1.78878			0.894392	−	0.447284i	$-0.147609\pi$
$659$	776835.			1.78878			0.894392	+	0.447284i	$0.147609\pi$
$660$		−	315266.i		−	0.723750i
$661$	−461004.	−	461004.i	−1.05512	−	1.05512i	−0.998389	−	0.0567313i	$-0.981932\pi$
$661$	−461004.	−	461004.i	−1.05512	−	1.05512i	−0.0567313	−	0.998389i	$-0.518068\pi$
$662$			151163.i			0.344928i
$663$		0			0
$664$	57711.2			0.130895
$665$	−10250.2	+	10250.2i	−0.0231788	+	0.0231788i
$666$	38694.5			0.0872370
$667$		−	424588.i		−	0.954368i
$668$	−222224.	+	222224.i	−0.498011	+	0.498011i
$669$	−326861.	+	326861.i	−0.730316	+	0.730316i
$670$	269941.	+	269941.i	0.601339	+	0.601339i
$671$	−159890.	−	159890.i	−0.355120	−	0.355120i
$672$	−150814.			−0.333968
$673$			734600.i			1.62189i	0.585124	+	0.810944i	$0.301046\pi$
$673$			734600.i			1.62189i	−0.585124	+	0.810944i	$0.698954\pi$
$674$	26180.8	+	26180.8i	0.0576319	+	0.0576319i
$675$		−	202949.i		−	0.445431i
$676$		0			0
$677$	−275668.			−0.601464			−0.300732	−	0.953709i	$-0.597231\pi$
$677$	−275668.			−0.601464			−0.300732	+	0.953709i	$0.597231\pi$
$678$	−70476.7	+	70476.7i	−0.153316	+	0.153316i
$679$	−1.02698e6			−2.22752
$680$			371084.i			0.802518i
$681$	509210.	−	509210.i	1.09800	−	1.09800i
$682$	−15996.9	+	15996.9i	−0.0343929	+	0.0343929i
$683$	−19994.2	−	19994.2i	−0.0428611	−	0.0428611i	0.685351	−	0.728213i	$-0.259649\pi$
$683$	−19994.2	−	19994.2i	−0.0428611	−	0.0428611i	−0.728213	+	0.685351i	$0.759649\pi$
$684$	−1878.57	−	1878.57i	−0.00401527	−	0.00401527i
$685$	−429228.			−0.914760
$686$		−	5896.62i		−	0.0125301i
$687$	345141.	+	345141.i	0.731279	+	0.731279i
$688$		−	90092.4i		−	0.190332i
$689$		0			0
$690$	616276.			1.29443
$691$	232076.	−	232076.i	0.486042	−	0.486042i	−0.421013	−	0.907055i	$-0.638325\pi$
$691$	232076.	−	232076.i	0.486042	−	0.486042i	0.907055	+	0.421013i	$0.138325\pi$
$692$	−52.1707			−0.000108947
$693$			363967.i			0.757871i
$694$	123598.	−	123598.i	0.256620	−	0.256620i
$695$	377932.	−	377932.i	0.782427	−	0.782427i
$696$	176210.	+	176210.i	0.363758	+	0.363758i
$697$	867623.	+	867623.i	1.78593	+	1.78593i
$698$	−525311.			−1.07822
$699$		−	652691.i		−	1.33584i
$700$	362884.	+	362884.i	0.740579	+	0.740579i
$701$			645341.i			1.31327i	0.754209	+	0.656634i	$0.228020\pi$
$701$			645341.i			1.31327i	−0.754209	+	0.656634i	$0.771980\pi$
$702$		0			0
$703$	1157.43			0.00234199
$704$	30256.8	−	30256.8i	0.0610489	−	0.0610489i
$705$	915767.			1.84250
$706$		−	548165.i		−	1.09977i
$707$	57229.4	−	57229.4i	0.114493	−	0.114493i
$708$	−37161.8	+	37161.8i	−0.0741362	+	0.0741362i
$709$	−66968.7	−	66968.7i	−0.133223	−	0.133223i	0.637351	−	0.770574i	$-0.280030\pi$
$709$	−66968.7	−	66968.7i	−0.133223	−	0.133223i	−0.770574	+	0.637351i	$0.780030\pi$
$710$	553454.	+	553454.i	1.09790	+	1.09790i
$711$	−206792.			−0.409067
$712$		−	93361.0i		−	0.184164i
$713$	−31270.6	−	31270.6i	−0.0615116	−	0.0615116i
$714$		−	982284.i		−	1.92682i
$715$		0			0
$716$	−202748.			−0.395485
$717$	−384608.	+	384608.i	−0.748135	+	0.748135i
$718$	−99095.8			−0.192223
$719$			667521.i			1.29124i	0.763659	+	0.645620i	$0.223401\pi$
$719$			667521.i			1.29124i	−0.763659	+	0.645620i	$0.776599\pi$
$720$	−111547.	+	111547.i	−0.215176	+	0.215176i
$721$	655413.	−	655413.i	1.26079	−	1.26079i
$722$	260586.	+	260586.i	0.499892	+	0.499892i
$723$	−831738.	−	831738.i	−1.59115	−	1.59115i
$724$	108080.			0.206190
$725$		−	847979.i		−	1.61328i
$726$	183533.	+	183533.i	0.348210	+	0.348210i
$727$			208700.i			0.394870i	0.980316	+	0.197435i	$0.0632611\pi$
$727$			208700.i			0.394870i	−0.980316	+	0.197435i	$0.936739\pi$
$728$		0			0
$729$	−269577.			−0.507257
$730$	134470.	−	134470.i	0.252336	−	0.252336i
$731$	586789.			1.09811
$732$			259425.i			0.484162i
$733$	351672.	−	351672.i	0.654530	−	0.654530i	−0.299550	−	0.954080i	$-0.596837\pi$
$733$	351672.	−	351672.i	0.654530	−	0.654530i	0.954080	+	0.299550i	$0.0968367\pi$
$734$	337352.	−	337352.i	0.626169	−	0.626169i
$735$	−810564.	−	810564.i	−1.50042	−	1.50042i
$736$	59145.5	+	59145.5i	0.109186	+	0.109186i
$737$	286710.			0.527847
$738$			521612.i			0.957712i
$739$	350623.	+	350623.i	0.642024	+	0.642024i	0.951053	−	0.309029i	$-0.100004\pi$
$739$	350623.	+	350623.i	0.642024	+	0.642024i	−0.309029	+	0.951053i	$0.600004\pi$
$740$		−	68727.0i		−	0.125506i
$741$		0			0
$742$	−495418.			−0.899837
$743$	−619690.	+	619690.i	−1.12253	+	1.12253i	−0.131166	+	0.991360i	$0.541872\pi$
$743$	−619690.	+	619690.i	−1.12253	+	1.12253i	−0.991360	+	0.131166i	$0.958128\pi$
$744$	25955.5			0.0468903
$745$			1.05228e6i			1.89592i
$746$	209405.	−	209405.i	0.376279	−	0.376279i
$747$	−112990.	+	112990.i	−0.202488	+	0.202488i
$748$	197068.	+	197068.i	0.352220	+	0.352220i
$749$	−801207.	−	801207.i	−1.42817	−	1.42817i
$750$	397242.			0.706209
$751$		−	256027.i		−	0.453947i	−0.973901	−	0.226974i	$-0.927117\pi$
$751$		−	256027.i		−	0.453947i	0.973901	−	0.226974i	$-0.0728831\pi$
$752$	87888.3	+	87888.3i	0.155416	+	0.155416i
$753$		−	356335.i		−	0.628446i
$754$		0			0
$755$	−586922.			−1.02964
$756$	−86476.2	+	86476.2i	−0.151305	+	0.151305i
$757$	963391.			1.68117			0.840583	−	0.541683i	$-0.182213\pi$
$757$	963391.			1.68117			0.840583	+	0.541683i	$0.182213\pi$
$758$		−	502002.i		−	0.873709i
$759$	327280.	−	327280.i	0.568115	−	0.568115i
$760$	−3336.61	+	3336.61i	−0.00577668	+	0.00577668i
$761$	276390.	+	276390.i	0.477257	+	0.477257i	0.904253	−	0.426996i	$-0.140428\pi$
$761$	276390.	+	276390.i	0.477257	+	0.477257i	−0.426996	+	0.904253i	$0.640428\pi$
$762$	708746.	+	708746.i	1.22062	+	1.22062i
$763$	−1.46875e6			−2.52290
$764$			103994.i			0.178165i
$765$	−726530.	−	726530.i	−1.24145	−	1.24145i
$766$		−	80928.3i		−	0.137925i
$767$		0			0
$768$	−49092.5			−0.0832324
$769$	−132903.	+	132903.i	−0.224740	+	0.224740i	−0.810491	−	0.585751i	$-0.800800\pi$
$769$	−132903.	+	132903.i	−0.224740	+	0.224740i	0.585751	+	0.810491i	$0.300800\pi$
$770$	646457.			1.09033
$771$		−	153854.i		−	0.258822i
$772$	−97011.9	+	97011.9i	−0.162776	+	0.162776i
$773$	372392.	−	372392.i	0.623221	−	0.623221i	−0.323133	−	0.946354i	$-0.604736\pi$
$773$	372392.	−	372392.i	0.623221	−	0.623221i	0.946354	+	0.323133i	$0.104736\pi$
$774$	176388.	+	176388.i	0.294433	+	0.294433i
$775$	−62453.0	−	62453.0i	−0.103980	−	0.103980i
$776$	−334297.			−0.555149
$777$			181925.i			0.301335i
$778$	−278516.	−	278516.i	−0.460141	−	0.460141i
$779$			15602.5i			0.0257110i
$780$		0			0
$781$	587835.			0.963725
$782$	−385226.	+	385226.i	−0.629944	+	0.629944i
$783$	202076.			0.329603
$784$		−	155583.i		−	0.253123i
$785$	118290.	−	118290.i	0.191960	−	0.191960i
$786$	384810.	−	384810.i	0.622875	−	0.622875i
$787$	−273464.	−	273464.i	−0.441521	−	0.441521i	0.451002	−	0.892523i	$-0.351067\pi$
$787$	−273464.	−	273464.i	−0.441521	−	0.441521i	−0.892523	+	0.451002i	$0.851067\pi$
$788$	−2798.79	−	2798.79i	−0.00450731	−	0.00450731i
$789$	−717330.			−1.15230
$790$			367293.i			0.588516i
$791$	−144514.	−	144514.i	−0.230970	−	0.230970i
$792$			118477.i			0.188879i
$793$		0			0
$794$	103182.			0.163668
$795$	−840170.	+	840170.i	−1.32933	+	1.32933i
$796$	−449148.			−0.708865
$797$		−	838853.i		−	1.32059i	−0.751005	−	0.660297i	$-0.770431\pi$
$797$		−	838853.i		−	1.32059i	0.751005	−	0.660297i	$-0.229569\pi$
$798$	8832.21	−	8832.21i	0.0138696	−	0.0138696i
$799$	−572433.	+	572433.i	−0.896668	+	0.896668i
$800$	118124.	+	118124.i	0.184569	+	0.184569i
$801$	182787.	+	182787.i	0.284892	+	0.284892i
$802$	−379823.			−0.590517
$803$		−	142823.i		−	0.221497i
$804$	−232598.	−	232598.i	−0.359826	−	0.359826i
$805$			1.26368e6i			1.95006i
$806$		0			0
$807$	966720.			1.48441
$808$	18629.1	−	18629.1i	0.0285344	−	0.0285344i
$809$	−560469.			−0.856356			−0.428178	−	0.903694i	$-0.640844\pi$
$809$	−560469.			−0.856356			−0.428178	+	0.903694i	$0.640844\pi$
$810$			858016.i			1.30775i
$811$	2176.16	−	2176.16i	0.00330863	−	0.00330863i	−0.705451	−	0.708759i	$-0.749255\pi$
$811$	2176.16	−	2176.16i	0.00330863	−	0.00330863i	0.708759	+	0.705451i	$0.249255\pi$
$812$	−361322.	+	361322.i	−0.548002	+	0.548002i
$813$	−655308.	−	655308.i	−0.991436	−	0.991436i
$814$	−36498.2	−	36498.2i	−0.0550837	−	0.0550837i
$815$	535993.			0.806945
$816$		−	319749.i		−	0.480207i
$817$	5276.12	+	5276.12i	0.00790443	+	0.00790443i
$818$		−	32465.6i		−	0.0485195i
$819$		0			0
$820$	926458.			1.37784
$821$	79463.3	−	79463.3i	0.117891	−	0.117891i	−0.645700	−	0.763591i	$-0.723434\pi$
$821$	79463.3	−	79463.3i	0.117891	−	0.117891i	0.763591	+	0.645700i	$0.223434\pi$
$822$	369849.			0.547370
$823$			651622.i			0.962047i	0.876708	+	0.481023i	$0.159735\pi$
$823$			651622.i			0.962047i	−0.876708	+	0.481023i	$0.840265\pi$
$824$	213347.	−	213347.i	0.314219	−	0.314219i
$825$	653638.	−	653638.i	0.960349	−	0.960349i
$826$	−76200.9	−	76200.9i	−0.111686	−	0.111686i
$827$	458846.	+	458846.i	0.670897	+	0.670897i	0.957923	−	0.287026i	$-0.0926665\pi$
$827$	458846.	+	458846.i	0.670897	+	0.670897i	−0.287026	+	0.957923i	$0.592667\pi$
$828$	−231597.			−0.337809
$829$			1.22076e6i			1.77631i	0.459540	+	0.888157i	$0.348014\pi$
$829$			1.22076e6i			1.77631i	−0.459540	+	0.888157i	$0.651986\pi$
$830$	200687.	+	200687.i	0.291315	+	0.291315i
$831$			658590.i			0.953703i
$832$		0			0
$833$	1.01334e6			1.46038
$834$	−325649.	+	325649.i	−0.468185	+	0.468185i
$835$	−1.54554e6			−2.21670
$836$			3543.88i			0.00507069i
$837$	14882.7	−	14882.7i	0.0212438	−	0.0212438i
$838$	−447639.	+	447639.i	−0.637441	+	0.637441i
$839$	−608696.	−	608696.i	−0.864722	−	0.864722i	0.127160	−	0.991882i	$-0.459414\pi$
$839$	−608696.	−	608696.i	−0.864722	−	0.864722i	−0.991882	+	0.127160i	$0.959414\pi$
$840$	−524447.	−	524447.i	−0.743264	−	0.743264i
$841$	137048.			0.193767
$842$			926990.i			1.30753i
$843$	298687.	+	298687.i	0.420302	+	0.420302i
$844$		−	84565.3i		−	0.118716i
$845$		0			0
$846$	−344145.			−0.480840
$847$	−376338.	+	376338.i	−0.524579	+	0.524579i
$848$	−161266.			−0.224260
$849$		−	488319.i		−	0.677467i
$850$	−769366.	+	769366.i	−1.06487	+	1.06487i
$851$	71346.1	−	71346.1i	0.0985171	−	0.0985171i
$852$	−476889.	−	476889.i	−0.656959	−	0.656959i
$853$	−835615.	−	835615.i	−1.14844	−	1.14844i	−0.986860	−	0.161579i	$-0.948341\pi$
$853$	−835615.	−	835615.i	−1.14844	−	1.14844i	−0.161579	−	0.986860i	$-0.551659\pi$
$854$	−531956.			−0.729390
$855$		−	13065.2i		−	0.0178724i
$856$	−260805.	−	260805.i	−0.355934	−	0.355934i
$857$		−	1.00820e6i		−	1.37273i	−0.727256	−	0.686366i	$-0.759205\pi$
$857$		−	1.00820e6i		−	1.37273i	0.727256	−	0.686366i	$-0.240795\pi$
$858$		0			0
$859$	−766118.			−1.03827			−0.519134	−	0.854693i	$-0.673745\pi$
$859$	−766118.			−1.03827			−0.519134	+	0.854693i	$0.673745\pi$
$860$	313290.	−	313290.i	0.423594	−	0.423594i
$861$	−2.45239e6			−3.30814
$862$			822466.i			1.10689i
$863$	−205886.	+	205886.i	−0.276442	+	0.276442i	−0.831687	−	0.555245i	$-0.812625\pi$
$863$	−205886.	+	205886.i	−0.276442	+	0.276442i	0.555245	+	0.831687i	$0.312625\pi$
$864$	−28149.4	+	28149.4i	−0.0377087	+	0.0377087i
$865$	−181.420	−	181.420i	−0.000242467	−	0.000242467i
$866$	444506.	+	444506.i	0.592710	+	0.592710i
$867$	1.08155e6			1.43882
$868$			53222.1i			0.0706403i
$869$	195055.	+	195055.i	0.258296	+	0.258296i
$870$			1.22552e6i			1.61913i
$871$		0			0
$872$	−478102.			−0.628764
$873$	654506.	−	654506.i	0.858786	−	0.858786i
$874$	−6927.53			−0.00906891
$875$			814552.i			1.06390i
$876$	−115867.	+	115867.i	−0.150991	+	0.150991i
$877$	888049.	−	888049.i	1.15462	−	1.15462i	0.169001	−	0.985616i	$-0.445946\pi$
$877$	888049.	−	888049.i	1.15462	−	1.15462i	0.985616	−	0.169001i	$-0.0540540\pi$
$878$	−607276.	−	607276.i	−0.787765	−	0.787765i
$879$	1.03504e6	+	1.03504e6i	1.33961	+	1.33961i
$880$	210432.			0.271736
$881$		−	375767.i		−	0.484136i	−0.970259	−	0.242068i	$-0.922174\pi$
$881$		−	375767.i		−	0.484136i	0.970259	−	0.242068i	$-0.0778256\pi$
$882$	304610.	+	304610.i	0.391568	+	0.391568i
$883$		−	420890.i		−	0.539818i	−0.962886	−	0.269909i	$-0.913006\pi$
$883$		−	420890.i		−	0.539818i	0.962886	−	0.269909i	$-0.0869936\pi$
$884$		0			0
$885$	−258456.			−0.329989
$886$	207655.	−	207655.i	0.264530	−	0.264530i
$887$	1.04182e6			1.32417			0.662086	−	0.749428i	$-0.269671\pi$
$887$	1.04182e6			1.32417			0.662086	+	0.749428i	$0.269671\pi$
$888$			59219.3i			0.0750996i
$889$	−1.45329e6	+	1.45329e6i	−1.83887	+	1.83887i
$890$	324657.	−	324657.i	0.409868	−	0.409868i
$891$	455658.	+	455658.i	0.573963	+	0.573963i
$892$	−218171.	−	218171.i	−0.274201	−	0.274201i
$893$	−10294.1			−0.0129088
$894$		−	906711.i		−	1.13447i
$895$	−705041.	−	705041.i	−0.880174	−	0.880174i
$896$		−	100665.i		−	0.125390i
$897$		0			0
$898$	−44925.1			−0.0557104
$899$	62184.2	−	62184.2i	0.0769415	−	0.0769415i
$900$	−462541.			−0.571038
$901$		−	1.05036e6i		−	1.29386i
$902$	492006.	−	492006.i	0.604724	−	0.604724i
$903$	−829299.	+	829299.i	−1.01703	+	1.01703i
$904$	−47041.5	−	47041.5i	−0.0575631	−	0.0575631i
$905$	375840.	+	375840.i	0.458888	+	0.458888i
$906$	505727.			0.616113
$907$			1.12762e6i			1.37071i	0.728207	+	0.685357i	$0.240354\pi$
$907$			1.12762e6i			1.37071i	−0.728207	+	0.685357i	$0.759646\pi$
$908$	339885.	+	339885.i	0.412250	+	0.412250i
$909$			72946.1i			0.0882824i
$910$		0			0
$911$	621926.			0.749380			0.374690	−	0.927150i	$-0.377749\pi$
$911$	621926.			0.749380			0.374690	+	0.927150i	$0.377749\pi$
$912$	2875.02	−	2875.02i	0.00345662	−	0.00345662i
$913$	213154.			0.255712
$914$			650068.i			0.778155i
$915$	−902135.	+	902135.i	−1.07753	+	1.07753i
$916$	−230373.	+	230373.i	−0.274562	+	0.274562i
$917$	789058.	+	789058.i	0.938362	+	0.938362i
$918$	−183342.	−	183342.i	−0.217559	−	0.217559i
$919$	212127.			0.251169			0.125584	−	0.992083i	$-0.459919\pi$
$919$	212127.			0.251169			0.125584	+	0.992083i	$0.459919\pi$
$920$			411349.i			0.485999i
$921$	201685.	+	201685.i	0.237768	+	0.237768i
$922$			149101.i			0.175396i
$923$		0			0
$924$	−557027.			−0.652427
$925$	142491.	−	142491.i	0.166535	−	0.166535i
$926$	−549757.			−0.641134
$927$			835406.i			0.972160i
$928$	−117616.	+	117616.i	−0.136575	+	0.136575i
$929$	329117.	−	329117.i	0.381346	−	0.381346i	−0.490241	−	0.871587i	$-0.663091\pi$
$929$	329117.	−	329117.i	0.381346	−	0.381346i	0.871587	+	0.490241i	$0.163091\pi$
$930$	90258.4	+	90258.4i	0.104357	+	0.104357i
$931$	9111.50	+	9111.50i	0.0105121	+	0.0105121i
$932$	435655.			0.501546
$933$			1.09861e6i			1.26206i
$934$	98240.0	+	98240.0i	0.112615	+	0.112615i
$935$			1.37058e6i			1.56777i
$936$		0			0
$937$	860175.			0.979733			0.489867	−	0.871797i	$-0.337045\pi$
$937$	860175.			0.979733			0.489867	+	0.871797i	$0.337045\pi$
$938$	476945.	−	476945.i	0.542079	−	0.542079i
$939$	1.17090e6			1.32797
$940$			611252.i			0.691774i
$941$	976348.	−	976348.i	1.10262	−	1.10262i	0.108525	−	0.994094i	$-0.465387\pi$
$941$	976348.	−	976348.i	1.10262	−	1.10262i	0.994094	−	0.108525i	$-0.0346128\pi$
$942$	−101926.	+	101926.i	−0.114864	+	0.114864i
$943$	961765.	+	961765.i	1.08155	+	1.08155i
$944$	−24804.6	−	24804.6i	−0.0278348	−	0.0278348i
$945$	−601431.			−0.673476
$946$		−	332752.i		−	0.371825i
$947$	−74686.8	−	74686.8i	−0.0832806	−	0.0832806i	0.664239	−	0.747520i	$-0.268756\pi$
$947$	−74686.8	−	74686.8i	−0.0832806	−	0.0832806i	−0.747520	+	0.664239i	$0.768756\pi$
$948$		−	316482.i		−	0.352153i
$949$		0			0
$950$	−13835.5			−0.0153302
$951$	−164996.	+	164996.i	−0.182437	+	0.182437i
$952$	655650.			0.723433
$953$			1.20235e6i			1.32387i	0.749562	+	0.661935i	$0.230264\pi$
$953$			1.20235e6i			1.32387i	−0.749562	+	0.661935i	$0.769736\pi$
$954$	315736.	−	315736.i	0.346918	−	0.346918i
$955$	−361632.	+	361632.i	−0.396515	+	0.396515i
$956$	−256716.	−	256716.i	−0.280891	−	0.280891i
$957$	650824.	+	650824.i	0.710624	+	0.710624i
$958$	1.04960e6			1.14365
$959$			758381.i			0.824614i
$960$	−170716.	−	170716.i	−0.185239	−	0.185239i
$961$			914361.i			0.990082i
$962$		0			0
$963$	1.02124e6			1.10122
$964$	555164.	−	555164.i	0.597403	−	0.597403i
$965$	−674705.			−0.724535
$966$		−	1.08887e6i		−	1.16686i
$967$	641718.	−	641718.i	0.686264	−	0.686264i	−0.275140	−	0.961404i	$-0.588724\pi$
$967$	641718.	−	641718.i	0.686264	−	0.686264i	0.961404	+	0.275140i	$0.0887241\pi$
$968$	−122504.	+	122504.i	−0.130737	+	0.130737i
$969$	18725.6	+	18725.6i	0.0199429	+	0.0199429i
$970$	−1.16250e6	−	1.16250e6i	−1.23552	−	1.23552i
$971$	−260542.			−0.276337			−0.138168	−	0.990409i	$-0.544121\pi$
$971$	−260542.			−0.276337			−0.138168	+	0.990409i	$0.544121\pi$
$972$		−	596812.i		−	0.631691i
$973$	−667748.	−	667748.i	−0.705321	−	0.705321i
$974$			211015.i			0.222431i
$975$		0			0
$976$	−173160.			−0.181781
$977$	1.20062e6	−	1.20062e6i	1.25781	−	1.25781i	0.305680	−	0.952134i	$-0.401116\pi$
$977$	1.20062e6	−	1.20062e6i	1.25781	−	1.25781i	0.952134	−	0.305680i	$-0.0988839\pi$
$978$	−461844.			−0.482856
$979$		−	344825.i		−	0.359777i
$980$	541031.	−	541031.i	0.563339	−	0.563339i
$981$	936055.	−	936055.i	0.972665	−	0.972665i
$982$	34631.7	+	34631.7i	0.0359129	+	0.0359129i
$983$	156188.	+	156188.i	0.161637	+	0.161637i	0.783291	−	0.621655i	$-0.213539\pi$
$983$	156188.	+	156188.i	0.161637	+	0.161637i	−0.621655	+	0.783291i	$0.713539\pi$
$984$	−798292.			−0.824464
$985$		−	19465.2i		−	0.0200625i
$986$	−766055.	−	766055.i	−0.787963	−	0.787963i
$987$		−	1.61802e6i		−	1.66092i
$988$		0			0
$989$	650459.			0.665009
$990$	−411995.	+	411995.i	−0.420361	+	0.420361i
$991$	440953.			0.448998			0.224499	−	0.974474i	$-0.427925\pi$
$991$	440953.			0.448998			0.224499	+	0.974474i	$0.427925\pi$
$992$			17324.6i			0.0176052i
$993$	452939.	−	452939.i	0.459347	−	0.459347i
$994$	977868.	−	977868.i	0.989709	−	0.989709i
$995$	−1.56188e6	−	1.56188e6i	−1.57762	−	1.57762i
$996$	−172924.	−	172924.i	−0.174316	−	0.174316i
$997$	−853876.			−0.859023			−0.429511	−	0.903061i	$-0.641314\pi$
$997$	−853876.			−0.859023			−0.429511	+	0.903061i	$0.641314\pi$
$998$			305215.i			0.306440i
$999$	33956.1	+	33956.1i	0.0340241	+	0.0340241i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.5.d.h.239.1		8
13.3	even	3		26.5.f.a.19.2	yes	8
13.5	odd	4		338.5.d.e.99.1		8
13.7	odd	12		26.5.f.a.11.2	✓	8
13.8	odd	4	inner	338.5.d.h.99.1		8
13.12	even	2		338.5.d.e.239.1		8
39.20	even	12		234.5.bb.c.37.1		8
39.29	odd	6		234.5.bb.c.19.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.5.f.a.11.2	✓	8	13.7	odd	12
26.5.f.a.19.2	yes	8	13.3	even	3
234.5.bb.c.19.1		8	39.29	odd	6
234.5.bb.c.37.1		8	39.20	even	12
338.5.d.e.99.1		8	13.5	odd	4
338.5.d.e.239.1		8	13.12	even	2
338.5.d.h.99.1		8	13.8	odd	4	inner
338.5.d.h.239.1		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} -11.9855 q^{3} -8.00000i q^{4} +(27.8195 - 27.8195i) q^{5} +(-23.9709 + 23.9709i) q^{6} +(-49.1528 - 49.1528i) q^{7} +(-16.0000 - 16.0000i) q^{8} +62.6514 q^{9} -111.278i q^{10} +(-59.0953 - 59.0953i) q^{11} +95.8837i q^{12} -196.611 q^{14} +(-333.429 + 333.429i) q^{15} -64.0000 q^{16} -416.844i q^{17} +(125.303 - 125.303i) q^{18} +(3.74806 - 3.74806i) q^{19} +(-222.556 - 222.556i) q^{20} +(589.119 + 589.119i) q^{21} -236.381 q^{22} -462.074i q^{23} +(191.767 + 191.767i) q^{24} -922.846i q^{25} +219.917 q^{27} +(-393.222 + 393.222i) q^{28} +918.874 q^{29} +1333.72i q^{30} +(67.6743 - 67.6743i) q^{31} +(-128.000 + 128.000i) q^{32} +(708.285 + 708.285i) q^{33} +(-833.689 - 833.689i) q^{34} -2734.81 q^{35} -501.211i q^{36} +(154.404 + 154.404i) q^{37} -14.9922i q^{38} -890.223 q^{40} +(-2081.41 + 2081.41i) q^{41} +2356.48 q^{42} +1407.69i q^{43} +(-472.762 + 472.762i) q^{44} +(1742.93 - 1742.93i) q^{45} +(-924.149 - 924.149i) q^{46} +(-1373.25 - 1373.25i) q^{47} +767.070 q^{48} +2430.99i q^{49} +(-1845.69 - 1845.69i) q^{50} +4996.07i q^{51} +2519.79 q^{53} +(439.834 - 439.834i) q^{54} -3288.00 q^{55} +1572.89i q^{56} +(-44.9222 + 44.9222i) q^{57} +(1837.75 - 1837.75i) q^{58} +(387.572 + 387.572i) q^{59} +(2667.43 + 2667.43i) q^{60} +2705.62 q^{61} -270.697i q^{62} +(-3079.49 - 3079.49i) q^{63} +512.000i q^{64} +2833.14 q^{66} +(-2425.83 + 2425.83i) q^{67} -3334.75 q^{68} +5538.18i q^{69} +(-5469.62 + 5469.62i) q^{70} +(-4973.62 + 4973.62i) q^{71} +(-1002.42 - 1002.42i) q^{72} +(1208.41 + 1208.41i) q^{73} +617.616 q^{74} +11060.7i q^{75} +(-29.9845 - 29.9845i) q^{76} +5809.40i q^{77} -3300.68 q^{79} +(-1780.45 + 1780.45i) q^{80} -7710.57 q^{81} +8325.63i q^{82} +(-1803.48 + 1803.48i) q^{83} +(4712.95 - 4712.95i) q^{84} +(-11596.4 - 11596.4i) q^{85} +(2815.39 + 2815.39i) q^{86} -11013.1 q^{87} +1891.05i q^{88} +(2917.53 + 2917.53i) q^{89} -6971.71i q^{90} -3696.59 q^{92} +(-811.109 + 811.109i) q^{93} -5493.02 q^{94} -208.538i q^{95} +(1534.14 - 1534.14i) q^{96} +(10446.8 - 10446.8i) q^{97} +(4861.98 + 4861.98i) q^{98} +(-3702.40 - 3702.40i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{2} + 30 q^{5} - 86 q^{7} - 128 q^{8} - 180 q^{9} - 474 q^{11} - 344 q^{14} - 1128 q^{15} - 512 q^{16} - 360 q^{18} - 518 q^{19} - 240 q^{20} + 1914 q^{21} - 1896 q^{22} - 2268 q^{27} - 688 q^{28}+ \cdots + 9504 q^{99}+O(q^{100}) \) 8 * q + 16 * q^2 + 30 * q^5 - 86 * q^7 - 128 * q^8 - 180 * q^9 - 474 * q^11 - 344 * q^14 - 1128 * q^15 - 512 * q^16 - 360 * q^18 - 518 * q^19 - 240 * q^20 + 1914 * q^21 - 1896 * q^22 - 2268 * q^27 - 688 * q^28 + 504 * q^29 - 3376 * q^31 - 1024 * q^32 - 246 * q^33 + 1080 * q^34 - 12948 * q^35 - 5056 * q^37 - 960 * q^40 - 4392 * q^41 + 7656 * q^42 - 3792 * q^44 + 2898 * q^45 + 24 * q^46 + 4848 * q^47 - 3152 * q^50 - 6840 * q^53 - 4536 * q^54 - 588 * q^55 - 4434 * q^57 + 1008 * q^58 - 3510 * q^59 + 9024 * q^60 - 17340 * q^61 - 3276 * q^63 - 984 * q^66 - 7334 * q^67 + 4320 * q^68 - 25896 * q^70 - 7998 * q^71 + 2880 * q^72 + 19294 * q^73 - 20224 * q^74 + 4144 * q^76 - 11328 * q^79 - 1920 * q^80 - 12096 * q^81 + 10344 * q^83 + 15312 * q^84 - 20670 * q^85 + 13536 * q^86 - 36744 * q^87 + 1998 * q^89 + 96 * q^92 - 12744 * q^93 + 19392 * q^94 + 13694 * q^97 + 24640 * q^98 + 9504 * q^99

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