LMFDB - Embedded newform 841.2.b.b.840.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [841,2,Mod(840,841)] 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("841.840"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(841, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 841 = 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	841.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$6.71541880999$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			840.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	841.840
Dual form			841.2.b.b.840.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.61803i q^{2} +1.61803i q^{3} -0.618034 q^{4} +2.85410 q^{5} +2.61803 q^{6} +2.23607 q^{7} -2.23607i q^{8} +0.381966 q^{9} -4.61803i q^{10} -3.61803i q^{11} -1.00000i q^{12} -4.23607 q^{13} -3.61803i q^{14} +4.61803i q^{15} -4.85410 q^{16} -6.61803i q^{17} -0.618034i q^{18} +1.85410i q^{19} -1.76393 q^{20} +3.61803i q^{21} -5.85410 q^{22} +3.23607 q^{23} +3.61803 q^{24} +3.14590 q^{25} +6.85410i q^{26} +5.47214i q^{27} -1.38197 q^{28} +7.47214 q^{30} +1.09017i q^{31} +3.38197i q^{32} +5.85410 q^{33} -10.7082 q^{34} +6.38197 q^{35} -0.236068 q^{36} +8.70820i q^{37} +3.00000 q^{38} -6.85410i q^{39} -6.38197i q^{40} +2.85410i q^{41} +5.85410 q^{42} -2.76393i q^{43} +2.23607i q^{44} +1.09017 q^{45} -5.23607i q^{46} +7.00000i q^{47} -7.85410i q^{48} -2.00000 q^{49} -5.09017i q^{50} +10.7082 q^{51} +2.61803 q^{52} -2.00000 q^{53} +8.85410 q^{54} -10.3262i q^{55} -5.00000i q^{56} -3.00000 q^{57} -5.09017 q^{59} -2.85410i q^{60} +1.61803i q^{61} +1.76393 q^{62} +0.854102 q^{63} -4.23607 q^{64} -12.0902 q^{65} -9.47214i q^{66} +10.4721 q^{67} +4.09017i q^{68} +5.23607i q^{69} -10.3262i q^{70} -1.52786 q^{71} -0.854102i q^{72} +0.291796i q^{73} +14.0902 q^{74} +5.09017i q^{75} -1.14590i q^{76} -8.09017i q^{77} -11.0902 q^{78} +5.09017i q^{79} -13.8541 q^{80} -7.70820 q^{81} +4.61803 q^{82} +7.94427 q^{83} -2.23607i q^{84} -18.8885i q^{85} -4.47214 q^{86} -8.09017 q^{88} -8.70820i q^{89} -1.76393i q^{90} -9.47214 q^{91} -2.00000 q^{92} -1.76393 q^{93} +11.3262 q^{94} +5.29180i q^{95} -5.47214 q^{96} +16.5623i q^{97} +3.23607i q^{98} -1.38197i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{5} + 6 q^{6} + 6 q^{9} - 8 q^{13} - 6 q^{16} - 16 q^{20} - 10 q^{22} + 4 q^{23} + 10 q^{24} + 26 q^{25} - 10 q^{28} + 12 q^{30} + 10 q^{33} - 16 q^{34} + 30 q^{35} + 8 q^{36} + 12 q^{38}+ \cdots - 4 q^{96}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^5 + 6 * q^6 + 6 * q^9 - 8 * q^13 - 6 * q^16 - 16 * q^20 - 10 * q^22 + 4 * q^23 + 10 * q^24 + 26 * q^25 - 10 * q^28 + 12 * q^30 + 10 * q^33 - 16 * q^34 + 30 * q^35 + 8 * q^36 + 12 * q^38 + 10 * q^42 - 18 * q^45 - 8 * q^49 + 16 * q^51 + 6 * q^52 - 8 * q^53 + 22 * q^54 - 12 * q^57 + 2 * q^59 + 16 * q^62 - 10 * q^63 - 8 * q^64 - 26 * q^65 + 24 * q^67 - 24 * q^71 + 34 * q^74 - 22 * q^78 - 42 * q^80 - 4 * q^81 + 14 * q^82 - 4 * q^83 - 10 * q^88 - 20 * q^91 - 8 * q^92 - 16 * q^93 + 14 * q^94 - 4 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/841\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 1.61803i	− 1.14412i	−0.820211	−	0.572061i	$-0.806144\pi$
$2$	− 1.61803i	− 1.14412i	0.820211	−	0.572061i	$-0.193856\pi$
$3$	1.61803i	0.934172i	0.884212	+	0.467086i	$0.154696\pi$
$3$	1.61803i	0.934172i	−0.884212	+	0.467086i	$0.845304\pi$
$4$	−0.618034	−0.309017
$5$	2.85410	1.27639	0.638197	−	0.769873i	$-0.279681\pi$
$5$	2.85410	1.27639	0.638197	+	0.769873i	$0.279681\pi$
$6$	2.61803	1.06881
$7$	2.23607	0.845154	0.422577	−	0.906327i	$-0.361126\pi$
$7$	2.23607	0.845154	0.422577	+	0.906327i	$0.361126\pi$
$8$	− 2.23607i	− 0.790569i
$9$	0.381966	0.127322
$10$	− 4.61803i	− 1.46035i
$11$	− 3.61803i	− 1.09088i	−0.838150	−	0.545439i	$-0.816363\pi$
$11$	− 3.61803i	− 1.09088i	0.838150	−	0.545439i	$-0.183637\pi$
$12$	− 1.00000i	− 0.288675i
$13$	−4.23607	−1.17487	−0.587437	−	0.809270i	$-0.699863\pi$
$13$	−4.23607	−1.17487	−0.587437	+	0.809270i	$0.699863\pi$
$14$	− 3.61803i	− 0.966960i
$15$	4.61803i	1.19237i
$16$	−4.85410	−1.21353
$17$	− 6.61803i	− 1.60511i	−0.596579	−	0.802555i	$-0.703474\pi$
$17$	− 6.61803i	− 1.60511i	0.596579	−	0.802555i	$-0.296526\pi$
$18$	− 0.618034i	− 0.145672i
$19$	1.85410i	0.425360i	0.977122	+	0.212680i	$0.0682192\pi$
$19$	1.85410i	0.425360i	−0.977122	+	0.212680i	$0.931781\pi$
$20$	−1.76393	−0.394427
$21$	3.61803i	0.789520i
$22$	−5.85410	−1.24810
$23$	3.23607	0.674767	0.337383	−	0.941367i	$-0.390458\pi$
$23$	3.23607	0.674767	0.337383	+	0.941367i	$0.390458\pi$
$24$	3.61803	0.738528
$25$	3.14590	0.629180
$26$	6.85410i	1.34420i
$27$	5.47214i	1.05311i
$28$	−1.38197	−0.261167
$29$	0	0
$29$	0	0
$30$	7.47214	1.36422
$31$	1.09017i	0.195800i	0.995196	+	0.0979002i	$0.0312126\pi$
$31$	1.09017i	0.195800i	−0.995196	+	0.0979002i	$0.968787\pi$
$32$	3.38197i	0.597853i
$33$	5.85410	1.01907
$34$	−10.7082	−1.83644
$35$	6.38197	1.07875
$36$	−0.236068	−0.0393447
$37$	8.70820i	1.43162i	0.698295	+	0.715810i	$0.253942\pi$
$37$	8.70820i	1.43162i	−0.698295	+	0.715810i	$0.746058\pi$
$38$	3.00000	0.486664
$39$	− 6.85410i	− 1.09753i
$40$	− 6.38197i	− 1.00908i
$41$	2.85410i	0.445736i	0.974849	+	0.222868i	$0.0715419\pi$
$41$	2.85410i	0.445736i	−0.974849	+	0.222868i	$0.928458\pi$
$42$	5.85410	0.903308
$43$	− 2.76393i	− 0.421496i	−0.977540	−	0.210748i	$-0.932410\pi$
$43$	− 2.76393i	− 0.421496i	0.977540	−	0.210748i	$-0.0675899\pi$
$44$	2.23607i	0.337100i
$45$	1.09017	0.162513
$46$	− 5.23607i	− 0.772016i
$47$	7.00000i	1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$	7.00000i	1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$	− 7.85410i	− 1.13364i
$49$	−2.00000	−0.285714
$50$	− 5.09017i	− 0.719859i
$51$	10.7082	1.49945
$52$	2.61803	0.363056
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	8.85410	1.20489
$55$	− 10.3262i	− 1.39239i
$56$	− 5.00000i	− 0.668153i
$57$	−3.00000	−0.397360
$58$	0	0
$59$	−5.09017	−0.662684	−0.331342	−	0.943511i	$-0.607501\pi$
$59$	−5.09017	−0.662684	−0.331342	+	0.943511i	$0.607501\pi$
$60$	− 2.85410i	− 0.368463i
$61$	1.61803i	0.207168i	0.994621	+	0.103584i	$0.0330311\pi$
$61$	1.61803i	0.207168i	−0.994621	+	0.103584i	$0.966969\pi$
$62$	1.76393	0.224020
$63$	0.854102	0.107607
$64$	−4.23607	−0.529508
$65$	−12.0902	−1.49960
$66$	− 9.47214i	− 1.16594i
$67$	10.4721	1.27938	0.639688	−	0.768635i	$-0.279064\pi$
$67$	10.4721	1.27938	0.639688	+	0.768635i	$0.279064\pi$
$68$	4.09017i	0.496006i
$69$	5.23607i	0.630349i
$70$	− 10.3262i	− 1.23422i
$71$	−1.52786	−0.181324	−0.0906621	−	0.995882i	$-0.528898\pi$
$71$	−1.52786	−0.181324	−0.0906621	+	0.995882i	$0.528898\pi$
$72$	− 0.854102i	− 0.100657i
$73$	0.291796i	0.0341521i	0.999854	+	0.0170761i	$0.00543575\pi$
$73$	0.291796i	0.0341521i	−0.999854	+	0.0170761i	$0.994564\pi$
$74$	14.0902	1.63795
$75$	5.09017i	0.587762i
$76$	− 1.14590i	− 0.131444i
$77$	− 8.09017i	− 0.921960i
$78$	−11.0902	−1.25571
$79$	5.09017i	0.572689i	0.958127	+	0.286344i	$0.0924402\pi$
$79$	5.09017i	0.572689i	−0.958127	+	0.286344i	$0.907560\pi$
$80$	−13.8541	−1.54894
$81$	−7.70820	−0.856467
$82$	4.61803	0.509977
$83$	7.94427	0.871997	0.435999	−	0.899947i	$-0.356395\pi$
$83$	7.94427	0.871997	0.435999	+	0.899947i	$0.356395\pi$
$84$	− 2.23607i	− 0.243975i
$85$	− 18.8885i	− 2.04875i
$86$	−4.47214	−0.482243
$87$	0	0
$88$	−8.09017	−0.862415
$89$	− 8.70820i	− 0.923068i	−0.887123	−	0.461534i	$-0.847299\pi$
$89$	− 8.70820i	− 0.923068i	0.887123	−	0.461534i	$-0.152701\pi$
$90$	− 1.76393i	− 0.185935i
$91$	−9.47214	−0.992950
$92$	−2.00000	−0.208514
$93$	−1.76393	−0.182911
$94$	11.3262	1.16821
$95$	5.29180i	0.542927i
$96$	−5.47214	−0.558498
$97$	16.5623i	1.68165i	0.541309	+	0.840824i	$0.317929\pi$
$97$	16.5623i	1.68165i	−0.541309	+	0.840824i	$0.682071\pi$
$98$	3.23607i	0.326892i
$99$	− 1.38197i	− 0.138893i
$100$	−1.94427	−0.194427
$101$	1.61803i	0.161000i	0.996755	+	0.0805002i	$0.0256518\pi$
$101$	1.61803i	0.161000i	−0.996755	+	0.0805002i	$0.974348\pi$
$102$	− 17.3262i	− 1.71555i
$103$	−13.1803	−1.29870	−0.649349	−	0.760491i	$-0.724958\pi$
$103$	−13.1803	−1.29870	−0.649349	+	0.760491i	$0.724958\pi$
$104$	9.47214i	0.928819i
$105$	10.3262i	1.00774i
$106$	3.23607i	0.314315i
$107$	11.2361	1.08623	0.543116	−	0.839658i	$-0.317244\pi$
$107$	11.2361	1.08623	0.543116	+	0.839658i	$0.317244\pi$
$108$	− 3.38197i	− 0.325430i
$109$	16.6180	1.59172	0.795859	−	0.605481i	$-0.207019\pi$
$109$	16.6180	1.59172	0.795859	+	0.605481i	$0.207019\pi$
$110$	−16.7082	−1.59306
$111$	−14.0902	−1.33738
$112$	−10.8541	−1.02562
$113$	9.94427i	0.935478i	0.883867	+	0.467739i	$0.154931\pi$
$113$	9.94427i	0.935478i	−0.883867	+	0.467739i	$0.845069\pi$
$114$	4.85410i	0.454628i
$115$	9.23607	0.861268
$116$	0	0
$117$	−1.61803	−0.149587
$118$	8.23607i	0.758192i
$119$	− 14.7984i	− 1.35656i
$120$	10.3262	0.942652
$121$	−2.09017	−0.190015
$122$	2.61803	0.237026
$123$	−4.61803	−0.416394
$124$	− 0.673762i	− 0.0605056i
$125$	−5.29180	−0.473313
$126$	− 1.38197i	− 0.123115i
$127$	− 1.94427i	− 0.172526i	−0.996272	−	0.0862631i	$-0.972507\pi$
$127$	− 1.94427i	− 0.172526i	0.996272	−	0.0862631i	$-0.0274926\pi$
$128$	13.6180i	1.20368i
$129$	4.47214	0.393750
$130$	19.5623i	1.71573i
$131$	1.32624i	0.115874i	0.998320	+	0.0579370i	$0.0184522\pi$
$131$	1.32624i	0.115874i	−0.998320	+	0.0579370i	$0.981548\pi$
$132$	−3.61803	−0.314909
$133$	4.14590i	0.359495i
$134$	− 16.9443i	− 1.46376i
$135$	15.6180i	1.34419i
$136$	−14.7984	−1.26895
$137$	13.8541i	1.18364i	0.806072	+	0.591818i	$0.201590\pi$
$137$	13.8541i	1.18364i	−0.806072	+	0.591818i	$0.798410\pi$
$138$	8.47214	0.721196
$139$	14.7082	1.24753	0.623767	−	0.781611i	$-0.285601\pi$
$139$	14.7082	1.24753	0.623767	+	0.781611i	$0.285601\pi$
$140$	−3.94427	−0.333352
$141$	−11.3262	−0.953841
$142$	2.47214i	0.207457i
$143$	15.3262i	1.28164i
$144$	−1.85410	−0.154508
$145$	0	0
$146$	0.472136	0.0390742
$147$	− 3.23607i	− 0.266906i
$148$	− 5.38197i	− 0.442395i
$149$	−7.38197	−0.604754	−0.302377	−	0.953188i	$-0.597780\pi$
$149$	−7.38197	−0.604754	−0.302377	+	0.953188i	$0.597780\pi$
$150$	8.23607	0.672472
$151$	−18.3262	−1.49137	−0.745684	−	0.666300i	$-0.767877\pi$
$151$	−18.3262	−1.49137	−0.745684	+	0.666300i	$0.767877\pi$
$152$	4.14590	0.336277
$153$	− 2.52786i	− 0.204366i
$154$	−13.0902	−1.05484
$155$	3.11146i	0.249918i
$156$	4.23607i	0.339157i
$157$	5.56231i	0.443920i	0.975056	+	0.221960i	$0.0712455\pi$
$157$	5.56231i	0.443920i	−0.975056	+	0.221960i	$0.928755\pi$
$158$	8.23607	0.655226
$159$	− 3.23607i	− 0.256637i
$160$	9.65248i	0.763095i
$161$	7.23607	0.570282
$162$	12.4721i	0.979904i
$163$	− 23.0344i	− 1.80420i	−0.431530	−	0.902098i	$-0.642026\pi$
$163$	− 23.0344i	− 1.80420i	0.431530	−	0.902098i	$-0.357974\pi$
$164$	− 1.76393i	− 0.137740i
$165$	16.7082	1.30073
$166$	− 12.8541i	− 0.997672i
$167$	−19.4721	−1.50680	−0.753400	−	0.657563i	$-0.771587\pi$
$167$	−19.4721	−1.50680	−0.753400	+	0.657563i	$0.771587\pi$
$168$	8.09017	0.624170
$169$	4.94427	0.380329
$170$	−30.5623	−2.34402
$171$	0.708204i	0.0541577i
$172$	1.70820i	0.130249i
$173$	7.09017	0.539056	0.269528	−	0.962993i	$-0.413132\pi$
$173$	7.09017	0.539056	0.269528	+	0.962993i	$0.413132\pi$
$174$	0	0
$175$	7.03444	0.531754
$176$	17.5623i	1.32381i
$177$	− 8.23607i	− 0.619061i
$178$	−14.0902	−1.05610
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	−0.673762	−0.0502193
$181$	−11.9443	−0.887811	−0.443905	−	0.896074i	$-0.646407\pi$
$181$	−11.9443	−0.887811	−0.443905	+	0.896074i	$0.646407\pi$
$182$	15.3262i	1.13606i
$183$	−2.61803	−0.193531
$184$	− 7.23607i	− 0.533450i
$185$	24.8541i	1.82731i
$186$	2.85410i	0.209273i
$187$	−23.9443	−1.75098
$188$	− 4.32624i	− 0.315523i
$189$	12.2361i	0.890043i
$190$	8.56231	0.621175
$191$	− 12.0344i	− 0.870782i	−0.900242	−	0.435391i	$-0.856610\pi$
$191$	− 12.0344i	− 0.870782i	0.900242	−	0.435391i	$-0.143390\pi$
$192$	− 6.85410i	− 0.494652i
$193$	3.52786i	0.253941i	0.991906	+	0.126971i	$0.0405254\pi$
$193$	3.52786i	0.253941i	−0.991906	+	0.126971i	$0.959475\pi$
$194$	26.7984	1.92401
$195$	− 19.5623i	− 1.40089i
$196$	1.23607	0.0882906
$197$	−19.7082	−1.40415	−0.702076	−	0.712102i	$-0.747743\pi$
$197$	−19.7082	−1.40415	−0.702076	+	0.712102i	$0.747743\pi$
$198$	−2.23607	−0.158910
$199$	0.854102	0.0605457	0.0302728	−	0.999542i	$-0.490362\pi$
$199$	0.854102	0.0605457	0.0302728	+	0.999542i	$0.490362\pi$
$200$	− 7.03444i	− 0.497410i
$201$	16.9443i	1.19516i
$202$	2.61803	0.184204
$203$	0	0
$204$	−6.61803	−0.463355
$205$	8.14590i	0.568934i
$206$	21.3262i	1.48587i
$207$	1.23607	0.0859127
$208$	20.5623	1.42574
$209$	6.70820	0.464016
$210$	16.7082	1.15298
$211$	− 19.6525i	− 1.35293i	−0.736474	−	0.676466i	$-0.763511\pi$
$211$	− 19.6525i	− 1.35293i	0.736474	−	0.676466i	$-0.236489\pi$
$212$	1.23607	0.0848935
$213$	− 2.47214i	− 0.169388i
$214$	− 18.1803i	− 1.24278i
$215$	− 7.88854i	− 0.537994i
$216$	12.2361	0.832559
$217$	2.43769i	0.165481i
$218$	− 26.8885i	− 1.82112i
$219$	−0.472136	−0.0319040
$220$	6.38197i	0.430272i
$221$	28.0344i	1.88580i
$222$	22.7984i	1.53013i
$223$	18.3262	1.22722	0.613608	−	0.789611i	$-0.289718\pi$
$223$	18.3262	1.22722	0.613608	+	0.789611i	$0.289718\pi$
$224$	7.56231i	0.505278i
$225$	1.20163	0.0801084
$226$	16.0902	1.07030
$227$	14.8885	0.988187	0.494094	−	0.869409i	$-0.335500\pi$
$227$	14.8885	0.988187	0.494094	+	0.869409i	$0.335500\pi$
$228$	1.85410	0.122791
$229$	− 15.7082i	− 1.03803i	−0.854766	−	0.519014i	$-0.826299\pi$
$229$	− 15.7082i	− 1.03803i	0.854766	−	0.519014i	$-0.173701\pi$
$230$	− 14.9443i	− 0.985396i
$231$	13.0902	0.861270
$232$	0	0
$233$	10.7639	0.705169	0.352584	−	0.935780i	$-0.385303\pi$
$233$	10.7639	0.705169	0.352584	+	0.935780i	$0.385303\pi$
$234$	2.61803i	0.171146i
$235$	19.9787i	1.30327i
$236$	3.14590	0.204781
$237$	−8.23607	−0.534990
$238$	−23.9443	−1.55208
$239$	14.7426	0.953622	0.476811	−	0.879006i	$-0.341793\pi$
$239$	14.7426	0.953622	0.476811	+	0.879006i	$0.341793\pi$
$240$	− 22.4164i	− 1.44697i
$241$	−26.6525	−1.71684	−0.858418	−	0.512950i	$-0.828553\pi$
$241$	−26.6525	−1.71684	−0.858418	+	0.512950i	$0.828553\pi$
$242$	3.38197i	0.217401i
$243$	3.94427i	0.253025i
$244$	− 1.00000i	− 0.0640184i
$245$	−5.70820	−0.364684
$246$	7.47214i	0.476406i
$247$	− 7.85410i	− 0.499745i
$248$	2.43769	0.154794
$249$	12.8541i	0.814596i
$250$	8.56231i	0.541528i
$251$	11.6525i	0.735498i	0.929925	+	0.367749i	$0.119871\pi$
$251$	11.6525i	0.735498i	−0.929925	+	0.367749i	$0.880129\pi$
$252$	−0.527864	−0.0332523
$253$	− 11.7082i	− 0.736088i
$254$	−3.14590	−0.197391
$255$	30.5623	1.91389
$256$	13.5623	0.847644
$257$	−0.819660	−0.0511290	−0.0255645	−	0.999673i	$-0.508138\pi$
$257$	−0.819660	−0.0511290	−0.0255645	+	0.999673i	$0.508138\pi$
$258$	− 7.23607i	− 0.450498i
$259$	19.4721i	1.20994i
$260$	7.47214	0.463402
$261$	0	0
$262$	2.14590	0.132574
$263$	− 3.29180i	− 0.202981i	−0.994837	−	0.101490i	$-0.967639\pi$
$263$	− 3.29180i	− 0.202981i	0.994837	−	0.101490i	$-0.0323611\pi$
$264$	− 13.0902i	− 0.805644i
$265$	−5.70820	−0.350652
$266$	6.70820	0.411306
$267$	14.0902	0.862304
$268$	−6.47214	−0.395349
$269$	6.00000i	0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$	6.00000i	0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$	25.2705	1.53791
$271$	12.1803i	0.739903i	0.929051	+	0.369951i	$0.120626\pi$
$271$	12.1803i	0.739903i	−0.929051	+	0.369951i	$0.879374\pi$
$272$	32.1246i	1.94784i
$273$	− 15.3262i	− 0.927586i
$274$	22.4164	1.35422
$275$	− 11.3820i	− 0.686358i
$276$	− 3.23607i	− 0.194788i
$277$	−23.6180	−1.41907	−0.709535	−	0.704670i	$-0.751095\pi$
$277$	−23.6180	−1.41907	−0.709535	+	0.704670i	$0.751095\pi$
$278$	− 23.7984i	− 1.42733i
$279$	0.416408i	0.0249297i
$280$	− 14.2705i	− 0.852826i
$281$	−17.1246	−1.02157	−0.510784	−	0.859709i	$-0.670645\pi$
$281$	−17.1246	−1.02157	−0.510784	+	0.859709i	$0.670645\pi$
$282$	18.3262i	1.09131i
$283$	0.763932	0.0454110	0.0227055	−	0.999742i	$-0.492772\pi$
$283$	0.763932	0.0454110	0.0227055	+	0.999742i	$0.492772\pi$
$284$	0.944272	0.0560322
$285$	−8.56231	−0.507187
$286$	24.7984	1.46636
$287$	6.38197i	0.376716i
$288$	1.29180i	0.0761198i
$289$	−26.7984	−1.57637
$290$	0	0
$291$	−26.7984	−1.57095
$292$	− 0.180340i	− 0.0105536i
$293$	17.4721i	1.02073i	0.859957	+	0.510367i	$0.170490\pi$
$293$	17.4721i	1.02073i	−0.859957	+	0.510367i	$0.829510\pi$
$294$	−5.23607	−0.305374
$295$	−14.5279	−0.845845
$296$	19.4721	1.13179
$297$	19.7984	1.14882
$298$	11.9443i	0.691913i
$299$	−13.7082	−0.792766
$300$	− 3.14590i	− 0.181629i
$301$	− 6.18034i	− 0.356229i
$302$	29.6525i	1.70631i
$303$	−2.61803	−0.150402
$304$	− 9.00000i	− 0.516185i
$305$	4.61803i	0.264428i
$306$	−4.09017	−0.233819
$307$	3.18034i	0.181512i	0.995873	+	0.0907558i	$0.0289283\pi$
$307$	3.18034i	0.181512i	−0.995873	+	0.0907558i	$0.971072\pi$
$308$	5.00000i	0.284901i
$309$	− 21.3262i	− 1.21321i
$310$	5.03444	0.285937
$311$	− 9.09017i	− 0.515456i	−0.966217	−	0.257728i	$-0.917026\pi$
$311$	− 9.09017i	− 0.515456i	0.966217	−	0.257728i	$-0.0829739\pi$
$312$	−15.3262	−0.867677
$313$	−24.0902	−1.36166	−0.680828	−	0.732443i	$-0.738380\pi$
$313$	−24.0902	−1.36166	−0.680828	+	0.732443i	$0.738380\pi$
$314$	9.00000	0.507899
$315$	2.43769	0.137349
$316$	− 3.14590i	− 0.176971i
$317$	− 14.2918i	− 0.802707i	−0.915923	−	0.401354i	$-0.868540\pi$
$317$	− 14.2918i	− 0.802707i	0.915923	−	0.401354i	$-0.131460\pi$
$318$	−5.23607	−0.293624
$319$	0	0
$320$	−12.0902	−0.675861
$321$	18.1803i	1.01473i
$322$	− 11.7082i	− 0.652473i
$323$	12.2705	0.682749
$324$	4.76393	0.264663
$325$	−13.3262	−0.739207
$326$	−37.2705	−2.06422
$327$	26.8885i	1.48694i
$328$	6.38197	0.352385
$329$	15.6525i	0.862949i
$330$	− 27.0344i	− 1.48820i
$331$	1.18034i	0.0648773i	0.999474	+	0.0324387i	$0.0103274\pi$
$331$	1.18034i	0.0648773i	−0.999474	+	0.0324387i	$0.989673\pi$
$332$	−4.90983	−0.269462
$333$	3.32624i	0.182277i
$334$	31.5066i	1.72396i
$335$	29.8885	1.63299
$336$	− 17.5623i	− 0.958102i
$337$	− 24.0689i	− 1.31112i	−0.755145	−	0.655558i	$-0.772434\pi$
$337$	− 24.0689i	− 1.31112i	0.755145	−	0.655558i	$-0.227566\pi$
$338$	− 8.00000i	− 0.435143i
$339$	−16.0902	−0.873898
$340$	11.6738i	0.633099i
$341$	3.94427	0.213594
$342$	1.14590	0.0619631
$343$	−20.1246	−1.08663
$344$	−6.18034	−0.333222
$345$	14.9443i	0.804573i
$346$	− 11.4721i	− 0.616746i
$347$	−8.12461	−0.436152	−0.218076	−	0.975932i	$-0.569978\pi$
$347$	−8.12461	−0.436152	−0.218076	+	0.975932i	$0.569978\pi$
$348$	0	0
$349$	13.4721	0.721147	0.360573	−	0.932731i	$-0.382581\pi$
$349$	13.4721	0.721147	0.360573	+	0.932731i	$0.382581\pi$
$350$	− 11.3820i	− 0.608392i
$351$	− 23.1803i	− 1.23728i
$352$	12.2361	0.652185
$353$	−21.1246	−1.12435	−0.562175	−	0.827018i	$-0.690035\pi$
$353$	−21.1246	−1.12435	−0.562175	+	0.827018i	$0.690035\pi$
$354$	−13.3262	−0.708282
$355$	−4.36068	−0.231441
$356$	5.38197i	0.285244i
$357$	23.9443	1.26727
$358$	− 25.8885i	− 1.36825i
$359$	− 28.2361i	− 1.49024i	−0.666929	−	0.745121i	$-0.732392\pi$
$359$	− 28.2361i	− 1.49024i	0.666929	−	0.745121i	$-0.267608\pi$
$360$	− 2.43769i	− 0.128478i
$361$	15.5623	0.819069
$362$	19.3262i	1.01576i
$363$	− 3.38197i	− 0.177507i
$364$	5.85410	0.306838
$365$	0.832816i	0.0435916i
$366$	4.23607i	0.221423i
$367$	− 6.27051i	− 0.327318i	−0.986517	−	0.163659i	$-0.947670\pi$
$367$	− 6.27051i	− 0.327318i	0.986517	−	0.163659i	$-0.0523297\pi$
$368$	−15.7082	−0.818847
$369$	1.09017i	0.0567520i
$370$	40.2148	2.09067
$371$	−4.47214	−0.232182
$372$	1.09017	0.0565227
$373$	18.3820	0.951782	0.475891	−	0.879504i	$-0.342126\pi$
$373$	18.3820	0.951782	0.475891	+	0.879504i	$0.342126\pi$
$374$	38.7426i	2.00333i
$375$	− 8.56231i	− 0.442156i
$376$	15.6525	0.807215
$377$	0	0
$378$	19.7984	1.01832
$379$	− 37.7082i	− 1.93694i	−0.249130	−	0.968470i	$-0.580145\pi$
$379$	− 37.7082i	− 1.93694i	0.249130	−	0.968470i	$-0.419855\pi$
$380$	− 3.27051i	− 0.167774i
$381$	3.14590	0.161169
$382$	−19.4721	−0.996281
$383$	22.1459	1.13160	0.565801	−	0.824542i	$-0.308567\pi$
$383$	22.1459	1.13160	0.565801	+	0.824542i	$0.308567\pi$
$384$	−22.0344	−1.12444
$385$	− 23.0902i	− 1.17678i
$386$	5.70820	0.290540
$387$	− 1.05573i	− 0.0536657i
$388$	− 10.2361i	− 0.519658i
$389$	− 21.1246i	− 1.07106i	−0.844516	−	0.535530i	$-0.820112\pi$
$389$	− 21.1246i	− 1.07106i	0.844516	−	0.535530i	$-0.179888\pi$
$390$	−31.6525	−1.60279
$391$	− 21.4164i	− 1.08307i
$392$	4.47214i	0.225877i
$393$	−2.14590	−0.108246
$394$	31.8885i	1.60652i
$395$	14.5279i	0.730976i
$396$	0.854102i	0.0429202i
$397$	−31.9443	−1.60324	−0.801619	−	0.597836i	$-0.796027\pi$
$397$	−31.9443	−1.60324	−0.801619	+	0.597836i	$0.796027\pi$
$398$	− 1.38197i	− 0.0692717i
$399$	−6.70820	−0.335830
$400$	−15.2705	−0.763525
$401$	−33.0689	−1.65138	−0.825691	−	0.564123i	$-0.809214\pi$
$401$	−33.0689	−1.65138	−0.825691	+	0.564123i	$0.809214\pi$
$402$	27.4164	1.36741
$403$	− 4.61803i	− 0.230041i
$404$	− 1.00000i	− 0.0497519i
$405$	−22.0000	−1.09319
$406$	0	0
$407$	31.5066	1.56172
$408$	− 23.9443i	− 1.18542i
$409$	0.583592i	0.0288568i	0.999896	+	0.0144284i	$0.00459286\pi$
$409$	0.583592i	0.0288568i	−0.999896	+	0.0144284i	$0.995407\pi$
$410$	13.1803	0.650931
$411$	−22.4164	−1.10572
$412$	8.14590	0.401320
$413$	−11.3820	−0.560070
$414$	− 2.00000i	− 0.0982946i
$415$	22.6738	1.11301
$416$	− 14.3262i	− 0.702402i
$417$	23.7984i	1.16541i
$418$	− 10.8541i	− 0.530891i
$419$	−2.56231	−0.125177	−0.0625884	−	0.998039i	$-0.519936\pi$
$419$	−2.56231	−0.125177	−0.0625884	+	0.998039i	$0.519936\pi$
$420$	− 6.38197i	− 0.311408i
$421$	1.96556i	0.0957954i	0.998852	+	0.0478977i	$0.0152522\pi$
$421$	1.96556i	0.0957954i	−0.998852	+	0.0478977i	$0.984748\pi$
$422$	−31.7984	−1.54792
$423$	2.67376i	0.130003i
$424$	4.47214i	0.217186i
$425$	− 20.8197i	− 1.00990i
$426$	−4.00000	−0.193801
$427$	3.61803i	0.175089i
$428$	−6.94427	−0.335664
$429$	−24.7984	−1.19728
$430$	−12.7639	−0.615531
$431$	−34.5967	−1.66647	−0.833233	−	0.552921i	$-0.813513\pi$
$431$	−34.5967	−1.66647	−0.833233	+	0.552921i	$0.813513\pi$
$432$	− 26.5623i	− 1.27798i
$433$	− 12.6180i	− 0.606384i	−0.952929	−	0.303192i	$-0.901948\pi$
$433$	− 12.6180i	− 0.606384i	0.952929	−	0.303192i	$-0.0980524\pi$
$434$	3.94427	0.189331
$435$	0	0
$436$	−10.2705	−0.491868
$437$	6.00000i	0.287019i
$438$	0.763932i	0.0365021i
$439$	3.05573	0.145842	0.0729210	−	0.997338i	$-0.476768\pi$
$439$	3.05573	0.145842	0.0729210	+	0.997338i	$0.476768\pi$
$440$	−23.0902	−1.10078
$441$	−0.763932	−0.0363777
$442$	45.3607	2.15759
$443$	13.0902i	0.621933i	0.950421	+	0.310966i	$0.100653\pi$
$443$	13.0902i	0.621933i	−0.950421	+	0.310966i	$0.899347\pi$
$444$	8.70820	0.413273
$445$	− 24.8541i	− 1.17820i
$446$	− 29.6525i	− 1.40409i
$447$	− 11.9443i	− 0.564945i
$448$	−9.47214	−0.447516
$449$	14.1246i	0.666582i	0.942824	+	0.333291i	$0.108159\pi$
$449$	14.1246i	0.666582i	−0.942824	+	0.333291i	$0.891841\pi$
$450$	− 1.94427i	− 0.0916539i
$451$	10.3262	0.486244
$452$	− 6.14590i	− 0.289079i
$453$	− 29.6525i	− 1.39319i
$454$	− 24.0902i	− 1.13061i
$455$	−27.0344	−1.26739
$456$	6.70820i	0.314140i
$457$	−5.29180	−0.247540	−0.123770	−	0.992311i	$-0.539498\pi$
$457$	−5.29180	−0.247540	−0.123770	+	0.992311i	$0.539498\pi$
$458$	−25.4164	−1.18763
$459$	36.2148	1.69036
$460$	−5.70820	−0.266146
$461$	− 7.97871i	− 0.371606i	−0.982587	−	0.185803i	$-0.940511\pi$
$461$	− 7.97871i	− 0.371606i	0.982587	−	0.185803i	$-0.0594886\pi$
$462$	− 21.1803i	− 0.985399i
$463$	2.70820	0.125861	0.0629305	−	0.998018i	$-0.479955\pi$
$463$	2.70820	0.125861	0.0629305	+	0.998018i	$0.479955\pi$
$464$	0	0
$465$	−5.03444	−0.233467
$466$	− 17.4164i	− 0.806800i
$467$	− 0.0557281i	− 0.00257879i	−0.999999	−	0.00128939i	$-0.999590\pi$
$467$	− 0.0557281i	− 0.00257879i	0.999999	−	0.00128939i	$-0.000410427\pi$
$468$	1.00000	0.0462250
$469$	23.4164	1.08127
$470$	32.3262	1.49110
$471$	−9.00000	−0.414698
$472$	11.3820i	0.523897i
$473$	−10.0000	−0.459800
$474$	13.3262i	0.612094i
$475$	5.83282i	0.267628i
$476$	9.14590i	0.419202i
$477$	−0.763932	−0.0349780
$478$	− 23.8541i	− 1.09106i
$479$	− 11.1803i	− 0.510843i	−0.966830	−	0.255421i	$-0.917786\pi$
$479$	− 11.1803i	− 0.510843i	0.966830	−	0.255421i	$-0.0822142\pi$
$480$	−15.6180	−0.712862
$481$	− 36.8885i	− 1.68197i
$482$	43.1246i	1.96427i
$483$	11.7082i	0.532742i
$484$	1.29180	0.0587180
$485$	47.2705i	2.14644i
$486$	6.38197	0.289492
$487$	22.4377	1.01675	0.508374	−	0.861136i	$-0.330247\pi$
$487$	22.4377	1.01675	0.508374	+	0.861136i	$0.330247\pi$
$488$	3.61803	0.163781
$489$	37.2705	1.68543
$490$	9.23607i	0.417243i
$491$	− 25.1246i	− 1.13386i	−0.823767	−	0.566929i	$-0.808131\pi$
$491$	− 25.1246i	− 1.13386i	0.823767	−	0.566929i	$-0.191869\pi$
$492$	2.85410	0.128673
$493$	0	0
$494$	−12.7082	−0.571769
$495$	− 3.94427i	− 0.177282i
$496$	− 5.29180i	− 0.237609i
$497$	−3.41641	−0.153247
$498$	20.7984	0.931997
$499$	35.6869	1.59757	0.798783	−	0.601619i	$-0.205478\pi$
$499$	35.6869	1.59757	0.798783	+	0.601619i	$0.205478\pi$
$500$	3.27051	0.146262
$501$	− 31.5066i	− 1.40761i
$502$	18.8541	0.841500
$503$	19.2705i	0.859230i	0.903012	+	0.429615i	$0.141351\pi$
$503$	19.2705i	0.859230i	−0.903012	+	0.429615i	$0.858649\pi$
$504$	− 1.90983i	− 0.0850706i
$505$	4.61803i	0.205500i
$506$	−18.9443	−0.842176
$507$	8.00000i	0.355292i
$508$	1.20163i	0.0533135i
$509$	11.4377	0.506967	0.253483	−	0.967340i	$-0.418424\pi$
$509$	11.4377	0.506967	0.253483	+	0.967340i	$0.418424\pi$
$510$	− 49.4508i	− 2.18972i
$511$	0.652476i	0.0288638i
$512$	5.29180i	0.233867i
$513$	−10.1459	−0.447952
$514$	1.32624i	0.0584978i
$515$	−37.6180	−1.65765
$516$	−2.76393	−0.121675
$517$	25.3262	1.11385
$518$	31.5066	1.38432
$519$	11.4721i	0.503571i
$520$	27.0344i	1.18554i
$521$	7.09017	0.310626	0.155313	−	0.987865i	$-0.450361\pi$
$521$	7.09017	0.310626	0.155313	+	0.987865i	$0.450361\pi$
$522$	0	0
$523$	−22.6180	−0.989018	−0.494509	−	0.869173i	$-0.664652\pi$
$523$	−22.6180	−0.989018	−0.494509	+	0.869173i	$0.664652\pi$
$524$	− 0.819660i	− 0.0358070i
$525$	11.3820i	0.496750i
$526$	−5.32624	−0.232235
$527$	7.21478	0.314281
$528$	−28.4164	−1.23667
$529$	−12.5279	−0.544690
$530$	9.23607i	0.401189i
$531$	−1.94427	−0.0843742
$532$	− 2.56231i	− 0.111090i
$533$	− 12.0902i	− 0.523683i
$534$	− 22.7984i	− 0.986582i
$535$	32.0689	1.38646
$536$	− 23.4164i	− 1.01143i
$537$	25.8885i	1.11717i
$538$	9.70820	0.418550
$539$	7.23607i	0.311680i
$540$	− 9.65248i	− 0.415376i
$541$	34.5967i	1.48743i	0.668497	+	0.743715i	$0.266938\pi$
$541$	34.5967i	1.48743i	−0.668497	+	0.743715i	$0.733062\pi$
$542$	19.7082	0.846540
$543$	− 19.3262i	− 0.829368i
$544$	22.3820	0.959619
$545$	47.4296	2.03166
$546$	−24.7984	−1.06127
$547$	9.61803	0.411237	0.205619	−	0.978632i	$-0.434079\pi$
$547$	9.61803	0.411237	0.205619	+	0.978632i	$0.434079\pi$
$548$	− 8.56231i	− 0.365764i
$549$	0.618034i	0.0263770i
$550$	−18.4164	−0.785278
$551$	0	0
$552$	11.7082	0.498334
$553$	11.3820i	0.484010i
$554$	38.2148i	1.62359i
$555$	−40.2148	−1.70702
$556$	−9.09017	−0.385509
$557$	−32.5066	−1.37735	−0.688674	−	0.725071i	$-0.741807\pi$
$557$	−32.5066	−1.37735	−0.688674	+	0.725071i	$0.741807\pi$
$558$	0.673762	0.0285226
$559$	11.7082i	0.495204i
$560$	−30.9787	−1.30909
$561$	− 38.7426i	− 1.63572i
$562$	27.7082i	1.16880i
$563$	45.3951i	1.91318i	0.291443	+	0.956588i	$0.405865\pi$
$563$	45.3951i	1.91318i	−0.291443	+	0.956588i	$0.594135\pi$
$564$	7.00000	0.294753
$565$	28.3820i	1.19404i
$566$	− 1.23607i	− 0.0519558i
$567$	−17.2361	−0.723847
$568$	3.41641i	0.143349i
$569$	− 15.9443i	− 0.668419i	−0.942499	−	0.334209i	$-0.891531\pi$
$569$	− 15.9443i	− 0.668419i	0.942499	−	0.334209i	$-0.108469\pi$
$570$	13.8541i	0.580284i
$571$	3.50658	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$571$	3.50658	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$572$	− 9.47214i	− 0.396050i
$573$	19.4721	0.813460
$574$	10.3262	0.431009
$575$	10.1803	0.424550
$576$	−1.61803	−0.0674181
$577$	7.23607i	0.301241i	0.988592	+	0.150621i	$0.0481272\pi$
$577$	7.23607i	0.301241i	−0.988592	+	0.150621i	$0.951873\pi$
$578$	43.3607i	1.80357i
$579$	−5.70820	−0.237225
$580$	0	0
$581$	17.7639	0.736972
$582$	43.3607i	1.79736i
$583$	7.23607i	0.299687i
$584$	0.652476	0.0269996
$585$	−4.61803	−0.190932
$586$	28.2705	1.16784
$587$	44.3820	1.83184	0.915920	−	0.401361i	$-0.131463\pi$
$587$	44.3820	1.83184	0.915920	+	0.401361i	$0.131463\pi$
$588$	2.00000i	0.0824786i
$589$	−2.02129	−0.0832856
$590$	23.5066i	0.967750i
$591$	− 31.8885i	− 1.31172i
$592$	− 42.2705i	− 1.73731i
$593$	34.5623	1.41930	0.709652	−	0.704552i	$-0.248852\pi$
$593$	34.5623	1.41930	0.709652	+	0.704552i	$0.248852\pi$
$594$	− 32.0344i	− 1.31439i
$595$	− 42.2361i	− 1.73151i
$596$	4.56231	0.186879
$597$	1.38197i	0.0565601i
$598$	22.1803i	0.907022i
$599$	− 45.0689i	− 1.84146i	−0.390195	−	0.920732i	$-0.627592\pi$
$599$	− 45.0689i	− 1.84146i	0.390195	−	0.920732i	$-0.372408\pi$
$600$	11.3820	0.464667
$601$	− 40.1591i	− 1.63812i	−0.573706	−	0.819061i	$-0.694495\pi$
$601$	− 40.1591i	− 1.63812i	0.573706	−	0.819061i	$-0.305505\pi$
$602$	−10.0000	−0.407570
$603$	4.00000	0.162893
$604$	11.3262	0.460858
$605$	−5.96556	−0.242534
$606$	4.23607i	0.172078i
$607$	35.9787i	1.46033i	0.683270	+	0.730165i	$0.260557\pi$
$607$	35.9787i	1.46033i	−0.683270	+	0.730165i	$0.739443\pi$
$608$	−6.27051	−0.254303
$609$	0	0
$610$	7.47214	0.302538
$611$	− 29.6525i	− 1.19961i
$612$	1.56231i	0.0631525i
$613$	39.5410	1.59705	0.798523	−	0.601964i	$-0.205615\pi$
$613$	39.5410	1.59705	0.798523	+	0.601964i	$0.205615\pi$
$614$	5.14590	0.207672
$615$	−13.1803	−0.531483
$616$	−18.0902	−0.728874
$617$	− 8.18034i	− 0.329328i	−0.986350	−	0.164664i	$-0.947346\pi$
$617$	− 8.18034i	− 0.329328i	0.986350	−	0.164664i	$-0.0526540\pi$
$618$	−34.5066	−1.38806
$619$	− 24.9443i	− 1.00259i	−0.865275	−	0.501297i	$-0.832856\pi$
$619$	− 24.9443i	− 1.00259i	0.865275	−	0.501297i	$-0.167144\pi$
$620$	− 1.92299i	− 0.0772290i
$621$	17.7082i	0.710606i
$622$	−14.7082	−0.589745
$623$	− 19.4721i	− 0.780135i
$624$	33.2705i	1.33189i
$625$	−30.8328	−1.23331
$626$	38.9787i	1.55790i
$627$	10.8541i	0.433471i
$628$	− 3.43769i	− 0.137179i
$629$	57.6312	2.29791
$630$	− 3.94427i	− 0.157144i
$631$	−23.2148	−0.924166	−0.462083	−	0.886837i	$-0.652898\pi$
$631$	−23.2148	−0.924166	−0.462083	+	0.886837i	$0.652898\pi$
$632$	11.3820	0.452750
$633$	31.7984	1.26387
$634$	−23.1246	−0.918396
$635$	− 5.54915i	− 0.220211i
$636$	2.00000i	0.0793052i
$637$	8.47214	0.335678
$638$	0	0
$639$	−0.583592	−0.0230865
$640$	38.8673i	1.53636i
$641$	28.9443i	1.14323i	0.820522	+	0.571615i	$0.193683\pi$
$641$	28.9443i	1.14323i	−0.820522	+	0.571615i	$0.806317\pi$
$642$	29.4164	1.16097
$643$	10.5836	0.417376	0.208688	−	0.977982i	$-0.433081\pi$
$643$	10.5836	0.417376	0.208688	+	0.977982i	$0.433081\pi$
$644$	−4.47214	−0.176227
$645$	12.7639	0.502579
$646$	− 19.8541i	− 0.781149i
$647$	39.4721	1.55181	0.775905	−	0.630850i	$-0.217294\pi$
$647$	39.4721	1.55181	0.775905	+	0.630850i	$0.217294\pi$
$648$	17.2361i	0.677097i
$649$	18.4164i	0.722907i
$650$	21.5623i	0.845743i
$651$	−3.94427	−0.154588
$652$	14.2361i	0.557527i
$653$	28.0132i	1.09624i	0.836400	+	0.548120i	$0.184656\pi$
$653$	28.0132i	1.09624i	−0.836400	+	0.548120i	$0.815344\pi$
$654$	43.5066	1.70124
$655$	3.78522i	0.147901i
$656$	− 13.8541i	− 0.540912i
$657$	0.111456i	0.00434832i
$658$	25.3262	0.987320
$659$	− 24.9443i	− 0.971691i	−0.874045	−	0.485845i	$-0.838512\pi$
$659$	− 24.9443i	− 0.971691i	0.874045	−	0.485845i	$-0.161488\pi$
$660$	−10.3262	−0.401948
$661$	18.4508	0.717655	0.358827	−	0.933404i	$-0.383177\pi$
$661$	18.4508	0.717655	0.358827	+	0.933404i	$0.383177\pi$
$662$	1.90983	0.0742277
$663$	−45.3607	−1.76166
$664$	− 17.7639i	− 0.689374i
$665$	11.8328i	0.458857i
$666$	5.38197	0.208547
$667$	0	0
$668$	12.0344	0.465627
$669$	29.6525i	1.14643i
$670$	− 48.3607i	− 1.86834i
$671$	5.85410	0.225995
$672$	−12.2361	−0.472017
$673$	−2.47214	−0.0952938	−0.0476469	−	0.998864i	$-0.515172\pi$
$673$	−2.47214	−0.0952938	−0.0476469	+	0.998864i	$0.515172\pi$
$674$	−38.9443	−1.50008
$675$	17.2148i	0.662597i
$676$	−3.05573	−0.117528
$677$	− 12.8328i	− 0.493205i	−0.969117	−	0.246603i	$-0.920686\pi$
$677$	− 12.8328i	− 0.493205i	0.969117	−	0.246603i	$-0.0793142\pi$
$678$	26.0344i	0.999847i
$679$	37.0344i	1.42125i
$680$	−42.2361	−1.61968
$681$	24.0902i	0.923137i
$682$	− 6.38197i	− 0.244378i
$683$	14.1459	0.541278	0.270639	−	0.962681i	$-0.412765\pi$
$683$	14.1459	0.541278	0.270639	+	0.962681i	$0.412765\pi$
$684$	− 0.437694i	− 0.0167357i
$685$	39.5410i	1.51078i
$686$	32.5623i	1.24323i
$687$	25.4164	0.969696
$688$	13.4164i	0.511496i
$689$	8.47214	0.322763
$690$	24.1803	0.920530
$691$	−41.8328	−1.59140	−0.795698	−	0.605694i	$-0.792896\pi$
$691$	−41.8328	−1.59140	−0.795698	+	0.605694i	$0.792896\pi$
$692$	−4.38197	−0.166577
$693$	− 3.09017i	− 0.117386i
$694$	13.1459i	0.499011i
$695$	41.9787	1.59234
$696$	0	0
$697$	18.8885	0.715455
$698$	− 21.7984i	− 0.825081i
$699$	17.4164i	0.658749i
$700$	−4.34752	−0.164321
$701$	38.9443	1.47090	0.735452	−	0.677576i	$-0.236970\pi$
$701$	38.9443	1.47090	0.735452	+	0.677576i	$0.236970\pi$
$702$	−37.5066	−1.41559
$703$	−16.1459	−0.608954
$704$	15.3262i	0.577629i
$705$	−32.3262	−1.21748
$706$	34.1803i	1.28639i
$707$	3.61803i	0.136070i
$708$	5.09017i	0.191300i
$709$	3.49342	0.131198	0.0655991	−	0.997846i	$-0.479104\pi$
$709$	3.49342	0.131198	0.0655991	+	0.997846i	$0.479104\pi$
$710$	7.05573i	0.264797i
$711$	1.94427i	0.0729159i
$712$	−19.4721	−0.729749
$713$	3.52786i	0.132120i
$714$	− 38.7426i	− 1.44991i
$715$	43.7426i	1.63588i
$716$	−9.88854	−0.369552
$717$	23.8541i	0.890848i
$718$	−45.6869	−1.70502
$719$	−29.5066	−1.10041	−0.550205	−	0.835030i	$-0.685450\pi$
$719$	−29.5066	−1.10041	−0.550205	+	0.835030i	$0.685450\pi$
$720$	−5.29180	−0.197214
$721$	−29.4721	−1.09760
$722$	− 25.1803i	− 0.937115i
$723$	− 43.1246i	− 1.60382i
$724$	7.38197	0.274349
$725$	0	0
$726$	−5.47214	−0.203090
$727$	45.9443i	1.70398i	0.523559	+	0.851989i	$0.324604\pi$
$727$	45.9443i	1.70398i	−0.523559	+	0.851989i	$0.675396\pi$
$728$	21.1803i	0.784996i
$729$	−29.5066	−1.09284
$730$	1.34752	0.0498741
$731$	−18.2918	−0.676547
$732$	1.61803	0.0598043
$733$	− 37.1803i	− 1.37329i	−0.726994	−	0.686644i	$-0.759083\pi$
$733$	− 37.1803i	− 1.37329i	0.726994	−	0.686644i	$-0.240917\pi$
$734$	−10.1459	−0.374492
$735$	− 9.23607i	− 0.340677i
$736$	10.9443i	0.403411i
$737$	− 37.8885i	− 1.39564i
$738$	1.76393	0.0649312
$739$	8.06888i	0.296819i	0.988926	+	0.148409i	$0.0474153\pi$
$739$	8.06888i	0.296819i	−0.988926	+	0.148409i	$0.952585\pi$
$740$	− 15.3607i	− 0.564670i
$741$	12.7082	0.466848
$742$	7.23607i	0.265644i
$743$	− 30.7639i	− 1.12862i	−0.825563	−	0.564310i	$-0.809142\pi$
$743$	− 30.7639i	− 1.12862i	0.825563	−	0.564310i	$-0.190858\pi$
$744$	3.94427i	0.144604i
$745$	−21.0689	−0.771904
$746$	− 29.7426i	− 1.08896i
$747$	3.03444	0.111024
$748$	14.7984	0.541082
$749$	25.1246	0.918033
$750$	−13.8541	−0.505880
$751$	27.4721i	1.00247i	0.865310	+	0.501236i	$0.167121\pi$
$751$	27.4721i	1.00247i	−0.865310	+	0.501236i	$0.832879\pi$
$752$	− 33.9787i	− 1.23908i
$753$	−18.8541	−0.687082
$754$	0	0
$755$	−52.3050	−1.90357
$756$	− 7.56231i	− 0.275038i
$757$	− 46.9787i	− 1.70747i	−0.520707	−	0.853735i	$-0.674332\pi$
$757$	− 46.9787i	− 1.70747i	0.520707	−	0.853735i	$-0.325668\pi$
$758$	−61.0132	−2.21610
$759$	18.9443	0.687634
$760$	11.8328	0.429221
$761$	49.7984	1.80519	0.902595	−	0.430491i	$-0.141660\pi$
$761$	49.7984	1.80519	0.902595	+	0.430491i	$0.141660\pi$
$762$	− 5.09017i	− 0.184397i
$763$	37.1591	1.34525
$764$	7.43769i	0.269086i
$765$	− 7.21478i	− 0.260851i
$766$	− 35.8328i	− 1.29469i
$767$	21.5623	0.778570
$768$	21.9443i	0.791846i
$769$	− 25.3607i	− 0.914530i	−0.889331	−	0.457265i	$-0.848829\pi$
$769$	− 25.3607i	− 0.914530i	0.889331	−	0.457265i	$-0.151171\pi$
$770$	−37.3607	−1.34639
$771$	− 1.32624i	− 0.0477633i
$772$	− 2.18034i	− 0.0784721i
$773$	− 14.0000i	− 0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$	− 14.0000i	− 0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$	−1.70820	−0.0614001
$775$	3.42956i	0.123194i
$776$	37.0344	1.32946
$777$	−31.5066	−1.13029
$778$	−34.1803	−1.22542
$779$	−5.29180	−0.189598
$780$	12.0902i	0.432898i
$781$	5.52786i	0.197803i
$782$	−34.6525	−1.23917
$783$	0	0
$784$	9.70820	0.346722
$785$	15.8754i	0.566617i
$786$	3.47214i	0.123847i
$787$	25.3607	0.904011	0.452005	−	0.892015i	$-0.350709\pi$
$787$	25.3607	0.904011	0.452005	+	0.892015i	$0.350709\pi$
$788$	12.1803	0.433907
$789$	5.32624	0.189619
$790$	23.5066	0.836327
$791$	22.2361i	0.790624i
$792$	−3.09017	−0.109804
$793$	− 6.85410i	− 0.243396i
$794$	51.6869i	1.83430i
$795$	− 9.23607i	− 0.327570i
$796$	−0.527864	−0.0187096
$797$	20.8197i	0.737470i	0.929535	+	0.368735i	$0.120209\pi$
$797$	20.8197i	0.737470i	−0.929535	+	0.368735i	$0.879791\pi$
$798$	10.8541i	0.384231i
$799$	46.3262	1.63890
$800$	10.6393i	0.376157i
$801$	− 3.32624i	− 0.117527i
$802$	53.5066i	1.88938i
$803$	1.05573	0.0372558
$804$	− 10.4721i	− 0.369324i
$805$	20.6525	0.727904
$806$	−7.47214	−0.263195
$807$	−9.70820	−0.341745
$808$	3.61803	0.127282
$809$	− 15.0689i	− 0.529794i	−0.964277	−	0.264897i	$-0.914662\pi$
$809$	− 15.0689i	− 0.529794i	0.964277	−	0.264897i	$-0.0853379\pi$
$810$	35.5967i	1.25074i
$811$	25.6525	0.900780	0.450390	−	0.892832i	$-0.351285\pi$
$811$	25.6525	0.900780	0.450390	+	0.892832i	$0.351285\pi$
$812$	0	0
$813$	−19.7082	−0.691197
$814$	− 50.9787i	− 1.78680i
$815$	− 65.7426i	− 2.30286i
$816$	−51.9787	−1.81962
$817$	5.12461	0.179287
$818$	0.944272	0.0330157
$819$	−3.61803	−0.126424
$820$	− 5.03444i	− 0.175810i
$821$	−35.4164	−1.23604	−0.618021	−	0.786162i	$-0.712065\pi$
$821$	−35.4164	−1.23604	−0.618021	+	0.786162i	$0.712065\pi$
$822$	36.2705i	1.26508i
$823$	3.47214i	0.121031i	0.998167	+	0.0605155i	$0.0192745\pi$
$823$	3.47214i	0.121031i	−0.998167	+	0.0605155i	$0.980726\pi$
$824$	29.4721i	1.02671i
$825$	18.4164	0.641177
$826$	18.4164i	0.640789i
$827$	− 56.0344i	− 1.94851i	−0.225452	−	0.974254i	$-0.572386\pi$
$827$	− 56.0344i	− 1.94851i	0.225452	−	0.974254i	$-0.427614\pi$
$828$	−0.763932	−0.0265485
$829$	8.20163i	0.284854i	0.989805	+	0.142427i	$0.0454907\pi$
$829$	8.20163i	0.284854i	−0.989805	+	0.142427i	$0.954509\pi$
$830$	− 36.6869i	− 1.27342i
$831$	− 38.2148i	− 1.32566i
$832$	17.9443	0.622106
$833$	13.2361i	0.458603i
$834$	38.5066	1.33337
$835$	−55.5755	−1.92327
$836$	−4.14590	−0.143389
$837$	−5.96556	−0.206200
$838$	4.14590i	0.143218i
$839$	− 3.21478i	− 0.110987i	−0.998459	−	0.0554933i	$-0.982327\pi$
$839$	− 3.21478i	− 0.110987i	0.998459	−	0.0554933i	$-0.0176731\pi$
$840$	23.0902	0.796687
$841$	0	0
$842$	3.18034	0.109602
$843$	− 27.7082i	− 0.954321i
$844$	12.1459i	0.418079i
$845$	14.1115	0.485449
$846$	4.32624	0.148739
$847$	−4.67376	−0.160592
$848$	9.70820	0.333381
$849$	1.23607i	0.0424217i
$850$	−33.6869	−1.15545
$851$	28.1803i	0.966010i
$852$	1.52786i	0.0523438i
$853$	45.0000i	1.54077i	0.637579	+	0.770385i	$0.279936\pi$
$853$	45.0000i	1.54077i	−0.637579	+	0.770385i	$0.720064\pi$
$854$	5.85410	0.200323
$855$	2.02129i	0.0691265i
$856$	− 25.1246i	− 0.858742i
$857$	−29.6738	−1.01364	−0.506818	−	0.862053i	$-0.669178\pi$
$857$	−29.6738	−1.01364	−0.506818	+	0.862053i	$0.669178\pi$
$858$	40.1246i	1.36983i
$859$	− 19.2918i	− 0.658228i	−0.944290	−	0.329114i	$-0.893250\pi$
$859$	− 19.2918i	− 0.658228i	0.944290	−	0.329114i	$-0.106750\pi$
$860$	4.87539i	0.166249i
$861$	−10.3262	−0.351917
$862$	55.9787i	1.90664i
$863$	24.7426	0.842249	0.421125	−	0.907003i	$-0.361635\pi$
$863$	24.7426	0.842249	0.421125	+	0.907003i	$0.361635\pi$
$864$	−18.5066	−0.629607
$865$	20.2361	0.688047
$866$	−20.4164	−0.693778
$867$	− 43.3607i	− 1.47261i
$868$	− 1.50658i	− 0.0511366i
$869$	18.4164	0.624734
$870$	0	0
$871$	−44.3607	−1.50310
$872$	− 37.1591i	− 1.25836i
$873$	6.32624i	0.214111i
$874$	9.70820	0.328385
$875$	−11.8328	−0.400022
$876$	0.291796	0.00985888
$877$	6.61803	0.223475	0.111738	−	0.993738i	$-0.464358\pi$
$877$	6.61803	0.223475	0.111738	+	0.993738i	$0.464358\pi$
$878$	− 4.94427i	− 0.166861i
$879$	−28.2705	−0.953541
$880$	50.1246i	1.68970i
$881$	− 29.2705i	− 0.986149i	−0.869987	−	0.493074i	$-0.835873\pi$
$881$	− 29.2705i	− 0.986149i	0.869987	−	0.493074i	$-0.164127\pi$
$882$	1.23607i	0.0416206i
$883$	−39.6869	−1.33557	−0.667786	−	0.744354i	$-0.732758\pi$
$883$	−39.6869	−1.33557	−0.667786	+	0.744354i	$0.732758\pi$
$884$	− 17.3262i	− 0.582744i
$885$	− 23.5066i	− 0.790165i
$886$	21.1803	0.711567
$887$	20.0689i	0.673847i	0.941532	+	0.336924i	$0.109386\pi$
$887$	20.0689i	0.673847i	−0.941532	+	0.336924i	$0.890614\pi$
$888$	31.5066i	1.05729i
$889$	− 4.34752i	− 0.145811i
$890$	−40.2148	−1.34800
$891$	27.8885i	0.934301i
$892$	−11.3262	−0.379230
$893$	−12.9787	−0.434316
$894$	−19.3262	−0.646366
$895$	45.6656	1.52643
$896$	30.4508i	1.01729i
$897$	− 22.1803i	− 0.740580i
$898$	22.8541	0.762651
$899$	0	0
$900$	−0.742646	−0.0247549
$901$	13.2361i	0.440957i
$902$	− 16.7082i	− 0.556322i
$903$	10.0000	0.332779
$904$	22.2361	0.739561
$905$	−34.0902	−1.13320
$906$	−47.9787	−1.59399
$907$	− 14.2361i	− 0.472701i	−0.971668	−	0.236350i	$-0.924049\pi$
$907$	− 14.2361i	− 0.472701i	0.971668	−	0.236350i	$-0.0759513\pi$
$908$	−9.20163	−0.305367
$909$	0.618034i	0.0204989i
$910$	43.7426i	1.45005i
$911$	− 6.94427i	− 0.230074i	−0.993361	−	0.115037i	$-0.963301\pi$
$911$	− 6.94427i	− 0.230074i	0.993361	−	0.115037i	$-0.0366987\pi$
$912$	14.5623	0.482206
$913$	− 28.7426i	− 0.951243i
$914$	8.56231i	0.283216i
$915$	−7.47214	−0.247021
$916$	9.70820i	0.320768i
$917$	2.96556i	0.0979314i
$918$	− 58.5967i	− 1.93398i
$919$	−31.3050	−1.03266	−0.516328	−	0.856391i	$-0.672701\pi$
$919$	−31.3050	−1.03266	−0.516328	+	0.856391i	$0.672701\pi$
$920$	− 20.6525i	− 0.680892i
$921$	−5.14590	−0.169563
$922$	−12.9098	−0.425163
$923$	6.47214	0.213033
$924$	−8.09017	−0.266147
$925$	27.3951i	0.900746i
$926$	− 4.38197i	− 0.144000i
$927$	−5.03444	−0.165353
$928$	0	0
$929$	27.6525	0.907248	0.453624	−	0.891193i	$-0.350131\pi$
$929$	27.6525	0.907248	0.453624	+	0.891193i	$0.350131\pi$
$930$	8.14590i	0.267115i
$931$	− 3.70820i	− 0.121531i
$932$	−6.65248	−0.217909
$933$	14.7082	0.481525
$934$	−0.0901699	−0.00295045
$935$	−68.3394	−2.23494
$936$	3.61803i	0.118259i
$937$	24.6525	0.805361	0.402681	−	0.915341i	$-0.368079\pi$
$937$	24.6525	0.805361	0.402681	+	0.915341i	$0.368079\pi$
$938$	− 37.8885i	− 1.23710i
$939$	− 38.9787i	− 1.27202i
$940$	− 12.3475i	− 0.402732i
$941$	−0.888544	−0.0289657	−0.0144829	−	0.999895i	$-0.504610\pi$
$941$	−0.888544	−0.0289657	−0.0144829	+	0.999895i	$0.504610\pi$
$942$	14.5623i	0.474466i
$943$	9.23607i	0.300768i
$944$	24.7082	0.804184
$945$	34.9230i	1.13604i
$946$	16.1803i	0.526068i
$947$	13.9656i	0.453820i	0.973916	+	0.226910i	$0.0728623\pi$
$947$	13.9656i	0.453820i	−0.973916	+	0.226910i	$0.927138\pi$
$948$	5.09017	0.165321
$949$	− 1.23607i	− 0.0401245i
$950$	9.43769	0.306199
$951$	23.1246	0.749867
$952$	−33.0902	−1.07246
$953$	−35.6312	−1.15421	−0.577104	−	0.816671i	$-0.695817\pi$
$953$	−35.6312	−1.15421	−0.577104	+	0.816671i	$0.695817\pi$
$954$	1.23607i	0.0400192i
$955$	− 34.3475i	− 1.11146i
$956$	−9.11146	−0.294686
$957$	0	0
$958$	−18.0902	−0.584467
$959$	30.9787i	1.00035i
$960$	− 19.5623i	− 0.631371i
$961$	29.8115	0.961662
$962$	−59.6869	−1.92438
$963$	4.29180	0.138301
$964$	16.4721	0.530532
$965$	10.0689i	0.324129i
$966$	18.9443	0.609522
$967$	16.5623i	0.532608i	0.963889	+	0.266304i	$0.0858025\pi$
$967$	16.5623i	0.532608i	−0.963889	+	0.266304i	$0.914198\pi$
$968$	4.67376i	0.150220i
$969$	19.8541i	0.637806i
$970$	76.4853	2.45579
$971$	18.5066i	0.593904i	0.954892	+	0.296952i	$0.0959702\pi$
$971$	18.5066i	0.593904i	−0.954892	+	0.296952i	$0.904030\pi$
$972$	− 2.43769i	− 0.0781891i
$973$	32.8885	1.05436
$974$	− 36.3050i	− 1.16329i
$975$	− 21.5623i	− 0.690546i
$976$	− 7.85410i	− 0.251404i
$977$	6.21478	0.198828	0.0994142	−	0.995046i	$-0.468303\pi$
$977$	6.21478	0.198828	0.0994142	+	0.995046i	$0.468303\pi$
$978$	− 60.3050i	− 1.92834i
$979$	−31.5066	−1.00695
$980$	3.52786	0.112693
$981$	6.34752	0.202661
$982$	−40.6525	−1.29727
$983$	− 6.94427i	− 0.221488i	−0.993849	−	0.110744i	$-0.964677\pi$
$983$	− 6.94427i	− 0.221488i	0.993849	−	0.110744i	$-0.0353233\pi$
$984$	10.3262i	0.329188i
$985$	−56.2492	−1.79225
$986$	0	0
$987$	−25.3262	−0.806143
$988$	4.85410i	0.154430i
$989$	− 8.94427i	− 0.284411i
$990$	−6.38197	−0.202832
$991$	−38.6525	−1.22784	−0.613918	−	0.789370i	$-0.710408\pi$
$991$	−38.6525	−1.22784	−0.613918	+	0.789370i	$0.710408\pi$
$992$	−3.68692	−0.117060
$993$	−1.90983	−0.0606066
$994$	5.52786i	0.175333i
$995$	2.43769	0.0772801
$996$	− 7.94427i	− 0.251724i
$997$	28.0902i	0.889625i	0.895624	+	0.444812i	$0.146730\pi$
$997$	28.0902i	0.889625i	−0.895624	+	0.444812i	$0.853270\pi$
$998$	− 57.7426i	− 1.82781i
$999$	−47.6525	−1.50766

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	841.2.b.b.840.1		4
29.2	odd	28		841.2.d.i.605.2		12
29.3	odd	28		841.2.d.i.571.2		12
29.4	even	14		841.2.e.j.651.4		24
29.5	even	14		841.2.e.j.236.1		24
29.6	even	14		841.2.e.j.196.4		24
29.7	even	7		841.2.e.j.270.4		24
29.8	odd	28		841.2.d.g.574.2		12
29.9	even	14		841.2.e.j.267.4		24
29.10	odd	28		841.2.d.g.190.1		12
29.11	odd	28		841.2.d.i.778.1		12
29.12	odd	4		841.2.a.c.1.2	yes	2
29.13	even	14		841.2.e.j.63.1		24
29.14	odd	28		841.2.d.i.645.2		12
29.15	odd	28		841.2.d.g.645.1		12
29.16	even	7		841.2.e.j.63.4		24
29.17	odd	4		841.2.a.a.1.1	✓	2
29.18	odd	28		841.2.d.g.778.2		12
29.19	odd	28		841.2.d.i.190.2		12
29.20	even	7		841.2.e.j.267.1		24
29.21	odd	28		841.2.d.i.574.1		12
29.22	even	14		841.2.e.j.270.1		24
29.23	even	7		841.2.e.j.196.1		24
29.24	even	7		841.2.e.j.236.4		24
29.25	even	7		841.2.e.j.651.1		24
29.26	odd	28		841.2.d.g.571.1		12
29.27	odd	28		841.2.d.g.605.1		12
29.28	even	2	inner	841.2.b.b.840.4		4
87.17	even	4		7569.2.a.l.1.2		2
87.41	even	4		7569.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
841.2.a.a.1.1	✓	2	29.17	odd	4
841.2.a.c.1.2	yes	2	29.12	odd	4
841.2.b.b.840.1		4	1.1	even	1	trivial
841.2.b.b.840.4		4	29.28	even	2	inner
841.2.d.g.190.1		12	29.10	odd	28
841.2.d.g.571.1		12	29.26	odd	28
841.2.d.g.574.2		12	29.8	odd	28
841.2.d.g.605.1		12	29.27	odd	28
841.2.d.g.645.1		12	29.15	odd	28
841.2.d.g.778.2		12	29.18	odd	28
841.2.d.i.190.2		12	29.19	odd	28
841.2.d.i.571.2		12	29.3	odd	28
841.2.d.i.574.1		12	29.21	odd	28
841.2.d.i.605.2		12	29.2	odd	28
841.2.d.i.645.2		12	29.14	odd	28
841.2.d.i.778.1		12	29.11	odd	28
841.2.e.j.63.1		24	29.13	even	14
841.2.e.j.63.4		24	29.16	even	7
841.2.e.j.196.1		24	29.23	even	7
841.2.e.j.196.4		24	29.6	even	14
841.2.e.j.236.1		24	29.5	even	14
841.2.e.j.236.4		24	29.24	even	7
841.2.e.j.267.1		24	29.20	even	7
841.2.e.j.267.4		24	29.9	even	14
841.2.e.j.270.1		24	29.22	even	14
841.2.e.j.270.4		24	29.7	even	7
841.2.e.j.651.1		24	29.25	even	7
841.2.e.j.651.4		24	29.4	even	14
7569.2.a.d.1.1		2	87.41	even	4
7569.2.a.l.1.2		2	87.17	even	4

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:	\( N \)	\(=\)	\( 841 = 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	841.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.61803i q^{2} +1.61803i q^{3} -0.618034 q^{4} +2.85410 q^{5} +2.61803 q^{6} +2.23607 q^{7} -2.23607i q^{8} +0.381966 q^{9} -4.61803i q^{10} -3.61803i q^{11} -1.00000i q^{12} -4.23607 q^{13} -3.61803i q^{14} +4.61803i q^{15} -4.85410 q^{16} -6.61803i q^{17} -0.618034i q^{18} +1.85410i q^{19} -1.76393 q^{20} +3.61803i q^{21} -5.85410 q^{22} +3.23607 q^{23} +3.61803 q^{24} +3.14590 q^{25} +6.85410i q^{26} +5.47214i q^{27} -1.38197 q^{28} +7.47214 q^{30} +1.09017i q^{31} +3.38197i q^{32} +5.85410 q^{33} -10.7082 q^{34} +6.38197 q^{35} -0.236068 q^{36} +8.70820i q^{37} +3.00000 q^{38} -6.85410i q^{39} -6.38197i q^{40} +2.85410i q^{41} +5.85410 q^{42} -2.76393i q^{43} +2.23607i q^{44} +1.09017 q^{45} -5.23607i q^{46} +7.00000i q^{47} -7.85410i q^{48} -2.00000 q^{49} -5.09017i q^{50} +10.7082 q^{51} +2.61803 q^{52} -2.00000 q^{53} +8.85410 q^{54} -10.3262i q^{55} -5.00000i q^{56} -3.00000 q^{57} -5.09017 q^{59} -2.85410i q^{60} +1.61803i q^{61} +1.76393 q^{62} +0.854102 q^{63} -4.23607 q^{64} -12.0902 q^{65} -9.47214i q^{66} +10.4721 q^{67} +4.09017i q^{68} +5.23607i q^{69} -10.3262i q^{70} -1.52786 q^{71} -0.854102i q^{72} +0.291796i q^{73} +14.0902 q^{74} +5.09017i q^{75} -1.14590i q^{76} -8.09017i q^{77} -11.0902 q^{78} +5.09017i q^{79} -13.8541 q^{80} -7.70820 q^{81} +4.61803 q^{82} +7.94427 q^{83} -2.23607i q^{84} -18.8885i q^{85} -4.47214 q^{86} -8.09017 q^{88} -8.70820i q^{89} -1.76393i q^{90} -9.47214 q^{91} -2.00000 q^{92} -1.76393 q^{93} +11.3262 q^{94} +5.29180i q^{95} -5.47214 q^{96} +16.5623i q^{97} +3.23607i q^{98} -1.38197i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{5} + 6 q^{6} + 6 q^{9} - 8 q^{13} - 6 q^{16} - 16 q^{20} - 10 q^{22} + 4 q^{23} + 10 q^{24} + 26 q^{25} - 10 q^{28} + 12 q^{30} + 10 q^{33} - 16 q^{34} + 30 q^{35} + 8 q^{36} + 12 q^{38}+ \cdots - 4 q^{96}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^5 + 6 * q^6 + 6 * q^9 - 8 * q^13 - 6 * q^16 - 16 * q^20 - 10 * q^22 + 4 * q^23 + 10 * q^24 + 26 * q^25 - 10 * q^28 + 12 * q^30 + 10 * q^33 - 16 * q^34 + 30 * q^35 + 8 * q^36 + 12 * q^38 + 10 * q^42 - 18 * q^45 - 8 * q^49 + 16 * q^51 + 6 * q^52 - 8 * q^53 + 22 * q^54 - 12 * q^57 + 2 * q^59 + 16 * q^62 - 10 * q^63 - 8 * q^64 - 26 * q^65 + 24 * q^67 - 24 * q^71 + 34 * q^74 - 22 * q^78 - 42 * q^80 - 4 * q^81 + 14 * q^82 - 4 * q^83 - 10 * q^88 - 20 * q^91 - 8 * q^92 - 16 * q^93 + 14 * q^94 - 4 * q^96

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