LMFDB - Embedded newform 7569.2.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7569,2,Mod(1,7569)] 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("7569.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7569, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,1,0,-1,-1,0,0,0,0,7,5,0,4,5,0,-3,11,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.4387692899$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 841)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} +2.85410 q^{5} +2.23607 q^{7} -2.23607 q^{8} +4.61803 q^{10} +3.61803 q^{11} +4.23607 q^{13} +3.61803 q^{14} -4.85410 q^{16} +6.61803 q^{17} +1.85410 q^{19} +1.76393 q^{20} +5.85410 q^{22} -3.23607 q^{23} +3.14590 q^{25} +6.85410 q^{26} +1.38197 q^{28} +1.09017 q^{31} -3.38197 q^{32} +10.7082 q^{34} +6.38197 q^{35} -8.70820 q^{37} +3.00000 q^{38} -6.38197 q^{40} +2.85410 q^{41} -2.76393 q^{43} +2.23607 q^{44} -5.23607 q^{46} +7.00000 q^{47} -2.00000 q^{49} +5.09017 q^{50} +2.61803 q^{52} +2.00000 q^{53} +10.3262 q^{55} -5.00000 q^{56} +5.09017 q^{59} +1.61803 q^{61} +1.76393 q^{62} +4.23607 q^{64} +12.0902 q^{65} -10.4721 q^{67} +4.09017 q^{68} +10.3262 q^{70} -1.52786 q^{71} -0.291796 q^{73} -14.0902 q^{74} +1.14590 q^{76} +8.09017 q^{77} +5.09017 q^{79} -13.8541 q^{80} +4.61803 q^{82} -7.94427 q^{83} +18.8885 q^{85} -4.47214 q^{86} -8.09017 q^{88} +8.70820 q^{89} +9.47214 q^{91} -2.00000 q^{92} +11.3262 q^{94} +5.29180 q^{95} -16.5623 q^{97} -3.23607 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - q^{5} + 7 q^{10} + 5 q^{11} + 4 q^{13} + 5 q^{14} - 3 q^{16} + 11 q^{17} - 3 q^{19} + 8 q^{20} + 5 q^{22} - 2 q^{23} + 13 q^{25} + 7 q^{26} + 5 q^{28} - 9 q^{31} - 9 q^{32} + 8 q^{34}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - q^5 + 7 * q^10 + 5 * q^11 + 4 * q^13 + 5 * q^14 - 3 * q^16 + 11 * q^17 - 3 * q^19 + 8 * q^20 + 5 * q^22 - 2 * q^23 + 13 * q^25 + 7 * q^26 + 5 * q^28 - 9 * q^31 - 9 * q^32 + 8 * q^34 + 15 * q^35 - 4 * q^37 + 6 * q^38 - 15 * q^40 - q^41 - 10 * q^43 - 6 * q^46 + 14 * q^47 - 4 * q^49 - q^50 + 3 * q^52 + 4 * q^53 + 5 * q^55 - 10 * q^56 - q^59 + q^61 + 8 * q^62 + 4 * q^64 + 13 * q^65 - 12 * q^67 - 3 * q^68 + 5 * q^70 - 12 * q^71 - 14 * q^73 - 17 * q^74 + 9 * q^76 + 5 * q^77 - q^79 - 21 * q^80 + 7 * q^82 + 2 * q^83 + 2 * q^85 - 5 * q^88 + 4 * q^89 + 10 * q^91 - 4 * q^92 + 7 * q^94 + 24 * q^95 - 13 * q^97 - 2 * q^98

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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	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	$0.279681\pi$
$6$	0	0
$7$	2.23607	0.845154	0.422577	−	0.906327i	$-0.361126\pi$
$7$	2.23607	0.845154	0.422577	+	0.906327i	$0.361126\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	4.61803	1.46035
$11$	3.61803	1.09088	0.545439	−	0.838150i	$-0.316363\pi$
$11$	3.61803	1.09088	0.545439	+	0.838150i	$0.316363\pi$
$12$	0	0
$13$	4.23607	1.17487	0.587437	−	0.809270i	$-0.300137\pi$
$13$	4.23607	1.17487	0.587437	+	0.809270i	$0.300137\pi$
$14$	3.61803	0.966960
$15$	0	0
$16$	−4.85410	−1.21353
$17$	6.61803	1.60511	0.802555	−	0.596579i	$-0.203474\pi$
$17$	6.61803	1.60511	0.802555	+	0.596579i	$0.203474\pi$
$18$	0	0
$19$	1.85410	0.425360	0.212680	−	0.977122i	$-0.431781\pi$
$19$	1.85410	0.425360	0.212680	+	0.977122i	$0.431781\pi$
$20$	1.76393	0.394427
$21$	0	0
$22$	5.85410	1.24810
$23$	−3.23607	−0.674767	−0.337383	−	0.941367i	$-0.609542\pi$
$23$	−3.23607	−0.674767	−0.337383	+	0.941367i	$0.609542\pi$
$24$	0	0
$25$	3.14590	0.629180
$26$	6.85410	1.34420
$27$	0	0
$28$	1.38197	0.261167
$29$	0	0
$29$	0	0
$30$	0	0
$31$	1.09017	0.195800	0.0979002	−	0.995196i	$-0.468787\pi$
$31$	1.09017	0.195800	0.0979002	+	0.995196i	$0.468787\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	10.7082	1.83644
$35$	6.38197	1.07875
$36$	0	0
$37$	−8.70820	−1.43162	−0.715810	−	0.698295i	$-0.753942\pi$
$37$	−8.70820	−1.43162	−0.715810	+	0.698295i	$0.753942\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	−6.38197	−1.00908
$41$	2.85410	0.445736	0.222868	−	0.974849i	$-0.428458\pi$
$41$	2.85410	0.445736	0.222868	+	0.974849i	$0.428458\pi$
$42$	0	0
$43$	−2.76393	−0.421496	−0.210748	−	0.977540i	$-0.567590\pi$
$43$	−2.76393	−0.421496	−0.210748	+	0.977540i	$0.567590\pi$
$44$	2.23607	0.337100
$45$	0	0
$46$	−5.23607	−0.772016
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	5.09017	0.719859
$51$	0	0
$52$	2.61803	0.363056
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	10.3262	1.39239
$56$	−5.00000	−0.668153
$57$	0	0
$58$	0	0
$59$	5.09017	0.662684	0.331342	−	0.943511i	$-0.392499\pi$
$59$	5.09017	0.662684	0.331342	+	0.943511i	$0.392499\pi$
$60$	0	0
$61$	1.61803	0.207168	0.103584	−	0.994621i	$-0.466969\pi$
$61$	1.61803	0.207168	0.103584	+	0.994621i	$0.466969\pi$
$62$	1.76393	0.224020
$63$	0	0
$64$	4.23607	0.529508
$65$	12.0902	1.49960
$66$	0	0
$67$	−10.4721	−1.27938	−0.639688	−	0.768635i	$-0.720936\pi$
$67$	−10.4721	−1.27938	−0.639688	+	0.768635i	$0.720936\pi$
$68$	4.09017	0.496006
$69$	0	0
$70$	10.3262	1.23422
$71$	−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	0
$73$	−0.291796	−0.0341521	−0.0170761	−	0.999854i	$-0.505436\pi$
$73$	−0.291796	−0.0341521	−0.0170761	+	0.999854i	$0.505436\pi$
$74$	−14.0902	−1.63795
$75$	0	0
$76$	1.14590	0.131444
$77$	8.09017	0.921960
$78$	0	0
$79$	5.09017	0.572689	0.286344	−	0.958127i	$-0.407560\pi$
$79$	5.09017	0.572689	0.286344	+	0.958127i	$0.407560\pi$
$80$	−13.8541	−1.54894
$81$	0	0
$82$	4.61803	0.509977
$83$	−7.94427	−0.871997	−0.435999	−	0.899947i	$-0.643605\pi$
$83$	−7.94427	−0.871997	−0.435999	+	0.899947i	$0.643605\pi$
$84$	0	0
$85$	18.8885	2.04875
$86$	−4.47214	−0.482243
$87$	0	0
$88$	−8.09017	−0.862415
$89$	8.70820	0.923068	0.461534	−	0.887123i	$-0.347299\pi$
$89$	8.70820	0.923068	0.461534	+	0.887123i	$0.347299\pi$
$90$	0	0
$91$	9.47214	0.992950
$92$	−2.00000	−0.208514
$93$	0	0
$94$	11.3262	1.16821
$95$	5.29180	0.542927
$96$	0	0
$97$	−16.5623	−1.68165	−0.840824	−	0.541309i	$-0.817929\pi$
$97$	−16.5623	−1.68165	−0.840824	+	0.541309i	$0.817929\pi$
$98$	−3.23607	−0.326892
$99$	0	0
$100$	1.94427	0.194427
$101$	−1.61803	−0.161000	−0.0805002	−	0.996755i	$-0.525652\pi$
$101$	−1.61803	−0.161000	−0.0805002	+	0.996755i	$0.525652\pi$
$102$	0	0
$103$	−13.1803	−1.29870	−0.649349	−	0.760491i	$-0.724958\pi$
$103$	−13.1803	−1.29870	−0.649349	+	0.760491i	$0.724958\pi$
$104$	−9.47214	−0.928819
$105$	0	0
$106$	3.23607	0.314315
$107$	−11.2361	−1.08623	−0.543116	−	0.839658i	$-0.682756\pi$
$107$	−11.2361	−1.08623	−0.543116	+	0.839658i	$0.682756\pi$
$108$	0	0
$109$	−16.6180	−1.59172	−0.795859	−	0.605481i	$-0.792981\pi$
$109$	−16.6180	−1.59172	−0.795859	+	0.605481i	$0.792981\pi$
$110$	16.7082	1.59306
$111$	0	0
$112$	−10.8541	−1.02562
$113$	9.94427	0.935478	0.467739	−	0.883867i	$-0.345069\pi$
$113$	9.94427	0.935478	0.467739	+	0.883867i	$0.345069\pi$
$114$	0	0
$115$	−9.23607	−0.861268
$116$	0	0
$117$	0	0
$118$	8.23607	0.758192
$119$	14.7984	1.35656
$120$	0	0
$121$	2.09017	0.190015
$122$	2.61803	0.237026
$123$	0	0
$124$	0.673762	0.0605056
$125$	−5.29180	−0.473313
$126$	0	0
$127$	−1.94427	−0.172526	−0.0862631	−	0.996272i	$-0.527493\pi$
$127$	−1.94427	−0.172526	−0.0862631	+	0.996272i	$0.527493\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	19.5623	1.71573
$131$	1.32624	0.115874	0.0579370	−	0.998320i	$-0.481548\pi$
$131$	1.32624	0.115874	0.0579370	+	0.998320i	$0.481548\pi$
$132$	0	0
$133$	4.14590	0.359495
$134$	−16.9443	−1.46376
$135$	0	0
$136$	−14.7984	−1.26895
$137$	−13.8541	−1.18364	−0.591818	−	0.806072i	$-0.701590\pi$
$137$	−13.8541	−1.18364	−0.591818	+	0.806072i	$0.701590\pi$
$138$	0	0
$139$	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
$141$	0	0
$142$	−2.47214	−0.207457
$143$	15.3262	1.28164
$144$	0	0
$145$	0	0
$146$	−0.472136	−0.0390742
$147$	0	0
$148$	−5.38197	−0.442395
$149$	−7.38197	−0.604754	−0.302377	−	0.953188i	$-0.597780\pi$
$149$	−7.38197	−0.604754	−0.302377	+	0.953188i	$0.597780\pi$
$150$	0	0
$151$	18.3262	1.49137	0.745684	−	0.666300i	$-0.232123\pi$
$151$	18.3262	1.49137	0.745684	+	0.666300i	$0.232123\pi$
$152$	−4.14590	−0.336277
$153$	0	0
$154$	13.0902	1.05484
$155$	3.11146	0.249918
$156$	0	0
$157$	−5.56231	−0.443920	−0.221960	−	0.975056i	$-0.571245\pi$
$157$	−5.56231	−0.443920	−0.221960	+	0.975056i	$0.571245\pi$
$158$	8.23607	0.655226
$159$	0	0
$160$	−9.65248	−0.763095
$161$	−7.23607	−0.570282
$162$	0	0
$163$	23.0344	1.80420	0.902098	−	0.431530i	$-0.142026\pi$
$163$	23.0344	1.80420	0.902098	+	0.431530i	$0.142026\pi$
$164$	1.76393	0.137740
$165$	0	0
$166$	−12.8541	−0.997672
$167$	−19.4721	−1.50680	−0.753400	−	0.657563i	$-0.771587\pi$
$167$	−19.4721	−1.50680	−0.753400	+	0.657563i	$0.771587\pi$
$168$	0	0
$169$	4.94427	0.380329
$170$	30.5623	2.34402
$171$	0	0
$172$	−1.70820	−0.130249
$173$	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
$176$	−17.5623	−1.32381
$177$	0	0
$178$	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	0
$181$	−11.9443	−0.887811	−0.443905	−	0.896074i	$-0.646407\pi$
$181$	−11.9443	−0.887811	−0.443905	+	0.896074i	$0.646407\pi$
$182$	15.3262	1.13606
$183$	0	0
$184$	7.23607	0.533450
$185$	−24.8541	−1.82731
$186$	0	0
$187$	23.9443	1.75098
$188$	4.32624	0.315523
$189$	0	0
$190$	8.56231	0.621175
$191$	12.0344	0.870782	0.435391	−	0.900242i	$-0.356610\pi$
$191$	12.0344	0.870782	0.435391	+	0.900242i	$0.356610\pi$
$192$	0	0
$193$	3.52786	0.253941	0.126971	−	0.991906i	$-0.459475\pi$
$193$	3.52786	0.253941	0.126971	+	0.991906i	$0.459475\pi$
$194$	−26.7984	−1.92401
$195$	0	0
$196$	−1.23607	−0.0882906
$197$	19.7082	1.40415	0.702076	−	0.712102i	$-0.252257\pi$
$197$	19.7082	1.40415	0.702076	+	0.712102i	$0.252257\pi$
$198$	0	0
$199$	0.854102	0.0605457	0.0302728	−	0.999542i	$-0.490362\pi$
$199$	0.854102	0.0605457	0.0302728	+	0.999542i	$0.490362\pi$
$200$	−7.03444	−0.497410
$201$	0	0
$202$	−2.61803	−0.184204
$203$	0	0
$204$	0	0
$205$	8.14590	0.568934
$206$	−21.3262	−1.48587
$207$	0	0
$208$	−20.5623	−1.42574
$209$	6.70820	0.464016
$210$	0	0
$211$	19.6525	1.35293	0.676466	−	0.736474i	$-0.263511\pi$
$211$	19.6525	1.35293	0.676466	+	0.736474i	$0.263511\pi$
$212$	1.23607	0.0848935
$213$	0	0
$214$	−18.1803	−1.24278
$215$	−7.88854	−0.537994
$216$	0	0
$217$	2.43769	0.165481
$218$	−26.8885	−1.82112
$219$	0	0
$220$	6.38197	0.430272
$221$	28.0344	1.88580
$222$	0	0
$223$	18.3262	1.22722	0.613608	−	0.789611i	$-0.289718\pi$
$223$	18.3262	1.22722	0.613608	+	0.789611i	$0.289718\pi$
$224$	−7.56231	−0.505278
$225$	0	0
$226$	16.0902	1.07030
$227$	−14.8885	−0.988187	−0.494094	−	0.869409i	$-0.664500\pi$
$227$	−14.8885	−0.988187	−0.494094	+	0.869409i	$0.664500\pi$
$228$	0	0
$229$	15.7082	1.03803	0.519014	−	0.854766i	$-0.326299\pi$
$229$	15.7082	1.03803	0.519014	+	0.854766i	$0.326299\pi$
$230$	−14.9443	−0.985396
$231$	0	0
$232$	0	0
$233$	−10.7639	−0.705169	−0.352584	−	0.935780i	$-0.614697\pi$
$233$	−10.7639	−0.705169	−0.352584	+	0.935780i	$0.614697\pi$
$234$	0	0
$235$	19.9787	1.30327
$236$	3.14590	0.204781
$237$	0	0
$238$	23.9443	1.55208
$239$	−14.7426	−0.953622	−0.476811	−	0.879006i	$-0.658207\pi$
$239$	−14.7426	−0.953622	−0.476811	+	0.879006i	$0.658207\pi$
$240$	0	0
$241$	26.6525	1.71684	0.858418	−	0.512950i	$-0.171447\pi$
$241$	26.6525	1.71684	0.858418	+	0.512950i	$0.171447\pi$
$242$	3.38197	0.217401
$243$	0	0
$244$	1.00000	0.0640184
$245$	−5.70820	−0.364684
$246$	0	0
$247$	7.85410	0.499745
$248$	−2.43769	−0.154794
$249$	0	0
$250$	−8.56231	−0.541528
$251$	−11.6525	−0.735498	−0.367749	−	0.929925i	$-0.619871\pi$
$251$	−11.6525	−0.735498	−0.367749	+	0.929925i	$0.619871\pi$
$252$	0	0
$253$	−11.7082	−0.736088
$254$	−3.14590	−0.197391
$255$	0	0
$256$	13.5623	0.847644
$257$	0.819660	0.0511290	0.0255645	−	0.999673i	$-0.491862\pi$
$257$	0.819660	0.0511290	0.0255645	+	0.999673i	$0.491862\pi$
$258$	0	0
$259$	−19.4721	−1.20994
$260$	7.47214	0.463402
$261$	0	0
$262$	2.14590	0.132574
$263$	3.29180	0.202981	0.101490	−	0.994837i	$-0.467639\pi$
$263$	3.29180	0.202981	0.101490	+	0.994837i	$0.467639\pi$
$264$	0	0
$265$	5.70820	0.350652
$266$	6.70820	0.411306
$267$	0	0
$268$	−6.47214	−0.395349
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−12.1803	−0.739903	−0.369951	−	0.929051i	$-0.620626\pi$
$271$	−12.1803	−0.739903	−0.369951	+	0.929051i	$0.620626\pi$
$272$	−32.1246	−1.94784
$273$	0	0
$274$	−22.4164	−1.35422
$275$	11.3820	0.686358
$276$	0	0
$277$	−23.6180	−1.41907	−0.709535	−	0.704670i	$-0.751095\pi$
$277$	−23.6180	−1.41907	−0.709535	+	0.704670i	$0.751095\pi$
$278$	23.7984	1.42733
$279$	0	0
$280$	−14.2705	−0.852826
$281$	17.1246	1.02157	0.510784	−	0.859709i	$-0.329355\pi$
$281$	17.1246	1.02157	0.510784	+	0.859709i	$0.329355\pi$
$282$	0	0
$283$	−0.763932	−0.0454110	−0.0227055	−	0.999742i	$-0.507228\pi$
$283$	−0.763932	−0.0454110	−0.0227055	+	0.999742i	$0.507228\pi$
$284$	−0.944272	−0.0560322
$285$	0	0
$286$	24.7984	1.46636
$287$	6.38197	0.376716
$288$	0	0
$289$	26.7984	1.57637
$290$	0	0
$291$	0	0
$292$	−0.180340	−0.0105536
$293$	−17.4721	−1.02073	−0.510367	−	0.859957i	$-0.670490\pi$
$293$	−17.4721	−1.02073	−0.510367	+	0.859957i	$0.670490\pi$
$294$	0	0
$295$	14.5279	0.845845
$296$	19.4721	1.13179
$297$	0	0
$298$	−11.9443	−0.691913
$299$	−13.7082	−0.792766
$300$	0	0
$301$	−6.18034	−0.356229
$302$	29.6525	1.70631
$303$	0	0
$304$	−9.00000	−0.516185
$305$	4.61803	0.264428
$306$	0	0
$307$	3.18034	0.181512	0.0907558	−	0.995873i	$-0.471072\pi$
$307$	3.18034	0.181512	0.0907558	+	0.995873i	$0.471072\pi$
$308$	5.00000	0.284901
$309$	0	0
$310$	5.03444	0.285937
$311$	9.09017	0.515456	0.257728	−	0.966217i	$-0.417026\pi$
$311$	9.09017	0.515456	0.257728	+	0.966217i	$0.417026\pi$
$312$	0	0
$313$	−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
$315$	0	0
$316$	3.14590	0.176971
$317$	−14.2918	−0.802707	−0.401354	−	0.915923i	$-0.631460\pi$
$317$	−14.2918	−0.802707	−0.401354	+	0.915923i	$0.631460\pi$
$318$	0	0
$319$	0	0
$320$	12.0902	0.675861
$321$	0	0
$322$	−11.7082	−0.652473
$323$	12.2705	0.682749
$324$	0	0
$325$	13.3262	0.739207
$326$	37.2705	2.06422
$327$	0	0
$328$	−6.38197	−0.352385
$329$	15.6525	0.862949
$330$	0	0
$331$	−1.18034	−0.0648773	−0.0324387	−	0.999474i	$-0.510327\pi$
$331$	−1.18034	−0.0648773	−0.0324387	+	0.999474i	$0.510327\pi$
$332$	−4.90983	−0.269462
$333$	0	0
$334$	−31.5066	−1.72396
$335$	−29.8885	−1.63299
$336$	0	0
$337$	24.0689	1.31112	0.655558	−	0.755145i	$-0.272434\pi$
$337$	24.0689	1.31112	0.655558	+	0.755145i	$0.272434\pi$
$338$	8.00000	0.435143
$339$	0	0
$340$	11.6738	0.633099
$341$	3.94427	0.213594
$342$	0	0
$343$	−20.1246	−1.08663
$344$	6.18034	0.333222
$345$	0	0
$346$	11.4721	0.616746
$347$	−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	$0.382581\pi$
$350$	11.3820	0.608392
$351$	0	0
$352$	−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	$0.690035\pi$
$354$	0	0
$355$	−4.36068	−0.231441
$356$	5.38197	0.285244
$357$	0	0
$358$	25.8885	1.36825
$359$	28.2361	1.49024	0.745121	−	0.666929i	$-0.232392\pi$
$359$	28.2361	1.49024	0.745121	+	0.666929i	$0.232392\pi$
$360$	0	0
$361$	−15.5623	−0.819069
$362$	−19.3262	−1.01576
$363$	0	0
$364$	5.85410	0.306838
$365$	−0.832816	−0.0435916
$366$	0	0
$367$	−6.27051	−0.327318	−0.163659	−	0.986517i	$-0.552330\pi$
$367$	−6.27051	−0.327318	−0.163659	+	0.986517i	$0.552330\pi$
$368$	15.7082	0.818847
$369$	0	0
$370$	−40.2148	−2.09067
$371$	4.47214	0.232182
$372$	0	0
$373$	18.3820	0.951782	0.475891	−	0.879504i	$-0.342126\pi$
$373$	18.3820	0.951782	0.475891	+	0.879504i	$0.342126\pi$
$374$	38.7426	2.00333
$375$	0	0
$376$	−15.6525	−0.807215
$377$	0	0
$378$	0	0
$379$	−37.7082	−1.93694	−0.968470	−	0.249130i	$-0.919855\pi$
$379$	−37.7082	−1.93694	−0.968470	+	0.249130i	$0.919855\pi$
$380$	3.27051	0.167774
$381$	0	0
$382$	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	$0.308567\pi$
$384$	0	0
$385$	23.0902	1.17678
$386$	5.70820	0.290540
$387$	0	0
$388$	−10.2361	−0.519658
$389$	−21.1246	−1.07106	−0.535530	−	0.844516i	$-0.679888\pi$
$389$	−21.1246	−1.07106	−0.535530	+	0.844516i	$0.679888\pi$
$390$	0	0
$391$	−21.4164	−1.08307
$392$	4.47214	0.225877
$393$	0	0
$394$	31.8885	1.60652
$395$	14.5279	0.730976
$396$	0	0
$397$	−31.9443	−1.60324	−0.801619	−	0.597836i	$-0.796027\pi$
$397$	−31.9443	−1.60324	−0.801619	+	0.597836i	$0.796027\pi$
$398$	1.38197	0.0692717
$399$	0	0
$400$	−15.2705	−0.763525
$401$	33.0689	1.65138	0.825691	−	0.564123i	$-0.190786\pi$
$401$	33.0689	1.65138	0.825691	+	0.564123i	$0.190786\pi$
$402$	0	0
$403$	4.61803	0.230041
$404$	−1.00000	−0.0497519
$405$	0	0
$406$	0	0
$407$	−31.5066	−1.56172
$408$	0	0
$409$	0.583592	0.0288568	0.0144284	−	0.999896i	$-0.495407\pi$
$409$	0.583592	0.0288568	0.0144284	+	0.999896i	$0.495407\pi$
$410$	13.1803	0.650931
$411$	0	0
$412$	−8.14590	−0.401320
$413$	11.3820	0.560070
$414$	0	0
$415$	−22.6738	−1.11301
$416$	−14.3262	−0.702402
$417$	0	0
$418$	10.8541	0.530891
$419$	−2.56231	−0.125177	−0.0625884	−	0.998039i	$-0.519936\pi$
$419$	−2.56231	−0.125177	−0.0625884	+	0.998039i	$0.519936\pi$
$420$	0	0
$421$	−1.96556	−0.0957954	−0.0478977	−	0.998852i	$-0.515252\pi$
$421$	−1.96556	−0.0957954	−0.0478977	+	0.998852i	$0.515252\pi$
$422$	31.7984	1.54792
$423$	0	0
$424$	−4.47214	−0.217186
$425$	20.8197	1.00990
$426$	0	0
$427$	3.61803	0.175089
$428$	−6.94427	−0.335664
$429$	0	0
$430$	−12.7639	−0.615531
$431$	34.5967	1.66647	0.833233	−	0.552921i	$-0.186487\pi$
$431$	34.5967	1.66647	0.833233	+	0.552921i	$0.186487\pi$
$432$	0	0
$433$	12.6180	0.606384	0.303192	−	0.952929i	$-0.401948\pi$
$433$	12.6180	0.606384	0.303192	+	0.952929i	$0.401948\pi$
$434$	3.94427	0.189331
$435$	0	0
$436$	−10.2705	−0.491868
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−3.05573	−0.145842	−0.0729210	−	0.997338i	$-0.523232\pi$
$439$	−3.05573	−0.145842	−0.0729210	+	0.997338i	$0.523232\pi$
$440$	−23.0902	−1.10078
$441$	0	0
$442$	45.3607	2.15759
$443$	13.0902	0.621933	0.310966	−	0.950421i	$-0.399347\pi$
$443$	13.0902	0.621933	0.310966	+	0.950421i	$0.399347\pi$
$444$	0	0
$445$	24.8541	1.17820
$446$	29.6525	1.40409
$447$	0	0
$448$	9.47214	0.447516
$449$	−14.1246	−0.666582	−0.333291	−	0.942824i	$-0.608159\pi$
$449$	−14.1246	−0.666582	−0.333291	+	0.942824i	$0.608159\pi$
$450$	0	0
$451$	10.3262	0.486244
$452$	6.14590	0.289079
$453$	0	0
$454$	−24.0902	−1.13061
$455$	27.0344	1.26739
$456$	0	0
$457$	5.29180	0.247540	0.123770	−	0.992311i	$-0.460502\pi$
$457$	5.29180	0.247540	0.123770	+	0.992311i	$0.460502\pi$
$458$	25.4164	1.18763
$459$	0	0
$460$	−5.70820	−0.266146
$461$	−7.97871	−0.371606	−0.185803	−	0.982587i	$-0.559489\pi$
$461$	−7.97871	−0.371606	−0.185803	+	0.982587i	$0.559489\pi$
$462$	0	0
$463$	−2.70820	−0.125861	−0.0629305	−	0.998018i	$-0.520045\pi$
$463$	−2.70820	−0.125861	−0.0629305	+	0.998018i	$0.520045\pi$
$464$	0	0
$465$	0	0
$466$	−17.4164	−0.806800
$467$	0.0557281	0.00257879	0.00128939	−	0.999999i	$-0.499590\pi$
$467$	0.0557281	0.00257879	0.00128939	+	0.999999i	$0.499590\pi$
$468$	0	0
$469$	−23.4164	−1.08127
$470$	32.3262	1.49110
$471$	0	0
$472$	−11.3820	−0.523897
$473$	−10.0000	−0.459800
$474$	0	0
$475$	5.83282	0.267628
$476$	9.14590	0.419202
$477$	0	0
$478$	−23.8541	−1.09106
$479$	−11.1803	−0.510843	−0.255421	−	0.966830i	$-0.582214\pi$
$479$	−11.1803	−0.510843	−0.255421	+	0.966830i	$0.582214\pi$
$480$	0	0
$481$	−36.8885	−1.68197
$482$	43.1246	1.96427
$483$	0	0
$484$	1.29180	0.0587180
$485$	−47.2705	−2.14644
$486$	0	0
$487$	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
$489$	0	0
$490$	−9.23607	−0.417243
$491$	−25.1246	−1.13386	−0.566929	−	0.823767i	$-0.691869\pi$
$491$	−25.1246	−1.13386	−0.566929	+	0.823767i	$0.691869\pi$
$492$	0	0
$493$	0	0
$494$	12.7082	0.571769
$495$	0	0
$496$	−5.29180	−0.237609
$497$	−3.41641	−0.153247
$498$	0	0
$499$	−35.6869	−1.59757	−0.798783	−	0.601619i	$-0.794522\pi$
$499$	−35.6869	−1.59757	−0.798783	+	0.601619i	$0.794522\pi$
$500$	−3.27051	−0.146262
$501$	0	0
$502$	−18.8541	−0.841500
$503$	19.2705	0.859230	0.429615	−	0.903012i	$-0.358649\pi$
$503$	19.2705	0.859230	0.429615	+	0.903012i	$0.358649\pi$
$504$	0	0
$505$	−4.61803	−0.205500
$506$	−18.9443	−0.842176
$507$	0	0
$508$	−1.20163	−0.0533135
$509$	−11.4377	−0.506967	−0.253483	−	0.967340i	$-0.581576\pi$
$509$	−11.4377	−0.506967	−0.253483	+	0.967340i	$0.581576\pi$
$510$	0	0
$511$	−0.652476	−0.0288638
$512$	−5.29180	−0.233867
$513$	0	0
$514$	1.32624	0.0584978
$515$	−37.6180	−1.65765
$516$	0	0
$517$	25.3262	1.11385
$518$	−31.5066	−1.38432
$519$	0	0
$520$	−27.0344	−1.18554
$521$	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	$0.664652\pi$
$524$	0.819660	0.0358070
$525$	0	0
$526$	5.32624	0.232235
$527$	7.21478	0.314281
$528$	0	0
$529$	−12.5279	−0.544690
$530$	9.23607	0.401189
$531$	0	0
$532$	2.56231	0.111090
$533$	12.0902	0.523683
$534$	0	0
$535$	−32.0689	−1.38646
$536$	23.4164	1.01143
$537$	0	0
$538$	9.70820	0.418550
$539$	−7.23607	−0.311680
$540$	0	0
$541$	34.5967	1.48743	0.743715	−	0.668497i	$-0.233062\pi$
$541$	34.5967	1.48743	0.743715	+	0.668497i	$0.233062\pi$
$542$	−19.7082	−0.846540
$543$	0	0
$544$	−22.3820	−0.959619
$545$	−47.4296	−2.03166
$546$	0	0
$547$	9.61803	0.411237	0.205619	−	0.978632i	$-0.434079\pi$
$547$	9.61803	0.411237	0.205619	+	0.978632i	$0.434079\pi$
$548$	−8.56231	−0.365764
$549$	0	0
$550$	18.4164	0.785278
$551$	0	0
$552$	0	0
$553$	11.3820	0.484010
$554$	−38.2148	−1.62359
$555$	0	0
$556$	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	0
$559$	−11.7082	−0.495204
$560$	−30.9787	−1.30909
$561$	0	0
$562$	27.7082	1.16880
$563$	45.3951	1.91318	0.956588	−	0.291443i	$-0.0941354\pi$
$563$	45.3951	1.91318	0.956588	+	0.291443i	$0.0941354\pi$
$564$	0	0
$565$	28.3820	1.19404
$566$	−1.23607	−0.0519558
$567$	0	0
$568$	3.41641	0.143349
$569$	−15.9443	−0.668419	−0.334209	−	0.942499i	$-0.608469\pi$
$569$	−15.9443	−0.668419	−0.334209	+	0.942499i	$0.608469\pi$
$570$	0	0
$571$	3.50658	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$571$	3.50658	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$572$	9.47214	0.396050
$573$	0	0
$574$	10.3262	0.431009
$575$	−10.1803	−0.424550
$576$	0	0
$577$	−7.23607	−0.301241	−0.150621	−	0.988592i	$-0.548127\pi$
$577$	−7.23607	−0.301241	−0.150621	+	0.988592i	$0.548127\pi$
$578$	43.3607	1.80357
$579$	0	0
$580$	0	0
$581$	−17.7639	−0.736972
$582$	0	0
$583$	7.23607	0.299687
$584$	0.652476	0.0269996
$585$	0	0
$586$	−28.2705	−1.16784
$587$	−44.3820	−1.83184	−0.915920	−	0.401361i	$-0.868537\pi$
$587$	−44.3820	−1.83184	−0.915920	+	0.401361i	$0.868537\pi$
$588$	0	0
$589$	2.02129	0.0832856
$590$	23.5066	0.967750
$591$	0	0
$592$	42.2705	1.73731
$593$	34.5623	1.41930	0.709652	−	0.704552i	$-0.248852\pi$
$593$	34.5623	1.41930	0.709652	+	0.704552i	$0.248852\pi$
$594$	0	0
$595$	42.2361	1.73151
$596$	−4.56231	−0.186879
$597$	0	0
$598$	−22.1803	−0.907022
$599$	45.0689	1.84146	0.920732	−	0.390195i	$-0.127592\pi$
$599$	45.0689	1.84146	0.920732	+	0.390195i	$0.127592\pi$
$600$	0	0
$601$	−40.1591	−1.63812	−0.819061	−	0.573706i	$-0.805505\pi$
$601$	−40.1591	−1.63812	−0.819061	+	0.573706i	$0.805505\pi$
$602$	−10.0000	−0.407570
$603$	0	0
$604$	11.3262	0.460858
$605$	5.96556	0.242534
$606$	0	0
$607$	−35.9787	−1.46033	−0.730165	−	0.683270i	$-0.760557\pi$
$607$	−35.9787	−1.46033	−0.730165	+	0.683270i	$0.760557\pi$
$608$	−6.27051	−0.254303
$609$	0	0
$610$	7.47214	0.302538
$611$	29.6525	1.19961
$612$	0	0
$613$	−39.5410	−1.59705	−0.798523	−	0.601964i	$-0.794385\pi$
$613$	−39.5410	−1.59705	−0.798523	+	0.601964i	$0.794385\pi$
$614$	5.14590	0.207672
$615$	0	0
$616$	−18.0902	−0.728874
$617$	−8.18034	−0.329328	−0.164664	−	0.986350i	$-0.552654\pi$
$617$	−8.18034	−0.329328	−0.164664	+	0.986350i	$0.552654\pi$
$618$	0	0
$619$	24.9443	1.00259	0.501297	−	0.865275i	$-0.332856\pi$
$619$	24.9443	1.00259	0.501297	+	0.865275i	$0.332856\pi$
$620$	1.92299	0.0772290
$621$	0	0
$622$	14.7082	0.589745
$623$	19.4721	0.780135
$624$	0	0
$625$	−30.8328	−1.23331
$626$	−38.9787	−1.55790
$627$	0	0
$628$	−3.43769	−0.137179
$629$	−57.6312	−2.29791
$630$	0	0
$631$	23.2148	0.924166	0.462083	−	0.886837i	$-0.347102\pi$
$631$	23.2148	0.924166	0.462083	+	0.886837i	$0.347102\pi$
$632$	−11.3820	−0.452750
$633$	0	0
$634$	−23.1246	−0.918396
$635$	−5.54915	−0.220211
$636$	0	0
$637$	−8.47214	−0.335678
$638$	0	0
$639$	0	0
$640$	38.8673	1.53636
$641$	−28.9443	−1.14323	−0.571615	−	0.820522i	$-0.693683\pi$
$641$	−28.9443	−1.14323	−0.571615	+	0.820522i	$0.693683\pi$
$642$	0	0
$643$	−10.5836	−0.417376	−0.208688	−	0.977982i	$-0.566919\pi$
$643$	−10.5836	−0.417376	−0.208688	+	0.977982i	$0.566919\pi$
$644$	−4.47214	−0.176227
$645$	0	0
$646$	19.8541	0.781149
$647$	39.4721	1.55181	0.775905	−	0.630850i	$-0.217294\pi$
$647$	39.4721	1.55181	0.775905	+	0.630850i	$0.217294\pi$
$648$	0	0
$649$	18.4164	0.722907
$650$	21.5623	0.845743
$651$	0	0
$652$	14.2361	0.557527
$653$	28.0132	1.09624	0.548120	−	0.836400i	$-0.315344\pi$
$653$	28.0132	1.09624	0.548120	+	0.836400i	$0.315344\pi$
$654$	0	0
$655$	3.78522	0.147901
$656$	−13.8541	−0.540912
$657$	0	0
$658$	25.3262	0.987320
$659$	24.9443	0.971691	0.485845	−	0.874045i	$-0.338512\pi$
$659$	24.9443	0.971691	0.485845	+	0.874045i	$0.338512\pi$
$660$	0	0
$661$	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
$663$	0	0
$664$	17.7639	0.689374
$665$	11.8328	0.458857
$666$	0	0
$667$	0	0
$668$	−12.0344	−0.465627
$669$	0	0
$670$	−48.3607	−1.86834
$671$	5.85410	0.225995
$672$	0	0
$673$	2.47214	0.0952938	0.0476469	−	0.998864i	$-0.484828\pi$
$673$	2.47214	0.0952938	0.0476469	+	0.998864i	$0.484828\pi$
$674$	38.9443	1.50008
$675$	0	0
$676$	3.05573	0.117528
$677$	−12.8328	−0.493205	−0.246603	−	0.969117i	$-0.579314\pi$
$677$	−12.8328	−0.493205	−0.246603	+	0.969117i	$0.579314\pi$
$678$	0	0
$679$	−37.0344	−1.42125
$680$	−42.2361	−1.61968
$681$	0	0
$682$	6.38197	0.244378
$683$	−14.1459	−0.541278	−0.270639	−	0.962681i	$-0.587235\pi$
$683$	−14.1459	−0.541278	−0.270639	+	0.962681i	$0.587235\pi$
$684$	0	0
$685$	−39.5410	−1.51078
$686$	−32.5623	−1.24323
$687$	0	0
$688$	13.4164	0.511496
$689$	8.47214	0.322763
$690$	0	0
$691$	−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
$693$	0	0
$694$	−13.1459	−0.499011
$695$	41.9787	1.59234
$696$	0	0
$697$	18.8885	0.715455
$698$	21.7984	0.825081
$699$	0	0
$700$	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	$0.236970\pi$
$702$	0	0
$703$	−16.1459	−0.608954
$704$	15.3262	0.577629
$705$	0	0
$706$	−34.1803	−1.28639
$707$	−3.61803	−0.136070
$708$	0	0
$709$	−3.49342	−0.131198	−0.0655991	−	0.997846i	$-0.520896\pi$
$709$	−3.49342	−0.131198	−0.0655991	+	0.997846i	$0.520896\pi$
$710$	−7.05573	−0.264797
$711$	0	0
$712$	−19.4721	−0.729749
$713$	−3.52786	−0.132120
$714$	0	0
$715$	43.7426	1.63588
$716$	9.88854	0.369552
$717$	0	0
$718$	45.6869	1.70502
$719$	29.5066	1.10041	0.550205	−	0.835030i	$-0.314550\pi$
$719$	29.5066	1.10041	0.550205	+	0.835030i	$0.314550\pi$
$720$	0	0
$721$	−29.4721	−1.09760
$722$	−25.1803	−0.937115
$723$	0	0
$724$	−7.38197	−0.274349
$725$	0	0
$726$	0	0
$727$	45.9443	1.70398	0.851989	−	0.523559i	$-0.175396\pi$
$727$	45.9443	1.70398	0.851989	+	0.523559i	$0.175396\pi$
$728$	−21.1803	−0.784996
$729$	0	0
$730$	−1.34752	−0.0498741
$731$	−18.2918	−0.676547
$732$	0	0
$733$	37.1803	1.37329	0.686644	−	0.726994i	$-0.259083\pi$
$733$	37.1803	1.37329	0.686644	+	0.726994i	$0.259083\pi$
$734$	−10.1459	−0.374492
$735$	0	0
$736$	10.9443	0.403411
$737$	−37.8885	−1.39564
$738$	0	0
$739$	8.06888	0.296819	0.148409	−	0.988926i	$-0.452585\pi$
$739$	8.06888	0.296819	0.148409	+	0.988926i	$0.452585\pi$
$740$	−15.3607	−0.564670
$741$	0	0
$742$	7.23607	0.265644
$743$	−30.7639	−1.12862	−0.564310	−	0.825563i	$-0.690858\pi$
$743$	−30.7639	−1.12862	−0.564310	+	0.825563i	$0.690858\pi$
$744$	0	0
$745$	−21.0689	−0.771904
$746$	29.7426	1.08896
$747$	0	0
$748$	14.7984	0.541082
$749$	−25.1246	−0.918033
$750$	0	0
$751$	−27.4721	−1.00247	−0.501236	−	0.865310i	$-0.667121\pi$
$751$	−27.4721	−1.00247	−0.501236	+	0.865310i	$0.667121\pi$
$752$	−33.9787	−1.23908
$753$	0	0
$754$	0	0
$755$	52.3050	1.90357
$756$	0	0
$757$	−46.9787	−1.70747	−0.853735	−	0.520707i	$-0.825668\pi$
$757$	−46.9787	−1.70747	−0.853735	+	0.520707i	$0.825668\pi$
$758$	−61.0132	−2.21610
$759$	0	0
$760$	−11.8328	−0.429221
$761$	−49.7984	−1.80519	−0.902595	−	0.430491i	$-0.858340\pi$
$761$	−49.7984	−1.80519	−0.902595	+	0.430491i	$0.858340\pi$
$762$	0	0
$763$	−37.1591	−1.34525
$764$	7.43769	0.269086
$765$	0	0
$766$	35.8328	1.29469
$767$	21.5623	0.778570
$768$	0	0
$769$	25.3607	0.914530	0.457265	−	0.889331i	$-0.348829\pi$
$769$	25.3607	0.914530	0.457265	+	0.889331i	$0.348829\pi$
$770$	37.3607	1.34639
$771$	0	0
$772$	2.18034	0.0784721
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	3.42956	0.123194
$776$	37.0344	1.32946
$777$	0	0
$778$	−34.1803	−1.22542
$779$	5.29180	0.189598
$780$	0	0
$781$	−5.52786	−0.197803
$782$	−34.6525	−1.23917
$783$	0	0
$784$	9.70820	0.346722
$785$	−15.8754	−0.566617
$786$	0	0
$787$	−25.3607	−0.904011	−0.452005	−	0.892015i	$-0.649291\pi$
$787$	−25.3607	−0.904011	−0.452005	+	0.892015i	$0.649291\pi$
$788$	12.1803	0.433907
$789$	0	0
$790$	23.5066	0.836327
$791$	22.2361	0.790624
$792$	0	0
$793$	6.85410	0.243396
$794$	−51.6869	−1.83430
$795$	0	0
$796$	0.527864	0.0187096
$797$	−20.8197	−0.737470	−0.368735	−	0.929535i	$-0.620209\pi$
$797$	−20.8197	−0.737470	−0.368735	+	0.929535i	$0.620209\pi$
$798$	0	0
$799$	46.3262	1.63890
$800$	−10.6393	−0.376157
$801$	0	0
$802$	53.5066	1.88938
$803$	−1.05573	−0.0372558
$804$	0	0
$805$	−20.6525	−0.727904
$806$	7.47214	0.263195
$807$	0	0
$808$	3.61803	0.127282
$809$	−15.0689	−0.529794	−0.264897	−	0.964277i	$-0.585338\pi$
$809$	−15.0689	−0.529794	−0.264897	+	0.964277i	$0.585338\pi$
$810$	0	0
$811$	−25.6525	−0.900780	−0.450390	−	0.892832i	$-0.648715\pi$
$811$	−25.6525	−0.900780	−0.450390	+	0.892832i	$0.648715\pi$
$812$	0	0
$813$	0	0
$814$	−50.9787	−1.78680
$815$	65.7426	2.30286
$816$	0	0
$817$	−5.12461	−0.179287
$818$	0.944272	0.0330157
$819$	0	0
$820$	5.03444	0.175810
$821$	−35.4164	−1.23604	−0.618021	−	0.786162i	$-0.712065\pi$
$821$	−35.4164	−1.23604	−0.618021	+	0.786162i	$0.712065\pi$
$822$	0	0
$823$	3.47214	0.121031	0.0605155	−	0.998167i	$-0.480726\pi$
$823$	3.47214	0.121031	0.0605155	+	0.998167i	$0.480726\pi$
$824$	29.4721	1.02671
$825$	0	0
$826$	18.4164	0.640789
$827$	−56.0344	−1.94851	−0.974254	−	0.225452i	$-0.927614\pi$
$827$	−56.0344	−1.94851	−0.974254	+	0.225452i	$0.927614\pi$
$828$	0	0
$829$	8.20163	0.284854	0.142427	−	0.989805i	$-0.454509\pi$
$829$	8.20163	0.284854	0.142427	+	0.989805i	$0.454509\pi$
$830$	−36.6869	−1.27342
$831$	0	0
$832$	17.9443	0.622106
$833$	−13.2361	−0.458603
$834$	0	0
$835$	−55.5755	−1.92327
$836$	4.14590	0.143389
$837$	0	0
$838$	−4.14590	−0.143218
$839$	−3.21478	−0.110987	−0.0554933	−	0.998459i	$-0.517673\pi$
$839$	−3.21478	−0.110987	−0.0554933	+	0.998459i	$0.517673\pi$
$840$	0	0
$841$	0	0
$842$	−3.18034	−0.109602
$843$	0	0
$844$	12.1459	0.418079
$845$	14.1115	0.485449
$846$	0	0
$847$	4.67376	0.160592
$848$	−9.70820	−0.333381
$849$	0	0
$850$	33.6869	1.15545
$851$	28.1803	0.966010
$852$	0	0
$853$	−45.0000	−1.54077	−0.770385	−	0.637579i	$-0.779936\pi$
$853$	−45.0000	−1.54077	−0.770385	+	0.637579i	$0.779936\pi$
$854$	5.85410	0.200323
$855$	0	0
$856$	25.1246	0.858742
$857$	29.6738	1.01364	0.506818	−	0.862053i	$-0.330822\pi$
$857$	29.6738	1.01364	0.506818	+	0.862053i	$0.330822\pi$
$858$	0	0
$859$	19.2918	0.658228	0.329114	−	0.944290i	$-0.393250\pi$
$859$	19.2918	0.658228	0.329114	+	0.944290i	$0.393250\pi$
$860$	−4.87539	−0.166249
$861$	0	0
$862$	55.9787	1.90664
$863$	24.7426	0.842249	0.421125	−	0.907003i	$-0.361635\pi$
$863$	24.7426	0.842249	0.421125	+	0.907003i	$0.361635\pi$
$864$	0	0
$865$	20.2361	0.688047
$866$	20.4164	0.693778
$867$	0	0
$868$	1.50658	0.0511366
$869$	18.4164	0.624734
$870$	0	0
$871$	−44.3607	−1.50310
$872$	37.1591	1.25836
$873$	0	0
$874$	−9.70820	−0.328385
$875$	−11.8328	−0.400022
$876$	0	0
$877$	6.61803	0.223475	0.111738	−	0.993738i	$-0.464358\pi$
$877$	6.61803	0.223475	0.111738	+	0.993738i	$0.464358\pi$
$878$	−4.94427	−0.166861
$879$	0	0
$880$	−50.1246	−1.68970
$881$	29.2705	0.986149	0.493074	−	0.869987i	$-0.335873\pi$
$881$	29.2705	0.986149	0.493074	+	0.869987i	$0.335873\pi$
$882$	0	0
$883$	39.6869	1.33557	0.667786	−	0.744354i	$-0.267242\pi$
$883$	39.6869	1.33557	0.667786	+	0.744354i	$0.267242\pi$
$884$	17.3262	0.582744
$885$	0	0
$886$	21.1803	0.711567
$887$	−20.0689	−0.673847	−0.336924	−	0.941532i	$-0.609386\pi$
$887$	−20.0689	−0.673847	−0.336924	+	0.941532i	$0.609386\pi$
$888$	0	0
$889$	−4.34752	−0.145811
$890$	40.2148	1.34800
$891$	0	0
$892$	11.3262	0.379230
$893$	12.9787	0.434316
$894$	0	0
$895$	45.6656	1.52643
$896$	30.4508	1.01729
$897$	0	0
$898$	−22.8541	−0.762651
$899$	0	0
$900$	0	0
$901$	13.2361	0.440957
$902$	16.7082	0.556322
$903$	0	0
$904$	−22.2361	−0.739561
$905$	−34.0902	−1.13320
$906$	0	0
$907$	14.2361	0.472701	0.236350	−	0.971668i	$-0.424049\pi$
$907$	14.2361	0.472701	0.236350	+	0.971668i	$0.424049\pi$
$908$	−9.20163	−0.305367
$909$	0	0
$910$	43.7426	1.45005
$911$	−6.94427	−0.230074	−0.115037	−	0.993361i	$-0.536699\pi$
$911$	−6.94427	−0.230074	−0.115037	+	0.993361i	$0.536699\pi$
$912$	0	0
$913$	−28.7426	−0.951243
$914$	8.56231	0.283216
$915$	0	0
$916$	9.70820	0.320768
$917$	2.96556	0.0979314
$918$	0	0
$919$	−31.3050	−1.03266	−0.516328	−	0.856391i	$-0.672701\pi$
$919$	−31.3050	−1.03266	−0.516328	+	0.856391i	$0.672701\pi$
$920$	20.6525	0.680892
$921$	0	0
$922$	−12.9098	−0.425163
$923$	−6.47214	−0.213033
$924$	0	0
$925$	−27.3951	−0.900746
$926$	−4.38197	−0.144000
$927$	0	0
$928$	0	0
$929$	−27.6525	−0.907248	−0.453624	−	0.891193i	$-0.649869\pi$
$929$	−27.6525	−0.907248	−0.453624	+	0.891193i	$0.649869\pi$
$930$	0	0
$931$	−3.70820	−0.121531
$932$	−6.65248	−0.217909
$933$	0	0
$934$	0.0901699	0.00295045
$935$	68.3394	2.23494
$936$	0	0
$937$	−24.6525	−0.805361	−0.402681	−	0.915341i	$-0.631921\pi$
$937$	−24.6525	−0.805361	−0.402681	+	0.915341i	$0.631921\pi$
$938$	−37.8885	−1.23710
$939$	0	0
$940$	12.3475	0.402732
$941$	−0.888544	−0.0289657	−0.0144829	−	0.999895i	$-0.504610\pi$
$941$	−0.888544	−0.0289657	−0.0144829	+	0.999895i	$0.504610\pi$
$942$	0	0
$943$	−9.23607	−0.300768
$944$	−24.7082	−0.804184
$945$	0	0
$946$	−16.1803	−0.526068
$947$	−13.9656	−0.453820	−0.226910	−	0.973916i	$-0.572862\pi$
$947$	−13.9656	−0.453820	−0.226910	+	0.973916i	$0.572862\pi$
$948$	0	0
$949$	−1.23607	−0.0401245
$950$	9.43769	0.306199
$951$	0	0
$952$	−33.0902	−1.07246
$953$	35.6312	1.15421	0.577104	−	0.816671i	$-0.304183\pi$
$953$	35.6312	1.15421	0.577104	+	0.816671i	$0.304183\pi$
$954$	0	0
$955$	34.3475	1.11146
$956$	−9.11146	−0.294686
$957$	0	0
$958$	−18.0902	−0.584467
$959$	−30.9787	−1.00035
$960$	0	0
$961$	−29.8115	−0.961662
$962$	−59.6869	−1.92438
$963$	0	0
$964$	16.4721	0.530532
$965$	10.0689	0.324129
$966$	0	0
$967$	−16.5623	−0.532608	−0.266304	−	0.963889i	$-0.585802\pi$
$967$	−16.5623	−0.532608	−0.266304	+	0.963889i	$0.585802\pi$
$968$	−4.67376	−0.150220
$969$	0	0
$970$	−76.4853	−2.45579
$971$	−18.5066	−0.593904	−0.296952	−	0.954892i	$-0.595970\pi$
$971$	−18.5066	−0.593904	−0.296952	+	0.954892i	$0.595970\pi$
$972$	0	0
$973$	32.8885	1.05436
$974$	36.3050	1.16329
$975$	0	0
$976$	−7.85410	−0.251404
$977$	−6.21478	−0.198828	−0.0994142	−	0.995046i	$-0.531697\pi$
$977$	−6.21478	−0.198828	−0.0994142	+	0.995046i	$0.531697\pi$
$978$	0	0
$979$	31.5066	1.00695
$980$	−3.52786	−0.112693
$981$	0	0
$982$	−40.6525	−1.29727
$983$	−6.94427	−0.221488	−0.110744	−	0.993849i	$-0.535323\pi$
$983$	−6.94427	−0.221488	−0.110744	+	0.993849i	$0.535323\pi$
$984$	0	0
$985$	56.2492	1.79225
$986$	0	0
$987$	0	0
$988$	4.85410	0.154430
$989$	8.94427	0.284411
$990$	0	0
$991$	38.6525	1.22784	0.613918	−	0.789370i	$-0.289592\pi$
$991$	38.6525	1.22784	0.613918	+	0.789370i	$0.289592\pi$
$992$	−3.68692	−0.117060
$993$	0	0
$994$	−5.52786	−0.175333
$995$	2.43769	0.0772801
$996$	0	0
$997$	28.0902	0.889625	0.444812	−	0.895624i	$-0.353270\pi$
$997$	28.0902	0.889625	0.444812	+	0.895624i	$0.353270\pi$
$998$	−57.7426	−1.82781
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.l.1.2		2
3.2	odd	2		841.2.a.a.1.1	✓	2
29.28	even	2		7569.2.a.d.1.1		2
87.2	even	28		841.2.e.j.236.1		24
87.5	odd	14		841.2.d.g.605.1		12
87.8	even	28		841.2.e.j.267.1		24
87.11	even	28		841.2.e.j.63.1		24
87.14	even	28		841.2.e.j.196.4		24
87.17	even	4		841.2.b.b.840.4		4
87.20	odd	14		841.2.d.i.574.1		12
87.23	odd	14		841.2.d.i.645.2		12
87.26	even	28		841.2.e.j.270.4		24
87.32	even	28		841.2.e.j.270.1		24
87.35	odd	14		841.2.d.g.645.1		12
87.38	odd	14		841.2.d.g.574.2		12
87.41	even	4		841.2.b.b.840.1		4
87.44	even	28		841.2.e.j.196.1		24
87.47	even	28		841.2.e.j.63.4		24
87.50	even	28		841.2.e.j.267.4		24
87.53	odd	14		841.2.d.i.605.2		12
87.56	even	28		841.2.e.j.236.4		24
87.62	odd	14		841.2.d.g.190.1		12
87.65	odd	14		841.2.d.i.571.2		12
87.68	even	28		841.2.e.j.651.1		24
87.71	odd	14		841.2.d.g.778.2		12
87.74	odd	14		841.2.d.i.778.1		12
87.77	even	28		841.2.e.j.651.4		24
87.80	odd	14		841.2.d.g.571.1		12
87.83	odd	14		841.2.d.i.190.2		12
87.86	odd	2		841.2.a.c.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
841.2.a.a.1.1	✓	2	3.2	odd	2
841.2.a.c.1.2	yes	2	87.86	odd	2
841.2.b.b.840.1		4	87.41	even	4
841.2.b.b.840.4		4	87.17	even	4
841.2.d.g.190.1		12	87.62	odd	14
841.2.d.g.571.1		12	87.80	odd	14
841.2.d.g.574.2		12	87.38	odd	14
841.2.d.g.605.1		12	87.5	odd	14
841.2.d.g.645.1		12	87.35	odd	14
841.2.d.g.778.2		12	87.71	odd	14
841.2.d.i.190.2		12	87.83	odd	14
841.2.d.i.571.2		12	87.65	odd	14
841.2.d.i.574.1		12	87.20	odd	14
841.2.d.i.605.2		12	87.53	odd	14
841.2.d.i.645.2		12	87.23	odd	14
841.2.d.i.778.1		12	87.74	odd	14
841.2.e.j.63.1		24	87.11	even	28
841.2.e.j.63.4		24	87.47	even	28
841.2.e.j.196.1		24	87.44	even	28
841.2.e.j.196.4		24	87.14	even	28
841.2.e.j.236.1		24	87.2	even	28
841.2.e.j.236.4		24	87.56	even	28
841.2.e.j.267.1		24	87.8	even	28
841.2.e.j.267.4		24	87.50	even	28
841.2.e.j.270.1		24	87.32	even	28
841.2.e.j.270.4		24	87.26	even	28
841.2.e.j.651.1		24	87.68	even	28
841.2.e.j.651.4		24	87.77	even	28
7569.2.a.d.1.1		2	29.28	even	2
7569.2.a.l.1.2		2	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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} +2.85410 q^{5} +2.23607 q^{7} -2.23607 q^{8} +4.61803 q^{10} +3.61803 q^{11} +4.23607 q^{13} +3.61803 q^{14} -4.85410 q^{16} +6.61803 q^{17} +1.85410 q^{19} +1.76393 q^{20} +5.85410 q^{22} -3.23607 q^{23} +3.14590 q^{25} +6.85410 q^{26} +1.38197 q^{28} +1.09017 q^{31} -3.38197 q^{32} +10.7082 q^{34} +6.38197 q^{35} -8.70820 q^{37} +3.00000 q^{38} -6.38197 q^{40} +2.85410 q^{41} -2.76393 q^{43} +2.23607 q^{44} -5.23607 q^{46} +7.00000 q^{47} -2.00000 q^{49} +5.09017 q^{50} +2.61803 q^{52} +2.00000 q^{53} +10.3262 q^{55} -5.00000 q^{56} +5.09017 q^{59} +1.61803 q^{61} +1.76393 q^{62} +4.23607 q^{64} +12.0902 q^{65} -10.4721 q^{67} +4.09017 q^{68} +10.3262 q^{70} -1.52786 q^{71} -0.291796 q^{73} -14.0902 q^{74} +1.14590 q^{76} +8.09017 q^{77} +5.09017 q^{79} -13.8541 q^{80} +4.61803 q^{82} -7.94427 q^{83} +18.8885 q^{85} -4.47214 q^{86} -8.09017 q^{88} +8.70820 q^{89} +9.47214 q^{91} -2.00000 q^{92} +11.3262 q^{94} +5.29180 q^{95} -16.5623 q^{97} -3.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - q^{5} + 7 q^{10} + 5 q^{11} + 4 q^{13} + 5 q^{14} - 3 q^{16} + 11 q^{17} - 3 q^{19} + 8 q^{20} + 5 q^{22} - 2 q^{23} + 13 q^{25} + 7 q^{26} + 5 q^{28} - 9 q^{31} - 9 q^{32} + 8 q^{34}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - q^5 + 7 * q^10 + 5 * q^11 + 4 * q^13 + 5 * q^14 - 3 * q^16 + 11 * q^17 - 3 * q^19 + 8 * q^20 + 5 * q^22 - 2 * q^23 + 13 * q^25 + 7 * q^26 + 5 * q^28 - 9 * q^31 - 9 * q^32 + 8 * q^34 + 15 * q^35 - 4 * q^37 + 6 * q^38 - 15 * q^40 - q^41 - 10 * q^43 - 6 * q^46 + 14 * q^47 - 4 * q^49 - q^50 + 3 * q^52 + 4 * q^53 + 5 * q^55 - 10 * q^56 - q^59 + q^61 + 8 * q^62 + 4 * q^64 + 13 * q^65 - 12 * q^67 - 3 * q^68 + 5 * q^70 - 12 * q^71 - 14 * q^73 - 17 * q^74 + 9 * q^76 + 5 * q^77 - q^79 - 21 * q^80 + 7 * q^82 + 2 * q^83 + 2 * q^85 - 5 * q^88 + 4 * q^89 + 10 * q^91 - 4 * q^92 + 7 * q^94 + 24 * q^95 - 13 * q^97 - 2 * q^98

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