LMFDB - Embedded newform 675.4.a.x.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,26,0,0,30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.3173728.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 29x^{2} + 118 $ x^4 - 29*x^2 + 118
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.21254$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21254 q^{2} -3.10469 q^{4} +17.1047 q^{7} -24.5695 q^{8} -66.8393 q^{11} +72.7328 q^{13} +37.8447 q^{14} -29.5234 q^{16} -40.2889 q^{17} +38.4766 q^{19} -147.884 q^{22} +204.480 q^{23} +160.924 q^{26} -53.1047 q^{28} +21.6621 q^{29} +128.372 q^{31} +131.234 q^{32} -89.1406 q^{34} +19.4187 q^{37} +85.1308 q^{38} -270.393 q^{41} +242.884 q^{43} +207.515 q^{44} +452.419 q^{46} +307.646 q^{47} -50.4297 q^{49} -225.813 q^{52} +289.019 q^{53} -420.254 q^{56} +47.9282 q^{58} -17.7003 q^{59} +764.851 q^{61} +284.027 q^{62} +526.548 q^{64} -532.176 q^{67} +125.084 q^{68} -409.886 q^{71} +220.581 q^{73} +42.9647 q^{74} -119.458 q^{76} -1143.27 q^{77} +1133.70 q^{79} -598.253 q^{82} +253.619 q^{83} +537.390 q^{86} +1642.21 q^{88} +1625.13 q^{89} +1244.07 q^{91} -634.846 q^{92} +680.678 q^{94} +457.573 q^{97} -111.578 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 26 q^{4} + 30 q^{7} + 22 q^{13} + 74 q^{16} + 346 q^{19} + 100 q^{22} - 174 q^{28} + 744 q^{31} + 796 q^{34} - 76 q^{37} + 280 q^{43} + 1656 q^{46} - 778 q^{49} - 2440 q^{52} + 2420 q^{58} + 178 q^{61}+ \cdots - 2050 q^{97}+O(q^{100}) $ 4 * q + 26 * q^4 + 30 * q^7 + 22 * q^13 + 74 * q^16 + 346 * q^19 + 100 * q^22 - 174 * q^28 + 744 * q^31 + 796 * q^34 - 76 * q^37 + 280 * q^43 + 1656 * q^46 - 778 * q^49 - 2440 * q^52 + 2420 * q^58 + 178 * q^61 + 70 * q^64 + 138 * q^67 + 1036 * q^73 + 4094 * q^76 + 2076 * q^79 - 5236 * q^82 + 6492 * q^88 + 2748 * q^91 - 1196 * q^94 - 2050 * q^97

Coefficient data

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

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

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

$2$	2.21254	0.782249	0.391125	−	0.920338i	$-0.372086\pi$
$2$	2.21254	0.782249	0.391125	+	0.920338i	$0.372086\pi$
$3$	0	0
$3$	0	0
$4$	−3.10469	−0.388086
$5$	0	0
$5$	0	0
$6$	0	0
$7$	17.1047	0.923566	0.461783	−	0.886993i	$-0.347210\pi$
$7$	17.1047	0.923566	0.461783	+	0.886993i	$0.347210\pi$
$8$	−24.5695	−1.08583
$9$	0	0
$10$	0	0
$11$	−66.8393	−1.83207	−0.916037	−	0.401094i	$-0.868630\pi$
$11$	−66.8393	−1.83207	−0.916037	+	0.401094i	$0.868630\pi$
$12$	0	0
$13$	72.7328	1.55173	0.775863	−	0.630901i	$-0.217315\pi$
$13$	72.7328	1.55173	0.775863	+	0.630901i	$0.217315\pi$
$14$	37.8447	0.722459
$15$	0	0
$16$	−29.5234	−0.461304
$17$	−40.2889	−0.574794	−0.287397	−	0.957812i	$-0.592790\pi$
$17$	−40.2889	−0.574794	−0.287397	+	0.957812i	$0.592790\pi$
$18$	0	0
$19$	38.4766	0.464586	0.232293	−	0.972646i	$-0.425377\pi$
$19$	38.4766	0.464586	0.232293	+	0.972646i	$0.425377\pi$
$20$	0	0
$21$	0	0
$22$	−147.884	−1.43314
$23$	204.480	1.85378	0.926891	−	0.375331i	$-0.122471\pi$
$23$	204.480	1.85378	0.926891	+	0.375331i	$0.122471\pi$
$24$	0	0
$25$	0	0
$26$	160.924	1.21384
$27$	0	0
$28$	−53.1047	−0.358423
$29$	21.6621	0.138709	0.0693544	−	0.997592i	$-0.477906\pi$
$29$	21.6621	0.138709	0.0693544	+	0.997592i	$0.477906\pi$
$30$	0	0
$31$	128.372	0.743751	0.371875	−	0.928283i	$-0.378715\pi$
$31$	128.372	0.743751	0.371875	+	0.928283i	$0.378715\pi$
$32$	131.234	0.724975
$33$	0	0
$34$	−89.1406	−0.449632
$35$	0	0
$36$	0	0
$37$	19.4187	0.0862817	0.0431408	−	0.999069i	$-0.486264\pi$
$37$	19.4187	0.0862817	0.0431408	+	0.999069i	$0.486264\pi$
$38$	85.1308	0.363422
$39$	0	0
$40$	0	0
$41$	−270.393	−1.02996	−0.514978	−	0.857203i	$-0.672200\pi$
$41$	−270.393	−1.02996	−0.514978	+	0.857203i	$0.672200\pi$
$42$	0	0
$43$	242.884	0.861384	0.430692	−	0.902499i	$-0.358270\pi$
$43$	242.884	0.861384	0.430692	+	0.902499i	$0.358270\pi$
$44$	207.515	0.711002
$45$	0	0
$46$	452.419	1.45012
$47$	307.646	0.954783	0.477391	−	0.878691i	$-0.341582\pi$
$47$	307.646	0.954783	0.477391	+	0.878691i	$0.341582\pi$
$48$	0	0
$49$	−50.4297	−0.147025
$50$	0	0
$51$	0	0
$52$	−225.813	−0.602203
$53$	289.019	0.749054	0.374527	−	0.927216i	$-0.377805\pi$
$53$	289.019	0.749054	0.374527	+	0.927216i	$0.377805\pi$
$54$	0	0
$55$	0	0
$56$	−420.254	−1.00284
$57$	0	0
$58$	47.9282	0.108505
$59$	−17.7003	−0.0390573	−0.0195287	−	0.999809i	$-0.506217\pi$
$59$	−17.7003	−0.0390573	−0.0195287	+	0.999809i	$0.506217\pi$
$60$	0	0
$61$	764.851	1.60540	0.802698	−	0.596385i	$-0.203397\pi$
$61$	764.851	1.60540	0.802698	+	0.596385i	$0.203397\pi$
$62$	284.027	0.581799
$63$	0	0
$64$	526.548	1.02841
$65$	0	0
$66$	0	0
$67$	−532.176	−0.970384	−0.485192	−	0.874408i	$-0.661250\pi$
$67$	−532.176	−0.970384	−0.485192	+	0.874408i	$0.661250\pi$
$68$	125.084	0.223069
$69$	0	0
$70$	0	0
$71$	−409.886	−0.685134	−0.342567	−	0.939493i	$-0.611296\pi$
$71$	−409.886	−0.685134	−0.342567	+	0.939493i	$0.611296\pi$
$72$	0	0
$73$	220.581	0.353659	0.176829	−	0.984242i	$-0.443416\pi$
$73$	220.581	0.353659	0.176829	+	0.984242i	$0.443416\pi$
$74$	42.9647	0.0674938
$75$	0	0
$76$	−119.458	−0.180299
$77$	−1143.27	−1.69204
$78$	0	0
$79$	1133.70	1.61457	0.807286	−	0.590160i	$-0.200935\pi$
$79$	1133.70	1.61457	0.807286	+	0.590160i	$0.200935\pi$
$80$	0	0
$81$	0	0
$82$	−598.253	−0.805683
$83$	253.619	0.335401	0.167700	−	0.985838i	$-0.446366\pi$
$83$	253.619	0.335401	0.167700	+	0.985838i	$0.446366\pi$
$84$	0	0
$85$	0	0
$86$	537.390	0.673817
$87$	0	0
$88$	1642.21	1.98932
$89$	1625.13	1.93555	0.967775	−	0.251816i	$-0.0810279\pi$
$89$	1625.13	1.93555	0.967775	+	0.251816i	$0.0810279\pi$
$90$	0	0
$91$	1244.07	1.43312
$92$	−634.846	−0.719426
$93$	0	0
$94$	680.678	0.746878
$95$	0	0
$96$	0	0
$97$	457.573	0.478964	0.239482	−	0.970901i	$-0.423022\pi$
$97$	457.573	0.478964	0.239482	+	0.970901i	$0.423022\pi$
$98$	−111.578	−0.115011
$99$	0	0
$100$	0	0
$101$	1262.95	1.24424	0.622120	−	0.782922i	$-0.286272\pi$
$101$	1262.95	1.24424	0.622120	+	0.782922i	$0.286272\pi$
$102$	0	0
$103$	433.419	0.414622	0.207311	−	0.978275i	$-0.433529\pi$
$103$	433.419	0.414622	0.207311	+	0.978275i	$0.433529\pi$
$104$	−1787.01	−1.68491
$105$	0	0
$106$	639.466	0.585947
$107$	1667.95	1.50698	0.753489	−	0.657461i	$-0.228370\pi$
$107$	1667.95	1.50698	0.753489	+	0.657461i	$0.228370\pi$
$108$	0	0
$109$	−1191.98	−1.04744	−0.523721	−	0.851890i	$-0.675456\pi$
$109$	−1191.98	−1.04744	−0.523721	+	0.851890i	$0.675456\pi$
$110$	0	0
$111$	0	0
$112$	−504.989	−0.426044
$113$	−506.311	−0.421502	−0.210751	−	0.977540i	$-0.567591\pi$
$113$	−506.311	−0.421502	−0.210751	+	0.977540i	$0.567591\pi$
$114$	0	0
$115$	0	0
$116$	−67.2541	−0.0538309
$117$	0	0
$118$	−39.1625	−0.0305526
$119$	−689.129	−0.530860
$120$	0	0
$121$	3136.49	2.35649
$122$	1692.26	1.25582
$123$	0	0
$124$	−398.554	−0.288639
$125$	0	0
$126$	0	0
$127$	−2042.84	−1.42735	−0.713673	−	0.700479i	$-0.752970\pi$
$127$	−2042.84	−1.42735	−0.713673	+	0.700479i	$0.752970\pi$
$128$	115.131	0.0795021
$129$	0	0
$130$	0	0
$131$	−1854.98	−1.23718	−0.618590	−	0.785714i	$-0.712296\pi$
$131$	−1854.98	−1.23718	−0.618590	+	0.785714i	$0.712296\pi$
$132$	0	0
$133$	658.130	0.429076
$134$	−1177.46	−0.759082
$135$	0	0
$136$	989.878	0.624128
$137$	−1698.04	−1.05893	−0.529466	−	0.848331i	$-0.677608\pi$
$137$	−1698.04	−1.05893	−0.529466	+	0.848331i	$0.677608\pi$
$138$	0	0
$139$	2386.02	1.45597	0.727986	−	0.685592i	$-0.240457\pi$
$139$	2386.02	1.45597	0.727986	+	0.685592i	$0.240457\pi$
$140$	0	0
$141$	0	0
$142$	−906.887	−0.535946
$143$	−4861.41	−2.84288
$144$	0	0
$145$	0	0
$146$	488.044	0.276649
$147$	0	0
$148$	−60.2891	−0.0334847
$149$	−1090.68	−0.599676	−0.299838	−	0.953990i	$-0.596933\pi$
$149$	−1090.68	−0.599676	−0.299838	+	0.953990i	$0.596933\pi$
$150$	0	0
$151$	−2779.07	−1.49773	−0.748865	−	0.662722i	$-0.769401\pi$
$151$	−2779.07	−1.49773	−0.748865	+	0.662722i	$0.769401\pi$
$152$	−945.351	−0.504461
$153$	0	0
$154$	−2529.52	−1.32360
$155$	0	0
$156$	0	0
$157$	−1277.14	−0.649218	−0.324609	−	0.945848i	$-0.605233\pi$
$157$	−1277.14	−0.649218	−0.324609	+	0.945848i	$0.605233\pi$
$158$	2508.35	1.26300
$159$	0	0
$160$	0	0
$161$	3497.56	1.71209
$162$	0	0
$163$	−1328.05	−0.638167	−0.319084	−	0.947727i	$-0.603375\pi$
$163$	−1328.05	−0.638167	−0.319084	+	0.947727i	$0.603375\pi$
$164$	839.484	0.399712
$165$	0	0
$166$	561.141	0.262367
$167$	−3995.28	−1.85128	−0.925641	−	0.378404i	$-0.876473\pi$
$167$	−3995.28	−1.85128	−0.925641	+	0.378404i	$0.876473\pi$
$168$	0	0
$169$	3093.06	1.40786
$170$	0	0
$171$	0	0
$172$	−754.080	−0.334291
$173$	−807.216	−0.354748	−0.177374	−	0.984143i	$-0.556760\pi$
$173$	−807.216	−0.354748	−0.177374	+	0.984143i	$0.556760\pi$
$174$	0	0
$175$	0	0
$176$	1973.33	0.845142
$177$	0	0
$178$	3595.67	1.51408
$179$	1708.08	0.713227	0.356614	−	0.934252i	$-0.383931\pi$
$179$	1708.08	0.713227	0.356614	+	0.934252i	$0.383931\pi$
$180$	0	0
$181$	1772.50	0.727895	0.363948	−	0.931419i	$-0.381429\pi$
$181$	1772.50	0.727895	0.363948	+	0.931419i	$0.381429\pi$
$182$	2752.55	1.12106
$183$	0	0
$184$	−5023.97	−2.01289
$185$	0	0
$186$	0	0
$187$	2692.88	1.05306
$188$	−955.145	−0.370538
$189$	0	0
$190$	0	0
$191$	−2644.91	−1.00199	−0.500993	−	0.865452i	$-0.667032\pi$
$191$	−2644.91	−1.00199	−0.500993	+	0.865452i	$0.667032\pi$
$192$	0	0
$193$	−531.799	−0.198340	−0.0991702	−	0.995070i	$-0.531619\pi$
$193$	−531.799	−0.198340	−0.0991702	+	0.995070i	$0.531619\pi$
$194$	1012.40	0.374670
$195$	0	0
$196$	156.568	0.0570585
$197$	−1331.90	−0.481694	−0.240847	−	0.970563i	$-0.577425\pi$
$197$	−1331.90	−0.481694	−0.240847	+	0.970563i	$0.577425\pi$
$198$	0	0
$199$	1775.44	0.632450	0.316225	−	0.948684i	$-0.397585\pi$
$199$	1775.44	0.632450	0.316225	+	0.948684i	$0.397585\pi$
$200$	0	0
$201$	0	0
$202$	2794.32	0.973306
$203$	370.524	0.128107
$204$	0	0
$205$	0	0
$206$	958.954	0.324337
$207$	0	0
$208$	−2147.32	−0.715817
$209$	−2571.75	−0.851155
$210$	0	0
$211$	−3344.62	−1.09125	−0.545623	−	0.838031i	$-0.683707\pi$
$211$	−3344.62	−1.09125	−0.545623	+	0.838031i	$0.683707\pi$
$212$	−897.314	−0.290697
$213$	0	0
$214$	3690.39	1.17883
$215$	0	0
$216$	0	0
$217$	2195.76	0.686903
$218$	−2637.30	−0.819360
$219$	0	0
$220$	0	0
$221$	−2930.32	−0.891923
$222$	0	0
$223$	4203.68	1.26233	0.631164	−	0.775649i	$-0.282578\pi$
$223$	4203.68	1.26233	0.631164	+	0.775649i	$0.282578\pi$
$224$	2244.72	0.669562
$225$	0	0
$226$	−1120.23	−0.329720
$227$	−305.634	−0.0893641	−0.0446821	−	0.999001i	$-0.514227\pi$
$227$	−305.634	−0.0893641	−0.0446821	+	0.999001i	$0.514227\pi$
$228$	0	0
$229$	−123.562	−0.0356560	−0.0178280	−	0.999841i	$-0.505675\pi$
$229$	−123.562	−0.0356560	−0.0178280	+	0.999841i	$0.505675\pi$
$230$	0	0
$231$	0	0
$232$	−532.228	−0.150614
$233$	5246.76	1.47522	0.737611	−	0.675226i	$-0.235954\pi$
$233$	5246.76	1.47522	0.737611	+	0.675226i	$0.235954\pi$
$234$	0	0
$235$	0	0
$236$	54.9538	0.0151576
$237$	0	0
$238$	−1524.72	−0.415265
$239$	−4300.72	−1.16398	−0.581988	−	0.813197i	$-0.697725\pi$
$239$	−4300.72	−1.16398	−0.581988	+	0.813197i	$0.697725\pi$
$240$	0	0
$241$	−5355.47	−1.43144	−0.715718	−	0.698390i	$-0.753900\pi$
$241$	−5355.47	−1.43144	−0.715718	+	0.698390i	$0.753900\pi$
$242$	6939.60	1.84337
$243$	0	0
$244$	−2374.62	−0.623032
$245$	0	0
$246$	0	0
$247$	2798.51	0.720910
$248$	−3154.03	−0.807586
$249$	0	0
$250$	0	0
$251$	−2587.28	−0.650627	−0.325314	−	0.945606i	$-0.605470\pi$
$251$	−2587.28	−0.650627	−0.325314	+	0.945606i	$0.605470\pi$
$252$	0	0
$253$	−13667.3	−3.39626
$254$	−4519.86	−1.11654
$255$	0	0
$256$	−3957.65	−0.966224
$257$	4909.53	1.19163	0.595813	−	0.803123i	$-0.296830\pi$
$257$	4909.53	1.19163	0.595813	+	0.803123i	$0.296830\pi$
$258$	0	0
$259$	332.152	0.0796868
$260$	0	0
$261$	0	0
$262$	−4104.22	−0.967783
$263$	4464.34	1.04670	0.523351	−	0.852117i	$-0.324682\pi$
$263$	4464.34	1.04670	0.523351	+	0.852117i	$0.324682\pi$
$264$	0	0
$265$	0	0
$266$	1456.14	0.335644
$267$	0	0
$268$	1652.24	0.376592
$269$	5957.80	1.35039	0.675193	−	0.737641i	$-0.264060\pi$
$269$	5957.80	1.35039	0.675193	+	0.737641i	$0.264060\pi$
$270$	0	0
$271$	4805.80	1.07724	0.538619	−	0.842549i	$-0.318946\pi$
$271$	4805.80	1.07724	0.538619	+	0.842549i	$0.318946\pi$
$272$	1189.47	0.265154
$273$	0	0
$274$	−3756.98	−0.828349
$275$	0	0
$276$	0	0
$277$	3531.98	0.766122	0.383061	−	0.923723i	$-0.374870\pi$
$277$	3531.98	0.766122	0.383061	+	0.923723i	$0.374870\pi$
$278$	5279.17	1.13893
$279$	0	0
$280$	0	0
$281$	−437.107	−0.0927958	−0.0463979	−	0.998923i	$-0.514774\pi$
$281$	−437.107	−0.0927958	−0.0463979	+	0.998923i	$0.514774\pi$
$282$	0	0
$283$	−6278.33	−1.31876	−0.659378	−	0.751811i	$-0.729180\pi$
$283$	−6278.33	−1.31876	−0.659378	+	0.751811i	$0.729180\pi$
$284$	1272.57	0.265891
$285$	0	0
$286$	−10756.0	−2.22384
$287$	−4624.98	−0.951233
$288$	0	0
$289$	−3289.81	−0.669612
$290$	0	0
$291$	0	0
$292$	−684.836	−0.137250
$293$	1065.21	0.212390	0.106195	−	0.994345i	$-0.466133\pi$
$293$	1065.21	0.212390	0.106195	+	0.994345i	$0.466133\pi$
$294$	0	0
$295$	0	0
$296$	−477.109	−0.0936872
$297$	0	0
$298$	−2413.16	−0.469096
$299$	14872.4	2.87656
$300$	0	0
$301$	4154.46	0.795545
$302$	−6148.78	−1.17160
$303$	0	0
$304$	−1135.96	−0.214315
$305$	0	0
$306$	0	0
$307$	10015.7	1.86197	0.930986	−	0.365055i	$-0.118950\pi$
$307$	10015.7	1.86197	0.930986	+	0.365055i	$0.118950\pi$
$308$	3549.48	0.656657
$309$	0	0
$310$	0	0
$311$	3600.63	0.656505	0.328253	−	0.944590i	$-0.393540\pi$
$311$	3600.63	0.656505	0.328253	+	0.944590i	$0.393540\pi$
$312$	0	0
$313$	1913.29	0.345512	0.172756	−	0.984965i	$-0.444733\pi$
$313$	1913.29	0.345512	0.172756	+	0.984965i	$0.444733\pi$
$314$	−2825.73	−0.507850
$315$	0	0
$316$	−3519.78	−0.626593
$317$	−4472.16	−0.792371	−0.396186	−	0.918170i	$-0.629666\pi$
$317$	−4472.16	−0.792371	−0.396186	+	0.918170i	$0.629666\pi$
$318$	0	0
$319$	−1447.88	−0.254125
$320$	0	0
$321$	0	0
$322$	7738.48	1.33928
$323$	−1550.18	−0.267041
$324$	0	0
$325$	0	0
$326$	−2938.37	−0.499206
$327$	0	0
$328$	6643.41	1.11836
$329$	5262.19	0.881805
$330$	0	0
$331$	−7693.11	−1.27750	−0.638748	−	0.769416i	$-0.720547\pi$
$331$	−7693.11	−1.27750	−0.638748	+	0.769416i	$0.720547\pi$
$332$	−787.407	−0.130164
$333$	0	0
$334$	−8839.70	−1.44816
$335$	0	0
$336$	0	0
$337$	−395.941	−0.0640009	−0.0320005	−	0.999488i	$-0.510188\pi$
$337$	−395.941	−0.0640009	−0.0320005	+	0.999488i	$0.510188\pi$
$338$	6843.51	1.10130
$339$	0	0
$340$	0	0
$341$	−8580.29	−1.36261
$342$	0	0
$343$	−6729.49	−1.05935
$344$	−5967.55	−0.935316
$345$	0	0
$346$	−1785.99	−0.277502
$347$	9074.49	1.40387	0.701937	−	0.712239i	$-0.252319\pi$
$347$	9074.49	1.40387	0.701937	+	0.712239i	$0.252319\pi$
$348$	0	0
$349$	8879.84	1.36197	0.680984	−	0.732298i	$-0.261552\pi$
$349$	8879.84	1.36197	0.680984	+	0.732298i	$0.261552\pi$
$350$	0	0
$351$	0	0
$352$	−8771.62	−1.32821
$353$	−5229.73	−0.788528	−0.394264	−	0.918997i	$-0.629000\pi$
$353$	−5229.73	−0.788528	−0.394264	+	0.918997i	$0.629000\pi$
$354$	0	0
$355$	0	0
$356$	−5045.53	−0.751160
$357$	0	0
$358$	3779.18	0.557922
$359$	−13431.1	−1.97456	−0.987278	−	0.159001i	$-0.949173\pi$
$359$	−13431.1	−1.97456	−0.987278	+	0.159001i	$0.949173\pi$
$360$	0	0
$361$	−5378.55	−0.784160
$362$	3921.72	0.569395
$363$	0	0
$364$	−3862.45	−0.556175
$365$	0	0
$366$	0	0
$367$	7283.77	1.03599	0.517997	−	0.855383i	$-0.326678\pi$
$367$	7283.77	1.03599	0.517997	+	0.855383i	$0.326678\pi$
$368$	−6036.94	−0.855156
$369$	0	0
$370$	0	0
$371$	4943.59	0.691801
$372$	0	0
$373$	−7344.75	−1.01956	−0.509781	−	0.860304i	$-0.670274\pi$
$373$	−7344.75	−1.01956	−0.509781	+	0.860304i	$0.670274\pi$
$374$	5958.10	0.823759
$375$	0	0
$376$	−7558.72	−1.03673
$377$	1575.55	0.215238
$378$	0	0
$379$	−1553.95	−0.210610	−0.105305	−	0.994440i	$-0.533582\pi$
$379$	−1553.95	−0.210610	−0.105305	+	0.994440i	$0.533582\pi$
$380$	0	0
$381$	0	0
$382$	−5851.96	−0.783802
$383$	−2923.68	−0.390060	−0.195030	−	0.980797i	$-0.562480\pi$
$383$	−2923.68	−0.390060	−0.195030	+	0.980797i	$0.562480\pi$
$384$	0	0
$385$	0	0
$386$	−1176.62	−0.155152
$387$	0	0
$388$	−1420.62	−0.185879
$389$	−397.904	−0.0518625	−0.0259312	−	0.999664i	$-0.508255\pi$
$389$	−397.904	−0.0518625	−0.0259312	+	0.999664i	$0.508255\pi$
$390$	0	0
$391$	−8238.26	−1.06554
$392$	1239.03	0.159644
$393$	0	0
$394$	−2946.87	−0.376805
$395$	0	0
$396$	0	0
$397$	−1333.10	−0.168530	−0.0842649	−	0.996443i	$-0.526854\pi$
$397$	−1333.10	−0.168530	−0.0842649	+	0.996443i	$0.526854\pi$
$398$	3928.22	0.494734
$399$	0	0
$400$	0	0
$401$	−6293.28	−0.783719	−0.391860	−	0.920025i	$-0.628168\pi$
$401$	−6293.28	−0.783719	−0.391860	+	0.920025i	$0.628168\pi$
$402$	0	0
$403$	9336.85	1.15410
$404$	−3921.06	−0.482872
$405$	0	0
$406$	819.797	0.100211
$407$	−1297.94	−0.158074
$408$	0	0
$409$	12522.2	1.51390	0.756948	−	0.653475i	$-0.226690\pi$
$409$	12522.2	1.51390	0.756948	+	0.653475i	$0.226690\pi$
$410$	0	0
$411$	0	0
$412$	−1345.63	−0.160909
$413$	−302.758	−0.0360720
$414$	0	0
$415$	0	0
$416$	9545.05	1.12496
$417$	0	0
$418$	−5690.08	−0.665816
$419$	5505.42	0.641903	0.320952	−	0.947096i	$-0.395997\pi$
$419$	5505.42	0.641903	0.320952	+	0.947096i	$0.395997\pi$
$420$	0	0
$421$	5957.76	0.689700	0.344850	−	0.938658i	$-0.387930\pi$
$421$	5957.76	0.689700	0.344850	+	0.938658i	$0.387930\pi$
$422$	−7400.08	−0.853627
$423$	0	0
$424$	−7101.07	−0.813345
$425$	0	0
$426$	0	0
$427$	13082.5	1.48269
$428$	−5178.45	−0.584836
$429$	0	0
$430$	0	0
$431$	12509.3	1.39804	0.699018	−	0.715104i	$-0.253621\pi$
$431$	12509.3	1.39804	0.699018	+	0.715104i	$0.253621\pi$
$432$	0	0
$433$	4345.52	0.482292	0.241146	−	0.970489i	$-0.422477\pi$
$433$	4345.52	0.482292	0.241146	+	0.970489i	$0.422477\pi$
$434$	4858.20	0.537330
$435$	0	0
$436$	3700.73	0.406497
$437$	7867.68	0.861241
$438$	0	0
$439$	7014.09	0.762561	0.381281	−	0.924459i	$-0.375483\pi$
$439$	7014.09	0.762561	0.381281	+	0.924459i	$0.375483\pi$
$440$	0	0
$441$	0	0
$442$	−6483.45	−0.697706
$443$	8727.29	0.935996	0.467998	−	0.883730i	$-0.344975\pi$
$443$	8727.29	0.935996	0.467998	+	0.883730i	$0.344975\pi$
$444$	0	0
$445$	0	0
$446$	9300.79	0.987455
$447$	0	0
$448$	9006.44	0.949809
$449$	−13100.4	−1.37694	−0.688472	−	0.725263i	$-0.741718\pi$
$449$	−13100.4	−1.37694	−0.688472	+	0.725263i	$0.741718\pi$
$450$	0	0
$451$	18072.9	1.88696
$452$	1571.94	0.163579
$453$	0	0
$454$	−676.227	−0.0699050
$455$	0	0
$456$	0	0
$457$	1063.47	0.108855	0.0544277	−	0.998518i	$-0.482667\pi$
$457$	1063.47	0.108855	0.0544277	+	0.998518i	$0.482667\pi$
$458$	−273.386	−0.0278919
$459$	0	0
$460$	0	0
$461$	8355.46	0.844149	0.422074	−	0.906561i	$-0.361302\pi$
$461$	8355.46	0.844149	0.422074	+	0.906561i	$0.361302\pi$
$462$	0	0
$463$	−11866.0	−1.19106	−0.595528	−	0.803335i	$-0.703057\pi$
$463$	−11866.0	−1.19106	−0.595528	+	0.803335i	$0.703057\pi$
$464$	−639.540	−0.0639868
$465$	0	0
$466$	11608.6	1.15399
$467$	−5687.16	−0.563534	−0.281767	−	0.959483i	$-0.590920\pi$
$467$	−5687.16	−0.563534	−0.281767	+	0.959483i	$0.590920\pi$
$468$	0	0
$469$	−9102.71	−0.896214
$470$	0	0
$471$	0	0
$472$	434.887	0.0424096
$473$	−16234.2	−1.57812
$474$	0	0
$475$	0	0
$476$	2139.53	0.206019
$477$	0	0
$478$	−9515.49	−0.910520
$479$	−10556.5	−1.00697	−0.503485	−	0.864004i	$-0.667949\pi$
$479$	−10556.5	−1.00697	−0.503485	+	0.864004i	$0.667949\pi$
$480$	0	0
$481$	1412.38	0.133886
$482$	−11849.2	−1.11974
$483$	0	0
$484$	−9737.83	−0.914522
$485$	0	0
$486$	0	0
$487$	−962.879	−0.0895938	−0.0447969	−	0.998996i	$-0.514264\pi$
$487$	−962.879	−0.0895938	−0.0447969	+	0.998996i	$0.514264\pi$
$488$	−18792.0	−1.74319
$489$	0	0
$490$	0	0
$491$	7787.16	0.715743	0.357871	−	0.933771i	$-0.383503\pi$
$491$	7787.16	0.715743	0.357871	+	0.933771i	$0.383503\pi$
$492$	0	0
$493$	−872.742	−0.0797289
$494$	6191.80	0.563932
$495$	0	0
$496$	−3789.98	−0.343095
$497$	−7010.97	−0.632767
$498$	0	0
$499$	4036.80	0.362148	0.181074	−	0.983469i	$-0.442043\pi$
$499$	4036.80	0.362148	0.181074	+	0.983469i	$0.442043\pi$
$500$	0	0
$501$	0	0
$502$	−5724.44	−0.508953
$503$	13228.9	1.17266	0.586329	−	0.810073i	$-0.300573\pi$
$503$	13228.9	1.17266	0.586329	+	0.810073i	$0.300573\pi$
$504$	0	0
$505$	0	0
$506$	−30239.4	−2.65673
$507$	0	0
$508$	6342.39	0.553933
$509$	−5516.83	−0.480411	−0.240206	−	0.970722i	$-0.577215\pi$
$509$	−5516.83	−0.480411	−0.240206	+	0.970722i	$0.577215\pi$
$510$	0	0
$511$	3772.97	0.326627
$512$	−9677.50	−0.835331
$513$	0	0
$514$	10862.5	0.932149
$515$	0	0
$516$	0	0
$517$	−20562.9	−1.74923
$518$	734.897	0.0623350
$519$	0	0
$520$	0	0
$521$	−12832.4	−1.07908	−0.539538	−	0.841961i	$-0.681401\pi$
$521$	−12832.4	−1.07908	−0.539538	+	0.841961i	$0.681401\pi$
$522$	0	0
$523$	1005.75	0.0840888	0.0420444	−	0.999116i	$-0.486613\pi$
$523$	1005.75	0.0840888	0.0420444	+	0.999116i	$0.486613\pi$
$524$	5759.14	0.480132
$525$	0	0
$526$	9877.50	0.818783
$527$	−5171.96	−0.427503
$528$	0	0
$529$	29645.0	2.43651
$530$	0	0
$531$	0	0
$532$	−2043.29	−0.166518
$533$	−19666.4	−1.59821
$534$	0	0
$535$	0	0
$536$	13075.3	1.05367
$537$	0	0
$538$	13181.9	1.05634
$539$	3370.69	0.269361
$540$	0	0
$541$	15446.1	1.22750	0.613751	−	0.789500i	$-0.289660\pi$
$541$	15446.1	1.22750	0.613751	+	0.789500i	$0.289660\pi$
$542$	10633.0	0.842669
$543$	0	0
$544$	−5287.29	−0.416711
$545$	0	0
$546$	0	0
$547$	−3115.98	−0.243564	−0.121782	−	0.992557i	$-0.538861\pi$
$547$	−3115.98	−0.243564	−0.121782	+	0.992557i	$0.538861\pi$
$548$	5271.90	0.410957
$549$	0	0
$550$	0	0
$551$	833.484	0.0644421
$552$	0	0
$553$	19391.6	1.49116
$554$	7814.62	0.599299
$555$	0	0
$556$	−7407.86	−0.565042
$557$	1843.45	0.140232	0.0701162	−	0.997539i	$-0.477663\pi$
$557$	1843.45	0.140232	0.0701162	+	0.997539i	$0.477663\pi$
$558$	0	0
$559$	17665.7	1.33663
$560$	0	0
$561$	0	0
$562$	−967.115	−0.0725895
$563$	−13957.3	−1.04481	−0.522407	−	0.852696i	$-0.674966\pi$
$563$	−13957.3	−1.04481	−0.522407	+	0.852696i	$0.674966\pi$
$564$	0	0
$565$	0	0
$566$	−13891.0	−1.03160
$567$	0	0
$568$	10070.7	0.743939
$569$	−20049.7	−1.47720	−0.738601	−	0.674143i	$-0.764513\pi$
$569$	−20049.7	−1.47720	−0.738601	+	0.674143i	$0.764513\pi$
$570$	0	0
$571$	−1270.87	−0.0931421	−0.0465711	−	0.998915i	$-0.514829\pi$
$571$	−1270.87	−0.0931421	−0.0465711	+	0.998915i	$0.514829\pi$
$572$	15093.2	1.10328
$573$	0	0
$574$	−10232.9	−0.744102
$575$	0	0
$576$	0	0
$577$	−211.623	−0.0152686	−0.00763431	−	0.999971i	$-0.502430\pi$
$577$	−211.623	−0.0152686	−0.00763431	+	0.999971i	$0.502430\pi$
$578$	−7278.81	−0.523804
$579$	0	0
$580$	0	0
$581$	4338.07	0.309765
$582$	0	0
$583$	−19317.9	−1.37232
$584$	−5419.57	−0.384013
$585$	0	0
$586$	2356.82	0.166142
$587$	11135.1	0.782956	0.391478	−	0.920188i	$-0.371964\pi$
$587$	11135.1	0.782956	0.391478	+	0.920188i	$0.371964\pi$
$588$	0	0
$589$	4939.31	0.345536
$590$	0	0
$591$	0	0
$592$	−573.308	−0.0398021
$593$	20593.7	1.42611	0.713054	−	0.701109i	$-0.247311\pi$
$593$	20593.7	1.42611	0.713054	+	0.701109i	$0.247311\pi$
$594$	0	0
$595$	0	0
$596$	3386.21	0.232726
$597$	0	0
$598$	32905.7	2.25019
$599$	−2510.98	−0.171278	−0.0856392	−	0.996326i	$-0.527293\pi$
$599$	−2510.98	−0.171278	−0.0856392	+	0.996326i	$0.527293\pi$
$600$	0	0
$601$	21276.4	1.44406	0.722031	−	0.691861i	$-0.243209\pi$
$601$	21276.4	1.44406	0.722031	+	0.691861i	$0.243209\pi$
$602$	9191.89	0.622315
$603$	0	0
$604$	8628.13	0.581248
$605$	0	0
$606$	0	0
$607$	−21993.1	−1.47063	−0.735316	−	0.677725i	$-0.762966\pi$
$607$	−21993.1	−1.47063	−0.735316	+	0.677725i	$0.762966\pi$
$608$	5049.45	0.336813
$609$	0	0
$610$	0	0
$611$	22376.0	1.48156
$612$	0	0
$613$	13319.8	0.877622	0.438811	−	0.898579i	$-0.355400\pi$
$613$	13319.8	0.877622	0.438811	+	0.898579i	$0.355400\pi$
$614$	22160.1	1.45653
$615$	0	0
$616$	28089.5	1.83727
$617$	7815.28	0.509938	0.254969	−	0.966949i	$-0.417935\pi$
$617$	7815.28	0.509938	0.254969	+	0.966949i	$0.417935\pi$
$618$	0	0
$619$	−18163.9	−1.17943	−0.589716	−	0.807611i	$-0.700760\pi$
$619$	−18163.9	−1.17943	−0.589716	+	0.807611i	$0.700760\pi$
$620$	0	0
$621$	0	0
$622$	7966.53	0.513551
$623$	27797.4	1.78761
$624$	0	0
$625$	0	0
$626$	4233.21	0.270277
$627$	0	0
$628$	3965.13	0.251952
$629$	−782.360	−0.0495941
$630$	0	0
$631$	2884.98	0.182011	0.0910057	−	0.995850i	$-0.470992\pi$
$631$	2884.98	0.182011	0.0910057	+	0.995850i	$0.470992\pi$
$632$	−27854.5	−1.75315
$633$	0	0
$634$	−9894.82	−0.619832
$635$	0	0
$636$	0	0
$637$	−3667.89	−0.228143
$638$	−3203.49	−0.198789
$639$	0	0
$640$	0	0
$641$	26713.6	1.64606	0.823029	−	0.567999i	$-0.192282\pi$
$641$	26713.6	1.64606	0.823029	+	0.567999i	$0.192282\pi$
$642$	0	0
$643$	−11316.8	−0.694079	−0.347039	−	0.937851i	$-0.612813\pi$
$643$	−11316.8	−0.694079	−0.347039	+	0.937851i	$0.612813\pi$
$644$	−10858.8	−0.664438
$645$	0	0
$646$	−3429.82	−0.208893
$647$	−11244.8	−0.683272	−0.341636	−	0.939832i	$-0.610981\pi$
$647$	−11244.8	−0.683272	−0.341636	+	0.939832i	$0.610981\pi$
$648$	0	0
$649$	1183.07	0.0715559
$650$	0	0
$651$	0	0
$652$	4123.19	0.247664
$653$	19898.7	1.19249	0.596244	−	0.802803i	$-0.296659\pi$
$653$	19898.7	1.19249	0.596244	+	0.802803i	$0.296659\pi$
$654$	0	0
$655$	0	0
$656$	7982.92	0.475123
$657$	0	0
$658$	11642.8	0.689792
$659$	21502.2	1.27103	0.635514	−	0.772089i	$-0.280788\pi$
$659$	21502.2	1.27103	0.635514	+	0.772089i	$0.280788\pi$
$660$	0	0
$661$	11903.3	0.700434	0.350217	−	0.936669i	$-0.386108\pi$
$661$	11903.3	0.700434	0.350217	+	0.936669i	$0.386108\pi$
$662$	−17021.3	−0.999321
$663$	0	0
$664$	−6231.29	−0.364188
$665$	0	0
$666$	0	0
$667$	4429.46	0.257136
$668$	12404.1	0.718456
$669$	0	0
$670$	0	0
$671$	−51122.1	−2.94121
$672$	0	0
$673$	11771.3	0.674221	0.337110	−	0.941465i	$-0.390550\pi$
$673$	11771.3	0.674221	0.337110	+	0.941465i	$0.390550\pi$
$674$	−876.035	−0.0500647
$675$	0	0
$676$	−9602.98	−0.546369
$677$	25739.9	1.46125	0.730624	−	0.682780i	$-0.239229\pi$
$677$	25739.9	1.46125	0.730624	+	0.682780i	$0.239229\pi$
$678$	0	0
$679$	7826.65	0.442355
$680$	0	0
$681$	0	0
$682$	−18984.2	−1.06590
$683$	−29200.8	−1.63592	−0.817962	−	0.575272i	$-0.804896\pi$
$683$	−29200.8	−1.63592	−0.817962	+	0.575272i	$0.804896\pi$
$684$	0	0
$685$	0	0
$686$	−14889.2	−0.828679
$687$	0	0
$688$	−7170.78	−0.397360
$689$	21021.2	1.16233
$690$	0	0
$691$	5160.72	0.284114	0.142057	−	0.989858i	$-0.454628\pi$
$691$	5160.72	0.284114	0.142057	+	0.989858i	$0.454628\pi$
$692$	2506.15	0.137673
$693$	0	0
$694$	20077.6	1.09818
$695$	0	0
$696$	0	0
$697$	10893.8	0.592012
$698$	19647.0	1.06540
$699$	0	0
$700$	0	0
$701$	−11437.2	−0.616229	−0.308115	−	0.951349i	$-0.599698\pi$
$701$	−11437.2	−0.616229	−0.308115	+	0.951349i	$0.599698\pi$
$702$	0	0
$703$	747.167	0.0400852
$704$	−35194.1	−1.88413
$705$	0	0
$706$	−11571.0	−0.616826
$707$	21602.4	1.14914
$708$	0	0
$709$	726.443	0.0384797	0.0192399	−	0.999815i	$-0.493875\pi$
$709$	726.443	0.0384797	0.0192399	+	0.999815i	$0.493875\pi$
$710$	0	0
$711$	0	0
$712$	−39928.8	−2.10168
$713$	26249.5	1.37875
$714$	0	0
$715$	0	0
$716$	−5303.04	−0.276793
$717$	0	0
$718$	−29716.8	−1.54460
$719$	2949.11	0.152967	0.0764835	−	0.997071i	$-0.475631\pi$
$719$	2949.11	0.152967	0.0764835	+	0.997071i	$0.475631\pi$
$720$	0	0
$721$	7413.49	0.382930
$722$	−11900.2	−0.613409
$723$	0	0
$724$	−5503.06	−0.282486
$725$	0	0
$726$	0	0
$727$	−34433.2	−1.75661	−0.878306	−	0.478100i	$-0.841326\pi$
$727$	−34433.2	−1.75661	−0.878306	+	0.478100i	$0.841326\pi$
$728$	−30566.2	−1.55613
$729$	0	0
$730$	0	0
$731$	−9785.54	−0.495118
$732$	0	0
$733$	−1382.56	−0.0696670	−0.0348335	−	0.999393i	$-0.511090\pi$
$733$	−1382.56	−0.0696670	−0.0348335	+	0.999393i	$0.511090\pi$
$734$	16115.6	0.810406
$735$	0	0
$736$	26834.8	1.34395
$737$	35570.3	1.77781
$738$	0	0
$739$	−10335.3	−0.514466	−0.257233	−	0.966349i	$-0.582811\pi$
$739$	−10335.3	−0.514466	−0.257233	+	0.966349i	$0.582811\pi$
$740$	0	0
$741$	0	0
$742$	10937.9	0.541161
$743$	4880.52	0.240981	0.120491	−	0.992714i	$-0.461553\pi$
$743$	4880.52	0.240981	0.120491	+	0.992714i	$0.461553\pi$
$744$	0	0
$745$	0	0
$746$	−16250.5	−0.797552
$747$	0	0
$748$	−8360.55	−0.408679
$749$	28529.7	1.39179
$750$	0	0
$751$	−21584.1	−1.04875	−0.524377	−	0.851486i	$-0.675702\pi$
$751$	−21584.1	−1.04875	−0.524377	+	0.851486i	$0.675702\pi$
$752$	−9082.77	−0.440445
$753$	0	0
$754$	3485.95	0.168370
$755$	0	0
$756$	0	0
$757$	39391.6	1.89130	0.945648	−	0.325191i	$-0.105429\pi$
$757$	39391.6	1.89130	0.945648	+	0.325191i	$0.105429\pi$
$758$	−3438.17	−0.164749
$759$	0	0
$760$	0	0
$761$	−256.813	−0.0122332	−0.00611660	−	0.999981i	$-0.501947\pi$
$761$	−256.813	−0.0122332	−0.00611660	+	0.999981i	$0.501947\pi$
$762$	0	0
$763$	−20388.5	−0.967381
$764$	8211.63	0.388856
$765$	0	0
$766$	−6468.74	−0.305124
$767$	−1287.39	−0.0606063
$768$	0	0
$769$	2034.58	0.0954081	0.0477041	−	0.998862i	$-0.484810\pi$
$769$	2034.58	0.0954081	0.0477041	+	0.998862i	$0.484810\pi$
$770$	0	0
$771$	0	0
$772$	1651.07	0.0769731
$773$	−2737.54	−0.127377	−0.0636884	−	0.997970i	$-0.520286\pi$
$773$	−2737.54	−0.127377	−0.0636884	+	0.997970i	$0.520286\pi$
$774$	0	0
$775$	0	0
$776$	−11242.4	−0.520073
$777$	0	0
$778$	−880.376	−0.0405694
$779$	−10403.8	−0.478503
$780$	0	0
$781$	27396.5	1.25522
$782$	−18227.4	−0.833519
$783$	0	0
$784$	1488.86	0.0678233
$785$	0	0
$786$	0	0
$787$	18449.1	0.835629	0.417814	−	0.908532i	$-0.362796\pi$
$787$	18449.1	0.835629	0.417814	+	0.908532i	$0.362796\pi$
$788$	4135.13	0.186939
$789$	0	0
$790$	0	0
$791$	−8660.29	−0.389285
$792$	0	0
$793$	55629.8	2.49114
$794$	−2949.53	−0.131832
$795$	0	0
$796$	−5512.18	−0.245445
$797$	−35731.8	−1.58806	−0.794031	−	0.607878i	$-0.792021\pi$
$797$	−35731.8	−1.58806	−0.794031	+	0.607878i	$0.792021\pi$
$798$	0	0
$799$	−12394.7	−0.548803
$800$	0	0
$801$	0	0
$802$	−13924.1	−0.613064
$803$	−14743.5	−0.647929
$804$	0	0
$805$	0	0
$806$	20658.1	0.902792
$807$	0	0
$808$	−31030.1	−1.35103
$809$	21421.1	0.930933	0.465467	−	0.885065i	$-0.345887\pi$
$809$	21421.1	0.930933	0.465467	+	0.885065i	$0.345887\pi$
$810$	0	0
$811$	−27879.1	−1.20711	−0.603555	−	0.797321i	$-0.706250\pi$
$811$	−27879.1	−1.20711	−0.603555	+	0.797321i	$0.706250\pi$
$812$	−1150.36	−0.0497164
$813$	0	0
$814$	−2871.73	−0.123654
$815$	0	0
$816$	0	0
$817$	9345.36	0.400187
$818$	27705.8	1.18424
$819$	0	0
$820$	0	0
$821$	26618.4	1.13153	0.565767	−	0.824565i	$-0.308580\pi$
$821$	26618.4	1.13153	0.565767	+	0.824565i	$0.308580\pi$
$822$	0	0
$823$	9804.65	0.415272	0.207636	−	0.978206i	$-0.433423\pi$
$823$	9804.65	0.415272	0.207636	+	0.978206i	$0.433423\pi$
$824$	−10648.9	−0.450208
$825$	0	0
$826$	−669.862	−0.0282173
$827$	21534.9	0.905491	0.452745	−	0.891640i	$-0.350445\pi$
$827$	21534.9	0.905491	0.452745	+	0.891640i	$0.350445\pi$
$828$	0	0
$829$	−2671.33	−0.111917	−0.0559585	−	0.998433i	$-0.517821\pi$
$829$	−2671.33	−0.111917	−0.0559585	+	0.998433i	$0.517821\pi$
$830$	0	0
$831$	0	0
$832$	38297.3	1.59582
$833$	2031.76	0.0845092
$834$	0	0
$835$	0	0
$836$	7984.47	0.330321
$837$	0	0
$838$	12180.9	0.502129
$839$	−20755.8	−0.854076	−0.427038	−	0.904234i	$-0.640443\pi$
$839$	−20755.8	−0.854076	−0.427038	+	0.904234i	$0.640443\pi$
$840$	0	0
$841$	−23919.8	−0.980760
$842$	13181.8	0.539517
$843$	0	0
$844$	10384.0	0.423497
$845$	0	0
$846$	0	0
$847$	53648.7	2.17638
$848$	−8532.84	−0.345541
$849$	0	0
$850$	0	0
$851$	3970.74	0.159947
$852$	0	0
$853$	−15242.4	−0.611831	−0.305915	−	0.952059i	$-0.598962\pi$
$853$	−15242.4	−0.611831	−0.305915	+	0.952059i	$0.598962\pi$
$854$	28945.6	1.15983
$855$	0	0
$856$	−40980.7	−1.63632
$857$	−30965.0	−1.23424	−0.617121	−	0.786868i	$-0.711701\pi$
$857$	−30965.0	−1.23424	−0.617121	+	0.786868i	$0.711701\pi$
$858$	0	0
$859$	15617.3	0.620320	0.310160	−	0.950684i	$-0.399617\pi$
$859$	15617.3	0.620320	0.310160	+	0.950684i	$0.399617\pi$
$860$	0	0
$861$	0	0
$862$	27677.4	1.09361
$863$	17853.6	0.704223	0.352112	−	0.935958i	$-0.385464\pi$
$863$	17853.6	0.704223	0.352112	+	0.935958i	$0.385464\pi$
$864$	0	0
$865$	0	0
$866$	9614.63	0.377273
$867$	0	0
$868$	−6817.15	−0.266577
$869$	−75775.7	−2.95802
$870$	0	0
$871$	−38706.7	−1.50577
$872$	29286.4	1.13734
$873$	0	0
$874$	17407.5	0.673705
$875$	0	0
$876$	0	0
$877$	−31782.1	−1.22372	−0.611862	−	0.790965i	$-0.709579\pi$
$877$	−31782.1	−1.22372	−0.611862	+	0.790965i	$0.709579\pi$
$878$	15518.9	0.596513
$879$	0	0
$880$	0	0
$881$	51478.0	1.96860	0.984301	−	0.176496i	$-0.0564763\pi$
$881$	51478.0	1.96860	0.984301	+	0.176496i	$0.0564763\pi$
$882$	0	0
$883$	−5259.06	−0.200432	−0.100216	−	0.994966i	$-0.531953\pi$
$883$	−5259.06	−0.200432	−0.100216	+	0.994966i	$0.531953\pi$
$884$	9097.74	0.346142
$885$	0	0
$886$	19309.4	0.732182
$887$	12791.9	0.484229	0.242114	−	0.970248i	$-0.422159\pi$
$887$	12791.9	0.484229	0.242114	+	0.970248i	$0.422159\pi$
$888$	0	0
$889$	−34942.2	−1.31825
$890$	0	0
$891$	0	0
$892$	−13051.1	−0.489891
$893$	11837.2	0.443579
$894$	0	0
$895$	0	0
$896$	1969.28	0.0734254
$897$	0	0
$898$	−28985.2	−1.07711
$899$	2780.81	0.103165
$900$	0	0
$901$	−11644.3	−0.430551
$902$	39986.8	1.47607
$903$	0	0
$904$	12439.8	0.457679
$905$	0	0
$906$	0	0
$907$	−19969.8	−0.731076	−0.365538	−	0.930796i	$-0.619115\pi$
$907$	−19969.8	−0.731076	−0.365538	+	0.930796i	$0.619115\pi$
$908$	948.899	0.0346810
$909$	0	0
$910$	0	0
$911$	−10692.5	−0.388868	−0.194434	−	0.980916i	$-0.562287\pi$
$911$	−10692.5	−0.388868	−0.194434	+	0.980916i	$0.562287\pi$
$912$	0	0
$913$	−16951.7	−0.614479
$914$	2352.96	0.0851521
$915$	0	0
$916$	383.622	0.0138376
$917$	−31728.9	−1.14262
$918$	0	0
$919$	−14818.3	−0.531895	−0.265947	−	0.963988i	$-0.585685\pi$
$919$	−14818.3	−0.531895	−0.265947	+	0.963988i	$0.585685\pi$
$920$	0	0
$921$	0	0
$922$	18486.8	0.660335
$923$	−29812.2	−1.06314
$924$	0	0
$925$	0	0
$926$	−26253.9	−0.931702
$927$	0	0
$928$	2842.82	0.100560
$929$	−2229.09	−0.0787234	−0.0393617	−	0.999225i	$-0.512532\pi$
$929$	−2229.09	−0.0787234	−0.0393617	+	0.999225i	$0.512532\pi$
$930$	0	0
$931$	−1940.36	−0.0683059
$932$	−16289.5	−0.572512
$933$	0	0
$934$	−12583.0	−0.440824
$935$	0	0
$936$	0	0
$937$	−17737.2	−0.618408	−0.309204	−	0.950996i	$-0.600063\pi$
$937$	−17737.2	−0.618408	−0.309204	+	0.950996i	$0.600063\pi$
$938$	−20140.1	−0.701063
$939$	0	0
$940$	0	0
$941$	−4291.21	−0.148660	−0.0743302	−	0.997234i	$-0.523682\pi$
$941$	−4291.21	−0.148660	−0.0743302	+	0.997234i	$0.523682\pi$
$942$	0	0
$943$	−55289.8	−1.90931
$944$	522.573	0.0180173
$945$	0	0
$946$	−35918.8	−1.23448
$947$	−17768.1	−0.609700	−0.304850	−	0.952400i	$-0.598606\pi$
$947$	−17768.1	−0.609700	−0.304850	+	0.952400i	$0.598606\pi$
$948$	0	0
$949$	16043.5	0.548782
$950$	0	0
$951$	0	0
$952$	16931.6	0.576423
$953$	−48689.0	−1.65497	−0.827487	−	0.561485i	$-0.810230\pi$
$953$	−48689.0	−1.65497	−0.827487	+	0.561485i	$0.810230\pi$
$954$	0	0
$955$	0	0
$956$	13352.4	0.451723
$957$	0	0
$958$	−23356.6	−0.787701
$959$	−29044.5	−0.977994
$960$	0	0
$961$	−13311.7	−0.446835
$962$	3124.94	0.104732
$963$	0	0
$964$	16627.0	0.555520
$965$	0	0
$966$	0	0
$967$	−25690.1	−0.854330	−0.427165	−	0.904174i	$-0.640488\pi$
$967$	−25690.1	−0.854330	−0.427165	+	0.904174i	$0.640488\pi$
$968$	−77062.1	−2.55875
$969$	0	0
$970$	0	0
$971$	−9697.83	−0.320513	−0.160257	−	0.987075i	$-0.551232\pi$
$971$	−9697.83	−0.320513	−0.160257	+	0.987075i	$0.551232\pi$
$972$	0	0
$973$	40812.2	1.34469
$974$	−2130.40	−0.0700847
$975$	0	0
$976$	−22581.0	−0.740575
$977$	−32608.3	−1.06779	−0.533896	−	0.845550i	$-0.679273\pi$
$977$	−32608.3	−1.06779	−0.533896	+	0.845550i	$0.679273\pi$
$978$	0	0
$979$	−108623.	−3.54607
$980$	0	0
$981$	0	0
$982$	17229.4	0.559890
$983$	−29544.1	−0.958606	−0.479303	−	0.877649i	$-0.659111\pi$
$983$	−29544.1	−0.958606	−0.479303	+	0.877649i	$0.659111\pi$
$984$	0	0
$985$	0	0
$986$	−1930.97	−0.0623679
$987$	0	0
$988$	−8688.49	−0.279775
$989$	49664.9	1.59682
$990$	0	0
$991$	−51930.3	−1.66460	−0.832301	−	0.554324i	$-0.812977\pi$
$991$	−51930.3	−1.66460	−0.832301	+	0.554324i	$0.812977\pi$
$992$	16846.8	0.539201
$993$	0	0
$994$	−15512.0	−0.494981
$995$	0	0
$996$	0	0
$997$	7361.72	0.233849	0.116925	−	0.993141i	$-0.462696\pi$
$997$	7361.72	0.233849	0.116925	+	0.993141i	$0.462696\pi$
$998$	8931.56	0.283290
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.x.1.3	yes	4
3.2	odd	2	inner	675.4.a.x.1.2	yes	4
5.2	odd	4		675.4.b.o.649.6		8
5.3	odd	4		675.4.b.o.649.3		8
5.4	even	2		675.4.a.w.1.2	✓	4
15.2	even	4		675.4.b.o.649.4		8
15.8	even	4		675.4.b.o.649.5		8
15.14	odd	2		675.4.a.w.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.4.a.w.1.2	✓	4	5.4	even	2
675.4.a.w.1.3	yes	4	15.14	odd	2
675.4.a.x.1.2	yes	4	3.2	odd	2	inner
675.4.a.x.1.3	yes	4	1.1	even	1	trivial
675.4.b.o.649.3		8	5.3	odd	4
675.4.b.o.649.4		8	15.2	even	4
675.4.b.o.649.5		8	15.8	even	4
675.4.b.o.649.6		8	5.2	odd	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q+2.21254 q^{2} -3.10469 q^{4} +17.1047 q^{7} -24.5695 q^{8} -66.8393 q^{11} +72.7328 q^{13} +37.8447 q^{14} -29.5234 q^{16} -40.2889 q^{17} +38.4766 q^{19} -147.884 q^{22} +204.480 q^{23} +160.924 q^{26} -53.1047 q^{28} +21.6621 q^{29} +128.372 q^{31} +131.234 q^{32} -89.1406 q^{34} +19.4187 q^{37} +85.1308 q^{38} -270.393 q^{41} +242.884 q^{43} +207.515 q^{44} +452.419 q^{46} +307.646 q^{47} -50.4297 q^{49} -225.813 q^{52} +289.019 q^{53} -420.254 q^{56} +47.9282 q^{58} -17.7003 q^{59} +764.851 q^{61} +284.027 q^{62} +526.548 q^{64} -532.176 q^{67} +125.084 q^{68} -409.886 q^{71} +220.581 q^{73} +42.9647 q^{74} -119.458 q^{76} -1143.27 q^{77} +1133.70 q^{79} -598.253 q^{82} +253.619 q^{83} +537.390 q^{86} +1642.21 q^{88} +1625.13 q^{89} +1244.07 q^{91} -634.846 q^{92} +680.678 q^{94} +457.573 q^{97} -111.578 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 26 q^{4} + 30 q^{7} + 22 q^{13} + 74 q^{16} + 346 q^{19} + 100 q^{22} - 174 q^{28} + 744 q^{31} + 796 q^{34} - 76 q^{37} + 280 q^{43} + 1656 q^{46} - 778 q^{49} - 2440 q^{52} + 2420 q^{58} + 178 q^{61}+ \cdots - 2050 q^{97}+O(q^{100}) \) 4 * q + 26 * q^4 + 30 * q^7 + 22 * q^13 + 74 * q^16 + 346 * q^19 + 100 * q^22 - 174 * q^28 + 744 * q^31 + 796 * q^34 - 76 * q^37 + 280 * q^43 + 1656 * q^46 - 778 * q^49 - 2440 * q^52 + 2420 * q^58 + 178 * q^61 + 70 * q^64 + 138 * q^67 + 1036 * q^73 + 4094 * q^76 + 2076 * q^79 - 5236 * q^82 + 6492 * q^88 + 2748 * q^91 - 1196 * q^94 - 2050 * q^97

Introduction

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