LMFDB - Embedded newform 675.4.b.o.649.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(649,675)] 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("675.649"); 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, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-52,0,0,0,0,0,0,0,0,0,0,0,148,0,0,-692] 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:	no
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} - 38x^{5} + 650x^{4} - 2138x^{3} + 3698x^{2} - 3182x + 1369 $ x^8 - 2x^7 + 2x^6 - 38x^5 + 650x^4 - 2138x^3 + 3698x^2 - 3182*x + 1369
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.3
Root			$0.744513 + 0.744513i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.o.649.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21254i q^{2} +3.10469 q^{4} -17.1047i q^{7} -24.5695i q^{8} -66.8393 q^{11} +72.7328i q^{13} -37.8447 q^{14} -29.5234 q^{16} +40.2889i q^{17} -38.4766 q^{19} +147.884i q^{22} +204.480i q^{23} +160.924 q^{26} -53.1047i q^{28} -21.6621 q^{29} +128.372 q^{31} -131.234i q^{32} +89.1406 q^{34} -19.4187i q^{37} +85.1308i q^{38} -270.393 q^{41} +242.884i q^{43} -207.515 q^{44} +452.419 q^{46} -307.646i q^{47} +50.4297 q^{49} +225.813i q^{52} +289.019i q^{53} -420.254 q^{56} +47.9282i q^{58} +17.7003 q^{59} +764.851 q^{61} -284.027i q^{62} -526.548 q^{64} +532.176i q^{67} +125.084i q^{68} -409.886 q^{71} +220.581i q^{73} -42.9647 q^{74} -119.458 q^{76} +1143.27i q^{77} -1133.70 q^{79} +598.253i q^{82} +253.619i q^{83} +537.390 q^{86} +1642.21i q^{88} -1625.13 q^{89} +1244.07 q^{91} +634.846i q^{92} -680.678 q^{94} -457.573i q^{97} -111.578i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 52 q^{4} + 148 q^{16} - 692 q^{19} + 1488 q^{31} - 1592 q^{34} + 3312 q^{46} + 1556 q^{49} + 356 q^{61} - 140 q^{64} + 8188 q^{76} - 4152 q^{79} + 5496 q^{91} + 2392 q^{94}+O(q^{100}) $ 8 * q - 52 * q^4 + 148 * q^16 - 692 * q^19 + 1488 * q^31 - 1592 * q^34 + 3312 * q^46 + 1556 * q^49 + 356 * q^61 - 140 * q^64 + 8188 * q^76 - 4152 * q^79 + 5496 * q^91 + 2392 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.21254i	− 0.782249i	−0.920338	−	0.391125i	$-0.872086\pi$
$2$	− 2.21254i	− 0.782249i	0.920338	−	0.391125i	$-0.127914\pi$
$3$	0	0
$3$	0	0
$4$	3.10469	0.388086
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 17.1047i	− 0.923566i	−0.886993	−	0.461783i	$-0.847210\pi$
$7$	− 17.1047i	− 0.923566i	0.886993	−	0.461783i	$-0.152790\pi$
$8$	− 24.5695i	− 1.08583i
$9$	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
$13$	72.7328i	1.55173i	0.630901	+	0.775863i	$0.282685\pi$
$13$	72.7328i	1.55173i	−0.630901	+	0.775863i	$0.717315\pi$
$14$	−37.8447	−0.722459
$15$	0	0
$16$	−29.5234	−0.461304
$17$	40.2889i	0.574794i	0.957812	+	0.287397i	$0.0927898\pi$
$17$	40.2889i	0.574794i	−0.957812	+	0.287397i	$0.907210\pi$
$18$	0	0
$19$	−38.4766	−0.464586	−0.232293	−	0.972646i	$-0.574623\pi$
$19$	−38.4766	−0.464586	−0.232293	+	0.972646i	$0.574623\pi$
$20$	0	0
$21$	0	0
$22$	147.884i	1.43314i
$23$	204.480i	1.85378i	0.375331	+	0.926891i	$0.377529\pi$
$23$	204.480i	1.85378i	−0.375331	+	0.926891i	$0.622471\pi$
$24$	0	0
$25$	0	0
$26$	160.924	1.21384
$27$	0	0
$28$	− 53.1047i	− 0.358423i
$29$	−21.6621	−0.138709	−0.0693544	−	0.997592i	$-0.522094\pi$
$29$	−21.6621	−0.138709	−0.0693544	+	0.997592i	$0.522094\pi$
$30$	0	0
$31$	128.372	0.743751	0.371875	−	0.928283i	$-0.378715\pi$
$31$	128.372	0.743751	0.371875	+	0.928283i	$0.378715\pi$
$32$	− 131.234i	− 0.724975i
$33$	0	0
$34$	89.1406	0.449632
$35$	0	0
$36$	0	0
$37$	− 19.4187i	− 0.0862817i	−0.999069	−	0.0431408i	$-0.986264\pi$
$37$	− 19.4187i	− 0.0862817i	0.999069	−	0.0431408i	$-0.0137364\pi$
$38$	85.1308i	0.363422i
$39$	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
$43$	242.884i	0.861384i	0.902499	+	0.430692i	$0.141730\pi$
$43$	242.884i	0.861384i	−0.902499	+	0.430692i	$0.858270\pi$
$44$	−207.515	−0.711002
$45$	0	0
$46$	452.419	1.45012
$47$	− 307.646i	− 0.954783i	−0.878691	−	0.477391i	$-0.841582\pi$
$47$	− 307.646i	− 0.954783i	0.878691	−	0.477391i	$-0.158418\pi$
$48$	0	0
$49$	50.4297	0.147025
$50$	0	0
$51$	0	0
$52$	225.813i	0.602203i
$53$	289.019i	0.749054i	0.927216	+	0.374527i	$0.122195\pi$
$53$	289.019i	0.749054i	−0.927216	+	0.374527i	$0.877805\pi$
$54$	0	0
$55$	0	0
$56$	−420.254	−1.00284
$57$	0	0
$58$	47.9282i	0.108505i
$59$	17.7003	0.0390573	0.0195287	−	0.999809i	$-0.493783\pi$
$59$	17.7003	0.0390573	0.0195287	+	0.999809i	$0.493783\pi$
$60$	0	0
$61$	764.851	1.60540	0.802698	−	0.596385i	$-0.203397\pi$
$61$	764.851	1.60540	0.802698	+	0.596385i	$0.203397\pi$
$62$	− 284.027i	− 0.581799i
$63$	0	0
$64$	−526.548	−1.02841
$65$	0	0
$66$	0	0
$67$	532.176i	0.970384i	0.874408	+	0.485192i	$0.161250\pi$
$67$	532.176i	0.970384i	−0.874408	+	0.485192i	$0.838750\pi$
$68$	125.084i	0.223069i
$69$	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
$73$	220.581i	0.353659i	0.984242	+	0.176829i	$0.0565841\pi$
$73$	220.581i	0.353659i	−0.984242	+	0.176829i	$0.943416\pi$
$74$	−42.9647	−0.0674938
$75$	0	0
$76$	−119.458	−0.180299
$77$	1143.27i	1.69204i
$78$	0	0
$79$	−1133.70	−1.61457	−0.807286	−	0.590160i	$-0.799065\pi$
$79$	−1133.70	−1.61457	−0.807286	+	0.590160i	$0.799065\pi$
$80$	0	0
$81$	0	0
$82$	598.253i	0.805683i
$83$	253.619i	0.335401i	0.985838	+	0.167700i	$0.0536341\pi$
$83$	253.619i	0.335401i	−0.985838	+	0.167700i	$0.946366\pi$
$84$	0	0
$85$	0	0
$86$	537.390	0.673817
$87$	0	0
$88$	1642.21i	1.98932i
$89$	−1625.13	−1.93555	−0.967775	−	0.251816i	$-0.918972\pi$
$89$	−1625.13	−1.93555	−0.967775	+	0.251816i	$0.918972\pi$
$90$	0	0
$91$	1244.07	1.43312
$92$	634.846i	0.719426i
$93$	0	0
$94$	−680.678	−0.746878
$95$	0	0
$96$	0	0
$97$	− 457.573i	− 0.478964i	−0.970901	−	0.239482i	$-0.923022\pi$
$97$	− 457.573i	− 0.478964i	0.970901	−	0.239482i	$-0.0769776\pi$
$98$	− 111.578i	− 0.115011i
$99$	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
$103$	433.419i	0.414622i	0.978275	+	0.207311i	$0.0664711\pi$
$103$	433.419i	0.414622i	−0.978275	+	0.207311i	$0.933529\pi$
$104$	1787.01	1.68491
$105$	0	0
$106$	639.466	0.585947
$107$	− 1667.95i	− 1.50698i	−0.657461	−	0.753489i	$-0.728370\pi$
$107$	− 1667.95i	− 1.50698i	0.657461	−	0.753489i	$-0.271630\pi$
$108$	0	0
$109$	1191.98	1.04744	0.523721	−	0.851890i	$-0.324544\pi$
$109$	1191.98	1.04744	0.523721	+	0.851890i	$0.324544\pi$
$110$	0	0
$111$	0	0
$112$	504.989i	0.426044i
$113$	− 506.311i	− 0.421502i	−0.977540	−	0.210751i	$-0.932409\pi$
$113$	− 506.311i	− 0.421502i	0.977540	−	0.210751i	$-0.0675909\pi$
$114$	0	0
$115$	0	0
$116$	−67.2541	−0.0538309
$117$	0	0
$118$	− 39.1625i	− 0.0305526i
$119$	689.129	0.530860
$120$	0	0
$121$	3136.49	2.35649
$122$	− 1692.26i	− 1.25582i
$123$	0	0
$124$	398.554	0.288639
$125$	0	0
$126$	0	0
$127$	2042.84i	1.42735i	0.700479	+	0.713673i	$0.252970\pi$
$127$	2042.84i	1.42735i	−0.700479	+	0.713673i	$0.747030\pi$
$128$	115.131i	0.0795021i
$129$	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
$133$	658.130i	0.429076i
$134$	1177.46	0.759082
$135$	0	0
$136$	989.878	0.624128
$137$	1698.04i	1.05893i	0.848331	+	0.529466i	$0.177608\pi$
$137$	1698.04i	1.05893i	−0.848331	+	0.529466i	$0.822392\pi$
$138$	0	0
$139$	−2386.02	−1.45597	−0.727986	−	0.685592i	$-0.759543\pi$
$139$	−2386.02	−1.45597	−0.727986	+	0.685592i	$0.759543\pi$
$140$	0	0
$141$	0	0
$142$	906.887i	0.535946i
$143$	− 4861.41i	− 2.84288i
$144$	0	0
$145$	0	0
$146$	488.044	0.276649
$147$	0	0
$148$	− 60.2891i	− 0.0334847i
$149$	1090.68	0.599676	0.299838	−	0.953990i	$-0.403067\pi$
$149$	1090.68	0.599676	0.299838	+	0.953990i	$0.403067\pi$
$150$	0	0
$151$	−2779.07	−1.49773	−0.748865	−	0.662722i	$-0.769401\pi$
$151$	−2779.07	−1.49773	−0.748865	+	0.662722i	$0.769401\pi$
$152$	945.351i	0.504461i
$153$	0	0
$154$	2529.52	1.32360
$155$	0	0
$156$	0	0
$157$	1277.14i	0.649218i	0.945848	+	0.324609i	$0.105233\pi$
$157$	1277.14i	0.649218i	−0.945848	+	0.324609i	$0.894767\pi$
$158$	2508.35i	1.26300i
$159$	0	0
$160$	0	0
$161$	3497.56	1.71209
$162$	0	0
$163$	− 1328.05i	− 0.638167i	−0.947727	−	0.319084i	$-0.896625\pi$
$163$	− 1328.05i	− 0.638167i	0.947727	−	0.319084i	$-0.103375\pi$
$164$	−839.484	−0.399712
$165$	0	0
$166$	561.141	0.262367
$167$	3995.28i	1.85128i	0.378404	+	0.925641i	$0.376473\pi$
$167$	3995.28i	1.85128i	−0.378404	+	0.925641i	$0.623527\pi$
$168$	0	0
$169$	−3093.06	−1.40786
$170$	0	0
$171$	0	0
$172$	754.080i	0.334291i
$173$	− 807.216i	− 0.354748i	−0.984143	−	0.177374i	$-0.943240\pi$
$173$	− 807.216i	− 0.354748i	0.984143	−	0.177374i	$-0.0567603\pi$
$174$	0	0
$175$	0	0
$176$	1973.33	0.845142
$177$	0	0
$178$	3595.67i	1.51408i
$179$	−1708.08	−0.713227	−0.356614	−	0.934252i	$-0.616069\pi$
$179$	−1708.08	−0.713227	−0.356614	+	0.934252i	$0.616069\pi$
$180$	0	0
$181$	1772.50	0.727895	0.363948	−	0.931419i	$-0.381429\pi$
$181$	1772.50	0.727895	0.363948	+	0.931419i	$0.381429\pi$
$182$	− 2752.55i	− 1.12106i
$183$	0	0
$184$	5023.97	2.01289
$185$	0	0
$186$	0	0
$187$	− 2692.88i	− 1.05306i
$188$	− 955.145i	− 0.370538i
$189$	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
$193$	− 531.799i	− 0.198340i	−0.995070	−	0.0991702i	$-0.968381\pi$
$193$	− 531.799i	− 0.198340i	0.995070	−	0.0991702i	$-0.0316188\pi$
$194$	−1012.40	−0.374670
$195$	0	0
$196$	156.568	0.0570585
$197$	1331.90i	0.481694i	0.970563	+	0.240847i	$0.0774253\pi$
$197$	1331.90i	0.481694i	−0.970563	+	0.240847i	$0.922575\pi$
$198$	0	0
$199$	−1775.44	−0.632450	−0.316225	−	0.948684i	$-0.602415\pi$
$199$	−1775.44	−0.632450	−0.316225	+	0.948684i	$0.602415\pi$
$200$	0	0
$201$	0	0
$202$	− 2794.32i	− 0.973306i
$203$	370.524i	0.128107i
$204$	0	0
$205$	0	0
$206$	958.954	0.324337
$207$	0	0
$208$	− 2147.32i	− 0.715817i
$209$	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	$0.683707\pi$
$212$	897.314i	0.290697i
$213$	0	0
$214$	−3690.39	−1.17883
$215$	0	0
$216$	0	0
$217$	− 2195.76i	− 0.686903i
$218$	− 2637.30i	− 0.819360i
$219$	0	0
$220$	0	0
$221$	−2930.32	−0.891923
$222$	0	0
$223$	4203.68i	1.26233i	0.775649	+	0.631164i	$0.217422\pi$
$223$	4203.68i	1.26233i	−0.775649	+	0.631164i	$0.782578\pi$
$224$	−2244.72	−0.669562
$225$	0	0
$226$	−1120.23	−0.329720
$227$	305.634i	0.0893641i	0.999001	+	0.0446821i	$0.0142275\pi$
$227$	305.634i	0.0893641i	−0.999001	+	0.0446821i	$0.985773\pi$
$228$	0	0
$229$	123.562	0.0356560	0.0178280	−	0.999841i	$-0.494325\pi$
$229$	123.562	0.0356560	0.0178280	+	0.999841i	$0.494325\pi$
$230$	0	0
$231$	0	0
$232$	532.228i	0.150614i
$233$	5246.76i	1.47522i	0.675226	+	0.737611i	$0.264046\pi$
$233$	5246.76i	1.47522i	−0.675226	+	0.737611i	$0.735954\pi$
$234$	0	0
$235$	0	0
$236$	54.9538	0.0151576
$237$	0	0
$238$	− 1524.72i	− 0.415265i
$239$	4300.72	1.16398	0.581988	−	0.813197i	$-0.302275\pi$
$239$	4300.72	1.16398	0.581988	+	0.813197i	$0.302275\pi$
$240$	0	0
$241$	−5355.47	−1.43144	−0.715718	−	0.698390i	$-0.753900\pi$
$241$	−5355.47	−1.43144	−0.715718	+	0.698390i	$0.753900\pi$
$242$	− 6939.60i	− 1.84337i
$243$	0	0
$244$	2374.62	0.623032
$245$	0	0
$246$	0	0
$247$	− 2798.51i	− 0.720910i
$248$	− 3154.03i	− 0.807586i
$249$	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
$253$	− 13667.3i	− 3.39626i
$254$	4519.86	1.11654
$255$	0	0
$256$	−3957.65	−0.966224
$257$	− 4909.53i	− 1.19163i	−0.803123	−	0.595813i	$-0.796830\pi$
$257$	− 4909.53i	− 1.19163i	0.803123	−	0.595813i	$-0.203170\pi$
$258$	0	0
$259$	−332.152	−0.0796868
$260$	0	0
$261$	0	0
$262$	4104.22i	0.967783i
$263$	4464.34i	1.04670i	0.852117	+	0.523351i	$0.175318\pi$
$263$	4464.34i	1.04670i	−0.852117	+	0.523351i	$0.824682\pi$
$264$	0	0
$265$	0	0
$266$	1456.14	0.335644
$267$	0	0
$268$	1652.24i	0.376592i
$269$	−5957.80	−1.35039	−0.675193	−	0.737641i	$-0.735940\pi$
$269$	−5957.80	−1.35039	−0.675193	+	0.737641i	$0.735940\pi$
$270$	0	0
$271$	4805.80	1.07724	0.538619	−	0.842549i	$-0.318946\pi$
$271$	4805.80	1.07724	0.538619	+	0.842549i	$0.318946\pi$
$272$	− 1189.47i	− 0.265154i
$273$	0	0
$274$	3756.98	0.828349
$275$	0	0
$276$	0	0
$277$	− 3531.98i	− 0.766122i	−0.923723	−	0.383061i	$-0.874870\pi$
$277$	− 3531.98i	− 0.766122i	0.923723	−	0.383061i	$-0.125130\pi$
$278$	5279.17i	1.13893i
$279$	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
$283$	− 6278.33i	− 1.31876i	−0.751811	−	0.659378i	$-0.770820\pi$
$283$	− 6278.33i	− 1.31876i	0.751811	−	0.659378i	$-0.229180\pi$
$284$	−1272.57	−0.265891
$285$	0	0
$286$	−10756.0	−2.22384
$287$	4624.98i	0.951233i
$288$	0	0
$289$	3289.81	0.669612
$290$	0	0
$291$	0	0
$292$	684.836i	0.137250i
$293$	1065.21i	0.212390i	0.994345	+	0.106195i	$0.0338668\pi$
$293$	1065.21i	0.212390i	−0.994345	+	0.106195i	$0.966133\pi$
$294$	0	0
$295$	0	0
$296$	−477.109	−0.0936872
$297$	0	0
$298$	− 2413.16i	− 0.469096i
$299$	−14872.4	−2.87656
$300$	0	0
$301$	4154.46	0.795545
$302$	6148.78i	1.17160i
$303$	0	0
$304$	1135.96	0.214315
$305$	0	0
$306$	0	0
$307$	− 10015.7i	− 1.86197i	−0.365055	−	0.930986i	$-0.618950\pi$
$307$	− 10015.7i	− 1.86197i	0.365055	−	0.930986i	$-0.381050\pi$
$308$	3549.48i	0.656657i
$309$	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
$313$	1913.29i	0.345512i	0.984965	+	0.172756i	$0.0552672\pi$
$313$	1913.29i	0.345512i	−0.984965	+	0.172756i	$0.944733\pi$
$314$	2825.73	0.507850
$315$	0	0
$316$	−3519.78	−0.626593
$317$	4472.16i	0.792371i	0.918170	+	0.396186i	$0.129666\pi$
$317$	4472.16i	0.792371i	−0.918170	+	0.396186i	$0.870334\pi$
$318$	0	0
$319$	1447.88	0.254125
$320$	0	0
$321$	0	0
$322$	− 7738.48i	− 1.33928i
$323$	− 1550.18i	− 0.267041i
$324$	0	0
$325$	0	0
$326$	−2938.37	−0.499206
$327$	0	0
$328$	6643.41i	1.11836i
$329$	−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	$0.720547\pi$
$332$	787.407i	0.130164i
$333$	0	0
$334$	8839.70	1.44816
$335$	0	0
$336$	0	0
$337$	395.941i	0.0640009i	0.999488	+	0.0320005i	$0.0101878\pi$
$337$	395.941i	0.0640009i	−0.999488	+	0.0320005i	$0.989812\pi$
$338$	6843.51i	1.10130i
$339$	0	0
$340$	0	0
$341$	−8580.29	−1.36261
$342$	0	0
$343$	− 6729.49i	− 1.05935i
$344$	5967.55	0.935316
$345$	0	0
$346$	−1785.99	−0.277502
$347$	− 9074.49i	− 1.40387i	−0.712239	−	0.701937i	$-0.752319\pi$
$347$	− 9074.49i	− 1.40387i	0.712239	−	0.701937i	$-0.247681\pi$
$348$	0	0
$349$	−8879.84	−1.36197	−0.680984	−	0.732298i	$-0.738448\pi$
$349$	−8879.84	−1.36197	−0.680984	+	0.732298i	$0.738448\pi$
$350$	0	0
$351$	0	0
$352$	8771.62i	1.32821i
$353$	− 5229.73i	− 0.788528i	−0.918997	−	0.394264i	$-0.871000\pi$
$353$	− 5229.73i	− 0.788528i	0.918997	−	0.394264i	$-0.129000\pi$
$354$	0	0
$355$	0	0
$356$	−5045.53	−0.751160
$357$	0	0
$358$	3779.18i	0.557922i
$359$	13431.1	1.97456	0.987278	−	0.159001i	$-0.0508272\pi$
$359$	13431.1	1.97456	0.987278	+	0.159001i	$0.0508272\pi$
$360$	0	0
$361$	−5378.55	−0.784160
$362$	− 3921.72i	− 0.569395i
$363$	0	0
$364$	3862.45	0.556175
$365$	0	0
$366$	0	0
$367$	− 7283.77i	− 1.03599i	−0.855383	−	0.517997i	$-0.826678\pi$
$367$	− 7283.77i	− 1.03599i	0.855383	−	0.517997i	$-0.173322\pi$
$368$	− 6036.94i	− 0.855156i
$369$	0	0
$370$	0	0
$371$	4943.59	0.691801
$372$	0	0
$373$	− 7344.75i	− 1.01956i	−0.860304	−	0.509781i	$-0.829726\pi$
$373$	− 7344.75i	− 1.01956i	0.860304	−	0.509781i	$-0.170274\pi$
$374$	−5958.10	−0.823759
$375$	0	0
$376$	−7558.72	−1.03673
$377$	− 1575.55i	− 0.215238i
$378$	0	0
$379$	1553.95	0.210610	0.105305	−	0.994440i	$-0.466418\pi$
$379$	1553.95	0.210610	0.105305	+	0.994440i	$0.466418\pi$
$380$	0	0
$381$	0	0
$382$	5851.96i	0.783802i
$383$	− 2923.68i	− 0.390060i	−0.980797	−	0.195030i	$-0.937520\pi$
$383$	− 2923.68i	− 0.390060i	0.980797	−	0.195030i	$-0.0624804\pi$
$384$	0	0
$385$	0	0
$386$	−1176.62	−0.155152
$387$	0	0
$388$	− 1420.62i	− 0.185879i
$389$	397.904	0.0518625	0.0259312	−	0.999664i	$-0.491745\pi$
$389$	397.904	0.0518625	0.0259312	+	0.999664i	$0.491745\pi$
$390$	0	0
$391$	−8238.26	−1.06554
$392$	− 1239.03i	− 0.159644i
$393$	0	0
$394$	2946.87	0.376805
$395$	0	0
$396$	0	0
$397$	1333.10i	0.168530i	0.996443	+	0.0842649i	$0.0268542\pi$
$397$	1333.10i	0.168530i	−0.996443	+	0.0842649i	$0.973146\pi$
$398$	3928.22i	0.494734i
$399$	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
$403$	9336.85i	1.15410i
$404$	3921.06	0.482872
$405$	0	0
$406$	819.797	0.100211
$407$	1297.94i	0.158074i
$408$	0	0
$409$	−12522.2	−1.51390	−0.756948	−	0.653475i	$-0.773310\pi$
$409$	−12522.2	−1.51390	−0.756948	+	0.653475i	$0.773310\pi$
$410$	0	0
$411$	0	0
$412$	1345.63i	0.160909i
$413$	− 302.758i	− 0.0360720i
$414$	0	0
$415$	0	0
$416$	9545.05	1.12496
$417$	0	0
$418$	− 5690.08i	− 0.665816i
$419$	−5505.42	−0.641903	−0.320952	−	0.947096i	$-0.604003\pi$
$419$	−5505.42	−0.641903	−0.320952	+	0.947096i	$0.604003\pi$
$420$	0	0
$421$	5957.76	0.689700	0.344850	−	0.938658i	$-0.387930\pi$
$421$	5957.76	0.689700	0.344850	+	0.938658i	$0.387930\pi$
$422$	7400.08i	0.853627i
$423$	0	0
$424$	7101.07	0.813345
$425$	0	0
$426$	0	0
$427$	− 13082.5i	− 1.48269i
$428$	− 5178.45i	− 0.584836i
$429$	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
$433$	4345.52i	0.482292i	0.970489	+	0.241146i	$0.0775233\pi$
$433$	4345.52i	0.482292i	−0.970489	+	0.241146i	$0.922477\pi$
$434$	−4858.20	−0.537330
$435$	0	0
$436$	3700.73	0.406497
$437$	− 7867.68i	− 0.861241i
$438$	0	0
$439$	−7014.09	−0.762561	−0.381281	−	0.924459i	$-0.624517\pi$
$439$	−7014.09	−0.762561	−0.381281	+	0.924459i	$0.624517\pi$
$440$	0	0
$441$	0	0
$442$	6483.45i	0.697706i
$443$	8727.29i	0.935996i	0.883730	+	0.467998i	$0.155025\pi$
$443$	8727.29i	0.935996i	−0.883730	+	0.467998i	$0.844975\pi$
$444$	0	0
$445$	0	0
$446$	9300.79	0.987455
$447$	0	0
$448$	9006.44i	0.949809i
$449$	13100.4	1.37694	0.688472	−	0.725263i	$-0.258282\pi$
$449$	13100.4	1.37694	0.688472	+	0.725263i	$0.258282\pi$
$450$	0	0
$451$	18072.9	1.88696
$452$	− 1571.94i	− 0.163579i
$453$	0	0
$454$	676.227	0.0699050
$455$	0	0
$456$	0	0
$457$	− 1063.47i	− 0.108855i	−0.998518	−	0.0544277i	$-0.982667\pi$
$457$	− 1063.47i	− 0.108855i	0.998518	−	0.0544277i	$-0.0173334\pi$
$458$	− 273.386i	− 0.0278919i
$459$	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
$463$	− 11866.0i	− 1.19106i	−0.803335	−	0.595528i	$-0.796943\pi$
$463$	− 11866.0i	− 1.19106i	0.803335	−	0.595528i	$-0.203057\pi$
$464$	639.540	0.0639868
$465$	0	0
$466$	11608.6	1.15399
$467$	5687.16i	0.563534i	0.959483	+	0.281767i	$0.0909205\pi$
$467$	5687.16i	0.563534i	−0.959483	+	0.281767i	$0.909080\pi$
$468$	0	0
$469$	9102.71	0.896214
$470$	0	0
$471$	0	0
$472$	− 434.887i	− 0.0424096i
$473$	− 16234.2i	− 1.57812i
$474$	0	0
$475$	0	0
$476$	2139.53	0.206019
$477$	0	0
$478$	− 9515.49i	− 0.910520i
$479$	10556.5	1.00697	0.503485	−	0.864004i	$-0.332051\pi$
$479$	10556.5	1.00697	0.503485	+	0.864004i	$0.332051\pi$
$480$	0	0
$481$	1412.38	0.133886
$482$	11849.2i	1.11974i
$483$	0	0
$484$	9737.83	0.914522
$485$	0	0
$486$	0	0
$487$	962.879i	0.0895938i	0.998996	+	0.0447969i	$0.0142641\pi$
$487$	962.879i	0.0895938i	−0.998996	+	0.0447969i	$0.985736\pi$
$488$	− 18792.0i	− 1.74319i
$489$	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
$493$	− 872.742i	− 0.0797289i
$494$	−6191.80	−0.563932
$495$	0	0
$496$	−3789.98	−0.343095
$497$	7010.97i	0.632767i
$498$	0	0
$499$	−4036.80	−0.362148	−0.181074	−	0.983469i	$-0.557957\pi$
$499$	−4036.80	−0.362148	−0.181074	+	0.983469i	$0.557957\pi$
$500$	0	0
$501$	0	0
$502$	5724.44i	0.508953i
$503$	13228.9i	1.17266i	0.810073	+	0.586329i	$0.199427\pi$
$503$	13228.9i	1.17266i	−0.810073	+	0.586329i	$0.800573\pi$
$504$	0	0
$505$	0	0
$506$	−30239.4	−2.65673
$507$	0	0
$508$	6342.39i	0.553933i
$509$	5516.83	0.480411	0.240206	−	0.970722i	$-0.422785\pi$
$509$	5516.83	0.480411	0.240206	+	0.970722i	$0.422785\pi$
$510$	0	0
$511$	3772.97	0.326627
$512$	9677.50i	0.835331i
$513$	0	0
$514$	−10862.5	−0.932149
$515$	0	0
$516$	0	0
$517$	20562.9i	1.74923i
$518$	734.897i	0.0623350i
$519$	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
$523$	1005.75i	0.0840888i	0.999116	+	0.0420444i	$0.0133871\pi$
$523$	1005.75i	0.0840888i	−0.999116	+	0.0420444i	$0.986613\pi$
$524$	−5759.14	−0.480132
$525$	0	0
$526$	9877.50	0.818783
$527$	5171.96i	0.427503i
$528$	0	0
$529$	−29645.0	−2.43651
$530$	0	0
$531$	0	0
$532$	2043.29i	0.166518i
$533$	− 19666.4i	− 1.59821i
$534$	0	0
$535$	0	0
$536$	13075.3	1.05367
$537$	0	0
$538$	13181.9i	1.05634i
$539$	−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	$0.289660\pi$
$542$	− 10633.0i	− 0.842669i
$543$	0	0
$544$	5287.29	0.416711
$545$	0	0
$546$	0	0
$547$	3115.98i	0.243564i	0.992557	+	0.121782i	$0.0388609\pi$
$547$	3115.98i	0.243564i	−0.992557	+	0.121782i	$0.961139\pi$
$548$	5271.90i	0.410957i
$549$	0	0
$550$	0	0
$551$	833.484	0.0644421
$552$	0	0
$553$	19391.6i	1.49116i
$554$	−7814.62	−0.599299
$555$	0	0
$556$	−7407.86	−0.565042
$557$	− 1843.45i	− 0.140232i	−0.997539	−	0.0701162i	$-0.977663\pi$
$557$	− 1843.45i	− 0.140232i	0.997539	−	0.0701162i	$-0.0223370\pi$
$558$	0	0
$559$	−17665.7	−1.33663
$560$	0	0
$561$	0	0
$562$	967.115i	0.0725895i
$563$	− 13957.3i	− 1.04481i	−0.852696	−	0.522407i	$-0.825034\pi$
$563$	− 13957.3i	− 1.04481i	0.852696	−	0.522407i	$-0.174966\pi$
$564$	0	0
$565$	0	0
$566$	−13891.0	−1.03160
$567$	0	0
$568$	10070.7i	0.743939i
$569$	20049.7	1.47720	0.738601	−	0.674143i	$-0.235487\pi$
$569$	20049.7	1.47720	0.738601	+	0.674143i	$0.235487\pi$
$570$	0	0
$571$	−1270.87	−0.0931421	−0.0465711	−	0.998915i	$-0.514829\pi$
$571$	−1270.87	−0.0931421	−0.0465711	+	0.998915i	$0.514829\pi$
$572$	− 15093.2i	− 1.10328i
$573$	0	0
$574$	10232.9	0.744102
$575$	0	0
$576$	0	0
$577$	211.623i	0.0152686i	0.999971	+	0.00763431i	$0.00243010\pi$
$577$	211.623i	0.0152686i	−0.999971	+	0.00763431i	$0.997570\pi$
$578$	− 7278.81i	− 0.523804i
$579$	0	0
$580$	0	0
$581$	4338.07	0.309765
$582$	0	0
$583$	− 19317.9i	− 1.37232i
$584$	5419.57	0.384013
$585$	0	0
$586$	2356.82	0.166142
$587$	− 11135.1i	− 0.782956i	−0.920188	−	0.391478i	$-0.871964\pi$
$587$	− 11135.1i	− 0.782956i	0.920188	−	0.391478i	$-0.128036\pi$
$588$	0	0
$589$	−4939.31	−0.345536
$590$	0	0
$591$	0	0
$592$	573.308i	0.0398021i
$593$	20593.7i	1.42611i	0.701109	+	0.713054i	$0.252689\pi$
$593$	20593.7i	1.42611i	−0.701109	+	0.713054i	$0.747311\pi$
$594$	0	0
$595$	0	0
$596$	3386.21	0.232726
$597$	0	0
$598$	32905.7i	2.25019i
$599$	2510.98	0.171278	0.0856392	−	0.996326i	$-0.472707\pi$
$599$	2510.98	0.171278	0.0856392	+	0.996326i	$0.472707\pi$
$600$	0	0
$601$	21276.4	1.44406	0.722031	−	0.691861i	$-0.243209\pi$
$601$	21276.4	1.44406	0.722031	+	0.691861i	$0.243209\pi$
$602$	− 9191.89i	− 0.622315i
$603$	0	0
$604$	−8628.13	−0.581248
$605$	0	0
$606$	0	0
$607$	21993.1i	1.47063i	0.677725	+	0.735316i	$0.262966\pi$
$607$	21993.1i	1.47063i	−0.677725	+	0.735316i	$0.737034\pi$
$608$	5049.45i	0.336813i
$609$	0	0
$610$	0	0
$611$	22376.0	1.48156
$612$	0	0
$613$	13319.8i	0.877622i	0.898579	+	0.438811i	$0.144600\pi$
$613$	13319.8i	0.877622i	−0.898579	+	0.438811i	$0.855400\pi$
$614$	−22160.1	−1.45653
$615$	0	0
$616$	28089.5	1.83727
$617$	− 7815.28i	− 0.509938i	−0.966949	−	0.254969i	$-0.917935\pi$
$617$	− 7815.28i	− 0.509938i	0.966949	−	0.254969i	$-0.0820652\pi$
$618$	0	0
$619$	18163.9	1.17943	0.589716	−	0.807611i	$-0.299240\pi$
$619$	18163.9	1.17943	0.589716	+	0.807611i	$0.299240\pi$
$620$	0	0
$621$	0	0
$622$	− 7966.53i	− 0.513551i
$623$	27797.4i	1.78761i
$624$	0	0
$625$	0	0
$626$	4233.21	0.270277
$627$	0	0
$628$	3965.13i	0.251952i
$629$	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	$0.470992\pi$
$632$	27854.5i	1.75315i
$633$	0	0
$634$	9894.82	0.619832
$635$	0	0
$636$	0	0
$637$	3667.89i	0.228143i
$638$	− 3203.49i	− 0.198789i
$639$	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
$643$	− 11316.8i	− 0.694079i	−0.937851	−	0.347039i	$-0.887187\pi$
$643$	− 11316.8i	− 0.694079i	0.937851	−	0.347039i	$-0.112813\pi$
$644$	10858.8	0.664438
$645$	0	0
$646$	−3429.82	−0.208893
$647$	11244.8i	0.683272i	0.939832	+	0.341636i	$0.110981\pi$
$647$	11244.8i	0.683272i	−0.939832	+	0.341636i	$0.889019\pi$
$648$	0	0
$649$	−1183.07	−0.0715559
$650$	0	0
$651$	0	0
$652$	− 4123.19i	− 0.247664i
$653$	19898.7i	1.19249i	0.802803	+	0.596244i	$0.203341\pi$
$653$	19898.7i	1.19249i	−0.802803	+	0.596244i	$0.796659\pi$
$654$	0	0
$655$	0	0
$656$	7982.92	0.475123
$657$	0	0
$658$	11642.8i	0.689792i
$659$	−21502.2	−1.27103	−0.635514	−	0.772089i	$-0.719212\pi$
$659$	−21502.2	−1.27103	−0.635514	+	0.772089i	$0.719212\pi$
$660$	0	0
$661$	11903.3	0.700434	0.350217	−	0.936669i	$-0.386108\pi$
$661$	11903.3	0.700434	0.350217	+	0.936669i	$0.386108\pi$
$662$	17021.3i	0.999321i
$663$	0	0
$664$	6231.29	0.364188
$665$	0	0
$666$	0	0
$667$	− 4429.46i	− 0.257136i
$668$	12404.1i	0.718456i
$669$	0	0
$670$	0	0
$671$	−51122.1	−2.94121
$672$	0	0
$673$	11771.3i	0.674221i	0.941465	+	0.337110i	$0.109450\pi$
$673$	11771.3i	0.674221i	−0.941465	+	0.337110i	$0.890550\pi$
$674$	876.035	0.0500647
$675$	0	0
$676$	−9602.98	−0.546369
$677$	− 25739.9i	− 1.46125i	−0.682780	−	0.730624i	$-0.739229\pi$
$677$	− 25739.9i	− 1.46125i	0.682780	−	0.730624i	$-0.260771\pi$
$678$	0	0
$679$	−7826.65	−0.442355
$680$	0	0
$681$	0	0
$682$	18984.2i	1.06590i
$683$	− 29200.8i	− 1.63592i	−0.575272	−	0.817962i	$-0.695104\pi$
$683$	− 29200.8i	− 1.63592i	0.575272	−	0.817962i	$-0.304896\pi$
$684$	0	0
$685$	0	0
$686$	−14889.2	−0.828679
$687$	0	0
$688$	− 7170.78i	− 0.397360i
$689$	−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	$0.454628\pi$
$692$	− 2506.15i	− 0.137673i
$693$	0	0
$694$	−20077.6	−1.09818
$695$	0	0
$696$	0	0
$697$	− 10893.8i	− 0.592012i
$698$	19647.0i	1.06540i
$699$	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
$703$	747.167i	0.0400852i
$704$	35194.1	1.88413
$705$	0	0
$706$	−11571.0	−0.616826
$707$	− 21602.4i	− 1.14914i
$708$	0	0
$709$	−726.443	−0.0384797	−0.0192399	−	0.999815i	$-0.506125\pi$
$709$	−726.443	−0.0384797	−0.0192399	+	0.999815i	$0.506125\pi$
$710$	0	0
$711$	0	0
$712$	39928.8i	2.10168i
$713$	26249.5i	1.37875i
$714$	0	0
$715$	0	0
$716$	−5303.04	−0.276793
$717$	0	0
$718$	− 29716.8i	− 1.54460i
$719$	−2949.11	−0.152967	−0.0764835	−	0.997071i	$-0.524369\pi$
$719$	−2949.11	−0.152967	−0.0764835	+	0.997071i	$0.524369\pi$
$720$	0	0
$721$	7413.49	0.382930
$722$	11900.2i	0.613409i
$723$	0	0
$724$	5503.06	0.282486
$725$	0	0
$726$	0	0
$727$	34433.2i	1.75661i	0.478100	+	0.878306i	$0.341326\pi$
$727$	34433.2i	1.75661i	−0.478100	+	0.878306i	$0.658674\pi$
$728$	− 30566.2i	− 1.55613i
$729$	0	0
$730$	0	0
$731$	−9785.54	−0.495118
$732$	0	0
$733$	− 1382.56i	− 0.0696670i	−0.999393	−	0.0348335i	$-0.988910\pi$
$733$	− 1382.56i	− 0.0696670i	0.999393	−	0.0348335i	$-0.0110901\pi$
$734$	−16115.6	−0.810406
$735$	0	0
$736$	26834.8	1.34395
$737$	− 35570.3i	− 1.77781i
$738$	0	0
$739$	10335.3	0.514466	0.257233	−	0.966349i	$-0.417189\pi$
$739$	10335.3	0.514466	0.257233	+	0.966349i	$0.417189\pi$
$740$	0	0
$741$	0	0
$742$	− 10937.9i	− 0.541161i
$743$	4880.52i	0.240981i	0.992714	+	0.120491i	$0.0384468\pi$
$743$	4880.52i	0.240981i	−0.992714	+	0.120491i	$0.961553\pi$
$744$	0	0
$745$	0	0
$746$	−16250.5	−0.797552
$747$	0	0
$748$	− 8360.55i	− 0.408679i
$749$	−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	$0.675702\pi$
$752$	9082.77i	0.440445i
$753$	0	0
$754$	−3485.95	−0.168370
$755$	0	0
$756$	0	0
$757$	− 39391.6i	− 1.89130i	−0.325191	−	0.945648i	$-0.605429\pi$
$757$	− 39391.6i	− 1.89130i	0.325191	−	0.945648i	$-0.394571\pi$
$758$	− 3438.17i	− 0.164749i
$759$	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
$763$	− 20388.5i	− 0.967381i
$764$	−8211.63	−0.388856
$765$	0	0
$766$	−6468.74	−0.305124
$767$	1287.39i	0.0606063i
$768$	0	0
$769$	−2034.58	−0.0954081	−0.0477041	−	0.998862i	$-0.515190\pi$
$769$	−2034.58	−0.0954081	−0.0477041	+	0.998862i	$0.515190\pi$
$770$	0	0
$771$	0	0
$772$	− 1651.07i	− 0.0769731i
$773$	− 2737.54i	− 0.127377i	−0.997970	−	0.0636884i	$-0.979714\pi$
$773$	− 2737.54i	− 0.127377i	0.997970	−	0.0636884i	$-0.0202864\pi$
$774$	0	0
$775$	0	0
$776$	−11242.4	−0.520073
$777$	0	0
$778$	− 880.376i	− 0.0405694i
$779$	10403.8	0.478503
$780$	0	0
$781$	27396.5	1.25522
$782$	18227.4i	0.833519i
$783$	0	0
$784$	−1488.86	−0.0678233
$785$	0	0
$786$	0	0
$787$	− 18449.1i	− 0.835629i	−0.908532	−	0.417814i	$-0.862796\pi$
$787$	− 18449.1i	− 0.835629i	0.908532	−	0.417814i	$-0.137204\pi$
$788$	4135.13i	0.186939i
$789$	0	0
$790$	0	0
$791$	−8660.29	−0.389285
$792$	0	0
$793$	55629.8i	2.49114i
$794$	2949.53	0.131832
$795$	0	0
$796$	−5512.18	−0.245445
$797$	35731.8i	1.58806i	0.607878	+	0.794031i	$0.292021\pi$
$797$	35731.8i	1.58806i	−0.607878	+	0.794031i	$0.707979\pi$
$798$	0	0
$799$	12394.7	0.548803
$800$	0	0
$801$	0	0
$802$	13924.1i	0.613064i
$803$	− 14743.5i	− 0.647929i
$804$	0	0
$805$	0	0
$806$	20658.1	0.902792
$807$	0	0
$808$	− 31030.1i	− 1.35103i
$809$	−21421.1	−0.930933	−0.465467	−	0.885065i	$-0.654113\pi$
$809$	−21421.1	−0.930933	−0.465467	+	0.885065i	$0.654113\pi$
$810$	0	0
$811$	−27879.1	−1.20711	−0.603555	−	0.797321i	$-0.706250\pi$
$811$	−27879.1	−1.20711	−0.603555	+	0.797321i	$0.706250\pi$
$812$	1150.36i	0.0497164i
$813$	0	0
$814$	2871.73	0.123654
$815$	0	0
$816$	0	0
$817$	− 9345.36i	− 0.400187i
$818$	27705.8i	1.18424i
$819$	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
$823$	9804.65i	0.415272i	0.978206	+	0.207636i	$0.0665769\pi$
$823$	9804.65i	0.415272i	−0.978206	+	0.207636i	$0.933423\pi$
$824$	10648.9	0.450208
$825$	0	0
$826$	−669.862	−0.0282173
$827$	− 21534.9i	− 0.905491i	−0.891640	−	0.452745i	$-0.850445\pi$
$827$	− 21534.9i	− 0.905491i	0.891640	−	0.452745i	$-0.149555\pi$
$828$	0	0
$829$	2671.33	0.111917	0.0559585	−	0.998433i	$-0.482179\pi$
$829$	2671.33	0.111917	0.0559585	+	0.998433i	$0.482179\pi$
$830$	0	0
$831$	0	0
$832$	− 38297.3i	− 1.59582i
$833$	2031.76i	0.0845092i
$834$	0	0
$835$	0	0
$836$	7984.47	0.330321
$837$	0	0
$838$	12180.9i	0.502129i
$839$	20755.8	0.854076	0.427038	−	0.904234i	$-0.359557\pi$
$839$	20755.8	0.854076	0.427038	+	0.904234i	$0.359557\pi$
$840$	0	0
$841$	−23919.8	−0.980760
$842$	− 13181.8i	− 0.539517i
$843$	0	0
$844$	−10384.0	−0.423497
$845$	0	0
$846$	0	0
$847$	− 53648.7i	− 2.17638i
$848$	− 8532.84i	− 0.345541i
$849$	0	0
$850$	0	0
$851$	3970.74	0.159947
$852$	0	0
$853$	− 15242.4i	− 0.611831i	−0.952059	−	0.305915i	$-0.901038\pi$
$853$	− 15242.4i	− 0.611831i	0.952059	−	0.305915i	$-0.0989624\pi$
$854$	−28945.6	−1.15983
$855$	0	0
$856$	−40980.7	−1.63632
$857$	30965.0i	1.23424i	0.786868	+	0.617121i	$0.211701\pi$
$857$	30965.0i	1.23424i	−0.786868	+	0.617121i	$0.788299\pi$
$858$	0	0
$859$	−15617.3	−0.620320	−0.310160	−	0.950684i	$-0.600383\pi$
$859$	−15617.3	−0.620320	−0.310160	+	0.950684i	$0.600383\pi$
$860$	0	0
$861$	0	0
$862$	− 27677.4i	− 1.09361i
$863$	17853.6i	0.704223i	0.935958	+	0.352112i	$0.114536\pi$
$863$	17853.6i	0.704223i	−0.935958	+	0.352112i	$0.885464\pi$
$864$	0	0
$865$	0	0
$866$	9614.63	0.377273
$867$	0	0
$868$	− 6817.15i	− 0.266577i
$869$	75775.7	2.95802
$870$	0	0
$871$	−38706.7	−1.50577
$872$	− 29286.4i	− 1.13734i
$873$	0	0
$874$	−17407.5	−0.673705
$875$	0	0
$876$	0	0
$877$	31782.1i	1.22372i	0.790965	+	0.611862i	$0.209579\pi$
$877$	31782.1i	1.22372i	−0.790965	+	0.611862i	$0.790421\pi$
$878$	15518.9i	0.596513i
$879$	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
$883$	− 5259.06i	− 0.200432i	−0.994966	−	0.100216i	$-0.968047\pi$
$883$	− 5259.06i	− 0.200432i	0.994966	−	0.100216i	$-0.0319534\pi$
$884$	−9097.74	−0.346142
$885$	0	0
$886$	19309.4	0.732182
$887$	− 12791.9i	− 0.484229i	−0.970248	−	0.242114i	$-0.922159\pi$
$887$	− 12791.9i	− 0.484229i	0.970248	−	0.242114i	$-0.0778410\pi$
$888$	0	0
$889$	34942.2	1.31825
$890$	0	0
$891$	0	0
$892$	13051.1i	0.489891i
$893$	11837.2i	0.443579i
$894$	0	0
$895$	0	0
$896$	1969.28	0.0734254
$897$	0	0
$898$	− 28985.2i	− 1.07711i
$899$	−2780.81	−0.103165
$900$	0	0
$901$	−11644.3	−0.430551
$902$	− 39986.8i	− 1.47607i
$903$	0	0
$904$	−12439.8	−0.457679
$905$	0	0
$906$	0	0
$907$	19969.8i	0.731076i	0.930796	+	0.365538i	$0.119115\pi$
$907$	19969.8i	0.731076i	−0.930796	+	0.365538i	$0.880885\pi$
$908$	948.899i	0.0346810i
$909$	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
$913$	− 16951.7i	− 0.614479i
$914$	−2352.96	−0.0851521
$915$	0	0
$916$	383.622	0.0138376
$917$	31728.9i	1.14262i
$918$	0	0
$919$	14818.3	0.531895	0.265947	−	0.963988i	$-0.414315\pi$
$919$	14818.3	0.531895	0.265947	+	0.963988i	$0.414315\pi$
$920$	0	0
$921$	0	0
$922$	− 18486.8i	− 0.660335i
$923$	− 29812.2i	− 1.06314i
$924$	0	0
$925$	0	0
$926$	−26253.9	−0.931702
$927$	0	0
$928$	2842.82i	0.100560i
$929$	2229.09	0.0787234	0.0393617	−	0.999225i	$-0.487468\pi$
$929$	2229.09	0.0787234	0.0393617	+	0.999225i	$0.487468\pi$
$930$	0	0
$931$	−1940.36	−0.0683059
$932$	16289.5i	0.572512i
$933$	0	0
$934$	12583.0	0.440824
$935$	0	0
$936$	0	0
$937$	17737.2i	0.618408i	0.950996	+	0.309204i	$0.100063\pi$
$937$	17737.2i	0.618408i	−0.950996	+	0.309204i	$0.899937\pi$
$938$	− 20140.1i	− 0.701063i
$939$	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
$943$	− 55289.8i	− 1.90931i
$944$	−522.573	−0.0180173
$945$	0	0
$946$	−35918.8	−1.23448
$947$	17768.1i	0.609700i	0.952400	+	0.304850i	$0.0986063\pi$
$947$	17768.1i	0.609700i	−0.952400	+	0.304850i	$0.901394\pi$
$948$	0	0
$949$	−16043.5	−0.548782
$950$	0	0
$951$	0	0
$952$	− 16931.6i	− 0.576423i
$953$	− 48689.0i	− 1.65497i	−0.561485	−	0.827487i	$-0.689770\pi$
$953$	− 48689.0i	− 1.65497i	0.561485	−	0.827487i	$-0.310230\pi$
$954$	0	0
$955$	0	0
$956$	13352.4	0.451723
$957$	0	0
$958$	− 23356.6i	− 0.787701i
$959$	29044.5	0.977994
$960$	0	0
$961$	−13311.7	−0.446835
$962$	− 3124.94i	− 0.104732i
$963$	0	0
$964$	−16627.0	−0.555520
$965$	0	0
$966$	0	0
$967$	25690.1i	0.854330i	0.904174	+	0.427165i	$0.140488\pi$
$967$	25690.1i	0.854330i	−0.904174	+	0.427165i	$0.859512\pi$
$968$	− 77062.1i	− 2.55875i
$969$	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
$973$	40812.2i	1.34469i
$974$	2130.40	0.0700847
$975$	0	0
$976$	−22581.0	−0.740575
$977$	32608.3i	1.06779i	0.845550	+	0.533896i	$0.179273\pi$
$977$	32608.3i	1.06779i	−0.845550	+	0.533896i	$0.820727\pi$
$978$	0	0
$979$	108623.	3.54607
$980$	0	0
$981$	0	0
$982$	− 17229.4i	− 0.559890i
$983$	− 29544.1i	− 0.958606i	−0.877649	−	0.479303i	$-0.840889\pi$
$983$	− 29544.1i	− 0.958606i	0.877649	−	0.479303i	$-0.159111\pi$
$984$	0	0
$985$	0	0
$986$	−1930.97	−0.0623679
$987$	0	0
$988$	− 8688.49i	− 0.279775i
$989$	−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	$0.812977\pi$
$992$	− 16846.8i	− 0.539201i
$993$	0	0
$994$	15512.0	0.494981
$995$	0	0
$996$	0	0
$997$	− 7361.72i	− 0.233849i	−0.993141	−	0.116925i	$-0.962696\pi$
$997$	− 7361.72i	− 0.233849i	0.993141	−	0.116925i	$-0.0373036\pi$
$998$	8931.56i	0.283290i
$999$	0	0

Twists

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

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

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.21254i q^{2} +3.10469 q^{4} -17.1047i q^{7} -24.5695i q^{8} -66.8393 q^{11} +72.7328i q^{13} -37.8447 q^{14} -29.5234 q^{16} +40.2889i q^{17} -38.4766 q^{19} +147.884i q^{22} +204.480i q^{23} +160.924 q^{26} -53.1047i q^{28} -21.6621 q^{29} +128.372 q^{31} -131.234i q^{32} +89.1406 q^{34} -19.4187i q^{37} +85.1308i q^{38} -270.393 q^{41} +242.884i q^{43} -207.515 q^{44} +452.419 q^{46} -307.646i q^{47} +50.4297 q^{49} +225.813i q^{52} +289.019i q^{53} -420.254 q^{56} +47.9282i q^{58} +17.7003 q^{59} +764.851 q^{61} -284.027i q^{62} -526.548 q^{64} +532.176i q^{67} +125.084i q^{68} -409.886 q^{71} +220.581i q^{73} -42.9647 q^{74} -119.458 q^{76} +1143.27i q^{77} -1133.70 q^{79} +598.253i q^{82} +253.619i q^{83} +537.390 q^{86} +1642.21i q^{88} -1625.13 q^{89} +1244.07 q^{91} +634.846i q^{92} -680.678 q^{94} -457.573i q^{97} -111.578i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 52 q^{4} + 148 q^{16} - 692 q^{19} + 1488 q^{31} - 1592 q^{34} + 3312 q^{46} + 1556 q^{49} + 356 q^{61} - 140 q^{64} + 8188 q^{76} - 4152 q^{79} + 5496 q^{91} + 2392 q^{94}+O(q^{100}) \) 8 * q - 52 * q^4 + 148 * q^16 - 692 * q^19 + 1488 * q^31 - 1592 * q^34 + 3312 * q^46 + 1556 * q^49 + 356 * q^61 - 140 * q^64 + 8188 * q^76 - 4152 * q^79 + 5496 * q^91 + 2392 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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