LMFDB - Embedded newform 324.5.c.b.161.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,5,Mod(161,324)] 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("324.161"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	324.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,52] 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:	no
Analytic conductor:	$33.4918680392$
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} - 4x^{7} - 130x^{6} + 404x^{5} + 7007x^{4} - 14692x^{3} - 164750x^{2} + 172164x + 1445046 $ x^8 - 4x^7 - 130x^6 + 404x^5 + 7007x^4 - 14692x^3 - 164750x^2 + 172164*x + 1445046
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{20} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$-4.23219 - 0.517638i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.5.c.b.161.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-16.4027i q^{5} -47.6885 q^{7} +109.967i q^{11} -34.6696 q^{13} +208.116i q^{17} +272.343 q^{19} -85.7457i q^{23} +355.950 q^{25} -637.603i q^{29} +1099.27 q^{31} +782.221i q^{35} +219.002 q^{37} -3310.80i q^{41} +2216.40 q^{43} +1823.78i q^{47} -126.810 q^{49} +4216.32i q^{53} +1803.77 q^{55} -4578.21i q^{59} +455.999 q^{61} +568.676i q^{65} +6977.24 q^{67} +7613.81i q^{71} +9842.32 q^{73} -5244.18i q^{77} +4735.00 q^{79} -4505.70i q^{83} +3413.67 q^{85} +5240.55i q^{89} +1653.34 q^{91} -4467.16i q^{95} +9041.72 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 52 q^{7} + 220 q^{13} - 308 q^{19} - 328 q^{25} - 1472 q^{31} + 1228 q^{37} + 388 q^{43} + 3432 q^{49} - 36 q^{55} + 1804 q^{61} - 2180 q^{67} - 4544 q^{73} - 5012 q^{79} - 5364 q^{85} - 1756 q^{91}+ \cdots - 2528 q^{97}+O(q^{100}) $ 8 * q + 52 * q^7 + 220 * q^13 - 308 * q^19 - 328 * q^25 - 1472 * q^31 + 1228 * q^37 + 388 * q^43 + 3432 * q^49 - 36 * q^55 + 1804 * q^61 - 2180 * q^67 - 4544 * q^73 - 5012 * q^79 - 5364 * q^85 - 1756 * q^91 - 2528 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 16.4027i	− 0.656109i	−0.944659	−	0.328055i	$-0.893607\pi$
$5$	− 16.4027i	− 0.656109i	0.944659	−	0.328055i	$-0.106393\pi$
$6$	0	0
$7$	−47.6885	−0.973234	−0.486617	−	0.873615i	$-0.661769\pi$
$7$	−47.6885	−0.973234	−0.486617	+	0.873615i	$0.661769\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	109.967i	0.908821i	0.890792	+	0.454411i	$0.150150\pi$
$11$	109.967i	0.908821i	−0.890792	+	0.454411i	$0.849850\pi$
$12$	0	0
$13$	−34.6696	−0.205146	−0.102573	−	0.994725i	$-0.532707\pi$
$13$	−34.6696	−0.205146	−0.102573	+	0.994725i	$0.532707\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	208.116i	0.720124i	0.932928	+	0.360062i	$0.117245\pi$
$17$	208.116i	0.720124i	−0.932928	+	0.360062i	$0.882755\pi$
$18$	0	0
$19$	272.343	0.754412	0.377206	−	0.926129i	$-0.376885\pi$
$19$	272.343	0.754412	0.377206	+	0.926129i	$0.376885\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 85.7457i	− 0.162090i	−0.996710	−	0.0810450i	$-0.974174\pi$
$23$	− 85.7457i	− 0.162090i	0.996710	−	0.0810450i	$-0.0258258\pi$
$24$	0	0
$25$	355.950	0.569520
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 637.603i	− 0.758148i	−0.925366	−	0.379074i	$-0.876243\pi$
$29$	− 637.603i	− 0.758148i	0.925366	−	0.379074i	$-0.123757\pi$
$30$	0	0
$31$	1099.27	1.14388	0.571941	−	0.820295i	$-0.306191\pi$
$31$	1099.27	1.14388	0.571941	+	0.820295i	$0.306191\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	782.221i	0.638548i
$36$	0	0
$37$	219.002	0.159972	0.0799860	−	0.996796i	$-0.474512\pi$
$37$	219.002	0.159972	0.0799860	+	0.996796i	$0.474512\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3310.80i	− 1.96954i	−0.173856	−	0.984771i	$-0.555623\pi$
$41$	− 3310.80i	− 1.96954i	0.173856	−	0.984771i	$-0.444377\pi$
$42$	0	0
$43$	2216.40	1.19870	0.599351	−	0.800486i	$-0.295425\pi$
$43$	2216.40	1.19870	0.599351	+	0.800486i	$0.295425\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1823.78i	0.825612i	0.910819	+	0.412806i	$0.135451\pi$
$47$	1823.78i	0.825612i	−0.910819	+	0.412806i	$0.864549\pi$
$48$	0	0
$49$	−126.810	−0.0528156
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4216.32i	1.50100i	0.660868	+	0.750502i	$0.270188\pi$
$53$	4216.32i	1.50100i	−0.660868	+	0.750502i	$0.729812\pi$
$54$	0	0
$55$	1803.77	0.596286
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4578.21i	− 1.31520i	−0.753368	−	0.657600i	$-0.771572\pi$
$59$	− 4578.21i	− 1.31520i	0.753368	−	0.657600i	$-0.228428\pi$
$60$	0	0
$61$	455.999	0.122547	0.0612737	−	0.998121i	$-0.480484\pi$
$61$	455.999	0.122547	0.0612737	+	0.998121i	$0.480484\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	568.676i	0.134598i
$66$	0	0
$67$	6977.24	1.55430	0.777148	−	0.629317i	$-0.216665\pi$
$67$	6977.24	1.55430	0.777148	+	0.629317i	$0.216665\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7613.81i	1.51038i	0.655508	+	0.755188i	$0.272455\pi$
$71$	7613.81i	1.51038i	−0.655508	+	0.755188i	$0.727545\pi$
$72$	0	0
$73$	9842.32	1.84694	0.923468	−	0.383675i	$-0.125342\pi$
$73$	9842.32	1.84694	0.923468	+	0.383675i	$0.125342\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5244.18i	− 0.884496i
$78$	0	0
$79$	4735.00	0.758693	0.379346	−	0.925255i	$-0.376149\pi$
$79$	4735.00	0.758693	0.379346	+	0.925255i	$0.376149\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4505.70i	− 0.654043i	−0.945017	−	0.327022i	$-0.893955\pi$
$83$	− 4505.70i	− 0.654043i	0.945017	−	0.327022i	$-0.106045\pi$
$84$	0	0
$85$	3413.67	0.472480
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5240.55i	0.661603i	0.943700	+	0.330801i	$0.107319\pi$
$89$	5240.55i	0.661603i	−0.943700	+	0.330801i	$0.892681\pi$
$90$	0	0
$91$	1653.34	0.199655
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 4467.16i	− 0.494977i
$96$	0	0
$97$	9041.72	0.960965	0.480482	−	0.877004i	$-0.340462\pi$
$97$	9041.72	0.960965	0.480482	+	0.877004i	$0.340462\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10842.4i	1.06287i	0.847098	+	0.531436i	$0.178347\pi$
$101$	10842.4i	1.06287i	−0.847098	+	0.531436i	$0.821653\pi$
$102$	0	0
$103$	−1145.32	−0.107957	−0.0539786	−	0.998542i	$-0.517190\pi$
$103$	−1145.32	−0.107957	−0.0539786	+	0.998542i	$0.517190\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6886.22i	0.601470i	0.953708	+	0.300735i	$0.0972319\pi$
$107$	6886.22i	0.601470i	−0.953708	+	0.300735i	$0.902768\pi$
$108$	0	0
$109$	−11759.8	−0.989802	−0.494901	−	0.868949i	$-0.664796\pi$
$109$	−11759.8	−0.989802	−0.494901	+	0.868949i	$0.664796\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 20938.8i	− 1.63982i	−0.572496	−	0.819908i	$-0.694025\pi$
$113$	− 20938.8i	− 1.63982i	0.572496	−	0.819908i	$-0.305975\pi$
$114$	0	0
$115$	−1406.46	−0.106349
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 9924.73i	− 0.700850i
$120$	0	0
$121$	2548.17	0.174044
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 16090.3i	− 1.02978i
$126$	0	0
$127$	8797.12	0.545422	0.272711	−	0.962096i	$-0.412080\pi$
$127$	8797.12	0.545422	0.272711	+	0.962096i	$0.412080\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15778.0i	0.919408i	0.888072	+	0.459704i	$0.152044\pi$
$131$	15778.0i	0.919408i	−0.888072	+	0.459704i	$0.847956\pi$
$132$	0	0
$133$	−12987.6	−0.734219
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7277.21i	0.387725i	0.981029	+	0.193863i	$0.0621016\pi$
$137$	7277.21i	0.387725i	−0.981029	+	0.193863i	$0.937898\pi$
$138$	0	0
$139$	−3991.03	−0.206564	−0.103282	−	0.994652i	$-0.532934\pi$
$139$	−3991.03	−0.206564	−0.103282	+	0.994652i	$0.532934\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 3812.53i	− 0.186441i
$144$	0	0
$145$	−10458.4	−0.497428
$146$	0	0
$147$	0	0
$148$	0	0
$149$	34396.9i	1.54934i	0.632365	+	0.774670i	$0.282084\pi$
$149$	34396.9i	1.54934i	−0.632365	+	0.774670i	$0.717916\pi$
$150$	0	0
$151$	4812.68	0.211073	0.105537	−	0.994415i	$-0.466344\pi$
$151$	4812.68	0.211073	0.105537	+	0.994415i	$0.466344\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 18031.0i	− 0.750512i
$156$	0	0
$157$	−12322.9	−0.499933	−0.249967	−	0.968254i	$-0.580420\pi$
$157$	−12322.9	−0.499933	−0.249967	+	0.968254i	$0.580420\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4089.08i	0.157752i
$162$	0	0
$163$	15529.6	0.584502	0.292251	−	0.956342i	$-0.405596\pi$
$163$	15529.6	0.584502	0.292251	+	0.956342i	$0.405596\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	43357.0i	1.55463i	0.629114	+	0.777313i	$0.283418\pi$
$167$	43357.0i	1.55463i	−0.629114	+	0.777313i	$0.716582\pi$
$168$	0	0
$169$	−27359.0	−0.957915
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18903.5i	0.631611i	0.948824	+	0.315806i	$0.102275\pi$
$173$	18903.5i	0.631611i	−0.948824	+	0.315806i	$0.897725\pi$
$174$	0	0
$175$	−16974.7	−0.554277
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 59520.4i	− 1.85763i	−0.370540	−	0.928817i	$-0.620827\pi$
$179$	− 59520.4i	− 1.85763i	0.370540	−	0.928817i	$-0.379173\pi$
$180$	0	0
$181$	−40098.7	−1.22398	−0.611988	−	0.790867i	$-0.709630\pi$
$181$	−40098.7	−1.22398	−0.611988	+	0.790867i	$0.709630\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 3592.23i	− 0.104959i
$186$	0	0
$187$	−22886.0	−0.654465
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 18607.8i	− 0.510068i	−0.966932	−	0.255034i	$-0.917913\pi$
$191$	− 18607.8i	− 0.510068i	0.966932	−	0.255034i	$-0.0820867\pi$
$192$	0	0
$193$	−48164.0	−1.29303	−0.646514	−	0.762902i	$-0.723774\pi$
$193$	−48164.0	−1.29303	−0.646514	+	0.762902i	$0.723774\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 37721.6i	− 0.971981i	−0.873964	−	0.485990i	$-0.838459\pi$
$197$	− 37721.6i	− 0.971981i	0.873964	−	0.485990i	$-0.161541\pi$
$198$	0	0
$199$	−35926.3	−0.907207	−0.453603	−	0.891204i	$-0.649862\pi$
$199$	−35926.3	−0.907207	−0.453603	+	0.891204i	$0.649862\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	30406.3i	0.737856i
$204$	0	0
$205$	−54306.2	−1.29223
$206$	0	0
$207$	0	0
$208$	0	0
$209$	29948.8i	0.685626i
$210$	0	0
$211$	44633.0	1.00252	0.501258	−	0.865298i	$-0.332871\pi$
$211$	44633.0	1.00252	0.501258	+	0.865298i	$0.332871\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 36355.0i	− 0.786480i
$216$	0	0
$217$	−52422.5	−1.11327
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 7215.30i	− 0.147730i
$222$	0	0
$223$	80998.9	1.62881	0.814403	−	0.580300i	$-0.197065\pi$
$223$	80998.9	1.62881	0.814403	+	0.580300i	$0.197065\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 45030.0i	− 0.873876i	−0.899492	−	0.436938i	$-0.856063\pi$
$227$	− 45030.0i	− 0.873876i	0.899492	−	0.436938i	$-0.143937\pi$
$228$	0	0
$229$	−73791.1	−1.40713	−0.703563	−	0.710633i	$-0.748409\pi$
$229$	−73791.1	−1.40713	−0.703563	+	0.710633i	$0.748409\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3288.93i	0.0605819i	0.999541	+	0.0302910i	$0.00964339\pi$
$233$	3288.93i	0.0605819i	−0.999541	+	0.0302910i	$0.990357\pi$
$234$	0	0
$235$	29914.9	0.541691
$236$	0	0
$237$	0	0
$238$	0	0
$239$	108501.i	1.89950i	0.313015	+	0.949748i	$0.398661\pi$
$239$	108501.i	1.89950i	−0.313015	+	0.949748i	$0.601339\pi$
$240$	0	0
$241$	−35305.4	−0.607864	−0.303932	−	0.952694i	$-0.598300\pi$
$241$	−35305.4	−0.607864	−0.303932	+	0.952694i	$0.598300\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2080.03i	0.0346528i
$246$	0	0
$247$	−9442.01	−0.154764
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11418.4i	0.181242i	0.995885	+	0.0906211i	$0.0288852\pi$
$251$	11418.4i	0.181242i	−0.995885	+	0.0906211i	$0.971115\pi$
$252$	0	0
$253$	9429.23	0.147311
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 45357.1i	− 0.686720i	−0.939204	−	0.343360i	$-0.888435\pi$
$257$	− 45357.1i	− 0.686720i	0.939204	−	0.343360i	$-0.111565\pi$
$258$	0	0
$259$	−10443.9	−0.155690
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 48250.3i	− 0.697571i	−0.937203	−	0.348785i	$-0.886594\pi$
$263$	− 48250.3i	− 0.697571i	0.937203	−	0.348785i	$-0.113406\pi$
$264$	0	0
$265$	69159.2	0.984823
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 35673.2i	− 0.492989i	−0.969144	−	0.246494i	$-0.920721\pi$
$269$	− 35673.2i	− 0.492989i	0.969144	−	0.246494i	$-0.0792787\pi$
$270$	0	0
$271$	54635.1	0.743931	0.371966	−	0.928247i	$-0.378684\pi$
$271$	54635.1	0.743931	0.371966	+	0.928247i	$0.378684\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	39142.9i	0.517592i
$276$	0	0
$277$	13626.5	0.177593	0.0887963	−	0.996050i	$-0.471698\pi$
$277$	13626.5	0.177593	0.0887963	+	0.996050i	$0.471698\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	91373.3i	1.15720i	0.815613	+	0.578598i	$0.196400\pi$
$281$	91373.3i	1.15720i	−0.815613	+	0.578598i	$0.803600\pi$
$282$	0	0
$283$	79167.6	0.988496	0.494248	−	0.869321i	$-0.335444\pi$
$283$	79167.6	0.988496	0.494248	+	0.869321i	$0.335444\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	157887.i	1.91683i
$288$	0	0
$289$	40208.7	0.481421
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 171342.i	− 1.99585i	−0.0643968	−	0.997924i	$-0.520512\pi$
$293$	− 171342.i	− 1.99585i	0.0643968	−	0.997924i	$-0.479488\pi$
$294$	0	0
$295$	−75095.1	−0.862915
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2972.77i	0.0332521i
$300$	0	0
$301$	−105697.	−1.16662
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 7479.63i	− 0.0804045i
$306$	0	0
$307$	−5479.22	−0.0581356	−0.0290678	−	0.999577i	$-0.509254\pi$
$307$	−5479.22	−0.0581356	−0.0290678	+	0.999577i	$0.509254\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 70200.0i	− 0.725799i	−0.931828	−	0.362900i	$-0.881787\pi$
$311$	− 70200.0i	− 0.725799i	0.931828	−	0.362900i	$-0.118213\pi$
$312$	0	0
$313$	77812.9	0.794260	0.397130	−	0.917762i	$-0.370006\pi$
$313$	77812.9	0.794260	0.397130	+	0.917762i	$0.370006\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	92341.9i	0.918925i	0.888197	+	0.459463i	$0.151958\pi$
$317$	92341.9i	0.918925i	−0.888197	+	0.459463i	$0.848042\pi$
$318$	0	0
$319$	70115.5	0.689021
$320$	0	0
$321$	0	0
$322$	0	0
$323$	56678.9i	0.543270i
$324$	0	0
$325$	−12340.7	−0.116835
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 86973.1i	− 0.803513i
$330$	0	0
$331$	190474.	1.73852	0.869261	−	0.494353i	$-0.164595\pi$
$331$	190474.	1.73852	0.869261	+	0.494353i	$0.164595\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 114446.i	− 1.01979i
$336$	0	0
$337$	5228.09	0.0460345	0.0230173	−	0.999735i	$-0.492673\pi$
$337$	5228.09	0.0460345	0.0230173	+	0.999735i	$0.492673\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	120884.i	1.03958i
$342$	0	0
$343$	120547.	1.02464
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 64714.7i	− 0.537457i	−0.963216	−	0.268729i	$-0.913396\pi$
$347$	− 64714.7i	− 0.537457i	0.963216	−	0.268729i	$-0.0866035\pi$
$348$	0	0
$349$	−938.071	−0.00770167	−0.00385084	−	0.999993i	$-0.501226\pi$
$349$	−938.071	−0.00770167	−0.00385084	+	0.999993i	$0.501226\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	74827.3i	0.600497i	0.953861	+	0.300248i	$0.0970696\pi$
$353$	74827.3i	0.600497i	−0.953861	+	0.300248i	$0.902930\pi$
$354$	0	0
$355$	124887.	0.990972
$356$	0	0
$357$	0	0
$358$	0	0
$359$	106644.i	0.827464i	0.910399	+	0.413732i	$0.135775\pi$
$359$	106644.i	0.827464i	−0.910399	+	0.413732i	$0.864225\pi$
$360$	0	0
$361$	−56150.5	−0.430863
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 161441.i	− 1.21179i
$366$	0	0
$367$	189532.	1.40718	0.703592	−	0.710604i	$-0.251578\pi$
$367$	189532.	1.40718	0.703592	+	0.710604i	$0.251578\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 201070.i	− 1.46083i
$372$	0	0
$373$	223453.	1.60608	0.803041	−	0.595924i	$-0.203214\pi$
$373$	223453.	1.60608	0.803041	+	0.595924i	$0.203214\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	22105.4i	0.155531i
$378$	0	0
$379$	82882.2	0.577010	0.288505	−	0.957478i	$-0.406842\pi$
$379$	82882.2	0.577010	0.288505	+	0.957478i	$0.406842\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20218.4i	0.137832i	0.997622	+	0.0689158i	$0.0219540\pi$
$383$	20218.4i	0.137832i	−0.997622	+	0.0689158i	$0.978046\pi$
$384$	0	0
$385$	−86018.8	−0.580326
$386$	0	0
$387$	0	0
$388$	0	0
$389$	59243.0i	0.391505i	0.980653	+	0.195753i	$0.0627150\pi$
$389$	59243.0i	0.391505i	−0.980653	+	0.195753i	$0.937285\pi$
$390$	0	0
$391$	17845.0	0.116725
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 77667.0i	− 0.497786i
$396$	0	0
$397$	226012.	1.43400	0.717002	−	0.697072i	$-0.245514\pi$
$397$	226012.	1.43400	0.717002	+	0.697072i	$0.245514\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 69340.7i	− 0.431220i	−0.976480	−	0.215610i	$-0.930826\pi$
$401$	− 69340.7i	− 0.431220i	0.976480	−	0.215610i	$-0.0691740\pi$
$402$	0	0
$403$	−38111.3	−0.234662
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24083.1i	0.145386i
$408$	0	0
$409$	−214964.	−1.28505	−0.642523	−	0.766266i	$-0.722112\pi$
$409$	−214964.	−1.28505	−0.642523	+	0.766266i	$0.722112\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	218328.i	1.28000i
$414$	0	0
$415$	−73905.9	−0.429124
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24509.0i	0.139604i	0.997561	+	0.0698021i	$0.0222368\pi$
$419$	24509.0i	0.139604i	−0.997561	+	0.0698021i	$0.977763\pi$
$420$	0	0
$421$	44234.3	0.249571	0.124786	−	0.992184i	$-0.460176\pi$
$421$	44234.3	0.249571	0.124786	+	0.992184i	$0.460176\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	74078.9i	0.410126i
$426$	0	0
$427$	−21745.9	−0.119267
$428$	0	0
$429$	0	0
$430$	0	0
$431$	72928.0i	0.392591i	0.980545	+	0.196295i	$0.0628911\pi$
$431$	72928.0i	0.392591i	−0.980545	+	0.196295i	$0.937109\pi$
$432$	0	0
$433$	−216643.	−1.15550	−0.577749	−	0.816215i	$-0.696069\pi$
$433$	−216643.	−1.15550	−0.577749	+	0.816215i	$0.696069\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 23352.2i	− 0.122283i
$438$	0	0
$439$	−11304.8	−0.0586591	−0.0293296	−	0.999570i	$-0.509337\pi$
$439$	−11304.8	−0.0586591	−0.0293296	+	0.999570i	$0.509337\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	305549.i	1.55695i	0.627678	+	0.778473i	$0.284005\pi$
$443$	305549.i	1.55695i	−0.627678	+	0.778473i	$0.715995\pi$
$444$	0	0
$445$	85959.4	0.434084
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 26446.7i	− 0.131184i	−0.997847	−	0.0655918i	$-0.979106\pi$
$449$	− 26446.7i	− 0.131184i	0.997847	−	0.0655918i	$-0.0208935\pi$
$450$	0	0
$451$	364080.	1.78996
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 27119.3i	− 0.130995i
$456$	0	0
$457$	−315288.	−1.50964	−0.754822	−	0.655929i	$-0.772277\pi$
$457$	−315288.	−1.50964	−0.754822	+	0.655929i	$0.772277\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6024.67i	0.0283486i	0.999900	+	0.0141743i	$0.00451197\pi$
$461$	6024.67i	0.0283486i	−0.999900	+	0.0141743i	$0.995488\pi$
$462$	0	0
$463$	−182943.	−0.853403	−0.426702	−	0.904392i	$-0.640325\pi$
$463$	−182943.	−0.853403	−0.426702	+	0.904392i	$0.640325\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 262279.i	− 1.20262i	−0.799014	−	0.601312i	$-0.794645\pi$
$467$	− 262279.i	− 1.20262i	0.799014	−	0.601312i	$-0.205355\pi$
$468$	0	0
$469$	−332734.	−1.51269
$470$	0	0
$471$	0	0
$472$	0	0
$473$	243732.i	1.08941i
$474$	0	0
$475$	96940.5	0.429653
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 244075.i	− 1.06378i	−0.846813	−	0.531891i	$-0.821482\pi$
$479$	− 244075.i	− 1.06378i	0.846813	−	0.531891i	$-0.178518\pi$
$480$	0	0
$481$	−7592.70	−0.0328176
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 148309.i	− 0.630498i
$486$	0	0
$487$	−184209.	−0.776701	−0.388350	−	0.921512i	$-0.626955\pi$
$487$	−184209.	−0.776701	−0.388350	+	0.921512i	$0.626955\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	162714.i	0.674937i	0.941337	+	0.337468i	$0.109571\pi$
$491$	162714.i	0.674937i	−0.941337	+	0.337468i	$0.890429\pi$
$492$	0	0
$493$	132695.	0.545961
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 363091.i	− 1.46995i
$498$	0	0
$499$	67122.8	0.269568	0.134784	−	0.990875i	$-0.456966\pi$
$499$	67122.8	0.269568	0.134784	+	0.990875i	$0.456966\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 263850.i	− 1.04285i	−0.853297	−	0.521425i	$-0.825401\pi$
$503$	− 263850.i	− 1.04285i	0.853297	−	0.521425i	$-0.174599\pi$
$504$	0	0
$505$	177844.	0.697360
$506$	0	0
$507$	0	0
$508$	0	0
$509$	169333.i	0.653592i	0.945095	+	0.326796i	$0.105969\pi$
$509$	169333.i	0.653592i	−0.945095	+	0.326796i	$0.894031\pi$
$510$	0	0
$511$	−469365.	−1.79750
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18786.3i	0.0708317i
$516$	0	0
$517$	−200556.	−0.750334
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 214222.i	− 0.789203i	−0.918852	−	0.394602i	$-0.870883\pi$
$521$	− 214222.i	− 0.789203i	0.918852	−	0.394602i	$-0.129117\pi$
$522$	0	0
$523$	−264574.	−0.967262	−0.483631	−	0.875272i	$-0.660682\pi$
$523$	−264574.	−0.967262	−0.483631	+	0.875272i	$0.660682\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	228776.i	0.823738i
$528$	0	0
$529$	272489.	0.973727
$530$	0	0
$531$	0	0
$532$	0	0
$533$	114784.i	0.404043i
$534$	0	0
$535$	112953.	0.394630
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 13945.0i	− 0.0479999i
$540$	0	0
$541$	−304222.	−1.03943	−0.519715	−	0.854340i	$-0.673962\pi$
$541$	−304222.	−1.03943	−0.519715	+	0.854340i	$0.673962\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	192894.i	0.649419i
$546$	0	0
$547$	237793.	0.794738	0.397369	−	0.917659i	$-0.369923\pi$
$547$	237793.	0.794738	0.397369	+	0.917659i	$0.369923\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 173646.i	− 0.571956i
$552$	0	0
$553$	−225805.	−0.738386
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 197058.i	− 0.635162i	−0.948231	−	0.317581i	$-0.897130\pi$
$557$	− 197058.i	− 0.635162i	0.948231	−	0.317581i	$-0.102870\pi$
$558$	0	0
$559$	−76841.8	−0.245909
$560$	0	0
$561$	0	0
$562$	0	0
$563$	380965.i	1.20190i	0.799287	+	0.600949i	$0.205211\pi$
$563$	380965.i	1.20190i	−0.799287	+	0.600949i	$0.794789\pi$
$564$	0	0
$565$	−343454.	−1.07590
$566$	0	0
$567$	0	0
$568$	0	0
$569$	376.317i	0.00116233i	1.00000		0.000581165i	$0.000184991\pi$
$569$	376.317i	0.00116233i	−1.00000		0.000581165i	$0.999815\pi$
$570$	0	0
$571$	−617575.	−1.89416	−0.947082	−	0.320993i	$-0.895983\pi$
$571$	−617575.	−1.89416	−0.947082	+	0.320993i	$0.895983\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 30521.2i	− 0.0923136i
$576$	0	0
$577$	−83037.0	−0.249414	−0.124707	−	0.992194i	$-0.539799\pi$
$577$	−83037.0	−0.249414	−0.124707	+	0.992194i	$0.539799\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	214870.i	0.636537i
$582$	0	0
$583$	−463658.	−1.36414
$584$	0	0
$585$	0	0
$586$	0	0
$587$	602180.i	1.74763i	0.486256	+	0.873816i	$0.338362\pi$
$587$	602180.i	1.74763i	−0.486256	+	0.873816i	$0.661638\pi$
$588$	0	0
$589$	299378.	0.862958
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 500703.i	− 1.42387i	−0.702245	−	0.711935i	$-0.747819\pi$
$593$	− 500703.i	− 1.42387i	0.702245	−	0.711935i	$-0.252181\pi$
$594$	0	0
$595$	−162793.	−0.459834
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 423568.i	− 1.18051i	−0.807217	−	0.590254i	$-0.799027\pi$
$599$	− 423568.i	− 1.18051i	0.807217	−	0.590254i	$-0.200973\pi$
$600$	0	0
$601$	−114621.	−0.317334	−0.158667	−	0.987332i	$-0.550720\pi$
$601$	−114621.	−0.317334	−0.158667	+	0.987332i	$0.550720\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 41797.0i	− 0.114192i
$606$	0	0
$607$	−306023.	−0.830571	−0.415286	−	0.909691i	$-0.636318\pi$
$607$	−306023.	−0.830571	−0.415286	+	0.909691i	$0.636318\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 63229.6i	− 0.169371i
$612$	0	0
$613$	648468.	1.72571	0.862854	−	0.505453i	$-0.168675\pi$
$613$	648468.	1.72571	0.862854	+	0.505453i	$0.168675\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	255392.i	0.670869i	0.942064	+	0.335434i	$0.108883\pi$
$617$	255392.i	0.670869i	−0.942064	+	0.335434i	$0.891117\pi$
$618$	0	0
$619$	−383024.	−0.999644	−0.499822	−	0.866128i	$-0.666601\pi$
$619$	−383024.	−0.999644	−0.499822	+	0.866128i	$0.666601\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 249914.i	− 0.643894i
$624$	0	0
$625$	−41455.4	−0.106126
$626$	0	0
$627$	0	0
$628$	0	0
$629$	45577.8i	0.115200i
$630$	0	0
$631$	−709379.	−1.78164	−0.890819	−	0.454358i	$-0.849869\pi$
$631$	−709379.	−1.78164	−0.890819	+	0.454358i	$0.849869\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 144297.i	− 0.357857i
$636$	0	0
$637$	4396.46	0.0108349
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 86994.3i	− 0.211726i	−0.994381	−	0.105863i	$-0.966239\pi$
$641$	− 86994.3i	− 0.211726i	0.994381	−	0.105863i	$-0.0337605\pi$
$642$	0	0
$643$	637468.	1.54183	0.770915	−	0.636938i	$-0.219799\pi$
$643$	637468.	1.54183	0.770915	+	0.636938i	$0.219799\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	567845.i	1.35651i	0.734829	+	0.678253i	$0.237262\pi$
$647$	567845.i	1.35651i	−0.734829	+	0.678253i	$0.762738\pi$
$648$	0	0
$649$	503454.	1.19528
$650$	0	0
$651$	0	0
$652$	0	0
$653$	418741.i	0.982018i	0.871155	+	0.491009i	$0.163372\pi$
$653$	418741.i	0.982018i	−0.871155	+	0.491009i	$0.836628\pi$
$654$	0	0
$655$	258802.	0.603232
$656$	0	0
$657$	0	0
$658$	0	0
$659$	408601.i	0.940867i	0.882435	+	0.470434i	$0.155902\pi$
$659$	408601.i	0.940867i	−0.882435	+	0.470434i	$0.844098\pi$
$660$	0	0
$661$	−119365.	−0.273197	−0.136598	−	0.990627i	$-0.543617\pi$
$661$	−119365.	−0.273197	−0.136598	+	0.990627i	$0.543617\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	213032.i	0.481728i
$666$	0	0
$667$	−54671.7	−0.122888
$668$	0	0
$669$	0	0
$670$	0	0
$671$	50145.0i	0.111374i
$672$	0	0
$673$	232363.	0.513024	0.256512	−	0.966541i	$-0.417427\pi$
$673$	232363.	0.513024	0.256512	+	0.966541i	$0.417427\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 726313.i	− 1.58470i	−0.610068	−	0.792349i	$-0.708858\pi$
$677$	− 726313.i	− 1.58470i	0.610068	−	0.792349i	$-0.291142\pi$
$678$	0	0
$679$	−431186.	−0.935243
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 688320.i	− 1.47553i	−0.675056	−	0.737766i	$-0.735881\pi$
$683$	− 688320.i	− 1.47553i	0.675056	−	0.737766i	$-0.264119\pi$
$684$	0	0
$685$	119366.	0.254390
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 146178.i	− 0.307924i
$690$	0	0
$691$	434535.	0.910058	0.455029	−	0.890477i	$-0.349629\pi$
$691$	434535.	0.910058	0.455029	+	0.890477i	$0.349629\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	65463.8i	0.135529i
$696$	0	0
$697$	689030.	1.41832
$698$	0	0
$699$	0	0
$700$	0	0
$701$	301886.i	0.614338i	0.951655	+	0.307169i	$0.0993817\pi$
$701$	301886.i	0.614338i	−0.951655	+	0.307169i	$0.900618\pi$
$702$	0	0
$703$	59643.5	0.120685
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 517056.i	− 1.03442i
$708$	0	0
$709$	930146.	1.85037	0.925185	−	0.379517i	$-0.123910\pi$
$709$	930146.	1.85037	0.925185	+	0.379517i	$0.123910\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 94257.7i	− 0.185412i
$714$	0	0
$715$	−62535.9	−0.122326
$716$	0	0
$717$	0	0
$718$	0	0
$719$	498988.i	0.965233i	0.875832	+	0.482617i	$0.160314\pi$
$719$	498988.i	0.965233i	−0.875832	+	0.482617i	$0.839686\pi$
$720$	0	0
$721$	54618.4	0.105068
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 226955.i	− 0.431781i
$726$	0	0
$727$	−817751.	−1.54722	−0.773610	−	0.633661i	$-0.781551\pi$
$727$	−817751.	−1.54722	−0.773610	+	0.633661i	$0.781551\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	461268.i	0.863215i
$732$	0	0
$733$	−201746.	−0.375488	−0.187744	−	0.982218i	$-0.560118\pi$
$733$	−201746.	−0.375488	−0.187744	+	0.982218i	$0.560118\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	767269.i	1.41258i
$738$	0	0
$739$	295134.	0.540418	0.270209	−	0.962802i	$-0.412907\pi$
$739$	295134.	0.540418	0.270209	+	0.962802i	$0.412907\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 943146.i	− 1.70845i	−0.519907	−	0.854223i	$-0.674033\pi$
$743$	− 943146.i	− 1.70845i	0.519907	−	0.854223i	$-0.325967\pi$
$744$	0	0
$745$	564203.	1.01654
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 328393.i	− 0.585371i
$750$	0	0
$751$	743629.	1.31849	0.659245	−	0.751929i	$-0.270876\pi$
$751$	743629.	1.31849	0.659245	+	0.751929i	$0.270876\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 78941.0i	− 0.138487i
$756$	0	0
$757$	−354723.	−0.619010	−0.309505	−	0.950898i	$-0.600163\pi$
$757$	−354723.	−0.619010	−0.309505	+	0.950898i	$0.600163\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 520653.i	− 0.899040i	−0.893270	−	0.449520i	$-0.851595\pi$
$761$	− 520653.i	− 0.899040i	0.893270	−	0.449520i	$-0.148405\pi$
$762$	0	0
$763$	560809.	0.963309
$764$	0	0
$765$	0	0
$766$	0	0
$767$	158725.i	0.269807i
$768$	0	0
$769$	760246.	1.28559	0.642794	−	0.766039i	$-0.277775\pi$
$769$	760246.	1.28559	0.642794	+	0.766039i	$0.277775\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 480432.i	− 0.804031i	−0.915633	−	0.402016i	$-0.868310\pi$
$773$	− 480432.i	− 0.804031i	0.915633	−	0.402016i	$-0.131690\pi$
$774$	0	0
$775$	391286.	0.651464
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 901672.i	− 1.48585i
$780$	0	0
$781$	−837271.	−1.37266
$782$	0	0
$783$	0	0
$784$	0	0
$785$	202129.i	0.328011i
$786$	0	0
$787$	−345949.	−0.558551	−0.279276	−	0.960211i	$-0.590094\pi$
$787$	−345949.	−0.558551	−0.279276	+	0.960211i	$0.590094\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	998539.i	1.59592i
$792$	0	0
$793$	−15809.3	−0.0251401
$794$	0	0
$795$	0	0
$796$	0	0
$797$	211316.i	0.332672i	0.986069	+	0.166336i	$0.0531937\pi$
$797$	211316.i	0.332672i	−0.986069	+	0.166336i	$0.946806\pi$
$798$	0	0
$799$	−379557.	−0.594543
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.08233e6i	1.67854i
$804$	0	0
$805$	67072.1	0.103502
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.05505e6i	1.61204i	0.591888	+	0.806020i	$0.298383\pi$
$809$	1.05505e6i	1.61204i	−0.591888	+	0.806020i	$0.701617\pi$
$810$	0	0
$811$	378708.	0.575788	0.287894	−	0.957662i	$-0.407045\pi$
$811$	378708.	0.575788	0.287894	+	0.957662i	$0.407045\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 254729.i	− 0.383498i
$816$	0	0
$817$	603621.	0.904315
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 712921.i	− 1.05768i	−0.848721	−	0.528841i	$-0.822627\pi$
$821$	− 712921.i	− 1.05768i	0.848721	−	0.528841i	$-0.177373\pi$
$822$	0	0
$823$	−637035.	−0.940511	−0.470255	−	0.882530i	$-0.655838\pi$
$823$	−637035.	−0.940511	−0.470255	+	0.882530i	$0.655838\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 149310.i	− 0.218313i	−0.994025	−	0.109156i	$-0.965185\pi$
$827$	− 149310.i	− 0.218313i	0.994025	−	0.109156i	$-0.0348149\pi$
$828$	0	0
$829$	1.19640e6	1.74088	0.870440	−	0.492274i	$-0.163834\pi$
$829$	1.19640e6	1.74088	0.870440	+	0.492274i	$0.163834\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 26391.2i	− 0.0380338i
$834$	0	0
$835$	711173.	1.02001
$836$	0	0
$837$	0	0
$838$	0	0
$839$	225799.i	0.320774i	0.987054	+	0.160387i	$0.0512741\pi$
$839$	225799.i	0.320774i	−0.987054	+	0.160387i	$0.948726\pi$
$840$	0	0
$841$	300744.	0.425211
$842$	0	0
$843$	0	0
$844$	0	0
$845$	448763.i	0.628497i
$846$	0	0
$847$	−121518.	−0.169385
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 18778.4i	− 0.0259299i
$852$	0	0
$853$	707190.	0.971936	0.485968	−	0.873977i	$-0.338467\pi$
$853$	707190.	0.971936	0.485968	+	0.873977i	$0.338467\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 970045.i	− 1.32078i	−0.750923	−	0.660390i	$-0.770391\pi$
$857$	− 970045.i	− 1.32078i	0.750923	−	0.660390i	$-0.229609\pi$
$858$	0	0
$859$	836206.	1.13325	0.566627	−	0.823975i	$-0.308248\pi$
$859$	836206.	1.13325	0.566627	+	0.823975i	$0.308248\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 103918.i	− 0.139531i	−0.997563	−	0.0697653i	$-0.977775\pi$
$863$	− 103918.i	− 0.139531i	0.997563	−	0.0697653i	$-0.0222250\pi$
$864$	0	0
$865$	310069.	0.414406
$866$	0	0
$867$	0	0
$868$	0	0
$869$	520696.i	0.689516i
$870$	0	0
$871$	−241898.	−0.318857
$872$	0	0
$873$	0	0
$874$	0	0
$875$	767320.i	1.00221i
$876$	0	0
$877$	842153.	1.09494	0.547472	−	0.836824i	$-0.315590\pi$
$877$	842153.	1.09494	0.547472	+	0.836824i	$0.315590\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	965973.i	1.24455i	0.782798	+	0.622276i	$0.213792\pi$
$881$	965973.i	1.24455i	−0.782798	+	0.622276i	$0.786208\pi$
$882$	0	0
$883$	854895.	1.09646	0.548228	−	0.836329i	$-0.315303\pi$
$883$	854895.	1.09646	0.548228	+	0.836329i	$0.315303\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 623326.i	− 0.792261i	−0.918194	−	0.396131i	$-0.870353\pi$
$887$	− 623326.i	− 0.792261i	0.918194	−	0.396131i	$-0.129647\pi$
$888$	0	0
$889$	−419521.	−0.530824
$890$	0	0
$891$	0	0
$892$	0	0
$893$	496692.i	0.622851i
$894$	0	0
$895$	−976298.	−1.21881
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 700898.i	− 0.867232i
$900$	0	0
$901$	−877483.	−1.08091
$902$	0	0
$903$	0	0
$904$	0	0
$905$	657728.i	0.803063i
$906$	0	0
$907$	−205672.	−0.250012	−0.125006	−	0.992156i	$-0.539895\pi$
$907$	−205672.	−0.250012	−0.125006	+	0.992156i	$0.539895\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1.63205e6i	− 1.96651i	−0.182222	−	0.983257i	$-0.558329\pi$
$911$	− 1.63205e6i	− 1.96651i	0.182222	−	0.983257i	$-0.441671\pi$
$912$	0	0
$913$	495481.	0.594409
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 752427.i	− 0.894799i
$918$	0	0
$919$	−1.53874e6	−1.82194	−0.910971	−	0.412471i	$-0.864666\pi$
$919$	−1.53874e6	−1.82194	−0.910971	+	0.412471i	$0.864666\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 263968.i	− 0.309847i
$924$	0	0
$925$	77953.7	0.0911074
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 158309.i	− 0.183432i	−0.995785	−	0.0917159i	$-0.970765\pi$
$929$	− 158309.i	− 0.183432i	0.995785	−	0.0917159i	$-0.0292352\pi$
$930$	0	0
$931$	−34535.8	−0.0398447
$932$	0	0
$933$	0	0
$934$	0	0
$935$	375392.i	0.429400i
$936$	0	0
$937$	518596.	0.590677	0.295339	−	0.955393i	$-0.404568\pi$
$937$	518596.	0.590677	0.295339	+	0.955393i	$0.404568\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 51485.9i	− 0.0581445i	−0.999577	−	0.0290723i	$-0.990745\pi$
$941$	− 51485.9i	− 0.0581445i	0.999577	−	0.0290723i	$-0.00925529\pi$
$942$	0	0
$943$	−283887.	−0.319243
$944$	0	0
$945$	0	0
$946$	0	0
$947$	377819.i	0.421293i	0.977562	+	0.210646i	$0.0675569\pi$
$947$	377819.i	0.421293i	−0.977562	+	0.210646i	$0.932443\pi$
$948$	0	0
$949$	−341229.	−0.378891
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.32438e6i	− 1.45823i	−0.684389	−	0.729117i	$-0.739931\pi$
$953$	− 1.32438e6i	− 1.45823i	0.684389	−	0.729117i	$-0.260069\pi$
$954$	0	0
$955$	−305219.	−0.334661
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 347039.i	− 0.377347i
$960$	0	0
$961$	284876.	0.308467
$962$	0	0
$963$	0	0
$964$	0	0
$965$	790021.i	0.848368i
$966$	0	0
$967$	−1.17148e6	−1.25280	−0.626402	−	0.779500i	$-0.715473\pi$
$967$	−1.17148e6	−1.25280	−0.626402	+	0.779500i	$0.715473\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 860593.i	− 0.912766i	−0.889783	−	0.456383i	$-0.849145\pi$
$971$	− 860593.i	− 0.912766i	0.889783	−	0.456383i	$-0.150855\pi$
$972$	0	0
$973$	190326.	0.201036
$974$	0	0
$975$	0	0
$976$	0	0
$977$	115797.i	0.121313i	0.998159	+	0.0606565i	$0.0193194\pi$
$977$	115797.i	0.121313i	−0.998159	+	0.0606565i	$0.980681\pi$
$978$	0	0
$979$	−576290.	−0.601279
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 74121.2i	− 0.0767071i	−0.999264	−	0.0383535i	$-0.987789\pi$
$983$	− 74121.2i	− 0.0767071i	0.999264	−	0.0383535i	$-0.0122113\pi$
$984$	0	0
$985$	−618737.	−0.637726
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 190047.i	− 0.194298i
$990$	0	0
$991$	−1.30451e6	−1.32831	−0.664154	−	0.747596i	$-0.731208\pi$
$991$	−1.30451e6	−1.32831	−0.664154	+	0.747596i	$0.731208\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	589289.i	0.595227i
$996$	0	0
$997$	−224787.	−0.226142	−0.113071	−	0.993587i	$-0.536069\pi$
$997$	−224787.	−0.226142	−0.113071	+	0.993587i	$0.536069\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.c.b.161.4	✓	8
3.2	odd	2	inner	324.5.c.b.161.5	yes	8
4.3	odd	2		1296.5.e.f.161.4		8
9.2	odd	6		324.5.g.f.53.4		16
9.4	even	3		324.5.g.f.269.4		16
9.5	odd	6		324.5.g.f.269.5		16
9.7	even	3		324.5.g.f.53.5		16
12.11	even	2		1296.5.e.f.161.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.5.c.b.161.4	✓	8	1.1	even	1	trivial
324.5.c.b.161.5	yes	8	3.2	odd	2	inner
324.5.g.f.53.4		16	9.2	odd	6
324.5.g.f.53.5		16	9.7	even	3
324.5.g.f.269.4		16	9.4	even	3
324.5.g.f.269.5		16	9.5	odd	6
1296.5.e.f.161.4		8	4.3	odd	2
1296.5.e.f.161.5		8	12.11	even	2

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-16.4027i q^{5} -47.6885 q^{7} +109.967i q^{11} -34.6696 q^{13} +208.116i q^{17} +272.343 q^{19} -85.7457i q^{23} +355.950 q^{25} -637.603i q^{29} +1099.27 q^{31} +782.221i q^{35} +219.002 q^{37} -3310.80i q^{41} +2216.40 q^{43} +1823.78i q^{47} -126.810 q^{49} +4216.32i q^{53} +1803.77 q^{55} -4578.21i q^{59} +455.999 q^{61} +568.676i q^{65} +6977.24 q^{67} +7613.81i q^{71} +9842.32 q^{73} -5244.18i q^{77} +4735.00 q^{79} -4505.70i q^{83} +3413.67 q^{85} +5240.55i q^{89} +1653.34 q^{91} -4467.16i q^{95} +9041.72 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 52 q^{7} + 220 q^{13} - 308 q^{19} - 328 q^{25} - 1472 q^{31} + 1228 q^{37} + 388 q^{43} + 3432 q^{49} - 36 q^{55} + 1804 q^{61} - 2180 q^{67} - 4544 q^{73} - 5012 q^{79} - 5364 q^{85} - 1756 q^{91}+ \cdots - 2528 q^{97}+O(q^{100}) \) 8 * q + 52 * q^7 + 220 * q^13 - 308 * q^19 - 328 * q^25 - 1472 * q^31 + 1228 * q^37 + 388 * q^43 + 3432 * q^49 - 36 * q^55 + 1804 * q^61 - 2180 * q^67 - 4544 * q^73 - 5012 * q^79 - 5364 * q^85 - 1756 * q^91 - 2528 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more