LMFDB - Embedded newform 1134.3.b.c.323.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1134,3,Mod(323,1134)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1134, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1134.323"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1134.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$30.8992619785$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.20
Character	$\chi$	$=$	1134.323
Dual form			1134.3.b.c.323.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} -2.00000 q^{4} +3.19697i q^{5} +2.64575 q^{7} -2.82843i q^{8} -4.52120 q^{10} +18.8556i q^{11} +10.3719 q^{13} +3.74166i q^{14} +4.00000 q^{16} -24.4251i q^{17} -26.5935 q^{19} -6.39395i q^{20} -26.6659 q^{22} +40.8754i q^{23} +14.7794 q^{25} +14.6680i q^{26} -5.29150 q^{28} +21.7328i q^{29} -2.75433 q^{31} +5.65685i q^{32} +34.5423 q^{34} +8.45840i q^{35} +45.6422 q^{37} -37.6088i q^{38} +9.04241 q^{40} -14.8264i q^{41} -28.9107 q^{43} -37.7112i q^{44} -57.8065 q^{46} +29.7697i q^{47} +7.00000 q^{49} +20.9012i q^{50} -20.7437 q^{52} +18.1401i q^{53} -60.2809 q^{55} -7.48331i q^{56} -30.7349 q^{58} -53.5911i q^{59} -27.8674 q^{61} -3.89520i q^{62} -8.00000 q^{64} +33.1585i q^{65} -92.5552 q^{67} +48.8502i q^{68} -11.9620 q^{70} +81.9005i q^{71} -81.2899 q^{73} +64.5478i q^{74} +53.1869 q^{76} +49.8873i q^{77} -84.6130 q^{79} +12.7879i q^{80} +20.9677 q^{82} +115.359i q^{83} +78.0865 q^{85} -40.8860i q^{86} +53.3317 q^{88} +13.7346i q^{89} +27.4413 q^{91} -81.7507i q^{92} -42.1007 q^{94} -85.0186i q^{95} -111.586 q^{97} +9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 48 q^{4} + 96 q^{16} + 24 q^{19} - 48 q^{22} - 144 q^{25} + 120 q^{31} + 96 q^{34} - 168 q^{37} - 120 q^{43} + 168 q^{49} + 264 q^{55} - 192 q^{64} - 144 q^{67} + 24 q^{73} - 48 q^{76} - 24 q^{79}+ \cdots - 96 q^{97}+O(q^{100}) $ 24 * q - 48 * q^4 + 96 * q^16 + 24 * q^19 - 48 * q^22 - 144 * q^25 + 120 * q^31 + 96 * q^34 - 168 * q^37 - 120 * q^43 + 168 * q^49 + 264 * q^55 - 192 * q^64 - 144 * q^67 + 24 * q^73 - 48 * q^76 - 24 * q^79 - 288 * q^82 + 312 * q^85 + 96 * q^88 - 168 * q^91 - 96 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	3.19697i	0.639395i	0.947520	+	0.319697i	$0.103581\pi$
$5$	3.19697i	0.639395i	−0.947520	+	0.319697i	$0.896419\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	−4.52120	−0.452120
$11$	18.8556i	1.71415i	0.515195	+	0.857073i	$0.327720\pi$
$11$	18.8556i	1.71415i	−0.515195	+	0.857073i	$0.672280\pi$
$12$	0	0
$13$	10.3719	0.797835	0.398917	−	0.916987i	$-0.369386\pi$
$13$	10.3719	0.797835	0.398917	+	0.916987i	$0.369386\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 24.4251i	− 1.43677i	−0.695645	−	0.718386i	$-0.744881\pi$
$17$	− 24.4251i	− 1.43677i	0.695645	−	0.718386i	$-0.255119\pi$
$18$	0	0
$19$	−26.5935	−1.39966	−0.699828	−	0.714312i	$-0.746740\pi$
$19$	−26.5935	−1.39966	−0.699828	+	0.714312i	$0.746740\pi$
$20$	− 6.39395i	− 0.319697i
$21$	0	0
$22$	−26.6659	−1.21208
$23$	40.8754i	1.77719i	0.458692	+	0.888595i	$0.348318\pi$
$23$	40.8754i	1.77719i	−0.458692	+	0.888595i	$0.651682\pi$
$24$	0	0
$25$	14.7794	0.591175
$26$	14.6680i	0.564154i
$27$	0	0
$28$	−5.29150	−0.188982
$29$	21.7328i	0.749408i	0.927145	+	0.374704i	$0.122256\pi$
$29$	21.7328i	0.749408i	−0.927145	+	0.374704i	$0.877744\pi$
$30$	0	0
$31$	−2.75433	−0.0888492	−0.0444246	−	0.999013i	$-0.514145\pi$
$31$	−2.75433	−0.0888492	−0.0444246	+	0.999013i	$0.514145\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	34.5423	1.01595
$35$	8.45840i	0.241668i
$36$	0	0
$37$	45.6422	1.23357	0.616786	−	0.787131i	$-0.288434\pi$
$37$	45.6422	1.23357	0.616786	+	0.787131i	$0.288434\pi$
$38$	− 37.6088i	− 0.989706i
$39$	0	0
$40$	9.04241	0.226060
$41$	− 14.8264i	− 0.361619i	−0.983518	−	0.180809i	$-0.942128\pi$
$41$	− 14.8264i	− 0.361619i	0.983518	−	0.180809i	$-0.0578717\pi$
$42$	0	0
$43$	−28.9107	−0.672343	−0.336171	−	0.941801i	$-0.609132\pi$
$43$	−28.9107	−0.672343	−0.336171	+	0.941801i	$0.609132\pi$
$44$	− 37.7112i	− 0.857073i
$45$	0	0
$46$	−57.8065	−1.25666
$47$	29.7697i	0.633398i	0.948526	+	0.316699i	$0.102575\pi$
$47$	29.7697i	0.633398i	−0.948526	+	0.316699i	$0.897425\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	20.9012i	0.418024i
$51$	0	0
$52$	−20.7437	−0.398917
$53$	18.1401i	0.342266i	0.985248	+	0.171133i	$0.0547428\pi$
$53$	18.1401i	0.342266i	−0.985248	+	0.171133i	$0.945257\pi$
$54$	0	0
$55$	−60.2809	−1.09602
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	−30.7349	−0.529911
$59$	− 53.5911i	− 0.908324i	−0.890919	−	0.454162i	$-0.849939\pi$
$59$	− 53.5911i	− 0.908324i	0.890919	−	0.454162i	$-0.150061\pi$
$60$	0	0
$61$	−27.8674	−0.456843	−0.228422	−	0.973562i	$-0.573356\pi$
$61$	−27.8674	−0.456843	−0.228422	+	0.973562i	$0.573356\pi$
$62$	− 3.89520i	− 0.0628259i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	33.1585i	0.510131i
$66$	0	0
$67$	−92.5552	−1.38142	−0.690711	−	0.723131i	$-0.742702\pi$
$67$	−92.5552	−1.38142	−0.690711	+	0.723131i	$0.742702\pi$
$68$	48.8502i	0.718386i
$69$	0	0
$70$	−11.9620	−0.170885
$71$	81.9005i	1.15353i	0.816911	+	0.576764i	$0.195685\pi$
$71$	81.9005i	1.15353i	−0.816911	+	0.576764i	$0.804315\pi$
$72$	0	0
$73$	−81.2899	−1.11356	−0.556780	−	0.830660i	$-0.687964\pi$
$73$	−81.2899	−1.11356	−0.556780	+	0.830660i	$0.687964\pi$
$74$	64.5478i	0.872268i
$75$	0	0
$76$	53.1869	0.699828
$77$	49.8873i	0.647886i
$78$	0	0
$79$	−84.6130	−1.07105	−0.535525	−	0.844519i	$-0.679886\pi$
$79$	−84.6130	−1.07105	−0.535525	+	0.844519i	$0.679886\pi$
$80$	12.7879i	0.159849i
$81$	0	0
$82$	20.9677	0.255703
$83$	115.359i	1.38987i	0.719073	+	0.694935i	$0.244567\pi$
$83$	115.359i	1.38987i	−0.719073	+	0.694935i	$0.755433\pi$
$84$	0	0
$85$	78.0865	0.918664
$86$	− 40.8860i	− 0.475418i
$87$	0	0
$88$	53.3317	0.606042
$89$	13.7346i	0.154321i	0.997019	+	0.0771606i	$0.0245854\pi$
$89$	13.7346i	0.154321i	−0.997019	+	0.0771606i	$0.975415\pi$
$90$	0	0
$91$	27.4413	0.301553
$92$	− 81.7507i	− 0.888595i
$93$	0	0
$94$	−42.1007	−0.447880
$95$	− 85.0186i	− 0.894932i
$96$	0	0
$97$	−111.586	−1.15037	−0.575184	−	0.818024i	$-0.695070\pi$
$97$	−111.586	−1.15037	−0.575184	+	0.818024i	$0.695070\pi$
$98$	9.89949i	0.101015i
$99$	0	0
$100$	−29.5587	−0.295587
$101$	− 144.932i	− 1.43497i	−0.696576	−	0.717483i	$-0.745294\pi$
$101$	− 144.932i	− 1.43497i	0.696576	−	0.717483i	$-0.254706\pi$
$102$	0	0
$103$	−44.8103	−0.435051	−0.217525	−	0.976055i	$-0.569799\pi$
$103$	−44.8103	−0.435051	−0.217525	+	0.976055i	$0.569799\pi$
$104$	− 29.3360i	− 0.282077i
$105$	0	0
$106$	−25.6540	−0.242019
$107$	− 100.220i	− 0.936634i	−0.883560	−	0.468317i	$-0.844861\pi$
$107$	− 100.220i	− 0.936634i	0.883560	−	0.468317i	$-0.155139\pi$
$108$	0	0
$109$	−60.7449	−0.557292	−0.278646	−	0.960394i	$-0.589886\pi$
$109$	−60.7449	−0.557292	−0.278646	+	0.960394i	$0.589886\pi$
$110$	− 85.2500i	− 0.775000i
$111$	0	0
$112$	10.5830	0.0944911
$113$	14.5640i	0.128885i	0.997921	+	0.0644424i	$0.0205269\pi$
$113$	14.5640i	0.128885i	−0.997921	+	0.0644424i	$0.979473\pi$
$114$	0	0
$115$	−130.677	−1.13633
$116$	− 43.4656i	− 0.374704i
$117$	0	0
$118$	75.7892	0.642282
$119$	− 64.6228i	− 0.543049i
$120$	0	0
$121$	−234.534	−1.93830
$122$	− 39.4105i	− 0.323037i
$123$	0	0
$124$	5.50865	0.0444246
$125$	127.174i	1.01739i
$126$	0	0
$127$	221.852	1.74687	0.873433	−	0.486944i	$-0.161888\pi$
$127$	221.852	1.74687	0.873433	+	0.486944i	$0.161888\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	−46.8932	−0.360717
$131$	3.38630i	0.0258496i	0.999916	+	0.0129248i	$0.00411421\pi$
$131$	3.38630i	0.0258496i	−0.999916	+	0.0129248i	$0.995886\pi$
$132$	0	0
$133$	−70.3597	−0.529020
$134$	− 130.893i	− 0.976812i
$135$	0	0
$136$	−69.0847	−0.507976
$137$	21.5878i	0.157575i	0.996891	+	0.0787877i	$0.0251049\pi$
$137$	21.5878i	0.157575i	−0.996891	+	0.0787877i	$0.974895\pi$
$138$	0	0
$139$	−60.3228	−0.433977	−0.216988	−	0.976174i	$-0.569623\pi$
$139$	−60.3228	−0.433977	−0.216988	+	0.976174i	$0.569623\pi$
$140$	− 16.9168i	− 0.120834i
$141$	0	0
$142$	−115.825	−0.815668
$143$	195.568i	1.36761i
$144$	0	0
$145$	−69.4792	−0.479167
$146$	− 114.961i	− 0.787406i
$147$	0	0
$148$	−91.2844	−0.616786
$149$	15.5380i	0.104282i	0.998640	+	0.0521409i	$0.0166045\pi$
$149$	15.5380i	0.104282i	−0.998640	+	0.0521409i	$0.983396\pi$
$150$	0	0
$151$	26.8234	0.177638	0.0888192	−	0.996048i	$-0.471691\pi$
$151$	26.8234	0.177638	0.0888192	+	0.996048i	$0.471691\pi$
$152$	75.2177i	0.494853i
$153$	0	0
$154$	−70.5512	−0.458125
$155$	− 8.80550i	− 0.0568097i
$156$	0	0
$157$	106.134	0.676012	0.338006	−	0.941144i	$-0.390248\pi$
$157$	106.134	0.676012	0.338006	+	0.941144i	$0.390248\pi$
$158$	− 119.661i	− 0.757347i
$159$	0	0
$160$	−18.0848	−0.113030
$161$	108.146i	0.671715i
$162$	0	0
$163$	11.8953	0.0729774	0.0364887	−	0.999334i	$-0.488383\pi$
$163$	11.8953	0.0729774	0.0364887	+	0.999334i	$0.488383\pi$
$164$	29.6527i	0.180809i
$165$	0	0
$166$	−163.143	−0.982786
$167$	139.394i	0.834696i	0.908747	+	0.417348i	$0.137040\pi$
$167$	139.394i	0.834696i	−0.908747	+	0.417348i	$0.862960\pi$
$168$	0	0
$169$	−61.4247	−0.363460
$170$	110.431i	0.649594i
$171$	0	0
$172$	57.8215	0.336171
$173$	229.396i	1.32599i	0.748624	+	0.662994i	$0.230715\pi$
$173$	229.396i	1.32599i	−0.748624	+	0.662994i	$0.769285\pi$
$174$	0	0
$175$	39.1025	0.223443
$176$	75.4224i	0.428537i
$177$	0	0
$178$	−19.4236	−0.109122
$179$	− 202.541i	− 1.13152i	−0.824571	−	0.565758i	$-0.808584\pi$
$179$	− 202.541i	− 1.13152i	0.824571	−	0.565758i	$-0.191416\pi$
$180$	0	0
$181$	4.20404	0.0232267	0.0116134	−	0.999933i	$-0.496303\pi$
$181$	4.20404	0.0232267	0.0116134	+	0.999933i	$0.496303\pi$
$182$	38.8079i	0.213230i
$183$	0	0
$184$	115.613	0.628332
$185$	145.917i	0.788740i
$186$	0	0
$187$	460.551	2.46284
$188$	− 59.5395i	− 0.316699i
$189$	0	0
$190$	120.234	0.632813
$191$	271.910i	1.42361i	0.702376	+	0.711806i	$0.252123\pi$
$191$	271.910i	1.42361i	−0.702376	+	0.711806i	$0.747877\pi$
$192$	0	0
$193$	−144.553	−0.748981	−0.374490	−	0.927231i	$-0.622182\pi$
$193$	−144.553	−0.748981	−0.374490	+	0.927231i	$0.622182\pi$
$194$	− 157.806i	− 0.813433i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	− 53.0627i	− 0.269354i	−0.990890	−	0.134677i	$-0.957000\pi$
$197$	− 53.0627i	− 0.269354i	0.990890	−	0.134677i	$-0.0429996\pi$
$198$	0	0
$199$	−114.808	−0.576923	−0.288461	−	0.957491i	$-0.593144\pi$
$199$	−114.808	−0.576923	−0.288461	+	0.957491i	$0.593144\pi$
$200$	− 41.8024i	− 0.209012i
$201$	0	0
$202$	204.964	1.01467
$203$	57.4996i	0.283249i
$204$	0	0
$205$	47.3995	0.231217
$206$	− 63.3713i	− 0.307628i
$207$	0	0
$208$	41.4874	0.199459
$209$	− 501.436i	− 2.39921i
$210$	0	0
$211$	360.959	1.71071	0.855353	−	0.518045i	$-0.173340\pi$
$211$	360.959	1.71071	0.855353	+	0.518045i	$0.173340\pi$
$212$	− 36.2802i	− 0.171133i
$213$	0	0
$214$	141.732	0.662300
$215$	− 92.4269i	− 0.429892i
$216$	0	0
$217$	−7.28726	−0.0335818
$218$	− 85.9062i	− 0.394065i
$219$	0	0
$220$	120.562	0.548008
$221$	− 253.334i	− 1.14631i
$222$	0	0
$223$	−206.857	−0.927610	−0.463805	−	0.885937i	$-0.653516\pi$
$223$	−206.857	−0.927610	−0.463805	+	0.885937i	$0.653516\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	−20.5966	−0.0911353
$227$	− 131.641i	− 0.579916i	−0.957039	−	0.289958i	$-0.906359\pi$
$227$	− 131.641i	− 0.579916i	0.957039	−	0.289958i	$-0.0936413\pi$
$228$	0	0
$229$	236.528	1.03287	0.516436	−	0.856326i	$-0.327258\pi$
$229$	236.528	1.03287	0.516436	+	0.856326i	$0.327258\pi$
$230$	− 184.806i	− 0.803504i
$231$	0	0
$232$	61.4697	0.264956
$233$	155.574i	0.667699i	0.942626	+	0.333849i	$0.108348\pi$
$233$	155.574i	0.667699i	−0.942626	+	0.333849i	$0.891652\pi$
$234$	0	0
$235$	−95.1730	−0.404992
$236$	107.182i	0.454162i
$237$	0	0
$238$	91.3904	0.383993
$239$	− 59.5962i	− 0.249357i	−0.992197	−	0.124678i	$-0.960210\pi$
$239$	− 59.5962i	− 0.249357i	0.992197	−	0.124678i	$-0.0397899\pi$
$240$	0	0
$241$	−119.491	−0.495811	−0.247906	−	0.968784i	$-0.579742\pi$
$241$	−119.491	−0.495811	−0.247906	+	0.968784i	$0.579742\pi$
$242$	− 331.681i	− 1.37058i
$243$	0	0
$244$	55.7349	0.228422
$245$	22.3788i	0.0913421i
$246$	0	0
$247$	−275.823	−1.11669
$248$	7.79041i	0.0314129i
$249$	0	0
$250$	−179.851	−0.719402
$251$	− 483.177i	− 1.92501i	−0.271267	−	0.962504i	$-0.587443\pi$
$251$	− 483.177i	− 1.92501i	0.271267	−	0.962504i	$-0.412557\pi$
$252$	0	0
$253$	−770.730	−3.04636
$254$	313.746i	1.23522i
$255$	0	0
$256$	16.0000	0.0625000
$257$	158.001i	0.614788i	0.951582	+	0.307394i	$0.0994570\pi$
$257$	158.001i	0.614788i	−0.951582	+	0.307394i	$0.900543\pi$
$258$	0	0
$259$	120.758	0.466247
$260$	− 66.3171i	− 0.255066i
$261$	0	0
$262$	−4.78895	−0.0182784
$263$	282.878i	1.07558i	0.843078	+	0.537792i	$0.180741\pi$
$263$	282.878i	1.07558i	−0.843078	+	0.537792i	$0.819259\pi$
$264$	0	0
$265$	−57.9934	−0.218843
$266$	− 99.5036i	− 0.374074i
$267$	0	0
$268$	185.110	0.690711
$269$	125.436i	0.466303i	0.972440	+	0.233152i	$0.0749039\pi$
$269$	125.436i	0.466303i	−0.972440	+	0.233152i	$0.925096\pi$
$270$	0	0
$271$	418.065	1.54268	0.771338	−	0.636426i	$-0.219588\pi$
$271$	418.065	1.54268	0.771338	+	0.636426i	$0.219588\pi$
$272$	− 97.7005i	− 0.359193i
$273$	0	0
$274$	−30.5298	−0.111423
$275$	278.674i	1.01336i
$276$	0	0
$277$	456.711	1.64878	0.824389	−	0.566024i	$-0.191519\pi$
$277$	456.711	1.64878	0.824389	+	0.566024i	$0.191519\pi$
$278$	− 85.3093i	− 0.306868i
$279$	0	0
$280$	23.9240	0.0854427
$281$	− 255.330i	− 0.908649i	−0.890836	−	0.454324i	$-0.849881\pi$
$281$	− 255.330i	− 0.908649i	0.890836	−	0.454324i	$-0.150119\pi$
$282$	0	0
$283$	8.23376	0.0290946	0.0145473	−	0.999894i	$-0.495369\pi$
$283$	8.23376	0.0290946	0.0145473	+	0.999894i	$0.495369\pi$
$284$	− 163.801i	− 0.576764i
$285$	0	0
$286$	−276.574	−0.967043
$287$	− 39.2269i	− 0.136679i
$288$	0	0
$289$	−307.587	−1.06431
$290$	− 98.2585i	− 0.338822i
$291$	0	0
$292$	162.580	0.556780
$293$	107.000i	0.365189i	0.983188	+	0.182594i	$0.0584495\pi$
$293$	107.000i	0.365189i	−0.983188	+	0.182594i	$0.941550\pi$
$294$	0	0
$295$	171.329	0.580777
$296$	− 129.096i	− 0.436134i
$297$	0	0
$298$	−21.9740	−0.0737384
$299$	423.953i	1.41790i
$300$	0	0
$301$	−76.4906	−0.254122
$302$	37.9340i	0.125609i
$303$	0	0
$304$	−106.374	−0.349914
$305$	− 89.0915i	− 0.292103i
$306$	0	0
$307$	540.799	1.76156	0.880780	−	0.473526i	$-0.157019\pi$
$307$	540.799	1.76156	0.880780	+	0.473526i	$0.157019\pi$
$308$	− 99.7745i	− 0.323943i
$309$	0	0
$310$	12.4529	0.0401705
$311$	262.025i	0.842523i	0.906939	+	0.421261i	$0.138413\pi$
$311$	262.025i	0.842523i	−0.906939	+	0.421261i	$0.861587\pi$
$312$	0	0
$313$	604.880	1.93252	0.966262	−	0.257561i	$-0.0829188\pi$
$313$	604.880	1.93252	0.966262	+	0.257561i	$0.0829188\pi$
$314$	150.096i	0.478013i
$315$	0	0
$316$	169.226	0.535525
$317$	− 87.4826i	− 0.275970i	−0.990434	−	0.137985i	$-0.955937\pi$
$317$	− 87.4826i	− 0.275970i	0.990434	−	0.137985i	$-0.0440626\pi$
$318$	0	0
$319$	−409.786	−1.28459
$320$	− 25.5758i	− 0.0799243i
$321$	0	0
$322$	−152.942	−0.474974
$323$	649.549i	2.01099i
$324$	0	0
$325$	153.289	0.471660
$326$	16.8225i	0.0516028i
$327$	0	0
$328$	−41.9353	−0.127852
$329$	78.7633i	0.239402i
$330$	0	0
$331$	136.276	0.411709	0.205855	−	0.978583i	$-0.434003\pi$
$331$	136.276	0.411709	0.205855	+	0.978583i	$0.434003\pi$
$332$	− 230.718i	− 0.694935i
$333$	0	0
$334$	−197.133	−0.590219
$335$	− 295.897i	− 0.883273i
$336$	0	0
$337$	242.483	0.719534	0.359767	−	0.933042i	$-0.382856\pi$
$337$	242.483	0.719534	0.359767	+	0.933042i	$0.382856\pi$
$338$	− 86.8676i	− 0.257005i
$339$	0	0
$340$	−156.173	−0.459332
$341$	− 51.9345i	− 0.152301i
$342$	0	0
$343$	18.5203	0.0539949
$344$	81.7719i	0.237709i
$345$	0	0
$346$	−324.415	−0.937616
$347$	644.674i	1.85785i	0.370269	+	0.928924i	$0.379265\pi$
$347$	644.674i	1.85785i	−0.370269	+	0.928924i	$0.620735\pi$
$348$	0	0
$349$	605.831	1.73590	0.867952	−	0.496648i	$-0.165436\pi$
$349$	605.831	1.73590	0.867952	+	0.496648i	$0.165436\pi$
$350$	55.2993i	0.157998i
$351$	0	0
$352$	−106.663	−0.303021
$353$	− 163.042i	− 0.461875i	−0.972969	−	0.230938i	$-0.925821\pi$
$353$	− 163.042i	− 0.461875i	0.972969	−	0.230938i	$-0.0741793\pi$
$354$	0	0
$355$	−261.834	−0.737560
$356$	− 27.4692i	− 0.0771606i
$357$	0	0
$358$	286.437	0.800103
$359$	227.325i	0.633217i	0.948556	+	0.316608i	$0.102544\pi$
$359$	227.325i	0.633217i	−0.948556	+	0.316608i	$0.897456\pi$
$360$	0	0
$361$	346.212	0.959036
$362$	5.94541i	0.0164238i
$363$	0	0
$364$	−54.8827	−0.150777
$365$	− 259.882i	− 0.712005i
$366$	0	0
$367$	−675.506	−1.84062	−0.920308	−	0.391194i	$-0.872062\pi$
$367$	−675.506	−1.84062	−0.920308	+	0.391194i	$0.872062\pi$
$368$	163.501i	0.444298i
$369$	0	0
$370$	−206.358	−0.557723
$371$	47.9942i	0.129364i
$372$	0	0
$373$	444.784	1.19245	0.596225	−	0.802817i	$-0.296667\pi$
$373$	444.784	1.19245	0.596225	+	0.802817i	$0.296667\pi$
$374$	651.317i	1.74149i
$375$	0	0
$376$	84.2015	0.223940
$377$	225.410i	0.597903i
$378$	0	0
$379$	586.241	1.54681	0.773405	−	0.633912i	$-0.218552\pi$
$379$	586.241	1.54681	0.773405	+	0.633912i	$0.218552\pi$
$380$	170.037i	0.447466i
$381$	0	0
$382$	−384.539	−1.00665
$383$	− 150.256i	− 0.392313i	−0.980573	−	0.196157i	$-0.937154\pi$
$383$	− 150.256i	− 0.392313i	0.980573	−	0.196157i	$-0.0628461\pi$
$384$	0	0
$385$	−159.488	−0.414255
$386$	− 204.429i	− 0.529609i
$387$	0	0
$388$	223.171	0.575184
$389$	539.970i	1.38810i	0.719928	+	0.694048i	$0.244175\pi$
$389$	539.970i	1.38810i	−0.719928	+	0.694048i	$0.755825\pi$
$390$	0	0
$391$	998.386	2.55342
$392$	− 19.7990i	− 0.0505076i
$393$	0	0
$394$	75.0419	0.190462
$395$	− 270.505i	− 0.684824i
$396$	0	0
$397$	491.223	1.23734	0.618669	−	0.785652i	$-0.287672\pi$
$397$	491.223	1.23734	0.618669	+	0.785652i	$0.287672\pi$
$398$	− 162.363i	− 0.407946i
$399$	0	0
$400$	59.1175	0.147794
$401$	− 216.027i	− 0.538721i	−0.963039	−	0.269361i	$-0.913188\pi$
$401$	− 216.027i	− 0.538721i	0.963039	−	0.269361i	$-0.0868123\pi$
$402$	0	0
$403$	−28.5675	−0.0708870
$404$	289.863i	0.717483i
$405$	0	0
$406$	−81.3168	−0.200288
$407$	860.611i	2.11452i
$408$	0	0
$409$	105.752	0.258563	0.129282	−	0.991608i	$-0.458733\pi$
$409$	105.752	0.258563	0.129282	+	0.991608i	$0.458733\pi$
$410$	67.0330i	0.163495i
$411$	0	0
$412$	89.6205	0.217525
$413$	− 141.789i	− 0.343314i
$414$	0	0
$415$	−368.800	−0.888675
$416$	58.6721i	0.141039i
$417$	0	0
$418$	709.137	1.69650
$419$	13.3505i	0.0318628i	0.999873	+	0.0159314i	$0.00507134\pi$
$419$	13.3505i	0.0318628i	−0.999873	+	0.0159314i	$0.994929\pi$
$420$	0	0
$421$	−475.240	−1.12884	−0.564419	−	0.825489i	$-0.690900\pi$
$421$	−475.240	−1.12884	−0.564419	+	0.825489i	$0.690900\pi$
$422$	510.473i	1.20965i
$423$	0	0
$424$	51.3080	0.121009
$425$	− 360.988i	− 0.849383i
$426$	0	0
$427$	−73.7303	−0.172671
$428$	200.440i	0.468317i
$429$	0	0
$430$	130.711	0.303980
$431$	235.192i	0.545690i	0.962058	+	0.272845i	$0.0879646\pi$
$431$	235.192i	0.545690i	−0.962058	+	0.272845i	$0.912035\pi$
$432$	0	0
$433$	−21.0400	−0.0485913	−0.0242957	−	0.999705i	$-0.507734\pi$
$433$	−21.0400	−0.0485913	−0.0242957	+	0.999705i	$0.507734\pi$
$434$	− 10.3057i	− 0.0237460i
$435$	0	0
$436$	121.490	0.278646
$437$	− 1087.02i	− 2.48745i
$438$	0	0
$439$	−249.151	−0.567543	−0.283772	−	0.958892i	$-0.591586\pi$
$439$	−249.151	−0.567543	−0.283772	+	0.958892i	$0.591586\pi$
$440$	170.500i	0.387500i
$441$	0	0
$442$	358.268	0.810561
$443$	− 326.742i	− 0.737566i	−0.929515	−	0.368783i	$-0.879775\pi$
$443$	− 326.742i	− 0.737566i	0.929515	−	0.368783i	$-0.120225\pi$
$444$	0	0
$445$	−43.9091	−0.0986722
$446$	− 292.540i	− 0.655920i
$447$	0	0
$448$	−21.1660	−0.0472456
$449$	510.632i	1.13727i	0.822592	+	0.568633i	$0.192527\pi$
$449$	510.632i	1.13727i	−0.822592	+	0.568633i	$0.807473\pi$
$450$	0	0
$451$	279.560	0.619868
$452$	− 29.1280i	− 0.0644424i
$453$	0	0
$454$	186.168	0.410063
$455$	87.7292i	0.192811i
$456$	0	0
$457$	−181.097	−0.396274	−0.198137	−	0.980174i	$-0.563489\pi$
$457$	−181.097	−0.396274	−0.198137	+	0.980174i	$0.563489\pi$
$458$	334.501i	0.730351i
$459$	0	0
$460$	261.355	0.568163
$461$	− 296.193i	− 0.642502i	−0.946994	−	0.321251i	$-0.895897\pi$
$461$	− 296.193i	− 0.642502i	0.946994	−	0.321251i	$-0.104103\pi$
$462$	0	0
$463$	536.310	1.15834	0.579169	−	0.815208i	$-0.303377\pi$
$463$	536.310	1.15834	0.579169	+	0.815208i	$0.303377\pi$
$464$	86.9313i	0.187352i
$465$	0	0
$466$	−220.015	−0.472134
$467$	− 87.9169i	− 0.188259i	−0.995560	−	0.0941294i	$-0.969993\pi$
$467$	− 87.9169i	− 0.188259i	0.995560	−	0.0941294i	$-0.0300068\pi$
$468$	0	0
$469$	−244.878	−0.522128
$470$	− 134.595i	− 0.286372i
$471$	0	0
$472$	−151.578	−0.321141
$473$	− 545.130i	− 1.15249i
$474$	0	0
$475$	−393.034	−0.827441
$476$	129.246i	0.271524i
$477$	0	0
$478$	84.2818	0.176322
$479$	297.878i	0.621874i	0.950431	+	0.310937i	$0.100643\pi$
$479$	297.878i	0.621874i	−0.950431	+	0.310937i	$0.899357\pi$
$480$	0	0
$481$	473.394	0.984187
$482$	− 168.985i	− 0.350592i
$483$	0	0
$484$	469.068	0.969149
$485$	− 356.737i	− 0.735539i
$486$	0	0
$487$	−441.814	−0.907215	−0.453608	−	0.891202i	$-0.649863\pi$
$487$	−441.814	−0.907215	−0.453608	+	0.891202i	$0.649863\pi$
$488$	78.8210i	0.161519i
$489$	0	0
$490$	−31.6484	−0.0645886
$491$	333.309i	0.678838i	0.940635	+	0.339419i	$0.110230\pi$
$491$	333.309i	0.678838i	−0.940635	+	0.339419i	$0.889770\pi$
$492$	0	0
$493$	530.827	1.07673
$494$	− 390.073i	− 0.789622i
$495$	0	0
$496$	−11.0173	−0.0222123
$497$	216.688i	0.435993i
$498$	0	0
$499$	−844.487	−1.69236	−0.846179	−	0.532899i	$-0.821103\pi$
$499$	−844.487	−1.69236	−0.846179	+	0.532899i	$0.821103\pi$
$500$	− 254.347i	− 0.508694i
$501$	0	0
$502$	683.316	1.36119
$503$	60.3243i	0.119929i	0.998201	+	0.0599645i	$0.0190987\pi$
$503$	60.3243i	0.119929i	−0.998201	+	0.0599645i	$0.980901\pi$
$504$	0	0
$505$	463.343	0.917510
$506$	− 1089.98i	− 2.15410i
$507$	0	0
$508$	−443.704	−0.873433
$509$	392.469i	0.771059i	0.922696	+	0.385529i	$0.125981\pi$
$509$	392.469i	0.771059i	−0.922696	+	0.385529i	$0.874019\pi$
$510$	0	0
$511$	−215.073	−0.420886
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	−223.447	−0.434721
$515$	− 143.257i	− 0.278169i
$516$	0	0
$517$	−561.326	−1.08574
$518$	170.777i	0.329686i
$519$	0	0
$520$	93.7865	0.180359
$521$	− 45.5654i	− 0.0874577i	−0.999043	−	0.0437288i	$-0.986076\pi$
$521$	− 45.5654i	− 0.0874577i	0.999043	−	0.0437288i	$-0.0139238\pi$
$522$	0	0
$523$	445.063	0.850980	0.425490	−	0.904963i	$-0.360102\pi$
$523$	445.063	0.850980	0.425490	+	0.904963i	$0.360102\pi$
$524$	− 6.77259i	− 0.0129248i
$525$	0	0
$526$	−400.051	−0.760552
$527$	67.2747i	0.127656i
$528$	0	0
$529$	−1141.80	−2.15840
$530$	− 82.0151i	− 0.154745i
$531$	0	0
$532$	140.719	0.264510
$533$	− 153.777i	− 0.288512i
$534$	0	0
$535$	320.400	0.598879
$536$	261.786i	0.488406i
$537$	0	0
$538$	−177.393	−0.329726
$539$	131.989i	0.244878i
$540$	0	0
$541$	302.399	0.558964	0.279482	−	0.960151i	$-0.409837\pi$
$541$	302.399	0.558964	0.279482	+	0.960151i	$0.409837\pi$
$542$	591.233i	1.09084i
$543$	0	0
$544$	138.169	0.253988
$545$	− 194.200i	− 0.356330i
$546$	0	0
$547$	338.352	0.618559	0.309279	−	0.950971i	$-0.399912\pi$
$547$	338.352	0.618559	0.309279	+	0.950971i	$0.399912\pi$
$548$	− 43.1756i	− 0.0787877i
$549$	0	0
$550$	−394.104	−0.716554
$551$	− 577.951i	− 1.04891i
$552$	0	0
$553$	−223.865	−0.404819
$554$	645.887i	1.16586i
$555$	0	0
$556$	120.646	0.216988
$557$	250.404i	0.449559i	0.974410	+	0.224780i	$0.0721662\pi$
$557$	250.404i	0.449559i	−0.974410	+	0.224780i	$0.927834\pi$
$558$	0	0
$559$	−299.858	−0.536418
$560$	33.8336i	0.0604171i
$561$	0	0
$562$	361.092	0.642512
$563$	− 833.366i	− 1.48022i	−0.672484	−	0.740112i	$-0.734773\pi$
$563$	− 833.366i	− 1.48022i	0.672484	−	0.740112i	$-0.265227\pi$
$564$	0	0
$565$	−46.5606	−0.0824082
$566$	11.6443i	0.0205730i
$567$	0	0
$568$	231.650	0.407834
$569$	237.538i	0.417466i	0.977973	+	0.208733i	$0.0669339\pi$
$569$	237.538i	0.417466i	−0.977973	+	0.208733i	$0.933066\pi$
$570$	0	0
$571$	793.494	1.38966	0.694829	−	0.719175i	$-0.255480\pi$
$571$	793.494	1.38966	0.694829	+	0.719175i	$0.255480\pi$
$572$	− 391.135i	− 0.683803i
$573$	0	0
$574$	55.4752	0.0966467
$575$	604.112i	1.05063i
$576$	0	0
$577$	−164.298	−0.284745	−0.142372	−	0.989813i	$-0.545473\pi$
$577$	−164.298	−0.284745	−0.142372	+	0.989813i	$0.545473\pi$
$578$	− 434.993i	− 0.752584i
$579$	0	0
$580$	138.958	0.239584
$581$	305.212i	0.525321i
$582$	0	0
$583$	−342.043	−0.586694
$584$	229.923i	0.393703i
$585$	0	0
$586$	−151.321	−0.258227
$587$	919.758i	1.56688i	0.621468	+	0.783439i	$0.286536\pi$
$587$	919.758i	1.56688i	−0.621468	+	0.783439i	$0.713464\pi$
$588$	0	0
$589$	73.2470	0.124358
$590$	242.296i	0.410671i
$591$	0	0
$592$	182.569	0.308393
$593$	− 1156.90i	− 1.95092i	−0.220178	−	0.975460i	$-0.570664\pi$
$593$	− 1156.90i	− 1.95092i	0.220178	−	0.975460i	$-0.429336\pi$
$594$	0	0
$595$	206.597	0.347222
$596$	− 31.0760i	− 0.0521409i
$597$	0	0
$598$	−599.561	−1.00261
$599$	− 906.097i	− 1.51268i	−0.654177	−	0.756342i	$-0.726985\pi$
$599$	− 906.097i	− 1.51268i	0.654177	−	0.756342i	$-0.273015\pi$
$600$	0	0
$601$	1058.44	1.76112	0.880562	−	0.473932i	$-0.157166\pi$
$601$	1058.44	1.76112	0.880562	+	0.473932i	$0.157166\pi$
$602$	− 108.174i	− 0.179691i
$603$	0	0
$604$	−53.6468	−0.0888192
$605$	− 749.799i	− 1.23934i
$606$	0	0
$607$	−923.151	−1.52084	−0.760421	−	0.649430i	$-0.775007\pi$
$607$	−923.151	−1.52084	−0.760421	+	0.649430i	$0.775007\pi$
$608$	− 150.435i	− 0.247426i
$609$	0	0
$610$	125.994	0.206548
$611$	308.767i	0.505347i
$612$	0	0
$613$	−613.947	−1.00154	−0.500772	−	0.865579i	$-0.666951\pi$
$613$	−613.947	−1.00154	−0.500772	+	0.865579i	$0.666951\pi$
$614$	764.805i	1.24561i
$615$	0	0
$616$	141.102	0.229062
$617$	531.192i	0.860927i	0.902608	+	0.430464i	$0.141650\pi$
$617$	531.192i	0.860927i	−0.902608	+	0.430464i	$0.858350\pi$
$618$	0	0
$619$	421.781	0.681391	0.340695	−	0.940174i	$-0.389337\pi$
$619$	421.781	0.681391	0.340695	+	0.940174i	$0.389337\pi$
$620$	17.6110i	0.0284049i
$621$	0	0
$622$	−370.559	−0.595754
$623$	36.3383i	0.0583279i
$624$	0	0
$625$	−37.0863	−0.0593381
$626$	855.430i	1.36650i
$627$	0	0
$628$	−212.268	−0.338006
$629$	− 1114.82i	− 1.77236i
$630$	0	0
$631$	−552.102	−0.874964	−0.437482	−	0.899227i	$-0.644130\pi$
$631$	−552.102	−0.874964	−0.437482	+	0.899227i	$0.644130\pi$
$632$	239.322i	0.378673i
$633$	0	0
$634$	123.719	0.195141
$635$	709.255i	1.11694i
$636$	0	0
$637$	72.6030	0.113976
$638$	− 579.524i	− 0.908346i
$639$	0	0
$640$	36.1696	0.0565150
$641$	− 451.659i	− 0.704616i	−0.935884	−	0.352308i	$-0.885397\pi$
$641$	− 451.659i	− 0.704616i	0.935884	−	0.352308i	$-0.114603\pi$
$642$	0	0
$643$	−819.014	−1.27374	−0.636869	−	0.770972i	$-0.719771\pi$
$643$	−819.014	−1.27374	−0.636869	+	0.770972i	$0.719771\pi$
$644$	− 216.292i	− 0.335857i
$645$	0	0
$646$	−918.600	−1.42198
$647$	− 908.400i	− 1.40402i	−0.712168	−	0.702009i	$-0.752286\pi$
$647$	− 908.400i	− 1.40402i	0.712168	−	0.702009i	$-0.247714\pi$
$648$	0	0
$649$	1010.49	1.55700
$650$	216.784i	0.333514i
$651$	0	0
$652$	−23.7906	−0.0364887
$653$	− 230.798i	− 0.353443i	−0.984261	−	0.176721i	$-0.943451\pi$
$653$	− 230.798i	− 0.353443i	0.984261	−	0.176721i	$-0.0565491\pi$
$654$	0	0
$655$	−10.8259	−0.0165281
$656$	− 59.3055i	− 0.0904047i
$657$	0	0
$658$	−111.388	−0.169283
$659$	45.9156i	0.0696747i	0.999393	+	0.0348373i	$0.0110913\pi$
$659$	45.9156i	0.0696747i	−0.999393	+	0.0348373i	$0.988909\pi$
$660$	0	0
$661$	339.503	0.513620	0.256810	−	0.966462i	$-0.417329\pi$
$661$	339.503	0.513620	0.256810	+	0.966462i	$0.417329\pi$
$662$	192.723i	0.291122i
$663$	0	0
$664$	326.285	0.491393
$665$	− 224.938i	− 0.338253i
$666$	0	0
$667$	−888.337	−1.33184
$668$	− 278.788i	− 0.417348i
$669$	0	0
$670$	418.461	0.624569
$671$	− 525.458i	− 0.783096i
$672$	0	0
$673$	−527.771	−0.784207	−0.392104	−	0.919921i	$-0.628252\pi$
$673$	−527.771	−0.784207	−0.392104	+	0.919921i	$0.628252\pi$
$674$	342.923i	0.508787i
$675$	0	0
$676$	122.849	0.181730
$677$	− 187.263i	− 0.276608i	−0.990390	−	0.138304i	$-0.955835\pi$
$677$	− 187.263i	− 0.276608i	0.990390	−	0.138304i	$-0.0441651\pi$
$678$	0	0
$679$	−295.228	−0.434798
$680$	− 220.862i	− 0.324797i
$681$	0	0
$682$	73.4465	0.107693
$683$	− 575.317i	− 0.842338i	−0.906982	−	0.421169i	$-0.861620\pi$
$683$	− 575.317i	− 0.842338i	0.906982	−	0.421169i	$-0.138380\pi$
$684$	0	0
$685$	−69.0157	−0.100753
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	−115.643	−0.168086
$689$	188.147i	0.273072i
$690$	0	0
$691$	538.231	0.778916	0.389458	−	0.921044i	$-0.372662\pi$
$691$	538.231	0.778916	0.389458	+	0.921044i	$0.372662\pi$
$692$	− 458.792i	− 0.662994i
$693$	0	0
$694$	−911.706	−1.31370
$695$	− 192.850i	− 0.277482i
$696$	0	0
$697$	−362.136	−0.519564
$698$	856.774i	1.22747i
$699$	0	0
$700$	−78.2050	−0.111721
$701$	− 356.636i	− 0.508753i	−0.967105	−	0.254376i	$-0.918130\pi$
$701$	− 356.636i	− 0.508753i	0.967105	−	0.254376i	$-0.0818702\pi$
$702$	0	0
$703$	−1213.78	−1.72658
$704$	− 150.845i	− 0.214268i
$705$	0	0
$706$	230.576	0.326595
$707$	− 383.453i	− 0.542367i
$708$	0	0
$709$	−401.615	−0.566452	−0.283226	−	0.959053i	$-0.591405\pi$
$709$	−401.615	−0.566452	−0.283226	+	0.959053i	$0.591405\pi$
$710$	− 370.289i	− 0.521533i
$711$	0	0
$712$	38.8473	0.0545608
$713$	− 112.584i	− 0.157902i
$714$	0	0
$715$	−625.224	−0.874440
$716$	405.083i	0.565758i
$717$	0	0
$718$	−321.486	−0.447752
$719$	916.413i	1.27457i	0.770630	+	0.637283i	$0.219942\pi$
$719$	916.413i	1.27457i	−0.770630	+	0.637283i	$0.780058\pi$
$720$	0	0
$721$	−118.557	−0.164434
$722$	489.618i	0.678141i
$723$	0	0
$724$	−8.40808	−0.0116134
$725$	321.197i	0.443031i
$726$	0	0
$727$	708.685	0.974808	0.487404	−	0.873177i	$-0.337944\pi$
$727$	708.685	0.974808	0.487404	+	0.873177i	$0.337944\pi$
$728$	− 77.6158i	− 0.106615i
$729$	0	0
$730$	367.528	0.503463
$731$	706.148i	0.966003i
$732$	0	0
$733$	1281.72	1.74860	0.874298	−	0.485389i	$-0.161322\pi$
$733$	1281.72	1.74860	0.874298	+	0.485389i	$0.161322\pi$
$734$	− 955.310i	− 1.30151i
$735$	0	0
$736$	−231.226	−0.314166
$737$	− 1745.19i	− 2.36796i
$738$	0	0
$739$	−1077.96	−1.45867	−0.729336	−	0.684155i	$-0.760171\pi$
$739$	−1077.96	−1.45867	−0.729336	+	0.684155i	$0.760171\pi$
$740$	− 291.834i	− 0.394370i
$741$	0	0
$742$	−67.8741	−0.0914745
$743$	− 264.701i	− 0.356259i	−0.984007	−	0.178130i	$-0.942995\pi$
$743$	− 264.701i	− 0.356259i	0.984007	−	0.178130i	$-0.0570046\pi$
$744$	0	0
$745$	−49.6745	−0.0666772
$746$	629.019i	0.843189i
$747$	0	0
$748$	−921.101	−1.23142
$749$	− 265.157i	− 0.354014i
$750$	0	0
$751$	−36.5467	−0.0486640	−0.0243320	−	0.999704i	$-0.507746\pi$
$751$	−36.5467	−0.0486640	−0.0243320	+	0.999704i	$0.507746\pi$
$752$	119.079i	0.158350i
$753$	0	0
$754$	−318.777	−0.422782
$755$	85.7536i	0.113581i
$756$	0	0
$757$	299.469	0.395600	0.197800	−	0.980242i	$-0.436620\pi$
$757$	299.469	0.395600	0.197800	+	0.980242i	$0.436620\pi$
$758$	829.070i	1.09376i
$759$	0	0
$760$	−240.469	−0.316406
$761$	− 504.968i	− 0.663558i	−0.943357	−	0.331779i	$-0.892351\pi$
$761$	− 504.968i	− 0.663558i	0.943357	−	0.331779i	$-0.107649\pi$
$762$	0	0
$763$	−160.716	−0.210637
$764$	− 543.820i	− 0.711806i
$765$	0	0
$766$	212.494	0.277407
$767$	− 555.839i	− 0.724692i
$768$	0	0
$769$	−752.647	−0.978735	−0.489368	−	0.872078i	$-0.662772\pi$
$769$	−752.647	−0.978735	−0.489368	+	0.872078i	$0.662772\pi$
$770$	− 225.550i	− 0.292923i
$771$	0	0
$772$	289.107	0.374490
$773$	4.76817i	0.00616840i	0.999995	+	0.00308420i	$0.000981732\pi$
$773$	4.76817i	0.00616840i	−0.999995	+	0.00308420i	$0.999018\pi$
$774$	0	0
$775$	−40.7072	−0.0525254
$776$	315.612i	0.406717i
$777$	0	0
$778$	−763.632	−0.981533
$779$	394.284i	0.506142i
$780$	0	0
$781$	−1544.28	−1.97732
$782$	1411.93i	1.80554i
$783$	0	0
$784$	28.0000	0.0357143
$785$	339.307i	0.432238i
$786$	0	0
$787$	−849.743	−1.07972	−0.539862	−	0.841754i	$-0.681524\pi$
$787$	−849.743	−1.07972	−0.539862	+	0.841754i	$0.681524\pi$
$788$	106.125i	0.134677i
$789$	0	0
$790$	382.552	0.484244
$791$	38.5327i	0.0487139i
$792$	0	0
$793$	−289.037	−0.364485
$794$	694.694i	0.874930i
$795$	0	0
$796$	229.615	0.288461
$797$	− 192.021i	− 0.240929i	−0.992718	−	0.120465i	$-0.961562\pi$
$797$	− 192.021i	− 0.240929i	0.992718	−	0.120465i	$-0.0384385\pi$
$798$	0	0
$799$	727.129	0.910049
$800$	83.6047i	0.104506i
$801$	0	0
$802$	305.508	0.380933
$803$	− 1532.77i	− 1.90881i
$804$	0	0
$805$	−345.740	−0.429491
$806$	− 40.4005i	− 0.0501247i
$807$	0	0
$808$	−409.929	−0.507337
$809$	1015.55i	1.25532i	0.778488	+	0.627660i	$0.215987\pi$
$809$	1015.55i	1.25532i	−0.778488	+	0.627660i	$0.784013\pi$
$810$	0	0
$811$	481.576	0.593805	0.296902	−	0.954908i	$-0.404046\pi$
$811$	481.576	0.593805	0.296902	+	0.954908i	$0.404046\pi$
$812$	− 114.999i	− 0.141625i
$813$	0	0
$814$	−1217.09	−1.49519
$815$	38.0290i	0.0466614i
$816$	0	0
$817$	768.837	0.941048
$818$	149.556i	0.182832i
$819$	0	0
$820$	−94.7990	−0.115609
$821$	− 480.333i	− 0.585058i	−0.956257	−	0.292529i	$-0.905503\pi$
$821$	− 480.333i	− 0.585058i	0.956257	−	0.292529i	$-0.0944968\pi$
$822$	0	0
$823$	588.804	0.715437	0.357718	−	0.933830i	$-0.383555\pi$
$823$	588.804	0.715437	0.357718	+	0.933830i	$0.383555\pi$
$824$	126.743i	0.153814i
$825$	0	0
$826$	200.519	0.242760
$827$	− 280.409i	− 0.339068i	−0.985524	−	0.169534i	$-0.945774\pi$
$827$	− 280.409i	− 0.339068i	0.985524	−	0.169534i	$-0.0542262\pi$
$828$	0	0
$829$	190.758	0.230106	0.115053	−	0.993359i	$-0.463296\pi$
$829$	190.758	0.230106	0.115053	+	0.993359i	$0.463296\pi$
$830$	− 521.562i	− 0.628388i
$831$	0	0
$832$	−82.9748	−0.0997293
$833$	− 170.976i	− 0.205253i
$834$	0	0
$835$	−445.639	−0.533700
$836$	1002.87i	1.19961i
$837$	0	0
$838$	−18.8805	−0.0225304
$839$	1073.07i	1.27899i	0.768797	+	0.639493i	$0.220856\pi$
$839$	1073.07i	1.27899i	−0.768797	+	0.639493i	$0.779144\pi$
$840$	0	0
$841$	368.684	0.438388
$842$	− 672.092i	− 0.798208i
$843$	0	0
$844$	−721.918	−0.855353
$845$	− 196.373i	− 0.232394i
$846$	0	0
$847$	−620.519	−0.732608
$848$	72.5604i	0.0855665i
$849$	0	0
$850$	510.514	0.600605
$851$	1865.64i	2.19229i
$852$	0	0
$853$	−1008.68	−1.18251	−0.591253	−	0.806486i	$-0.701366\pi$
$853$	−1008.68	−1.18251	−0.591253	+	0.806486i	$0.701366\pi$
$854$	− 104.270i	− 0.122097i
$855$	0	0
$856$	−283.465	−0.331150
$857$	301.759i	0.352111i	0.984380	+	0.176055i	$0.0563338\pi$
$857$	301.759i	0.352111i	−0.984380	+	0.176055i	$0.943666\pi$
$858$	0	0
$859$	1088.50	1.26717	0.633587	−	0.773672i	$-0.281582\pi$
$859$	1088.50	1.26717	0.633587	+	0.773672i	$0.281582\pi$
$860$	184.854i	0.214946i
$861$	0	0
$862$	−332.612	−0.385861
$863$	1120.50i	1.29838i	0.760627	+	0.649189i	$0.224892\pi$
$863$	1120.50i	1.29838i	−0.760627	+	0.649189i	$0.775108\pi$
$864$	0	0
$865$	−733.373	−0.847830
$866$	− 29.7551i	− 0.0343593i
$867$	0	0
$868$	14.5745	0.0167909
$869$	− 1595.43i	− 1.83594i
$870$	0	0
$871$	−959.969	−1.10215
$872$	171.812i	0.197033i
$873$	0	0
$874$	1537.27	1.75890
$875$	336.470i	0.384537i
$876$	0	0
$877$	45.9628	0.0524091	0.0262046	−	0.999657i	$-0.491658\pi$
$877$	45.9628	0.0524091	0.0262046	+	0.999657i	$0.491658\pi$
$878$	− 352.353i	− 0.401314i
$879$	0	0
$880$	−241.124	−0.274004
$881$	881.775i	1.00088i	0.865771	+	0.500440i	$0.166828\pi$
$881$	881.775i	1.00088i	−0.865771	+	0.500440i	$0.833172\pi$
$882$	0	0
$883$	216.488	0.245174	0.122587	−	0.992458i	$-0.460881\pi$
$883$	216.488	0.245174	0.122587	+	0.992458i	$0.460881\pi$
$884$	506.668i	0.573153i
$885$	0	0
$886$	462.083	0.521538
$887$	142.061i	0.160159i	0.996788	+	0.0800796i	$0.0255175\pi$
$887$	142.061i	0.160159i	−0.996788	+	0.0800796i	$0.974483\pi$
$888$	0	0
$889$	586.965	0.660254
$890$	− 62.0969i	− 0.0697718i
$891$	0	0
$892$	413.714	0.463805
$893$	− 791.680i	− 0.886540i
$894$	0	0
$895$	647.519	0.723485
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	−722.143	−0.804168
$899$	− 59.8593i	− 0.0665843i
$900$	0	0
$901$	443.074	0.491758
$902$	395.358i	0.438313i
$903$	0	0
$904$	41.1931	0.0455676
$905$	13.4402i	0.0148510i
$906$	0	0
$907$	−451.060	−0.497309	−0.248655	−	0.968592i	$-0.579988\pi$
$907$	−451.060	−0.497309	−0.248655	+	0.968592i	$0.579988\pi$
$908$	263.282i	0.289958i
$909$	0	0
$910$	−124.068	−0.136338
$911$	− 989.545i	− 1.08622i	−0.839662	−	0.543109i	$-0.817247\pi$
$911$	− 989.545i	− 1.08622i	0.839662	−	0.543109i	$-0.182753\pi$
$912$	0	0
$913$	−2175.17	−2.38244
$914$	− 256.110i	− 0.280208i
$915$	0	0
$916$	−473.056	−0.516436
$917$	8.95930i	0.00977023i
$918$	0	0
$919$	942.500	1.02557	0.512786	−	0.858517i	$-0.328614\pi$
$919$	942.500	1.02557	0.512786	+	0.858517i	$0.328614\pi$
$920$	369.612i	0.401752i
$921$	0	0
$922$	418.881	0.454318
$923$	849.460i	0.920325i
$924$	0	0
$925$	674.562	0.729257
$926$	758.457i	0.819068i
$927$	0	0
$928$	−122.939	−0.132478
$929$	630.378i	0.678555i	0.940686	+	0.339278i	$0.110183\pi$
$929$	630.378i	0.678555i	−0.940686	+	0.339278i	$0.889817\pi$
$930$	0	0
$931$	−186.154	−0.199951
$932$	− 311.148i	− 0.333849i
$933$	0	0
$934$	124.333	0.133119
$935$	1472.37i	1.57473i
$936$	0	0
$937$	236.083	0.251956	0.125978	−	0.992033i	$-0.459793\pi$
$937$	236.083	0.251956	0.125978	+	0.992033i	$0.459793\pi$
$938$	− 346.310i	− 0.369200i
$939$	0	0
$940$	190.346	0.202496
$941$	833.633i	0.885901i	0.896546	+	0.442950i	$0.146068\pi$
$941$	833.633i	0.885901i	−0.896546	+	0.442950i	$0.853932\pi$
$942$	0	0
$943$	606.033	0.642665
$944$	− 214.364i	− 0.227081i
$945$	0	0
$946$	770.930	0.814936
$947$	− 785.821i	− 0.829801i	−0.909867	−	0.414900i	$-0.863816\pi$
$947$	− 785.821i	− 0.829801i	0.909867	−	0.414900i	$-0.136184\pi$
$948$	0	0
$949$	−843.127	−0.888438
$950$	− 555.835i	− 0.585089i
$951$	0	0
$952$	−182.781	−0.191997
$953$	− 1096.70i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$953$	− 1096.70i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$954$	0	0
$955$	−869.289	−0.910250
$956$	119.192i	0.124678i
$957$	0	0
$958$	−421.263	−0.439731
$959$	57.1160i	0.0595579i
$960$	0	0
$961$	−953.414	−0.992106
$962$	669.480i	0.695925i
$963$	0	0
$964$	238.981	0.247906
$965$	− 462.133i	− 0.478894i
$966$	0	0
$967$	1295.95	1.34018	0.670088	−	0.742282i	$-0.266256\pi$
$967$	1295.95	1.34018	0.670088	+	0.742282i	$0.266256\pi$
$968$	663.363i	0.685292i
$969$	0	0
$970$	504.502	0.520105
$971$	830.704i	0.855514i	0.903894	+	0.427757i	$0.140696\pi$
$971$	830.704i	0.855514i	−0.903894	+	0.427757i	$0.859304\pi$
$972$	0	0
$973$	−159.599	−0.164028
$974$	− 624.819i	− 0.641498i
$975$	0	0
$976$	−111.470	−0.114211
$977$	− 513.423i	− 0.525510i	−0.964863	−	0.262755i	$-0.915369\pi$
$977$	− 513.423i	− 0.525510i	0.964863	−	0.262755i	$-0.0846311\pi$
$978$	0	0
$979$	−258.974	−0.264529
$980$	− 44.7576i	− 0.0456710i
$981$	0	0
$982$	−471.371	−0.480011
$983$	− 883.906i	− 0.899192i	−0.893232	−	0.449596i	$-0.851568\pi$
$983$	− 883.906i	− 0.899192i	0.893232	−	0.449596i	$-0.148432\pi$
$984$	0	0
$985$	169.640	0.172223
$986$	750.703i	0.761362i
$987$	0	0
$988$	551.647	0.558347
$989$	− 1181.74i	− 1.19488i
$990$	0	0
$991$	1014.86	1.02407	0.512036	−	0.858964i	$-0.328891\pi$
$991$	1014.86	1.02407	0.512036	+	0.858964i	$0.328891\pi$
$992$	− 15.5808i	− 0.0157065i
$993$	0	0
$994$	−306.444	−0.308293
$995$	− 367.037i	− 0.368881i
$996$	0	0
$997$	1234.91	1.23862	0.619312	−	0.785145i	$-0.287411\pi$
$997$	1234.91	1.23862	0.619312	+	0.785145i	$0.287411\pi$
$998$	− 1194.28i	− 1.19668i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.3.b.c.323.20		24
3.2	odd	2	inner	1134.3.b.c.323.5		24
9.2	odd	6		126.3.q.a.113.1	yes	24
9.4	even	3		126.3.q.a.29.1	✓	24
9.5	odd	6		378.3.q.a.197.9		24
9.7	even	3		378.3.q.a.71.9		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.q.a.29.1	✓	24	9.4	even	3
126.3.q.a.113.1	yes	24	9.2	odd	6
378.3.q.a.71.9		24	9.7	even	3
378.3.q.a.197.9		24	9.5	odd	6
1134.3.b.c.323.5		24	3.2	odd	2	inner
1134.3.b.c.323.20		24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1134.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +3.19697i q^{5} +2.64575 q^{7} -2.82843i q^{8} -4.52120 q^{10} +18.8556i q^{11} +10.3719 q^{13} +3.74166i q^{14} +4.00000 q^{16} -24.4251i q^{17} -26.5935 q^{19} -6.39395i q^{20} -26.6659 q^{22} +40.8754i q^{23} +14.7794 q^{25} +14.6680i q^{26} -5.29150 q^{28} +21.7328i q^{29} -2.75433 q^{31} +5.65685i q^{32} +34.5423 q^{34} +8.45840i q^{35} +45.6422 q^{37} -37.6088i q^{38} +9.04241 q^{40} -14.8264i q^{41} -28.9107 q^{43} -37.7112i q^{44} -57.8065 q^{46} +29.7697i q^{47} +7.00000 q^{49} +20.9012i q^{50} -20.7437 q^{52} +18.1401i q^{53} -60.2809 q^{55} -7.48331i q^{56} -30.7349 q^{58} -53.5911i q^{59} -27.8674 q^{61} -3.89520i q^{62} -8.00000 q^{64} +33.1585i q^{65} -92.5552 q^{67} +48.8502i q^{68} -11.9620 q^{70} +81.9005i q^{71} -81.2899 q^{73} +64.5478i q^{74} +53.1869 q^{76} +49.8873i q^{77} -84.6130 q^{79} +12.7879i q^{80} +20.9677 q^{82} +115.359i q^{83} +78.0865 q^{85} -40.8860i q^{86} +53.3317 q^{88} +13.7346i q^{89} +27.4413 q^{91} -81.7507i q^{92} -42.1007 q^{94} -85.0186i q^{95} -111.586 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{4} + 96 q^{16} + 24 q^{19} - 48 q^{22} - 144 q^{25} + 120 q^{31} + 96 q^{34} - 168 q^{37} - 120 q^{43} + 168 q^{49} + 264 q^{55} - 192 q^{64} - 144 q^{67} + 24 q^{73} - 48 q^{76} - 24 q^{79}+ \cdots - 96 q^{97}+O(q^{100}) \) 24 * q - 48 * q^4 + 96 * q^16 + 24 * q^19 - 48 * q^22 - 144 * q^25 + 120 * q^31 + 96 * q^34 - 168 * q^37 - 120 * q^43 + 168 * q^49 + 264 * q^55 - 192 * q^64 - 144 * q^67 + 24 * q^73 - 48 * q^76 - 24 * q^79 - 288 * q^82 + 312 * q^85 + 96 * q^88 - 168 * q^91 - 96 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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