LMFDB - Embedded newform 1134.3.b.c.323.18

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			323.18
Character	$\chi$	$=$	1134.323
Dual form			1134.3.b.c.323.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} -2.00000 q^{4} -0.768169i q^{5} +2.64575 q^{7} -2.82843i q^{8} +1.08636 q^{10} -3.64413i q^{11} +4.36322 q^{13} +3.74166i q^{14} +4.00000 q^{16} +8.55474i q^{17} -18.7989 q^{19} +1.53634i q^{20} +5.15358 q^{22} -12.5264i q^{23} +24.4099 q^{25} +6.17053i q^{26} -5.29150 q^{28} -16.4328i q^{29} +13.7654 q^{31} +5.65685i q^{32} -12.0982 q^{34} -2.03238i q^{35} -28.3620 q^{37} -26.5857i q^{38} -2.17271 q^{40} -59.2662i q^{41} +51.4940 q^{43} +7.28827i q^{44} +17.7150 q^{46} -55.5357i q^{47} +7.00000 q^{49} +34.5208i q^{50} -8.72644 q^{52} +51.2740i q^{53} -2.79931 q^{55} -7.48331i q^{56} +23.2395 q^{58} -64.3746i q^{59} +49.2644 q^{61} +19.4672i q^{62} -8.00000 q^{64} -3.35169i q^{65} -49.3916 q^{67} -17.1095i q^{68} +2.87423 q^{70} -81.1558i q^{71} +140.515 q^{73} -40.1099i q^{74} +37.5978 q^{76} -9.64147i q^{77} +139.430 q^{79} -3.07268i q^{80} +83.8150 q^{82} +134.036i q^{83} +6.57148 q^{85} +72.8236i q^{86} -10.3072 q^{88} -8.94002i q^{89} +11.5440 q^{91} +25.0528i q^{92} +78.5393 q^{94} +14.4407i q^{95} -90.5001 q^{97} +9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 0.768169i	− 0.153634i	−0.997045	−	0.0768169i	$-0.975524\pi$
$5$	− 0.768169i	− 0.153634i	0.997045	−	0.0768169i	$-0.0244757\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	1.08636	0.108636
$11$	− 3.64413i	− 0.331285i	−0.986186	−	0.165642i	$-0.947030\pi$
$11$	− 3.64413i	− 0.331285i	0.986186	−	0.165642i	$-0.0529698\pi$
$12$	0	0
$13$	4.36322	0.335632	0.167816	−	0.985818i	$-0.446328\pi$
$13$	4.36322	0.335632	0.167816	+	0.985818i	$0.446328\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	8.55474i	0.503220i	0.967829	+	0.251610i	$0.0809600\pi$
$17$	8.55474i	0.503220i	−0.967829	+	0.251610i	$0.919040\pi$
$18$	0	0
$19$	−18.7989	−0.989416	−0.494708	−	0.869059i	$-0.664725\pi$
$19$	−18.7989	−0.989416	−0.494708	+	0.869059i	$0.664725\pi$
$20$	1.53634i	0.0768169i
$21$	0	0
$22$	5.15358	0.234254
$23$	− 12.5264i	− 0.544625i	−0.962209	−	0.272313i	$-0.912211\pi$
$23$	− 12.5264i	− 0.544625i	0.962209	−	0.272313i	$-0.0877885\pi$
$24$	0	0
$25$	24.4099	0.976397
$26$	6.17053i	0.237328i
$27$	0	0
$28$	−5.29150	−0.188982
$29$	− 16.4328i	− 0.566649i	−0.959024	−	0.283324i	$-0.908563\pi$
$29$	− 16.4328i	− 0.566649i	0.959024	−	0.283324i	$-0.0914373\pi$
$30$	0	0
$31$	13.7654	0.444045	0.222023	−	0.975042i	$-0.428734\pi$
$31$	13.7654	0.444045	0.222023	+	0.975042i	$0.428734\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	−12.0982	−0.355830
$35$	− 2.03238i	− 0.0580681i
$36$	0	0
$37$	−28.3620	−0.766541	−0.383270	−	0.923636i	$-0.625202\pi$
$37$	−28.3620	−0.766541	−0.383270	+	0.923636i	$0.625202\pi$
$38$	− 26.5857i	− 0.699622i
$39$	0	0
$40$	−2.17271	−0.0543178
$41$	− 59.2662i	− 1.44552i	−0.691101	−	0.722758i	$-0.742874\pi$
$41$	− 59.2662i	− 1.44552i	0.691101	−	0.722758i	$-0.257126\pi$
$42$	0	0
$43$	51.4940	1.19754	0.598768	−	0.800923i	$-0.295657\pi$
$43$	51.4940	1.19754	0.598768	+	0.800923i	$0.295657\pi$
$44$	7.28827i	0.165642i
$45$	0	0
$46$	17.7150	0.385108
$47$	− 55.5357i	− 1.18161i	−0.806814	−	0.590805i	$-0.798810\pi$
$47$	− 55.5357i	− 1.18161i	0.806814	−	0.590805i	$-0.201190\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	34.5208i	0.690417i
$51$	0	0
$52$	−8.72644	−0.167816
$53$	51.2740i	0.967434i	0.875224	+	0.483717i	$0.160714\pi$
$53$	51.2740i	0.967434i	−0.875224	+	0.483717i	$0.839286\pi$
$54$	0	0
$55$	−2.79931	−0.0508966
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	23.2395	0.400681
$59$	− 64.3746i	− 1.09109i	−0.838080	−	0.545547i	$-0.816322\pi$
$59$	− 64.3746i	− 1.09109i	0.838080	−	0.545547i	$-0.183678\pi$
$60$	0	0
$61$	49.2644	0.807614	0.403807	−	0.914844i	$-0.367687\pi$
$61$	49.2644	0.807614	0.403807	+	0.914844i	$0.367687\pi$
$62$	19.4672i	0.313987i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 3.35169i	− 0.0515645i
$66$	0	0
$67$	−49.3916	−0.737188	−0.368594	−	0.929590i	$-0.620161\pi$
$67$	−49.3916	−0.737188	−0.368594	+	0.929590i	$0.620161\pi$
$68$	− 17.1095i	− 0.251610i
$69$	0	0
$70$	2.87423	0.0410604
$71$	− 81.1558i	− 1.14304i	−0.820588	−	0.571520i	$-0.806354\pi$
$71$	− 81.1558i	− 1.14304i	0.820588	−	0.571520i	$-0.193646\pi$
$72$	0	0
$73$	140.515	1.92486	0.962431	−	0.271527i	$-0.0875285\pi$
$73$	140.515	1.92486	0.962431	+	0.271527i	$0.0875285\pi$
$74$	− 40.1099i	− 0.542026i
$75$	0	0
$76$	37.5978	0.494708
$77$	− 9.64147i	− 0.125214i
$78$	0	0
$79$	139.430	1.76493	0.882466	−	0.470377i	$-0.155882\pi$
$79$	139.430	1.76493	0.882466	+	0.470377i	$0.155882\pi$
$80$	− 3.07268i	− 0.0384085i
$81$	0	0
$82$	83.8150	1.02213
$83$	134.036i	1.61490i	0.589938	+	0.807449i	$0.299152\pi$
$83$	134.036i	1.61490i	−0.589938	+	0.807449i	$0.700848\pi$
$84$	0	0
$85$	6.57148	0.0773116
$86$	72.8236i	0.846786i
$87$	0	0
$88$	−10.3072	−0.117127
$89$	− 8.94002i	− 0.100450i	−0.998738	−	0.0502248i	$-0.984006\pi$
$89$	− 8.94002i	− 0.100450i	0.998738	−	0.0502248i	$-0.0159938\pi$
$90$	0	0
$91$	11.5440	0.126857
$92$	25.0528i	0.272313i
$93$	0	0
$94$	78.5393	0.835525
$95$	14.4407i	0.152008i
$96$	0	0
$97$	−90.5001	−0.932990	−0.466495	−	0.884524i	$-0.654484\pi$
$97$	−90.5001	−0.932990	−0.466495	+	0.884524i	$0.654484\pi$
$98$	9.89949i	0.101015i
$99$	0	0
$100$	−48.8198	−0.488198
$101$	65.9200i	0.652673i	0.945254	+	0.326336i	$0.105814\pi$
$101$	65.9200i	0.652673i	−0.945254	+	0.326336i	$0.894186\pi$
$102$	0	0
$103$	10.9597	0.106404	0.0532022	−	0.998584i	$-0.483057\pi$
$103$	10.9597	0.106404	0.0532022	+	0.998584i	$0.483057\pi$
$104$	− 12.3411i	− 0.118664i
$105$	0	0
$106$	−72.5124	−0.684079
$107$	− 195.963i	− 1.83143i	−0.401826	−	0.915716i	$-0.631624\pi$
$107$	− 195.963i	− 1.83143i	0.401826	−	0.915716i	$-0.368376\pi$
$108$	0	0
$109$	181.440	1.66459	0.832295	−	0.554333i	$-0.187027\pi$
$109$	181.440	1.66459	0.832295	+	0.554333i	$0.187027\pi$
$110$	− 3.95882i	− 0.0359893i
$111$	0	0
$112$	10.5830	0.0944911
$113$	− 68.0545i	− 0.602252i	−0.953584	−	0.301126i	$-0.902637\pi$
$113$	− 68.0545i	− 0.602252i	0.953584	−	0.301126i	$-0.0973625\pi$
$114$	0	0
$115$	−9.62238	−0.0836729
$116$	32.8656i	0.283324i
$117$	0	0
$118$	91.0394	0.771520
$119$	22.6337i	0.190199i
$120$	0	0
$121$	107.720	0.890250
$122$	69.6704i	0.571069i
$123$	0	0
$124$	−27.5308	−0.222023
$125$	− 37.9552i	− 0.303641i
$126$	0	0
$127$	−71.4363	−0.562490	−0.281245	−	0.959636i	$-0.590747\pi$
$127$	−71.4363	−0.562490	−0.281245	+	0.959636i	$0.590747\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	4.74001	0.0364616
$131$	143.179i	1.09297i	0.837469	+	0.546484i	$0.184034\pi$
$131$	143.179i	1.09297i	−0.837469	+	0.546484i	$0.815966\pi$
$132$	0	0
$133$	−49.7372	−0.373964
$134$	− 69.8503i	− 0.521271i
$135$	0	0
$136$	24.1964	0.177915
$137$	− 132.721i	− 0.968767i	−0.874856	−	0.484384i	$-0.839044\pi$
$137$	− 132.721i	− 0.968767i	0.874856	−	0.484384i	$-0.160956\pi$
$138$	0	0
$139$	197.508	1.42092	0.710459	−	0.703739i	$-0.248487\pi$
$139$	197.508	1.42092	0.710459	+	0.703739i	$0.248487\pi$
$140$	4.06477i	0.0290341i
$141$	0	0
$142$	114.772	0.808251
$143$	− 15.9002i	− 0.111190i
$144$	0	0
$145$	−12.6232	−0.0870564
$146$	198.718i	1.36108i
$147$	0	0
$148$	56.7240	0.383270
$149$	10.8727i	0.0729711i	0.999334	+	0.0364856i	$0.0116163\pi$
$149$	10.8727i	0.0729711i	−0.999334	+	0.0364856i	$0.988384\pi$
$150$	0	0
$151$	81.6331	0.540616	0.270308	−	0.962774i	$-0.412874\pi$
$151$	81.6331	0.540616	0.270308	+	0.962774i	$0.412874\pi$
$152$	53.1713i	0.349811i
$153$	0	0
$154$	13.6351	0.0885396
$155$	− 10.5742i	− 0.0682204i
$156$	0	0
$157$	124.898	0.795526	0.397763	−	0.917488i	$-0.369787\pi$
$157$	124.898	0.795526	0.397763	+	0.917488i	$0.369787\pi$
$158$	197.183i	1.24799i
$159$	0	0
$160$	4.34542	0.0271589
$161$	− 33.1417i	− 0.205849i
$162$	0	0
$163$	74.3495	0.456132	0.228066	−	0.973646i	$-0.426760\pi$
$163$	74.3495	0.456132	0.228066	+	0.973646i	$0.426760\pi$
$164$	118.532i	0.722758i
$165$	0	0
$166$	−189.556	−1.14190
$167$	52.2889i	0.313107i	0.987669	+	0.156554i	$0.0500384\pi$
$167$	52.2889i	0.313107i	−0.987669	+	0.156554i	$0.949962\pi$
$168$	0	0
$169$	−149.962	−0.887351
$170$	9.29348i	0.0546675i
$171$	0	0
$172$	−102.988	−0.598768
$173$	− 293.116i	− 1.69431i	−0.531343	−	0.847157i	$-0.678312\pi$
$173$	− 293.116i	− 1.69431i	0.531343	−	0.847157i	$-0.321688\pi$
$174$	0	0
$175$	64.5826	0.369043
$176$	− 14.5765i	− 0.0828212i
$177$	0	0
$178$	12.6431	0.0710286
$179$	140.420i	0.784470i	0.919865	+	0.392235i	$0.128298\pi$
$179$	140.420i	0.784470i	−0.919865	+	0.392235i	$0.871702\pi$
$180$	0	0
$181$	−83.3787	−0.460656	−0.230328	−	0.973113i	$-0.573980\pi$
$181$	−83.3787	−0.460656	−0.230328	+	0.973113i	$0.573980\pi$
$182$	16.3257i	0.0897015i
$183$	0	0
$184$	−35.4300	−0.192554
$185$	21.7868i	0.117767i
$186$	0	0
$187$	31.1746	0.166709
$188$	111.071i	0.590805i
$189$	0	0
$190$	−20.4223	−0.107486
$191$	110.975i	0.581019i	0.956872	+	0.290510i	$0.0938248\pi$
$191$	110.975i	0.581019i	−0.956872	+	0.290510i	$0.906175\pi$
$192$	0	0
$193$	−317.941	−1.64736	−0.823682	−	0.567052i	$-0.808084\pi$
$193$	−317.941	−1.64736	−0.823682	+	0.567052i	$0.808084\pi$
$194$	− 127.986i	− 0.659724i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	124.246i	0.630690i	0.948977	+	0.315345i	$0.102120\pi$
$197$	124.246i	0.630690i	−0.948977	+	0.315345i	$0.897880\pi$
$198$	0	0
$199$	13.0169	0.0654116	0.0327058	−	0.999465i	$-0.489588\pi$
$199$	13.0169	0.0654116	0.0327058	+	0.999465i	$0.489588\pi$
$200$	− 69.0417i	− 0.345208i
$201$	0	0
$202$	−93.2249	−0.461509
$203$	− 43.4771i	− 0.214173i
$204$	0	0
$205$	−45.5264	−0.222080
$206$	15.4993i	0.0752393i
$207$	0	0
$208$	17.4529	0.0839081
$209$	68.5057i	0.327778i
$210$	0	0
$211$	−15.9339	−0.0755160	−0.0377580	−	0.999287i	$-0.512022\pi$
$211$	−15.9339	−0.0755160	−0.0377580	+	0.999287i	$0.512022\pi$
$212$	− 102.548i	− 0.483717i
$213$	0	0
$214$	277.134	1.29502
$215$	− 39.5561i	− 0.183982i
$216$	0	0
$217$	36.4198	0.167833
$218$	256.595i	1.17704i
$219$	0	0
$220$	5.59862	0.0254483
$221$	37.3262i	0.168897i
$222$	0	0
$223$	−94.0784	−0.421876	−0.210938	−	0.977499i	$-0.567652\pi$
$223$	−94.0784	−0.421876	−0.210938	+	0.977499i	$0.567652\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	96.2436	0.425857
$227$	43.3132i	0.190807i	0.995439	+	0.0954035i	$0.0304141\pi$
$227$	43.3132i	0.190807i	−0.995439	+	0.0954035i	$0.969586\pi$
$228$	0	0
$229$	−219.826	−0.959938	−0.479969	−	0.877285i	$-0.659352\pi$
$229$	−219.826	−0.959938	−0.479969	+	0.877285i	$0.659352\pi$
$230$	− 13.6081i	− 0.0591657i
$231$	0	0
$232$	−46.4790	−0.200341
$233$	237.106i	1.01762i	0.860879	+	0.508811i	$0.169915\pi$
$233$	237.106i	1.01762i	−0.860879	+	0.508811i	$0.830085\pi$
$234$	0	0
$235$	−42.6608	−0.181535
$236$	128.749i	0.545547i
$237$	0	0
$238$	−32.0089	−0.134491
$239$	− 451.470i	− 1.88900i	−0.328517	−	0.944498i	$-0.606549\pi$
$239$	− 451.470i	− 1.88900i	0.328517	−	0.944498i	$-0.393451\pi$
$240$	0	0
$241$	−34.9949	−0.145207	−0.0726036	−	0.997361i	$-0.523131\pi$
$241$	−34.9949	−0.145207	−0.0726036	+	0.997361i	$0.523131\pi$
$242$	152.339i	0.629502i
$243$	0	0
$244$	−98.5289	−0.403807
$245$	− 5.37718i	− 0.0219477i
$246$	0	0
$247$	−82.0237	−0.332080
$248$	− 38.9344i	− 0.156994i
$249$	0	0
$250$	53.6767	0.214707
$251$	− 201.405i	− 0.802409i	−0.915988	−	0.401205i	$-0.868592\pi$
$251$	− 201.405i	− 0.802409i	0.915988	−	0.401205i	$-0.131408\pi$
$252$	0	0
$253$	−45.6478	−0.180426
$254$	− 101.026i	− 0.397741i
$255$	0	0
$256$	16.0000	0.0625000
$257$	159.959i	0.622408i	0.950343	+	0.311204i	$0.100732\pi$
$257$	159.959i	0.622408i	−0.950343	+	0.311204i	$0.899268\pi$
$258$	0	0
$259$	−75.0388	−0.289725
$260$	6.70338i	0.0257822i
$261$	0	0
$262$	−202.486	−0.772846
$263$	− 447.858i	− 1.70288i	−0.524450	−	0.851441i	$-0.675729\pi$
$263$	− 447.858i	− 1.70288i	0.524450	−	0.851441i	$-0.324271\pi$
$264$	0	0
$265$	39.3871	0.148631
$266$	− 70.3390i	− 0.264432i
$267$	0	0
$268$	98.7833	0.368594
$269$	177.841i	0.661118i	0.943785	+	0.330559i	$0.107237\pi$
$269$	177.841i	0.661118i	−0.943785	+	0.330559i	$0.892763\pi$
$270$	0	0
$271$	−274.571	−1.01318	−0.506588	−	0.862188i	$-0.669093\pi$
$271$	−274.571	−1.01318	−0.506588	+	0.862188i	$0.669093\pi$
$272$	34.2189i	0.125805i
$273$	0	0
$274$	187.696	0.685022
$275$	− 88.9530i	− 0.323465i
$276$	0	0
$277$	517.748	1.86912	0.934562	−	0.355799i	$-0.115791\pi$
$277$	517.748	1.86912	0.934562	+	0.355799i	$0.115791\pi$
$278$	279.318i	1.00474i
$279$	0	0
$280$	−5.74845	−0.0205302
$281$	− 132.477i	− 0.471450i	−0.971820	−	0.235725i	$-0.924254\pi$
$281$	− 132.477i	− 0.471450i	0.971820	−	0.235725i	$-0.0757464\pi$
$282$	0	0
$283$	256.768	0.907309	0.453654	−	0.891178i	$-0.350120\pi$
$283$	256.768	0.907309	0.453654	+	0.891178i	$0.350120\pi$
$284$	162.312i	0.571520i
$285$	0	0
$286$	22.4862	0.0786232
$287$	− 156.804i	− 0.546354i
$288$	0	0
$289$	215.816	0.746770
$290$	− 17.8519i	− 0.0615582i
$291$	0	0
$292$	−281.030	−0.962431
$293$	481.204i	1.64234i	0.570687	+	0.821168i	$0.306677\pi$
$293$	481.204i	1.64234i	−0.570687	+	0.821168i	$0.693323\pi$
$294$	0	0
$295$	−49.4506	−0.167629
$296$	80.2199i	0.271013i
$297$	0	0
$298$	−15.3763	−0.0515984
$299$	− 54.6554i	− 0.182794i
$300$	0	0
$301$	136.240	0.452626
$302$	115.447i	0.382273i
$303$	0	0
$304$	−75.1956	−0.247354
$305$	− 37.8434i	− 0.124077i
$306$	0	0
$307$	−593.798	−1.93420	−0.967098	−	0.254406i	$-0.918120\pi$
$307$	−593.798	−1.93420	−0.967098	+	0.254406i	$0.918120\pi$
$308$	19.2829i	0.0626070i
$309$	0	0
$310$	14.9541	0.0482391
$311$	− 349.430i	− 1.12357i	−0.827284	−	0.561784i	$-0.810115\pi$
$311$	− 349.430i	− 1.12357i	0.827284	−	0.561784i	$-0.189885\pi$
$312$	0	0
$313$	−370.955	−1.18516	−0.592580	−	0.805511i	$-0.701891\pi$
$313$	−370.955	−1.18516	−0.592580	+	0.805511i	$0.701891\pi$
$314$	176.632i	0.562522i
$315$	0	0
$316$	−278.859	−0.882466
$317$	− 523.174i	− 1.65039i	−0.564846	−	0.825196i	$-0.691065\pi$
$317$	− 523.174i	− 1.65039i	0.564846	−	0.825196i	$-0.308935\pi$
$318$	0	0
$319$	−59.8834	−0.187722
$320$	6.14535i	0.0192042i
$321$	0	0
$322$	46.8694	0.145557
$323$	− 160.820i	− 0.497893i
$324$	0	0
$325$	106.506	0.327710
$326$	105.146i	0.322534i
$327$	0	0
$328$	−167.630	−0.511067
$329$	− 146.934i	− 0.446607i
$330$	0	0
$331$	−95.7996	−0.289425	−0.144712	−	0.989474i	$-0.546226\pi$
$331$	−95.7996	−0.289425	−0.144712	+	0.989474i	$0.546226\pi$
$332$	− 268.073i	− 0.807449i
$333$	0	0
$334$	−73.9476	−0.221400
$335$	37.9411i	0.113257i
$336$	0	0
$337$	−377.121	−1.11905	−0.559526	−	0.828813i	$-0.689017\pi$
$337$	−377.121	−1.11905	−0.559526	+	0.828813i	$0.689017\pi$
$338$	− 212.079i	− 0.627452i
$339$	0	0
$340$	−13.1430	−0.0386558
$341$	− 50.1630i	− 0.147105i
$342$	0	0
$343$	18.5203	0.0539949
$344$	− 145.647i	− 0.423393i
$345$	0	0
$346$	414.529	1.19806
$347$	415.106i	1.19627i	0.801395	+	0.598135i	$0.204091\pi$
$347$	415.106i	1.19627i	−0.801395	+	0.598135i	$0.795909\pi$
$348$	0	0
$349$	−215.535	−0.617579	−0.308789	−	0.951130i	$-0.599924\pi$
$349$	−215.535	−0.617579	−0.308789	+	0.951130i	$0.599924\pi$
$350$	91.3335i	0.260953i
$351$	0	0
$352$	20.6143	0.0585634
$353$	396.462i	1.12312i	0.827436	+	0.561561i	$0.189799\pi$
$353$	396.462i	1.12312i	−0.827436	+	0.561561i	$0.810201\pi$
$354$	0	0
$355$	−62.3414	−0.175610
$356$	17.8800i	0.0502248i
$357$	0	0
$358$	−198.584	−0.554704
$359$	− 260.223i	− 0.724854i	−0.932012	−	0.362427i	$-0.881948\pi$
$359$	− 260.223i	− 0.724854i	0.932012	−	0.362427i	$-0.118052\pi$
$360$	0	0
$361$	−7.60154	−0.0210569
$362$	− 117.915i	− 0.325733i
$363$	0	0
$364$	−23.0880	−0.0634286
$365$	− 107.939i	− 0.295724i
$366$	0	0
$367$	28.3470	0.0772399	0.0386199	−	0.999254i	$-0.487704\pi$
$367$	28.3470	0.0772399	0.0386199	+	0.999254i	$0.487704\pi$
$368$	− 50.1055i	− 0.136156i
$369$	0	0
$370$	−30.8112	−0.0832736
$371$	135.658i	0.365656i
$372$	0	0
$373$	146.919	0.393883	0.196942	−	0.980415i	$-0.436899\pi$
$373$	146.919	0.393883	0.196942	+	0.980415i	$0.436899\pi$
$374$	44.0875i	0.117881i
$375$	0	0
$376$	−157.079	−0.417762
$377$	− 71.7000i	− 0.190186i
$378$	0	0
$379$	236.871	0.624991	0.312495	−	0.949919i	$-0.398835\pi$
$379$	236.871	0.624991	0.312495	+	0.949919i	$0.398835\pi$
$380$	− 28.8815i	− 0.0760038i
$381$	0	0
$382$	−156.942	−0.410843
$383$	− 278.010i	− 0.725876i	−0.931813	−	0.362938i	$-0.881774\pi$
$383$	− 278.010i	− 0.725876i	0.931813	−	0.362938i	$-0.118226\pi$
$384$	0	0
$385$	−7.40628	−0.0192371
$386$	− 449.637i	− 1.16486i
$387$	0	0
$388$	181.000	0.466495
$389$	18.6572i	0.0479620i	0.999712	+	0.0239810i	$0.00763412\pi$
$389$	18.6572i	0.0479620i	−0.999712	+	0.0239810i	$0.992366\pi$
$390$	0	0
$391$	107.160	0.274066
$392$	− 19.7990i	− 0.0505076i
$393$	0	0
$394$	−175.710	−0.445965
$395$	− 107.105i	− 0.271153i
$396$	0	0
$397$	−467.828	−1.17841	−0.589205	−	0.807984i	$-0.700559\pi$
$397$	−467.828	−1.17841	−0.589205	+	0.807984i	$0.700559\pi$
$398$	18.4087i	0.0462530i
$399$	0	0
$400$	97.6397	0.244099
$401$	29.3110i	0.0730947i	0.999332	+	0.0365473i	$0.0116360\pi$
$401$	29.3110i	0.0730947i	−0.999332	+	0.0365473i	$0.988364\pi$
$402$	0	0
$403$	60.0615	0.149036
$404$	− 131.840i	− 0.326336i
$405$	0	0
$406$	61.4860	0.151443
$407$	103.355i	0.253943i
$408$	0	0
$409$	−345.330	−0.844327	−0.422163	−	0.906520i	$-0.638729\pi$
$409$	−345.330	−0.844327	−0.422163	+	0.906520i	$0.638729\pi$
$410$	− 64.3841i	− 0.157034i
$411$	0	0
$412$	−21.9193	−0.0532022
$413$	− 170.319i	− 0.412395i
$414$	0	0
$415$	102.963	0.248103
$416$	24.6821i	0.0593320i
$417$	0	0
$418$	−96.8817	−0.231774
$419$	− 153.377i	− 0.366054i	−0.983108	−	0.183027i	$-0.941410\pi$
$419$	− 153.377i	− 0.366054i	0.983108	−	0.183027i	$-0.0585897\pi$
$420$	0	0
$421$	−621.810	−1.47698	−0.738492	−	0.674263i	$-0.764462\pi$
$421$	−621.810	−1.47698	−0.738492	+	0.674263i	$0.764462\pi$
$422$	− 22.5339i	− 0.0533979i
$423$	0	0
$424$	145.025	0.342040
$425$	208.820i	0.491342i
$426$	0	0
$427$	130.341	0.305249
$428$	391.927i	0.915716i
$429$	0	0
$430$	55.9408	0.130095
$431$	− 45.0311i	− 0.104480i	−0.998635	−	0.0522402i	$-0.983364\pi$
$431$	− 45.0311i	− 0.104480i	0.998635	−	0.0522402i	$-0.0166362\pi$
$432$	0	0
$433$	470.931	1.08760	0.543800	−	0.839215i	$-0.316985\pi$
$433$	470.931	1.08760	0.543800	+	0.839215i	$0.316985\pi$
$434$	51.5054i	0.118676i
$435$	0	0
$436$	−362.881	−0.832295
$437$	235.482i	0.538861i
$438$	0	0
$439$	−397.109	−0.904575	−0.452288	−	0.891872i	$-0.649392\pi$
$439$	−397.109	−0.904575	−0.452288	+	0.891872i	$0.649392\pi$
$440$	7.91765i	0.0179947i
$441$	0	0
$442$	−52.7872	−0.119428
$443$	− 279.907i	− 0.631844i	−0.948785	−	0.315922i	$-0.897686\pi$
$443$	− 279.907i	− 0.631844i	0.948785	−	0.315922i	$-0.102314\pi$
$444$	0	0
$445$	−6.86744	−0.0154325
$446$	− 133.047i	− 0.298312i
$447$	0	0
$448$	−21.1660	−0.0472456
$449$	438.684i	0.977023i	0.872557	+	0.488512i	$0.162460\pi$
$449$	438.684i	0.977023i	−0.872557	+	0.488512i	$0.837540\pi$
$450$	0	0
$451$	−215.974	−0.478878
$452$	136.109i	0.301126i
$453$	0	0
$454$	−61.2541	−0.134921
$455$	− 8.86774i	− 0.0194895i
$456$	0	0
$457$	44.0886	0.0964739	0.0482369	−	0.998836i	$-0.484640\pi$
$457$	44.0886	0.0964739	0.0482369	+	0.998836i	$0.484640\pi$
$458$	− 310.881i	− 0.678779i
$459$	0	0
$460$	19.2448	0.0418364
$461$	691.817i	1.50069i	0.661048	+	0.750343i	$0.270112\pi$
$461$	691.817i	1.50069i	−0.661048	+	0.750343i	$0.729888\pi$
$462$	0	0
$463$	−922.672	−1.99281	−0.996406	−	0.0847096i	$-0.973004\pi$
$463$	−922.672	−1.99281	−0.996406	+	0.0847096i	$0.973004\pi$
$464$	− 65.7313i	− 0.141662i
$465$	0	0
$466$	−335.318	−0.719567
$467$	− 801.602i	− 1.71649i	−0.513238	−	0.858246i	$-0.671554\pi$
$467$	− 801.602i	− 1.71649i	0.513238	−	0.858246i	$-0.328446\pi$
$468$	0	0
$469$	−130.678	−0.278631
$470$	− 60.3315i	− 0.128365i
$471$	0	0
$472$	−182.079	−0.385760
$473$	− 187.651i	− 0.396726i
$474$	0	0
$475$	−458.879	−0.966062
$476$	− 45.2674i	− 0.0950996i
$477$	0	0
$478$	638.475	1.33572
$479$	222.279i	0.464047i	0.972710	+	0.232024i	$0.0745347\pi$
$479$	222.279i	0.464047i	−0.972710	+	0.232024i	$0.925465\pi$
$480$	0	0
$481$	−123.750	−0.257276
$482$	− 49.4903i	− 0.102677i
$483$	0	0
$484$	−215.441	−0.445125
$485$	69.5194i	0.143339i
$486$	0	0
$487$	230.860	0.474045	0.237023	−	0.971504i	$-0.423828\pi$
$487$	230.860	0.474045	0.237023	+	0.971504i	$0.423828\pi$
$488$	− 139.341i	− 0.285535i
$489$	0	0
$490$	7.60449	0.0155194
$491$	− 663.647i	− 1.35162i	−0.737075	−	0.675811i	$-0.763793\pi$
$491$	− 663.647i	− 1.35162i	0.737075	−	0.675811i	$-0.236207\pi$
$492$	0	0
$493$	140.578	0.285149
$494$	− 115.999i	− 0.234816i
$495$	0	0
$496$	55.0616	0.111011
$497$	− 214.718i	− 0.432028i
$498$	0	0
$499$	11.7195	0.0234860	0.0117430	−	0.999931i	$-0.496262\pi$
$499$	11.7195	0.0234860	0.0117430	+	0.999931i	$0.496262\pi$
$500$	75.9103i	0.151821i
$501$	0	0
$502$	284.829	0.567389
$503$	700.769i	1.39318i	0.717470	+	0.696589i	$0.245300\pi$
$503$	700.769i	1.39318i	−0.717470	+	0.696589i	$0.754700\pi$
$504$	0	0
$505$	50.6377	0.100273
$506$	− 64.5558i	− 0.127581i
$507$	0	0
$508$	142.873	0.281245
$509$	192.627i	0.378442i	0.981935	+	0.189221i	$0.0605963\pi$
$509$	192.627i	0.378442i	−0.981935	+	0.189221i	$0.939404\pi$
$510$	0	0
$511$	371.768	0.727529
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	−226.216	−0.440109
$515$	− 8.41887i	− 0.0163473i
$516$	0	0
$517$	−202.380	−0.391450
$518$	− 106.121i	− 0.204867i
$519$	0	0
$520$	−9.48002	−0.0182308
$521$	76.6618i	0.147144i	0.997290	+	0.0735718i	$0.0234398\pi$
$521$	76.6618i	0.147144i	−0.997290	+	0.0735718i	$0.976560\pi$
$522$	0	0
$523$	−664.017	−1.26963	−0.634815	−	0.772664i	$-0.718924\pi$
$523$	−664.017	−1.26963	−0.634815	+	0.772664i	$0.718924\pi$
$524$	− 286.358i	− 0.546484i
$525$	0	0
$526$	633.367	1.20412
$527$	117.759i	0.223452i
$528$	0	0
$529$	372.090	0.703383
$530$	55.7018i	0.105098i
$531$	0	0
$532$	99.4744	0.186982
$533$	− 258.591i	− 0.485162i
$534$	0	0
$535$	−150.533	−0.281370
$536$	139.701i	0.260635i
$537$	0	0
$538$	−251.505	−0.467481
$539$	− 25.5089i	− 0.0473264i
$540$	0	0
$541$	−214.795	−0.397034	−0.198517	−	0.980097i	$-0.563613\pi$
$541$	−214.795	−0.397034	−0.198517	+	0.980097i	$0.563613\pi$
$542$	− 388.301i	− 0.716423i
$543$	0	0
$544$	−48.3929	−0.0889575
$545$	− 139.377i	− 0.255737i
$546$	0	0
$547$	414.077	0.756996	0.378498	−	0.925602i	$-0.376441\pi$
$547$	414.077	0.756996	0.378498	+	0.925602i	$0.376441\pi$
$548$	265.442i	0.484384i
$549$	0	0
$550$	125.799	0.228725
$551$	308.919i	0.560651i
$552$	0	0
$553$	368.896	0.667081
$554$	732.206i	1.32167i
$555$	0	0
$556$	−395.015	−0.710459
$557$	421.048i	0.755921i	0.925822	+	0.377961i	$0.123375\pi$
$557$	421.048i	0.755921i	−0.925822	+	0.377961i	$0.876625\pi$
$558$	0	0
$559$	224.680	0.401932
$560$	− 8.12954i	− 0.0145170i
$561$	0	0
$562$	187.351	0.333365
$563$	582.732i	1.03505i	0.855669	+	0.517524i	$0.173146\pi$
$563$	582.732i	1.03505i	−0.855669	+	0.517524i	$0.826854\pi$
$564$	0	0
$565$	−52.2774	−0.0925263
$566$	363.125i	0.641564i
$567$	0	0
$568$	−229.543	−0.404126
$569$	342.669i	0.602231i	0.953588	+	0.301115i	$0.0973590\pi$
$569$	342.669i	0.602231i	−0.953588	+	0.301115i	$0.902641\pi$
$570$	0	0
$571$	996.251	1.74475	0.872374	−	0.488839i	$-0.162579\pi$
$571$	996.251	1.74475	0.872374	+	0.488839i	$0.162579\pi$
$572$	31.8003i	0.0555950i
$573$	0	0
$574$	221.754	0.386331
$575$	− 305.768i	− 0.531771i
$576$	0	0
$577$	−46.3061	−0.0802532	−0.0401266	−	0.999195i	$-0.512776\pi$
$577$	−46.3061	−0.0802532	−0.0401266	+	0.999195i	$0.512776\pi$
$578$	305.211i	0.528046i
$579$	0	0
$580$	25.2464	0.0435282
$581$	354.627i	0.610374i
$582$	0	0
$583$	186.849	0.320496
$584$	− 397.436i	− 0.680541i
$585$	0	0
$586$	−680.526	−1.16131
$587$	− 381.323i	− 0.649614i	−0.945780	−	0.324807i	$-0.894701\pi$
$587$	− 381.323i	− 0.649614i	0.945780	−	0.324807i	$-0.105299\pi$
$588$	0	0
$589$	−258.774	−0.439345
$590$	− 69.9337i	− 0.118532i
$591$	0	0
$592$	−113.448	−0.191635
$593$	396.610i	0.668820i	0.942428	+	0.334410i	$0.108537\pi$
$593$	396.610i	0.668820i	−0.942428	+	0.334410i	$0.891463\pi$
$594$	0	0
$595$	17.3865	0.0292210
$596$	− 21.7454i	− 0.0364856i
$597$	0	0
$598$	77.2944	0.129255
$599$	− 610.323i	− 1.01890i	−0.860499	−	0.509452i	$-0.829848\pi$
$599$	− 610.323i	− 1.01890i	0.860499	−	0.509452i	$-0.170152\pi$
$600$	0	0
$601$	−94.6287	−0.157452	−0.0787261	−	0.996896i	$-0.525085\pi$
$601$	−94.6287	−0.157452	−0.0787261	+	0.996896i	$0.525085\pi$
$602$	192.673i	0.320055i
$603$	0	0
$604$	−163.266	−0.270308
$605$	− 82.7474i	− 0.136773i
$606$	0	0
$607$	−916.191	−1.50938	−0.754688	−	0.656084i	$-0.772212\pi$
$607$	−916.191	−1.50938	−0.754688	+	0.656084i	$0.772212\pi$
$608$	− 106.343i	− 0.174906i
$609$	0	0
$610$	53.5187	0.0877355
$611$	− 242.315i	− 0.396587i
$612$	0	0
$613$	300.794	0.490691	0.245346	−	0.969436i	$-0.421099\pi$
$613$	300.794	0.490691	0.245346	+	0.969436i	$0.421099\pi$
$614$	− 839.757i	− 1.36768i
$615$	0	0
$616$	−27.2702	−0.0442698
$617$	276.439i	0.448037i	0.974585	+	0.224018i	$0.0719175\pi$
$617$	276.439i	0.448037i	−0.974585	+	0.224018i	$0.928082\pi$
$618$	0	0
$619$	937.840	1.51509	0.757545	−	0.652783i	$-0.226399\pi$
$619$	937.840	1.51509	0.757545	+	0.652783i	$0.226399\pi$
$620$	21.1483i	0.0341102i
$621$	0	0
$622$	494.168	0.794483
$623$	− 23.6531i	− 0.0379664i
$624$	0	0
$625$	581.092	0.929747
$626$	− 524.610i	− 0.838035i
$627$	0	0
$628$	−249.795	−0.397763
$629$	− 242.629i	− 0.385738i
$630$	0	0
$631$	−143.248	−0.227018	−0.113509	−	0.993537i	$-0.536209\pi$
$631$	−143.248	−0.227018	−0.113509	+	0.993537i	$0.536209\pi$
$632$	− 394.366i	− 0.623997i
$633$	0	0
$634$	739.880	1.16700
$635$	54.8751i	0.0864175i
$636$	0	0
$637$	30.5426	0.0479475
$638$	− 84.6879i	− 0.132740i
$639$	0	0
$640$	−8.69084	−0.0135794
$641$	1263.43i	1.97102i	0.169603	+	0.985512i	$0.445751\pi$
$641$	1263.43i	1.97102i	−0.169603	+	0.985512i	$0.554249\pi$
$642$	0	0
$643$	240.885	0.374626	0.187313	−	0.982300i	$-0.440022\pi$
$643$	240.885	0.374626	0.187313	+	0.982300i	$0.440022\pi$
$644$	66.2834i	0.102925i
$645$	0	0
$646$	227.433	0.352064
$647$	1080.88i	1.67060i	0.549797	+	0.835298i	$0.314705\pi$
$647$	1080.88i	1.67060i	−0.549797	+	0.835298i	$0.685295\pi$
$648$	0	0
$649$	−234.590	−0.361463
$650$	150.622i	0.231726i
$651$	0	0
$652$	−148.699	−0.228066
$653$	378.542i	0.579698i	0.957072	+	0.289849i	$0.0936050\pi$
$653$	378.542i	0.579698i	−0.957072	+	0.289849i	$0.906395\pi$
$654$	0	0
$655$	109.986	0.167917
$656$	− 237.065i	− 0.361379i
$657$	0	0
$658$	207.796	0.315799
$659$	− 280.670i	− 0.425902i	−0.977063	−	0.212951i	$-0.931692\pi$
$659$	− 280.670i	− 0.425902i	0.977063	−	0.212951i	$-0.0683075\pi$
$660$	0	0
$661$	200.005	0.302580	0.151290	−	0.988489i	$-0.451657\pi$
$661$	200.005	0.302580	0.151290	+	0.988489i	$0.451657\pi$
$662$	− 135.481i	− 0.204654i
$663$	0	0
$664$	379.112	0.570952
$665$	38.2066i	0.0574535i
$666$	0	0
$667$	−205.844	−0.308611
$668$	− 104.578i	− 0.156554i
$669$	0	0
$670$	−53.6568	−0.0800848
$671$	− 179.526i	− 0.267550i
$672$	0	0
$673$	909.136	1.35087	0.675435	−	0.737419i	$-0.263956\pi$
$673$	909.136	1.35087	0.675435	+	0.737419i	$0.263956\pi$
$674$	− 533.329i	− 0.791289i
$675$	0	0
$676$	299.925	0.443675
$677$	1006.23i	1.48630i	0.669125	+	0.743150i	$0.266669\pi$
$677$	1006.23i	1.48630i	−0.669125	+	0.743150i	$0.733331\pi$
$678$	0	0
$679$	−239.441	−0.352637
$680$	− 18.5870i	− 0.0273338i
$681$	0	0
$682$	70.9411	0.104019
$683$	− 321.974i	− 0.471412i	−0.971824	−	0.235706i	$-0.924260\pi$
$683$	− 321.974i	− 0.471412i	0.971824	−	0.235706i	$-0.0757402\pi$
$684$	0	0
$685$	−101.952	−0.148835
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	205.976	0.299384
$689$	223.720i	0.324702i
$690$	0	0
$691$	−1159.92	−1.67862	−0.839308	−	0.543656i	$-0.817040\pi$
$691$	−1159.92	−1.67862	−0.839308	+	0.543656i	$0.817040\pi$
$692$	586.232i	0.847157i
$693$	0	0
$694$	−587.048	−0.845891
$695$	− 151.719i	− 0.218301i
$696$	0	0
$697$	507.006	0.727412
$698$	− 304.812i	− 0.436694i
$699$	0	0
$700$	−129.165	−0.184522
$701$	943.410i	1.34581i	0.739731	+	0.672903i	$0.234953\pi$
$701$	943.410i	1.34581i	−0.739731	+	0.672903i	$0.765047\pi$
$702$	0	0
$703$	533.174	0.758427
$704$	29.1531i	0.0414106i
$705$	0	0
$706$	−560.682	−0.794167
$707$	174.408i	0.246687i
$708$	0	0
$709$	966.545	1.36325	0.681625	−	0.731701i	$-0.261273\pi$
$709$	966.545	1.36325	0.681625	+	0.731701i	$0.261273\pi$
$710$	− 88.1640i	− 0.124175i
$711$	0	0
$712$	−25.2862	−0.0355143
$713$	− 172.431i	− 0.241838i
$714$	0	0
$715$	−12.2140	−0.0170825
$716$	− 280.840i	− 0.392235i
$717$	0	0
$718$	368.010	0.512549
$719$	151.566i	0.210801i	0.994430	+	0.105400i	$0.0336124\pi$
$719$	151.566i	0.210801i	−0.994430	+	0.105400i	$0.966388\pi$
$720$	0	0
$721$	28.9965	0.0402171
$722$	− 10.7502i	− 0.0148895i
$723$	0	0
$724$	166.757	0.230328
$725$	− 401.124i	− 0.553274i
$726$	0	0
$727$	770.708	1.06012	0.530061	−	0.847960i	$-0.322169\pi$
$727$	770.708	1.06012	0.530061	+	0.847960i	$0.322169\pi$
$728$	− 32.6514i	− 0.0448508i
$729$	0	0
$730$	152.649	0.209108
$731$	440.518i	0.602624i
$732$	0	0
$733$	382.489	0.521813	0.260906	−	0.965364i	$-0.415979\pi$
$733$	382.489	0.521813	0.260906	+	0.965364i	$0.415979\pi$
$734$	40.0888i	0.0546169i
$735$	0	0
$736$	70.8599	0.0962771
$737$	179.990i	0.244219i
$738$	0	0
$739$	773.926	1.04726	0.523630	−	0.851945i	$-0.324577\pi$
$739$	773.926	1.04726	0.523630	+	0.851945i	$0.324577\pi$
$740$	− 43.5736i	− 0.0588833i
$741$	0	0
$742$	−191.850	−0.258558
$743$	33.8300i	0.0455316i	0.999741	+	0.0227658i	$0.00724721\pi$
$743$	33.8300i	0.0455316i	−0.999741	+	0.0227658i	$0.992753\pi$
$744$	0	0
$745$	8.35207	0.0112108
$746$	207.774i	0.278518i
$747$	0	0
$748$	−62.3492	−0.0833545
$749$	− 518.470i	− 0.692216i
$750$	0	0
$751$	583.796	0.777359	0.388679	−	0.921373i	$-0.372931\pi$
$751$	583.796	0.777359	0.388679	+	0.921373i	$0.372931\pi$
$752$	− 222.143i	− 0.295403i
$753$	0	0
$754$	101.399	0.134482
$755$	− 62.7080i	− 0.0830569i
$756$	0	0
$757$	−50.5749	−0.0668097	−0.0334048	−	0.999442i	$-0.510635\pi$
$757$	−50.5749	−0.0668097	−0.0334048	+	0.999442i	$0.510635\pi$
$758$	334.987i	0.441935i
$759$	0	0
$760$	40.8446	0.0537428
$761$	− 366.129i	− 0.481116i	−0.970635	−	0.240558i	$-0.922670\pi$
$761$	− 366.129i	− 0.481116i	0.970635	−	0.240558i	$-0.0773304\pi$
$762$	0	0
$763$	480.046	0.629156
$764$	− 221.949i	− 0.290510i
$765$	0	0
$766$	393.166	0.513272
$767$	− 280.881i	− 0.366207i
$768$	0	0
$769$	−652.530	−0.848544	−0.424272	−	0.905535i	$-0.639470\pi$
$769$	−652.530	−0.848544	−0.424272	+	0.905535i	$0.639470\pi$
$770$	− 10.4741i	− 0.0136027i
$771$	0	0
$772$	635.882	0.823682
$773$	1453.09i	1.87980i	0.341447	+	0.939901i	$0.389083\pi$
$773$	1453.09i	1.87980i	−0.341447	+	0.939901i	$0.610917\pi$
$774$	0	0
$775$	336.012	0.433564
$776$	255.973i	0.329862i
$777$	0	0
$778$	−26.3853	−0.0339143
$779$	1114.14i	1.43022i
$780$	0	0
$781$	−295.743	−0.378672
$782$	151.547i	0.193794i
$783$	0	0
$784$	28.0000	0.0357143
$785$	− 95.9424i	− 0.122220i
$786$	0	0
$787$	−370.660	−0.470978	−0.235489	−	0.971877i	$-0.575669\pi$
$787$	−370.660	−0.470978	−0.235489	+	0.971877i	$0.575669\pi$
$788$	− 248.492i	− 0.315345i
$789$	0	0
$790$	151.470	0.191734
$791$	− 180.055i	− 0.227630i
$792$	0	0
$793$	214.952	0.271061
$794$	− 661.609i	− 0.833261i
$795$	0	0
$796$	−26.0338	−0.0327058
$797$	1107.06i	1.38903i	0.719476	+	0.694517i	$0.244382\pi$
$797$	1107.06i	1.38903i	−0.719476	+	0.694517i	$0.755618\pi$
$798$	0	0
$799$	475.093	0.594610
$800$	138.083i	0.172604i
$801$	0	0
$802$	−41.4520	−0.0516858
$803$	− 512.055i	− 0.637678i
$804$	0	0
$805$	−25.4584	−0.0316254
$806$	84.9398i	0.105384i
$807$	0	0
$808$	186.450	0.230755
$809$	− 256.032i	− 0.316480i	−0.987401	−	0.158240i	$-0.949418\pi$
$809$	− 256.032i	− 0.316480i	0.987401	−	0.158240i	$-0.0505819\pi$
$810$	0	0
$811$	−398.768	−0.491699	−0.245849	−	0.969308i	$-0.579067\pi$
$811$	−398.768	−0.491699	−0.245849	+	0.969308i	$0.579067\pi$
$812$	86.9543i	0.107087i
$813$	0	0
$814$	−146.166	−0.179565
$815$	− 57.1130i	− 0.0700773i
$816$	0	0
$817$	−968.031	−1.18486
$818$	− 488.370i	− 0.597029i
$819$	0	0
$820$	91.0529	0.111040
$821$	1323.89i	1.61253i	0.591553	+	0.806266i	$0.298515\pi$
$821$	1323.89i	1.61253i	−0.591553	+	0.806266i	$0.701485\pi$
$822$	0	0
$823$	1121.87	1.36315	0.681573	−	0.731750i	$-0.261296\pi$
$823$	1121.87	1.36315	0.681573	+	0.731750i	$0.261296\pi$
$824$	− 30.9986i	− 0.0376197i
$825$	0	0
$826$	240.868	0.291607
$827$	65.6004i	0.0793233i	0.999213	+	0.0396617i	$0.0126280\pi$
$827$	65.6004i	0.0793233i	−0.999213	+	0.0396617i	$0.987372\pi$
$828$	0	0
$829$	923.368	1.11383	0.556917	−	0.830568i	$-0.311984\pi$
$829$	923.368	1.11383	0.556917	+	0.830568i	$0.311984\pi$
$830$	145.611i	0.175435i
$831$	0	0
$832$	−34.9058	−0.0419541
$833$	59.8832i	0.0718885i
$834$	0	0
$835$	40.1667	0.0481038
$836$	− 137.011i	− 0.163889i
$837$	0	0
$838$	216.908	0.258840
$839$	− 1414.94i	− 1.68646i	−0.537550	−	0.843232i	$-0.680650\pi$
$839$	− 1414.94i	− 1.68646i	0.537550	−	0.843232i	$-0.319350\pi$
$840$	0	0
$841$	570.963	0.678909
$842$	− 879.372i	− 1.04438i
$843$	0	0
$844$	31.8678	0.0377580
$845$	115.196i	0.136327i
$846$	0	0
$847$	285.001	0.336483
$848$	205.096i	0.241859i
$849$	0	0
$850$	−295.317	−0.347431
$851$	355.273i	0.417478i
$852$	0	0
$853$	559.586	0.656021	0.328010	−	0.944674i	$-0.393622\pi$
$853$	559.586	0.656021	0.328010	+	0.944674i	$0.393622\pi$
$854$	184.331i	0.215844i
$855$	0	0
$856$	−554.268	−0.647509
$857$	− 1088.85i	− 1.27054i	−0.772292	−	0.635268i	$-0.780890\pi$
$857$	− 1088.85i	− 1.27054i	0.772292	−	0.635268i	$-0.219110\pi$
$858$	0	0
$859$	193.871	0.225694	0.112847	−	0.993612i	$-0.464003\pi$
$859$	193.871	0.225694	0.112847	+	0.993612i	$0.464003\pi$
$860$	79.1123i	0.0919910i
$861$	0	0
$862$	63.6836	0.0738789
$863$	514.067i	0.595674i	0.954617	+	0.297837i	$0.0962652\pi$
$863$	514.067i	0.595674i	−0.954617	+	0.297837i	$0.903735\pi$
$864$	0	0
$865$	−225.163	−0.260304
$866$	665.997i	0.769050i
$867$	0	0
$868$	−72.8397	−0.0839167
$869$	− 508.100i	− 0.584695i
$870$	0	0
$871$	−215.507	−0.247424
$872$	− 513.191i	− 0.588521i
$873$	0	0
$874$	−333.022	−0.381032
$875$	− 100.420i	− 0.114766i
$876$	0	0
$877$	1098.75	1.25285	0.626425	−	0.779482i	$-0.284518\pi$
$877$	1098.75	1.25285	0.626425	+	0.779482i	$0.284518\pi$
$878$	− 561.596i	− 0.639631i
$879$	0	0
$880$	−11.1972	−0.0127241
$881$	− 532.250i	− 0.604143i	−0.953285	−	0.302072i	$-0.902322\pi$
$881$	− 532.250i	− 0.604143i	0.953285	−	0.302072i	$-0.0976782\pi$
$882$	0	0
$883$	516.111	0.584498	0.292249	−	0.956342i	$-0.405596\pi$
$883$	516.111	0.584498	0.292249	+	0.956342i	$0.405596\pi$
$884$	− 74.6524i	− 0.0844484i
$885$	0	0
$886$	395.848	0.446781
$887$	761.192i	0.858164i	0.903265	+	0.429082i	$0.141163\pi$
$887$	761.192i	0.858164i	−0.903265	+	0.429082i	$0.858837\pi$
$888$	0	0
$889$	−189.003	−0.212601
$890$	− 9.71203i	− 0.0109124i
$891$	0	0
$892$	188.157	0.210938
$893$	1044.01i	1.16910i
$894$	0	0
$895$	107.866	0.120521
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	−620.392	−0.690860
$899$	− 226.204i	− 0.251618i
$900$	0	0
$901$	−438.636	−0.486832
$902$	− 305.433i	− 0.338618i
$903$	0	0
$904$	−192.487	−0.212928
$905$	64.0489i	0.0707723i
$906$	0	0
$907$	−1599.51	−1.76352	−0.881760	−	0.471698i	$-0.843641\pi$
$907$	−1599.51	−1.76352	−0.881760	+	0.471698i	$0.843641\pi$
$908$	− 86.6264i	− 0.0954035i
$909$	0	0
$910$	12.5409	0.0137812
$911$	− 667.717i	− 0.732949i	−0.930428	−	0.366475i	$-0.880565\pi$
$911$	− 667.717i	− 0.732949i	0.930428	−	0.366475i	$-0.119435\pi$
$912$	0	0
$913$	488.447	0.534991
$914$	62.3506i	0.0682173i
$915$	0	0
$916$	439.652	0.479969
$917$	378.816i	0.413103i
$918$	0	0
$919$	971.428	1.05705	0.528524	−	0.848918i	$-0.322746\pi$
$919$	971.428	1.05705	0.528524	+	0.848918i	$0.322746\pi$
$920$	27.2162i	0.0295828i
$921$	0	0
$922$	−978.376	−1.06115
$923$	− 354.101i	− 0.383641i
$924$	0	0
$925$	−692.314	−0.748448
$926$	− 1304.85i	− 1.40913i
$927$	0	0
$928$	92.9580	0.100170
$929$	308.752i	0.332348i	0.986096	+	0.166174i	$0.0531414\pi$
$929$	308.752i	0.332348i	−0.986096	+	0.166174i	$0.946859\pi$
$930$	0	0
$931$	−131.592	−0.141345
$932$	− 474.211i	− 0.508811i
$933$	0	0
$934$	1133.64	1.21374
$935$	− 23.9474i	− 0.0256122i
$936$	0	0
$937$	−18.3558	−0.0195900	−0.00979498	−	0.999952i	$-0.503118\pi$
$937$	−18.3558	−0.0195900	−0.00979498	+	0.999952i	$0.503118\pi$
$938$	− 184.807i	− 0.197022i
$939$	0	0
$940$	85.3216	0.0907677
$941$	1341.24i	1.42533i	0.701502	+	0.712667i	$0.252513\pi$
$941$	1341.24i	1.42533i	−0.701502	+	0.712667i	$0.747487\pi$
$942$	0	0
$943$	−742.391	−0.787265
$944$	− 257.498i	− 0.272774i
$945$	0	0
$946$	265.379	0.280527
$947$	1404.01i	1.48259i	0.671180	+	0.741294i	$0.265788\pi$
$947$	1404.01i	1.48259i	−0.671180	+	0.741294i	$0.734212\pi$
$948$	0	0
$949$	613.098	0.646046
$950$	− 648.954i	− 0.683109i
$951$	0	0
$952$	64.0178	0.0672456
$953$	− 1299.63i	− 1.36372i	−0.731483	−	0.681860i	$-0.761171\pi$
$953$	− 1299.63i	− 1.36372i	0.731483	−	0.681860i	$-0.238829\pi$
$954$	0	0
$955$	85.2473	0.0892642
$956$	902.940i	0.944498i
$957$	0	0
$958$	−314.350	−0.328131
$959$	− 351.147i	− 0.366160i
$960$	0	0
$961$	−771.514	−0.802824
$962$	− 175.009i	− 0.181922i
$963$	0	0
$964$	69.9898	0.0726036
$965$	244.233i	0.253091i
$966$	0	0
$967$	−1331.90	−1.37735	−0.688676	−	0.725069i	$-0.741808\pi$
$967$	−1331.90	−1.37735	−0.688676	+	0.725069i	$0.741808\pi$
$968$	− 304.679i	− 0.314751i
$969$	0	0
$970$	−98.3152	−0.101356
$971$	− 611.052i	− 0.629301i	−0.949208	−	0.314651i	$-0.898113\pi$
$971$	− 611.052i	− 0.629301i	0.949208	−	0.314651i	$-0.101887\pi$
$972$	0	0
$973$	522.556	0.537057
$974$	326.485i	0.335200i
$975$	0	0
$976$	197.058	0.201903
$977$	− 946.973i	− 0.969266i	−0.874718	−	0.484633i	$-0.838953\pi$
$977$	− 946.973i	− 0.969266i	0.874718	−	0.484633i	$-0.161047\pi$
$978$	0	0
$979$	−32.5786	−0.0332774
$980$	10.7544i	0.0109738i
$981$	0	0
$982$	938.538	0.955741
$983$	− 609.254i	− 0.619790i	−0.950771	−	0.309895i	$-0.899706\pi$
$983$	− 609.254i	− 0.619790i	0.950771	−	0.309895i	$-0.100294\pi$
$984$	0	0
$985$	95.4419	0.0968953
$986$	198.808i	0.201631i
$987$	0	0
$988$	164.047	0.166040
$989$	− 645.034i	− 0.652209i
$990$	0	0
$991$	1001.54	1.01064	0.505319	−	0.862933i	$-0.331375\pi$
$991$	1001.54	1.01064	0.505319	+	0.862933i	$0.331375\pi$
$992$	77.8689i	0.0784968i
$993$	0	0
$994$	303.657	0.305490
$995$	− 9.99918i	− 0.0100494i
$996$	0	0
$997$	−73.9153	−0.0741377	−0.0370688	−	0.999313i	$-0.511802\pi$
$997$	−73.9153	−0.0741377	−0.0370688	+	0.999313i	$0.511802\pi$
$998$	16.5739i	0.0166071i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.3.b.c.323.18		24
3.2	odd	2	inner	1134.3.b.c.323.7		24
9.2	odd	6		378.3.q.a.71.4		24
9.4	even	3		378.3.q.a.197.4		24
9.5	odd	6		126.3.q.a.29.12	✓	24
9.7	even	3		126.3.q.a.113.12	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.q.a.29.12	✓	24	9.5	odd	6
126.3.q.a.113.12	yes	24	9.7	even	3
378.3.q.a.71.4		24	9.2	odd	6
378.3.q.a.197.4		24	9.4	even	3
1134.3.b.c.323.7		24	3.2	odd	2	inner
1134.3.b.c.323.18		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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -0.768169i q^{5} +2.64575 q^{7} -2.82843i q^{8} +1.08636 q^{10} -3.64413i q^{11} +4.36322 q^{13} +3.74166i q^{14} +4.00000 q^{16} +8.55474i q^{17} -18.7989 q^{19} +1.53634i q^{20} +5.15358 q^{22} -12.5264i q^{23} +24.4099 q^{25} +6.17053i q^{26} -5.29150 q^{28} -16.4328i q^{29} +13.7654 q^{31} +5.65685i q^{32} -12.0982 q^{34} -2.03238i q^{35} -28.3620 q^{37} -26.5857i q^{38} -2.17271 q^{40} -59.2662i q^{41} +51.4940 q^{43} +7.28827i q^{44} +17.7150 q^{46} -55.5357i q^{47} +7.00000 q^{49} +34.5208i q^{50} -8.72644 q^{52} +51.2740i q^{53} -2.79931 q^{55} -7.48331i q^{56} +23.2395 q^{58} -64.3746i q^{59} +49.2644 q^{61} +19.4672i q^{62} -8.00000 q^{64} -3.35169i q^{65} -49.3916 q^{67} -17.1095i q^{68} +2.87423 q^{70} -81.1558i q^{71} +140.515 q^{73} -40.1099i q^{74} +37.5978 q^{76} -9.64147i q^{77} +139.430 q^{79} -3.07268i q^{80} +83.8150 q^{82} +134.036i q^{83} +6.57148 q^{85} +72.8236i q^{86} -10.3072 q^{88} -8.94002i q^{89} +11.5440 q^{91} +25.0528i q^{92} +78.5393 q^{94} +14.4407i q^{95} -90.5001 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{4} + 96 q^{16} + 24 q^{19} - 48 q^{22} - 144 q^{25} + 120 q^{31} + 96 q^{34} - 168 q^{37} - 120 q^{43} + 168 q^{49} + 264 q^{55} - 192 q^{64} - 144 q^{67} + 24 q^{73} - 48 q^{76} - 24 q^{79}+ \cdots - 96 q^{97}+O(q^{100}) \) 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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