LMFDB - Embedded newform 1950.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1,1950)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$15.5708283941$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} -10.0000 q^{29} +1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} +1.00000 q^{39} +2.00000 q^{41} +4.00000 q^{43} +4.00000 q^{46} +1.00000 q^{48} -7.00000 q^{49} +6.00000 q^{51} +1.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -10.0000 q^{58} +6.00000 q^{61} +1.00000 q^{64} -4.00000 q^{67} +6.00000 q^{68} +4.00000 q^{69} +16.0000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +6.00000 q^{74} +1.00000 q^{78} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} +4.00000 q^{86} -10.0000 q^{87} -6.00000 q^{89} +4.00000 q^{92} +1.00000 q^{96} -14.0000 q^{97} -7.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	−7.00000	−1.00000
$50$	0	0
$51$	6.00000	0.840168
$52$	1.00000	0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−10.0000	−1.31306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	4.00000	0.481543
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	0	0
$77$	0	0
$78$	1.00000	0.113228
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	−10.0000	−1.07211
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	4.00000	0.417029
$93$	0	0
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	6.00000	0.594089
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	6.00000	0.582772
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	1.00000	0.0962250
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	0	0
$116$	−10.0000	−0.928477
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	6.00000	0.543214
$123$	2.00000	0.180334
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	4.00000	0.340503
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	16.0000	1.34269
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	−7.00000	−0.577350
$148$	6.00000	0.493197
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	1.00000	0.0800641
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−10.0000	−0.758098
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−6.00000	−0.449719
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	1.00000	0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	6.00000	0.422159
$203$	0	0
$204$	6.00000	0.420084
$205$	0	0
$206$	−12.0000	−0.836080
$207$	4.00000	0.278019
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.00000	0.412082
$213$	16.0000	1.09630
$214$	4.00000	0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	−14.0000	−0.948200
$219$	2.00000	0.135147
$220$	0	0
$221$	6.00000	0.403604
$222$	6.00000	0.402694
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	−10.0000	−0.665190
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	−10.0000	−0.656532
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−11.0000	−0.707107
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	2.00000	0.127515
$247$	0	0
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	0	0
$261$	−10.0000	−0.618984
$262$	12.0000	0.741362
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.00000	−0.367194
$268$	−4.00000	−0.244339
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	4.00000	0.240772
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	−14.0000	−0.820695
$292$	2.00000	0.117041
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	−14.0000	−0.810998
$299$	4.00000	0.231326
$300$	0	0
$301$	0	0
$302$	−16.0000	−0.920697
$303$	6.00000	0.344691
$304$	0	0
$305$	0	0
$306$	6.00000	0.342997
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	1.00000	0.0566139
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	−14.0000	−0.774202
$328$	2.00000	0.110432
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−4.00000	−0.219529
$333$	6.00000	0.328798
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	1.00000	0.0543928
$339$	−10.0000	−0.543125
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−10.0000	−0.536056
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−20.0000	−1.05703
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−10.0000	−0.525588
$363$	−11.0000	−0.577350
$364$	0	0
$365$	0	0
$366$	6.00000	0.313625
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	4.00000	0.208514
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.0000	−0.515026
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	24.0000	1.22795
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	4.00000	0.203331
$388$	−14.0000	−0.710742
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	−7.00000	−0.353553
$393$	12.0000	0.605320
$394$	22.0000	1.10834
$395$	0	0
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−4.00000	−0.199502
$403$	0	0
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	0	0
$408$	6.00000	0.297044
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	−12.0000	−0.591198
$413$	0	0
$414$	4.00000	0.196589
$415$	0	0
$416$	1.00000	0.0490290
$417$	4.00000	0.195881
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	16.0000	0.775203
$427$	0	0
$428$	4.00000	0.193347
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	1.00000	0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	0	0
$436$	−14.0000	−0.670478
$437$	0	0
$438$	2.00000	0.0955637
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	6.00000	0.285391
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	−16.0000	−0.757622
$447$	−14.0000	−0.662177
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	0	0
$452$	−10.0000	−0.470360
$453$	−16.0000	−0.751746
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	2.00000	0.0934539
$459$	6.00000	0.280056
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	−18.0000	−0.833834
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.00000	0.274721
$478$	16.0000	0.731823
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	−14.0000	−0.637683
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	6.00000	0.271607
$489$	4.00000	0.180886
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	2.00000	0.0901670
$493$	−60.0000	−2.70226
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	−4.00000	−0.178529
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	12.0000	0.532414
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	−10.0000	−0.437688
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	28.0000	1.22086
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.00000	0.0866296
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−4.00000	−0.172774
$537$	−20.0000	−0.863064
$538$	14.0000	0.603583
$539$	0	0
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−24.0000	−1.03089
$543$	−10.0000	−0.429141
$544$	6.00000	0.257248
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−6.00000	−0.256307
$549$	6.00000	0.256074
$550$	0	0
$551$	0	0
$552$	4.00000	0.170251
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	4.00000	0.169638
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	0	0
$566$	12.0000	0.504398
$567$	0	0
$568$	16.0000	0.671345
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	19.0000	0.790296
$579$	−14.0000	−0.581820
$580$	0	0
$581$	0	0
$582$	−14.0000	−0.580319
$583$	0	0
$584$	2.00000	0.0827606
$585$	0	0
$586$	−26.0000	−1.07405
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−7.00000	−0.288675
$589$	0	0
$590$	0	0
$591$	22.0000	0.904959
$592$	6.00000	0.246598
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	0	0
$596$	−14.0000	−0.573462
$597$	−16.0000	−0.654836
$598$	4.00000	0.163572
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	−16.0000	−0.651031
$605$	0	0
$606$	6.00000	0.243733
$607$	36.0000	1.46119	0.730597	−	0.682808i	$-0.239242\pi$
$607$	36.0000	1.46119	0.730597	+	0.682808i	$0.239242\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	6.00000	0.242536
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	−12.0000	−0.482711
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−16.0000	−0.641542
$623$	0	0
$624$	1.00000	0.0400320
$625$	0	0
$626$	−26.0000	−1.03917
$627$	0	0
$628$	14.0000	0.558661
$629$	36.0000	1.43541
$630$	0	0
$631$	−48.0000	−1.91085	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000	−1.91085	−0.955425	+	0.295234i	$0.904602\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	6.00000	0.237915
$637$	−7.00000	−0.277350
$638$	0	0
$639$	16.0000	0.632950
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	4.00000	0.157867
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.00000	0.156652
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	2.00000	0.0780869
$657$	2.00000	0.0780274
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	8.00000	0.310929
$663$	6.00000	0.233021
$664$	−4.00000	−0.155230
$665$	0	0
$666$	6.00000	0.232495
$667$	−40.0000	−1.54881
$668$	−24.0000	−0.928588
$669$	−16.0000	−0.618596
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−42.0000	−1.61898	−0.809491	−	0.587133i	$-0.800257\pi$
$673$	−42.0000	−1.61898	−0.809491	+	0.587133i	$0.800257\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	1.00000	0.0384615
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	−10.0000	−0.384048
$679$	0	0
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.00000	0.0763048
$688$	4.00000	0.152499
$689$	6.00000	0.228582
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−10.0000	−0.379049
$697$	12.0000	0.454532
$698$	34.0000	1.28692
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	1.00000	0.0377426
$703$	0	0
$704$	0	0
$705$	0	0
$706$	26.0000	0.978523
$707$	0	0
$708$	0	0
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	0	0
$711$	0	0
$712$	−6.00000	−0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−20.0000	−0.747435
$717$	16.0000	0.597531
$718$	24.0000	0.895672
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	−19.0000	−0.707107
$723$	−14.0000	−0.520666
$724$	−10.0000	−0.371647
$725$	0	0
$726$	−11.0000	−0.408248
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	6.00000	0.221766
$733$	−50.0000	−1.84679	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000	−1.84679	−0.923396	+	0.383849i	$0.874598\pi$
$734$	−20.0000	−0.738213
$735$	0	0
$736$	4.00000	0.147442
$737$	0	0
$738$	2.00000	0.0736210
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	0	0
$746$	−10.0000	−0.366126
$747$	−4.00000	−0.146352
$748$	0	0
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	−10.0000	−0.364179
$755$	0	0
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	24.0000	0.871719
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	12.0000	0.434714
$763$	0	0
$764$	24.0000	0.868290
$765$	0	0
$766$	8.00000	0.289052
$767$	0	0
$768$	1.00000	0.0360844
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−14.0000	−0.503871
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	−18.0000	−0.645331
$779$	0	0
$780$	0	0
$781$	0	0
$782$	24.0000	0.858238
$783$	−10.0000	−0.357371
$784$	−7.00000	−0.250000
$785$	0	0
$786$	12.0000	0.428026
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	22.0000	0.783718
$789$	28.0000	0.996826
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.00000	0.213066
$794$	14.0000	0.496841
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	18.0000	0.635602
$803$	0	0
$804$	−4.00000	−0.141069
$805$	0	0
$806$	0	0
$807$	14.0000	0.492823
$808$	6.00000	0.211079
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	0	0
$815$	0	0
$816$	6.00000	0.210042
$817$	0	0
$818$	2.00000	0.0699284
$819$	0	0
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	−6.00000	−0.209274
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	4.00000	0.139010
$829$	54.0000	1.87550	0.937749	−	0.347314i	$-0.112906\pi$
$829$	54.0000	1.87550	0.937749	+	0.347314i	$0.112906\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	1.00000	0.0346688
$833$	−42.0000	−1.45521
$834$	4.00000	0.138509
$835$	0	0
$836$	0	0
$837$	0	0
$838$	12.0000	0.414533
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−22.0000	−0.758170
$843$	−6.00000	−0.206651
$844$	−12.0000	−0.413057
$845$	0	0
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	12.0000	0.411839
$850$	0	0
$851$	24.0000	0.822709
$852$	16.0000	0.548151
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	0	0
$855$	0	0
$856$	4.00000	0.136717
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	19.0000	0.645274
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	−14.0000	−0.474100
$873$	−14.0000	−0.473828
$874$	0	0
$875$	0	0
$876$	2.00000	0.0675737
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−24.0000	−0.809961
$879$	−26.0000	−0.876958
$880$	0	0
$881$	34.0000	1.14549	0.572745	−	0.819734i	$-0.305879\pi$
$881$	34.0000	1.14549	0.572745	+	0.819734i	$0.305879\pi$
$882$	−7.00000	−0.235702
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	4.00000	0.134383
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−16.0000	−0.535720
$893$	0	0
$894$	−14.0000	−0.468230
$895$	0	0
$896$	0	0
$897$	4.00000	0.133556
$898$	−14.0000	−0.467186
$899$	0	0
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	−10.0000	−0.332595
$905$	0	0
$906$	−16.0000	−0.531564
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−20.0000	−0.663723
$909$	6.00000	0.199007
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	0	0
$914$	−6.00000	−0.198462
$915$	0	0
$916$	2.00000	0.0660819
$917$	0	0
$918$	6.00000	0.198030
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	−14.0000	−0.461065
$923$	16.0000	0.526646
$924$	0	0
$925$	0	0
$926$	24.0000	0.788689
$927$	−12.0000	−0.394132
$928$	−10.0000	−0.328266
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	0	0
$931$	0	0
$932$	−18.0000	−0.589610
$933$	−16.0000	−0.523816
$934$	28.0000	0.916188
$935$	0	0
$936$	1.00000	0.0326860
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	−26.0000	−0.848478
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	14.0000	0.456145
$943$	8.00000	0.260516
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	6.00000	0.193448
$963$	4.00000	0.128898
$964$	−14.0000	−0.450910
$965$	0	0
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−8.00000	−0.256337
$975$	0	0
$976$	6.00000	0.192055
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	4.00000	0.127906
$979$	0	0
$980$	0	0
$981$	−14.0000	−0.446986
$982$	20.0000	0.638226
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	−60.0000	−1.91079
$987$	0	0
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	8.00000	0.253872
$994$	0	0
$995$	0	0
$996$	−4.00000	−0.126745
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	−40.0000	−1.26618
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.a.y.1.1		1
3.2	odd	2		5850.2.a.m.1.1		1
5.2	odd	4		1950.2.e.l.1249.2		2
5.3	odd	4		1950.2.e.l.1249.1		2
5.4	even	2		390.2.a.a.1.1	✓	1
15.2	even	4		5850.2.e.p.5149.1		2
15.8	even	4		5850.2.e.p.5149.2		2
15.14	odd	2		1170.2.a.m.1.1		1
20.19	odd	2		3120.2.a.q.1.1		1
60.59	even	2		9360.2.a.bn.1.1		1
65.34	odd	4		5070.2.b.c.1351.1		2
65.44	odd	4		5070.2.b.c.1351.2		2
65.64	even	2		5070.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.a.1.1	✓	1	5.4	even	2
1170.2.a.m.1.1		1	15.14	odd	2
1950.2.a.y.1.1		1	1.1	even	1	trivial
1950.2.e.l.1249.1		2	5.3	odd	4
1950.2.e.l.1249.2		2	5.2	odd	4
3120.2.a.q.1.1		1	20.19	odd	2
5070.2.a.s.1.1		1	65.64	even	2
5070.2.b.c.1351.1		2	65.34	odd	4
5070.2.b.c.1351.2		2	65.44	odd	4
5850.2.a.m.1.1		1	3.2	odd	2
5850.2.e.p.5149.1		2	15.2	even	4
5850.2.e.p.5149.2		2	15.8	even	4
9360.2.a.bn.1.1		1	60.59	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more