LMFDB - Embedded newform 4950.2.a.ba.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4950,2,Mod(1,4950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4950.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4950.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,0,0,-3,1,0,0,-1,0,-4,-3,0,1,3,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.5259490005$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4950.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{7} +1.00000 q^{8} -1.00000 q^{11} -4.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -5.00000 q^{19} -1.00000 q^{22} +4.00000 q^{23} -4.00000 q^{26} -3.00000 q^{28} -5.00000 q^{29} +7.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} +7.00000 q^{37} -5.00000 q^{38} +8.00000 q^{41} +6.00000 q^{43} -1.00000 q^{44} +4.00000 q^{46} +8.00000 q^{47} +2.00000 q^{49} -4.00000 q^{52} +9.00000 q^{53} -3.00000 q^{56} -5.00000 q^{58} -13.0000 q^{61} +7.00000 q^{62} +1.00000 q^{64} +12.0000 q^{67} +3.00000 q^{68} +3.00000 q^{71} +6.00000 q^{73} +7.00000 q^{74} -5.00000 q^{76} +3.00000 q^{77} +8.00000 q^{82} +4.00000 q^{83} +6.00000 q^{86} -1.00000 q^{88} +15.0000 q^{89} +12.0000 q^{91} +4.00000 q^{92} +8.00000 q^{94} +12.0000 q^{97} +2.00000 q^{98} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	0	0
$26$	−4.00000	−0.784465
$27$	0	0
$28$	−3.00000	−0.566947
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	−5.00000	−0.811107
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	4.00000	0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	−4.00000	−0.554700
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−5.00000	−0.656532
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	7.00000	0.889001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	−5.00000	−0.573539
$77$	3.00000	0.341882
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	8.00000	0.883452
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	6.00000	0.646997
$87$	0	0
$88$	−1.00000	−0.106600
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	4.00000	0.417029
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	2.00000	0.202031
$99$	0	0
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	9.00000	0.874157
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	−3.00000	−0.283473
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	−5.00000	−0.464238
$117$	0	0
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	1.00000	0.0909091
$122$	−13.0000	−1.17696
$123$	0	0
$124$	7.00000	0.628619
$125$	0	0
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−7.00000	−0.611593	−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000	−0.611593	−0.305796	+	0.952097i	$0.598923\pi$
$132$	0	0
$133$	15.0000	1.30066
$134$	12.0000	1.03664
$135$	0	0
$136$	3.00000	0.257248
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	0	0
$142$	3.00000	0.251754
$143$	4.00000	0.334497
$144$	0	0
$145$	0	0
$146$	6.00000	0.496564
$147$	0	0
$148$	7.00000	0.575396
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−5.00000	−0.405554
$153$	0	0
$154$	3.00000	0.241747
$155$	0	0
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	21.0000	1.64485	0.822423	−	0.568876i	$-0.192621\pi$
$163$	21.0000	1.64485	0.822423	+	0.568876i	$0.192621\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	4.00000	0.310460
$167$	23.0000	1.77979	0.889897	−	0.456162i	$-0.150776\pi$
$167$	23.0000	1.77979	0.889897	+	0.456162i	$0.150776\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	6.00000	0.457496
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	15.0000	1.12430
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	12.0000	0.889499
$183$	0	0
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	−3.00000	−0.219382
$188$	8.00000	0.583460
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	1.00000	0.0719816	0.0359908	−	0.999352i	$-0.488541\pi$
$193$	1.00000	0.0719816	0.0359908	+	0.999352i	$0.488541\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	2.00000	0.142857
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	25.0000	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$200$	0	0
$201$	0	0
$202$	−2.00000	−0.140720
$203$	15.0000	1.05279
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−4.00000	−0.277350
$209$	5.00000	0.345857
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	−21.0000	−1.42557
$218$	−10.0000	−0.677285
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	−5.00000	−0.328266
$233$	19.0000	1.24473	0.622366	−	0.782727i	$-0.286172\pi$
$233$	19.0000	1.24473	0.622366	+	0.782727i	$0.286172\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−9.00000	−0.583383
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−13.0000	−0.832240
$245$	0	0
$246$	0	0
$247$	20.0000	1.27257
$248$	7.00000	0.444500
$249$	0	0
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.0000	1.74659	0.873296	−	0.487190i	$-0.161978\pi$
$257$	28.0000	1.74659	0.873296	+	0.487190i	$0.161978\pi$
$258$	0	0
$259$	−21.0000	−1.30488
$260$	0	0
$261$	0	0
$262$	−7.00000	−0.432461
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	0	0
$265$	0	0
$266$	15.0000	0.919709
$267$	0	0
$268$	12.0000	0.733017
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	20.0000	1.19952
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	4.00000	0.236525
$287$	−24.0000	−1.41668
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	6.00000	0.351123
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	0	0
$295$	0	0
$296$	7.00000	0.406867
$297$	0	0
$298$	5.00000	0.289642
$299$	−16.0000	−0.925304
$300$	0	0
$301$	−18.0000	−1.03750
$302$	2.00000	0.115087
$303$	0	0
$304$	−5.00000	−0.286770
$305$	0	0
$306$	0	0
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	0	0
$311$	13.0000	0.737162	0.368581	−	0.929596i	$-0.379844\pi$
$311$	13.0000	0.737162	0.368581	+	0.929596i	$0.379844\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	0	0
$317$	23.0000	1.29181	0.645904	−	0.763418i	$-0.276480\pi$
$317$	23.0000	1.29181	0.645904	+	0.763418i	$0.276480\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	0	0
$321$	0	0
$322$	−12.0000	−0.668734
$323$	−15.0000	−0.834622
$324$	0	0
$325$	0	0
$326$	21.0000	1.16308
$327$	0	0
$328$	8.00000	0.441726
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	23.0000	1.25850
$335$	0	0
$336$	0	0
$337$	7.00000	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	0	0
$341$	−7.00000	−0.379071
$342$	0	0
$343$	15.0000	0.809924
$344$	6.00000	0.323498
$345$	0	0
$346$	14.0000	0.752645
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	15.0000	0.794998
$357$	0	0
$358$	−10.0000	−0.528516
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−18.0000	−0.946059
$363$	0	0
$364$	12.0000	0.628971
$365$	0	0
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	0	0
$371$	−27.0000	−1.40177
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	−3.00000	−0.155126
$375$	0	0
$376$	8.00000	0.412568
$377$	20.0000	1.03005
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	0	0
$382$	8.00000	0.409316
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	0	0
$386$	1.00000	0.0508987
$387$	0	0
$388$	12.0000	0.609208
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	2.00000	0.101015
$393$	0	0
$394$	−22.0000	−1.10834
$395$	0	0
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	25.0000	1.25314
$399$	0	0
$400$	0	0
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	0	0
$403$	−28.0000	−1.39478
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	15.0000	0.744438
$407$	−7.00000	−0.346977
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	0	0
$412$	−4.00000	−0.197066
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−4.00000	−0.196116
$417$	0	0
$418$	5.00000	0.244558
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	−13.0000	−0.632830
$423$	0	0
$424$	9.00000	0.437079
$425$	0	0
$426$	0	0
$427$	39.0000	1.88734
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	−21.0000	−1.00803
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−20.0000	−0.956730
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	−12.0000	−0.570782
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	0	0
$446$	−14.0000	−0.662919
$447$	0	0
$448$	−3.00000	−0.141737
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	−6.00000	−0.282216
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	27.0000	1.26301	0.631503	−	0.775373i	$-0.282438\pi$
$457$	27.0000	1.26301	0.631503	+	0.775373i	$0.282438\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.0000	0.605470	0.302735	−	0.953075i	$-0.402100\pi$
$461$	13.0000	0.605470	0.302735	+	0.953075i	$0.402100\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−5.00000	−0.232119
$465$	0	0
$466$	19.0000	0.880158
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	0	0
$469$	−36.0000	−1.66233
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	0	0
$476$	−9.00000	−0.412514
$477$	0	0
$478$	0	0
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	−28.0000	−1.27669
$482$	−8.00000	−0.364390
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	−13.0000	−0.588482
$489$	0	0
$490$	0	0
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	0	0
$493$	−15.0000	−0.675566
$494$	20.0000	0.899843
$495$	0	0
$496$	7.00000	0.314309
$497$	−9.00000	−0.403705
$498$	0	0
$499$	−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
$501$	0	0
$502$	−2.00000	−0.0892644
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	0	0
$506$	−4.00000	−0.177822
$507$	0	0
$508$	−8.00000	−0.354943
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	1.00000	0.0441942
$513$	0	0
$514$	28.0000	1.23503
$515$	0	0
$516$	0	0
$517$	−8.00000	−0.351840
$518$	−21.0000	−0.922687
$519$	0	0
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−7.00000	−0.305796
$525$	0	0
$526$	9.00000	0.392419
$527$	21.0000	0.914774
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	15.0000	0.650332
$533$	−32.0000	−1.38607
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	22.0000	0.944981
$543$	0	0
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	0	0
$551$	25.0000	1.06504
$552$	0	0
$553$	0	0
$554$	−28.0000	−1.18961
$555$	0	0
$556$	20.0000	0.848189
$557$	28.0000	1.18640	0.593199	−	0.805056i	$-0.297865\pi$
$557$	28.0000	1.18640	0.593199	+	0.805056i	$0.297865\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	0	0
$566$	16.0000	0.672530
$567$	0	0
$568$	3.00000	0.125877
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	−3.00000	−0.125546	−0.0627730	−	0.998028i	$-0.519994\pi$
$571$	−3.00000	−0.125546	−0.0627730	+	0.998028i	$0.519994\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	−24.0000	−1.00174
$575$	0	0
$576$	0	0
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−9.00000	−0.372742
$584$	6.00000	0.248282
$585$	0	0
$586$	−16.0000	−0.660954
$587$	−7.00000	−0.288921	−0.144460	−	0.989511i	$-0.546145\pi$
$587$	−7.00000	−0.288921	−0.144460	+	0.989511i	$0.546145\pi$
$588$	0	0
$589$	−35.0000	−1.44215
$590$	0	0
$591$	0	0
$592$	7.00000	0.287698
$593$	−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
$596$	5.00000	0.204808
$597$	0	0
$598$	−16.0000	−0.654289
$599$	5.00000	0.204294	0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000	0.204294	0.102147	+	0.994769i	$0.467429\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−18.0000	−0.733625
$603$	0	0
$604$	2.00000	0.0813788
$605$	0	0
$606$	0	0
$607$	7.00000	0.284121	0.142061	−	0.989858i	$-0.454627\pi$
$607$	7.00000	0.284121	0.142061	+	0.989858i	$0.454627\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	0	0
$611$	−32.0000	−1.29458
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	3.00000	0.120873
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	0	0
$622$	13.0000	0.521253
$623$	−45.0000	−1.80289
$624$	0	0
$625$	0	0
$626$	−14.0000	−0.559553
$627$	0	0
$628$	−13.0000	−0.518756
$629$	21.0000	0.837325
$630$	0	0
$631$	−43.0000	−1.71180	−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000	−1.71180	−0.855901	+	0.517139i	$0.826997\pi$
$632$	0	0
$633$	0	0
$634$	23.0000	0.913447
$635$	0	0
$636$	0	0
$637$	−8.00000	−0.316972
$638$	5.00000	0.197952
$639$	0	0
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	1.00000	0.0394362	0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000	0.0394362	0.0197181	+	0.999806i	$0.493723\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−15.0000	−0.590167
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	21.0000	0.822423
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	0	0
$655$	0	0
$656$	8.00000	0.312348
$657$	0	0
$658$	−24.0000	−0.935617
$659$	35.0000	1.36341	0.681703	−	0.731629i	$-0.261240\pi$
$659$	35.0000	1.36341	0.681703	+	0.731629i	$0.261240\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	−18.0000	−0.699590
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	0	0
$667$	−20.0000	−0.774403
$668$	23.0000	0.889897
$669$	0	0
$670$	0	0
$671$	13.0000	0.501859
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	7.00000	0.269630
$675$	0	0
$676$	3.00000	0.115385
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	0	0
$679$	−36.0000	−1.38155
$680$	0	0
$681$	0	0
$682$	−7.00000	−0.268044
$683$	49.0000	1.87493	0.937466	−	0.348076i	$-0.113165\pi$
$683$	49.0000	1.87493	0.937466	+	0.348076i	$0.113165\pi$
$684$	0	0
$685$	0	0
$686$	15.0000	0.572703
$687$	0	0
$688$	6.00000	0.228748
$689$	−36.0000	−1.37149
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	18.0000	0.683271
$695$	0	0
$696$	0	0
$697$	24.0000	0.909065
$698$	−10.0000	−0.378506
$699$	0	0
$700$	0	0
$701$	−27.0000	−1.01978	−0.509888	−	0.860241i	$-0.670313\pi$
$701$	−27.0000	−1.01978	−0.509888	+	0.860241i	$0.670313\pi$
$702$	0	0
$703$	−35.0000	−1.32005
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	14.0000	0.526897
$707$	6.00000	0.225653
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	15.0000	0.562149
$713$	28.0000	1.04861
$714$	0	0
$715$	0	0
$716$	−10.0000	−0.373718
$717$	0	0
$718$	0	0
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	6.00000	0.223297
$723$	0	0
$724$	−18.0000	−0.668965
$725$	0	0
$726$	0	0
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	12.0000	0.444750
$729$	0	0
$730$	0	0
$731$	18.0000	0.665754
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	4.00000	0.147442
$737$	−12.0000	−0.442026
$738$	0	0
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	0	0
$742$	−27.0000	−0.991201
$743$	−1.00000	−0.0366864	−0.0183432	−	0.999832i	$-0.505839\pi$
$743$	−1.00000	−0.0366864	−0.0183432	+	0.999832i	$0.505839\pi$
$744$	0	0
$745$	0	0
$746$	−4.00000	−0.146450
$747$	0	0
$748$	−3.00000	−0.109691
$749$	36.0000	1.31541
$750$	0	0
$751$	27.0000	0.985244	0.492622	−	0.870243i	$-0.336039\pi$
$751$	27.0000	0.985244	0.492622	+	0.870243i	$0.336039\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	20.0000	0.728357
$755$	0	0
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−10.0000	−0.363216
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	30.0000	1.08607
$764$	8.00000	0.289430
$765$	0	0
$766$	−6.00000	−0.216789
$767$	0	0
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	0	0
$772$	1.00000	0.0359908
$773$	−11.0000	−0.395643	−0.197821	−	0.980238i	$-0.563387\pi$
$773$	−11.0000	−0.395643	−0.197821	+	0.980238i	$0.563387\pi$
$774$	0	0
$775$	0	0
$776$	12.0000	0.430775
$777$	0	0
$778$	30.0000	1.07555
$779$	−40.0000	−1.43315
$780$	0	0
$781$	−3.00000	−0.107348
$782$	12.0000	0.429119
$783$	0	0
$784$	2.00000	0.0714286
$785$	0	0
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	−22.0000	−0.783718
$789$	0	0
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	52.0000	1.84657
$794$	22.0000	0.780751
$795$	0	0
$796$	25.0000	0.886102
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	0	0
$802$	−27.0000	−0.953403
$803$	−6.00000	−0.211735
$804$	0	0
$805$	0	0
$806$	−28.0000	−0.986258
$807$	0	0
$808$	−2.00000	−0.0703598
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	37.0000	1.29925	0.649623	−	0.760257i	$-0.274927\pi$
$811$	37.0000	1.29925	0.649623	+	0.760257i	$0.274927\pi$
$812$	15.0000	0.526397
$813$	0	0
$814$	−7.00000	−0.245350
$815$	0	0
$816$	0	0
$817$	−30.0000	−1.04957
$818$	20.0000	0.699284
$819$	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	$0.625420\pi$
$822$	0	0
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	0	0
$832$	−4.00000	−0.138675
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	5.00000	0.172929
$837$	0	0
$838$	20.0000	0.690889
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−28.0000	−0.964944
$843$	0	0
$844$	−13.0000	−0.447478
$845$	0	0
$846$	0	0
$847$	−3.00000	−0.103081
$848$	9.00000	0.309061
$849$	0	0
$850$	0	0
$851$	28.0000	0.959828
$852$	0	0
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	39.0000	1.33455
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−17.0000	−0.580709	−0.290354	−	0.956919i	$-0.593773\pi$
$857$	−17.0000	−0.580709	−0.290354	+	0.956919i	$0.593773\pi$
$858$	0	0
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	0	0
$861$	0	0
$862$	−32.0000	−1.08992
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	0	0
$865$	0	0
$866$	−14.0000	−0.475739
$867$	0	0
$868$	−21.0000	−0.712786
$869$	0	0
$870$	0	0
$871$	−48.0000	−1.62642
$872$	−10.0000	−0.338643
$873$	0	0
$874$	−20.0000	−0.676510
$875$	0	0
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	10.0000	0.337484
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−9.00000	−0.302874	−0.151437	−	0.988467i	$-0.548390\pi$
$883$	−9.00000	−0.302874	−0.151437	+	0.988467i	$0.548390\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	4.00000	0.134383
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	0	0
$892$	−14.0000	−0.468755
$893$	−40.0000	−1.33855
$894$	0	0
$895$	0	0
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−30.0000	−1.00111
$899$	−35.0000	−1.16732
$900$	0	0
$901$	27.0000	0.899500
$902$	−8.00000	−0.266371
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	−23.0000	−0.763702	−0.381851	−	0.924224i	$-0.624713\pi$
$907$	−23.0000	−0.763702	−0.381851	+	0.924224i	$0.624713\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	0	0
$911$	−37.0000	−1.22586	−0.612932	−	0.790135i	$-0.710010\pi$
$911$	−37.0000	−1.22586	−0.612932	+	0.790135i	$0.710010\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	27.0000	0.893081
$915$	0	0
$916$	0	0
$917$	21.0000	0.693481
$918$	0	0
$919$	50.0000	1.64935	0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000	1.64935	0.824674	+	0.565608i	$0.191359\pi$
$920$	0	0
$921$	0	0
$922$	13.0000	0.428132
$923$	−12.0000	−0.394985
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	0	0
$928$	−5.00000	−0.164133
$929$	−15.0000	−0.492134	−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000	−0.492134	−0.246067	+	0.969253i	$0.579138\pi$
$930$	0	0
$931$	−10.0000	−0.327737
$932$	19.0000	0.622366
$933$	0	0
$934$	3.00000	0.0981630
$935$	0	0
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	−36.0000	−1.17544
$939$	0	0
$940$	0	0
$941$	−17.0000	−0.554184	−0.277092	−	0.960843i	$-0.589371\pi$
$941$	−17.0000	−0.554184	−0.277092	+	0.960843i	$0.589371\pi$
$942$	0	0
$943$	32.0000	1.04206
$944$	0	0
$945$	0	0
$946$	−6.00000	−0.195077
$947$	53.0000	1.72227	0.861134	−	0.508378i	$-0.169755\pi$
$947$	53.0000	1.72227	0.861134	+	0.508378i	$0.169755\pi$
$948$	0	0
$949$	−24.0000	−0.779073
$950$	0	0
$951$	0	0
$952$	−9.00000	−0.291692
$953$	29.0000	0.939402	0.469701	−	0.882826i	$-0.344362\pi$
$953$	29.0000	0.939402	0.469701	+	0.882826i	$0.344362\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	10.0000	0.323085
$959$	36.0000	1.16250
$960$	0	0
$961$	18.0000	0.580645
$962$	−28.0000	−0.902756
$963$	0	0
$964$	−8.00000	−0.257663
$965$	0	0
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	−60.0000	−1.92351
$974$	22.0000	0.704925
$975$	0	0
$976$	−13.0000	−0.416120
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	0	0
$979$	−15.0000	−0.479402
$980$	0	0
$981$	0	0
$982$	33.0000	1.05307
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	0	0
$986$	−15.0000	−0.477697
$987$	0	0
$988$	20.0000	0.636285
$989$	24.0000	0.763156
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	7.00000	0.222250
$993$	0	0
$994$	−9.00000	−0.285463
$995$	0	0
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	−40.0000	−1.26618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.a.ba.1.1		1
3.2	odd	2		550.2.a.e.1.1		1
5.2	odd	4		990.2.c.d.199.2		2
5.3	odd	4		990.2.c.d.199.1		2
5.4	even	2		4950.2.a.q.1.1		1
12.11	even	2		4400.2.a.k.1.1		1
15.2	even	4		110.2.b.a.89.1	✓	2
15.8	even	4		110.2.b.a.89.2	yes	2
15.14	odd	2		550.2.a.j.1.1		1
33.32	even	2		6050.2.a.bk.1.1		1
60.23	odd	4		880.2.b.a.529.1		2
60.47	odd	4		880.2.b.a.529.2		2
60.59	even	2		4400.2.a.s.1.1		1
165.32	odd	4		1210.2.b.a.969.2		2
165.98	odd	4		1210.2.b.a.969.1		2
165.164	even	2		6050.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.b.a.89.1	✓	2	15.2	even	4
110.2.b.a.89.2	yes	2	15.8	even	4
550.2.a.e.1.1		1	3.2	odd	2
550.2.a.j.1.1		1	15.14	odd	2
880.2.b.a.529.1		2	60.23	odd	4
880.2.b.a.529.2		2	60.47	odd	4
990.2.c.d.199.1		2	5.3	odd	4
990.2.c.d.199.2		2	5.2	odd	4
1210.2.b.a.969.1		2	165.98	odd	4
1210.2.b.a.969.2		2	165.32	odd	4
4400.2.a.k.1.1		1	12.11	even	2
4400.2.a.s.1.1		1	60.59	even	2
4950.2.a.q.1.1		1	5.4	even	2
4950.2.a.ba.1.1		1	1.1	even	1	trivial
6050.2.a.f.1.1		1	165.164	even	2
6050.2.a.bk.1.1		1	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4950.a (trivial)

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