LMFDB - Embedded newform 1216.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.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:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +4.00000 q^{5} -3.00000 q^{7} -2.00000 q^{9} +2.00000 q^{11} +1.00000 q^{13} -4.00000 q^{15} +3.00000 q^{17} -1.00000 q^{19} +3.00000 q^{21} +1.00000 q^{23} +11.0000 q^{25} +5.00000 q^{27} +5.00000 q^{29} +8.00000 q^{31} -2.00000 q^{33} -12.0000 q^{35} +2.00000 q^{37} -1.00000 q^{39} -8.00000 q^{41} +4.00000 q^{43} -8.00000 q^{45} -8.00000 q^{47} +2.00000 q^{49} -3.00000 q^{51} +1.00000 q^{53} +8.00000 q^{55} +1.00000 q^{57} +15.0000 q^{59} -2.00000 q^{61} +6.00000 q^{63} +4.00000 q^{65} +3.00000 q^{67} -1.00000 q^{69} -2.00000 q^{71} +9.00000 q^{73} -11.0000 q^{75} -6.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +12.0000 q^{85} -5.00000 q^{87} -3.00000 q^{91} -8.00000 q^{93} -4.00000 q^{95} -2.00000 q^{97} -4.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	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
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−12.0000	−2.02837
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	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
$45$	−8.00000	−1.19257
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	15.0000	1.95283	0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000	1.95283	0.976417	+	0.215894i	$0.0692665\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0	0
$75$	−11.0000	−1.27017
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	0	0
$87$	−5.00000	−0.536056
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	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
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	12.0000	1.17108
$106$	0	0
$107$	−7.00000	−0.676716	−0.338358	−	0.941018i	$-0.609871\pi$
$107$	−7.00000	−0.676716	−0.338358	+	0.941018i	$0.609871\pi$
$108$	0	0
$109$	15.0000	1.43674	0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000	1.43674	0.718370	+	0.695662i	$0.244889\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	8.00000	0.721336
$124$	0	0
$125$	24.0000	2.14663
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	0	0
$129$	−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
$133$	3.00000	0.260133
$134$	0	0
$135$	20.0000	1.72133
$136$	0	0
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	20.0000	1.66091
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	32.0000	2.57030
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	−8.00000	−0.622799
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−33.0000	−2.49457
$176$	0	0
$177$	−15.0000	−1.12747
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	6.00000	0.438763
$188$	0	0
$189$	−15.0000	−1.09109
$190$	0	0
$191$	−7.00000	−0.506502	−0.253251	−	0.967401i	$-0.581500\pi$
$191$	−7.00000	−0.506502	−0.253251	+	0.967401i	$0.581500\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	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
$201$	−3.00000	−0.211604
$202$	0	0
$203$	−15.0000	−1.05279
$204$	0	0
$205$	−32.0000	−2.23498
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	27.0000	1.85876	0.929378	−	0.369129i	$-0.120344\pi$
$211$	27.0000	1.85876	0.929378	+	0.369129i	$0.120344\pi$
$212$	0	0
$213$	2.00000	0.137038
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	−9.00000	−0.608164
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	−22.0000	−1.46667
$226$	0	0
$227$	−17.0000	−1.12833	−0.564165	−	0.825662i	$-0.690802\pi$
$227$	−17.0000	−1.12833	−0.564165	+	0.825662i	$0.690802\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−32.0000	−2.08745
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	8.00000	0.511101
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−10.0000	−0.618984
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	22.0000	1.32665
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	−16.0000	−0.957895
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	60.0000	3.49334
$296$	0	0
$297$	10.0000	0.580259
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	−12.0000	−0.691669
$302$	0	0
$303$	2.00000	0.114897
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−7.00000	−0.396934	−0.198467	−	0.980108i	$-0.563596\pi$
$311$	−7.00000	−0.396934	−0.198467	+	0.980108i	$0.563596\pi$
$312$	0	0
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	0	0
$315$	24.0000	1.35225
$316$	0	0
$317$	27.0000	1.51647	0.758236	−	0.651981i	$-0.226062\pi$
$317$	27.0000	1.51647	0.758236	+	0.651981i	$0.226062\pi$
$318$	0	0
$319$	10.0000	0.559893
$320$	0	0
$321$	7.00000	0.390702
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	11.0000	0.610170
$326$	0	0
$327$	−15.0000	−0.829502
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	−14.0000	−0.760376
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	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
$351$	5.00000	0.266880
$352$	0	0
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	9.00000	0.476331
$358$	0	0
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	36.0000	1.88433
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	16.0000	0.832927
$370$	0	0
$371$	−3.00000	−0.155752
$372$	0	0
$373$	−29.0000	−1.50156	−0.750782	−	0.660551i	$-0.770323\pi$
$373$	−29.0000	−1.50156	−0.750782	+	0.660551i	$0.770323\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	0	0
$377$	5.00000	0.257513
$378$	0	0
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	0	0
$383$	26.0000	1.32854	0.664269	−	0.747494i	$-0.268743\pi$
$383$	26.0000	1.32854	0.664269	+	0.747494i	$0.268743\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	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
$391$	3.00000	0.151717
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	40.0000	2.01262
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	4.00000	0.198762
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	0	0
$411$	17.0000	0.838548
$412$	0	0
$413$	−45.0000	−2.21431
$414$	0	0
$415$	−24.0000	−1.17811
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	0	0
$423$	16.0000	0.777947
$424$	0	0
$425$	33.0000	1.60074
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	−20.0000	−0.958927
$436$	0	0
$437$	−1.00000	−0.0478365
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	−26.0000	−1.23530	−0.617649	−	0.786454i	$-0.711915\pi$
$443$	−26.0000	−1.23530	−0.617649	+	0.786454i	$0.711915\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	0	0
$453$	2.00000	0.0939682
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−7.00000	−0.327446	−0.163723	−	0.986506i	$-0.552350\pi$
$457$	−7.00000	−0.327446	−0.163723	+	0.986506i	$0.552350\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	28.0000	1.30409	0.652045	−	0.758180i	$-0.273911\pi$
$461$	28.0000	1.30409	0.652045	+	0.758180i	$0.273911\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	−32.0000	−1.48396
$466$	0	0
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	−11.0000	−0.504715
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	3.00000	0.136505
$484$	0	0
$485$	−8.00000	−0.363261
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	15.0000	0.675566
$494$	0	0
$495$	−16.0000	−0.719147
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	−39.0000	−1.73892	−0.869462	−	0.494000i	$-0.835534\pi$
$503$	−39.0000	−1.73892	−0.869462	+	0.494000i	$0.835534\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	−27.0000	−1.19441
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	24.0000	1.05757
$516$	0	0
$517$	−16.0000	−0.703679
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	0	0
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	0	0
$525$	33.0000	1.44024
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−30.0000	−1.30189
$532$	0	0
$533$	−8.00000	−0.346518
$534$	0	0
$535$	−28.0000	−1.21055
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	60.0000	2.57012
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	4.00000	0.170716
$550$	0	0
$551$	−5.00000	−0.213007
$552$	0	0
$553$	−30.0000	−1.27573
$554$	0	0
$555$	−8.00000	−0.339581
$556$	0	0
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	−6.00000	−0.253320
$562$	0	0
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	56.0000	2.35594
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	40.0000	1.67689	0.838444	−	0.544988i	$-0.183466\pi$
$569$	40.0000	1.67689	0.838444	+	0.544988i	$0.183466\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	7.00000	0.292429
$574$	0	0
$575$	11.0000	0.458732
$576$	0	0
$577$	−37.0000	−1.54033	−0.770165	−	0.637845i	$-0.779826\pi$
$577$	−37.0000	−1.54033	−0.770165	+	0.637845i	$0.779826\pi$
$578$	0	0
$579$	6.00000	0.249351
$580$	0	0
$581$	18.0000	0.746766
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	−8.00000	−0.330759
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	8.00000	0.329076
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	−36.0000	−1.47586
$596$	0	0
$597$	−25.0000	−1.02318
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	0	0
$603$	−6.00000	−0.244339
$604$	0	0
$605$	−28.0000	−1.13836
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	0	0
$609$	15.0000	0.607831
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	0	0
$615$	32.0000	1.29036
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	0	0
$623$	0	0
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	6.00000	0.239236
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	−27.0000	−1.07315
$634$	0	0
$635$	−72.0000	−2.85723
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	0	0
$647$	−23.0000	−0.904223	−0.452112	−	0.891961i	$-0.649329\pi$
$647$	−23.0000	−0.904223	−0.452112	+	0.891961i	$0.649329\pi$
$648$	0	0
$649$	30.0000	1.17760
$650$	0	0
$651$	24.0000	0.940634
$652$	0	0
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	48.0000	1.87552
$656$	0	0
$657$	−18.0000	−0.702247
$658$	0	0
$659$	5.00000	0.194772	0.0973862	−	0.995247i	$-0.468952\pi$
$659$	5.00000	0.194772	0.0973862	+	0.995247i	$0.468952\pi$
$660$	0	0
$661$	23.0000	0.894596	0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000	0.894596	0.447298	+	0.894385i	$0.352386\pi$
$662$	0	0
$663$	−3.00000	−0.116510
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	5.00000	0.193601
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	0	0
$675$	55.0000	2.11695
$676$	0	0
$677$	−13.0000	−0.499631	−0.249815	−	0.968294i	$-0.580370\pi$
$677$	−13.0000	−0.499631	−0.249815	+	0.968294i	$0.580370\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	17.0000	0.651441
$682$	0	0
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	−68.0000	−2.59815
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	1.00000	0.0380970
$690$	0	0
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	0	0
$693$	12.0000	0.455842
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	28.0000	1.05755	0.528773	−	0.848763i	$-0.322652\pi$
$701$	28.0000	1.05755	0.528773	+	0.848763i	$0.322652\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	32.0000	1.20519
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	0	0
$711$	−20.0000	−0.750059
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	15.0000	0.560185
$718$	0	0
$719$	5.00000	0.186469	0.0932343	−	0.995644i	$-0.470279\pi$
$719$	5.00000	0.186469	0.0932343	+	0.995644i	$0.470279\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	0	0
$723$	8.00000	0.297523
$724$	0	0
$725$	55.0000	2.04265
$726$	0	0
$727$	17.0000	0.630495	0.315248	−	0.949009i	$-0.397912\pi$
$727$	17.0000	0.630495	0.315248	+	0.949009i	$0.397912\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	36.0000	1.32969	0.664845	−	0.746981i	$-0.268498\pi$
$733$	36.0000	1.32969	0.664845	+	0.746981i	$0.268498\pi$
$734$	0	0
$735$	−8.00000	−0.295084
$736$	0	0
$737$	6.00000	0.221013
$738$	0	0
$739$	−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
$741$	1.00000	0.0367359
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	21.0000	0.767323
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	−2.00000	−0.0728841
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	−45.0000	−1.62911
$764$	0	0
$765$	−24.0000	−0.867722
$766$	0	0
$767$	15.0000	0.541619
$768$	0	0
$769$	−35.0000	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$770$	0	0
$771$	−8.00000	−0.288113
$772$	0	0
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	0	0
$775$	88.0000	3.16105
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	8.00000	0.286630
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	25.0000	0.893427
$784$	0	0
$785$	8.00000	0.285532
$786$	0	0
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	0	0
$791$	−42.0000	−1.49335
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	−4.00000	−0.141865
$796$	0	0
$797$	−3.00000	−0.106265	−0.0531327	−	0.998587i	$-0.516921\pi$
$797$	−3.00000	−0.106265	−0.0531327	+	0.998587i	$0.516921\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	0	0
$802$	0	0
$803$	18.0000	0.635206
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	30.0000	1.05605
$808$	0	0
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	0	0
$811$	−3.00000	−0.105344	−0.0526721	−	0.998612i	$-0.516774\pi$
$811$	−3.00000	−0.105344	−0.0526721	+	0.998612i	$0.516774\pi$
$812$	0	0
$813$	7.00000	0.245501
$814$	0	0
$815$	−64.0000	−2.24182
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	0	0
$823$	−29.0000	−1.01088	−0.505438	−	0.862863i	$-0.668669\pi$
$823$	−29.0000	−1.01088	−0.505438	+	0.862863i	$0.668669\pi$
$824$	0	0
$825$	−22.0000	−0.765942
$826$	0	0
$827$	23.0000	0.799788	0.399894	−	0.916561i	$-0.369047\pi$
$827$	23.0000	0.799788	0.399894	+	0.916561i	$0.369047\pi$
$828$	0	0
$829$	15.0000	0.520972	0.260486	−	0.965478i	$-0.416117\pi$
$829$	15.0000	0.520972	0.260486	+	0.965478i	$0.416117\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	48.0000	1.66111
$836$	0	0
$837$	40.0000	1.38260
$838$	0	0
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	8.00000	0.275535
$844$	0	0
$845$	−48.0000	−1.65125
$846$	0	0
$847$	21.0000	0.721569
$848$	0	0
$849$	6.00000	0.205919
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	0	0
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	0	0
$865$	24.0000	0.816024
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	20.0000	0.678454
$870$	0	0
$871$	3.00000	0.101651
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	−72.0000	−2.43404
$876$	0	0
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	0	0
$879$	9.00000	0.303562
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	0	0
$885$	−60.0000	−2.01688
$886$	0	0
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	0	0
$889$	54.0000	1.81110
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.00000	−0.0333890
$898$	0	0
$899$	40.0000	1.33407
$900$	0	0
$901$	3.00000	0.0999445
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	−88.0000	−2.92522
$906$	0	0
$907$	53.0000	1.75984	0.879918	−	0.475125i	$-0.157597\pi$
$907$	53.0000	1.75984	0.879918	+	0.475125i	$0.157597\pi$
$908$	0	0
$909$	4.00000	0.132672
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	8.00000	0.264472
$916$	0	0
$917$	−36.0000	−1.18882
$918$	0	0
$919$	−5.00000	−0.164935	−0.0824674	−	0.996594i	$-0.526280\pi$
$919$	−5.00000	−0.164935	−0.0824674	+	0.996594i	$0.526280\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	−2.00000	−0.0658308
$924$	0	0
$925$	22.0000	0.723356
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	−55.0000	−1.80449	−0.902246	−	0.431222i	$-0.858082\pi$
$929$	−55.0000	−1.80449	−0.902246	+	0.431222i	$0.858082\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	0	0
$933$	7.00000	0.229170
$934$	0	0
$935$	24.0000	0.784884
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	0	0
$939$	−29.0000	−0.946379
$940$	0	0
$941$	−7.00000	−0.228193	−0.114097	−	0.993470i	$-0.536397\pi$
$941$	−7.00000	−0.228193	−0.114097	+	0.993470i	$0.536397\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	−60.0000	−1.95180
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	9.00000	0.292152
$950$	0	0
$951$	−27.0000	−0.875535
$952$	0	0
$953$	−46.0000	−1.49009	−0.745043	−	0.667016i	$-0.767571\pi$
$953$	−46.0000	−1.49009	−0.745043	+	0.667016i	$0.767571\pi$
$954$	0	0
$955$	−28.0000	−0.906059
$956$	0	0
$957$	−10.0000	−0.323254
$958$	0	0
$959$	51.0000	1.64688
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	14.0000	0.451144
$964$	0	0
$965$	−24.0000	−0.772587
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	0	0
$969$	3.00000	0.0963739
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−11.0000	−0.352282
$976$	0	0
$977$	8.00000	0.255943	0.127971	−	0.991778i	$-0.459153\pi$
$977$	8.00000	0.255943	0.127971	+	0.991778i	$0.459153\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−30.0000	−0.957826
$982$	0	0
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	−32.0000	−1.01960
$986$	0	0
$987$	−24.0000	−0.763928
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	−17.0000	−0.539479
$994$	0	0
$995$	100.000	3.17021
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.a.g.1.1		1
4.3	odd	2		1216.2.a.n.1.1		1
8.3	odd	2		38.2.a.b.1.1	✓	1
8.5	even	2		304.2.a.d.1.1		1
24.5	odd	2		2736.2.a.w.1.1		1
24.11	even	2		342.2.a.d.1.1		1
40.3	even	4		950.2.b.c.799.1		2
40.19	odd	2		950.2.a.b.1.1		1
40.27	even	4		950.2.b.c.799.2		2
40.29	even	2		7600.2.a.h.1.1		1
56.27	even	2		1862.2.a.f.1.1		1
88.43	even	2		4598.2.a.a.1.1		1
104.51	odd	2		6422.2.a.b.1.1		1
120.59	even	2		8550.2.a.u.1.1		1
152.3	even	18		722.2.e.d.389.1		6
152.11	odd	6		722.2.c.d.653.1		2
152.27	even	6		722.2.c.f.653.1		2
152.35	odd	18		722.2.e.c.389.1		6
152.37	odd	2		5776.2.a.d.1.1		1
152.43	odd	18		722.2.e.c.595.1		6
152.51	even	18		722.2.e.d.245.1		6
152.59	even	18		722.2.e.d.99.1		6
152.67	even	18		722.2.e.d.423.1		6
152.75	even	2		722.2.a.b.1.1		1
152.83	odd	6		722.2.c.d.429.1		2
152.91	even	18		722.2.e.d.415.1		6
152.99	odd	18		722.2.e.c.415.1		6
152.107	even	6		722.2.c.f.429.1		2
152.123	odd	18		722.2.e.c.423.1		6
152.131	odd	18		722.2.e.c.99.1		6
152.139	odd	18		722.2.e.c.245.1		6
152.147	even	18		722.2.e.d.595.1		6
456.227	odd	2		6498.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.a.b.1.1	✓	1	8.3	odd	2
304.2.a.d.1.1		1	8.5	even	2
342.2.a.d.1.1		1	24.11	even	2
722.2.a.b.1.1		1	152.75	even	2
722.2.c.d.429.1		2	152.83	odd	6
722.2.c.d.653.1		2	152.11	odd	6
722.2.c.f.429.1		2	152.107	even	6
722.2.c.f.653.1		2	152.27	even	6
722.2.e.c.99.1		6	152.131	odd	18
722.2.e.c.245.1		6	152.139	odd	18
722.2.e.c.389.1		6	152.35	odd	18
722.2.e.c.415.1		6	152.99	odd	18
722.2.e.c.423.1		6	152.123	odd	18
722.2.e.c.595.1		6	152.43	odd	18
722.2.e.d.99.1		6	152.59	even	18
722.2.e.d.245.1		6	152.51	even	18
722.2.e.d.389.1		6	152.3	even	18
722.2.e.d.415.1		6	152.91	even	18
722.2.e.d.423.1		6	152.67	even	18
722.2.e.d.595.1		6	152.147	even	18
950.2.a.b.1.1		1	40.19	odd	2
950.2.b.c.799.1		2	40.3	even	4
950.2.b.c.799.2		2	40.27	even	4
1216.2.a.g.1.1		1	1.1	even	1	trivial
1216.2.a.n.1.1		1	4.3	odd	2
1862.2.a.f.1.1		1	56.27	even	2
2736.2.a.w.1.1		1	24.5	odd	2
4598.2.a.a.1.1		1	88.43	even	2
5776.2.a.d.1.1		1	152.37	odd	2
6422.2.a.b.1.1		1	104.51	odd	2
6498.2.a.y.1.1		1	456.227	odd	2
7600.2.a.h.1.1		1	40.29	even	2
8550.2.a.u.1.1		1	120.59	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

Introduction

L-functions

Modular forms

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Representations

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