LMFDB - Embedded newform 9216.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9216,2,Mod(1,9216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9216, 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("9216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$9216 = 2^{10} \cdot 3^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9216.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:	$73.5901305028$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 2$ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 4608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9216.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{5} +4.00000 q^{7} +5.65685 q^{11} -4.24264 q^{13} +6.00000 q^{17} +5.65685 q^{19} -8.00000 q^{23} -3.00000 q^{25} -4.24264 q^{29} +4.00000 q^{31} +5.65685 q^{35} -1.41421 q^{37} +2.00000 q^{41} -5.65685 q^{43} +8.00000 q^{47} +9.00000 q^{49} +9.89949 q^{53} +8.00000 q^{55} -4.24264 q^{61} -6.00000 q^{65} +11.3137 q^{67} +10.0000 q^{73} +22.6274 q^{77} +12.0000 q^{79} +5.65685 q^{83} +8.48528 q^{85} -16.0000 q^{89} -16.9706 q^{91} +8.00000 q^{95} +8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 8 q^{7} + 12 q^{17} - 16 q^{23} - 6 q^{25} + 8 q^{31} + 4 q^{41} + 16 q^{47} + 18 q^{49} + 16 q^{55} - 12 q^{65} + 20 q^{73} + 24 q^{79} - 32 q^{89} + 16 q^{95} + 16 q^{97}+O(q^{100})$ 2 * q + 8 * q^7 + 12 * q^17 - 16 * q^23 - 6 * q^25 + 8 * q^31 + 4 * q^41 + 16 * q^47 + 18 * q^49 + 16 * q^55 - 12 * q^65 + 20 * q^73 + 24 * q^79 - 32 * q^89 + 16 * q^95 + 16 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	5.65685	1.29777	0.648886	−	0.760886i	$-0.275235\pi$
$19$	5.65685	1.29777	0.648886	+	0.760886i	$0.275235\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.24264	−0.787839	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	−4.24264	−0.787839	−0.393919	+	0.919145i	$0.628881\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.65685	0.956183
$36$	0	0
$37$	−1.41421	−0.232495	−0.116248	−	0.993220i	$-0.537087\pi$
$37$	−1.41421	−0.232495	−0.116248	+	0.993220i	$0.537087\pi$
$38$	0	0
$39$	0	0
$40$	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
$43$	−5.65685	−0.862662	−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685	−0.862662	−0.431331	+	0.902194i	$0.641956\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	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
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.89949	1.35980	0.679900	−	0.733305i	$-0.262023\pi$
$53$	9.89949	1.35980	0.679900	+	0.733305i	$0.262023\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−4.24264	−0.543214	−0.271607	−	0.962408i	$-0.587555\pi$
$61$	−4.24264	−0.543214	−0.271607	+	0.962408i	$0.587555\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	22.6274	2.57863
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.65685	0.620920	0.310460	−	0.950586i	$-0.399517\pi$
$83$	5.65685	0.620920	0.310460	+	0.950586i	$0.399517\pi$
$84$	0	0
$85$	8.48528	0.920358
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	−16.9706	−1.77900
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.89949	−0.985037	−0.492518	−	0.870302i	$-0.663924\pi$
$101$	−9.89949	−0.985037	−0.492518	+	0.870302i	$0.663924\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.3137	−1.09374	−0.546869	−	0.837218i	$-0.684180\pi$
$107$	−11.3137	−1.09374	−0.546869	+	0.837218i	$0.684180\pi$
$108$	0	0
$109$	−7.07107	−0.677285	−0.338643	−	0.940915i	$-0.609968\pi$
$109$	−7.07107	−0.677285	−0.338643	+	0.940915i	$0.609968\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	0	0
$115$	−11.3137	−1.05501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	24.0000	2.20008
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	22.6274	1.96205
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−24.0000	−2.00698
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.41421	−0.115857	−0.0579284	−	0.998321i	$-0.518450\pi$
$149$	−1.41421	−0.115857	−0.0579284	+	0.998321i	$0.518450\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.65685	0.454369
$156$	0	0
$157$	−4.24264	−0.338600	−0.169300	−	0.985565i	$-0.554151\pi$
$157$	−4.24264	−0.338600	−0.169300	+	0.985565i	$0.554151\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−32.0000	−2.52195
$162$	0	0
$163$	−16.9706	−1.32924	−0.664619	−	0.747183i	$-0.731406\pi$
$163$	−16.9706	−1.32924	−0.664619	+	0.747183i	$0.731406\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.5563	1.18273	0.591364	−	0.806405i	$-0.298590\pi$
$173$	15.5563	1.18273	0.591364	+	0.806405i	$0.298590\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.3137	−0.845626	−0.422813	−	0.906217i	$-0.638957\pi$
$179$	−11.3137	−0.845626	−0.422813	+	0.906217i	$0.638957\pi$
$180$	0	0
$181$	12.7279	0.946059	0.473029	−	0.881047i	$-0.343160\pi$
$181$	12.7279	0.946059	0.473029	+	0.881047i	$0.343160\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	33.9411	2.48202
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.41421	−0.100759	−0.0503793	−	0.998730i	$-0.516043\pi$
$197$	−1.41421	−0.100759	−0.0503793	+	0.998730i	$0.516043\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−16.9706	−1.19110
$204$	0	0
$205$	2.82843	0.197546
$206$	0	0
$207$	0	0
$208$	0	0
$209$	32.0000	2.21349
$210$	0	0
$211$	22.6274	1.55774	0.778868	−	0.627188i	$-0.215794\pi$
$211$	22.6274	1.55774	0.778868	+	0.627188i	$0.215794\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−25.4558	−1.71235
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.9706	−1.12638	−0.563188	−	0.826329i	$-0.690425\pi$
$227$	−16.9706	−1.12638	−0.563188	+	0.826329i	$0.690425\pi$
$228$	0	0
$229$	9.89949	0.654177	0.327089	−	0.944994i	$-0.393932\pi$
$229$	9.89949	0.654177	0.327089	+	0.944994i	$0.393932\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	0	0
$235$	11.3137	0.738025
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	24.0000	1.54598	0.772988	−	0.634421i	$-0.218761\pi$
$241$	24.0000	1.54598	0.772988	+	0.634421i	$0.218761\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.7279	0.813157
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.65685	−0.357057	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685	−0.357057	−0.178529	+	0.983935i	$0.557134\pi$
$252$	0	0
$253$	−45.2548	−2.84515
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−5.65685	−0.351500
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	14.0000	0.860013
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.24264	0.258678	0.129339	−	0.991600i	$-0.458714\pi$
$269$	4.24264	0.258678	0.129339	+	0.991600i	$0.458714\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−16.9706	−1.02336
$276$	0	0
$277$	21.2132	1.27458	0.637289	−	0.770625i	$-0.280056\pi$
$277$	21.2132	1.27458	0.637289	+	0.770625i	$0.280056\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−11.3137	−0.672530	−0.336265	−	0.941767i	$-0.609164\pi$
$283$	−11.3137	−0.672530	−0.336265	+	0.941767i	$0.609164\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.41421	−0.0826192	−0.0413096	−	0.999146i	$-0.513153\pi$
$293$	−1.41421	−0.0826192	−0.0413096	+	0.999146i	$0.513153\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	33.9411	1.96287
$300$	0	0
$301$	−22.6274	−1.30422
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	−33.9411	−1.93712	−0.968561	−	0.248776i	$-0.919972\pi$
$307$	−33.9411	−1.93712	−0.968561	+	0.248776i	$0.919972\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.8701	−1.50917	−0.754586	−	0.656201i	$-0.772162\pi$
$317$	−26.8701	−1.50917	−0.754586	+	0.656201i	$0.772162\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	0	0
$323$	33.9411	1.88853
$324$	0	0
$325$	12.7279	0.706018
$326$	0	0
$327$	0	0
$328$	0	0
$329$	32.0000	1.76422
$330$	0	0
$331$	22.6274	1.24372	0.621858	−	0.783130i	$-0.286378\pi$
$331$	22.6274	1.24372	0.621858	+	0.783130i	$0.286378\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	16.0000	0.874173
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	22.6274	1.22534
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.2843	−1.51838	−0.759190	−	0.650870i	$-0.774404\pi$
$347$	−28.2843	−1.51838	−0.759190	+	0.650870i	$0.774404\pi$
$348$	0	0
$349$	15.5563	0.832712	0.416356	−	0.909202i	$-0.363307\pi$
$349$	15.5563	0.832712	0.416356	+	0.909202i	$0.363307\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.0000	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.1421	0.740233
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	39.5980	2.05582
$372$	0	0
$373$	−21.2132	−1.09838	−0.549189	−	0.835698i	$-0.685063\pi$
$373$	−21.2132	−1.09838	−0.549189	+	0.835698i	$0.685063\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.0000	0.927047
$378$	0	0
$379$	16.9706	0.871719	0.435860	−	0.900015i	$-0.356444\pi$
$379$	16.9706	0.871719	0.435860	+	0.900015i	$0.356444\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	32.0000	1.63087
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−35.3553	−1.79259	−0.896293	−	0.443461i	$-0.853750\pi$
$389$	−35.3553	−1.79259	−0.896293	+	0.443461i	$0.853750\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.9706	0.853882
$396$	0	0
$397$	−38.1838	−1.91639	−0.958194	−	0.286119i	$-0.907635\pi$
$397$	−38.1838	−1.91639	−0.958194	+	0.286119i	$0.907635\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−16.9706	−0.845364
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.9706	0.829066	0.414533	−	0.910034i	$-0.363945\pi$
$419$	16.9706	0.829066	0.414533	+	0.910034i	$0.363945\pi$
$420$	0	0
$421$	−12.7279	−0.620321	−0.310160	−	0.950684i	$-0.600383\pi$
$421$	−12.7279	−0.620321	−0.310160	+	0.950684i	$0.600383\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−18.0000	−0.873128
$426$	0	0
$427$	−16.9706	−0.821263
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−45.2548	−2.16483
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.9706	0.806296	0.403148	−	0.915135i	$-0.367916\pi$
$443$	16.9706	0.806296	0.403148	+	0.915135i	$0.367916\pi$
$444$	0	0
$445$	−22.6274	−1.07264
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	11.3137	0.532742
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−24.0000	−1.12514
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−26.8701	−1.25146	−0.625732	−	0.780038i	$-0.715200\pi$
$461$	−26.8701	−1.25146	−0.625732	+	0.780038i	$0.715200\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.65685	0.261768	0.130884	−	0.991398i	$-0.458218\pi$
$467$	5.65685	0.261768	0.130884	+	0.991398i	$0.458218\pi$
$468$	0	0
$469$	45.2548	2.08967
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−32.0000	−1.47136
$474$	0	0
$475$	−16.9706	−0.778663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.3137	0.513729
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−11.3137	−0.510581	−0.255290	−	0.966864i	$-0.582171\pi$
$491$	−11.3137	−0.510581	−0.255290	+	0.966864i	$0.582171\pi$
$492$	0	0
$493$	−25.4558	−1.14647
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	11.3137	0.506471	0.253236	−	0.967405i	$-0.418505\pi$
$499$	11.3137	0.506471	0.253236	+	0.967405i	$0.418505\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.5563	0.689523	0.344762	−	0.938690i	$-0.387960\pi$
$509$	15.5563	0.689523	0.344762	+	0.938690i	$0.387960\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.65685	−0.249271
$516$	0	0
$517$	45.2548	1.99031
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	−16.9706	−0.742071	−0.371035	−	0.928619i	$-0.620997\pi$
$523$	−16.9706	−0.742071	−0.371035	+	0.928619i	$0.620997\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.48528	−0.367538
$534$	0	0
$535$	−16.0000	−0.691740
$536$	0	0
$537$	0	0
$538$	0	0
$539$	50.9117	2.19292
$540$	0	0
$541$	4.24264	0.182405	0.0912027	−	0.995832i	$-0.470929\pi$
$541$	4.24264	0.182405	0.0912027	+	0.995832i	$0.470929\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−16.9706	−0.725609	−0.362804	−	0.931865i	$-0.618181\pi$
$547$	−16.9706	−0.725609	−0.362804	+	0.931865i	$0.618181\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	48.0000	2.04117
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.5563	0.659144	0.329572	−	0.944131i	$-0.393096\pi$
$557$	15.5563	0.659144	0.329572	+	0.944131i	$0.393096\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.2843	−1.19204	−0.596020	−	0.802970i	$-0.703252\pi$
$563$	−28.2843	−1.19204	−0.596020	+	0.802970i	$0.703252\pi$
$564$	0	0
$565$	22.6274	0.951943
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−33.9411	−1.42039	−0.710196	−	0.704004i	$-0.751394\pi$
$571$	−33.9411	−1.42039	−0.710196	+	0.704004i	$0.751394\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24.0000	1.00087
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	22.6274	0.938743
$582$	0	0
$583$	56.0000	2.31928
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.6274	−0.933933	−0.466967	−	0.884275i	$-0.654653\pi$
$587$	−22.6274	−0.933933	−0.466967	+	0.884275i	$0.654653\pi$
$588$	0	0
$589$	22.6274	0.932346
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	0	0
$595$	33.9411	1.39145
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	29.6985	1.20742
$606$	0	0
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−33.9411	−1.37311
$612$	0	0
$613$	−9.89949	−0.399837	−0.199918	−	0.979813i	$-0.564068\pi$
$613$	−9.89949	−0.399837	−0.199918	+	0.979813i	$0.564068\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.0000	1.28827	0.644136	−	0.764911i	$-0.277217\pi$
$617$	32.0000	1.28827	0.644136	+	0.764911i	$0.277217\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−64.0000	−2.56411
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.48528	−0.338330
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.65685	0.224485
$636$	0	0
$637$	−38.1838	−1.51290
$638$	0	0
$639$	0	0
$640$	0	0
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	0	0
$643$	16.9706	0.669254	0.334627	−	0.942351i	$-0.391390\pi$
$643$	16.9706	0.669254	0.334627	+	0.942351i	$0.391390\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26.8701	−1.05151	−0.525753	−	0.850637i	$-0.676216\pi$
$653$	−26.8701	−1.05151	−0.525753	+	0.850637i	$0.676216\pi$
$654$	0	0
$655$	−16.0000	−0.625172
$656$	0	0
$657$	0	0
$658$	0	0
$659$	11.3137	0.440720	0.220360	−	0.975419i	$-0.429277\pi$
$659$	11.3137	0.440720	0.220360	+	0.975419i	$0.429277\pi$
$660$	0	0
$661$	9.89949	0.385046	0.192523	−	0.981292i	$-0.438333\pi$
$661$	9.89949	0.385046	0.192523	+	0.981292i	$0.438333\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	32.0000	1.24091
$666$	0	0
$667$	33.9411	1.31421
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.7279	−0.489174	−0.244587	−	0.969627i	$-0.578652\pi$
$677$	−12.7279	−0.489174	−0.244587	+	0.969627i	$0.578652\pi$
$678$	0	0
$679$	32.0000	1.22805
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16.9706	−0.649361	−0.324680	−	0.945824i	$-0.605257\pi$
$683$	−16.9706	−0.649361	−0.324680	+	0.945824i	$0.605257\pi$
$684$	0	0
$685$	−2.82843	−0.108069
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−42.0000	−1.60007
$690$	0	0
$691$	50.9117	1.93677	0.968386	−	0.249457i	$-0.0802520\pi$
$691$	50.9117	1.93677	0.968386	+	0.249457i	$0.0802520\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15.5563	0.587555	0.293778	−	0.955874i	$-0.405087\pi$
$701$	15.5563	0.587555	0.293778	+	0.955874i	$0.405087\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−39.5980	−1.48924
$708$	0	0
$709$	46.6690	1.75269	0.876346	−	0.481681i	$-0.159974\pi$
$709$	46.6690	1.75269	0.876346	+	0.481681i	$0.159974\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	−33.9411	−1.26933
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	12.7279	0.472703
$726$	0	0
$727$	52.0000	1.92857	0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000	1.92857	0.964287	+	0.264861i	$0.0853260\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−33.9411	−1.25536
$732$	0	0
$733$	52.3259	1.93270	0.966351	−	0.257228i	$-0.0828093\pi$
$733$	52.3259	1.93270	0.966351	+	0.257228i	$0.0828093\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	64.0000	2.35747
$738$	0	0
$739$	33.9411	1.24854	0.624272	−	0.781207i	$-0.285396\pi$
$739$	33.9411	1.24854	0.624272	+	0.781207i	$0.285396\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−45.2548	−1.65358
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.65685	−0.205874
$756$	0	0
$757$	−9.89949	−0.359803	−0.179902	−	0.983685i	$-0.557578\pi$
$757$	−9.89949	−0.359803	−0.179902	+	0.983685i	$0.557578\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	−28.2843	−1.02396
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	12.7279	0.457792	0.228896	−	0.973451i	$-0.426489\pi$
$773$	12.7279	0.457792	0.228896	+	0.973451i	$0.426489\pi$
$774$	0	0
$775$	−12.0000	−0.431053
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.3137	0.405356
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.00000	−0.214149
$786$	0	0
$787$	−16.9706	−0.604935	−0.302468	−	0.953160i	$-0.597810\pi$
$787$	−16.9706	−0.604935	−0.302468	+	0.953160i	$0.597810\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	64.0000	2.27558
$792$	0	0
$793$	18.0000	0.639199
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.07107	0.250470	0.125235	−	0.992127i	$-0.460032\pi$
$797$	7.07107	0.250470	0.125235	+	0.992127i	$0.460032\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	0	0
$801$	0	0
$802$	0	0
$803$	56.5685	1.99626
$804$	0	0
$805$	−45.2548	−1.59502
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	5.65685	0.198639	0.0993195	−	0.995056i	$-0.468333\pi$
$811$	5.65685	0.198639	0.0993195	+	0.995056i	$0.468333\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−24.0000	−0.840683
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	0	0
$820$	0	0
$821$	55.1543	1.92490	0.962450	−	0.271460i	$-0.0875065\pi$
$821$	55.1543	1.92490	0.962450	+	0.271460i	$0.0875065\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−45.2548	−1.57366	−0.786832	−	0.617167i	$-0.788280\pi$
$827$	−45.2548	−1.57366	−0.786832	+	0.617167i	$0.788280\pi$
$828$	0	0
$829$	−41.0122	−1.42441	−0.712206	−	0.701970i	$-0.752304\pi$
$829$	−41.0122	−1.42441	−0.712206	+	0.701970i	$0.752304\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	54.0000	1.87099
$834$	0	0
$835$	−11.3137	−0.391527
$836$	0	0
$837$	0	0
$838$	0	0
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.07107	0.243252
$846$	0	0
$847$	84.0000	2.88627
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.3137	0.387829
$852$	0	0
$853$	1.41421	0.0484218	0.0242109	−	0.999707i	$-0.492293\pi$
$853$	1.41421	0.0484218	0.0242109	+	0.999707i	$0.492293\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	16.9706	0.579028	0.289514	−	0.957174i	$-0.406506\pi$
$859$	16.9706	0.579028	0.289514	+	0.957174i	$0.406506\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	0	0
$867$	0	0
$868$	0	0
$869$	67.8823	2.30275
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−45.2548	−1.52989
$876$	0	0
$877$	7.07107	0.238773	0.119386	−	0.992848i	$-0.461907\pi$
$877$	7.07107	0.238773	0.119386	+	0.992848i	$0.461907\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−5.65685	−0.190368	−0.0951842	−	0.995460i	$-0.530344\pi$
$883$	−5.65685	−0.190368	−0.0951842	+	0.995460i	$0.530344\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	0	0
$893$	45.2548	1.51440
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16.9706	−0.566000
$900$	0	0
$901$	59.3970	1.97880
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	39.5980	1.31483	0.657415	−	0.753529i	$-0.271650\pi$
$907$	39.5980	1.31483	0.657415	+	0.753529i	$0.271650\pi$
$908$	0	0
$909$	0	0
$910$	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
$913$	32.0000	1.05905
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−45.2548	−1.49445
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	4.24264	0.139497
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	50.9117	1.66856
$932$	0	0
$933$	0	0
$934$	0	0
$935$	48.0000	1.56977
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−49.4975	−1.61357	−0.806786	−	0.590844i	$-0.798795\pi$
$941$	−49.4975	−1.61357	−0.806786	+	0.590844i	$0.798795\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−56.5685	−1.83823	−0.919115	−	0.393989i	$-0.871095\pi$
$947$	−56.5685	−1.83823	−0.919115	+	0.393989i	$0.871095\pi$
$948$	0	0
$949$	−42.4264	−1.37722
$950$	0	0
$951$	0	0
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	−33.9411	−1.09831
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.3137	0.364201
$966$	0	0
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	39.5980	1.27076	0.635380	−	0.772200i	$-0.280844\pi$
$971$	39.5980	1.27076	0.635380	+	0.772200i	$0.280844\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	0	0
$979$	−90.5097	−2.89270
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	45.2548	1.43902
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.65685	0.179334
$996$	0	0
$997$	−46.6690	−1.47802	−0.739012	−	0.673693i	$-0.764707\pi$
$997$	−46.6690	−1.47802	−0.739012	+	0.673693i	$0.764707\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9216.2.a.v.1.2		2
3.2	odd	2		9216.2.a.t.1.1		2
4.3	odd	2		9216.2.a.c.1.2		2
8.3	odd	2		9216.2.a.c.1.1		2
8.5	even	2	inner	9216.2.a.v.1.1		2
12.11	even	2		9216.2.a.a.1.1		2
24.5	odd	2		9216.2.a.t.1.2		2
24.11	even	2		9216.2.a.a.1.2		2
32.3	odd	8		4608.2.k.e.1153.1	yes	2
32.5	even	8		4608.2.k.m.3457.1	yes	2
32.11	odd	8		4608.2.k.e.3457.1	yes	2
32.13	even	8		4608.2.k.m.1153.1	yes	2
32.19	odd	8		4608.2.k.t.1153.1	yes	2
32.21	even	8		4608.2.k.l.3457.1	yes	2
32.27	odd	8		4608.2.k.t.3457.1	yes	2
32.29	even	8		4608.2.k.l.1153.1	yes	2
96.5	odd	8		4608.2.k.k.3457.1	yes	2
96.11	even	8		4608.2.k.u.3457.1	yes	2
96.29	odd	8		4608.2.k.n.1153.1	yes	2
96.35	even	8		4608.2.k.u.1153.1	yes	2
96.53	odd	8		4608.2.k.n.3457.1	yes	2
96.59	even	8		4608.2.k.d.3457.1	yes	2
96.77	odd	8		4608.2.k.k.1153.1	yes	2
96.83	even	8		4608.2.k.d.1153.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.k.d.1153.1	✓	2	96.83	even	8
4608.2.k.d.3457.1	yes	2	96.59	even	8
4608.2.k.e.1153.1	yes	2	32.3	odd	8
4608.2.k.e.3457.1	yes	2	32.11	odd	8
4608.2.k.k.1153.1	yes	2	96.77	odd	8
4608.2.k.k.3457.1	yes	2	96.5	odd	8
4608.2.k.l.1153.1	yes	2	32.29	even	8
4608.2.k.l.3457.1	yes	2	32.21	even	8
4608.2.k.m.1153.1	yes	2	32.13	even	8
4608.2.k.m.3457.1	yes	2	32.5	even	8
4608.2.k.n.1153.1	yes	2	96.29	odd	8
4608.2.k.n.3457.1	yes	2	96.53	odd	8
4608.2.k.t.1153.1	yes	2	32.19	odd	8
4608.2.k.t.3457.1	yes	2	32.27	odd	8
4608.2.k.u.1153.1	yes	2	96.35	even	8
4608.2.k.u.3457.1	yes	2	96.11	even	8
9216.2.a.a.1.1		2	12.11	even	2
9216.2.a.a.1.2		2	24.11	even	2
9216.2.a.c.1.1		2	8.3	odd	2
9216.2.a.c.1.2		2	4.3	odd	2
9216.2.a.t.1.1		2	3.2	odd	2
9216.2.a.t.1.2		2	24.5	odd	2
9216.2.a.v.1.1		2	8.5	even	2	inner
9216.2.a.v.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	9216.2.a.v.1.2

Level	$9216$
Weight	$2$
Character	9216.1
Self dual	yes
Analytic conductor	$73.590$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$