LMFDB - Embedded newform 9216.2.a.bs.1.6

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:	$8$
Coefficient field:	8.8.3288334336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} - 8x^{5} + 24x^{4} + 8x^{3} - 16x^{2} + 2 $ x^8 - 12x^6 - 8x^5 + 24x^4 + 8x^3 - 16*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 4608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.724535$ 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)

$f(q)$	$=$	$q+1.78089 q^{5} +3.29066 q^{7} +O(q^{10})$	$q+1.78089 q^{5} +3.29066 q^{7} +1.53073 q^{11} +0.585786 q^{13} +6.08034 q^{17} +1.92762 q^{19} +5.22625 q^{23} -1.82843 q^{25} +6.81801 q^{29} -6.01673 q^{31} +5.86030 q^{35} +3.41421 q^{37} +1.04322 q^{41} +11.2350 q^{43} +0.896683 q^{47} +3.82843 q^{49} -6.81801 q^{53} +2.72607 q^{55} -10.4525 q^{59} +4.58579 q^{61} +1.04322 q^{65} +9.30739 q^{67} -14.7821 q^{71} -6.48528 q^{73} +5.03712 q^{77} -3.29066 q^{79} -13.2513 q^{83} +10.8284 q^{85} +7.12356 q^{89} +1.92762 q^{91} +3.43289 q^{95} +7.31371 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{13} + 8 q^{25} + 16 q^{37} + 8 q^{49} + 48 q^{61} + 16 q^{73} + 64 q^{85} - 32 q^{97}+O(q^{100}) $ 8 * q + 16 * q^13 + 8 * q^25 + 16 * q^37 + 8 * q^49 + 48 * q^61 + 16 * q^73 + 64 * q^85 - 32 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.78089	0.796439	0.398219	−	0.917290i	$-0.369628\pi$
$5$	1.78089	0.796439	0.398219	+	0.917290i	$0.369628\pi$
$6$	0	0
$7$	3.29066	1.24375	0.621876	−	0.783116i	$-0.286371\pi$
$7$	3.29066	1.24375	0.621876	+	0.783116i	$0.286371\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.53073	0.461534	0.230767	−	0.973009i	$-0.425877\pi$
$11$	1.53073	0.461534	0.230767	+	0.973009i	$0.425877\pi$
$12$	0	0
$13$	0.585786	0.162468	0.0812340	−	0.996695i	$-0.474114\pi$
$13$	0.585786	0.162468	0.0812340	+	0.996695i	$0.474114\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.08034	1.47470	0.737350	−	0.675511i	$-0.236077\pi$
$17$	6.08034	1.47470	0.737350	+	0.675511i	$0.236077\pi$
$18$	0	0
$19$	1.92762	0.442227	0.221113	−	0.975248i	$-0.429031\pi$
$19$	1.92762	0.442227	0.221113	+	0.975248i	$0.429031\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.22625	1.08975	0.544874	−	0.838518i	$-0.316577\pi$
$23$	5.22625	1.08975	0.544874	+	0.838518i	$0.316577\pi$
$24$	0	0
$25$	−1.82843	−0.365685
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.81801	1.26607	0.633036	−	0.774122i	$-0.281808\pi$
$29$	6.81801	1.26607	0.633036	+	0.774122i	$0.281808\pi$
$30$	0	0
$31$	−6.01673	−1.08064	−0.540318	−	0.841461i	$-0.681696\pi$
$31$	−6.01673	−1.08064	−0.540318	+	0.841461i	$0.681696\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.86030	0.990572
$36$	0	0
$37$	3.41421	0.561293	0.280647	−	0.959811i	$-0.409451\pi$
$37$	3.41421	0.561293	0.280647	+	0.959811i	$0.409451\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.04322	0.162924	0.0814619	−	0.996676i	$-0.474041\pi$
$41$	1.04322	0.162924	0.0814619	+	0.996676i	$0.474041\pi$
$42$	0	0
$43$	11.2350	1.71332	0.856661	−	0.515879i	$-0.172535\pi$
$43$	11.2350	1.71332	0.856661	+	0.515879i	$0.172535\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.896683	0.130795	0.0653973	−	0.997859i	$-0.479169\pi$
$47$	0.896683	0.130795	0.0653973	+	0.997859i	$0.479169\pi$
$48$	0	0
$49$	3.82843	0.546918
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.81801	−0.936526	−0.468263	−	0.883589i	$-0.655120\pi$
$53$	−6.81801	−0.936526	−0.468263	+	0.883589i	$0.655120\pi$
$54$	0	0
$55$	2.72607	0.367583
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.4525	−1.36080	−0.680400	−	0.732841i	$-0.738194\pi$
$59$	−10.4525	−1.36080	−0.680400	+	0.732841i	$0.738194\pi$
$60$	0	0
$61$	4.58579	0.587150	0.293575	−	0.955936i	$-0.405155\pi$
$61$	4.58579	0.587150	0.293575	+	0.955936i	$0.405155\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.04322	0.129396
$66$	0	0
$67$	9.30739	1.13708	0.568539	−	0.822656i	$-0.307509\pi$
$67$	9.30739	1.13708	0.568539	+	0.822656i	$0.307509\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.7821	−1.75431	−0.877155	−	0.480208i	$-0.840561\pi$
$71$	−14.7821	−1.75431	−0.877155	+	0.480208i	$0.840561\pi$
$72$	0	0
$73$	−6.48528	−0.759045	−0.379522	−	0.925183i	$-0.623912\pi$
$73$	−6.48528	−0.759045	−0.379522	+	0.925183i	$0.623912\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.03712	0.574033
$78$	0	0
$79$	−3.29066	−0.370228	−0.185114	−	0.982717i	$-0.559265\pi$
$79$	−3.29066	−0.370228	−0.185114	+	0.982717i	$0.559265\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.2513	−1.45452	−0.727262	−	0.686360i	$-0.759207\pi$
$83$	−13.2513	−1.45452	−0.727262	+	0.686360i	$0.759207\pi$
$84$	0	0
$85$	10.8284	1.17451
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.12356	0.755096	0.377548	−	0.925990i	$-0.376767\pi$
$89$	7.12356	0.755096	0.377548	+	0.925990i	$0.376767\pi$
$90$	0	0
$91$	1.92762	0.202070
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.43289	0.352207
$96$	0	0
$97$	7.31371	0.742595	0.371297	−	0.928514i	$-0.378913\pi$
$97$	7.31371	0.742595	0.371297	+	0.928514i	$0.378913\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.4169	−1.53404	−0.767020	−	0.641623i	$-0.778261\pi$
$101$	−15.4169	−1.53404	−0.767020	+	0.641623i	$0.778261\pi$
$102$	0	0
$103$	6.01673	0.592846	0.296423	−	0.955057i	$-0.404206\pi$
$103$	6.01673	0.592846	0.296423	+	0.955057i	$0.404206\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.5140	1.30644	0.653222	−	0.757166i	$-0.273417\pi$
$107$	13.5140	1.30644	0.653222	+	0.757166i	$0.273417\pi$
$108$	0	0
$109$	13.0711	1.25198	0.625991	−	0.779831i	$-0.284695\pi$
$109$	13.0711	1.25198	0.625991	+	0.779831i	$0.284695\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.0742	−0.947705	−0.473852	−	0.880604i	$-0.657137\pi$
$113$	−10.0742	−0.947705	−0.473852	+	0.880604i	$0.657137\pi$
$114$	0	0
$115$	9.30739	0.867918
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20.0083	1.83416
$120$	0	0
$121$	−8.65685	−0.786987
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.1607	−1.08768
$126$	0	0
$127$	−0.564588	−0.0500990	−0.0250495	−	0.999686i	$-0.507974\pi$
$127$	−0.564588	−0.0500990	−0.0250495	+	0.999686i	$0.507974\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.7206	1.02403	0.512017	−	0.858975i	$-0.328898\pi$
$131$	11.7206	1.02403	0.512017	+	0.858975i	$0.328898\pi$
$132$	0	0
$133$	6.34315	0.550020
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.1546	1.38018	0.690090	−	0.723724i	$-0.257571\pi$
$137$	16.1546	1.38018	0.690090	+	0.723724i	$0.257571\pi$
$138$	0	0
$139$	−18.6148	−1.57888	−0.789442	−	0.613825i	$-0.789630\pi$
$139$	−18.6148	−1.57888	−0.789442	+	0.613825i	$0.789630\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.896683	0.0749844
$144$	0	0
$145$	12.1421	1.00835
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.34267	0.437689	0.218844	−	0.975760i	$-0.429771\pi$
$149$	5.34267	0.437689	0.218844	+	0.975760i	$0.429771\pi$
$150$	0	0
$151$	0.564588	0.0459455	0.0229727	−	0.999736i	$-0.492687\pi$
$151$	0.564588	0.0459455	0.0229727	+	0.999736i	$0.492687\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.7151	−0.860660
$156$	0	0
$157$	14.2426	1.13669	0.568343	−	0.822792i	$-0.307585\pi$
$157$	14.2426	1.13669	0.568343	+	0.822792i	$0.307585\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	17.1978	1.35538
$162$	0	0
$163$	−16.6871	−1.30704	−0.653519	−	0.756910i	$-0.726708\pi$
$163$	−16.6871	−1.30704	−0.653519	+	0.756910i	$0.726708\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.8017	−1.68707	−0.843533	−	0.537078i	$-0.819528\pi$
$167$	−21.8017	−1.68707	−0.843533	+	0.537078i	$0.819528\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.25623	0.247567	0.123783	−	0.992309i	$-0.460497\pi$
$173$	3.25623	0.247567	0.123783	+	0.992309i	$0.460497\pi$
$174$	0	0
$175$	−6.01673	−0.454822
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−16.0502	−1.19965	−0.599823	−	0.800133i	$-0.704762\pi$
$179$	−16.0502	−1.19965	−0.599823	+	0.800133i	$0.704762\pi$
$180$	0	0
$181$	1.27208	0.0945528	0.0472764	−	0.998882i	$-0.484946\pi$
$181$	1.27208	0.0945528	0.0472764	+	0.998882i	$0.484946\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.08034	0.447036
$186$	0	0
$187$	9.30739	0.680623
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.3492	−0.821198	−0.410599	−	0.911816i	$-0.634680\pi$
$191$	−11.3492	−0.821198	−0.410599	+	0.911816i	$0.634680\pi$
$192$	0	0
$193$	12.4853	0.898710	0.449355	−	0.893353i	$-0.351654\pi$
$193$	12.4853	0.898710	0.449355	+	0.893353i	$0.351654\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.25623	0.231997	0.115998	−	0.993249i	$-0.462993\pi$
$197$	3.25623	0.231997	0.115998	+	0.993249i	$0.462993\pi$
$198$	0	0
$199$	−19.1794	−1.35959	−0.679794	−	0.733403i	$-0.737931\pi$
$199$	−19.1794	−1.35959	−0.679794	+	0.733403i	$0.737931\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	22.4357	1.57468
$204$	0	0
$205$	1.85786	0.129759
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.95068	0.204103
$210$	0	0
$211$	−26.3253	−1.81231	−0.906153	−	0.422950i	$-0.860994\pi$
$211$	−26.3253	−1.81231	−0.906153	+	0.422950i	$0.860994\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	20.0083	1.36456
$216$	0	0
$217$	−19.7990	−1.34404
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.56178	0.239591
$222$	0	0
$223$	21.9054	1.46690	0.733448	−	0.679746i	$-0.237910\pi$
$223$	21.9054	1.46690	0.733448	+	0.679746i	$0.237910\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.3128	1.08272	0.541359	−	0.840791i	$-0.317910\pi$
$227$	16.3128	1.08272	0.541359	+	0.840791i	$0.317910\pi$
$228$	0	0
$229$	25.0711	1.65674	0.828371	−	0.560179i	$-0.189268\pi$
$229$	25.0711	1.65674	0.828371	+	0.560179i	$0.189268\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−27.2720	−1.78665	−0.893326	−	0.449410i	$-0.851634\pi$
$233$	−27.2720	−1.78665	−0.893326	+	0.449410i	$0.851634\pi$
$234$	0	0
$235$	1.59689	0.104170
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.8854	−0.898171	−0.449086	−	0.893489i	$-0.648250\pi$
$239$	−13.8854	−0.898171	−0.449086	+	0.893489i	$0.648250\pi$
$240$	0	0
$241$	10.1421	0.653312	0.326656	−	0.945143i	$-0.394078\pi$
$241$	10.1421	0.653312	0.326656	+	0.945143i	$0.394078\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.81801	0.435587
$246$	0	0
$247$	1.12918	0.0718477
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.65367	−0.483095	−0.241548	−	0.970389i	$-0.577655\pi$
$251$	−7.65367	−0.483095	−0.241548	+	0.970389i	$0.577655\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.1978	−1.07277	−0.536385	−	0.843974i	$-0.680210\pi$
$257$	−17.1978	−1.07277	−0.536385	+	0.843974i	$0.680210\pi$
$258$	0	0
$259$	11.2350	0.698109
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.1116	−1.17847	−0.589237	−	0.807960i	$-0.700572\pi$
$263$	−19.1116	−1.17847	−0.589237	+	0.807960i	$0.700572\pi$
$264$	0	0
$265$	−12.1421	−0.745885
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.73157	−0.288489	−0.144244	−	0.989542i	$-0.546075\pi$
$269$	−4.73157	−0.288489	−0.144244	+	0.989542i	$0.546075\pi$
$270$	0	0
$271$	16.4533	0.999466	0.499733	−	0.866179i	$-0.333431\pi$
$271$	16.4533	0.999466	0.499733	+	0.866179i	$0.333431\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.79884	−0.168776
$276$	0	0
$277$	28.3848	1.70548	0.852738	−	0.522339i	$-0.174940\pi$
$277$	28.3848	1.70548	0.852738	+	0.522339i	$0.174940\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−31.4449	−1.87585	−0.937924	−	0.346842i	$-0.887254\pi$
$281$	−31.4449	−1.87585	−0.937924	+	0.346842i	$0.887254\pi$
$282$	0	0
$283$	−17.0179	−1.01161	−0.505804	−	0.862649i	$-0.668804\pi$
$283$	−17.0179	−1.01161	−0.505804	+	0.862649i	$0.668804\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.43289	0.202637
$288$	0	0
$289$	19.9706	1.17474
$290$	0	0
$291$	0	0
$292$	0	0
$293$	27.5776	1.61110	0.805550	−	0.592527i	$-0.201870\pi$
$293$	27.5776	1.61110	0.805550	+	0.592527i	$0.201870\pi$
$294$	0	0
$295$	−18.6148	−1.08379
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.06147	0.177049
$300$	0	0
$301$	36.9706	2.13095
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.16679	0.467629
$306$	0	0
$307$	9.30739	0.531201	0.265600	−	0.964083i	$-0.414430\pi$
$307$	9.30739	0.531201	0.265600	+	0.964083i	$0.414430\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10.4525	0.592707	0.296354	−	0.955078i	$-0.404229\pi$
$311$	10.4525	0.592707	0.296354	+	0.955078i	$0.404229\pi$
$312$	0	0
$313$	−10.1421	−0.573267	−0.286634	−	0.958040i	$-0.592536\pi$
$313$	−10.1421	−0.573267	−0.286634	+	0.958040i	$0.592536\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.1023	1.46605	0.733024	−	0.680202i	$-0.238108\pi$
$317$	26.1023	1.46605	0.733024	+	0.680202i	$0.238108\pi$
$318$	0	0
$319$	10.4366	0.584335
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11.7206	0.652152
$324$	0	0
$325$	−1.07107	−0.0594122
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.95068	0.162676
$330$	0	0
$331$	−18.6148	−1.02316	−0.511580	−	0.859236i	$-0.670940\pi$
$331$	−18.6148	−1.02316	−0.511580	+	0.859236i	$0.670940\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	16.5754	0.905613
$336$	0	0
$337$	14.4853	0.789064	0.394532	−	0.918882i	$-0.370907\pi$
$337$	14.4853	0.789064	0.394532	+	0.918882i	$0.370907\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9.21001	−0.498750
$342$	0	0
$343$	−10.4366	−0.563521
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.9719	−1.34056	−0.670282	−	0.742106i	$-0.733827\pi$
$347$	−24.9719	−1.34056	−0.670282	+	0.742106i	$0.733827\pi$
$348$	0	0
$349$	21.7574	1.16464	0.582322	−	0.812958i	$-0.302144\pi$
$349$	21.7574	1.16464	0.582322	+	0.812958i	$0.302144\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.3707	−1.13745	−0.568724	−	0.822529i	$-0.692563\pi$
$353$	−21.3707	−1.13745	−0.568724	+	0.822529i	$0.692563\pi$
$354$	0	0
$355$	−26.3253	−1.39720
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.32957	−0.228506	−0.114253	−	0.993452i	$-0.536447\pi$
$359$	−4.32957	−0.228506	−0.114253	+	0.993452i	$0.536447\pi$
$360$	0	0
$361$	−15.2843	−0.804435
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11.5496	−0.604533
$366$	0	0
$367$	18.0502	0.942212	0.471106	−	0.882077i	$-0.343855\pi$
$367$	18.0502	0.942212	0.471106	+	0.882077i	$0.343855\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−22.4357	−1.16481
$372$	0	0
$373$	5.27208	0.272978	0.136489	−	0.990642i	$-0.456418\pi$
$373$	5.27208	0.272978	0.136489	+	0.990642i	$0.456418\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.99390	0.205696
$378$	0	0
$379$	−7.37976	−0.379073	−0.189536	−	0.981874i	$-0.560699\pi$
$379$	−7.37976	−0.379073	−0.189536	+	0.981874i	$0.560699\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.0083	1.02238	0.511189	−	0.859468i	$-0.329205\pi$
$383$	20.0083	1.02238	0.511189	+	0.859468i	$0.329205\pi$
$384$	0	0
$385$	8.97056	0.457182
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0.305553	0.0154921	0.00774607	−	0.999970i	$-0.497534\pi$
$389$	0.305553	0.0154921	0.00774607	+	0.999970i	$0.497534\pi$
$390$	0	0
$391$	31.7774	1.60705
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.86030	−0.294864
$396$	0	0
$397$	−16.3848	−0.822328	−0.411164	−	0.911561i	$-0.634878\pi$
$397$	−16.3848	−0.822328	−0.411164	+	0.911561i	$0.634878\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.0681	0.702529	0.351265	−	0.936276i	$-0.385752\pi$
$401$	14.0681	0.702529	0.351265	+	0.936276i	$0.385752\pi$
$402$	0	0
$403$	−3.52452	−0.175569
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.22625	0.259056
$408$	0	0
$409$	−1.65685	−0.0819262	−0.0409631	−	0.999161i	$-0.513043\pi$
$409$	−1.65685	−0.0819262	−0.0409631	+	0.999161i	$0.513043\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−34.3956	−1.69250
$414$	0	0
$415$	−23.5992	−1.15844
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13.7766	−0.673031	−0.336516	−	0.941678i	$-0.609248\pi$
$419$	−13.7766	−0.673031	−0.336516	+	0.941678i	$0.609248\pi$
$420$	0	0
$421$	28.3848	1.38339	0.691695	−	0.722190i	$-0.256864\pi$
$421$	28.3848	1.38339	0.691695	+	0.722190i	$0.256864\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−11.1175	−0.539276
$426$	0	0
$427$	15.0903	0.730269
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.6788	0.755219	0.377610	−	0.925965i	$-0.376746\pi$
$431$	15.6788	0.755219	0.377610	+	0.925965i	$0.376746\pi$
$432$	0	0
$433$	28.9706	1.39224	0.696118	−	0.717927i	$-0.254909\pi$
$433$	28.9706	1.39224	0.696118	+	0.717927i	$0.254909\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.0742	0.481916
$438$	0	0
$439$	−32.3420	−1.54360	−0.771799	−	0.635866i	$-0.780643\pi$
$439$	−32.3420	−1.54360	−0.771799	+	0.635866i	$0.780643\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.2513	0.629590	0.314795	−	0.949160i	$-0.398064\pi$
$443$	13.2513	0.629590	0.314795	+	0.949160i	$0.398064\pi$
$444$	0	0
$445$	12.6863	0.601388
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−11.1175	−0.524666	−0.262333	−	0.964977i	$-0.584492\pi$
$449$	−11.1175	−0.524666	−0.262333	+	0.964977i	$0.584492\pi$
$450$	0	0
$451$	1.59689	0.0751948
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.43289	0.160936
$456$	0	0
$457$	−15.3137	−0.716345	−0.358173	−	0.933655i	$-0.616600\pi$
$457$	−15.3137	−0.716345	−0.358173	+	0.933655i	$0.616600\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−24.6269	−1.14699	−0.573495	−	0.819209i	$-0.694413\pi$
$461$	−24.6269	−1.14699	−0.573495	+	0.819209i	$0.694413\pi$
$462$	0	0
$463$	−26.8898	−1.24968	−0.624839	−	0.780754i	$-0.714835\pi$
$463$	−26.8898	−1.24968	−0.624839	+	0.780754i	$0.714835\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.1899	0.471531	0.235766	−	0.971810i	$-0.424240\pi$
$467$	10.1899	0.471531	0.235766	+	0.971810i	$0.424240\pi$
$468$	0	0
$469$	30.6274	1.41424
$470$	0	0
$471$	0	0
$472$	0	0
$473$	17.1978	0.790756
$474$	0	0
$475$	−3.52452	−0.161716
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.9887	−0.593469	−0.296735	−	0.954960i	$-0.595898\pi$
$479$	−12.9887	−0.593469	−0.296735	+	0.954960i	$0.595898\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.0249	0.591431
$486$	0	0
$487$	4.41983	0.200282	0.100141	−	0.994973i	$-0.468071\pi$
$487$	4.41983	0.200282	0.100141	+	0.994973i	$0.468071\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.06147	0.138162	0.0690810	−	0.997611i	$-0.477993\pi$
$491$	3.06147	0.138162	0.0690810	+	0.997611i	$0.477993\pi$
$492$	0	0
$493$	41.4558	1.86708
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−48.6427	−2.18193
$498$	0	0
$499$	−9.30739	−0.416656	−0.208328	−	0.978059i	$-0.566802\pi$
$499$	−9.30739	−0.416656	−0.208328	+	0.978059i	$0.566802\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.32957	−0.193046	−0.0965230	−	0.995331i	$-0.530772\pi$
$503$	−4.32957	−0.193046	−0.0965230	+	0.995331i	$0.530772\pi$
$504$	0	0
$505$	−27.4558	−1.22177
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.9416	−0.617949	−0.308975	−	0.951070i	$-0.599986\pi$
$509$	−13.9416	−0.617949	−0.308975	+	0.951070i	$0.599986\pi$
$510$	0	0
$511$	−21.3408	−0.944063
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.7151	0.472165
$516$	0	0
$517$	1.37258	0.0603661
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21.1917	0.928425	0.464213	−	0.885724i	$-0.346337\pi$
$521$	21.1917	0.928425	0.464213	+	0.885724i	$0.346337\pi$
$522$	0	0
$523$	20.5424	0.898256	0.449128	−	0.893467i	$-0.351735\pi$
$523$	20.5424	0.898256	0.449128	+	0.893467i	$0.351735\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−36.5838	−1.59361
$528$	0	0
$529$	4.31371	0.187553
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.611105	0.0264699
$534$	0	0
$535$	24.0669	1.04050
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.86030	0.252421
$540$	0	0
$541$	6.72792	0.289256	0.144628	−	0.989486i	$-0.453801\pi$
$541$	6.72792	0.289256	0.144628	+	0.989486i	$0.453801\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	23.2781	0.997126
$546$	0	0
$547$	33.7050	1.44112	0.720561	−	0.693391i	$-0.243884\pi$
$547$	33.7050	1.44112	0.720561	+	0.693391i	$0.243884\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.1426	0.559892
$552$	0	0
$553$	−10.8284	−0.460472
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.78089	−0.0754588	−0.0377294	−	0.999288i	$-0.512012\pi$
$557$	−1.78089	−0.0754588	−0.0377294	+	0.999288i	$0.512012\pi$
$558$	0	0
$559$	6.58132	0.278360
$560$	0	0
$561$	0	0
$562$	0	0
$563$	40.2793	1.69757	0.848785	−	0.528739i	$-0.177335\pi$
$563$	40.2793	1.69757	0.848785	+	0.528739i	$0.177335\pi$
$564$	0	0
$565$	−17.9411	−0.754789
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13.2039	−0.553537	−0.276768	−	0.960937i	$-0.589263\pi$
$569$	−13.2039	−0.553537	−0.276768	+	0.960937i	$0.589263\pi$
$570$	0	0
$571$	−17.0179	−0.712176	−0.356088	−	0.934452i	$-0.615890\pi$
$571$	−17.0179	−0.712176	−0.356088	+	0.934452i	$0.615890\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.55582	−0.398505
$576$	0	0
$577$	37.1127	1.54502	0.772511	−	0.635001i	$-0.219001\pi$
$577$	37.1127	1.54502	0.772511	+	0.635001i	$0.219001\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−43.6056	−1.80907
$582$	0	0
$583$	−10.4366	−0.432238
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.7821	−0.610121	−0.305061	−	0.952333i	$-0.598677\pi$
$587$	−14.7821	−0.610121	−0.305061	+	0.952333i	$0.598677\pi$
$588$	0	0
$589$	−11.5980	−0.477886
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−10.0742	−0.413699	−0.206850	−	0.978373i	$-0.566321\pi$
$593$	−10.0742	−0.413699	−0.206850	+	0.978373i	$0.566321\pi$
$594$	0	0
$595$	35.6326	1.46080
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13.8854	−0.567342	−0.283671	−	0.958922i	$-0.591552\pi$
$599$	−13.8854	−0.567342	−0.283671	+	0.958922i	$0.591552\pi$
$600$	0	0
$601$	−20.8284	−0.849609	−0.424805	−	0.905285i	$-0.639657\pi$
$601$	−20.8284	−0.849609	−0.424805	+	0.905285i	$0.639657\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−15.4169	−0.626787
$606$	0	0
$607$	1.69376	0.0687477	0.0343739	−	0.999409i	$-0.489056\pi$
$607$	1.69376	0.0687477	0.0343739	+	0.999409i	$0.489056\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.525265	0.0212499
$612$	0	0
$613$	19.8995	0.803733	0.401867	−	0.915698i	$-0.368362\pi$
$613$	19.8995	0.803733	0.401867	+	0.915698i	$0.368362\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	26.3253	1.05810	0.529051	−	0.848590i	$-0.322548\pi$
$619$	26.3253	1.05810	0.529051	+	0.848590i	$0.322548\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	23.4412	0.939152
$624$	0	0
$625$	−12.5147	−0.500589
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.7596	0.827739
$630$	0	0
$631$	41.6494	1.65804	0.829018	−	0.559222i	$-0.188900\pi$
$631$	41.6494	1.65804	0.829018	+	0.559222i	$0.188900\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.00547	−0.0399008
$636$	0	0
$637$	2.24264	0.0888567
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.08034	0.240159	0.120080	−	0.992764i	$-0.461685\pi$
$641$	6.08034	0.240159	0.120080	+	0.992764i	$0.461685\pi$
$642$	0	0
$643$	5.78287	0.228054	0.114027	−	0.993478i	$-0.463625\pi$
$643$	5.78287	0.228054	0.114027	+	0.993478i	$0.463625\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.9970	1.29725	0.648624	−	0.761109i	$-0.275345\pi$
$647$	32.9970	1.29725	0.648624	+	0.761109i	$0.275345\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.2629	1.49734	0.748672	−	0.662940i	$-0.230692\pi$
$653$	38.2629	1.49734	0.748672	+	0.662940i	$0.230692\pi$
$654$	0	0
$655$	20.8731	0.815580
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.06147	−0.119258	−0.0596289	−	0.998221i	$-0.518992\pi$
$659$	−3.06147	−0.119258	−0.0596289	+	0.998221i	$0.518992\pi$
$660$	0	0
$661$	40.3848	1.57079	0.785393	−	0.618998i	$-0.212461\pi$
$661$	40.3848	1.57079	0.785393	+	0.618998i	$0.212461\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.2965	0.438058
$666$	0	0
$667$	35.6326	1.37970
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.01962	0.270989
$672$	0	0
$673$	6.34315	0.244510	0.122255	−	0.992499i	$-0.460987\pi$
$673$	6.34315	0.244510	0.122255	+	0.992499i	$0.460987\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.8058	−0.569033	−0.284517	−	0.958671i	$-0.591833\pi$
$677$	−14.8058	−0.569033	−0.284517	+	0.958671i	$0.591833\pi$
$678$	0	0
$679$	24.0669	0.923603
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16.8381	−0.644291	−0.322145	−	0.946690i	$-0.604404\pi$
$683$	−16.8381	−0.644291	−0.322145	+	0.946690i	$0.604404\pi$
$684$	0	0
$685$	28.7696	1.09923
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3.99390	−0.152155
$690$	0	0
$691$	1.92762	0.0733302	0.0366651	−	0.999328i	$-0.488327\pi$
$691$	1.92762	0.0733302	0.0366651	+	0.999328i	$0.488327\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−33.1509	−1.25748
$696$	0	0
$697$	6.34315	0.240264
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19.5898	0.739897	0.369948	−	0.929052i	$-0.379375\pi$
$701$	19.5898	0.739897	0.369948	+	0.929052i	$0.379375\pi$
$702$	0	0
$703$	6.58132	0.248219
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−50.7318	−1.90797
$708$	0	0
$709$	−6.72792	−0.252672	−0.126336	−	0.991987i	$-0.540322\pi$
$709$	−6.72792	−0.252672	−0.126336	+	0.991987i	$0.540322\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−31.4449	−1.17762
$714$	0	0
$715$	1.59689	0.0597205
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.12293	−0.228347	−0.114173	−	0.993461i	$-0.536422\pi$
$719$	−6.12293	−0.228347	−0.114173	+	0.993461i	$0.536422\pi$
$720$	0	0
$721$	19.7990	0.737353
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.4662	−0.462984
$726$	0	0
$727$	−11.4689	−0.425357	−0.212678	−	0.977122i	$-0.568219\pi$
$727$	−11.4689	−0.425357	−0.212678	+	0.977122i	$0.568219\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	68.3127	2.52664
$732$	0	0
$733$	29.5563	1.09169	0.545844	−	0.837887i	$-0.316209\pi$
$733$	29.5563	1.09169	0.545844	+	0.837887i	$0.316209\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.2471	0.524800
$738$	0	0
$739$	17.0179	0.626013	0.313006	−	0.949751i	$-0.398664\pi$
$739$	17.0179	0.626013	0.313006	+	0.949751i	$0.398664\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.8854	0.509406	0.254703	−	0.967019i	$-0.418022\pi$
$743$	13.8854	0.509406	0.254703	+	0.967019i	$0.418022\pi$
$744$	0	0
$745$	9.51472	0.348592
$746$	0	0
$747$	0	0
$748$	0	0
$749$	44.4699	1.62489
$750$	0	0
$751$	−52.5537	−1.91771	−0.958855	−	0.283896i	$-0.908373\pi$
$751$	−52.5537	−1.91771	−0.958855	+	0.283896i	$0.908373\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.00547	0.0365928
$756$	0	0
$757$	−50.0416	−1.81879	−0.909397	−	0.415929i	$-0.863456\pi$
$757$	−50.0416	−1.81879	−0.909397	+	0.415929i	$0.863456\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.1917	−0.768199	−0.384099	−	0.923292i	$-0.625488\pi$
$761$	−21.1917	−0.768199	−0.384099	+	0.923292i	$0.625488\pi$
$762$	0	0
$763$	43.0124	1.55715
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.12293	−0.221086
$768$	0	0
$769$	−25.1716	−0.907710	−0.453855	−	0.891076i	$-0.649952\pi$
$769$	−25.1716	−0.907710	−0.453855	+	0.891076i	$0.649952\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	33.2258	1.19505	0.597525	−	0.801850i	$-0.296151\pi$
$773$	33.2258	1.19505	0.597525	+	0.801850i	$0.296151\pi$
$774$	0	0
$775$	11.0011	0.395173
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.01094	0.0720493
$780$	0	0
$781$	−22.6274	−0.809673
$782$	0	0
$783$	0	0
$784$	0	0
$785$	25.3646	0.905301
$786$	0	0
$787$	7.37976	0.263060	0.131530	−	0.991312i	$-0.458011\pi$
$787$	7.37976	0.263060	0.131530	+	0.991312i	$0.458011\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−33.1509	−1.17871
$792$	0	0
$793$	2.68629	0.0953930
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.0529	1.02911	0.514554	−	0.857458i	$-0.327958\pi$
$797$	29.0529	1.02911	0.514554	+	0.857458i	$0.327958\pi$
$798$	0	0
$799$	5.45214	0.192883
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.92724	−0.350325
$804$	0	0
$805$	30.6274	1.07947
$806$	0	0
$807$	0	0
$808$	0	0
$809$	27.4510	0.965127	0.482563	−	0.875861i	$-0.339706\pi$
$809$	27.4510	0.965127	0.482563	+	0.875861i	$0.339706\pi$
$810$	0	0
$811$	−48.4645	−1.70182	−0.850910	−	0.525311i	$-0.823949\pi$
$811$	−48.4645	−1.70182	−0.850910	+	0.525311i	$0.823949\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−29.7180	−1.04098
$816$	0	0
$817$	21.6569	0.757677
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.9294	−0.765340	−0.382670	−	0.923885i	$-0.624995\pi$
$821$	−21.9294	−0.765340	−0.382670	+	0.923885i	$0.624995\pi$
$822$	0	0
$823$	−3.29066	−0.114705	−0.0573526	−	0.998354i	$-0.518266\pi$
$823$	−3.29066	−0.114705	−0.0573526	+	0.998354i	$0.518266\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.79337	0.0623615	0.0311807	−	0.999514i	$-0.490073\pi$
$827$	1.79337	0.0623615	0.0311807	+	0.999514i	$0.490073\pi$
$828$	0	0
$829$	−9.55635	−0.331906	−0.165953	−	0.986134i	$-0.553070\pi$
$829$	−9.55635	−0.331906	−0.165953	+	0.986134i	$0.553070\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	23.2781	0.806540
$834$	0	0
$835$	−38.8264	−1.34364
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.3575	1.08258	0.541291	−	0.840835i	$-0.317936\pi$
$839$	31.3575	1.08258	0.541291	+	0.840835i	$0.317936\pi$
$840$	0	0
$841$	17.4853	0.602941
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−22.5405	−0.775416
$846$	0	0
$847$	−28.4867	−0.978816
$848$	0	0
$849$	0	0
$850$	0	0
$851$	17.8435	0.611669
$852$	0	0
$853$	50.5269	1.73001	0.865004	−	0.501765i	$-0.167316\pi$
$853$	50.5269	1.73001	0.865004	+	0.501765i	$0.167316\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.99390	0.136429	0.0682145	−	0.997671i	$-0.478270\pi$
$857$	3.99390	0.136429	0.0682145	+	0.997671i	$0.478270\pi$
$858$	0	0
$859$	16.6871	0.569358	0.284679	−	0.958623i	$-0.408113\pi$
$859$	16.6871	0.569358	0.284679	+	0.958623i	$0.408113\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−7.76245	−0.264237	−0.132119	−	0.991234i	$-0.542178\pi$
$863$	−7.76245	−0.264237	−0.132119	+	0.991234i	$0.542178\pi$
$864$	0	0
$865$	5.79899	0.197172
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5.03712	−0.170873
$870$	0	0
$871$	5.45214	0.184739
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−40.0166	−1.35281
$876$	0	0
$877$	41.8406	1.41286	0.706429	−	0.707784i	$-0.250305\pi$
$877$	41.8406	1.41286	0.706429	+	0.707784i	$0.250305\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	51.5934	1.73823	0.869113	−	0.494613i	$-0.164690\pi$
$881$	51.5934	1.73823	0.869113	+	0.494613i	$0.164690\pi$
$882$	0	0
$883$	−15.0903	−0.507827	−0.253914	−	0.967227i	$-0.581718\pi$
$883$	−15.0903	−0.507827	−0.253914	+	0.967227i	$0.581718\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−42.5529	−1.42878	−0.714392	−	0.699745i	$-0.753297\pi$
$887$	−42.5529	−1.42878	−0.714392	+	0.699745i	$0.753297\pi$
$888$	0	0
$889$	−1.85786	−0.0623108
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.72847	0.0578409
$894$	0	0
$895$	−28.5836	−0.955445
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−41.0221	−1.36816
$900$	0	0
$901$	−41.4558	−1.38109
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.26543	0.0753055
$906$	0	0
$907$	24.3976	0.810110	0.405055	−	0.914292i	$-0.367252\pi$
$907$	24.3976	0.810110	0.405055	+	0.914292i	$0.367252\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	53.9020	1.78585	0.892927	−	0.450201i	$-0.148648\pi$
$911$	53.9020	1.78585	0.892927	+	0.450201i	$0.148648\pi$
$912$	0	0
$913$	−20.2843	−0.671311
$914$	0	0
$915$	0	0
$916$	0	0
$917$	38.5685	1.27364
$918$	0	0
$919$	12.1303	0.400142	0.200071	−	0.979781i	$-0.435883\pi$
$919$	12.1303	0.400142	0.200071	+	0.979781i	$0.435883\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.65914	−0.285019
$924$	0	0
$925$	−6.24264	−0.205257
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−23.2781	−0.763731	−0.381866	−	0.924218i	$-0.624718\pi$
$929$	−23.2781	−0.763731	−0.381866	+	0.924218i	$0.624718\pi$
$930$	0	0
$931$	7.37976	0.241862
$932$	0	0
$933$	0	0
$934$	0	0
$935$	16.5754	0.542075
$936$	0	0
$937$	−51.9411	−1.69684	−0.848421	−	0.529322i	$-0.822447\pi$
$937$	−51.9411	−1.69684	−0.848421	+	0.529322i	$0.822447\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−48.9483	−1.59567	−0.797834	−	0.602877i	$-0.794021\pi$
$941$	−48.9483	−1.59567	−0.797834	+	0.602877i	$0.794021\pi$
$942$	0	0
$943$	5.45214	0.177546
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.8322	1.00191	0.500957	−	0.865472i	$-0.332982\pi$
$947$	30.8322	1.00191	0.500957	+	0.865472i	$0.332982\pi$
$948$	0	0
$949$	−3.79899	−0.123320
$950$	0	0
$951$	0	0
$952$	0	0
$953$	35.4388	1.14798	0.573988	−	0.818864i	$-0.305396\pi$
$953$	35.4388	1.14798	0.573988	+	0.818864i	$0.305396\pi$
$954$	0	0
$955$	−20.2117	−0.654034
$956$	0	0
$957$	0	0
$958$	0	0
$959$	53.1592	1.71660
$960$	0	0
$961$	5.20101	0.167775
$962$	0	0
$963$	0	0
$964$	0	0
$965$	22.2349	0.715768
$966$	0	0
$967$	−49.3599	−1.58731	−0.793653	−	0.608371i	$-0.791823\pi$
$967$	−49.3599	−1.58731	−0.793653	+	0.608371i	$0.791823\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−45.8770	−1.47226	−0.736131	−	0.676839i	$-0.763349\pi$
$971$	−45.8770	−1.47226	−0.736131	+	0.676839i	$0.763349\pi$
$972$	0	0
$973$	−61.2548	−1.96374
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.3646	0.811485	0.405743	−	0.913987i	$-0.367013\pi$
$977$	25.3646	0.811485	0.405743	+	0.913987i	$0.367013\pi$
$978$	0	0
$979$	10.9043	0.348502
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−41.8100	−1.33353	−0.666766	−	0.745267i	$-0.732322\pi$
$983$	−41.8100	−1.33353	−0.666766	+	0.745267i	$0.732322\pi$
$984$	0	0
$985$	5.79899	0.184771
$986$	0	0
$987$	0	0
$988$	0	0
$989$	58.7170	1.86709
$990$	0	0
$991$	24.1638	0.767588	0.383794	−	0.923419i	$-0.374617\pi$
$991$	24.1638	0.767588	0.383794	+	0.923419i	$0.374617\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−34.1563	−1.08283
$996$	0	0
$997$	10.0416	0.318022	0.159011	−	0.987277i	$-0.449170\pi$
$997$	10.0416	0.318022	0.159011	+	0.987277i	$0.449170\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9216.2.a.bs.1.6		8
3.2	odd	2	inner	9216.2.a.bs.1.4		8
4.3	odd	2	inner	9216.2.a.bs.1.5		8
8.3	odd	2		9216.2.a.br.1.3		8
8.5	even	2		9216.2.a.br.1.4		8
12.11	even	2	inner	9216.2.a.bs.1.3		8
24.5	odd	2		9216.2.a.br.1.6		8
24.11	even	2		9216.2.a.br.1.5		8
32.3	odd	8		4608.2.k.bl.1153.4	yes	16
32.5	even	8		4608.2.k.bk.3457.6	yes	16
32.11	odd	8		4608.2.k.bl.3457.3	yes	16
32.13	even	8		4608.2.k.bk.1153.5	yes	16
32.19	odd	8		4608.2.k.bk.1153.6	yes	16
32.21	even	8		4608.2.k.bl.3457.4	yes	16
32.27	odd	8		4608.2.k.bk.3457.5	yes	16
32.29	even	8		4608.2.k.bl.1153.3	yes	16
96.5	odd	8		4608.2.k.bk.3457.4	yes	16
96.11	even	8		4608.2.k.bl.3457.5	yes	16
96.29	odd	8		4608.2.k.bl.1153.5	yes	16
96.35	even	8		4608.2.k.bl.1153.6	yes	16
96.53	odd	8		4608.2.k.bl.3457.6	yes	16
96.59	even	8		4608.2.k.bk.3457.3	yes	16
96.77	odd	8		4608.2.k.bk.1153.3	✓	16
96.83	even	8		4608.2.k.bk.1153.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.k.bk.1153.3	✓	16	96.77	odd	8
4608.2.k.bk.1153.4	yes	16	96.83	even	8
4608.2.k.bk.1153.5	yes	16	32.13	even	8
4608.2.k.bk.1153.6	yes	16	32.19	odd	8
4608.2.k.bk.3457.3	yes	16	96.59	even	8
4608.2.k.bk.3457.4	yes	16	96.5	odd	8
4608.2.k.bk.3457.5	yes	16	32.27	odd	8
4608.2.k.bk.3457.6	yes	16	32.5	even	8
4608.2.k.bl.1153.3	yes	16	32.29	even	8
4608.2.k.bl.1153.4	yes	16	32.3	odd	8
4608.2.k.bl.1153.5	yes	16	96.29	odd	8
4608.2.k.bl.1153.6	yes	16	96.35	even	8
4608.2.k.bl.3457.3	yes	16	32.11	odd	8
4608.2.k.bl.3457.4	yes	16	32.21	even	8
4608.2.k.bl.3457.5	yes	16	96.11	even	8
4608.2.k.bl.3457.6	yes	16	96.53	odd	8
9216.2.a.br.1.3		8	8.3	odd	2
9216.2.a.br.1.4		8	8.5	even	2
9216.2.a.br.1.5		8	24.11	even	2
9216.2.a.br.1.6		8	24.5	odd	2
9216.2.a.bs.1.3		8	12.11	even	2	inner
9216.2.a.bs.1.4		8	3.2	odd	2	inner
9216.2.a.bs.1.5		8	4.3	odd	2	inner
9216.2.a.bs.1.6		8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9216 = 2^{10} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9216.a (trivial)

\(f(q)\)	\(=\)	\(q+1.78089 q^{5} +3.29066 q^{7} +O(q^{10})\)	\(q+1.78089 q^{5} +3.29066 q^{7} +1.53073 q^{11} +0.585786 q^{13} +6.08034 q^{17} +1.92762 q^{19} +5.22625 q^{23} -1.82843 q^{25} +6.81801 q^{29} -6.01673 q^{31} +5.86030 q^{35} +3.41421 q^{37} +1.04322 q^{41} +11.2350 q^{43} +0.896683 q^{47} +3.82843 q^{49} -6.81801 q^{53} +2.72607 q^{55} -10.4525 q^{59} +4.58579 q^{61} +1.04322 q^{65} +9.30739 q^{67} -14.7821 q^{71} -6.48528 q^{73} +5.03712 q^{77} -3.29066 q^{79} -13.2513 q^{83} +10.8284 q^{85} +7.12356 q^{89} +1.92762 q^{91} +3.43289 q^{95} +7.31371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{13} + 8 q^{25} + 16 q^{37} + 8 q^{49} + 48 q^{61} + 16 q^{73} + 64 q^{85} - 32 q^{97}+O(q^{100}) \) 8 * q + 16 * q^13 + 8 * q^25 + 16 * q^37 + 8 * q^49 + 48 * q^61 + 16 * q^73 + 64 * q^85 - 32 * q^97

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