LMFDB - Embedded newform 8649.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1389928896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 36x^{2} - 31 $ x^6 - 12x^4 + 36x^2 - 31
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.3
Root			$-1.26672$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.26672 q^{2} -0.395426 q^{4} -4.47014 q^{5} +4.92434 q^{7} +3.03433 q^{8} +O(q^{10})$	$q-1.26672 q^{2} -0.395426 q^{4} -4.47014 q^{5} +4.92434 q^{7} +3.03433 q^{8} +5.66241 q^{10} -6.23775 q^{14} -3.05279 q^{16} +5.19133 q^{19} +1.76761 q^{20} +14.9822 q^{25} -1.94721 q^{28} -2.20164 q^{32} -22.0125 q^{35} -6.57595 q^{38} -13.5639 q^{40} +8.34356 q^{41} +11.1355 q^{47} +17.2492 q^{49} -18.9782 q^{50} +14.9421 q^{56} -3.13016 q^{59} +8.89443 q^{64} +12.0000 q^{67} +27.8837 q^{70} +15.9439 q^{71} -2.05279 q^{76} +13.6464 q^{80} -10.5689 q^{82} -14.1056 q^{94} -23.2060 q^{95} +9.44821 q^{97} -21.8498 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{4}+O(q^{10}) $ 6 * q + 12 * q^4	$ 6 q + 12 q^{4} + 18 q^{10} + 24 q^{16} + 30 q^{25} - 54 q^{28} + 36 q^{40} + 42 q^{49} + 138 q^{64} + 72 q^{67} + 6 q^{70} + 30 q^{76} + 42 q^{82}+O(q^{100}) $ 6 * q + 12 * q^4 + 18 * q^10 + 24 * q^16 + 30 * q^25 - 54 * q^28 + 36 * q^40 + 42 * q^49 + 138 * q^64 + 72 * q^67 + 6 * q^70 + 30 * q^76 + 42 * q^82

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.26672	−0.895705	−0.447852	−	0.894108i	$-0.647811\pi$
$2$	−1.26672	−0.895705	−0.447852	+	0.894108i	$0.647811\pi$
$3$	0	0
$3$	0	0
$4$	−0.395426	−0.197713
$5$	−4.47014	−1.99911	−0.999554	−	0.0298497i	$-0.990497\pi$
$5$	−4.47014	−1.99911	−0.999554	+	0.0298497i	$0.990497\pi$
$6$	0	0
$7$	4.92434	1.86123	0.930614	−	0.366003i	$-0.119274\pi$
$7$	4.92434	1.86123	0.930614	+	0.366003i	$0.119274\pi$
$8$	3.03433	1.07280
$9$	0	0
$10$	5.66241	1.79061
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−6.23775	−1.66711
$15$	0	0
$16$	−3.05279	−0.763197
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	5.19133	1.19097	0.595486	−	0.803366i	$-0.296959\pi$
$19$	5.19133	1.19097	0.595486	+	0.803366i	$0.296959\pi$
$20$	1.76761	0.395250
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	14.9822	2.99644
$26$	0	0
$27$	0	0
$28$	−1.94721	−0.367989
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−2.20164	−0.389198
$33$	0	0
$34$	0	0
$35$	−22.0125	−3.72080
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−6.57595	−1.06676
$39$	0	0
$40$	−13.5639	−2.14464
$41$	8.34356	1.30304	0.651522	−	0.758629i	$-0.274131\pi$
$41$	8.34356	1.30304	0.651522	+	0.758629i	$0.274131\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.1355	1.62428	0.812142	−	0.583460i	$-0.198301\pi$
$47$	11.1355	1.62428	0.812142	+	0.583460i	$0.198301\pi$
$48$	0	0
$49$	17.2492	2.46417
$50$	−18.9782	−2.68392
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	14.9421	1.99672
$57$	0	0
$58$	0	0
$59$	−3.13016	−0.407513	−0.203756	−	0.979022i	$-0.565315\pi$
$59$	−3.13016	−0.407513	−0.203756	+	0.979022i	$0.565315\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	8.89443	1.11180
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	0	0
$70$	27.8837	3.33273
$71$	15.9439	1.89219	0.946094	−	0.323891i	$-0.104991\pi$
$71$	15.9439	1.89219	0.946094	+	0.323891i	$0.104991\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	−2.05279	−0.235471
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	13.6464	1.52571
$81$	0	0
$82$	−10.5689	−1.16714
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−14.1056	−1.45488
$95$	−23.2060	−2.38088
$96$	0	0
$97$	9.44821	0.959321	0.479660	−	0.877454i	$-0.340760\pi$
$97$	9.44821	0.959321	0.479660	+	0.877454i	$0.340760\pi$
$98$	−21.8498	−2.20717
$99$	0	0
$100$	−5.92434	−0.592434
$101$	−3.27669	−0.326043	−0.163021	−	0.986623i	$-0.552124\pi$
$101$	−3.27669	−0.326043	−0.163021	+	0.986623i	$0.552124\pi$
$102$	0	0
$103$	4.39038	0.432597	0.216298	−	0.976327i	$-0.430602\pi$
$103$	4.39038	0.432597	0.216298	+	0.976327i	$0.430602\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.7504	−1.42598	−0.712988	−	0.701176i	$-0.752659\pi$
$107$	−14.7504	−1.42598	−0.712988	+	0.701176i	$0.752659\pi$
$108$	0	0
$109$	−20.0979	−1.92503	−0.962513	−	0.271237i	$-0.912567\pi$
$109$	−20.0979	−1.92503	−0.962513	+	0.271237i	$0.912567\pi$
$110$	0	0
$111$	0	0
$112$	−15.0330	−1.42048
$113$	−18.4773	−1.73820	−0.869099	−	0.494638i	$-0.835301\pi$
$113$	−18.4773	−1.73820	−0.869099	+	0.494638i	$0.835301\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	3.96503	0.365011
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−44.6218	−3.99109
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−6.86345	−0.606649
$129$	0	0
$130$	0	0
$131$	11.1355	0.972916	0.486458	−	0.873704i	$-0.338289\pi$
$131$	11.1355	0.972916	0.486458	+	0.873704i	$0.338289\pi$
$132$	0	0
$133$	25.5639	2.21667
$134$	−15.2006	−1.31313
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	8.70432	0.735650
$141$	0	0
$142$	−20.1964	−1.69484
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.2711	1.82452	0.912258	−	0.409616i	$-0.134337\pi$
$149$	22.2711	1.82452	0.912258	+	0.409616i	$0.134337\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	15.7522	1.27767
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−20.3648	−1.62529	−0.812645	−	0.582758i	$-0.801973\pi$
$157$	−20.3648	−1.62529	−0.812645	+	0.582758i	$0.801973\pi$
$158$	0	0
$159$	0	0
$160$	9.84164	0.778050
$161$	0	0
$162$	0	0
$163$	−5.72530	−0.448440	−0.224220	−	0.974539i	$-0.571983\pi$
$163$	−5.72530	−0.448440	−0.224220	+	0.974539i	$0.571983\pi$
$164$	−3.29926	−0.257629
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.2711	−1.69324	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	−22.2711	−1.69324	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	73.7774	5.57705
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−4.40328	−0.321142
$189$	0	0
$190$	29.3954	2.13257
$191$	23.6907	1.71420	0.857099	−	0.515151i	$-0.172264\pi$
$191$	23.6907	1.71420	0.857099	+	0.515151i	$0.172264\pi$
$192$	0	0
$193$	9.18123	0.660879	0.330440	−	0.943827i	$-0.392803\pi$
$193$	9.18123	0.660879	0.330440	+	0.943827i	$0.392803\pi$
$194$	−11.9682	−0.859268
$195$	0	0
$196$	−6.82077	−0.487198
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	45.4609	3.21457
$201$	0	0
$202$	4.15064	0.292038
$203$	0	0
$204$	0	0
$205$	−37.2969	−2.60493
$206$	−5.56137	−0.387479
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.12339	0.283866	0.141933	−	0.989876i	$-0.454668\pi$
$211$	4.12339	0.283866	0.141933	+	0.989876i	$0.454668\pi$
$212$	0	0
$213$	0	0
$214$	18.6846	1.27725
$215$	0	0
$216$	0	0
$217$	0	0
$218$	25.4583	1.72425
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−10.8416	−0.724387
$225$	0	0
$226$	23.4055	1.55691
$227$	−11.1355	−0.739091	−0.369546	−	0.929213i	$-0.620487\pi$
$227$	−11.1355	−0.739091	−0.369546	+	0.929213i	$0.620487\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.15010	−0.468419	−0.234209	−	0.972186i	$-0.575250\pi$
$233$	−7.15010	−0.468419	−0.234209	+	0.972186i	$0.575250\pi$
$234$	0	0
$235$	−49.7774	−3.24712
$236$	1.23775	0.0805705
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	13.9339	0.895705
$243$	0	0
$244$	0	0
$245$	−77.1062	−4.92614
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	56.5232	3.57484
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−9.09480	−0.568425
$257$	10.7305	0.669348	0.334674	−	0.942334i	$-0.391374\pi$
$257$	10.7305	0.669348	0.334674	+	0.942334i	$0.391374\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−14.1056	−0.871445
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	−32.3822	−1.98548
$267$	0	0
$268$	−4.74511	−0.289854
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	−66.7932	−3.99166
$281$	16.0904	0.959872	0.479936	−	0.877303i	$-0.340660\pi$
$281$	16.0904	0.959872	0.479936	+	0.877303i	$0.340660\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−6.30462	−0.374110
$285$	0	0
$286$	0	0
$287$	41.0866	2.42526
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.2711	1.30109	0.650545	−	0.759468i	$-0.274541\pi$
$293$	22.2711	1.30109	0.650545	+	0.759468i	$0.274541\pi$
$294$	0	0
$295$	13.9923	0.814662
$296$	0	0
$297$	0	0
$298$	−28.2111	−1.63423
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−15.8480	−0.908946
$305$	0	0
$306$	0	0
$307$	−34.7374	−1.98257	−0.991284	−	0.131744i	$-0.957942\pi$
$307$	−34.7374	−1.98257	−0.991284	+	0.131744i	$0.957942\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.32362	−0.245170	−0.122585	−	0.992458i	$-0.539118\pi$
$311$	−4.32362	−0.245170	−0.122585	+	0.992458i	$0.539118\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	25.7965	1.45578
$315$	0	0
$316$	0	0
$317$	30.0975	1.69045	0.845223	−	0.534413i	$-0.179467\pi$
$317$	30.0975	1.69045	0.845223	+	0.534413i	$0.179467\pi$
$318$	0	0
$319$	0	0
$320$	−39.7594	−2.22262
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	7.25233	0.401670
$327$	0	0
$328$	25.3171	1.39790
$329$	54.8352	3.02316
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−53.6417	−2.93076
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	16.4673	0.895705
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	50.4704	2.72515
$344$	0	0
$345$	0	0
$346$	28.2111	1.51664
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	−93.4552	−4.99539
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−71.2714	−3.78269
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−37.6979	−1.98962	−0.994808	−	0.101767i	$-0.967550\pi$
$359$	−37.6979	−1.98962	−0.994808	+	0.101767i	$0.967550\pi$
$360$	0	0
$361$	7.94988	0.418415
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	20.8988	1.08210	0.541050	−	0.840991i	$-0.318027\pi$
$373$	20.8988	1.08210	0.541050	+	0.840991i	$0.318027\pi$
$374$	0	0
$375$	0	0
$376$	33.7889	1.74253
$377$	0	0
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	9.17625	0.470732
$381$	0	0
$382$	−30.0094	−1.53542
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−11.6300	−0.591953
$387$	0	0
$388$	−3.73607	−0.189670
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	52.3396	2.64355
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	39.7952	1.99727	0.998633	−	0.0522772i	$-0.0166479\pi$
$397$	39.7952	1.99727	0.998633	+	0.0522772i	$0.0166479\pi$
$398$	0	0
$399$	0	0
$400$	−45.7374	−2.28687
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	1.29569	0.0644629
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	47.2446	2.33325
$411$	0	0
$412$	−1.73607	−0.0855300
$413$	−15.4140	−0.758473
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−22.4972	−1.09906	−0.549531	−	0.835473i	$-0.685194\pi$
$419$	−22.4972	−1.09906	−0.549531	+	0.835473i	$0.685194\pi$
$420$	0	0
$421$	39.5282	1.92649	0.963244	−	0.268627i	$-0.0865698\pi$
$421$	39.5282	1.92649	0.963244	+	0.268627i	$0.0865698\pi$
$422$	−5.22317	−0.254260
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	5.83270	0.281934
$429$	0	0
$430$	0	0
$431$	−11.1355	−0.536380	−0.268190	−	0.963366i	$-0.586425\pi$
$431$	−11.1355	−0.536380	−0.268190	+	0.963366i	$0.586425\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	7.94721	0.380603
$437$	0	0
$438$	0	0
$439$	−35.5383	−1.69615	−0.848076	−	0.529874i	$-0.822239\pi$
$439$	−35.5383	−1.69615	−0.848076	+	0.529874i	$0.822239\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	38.8913	1.84778	0.923891	−	0.382656i	$-0.124991\pi$
$443$	38.8913	1.84778	0.923891	+	0.382656i	$0.124991\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	43.7992	2.06932
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	7.30641	0.343664
$453$	0	0
$454$	14.1056	0.662007
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	9.05716	0.419565
$467$	−42.7647	−1.97892	−0.989458	−	0.144822i	$-0.953739\pi$
$467$	−42.7647	−1.97892	−0.989458	+	0.144822i	$0.953739\pi$
$468$	0	0
$469$	59.0921	2.72862
$470$	63.0539	2.90846
$471$	0	0
$472$	−9.49795	−0.437178
$473$	0	0
$474$	0	0
$475$	77.7774	3.56867
$476$	0	0
$477$	0	0
$478$	0	0
$479$	41.5713	1.89944	0.949720	−	0.313101i	$-0.101368\pi$
$479$	41.5713	1.89944	0.949720	+	0.313101i	$0.101368\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	4.34969	0.197713
$485$	−42.2349	−1.91779
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	97.6718	4.41236
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	78.5131	3.52179
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	17.6446	0.789091
$501$	0	0
$502$	0	0
$503$	−0.450204	−0.0200736	−0.0100368	−	0.999950i	$-0.503195\pi$
$503$	−0.450204	−0.0200736	−0.0100368	+	0.999950i	$0.503195\pi$
$504$	0	0
$505$	14.6473	0.651795
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	25.2474	1.11579
$513$	0	0
$514$	−13.5925	−0.599538
$515$	−19.6256	−0.864808
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−44.5421	−1.95143	−0.975713	−	0.219054i	$-0.929703\pi$
$521$	−44.5421	−1.95143	−0.975713	+	0.219054i	$0.929703\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−4.40328	−0.192358
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	−10.1086	−0.438264
$533$	0	0
$534$	0	0
$535$	65.9364	2.85068
$536$	36.4119	1.57276
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	21.1658	0.909988	0.454994	−	0.890494i	$-0.349641\pi$
$541$	21.1658	0.909988	0.454994	+	0.890494i	$0.349641\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	89.8403	3.84833
$546$	0	0
$547$	3.58942	0.153473	0.0767364	−	0.997051i	$-0.475550\pi$
$547$	3.58942	0.153473	0.0767364	+	0.997051i	$0.475550\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	67.1995	2.83970
$561$	0	0
$562$	−20.3820	−0.859762
$563$	31.1445	1.31258	0.656292	−	0.754507i	$-0.272124\pi$
$563$	31.1445	1.31258	0.656292	+	0.754507i	$0.272124\pi$
$564$	0	0
$565$	82.5962	3.47485
$566$	−5.06687	−0.212976
$567$	0	0
$568$	48.3789	2.02993
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	−52.0451	−2.17232
$575$	0	0
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	21.5342	0.895705
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−28.2111	−1.16539
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	−17.7243	−0.729697
$591$	0	0
$592$	0	0
$593$	37.5513	1.54205	0.771024	−	0.636806i	$-0.219745\pi$
$593$	37.5513	1.54205	0.771024	+	0.636806i	$0.219745\pi$
$594$	0	0
$595$	0	0
$596$	−8.80656	−0.360731
$597$	0	0
$598$	0	0
$599$	18.3308	0.748975	0.374488	−	0.927232i	$-0.377819\pi$
$599$	18.3308	0.748975	0.374488	+	0.927232i	$0.377819\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	49.1716	1.99911
$606$	0	0
$607$	48.0000	1.94826	0.974130	−	0.225989i	$-0.0725612\pi$
$607$	48.0000	1.94826	0.974130	+	0.225989i	$0.0725612\pi$
$608$	−11.4294	−0.463525
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	44.0025	1.77580
$615$	0	0
$616$	0	0
$617$	44.5421	1.79320	0.896599	−	0.442843i	$-0.146030\pi$
$617$	44.5421	1.79320	0.896599	+	0.442843i	$0.146030\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	5.47681	0.219600
$623$	0	0
$624$	0	0
$625$	124.555	4.98219
$626$	0	0
$627$	0	0
$628$	8.05279	0.321341
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	−38.1251	−1.51414
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	30.6806	1.21276
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	2.26393	0.0886624
$653$	−22.2711	−0.871534	−0.435767	−	0.900060i	$-0.643523\pi$
$653$	−22.2711	−0.871534	−0.435767	+	0.900060i	$0.643523\pi$
$654$	0	0
$655$	−49.7774	−1.94496
$656$	−25.4711	−0.994479
$657$	0	0
$658$	−69.4607	−2.70786
$659$	−26.3707	−1.02725	−0.513627	−	0.858013i	$-0.671699\pi$
$659$	−26.3707	−1.02725	−0.513627	+	0.858013i	$0.671699\pi$
$660$	0	0
$661$	−18.4959	−0.719409	−0.359705	−	0.933066i	$-0.617123\pi$
$661$	−18.4959	−0.719409	−0.359705	+	0.933066i	$0.617123\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−114.274	−4.43136
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	67.9489	2.62510
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	5.14054	0.197713
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	46.5262	1.78551
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−50.5116	−1.93277	−0.966385	−	0.257098i	$-0.917234\pi$
$683$	−50.5116	−1.93277	−0.966385	+	0.257098i	$0.917234\pi$
$684$	0	0
$685$	0	0
$686$	−63.9318	−2.44093
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−33.6695	−1.28085	−0.640423	−	0.768022i	$-0.721241\pi$
$691$	−33.6695	−1.28085	−0.640423	+	0.768022i	$0.721241\pi$
$692$	8.80656	0.334775
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−38.0015	−1.43838
$699$	0	0
$700$	−29.1735	−1.10265
$701$	33.6779	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$701$	33.6779	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.1355	−0.606839
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	90.2807	3.38817
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	47.7525	1.78211
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	21.6197	0.805161
$722$	−10.0703	−0.374776
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	36.0723	1.33785	0.668924	−	0.743331i	$-0.266755\pi$
$727$	36.0723	1.33785	0.668924	+	0.743331i	$0.266755\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−38.7273	−1.43043	−0.715213	−	0.698907i	$-0.753670\pi$
$733$	−38.7273	−1.43043	−0.715213	+	0.698907i	$0.753670\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−99.5548	−3.64741
$746$	−26.4729	−0.969241
$747$	0	0
$748$	0	0
$749$	−72.6361	−2.65406
$750$	0	0
$751$	−54.7018	−1.99610	−0.998048	−	0.0624574i	$-0.980106\pi$
$751$	−54.7018	−1.99610	−0.998048	+	0.0624574i	$0.980106\pi$
$752$	−33.9944	−1.23965
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	−25.3344	−0.920185
$759$	0	0
$760$	−70.4146	−2.55420
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−98.9687	−3.58291
$764$	−9.36792	−0.338919
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−11.5841	−0.417733	−0.208866	−	0.977944i	$-0.566977\pi$
$769$	−11.5841	−0.417733	−0.208866	+	0.977944i	$0.566977\pi$
$770$	0	0
$771$	0	0
$772$	−3.63050	−0.130664
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	28.6690	1.02916
$777$	0	0
$778$	0	0
$779$	43.3141	1.55189
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−52.6580	−1.88064
$785$	91.0337	3.24913
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−90.9886	−3.23518
$792$	0	0
$793$	0	0
$794$	−50.4093	−1.78896
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	−32.9854	−1.16621
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−9.94255	−0.349778
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	25.5929	0.896480
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	14.7482	0.515028
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	13.3218	0.464088
$825$	0	0
$826$	19.5252	0.679368
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	28.4977	0.984435
$839$	55.6776	1.92221	0.961103	−	0.276191i	$-0.0890721\pi$
$839$	55.6776	1.92221	0.961103	+	0.276191i	$0.0890721\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−50.0711	−1.72557
$843$	0	0
$844$	−1.63050	−0.0561240
$845$	58.1119	1.99911
$846$	0	0
$847$	−54.1678	−1.86123
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	0	0
$855$	0	0
$856$	−44.7576	−1.52978
$857$	44.5421	1.52153	0.760765	−	0.649028i	$-0.224824\pi$
$857$	44.5421	1.52153	0.760765	+	0.649028i	$0.224824\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	14.1056	0.480438
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	99.5548	3.38497
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−60.9835	−2.06516
$873$	0	0
$874$	0	0
$875$	−219.733	−7.42833
$876$	0	0
$877$	41.1301	1.38887	0.694433	−	0.719557i	$-0.255655\pi$
$877$	41.1301	1.38887	0.694433	+	0.719557i	$0.255655\pi$
$878$	45.0171	1.51925
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	−49.2643	−1.65507
$887$	−49.3181	−1.65594	−0.827970	−	0.560773i	$-0.810504\pi$
$887$	−49.3181	−1.65594	−0.827970	+	0.560773i	$0.810504\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	57.8082	1.93448
$894$	0	0
$895$	0	0
$896$	−33.7980	−1.12911
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−56.0662	−1.86473
$905$	0	0
$906$	0	0
$907$	55.5027	1.84294	0.921469	−	0.388453i	$-0.126990\pi$
$907$	55.5027	1.84294	0.921469	+	0.388453i	$0.126990\pi$
$908$	4.40328	0.146128
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	54.8352	1.81082
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	89.5461	2.93475
$932$	2.82734	0.0926125
$933$	0	0
$934$	54.1708	1.77252
$935$	0	0
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	−74.8530	−2.44404
$939$	0	0
$940$	19.6833	0.641998
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	9.55572	0.311012
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	−98.5220	−3.19648
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	−105.901	−3.42687
$956$	0	0
$957$	0	0
$958$	−52.6591	−1.70134
$959$	0	0
$960$	0	0
$961$	0	0
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−41.0414	−1.32117
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−33.3776	−1.07280
$969$	0	0
$970$	53.4997	1.71777
$971$	55.6776	1.78678	0.893390	−	0.449281i	$-0.148320\pi$
$971$	55.6776	1.78678	0.893390	+	0.449281i	$0.148320\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	61.9853	1.98308	0.991542	−	0.129784i	$-0.0414283\pi$
$977$	61.9853	1.98308	0.991542	+	0.129784i	$0.0414283\pi$
$978$	0	0
$979$	0	0
$980$	30.4898	0.973961
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	−99.4539	−3.15449
$995$	0	0
$996$	0	0
$997$	−21.6998	−0.687238	−0.343619	−	0.939109i	$-0.611653\pi$
$997$	−21.6998	−0.687238	−0.343619	+	0.939109i	$0.611653\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.ba.1.3	✓	6
3.2	odd	2	inner	8649.2.a.ba.1.4	yes	6
31.30	odd	2	CM	8649.2.a.ba.1.3	✓	6
93.92	even	2	inner	8649.2.a.ba.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8649.2.a.ba.1.3	✓	6	1.1	even	1	trivial
8649.2.a.ba.1.3	✓	6	31.30	odd	2	CM
8649.2.a.ba.1.4	yes	6	3.2	odd	2	inner
8649.2.a.ba.1.4	yes	6	93.92	even	2	inner

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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-1.26672 q^{2} -0.395426 q^{4} -4.47014 q^{5} +4.92434 q^{7} +3.03433 q^{8} +O(q^{10})\)	\(q-1.26672 q^{2} -0.395426 q^{4} -4.47014 q^{5} +4.92434 q^{7} +3.03433 q^{8} +5.66241 q^{10} -6.23775 q^{14} -3.05279 q^{16} +5.19133 q^{19} +1.76761 q^{20} +14.9822 q^{25} -1.94721 q^{28} -2.20164 q^{32} -22.0125 q^{35} -6.57595 q^{38} -13.5639 q^{40} +8.34356 q^{41} +11.1355 q^{47} +17.2492 q^{49} -18.9782 q^{50} +14.9421 q^{56} -3.13016 q^{59} +8.89443 q^{64} +12.0000 q^{67} +27.8837 q^{70} +15.9439 q^{71} -2.05279 q^{76} +13.6464 q^{80} -10.5689 q^{82} -14.1056 q^{94} -23.2060 q^{95} +9.44821 q^{97} -21.8498 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{4}+O(q^{10}) \) 6 * q + 12 * q^4	\( 6 q + 12 q^{4} + 18 q^{10} + 24 q^{16} + 30 q^{25} - 54 q^{28} + 36 q^{40} + 42 q^{49} + 138 q^{64} + 72 q^{67} + 6 q^{70} + 30 q^{76} + 42 q^{82}+O(q^{100}) \) 6 * q + 12 * q^4 + 18 * q^10 + 24 * q^16 + 30 * q^25 - 54 * q^28 + 36 * q^40 + 42 * q^49 + 138 * q^64 + 72 * q^67 + 6 * q^70 + 30 * q^76 + 42 * q^82

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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