LMFDB - Embedded newform 196.3.c.c.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [196,3,Mod(99,196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(196, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("196.99");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 196 = 2^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	196.c (of order $2$, degree $1$, minimal)

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:	$5.34061318146$
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:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			99.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	196.99

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +1.41421 q^{5} +8.00000 q^{8} +9.00000 q^{9} +2.82843 q^{10} -24.0416 q^{13} +16.0000 q^{16} +32.5269 q^{17} +18.0000 q^{18} +5.65685 q^{20} -23.0000 q^{25} -48.0833 q^{26} -40.0000 q^{29} +32.0000 q^{32} +65.0538 q^{34} +36.0000 q^{36} -24.0000 q^{37} +11.3137 q^{40} -43.8406 q^{41} +12.7279 q^{45} -46.0000 q^{50} -96.1665 q^{52} -90.0000 q^{53} -80.0000 q^{58} +100.409 q^{61} +64.0000 q^{64} -34.0000 q^{65} +130.108 q^{68} +72.0000 q^{72} -145.664 q^{73} -48.0000 q^{74} +22.6274 q^{80} +81.0000 q^{81} -87.6812 q^{82} +46.0000 q^{85} -57.9828 q^{89} +25.4558 q^{90} +193.747 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 18 q^{9} + 32 q^{16} + 36 q^{18} - 46 q^{25} - 80 q^{29} + 64 q^{32} + 72 q^{36} - 48 q^{37} - 92 q^{50} - 180 q^{53} - 160 q^{58} + 128 q^{64} - 68 q^{65} + 144 q^{72}+ \cdots + 92 q^{85}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 + 18 * q^9 + 32 * q^16 + 36 * q^18 - 46 * q^25 - 80 * q^29 + 64 * q^32 + 72 * q^36 - 48 * q^37 - 92 * q^50 - 180 * q^53 - 160 * q^58 + 128 * q^64 - 68 * q^65 + 144 * q^72 - 96 * q^74 + 162 * q^81 + 92 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/196\mathbb{Z}\right)^\times$.

$n$	$99$	$101$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.00000	1.00000
$2$	2.00000	1.00000
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	4.00000	1.00000
$5$	1.41421	0.282843	0.141421	−	0.989949i	$-0.454833\pi$
$5$	1.41421	0.282843	0.141421	+	0.989949i	$0.454833\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.00000	1.00000
$9$	9.00000	1.00000
$10$	2.82843	0.282843
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	−24.0416	−1.84936	−0.924678	−	0.380750i	$-0.875666\pi$
$13$	−24.0416	−1.84936	−0.924678	+	0.380750i	$0.875666\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	32.5269	1.91335	0.956674	−	0.291162i	$-0.0940417\pi$
$17$	32.5269	1.91335	0.956674	+	0.291162i	$0.0940417\pi$
$18$	18.0000	1.00000
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	5.65685	0.282843
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−23.0000	−0.920000
$26$	−48.0833	−1.84936
$27$	0	0
$28$	0	0
$29$	−40.0000	−1.37931	−0.689655	−	0.724138i	$-0.742238\pi$
$29$	−40.0000	−1.37931	−0.689655	+	0.724138i	$0.742238\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	32.0000	1.00000
$33$	0	0
$34$	65.0538	1.91335
$35$	0	0
$36$	36.0000	1.00000
$37$	−24.0000	−0.648649	−0.324324	−	0.945946i	$-0.605137\pi$
$37$	−24.0000	−0.648649	−0.324324	+	0.945946i	$0.605137\pi$
$38$	0	0
$39$	0	0
$40$	11.3137	0.282843
$41$	−43.8406	−1.06928	−0.534642	−	0.845079i	$-0.679553\pi$
$41$	−43.8406	−1.06928	−0.534642	+	0.845079i	$0.679553\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	12.7279	0.282843
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	0	0
$50$	−46.0000	−0.920000
$51$	0	0
$52$	−96.1665	−1.84936
$53$	−90.0000	−1.69811	−0.849057	−	0.528302i	$-0.822829\pi$
$53$	−90.0000	−1.69811	−0.849057	+	0.528302i	$0.822829\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−80.0000	−1.37931
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	100.409	1.64605	0.823026	−	0.568004i	$-0.192284\pi$
$61$	100.409	1.64605	0.823026	+	0.568004i	$0.192284\pi$
$62$	0	0
$63$	0	0
$64$	64.0000	1.00000
$65$	−34.0000	−0.523077
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	130.108	1.91335
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	72.0000	1.00000
$73$	−145.664	−1.99540	−0.997699	−	0.0678048i	$-0.978401\pi$
$73$	−145.664	−1.99540	−0.997699	+	0.0678048i	$0.978401\pi$
$74$	−48.0000	−0.648649
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	22.6274	0.282843
$81$	81.0000	1.00000
$82$	−87.6812	−1.06928
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	46.0000	0.541176
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−57.9828	−0.651492	−0.325746	−	0.945457i	$-0.605615\pi$
$89$	−57.9828	−0.651492	−0.325746	+	0.945457i	$0.605615\pi$
$90$	25.4558	0.282843
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	193.747	1.99739	0.998697	−	0.0510283i	$-0.0162499\pi$
$97$	193.747	1.99739	0.998697	+	0.0510283i	$0.0162499\pi$
$98$	0	0
$99$	0	0
$100$	−92.0000	−0.920000
$101$	111.723	1.10617	0.553084	−	0.833126i	$-0.313451\pi$
$101$	111.723	1.10617	0.553084	+	0.833126i	$0.313451\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−192.333	−1.84936
$105$	0	0
$106$	−180.000	−1.69811
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	120.000	1.10092	0.550459	−	0.834862i	$-0.314453\pi$
$109$	120.000	1.10092	0.550459	+	0.834862i	$0.314453\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	30.0000	0.265487	0.132743	−	0.991150i	$-0.457621\pi$
$113$	30.0000	0.265487	0.132743	+	0.991150i	$0.457621\pi$
$114$	0	0
$115$	0	0
$116$	−160.000	−1.37931
$117$	−216.375	−1.84936
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	200.818	1.64605
$123$	0	0
$124$	0	0
$125$	−67.8823	−0.543058
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	128.000	1.00000
$129$	0	0
$130$	−68.0000	−0.523077
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	260.215	1.91335
$137$	176.000	1.28467	0.642336	−	0.766423i	$-0.277965\pi$
$137$	176.000	1.28467	0.642336	+	0.766423i	$0.277965\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	144.000	1.00000
$145$	−56.5685	−0.390128
$146$	−291.328	−1.99540
$147$	0	0
$148$	−96.0000	−0.648649
$149$	102.000	0.684564	0.342282	−	0.939597i	$-0.388800\pi$
$149$	102.000	0.684564	0.342282	+	0.939597i	$0.388800\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	292.742	1.91335
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−66.4680	−0.423363	−0.211682	−	0.977339i	$-0.567894\pi$
$157$	−66.4680	−0.423363	−0.211682	+	0.977339i	$0.567894\pi$
$158$	0	0
$159$	0	0
$160$	45.2548	0.282843
$161$	0	0
$162$	162.000	1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−175.362	−1.06928
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	409.000	2.42012
$170$	92.0000	0.541176
$171$	0	0
$172$	0	0
$173$	159.806	0.923735	0.461867	−	0.886949i	$-0.347180\pi$
$173$	159.806	0.923735	0.461867	+	0.886949i	$0.347180\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−115.966	−0.651492
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	50.9117	0.282843
$181$	−281.428	−1.55485	−0.777427	−	0.628973i	$-0.783475\pi$
$181$	−281.428	−1.55485	−0.777427	+	0.628973i	$0.783475\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−33.9411	−0.183466
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	190.000	0.984456	0.492228	−	0.870466i	$-0.336183\pi$
$193$	190.000	0.984456	0.492228	+	0.870466i	$0.336183\pi$
$194$	387.495	1.99739
$195$	0	0
$196$	0	0
$197$	−390.000	−1.97970	−0.989848	−	0.142132i	$-0.954604\pi$
$197$	−390.000	−1.97970	−0.989848	+	0.142132i	$0.954604\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−184.000	−0.920000
$201$	0	0
$202$	223.446	1.10617
$203$	0	0
$204$	0	0
$205$	−62.0000	−0.302439
$206$	0	0
$207$	0	0
$208$	−384.666	−1.84936
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−360.000	−1.69811
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	240.000	1.10092
$219$	0	0
$220$	0	0
$221$	−782.000	−3.53846
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−207.000	−0.920000
$226$	60.0000	0.265487
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	397.394	1.73535	0.867673	−	0.497136i	$-0.165615\pi$
$229$	397.394	1.73535	0.867673	+	0.497136i	$0.165615\pi$
$230$	0	0
$231$	0	0
$232$	−320.000	−1.37931
$233$	−416.000	−1.78541	−0.892704	−	0.450644i	$-0.851194\pi$
$233$	−416.000	−1.78541	−0.892704	+	0.450644i	$0.851194\pi$
$234$	−432.749	−1.84936
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−125.865	−0.522261	−0.261131	−	0.965303i	$-0.584095\pi$
$241$	−125.865	−0.522261	−0.261131	+	0.965303i	$0.584095\pi$
$242$	242.000	1.00000
$243$	0	0
$244$	401.637	1.64605
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−135.765	−0.543058
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	−315.370	−1.22712	−0.613560	−	0.789648i	$-0.710263\pi$
$257$	−315.370	−1.22712	−0.613560	+	0.789648i	$0.710263\pi$
$258$	0	0
$259$	0	0
$260$	−136.000	−0.523077
$261$	−360.000	−1.37931
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−127.279	−0.480299
$266$	0	0
$267$	0	0
$268$	0	0
$269$	270.115	1.00414	0.502072	−	0.864826i	$-0.332571\pi$
$269$	270.115	1.00414	0.502072	+	0.864826i	$0.332571\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	520.431	1.91335
$273$	0	0
$274$	352.000	1.28467
$275$	0	0
$276$	0	0
$277$	−230.000	−0.830325	−0.415162	−	0.909747i	$-0.636275\pi$
$277$	−230.000	−0.830325	−0.415162	+	0.909747i	$0.636275\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−320.000	−1.13879	−0.569395	−	0.822064i	$-0.692822\pi$
$281$	−320.000	−1.13879	−0.569395	+	0.822064i	$0.692822\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	288.000	1.00000
$289$	769.000	2.66090
$290$	−113.137	−0.390128
$291$	0	0
$292$	−582.656	−1.99540
$293$	−499.217	−1.70381	−0.851907	−	0.523693i	$-0.824554\pi$
$293$	−499.217	−1.70381	−0.851907	+	0.523693i	$0.824554\pi$
$294$	0	0
$295$	0	0
$296$	−192.000	−0.648649
$297$	0	0
$298$	204.000	0.684564
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	142.000	0.465574
$306$	585.484	1.91335
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	476.590	1.52265	0.761326	−	0.648369i	$-0.224549\pi$
$313$	476.590	1.52265	0.761326	+	0.648369i	$0.224549\pi$
$314$	−132.936	−0.423363
$315$	0	0
$316$	0	0
$317$	−150.000	−0.473186	−0.236593	−	0.971609i	$-0.576031\pi$
$317$	−150.000	−0.473186	−0.236593	+	0.971609i	$0.576031\pi$
$318$	0	0
$319$	0	0
$320$	90.5097	0.282843
$321$	0	0
$322$	0	0
$323$	0	0
$324$	324.000	1.00000
$325$	552.958	1.70141
$326$	0	0
$327$	0	0
$328$	−350.725	−1.06928
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	−216.000	−0.648649
$334$	0	0
$335$	0	0
$336$	0	0
$337$	576.000	1.70920	0.854599	−	0.519288i	$-0.173803\pi$
$337$	576.000	1.70920	0.854599	+	0.519288i	$0.173803\pi$
$338$	818.000	2.42012
$339$	0	0
$340$	184.000	0.541176
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	319.612	0.923735
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−677.408	−1.94100	−0.970499	−	0.241105i	$-0.922490\pi$
$349$	−677.408	−1.94100	−0.970499	+	0.241105i	$0.922490\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−66.4680	−0.188295	−0.0941474	−	0.995558i	$-0.530012\pi$
$353$	−66.4680	−0.188295	−0.0941474	+	0.995558i	$0.530012\pi$
$354$	0	0
$355$	0	0
$356$	−231.931	−0.651492
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	101.823	0.282843
$361$	361.000	1.00000
$362$	−562.857	−1.55485
$363$	0	0
$364$	0	0
$365$	−206.000	−0.564384
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−394.566	−1.06928
$370$	−67.8823	−0.183466
$371$	0	0
$372$	0	0
$373$	550.000	1.47453	0.737265	−	0.675603i	$-0.236117\pi$
$373$	550.000	1.47453	0.737265	+	0.675603i	$0.236117\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	961.665	2.55084
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	380.000	0.984456
$387$	0	0
$388$	774.989	1.99739
$389$	680.000	1.74807	0.874036	−	0.485861i	$-0.161494\pi$
$389$	680.000	1.74807	0.874036	+	0.485861i	$0.161494\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−780.000	−1.97970
$395$	0	0
$396$	0	0
$397$	−137.179	−0.345538	−0.172769	−	0.984962i	$-0.555271\pi$
$397$	−137.179	−0.345538	−0.172769	+	0.984962i	$0.555271\pi$
$398$	0	0
$399$	0	0
$400$	−368.000	−0.920000
$401$	−80.0000	−0.199501	−0.0997506	−	0.995012i	$-0.531805\pi$
$401$	−80.0000	−0.199501	−0.0997506	+	0.995012i	$0.531805\pi$
$402$	0	0
$403$	0	0
$404$	446.891	1.10617
$405$	114.551	0.282843
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−383.252	−0.937046	−0.468523	−	0.883451i	$-0.655214\pi$
$409$	−383.252	−0.937046	−0.468523	+	0.883451i	$0.655214\pi$
$410$	−124.000	−0.302439
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−769.332	−1.84936
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	58.0000	0.137767	0.0688836	−	0.997625i	$-0.478056\pi$
$421$	58.0000	0.137767	0.0688836	+	0.997625i	$0.478056\pi$
$422$	0	0
$423$	0	0
$424$	−720.000	−1.69811
$425$	−748.119	−1.76028
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	371.938	0.858980	0.429490	−	0.903072i	$-0.358693\pi$
$433$	371.938	0.858980	0.429490	+	0.903072i	$0.358693\pi$
$434$	0	0
$435$	0	0
$436$	480.000	1.10092
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	−1564.00	−3.53846
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−82.0000	−0.184270
$446$	0	0
$447$	0	0
$448$	0	0
$449$	702.000	1.56347	0.781737	−	0.623608i	$-0.214334\pi$
$449$	702.000	1.56347	0.781737	+	0.623608i	$0.214334\pi$
$450$	−414.000	−0.920000
$451$	0	0
$452$	120.000	0.265487
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−850.000	−1.85996	−0.929978	−	0.367615i	$-0.880174\pi$
$457$	−850.000	−1.85996	−0.929978	+	0.367615i	$0.880174\pi$
$458$	794.788	1.73535
$459$	0	0
$460$	0	0
$461$	906.511	1.96640	0.983201	−	0.182529i	$-0.0584282\pi$
$461$	906.511	1.96640	0.983201	+	0.182529i	$0.0584282\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−640.000	−1.37931
$465$	0	0
$466$	−832.000	−1.78541
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−865.499	−1.84936
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−810.000	−1.69811
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	576.999	1.19958
$482$	−251.730	−0.522261
$483$	0	0
$484$	484.000	1.00000
$485$	274.000	0.564948
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	803.273	1.64605
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	−1301.08	−2.63910
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−271.529	−0.543058
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	158.000	0.312871
$506$	0	0
$507$	0	0
$508$	0	0
$509$	337.997	0.664041	0.332021	−	0.943272i	$-0.392270\pi$
$509$	337.997	0.664041	0.332021	+	0.943272i	$0.392270\pi$
$510$	0	0
$511$	0	0
$512$	512.000	1.00000
$513$	0	0
$514$	−630.739	−1.22712
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−272.000	−0.523077
$521$	−1016.82	−1.95167	−0.975835	−	0.218511i	$-0.929880\pi$
$521$	−1016.82	−1.95167	−0.975835	+	0.218511i	$0.929880\pi$
$522$	−720.000	−1.37931
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	−254.558	−0.480299
$531$	0	0
$532$	0	0
$533$	1054.00	1.97749
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	540.230	1.00414
$539$	0	0
$540$	0	0
$541$	−682.000	−1.26063	−0.630314	−	0.776340i	$-0.717074\pi$
$541$	−682.000	−1.26063	−0.630314	+	0.776340i	$0.717074\pi$
$542$	0	0
$543$	0	0
$544$	1040.86	1.91335
$545$	169.706	0.311386
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	704.000	1.28467
$549$	903.682	1.64605
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−460.000	−0.830325
$555$	0	0
$556$	0	0
$557$	330.000	0.592460	0.296230	−	0.955117i	$-0.404271\pi$
$557$	330.000	0.592460	0.296230	+	0.955117i	$0.404271\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−640.000	−1.13879
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	42.4264	0.0750910
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1040.00	1.82777	0.913884	−	0.405975i	$-0.133068\pi$
$569$	1040.00	1.82777	0.913884	+	0.405975i	$0.133068\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	576.000	1.00000
$577$	745.291	1.29166	0.645832	−	0.763479i	$-0.276510\pi$
$577$	745.291	1.29166	0.645832	+	0.763479i	$0.276510\pi$
$578$	1538.00	2.66090
$579$	0	0
$580$	−226.274	−0.390128
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−1165.31	−1.99540
$585$	−306.000	−0.523077
$586$	−998.435	−1.70381
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−384.000	−0.648649
$593$	−137.179	−0.231330	−0.115665	−	0.993288i	$-0.536900\pi$
$593$	−137.179	−0.231330	−0.115665	+	0.993288i	$0.536900\pi$
$594$	0	0
$595$	0	0
$596$	408.000	0.684564
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−439.820	−0.731814	−0.365907	−	0.930651i	$-0.619241\pi$
$601$	−439.820	−0.731814	−0.365907	+	0.930651i	$0.619241\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	171.120	0.282843
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	284.000	0.465574
$611$	0	0
$612$	1170.97	1.91335
$613$	−1224.00	−1.99674	−0.998369	−	0.0570962i	$-0.981816\pi$
$613$	−1224.00	−1.99674	−0.998369	+	0.0570962i	$0.981816\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1216.00	−1.97083	−0.985413	−	0.170178i	$-0.945566\pi$
$617$	−1216.00	−1.97083	−0.985413	+	0.170178i	$0.945566\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	479.000	0.766400
$626$	953.180	1.52265
$627$	0	0
$628$	−265.872	−0.423363
$629$	−780.646	−1.24109
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−300.000	−0.473186
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	181.019	0.282843
$641$	400.000	0.624025	0.312012	−	0.950078i	$-0.398997\pi$
$641$	400.000	0.624025	0.312012	+	0.950078i	$0.398997\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	648.000	1.00000
$649$	0	0
$650$	1105.92	1.70141
$651$	0	0
$652$	0	0
$653$	−1144.00	−1.75191	−0.875957	−	0.482389i	$-0.839769\pi$
$653$	−1144.00	−1.75191	−0.875957	+	0.482389i	$0.839769\pi$
$654$	0	0
$655$	0	0
$656$	−701.450	−1.06928
$657$	−1310.98	−1.99540
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	408.708	0.618317	0.309159	−	0.951010i	$-0.399953\pi$
$661$	408.708	0.618317	0.309159	+	0.951010i	$0.399953\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−432.000	−0.648649
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1104.00	−1.64042	−0.820208	−	0.572065i	$-0.806142\pi$
$673$	−1104.00	−1.64042	−0.820208	+	0.572065i	$0.806142\pi$
$674$	1152.00	1.70920
$675$	0	0
$676$	1636.00	2.42012
$677$	−1028.13	−1.51866	−0.759330	−	0.650705i	$-0.774473\pi$
$677$	−1028.13	−1.51866	−0.759330	+	0.650705i	$0.774473\pi$
$678$	0	0
$679$	0	0
$680$	368.000	0.541176
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	248.902	0.363360
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2163.75	3.14042
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	639.225	0.923735
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1426.00	−2.04591
$698$	−1354.82	−1.94100
$699$	0	0
$700$	0	0
$701$	520.000	0.741797	0.370899	−	0.928673i	$-0.379050\pi$
$701$	520.000	0.741797	0.370899	+	0.928673i	$0.379050\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−132.936	−0.188295
$707$	0	0
$708$	0	0
$709$	1320.00	1.86178	0.930889	−	0.365303i	$-0.119035\pi$
$709$	1320.00	1.86178	0.930889	+	0.365303i	$0.119035\pi$
$710$	0	0
$711$	0	0
$712$	−463.862	−0.651492
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	203.647	0.282843
$721$	0	0
$722$	722.000	1.00000
$723$	0	0
$724$	−1125.71	−1.55485
$725$	920.000	1.26897
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	729.000	1.00000
$730$	−412.000	−0.564384
$731$	0	0
$732$	0	0
$733$	872.570	1.19041	0.595204	−	0.803574i	$-0.297071\pi$
$733$	872.570	1.19041	0.595204	+	0.803574i	$0.297071\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	−789.131	−1.06928
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	−135.765	−0.183466
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	144.250	0.193624
$746$	1100.00	1.47453
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	1923.33	2.55084
$755$	0	0
$756$	0	0
$757$	−936.000	−1.23646	−0.618230	−	0.785997i	$-0.712150\pi$
$757$	−936.000	−1.23646	−0.618230	+	0.785997i	$0.712150\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1129.96	1.48483	0.742416	−	0.669940i	$-0.233680\pi$
$761$	1129.96	1.48483	0.742416	+	0.669940i	$0.233680\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	414.000	0.541176
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1528.76	−1.98799	−0.993995	−	0.109422i	$-0.965100\pi$
$769$	−1528.76	−1.98799	−0.993995	+	0.109422i	$0.965100\pi$
$770$	0	0
$771$	0	0
$772$	760.000	0.984456
$773$	−1333.60	−1.72523	−0.862615	−	0.505860i	$-0.831175\pi$
$773$	−1333.60	−1.72523	−0.862615	+	0.505860i	$0.831175\pi$
$774$	0	0
$775$	0	0
$776$	1549.98	1.99739
$777$	0	0
$778$	1360.00	1.74807
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−94.0000	−0.119745
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−1560.00	−1.97970
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2414.00	−3.04414
$794$	−274.357	−0.345538
$795$	0	0
$796$	0	0
$797$	−24.0416	−0.0301652	−0.0150826	−	0.999886i	$-0.504801\pi$
$797$	−24.0416	−0.0301652	−0.0150826	+	0.999886i	$0.504801\pi$
$798$	0	0
$799$	0	0
$800$	−736.000	−0.920000
$801$	−521.845	−0.651492
$802$	−160.000	−0.199501
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	893.783	1.10617
$809$	−1518.00	−1.87639	−0.938195	−	0.346106i	$-0.887504\pi$
$809$	−1518.00	−1.87639	−0.938195	+	0.346106i	$0.887504\pi$
$810$	229.103	0.282843
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−766.504	−0.937046
$819$	0	0
$820$	−248.000	−0.302439
$821$	858.000	1.04507	0.522533	−	0.852619i	$-0.324987\pi$
$821$	858.000	1.04507	0.522533	+	0.852619i	$0.324987\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−125.865	−0.151828	−0.0759138	−	0.997114i	$-0.524187\pi$
$829$	−125.865	−0.151828	−0.0759138	+	0.997114i	$0.524187\pi$
$830$	0	0
$831$	0	0
$832$	−1538.66	−1.84936
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	759.000	0.902497
$842$	116.000	0.137767
$843$	0	0
$844$	0	0
$845$	578.413	0.684513
$846$	0	0
$847$	0	0
$848$	−1440.00	−1.69811
$849$	0	0
$850$	−1496.24	−1.76028
$851$	0	0
$852$	0	0
$853$	1460.88	1.71264	0.856320	−	0.516445i	$-0.172745\pi$
$853$	1460.88	1.71264	0.856320	+	0.516445i	$0.172745\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−838.629	−0.978563	−0.489282	−	0.872126i	$-0.662741\pi$
$857$	−838.629	−0.978563	−0.489282	+	0.872126i	$0.662741\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	226.000	0.261272
$866$	743.876	0.858980
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	960.000	1.10092
$873$	1743.73	1.99739
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−696.000	−0.793615	−0.396807	−	0.917902i	$-0.629882\pi$
$877$	−696.000	−0.793615	−0.396807	+	0.917902i	$0.629882\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	609.526	0.691857	0.345929	−	0.938261i	$-0.387564\pi$
$881$	609.526	0.691857	0.345929	+	0.938261i	$0.387564\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−3128.00	−3.53846
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	−164.000	−0.184270
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	1404.00	1.56347
$899$	0	0
$900$	−828.000	−0.920000
$901$	−2927.42	−3.24908
$902$	0	0
$903$	0	0
$904$	240.000	0.265487
$905$	−398.000	−0.439779
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	1005.51	1.10617
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	−1700.00	−1.85996
$915$	0	0
$916$	1589.58	1.73535
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	1813.02	1.96640
$923$	0	0
$924$	0	0
$925$	552.000	0.596757
$926$	0	0
$927$	0	0
$928$	−1280.00	−1.37931
$929$	−1483.51	−1.59689	−0.798445	−	0.602068i	$-0.794343\pi$
$929$	−1483.51	−1.59689	−0.798445	+	0.602068i	$0.794343\pi$
$930$	0	0
$931$	0	0
$932$	−1664.00	−1.78541
$933$	0	0
$934$	0	0
$935$	0	0
$936$	−1731.00	−1.84936
$937$	985.707	1.05198	0.525991	−	0.850490i	$-0.323695\pi$
$937$	985.707	1.05198	0.525991	+	0.850490i	$0.323695\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1868.18	−1.98531	−0.992655	−	0.120982i	$-0.961396\pi$
$941$	−1868.18	−1.98531	−0.992655	+	0.120982i	$0.961396\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	3502.00	3.69020
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1230.00	−1.29066	−0.645331	−	0.763903i	$-0.723280\pi$
$953$	−1230.00	−1.29066	−0.645331	+	0.763903i	$0.723280\pi$
$954$	−1620.00	−1.69811
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	1154.00	1.19958
$963$	0	0
$964$	−503.460	−0.522261
$965$	268.701	0.278446
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	968.000	1.00000
$969$	0	0
$970$	548.000	0.564948
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	1606.55	1.64605
$977$	−496.000	−0.507677	−0.253838	−	0.967247i	$-0.581693\pi$
$977$	−496.000	−0.507677	−0.253838	+	0.967247i	$0.581693\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1080.00	1.10092
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−551.543	−0.559942
$986$	−2602.15	−2.63910
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1834.23	1.83975	0.919877	−	0.392207i	$-0.128288\pi$
$997$	1834.23	1.83975	0.919877	+	0.392207i	$0.128288\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.3.c.c.99.2	yes	2
4.3	odd	2	CM	196.3.c.c.99.2	yes	2
7.2	even	3		196.3.g.a.67.1		4
7.3	odd	6		196.3.g.a.79.2		4
7.4	even	3		196.3.g.a.79.1		4
7.5	odd	6		196.3.g.a.67.2		4
7.6	odd	2	inner	196.3.c.c.99.1	✓	2
28.3	even	6		196.3.g.a.79.2		4
28.11	odd	6		196.3.g.a.79.1		4
28.19	even	6		196.3.g.a.67.2		4
28.23	odd	6		196.3.g.a.67.1		4
28.27	even	2	inner	196.3.c.c.99.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.3.c.c.99.1	✓	2	7.6	odd	2	inner
196.3.c.c.99.1	✓	2	28.27	even	2	inner
196.3.c.c.99.2	yes	2	1.1	even	1	trivial
196.3.c.c.99.2	yes	2	4.3	odd	2	CM
196.3.g.a.67.1		4	7.2	even	3
196.3.g.a.67.1		4	28.23	odd	6
196.3.g.a.67.2		4	7.5	odd	6
196.3.g.a.67.2		4	28.19	even	6
196.3.g.a.79.1		4	7.4	even	3
196.3.g.a.79.1		4	28.11	odd	6
196.3.g.a.79.2		4	7.3	odd	6
196.3.g.a.79.2		4	28.3	even	6

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:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	196.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +1.41421 q^{5} +8.00000 q^{8} +9.00000 q^{9} +2.82843 q^{10} -24.0416 q^{13} +16.0000 q^{16} +32.5269 q^{17} +18.0000 q^{18} +5.65685 q^{20} -23.0000 q^{25} -48.0833 q^{26} -40.0000 q^{29} +32.0000 q^{32} +65.0538 q^{34} +36.0000 q^{36} -24.0000 q^{37} +11.3137 q^{40} -43.8406 q^{41} +12.7279 q^{45} -46.0000 q^{50} -96.1665 q^{52} -90.0000 q^{53} -80.0000 q^{58} +100.409 q^{61} +64.0000 q^{64} -34.0000 q^{65} +130.108 q^{68} +72.0000 q^{72} -145.664 q^{73} -48.0000 q^{74} +22.6274 q^{80} +81.0000 q^{81} -87.6812 q^{82} +46.0000 q^{85} -57.9828 q^{89} +25.4558 q^{90} +193.747 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 18 q^{9} + 32 q^{16} + 36 q^{18} - 46 q^{25} - 80 q^{29} + 64 q^{32} + 72 q^{36} - 48 q^{37} - 92 q^{50} - 180 q^{53} - 160 q^{58} + 128 q^{64} - 68 q^{65} + 144 q^{72}+ \cdots + 92 q^{85}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 + 18 * q^9 + 32 * q^16 + 36 * q^18 - 46 * q^25 - 80 * q^29 + 64 * q^32 + 72 * q^36 - 48 * q^37 - 92 * q^50 - 180 * q^53 - 160 * q^58 + 128 * q^64 - 68 * q^65 + 144 * q^72 - 96 * q^74 + 162 * q^81 + 92 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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