LMFDB - Embedded newform 196.3.c.a.99.1

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:	$ 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			99.1
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)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -9.89949 q^{5} -8.00000 q^{8} +9.00000 q^{9} +19.7990 q^{10} +9.89949 q^{13} +16.0000 q^{16} +9.89949 q^{17} -18.0000 q^{18} -39.5980 q^{20} +73.0000 q^{25} -19.7990 q^{26} +40.0000 q^{29} -32.0000 q^{32} -19.7990 q^{34} +36.0000 q^{36} +24.0000 q^{37} +79.1960 q^{40} +69.2965 q^{41} -89.0955 q^{45} -146.000 q^{50} +39.5980 q^{52} -90.0000 q^{53} -80.0000 q^{58} -69.2965 q^{61} +64.0000 q^{64} -98.0000 q^{65} +39.5980 q^{68} -72.0000 q^{72} -9.89949 q^{73} -48.0000 q^{74} -158.392 q^{80} +81.0000 q^{81} -138.593 q^{82} -98.0000 q^{85} +168.291 q^{89} +178.191 q^{90} -9.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 18 q^{9} + 32 q^{16} - 36 q^{18} + 146 q^{25} + 80 q^{29} - 64 q^{32} + 72 q^{36} + 48 q^{37} - 292 q^{50} - 180 q^{53} - 160 q^{58} + 128 q^{64} - 196 q^{65} - 144 q^{72}+ \cdots - 196 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 + 146 * q^25 + 80 * q^29 - 64 * q^32 + 72 * q^36 + 48 * q^37 - 292 * q^50 - 180 * q^53 - 160 * q^58 + 128 * q^64 - 196 * q^65 - 144 * q^72 - 96 * q^74 + 162 * q^81 - 196 * 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
$5$	−9.89949	−1.97990	−0.989949	−	0.141421i	$-0.954833\pi$
$5$	−9.89949	−1.97990	−0.989949	+	0.141421i	$0.954833\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.00000	−1.00000
$9$	9.00000	1.00000
$10$	19.7990	1.97990
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	9.89949	0.761500	0.380750	−	0.924678i	$-0.375666\pi$
$13$	9.89949	0.761500	0.380750	+	0.924678i	$0.375666\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	9.89949	0.582323	0.291162	−	0.956674i	$-0.405958\pi$
$17$	9.89949	0.582323	0.291162	+	0.956674i	$0.405958\pi$
$18$	−18.0000	−1.00000
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−39.5980	−1.97990
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	73.0000	2.92000
$26$	−19.7990	−0.761500
$27$	0	0
$28$	0	0
$29$	40.0000	1.37931	0.689655	−	0.724138i	$-0.257762\pi$
$29$	40.0000	1.37931	0.689655	+	0.724138i	$0.257762\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−32.0000	−1.00000
$33$	0	0
$34$	−19.7990	−0.582323
$35$	0	0
$36$	36.0000	1.00000
$37$	24.0000	0.648649	0.324324	−	0.945946i	$-0.394863\pi$
$37$	24.0000	0.648649	0.324324	+	0.945946i	$0.394863\pi$
$38$	0	0
$39$	0	0
$40$	79.1960	1.97990
$41$	69.2965	1.69016	0.845079	−	0.534642i	$-0.179553\pi$
$41$	69.2965	1.69016	0.845079	+	0.534642i	$0.179553\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	−89.0955	−1.97990
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	0	0
$50$	−146.000	−2.92000
$51$	0	0
$52$	39.5980	0.761500
$53$	−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
$61$	−69.2965	−1.13601	−0.568004	−	0.823026i	$-0.692284\pi$
$61$	−69.2965	−1.13601	−0.568004	+	0.823026i	$0.692284\pi$
$62$	0	0
$63$	0	0
$64$	64.0000	1.00000
$65$	−98.0000	−1.50769
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	39.5980	0.582323
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−72.0000	−1.00000
$73$	−9.89949	−0.135610	−0.0678048	−	0.997699i	$-0.521599\pi$
$73$	−9.89949	−0.135610	−0.0678048	+	0.997699i	$0.521599\pi$
$74$	−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			$\pi$
$80$	−158.392	−1.97990
$81$	81.0000	1.00000
$82$	−138.593	−1.69016
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	−98.0000	−1.15294
$86$	0	0
$87$	0	0
$88$	0	0
$89$	168.291	1.89091	0.945457	−	0.325746i	$-0.105615\pi$
$89$	168.291	1.89091	0.945457	+	0.325746i	$0.105615\pi$
$90$	178.191	1.97990
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.89949	−0.102057	−0.0510283	−	0.998697i	$-0.516250\pi$
$97$	−9.89949	−0.102057	−0.0510283	+	0.998697i	$0.516250\pi$
$98$	0	0
$99$	0	0
$100$	292.000	2.92000
$101$	168.291	1.66625	0.833126	−	0.553084i	$-0.186549\pi$
$101$	168.291	1.66625	0.833126	+	0.553084i	$0.186549\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−79.1960	−0.761500
$105$	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	$-0.685547\pi$
$109$	−120.000	−1.10092	−0.550459	+	0.834862i	$0.685547\pi$
$110$	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
$117$	89.0955	0.761500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	138.593	1.13601
$123$	0	0
$124$	0	0
$125$	−475.176	−3.80141
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−128.000	−1.00000
$129$	0	0
$130$	196.000	1.50769
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	−79.1960	−0.582323
$137$	−176.000	−1.28467	−0.642336	−	0.766423i	$-0.722035\pi$
$137$	−176.000	−1.28467	−0.642336	+	0.766423i	$0.722035\pi$
$138$	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
$145$	−395.980	−2.73090
$146$	19.7990	0.135610
$147$	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
$153$	89.0955	0.582323
$154$	0	0
$155$	0	0
$156$	0	0
$157$	306.884	1.95468	0.977339	−	0.211682i	$-0.0678940\pi$
$157$	306.884	1.95468	0.977339	+	0.211682i	$0.0678940\pi$
$158$	0	0
$159$	0	0
$160$	316.784	1.97990
$161$	0	0
$162$	−162.000	−1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	277.186	1.69016
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−71.0000	−0.420118
$170$	196.000	1.15294
$171$	0	0
$172$	0	0
$173$	306.884	1.77390	0.886949	−	0.461867i	$-0.152820\pi$
$173$	306.884	1.77390	0.886949	+	0.461867i	$0.152820\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−336.583	−1.89091
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−356.382	−1.97990
$181$	227.688	1.25795	0.628973	−	0.777427i	$-0.283475\pi$
$181$	227.688	1.25795	0.628973	+	0.777427i	$0.283475\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−237.588	−1.28426
$186$	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	$0.336183\pi$
$194$	19.7990	0.102057
$195$	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			$\pi$
$200$	−584.000	−2.92000
$201$	0	0
$202$	−336.583	−1.66625
$203$	0	0
$204$	0	0
$205$	−686.000	−3.34634
$206$	0	0
$207$	0	0
$208$	158.392	0.761500
$209$	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
$221$	98.0000	0.443439
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	657.000	2.92000
$226$	−60.0000	−0.265487
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	227.688	0.994272	0.497136	−	0.867673i	$-0.334385\pi$
$229$	227.688	0.994272	0.497136	+	0.867673i	$0.334385\pi$
$230$	0	0
$231$	0	0
$232$	−320.000	−1.37931
$233$	416.000	1.78541	0.892704	−	0.450644i	$-0.148806\pi$
$233$	416.000	1.78541	0.892704	+	0.450644i	$0.148806\pi$
$234$	−178.191	−0.761500
$235$	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
$241$	−465.276	−1.93061	−0.965303	−	0.261131i	$-0.915905\pi$
$241$	−465.276	−1.93061	−0.965303	+	0.261131i	$0.915905\pi$
$242$	−242.000	−1.00000
$243$	0	0
$244$	−277.186	−1.13601
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	950.352	3.80141
$251$	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
$257$	−405.879	−1.57930	−0.789648	−	0.613560i	$-0.789737\pi$
$257$	−405.879	−1.57930	−0.789648	+	0.613560i	$0.789737\pi$
$258$	0	0
$259$	0	0
$260$	−392.000	−1.50769
$261$	360.000	1.37931
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	890.955	3.36209
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−465.276	−1.72965	−0.864826	−	0.502072i	$-0.832571\pi$
$269$	−465.276	−1.72965	−0.864826	+	0.502072i	$0.832571\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	158.392	0.582323
$273$	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	$-0.307178\pi$
$281$	320.000	1.13879	0.569395	+	0.822064i	$0.307178\pi$
$282$	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
$289$	−191.000	−0.660900
$290$	791.960	2.73090
$291$	0	0
$292$	−39.5980	−0.135610
$293$	−306.884	−1.04739	−0.523693	−	0.851907i	$-0.675446\pi$
$293$	−306.884	−1.04739	−0.523693	+	0.851907i	$0.675446\pi$
$294$	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
$305$	686.000	2.24918
$306$	−178.191	−0.582323
$307$	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
$313$	−405.879	−1.29674	−0.648369	−	0.761326i	$-0.724549\pi$
$313$	−405.879	−1.29674	−0.648369	+	0.761326i	$0.724549\pi$
$314$	−613.769	−1.95468
$315$	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
$320$	−633.568	−1.97990
$321$	0	0
$322$	0	0
$323$	0	0
$324$	324.000	1.00000
$325$	722.663	2.22358
$326$	0	0
$327$	0	0
$328$	−554.372	−1.69016
$329$	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	$-0.826197\pi$
$337$	−576.000	−1.70920	−0.854599	+	0.519288i	$0.826197\pi$
$338$	142.000	0.420118
$339$	0	0
$340$	−392.000	−1.15294
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−613.769	−1.77390
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−168.291	−0.482210	−0.241105	−	0.970499i	$-0.577510\pi$
$349$	−168.291	−0.482210	−0.241105	+	0.970499i	$0.577510\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	702.864	1.99112	0.995558	−	0.0941474i	$-0.0300125\pi$
$353$	702.864	1.99112	0.995558	+	0.0941474i	$0.0300125\pi$
$354$	0	0
$355$	0	0
$356$	673.166	1.89091
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	712.764	1.97990
$361$	361.000	1.00000
$362$	−455.377	−1.25795
$363$	0	0
$364$	0	0
$365$	98.0000	0.268493
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	623.668	1.69016
$370$	475.176	1.28426
$371$	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
$377$	395.980	1.05034
$378$	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
$388$	−39.5980	−0.102057
$389$	−680.000	−1.74807	−0.874036	−	0.485861i	$-0.838506\pi$
$389$	−680.000	−1.74807	−0.874036	+	0.485861i	$0.838506\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	780.000	1.97970
$395$	0	0
$396$	0	0
$397$	−782.060	−1.96992	−0.984962	−	0.172769i	$-0.944729\pi$
$397$	−782.060	−1.96992	−0.984962	+	0.172769i	$0.944729\pi$
$398$	0	0
$399$	0	0
$400$	1168.00	2.92000
$401$	80.0000	0.199501	0.0997506	−	0.995012i	$-0.468195\pi$
$401$	80.0000	0.199501	0.0997506	+	0.995012i	$0.468195\pi$
$402$	0	0
$403$	0	0
$404$	673.166	1.66625
$405$	−801.859	−1.97990
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−722.663	−1.76690	−0.883451	−	0.468523i	$-0.844786\pi$
$409$	−722.663	−1.76690	−0.883451	+	0.468523i	$0.844786\pi$
$410$	1372.00	3.34634
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−316.784	−0.761500
$417$	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
$425$	722.663	1.70038
$426$	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
$433$	−782.060	−1.80614	−0.903072	−	0.429490i	$-0.858693\pi$
$433$	−782.060	−1.80614	−0.903072	+	0.429490i	$0.858693\pi$
$434$	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
$442$	−196.000	−0.443439
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−1666.00	−3.74382
$446$	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	$0.214334\pi$
$450$	−1314.00	−2.92000
$451$	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	$0.880174\pi$
$458$	−455.377	−0.994272
$459$	0	0
$460$	0	0
$461$	−168.291	−0.365057	−0.182529	−	0.983201i	$-0.558428\pi$
$461$	−168.291	−0.365057	−0.182529	+	0.983201i	$0.558428\pi$
$462$	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			$\pi$
$468$	356.382	0.761500
$469$	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
$481$	237.588	0.493946
$482$	930.553	1.93061
$483$	0	0
$484$	484.000	1.00000
$485$	98.0000	0.202062
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	554.372	1.13601
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	395.980	0.803204
$494$	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			$\pi$
$500$	−1900.70	−3.80141
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−1666.00	−3.29901
$506$	0	0
$507$	0	0
$508$	0	0
$509$	960.251	1.88654	0.943272	−	0.332021i	$-0.107730\pi$
$509$	960.251	1.88654	0.943272	+	0.332021i	$0.107730\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−1.00000
$513$	0	0
$514$	811.759	1.57930
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	784.000	1.50769
$521$	227.688	0.437022	0.218511	−	0.975835i	$-0.429880\pi$
$521$	227.688	0.437022	0.218511	+	0.975835i	$0.429880\pi$
$522$	−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
$530$	−1781.91	−3.36209
$531$	0	0
$532$	0	0
$533$	686.000	1.28705
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	930.553	1.72965
$539$	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
$544$	−316.784	−0.582323
$545$	1187.94	2.17971
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−704.000	−1.28467
$549$	−623.668	−1.13601
$550$	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
$565$	−296.985	−0.525637
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1040.00	−1.82777	−0.913884	−	0.405975i	$-0.866932\pi$
$569$	−1040.00	−1.82777	−0.913884	+	0.405975i	$0.866932\pi$
$570$	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
$577$	881.055	1.52696	0.763479	−	0.645832i	$-0.223490\pi$
$577$	881.055	1.52696	0.763479	+	0.645832i	$0.223490\pi$
$578$	382.000	0.660900
$579$	0	0
$580$	−1583.92	−2.73090
$581$	0	0
$582$	0	0
$583$	0	0
$584$	79.1960	0.135610
$585$	−882.000	−1.50769
$586$	613.769	1.04739
$587$	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
$593$	−1178.04	−1.98658	−0.993288	−	0.115665i	$-0.963100\pi$
$593$	−1178.04	−1.98658	−0.993288	+	0.115665i	$0.963100\pi$
$594$	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
$601$	−1118.64	−1.86130	−0.930651	−	0.365907i	$-0.880759\pi$
$601$	−1118.64	−1.86130	−0.930651	+	0.365907i	$0.880759\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1197.84	−1.97990
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	−1372.00	−2.24918
$611$	0	0
$612$	356.382	0.582323
$613$	1224.00	1.99674	0.998369	−	0.0570962i	$-0.0181842\pi$
$613$	1224.00	1.99674	0.998369	+	0.0570962i	$0.0181842\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1216.00	1.97083	0.985413	−	0.170178i	$-0.0544344\pi$
$617$	1216.00	1.97083	0.985413	+	0.170178i	$0.0544344\pi$
$618$	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
$625$	2879.00	4.60640
$626$	811.759	1.29674
$627$	0	0
$628$	1227.54	1.95468
$629$	237.588	0.377723
$630$	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
$640$	1267.14	1.97990
$641$	−400.000	−0.624025	−0.312012	−	0.950078i	$-0.601003\pi$
$641$	−400.000	−0.624025	−0.312012	+	0.950078i	$0.601003\pi$
$642$	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
$650$	−1445.33	−2.22358
$651$	0	0
$652$	0	0
$653$	1144.00	1.75191	0.875957	−	0.482389i	$-0.160231\pi$
$653$	1144.00	1.75191	0.875957	+	0.482389i	$0.160231\pi$
$654$	0	0
$655$	0	0
$656$	1108.74	1.69016
$657$	−89.0955	−0.135610
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	1257.24	1.90202	0.951010	−	0.309159i	$-0.100047\pi$
$661$	1257.24	1.90202	0.951010	+	0.309159i	$0.100047\pi$
$662$	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	$-0.193858\pi$
$673$	1104.00	1.64042	0.820208	+	0.572065i	$0.193858\pi$
$674$	1152.00	1.70920
$675$	0	0
$676$	−284.000	−0.420118
$677$	−881.055	−1.30141	−0.650705	−	0.759330i	$-0.725527\pi$
$677$	−881.055	−1.30141	−0.650705	+	0.759330i	$0.725527\pi$
$678$	0	0
$679$	0	0
$680$	784.000	1.15294
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	1742.31	2.54352
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−890.955	−1.29311
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	1227.54	1.77390
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	686.000	0.984218
$698$	336.583	0.482210
$699$	0	0
$700$	0	0
$701$	−520.000	−0.741797	−0.370899	−	0.928673i	$-0.620950\pi$
$701$	−520.000	−0.741797	−0.370899	+	0.928673i	$0.620950\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−1405.73	−1.99112
$707$	0	0
$708$	0	0
$709$	−1320.00	−1.86178	−0.930889	−	0.365303i	$-0.880965\pi$
$709$	−1320.00	−1.86178	−0.930889	+	0.365303i	$0.880965\pi$
$710$	0	0
$711$	0	0
$712$	−1346.33	−1.89091
$713$	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			$\pi$
$720$	−1425.53	−1.97990
$721$	0	0
$722$	−722.000	−1.00000
$723$	0	0
$724$	910.754	1.25795
$725$	2920.00	4.02759
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	729.000	1.00000
$730$	−196.000	−0.268493
$731$	0	0
$732$	0	0
$733$	1178.04	1.60715	0.803574	−	0.595204i	$-0.202929\pi$
$733$	1178.04	1.60715	0.803574	+	0.595204i	$0.202929\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	−1247.34	−1.69016
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	−950.352	−1.28426
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−1009.75	−1.35537
$746$	−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
$754$	−791.960	−1.05034
$755$	0	0
$756$	0	0
$757$	936.000	1.23646	0.618230	−	0.785997i	$-0.287850\pi$
$757$	936.000	1.23646	0.618230	+	0.785997i	$0.287850\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1019.65	−1.33988	−0.669940	−	0.742416i	$-0.733680\pi$
$761$	−1019.65	−1.33988	−0.669940	+	0.742416i	$0.733680\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−882.000	−1.15294
$766$	0	0
$767$	0	0
$768$	0	0
$769$	168.291	0.218844	0.109422	−	0.993995i	$-0.465100\pi$
$769$	168.291	0.218844	0.109422	+	0.993995i	$0.465100\pi$
$770$	0	0
$771$	0	0
$772$	760.000	0.984456
$773$	782.060	1.01172	0.505860	−	0.862615i	$-0.331175\pi$
$773$	782.060	1.01172	0.505860	+	0.862615i	$0.331175\pi$
$774$	0	0
$775$	0	0
$776$	79.1960	0.102057
$777$	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
$785$	−3038.00	−3.87006
$786$	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
$793$	−686.000	−0.865069
$794$	1564.12	1.96992
$795$	0	0
$796$	0	0
$797$	1593.82	1.99977	0.999886	−	0.0150826i	$-0.00480112\pi$
$797$	1593.82	1.99977	0.999886	+	0.0150826i	$0.00480112\pi$
$798$	0	0
$799$	0	0
$800$	−2336.00	−2.92000
$801$	1514.62	1.89091
$802$	−160.000	−0.199501
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−1346.33	−1.66625
$809$	−1518.00	−1.87639	−0.938195	−	0.346106i	$-0.887504\pi$
$809$	−1518.00	−1.87639	−0.938195	+	0.346106i	$0.887504\pi$
$810$	1603.72	1.97990
$811$	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
$818$	1445.33	1.76690
$819$	0	0
$820$	−2744.00	−3.34634
$821$	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
$829$	−1653.22	−1.99423	−0.997114	−	0.0759138i	$-0.975813\pi$
$829$	−1653.22	−1.99423	−0.997114	+	0.0759138i	$0.975813\pi$
$830$	0	0
$831$	0	0
$832$	633.568	0.761500
$833$	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
$845$	702.864	0.831792
$846$	0	0
$847$	0	0
$848$	−1440.00	−1.69811
$849$	0	0
$850$	−1445.33	−1.70038
$851$	0	0
$852$	0	0
$853$	−881.055	−1.03289	−0.516445	−	0.856320i	$-0.672745\pi$
$853$	−881.055	−1.03289	−0.516445	+	0.856320i	$0.672745\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1494.82	−1.74425	−0.872126	−	0.489282i	$-0.837259\pi$
$857$	−1494.82	−1.74425	−0.872126	+	0.489282i	$0.837259\pi$
$858$	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
$865$	−3038.00	−3.51214
$866$	1564.12	1.80614
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	960.000	1.10092
$873$	−89.0955	−0.102057
$874$	0	0
$875$	0	0
$876$	0	0
$877$	696.000	0.793615	0.396807	−	0.917902i	$-0.370118\pi$
$877$	696.000	0.793615	0.396807	+	0.917902i	$0.370118\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1653.22	−1.87652	−0.938261	−	0.345929i	$-0.887564\pi$
$881$	−1653.22	−1.87652	−0.938261	+	0.345929i	$0.887564\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	392.000	0.443439
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	3332.00	3.74382
$891$	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
$900$	2628.00	2.92000
$901$	−890.955	−0.988851
$902$	0	0
$903$	0	0
$904$	−240.000	−0.265487
$905$	−2254.00	−2.49061
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	1514.62	1.66625
$910$	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
$916$	910.754	0.994272
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	336.583	0.365057
$923$	0	0
$924$	0	0
$925$	1752.00	1.89405
$926$	0	0
$927$	0	0
$928$	−1280.00	−1.37931
$929$	1118.64	1.20414	0.602068	−	0.798445i	$-0.294343\pi$
$929$	1118.64	1.20414	0.602068	+	0.798445i	$0.294343\pi$
$930$	0	0
$931$	0	0
$932$	1664.00	1.78541
$933$	0	0
$934$	0	0
$935$	0	0
$936$	−712.764	−0.761500
$937$	−1593.82	−1.70098	−0.850490	−	0.525991i	$-0.823695\pi$
$937$	−1593.82	−1.70098	−0.850490	+	0.525991i	$0.823695\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−227.688	−0.241964	−0.120982	−	0.992655i	$-0.538604\pi$
$941$	−227.688	−0.241964	−0.120982	+	0.992655i	$0.538604\pi$
$942$	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
$949$	−98.0000	−0.103267
$950$	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
$962$	−475.176	−0.493946
$963$	0	0
$964$	−1861.11	−1.93061
$965$	−1880.90	−1.94912
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−968.000	−1.00000
$969$	0	0
$970$	−196.000	−0.202062
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−1108.74	−1.13601
$977$	496.000	0.507677	0.253838	−	0.967247i	$-0.418307\pi$
$977$	496.000	0.507677	0.253838	+	0.967247i	$0.418307\pi$
$978$	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
$985$	3860.80	3.91960
$986$	−791.960	−0.803204
$987$	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
$997$	782.060	0.784413	0.392207	−	0.919877i	$-0.371712\pi$
$997$	782.060	0.784413	0.392207	+	0.919877i	$0.371712\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -9.89949 q^{5} -8.00000 q^{8} +9.00000 q^{9} +19.7990 q^{10} +9.89949 q^{13} +16.0000 q^{16} +9.89949 q^{17} -18.0000 q^{18} -39.5980 q^{20} +73.0000 q^{25} -19.7990 q^{26} +40.0000 q^{29} -32.0000 q^{32} -19.7990 q^{34} +36.0000 q^{36} +24.0000 q^{37} +79.1960 q^{40} +69.2965 q^{41} -89.0955 q^{45} -146.000 q^{50} +39.5980 q^{52} -90.0000 q^{53} -80.0000 q^{58} -69.2965 q^{61} +64.0000 q^{64} -98.0000 q^{65} +39.5980 q^{68} -72.0000 q^{72} -9.89949 q^{73} -48.0000 q^{74} -158.392 q^{80} +81.0000 q^{81} -138.593 q^{82} -98.0000 q^{85} +168.291 q^{89} +178.191 q^{90} -9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 18 q^{9} + 32 q^{16} - 36 q^{18} + 146 q^{25} + 80 q^{29} - 64 q^{32} + 72 q^{36} + 48 q^{37} - 292 q^{50} - 180 q^{53} - 160 q^{58} + 128 q^{64} - 196 q^{65} - 144 q^{72}+ \cdots - 196 q^{85}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 + 18 * q^9 + 32 * q^16 - 36 * q^18 + 146 * q^25 + 80 * q^29 - 64 * q^32 + 72 * q^36 + 48 * q^37 - 292 * q^50 - 180 * q^53 - 160 * q^58 + 128 * q^64 - 196 * q^65 - 144 * q^72 - 96 * q^74 + 162 * q^81 - 196 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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