LMFDB - Embedded newform 891.2.d.a.890.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [891,2,Mod(890,891)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(891, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("891.890");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 891 = 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	891.d (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:	no
Analytic conductor:	$7.11467082010$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			890.1
Root			$-1.18614 - 1.26217i$ of defining polynomial
Character	$\chi$	$=$	891.890
Dual form			891.2.d.a.890.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} -4.25639i q^{5} -3.31662i q^{11} +4.00000 q^{16} +8.51278i q^{20} +3.31662i q^{23} -13.1168 q^{25} -11.1168 q^{31} -5.11684 q^{37} +6.63325i q^{44} -7.07568i q^{47} +7.00000 q^{49} -1.43710i q^{53} -14.1168 q^{55} +11.3321i q^{59} -8.00000 q^{64} -2.11684 q^{67} +5.69349i q^{71} -17.0256i q^{80} -16.5831i q^{89} -6.63325i q^{92} -17.1168 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 16 q^{16} - 18 q^{25} - 10 q^{31} + 14 q^{37} + 28 q^{49} - 22 q^{55} - 32 q^{64} + 26 q^{67} - 34 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 16 * q^16 - 18 * q^25 - 10 * q^31 + 14 * q^37 + 28 * q^49 - 22 * q^55 - 32 * q^64 + 26 * q^67 - 34 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/891\mathbb{Z}\right)^\times$.

$n$	$244$	$650$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	− 4.25639i	− 1.90351i	−0.306851	−	0.951757i	$-0.599275\pi$
$5$	− 4.25639i	− 1.90351i	0.306851	−	0.951757i	$-0.400725\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.31662i	− 1.00000i
$11$	− 3.31662i	− 1.00000i
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	8.51278i	1.90351i
$21$	0	0
$22$	0	0
$23$	3.31662i	0.691564i	0.938315	+	0.345782i	$0.112386\pi$
$23$	3.31662i	0.691564i	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	−13.1168	−2.62337
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−11.1168	−1.99664	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$	−11.1168	−1.99664	−0.998322	+	0.0579057i	$0.981558\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.11684	−0.841204	−0.420602	−	0.907245i	$-0.638181\pi$
$37$	−5.11684	−0.841204	−0.420602	+	0.907245i	$0.638181\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	6.63325i	1.00000i
$45$	0	0
$46$	0	0
$47$	− 7.07568i	− 1.03209i	−0.856560	−	0.516047i	$-0.827403\pi$
$47$	− 7.07568i	− 1.03209i	0.856560	−	0.516047i	$-0.172597\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.43710i	− 0.197400i	−0.995117	−	0.0987002i	$-0.968532\pi$
$53$	− 1.43710i	− 0.197400i	0.995117	−	0.0987002i	$-0.0314685\pi$
$54$	0	0
$55$	−14.1168	−1.90351
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3321i	1.47531i	0.675178	+	0.737655i	$0.264067\pi$
$59$	11.3321i	1.47531i	−0.675178	+	0.737655i	$0.735933\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−2.11684	−0.258614	−0.129307	−	0.991605i	$-0.541275\pi$
$67$	−2.11684	−0.258614	−0.129307	+	0.991605i	$0.541275\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.69349i	0.675692i	0.941201	+	0.337846i	$0.109698\pi$
$71$	5.69349i	0.675692i	−0.941201	+	0.337846i	$0.890302\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	− 17.0256i	− 1.90351i
$81$	0	0
$82$	0	0
$83$	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
$89$	− 16.5831i	− 1.75781i	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	− 16.5831i	− 1.75781i	0.476999	−	0.878904i	$-0.341725\pi$
$90$	0	0
$91$	0	0
$92$	− 6.63325i	− 0.691564i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−17.1168	−1.73795	−0.868976	−	0.494854i	$-0.835222\pi$
$97$	−17.1168	−1.73795	−0.868976	+	0.494854i	$0.835222\pi$
$98$	0	0
$99$	0	0
$100$	26.2337	2.62337
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	19.2337	1.89515	0.947576	−	0.319531i	$-0.103525\pi$
$103$	19.2337	1.89515	0.947576	+	0.319531i	$0.103525\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 19.8448i	− 1.86685i	−0.358778	−	0.933423i	$-0.616806\pi$
$113$	− 19.8448i	− 1.86685i	0.358778	−	0.933423i	$-0.383194\pi$
$114$	0	0
$115$	14.1168	1.31640
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	22.2337	1.99664
$125$	34.5484i	3.09011i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.2063i	− 1.21372i	−0.794808	−	0.606861i	$-0.792428\pi$
$137$	− 14.2063i	− 1.21372i	0.794808	−	0.606861i	$-0.207572\pi$
$138$	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
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	10.2337	0.841204
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	47.3176i	3.80064i
$156$	0	0
$157$	−20.1168	−1.60550	−0.802749	−	0.596316i	$-0.796630\pi$
$157$	−20.1168	−1.60550	−0.802749	+	0.596316i	$0.796630\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	25.2337	1.97645	0.988227	−	0.152992i	$-0.0488907\pi$
$163$	25.2337	1.97645	0.988227	+	0.152992i	$0.0488907\pi$
$164$	0	0
$165$	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
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	− 13.2665i	− 1.00000i
$177$	0	0
$178$	0	0
$179$	− 9.89497i	− 0.739585i	−0.929114	−	0.369792i	$-0.879429\pi$
$179$	− 9.89497i	− 0.739585i	0.929114	−	0.369792i	$-0.120571\pi$
$180$	0	0
$181$	3.88316	0.288633	0.144316	−	0.989532i	$-0.453902\pi$
$181$	3.88316	0.288633	0.144316	+	0.989532i	$0.453902\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	21.7793i	1.60124i
$186$	0	0
$187$	0	0
$188$	14.1514i	1.03209i
$189$	0	0
$190$	0	0
$191$	1.38219i	0.100012i	0.998749	+	0.0500060i	$0.0159241\pi$
$191$	1.38219i	0.100012i	−0.998749	+	0.0500060i	$0.984076\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−14.0000	−1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	7.23369	0.512783	0.256391	−	0.966573i	$-0.417466\pi$
$199$	7.23369	0.512783	0.256391	+	0.966573i	$0.417466\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	2.87419i	0.197400i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	28.2337	1.90351
$221$	0	0
$222$	0	0
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−30.1168	−1.96461
$236$	− 22.6641i	− 1.47531i
$237$	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	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 29.7947i	− 1.90351i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 16.5831i	− 1.04672i	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	− 16.5831i	− 1.04672i	0.852112	−	0.523359i	$-0.175321\pi$
$252$	0	0
$253$	11.0000	0.691564
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	− 26.5330i	− 1.65508i	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	− 26.5330i	− 1.65508i	0.561405	−	0.827541i	$-0.310261\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−6.11684	−0.375755
$266$	0	0
$267$	0	0
$268$	4.23369	0.258614
$269$	− 32.6140i	− 1.98851i	−0.107031	−	0.994256i	$-0.534134\pi$
$269$	− 32.6140i	− 1.98851i	0.107031	−	0.994256i	$-0.465866\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	43.5036i	2.62337i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	− 11.3870i	− 0.675692i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	48.2337	2.80827
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 26.9754i	− 1.52964i	−0.644246	−	0.764818i	$-0.722829\pi$
$311$	− 26.9754i	− 1.52964i	0.644246	−	0.764818i	$-0.277171\pi$
$312$	0	0
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.2164i	1.30396i	0.758236	+	0.651981i	$0.226062\pi$
$317$	23.2164i	1.30396i	−0.758236	+	0.651981i	$0.773938\pi$
$318$	0	0
$319$	0	0
$320$	34.0511i	1.90351i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−26.1168	−1.43551	−0.717756	−	0.696295i	$-0.754831\pi$
$331$	−26.1168	−1.43551	−0.717756	+	0.696295i	$0.754831\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.01011i	0.492275i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	36.8704i	1.99664i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 36.4829i	− 1.94179i	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	− 36.4829i	− 1.94179i	0.239511	−	0.970894i	$-0.423013\pi$
$354$	0	0
$355$	24.2337	1.28619
$356$	33.1662i	1.75781i
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.88316	0.515897	0.257948	−	0.966159i	$-0.416954\pi$
$367$	9.88316	0.515897	0.257948	+	0.966159i	$0.416954\pi$
$368$	13.2665i	0.691564i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−25.0000	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 35.4333i	− 1.81056i	−0.424818	−	0.905279i	$-0.639662\pi$
$383$	− 35.4333i	− 1.81056i	0.424818	−	0.905279i	$-0.360338\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	34.2337	1.73795
$389$	31.2318i	1.58352i	0.610835	+	0.791758i	$0.290834\pi$
$389$	31.2318i	1.58352i	−0.610835	+	0.791758i	$0.709166\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−35.4674	−1.78006	−0.890028	−	0.455905i	$-0.849316\pi$
$397$	−35.4674	−1.78006	−0.890028	+	0.455905i	$0.849316\pi$
$398$	0	0
$399$	0	0
$400$	−52.4674	−2.62337
$401$	− 12.7143i	− 0.634920i	−0.948272	−	0.317460i	$-0.897170\pi$
$401$	− 12.7143i	− 0.634920i	0.948272	−	0.317460i	$-0.102830\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.9707i	0.841204i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	−38.4674	−1.89515
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.20149i	0.205256i	0.994720	+	0.102628i	$0.0327251\pi$
$419$	4.20149i	0.205256i	−0.994720	+	0.102628i	$0.967275\pi$
$420$	0	0
$421$	−29.4674	−1.43615	−0.718076	−	0.695965i	$-0.754977\pi$
$421$	−29.4674	−1.43615	−0.718076	+	0.695965i	$0.754977\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	29.0000	1.39365	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000	1.39365	0.696826	+	0.717241i	$0.254595\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.0549029i	0.00260851i	0.999999	+	0.00130426i	$0.000415158\pi$
$443$	0.0549029i	0.00260851i	−0.999999	+	0.00130426i	$0.999585\pi$
$444$	0	0
$445$	−70.5842	−3.34601
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 25.4834i	− 1.20264i	−0.799009	−	0.601319i	$-0.794642\pi$
$449$	− 25.4834i	− 1.20264i	0.799009	−	0.601319i	$-0.205358\pi$
$450$	0	0
$451$	0	0
$452$	39.6897i	1.86685i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	−28.2337	−1.31640
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 24.1561i	− 1.11781i	−0.829231	−	0.558906i	$-0.811221\pi$
$467$	− 24.1561i	− 1.11781i	0.829231	−	0.558906i	$-0.188779\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	72.8559i	3.30822i
$486$	0	0
$487$	12.8832	0.583792	0.291896	−	0.956450i	$-0.405714\pi$
$487$	12.8832	0.583792	0.291896	+	0.956450i	$0.405714\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	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
$496$	−44.4674	−1.99664
$497$	0	0
$498$	0	0
$499$	37.2337	1.66681	0.833404	−	0.552664i	$-0.186389\pi$
$499$	37.2337	1.66681	0.833404	+	0.552664i	$0.186389\pi$
$500$	− 69.0969i	− 3.09011i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.31662i	0.147007i	0.997295	+	0.0735034i	$0.0234180\pi$
$509$	3.31662i	0.147007i	−0.997295	+	0.0735034i	$0.976582\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 81.8661i	− 3.60745i
$516$	0	0
$517$	−23.4674	−1.03209
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 8.56768i	− 0.375357i	−0.982231	−	0.187678i	$-0.939904\pi$
$521$	− 8.56768i	− 0.375357i	0.982231	−	0.187678i	$-0.0600963\pi$
$522$	0	0
$523$	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
$529$	12.0000	0.521739
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 23.2164i	− 1.00000i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	28.4125i	1.21372i
$549$	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
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−84.4674	−3.55357
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 43.5036i	− 1.81423i
$576$	0	0
$577$	−32.1168	−1.33704	−0.668521	−	0.743693i	$-0.733072\pi$
$577$	−32.1168	−1.33704	−0.668521	+	0.743693i	$0.733072\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.76631	−0.197400
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 38.2526i	− 1.57885i	−0.613845	−	0.789427i	$-0.710378\pi$
$587$	− 38.2526i	− 1.57885i	0.613845	−	0.789427i	$-0.289622\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−20.4674	−0.841204
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	33.1662i	1.35514i	0.735460	+	0.677568i	$0.236966\pi$
$599$	33.1662i	1.35514i	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	46.8203i	1.90351i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	49.6396i	1.99841i	0.0398207	+	0.999207i	$0.487321\pi$
$617$	49.6396i	1.99841i	−0.0398207	+	0.999207i	$0.512679\pi$
$618$	0	0
$619$	43.5842	1.75180	0.875899	−	0.482495i	$-0.160269\pi$
$619$	43.5842	1.75180	0.875899	+	0.482495i	$0.160269\pi$
$620$	− 94.6352i	− 3.80064i
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	81.4674	3.25870
$626$	0	0
$627$	0	0
$628$	40.2337	1.60550
$629$	0	0
$630$	0	0
$631$	46.5842	1.85449	0.927244	−	0.374457i	$-0.122171\pi$
$631$	46.5842	1.85449	0.927244	+	0.374457i	$0.122171\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.2164i	0.916992i	0.888697	+	0.458496i	$0.151612\pi$
$641$	23.2164i	0.916992i	−0.888697	+	0.458496i	$0.848388\pi$
$642$	0	0
$643$	41.0000	1.61688	0.808441	−	0.588577i	$-0.200312\pi$
$643$	41.0000	1.61688	0.808441	+	0.588577i	$0.200312\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	43.1161i	1.69507i	0.530740	+	0.847535i	$0.321914\pi$
$647$	43.1161i	1.69507i	−0.530740	+	0.847535i	$0.678086\pi$
$648$	0	0
$649$	37.5842	1.47531
$650$	0	0
$651$	0	0
$652$	−50.4674	−1.97645
$653$	42.5090i	1.66351i	0.555147	+	0.831753i	$0.312662\pi$
$653$	42.5090i	1.66351i	−0.555147	+	0.831753i	$0.687338\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	49.5842	1.92860	0.964301	−	0.264807i	$-0.0853084\pi$
$661$	49.5842	1.92860	0.964301	+	0.264807i	$0.0853084\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−26.0000	−1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.0010i	1.68365i	0.539750	+	0.841825i	$0.318519\pi$
$683$	44.0010i	1.68365i	−0.539750	+	0.841825i	$0.681481\pi$
$684$	0	0
$685$	−60.4674	−2.31034
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	34.5842	1.31565	0.657823	−	0.753173i	$-0.271478\pi$
$691$	34.5842	1.31565	0.657823	+	0.753173i	$0.271478\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	26.5330i	1.00000i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	52.5842	1.97484	0.987421	−	0.158114i	$-0.0505412\pi$
$709$	52.5842	1.97484	0.987421	+	0.158114i	$0.0505412\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 36.8704i	− 1.38081i
$714$	0	0
$715$	0	0
$716$	19.7899i	0.739585i
$717$	0	0
$718$	0	0
$719$	52.4589i	1.95639i	0.207700	+	0.978193i	$0.433402\pi$
$719$	52.4589i	1.95639i	−0.207700	+	0.978193i	$0.566598\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	−7.76631	−0.288633
$725$	0	0
$726$	0	0
$727$	−35.1168	−1.30241	−0.651206	−	0.758901i	$-0.725737\pi$
$727$	−35.1168	−1.30241	−0.651206	+	0.758901i	$0.725737\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.02078i	0.258614i
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	− 43.5586i	− 1.60124i
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	31.5842	1.15252	0.576262	−	0.817265i	$-0.304511\pi$
$751$	31.5842	1.15252	0.576262	+	0.817265i	$0.304511\pi$
$752$	− 28.3027i	− 1.03209i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−53.4674	−1.94330	−0.971652	−	0.236414i	$-0.924028\pi$
$757$	−53.4674	−1.94330	−0.971652	+	0.236414i	$0.924028\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	− 2.76439i	− 0.100012i
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13.2665i	0.477163i	0.971123	+	0.238581i	$0.0766824\pi$
$773$	13.2665i	0.477163i	−0.971123	+	0.238581i	$0.923318\pi$
$774$	0	0
$775$	145.818	5.23793
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	18.8832	0.675692
$782$	0	0
$783$	0	0
$784$	28.0000	1.00000
$785$	85.6251i	3.05609i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−14.4674	−0.512783
$797$	25.5932i	0.906559i	0.891368	+	0.453279i	$0.149746\pi$
$797$	25.5932i	0.906559i	−0.891368	+	0.453279i	$0.850254\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 107.404i	− 3.76221i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	−49.0000	−1.70803	−0.854016	−	0.520246i	$-0.825840\pi$
$823$	−49.0000	−1.70803	−0.854016	+	0.520246i	$0.825840\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	28.5842	0.992771	0.496385	−	0.868102i	$-0.334660\pi$
$829$	28.5842	0.992771	0.496385	+	0.868102i	$0.334660\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 36.4829i	− 1.25953i	−0.776786	−	0.629764i	$-0.783151\pi$
$839$	− 36.4829i	− 1.25953i	0.776786	−	0.629764i	$-0.216849\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 55.3331i	− 1.90351i
$846$	0	0
$847$	0	0
$848$	− 5.74839i	− 0.197400i
$849$	0	0
$850$	0	0
$851$	− 16.9707i	− 0.581746i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	58.5842	1.99887	0.999434	−	0.0336436i	$-0.0107111\pi$
$859$	58.5842	1.99887	0.999434	+	0.0336436i	$0.0107111\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 46.4327i	− 1.58059i	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	− 46.4327i	− 1.58059i	0.612727	−	0.790295i	$-0.290072\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	−56.4674	−1.90351
$881$	− 41.0719i	− 1.38375i	−0.722019	−	0.691873i	$-0.756786\pi$
$881$	− 41.0719i	− 1.38375i	0.722019	−	0.691873i	$-0.243214\pi$
$882$	0	0
$883$	−10.7663	−0.362315	−0.181158	−	0.983454i	$-0.557984\pi$
$883$	−10.7663	−0.362315	−0.181158	+	0.983454i	$0.557984\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	2.00000	0.0669650
$893$	0	0
$894$	0	0
$895$	−42.1168	−1.40781
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 16.5282i	− 0.549417i
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	55.2781i	1.83145i	0.401809	+	0.915723i	$0.368381\pi$
$911$	55.2781i	1.83145i	−0.401809	+	0.915723i	$0.631619\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−10.0000	−0.330409
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	67.1168	2.20679
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 52.5138i	− 1.72292i	−0.507825	−	0.861460i	$-0.669550\pi$
$929$	− 52.5138i	− 1.72292i	0.507825	−	0.861460i	$-0.330450\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	60.2337	1.96461
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	45.3283i	1.47531i
$945$	0	0
$946$	0	0
$947$	− 60.9716i	− 1.98131i	−0.136385	−	0.990656i	$-0.543548\pi$
$947$	− 60.9716i	− 1.98131i	0.136385	−	0.990656i	$-0.456452\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	5.88316	0.190374
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	92.5842	2.98659
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	43.1161i	1.38366i	0.722059	+	0.691831i	$0.243196\pi$
$971$	43.1161i	1.38366i	−0.722059	+	0.691831i	$0.756804\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 56.3826i	− 1.80384i	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	− 56.3826i	− 1.80384i	0.431903	−	0.901920i	$-0.357842\pi$
$978$	0	0
$979$	−55.0000	−1.75781
$980$	59.5894i	1.90351i
$981$	0	0
$982$	0	0
$983$	62.4087i	1.99053i	0.0972017	+	0.995265i	$0.469011\pi$
$983$	62.4087i	1.99053i	−0.0972017	+	0.995265i	$0.530989\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 30.7894i	− 0.976089i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.2.d.a.890.1		4
3.2	odd	2	inner	891.2.d.a.890.4		4
9.2	odd	6		99.2.g.a.32.1	✓	4
9.4	even	3		99.2.g.a.65.1	yes	4
9.5	odd	6		297.2.g.a.197.2		4
9.7	even	3		297.2.g.a.98.2		4
11.10	odd	2	CM	891.2.d.a.890.1		4
33.32	even	2	inner	891.2.d.a.890.4		4
99.32	even	6		297.2.g.a.197.2		4
99.43	odd	6		297.2.g.a.98.2		4
99.65	even	6		99.2.g.a.32.1	✓	4
99.76	odd	6		99.2.g.a.65.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.g.a.32.1	✓	4	9.2	odd	6
99.2.g.a.32.1	✓	4	99.65	even	6
99.2.g.a.65.1	yes	4	9.4	even	3
99.2.g.a.65.1	yes	4	99.76	odd	6
297.2.g.a.98.2		4	9.7	even	3
297.2.g.a.98.2		4	99.43	odd	6
297.2.g.a.197.2		4	9.5	odd	6
297.2.g.a.197.2		4	99.32	even	6
891.2.d.a.890.1		4	1.1	even	1	trivial
891.2.d.a.890.1		4	11.10	odd	2	CM
891.2.d.a.890.4		4	3.2	odd	2	inner
891.2.d.a.890.4		4	33.32	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	891.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} -4.25639i q^{5} -3.31662i q^{11} +4.00000 q^{16} +8.51278i q^{20} +3.31662i q^{23} -13.1168 q^{25} -11.1168 q^{31} -5.11684 q^{37} +6.63325i q^{44} -7.07568i q^{47} +7.00000 q^{49} -1.43710i q^{53} -14.1168 q^{55} +11.3321i q^{59} -8.00000 q^{64} -2.11684 q^{67} +5.69349i q^{71} -17.0256i q^{80} -16.5831i q^{89} -6.63325i q^{92} -17.1168 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 16 q^{16} - 18 q^{25} - 10 q^{31} + 14 q^{37} + 28 q^{49} - 22 q^{55} - 32 q^{64} + 26 q^{67} - 34 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^16 - 18 * q^25 - 10 * q^31 + 14 * q^37 + 28 * q^49 - 22 * q^55 - 32 * q^64 + 26 * q^67 - 34 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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