LMFDB - Embedded newform 4140.2.n.a.2069.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(2069,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("4140.2069");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.n (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1352 x^{12} - 4836 x^{11} + 18782 x^{10} - 55300 x^{9} + \cdots + 11064600 $ x^16 - 8x^15 + 60x^14 - 280x^13 + 1352x^12 - 4836x^11 + 18782x^10 - 55300x^9 + 177369x^8 - 421148x^7 + 1135954x^6 - 2123100x^5 + 5223234x^4 - 7310160x^3 + 12160080x^2 - 8802000*x + 11064600
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2069.3
Root			$1.07874 - 1.43199i$ of defining polynomial
Character	$\chi$	$=$	4140.2069
Dual form			4140.2.n.a.2069.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} -3.40113 q^{7} +O(q^{10})$	$q-2.23607i q^{5} -3.40113 q^{7} +8.20739i q^{17} +4.79583i q^{23} -5.00000 q^{25} -10.0996i q^{29} +7.95998 q^{31} +7.60515i q^{35} +1.73873 q^{37} -3.98596i q^{41} +13.1069 q^{43} +4.56767 q^{49} +11.3419i q^{53} -13.2054i q^{59} +8.68358 q^{67} -6.99375i q^{71} -6.86977i q^{83} +18.3523 q^{85} -16.7269 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 80 q^{25} + 112 q^{49}+O(q^{100}) $ 16 * q - 80 * q^25 + 112 * q^49

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	0	0
$7$	−3.40113	−1.28551	−0.642753	−	0.766074i	$-0.722208\pi$
$7$	−3.40113	−1.28551	−0.642753	+	0.766074i	$0.722208\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	8.20739i	1.99058i	0.0969184	+	0.995292i	$0.469101\pi$
$17$	8.20739i	1.99058i	−0.0969184	+	0.995292i	$0.530899\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.79583i	1.00000i
$23$	4.79583i	1.00000i
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 10.0996i	− 1.87545i	−0.347385	−	0.937723i	$-0.612930\pi$
$29$	− 10.0996i	− 1.87545i	0.347385	−	0.937723i	$-0.387070\pi$
$30$	0	0
$31$	7.95998	1.42965	0.714827	−	0.699301i	$-0.246505\pi$
$31$	7.95998	1.42965	0.714827	+	0.699301i	$0.246505\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	7.60515i	1.28551i
$36$	0	0
$37$	1.73873	0.285846	0.142923	−	0.989734i	$-0.454350\pi$
$37$	1.73873	0.285846	0.142923	+	0.989734i	$0.454350\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.98596i	− 0.622502i	−0.950328	−	0.311251i	$-0.899252\pi$
$41$	− 3.98596i	− 0.622502i	0.950328	−	0.311251i	$-0.100748\pi$
$42$	0	0
$43$	13.1069	1.99878	0.999391	−	0.0349050i	$-0.0111129\pi$
$43$	13.1069	1.99878	0.999391	+	0.0349050i	$0.0111129\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	4.56767	0.652525
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.3419i	1.55793i	0.627067	+	0.778965i	$0.284255\pi$
$53$	11.3419i	1.55793i	−0.627067	+	0.778965i	$0.715745\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 13.2054i	− 1.71920i	−0.510969	−	0.859599i	$-0.670713\pi$
$59$	− 13.2054i	− 1.71920i	0.510969	−	0.859599i	$-0.329287\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.68358	1.06087	0.530434	−	0.847726i	$-0.322029\pi$
$67$	8.68358	1.06087	0.530434	+	0.847726i	$0.322029\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 6.99375i	− 0.830006i	−0.909820	−	0.415003i	$-0.863781\pi$
$71$	− 6.99375i	− 0.830006i	0.909820	−	0.415003i	$-0.136219\pi$
$72$	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			$\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.86977i	− 0.754055i	−0.926202	−	0.377027i	$-0.876946\pi$
$83$	− 6.86977i	− 0.754055i	0.926202	−	0.377027i	$-0.123054\pi$
$84$	0	0
$85$	18.3523	1.99058
$86$	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
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.7269	−1.69836	−0.849182	−	0.528101i	$-0.822904\pi$
$97$	−16.7269	−1.69836	−0.849182	+	0.528101i	$0.822904\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	19.1963i	1.91010i	0.296445	+	0.955050i	$0.404199\pi$
$101$	19.1963i	1.91010i	−0.296445	+	0.955050i	$0.595801\pi$
$102$	0	0
$103$	−5.86678	−0.578071	−0.289036	−	0.957318i	$-0.593335\pi$
$103$	−5.86678	−0.578071	−0.289036	+	0.957318i	$0.593335\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15.0305i	− 1.45305i	−0.687138	−	0.726527i	$-0.741133\pi$
$107$	− 15.0305i	− 1.45305i	0.687138	−	0.726527i	$-0.258867\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 20.2862i	− 1.90836i	−0.299229	−	0.954181i	$-0.596729\pi$
$113$	− 20.2862i	− 1.90836i	0.299229	−	0.954181i	$-0.403271\pi$
$114$	0	0
$115$	10.7238	1.00000
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 27.9144i	− 2.55891i
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803i	1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 20.8226i	− 1.81928i	−0.415398	−	0.909640i	$-0.636358\pi$
$131$	− 20.8226i	− 1.81928i	0.415398	−	0.909640i	$-0.363642\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 19.1833i	− 1.63894i	−0.573121	−	0.819471i	$-0.694267\pi$
$137$	− 19.1833i	− 1.63894i	0.573121	−	0.819471i	$-0.305733\pi$
$138$	0	0
$139$	−12.8246	−1.08777	−0.543885	−	0.839159i	$-0.683047\pi$
$139$	−12.8246	−1.08777	−0.543885	+	0.839159i	$0.683047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−22.5834	−1.87545
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	21.4476	1.74538	0.872691	−	0.488273i	$-0.162373\pi$
$151$	21.4476	1.74538	0.872691	+	0.488273i	$0.162373\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 17.7991i	− 1.42965i
$156$	0	0
$157$	22.5174	1.79708	0.898541	−	0.438889i	$-0.144628\pi$
$157$	22.5174	1.79708	0.898541	+	0.438889i	$0.144628\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 16.3112i	− 1.28551i
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	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
$175$	17.0056	1.28551
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3.85204i	− 0.287915i	−0.989584	−	0.143958i	$-0.954017\pi$
$179$	− 3.85204i	− 0.287915i	0.989584	−	0.143958i	$-0.0459829\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 3.88792i	− 0.285846i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	34.3500i	2.41090i
$204$	0	0
$205$	−8.91287	−0.622502
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	28.7446	1.97886	0.989430	−	0.145014i	$-0.0463229\pi$
$211$	28.7446	1.97886	0.989430	+	0.145014i	$0.0463229\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 29.3079i	− 1.99878i
$216$	0	0
$217$	−27.0729	−1.83783
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.94427i	0.593652i	0.954932	+	0.296826i	$0.0959282\pi$
$227$	8.94427i	0.593652i	−0.954932	+	0.296826i	$0.904072\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	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
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	30.7874i	1.99147i	0.0922612	+	0.995735i	$0.470591\pi$
$239$	30.7874i	1.99147i	−0.0922612	+	0.995735i	$0.529409\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 10.2136i	− 0.652525i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	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
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−5.91365	−0.367457
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.07451i	− 0.127920i	−0.997952	−	0.0639598i	$-0.979627\pi$
$263$	− 2.07451i	− 0.127920i	0.997952	−	0.0639598i	$-0.0203729\pi$
$264$	0	0
$265$	25.3613	1.55793
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 27.1682i	− 1.65647i	−0.560379	−	0.828236i	$-0.689344\pi$
$269$	− 27.1682i	− 1.65647i	0.560379	−	0.828236i	$-0.310656\pi$
$270$	0	0
$271$	31.3613	1.90506	0.952531	−	0.304443i	$-0.0984703\pi$
$271$	31.3613	1.90506	0.952531	+	0.304443i	$0.0984703\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	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
$283$	33.0057	1.96198	0.980991	−	0.194051i	$-0.0621628\pi$
$283$	33.0057	1.96198	0.980991	+	0.194051i	$0.0621628\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.5567i	0.800229i
$288$	0	0
$289$	−50.3613	−2.96243
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.9759i	0.641221i	0.947211	+	0.320610i	$0.103888\pi$
$293$	10.9759i	0.641221i	−0.947211	+	0.320610i	$0.896112\pi$
$294$	0	0
$295$	−29.5282	−1.71920
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−44.5782	−2.56944
$302$	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
$311$	4.63324i	0.262727i	0.991334	+	0.131363i	$0.0419355\pi$
$311$	4.63324i	0.262727i	−0.991334	+	0.131363i	$0.958065\pi$
$312$	0	0
$313$	2.11061	0.119299	0.0596494	−	0.998219i	$-0.481002\pi$
$313$	2.11061	0.119299	0.0596494	+	0.998219i	$0.481002\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	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
$331$	19.3613	1.06419	0.532096	−	0.846684i	$-0.321405\pi$
$331$	19.3613	1.06419	0.532096	+	0.846684i	$0.321405\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 19.4171i	− 1.06087i
$336$	0	0
$337$	−35.7006	−1.94474	−0.972368	−	0.233454i	$-0.924997\pi$
$337$	−35.7006	−1.94474	−0.972368	+	0.233454i	$0.924997\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8.27265	0.446681
$344$	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
$349$	37.3613	1.99990	0.999951	−	0.00987003i	$-0.00314178\pi$
$349$	37.3613	1.99990	0.999951	+	0.00987003i	$0.00314178\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−15.6385	−0.830006
$356$	0	0
$357$	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
$367$	−5.20611	−0.271757	−0.135878	−	0.990726i	$-0.543386\pi$
$367$	−5.20611	−0.271757	−0.135878	+	0.990726i	$0.543386\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 38.5753i	− 2.00273i
$372$	0	0
$373$	23.9670	1.24097	0.620483	−	0.784220i	$-0.286937\pi$
$373$	23.9670	1.24097	0.620483	+	0.784220i	$0.286937\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	34.2138i	1.74825i	0.485705	+	0.874123i	$0.338563\pi$
$383$	34.2138i	1.74825i	−0.485705	+	0.874123i	$0.661437\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−39.3613	−1.99058
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	39.1369	1.93519	0.967597	−	0.252498i	$-0.0812521\pi$
$409$	39.1369	1.93519	0.967597	+	0.252498i	$0.0812521\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	44.9133i	2.21004i
$414$	0	0
$415$	−15.3613	−0.754055
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 41.0370i	− 1.99058i
$426$	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
$433$	26.0608	1.25240	0.626201	−	0.779661i	$-0.284609\pi$
$433$	26.0608	1.25240	0.626201	+	0.779661i	$0.284609\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	21.4476	1.02364	0.511819	−	0.859093i	$-0.328972\pi$
$439$	21.4476	1.02364	0.511819	+	0.859093i	$0.328972\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	42.3785i	1.99996i	0.00597416	+	0.999982i	$0.498098\pi$
$449$	42.3785i	1.99996i	−0.00597416	+	0.999982i	$0.501902\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.6284	0.731067	0.365533	−	0.930798i	$-0.380887\pi$
$457$	15.6284	0.731067	0.365533	+	0.930798i	$0.380887\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.9173i	1.34681i	0.739273	+	0.673406i	$0.235169\pi$
$461$	28.9173i	1.34681i	−0.739273	+	0.673406i	$0.764831\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 38.4978i	− 1.78147i	−0.454525	−	0.890734i	$-0.650191\pi$
$467$	− 38.4978i	− 1.78147i	0.454525	−	0.890734i	$-0.349809\pi$
$468$	0	0
$469$	−29.5340	−1.36375
$470$	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
$484$	0	0
$485$	37.4026i	1.69836i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 38.7593i	− 1.74918i	−0.484861	−	0.874591i	$-0.661130\pi$
$491$	− 38.7593i	− 1.74918i	0.484861	−	0.874591i	$-0.338870\pi$
$492$	0	0
$493$	82.8912	3.73323
$494$	0	0
$495$	0	0
$496$	0	0
$497$	23.7867i	1.06698i
$498$	0	0
$499$	43.3613	1.94112	0.970558	−	0.240866i	$-0.0774314\pi$
$499$	43.3613	1.94112	0.970558	+	0.240866i	$0.0774314\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 33.7026i	− 1.50272i	−0.659890	−	0.751362i	$-0.729397\pi$
$503$	− 33.7026i	− 1.50272i	0.659890	−	0.751362i	$-0.270603\pi$
$504$	0	0
$505$	42.9242	1.91010
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.4320i	0.905633i	0.891604	+	0.452816i	$0.149581\pi$
$509$	20.4320i	0.905633i	−0.891604	+	0.452816i	$0.850419\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.1185i	0.578071i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−24.8404	−1.08620	−0.543098	−	0.839669i	$-0.682749\pi$
$523$	−24.8404	−1.08620	−0.543098	+	0.839669i	$0.682749\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	65.3307i	2.84585i
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−33.6092	−1.45305
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−42.8952	−1.84421	−0.922105	−	0.386940i	$-0.873532\pi$
$541$	−42.8952	−1.84421	−0.922105	+	0.386940i	$0.873532\pi$
$542$	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			$\pi$
$548$	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
$557$	− 38.2684i	− 1.62148i	−0.585403	−	0.810742i	$-0.699064\pi$
$557$	− 38.2684i	− 1.62148i	0.585403	−	0.810742i	$-0.300936\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 12.2620i	− 0.516780i	−0.966041	−	0.258390i	$-0.916808\pi$
$563$	− 12.2620i	− 0.516780i	0.966041	−	0.258390i	$-0.0831920\pi$
$564$	0	0
$565$	−45.3613	−1.90836
$566$	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
$575$	− 23.9792i	− 1.00000i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	23.3650i	0.969342i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	−62.4185	−2.55891
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 46.2784i	− 1.89089i	−0.325787	−	0.945443i	$-0.605629\pi$
$599$	− 46.2784i	− 1.89089i	0.325787	−	0.945443i	$-0.394371\pi$
$600$	0	0
$601$	−9.36126	−0.381854	−0.190927	−	0.981604i	$-0.561149\pi$
$601$	−9.36126	−0.381854	−0.190927	+	0.981604i	$0.561149\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24.5967i	1.00000i
$606$	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
$613$	4.99338	0.201681	0.100840	−	0.994903i	$-0.467847\pi$
$613$	4.99338	0.201681	0.100840	+	0.994903i	$0.467847\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 25.0814i	− 1.00974i	−0.863195	−	0.504870i	$-0.831540\pi$
$617$	− 25.0814i	− 1.00974i	0.863195	−	0.504870i	$-0.168460\pi$
$618$	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
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.2705i	0.569000i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	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
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	28.0291	1.10536	0.552680	−	0.833393i	$-0.313605\pi$
$643$	28.0291	1.10536	0.552680	+	0.833393i	$0.313605\pi$
$644$	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
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	−46.5608	−1.81928
$656$	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
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	48.4359	1.87545
$668$	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
$676$	0	0
$677$	− 51.9142i	− 1.99523i	−0.0690480	−	0.997613i	$-0.521996\pi$
$677$	− 51.9142i	− 1.99523i	0.0690480	−	0.997613i	$-0.478004\pi$
$678$	0	0
$679$	56.8905	2.18326
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	−42.8952	−1.63894
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	21.4476	0.815906	0.407953	−	0.913003i	$-0.366243\pi$
$691$	21.4476	0.815906	0.407953	+	0.913003i	$0.366243\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28.6767i	1.08777i
$696$	0	0
$697$	32.7143	1.23914
$698$	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
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 65.2890i	− 2.45544i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	38.1747i	1.42965i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 14.8436i	− 0.553571i	−0.960932	−	0.276786i	$-0.910731\pi$
$719$	− 14.8436i	− 0.553571i	0.960932	−	0.276786i	$-0.0892692\pi$
$720$	0	0
$721$	19.9537	0.743114
$722$	0	0
$723$	0	0
$724$	0	0
$725$	50.4979i	1.87545i
$726$	0	0
$727$	−53.8503	−1.99720	−0.998598	−	0.0529319i	$-0.983143\pi$
$727$	−53.8503	−1.99720	−0.998598	+	0.0529319i	$0.983143\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	107.573i	3.97874i
$732$	0	0
$733$	−49.7264	−1.83669	−0.918343	−	0.395785i	$-0.870473\pi$
$733$	−49.7264	−1.83669	−0.918343	+	0.395785i	$0.870473\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−19.0153	−0.699489	−0.349744	−	0.936845i	$-0.613732\pi$
$739$	−19.0153	−0.699489	−0.349744	+	0.936845i	$0.613732\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.59166i	0.351884i	0.984401	+	0.175942i	$0.0562971\pi$
$743$	9.59166i	0.351884i	−0.984401	+	0.175942i	$0.943703\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	51.1207i	1.86791i
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 47.9583i	− 1.74538i
$756$	0	0
$757$	−46.9054	−1.70481	−0.852403	−	0.522885i	$-0.824856\pi$
$757$	−46.9054	−1.70481	−0.852403	+	0.522885i	$0.824856\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 26.4347i	− 0.958256i	−0.877745	−	0.479128i	$-0.840953\pi$
$761$	− 26.4347i	− 0.958256i	0.877745	−	0.479128i	$-0.159047\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	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
$773$	38.3667i	1.37995i	0.723832	+	0.689976i	$0.242379\pi$
$773$	38.3667i	1.37995i	−0.723832	+	0.689976i	$0.757621\pi$
$774$	0	0
$775$	−39.7999	−1.42965
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 50.3504i	− 1.79708i
$786$	0	0
$787$	−55.2381	−1.96903	−0.984514	−	0.175308i	$-0.943908\pi$
$787$	−55.2381	−1.96903	−0.984514	+	0.175308i	$0.943908\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	68.9959i	2.45321i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 43.8055i	− 1.55167i	−0.630936	−	0.775835i	$-0.717329\pi$
$797$	− 43.8055i	− 1.55167i	0.630936	−	0.775835i	$-0.282671\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−36.4730	−1.28551
$806$	0	0
$807$	0	0
$808$	0	0
$809$	56.7096i	1.99380i	0.0786657	+	0.996901i	$0.474934\pi$
$809$	56.7096i	1.99380i	−0.0786657	+	0.996901i	$0.525066\pi$
$810$	0	0
$811$	−51.3613	−1.80354	−0.901769	−	0.432218i	$-0.857731\pi$
$811$	−51.3613	−1.80354	−0.901769	+	0.432218i	$0.857731\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.46144i	0.120805i	0.998174	+	0.0604026i	$0.0192385\pi$
$821$	3.46144i	0.120805i	−0.998174	+	0.0604026i	$0.980762\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	56.3864i	1.96075i	0.197150	+	0.980373i	$0.436831\pi$
$827$	56.3864i	1.96075i	−0.197150	+	0.980373i	$0.563169\pi$
$828$	0	0
$829$	−8.62298	−0.299488	−0.149744	−	0.988725i	$-0.547845\pi$
$829$	−8.62298	−0.299488	−0.149744	+	0.988725i	$0.547845\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	37.4887i	1.29891i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−73.0015	−2.51729
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 29.0689i	− 1.00000i
$846$	0	0
$847$	37.4124	1.28551
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.33867i	0.285846i
$852$	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
$859$	−3.36126	−0.114685	−0.0573424	−	0.998355i	$-0.518263\pi$
$859$	−3.36126	−0.114685	−0.0573424	+	0.998355i	$0.518263\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	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
$875$	− 38.0258i	− 1.28551i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	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	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	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
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−8.61343	−0.287915
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 80.3925i	− 2.68124i
$900$	0	0
$901$	−93.0874	−3.10119
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−14.4246	−0.478961	−0.239481	−	0.970901i	$-0.576977\pi$
$907$	−14.4246	−0.478961	−0.239481	+	0.970901i	$0.576977\pi$
$908$	0	0
$909$	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
$917$	70.8204i	2.33869i
$918$	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
$925$	−8.69366	−0.285846
$926$	0	0
$927$	0	0
$928$	0	0
$929$	44.2863i	1.45298i	0.687175	+	0.726492i	$0.258851\pi$
$929$	44.2863i	1.45298i	−0.687175	+	0.726492i	$0.741149\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−54.6743	−1.78613	−0.893065	−	0.449927i	$-0.851450\pi$
$937$	−54.6743	−1.78613	−0.893065	+	0.449927i	$0.851450\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	19.1160	0.622502
$944$	0	0
$945$	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
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.3607i	0.724333i	0.932113	+	0.362167i	$0.117963\pi$
$953$	22.3607i	0.724333i	−0.932113	+	0.362167i	$0.882037\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	65.2450i	2.10687i
$960$	0	0
$961$	32.3613	1.04391
$962$	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
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	43.6182	1.39834
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.67030i	0.0854305i	0.999087	+	0.0427153i	$0.0136008\pi$
$977$	2.67030i	0.0854305i	−0.999087	+	0.0427153i	$0.986399\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	36.9824i	1.17955i	0.807566	+	0.589777i	$0.200785\pi$
$983$	36.9824i	1.17955i	−0.807566	+	0.589777i	$0.799215\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	62.8584i	1.99878i
$990$	0	0
$991$	1.76933	0.0562045	0.0281022	−	0.999605i	$-0.491054\pi$
$991$	1.76933	0.0562045	0.0281022	+	0.999605i	$0.491054\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	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	4140.2.n.a.2069.3	✓	16
3.2	odd	2	inner	4140.2.n.a.2069.11	yes	16
5.4	even	2	inner	4140.2.n.a.2069.14	yes	16
15.14	odd	2	inner	4140.2.n.a.2069.6	yes	16
23.22	odd	2	inner	4140.2.n.a.2069.14	yes	16
69.68	even	2	inner	4140.2.n.a.2069.6	yes	16
115.114	odd	2	CM	4140.2.n.a.2069.3	✓	16
345.344	even	2	inner	4140.2.n.a.2069.11	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.n.a.2069.3	✓	16	1.1	even	1	trivial
4140.2.n.a.2069.3	✓	16	115.114	odd	2	CM
4140.2.n.a.2069.6	yes	16	15.14	odd	2	inner
4140.2.n.a.2069.6	yes	16	69.68	even	2	inner
4140.2.n.a.2069.11	yes	16	3.2	odd	2	inner
4140.2.n.a.2069.11	yes	16	345.344	even	2	inner
4140.2.n.a.2069.14	yes	16	5.4	even	2	inner
4140.2.n.a.2069.14	yes	16	23.22	odd	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} -3.40113 q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} -3.40113 q^{7} +8.20739i q^{17} +4.79583i q^{23} -5.00000 q^{25} -10.0996i q^{29} +7.95998 q^{31} +7.60515i q^{35} +1.73873 q^{37} -3.98596i q^{41} +13.1069 q^{43} +4.56767 q^{49} +11.3419i q^{53} -13.2054i q^{59} +8.68358 q^{67} -6.99375i q^{71} -6.86977i q^{83} +18.3523 q^{85} -16.7269 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 80 q^{25} + 112 q^{49}+O(q^{100}) \) 16 * q - 80 * q^25 + 112 * q^49

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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