LMFDB - Embedded newform 4140.2.n.a.2069.2

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			2069.2
Root			$-1.65988 + 2.13910i$ of defining polynomial
Character	$\chi$	$=$	4140.2069
Dual form			4140.2.n.a.2069.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} -4.05368 q^{7} +O(q^{10})$	$q-2.23607i q^{5} -4.05368 q^{7} -8.20739i q^{17} -4.79583i q^{23} -5.00000 q^{25} -3.74145i q^{29} -7.95998 q^{31} +9.06431i q^{35} +12.0406 q^{37} +12.1701i q^{41} -0.457775 q^{43} +9.43233 q^{49} +11.3419i q^{53} +7.84966i q^{59} -13.8779 q^{67} -15.3326i q^{71} -6.86977i q^{83} -18.3523 q^{85} +10.4024 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−4.05368	−1.53215	−0.766074	−	0.642753i	$-0.777792\pi$
$7$	−4.05368	−1.53215	−0.766074	+	0.642753i	$0.777792\pi$
$8$	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	$-0.530899\pi$
$17$	− 8.20739i	− 1.99058i	0.0969184	−	0.995292i	$-0.469101\pi$
$18$	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
$29$	− 3.74145i	− 0.694770i	−0.937723	−	0.347385i	$-0.887070\pi$
$29$	− 3.74145i	− 0.694770i	0.937723	−	0.347385i	$-0.112930\pi$
$30$	0	0
$31$	−7.95998	−1.42965	−0.714827	−	0.699301i	$-0.753495\pi$
$31$	−7.95998	−1.42965	−0.714827	+	0.699301i	$0.753495\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.06431i	1.53215i
$36$	0	0
$37$	12.0406	1.97947	0.989734	−	0.142923i	$-0.0456501\pi$
$37$	12.0406	1.97947	0.989734	+	0.142923i	$0.0456501\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12.1701i	1.90066i	0.311251	+	0.950328i	$0.399252\pi$
$41$	12.1701i	1.90066i	−0.311251	+	0.950328i	$0.600748\pi$
$42$	0	0
$43$	−0.457775	−0.0698100	−0.0349050	−	0.999391i	$-0.511113\pi$
$43$	−0.457775	−0.0698100	−0.0349050	+	0.999391i	$0.511113\pi$
$44$	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
$49$	9.43233	1.34748
$50$	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
$59$	7.84966i	1.02194i	0.859599	+	0.510969i	$0.170713\pi$
$59$	7.84966i	1.02194i	−0.859599	+	0.510969i	$0.829287\pi$
$60$	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
$67$	−13.8779	−1.69545	−0.847726	−	0.530434i	$-0.822029\pi$
$67$	−13.8779	−1.69545	−0.847726	+	0.530434i	$0.822029\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 15.3326i	− 1.81964i	−0.415003	−	0.909820i	$-0.636219\pi$
$71$	− 15.3326i	− 1.81964i	0.415003	−	0.909820i	$-0.363781\pi$
$72$	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
$97$	10.4024	1.05620	0.528101	−	0.849182i	$-0.322904\pi$
$97$	10.4024	1.05620	0.528101	+	0.849182i	$0.322904\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.95848i	0.592891i	0.955050	+	0.296445i	$0.0958012\pi$
$101$	5.95848i	0.592891i	−0.955050	+	0.296445i	$0.904199\pi$
$102$	0	0
$103$	−19.4314	−1.91464	−0.957318	−	0.289036i	$-0.906665\pi$
$103$	−19.4314	−1.91464	−0.957318	+	0.289036i	$0.906665\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.0305i	1.45305i	0.687138	+	0.726527i	$0.258867\pi$
$107$	15.0305i	1.45305i	−0.687138	+	0.726527i	$0.741133\pi$
$108$	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
$119$	33.2701i	3.04987i
$120$	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
$131$	9.50890i	0.830796i	0.909640	+	0.415398i	$0.136358\pi$
$131$	9.50890i	0.830796i	−0.909640	+	0.415398i	$0.863642\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.1833i	1.63894i	0.573121	+	0.819471i	$0.305733\pi$
$137$	19.1833i	1.63894i	−0.573121	+	0.819471i	$0.694267\pi$
$138$	0	0
$139$	12.8246	1.08777	0.543885	−	0.839159i	$-0.316953\pi$
$139$	12.8246	1.08777	0.543885	+	0.839159i	$0.316953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−8.36614	−0.694770
$146$	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	$-0.837627\pi$
$151$	−21.4476	−1.74538	−0.872691	+	0.488273i	$0.837627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	17.7991i	1.42965i
$156$	0	0
$157$	−10.9985	−0.877778	−0.438889	−	0.898541i	$-0.644628\pi$
$157$	−10.9985	−0.877778	−0.438889	+	0.898541i	$0.644628\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	19.4408i	1.53215i
$162$	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
$175$	20.2684	1.53215
$176$	0	0
$177$	0	0
$178$	0	0
$179$	26.4795i	1.97917i	0.143958	+	0.989584i	$0.454017\pi$
$179$	26.4795i	1.97917i	−0.143958	+	0.989584i	$0.545983\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 26.9237i	− 1.97947i
$186$	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
$203$	15.1666i	1.06449i
$204$	0	0
$205$	27.2132	1.90066
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−28.7446	−1.97886	−0.989430	−	0.145014i	$-0.953677\pi$
$211$	−28.7446	−1.97886	−0.989430	+	0.145014i	$0.953677\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.02362i	0.0698100i
$216$	0	0
$217$	32.2672	2.19044
$218$	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
$239$	2.85265i	0.184522i	0.995735	+	0.0922612i	$0.0294095\pi$
$239$	2.85265i	0.184522i	−0.995735	+	0.0922612i	$0.970591\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 21.0913i	− 1.34748i
$246$	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
$259$	−48.8089	−3.03284
$260$	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
$269$	18.3818i	1.12076i	0.828236	+	0.560379i	$0.189344\pi$
$269$	18.3818i	1.12076i	−0.828236	+	0.560379i	$0.810656\pi$
$270$	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
$283$	6.52889	0.388102	0.194051	−	0.980991i	$-0.437837\pi$
$283$	6.52889	0.388102	0.194051	+	0.980991i	$0.437837\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 49.3338i	− 2.91208i
$288$	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	$-0.896112\pi$
$293$	− 10.9759i	− 0.641221i	0.947211	−	0.320610i	$-0.103888\pi$
$294$	0	0
$295$	17.5524	1.02194
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.85567	0.106959
$302$	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
$311$	34.9647i	1.98267i	0.131363	+	0.991334i	$0.458065\pi$
$311$	34.9647i	1.98267i	−0.131363	+	0.991334i	$0.541935\pi$
$312$	0	0
$313$	−35.3206	−1.99644	−0.998219	−	0.0596494i	$-0.981002\pi$
$313$	−35.3206	−1.99644	−0.998219	+	0.0596494i	$0.981002\pi$
$314$	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
$335$	31.0319i	1.69545i
$336$	0	0
$337$	−8.57128	−0.466908	−0.233454	−	0.972368i	$-0.575003\pi$
$337$	−8.57128	−0.466908	−0.233454	+	0.972368i	$0.575003\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−9.85987	−0.532383
$344$	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
$355$	−34.2846	−1.81964
$356$	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
$367$	37.9591	1.98145	0.990726	−	0.135878i	$-0.0433856\pi$
$367$	37.9591	1.98145	0.990726	+	0.135878i	$0.0433856\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 45.9764i	− 2.38698i
$372$	0	0
$373$	−30.2916	−1.56844	−0.784220	−	0.620483i	$-0.786937\pi$
$373$	−30.2916	−1.56844	−0.784220	+	0.620483i	$0.786937\pi$
$374$	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	$-0.661437\pi$
$383$	− 34.2138i	− 1.74825i	0.485705	−	0.874123i	$-0.338563\pi$
$384$	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	$-0.918748\pi$
$409$	−39.1369	−1.93519	−0.967597	+	0.252498i	$0.918748\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 31.8200i	− 1.56576i
$414$	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
$433$	32.4474	1.55932	0.779661	−	0.626201i	$-0.215391\pi$
$433$	32.4474	1.55932	0.779661	+	0.626201i	$0.215391\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−21.4476	−1.02364	−0.511819	−	0.859093i	$-0.671028\pi$
$439$	−21.4476	−1.02364	−0.511819	+	0.859093i	$0.671028\pi$
$440$	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
$449$	− 0.253180i	− 0.0119483i	−0.999982	−	0.00597416i	$-0.998098\pi$
$449$	− 0.253180i	− 0.0119483i	0.999982	−	0.00597416i	$-0.00190165\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−39.7964	−1.86160	−0.930798	−	0.365533i	$-0.880887\pi$
$457$	−39.7964	−1.86160	−0.930798	+	0.365533i	$0.880887\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 31.7457i	− 1.47855i	−0.673406	−	0.739273i	$-0.735169\pi$
$461$	− 31.7457i	− 1.47855i	0.673406	−	0.739273i	$-0.264831\pi$
$462$	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
$469$	56.2565	2.59768
$470$	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
$485$	− 23.2604i	− 1.05620i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.4876i	0.969723i	0.874591	+	0.484861i	$0.161130\pi$
$491$	21.4876i	0.969723i	−0.874591	+	0.484861i	$0.838870\pi$
$492$	0	0
$493$	−30.7075	−1.38300
$494$	0	0
$495$	0	0
$496$	0	0
$497$	62.1533i	2.78796i
$498$	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
$505$	13.3236	0.592891
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 40.2310i	− 1.78321i	−0.452816	−	0.891604i	$-0.649581\pi$
$509$	− 40.2310i	− 1.78321i	0.452816	−	0.891604i	$-0.350419\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	43.4500i	1.91464i
$516$	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
$523$	−38.4051	−1.67934	−0.839669	−	0.543098i	$-0.817251\pi$
$523$	−38.4051	−1.67934	−0.839669	+	0.543098i	$0.817251\pi$
$524$	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	$-0.126468\pi$
$541$	42.8952	1.84421	0.922105	+	0.386940i	$0.126468\pi$
$542$	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	$0.300936\pi$
$557$	38.2684i	1.62148i	−0.585403	+	0.810742i	$0.699064\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.2620i	0.516780i	0.966041	+	0.258390i	$0.0831920\pi$
$563$	12.2620i	0.516780i	−0.966041	+	0.258390i	$0.916808\pi$
$564$	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
$581$	27.8478i	1.15532i
$582$	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
$595$	74.3943	3.04987
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 15.9469i	− 0.651575i	−0.945443	−	0.325787i	$-0.894371\pi$
$599$	− 15.9469i	− 0.651575i	0.945443	−	0.325787i	$-0.105629\pi$
$600$	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
$613$	−49.2653	−1.98981	−0.994903	−	0.100840i	$-0.967847\pi$
$613$	−49.2653	−1.98981	−0.994903	+	0.100840i	$0.967847\pi$
$614$	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
$629$	− 98.8222i	− 3.94030i
$630$	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
$643$	−42.2655	−1.66679	−0.833393	−	0.552680i	$-0.813605\pi$
$643$	−42.2655	−1.66679	−0.833393	+	0.552680i	$0.813605\pi$
$644$	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
$655$	21.2625	0.830796
$656$	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
$667$	−17.9434	−0.694770
$668$	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
$679$	−42.1679	−1.61826
$680$	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	$-0.633757\pi$
$691$	−21.4476	−0.815906	−0.407953	+	0.913003i	$0.633757\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 28.6767i	− 1.08777i
$696$	0	0
$697$	99.8850	3.78342
$698$	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
$707$	− 24.1538i	− 0.908396i
$708$	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
$719$	− 51.5332i	− 1.92186i	−0.276786	−	0.960932i	$-0.589269\pi$
$719$	− 51.5332i	− 1.92186i	0.276786	−	0.960932i	$-0.410731\pi$
$720$	0	0
$721$	78.7689	2.93351
$722$	0	0
$723$	0	0
$724$	0	0
$725$	18.7073i	0.694770i
$726$	0	0
$727$	−2.85440	−0.105864	−0.0529319	−	0.998598i	$-0.516857\pi$
$727$	−2.85440	−0.105864	−0.0529319	+	0.998598i	$0.516857\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.75714i	0.138963i
$732$	0	0
$733$	−21.4309	−0.791569	−0.395785	−	0.918343i	$-0.629527\pi$
$733$	−21.4309	−0.791569	−0.395785	+	0.918343i	$0.629527\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	19.0153	0.699489	0.349744	−	0.936845i	$-0.386268\pi$
$739$	19.0153	0.699489	0.349744	+	0.936845i	$0.386268\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 9.59166i	− 0.351884i	−0.984401	−	0.175942i	$-0.943703\pi$
$743$	− 9.59166i	− 0.351884i	0.984401	−	0.175942i	$-0.0562971\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 60.9289i	− 2.22629i
$750$	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
$757$	−28.7729	−1.04577	−0.522885	−	0.852403i	$-0.675144\pi$
$757$	−28.7729	−1.04577	−0.522885	+	0.852403i	$0.675144\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 48.4274i	− 1.75549i	−0.479128	−	0.877745i	$-0.659047\pi$
$761$	− 48.4274i	− 1.75549i	0.479128	−	0.877745i	$-0.340953\pi$
$762$	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	$-0.757621\pi$
$773$	− 38.3667i	− 1.37995i	0.723832	−	0.689976i	$-0.242379\pi$
$774$	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
$785$	24.5934i	0.877778i
$786$	0	0
$787$	9.83601	0.350616	0.175308	−	0.984514i	$-0.443908\pi$
$787$	9.83601	0.350616	0.175308	+	0.984514i	$0.443908\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	82.2337i	2.92389i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	43.8055i	1.55167i	0.630936	+	0.775835i	$0.282671\pi$
$797$	43.8055i	1.55167i	−0.630936	+	0.775835i	$0.717329\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	43.4709	1.53215
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.47497i	− 0.157331i	−0.996901	−	0.0786657i	$-0.974934\pi$
$809$	− 4.47497i	− 0.157331i	0.996901	−	0.0786657i	$-0.0250660\pi$
$810$	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
$821$	− 57.2016i	− 1.99635i	−0.0604026	−	0.998174i	$-0.519238\pi$
$821$	− 57.2016i	− 1.99635i	0.0604026	−	0.998174i	$-0.480762\pi$
$822$	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	$-0.452155\pi$
$829$	8.62298	0.299488	0.149744	+	0.988725i	$0.452155\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 77.4148i	− 2.68226i
$834$	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
$841$	15.0015	0.517295
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 29.0689i	− 1.00000i
$846$	0	0
$847$	44.5905	1.53215
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 57.7448i	− 1.97947i
$852$	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
$875$	− 45.3215i	− 1.53215i
$876$	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
$895$	59.2099	1.97917
$896$	0	0
$897$	0	0
$898$	0	0
$899$	29.7819i	0.993281i
$900$	0	0
$901$	93.0874	3.10119
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	58.4802	1.94180	0.970901	−	0.239481i	$-0.0769772\pi$
$907$	58.4802	1.94180	0.970901	+	0.239481i	$0.0769772\pi$
$908$	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
$917$	− 38.5460i	− 1.27290i
$918$	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
$925$	−60.2032	−1.97947
$926$	0	0
$927$	0	0
$928$	0	0
$929$	41.8895i	1.37435i	0.726492	+	0.687175i	$0.241149\pi$
$929$	41.8895i	1.37435i	−0.726492	+	0.687175i	$0.758851\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−27.5449	−0.899854	−0.449927	−	0.893065i	$-0.648550\pi$
$937$	−27.5449	−0.899854	−0.449927	+	0.893065i	$0.648550\pi$
$938$	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
$943$	58.3659	1.90066
$944$	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
$959$	− 77.7631i	− 2.51110i
$960$	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
$973$	−51.9870	−1.66663
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.67030i	− 0.0854305i	−0.999087	−	0.0427153i	$-0.986399\pi$
$977$	− 2.67030i	− 0.0854305i	0.999087	−	0.0427153i	$-0.0136008\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 36.9824i	− 1.17955i	−0.807566	−	0.589777i	$-0.799215\pi$
$983$	− 36.9824i	− 1.17955i	0.807566	−	0.589777i	$-0.200785\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.19541i	0.0698100i
$990$	0	0
$991$	−1.76933	−0.0562045	−0.0281022	−	0.999605i	$-0.508946\pi$
$991$	−1.76933	−0.0562045	−0.0281022	+	0.999605i	$0.508946\pi$
$992$	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.2069.2	✓	16
3.2	odd	2	inner	4140.2.n.a.2069.10	yes	16
5.4	even	2	inner	4140.2.n.a.2069.15	yes	16
15.14	odd	2	inner	4140.2.n.a.2069.7	yes	16
23.22	odd	2	inner	4140.2.n.a.2069.15	yes	16
69.68	even	2	inner	4140.2.n.a.2069.7	yes	16
115.114	odd	2	CM	4140.2.n.a.2069.2	✓	16
345.344	even	2	inner	4140.2.n.a.2069.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.n.a.2069.2	✓	16	1.1	even	1	trivial
4140.2.n.a.2069.2	✓	16	115.114	odd	2	CM
4140.2.n.a.2069.7	yes	16	15.14	odd	2	inner
4140.2.n.a.2069.7	yes	16	69.68	even	2	inner
4140.2.n.a.2069.10	yes	16	3.2	odd	2	inner
4140.2.n.a.2069.10	yes	16	345.344	even	2	inner
4140.2.n.a.2069.15	yes	16	5.4	even	2	inner
4140.2.n.a.2069.15	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)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} -4.05368 q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} -4.05368 q^{7} -8.20739i q^{17} -4.79583i q^{23} -5.00000 q^{25} -3.74145i q^{29} -7.95998 q^{31} +9.06431i q^{35} +12.0406 q^{37} +12.1701i q^{41} -0.457775 q^{43} +9.43233 q^{49} +11.3419i q^{53} +7.84966i q^{59} -13.8779 q^{67} -15.3326i q^{71} -6.86977i q^{83} -18.3523 q^{85} +10.4024 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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