LMFDB - Embedded newform 2601.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.a (trivial)

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:	$20.7690895657$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} +4.94356 q^{7} +O(q^{10})$	$q-2.00000 q^{4} +4.94356 q^{7} +7.17024 q^{13} +4.00000 q^{16} +3.90167 q^{19} -5.00000 q^{25} -9.88713 q^{28} +3.61587 q^{31} -8.55943 q^{37} -11.0719 q^{43} +17.4388 q^{49} -14.3405 q^{52} +6.27126 q^{61} -8.00000 q^{64} +0.633103 q^{67} +17.0000 q^{73} -7.80335 q^{76} +17.0000 q^{79} +35.4466 q^{91} -11.2148 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{4}+O(q^{10}) $ 3 * q - 6 * q^4	$ 3 q - 6 q^{4} + 12 q^{16} - 15 q^{25} + 21 q^{49} - 24 q^{64} + 51 q^{73} + 51 q^{79} + 51 q^{91}+O(q^{100}) $ 3 * q - 6 * q^4 + 12 * q^16 - 15 * q^25 + 21 * q^49 - 24 * q^64 + 51 * q^73 + 51 * q^79 + 51 * q^91

Display 100 coefficients 10 coefficients

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	4.94356	1.86849	0.934246	−	0.356630i	$-0.116074\pi$
$7$	4.94356	1.86849	0.934246	+	0.356630i	$0.116074\pi$
$8$	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
$13$	7.17024	1.98867	0.994334	−	0.106301i	$-0.0339006\pi$
$13$	7.17024	1.98867	0.994334	+	0.106301i	$0.0339006\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	3.90167	0.895105	0.447553	−	0.894258i	$-0.352296\pi$
$19$	3.90167	0.895105	0.447553	+	0.894258i	$0.352296\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	−9.88713	−1.86849
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	3.61587	0.649429	0.324714	−	0.945812i	$-0.394732\pi$
$31$	3.61587	0.649429	0.324714	+	0.945812i	$0.394732\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.55943	−1.40716	−0.703581	−	0.710615i	$-0.748417\pi$
$37$	−8.55943	−1.40716	−0.703581	+	0.710615i	$0.748417\pi$
$38$	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
$43$	−11.0719	−1.68845	−0.844226	−	0.535988i	$-0.819939\pi$
$43$	−11.0719	−1.68845	−0.844226	+	0.535988i	$0.819939\pi$
$44$	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
$49$	17.4388	2.49126
$50$	0	0
$51$	0	0
$52$	−14.3405	−1.98867
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	6.27126	0.802952	0.401476	−	0.915869i	$-0.368497\pi$
$61$	6.27126	0.802952	0.401476	+	0.915869i	$0.368497\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	0.633103	0.0773459	0.0386729	−	0.999252i	$-0.487687\pi$
$67$	0.633103	0.0773459	0.0386729	+	0.999252i	$0.487687\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	17.0000	1.98970	0.994850	−	0.101361i	$-0.0323196\pi$
$73$	17.0000	1.98970	0.994850	+	0.101361i	$0.0323196\pi$
$74$	0	0
$75$	0	0
$76$	−7.80335	−0.895105
$77$	0	0
$78$	0	0
$79$	17.0000	1.91265	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	17.0000	1.91265	0.956325	+	0.292306i	$0.0944227\pi$
$80$	0	0
$81$	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
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	35.4466	3.71581
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.2148	−1.13869	−0.569346	−	0.822098i	$-0.692804\pi$
$97$	−11.2148	−1.13869	−0.569346	+	0.822098i	$0.692804\pi$
$98$	0	0
$99$	0	0
$100$	10.0000	1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−17.6091	−1.73507	−0.867536	−	0.497374i	$-0.834298\pi$
$103$	−17.6091	−1.73507	−0.867536	+	0.497374i	$0.834298\pi$
$104$	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
$109$	17.0000	1.62830	0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000	1.62830	0.814152	+	0.580651i	$0.197202\pi$
$110$	0	0
$111$	0	0
$112$	19.7743	1.86849
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	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
$124$	−7.23173	−0.649429
$125$	0	0
$126$	0	0
$127$	13.7074	1.21633	0.608167	−	0.793809i	$-0.291905\pi$
$127$	13.7074	1.21633	0.608167	+	0.793809i	$0.291905\pi$
$128$	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
$133$	19.2882	1.67250
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−23.3901	−1.98392	−0.991962	−	0.126536i	$-0.959614\pi$
$139$	−23.3901	−1.98392	−0.991962	+	0.126536i	$0.959614\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	17.1189	1.40716
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−4.53478	−0.369035	−0.184517	−	0.982829i	$-0.559072\pi$
$151$	−4.53478	−0.369035	−0.184517	+	0.982829i	$0.559072\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.63547	−0.210333	−0.105167	−	0.994455i	$-0.533538\pi$
$157$	−2.63547	−0.210333	−0.105167	+	0.994455i	$0.533538\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	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
$169$	38.4124	2.95480
$170$	0	0
$171$	0	0
$172$	22.1438	1.68845
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−24.7178	−1.86849
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−22.0624	−1.63989	−0.819943	−	0.572444i	$-0.805995\pi$
$181$	−22.0624	−1.63989	−0.819943	+	0.572444i	$0.805995\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	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
$193$	7.59896	0.546985	0.273492	−	0.961874i	$-0.411821\pi$
$193$	7.59896	0.546985	0.273492	+	0.961874i	$0.411821\pi$
$194$	0	0
$195$	0	0
$196$	−34.8776	−2.49126
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	28.6810	1.98867
$209$	0	0
$210$	0	0
$211$	21.1019	1.45272	0.726359	−	0.687315i	$-0.241211\pi$
$211$	21.1019	1.45272	0.726359	+	0.687315i	$0.241211\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	17.8753	1.21345
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.00000	0.334825	0.167412	−	0.985887i	$-0.446459\pi$
$223$	5.00000	0.334825	0.167412	+	0.985887i	$0.446459\pi$
$224$	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
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	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
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	0	0
$244$	−12.5425	−0.802952
$245$	0	0
$246$	0	0
$247$	27.9760	1.78007
$248$	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
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−42.3141	−2.62927
$260$	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
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−1.26621	−0.0773459
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−32.5827	−1.97926	−0.989628	−	0.143657i	$-0.954114\pi$
$271$	−32.5827	−1.97926	−0.989628	+	0.143657i	$0.954114\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	33.2772	1.99944	0.999718	−	0.0237496i	$-0.00756043\pi$
$277$	33.2772	1.99944	0.999718	+	0.0237496i	$0.00756043\pi$
$278$	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
$283$	15.7912	0.938687	0.469344	−	0.883016i	$-0.344491\pi$
$283$	15.7912	0.938687	0.469344	+	0.883016i	$0.344491\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0	0
$290$	0	0
$291$	0	0
$292$	−34.0000	−1.98970
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−54.7347	−3.15486
$302$	0	0
$303$	0	0
$304$	15.6067	0.895105
$305$	0	0
$306$	0	0
$307$	−19.0000	−1.08439	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−19.0000	−1.08439	−0.542194	+	0.840254i	$0.682406\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0.960475	0.0542893	0.0271446	−	0.999632i	$-0.491359\pi$
$313$	0.960475	0.0542893	0.0271446	+	0.999632i	$0.491359\pi$
$314$	0	0
$315$	0	0
$316$	−34.0000	−1.91265
$317$	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
$325$	−35.8512	−1.98867
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−24.1462	−1.32720	−0.663598	−	0.748090i	$-0.730971\pi$
$331$	−24.1462	−1.32720	−0.663598	+	0.748090i	$0.730971\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	51.6049	2.78640
$344$	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
$349$	23.0000	1.23116	0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000	1.23116	0.615581	+	0.788074i	$0.288921\pi$
$350$	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
$355$	0	0
$356$	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
$361$	−3.77694	−0.198786
$362$	0	0
$363$	0	0
$364$	−70.8931	−3.71581
$365$	0	0
$366$	0	0
$367$	−38.2208	−1.99511	−0.997555	−	0.0698862i	$-0.977736\pi$
$367$	−38.2208	−1.99511	−0.997555	+	0.0698862i	$0.977736\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−25.0000	−1.29445	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000	−1.29445	−0.647225	+	0.762299i	$0.724071\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	30.6219	1.57294	0.786469	−	0.617629i	$-0.211907\pi$
$379$	30.6219	1.57294	0.786469	+	0.617629i	$0.211907\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	22.4296	1.13869
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	25.9267	1.29150
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	20.2445	1.00103	0.500514	−	0.865729i	$-0.333144\pi$
$409$	20.2445	1.00103	0.500514	+	0.865729i	$0.333144\pi$
$410$	0	0
$411$	0	0
$412$	35.2181	1.73507
$413$	0	0
$414$	0	0
$415$	0	0
$416$	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
$421$	−39.1198	−1.90658	−0.953291	−	0.302053i	$-0.902328\pi$
$421$	−39.1198	−1.90658	−0.953291	+	0.302053i	$0.902328\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	31.0024	1.50031
$428$	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
$433$	12.3381	0.592932	0.296466	−	0.955043i	$-0.404192\pi$
$433$	12.3381	0.592932	0.296466	+	0.955043i	$0.404192\pi$
$434$	0	0
$435$	0	0
$436$	−34.0000	−1.62830
$437$	0	0
$438$	0	0
$439$	−35.5654	−1.69744	−0.848722	−	0.528839i	$-0.822628\pi$
$439$	−35.5654	−1.69744	−0.848722	+	0.528839i	$0.822628\pi$
$440$	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
$448$	−39.5485	−1.86849
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−9.17261	−0.426287	−0.213144	−	0.977021i	$-0.568370\pi$
$463$	−9.17261	−0.426287	−0.213144	+	0.977021i	$0.568370\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	3.12979	0.144520
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−19.5084	−0.895105
$476$	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
$481$	−61.3732	−2.79838
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	0	0
$487$	35.9326	1.62826	0.814132	−	0.580680i	$-0.197213\pi$
$487$	35.9326	1.62826	0.814132	+	0.580680i	$0.197213\pi$
$488$	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
$496$	14.4635	0.649429
$497$	0	0
$498$	0	0
$499$	−19.4070	−0.868778	−0.434389	−	0.900725i	$-0.643036\pi$
$499$	−19.4070	−0.868778	−0.434389	+	0.900725i	$0.643036\pi$
$500$	0	0
$501$	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
$508$	−27.4148	−1.21633
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	84.0406	3.71774
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	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
$523$	35.0000	1.53044	0.765222	−	0.643767i	$-0.222629\pi$
$523$	35.0000	1.53044	0.765222	+	0.643767i	$0.222629\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	−38.5763	−1.67250
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	29.2942	1.25253	0.626264	−	0.779611i	$-0.284583\pi$
$547$	29.2942	1.25253	0.626264	+	0.779611i	$0.284583\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	84.0406	3.57377
$554$	0	0
$555$	0	0
$556$	46.7802	1.98392
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−79.3884	−3.35777
$560$	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
$565$	0	0
$566$	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
$571$	−0.367221	−0.0153677	−0.00768386	−	0.999970i	$-0.502446\pi$
$571$	−0.367221	−0.0153677	−0.00768386	+	0.999970i	$0.502446\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	38.4867	1.60222	0.801111	−	0.598516i	$-0.204243\pi$
$577$	38.4867	1.60222	0.801111	+	0.598516i	$0.204243\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	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
$589$	14.1079	0.581307
$590$	0	0
$591$	0	0
$592$	−34.2377	−1.40716
$593$	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
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	45.4525	1.85405	0.927024	−	0.375002i	$-0.122358\pi$
$601$	45.4525	1.85405	0.927024	+	0.375002i	$0.122358\pi$
$602$	0	0
$603$	0	0
$604$	9.06955	0.369035
$605$	0	0
$606$	0	0
$607$	−40.8762	−1.65911	−0.829557	−	0.558422i	$-0.811407\pi$
$607$	−40.8762	−1.65911	−0.829557	+	0.558422i	$0.811407\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	23.5131	0.949685	0.474843	−	0.880071i	$-0.342505\pi$
$613$	23.5131	0.949685	0.474843	+	0.880071i	$0.342505\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	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
$628$	5.27093	0.210333
$629$	0	0
$630$	0	0
$631$	−16.2398	−0.646496	−0.323248	−	0.946314i	$-0.604775\pi$
$631$	−16.2398	−0.646496	−0.323248	+	0.946314i	$0.604775\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	125.041	4.95429
$638$	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
$643$	23.7573	0.936898	0.468449	−	0.883491i	$-0.344813\pi$
$643$	23.7573	0.936898	0.468449	+	0.883491i	$0.344813\pi$
$644$	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
$652$	−34.0000	−1.33154
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	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
$661$	−47.5562	−1.84972	−0.924862	−	0.380303i	$-0.875820\pi$
$661$	−47.5562	−1.84972	−0.924862	+	0.380303i	$0.875820\pi$
$662$	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
$673$	−51.7238	−1.99381	−0.996903	−	0.0786409i	$-0.974942\pi$
$673$	−51.7238	−1.99381	−0.996903	+	0.0786409i	$0.974942\pi$
$674$	0	0
$675$	0	0
$676$	−76.8248	−2.95480
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−55.4412	−2.12764
$680$	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
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−44.2877	−1.68845
$689$	0	0
$690$	0	0
$691$	10.2543	0.390094	0.195047	−	0.980794i	$-0.437514\pi$
$691$	10.2543	0.390094	0.195047	+	0.980794i	$0.437514\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	49.4356	1.86849
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−33.3961	−1.25956
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	37.2603	1.39934	0.699671	−	0.714466i	$-0.253330\pi$
$709$	37.2603	1.39934	0.699671	+	0.714466i	$0.253330\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−87.0515	−3.24197
$722$	0	0
$723$	0	0
$724$	44.1248	1.63989
$725$	0	0
$726$	0	0
$727$	−30.6833	−1.13798	−0.568991	−	0.822344i	$-0.692666\pi$
$727$	−30.6833	−1.13798	−0.568991	+	0.822344i	$0.692666\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−54.0934	−1.99798	−0.998992	−	0.0448796i	$-0.985710\pi$
$733$	−54.0934	−1.99798	−0.998992	+	0.0448796i	$0.985710\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−45.6569	−1.67952	−0.839759	−	0.542960i	$-0.817304\pi$
$739$	−45.6569	−1.67952	−0.839759	+	0.542960i	$0.817304\pi$
$740$	0	0
$741$	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
$751$	−28.7009	−1.04731	−0.523655	−	0.851930i	$-0.675432\pi$
$751$	−28.7009	−1.04731	−0.523655	+	0.851930i	$0.675432\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−41.0191	−1.49086	−0.745432	−	0.666581i	$-0.767757\pi$
$757$	−41.0191	−1.49086	−0.745432	+	0.666581i	$0.767757\pi$
$758$	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
$763$	84.0406	3.04247
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	47.0000	1.69486	0.847432	−	0.530904i	$-0.178148\pi$
$769$	47.0000	1.69486	0.847432	+	0.530904i	$0.178148\pi$
$770$	0	0
$771$	0	0
$772$	−15.1979	−0.546985
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−18.0793	−0.649429
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	69.7553	2.49126
$785$	0	0
$786$	0	0
$787$	13.1358	0.468240	0.234120	−	0.972208i	$-0.424779\pi$
$787$	13.1358	0.468240	0.234120	+	0.972208i	$0.424779\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	44.9665	1.59681
$794$	0	0
$795$	0	0
$796$	−34.0000	−1.20510
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	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
$811$	−49.0684	−1.72302	−0.861512	−	0.507736i	$-0.830482\pi$
$811$	−49.0684	−1.72302	−0.861512	+	0.507736i	$0.830482\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−43.1990	−1.51134
$818$	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
$823$	−32.9100	−1.14717	−0.573586	−	0.819146i	$-0.694448\pi$
$823$	−32.9100	−1.14717	−0.573586	+	0.819146i	$0.694448\pi$
$824$	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
$829$	−12.9712	−0.450509	−0.225255	−	0.974300i	$-0.572321\pi$
$829$	−12.9712	−0.450509	−0.225255	+	0.974300i	$0.572321\pi$
$830$	0	0
$831$	0	0
$832$	−57.3620	−1.98867
$833$	0	0
$834$	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
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	−42.2039	−1.45272
$845$	0	0
$846$	0	0
$847$	−54.3792	−1.86849
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−3.24865	−0.111232	−0.0556158	−	0.998452i	$-0.517712\pi$
$853$	−3.24865	−0.111232	−0.0556158	+	0.998452i	$0.517712\pi$
$854$	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
$859$	26.7817	0.913779	0.456889	−	0.889524i	$-0.348964\pi$
$859$	26.7817	0.913779	0.456889	+	0.889524i	$0.348964\pi$
$860$	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
$868$	−35.7505	−1.21345
$869$	0	0
$870$	0	0
$871$	4.53951	0.153815
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	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
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	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
$889$	67.7633	2.27271
$890$	0	0
$891$	0	0
$892$	−10.0000	−0.334825
$893$	0	0
$894$	0	0
$895$	0	0
$896$	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
$905$	0	0
$906$	0	0
$907$	−1.69492	−0.0562788	−0.0281394	−	0.999604i	$-0.508958\pi$
$907$	−1.69492	−0.0562788	−0.0281394	+	0.999604i	$0.508958\pi$
$908$	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
$916$	14.0000	0.462573
$917$	0	0
$918$	0	0
$919$	53.0000	1.74831	0.874154	−	0.485648i	$-0.161416\pi$
$919$	53.0000	1.74831	0.874154	+	0.485648i	$0.161416\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	42.7972	1.40716
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	68.0406	2.22994
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−15.7098	−0.513215	−0.256608	−	0.966516i	$-0.582605\pi$
$937$	−15.7098	−0.513215	−0.256608	+	0.966516i	$0.582605\pi$
$938$	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
$943$	0	0
$944$	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
$949$	121.894	3.95685
$950$	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
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−17.9255	−0.578242
$962$	0	0
$963$	0	0
$964$	−34.0000	−1.09507
$965$	0	0
$966$	0	0
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	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
$973$	−115.631	−3.70694
$974$	0	0
$975$	0	0
$976$	25.0850	0.802952
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−55.9519	−1.78007
$989$	0	0
$990$	0	0
$991$	17.0000	0.540023	0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000	0.540023	0.270011	+	0.962857i	$0.412973\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−47.7407	−1.51196	−0.755982	−	0.654593i	$-0.772840\pi$
$997$	−47.7407	−1.51196	−0.755982	+	0.654593i	$0.772840\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.v.1.3	yes	3
3.2	odd	2	CM	2601.2.a.v.1.3	yes	3
17.16	even	2		2601.2.a.u.1.1	✓	3
51.50	odd	2		2601.2.a.u.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2601.2.a.u.1.1	✓	3	17.16	even	2
2601.2.a.u.1.1	✓	3	51.50	odd	2
2601.2.a.v.1.3	yes	3	1.1	even	1	trivial
2601.2.a.v.1.3	yes	3	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} +4.94356 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} +4.94356 q^{7} +7.17024 q^{13} +4.00000 q^{16} +3.90167 q^{19} -5.00000 q^{25} -9.88713 q^{28} +3.61587 q^{31} -8.55943 q^{37} -11.0719 q^{43} +17.4388 q^{49} -14.3405 q^{52} +6.27126 q^{61} -8.00000 q^{64} +0.633103 q^{67} +17.0000 q^{73} -7.80335 q^{76} +17.0000 q^{79} +35.4466 q^{91} -11.2148 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{4}+O(q^{10}) \) 3 * q - 6 * q^4	\( 3 q - 6 q^{4} + 12 q^{16} - 15 q^{25} + 21 q^{49} - 24 q^{64} + 51 q^{73} + 51 q^{79} + 51 q^{91}+O(q^{100}) \) 3 * q - 6 * q^4 + 12 * q^16 - 15 * q^25 + 21 * q^49 - 24 * q^64 + 51 * q^73 + 51 * q^79 + 51 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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