LMFDB - Embedded newform 612.1.e.a.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [612,1,Mod(271,612)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(612, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("612.271");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$612 = 2^{2} \cdot 3^{2} \cdot 17$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	612.e (of order $2$ , degree $1$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.305427787731$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 68)
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(i, \sqrt{17})$
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.2448.1
Stark unit:	Root of $x^{4} - 640x^{3} - 1026x^{2} - 640x + 1$

Embedding invariants

Embedding label			271.1
Character	$\chi$	$=$	612.271

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -2.00000 q^{13} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{32} -1.00000 q^{34} -1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{64} -1.00000 q^{68} +2.00000 q^{89} -1.00000 q^{98} +O(q^{100})

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/612\mathbb{Z}\right)^\times$ .

$n$	$37$	$137$	$307$
$\chi(n)$	$-1$	$1$	$-1$

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.00000	1.00000
$2$	1.00000	1.00000
$3$	0	0
$3$	0	0
$4$	1.00000	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	1.00000
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−2.00000	−2.00000	−1.00000			$\pi$
$13$	−2.00000	−2.00000	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	1.00000
$17$	−1.00000	−1.00000
$17$	−1.00000	−1.00000
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	1.00000
$26$	−2.00000	−2.00000
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	1.00000
$33$	0	0
$34$	−1.00000	−1.00000
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−1.00000	−1.00000
$50$	1.00000	1.00000
$51$	0	0
$52$	−2.00000	−2.00000
$53$	−2.00000	−2.00000	−1.00000			$\pi$
$53$	−2.00000	−2.00000	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.00000	−1.00000
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	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		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.00000	2.00000	1.00000			$0$
$89$	2.00000	2.00000	1.00000			$0$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.00000	−1.00000
$99$	0	0
$100$	1.00000	1.00000
$101$	2.00000	2.00000	1.00000			$0$
$101$	2.00000	2.00000	1.00000			$0$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−2.00000	−2.00000
$105$	0	0
$106$	−2.00000	−2.00000
$107$	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
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000	1.00000
$129$	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
$136$	−1.00000	−1.00000
$137$	2.00000	2.00000	1.00000			$0$
$137$	2.00000	2.00000	1.00000			$0$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.00000	−2.00000	−1.00000			$\pi$
$149$	−2.00000	−2.00000	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	2.00000	1.00000			$0$
$157$	2.00000	2.00000	1.00000			$0$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	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
$169$	3.00000	3.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	2.00000	2.00000
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−1.00000	−1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	1.00000	1.00000
$201$	0	0
$202$	2.00000	2.00000
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−2.00000	−2.00000
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−2.00000	−2.00000
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.00000	2.00000
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−2.00000	−2.00000	−1.00000			$\pi$
$229$	−2.00000	−2.00000	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−1.00000	−1.00000
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	2.00000	2.00000	1.00000			$0$
$257$	2.00000	2.00000	1.00000			$0$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−1.00000	−1.00000
$273$	0	0
$274$	2.00000	2.00000
$275$	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
$281$	−2.00000	−2.00000	−1.00000			$\pi$
$281$	−2.00000	−2.00000	−1.00000			$\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.00000	−2.00000	−1.00000			$\pi$
$293$	−2.00000	−2.00000	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−2.00000	−2.00000
$299$	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
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	2.00000	2.00000
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−2.00000	−2.00000
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	3.00000	3.00000
$339$	0	0
$340$	0	0
$341$	0	0
$342$	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
$349$	2.00000	2.00000	1.00000			$0$
$349$	2.00000	2.00000	1.00000			$0$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.00000	−2.00000	−1.00000			$\pi$
$353$	−2.00000	−2.00000	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	2.00000	2.00000
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.00000	−2.00000	−1.00000			$\pi$
$373$	−2.00000	−2.00000	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.00000	2.00000	1.00000			$0$
$389$	2.00000	2.00000	1.00000			$0$
$390$	0	0
$391$	0	0
$392$	−1.00000	−1.00000
$393$	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
$400$	1.00000	1.00000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	2.00000	2.00000
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.00000	2.00000	1.00000			$0$
$409$	2.00000	2.00000	1.00000			$0$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.00000	−2.00000
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−2.00000	−2.00000	−1.00000			$\pi$
$421$	−2.00000	−2.00000	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	−2.00000	−2.00000
$425$	−1.00000	−1.00000
$426$	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
$433$	−2.00000	−2.00000	−1.00000			$\pi$
$433$	−2.00000	−2.00000	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	2.00000	2.00000
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.00000	−2.00000	−1.00000			$\pi$
$457$	−2.00000	−2.00000	−1.00000			$\pi$
$458$	−2.00000	−2.00000
$459$	0	0
$460$	0	0
$461$	−2.00000	−2.00000	−1.00000			$\pi$
$461$	−2.00000	−2.00000	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	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
$484$	−1.00000	−1.00000
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	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
$508$	0	0
$509$	−2.00000	−2.00000	−1.00000			$\pi$
$509$	−2.00000	−2.00000	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	1.00000
$513$	0	0
$514$	2.00000	2.00000
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	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
$529$	−1.00000	−1.00000
$530$	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
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−1.00000	−1.00000
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	2.00000	2.00000
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.00000	2.00000	1.00000			$0$
$557$	2.00000	2.00000	1.00000			$0$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−2.00000	−2.00000
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.00000	−2.00000	−1.00000			$\pi$
$569$	−2.00000	−2.00000	−1.00000			$\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.00000	−2.00000	−1.00000			$\pi$
$577$	−2.00000	−2.00000	−1.00000			$\pi$
$578$	1.00000	1.00000
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−2.00000	−2.00000
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.00000	−2.00000	−1.00000			$\pi$
$593$	−2.00000	−2.00000	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	−2.00000	−2.00000
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.00000	2.00000	1.00000			$0$
$613$	2.00000	2.00000	1.00000			$0$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	0	0
$627$	0	0
$628$	2.00000	2.00000
$629$	0	0
$630$	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
$637$	2.00000	2.00000
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	−2.00000	−2.00000
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	2.00000	2.00000	1.00000			$0$
$661$	2.00000	2.00000	1.00000			$0$
$662$	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
$676$	3.00000	3.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	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
$688$	0	0
$689$	4.00000	4.00000
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	2.00000	2.00000
$699$	0	0
$700$	0	0
$701$	2.00000	2.00000	1.00000			$0$
$701$	2.00000	2.00000	1.00000			$0$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−2.00000	−2.00000
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	2.00000	2.00000
$713$	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
$721$	0	0
$722$	1.00000	1.00000
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	2.00000	2.00000	1.00000			$0$
$733$	2.00000	2.00000	1.00000			$0$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	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
$746$	−2.00000	−2.00000
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.00000	−2.00000	−1.00000			$\pi$
$757$	−2.00000	−2.00000	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.00000	2.00000	1.00000			$0$
$761$	2.00000	2.00000	1.00000			$0$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−2.00000	−2.00000	−1.00000			$\pi$
$769$	−2.00000	−2.00000	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.00000	2.00000	1.00000			$0$
$773$	2.00000	2.00000	1.00000			$0$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	2.00000	2.00000
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1.00000	−1.00000
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.00000	−2.00000	−1.00000			$\pi$
$797$	−2.00000	−2.00000	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	1.00000
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	2.00000	2.00000
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	2.00000	2.00000
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	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
$829$	2.00000	2.00000	1.00000			$0$
$829$	2.00000	2.00000	1.00000			$0$
$830$	0	0
$831$	0	0
$832$	−2.00000	−2.00000
$833$	1.00000	1.00000
$834$	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
$841$	1.00000	1.00000
$842$	−2.00000	−2.00000
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−2.00000	−2.00000
$849$	0	0
$850$	−1.00000	−1.00000
$851$	0	0
$852$	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	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−2.00000
$867$	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
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	2.00000	2.00000
$885$	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
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	2.00000	2.00000
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	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
$914$	−2.00000	−2.00000
$915$	0	0
$916$	−2.00000	−2.00000
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	−2.00000	−2.00000
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.00000	2.00000	1.00000			$0$
$937$	2.00000	2.00000	1.00000			$0$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	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
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.00000	2.00000	1.00000			$0$
$953$	2.00000	2.00000	1.00000			$0$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.00000	−1.00000
$962$	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			$\pi$
$968$	−1.00000	−1.00000
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.00000	−2.00000	−1.00000			$\pi$
$977$	−2.00000	−2.00000	−1.00000			$\pi$
$978$	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
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	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

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	612.1.e.a.271.1		1
3.2	odd	2		68.1.d.a.67.1	✓	1
4.3	odd	2	CM	612.1.e.a.271.1		1
12.11	even	2		68.1.d.a.67.1	✓	1
15.2	even	4		1700.1.d.b.1699.1		2
15.8	even	4		1700.1.d.b.1699.2		2
15.14	odd	2		1700.1.h.d.951.1		1
17.16	even	2	RM	612.1.e.a.271.1		1
21.2	odd	6		3332.1.o.c.67.1		2
21.5	even	6		3332.1.o.d.67.1		2
21.11	odd	6		3332.1.o.c.2039.1		2
21.17	even	6		3332.1.o.d.2039.1		2
21.20	even	2		3332.1.g.a.883.1		1
24.5	odd	2		1088.1.g.a.1087.1		1
24.11	even	2		1088.1.g.a.1087.1		1
51.2	odd	8		1156.1.f.a.251.1		2
51.5	even	16		1156.1.g.a.179.1		4
51.8	odd	8		1156.1.f.a.327.1		2
51.11	even	16		1156.1.g.a.423.1		4
51.14	even	16		1156.1.g.a.399.1		4
51.20	even	16		1156.1.g.a.399.1		4
51.23	even	16		1156.1.g.a.423.1		4
51.26	odd	8		1156.1.f.a.327.1		2
51.29	even	16		1156.1.g.a.179.1		4
51.32	odd	8		1156.1.f.a.251.1		2
51.38	odd	4		1156.1.c.a.579.1		1
51.41	even	16		1156.1.g.a.155.1		4
51.44	even	16		1156.1.g.a.155.1		4
51.47	odd	4		1156.1.c.a.579.1		1
51.50	odd	2		68.1.d.a.67.1	✓	1
60.23	odd	4		1700.1.d.b.1699.2		2
60.47	odd	4		1700.1.d.b.1699.1		2
60.59	even	2		1700.1.h.d.951.1		1
68.67	odd	2	CM	612.1.e.a.271.1		1
84.11	even	6		3332.1.o.c.2039.1		2
84.23	even	6		3332.1.o.c.67.1		2
84.47	odd	6		3332.1.o.d.67.1		2
84.59	odd	6		3332.1.o.d.2039.1		2
84.83	odd	2		3332.1.g.a.883.1		1
204.11	odd	16		1156.1.g.a.423.1		4
204.23	odd	16		1156.1.g.a.423.1		4
204.47	even	4		1156.1.c.a.579.1		1
204.59	even	8		1156.1.f.a.327.1		2
204.71	odd	16		1156.1.g.a.399.1		4
204.83	even	8		1156.1.f.a.251.1		2
204.95	odd	16		1156.1.g.a.155.1		4
204.107	odd	16		1156.1.g.a.179.1		4
204.131	odd	16		1156.1.g.a.179.1		4
204.143	odd	16		1156.1.g.a.155.1		4
204.155	even	8		1156.1.f.a.251.1		2
204.167	odd	16		1156.1.g.a.399.1		4
204.179	even	8		1156.1.f.a.327.1		2
204.191	even	4		1156.1.c.a.579.1		1
204.203	even	2		68.1.d.a.67.1	✓	1
255.152	even	4		1700.1.d.b.1699.1		2
255.203	even	4		1700.1.d.b.1699.2		2
255.254	odd	2		1700.1.h.d.951.1		1
357.101	even	6		3332.1.o.d.2039.1		2
357.152	even	6		3332.1.o.d.67.1		2
357.254	odd	6		3332.1.o.c.67.1		2
357.305	odd	6		3332.1.o.c.2039.1		2
357.356	even	2		3332.1.g.a.883.1		1
408.101	odd	2		1088.1.g.a.1087.1		1
408.203	even	2		1088.1.g.a.1087.1		1
1020.203	odd	4		1700.1.d.b.1699.2		2
1020.407	odd	4		1700.1.d.b.1699.1		2
1020.1019	even	2		1700.1.h.d.951.1		1
1428.611	even	6		3332.1.o.c.67.1		2
1428.815	odd	6		3332.1.o.d.2039.1		2
1428.1019	even	6		3332.1.o.c.2039.1		2
1428.1223	odd	6		3332.1.o.d.67.1		2
1428.1427	odd	2		3332.1.g.a.883.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.1.d.a.67.1	✓	1	3.2	odd	2
68.1.d.a.67.1	✓	1	12.11	even	2
68.1.d.a.67.1	✓	1	51.50	odd	2
68.1.d.a.67.1	✓	1	204.203	even	2
612.1.e.a.271.1		1	1.1	even	1	trivial
612.1.e.a.271.1		1	4.3	odd	2	CM
612.1.e.a.271.1		1	17.16	even	2	RM
612.1.e.a.271.1		1	68.67	odd	2	CM
1088.1.g.a.1087.1		1	24.5	odd	2
1088.1.g.a.1087.1		1	24.11	even	2
1088.1.g.a.1087.1		1	408.101	odd	2
1088.1.g.a.1087.1		1	408.203	even	2
1156.1.c.a.579.1		1	51.38	odd	4
1156.1.c.a.579.1		1	51.47	odd	4
1156.1.c.a.579.1		1	204.47	even	4
1156.1.c.a.579.1		1	204.191	even	4
1156.1.f.a.251.1		2	51.2	odd	8
1156.1.f.a.251.1		2	51.32	odd	8
1156.1.f.a.251.1		2	204.83	even	8
1156.1.f.a.251.1		2	204.155	even	8
1156.1.f.a.327.1		2	51.8	odd	8
1156.1.f.a.327.1		2	51.26	odd	8
1156.1.f.a.327.1		2	204.59	even	8
1156.1.f.a.327.1		2	204.179	even	8
1156.1.g.a.155.1		4	51.41	even	16
1156.1.g.a.155.1		4	51.44	even	16
1156.1.g.a.155.1		4	204.95	odd	16
1156.1.g.a.155.1		4	204.143	odd	16
1156.1.g.a.179.1		4	51.5	even	16
1156.1.g.a.179.1		4	51.29	even	16
1156.1.g.a.179.1		4	204.107	odd	16
1156.1.g.a.179.1		4	204.131	odd	16
1156.1.g.a.399.1		4	51.14	even	16
1156.1.g.a.399.1		4	51.20	even	16
1156.1.g.a.399.1		4	204.71	odd	16
1156.1.g.a.399.1		4	204.167	odd	16
1156.1.g.a.423.1		4	51.11	even	16
1156.1.g.a.423.1		4	51.23	even	16
1156.1.g.a.423.1		4	204.11	odd	16
1156.1.g.a.423.1		4	204.23	odd	16
1700.1.d.b.1699.1		2	15.2	even	4
1700.1.d.b.1699.1		2	60.47	odd	4
1700.1.d.b.1699.1		2	255.152	even	4
1700.1.d.b.1699.1		2	1020.407	odd	4
1700.1.d.b.1699.2		2	15.8	even	4
1700.1.d.b.1699.2		2	60.23	odd	4
1700.1.d.b.1699.2		2	255.203	even	4
1700.1.d.b.1699.2		2	1020.203	odd	4
1700.1.h.d.951.1		1	15.14	odd	2
1700.1.h.d.951.1		1	60.59	even	2
1700.1.h.d.951.1		1	255.254	odd	2
1700.1.h.d.951.1		1	1020.1019	even	2
3332.1.g.a.883.1		1	21.20	even	2
3332.1.g.a.883.1		1	84.83	odd	2
3332.1.g.a.883.1		1	357.356	even	2
3332.1.g.a.883.1		1	1428.1427	odd	2
3332.1.o.c.67.1		2	21.2	odd	6
3332.1.o.c.67.1		2	84.23	even	6
3332.1.o.c.67.1		2	357.254	odd	6
3332.1.o.c.67.1		2	1428.611	even	6
3332.1.o.c.2039.1		2	21.11	odd	6
3332.1.o.c.2039.1		2	84.11	even	6
3332.1.o.c.2039.1		2	357.305	odd	6
3332.1.o.c.2039.1		2	1428.1019	even	6
3332.1.o.d.67.1		2	21.5	even	6
3332.1.o.d.67.1		2	84.47	odd	6
3332.1.o.d.67.1		2	357.152	even	6
3332.1.o.d.67.1		2	1428.1223	odd	6
3332.1.o.d.2039.1		2	21.17	even	6
3332.1.o.d.2039.1		2	84.59	odd	6
3332.1.o.d.2039.1		2	357.101	even	6
3332.1.o.d.2039.1		2	1428.815	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	612.1.e.a.271.1

Level	$612$
Weight	$1$
Character	612.271
Self dual	yes
Analytic conductor	$0.305$
Analytic rank	$0$
Dimension	$1$
Projective image	$D_{2}$
CM/RM discs	-4, -68, 17
Inner twists	$4$