LMFDB - Embedded newform 3960.1.x.g.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,1,Mod(109,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3960.109");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3960.x (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:	$1.97629745003$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(\sqrt{-6}, \sqrt{-55})$
Artin image:	$D_4:C_2$
Artin field:	Galois closure of 8.0.9032601600.3

Embedding invariants

Embedding label			109.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3960.109
Dual form			3960.1.x.g.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +2.00000 q^{7} +1.00000i q^{8} -1.00000 q^{10} +1.00000i q^{11} -2.00000i q^{14} +1.00000 q^{16} +1.00000i q^{20} +1.00000 q^{22} -1.00000 q^{25} -2.00000 q^{28} +2.00000 q^{31} -1.00000i q^{32} -2.00000i q^{35} +1.00000 q^{40} -1.00000i q^{44} +3.00000 q^{49} +1.00000i q^{50} +1.00000 q^{55} +2.00000i q^{56} +2.00000i q^{59} -2.00000i q^{62} -1.00000 q^{64} -2.00000 q^{70} -2.00000 q^{73} +2.00000i q^{77} -1.00000i q^{80} -2.00000i q^{83} -1.00000 q^{88} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{7} - 2 q^{10} + 2 q^{16} + 2 q^{22} - 2 q^{25} - 4 q^{28} + 4 q^{31} + 2 q^{40} + 6 q^{49} + 2 q^{55} - 2 q^{64} - 4 q^{70} - 4 q^{73} - 2 q^{88}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^7 - 2 * q^10 + 2 * q^16 + 2 * q^22 - 2 * q^25 - 4 * q^28 + 4 * q^31 + 2 * q^40 + 6 * q^49 + 2 * q^55 - 2 * q^64 - 4 * q^70 - 4 * q^73 - 2 * q^88

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times$.

$n$	$991$	$1981$	$2377$	$2521$	$3521$
$\chi(n)$	$1$	$-1$	$-1$	$-1$	$1$

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 $

$2$	− 1.00000i	− 1.00000i
$2$	− 1.00000i	− 1.00000i
$3$	0	0
$3$	0	0
$4$	−1.00000	−1.00000
$5$	− 1.00000i	− 1.00000i
$5$	− 1.00000i	− 1.00000i
$6$	0	0
$7$	2.00000	2.00000	1.00000			$0$
$7$	2.00000	2.00000	1.00000			$0$
$8$	1.00000i	1.00000i
$9$	0	0
$10$	−1.00000	−1.00000
$11$	1.00000i	1.00000i
$11$	1.00000i	1.00000i
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	− 2.00000i	− 2.00000i
$15$	0	0
$16$	1.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1.00000i	1.00000i
$21$	0	0
$22$	1.00000	1.00000
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−1.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	−2.00000	−2.00000
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	2.00000	2.00000	1.00000			$0$
$31$	2.00000	2.00000	1.00000			$0$
$32$	− 1.00000i	− 1.00000i
$33$	0	0
$34$	0	0
$35$	− 2.00000i	− 2.00000i
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	1.00000	1.00000
$41$	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			$\pi$
$44$	− 1.00000i	− 1.00000i
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	3.00000	3.00000
$50$	1.00000i	1.00000i
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	1.00000	1.00000
$56$	2.00000i	2.00000i
$57$	0	0
$58$	0	0
$59$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$59$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	− 2.00000i	− 2.00000i
$63$	0	0
$64$	−1.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	−2.00000	−2.00000
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−2.00000	−2.00000	−1.00000			$\pi$
$73$	−2.00000	−2.00000	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000i	2.00000i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	− 1.00000i	− 1.00000i
$81$	0	0
$82$	0	0
$83$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$83$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−1.00000	−1.00000
$89$	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
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 3.00000i	− 3.00000i
$99$	0	0
$100$	1.00000	1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$107$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	− 1.00000i	− 1.00000i
$111$	0	0
$112$	2.00000	2.00000
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	2.00000	2.00000
$119$	0	0
$120$	0	0
$121$	−1.00000	−1.00000
$122$	0	0
$123$	0	0
$124$	−2.00000	−2.00000
$125$	1.00000i	1.00000i
$126$	0	0
$127$	−2.00000	−2.00000	−1.00000			$\pi$
$127$	−2.00000	−2.00000	−1.00000			$\pi$
$128$	1.00000i	1.00000i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	2.00000i	2.00000i
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	2.00000i	2.00000i
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	2.00000	2.00000
$155$	− 2.00000i	− 2.00000i
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	−1.00000	−1.00000
$161$	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
$166$	−2.00000	−2.00000
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$173$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$174$	0	0
$175$	−2.00000	−2.00000
$176$	1.00000i	1.00000i
$177$	0	0
$178$	0	0
$179$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$179$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$180$	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		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−2.00000	−2.00000	−1.00000			$\pi$
$193$	−2.00000	−2.00000	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−3.00000	−3.00000
$197$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$197$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$198$	0	0
$199$	−2.00000	−2.00000	−1.00000			$\pi$
$199$	−2.00000	−2.00000	−1.00000			$\pi$
$200$	− 1.00000i	− 1.00000i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	−2.00000	−2.00000
$215$	0	0
$216$	0	0
$217$	4.00000	4.00000
$218$	0	0
$219$	0	0
$220$	−1.00000	−1.00000
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	− 2.00000i	− 2.00000i
$225$	0	0
$226$	0	0
$227$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$227$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$228$	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
$236$	− 2.00000i	− 2.00000i
$237$	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			$\pi$
$242$	1.00000i	1.00000i
$243$	0	0
$244$	0	0
$245$	− 3.00000i	− 3.00000i
$246$	0	0
$247$	0	0
$248$	2.00000i	2.00000i
$249$	0	0
$250$	1.00000	1.00000
$251$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$251$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$252$	0	0
$253$	0	0
$254$	2.00000i	2.00000i
$255$	0	0
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$269$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.00000i	− 1.00000i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	2.00000	2.00000
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	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
$292$	2.00000	2.00000
$293$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$293$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$294$	0	0
$295$	2.00000	2.00000
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 2.00000i	− 2.00000i
$309$	0	0
$310$	−2.00000	−2.00000
$311$	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			$\pi$
$314$	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
$320$	1.00000i	1.00000i
$321$	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
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	2.00000i	2.00000i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00000	2.00000	1.00000			$0$
$337$	2.00000	2.00000	1.00000			$0$
$338$	− 1.00000i	− 1.00000i
$339$	0	0
$340$	0	0
$341$	2.00000i	2.00000i
$342$	0	0
$343$	4.00000	4.00000
$344$	0	0
$345$	0	0
$346$	−2.00000	−2.00000
$347$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$347$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	2.00000i	2.00000i
$351$	0	0
$352$	1.00000	1.00000
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−2.00000	−2.00000
$359$	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
$365$	2.00000i	2.00000i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	2.00000	2.00000
$386$	2.00000i	2.00000i
$387$	0	0
$388$	0	0
$389$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$389$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	3.00000i	3.00000i
$393$	0	0
$394$	2.00000	2.00000
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	2.00000i	2.00000i
$399$	0	0
$400$	−1.00000	−1.00000
$401$	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
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000i	4.00000i
$414$	0	0
$415$	−2.00000	−2.00000
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$419$	2.00000i	2.00000i			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
$425$	0	0
$426$	0	0
$427$	0	0
$428$	2.00000i	2.00000i
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	− 4.00000i	− 4.00000i
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	1.00000i	1.00000i
$441$	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
$448$	−2.00000	−2.00000
$449$	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
$454$	2.00000	2.00000
$455$	0	0
$456$	0	0
$457$	−2.00000	−2.00000	−1.00000			$\pi$
$457$	−2.00000	−2.00000	−1.00000			$\pi$
$458$	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
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	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
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−2.00000	−2.00000
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	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	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−3.00000	−3.00000
$491$	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
$496$	2.00000	2.00000
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 1.00000i	− 1.00000i
$501$	0	0
$502$	−2.00000	−2.00000
$503$	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
$508$	2.00000	2.00000
$509$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$509$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$510$	0	0
$511$	−4.00000	−4.00000
$512$	− 1.00000i	− 1.00000i
$513$	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
$523$	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
$535$	−2.00000	−2.00000
$536$	0	0
$537$	0	0
$538$	2.00000	2.00000
$539$	3.00000i	3.00000i
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	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
$550$	−1.00000	−1.00000
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$557$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$558$	0	0
$559$	0	0
$560$	− 2.00000i	− 2.00000i
$561$	0	0
$562$	0	0
$563$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$563$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−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
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	1.00000i	1.00000i
$579$	0	0
$580$	0	0
$581$	− 4.00000i	− 4.00000i
$582$	0	0
$583$	0	0
$584$	− 2.00000i	− 2.00000i
$585$	0	0
$586$	2.00000	2.00000
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	− 2.00000i	− 2.00000i
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	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
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.00000i	1.00000i
$606$	0	0
$607$	2.00000	2.00000	1.00000			$0$
$607$	2.00000	2.00000	1.00000			$0$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	−2.00000	−2.00000
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	2.00000i	2.00000i
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	2.00000	2.00000	1.00000			$0$
$631$	2.00000	2.00000	1.00000			$0$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.00000i	2.00000i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	1.00000	1.00000
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	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
$649$	−2.00000	−2.00000
$650$	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
$655$	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
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	2.00000	2.00000
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−2.00000	−2.00000	−1.00000			$\pi$
$673$	−2.00000	−2.00000	−1.00000			$\pi$
$674$	− 2.00000i	− 2.00000i
$675$	0	0
$676$	−1.00000	−1.00000
$677$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$677$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	2.00000	2.00000
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	− 4.00000i	− 4.00000i
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	2.00000i	2.00000i
$693$	0	0
$694$	−2.00000	−2.00000
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	2.00000	2.00000
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	− 1.00000i	− 1.00000i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	2.00000i	2.00000i
$717$	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
$722$	1.00000i	1.00000i
$723$	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
$730$	2.00000	2.00000
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	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
$749$	− 4.00000i	− 4.00000i
$750$	0	0
$751$	2.00000	2.00000	1.00000			$0$
$751$	2.00000	2.00000	1.00000			$0$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	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			$\pi$
$770$	− 2.00000i	− 2.00000i
$771$	0	0
$772$	2.00000	2.00000
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−2.00000	−2.00000
$776$	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
$784$	3.00000	3.00000
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	− 2.00000i	− 2.00000i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	2.00000	2.00000
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	1.00000i	1.00000i
$801$	0	0
$802$	0	0
$803$	− 2.00000i	− 2.00000i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	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
$818$	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
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	4.00000	4.00000
$827$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$827$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	2.00000i	2.00000i
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	2.00000	2.00000
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−1.00000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.00000i	− 1.00000i
$846$	0	0
$847$	−2.00000	−2.00000
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	2.00000	2.00000
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	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
$865$	−2.00000	−2.00000
$866$	0	0
$867$	0	0
$868$	−4.00000	−4.00000
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.00000i	2.00000i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	1.00000	1.00000
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	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
$889$	−4.00000	−4.00000
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−2.00000	−2.00000
$896$	2.00000i	2.00000i
$897$	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
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	− 2.00000i	− 2.00000i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	2.00000	2.00000
$914$	2.00000i	2.00000i
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	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
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.00000	−2.00000	−1.00000			$\pi$
$937$	−2.00000	−2.00000	−1.00000			$\pi$
$938$	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
$944$	2.00000i	2.00000i
$945$	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
$953$	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
$961$	3.00000	3.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.00000i	2.00000i
$966$	0	0
$967$	−2.00000	−2.00000	−1.00000			$\pi$
$967$	−2.00000	−2.00000	−1.00000			$\pi$
$968$	− 1.00000i	− 1.00000i
$969$	0	0
$970$	0	0
$971$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$971$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	3.00000i	3.00000i
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	2.00000	2.00000
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−2.00000	−2.00000	−1.00000			$\pi$
$991$	−2.00000	−2.00000	−1.00000			$\pi$
$992$	− 2.00000i	− 2.00000i
$993$	0	0
$994$	0	0
$995$	2.00000i	2.00000i
$996$	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	3960.1.x.g.109.1	yes	2
3.2	odd	2	inner	3960.1.x.g.109.2	yes	2
5.4	even	2		3960.1.x.f.109.2	yes	2
8.5	even	2	inner	3960.1.x.g.109.2	yes	2
11.10	odd	2		3960.1.x.f.109.2	yes	2
15.14	odd	2		3960.1.x.f.109.1	✓	2
24.5	odd	2	CM	3960.1.x.g.109.1	yes	2
33.32	even	2		3960.1.x.f.109.1	✓	2
40.29	even	2		3960.1.x.f.109.1	✓	2
55.54	odd	2	CM	3960.1.x.g.109.1	yes	2
88.21	odd	2		3960.1.x.f.109.1	✓	2
120.29	odd	2		3960.1.x.f.109.2	yes	2
165.164	even	2	inner	3960.1.x.g.109.2	yes	2
264.197	even	2		3960.1.x.f.109.2	yes	2
440.109	odd	2	inner	3960.1.x.g.109.2	yes	2
1320.989	even	2	RM	3960.1.x.g.109.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.1.x.f.109.1	✓	2	15.14	odd	2
3960.1.x.f.109.1	✓	2	33.32	even	2
3960.1.x.f.109.1	✓	2	40.29	even	2
3960.1.x.f.109.1	✓	2	88.21	odd	2
3960.1.x.f.109.2	yes	2	5.4	even	2
3960.1.x.f.109.2	yes	2	11.10	odd	2
3960.1.x.f.109.2	yes	2	120.29	odd	2
3960.1.x.f.109.2	yes	2	264.197	even	2
3960.1.x.g.109.1	yes	2	1.1	even	1	trivial
3960.1.x.g.109.1	yes	2	24.5	odd	2	CM
3960.1.x.g.109.1	yes	2	55.54	odd	2	CM
3960.1.x.g.109.1	yes	2	1320.989	even	2	RM
3960.1.x.g.109.2	yes	2	3.2	odd	2	inner
3960.1.x.g.109.2	yes	2	8.5	even	2	inner
3960.1.x.g.109.2	yes	2	165.164	even	2	inner
3960.1.x.g.109.2	yes	2	440.109	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:	\( N \)	\(=\)	\( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3960.x (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +2.00000 q^{7} +1.00000i q^{8} -1.00000 q^{10} +1.00000i q^{11} -2.00000i q^{14} +1.00000 q^{16} +1.00000i q^{20} +1.00000 q^{22} -1.00000 q^{25} -2.00000 q^{28} +2.00000 q^{31} -1.00000i q^{32} -2.00000i q^{35} +1.00000 q^{40} -1.00000i q^{44} +3.00000 q^{49} +1.00000i q^{50} +1.00000 q^{55} +2.00000i q^{56} +2.00000i q^{59} -2.00000i q^{62} -1.00000 q^{64} -2.00000 q^{70} -2.00000 q^{73} +2.00000i q^{77} -1.00000i q^{80} -2.00000i q^{83} -1.00000 q^{88} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{7} - 2 q^{10} + 2 q^{16} + 2 q^{22} - 2 q^{25} - 4 q^{28} + 4 q^{31} + 2 q^{40} + 6 q^{49} + 2 q^{55} - 2 q^{64} - 4 q^{70} - 4 q^{73} - 2 q^{88}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^7 - 2 * q^10 + 2 * q^16 + 2 * q^22 - 2 * q^25 - 4 * q^28 + 4 * q^31 + 2 * q^40 + 6 * q^49 + 2 * q^55 - 2 * q^64 - 4 * q^70 - 4 * q^73 - 2 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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