LMFDB - Embedded newform 2280.1.t.f.1139.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2280,1,Mod(1139,2280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2280.1139"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2280.t (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,2,0,0,-2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$1.13786822880$
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 $\Q(\sqrt{-15}, \sqrt{-114})$
Artin image:	$D_4:C_2$
Artin field:	Galois closure of 8.0.1169640000.6

Embedding invariants

Embedding label			1139.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2280.1139
Dual form			2280.1.t.f.1139.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{10} +1.00000i q^{12} +1.00000 q^{15} +1.00000 q^{16} -1.00000i q^{18} +1.00000 q^{19} -1.00000i q^{20} +2.00000i q^{23} -1.00000 q^{24} -1.00000 q^{25} +1.00000i q^{27} +1.00000i q^{30} +2.00000 q^{31} +1.00000i q^{32} +1.00000 q^{36} +1.00000i q^{38} +1.00000 q^{40} -1.00000i q^{45} -2.00000 q^{46} +2.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} -1.00000i q^{50} -1.00000 q^{54} -1.00000i q^{57} -1.00000 q^{60} +2.00000i q^{62} -1.00000 q^{64} +2.00000 q^{69} +1.00000i q^{72} +1.00000i q^{75} -1.00000 q^{76} +2.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +1.00000 q^{90} -2.00000i q^{92} -2.00000i q^{93} -2.00000 q^{94} +1.00000i q^{95} +1.00000 q^{96} -1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{10} + 2 q^{15} + 2 q^{16} + 2 q^{19} - 2 q^{24} - 2 q^{25} + 4 q^{31} + 2 q^{36} + 2 q^{40} - 4 q^{46} - 2 q^{49} - 2 q^{54} - 2 q^{60} - 2 q^{64} + 4 q^{69} - 2 q^{76}+ \cdots + 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^10 + 2 * q^15 + 2 * q^16 + 2 * q^19 - 2 * q^24 - 2 * q^25 + 4 * q^31 + 2 * q^36 + 2 * q^40 - 4 * q^46 - 2 * q^49 - 2 * q^54 - 2 * q^60 - 2 * q^64 + 4 * q^69 - 2 * q^76 + 4 * q^79 + 2 * q^81 + 2 * q^90 - 4 * q^94 + 2 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2280\mathbb{Z}\right)^\times$.

$n$	$457$	$761$	$1141$	$1711$	$1921$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$	$-1$

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
$3$	− 1.00000i	− 1.00000i
$3$	− 1.00000i	− 1.00000i
$4$	−1.00000	−1.00000
$5$	1.00000i	1.00000i
$5$	1.00000i	1.00000i
$6$	1.00000	1.00000
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	− 1.00000i	− 1.00000i
$9$	−1.00000	−1.00000
$10$	−1.00000	−1.00000
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.00000i	1.00000i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	1.00000	1.00000
$16$	1.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 1.00000i	− 1.00000i
$19$	1.00000	1.00000
$19$	1.00000	1.00000
$20$	− 1.00000i	− 1.00000i
$21$	0	0
$22$	0	0
$23$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$23$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$24$	−1.00000	−1.00000
$25$	−1.00000	−1.00000
$26$	0	0
$27$	1.00000i	1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	1.00000i	1.00000i
$31$	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
$35$	0	0
$36$	1.00000	1.00000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	1.00000i	1.00000i
$39$	0	0
$40$	1.00000	1.00000
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	− 1.00000i	− 1.00000i
$46$	−2.00000	−2.00000
$47$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$47$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$48$	− 1.00000i	− 1.00000i
$49$	−1.00000	−1.00000
$50$	− 1.00000i	− 1.00000i
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−1.00000	−1.00000
$55$	0	0
$56$	0	0
$57$	− 1.00000i	− 1.00000i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−1.00000	−1.00000
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	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
$69$	2.00000	2.00000
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.00000i	1.00000i
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	1.00000i	1.00000i
$76$	−1.00000	−1.00000
$77$	0	0
$78$	0	0
$79$	2.00000	2.00000	1.00000			$0$
$79$	2.00000	2.00000	1.00000			$0$
$80$	1.00000i	1.00000i
$81$	1.00000	1.00000
$82$	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	$0.5\pi$
$90$	1.00000	1.00000
$91$	0	0
$92$	− 2.00000i	− 2.00000i
$93$	− 2.00000i	− 2.00000i
$94$	−2.00000	−2.00000
$95$	1.00000i	1.00000i
$96$	1.00000	1.00000
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	− 1.00000i	− 1.00000i
$99$	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	$-0.5\pi$
$108$	− 1.00000i	− 1.00000i
$109$	−2.00000	−2.00000	−1.00000			$\pi$
$109$	−2.00000	−2.00000	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$113$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$114$	1.00000	1.00000
$115$	−2.00000	−2.00000
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	− 1.00000i	− 1.00000i
$121$	1.00000	1.00000
$122$	0	0
$123$	0	0
$124$	−2.00000	−2.00000
$125$	− 1.00000i	− 1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 1.00000i	− 1.00000i
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−1.00000
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	2.00000i	2.00000i
$139$	2.00000	2.00000	1.00000			$0$
$139$	2.00000	2.00000	1.00000			$0$
$140$	0	0
$141$	2.00000	2.00000
$142$	0	0
$143$	0	0
$144$	−1.00000	−1.00000
$145$	0	0
$146$	0	0
$147$	1.00000i	1.00000i
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	−1.00000	−1.00000
$151$	−2.00000	−2.00000	−1.00000			$\pi$
$151$	−2.00000	−2.00000	−1.00000			$\pi$
$152$	− 1.00000i	− 1.00000i
$153$	0	0
$154$	0	0
$155$	2.00000i	2.00000i
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	2.00000i	2.00000i
$159$	0	0
$160$	−1.00000	−1.00000
$161$	0	0
$162$	1.00000i	1.00000i
$163$	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
$169$	1.00000	1.00000
$170$	0	0
$171$	−1.00000	−1.00000
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	1.00000i	1.00000i
$181$	−2.00000	−2.00000	−1.00000			$\pi$
$181$	−2.00000	−2.00000	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	2.00000	2.00000
$185$	0	0
$186$	2.00000	2.00000
$187$	0	0
$188$	− 2.00000i	− 2.00000i
$189$	0	0
$190$	−1.00000	−1.00000
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000i	1.00000i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	1.00000	1.00000
$197$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$197$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−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
$207$	− 2.00000i	− 2.00000i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	2.00000	2.00000
$215$	0	0
$216$	1.00000	1.00000
$217$	0	0
$218$	− 2.00000i	− 2.00000i
$219$	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
$225$	1.00000	1.00000
$226$	−2.00000	−2.00000
$227$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$227$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$228$	1.00000i	1.00000i
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	− 2.00000i	− 2.00000i
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−2.00000	−2.00000
$236$	0	0
$237$	− 2.00000i	− 2.00000i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	1.00000	1.00000
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.00000i	1.00000i
$243$	− 1.00000i	− 1.00000i
$244$	0	0
$245$	− 1.00000i	− 1.00000i
$246$	0	0
$247$	0	0
$248$	− 2.00000i	− 2.00000i
$249$	0	0
$250$	1.00000	1.00000
$251$	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
$257$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$257$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$263$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$264$	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			$\pi$
$270$	− 1.00000i	− 1.00000i
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−2.00000	−2.00000
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	2.00000i	2.00000i
$279$	−2.00000	−2.00000
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	2.00000i	2.00000i
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	1.00000	1.00000
$286$	0	0
$287$	0	0
$288$	− 1.00000i	− 1.00000i
$289$	−1.00000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	−1.00000	−1.00000
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	− 1.00000i	− 1.00000i
$301$	0	0
$302$	− 2.00000i	− 2.00000i
$303$	0	0
$304$	1.00000	1.00000
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	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
$316$	−2.00000	−2.00000
$317$	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
$321$	−2.00000	−2.00000
$322$	0	0
$323$	0	0
$324$	−1.00000	−1.00000
$325$	0	0
$326$	0	0
$327$	2.00000i	2.00000i
$328$	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		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	1.00000i	1.00000i
$339$	2.00000	2.00000
$340$	0	0
$341$	0	0
$342$	− 1.00000i	− 1.00000i
$343$	0	0
$344$	0	0
$345$	2.00000i	2.00000i
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	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	$0.5\pi$
$360$	−1.00000	−1.00000
$361$	1.00000	1.00000
$362$	− 2.00000i	− 2.00000i
$363$	− 1.00000i	− 1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	2.00000i	2.00000i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	2.00000i	2.00000i
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−1.00000	−1.00000
$376$	2.00000	2.00000
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	− 1.00000i	− 1.00000i
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−1.00000	−1.00000
$385$	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
$391$	0	0
$392$	1.00000i	1.00000i
$393$	0	0
$394$	2.00000	2.00000
$395$	2.00000i	2.00000i
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	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
$405$	1.00000i	1.00000i
$406$	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
$413$	0	0
$414$	2.00000	2.00000
$415$	0	0
$416$	0	0
$417$	− 2.00000i	− 2.00000i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	2.00000	2.00000	1.00000			$0$
$421$	2.00000	2.00000	1.00000			$0$
$422$	0	0
$423$	− 2.00000i	− 2.00000i
$424$	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			$\pi$
$432$	1.00000i	1.00000i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	2.00000
$437$	2.00000i	2.00000i
$438$	0	0
$439$	−2.00000	−2.00000	−1.00000			$\pi$
$439$	−2.00000	−2.00000	−1.00000			$\pi$
$440$	0	0
$441$	1.00000	1.00000
$442$	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	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	1.00000i	1.00000i
$451$	0	0
$452$	− 2.00000i	− 2.00000i
$453$	2.00000i	2.00000i
$454$	2.00000	2.00000
$455$	0	0
$456$	−1.00000	−1.00000
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	2.00000	2.00000
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	2.00000	2.00000
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	− 2.00000i	− 2.00000i
$471$	0	0
$472$	0	0
$473$	0	0
$474$	2.00000	2.00000
$475$	−1.00000	−1.00000
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	1.00000i	1.00000i
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−1.00000	−1.00000
$485$	0	0
$486$	1.00000	1.00000
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	1.00000	1.00000
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	2.00000	2.00000
$497$	0	0
$498$	0	0
$499$	−2.00000	−2.00000	−1.00000			$\pi$
$499$	−2.00000	−2.00000	−1.00000			$\pi$
$500$	1.00000i	1.00000i
$501$	0	0
$502$	0	0
$503$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$503$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.00000i	− 1.00000i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	1.00000i
$513$	1.00000i	1.00000i
$514$	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		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	2.00000	2.00000
$527$	0	0
$528$	0	0
$529$	−3.00000	−3.00000
$530$	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
$538$	0	0
$539$	0	0
$540$	1.00000	1.00000
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	2.00000i	2.00000i
$544$	0	0
$545$	− 2.00000i	− 2.00000i
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	− 2.00000i	− 2.00000i
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−2.00000	−2.00000
$557$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$557$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$558$	− 2.00000i	− 2.00000i
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$563$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$564$	−2.00000	−2.00000
$565$	−2.00000	−2.00000
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	1.00000i	1.00000i
$571$	2.00000	2.00000	1.00000			$0$
$571$	2.00000	2.00000	1.00000			$0$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 2.00000i	− 2.00000i
$576$	1.00000	1.00000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 1.00000i	− 1.00000i
$579$	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	$0.5\pi$
$588$	− 1.00000i	− 1.00000i
$589$	2.00000	2.00000
$590$	0	0
$591$	−2.00000	−2.00000
$592$	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	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	1.00000	1.00000
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	2.00000	2.00000
$605$	1.00000i	1.00000i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.00000i	1.00000i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	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
$619$	2.00000	2.00000	1.00000			$0$
$619$	2.00000	2.00000	1.00000			$0$
$620$	− 2.00000i	− 2.00000i
$621$	−2.00000	−2.00000
$622$	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
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	− 2.00000i	− 2.00000i
$633$	0	0
$634$	0	0
$635$	0	0
$636$	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	$0.5\pi$
$642$	− 2.00000i	− 2.00000i
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$647$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$648$	− 1.00000i	− 1.00000i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$653$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$654$	−2.00000	−2.00000
$655$	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
$661$	−2.00000	−2.00000	−1.00000			$\pi$
$661$	−2.00000	−2.00000	−1.00000			$\pi$
$662$	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
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	− 1.00000i	− 1.00000i
$676$	−1.00000	−1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	2.00000i	2.00000i
$679$	0	0
$680$	0	0
$681$	−2.00000	−2.00000
$682$	0	0
$683$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$683$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$684$	1.00000	1.00000
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	−2.00000	−2.00000
$691$	2.00000	2.00000	1.00000			$0$
$691$	2.00000	2.00000	1.00000			$0$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.00000i	2.00000i
$696$	0	0
$697$	0	0
$698$	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
$705$	2.00000i	2.00000i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−2.00000	−2.00000
$712$	0	0
$713$	4.00000i	4.00000i
$714$	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	$0.5\pi$
$720$	− 1.00000i	− 1.00000i
$721$	0	0
$722$	1.00000i	1.00000i
$723$	0	0
$724$	2.00000	2.00000
$725$	0	0
$726$	1.00000	1.00000
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−1.00000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	−1.00000	−1.00000
$736$	−2.00000	−2.00000
$737$	0	0
$738$	0	0
$739$	2.00000	2.00000	1.00000			$0$
$739$	2.00000	2.00000	1.00000			$0$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−2.00000	−2.00000
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	− 1.00000i	− 1.00000i
$751$	2.00000	2.00000	1.00000			$0$
$751$	2.00000	2.00000	1.00000			$0$
$752$	2.00000i	2.00000i
$753$	0	0
$754$	0	0
$755$	− 2.00000i	− 2.00000i
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	1.00000	1.00000
$761$	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
$768$	− 1.00000i	− 1.00000i
$769$	−2.00000	−2.00000	−1.00000			$\pi$
$769$	−2.00000	−2.00000	−1.00000			$\pi$
$770$	0	0
$771$	−2.00000	−2.00000
$772$	0	0
$773$	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
$778$	0	0
$779$	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	$0.5\pi$
$788$	2.00000i	2.00000i
$789$	−2.00000	−2.00000
$790$	−2.00000	−2.00000
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	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
$803$	0	0
$804$	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			$\pi$
$810$	−1.00000	−1.00000
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	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		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$827$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$828$	2.00000i	2.00000i
$829$	−2.00000	−2.00000	−1.00000			$\pi$
$829$	−2.00000	−2.00000	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	2.00000	2.00000
$835$	0	0
$836$	0	0
$837$	2.00000i	2.00000i
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	2.00000i	2.00000i
$843$	0	0
$844$	0	0
$845$	1.00000i	1.00000i
$846$	2.00000	2.00000
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	− 1.00000i	− 1.00000i
$856$	−2.00000	−2.00000
$857$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$857$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$858$	0	0
$859$	−2.00000	−2.00000	−1.00000			$\pi$
$859$	−2.00000	−2.00000	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−1.00000	−1.00000
$865$	0	0
$866$	0	0
$867$	1.00000i	1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	2.00000i	2.00000i
$873$	0	0
$874$	−2.00000	−2.00000
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	− 2.00000i	− 2.00000i
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.00000i	1.00000i
$883$	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
$893$	2.00000i	2.00000i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−1.00000	−1.00000
$901$	0	0
$902$	0	0
$903$	0	0
$904$	2.00000	2.00000
$905$	− 2.00000i	− 2.00000i
$906$	−2.00000	−2.00000
$907$	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	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	− 1.00000i	− 1.00000i
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	2.00000i	2.00000i
$921$	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	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	2.00000i	2.00000i
$931$	−1.00000	−1.00000
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	2.00000	2.00000
$941$	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	$0.5\pi$
$948$	2.00000i	2.00000i
$949$	0	0
$950$	− 1.00000i	− 1.00000i
$951$	0	0
$952$	0	0
$953$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$953$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−1.00000	−1.00000
$961$	3.00000	3.00000
$962$	0	0
$963$	2.00000i	2.00000i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	− 1.00000i	− 1.00000i
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	1.00000i	1.00000i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$977$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	1.00000i	1.00000i
$981$	2.00000	2.00000
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	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			$0$
$991$	2.00000	2.00000	1.00000			$0$
$992$	2.00000i	2.00000i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	− 2.00000i	− 2.00000i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2280.1.t.f.1139.2	yes	2
3.2	odd	2	inner	2280.1.t.f.1139.1	✓	2
5.4	even	2	inner	2280.1.t.f.1139.1	✓	2
8.3	odd	2		2280.1.t.g.1139.2	yes	2
15.14	odd	2	CM	2280.1.t.f.1139.2	yes	2
19.18	odd	2		2280.1.t.g.1139.1	yes	2
24.11	even	2		2280.1.t.g.1139.1	yes	2
40.19	odd	2		2280.1.t.g.1139.1	yes	2
57.56	even	2		2280.1.t.g.1139.2	yes	2
95.94	odd	2		2280.1.t.g.1139.2	yes	2
120.59	even	2		2280.1.t.g.1139.2	yes	2
152.75	even	2	inner	2280.1.t.f.1139.1	✓	2
285.284	even	2		2280.1.t.g.1139.1	yes	2
456.227	odd	2	CM	2280.1.t.f.1139.2	yes	2
760.379	even	2	RM	2280.1.t.f.1139.2	yes	2
2280.1139	odd	2	inner	2280.1.t.f.1139.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.1.t.f.1139.1	✓	2	3.2	odd	2	inner
2280.1.t.f.1139.1	✓	2	5.4	even	2	inner
2280.1.t.f.1139.1	✓	2	152.75	even	2	inner
2280.1.t.f.1139.1	✓	2	2280.1139	odd	2	inner
2280.1.t.f.1139.2	yes	2	1.1	even	1	trivial
2280.1.t.f.1139.2	yes	2	15.14	odd	2	CM
2280.1.t.f.1139.2	yes	2	456.227	odd	2	CM
2280.1.t.f.1139.2	yes	2	760.379	even	2	RM
2280.1.t.g.1139.1	yes	2	19.18	odd	2
2280.1.t.g.1139.1	yes	2	24.11	even	2
2280.1.t.g.1139.1	yes	2	40.19	odd	2
2280.1.t.g.1139.1	yes	2	285.284	even	2
2280.1.t.g.1139.2	yes	2	8.3	odd	2
2280.1.t.g.1139.2	yes	2	57.56	even	2
2280.1.t.g.1139.2	yes	2	95.94	odd	2
2280.1.t.g.1139.2	yes	2	120.59	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2280.t (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{10} +1.00000i q^{12} +1.00000 q^{15} +1.00000 q^{16} -1.00000i q^{18} +1.00000 q^{19} -1.00000i q^{20} +2.00000i q^{23} -1.00000 q^{24} -1.00000 q^{25} +1.00000i q^{27} +1.00000i q^{30} +2.00000 q^{31} +1.00000i q^{32} +1.00000 q^{36} +1.00000i q^{38} +1.00000 q^{40} -1.00000i q^{45} -2.00000 q^{46} +2.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} -1.00000i q^{50} -1.00000 q^{54} -1.00000i q^{57} -1.00000 q^{60} +2.00000i q^{62} -1.00000 q^{64} +2.00000 q^{69} +1.00000i q^{72} +1.00000i q^{75} -1.00000 q^{76} +2.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +1.00000 q^{90} -2.00000i q^{92} -2.00000i q^{93} -2.00000 q^{94} +1.00000i q^{95} +1.00000 q^{96} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{10} + 2 q^{15} + 2 q^{16} + 2 q^{19} - 2 q^{24} - 2 q^{25} + 4 q^{31} + 2 q^{36} + 2 q^{40} - 4 q^{46} - 2 q^{49} - 2 q^{54} - 2 q^{60} - 2 q^{64} + 4 q^{69} - 2 q^{76}+ \cdots + 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^10 + 2 * q^15 + 2 * q^16 + 2 * q^19 - 2 * q^24 - 2 * q^25 + 4 * q^31 + 2 * q^36 + 2 * q^40 - 4 * q^46 - 2 * q^49 - 2 * q^54 - 2 * q^60 - 2 * q^64 + 4 * q^69 - 2 * q^76 + 4 * q^79 + 2 * q^81 + 2 * q^90 - 4 * q^94 + 2 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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