LMFDB - Embedded newform 3751.1.d.e.1332.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3751,1,Mod(1332,3751)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3751.1332"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3751, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3751 = 11^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3751.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [3,0,0,3,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.87199286239$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.1636073786281.1
Artin image:	$D_{18}$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

Embedding invariants

Embedding label			1332.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	3751.1332

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.347296 q^{2} -0.879385 q^{4} +0.347296 q^{5} -1.53209 q^{7} +0.652704 q^{8} +1.00000 q^{9} -0.120615 q^{10} +0.532089 q^{14} +0.652704 q^{16} -0.347296 q^{18} -0.347296 q^{19} -0.305407 q^{20} -0.879385 q^{25} +1.34730 q^{28} +1.00000 q^{31} -0.879385 q^{32} -0.532089 q^{35} -0.879385 q^{36} +0.120615 q^{38} +0.226682 q^{40} +1.87939 q^{41} +0.347296 q^{45} -1.00000 q^{47} +1.34730 q^{49} +0.305407 q^{50} -1.00000 q^{56} +1.53209 q^{59} -0.347296 q^{62} -1.53209 q^{63} -0.347296 q^{64} -1.00000 q^{67} +0.184793 q^{70} +1.53209 q^{71} +0.652704 q^{72} +0.305407 q^{76} +0.226682 q^{80} +1.00000 q^{81} -0.652704 q^{82} -0.120615 q^{90} +0.347296 q^{94} -0.120615 q^{95} +1.53209 q^{97} -0.467911 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{4} + 3 q^{8} + 3 q^{9} - 6 q^{10} - 3 q^{14} + 3 q^{16} - 3 q^{20} + 3 q^{25} + 3 q^{28} + 3 q^{31} + 3 q^{32} + 3 q^{35} + 3 q^{36} + 6 q^{38} - 6 q^{40} - 3 q^{47} + 3 q^{49} + 3 q^{50} - 3 q^{56}+ \cdots - 6 q^{98}+O(q^{100}) $ 3 * q + 3 * q^4 + 3 * q^8 + 3 * q^9 - 6 * q^10 - 3 * q^14 + 3 * q^16 - 3 * q^20 + 3 * q^25 + 3 * q^28 + 3 * q^31 + 3 * q^32 + 3 * q^35 + 3 * q^36 + 6 * q^38 - 6 * q^40 - 3 * q^47 + 3 * q^49 + 3 * q^50 - 3 * q^56 - 3 * q^67 - 3 * q^70 + 3 * q^72 + 3 * q^76 - 6 * q^80 + 3 * q^81 - 3 * q^82 - 6 * q^90 - 6 * q^95 - 6 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3751\mathbb{Z}\right)^\times$.

$n$	$2421$	$2543$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$2$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−0.879385	−0.879385
$5$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$5$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$6$	0	0
$7$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$7$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$8$	0.652704	0.652704
$9$	1.00000	1.00000
$10$	−0.120615	−0.120615
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0.532089	0.532089
$15$	0	0
$16$	0.652704	0.652704
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−0.347296	−0.347296
$19$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$19$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$20$	−0.305407	−0.305407
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−0.879385	−0.879385
$26$	0	0
$27$	0	0
$28$	1.34730	1.34730
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	1.00000	1.00000
$31$	1.00000	1.00000
$32$	−0.879385	−0.879385
$33$	0	0
$34$	0	0
$35$	−0.532089	−0.532089
$36$	−0.879385	−0.879385
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0.120615	0.120615
$39$	0	0
$40$	0.226682	0.226682
$41$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$41$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0.347296	0.347296
$46$	0	0
$47$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$47$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$48$	0	0
$49$	1.34730	1.34730
$50$	0.305407	0.305407
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−1.00000
$57$	0	0
$58$	0	0
$59$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$59$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−0.347296	−0.347296
$63$	−1.53209	−1.53209
$64$	−0.347296	−0.347296
$65$	0	0
$66$	0	0
$67$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$67$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$68$	0	0
$69$	0	0
$70$	0.184793	0.184793
$71$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$71$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$72$	0.652704	0.652704
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0.305407	0.305407
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0.226682	0.226682
$81$	1.00000	1.00000
$82$	−0.652704	−0.652704
$83$	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
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−0.120615	−0.120615
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0.347296	0.347296
$95$	−0.120615	−0.120615
$96$	0	0
$97$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$97$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$98$	−0.467911	−0.467911
$99$	0	0
$100$	0.773318	0.773318
$101$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$101$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$102$	0	0
$103$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$103$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$107$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$108$	0	0
$109$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$109$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−1.00000
$113$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$113$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−0.532089	−0.532089
$119$	0	0
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	−0.879385	−0.879385
$125$	−0.652704	−0.652704
$126$	0.532089	0.532089
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000	1.00000
$129$	0	0
$130$	0	0
$131$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$131$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$132$	0	0
$133$	0.532089	0.532089
$134$	0.347296	0.347296
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0.467911	0.467911
$141$	0	0
$142$	−0.532089	−0.532089
$143$	0	0
$144$	0.652704	0.652704
$145$	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			$\pi$
$152$	−0.226682	−0.226682
$153$	0	0
$154$	0	0
$155$	0.347296	0.347296
$156$	0	0
$157$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$157$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$158$	0	0
$159$	0	0
$160$	−0.305407	−0.305407
$161$	0	0
$162$	−0.347296	−0.347296
$163$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$163$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$164$	−1.65270	−1.65270
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	−0.347296	−0.347296
$172$	0	0
$173$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$173$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$174$	0	0
$175$	1.34730	1.34730
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−0.305407	−0.305407
$181$	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
$188$	0.879385	0.879385
$189$	0	0
$190$	0.0418891	0.0418891
$191$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$191$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$192$	0	0
$193$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$193$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$194$	−0.532089	−0.532089
$195$	0	0
$196$	−1.18479	−1.18479
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−0.573978	−0.573978
$201$	0	0
$202$	−0.652704	−0.652704
$203$	0	0
$204$	0	0
$205$	0.652704	0.652704
$206$	−0.120615	−0.120615
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$211$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$212$	0	0
$213$	0	0
$214$	−0.652704	−0.652704
$215$	0	0
$216$	0	0
$217$	−1.53209	−1.53209
$218$	0.532089	0.532089
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	1.34730	1.34730
$225$	−0.879385	−0.879385
$226$	−0.532089	−0.532089
$227$	−2.00000	−2.00000	−1.00000			$\pi$
$227$	−2.00000	−2.00000	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$233$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$234$	0	0
$235$	−0.347296	−0.347296
$236$	−1.34730	−1.34730
$237$	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			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.467911	0.467911
$246$	0	0
$247$	0	0
$248$	0.652704	0.652704
$249$	0	0
$250$	0.226682	0.226682
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	1.34730	1.34730
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$257$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−0.347296	−0.347296
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−0.184793	−0.184793
$267$	0	0
$268$	0.879385	0.879385
$269$	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			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	1.00000	1.00000
$280$	−0.347296	−0.347296
$281$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$281$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$282$	0	0
$283$	−2.00000	−2.00000	−1.00000			$\pi$
$283$	−2.00000	−2.00000	−1.00000			$\pi$
$284$	−1.34730	−1.34730
$285$	0	0
$286$	0	0
$287$	−2.87939	−2.87939
$288$	−0.879385	−0.879385
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$293$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$294$	0	0
$295$	0.532089	0.532089
$296$	0	0
$297$	0	0
$298$	0.694593	0.694593
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−0.226682	−0.226682
$305$	0	0
$306$	0	0
$307$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$307$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$308$	0	0
$309$	0	0
$310$	−0.120615	−0.120615
$311$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$311$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	−0.532089	−0.532089
$315$	−0.532089	−0.532089
$316$	0	0
$317$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$317$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$318$	0	0
$319$	0	0
$320$	−0.120615	−0.120615
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.879385	−0.879385
$325$	0	0
$326$	0.652704	0.652704
$327$	0	0
$328$	1.22668	1.22668
$329$	1.53209	1.53209
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.347296	−0.347296
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−0.347296	−0.347296
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0.120615	0.120615
$343$	−0.532089	−0.532089
$344$	0	0
$345$	0	0
$346$	−0.347296	−0.347296
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$349$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$350$	−0.467911	−0.467911
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0.532089	0.532089
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$359$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$360$	0.226682	0.226682
$361$	−0.879385	−0.879385
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	1.87939	1.87939
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$373$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$374$	0	0
$375$	0	0
$376$	−0.652704	−0.652704
$377$	0	0
$378$	0	0
$379$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$380$	0.106067	0.106067
$381$	0	0
$382$	0.652704	0.652704
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−0.652704	−0.652704
$387$	0	0
$388$	−1.34730	−1.34730
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0.879385	0.879385
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$397$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$398$	0	0
$399$	0	0
$400$	−0.573978	−0.573978
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	−1.65270	−1.65270
$405$	0.347296	0.347296
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−0.226682	−0.226682
$411$	0	0
$412$	−0.305407	−0.305407
$413$	−2.34730	−2.34730
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$419$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$420$	0	0
$421$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$421$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$422$	−0.652704	−0.652704
$423$	−1.00000	−1.00000
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−1.65270	−1.65270
$429$	0	0
$430$	0	0
$431$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$431$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0.532089	0.532089
$435$	0	0
$436$	1.34730	1.34730
$437$	0	0
$438$	0	0
$439$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$439$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$440$	0	0
$441$	1.34730	1.34730
$442$	0	0
$443$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$443$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0.532089	0.532089
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0.305407	0.305407
$451$	0	0
$452$	−1.34730	−1.34730
$453$	0	0
$454$	0.694593	0.694593
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0.120615	0.120615
$467$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$467$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$468$	0	0
$469$	1.53209	1.53209
$470$	0.120615	0.120615
$471$	0	0
$472$	1.00000	1.00000
$473$	0	0
$474$	0	0
$475$	0.305407	0.305407
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$479$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.532089	0.532089
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−0.162504	−0.162504
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0.652704	0.652704
$497$	−2.34730	−2.34730
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0.573978	0.573978
$501$	0	0
$502$	0	0
$503$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$503$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$504$	−1.00000	−1.00000
$505$	0.652704	0.652704
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−1.00000
$513$	0	0
$514$	0.652704	0.652704
$515$	0.120615	0.120615
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−0.879385	−0.879385
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	1.53209	1.53209
$532$	−0.467911	−0.467911
$533$	0	0
$534$	0	0
$535$	0.652704	0.652704
$536$	−0.652704	−0.652704
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$541$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.532089	−0.532089
$546$	0	0
$547$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$547$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	−0.347296	−0.347296
$559$	0	0
$560$	−0.347296	−0.347296
$561$	0	0
$562$	0.532089	0.532089
$563$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$563$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$564$	0	0
$565$	0.532089	0.532089
$566$	0.694593	0.694593
$567$	−1.53209	−1.53209
$568$	1.00000	1.00000
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	1.00000	1.00000
$575$	0	0
$576$	−0.347296	−0.347296
$577$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$577$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$578$	−0.347296	−0.347296
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−0.347296	−0.347296
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	−0.347296	−0.347296
$590$	−0.184793	−0.184793
$591$	0	0
$592$	0	0
$593$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$593$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$594$	0	0
$595$	0	0
$596$	1.75877	1.75877
$597$	0	0
$598$	0	0
$599$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$599$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	−1.00000	−1.00000
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$607$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$608$	0.305407	0.305407
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−0.652704	−0.652704
$615$	0	0
$616$	0	0
$617$	2.00000	2.00000	1.00000			$0$
$617$	2.00000	2.00000	1.00000			$0$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	−0.305407	−0.305407
$621$	0	0
$622$	0.652704	0.652704
$623$	0	0
$624$	0	0
$625$	0.652704	0.652704
$626$	0	0
$627$	0	0
$628$	−1.34730	−1.34730
$629$	0	0
$630$	0.184793	0.184793
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−0.120615	−0.120615
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.53209	1.53209
$640$	0.347296	0.347296
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.652704	0.652704
$649$	0	0
$650$	0	0
$651$	0	0
$652$	1.65270	1.65270
$653$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$653$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$654$	0	0
$655$	0.347296	0.347296
$656$	1.22668	1.22668
$657$	0	0
$658$	−0.532089	−0.532089
$659$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$659$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$660$	0	0
$661$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$661$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.184793	0.184793
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0.120615	0.120615
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−0.879385	−0.879385
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	−2.34730	−2.34730
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$683$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$684$	0.305407	0.305407
$685$	0	0
$686$	0.184793	0.184793
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$691$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$692$	−0.879385	−0.879385
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−0.347296	−0.347296
$699$	0	0
$700$	−1.18479	−1.18479
$701$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$701$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.87939	−2.87939
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	−0.184793	−0.184793
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−0.652704	−0.652704
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0.226682	0.226682
$721$	−0.532089	−0.532089
$722$	0.305407	0.305407
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$727$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$733$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	−0.652704	−0.652704
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−0.694593	−0.694593
$746$	−0.652704	−0.652704
$747$	0	0
$748$	0	0
$749$	−2.87939	−2.87939
$750$	0	0
$751$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$751$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$752$	−0.652704	−0.652704
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0.347296	0.347296
$759$	0	0
$760$	−0.0787257	−0.0787257
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	2.34730	2.34730
$764$	1.65270	1.65270
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$769$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$770$	0	0
$771$	0	0
$772$	−1.65270	−1.65270
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−0.879385	−0.879385
$776$	1.00000	1.00000
$777$	0	0
$778$	0	0
$779$	−0.652704	−0.652704
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0.879385	0.879385
$785$	0.532089	0.532089
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.34730	−2.34730
$792$	0	0
$793$	0	0
$794$	0.652704	0.652704
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0.773318	0.773318
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	1.22668	1.22668
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−0.120615	−0.120615
$811$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$811$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.652704	−0.652704
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−0.573978	−0.573978
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0.226682	0.226682
$825$	0	0
$826$	0.815207	0.815207
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0.652704	0.652704
$839$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$839$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	−0.120615	−0.120615
$843$	0	0
$844$	−1.65270	−1.65270
$845$	0.347296	0.347296
$846$	0.347296	0.347296
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−2.00000	−2.00000	−1.00000			$\pi$
$853$	−2.00000	−2.00000	−1.00000			$\pi$
$854$	0	0
$855$	−0.120615	−0.120615
$856$	1.22668	1.22668
$857$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$857$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	−0.347296	−0.347296
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0.347296	0.347296
$866$	0	0
$867$	0	0
$868$	1.34730	1.34730
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−1.00000	−1.00000
$873$	1.53209	1.53209
$874$	0	0
$875$	1.00000	1.00000
$876$	0	0
$877$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$877$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$878$	0.532089	0.532089
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−0.467911	−0.467911
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−0.120615	−0.120615
$887$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$887$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.347296	0.347296
$894$	0	0
$895$	0	0
$896$	−1.53209	−1.53209
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0.773318	0.773318
$901$	0	0
$902$	0	0
$903$	0	0
$904$	1.00000	1.00000
$905$	0	0
$906$	0	0
$907$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$907$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$908$	1.75877	1.75877
$909$	1.87939	1.87939
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.53209	−1.53209
$918$	0	0
$919$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$919$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.347296	0.347296
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−0.467911	−0.467911
$932$	0.305407	0.305407
$933$	0	0
$934$	0.652704	0.652704
$935$	0	0
$936$	0	0
$937$	−2.00000	−2.00000	−1.00000			$\pi$
$937$	−2.00000	−2.00000	−1.00000			$\pi$
$938$	−0.532089	−0.532089
$939$	0	0
$940$	0.305407	0.305407
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	1.00000	1.00000
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	−0.106067	−0.106067
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	−0.652704	−0.652704
$956$	0	0
$957$	0	0
$958$	0.120615	0.120615
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	1.87939	1.87939
$964$	0	0
$965$	0.652704	0.652704
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	−0.184793	−0.184793
$971$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$977$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$978$	0	0
$979$	0	0
$980$	−0.411474	−0.411474
$981$	−1.53209	−1.53209
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−0.879385	−0.879385
$993$	0	0
$994$	0.815207	0.815207
$995$	0	0
$996$	0	0
$997$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$997$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3751.1.d.e.1332.2	yes	3
11.2	odd	10		3751.1.t.f.2138.2		12
11.3	even	5		3751.1.t.e.2913.2		12
11.4	even	5		3751.1.t.e.3657.2		12
11.5	even	5		3751.1.t.e.2665.2		12
11.6	odd	10		3751.1.t.f.2665.2		12
11.7	odd	10		3751.1.t.f.3657.2		12
11.8	odd	10		3751.1.t.f.2913.2		12
11.9	even	5		3751.1.t.e.2138.2		12
11.10	odd	2		3751.1.d.d.1332.2	✓	3
31.30	odd	2	CM	3751.1.d.e.1332.2	yes	3
341.30	even	10		3751.1.t.f.2913.2		12
341.61	even	10		3751.1.t.f.2665.2		12
341.92	odd	10		3751.1.t.e.3657.2		12
341.123	even	10		3751.1.t.f.2138.2		12
341.185	odd	10		3751.1.t.e.2138.2		12
341.216	even	10		3751.1.t.f.3657.2		12
341.247	odd	10		3751.1.t.e.2665.2		12
341.278	odd	10		3751.1.t.e.2913.2		12
341.340	even	2		3751.1.d.d.1332.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3751.1.d.d.1332.2	✓	3	11.10	odd	2
3751.1.d.d.1332.2	✓	3	341.340	even	2
3751.1.d.e.1332.2	yes	3	1.1	even	1	trivial
3751.1.d.e.1332.2	yes	3	31.30	odd	2	CM
3751.1.t.e.2138.2		12	11.9	even	5
3751.1.t.e.2138.2		12	341.185	odd	10
3751.1.t.e.2665.2		12	11.5	even	5
3751.1.t.e.2665.2		12	341.247	odd	10
3751.1.t.e.2913.2		12	11.3	even	5
3751.1.t.e.2913.2		12	341.278	odd	10
3751.1.t.e.3657.2		12	11.4	even	5
3751.1.t.e.3657.2		12	341.92	odd	10
3751.1.t.f.2138.2		12	11.2	odd	10
3751.1.t.f.2138.2		12	341.123	even	10
3751.1.t.f.2665.2		12	11.6	odd	10
3751.1.t.f.2665.2		12	341.61	even	10
3751.1.t.f.2913.2		12	11.8	odd	10
3751.1.t.f.2913.2		12	341.30	even	10
3751.1.t.f.3657.2		12	11.7	odd	10
3751.1.t.f.3657.2		12	341.216	even	10

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:	\( N \)	\(=\)	\( 3751 = 11^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3751.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.347296 q^{2} -0.879385 q^{4} +0.347296 q^{5} -1.53209 q^{7} +0.652704 q^{8} +1.00000 q^{9} -0.120615 q^{10} +0.532089 q^{14} +0.652704 q^{16} -0.347296 q^{18} -0.347296 q^{19} -0.305407 q^{20} -0.879385 q^{25} +1.34730 q^{28} +1.00000 q^{31} -0.879385 q^{32} -0.532089 q^{35} -0.879385 q^{36} +0.120615 q^{38} +0.226682 q^{40} +1.87939 q^{41} +0.347296 q^{45} -1.00000 q^{47} +1.34730 q^{49} +0.305407 q^{50} -1.00000 q^{56} +1.53209 q^{59} -0.347296 q^{62} -1.53209 q^{63} -0.347296 q^{64} -1.00000 q^{67} +0.184793 q^{70} +1.53209 q^{71} +0.652704 q^{72} +0.305407 q^{76} +0.226682 q^{80} +1.00000 q^{81} -0.652704 q^{82} -0.120615 q^{90} +0.347296 q^{94} -0.120615 q^{95} +1.53209 q^{97} -0.467911 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{4} + 3 q^{8} + 3 q^{9} - 6 q^{10} - 3 q^{14} + 3 q^{16} - 3 q^{20} + 3 q^{25} + 3 q^{28} + 3 q^{31} + 3 q^{32} + 3 q^{35} + 3 q^{36} + 6 q^{38} - 6 q^{40} - 3 q^{47} + 3 q^{49} + 3 q^{50} - 3 q^{56}+ \cdots - 6 q^{98}+O(q^{100}) \) 3 * q + 3 * q^4 + 3 * q^8 + 3 * q^9 - 6 * q^10 - 3 * q^14 + 3 * q^16 - 3 * q^20 + 3 * q^25 + 3 * q^28 + 3 * q^31 + 3 * q^32 + 3 * q^35 + 3 * q^36 + 6 * q^38 - 6 * q^40 - 3 * q^47 + 3 * q^49 + 3 * q^50 - 3 * q^56 - 3 * q^67 - 3 * q^70 + 3 * q^72 + 3 * q^76 - 6 * q^80 + 3 * q^81 - 3 * q^82 - 6 * q^90 - 6 * q^95 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more