LMFDB - Embedded newform 1944.1.h.b.485.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1944,1,Mod(485,1944)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1944, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1944.485");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1944 = 2^{3} \cdot 3^{5} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1944.h (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:	$0.970182384559$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.1586874322944.5
Artin image:	$D_9$
Artin field:	Galois closure of 9.1.1586874322944.5

Embedding invariants

Embedding label			485.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	1944.485

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.347296 q^{5} +1.53209 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +0.347296 q^{5} +1.53209 q^{7} +1.00000 q^{8} +0.347296 q^{10} -1.87939 q^{11} +1.53209 q^{14} +1.00000 q^{16} +0.347296 q^{20} -1.87939 q^{22} -0.879385 q^{25} +1.53209 q^{28} -1.00000 q^{29} -1.87939 q^{31} +1.00000 q^{32} +0.532089 q^{35} +0.347296 q^{40} -1.87939 q^{44} +1.34730 q^{49} -0.879385 q^{50} +1.53209 q^{53} -0.652704 q^{55} +1.53209 q^{56} -1.00000 q^{58} -1.00000 q^{59} -1.87939 q^{62} +1.00000 q^{64} +0.532089 q^{70} +1.53209 q^{73} -2.87939 q^{77} -1.00000 q^{79} +0.347296 q^{80} +1.53209 q^{83} -1.87939 q^{88} +0.347296 q^{97} +1.34730 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{16} + 3 q^{25} - 3 q^{29} + 3 q^{32} - 3 q^{35} + 3 q^{49} + 3 q^{50} - 3 q^{55} - 3 q^{58} - 3 q^{59} + 3 q^{64} - 3 q^{70} - 3 q^{77} - 3 q^{79} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 3 * q^16 + 3 * q^25 - 3 * q^29 + 3 * q^32 - 3 * q^35 + 3 * q^49 + 3 * q^50 - 3 * q^55 - 3 * q^58 - 3 * q^59 + 3 * q^64 - 3 * q^70 - 3 * q^77 - 3 * q^79 + 3 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1944\mathbb{Z}\right)^\times$.

$n$	$487$	$973$	$1217$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	0.347296	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	$-0.222222\pi$
$7$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$8$	1.00000	1.00000
$9$	0	0
$10$	0.347296	0.347296
$11$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$11$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	1.53209	1.53209
$15$	0	0
$16$	1.00000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0.347296	0.347296
$21$	0	0
$22$	−1.87939	−1.87939
$23$	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
$28$	1.53209	1.53209
$29$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$29$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$30$	0	0
$31$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$31$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$32$	1.00000	1.00000
$33$	0	0
$34$	0	0
$35$	0.532089	0.532089
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0.347296	0.347296
$41$	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			$\pi$
$44$	−1.87939	−1.87939
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	1.34730	1.34730
$50$	−0.879385	−0.879385
$51$	0	0
$52$	0	0
$53$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$53$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$54$	0	0
$55$	−0.652704	−0.652704
$56$	1.53209	1.53209
$57$	0	0
$58$	−1.00000	−1.00000
$59$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−1.87939	−1.87939
$63$	0	0
$64$	1.00000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0.532089	0.532089
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$73$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.87939	−2.87939
$78$	0	0
$79$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$79$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$80$	0.347296	0.347296
$81$	0	0
$82$	0	0
$83$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$83$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−1.87939	−1.87939
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$97$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$98$	1.34730	1.34730
$99$	0	0
$100$	−0.879385	−0.879385
$101$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$101$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$102$	0	0
$103$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$103$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$104$	0	0
$105$	0	0
$106$	1.53209	1.53209
$107$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$107$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−0.652704	−0.652704
$111$	0	0
$112$	1.53209	1.53209
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−1.00000	−1.00000
$117$	0	0
$118$	−1.00000	−1.00000
$119$	0	0
$120$	0	0
$121$	2.53209	2.53209
$122$	0	0
$123$	0	0
$124$	−1.87939	−1.87939
$125$	−0.652704	−0.652704
$126$	0	0
$127$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$127$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$128$	1.00000	1.00000
$129$	0	0
$130$	0	0
$131$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$131$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$132$	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	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0.532089	0.532089
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.347296	−0.347296
$146$	1.53209	1.53209
$147$	0	0
$148$	0	0
$149$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$149$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$150$	0	0
$151$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$151$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$152$	0	0
$153$	0	0
$154$	−2.87939	−2.87939
$155$	−0.652704	−0.652704
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−1.00000	−1.00000
$159$	0	0
$160$	0.347296	0.347296
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	1.53209	1.53209
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$173$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$174$	0	0
$175$	−1.34730	−1.34730
$176$	−1.87939	−1.87939
$177$	0	0
$178$	0	0
$179$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$179$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$180$	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
$193$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$193$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$194$	0.347296	0.347296
$195$	0	0
$196$	1.34730	1.34730
$197$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$197$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$198$	0	0
$199$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$199$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$200$	−0.879385	−0.879385
$201$	0	0
$202$	−1.87939	−1.87939
$203$	−1.53209	−1.53209
$204$	0	0
$205$	0	0
$206$	−1.00000	−1.00000
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	1.53209	1.53209
$213$	0	0
$214$	0.347296	0.347296
$215$	0	0
$216$	0	0
$217$	−2.87939	−2.87939
$218$	0	0
$219$	0	0
$220$	−0.652704	−0.652704
$221$	0	0
$222$	0	0
$223$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$223$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$224$	1.53209	1.53209
$225$	0	0
$226$	0	0
$227$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$227$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	−1.00000	−1.00000
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	−1.00000	−1.00000
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$242$	2.53209	2.53209
$243$	0	0
$244$	0	0
$245$	0.467911	0.467911
$246$	0	0
$247$	0	0
$248$	−1.87939	−1.87939
$249$	0	0
$250$	−0.652704	−0.652704
$251$	2.00000	2.00000	1.00000			$0$
$251$	2.00000	2.00000	1.00000			$0$
$252$	0	0
$253$	0	0
$254$	0.347296	0.347296
$255$	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
$262$	0.347296	0.347296
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0.532089	0.532089
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$270$	0	0
$271$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$271$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.65270	1.65270
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0.532089	0.532089
$281$	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
$290$	−0.347296	−0.347296
$291$	0	0
$292$	1.53209	1.53209
$293$	2.00000	2.00000	1.00000			$0$
$293$	2.00000	2.00000	1.00000			$0$
$294$	0	0
$295$	−0.347296	−0.347296
$296$	0	0
$297$	0	0
$298$	1.53209	1.53209
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0.347296	0.347296
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	−2.87939	−2.87939
$309$	0	0
$310$	−0.652704	−0.652704
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$313$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$314$	0	0
$315$	0	0
$316$	−1.00000	−1.00000
$317$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$317$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$318$	0	0
$319$	1.87939	1.87939
$320$	0.347296	0.347296
$321$	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			$\pi$
$332$	1.53209	1.53209
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$338$	1.00000	1.00000
$339$	0	0
$340$	0	0
$341$	3.53209	3.53209
$342$	0	0
$343$	0.532089	0.532089
$344$	0	0
$345$	0	0
$346$	1.53209	1.53209
$347$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$347$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−1.34730	−1.34730
$351$	0	0
$352$	−1.87939	−1.87939
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−1.87939	−1.87939
$359$	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
$365$	0.532089	0.532089
$366$	0	0
$367$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$367$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.34730	2.34730
$372$	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
$385$	−1.00000	−1.00000
$386$	0.347296	0.347296
$387$	0	0
$388$	0.347296	0.347296
$389$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$389$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$390$	0	0
$391$	0	0
$392$	1.34730	1.34730
$393$	0	0
$394$	−1.87939	−1.87939
$395$	−0.347296	−0.347296
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−1.87939	−1.87939
$399$	0	0
$400$	−0.879385	−0.879385
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	−1.87939	−1.87939
$405$	0	0
$406$	−1.53209	−1.53209
$407$	0	0
$408$	0	0
$409$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$409$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$410$	0	0
$411$	0	0
$412$	−1.00000	−1.00000
$413$	−1.53209	−1.53209
$414$	0	0
$415$	0.532089	0.532089
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	1.53209	1.53209
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0.347296	0.347296
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$433$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$434$	−2.87939	−2.87939
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$439$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$440$	−0.652704	−0.652704
$441$	0	0
$442$	0	0
$443$	2.00000	2.00000	1.00000			$0$
$443$	2.00000	2.00000	1.00000			$0$
$444$	0	0
$445$	0	0
$446$	−1.00000	−1.00000
$447$	0	0
$448$	1.53209	1.53209
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−1.00000	−1.00000
$455$	0	0
$456$	0	0
$457$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$457$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$461$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$462$	0	0
$463$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$463$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$464$	−1.00000	−1.00000
$465$	0	0
$466$	0	0
$467$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$467$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.00000	−1.00000
$473$	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
$482$	−1.00000	−1.00000
$483$	0	0
$484$	2.53209	2.53209
$485$	0.120615	0.120615
$486$	0	0
$487$	2.00000	2.00000	1.00000			$0$
$487$	2.00000	2.00000	1.00000			$0$
$488$	0	0
$489$	0	0
$490$	0.467911	0.467911
$491$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$491$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−1.87939	−1.87939
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−0.652704	−0.652704
$501$	0	0
$502$	2.00000	2.00000
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−0.652704	−0.652704
$506$	0	0
$507$	0	0
$508$	0.347296	0.347296
$509$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$509$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$510$	0	0
$511$	2.34730	2.34730
$512$	1.00000	1.00000
$513$	0	0
$514$	0	0
$515$	−0.347296	−0.347296
$516$	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			$\pi$
$524$	0.347296	0.347296
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0.532089	0.532089
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0.120615	0.120615
$536$	0	0
$537$	0	0
$538$	−1.00000	−1.00000
$539$	−2.53209	−2.53209
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−1.87939	−1.87939
$543$	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
$550$	1.65270	1.65270
$551$	0	0
$552$	0	0
$553$	−1.53209	−1.53209
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$557$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$558$	0	0
$559$	0	0
$560$	0.532089	0.532089
$561$	0	0
$562$	0	0
$563$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$563$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$564$	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	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$577$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$578$	1.00000	1.00000
$579$	0	0
$580$	−0.347296	−0.347296
$581$	2.34730	2.34730
$582$	0	0
$583$	−2.87939	−2.87939
$584$	1.53209	1.53209
$585$	0	0
$586$	2.00000	2.00000
$587$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$587$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$588$	0	0
$589$	0	0
$590$	−0.347296	−0.347296
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	1.53209	1.53209
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$601$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$602$	0	0
$603$	0	0
$604$	0.347296	0.347296
$605$	0.879385	0.879385
$606$	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
$616$	−2.87939	−2.87939
$617$	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			$\pi$
$620$	−0.652704	−0.652704
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0.652704	0.652704
$626$	−1.87939	−1.87939
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$631$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$632$	−1.00000	−1.00000
$633$	0	0
$634$	1.53209	1.53209
$635$	0.120615	0.120615
$636$	0	0
$637$	0	0
$638$	1.87939	1.87939
$639$	0	0
$640$	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	0
$649$	1.87939	1.87939
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$653$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$654$	0	0
$655$	0.120615	0.120615
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$659$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	1.53209	1.53209
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$673$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$674$	−1.00000	−1.00000
$675$	0	0
$676$	1.00000	1.00000
$677$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$677$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$678$	0	0
$679$	0.532089	0.532089
$680$	0	0
$681$	0	0
$682$	3.53209	3.53209
$683$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$684$	0	0
$685$	0	0
$686$	0.532089	0.532089
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	1.53209	1.53209
$693$	0	0
$694$	1.53209	1.53209
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−1.34730	−1.34730
$701$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$701$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$702$	0	0
$703$	0	0
$704$	−1.87939	−1.87939
$705$	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	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−1.87939	−1.87939
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−1.53209	−1.53209
$722$	1.00000	1.00000
$723$	0	0
$724$	0	0
$725$	0.879385	0.879385
$726$	0	0
$727$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$727$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$728$	0	0
$729$	0	0
$730$	0.532089	0.532089
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	1.53209	1.53209
$735$	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
$742$	2.34730	2.34730
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0.532089	0.532089
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.532089	0.532089
$750$	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	0
$753$	0	0
$754$	0	0
$755$	0.120615	0.120615
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	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
$769$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$769$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$770$	−1.00000	−1.00000
$771$	0	0
$772$	0.347296	0.347296
$773$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$773$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$774$	0	0
$775$	1.65270	1.65270
$776$	0.347296	0.347296
$777$	0	0
$778$	0.347296	0.347296
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.34730	1.34730
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−1.87939	−1.87939
$789$	0	0
$790$	−0.347296	−0.347296
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−1.87939	−1.87939
$797$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$797$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$798$	0	0
$799$	0	0
$800$	−0.879385	−0.879385
$801$	0	0
$802$	0	0
$803$	−2.87939	−2.87939
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−1.87939	−1.87939
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	−1.53209	−1.53209
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−1.87939	−1.87939
$819$	0	0
$820$	0	0
$821$	2.00000	2.00000	1.00000			$0$
$821$	2.00000	2.00000	1.00000			$0$
$822$	0	0
$823$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$823$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$824$	−1.00000	−1.00000
$825$	0	0
$826$	−1.53209	−1.53209
$827$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$827$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0.532089	0.532089
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−1.00000	−1.00000
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.347296	0.347296
$846$	0	0
$847$	3.87939	3.87939
$848$	1.53209	1.53209
$849$	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
$856$	0.347296	0.347296
$857$	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
$865$	0.532089	0.532089
$866$	0.347296	0.347296
$867$	0	0
$868$	−2.87939	−2.87939
$869$	1.87939	1.87939
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.00000	−1.00000
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0.347296	0.347296
$879$	0	0
$880$	−0.652704	−0.652704
$881$	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			$\pi$
$884$	0	0
$885$	0	0
$886$	2.00000	2.00000
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0.532089	0.532089
$890$	0	0
$891$	0	0
$892$	−1.00000	−1.00000
$893$	0	0
$894$	0	0
$895$	−0.652704	−0.652704
$896$	1.53209	1.53209
$897$	0	0
$898$	0	0
$899$	1.87939	1.87939
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−1.00000	−1.00000
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−2.87939	−2.87939
$914$	0.347296	0.347296
$915$	0	0
$916$	0	0
$917$	0.532089	0.532089
$918$	0	0
$919$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$919$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$920$	0	0
$921$	0	0
$922$	0.347296	0.347296
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0.347296	0.347296
$927$	0	0
$928$	−1.00000	−1.00000
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0.347296	0.347296
$935$	0	0
$936$	0	0
$937$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$937$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$941$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$942$	0	0
$943$	0	0
$944$	−1.00000	−1.00000
$945$	0	0
$946$	0	0
$947$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$947$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.53209	2.53209
$962$	0	0
$963$	0	0
$964$	−1.00000	−1.00000
$965$	0.120615	0.120615
$966$	0	0
$967$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$967$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$968$	2.53209	2.53209
$969$	0	0
$970$	0.120615	0.120615
$971$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$971$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$972$	0	0
$973$	0	0
$974$	2.00000	2.00000
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0.467911	0.467911
$981$	0	0
$982$	1.53209	1.53209
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−0.652704	−0.652704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$991$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$992$	−1.87939	−1.87939
$993$	0	0
$994$	0	0
$995$	−0.652704	−0.652704
$996$	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	1944.1.h.b.485.2	yes	3
3.2	odd	2		1944.1.h.a.485.2	✓	3
8.5	even	2		1944.1.h.a.485.2	✓	3
9.2	odd	6		1944.1.j.b.1133.2		6
9.4	even	3		1944.1.j.a.1781.2		6
9.5	odd	6		1944.1.j.b.1781.2		6
9.7	even	3		1944.1.j.a.1133.2		6
24.5	odd	2	CM	1944.1.h.b.485.2	yes	3
72.5	odd	6		1944.1.j.a.1781.2		6
72.13	even	6		1944.1.j.b.1781.2		6
72.29	odd	6		1944.1.j.a.1133.2		6
72.61	even	6		1944.1.j.b.1133.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1944.1.h.a.485.2	✓	3	3.2	odd	2
1944.1.h.a.485.2	✓	3	8.5	even	2
1944.1.h.b.485.2	yes	3	1.1	even	1	trivial
1944.1.h.b.485.2	yes	3	24.5	odd	2	CM
1944.1.j.a.1133.2		6	9.7	even	3
1944.1.j.a.1133.2		6	72.29	odd	6
1944.1.j.a.1781.2		6	9.4	even	3
1944.1.j.a.1781.2		6	72.5	odd	6
1944.1.j.b.1133.2		6	9.2	odd	6
1944.1.j.b.1133.2		6	72.61	even	6
1944.1.j.b.1781.2		6	9.5	odd	6
1944.1.j.b.1781.2		6	72.13	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1944 = 2^{3} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1944.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.347296 q^{5} +1.53209 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.347296 q^{5} +1.53209 q^{7} +1.00000 q^{8} +0.347296 q^{10} -1.87939 q^{11} +1.53209 q^{14} +1.00000 q^{16} +0.347296 q^{20} -1.87939 q^{22} -0.879385 q^{25} +1.53209 q^{28} -1.00000 q^{29} -1.87939 q^{31} +1.00000 q^{32} +0.532089 q^{35} +0.347296 q^{40} -1.87939 q^{44} +1.34730 q^{49} -0.879385 q^{50} +1.53209 q^{53} -0.652704 q^{55} +1.53209 q^{56} -1.00000 q^{58} -1.00000 q^{59} -1.87939 q^{62} +1.00000 q^{64} +0.532089 q^{70} +1.53209 q^{73} -2.87939 q^{77} -1.00000 q^{79} +0.347296 q^{80} +1.53209 q^{83} -1.87939 q^{88} +0.347296 q^{97} +1.34730 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{16} + 3 q^{25} - 3 q^{29} + 3 q^{32} - 3 q^{35} + 3 q^{49} + 3 q^{50} - 3 q^{55} - 3 q^{58} - 3 q^{59} + 3 q^{64} - 3 q^{70} - 3 q^{77} - 3 q^{79} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 3 * q^16 + 3 * q^25 - 3 * q^29 + 3 * q^32 - 3 * q^35 + 3 * q^49 + 3 * q^50 - 3 * q^55 - 3 * q^58 - 3 * q^59 + 3 * q^64 - 3 * q^70 - 3 * q^77 - 3 * q^79 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

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