LMFDB - Embedded newform 1944.1.h.a.485.1

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

Embedding invariants

Embedding label			485.1
Root			$-1.53209$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.53209 q^{5} -1.87939 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.53209 q^{5} -1.87939 q^{7} -1.00000 q^{8} +1.53209 q^{10} -0.347296 q^{11} +1.87939 q^{14} +1.00000 q^{16} -1.53209 q^{20} +0.347296 q^{22} +1.34730 q^{25} -1.87939 q^{28} +1.00000 q^{29} +0.347296 q^{31} -1.00000 q^{32} +2.87939 q^{35} +1.53209 q^{40} -0.347296 q^{44} +2.53209 q^{49} -1.34730 q^{50} +1.87939 q^{53} +0.532089 q^{55} +1.87939 q^{56} -1.00000 q^{58} +1.00000 q^{59} -0.347296 q^{62} +1.00000 q^{64} -2.87939 q^{70} -1.87939 q^{73} +0.652704 q^{77} -1.00000 q^{79} -1.53209 q^{80} +1.87939 q^{83} +0.347296 q^{88} +1.53209 q^{97} -2.53209 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$5$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$6$	0	0
$7$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$7$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$8$	−1.00000	−1.00000
$9$	0	0
$10$	1.53209	1.53209
$11$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$11$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	1.87939	1.87939
$15$	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			$\pi$
$20$	−1.53209	−1.53209
$21$	0	0
$22$	0.347296	0.347296
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	1.34730	1.34730
$26$	0	0
$27$	0	0
$28$	−1.87939	−1.87939
$29$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$29$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$30$	0	0
$31$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$31$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$32$	−1.00000	−1.00000
$33$	0	0
$34$	0	0
$35$	2.87939	2.87939
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	1.53209	1.53209
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−0.347296	−0.347296
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	2.53209	2.53209
$50$	−1.34730	−1.34730
$51$	0	0
$52$	0	0
$53$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$53$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$54$	0	0
$55$	0.532089	0.532089
$56$	1.87939	1.87939
$57$	0	0
$58$	−1.00000	−1.00000
$59$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$59$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−0.347296	−0.347296
$63$	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
$70$	−2.87939	−2.87939
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$73$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.652704	0.652704
$78$	0	0
$79$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$79$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$80$	−1.53209	−1.53209
$81$	0	0
$82$	0	0
$83$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$83$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0.347296	0.347296
$89$	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
$97$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$97$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$98$	−2.53209	−2.53209
$99$	0	0
$100$	1.34730	1.34730
$101$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$101$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$102$	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
$106$	−1.87939	−1.87939
$107$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$107$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−0.532089	−0.532089
$111$	0	0
$112$	−1.87939	−1.87939
$113$	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
$121$	−0.879385	−0.879385
$122$	0	0
$123$	0	0
$124$	0.347296	0.347296
$125$	−0.532089	−0.532089
$126$	0	0
$127$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$127$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$128$	−1.00000	−1.00000
$129$	0	0
$130$	0	0
$131$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$131$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$132$	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			$\pi$
$140$	2.87939	2.87939
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.53209	−1.53209
$146$	1.87939	1.87939
$147$	0	0
$148$	0	0
$149$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$149$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$150$	0	0
$151$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$151$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$152$	0	0
$153$	0	0
$154$	−0.652704	−0.652704
$155$	−0.532089	−0.532089
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	1.00000	1.00000
$159$	0	0
$160$	1.53209	1.53209
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−1.87939	−1.87939
$167$	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
$173$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$173$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$174$	0	0
$175$	−2.53209	−2.53209
$176$	−0.347296	−0.347296
$177$	0	0
$178$	0	0
$179$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$179$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$180$	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
$193$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$193$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$194$	−1.53209	−1.53209
$195$	0	0
$196$	2.53209	2.53209
$197$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$197$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$198$	0	0
$199$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$199$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$200$	−1.34730	−1.34730
$201$	0	0
$202$	0.347296	0.347296
$203$	−1.87939	−1.87939
$204$	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			$\pi$
$212$	1.87939	1.87939
$213$	0	0
$214$	1.53209	1.53209
$215$	0	0
$216$	0	0
$217$	−0.652704	−0.652704
$218$	0	0
$219$	0	0
$220$	0.532089	0.532089
$221$	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	$0.666667\pi$
$224$	1.87939	1.87939
$225$	0	0
$226$	0	0
$227$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$227$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$228$	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	$0.666667\pi$
$242$	0.879385	0.879385
$243$	0	0
$244$	0	0
$245$	−3.87939	−3.87939
$246$	0	0
$247$	0	0
$248$	−0.347296	−0.347296
$249$	0	0
$250$	0.532089	0.532089
$251$	−2.00000	−2.00000	−1.00000			$\pi$
$251$	−2.00000	−2.00000	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	−1.53209	−1.53209
$255$	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
$262$	1.53209	1.53209
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−2.87939	−2.87939
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$269$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$270$	0	0
$271$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$271$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.467911	−0.467911
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	−2.87939	−2.87939
$281$	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
$290$	1.53209	1.53209
$291$	0	0
$292$	−1.87939	−1.87939
$293$	−2.00000	−2.00000	−1.00000			$\pi$
$293$	−2.00000	−2.00000	−1.00000			$\pi$
$294$	0	0
$295$	−1.53209	−1.53209
$296$	0	0
$297$	0	0
$298$	−1.87939	−1.87939
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−1.53209	−1.53209
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0.652704	0.652704
$309$	0	0
$310$	0.532089	0.532089
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$313$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$314$	0	0
$315$	0	0
$316$	−1.00000	−1.00000
$317$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$317$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$318$	0	0
$319$	−0.347296	−0.347296
$320$	−1.53209	−1.53209
$321$	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			$\pi$
$332$	1.87939	1.87939
$333$	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
$341$	−0.120615	−0.120615
$342$	0	0
$343$	−2.87939	−2.87939
$344$	0	0
$345$	0	0
$346$	−1.87939	−1.87939
$347$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$347$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	2.53209	2.53209
$351$	0	0
$352$	0.347296	0.347296
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0.347296	0.347296
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.87939	2.87939
$366$	0	0
$367$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$367$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.53209	−3.53209
$372$	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
$386$	−1.53209	−1.53209
$387$	0	0
$388$	1.53209	1.53209
$389$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$389$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$390$	0	0
$391$	0	0
$392$	−2.53209	−2.53209
$393$	0	0
$394$	0.347296	0.347296
$395$	1.53209	1.53209
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−0.347296	−0.347296
$399$	0	0
$400$	1.34730	1.34730
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	−0.347296	−0.347296
$405$	0	0
$406$	1.87939	1.87939
$407$	0	0
$408$	0	0
$409$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$409$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$410$	0	0
$411$	0	0
$412$	−1.00000	−1.00000
$413$	−1.87939	−1.87939
$414$	0	0
$415$	−2.87939	−2.87939
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$419$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	−1.87939	−1.87939
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−1.53209	−1.53209
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$433$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$434$	0.652704	0.652704
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$439$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$440$	−0.532089	−0.532089
$441$	0	0
$442$	0	0
$443$	−2.00000	−2.00000	−1.00000			$\pi$
$443$	−2.00000	−2.00000	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	1.00000	1.00000
$447$	0	0
$448$	−1.87939	−1.87939
$449$	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
$457$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$457$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$461$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$462$	0	0
$463$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$463$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$464$	1.00000	1.00000
$465$	0	0
$466$	0	0
$467$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$467$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$468$	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
$484$	−0.879385	−0.879385
$485$	−2.34730	−2.34730
$486$	0	0
$487$	2.00000	2.00000	1.00000			$0$
$487$	2.00000	2.00000	1.00000			$0$
$488$	0	0
$489$	0	0
$490$	3.87939	3.87939
$491$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$491$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0.347296	0.347296
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−0.532089	−0.532089
$501$	0	0
$502$	2.00000	2.00000
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0.532089	0.532089
$506$	0	0
$507$	0	0
$508$	1.53209	1.53209
$509$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$509$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$510$	0	0
$511$	3.53209	3.53209
$512$	−1.00000	−1.00000
$513$	0	0
$514$	0	0
$515$	1.53209	1.53209
$516$	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			$\pi$
$524$	−1.53209	−1.53209
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	2.87939	2.87939
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	2.34730	2.34730
$536$	0	0
$537$	0	0
$538$	−1.00000	−1.00000
$539$	−0.879385	−0.879385
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−0.347296	−0.347296
$543$	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
$550$	0.467911	0.467911
$551$	0	0
$552$	0	0
$553$	1.87939	1.87939
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$557$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$558$	0	0
$559$	0	0
$560$	2.87939	2.87939
$561$	0	0
$562$	0	0
$563$	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	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
$580$	−1.53209	−1.53209
$581$	−3.53209	−3.53209
$582$	0	0
$583$	−0.652704	−0.652704
$584$	1.87939	1.87939
$585$	0	0
$586$	2.00000	2.00000
$587$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$587$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$588$	0	0
$589$	0	0
$590$	1.53209	1.53209
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	1.87939	1.87939
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$601$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$602$	0	0
$603$	0	0
$604$	1.53209	1.53209
$605$	1.34730	1.34730
$606$	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
$616$	−0.652704	−0.652704
$617$	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			$\pi$
$620$	−0.532089	−0.532089
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−0.532089	−0.532089
$626$	−0.347296	−0.347296
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$631$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$632$	1.00000	1.00000
$633$	0	0
$634$	−1.87939	−1.87939
$635$	−2.34730	−2.34730
$636$	0	0
$637$	0	0
$638$	0.347296	0.347296
$639$	0	0
$640$	1.53209	1.53209
$641$	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
$649$	−0.347296	−0.347296
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$653$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$654$	0	0
$655$	2.34730	2.34730
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$659$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	−1.87939	−1.87939
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$673$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$674$	1.00000	1.00000
$675$	0	0
$676$	1.00000	1.00000
$677$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$677$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$678$	0	0
$679$	−2.87939	−2.87939
$680$	0	0
$681$	0	0
$682$	0.120615	0.120615
$683$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$683$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$684$	0	0
$685$	0	0
$686$	2.87939	2.87939
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	1.87939	1.87939
$693$	0	0
$694$	−1.87939	−1.87939
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−2.53209	−2.53209
$701$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$701$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$702$	0	0
$703$	0	0
$704$	−0.347296	−0.347296
$705$	0	0
$706$	0	0
$707$	0.652704	0.652704
$708$	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
$716$	−0.347296	−0.347296
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	1.87939	1.87939
$722$	−1.00000	−1.00000
$723$	0	0
$724$	0	0
$725$	1.34730	1.34730
$726$	0	0
$727$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$727$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$728$	0	0
$729$	0	0
$730$	−2.87939	−2.87939
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	1.87939	1.87939
$735$	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
$742$	3.53209	3.53209
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−2.87939	−2.87939
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.87939	2.87939
$750$	0	0
$751$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$751$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.34730	−2.34730
$756$	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
$769$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$769$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$770$	1.00000	1.00000
$771$	0	0
$772$	1.53209	1.53209
$773$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$773$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$774$	0	0
$775$	0.467911	0.467911
$776$	−1.53209	−1.53209
$777$	0	0
$778$	1.53209	1.53209
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	2.53209	2.53209
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−0.347296	−0.347296
$789$	0	0
$790$	−1.53209	−1.53209
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0.347296	0.347296
$797$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$797$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$798$	0	0
$799$	0	0
$800$	−1.34730	−1.34730
$801$	0	0
$802$	0	0
$803$	0.652704	0.652704
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0.347296	0.347296
$809$	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			$\pi$
$812$	−1.87939	−1.87939
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−0.347296	−0.347296
$819$	0	0
$820$	0	0
$821$	−2.00000	−2.00000	−1.00000			$\pi$
$821$	−2.00000	−2.00000	−1.00000			$\pi$
$822$	0	0
$823$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$823$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$824$	1.00000	1.00000
$825$	0	0
$826$	1.87939	1.87939
$827$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$827$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	2.87939	2.87939
$831$	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
$845$	−1.53209	−1.53209
$846$	0	0
$847$	1.65270	1.65270
$848$	1.87939	1.87939
$849$	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
$856$	1.53209	1.53209
$857$	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
$865$	−2.87939	−2.87939
$866$	−1.53209	−1.53209
$867$	0	0
$868$	−0.652704	−0.652704
$869$	0.347296	0.347296
$870$	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			$\pi$
$878$	−1.53209	−1.53209
$879$	0	0
$880$	0.532089	0.532089
$881$	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
$889$	−2.87939	−2.87939
$890$	0	0
$891$	0	0
$892$	−1.00000	−1.00000
$893$	0	0
$894$	0	0
$895$	0.532089	0.532089
$896$	1.87939	1.87939
$897$	0	0
$898$	0	0
$899$	0.347296	0.347296
$900$	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
$913$	−0.652704	−0.652704
$914$	−1.53209	−1.53209
$915$	0	0
$916$	0	0
$917$	2.87939	2.87939
$918$	0	0
$919$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$919$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$920$	0	0
$921$	0	0
$922$	1.53209	1.53209
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−1.53209	−1.53209
$927$	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
$934$	1.53209	1.53209
$935$	0	0
$936$	0	0
$937$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$937$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$941$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$942$	0	0
$943$	0	0
$944$	1.00000	1.00000
$945$	0	0
$946$	0	0
$947$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$947$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$948$	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
$961$	−0.879385	−0.879385
$962$	0	0
$963$	0	0
$964$	−1.00000	−1.00000
$965$	−2.34730	−2.34730
$966$	0	0
$967$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$967$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$968$	0.879385	0.879385
$969$	0	0
$970$	2.34730	2.34730
$971$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$971$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$972$	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
$980$	−3.87939	−3.87939
$981$	0	0
$982$	−1.87939	−1.87939
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0.532089	0.532089
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$991$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$992$	−0.347296	−0.347296
$993$	0	0
$994$	0	0
$995$	−0.532089	−0.532089
$996$	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.a.485.1	✓	3
3.2	odd	2		1944.1.h.b.485.3	yes	3
8.5	even	2		1944.1.h.b.485.3	yes	3
9.2	odd	6		1944.1.j.a.1133.1		6
9.4	even	3		1944.1.j.b.1781.3		6
9.5	odd	6		1944.1.j.a.1781.1		6
9.7	even	3		1944.1.j.b.1133.3		6
24.5	odd	2	CM	1944.1.h.a.485.1	✓	3
72.5	odd	6		1944.1.j.b.1781.3		6
72.13	even	6		1944.1.j.a.1781.1		6
72.29	odd	6		1944.1.j.b.1133.3		6
72.61	even	6		1944.1.j.a.1133.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1944.1.h.a.485.1	✓	3	1.1	even	1	trivial
1944.1.h.a.485.1	✓	3	24.5	odd	2	CM
1944.1.h.b.485.3	yes	3	3.2	odd	2
1944.1.h.b.485.3	yes	3	8.5	even	2
1944.1.j.a.1133.1		6	9.2	odd	6
1944.1.j.a.1133.1		6	72.61	even	6
1944.1.j.a.1781.1		6	9.5	odd	6
1944.1.j.a.1781.1		6	72.13	even	6
1944.1.j.b.1133.3		6	9.7	even	3
1944.1.j.b.1133.3		6	72.29	odd	6
1944.1.j.b.1781.3		6	9.4	even	3
1944.1.j.b.1781.3		6	72.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.53209 q^{5} -1.87939 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.53209 q^{5} -1.87939 q^{7} -1.00000 q^{8} +1.53209 q^{10} -0.347296 q^{11} +1.87939 q^{14} +1.00000 q^{16} -1.53209 q^{20} +0.347296 q^{22} +1.34730 q^{25} -1.87939 q^{28} +1.00000 q^{29} +0.347296 q^{31} -1.00000 q^{32} +2.87939 q^{35} +1.53209 q^{40} -0.347296 q^{44} +2.53209 q^{49} -1.34730 q^{50} +1.87939 q^{53} +0.532089 q^{55} +1.87939 q^{56} -1.00000 q^{58} +1.00000 q^{59} -0.347296 q^{62} +1.00000 q^{64} -2.87939 q^{70} -1.87939 q^{73} +0.652704 q^{77} -1.00000 q^{79} -1.53209 q^{80} +1.87939 q^{83} +0.347296 q^{88} +1.53209 q^{97} -2.53209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) \) 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

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Modular forms

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