LMFDB - Embedded newform 3343.1.b.a.3342.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3343,1,Mod(3342,3343)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3343, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3343.3342");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3343 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3343.b (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:	$1.66837433723$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\Q(\zeta_{38})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1 $ x^9 - x^8 - 8x^7 + 7x^6 + 21x^5 - 15x^4 - 20x^3 + 10x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{19}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{19} - \cdots)$

Embedding invariants

Embedding label			3342.4
Root			$1.35456$ of defining polynomial
Character	$\chi$	$=$	3343.3342

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.803391 q^{2} -0.354563 q^{4} +1.08824 q^{8} +1.00000 q^{9} -1.97272 q^{11} -0.519722 q^{16} -0.803391 q^{18} +1.57828 q^{19} +1.58487 q^{22} +1.00000 q^{25} -1.75895 q^{31} -0.670704 q^{32} -0.354563 q^{36} -1.26798 q^{38} +1.09390 q^{43} +0.699455 q^{44} +1.00000 q^{49} -0.803391 q^{50} -0.165159 q^{53} +0.490971 q^{59} +0.490971 q^{61} +1.41312 q^{62} +1.05856 q^{64} +1.08824 q^{72} -0.559600 q^{76} +1.00000 q^{81} -0.878826 q^{86} -2.14680 q^{88} +1.09390 q^{89} -0.803391 q^{98} -1.97272 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{2} + 8 q^{4} - 2 q^{8} + 9 q^{9} - q^{11} + 7 q^{16} - q^{18} - q^{19} - 2 q^{22} + 9 q^{25} - q^{31} - 3 q^{32} + 8 q^{36} - 2 q^{38} - q^{43} - 3 q^{44} + 9 q^{49} - q^{50} - q^{53} - q^{59}+ \cdots - q^{99}+O(q^{100}) $ 9 * q - q^2 + 8 * q^4 - 2 * q^8 + 9 * q^9 - q^11 + 7 * q^16 - q^18 - q^19 - 2 * q^22 + 9 * q^25 - q^31 - 3 * q^32 + 8 * q^36 - 2 * q^38 - q^43 - 3 * q^44 + 9 * q^49 - q^50 - q^53 - q^59 - q^61 - 2 * q^62 + 6 * q^64 - 2 * q^72 - 3 * q^76 + 9 * q^81 - 2 * q^86 - 4 * q^88 - q^89 - q^98 - q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3343\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

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 $

$2$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$2$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−0.354563	−0.354563
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.08824	1.08824
$9$	1.00000	1.00000
$10$	0	0
$11$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$11$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	−0.519722	−0.519722
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−0.803391	−0.803391
$19$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$19$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$20$	0	0
$21$	0	0
$22$	1.58487	1.58487
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	1.00000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$31$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$32$	−0.670704	−0.670704
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−0.354563	−0.354563
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−1.26798	−1.26798
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$43$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$44$	0.699455	0.699455
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	1.00000	1.00000
$50$	−0.803391	−0.803391
$51$	0	0
$52$	0	0
$53$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$53$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$59$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$60$	0	0
$61$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$61$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$62$	1.41312	1.41312
$63$	0	0
$64$	1.05856	1.05856
$65$	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	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.08824	1.08824
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	−0.559600	−0.559600
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.00000	1.00000
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	−0.878826	−0.878826
$87$	0	0
$88$	−2.14680	−2.14680
$89$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$89$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−0.803391	−0.803391
$99$	−1.97272	−1.97272
$100$	−0.354563	−0.354563
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$103$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$104$	0	0
$105$	0	0
$106$	0.132687	0.132687
$107$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$107$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$108$	0	0
$109$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$109$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−0.394442	−0.394442
$119$	0	0
$120$	0	0
$121$	2.89163	2.89163
$122$	−0.394442	−0.394442
$123$	0	0
$124$	0.623658	0.623658
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.179733	−0.179733
$129$	0	0
$130$	0	0
$131$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$131$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$132$	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	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−0.519722	−0.519722
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	1.71755	1.71755
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$157$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−0.803391	−0.803391
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	1.57828	1.57828
$172$	−0.387855	−0.387855
$173$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$173$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$174$	0	0
$175$	0	0
$176$	1.02527	1.02527
$177$	0	0
$178$	−0.878826	−0.878826
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	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
$193$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$193$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$194$	0	0
$195$	0	0
$196$	−0.354563	−0.354563
$197$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$197$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$198$	1.58487	1.58487
$199$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$199$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$200$	1.08824	1.08824
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0.132687	0.132687
$207$	0	0
$208$	0	0
$209$	−3.11351	−3.11351
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0.0585592	0.0585592
$213$	0	0
$214$	−1.51972	−1.51972
$215$	0	0
$216$	0	0
$217$	0	0
$218$	1.41312	1.41312
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$223$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$224$	0	0
$225$	1.00000	1.00000
$226$	0	0
$227$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$227$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$233$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$234$	0	0
$235$	0	0
$236$	−0.174080	−0.174080
$237$	0	0
$238$	0	0
$239$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$239$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−2.32311	−2.32311
$243$	0	0
$244$	−0.174080	−0.174080
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−1.91416	−1.91416
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.914163	−0.914163
$257$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$257$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0.132687	0.132687
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$269$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$270$	0	0
$271$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$271$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.97272	−1.97272
$276$	0	0
$277$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$277$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$278$	0	0
$279$	−1.75895	−1.75895
$280$	0	0
$281$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$281$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$282$	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
$288$	−0.670704	−0.670704
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−0.820267	−0.820267
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$311$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$312$	0	0
$313$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$313$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$314$	−1.51972	−1.51972
$315$	0	0
$316$	0	0
$317$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$317$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.354563	−0.354563
$325$	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			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$337$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$338$	−0.803391	−0.803391
$339$	0	0
$340$	0	0
$341$	3.46992	3.46992
$342$	−1.26798	−1.26798
$343$	0	0
$344$	1.19043	1.19043
$345$	0	0
$346$	−0.878826	−0.878826
$347$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$347$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$348$	0	0
$349$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$349$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$350$	0	0
$351$	0	0
$352$	1.32311	1.32311
$353$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$353$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$354$	0	0
$355$	0	0
$356$	−0.387855	−0.387855
$357$	0	0
$358$	0	0
$359$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$359$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$360$	0	0
$361$	1.49097	1.49097
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$367$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−1.26798	−1.26798
$387$	1.09390	1.09390
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	1.08824	1.08824
$393$	0	0
$394$	−1.26798	−1.26798
$395$	0	0
$396$	0.699455	0.699455
$397$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$397$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$398$	−1.51972	−1.51972
$399$	0	0
$400$	−0.519722	−0.519722
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$409$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$410$	0	0
$411$	0	0
$412$	0.0585592	0.0585592
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	2.50137	2.50137
$419$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$419$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$420$	0	0
$421$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$421$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$422$	0	0
$423$	0	0
$424$	−0.179733	−0.179733
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−0.670704	−0.670704
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$433$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$434$	0	0
$435$	0	0
$436$	0.623658	0.623658
$437$	0	0
$438$	0	0
$439$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$439$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$440$	0	0
$441$	1.00000	1.00000
$442$	0	0
$443$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$443$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$444$	0	0
$445$	0	0
$446$	1.08824	1.08824
$447$	0	0
$448$	0	0
$449$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$449$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$450$	−0.803391	−0.803391
$451$	0	0
$452$	0	0
$453$	0	0
$454$	1.08824	1.08824
$455$	0	0
$456$	0	0
$457$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$457$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$463$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$464$	0	0
$465$	0	0
$466$	−0.394442	−0.394442
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0.534296	0.534296
$473$	−2.15795	−2.15795
$474$	0	0
$475$	1.57828	1.57828
$476$	0	0
$477$	−0.165159	−0.165159
$478$	−0.394442	−0.394442
$479$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$479$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−1.02527	−1.02527
$485$	0	0
$486$	0	0
$487$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$487$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$488$	0.534296	0.534296
$489$	0	0
$490$	0	0
$491$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$491$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0.914163	0.914163
$497$	0	0
$498$	0	0
$499$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$499$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$509$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$510$	0	0
$511$	0	0
$512$	0.914163	0.914163
$513$	0	0
$514$	1.41312	1.41312
$515$	0	0
$516$	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
$523$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$523$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$524$	0.0585592	0.0585592
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	0.490971	0.490971
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	1.08824	1.08824
$539$	−1.97272	−1.97272
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−1.51972	−1.51972
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$547$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$548$	0	0
$549$	0.490971	0.490971
$550$	1.58487	1.58487
$551$	0	0
$552$	0	0
$553$	0	0
$554$	1.08824	1.08824
$555$	0	0
$556$	0	0
$557$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$557$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$558$	1.41312	1.41312
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0.132687	0.132687
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$569$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$570$	0	0
$571$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$571$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.05856	1.05856
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−0.803391	−0.803391
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.325812	0.325812
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$587$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$588$	0	0
$589$	−2.77611	−2.77611
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$593$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$599$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$600$	0	0
$601$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$601$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−1.05856	−1.05856
$609$	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	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$619$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$620$	0	0
$621$	0	0
$622$	−0.878826	−0.878826
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	1.58487	1.58487
$627$	0	0
$628$	−0.670704	−0.670704
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	1.58487	1.58487
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$641$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$642$	0	0
$643$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$643$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	1.08824	1.08824
$649$	−0.968550	−0.968550
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$653$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$661$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.968550	−0.968550
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0.132687	0.132687
$675$	0	0
$676$	−0.354563	−0.354563
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	−2.78770	−2.78770
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	−0.559600	−0.559600
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−0.568522	−0.568522
$689$	0	0
$690$	0	0
$691$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$691$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$692$	−0.387855	−0.387855
$693$	0	0
$694$	1.08824	1.08824
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−0.878826	−0.878826
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−2.08824	−2.08824
$705$	0	0
$706$	−1.26798	−1.26798
$707$	0	0
$708$	0	0
$709$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$709$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$710$	0	0
$711$	0	0
$712$	1.19043	1.19043
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	1.08824	1.08824
$719$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$719$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$720$	0	0
$721$	0	0
$722$	−1.19783	−1.19783
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$727$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−0.394442	−0.394442
$735$	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
$742$	0	0
$743$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$743$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	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	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
$769$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$769$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$770$	0	0
$771$	0	0
$772$	−0.559600	−0.559600
$773$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$773$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$774$	−0.878826	−0.878826
$775$	−1.75895	−1.75895
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−0.519722	−0.519722
$785$	0	0
$786$	0	0
$787$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$787$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$788$	−0.559600	−0.559600
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−2.14680	−2.14680
$793$	0	0
$794$	1.58487	1.58487
$795$	0	0
$796$	−0.670704	−0.670704
$797$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$797$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$798$	0	0
$799$	0	0
$800$	−0.670704	−0.670704
$801$	1.09390	1.09390
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.72648	1.72648
$818$	1.58487	1.58487
$819$	0	0
$820$	0	0
$821$	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			$\pi$
$824$	−0.179733	−0.179733
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$829$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	1.10394	1.10394
$837$	0	0
$838$	0.645437	0.645437
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0.645437	0.645437
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0.0858366	0.0858366
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$853$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$854$	0	0
$855$	0	0
$856$	2.05856	2.05856
$857$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$857$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$863$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$864$	0	0
$865$	0	0
$866$	0.645437	0.645437
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−1.91416	−1.91416
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$877$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$878$	0.645437	0.645437
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−0.803391	−0.803391
$883$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$883$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$884$	0	0
$885$	0	0
$886$	1.41312	1.41312
$887$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$887$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.97272	−1.97272
$892$	0.480278	0.480278
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−1.51972	−1.51972
$899$	0	0
$900$	−0.354563	−0.354563
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$907$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$908$	0.480278	0.480278
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	1.08824	1.08824
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0.645437	0.645437
$927$	−0.165159	−0.165159
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	1.57828	1.57828
$932$	−0.174080	−0.174080
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$941$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$942$	0	0
$943$	0	0
$944$	−0.255168	−0.255168
$945$	0	0
$946$	1.73368	1.73368
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	−1.26798	−1.26798
$951$	0	0
$952$	0	0
$953$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$953$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$954$	0.132687	0.132687
$955$	0	0
$956$	−0.174080	−0.174080
$957$	0	0
$958$	−0.878826	−0.878826
$959$	0	0
$960$	0	0
$961$	2.09390	2.09390
$962$	0	0
$963$	1.89163	1.89163
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$967$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$968$	3.14680	3.14680
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	−0.878826	−0.878826
$975$	0	0
$976$	−0.255168	−0.255168
$977$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$977$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$978$	0	0
$979$	−2.15795	−2.15795
$980$	0	0
$981$	−1.75895	−1.75895
$982$	1.41312	1.41312
$983$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$983$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$991$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$992$	1.17973	1.17973
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$997$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$998$	0.132687	0.132687
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3343.1.b.a.3342.4	✓	9
3343.3342	odd	2	CM	3343.1.b.a.3342.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3343.1.b.a.3342.4	✓	9	1.1	even	1	trivial
3343.1.b.a.3342.4	✓	9	3343.3342	odd	2	CM

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:	\( N \)	\(=\)	\( 3343 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3343.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.803391 q^{2} -0.354563 q^{4} +1.08824 q^{8} +1.00000 q^{9} -1.97272 q^{11} -0.519722 q^{16} -0.803391 q^{18} +1.57828 q^{19} +1.58487 q^{22} +1.00000 q^{25} -1.75895 q^{31} -0.670704 q^{32} -0.354563 q^{36} -1.26798 q^{38} +1.09390 q^{43} +0.699455 q^{44} +1.00000 q^{49} -0.803391 q^{50} -0.165159 q^{53} +0.490971 q^{59} +0.490971 q^{61} +1.41312 q^{62} +1.05856 q^{64} +1.08824 q^{72} -0.559600 q^{76} +1.00000 q^{81} -0.878826 q^{86} -2.14680 q^{88} +1.09390 q^{89} -0.803391 q^{98} -1.97272 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} + 8 q^{4} - 2 q^{8} + 9 q^{9} - q^{11} + 7 q^{16} - q^{18} - q^{19} - 2 q^{22} + 9 q^{25} - q^{31} - 3 q^{32} + 8 q^{36} - 2 q^{38} - q^{43} - 3 q^{44} + 9 q^{49} - q^{50} - q^{53} - q^{59}+ \cdots - q^{99}+O(q^{100}) \) 9 * q - q^2 + 8 * q^4 - 2 * q^8 + 9 * q^9 - q^11 + 7 * q^16 - q^18 - q^19 - 2 * q^22 + 9 * q^25 - q^31 - 3 * q^32 + 8 * q^36 - 2 * q^38 - q^43 - 3 * q^44 + 9 * q^49 - q^50 - q^53 - q^59 - q^61 - 2 * q^62 + 6 * q^64 - 2 * q^72 - 3 * q^76 + 9 * q^81 - 2 * q^86 - 4 * q^88 - q^89 - q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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