LMFDB - Embedded newform 507.2.f.b.437.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(239,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("507.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.f (of order $4$, degree $2$, 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:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			437.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	507.437
Dual form			507.2.f.b.239.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +2.00000i q^{4} +(-3.09808 + 3.09808i) q^{7} +3.00000 q^{9} +3.46410i q^{12} -4.00000 q^{16} +(2.26795 + 2.26795i) q^{19} +(-5.36603 + 5.36603i) q^{21} +5.00000i q^{25} +5.19615 q^{27} +(-6.19615 - 6.19615i) q^{28} +(-0.830127 - 0.830127i) q^{31} +6.00000i q^{36} +(8.46410 - 8.46410i) q^{37} -1.73205i q^{43} -6.92820 q^{48} -12.1962i q^{49} +(3.92820 + 3.92820i) q^{57} +8.66025 q^{61} +(-9.29423 + 9.29423i) q^{63} -8.00000i q^{64} +(11.5622 + 11.5622i) q^{67} +(7.63397 - 7.63397i) q^{73} +8.66025i q^{75} +(-4.53590 + 4.53590i) q^{76} -12.1244 q^{79} +9.00000 q^{81} +(-10.7321 - 10.7321i) q^{84} +(-1.43782 - 1.43782i) q^{93} +(7.02628 + 7.02628i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{7} + 12 q^{9} - 16 q^{16} + 16 q^{19} - 18 q^{21} - 4 q^{28} + 14 q^{31} + 20 q^{37} - 12 q^{57} - 6 q^{63} + 22 q^{67} + 34 q^{73} - 32 q^{76} + 36 q^{81} - 36 q^{84} - 30 q^{93} - 10 q^{97}+O(q^{100}) $ 4 * q - 2 * q^7 + 12 * q^9 - 16 * q^16 + 16 * q^19 - 18 * q^21 - 4 * q^28 + 14 * q^31 + 20 * q^37 - 12 * q^57 - 6 * q^63 + 22 * q^67 + 34 * q^73 - 32 * q^76 + 36 * q^81 - 36 * q^84 - 30 * q^93 - 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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			0		−0.707107	−	0.707107i	$-0.750000\pi$
$2$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$3$	1.73205			1.00000
$3$	1.73205			1.00000
$4$			2.00000i			1.00000i
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$		0			0
$7$	−3.09808	+	3.09808i	−1.17096	+	1.17096i	−0.188982	+	0.981981i	$0.560519\pi$
$7$	−3.09808	+	3.09808i	−1.17096	+	1.17096i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$			3.46410i			1.00000i
$13$		0			0
$13$		0			0
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	2.26795	+	2.26795i	0.520303	+	0.520303i	0.917663	−	0.397360i	$-0.130073\pi$
$19$	2.26795	+	2.26795i	0.520303	+	0.520303i	−0.397360	+	0.917663i	$0.630073\pi$
$20$		0			0
$21$	−5.36603	+	5.36603i	−1.17096	+	1.17096i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$			5.00000i			1.00000i
$26$		0			0
$27$	5.19615			1.00000
$28$	−6.19615	−	6.19615i	−1.17096	−	1.17096i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−0.830127	−	0.830127i	−0.149095	−	0.149095i	0.628619	−	0.777714i	$-0.283621\pi$
$31$	−0.830127	−	0.830127i	−0.149095	−	0.149095i	−0.777714	+	0.628619i	$0.783621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$			6.00000i			1.00000i
$37$	8.46410	−	8.46410i	1.39149	−	1.39149i	0.569495	−	0.821995i	$-0.307139\pi$
$37$	8.46410	−	8.46410i	1.39149	−	1.39149i	0.821995	−	0.569495i	$-0.192861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$41$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$		−	1.73205i		−	0.264135i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$		−	1.73205i		−	0.264135i	0.991241	−	0.132068i	$-0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	−6.92820			−1.00000
$49$		−	12.1962i		−	1.74231i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.92820	+	3.92820i	0.520303	+	0.520303i
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	8.66025			1.10883			0.554416	−	0.832240i	$-0.312942\pi$
$61$	8.66025			1.10883			0.554416	+	0.832240i	$0.312942\pi$
$62$		0			0
$63$	−9.29423	+	9.29423i	−1.17096	+	1.17096i
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	11.5622	+	11.5622i	1.41254	+	1.41254i	0.740613	+	0.671932i	$0.234535\pi$
$67$	11.5622	+	11.5622i	1.41254	+	1.41254i	0.671932	+	0.740613i	$0.265465\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$71$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$72$		0			0
$73$	7.63397	−	7.63397i	0.893489	−	0.893489i	−0.101361	−	0.994850i	$-0.532320\pi$
$73$	7.63397	−	7.63397i	0.893489	−	0.893489i	0.994850	+	0.101361i	$0.0323196\pi$
$74$		0			0
$75$			8.66025i			1.00000i
$76$	−4.53590	+	4.53590i	−0.520303	+	0.520303i
$77$		0			0
$78$		0			0
$79$	−12.1244			−1.36410			−0.682048	−	0.731307i	$-0.738911\pi$
$79$	−12.1244			−1.36410			−0.682048	+	0.731307i	$0.738911\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$	−10.7321	−	10.7321i	−1.17096	−	1.17096i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−1.43782	−	1.43782i	−0.149095	−	0.149095i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	7.02628	+	7.02628i	0.713411	+	0.713411i	0.967247	−	0.253837i	$-0.0816925\pi$
$97$	7.02628	+	7.02628i	0.713411	+	0.713411i	−0.253837	+	0.967247i	$0.581693\pi$
$98$		0			0
$99$		0			0
$100$	−10.0000			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		−	15.5885i		−	1.53598i	−0.640464	−	0.767988i	$-0.721258\pi$
$103$		−	15.5885i		−	1.53598i	0.640464	−	0.767988i	$-0.278742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$			10.3923i			1.00000i
$109$	−13.8301	−	13.8301i	−1.32469	−	1.32469i	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−13.8301	−	13.8301i	−1.32469	−	1.32469i	−0.414751	−	0.909935i	$-0.636131\pi$
$110$		0			0
$111$	14.6603	−	14.6603i	1.39149	−	1.39149i
$112$	12.3923	−	12.3923i	1.17096	−	1.17096i
$113$		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			0
$119$		0			0
$120$		0			0
$121$		−	11.0000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$	1.66025	−	1.66025i	0.149095	−	0.149095i
$125$		0			0
$126$		0			0
$127$		−	1.00000i		−	0.0887357i	−0.999015	−	0.0443678i	$-0.985873\pi$
$127$		−	1.00000i		−	0.0887357i	0.999015	−	0.0443678i	$-0.0141274\pi$
$128$		0			0
$129$		−	3.00000i		−	0.264135i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	−14.0526			−1.21851
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$	−7.00000			−0.593732			−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000			−0.593732			−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$		−	21.1244i		−	1.74231i
$148$	16.9282	+	16.9282i	1.39149	+	1.39149i
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$	−10.1244	+	10.1244i	−0.823908	+	0.823908i	−0.986666	−	0.162758i	$-0.947961\pi$
$151$	−10.1244	+	10.1244i	−0.823908	+	0.823908i	0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−11.0000			−0.877896			−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000			−0.877896			−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	9.90192	−	9.90192i	0.775579	−	0.775579i	−0.203497	−	0.979076i	$-0.565231\pi$
$163$	9.90192	−	9.90192i	0.775579	−	0.775579i	0.979076	+	0.203497i	$0.0652307\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	6.80385	+	6.80385i	0.520303	+	0.520303i
$172$	3.46410			0.264135
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−15.4904	−	15.4904i	−1.17096	−	1.17096i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$			6.92820i			0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$181$			6.92820i			0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$	15.0000			1.10883
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−16.0981	+	16.0981i	−1.17096	+	1.17096i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		−	13.8564i		−	1.00000i
$193$	3.70577	−	3.70577i	0.266747	−	0.266747i	−0.561041	−	0.827788i	$-0.689599\pi$
$193$	3.70577	−	3.70577i	0.266747	−	0.266747i	0.827788	+	0.561041i	$0.189599\pi$
$194$		0			0
$195$		0			0
$196$	24.3923			1.74231
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$			17.0000i			1.20510i	0.798082	+	0.602549i	$0.205848\pi$
$199$			17.0000i			1.20510i	−0.798082	+	0.602549i	$0.794152\pi$
$200$		0			0
$201$	20.0263	+	20.0263i	1.41254	+	1.41254i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	25.9808			1.78859			0.894295	−	0.447478i	$-0.147678\pi$
$211$	25.9808			1.78859			0.894295	+	0.447478i	$0.147678\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	5.14359			0.349170
$218$		0			0
$219$	13.2224	−	13.2224i	0.893489	−	0.893489i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−19.1962	−	19.1962i	−1.28547	−	1.28547i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−19.1962	−	19.1962i	−1.28547	−	1.28547i	−0.347960	−	0.937509i	$-0.613126\pi$
$224$		0			0
$225$			15.0000i			1.00000i
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$	−7.85641	+	7.85641i	−0.520303	+	0.520303i
$229$	−0.607695	+	0.607695i	−0.0401576	+	0.0401576i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−0.607695	+	0.607695i	−0.0401576	+	0.0401576i	0.686743	+	0.726900i	$0.259040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−21.0000			−1.36410
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−20.8564	+	20.8564i	−1.34348	+	1.34348i	−0.450910	+	0.892570i	$0.648900\pi$
$241$	−20.8564	+	20.8564i	−1.34348	+	1.34348i	−0.892570	+	0.450910i	$0.851100\pi$
$242$		0			0
$243$	15.5885			1.00000
$244$			17.3205i			1.10883i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−18.5885	−	18.5885i	−1.17096	−	1.17096i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$			52.4449i			3.25877i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−23.1244	+	23.1244i	−1.41254	+	1.41254i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−22.2942	+	22.2942i	−1.35428	+	1.35428i	−0.473466	+	0.880812i	$0.656997\pi$
$271$	−22.2942	+	22.2942i	−1.35428	+	1.35428i	−0.880812	+	0.473466i	$0.843003\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$			20.7846i			1.24883i	0.781094	+	0.624413i	$0.214662\pi$
$277$			20.7846i			1.24883i	−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$	−2.49038	−	2.49038i	−0.149095	−	0.149095i
$280$		0			0
$281$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$281$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$282$		0			0
$283$			25.0000i			1.48610i	0.669238	+	0.743048i	$0.266621\pi$
$283$			25.0000i			1.48610i	−0.669238	+	0.743048i	$0.733379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	12.1699	+	12.1699i	0.713411	+	0.713411i
$292$	15.2679	+	15.2679i	0.893489	+	0.893489i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−17.3205			−1.00000
$301$	5.36603	+	5.36603i	0.309293	+	0.309293i
$302$		0			0
$303$		0			0
$304$	−9.07180	−	9.07180i	−0.520303	−	0.520303i
$305$		0			0
$306$		0			0
$307$	−18.3660	+	18.3660i	−1.04820	+	1.04820i	−0.0494267	+	0.998778i	$0.515739\pi$
$307$	−18.3660	+	18.3660i	−1.04820	+	1.04820i	−0.998778	+	0.0494267i	$0.984261\pi$
$308$		0			0
$309$		−	27.0000i		−	1.53598i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	32.9090			1.86012			0.930062	−	0.367402i	$-0.119753\pi$
$313$	32.9090			1.86012			0.930062	+	0.367402i	$0.119753\pi$
$314$		0			0
$315$		0			0
$316$		−	24.2487i		−	1.36410i
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$		0			0
$326$		0			0
$327$	−23.9545	−	23.9545i	−1.32469	−	1.32469i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5.97372	−	5.97372i	−0.328345	−	0.328345i	0.523612	−	0.851957i	$-0.324584\pi$
$331$	−5.97372	−	5.97372i	−0.328345	−	0.328345i	−0.851957	+	0.523612i	$0.824584\pi$
$332$		0			0
$333$	25.3923	−	25.3923i	1.39149	−	1.39149i
$334$		0			0
$335$		0			0
$336$	21.4641	−	21.4641i	1.17096	−	1.17096i
$337$		−	29.0000i		−	1.57973i	−0.613280	−	0.789865i	$-0.710150\pi$
$337$		−	29.0000i		−	1.57973i	0.613280	−	0.789865i	$-0.289850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	16.0981	+	16.0981i	0.869214	+	0.869214i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	26.2224	−	26.2224i	1.40365	−	1.40365i	0.615581	−	0.788074i	$-0.288921\pi$
$349$	26.2224	−	26.2224i	1.40365	−	1.40365i	0.788074	−	0.615581i	$-0.211079\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$		0			0
$361$		−	8.71281i		−	0.458569i
$362$		0			0
$363$		−	19.0526i		−	1.00000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	31.0000			1.61819			0.809093	−	0.587680i	$-0.199959\pi$
$367$	31.0000			1.61819			0.809093	+	0.587680i	$0.199959\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	2.87564	−	2.87564i	0.149095	−	0.149095i
$373$	−36.3731			−1.88333			−0.941663	−	0.336557i	$-0.890737\pi$
$373$	−36.3731			−1.88333			−0.941663	+	0.336557i	$0.890737\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	24.5622	+	24.5622i	1.26167	+	1.26167i	0.950281	+	0.311393i	$0.100796\pi$
$379$	24.5622	+	24.5622i	1.26167	+	1.26167i	0.311393	+	0.950281i	$0.399204\pi$
$380$		0			0
$381$		−	1.73205i		−	0.0887357i
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	5.19615i		−	0.264135i
$388$	−14.0526	+	14.0526i	−0.713411	+	0.713411i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	19.4186	−	19.4186i	0.974591	−	0.974591i	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	19.4186	−	19.4186i	0.974591	−	0.974591i	0.999685	+	0.0250943i	$0.00798860\pi$
$398$		0			0
$399$	−24.3397			−1.21851
$400$		−	20.0000i		−	1.00000i
$401$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$401$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−28.4904	−	28.4904i	−1.40876	−	1.40876i	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	−28.4904	−	28.4904i	−1.40876	−	1.40876i	−0.642333	−	0.766426i	$-0.722033\pi$
$410$		0			0
$411$		0			0
$412$	31.1769			1.53598
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−12.1244			−0.593732
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	8.68653	+	8.68653i	0.423356	+	0.423356i	0.886357	−	0.463002i	$-0.153228\pi$
$421$	8.68653	+	8.68653i	0.423356	+	0.423356i	−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−26.8301	+	26.8301i	−1.29840	+	1.29840i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$	−20.7846			−1.00000
$433$		−	35.0000i		−	1.68199i	−0.541041	−	0.840996i	$-0.681970\pi$
$433$		−	35.0000i		−	1.68199i	0.541041	−	0.840996i	$-0.318030\pi$
$434$		0			0
$435$		0			0
$436$	27.6603	−	27.6603i	1.32469	−	1.32469i
$437$		0			0
$438$		0			0
$439$			39.8372i			1.90132i	0.310228	+	0.950662i	$0.399595\pi$
$439$			39.8372i			1.90132i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$		−	36.5885i		−	1.74231i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	29.3205	+	29.3205i	1.39149	+	1.39149i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	24.7846	+	24.7846i	1.17096	+	1.17096i
$449$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$449$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−17.5359	+	17.5359i	−0.823908	+	0.823908i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−14.4378	−	14.4378i	−0.675373	−	0.675373i	0.283577	−	0.958950i	$-0.408479\pi$
$457$	−14.4378	−	14.4378i	−0.675373	−	0.675373i	−0.958950	+	0.283577i	$0.908479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$	20.6340	−	20.6340i	0.958942	−	0.958942i	−0.0402476	−	0.999190i	$-0.512815\pi$
$463$	20.6340	−	20.6340i	0.958942	−	0.958942i	0.999190	+	0.0402476i	$0.0128147\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−71.6410			−3.30807
$470$		0			0
$471$	−19.0526			−0.877896
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−11.3397	+	11.3397i	−0.520303	+	0.520303i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	−23.7321	−	23.7321i	−1.07540	−	1.07540i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−23.7321	−	23.7321i	−1.07540	−	1.07540i	−0.0784867	−	0.996915i	$-0.525009\pi$
$488$		0			0
$489$	17.1506	−	17.1506i	0.775579	−	0.775579i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	3.32051	+	3.32051i	0.149095	+	0.149095i
$497$		0			0
$498$		0			0
$499$	−31.5885	−	31.5885i	−1.41409	−	1.41409i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−31.5885	−	31.5885i	−1.41409	−	1.41409i	−0.697835	−	0.716258i	$-0.745853\pi$
$500$		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
$508$	2.00000			0.0887357
$509$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$509$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$			47.3013i			2.09248i
$512$		0			0
$513$	11.7846	+	11.7846i	0.520303	+	0.520303i
$514$		0			0
$515$		0			0
$516$	6.00000			0.264135
$517$		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
$523$	−8.00000			−0.349816			−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000			−0.349816			−0.174908	+	0.984585i	$0.555963\pi$
$524$		0			0
$525$	−26.8301	−	26.8301i	−1.17096	−	1.17096i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$		−	28.1051i		−	1.21851i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	30.1506	−	30.1506i	1.29628	−	1.29628i	0.365444	−	0.930834i	$-0.380917\pi$
$541$	30.1506	−	30.1506i	1.29628	−	1.29628i	0.930834	−	0.365444i	$-0.119083\pi$
$542$		0			0
$543$			12.0000i			0.514969i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	41.0000			1.75303			0.876517	−	0.481371i	$-0.159861\pi$
$547$	41.0000			1.75303			0.876517	+	0.481371i	$0.159861\pi$
$548$		0			0
$549$	25.9808			1.10883
$550$		0			0
$551$		0			0
$552$		0			0
$553$	37.5622	−	37.5622i	1.59731	−	1.59731i
$554$		0			0
$555$		0			0
$556$		−	14.0000i		−	0.593732i
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−27.8827	+	27.8827i	−1.17096	+	1.17096i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		−	16.0000i		−	0.669579i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$		−	16.0000i		−	0.669579i	0.942293	−	0.334790i	$-0.108665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		−	24.0000i		−	1.00000i
$577$	29.9282	+	29.9282i	1.24593	+	1.24593i	0.957503	+	0.288425i	$0.0931316\pi$
$577$	29.9282	+	29.9282i	1.24593	+	1.24593i	0.288425	+	0.957503i	$0.406868\pi$
$578$		0			0
$579$	6.41858	−	6.41858i	0.266747	−	0.266747i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$	42.2487			1.74231
$589$		−	3.76537i		−	0.155149i
$590$		0			0
$591$		0			0
$592$	−33.8564	+	33.8564i	−1.39149	+	1.39149i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			29.4449i			1.20510i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−41.5692			−1.69564			−0.847822	−	0.530281i	$-0.822086\pi$
$601$	−41.5692			−1.69564			−0.847822	+	0.530281i	$0.822086\pi$
$602$		0			0
$603$	34.6865	+	34.6865i	1.41254	+	1.41254i
$604$	−20.2487	−	20.2487i	−0.823908	−	0.823908i
$605$		0			0
$606$		0			0
$607$	−20.0000			−0.811775			−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000			−0.811775			−0.405887	+	0.913923i	$0.633038\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	2.04552	+	2.04552i	0.0826177	+	0.0826177i	0.747208	−	0.664590i	$-0.231394\pi$
$613$	2.04552	+	2.04552i	0.0826177	+	0.0826177i	−0.664590	+	0.747208i	$0.731394\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$	−14.8827	+	14.8827i	−0.598186	+	0.598186i	−0.939829	−	0.341644i	$-0.889016\pi$
$619$	−14.8827	+	14.8827i	−0.598186	+	0.598186i	0.341644	+	0.939829i	$0.389016\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$		−	22.0000i		−	0.877896i
$629$		0			0
$630$		0			0
$631$	−25.6147	+	25.6147i	−1.01971	+	1.01971i	−0.0199047	+	0.999802i	$0.506336\pi$
$631$	−25.6147	+	25.6147i	−1.01971	+	1.01971i	−0.999802	+	0.0199047i	$0.993664\pi$
$632$		0			0
$633$	45.0000			1.78859
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	33.0263	+	33.0263i	1.30243	+	1.30243i	0.926750	+	0.375680i	$0.122591\pi$
$643$	33.0263	+	33.0263i	1.30243	+	1.30243i	0.375680	+	0.926750i	$0.377409\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	8.90897			0.349170
$652$	19.8038	+	19.8038i	0.775579	+	0.775579i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	22.9019	−	22.9019i	0.893489	−	0.893489i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	16.7058	−	16.7058i	0.649779	−	0.649779i	−0.303160	−	0.952940i	$-0.598042\pi$
$661$	16.7058	−	16.7058i	0.649779	−	0.649779i	0.952940	+	0.303160i	$0.0980418\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−33.2487	−	33.2487i	−1.28547	−	1.28547i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	50.2295i		−	1.93620i	−0.250557	−	0.968102i	$-0.580614\pi$
$673$		−	50.2295i		−	1.93620i	0.250557	−	0.968102i	$-0.419386\pi$
$674$		0			0
$675$			25.9808i			1.00000i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−43.5359			−1.67075
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$	−13.6077	+	13.6077i	−0.520303	+	0.520303i
$685$		0			0
$686$		0			0
$687$	−1.05256	+	1.05256i	−0.0401576	+	0.0401576i
$688$			6.92820i			0.264135i
$689$		0			0
$690$		0			0
$691$	−36.9545	−	36.9545i	−1.40581	−	1.40581i	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	−36.9545	−	36.9545i	−1.40581	−	1.40581i	−0.625958	−	0.779857i	$-0.715292\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	30.9808	−	30.9808i	1.17096	−	1.17096i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	38.3923			1.44799
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−29.0981	+	29.0981i	−1.09280	+	1.09280i	−0.0975728	+	0.995228i	$0.531108\pi$
$709$	−29.0981	+	29.0981i	−1.09280	+	1.09280i	−0.995228	+	0.0975728i	$0.968892\pi$
$710$		0			0
$711$	−36.3731			−1.36410
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	48.2942	+	48.2942i	1.79857	+	1.79857i
$722$		0			0
$723$	−36.1244	+	36.1244i	−1.34348	+	1.34348i
$724$	−13.8564			−0.514969
$725$		0			0
$726$		0			0
$727$			49.0000i			1.81731i	0.417548	+	0.908655i	$0.362889\pi$
$727$			49.0000i			1.81731i	−0.417548	+	0.908655i	$0.637111\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$			30.0000i			1.10883i
$733$	−23.3468	−	23.3468i	−0.862333	−	0.862333i	0.129275	−	0.991609i	$-0.458735\pi$
$733$	−23.3468	−	23.3468i	−0.862333	−	0.862333i	−0.991609	+	0.129275i	$0.958735\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	17.9808	−	17.9808i	0.661433	−	0.661433i	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	17.9808	−	17.9808i	0.661433	−	0.661433i	0.955718	+	0.294285i	$0.0950814\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$			17.3205i			0.632034i	0.948753	+	0.316017i	$0.102346\pi$
$751$			17.3205i			0.632034i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−32.1962	−	32.1962i	−1.17096	−	1.17096i
$757$	48.4974			1.76267			0.881334	−	0.472493i	$-0.156646\pi$
$757$	48.4974			1.76267			0.881334	+	0.472493i	$0.156646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$761$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$762$		0			0
$763$	85.6936			3.10232
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	27.7128			1.00000
$769$	−28.7128	−	28.7128i	−1.03541	−	1.03541i	−0.999350	−	0.0360609i	$-0.988519\pi$
$769$	−28.7128	−	28.7128i	−1.03541	−	1.03541i	−0.0360609	−	0.999350i	$-0.511481\pi$
$770$		0			0
$771$		0			0
$772$	7.41154	+	7.41154i	0.266747	+	0.266747i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$	4.15064	−	4.15064i	0.149095	−	0.149095i
$776$		0			0
$777$			90.8372i			3.25877i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			48.7846i			1.74231i
$785$		0			0
$786$		0			0
$787$	−12.6147	+	12.6147i	−0.449667	+	0.449667i	−0.895244	−	0.445577i	$-0.852999\pi$
$787$	−12.6147	+	12.6147i	−0.449667	+	0.449667i	0.445577	+	0.895244i	$0.352999\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−34.0000			−1.20510
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0

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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +2.00000i q^{4} +(-3.09808 + 3.09808i) q^{7} +3.00000 q^{9} +3.46410i q^{12} -4.00000 q^{16} +(2.26795 + 2.26795i) q^{19} +(-5.36603 + 5.36603i) q^{21} +5.00000i q^{25} +5.19615 q^{27} +(-6.19615 - 6.19615i) q^{28} +(-0.830127 - 0.830127i) q^{31} +6.00000i q^{36} +(8.46410 - 8.46410i) q^{37} -1.73205i q^{43} -6.92820 q^{48} -12.1962i q^{49} +(3.92820 + 3.92820i) q^{57} +8.66025 q^{61} +(-9.29423 + 9.29423i) q^{63} -8.00000i q^{64} +(11.5622 + 11.5622i) q^{67} +(7.63397 - 7.63397i) q^{73} +8.66025i q^{75} +(-4.53590 + 4.53590i) q^{76} -12.1244 q^{79} +9.00000 q^{81} +(-10.7321 - 10.7321i) q^{84} +(-1.43782 - 1.43782i) q^{93} +(7.02628 + 7.02628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{7} + 12 q^{9} - 16 q^{16} + 16 q^{19} - 18 q^{21} - 4 q^{28} + 14 q^{31} + 20 q^{37} - 12 q^{57} - 6 q^{63} + 22 q^{67} + 34 q^{73} - 32 q^{76} + 36 q^{81} - 36 q^{84} - 30 q^{93} - 10 q^{97}+O(q^{100}) \) 4 * q - 2 * q^7 + 12 * q^9 - 16 * q^16 + 16 * q^19 - 18 * q^21 - 4 * q^28 + 14 * q^31 + 20 * q^37 - 12 * q^57 - 6 * q^63 + 22 * q^67 + 34 * q^73 - 32 * q^76 + 36 * q^81 - 36 * q^84 - 30 * q^93 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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