LMFDB - Embedded newform 75.2.e.b.68.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,2,Mod(32,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("75.32");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	75.e (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:	$0.598878015160$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			68.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	75.68
Dual form			75.2.e.b.32.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 + 1.22474i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 + 1.22474i) q^{7} +3.00000i q^{9} +(2.44949 - 2.44949i) q^{12} +(-3.67423 - 3.67423i) q^{13} -4.00000 q^{16} +1.00000i q^{19} -3.00000 q^{21} +(-3.67423 + 3.67423i) q^{27} +(2.44949 + 2.44949i) q^{28} +7.00000 q^{31} +6.00000 q^{36} +(4.89898 - 4.89898i) q^{37} -9.00000i q^{39} +(8.57321 + 8.57321i) q^{43} +(-4.89898 - 4.89898i) q^{48} +4.00000i q^{49} +(-7.34847 + 7.34847i) q^{52} +(-1.22474 + 1.22474i) q^{57} -13.0000 q^{61} +(-3.67423 - 3.67423i) q^{63} +8.00000i q^{64} +(11.0227 - 11.0227i) q^{67} +(-9.79796 - 9.79796i) q^{73} +2.00000 q^{76} -4.00000i q^{79} -9.00000 q^{81} +6.00000i q^{84} +9.00000 q^{91} +(8.57321 + 8.57321i) q^{93} +(-13.4722 + 13.4722i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 16 q^{16} - 12 q^{21} + 28 q^{31} + 24 q^{36} - 52 q^{61} + 8 q^{76} - 36 q^{81} + 36 q^{91}+O(q^{100}) $ 4 * q - 16 * q^16 - 12 * q^21 + 28 * q^31 + 24 * q^36 - 52 * q^61 + 8 * q^76 - 36 * q^81 + 36 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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	$-0.250000\pi$
$2$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$4$		−	2.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.22474	+	1.22474i	−0.462910	+	0.462910i	−0.899608	−	0.436698i	$-0.856148\pi$
$7$	−1.22474	+	1.22474i	−0.462910	+	0.462910i	0.436698	+	0.899608i	$0.356148\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	2.44949	−	2.44949i	0.707107	−	0.707107i
$13$	−3.67423	−	3.67423i	−1.01905	−	1.01905i	−0.999815	−	0.0192343i	$-0.993877\pi$
$13$	−3.67423	−	3.67423i	−1.01905	−	1.01905i	−0.0192343	−	0.999815i	$-0.506123\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$			1.00000i			0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$			1.00000i			0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$		0			0
$21$	−3.00000			−0.654654
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.707107	+	0.707107i
$28$	2.44949	+	2.44949i	0.462910	+	0.462910i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	7.00000			1.25724			0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000			1.25724			0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	6.00000			1.00000
$37$	4.89898	−	4.89898i	0.805387	−	0.805387i	−0.178545	−	0.983932i	$-0.557139\pi$
$37$	4.89898	−	4.89898i	0.805387	−	0.805387i	0.983932	+	0.178545i	$0.0571389\pi$
$38$		0			0
$39$		−	9.00000i		−	1.44115i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	8.57321	+	8.57321i	1.30740	+	1.30740i	0.923283	+	0.384120i	$0.125495\pi$
$43$	8.57321	+	8.57321i	1.30740	+	1.30740i	0.384120	+	0.923283i	$0.374505\pi$
$44$		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	$0.750000\pi$
$48$	−4.89898	−	4.89898i	−0.707107	−	0.707107i
$49$			4.00000i			0.571429i
$50$		0			0
$51$		0			0
$52$	−7.34847	+	7.34847i	−1.01905	+	1.01905i
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.22474	+	1.22474i	−0.162221	+	0.162221i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−13.0000			−1.66448			−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000			−1.66448			−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$	−3.67423	−	3.67423i	−0.462910	−	0.462910i
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	11.0227	−	11.0227i	1.34664	−	1.34664i	0.457352	−	0.889286i	$-0.348798\pi$
$67$	11.0227	−	11.0227i	1.34664	−	1.34664i	0.889286	−	0.457352i	$-0.151202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−9.79796	−	9.79796i	−1.14676	−	1.14676i	−0.987185	−	0.159579i	$-0.948986\pi$
$73$	−9.79796	−	9.79796i	−1.14676	−	1.14676i	−0.159579	−	0.987185i	$-0.551014\pi$
$74$		0			0
$75$		0			0
$76$	2.00000			0.229416
$77$		0			0
$78$		0			0
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		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	$0.250000\pi$
$84$			6.00000i			0.654654i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	9.00000			0.943456
$92$		0			0
$93$	8.57321	+	8.57321i	0.889001	+	0.889001i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−13.4722	+	13.4722i	−1.36789	+	1.36789i	−0.504457	+	0.863437i	$0.668307\pi$
$97$	−13.4722	+	13.4722i	−1.36789	+	1.36789i	−0.863437	+	0.504457i	$0.831693\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	2.44949	+	2.44949i	0.241355	+	0.241355i	0.817411	−	0.576055i	$-0.195409\pi$
$103$	2.44949	+	2.44949i	0.241355	+	0.241355i	−0.576055	+	0.817411i	$0.695409\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$	7.34847	+	7.34847i	0.707107	+	0.707107i
$109$		−	19.0000i		−	1.81987i	−0.414751	−	0.909935i	$-0.636131\pi$
$109$		−	19.0000i		−	1.81987i	0.414751	−	0.909935i	$-0.363869\pi$
$110$		0			0
$111$	12.0000			1.13899
$112$	4.89898	−	4.89898i	0.462910	−	0.462910i
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	11.0227	−	11.0227i	1.01905	−	1.01905i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		−	14.0000i		−	1.25724i
$125$		0			0
$126$		0			0
$127$	−7.34847	+	7.34847i	−0.652071	+	0.652071i	−0.953491	−	0.301420i	$-0.902539\pi$
$127$	−7.34847	+	7.34847i	−0.652071	+	0.652071i	0.301420	+	0.953491i	$0.402539\pi$
$128$		0			0
$129$			21.0000i			1.84895i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	−1.22474	−	1.22474i	−0.106199	−	0.106199i
$134$		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
$139$			16.0000i			1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$			16.0000i			1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		−	12.0000i		−	1.00000i
$145$		0			0
$146$		0			0
$147$	−4.89898	+	4.89898i	−0.404061	+	0.404061i
$148$	−9.79796	−	9.79796i	−0.805387	−	0.805387i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−23.0000			−1.87171			−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−23.0000			−1.87171			−0.935857	+	0.352381i	$0.885372\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−18.0000			−1.44115
$157$	−1.22474	+	1.22474i	−0.0977453	+	0.0977453i	−0.754288	−	0.656543i	$-0.772018\pi$
$157$	−1.22474	+	1.22474i	−0.0977453	+	0.0977453i	0.656543	+	0.754288i	$0.272018\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3.67423	−	3.67423i	−0.287788	−	0.287788i	0.548417	−	0.836205i	$-0.315231\pi$
$163$	−3.67423	−	3.67423i	−0.287788	−	0.287788i	−0.836205	+	0.548417i	$0.815231\pi$
$164$		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
$169$			14.0000i			1.07692i
$170$		0			0
$171$	−3.00000			−0.229416
$172$	17.1464	−	17.1464i	1.30740	−	1.30740i
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$		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
$181$	7.00000			0.520306			0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000			0.520306			0.260153	+	0.965567i	$0.416227\pi$
$182$		0			0
$183$	−15.9217	−	15.9217i	−1.17696	−	1.17696i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	9.00000i		−	0.654654i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−9.79796	+	9.79796i	−0.707107	+	0.707107i
$193$	8.57321	+	8.57321i	0.617113	+	0.617113i	0.944790	−	0.327677i	$-0.106266\pi$
$193$	8.57321	+	8.57321i	0.617113	+	0.617113i	−0.327677	+	0.944790i	$0.606266\pi$
$194$		0			0
$195$		0			0
$196$	8.00000			0.571429
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$			11.0000i			0.779769i	0.920864	+	0.389885i	$0.127485\pi$
$199$			11.0000i			0.779769i	−0.920864	+	0.389885i	$0.872515\pi$
$200$		0			0
$201$	27.0000			1.90443
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	14.6969	+	14.6969i	1.01905	+	1.01905i
$209$		0			0
$210$		0			0
$211$	−13.0000			−0.894957			−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000			−0.894957			−0.447478	+	0.894295i	$0.647678\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−8.57321	+	8.57321i	−0.581988	+	0.581988i
$218$		0			0
$219$		−	24.0000i		−	1.62177i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	20.8207	+	20.8207i	1.39425	+	1.39425i	0.815506	+	0.578749i	$0.196459\pi$
$223$	20.8207	+	20.8207i	1.39425	+	1.39425i	0.578749	+	0.815506i	$0.303541\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$	2.44949	+	2.44949i	0.162221	+	0.162221i
$229$		−	29.0000i		−	1.91637i	−0.286143	−	0.958187i	$-0.592373\pi$
$229$		−	29.0000i		−	1.91637i	0.286143	−	0.958187i	$-0.407627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	4.89898	−	4.89898i	0.318223	−	0.318223i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	17.0000			1.09507			0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000			1.09507			0.547533	+	0.836784i	$0.315567\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.707107	−	0.707107i
$244$			26.0000i			1.66448i
$245$		0			0
$246$		0			0
$247$	3.67423	−	3.67423i	0.233786	−	0.233786i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−7.34847	+	7.34847i	−0.462910	+	0.462910i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$			12.0000i			0.745644i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−22.0454	−	22.0454i	−1.34664	−	1.34664i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$		0			0
$273$	11.0227	+	11.0227i	0.667124	+	0.667124i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	23.2702	−	23.2702i	1.39817	−	1.39817i	0.592869	−	0.805299i	$-0.297995\pi$
$277$	23.2702	−	23.2702i	1.39817	−	1.39817i	0.805299	−	0.592869i	$-0.202005\pi$
$278$		0			0
$279$			21.0000i			1.25724i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−15.9217	−	15.9217i	−0.946446	−	0.946446i	0.0521913	−	0.998637i	$-0.483379\pi$
$283$	−15.9217	−	15.9217i	−0.946446	−	0.946446i	−0.998637	+	0.0521913i	$0.983379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$	−33.0000			−1.93449
$292$	−19.5959	+	19.5959i	−1.14676	+	1.14676i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−21.0000			−1.21042
$302$		0			0
$303$		0			0
$304$		−	4.00000i		−	0.229416i
$305$		0			0
$306$		0			0
$307$	−1.22474	+	1.22474i	−0.0698999	+	0.0698999i	−0.741192	−	0.671293i	$-0.765739\pi$
$307$	−1.22474	+	1.22474i	−0.0698999	+	0.0698999i	0.671293	+	0.741192i	$0.265739\pi$
$308$		0			0
$309$			6.00000i			0.341328i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−3.67423	−	3.67423i	−0.207680	−	0.207680i	0.595601	−	0.803281i	$-0.296914\pi$
$313$	−3.67423	−	3.67423i	−0.207680	−	0.207680i	−0.803281	+	0.595601i	$0.796914\pi$
$314$		0			0
$315$		0			0
$316$	−8.00000			−0.450035
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		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
$327$	23.2702	−	23.2702i	1.28684	−	1.28684i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	32.0000			1.75888			0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000			1.75888			0.879440	+	0.476011i	$0.157918\pi$
$332$		0			0
$333$	14.6969	+	14.6969i	0.805387	+	0.805387i
$334$		0			0
$335$		0			0
$336$	12.0000			0.654654
$337$	−25.7196	+	25.7196i	−1.40104	+	1.40104i	−0.604223	+	0.796815i	$0.706516\pi$
$337$	−25.7196	+	25.7196i	−1.40104	+	1.40104i	−0.796815	+	0.604223i	$0.793484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−13.4722	−	13.4722i	−0.727430	−	0.727430i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	14.0000i		−	0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$		−	14.0000i		−	0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$		0			0
$351$	27.0000			1.44115
$352$		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			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$	13.4722	+	13.4722i	0.707107	+	0.707107i
$364$		−	18.0000i		−	0.943456i
$365$		0			0
$366$		0			0
$367$	11.0227	−	11.0227i	0.575380	−	0.575380i	−0.358247	−	0.933627i	$-0.616625\pi$
$367$	11.0227	−	11.0227i	0.575380	−	0.575380i	0.933627	+	0.358247i	$0.116625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	17.1464	−	17.1464i	0.889001	−	0.889001i
$373$	20.8207	+	20.8207i	1.07805	+	1.07805i	0.996684	+	0.0813690i	$0.0259292\pi$
$373$	20.8207	+	20.8207i	1.07805	+	1.07805i	0.0813690	+	0.996684i	$0.474071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	29.0000i		−	1.48963i	−0.667271	−	0.744815i	$-0.732538\pi$
$379$		−	29.0000i		−	1.48963i	0.667271	−	0.744815i	$-0.267462\pi$
$380$		0			0
$381$	−18.0000			−0.922168
$382$		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
$387$	−25.7196	+	25.7196i	−1.30740	+	1.30740i
$388$	26.9444	+	26.9444i	1.36789	+	1.36789i
$389$		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
$397$	−13.4722	+	13.4722i	−0.676150	+	0.676150i	−0.959127	−	0.282977i	$-0.908678\pi$
$397$	−13.4722	+	13.4722i	−0.676150	+	0.676150i	0.282977	+	0.959127i	$0.408678\pi$
$398$		0			0
$399$		−	3.00000i		−	0.150188i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−25.7196	−	25.7196i	−1.28119	−	1.28119i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			31.0000i			1.53285i	0.642333	+	0.766426i	$0.277967\pi$
$409$			31.0000i			1.53285i	−0.642333	+	0.766426i	$0.722033\pi$
$410$		0			0
$411$		0			0
$412$	4.89898	−	4.89898i	0.241355	−	0.241355i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−19.5959	+	19.5959i	−0.959616	+	0.959616i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	15.9217	−	15.9217i	0.770504	−	0.770504i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	14.6969	−	14.6969i	0.707107	−	0.707107i
$433$	−15.9217	−	15.9217i	−0.765147	−	0.765147i	0.212101	−	0.977248i	$-0.431970\pi$
$433$	−15.9217	−	15.9217i	−0.765147	−	0.765147i	−0.977248	+	0.212101i	$0.931970\pi$
$434$		0			0
$435$		0			0
$436$	−38.0000			−1.81987
$437$		0			0
$438$		0			0
$439$			41.0000i			1.95682i	0.206666	+	0.978412i	$0.433739\pi$
$439$			41.0000i			1.95682i	−0.206666	+	0.978412i	$0.566261\pi$
$440$		0			0
$441$	−12.0000			−0.571429
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		−	24.0000i		−	1.13899i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−9.79796	−	9.79796i	−0.462910	−	0.462910i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−28.1691	−	28.1691i	−1.32350	−	1.32350i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	29.3939	−	29.3939i	1.37499	−	1.37499i	0.522108	−	0.852879i	$-0.325146\pi$
$457$	29.3939	−	29.3939i	1.37499	−	1.37499i	0.852879	−	0.522108i	$-0.174854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	26.9444	+	26.9444i	1.25221	+	1.25221i	0.954726	+	0.297486i	$0.0961480\pi$
$463$	26.9444	+	26.9444i	1.25221	+	1.25221i	0.297486	+	0.954726i	$0.403852\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	−22.0454	−	22.0454i	−1.01905	−	1.01905i
$469$			27.0000i			1.24674i
$470$		0			0
$471$	−3.00000			−0.138233
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−36.0000			−1.64146
$482$		0			0
$483$		0			0
$484$		−	22.0000i		−	1.00000i
$485$		0			0
$486$		0			0
$487$	−25.7196	+	25.7196i	−1.16547	+	1.16547i	−0.182208	+	0.983260i	$0.558325\pi$
$487$	−25.7196	+	25.7196i	−1.16547	+	1.16547i	−0.983260	+	0.182208i	$0.941675\pi$
$488$		0			0
$489$		−	9.00000i		−	0.406994i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−28.0000			−1.25724
$497$		0			0
$498$		0			0
$499$			11.0000i			0.492428i	0.969216	+	0.246214i	$0.0791865\pi$
$499$			11.0000i			0.492428i	−0.969216	+	0.246214i	$0.920813\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−17.1464	+	17.1464i	−0.761500	+	0.761500i
$508$	14.6969	+	14.6969i	0.652071	+	0.652071i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	24.0000			1.06170
$512$		0			0
$513$	−3.67423	−	3.67423i	−0.162221	−	0.162221i
$514$		0			0
$515$		0			0
$516$	42.0000			1.84895
$517$		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
$523$	20.8207	+	20.8207i	0.910424	+	0.910424i	0.996305	−	0.0858814i	$-0.0273706\pi$
$523$	20.8207	+	20.8207i	0.910424	+	0.910424i	−0.0858814	+	0.996305i	$0.527371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			23.0000i			1.00000i
$530$		0			0
$531$		0			0
$532$	−2.44949	+	2.44949i	−0.106199	+	0.106199i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$	8.57321	+	8.57321i	0.367912	+	0.367912i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	17.1464	−	17.1464i	0.733128	−	0.733128i	−0.238110	−	0.971238i	$-0.576528\pi$
$547$	17.1464	−	17.1464i	0.733128	−	0.733128i	0.971238	+	0.238110i	$0.0765278\pi$
$548$		0			0
$549$		−	39.0000i		−	1.66448i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	4.89898	+	4.89898i	0.208326	+	0.208326i
$554$		0			0
$555$		0			0
$556$	32.0000			1.35710
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		−	63.0000i		−	2.66462i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	11.0227	−	11.0227i	0.462910	−	0.462910i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	47.0000			1.96689			0.983444	−	0.181210i	$-0.0580014\pi$
$571$	47.0000			1.96689			0.983444	+	0.181210i	$0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.0000			−1.00000
$577$	23.2702	−	23.2702i	0.968749	−	0.968749i	−0.0307771	−	0.999526i	$-0.509798\pi$
$577$	23.2702	−	23.2702i	0.968749	−	0.968749i	0.999526	+	0.0307771i	$0.00979822\pi$
$578$		0			0
$579$			21.0000i			0.872730i
$580$		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	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$	9.79796	+	9.79796i	0.404061	+	0.404061i
$589$			7.00000i			0.288430i
$590$		0			0
$591$		0			0
$592$	−19.5959	+	19.5959i	−0.805387	+	0.805387i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−13.4722	+	13.4722i	−0.551380	+	0.551380i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−23.0000			−0.938190			−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000			−0.938190			−0.469095	+	0.883148i	$0.655420\pi$
$602$		0			0
$603$	33.0681	+	33.0681i	1.34664	+	1.34664i
$604$			46.0000i			1.87171i
$605$		0			0
$606$		0			0
$607$	−31.8434	+	31.8434i	−1.29248	+	1.29248i	−0.359235	+	0.933247i	$0.616962\pi$
$607$	−31.8434	+	31.8434i	−1.29248	+	1.29248i	−0.933247	+	0.359235i	$0.883038\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−34.2929	−	34.2929i	−1.38508	−	1.38508i	−0.835337	−	0.549739i	$-0.814727\pi$
$613$	−34.2929	−	34.2929i	−1.38508	−	1.38508i	−0.549739	−	0.835337i	$-0.685273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		−	49.0000i		−	1.96948i	−0.174042	−	0.984738i	$-0.555683\pi$
$619$		−	49.0000i		−	1.96948i	0.174042	−	0.984738i	$-0.444317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$			36.0000i			1.44115i
$625$		0			0
$626$		0			0
$627$		0			0
$628$	2.44949	+	2.44949i	0.0977453	+	0.0977453i
$629$		0			0
$630$		0			0
$631$	−43.0000			−1.71180			−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000			−1.71180			−0.855901	+	0.517139i	$0.826997\pi$
$632$		0			0
$633$	−15.9217	−	15.9217i	−0.632830	−	0.632830i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	14.6969	−	14.6969i	0.582314	−	0.582314i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	−22.0454	−	22.0454i	−0.869386	−	0.869386i	0.123018	−	0.992404i	$-0.460743\pi$
$643$	−22.0454	−	22.0454i	−0.869386	−	0.869386i	−0.992404	+	0.123018i	$0.960743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−21.0000			−0.823055
$652$	−7.34847	+	7.34847i	−0.287788	+	0.287788i
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	29.3939	−	29.3939i	1.14676	−	1.14676i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−38.0000			−1.47803			−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000			−1.47803			−0.739014	+	0.673690i	$0.764708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			51.0000i			1.97177i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−9.79796	−	9.79796i	−0.377684	−	0.377684i	0.492582	−	0.870266i	$-0.336053\pi$
$673$	−9.79796	−	9.79796i	−0.377684	−	0.377684i	−0.870266	+	0.492582i	$0.836053\pi$
$674$		0			0
$675$		0			0
$676$	28.0000			1.07692
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		−	33.0000i		−	1.26642i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$			6.00000i			0.229416i
$685$		0			0
$686$		0			0
$687$	35.5176	−	35.5176i	1.35508	−	1.35508i
$688$	−34.2929	−	34.2929i	−1.30740	−	1.30740i
$689$		0			0
$690$		0			0
$691$	−8.00000			−0.304334			−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000			−0.304334			−0.152167	+	0.988355i	$0.548625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	4.89898	+	4.89898i	0.184769	+	0.184769i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			31.0000i			1.16423i	0.813107	+	0.582115i	$0.197775\pi$
$709$			31.0000i			1.16423i	−0.813107	+	0.582115i	$0.802225\pi$
$710$		0			0
$711$	12.0000			0.450035
$712$		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
$721$	−6.00000			−0.223452
$722$		0			0
$723$	20.8207	+	20.8207i	0.774329	+	0.774329i
$724$		−	14.0000i		−	0.520306i
$725$		0			0
$726$		0			0
$727$	−37.9671	+	37.9671i	−1.40812	+	1.40812i	−0.638498	+	0.769624i	$0.720444\pi$
$727$	−37.9671	+	37.9671i	−1.40812	+	1.40812i	−0.769624	+	0.638498i	$0.779556\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$	−31.8434	+	31.8434i	−1.17696	+	1.17696i
$733$	14.6969	+	14.6969i	0.542844	+	0.542844i	0.924362	−	0.381518i	$-0.124598\pi$
$733$	14.6969	+	14.6969i	0.542844	+	0.542844i	−0.381518	+	0.924362i	$0.624598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$			16.0000i			0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$			16.0000i			0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$		0			0
$741$	9.00000			0.330623
$742$		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
$751$	52.0000			1.89751			0.948753	−	0.316017i	$-0.102346\pi$
$751$	52.0000			1.89751			0.948753	+	0.316017i	$0.102346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−18.0000			−0.654654
$757$	−1.22474	+	1.22474i	−0.0445141	+	0.0445141i	−0.729013	−	0.684499i	$-0.760021\pi$
$757$	−1.22474	+	1.22474i	−0.0445141	+	0.0445141i	0.684499	+	0.729013i	$0.260021\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	23.2702	+	23.2702i	0.842436	+	0.842436i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	19.5959	+	19.5959i	0.707107	+	0.707107i
$769$		−	49.0000i		−	1.76699i	−0.468445	−	0.883493i	$-0.655186\pi$
$769$		−	49.0000i		−	1.76699i	0.468445	−	0.883493i	$-0.344814\pi$
$770$		0			0
$771$		0			0
$772$	17.1464	−	17.1464i	0.617113	−	0.617113i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−14.6969	+	14.6969i	−0.527250	+	0.527250i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	16.0000i		−	0.571429i
$785$		0			0
$786$		0			0
$787$	35.5176	−	35.5176i	1.26607	−	1.26607i	0.317962	−	0.948103i	$-0.397001\pi$
$787$	35.5176	−	35.5176i	1.26607	−	1.26607i	0.948103	−	0.317962i	$-0.102999\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	47.7650	+	47.7650i	1.69619	+	1.69619i
$794$		0			0
$795$		0			0
$796$	22.0000			0.779769
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−

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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 + 1.22474i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(-1.22474 + 1.22474i) q^{7} +3.00000i q^{9} +(2.44949 - 2.44949i) q^{12} +(-3.67423 - 3.67423i) q^{13} -4.00000 q^{16} +1.00000i q^{19} -3.00000 q^{21} +(-3.67423 + 3.67423i) q^{27} +(2.44949 + 2.44949i) q^{28} +7.00000 q^{31} +6.00000 q^{36} +(4.89898 - 4.89898i) q^{37} -9.00000i q^{39} +(8.57321 + 8.57321i) q^{43} +(-4.89898 - 4.89898i) q^{48} +4.00000i q^{49} +(-7.34847 + 7.34847i) q^{52} +(-1.22474 + 1.22474i) q^{57} -13.0000 q^{61} +(-3.67423 - 3.67423i) q^{63} +8.00000i q^{64} +(11.0227 - 11.0227i) q^{67} +(-9.79796 - 9.79796i) q^{73} +2.00000 q^{76} -4.00000i q^{79} -9.00000 q^{81} +6.00000i q^{84} +9.00000 q^{91} +(8.57321 + 8.57321i) q^{93} +(-13.4722 + 13.4722i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 16 q^{16} - 12 q^{21} + 28 q^{31} + 24 q^{36} - 52 q^{61} + 8 q^{76} - 36 q^{81} + 36 q^{91}+O(q^{100}) \) 4 * q - 16 * q^16 - 12 * q^21 + 28 * q^31 + 24 * q^36 - 52 * q^61 + 8 * q^76 - 36 * q^81 + 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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