LMFDB - Embedded newform 63.6.e.b.46.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,6,Mod(37,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("63.37");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	63.e (of order $3$, 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:	$10.1041806482$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			46.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	63.46
Dual form			63.6.e.b.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(16.0000 - 27.7128i) q^{4} +(-105.500 - 75.3442i) q^{7} +O(q^{10})$	$q+(16.0000 - 27.7128i) q^{4} +(-105.500 - 75.3442i) q^{7} -427.000 q^{13} +(-512.000 - 886.810i) q^{16} +(-1571.50 - 2721.92i) q^{19} +(1562.50 - 2706.33i) q^{25} +(-3776.00 + 1718.19i) q^{28} +(-1361.50 + 2358.19i) q^{31} +(3330.50 + 5768.60i) q^{37} +22475.0 q^{43} +(5453.50 + 15897.6i) q^{49} +(-6832.00 + 11833.4i) q^{52} +(19313.0 + 33451.1i) q^{61} -32768.0 q^{64} +(18969.5 - 32856.1i) q^{67} +(39063.5 - 67660.0i) q^{73} -100576. q^{76} +(-45428.5 - 78684.5i) q^{79} +(45048.5 + 32172.0i) q^{91} -134386. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{4} - 211 q^{7}+O(q^{10}) $ 2 * q + 32 * q^4 - 211 * q^7	$ 2 q + 32 q^{4} - 211 q^{7} - 854 q^{13} - 1024 q^{16} - 3143 q^{19} + 3125 q^{25} - 7552 q^{28} - 2723 q^{31} + 6661 q^{37} + 44950 q^{43} + 10907 q^{49} - 13664 q^{52} + 38626 q^{61} - 65536 q^{64} + 37939 q^{67} + 78127 q^{73} - 201152 q^{76} - 90857 q^{79} + 90097 q^{91} - 268772 q^{97}+O(q^{100}) $ 2 * q + 32 * q^4 - 211 * q^7 - 854 * q^13 - 1024 * q^16 - 3143 * q^19 + 3125 * q^25 - 7552 * q^28 - 2723 * q^31 + 6661 * q^37 + 44950 * q^43 + 10907 * q^49 - 13664 * q^52 + 38626 * q^61 - 65536 * q^64 + 37939 * q^67 + 78127 * q^73 - 201152 * q^76 - 90857 * q^79 + 90097 * q^91 - 268772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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			0		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$		0			0
$3$		0			0
$4$	16.0000	−	27.7128i	0.500000	−	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−105.500	−	75.3442i	−0.813781	−	0.581172i
$7$	−105.500	−	75.3442i	−0.813781	−	0.581172i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	−427.000			−0.700760			−0.350380	−	0.936608i	$-0.613948\pi$
$13$	−427.000			−0.700760			−0.350380	+	0.936608i	$0.613948\pi$
$14$		0			0
$15$		0			0
$16$	−512.000	−	886.810i	−0.500000	−	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−1571.50	−	2721.92i	−0.998689	−	1.72978i	−0.543671	−	0.839299i	$-0.682966\pi$
$19$	−1571.50	−	2721.92i	−0.998689	−	1.72978i	−0.455018	−	0.890482i	$-0.650367\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	1562.50	−	2706.33i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$	−3776.00	+	1718.19i	−0.910200	+	0.414169i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−1361.50	+	2358.19i	−0.254456	+	0.440731i	−0.964748	−	0.263176i	$-0.915230\pi$
$31$	−1361.50	+	2358.19i	−0.254456	+	0.440731i	0.710291	+	0.703908i	$0.248563\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3330.50	+	5768.60i	0.399949	+	0.692733i	0.993719	−	0.111902i	$-0.0356943\pi$
$37$	3330.50	+	5768.60i	0.399949	+	0.692733i	−0.593770	+	0.804635i	$0.702361\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	22475.0			1.85365			0.926827	−	0.375489i	$-0.122525\pi$
$43$	22475.0			1.85365			0.926827	+	0.375489i	$0.122525\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	5453.50	+	15897.6i	0.324478	+	0.945893i
$50$		0			0
$51$		0			0
$52$	−6832.00	+	11833.4i	−0.350380	+	0.606876i
$53$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$53$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	19313.0	+	33451.1i	0.664546	+	1.15103i	0.979408	+	0.201890i	$0.0647084\pi$
$61$	19313.0	+	33451.1i	0.664546	+	1.15103i	−0.314862	+	0.949137i	$0.601958\pi$
$62$		0			0
$63$		0			0
$64$	−32768.0			−1.00000
$65$		0			0
$66$		0			0
$67$	18969.5	−	32856.1i	0.516260	−	0.894189i	−0.483561	−	0.875310i	$-0.660657\pi$
$67$	18969.5	−	32856.1i	0.516260	−	0.894189i	0.999822	−	0.0188789i	$-0.00600969\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	39063.5	−	67660.0i	0.857954	−	1.48602i	−0.0159232	−	0.999873i	$-0.505069\pi$
$73$	39063.5	−	67660.0i	0.857954	−	1.48602i	0.873877	−	0.486147i	$-0.161598\pi$
$74$		0			0
$75$		0			0
$76$	−100576.			−1.99738
$77$		0			0
$78$		0			0
$79$	−45428.5	−	78684.5i	−0.818956	−	1.41847i	−0.906452	−	0.422309i	$-0.861220\pi$
$79$	−45428.5	−	78684.5i	−0.818956	−	1.41847i	0.0874958	−	0.996165i	$-0.472114\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	45048.5	+	32172.0i	0.570265	+	0.407262i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−134386.			−1.45019			−0.725095	−	0.688649i	$-0.758204\pi$
$97$	−134386.			−1.45019			−0.725095	+	0.688649i	$0.758204\pi$
$98$		0			0
$99$		0			0
$100$	−50000.0	−	86602.5i	−0.500000	−	0.866025i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	105738.	+	183144.i	0.982065	+	1.70099i	0.654317	+	0.756221i	$0.272956\pi$
$103$	105738.	+	183144.i	0.982065	+	1.70099i	0.327748	+	0.944765i	$0.393710\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$	123922.	−	214638.i	0.999034	−	1.73038i	0.461467	−	0.887157i	$-0.347323\pi$
$109$	123922.	−	214638.i	0.999034	−	1.73038i	0.537567	−	0.843221i	$-0.319344\pi$
$110$		0			0
$111$		0			0
$112$	−12800.0	+	132135.i	−0.0964195	+	0.995341i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	80525.5	+	139474.i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	43568.0	+	75462.0i	0.254456	+	0.440731i
$125$		0			0
$126$		0			0
$127$	347111.			1.90967			0.954837	−	0.297131i	$-0.0960299\pi$
$127$	347111.			1.90967			0.954837	+	0.297131i	$0.0960299\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	−39287.5	+	405566.i	−0.192586	+	1.98807i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	−454657.			−1.99594			−0.997969	−	0.0637074i	$-0.979708\pi$
$139$	−454657.			−1.99594			−0.997969	+	0.0637074i	$0.979708\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	213152.			0.799899
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	204362.	−	353965.i	0.729387	−	1.26333i	−0.227756	−	0.973718i	$-0.573139\pi$
$151$	204362.	−	353965.i	0.729387	−	1.26333i	0.957143	−	0.289616i	$-0.0935277\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−54607.0	+	94582.1i	−0.176807	+	0.306239i	−0.940785	−	0.339004i	$-0.889910\pi$
$157$	−54607.0	+	94582.1i	−0.176807	+	0.306239i	0.763978	+	0.645242i	$0.223243\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−339124.	−	587380.i	−0.999746	−	1.73161i	−0.519405	−	0.854528i	$-0.673846\pi$
$163$	−339124.	−	587380.i	−0.999746	−	1.73161i	−0.480341	−	0.877082i	$-0.659487\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−188964.			−0.508935
$170$		0			0
$171$		0			0
$172$	359600.	−	622845.i	0.926827	−	1.60531i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−368750.	+	167792.i	−0.910200	+	0.414169i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−853027.			−1.93538			−0.967690	−	0.252142i	$-0.918865\pi$
$181$	−853027.			−1.93538			−0.967690	+	0.252142i	$0.918865\pi$
$182$		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		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	−328188.	+	568437.i	−0.634204	+	1.09847i	0.352480	+	0.935820i	$0.385339\pi$
$193$	−328188.	+	568437.i	−0.634204	+	1.09847i	−0.986683	+	0.162653i	$0.947995\pi$
$194$		0			0
$195$		0			0
$196$	527824.	+	103230.i	0.981407	+	0.191941i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	456134.	−	790047.i	0.816507	−	1.41423i	−0.0917343	−	0.995784i	$-0.529241\pi$
$199$	456134.	−	790047.i	0.816507	−	1.41423i	0.908241	−	0.418448i	$-0.137426\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	218624.	+	378668.i	0.350380	+	0.606876i
$209$		0			0
$210$		0			0
$211$	−288976.			−0.446844			−0.223422	−	0.974722i	$-0.571723\pi$
$211$	−288976.			−0.446844			−0.223422	+	0.974722i	$0.571723\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	321314.	−	146208.i	0.463213	−	0.210776i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	304052.			0.409436			0.204718	−	0.978821i	$-0.434372\pi$
$223$	304052.			0.409436			0.204718	+	0.978821i	$0.434372\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	729754.	+	1.26397e6i	0.919576	+	1.59275i	0.800060	+	0.599919i	$0.204801\pi$
$229$	729754.	+	1.26397e6i	0.919576	+	1.59275i	0.119515	+	0.992832i	$0.461866\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−648487.	+	1.12321e6i	−0.719215	+	1.24572i	0.242096	+	0.970252i	$0.422165\pi$
$241$	−648487.	+	1.12321e6i	−0.719215	+	1.24572i	−0.961311	+	0.275465i	$0.911168\pi$
$242$		0			0
$243$		0			0
$244$	1.23603e6			1.32909
$245$		0			0
$246$		0			0
$247$	671030.	+	1.16226e6i	0.699842	+	1.21216i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−524288.	+	908093.i	−0.500000	+	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	83262.5	−	859521.i	0.0771259	−	0.796172i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−607024.	−	1.05140e6i	−0.516260	−	0.894189i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	−1.12643e6	−	1.95103e6i	−0.931707	−	1.61376i	−0.780403	−	0.625277i	$-0.784986\pi$
$271$	−1.12643e6	−	1.95103e6i	−0.931707	−	1.61376i	−0.151304	−	0.988487i	$-0.548347\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−369194.	+	639464.i	−0.289105	+	0.500745i	−0.973596	−	0.228276i	$-0.926691\pi$
$277$	−369194.	+	639464.i	−0.289105	+	0.500745i	0.684491	+	0.729021i	$0.260024\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	1.16729e6	−	2.02180e6i	0.866387	−	1.50063i	0.000724409	−	1.00000i	$-0.499769\pi$
$283$	1.16729e6	−	2.02180e6i	0.866387	−	1.50063i	0.865663	−	0.500627i	$-0.166897\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	709928.	+	1.22963e6i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$	−1.25003e6	−	2.16512e6i	−0.857954	−	1.48602i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−2.37111e6	−	1.69336e6i	−1.50847	−	1.07729i
$302$		0			0
$303$		0			0
$304$	−1.60922e6	+	2.78724e6i	−0.998689	+	1.72978i
$305$		0			0
$306$		0			0
$307$	3.20232e6			1.93919			0.969593	−	0.244723i	$-0.0786971\pi$
$307$	3.20232e6			1.93919			0.969593	+	0.244723i	$0.0786971\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−1.28354e6	−	2.22315e6i	−0.740539	−	1.28265i	−0.952250	−	0.305318i	$-0.901237\pi$
$313$	−1.28354e6	−	2.22315e6i	−0.740539	−	1.28265i	0.211712	−	0.977332i	$-0.432096\pi$
$314$		0			0
$315$		0			0
$316$	−2.90742e6			−1.63791
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−667188.	+	1.15560e6i	−0.350380	+	0.606876i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	273076.	+	472981.i	0.136998	+	0.237287i	0.926359	−	0.376642i	$-0.122921\pi$
$331$	273076.	+	472981.i	0.136998	+	0.237287i	−0.789361	+	0.613929i	$0.789588\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.63172e6			1.26231			0.631155	−	0.775657i	$-0.282581\pi$
$337$	2.63172e6			1.26231			0.631155	+	0.775657i	$0.282581\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	622450.	−	2.08809e6i	0.285673	−	0.958327i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	4.27561e6			1.87904			0.939518	−	0.342501i	$-0.111274\pi$
$349$	4.27561e6			1.87904			0.939518	+	0.342501i	$0.111274\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	−3.70118e6	+	6.41062e6i	−1.49476	+	2.58900i
$362$		0			0
$363$		0			0
$364$	1.61235e6	−	733669.i	0.637832	−	0.290233i
$365$		0			0
$366$		0			0
$367$	1.29159e6	−	2.23710e6i	0.500563	−	0.867000i	−0.499437	−	0.866350i	$-0.666460\pi$
$367$	1.29159e6	−	2.23710e6i	0.500563	−	0.867000i	1.00000		0.000650122i	$-0.000206940\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	2.65864e6	+	4.60489e6i	0.989434	+	1.71375i	0.620276	+	0.784384i	$0.287021\pi$
$373$	2.65864e6	+	4.60489e6i	0.989434	+	1.71375i	0.369158	+	0.929367i	$0.379646\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	69893.0			0.0249940			0.0124970	−	0.999922i	$-0.496022\pi$
$379$	69893.0			0.0249940			0.0124970	+	0.999922i	$0.496022\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−2.15018e6	+	3.72421e6i	−0.725095	+	1.25590i
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	2.50171e6	+	4.33309e6i	0.796638	+	1.37982i	0.921794	+	0.387681i	$0.126724\pi$
$397$	2.50171e6	+	4.33309e6i	0.796638	+	1.37982i	−0.125156	+	0.992137i	$0.539943\pi$
$398$		0			0
$399$		0			0
$400$	−3.20000e6			−1.00000
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	581361.	−	1.00695e6i	0.178313	−	0.308847i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	2.58510e6	−	4.47753e6i	0.764134	−	1.32352i	−0.176569	−	0.984288i	$-0.556500\pi$
$409$	2.58510e6	−	4.47753e6i	0.764134	−	1.32352i	0.940703	−	0.339231i	$-0.110167\pi$
$410$		0			0
$411$		0			0
$412$	6.76726e6			1.96413
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−4.87580e6			−1.34073			−0.670364	−	0.742033i	$-0.733862\pi$
$421$	−4.87580e6			−1.34073			−0.670364	+	0.742033i	$0.733862\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	482825.	−	4.98421e6i	0.128150	−	1.32290i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		0			0
$433$	−7.49192e6			−1.92032			−0.960160	−	0.279450i	$-0.909848\pi$
$433$	−7.49192e6			−1.92032			−0.960160	+	0.279450i	$0.909848\pi$
$434$		0			0
$435$		0			0
$436$	−3.96549e6	−	6.86843e6i	−0.999034	−	1.73038i
$437$		0			0
$438$		0			0
$439$	−1.99667e6	−	3.45833e6i	−0.494475	−	0.856455i	0.505505	−	0.862824i	$-0.331306\pi$
$439$	−1.99667e6	−	3.45833e6i	−0.494475	−	0.856455i	−0.999980	+	0.00636830i	$0.997973\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	3.45702e6	+	2.46888e6i	0.813781	+	0.581172i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−592880.	−	1.02690e6i	−0.132793	−	0.230005i	0.791959	−	0.610574i	$-0.209061\pi$
$457$	−592880.	−	1.02690e6i	−0.132793	−	0.230005i	−0.924752	+	0.380569i	$0.875728\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	2.92217e6			0.633510			0.316755	−	0.948507i	$-0.397407\pi$
$463$	2.92217e6			0.633510			0.316755	+	0.948507i	$0.397407\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	−4.47680e6	+	2.03708e6i	−0.939801	+	0.427638i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−9.82188e6			−1.99738
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	−1.42212e6	−	2.46319e6i	−0.280269	−	0.485440i
$482$		0			0
$483$		0			0
$484$	5.15363e6			1.00000
$485$		0			0
$486$		0			0
$487$	682437.	−	1.18202e6i	0.130389	−	0.225840i	−0.793438	−	0.608652i	$-0.791711\pi$
$487$	682437.	−	1.18202e6i	0.130389	−	0.225840i	0.923827	+	0.382811i	$0.125044\pi$
$488$		0			0
$489$		0			0
$490$		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
$496$	2.78835e6			0.508913
$497$		0			0
$498$		0			0
$499$	1.08685e6	+	1.88247e6i	0.195397	+	0.338437i	0.947030	−	0.321144i	$-0.104067\pi$
$499$	1.08685e6	+	1.88247e6i	0.195397	+	0.338437i	−0.751634	+	0.659581i	$0.770734\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	5.55378e6	−	9.61942e6i	0.954837	−	1.65383i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−9.21899e6	+	4.19492e6i	−1.56182	+	0.710675i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	5.86434e6	+	1.01573e7i	0.937486	+	1.62377i	0.770140	+	0.637875i	$0.220186\pi$
$523$	5.86434e6	+	1.01573e7i	0.937486	+	1.62377i	0.167346	+	0.985898i	$0.446480\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.21817e6	−	5.57404e6i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	1.06108e7	+	7.57782e6i	1.62543	+	1.16082i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.51118e6	+	2.61743e6i	0.221984	+	0.384488i	0.955410	−	0.295281i	$-0.0954134\pi$
$541$	1.51118e6	+	2.61743e6i	0.221984	+	0.384488i	−0.733426	+	0.679769i	$0.762080\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.27982e7			1.82886			0.914430	−	0.404744i	$-0.132639\pi$
$547$	1.27982e7			1.82886			0.914430	+	0.404744i	$0.132639\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.13571e6	+	1.17240e7i	−0.157927	+	1.63028i
$554$		0			0
$555$		0			0
$556$	−7.27451e6	+	1.25998e7i	−0.997969	+	1.72853i
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$	−9.59682e6			−1.29897
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	−2.94390e6	+	5.09898e6i	−0.377862	+	0.654475i	−0.990751	−	0.135693i	$-0.956674\pi$
$571$	−2.94390e6	+	5.09898e6i	−0.377862	+	0.654475i	0.612889	+	0.790169i	$0.290007\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	6.45496e6	−	1.11803e7i	0.807150	−	1.39802i	−0.107680	−	0.994186i	$-0.534342\pi$
$577$	6.45496e6	−	1.11803e7i	0.807150	−	1.39802i	0.914830	−	0.403839i	$-0.132324\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	8.55839e6			1.01649
$590$		0			0
$591$		0			0
$592$	3.41043e6	−	5.90704e6i	0.399949	−	0.692733i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−1.74342e7			−1.96887			−0.984435	−	0.175749i	$-0.943765\pi$
$601$	−1.74342e7			−1.96887			−0.984435	+	0.175749i	$0.943765\pi$
$602$		0			0
$603$		0			0
$604$	−6.53958e6	−	1.13269e7i	−0.729387	−	1.26333i
$605$		0			0
$606$		0			0
$607$	7.84027e6	+	1.35797e7i	0.863693	+	1.49596i	0.868339	+	0.495970i	$0.165188\pi$
$607$	7.84027e6	+	1.35797e7i	0.863693	+	1.49596i	−0.00464665	+	0.999989i	$0.501479\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	7.91172e6	−	1.37035e7i	0.850394	−	1.47292i	−0.0304599	−	0.999536i	$-0.509697\pi$
$613$	7.91172e6	−	1.37035e7i	0.850394	−	1.47292i	0.880853	−	0.473389i	$-0.156969\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	6.11302e6	−	1.05881e7i	0.641253	−	1.11068i	−0.343901	−	0.939006i	$-0.611748\pi$
$619$	6.11302e6	−	1.05881e7i	0.641253	−	1.11068i	0.985153	−	0.171676i	$-0.0549183\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−4.88281e6	−	8.45728e6i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	1.74742e6	+	3.02663e6i	0.176807	+	0.306239i
$629$		0			0
$630$		0			0
$631$	−1.62439e7			−1.62411			−0.812057	−	0.583579i	$-0.801652\pi$
$631$	−1.62439e7			−1.62411			−0.812057	+	0.583579i	$0.801652\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.32864e6	−	6.78829e6i	−0.227381	−	0.662844i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	−1.33848e7			−1.27668			−0.638342	−	0.769753i	$-0.720380\pi$
$643$	−1.33848e7			−1.27668			−0.638342	+	0.769753i	$0.720380\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−2.17039e7			−1.99949
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	345474.	−	598379.i	0.0307548	−	0.0532688i	−0.850238	−	0.526398i	$-0.823542\pi$
$661$	345474.	−	598379.i	0.0307548	−	0.0532688i	0.880993	+	0.473129i	$0.156876\pi$
$662$		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			0
$672$		0			0
$673$	1.73115e7			1.47332			0.736661	−	0.676262i	$-0.236401\pi$
$673$	1.73115e7			1.47332			0.736661	+	0.676262i	$0.236401\pi$
$674$		0			0
$675$		0			0
$676$	−3.02342e6	+	5.23672e6i	−0.254467	+	0.440751i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	1.41777e7	+	1.01252e7i	1.18014	+	0.842810i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$683$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.15072e7	−	1.99311e7i	−0.926827	−	1.60531i
$689$		0			0
$690$		0			0
$691$	−3.50960e6	−	6.07881e6i	−0.279616	−	0.484310i	0.691673	−	0.722211i	$-0.256874\pi$
$691$	−3.50960e6	−	6.07881e6i	−0.279616	−	0.484310i	−0.971289	+	0.237901i	$0.923541\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.25000e6	+	1.29038e7i	−0.0964195	+	0.995341i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	1.04678e7	−	1.81307e7i	0.798850	−	1.38365i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.13507e7	+	1.96600e7i	0.848021	+	1.46882i	0.882971	+	0.469427i	$0.155539\pi$
$709$	1.13507e7	+	1.96600e7i	0.848021	+	1.46882i	−0.0349502	+	0.999389i	$0.511127\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	2.64346e6	−	2.72885e7i	0.189380	−	1.95498i
$722$		0			0
$723$		0			0
$724$	−1.36484e7	+	2.36398e7i	−0.967690	+	1.67609i
$725$		0			0
$726$		0			0
$727$	1.56893e7			1.10095			0.550474	−	0.834853i	$-0.314447\pi$
$727$	1.56893e7			1.10095			0.550474	+	0.834853i	$0.314447\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	8.78241e6	+	1.52116e7i	0.603746	+	1.04572i	0.992248	+	0.124270i	$0.0396590\pi$
$733$	8.78241e6	+	1.52116e7i	0.603746	+	1.04572i	−0.388503	+	0.921448i	$0.627008\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−6.40852e6	+	1.10999e7i	−0.431665	+	0.747666i	−0.997017	−	0.0771842i	$-0.975407\pi$
$739$	−6.40852e6	+	1.10999e7i	−0.431665	+	0.747666i	0.565352	+	0.824850i	$0.308740\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.36610e7	+	2.36615e7i	0.883857	+	1.53088i	0.847019	+	0.531563i	$0.178395\pi$
$751$	1.36610e7	+	2.36615e7i	0.883857	+	1.53088i	0.0368381	+	0.999321i	$0.488271\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.98526e7			1.25915			0.629575	−	0.776940i	$-0.283229\pi$
$757$	1.98526e7			1.25915			0.629575	+	0.776940i	$0.283229\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	−2.92455e7	+	1.33076e7i	−1.81864	+	0.827537i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.50017e7			1.52459			0.762296	−	0.647228i	$-0.224072\pi$
$769$	2.50017e7			1.52459			0.762296	+	0.647228i	$0.224072\pi$
$770$		0			0
$771$		0			0
$772$	1.05020e7	+	1.81900e7i	0.634204	+	1.09847i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	4.25469e6	+	7.36933e6i	0.254456	+	0.440731i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.13060e7	−	1.29758e7i	0.656929	−	0.753953i
$785$		0			0
$786$		0			0
$787$	−1.65531e7	+	2.86708e7i	−0.952668	+	1.65007i	−0.213052	+	0.977041i	$0.568340\pi$
$787$	−1.65531e7	+	2.86708e7i	−0.952668	+	1.65007i	−0.739616	+	0.673029i	$0.764993\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−8.24665e6	−	1.42836e7i	−0.465688	−	0.806595i
$794$		0			0
$795$		0			0
$796$	−1.45963e7	−	2.52815e7i	−0.816507	−	1.41423i
$797$		0			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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(16.0000 - 27.7128i) q^{4} +(-105.500 - 75.3442i) q^{7} +O(q^{10})\)	\(q+(16.0000 - 27.7128i) q^{4} +(-105.500 - 75.3442i) q^{7} -427.000 q^{13} +(-512.000 - 886.810i) q^{16} +(-1571.50 - 2721.92i) q^{19} +(1562.50 - 2706.33i) q^{25} +(-3776.00 + 1718.19i) q^{28} +(-1361.50 + 2358.19i) q^{31} +(3330.50 + 5768.60i) q^{37} +22475.0 q^{43} +(5453.50 + 15897.6i) q^{49} +(-6832.00 + 11833.4i) q^{52} +(19313.0 + 33451.1i) q^{61} -32768.0 q^{64} +(18969.5 - 32856.1i) q^{67} +(39063.5 - 67660.0i) q^{73} -100576. q^{76} +(-45428.5 - 78684.5i) q^{79} +(45048.5 + 32172.0i) q^{91} -134386. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{4} - 211 q^{7}+O(q^{10}) \) 2 * q + 32 * q^4 - 211 * q^7	\( 2 q + 32 q^{4} - 211 q^{7} - 854 q^{13} - 1024 q^{16} - 3143 q^{19} + 3125 q^{25} - 7552 q^{28} - 2723 q^{31} + 6661 q^{37} + 44950 q^{43} + 10907 q^{49} - 13664 q^{52} + 38626 q^{61} - 65536 q^{64} + 37939 q^{67} + 78127 q^{73} - 201152 q^{76} - 90857 q^{79} + 90097 q^{91} - 268772 q^{97}+O(q^{100}) \) 2 * q + 32 * q^4 - 211 * q^7 - 854 * q^13 - 1024 * q^16 - 3143 * q^19 + 3125 * q^25 - 7552 * q^28 - 2723 * q^31 + 6661 * q^37 + 44950 * q^43 + 10907 * q^49 - 13664 * q^52 + 38626 * q^61 - 65536 * q^64 + 37939 * q^67 + 78127 * q^73 - 201152 * q^76 - 90857 * q^79 + 90097 * q^91 - 268772 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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