LMFDB - Embedded newform 3744.2.m.b.1585.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1585,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("3744.1585");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.m (of order $2$, degree $1$, not 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:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3744.1585
Dual form			3744.2.m.b.1585.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.00000i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.00000i q^{7} +2.00000 q^{11} +(-3.00000 - 2.00000i) q^{13} -3.00000 q^{17} -6.00000 q^{23} -4.00000 q^{25} +6.00000i q^{29} +3.00000i q^{35} +3.00000 q^{37} +10.0000i q^{41} +9.00000i q^{43} -7.00000i q^{47} -2.00000 q^{49} -6.00000i q^{53} -2.00000 q^{55} +10.0000 q^{59} +10.0000i q^{61} +(3.00000 + 2.00000i) q^{65} +12.0000 q^{67} -5.00000i q^{71} +6.00000i q^{73} -6.00000i q^{77} -16.0000 q^{83} +3.00000 q^{85} -4.00000i q^{89} +(-6.00000 + 9.00000i) q^{91} +18.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} + 4 q^{11} - 6 q^{13} - 6 q^{17} - 12 q^{23} - 8 q^{25} + 6 q^{37} - 4 q^{49} - 4 q^{55} + 20 q^{59} + 6 q^{65} + 24 q^{67} - 32 q^{83} + 6 q^{85} - 12 q^{91}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^11 - 6 * q^13 - 6 * q^17 - 12 * q^23 - 8 * q^25 + 6 * q^37 - 4 * q^49 - 4 * q^55 + 20 * q^59 + 6 * q^65 + 24 * q^67 - 32 * q^83 + 6 * q^85 - 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$.

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000			−0.447214			−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000			−0.447214			−0.223607	+	0.974679i	$0.571783\pi$
$6$		0			0
$7$		−	3.00000i		−	1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$		−	3.00000i		−	1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.00000			−1.25109			−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000			−1.25109			−0.625543	+	0.780189i	$0.715123\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			3.00000i			0.507093i
$36$		0			0
$37$	3.00000			0.493197			0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000			0.493197			0.246598	+	0.969118i	$0.420687\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$			9.00000i			1.37249i	0.727372	+	0.686244i	$0.240742\pi$
$43$			9.00000i			1.37249i	−0.727372	+	0.686244i	$0.759258\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	7.00000i		−	1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$		−	7.00000i		−	1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$		0			0
$49$	−2.00000			−0.285714
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$		0			0
$61$			10.0000i			1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$			10.0000i			1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.00000	+	2.00000i	0.372104	+	0.248069i
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.00000i		−	0.593391i	−0.954972	−	0.296695i	$-0.904115\pi$
$71$		−	5.00000i		−	0.593391i	0.954972	−	0.296695i	$-0.0958846\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	6.00000i		−	0.683763i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$	3.00000			0.325396
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.00000i		−	0.423999i	−0.977270	−	0.212000i	$-0.932002\pi$
$89$		−	4.00000i		−	0.423999i	0.977270	−	0.212000i	$-0.0679975\pi$
$90$		0			0
$91$	−6.00000	+	9.00000i	−0.628971	+	0.943456i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			18.0000i			1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$			18.0000i			1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	15.0000			1.43674			0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000			1.43674			0.718370	+	0.695662i	$0.244889\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	6.00000			0.559503
$116$		0			0
$117$		0			0
$118$		0			0
$119$			9.00000i			0.825029i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	−18.0000			−1.59724			−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000			−1.59724			−0.798621	+	0.601834i	$0.794437\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			15.0000i			1.31056i	0.755388	+	0.655278i	$0.227449\pi$
$131$			15.0000i			1.31056i	−0.755388	+	0.655278i	$0.772551\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		0			0
$139$		−	9.00000i		−	0.763370i	−0.924292	−	0.381685i	$-0.875344\pi$
$139$		−	9.00000i		−	0.763370i	0.924292	−	0.381685i	$-0.124656\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.00000	−	4.00000i	−0.501745	−	0.334497i
$144$		0			0
$145$		−	6.00000i		−	0.498273i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$			15.0000i			1.22068i	0.792139	+	0.610341i	$0.208968\pi$
$151$			15.0000i			1.22068i	−0.792139	+	0.610341i	$0.791032\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			18.0000i			1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$			18.0000i			1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			18.0000i			1.41860i
$162$		0			0
$163$	6.00000			0.469956			0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000			0.469956			0.234978	+	0.972001i	$0.424498\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$			24.0000i			1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$			24.0000i			1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$		0			0
$175$			12.0000i			0.907115i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			9.00000i			0.672692i	0.941739	+	0.336346i	$0.109191\pi$
$179$			9.00000i			0.672692i	−0.941739	+	0.336346i	$0.890809\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.00000			−0.220564
$186$		0			0
$187$	−6.00000			−0.438763
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−18.0000			−1.30243			−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000			−1.30243			−0.651217	+	0.758891i	$0.725741\pi$
$192$		0			0
$193$		−	24.0000i		−	1.72756i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$		−	24.0000i		−	1.72756i	0.503871	−	0.863779i	$-0.331909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−23.0000			−1.63868			−0.819341	−	0.573306i	$-0.805660\pi$
$197$	−23.0000			−1.63868			−0.819341	+	0.573306i	$0.805660\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	18.0000			1.26335
$204$		0			0
$205$		−	10.0000i		−	0.698430i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			5.00000i			0.344214i	0.985078	+	0.172107i	$0.0550575\pi$
$211$			5.00000i			0.344214i	−0.985078	+	0.172107i	$0.944942\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	9.00000i		−	0.613795i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9.00000	+	6.00000i	0.605406	+	0.403604i
$222$		0			0
$223$		−	21.0000i		−	1.40626i	−0.711059	−	0.703132i	$-0.751784\pi$
$223$		−	21.0000i		−	1.40626i	0.711059	−	0.703132i	$-0.248216\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.00000			−0.132745			−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000			−0.132745			−0.0663723	+	0.997795i	$0.521143\pi$
$228$		0			0
$229$	−15.0000			−0.991228			−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000			−0.991228			−0.495614	+	0.868543i	$0.665057\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.00000			−0.589610			−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000			−0.589610			−0.294805	+	0.955557i	$0.595255\pi$
$234$		0			0
$235$			7.00000i			0.456630i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			19.0000i			1.22901i	0.788914	+	0.614504i	$0.210644\pi$
$239$			19.0000i			1.22901i	−0.788914	+	0.614504i	$0.789356\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.00000			0.127775
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.00000			−0.187135			−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000			−0.187135			−0.0935674	+	0.995613i	$0.529827\pi$
$258$		0			0
$259$		−	9.00000i		−	0.559233i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000			−0.369976			−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000			−0.369976			−0.184988	+	0.982741i	$0.559225\pi$
$264$		0			0
$265$			6.00000i			0.368577i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			6.00000i			0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$			6.00000i			0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$		0			0
$271$			15.0000i			0.911185i	0.890188	+	0.455593i	$0.150573\pi$
$271$			15.0000i			0.911185i	−0.890188	+	0.455593i	$0.849427\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.00000			−0.482418
$276$		0			0
$277$			18.0000i			1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$			18.0000i			1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	20.0000i		−	1.19310i	−0.802576	−	0.596550i	$-0.796538\pi$
$281$		−	20.0000i		−	1.19310i	0.802576	−	0.596550i	$-0.203462\pi$
$282$		0			0
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	30.0000			1.77084
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.00000			0.0584206			0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000			0.0584206			0.0292103	+	0.999573i	$0.490701\pi$
$294$		0			0
$295$	−10.0000			−0.582223
$296$		0			0
$297$		0			0
$298$		0			0
$299$	18.0000	+	12.0000i	1.04097	+	0.693978i
$300$		0			0
$301$	27.0000			1.55625
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	10.0000i		−	0.572598i
$306$		0			0
$307$	−18.0000			−1.02731			−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000			−1.02731			−0.513657	+	0.857996i	$0.671710\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$	9.00000			0.508710			0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000			0.508710			0.254355	+	0.967111i	$0.418137\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.00000			0.112331			0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000			0.112331			0.0561656	+	0.998421i	$0.482113\pi$
$318$		0			0
$319$			12.0000i			0.671871i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	12.0000	+	8.00000i	0.665640	+	0.443760i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−21.0000			−1.15777
$330$		0			0
$331$	−12.0000			−0.659580			−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000			−0.659580			−0.329790	+	0.944054i	$0.606978\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$	−27.0000			−1.47078			−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000			−1.47078			−0.735392	+	0.677642i	$0.763002\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	15.0000i		−	0.809924i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	27.0000i		−	1.44944i	−0.689046	−	0.724718i	$-0.741970\pi$
$347$		−	27.0000i		−	1.44944i	0.689046	−	0.724718i	$-0.258030\pi$
$348$		0			0
$349$	−15.0000			−0.802932			−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000			−0.802932			−0.401466	+	0.915874i	$0.631499\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	16.0000i		−	0.851594i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$		−	16.0000i		−	0.851594i	0.904819	−	0.425797i	$-0.140006\pi$
$354$		0			0
$355$			5.00000i			0.265372i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.00000i			0.211112i	0.994413	+	0.105556i	$0.0336622\pi$
$359$			4.00000i			0.211112i	−0.994413	+	0.105556i	$0.966338\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	6.00000i		−	0.314054i
$366$		0			0
$367$	−28.0000			−1.46159			−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000			−1.46159			−0.730794	+	0.682598i	$0.760850\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−18.0000			−0.934513
$372$		0			0
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.0000	−	18.0000i	0.618031	−	0.927047i
$378$		0			0
$379$	30.0000			1.54100			0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000			1.54100			0.770498	+	0.637442i	$0.220007\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			1.00000i			0.0510976i	0.999674	+	0.0255488i	$0.00813332\pi$
$383$			1.00000i			0.0510976i	−0.999674	+	0.0255488i	$0.991867\pi$
$384$		0			0
$385$			6.00000i			0.305788i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			6.00000i			0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$			6.00000i			0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	18.0000			0.903394			0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000			0.903394			0.451697	+	0.892171i	$0.350819\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			10.0000i			0.499376i	0.968326	+	0.249688i	$0.0803281\pi$
$401$			10.0000i			0.499376i	−0.968326	+	0.249688i	$0.919672\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.00000			0.297409
$408$		0			0
$409$		−	6.00000i		−	0.296681i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$		−	6.00000i		−	0.296681i	0.988936	−	0.148340i	$-0.0473931\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	30.0000i		−	1.47620i
$414$		0			0
$415$	16.0000			0.785409
$416$		0			0
$417$		0			0
$418$		0			0
$419$			9.00000i			0.439679i	0.975536	+	0.219839i	$0.0705533\pi$
$419$			9.00000i			0.439679i	−0.975536	+	0.219839i	$0.929447\pi$
$420$		0			0
$421$	−3.00000			−0.146211			−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000			−0.146211			−0.0731055	+	0.997324i	$0.523291\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.0000			0.582086
$426$		0			0
$427$	30.0000			1.45180
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	5.00000i		−	0.240842i	−0.992723	−	0.120421i	$-0.961576\pi$
$431$		−	5.00000i		−	0.240842i	0.992723	−	0.120421i	$-0.0384244\pi$
$432$		0			0
$433$	9.00000			0.432512			0.216256	−	0.976337i	$-0.430615\pi$
$433$	9.00000			0.432512			0.216256	+	0.976337i	$0.430615\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	10.0000			0.477274			0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000			0.477274			0.238637	+	0.971109i	$0.423299\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	9.00000i		−	0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$		−	9.00000i		−	0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$		0			0
$445$			4.00000i			0.189618i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	14.0000i		−	0.660701i	−0.943858	−	0.330350i	$-0.892833\pi$
$449$		−	14.0000i		−	0.660701i	0.943858	−	0.330350i	$-0.107167\pi$
$450$		0			0
$451$			20.0000i			0.941763i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.00000	−	9.00000i	0.281284	−	0.421927i
$456$		0			0
$457$		−	12.0000i		−	0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$457$		−	12.0000i		−	0.561336i	0.959805	−	0.280668i	$-0.0905560\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.00000			−0.326023			−0.163011	−	0.986624i	$-0.552121\pi$
$461$	−7.00000			−0.326023			−0.163011	+	0.986624i	$0.552121\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	12.0000i		−	0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$		−	12.0000i		−	0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$		0			0
$469$		−	36.0000i		−	1.66233i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			18.0000i			0.827641i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$			29.0000i			1.32504i	0.749043	+	0.662522i	$0.230514\pi$
$479$			29.0000i			1.32504i	−0.749043	+	0.662522i	$0.769486\pi$
$480$		0			0
$481$	−9.00000	−	6.00000i	−0.410365	−	0.273576i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	18.0000i		−	0.817338i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			15.0000i			0.676941i	0.940977	+	0.338470i	$0.109909\pi$
$491$			15.0000i			0.676941i	−0.940977	+	0.338470i	$0.890091\pi$
$492$		0			0
$493$		−	18.0000i		−	0.810679i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−15.0000			−0.672842
$498$		0			0
$499$	−30.0000			−1.34298			−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000			−1.34298			−0.671492	+	0.741012i	$0.734346\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.0000			0.443242			0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000			0.443242			0.221621	+	0.975133i	$0.428865\pi$
$510$		0			0
$511$	18.0000			0.796273
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.00000			0.176261
$516$		0			0
$517$		−	14.0000i		−	0.615719i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$		−	36.0000i		−	1.57417i	−0.616844	−	0.787085i	$-0.711589\pi$
$523$		−	36.0000i		−	1.57417i	0.616844	−	0.787085i	$-0.288411\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$	20.0000	−	30.0000i	0.866296	−	1.29944i
$534$		0			0
$535$			12.0000i			0.518805i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.00000			−0.172292
$540$		0			0
$541$	−33.0000			−1.41878			−0.709390	−	0.704816i	$-0.751030\pi$
$541$	−33.0000			−1.41878			−0.709390	+	0.704816i	$0.751030\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.0000			−0.642529
$546$		0			0
$547$			27.0000i			1.15444i	0.816590	+	0.577218i	$0.195862\pi$
$547$			27.0000i			1.15444i	−0.816590	+	0.577218i	$0.804138\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−23.0000			−0.974541			−0.487271	−	0.873251i	$-0.662007\pi$
$557$	−23.0000			−0.974541			−0.487271	+	0.873251i	$0.662007\pi$
$558$		0			0
$559$	18.0000	−	27.0000i	0.761319	−	1.14198i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			21.0000i			0.885044i	0.896758	+	0.442522i	$0.145916\pi$
$563$			21.0000i			0.885044i	−0.896758	+	0.442522i	$0.854084\pi$
$564$		0			0
$565$	−6.00000			−0.252422
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.0000			−0.628833			−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000			−0.628833			−0.314416	+	0.949285i	$0.601809\pi$
$570$		0			0
$571$		−	5.00000i		−	0.209243i	−0.994512	−	0.104622i	$-0.966637\pi$
$571$		−	5.00000i		−	0.209243i	0.994512	−	0.104622i	$-0.0333632\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.0000			1.00087
$576$		0			0
$577$			18.0000i			0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$			18.0000i			0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			48.0000i			1.99138i
$582$		0			0
$583$		−	12.0000i		−	0.496989i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.00000			−0.0825488			−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000			−0.0825488			−0.0412744	+	0.999148i	$0.513142\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			14.0000i			0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$			14.0000i			0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$		0			0
$595$		−	9.00000i		−	0.368964i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$		0			0
$601$	27.0000			1.10135			0.550676	−	0.834719i	$-0.314370\pi$
$601$	27.0000			1.10135			0.550676	+	0.834719i	$0.314370\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.00000			0.284590
$606$		0			0
$607$	22.0000			0.892952			0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000			0.892952			0.446476	+	0.894795i	$0.352679\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14.0000	+	21.0000i	−0.566379	+	0.849569i
$612$		0			0
$613$	−6.00000			−0.242338			−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000			−0.242338			−0.121169	+	0.992632i	$0.538664\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	8.00000i		−	0.322068i	−0.986949	−	0.161034i	$-0.948517\pi$
$617$		−	8.00000i		−	0.322068i	0.986949	−	0.161034i	$-0.0514829\pi$
$618$		0			0
$619$	30.0000			1.20580			0.602901	−	0.797816i	$-0.294011\pi$
$619$	30.0000			1.20580			0.602901	+	0.797816i	$0.294011\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.0000			−0.480770
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.00000			−0.358854
$630$		0			0
$631$		−	45.0000i		−	1.79142i	−0.444637	−	0.895711i	$-0.646667\pi$
$631$		−	45.0000i		−	1.79142i	0.444637	−	0.895711i	$-0.353333\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.0000			0.714308
$636$		0			0
$637$	6.00000	+	4.00000i	0.237729	+	0.158486i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$		0			0
$643$	−24.0000			−0.946468			−0.473234	−	0.880937i	$-0.656913\pi$
$643$	−24.0000			−0.946468			−0.473234	+	0.880937i	$0.656913\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18.0000			0.707653			0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000			0.707653			0.353827	+	0.935311i	$0.384880\pi$
$648$		0			0
$649$	20.0000			0.785069
$650$		0			0
$651$		0			0
$652$		0			0
$653$			24.0000i			0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$			24.0000i			0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$		0			0
$655$		−	15.0000i		−	0.586098i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	36.0000i		−	1.40236i	−0.712984	−	0.701180i	$-0.752657\pi$
$659$		−	36.0000i		−	1.40236i	0.712984	−	0.701180i	$-0.247343\pi$
$660$		0			0
$661$	−18.0000			−0.700119			−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000			−0.700119			−0.350059	+	0.936727i	$0.613839\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	36.0000i		−	1.39393i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			20.0000i			0.772091i
$672$		0			0
$673$	19.0000			0.732396			0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000			0.732396			0.366198	+	0.930537i	$0.380659\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			12.0000i			0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$			12.0000i			0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$		0			0
$679$	54.0000			2.07233
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4.00000			0.153056			0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000			0.153056			0.0765279	+	0.997067i	$0.475617\pi$
$684$		0			0
$685$		−	2.00000i		−	0.0764161i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12.0000	+	18.0000i	−0.457164	+	0.685745i
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			9.00000i			0.341389i
$696$		0			0
$697$		−	30.0000i		−	1.13633i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	30.0000i		−	1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$		−	30.0000i		−	1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	30.0000			1.12667			0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000			1.12667			0.563337	+	0.826227i	$0.309517\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	6.00000	+	4.00000i	0.224387	+	0.149592i
$716$		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
$721$			12.0000i			0.446903i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	24.0000i		−	0.891338i
$726$		0			0
$727$	−18.0000			−0.667583			−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000			−0.667583			−0.333792	+	0.942647i	$0.608328\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	27.0000i		−	0.998631i
$732$		0			0
$733$	−51.0000			−1.88373			−0.941864	−	0.335994i	$-0.890928\pi$
$733$	−51.0000			−1.88373			−0.941864	+	0.335994i	$0.890928\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.0000			0.884051
$738$		0			0
$739$	−30.0000			−1.10357			−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000			−1.10357			−0.551784	+	0.833987i	$0.686053\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	19.0000i		−	0.697042i	−0.937301	−	0.348521i	$-0.886684\pi$
$743$		−	19.0000i		−	0.697042i	0.937301	−	0.348521i	$-0.113316\pi$
$744$		0			0
$745$	−10.0000			−0.366372
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−36.0000			−1.31541
$750$		0			0
$751$	−22.0000			−0.802791			−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000			−0.802791			−0.401396	+	0.915905i	$0.631475\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	15.0000i		−	0.545906i
$756$		0			0
$757$			38.0000i			1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$			38.0000i			1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	10.0000i		−	0.362500i	−0.983437	−	0.181250i	$-0.941986\pi$
$761$		−	10.0000i		−	0.362500i	0.983437	−	0.181250i	$-0.0580143\pi$
$762$		0			0
$763$		−	45.0000i		−	1.62911i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−30.0000	−	20.0000i	−1.08324	−	0.722158i
$768$		0			0
$769$		−	36.0000i		−	1.29819i	−0.760706	−	0.649097i	$-0.775147\pi$
$769$		−	36.0000i		−	1.29819i	0.760706	−	0.649097i	$-0.224853\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−49.0000			−1.76241			−0.881204	−	0.472737i	$-0.843266\pi$
$773$	−49.0000			−1.76241			−0.881204	+	0.472737i	$0.843266\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		−	10.0000i		−	0.357828i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	18.0000i		−	0.642448i
$786$		0			0
$787$	−18.0000			−0.641631			−0.320815	−	0.947142i	$-0.603957\pi$
$787$	−18.0000			−0.641631			−0.320815	+	0.947142i	$0.603957\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	18.0000i		−	0.640006i
$792$		0			0
$793$	20.0000	−	30.0000i	0.710221	−	1.06533i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			12.0000i			0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$			12.0000i			0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$		0			0
$799$			21.0000i			0.742927i
$800$		0			0
$801$		0			0
$802$		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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.00000i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.00000i q^{7} +2.00000 q^{11} +(-3.00000 - 2.00000i) q^{13} -3.00000 q^{17} -6.00000 q^{23} -4.00000 q^{25} +6.00000i q^{29} +3.00000i q^{35} +3.00000 q^{37} +10.0000i q^{41} +9.00000i q^{43} -7.00000i q^{47} -2.00000 q^{49} -6.00000i q^{53} -2.00000 q^{55} +10.0000 q^{59} +10.0000i q^{61} +(3.00000 + 2.00000i) q^{65} +12.0000 q^{67} -5.00000i q^{71} +6.00000i q^{73} -6.00000i q^{77} -16.0000 q^{83} +3.00000 q^{85} -4.00000i q^{89} +(-6.00000 + 9.00000i) q^{91} +18.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} + 4 q^{11} - 6 q^{13} - 6 q^{17} - 12 q^{23} - 8 q^{25} + 6 q^{37} - 4 q^{49} - 4 q^{55} + 20 q^{59} + 6 q^{65} + 24 q^{67} - 32 q^{83} + 6 q^{85} - 12 q^{91}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^11 - 6 * q^13 - 6 * q^17 - 12 * q^23 - 8 * q^25 + 6 * q^37 - 4 * q^49 - 4 * q^55 + 20 * q^59 + 6 * q^65 + 24 * q^67 - 32 * q^83 + 6 * q^85 - 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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