LMFDB - Embedded newform 3744.2.m.a.1585.2

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:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{5} +O(q^{10})$	$q-2.00000 q^{5} +4.00000 q^{11} +(-3.00000 + 2.00000i) q^{13} +6.00000 q^{17} -6.00000 q^{19} -1.00000 q^{25} -6.00000i q^{29} +6.00000 q^{37} -2.00000i q^{41} +12.0000i q^{43} +8.00000i q^{47} +7.00000 q^{49} -6.00000i q^{53} -8.00000 q^{55} -4.00000 q^{59} +8.00000i q^{61} +(6.00000 - 4.00000i) q^{65} -6.00000 q^{67} +4.00000i q^{71} -12.0000i q^{73} -6.00000 q^{79} -8.00000 q^{83} -12.0000 q^{85} +2.00000i q^{89} +12.0000 q^{95} +12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} + 8 q^{11} - 6 q^{13} + 12 q^{17} - 12 q^{19} - 2 q^{25} + 12 q^{37} + 14 q^{49} - 16 q^{55} - 8 q^{59} + 12 q^{65} - 12 q^{67} - 12 q^{79} - 16 q^{83} - 24 q^{85} + 24 q^{95}+O(q^{100}) $ 2 * q - 4 * q^5 + 8 * q^11 - 6 * q^13 + 12 * q^17 - 12 * q^19 - 2 * q^25 + 12 * q^37 + 14 * q^49 - 16 * q^55 - 8 * q^59 + 12 * q^65 - 12 * q^67 - 12 * q^79 - 16 * q^83 - 24 * q^85 + 24 * q^95

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
$5$	−2.00000			−0.894427			−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000			−0.894427			−0.447214	+	0.894427i	$0.647584\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.00000			1.20605			0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000			1.20605			0.603023	+	0.797724i	$0.293963\pi$
$12$		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
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	2.00000i		−	0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$		−	2.00000i		−	0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$		0			0
$43$			12.0000i			1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$			12.0000i			1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		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
$55$	−8.00000			−1.07872
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$			8.00000i			1.02430i	0.858898	+	0.512148i	$0.171150\pi$
$61$			8.00000i			1.02430i	−0.858898	+	0.512148i	$0.828850\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000	−	4.00000i	0.744208	−	0.496139i
$66$		0			0
$67$	−6.00000			−0.733017			−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000			−0.733017			−0.366508	+	0.930415i	$0.619447\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.00000i			0.474713i	0.971423	+	0.237356i	$0.0762809\pi$
$71$			4.00000i			0.474713i	−0.971423	+	0.237356i	$0.923719\pi$
$72$		0			0
$73$		−	12.0000i		−	1.40449i	−0.711934	−	0.702247i	$-0.752180\pi$
$73$		−	12.0000i		−	1.40449i	0.711934	−	0.702247i	$-0.247820\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.00000			−0.675053			−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000			−0.675053			−0.337526	+	0.941316i	$0.609590\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.00000			−0.878114			−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000			−0.878114			−0.439057	+	0.898459i	$0.644687\pi$
$84$		0			0
$85$	−12.0000			−1.30158
$86$		0			0
$87$		0			0
$88$		0			0
$89$			2.00000i			0.212000i	0.994366	+	0.106000i	$0.0338043\pi$
$89$			2.00000i			0.212000i	−0.994366	+	0.106000i	$0.966196\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	12.0000			1.23117
$96$		0			0
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			18.0000i			1.79107i	0.444994	+	0.895533i	$0.353206\pi$
$101$			18.0000i			1.79107i	−0.444994	+	0.895533i	$0.646794\pi$
$102$		0			0
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−6.00000			−0.574696			−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000			−0.574696			−0.287348	+	0.957826i	$0.592774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	18.0000			1.69330			0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000			1.69330			0.846649	+	0.532152i	$0.178617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	6.00000			0.532414			0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000			0.532414			0.266207	+	0.963916i	$0.414230\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		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
$139$			12.0000i			1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$			12.0000i			1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.0000	+	8.00000i	−1.00349	+	0.668994i
$144$		0			0
$145$			12.0000i			0.996546i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.00000			0.163846			0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000			0.163846			0.0819232	+	0.996639i	$0.473894\pi$
$150$		0			0
$151$			24.0000i			1.95309i	0.215308	+	0.976546i	$0.430924\pi$
$151$			24.0000i			1.95309i	−0.215308	+	0.976546i	$0.569076\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.00000			−0.469956			−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000			−0.469956			−0.234978	+	0.972001i	$0.575502\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			20.0000i			1.54765i	0.633402	+	0.773823i	$0.281658\pi$
$167$			20.0000i			1.54765i	−0.633402	+	0.773823i	$0.718342\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$			18.0000i			1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$			18.0000i			1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$			12.0000i			0.891953i	0.895045	+	0.445976i	$0.147144\pi$
$181$			12.0000i			0.891953i	−0.895045	+	0.445976i	$0.852856\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.0000			−0.882258
$186$		0			0
$187$	24.0000			1.75505
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	18.0000			1.27599			0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000			1.27599			0.637993	+	0.770042i	$0.279765\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$			4.00000i			0.279372i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−24.0000			−1.66011
$210$		0			0
$211$			4.00000i			0.275371i	0.990476	+	0.137686i	$0.0439664\pi$
$211$			4.00000i			0.275371i	−0.990476	+	0.137686i	$0.956034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	24.0000i		−	1.63679i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−18.0000	+	12.0000i	−1.21081	+	0.807207i
$222$		0			0
$223$			24.0000i			1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$			24.0000i			1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	20.0000			1.32745			0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000			1.32745			0.663723	+	0.747978i	$0.268975\pi$
$228$		0			0
$229$	−6.00000			−0.396491			−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000			−0.396491			−0.198246	+	0.980152i	$0.563524\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$		−	16.0000i		−	1.04372i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	8.00000i		−	0.517477i	−0.965947	−	0.258738i	$-0.916693\pi$
$239$		−	8.00000i		−	0.517477i	0.965947	−	0.258738i	$-0.0833068\pi$
$240$		0			0
$241$			12.0000i			0.772988i	0.922292	+	0.386494i	$0.126314\pi$
$241$			12.0000i			0.772988i	−0.922292	+	0.386494i	$0.873686\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−14.0000			−0.894427
$246$		0			0
$247$	18.0000	−	12.0000i	1.14531	−	0.763542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			24.0000i			1.51487i	0.652913	+	0.757433i	$0.273547\pi$
$251$			24.0000i			1.51487i	−0.652913	+	0.757433i	$0.726453\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$			12.0000i			0.737154i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	30.0000i		−	1.82913i	−0.404436	−	0.914566i	$-0.632532\pi$
$269$		−	30.0000i		−	1.82913i	0.404436	−	0.914566i	$-0.367468\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.00000			−0.241209
$276$		0			0
$277$			12.0000i			0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$			12.0000i			0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			22.0000i			1.31241i	0.754583	+	0.656205i	$0.227839\pi$
$281$			22.0000i			1.31241i	−0.754583	+	0.656205i	$0.772161\pi$
$282$		0			0
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.00000			0.116841			0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000			0.116841			0.0584206	+	0.998292i	$0.481394\pi$
$294$		0			0
$295$	8.00000			0.465778
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	16.0000i		−	0.916157i
$306$		0			0
$307$	−6.00000			−0.342438			−0.171219	−	0.985233i	$-0.554771\pi$
$307$	−6.00000			−0.342438			−0.171219	+	0.985233i	$0.554771\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	18.0000			1.01742			0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000			1.01742			0.508710	+	0.860938i	$0.330123\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−26.0000			−1.46031			−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000			−1.46031			−0.730153	+	0.683284i	$0.760551\pi$
$318$		0			0
$319$		−	24.0000i		−	1.34374i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−36.0000			−2.00309
$324$		0			0
$325$	3.00000	−	2.00000i	0.166410	−	0.110940i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−18.0000			−0.989369			−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000			−0.989369			−0.494685	+	0.869072i	$0.664716\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.0000			0.655630
$336$		0			0
$337$	−30.0000			−1.63420			−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000			−1.63420			−0.817102	+	0.576493i	$0.804421\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$			24.0000i			1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$			24.0000i			1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$		0			0
$349$	18.0000			0.963518			0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000			0.963518			0.481759	+	0.876304i	$0.339998\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	34.0000i		−	1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$		−	34.0000i		−	1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$		0			0
$355$		−	8.00000i		−	0.424596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	8.00000i		−	0.422224i	−0.977462	−	0.211112i	$-0.932292\pi$
$359$		−	8.00000i		−	0.422224i	0.977462	−	0.211112i	$-0.0677085\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$			24.0000i			1.25622i
$366$		0			0
$367$	−10.0000			−0.521996			−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000			−0.521996			−0.260998	+	0.965339i	$0.584052\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	32.0000i		−	1.65690i	−0.560065	−	0.828449i	$-0.689224\pi$
$373$		−	32.0000i		−	1.65690i	0.560065	−	0.828449i	$-0.310776\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.0000	+	18.0000i	0.618031	+	0.927047i
$378$		0			0
$379$	−18.0000			−0.924598			−0.462299	−	0.886724i	$-0.652975\pi$
$379$	−18.0000			−0.924598			−0.462299	+	0.886724i	$0.652975\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	20.0000i		−	1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$		−	20.0000i		−	1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	6.00000i		−	0.304212i	−0.988364	−	0.152106i	$-0.951394\pi$
$389$		−	6.00000i		−	0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.0000			0.603786
$396$		0			0
$397$	−18.0000			−0.903394			−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000			−0.903394			−0.451697	+	0.892171i	$0.649181\pi$
$398$		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
$407$	24.0000			1.18964
$408$		0			0
$409$			36.0000i			1.78009i	0.455877	+	0.890043i	$0.349326\pi$
$409$			36.0000i			1.78009i	−0.455877	+	0.890043i	$0.650674\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	16.0000			0.785409
$416$		0			0
$417$		0			0
$418$		0			0
$419$			12.0000i			0.586238i	0.956076	+	0.293119i	$0.0946933\pi$
$419$			12.0000i			0.586238i	−0.956076	+	0.293119i	$0.905307\pi$
$420$		0			0
$421$	30.0000			1.46211			0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000			1.46211			0.731055	+	0.682318i	$0.239028\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.00000			−0.291043
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	32.0000i		−	1.54139i	−0.637207	−	0.770693i	$-0.719910\pi$
$431$		−	32.0000i		−	1.54139i	0.637207	−	0.770693i	$-0.280090\pi$
$432$		0			0
$433$	−6.00000			−0.288342			−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000			−0.288342			−0.144171	+	0.989553i	$0.546051\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−26.0000			−1.24091			−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000			−1.24091			−0.620456	+	0.784241i	$0.713053\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	36.0000i		−	1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$		−	36.0000i		−	1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$		0			0
$445$		−	4.00000i		−	0.189618i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			10.0000i			0.471929i	0.971762	+	0.235965i	$0.0758249\pi$
$449$			10.0000i			0.471929i	−0.971762	+	0.235965i	$0.924175\pi$
$450$		0			0
$451$		−	8.00000i		−	0.376705i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	36.0000i		−	1.68401i	−0.539471	−	0.842004i	$-0.681376\pi$
$457$		−	36.0000i		−	1.68401i	0.539471	−	0.842004i	$-0.318624\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	34.0000			1.58354			0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000			1.58354			0.791769	+	0.610821i	$0.209160\pi$
$462$		0			0
$463$		−	24.0000i		−	1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$		−	24.0000i		−	1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	24.0000i		−	1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$		−	24.0000i		−	1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			48.0000i			2.20704i
$474$		0			0
$475$	6.00000			0.275299
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	4.00000i		−	0.182765i	−0.995816	−	0.0913823i	$-0.970871\pi$
$479$		−	4.00000i		−	0.182765i	0.995816	−	0.0913823i	$-0.0291285\pi$
$480$		0			0
$481$	−18.0000	+	12.0000i	−0.820729	+	0.547153i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	24.0000i		−	1.08978i
$486$		0			0
$487$			24.0000i			1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$			24.0000i			1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	12.0000i		−	0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$		−	12.0000i		−	0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$		0			0
$493$		−	36.0000i		−	1.62136i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	18.0000			0.805791			0.402895	−	0.915246i	$-0.368004\pi$
$499$	18.0000			0.805791			0.402895	+	0.915246i	$0.368004\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−36.0000			−1.60516			−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000			−1.60516			−0.802580	+	0.596544i	$0.796540\pi$
$504$		0			0
$505$		−	36.0000i		−	1.60198i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.0000			0.620539			0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000			0.620539			0.310270	+	0.950649i	$0.399581\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−28.0000			−1.23383
$516$		0			0
$517$			32.0000i			1.40736i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−30.0000			−1.31432			−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000			−1.31432			−0.657162	+	0.753749i	$0.728243\pi$
$522$		0			0
$523$			36.0000i			1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$			36.0000i			1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.00000	+	6.00000i	0.173259	+	0.259889i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	28.0000			1.20605
$540$		0			0
$541$	30.0000			1.28980			0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000			1.28980			0.644900	+	0.764267i	$0.276899\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12.0000			0.514024
$546$		0			0
$547$			12.0000i			0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$			12.0000i			0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			36.0000i			1.53365i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.00000			0.0847427			0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000			0.0847427			0.0423714	+	0.999102i	$0.486509\pi$
$558$		0			0
$559$	−24.0000	−	36.0000i	−1.01509	−	1.52264i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			12.0000i			0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$			12.0000i			0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$		0			0
$565$	−36.0000			−1.51453
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$		−	4.00000i		−	0.167395i	−0.996491	−	0.0836974i	$-0.973327\pi$
$571$		−	4.00000i		−	0.167395i	0.996491	−	0.0836974i	$-0.0266729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			24.0000i			0.999133i	0.866276	+	0.499567i	$0.166507\pi$
$577$			24.0000i			0.999133i	−0.866276	+	0.499567i	$0.833493\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	24.0000i		−	0.993978i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.0000			0.825488			0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000			0.825488			0.412744	+	0.910847i	$0.364570\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	34.0000i		−	1.39621i	−0.715994	−	0.698106i	$-0.754026\pi$
$593$		−	34.0000i		−	1.39621i	0.715994	−	0.698106i	$-0.245974\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	36.0000			1.47092			0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000			1.47092			0.735460	+	0.677568i	$0.236966\pi$
$600$		0			0
$601$	30.0000			1.22373			0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000			1.22373			0.611863	+	0.790964i	$0.290420\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.0000			−0.406558
$606$		0			0
$607$	−14.0000			−0.568242			−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000			−0.568242			−0.284121	+	0.958788i	$0.591702\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.0000	−	24.0000i	−0.647291	−	0.970936i
$612$		0			0
$613$	−42.0000			−1.69636			−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000			−1.69636			−0.848182	+	0.529705i	$0.822303\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			10.0000i			0.402585i	0.979531	+	0.201292i	$0.0645141\pi$
$617$			10.0000i			0.402585i	−0.979531	+	0.201292i	$0.935486\pi$
$618$		0			0
$619$	6.00000			0.241160			0.120580	−	0.992704i	$-0.461525\pi$
$619$	6.00000			0.241160			0.120580	+	0.992704i	$0.461525\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	36.0000			1.43541
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.0000			−0.476205
$636$		0			0
$637$	−21.0000	+	14.0000i	−0.832050	+	0.554700i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.00000			0.236986			0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000			0.236986			0.118493	+	0.992955i	$0.462194\pi$
$642$		0			0
$643$	30.0000			1.18308			0.591542	−	0.806274i	$-0.298519\pi$
$643$	30.0000			1.18308			0.591542	+	0.806274i	$0.298519\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	18.0000i		−	0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$		−	18.0000i		−	0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	12.0000i		−	0.467454i	−0.972302	−	0.233727i	$-0.924908\pi$
$659$		−	12.0000i		−	0.467454i	0.972302	−	0.233727i	$-0.0750921\pi$
$660$		0			0
$661$	42.0000			1.63361			0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000			1.63361			0.816805	+	0.576913i	$0.195743\pi$
$662$		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
$671$			32.0000i			1.23535i
$672$		0			0
$673$	−26.0000			−1.00223			−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000			−1.00223			−0.501113	+	0.865382i	$0.667076\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	18.0000i		−	0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$		−	18.0000i		−	0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	44.0000			1.68361			0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000			1.68361			0.841807	+	0.539779i	$0.181492\pi$
$684$		0			0
$685$		−	4.00000i		−	0.152832i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.0000	+	18.0000i	0.457164	+	0.685745i
$690$		0			0
$691$	−18.0000			−0.684752			−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000			−0.684752			−0.342376	+	0.939563i	$0.611232\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	24.0000i		−	0.910372i
$696$		0			0
$697$		−	12.0000i		−	0.454532i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	6.00000i		−	0.226617i	−0.993560	−	0.113308i	$-0.963855\pi$
$701$		−	6.00000i		−	0.226617i	0.993560	−	0.113308i	$-0.0361448\pi$
$702$		0			0
$703$	−36.0000			−1.35777
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−18.0000			−0.676004			−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000			−0.676004			−0.338002	+	0.941145i	$0.609751\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	24.0000	−	16.0000i	0.897549	−	0.598366i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$			6.00000i			0.222834i
$726$		0			0
$727$	−42.0000			−1.55769			−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000			−1.55769			−0.778847	+	0.627214i	$0.784195\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			72.0000i			2.66302i
$732$		0			0
$733$	6.00000			0.221615			0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000			0.221615			0.110808	+	0.993842i	$0.464656\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.0000			−0.884051
$738$		0			0
$739$	−6.00000			−0.220714			−0.110357	−	0.993892i	$-0.535199\pi$
$739$	−6.00000			−0.220714			−0.110357	+	0.993892i	$0.535199\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			8.00000i			0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$			8.00000i			0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$		0			0
$745$	−4.00000			−0.146549
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	14.0000			0.510867			0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000			0.510867			0.255434	+	0.966827i	$0.417782\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	48.0000i		−	1.74690i
$756$		0			0
$757$		−	20.0000i		−	0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$		−	20.0000i		−	0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	46.0000i		−	1.66750i	−0.552143	−	0.833749i	$-0.686190\pi$
$761$		−	46.0000i		−	1.66750i	0.552143	−	0.833749i	$-0.313810\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.0000	−	8.00000i	0.433295	−	0.288863i
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.0000			0.791285			0.395643	−	0.918405i	$-0.370522\pi$
$773$	22.0000			0.791285			0.395643	+	0.918405i	$0.370522\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			12.0000i			0.429945i
$780$		0			0
$781$			16.0000i			0.572525i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−42.0000			−1.49714			−0.748569	−	0.663057i	$-0.769259\pi$
$787$	−42.0000			−1.49714			−0.748569	+	0.663057i	$0.769259\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−16.0000	−	24.0000i	−0.568177	−	0.852265i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	18.0000i		−	0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$		−	18.0000i		−	0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$		0			0
$799$			48.0000i			1.69812i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	48.0000i		−	1.69388i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	54.0000			1.89620			0.948098	−	0.317978i	$-0.103004\pi$
$811$	54.0000			1.89620			0.948098	+	0.317978i	$0.103004\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.0000			0.420342
$816$		0			0
$817$		−	72.0000i		−	2.51896i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.0000			−0.349002			−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000			−0.349002			−0.174501	+	0.984657i	$0.555831\pi$
$822$		0			0
$823$	−14.0000			−0.488009			−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000			−0.488009			−0.244005	+	0.969774i	$0.578461\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	56.0000			1.94731			0.973655	−	0.228024i	$-0.0732266\pi$
$827$	56.0000			1.94731			0.973655	+	0.228024i	$0.0732266\pi$
$828$		0			0
$829$		−	20.0000i		−	0.694629i	−0.937749	−	0.347314i	$-0.887094\pi$
$829$		−	20.0000i		−	0.694629i	0.937749	−	0.347314i	$-0.112906\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	42.0000			1.45521
$834$		0			0
$835$		−	40.0000i		−	1.38426i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			20.0000i			0.690477i	0.938515	+	0.345238i	$0.112202\pi$
$839$			20.0000i			0.690477i	−0.938515	+	0.345238i	$0.887798\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−10.0000	+	24.0000i	−0.344010	+	0.825625i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	42.0000			1.43805			0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000			1.43805			0.719026	+	0.694983i	$0.244588\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.00000			0.204956			0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000			0.204956			0.102478	+	0.994735i	$0.467323\pi$
$858$		0			0
$859$			4.00000i			0.136478i	0.997669	+	0.0682391i	$0.0217381\pi$
$859$			4.00000i			0.136478i	−0.997669	+	0.0682391i	$0.978262\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			20.0000i			0.680808i	0.940279	+	0.340404i	$0.110564\pi$
$863$			20.0000i			0.680808i	−0.940279	+	0.340404i	$0.889436\pi$
$864$		0			0
$865$		−	36.0000i		−	1.22404i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−24.0000			−0.814144
$870$		0			0
$871$	18.0000	−	12.0000i	0.609907	−	0.406604i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	30.0000			1.01303			0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000			1.01303			0.506514	+	0.862232i	$0.330934\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$		−	20.0000i		−	0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$		−	20.0000i		−	0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.0000			0.402921			0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000			0.402921			0.201460	+	0.979497i	$0.435431\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	48.0000i		−	1.60626i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	36.0000i		−	1.19933i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	24.0000i		−	0.797787i
$906$		0			0
$907$		−	44.0000i		−	1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$		−	44.0000i		−	1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−36.0000			−1.19273			−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000			−1.19273			−0.596367	+	0.802712i	$0.703390\pi$
$912$		0			0
$913$	−32.0000			−1.05905
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	6.00000			0.197922			0.0989609	−	0.995091i	$-0.468448\pi$
$919$	6.00000			0.197922			0.0989609	+	0.995091i	$0.468448\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.00000	−	12.0000i	−0.263323	−	0.394985i
$924$		0			0
$925$	−6.00000			−0.197279
$926$		0			0
$927$		0			0
$928$		0			0
$929$			2.00000i			0.0656179i	0.999462	+	0.0328089i	$0.0104453\pi$
$929$			2.00000i			0.0656179i	−0.999462	+	0.0328089i	$0.989555\pi$
$930$		0			0
$931$	−42.0000			−1.37649
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−48.0000			−1.56977
$936$		0			0
$937$	−30.0000			−0.980057			−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000			−0.980057			−0.490029	+	0.871706i	$0.663014\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.0000			−0.325991			−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000			−0.325991			−0.162995	+	0.986627i	$0.552116\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−28.0000			−0.909878			−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000			−0.909878			−0.454939	+	0.890523i	$0.650339\pi$
$948$		0			0
$949$	24.0000	+	36.0000i	0.779073	+	1.16861i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−18.0000			−0.583077			−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000			−0.583077			−0.291539	+	0.956559i	$0.594167\pi$
$954$		0			0
$955$	24.0000			0.776622
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	48.0000i		−	1.54358i	−0.635880	−	0.771788i	$-0.719363\pi$
$967$		−	48.0000i		−	1.54358i	0.635880	−	0.771788i	$-0.280637\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			24.0000i			0.770197i	0.922876	+	0.385098i	$0.125832\pi$
$971$			24.0000i			0.770197i	−0.922876	+	0.385098i	$0.874168\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	34.0000i		−	1.08776i	−0.839164	−	0.543878i	$-0.816955\pi$
$977$		−	34.0000i		−	1.08776i	0.839164	−	0.543878i	$-0.183045\pi$
$978$		0			0
$979$			8.00000i			0.255681i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	16.0000i		−	0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$		−	16.0000i		−	0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$		0			0
$985$	−4.00000			−0.127451
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	38.0000			1.20711			0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000			1.20711			0.603555	+	0.797321i	$0.293750\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−36.0000			−1.14128
$996$		0			0
$997$		−	48.0000i		−	1.52018i	−0.649821	−	0.760088i	$-0.725156\pi$
$997$		−	48.0000i		−	1.52018i	0.649821	−	0.760088i	$-0.274844\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.m.a.1585.2		2
3.2	odd	2		1248.2.m.b.337.2		2
4.3	odd	2		936.2.m.a.181.2		2
8.3	odd	2		936.2.m.d.181.2		2
8.5	even	2		3744.2.m.d.1585.1		2
12.11	even	2		312.2.m.b.181.1	yes	2
13.12	even	2		3744.2.m.d.1585.2		2
24.5	odd	2		1248.2.m.a.337.1		2
24.11	even	2		312.2.m.a.181.1	✓	2
39.38	odd	2		1248.2.m.a.337.2		2
52.51	odd	2		936.2.m.d.181.1		2
104.51	odd	2		936.2.m.a.181.1		2
104.77	even	2	inner	3744.2.m.a.1585.1		2
156.155	even	2		312.2.m.a.181.2	yes	2
312.77	odd	2		1248.2.m.b.337.1		2
312.155	even	2		312.2.m.b.181.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.m.a.181.1	✓	2	24.11	even	2
312.2.m.a.181.2	yes	2	156.155	even	2
312.2.m.b.181.1	yes	2	12.11	even	2
312.2.m.b.181.2	yes	2	312.155	even	2
936.2.m.a.181.1		2	104.51	odd	2
936.2.m.a.181.2		2	4.3	odd	2
936.2.m.d.181.1		2	52.51	odd	2
936.2.m.d.181.2		2	8.3	odd	2
1248.2.m.a.337.1		2	24.5	odd	2
1248.2.m.a.337.2		2	39.38	odd	2
1248.2.m.b.337.1		2	312.77	odd	2
1248.2.m.b.337.2		2	3.2	odd	2
3744.2.m.a.1585.1		2	104.77	even	2	inner
3744.2.m.a.1585.2		2	1.1	even	1	trivial
3744.2.m.d.1585.1		2	8.5	even	2
3744.2.m.d.1585.2		2	13.12	even	2

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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{5} +O(q^{10})\)	\(q-2.00000 q^{5} +4.00000 q^{11} +(-3.00000 + 2.00000i) q^{13} +6.00000 q^{17} -6.00000 q^{19} -1.00000 q^{25} -6.00000i q^{29} +6.00000 q^{37} -2.00000i q^{41} +12.0000i q^{43} +8.00000i q^{47} +7.00000 q^{49} -6.00000i q^{53} -8.00000 q^{55} -4.00000 q^{59} +8.00000i q^{61} +(6.00000 - 4.00000i) q^{65} -6.00000 q^{67} +4.00000i q^{71} -12.0000i q^{73} -6.00000 q^{79} -8.00000 q^{83} -12.0000 q^{85} +2.00000i q^{89} +12.0000 q^{95} +12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} + 8 q^{11} - 6 q^{13} + 12 q^{17} - 12 q^{19} - 2 q^{25} + 12 q^{37} + 14 q^{49} - 16 q^{55} - 8 q^{59} + 12 q^{65} - 12 q^{67} - 12 q^{79} - 16 q^{83} - 24 q^{85} + 24 q^{95}+O(q^{100}) \) 2 * q - 4 * q^5 + 8 * q^11 - 6 * q^13 + 12 * q^17 - 12 * q^19 - 2 * q^25 + 12 * q^37 + 14 * q^49 - 16 * q^55 - 8 * q^59 + 12 * q^65 - 12 * q^67 - 12 * q^79 - 16 * q^83 - 24 * q^85 + 24 * q^95

Introduction

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Modular forms

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