LMFDB - Embedded newform 2736.2.k.k.2431.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2431,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("2736.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.k (of order $2$, degree $1$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 304)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.4
Root			$1.32288 - 2.29129i$ of defining polynomial
Character	$\chi$	$=$	2736.2431
Dual form			2736.2.k.k.2431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{7} +O(q^{10})$	$q+1.73205i q^{7} +3.46410i q^{11} +4.58258i q^{13} -3.00000 q^{17} +(-2.64575 - 3.46410i) q^{19} +5.19615i q^{23} -5.00000 q^{25} -4.58258i q^{29} -5.29150 q^{31} -9.16515i q^{37} -9.16515i q^{41} -3.46410i q^{47} +4.00000 q^{49} +4.58258i q^{53} -7.93725 q^{59} -8.00000 q^{61} +2.64575 q^{67} +15.8745 q^{71} -11.0000 q^{73} -6.00000 q^{77} +5.29150 q^{79} +6.92820i q^{83} -18.3303i q^{89} -7.93725 q^{91} +9.16515i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 12 q^{17} - 20 q^{25} + 16 q^{49} - 32 q^{61} - 44 q^{73} - 24 q^{77}+O(q^{100}) $ 4 * q - 12 * q^17 - 20 * q^25 + 16 * q^49 - 32 * q^61 - 44 * q^73 - 24 * q^77

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$			1.73205i			0.654654i	0.944911	+	0.327327i	$0.106148\pi$
$7$			1.73205i			0.654654i	−0.944911	+	0.327327i	$0.893852\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$			4.58258i			1.27098i	0.772110	+	0.635489i	$0.219201\pi$
$13$			4.58258i			1.27098i	−0.772110	+	0.635489i	$0.780799\pi$
$14$		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
$19$	−2.64575	−	3.46410i	−0.606977	−	0.794719i
$19$	−2.64575	−	3.46410i	−0.606977	−	0.794719i
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.19615i			1.08347i	0.840548	+	0.541736i	$0.182233\pi$
$23$			5.19615i			1.08347i	−0.840548	+	0.541736i	$0.817767\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.58258i		−	0.850963i	−0.904967	−	0.425481i	$-0.860105\pi$
$29$		−	4.58258i		−	0.850963i	0.904967	−	0.425481i	$-0.139895\pi$
$30$		0			0
$31$	−5.29150			−0.950382			−0.475191	−	0.879883i	$-0.657621\pi$
$31$	−5.29150			−0.950382			−0.475191	+	0.879883i	$0.657621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	9.16515i		−	1.50674i	−0.657596	−	0.753371i	$-0.728427\pi$
$37$		−	9.16515i		−	1.50674i	0.657596	−	0.753371i	$-0.271573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	9.16515i		−	1.43136i	−0.698430	−	0.715678i	$-0.746118\pi$
$41$		−	9.16515i		−	1.43136i	0.698430	−	0.715678i	$-0.253882\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.46410i		−	0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$		−	3.46410i		−	0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$		0			0
$49$	4.00000			0.571429
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4.58258i			0.629465i	0.949180	+	0.314733i	$0.101915\pi$
$53$			4.58258i			0.629465i	−0.949180	+	0.314733i	$0.898085\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.93725			−1.03334			−0.516671	−	0.856184i	$-0.672829\pi$
$59$	−7.93725			−1.03334			−0.516671	+	0.856184i	$0.672829\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.64575			0.323230			0.161615	−	0.986854i	$-0.448330\pi$
$67$	2.64575			0.323230			0.161615	+	0.986854i	$0.448330\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	15.8745			1.88396			0.941979	−	0.335673i	$-0.108964\pi$
$71$	15.8745			1.88396			0.941979	+	0.335673i	$0.108964\pi$
$72$		0			0
$73$	−11.0000			−1.28745			−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000			−1.28745			−0.643726	+	0.765256i	$0.722612\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000			−0.683763
$78$		0			0
$79$	5.29150			0.595341			0.297670	−	0.954669i	$-0.403790\pi$
$79$	5.29150			0.595341			0.297670	+	0.954669i	$0.403790\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			6.92820i			0.760469i	0.924890	+	0.380235i	$0.124157\pi$
$83$			6.92820i			0.760469i	−0.924890	+	0.380235i	$0.875843\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	18.3303i		−	1.94301i	−0.237023	−	0.971504i	$-0.576172\pi$
$89$		−	18.3303i		−	1.94301i	0.237023	−	0.971504i	$-0.423828\pi$
$90$		0			0
$91$	−7.93725			−0.832050
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			9.16515i			0.930580i	0.885158	+	0.465290i	$0.154050\pi$
$97$			9.16515i			0.930580i	−0.885158	+	0.465290i	$0.845950\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−18.0000			−1.79107			−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000			−1.79107			−0.895533	+	0.444994i	$0.853206\pi$
$102$		0			0
$103$	−10.5830			−1.04277			−0.521387	−	0.853320i	$-0.674585\pi$
$103$	−10.5830			−1.04277			−0.521387	+	0.853320i	$0.674585\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.93725			−0.767323			−0.383662	−	0.923474i	$-0.625337\pi$
$107$	−7.93725			−0.767323			−0.383662	+	0.923474i	$0.625337\pi$
$108$		0			0
$109$		−	13.7477i		−	1.31679i	−0.752671	−	0.658397i	$-0.771235\pi$
$109$		−	13.7477i		−	1.31679i	0.752671	−	0.658397i	$-0.228765\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			9.16515i			0.862185i	0.902308	+	0.431092i	$0.141872\pi$
$113$			9.16515i			0.862185i	−0.902308	+	0.431092i	$0.858128\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	5.19615i		−	0.476331i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	10.5830			0.939090			0.469545	−	0.882909i	$-0.344418\pi$
$127$	10.5830			0.939090			0.469545	+	0.882909i	$0.344418\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			20.7846i			1.81596i	0.419014	+	0.907980i	$0.362376\pi$
$131$			20.7846i			1.81596i	−0.419014	+	0.907980i	$0.637624\pi$
$132$		0			0
$133$	6.00000	−	4.58258i	0.520266	−	0.397360i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.00000			−0.768922			−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000			−0.768922			−0.384461	+	0.923141i	$0.625613\pi$
$138$		0			0
$139$			17.3205i			1.46911i	0.678551	+	0.734553i	$0.262608\pi$
$139$			17.3205i			1.46911i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−15.8745			−1.32749
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	5.29150			0.430616			0.215308	−	0.976546i	$-0.430924\pi$
$151$	5.29150			0.430616			0.215308	+	0.976546i	$0.430924\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	10.0000			0.798087			0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000			0.798087			0.399043	+	0.916932i	$0.369342\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.00000			−0.709299
$162$		0			0
$163$			17.3205i			1.35665i	0.734763	+	0.678323i	$0.237293\pi$
$163$			17.3205i			1.35665i	−0.734763	+	0.678323i	$0.762707\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−15.8745			−1.22841			−0.614203	−	0.789148i	$-0.710522\pi$
$167$	−15.8745			−1.22841			−0.614203	+	0.789148i	$0.710522\pi$
$168$		0			0
$169$	−8.00000			−0.615385
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	9.16515i		−	0.696814i	−0.937343	−	0.348407i	$-0.886723\pi$
$173$		−	9.16515i		−	0.696814i	0.937343	−	0.348407i	$-0.113277\pi$
$174$		0			0
$175$		−	8.66025i		−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.8745			−1.18652			−0.593258	−	0.805012i	$-0.702159\pi$
$179$	−15.8745			−1.18652			−0.593258	+	0.805012i	$0.702159\pi$
$180$		0			0
$181$			9.16515i			0.681240i	0.940201	+	0.340620i	$0.110637\pi$
$181$			9.16515i			0.681240i	−0.940201	+	0.340620i	$0.889363\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	10.3923i		−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.1244i		−	0.877288i	−0.898661	−	0.438644i	$-0.855459\pi$
$191$		−	12.1244i		−	0.877288i	0.898661	−	0.438644i	$-0.144541\pi$
$192$		0			0
$193$		−	9.16515i		−	0.659722i	−0.944030	−	0.329861i	$-0.892998\pi$
$193$		−	9.16515i		−	0.659722i	0.944030	−	0.329861i	$-0.107002\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$			8.66025i			0.613909i	0.951724	+	0.306955i	$0.0993100\pi$
$199$			8.66025i			0.613909i	−0.951724	+	0.306955i	$0.900690\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.93725			0.557086
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	12.0000	−	9.16515i	0.830057	−	0.633967i
$210$		0			0
$211$	2.64575			0.182141			0.0910705	−	0.995844i	$-0.470971\pi$
$211$	2.64575			0.182141			0.0910705	+	0.995844i	$0.470971\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	9.16515i		−	0.622171i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	13.7477i		−	0.924772i
$222$		0			0
$223$	−10.5830			−0.708690			−0.354345	−	0.935115i	$-0.615296\pi$
$223$	−10.5830			−0.708690			−0.354345	+	0.935115i	$0.615296\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.93725			0.526814			0.263407	−	0.964685i	$-0.415154\pi$
$227$	7.93725			0.526814			0.263407	+	0.964685i	$0.415154\pi$
$228$		0			0
$229$	8.00000			0.528655			0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000			0.528655			0.264327	+	0.964433i	$0.414850\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1.73205i			0.112037i	0.998430	+	0.0560185i	$0.0178406\pi$
$239$			1.73205i			0.112037i	−0.998430	+	0.0560185i	$0.982159\pi$
$240$		0			0
$241$		−	9.16515i		−	0.590379i	−0.955439	−	0.295190i	$-0.904617\pi$
$241$		−	9.16515i		−	0.590379i	0.955439	−	0.295190i	$-0.0953828\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	15.8745	−	12.1244i	1.01007	−	0.771454i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			10.3923i			0.655956i	0.944685	+	0.327978i	$0.106367\pi$
$251$			10.3923i			0.655956i	−0.944685	+	0.327978i	$0.893633\pi$
$252$		0			0
$253$	−18.0000			−1.13165
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	15.8745			0.986394
$260$		0			0
$261$		0			0
$262$		0			0
$263$			3.46410i			0.213606i	0.994280	+	0.106803i	$0.0340614\pi$
$263$			3.46410i			0.213606i	−0.994280	+	0.106803i	$0.965939\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			27.4955i			1.67643i	0.545342	+	0.838214i	$0.316400\pi$
$269$			27.4955i			1.67643i	−0.545342	+	0.838214i	$0.683600\pi$
$270$		0			0
$271$		−	8.66025i		−	0.526073i	−0.964786	−	0.263036i	$-0.915276\pi$
$271$		−	8.66025i		−	0.526073i	0.964786	−	0.263036i	$-0.0847240\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	17.3205i		−	1.04447i
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$			20.7846i			1.23552i	0.786368	+	0.617758i	$0.211959\pi$
$283$			20.7846i			1.23552i	−0.786368	+	0.617758i	$0.788041\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.8745			0.937043
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$			22.9129i			1.33858i	0.742999	+	0.669292i	$0.233403\pi$
$293$			22.9129i			1.33858i	−0.742999	+	0.669292i	$0.766597\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−23.8118			−1.37707
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−26.4575			−1.51001			−0.755005	−	0.655719i	$-0.772366\pi$
$307$	−26.4575			−1.51001			−0.755005	+	0.655719i	$0.772366\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	22.5167i		−	1.27680i	−0.769704	−	0.638401i	$-0.779596\pi$
$311$		−	22.5167i		−	1.27680i	0.769704	−	0.638401i	$-0.220404\pi$
$312$		0			0
$313$	1.00000			0.0565233			0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000			0.0565233			0.0282617	+	0.999601i	$0.491003\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			13.7477i			0.772149i	0.922468	+	0.386075i	$0.126169\pi$
$317$			13.7477i			0.772149i	−0.922468	+	0.386075i	$0.873831\pi$
$318$		0			0
$319$	15.8745			0.888802
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.93725	+	10.3923i	0.441641	+	0.578243i
$324$		0			0
$325$		−	22.9129i		−	1.27098i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.00000			0.330791
$330$		0			0
$331$	−2.64575			−0.145424			−0.0727118	−	0.997353i	$-0.523165\pi$
$331$	−2.64575			−0.145424			−0.0727118	+	0.997353i	$0.523165\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$			18.3303i			0.998515i	0.866454	+	0.499258i	$0.166394\pi$
$337$			18.3303i			0.998515i	−0.866454	+	0.499258i	$0.833606\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	18.3303i		−	0.992642i
$342$		0			0
$343$			19.0526i			1.02874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	21.0000			1.11772			0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000			1.11772			0.558859	+	0.829263i	$0.311239\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			5.19615i			0.274242i	0.990554	+	0.137121i	$0.0437850\pi$
$359$			5.19615i			0.274242i	−0.990554	+	0.137121i	$0.956215\pi$
$360$		0			0
$361$	−5.00000	+	18.3303i	−0.263158	+	0.964753i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$			3.46410i			0.180825i	0.995904	+	0.0904123i	$0.0288185\pi$
$367$			3.46410i			0.180825i	−0.995904	+	0.0904123i	$0.971182\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.93725			−0.412082
$372$		0			0
$373$			32.0780i			1.66094i	0.557065	+	0.830469i	$0.311927\pi$
$373$			32.0780i			1.66094i	−0.557065	+	0.830469i	$0.688073\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21.0000			1.08156
$378$		0			0
$379$	−34.3948			−1.76674			−0.883370	−	0.468676i	$-0.844731\pi$
$379$	−34.3948			−1.76674			−0.883370	+	0.468676i	$0.844731\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	31.7490			1.62230			0.811149	−	0.584839i	$-0.198842\pi$
$383$	31.7490			1.62230			0.811149	+	0.584839i	$0.198842\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$		−	15.5885i		−	0.788342i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−22.0000			−1.10415			−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000			−1.10415			−0.552074	+	0.833795i	$0.686163\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		−	24.2487i		−	1.20791i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	31.7490			1.57374
$408$		0			0
$409$			9.16515i			0.453188i	0.973989	+	0.226594i	$0.0727590\pi$
$409$			9.16515i			0.453188i	−0.973989	+	0.226594i	$0.927241\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	13.7477i		−	0.676481i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			6.92820i			0.338465i	0.985576	+	0.169232i	$0.0541289\pi$
$419$			6.92820i			0.338465i	−0.985576	+	0.169232i	$0.945871\pi$
$420$		0			0
$421$		−	22.9129i		−	1.11671i	−0.829604	−	0.558353i	$-0.811434\pi$
$421$		−	22.9129i		−	1.11671i	0.829604	−	0.558353i	$-0.188566\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	15.0000			0.727607
$426$		0			0
$427$		−	13.8564i		−	0.670559i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18.0000	−	13.7477i	0.861057	−	0.657643i
$438$		0			0
$439$	−10.5830			−0.505099			−0.252550	−	0.967584i	$-0.581269\pi$
$439$	−10.5830			−0.505099			−0.252550	+	0.967584i	$0.581269\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			17.3205i			0.822922i	0.911427	+	0.411461i	$0.134981\pi$
$443$			17.3205i			0.822922i	−0.911427	+	0.411461i	$0.865019\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	9.16515i		−	0.432530i	−0.976335	−	0.216265i	$-0.930612\pi$
$449$		−	9.16515i		−	0.432530i	0.976335	−	0.216265i	$-0.0693876\pi$
$450$		0			0
$451$	31.7490			1.49500
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	25.0000			1.16945			0.584725	−	0.811231i	$-0.301202\pi$
$457$	25.0000			1.16945			0.584725	+	0.811231i	$0.301202\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12.0000			0.558896			0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000			0.558896			0.279448	+	0.960161i	$0.409849\pi$
$462$		0			0
$463$		−	38.1051i		−	1.77090i	−0.464739	−	0.885448i	$-0.653852\pi$
$463$		−	38.1051i		−	1.77090i	0.464739	−	0.885448i	$-0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	3.46410i		−	0.160300i	−0.996783	−	0.0801498i	$-0.974460\pi$
$467$		−	3.46410i		−	0.160300i	0.996783	−	0.0801498i	$-0.0255399\pi$
$468$		0			0
$469$			4.58258i			0.211604i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	13.2288	+	17.3205i	0.606977	+	0.794719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	10.3923i		−	0.474837i	−0.971408	−	0.237418i	$-0.923699\pi$
$479$		−	10.3923i		−	0.474837i	0.971408	−	0.237418i	$-0.0763012\pi$
$480$		0			0
$481$	42.0000			1.91504
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	10.5830			0.479562			0.239781	−	0.970827i	$-0.422924\pi$
$487$	10.5830			0.479562			0.239781	+	0.970827i	$0.422924\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	13.8564i		−	0.625331i	−0.949863	−	0.312665i	$-0.898778\pi$
$491$		−	13.8564i		−	0.625331i	0.949863	−	0.312665i	$-0.101222\pi$
$492$		0			0
$493$			13.7477i			0.619166i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			27.4955i			1.23334i
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	43.3013i		−	1.93071i	−0.260943	−	0.965354i	$-0.584034\pi$
$503$		−	43.3013i		−	1.93071i	0.260943	−	0.965354i	$-0.415966\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	9.16515i		−	0.406238i	−0.979154	−	0.203119i	$-0.934892\pi$
$509$		−	9.16515i		−	0.406238i	0.979154	−	0.203119i	$-0.0651079\pi$
$510$		0			0
$511$		−	19.0526i		−	0.842836i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	12.0000			0.527759
$518$		0			0
$519$		0			0
$520$		0			0
$521$			27.4955i			1.20460i	0.798271	+	0.602299i	$0.205748\pi$
$521$			27.4955i			1.20460i	−0.798271	+	0.602299i	$0.794252\pi$
$522$		0			0
$523$	−13.2288			−0.578453			−0.289227	−	0.957261i	$-0.593398\pi$
$523$	−13.2288			−0.578453			−0.289227	+	0.957261i	$0.593398\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	15.8745			0.691504
$528$		0			0
$529$	−4.00000			−0.173913
$530$		0			0
$531$		0			0
$532$		0			0
$533$	42.0000			1.81922
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			13.8564i			0.596838i
$540$		0			0
$541$	−20.0000			−0.859867			−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000			−0.859867			−0.429934	+	0.902861i	$0.641463\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−26.4575			−1.13124			−0.565621	−	0.824665i	$-0.691363\pi$
$547$	−26.4575			−1.13124			−0.565621	+	0.824665i	$0.691363\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−15.8745	+	12.1244i	−0.676277	+	0.516515i
$552$		0			0
$553$			9.16515i			0.389742i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.0000			1.01691			0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000			1.01691			0.508456	+	0.861088i	$0.330216\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−15.8745			−0.669031			−0.334515	−	0.942390i	$-0.608573\pi$
$563$	−15.8745			−0.669031			−0.334515	+	0.942390i	$0.608573\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			27.4955i			1.15267i	0.817214	+	0.576335i	$0.195518\pi$
$569$			27.4955i			1.15267i	−0.817214	+	0.576335i	$0.804482\pi$
$570$		0			0
$571$		−	3.46410i		−	0.144968i	−0.997370	−	0.0724841i	$-0.976907\pi$
$571$		−	3.46410i		−	0.144968i	0.997370	−	0.0724841i	$-0.0230926\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	25.9808i		−	1.08347i
$576$		0			0
$577$	−17.0000			−0.707719			−0.353860	−	0.935299i	$-0.615131\pi$
$577$	−17.0000			−0.707719			−0.353860	+	0.935299i	$0.615131\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.0000			−0.497844
$582$		0			0
$583$	−15.8745			−0.657455
$584$		0			0
$585$		0			0
$586$		0			0
$587$			41.5692i			1.71575i	0.513862	+	0.857873i	$0.328214\pi$
$587$			41.5692i			1.71575i	−0.513862	+	0.857873i	$0.671786\pi$
$588$		0			0
$589$	14.0000	+	18.3303i	0.576860	+	0.755287i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	31.7490			1.29723			0.648615	−	0.761117i	$-0.275349\pi$
$599$	31.7490			1.29723			0.648615	+	0.761117i	$0.275349\pi$
$600$		0			0
$601$			27.4955i			1.12156i	0.827964	+	0.560781i	$0.189499\pi$
$601$			27.4955i			1.12156i	−0.827964	+	0.560781i	$0.810501\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	26.4575			1.07388			0.536939	−	0.843621i	$-0.319581\pi$
$607$	26.4575			1.07388			0.536939	+	0.843621i	$0.319581\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.8745			0.642214
$612$		0			0
$613$	−44.0000			−1.77714			−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000			−1.77714			−0.888572	+	0.458738i	$0.848302\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	30.0000			1.20775			0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000			1.20775			0.603877	+	0.797077i	$0.293622\pi$
$618$		0			0
$619$			13.8564i			0.556936i	0.960446	+	0.278468i	$0.0898266\pi$
$619$			13.8564i			0.556936i	−0.960446	+	0.278468i	$0.910173\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	31.7490			1.27200
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			27.4955i			1.09632i
$630$		0			0
$631$			24.2487i			0.965326i	0.875806	+	0.482663i	$0.160330\pi$
$631$			24.2487i			0.965326i	−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$			18.3303i			0.726273i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			36.6606i			1.44801i	0.689796	+	0.724003i	$0.257700\pi$
$641$			36.6606i			1.44801i	−0.689796	+	0.724003i	$0.742300\pi$
$642$		0			0
$643$		−	31.1769i		−	1.22950i	−0.788723	−	0.614749i	$-0.789257\pi$
$643$		−	31.1769i		−	1.22950i	0.788723	−	0.614749i	$-0.210743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			32.9090i			1.29378i	0.762581	+	0.646892i	$0.223932\pi$
$647$			32.9090i			1.29378i	−0.762581	+	0.646892i	$0.776068\pi$
$648$		0			0
$649$		−	27.4955i		−	1.07929i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23.8118			−0.927575			−0.463787	−	0.885947i	$-0.653510\pi$
$659$	−23.8118			−0.927575			−0.463787	+	0.885947i	$0.653510\pi$
$660$		0			0
$661$		−	4.58258i		−	0.178242i	−0.996021	−	0.0891208i	$-0.971594\pi$
$661$		−	4.58258i		−	0.178242i	0.996021	−	0.0891208i	$-0.0284057\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	23.8118			0.921995
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	27.7128i		−	1.06984i
$672$		0			0
$673$		−	45.8258i		−	1.76645i	−0.468947	−	0.883227i	$-0.655366\pi$
$673$		−	45.8258i		−	1.76645i	0.468947	−	0.883227i	$-0.344634\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	13.7477i		−	0.528368i	−0.964472	−	0.264184i	$-0.914897\pi$
$677$		−	13.7477i		−	0.528368i	0.964472	−	0.264184i	$-0.0851026\pi$
$678$		0			0
$679$	−15.8745			−0.609208
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.8745			0.607421			0.303711	−	0.952764i	$-0.401774\pi$
$683$	15.8745			0.607421			0.303711	+	0.952764i	$0.401774\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21.0000			−0.800036
$690$		0			0
$691$			20.7846i			0.790684i	0.918534	+	0.395342i	$0.129374\pi$
$691$			20.7846i			0.790684i	−0.918534	+	0.395342i	$0.870626\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$			27.4955i			1.04146i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	36.0000			1.35970			0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000			1.35970			0.679851	+	0.733351i	$0.262045\pi$
$702$		0			0
$703$	−31.7490	+	24.2487i	−1.19744	+	0.914557i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	31.1769i		−	1.17253i
$708$		0			0
$709$	4.00000			0.150223			0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000			0.150223			0.0751116	+	0.997175i	$0.476069\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	27.4955i		−	1.02971i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	39.8372i		−	1.48568i	−0.669471	−	0.742838i	$-0.733479\pi$
$719$		−	39.8372i		−	1.48568i	0.669471	−	0.742838i	$-0.266521\pi$
$720$		0			0
$721$		−	18.3303i		−	0.682656i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			22.9129i			0.850963i
$726$		0			0
$727$		−	25.9808i		−	0.963573i	−0.876289	−	0.481787i	$-0.839988\pi$
$727$		−	25.9808i		−	0.963573i	0.876289	−	0.481787i	$-0.160012\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	28.0000			1.03420			0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000			1.03420			0.517102	+	0.855924i	$0.327011\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			9.16515i			0.337603i
$738$		0			0
$739$			20.7846i			0.764574i	0.924044	+	0.382287i	$0.124863\pi$
$739$			20.7846i			0.764574i	−0.924044	+	0.382287i	$0.875137\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.8745			−0.582379			−0.291190	−	0.956665i	$-0.594051\pi$
$743$	−15.8745			−0.582379			−0.291190	+	0.956665i	$0.594051\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	13.7477i		−	0.502331i
$750$		0			0
$751$	37.0405			1.35163			0.675814	−	0.737072i	$-0.263792\pi$
$751$	37.0405			1.35163			0.675814	+	0.737072i	$0.263792\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	20.0000			0.726912			0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000			0.726912			0.363456	+	0.931611i	$0.381597\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−21.0000			−0.761249			−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000			−0.761249			−0.380625	+	0.924730i	$0.624291\pi$
$762$		0			0
$763$	23.8118			0.862044
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	36.3731i		−	1.31336i
$768$		0			0
$769$	7.00000			0.252426			0.126213	−	0.992003i	$-0.459718\pi$
$769$	7.00000			0.252426			0.126213	+	0.992003i	$0.459718\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			4.58258i			0.164824i	0.996598	+	0.0824119i	$0.0262623\pi$
$773$			4.58258i			0.164824i	−0.996598	+	0.0824119i	$0.973738\pi$
$774$		0			0
$775$	26.4575			0.950382
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−31.7490	+	24.2487i	−1.13753	+	0.868800i
$780$		0			0
$781$			54.9909i			1.96773i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	2.64575			0.0943108			0.0471554	−	0.998888i	$-0.484984\pi$
$787$	2.64575			0.0943108			0.0471554	+	0.998888i	$0.484984\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−15.8745			−0.564433
$792$		0			0
$793$		−	36.6606i		−	1.30186i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			32.0780i			1.13626i	0.822938	+	0.568131i	$0.192333\pi$
$797$			32.0780i			1.13626i	−0.822938	+	0.568131i	$0.807667\pi$
$798$		0			0
$799$			10.3923i			0.367653i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	38.1051i		−	1.34470i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	39.0000			1.37117			0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000			1.37117			0.685583	+	0.727994i	$0.259547\pi$
$810$		0			0
$811$	−2.64575			−0.0929049			−0.0464524	−	0.998921i	$-0.514792\pi$
$811$	−2.64575			−0.0929049			−0.0464524	+	0.998921i	$0.514792\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.0000			0.628204			0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000			0.628204			0.314102	+	0.949389i	$0.398297\pi$
$822$		0			0
$823$		−	32.9090i		−	1.14713i	−0.819159	−	0.573567i	$-0.805559\pi$
$823$		−	32.9090i		−	1.14713i	0.819159	−	0.573567i	$-0.194441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−7.93725			−0.276005			−0.138003	−	0.990432i	$-0.544068\pi$
$827$	−7.93725			−0.276005			−0.138003	+	0.990432i	$0.544068\pi$
$828$		0			0
$829$		−	32.0780i		−	1.11412i	−0.830474	−	0.557058i	$-0.811930\pi$
$829$		−	32.0780i		−	1.11412i	0.830474	−	0.557058i	$-0.188070\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.0000			−0.415775
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−31.7490			−1.09610			−0.548049	−	0.836446i	$-0.684629\pi$
$839$	−31.7490			−1.09610			−0.548049	+	0.836446i	$0.684629\pi$
$840$		0			0
$841$	8.00000			0.275862
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	1.73205i		−	0.0595140i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	47.6235			1.63251
$852$		0			0
$853$	44.0000			1.50653			0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000			1.50653			0.753266	+	0.657716i	$0.228477\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	18.3303i		−	0.626151i	−0.949728	−	0.313076i	$-0.898641\pi$
$857$		−	18.3303i		−	0.626151i	0.949728	−	0.313076i	$-0.101359\pi$
$858$		0			0
$859$			6.92820i			0.236387i	0.992991	+	0.118194i	$0.0377103\pi$
$859$			6.92820i			0.236387i	−0.992991	+	0.118194i	$0.962290\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	31.7490			1.08075			0.540375	−	0.841425i	$-0.318283\pi$
$863$	31.7490			1.08075			0.540375	+	0.841425i	$0.318283\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			18.3303i			0.621813i
$870$		0			0
$871$			12.1244i			0.410818i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			22.9129i			0.773713i	0.922140	+	0.386856i	$0.126439\pi$
$877$			22.9129i			0.773713i	−0.922140	+	0.386856i	$0.873561\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$		−	13.8564i		−	0.466305i	−0.972440	−	0.233153i	$-0.925096\pi$
$883$		−	13.8564i		−	0.466305i	0.972440	−	0.233153i	$-0.0749042\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−31.7490			−1.06603			−0.533014	−	0.846107i	$-0.678941\pi$
$887$	−31.7490			−1.06603			−0.533014	+	0.846107i	$0.678941\pi$
$888$		0			0
$889$			18.3303i			0.614779i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12.0000	+	9.16515i	−0.401565	+	0.306700i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			24.2487i			0.808740i
$900$		0			0
$901$		−	13.7477i		−	0.458003i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−44.9778			−1.49346			−0.746731	−	0.665126i	$-0.768378\pi$
$907$	−44.9778			−1.49346			−0.746731	+	0.665126i	$0.768378\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−24.0000			−0.794284
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−36.0000			−1.18882
$918$		0			0
$919$		−	5.19615i		−	0.171405i	−0.996321	−	0.0857026i	$-0.972687\pi$
$919$		−	5.19615i		−	0.171405i	0.996321	−	0.0857026i	$-0.0273135\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			72.7461i			2.39447i
$924$		0			0
$925$			45.8258i			1.50674i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15.0000			−0.492134			−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000			−0.492134			−0.246067	+	0.969253i	$0.579138\pi$
$930$		0			0
$931$	−10.5830	−	13.8564i	−0.346844	−	0.454125i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−7.00000			−0.228680			−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000			−0.228680			−0.114340	+	0.993442i	$0.536475\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	41.2432i		−	1.34449i	−0.740329	−	0.672245i	$-0.765330\pi$
$941$		−	41.2432i		−	1.34449i	0.740329	−	0.672245i	$-0.234670\pi$
$942$		0			0
$943$	47.6235			1.55084
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	48.4974i		−	1.57595i	−0.615704	−	0.787977i	$-0.711128\pi$
$947$		−	48.4974i		−	1.57595i	0.615704	−	0.787977i	$-0.288872\pi$
$948$		0			0
$949$		−	50.4083i		−	1.63632i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	36.6606i		−	1.18755i	−0.804630	−	0.593777i	$-0.797636\pi$
$953$		−	36.6606i		−	1.18755i	0.804630	−	0.593777i	$-0.202364\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	15.5885i		−	0.503378i
$960$		0			0
$961$	−3.00000			−0.0967742
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			24.2487i			0.779786i	0.920860	+	0.389893i	$0.127488\pi$
$967$			24.2487i			0.779786i	−0.920860	+	0.389893i	$0.872512\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−47.6235			−1.52831			−0.764156	−	0.645032i	$-0.776844\pi$
$971$	−47.6235			−1.52831			−0.764156	+	0.645032i	$0.776844\pi$
$972$		0			0
$973$	−30.0000			−0.961756
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	9.16515i		−	0.293219i	−0.989194	−	0.146610i	$-0.953164\pi$
$977$		−	9.16515i		−	0.293219i	0.989194	−	0.146610i	$-0.0468361\pi$
$978$		0			0
$979$	63.4980			2.02941
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−52.9150			−1.68090			−0.840451	−	0.541888i	$-0.817710\pi$
$991$	−52.9150			−1.68090			−0.840451	+	0.541888i	$0.817710\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	10.0000			0.316703			0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000			0.316703			0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.k.k.2431.4		4
3.2	odd	2		304.2.h.b.303.4	yes	4
4.3	odd	2	inner	2736.2.k.k.2431.2		4
12.11	even	2		304.2.h.b.303.1	✓	4
19.18	odd	2	inner	2736.2.k.k.2431.3		4
24.5	odd	2		1216.2.h.c.1215.2		4
24.11	even	2		1216.2.h.c.1215.3		4
57.56	even	2		304.2.h.b.303.2	yes	4
76.75	even	2	inner	2736.2.k.k.2431.1		4
228.227	odd	2		304.2.h.b.303.3	yes	4
456.227	odd	2		1216.2.h.c.1215.1		4
456.341	even	2		1216.2.h.c.1215.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.h.b.303.1	✓	4	12.11	even	2
304.2.h.b.303.2	yes	4	57.56	even	2
304.2.h.b.303.3	yes	4	228.227	odd	2
304.2.h.b.303.4	yes	4	3.2	odd	2
1216.2.h.c.1215.1		4	456.227	odd	2
1216.2.h.c.1215.2		4	24.5	odd	2
1216.2.h.c.1215.3		4	24.11	even	2
1216.2.h.c.1215.4		4	456.341	even	2
2736.2.k.k.2431.1		4	76.75	even	2	inner
2736.2.k.k.2431.2		4	4.3	odd	2	inner
2736.2.k.k.2431.3		4	19.18	odd	2	inner
2736.2.k.k.2431.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{7} +O(q^{10})\)	\(q+1.73205i q^{7} +3.46410i q^{11} +4.58258i q^{13} -3.00000 q^{17} +(-2.64575 - 3.46410i) q^{19} +5.19615i q^{23} -5.00000 q^{25} -4.58258i q^{29} -5.29150 q^{31} -9.16515i q^{37} -9.16515i q^{41} -3.46410i q^{47} +4.00000 q^{49} +4.58258i q^{53} -7.93725 q^{59} -8.00000 q^{61} +2.64575 q^{67} +15.8745 q^{71} -11.0000 q^{73} -6.00000 q^{77} +5.29150 q^{79} +6.92820i q^{83} -18.3303i q^{89} -7.93725 q^{91} +9.16515i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{17} - 20 q^{25} + 16 q^{49} - 32 q^{61} - 44 q^{73} - 24 q^{77}+O(q^{100}) \) 4 * q - 12 * q^17 - 20 * q^25 + 16 * q^49 - 32 * q^61 - 44 * q^73 - 24 * q^77

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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