LMFDB - Embedded newform 1656.2.m.a.1241.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1656,2,Mod(1241,1656)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1656, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1656.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1656 = 2^{3} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1656.m (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:	$13.2232265747$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.1
Character	$\chi$	$=$	1656.1241
Dual form			1656.2.m.a.1241.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.10131 q^{5} +1.73842i q^{7} +O(q^{10})$	$q-4.10131 q^{5} +1.73842i q^{7} +3.47914 q^{11} -4.27673 q^{13} -5.54242 q^{17} +4.56368i q^{19} +(0.257710 - 4.78890i) q^{23} +11.8207 q^{25} -7.63505i q^{29} +7.08201 q^{31} -7.12980i q^{35} +1.02065i q^{37} +3.66779i q^{41} +0.721032i q^{43} +2.63400i q^{47} +3.97790 q^{49} -1.12413 q^{53} -14.2690 q^{55} -12.4635i q^{59} -11.1775i q^{61} +17.5402 q^{65} -2.39135i q^{67} -0.495799i q^{71} +5.00105 q^{73} +6.04821i q^{77} -15.0155i q^{79} +15.2739 q^{83} +22.7312 q^{85} +2.94963 q^{89} -7.43475i q^{91} -18.7171i q^{95} +1.86805i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 40 q^{25} - 16 q^{31} - 40 q^{49} - 64 q^{55} + 48 q^{85}+O(q^{100}) $ 24 * q + 40 * q^25 - 16 * q^31 - 40 * q^49 - 64 * q^55 + 48 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1656\mathbb{Z}\right)^\times$.

$n$	$415$	$649$	$829$	$1289$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	−4.10131			−1.83416			−0.917081	−	0.398701i	$-0.869461\pi$
$5$	−4.10131			−1.83416			−0.917081	+	0.398701i	$0.869461\pi$
$6$		0			0
$7$			1.73842i			0.657061i	0.944493	+	0.328530i	$0.106553\pi$
$7$			1.73842i			0.657061i	−0.944493	+	0.328530i	$0.893447\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.47914			1.04900			0.524501	−	0.851410i	$-0.324252\pi$
$11$	3.47914			1.04900			0.524501	+	0.851410i	$0.324252\pi$
$12$		0			0
$13$	−4.27673			−1.18615			−0.593076	−	0.805147i	$-0.702087\pi$
$13$	−4.27673			−1.18615			−0.593076	+	0.805147i	$0.702087\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.54242			−1.34423			−0.672117	−	0.740445i	$-0.734615\pi$
$17$	−5.54242			−1.34423			−0.672117	+	0.740445i	$0.734615\pi$
$18$		0			0
$19$			4.56368i			1.04698i	0.852032	+	0.523490i	$0.175370\pi$
$19$			4.56368i			1.04698i	−0.852032	+	0.523490i	$0.824630\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.257710	−	4.78890i	0.0537362	−	0.998555i
$23$	0.257710	−	4.78890i	0.0537362	−	0.998555i
$24$		0			0
$25$	11.8207			2.36415
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	7.63505i		−	1.41779i	−0.705313	−	0.708896i	$-0.749193\pi$
$29$		−	7.63505i		−	1.41779i	0.705313	−	0.708896i	$-0.250807\pi$
$30$		0			0
$31$	7.08201			1.27197			0.635983	−	0.771703i	$-0.280595\pi$
$31$	7.08201			1.27197			0.635983	+	0.771703i	$0.280595\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	7.12980i		−	1.20516i
$36$		0			0
$37$			1.02065i			0.167793i	0.996474	+	0.0838967i	$0.0267366\pi$
$37$			1.02065i			0.167793i	−0.996474	+	0.0838967i	$0.973263\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.66779i			0.572813i	0.958108	+	0.286406i	$0.0924608\pi$
$41$			3.66779i			0.572813i	−0.958108	+	0.286406i	$0.907539\pi$
$42$		0			0
$43$			0.721032i			0.109956i	0.998488	+	0.0549781i	$0.0175089\pi$
$43$			0.721032i			0.109956i	−0.998488	+	0.0549781i	$0.982491\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.63400i			0.384208i	0.981375	+	0.192104i	$0.0615311\pi$
$47$			2.63400i			0.384208i	−0.981375	+	0.192104i	$0.938469\pi$
$48$		0			0
$49$	3.97790			0.568271
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.12413			−0.154412			−0.0772059	−	0.997015i	$-0.524600\pi$
$53$	−1.12413			−0.154412			−0.0772059	+	0.997015i	$0.524600\pi$
$54$		0			0
$55$	−14.2690			−1.92404
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	12.4635i		−	1.62261i	−0.584626	−	0.811303i	$-0.698759\pi$
$59$		−	12.4635i		−	1.62261i	0.584626	−	0.811303i	$-0.301241\pi$
$60$		0			0
$61$		−	11.1775i		−	1.43113i	−0.698546	−	0.715565i	$-0.746169\pi$
$61$		−	11.1775i		−	1.43113i	0.698546	−	0.715565i	$-0.253831\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	17.5402			2.17559
$66$		0			0
$67$		−	2.39135i		−	0.292150i	−0.989274	−	0.146075i	$-0.953336\pi$
$67$		−	2.39135i		−	0.292150i	0.989274	−	0.146075i	$-0.0466641\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	0.495799i		−	0.0588405i	−0.999567	−	0.0294203i	$-0.990634\pi$
$71$		−	0.495799i		−	0.0588405i	0.999567	−	0.0294203i	$-0.00936611\pi$
$72$		0			0
$73$	5.00105			0.585329			0.292664	−	0.956215i	$-0.405458\pi$
$73$	5.00105			0.585329			0.292664	+	0.956215i	$0.405458\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			6.04821i			0.689258i
$78$		0			0
$79$		−	15.0155i		−	1.68938i	−0.535258	−	0.844689i	$-0.679786\pi$
$79$		−	15.0155i		−	1.68938i	0.535258	−	0.844689i	$-0.320214\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	15.2739			1.67653			0.838263	−	0.545267i	$-0.183572\pi$
$83$	15.2739			1.67653			0.838263	+	0.545267i	$0.183572\pi$
$84$		0			0
$85$	22.7312			2.46554
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.94963			0.312660			0.156330	−	0.987705i	$-0.450034\pi$
$89$	2.94963			0.312660			0.156330	+	0.987705i	$0.450034\pi$
$90$		0			0
$91$		−	7.43475i		−	0.779374i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	18.7171i		−	1.92033i
$96$		0			0
$97$			1.86805i			0.189672i	0.995493	+	0.0948359i	$0.0302326\pi$
$97$			1.86805i			0.189672i	−0.995493	+	0.0948359i	$0.969767\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	16.5117i		−	1.64297i	−0.570227	−	0.821487i	$-0.693145\pi$
$101$		−	16.5117i		−	1.64297i	0.570227	−	0.821487i	$-0.306855\pi$
$102$		0			0
$103$		−	3.13371i		−	0.308774i	−0.988010	−	0.154387i	$-0.950660\pi$
$103$		−	3.13371i		−	0.308774i	0.988010	−	0.154387i	$-0.0493403\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−20.3192			−1.96434			−0.982168	−	0.188005i	$-0.939798\pi$
$107$	−20.3192			−1.96434			−0.982168	+	0.188005i	$0.939798\pi$
$108$		0			0
$109$		−	13.0659i		−	1.25149i	−0.780029	−	0.625744i	$-0.784796\pi$
$109$		−	13.0659i		−	1.25149i	0.780029	−	0.625744i	$-0.215204\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.11096			0.857087			0.428543	−	0.903521i	$-0.359027\pi$
$113$	9.11096			0.857087			0.428543	+	0.903521i	$0.359027\pi$
$114$		0			0
$115$	−1.05695	+	19.6408i	−0.0985608	+	1.83151i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	9.63505i		−	0.883243i
$120$		0			0
$121$	1.10444			0.100404
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−27.9740			−2.50207
$126$		0			0
$127$	7.26800			0.644930			0.322465	−	0.946581i	$-0.395488\pi$
$127$	7.26800			0.644930			0.322465	+	0.946581i	$0.395488\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			18.6017i			1.62524i	0.582797	+	0.812618i	$0.301958\pi$
$131$			18.6017i			1.62524i	−0.582797	+	0.812618i	$0.698042\pi$
$132$		0			0
$133$	−7.93359			−0.687929
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.45086			0.807442			0.403721	−	0.914882i	$-0.367717\pi$
$137$	9.45086			0.807442			0.403721	+	0.914882i	$0.367717\pi$
$138$		0			0
$139$	7.96916			0.675936			0.337968	−	0.941158i	$-0.390261\pi$
$139$	7.96916			0.675936			0.337968	+	0.941158i	$0.390261\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.8794			−1.24427
$144$		0			0
$145$			31.3137i			2.60046i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.36453			−0.111787			−0.0558933	−	0.998437i	$-0.517801\pi$
$149$	−1.36453			−0.111787			−0.0558933	+	0.998437i	$0.517801\pi$
$150$		0			0
$151$	−6.54401			−0.532544			−0.266272	−	0.963898i	$-0.585792\pi$
$151$	−6.54401			−0.532544			−0.266272	+	0.963898i	$0.585792\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−29.0455			−2.33299
$156$		0			0
$157$			13.6216i			1.08712i	0.839370	+	0.543560i	$0.182924\pi$
$157$			13.6216i			1.08712i	−0.839370	+	0.543560i	$0.817076\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.32512	+	0.448007i	0.656111	+	0.0353079i
$162$		0			0
$163$	−3.88821			−0.304548			−0.152274	−	0.988338i	$-0.548660\pi$
$163$	−3.88821			−0.304548			−0.152274	+	0.988338i	$0.548660\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1.34420i			0.104017i	0.998647	+	0.0520085i	$0.0165623\pi$
$167$			1.34420i			0.104017i	−0.998647	+	0.0520085i	$0.983438\pi$
$168$		0			0
$169$	5.29043			0.406956
$170$		0			0
$171$		0			0
$172$		0			0
$173$			16.9030i			1.28511i	0.766238	+	0.642557i	$0.222127\pi$
$173$			16.9030i			1.28511i	−0.766238	+	0.642557i	$0.777873\pi$
$174$		0			0
$175$			20.5494i			1.55339i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	24.9706i		−	1.86639i	−0.359367	−	0.933196i	$-0.617007\pi$
$179$		−	24.9706i		−	1.86639i	0.359367	−	0.933196i	$-0.382993\pi$
$180$		0			0
$181$		−	1.57632i		−	0.117167i	−0.998283	−	0.0585833i	$-0.981342\pi$
$181$		−	1.57632i		−	0.117167i	0.998283	−	0.0585833i	$-0.0186583\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	4.18599i		−	0.307760i
$186$		0			0
$187$	−19.2829			−1.41010
$188$		0			0
$189$		0			0
$190$		0			0
$191$	14.7923			1.07033			0.535166	−	0.844747i	$-0.320249\pi$
$191$	14.7923			1.07033			0.535166	+	0.844747i	$0.320249\pi$
$192$		0			0
$193$	25.4786			1.83399			0.916997	−	0.398895i	$-0.130606\pi$
$193$	25.4786			1.83399			0.916997	+	0.398895i	$0.130606\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			13.2919i			0.947009i	0.880791	+	0.473504i	$0.157011\pi$
$197$			13.2919i			0.947009i	−0.880791	+	0.473504i	$0.842989\pi$
$198$		0			0
$199$		−	9.43023i		−	0.668491i	−0.942486	−	0.334246i	$-0.891518\pi$
$199$		−	9.43023i		−	0.668491i	0.942486	−	0.334246i	$-0.108482\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.2729			0.931576
$204$		0			0
$205$		−	15.0428i		−	1.05063i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			15.8777i			1.09828i
$210$		0			0
$211$	−20.5689			−1.41602			−0.708012	−	0.706201i	$-0.750408\pi$
$211$	−20.5689			−1.41602			−0.708012	+	0.706201i	$0.750408\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	2.95717i		−	0.201678i
$216$		0			0
$217$			12.3115i			0.835759i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	23.7034			1.59447
$222$		0			0
$223$	−9.31336			−0.623669			−0.311834	−	0.950136i	$-0.600943\pi$
$223$	−9.31336			−0.623669			−0.311834	+	0.950136i	$0.600943\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.97500			−0.462947			−0.231473	−	0.972841i	$-0.574355\pi$
$227$	−6.97500			−0.462947			−0.231473	+	0.972841i	$0.574355\pi$
$228$		0			0
$229$			19.1532i			1.26568i	0.774283	+	0.632840i	$0.218111\pi$
$229$			19.1532i			1.26568i	−0.774283	+	0.632840i	$0.781889\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			15.7896i			1.03441i	0.855861	+	0.517206i	$0.173028\pi$
$233$			15.7896i			1.03441i	−0.855861	+	0.517206i	$0.826972\pi$
$234$		0			0
$235$		−	10.8028i		−	0.704700i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			25.7533i			1.66584i	0.553393	+	0.832921i	$0.313333\pi$
$239$			25.7533i			1.66584i	−0.553393	+	0.832921i	$0.686667\pi$
$240$		0			0
$241$		−	9.57101i		−	0.616523i	−0.951302	−	0.308262i	$-0.900253\pi$
$241$		−	9.57101i		−	0.616523i	0.951302	−	0.308262i	$-0.0997472\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−16.3146			−1.04230
$246$		0			0
$247$		−	19.5176i		−	1.24188i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.41583			−0.594322			−0.297161	−	0.954827i	$-0.596040\pi$
$251$	−9.41583			−0.594322			−0.297161	+	0.954827i	$0.596040\pi$
$252$		0			0
$253$	0.896609	−	16.6613i	0.0563693	−	1.04749i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	14.3485i		−	0.895036i	−0.894275	−	0.447518i	$-0.852308\pi$
$257$		−	14.3485i		−	0.895036i	0.894275	−	0.447518i	$-0.147692\pi$
$258$		0			0
$259$	−1.77431			−0.110250
$260$		0			0
$261$		0			0
$262$		0			0
$263$	18.8637			1.16319			0.581594	−	0.813479i	$-0.302429\pi$
$263$	18.8637			1.16319			0.581594	+	0.813479i	$0.302429\pi$
$264$		0			0
$265$	4.61043			0.283216
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	14.5599i		−	0.887733i	−0.896093	−	0.443866i	$-0.853606\pi$
$269$		−	14.5599i		−	0.887733i	0.896093	−	0.443866i	$-0.146394\pi$
$270$		0			0
$271$	−11.5887			−0.703962			−0.351981	−	0.936007i	$-0.614492\pi$
$271$	−11.5887			−0.703962			−0.351981	+	0.936007i	$0.614492\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	41.1261			2.47999
$276$		0			0
$277$	−19.5578			−1.17512			−0.587558	−	0.809182i	$-0.699911\pi$
$277$	−19.5578			−1.17512			−0.587558	+	0.809182i	$0.699911\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.33083			−0.318011			−0.159005	−	0.987278i	$-0.550829\pi$
$281$	−5.33083			−0.318011			−0.159005	+	0.987278i	$0.550829\pi$
$282$		0			0
$283$		−	20.6781i		−	1.22919i	−0.788844	−	0.614594i	$-0.789320\pi$
$283$		−	20.6781i		−	1.22919i	0.788844	−	0.614594i	$-0.210680\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.37616			−0.376373
$288$		0			0
$289$	13.7184			0.806965
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−25.6514			−1.49857			−0.749286	−	0.662246i	$-0.769603\pi$
$293$	−25.6514			−1.49857			−0.749286	+	0.662246i	$0.769603\pi$
$294$		0			0
$295$			51.1166i			2.97612i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.10215	+	20.4808i	−0.0637393	+	1.18444i
$300$		0			0
$301$	−1.25346			−0.0722480
$302$		0			0
$303$		0			0
$304$		0			0
$305$			45.8423i			2.62493i
$306$		0			0
$307$	−5.48329			−0.312948			−0.156474	−	0.987682i	$-0.550013\pi$
$307$	−5.48329			−0.312948			−0.156474	+	0.987682i	$0.550013\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			0.200445i			0.0113662i	0.999984	+	0.00568310i	$0.00180900\pi$
$311$			0.200445i			0.0113662i	−0.999984	+	0.00568310i	$0.998191\pi$
$312$		0			0
$313$		−	16.9220i		−	0.956485i	−0.878228	−	0.478243i	$-0.841274\pi$
$313$		−	16.9220i		−	0.956485i	0.878228	−	0.478243i	$-0.158726\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	7.30918i		−	0.410524i	−0.978707	−	0.205262i	$-0.934195\pi$
$317$		−	7.30918i		−	0.410524i	0.978707	−	0.205262i	$-0.0658047\pi$
$318$		0			0
$319$		−	26.5634i		−	1.48727i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	25.2938i		−	1.40739i
$324$		0			0
$325$	−50.5541			−2.80424
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.57899			−0.252448
$330$		0			0
$331$	31.8518			1.75073			0.875367	−	0.483460i	$-0.160620\pi$
$331$	31.8518			1.75073			0.875367	+	0.483460i	$0.160620\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			9.80767i			0.535850i
$336$		0			0
$337$		−	26.7321i		−	1.45619i	−0.685477	−	0.728094i	$-0.740406\pi$
$337$		−	26.7321i		−	1.45619i	0.685477	−	0.728094i	$-0.259594\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	24.6393			1.33429
$342$		0			0
$343$			19.0842i			1.03045i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			10.1414i			0.544417i	0.962238	+	0.272209i	$0.0877540\pi$
$347$			10.1414i			0.544417i	−0.962238	+	0.272209i	$0.912246\pi$
$348$		0			0
$349$	−12.5230			−0.670339			−0.335169	−	0.942158i	$-0.608794\pi$
$349$	−12.5230			−0.670339			−0.335169	+	0.942158i	$0.608794\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			9.41421i			0.501068i	0.968108	+	0.250534i	$0.0806062\pi$
$353$			9.41421i			0.501068i	−0.968108	+	0.250534i	$0.919394\pi$
$354$		0			0
$355$			2.03343i			0.107923i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	34.1320			1.80142			0.900710	−	0.434421i	$-0.143047\pi$
$359$	34.1320			1.80142			0.900710	+	0.434421i	$0.143047\pi$
$360$		0			0
$361$	−1.82715			−0.0961658
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−20.5109			−1.07359
$366$		0			0
$367$		−	3.33735i		−	0.174208i	−0.996199	−	0.0871042i	$-0.972239\pi$
$367$		−	3.33735i		−	0.174208i	0.996199	−	0.0871042i	$-0.0277613\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	1.95422i		−	0.101458i
$372$		0			0
$373$		−	3.81450i		−	0.197507i	−0.995112	−	0.0987536i	$-0.968514\pi$
$373$		−	3.81450i		−	0.197507i	0.995112	−	0.0987536i	$-0.0314856\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			32.6531i			1.68172i
$378$		0			0
$379$		−	23.9373i		−	1.22958i	−0.788692	−	0.614788i	$-0.789241\pi$
$379$		−	23.9373i		−	1.22958i	0.788692	−	0.614788i	$-0.210759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−15.8250			−0.808621			−0.404310	−	0.914622i	$-0.632488\pi$
$383$	−15.8250			−0.808621			−0.404310	+	0.914622i	$0.632488\pi$
$384$		0			0
$385$		−	24.8056i		−	1.26421i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	29.3097			1.48606			0.743030	−	0.669258i	$-0.233388\pi$
$389$	29.3097			1.48606			0.743030	+	0.669258i	$0.233388\pi$
$390$		0			0
$391$	−1.42833	+	26.5421i	−0.0722340	+	1.34229i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			61.5833i			3.09859i
$396$		0			0
$397$	3.86507			0.193982			0.0969912	−	0.995285i	$-0.469078\pi$
$397$	3.86507			0.193982			0.0969912	+	0.995285i	$0.469078\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−17.6293			−0.880364			−0.440182	−	0.897908i	$-0.645086\pi$
$401$	−17.6293			−0.880364			−0.440182	+	0.897908i	$0.645086\pi$
$402$		0			0
$403$	−30.2878			−1.50874
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.55098i			0.176015i
$408$		0			0
$409$	−2.33581			−0.115498			−0.0577492	−	0.998331i	$-0.518392\pi$
$409$	−2.33581			−0.115498			−0.0577492	+	0.998331i	$0.518392\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	21.6668			1.06615
$414$		0			0
$415$	−62.6429			−3.07502
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.27429			−0.306519			−0.153260	−	0.988186i	$-0.548977\pi$
$419$	−6.27429			−0.306519			−0.153260	+	0.988186i	$0.548977\pi$
$420$		0			0
$421$			24.5189i			1.19498i	0.801877	+	0.597489i	$0.203835\pi$
$421$			24.5189i			1.19498i	−0.801877	+	0.597489i	$0.796165\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−65.5155			−3.17797
$426$		0			0
$427$	19.4312			0.940340
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.3822			−0.548259			−0.274130	−	0.961693i	$-0.588390\pi$
$431$	−11.3822			−0.548259			−0.274130	+	0.961693i	$0.588390\pi$
$432$		0			0
$433$		−	1.26137i		−	0.0606174i	−0.999541	−	0.0303087i	$-0.990351\pi$
$433$		−	1.26137i		−	0.0606174i	0.999541	−	0.0303087i	$-0.00964903\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	21.8550	+	1.17610i	1.04547	+	0.0562607i
$438$		0			0
$439$	18.6639			0.890780			0.445390	−	0.895337i	$-0.353065\pi$
$439$	18.6639			0.890780			0.445390	+	0.895337i	$0.353065\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	14.1949i		−	0.674422i	−0.941429	−	0.337211i	$-0.890516\pi$
$443$		−	14.1949i		−	0.674422i	0.941429	−	0.337211i	$-0.109484\pi$
$444$		0			0
$445$	−12.0974			−0.573469
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	22.0527i		−	1.04073i	−0.853943	−	0.520367i	$-0.825795\pi$
$449$		−	22.0527i		−	1.04073i	0.853943	−	0.520367i	$-0.174205\pi$
$450$		0			0
$451$			12.7608i			0.600881i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			30.4922i			1.42950i
$456$		0			0
$457$		−	34.6071i		−	1.61885i	−0.587221	−	0.809427i	$-0.699778\pi$
$457$		−	34.6071i		−	1.61885i	0.587221	−	0.809427i	$-0.300222\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	9.65232i		−	0.449554i	−0.974410	−	0.224777i	$-0.927835\pi$
$461$		−	9.65232i		−	0.449554i	0.974410	−	0.224777i	$-0.0721653\pi$
$462$		0			0
$463$	10.2815			0.477820			0.238910	−	0.971042i	$-0.423210\pi$
$463$	10.2815			0.477820			0.238910	+	0.971042i	$0.423210\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	34.6720			1.60443			0.802216	−	0.597035i	$-0.203654\pi$
$467$	34.6720			1.60443			0.802216	+	0.597035i	$0.203654\pi$
$468$		0			0
$469$	4.15717			0.191960
$470$		0			0
$471$		0			0
$472$		0			0
$473$			2.50857i			0.115344i
$474$		0			0
$475$			53.9461i			2.47521i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−39.1250			−1.78767			−0.893833	−	0.448401i	$-0.851994\pi$
$479$	−39.1250			−1.78767			−0.893833	+	0.448401i	$0.851994\pi$
$480$		0			0
$481$		−	4.36503i		−	0.199028i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	7.66145i		−	0.347889i
$486$		0			0
$487$	11.7868			0.534111			0.267056	−	0.963681i	$-0.413949\pi$
$487$	11.7868			0.534111			0.267056	+	0.963681i	$0.413949\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	17.0851i		−	0.771041i	−0.922699	−	0.385520i	$-0.874022\pi$
$491$		−	17.0851i		−	0.771041i	0.922699	−	0.385520i	$-0.125978\pi$
$492$		0			0
$493$			42.3166i			1.90585i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.861907			0.0386618
$498$		0			0
$499$	23.5727			1.05526			0.527630	−	0.849474i	$-0.323081\pi$
$499$	23.5727			1.05526			0.527630	+	0.849474i	$0.323081\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.55671			−0.113998			−0.0569991	−	0.998374i	$-0.518153\pi$
$503$	−2.55671			−0.113998			−0.0569991	+	0.998374i	$0.518153\pi$
$504$		0			0
$505$			67.7195i			3.01348i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	36.2391i		−	1.60627i	−0.595797	−	0.803135i	$-0.703164\pi$
$509$		−	36.2391i		−	1.60627i	0.595797	−	0.803135i	$-0.296836\pi$
$510$		0			0
$511$			8.69392i			0.384597i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			12.8523i			0.566341i
$516$		0			0
$517$			9.16406i			0.403035i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.5963			−0.508042			−0.254021	−	0.967199i	$-0.581753\pi$
$521$	−11.5963			−0.508042			−0.254021	+	0.967199i	$0.581753\pi$
$522$		0			0
$523$		−	23.5715i		−	1.03071i	−0.856977	−	0.515355i	$-0.827660\pi$
$523$		−	23.5715i		−	1.03071i	0.856977	−	0.515355i	$-0.172340\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−39.2514			−1.70982
$528$		0			0
$529$	−22.8672	−	2.46829i	−0.994225	−	0.107317i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	15.6862i		−	0.679443i
$534$		0			0
$535$	83.3355			3.60291
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13.8397			0.596117
$540$		0			0
$541$	−28.6785			−1.23298			−0.616492	−	0.787361i	$-0.711447\pi$
$541$	−28.6785			−1.23298			−0.616492	+	0.787361i	$0.711447\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			53.5874i			2.29543i
$546$		0			0
$547$	−31.0536			−1.32776			−0.663878	−	0.747841i	$-0.731091\pi$
$547$	−31.0536			−1.32776			−0.663878	+	0.747841i	$0.731091\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	34.8439			1.48440
$552$		0			0
$553$	26.1033			1.11002
$554$		0			0
$555$		0			0
$556$		0			0
$557$	30.9338			1.31071			0.655354	−	0.755322i	$-0.272520\pi$
$557$	30.9338			1.31071			0.655354	+	0.755322i	$0.272520\pi$
$558$		0			0
$559$		−	3.08366i		−	0.130425i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−27.5729			−1.16206			−0.581029	−	0.813883i	$-0.697350\pi$
$563$	−27.5729			−1.16206			−0.581029	+	0.813883i	$0.697350\pi$
$564$		0			0
$565$	−37.3669			−1.57204
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.37987			0.225536			0.112768	−	0.993621i	$-0.464028\pi$
$569$	5.37987			0.225536			0.112768	+	0.993621i	$0.464028\pi$
$570$		0			0
$571$		−	33.6403i		−	1.40780i	−0.710299	−	0.703901i	$-0.751440\pi$
$571$		−	33.6403i		−	1.40780i	0.710299	−	0.703901i	$-0.248560\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.04632	−	56.6084i	0.127040	−	2.36073i
$576$		0			0
$577$	−28.4686			−1.18516			−0.592582	−	0.805510i	$-0.701892\pi$
$577$	−28.4686			−1.18516			−0.592582	+	0.805510i	$0.701892\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			26.5524i			1.10158i
$582$		0			0
$583$	−3.91103			−0.161978
$584$		0			0
$585$		0			0
$586$		0			0
$587$			35.8030i			1.47775i	0.673844	+	0.738874i	$0.264642\pi$
$587$			35.8030i			1.47775i	−0.673844	+	0.738874i	$0.735358\pi$
$588$		0			0
$589$			32.3200i			1.33172i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	31.8762i		−	1.30900i	−0.756063	−	0.654499i	$-0.772879\pi$
$593$		−	31.8762i		−	1.30900i	0.756063	−	0.654499i	$-0.227121\pi$
$594$		0			0
$595$			39.5163i			1.62001i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			35.3703i			1.44519i	0.691270	+	0.722596i	$0.257051\pi$
$599$			35.3703i			1.44519i	−0.691270	+	0.722596i	$0.742949\pi$
$600$		0			0
$601$	21.7547			0.887393			0.443696	−	0.896177i	$-0.353667\pi$
$601$	21.7547			0.887393			0.443696	+	0.896177i	$0.353667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.52966			−0.184157
$606$		0			0
$607$	−9.02352			−0.366253			−0.183127	−	0.983089i	$-0.558622\pi$
$607$	−9.02352			−0.366253			−0.183127	+	0.983089i	$0.558622\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	11.2649i		−	0.455729i
$612$		0			0
$613$		−	37.0800i		−	1.49765i	−0.662769	−	0.748824i	$-0.730619\pi$
$613$		−	37.0800i		−	1.49765i	0.662769	−	0.748824i	$-0.269381\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.11992			0.0450863			0.0225432	−	0.999746i	$-0.492824\pi$
$617$	1.11992			0.0450863			0.0225432	+	0.999746i	$0.492824\pi$
$618$		0			0
$619$			8.98979i			0.361330i	0.983545	+	0.180665i	$0.0578250\pi$
$619$			8.98979i			0.361330i	−0.983545	+	0.180665i	$0.942175\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			5.12770i			0.205437i
$624$		0			0
$625$	55.6262			2.22505
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	5.65685i		−	0.225554i
$630$		0			0
$631$		−	17.4018i		−	0.692754i	−0.938095	−	0.346377i	$-0.887412\pi$
$631$		−	17.4018i		−	0.692754i	0.938095	−	0.346377i	$-0.112588\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−29.8083			−1.18291
$636$		0			0
$637$	−17.0124			−0.674056
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.14756			0.242814			0.121407	−	0.992603i	$-0.461259\pi$
$641$	6.14756			0.242814			0.121407	+	0.992603i	$0.461259\pi$
$642$		0			0
$643$		−	32.8295i		−	1.29467i	−0.762206	−	0.647335i	$-0.775884\pi$
$643$		−	32.8295i		−	1.29467i	0.762206	−	0.647335i	$-0.224116\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			21.4038i			0.841470i	0.907184	+	0.420735i	$0.138228\pi$
$647$			21.4038i			0.841470i	−0.907184	+	0.420735i	$0.861772\pi$
$648$		0			0
$649$		−	43.3622i		−	1.70212i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	12.5867i		−	0.492556i	−0.969199	−	0.246278i	$-0.920792\pi$
$653$		−	12.5867i		−	0.492556i	0.969199	−	0.246278i	$-0.0792076\pi$
$654$		0			0
$655$		−	76.2912i		−	2.98095i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−32.7368			−1.27525			−0.637623	−	0.770348i	$-0.720082\pi$
$659$	−32.7368			−1.27525			−0.637623	+	0.770348i	$0.720082\pi$
$660$		0			0
$661$			44.7895i			1.74211i	0.491186	+	0.871055i	$0.336563\pi$
$661$			44.7895i			1.74211i	−0.491186	+	0.871055i	$0.663437\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	32.5381			1.26177
$666$		0			0
$667$	−36.5635	−	1.96763i	−1.41574	−	0.0761868i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	38.8881i		−	1.50126i
$672$		0			0
$673$	22.0817			0.851188			0.425594	−	0.904914i	$-0.360065\pi$
$673$	22.0817			0.851188			0.425594	+	0.904914i	$0.360065\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	29.6083			1.13794			0.568970	−	0.822359i	$-0.307342\pi$
$677$	29.6083			1.13794			0.568970	+	0.822359i	$0.307342\pi$
$678$		0			0
$679$	−3.24746			−0.124626
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	8.60378i		−	0.329214i	−0.986359	−	0.164607i	$-0.947364\pi$
$683$		−	8.60378i		−	0.329214i	0.986359	−	0.164607i	$-0.0526356\pi$
$684$		0			0
$685$	−38.7609			−1.48098
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.80762			0.183156
$690$		0			0
$691$	27.3074			1.03882			0.519412	−	0.854524i	$-0.326151\pi$
$691$	27.3074			1.03882			0.519412	+	0.854524i	$0.326151\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−32.6840			−1.23977
$696$		0			0
$697$		−	20.3284i		−	0.769994i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8.34822			0.315308			0.157654	−	0.987494i	$-0.449607\pi$
$701$	8.34822			0.315308			0.157654	+	0.987494i	$0.449607\pi$
$702$		0			0
$703$	−4.65790			−0.175676
$704$		0			0
$705$		0			0
$706$		0			0
$707$	28.7042			1.07953
$708$		0			0
$709$			16.0843i			0.604060i	0.953298	+	0.302030i	$0.0976642\pi$
$709$			16.0843i			0.604060i	−0.953298	+	0.302030i	$0.902336\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.82510	−	33.9150i	0.0683506	−	1.27013i
$714$		0			0
$715$	61.0249			2.28220
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	12.1660i		−	0.453717i	−0.973928	−	0.226858i	$-0.927155\pi$
$719$		−	12.1660i		−	0.453717i	0.973928	−	0.226858i	$-0.0728455\pi$
$720$		0			0
$721$	5.44771			0.202883
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	90.2519i		−	3.35187i
$726$		0			0
$727$		−	20.0891i		−	0.745063i	−0.928020	−	0.372531i	$-0.878490\pi$
$727$		−	20.0891i		−	0.745063i	0.928020	−	0.372531i	$-0.121510\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	3.99626i		−	0.147807i
$732$		0			0
$733$		−	12.8386i		−	0.474206i	−0.971484	−	0.237103i	$-0.923802\pi$
$733$		−	12.8386i		−	0.474206i	0.971484	−	0.237103i	$-0.0761979\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	8.31985i		−	0.306466i
$738$		0			0
$739$	16.6435			0.612242			0.306121	−	0.951993i	$-0.400969\pi$
$739$	16.6435			0.612242			0.306121	+	0.951993i	$0.400969\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	10.1175			0.371175			0.185588	−	0.982628i	$-0.440581\pi$
$743$	10.1175			0.371175			0.185588	+	0.982628i	$0.440581\pi$
$744$		0			0
$745$	5.59636			0.205035
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	35.3234i		−	1.29069i
$750$		0			0
$751$		−	35.1750i		−	1.28356i	−0.766891	−	0.641778i	$-0.778197\pi$
$751$		−	35.1750i		−	1.28356i	0.766891	−	0.641778i	$-0.221803\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	26.8390			0.976772
$756$		0			0
$757$		−	44.6556i		−	1.62304i	−0.584327	−	0.811518i	$-0.698642\pi$
$757$		−	44.6556i		−	1.62304i	0.584327	−	0.811518i	$-0.301358\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			2.62940i			0.0953156i	0.998864	+	0.0476578i	$0.0151757\pi$
$761$			2.62940i			0.0953156i	−0.998864	+	0.0476578i	$0.984824\pi$
$762$		0			0
$763$	22.7140			0.822303
$764$		0			0
$765$		0			0
$766$		0			0
$767$			53.3029i			1.92466i
$768$		0			0
$769$			11.9225i			0.429938i	0.976621	+	0.214969i	$0.0689650\pi$
$769$			11.9225i			0.429938i	−0.976621	+	0.214969i	$0.931035\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.28581			0.0462474			0.0231237	−	0.999733i	$-0.492639\pi$
$773$	1.28581			0.0462474			0.0231237	+	0.999733i	$0.492639\pi$
$774$		0			0
$775$	83.7146			3.00712
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−16.7386			−0.599723
$780$		0			0
$781$		−	1.72496i		−	0.0617238i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	55.8663i		−	1.99395i
$786$		0			0
$787$			1.91814i			0.0683745i	0.999415	+	0.0341872i	$0.0108843\pi$
$787$			1.91814i			0.0683745i	−0.999415	+	0.0341872i	$0.989116\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			15.8387i			0.563158i
$792$		0			0
$793$			47.8031i			1.69754i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.7540			0.628879			0.314439	−	0.949278i	$-0.398183\pi$
$797$	17.7540			0.628879			0.314439	+	0.949278i	$0.398183\pi$
$798$		0			0
$799$		−	14.5987i		−	0.516466i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	17.3994			0.614011
$804$		0			0
$805$	−34.1439	−	1.83742i	−1.20341	−	0.0647604i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			46.3445i			1.62939i	0.579892	+	0.814693i	$0.303095\pi$
$809$			46.3445i			1.62939i	−0.579892	+	0.814693i	$0.696905\pi$
$810$		0			0
$811$	21.5823			0.757856			0.378928	−	0.925426i	$-0.376293\pi$
$811$	21.5823			0.757856			0.378928	+	0.925426i	$0.376293\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	15.9467			0.558590
$816$		0			0
$817$	−3.29056			−0.115122
$818$		0			0
$819$		0			0
$820$		0			0
$821$			10.0243i			0.349851i	0.984582	+	0.174925i	$0.0559684\pi$
$821$			10.0243i			0.349851i	−0.984582	+	0.174925i	$0.944032\pi$
$822$		0			0
$823$	−26.5693			−0.926147			−0.463073	−	0.886320i	$-0.653253\pi$
$823$	−26.5693			−0.926147			−0.463073	+	0.886320i	$0.653253\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	17.6801			0.614796			0.307398	−	0.951581i	$-0.400542\pi$
$827$	17.6801			0.614796			0.307398	+	0.951581i	$0.400542\pi$
$828$		0			0
$829$	−29.6388			−1.02940			−0.514700	−	0.857370i	$-0.672097\pi$
$829$	−29.6388			−1.02940			−0.514700	+	0.857370i	$0.672097\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−22.0472			−0.763889
$834$		0			0
$835$		−	5.51297i		−	0.190784i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	27.3095			0.942827			0.471413	−	0.881912i	$-0.343744\pi$
$839$	27.3095			0.942827			0.471413	+	0.881912i	$0.343744\pi$
$840$		0			0
$841$	−29.2940			−1.01014
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−21.6977			−0.746424
$846$		0			0
$847$			1.91998i			0.0659714i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.88778	+	0.263031i	0.167551	+	0.00901657i
$852$		0			0
$853$	4.68013			0.160245			0.0801223	−	0.996785i	$-0.474469\pi$
$853$	4.68013			0.160245			0.0801223	+	0.996785i	$0.474469\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			32.8618i			1.12254i	0.827634	+	0.561269i	$0.189687\pi$
$857$			32.8618i			1.12254i	−0.827634	+	0.561269i	$0.810313\pi$
$858$		0			0
$859$	45.5440			1.55394			0.776970	−	0.629537i	$-0.216756\pi$
$859$	45.5440			1.55394			0.776970	+	0.629537i	$0.216756\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	37.6556i		−	1.28181i	−0.767620	−	0.640906i	$-0.778559\pi$
$863$		−	37.6556i		−	1.28181i	0.767620	−	0.640906i	$-0.221441\pi$
$864$		0			0
$865$		−	69.3246i		−	2.35711i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	52.2411i		−	1.77216i
$870$		0			0
$871$			10.2272i			0.346534i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	48.6305i		−	1.64401i
$876$		0			0
$877$	8.88765			0.300114			0.150057	−	0.988677i	$-0.452054\pi$
$877$	8.88765			0.300114			0.150057	+	0.988677i	$0.452054\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−53.5249			−1.80330			−0.901650	−	0.432466i	$-0.857644\pi$
$881$	−53.5249			−1.80330			−0.901650	+	0.432466i	$0.857644\pi$
$882$		0			0
$883$	14.1704			0.476871			0.238436	−	0.971158i	$-0.423365\pi$
$883$	14.1704			0.476871			0.238436	+	0.971158i	$0.423365\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	22.8230i		−	0.766322i	−0.923681	−	0.383161i	$-0.874835\pi$
$887$		−	22.8230i		−	0.766322i	0.923681	−	0.383161i	$-0.125165\pi$
$888$		0			0
$889$			12.6348i			0.423758i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12.0207			−0.402258
$894$		0			0
$895$			102.412i			3.42327i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	54.0715i		−	1.80338i
$900$		0			0
$901$	6.23043			0.207566
$902$		0			0
$903$		0			0
$904$		0			0
$905$			6.46496i			0.214903i
$906$		0			0
$907$		−	4.55446i		−	0.151228i	−0.997137	−	0.0756142i	$-0.975908\pi$
$907$		−	4.55446i		−	0.151228i	0.997137	−	0.0756142i	$-0.0240917\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−46.4283			−1.53824			−0.769120	−	0.639105i	$-0.779305\pi$
$911$	−46.4283			−1.53824			−0.769120	+	0.639105i	$0.779305\pi$
$912$		0			0
$913$	53.1400			1.75868
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−32.3375			−1.06788
$918$		0			0
$919$			45.7692i			1.50979i	0.655847	+	0.754894i	$0.272312\pi$
$919$			45.7692i			1.50979i	−0.655847	+	0.754894i	$0.727688\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.12040i			0.0697938i
$924$		0			0
$925$			12.0648i			0.396688i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	36.4759i		−	1.19674i	−0.801221	−	0.598369i	$-0.795816\pi$
$929$		−	36.4759i		−	1.19674i	0.801221	−	0.598369i	$-0.204184\pi$
$930$		0			0
$931$			18.1538i			0.594968i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	79.0850			2.58636
$936$		0			0
$937$			42.9529i			1.40321i	0.712566	+	0.701605i	$0.247533\pi$
$937$			42.9529i			1.40321i	−0.712566	+	0.701605i	$0.752467\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	15.5482			0.506858			0.253429	−	0.967354i	$-0.418442\pi$
$941$	15.5482			0.506858			0.253429	+	0.967354i	$0.418442\pi$
$942$		0			0
$943$	17.5647	+	0.945225i	0.571985	+	0.0307808i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			14.6711i			0.476747i	0.971174	+	0.238373i	$0.0766142\pi$
$947$			14.6711i			0.476747i	−0.971174	+	0.238373i	$0.923386\pi$
$948$		0			0
$949$	−21.3882			−0.694289
$950$		0			0
$951$		0			0
$952$		0			0
$953$	22.6641			0.734161			0.367080	−	0.930189i	$-0.380357\pi$
$953$	22.6641			0.734161			0.367080	+	0.930189i	$0.380357\pi$
$954$		0			0
$955$	−60.6677			−1.96316
$956$		0			0
$957$		0			0
$958$		0			0
$959$			16.4296i			0.530538i
$960$		0			0
$961$	19.1548			0.617897
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−104.496			−3.36384
$966$		0			0
$967$	−4.02919			−0.129570			−0.0647849	−	0.997899i	$-0.520636\pi$
$967$	−4.02919			−0.129570			−0.0647849	+	0.997899i	$0.520636\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−41.9697			−1.34687			−0.673435	−	0.739246i	$-0.735182\pi$
$971$	−41.9697			−1.34687			−0.673435	+	0.739246i	$0.735182\pi$
$972$		0			0
$973$			13.8537i			0.444131i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.5034			−0.623970			−0.311985	−	0.950087i	$-0.600994\pi$
$977$	−19.5034			−0.623970			−0.311985	+	0.950087i	$0.600994\pi$
$978$		0			0
$979$	10.2622			0.327981
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.81539			0.121692			0.0608460	−	0.998147i	$-0.480620\pi$
$983$	3.81539			0.121692			0.0608460	+	0.998147i	$0.480620\pi$
$984$		0			0
$985$		−	54.5142i		−	1.73697i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.45295	+	0.185817i	0.109797	+	0.00590863i
$990$		0			0
$991$	−0.404570			−0.0128516			−0.00642580	−	0.999979i	$-0.502045\pi$
$991$	−0.404570			−0.0128516			−0.00642580	+	0.999979i	$0.502045\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			38.6763i			1.22612i
$996$		0			0
$997$	51.9132			1.64411			0.822054	−	0.569409i	$-0.192828\pi$
$997$	51.9132			1.64411			0.822054	+	0.569409i	$0.192828\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1656.2.m.a.1241.1	✓	24
3.2	odd	2	inner	1656.2.m.a.1241.23	yes	24
4.3	odd	2		3312.2.m.d.2897.1		24
12.11	even	2		3312.2.m.d.2897.23		24
23.22	odd	2	inner	1656.2.m.a.1241.24	yes	24
69.68	even	2	inner	1656.2.m.a.1241.2	yes	24
92.91	even	2		3312.2.m.d.2897.24		24
276.275	odd	2		3312.2.m.d.2897.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1656.2.m.a.1241.1	✓	24	1.1	even	1	trivial
1656.2.m.a.1241.2	yes	24	69.68	even	2	inner
1656.2.m.a.1241.23	yes	24	3.2	odd	2	inner
1656.2.m.a.1241.24	yes	24	23.22	odd	2	inner
3312.2.m.d.2897.1		24	4.3	odd	2
3312.2.m.d.2897.2		24	276.275	odd	2
3312.2.m.d.2897.23		24	12.11	even	2
3312.2.m.d.2897.24		24	92.91	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:	\( N \)	\(=\)	\( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1656.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.10131 q^{5} +1.73842i q^{7} +O(q^{10})\)	\(q-4.10131 q^{5} +1.73842i q^{7} +3.47914 q^{11} -4.27673 q^{13} -5.54242 q^{17} +4.56368i q^{19} +(0.257710 - 4.78890i) q^{23} +11.8207 q^{25} -7.63505i q^{29} +7.08201 q^{31} -7.12980i q^{35} +1.02065i q^{37} +3.66779i q^{41} +0.721032i q^{43} +2.63400i q^{47} +3.97790 q^{49} -1.12413 q^{53} -14.2690 q^{55} -12.4635i q^{59} -11.1775i q^{61} +17.5402 q^{65} -2.39135i q^{67} -0.495799i q^{71} +5.00105 q^{73} +6.04821i q^{77} -15.0155i q^{79} +15.2739 q^{83} +22.7312 q^{85} +2.94963 q^{89} -7.43475i q^{91} -18.7171i q^{95} +1.86805i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 40 q^{25} - 16 q^{31} - 40 q^{49} - 64 q^{55} + 48 q^{85}+O(q^{100}) \) 24 * q + 40 * q^25 - 16 * q^31 - 40 * q^49 - 64 * q^55 + 48 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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