LMFDB - Embedded newform 8820.2.f.a.4409.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8820,2,Mod(4409,8820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8820.4409"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8820, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8820.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$70.4280545828$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4409.14
Character	$\chi$	$=$	8820.4409
Dual form			8820.2.f.a.4409.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.210848 - 2.22610i) q^{5} -1.53778i q^{11} -0.219560 q^{13} +1.42614i q^{17} +4.38919i q^{19} +0.778063 q^{23} +(-4.91109 + 0.938738i) q^{25} +6.82176i q^{29} -6.14729i q^{31} -5.99472i q^{37} +8.90761 q^{41} +1.91655i q^{43} +11.2038i q^{47} +4.82775 q^{53} +(-3.42325 + 0.324237i) q^{55} -0.672559 q^{59} +11.1329i q^{61} +(0.0462936 + 0.488763i) q^{65} -11.3025i q^{67} +8.48310i q^{71} +2.18092 q^{73} -6.39462 q^{79} +5.09349i q^{83} +(3.17475 - 0.300699i) q^{85} -5.95639 q^{89} +(9.77080 - 0.925451i) q^{95} +12.7593 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 24 q^{25} - 64 q^{79} + 32 q^{85}+O(q^{100}) $ 32 * q + 24 * q^25 - 64 * q^79 + 32 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8820\mathbb{Z}\right)^\times$.

$n$	$1081$	$4411$	$7057$	$7841$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$5$	−0.210848	−	2.22610i	−0.0942939	−	0.995544i
$5$	−0.210848	−	2.22610i	−0.0942939	−	0.995544i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.53778i		−	0.463657i	−0.972757	−	0.231829i	$-0.925529\pi$
$11$		−	1.53778i		−	0.463657i	0.972757	−	0.231829i	$-0.0744709\pi$
$12$		0			0
$13$	−0.219560			−0.0608949			−0.0304475	−	0.999536i	$-0.509693\pi$
$13$	−0.219560			−0.0608949			−0.0304475	+	0.999536i	$0.509693\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.42614i			0.345891i	0.984931	+	0.172945i	$0.0553284\pi$
$17$			1.42614i			0.345891i	−0.984931	+	0.172945i	$0.944672\pi$
$18$		0			0
$19$			4.38919i			1.00695i	0.864010	+	0.503475i	$0.167945\pi$
$19$			4.38919i			1.00695i	−0.864010	+	0.503475i	$0.832055\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.778063			0.162237			0.0811187	−	0.996704i	$-0.474151\pi$
$23$	0.778063			0.162237			0.0811187	+	0.996704i	$0.474151\pi$
$24$		0			0
$25$	−4.91109	+	0.938738i	−0.982217	+	0.187748i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.82176i			1.26677i	0.773837	+	0.633385i	$0.218335\pi$
$29$			6.82176i			1.26677i	−0.773837	+	0.633385i	$0.781665\pi$
$30$		0			0
$31$		−	6.14729i		−	1.10409i	−0.833815	−	0.552043i	$-0.813848\pi$
$31$		−	6.14729i		−	1.10409i	0.833815	−	0.552043i	$-0.186152\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	5.99472i		−	0.985526i	−0.870164	−	0.492763i	$-0.835987\pi$
$37$		−	5.99472i		−	0.985526i	0.870164	−	0.492763i	$-0.164013\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.90761			1.39113			0.695567	−	0.718461i	$-0.255153\pi$
$41$	8.90761			1.39113			0.695567	+	0.718461i	$0.255153\pi$
$42$		0			0
$43$			1.91655i			0.292271i	0.989265	+	0.146135i	$0.0466835\pi$
$43$			1.91655i			0.292271i	−0.989265	+	0.146135i	$0.953316\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			11.2038i			1.63425i	0.576461	+	0.817125i	$0.304433\pi$
$47$			11.2038i			1.63425i	−0.576461	+	0.817125i	$0.695567\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.82775			0.663143			0.331571	−	0.943430i	$-0.392421\pi$
$53$	4.82775			0.663143			0.331571	+	0.943430i	$0.392421\pi$
$54$		0			0
$55$	−3.42325	+	0.324237i	−0.461592	+	0.0437201i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.672559			−0.0875598			−0.0437799	−	0.999041i	$-0.513940\pi$
$59$	−0.672559			−0.0875598			−0.0437799	+	0.999041i	$0.513940\pi$
$60$		0			0
$61$			11.1329i			1.42542i	0.701461	+	0.712708i	$0.252531\pi$
$61$			11.1329i			1.42542i	−0.701461	+	0.712708i	$0.747469\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.0462936	+	0.488763i	0.00574202	+	0.0606236i
$66$		0			0
$67$		−	11.3025i		−	1.38082i	−0.723419	−	0.690409i	$-0.757431\pi$
$67$		−	11.3025i		−	1.38082i	0.723419	−	0.690409i	$-0.242569\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			8.48310i			1.00676i	0.864065	+	0.503380i	$0.167910\pi$
$71$			8.48310i			1.00676i	−0.864065	+	0.503380i	$0.832090\pi$
$72$		0			0
$73$	2.18092			0.255258			0.127629	−	0.991822i	$-0.459263\pi$
$73$	2.18092			0.255258			0.127629	+	0.991822i	$0.459263\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.39462			−0.719451			−0.359725	−	0.933058i	$-0.617130\pi$
$79$	−6.39462			−0.719451			−0.359725	+	0.933058i	$0.617130\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			5.09349i			0.559083i	0.960134	+	0.279542i	$0.0901825\pi$
$83$			5.09349i			0.559083i	−0.960134	+	0.279542i	$0.909818\pi$
$84$		0			0
$85$	3.17475	−	0.300699i	0.344350	−	0.0326154i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.95639			−0.631376			−0.315688	−	0.948863i	$-0.602235\pi$
$89$	−5.95639			−0.631376			−0.315688	+	0.948863i	$0.602235\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9.77080	−	0.925451i	1.00246	−	0.0949492i
$96$		0			0
$97$	12.7593			1.29551			0.647754	−	0.761850i	$-0.275709\pi$
$97$	12.7593			1.29551			0.647754	+	0.761850i	$0.275709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.12205			0.310655			0.155328	−	0.987863i	$-0.450357\pi$
$101$	3.12205			0.310655			0.155328	+	0.987863i	$0.450357\pi$
$102$		0			0
$103$	12.7568			1.25697			0.628483	−	0.777824i	$-0.283676\pi$
$103$	12.7568			1.25697			0.628483	+	0.777824i	$0.283676\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.4349			1.20213			0.601064	−	0.799201i	$-0.294744\pi$
$107$	12.4349			1.20213			0.601064	+	0.799201i	$0.294744\pi$
$108$		0			0
$109$	−10.5694			−1.01236			−0.506181	−	0.862427i	$-0.668943\pi$
$109$	−10.5694			−1.01236			−0.506181	+	0.862427i	$0.668943\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.57684			−0.900913			−0.450457	−	0.892798i	$-0.648739\pi$
$113$	−9.57684			−0.900913			−0.450457	+	0.892798i	$0.648739\pi$
$114$		0			0
$115$	−0.164053	−	1.73205i	−0.0152980	−	0.161515i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.63524			0.785022
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.12522	+	10.7347i	0.279528	+	0.960137i
$126$		0			0
$127$			10.0099i			0.888231i	0.895970	+	0.444115i	$0.146482\pi$
$127$			10.0099i			0.888231i	−0.895970	+	0.444115i	$0.853518\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	11.1494			0.974129			0.487065	−	0.873366i	$-0.338068\pi$
$131$	11.1494			0.974129			0.487065	+	0.873366i	$0.338068\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.8418			−1.01172			−0.505859	−	0.862616i	$-0.668824\pi$
$137$	−11.8418			−1.01172			−0.505859	+	0.862616i	$0.668824\pi$
$138$		0			0
$139$			11.6201i			0.985601i	0.870142	+	0.492800i	$0.164027\pi$
$139$			11.6201i			0.985601i	−0.870142	+	0.492800i	$0.835973\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			0.337634i			0.0282344i
$144$		0			0
$145$	15.1860	−	1.43835i	1.26113	−	0.119449i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			11.2345i			0.920370i	0.887823	+	0.460185i	$0.152217\pi$
$149$			11.2345i			0.920370i	−0.887823	+	0.460185i	$0.847783\pi$
$150$		0			0
$151$	9.46049			0.769884			0.384942	−	0.922941i	$-0.374221\pi$
$151$	9.46049			0.769884			0.384942	+	0.922941i	$0.374221\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−13.6845	+	1.29614i	−1.09917	+	0.104109i
$156$		0			0
$157$	6.18536			0.493646			0.246823	−	0.969061i	$-0.420613\pi$
$157$	6.18536			0.493646			0.246823	+	0.969061i	$0.420613\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	18.5102i		−	1.44983i	−0.688836	−	0.724917i	$-0.741878\pi$
$163$		−	18.5102i		−	1.44983i	0.688836	−	0.724917i	$-0.258122\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	13.3700i		−	1.03460i	−0.855803	−	0.517302i	$-0.826936\pi$
$167$		−	13.3700i		−	1.03460i	0.855803	−	0.517302i	$-0.173064\pi$
$168$		0			0
$169$	−12.9518			−0.996292
$170$		0			0
$171$		0			0
$172$		0			0
$173$			6.83700i			0.519807i	0.965635	+	0.259904i	$0.0836908\pi$
$173$			6.83700i			0.519807i	−0.965635	+	0.259904i	$0.916309\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			11.3603i			0.849108i	0.905403	+	0.424554i	$0.139569\pi$
$179$			11.3603i			0.849108i	−0.905403	+	0.424554i	$0.860431\pi$
$180$		0			0
$181$		−	2.24531i		−	0.166893i	−0.996512	−	0.0834464i	$-0.973407\pi$
$181$		−	2.24531i		−	0.166893i	0.996512	−	0.0834464i	$-0.0265927\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.3449	+	1.26397i	−0.981135	+	0.0929291i
$186$		0			0
$187$	2.19309			0.160375
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.1189i		−	0.876894i	−0.898757	−	0.438447i	$-0.855529\pi$
$191$		−	12.1189i		−	0.876894i	0.898757	−	0.438447i	$-0.144471\pi$
$192$		0			0
$193$		−	14.6831i		−	1.05691i	−0.848960	−	0.528457i	$-0.822771\pi$
$193$		−	14.6831i		−	1.05691i	0.848960	−	0.528457i	$-0.177229\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.8181			1.05575			0.527874	−	0.849322i	$-0.322989\pi$
$197$	14.8181			1.05575			0.527874	+	0.849322i	$0.322989\pi$
$198$		0			0
$199$			1.10066i			0.0780237i	0.999239	+	0.0390119i	$0.0124210\pi$
$199$			1.10066i			0.0780237i	−0.999239	+	0.0390119i	$0.987579\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.87815	−	19.8293i	−0.131176	−	1.38494i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.74960			0.466880
$210$		0			0
$211$	−7.28575			−0.501572			−0.250786	−	0.968043i	$-0.580689\pi$
$211$	−7.28575			−0.501572			−0.250786	+	0.968043i	$0.580689\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.26644	−	0.404100i	0.290969	−	0.0275594i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	0.313124i		−	0.0210630i
$222$		0			0
$223$	−4.06241			−0.272039			−0.136020	−	0.990706i	$-0.543431\pi$
$223$	−4.06241			−0.272039			−0.136020	+	0.990706i	$0.543431\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	5.56436i		−	0.369320i	−0.982803	−	0.184660i	$-0.940882\pi$
$227$		−	5.56436i		−	0.369320i	0.982803	−	0.184660i	$-0.0591183\pi$
$228$		0			0
$229$		−	12.2375i		−	0.808679i	−0.914609	−	0.404340i	$-0.867501\pi$
$229$		−	12.2375i		−	0.808679i	0.914609	−	0.404340i	$-0.132499\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.0584			0.986508			0.493254	−	0.869885i	$-0.335807\pi$
$233$	15.0584			0.986508			0.493254	+	0.869885i	$0.335807\pi$
$234$		0			0
$235$	24.9409	−	2.36231i	1.62697	−	0.154100i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	26.7865i		−	1.73268i	−0.499457	−	0.866339i	$-0.666467\pi$
$239$		−	26.7865i		−	1.73268i	0.499457	−	0.866339i	$-0.333533\pi$
$240$		0			0
$241$			19.9633i			1.28595i	0.765886	+	0.642976i	$0.222300\pi$
$241$			19.9633i			1.28595i	−0.765886	+	0.642976i	$0.777700\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	0.963689i		−	0.0613181i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−26.2579			−1.65739			−0.828693	−	0.559703i	$-0.810915\pi$
$251$	−26.2579			−1.65739			−0.828693	+	0.559703i	$0.810915\pi$
$252$		0			0
$253$		−	1.19649i		−	0.0752226i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			30.4794i			1.90125i	0.310340	+	0.950626i	$0.399557\pi$
$257$			30.4794i			1.90125i	−0.310340	+	0.950626i	$0.600443\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	24.6586			1.52052			0.760258	−	0.649621i	$-0.225072\pi$
$263$	24.6586			1.52052			0.760258	+	0.649621i	$0.225072\pi$
$264$		0			0
$265$	−1.01792	−	10.7471i	−0.0625303	−	0.660188i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	23.4337			1.42878			0.714389	−	0.699748i	$-0.246705\pi$
$269$	23.4337			1.42878			0.714389	+	0.699748i	$0.246705\pi$
$270$		0			0
$271$		−	24.8087i		−	1.50702i	−0.657434	−	0.753512i	$-0.728358\pi$
$271$		−	24.8087i		−	1.50702i	0.657434	−	0.753512i	$-0.271642\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.44357	+	7.55216i	0.0870506	+	0.455412i
$276$		0			0
$277$		−	27.1596i		−	1.63186i	−0.578151	−	0.815930i	$-0.696225\pi$
$277$		−	27.1596i		−	1.63186i	0.578151	−	0.815930i	$-0.303775\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	16.0353i		−	0.956584i	−0.878201	−	0.478292i	$-0.841256\pi$
$281$		−	16.0353i		−	0.956584i	0.878201	−	0.478292i	$-0.158744\pi$
$282$		0			0
$283$	3.55562			0.211360			0.105680	−	0.994400i	$-0.466298\pi$
$283$	3.55562			0.211360			0.105680	+	0.994400i	$0.466298\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	14.9661			0.880360
$290$		0			0
$291$		0			0
$292$		0			0
$293$			9.76778i			0.570640i	0.958432	+	0.285320i	$0.0920999\pi$
$293$			9.76778i			0.570640i	−0.958432	+	0.285320i	$0.907900\pi$
$294$		0			0
$295$	0.141808	+	1.49719i	0.00825636	+	0.0871696i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.170831			−0.00987943
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	24.7829	−	2.34734i	1.41906	−	0.134408i
$306$		0			0
$307$	12.6916			0.724348			0.362174	−	0.932111i	$-0.382035\pi$
$307$	12.6916			0.724348			0.362174	+	0.932111i	$0.382035\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−27.8208			−1.57758			−0.788788	−	0.614666i	$-0.789291\pi$
$311$	−27.8208			−1.57758			−0.788788	+	0.614666i	$0.789291\pi$
$312$		0			0
$313$	32.3357			1.82772			0.913862	−	0.406024i	$-0.133085\pi$
$313$	32.3357			1.82772			0.913862	+	0.406024i	$0.133085\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	28.6476			1.60901			0.804504	−	0.593947i	$-0.202431\pi$
$317$	28.6476			1.60901			0.804504	+	0.593947i	$0.202431\pi$
$318$		0			0
$319$	10.4904			0.587347
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.25962			−0.348294
$324$		0			0
$325$	1.07828	−	0.206109i	0.0598120	−	0.0114329i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−8.13873			−0.447345			−0.223672	−	0.974664i	$-0.571805\pi$
$331$	−8.13873			−0.447345			−0.223672	+	0.974664i	$0.571805\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−25.1605	+	2.38310i	−1.37467	+	0.130203i
$336$		0			0
$337$			15.0409i			0.819329i	0.912236	+	0.409665i	$0.134354\pi$
$337$			15.0409i			0.819329i	−0.912236	+	0.409665i	$0.865646\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−9.45317			−0.511918
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.95018			−0.319423			−0.159711	−	0.987164i	$-0.551056\pi$
$347$	−5.95018			−0.319423			−0.159711	+	0.987164i	$0.551056\pi$
$348$		0			0
$349$			7.16980i			0.383790i	0.981415	+	0.191895i	$0.0614634\pi$
$349$			7.16980i			0.383790i	−0.981415	+	0.191895i	$0.938537\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			16.6863i			0.888124i	0.895996	+	0.444062i	$0.146463\pi$
$353$			16.6863i			0.888124i	−0.895996	+	0.444062i	$0.853537\pi$
$354$		0			0
$355$	18.8843	−	1.78864i	1.00227	−	0.0949313i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	18.1311i		−	0.956924i	−0.878108	−	0.478462i	$-0.841195\pi$
$359$		−	18.1311i		−	0.956924i	0.878108	−	0.478462i	$-0.158805\pi$
$360$		0			0
$361$	−0.264997			−0.0139472
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.459843	−	4.85497i	−0.0240693	−	0.254121i
$366$		0			0
$367$	3.94893			0.206132			0.103066	−	0.994674i	$-0.467135\pi$
$367$	3.94893			0.206132			0.103066	+	0.994674i	$0.467135\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	24.7320i		−	1.28058i	−0.768135	−	0.640288i	$-0.778815\pi$
$373$		−	24.7320i		−	1.28058i	0.768135	−	0.640288i	$-0.221185\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.49778i		−	0.0771398i
$378$		0			0
$379$	2.75630			0.141581			0.0707907	−	0.997491i	$-0.477448\pi$
$379$	2.75630			0.141581			0.0707907	+	0.997491i	$0.477448\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			4.54106i			0.232037i	0.993247	+	0.116019i	$0.0370132\pi$
$383$			4.54106i			0.232037i	−0.993247	+	0.116019i	$0.962987\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			22.1310i			1.12209i	0.827787	+	0.561043i	$0.189600\pi$
$389$			22.1310i			1.12209i	−0.827787	+	0.561043i	$0.810400\pi$
$390$		0			0
$391$			1.10963i			0.0561164i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.34829	+	14.2351i	0.0678398	+	0.716245i
$396$		0			0
$397$	19.4923			0.978290			0.489145	−	0.872202i	$-0.337309\pi$
$397$	19.4923			0.978290			0.489145	+	0.872202i	$0.337309\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			28.6202i			1.42922i	0.699521	+	0.714612i	$0.253397\pi$
$401$			28.6202i			1.42922i	−0.699521	+	0.714612i	$0.746603\pi$
$402$		0			0
$403$			1.34970i			0.0672333i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.21855			−0.456946
$408$		0			0
$409$			25.2659i			1.24932i	0.780898	+	0.624659i	$0.214762\pi$
$409$			25.2659i			1.24932i	−0.780898	+	0.624659i	$0.785238\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	11.3386	−	1.07395i	0.556592	−	0.0527182i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11.7457			0.573816			0.286908	−	0.957958i	$-0.407373\pi$
$419$	11.7457			0.573816			0.286908	+	0.957958i	$0.407373\pi$
$420$		0			0
$421$	9.60230			0.467988			0.233994	−	0.972238i	$-0.424820\pi$
$421$	9.60230			0.467988			0.233994	+	0.972238i	$0.424820\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.33878	−	7.00392i	−0.0649402	−	0.339740i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	14.1704i		−	0.682562i	−0.939961	−	0.341281i	$-0.889139\pi$
$431$		−	14.1704i		−	0.682562i	0.939961	−	0.341281i	$-0.110861\pi$
$432$		0			0
$433$	26.9828			1.29671			0.648356	−	0.761338i	$-0.275457\pi$
$433$	26.9828			1.29671			0.648356	+	0.761338i	$0.275457\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			3.41507i			0.163365i
$438$		0			0
$439$			5.39738i			0.257603i	0.991670	+	0.128802i	$0.0411130\pi$
$439$			5.39738i			0.257603i	−0.991670	+	0.128802i	$0.958887\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.67991			−0.222349			−0.111175	−	0.993801i	$-0.535461\pi$
$443$	−4.67991			−0.222349			−0.111175	+	0.993801i	$0.535461\pi$
$444$		0			0
$445$	1.25589	+	13.2595i	0.0595349	+	0.628563i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			8.77899i			0.414306i	0.978309	+	0.207153i	$0.0664198\pi$
$449$			8.77899i			0.414306i	−0.978309	+	0.207153i	$0.933580\pi$
$450$		0			0
$451$		−	13.6979i		−	0.645010i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	37.7768i		−	1.76712i	−0.468315	−	0.883562i	$-0.655139\pi$
$457$		−	37.7768i		−	1.76712i	0.468315	−	0.883562i	$-0.344861\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.85808			−0.412562			−0.206281	−	0.978493i	$-0.566136\pi$
$461$	−8.85808			−0.412562			−0.206281	+	0.978493i	$0.566136\pi$
$462$		0			0
$463$			31.8659i			1.48094i	0.672092	+	0.740468i	$0.265396\pi$
$463$			31.8659i			1.48094i	−0.672092	+	0.740468i	$0.734604\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			34.3179i			1.58804i	0.607889	+	0.794022i	$0.292016\pi$
$467$			34.3179i			1.58804i	−0.607889	+	0.794022i	$0.707984\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.94722			0.135514
$474$		0			0
$475$	−4.12030	−	21.5557i	−0.189052	−	0.989043i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	24.8797			1.13678			0.568392	−	0.822758i	$-0.307566\pi$
$479$	24.8797			1.13678			0.568392	+	0.822758i	$0.307566\pi$
$480$		0			0
$481$			1.31620i			0.0600135i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.69026	−	28.4035i	−0.122159	−	1.28974i
$486$		0			0
$487$		−	38.8243i		−	1.75930i	−0.475623	−	0.879649i	$-0.657777\pi$
$487$		−	38.8243i		−	1.75930i	0.475623	−	0.879649i	$-0.342223\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	23.4000i		−	1.05603i	−0.849235	−	0.528014i	$-0.822937\pi$
$491$		−	23.4000i		−	1.05603i	0.849235	−	0.528014i	$-0.177063\pi$
$492$		0			0
$493$	−9.72881			−0.438164
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−3.61757			−0.161945			−0.0809723	−	0.996716i	$-0.525803\pi$
$499$	−3.61757			−0.161945			−0.0809723	+	0.996716i	$0.525803\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			9.87647i			0.440370i	0.975458	+	0.220185i	$0.0706661\pi$
$503$			9.87647i			0.440370i	−0.975458	+	0.220185i	$0.929334\pi$
$504$		0			0
$505$	−0.658277	−	6.95001i	−0.0292929	−	0.309271i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−22.6647			−1.00459			−0.502297	−	0.864695i	$-0.667511\pi$
$509$	−22.6647			−1.00459			−0.502297	+	0.864695i	$0.667511\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.68974	−	28.3980i	−0.118524	−	1.25136i
$516$		0			0
$517$	17.2290			0.757732
$518$		0			0
$519$		0			0
$520$		0			0
$521$	13.6421			0.597671			0.298836	−	0.954305i	$-0.403402\pi$
$521$	13.6421			0.597671			0.298836	+	0.954305i	$0.403402\pi$
$522$		0			0
$523$	10.6844			0.467197			0.233599	−	0.972333i	$-0.424950\pi$
$523$	10.6844			0.467197			0.233599	+	0.972333i	$0.424950\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.76693			0.381893
$528$		0			0
$529$	−22.3946			−0.973679
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.95575			−0.0847130
$534$		0			0
$535$	−2.62187	−	27.6814i	−0.113353	−	1.19677i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−12.5816			−0.540923			−0.270462	−	0.962731i	$-0.587176\pi$
$541$	−12.5816			−0.540923			−0.270462	+	0.962731i	$0.587176\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.22853	+	23.5285i	0.0954595	+	1.00785i
$546$		0			0
$547$			36.2103i			1.54824i	0.633040	+	0.774119i	$0.281807\pi$
$547$			36.2103i			1.54824i	−0.633040	+	0.774119i	$0.718193\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−29.9420			−1.27557
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.9465			0.802789			0.401394	−	0.915905i	$-0.368526\pi$
$557$	18.9465			0.802789			0.401394	+	0.915905i	$0.368526\pi$
$558$		0			0
$559$		−	0.420797i		−	0.0177978i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	30.0736i		−	1.26745i	−0.773558	−	0.633725i	$-0.781525\pi$
$563$		−	30.0736i		−	1.26745i	0.773558	−	0.633725i	$-0.218475\pi$
$564$		0			0
$565$	2.01925	+	21.3190i	0.0849507	+	0.896899i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			14.9169i			0.625351i	0.949860	+	0.312675i	$0.101225\pi$
$569$			14.9169i			0.625351i	−0.949860	+	0.312675i	$0.898775\pi$
$570$		0			0
$571$	−9.26835			−0.387868			−0.193934	−	0.981015i	$-0.562125\pi$
$571$	−9.26835			−0.387868			−0.193934	+	0.981015i	$0.562125\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.82114	+	0.730398i	−0.159352	+	0.0304597i
$576$		0			0
$577$	4.93818			0.205579			0.102790	−	0.994703i	$-0.467223\pi$
$577$	4.93818			0.205579			0.102790	+	0.994703i	$0.467223\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	7.42401i		−	0.307471i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	24.6187i		−	1.01612i	−0.861321	−	0.508061i	$-0.830362\pi$
$587$		−	24.6187i		−	1.01612i	0.861321	−	0.508061i	$-0.169638\pi$
$588$		0			0
$589$	26.9817			1.11176
$590$		0			0
$591$		0			0
$592$		0			0
$593$			12.3160i			0.505757i	0.967498	+	0.252879i	$0.0813774\pi$
$593$			12.3160i			0.505757i	−0.967498	+	0.252879i	$0.918623\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			14.2041i			0.580362i	0.956972	+	0.290181i	$0.0937155\pi$
$599$			14.2041i			0.580362i	−0.956972	+	0.290181i	$0.906284\pi$
$600$		0			0
$601$		−	21.1744i		−	0.863722i	−0.901940	−	0.431861i	$-0.857857\pi$
$601$		−	21.1744i		−	0.863722i	0.901940	−	0.431861i	$-0.142143\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.82072	−	19.2230i	−0.0740228	−	0.781524i
$606$		0			0
$607$	11.7023			0.474984			0.237492	−	0.971390i	$-0.423675\pi$
$607$	11.7023			0.474984			0.237492	+	0.971390i	$0.423675\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	2.45991i		−	0.0995175i
$612$		0			0
$613$		−	11.0349i		−	0.445695i	−0.974853	−	0.222847i	$-0.928465\pi$
$613$		−	11.0349i		−	0.445695i	0.974853	−	0.222847i	$-0.0715352\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.14075			0.0861833			0.0430917	−	0.999071i	$-0.486279\pi$
$617$	2.14075			0.0861833			0.0430917	+	0.999071i	$0.486279\pi$
$618$		0			0
$619$			26.9930i			1.08494i	0.840075	+	0.542470i	$0.182511\pi$
$619$			26.9930i			1.08494i	−0.840075	+	0.542470i	$0.817489\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	23.2375	−	9.22045i	0.929502	−	0.368818i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.54933			0.340884
$630$		0			0
$631$	24.0872			0.958895			0.479448	−	0.877571i	$-0.340837\pi$
$631$	24.0872			0.958895			0.479448	+	0.877571i	$0.340837\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	22.2830	−	2.11055i	0.884273	−	0.0837548i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	0.732576i		−	0.0289350i	−0.999895	−	0.0144675i	$-0.995395\pi$
$641$		−	0.732576i		−	0.0289350i	0.999895	−	0.0144675i	$-0.00460531\pi$
$642$		0			0
$643$	−35.1704			−1.38699			−0.693493	−	0.720463i	$-0.743929\pi$
$643$	−35.1704			−1.38699			−0.693493	+	0.720463i	$0.743929\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			28.5353i			1.12184i	0.827871	+	0.560919i	$0.189552\pi$
$647$			28.5353i			1.12184i	−0.827871	+	0.560919i	$0.810448\pi$
$648$		0			0
$649$			1.03425i			0.0405977i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	48.9555			1.91578			0.957889	−	0.287138i	$-0.0927037\pi$
$653$	48.9555			1.91578			0.957889	+	0.287138i	$0.0927037\pi$
$654$		0			0
$655$	−2.35083	−	24.8198i	−0.0918545	−	0.969789i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			51.1673i			1.99320i	0.0824134	+	0.996598i	$0.473737\pi$
$659$			51.1673i			1.99320i	−0.0824134	+	0.996598i	$0.526263\pi$
$660$		0			0
$661$		−	43.7428i		−	1.70140i	−0.525654	−	0.850699i	$-0.676179\pi$
$661$		−	43.7428i		−	1.70140i	0.525654	−	0.850699i	$-0.323821\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			5.30776i			0.205517i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	17.1199			0.660905
$672$		0			0
$673$		−	46.7354i		−	1.80152i	−0.434322	−	0.900758i	$-0.643012\pi$
$673$		−	46.7354i		−	1.80152i	0.434322	−	0.900758i	$-0.356988\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	2.70870i		−	0.104104i	−0.998644	−	0.0520520i	$-0.983424\pi$
$677$		−	2.70870i		−	0.104104i	0.998644	−	0.0520520i	$-0.0165761\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	28.4364			1.08809			0.544044	−	0.839056i	$-0.316892\pi$
$683$	28.4364			1.08809			0.544044	+	0.839056i	$0.316892\pi$
$684$		0			0
$685$	2.49683	+	26.3612i	0.0953988	+	1.00721i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.05998			−0.0403820
$690$		0			0
$691$			45.1839i			1.71888i	0.511239	+	0.859439i	$0.329187\pi$
$691$			45.1839i			1.71888i	−0.511239	+	0.859439i	$0.670813\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	25.8675	−	2.45006i	0.981209	−	0.0929362i
$696$		0			0
$697$			12.7035i			0.481180i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			36.4833i			1.37796i	0.724782	+	0.688978i	$0.241940\pi$
$701$			36.4833i			1.37796i	−0.724782	+	0.688978i	$0.758060\pi$
$702$		0			0
$703$	26.3120			0.992375
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−37.3040			−1.40098			−0.700490	−	0.713662i	$-0.747035\pi$
$709$	−37.3040			−1.40098			−0.700490	+	0.713662i	$0.747035\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	4.78299i		−	0.179124i
$714$		0			0
$715$	0.751609	−	0.0711893i	0.0281086	−	0.00266233i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−22.2061			−0.828149			−0.414074	−	0.910243i	$-0.635895\pi$
$719$	−22.2061			−0.828149			−0.414074	+	0.910243i	$0.635895\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.40385	−	33.5023i	−0.237833	−	1.24424i
$726$		0			0
$727$	−16.0910			−0.596783			−0.298392	−	0.954443i	$-0.596450\pi$
$727$	−16.0910			−0.596783			−0.298392	+	0.954443i	$0.596450\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.73327			−0.101094
$732$		0			0
$733$	−3.17201			−0.117161			−0.0585803	−	0.998283i	$-0.518657\pi$
$733$	−3.17201			−0.117161			−0.0585803	+	0.998283i	$0.518657\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−17.3807			−0.640226
$738$		0			0
$739$	−41.4245			−1.52382			−0.761912	−	0.647681i	$-0.775739\pi$
$739$	−41.4245			−1.52382			−0.761912	+	0.647681i	$0.775739\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14.7140			−0.539805			−0.269903	−	0.962888i	$-0.586992\pi$
$743$	−14.7140			−0.539805			−0.269903	+	0.962888i	$0.586992\pi$
$744$		0			0
$745$	25.0093	−	2.36878i	0.916269	−	0.0867853i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	35.2588			1.28661			0.643307	−	0.765609i	$-0.277562\pi$
$751$	35.2588			1.28661			0.643307	+	0.765609i	$0.277562\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.99472	−	21.0601i	−0.0725954	−	0.766454i
$756$		0			0
$757$		−	32.1067i		−	1.16694i	−0.812135	−	0.583470i	$-0.801695\pi$
$757$		−	32.1067i		−	1.16694i	0.812135	−	0.583470i	$-0.198305\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	23.6109			0.855894			0.427947	−	0.903804i	$-0.359237\pi$
$761$	23.6109			0.855894			0.427947	+	0.903804i	$0.359237\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.147667			0.00533194
$768$		0			0
$769$			16.3200i			0.588516i	0.955726	+	0.294258i	$0.0950724\pi$
$769$			16.3200i			0.588516i	−0.955726	+	0.294258i	$0.904928\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			44.4985i			1.60050i	0.599668	+	0.800249i	$0.295299\pi$
$773$			44.4985i			1.60050i	−0.599668	+	0.800249i	$0.704701\pi$
$774$		0			0
$775$	5.77070	+	30.1899i	0.207290	+	1.08445i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			39.0972i			1.40080i
$780$		0			0
$781$	13.0451			0.466791
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.30417	−	13.7693i	−0.0465478	−	0.491446i
$786$		0			0
$787$	−17.2290			−0.614149			−0.307074	−	0.951686i	$-0.599350\pi$
$787$	−17.2290			−0.614149			−0.307074	+	0.951686i	$0.599350\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	2.44433i		−	0.0868006i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			35.5457i			1.25909i	0.776962	+	0.629547i	$0.216760\pi$
$797$			35.5457i			1.25909i	−0.776962	+	0.629547i	$0.783240\pi$
$798$		0			0
$799$	−15.9783			−0.565272
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	3.35378i		−	0.118352i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			24.2140i			0.851317i	0.904884	+	0.425659i	$0.139958\pi$
$809$			24.2140i			0.851317i	−0.904884	+	0.425659i	$0.860042\pi$
$810$		0			0
$811$		−	10.7691i		−	0.378155i	−0.981962	−	0.189077i	$-0.939450\pi$
$811$		−	10.7691i		−	0.378155i	0.981962	−	0.189077i	$-0.0605497\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−41.2057	+	3.90284i	−1.44337	+	0.136711i
$816$		0			0
$817$	−8.41209			−0.294302
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	32.5314i		−	1.13535i	−0.823252	−	0.567676i	$-0.807843\pi$
$821$		−	32.5314i		−	1.13535i	0.823252	−	0.567676i	$-0.192157\pi$
$822$		0			0
$823$		−	28.9272i		−	1.00834i	−0.863605	−	0.504169i	$-0.831799\pi$
$823$		−	28.9272i		−	1.00834i	0.863605	−	0.504169i	$-0.168201\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−22.1968			−0.771857			−0.385928	−	0.922529i	$-0.626119\pi$
$827$	−22.1968			−0.771857			−0.385928	+	0.922529i	$0.626119\pi$
$828$		0			0
$829$		−	12.2632i		−	0.425919i	−0.977061	−	0.212959i	$-0.931690\pi$
$829$		−	12.2632i		−	0.425919i	0.977061	−	0.212959i	$-0.0683102\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−29.7631	+	2.81904i	−1.03000	+	0.0975570i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	53.7542			1.85580			0.927901	−	0.372826i	$-0.121611\pi$
$839$	53.7542			1.85580			0.927901	+	0.372826i	$0.121611\pi$
$840$		0			0
$841$	−17.5364			−0.604704
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.73086	+	28.8321i	0.0939443	+	0.991853i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	4.66427i		−	0.159889i
$852$		0			0
$853$	50.9882			1.74580			0.872902	−	0.487896i	$-0.162236\pi$
$853$	50.9882			1.74580			0.872902	+	0.487896i	$0.162236\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	37.8150i		−	1.29173i	−0.763450	−	0.645867i	$-0.776496\pi$
$857$		−	37.8150i		−	1.29173i	0.763450	−	0.645867i	$-0.223504\pi$
$858$		0			0
$859$		−	16.8162i		−	0.573762i	−0.957966	−	0.286881i	$-0.907382\pi$
$859$		−	16.8162i		−	0.573762i	0.957966	−	0.286881i	$-0.0926184\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−53.0426			−1.80559			−0.902795	−	0.430071i	$-0.858488\pi$
$863$	−53.0426			−1.80559			−0.902795	+	0.430071i	$0.858488\pi$
$864$		0			0
$865$	15.2199	−	1.44156i	0.517491	−	0.0490147i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			9.83350i			0.333579i
$870$		0			0
$871$			2.48157i			0.0840848i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	1.04755i		−	0.0353733i	−0.999844	−	0.0176867i	$-0.994370\pi$
$877$		−	1.04755i		−	0.0353733i	0.999844	−	0.0176867i	$-0.00563014\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.68108			−0.326164			−0.163082	−	0.986613i	$-0.552143\pi$
$881$	−9.68108			−0.326164			−0.163082	+	0.986613i	$0.552143\pi$
$882$		0			0
$883$			48.8068i			1.64248i	0.570583	+	0.821240i	$0.306717\pi$
$883$			48.8068i			1.64248i	−0.570583	+	0.821240i	$0.693283\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	40.0428i		−	1.34450i	−0.740322	−	0.672252i	$-0.765327\pi$
$887$		−	40.0428i		−	1.34450i	0.740322	−	0.672252i	$-0.234673\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−49.1758			−1.64561
$894$		0			0
$895$	25.2892	−	2.39529i	0.845325	−	0.0800657i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	41.9354			1.39862
$900$		0			0
$901$			6.88507i			0.229375i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.99830	+	0.473419i	−0.166149	+	0.0157370i
$906$		0			0
$907$			19.9113i			0.661143i	0.943781	+	0.330572i	$0.107242\pi$
$907$			19.9113i			0.661143i	−0.943781	+	0.330572i	$0.892758\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			3.57388i			0.118408i	0.998246	+	0.0592040i	$0.0188562\pi$
$911$			3.57388i			0.118408i	−0.998246	+	0.0592040i	$0.981144\pi$
$912$		0			0
$913$	7.83265			0.259223
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	12.7532			0.420690			0.210345	−	0.977627i	$-0.432541\pi$
$919$	12.7532			0.420690			0.210345	+	0.977627i	$0.432541\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	1.86255i		−	0.0613065i
$924$		0			0
$925$	5.62747	+	29.4406i	0.185030	+	0.968001i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	24.0124			0.787820			0.393910	−	0.919149i	$-0.371122\pi$
$929$	24.0124			0.787820			0.393910	+	0.919149i	$0.371122\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.462408	−	4.88205i	−0.0151224	−	0.159660i
$936$		0			0
$937$	−14.2712			−0.466219			−0.233110	−	0.972450i	$-0.574890\pi$
$937$	−14.2712			−0.466219			−0.233110	+	0.972450i	$0.574890\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.48862			−0.211523			−0.105761	−	0.994392i	$-0.533728\pi$
$941$	−6.48862			−0.211523			−0.105761	+	0.994392i	$0.533728\pi$
$942$		0			0
$943$	6.93068			0.225694
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11.4634			−0.372511			−0.186255	−	0.982501i	$-0.559635\pi$
$947$	−11.4634			−0.372511			−0.186255	+	0.982501i	$0.559635\pi$
$948$		0			0
$949$	−0.478843			−0.0155439
$950$		0			0
$951$		0			0
$952$		0			0
$953$	37.3915			1.21123			0.605614	−	0.795758i	$-0.292928\pi$
$953$	37.3915			1.21123			0.605614	+	0.795758i	$0.292928\pi$
$954$		0			0
$955$	−26.9780	+	2.55524i	−0.872986	+	0.0826857i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−6.78923			−0.219008
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−32.6862	+	3.09590i	−1.05220	+	0.0996606i
$966$		0			0
$967$		−	23.8585i		−	0.767238i	−0.923491	−	0.383619i	$-0.874678\pi$
$967$		−	23.8585i		−	0.767238i	0.923491	−	0.383619i	$-0.125322\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.9732			0.640971			0.320486	−	0.947253i	$-0.396154\pi$
$971$	19.9732			0.640971			0.320486	+	0.947253i	$0.396154\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.30234			0.0736584			0.0368292	−	0.999322i	$-0.488274\pi$
$977$	2.30234			0.0736584			0.0368292	+	0.999322i	$0.488274\pi$
$978$		0			0
$979$			9.15960i			0.292742i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	38.3195i		−	1.22220i	−0.791552	−	0.611101i	$-0.790727\pi$
$983$		−	38.3195i		−	1.22220i	0.791552	−	0.611101i	$-0.209273\pi$
$984$		0			0
$985$	−3.12437	−	32.9867i	−0.0995507	−	1.05104i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1.49120i			0.0474173i
$990$		0			0
$991$	26.1079			0.829345			0.414673	−	0.909971i	$-0.363896\pi$
$991$	26.1079			0.829345			0.414673	+	0.909971i	$0.363896\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.45018	−	0.232072i	0.0776761	−	0.00735716i
$996$		0			0
$997$	19.5259			0.618392			0.309196	−	0.950998i	$-0.399940\pi$
$997$	19.5259			0.618392			0.309196	+	0.950998i	$0.399940\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.f.a.4409.14		32
3.2	odd	2	inner	8820.2.f.a.4409.19		32
5.4	even	2	inner	8820.2.f.a.4409.15		32
7.2	even	3		1260.2.dc.a.269.4	yes	32
7.3	odd	6		1260.2.dc.a.89.2	✓	32
7.6	odd	2	inner	8820.2.f.a.4409.20		32
15.14	odd	2	inner	8820.2.f.a.4409.18		32
21.2	odd	6		1260.2.dc.a.269.13	yes	32
21.17	even	6		1260.2.dc.a.89.15	yes	32
21.20	even	2	inner	8820.2.f.a.4409.13		32
35.2	odd	12		6300.2.ch.f.4301.11		32
35.3	even	12		6300.2.ch.f.1601.6		32
35.9	even	6		1260.2.dc.a.269.15	yes	32
35.17	even	12		6300.2.ch.f.1601.12		32
35.23	odd	12		6300.2.ch.f.4301.5		32
35.24	odd	6		1260.2.dc.a.89.13	yes	32
35.34	odd	2	inner	8820.2.f.a.4409.17		32
105.2	even	12		6300.2.ch.f.4301.12		32
105.17	odd	12		6300.2.ch.f.1601.11		32
105.23	even	12		6300.2.ch.f.4301.6		32
105.38	odd	12		6300.2.ch.f.1601.5		32
105.44	odd	6		1260.2.dc.a.269.2	yes	32
105.59	even	6		1260.2.dc.a.89.4	yes	32
105.104	even	2	inner	8820.2.f.a.4409.16		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.dc.a.89.2	✓	32	7.3	odd	6
1260.2.dc.a.89.4	yes	32	105.59	even	6
1260.2.dc.a.89.13	yes	32	35.24	odd	6
1260.2.dc.a.89.15	yes	32	21.17	even	6
1260.2.dc.a.269.2	yes	32	105.44	odd	6
1260.2.dc.a.269.4	yes	32	7.2	even	3
1260.2.dc.a.269.13	yes	32	21.2	odd	6
1260.2.dc.a.269.15	yes	32	35.9	even	6
6300.2.ch.f.1601.5		32	105.38	odd	12
6300.2.ch.f.1601.6		32	35.3	even	12
6300.2.ch.f.1601.11		32	105.17	odd	12
6300.2.ch.f.1601.12		32	35.17	even	12
6300.2.ch.f.4301.5		32	35.23	odd	12
6300.2.ch.f.4301.6		32	105.23	even	12
6300.2.ch.f.4301.11		32	35.2	odd	12
6300.2.ch.f.4301.12		32	105.2	even	12
8820.2.f.a.4409.13		32	21.20	even	2	inner
8820.2.f.a.4409.14		32	1.1	even	1	trivial
8820.2.f.a.4409.15		32	5.4	even	2	inner
8820.2.f.a.4409.16		32	105.104	even	2	inner
8820.2.f.a.4409.17		32	35.34	odd	2	inner
8820.2.f.a.4409.18		32	15.14	odd	2	inner
8820.2.f.a.4409.19		32	3.2	odd	2	inner
8820.2.f.a.4409.20		32	7.6	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.210848 - 2.22610i) q^{5} -1.53778i q^{11} -0.219560 q^{13} +1.42614i q^{17} +4.38919i q^{19} +0.778063 q^{23} +(-4.91109 + 0.938738i) q^{25} +6.82176i q^{29} -6.14729i q^{31} -5.99472i q^{37} +8.90761 q^{41} +1.91655i q^{43} +11.2038i q^{47} +4.82775 q^{53} +(-3.42325 + 0.324237i) q^{55} -0.672559 q^{59} +11.1329i q^{61} +(0.0462936 + 0.488763i) q^{65} -11.3025i q^{67} +8.48310i q^{71} +2.18092 q^{73} -6.39462 q^{79} +5.09349i q^{83} +(3.17475 - 0.300699i) q^{85} -5.95639 q^{89} +(9.77080 - 0.925451i) q^{95} +12.7593 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 24 q^{25} - 64 q^{79} + 32 q^{85}+O(q^{100}) \) 32 * q + 24 * q^25 - 64 * q^79 + 32 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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