LMFDB - Embedded newform 8820.2.f.a.4409.11

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			4409.11
Character	$\chi$	$=$	8820.4409
Dual form			8820.2.f.a.4409.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.61232 + 1.54934i) q^{5} +5.55965i q^{11} -6.30365 q^{13} +4.42934i q^{17} +6.93455i q^{19} +1.11793 q^{23} +(0.199121 - 4.99603i) q^{25} +6.69226i q^{29} +6.04156i q^{31} +0.546142i q^{37} +3.53503 q^{41} +6.28394i q^{43} +6.04039i q^{47} +10.1966 q^{53} +(-8.61376 - 8.96391i) q^{55} +0.597794 q^{59} +5.45331i q^{61} +(10.1635 - 9.76647i) q^{65} -6.93535i q^{67} -5.84125i q^{71} +8.06649 q^{73} -5.75023 q^{79} +2.16066i q^{83} +(-6.86253 - 7.14149i) q^{85} -13.6294 q^{89} +(-10.7440 - 11.1807i) q^{95} +4.73923 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.61232	+	1.54934i	−0.721049	+	0.692884i
$5$	−1.61232	+	1.54934i	−0.721049	+	0.692884i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.55965i			1.67630i	0.545442	+	0.838149i	$0.316362\pi$
$11$			5.55965i			1.67630i	−0.545442	+	0.838149i	$0.683638\pi$
$12$		0			0
$13$	−6.30365			−1.74832			−0.874160	−	0.485639i	$-0.838587\pi$
$13$	−6.30365			−1.74832			−0.874160	+	0.485639i	$0.838587\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.42934i			1.07427i	0.843495	+	0.537136i	$0.180494\pi$
$17$			4.42934i			1.07427i	−0.843495	+	0.537136i	$0.819506\pi$
$18$		0			0
$19$			6.93455i			1.59090i	0.606022	+	0.795448i	$0.292764\pi$
$19$			6.93455i			1.59090i	−0.606022	+	0.795448i	$0.707236\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.11793			0.233105			0.116552	−	0.993185i	$-0.462816\pi$
$23$	1.11793			0.233105			0.116552	+	0.993185i	$0.462816\pi$
$24$		0			0
$25$	0.199121	−	4.99603i	0.0398241	−	0.999207i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.69226i			1.24272i	0.783524	+	0.621361i	$0.213420\pi$
$29$			6.69226i			1.24272i	−0.783524	+	0.621361i	$0.786580\pi$
$30$		0			0
$31$			6.04156i			1.08510i	0.840025	+	0.542548i	$0.182540\pi$
$31$			6.04156i			1.08510i	−0.840025	+	0.542548i	$0.817460\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			0.546142i			0.0897852i	0.998992	+	0.0448926i	$0.0142946\pi$
$37$			0.546142i			0.0897852i	−0.998992	+	0.0448926i	$0.985705\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.53503			0.552079			0.276039	−	0.961146i	$-0.410978\pi$
$41$	3.53503			0.552079			0.276039	+	0.961146i	$0.410978\pi$
$42$		0			0
$43$			6.28394i			0.958291i	0.877735	+	0.479146i	$0.159053\pi$
$43$			6.28394i			0.958291i	−0.877735	+	0.479146i	$0.840947\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.04039i			0.881082i	0.897733	+	0.440541i	$0.145213\pi$
$47$			6.04039i			0.881082i	−0.897733	+	0.440541i	$0.854787\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	10.1966			1.40062			0.700309	−	0.713840i	$-0.253046\pi$
$53$	10.1966			1.40062			0.700309	+	0.713840i	$0.253046\pi$
$54$		0			0
$55$	−8.61376	−	8.96391i	−1.16148	−	1.20869i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.597794			0.0778262			0.0389131	−	0.999243i	$-0.487610\pi$
$59$	0.597794			0.0778262			0.0389131	+	0.999243i	$0.487610\pi$
$60$		0			0
$61$			5.45331i			0.698225i	0.937081	+	0.349112i	$0.113517\pi$
$61$			5.45331i			0.698225i	−0.937081	+	0.349112i	$0.886483\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.1635	−	9.76647i	1.26062	−	1.21138i
$66$		0			0
$67$		−	6.93535i		−	0.847288i	−0.905829	−	0.423644i	$-0.860751\pi$
$67$		−	6.93535i		−	0.847288i	0.905829	−	0.423644i	$-0.139249\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.84125i		−	0.693229i	−0.938008	−	0.346614i	$-0.887331\pi$
$71$		−	5.84125i		−	0.693229i	0.938008	−	0.346614i	$-0.112669\pi$
$72$		0			0
$73$	8.06649			0.944111			0.472056	−	0.881569i	$-0.343512\pi$
$73$	8.06649			0.944111			0.472056	+	0.881569i	$0.343512\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.75023			−0.646951			−0.323476	−	0.946236i	$-0.604851\pi$
$79$	−5.75023			−0.646951			−0.323476	+	0.946236i	$0.604851\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			2.16066i			0.237164i	0.992944	+	0.118582i	$0.0378348\pi$
$83$			2.16066i			0.237164i	−0.992944	+	0.118582i	$0.962165\pi$
$84$		0			0
$85$	−6.86253	−	7.14149i	−0.744346	−	0.774603i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.6294			−1.44471			−0.722355	−	0.691522i	$-0.756940\pi$
$89$	−13.6294			−1.44471			−0.722355	+	0.691522i	$0.756940\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.7440	−	11.1807i	−1.10231	−	1.14711i
$96$		0			0
$97$	4.73923			0.481196			0.240598	−	0.970625i	$-0.422657\pi$
$97$	4.73923			0.481196			0.240598	+	0.970625i	$0.422657\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.04728			−0.303216			−0.151608	−	0.988441i	$-0.548445\pi$
$101$	−3.04728			−0.303216			−0.151608	+	0.988441i	$0.548445\pi$
$102$		0			0
$103$	−6.96191			−0.685977			−0.342988	−	0.939340i	$-0.611439\pi$
$103$	−6.96191			−0.685977			−0.342988	+	0.939340i	$0.611439\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0206			−1.16208			−0.581038	−	0.813877i	$-0.697353\pi$
$107$	−12.0206			−1.16208			−0.581038	+	0.813877i	$0.697353\pi$
$108$		0			0
$109$	0.112302			0.0107566			0.00537831	−	0.999986i	$-0.498288\pi$
$109$	0.112302			0.0107566			0.00537831	+	0.999986i	$0.498288\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	17.1391			1.61231			0.806157	−	0.591701i	$-0.201543\pi$
$113$	17.1391			1.61231			0.806157	+	0.591701i	$0.201543\pi$
$114$		0			0
$115$	−1.80246	+	1.73205i	−0.168080	+	0.161515i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−19.9097			−1.80997
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.41949	+	8.36369i	0.663619	+	0.748071i
$126$		0			0
$127$			20.9470i			1.85874i	0.369147	+	0.929371i	$0.379650\pi$
$127$			20.9470i			1.85874i	−0.369147	+	0.929371i	$0.620350\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.93999			0.868461			0.434231	−	0.900802i	$-0.357020\pi$
$131$	9.93999			0.868461			0.434231	+	0.900802i	$0.357020\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	19.5976			1.67434			0.837168	−	0.546945i	$-0.184209\pi$
$137$	19.5976			1.67434			0.837168	+	0.546945i	$0.184209\pi$
$138$		0			0
$139$			17.0888i			1.44945i	0.689038	+	0.724725i	$0.258033\pi$
$139$			17.0888i			1.44945i	−0.689038	+	0.724725i	$0.741967\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	35.0461i		−	2.93070i
$144$		0			0
$145$	−10.3686	−	10.7900i	−0.861062	−	0.896064i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	8.62362i		−	0.706474i	−0.935534	−	0.353237i	$-0.885081\pi$
$149$		−	8.62362i		−	0.706474i	0.935534	−	0.353237i	$-0.114919\pi$
$150$		0			0
$151$	−9.04717			−0.736248			−0.368124	−	0.929777i	$-0.620000\pi$
$151$	−9.04717			−0.736248			−0.368124	+	0.929777i	$0.620000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.36040	−	9.74090i	−0.751846	−	0.782408i
$156$		0			0
$157$	−13.3846			−1.06821			−0.534103	−	0.845420i	$-0.679350\pi$
$157$	−13.3846			−1.06821			−0.534103	+	0.845420i	$0.679350\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	10.4373i		−	0.817516i	−0.912643	−	0.408758i	$-0.865962\pi$
$163$		−	10.4373i		−	0.817516i	0.912643	−	0.408758i	$-0.134038\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	9.52462i		−	0.737037i	−0.929620	−	0.368519i	$-0.879865\pi$
$167$		−	9.52462i		−	0.737037i	0.929620	−	0.368519i	$-0.120135\pi$
$168$		0			0
$169$	26.7361			2.05662
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	15.8860i		−	1.20779i	−0.797063	−	0.603896i	$-0.793614\pi$
$173$		−	15.8860i		−	1.20779i	0.797063	−	0.603896i	$-0.206386\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	1.27095i		−	0.0949954i	−0.998871	−	0.0474977i	$-0.984875\pi$
$179$		−	1.27095i		−	0.0949954i	0.998871	−	0.0474977i	$-0.0151247\pi$
$180$		0			0
$181$			1.34067i			0.0996511i	0.998758	+	0.0498256i	$0.0158665\pi$
$181$			1.34067i			0.0996511i	−0.998758	+	0.0498256i	$0.984133\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.846157	−	0.880553i	−0.0622107	−	0.0647396i
$186$		0			0
$187$	−24.6256			−1.80080
$188$		0			0
$189$		0			0
$190$		0			0
$191$			2.98725i			0.216150i	0.994143	+	0.108075i	$0.0344686\pi$
$191$			2.98725i			0.216150i	−0.994143	+	0.108075i	$0.965531\pi$
$192$		0			0
$193$			6.38141i			0.459344i	0.973268	+	0.229672i	$0.0737653\pi$
$193$			6.38141i			0.459344i	−0.973268	+	0.229672i	$0.926235\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.10025			−0.292130			−0.146065	−	0.989275i	$-0.546661\pi$
$197$	−4.10025			−0.292130			−0.146065	+	0.989275i	$0.546661\pi$
$198$		0			0
$199$		−	24.6528i		−	1.74759i	−0.486294	−	0.873795i	$-0.661652\pi$
$199$		−	24.6528i		−	1.74759i	0.486294	−	0.873795i	$-0.338348\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−5.69958	+	5.47694i	−0.398076	+	0.382526i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−38.5537			−2.66681
$210$		0			0
$211$	1.18464			0.0815538			0.0407769	−	0.999168i	$-0.487017\pi$
$211$	1.18464			0.0815538			0.0407769	+	0.999168i	$0.487017\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.73593	−	10.1317i	−0.663985	−	0.690975i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	27.9210i		−	1.87817i
$222$		0			0
$223$	5.24874			0.351481			0.175741	−	0.984436i	$-0.443768\pi$
$223$	5.24874			0.351481			0.175741	+	0.984436i	$0.443768\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			24.1319i			1.60169i	0.598869	+	0.800847i	$0.295617\pi$
$227$			24.1319i			1.60169i	−0.598869	+	0.800847i	$0.704383\pi$
$228$		0			0
$229$			27.4961i			1.81699i	0.417891	+	0.908497i	$0.362770\pi$
$229$			27.4961i			1.81699i	−0.417891	+	0.908497i	$0.637230\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.4584			0.816178			0.408089	−	0.912942i	$-0.366195\pi$
$233$	12.4584			0.816178			0.408089	+	0.912942i	$0.366195\pi$
$234$		0			0
$235$	−9.35859	−	9.73902i	−0.610487	−	0.635303i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			21.7203i			1.40497i	0.711698	+	0.702486i	$0.247926\pi$
$239$			21.7203i			1.40497i	−0.711698	+	0.702486i	$0.752074\pi$
$240$		0			0
$241$		−	3.16571i		−	0.203921i	−0.994788	−	0.101961i	$-0.967488\pi$
$241$		−	3.16571i		−	0.203921i	0.994788	−	0.101961i	$-0.0325116\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	43.7130i		−	2.78139i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.34842			−0.148231			−0.0741155	−	0.997250i	$-0.523613\pi$
$251$	−2.34842			−0.148231			−0.0741155	+	0.997250i	$0.523613\pi$
$252$		0			0
$253$			6.21531i			0.390753i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	27.0651i		−	1.68828i	−0.536125	−	0.844139i	$-0.680112\pi$
$257$		−	27.0651i		−	1.68828i	0.536125	−	0.844139i	$-0.319888\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.83398			0.606389			0.303195	−	0.952929i	$-0.401947\pi$
$263$	9.83398			0.606389			0.303195	+	0.952929i	$0.401947\pi$
$264$		0			0
$265$	−16.4402	+	15.7980i	−1.00991	+	0.970465i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18.2090			−1.11022			−0.555110	−	0.831777i	$-0.687324\pi$
$269$	−18.2090			−1.11022			−0.555110	+	0.831777i	$0.687324\pi$
$270$		0			0
$271$			11.2528i			0.683557i	0.939780	+	0.341779i	$0.111029\pi$
$271$			11.2528i			0.683557i	−0.939780	+	0.341779i	$0.888971\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	27.7762	+	1.10704i	1.67497	+	0.0667570i
$276$		0			0
$277$			11.6739i			0.701418i	0.936484	+	0.350709i	$0.114059\pi$
$277$			11.6739i			0.701418i	−0.936484	+	0.350709i	$0.885941\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			4.73421i			0.282419i	0.989980	+	0.141210i	$0.0450991\pi$
$281$			4.73421i			0.282419i	−0.989980	+	0.141210i	$0.954901\pi$
$282$		0			0
$283$	10.3348			0.614341			0.307171	−	0.951654i	$-0.400618\pi$
$283$	10.3348			0.614341			0.307171	+	0.951654i	$0.400618\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.61905			−0.154062
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	0.661681i		−	0.0386558i	−0.999813	−	0.0193279i	$-0.993847\pi$
$293$		−	0.661681i		−	0.0386558i	0.999813	−	0.0193279i	$-0.00615265\pi$
$294$		0			0
$295$	−0.963832	+	0.926183i	−0.0561165	+	0.0539245i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.04705			−0.407542
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.44901	−	8.79245i	−0.483789	−	0.503454i
$306$		0			0
$307$	32.9301			1.87942			0.939710	−	0.341973i	$-0.111095\pi$
$307$	32.9301			1.87942			0.939710	+	0.341973i	$0.111095\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.5824			0.656775			0.328388	−	0.944543i	$-0.393495\pi$
$311$	11.5824			0.656775			0.328388	+	0.944543i	$0.393495\pi$
$312$		0			0
$313$	−7.27315			−0.411103			−0.205551	−	0.978646i	$-0.565899\pi$
$313$	−7.27315			−0.411103			−0.205551	+	0.978646i	$0.565899\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.08879			0.398146			0.199073	−	0.979985i	$-0.436207\pi$
$317$	7.08879			0.398146			0.199073	+	0.979985i	$0.436207\pi$
$318$		0			0
$319$	−37.2066			−2.08317
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−30.7155			−1.70906
$324$		0			0
$325$	−1.25519	+	31.4933i	−0.0696253	+	1.74693i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	13.2246			0.726890			0.363445	−	0.931616i	$-0.381600\pi$
$331$	13.2246			0.726890			0.363445	+	0.931616i	$0.381600\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10.7452	+	11.1820i	0.587072	+	0.610936i
$336$		0			0
$337$		−	23.3122i		−	1.26990i	−0.772554	−	0.634949i	$-0.781021\pi$
$337$		−	23.3122i		−	1.26990i	0.772554	−	0.634949i	$-0.218979\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−33.5890			−1.81894
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.34457			−0.447960			−0.223980	−	0.974594i	$-0.571905\pi$
$347$	−8.34457			−0.447960			−0.223980	+	0.974594i	$0.571905\pi$
$348$		0			0
$349$			3.03784i			0.162612i	0.996689	+	0.0813059i	$0.0259091\pi$
$349$			3.03784i			0.162612i	−0.996689	+	0.0813059i	$0.974091\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			23.4814i			1.24979i	0.780709	+	0.624895i	$0.214858\pi$
$353$			23.4814i			1.24979i	−0.780709	+	0.624895i	$0.785142\pi$
$354$		0			0
$355$	9.05005	+	9.41793i	0.480327	+	0.499852i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			10.6471i			0.561932i	0.959718	+	0.280966i	$0.0906548\pi$
$359$			10.6471i			0.561932i	−0.959718	+	0.280966i	$0.909345\pi$
$360$		0			0
$361$	−29.0881			−1.53095
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−13.0057	+	12.4977i	−0.680751	+	0.654159i
$366$		0			0
$367$	−25.9523			−1.35470			−0.677349	−	0.735662i	$-0.736871\pi$
$367$	−25.9523			−1.35470			−0.677349	+	0.735662i	$0.736871\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	30.8416i		−	1.59692i	−0.602051	−	0.798458i	$-0.705650\pi$
$373$		−	30.8416i		−	1.59692i	0.602051	−	0.798458i	$-0.294350\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	42.1857i		−	2.17268i
$378$		0			0
$379$	10.3992			0.534169			0.267084	−	0.963673i	$-0.413940\pi$
$379$	10.3992			0.534169			0.267084	+	0.963673i	$0.413940\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	11.5758i		−	0.591498i	−0.955266	−	0.295749i	$-0.904431\pi$
$383$		−	11.5758i		−	0.591498i	0.955266	−	0.295749i	$-0.0955692\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			36.1963i			1.83523i	0.397474	+	0.917614i	$0.369887\pi$
$389$			36.1963i			1.83523i	−0.397474	+	0.917614i	$0.630113\pi$
$390$		0			0
$391$			4.95170i			0.250418i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.27118	−	8.90903i	0.466484	−	0.448262i
$396$		0			0
$397$	1.26322			0.0633993			0.0316997	−	0.999497i	$-0.489908\pi$
$397$	1.26322			0.0633993			0.0316997	+	0.999497i	$0.489908\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	39.4858i		−	1.97183i	−0.167252	−	0.985914i	$-0.553489\pi$
$401$		−	39.4858i		−	1.97183i	0.167252	−	0.985914i	$-0.446511\pi$
$402$		0			0
$403$		−	38.0839i		−	1.89709i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.03636			−0.150507
$408$		0			0
$409$			19.9234i			0.985147i	0.870271	+	0.492574i	$0.163944\pi$
$409$			19.9234i			0.985147i	−0.870271	+	0.492574i	$0.836056\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−3.34759	−	3.48367i	−0.164327	−	0.171007i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.60260			−0.224852			−0.112426	−	0.993660i	$-0.535862\pi$
$419$	−4.60260			−0.224852			−0.112426	+	0.993660i	$0.535862\pi$
$420$		0			0
$421$	−10.0110			−0.487906			−0.243953	−	0.969787i	$-0.578444\pi$
$421$	−10.0110			−0.487906			−0.243953	+	0.969787i	$0.578444\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.1291	+	0.881972i	1.07342	+	0.0427819i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	18.0111i		−	0.867562i	−0.901018	−	0.433781i	$-0.857179\pi$
$431$		−	18.0111i		−	0.867562i	0.901018	−	0.433781i	$-0.142821\pi$
$432$		0			0
$433$	−17.5684			−0.844281			−0.422141	−	0.906530i	$-0.638721\pi$
$433$	−17.5684			−0.844281			−0.422141	+	0.906530i	$0.638721\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			7.75236i			0.370846i
$438$		0			0
$439$			21.5021i			1.02624i	0.858316	+	0.513121i	$0.171511\pi$
$439$			21.5021i			1.02624i	−0.858316	+	0.513121i	$0.828489\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	33.1481			1.57491			0.787456	−	0.616371i	$-0.211398\pi$
$443$	33.1481			1.57491			0.787456	+	0.616371i	$0.211398\pi$
$444$		0			0
$445$	21.9748	−	21.1165i	1.04171	−	1.00102i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	18.0471i		−	0.851696i	−0.904795	−	0.425848i	$-0.859976\pi$
$449$		−	18.0471i		−	0.851696i	0.904795	−	0.425848i	$-0.140024\pi$
$450$		0			0
$451$			19.6535i			0.925448i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	19.3051i		−	0.903056i	−0.892257	−	0.451528i	$-0.850879\pi$
$457$		−	19.3051i		−	0.903056i	0.892257	−	0.451528i	$-0.149121\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	22.4305			1.04469			0.522347	−	0.852733i	$-0.325056\pi$
$461$	22.4305			1.04469			0.522347	+	0.852733i	$0.325056\pi$
$462$		0			0
$463$		−	18.4754i		−	0.858625i	−0.903156	−	0.429313i	$-0.858756\pi$
$463$		−	18.4754i		−	0.858625i	0.903156	−	0.429313i	$-0.141244\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			10.9255i			0.505573i	0.967522	+	0.252787i	$0.0813470\pi$
$467$			10.9255i			0.505573i	−0.967522	+	0.252787i	$0.918653\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−34.9365			−1.60638
$474$		0			0
$475$	34.6453	+	1.38081i	1.58963	+	0.0633560i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.36876			−0.245305			−0.122652	−	0.992450i	$-0.539140\pi$
$479$	−5.36876			−0.245305			−0.122652	+	0.992450i	$0.539140\pi$
$480$		0			0
$481$		−	3.44269i		−	0.156973i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.64113	+	7.34265i	−0.346966	+	0.333413i
$486$		0			0
$487$		−	24.6915i		−	1.11888i	−0.828872	−	0.559439i	$-0.811017\pi$
$487$		−	24.6915i		−	1.11888i	0.828872	−	0.559439i	$-0.188983\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			24.1955i			1.09193i	0.837808	+	0.545964i	$0.183837\pi$
$491$			24.1955i			1.09193i	−0.837808	+	0.545964i	$0.816163\pi$
$492$		0			0
$493$	−29.6423			−1.33502
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−32.6238			−1.46044			−0.730220	−	0.683212i	$-0.760582\pi$
$499$	−32.6238			−1.46044			−0.730220	+	0.683212i	$0.760582\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	4.59339i		−	0.204809i	−0.994743	−	0.102405i	$-0.967346\pi$
$503$		−	4.59339i		−	0.204809i	0.994743	−	0.102405i	$-0.0326536\pi$
$504$		0			0
$505$	4.91318	−	4.72126i	0.218634	−	0.210094i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	23.7945			1.05467			0.527336	−	0.849657i	$-0.323191\pi$
$509$	23.7945			1.05467			0.527336	+	0.849657i	$0.323191\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	11.2248	−	10.7863i	0.494623	−	0.475302i
$516$		0			0
$517$	−33.5825			−1.47695
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.5225			0.548622			0.274311	−	0.961641i	$-0.411550\pi$
$521$	12.5225			0.548622			0.274311	+	0.961641i	$0.411550\pi$
$522$		0			0
$523$	−17.7271			−0.775153			−0.387577	−	0.921837i	$-0.626688\pi$
$523$	−17.7271			−0.775153			−0.387577	+	0.921837i	$0.626688\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−26.7601			−1.16569
$528$		0			0
$529$	−21.7502			−0.945662
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−22.2836			−0.965210
$534$		0			0
$535$	19.3810	−	18.6239i	0.837914	−	0.805183i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−30.2617			−1.30105			−0.650526	−	0.759484i	$-0.725451\pi$
$541$	−30.2617			−1.30105			−0.650526	+	0.759484i	$0.725451\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.181067	+	0.173994i	−0.00775606	+	0.00745309i
$546$		0			0
$547$			11.9209i			0.509699i	0.966981	+	0.254850i	$0.0820260\pi$
$547$			11.9209i			0.509699i	−0.966981	+	0.254850i	$0.917974\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−46.4079			−1.97704
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	42.5109			1.80125			0.900623	−	0.434602i	$-0.143111\pi$
$557$	42.5109			1.80125			0.900623	+	0.434602i	$0.143111\pi$
$558$		0			0
$559$		−	39.6118i		−	1.67540i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			13.2275i			0.557473i	0.960368	+	0.278737i	$0.0899157\pi$
$563$			13.2275i			0.557473i	−0.960368	+	0.278737i	$0.910084\pi$
$564$		0			0
$565$	−27.6337	+	26.5543i	−1.16256	+	1.11715i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	18.3543i		−	0.769451i	−0.923031	−	0.384726i	$-0.874296\pi$
$569$		−	18.3543i		−	0.769451i	0.923031	−	0.384726i	$-0.125704\pi$
$570$		0			0
$571$	41.5624			1.73933			0.869667	−	0.493639i	$-0.164334\pi$
$571$	41.5624			1.73933			0.869667	+	0.493639i	$0.164334\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.222603	−	5.58522i	0.00928319	−	0.232920i
$576$		0			0
$577$	8.26475			0.344066			0.172033	−	0.985091i	$-0.444966\pi$
$577$	8.26475			0.344066			0.172033	+	0.985091i	$0.444966\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			56.6898i			2.34785i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	15.6801i		−	0.647188i	−0.946196	−	0.323594i	$-0.895109\pi$
$587$		−	15.6801i		−	0.647188i	0.946196	−	0.323594i	$-0.104891\pi$
$588$		0			0
$589$	−41.8955			−1.72628
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	14.9929i		−	0.615684i	−0.951438	−	0.307842i	$-0.900393\pi$
$593$		−	14.9929i		−	0.615684i	0.951438	−	0.307842i	$-0.0996068\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	6.04742i		−	0.247091i	−0.992339	−	0.123545i	$-0.960574\pi$
$599$		−	6.04742i		−	0.247091i	0.992339	−	0.123545i	$-0.0394265\pi$
$600$		0			0
$601$			7.82990i			0.319388i	0.987167	+	0.159694i	$0.0510508\pi$
$601$			7.82990i			0.319388i	−0.987167	+	0.159694i	$0.948949\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	32.1007	−	30.8468i	1.30508	−	1.25410i
$606$		0			0
$607$	−1.28692			−0.0522345			−0.0261172	−	0.999659i	$-0.508314\pi$
$607$	−1.28692			−0.0522345			−0.0261172	+	0.999659i	$0.508314\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	38.0765i		−	1.54041i
$612$		0			0
$613$		−	20.3207i		−	0.820747i	−0.911918	−	0.410373i	$-0.865398\pi$
$613$		−	20.3207i		−	0.820747i	0.911918	−	0.410373i	$-0.134602\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.8245			0.637071			0.318536	−	0.947911i	$-0.396809\pi$
$617$	15.8245			0.637071			0.318536	+	0.947911i	$0.396809\pi$
$618$		0			0
$619$		−	19.7098i		−	0.792204i	−0.918207	−	0.396102i	$-0.870363\pi$
$619$		−	19.7098i		−	0.792204i	0.918207	−	0.396102i	$-0.129637\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.9207	−	1.98963i	−0.996828	−	0.0795850i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.41905			−0.0964538
$630$		0			0
$631$	42.6898			1.69945			0.849727	−	0.527222i	$-0.176767\pi$
$631$	42.6898			1.69945			0.849727	+	0.527222i	$0.176767\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−32.4539	−	33.7731i	−1.28789	−	1.34024i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	8.74131i		−	0.345261i	−0.984987	−	0.172630i	$-0.944773\pi$
$641$		−	8.74131i		−	0.345261i	0.984987	−	0.172630i	$-0.0552266\pi$
$642$		0			0
$643$	3.69796			0.145833			0.0729167	−	0.997338i	$-0.476769\pi$
$643$	3.69796			0.145833			0.0729167	+	0.997338i	$0.476769\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	30.2730i		−	1.19016i	−0.803668	−	0.595078i	$-0.797121\pi$
$647$		−	30.2730i		−	1.19016i	0.803668	−	0.595078i	$-0.202879\pi$
$648$		0			0
$649$			3.32353i			0.130460i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−20.5385			−0.803735			−0.401867	−	0.915698i	$-0.631639\pi$
$653$	−20.5385			−0.803735			−0.401867	+	0.915698i	$0.631639\pi$
$654$		0			0
$655$	−16.0264	+	15.4004i	−0.626203	+	0.601743i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			19.7847i			0.770704i	0.922770	+	0.385352i	$0.125920\pi$
$659$			19.7847i			0.770704i	−0.922770	+	0.385352i	$0.874080\pi$
$660$		0			0
$661$		−	20.1591i		−	0.784099i	−0.919944	−	0.392049i	$-0.871766\pi$
$661$		−	20.1591i		−	0.784099i	0.919944	−	0.392049i	$-0.128234\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			7.48149i			0.289685i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−30.3185			−1.17043
$672$		0			0
$673$			36.1986i			1.39535i	0.716414	+	0.697676i	$0.245782\pi$
$673$			36.1986i			1.39535i	−0.716414	+	0.697676i	$0.754218\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	14.9480i		−	0.574499i	−0.957856	−	0.287249i	$-0.907259\pi$
$677$		−	14.9480i		−	0.574499i	0.957856	−	0.287249i	$-0.0927408\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	40.9640			1.56744			0.783721	−	0.621113i	$-0.213319\pi$
$683$	40.9640			1.56744			0.783721	+	0.621113i	$0.213319\pi$
$684$		0			0
$685$	−31.5975	+	30.3633i	−1.20728	+	1.16012i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−64.2762			−2.44873
$690$		0			0
$691$		−	4.41121i		−	0.167810i	−0.996474	−	0.0839051i	$-0.973261\pi$
$691$		−	4.41121i		−	0.167810i	0.996474	−	0.0839051i	$-0.0267393\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−26.4762	−	27.5525i	−1.00430	−	1.04512i
$696$		0			0
$697$			15.6578i			0.593083i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	24.7843i		−	0.936090i	−0.883705	−	0.468045i	$-0.844959\pi$
$701$		−	24.7843i		−	0.936090i	0.883705	−	0.468045i	$-0.155041\pi$
$702$		0			0
$703$	−3.78725			−0.142839
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−45.0418			−1.69158			−0.845790	−	0.533516i	$-0.820871\pi$
$709$	−45.0418			−1.69158			−0.845790	+	0.533516i	$0.820871\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			6.75405i			0.252941i
$714$		0			0
$715$	54.2982	+	56.5054i	2.03064	+	2.11318i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	11.1176			0.414617			0.207309	−	0.978276i	$-0.433530\pi$
$719$	11.1176			0.414617			0.207309	+	0.978276i	$0.433530\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	33.4348	+	1.33257i	1.24174	+	0.0494903i
$726$		0			0
$727$	−37.1241			−1.37685			−0.688427	−	0.725305i	$-0.741699\pi$
$727$	−37.1241			−1.37685			−0.688427	+	0.725305i	$0.741699\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−27.8337			−1.02947
$732$		0			0
$733$	−2.22871			−0.0823192			−0.0411596	−	0.999153i	$-0.513105\pi$
$733$	−2.22871			−0.0823192			−0.0411596	+	0.999153i	$0.513105\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	38.5581			1.42031
$738$		0			0
$739$	−11.5908			−0.426373			−0.213186	−	0.977012i	$-0.568384\pi$
$739$	−11.5908			−0.426373			−0.213186	+	0.977012i	$0.568384\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.4284			−0.566015			−0.283007	−	0.959118i	$-0.591332\pi$
$743$	−15.4284			−0.566015			−0.283007	+	0.959118i	$0.591332\pi$
$744$		0			0
$745$	13.3609	+	13.9040i	0.489505	+	0.509403i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	23.5665			0.859954			0.429977	−	0.902840i	$-0.358522\pi$
$751$	23.5665			0.859954			0.429977	+	0.902840i	$0.358522\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14.5869	−	14.0171i	0.530871	−	0.510134i
$756$		0			0
$757$			17.6064i			0.639916i	0.947432	+	0.319958i	$0.103669\pi$
$757$			17.6064i			0.639916i	−0.947432	+	0.319958i	$0.896331\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9.77163			0.354221			0.177111	−	0.984191i	$-0.443325\pi$
$761$	9.77163			0.354221			0.177111	+	0.984191i	$0.443325\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.76829			−0.136065
$768$		0			0
$769$		−	29.3473i		−	1.05829i	−0.848531	−	0.529146i	$-0.822512\pi$
$769$		−	29.3473i		−	1.05829i	0.848531	−	0.529146i	$-0.177488\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			29.5220i			1.06183i	0.847424	+	0.530917i	$0.178152\pi$
$773$			29.5220i			1.06183i	−0.847424	+	0.530917i	$0.821848\pi$
$774$		0			0
$775$	30.1838	+	1.20300i	1.08424	+	0.0432130i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			24.5138i			0.878300i
$780$		0			0
$781$	32.4753			1.16206
$782$		0			0
$783$		0			0
$784$		0			0
$785$	21.5802	−	20.7372i	0.770229	−	0.740142i
$786$		0			0
$787$	33.5825			1.19709			0.598543	−	0.801091i	$-0.295747\pi$
$787$	33.5825			1.19709			0.598543	+	0.801091i	$0.295747\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	34.3758i		−	1.22072i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			19.0563i			0.675010i	0.941324	+	0.337505i	$0.109583\pi$
$797$			19.0563i			0.675010i	−0.941324	+	0.337505i	$0.890417\pi$
$798$		0			0
$799$	−26.7549			−0.946522
$800$		0			0
$801$		0			0
$802$		0			0
$803$			44.8469i			1.58261i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			18.8096i			0.661312i	0.943751	+	0.330656i	$0.107270\pi$
$809$			18.8096i			0.661312i	−0.943751	+	0.330656i	$0.892730\pi$
$810$		0			0
$811$			18.7455i			0.658242i	0.944288	+	0.329121i	$0.106752\pi$
$811$			18.7455i			0.658242i	−0.944288	+	0.329121i	$0.893248\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16.1709	+	16.8283i	0.566443	+	0.589469i
$816$		0			0
$817$	−43.5763			−1.52454
$818$		0			0
$819$		0			0
$820$		0			0
$821$			46.7106i			1.63021i	0.579312	+	0.815106i	$0.303321\pi$
$821$			46.7106i			1.63021i	−0.579312	+	0.815106i	$0.696679\pi$
$822$		0			0
$823$		−	6.21166i		−	0.216525i	−0.994122	−	0.108262i	$-0.965471\pi$
$823$		−	6.21166i		−	0.216525i	0.994122	−	0.108262i	$-0.0345287\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	35.6188			1.23859			0.619294	−	0.785159i	$-0.287419\pi$
$827$	35.6188			1.23859			0.619294	+	0.785159i	$0.287419\pi$
$828$		0			0
$829$		−	34.2536i		−	1.18968i	−0.803845	−	0.594839i	$-0.797216\pi$
$829$		−	34.2536i		−	1.18968i	0.803845	−	0.594839i	$-0.202784\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	14.7568	+	15.3567i	0.510681	+	0.531440i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	26.3915			0.911136			0.455568	−	0.890201i	$-0.349436\pi$
$839$	26.3915			0.911136			0.455568	+	0.890201i	$0.349436\pi$
$840$		0			0
$841$	−15.7864			−0.544359
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−43.1070	+	41.4231i	−1.48292	+	1.42500i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$			0.610550i			0.0209294i
$852$		0			0
$853$	−17.4336			−0.596914			−0.298457	−	0.954423i	$-0.596472\pi$
$853$	−17.4336			−0.596914			−0.298457	+	0.954423i	$0.596472\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	41.5914i		−	1.42074i	−0.703830	−	0.710368i	$-0.748528\pi$
$857$		−	41.5914i		−	1.42074i	0.703830	−	0.710368i	$-0.251472\pi$
$858$		0			0
$859$		−	11.8926i		−	0.405770i	−0.979203	−	0.202885i	$-0.934968\pi$
$859$		−	11.8926i		−	0.405770i	0.979203	−	0.202885i	$-0.0650318\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14.1506			0.481692			0.240846	−	0.970563i	$-0.422575\pi$
$863$	14.1506			0.481692			0.240846	+	0.970563i	$0.422575\pi$
$864$		0			0
$865$	24.6128	+	25.6133i	0.836859	+	0.870877i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	31.9693i		−	1.08448i
$870$		0			0
$871$			43.7181i			1.48133i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	5.38634i		−	0.181884i	−0.995856	−	0.0909420i	$-0.971012\pi$
$877$		−	5.38634i		−	0.181884i	0.995856	−	0.0909420i	$-0.0289878\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−16.9730			−0.571836			−0.285918	−	0.958254i	$-0.592298\pi$
$881$	−16.9730			−0.571836			−0.285918	+	0.958254i	$0.592298\pi$
$882$		0			0
$883$		−	45.7671i		−	1.54019i	−0.637931	−	0.770094i	$-0.720210\pi$
$883$		−	45.7671i		−	1.54019i	0.637931	−	0.770094i	$-0.279790\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	16.0728i		−	0.539671i	−0.962906	−	0.269836i	$-0.913031\pi$
$887$		−	16.0728i		−	0.539671i	0.962906	−	0.269836i	$-0.0869694\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−41.8874			−1.40171
$894$		0			0
$895$	1.96913	+	2.04918i	0.0658208	+	0.0684964i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−40.4317			−1.34847
$900$		0			0
$901$			45.1644i			1.50465i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.07715	−	2.16158i	−0.0690466	−	0.0718534i
$906$		0			0
$907$			19.1598i			0.636192i	0.948059	+	0.318096i	$0.103043\pi$
$907$			19.1598i			0.636192i	−0.948059	+	0.318096i	$0.896957\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			23.0436i			0.763467i	0.924272	+	0.381734i	$0.124673\pi$
$911$			23.0436i			0.763467i	−0.924272	+	0.381734i	$0.875327\pi$
$912$		0			0
$913$	−12.0125			−0.397557
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.138384			0.00456488			0.00228244	−	0.999997i	$-0.499273\pi$
$919$	0.138384			0.00456488			0.00228244	+	0.999997i	$0.499273\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			36.8212i			1.21198i
$924$		0			0
$925$	2.72854	+	0.108748i	0.0897140	+	0.00357562i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20.7577			0.681039			0.340520	−	0.940237i	$-0.389397\pi$
$929$	20.7577			0.681039			0.340520	+	0.940237i	$0.389397\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	39.7042	−	38.1533i	1.29847	−	1.24775i
$936$		0			0
$937$	13.4680			0.439980			0.219990	−	0.975502i	$-0.429398\pi$
$937$	13.4680			0.439980			0.219990	+	0.975502i	$0.429398\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	30.8505			1.00570			0.502849	−	0.864374i	$-0.332285\pi$
$941$	30.8505			1.00570			0.502849	+	0.864374i	$0.332285\pi$
$942$		0			0
$943$	3.95192			0.128692
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.667809			0.0217009			0.0108504	−	0.999941i	$-0.496546\pi$
$947$	0.667809			0.0217009			0.0108504	+	0.999941i	$0.496546\pi$
$948$		0			0
$949$	−50.8484			−1.65061
$950$		0			0
$951$		0			0
$952$		0			0
$953$	60.0400			1.94488			0.972442	−	0.233143i	$-0.0749011\pi$
$953$	60.0400			1.94488			0.972442	+	0.233143i	$0.0749011\pi$
$954$		0			0
$955$	−4.62825	−	4.81638i	−0.149767	−	0.155855i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−5.50046			−0.177434
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.88694	−	10.2888i	−0.318272	−	0.331210i
$966$		0			0
$967$			16.4169i			0.527932i	0.964532	+	0.263966i	$0.0850307\pi$
$967$			16.4169i			0.527932i	−0.964532	+	0.263966i	$0.914969\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	49.1588			1.57758			0.788790	−	0.614663i	$-0.210708\pi$
$971$	49.1588			1.57758			0.788790	+	0.614663i	$0.210708\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.14925			−0.164739			−0.0823696	−	0.996602i	$-0.526249\pi$
$977$	−5.14925			−0.164739			−0.0823696	+	0.996602i	$0.526249\pi$
$978$		0			0
$979$		−	75.7745i		−	2.42176i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	21.5494i		−	0.687318i	−0.939094	−	0.343659i	$-0.888334\pi$
$983$		−	21.5494i		−	0.687318i	0.939094	−	0.343659i	$-0.111666\pi$
$984$		0			0
$985$	6.61089	−	6.35266i	0.210640	−	0.202412i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			7.02501i			0.223382i
$990$		0			0
$991$	7.41712			0.235613			0.117806	−	0.993037i	$-0.462414\pi$
$991$	7.41712			0.235613			0.117806	+	0.993037i	$0.462414\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	38.1954	+	39.7481i	1.21088	+	1.26010i
$996$		0			0
$997$	6.88869			0.218167			0.109083	−	0.994033i	$-0.465208\pi$
$997$	6.88869			0.218167			0.109083	+	0.994033i	$0.465208\pi$
$998$		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.4409.11		32
3.2	odd	2	inner	8820.2.f.a.4409.21		32
5.4	even	2	inner	8820.2.f.a.4409.10		32
7.4	even	3		1260.2.dc.a.89.7	yes	32
7.5	odd	6		1260.2.dc.a.269.1	yes	32
7.6	odd	2	inner	8820.2.f.a.4409.22		32
15.14	odd	2	inner	8820.2.f.a.4409.24		32
21.5	even	6		1260.2.dc.a.269.16	yes	32
21.11	odd	6		1260.2.dc.a.89.10	yes	32
21.20	even	2	inner	8820.2.f.a.4409.12		32
35.4	even	6		1260.2.dc.a.89.16	yes	32
35.12	even	12		6300.2.ch.f.4301.10		32
35.18	odd	12		6300.2.ch.f.1601.7		32
35.19	odd	6		1260.2.dc.a.269.10	yes	32
35.32	odd	12		6300.2.ch.f.1601.9		32
35.33	even	12		6300.2.ch.f.4301.8		32
35.34	odd	2	inner	8820.2.f.a.4409.23		32
105.32	even	12		6300.2.ch.f.1601.10		32
105.47	odd	12		6300.2.ch.f.4301.9		32
105.53	even	12		6300.2.ch.f.1601.8		32
105.68	odd	12		6300.2.ch.f.4301.7		32
105.74	odd	6		1260.2.dc.a.89.1	✓	32
105.89	even	6		1260.2.dc.a.269.7	yes	32
105.104	even	2	inner	8820.2.f.a.4409.9		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.dc.a.89.1	✓	32	105.74	odd	6
1260.2.dc.a.89.7	yes	32	7.4	even	3
1260.2.dc.a.89.10	yes	32	21.11	odd	6
1260.2.dc.a.89.16	yes	32	35.4	even	6
1260.2.dc.a.269.1	yes	32	7.5	odd	6
1260.2.dc.a.269.7	yes	32	105.89	even	6
1260.2.dc.a.269.10	yes	32	35.19	odd	6
1260.2.dc.a.269.16	yes	32	21.5	even	6
6300.2.ch.f.1601.7		32	35.18	odd	12
6300.2.ch.f.1601.8		32	105.53	even	12
6300.2.ch.f.1601.9		32	35.32	odd	12
6300.2.ch.f.1601.10		32	105.32	even	12
6300.2.ch.f.4301.7		32	105.68	odd	12
6300.2.ch.f.4301.8		32	35.33	even	12
6300.2.ch.f.4301.9		32	105.47	odd	12
6300.2.ch.f.4301.10		32	35.12	even	12
8820.2.f.a.4409.9		32	105.104	even	2	inner
8820.2.f.a.4409.10		32	5.4	even	2	inner
8820.2.f.a.4409.11		32	1.1	even	1	trivial
8820.2.f.a.4409.12		32	21.20	even	2	inner
8820.2.f.a.4409.21		32	3.2	odd	2	inner
8820.2.f.a.4409.22		32	7.6	odd	2	inner
8820.2.f.a.4409.23		32	35.34	odd	2	inner
8820.2.f.a.4409.24		32	15.14	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)

\(f(q)\)	\(=\)	\(q+(-1.61232 + 1.54934i) q^{5} +5.55965i q^{11} -6.30365 q^{13} +4.42934i q^{17} +6.93455i q^{19} +1.11793 q^{23} +(0.199121 - 4.99603i) q^{25} +6.69226i q^{29} +6.04156i q^{31} +0.546142i q^{37} +3.53503 q^{41} +6.28394i q^{43} +6.04039i q^{47} +10.1966 q^{53} +(-8.61376 - 8.96391i) q^{55} +0.597794 q^{59} +5.45331i q^{61} +(10.1635 - 9.76647i) q^{65} -6.93535i q^{67} -5.84125i q^{71} +8.06649 q^{73} -5.75023 q^{79} +2.16066i q^{83} +(-6.86253 - 7.14149i) q^{85} -13.6294 q^{89} +(-10.7440 - 11.1807i) q^{95} +4.73923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 24 q^{25} - 64 q^{79} + 32 q^{85}+O(q^{100}) \) 32 * q + 24 * q^25 - 64 * q^79 + 32 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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