LMFDB - Embedded newform 8820.2.f.a.4409.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8820,2,Mod(4409,8820)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			4409.1
Character	$\chi$	$=$	8820.4409
Dual form			8820.2.f.a.4409.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.19078 - 0.447731i) q^{5} +O(q^{10})$	$q+(-2.19078 - 0.447731i) q^{5} -0.694412i q^{11} +3.76549 q^{13} +0.0400982i q^{17} +0.124025i q^{19} -3.86851 q^{23} +(4.59907 + 1.96176i) q^{25} +4.39545i q^{29} -3.01152i q^{31} -11.1279i q^{37} -7.91118 q^{41} -3.73746i q^{43} +6.59624i q^{47} +8.96039 q^{53} +(-0.310909 + 1.52131i) q^{55} +5.80370 q^{59} -3.06093i q^{61} +(-8.24938 - 1.68592i) q^{65} +5.87597i q^{67} +7.19849i q^{71} -8.53178 q^{73} +7.96538 q^{79} -6.32489i q^{83} +(0.0179532 - 0.0878466i) q^{85} -10.0099 q^{89} +(0.0555296 - 0.271711i) q^{95} +11.5960 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * 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

Display 100 coefficients 10 coefficients

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
$5$	−2.19078	−	0.447731i	−0.979749	−	0.200231i
$5$	−2.19078	−	0.447731i	−0.979749	−	0.200231i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.694412i		−	0.209373i	−0.994505	−	0.104687i	$-0.966616\pi$
$11$		−	0.694412i		−	0.209373i	0.994505	−	0.104687i	$-0.0333839\pi$
$12$		0			0
$13$	3.76549			1.04436			0.522179	−	0.852836i	$-0.325119\pi$
$13$	3.76549			1.04436			0.522179	+	0.852836i	$0.325119\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.0400982i			0.00972525i	0.999988	+	0.00486263i	$0.00154783\pi$
$17$			0.0400982i			0.00972525i	−0.999988	+	0.00486263i	$0.998452\pi$
$18$		0			0
$19$			0.124025i			0.0284532i	0.999899	+	0.0142266i	$0.00452862\pi$
$19$			0.124025i			0.0284532i	−0.999899	+	0.0142266i	$0.995471\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.86851			−0.806640			−0.403320	−	0.915059i	$-0.632144\pi$
$23$	−3.86851			−0.806640			−0.403320	+	0.915059i	$0.632144\pi$
$24$		0			0
$25$	4.59907	+	1.96176i	0.919815	+	0.392353i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.39545i			0.816214i	0.912934	+	0.408107i	$0.133811\pi$
$29$			4.39545i			0.816214i	−0.912934	+	0.408107i	$0.866189\pi$
$30$		0			0
$31$		−	3.01152i		−	0.540885i	−0.962736	−	0.270442i	$-0.912830\pi$
$31$		−	3.01152i		−	0.540885i	0.962736	−	0.270442i	$-0.0871699\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	11.1279i		−	1.82941i	−0.404122	−	0.914705i	$-0.632423\pi$
$37$		−	11.1279i		−	1.82941i	0.404122	−	0.914705i	$-0.367577\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.91118			−1.23552			−0.617759	−	0.786367i	$-0.711959\pi$
$41$	−7.91118			−1.23552			−0.617759	+	0.786367i	$0.711959\pi$
$42$		0			0
$43$		−	3.73746i		−	0.569957i	−0.958534	−	0.284979i	$-0.908014\pi$
$43$		−	3.73746i		−	0.569957i	0.958534	−	0.284979i	$-0.0919865\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.59624i			0.962160i	0.876677	+	0.481080i	$0.159755\pi$
$47$			6.59624i			0.962160i	−0.876677	+	0.481080i	$0.840245\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.96039			1.23080			0.615402	−	0.788213i	$-0.288994\pi$
$53$	8.96039			1.23080			0.615402	+	0.788213i	$0.288994\pi$
$54$		0			0
$55$	−0.310909	+	1.52131i	−0.0419230	+	0.205133i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.80370			0.755577			0.377789	−	0.925892i	$-0.376685\pi$
$59$	5.80370			0.755577			0.377789	+	0.925892i	$0.376685\pi$
$60$		0			0
$61$		−	3.06093i		−	0.391912i	−0.980613	−	0.195956i	$-0.937219\pi$
$61$		−	3.06093i		−	0.391912i	0.980613	−	0.195956i	$-0.0627810\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.24938	−	1.68592i	−1.02321	−	0.209113i
$66$		0			0
$67$			5.87597i			0.717864i	0.933364	+	0.358932i	$0.116859\pi$
$67$			5.87597i			0.717864i	−0.933364	+	0.358932i	$0.883141\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.19849i			0.854303i	0.904180	+	0.427152i	$0.140483\pi$
$71$			7.19849i			0.854303i	−0.904180	+	0.427152i	$0.859517\pi$
$72$		0			0
$73$	−8.53178			−0.998570			−0.499285	−	0.866438i	$-0.666404\pi$
$73$	−8.53178			−0.998570			−0.499285	+	0.866438i	$0.666404\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	7.96538			0.896175			0.448088	−	0.893990i	$-0.352105\pi$
$79$	7.96538			0.896175			0.448088	+	0.893990i	$0.352105\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	6.32489i		−	0.694246i	−0.937820	−	0.347123i	$-0.887159\pi$
$83$		−	6.32489i		−	0.694246i	0.937820	−	0.347123i	$-0.112841\pi$
$84$		0			0
$85$	0.0179532	−	0.0878466i	0.00194730	−	0.00952830i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.0099			−1.06104			−0.530522	−	0.847671i	$-0.678004\pi$
$89$	−10.0099			−1.06104			−0.530522	+	0.847671i	$0.678004\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.0555296	−	0.271711i	0.00569722	−	0.0278770i
$96$		0			0
$97$	11.5960			1.17740			0.588698	−	0.808353i	$-0.299641\pi$
$97$	11.5960			1.17740			0.588698	+	0.808353i	$0.299641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.35421			−0.333756			−0.166878	−	0.985978i	$-0.553369\pi$
$101$	−3.35421			−0.333756			−0.166878	+	0.985978i	$0.553369\pi$
$102$		0			0
$103$	−11.0078			−1.08463			−0.542316	−	0.840174i	$-0.682453\pi$
$103$	−11.0078			−1.08463			−0.542316	+	0.840174i	$0.682453\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.5905			1.31384			0.656920	−	0.753960i	$-0.271859\pi$
$107$	13.5905			1.31384			0.656920	+	0.753960i	$0.271859\pi$
$108$		0			0
$109$	6.94743			0.665443			0.332721	−	0.943025i	$-0.392033\pi$
$109$	6.94743			0.665443			0.332721	+	0.943025i	$0.392033\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13.5489			1.27457			0.637285	−	0.770628i	$-0.280057\pi$
$113$	13.5489			1.27457			0.637285	+	0.770628i	$0.280057\pi$
$114$		0			0
$115$	8.47507	+	1.73205i	0.790305	+	0.161515i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.5178			0.956163
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.19724	−	6.35695i	−0.822626	−	0.568583i
$126$		0			0
$127$			4.96761i			0.440804i	0.975409	+	0.220402i	$0.0707369\pi$
$127$			4.96761i			0.440804i	−0.975409	+	0.220402i	$0.929263\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−13.4955			−1.17911			−0.589554	−	0.807729i	$-0.700696\pi$
$131$	−13.4955			−1.17911			−0.589554	+	0.807729i	$0.700696\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.9797			−1.02350			−0.511749	−	0.859135i	$-0.671002\pi$
$137$	−11.9797			−1.02350			−0.511749	+	0.859135i	$0.671002\pi$
$138$		0			0
$139$		−	1.63912i		−	0.139029i	−0.997581	−	0.0695143i	$-0.977855\pi$
$139$		−	1.63912i		−	0.139029i	0.997581	−	0.0695143i	$-0.0221449\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	2.61480i		−	0.218661i
$144$		0			0
$145$	1.96798	−	9.62948i	0.163432	−	0.799685i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	11.5026i		−	0.942326i	−0.882046	−	0.471163i	$-0.843834\pi$
$149$		−	11.5026i		−	0.942326i	0.882046	−	0.471163i	$-0.156166\pi$
$150$		0			0
$151$	14.4998			1.17998			0.589990	−	0.807410i	$-0.299132\pi$
$151$	14.4998			1.17998			0.589990	+	0.807410i	$0.299132\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.34835	+	6.59759i	−0.108302	+	0.529931i
$156$		0			0
$157$	−10.6855			−0.852799			−0.426400	−	0.904535i	$-0.640218\pi$
$157$	−10.6855			−0.852799			−0.426400	+	0.904535i	$0.640218\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			12.0259i			0.941938i	0.882150	+	0.470969i	$0.156096\pi$
$163$			12.0259i			0.941938i	−0.882150	+	0.470969i	$0.843904\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			25.1151i			1.94347i	0.236081	+	0.971733i	$0.424137\pi$
$167$			25.1151i			1.94347i	−0.236081	+	0.971733i	$0.575863\pi$
$168$		0			0
$169$	1.17891			0.0906854
$170$		0			0
$171$		0			0
$172$		0			0
$173$			8.97117i			0.682065i	0.940051	+	0.341033i	$0.110777\pi$
$173$			8.97117i			0.682065i	−0.940051	+	0.341033i	$0.889223\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	25.0777i		−	1.87440i	−0.348799	−	0.937198i	$-0.613410\pi$
$179$		−	25.0777i		−	1.87440i	0.348799	−	0.937198i	$-0.386590\pi$
$180$		0			0
$181$		−	4.30930i		−	0.320308i	−0.987092	−	0.160154i	$-0.948801\pi$
$181$		−	4.30930i		−	0.320308i	0.987092	−	0.160154i	$-0.0511991\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.98229	+	24.3788i	−0.366305	+	1.79236i
$186$		0			0
$187$	0.0278447			0.00203621
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.6536i		−	0.915580i	−0.889060	−	0.457790i	$-0.848641\pi$
$191$		−	12.6536i		−	0.915580i	0.889060	−	0.457790i	$-0.151359\pi$
$192$		0			0
$193$		−	16.9709i		−	1.22159i	−0.791788	−	0.610796i	$-0.790849\pi$
$193$		−	16.9709i		−	1.22159i	0.791788	−	0.610796i	$-0.209151\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.0304			−1.07087			−0.535436	−	0.844576i	$-0.679853\pi$
$197$	−15.0304			−1.07087			−0.535436	+	0.844576i	$0.679853\pi$
$198$		0			0
$199$		−	16.6817i		−	1.18253i	−0.806476	−	0.591267i	$-0.798628\pi$
$199$		−	16.6817i		−	1.18253i	0.806476	−	0.591267i	$-0.201372\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	17.3317	+	3.54208i	1.21050	+	0.247389i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.0861242			0.00595733
$210$		0			0
$211$	−15.4819			−1.06582			−0.532908	−	0.846173i	$-0.678901\pi$
$211$	−15.4819			−1.06582			−0.532908	+	0.846173i	$0.678901\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.67337	+	8.18797i	−0.114123	+	0.558415i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$			0.150989i			0.0101567i
$222$		0			0
$223$	−25.1597			−1.68482			−0.842408	−	0.538841i	$-0.818862\pi$
$223$	−25.1597			−1.68482			−0.842408	+	0.538841i	$0.818862\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	27.5770i		−	1.83035i	−0.403057	−	0.915175i	$-0.632052\pi$
$227$		−	27.5770i		−	1.83035i	0.403057	−	0.915175i	$-0.367948\pi$
$228$		0			0
$229$		−	22.8804i		−	1.51198i	−0.654583	−	0.755990i	$-0.727156\pi$
$229$		−	22.8804i		−	1.51198i	0.654583	−	0.755990i	$-0.272844\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.67538			−0.633855			−0.316928	−	0.948450i	$-0.602651\pi$
$233$	−9.67538			−0.633855			−0.316928	+	0.948450i	$0.602651\pi$
$234$		0			0
$235$	2.95334	−	14.4509i	0.192655	−	0.942675i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4.30863i		−	0.278702i	−0.990243	−	0.139351i	$-0.955498\pi$
$239$		−	4.30863i		−	0.278702i	0.990243	−	0.139351i	$-0.0445016\pi$
$240$		0			0
$241$			6.42621i			0.413948i	0.978346	+	0.206974i	$0.0663616\pi$
$241$			6.42621i			0.413948i	−0.978346	+	0.206974i	$0.933638\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$			0.467013i			0.0297154i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.89601			0.624631			0.312315	−	0.949978i	$-0.398895\pi$
$251$	9.89601			0.624631			0.312315	+	0.949978i	$0.398895\pi$
$252$		0			0
$253$			2.68634i			0.168889i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	25.5233i		−	1.59210i	−0.605230	−	0.796051i	$-0.706919\pi$
$257$		−	25.5233i		−	1.59210i	0.605230	−	0.796051i	$-0.293081\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.58887			−0.159636			−0.0798182	−	0.996809i	$-0.525434\pi$
$263$	−2.58887			−0.159636			−0.0798182	+	0.996809i	$0.525434\pi$
$264$		0			0
$265$	−19.6303	−	4.01184i	−1.20588	−	0.246445i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.78108			−0.474421			−0.237210	−	0.971458i	$-0.576233\pi$
$269$	−7.78108			−0.474421			−0.237210	+	0.971458i	$0.576233\pi$
$270$		0			0
$271$		−	29.4490i		−	1.78890i	−0.447171	−	0.894448i	$-0.647569\pi$
$271$		−	29.4490i		−	1.78890i	0.447171	−	0.894448i	$-0.352431\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.36227	−	3.19365i	0.0821481	−	0.192584i
$276$		0			0
$277$		−	1.47816i		−	0.0888141i	−0.999014	−	0.0444070i	$-0.985860\pi$
$277$		−	1.47816i		−	0.0888141i	0.999014	−	0.0444070i	$-0.0141399\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4.00483i		−	0.238908i	−0.992840	−	0.119454i	$-0.961886\pi$
$281$		−	4.00483i		−	0.238908i	0.992840	−	0.119454i	$-0.0381145\pi$
$282$		0			0
$283$	19.4214			1.15448			0.577242	−	0.816573i	$-0.304129\pi$
$283$	19.4214			1.15448			0.577242	+	0.816573i	$0.304129\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.9984			0.999905
$290$		0			0
$291$		0			0
$292$		0			0
$293$			2.39101i			0.139684i	0.997558	+	0.0698421i	$0.0222495\pi$
$293$			2.39101i			0.139684i	−0.997558	+	0.0698421i	$0.977750\pi$
$294$		0			0
$295$	−12.7147	−	2.59849i	−0.740276	−	0.151290i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−14.5668			−0.842422
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.37047	+	6.70584i	−0.0784730	+	0.383975i
$306$		0			0
$307$	−1.67320			−0.0954947			−0.0477474	−	0.998859i	$-0.515204\pi$
$307$	−1.67320			−0.0954947			−0.0477474	+	0.998859i	$0.515204\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.00942			−0.113944			−0.0569719	−	0.998376i	$-0.518145\pi$
$311$	−2.00942			−0.113944			−0.0569719	+	0.998376i	$0.518145\pi$
$312$		0			0
$313$	21.7735			1.23071			0.615355	−	0.788250i	$-0.289013\pi$
$313$	21.7735			1.23071			0.615355	+	0.788250i	$0.289013\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.9377			0.951315			0.475657	−	0.879631i	$-0.342210\pi$
$317$	16.9377			0.951315			0.475657	+	0.879631i	$0.342210\pi$
$318$		0			0
$319$	3.05225			0.170893
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.00497317			−0.000276715
$324$		0			0
$325$	17.3178	+	7.38700i	0.960617	+	0.409757i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	26.8949			1.47827			0.739137	−	0.673555i	$-0.235233\pi$
$331$	26.8949			1.47827			0.739137	+	0.673555i	$0.235233\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.63085	−	12.8730i	0.143739	−	0.703327i
$336$		0			0
$337$			16.5625i			0.902216i	0.892469	+	0.451108i	$0.148971\pi$
$337$			16.5625i			0.902216i	−0.892469	+	0.451108i	$0.851029\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.09123			−0.113247
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−31.7841			−1.70626			−0.853130	−	0.521699i	$-0.825298\pi$
$347$	−31.7841			−1.70626			−0.853130	+	0.521699i	$0.825298\pi$
$348$		0			0
$349$		−	27.5288i		−	1.47358i	−0.676119	−	0.736792i	$-0.736340\pi$
$349$		−	27.5288i		−	1.47358i	0.676119	−	0.736792i	$-0.263660\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	1.06380i		−	0.0566203i	−0.999599	−	0.0283102i	$-0.990987\pi$
$353$		−	1.06380i		−	0.0566203i	0.999599	−	0.0283102i	$-0.00901261\pi$
$354$		0			0
$355$	3.22298	−	15.7703i	0.171058	−	0.837002i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	24.4493i		−	1.29038i	−0.764021	−	0.645192i	$-0.776778\pi$
$359$		−	24.4493i		−	1.29038i	0.764021	−	0.645192i	$-0.223222\pi$
$360$		0			0
$361$	18.9846			0.999190
$362$		0			0
$363$		0			0
$364$		0			0
$365$	18.6913	+	3.81994i	0.978347	+	0.199945i
$366$		0			0
$367$	−6.86335			−0.358264			−0.179132	−	0.983825i	$-0.557329\pi$
$367$	−6.86335			−0.358264			−0.179132	+	0.983825i	$0.557329\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	14.2775i		−	0.739262i	−0.929179	−	0.369631i	$-0.879484\pi$
$373$		−	14.2775i		−	0.739262i	0.929179	−	0.369631i	$-0.120516\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			16.5510i			0.852421i
$378$		0			0
$379$	−35.6634			−1.83190			−0.915952	−	0.401288i	$-0.868563\pi$
$379$	−35.6634			−1.83190			−0.915952	+	0.401288i	$0.868563\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	27.8008i		−	1.42055i	−0.703922	−	0.710277i	$-0.748570\pi$
$383$		−	27.8008i		−	1.42055i	0.703922	−	0.710277i	$-0.251430\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	15.3782i		−	0.779703i	−0.920878	−	0.389852i	$-0.872526\pi$
$389$		−	15.3782i		−	0.779703i	0.920878	−	0.389852i	$-0.127474\pi$
$390$		0			0
$391$		−	0.155120i		−	0.00784478i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−17.4504	−	3.56634i	−0.878026	−	0.179442i
$396$		0			0
$397$	−19.8052			−0.993993			−0.496997	−	0.867752i	$-0.665564\pi$
$397$	−19.8052			−0.993993			−0.496997	+	0.867752i	$0.665564\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	20.2275i		−	1.01011i	−0.863087	−	0.505056i	$-0.831472\pi$
$401$		−	20.2275i		−	1.01011i	0.863087	−	0.505056i	$-0.168528\pi$
$402$		0			0
$403$		−	11.3398i		−	0.564878i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.72733			−0.383029
$408$		0			0
$409$		−	16.3718i		−	0.809535i	−0.914420	−	0.404767i	$-0.867353\pi$
$409$		−	16.3718i		−	0.809535i	0.914420	−	0.404767i	$-0.132647\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.83184	+	13.8565i	−0.139010	+	0.680187i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	32.6180			1.59349			0.796746	−	0.604314i	$-0.206553\pi$
$419$	32.6180			1.59349			0.796746	+	0.604314i	$0.206553\pi$
$420$		0			0
$421$	1.78518			0.0870045			0.0435022	−	0.999053i	$-0.486148\pi$
$421$	1.78518			0.0870045			0.0435022	+	0.999053i	$0.486148\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.0786632	+	0.184415i	−0.00381573	+	0.00894543i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	31.1782i		−	1.50180i	−0.660414	−	0.750901i	$-0.729619\pi$
$431$		−	31.1782i		−	1.50180i	0.660414	−	0.750901i	$-0.270381\pi$
$432$		0			0
$433$	−1.35671			−0.0651994			−0.0325997	−	0.999468i	$-0.510379\pi$
$433$	−1.35671			−0.0651994			−0.0325997	+	0.999468i	$0.510379\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	0.479791i		−	0.0229515i
$438$		0			0
$439$		−	8.52568i		−	0.406908i	−0.979084	−	0.203454i	$-0.934783\pi$
$439$		−	8.52568i		−	0.406908i	0.979084	−	0.203454i	$-0.0652168\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.6249			−0.694849			−0.347424	−	0.937708i	$-0.612944\pi$
$443$	−14.6249			−0.694849			−0.347424	+	0.937708i	$0.612944\pi$
$444$		0			0
$445$	21.9295	+	4.48173i	1.03956	+	0.212454i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	18.0320i		−	0.850984i	−0.904962	−	0.425492i	$-0.860101\pi$
$449$		−	18.0320i		−	0.850984i	0.904962	−	0.425492i	$-0.139899\pi$
$450$		0			0
$451$			5.49362i			0.258684i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	28.1433i		−	1.31649i	−0.752804	−	0.658244i	$-0.771299\pi$
$457$		−	28.1433i		−	1.31649i	0.752804	−	0.658244i	$-0.228701\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−21.9940			−1.02436			−0.512181	−	0.858878i	$-0.671162\pi$
$461$	−21.9940			−1.02436			−0.512181	+	0.858878i	$0.671162\pi$
$462$		0			0
$463$		−	15.7014i		−	0.729705i	−0.931065	−	0.364852i	$-0.881119\pi$
$463$		−	15.7014i		−	0.729705i	0.931065	−	0.364852i	$-0.118881\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			33.7095i			1.55989i	0.625848	+	0.779945i	$0.284753\pi$
$467$			33.7095i			1.55989i	−0.625848	+	0.779945i	$0.715247\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.59534			−0.119334
$474$		0			0
$475$	−0.243307	+	0.570398i	−0.0111637	+	0.0261717i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	35.4095			1.61790			0.808950	−	0.587877i	$-0.200036\pi$
$479$	35.4095			1.61790			0.808950	+	0.587877i	$0.200036\pi$
$480$		0			0
$481$		−	41.9019i		−	1.91056i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−25.4044	−	5.19189i	−1.15355	−	0.235752i
$486$		0			0
$487$			5.04052i			0.228408i	0.993457	+	0.114204i	$0.0364317\pi$
$487$			5.04052i			0.228408i	−0.993457	+	0.114204i	$0.963568\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			12.6130i			0.569215i	0.958644	+	0.284608i	$0.0918633\pi$
$491$			12.6130i			0.569215i	−0.958644	+	0.284608i	$0.908137\pi$
$492$		0			0
$493$	−0.176250			−0.00793789
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.231485			−0.0103627			−0.00518134	−	0.999987i	$-0.501649\pi$
$499$	−0.231485			−0.0103627			−0.00518134	+	0.999987i	$0.501649\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	17.9750i		−	0.801465i	−0.916195	−	0.400732i	$-0.868756\pi$
$503$		−	17.9750i		−	0.801465i	0.916195	−	0.400732i	$-0.131244\pi$
$504$		0			0
$505$	7.34835	+	1.50178i	0.326997	+	0.0668284i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.9818			−0.841352			−0.420676	−	0.907211i	$-0.638207\pi$
$509$	−18.9818			−0.841352			−0.420676	+	0.907211i	$0.638207\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.1158	+	4.92854i	1.06267	+	0.217177i
$516$		0			0
$517$	4.58051			0.201450
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−29.2651			−1.28213			−0.641064	−	0.767488i	$-0.721507\pi$
$521$	−29.2651			−1.28213			−0.641064	+	0.767488i	$0.721507\pi$
$522$		0			0
$523$	−15.6607			−0.684795			−0.342397	−	0.939555i	$-0.611239\pi$
$523$	−15.6607			−0.684795			−0.342397	+	0.939555i	$0.611239\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.120757			0.00526024
$528$		0			0
$529$	−8.03462			−0.349331
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−29.7895			−1.29032
$534$		0			0
$535$	−29.7738	−	6.08487i	−1.28723	−	0.263072i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	22.6813			0.975146			0.487573	−	0.873082i	$-0.337882\pi$
$541$	22.6813			0.975146			0.487573	+	0.873082i	$0.337882\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.2203	−	3.11058i	−0.651967	−	0.133242i
$546$		0			0
$547$			6.67849i			0.285552i	0.989755	+	0.142776i	$0.0456028\pi$
$547$			6.67849i			0.285552i	−0.989755	+	0.142776i	$0.954397\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.545144			−0.0232239
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	29.2681			1.24013			0.620066	−	0.784550i	$-0.287106\pi$
$557$	29.2681			1.24013			0.620066	+	0.784550i	$0.287106\pi$
$558$		0			0
$559$		−	14.0734i		−	0.595240i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			38.2841i			1.61348i	0.590906	+	0.806741i	$0.298770\pi$
$563$			38.2841i			1.61348i	−0.590906	+	0.806741i	$0.701230\pi$
$564$		0			0
$565$	−29.6826	−	6.06624i	−1.24876	−	0.255209i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	19.8115i		−	0.830539i	−0.909698	−	0.415270i	$-0.863687\pi$
$569$		−	19.8115i		−	0.830539i	0.909698	−	0.415270i	$-0.136313\pi$
$570$		0			0
$571$	20.8756			0.873617			0.436809	−	0.899554i	$-0.356109\pi$
$571$	20.8756			0.873617			0.436809	+	0.899554i	$0.356109\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−17.7916	−	7.58910i	−0.741960	−	0.316487i
$576$		0			0
$577$	−12.5061			−0.520637			−0.260319	−	0.965523i	$-0.583828\pi$
$577$	−12.5061			−0.520637			−0.260319	+	0.965523i	$0.583828\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	6.22220i		−	0.257697i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	20.4930i		−	0.845838i	−0.906168	−	0.422919i	$-0.861006\pi$
$587$		−	20.4930i		−	0.845838i	0.906168	−	0.422919i	$-0.138994\pi$
$588$		0			0
$589$	0.373502			0.0153899
$590$		0			0
$591$		0			0
$592$		0			0
$593$			33.2778i			1.36655i	0.730159	+	0.683277i	$0.239446\pi$
$593$			33.2778i			1.36655i	−0.730159	+	0.683277i	$0.760554\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	39.0328i		−	1.59484i	−0.603427	−	0.797419i	$-0.706198\pi$
$599$		−	39.0328i		−	1.59484i	0.603427	−	0.797419i	$-0.293802\pi$
$600$		0			0
$601$		−	14.7657i		−	0.602304i	−0.953576	−	0.301152i	$-0.902629\pi$
$601$		−	14.7657i		−	0.602304i	0.953576	−	0.301152i	$-0.0973712\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−23.0422	−	4.70914i	−0.936799	−	0.191454i
$606$		0			0
$607$	3.96958			0.161120			0.0805602	−	0.996750i	$-0.474329\pi$
$607$	3.96958			0.161120			0.0805602	+	0.996750i	$0.474329\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			24.8381i			1.00484i
$612$		0			0
$613$			35.1466i			1.41956i	0.704424	+	0.709780i	$0.251206\pi$
$613$			35.1466i			1.41956i	−0.704424	+	0.709780i	$0.748794\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.3102			0.978692			0.489346	−	0.872090i	$-0.337236\pi$
$617$	24.3102			0.978692			0.489346	+	0.872090i	$0.337236\pi$
$618$		0			0
$619$			7.99259i			0.321249i	0.987016	+	0.160625i	$0.0513508\pi$
$619$			7.99259i			0.321249i	−0.987016	+	0.160625i	$0.948649\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	17.3030	+	18.0446i	0.692119	+	0.721783i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.446208			0.0177915
$630$		0			0
$631$	−14.1828			−0.564607			−0.282304	−	0.959325i	$-0.591099\pi$
$631$	−14.1828			−0.564607			−0.282304	+	0.959325i	$0.591099\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.22415	−	10.8830i	0.0882627	−	0.431877i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			23.2729i			0.919223i	0.888120	+	0.459612i	$0.152011\pi$
$641$			23.2729i			0.919223i	−0.888120	+	0.459612i	$0.847989\pi$
$642$		0			0
$643$	43.2624			1.70610			0.853052	−	0.521826i	$-0.174749\pi$
$643$	43.2624			1.70610			0.853052	+	0.521826i	$0.174749\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			14.0085i			0.550730i	0.961340	+	0.275365i	$0.0887986\pi$
$647$			14.0085i			0.550730i	−0.961340	+	0.275365i	$0.911201\pi$
$648$		0			0
$649$		−	4.03016i		−	0.158198i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.86254			0.112020			0.0560099	−	0.998430i	$-0.482162\pi$
$653$	2.86254			0.112020			0.0560099	+	0.998430i	$0.482162\pi$
$654$		0			0
$655$	29.5657	+	6.04235i	1.15523	+	0.236094i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			41.1201i			1.60181i	0.598789	+	0.800907i	$0.295649\pi$
$659$			41.1201i			1.60181i	−0.598789	+	0.800907i	$0.704351\pi$
$660$		0			0
$661$		−	13.8089i		−	0.537102i	−0.963265	−	0.268551i	$-0.913455\pi$
$661$		−	13.8089i		−	0.537102i	0.963265	−	0.268551i	$-0.0865449\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	17.0038i		−	0.658391i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.12555			−0.0820559
$672$		0			0
$673$			36.5668i			1.40955i	0.709433	+	0.704773i	$0.248951\pi$
$673$			36.5668i			1.40955i	−0.709433	+	0.704773i	$0.751049\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			14.4006i			0.553460i	0.960948	+	0.276730i	$0.0892507\pi$
$677$			14.4006i			0.553460i	−0.960948	+	0.276730i	$0.910749\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.8321			−0.491008			−0.245504	−	0.969396i	$-0.578953\pi$
$683$	−12.8321			−0.491008			−0.245504	+	0.969396i	$0.578953\pi$
$684$		0			0
$685$	26.2450	+	5.36369i	1.00277	+	0.204936i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	33.7403			1.28540
$690$		0			0
$691$			33.4219i			1.27143i	0.771924	+	0.635715i	$0.219295\pi$
$691$			33.4219i			1.27143i	−0.771924	+	0.635715i	$0.780705\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.733885	+	3.59096i	−0.0278378	+	0.136213i
$696$		0			0
$697$		−	0.317224i		−	0.0120157i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	7.88834i		−	0.297938i	−0.988842	−	0.148969i	$-0.952404\pi$
$701$		−	7.88834i		−	0.297938i	0.988842	−	0.148969i	$-0.0475955\pi$
$702$		0			0
$703$	1.38013			0.0520526
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	34.3463			1.28990			0.644951	−	0.764224i	$-0.276878\pi$
$709$	34.3463			1.28990			0.644951	+	0.764224i	$0.276878\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			11.6501i			0.436299i
$714$		0			0
$715$	−1.17073	+	5.72847i	−0.0437827	+	0.214232i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.1332			−0.788136			−0.394068	−	0.919081i	$-0.628933\pi$
$719$	−21.1332			−0.788136			−0.394068	+	0.919081i	$0.628933\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.62283	+	20.2150i	−0.320244	+	0.750766i
$726$		0			0
$727$	33.6575			1.24829			0.624143	−	0.781310i	$-0.285448\pi$
$727$	33.6575			1.24829			0.624143	+	0.781310i	$0.285448\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.149865			0.00554298
$732$		0			0
$733$	−31.6620			−1.16946			−0.584731	−	0.811227i	$-0.698800\pi$
$733$	−31.6620			−1.16946			−0.584731	+	0.811227i	$0.698800\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.08035			0.150301
$738$		0			0
$739$	−14.5870			−0.536593			−0.268296	−	0.963336i	$-0.586461\pi$
$739$	−14.5870			−0.536593			−0.268296	+	0.963336i	$0.586461\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	36.3676			1.33420			0.667099	−	0.744969i	$-0.267536\pi$
$743$	36.3676			1.33420			0.667099	+	0.744969i	$0.267536\pi$
$744$		0			0
$745$	−5.15005	+	25.1996i	−0.188683	+	0.923243i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−28.3450			−1.03432			−0.517162	−	0.855887i	$-0.673012\pi$
$751$	−28.3450			−1.03432			−0.517162	+	0.855887i	$0.673012\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−31.7660	−	6.49202i	−1.15608	−	0.236269i
$756$		0			0
$757$		−	45.7899i		−	1.66426i	−0.554580	−	0.832130i	$-0.687121\pi$
$757$		−	45.7899i		−	1.66426i	0.554580	−	0.832130i	$-0.312879\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.90030			−0.177636			−0.0888179	−	0.996048i	$-0.528309\pi$
$761$	−4.90030			−0.177636			−0.0888179	+	0.996048i	$0.528309\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	21.8538			0.789094
$768$		0			0
$769$		−	24.7837i		−	0.893722i	−0.894603	−	0.446861i	$-0.852542\pi$
$769$		−	24.7837i		−	0.893722i	0.894603	−	0.446861i	$-0.147458\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	15.0720i		−	0.542103i	−0.962565	−	0.271052i	$-0.912629\pi$
$773$		−	15.0720i		−	0.542103i	0.962565	−	0.271052i	$-0.0873714\pi$
$774$		0			0
$775$	5.90788	−	13.8502i	0.212217	−	0.497514i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	0.981181i		−	0.0351544i
$780$		0			0
$781$	4.99871			0.178868
$782$		0			0
$783$		0			0
$784$		0			0
$785$	23.4097	+	4.78425i	0.835529	+	0.170757i
$786$		0			0
$787$	−4.58051			−0.163277			−0.0816387	−	0.996662i	$-0.526015\pi$
$787$	−4.58051			−0.163277			−0.0816387	+	0.996662i	$0.526015\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	11.5259i		−	0.409297i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	23.9082i		−	0.846872i	−0.905926	−	0.423436i	$-0.860824\pi$
$797$		−	23.9082i		−	0.846872i	0.905926	−	0.423436i	$-0.139176\pi$
$798$		0			0
$799$	−0.264498			−0.00935725
$800$		0			0
$801$		0			0
$802$		0			0
$803$			5.92457i			0.209074i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	51.3808i		−	1.80645i	−0.429166	−	0.903226i	$-0.641192\pi$
$809$		−	51.3808i		−	1.80645i	0.429166	−	0.903226i	$-0.358808\pi$
$810$		0			0
$811$			33.9917i			1.19361i	0.802387	+	0.596804i	$0.203563\pi$
$811$			33.9917i			1.19361i	−0.802387	+	0.596804i	$0.796437\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.38434	−	26.3461i	0.188605	−	0.922862i
$816$		0			0
$817$	0.463537			0.0162171
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	34.7652i		−	1.21332i	−0.794963	−	0.606658i	$-0.792510\pi$
$821$		−	34.7652i		−	1.21332i	0.794963	−	0.606658i	$-0.207490\pi$
$822$		0			0
$823$			32.7568i			1.14183i	0.821009	+	0.570915i	$0.193411\pi$
$823$			32.7568i			1.14183i	−0.821009	+	0.570915i	$0.806589\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5.43764			0.189085			0.0945427	−	0.995521i	$-0.469861\pi$
$827$	5.43764			0.189085			0.0945427	+	0.995521i	$0.469861\pi$
$828$		0			0
$829$		−	39.8207i		−	1.38303i	−0.722362	−	0.691515i	$-0.756943\pi$
$829$		−	39.8207i		−	1.38303i	0.722362	−	0.691515i	$-0.243057\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	11.2448	−	55.0218i	0.389143	−	1.90411i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.3647			0.530449			0.265225	−	0.964187i	$-0.414554\pi$
$839$	15.3647			0.530449			0.265225	+	0.964187i	$0.414554\pi$
$840$		0			0
$841$	9.68003			0.333794
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.58274	−	0.527834i	−0.0888489	−	0.0181581i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$			43.0483i			1.47568i
$852$		0			0
$853$	−32.0274			−1.09660			−0.548299	−	0.836282i	$-0.684724\pi$
$853$	−32.0274			−1.09660			−0.548299	+	0.836282i	$0.684724\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			5.39727i			0.184367i	0.995742	+	0.0921836i	$0.0293847\pi$
$857$			5.39727i			0.184367i	−0.995742	+	0.0921836i	$0.970615\pi$
$858$		0			0
$859$			6.83527i			0.233217i	0.993178	+	0.116608i	$0.0372022\pi$
$859$			6.83527i			0.233217i	−0.993178	+	0.116608i	$0.962798\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11.6416			0.396284			0.198142	−	0.980173i	$-0.436509\pi$
$863$	11.6416			0.396284			0.198142	+	0.980173i	$0.436509\pi$
$864$		0			0
$865$	4.01667	−	19.6539i	0.136571	−	0.668253i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	5.53125i		−	0.187635i
$870$		0			0
$871$			22.1259i			0.749708i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			33.1838i			1.12054i	0.828311	+	0.560269i	$0.189302\pi$
$877$			33.1838i			1.12054i	−0.828311	+	0.560269i	$0.810698\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−41.0831			−1.38412			−0.692062	−	0.721838i	$-0.743298\pi$
$881$	−41.0831			−1.38412			−0.692062	+	0.721838i	$0.743298\pi$
$882$		0			0
$883$			11.7151i			0.394244i	0.980379	+	0.197122i	$0.0631595\pi$
$883$			11.7151i			0.394244i	−0.980379	+	0.197122i	$0.936841\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	15.9992i		−	0.537199i	−0.963252	−	0.268600i	$-0.913439\pi$
$887$		−	15.9992i		−	0.537199i	0.963252	−	0.268600i	$-0.0865608\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.818096			−0.0273765
$894$		0			0
$895$	−11.2281	+	54.9398i	−0.375312	+	1.83644i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	13.2370			0.441478
$900$		0			0
$901$			0.359296i			0.0119699i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.92941	+	9.44076i	−0.0641357	+	0.313821i
$906$		0			0
$907$		−	6.46711i		−	0.214737i	−0.994219	−	0.107368i	$-0.965758\pi$
$907$		−	6.46711i		−	0.214737i	0.994219	−	0.107368i	$-0.0342424\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			30.3458i			1.00540i	0.864461	+	0.502700i	$0.167660\pi$
$911$			30.3458i			1.00540i	−0.864461	+	0.502700i	$0.832340\pi$
$912$		0			0
$913$	−4.39208			−0.145357
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−47.8436			−1.57821			−0.789107	−	0.614256i	$-0.789456\pi$
$919$	−47.8436			−1.57821			−0.789107	+	0.614256i	$0.789456\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			27.1058i			0.892199i
$924$		0			0
$925$	21.8302	−	51.1779i	0.717774	−	1.68272i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−31.4768			−1.03272			−0.516360	−	0.856372i	$-0.672713\pi$
$929$	−31.4768			−1.03272			−0.516360	+	0.856372i	$0.672713\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.0610017	−	0.0124669i	−0.00199497	−	0.000407712i
$936$		0			0
$937$	−51.6376			−1.68693			−0.843464	−	0.537186i	$-0.819487\pi$
$937$	−51.6376			−1.68693			−0.843464	+	0.537186i	$0.819487\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.20763			0.169764			0.0848819	−	0.996391i	$-0.472949\pi$
$941$	5.20763			0.169764			0.0848819	+	0.996391i	$0.472949\pi$
$942$		0			0
$943$	30.6045			0.996619
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−53.3747			−1.73445			−0.867223	−	0.497921i	$-0.834097\pi$
$947$	−53.3747			−1.73445			−0.867223	+	0.497921i	$0.834097\pi$
$948$		0			0
$949$	−32.1263			−1.04287
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−58.6135			−1.89868			−0.949338	−	0.314257i	$-0.898245\pi$
$953$	−58.6135			−1.89868			−0.949338	+	0.314257i	$0.898245\pi$
$954$		0			0
$955$	−5.66539	+	27.7212i	−0.183328	+	0.897039i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	21.9308			0.707444
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.59839	+	37.1796i	−0.244601	+	1.19685i
$966$		0			0
$967$			14.9345i			0.480262i	0.970740	+	0.240131i	$0.0771905\pi$
$967$			14.9345i			0.480262i	−0.970740	+	0.240131i	$0.922810\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−23.8197			−0.764409			−0.382205	−	0.924078i	$-0.624835\pi$
$971$	−23.8197			−0.764409			−0.382205	+	0.924078i	$0.624835\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.80443			−0.0577287			−0.0288643	−	0.999583i	$-0.509189\pi$
$977$	−1.80443			−0.0577287			−0.0288643	+	0.999583i	$0.509189\pi$
$978$		0			0
$979$			6.95097i			0.222154i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			17.1677i			0.547563i	0.961792	+	0.273782i	$0.0882746\pi$
$983$			17.1677i			0.547563i	−0.961792	+	0.273782i	$0.911725\pi$
$984$		0			0
$985$	32.9284	+	6.72957i	1.04918	+	0.214422i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			14.4584i			0.459750i
$990$		0			0
$991$	15.2837			0.485504			0.242752	−	0.970088i	$-0.421950\pi$
$991$	15.2837			0.485504			0.242752	+	0.970088i	$0.421950\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7.46891	+	36.5460i	−0.236780	+	1.15859i
$996$		0			0
$997$	50.6756			1.60491			0.802456	−	0.596712i	$-0.203526\pi$
$997$	50.6756			1.60491			0.802456	+	0.596712i	$0.203526\pi$
$998$		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.4409.1		32
3.2	odd	2	inner	8820.2.f.a.4409.32		32
5.4	even	2	inner	8820.2.f.a.4409.4		32
7.4	even	3		1260.2.dc.a.89.12	yes	32
7.5	odd	6		1260.2.dc.a.269.6	yes	32
7.6	odd	2	inner	8820.2.f.a.4409.31		32
15.14	odd	2	inner	8820.2.f.a.4409.29		32
21.5	even	6		1260.2.dc.a.269.11	yes	32
21.11	odd	6		1260.2.dc.a.89.5	✓	32
21.20	even	2	inner	8820.2.f.a.4409.2		32
35.4	even	6		1260.2.dc.a.89.11	yes	32
35.12	even	12		6300.2.ch.f.4301.13		32
35.18	odd	12		6300.2.ch.f.1601.4		32
35.19	odd	6		1260.2.dc.a.269.5	yes	32
35.32	odd	12		6300.2.ch.f.1601.14		32
35.33	even	12		6300.2.ch.f.4301.3		32
35.34	odd	2	inner	8820.2.f.a.4409.30		32
105.32	even	12		6300.2.ch.f.1601.13		32
105.47	odd	12		6300.2.ch.f.4301.14		32
105.53	even	12		6300.2.ch.f.1601.3		32
105.68	odd	12		6300.2.ch.f.4301.4		32
105.74	odd	6		1260.2.dc.a.89.6	yes	32
105.89	even	6		1260.2.dc.a.269.12	yes	32
105.104	even	2	inner	8820.2.f.a.4409.3		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.dc.a.89.5	✓	32	21.11	odd	6
1260.2.dc.a.89.6	yes	32	105.74	odd	6
1260.2.dc.a.89.11	yes	32	35.4	even	6
1260.2.dc.a.89.12	yes	32	7.4	even	3
1260.2.dc.a.269.5	yes	32	35.19	odd	6
1260.2.dc.a.269.6	yes	32	7.5	odd	6
1260.2.dc.a.269.11	yes	32	21.5	even	6
1260.2.dc.a.269.12	yes	32	105.89	even	6
6300.2.ch.f.1601.3		32	105.53	even	12
6300.2.ch.f.1601.4		32	35.18	odd	12
6300.2.ch.f.1601.13		32	105.32	even	12
6300.2.ch.f.1601.14		32	35.32	odd	12
6300.2.ch.f.4301.3		32	35.33	even	12
6300.2.ch.f.4301.4		32	105.68	odd	12
6300.2.ch.f.4301.13		32	35.12	even	12
6300.2.ch.f.4301.14		32	105.47	odd	12
8820.2.f.a.4409.1		32	1.1	even	1	trivial
8820.2.f.a.4409.2		32	21.20	even	2	inner
8820.2.f.a.4409.3		32	105.104	even	2	inner
8820.2.f.a.4409.4		32	5.4	even	2	inner
8820.2.f.a.4409.29		32	15.14	odd	2	inner
8820.2.f.a.4409.30		32	35.34	odd	2	inner
8820.2.f.a.4409.31		32	7.6	odd	2	inner
8820.2.f.a.4409.32		32	3.2	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+(-2.19078 - 0.447731i) q^{5} +O(q^{10})\)	\(q+(-2.19078 - 0.447731i) q^{5} -0.694412i q^{11} +3.76549 q^{13} +0.0400982i q^{17} +0.124025i q^{19} -3.86851 q^{23} +(4.59907 + 1.96176i) q^{25} +4.39545i q^{29} -3.01152i q^{31} -11.1279i q^{37} -7.91118 q^{41} -3.73746i q^{43} +6.59624i q^{47} +8.96039 q^{53} +(-0.310909 + 1.52131i) q^{55} +5.80370 q^{59} -3.06093i q^{61} +(-8.24938 - 1.68592i) q^{65} +5.87597i q^{67} +7.19849i q^{71} -8.53178 q^{73} +7.96538 q^{79} -6.32489i q^{83} +(0.0179532 - 0.0878466i) q^{85} -10.0099 q^{89} +(0.0555296 - 0.271711i) q^{95} +11.5960 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * 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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