LMFDB - Embedded newform 8820.2.d.b.881.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8820,2,Mod(881,8820)]

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, 0, 1]))

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

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

chi := DirichletCharacter("8820.881");

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.d (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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + \cdots + 117649 $ x^12 - 2x^11 - 9x^10 + 58x^9 - 78x^8 - 298x^7 + 1341x^6 - 2086x^5 - 3822x^4 + 19894x^3 - 21609x^2 - 33614*x + 117649
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.5
Root			$1.75207 + 1.98249i$ of defining polynomial
Character	$\chi$	$=$	8820.881
Dual form			8820.2.d.b.881.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +O(q^{10})$	$q+1.00000 q^{5} -2.23292i q^{11} +4.92905i q^{13} +3.83953 q^{17} -5.96386i q^{19} -4.24918i q^{23} +1.00000 q^{25} +6.73838i q^{29} +1.80933i q^{31} -4.32755 q^{37} +2.50414 q^{41} +4.96414 q^{43} +10.8837 q^{47} +4.50545i q^{53} -2.23292i q^{55} -3.99215 q^{59} -6.25890i q^{61} +4.92905i q^{65} -11.1832 q^{67} -8.87665i q^{71} +5.01716i q^{73} +11.2041 q^{79} +0.295092 q^{83} +3.83953 q^{85} +14.1360 q^{89} -5.96386i q^{95} -1.05218i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{5}+O(q^{10}) $ 12 * q + 12 * q^5	$ 12 q + 12 q^{5} + 12 q^{25} + 4 q^{37} - 8 q^{41} + 36 q^{43} + 32 q^{47} - 4 q^{67} - 28 q^{79} + 40 q^{83} + 40 q^{89}+O(q^{100}) $ 12 * q + 12 * q^5 + 12 * q^25 + 4 * q^37 - 8 * q^41 + 36 * q^43 + 32 * q^47 - 4 * q^67 - 28 * q^79 + 40 * q^83 + 40 * q^89

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.23292i	− 0.673252i	−0.941638	−	0.336626i	$-0.890714\pi$
$11$	− 2.23292i	− 0.673252i	0.941638	−	0.336626i	$-0.109286\pi$
$12$	0	0
$13$	4.92905i	1.36707i	0.729917	+	0.683536i	$0.239559\pi$
$13$	4.92905i	1.36707i	−0.729917	+	0.683536i	$0.760441\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.83953	0.931224	0.465612	−	0.884989i	$-0.345834\pi$
$17$	3.83953	0.931224	0.465612	+	0.884989i	$0.345834\pi$
$18$	0	0
$19$	− 5.96386i	− 1.36820i	−0.729386	−	0.684102i	$-0.760194\pi$
$19$	− 5.96386i	− 1.36820i	0.729386	−	0.684102i	$-0.239806\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.24918i	− 0.886015i	−0.896518	−	0.443008i	$-0.853911\pi$
$23$	− 4.24918i	− 0.886015i	0.896518	−	0.443008i	$-0.146089\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.73838i	1.25129i	0.780110	+	0.625643i	$0.215163\pi$
$29$	6.73838i	1.25129i	−0.780110	+	0.625643i	$0.784837\pi$
$30$	0	0
$31$	1.80933i	0.324966i	0.986711	+	0.162483i	$0.0519502\pi$
$31$	1.80933i	0.324966i	−0.986711	+	0.162483i	$0.948050\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.32755	−0.711444	−0.355722	−	0.934592i	$-0.615765\pi$
$37$	−4.32755	−0.711444	−0.355722	+	0.934592i	$0.615765\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.50414	0.391080	0.195540	−	0.980696i	$-0.437354\pi$
$41$	2.50414	0.391080	0.195540	+	0.980696i	$0.437354\pi$
$42$	0	0
$43$	4.96414	0.757025	0.378512	−	0.925596i	$-0.376436\pi$
$43$	4.96414	0.757025	0.378512	+	0.925596i	$0.376436\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.8837	1.58755	0.793773	−	0.608214i	$-0.208114\pi$
$47$	10.8837	1.58755	0.793773	+	0.608214i	$0.208114\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.50545i	0.618872i	0.950920	+	0.309436i	$0.100140\pi$
$53$	4.50545i	0.618872i	−0.950920	+	0.309436i	$0.899860\pi$
$54$	0	0
$55$	− 2.23292i	− 0.301088i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.99215	−0.519733	−0.259867	−	0.965645i	$-0.583679\pi$
$59$	−3.99215	−0.519733	−0.259867	+	0.965645i	$0.583679\pi$
$60$	0	0
$61$	− 6.25890i	− 0.801369i	−0.916216	−	0.400685i	$-0.868772\pi$
$61$	− 6.25890i	− 0.801369i	0.916216	−	0.400685i	$-0.131228\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.92905i	0.611373i
$66$	0	0
$67$	−11.1832	−1.36625	−0.683123	−	0.730303i	$-0.739379\pi$
$67$	−11.1832	−1.36625	−0.683123	+	0.730303i	$0.739379\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 8.87665i	− 1.05347i	−0.850031	−	0.526733i	$-0.823417\pi$
$71$	− 8.87665i	− 1.05347i	0.850031	−	0.526733i	$-0.176583\pi$
$72$	0	0
$73$	5.01716i	0.587214i	0.955926	+	0.293607i	$0.0948557\pi$
$73$	5.01716i	0.587214i	−0.955926	+	0.293607i	$0.905144\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.2041	1.26056	0.630281	−	0.776367i	$-0.282939\pi$
$79$	11.2041	1.26056	0.630281	+	0.776367i	$0.282939\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.295092	0.0323906	0.0161953	−	0.999869i	$-0.494845\pi$
$83$	0.295092	0.0323906	0.0161953	+	0.999869i	$0.494845\pi$
$84$	0	0
$85$	3.83953	0.416456
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.1360	1.49842	0.749208	−	0.662335i	$-0.230435\pi$
$89$	14.1360	1.49842	0.749208	+	0.662335i	$0.230435\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 5.96386i	− 0.611879i
$96$	0	0
$97$	− 1.05218i	− 0.106833i	−0.998572	−	0.0534165i	$-0.982989\pi$
$97$	− 1.05218i	− 0.106833i	0.998572	−	0.0534165i	$-0.0170111\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.4071	−1.73207	−0.866034	−	0.499985i	$-0.833339\pi$
$101$	−17.4071	−1.73207	−0.866034	+	0.499985i	$0.833339\pi$
$102$	0	0
$103$	− 12.5384i	− 1.23544i	−0.786397	−	0.617721i	$-0.788056\pi$
$103$	− 12.5384i	− 1.23544i	0.786397	−	0.617721i	$-0.211944\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.92932i	0.476535i	0.971200	+	0.238268i	$0.0765795\pi$
$107$	4.92932i	0.476535i	−0.971200	+	0.238268i	$0.923420\pi$
$108$	0	0
$109$	13.0223	1.24731	0.623656	−	0.781699i	$-0.285647\pi$
$109$	13.0223	1.24731	0.623656	+	0.781699i	$0.285647\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 7.82363i	− 0.735985i	−0.929829	−	0.367993i	$-0.880045\pi$
$113$	− 7.82363i	− 0.735985i	0.929829	−	0.367993i	$-0.119955\pi$
$114$	0	0
$115$	− 4.24918i	− 0.396238i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.01405	0.546731
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.53736	−0.580097	−0.290048	−	0.957012i	$-0.593671\pi$
$127$	−6.53736	−0.580097	−0.290048	+	0.957012i	$0.593671\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.315665	−0.0275798	−0.0137899	−	0.999905i	$-0.504390\pi$
$131$	−0.315665	−0.0275798	−0.0137899	+	0.999905i	$0.504390\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.2097i	1.89750i	0.316026	+	0.948751i	$0.397651\pi$
$137$	22.2097i	1.89750i	−0.316026	+	0.948751i	$0.602349\pi$
$138$	0	0
$139$	− 14.8183i	− 1.25687i	−0.777862	−	0.628435i	$-0.783696\pi$
$139$	− 14.8183i	− 1.25687i	0.777862	−	0.628435i	$-0.216304\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.0062	0.920384
$144$	0	0
$145$	6.73838i	0.559592i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.10602i	0.336379i	0.985755	+	0.168189i	$0.0537920\pi$
$149$	4.10602i	0.336379i	−0.985755	+	0.168189i	$0.946208\pi$
$150$	0	0
$151$	11.5675	0.941352	0.470676	−	0.882306i	$-0.344010\pi$
$151$	11.5675	0.941352	0.470676	+	0.882306i	$0.344010\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.80933i	0.145329i
$156$	0	0
$157$	1.92814i	0.153883i	0.997036	+	0.0769413i	$0.0245154\pi$
$157$	1.92814i	0.153883i	−0.997036	+	0.0769413i	$0.975485\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.3345	1.27942	0.639709	−	0.768617i	$-0.279055\pi$
$163$	16.3345	1.27942	0.639709	+	0.768617i	$0.279055\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.2112	1.25446	0.627231	−	0.778834i	$-0.284188\pi$
$167$	16.2112	1.25446	0.627231	+	0.778834i	$0.284188\pi$
$168$	0	0
$169$	−11.2955	−0.868885
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−21.5331	−1.63713	−0.818564	−	0.574415i	$-0.805230\pi$
$173$	−21.5331	−1.63713	−0.818564	+	0.574415i	$0.805230\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 21.7519i	− 1.62581i	−0.582393	−	0.812907i	$-0.697884\pi$
$179$	− 21.7519i	− 1.62581i	0.582393	−	0.812907i	$-0.302116\pi$
$180$	0	0
$181$	− 10.4316i	− 0.775378i	−0.921790	−	0.387689i	$-0.873274\pi$
$181$	− 10.4316i	− 0.775378i	0.921790	−	0.387689i	$-0.126726\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.32755	−0.318167
$186$	0	0
$187$	− 8.57339i	− 0.626949i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.6063i	1.70809i	0.520201	+	0.854044i	$0.325857\pi$
$191$	23.6063i	1.70809i	−0.520201	+	0.854044i	$0.674143\pi$
$192$	0	0
$193$	16.3226	1.17493	0.587464	−	0.809250i	$-0.300126\pi$
$193$	16.3226	1.17493	0.587464	+	0.809250i	$0.300126\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.03368i	− 0.501129i	−0.968100	−	0.250565i	$-0.919384\pi$
$197$	− 7.03368i	− 0.501129i	0.968100	−	0.250565i	$-0.0806162\pi$
$198$	0	0
$199$	− 6.43512i	− 0.456173i	−0.973641	−	0.228087i	$-0.926753\pi$
$199$	− 6.43512i	− 0.456173i	0.973641	−	0.228087i	$-0.0732470\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.50414	0.174896
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−13.3169	−0.921146
$210$	0	0
$211$	−1.59463	−0.109779	−0.0548895	−	0.998492i	$-0.517481\pi$
$211$	−1.59463	−0.109779	−0.0548895	+	0.998492i	$0.517481\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.96414	0.338552
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	18.9252i	1.27305i
$222$	0	0
$223$	20.8137i	1.39379i	0.717174	+	0.696894i	$0.245435\pi$
$223$	20.8137i	1.39379i	−0.717174	+	0.696894i	$0.754565\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.89554	0.656790	0.328395	−	0.944540i	$-0.393492\pi$
$227$	9.89554	0.656790	0.328395	+	0.944540i	$0.393492\pi$
$228$	0	0
$229$	20.2219i	1.33630i	0.744025	+	0.668152i	$0.232914\pi$
$229$	20.2219i	1.33630i	−0.744025	+	0.668152i	$0.767086\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.65346i	0.566907i	0.958986	+	0.283454i	$0.0914802\pi$
$233$	8.65346i	0.566907i	−0.958986	+	0.283454i	$0.908520\pi$
$234$	0	0
$235$	10.8837	0.709972
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.9691i	1.16233i	0.813787	+	0.581163i	$0.197402\pi$
$239$	17.9691i	1.16233i	−0.813787	+	0.581163i	$0.802598\pi$
$240$	0	0
$241$	− 26.0955i	− 1.68096i	−0.541846	−	0.840478i	$-0.682274\pi$
$241$	− 26.0955i	− 1.68096i	0.541846	−	0.840478i	$-0.317726\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	29.3962	1.87043
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3.97721	−0.251039	−0.125520	−	0.992091i	$-0.540060\pi$
$251$	−3.97721	−0.251039	−0.125520	+	0.992091i	$0.540060\pi$
$252$	0	0
$253$	−9.48810	−0.596512
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.31927	0.394185	0.197093	−	0.980385i	$-0.436850\pi$
$257$	6.31927	0.394185	0.197093	+	0.980385i	$0.436850\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.4861i	1.13990i	0.821678	+	0.569952i	$0.193038\pi$
$263$	18.4861i	1.13990i	−0.821678	+	0.569952i	$0.806962\pi$
$264$	0	0
$265$	4.50545i	0.276768i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.5950	0.889872	0.444936	−	0.895562i	$-0.353226\pi$
$269$	14.5950	0.889872	0.444936	+	0.895562i	$0.353226\pi$
$270$	0	0
$271$	− 20.6470i	− 1.25422i	−0.778931	−	0.627109i	$-0.784238\pi$
$271$	− 20.6470i	− 1.25422i	0.778931	−	0.627109i	$-0.215762\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.23292i	− 0.134650i
$276$	0	0
$277$	24.3484	1.46295	0.731477	−	0.681866i	$-0.238831\pi$
$277$	24.3484	1.46295	0.731477	+	0.681866i	$0.238831\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 30.6625i	− 1.82917i	−0.404393	−	0.914585i	$-0.632517\pi$
$281$	− 30.6625i	− 1.82917i	0.404393	−	0.914585i	$-0.367483\pi$
$282$	0	0
$283$	− 4.44821i	− 0.264419i	−0.991222	−	0.132209i	$-0.957793\pi$
$283$	− 4.44821i	− 0.264419i	0.991222	−	0.132209i	$-0.0422071\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2.25797	−0.132822
$290$	0	0
$291$	0	0
$292$	0	0
$293$	32.7139	1.91117	0.955583	−	0.294721i	$-0.0952268\pi$
$293$	32.7139	1.91117	0.955583	+	0.294721i	$0.0952268\pi$
$294$	0	0
$295$	−3.99215	−0.232432
$296$	0	0
$297$	0	0
$298$	0	0
$299$	20.9444	1.21125
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 6.25890i	− 0.358383i
$306$	0	0
$307$	− 2.37826i	− 0.135735i	−0.997694	−	0.0678673i	$-0.978381\pi$
$307$	− 2.37826i	− 0.135735i	0.997694	−	0.0678673i	$-0.0216195\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.47133	0.423660	0.211830	−	0.977306i	$-0.432058\pi$
$311$	7.47133	0.423660	0.211830	+	0.977306i	$0.432058\pi$
$312$	0	0
$313$	− 19.3497i	− 1.09371i	−0.837228	−	0.546854i	$-0.815825\pi$
$313$	− 19.3497i	− 1.09371i	0.837228	−	0.546854i	$-0.184175\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.38007i	0.358341i	0.983818	+	0.179170i	$0.0573413\pi$
$317$	6.38007i	0.358341i	−0.983818	+	0.179170i	$0.942659\pi$
$318$	0	0
$319$	15.0463	0.842431
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 22.8985i	− 1.27410i
$324$	0	0
$325$	4.92905i	0.273414i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0.654466	0.0359727	0.0179863	−	0.999838i	$-0.494274\pi$
$331$	0.654466	0.0359727	0.0179863	+	0.999838i	$0.494274\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.1832	−0.611004
$336$	0	0
$337$	−20.4752	−1.11536	−0.557679	−	0.830057i	$-0.688308\pi$
$337$	−20.4752	−1.11536	−0.557679	+	0.830057i	$0.688308\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.04010	0.218784
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.6077i	0.837863i	0.908018	+	0.418931i	$0.137595\pi$
$347$	15.6077i	0.837863i	−0.908018	+	0.418931i	$0.862405\pi$
$348$	0	0
$349$	− 22.4320i	− 1.20076i	−0.799715	−	0.600380i	$-0.795016\pi$
$349$	− 22.4320i	− 1.20076i	0.799715	−	0.600380i	$-0.204984\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.5208	1.73091	0.865453	−	0.500990i	$-0.167031\pi$
$353$	32.5208	1.73091	0.865453	+	0.500990i	$0.167031\pi$
$354$	0	0
$355$	− 8.87665i	− 0.471124i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.6380i	0.825344i	0.910880	+	0.412672i	$0.135405\pi$
$359$	15.6380i	0.825344i	−0.910880	+	0.412672i	$0.864595\pi$
$360$	0	0
$361$	−16.5676	−0.871981
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.01716i	0.262610i
$366$	0	0
$367$	3.05166i	0.159295i	0.996823	+	0.0796476i	$0.0253795\pi$
$367$	3.05166i	0.159295i	−0.996823	+	0.0796476i	$0.974620\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.9475	0.618619	0.309310	−	0.950961i	$-0.399902\pi$
$373$	11.9475	0.618619	0.309310	+	0.950961i	$0.399902\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−33.2138	−1.71060
$378$	0	0
$379$	−34.8181	−1.78848	−0.894242	−	0.447584i	$-0.852285\pi$
$379$	−34.8181	−1.78848	−0.894242	+	0.447584i	$0.852285\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.45814	−0.0745076	−0.0372538	−	0.999306i	$-0.511861\pi$
$383$	−1.45814	−0.0745076	−0.0372538	+	0.999306i	$0.511861\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.1509i	1.02169i	0.859673	+	0.510845i	$0.170667\pi$
$389$	20.1509i	1.02169i	−0.859673	+	0.510845i	$0.829333\pi$
$390$	0	0
$391$	− 16.3149i	− 0.825079i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.2041	0.563741
$396$	0	0
$397$	− 19.3115i	− 0.969215i	−0.874732	−	0.484607i	$-0.838962\pi$
$397$	− 19.3115i	− 0.969215i	0.874732	−	0.484607i	$-0.161038\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 28.3691i	− 1.41669i	−0.705868	−	0.708343i	$-0.749443\pi$
$401$	− 28.3691i	− 1.41669i	0.705868	−	0.708343i	$-0.250557\pi$
$402$	0	0
$403$	−8.91828	−0.444251
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.66308i	0.478981i
$408$	0	0
$409$	− 15.8762i	− 0.785026i	−0.919747	−	0.392513i	$-0.871606\pi$
$409$	− 15.8762i	− 0.785026i	0.919747	−	0.392513i	$-0.128394\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0.295092	0.0144855
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.98073	0.0967652	0.0483826	−	0.998829i	$-0.484593\pi$
$419$	1.98073	0.0967652	0.0483826	+	0.998829i	$0.484593\pi$
$420$	0	0
$421$	−31.1448	−1.51790	−0.758952	−	0.651147i	$-0.774288\pi$
$421$	−31.1448	−1.51790	−0.758952	+	0.651147i	$0.774288\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.83953	0.186245
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.5894i	1.23260i	0.787512	+	0.616300i	$0.211369\pi$
$431$	25.5894i	1.23260i	−0.787512	+	0.616300i	$0.788631\pi$
$432$	0	0
$433$	− 6.90795i	− 0.331975i	−0.986128	−	0.165987i	$-0.946919\pi$
$433$	− 6.90795i	− 0.331975i	0.986128	−	0.165987i	$-0.0530811\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−25.3415	−1.21225
$438$	0	0
$439$	− 18.5663i	− 0.886120i	−0.896492	−	0.443060i	$-0.853893\pi$
$439$	− 18.5663i	− 0.886120i	0.896492	−	0.443060i	$-0.146107\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0525i	0.572633i	0.958135	+	0.286317i	$0.0924309\pi$
$443$	12.0525i	0.572633i	−0.958135	+	0.286317i	$0.907569\pi$
$444$	0	0
$445$	14.1360	0.670112
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 18.2147i	− 0.859604i	−0.902923	−	0.429802i	$-0.858583\pi$
$449$	− 18.2147i	− 0.859604i	0.902923	−	0.429802i	$-0.141417\pi$
$450$	0	0
$451$	− 5.59155i	− 0.263296i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.886209	−0.0414551	−0.0207276	−	0.999785i	$-0.506598\pi$
$457$	−0.886209	−0.0414551	−0.0207276	+	0.999785i	$0.506598\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.24276	0.0578811	0.0289405	−	0.999581i	$-0.490787\pi$
$461$	1.24276	0.0578811	0.0289405	+	0.999581i	$0.490787\pi$
$462$	0	0
$463$	35.8986	1.66835	0.834176	−	0.551499i	$-0.185944\pi$
$463$	35.8986	1.66835	0.834176	+	0.551499i	$0.185944\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.5952	−0.814208	−0.407104	−	0.913382i	$-0.633461\pi$
$467$	−17.5952	−0.814208	−0.407104	+	0.913382i	$0.633461\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 11.0846i	− 0.509668i
$474$	0	0
$475$	− 5.96386i	− 0.273641i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−23.5746	−1.07715	−0.538575	−	0.842578i	$-0.681037\pi$
$479$	−23.5746	−1.07715	−0.538575	+	0.842578i	$0.681037\pi$
$480$	0	0
$481$	− 21.3307i	− 0.972595i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.05218i	− 0.0477772i
$486$	0	0
$487$	3.54793	0.160772	0.0803861	−	0.996764i	$-0.474385\pi$
$487$	3.54793	0.160772	0.0803861	+	0.996764i	$0.474385\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	41.3330i	1.86533i	0.360739	+	0.932667i	$0.382525\pi$
$491$	41.3330i	1.86533i	−0.360739	+	0.932667i	$0.617475\pi$
$492$	0	0
$493$	25.8722i	1.16523i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	23.7292	1.06226	0.531132	−	0.847289i	$-0.321767\pi$
$499$	23.7292	1.06226	0.531132	+	0.847289i	$0.321767\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.4784	1.09144	0.545719	−	0.837968i	$-0.316257\pi$
$503$	24.4784	1.09144	0.545719	+	0.837968i	$0.316257\pi$
$504$	0	0
$505$	−17.4071	−0.774604
$506$	0	0
$507$	0	0
$508$	0	0
$509$	43.6140	1.93316	0.966579	−	0.256369i	$-0.0825262\pi$
$509$	43.6140	1.93316	0.966579	+	0.256369i	$0.0825262\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 12.5384i	− 0.552507i
$516$	0	0
$517$	− 24.3024i	− 1.06882i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.1104	−0.442947	−0.221473	−	0.975166i	$-0.571087\pi$
$521$	−10.1104	−0.442947	−0.221473	+	0.975166i	$0.571087\pi$
$522$	0	0
$523$	− 19.1213i	− 0.836118i	−0.908420	−	0.418059i	$-0.862711\pi$
$523$	− 19.1213i	− 0.836118i	0.908420	−	0.418059i	$-0.137289\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.94699i	0.302616i
$528$	0	0
$529$	4.94447	0.214977
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.3430i	0.534635i
$534$	0	0
$535$	4.92932i	0.213113i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16.3125	0.701331	0.350665	−	0.936501i	$-0.385955\pi$
$541$	16.3125	0.701331	0.350665	+	0.936501i	$0.385955\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.0223	0.557815
$546$	0	0
$547$	21.0498	0.900023	0.450012	−	0.893023i	$-0.351420\pi$
$547$	21.0498	0.900023	0.450012	+	0.893023i	$0.351420\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	40.1868	1.71201
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 21.7778i	− 0.922754i	−0.887204	−	0.461377i	$-0.847356\pi$
$557$	− 21.7778i	− 0.922754i	0.887204	−	0.461377i	$-0.152644\pi$
$558$	0	0
$559$	24.4685i	1.03491i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.8316	0.793656	0.396828	−	0.917893i	$-0.370111\pi$
$563$	18.8316	0.793656	0.396828	+	0.917893i	$0.370111\pi$
$564$	0	0
$565$	− 7.82363i	− 0.329143i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.9438i	1.33915i	0.742742	+	0.669577i	$0.233525\pi$
$569$	31.9438i	1.33915i	−0.742742	+	0.669577i	$0.766475\pi$
$570$	0	0
$571$	−19.7088	−0.824789	−0.412394	−	0.911005i	$-0.635307\pi$
$571$	−19.7088	−0.824789	−0.412394	+	0.911005i	$0.635307\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 4.24918i	− 0.177203i
$576$	0	0
$577$	− 15.7729i	− 0.656633i	−0.944568	−	0.328317i	$-0.893519\pi$
$577$	− 15.7729i	− 0.656633i	0.944568	−	0.328317i	$-0.106481\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.0603	0.416657
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.89811	0.243441	0.121721	−	0.992564i	$-0.461159\pi$
$587$	5.89811	0.243441	0.121721	+	0.992564i	$0.461159\pi$
$588$	0	0
$589$	10.7906	0.444619
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.36373	0.0970669	0.0485335	−	0.998822i	$-0.484545\pi$
$593$	2.36373	0.0970669	0.0485335	+	0.998822i	$0.484545\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16.9286i	0.691684i	0.938293	+	0.345842i	$0.112407\pi$
$599$	16.9286i	0.691684i	−0.938293	+	0.345842i	$0.887593\pi$
$600$	0	0
$601$	20.7753i	0.847443i	0.905793	+	0.423721i	$0.139276\pi$
$601$	20.7753i	0.847443i	−0.905793	+	0.423721i	$0.860724\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.01405	0.244506
$606$	0	0
$607$	− 2.75183i	− 0.111693i	−0.998439	−	0.0558466i	$-0.982214\pi$
$607$	− 2.75183i	− 0.111693i	0.998439	−	0.0558466i	$-0.0177858\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	53.6461i	2.17029i
$612$	0	0
$613$	15.5628	0.628575	0.314287	−	0.949328i	$-0.398234\pi$
$613$	15.5628	0.628575	0.314287	+	0.949328i	$0.398234\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.50012i	0.0603925i	0.999544	+	0.0301963i	$0.00961323\pi$
$617$	1.50012i	0.0603925i	−0.999544	+	0.0301963i	$0.990387\pi$
$618$	0	0
$619$	10.4623i	0.420513i	0.977646	+	0.210257i	$0.0674300\pi$
$619$	10.4623i	0.420513i	−0.977646	+	0.210257i	$0.932570\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.6158	−0.662514
$630$	0	0
$631$	42.9480	1.70973	0.854866	−	0.518849i	$-0.173639\pi$
$631$	42.9480	1.70973	0.854866	+	0.518849i	$0.173639\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.53736	−0.259427
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	43.1330i	1.70365i	0.523826	+	0.851825i	$0.324504\pi$
$641$	43.1330i	1.70365i	−0.523826	+	0.851825i	$0.675496\pi$
$642$	0	0
$643$	− 11.6402i	− 0.459045i	−0.973303	−	0.229523i	$-0.926284\pi$
$643$	− 11.6402i	− 0.459045i	0.973303	−	0.229523i	$-0.0737165\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−37.0432	−1.45632	−0.728159	−	0.685408i	$-0.759624\pi$
$647$	−37.0432	−1.45632	−0.728159	+	0.685408i	$0.759624\pi$
$648$	0	0
$649$	8.91416i	0.349912i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 31.7308i	− 1.24172i	−0.783921	−	0.620861i	$-0.786783\pi$
$653$	− 31.7308i	− 1.24172i	0.783921	−	0.620861i	$-0.213217\pi$
$654$	0	0
$655$	−0.315665	−0.0123341
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1.01969i	− 0.0397216i	−0.999803	−	0.0198608i	$-0.993678\pi$
$659$	− 1.01969i	− 0.0397216i	0.999803	−	0.0198608i	$-0.00632231\pi$
$660$	0	0
$661$	40.9457i	1.59260i	0.604899	+	0.796302i	$0.293213\pi$
$661$	40.9457i	1.59260i	−0.604899	+	0.796302i	$0.706787\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28.6326	1.10866
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−13.9756	−0.539524
$672$	0	0
$673$	−16.2023	−0.624552	−0.312276	−	0.949992i	$-0.601091\pi$
$673$	−16.2023	−0.624552	−0.312276	+	0.949992i	$0.601091\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.48419	0.326074	0.163037	−	0.986620i	$-0.447871\pi$
$677$	8.48419	0.326074	0.163037	+	0.986620i	$0.447871\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 32.8128i	− 1.25555i	−0.778396	−	0.627773i	$-0.783967\pi$
$683$	− 32.8128i	− 1.25555i	0.778396	−	0.627773i	$-0.216033\pi$
$684$	0	0
$685$	22.2097i	0.848588i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−22.2076	−0.846042
$690$	0	0
$691$	34.5459i	1.31419i	0.753809	+	0.657094i	$0.228214\pi$
$691$	34.5459i	1.31419i	−0.753809	+	0.657094i	$0.771786\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 14.8183i	− 0.562089i
$696$	0	0
$697$	9.61472	0.364183
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14.0470i	0.530547i	0.964173	+	0.265274i	$0.0854623\pi$
$701$	14.0470i	0.530547i	−0.964173	+	0.265274i	$0.914538\pi$
$702$	0	0
$703$	25.8089i	0.973401i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−42.3730	−1.59135	−0.795676	−	0.605722i	$-0.792884\pi$
$709$	−42.3730	−1.59135	−0.795676	+	0.605722i	$0.792884\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.68818	0.287924
$714$	0	0
$715$	11.0062	0.411608
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−28.3385	−1.05685	−0.528424	−	0.848981i	$-0.677217\pi$
$719$	−28.3385	−1.05685	−0.528424	+	0.848981i	$0.677217\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.73838i	0.250257i
$726$	0	0
$727$	− 14.2471i	− 0.528395i	−0.964469	−	0.264197i	$-0.914893\pi$
$727$	− 14.2471i	− 0.528395i	0.964469	−	0.264197i	$-0.0851070\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	19.0600	0.704959
$732$	0	0
$733$	− 28.7599i	− 1.06227i	−0.847287	−	0.531136i	$-0.821766\pi$
$733$	− 28.7599i	− 1.06227i	0.847287	−	0.531136i	$-0.178234\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.9713i	0.919828i
$738$	0	0
$739$	21.3948	0.787022	0.393511	−	0.919320i	$-0.371260\pi$
$739$	21.3948	0.787022	0.393511	+	0.919320i	$0.371260\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 11.2869i	− 0.414077i	−0.978333	−	0.207039i	$-0.933617\pi$
$743$	− 11.2869i	− 0.414077i	0.978333	−	0.207039i	$-0.0663826\pi$
$744$	0	0
$745$	4.10602i	0.150433i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.85438	−0.0676672	−0.0338336	−	0.999427i	$-0.510772\pi$
$751$	−1.85438	−0.0676672	−0.0338336	+	0.999427i	$0.510772\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.5675	0.420986
$756$	0	0
$757$	25.3114	0.919958	0.459979	−	0.887930i	$-0.347857\pi$
$757$	25.3114	0.919958	0.459979	+	0.887930i	$0.347857\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.45702	−0.234067	−0.117033	−	0.993128i	$-0.537338\pi$
$761$	−6.45702	−0.234067	−0.117033	+	0.993128i	$0.537338\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 19.6775i	− 0.710513i
$768$	0	0
$769$	28.0248i	1.01060i	0.862943	+	0.505301i	$0.168618\pi$
$769$	28.0248i	1.01060i	−0.862943	+	0.505301i	$0.831382\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−11.2519	−0.404703	−0.202352	−	0.979313i	$-0.564858\pi$
$773$	−11.2519	−0.404703	−0.202352	+	0.979313i	$0.564858\pi$
$774$	0	0
$775$	1.80933i	0.0649931i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 14.9343i	− 0.535078i
$780$	0	0
$781$	−19.8209	−0.709248
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.92814i	0.0688184i
$786$	0	0
$787$	− 45.7781i	− 1.63181i	−0.578185	−	0.815906i	$-0.696239\pi$
$787$	− 45.7781i	− 1.63181i	0.578185	−	0.815906i	$-0.303761\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	30.8504	1.09553
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.50164	−0.194878	−0.0974390	−	0.995241i	$-0.531065\pi$
$797$	−5.50164	−0.194878	−0.0974390	+	0.995241i	$0.531065\pi$
$798$	0	0
$799$	41.7882	1.47836
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.2029	0.395343
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 28.9201i	− 1.01678i	−0.861128	−	0.508389i	$-0.830241\pi$
$809$	− 28.9201i	− 1.01678i	0.861128	−	0.508389i	$-0.169759\pi$
$810$	0	0
$811$	− 39.1672i	− 1.37535i	−0.726021	−	0.687673i	$-0.758632\pi$
$811$	− 39.1672i	− 1.37535i	0.726021	−	0.687673i	$-0.241368\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.3345	0.572173
$816$	0	0
$817$	− 29.6055i	− 1.03576i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 19.5583i	− 0.682589i	−0.939956	−	0.341295i	$-0.889135\pi$
$821$	− 19.5583i	− 0.682589i	0.939956	−	0.341295i	$-0.110865\pi$
$822$	0	0
$823$	51.2087	1.78502	0.892511	−	0.451026i	$-0.148942\pi$
$823$	51.2087	1.78502	0.892511	+	0.451026i	$0.148942\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 21.2391i	− 0.738555i	−0.929319	−	0.369277i	$-0.879605\pi$
$827$	− 21.2391i	− 0.738555i	0.929319	−	0.369277i	$-0.120395\pi$
$828$	0	0
$829$	22.5901i	0.784585i	0.919841	+	0.392292i	$0.128318\pi$
$829$	22.5901i	0.784585i	−0.919841	+	0.392292i	$0.871682\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.2112	0.561012
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7.06207	0.243810	0.121905	−	0.992542i	$-0.461100\pi$
$839$	7.06207	0.243810	0.121905	+	0.992542i	$0.461100\pi$
$840$	0	0
$841$	−16.4058	−0.565716
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.2955	−0.388577
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.3885i	0.630350i
$852$	0	0
$853$	32.2255i	1.10338i	0.834050	+	0.551689i	$0.186017\pi$
$853$	32.2255i	1.10338i	−0.834050	+	0.551689i	$0.813983\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.4202	0.560902	0.280451	−	0.959868i	$-0.409516\pi$
$857$	16.4202	0.560902	0.280451	+	0.959868i	$0.409516\pi$
$858$	0	0
$859$	− 28.1679i	− 0.961076i	−0.876974	−	0.480538i	$-0.840441\pi$
$859$	− 28.1679i	− 0.961076i	0.876974	−	0.480538i	$-0.159559\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34.1063i	1.16099i	0.814264	+	0.580495i	$0.197141\pi$
$863$	34.1063i	1.16099i	−0.814264	+	0.580495i	$0.802859\pi$
$864$	0	0
$865$	−21.5331	−0.732146
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 25.0180i	− 0.848677i
$870$	0	0
$871$	− 55.1225i	− 1.86776i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.7964	−1.17499	−0.587495	−	0.809227i	$-0.699886\pi$
$877$	−34.7964	−1.17499	−0.587495	+	0.809227i	$0.699886\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−39.6838	−1.33698	−0.668491	−	0.743721i	$-0.733059\pi$
$881$	−39.6838	−1.33698	−0.668491	+	0.743721i	$0.733059\pi$
$882$	0	0
$883$	4.50323	0.151546	0.0757728	−	0.997125i	$-0.475858\pi$
$883$	4.50323	0.151546	0.0757728	+	0.997125i	$0.475858\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.2075	0.879961	0.439981	−	0.898007i	$-0.354985\pi$
$887$	26.2075	0.879961	0.439981	+	0.898007i	$0.354985\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 64.9087i	− 2.17209i
$894$	0	0
$895$	− 21.7519i	− 0.727086i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.1920	−0.406625
$900$	0	0
$901$	17.2988i	0.576308i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 10.4316i	− 0.346759i
$906$	0	0
$907$	−45.1884	−1.50046	−0.750229	−	0.661178i	$-0.770057\pi$
$907$	−45.1884	−1.50046	−0.750229	+	0.661178i	$0.770057\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	21.4888i	0.711956i	0.934494	+	0.355978i	$0.115852\pi$
$911$	21.4888i	0.711956i	−0.934494	+	0.355978i	$0.884148\pi$
$912$	0	0
$913$	− 0.658919i	− 0.0218070i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−9.56919	−0.315659	−0.157829	−	0.987466i	$-0.550450\pi$
$919$	−9.56919	−0.315659	−0.157829	+	0.987466i	$0.550450\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	43.7535	1.44016
$924$	0	0
$925$	−4.32755	−0.142289
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−18.2534	−0.598874	−0.299437	−	0.954116i	$-0.596799\pi$
$929$	−18.2534	−0.598874	−0.299437	+	0.954116i	$0.596799\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 8.57339i	− 0.280380i
$936$	0	0
$937$	30.4878i	0.995992i	0.867179	+	0.497996i	$0.165931\pi$
$937$	30.4878i	0.995992i	−0.867179	+	0.497996i	$0.834069\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−51.0256	−1.66339	−0.831694	−	0.555234i	$-0.812629\pi$
$941$	−51.0256	−1.66339	−0.831694	+	0.555234i	$0.812629\pi$
$942$	0	0
$943$	− 10.6405i	− 0.346503i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 55.5343i	− 1.80462i	−0.431087	−	0.902311i	$-0.641870\pi$
$947$	− 55.5343i	− 1.80462i	0.431087	−	0.902311i	$-0.358130\pi$
$948$	0	0
$949$	−24.7298	−0.802764
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 40.0901i	− 1.29865i	−0.760512	−	0.649323i	$-0.775052\pi$
$953$	− 40.0901i	− 1.29865i	0.760512	−	0.649323i	$-0.224948\pi$
$954$	0	0
$955$	23.6063i	0.763880i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	27.7263	0.894397
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.3226	0.525444
$966$	0	0
$967$	−37.7974	−1.21548	−0.607740	−	0.794136i	$-0.707924\pi$
$967$	−37.7974	−1.21548	−0.607740	+	0.794136i	$0.707924\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.7876	1.05220	0.526102	−	0.850422i	$-0.323653\pi$
$971$	32.7876	1.05220	0.526102	+	0.850422i	$0.323653\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.4856i	0.943328i	0.881778	+	0.471664i	$0.156346\pi$
$977$	29.4856i	0.943328i	−0.881778	+	0.471664i	$0.843654\pi$
$978$	0	0
$979$	− 31.5647i	− 1.00881i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	38.3639	1.22362	0.611810	−	0.791005i	$-0.290442\pi$
$983$	38.3639	1.22362	0.611810	+	0.791005i	$0.290442\pi$
$984$	0	0
$985$	− 7.03368i	− 0.224112i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 21.0935i	− 0.670735i
$990$	0	0
$991$	9.63794	0.306159	0.153080	−	0.988214i	$-0.451081\pi$
$991$	9.63794	0.306159	0.153080	+	0.988214i	$0.451081\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 6.43512i	− 0.204007i
$996$	0	0
$997$	− 8.07296i	− 0.255673i	−0.991795	−	0.127837i	$-0.959197\pi$
$997$	− 8.07296i	− 0.255673i	0.991795	−	0.127837i	$-0.0408033\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.d.b.881.5		12
3.2	odd	2		8820.2.d.a.881.8		12
7.4	even	3		1260.2.cg.a.341.5	✓	12
7.5	odd	6		1260.2.cg.b.521.5	yes	12
7.6	odd	2		8820.2.d.a.881.5		12
21.5	even	6		1260.2.cg.a.521.5	yes	12
21.11	odd	6		1260.2.cg.b.341.5	yes	12
21.20	even	2	inner	8820.2.d.b.881.8		12
35.4	even	6		6300.2.ch.c.1601.2		12
35.12	even	12		6300.2.dd.b.4049.3		24
35.18	odd	12		6300.2.dd.c.1349.3		24
35.19	odd	6		6300.2.ch.b.4301.2		12
35.32	odd	12		6300.2.dd.c.1349.10		24
35.33	even	12		6300.2.dd.b.4049.10		24
105.32	even	12		6300.2.dd.b.1349.10		24
105.47	odd	12		6300.2.dd.c.4049.3		24
105.53	even	12		6300.2.dd.b.1349.3		24
105.68	odd	12		6300.2.dd.c.4049.10		24
105.74	odd	6		6300.2.ch.b.1601.2		12
105.89	even	6		6300.2.ch.c.4301.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.cg.a.341.5	✓	12	7.4	even	3
1260.2.cg.a.521.5	yes	12	21.5	even	6
1260.2.cg.b.341.5	yes	12	21.11	odd	6
1260.2.cg.b.521.5	yes	12	7.5	odd	6
6300.2.ch.b.1601.2		12	105.74	odd	6
6300.2.ch.b.4301.2		12	35.19	odd	6
6300.2.ch.c.1601.2		12	35.4	even	6
6300.2.ch.c.4301.2		12	105.89	even	6
6300.2.dd.b.1349.3		24	105.53	even	12
6300.2.dd.b.1349.10		24	105.32	even	12
6300.2.dd.b.4049.3		24	35.12	even	12
6300.2.dd.b.4049.10		24	35.33	even	12
6300.2.dd.c.1349.3		24	35.18	odd	12
6300.2.dd.c.1349.10		24	35.32	odd	12
6300.2.dd.c.4049.3		24	105.47	odd	12
6300.2.dd.c.4049.10		24	105.68	odd	12
8820.2.d.a.881.5		12	7.6	odd	2
8820.2.d.a.881.8		12	3.2	odd	2
8820.2.d.b.881.5		12	1.1	even	1	trivial
8820.2.d.b.881.8		12	21.20	even	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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +O(q^{10})\)	\(q+1.00000 q^{5} -2.23292i q^{11} +4.92905i q^{13} +3.83953 q^{17} -5.96386i q^{19} -4.24918i q^{23} +1.00000 q^{25} +6.73838i q^{29} +1.80933i q^{31} -4.32755 q^{37} +2.50414 q^{41} +4.96414 q^{43} +10.8837 q^{47} +4.50545i q^{53} -2.23292i q^{55} -3.99215 q^{59} -6.25890i q^{61} +4.92905i q^{65} -11.1832 q^{67} -8.87665i q^{71} +5.01716i q^{73} +11.2041 q^{79} +0.295092 q^{83} +3.83953 q^{85} +14.1360 q^{89} -5.96386i q^{95} -1.05218i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{5}+O(q^{10}) \) 12 * q + 12 * q^5	\( 12 q + 12 q^{5} + 12 q^{25} + 4 q^{37} - 8 q^{41} + 36 q^{43} + 32 q^{47} - 4 q^{67} - 28 q^{79} + 40 q^{83} + 40 q^{89}+O(q^{100}) \) 12 * q + 12 * q^5 + 12 * q^25 + 4 * q^37 - 8 * q^41 + 36 * q^43 + 32 * q^47 - 4 * q^67 - 28 * q^79 + 40 * q^83 + 40 * q^89

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