LMFDB - Embedded newform 8820.2.d.b.881.10

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			881.10
Root			$-2.61674 - 0.390758i$ of defining polynomial
Character	$\chi$	$=$	8820.881
Dual form			8820.2.d.b.881.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +O(q^{10})$	$q+1.00000 q^{5} +2.51357i q^{11} -2.11077i q^{13} +4.49464 q^{17} -5.65308i q^{19} +7.98438i q^{23} +1.00000 q^{25} +4.97264i q^{29} +7.08341i q^{31} -9.35170 q^{37} -6.23347 q^{41} +9.47185 q^{43} -10.4442 q^{47} +7.48621i q^{53} +2.51357i q^{55} +0.376420 q^{59} +9.71345i q^{61} -2.11077i q^{65} -3.75582 q^{67} -9.01312i q^{71} -4.92308i q^{73} -15.9443 q^{79} +2.29871 q^{83} +4.49464 q^{85} +17.7086 q^{89} -5.65308i q^{95} +4.14156i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	2.51357i	0.757869i	0.925423	+	0.378935i	$0.123709\pi$
$11$	2.51357i	0.757869i	−0.925423	+	0.378935i	$0.876291\pi$
$12$	0	0
$13$	− 2.11077i	− 0.585422i	−0.956201	−	0.292711i	$-0.905443\pi$
$13$	− 2.11077i	− 0.585422i	0.956201	−	0.292711i	$-0.0945574\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.49464	1.09011	0.545056	−	0.838400i	$-0.316509\pi$
$17$	4.49464	1.09011	0.545056	+	0.838400i	$0.316509\pi$
$18$	0	0
$19$	− 5.65308i	− 1.29691i	−0.761255	−	0.648453i	$-0.775416\pi$
$19$	− 5.65308i	− 1.29691i	0.761255	−	0.648453i	$-0.224584\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.98438i	1.66486i	0.554131	+	0.832429i	$0.313051\pi$
$23$	7.98438i	1.66486i	−0.554131	+	0.832429i	$0.686949\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.97264i	0.923396i	0.887037	+	0.461698i	$0.152760\pi$
$29$	4.97264i	0.923396i	−0.887037	+	0.461698i	$0.847240\pi$
$30$	0	0
$31$	7.08341i	1.27222i	0.771599	+	0.636109i	$0.219457\pi$
$31$	7.08341i	1.27222i	−0.771599	+	0.636109i	$0.780543\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.35170	−1.53741	−0.768705	−	0.639604i	$-0.779098\pi$
$37$	−9.35170	−1.53741	−0.768705	+	0.639604i	$0.779098\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.23347	−0.973505	−0.486752	−	0.873540i	$-0.661819\pi$
$41$	−6.23347	−0.973505	−0.486752	+	0.873540i	$0.661819\pi$
$42$	0	0
$43$	9.47185	1.44444	0.722222	−	0.691661i	$-0.243121\pi$
$43$	9.47185	1.44444	0.722222	+	0.691661i	$0.243121\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.4442	−1.52344	−0.761718	−	0.647909i	$-0.775644\pi$
$47$	−10.4442	−1.52344	−0.761718	+	0.647909i	$0.775644\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.48621i	1.02831i	0.857697	+	0.514155i	$0.171894\pi$
$53$	7.48621i	1.02831i	−0.857697	+	0.514155i	$0.828106\pi$
$54$	0	0
$55$	2.51357i	0.338929i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.376420	0.0490057	0.0245028	−	0.999700i	$-0.492200\pi$
$59$	0.376420	0.0490057	0.0245028	+	0.999700i	$0.492200\pi$
$60$	0	0
$61$	9.71345i	1.24368i	0.783144	+	0.621840i	$0.213615\pi$
$61$	9.71345i	1.24368i	−0.783144	+	0.621840i	$0.786385\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.11077i	− 0.261809i
$66$	0	0
$67$	−3.75582	−0.458846	−0.229423	−	0.973327i	$-0.573684\pi$
$67$	−3.75582	−0.458846	−0.229423	+	0.973327i	$0.573684\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 9.01312i	− 1.06966i	−0.844959	−	0.534831i	$-0.820375\pi$
$71$	− 9.01312i	− 1.06966i	0.844959	−	0.534831i	$-0.179625\pi$
$72$	0	0
$73$	− 4.92308i	− 0.576203i	−0.957600	−	0.288101i	$-0.906976\pi$
$73$	− 4.92308i	− 0.576203i	0.957600	−	0.288101i	$-0.0930240\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−15.9443	−1.79388	−0.896940	−	0.442153i	$-0.854215\pi$
$79$	−15.9443	−1.79388	−0.896940	+	0.442153i	$0.854215\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.29871	0.252316	0.126158	−	0.992010i	$-0.459735\pi$
$83$	2.29871	0.252316	0.126158	+	0.992010i	$0.459735\pi$
$84$	0	0
$85$	4.49464	0.487513
$86$	0	0
$87$	0	0
$88$	0	0
$89$	17.7086	1.87710	0.938551	−	0.345140i	$-0.112168\pi$
$89$	17.7086	1.87710	0.938551	+	0.345140i	$0.112168\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 5.65308i	− 0.579994i
$96$	0	0
$97$	4.14156i	0.420512i	0.977646	+	0.210256i	$0.0674297\pi$
$97$	4.14156i	0.420512i	−0.977646	+	0.210256i	$0.932570\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.56986	0.852733	0.426366	−	0.904551i	$-0.359793\pi$
$101$	8.56986	0.852733	0.426366	+	0.904551i	$0.359793\pi$
$102$	0	0
$103$	12.0791i	1.19019i	0.803655	+	0.595095i	$0.202886\pi$
$103$	12.0791i	1.19019i	−0.803655	+	0.595095i	$0.797114\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.493607i	0.0477188i	0.999715	+	0.0238594i	$0.00759540\pi$
$107$	0.493607i	0.0477188i	−0.999715	+	0.0238594i	$0.992405\pi$
$108$	0	0
$109$	−5.78497	−0.554099	−0.277050	−	0.960856i	$-0.589357\pi$
$109$	−5.78497	−0.554099	−0.277050	+	0.960856i	$0.589357\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 17.0862i	− 1.60733i	−0.595081	−	0.803666i	$-0.702880\pi$
$113$	− 17.0862i	− 1.60733i	0.595081	−	0.803666i	$-0.297120\pi$
$114$	0	0
$115$	7.98438i	0.744547i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	4.68198	0.425634
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.65596	0.501885	0.250943	−	0.968002i	$-0.419259\pi$
$127$	5.65596	0.501885	0.250943	+	0.968002i	$0.419259\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.10944	−0.0969324	−0.0484662	−	0.998825i	$-0.515433\pi$
$131$	−1.10944	−0.0969324	−0.0484662	+	0.998825i	$0.515433\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.0920200i	0.00786180i	0.999992	+	0.00393090i	$0.00125125\pi$
$137$	0.0920200i	0.00786180i	−0.999992	+	0.00393090i	$0.998749\pi$
$138$	0	0
$139$	9.75255i	0.827200i	0.910459	+	0.413600i	$0.135729\pi$
$139$	9.75255i	0.827200i	−0.910459	+	0.413600i	$0.864271\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.30556	0.443673
$144$	0	0
$145$	4.97264i	0.412955i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.8953i	1.13835i	0.822217	+	0.569175i	$0.192737\pi$
$149$	13.8953i	1.13835i	−0.822217	+	0.569175i	$0.807263\pi$
$150$	0	0
$151$	−15.0645	−1.22593	−0.612966	−	0.790109i	$-0.710024\pi$
$151$	−15.0645	−1.22593	−0.612966	+	0.790109i	$0.710024\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.08341i	0.568953i
$156$	0	0
$157$	− 2.65850i	− 0.212172i	−0.994357	−	0.106086i	$-0.966168\pi$
$157$	− 2.65850i	− 0.212172i	0.994357	−	0.106086i	$-0.0338318\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.18178	0.719172	0.359586	−	0.933112i	$-0.382918\pi$
$163$	9.18178	0.719172	0.359586	+	0.933112i	$0.382918\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.0924548	−0.00715437	−0.00357718	−	0.999994i	$-0.501139\pi$
$167$	−0.0924548	−0.00715437	−0.00357718	+	0.999994i	$0.501139\pi$
$168$	0	0
$169$	8.54466	0.657281
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.322546	−0.0245227	−0.0122614	−	0.999925i	$-0.503903\pi$
$173$	−0.322546	−0.0245227	−0.0122614	+	0.999925i	$0.503903\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.23898i	0.391580i	0.980646	+	0.195790i	$0.0627271\pi$
$179$	5.23898i	0.391580i	−0.980646	+	0.195790i	$0.937273\pi$
$180$	0	0
$181$	− 20.3013i	− 1.50898i	−0.656310	−	0.754491i	$-0.727884\pi$
$181$	− 20.3013i	− 1.50898i	0.656310	−	0.754491i	$-0.272116\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.35170	−0.687550
$186$	0	0
$187$	11.2976i	0.826162i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.60874i	0.622906i	0.950261	+	0.311453i	$0.100816\pi$
$191$	8.60874i	0.622906i	−0.950261	+	0.311453i	$0.899184\pi$
$192$	0	0
$193$	4.93953	0.355555	0.177777	−	0.984071i	$-0.443109\pi$
$193$	4.93953	0.355555	0.177777	+	0.984071i	$0.443109\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.78952i	0.554980i	0.960729	+	0.277490i	$0.0895025\pi$
$197$	7.78952i	0.554980i	−0.960729	+	0.277490i	$0.910497\pi$
$198$	0	0
$199$	15.3381i	1.08729i	0.839316	+	0.543644i	$0.182956\pi$
$199$	15.3381i	1.08729i	−0.839316	+	0.543644i	$0.817044\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.23347	−0.435365
$206$	0	0
$207$	0	0
$208$	0	0
$209$	14.2094	0.982885
$210$	0	0
$211$	−17.4987	−1.20466	−0.602329	−	0.798248i	$-0.705761\pi$
$211$	−17.4987	−1.20466	−0.602329	+	0.798248i	$0.705761\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.47185	0.645975
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 9.48715i	− 0.638175i
$222$	0	0
$223$	22.4665i	1.50447i	0.658897	+	0.752234i	$0.271023\pi$
$223$	22.4665i	1.50447i	−0.658897	+	0.752234i	$0.728977\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.20190	−0.478007	−0.239003	−	0.971019i	$-0.576821\pi$
$227$	−7.20190	−0.478007	−0.239003	+	0.971019i	$0.576821\pi$
$228$	0	0
$229$	13.7806i	0.910645i	0.890327	+	0.455323i	$0.150476\pi$
$229$	13.7806i	0.910645i	−0.890327	+	0.455323i	$0.849524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 0.0572919i	− 0.00375332i	−0.999998	−	0.00187666i	$-0.999403\pi$
$233$	− 0.0572919i	− 0.00375332i	0.999998	−	0.00187666i	$-0.000597359\pi$
$234$	0	0
$235$	−10.4442	−0.681301
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.9150i	1.61162i	0.592177	+	0.805808i	$0.298269\pi$
$239$	24.9150i	1.61162i	−0.592177	+	0.805808i	$0.701731\pi$
$240$	0	0
$241$	− 21.5658i	− 1.38917i	−0.719409	−	0.694586i	$-0.755587\pi$
$241$	− 21.5658i	− 1.38917i	0.719409	−	0.694586i	$-0.244413\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11.9323	−0.759237
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.6047	0.984962	0.492481	−	0.870323i	$-0.336090\pi$
$251$	15.6047	0.984962	0.492481	+	0.870323i	$0.336090\pi$
$252$	0	0
$253$	−20.0693	−1.26174
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.8186	1.79766	0.898829	−	0.438300i	$-0.144419\pi$
$257$	28.8186	1.79766	0.898829	+	0.438300i	$0.144419\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	31.5341i	1.94448i	0.233994	+	0.972238i	$0.424820\pi$
$263$	31.5341i	1.94448i	−0.233994	+	0.972238i	$0.575180\pi$
$264$	0	0
$265$	7.48621i	0.459874i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−31.6359	−1.92888	−0.964438	−	0.264310i	$-0.914856\pi$
$269$	−31.6359	−1.92888	−0.964438	+	0.264310i	$0.914856\pi$
$270$	0	0
$271$	6.12729i	0.372206i	0.982530	+	0.186103i	$0.0595858\pi$
$271$	6.12729i	0.372206i	−0.982530	+	0.186103i	$0.940414\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.51357i	0.151574i
$276$	0	0
$277$	−16.7027	−1.00357	−0.501785	−	0.864992i	$-0.667323\pi$
$277$	−16.7027	−1.00357	−0.501785	+	0.864992i	$0.667323\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 6.24284i	− 0.372416i	−0.982510	−	0.186208i	$-0.940380\pi$
$281$	− 6.24284i	− 0.372416i	0.982510	−	0.186208i	$-0.0596199\pi$
$282$	0	0
$283$	− 4.27390i	− 0.254057i	−0.991899	−	0.127028i	$-0.959456\pi$
$283$	− 4.27390i	− 0.254057i	0.991899	−	0.127028i	$-0.0405440\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.20183	0.188343
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−19.5954	−1.14478	−0.572388	−	0.819983i	$-0.693983\pi$
$293$	−19.5954	−1.14478	−0.572388	+	0.819983i	$0.693983\pi$
$294$	0	0
$295$	0.376420	0.0219160
$296$	0	0
$297$	0	0
$298$	0	0
$299$	16.8532	0.974644
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.71345i	0.556191i
$306$	0	0
$307$	− 7.74134i	− 0.441822i	−0.975294	−	0.220911i	$-0.929097\pi$
$307$	− 7.74134i	− 0.441822i	0.975294	−	0.220911i	$-0.0709030\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.2244	0.806594	0.403297	−	0.915069i	$-0.367864\pi$
$311$	14.2244	0.806594	0.403297	+	0.915069i	$0.367864\pi$
$312$	0	0
$313$	23.0156i	1.30092i	0.759541	+	0.650460i	$0.225424\pi$
$313$	23.0156i	1.30092i	−0.759541	+	0.650460i	$0.774576\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.72932i	0.209460i	0.994501	+	0.104730i	$0.0333978\pi$
$317$	3.72932i	0.209460i	−0.994501	+	0.104730i	$0.966602\pi$
$318$	0	0
$319$	−12.4991	−0.699814
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 25.4086i	− 1.41377i
$324$	0	0
$325$	− 2.11077i	− 0.117084i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−30.5661	−1.68006	−0.840032	−	0.542537i	$-0.817464\pi$
$331$	−30.5661	−1.68006	−0.840032	+	0.542537i	$0.817464\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.75582	−0.205202
$336$	0	0
$337$	−4.06846	−0.221623	−0.110812	−	0.993841i	$-0.535345\pi$
$337$	−4.06846	−0.221623	−0.110812	+	0.993841i	$0.535345\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−17.8046	−0.964175
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.6590i	1.16272i	0.813648	+	0.581358i	$0.197478\pi$
$347$	21.6590i	1.16272i	−0.813648	+	0.581358i	$0.802522\pi$
$348$	0	0
$349$	− 3.23900i	− 0.173380i	−0.996235	−	0.0866900i	$-0.972371\pi$
$349$	− 3.23900i	− 0.173380i	0.996235	−	0.0866900i	$-0.0276290\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.95513	−0.263735	−0.131868	−	0.991267i	$-0.542097\pi$
$353$	−4.95513	−0.263735	−0.131868	+	0.991267i	$0.542097\pi$
$354$	0	0
$355$	− 9.01312i	− 0.478367i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 23.4123i	− 1.23566i	−0.786313	−	0.617828i	$-0.788013\pi$
$359$	− 23.4123i	− 1.23566i	0.786313	−	0.617828i	$-0.211987\pi$
$360$	0	0
$361$	−12.9573	−0.681964
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 4.92308i	− 0.257686i
$366$	0	0
$367$	− 14.0977i	− 0.735896i	−0.929846	−	0.367948i	$-0.880061\pi$
$367$	− 14.0977i	− 0.735896i	0.929846	−	0.367948i	$-0.119939\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−22.7643	−1.17869	−0.589345	−	0.807882i	$-0.700614\pi$
$373$	−22.7643	−1.17869	−0.589345	+	0.807882i	$0.700614\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.4961	0.540576
$378$	0	0
$379$	23.4861	1.20640	0.603200	−	0.797590i	$-0.293892\pi$
$379$	23.4861	1.20640	0.603200	+	0.797590i	$0.293892\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.5996	1.25698	0.628490	−	0.777818i	$-0.283673\pi$
$383$	24.5996	1.25698	0.628490	+	0.777818i	$0.283673\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 23.5993i	− 1.19653i	−0.801297	−	0.598267i	$-0.795856\pi$
$389$	− 23.5993i	− 1.19653i	0.801297	−	0.598267i	$-0.204144\pi$
$390$	0	0
$391$	35.8869i	1.81488i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.9443	−0.802247
$396$	0	0
$397$	21.0624i	1.05709i	0.848905	+	0.528545i	$0.177262\pi$
$397$	21.0624i	1.05709i	−0.848905	+	0.528545i	$0.822738\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.7091i	0.684601i	0.939591	+	0.342300i	$0.111206\pi$
$401$	13.7091i	0.684601i	−0.939591	+	0.342300i	$0.888794\pi$
$402$	0	0
$403$	14.9514	0.744784
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 23.5061i	− 1.16516i
$408$	0	0
$409$	13.7044i	0.677638i	0.940852	+	0.338819i	$0.110027\pi$
$409$	13.7044i	0.677638i	−0.940852	+	0.338819i	$0.889973\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.29871	0.112839
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.10776	−0.102971	−0.0514855	−	0.998674i	$-0.516396\pi$
$419$	−2.10776	−0.102971	−0.0514855	+	0.998674i	$0.516396\pi$
$420$	0	0
$421$	12.3265	0.600756	0.300378	−	0.953820i	$-0.402887\pi$
$421$	12.3265	0.600756	0.300378	+	0.953820i	$0.402887\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.49464	0.218022
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 3.26225i	− 0.157137i	−0.996909	−	0.0785685i	$-0.974965\pi$
$431$	− 3.26225i	− 0.157137i	0.996909	−	0.0785685i	$-0.0250349\pi$
$432$	0	0
$433$	19.4147i	0.933013i	0.884518	+	0.466506i	$0.154488\pi$
$433$	19.4147i	0.933013i	−0.884518	+	0.466506i	$0.845512\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	45.1363	2.15916
$438$	0	0
$439$	− 5.50383i	− 0.262683i	−0.991337	−	0.131342i	$-0.958071\pi$
$439$	− 5.50383i	− 0.262683i	0.991337	−	0.131342i	$-0.0419285\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.7064i	1.26886i	0.772981	+	0.634430i	$0.218765\pi$
$443$	26.7064i	1.26886i	−0.772981	+	0.634430i	$0.781235\pi$
$444$	0	0
$445$	17.7086	0.839466
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 2.96668i	− 0.140006i	−0.997547	−	0.0700031i	$-0.977699\pi$
$449$	− 2.96668i	− 0.140006i	0.997547	−	0.0700031i	$-0.0223009\pi$
$450$	0	0
$451$	− 15.6683i	− 0.737789i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.7057	1.29602	0.648010	−	0.761632i	$-0.275601\pi$
$457$	27.7057	1.29602	0.648010	+	0.761632i	$0.275601\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	39.3720	1.83374	0.916868	−	0.399190i	$-0.130709\pi$
$461$	39.3720	1.83374	0.916868	+	0.399190i	$0.130709\pi$
$462$	0	0
$463$	9.64212	0.448107	0.224054	−	0.974577i	$-0.428071\pi$
$463$	9.64212	0.448107	0.224054	+	0.974577i	$0.428071\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.598126	−0.0276780	−0.0138390	−	0.999904i	$-0.504405\pi$
$467$	−0.598126	−0.0276780	−0.0138390	+	0.999904i	$0.504405\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	23.8081i	1.09470i
$474$	0	0
$475$	− 5.65308i	− 0.259381i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18.5669	0.848343	0.424171	−	0.905582i	$-0.360565\pi$
$479$	18.5669	0.848343	0.424171	+	0.905582i	$0.360565\pi$
$480$	0	0
$481$	19.7393i	0.900033i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.14156i	0.188059i
$486$	0	0
$487$	0.883324	0.0400272	0.0200136	−	0.999800i	$-0.493629\pi$
$487$	0.883324	0.0400272	0.0200136	+	0.999800i	$0.493629\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.63816i	0.299576i	0.988718	+	0.149788i	$0.0478591\pi$
$491$	6.63816i	0.299576i	−0.988718	+	0.149788i	$0.952141\pi$
$492$	0	0
$493$	22.3503i	1.00660i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−31.8780	−1.42705	−0.713527	−	0.700627i	$-0.752904\pi$
$499$	−31.8780	−1.42705	−0.713527	+	0.700627i	$0.752904\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	25.2668	1.12659	0.563294	−	0.826256i	$-0.309534\pi$
$503$	25.2668	1.12659	0.563294	+	0.826256i	$0.309534\pi$
$504$	0	0
$505$	8.56986	0.381354
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.6705	0.827554	0.413777	−	0.910378i	$-0.364209\pi$
$509$	18.6705	0.827554	0.413777	+	0.910378i	$0.364209\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.0791i	0.532269i
$516$	0	0
$517$	− 26.2521i	− 1.15457i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.72658	0.338508	0.169254	−	0.985572i	$-0.445864\pi$
$521$	7.72658	0.338508	0.169254	+	0.985572i	$0.445864\pi$
$522$	0	0
$523$	13.8777i	0.606830i	0.952859	+	0.303415i	$0.0981268\pi$
$523$	13.8777i	0.606830i	−0.952859	+	0.303415i	$0.901873\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31.8374i	1.38686i
$528$	0	0
$529$	−40.7503	−1.77175
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13.1574i	0.569911i
$534$	0	0
$535$	0.493607i	0.0213405i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−34.1678	−1.46899	−0.734495	−	0.678614i	$-0.762581\pi$
$541$	−34.1678	−1.46899	−0.734495	+	0.678614i	$0.762581\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.78497	−0.247801
$546$	0	0
$547$	−17.3564	−0.742105	−0.371052	−	0.928612i	$-0.621003\pi$
$547$	−17.3564	−0.742105	−0.371052	+	0.928612i	$0.621003\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	28.1107	1.19756
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 21.1372i	− 0.895613i	−0.894130	−	0.447807i	$-0.852205\pi$
$557$	− 21.1372i	− 0.895613i	0.894130	−	0.447807i	$-0.147795\pi$
$558$	0	0
$559$	− 19.9929i	− 0.845609i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.90359	−0.333096	−0.166548	−	0.986033i	$-0.553262\pi$
$563$	−7.90359	−0.333096	−0.166548	+	0.986033i	$0.553262\pi$
$564$	0	0
$565$	− 17.0862i	− 0.718820i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 33.3161i	− 1.39669i	−0.715764	−	0.698343i	$-0.753921\pi$
$569$	− 33.3161i	− 1.39669i	0.715764	−	0.698343i	$-0.246079\pi$
$570$	0	0
$571$	−28.4241	−1.18951	−0.594756	−	0.803906i	$-0.702751\pi$
$571$	−28.4241	−1.18951	−0.594756	+	0.803906i	$0.702751\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.98438i	0.332972i
$576$	0	0
$577$	23.9889i	0.998671i	0.866409	+	0.499335i	$0.166422\pi$
$577$	23.9889i	0.998671i	−0.866409	+	0.499335i	$0.833578\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−18.8171	−0.779324
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.3522	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$587$	29.3522	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$588$	0	0
$589$	40.0431	1.64995
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25.1757	−1.03384	−0.516922	−	0.856033i	$-0.672922\pi$
$593$	−25.1757	−1.03384	−0.516922	+	0.856033i	$0.672922\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 15.2416i	− 0.622753i	−0.950287	−	0.311377i	$-0.899210\pi$
$599$	− 15.2416i	− 0.622753i	0.950287	−	0.311377i	$-0.100790\pi$
$600$	0	0
$601$	− 35.1651i	− 1.43441i	−0.696861	−	0.717206i	$-0.745421\pi$
$601$	− 35.1651i	− 1.43441i	0.696861	−	0.717206i	$-0.254579\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.68198	0.190349
$606$	0	0
$607$	25.0440i	1.01650i	0.861208	+	0.508252i	$0.169708\pi$
$607$	25.0440i	1.01650i	−0.861208	+	0.508252i	$0.830292\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	22.0452i	0.891853i
$612$	0	0
$613$	−11.9794	−0.483843	−0.241921	−	0.970296i	$-0.577778\pi$
$613$	−11.9794	−0.483843	−0.241921	+	0.970296i	$0.577778\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 25.8763i	− 1.04174i	−0.853635	−	0.520871i	$-0.825607\pi$
$617$	− 25.8763i	− 1.04174i	0.853635	−	0.520871i	$-0.174393\pi$
$618$	0	0
$619$	1.81737i	0.0730462i	0.999333	+	0.0365231i	$0.0116282\pi$
$619$	1.81737i	0.0730462i	−0.999333	+	0.0365231i	$0.988372\pi$
$620$	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
$629$	−42.0325	−1.67595
$630$	0	0
$631$	0.129106	0.00513962	0.00256981	−	0.999997i	$-0.499182\pi$
$631$	0.129106	0.00513962	0.00256981	+	0.999997i	$0.499182\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.65596	0.224450
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.1739i	1.31029i	0.755503	+	0.655145i	$0.227392\pi$
$641$	33.1739i	1.31029i	−0.755503	+	0.655145i	$0.772608\pi$
$642$	0	0
$643$	2.44358i	0.0963653i	0.998839	+	0.0481827i	$0.0153430\pi$
$643$	2.44358i	0.0963653i	−0.998839	+	0.0481827i	$0.984657\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.03938	−0.237433	−0.118716	−	0.992928i	$-0.537878\pi$
$647$	−6.03938	−0.237433	−0.118716	+	0.992928i	$0.537878\pi$
$648$	0	0
$649$	0.946156i	0.0371399i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.8589i	1.40327i	0.712539	+	0.701633i	$0.247545\pi$
$653$	35.8589i	1.40327i	−0.712539	+	0.701633i	$0.752455\pi$
$654$	0	0
$655$	−1.10944	−0.0433495
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 20.3494i	− 0.792702i	−0.918099	−	0.396351i	$-0.870276\pi$
$659$	− 20.3494i	− 0.792702i	0.918099	−	0.396351i	$-0.129724\pi$
$660$	0	0
$661$	7.79588i	0.303225i	0.988440	+	0.151612i	$0.0484465\pi$
$661$	7.79588i	0.303225i	−0.988440	+	0.151612i	$0.951553\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−39.7035	−1.53732
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.4154	−0.942547
$672$	0	0
$673$	16.3705	0.631035	0.315517	−	0.948920i	$-0.397822\pi$
$673$	16.3705	0.631035	0.315517	+	0.948920i	$0.397822\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.34647	0.320781	0.160390	−	0.987054i	$-0.448725\pi$
$677$	8.34647	0.320781	0.160390	+	0.987054i	$0.448725\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 10.8039i	− 0.413401i	−0.978404	−	0.206700i	$-0.933728\pi$
$683$	− 10.8039i	− 0.413401i	0.978404	−	0.206700i	$-0.0662725\pi$
$684$	0	0
$685$	0.0920200i	0.00351590i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15.8017	0.601995
$690$	0	0
$691$	4.53189i	0.172401i	0.996278	+	0.0862006i	$0.0274726\pi$
$691$	4.53189i	0.172401i	−0.996278	+	0.0862006i	$0.972527\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.75255i	0.369935i
$696$	0	0
$697$	−28.0172	−1.06123
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 33.7665i	− 1.27534i	−0.770309	−	0.637671i	$-0.779898\pi$
$701$	− 33.7665i	− 1.27534i	0.770309	−	0.637671i	$-0.220102\pi$
$702$	0	0
$703$	52.8659i	1.99387i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0.0261738	0.000982978 0	0.000491489	−	1.00000i	$-0.499844\pi$
$709$	0.0261738	0.000982978 0	0.000491489		1.00000i	$0.499844\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−56.5566	−2.11806
$714$	0	0
$715$	5.30556	0.198417
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−28.5721	−1.06556	−0.532780	−	0.846254i	$-0.678853\pi$
$719$	−28.5721	−1.06556	−0.532780	+	0.846254i	$0.678853\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.97264i	0.184679i
$726$	0	0
$727$	33.4387i	1.24017i	0.784533	+	0.620087i	$0.212903\pi$
$727$	33.4387i	1.24017i	−0.784533	+	0.620087i	$0.787097\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	42.5726	1.57460
$732$	0	0
$733$	− 20.1801i	− 0.745367i	−0.927958	−	0.372684i	$-0.878438\pi$
$733$	− 20.1801i	− 0.745367i	0.927958	−	0.372684i	$-0.121562\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 9.44050i	− 0.347745i
$738$	0	0
$739$	9.69879	0.356776	0.178388	−	0.983960i	$-0.442912\pi$
$739$	9.69879	0.356776	0.178388	+	0.983960i	$0.442912\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.3997i	1.62887i	0.580256	+	0.814434i	$0.302952\pi$
$743$	44.3997i	1.62887i	−0.580256	+	0.814434i	$0.697048\pi$
$744$	0	0
$745$	13.8953i	0.509085i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	22.3681	0.816222	0.408111	−	0.912932i	$-0.366188\pi$
$751$	22.3681	0.816222	0.408111	+	0.912932i	$0.366188\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.0645	−0.548253
$756$	0	0
$757$	50.9667	1.85242	0.926209	−	0.377011i	$-0.123048\pi$
$757$	50.9667	1.85242	0.926209	+	0.377011i	$0.123048\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−39.2656	−1.42338	−0.711688	−	0.702495i	$-0.752069\pi$
$761$	−39.2656	−1.42338	−0.711688	+	0.702495i	$0.752069\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 0.794535i	− 0.0286890i
$768$	0	0
$769$	− 3.86290i	− 0.139300i	−0.997572	−	0.0696498i	$-0.977812\pi$
$769$	− 3.86290i	− 0.139300i	0.997572	−	0.0696498i	$-0.0221882\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−48.4514	−1.74267	−0.871337	−	0.490684i	$-0.836747\pi$
$773$	−48.4514	−1.74267	−0.871337	+	0.490684i	$0.836747\pi$
$774$	0	0
$775$	7.08341i	0.254444i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	35.2383i	1.26254i
$780$	0	0
$781$	22.6551	0.810663
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2.65850i	− 0.0948860i
$786$	0	0
$787$	18.1961i	0.648619i	0.945951	+	0.324310i	$0.105132\pi$
$787$	18.1961i	0.648619i	−0.945951	+	0.324310i	$0.894868\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	20.5029	0.728078
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.3824	−1.07620	−0.538100	−	0.842881i	$-0.680858\pi$
$797$	−30.3824	−1.07620	−0.538100	+	0.842881i	$0.680858\pi$
$798$	0	0
$799$	−46.9427	−1.66071
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.3745	0.436686
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	51.7468i	1.81932i	0.415354	+	0.909660i	$0.363658\pi$
$809$	51.7468i	1.81932i	−0.415354	+	0.909660i	$0.636342\pi$
$810$	0	0
$811$	19.2500i	0.675958i	0.941154	+	0.337979i	$0.109743\pi$
$811$	19.2500i	0.675958i	−0.941154	+	0.337979i	$0.890257\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.18178	0.321624
$816$	0	0
$817$	− 53.5451i	− 1.87331i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 51.1311i	− 1.78449i	−0.451554	−	0.892244i	$-0.649130\pi$
$821$	− 51.1311i	− 1.78449i	0.451554	−	0.892244i	$-0.350870\pi$
$822$	0	0
$823$	42.1691	1.46992	0.734962	−	0.678108i	$-0.237200\pi$
$823$	42.1691	1.46992	0.734962	+	0.678108i	$0.237200\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5.16301i	− 0.179536i	−0.995963	−	0.0897678i	$-0.971388\pi$
$827$	− 5.16301i	− 0.179536i	0.995963	−	0.0897678i	$-0.0286125\pi$
$828$	0	0
$829$	− 33.8307i	− 1.17499i	−0.809229	−	0.587494i	$-0.800115\pi$
$829$	− 33.8307i	− 1.17499i	0.809229	−	0.587494i	$-0.199885\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−0.0924548	−0.00319953
$836$	0	0
$837$	0	0
$838$	0	0
$839$	28.5450	0.985483	0.492741	−	0.870176i	$-0.335995\pi$
$839$	28.5450	0.985483	0.492741	+	0.870176i	$0.335995\pi$
$840$	0	0
$841$	4.27283	0.147339
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.54466	0.293945
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 74.6675i	− 2.55957i
$852$	0	0
$853$	− 22.2041i	− 0.760252i	−0.924935	−	0.380126i	$-0.875881\pi$
$853$	− 22.2041i	− 0.760252i	0.924935	−	0.380126i	$-0.124119\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.8369	−0.575136	−0.287568	−	0.957760i	$-0.592847\pi$
$857$	−16.8369	−0.575136	−0.287568	+	0.957760i	$0.592847\pi$
$858$	0	0
$859$	48.2471i	1.64617i	0.567918	+	0.823085i	$0.307749\pi$
$859$	48.2471i	1.64617i	−0.567918	+	0.823085i	$0.692251\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14.8184i	0.504425i	0.967672	+	0.252213i	$0.0811582\pi$
$863$	14.8184i	0.504425i	−0.967672	+	0.252213i	$0.918842\pi$
$864$	0	0
$865$	−0.322546	−0.0109669
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 40.0772i	− 1.35953i
$870$	0	0
$871$	7.92766i	0.268618i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.2342	−0.413120	−0.206560	−	0.978434i	$-0.566227\pi$
$877$	−12.2342	−0.413120	−0.206560	+	0.978434i	$0.566227\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	39.7904	1.34057	0.670286	−	0.742103i	$-0.266172\pi$
$881$	39.7904	1.34057	0.670286	+	0.742103i	$0.266172\pi$
$882$	0	0
$883$	−34.7168	−1.16831	−0.584156	−	0.811641i	$-0.698574\pi$
$883$	−34.7168	−1.16831	−0.584156	+	0.811641i	$0.698574\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.944675	−0.0317191	−0.0158595	−	0.999874i	$-0.505048\pi$
$887$	−0.944675	−0.0317191	−0.0158595	+	0.999874i	$0.505048\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	59.0416i	1.97575i
$894$	0	0
$895$	5.23898i	0.175120i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−35.2233	−1.17476
$900$	0	0
$901$	33.6478i	1.12097i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 20.3013i	− 0.674837i
$906$	0	0
$907$	24.7762	0.822680	0.411340	−	0.911482i	$-0.365061\pi$
$907$	24.7762	0.822680	0.411340	+	0.911482i	$0.365061\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14.5204i	0.481083i	0.970639	+	0.240542i	$0.0773251\pi$
$911$	14.5204i	0.481083i	−0.970639	+	0.240542i	$0.922675\pi$
$912$	0	0
$913$	5.77795i	0.191222i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	6.99364	0.230699	0.115350	−	0.993325i	$-0.463201\pi$
$919$	6.99364	0.230699	0.115350	+	0.993325i	$0.463201\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−19.0246	−0.626203
$924$	0	0
$925$	−9.35170	−0.307482
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−14.7631	−0.484360	−0.242180	−	0.970231i	$-0.577863\pi$
$929$	−14.7631	−0.484360	−0.242180	+	0.970231i	$0.577863\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.2976i	0.369471i
$936$	0	0
$937$	48.8385i	1.59549i	0.602998	+	0.797743i	$0.293973\pi$
$937$	48.8385i	1.59549i	−0.602998	+	0.797743i	$0.706027\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	28.0184	0.913372	0.456686	−	0.889628i	$-0.349036\pi$
$941$	28.0184	0.913372	0.456686	+	0.889628i	$0.349036\pi$
$942$	0	0
$943$	− 49.7704i	− 1.62075i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2.56957i	− 0.0834997i	−0.999128	−	0.0417499i	$-0.986707\pi$
$947$	− 2.56957i	− 0.0834997i	0.999128	−	0.0417499i	$-0.0132933\pi$
$948$	0	0
$949$	−10.3915	−0.337322
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.9268i	0.418740i	0.977837	+	0.209370i	$0.0671413\pi$
$953$	12.9268i	0.418740i	−0.977837	+	0.209370i	$0.932859\pi$
$954$	0	0
$955$	8.60874i	0.278572i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−19.1747	−0.618539
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.93953	0.159009
$966$	0	0
$967$	−28.4481	−0.914829	−0.457414	−	0.889254i	$-0.651224\pi$
$967$	−28.4481	−0.914829	−0.457414	+	0.889254i	$0.651224\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30.9150	0.992109	0.496054	−	0.868291i	$-0.334782\pi$
$971$	30.9150	0.992109	0.496054	+	0.868291i	$0.334782\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 50.5984i	− 1.61879i	−0.587267	−	0.809393i	$-0.699796\pi$
$977$	− 50.5984i	− 1.61879i	0.587267	−	0.809393i	$-0.300204\pi$
$978$	0	0
$979$	44.5116i	1.42260i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−23.1839	−0.739453	−0.369726	−	0.929141i	$-0.620549\pi$
$983$	−23.1839	−0.739453	−0.369726	+	0.929141i	$0.620549\pi$
$984$	0	0
$985$	7.78952i	0.248195i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	75.6269i	2.40479i
$990$	0	0
$991$	−16.3189	−0.518388	−0.259194	−	0.965825i	$-0.583457\pi$
$991$	−16.3189	−0.518388	−0.259194	+	0.965825i	$0.583457\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15.3381i	0.486250i
$996$	0	0
$997$	− 55.1372i	− 1.74621i	−0.487531	−	0.873106i	$-0.662102\pi$
$997$	− 55.1372i	− 1.74621i	0.487531	−	0.873106i	$-0.337898\pi$
$998$	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.881.10		12
3.2	odd	2		8820.2.d.a.881.3		12
7.4	even	3		1260.2.cg.a.341.2	✓	12
7.5	odd	6		1260.2.cg.b.521.2	yes	12
7.6	odd	2		8820.2.d.a.881.10		12
21.5	even	6		1260.2.cg.a.521.2	yes	12
21.11	odd	6		1260.2.cg.b.341.2	yes	12
21.20	even	2	inner	8820.2.d.b.881.3		12
35.4	even	6		6300.2.ch.c.1601.5		12
35.12	even	12		6300.2.dd.b.4049.9		24
35.18	odd	12		6300.2.dd.c.1349.9		24
35.19	odd	6		6300.2.ch.b.4301.5		12
35.32	odd	12		6300.2.dd.c.1349.4		24
35.33	even	12		6300.2.dd.b.4049.4		24
105.32	even	12		6300.2.dd.b.1349.4		24
105.47	odd	12		6300.2.dd.c.4049.9		24
105.53	even	12		6300.2.dd.b.1349.9		24
105.68	odd	12		6300.2.dd.c.4049.4		24
105.74	odd	6		6300.2.ch.b.1601.5		12
105.89	even	6		6300.2.ch.c.4301.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.cg.a.341.2	✓	12	7.4	even	3
1260.2.cg.a.521.2	yes	12	21.5	even	6
1260.2.cg.b.341.2	yes	12	21.11	odd	6
1260.2.cg.b.521.2	yes	12	7.5	odd	6
6300.2.ch.b.1601.5		12	105.74	odd	6
6300.2.ch.b.4301.5		12	35.19	odd	6
6300.2.ch.c.1601.5		12	35.4	even	6
6300.2.ch.c.4301.5		12	105.89	even	6
6300.2.dd.b.1349.4		24	105.32	even	12
6300.2.dd.b.1349.9		24	105.53	even	12
6300.2.dd.b.4049.4		24	35.33	even	12
6300.2.dd.b.4049.9		24	35.12	even	12
6300.2.dd.c.1349.4		24	35.32	odd	12
6300.2.dd.c.1349.9		24	35.18	odd	12
6300.2.dd.c.4049.4		24	105.68	odd	12
6300.2.dd.c.4049.9		24	105.47	odd	12
8820.2.d.a.881.3		12	3.2	odd	2
8820.2.d.a.881.10		12	7.6	odd	2
8820.2.d.b.881.3		12	21.20	even	2	inner
8820.2.d.b.881.10		12	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +O(q^{10})\)	\(q+1.00000 q^{5} +2.51357i q^{11} -2.11077i q^{13} +4.49464 q^{17} -5.65308i q^{19} +7.98438i q^{23} +1.00000 q^{25} +4.97264i q^{29} +7.08341i q^{31} -9.35170 q^{37} -6.23347 q^{41} +9.47185 q^{43} -10.4442 q^{47} +7.48621i q^{53} +2.51357i q^{55} +0.376420 q^{59} +9.71345i q^{61} -2.11077i q^{65} -3.75582 q^{67} -9.01312i q^{71} -4.92308i q^{73} -15.9443 q^{79} +2.29871 q^{83} +4.49464 q^{85} +17.7086 q^{89} -5.65308i q^{95} +4.14156i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{5}+O(q^{10}) \) 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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