LMFDB - Embedded newform 8820.2.d.c.881.7

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1134 x^{12} - 3528 x^{11} + 9316 x^{10} - 19960 x^{9} + \cdots + 68 $ x^16 - 8x^15 + 60x^14 - 280x^13 + 1134x^12 - 3528x^11 + 9316x^10 - 19960x^9 + 35421x^8 - 51008x^7 + 58384x^6 - 51840x^5 + 34228x^4 - 15904x^3 + 4848x^2 - 864*x + 68
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.7
Root			$0.500000 - 0.199114i$ of defining polynomial
Character	$\chi$	$=$	8820.881
Dual form			8820.2.d.c.881.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +O(q^{10})$	$q-1.00000 q^{5} -0.471930i q^{11} -3.01212i q^{13} +5.79812 q^{17} +5.14928i q^{19} +1.83640i q^{23} +1.00000 q^{25} -3.94109i q^{29} -3.95583i q^{31} -2.74217 q^{37} -4.82150 q^{41} +5.86036 q^{43} +6.56645 q^{47} +2.80994i q^{53} +0.471930i q^{55} -5.98448 q^{59} -1.15420i q^{61} +3.01212i q^{65} -4.25104 q^{67} +9.68753i q^{71} -11.5895i q^{73} +11.9124 q^{79} +4.45101 q^{83} -5.79812 q^{85} -4.41508 q^{89} -5.14928i q^{95} +3.56574i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5}+O(q^{10}) $ 16 * q - 16 * q^5	$ 16 q - 16 q^{5} + 16 q^{25} - 32 q^{41} - 32 q^{43} + 32 q^{47} + 32 q^{59} - 32 q^{67} + 64 q^{89}+O(q^{100}) $ 16 * q - 16 * q^5 + 16 * q^25 - 32 * q^41 - 32 * q^43 + 32 * q^47 + 32 * q^59 - 32 * q^67 + 64 * 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
$11$	− 0.471930i	− 0.142292i	−0.997466	−	0.0711462i	$-0.977334\pi$
$11$	− 0.471930i	− 0.142292i	0.997466	−	0.0711462i	$-0.0226657\pi$
$12$	0	0
$13$	− 3.01212i	− 0.835413i	−0.908582	−	0.417706i	$-0.862834\pi$
$13$	− 3.01212i	− 0.835413i	0.908582	−	0.417706i	$-0.137166\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.79812	1.40625	0.703126	−	0.711066i	$-0.251787\pi$
$17$	5.79812	1.40625	0.703126	+	0.711066i	$0.251787\pi$
$18$	0	0
$19$	5.14928i	1.18132i	0.806919	+	0.590662i	$0.201134\pi$
$19$	5.14928i	1.18132i	−0.806919	+	0.590662i	$0.798866\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.83640i	0.382917i	0.981501	+	0.191458i	$0.0613217\pi$
$23$	1.83640i	0.382917i	−0.981501	+	0.191458i	$0.938678\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.94109i	− 0.731841i	−0.930646	−	0.365921i	$-0.880754\pi$
$29$	− 3.94109i	− 0.731841i	0.930646	−	0.365921i	$-0.119246\pi$
$30$	0	0
$31$	− 3.95583i	− 0.710487i	−0.934774	−	0.355244i	$-0.884398\pi$
$31$	− 3.95583i	− 0.710487i	0.934774	−	0.355244i	$-0.115602\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.74217	−0.450810	−0.225405	−	0.974265i	$-0.572371\pi$
$37$	−2.74217	−0.450810	−0.225405	+	0.974265i	$0.572371\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.82150	−0.752991	−0.376496	−	0.926418i	$-0.622871\pi$
$41$	−4.82150	−0.752991	−0.376496	+	0.926418i	$0.622871\pi$
$42$	0	0
$43$	5.86036	0.893696	0.446848	−	0.894610i	$-0.352546\pi$
$43$	5.86036	0.893696	0.446848	+	0.894610i	$0.352546\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.56645	0.957814	0.478907	−	0.877866i	$-0.341033\pi$
$47$	6.56645	0.957814	0.478907	+	0.877866i	$0.341033\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.80994i	0.385974i	0.981201	+	0.192987i	$0.0618176\pi$
$53$	2.80994i	0.385974i	−0.981201	+	0.192987i	$0.938182\pi$
$54$	0	0
$55$	0.471930i	0.0636351i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.98448	−0.779113	−0.389557	−	0.921003i	$-0.627372\pi$
$59$	−5.98448	−0.779113	−0.389557	+	0.921003i	$0.627372\pi$
$60$	0	0
$61$	− 1.15420i	− 0.147780i	−0.997266	−	0.0738900i	$-0.976459\pi$
$61$	− 1.15420i	− 0.147780i	0.997266	−	0.0738900i	$-0.0235414\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.01212i	0.373608i
$66$	0	0
$67$	−4.25104	−0.519347	−0.259674	−	0.965696i	$-0.583615\pi$
$67$	−4.25104	−0.519347	−0.259674	+	0.965696i	$0.583615\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.68753i	1.14970i	0.818259	+	0.574849i	$0.194939\pi$
$71$	9.68753i	1.14970i	−0.818259	+	0.574849i	$0.805061\pi$
$72$	0	0
$73$	− 11.5895i	− 1.35645i	−0.734854	−	0.678225i	$-0.762749\pi$
$73$	− 11.5895i	− 1.35645i	0.734854	−	0.678225i	$-0.237251\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.9124	1.34025	0.670127	−	0.742246i	$-0.266239\pi$
$79$	11.9124	1.34025	0.670127	+	0.742246i	$0.266239\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.45101	0.488562	0.244281	−	0.969704i	$-0.421448\pi$
$83$	4.45101	0.488562	0.244281	+	0.969704i	$0.421448\pi$
$84$	0	0
$85$	−5.79812	−0.628895
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.41508	−0.467998	−0.233999	−	0.972237i	$-0.575181\pi$
$89$	−4.41508	−0.467998	−0.233999	+	0.972237i	$0.575181\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 5.14928i	− 0.528305i
$96$	0	0
$97$	3.56574i	0.362046i	0.983479	+	0.181023i	$0.0579408\pi$
$97$	3.56574i	0.362046i	−0.983479	+	0.181023i	$0.942059\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.80713	0.577831	0.288915	−	0.957355i	$-0.406705\pi$
$101$	5.80713	0.577831	0.288915	+	0.957355i	$0.406705\pi$
$102$	0	0
$103$	− 14.1780i	− 1.39700i	−0.715609	−	0.698501i	$-0.753851\pi$
$103$	− 14.1780i	− 1.39700i	0.715609	−	0.698501i	$-0.246149\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.9076i	1.44117i	0.693367	+	0.720584i	$0.256126\pi$
$107$	14.9076i	1.44117i	−0.693367	+	0.720584i	$0.743874\pi$
$108$	0	0
$109$	−18.2664	−1.74961	−0.874803	−	0.484479i	$-0.839009\pi$
$109$	−18.2664	−1.74961	−0.874803	+	0.484479i	$0.839009\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.1398i	0.953874i	0.878937	+	0.476937i	$0.158253\pi$
$113$	10.1398i	0.953874i	−0.878937	+	0.476937i	$0.841747\pi$
$114$	0	0
$115$	− 1.83640i	− 0.171246i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.7773	0.979753
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.90227	0.346271	0.173135	−	0.984898i	$-0.444610\pi$
$127$	3.90227	0.346271	0.173135	+	0.984898i	$0.444610\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.73674	0.675962	0.337981	−	0.941153i	$-0.390256\pi$
$131$	7.73674	0.675962	0.337981	+	0.941153i	$0.390256\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 9.39720i	− 0.802857i	−0.915890	−	0.401429i	$-0.868514\pi$
$137$	− 9.39720i	− 0.802857i	0.915890	−	0.401429i	$-0.131486\pi$
$138$	0	0
$139$	− 4.97575i	− 0.422037i	−0.977482	−	0.211019i	$-0.932322\pi$
$139$	− 4.97575i	− 0.422037i	0.977482	−	0.211019i	$-0.0676781\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.42151	−0.118873
$144$	0	0
$145$	3.94109i	0.327289i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.3626i	1.09470i	0.836903	+	0.547352i	$0.184364\pi$
$149$	13.3626i	1.09470i	−0.836903	+	0.547352i	$0.815636\pi$
$150$	0	0
$151$	2.02851	0.165077	0.0825387	−	0.996588i	$-0.473697\pi$
$151$	2.02851	0.165077	0.0825387	+	0.996588i	$0.473697\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.95583i	0.317740i
$156$	0	0
$157$	− 2.10473i	− 0.167976i	−0.996467	−	0.0839878i	$-0.973234\pi$
$157$	− 2.10473i	− 0.167976i	0.996467	−	0.0839878i	$-0.0267657\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−5.10548	−0.399892	−0.199946	−	0.979807i	$-0.564077\pi$
$163$	−5.10548	−0.399892	−0.199946	+	0.979807i	$0.564077\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.9128	0.999221	0.499610	−	0.866250i	$-0.333477\pi$
$167$	12.9128	0.999221	0.499610	+	0.866250i	$0.333477\pi$
$168$	0	0
$169$	3.92712	0.302086
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.67115	0.279113	0.139556	−	0.990214i	$-0.455432\pi$
$173$	3.67115	0.279113	0.139556	+	0.990214i	$0.455432\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3.32381i	− 0.248433i	−0.992255	−	0.124217i	$-0.960358\pi$
$179$	− 3.32381i	− 0.248433i	0.992255	−	0.124217i	$-0.0396418\pi$
$180$	0	0
$181$	− 22.5441i	− 1.67569i	−0.545906	−	0.837846i	$-0.683815\pi$
$181$	− 22.5441i	− 1.67569i	0.545906	−	0.837846i	$-0.316185\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.74217	0.201609
$186$	0	0
$187$	− 2.73631i	− 0.200099i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 6.18079i	− 0.447226i	−0.974678	−	0.223613i	$-0.928215\pi$
$191$	− 6.18079i	− 0.447226i	0.974678	−	0.223613i	$-0.0717852\pi$
$192$	0	0
$193$	−11.1357	−0.801565	−0.400782	−	0.916173i	$-0.631262\pi$
$193$	−11.1357	−0.801565	−0.400782	+	0.916173i	$0.631262\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.374400i	0.0266749i	0.999911	+	0.0133374i	$0.00424557\pi$
$197$	0.374400i	0.0266749i	−0.999911	+	0.0133374i	$0.995754\pi$
$198$	0	0
$199$	− 20.2381i	− 1.43464i	−0.696744	−	0.717320i	$-0.745368\pi$
$199$	− 20.2381i	− 1.43464i	0.696744	−	0.717320i	$-0.254632\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.82150	0.336748
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.43010	0.168094
$210$	0	0
$211$	17.9254	1.23404	0.617018	−	0.786949i	$-0.288341\pi$
$211$	17.9254	1.23404	0.617018	+	0.786949i	$0.288341\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.86036	−0.399673
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 17.4647i	− 1.17480i
$222$	0	0
$223$	4.71666i	0.315851i	0.987451	+	0.157926i	$0.0504806\pi$
$223$	4.71666i	0.315851i	−0.987451	+	0.157926i	$0.949519\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.4605	1.29164	0.645820	−	0.763490i	$-0.276516\pi$
$227$	19.4605	1.29164	0.645820	+	0.763490i	$0.276516\pi$
$228$	0	0
$229$	2.22412i	0.146974i	0.997296	+	0.0734870i	$0.0234127\pi$
$229$	2.22412i	0.146974i	−0.997296	+	0.0734870i	$0.976587\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2.08879i	− 0.136841i	−0.997657	−	0.0684206i	$-0.978204\pi$
$233$	− 2.08879i	− 0.136841i	0.997657	−	0.0684206i	$-0.0217960\pi$
$234$	0	0
$235$	−6.56645	−0.428348
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 2.21852i	− 0.143504i	−0.997422	−	0.0717522i	$-0.977141\pi$
$239$	− 2.21852i	− 0.143504i	0.997422	−	0.0717522i	$-0.0228591\pi$
$240$	0	0
$241$	− 9.56825i	− 0.616345i	−0.951330	−	0.308173i	$-0.900283\pi$
$241$	− 9.56825i	− 0.616345i	0.951330	−	0.308173i	$-0.0997174\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	15.5103	0.986894
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−16.2109	−1.02322	−0.511612	−	0.859217i	$-0.670951\pi$
$251$	−16.2109	−1.02322	−0.511612	+	0.859217i	$0.670951\pi$
$252$	0	0
$253$	0.866655	0.0544861
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.675924	−0.0421630	−0.0210815	−	0.999778i	$-0.506711\pi$
$257$	−0.675924	−0.0421630	−0.0210815	+	0.999778i	$0.506711\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 9.76217i	− 0.601961i	−0.953630	−	0.300981i	$-0.902686\pi$
$263$	− 9.76217i	− 0.601961i	0.953630	−	0.300981i	$-0.0973140\pi$
$264$	0	0
$265$	− 2.80994i	− 0.172613i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27.4988	1.67663	0.838315	−	0.545187i	$-0.183541\pi$
$269$	27.4988	1.67663	0.838315	+	0.545187i	$0.183541\pi$
$270$	0	0
$271$	− 6.11725i	− 0.371597i	−0.982588	−	0.185798i	$-0.940513\pi$
$271$	− 6.11725i	− 0.371597i	0.982588	−	0.185798i	$-0.0594871\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 0.471930i	− 0.0284585i
$276$	0	0
$277$	−12.4334	−0.747050	−0.373525	−	0.927620i	$-0.621851\pi$
$277$	−12.4334	−0.747050	−0.373525	+	0.927620i	$0.621851\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 27.0142i	− 1.61153i	−0.592234	−	0.805766i	$-0.701754\pi$
$281$	− 27.0142i	− 1.61153i	0.592234	−	0.805766i	$-0.298246\pi$
$282$	0	0
$283$	− 23.2917i	− 1.38455i	−0.721636	−	0.692273i	$-0.756610\pi$
$283$	− 23.2917i	− 1.38455i	0.721636	−	0.692273i	$-0.243390\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.6182	0.977543
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.7901	1.03931	0.519654	−	0.854377i	$-0.326061\pi$
$293$	17.7901	1.03931	0.519654	+	0.854377i	$0.326061\pi$
$294$	0	0
$295$	5.98448	0.348430
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.53148	0.319894
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.15420i	0.0660892i
$306$	0	0
$307$	− 15.2079i	− 0.867959i	−0.900923	−	0.433980i	$-0.857109\pi$
$307$	− 15.2079i	− 0.867959i	0.900923	−	0.433980i	$-0.142891\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	30.9844	1.75696	0.878482	−	0.477775i	$-0.158556\pi$
$311$	30.9844	1.75696	0.878482	+	0.477775i	$0.158556\pi$
$312$	0	0
$313$	15.4504i	0.873307i	0.899630	+	0.436653i	$0.143836\pi$
$313$	15.4504i	0.873307i	−0.899630	+	0.436653i	$0.856164\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.546417i	0.0306899i	0.999882	+	0.0153449i	$0.00488463\pi$
$317$	0.546417i	0.0306899i	−0.999882	+	0.0153449i	$0.995115\pi$
$318$	0	0
$319$	−1.85992	−0.104135
$320$	0	0
$321$	0	0
$322$	0	0
$323$	29.8561i	1.66124i
$324$	0	0
$325$	− 3.01212i	− 0.167083i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	15.2168	0.836390	0.418195	−	0.908357i	$-0.362663\pi$
$331$	15.2168	0.836390	0.418195	+	0.908357i	$0.362663\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.25104	0.232259
$336$	0	0
$337$	30.0015	1.63428	0.817142	−	0.576436i	$-0.195557\pi$
$337$	30.0015	1.63428	0.817142	+	0.576436i	$0.195557\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.86687	−0.101097
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.5526i	0.942272i	0.882061	+	0.471136i	$0.156156\pi$
$347$	17.5526i	0.942272i	−0.882061	+	0.471136i	$0.843844\pi$
$348$	0	0
$349$	8.52171i	0.456157i	0.973643	+	0.228078i	$0.0732442\pi$
$349$	8.52171i	0.456157i	−0.973643	+	0.228078i	$0.926756\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.25878	0.173447	0.0867237	−	0.996232i	$-0.472360\pi$
$353$	3.25878	0.173447	0.0867237	+	0.996232i	$0.472360\pi$
$354$	0	0
$355$	− 9.68753i	− 0.514161i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 4.60306i	− 0.242940i	−0.992595	−	0.121470i	$-0.961239\pi$
$359$	− 4.60306i	− 0.242940i	0.992595	−	0.121470i	$-0.0387609\pi$
$360$	0	0
$361$	−7.51504	−0.395528
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.5895i	0.606623i
$366$	0	0
$367$	− 23.2681i	− 1.21459i	−0.794478	−	0.607293i	$-0.792255\pi$
$367$	− 23.2681i	− 1.21459i	0.794478	−	0.607293i	$-0.207745\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	33.0520	1.71137	0.855683	−	0.517500i	$-0.173137\pi$
$373$	33.0520	1.71137	0.855683	+	0.517500i	$0.173137\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11.8710	−0.611389
$378$	0	0
$379$	9.86412	0.506686	0.253343	−	0.967377i	$-0.418470\pi$
$379$	9.86412	0.506686	0.253343	+	0.967377i	$0.418470\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.1216	−0.772677	−0.386339	−	0.922357i	$-0.626260\pi$
$383$	−15.1216	−0.772677	−0.386339	+	0.922357i	$0.626260\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 15.4354i	− 0.782604i	−0.920262	−	0.391302i	$-0.872025\pi$
$389$	− 15.4354i	− 0.782604i	0.920262	−	0.391302i	$-0.127975\pi$
$390$	0	0
$391$	10.6477i	0.538477i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.9124	−0.599380
$396$	0	0
$397$	10.4566i	0.524804i	0.964959	+	0.262402i	$0.0845146\pi$
$397$	10.4566i	0.524804i	−0.964959	+	0.262402i	$0.915485\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.2265i	1.45950i	0.683714	+	0.729750i	$0.260364\pi$
$401$	29.2265i	1.45950i	−0.683714	+	0.729750i	$0.739636\pi$
$402$	0	0
$403$	−11.9154	−0.593550
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.29411i	0.0641469i
$408$	0	0
$409$	0.984426i	0.0486767i	0.999704	+	0.0243384i	$0.00774791\pi$
$409$	0.984426i	0.0486767i	−0.999704	+	0.0243384i	$0.992252\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.45101	−0.218492
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.90149	0.483719	0.241860	−	0.970311i	$-0.422243\pi$
$419$	9.90149	0.483719	0.241860	+	0.970311i	$0.422243\pi$
$420$	0	0
$421$	8.02682	0.391203	0.195602	−	0.980683i	$-0.437334\pi$
$421$	8.02682	0.391203	0.195602	+	0.980683i	$0.437334\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.79812	0.281250
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 16.3664i	− 0.788342i	−0.919037	−	0.394171i	$-0.871032\pi$
$431$	− 16.3664i	− 0.788342i	0.919037	−	0.394171i	$-0.128968\pi$
$432$	0	0
$433$	17.5302i	0.842446i	0.906957	+	0.421223i	$0.138399\pi$
$433$	17.5302i	0.842446i	−0.906957	+	0.421223i	$0.861601\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.45615	−0.452349
$438$	0	0
$439$	− 28.5095i	− 1.36069i	−0.732894	−	0.680343i	$-0.761831\pi$
$439$	− 28.5095i	− 1.36069i	0.732894	−	0.680343i	$-0.238169\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	41.0187i	1.94886i	0.224699	+	0.974428i	$0.427860\pi$
$443$	41.0187i	1.94886i	−0.224699	+	0.974428i	$0.572140\pi$
$444$	0	0
$445$	4.41508	0.209295
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.9589i	1.08350i	0.840540	+	0.541750i	$0.182238\pi$
$449$	22.9589i	1.08350i	−0.840540	+	0.541750i	$0.817762\pi$
$450$	0	0
$451$	2.27541i	0.107145i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.6669	1.48132	0.740658	−	0.671882i	$-0.234514\pi$
$457$	31.6669	1.48132	0.740658	+	0.671882i	$0.234514\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.7826	0.688492	0.344246	−	0.938879i	$-0.388135\pi$
$461$	14.7826	0.688492	0.344246	+	0.938879i	$0.388135\pi$
$462$	0	0
$463$	−30.0083	−1.39460	−0.697302	−	0.716778i	$-0.745616\pi$
$463$	−30.0083	−1.39460	−0.697302	+	0.716778i	$0.745616\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.29367	−0.106138	−0.0530691	−	0.998591i	$-0.516900\pi$
$467$	−2.29367	−0.106138	−0.0530691	+	0.998591i	$0.516900\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 2.76568i	− 0.127166i
$474$	0	0
$475$	5.14928i	0.236265i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.2521	1.01673	0.508363	−	0.861143i	$-0.330251\pi$
$479$	22.2521	1.01673	0.508363	+	0.861143i	$0.330251\pi$
$480$	0	0
$481$	8.25976i	0.376613i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 3.56574i	− 0.161912i
$486$	0	0
$487$	−0.196647	−0.00891091	−0.00445546	−	0.999990i	$-0.501418\pi$
$487$	−0.196647	−0.00891091	−0.00445546	+	0.999990i	$0.501418\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11.6099i	0.523946i	0.965075	+	0.261973i	$0.0843731\pi$
$491$	11.6099i	0.523946i	−0.965075	+	0.261973i	$0.915627\pi$
$492$	0	0
$493$	− 22.8509i	− 1.02915i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−24.2273	−1.08456	−0.542282	−	0.840197i	$-0.682440\pi$
$499$	−24.2273	−1.08456	−0.542282	+	0.840197i	$0.682440\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	25.7060	1.14617	0.573087	−	0.819494i	$-0.305746\pi$
$503$	25.7060	1.14617	0.573087	+	0.819494i	$0.305746\pi$
$504$	0	0
$505$	−5.80713	−0.258414
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.78594	0.389430	0.194715	−	0.980860i	$-0.437622\pi$
$509$	8.78594	0.389430	0.194715	+	0.980860i	$0.437622\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14.1780i	0.624758i
$516$	0	0
$517$	− 3.09891i	− 0.136290i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.90004	0.389918	0.194959	−	0.980811i	$-0.437543\pi$
$521$	8.90004	0.389918	0.194959	+	0.980811i	$0.437543\pi$
$522$	0	0
$523$	− 27.8702i	− 1.21868i	−0.792909	−	0.609340i	$-0.791435\pi$
$523$	− 27.8702i	− 1.21868i	0.792909	−	0.609340i	$-0.208565\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 22.9364i	− 0.999124i
$528$	0	0
$529$	19.6276	0.853375
$530$	0	0
$531$	0	0
$532$	0	0
$533$	14.5229i	0.629058i
$534$	0	0
$535$	− 14.9076i	− 0.644510i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	29.1194	1.25194	0.625969	−	0.779848i	$-0.284703\pi$
$541$	29.1194	1.25194	0.625969	+	0.779848i	$0.284703\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.2664	0.782447
$546$	0	0
$547$	17.9465	0.767335	0.383668	−	0.923471i	$-0.374661\pi$
$547$	17.9465	0.767335	0.383668	+	0.923471i	$0.374661\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.2937	0.864542
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 45.5133i	− 1.92846i	−0.265070	−	0.964229i	$-0.585395\pi$
$557$	− 45.5133i	− 1.92846i	0.265070	−	0.964229i	$-0.414605\pi$
$558$	0	0
$559$	− 17.6521i	− 0.746605i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.23256	−0.0940911	−0.0470456	−	0.998893i	$-0.514981\pi$
$563$	−2.23256	−0.0940911	−0.0470456	+	0.998893i	$0.514981\pi$
$564$	0	0
$565$	− 10.1398i	− 0.426586i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.0297i	0.713923i	0.934119	+	0.356962i	$0.116187\pi$
$569$	17.0297i	0.713923i	−0.934119	+	0.356962i	$0.883813\pi$
$570$	0	0
$571$	−46.1083	−1.92957	−0.964787	−	0.263033i	$-0.915277\pi$
$571$	−46.1083	−1.92957	−0.964787	+	0.263033i	$0.915277\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.83640i	0.0765834i
$576$	0	0
$577$	− 13.4906i	− 0.561621i	−0.959763	−	0.280811i	$-0.909397\pi$
$577$	− 13.4906i	− 0.561621i	0.959763	−	0.280811i	$-0.0906033\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.32609	0.0549212
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.9244	−0.615994	−0.307997	−	0.951387i	$-0.599659\pi$
$587$	−14.9244	−0.615994	−0.307997	+	0.951387i	$0.599659\pi$
$588$	0	0
$589$	20.3696	0.839316
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−39.2199	−1.61057	−0.805284	−	0.592889i	$-0.797987\pi$
$593$	−39.2199	−1.61057	−0.805284	+	0.592889i	$0.797987\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.37466i	0.0970262i	0.998823	+	0.0485131i	$0.0154483\pi$
$599$	2.37466i	0.0970262i	−0.998823	+	0.0485131i	$0.984552\pi$
$600$	0	0
$601$	− 19.7055i	− 0.803806i	−0.915682	−	0.401903i	$-0.868349\pi$
$601$	− 19.7055i	− 0.803806i	0.915682	−	0.401903i	$-0.131651\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.7773	−0.438159
$606$	0	0
$607$	− 14.3286i	− 0.581580i	−0.956787	−	0.290790i	$-0.906082\pi$
$607$	− 14.3286i	− 0.581580i	0.956787	−	0.290790i	$-0.0939180\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 19.7789i	− 0.800170i
$612$	0	0
$613$	−35.5477	−1.43576	−0.717880	−	0.696167i	$-0.754887\pi$
$613$	−35.5477	−1.43576	−0.717880	+	0.696167i	$0.754887\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 35.6434i	− 1.43495i	−0.696584	−	0.717475i	$-0.745298\pi$
$617$	− 35.6434i	− 1.43495i	0.696584	−	0.717475i	$-0.254702\pi$
$618$	0	0
$619$	− 18.2905i	− 0.735158i	−0.929992	−	0.367579i	$-0.880187\pi$
$619$	− 18.2905i	− 0.735158i	0.929992	−	0.367579i	$-0.119813\pi$
$620$	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
$629$	−15.8995	−0.633953
$630$	0	0
$631$	32.1588	1.28022	0.640111	−	0.768282i	$-0.278888\pi$
$631$	32.1588	1.28022	0.640111	+	0.768282i	$0.278888\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.90227	−0.154857
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17.0039i	0.671615i	0.941931	+	0.335807i	$0.109009\pi$
$641$	17.0039i	0.671615i	−0.941931	+	0.335807i	$0.890991\pi$
$642$	0	0
$643$	− 11.2881i	− 0.445160i	−0.974914	−	0.222580i	$-0.928552\pi$
$643$	− 11.2881i	− 0.445160i	0.974914	−	0.222580i	$-0.0714479\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.9921	0.864600	0.432300	−	0.901730i	$-0.357702\pi$
$647$	21.9921	0.864600	0.432300	+	0.901730i	$0.357702\pi$
$648$	0	0
$649$	2.82426i	0.110862i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21.7576i	0.851440i	0.904855	+	0.425720i	$0.139979\pi$
$653$	21.7576i	0.851440i	−0.904855	+	0.425720i	$0.860021\pi$
$654$	0	0
$655$	−7.73674	−0.302299
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.4506i	0.718731i	0.933197	+	0.359366i	$0.117007\pi$
$659$	18.4506i	0.718731i	−0.933197	+	0.359366i	$0.882993\pi$
$660$	0	0
$661$	14.7162i	0.572395i	0.958171	+	0.286197i	$0.0923913\pi$
$661$	14.7162i	0.572395i	−0.958171	+	0.286197i	$0.907609\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.23743	0.280234
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.544701	−0.0210280
$672$	0	0
$673$	−41.2229	−1.58903	−0.794514	−	0.607246i	$-0.792274\pi$
$673$	−41.2229	−1.58903	−0.794514	+	0.607246i	$0.792274\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.52722	0.0971290	0.0485645	−	0.998820i	$-0.484535\pi$
$677$	2.52722	0.0971290	0.0485645	+	0.998820i	$0.484535\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 28.1689i	− 1.07785i	−0.842352	−	0.538927i	$-0.818830\pi$
$683$	− 28.1689i	− 1.07785i	0.842352	−	0.538927i	$-0.181170\pi$
$684$	0	0
$685$	9.39720i	0.359049i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.46387	0.322448
$690$	0	0
$691$	37.5594i	1.42883i	0.699723	+	0.714414i	$0.253307\pi$
$691$	37.5594i	1.42883i	−0.699723	+	0.714414i	$0.746693\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.97575i	0.188741i
$696$	0	0
$697$	−27.9556	−1.05889
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 47.9199i	− 1.80991i	−0.425507	−	0.904955i	$-0.639904\pi$
$701$	− 47.9199i	− 1.80991i	0.425507	−	0.904955i	$-0.360096\pi$
$702$	0	0
$703$	− 14.1202i	− 0.532554i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5.95783	−0.223751	−0.111876	−	0.993722i	$-0.535686\pi$
$709$	−5.95783	−0.223751	−0.111876	+	0.993722i	$0.535686\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.26450	0.272058
$714$	0	0
$715$	1.42151	0.0531615
$716$	0	0
$717$	0	0
$718$	0	0
$719$	41.3866	1.54346	0.771729	−	0.635951i	$-0.219392\pi$
$719$	41.3866	1.54346	0.771729	+	0.635951i	$0.219392\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.94109i	− 0.146368i
$726$	0	0
$727$	− 36.2402i	− 1.34408i	−0.740517	−	0.672038i	$-0.765419\pi$
$727$	− 36.2402i	− 1.34408i	0.740517	−	0.672038i	$-0.234581\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	33.9791	1.25676
$732$	0	0
$733$	5.88971i	0.217541i	0.994067	+	0.108771i	$0.0346914\pi$
$733$	5.88971i	0.217541i	−0.994067	+	0.108771i	$0.965309\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.00619i	0.0738991i
$738$	0	0
$739$	7.78812	0.286491	0.143245	−	0.989687i	$-0.454246\pi$
$739$	7.78812	0.286491	0.143245	+	0.989687i	$0.454246\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.1412i	0.738908i	0.929249	+	0.369454i	$0.120455\pi$
$743$	20.1412i	0.738908i	−0.929249	+	0.369454i	$0.879545\pi$
$744$	0	0
$745$	− 13.3626i	− 0.489566i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−27.3582	−0.998316	−0.499158	−	0.866511i	$-0.666357\pi$
$751$	−27.3582	−0.998316	−0.499158	+	0.866511i	$0.666357\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.02851	−0.0738249
$756$	0	0
$757$	−28.5093	−1.03619	−0.518095	−	0.855323i	$-0.673359\pi$
$757$	−28.5093	−1.03619	−0.518095	+	0.855323i	$0.673359\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.6663	0.531652	0.265826	−	0.964021i	$-0.414355\pi$
$761$	14.6663	0.531652	0.265826	+	0.964021i	$0.414355\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.0260i	0.650881i
$768$	0	0
$769$	− 23.0346i	− 0.830649i	−0.909673	−	0.415324i	$-0.863668\pi$
$769$	− 23.0346i	− 0.830649i	0.909673	−	0.415324i	$-0.136332\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−47.3865	−1.70437	−0.852187	−	0.523237i	$-0.824724\pi$
$773$	−47.3865	−1.70437	−0.852187	+	0.523237i	$0.824724\pi$
$774$	0	0
$775$	− 3.95583i	− 0.142097i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 24.8272i	− 0.889527i
$780$	0	0
$781$	4.57184	0.163593
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.10473i	0.0751210i
$786$	0	0
$787$	14.5678i	0.519286i	0.965705	+	0.259643i	$0.0836049\pi$
$787$	14.5678i	0.519286i	−0.965705	+	0.259643i	$0.916395\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−3.47659	−0.123457
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38.3540	−1.35857	−0.679283	−	0.733876i	$-0.737709\pi$
$797$	−38.3540	−1.35857	−0.679283	+	0.733876i	$0.737709\pi$
$798$	0	0
$799$	38.0731	1.34693
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.46945	−0.193013
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 28.0522i	− 0.986263i	−0.869955	−	0.493131i	$-0.835852\pi$
$809$	− 28.0522i	− 0.986263i	0.869955	−	0.493131i	$-0.164148\pi$
$810$	0	0
$811$	− 7.87999i	− 0.276704i	−0.990383	−	0.138352i	$-0.955819\pi$
$811$	− 7.87999i	− 0.276704i	0.990383	−	0.138352i	$-0.0441805\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.10548	0.178837
$816$	0	0
$817$	30.1766i	1.05575i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.368813i	0.0128717i	0.999979	+	0.00643584i	$0.00204860\pi$
$821$	0.368813i	0.0128717i	−0.999979	+	0.00643584i	$0.997951\pi$
$822$	0	0
$823$	43.0438	1.50041	0.750206	−	0.661205i	$-0.229955\pi$
$823$	43.0438	1.50041	0.750206	+	0.661205i	$0.229955\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.9937i	0.903891i	0.892046	+	0.451945i	$0.149270\pi$
$827$	25.9937i	0.903891i	−0.892046	+	0.451945i	$0.850730\pi$
$828$	0	0
$829$	19.3160i	0.670871i	0.942063	+	0.335436i	$0.108884\pi$
$829$	19.3160i	0.670871i	−0.942063	+	0.335436i	$0.891116\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−12.9128	−0.446865
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17.4145	0.601216	0.300608	−	0.953748i	$-0.402810\pi$
$839$	17.4145	0.601216	0.300608	+	0.953748i	$0.402810\pi$
$840$	0	0
$841$	13.4678	0.464409
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.92712	−0.135097
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 5.03574i	− 0.172623i
$852$	0	0
$853$	27.0875i	0.927458i	0.885977	+	0.463729i	$0.153489\pi$
$853$	27.0875i	0.927458i	−0.885977	+	0.463729i	$0.846511\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.5813	0.600566	0.300283	−	0.953850i	$-0.402919\pi$
$857$	17.5813	0.600566	0.300283	+	0.953850i	$0.402919\pi$
$858$	0	0
$859$	− 8.37944i	− 0.285903i	−0.989730	−	0.142951i	$-0.954341\pi$
$859$	− 8.37944i	− 0.285903i	0.989730	−	0.142951i	$-0.0456592\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.89948i	0.166780i	0.996517	+	0.0833900i	$0.0265747\pi$
$863$	4.89948i	0.166780i	−0.996517	+	0.0833900i	$0.973425\pi$
$864$	0	0
$865$	−3.67115	−0.124823
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 5.62185i	− 0.190708i
$870$	0	0
$871$	12.8047i	0.433869i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.8854	0.367575	0.183788	−	0.982966i	$-0.441164\pi$
$877$	10.8854	0.367575	0.183788	+	0.982966i	$0.441164\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−43.1066	−1.45230	−0.726150	−	0.687537i	$-0.758692\pi$
$881$	−43.1066	−1.45230	−0.726150	+	0.687537i	$0.758692\pi$
$882$	0	0
$883$	34.7692	1.17008	0.585038	−	0.811006i	$-0.301080\pi$
$883$	34.7692	1.17008	0.585038	+	0.811006i	$0.301080\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.1007	1.04426	0.522129	−	0.852866i	$-0.325138\pi$
$887$	31.1007	1.04426	0.522129	+	0.852866i	$0.325138\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	33.8124i	1.13149i
$894$	0	0
$895$	3.32381i	0.111103i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15.5902	−0.519964
$900$	0	0
$901$	16.2924i	0.542777i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.5441i	0.749392i
$906$	0	0
$907$	23.9324	0.794661	0.397330	−	0.917676i	$-0.369937\pi$
$907$	23.9324	0.794661	0.397330	+	0.917676i	$0.369937\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 53.6875i	− 1.77875i	−0.457181	−	0.889374i	$-0.651141\pi$
$911$	− 53.6875i	− 1.77875i	0.457181	−	0.889374i	$-0.348859\pi$
$912$	0	0
$913$	− 2.10057i	− 0.0695186i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	49.3219	1.62698	0.813490	−	0.581579i	$-0.197565\pi$
$919$	49.3219	1.62698	0.813490	+	0.581579i	$0.197565\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	29.1800	0.960473
$924$	0	0
$925$	−2.74217	−0.0901621
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−31.7417	−1.04141	−0.520707	−	0.853736i	$-0.674331\pi$
$929$	−31.7417	−1.04141	−0.520707	+	0.853736i	$0.674331\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.73631i	0.0894869i
$936$	0	0
$937$	9.19801i	0.300486i	0.988649	+	0.150243i	$0.0480056\pi$
$937$	9.19801i	0.300486i	−0.988649	+	0.150243i	$0.951994\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	54.9553	1.79149	0.895746	−	0.444566i	$-0.146642\pi$
$941$	54.9553	1.79149	0.895746	+	0.444566i	$0.146642\pi$
$942$	0	0
$943$	− 8.85422i	− 0.288333i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.90681i	0.0944585i	0.998884	+	0.0472293i	$0.0150391\pi$
$947$	2.90681i	0.0944585i	−0.998884	+	0.0472293i	$0.984961\pi$
$948$	0	0
$949$	−34.9091	−1.13320
$950$	0	0
$951$	0	0
$952$	0	0
$953$	59.7695i	1.93612i	0.250712	+	0.968062i	$0.419335\pi$
$953$	59.7695i	1.93612i	−0.250712	+	0.968062i	$0.580665\pi$
$954$	0	0
$955$	6.18079i	0.200006i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	15.3514	0.495208
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.1357	0.358471
$966$	0	0
$967$	−8.11334	−0.260908	−0.130454	−	0.991454i	$-0.541643\pi$
$967$	−8.11334	−0.260908	−0.130454	+	0.991454i	$0.541643\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.0990	1.15847	0.579237	−	0.815159i	$-0.303351\pi$
$971$	36.0990	1.15847	0.579237	+	0.815159i	$0.303351\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.68933i	− 0.0860394i	−0.999074	−	0.0430197i	$-0.986302\pi$
$977$	− 2.68933i	− 0.0860394i	0.999074	−	0.0430197i	$-0.0136978\pi$
$978$	0	0
$979$	2.08361i	0.0665926i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−55.9516	−1.78458	−0.892289	−	0.451465i	$-0.850901\pi$
$983$	−55.9516	−1.78458	−0.892289	+	0.451465i	$0.850901\pi$
$984$	0	0
$985$	− 0.374400i	− 0.0119294i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.7620i	0.342211i
$990$	0	0
$991$	52.2349	1.65930	0.829648	−	0.558287i	$-0.188541\pi$
$991$	52.2349	1.65930	0.829648	+	0.558287i	$0.188541\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	20.2381i	0.641591i
$996$	0	0
$997$	− 2.44304i	− 0.0773720i	−0.999251	−	0.0386860i	$-0.987683\pi$
$997$	− 2.44304i	− 0.0773720i	0.999251	−	0.0386860i	$-0.0123172\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.d.c.881.7	✓	16
3.2	odd	2		8820.2.d.d.881.10	yes	16
7.6	odd	2		8820.2.d.d.881.7	yes	16
21.20	even	2	inner	8820.2.d.c.881.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8820.2.d.c.881.7	✓	16	1.1	even	1	trivial
8820.2.d.c.881.10	yes	16	21.20	even	2	inner
8820.2.d.d.881.7	yes	16	7.6	odd	2
8820.2.d.d.881.10	yes	16	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +O(q^{10})\)	\(q-1.00000 q^{5} -0.471930i q^{11} -3.01212i q^{13} +5.79812 q^{17} +5.14928i q^{19} +1.83640i q^{23} +1.00000 q^{25} -3.94109i q^{29} -3.95583i q^{31} -2.74217 q^{37} -4.82150 q^{41} +5.86036 q^{43} +6.56645 q^{47} +2.80994i q^{53} +0.471930i q^{55} -5.98448 q^{59} -1.15420i q^{61} +3.01212i q^{65} -4.25104 q^{67} +9.68753i q^{71} -11.5895i q^{73} +11.9124 q^{79} +4.45101 q^{83} -5.79812 q^{85} -4.41508 q^{89} -5.14928i q^{95} +3.56574i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5}+O(q^{10}) \) 16 * q - 16 * q^5	\( 16 q - 16 q^{5} + 16 q^{25} - 32 q^{41} - 32 q^{43} + 32 q^{47} + 32 q^{59} - 32 q^{67} + 64 q^{89}+O(q^{100}) \) 16 * q - 16 * q^5 + 16 * q^25 - 32 * q^41 - 32 * q^43 + 32 * q^47 + 32 * q^59 - 32 * q^67 + 64 * q^89

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