LMFDB - Embedded newform 8820.2.d.a.881.12

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			881.12
Root			$1.63107 - 2.08318i$ of defining polynomial
Character	$\chi$	$=$	8820.881
Dual form			8820.2.d.a.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +O(q^{10})$	$q-1.00000 q^{5} +5.89840i q^{11} -4.65726i q^{13} -2.10237 q^{17} -5.25849i q^{19} -7.83347i q^{23} +1.00000 q^{25} -0.273932i q^{29} +4.93119i q^{31} +5.83897 q^{37} -2.26213 q^{41} +8.63950 q^{43} -4.98713 q^{47} +5.62447i q^{53} -5.89840i q^{55} -4.67920 q^{59} +9.48923i q^{61} +4.65726i q^{65} -7.46687 q^{67} +14.3331i q^{71} -0.741912i q^{73} +2.95737 q^{79} -6.84497 q^{83} +2.10237 q^{85} +6.81007 q^{89} +5.25849i q^{95} +4.90827i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	5.89840i	1.77844i	0.457484	+	0.889218i	$0.348751\pi$
$11$	5.89840i	1.77844i	−0.457484	+	0.889218i	$0.651249\pi$
$12$	0	0
$13$	− 4.65726i	− 1.29169i	−0.763468	−	0.645845i	$-0.776505\pi$
$13$	− 4.65726i	− 1.29169i	0.763468	−	0.645845i	$-0.223495\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.10237	−0.509900	−0.254950	−	0.966954i	$-0.582059\pi$
$17$	−2.10237	−0.509900	−0.254950	+	0.966954i	$0.582059\pi$
$18$	0	0
$19$	− 5.25849i	− 1.20638i	−0.797598	−	0.603190i	$-0.793896\pi$
$19$	− 5.25849i	− 1.20638i	0.797598	−	0.603190i	$-0.206104\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 7.83347i	− 1.63339i	−0.577069	−	0.816696i	$-0.695803\pi$
$23$	− 7.83347i	− 1.63339i	0.577069	−	0.816696i	$-0.304197\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.273932i	− 0.0508680i	−0.999677	−	0.0254340i	$-0.991903\pi$
$29$	− 0.273932i	− 0.0508680i	0.999677	−	0.0254340i	$-0.00809676\pi$
$30$	0	0
$31$	4.93119i	0.885667i	0.896604	+	0.442834i	$0.146027\pi$
$31$	4.93119i	0.885667i	−0.896604	+	0.442834i	$0.853973\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.83897	0.959920	0.479960	−	0.877290i	$-0.340651\pi$
$37$	5.83897	0.959920	0.479960	+	0.877290i	$0.340651\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.26213	−0.353286	−0.176643	−	0.984275i	$-0.556524\pi$
$41$	−2.26213	−0.353286	−0.176643	+	0.984275i	$0.556524\pi$
$42$	0	0
$43$	8.63950	1.31751	0.658756	−	0.752357i	$-0.271083\pi$
$43$	8.63950	1.31751	0.658756	+	0.752357i	$0.271083\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.98713	−0.727448	−0.363724	−	0.931507i	$-0.618495\pi$
$47$	−4.98713	−0.727448	−0.363724	+	0.931507i	$0.618495\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.62447i	0.772581i	0.922377	+	0.386290i	$0.126244\pi$
$53$	5.62447i	0.772581i	−0.922377	+	0.386290i	$0.873756\pi$
$54$	0	0
$55$	− 5.89840i	− 0.795341i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.67920	−0.609180	−0.304590	−	0.952483i	$-0.598520\pi$
$59$	−4.67920	−0.609180	−0.304590	+	0.952483i	$0.598520\pi$
$60$	0	0
$61$	9.48923i	1.21497i	0.794330	+	0.607486i	$0.207822\pi$
$61$	9.48923i	1.21497i	−0.794330	+	0.607486i	$0.792178\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.65726i	0.577661i
$66$	0	0
$67$	−7.46687	−0.912223	−0.456112	−	0.889923i	$-0.650758\pi$
$67$	−7.46687	−0.912223	−0.456112	+	0.889923i	$0.650758\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.3331i	1.70103i	0.525954	+	0.850513i	$0.323709\pi$
$71$	14.3331i	1.70103i	−0.525954	+	0.850513i	$0.676291\pi$
$72$	0	0
$73$	− 0.741912i	− 0.0868342i	−0.999057	−	0.0434171i	$-0.986176\pi$
$73$	− 0.741912i	− 0.0868342i	0.999057	−	0.0434171i	$-0.0138244\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.95737	0.332730	0.166365	−	0.986064i	$-0.446797\pi$
$79$	2.95737	0.332730	0.166365	+	0.986064i	$0.446797\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.84497	−0.751333	−0.375666	−	0.926755i	$-0.622586\pi$
$83$	−6.84497	−0.751333	−0.375666	+	0.926755i	$0.622586\pi$
$84$	0	0
$85$	2.10237	0.228034
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.81007	0.721865	0.360933	−	0.932592i	$-0.382458\pi$
$89$	6.81007	0.721865	0.360933	+	0.932592i	$0.382458\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.25849i	0.539509i
$96$	0	0
$97$	4.90827i	0.498359i	0.968457	+	0.249179i	$0.0801609\pi$
$97$	4.90827i	0.498359i	−0.968457	+	0.249179i	$0.919839\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.41873	−0.638688	−0.319344	−	0.947639i	$-0.603462\pi$
$101$	−6.41873	−0.638688	−0.319344	+	0.947639i	$0.603462\pi$
$102$	0	0
$103$	− 16.5670i	− 1.63239i	−0.577775	−	0.816196i	$-0.696079\pi$
$103$	− 16.5670i	− 1.63239i	0.577775	−	0.816196i	$-0.303921\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.02034i	0.775355i	0.921795	+	0.387678i	$0.126723\pi$
$107$	8.02034i	0.775355i	−0.921795	+	0.387678i	$0.873277\pi$
$108$	0	0
$109$	−17.2669	−1.65387	−0.826935	−	0.562298i	$-0.809917\pi$
$109$	−17.2669	−1.65387	−0.826935	+	0.562298i	$0.809917\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 9.97950i	− 0.938793i	−0.882988	−	0.469396i	$-0.844472\pi$
$113$	− 9.97950i	− 0.938793i	0.882988	−	0.469396i	$-0.155528\pi$
$114$	0	0
$115$	7.83347i	0.730475i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−23.7912	−2.16283
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.06660	−0.538324	−0.269162	−	0.963095i	$-0.586747\pi$
$127$	−6.06660	−0.538324	−0.269162	+	0.963095i	$0.586747\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.6832	0.933396	0.466698	−	0.884417i	$-0.345444\pi$
$131$	10.6832	0.933396	0.466698	+	0.884417i	$0.345444\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.19340i	0.101959i	0.998700	+	0.0509797i	$0.0162344\pi$
$137$	1.19340i	0.101959i	−0.998700	+	0.0509797i	$0.983766\pi$
$138$	0	0
$139$	− 22.1889i	− 1.88203i	−0.338358	−	0.941017i	$-0.609872\pi$
$139$	− 22.1889i	− 1.88203i	0.338358	−	0.941017i	$-0.390128\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	27.4704	2.29719
$144$	0	0
$145$	0.273932i	0.0227488i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.0093i	1.72115i	0.509323	+	0.860575i	$0.329896\pi$
$149$	21.0093i	1.72115i	−0.509323	+	0.860575i	$0.670104\pi$
$150$	0	0
$151$	−10.5211	−0.856195	−0.428097	−	0.903733i	$-0.640816\pi$
$151$	−10.5211	−0.856195	−0.428097	+	0.903733i	$0.640816\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 4.93119i	− 0.396083i
$156$	0	0
$157$	− 17.6472i	− 1.40840i	−0.710001	−	0.704201i	$-0.751306\pi$
$157$	− 17.6472i	− 1.40840i	0.710001	−	0.704201i	$-0.248694\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−15.8209	−1.23919	−0.619596	−	0.784921i	$-0.712703\pi$
$163$	−15.8209	−1.23919	−0.619596	+	0.784921i	$0.712703\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.148166	−0.0114654	−0.00573270	−	0.999984i	$-0.501825\pi$
$167$	−0.148166	−0.0114654	−0.00573270	+	0.999984i	$0.501825\pi$
$168$	0	0
$169$	−8.69003	−0.668464
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.6390	−1.72121	−0.860606	−	0.509272i	$-0.829915\pi$
$173$	−22.6390	−1.72121	−0.860606	+	0.509272i	$0.829915\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 22.6732i	− 1.69468i	−0.531053	−	0.847339i	$-0.678203\pi$
$179$	− 22.6732i	− 1.69468i	0.531053	−	0.847339i	$-0.321797\pi$
$180$	0	0
$181$	24.4505i	1.81739i	0.417459	+	0.908696i	$0.362921\pi$
$181$	24.4505i	1.81739i	−0.417459	+	0.908696i	$0.637079\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.83897	−0.429289
$186$	0	0
$187$	− 12.4006i	− 0.906823i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 0.0226722i	− 0.00164050i	−1.00000		0.000820252i	$-0.999739\pi$
$191$	− 0.0226722i	− 0.00164050i	1.00000		0.000820252i	$-0.000261095\pi$
$192$	0	0
$193$	4.70124	0.338403	0.169201	−	0.985582i	$-0.445881\pi$
$193$	4.70124	0.338403	0.169201	+	0.985582i	$0.445881\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.8369i	− 1.27082i	−0.772173	−	0.635412i	$-0.780830\pi$
$197$	− 17.8369i	− 1.27082i	0.772173	−	0.635412i	$-0.219170\pi$
$198$	0	0
$199$	1.65855i	0.117571i	0.998271	+	0.0587856i	$0.0187228\pi$
$199$	1.65855i	0.117571i	−0.998271	+	0.0587856i	$0.981277\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.26213	0.157994
$206$	0	0
$207$	0	0
$208$	0	0
$209$	31.0167	2.14547
$210$	0	0
$211$	−10.3247	−0.710782	−0.355391	−	0.934718i	$-0.615652\pi$
$211$	−10.3247	−0.710782	−0.355391	+	0.934718i	$0.615652\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.63950	−0.589209
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.79127i	0.658632i
$222$	0	0
$223$	− 15.3565i	− 1.02835i	−0.857686	−	0.514174i	$-0.828098\pi$
$223$	− 15.3565i	− 1.02835i	0.857686	−	0.514174i	$-0.171902\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.5350	1.09747	0.548735	−	0.835997i	$-0.315110\pi$
$227$	16.5350	1.09747	0.548735	+	0.835997i	$0.315110\pi$
$228$	0	0
$229$	− 14.6174i	− 0.965945i	−0.875636	−	0.482972i	$-0.839557\pi$
$229$	− 14.6174i	− 0.965945i	0.875636	−	0.482972i	$-0.160443\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.06662i	0.593974i	0.954881	+	0.296987i	$0.0959818\pi$
$233$	9.06662i	0.593974i	−0.954881	+	0.296987i	$0.904018\pi$
$234$	0	0
$235$	4.98713	0.325325
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 19.0756i	− 1.23390i	−0.787003	−	0.616949i	$-0.788368\pi$
$239$	− 19.0756i	− 1.23390i	0.787003	−	0.616949i	$-0.211632\pi$
$240$	0	0
$241$	− 8.13007i	− 0.523704i	−0.965108	−	0.261852i	$-0.915667\pi$
$241$	− 8.13007i	− 0.523704i	0.965108	−	0.261852i	$-0.0843333\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−24.4901	−1.55827
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.54921	−0.350263	−0.175131	−	0.984545i	$-0.556035\pi$
$251$	−5.54921	−0.350263	−0.175131	+	0.984545i	$0.556035\pi$
$252$	0	0
$253$	46.2050	2.90488
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.36323	0.209792	0.104896	−	0.994483i	$-0.466549\pi$
$257$	3.36323	0.209792	0.104896	+	0.994483i	$0.466549\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.0942i	0.807420i	0.914887	+	0.403710i	$0.132280\pi$
$263$	13.0942i	0.807420i	−0.914887	+	0.403710i	$0.867720\pi$
$264$	0	0
$265$	− 5.62447i	− 0.345509i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−21.5988	−1.31690	−0.658451	−	0.752623i	$-0.728788\pi$
$269$	−21.5988	−1.31690	−0.658451	+	0.752623i	$0.728788\pi$
$270$	0	0
$271$	− 23.8085i	− 1.44626i	−0.690709	−	0.723132i	$-0.742702\pi$
$271$	− 23.8085i	− 1.44626i	0.690709	−	0.723132i	$-0.257298\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.89840i	0.355687i
$276$	0	0
$277$	−5.32844	−0.320155	−0.160077	−	0.987104i	$-0.551174\pi$
$277$	−5.32844	−0.320155	−0.160077	+	0.987104i	$0.551174\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 0.309995i	− 0.0184928i	−0.999957	−	0.00924639i	$-0.997057\pi$
$281$	− 0.309995i	− 0.0184928i	0.999957	−	0.00924639i	$-0.00294326\pi$
$282$	0	0
$283$	− 29.9839i	− 1.78236i	−0.453652	−	0.891179i	$-0.649879\pi$
$283$	− 29.9839i	− 1.78236i	0.453652	−	0.891179i	$-0.350121\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−12.5800	−0.740002
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−27.1877	−1.58832	−0.794162	−	0.607706i	$-0.792090\pi$
$293$	−27.1877	−1.58832	−0.794162	+	0.607706i	$0.792090\pi$
$294$	0	0
$295$	4.67920	0.272434
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−36.4825	−2.10984
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 9.48923i	− 0.543352i
$306$	0	0
$307$	9.94906i	0.567823i	0.958851	+	0.283911i	$0.0916322\pi$
$307$	9.94906i	0.567823i	−0.958851	+	0.283911i	$0.908368\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−32.2938	−1.83121	−0.915607	−	0.402075i	$-0.868289\pi$
$311$	−32.2938	−1.83121	−0.915607	+	0.402075i	$0.868289\pi$
$312$	0	0
$313$	− 20.3968i	− 1.15290i	−0.817134	−	0.576448i	$-0.804438\pi$
$313$	− 20.3968i	− 1.15290i	0.817134	−	0.576448i	$-0.195562\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.9191i	0.613275i	0.951826	+	0.306638i	$0.0992040\pi$
$317$	10.9191i	0.613275i	−0.951826	+	0.306638i	$0.900796\pi$
$318$	0	0
$319$	1.61576	0.0904654
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11.0553i	0.615132i
$324$	0	0
$325$	− 4.65726i	− 0.258338i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−11.4232	−0.627876	−0.313938	−	0.949443i	$-0.601648\pi$
$331$	−11.4232	−0.627876	−0.313938	+	0.949443i	$0.601648\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.46687	0.407959
$336$	0	0
$337$	−1.91966	−0.104571	−0.0522854	−	0.998632i	$-0.516651\pi$
$337$	−1.91966	−0.104571	−0.0522854	+	0.998632i	$0.516651\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−29.0861	−1.57510
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.53772i	0.136232i	0.997677	+	0.0681161i	$0.0216988\pi$
$347$	2.53772i	0.136232i	−0.997677	+	0.0681161i	$0.978301\pi$
$348$	0	0
$349$	22.8601i	1.22367i	0.790984	+	0.611837i	$0.209569\pi$
$349$	22.8601i	1.22367i	−0.790984	+	0.611837i	$0.790431\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.9530	0.849091	0.424546	−	0.905407i	$-0.360434\pi$
$353$	15.9530	0.849091	0.424546	+	0.905407i	$0.360434\pi$
$354$	0	0
$355$	− 14.3331i	− 0.760722i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.2931i	1.86270i	0.364129	+	0.931348i	$0.381367\pi$
$359$	35.2931i	1.86270i	−0.364129	+	0.931348i	$0.618633\pi$
$360$	0	0
$361$	−8.65167	−0.455351
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.741912i	0.0388335i
$366$	0	0
$367$	12.9223i	0.674536i	0.941409	+	0.337268i	$0.109503\pi$
$367$	12.9223i	0.674536i	−0.941409	+	0.337268i	$0.890497\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−9.97107	−0.516283	−0.258141	−	0.966107i	$-0.583110\pi$
$373$	−9.97107	−0.516283	−0.258141	+	0.966107i	$0.583110\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.27577	−0.0657057
$378$	0	0
$379$	−5.22480	−0.268380	−0.134190	−	0.990956i	$-0.542843\pi$
$379$	−5.22480	−0.268380	−0.134190	+	0.990956i	$0.542843\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.68134	−0.443596	−0.221798	−	0.975093i	$-0.571193\pi$
$383$	−8.68134	−0.443596	−0.221798	+	0.975093i	$0.571193\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.7728i	1.20533i	0.797995	+	0.602664i	$0.205894\pi$
$389$	23.7728i	1.20533i	−0.797995	+	0.602664i	$0.794106\pi$
$390$	0	0
$391$	16.4688i	0.832865i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.95737	−0.148801
$396$	0	0
$397$	− 3.03846i	− 0.152496i	−0.997089	−	0.0762479i	$-0.975706\pi$
$397$	− 3.03846i	− 0.152496i	0.997089	−	0.0762479i	$-0.0242940\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 20.9695i	− 1.04717i	−0.851975	−	0.523583i	$-0.824595\pi$
$401$	− 20.9695i	− 1.04717i	0.851975	−	0.523583i	$-0.175405\pi$
$402$	0	0
$403$	22.9658	1.14401
$404$	0	0
$405$	0	0
$406$	0	0
$407$	34.4406i	1.70716i
$408$	0	0
$409$	24.0164i	1.18753i	0.804637	+	0.593767i	$0.202360\pi$
$409$	24.0164i	1.18753i	−0.804637	+	0.593767i	$0.797640\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6.84497	0.336006
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−19.6680	−0.960844	−0.480422	−	0.877037i	$-0.659517\pi$
$419$	−19.6680	−0.960844	−0.480422	+	0.877037i	$0.659517\pi$
$420$	0	0
$421$	24.2954	1.18408	0.592042	−	0.805907i	$-0.298322\pi$
$421$	24.2954	1.18408	0.592042	+	0.805907i	$0.298322\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.10237	−0.101980
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 7.48153i	− 0.360372i	−0.983633	−	0.180186i	$-0.942330\pi$
$431$	− 7.48153i	− 0.360372i	0.983633	−	0.180186i	$-0.0576700\pi$
$432$	0	0
$433$	22.3722i	1.07514i	0.843220	+	0.537569i	$0.180657\pi$
$433$	22.3722i	1.07514i	−0.843220	+	0.537569i	$0.819343\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−41.1922	−1.97049
$438$	0	0
$439$	13.3765i	0.638427i	0.947683	+	0.319214i	$0.103419\pi$
$439$	13.3765i	0.638427i	−0.947683	+	0.319214i	$0.896581\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 10.1775i	− 0.483549i	−0.970332	−	0.241775i	$-0.922271\pi$
$443$	− 10.1775i	− 0.483549i	0.970332	−	0.241775i	$-0.0777294\pi$
$444$	0	0
$445$	−6.81007	−0.322828
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 15.6626i	− 0.739164i	−0.929198	−	0.369582i	$-0.879501\pi$
$449$	− 15.6626i	− 0.739164i	0.929198	−	0.369582i	$-0.120499\pi$
$450$	0	0
$451$	− 13.3430i	− 0.628296i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−38.8068	−1.81531	−0.907653	−	0.419722i	$-0.862127\pi$
$457$	−38.8068	−1.81531	−0.907653	+	0.419722i	$0.862127\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.6287	−0.541605	−0.270802	−	0.962635i	$-0.587289\pi$
$461$	−11.6287	−0.541605	−0.270802	+	0.962635i	$0.587289\pi$
$462$	0	0
$463$	−31.3865	−1.45865	−0.729326	−	0.684166i	$-0.760166\pi$
$463$	−31.3865	−1.45865	−0.729326	+	0.684166i	$0.760166\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.49206	−0.392966	−0.196483	−	0.980507i	$-0.562952\pi$
$467$	−8.49206	−0.392966	−0.196483	+	0.980507i	$0.562952\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	50.9593i	2.34311i
$474$	0	0
$475$	− 5.25849i	− 0.241276i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−21.3103	−0.973691	−0.486845	−	0.873488i	$-0.661853\pi$
$479$	−21.3103	−0.973691	−0.486845	+	0.873488i	$0.661853\pi$
$480$	0	0
$481$	− 27.1936i	− 1.23992i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 4.90827i	− 0.222873i
$486$	0	0
$487$	−38.9618	−1.76553	−0.882764	−	0.469817i	$-0.844320\pi$
$487$	−38.9618	−1.76553	−0.882764	+	0.469817i	$0.844320\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 21.9579i	− 0.990948i	−0.868623	−	0.495474i	$-0.834994\pi$
$491$	− 21.9579i	− 0.990948i	0.868623	−	0.495474i	$-0.165006\pi$
$492$	0	0
$493$	0.575907i	0.0259376i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	10.7100	0.479445	0.239723	−	0.970841i	$-0.422944\pi$
$499$	10.7100	0.479445	0.239723	+	0.970841i	$0.422944\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19.9518	0.889607	0.444803	−	0.895628i	$-0.353274\pi$
$503$	19.9518	0.889607	0.444803	+	0.895628i	$0.353274\pi$
$504$	0	0
$505$	6.41873	0.285630
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.9962	1.10794	0.553969	−	0.832538i	$-0.313113\pi$
$509$	24.9962	1.10794	0.553969	+	0.832538i	$0.313113\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.5670i	0.730028i
$516$	0	0
$517$	− 29.4161i	− 1.29372i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	31.1555	1.36495	0.682474	−	0.730910i	$-0.260904\pi$
$521$	31.1555	1.36495	0.682474	+	0.730910i	$0.260904\pi$
$522$	0	0
$523$	25.7064i	1.12406i	0.827115	+	0.562032i	$0.189980\pi$
$523$	25.7064i	1.12406i	−0.827115	+	0.562032i	$0.810020\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 10.3672i	− 0.451601i
$528$	0	0
$529$	−38.3632	−1.66797
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.5353i	0.456336i
$534$	0	0
$535$	− 8.02034i	− 0.346749i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.7201	0.546878	0.273439	−	0.961889i	$-0.411839\pi$
$541$	12.7201	0.546878	0.273439	+	0.961889i	$0.411839\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	17.2669	0.739633
$546$	0	0
$547$	19.8530	0.848853	0.424426	−	0.905462i	$-0.360476\pi$
$547$	19.8530	0.848853	0.424426	+	0.905462i	$0.360476\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.44047	−0.0613661
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.1801i	− 1.02455i	−0.858823	−	0.512273i	$-0.828804\pi$
$557$	− 24.1801i	− 1.02455i	0.858823	−	0.512273i	$-0.171196\pi$
$558$	0	0
$559$	− 40.2364i	− 1.70182i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.74960	0.242317	0.121158	−	0.992633i	$-0.461339\pi$
$563$	5.74960	0.242317	0.121158	+	0.992633i	$0.461339\pi$
$564$	0	0
$565$	9.97950i	0.419841i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 3.20684i	− 0.134438i	−0.997738	−	0.0672188i	$-0.978587\pi$
$569$	− 3.20684i	− 0.134438i	0.997738	−	0.0672188i	$-0.0214126\pi$
$570$	0	0
$571$	−3.77198	−0.157852	−0.0789262	−	0.996880i	$-0.525149\pi$
$571$	−3.77198	−0.157852	−0.0789262	+	0.996880i	$0.525149\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 7.83347i	− 0.326678i
$576$	0	0
$577$	28.2912i	1.17778i	0.808214	+	0.588888i	$0.200434\pi$
$577$	28.2912i	1.17778i	−0.808214	+	0.588888i	$0.799566\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−33.1754	−1.37398
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.8667	−1.43910	−0.719551	−	0.694439i	$-0.755653\pi$
$587$	−34.8667	−1.43910	−0.719551	+	0.694439i	$0.755653\pi$
$588$	0	0
$589$	25.9306	1.06845
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26.6302	−1.09357	−0.546786	−	0.837272i	$-0.684149\pi$
$593$	−26.6302	−1.09357	−0.546786	+	0.837272i	$0.684149\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.2877i	1.31924i	0.751599	+	0.659620i	$0.229283\pi$
$599$	32.2877i	1.31924i	−0.751599	+	0.659620i	$0.770717\pi$
$600$	0	0
$601$	− 39.8580i	− 1.62584i	−0.582373	−	0.812922i	$-0.697876\pi$
$601$	− 39.8580i	− 1.62584i	0.582373	−	0.812922i	$-0.302124\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	23.7912	0.967249
$606$	0	0
$607$	− 6.94268i	− 0.281795i	−0.990024	−	0.140897i	$-0.955001\pi$
$607$	− 6.94268i	− 0.281795i	0.990024	−	0.140897i	$-0.0449988\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	23.2263i	0.939637i
$612$	0	0
$613$	−12.8193	−0.517765	−0.258882	−	0.965909i	$-0.583354\pi$
$613$	−12.8193	−0.517765	−0.258882	+	0.965909i	$0.583354\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 31.8132i	− 1.28075i	−0.768062	−	0.640376i	$-0.778779\pi$
$617$	− 31.8132i	− 1.28075i	0.768062	−	0.640376i	$-0.221221\pi$
$618$	0	0
$619$	2.59073i	0.104130i	0.998644	+	0.0520650i	$0.0165803\pi$
$619$	2.59073i	0.104130i	−0.998644	+	0.0520650i	$0.983420\pi$
$620$	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
$629$	−12.2757	−0.489463
$630$	0	0
$631$	−16.3785	−0.652019	−0.326009	−	0.945367i	$-0.605704\pi$
$631$	−16.3785	−0.652019	−0.326009	+	0.945367i	$0.605704\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.06660	0.240746
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 43.6312i	− 1.72333i	−0.507480	−	0.861664i	$-0.669423\pi$
$641$	− 43.6312i	− 1.72333i	0.507480	−	0.861664i	$-0.330577\pi$
$642$	0	0
$643$	− 41.5279i	− 1.63770i	−0.574008	−	0.818850i	$-0.694612\pi$
$643$	− 41.5279i	− 1.63770i	0.574008	−	0.818850i	$-0.305388\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.2347	0.834823	0.417411	−	0.908718i	$-0.362937\pi$
$647$	21.2347	0.834823	0.417411	+	0.908718i	$0.362937\pi$
$648$	0	0
$649$	− 27.5998i	− 1.08339i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.7014i	1.16230i	0.813795	+	0.581152i	$0.197398\pi$
$653$	29.7014i	1.16230i	−0.813795	+	0.581152i	$0.802602\pi$
$654$	0	0
$655$	−10.6832	−0.417427
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.33542i	0.129930i	0.997888	+	0.0649648i	$0.0206935\pi$
$659$	3.33542i	0.129930i	−0.997888	+	0.0649648i	$0.979306\pi$
$660$	0	0
$661$	22.5805i	0.878281i	0.898418	+	0.439140i	$0.144717\pi$
$661$	22.5805i	0.878281i	−0.898418	+	0.439140i	$0.855283\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.14584	−0.0830873
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−55.9713	−2.16075
$672$	0	0
$673$	−27.8897	−1.07507	−0.537535	−	0.843241i	$-0.680645\pi$
$673$	−27.8897	−1.07507	−0.537535	+	0.843241i	$0.680645\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.8219	1.45361	0.726807	−	0.686842i	$-0.241003\pi$
$677$	37.8219	1.45361	0.726807	+	0.686842i	$0.241003\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28.3585i	1.08511i	0.840021	+	0.542554i	$0.182543\pi$
$683$	28.3585i	1.08511i	−0.840021	+	0.542554i	$0.817457\pi$
$684$	0	0
$685$	− 1.19340i	− 0.0455976i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	26.1946	0.997935
$690$	0	0
$691$	− 12.1351i	− 0.461641i	−0.972996	−	0.230820i	$-0.925859\pi$
$691$	− 12.1351i	− 0.461641i	0.972996	−	0.230820i	$-0.0741410\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22.1889i	0.841672i
$696$	0	0
$697$	4.75584	0.180140
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 34.1084i	− 1.28826i	−0.764918	−	0.644128i	$-0.777220\pi$
$701$	− 34.1084i	− 1.28826i	0.764918	−	0.644128i	$-0.222780\pi$
$702$	0	0
$703$	− 30.7041i	− 1.15803i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12.6916	0.476642	0.238321	−	0.971186i	$-0.423403\pi$
$709$	12.6916	0.476642	0.238321	+	0.971186i	$0.423403\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	38.6283	1.44664
$714$	0	0
$715$	−27.4704	−1.02733
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−14.0112	−0.522529	−0.261265	−	0.965267i	$-0.584140\pi$
$719$	−14.0112	−0.522529	−0.261265	+	0.965267i	$0.584140\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 0.273932i	− 0.0101736i
$726$	0	0
$727$	8.42559i	0.312488i	0.987719	+	0.156244i	$0.0499386\pi$
$727$	8.42559i	0.312488i	−0.987719	+	0.156244i	$0.950061\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−18.1634	−0.671798
$732$	0	0
$733$	31.8601i	1.17678i	0.808578	+	0.588389i	$0.200238\pi$
$733$	31.8601i	1.17678i	−0.808578	+	0.588389i	$0.799762\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 44.0426i	− 1.62233i
$738$	0	0
$739$	26.4512	0.973022	0.486511	−	0.873674i	$-0.338269\pi$
$739$	26.4512	0.973022	0.486511	+	0.873674i	$0.338269\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.4855i	0.384676i	0.981329	+	0.192338i	$0.0616070\pi$
$743$	10.4855i	0.384676i	−0.981329	+	0.192338i	$0.938393\pi$
$744$	0	0
$745$	− 21.0093i	− 0.769722i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−14.2449	−0.519804	−0.259902	−	0.965635i	$-0.583690\pi$
$751$	−14.2449	−0.519804	−0.259902	+	0.965635i	$0.583690\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10.5211	0.382902
$756$	0	0
$757$	25.9687	0.943848	0.471924	−	0.881639i	$-0.343560\pi$
$757$	25.9687	0.943848	0.471924	+	0.881639i	$0.343560\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.1691	0.803629	0.401814	−	0.915721i	$-0.368380\pi$
$761$	22.1691	0.803629	0.401814	+	0.915721i	$0.368380\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.7922i	0.786872i
$768$	0	0
$769$	15.4591i	0.557469i	0.960368	+	0.278734i	$0.0899149\pi$
$769$	15.4591i	0.557469i	−0.960368	+	0.278734i	$0.910085\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	51.8142	1.86363	0.931813	−	0.362938i	$-0.118226\pi$
$773$	51.8142	1.86363	0.931813	+	0.362938i	$0.118226\pi$
$774$	0	0
$775$	4.93119i	0.177133i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.8954i	0.426196i
$780$	0	0
$781$	−84.5424	−3.02517
$782$	0	0
$783$	0	0
$784$	0	0
$785$	17.6472i	0.629856i
$786$	0	0
$787$	14.2732i	0.508784i	0.967101	+	0.254392i	$0.0818753\pi$
$787$	14.2732i	0.508784i	−0.967101	+	0.254392i	$0.918125\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	44.1938	1.56937
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−23.5776	−0.835160	−0.417580	−	0.908640i	$-0.637122\pi$
$797$	−23.5776	−0.835160	−0.417580	+	0.908640i	$0.637122\pi$
$798$	0	0
$799$	10.4848	0.370925
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.37610	0.154429
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 39.7698i	− 1.39823i	−0.715008	−	0.699116i	$-0.753577\pi$
$809$	− 39.7698i	− 1.39823i	0.715008	−	0.699116i	$-0.246423\pi$
$810$	0	0
$811$	− 24.0838i	− 0.845697i	−0.906200	−	0.422849i	$-0.861030\pi$
$811$	− 24.0838i	− 0.845697i	0.906200	−	0.422849i	$-0.138970\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	15.8209	0.554183
$816$	0	0
$817$	− 45.4307i	− 1.58942i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.804628i	0.0280817i	0.999901	+	0.0140409i	$0.00446949\pi$
$821$	0.804628i	0.0280817i	−0.999901	+	0.0140409i	$0.995531\pi$
$822$	0	0
$823$	−8.67150	−0.302269	−0.151135	−	0.988513i	$-0.548293\pi$
$823$	−8.67150	−0.302269	−0.151135	+	0.988513i	$0.548293\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.7063i	0.685256i	0.939471	+	0.342628i	$0.111317\pi$
$827$	19.7063i	0.685256i	−0.939471	+	0.342628i	$0.888683\pi$
$828$	0	0
$829$	0.259148i	0.00900057i	0.999990	+	0.00450029i	$0.00143249\pi$
$829$	0.259148i	0.00900057i	−0.999990	+	0.00450029i	$0.998568\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0.148166	0.00512748
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.54513	−0.122392	−0.0611958	−	0.998126i	$-0.519491\pi$
$839$	−3.54513	−0.122392	−0.0611958	+	0.998126i	$0.519491\pi$
$840$	0	0
$841$	28.9250	0.997412
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.69003	0.298946
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 45.7394i	− 1.56793i
$852$	0	0
$853$	− 7.49013i	− 0.256457i	−0.991745	−	0.128229i	$-0.959071\pi$
$853$	− 7.49013i	− 0.256457i	0.991745	−	0.128229i	$-0.0409291\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−40.8290	−1.39469	−0.697345	−	0.716735i	$-0.745636\pi$
$857$	−40.8290	−1.39469	−0.697345	+	0.716735i	$0.745636\pi$
$858$	0	0
$859$	13.9180i	0.474875i	0.971403	+	0.237437i	$0.0763075\pi$
$859$	13.9180i	0.474875i	−0.971403	+	0.237437i	$0.923693\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.8371i	0.913544i	0.889584	+	0.456772i	$0.150995\pi$
$863$	26.8371i	0.913544i	−0.889584	+	0.456772i	$0.849005\pi$
$864$	0	0
$865$	22.6390	0.769749
$866$	0	0
$867$	0	0
$868$	0	0
$869$	17.4437i	0.591739i
$870$	0	0
$871$	34.7751i	1.17831i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	53.2105	1.79679	0.898396	−	0.439186i	$-0.144733\pi$
$877$	53.2105	1.79679	0.898396	+	0.439186i	$0.144733\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3.07028	−0.103440	−0.0517202	−	0.998662i	$-0.516470\pi$
$881$	−3.07028	−0.103440	−0.0517202	+	0.998662i	$0.516470\pi$
$882$	0	0
$883$	−3.99004	−0.134275	−0.0671377	−	0.997744i	$-0.521387\pi$
$883$	−3.99004	−0.134275	−0.0671377	+	0.997744i	$0.521387\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−50.0129	−1.67927	−0.839635	−	0.543151i	$-0.817231\pi$
$887$	−50.0129	−1.67927	−0.839635	+	0.543151i	$0.817231\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26.2248i	0.877578i
$894$	0	0
$895$	22.6732i	0.757883i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.35081	0.0450521
$900$	0	0
$901$	− 11.8247i	− 0.393938i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 24.4505i	− 0.812762i
$906$	0	0
$907$	20.2356	0.671913	0.335957	−	0.941877i	$-0.390940\pi$
$907$	20.2356	0.671913	0.335957	+	0.941877i	$0.390940\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 18.1459i	− 0.601199i	−0.953750	−	0.300600i	$-0.902813\pi$
$911$	− 18.1459i	− 0.601199i	0.953750	−	0.300600i	$-0.0971868\pi$
$912$	0	0
$913$	− 40.3744i	− 1.33620i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−34.6285	−1.14229	−0.571144	−	0.820850i	$-0.693500\pi$
$919$	−34.6285	−1.14229	−0.571144	+	0.820850i	$0.693500\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	66.7529	2.19720
$924$	0	0
$925$	5.83897	0.191984
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−25.7889	−0.846108	−0.423054	−	0.906105i	$-0.639042\pi$
$929$	−25.7889	−0.846108	−0.423054	+	0.906105i	$0.639042\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.4006i	0.405544i
$936$	0	0
$937$	− 0.367415i	− 0.0120029i	−0.999982	−	0.00600146i	$-0.998090\pi$
$937$	− 0.367415i	− 0.0120029i	0.999982	−	0.00600146i	$-0.00191034\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.81759	0.124450	0.0622250	−	0.998062i	$-0.480180\pi$
$941$	3.81759	0.124450	0.0622250	+	0.998062i	$0.480180\pi$
$942$	0	0
$943$	17.7203i	0.577054i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49.2913i	1.60175i	0.598829	+	0.800877i	$0.295633\pi$
$947$	49.2913i	1.60175i	−0.598829	+	0.800877i	$0.704367\pi$
$948$	0	0
$949$	−3.45527	−0.112163
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.0684i	0.358539i	0.983800	+	0.179270i	$0.0573734\pi$
$953$	11.0684i	0.358539i	−0.983800	+	0.179270i	$0.942627\pi$
$954$	0	0
$955$	0.0226722i	0 0.000733656i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	6.68339	0.215593
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.70124	−0.151338
$966$	0	0
$967$	20.3566	0.654623	0.327312	−	0.944916i	$-0.393857\pi$
$967$	20.3566	0.654623	0.327312	+	0.944916i	$0.393857\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	35.7052	1.14583	0.572917	−	0.819613i	$-0.305812\pi$
$971$	35.7052	1.14583	0.572917	+	0.819613i	$0.305812\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34.7574i	1.11199i	0.831186	+	0.555994i	$0.187662\pi$
$977$	34.7574i	1.11199i	−0.831186	+	0.555994i	$0.812338\pi$
$978$	0	0
$979$	40.1685i	1.28379i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−59.1208	−1.88566	−0.942831	−	0.333271i	$-0.891847\pi$
$983$	−59.1208	−1.88566	−0.942831	+	0.333271i	$0.891847\pi$
$984$	0	0
$985$	17.8369i	0.568330i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 67.6773i	− 2.15201i
$990$	0	0
$991$	−17.2589	−0.548247	−0.274124	−	0.961694i	$-0.588388\pi$
$991$	−17.2589	−0.548247	−0.274124	+	0.961694i	$0.588388\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.65855i	− 0.0525795i
$996$	0	0
$997$	− 8.55026i	− 0.270789i	−0.990792	−	0.135395i	$-0.956770\pi$
$997$	− 8.55026i	− 0.270789i	0.990792	−	0.135395i	$-0.0432302\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.d.a.881.12		12
3.2	odd	2		8820.2.d.b.881.1		12
7.2	even	3		1260.2.cg.b.521.4	yes	12
7.3	odd	6		1260.2.cg.a.341.4	✓	12
7.6	odd	2		8820.2.d.b.881.12		12
21.2	odd	6		1260.2.cg.a.521.4	yes	12
21.17	even	6		1260.2.cg.b.341.4	yes	12
21.20	even	2	inner	8820.2.d.a.881.1		12
35.2	odd	12		6300.2.dd.b.4049.11		24
35.3	even	12		6300.2.dd.c.1349.11		24
35.9	even	6		6300.2.ch.b.4301.3		12
35.17	even	12		6300.2.dd.c.1349.2		24
35.23	odd	12		6300.2.dd.b.4049.2		24
35.24	odd	6		6300.2.ch.c.1601.3		12
105.2	even	12		6300.2.dd.c.4049.11		24
105.17	odd	12		6300.2.dd.b.1349.2		24
105.23	even	12		6300.2.dd.c.4049.2		24
105.38	odd	12		6300.2.dd.b.1349.11		24
105.44	odd	6		6300.2.ch.c.4301.3		12
105.59	even	6		6300.2.ch.b.1601.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.cg.a.341.4	✓	12	7.3	odd	6
1260.2.cg.a.521.4	yes	12	21.2	odd	6
1260.2.cg.b.341.4	yes	12	21.17	even	6
1260.2.cg.b.521.4	yes	12	7.2	even	3
6300.2.ch.b.1601.3		12	105.59	even	6
6300.2.ch.b.4301.3		12	35.9	even	6
6300.2.ch.c.1601.3		12	35.24	odd	6
6300.2.ch.c.4301.3		12	105.44	odd	6
6300.2.dd.b.1349.2		24	105.17	odd	12
6300.2.dd.b.1349.11		24	105.38	odd	12
6300.2.dd.b.4049.2		24	35.23	odd	12
6300.2.dd.b.4049.11		24	35.2	odd	12
6300.2.dd.c.1349.2		24	35.17	even	12
6300.2.dd.c.1349.11		24	35.3	even	12
6300.2.dd.c.4049.2		24	105.23	even	12
6300.2.dd.c.4049.11		24	105.2	even	12
8820.2.d.a.881.1		12	21.20	even	2	inner
8820.2.d.a.881.12		12	1.1	even	1	trivial
8820.2.d.b.881.1		12	3.2	odd	2
8820.2.d.b.881.12		12	7.6	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +O(q^{10})\)	\(q-1.00000 q^{5} +5.89840i q^{11} -4.65726i q^{13} -2.10237 q^{17} -5.25849i q^{19} -7.83347i q^{23} +1.00000 q^{25} -0.273932i q^{29} +4.93119i q^{31} +5.83897 q^{37} -2.26213 q^{41} +8.63950 q^{43} -4.98713 q^{47} +5.62447i q^{53} -5.89840i q^{55} -4.67920 q^{59} +9.48923i q^{61} +4.65726i q^{65} -7.46687 q^{67} +14.3331i q^{71} -0.741912i q^{73} +2.95737 q^{79} -6.84497 q^{83} +2.10237 q^{85} +6.81007 q^{89} +5.25849i q^{95} +4.90827i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{5}+O(q^{10}) \) 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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