LMFDB - Embedded newform 8820.2.d.c.881.11

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			881.11
Root			$0.500000 + 2.81224i$ of defining polynomial
Character	$\chi$	$=$	8820.881
Dual form			8820.2.d.c.881.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +O(q^{10})$	$q-1.00000 q^{5} +0.941197i q^{11} +1.01212i q^{13} -7.32886 q^{17} -7.97770i q^{19} +7.85911i q^{23} +1.00000 q^{25} +6.10587i q^{29} -5.03739i q^{31} +7.30478 q^{37} -2.21038 q^{41} -5.26816 q^{43} +1.12907 q^{47} -9.73171i q^{53} -0.941197i q^{55} +12.5799 q^{59} +14.5839i q^{61} -1.01212i q^{65} -14.2980 q^{67} -1.77255i q^{71} -3.40369i q^{73} -12.3430 q^{79} +1.83989 q^{83} +7.32886 q^{85} +14.1493 q^{89} +7.97770i q^{95} -16.1966i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	0.941197i	0.283781i	0.989882	+	0.141891i	$0.0453182\pi$
$11$	0.941197i	0.283781i	−0.989882	+	0.141891i	$0.954682\pi$
$12$	0	0
$13$	1.01212i	0.280712i	0.990101	+	0.140356i	$0.0448248\pi$
$13$	1.01212i	0.280712i	−0.990101	+	0.140356i	$0.955175\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.32886	−1.77751	−0.888754	−	0.458384i	$-0.848429\pi$
$17$	−7.32886	−1.77751	−0.888754	+	0.458384i	$0.848429\pi$
$18$	0	0
$19$	− 7.97770i	− 1.83021i	−0.403215	−	0.915105i	$-0.632107\pi$
$19$	− 7.97770i	− 1.83021i	0.403215	−	0.915105i	$-0.367893\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.85911i	1.63874i	0.573266	+	0.819369i	$0.305676\pi$
$23$	7.85911i	1.63874i	−0.573266	+	0.819369i	$0.694324\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.10587i	1.13383i	0.823776	+	0.566916i	$0.191864\pi$
$29$	6.10587i	1.13383i	−0.823776	+	0.566916i	$0.808136\pi$
$30$	0	0
$31$	− 5.03739i	− 0.904741i	−0.891830	−	0.452371i	$-0.850578\pi$
$31$	− 5.03739i	− 0.904741i	0.891830	−	0.452371i	$-0.149422\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.30478	1.20090	0.600449	−	0.799663i	$-0.294988\pi$
$37$	7.30478	1.20090	0.600449	+	0.799663i	$0.294988\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.21038	−0.345203	−0.172602	−	0.984992i	$-0.555217\pi$
$41$	−2.21038	−0.345203	−0.172602	+	0.984992i	$0.555217\pi$
$42$	0	0
$43$	−5.26816	−0.803386	−0.401693	−	0.915774i	$-0.631578\pi$
$43$	−5.26816	−0.803386	−0.401693	+	0.915774i	$0.631578\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.12907	0.164692	0.0823461	−	0.996604i	$-0.473759\pi$
$47$	1.12907	0.164692	0.0823461	+	0.996604i	$0.473759\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.73171i	− 1.33675i	−0.743823	−	0.668376i	$-0.766990\pi$
$53$	− 9.73171i	− 1.33675i	0.743823	−	0.668376i	$-0.233010\pi$
$54$	0	0
$55$	− 0.941197i	− 0.126911i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.5799	1.63776	0.818880	−	0.573965i	$-0.194596\pi$
$59$	12.5799	1.63776	0.818880	+	0.573965i	$0.194596\pi$
$60$	0	0
$61$	14.5839i	1.86728i	0.358217	+	0.933638i	$0.383385\pi$
$61$	14.5839i	1.86728i	−0.358217	+	0.933638i	$0.616615\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.01212i	− 0.125538i
$66$	0	0
$67$	−14.2980	−1.74678	−0.873389	−	0.487023i	$-0.838083\pi$
$67$	−14.2980	−1.74678	−0.873389	+	0.487023i	$0.838083\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 1.77255i	− 0.210363i	−0.994453	−	0.105181i	$-0.966458\pi$
$71$	− 1.77255i	− 0.210363i	0.994453	−	0.105181i	$-0.0335424\pi$
$72$	0	0
$73$	− 3.40369i	− 0.398372i	−0.979962	−	0.199186i	$-0.936170\pi$
$73$	− 3.40369i	− 0.398372i	0.979962	−	0.199186i	$-0.0638298\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.3430	−1.38870	−0.694351	−	0.719637i	$-0.744308\pi$
$79$	−12.3430	−1.38870	−0.694351	+	0.719637i	$0.744308\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.83989	0.201955	0.100977	−	0.994889i	$-0.467803\pi$
$83$	1.83989	0.201955	0.100977	+	0.994889i	$0.467803\pi$
$84$	0	0
$85$	7.32886	0.794926
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.1493	1.49982	0.749910	−	0.661540i	$-0.230097\pi$
$89$	14.1493	1.49982	0.749910	+	0.661540i	$0.230097\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.97770i	0.818495i
$96$	0	0
$97$	− 16.1966i	− 1.64452i	−0.569115	−	0.822258i	$-0.692714\pi$
$97$	− 16.1966i	− 1.64452i	0.569115	−	0.822258i	$-0.307286\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.1079	1.60279	0.801396	−	0.598135i	$-0.204091\pi$
$101$	16.1079	1.60279	0.801396	+	0.598135i	$0.204091\pi$
$102$	0	0
$103$	4.85524i	0.478401i	0.970970	+	0.239200i	$0.0768853\pi$
$103$	4.85524i	0.478401i	−0.970970	+	0.239200i	$0.923115\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.32215i	− 0.707859i	−0.935272	−	0.353929i	$-0.884845\pi$
$107$	− 7.32215i	− 0.707859i	0.935272	−	0.353929i	$-0.115155\pi$
$108$	0	0
$109$	1.82749	0.175042	0.0875210	−	0.996163i	$-0.472106\pi$
$109$	1.82749	0.175042	0.0875210	+	0.996163i	$0.472106\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.57403i	− 0.148072i	−0.997256	−	0.0740361i	$-0.976412\pi$
$113$	− 1.57403i	− 0.148072i	0.997256	−	0.0740361i	$-0.0235880\pi$
$114$	0	0
$115$	− 7.85911i	− 0.732866i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.1141	0.919468
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.0323	1.15643	0.578216	−	0.815884i	$-0.303749\pi$
$127$	13.0323	1.15643	0.578216	+	0.815884i	$0.303749\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.03443	−0.0903786	−0.0451893	−	0.998978i	$-0.514389\pi$
$131$	−1.03443	−0.0903786	−0.0451893	+	0.998978i	$0.514389\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.68028i	− 0.143556i	−0.997421	−	0.0717780i	$-0.977133\pi$
$137$	− 1.68028i	− 0.143556i	0.997421	−	0.0717780i	$-0.0228673\pi$
$138$	0	0
$139$	− 7.58686i	− 0.643509i	−0.946823	−	0.321755i	$-0.895727\pi$
$139$	− 7.58686i	− 0.643509i	0.946823	−	0.321755i	$-0.104273\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.952607	−0.0796610
$144$	0	0
$145$	− 6.10587i	− 0.507065i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.3641i	0.930983i	0.885052	+	0.465491i	$0.154122\pi$
$149$	11.3641i	0.930983i	−0.885052	+	0.465491i	$0.845878\pi$
$150$	0	0
$151$	−0.797749	−0.0649199	−0.0324599	−	0.999473i	$-0.510334\pi$
$151$	−0.797749	−0.0649199	−0.0324599	+	0.999473i	$0.510334\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.03739i	0.404613i
$156$	0	0
$157$	− 14.6464i	− 1.16891i	−0.811427	−	0.584454i	$-0.801309\pi$
$157$	− 14.6464i	− 1.16891i	0.811427	−	0.584454i	$-0.198691\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	21.3636	1.67333	0.836664	−	0.547717i	$-0.184503\pi$
$163$	21.3636	1.67333	0.836664	+	0.547717i	$0.184503\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.82532	−0.218630	−0.109315	−	0.994007i	$-0.534866\pi$
$167$	−2.82532	−0.218630	−0.109315	+	0.994007i	$0.534866\pi$
$168$	0	0
$169$	11.9756	0.921201
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15.8101	−1.20202	−0.601010	−	0.799242i	$-0.705235\pi$
$173$	−15.8101	−1.20202	−0.601010	+	0.799242i	$0.705235\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 16.7824i	− 1.25437i	−0.778869	−	0.627186i	$-0.784207\pi$
$179$	− 16.7824i	− 1.25437i	0.778869	−	0.627186i	$-0.215793\pi$
$180$	0	0
$181$	3.70982i	0.275749i	0.990450	+	0.137874i	$0.0440270\pi$
$181$	3.70982i	0.275749i	−0.990450	+	0.137874i	$0.955973\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.30478	−0.537058
$186$	0	0
$187$	− 6.89790i	− 0.504424i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.8232i	1.50671i	0.657613	+	0.753356i	$0.271566\pi$
$191$	20.8232i	1.50671i	−0.657613	+	0.753356i	$0.728434\pi$
$192$	0	0
$193$	−22.9660	−1.65313	−0.826564	−	0.562843i	$-0.809707\pi$
$193$	−22.9660	−1.65313	−0.826564	+	0.562843i	$0.809707\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.34252i	− 0.523133i	−0.965185	−	0.261567i	$-0.915761\pi$
$197$	− 7.34252i	− 0.523133i	0.965185	−	0.261567i	$-0.0842391\pi$
$198$	0	0
$199$	− 14.8007i	− 1.04920i	−0.851350	−	0.524598i	$-0.824216\pi$
$199$	− 14.8007i	− 1.04920i	0.851350	−	0.524598i	$-0.175784\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.21038	0.154380
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.50859	0.519380
$210$	0	0
$211$	10.7047	0.736943	0.368471	−	0.929639i	$-0.379881\pi$
$211$	10.7047	0.736943	0.368471	+	0.929639i	$0.379881\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.26816	0.359285
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 7.41770i	− 0.498969i
$222$	0	0
$223$	10.4078i	0.696958i	0.937317	+	0.348479i	$0.113302\pi$
$223$	10.4078i	0.696958i	−0.937317	+	0.348479i	$0.886698\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.2989	1.01542	0.507712	−	0.861527i	$-0.330491\pi$
$227$	15.2989	1.01542	0.507712	+	0.861527i	$0.330491\pi$
$228$	0	0
$229$	− 4.29481i	− 0.283809i	−0.989880	−	0.141905i	$-0.954677\pi$
$229$	− 4.29481i	− 0.283809i	0.989880	−	0.141905i	$-0.0453226\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.9687i	1.37371i	0.726796	+	0.686853i	$0.241008\pi$
$233$	20.9687i	1.37371i	−0.726796	+	0.686853i	$0.758992\pi$
$234$	0	0
$235$	−1.12907	−0.0736526
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.4016i	0.996247i	0.867106	+	0.498124i	$0.165977\pi$
$239$	15.4016i	0.996247i	−0.867106	+	0.498124i	$0.834023\pi$
$240$	0	0
$241$	− 5.21244i	− 0.335763i	−0.985807	−	0.167881i	$-0.946307\pi$
$241$	− 5.21244i	− 0.335763i	0.985807	−	0.167881i	$-0.0536926\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.07441	0.513763
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.04612	0.381628	0.190814	−	0.981626i	$-0.438887\pi$
$251$	6.04612	0.381628	0.190814	+	0.981626i	$0.438887\pi$
$252$	0	0
$253$	−7.39697	−0.465044
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.14879	0.258795	0.129397	−	0.991593i	$-0.458696\pi$
$257$	4.14879	0.258795	0.129397	+	0.991593i	$0.458696\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.13375i	0.563211i	0.959530	+	0.281605i	$0.0908669\pi$
$263$	9.13375i	0.563211i	−0.959530	+	0.281605i	$0.909133\pi$
$264$	0	0
$265$	9.73171i	0.597814i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.85286	0.478797	0.239399	−	0.970921i	$-0.423050\pi$
$269$	7.85286	0.478797	0.239399	+	0.970921i	$0.423050\pi$
$270$	0	0
$271$	− 22.0705i	− 1.34069i	−0.742051	−	0.670343i	$-0.766147\pi$
$271$	− 22.0705i	− 1.34069i	0.742051	−	0.670343i	$-0.233853\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.941197i	0.0567563i
$276$	0	0
$277$	4.22161	0.253652	0.126826	−	0.991925i	$-0.459521\pi$
$277$	4.22161	0.253652	0.126826	+	0.991925i	$0.459521\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 5.58495i	− 0.333170i	−0.986027	−	0.166585i	$-0.946726\pi$
$281$	− 5.58495i	− 0.333170i	0.986027	−	0.166585i	$-0.0532741\pi$
$282$	0	0
$283$	− 17.6005i	− 1.04624i	−0.852258	−	0.523121i	$-0.824768\pi$
$283$	− 17.6005i	− 1.04624i	0.852258	−	0.523121i	$-0.175232\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	36.7121	2.15954
$290$	0	0
$291$	0	0
$292$	0	0
$293$	13.6285	0.796185	0.398092	−	0.917345i	$-0.369672\pi$
$293$	13.6285	0.796185	0.398092	+	0.917345i	$0.369672\pi$
$294$	0	0
$295$	−12.5799	−0.732428
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.95439	−0.460014
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 14.5839i	− 0.835072i
$306$	0	0
$307$	14.9330i	0.852271i	0.904659	+	0.426135i	$0.140125\pi$
$307$	14.9330i	0.852271i	−0.904659	+	0.426135i	$0.859875\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.64889	0.206910	0.103455	−	0.994634i	$-0.467010\pi$
$311$	3.64889	0.206910	0.103455	+	0.994634i	$0.467010\pi$
$312$	0	0
$313$	− 14.8278i	− 0.838119i	−0.907959	−	0.419059i	$-0.862360\pi$
$313$	− 14.8278i	− 0.838119i	0.907959	−	0.419059i	$-0.137640\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.7883i	0.886759i	0.896334	+	0.443379i	$0.146221\pi$
$317$	15.7883i	0.886759i	−0.896334	+	0.443379i	$0.853779\pi$
$318$	0	0
$319$	−5.74682	−0.321760
$320$	0	0
$321$	0	0
$322$	0	0
$323$	58.4674i	3.25322i
$324$	0	0
$325$	1.01212i	0.0561425i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.22894	0.452303	0.226152	−	0.974092i	$-0.427386\pi$
$331$	8.22894	0.452303	0.226152	+	0.974092i	$0.427386\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14.2980	0.781183
$336$	0	0
$337$	−9.07521	−0.494358	−0.247179	−	0.968970i	$-0.579504\pi$
$337$	−9.07521	−0.494358	−0.247179	+	0.968970i	$0.579504\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.74117	0.256749
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.7756i	− 1.00793i	−0.863724	−	0.503965i	$-0.831874\pi$
$347$	− 18.7756i	− 1.00793i	0.863724	−	0.503965i	$-0.168126\pi$
$348$	0	0
$349$	− 0.464593i	− 0.0248691i	−0.999923	−	0.0124345i	$-0.996042\pi$
$349$	− 0.464593i	− 0.0248691i	0.999923	−	0.0124345i	$-0.00395814\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.78079	0.307680	0.153840	−	0.988096i	$-0.450836\pi$
$353$	5.78079	0.307680	0.153840	+	0.988096i	$0.450836\pi$
$354$	0	0
$355$	1.77255i	0.0940772i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 16.2278i	− 0.856471i	−0.903667	−	0.428235i	$-0.859135\pi$
$359$	− 16.2278i	− 0.856471i	0.903667	−	0.428235i	$-0.140865\pi$
$360$	0	0
$361$	−44.6437	−2.34967
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.40369i	0.178157i
$366$	0	0
$367$	− 18.4434i	− 0.962738i	−0.876518	−	0.481369i	$-0.840140\pi$
$367$	− 18.4434i	− 0.962738i	0.876518	−	0.481369i	$-0.159860\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7.41051	−0.383702	−0.191851	−	0.981424i	$-0.561449\pi$
$373$	−7.41051	−0.383702	−0.191851	+	0.981424i	$0.561449\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.17989	−0.318280
$378$	0	0
$379$	28.0696	1.44184	0.720919	−	0.693020i	$-0.243720\pi$
$379$	28.0696	1.44184	0.720919	+	0.693020i	$0.243720\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.2609	1.13748	0.568739	−	0.822518i	$-0.307432\pi$
$383$	22.2609	1.13748	0.568739	+	0.822518i	$0.307432\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11.4817i	0.582146i	0.956701	+	0.291073i	$0.0940123\pi$
$389$	11.4817i	0.582146i	−0.956701	+	0.291073i	$0.905988\pi$
$390$	0	0
$391$	− 57.5983i	− 2.91287i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.3430	0.621046
$396$	0	0
$397$	23.1629i	1.16251i	0.813720	+	0.581257i	$0.197439\pi$
$397$	23.1629i	1.16251i	−0.813720	+	0.581257i	$0.802561\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 33.4930i	− 1.67256i	−0.548300	−	0.836281i	$-0.684725\pi$
$401$	− 33.4930i	− 1.67256i	0.548300	−	0.836281i	$-0.315275\pi$
$402$	0	0
$403$	5.09845	0.253972
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.87524i	0.340793i
$408$	0	0
$409$	− 13.5830i	− 0.671636i	−0.941927	−	0.335818i	$-0.890987\pi$
$409$	− 13.5830i	− 0.671636i	0.941927	−	0.335818i	$-0.109013\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.83989	−0.0903168
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.43412	0.363181	0.181590	−	0.983374i	$-0.441876\pi$
$419$	7.43412	0.363181	0.181590	+	0.983374i	$0.441876\pi$
$420$	0	0
$421$	30.2839	1.47594	0.737972	−	0.674831i	$-0.235783\pi$
$421$	30.2839	1.47594	0.737972	+	0.674831i	$0.235783\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.32886	−0.355502
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 27.8265i	− 1.34035i	−0.742201	−	0.670177i	$-0.766218\pi$
$431$	− 27.8265i	− 1.34035i	0.742201	−	0.670177i	$-0.233782\pi$
$432$	0	0
$433$	− 17.5728i	− 0.844493i	−0.906481	−	0.422246i	$-0.861242\pi$
$433$	− 17.5728i	− 0.844493i	0.906481	−	0.422246i	$-0.138758\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	62.6977	2.99924
$438$	0	0
$439$	27.6020i	1.31737i	0.752419	+	0.658685i	$0.228887\pi$
$439$	27.6020i	1.31737i	−0.752419	+	0.658685i	$0.771113\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 22.0324i	− 1.04679i	−0.852090	−	0.523396i	$-0.824665\pi$
$443$	− 22.0324i	− 1.04679i	0.852090	−	0.523396i	$-0.175335\pi$
$444$	0	0
$445$	−14.1493	−0.670740
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 34.3737i	− 1.62219i	−0.584911	−	0.811097i	$-0.698871\pi$
$449$	− 34.3737i	− 1.62219i	0.584911	−	0.811097i	$-0.301129\pi$
$450$	0	0
$451$	− 2.08040i	− 0.0979623i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.4462	1.14355	0.571773	−	0.820412i	$-0.306256\pi$
$457$	24.4462	1.14355	0.571773	+	0.820412i	$0.306256\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.0010	−0.605515	−0.302758	−	0.953068i	$-0.597907\pi$
$461$	−13.0010	−0.605515	−0.302758	+	0.953068i	$0.597907\pi$
$462$	0	0
$463$	24.6032	1.14341	0.571704	−	0.820460i	$-0.306283\pi$
$463$	24.6032	1.14341	0.571704	+	0.820460i	$0.306283\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.4522	−0.483669	−0.241835	−	0.970317i	$-0.577749\pi$
$467$	−10.4522	−0.483669	−0.241835	+	0.970317i	$0.577749\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 4.95837i	− 0.227986i
$474$	0	0
$475$	− 7.97770i	− 0.366042i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17.7919	−0.812934	−0.406467	−	0.913665i	$-0.633239\pi$
$479$	−17.7919	−0.812934	−0.406467	+	0.913665i	$0.633239\pi$
$480$	0	0
$481$	7.39334i	0.337107i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.1966i	0.735450i
$486$	0	0
$487$	7.59807	0.344302	0.172151	−	0.985071i	$-0.444928\pi$
$487$	7.59807	0.344302	0.172151	+	0.985071i	$0.444928\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 39.6496i	− 1.78936i	−0.446707	−	0.894680i	$-0.647403\pi$
$491$	− 39.6496i	− 1.78936i	0.446707	−	0.894680i	$-0.352597\pi$
$492$	0	0
$493$	− 44.7490i	− 2.01540i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.4468	1.18392	0.591960	−	0.805967i	$-0.298354\pi$
$499$	26.4468	1.18392	0.591960	+	0.805967i	$0.298354\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	25.9598	1.15749	0.578745	−	0.815509i	$-0.303543\pi$
$503$	25.9598	1.15749	0.578745	+	0.815509i	$0.303543\pi$
$504$	0	0
$505$	−16.1079	−0.716790
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−31.8917	−1.41357	−0.706787	−	0.707426i	$-0.749856\pi$
$509$	−31.8917	−1.41357	−0.706787	+	0.707426i	$0.749856\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 4.85524i	− 0.213947i
$516$	0	0
$517$	1.06268i	0.0467366i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.7280	1.56527	0.782636	−	0.622479i	$-0.213874\pi$
$521$	35.7280	1.56527	0.782636	+	0.622479i	$0.213874\pi$
$522$	0	0
$523$	− 35.5212i	− 1.55323i	−0.629974	−	0.776617i	$-0.716934\pi$
$523$	− 35.5212i	− 1.55323i	0.629974	−	0.776617i	$-0.283066\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36.9183i	1.60819i
$528$	0	0
$529$	−38.7657	−1.68546
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2.23717i	− 0.0969028i
$534$	0	0
$535$	7.32215i	0.316564i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.52546	0.323545	0.161772	−	0.986828i	$-0.448279\pi$
$541$	7.52546	0.323545	0.161772	+	0.986828i	$0.448279\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.82749	−0.0782811
$546$	0	0
$547$	32.8568	1.40485	0.702427	−	0.711756i	$-0.252100\pi$
$547$	32.8568	1.40485	0.702427	+	0.711756i	$0.252100\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	48.7108	2.07515
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11.7457i	− 0.497680i	−0.968545	−	0.248840i	$-0.919951\pi$
$557$	− 11.7457i	− 0.497680i	0.968545	−	0.248840i	$-0.0800493\pi$
$558$	0	0
$559$	− 5.33202i	− 0.225520i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−26.3989	−1.11258	−0.556291	−	0.830987i	$-0.687776\pi$
$563$	−26.3989	−1.11258	−0.556291	+	0.830987i	$0.687776\pi$
$564$	0	0
$565$	1.57403i	0.0662199i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.0676i	1.26050i	0.776393	+	0.630250i	$0.217047\pi$
$569$	30.0676i	1.26050i	−0.776393	+	0.630250i	$0.782953\pi$
$570$	0	0
$571$	28.1491	1.17800	0.589001	−	0.808132i	$-0.299521\pi$
$571$	28.1491	1.17800	0.589001	+	0.808132i	$0.299521\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.85911i	0.327748i
$576$	0	0
$577$	20.0114i	0.833084i	0.909116	+	0.416542i	$0.136758\pi$
$577$	20.0114i	0.833084i	−0.909116	+	0.416542i	$0.863242\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.15945	0.379346
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.433942	0.0179107	0.00895534	−	0.999960i	$-0.497149\pi$
$587$	0.433942	0.0179107	0.00895534	+	0.999960i	$0.497149\pi$
$588$	0	0
$589$	−40.1868	−1.65587
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−37.3861	−1.53526	−0.767631	−	0.640892i	$-0.778565\pi$
$593$	−37.3861	−1.53526	−0.767631	+	0.640892i	$0.778565\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1.03693i	− 0.0423677i	−0.999776	−	0.0211838i	$-0.993256\pi$
$599$	− 1.03693i	− 0.0423677i	0.999776	−	0.0211838i	$-0.00674353\pi$
$600$	0	0
$601$	17.0643i	0.696067i	0.937482	+	0.348033i	$0.113150\pi$
$601$	17.0643i	0.696067i	−0.937482	+	0.348033i	$0.886850\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.1141	−0.411199
$606$	0	0
$607$	− 27.7253i	− 1.12534i	−0.826683	−	0.562668i	$-0.809775\pi$
$607$	− 27.7253i	− 1.12534i	0.826683	−	0.562668i	$-0.190225\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.14276i	0.0462311i
$612$	0	0
$613$	7.56451	0.305528	0.152764	−	0.988263i	$-0.451183\pi$
$613$	7.56451	0.305528	0.152764	+	0.988263i	$0.451183\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.6945i	0.833129i	0.909106	+	0.416565i	$0.136766\pi$
$617$	20.6945i	0.833129i	−0.909106	+	0.416565i	$0.863234\pi$
$618$	0	0
$619$	− 19.1688i	− 0.770458i	−0.922821	−	0.385229i	$-0.874122\pi$
$619$	− 19.1688i	− 0.770458i	0.922821	−	0.385229i	$-0.125878\pi$
$620$	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
$629$	−53.5357	−2.13461
$630$	0	0
$631$	30.3932	1.20993	0.604967	−	0.796250i	$-0.293186\pi$
$631$	30.3932	1.20993	0.604967	+	0.796250i	$0.293186\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.0323	−0.517172
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 17.9617i	− 0.709443i	−0.934972	−	0.354721i	$-0.884576\pi$
$641$	− 17.9617i	− 0.709443i	0.934972	−	0.354721i	$-0.115424\pi$
$642$	0	0
$643$	40.3534i	1.59138i	0.605703	+	0.795691i	$0.292892\pi$
$643$	40.3534i	1.59138i	−0.605703	+	0.795691i	$0.707108\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.9732	0.667285	0.333642	−	0.942700i	$-0.391722\pi$
$647$	16.9732	0.667285	0.333642	+	0.942700i	$0.391722\pi$
$648$	0	0
$649$	11.8401i	0.464766i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.5998i	0.727865i	0.931425	+	0.363933i	$0.118566\pi$
$653$	18.5998i	0.727865i	−0.931425	+	0.363933i	$0.881434\pi$
$654$	0	0
$655$	1.03443	0.0404185
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.28333i	− 0.0889459i	−0.999011	−	0.0444730i	$-0.985839\pi$
$659$	− 2.28333i	− 0.0889459i	0.999011	−	0.0444730i	$-0.0141609\pi$
$660$	0	0
$661$	23.3983i	0.910087i	0.890469	+	0.455044i	$0.150376\pi$
$661$	23.3983i	0.910087i	−0.890469	+	0.455044i	$0.849624\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−47.9867	−1.85805
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−13.7263	−0.529899
$672$	0	0
$673$	−5.45915	−0.210435	−0.105217	−	0.994449i	$-0.533554\pi$
$673$	−5.45915	−0.210435	−0.105217	+	0.994449i	$0.533554\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.9943	1.42181	0.710904	−	0.703289i	$-0.248286\pi$
$677$	36.9943	1.42181	0.710904	+	0.703289i	$0.248286\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.8489i	0.606442i	0.952920	+	0.303221i	$0.0980620\pi$
$683$	15.8489i	0.606442i	−0.952920	+	0.303221i	$0.901938\pi$
$684$	0	0
$685$	1.68028i	0.0642002i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.84968	0.375243
$690$	0	0
$691$	− 22.7819i	− 0.866662i	−0.901235	−	0.433331i	$-0.857338\pi$
$691$	− 22.7819i	− 0.866662i	0.901235	−	0.433331i	$-0.142662\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.58686i	0.287786i
$696$	0	0
$697$	16.1995	0.613602
$698$	0	0
$699$	0	0
$700$	0	0
$701$	35.3238i	1.33416i	0.744986	+	0.667081i	$0.232456\pi$
$701$	35.3238i	1.33416i	−0.744986	+	0.667081i	$0.767544\pi$
$702$	0	0
$703$	− 58.2754i	− 2.19790i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	32.6162	1.22493	0.612464	−	0.790499i	$-0.290178\pi$
$709$	32.6162	1.22493	0.612464	+	0.790499i	$0.290178\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	39.5894	1.48263
$714$	0	0
$715$	0.952607	0.0356255
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.07656	0.301205	0.150602	−	0.988594i	$-0.451879\pi$
$719$	8.07656	0.301205	0.150602	+	0.988594i	$0.451879\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.10587i	0.226766i
$726$	0	0
$727$	− 11.1183i	− 0.412356i	−0.978515	−	0.206178i	$-0.933897\pi$
$727$	− 11.1183i	− 0.412356i	0.978515	−	0.206178i	$-0.0661026\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	38.6096	1.42803
$732$	0	0
$733$	− 16.1458i	− 0.596358i	−0.954510	−	0.298179i	$-0.903621\pi$
$733$	− 16.1458i	− 0.596358i	0.954510	−	0.298179i	$-0.0963792\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 13.4572i	− 0.495703i
$738$	0	0
$739$	−30.8406	−1.13449	−0.567244	−	0.823549i	$-0.691991\pi$
$739$	−30.8406	−1.13449	−0.567244	+	0.823549i	$0.691991\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.3861i	1.15144i	0.817645	+	0.575722i	$0.195279\pi$
$743$	31.3861i	1.15144i	−0.817645	+	0.575722i	$0.804721\pi$
$744$	0	0
$745$	− 11.3641i	− 0.416348i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	34.4562	1.25733	0.628663	−	0.777678i	$-0.283603\pi$
$751$	34.4562	1.25733	0.628663	+	0.777678i	$0.283603\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.797749	0.0290330
$756$	0	0
$757$	12.5272	0.455308	0.227654	−	0.973742i	$-0.426895\pi$
$757$	12.5272	0.455308	0.227654	+	0.973742i	$0.426895\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.9925	0.688478	0.344239	−	0.938882i	$-0.388137\pi$
$761$	18.9925	0.688478	0.344239	+	0.938882i	$0.388137\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.7324i	0.459739i
$768$	0	0
$769$	− 2.58180i	− 0.0931019i	−0.998916	−	0.0465510i	$-0.985177\pi$
$769$	− 2.58180i	− 0.0931019i	0.998916	−	0.0465510i	$-0.0148230\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−53.9263	−1.93960	−0.969798	−	0.243910i	$-0.921570\pi$
$773$	−53.9263	−1.93960	−0.969798	+	0.243910i	$0.921570\pi$
$774$	0	0
$775$	− 5.03739i	− 0.180948i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	17.6337i	0.631794i
$780$	0	0
$781$	1.66832	0.0596971
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.6464i	0.522751i
$786$	0	0
$787$	18.2218i	0.649539i	0.945793	+	0.324769i	$0.105287\pi$
$787$	18.2218i	0.649539i	−0.945793	+	0.324769i	$0.894713\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−14.7607	−0.524168
$794$	0	0
$795$	0	0
$796$	0	0
$797$	47.6655	1.68840	0.844200	−	0.536029i	$-0.180076\pi$
$797$	47.6655	1.68840	0.844200	+	0.536029i	$0.180076\pi$
$798$	0	0
$799$	−8.27481	−0.292742
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.20354	0.113051
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 5.79516i	− 0.203747i	−0.994797	−	0.101874i	$-0.967516\pi$
$809$	− 5.79516i	− 0.203747i	0.994797	−	0.101874i	$-0.0324837\pi$
$810$	0	0
$811$	13.2955i	0.466867i	0.972373	+	0.233434i	$0.0749962\pi$
$811$	13.2955i	0.466867i	−0.972373	+	0.233434i	$0.925004\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−21.3636	−0.748335
$816$	0	0
$817$	42.0278i	1.47037i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.3375i	1.09369i	0.837235	+	0.546843i	$0.184170\pi$
$821$	31.3375i	1.09369i	−0.837235	+	0.546843i	$0.815830\pi$
$822$	0	0
$823$	−38.1510	−1.32986	−0.664930	−	0.746905i	$-0.731539\pi$
$823$	−38.1510	−1.32986	−0.664930	+	0.746905i	$0.731539\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.40514i	0.118408i	0.998246	+	0.0592042i	$0.0188563\pi$
$827$	3.40514i	0.118408i	−0.998246	+	0.0592042i	$0.981144\pi$
$828$	0	0
$829$	9.89352i	0.343616i	0.985130	+	0.171808i	$0.0549609\pi$
$829$	9.89352i	0.343616i	−0.985130	+	0.171808i	$0.945039\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.82532	0.0977742
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.5340	1.08867	0.544337	−	0.838867i	$-0.316781\pi$
$839$	31.5340	1.08867	0.544337	+	0.838867i	$0.316781\pi$
$840$	0	0
$841$	−8.28164	−0.285574
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.9756	−0.411973
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	57.4091i	1.96796i
$852$	0	0
$853$	− 0.887988i	− 0.0304041i	−0.999884	−	0.0152021i	$-0.995161\pi$
$853$	− 0.887988i	− 0.0304041i	0.999884	−	0.0152021i	$-0.00483915\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−49.8024	−1.70122	−0.850610	−	0.525798i	$-0.823767\pi$
$857$	−49.8024	−1.70122	−0.850610	+	0.525798i	$0.823767\pi$
$858$	0	0
$859$	− 7.96101i	− 0.271626i	−0.990734	−	0.135813i	$-0.956635\pi$
$859$	− 7.96101i	− 0.271626i	0.990734	−	0.135813i	$-0.0433647\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.26059i	0.281194i	0.990067	+	0.140597i	$0.0449021\pi$
$863$	8.26059i	0.281194i	−0.990067	+	0.140597i	$0.955098\pi$
$864$	0	0
$865$	15.8101	0.537560
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 11.6172i	− 0.394088i
$870$	0	0
$871$	− 14.4713i	− 0.490342i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.9341	1.51732	0.758659	−	0.651488i	$-0.225855\pi$
$877$	44.9341	1.51732	0.758659	+	0.651488i	$0.225855\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−56.3478	−1.89841	−0.949203	−	0.314666i	$-0.898108\pi$
$881$	−56.3478	−1.89841	−0.949203	+	0.314666i	$0.898108\pi$
$882$	0	0
$883$	−49.8781	−1.67853	−0.839265	−	0.543722i	$-0.817014\pi$
$883$	−49.8781	−1.67853	−0.839265	+	0.543722i	$0.817014\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.5002	0.789059	0.394529	−	0.918883i	$-0.370908\pi$
$887$	23.5002	0.789059	0.394529	+	0.918883i	$0.370908\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 9.00741i	− 0.301421i
$894$	0	0
$895$	16.7824i	0.560972i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.7576	1.02582
$900$	0	0
$901$	71.3223i	2.37609i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 3.70982i	− 0.123319i
$906$	0	0
$907$	−44.9768	−1.49343	−0.746715	−	0.665144i	$-0.768370\pi$
$907$	−44.9768	−1.49343	−0.746715	+	0.665144i	$0.768370\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 26.5280i	− 0.878912i	−0.898264	−	0.439456i	$-0.855171\pi$
$911$	− 26.5280i	− 0.878912i	0.898264	−	0.439456i	$-0.144829\pi$
$912$	0	0
$913$	1.73170i	0.0573110i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−45.6470	−1.50575	−0.752877	−	0.658161i	$-0.771335\pi$
$919$	−45.6470	−1.50575	−0.752877	+	0.658161i	$0.771335\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.79404	0.0590515
$924$	0	0
$925$	7.30478	0.240180
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.06071	0.0348008	0.0174004	−	0.999849i	$-0.494461\pi$
$929$	1.06071	0.0348008	0.0174004	+	0.999849i	$0.494461\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.89790i	0.225585i
$936$	0	0
$937$	− 50.1604i	− 1.63867i	−0.573316	−	0.819334i	$-0.694343\pi$
$937$	− 50.1604i	− 1.63867i	0.573316	−	0.819334i	$-0.305657\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−21.6434	−0.705555	−0.352778	−	0.935707i	$-0.614763\pi$
$941$	−21.6434	−0.705555	−0.352778	+	0.935707i	$0.614763\pi$
$942$	0	0
$943$	− 17.3716i	− 0.565698i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 38.6050i	− 1.25449i	−0.778820	−	0.627247i	$-0.784181\pi$
$947$	− 38.6050i	− 1.25449i	0.778820	−	0.627247i	$-0.215819\pi$
$948$	0	0
$949$	3.44495	0.111828
$950$	0	0
$951$	0	0
$952$	0	0
$953$	57.2748i	1.85531i	0.373435	+	0.927656i	$0.378180\pi$
$953$	57.2748i	1.85531i	−0.373435	+	0.927656i	$0.621820\pi$
$954$	0	0
$955$	− 20.8232i	− 0.673822i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.62474	0.181443
$962$	0	0
$963$	0	0
$964$	0	0
$965$	22.9660	0.739301
$966$	0	0
$967$	15.1201	0.486231	0.243115	−	0.969997i	$-0.421831\pi$
$967$	15.1201	0.486231	0.243115	+	0.969997i	$0.421831\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.2051	−0.712596	−0.356298	−	0.934372i	$-0.615961\pi$
$971$	−22.2051	−0.712596	−0.356298	+	0.934372i	$0.615961\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 25.6755i	− 0.821430i	−0.911764	−	0.410715i	$-0.865279\pi$
$977$	− 25.6755i	− 0.821430i	0.911764	−	0.410715i	$-0.134721\pi$
$978$	0	0
$979$	13.3172i	0.425621i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15.0107	0.478767	0.239383	−	0.970925i	$-0.423055\pi$
$983$	15.0107	0.478767	0.239383	+	0.970925i	$0.423055\pi$
$984$	0	0
$985$	7.34252i	0.233952i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 41.4030i	− 1.31654i
$990$	0	0
$991$	36.3025	1.15319	0.576594	−	0.817031i	$-0.304381\pi$
$991$	36.3025	1.15319	0.576594	+	0.817031i	$0.304381\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.8007i	0.469214i
$996$	0	0
$997$	7.43700i	0.235532i	0.993041	+	0.117766i	$0.0375733\pi$
$997$	7.43700i	0.235532i	−0.993041	+	0.117766i	$0.962427\pi$
$998$	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.881.11	yes	16
3.2	odd	2		8820.2.d.d.881.6	yes	16
7.6	odd	2		8820.2.d.d.881.11	yes	16
21.20	even	2	inner	8820.2.d.c.881.6	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8820.2.d.c.881.6	✓	16	21.20	even	2	inner
8820.2.d.c.881.11	yes	16	1.1	even	1	trivial
8820.2.d.d.881.6	yes	16	3.2	odd	2
8820.2.d.d.881.11	yes	16	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +O(q^{10})\)	\(q-1.00000 q^{5} +0.941197i q^{11} +1.01212i q^{13} -7.32886 q^{17} -7.97770i q^{19} +7.85911i q^{23} +1.00000 q^{25} +6.10587i q^{29} -5.03739i q^{31} +7.30478 q^{37} -2.21038 q^{41} -5.26816 q^{43} +1.12907 q^{47} -9.73171i q^{53} -0.941197i q^{55} +12.5799 q^{59} +14.5839i q^{61} -1.01212i q^{65} -14.2980 q^{67} -1.77255i q^{71} -3.40369i q^{73} -12.3430 q^{79} +1.83989 q^{83} +7.32886 q^{85} +14.1493 q^{89} +7.97770i q^{95} -16.1966i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5}+O(q^{10}) \) 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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