LMFDB - Embedded newform 8820.2.d.b.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:	$12$
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			881.7
Root			$-2.64559 - 0.0290059i$ of defining polynomial
Character	$\chi$	$=$	8820.881
Dual form			8820.2.d.b.881.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +O(q^{10})$	$q+1.00000 q^{5} +1.79006i q^{11} +2.19816i q^{13} -0.936178 q^{17} -0.422975i q^{19} +4.08719i q^{23} +1.00000 q^{25} -8.43097i q^{29} -10.6291i q^{31} +8.37551 q^{37} -6.29118 q^{41} -9.56621 q^{43} +3.04767 q^{47} -6.64091i q^{53} +1.79006i q^{55} +12.7305 q^{59} -1.94552i q^{61} +2.19816i q^{65} +7.16354 q^{67} -4.91826i q^{71} -4.61131i q^{73} +5.56044 q^{79} +15.1448 q^{83} -0.936178 q^{85} -4.30207 q^{89} -0.422975i q^{95} +4.55329i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	1.79006i	0.539724i	0.962899	+	0.269862i	$0.0869781\pi$
$11$	1.79006i	0.539724i	−0.962899	+	0.269862i	$0.913022\pi$
$12$	0	0
$13$	2.19816i	0.609659i	0.952407	+	0.304830i	$0.0985995\pi$
$13$	2.19816i	0.609659i	−0.952407	+	0.304830i	$0.901400\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.936178	−0.227057	−0.113528	−	0.993535i	$-0.536215\pi$
$17$	−0.936178	−0.227057	−0.113528	+	0.993535i	$0.536215\pi$
$18$	0	0
$19$	− 0.422975i	− 0.0970371i	−0.998822	−	0.0485186i	$-0.984550\pi$
$19$	− 0.422975i	− 0.0970371i	0.998822	−	0.0485186i	$-0.0154500\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.08719i	0.852238i	0.904667	+	0.426119i	$0.140119\pi$
$23$	4.08719i	0.852238i	−0.904667	+	0.426119i	$0.859881\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.43097i	− 1.56559i	−0.622278	−	0.782796i	$-0.713793\pi$
$29$	− 8.43097i	− 1.56559i	0.622278	−	0.782796i	$-0.286207\pi$
$30$	0	0
$31$	− 10.6291i	− 1.90905i	−0.298134	−	0.954524i	$-0.596364\pi$
$31$	− 10.6291i	− 1.90905i	0.298134	−	0.954524i	$-0.403636\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.37551	1.37693	0.688463	−	0.725272i	$-0.258286\pi$
$37$	8.37551	1.37693	0.688463	+	0.725272i	$0.258286\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.29118	−0.982518	−0.491259	−	0.871014i	$-0.663463\pi$
$41$	−6.29118	−0.982518	−0.491259	+	0.871014i	$0.663463\pi$
$42$	0	0
$43$	−9.56621	−1.45883	−0.729417	−	0.684069i	$-0.760209\pi$
$43$	−9.56621	−1.45883	−0.729417	+	0.684069i	$0.760209\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.04767	0.444548	0.222274	−	0.974984i	$-0.428652\pi$
$47$	3.04767	0.444548	0.222274	+	0.974984i	$0.428652\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.64091i	− 0.912199i	−0.889929	−	0.456100i	$-0.849246\pi$
$53$	− 6.64091i	− 0.912199i	0.889929	−	0.456100i	$-0.150754\pi$
$54$	0	0
$55$	1.79006i	0.241372i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.7305	1.65737	0.828686	−	0.559714i	$-0.189089\pi$
$59$	12.7305	1.65737	0.828686	+	0.559714i	$0.189089\pi$
$60$	0	0
$61$	− 1.94552i	− 0.249098i	−0.992213	−	0.124549i	$-0.960252\pi$
$61$	− 1.94552i	− 0.249098i	0.992213	−	0.124549i	$-0.0397484\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.19816i	0.272648i
$66$	0	0
$67$	7.16354	0.875166	0.437583	−	0.899178i	$-0.355835\pi$
$67$	7.16354	0.875166	0.437583	+	0.899178i	$0.355835\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 4.91826i	− 0.583690i	−0.956466	−	0.291845i	$-0.905731\pi$
$71$	− 4.91826i	− 0.583690i	0.956466	−	0.291845i	$-0.0942691\pi$
$72$	0	0
$73$	− 4.61131i	− 0.539713i	−0.962901	−	0.269856i	$-0.913024\pi$
$73$	− 4.61131i	− 0.539713i	0.962901	−	0.269856i	$-0.0869762\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.56044	0.625598	0.312799	−	0.949819i	$-0.398733\pi$
$79$	5.56044	0.625598	0.312799	+	0.949819i	$0.398733\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.1448	1.66236	0.831178	−	0.556007i	$-0.187667\pi$
$83$	15.1448	1.66236	0.831178	+	0.556007i	$0.187667\pi$
$84$	0	0
$85$	−0.936178	−0.101543
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.30207	−0.456018	−0.228009	−	0.973659i	$-0.573222\pi$
$89$	−4.30207	−0.456018	−0.228009	+	0.973659i	$0.573222\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 0.422975i	− 0.0433963i
$96$	0	0
$97$	4.55329i	0.462317i	0.972916	+	0.231158i	$0.0742516\pi$
$97$	4.55329i	0.462317i	−0.972916	+	0.231158i	$0.925748\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.81496	−0.877122	−0.438561	−	0.898701i	$-0.644512\pi$
$101$	−8.81496	−0.877122	−0.438561	+	0.898701i	$0.644512\pi$
$102$	0	0
$103$	− 1.61781i	− 0.159407i	−0.996819	−	0.0797036i	$-0.974603\pi$
$103$	− 1.61781i	− 0.159407i	0.996819	−	0.0797036i	$-0.0253974\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.44722i	0.623276i	0.950201	+	0.311638i	$0.100878\pi$
$107$	6.44722i	0.623276i	−0.950201	+	0.311638i	$0.899122\pi$
$108$	0	0
$109$	−2.78669	−0.266917	−0.133458	−	0.991054i	$-0.542608\pi$
$109$	−2.78669	−0.266917	−0.133458	+	0.991054i	$0.542608\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.71522i	0.725787i	0.931831	+	0.362894i	$0.118211\pi$
$113$	7.71522i	0.725787i	−0.931831	+	0.362894i	$0.881789\pi$
$114$	0	0
$115$	4.08719i	0.381132i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.79568	0.708698
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.80732	−0.160374	−0.0801869	−	0.996780i	$-0.525552\pi$
$127$	−1.80732	−0.160374	−0.0801869	+	0.996780i	$0.525552\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.0631	0.966584	0.483292	−	0.875459i	$-0.339441\pi$
$131$	11.0631	0.966584	0.483292	+	0.875459i	$0.339441\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.3304i	1.56607i	0.621977	+	0.783035i	$0.286330\pi$
$137$	18.3304i	1.56607i	−0.621977	+	0.783035i	$0.713670\pi$
$138$	0	0
$139$	− 5.62390i	− 0.477013i	−0.971141	−	0.238507i	$-0.923342\pi$
$139$	− 5.62390i	− 0.477013i	0.971141	−	0.238507i	$-0.0766579\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.93484	−0.329048
$144$	0	0
$145$	− 8.43097i	− 0.700154i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.8008i	1.21253i	0.795264	+	0.606264i	$0.207332\pi$
$149$	14.8008i	1.21253i	−0.795264	+	0.606264i	$0.792668\pi$
$150$	0	0
$151$	7.75114	0.630779	0.315390	−	0.948962i	$-0.397865\pi$
$151$	7.75114	0.630779	0.315390	+	0.948962i	$0.397865\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 10.6291i	− 0.853752i
$156$	0	0
$157$	4.51234i	0.360124i	0.983655	+	0.180062i	$0.0576298\pi$
$157$	4.51234i	0.360124i	−0.983655	+	0.180062i	$0.942370\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	20.3084	1.59068	0.795340	−	0.606163i	$-0.207292\pi$
$163$	20.3084	1.59068	0.795340	+	0.606163i	$0.207292\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.32784	−0.334898	−0.167449	−	0.985881i	$-0.553553\pi$
$167$	−4.32784	−0.334898	−0.167449	+	0.985881i	$0.553553\pi$
$168$	0	0
$169$	8.16810	0.628316
$170$	0	0
$171$	0	0
$172$	0	0
$173$	25.3257	1.92548	0.962740	−	0.270428i	$-0.0871651\pi$
$173$	25.3257	1.92548	0.962740	+	0.270428i	$0.0871651\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.2136i	1.28660i	0.765614	+	0.643301i	$0.222435\pi$
$179$	17.2136i	1.28660i	−0.765614	+	0.643301i	$0.777565\pi$
$180$	0	0
$181$	− 3.08246i	− 0.229118i	−0.993416	−	0.114559i	$-0.963455\pi$
$181$	− 3.08246i	− 0.229118i	0.993416	−	0.114559i	$-0.0365454\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.37551	0.615780
$186$	0	0
$187$	− 1.67582i	− 0.122548i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.41391i	0.319379i	0.987167	+	0.159690i	$0.0510494\pi$
$191$	4.41391i	0.319379i	−0.987167	+	0.159690i	$0.948951\pi$
$192$	0	0
$193$	21.6209	1.55631	0.778153	−	0.628075i	$-0.216157\pi$
$193$	21.6209	1.55631	0.778153	+	0.628075i	$0.216157\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.65937i	0.189472i	0.995502	+	0.0947362i	$0.0302008\pi$
$197$	2.65937i	0.189472i	−0.995502	+	0.0947362i	$0.969799\pi$
$198$	0	0
$199$	11.6734i	0.827506i	0.910389	+	0.413753i	$0.135782\pi$
$199$	11.6734i	0.827506i	−0.910389	+	0.413753i	$0.864218\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.29118	−0.439395
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.757152	0.0523733
$210$	0	0
$211$	−5.93357	−0.408484	−0.204242	−	0.978920i	$-0.565473\pi$
$211$	−5.93357	−0.408484	−0.204242	+	0.978920i	$0.565473\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.56621	−0.652410
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.05787i	− 0.138427i
$222$	0	0
$223$	− 16.2789i	− 1.09011i	−0.838399	−	0.545057i	$-0.816508\pi$
$223$	− 16.2789i	− 1.09011i	0.838399	−	0.545057i	$-0.183492\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.735219	0.0487982	0.0243991	−	0.999702i	$-0.492233\pi$
$227$	0.735219	0.0487982	0.0243991	+	0.999702i	$0.492233\pi$
$228$	0	0
$229$	− 29.3737i	− 1.94107i	−0.240965	−	0.970534i	$-0.577464\pi$
$229$	− 29.3737i	− 1.94107i	0.240965	−	0.970534i	$-0.422536\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 24.3986i	− 1.59841i	−0.601061	−	0.799203i	$-0.705255\pi$
$233$	− 24.3986i	− 1.59841i	0.601061	−	0.799203i	$-0.294745\pi$
$234$	0	0
$235$	3.04767	0.198808
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 6.84320i	− 0.442650i	−0.975200	−	0.221325i	$-0.928962\pi$
$239$	− 6.84320i	− 0.442650i	0.975200	−	0.221325i	$-0.0710381\pi$
$240$	0	0
$241$	− 0.0701267i	− 0.00451726i	−0.999997	−	0.00225863i	$-0.999281\pi$
$241$	− 0.0701267i	− 0.00451726i	0.999997	−	0.00225863i	$-0.000718945\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.929765	0.0591596
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.5728	0.919824	0.459912	−	0.887965i	$-0.347881\pi$
$251$	14.5728	0.919824	0.459912	+	0.887965i	$0.347881\pi$
$252$	0	0
$253$	−7.31632	−0.459973
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.2069	0.699065	0.349532	−	0.936924i	$-0.386340\pi$
$257$	11.2069	0.699065	0.349532	+	0.936924i	$0.386340\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.650478i	0.0401102i	0.999799	+	0.0200551i	$0.00638417\pi$
$263$	0.650478i	0.0401102i	−0.999799	+	0.0200551i	$0.993616\pi$
$264$	0	0
$265$	− 6.64091i	− 0.407948i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.52840	−0.276101	−0.138051	−	0.990425i	$-0.544084\pi$
$269$	−4.52840	−0.276101	−0.138051	+	0.990425i	$0.544084\pi$
$270$	0	0
$271$	− 7.06018i	− 0.428876i	−0.976738	−	0.214438i	$-0.931208\pi$
$271$	− 7.06018i	− 0.428876i	0.976738	−	0.214438i	$-0.0687919\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.79006i	0.107945i
$276$	0	0
$277$	−17.3240	−1.04090	−0.520449	−	0.853893i	$-0.674235\pi$
$277$	−17.3240	−1.04090	−0.520449	+	0.853893i	$0.674235\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.9791i	1.60944i	0.593655	+	0.804720i	$0.297684\pi$
$281$	26.9791i	1.60944i	−0.593655	+	0.804720i	$0.702316\pi$
$282$	0	0
$283$	5.05107i	0.300255i	0.988667	+	0.150128i	$0.0479684\pi$
$283$	5.05107i	0.300255i	−0.988667	+	0.150128i	$0.952032\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.1236	−0.948445
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.78083	−0.571402	−0.285701	−	0.958319i	$-0.592226\pi$
$293$	−9.78083	−0.571402	−0.285701	+	0.958319i	$0.592226\pi$
$294$	0	0
$295$	12.7305	0.741199
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.98428	−0.519574
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.94552i	− 0.111400i
$306$	0	0
$307$	− 26.9503i	− 1.53813i	−0.639168	−	0.769067i	$-0.720721\pi$
$307$	− 26.9503i	− 1.53813i	0.639168	−	0.769067i	$-0.279279\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.7027	−0.720305	−0.360152	−	0.932893i	$-0.617275\pi$
$311$	−12.7027	−0.720305	−0.360152	+	0.932893i	$0.617275\pi$
$312$	0	0
$313$	− 23.9680i	− 1.35475i	−0.735638	−	0.677375i	$-0.763117\pi$
$313$	− 23.9680i	− 1.35475i	0.735638	−	0.677375i	$-0.236883\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.405096i	0.0227525i	0.999935	+	0.0113762i	$0.00362125\pi$
$317$	0.405096i	0.0227525i	−0.999935	+	0.0113762i	$0.996379\pi$
$318$	0	0
$319$	15.0920	0.844988
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.395980i	0.0220329i
$324$	0	0
$325$	2.19816i	0.121932i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.37859	0.460529	0.230264	−	0.973128i	$-0.426041\pi$
$331$	8.37859	0.460529	0.230264	+	0.973128i	$0.426041\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.16354	0.391386
$336$	0	0
$337$	−13.8266	−0.753181	−0.376590	−	0.926380i	$-0.622904\pi$
$337$	−13.8266	−0.753181	−0.376590	+	0.926380i	$0.622904\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.0268	1.03036
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.3697i	− 0.664038i	−0.943273	−	0.332019i	$-0.892270\pi$
$347$	− 12.3697i	− 0.664038i	0.943273	−	0.332019i	$-0.107730\pi$
$348$	0	0
$349$	6.67915i	0.357527i	0.983892	+	0.178763i	$0.0572096\pi$
$349$	6.67915i	0.357527i	−0.983892	+	0.178763i	$0.942790\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.70027	−0.303395	−0.151697	−	0.988427i	$-0.548474\pi$
$353$	−5.70027	−0.303395	−0.151697	+	0.988427i	$0.548474\pi$
$354$	0	0
$355$	− 4.91826i	− 0.261034i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.2164i	1.33087i	0.746455	+	0.665436i	$0.231754\pi$
$359$	25.2164i	1.33087i	−0.746455	+	0.665436i	$0.768246\pi$
$360$	0	0
$361$	18.8211	0.990584
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 4.61131i	− 0.241367i
$366$	0	0
$367$	− 28.2742i	− 1.47590i	−0.674854	−	0.737952i	$-0.735793\pi$
$367$	− 28.2742i	− 1.47590i	0.674854	−	0.737952i	$-0.264207\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.4496	0.851727	0.425863	−	0.904787i	$-0.359970\pi$
$373$	16.4496	0.851727	0.425863	+	0.904787i	$0.359970\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.5326	0.954478
$378$	0	0
$379$	6.16869	0.316864	0.158432	−	0.987370i	$-0.449356\pi$
$379$	6.16869	0.316864	0.158432	+	0.987370i	$0.449356\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10.1238	0.517302	0.258651	−	0.965971i	$-0.416722\pi$
$383$	10.1238	0.517302	0.258651	+	0.965971i	$0.416722\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.4325i	1.39088i	0.718583	+	0.695441i	$0.244791\pi$
$389$	27.4325i	1.39088i	−0.718583	+	0.695441i	$0.755209\pi$
$390$	0	0
$391$	− 3.82634i	− 0.193506i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.56044	0.279776
$396$	0	0
$397$	− 24.7703i	− 1.24318i	−0.783341	−	0.621592i	$-0.786486\pi$
$397$	− 24.7703i	− 1.24318i	0.783341	−	0.621592i	$-0.213514\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.1041i	1.20370i	0.798609	+	0.601851i	$0.205570\pi$
$401$	24.1041i	1.20370i	−0.798609	+	0.601851i	$0.794430\pi$
$402$	0	0
$403$	23.3645	1.16387
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.9927i	0.743160i
$408$	0	0
$409$	24.4460i	1.20878i	0.796690	+	0.604388i	$0.206582\pi$
$409$	24.4460i	1.20878i	−0.796690	+	0.604388i	$0.793418\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	15.1448	0.743428
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.05848	−0.295976	−0.147988	−	0.988989i	$-0.547280\pi$
$419$	−6.05848	−0.295976	−0.147988	+	0.988989i	$0.547280\pi$
$420$	0	0
$421$	−19.0817	−0.929986	−0.464993	−	0.885314i	$-0.653943\pi$
$421$	−19.0817	−0.929986	−0.464993	+	0.885314i	$0.653943\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.936178	−0.0454113
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	33.9874i	1.63711i	0.574425	+	0.818557i	$0.305226\pi$
$431$	33.9874i	1.63711i	−0.574425	+	0.818557i	$0.694774\pi$
$432$	0	0
$433$	19.2496i	0.925075i	0.886600	+	0.462537i	$0.153061\pi$
$433$	19.2496i	0.925075i	−0.886600	+	0.462537i	$0.846939\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.72878	0.0826987
$438$	0	0
$439$	− 33.5098i	− 1.59934i	−0.600442	−	0.799668i	$-0.705009\pi$
$439$	− 33.5098i	− 1.59934i	0.600442	−	0.799668i	$-0.294991\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.77905i	0.132037i	0.997818	+	0.0660183i	$0.0210296\pi$
$443$	2.77905i	0.132037i	−0.997818	+	0.0660183i	$0.978970\pi$
$444$	0	0
$445$	−4.30207	−0.203938
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 40.7948i	− 1.92523i	−0.270881	−	0.962613i	$-0.587315\pi$
$449$	− 40.7948i	− 1.92523i	0.270881	−	0.962613i	$-0.412685\pi$
$450$	0	0
$451$	− 11.2616i	− 0.530289i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.24028	0.104796	0.0523979	−	0.998626i	$-0.483314\pi$
$457$	2.24028	0.104796	0.0523979	+	0.998626i	$0.483314\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−36.5014	−1.70004	−0.850020	−	0.526751i	$-0.823410\pi$
$461$	−36.5014	−1.70004	−0.850020	+	0.526751i	$0.823410\pi$
$462$	0	0
$463$	−6.02692	−0.280095	−0.140047	−	0.990145i	$-0.544725\pi$
$463$	−6.02692	−0.280095	−0.140047	+	0.990145i	$0.544725\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.3450	1.26537	0.632687	−	0.774407i	$-0.281952\pi$
$467$	27.3450	1.26537	0.632687	+	0.774407i	$0.281952\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 17.1241i	− 0.787368i
$474$	0	0
$475$	− 0.422975i	− 0.0194074i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.8096	1.68187	0.840937	−	0.541133i	$-0.182004\pi$
$479$	36.8096	1.68187	0.840937	+	0.541133i	$0.182004\pi$
$480$	0	0
$481$	18.4107i	0.839455i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.55329i	0.206754i
$486$	0	0
$487$	31.7612	1.43924	0.719619	−	0.694370i	$-0.244317\pi$
$487$	31.7612	1.43924	0.719619	+	0.694370i	$0.244317\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.651553i	0.0294042i	0.999892	+	0.0147021i	$0.00467999\pi$
$491$	0.651553i	0.0294042i	−0.999892	+	0.0147021i	$0.995320\pi$
$492$	0	0
$493$	7.89289i	0.355478i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	21.9932	0.984552	0.492276	−	0.870439i	$-0.336165\pi$
$499$	21.9932	0.984552	0.492276	+	0.870439i	$0.336165\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	26.7369	1.19214	0.596070	−	0.802933i	$-0.296728\pi$
$503$	26.7369	1.19214	0.596070	+	0.802933i	$0.296728\pi$
$504$	0	0
$505$	−8.81496	−0.392261
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.0265	0.621715	0.310857	−	0.950457i	$-0.399384\pi$
$509$	14.0265	0.621715	0.310857	+	0.950457i	$0.399384\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1.61781i	− 0.0712890i
$516$	0	0
$517$	5.45551i	0.239933i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−30.1465	−1.32074	−0.660372	−	0.750939i	$-0.729601\pi$
$521$	−30.1465	−1.32074	−0.660372	+	0.750939i	$0.729601\pi$
$522$	0	0
$523$	− 0.722421i	− 0.0315892i	−0.999875	−	0.0157946i	$-0.994972\pi$
$523$	− 0.722421i	− 0.0315892i	0.999875	−	0.0157946i	$-0.00502779\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.95076i	0.433462i
$528$	0	0
$529$	6.29490	0.273691
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 13.8290i	− 0.599001i
$534$	0	0
$535$	6.44722i	0.278737i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.6332	0.543142	0.271571	−	0.962418i	$-0.412457\pi$
$541$	12.6332	0.543142	0.271571	+	0.962418i	$0.412457\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.78669	−0.119369
$546$	0	0
$547$	−10.0662	−0.430401	−0.215201	−	0.976570i	$-0.569041\pi$
$547$	−10.0662	−0.430401	−0.215201	+	0.976570i	$0.569041\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.56609	−0.151921
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.0227i	0.594161i	0.954852	+	0.297081i	$0.0960130\pi$
$557$	14.0227i	0.594161i	−0.954852	+	0.297081i	$0.903987\pi$
$558$	0	0
$559$	− 21.0280i	− 0.889392i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.9055	1.21822	0.609112	−	0.793084i	$-0.291526\pi$
$563$	28.9055	1.21822	0.609112	+	0.793084i	$0.291526\pi$
$564$	0	0
$565$	7.71522i	0.324582i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0.840739i	0.0352456i	0.999845	+	0.0176228i	$0.00560980\pi$
$569$	0.840739i	0.0352456i	−0.999845	+	0.0176228i	$0.994390\pi$
$570$	0	0
$571$	−45.3843	−1.89928	−0.949638	−	0.313349i	$-0.898549\pi$
$571$	−45.3843	−1.89928	−0.949638	+	0.313349i	$0.898549\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.08719i	0.170448i
$576$	0	0
$577$	− 19.9541i	− 0.830699i	−0.909662	−	0.415350i	$-0.863659\pi$
$577$	− 19.9541i	− 0.830699i	0.909662	−	0.415350i	$-0.136341\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	11.8876	0.492336
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.0763	−0.993733	−0.496867	−	0.867827i	$-0.665516\pi$
$587$	−24.0763	−0.993733	−0.496867	+	0.867827i	$0.665516\pi$
$588$	0	0
$589$	−4.49586	−0.185248
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−35.8778	−1.47333	−0.736663	−	0.676260i	$-0.763600\pi$
$593$	−35.8778	−1.47333	−0.736663	+	0.676260i	$0.763600\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 24.7502i	− 1.01127i	−0.862748	−	0.505633i	$-0.831259\pi$
$599$	− 24.7502i	− 1.01127i	0.862748	−	0.505633i	$-0.168741\pi$
$600$	0	0
$601$	− 35.0712i	− 1.43058i	−0.698826	−	0.715291i	$-0.746294\pi$
$601$	− 35.0712i	− 1.43058i	0.698826	−	0.715291i	$-0.253706\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.79568	0.316939
$606$	0	0
$607$	− 2.31774i	− 0.0940741i	−0.998893	−	0.0470371i	$-0.985022\pi$
$607$	− 2.31774i	− 0.0940741i	0.998893	−	0.0470371i	$-0.0149779\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.69925i	0.271023i
$612$	0	0
$613$	12.8479	0.518923	0.259461	−	0.965753i	$-0.416455\pi$
$613$	12.8479	0.518923	0.259461	+	0.965753i	$0.416455\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 29.7437i	− 1.19743i	−0.800960	−	0.598717i	$-0.795677\pi$
$617$	− 29.7437i	− 1.19743i	0.800960	−	0.598717i	$-0.204323\pi$
$618$	0	0
$619$	− 43.5259i	− 1.74945i	−0.484618	−	0.874726i	$-0.661041\pi$
$619$	− 43.5259i	− 1.74945i	0.484618	−	0.874726i	$-0.338959\pi$
$620$	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
$629$	−7.84097	−0.312640
$630$	0	0
$631$	4.21974	0.167985	0.0839925	−	0.996466i	$-0.473233\pi$
$631$	4.21974	0.167985	0.0839925	+	0.996466i	$0.473233\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.80732	−0.0717213
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	44.7656i	1.76813i	0.467361	+	0.884066i	$0.345205\pi$
$641$	44.7656i	1.76813i	−0.467361	+	0.884066i	$0.654795\pi$
$642$	0	0
$643$	27.0494i	1.06672i	0.845887	+	0.533362i	$0.179072\pi$
$643$	27.0494i	1.06672i	−0.845887	+	0.533362i	$0.820928\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.2647	−0.757372	−0.378686	−	0.925525i	$-0.623624\pi$
$647$	−19.2647	−0.757372	−0.378686	+	0.925525i	$0.623624\pi$
$648$	0	0
$649$	22.7884i	0.894524i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.1421i	1.17955i	0.807567	+	0.589776i	$0.200784\pi$
$653$	30.1421i	1.17955i	−0.807567	+	0.589776i	$0.799216\pi$
$654$	0	0
$655$	11.0631	0.432270
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.2084i	0.826164i	0.910694	+	0.413082i	$0.135548\pi$
$659$	21.2084i	0.826164i	−0.910694	+	0.413082i	$0.864452\pi$
$660$	0	0
$661$	− 43.1728i	− 1.67923i	−0.543184	−	0.839614i	$-0.682781\pi$
$661$	− 43.1728i	− 1.67923i	0.543184	−	0.839614i	$-0.317219\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	34.4590	1.33426
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.48259	0.134444
$672$	0	0
$673$	−31.7115	−1.22239	−0.611195	−	0.791480i	$-0.709311\pi$
$673$	−31.7115	−1.22239	−0.611195	+	0.791480i	$0.709311\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.2128	−1.31491	−0.657453	−	0.753496i	$-0.728366\pi$
$677$	−34.2128	−1.31491	−0.657453	+	0.753496i	$0.728366\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 47.9676i	− 1.83543i	−0.397240	−	0.917715i	$-0.630032\pi$
$683$	− 47.9676i	− 1.83543i	0.397240	−	0.917715i	$-0.369968\pi$
$684$	0	0
$685$	18.3304i	0.700368i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14.5978	0.556131
$690$	0	0
$691$	− 28.5575i	− 1.08638i	−0.839610	−	0.543189i	$-0.817217\pi$
$691$	− 28.5575i	− 1.08638i	0.839610	−	0.543189i	$-0.182783\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 5.62390i	− 0.213327i
$696$	0	0
$697$	5.88967	0.223087
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 25.7765i	− 0.973564i	−0.873523	−	0.486782i	$-0.838171\pi$
$701$	− 25.7765i	− 0.973564i	0.873523	−	0.486782i	$-0.161829\pi$
$702$	0	0
$703$	− 3.54263i	− 0.133613i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	43.5748	1.63649	0.818243	−	0.574872i	$-0.194948\pi$
$709$	43.5748	1.63649	0.818243	+	0.574872i	$0.194948\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	43.4432	1.62696
$714$	0	0
$715$	−3.93484	−0.147155
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0.407128	0.0151833	0.00759166	−	0.999971i	$-0.497583\pi$
$719$	0.407128	0.0151833	0.00759166	+	0.999971i	$0.497583\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 8.43097i	− 0.313118i
$726$	0	0
$727$	17.8033i	0.660288i	0.943931	+	0.330144i	$0.107097\pi$
$727$	17.8033i	0.660288i	−0.943931	+	0.330144i	$0.892903\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.95568	0.331238
$732$	0	0
$733$	3.46920i	0.128138i	0.997945	+	0.0640690i	$0.0204078\pi$
$733$	3.46920i	0.128138i	−0.997945	+	0.0640690i	$0.979592\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.8232i	0.472348i
$738$	0	0
$739$	−24.8044	−0.912445	−0.456222	−	0.889866i	$-0.650798\pi$
$739$	−24.8044	−0.912445	−0.456222	+	0.889866i	$0.650798\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.8141i	1.09377i	0.837206	+	0.546887i	$0.184187\pi$
$743$	29.8141i	1.09377i	−0.837206	+	0.546887i	$0.815813\pi$
$744$	0	0
$745$	14.8008i	0.542259i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	33.8495	1.23518	0.617592	−	0.786499i	$-0.288108\pi$
$751$	33.8495	1.23518	0.617592	+	0.786499i	$0.288108\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.75114	0.282093
$756$	0	0
$757$	−1.19627	−0.0434791	−0.0217395	−	0.999764i	$-0.506920\pi$
$757$	−1.19627	−0.0434791	−0.0217395	+	0.999764i	$0.506920\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	48.6746	1.76445	0.882226	−	0.470827i	$-0.156044\pi$
$761$	48.6746	1.76445	0.882226	+	0.470827i	$0.156044\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	27.9837i	1.01043i
$768$	0	0
$769$	− 9.21143i	− 0.332173i	−0.986111	−	0.166086i	$-0.946887\pi$
$769$	− 9.21143i	− 0.332173i	0.986111	−	0.166086i	$-0.0531131\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	44.0372	1.58391	0.791954	−	0.610581i	$-0.209064\pi$
$773$	44.0372	1.58391	0.791954	+	0.610581i	$0.209064\pi$
$774$	0	0
$775$	− 10.6291i	− 0.381810i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.66101i	0.0953407i
$780$	0	0
$781$	8.80399	0.315031
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.51234i	0.161052i
$786$	0	0
$787$	42.0440i	1.49871i	0.662169	+	0.749354i	$0.269636\pi$
$787$	42.0440i	1.49871i	−0.662169	+	0.749354i	$0.730364\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.27655	0.151865
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33.6692	−1.19263	−0.596313	−	0.802752i	$-0.703368\pi$
$797$	−33.6692	−1.19263	−0.596313	+	0.802752i	$0.703368\pi$
$798$	0	0
$799$	−2.85316	−0.100938
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.25453	0.291296
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.50012i	0.334007i	0.985956	+	0.167003i	$0.0534090\pi$
$809$	9.50012i	0.334007i	−0.985956	+	0.167003i	$0.946591\pi$
$810$	0	0
$811$	− 14.8210i	− 0.520435i	−0.965550	−	0.260217i	$-0.916206\pi$
$811$	− 14.8210i	− 0.520435i	0.965550	−	0.260217i	$-0.0837942\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.3084	0.711374
$816$	0	0
$817$	4.04627i	0.141561i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 18.0259i	− 0.629108i	−0.949240	−	0.314554i	$-0.898145\pi$
$821$	− 18.0259i	− 0.629108i	0.949240	−	0.314554i	$-0.101855\pi$
$822$	0	0
$823$	25.9601	0.904912	0.452456	−	0.891787i	$-0.350548\pi$
$823$	25.9601	0.904912	0.452456	+	0.891787i	$0.350548\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 24.4040i	− 0.848612i	−0.905519	−	0.424306i	$-0.860518\pi$
$827$	− 24.4040i	− 0.848612i	0.905519	−	0.424306i	$-0.139482\pi$
$828$	0	0
$829$	− 10.3829i	− 0.360614i	−0.983610	−	0.180307i	$-0.942291\pi$
$829$	− 10.3829i	− 0.360614i	0.983610	−	0.180307i	$-0.0577091\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.32784	−0.149771
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.7021	1.02543	0.512715	−	0.858559i	$-0.328640\pi$
$839$	29.7021	1.02543	0.512715	+	0.858559i	$0.328640\pi$
$840$	0	0
$841$	−42.0813	−1.45108
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.16810	0.280991
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	34.2323i	1.17347i
$852$	0	0
$853$	47.4739i	1.62548i	0.582629	+	0.812738i	$0.302024\pi$
$853$	47.4739i	1.62548i	−0.582629	+	0.812738i	$0.697976\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−33.5195	−1.14500	−0.572501	−	0.819904i	$-0.694027\pi$
$857$	−33.5195	−1.14500	−0.572501	+	0.819904i	$0.694027\pi$
$858$	0	0
$859$	33.6289i	1.14740i	0.819064	+	0.573702i	$0.194493\pi$
$859$	33.6289i	1.14740i	−0.819064	+	0.573702i	$0.805507\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.1469i	0.685809i	0.939370	+	0.342904i	$0.111411\pi$
$863$	20.1469i	0.685809i	−0.939370	+	0.342904i	$0.888589\pi$
$864$	0	0
$865$	25.3257	0.861101
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.95353i	0.337650i
$870$	0	0
$871$	15.7466i	0.533553i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28.4889	−0.962002	−0.481001	−	0.876720i	$-0.659727\pi$
$877$	−28.4889	−0.962002	−0.481001	+	0.876720i	$0.659727\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−27.5545	−0.928336	−0.464168	−	0.885747i	$-0.653647\pi$
$881$	−27.5545	−0.928336	−0.464168	+	0.885747i	$0.653647\pi$
$882$	0	0
$883$	48.1856	1.62158	0.810788	−	0.585340i	$-0.199039\pi$
$883$	48.1856	1.62158	0.810788	+	0.585340i	$0.199039\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.9743	1.61082	0.805410	−	0.592718i	$-0.201945\pi$
$887$	47.9743	1.61082	0.805410	+	0.592718i	$0.201945\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.28909i	− 0.0431376i
$894$	0	0
$895$	17.2136i	0.575386i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−89.6139	−2.98879
$900$	0	0
$901$	6.21707i	0.207121i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 3.08246i	− 0.102464i
$906$	0	0
$907$	−22.7060	−0.753939	−0.376970	−	0.926226i	$-0.623034\pi$
$907$	−22.7060	−0.753939	−0.376970	+	0.926226i	$0.623034\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 6.95948i	− 0.230578i	−0.993332	−	0.115289i	$-0.963221\pi$
$911$	− 6.95948i	− 0.230578i	0.993332	−	0.115289i	$-0.0367794\pi$
$912$	0	0
$913$	27.1101i	0.897214i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−37.4165	−1.23426	−0.617128	−	0.786863i	$-0.711704\pi$
$919$	−37.4165	−1.23426	−0.617128	+	0.786863i	$0.711704\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10.8111	0.355852
$924$	0	0
$925$	8.37551	0.275385
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34.2687	1.12432	0.562160	−	0.827029i	$-0.309971\pi$
$929$	34.2687	1.12432	0.562160	+	0.827029i	$0.309971\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1.67582i	− 0.0548051i
$936$	0	0
$937$	− 10.0892i	− 0.329600i	−0.986327	−	0.164800i	$-0.947302\pi$
$937$	− 10.0892i	− 0.329600i	0.986327	−	0.164800i	$-0.0526979\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6.04513	0.197066	0.0985328	−	0.995134i	$-0.468585\pi$
$941$	6.04513	0.197066	0.0985328	+	0.995134i	$0.468585\pi$
$942$	0	0
$943$	− 25.7133i	− 0.837339i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.3963i	0.695286i	0.937627	+	0.347643i	$0.113018\pi$
$947$	21.3963i	0.695286i	−0.937627	+	0.347643i	$0.886982\pi$
$948$	0	0
$949$	10.1364	0.329041
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 29.6828i	− 0.961519i	−0.876852	−	0.480760i	$-0.840361\pi$
$953$	− 29.6828i	− 0.961519i	0.876852	−	0.480760i	$-0.159639\pi$
$954$	0	0
$955$	4.41391i	0.142831i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−81.9784	−2.64446
$962$	0	0
$963$	0	0
$964$	0	0
$965$	21.6209	0.696001
$966$	0	0
$967$	41.7092	1.34128	0.670639	−	0.741783i	$-0.266020\pi$
$967$	41.7092	1.34128	0.670639	+	0.741783i	$0.266020\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21.5674	−0.692129	−0.346065	−	0.938211i	$-0.612482\pi$
$971$	−21.5674	−0.692129	−0.346065	+	0.938211i	$0.612482\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.3845i	1.00408i	0.864845	+	0.502040i	$0.167417\pi$
$977$	31.3845i	1.00408i	−0.864845	+	0.502040i	$0.832583\pi$
$978$	0	0
$979$	− 7.70097i	− 0.246124i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	55.9536	1.78464	0.892322	−	0.451400i	$-0.149075\pi$
$983$	55.9536	1.78464	0.892322	+	0.451400i	$0.149075\pi$
$984$	0	0
$985$	2.65937i	0.0847346i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 39.0989i	− 1.24327i
$990$	0	0
$991$	−62.3738	−1.98137	−0.990685	−	0.136171i	$-0.956520\pi$
$991$	−62.3738	−1.98137	−0.990685	+	0.136171i	$0.956520\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.6734i	0.370072i
$996$	0	0
$997$	− 29.0416i	− 0.919755i	−0.887982	−	0.459878i	$-0.847893\pi$
$997$	− 29.0416i	− 0.919755i	0.887982	−	0.459878i	$-0.152107\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.d.b.881.7		12
3.2	odd	2		8820.2.d.a.881.6		12
7.4	even	3		1260.2.cg.a.341.1	✓	12
7.5	odd	6		1260.2.cg.b.521.1	yes	12
7.6	odd	2		8820.2.d.a.881.7		12
21.5	even	6		1260.2.cg.a.521.1	yes	12
21.11	odd	6		1260.2.cg.b.341.1	yes	12
21.20	even	2	inner	8820.2.d.b.881.6		12
35.4	even	6		6300.2.ch.c.1601.6		12
35.12	even	12		6300.2.dd.b.4049.7		24
35.18	odd	12		6300.2.dd.c.1349.7		24
35.19	odd	6		6300.2.ch.b.4301.6		12
35.32	odd	12		6300.2.dd.c.1349.6		24
35.33	even	12		6300.2.dd.b.4049.6		24
105.32	even	12		6300.2.dd.b.1349.6		24
105.47	odd	12		6300.2.dd.c.4049.7		24
105.53	even	12		6300.2.dd.b.1349.7		24
105.68	odd	12		6300.2.dd.c.4049.6		24
105.74	odd	6		6300.2.ch.b.1601.6		12
105.89	even	6		6300.2.ch.c.4301.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.cg.a.341.1	✓	12	7.4	even	3
1260.2.cg.a.521.1	yes	12	21.5	even	6
1260.2.cg.b.341.1	yes	12	21.11	odd	6
1260.2.cg.b.521.1	yes	12	7.5	odd	6
6300.2.ch.b.1601.6		12	105.74	odd	6
6300.2.ch.b.4301.6		12	35.19	odd	6
6300.2.ch.c.1601.6		12	35.4	even	6
6300.2.ch.c.4301.6		12	105.89	even	6
6300.2.dd.b.1349.6		24	105.32	even	12
6300.2.dd.b.1349.7		24	105.53	even	12
6300.2.dd.b.4049.6		24	35.33	even	12
6300.2.dd.b.4049.7		24	35.12	even	12
6300.2.dd.c.1349.6		24	35.32	odd	12
6300.2.dd.c.1349.7		24	35.18	odd	12
6300.2.dd.c.4049.6		24	105.68	odd	12
6300.2.dd.c.4049.7		24	105.47	odd	12
8820.2.d.a.881.6		12	3.2	odd	2
8820.2.d.a.881.7		12	7.6	odd	2
8820.2.d.b.881.6		12	21.20	even	2	inner
8820.2.d.b.881.7		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +O(q^{10})\)	\(q+1.00000 q^{5} +1.79006i q^{11} +2.19816i q^{13} -0.936178 q^{17} -0.422975i q^{19} +4.08719i q^{23} +1.00000 q^{25} -8.43097i q^{29} -10.6291i q^{31} +8.37551 q^{37} -6.29118 q^{41} -9.56621 q^{43} +3.04767 q^{47} -6.64091i q^{53} +1.79006i q^{55} +12.7305 q^{59} -1.94552i q^{61} +2.19816i q^{65} +7.16354 q^{67} -4.91826i q^{71} -4.61131i q^{73} +5.56044 q^{79} +15.1448 q^{83} -0.936178 q^{85} -4.30207 q^{89} -0.422975i q^{95} +4.55329i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{5}+O(q^{10}) \) 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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