LMFDB - Embedded newform 4320.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(1,4320)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.a (trivial)

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:	yes
Analytic conductor:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -1.41421 q^{7} +1.41421 q^{11} +2.24264 q^{13} -5.24264 q^{17} -1.24264 q^{19} +8.65685 q^{23} +1.00000 q^{25} -4.24264 q^{29} +4.07107 q^{31} -1.41421 q^{35} -6.48528 q^{37} +6.24264 q^{41} +10.2426 q^{43} +0.343146 q^{47} -5.00000 q^{49} +1.24264 q^{53} +1.41421 q^{55} -1.07107 q^{59} +1.00000 q^{61} +2.24264 q^{65} +0.343146 q^{67} +10.5858 q^{71} +4.24264 q^{73} -2.00000 q^{77} -1.58579 q^{79} +6.17157 q^{83} -5.24264 q^{85} +2.24264 q^{89} -3.17157 q^{91} -1.24264 q^{95} +16.4853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 4 q^{13} - 2 q^{17} + 6 q^{19} + 6 q^{23} + 2 q^{25} - 6 q^{31} + 4 q^{37} + 4 q^{41} + 12 q^{43} + 12 q^{47} - 10 q^{49} - 6 q^{53} + 12 q^{59} + 2 q^{61} - 4 q^{65} + 12 q^{67} + 24 q^{71}+ \cdots + 16 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^13 - 2 * q^17 + 6 * q^19 + 6 * q^23 + 2 * q^25 - 6 * q^31 + 4 * q^37 + 4 * q^41 + 12 * q^43 + 12 * q^47 - 10 * q^49 - 6 * q^53 + 12 * q^59 + 2 * q^61 - 4 * q^65 + 12 * q^67 + 24 * q^71 - 4 * q^77 - 6 * q^79 + 18 * q^83 - 2 * q^85 - 4 * q^89 - 12 * q^91 + 6 * q^95 + 16 * q^97

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
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	2.24264	0.621997	0.310998	−	0.950410i	$-0.399337\pi$
$13$	2.24264	0.621997	0.310998	+	0.950410i	$0.399337\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.24264	−1.27153	−0.635764	−	0.771884i	$-0.719315\pi$
$17$	−5.24264	−1.27153	−0.635764	+	0.771884i	$0.719315\pi$
$18$	0	0
$19$	−1.24264	−0.285081	−0.142541	−	0.989789i	$-0.545527\pi$
$19$	−1.24264	−0.285081	−0.142541	+	0.989789i	$0.545527\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.65685	1.80508	0.902539	−	0.430607i	$-0.141701\pi$
$23$	8.65685	1.80508	0.902539	+	0.430607i	$0.141701\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.24264	−0.787839	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	−4.24264	−0.787839	−0.393919	+	0.919145i	$0.628881\pi$
$30$	0	0
$31$	4.07107	0.731185	0.365593	−	0.930775i	$-0.380866\pi$
$31$	4.07107	0.731185	0.365593	+	0.930775i	$0.380866\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.41421	−0.239046
$36$	0	0
$37$	−6.48528	−1.06617	−0.533087	−	0.846061i	$-0.678968\pi$
$37$	−6.48528	−1.06617	−0.533087	+	0.846061i	$0.678968\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.24264	0.974937	0.487468	−	0.873141i	$-0.337920\pi$
$41$	6.24264	0.974937	0.487468	+	0.873141i	$0.337920\pi$
$42$	0	0
$43$	10.2426	1.56199	0.780994	−	0.624538i	$-0.214713\pi$
$43$	10.2426	1.56199	0.780994	+	0.624538i	$0.214713\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.343146	0.0500530	0.0250265	−	0.999687i	$-0.492033\pi$
$47$	0.343146	0.0500530	0.0250265	+	0.999687i	$0.492033\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.24264	0.170690	0.0853449	−	0.996351i	$-0.472801\pi$
$53$	1.24264	0.170690	0.0853449	+	0.996351i	$0.472801\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.07107	−0.139441	−0.0697206	−	0.997567i	$-0.522211\pi$
$59$	−1.07107	−0.139441	−0.0697206	+	0.997567i	$0.522211\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.24264	0.278165
$66$	0	0
$67$	0.343146	0.0419219	0.0209610	−	0.999780i	$-0.493327\pi$
$67$	0.343146	0.0419219	0.0209610	+	0.999780i	$0.493327\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.5858	1.25630	0.628151	−	0.778092i	$-0.283812\pi$
$71$	10.5858	1.25630	0.628151	+	0.778092i	$0.283812\pi$
$72$	0	0
$73$	4.24264	0.496564	0.248282	−	0.968688i	$-0.420134\pi$
$73$	4.24264	0.496564	0.248282	+	0.968688i	$0.420134\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−1.58579	−0.178415	−0.0892075	−	0.996013i	$-0.528433\pi$
$79$	−1.58579	−0.178415	−0.0892075	+	0.996013i	$0.528433\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.17157	0.677418	0.338709	−	0.940891i	$-0.390010\pi$
$83$	6.17157	0.677418	0.338709	+	0.940891i	$0.390010\pi$
$84$	0	0
$85$	−5.24264	−0.568644
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.24264	0.237719	0.118860	−	0.992911i	$-0.462076\pi$
$89$	2.24264	0.237719	0.118860	+	0.992911i	$0.462076\pi$
$90$	0	0
$91$	−3.17157	−0.332471
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.24264	−0.127492
$96$	0	0
$97$	16.4853	1.67383	0.836913	−	0.547335i	$-0.184358\pi$
$97$	16.4853	1.67383	0.836913	+	0.547335i	$0.184358\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.48528	−0.446302	−0.223151	−	0.974784i	$-0.571634\pi$
$101$	−4.48528	−0.446302	−0.223151	+	0.974784i	$0.571634\pi$
$102$	0	0
$103$	−14.1421	−1.39347	−0.696733	−	0.717331i	$-0.745364\pi$
$103$	−14.1421	−1.39347	−0.696733	+	0.717331i	$0.745364\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.1421	1.94721	0.973607	−	0.228232i	$-0.0732943\pi$
$107$	20.1421	1.94721	0.973607	+	0.228232i	$0.0732943\pi$
$108$	0	0
$109$	3.00000	0.287348	0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000	0.287348	0.143674	+	0.989625i	$0.454108\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.485281	−0.0456514	−0.0228257	−	0.999739i	$-0.507266\pi$
$113$	−0.485281	−0.0456514	−0.0228257	+	0.999739i	$0.507266\pi$
$114$	0	0
$115$	8.65685	0.807256
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.41421	0.679660
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.6569	−1.03438	−0.517189	−	0.855871i	$-0.673022\pi$
$127$	−11.6569	−1.03438	−0.517189	+	0.855871i	$0.673022\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.82843	0.771343	0.385672	−	0.922636i	$-0.373970\pi$
$131$	8.82843	0.771343	0.385672	+	0.922636i	$0.373970\pi$
$132$	0	0
$133$	1.75736	0.152382
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.24264	0.618781	0.309390	−	0.950935i	$-0.399875\pi$
$137$	7.24264	0.618781	0.309390	+	0.950935i	$0.399875\pi$
$138$	0	0
$139$	5.65685	0.479808	0.239904	−	0.970797i	$-0.422884\pi$
$139$	5.65685	0.479808	0.239904	+	0.970797i	$0.422884\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.17157	0.265220
$144$	0	0
$145$	−4.24264	−0.352332
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.24264	0.511417	0.255709	−	0.966754i	$-0.417691\pi$
$149$	6.24264	0.511417	0.255709	+	0.966754i	$0.417691\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.07107	0.326996
$156$	0	0
$157$	−7.75736	−0.619105	−0.309552	−	0.950882i	$-0.600179\pi$
$157$	−7.75736	−0.619105	−0.309552	+	0.950882i	$0.600179\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−12.2426	−0.964855
$162$	0	0
$163$	20.8284	1.63141	0.815704	−	0.578469i	$-0.196350\pi$
$163$	20.8284	1.63141	0.815704	+	0.578469i	$0.196350\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.17157	0.477571	0.238785	−	0.971072i	$-0.423251\pi$
$167$	6.17157	0.477571	0.238785	+	0.971072i	$0.423251\pi$
$168$	0	0
$169$	−7.97056	−0.613120
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.2132	1.68884	0.844419	−	0.535683i	$-0.179946\pi$
$173$	22.2132	1.68884	0.844419	+	0.535683i	$0.179946\pi$
$174$	0	0
$175$	−1.41421	−0.106904
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.17157	0.685516	0.342758	−	0.939424i	$-0.388639\pi$
$179$	9.17157	0.685516	0.342758	+	0.939424i	$0.388639\pi$
$180$	0	0
$181$	15.9706	1.18708	0.593541	−	0.804804i	$-0.297729\pi$
$181$	15.9706	1.18708	0.593541	+	0.804804i	$0.297729\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.48528	−0.476807
$186$	0	0
$187$	−7.41421	−0.542181
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.4853	1.04812	0.524059	−	0.851682i	$-0.324417\pi$
$191$	14.4853	1.04812	0.524059	+	0.851682i	$0.324417\pi$
$192$	0	0
$193$	−10.2426	−0.737281	−0.368641	−	0.929572i	$-0.620177\pi$
$193$	−10.2426	−0.737281	−0.368641	+	0.929572i	$0.620177\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.24264	0.516017	0.258008	−	0.966143i	$-0.416934\pi$
$197$	7.24264	0.516017	0.258008	+	0.966143i	$0.416934\pi$
$198$	0	0
$199$	−14.8284	−1.05116	−0.525580	−	0.850744i	$-0.676152\pi$
$199$	−14.8284	−1.05116	−0.525580	+	0.850744i	$0.676152\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	6.24264	0.436005
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.75736	−0.121559
$210$	0	0
$211$	−1.24264	−0.0855469	−0.0427735	−	0.999085i	$-0.513619\pi$
$211$	−1.24264	−0.0855469	−0.0427735	+	0.999085i	$0.513619\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.2426	0.698542
$216$	0	0
$217$	−5.75736	−0.390835
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11.7574	−0.790886
$222$	0	0
$223$	14.4853	0.970006	0.485003	−	0.874512i	$-0.338818\pi$
$223$	14.4853	0.970006	0.485003	+	0.874512i	$0.338818\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.1716	0.807856	0.403928	−	0.914791i	$-0.367645\pi$
$227$	12.1716	0.807856	0.403928	+	0.914791i	$0.367645\pi$
$228$	0	0
$229$	−15.4853	−1.02330	−0.511648	−	0.859195i	$-0.670965\pi$
$229$	−15.4853	−1.02330	−0.511648	+	0.859195i	$0.670965\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	0.343146	0.0223844
$236$	0	0
$237$	0	0
$238$	0	0
$239$	25.7990	1.66880	0.834399	−	0.551161i	$-0.185815\pi$
$239$	25.7990	1.66880	0.834399	+	0.551161i	$0.185815\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.00000	−0.319438
$246$	0	0
$247$	−2.78680	−0.177320
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.7990	−1.24970	−0.624851	−	0.780744i	$-0.714840\pi$
$251$	−19.7990	−1.24970	−0.624851	+	0.780744i	$0.714840\pi$
$252$	0	0
$253$	12.2426	0.769688
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.7279	1.23059	0.615297	−	0.788295i	$-0.289036\pi$
$257$	19.7279	1.23059	0.615297	+	0.788295i	$0.289036\pi$
$258$	0	0
$259$	9.17157	0.569894
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.343146	0.0211593	0.0105796	−	0.999944i	$-0.496632\pi$
$263$	0.343146	0.0211593	0.0105796	+	0.999944i	$0.496632\pi$
$264$	0	0
$265$	1.24264	0.0763348
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.51472	0.580123	0.290061	−	0.957008i	$-0.406324\pi$
$269$	9.51472	0.580123	0.290061	+	0.957008i	$0.406324\pi$
$270$	0	0
$271$	−19.2426	−1.16891	−0.584454	−	0.811427i	$-0.698691\pi$
$271$	−19.2426	−1.16891	−0.584454	+	0.811427i	$0.698691\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.41421	0.0852803
$276$	0	0
$277$	20.7279	1.24542	0.622710	−	0.782453i	$-0.286032\pi$
$277$	20.7279	1.24542	0.622710	+	0.782453i	$0.286032\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−0.727922	−0.0434242	−0.0217121	−	0.999764i	$-0.506912\pi$
$281$	−0.727922	−0.0434242	−0.0217121	+	0.999764i	$0.506912\pi$
$282$	0	0
$283$	22.2426	1.32219	0.661094	−	0.750303i	$-0.270092\pi$
$283$	22.2426	1.32219	0.661094	+	0.750303i	$0.270092\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.82843	−0.521126
$288$	0	0
$289$	10.4853	0.616781
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−22.2132	−1.29771	−0.648855	−	0.760912i	$-0.724752\pi$
$293$	−22.2132	−1.29771	−0.648855	+	0.760912i	$0.724752\pi$
$294$	0	0
$295$	−1.07107	−0.0623600
$296$	0	0
$297$	0	0
$298$	0	0
$299$	19.4142	1.12275
$300$	0	0
$301$	−14.4853	−0.834918
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.00000	0.0572598
$306$	0	0
$307$	17.3137	0.988146	0.494073	−	0.869421i	$-0.335508\pi$
$307$	17.3137	0.988146	0.494073	+	0.869421i	$0.335508\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.10051	0.459338	0.229669	−	0.973269i	$-0.426236\pi$
$311$	8.10051	0.459338	0.229669	+	0.973269i	$0.426236\pi$
$312$	0	0
$313$	20.4853	1.15790	0.578948	−	0.815364i	$-0.303463\pi$
$313$	20.4853	1.15790	0.578948	+	0.815364i	$0.303463\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.2132	−1.02296	−0.511478	−	0.859297i	$-0.670902\pi$
$317$	−18.2132	−1.02296	−0.511478	+	0.859297i	$0.670902\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.51472	0.362489
$324$	0	0
$325$	2.24264	0.124399
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.485281	−0.0267544
$330$	0	0
$331$	−14.1421	−0.777322	−0.388661	−	0.921381i	$-0.627062\pi$
$331$	−14.1421	−0.777322	−0.388661	+	0.921381i	$0.627062\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.343146	0.0187481
$336$	0	0
$337$	−30.7279	−1.67386	−0.836928	−	0.547313i	$-0.815651\pi$
$337$	−30.7279	−1.67386	−0.836928	+	0.547313i	$0.815651\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.75736	0.311778
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.97056	−0.266834	−0.133417	−	0.991060i	$-0.542595\pi$
$347$	−4.97056	−0.266834	−0.133417	+	0.991060i	$0.542595\pi$
$348$	0	0
$349$	0.514719	0.0275523	0.0137761	−	0.999905i	$-0.495615\pi$
$349$	0.514719	0.0275523	0.0137761	+	0.999905i	$0.495615\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.48528	−0.132278	−0.0661391	−	0.997810i	$-0.521068\pi$
$353$	−2.48528	−0.132278	−0.0661391	+	0.997810i	$0.521068\pi$
$354$	0	0
$355$	10.5858	0.561835
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.6569	−0.615225	−0.307613	−	0.951512i	$-0.599530\pi$
$359$	−11.6569	−0.615225	−0.307613	+	0.951512i	$0.599530\pi$
$360$	0	0
$361$	−17.4558	−0.918729
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.24264	0.222070
$366$	0	0
$367$	−16.2426	−0.847859	−0.423929	−	0.905695i	$-0.639350\pi$
$367$	−16.2426	−0.847859	−0.423929	+	0.905695i	$0.639350\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.75736	−0.0912375
$372$	0	0
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.51472	−0.490033
$378$	0	0
$379$	4.41421	0.226743	0.113371	−	0.993553i	$-0.463835\pi$
$379$	4.41421	0.226743	0.113371	+	0.993553i	$0.463835\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.3137	−0.731396	−0.365698	−	0.930734i	$-0.619170\pi$
$383$	−14.3137	−0.731396	−0.365698	+	0.930734i	$0.619170\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.7574	−1.30595	−0.652975	−	0.757379i	$-0.726479\pi$
$389$	−25.7574	−1.30595	−0.652975	+	0.757379i	$0.726479\pi$
$390$	0	0
$391$	−45.3848	−2.29521
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.58579	−0.0797896
$396$	0	0
$397$	9.75736	0.489708	0.244854	−	0.969560i	$-0.421260\pi$
$397$	9.75736	0.489708	0.244854	+	0.969560i	$0.421260\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.4558	1.37108	0.685540	−	0.728035i	$-0.259566\pi$
$401$	27.4558	1.37108	0.685540	+	0.728035i	$0.259566\pi$
$402$	0	0
$403$	9.12994	0.454795
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.17157	−0.454618
$408$	0	0
$409$	−4.51472	−0.223238	−0.111619	−	0.993751i	$-0.535604\pi$
$409$	−4.51472	−0.223238	−0.111619	+	0.993751i	$0.535604\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.51472	0.0745344
$414$	0	0
$415$	6.17157	0.302951
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.0416	1.17451	0.587255	−	0.809402i	$-0.300208\pi$
$419$	24.0416	1.17451	0.587255	+	0.809402i	$0.300208\pi$
$420$	0	0
$421$	2.51472	0.122560	0.0612799	−	0.998121i	$-0.480482\pi$
$421$	2.51472	0.122560	0.0612799	+	0.998121i	$0.480482\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.24264	−0.254305
$426$	0	0
$427$	−1.41421	−0.0684386
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.8995	0.765852	0.382926	−	0.923779i	$-0.374916\pi$
$431$	15.8995	0.765852	0.382926	+	0.923779i	$0.374916\pi$
$432$	0	0
$433$	−6.24264	−0.300002	−0.150001	−	0.988686i	$-0.547928\pi$
$433$	−6.24264	−0.300002	−0.150001	+	0.988686i	$0.547928\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.7574	−0.514594
$438$	0	0
$439$	−28.4142	−1.35614	−0.678068	−	0.734999i	$-0.737183\pi$
$439$	−28.4142	−1.35614	−0.678068	+	0.734999i	$0.737183\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.62742	0.362390	0.181195	−	0.983447i	$-0.442004\pi$
$443$	7.62742	0.362390	0.181195	+	0.983447i	$0.442004\pi$
$444$	0	0
$445$	2.24264	0.106311
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.75736	0.177321	0.0886604	−	0.996062i	$-0.471741\pi$
$449$	3.75736	0.177321	0.0886604	+	0.996062i	$0.471741\pi$
$450$	0	0
$451$	8.82843	0.415714
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.17157	−0.148686
$456$	0	0
$457$	3.02944	0.141711	0.0708555	−	0.997487i	$-0.477427\pi$
$457$	3.02944	0.141711	0.0708555	+	0.997487i	$0.477427\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.9706	1.06985	0.534923	−	0.844901i	$-0.320341\pi$
$461$	22.9706	1.06985	0.534923	+	0.844901i	$0.320341\pi$
$462$	0	0
$463$	38.4853	1.78856	0.894281	−	0.447505i	$-0.147687\pi$
$463$	38.4853	1.78856	0.894281	+	0.447505i	$0.147687\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.1421	1.34854	0.674269	−	0.738486i	$-0.264459\pi$
$467$	29.1421	1.34854	0.674269	+	0.738486i	$0.264459\pi$
$468$	0	0
$469$	−0.485281	−0.0224082
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14.4853	0.666034
$474$	0	0
$475$	−1.24264	−0.0570163
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.2426	−0.742145	−0.371073	−	0.928604i	$-0.621010\pi$
$479$	−16.2426	−0.742145	−0.371073	+	0.928604i	$0.621010\pi$
$480$	0	0
$481$	−14.5442	−0.663156
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.4853	0.748558
$486$	0	0
$487$	−10.9289	−0.495237	−0.247619	−	0.968858i	$-0.579648\pi$
$487$	−10.9289	−0.495237	−0.247619	+	0.968858i	$0.579648\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.68629	−0.301748	−0.150874	−	0.988553i	$-0.548209\pi$
$491$	−6.68629	−0.301748	−0.150874	+	0.988553i	$0.548209\pi$
$492$	0	0
$493$	22.2426	1.00176
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.9706	−0.671522
$498$	0	0
$499$	21.7279	0.972675	0.486338	−	0.873771i	$-0.338332\pi$
$499$	21.7279	0.972675	0.486338	+	0.873771i	$0.338332\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.97056	0.355390	0.177695	−	0.984086i	$-0.443136\pi$
$503$	7.97056	0.355390	0.177695	+	0.984086i	$0.443136\pi$
$504$	0	0
$505$	−4.48528	−0.199592
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.02944	−0.134277	−0.0671387	−	0.997744i	$-0.521387\pi$
$509$	−3.02944	−0.134277	−0.0671387	+	0.997744i	$0.521387\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.1421	−0.623177
$516$	0	0
$517$	0.485281	0.0213427
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.21320	0.316016	0.158008	−	0.987438i	$-0.449493\pi$
$521$	7.21320	0.316016	0.158008	+	0.987438i	$0.449493\pi$
$522$	0	0
$523$	19.7574	0.863929	0.431965	−	0.901891i	$-0.357821\pi$
$523$	19.7574	0.863929	0.431965	+	0.901891i	$0.357821\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21.3431	−0.929722
$528$	0	0
$529$	51.9411	2.25831
$530$	0	0
$531$	0	0
$532$	0	0
$533$	14.0000	0.606407
$534$	0	0
$535$	20.1421	0.870820
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7.07107	−0.304572
$540$	0	0
$541$	−12.4853	−0.536784	−0.268392	−	0.963310i	$-0.586492\pi$
$541$	−12.4853	−0.536784	−0.268392	+	0.963310i	$0.586492\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.00000	0.128506
$546$	0	0
$547$	35.6985	1.52636	0.763178	−	0.646188i	$-0.223638\pi$
$547$	35.6985	1.52636	0.763178	+	0.646188i	$0.223638\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.27208	0.224598
$552$	0	0
$553$	2.24264	0.0953668
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.5147	0.742122	0.371061	−	0.928608i	$-0.378994\pi$
$557$	17.5147	0.742122	0.371061	+	0.928608i	$0.378994\pi$
$558$	0	0
$559$	22.9706	0.971551
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.14214	−0.343150	−0.171575	−	0.985171i	$-0.554886\pi$
$563$	−8.14214	−0.343150	−0.171575	+	0.985171i	$0.554886\pi$
$564$	0	0
$565$	−0.485281	−0.0204159
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.75736	−0.157517	−0.0787583	−	0.996894i	$-0.525096\pi$
$569$	−3.75736	−0.157517	−0.0787583	+	0.996894i	$0.525096\pi$
$570$	0	0
$571$	−25.5858	−1.07073	−0.535366	−	0.844620i	$-0.679826\pi$
$571$	−25.5858	−1.07073	−0.535366	+	0.844620i	$0.679826\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.65685	0.361016
$576$	0	0
$577$	28.7279	1.19596	0.597980	−	0.801511i	$-0.295970\pi$
$577$	28.7279	1.19596	0.597980	+	0.801511i	$0.295970\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.72792	−0.362095
$582$	0	0
$583$	1.75736	0.0727824
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.48528	−0.226402	−0.113201	−	0.993572i	$-0.536110\pi$
$587$	−5.48528	−0.226402	−0.113201	+	0.993572i	$0.536110\pi$
$588$	0	0
$589$	−5.05887	−0.208447
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−37.2426	−1.52937	−0.764686	−	0.644403i	$-0.777106\pi$
$593$	−37.2426	−1.52937	−0.764686	+	0.644403i	$0.777106\pi$
$594$	0	0
$595$	7.41421	0.303953
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−42.0416	−1.71777	−0.858887	−	0.512165i	$-0.828844\pi$
$599$	−42.0416	−1.71777	−0.858887	+	0.512165i	$0.828844\pi$
$600$	0	0
$601$	27.9706	1.14094	0.570472	−	0.821317i	$-0.306760\pi$
$601$	27.9706	1.14094	0.570472	+	0.821317i	$0.306760\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.00000	−0.365902
$606$	0	0
$607$	−41.3553	−1.67856	−0.839281	−	0.543698i	$-0.817024\pi$
$607$	−41.3553	−1.67856	−0.839281	+	0.543698i	$0.817024\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.769553	0.0311328
$612$	0	0
$613$	−12.9706	−0.523876	−0.261938	−	0.965085i	$-0.584362\pi$
$613$	−12.9706	−0.523876	−0.261938	+	0.965085i	$0.584362\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.2132	−0.813753	−0.406876	−	0.913483i	$-0.633382\pi$
$617$	−20.2132	−0.813753	−0.406876	+	0.913483i	$0.633382\pi$
$618$	0	0
$619$	19.7990	0.795789	0.397894	−	0.917431i	$-0.369741\pi$
$619$	19.7990	0.795789	0.397894	+	0.917431i	$0.369741\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.17157	−0.127066
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	34.0000	1.35567
$630$	0	0
$631$	−43.2426	−1.72146	−0.860731	−	0.509060i	$-0.829993\pi$
$631$	−43.2426	−1.72146	−0.860731	+	0.509060i	$0.829993\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.6569	−0.462588
$636$	0	0
$637$	−11.2132	−0.444283
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.9706	0.828287	0.414144	−	0.910212i	$-0.364081\pi$
$641$	20.9706	0.828287	0.414144	+	0.910212i	$0.364081\pi$
$642$	0	0
$643$	45.8995	1.81010	0.905050	−	0.425306i	$-0.139833\pi$
$643$	45.8995	1.81010	0.905050	+	0.425306i	$0.139833\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.34315	−0.367317	−0.183658	−	0.982990i	$-0.558794\pi$
$647$	−9.34315	−0.367317	−0.183658	+	0.982990i	$0.558794\pi$
$648$	0	0
$649$	−1.51472	−0.0594579
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.7574	−1.28189	−0.640947	−	0.767585i	$-0.721458\pi$
$653$	−32.7574	−1.28189	−0.640947	+	0.767585i	$0.721458\pi$
$654$	0	0
$655$	8.82843	0.344955
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−39.2132	−1.52753	−0.763765	−	0.645495i	$-0.776651\pi$
$659$	−39.2132	−1.52753	−0.763765	+	0.645495i	$0.776651\pi$
$660$	0	0
$661$	−4.48528	−0.174457	−0.0872286	−	0.996188i	$-0.527801\pi$
$661$	−4.48528	−0.174457	−0.0872286	+	0.996188i	$0.527801\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.75736	0.0681475
$666$	0	0
$667$	−36.7279	−1.42211
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.41421	0.0545951
$672$	0	0
$673$	−27.6985	−1.06770	−0.533849	−	0.845580i	$-0.679255\pi$
$673$	−27.6985	−1.06770	−0.533849	+	0.845580i	$0.679255\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.4264	1.47685	0.738423	−	0.674337i	$-0.235571\pi$
$677$	38.4264	1.47685	0.738423	+	0.674337i	$0.235571\pi$
$678$	0	0
$679$	−23.3137	−0.894698
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−26.6569	−1.02000	−0.509998	−	0.860176i	$-0.670354\pi$
$683$	−26.6569	−1.02000	−0.509998	+	0.860176i	$0.670354\pi$
$684$	0	0
$685$	7.24264	0.276727
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.78680	0.106168
$690$	0	0
$691$	3.04163	0.115709	0.0578545	−	0.998325i	$-0.481574\pi$
$691$	3.04163	0.115709	0.0578545	+	0.998325i	$0.481574\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.65685	0.214577
$696$	0	0
$697$	−32.7279	−1.23966
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.51472	0.0572101	0.0286051	−	0.999591i	$-0.490893\pi$
$701$	1.51472	0.0572101	0.0286051	+	0.999591i	$0.490893\pi$
$702$	0	0
$703$	8.05887	0.303946
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.34315	0.238559
$708$	0	0
$709$	33.4558	1.25646	0.628230	−	0.778027i	$-0.283780\pi$
$709$	33.4558	1.25646	0.628230	+	0.778027i	$0.283780\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	35.2426	1.31985
$714$	0	0
$715$	3.17157	0.118610
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33.5980	−1.25299	−0.626497	−	0.779424i	$-0.715512\pi$
$719$	−33.5980	−1.25299	−0.626497	+	0.779424i	$0.715512\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.24264	−0.157568
$726$	0	0
$727$	−32.4853	−1.20481	−0.602406	−	0.798190i	$-0.705791\pi$
$727$	−32.4853	−1.20481	−0.602406	+	0.798190i	$0.705791\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−53.6985	−1.98611
$732$	0	0
$733$	−45.4558	−1.67895	−0.839475	−	0.543398i	$-0.817137\pi$
$733$	−45.4558	−1.67895	−0.839475	+	0.543398i	$0.817137\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.485281	0.0178756
$738$	0	0
$739$	−32.3553	−1.19021	−0.595105	−	0.803648i	$-0.702890\pi$
$739$	−32.3553	−1.19021	−0.595105	+	0.803648i	$0.702890\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	28.2843	1.03765	0.518825	−	0.854881i	$-0.326370\pi$
$743$	28.2843	1.03765	0.518825	+	0.854881i	$0.326370\pi$
$744$	0	0
$745$	6.24264	0.228713
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−28.4853	−1.04083
$750$	0	0
$751$	−31.2426	−1.14006	−0.570030	−	0.821624i	$-0.693068\pi$
$751$	−31.2426	−1.14006	−0.570030	+	0.821624i	$0.693068\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−18.0000	−0.655087
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.6985	−0.786569	−0.393285	−	0.919417i	$-0.628661\pi$
$761$	−21.6985	−0.786569	−0.393285	+	0.919417i	$0.628661\pi$
$762$	0	0
$763$	−4.24264	−0.153594
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.40202	−0.0867319
$768$	0	0
$769$	−51.4264	−1.85448	−0.927242	−	0.374463i	$-0.877827\pi$
$769$	−51.4264	−1.85448	−0.927242	+	0.374463i	$0.877827\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24.2132	0.870888	0.435444	−	0.900216i	$-0.356591\pi$
$773$	24.2132	0.870888	0.435444	+	0.900216i	$0.356591\pi$
$774$	0	0
$775$	4.07107	0.146237
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.75736	−0.277936
$780$	0	0
$781$	14.9706	0.535689
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.75736	−0.276872
$786$	0	0
$787$	−6.72792	−0.239825	−0.119912	−	0.992784i	$-0.538261\pi$
$787$	−6.72792	−0.239825	−0.119912	+	0.992784i	$0.538261\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.686292	0.0244017
$792$	0	0
$793$	2.24264	0.0796385
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.2426	−0.894140	−0.447070	−	0.894499i	$-0.647533\pi$
$797$	−25.2426	−0.894140	−0.447070	+	0.894499i	$0.647533\pi$
$798$	0	0
$799$	−1.79899	−0.0636437
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.00000	0.211735
$804$	0	0
$805$	−12.2426	−0.431496
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.2426	0.360112	0.180056	−	0.983656i	$-0.442372\pi$
$809$	10.2426	0.360112	0.180056	+	0.983656i	$0.442372\pi$
$810$	0	0
$811$	−7.37258	−0.258886	−0.129443	−	0.991587i	$-0.541319\pi$
$811$	−7.37258	−0.258886	−0.129443	+	0.991587i	$0.541319\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.8284	0.729588
$816$	0	0
$817$	−12.7279	−0.445294
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.9706	−0.941279	−0.470640	−	0.882326i	$-0.655977\pi$
$821$	−26.9706	−0.941279	−0.470640	+	0.882326i	$0.655977\pi$
$822$	0	0
$823$	35.6985	1.24437	0.622185	−	0.782870i	$-0.286245\pi$
$823$	35.6985	1.24437	0.622185	+	0.782870i	$0.286245\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22.1127	−0.768934	−0.384467	−	0.923139i	$-0.625615\pi$
$827$	−22.1127	−0.768934	−0.384467	+	0.923139i	$0.625615\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	26.2132	0.908234
$834$	0	0
$835$	6.17157	0.213576
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.85786	−0.340331	−0.170166	−	0.985415i	$-0.554430\pi$
$839$	−9.85786	−0.340331	−0.170166	+	0.985415i	$0.554430\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.97056	−0.274196
$846$	0	0
$847$	12.7279	0.437337
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−56.1421	−1.92453
$852$	0	0
$853$	48.4853	1.66010	0.830052	−	0.557686i	$-0.188311\pi$
$853$	48.4853	1.66010	0.830052	+	0.557686i	$0.188311\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	36.2132	1.23702	0.618510	−	0.785777i	$-0.287737\pi$
$857$	36.2132	1.23702	0.618510	+	0.785777i	$0.287737\pi$
$858$	0	0
$859$	−50.6985	−1.72981	−0.864905	−	0.501936i	$-0.832621\pi$
$859$	−50.6985	−1.72981	−0.864905	+	0.501936i	$0.832621\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46.4558	−1.58138	−0.790688	−	0.612220i	$-0.790277\pi$
$863$	−46.4558	−1.58138	−0.790688	+	0.612220i	$0.790277\pi$
$864$	0	0
$865$	22.2132	0.755272
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.24264	−0.0760764
$870$	0	0
$871$	0.769553	0.0260753
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.41421	−0.0478091
$876$	0	0
$877$	−3.75736	−0.126877	−0.0634385	−	0.997986i	$-0.520207\pi$
$877$	−3.75736	−0.126877	−0.0634385	+	0.997986i	$0.520207\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−57.2132	−1.92756	−0.963781	−	0.266695i	$-0.914068\pi$
$881$	−57.2132	−1.92756	−0.963781	+	0.266695i	$0.914068\pi$
$882$	0	0
$883$	50.8701	1.71191	0.855957	−	0.517047i	$-0.172969\pi$
$883$	50.8701	1.71191	0.855957	+	0.517047i	$0.172969\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.1421	−1.58288	−0.791439	−	0.611248i	$-0.790668\pi$
$887$	−47.1421	−1.58288	−0.791439	+	0.611248i	$0.790668\pi$
$888$	0	0
$889$	16.4853	0.552899
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.426407	−0.0142692
$894$	0	0
$895$	9.17157	0.306572
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−17.2721	−0.576056
$900$	0	0
$901$	−6.51472	−0.217037
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.9706	0.530879
$906$	0	0
$907$	−42.4264	−1.40875	−0.704373	−	0.709830i	$-0.748772\pi$
$907$	−42.4264	−1.40875	−0.704373	+	0.709830i	$0.748772\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25.0711	0.830642	0.415321	−	0.909675i	$-0.363669\pi$
$911$	25.0711	0.830642	0.415321	+	0.909675i	$0.363669\pi$
$912$	0	0
$913$	8.72792	0.288852
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12.4853	−0.412300
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	23.7401	0.781415
$924$	0	0
$925$	−6.48528	−0.213235
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−50.2426	−1.64841	−0.824204	−	0.566293i	$-0.808377\pi$
$929$	−50.2426	−1.64841	−0.824204	+	0.566293i	$0.808377\pi$
$930$	0	0
$931$	6.21320	0.203630
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.41421	−0.242471
$936$	0	0
$937$	37.9411	1.23948	0.619741	−	0.784806i	$-0.287238\pi$
$937$	37.9411	1.23948	0.619741	+	0.784806i	$0.287238\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−38.9706	−1.27040	−0.635202	−	0.772346i	$-0.719083\pi$
$941$	−38.9706	−1.27040	−0.635202	+	0.772346i	$0.719083\pi$
$942$	0	0
$943$	54.0416	1.75984
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.51472	−0.211700	−0.105850	−	0.994382i	$-0.533756\pi$
$947$	−6.51472	−0.211700	−0.105850	+	0.994382i	$0.533756\pi$
$948$	0	0
$949$	9.51472	0.308861
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−46.9706	−1.52153	−0.760763	−	0.649030i	$-0.775175\pi$
$953$	−46.9706	−1.52153	−0.760763	+	0.649030i	$0.775175\pi$
$954$	0	0
$955$	14.4853	0.468733
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.2426	−0.330752
$960$	0	0
$961$	−14.4264	−0.465368
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.2426	−0.329722
$966$	0	0
$967$	2.48528	0.0799213	0.0399606	−	0.999201i	$-0.487277\pi$
$967$	2.48528	0.0799213	0.0399606	+	0.999201i	$0.487277\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−15.8579	−0.508903	−0.254452	−	0.967086i	$-0.581895\pi$
$971$	−15.8579	−0.508903	−0.254452	+	0.967086i	$0.581895\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.4853	1.61517	0.807584	−	0.589753i	$-0.200775\pi$
$977$	50.4853	1.61517	0.807584	+	0.589753i	$0.200775\pi$
$978$	0	0
$979$	3.17157	0.101364
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−40.7990	−1.30129	−0.650643	−	0.759384i	$-0.725500\pi$
$983$	−40.7990	−1.30129	−0.650643	+	0.759384i	$0.725500\pi$
$984$	0	0
$985$	7.24264	0.230770
$986$	0	0
$987$	0	0
$988$	0	0
$989$	88.6690	2.81951
$990$	0	0
$991$	−2.27208	−0.0721749	−0.0360875	−	0.999349i	$-0.511489\pi$
$991$	−2.27208	−0.0721749	−0.0360875	+	0.999349i	$0.511489\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14.8284	−0.470093
$996$	0	0
$997$	−8.24264	−0.261047	−0.130524	−	0.991445i	$-0.541666\pi$
$997$	−8.24264	−0.261047	−0.130524	+	0.991445i	$0.541666\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.a.bb.1.1	yes	2
3.2	odd	2		4320.2.a.r.1.1	yes	2
4.3	odd	2		4320.2.a.ba.1.2	yes	2
8.3	odd	2		8640.2.a.cp.1.2		2
8.5	even	2		8640.2.a.co.1.1		2
12.11	even	2		4320.2.a.q.1.2	✓	2
24.5	odd	2		8640.2.a.dc.1.1		2
24.11	even	2		8640.2.a.dd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4320.2.a.q.1.2	✓	2	12.11	even	2
4320.2.a.r.1.1	yes	2	3.2	odd	2
4320.2.a.ba.1.2	yes	2	4.3	odd	2
4320.2.a.bb.1.1	yes	2	1.1	even	1	trivial
8640.2.a.co.1.1		2	8.5	even	2
8640.2.a.cp.1.2		2	8.3	odd	2
8640.2.a.dc.1.1		2	24.5	odd	2
8640.2.a.dd.1.2		2	24.11	even	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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.41421 q^{7} +1.41421 q^{11} +2.24264 q^{13} -5.24264 q^{17} -1.24264 q^{19} +8.65685 q^{23} +1.00000 q^{25} -4.24264 q^{29} +4.07107 q^{31} -1.41421 q^{35} -6.48528 q^{37} +6.24264 q^{41} +10.2426 q^{43} +0.343146 q^{47} -5.00000 q^{49} +1.24264 q^{53} +1.41421 q^{55} -1.07107 q^{59} +1.00000 q^{61} +2.24264 q^{65} +0.343146 q^{67} +10.5858 q^{71} +4.24264 q^{73} -2.00000 q^{77} -1.58579 q^{79} +6.17157 q^{83} -5.24264 q^{85} +2.24264 q^{89} -3.17157 q^{91} -1.24264 q^{95} +16.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{13} - 2 q^{17} + 6 q^{19} + 6 q^{23} + 2 q^{25} - 6 q^{31} + 4 q^{37} + 4 q^{41} + 12 q^{43} + 12 q^{47} - 10 q^{49} - 6 q^{53} + 12 q^{59} + 2 q^{61} - 4 q^{65} + 12 q^{67} + 24 q^{71}+ \cdots + 16 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^13 - 2 * q^17 + 6 * q^19 + 6 * q^23 + 2 * q^25 - 6 * q^31 + 4 * q^37 + 4 * q^41 + 12 * q^43 + 12 * q^47 - 10 * q^49 - 6 * q^53 + 12 * q^59 + 2 * q^61 - 4 * q^65 + 12 * q^67 + 24 * q^71 - 4 * q^77 - 6 * q^79 + 18 * q^83 - 2 * q^85 - 4 * q^89 - 12 * q^91 + 6 * q^95 + 16 * q^97

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