LMFDB - Embedded newform 4320.2.a.ba.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:	$1$
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)

$f(q)$	$=$	$q+1.00000 q^{5} -1.41421 q^{7} +1.41421 q^{11} -6.24264 q^{13} +3.24264 q^{17} -7.24264 q^{19} +2.65685 q^{23} +1.00000 q^{25} +4.24264 q^{29} +10.0711 q^{31} -1.41421 q^{35} +10.4853 q^{37} -2.24264 q^{41} -1.75736 q^{43} -11.6569 q^{47} -5.00000 q^{49} -7.24264 q^{53} +1.41421 q^{55} -13.0711 q^{59} +1.00000 q^{61} -6.24264 q^{65} -11.6569 q^{67} -13.4142 q^{71} -4.24264 q^{73} -2.00000 q^{77} +4.41421 q^{79} -11.8284 q^{83} +3.24264 q^{85} -6.24264 q^{89} +8.82843 q^{91} -7.24264 q^{95} -0.485281 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	−6.24264	−1.73140	−0.865699	−	0.500566i	$-0.833125\pi$
$13$	−6.24264	−1.73140	−0.865699	+	0.500566i	$0.833125\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.24264	0.786456	0.393228	−	0.919441i	$-0.371358\pi$
$17$	3.24264	0.786456	0.393228	+	0.919441i	$0.371358\pi$
$18$	0	0
$19$	−7.24264	−1.66158	−0.830788	−	0.556589i	$-0.812110\pi$
$19$	−7.24264	−1.66158	−0.830788	+	0.556589i	$0.812110\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.65685	0.553992	0.276996	−	0.960871i	$-0.410661\pi$
$23$	2.65685	0.553992	0.276996	+	0.960871i	$0.410661\pi$
$24$	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	$-0.371119\pi$
$29$	4.24264	0.787839	0.393919	+	0.919145i	$0.371119\pi$
$30$	0	0
$31$	10.0711	1.80882	0.904409	−	0.426667i	$-0.140313\pi$
$31$	10.0711	1.80882	0.904409	+	0.426667i	$0.140313\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.41421	−0.239046
$36$	0	0
$37$	10.4853	1.72377	0.861885	−	0.507104i	$-0.169284\pi$
$37$	10.4853	1.72377	0.861885	+	0.507104i	$0.169284\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.24264	−0.350242	−0.175121	−	0.984547i	$-0.556032\pi$
$41$	−2.24264	−0.350242	−0.175121	+	0.984547i	$0.556032\pi$
$42$	0	0
$43$	−1.75736	−0.267995	−0.133997	−	0.990982i	$-0.542781\pi$
$43$	−1.75736	−0.267995	−0.133997	+	0.990982i	$0.542781\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.6569	−1.70033	−0.850163	−	0.526519i	$-0.823497\pi$
$47$	−11.6569	−1.70033	−0.850163	+	0.526519i	$0.823497\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.24264	−0.994853	−0.497427	−	0.867506i	$-0.665722\pi$
$53$	−7.24264	−0.994853	−0.497427	+	0.867506i	$0.665722\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.0711	−1.70171	−0.850854	−	0.525402i	$-0.823915\pi$
$59$	−13.0711	−1.70171	−0.850854	+	0.525402i	$0.823915\pi$
$60$	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
$65$	−6.24264	−0.774304
$66$	0	0
$67$	−11.6569	−1.42411	−0.712056	−	0.702123i	$-0.752236\pi$
$67$	−11.6569	−1.42411	−0.712056	+	0.702123i	$0.752236\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.4142	−1.59197	−0.795987	−	0.605314i	$-0.793048\pi$
$71$	−13.4142	−1.59197	−0.795987	+	0.605314i	$0.793048\pi$
$72$	0	0
$73$	−4.24264	−0.496564	−0.248282	−	0.968688i	$-0.579866\pi$
$73$	−4.24264	−0.496564	−0.248282	+	0.968688i	$0.579866\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	4.41421	0.496638	0.248319	−	0.968678i	$-0.420122\pi$
$79$	4.41421	0.496638	0.248319	+	0.968678i	$0.420122\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.8284	−1.29834	−0.649169	−	0.760644i	$-0.724883\pi$
$83$	−11.8284	−1.29834	−0.649169	+	0.760644i	$0.724883\pi$
$84$	0	0
$85$	3.24264	0.351714
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.24264	−0.661719	−0.330859	−	0.943680i	$-0.607339\pi$
$89$	−6.24264	−0.661719	−0.330859	+	0.943680i	$0.607339\pi$
$90$	0	0
$91$	8.82843	0.925471
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.24264	−0.743079
$96$	0	0
$97$	−0.485281	−0.0492729	−0.0246364	−	0.999696i	$-0.507843\pi$
$97$	−0.485281	−0.0492729	−0.0246364	+	0.999696i	$0.507843\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.4853	1.24233	0.621166	−	0.783679i	$-0.286659\pi$
$101$	12.4853	1.24233	0.621166	+	0.783679i	$0.286659\pi$
$102$	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
$107$	8.14214	0.787130	0.393565	−	0.919297i	$-0.371242\pi$
$107$	8.14214	0.787130	0.393565	+	0.919297i	$0.371242\pi$
$108$	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
$113$	16.4853	1.55080	0.775402	−	0.631467i	$-0.217547\pi$
$113$	16.4853	1.55080	0.775402	+	0.631467i	$0.217547\pi$
$114$	0	0
$115$	2.65685	0.247753
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.58579	−0.420378
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.343146	0.0304493	0.0152246	−	0.999884i	$-0.495154\pi$
$127$	0.343146	0.0304493	0.0152246	+	0.999884i	$0.495154\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.17157	−0.277102	−0.138551	−	0.990355i	$-0.544244\pi$
$131$	−3.17157	−0.277102	−0.138551	+	0.990355i	$0.544244\pi$
$132$	0	0
$133$	10.2426	0.888150
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.24264	−0.106166	−0.0530830	−	0.998590i	$-0.516905\pi$
$137$	−1.24264	−0.106166	−0.0530830	+	0.998590i	$0.516905\pi$
$138$	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
$143$	−8.82843	−0.738270
$144$	0	0
$145$	4.24264	0.352332
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.24264	−0.183724	−0.0918621	−	0.995772i	$-0.529282\pi$
$149$	−2.24264	−0.183724	−0.0918621	+	0.995772i	$0.529282\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.0711	0.808928
$156$	0	0
$157$	−16.2426	−1.29630	−0.648152	−	0.761511i	$-0.724458\pi$
$157$	−16.2426	−1.29630	−0.648152	+	0.761511i	$0.724458\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.75736	−0.296121
$162$	0	0
$163$	−15.1716	−1.18833	−0.594165	−	0.804343i	$-0.702517\pi$
$163$	−15.1716	−1.18833	−0.594165	+	0.804343i	$0.702517\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.8284	−0.915311	−0.457656	−	0.889130i	$-0.651311\pi$
$167$	−11.8284	−0.915311	−0.457656	+	0.889130i	$0.651311\pi$
$168$	0	0
$169$	25.9706	1.99774
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.2132	−1.53678	−0.768391	−	0.639981i	$-0.778942\pi$
$173$	−20.2132	−1.53678	−0.768391	+	0.639981i	$0.778942\pi$
$174$	0	0
$175$	−1.41421	−0.106904
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.8284	−1.10833	−0.554164	−	0.832407i	$-0.686962\pi$
$179$	−14.8284	−1.10833	−0.554164	+	0.832407i	$0.686962\pi$
$180$	0	0
$181$	−17.9706	−1.33574	−0.667871	−	0.744277i	$-0.732794\pi$
$181$	−17.9706	−1.33574	−0.667871	+	0.744277i	$0.732794\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.4853	0.770893
$186$	0	0
$187$	4.58579	0.335346
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.48528	0.179829	0.0899143	−	0.995950i	$-0.471341\pi$
$191$	2.48528	0.179829	0.0899143	+	0.995950i	$0.471341\pi$
$192$	0	0
$193$	−1.75736	−0.126497	−0.0632487	−	0.997998i	$-0.520146\pi$
$193$	−1.75736	−0.126497	−0.0632487	+	0.997998i	$0.520146\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.24264	−0.0885345	−0.0442672	−	0.999020i	$-0.514095\pi$
$197$	−1.24264	−0.0885345	−0.0442672	+	0.999020i	$0.514095\pi$
$198$	0	0
$199$	9.17157	0.650156	0.325078	−	0.945687i	$-0.394610\pi$
$199$	9.17157	0.650156	0.325078	+	0.945687i	$0.394610\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−2.24264	−0.156633
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.2426	−0.708498
$210$	0	0
$211$	−7.24264	−0.498604	−0.249302	−	0.968426i	$-0.580201\pi$
$211$	−7.24264	−0.498604	−0.249302	+	0.968426i	$0.580201\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.75736	−0.119851
$216$	0	0
$217$	−14.2426	−0.966853
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−20.2426	−1.36167
$222$	0	0
$223$	2.48528	0.166427	0.0832134	−	0.996532i	$-0.473482\pi$
$223$	2.48528	0.166427	0.0832134	+	0.996532i	$0.473482\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.8284	−1.18331	−0.591657	−	0.806190i	$-0.701526\pi$
$227$	−17.8284	−1.18331	−0.591657	+	0.806190i	$0.701526\pi$
$228$	0	0
$229$	1.48528	0.0981502	0.0490751	−	0.998795i	$-0.484373\pi$
$229$	1.48528	0.0981502	0.0490751	+	0.998795i	$0.484373\pi$
$230$	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
$235$	−11.6569	−0.760409
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.7990	0.892582	0.446291	−	0.894888i	$-0.352745\pi$
$239$	13.7990	0.892582	0.446291	+	0.894888i	$0.352745\pi$
$240$	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
$247$	45.2132	2.87685
$248$	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
$253$	3.75736	0.236223
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.72792	−0.357298	−0.178649	−	0.983913i	$-0.557173\pi$
$257$	−5.72792	−0.357298	−0.178649	+	0.983913i	$0.557173\pi$
$258$	0	0
$259$	−14.8284	−0.921394
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.6569	−0.718792	−0.359396	−	0.933185i	$-0.617017\pi$
$263$	−11.6569	−0.718792	−0.359396	+	0.933185i	$0.617017\pi$
$264$	0	0
$265$	−7.24264	−0.444912
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.4853	1.61484	0.807418	−	0.589980i	$-0.200864\pi$
$269$	26.4853	1.61484	0.807418	+	0.589980i	$0.200864\pi$
$270$	0	0
$271$	10.7574	0.653463	0.326732	−	0.945117i	$-0.394053\pi$
$271$	10.7574	0.653463	0.326732	+	0.945117i	$0.394053\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.41421	0.0852803
$276$	0	0
$277$	−4.72792	−0.284073	−0.142037	−	0.989861i	$-0.545365\pi$
$277$	−4.72792	−0.284073	−0.142037	+	0.989861i	$0.545365\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.7279	1.47514	0.737572	−	0.675268i	$-0.235972\pi$
$281$	24.7279	1.47514	0.737572	+	0.675268i	$0.235972\pi$
$282$	0	0
$283$	−13.7574	−0.817790	−0.408895	−	0.912581i	$-0.634086\pi$
$283$	−13.7574	−0.817790	−0.408895	+	0.912581i	$0.634086\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.17157	0.187212
$288$	0	0
$289$	−6.48528	−0.381487
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.2132	1.18087	0.590434	−	0.807086i	$-0.298957\pi$
$293$	20.2132	1.18087	0.590434	+	0.807086i	$0.298957\pi$
$294$	0	0
$295$	−13.0711	−0.761027
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−16.5858	−0.959181
$300$	0	0
$301$	2.48528	0.143249
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.00000	0.0572598
$306$	0	0
$307$	5.31371	0.303269	0.151635	−	0.988437i	$-0.451546\pi$
$307$	5.31371	0.303269	0.151635	+	0.988437i	$0.451546\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.8995	−1.58204	−0.791018	−	0.611793i	$-0.790448\pi$
$311$	−27.8995	−1.58204	−0.791018	+	0.611793i	$0.790448\pi$
$312$	0	0
$313$	3.51472	0.198664	0.0993318	−	0.995054i	$-0.468329\pi$
$313$	3.51472	0.198664	0.0993318	+	0.995054i	$0.468329\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.2132	1.35995	0.679974	−	0.733236i	$-0.261991\pi$
$317$	24.2132	1.35995	0.679974	+	0.733236i	$0.261991\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−23.4853	−1.30676
$324$	0	0
$325$	−6.24264	−0.346279
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16.4853	0.908863
$330$	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
$335$	−11.6569	−0.636882
$336$	0	0
$337$	−5.27208	−0.287188	−0.143594	−	0.989637i	$-0.545866\pi$
$337$	−5.27208	−0.287188	−0.143594	+	0.989637i	$0.545866\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.2426	0.771282
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.9706	−1.55522	−0.777611	−	0.628746i	$-0.783568\pi$
$347$	−28.9706	−1.55522	−0.777611	+	0.628746i	$0.783568\pi$
$348$	0	0
$349$	17.4853	0.935966	0.467983	−	0.883738i	$-0.344981\pi$
$349$	17.4853	0.935966	0.467983	+	0.883738i	$0.344981\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.4853	0.770974	0.385487	−	0.922713i	$-0.374034\pi$
$353$	14.4853	0.770974	0.385487	+	0.922713i	$0.374034\pi$
$354$	0	0
$355$	−13.4142	−0.711953
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.343146	0.0181105	0.00905527	−	0.999959i	$-0.497118\pi$
$359$	0.343146	0.0181105	0.00905527	+	0.999959i	$0.497118\pi$
$360$	0	0
$361$	33.4558	1.76083
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.24264	−0.222070
$366$	0	0
$367$	7.75736	0.404931	0.202465	−	0.979289i	$-0.435105\pi$
$367$	7.75736	0.404931	0.202465	+	0.979289i	$0.435105\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.2426	0.531771
$372$	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
$377$	−26.4853	−1.36406
$378$	0	0
$379$	−1.58579	−0.0814564	−0.0407282	−	0.999170i	$-0.512968\pi$
$379$	−1.58579	−0.0814564	−0.0407282	+	0.999170i	$0.512968\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.31371	−0.424811	−0.212405	−	0.977182i	$-0.568130\pi$
$383$	−8.31371	−0.424811	−0.212405	+	0.977182i	$0.568130\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−34.2426	−1.73617	−0.868085	−	0.496415i	$-0.834649\pi$
$389$	−34.2426	−1.73617	−0.868085	+	0.496415i	$0.834649\pi$
$390$	0	0
$391$	8.61522	0.435691
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.41421	0.222103
$396$	0	0
$397$	18.2426	0.915572	0.457786	−	0.889062i	$-0.348643\pi$
$397$	18.2426	0.915572	0.457786	+	0.889062i	$0.348643\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.4558	−1.17133	−0.585664	−	0.810554i	$-0.699166\pi$
$401$	−23.4558	−1.17133	−0.585664	+	0.810554i	$0.699166\pi$
$402$	0	0
$403$	−62.8701	−3.13178
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.8284	0.735018
$408$	0	0
$409$	−21.4853	−1.06238	−0.531189	−	0.847253i	$-0.678255\pi$
$409$	−21.4853	−1.06238	−0.531189	+	0.847253i	$0.678255\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	18.4853	0.909601
$414$	0	0
$415$	−11.8284	−0.580635
$416$	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
$421$	19.4853	0.949655	0.474827	−	0.880079i	$-0.342511\pi$
$421$	19.4853	0.949655	0.474827	+	0.880079i	$0.342511\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.24264	0.157291
$426$	0	0
$427$	−1.41421	−0.0684386
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.89949	0.187832	0.0939160	−	0.995580i	$-0.470061\pi$
$431$	3.89949	0.187832	0.0939160	+	0.995580i	$0.470061\pi$
$432$	0	0
$433$	2.24264	0.107774	0.0538872	−	0.998547i	$-0.482839\pi$
$433$	2.24264	0.107774	0.0538872	+	0.998547i	$0.482839\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−19.2426	−0.920500
$438$	0	0
$439$	25.5858	1.22114	0.610571	−	0.791961i	$-0.290940\pi$
$439$	25.5858	1.22114	0.610571	+	0.791961i	$0.290940\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	37.6274	1.78773	0.893866	−	0.448334i	$-0.147982\pi$
$443$	37.6274	1.78773	0.893866	+	0.448334i	$0.147982\pi$
$444$	0	0
$445$	−6.24264	−0.295930
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.2426	0.577766	0.288883	−	0.957364i	$-0.406716\pi$
$449$	12.2426	0.577766	0.288883	+	0.957364i	$0.406716\pi$
$450$	0	0
$451$	−3.17157	−0.149344
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8.82843	0.413883
$456$	0	0
$457$	36.9706	1.72941	0.864705	−	0.502280i	$-0.167505\pi$
$457$	36.9706	1.72941	0.864705	+	0.502280i	$0.167505\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.9706	−0.510950	−0.255475	−	0.966816i	$-0.582232\pi$
$461$	−10.9706	−0.510950	−0.255475	+	0.966816i	$0.582232\pi$
$462$	0	0
$463$	−21.5147	−0.999874	−0.499937	−	0.866062i	$-0.666643\pi$
$463$	−21.5147	−0.999874	−0.499937	+	0.866062i	$0.666643\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.857864	−0.0396972	−0.0198486	−	0.999803i	$-0.506318\pi$
$467$	−0.857864	−0.0396972	−0.0198486	+	0.999803i	$0.506318\pi$
$468$	0	0
$469$	16.4853	0.761220
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.48528	−0.114273
$474$	0	0
$475$	−7.24264	−0.332315
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.75736	0.354443	0.177221	−	0.984171i	$-0.443289\pi$
$479$	7.75736	0.354443	0.177221	+	0.984171i	$0.443289\pi$
$480$	0	0
$481$	−65.4558	−2.98453
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.485281	−0.0220355
$486$	0	0
$487$	25.0711	1.13608	0.568039	−	0.823001i	$-0.307702\pi$
$487$	25.0711	1.13608	0.568039	+	0.823001i	$0.307702\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	29.3137	1.32291	0.661455	−	0.749985i	$-0.269939\pi$
$491$	29.3137	1.32291	0.661455	+	0.749985i	$0.269939\pi$
$492$	0	0
$493$	13.7574	0.619600
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.9706	0.850946
$498$	0	0
$499$	3.72792	0.166885	0.0834424	−	0.996513i	$-0.473409\pi$
$499$	3.72792	0.166885	0.0834424	+	0.996513i	$0.473409\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	25.9706	1.15797	0.578985	−	0.815338i	$-0.303449\pi$
$503$	25.9706	1.15797	0.578985	+	0.815338i	$0.303449\pi$
$504$	0	0
$505$	12.4853	0.555588
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.9706	−1.63869	−0.819346	−	0.573300i	$-0.805663\pi$
$509$	−36.9706	−1.63869	−0.819346	+	0.573300i	$0.805663\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.1421	−0.623177
$516$	0	0
$517$	−16.4853	−0.725022
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−35.2132	−1.54272	−0.771359	−	0.636400i	$-0.780423\pi$
$521$	−35.2132	−1.54272	−0.771359	+	0.636400i	$0.780423\pi$
$522$	0	0
$523$	−28.2426	−1.23496	−0.617482	−	0.786585i	$-0.711847\pi$
$523$	−28.2426	−1.23496	−0.617482	+	0.786585i	$0.711847\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.6569	1.42255
$528$	0	0
$529$	−15.9411	−0.693092
$530$	0	0
$531$	0	0
$532$	0	0
$533$	14.0000	0.606407
$534$	0	0
$535$	8.14214	0.352015
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7.07107	−0.304572
$540$	0	0
$541$	4.48528	0.192837	0.0964187	−	0.995341i	$-0.469261\pi$
$541$	4.48528	0.192837	0.0964187	+	0.995341i	$0.469261\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.00000	0.128506
$546$	0	0
$547$	23.6985	1.01327	0.506637	−	0.862159i	$-0.330888\pi$
$547$	23.6985	1.01327	0.506637	+	0.862159i	$0.330888\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−30.7279	−1.30905
$552$	0	0
$553$	−6.24264	−0.265464
$554$	0	0
$555$	0	0
$556$	0	0
$557$	34.4853	1.46119	0.730594	−	0.682812i	$-0.239243\pi$
$557$	34.4853	1.46119	0.730594	+	0.682812i	$0.239243\pi$
$558$	0	0
$559$	10.9706	0.464005
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.1421	−0.848890	−0.424445	−	0.905454i	$-0.639531\pi$
$563$	−20.1421	−0.848890	−0.424445	+	0.905454i	$0.639531\pi$
$564$	0	0
$565$	16.4853	0.693541
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.2426	−0.513238	−0.256619	−	0.966513i	$-0.582609\pi$
$569$	−12.2426	−0.513238	−0.256619	+	0.966513i	$0.582609\pi$
$570$	0	0
$571$	28.4142	1.18910	0.594549	−	0.804059i	$-0.297331\pi$
$571$	28.4142	1.18910	0.594549	+	0.804059i	$0.297331\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.65685	0.110798
$576$	0	0
$577$	3.27208	0.136218	0.0681092	−	0.997678i	$-0.478303\pi$
$577$	3.27208	0.136218	0.0681092	+	0.997678i	$0.478303\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16.7279	0.693991
$582$	0	0
$583$	−10.2426	−0.424207
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.4853	−0.474048	−0.237024	−	0.971504i	$-0.576172\pi$
$587$	−11.4853	−0.474048	−0.237024	+	0.971504i	$0.576172\pi$
$588$	0	0
$589$	−72.9411	−3.00549
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−28.7574	−1.18092	−0.590462	−	0.807066i	$-0.701054\pi$
$593$	−28.7574	−1.18092	−0.590462	+	0.807066i	$0.701054\pi$
$594$	0	0
$595$	−4.58579	−0.187999
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.04163	−0.246854	−0.123427	−	0.992354i	$-0.539389\pi$
$599$	−6.04163	−0.246854	−0.123427	+	0.992354i	$0.539389\pi$
$600$	0	0
$601$	−5.97056	−0.243544	−0.121772	−	0.992558i	$-0.538858\pi$
$601$	−5.97056	−0.243544	−0.121772	+	0.992558i	$0.538858\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.00000	−0.365902
$606$	0	0
$607$	−29.3553	−1.19150	−0.595748	−	0.803171i	$-0.703144\pi$
$607$	−29.3553	−1.19150	−0.595748	+	0.803171i	$0.703144\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	72.7696	2.94394
$612$	0	0
$613$	20.9706	0.846993	0.423497	−	0.905898i	$-0.360803\pi$
$613$	20.9706	0.846993	0.423497	+	0.905898i	$0.360803\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.2132	0.894270	0.447135	−	0.894467i	$-0.352444\pi$
$617$	22.2132	0.894270	0.447135	+	0.894467i	$0.352444\pi$
$618$	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
$623$	8.82843	0.353703
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	34.0000	1.35567
$630$	0	0
$631$	34.7574	1.38367	0.691834	−	0.722056i	$-0.256803\pi$
$631$	34.7574	1.38367	0.691834	+	0.722056i	$0.256803\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.343146	0.0136173
$636$	0	0
$637$	31.2132	1.23671
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.9706	−0.512306	−0.256153	−	0.966636i	$-0.582455\pi$
$641$	−12.9706	−0.512306	−0.256153	+	0.966636i	$0.582455\pi$
$642$	0	0
$643$	−26.1005	−1.02930	−0.514652	−	0.857399i	$-0.672079\pi$
$643$	−26.1005	−1.02930	−0.514652	+	0.857399i	$0.672079\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.6569	0.812105	0.406052	−	0.913850i	$-0.366905\pi$
$647$	20.6569	0.812105	0.406052	+	0.913850i	$0.366905\pi$
$648$	0	0
$649$	−18.4853	−0.725611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−41.2426	−1.61395	−0.806975	−	0.590586i	$-0.798897\pi$
$653$	−41.2426	−1.61395	−0.806975	+	0.590586i	$0.798897\pi$
$654$	0	0
$655$	−3.17157	−0.123924
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.21320	−0.125169	−0.0625843	−	0.998040i	$-0.519934\pi$
$659$	−3.21320	−0.125169	−0.0625843	+	0.998040i	$0.519934\pi$
$660$	0	0
$661$	12.4853	0.485621	0.242811	−	0.970074i	$-0.421931\pi$
$661$	12.4853	0.485621	0.242811	+	0.970074i	$0.421931\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.2426	0.397193
$666$	0	0
$667$	11.2721	0.436457
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.41421	0.0545951
$672$	0	0
$673$	31.6985	1.22189	0.610943	−	0.791674i	$-0.290790\pi$
$673$	31.6985	1.22189	0.610943	+	0.791674i	$0.290790\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−46.4264	−1.78431	−0.892156	−	0.451728i	$-0.850808\pi$
$677$	−46.4264	−1.78431	−0.892156	+	0.451728i	$0.850808\pi$
$678$	0	0
$679$	0.686292	0.0263375
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.3431	0.587089	0.293545	−	0.955945i	$-0.405165\pi$
$683$	15.3431	0.587089	0.293545	+	0.955945i	$0.405165\pi$
$684$	0	0
$685$	−1.24264	−0.0474789
$686$	0	0
$687$	0	0
$688$	0	0
$689$	45.2132	1.72249
$690$	0	0
$691$	45.0416	1.71346	0.856732	−	0.515762i	$-0.172491\pi$
$691$	45.0416	1.71346	0.856732	+	0.515762i	$0.172491\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.65685	0.214577
$696$	0	0
$697$	−7.27208	−0.275450
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.4853	0.698179	0.349090	−	0.937089i	$-0.386491\pi$
$701$	18.4853	0.698179	0.349090	+	0.937089i	$0.386491\pi$
$702$	0	0
$703$	−75.9411	−2.86417
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.6569	−0.664054
$708$	0	0
$709$	−17.4558	−0.655568	−0.327784	−	0.944753i	$-0.606302\pi$
$709$	−17.4558	−0.655568	−0.327784	+	0.944753i	$0.606302\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	26.7574	1.00207
$714$	0	0
$715$	−8.82843	−0.330164
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−45.5980	−1.70052	−0.850259	−	0.526365i	$-0.823555\pi$
$719$	−45.5980	−1.70052	−0.850259	+	0.526365i	$0.823555\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.24264	0.157568
$726$	0	0
$727$	15.5147	0.575409	0.287705	−	0.957719i	$-0.407108\pi$
$727$	15.5147	0.575409	0.287705	+	0.957719i	$0.407108\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.69848	−0.210766
$732$	0	0
$733$	5.45584	0.201516	0.100758	−	0.994911i	$-0.467873\pi$
$733$	5.45584	0.201516	0.100758	+	0.994911i	$0.467873\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.4853	−0.607243
$738$	0	0
$739$	−38.3553	−1.41092	−0.705462	−	0.708748i	$-0.749260\pi$
$739$	−38.3553	−1.41092	−0.705462	+	0.708748i	$0.749260\pi$
$740$	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
$745$	−2.24264	−0.0821640
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−11.5147	−0.420739
$750$	0	0
$751$	22.7574	0.830428	0.415214	−	0.909724i	$-0.363707\pi$
$751$	22.7574	0.830428	0.415214	+	0.909724i	$0.363707\pi$
$752$	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
$761$	37.6985	1.36657	0.683285	−	0.730152i	$-0.260551\pi$
$761$	37.6985	1.36657	0.683285	+	0.730152i	$0.260551\pi$
$762$	0	0
$763$	−4.24264	−0.153594
$764$	0	0
$765$	0	0
$766$	0	0
$767$	81.5980	2.94633
$768$	0	0
$769$	33.4264	1.20539	0.602694	−	0.797973i	$-0.294094\pi$
$769$	33.4264	1.20539	0.602694	+	0.797973i	$0.294094\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18.2132	−0.655083	−0.327542	−	0.944837i	$-0.606220\pi$
$773$	−18.2132	−0.655083	−0.327542	+	0.944837i	$0.606220\pi$
$774$	0	0
$775$	10.0711	0.361763
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.2426	0.581953
$780$	0	0
$781$	−18.9706	−0.678820
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.2426	−0.579725
$786$	0	0
$787$	−18.7279	−0.667578	−0.333789	−	0.942648i	$-0.608327\pi$
$787$	−18.7279	−0.667578	−0.333789	+	0.942648i	$0.608327\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−23.3137	−0.828940
$792$	0	0
$793$	−6.24264	−0.221683
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−16.7574	−0.593576	−0.296788	−	0.954943i	$-0.595915\pi$
$797$	−16.7574	−0.593576	−0.296788	+	0.954943i	$0.595915\pi$
$798$	0	0
$799$	−37.7990	−1.33723
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	−3.75736	−0.132430
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.75736	0.0617855	0.0308927	−	0.999523i	$-0.490165\pi$
$809$	1.75736	0.0617855	0.0308927	+	0.999523i	$0.490165\pi$
$810$	0	0
$811$	52.6274	1.84800	0.923999	−	0.382394i	$-0.124900\pi$
$811$	52.6274	1.84800	0.923999	+	0.382394i	$0.124900\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−15.1716	−0.531437
$816$	0	0
$817$	12.7279	0.445294
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.97056	0.243274	0.121637	−	0.992575i	$-0.461186\pi$
$821$	6.97056	0.243274	0.121637	+	0.992575i	$0.461186\pi$
$822$	0	0
$823$	23.6985	0.826077	0.413039	−	0.910714i	$-0.364467\pi$
$823$	23.6985	0.826077	0.413039	+	0.910714i	$0.364467\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−40.1127	−1.39486	−0.697428	−	0.716655i	$-0.745672\pi$
$827$	−40.1127	−1.39486	−0.697428	+	0.716655i	$0.745672\pi$
$828$	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
$833$	−16.2132	−0.561754
$834$	0	0
$835$	−11.8284	−0.409340
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38.1421	1.31681	0.658406	−	0.752663i	$-0.271231\pi$
$839$	38.1421	1.31681	0.658406	+	0.752663i	$0.271231\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25.9706	0.893415
$846$	0	0
$847$	12.7279	0.437337
$848$	0	0
$849$	0	0
$850$	0	0
$851$	27.8579	0.954955
$852$	0	0
$853$	31.5147	1.07904	0.539522	−	0.841972i	$-0.318605\pi$
$853$	31.5147	1.07904	0.539522	+	0.841972i	$0.318605\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.21320	−0.212239	−0.106119	−	0.994353i	$-0.533843\pi$
$857$	−6.21320	−0.212239	−0.106119	+	0.994353i	$0.533843\pi$
$858$	0	0
$859$	−8.69848	−0.296788	−0.148394	−	0.988928i	$-0.547410\pi$
$859$	−8.69848	−0.296788	−0.148394	+	0.988928i	$0.547410\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.45584	−0.151679	−0.0758393	−	0.997120i	$-0.524164\pi$
$863$	−4.45584	−0.151679	−0.0758393	+	0.997120i	$0.524164\pi$
$864$	0	0
$865$	−20.2132	−0.687270
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.24264	0.211767
$870$	0	0
$871$	72.7696	2.46570
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.41421	−0.0478091
$876$	0	0
$877$	−12.2426	−0.413405	−0.206702	−	0.978404i	$-0.566273\pi$
$877$	−12.2426	−0.413405	−0.206702	+	0.978404i	$0.566273\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.7868	−0.498180	−0.249090	−	0.968480i	$-0.580131\pi$
$881$	−14.7868	−0.498180	−0.249090	+	0.968480i	$0.580131\pi$
$882$	0	0
$883$	2.87006	0.0965851	0.0482926	−	0.998833i	$-0.484622\pi$
$883$	2.87006	0.0965851	0.0482926	+	0.998833i	$0.484622\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.8579	0.633185	0.316593	−	0.948562i	$-0.397461\pi$
$887$	18.8579	0.633185	0.316593	+	0.948562i	$0.397461\pi$
$888$	0	0
$889$	−0.485281	−0.0162758
$890$	0	0
$891$	0	0
$892$	0	0
$893$	84.4264	2.82522
$894$	0	0
$895$	−14.8284	−0.495660
$896$	0	0
$897$	0	0
$898$	0	0
$899$	42.7279	1.42506
$900$	0	0
$901$	−23.4853	−0.782408
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.9706	−0.597362
$906$	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
$911$	−10.9289	−0.362092	−0.181046	−	0.983475i	$-0.557948\pi$
$911$	−10.9289	−0.362092	−0.181046	+	0.983475i	$0.557948\pi$
$912$	0	0
$913$	−16.7279	−0.553613
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.48528	0.148117
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	83.7401	2.75634
$924$	0	0
$925$	10.4853	0.344754
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−41.7574	−1.37001	−0.685007	−	0.728536i	$-0.740201\pi$
$929$	−41.7574	−1.37001	−0.685007	+	0.728536i	$0.740201\pi$
$930$	0	0
$931$	36.2132	1.18684
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.58579	0.149971
$936$	0	0
$937$	−29.9411	−0.978134	−0.489067	−	0.872246i	$-0.662663\pi$
$937$	−29.9411	−0.978134	−0.489067	+	0.872246i	$0.662663\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.02944	−0.163955	−0.0819775	−	0.996634i	$-0.526124\pi$
$941$	−5.02944	−0.163955	−0.0819775	+	0.996634i	$0.526124\pi$
$942$	0	0
$943$	−5.95837	−0.194031
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.4853	0.763169	0.381585	−	0.924334i	$-0.375379\pi$
$947$	23.4853	0.763169	0.381585	+	0.924334i	$0.375379\pi$
$948$	0	0
$949$	26.4853	0.859749
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−13.0294	−0.422065	−0.211032	−	0.977479i	$-0.567683\pi$
$953$	−13.0294	−0.422065	−0.211032	+	0.977479i	$0.567683\pi$
$954$	0	0
$955$	2.48528	0.0804218
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.75736	0.0567481
$960$	0	0
$961$	70.4264	2.27182
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.75736	−0.0565714
$966$	0	0
$967$	14.4853	0.465815	0.232908	−	0.972499i	$-0.425176\pi$
$967$	14.4853	0.465815	0.232908	+	0.972499i	$0.425176\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44.1421	1.41659	0.708294	−	0.705917i	$-0.249465\pi$
$971$	44.1421	1.41659	0.708294	+	0.705917i	$0.249465\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.5147	1.07223	0.536115	−	0.844145i	$-0.319891\pi$
$977$	33.5147	1.07223	0.536115	+	0.844145i	$0.319891\pi$
$978$	0	0
$979$	−8.82843	−0.282158
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.20101	0.0383063	0.0191531	−	0.999817i	$-0.493903\pi$
$983$	1.20101	0.0383063	0.0191531	+	0.999817i	$0.493903\pi$
$984$	0	0
$985$	−1.24264	−0.0395938
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.66905	−0.148467
$990$	0	0
$991$	27.7279	0.880806	0.440403	−	0.897800i	$-0.354835\pi$
$991$	27.7279	0.880806	0.440403	+	0.897800i	$0.354835\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.17157	0.290758
$996$	0	0
$997$	0.242641	0.00768451	0.00384225	−	0.999993i	$-0.498777\pi$
$997$	0.242641	0.00768451	0.00384225	+	0.999993i	$0.498777\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.41421 q^{7} +1.41421 q^{11} -6.24264 q^{13} +3.24264 q^{17} -7.24264 q^{19} +2.65685 q^{23} +1.00000 q^{25} +4.24264 q^{29} +10.0711 q^{31} -1.41421 q^{35} +10.4853 q^{37} -2.24264 q^{41} -1.75736 q^{43} -11.6569 q^{47} -5.00000 q^{49} -7.24264 q^{53} +1.41421 q^{55} -13.0711 q^{59} +1.00000 q^{61} -6.24264 q^{65} -11.6569 q^{67} -13.4142 q^{71} -4.24264 q^{73} -2.00000 q^{77} +4.41421 q^{79} -11.8284 q^{83} +3.24264 q^{85} -6.24264 q^{89} +8.82843 q^{91} -7.24264 q^{95} -0.485281 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{13} - 2 q^{17} - 6 q^{19} - 6 q^{23} + 2 q^{25} + 6 q^{31} + 4 q^{37} + 4 q^{41} - 12 q^{43} - 12 q^{47} - 10 q^{49} - 6 q^{53} - 12 q^{59} + 2 q^{61} - 4 q^{65} - 12 q^{67} - 24 q^{71}+ \cdots + 16 q^{97}+O(q^{100}) \) 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

Introduction

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