LMFDB - Embedded newform 4032.2.c.p.2017.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(2017,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("4032.2017");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.c (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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1344)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2017.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	4032.2017
Dual form			4032.2.c.p.2017.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051i q^{5} +1.00000 q^{7} +O(q^{10})$	$q-0.732051i q^{5} +1.00000 q^{7} -4.19615i q^{11} +1.26795 q^{17} +7.46410i q^{19} +8.73205 q^{23} +4.46410 q^{25} +3.46410i q^{29} -4.92820 q^{31} -0.732051i q^{35} +10.0000i q^{37} -1.26795 q^{41} +0.928203i q^{43} -6.92820 q^{47} +1.00000 q^{49} -3.46410i q^{53} -3.07180 q^{55} -10.9282i q^{59} -1.46410i q^{61} -7.46410i q^{67} +14.1962 q^{71} +10.3923 q^{73} -4.19615i q^{77} +9.46410 q^{79} -8.39230i q^{83} -0.928203i q^{85} +1.26795 q^{89} +5.46410 q^{95} +6.39230 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} + 12 q^{17} + 28 q^{23} + 4 q^{25} + 8 q^{31} - 12 q^{41} + 4 q^{49} - 40 q^{55} + 36 q^{71} + 24 q^{79} + 12 q^{89} + 8 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 12 * q^17 + 28 * q^23 + 4 * q^25 + 8 * q^31 - 12 * q^41 + 4 * q^49 - 40 * q^55 + 36 * q^71 + 24 * q^79 + 12 * q^89 + 8 * q^95 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	− 0.732051i	− 0.327383i	−0.986512	−	0.163692i	$-0.947660\pi$
$5$	− 0.732051i	− 0.327383i	0.986512	−	0.163692i	$-0.0523402\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.19615i	− 1.26519i	−0.774484	−	0.632594i	$-0.781990\pi$
$11$	− 4.19615i	− 1.26519i	0.774484	−	0.632594i	$-0.218010\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.26795	0.307523	0.153761	−	0.988108i	$-0.450861\pi$
$17$	1.26795	0.307523	0.153761	+	0.988108i	$0.450861\pi$
$18$	0	0
$19$	7.46410i	1.71238i	0.516659	+	0.856191i	$0.327175\pi$
$19$	7.46410i	1.71238i	−0.516659	+	0.856191i	$0.672825\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.73205	1.82076	0.910379	−	0.413775i	$-0.135790\pi$
$23$	8.73205	1.82076	0.910379	+	0.413775i	$0.135790\pi$
$24$	0	0
$25$	4.46410	0.892820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.46410i	0.643268i	0.946864	+	0.321634i	$0.104232\pi$
$29$	3.46410i	0.643268i	−0.946864	+	0.321634i	$0.895768\pi$
$30$	0	0
$31$	−4.92820	−0.885131	−0.442566	−	0.896736i	$-0.645932\pi$
$31$	−4.92820	−0.885131	−0.442566	+	0.896736i	$0.645932\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.732051i	− 0.123739i
$36$	0	0
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.26795	−0.198020	−0.0990102	−	0.995086i	$-0.531568\pi$
$41$	−1.26795	−0.198020	−0.0990102	+	0.995086i	$0.531568\pi$
$42$	0	0
$43$	0.928203i	0.141550i	0.997492	+	0.0707748i	$0.0225472\pi$
$43$	0.928203i	0.141550i	−0.997492	+	0.0707748i	$0.977453\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 3.46410i	− 0.475831i	−0.971286	−	0.237915i	$-0.923536\pi$
$53$	− 3.46410i	− 0.475831i	0.971286	−	0.237915i	$-0.0764641\pi$
$54$	0	0
$55$	−3.07180	−0.414201
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 10.9282i	− 1.42273i	−0.702822	−	0.711365i	$-0.748077\pi$
$59$	− 10.9282i	− 1.42273i	0.702822	−	0.711365i	$-0.251923\pi$
$60$	0	0
$61$	− 1.46410i	− 0.187459i	−0.995598	−	0.0937295i	$-0.970121\pi$
$61$	− 1.46410i	− 0.187459i	0.995598	−	0.0937295i	$-0.0298789\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.46410i	− 0.911885i	−0.890009	−	0.455943i	$-0.849302\pi$
$67$	− 7.46410i	− 0.911885i	0.890009	−	0.455943i	$-0.150698\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.1962	1.68477	0.842387	−	0.538874i	$-0.181150\pi$
$71$	14.1962	1.68477	0.842387	+	0.538874i	$0.181150\pi$
$72$	0	0
$73$	10.3923	1.21633	0.608164	−	0.793812i	$-0.291906\pi$
$73$	10.3923	1.21633	0.608164	+	0.793812i	$0.291906\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.19615i	− 0.478196i
$78$	0	0
$79$	9.46410	1.06479	0.532397	−	0.846495i	$-0.321291\pi$
$79$	9.46410	1.06479	0.532397	+	0.846495i	$0.321291\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.39230i	− 0.921175i	−0.887614	−	0.460588i	$-0.847639\pi$
$83$	− 8.39230i	− 0.921175i	0.887614	−	0.460588i	$-0.152361\pi$
$84$	0	0
$85$	− 0.928203i	− 0.100678i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.26795	0.134402	0.0672012	−	0.997739i	$-0.478593\pi$
$89$	1.26795	0.134402	0.0672012	+	0.997739i	$0.478593\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.46410	0.560605
$96$	0	0
$97$	6.39230	0.649040	0.324520	−	0.945879i	$-0.394797\pi$
$97$	6.39230	0.649040	0.324520	+	0.945879i	$0.394797\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.12436i	0.907907i	0.891025	+	0.453954i	$0.149987\pi$
$101$	9.12436i	0.907907i	−0.891025	+	0.453954i	$0.850013\pi$
$102$	0	0
$103$	8.92820	0.879722	0.439861	−	0.898066i	$-0.355028\pi$
$103$	8.92820	0.879722	0.439861	+	0.898066i	$0.355028\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6.73205i	− 0.650812i	−0.945574	−	0.325406i	$-0.894499\pi$
$107$	− 6.73205i	− 0.650812i	0.945574	−	0.325406i	$-0.105501\pi$
$108$	0	0
$109$	2.92820i	0.280471i	0.990118	+	0.140236i	$0.0447860\pi$
$109$	2.92820i	0.280471i	−0.990118	+	0.140236i	$0.955214\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.53590	−0.802990	−0.401495	−	0.915861i	$-0.631509\pi$
$113$	−8.53590	−0.802990	−0.401495	+	0.915861i	$0.631509\pi$
$114$	0	0
$115$	− 6.39230i	− 0.596086i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.26795	0.116233
$120$	0	0
$121$	−6.60770	−0.600700
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 6.92820i	− 0.619677i
$126$	0	0
$127$	−0.392305	−0.0348114	−0.0174057	−	0.999849i	$-0.505541\pi$
$127$	−0.392305	−0.0348114	−0.0174057	+	0.999849i	$0.505541\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 20.3923i	− 1.78168i	−0.454313	−	0.890842i	$-0.650115\pi$
$131$	− 20.3923i	− 1.78168i	0.454313	−	0.890842i	$-0.349885\pi$
$132$	0	0
$133$	7.46410i	0.647220i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.3923	1.22962	0.614809	−	0.788676i	$-0.289233\pi$
$137$	14.3923	1.22962	0.614809	+	0.788676i	$0.289233\pi$
$138$	0	0
$139$	− 2.92820i	− 0.248367i	−0.992259	−	0.124183i	$-0.960369\pi$
$139$	− 2.92820i	− 0.248367i	0.992259	−	0.124183i	$-0.0396311\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.53590	0.210595
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.5359i	1.02698i	0.858095	+	0.513490i	$0.171648\pi$
$149$	12.5359i	1.02698i	−0.858095	+	0.513490i	$0.828352\pi$
$150$	0	0
$151$	−14.9282	−1.21484	−0.607420	−	0.794381i	$-0.707795\pi$
$151$	−14.9282	−1.21484	−0.607420	+	0.794381i	$0.707795\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.60770i	0.289777i
$156$	0	0
$157$	1.46410i	0.116848i	0.998292	+	0.0584240i	$0.0186075\pi$
$157$	1.46410i	0.116848i	−0.998292	+	0.0584240i	$0.981392\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.73205	0.688182
$162$	0	0
$163$	2.39230i	0.187380i	0.995601	+	0.0936899i	$0.0298662\pi$
$163$	2.39230i	0.187380i	−0.995601	+	0.0936899i	$0.970134\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.46410	−0.422825	−0.211412	−	0.977397i	$-0.567806\pi$
$167$	−5.46410	−0.422825	−0.211412	+	0.977397i	$0.567806\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.66025i	0.278284i	0.990272	+	0.139142i	$0.0444344\pi$
$173$	3.66025i	0.278284i	−0.990272	+	0.139142i	$0.955566\pi$
$174$	0	0
$175$	4.46410	0.337454
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.80385i	0.583287i	0.956527	+	0.291643i	$0.0942021\pi$
$179$	7.80385i	0.583287i	−0.956527	+	0.291643i	$0.905798\pi$
$180$	0	0
$181$	− 9.07180i	− 0.674301i	−0.941451	−	0.337151i	$-0.890537\pi$
$181$	− 9.07180i	− 0.674301i	0.941451	−	0.337151i	$-0.109463\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.32051	0.538214
$186$	0	0
$187$	− 5.32051i	− 0.389074i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.7321	−0.921259	−0.460629	−	0.887593i	$-0.652376\pi$
$191$	−12.7321	−0.921259	−0.460629	+	0.887593i	$0.652376\pi$
$192$	0	0
$193$	13.4641	0.969167	0.484584	−	0.874745i	$-0.338971\pi$
$193$	13.4641	0.969167	0.484584	+	0.874745i	$0.338971\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.9282i	1.77606i	0.459785	+	0.888030i	$0.347927\pi$
$197$	24.9282i	1.77606i	−0.459785	+	0.888030i	$0.652073\pi$
$198$	0	0
$199$	−21.8564	−1.54936	−0.774680	−	0.632354i	$-0.782089\pi$
$199$	−21.8564	−1.54936	−0.774680	+	0.632354i	$0.782089\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.46410i	0.243132i
$204$	0	0
$205$	0.928203i	0.0648285i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	31.3205	2.16648
$210$	0	0
$211$	− 15.8564i	− 1.09160i	−0.837915	−	0.545800i	$-0.816226\pi$
$211$	− 15.8564i	− 1.09160i	0.837915	−	0.545800i	$-0.183774\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.679492	0.0463410
$216$	0	0
$217$	−4.92820	−0.334548
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	24.7846	1.65970	0.829850	−	0.557986i	$-0.188426\pi$
$223$	24.7846	1.65970	0.829850	+	0.557986i	$0.188426\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.4641i	0.893644i	0.894623	+	0.446822i	$0.147444\pi$
$227$	13.4641i	0.893644i	−0.894623	+	0.446822i	$0.852556\pi$
$228$	0	0
$229$	28.7846i	1.90214i	0.308975	+	0.951070i	$0.400014\pi$
$229$	28.7846i	1.90214i	−0.308975	+	0.951070i	$0.599986\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.8564	0.776739	0.388370	−	0.921504i	$-0.373038\pi$
$233$	11.8564	0.776739	0.388370	+	0.921504i	$0.373038\pi$
$234$	0	0
$235$	5.07180i	0.330848i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.0526	0.779615	0.389808	−	0.920896i	$-0.372542\pi$
$239$	12.0526	0.779615	0.389808	+	0.920896i	$0.372542\pi$
$240$	0	0
$241$	−24.2487	−1.56200	−0.780998	−	0.624533i	$-0.785289\pi$
$241$	−24.2487	−1.56200	−0.780998	+	0.624533i	$0.785289\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 0.732051i	− 0.0467690i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 14.5359i	− 0.917498i	−0.888566	−	0.458749i	$-0.848298\pi$
$251$	− 14.5359i	− 0.917498i	0.888566	−	0.458749i	$-0.151702\pi$
$252$	0	0
$253$	− 36.6410i	− 2.30360i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.5885	1.28427	0.642136	−	0.766590i	$-0.278048\pi$
$257$	20.5885	1.28427	0.642136	+	0.766590i	$0.278048\pi$
$258$	0	0
$259$	10.0000i	0.621370i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.0526	1.48314	0.741572	−	0.670873i	$-0.234080\pi$
$263$	24.0526	1.48314	0.741572	+	0.670873i	$0.234080\pi$
$264$	0	0
$265$	−2.53590	−0.155779
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.5885i	1.13336i	0.823939	+	0.566679i	$0.191772\pi$
$269$	18.5885i	1.13336i	−0.823939	+	0.566679i	$0.808228\pi$
$270$	0	0
$271$	27.8564	1.69216	0.846078	−	0.533059i	$-0.178958\pi$
$271$	27.8564	1.69216	0.846078	+	0.533059i	$0.178958\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 18.7321i	− 1.12959i
$276$	0	0
$277$	− 22.7846i	− 1.36899i	−0.729015	−	0.684497i	$-0.760022\pi$
$277$	− 22.7846i	− 1.36899i	0.729015	−	0.684497i	$-0.239978\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.8564	0.945914	0.472957	−	0.881086i	$-0.343187\pi$
$281$	15.8564	0.945914	0.472957	+	0.881086i	$0.343187\pi$
$282$	0	0
$283$	− 15.4641i	− 0.919245i	−0.888114	−	0.459623i	$-0.847985\pi$
$283$	− 15.4641i	− 0.919245i	0.888114	−	0.459623i	$-0.152015\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.26795	−0.0748447
$288$	0	0
$289$	−15.3923	−0.905430
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 16.0526i	− 0.937801i	−0.883251	−	0.468900i	$-0.844650\pi$
$293$	− 16.0526i	− 0.937801i	0.883251	−	0.468900i	$-0.155350\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0.928203i	0.0535007i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.07180	−0.0613709
$306$	0	0
$307$	− 7.46410i	− 0.425999i	−0.977052	−	0.212999i	$-0.931677\pi$
$307$	− 7.46410i	− 0.425999i	0.977052	−	0.212999i	$-0.0683232\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.07180	0.514414	0.257207	−	0.966356i	$-0.417198\pi$
$311$	9.07180	0.514414	0.257207	+	0.966356i	$0.417198\pi$
$312$	0	0
$313$	−3.07180	−0.173628	−0.0868141	−	0.996225i	$-0.527669\pi$
$313$	−3.07180	−0.173628	−0.0868141	+	0.996225i	$0.527669\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 25.3205i	− 1.42214i	−0.703121	−	0.711071i	$-0.748211\pi$
$317$	− 25.3205i	− 1.42214i	0.703121	−	0.711071i	$-0.251789\pi$
$318$	0	0
$319$	14.5359	0.813854
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.46410i	0.526597i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.92820	−0.381964
$330$	0	0
$331$	− 28.9282i	− 1.59004i	−0.606585	−	0.795019i	$-0.707461\pi$
$331$	− 28.9282i	− 1.59004i	0.606585	−	0.795019i	$-0.292539\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.46410	−0.298536
$336$	0	0
$337$	15.8564	0.863753	0.431877	−	0.901933i	$-0.357852\pi$
$337$	15.8564	0.863753	0.431877	+	0.901933i	$0.357852\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	20.6795i	1.11986i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.6603i	0.948052i	0.880511	+	0.474026i	$0.157200\pi$
$347$	17.6603i	0.948052i	−0.880511	+	0.474026i	$0.842800\pi$
$348$	0	0
$349$	21.4641i	1.14895i	0.818523	+	0.574474i	$0.194793\pi$
$349$	21.4641i	1.14895i	−0.818523	+	0.574474i	$0.805207\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−32.5885	−1.73451	−0.867254	−	0.497865i	$-0.834117\pi$
$353$	−32.5885	−1.73451	−0.867254	+	0.497865i	$0.834117\pi$
$354$	0	0
$355$	− 10.3923i	− 0.551566i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.9090	−1.78965	−0.894823	−	0.446420i	$-0.852699\pi$
$359$	−33.9090	−1.78965	−0.894823	+	0.446420i	$0.852699\pi$
$360$	0	0
$361$	−36.7128	−1.93225
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 7.60770i	− 0.398205i
$366$	0	0
$367$	10.1436	0.529491	0.264746	−	0.964318i	$-0.414712\pi$
$367$	10.1436	0.529491	0.264746	+	0.964318i	$0.414712\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 3.46410i	− 0.179847i
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 16.9282i	− 0.869543i	−0.900541	−	0.434772i	$-0.856829\pi$
$379$	− 16.9282i	− 0.869543i	0.900541	−	0.434772i	$-0.143171\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−28.3923	−1.45078	−0.725390	−	0.688339i	$-0.758340\pi$
$383$	−28.3923	−1.45078	−0.725390	+	0.688339i	$0.758340\pi$
$384$	0	0
$385$	−3.07180	−0.156553
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.4641i	0.784061i	0.919952	+	0.392031i	$0.128227\pi$
$389$	15.4641i	0.784061i	−0.919952	+	0.392031i	$0.871773\pi$
$390$	0	0
$391$	11.0718	0.559925
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 6.92820i	− 0.348596i
$396$	0	0
$397$	3.32051i	0.166652i	0.996522	+	0.0833258i	$0.0265542\pi$
$397$	3.32051i	0.166652i	−0.996522	+	0.0833258i	$0.973446\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.39230	0.119466	0.0597330	−	0.998214i	$-0.480975\pi$
$401$	2.39230	0.119466	0.0597330	+	0.998214i	$0.480975\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	41.9615	2.07996
$408$	0	0
$409$	38.3923	1.89838	0.949189	−	0.314708i	$-0.101906\pi$
$409$	38.3923	1.89838	0.949189	+	0.314708i	$0.101906\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 10.9282i	− 0.537742i
$414$	0	0
$415$	−6.14359	−0.301577
$416$	0	0
$417$	0	0
$418$	0	0
$419$	17.8564i	0.872343i	0.899864	+	0.436171i	$0.143666\pi$
$419$	17.8564i	0.872343i	−0.899864	+	0.436171i	$0.856334\pi$
$420$	0	0
$421$	− 10.7846i	− 0.525610i	−0.964849	−	0.262805i	$-0.915352\pi$
$421$	− 10.7846i	− 0.525610i	0.964849	−	0.262805i	$-0.0846475\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.66025	0.274563
$426$	0	0
$427$	− 1.46410i	− 0.0708528i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.05256	0.387878	0.193939	−	0.981014i	$-0.437874\pi$
$431$	8.05256	0.387878	0.193939	+	0.981014i	$0.437874\pi$
$432$	0	0
$433$	−24.9282	−1.19797	−0.598986	−	0.800759i	$-0.704430\pi$
$433$	−24.9282	−1.19797	−0.598986	+	0.800759i	$0.704430\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	65.1769i	3.11783i
$438$	0	0
$439$	−13.8564	−0.661330	−0.330665	−	0.943748i	$-0.607273\pi$
$439$	−13.8564	−0.661330	−0.330665	+	0.943748i	$0.607273\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 5.26795i	− 0.250288i	−0.992139	−	0.125144i	$-0.960061\pi$
$443$	− 5.26795i	− 0.250288i	0.992139	−	0.125144i	$-0.0399393\pi$
$444$	0	0
$445$	− 0.928203i	− 0.0440011i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.53590	0.402834	0.201417	−	0.979506i	$-0.435445\pi$
$449$	8.53590	0.402834	0.201417	+	0.979506i	$0.435445\pi$
$450$	0	0
$451$	5.32051i	0.250533i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.928203	0.0434195	0.0217098	−	0.999764i	$-0.493089\pi$
$457$	0.928203	0.0434195	0.0217098	+	0.999764i	$0.493089\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 18.5885i	− 0.865751i	−0.901454	−	0.432875i	$-0.857499\pi$
$461$	− 18.5885i	− 0.865751i	0.901454	−	0.432875i	$-0.142501\pi$
$462$	0	0
$463$	−26.2487	−1.21988	−0.609941	−	0.792447i	$-0.708807\pi$
$463$	−26.2487	−1.21988	−0.609941	+	0.792447i	$0.708807\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.32051i	0.338753i	0.985551	+	0.169376i	$0.0541753\pi$
$467$	7.32051i	0.338753i	−0.985551	+	0.169376i	$0.945825\pi$
$468$	0	0
$469$	− 7.46410i	− 0.344660i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.89488	0.179087
$474$	0	0
$475$	33.3205i	1.52885i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−35.7128	−1.63176	−0.815880	−	0.578221i	$-0.803747\pi$
$479$	−35.7128	−1.63176	−0.815880	+	0.578221i	$0.803747\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 4.67949i	− 0.212485i
$486$	0	0
$487$	21.8564	0.990408	0.495204	−	0.868777i	$-0.335093\pi$
$487$	21.8564	0.990408	0.495204	+	0.868777i	$0.335093\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	34.0526i	1.53677i	0.639988	+	0.768385i	$0.278939\pi$
$491$	34.0526i	1.53677i	−0.639988	+	0.768385i	$0.721061\pi$
$492$	0	0
$493$	4.39230i	0.197819i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.1962	0.636784
$498$	0	0
$499$	8.14359i	0.364557i	0.983247	+	0.182279i	$0.0583473\pi$
$499$	8.14359i	0.364557i	−0.983247	+	0.182279i	$0.941653\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.6077	0.695913	0.347956	−	0.937511i	$-0.386876\pi$
$503$	15.6077	0.695913	0.347956	+	0.937511i	$0.386876\pi$
$504$	0	0
$505$	6.67949	0.297233
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 40.7321i	− 1.80542i	−0.430253	−	0.902708i	$-0.641576\pi$
$509$	− 40.7321i	− 1.80542i	0.430253	−	0.902708i	$-0.358424\pi$
$510$	0	0
$511$	10.3923	0.459728
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 6.53590i	− 0.288006i
$516$	0	0
$517$	29.0718i	1.27858i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−25.2679	−1.10701	−0.553504	−	0.832846i	$-0.686710\pi$
$521$	−25.2679	−1.10701	−0.553504	+	0.832846i	$0.686710\pi$
$522$	0	0
$523$	42.6410i	1.86456i	0.361736	+	0.932281i	$0.382184\pi$
$523$	42.6410i	1.86456i	−0.361736	+	0.932281i	$0.617816\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.24871	−0.272198
$528$	0	0
$529$	53.2487	2.31516
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−4.92820	−0.213065
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.19615i	− 0.180741i
$540$	0	0
$541$	− 0.143594i	− 0.00617357i	−0.999995	−	0.00308678i	$-0.999017\pi$
$541$	− 0.143594i	− 0.00617357i	0.999995	−	0.00308678i	$-0.000982555\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.14359	0.0918215
$546$	0	0
$547$	6.39230i	0.273315i	0.990618	+	0.136658i	$0.0436360\pi$
$547$	6.39230i	0.273315i	−0.990618	+	0.136658i	$0.956364\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−25.8564	−1.10152
$552$	0	0
$553$	9.46410	0.402455
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 29.3205i	− 1.24235i	−0.783672	−	0.621175i	$-0.786656\pi$
$557$	− 29.3205i	− 1.24235i	0.783672	−	0.621175i	$-0.213344\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 21.8564i	− 0.921138i	−0.887624	−	0.460569i	$-0.847645\pi$
$563$	− 21.8564i	− 0.921138i	0.887624	−	0.460569i	$-0.152355\pi$
$564$	0	0
$565$	6.24871i	0.262885i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.9282	−0.709667	−0.354834	−	0.934929i	$-0.615462\pi$
$569$	−16.9282	−0.709667	−0.354834	+	0.934929i	$0.615462\pi$
$570$	0	0
$571$	7.46410i	0.312363i	0.987728	+	0.156181i	$0.0499185\pi$
$571$	7.46410i	0.312363i	−0.987728	+	0.156181i	$0.950082\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	38.9808	1.62561
$576$	0	0
$577$	−18.7846	−0.782014	−0.391007	−	0.920388i	$-0.627873\pi$
$577$	−18.7846	−0.782014	−0.391007	+	0.920388i	$0.627873\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 8.39230i	− 0.348171i
$582$	0	0
$583$	−14.5359	−0.602015
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.53590i	0.104668i	0.998630	+	0.0523339i	$0.0166660\pi$
$587$	2.53590i	0.104668i	−0.998630	+	0.0523339i	$0.983334\pi$
$588$	0	0
$589$	− 36.7846i	− 1.51568i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−34.8372	−1.43059	−0.715296	−	0.698822i	$-0.753708\pi$
$593$	−34.8372	−1.43059	−0.715296	+	0.698822i	$0.753708\pi$
$594$	0	0
$595$	− 0.928203i	− 0.0380526i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.33975	0.340753	0.170376	−	0.985379i	$-0.445502\pi$
$599$	8.33975	0.340753	0.170376	+	0.985379i	$0.445502\pi$
$600$	0	0
$601$	−13.7128	−0.559357	−0.279679	−	0.960094i	$-0.590228\pi$
$601$	−13.7128	−0.559357	−0.279679	+	0.960094i	$0.590228\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.83717i	0.196659i
$606$	0	0
$607$	−21.0718	−0.855278	−0.427639	−	0.903950i	$-0.640655\pi$
$607$	−21.0718	−0.855278	−0.427639	+	0.903950i	$0.640655\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	28.0000i	1.13091i	0.824779	+	0.565455i	$0.191299\pi$
$613$	28.0000i	1.13091i	−0.824779	+	0.565455i	$0.808701\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.2487	−1.45932	−0.729659	−	0.683811i	$-0.760321\pi$
$617$	−36.2487	−1.45932	−0.729659	+	0.683811i	$0.760321\pi$
$618$	0	0
$619$	− 4.00000i	− 0.160774i	−0.996764	−	0.0803868i	$-0.974384\pi$
$619$	− 4.00000i	− 0.160774i	0.996764	−	0.0803868i	$-0.0256155\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.26795	0.0507993
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.6795i	0.505564i
$630$	0	0
$631$	−26.5359	−1.05638	−0.528189	−	0.849127i	$-0.677129\pi$
$631$	−26.5359	−1.05638	−0.528189	+	0.849127i	$0.677129\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.287187i	0.0113967i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−25.6077	−1.01144	−0.505722	−	0.862697i	$-0.668774\pi$
$641$	−25.6077	−1.01144	−0.505722	+	0.862697i	$0.668774\pi$
$642$	0	0
$643$	6.39230i	0.252088i	0.992025	+	0.126044i	$0.0402280\pi$
$643$	6.39230i	0.252088i	−0.992025	+	0.126044i	$0.959772\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.3923	−1.27347	−0.636736	−	0.771082i	$-0.719716\pi$
$647$	−32.3923	−1.27347	−0.636736	+	0.771082i	$0.719716\pi$
$648$	0	0
$649$	−45.8564	−1.80002
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.7846i	0.735099i	0.930004	+	0.367549i	$0.119803\pi$
$653$	18.7846i	0.735099i	−0.930004	+	0.367549i	$0.880197\pi$
$654$	0	0
$655$	−14.9282	−0.583293
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 10.0526i	− 0.391592i	−0.980645	−	0.195796i	$-0.937271\pi$
$659$	− 10.0526i	− 0.391592i	0.980645	−	0.195796i	$-0.0627291\pi$
$660$	0	0
$661$	21.4641i	0.834857i	0.908710	+	0.417428i	$0.137069\pi$
$661$	21.4641i	0.834857i	−0.908710	+	0.417428i	$0.862931\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.46410	0.211889
$666$	0	0
$667$	30.2487i	1.17123i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.14359	−0.237171
$672$	0	0
$673$	3.32051	0.127996	0.0639981	−	0.997950i	$-0.479615\pi$
$673$	3.32051	0.127996	0.0639981	+	0.997950i	$0.479615\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 46.5885i	− 1.79054i	−0.445524	−	0.895270i	$-0.646983\pi$
$677$	− 46.5885i	− 1.79054i	0.445524	−	0.895270i	$-0.353017\pi$
$678$	0	0
$679$	6.39230	0.245314
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 27.1244i	− 1.03788i	−0.854809	−	0.518942i	$-0.826326\pi$
$683$	− 27.1244i	− 1.03788i	0.854809	−	0.518942i	$-0.173674\pi$
$684$	0	0
$685$	− 10.5359i	− 0.402556i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 5.07180i	− 0.192940i	−0.995336	−	0.0964701i	$-0.969245\pi$
$691$	− 5.07180i	− 0.192940i	0.995336	−	0.0964701i	$-0.0307552\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.14359	−0.0813111
$696$	0	0
$697$	−1.60770	−0.0608958
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 24.9282i	− 0.941525i	−0.882260	−	0.470763i	$-0.843979\pi$
$701$	− 24.9282i	− 0.941525i	0.882260	−	0.470763i	$-0.156021\pi$
$702$	0	0
$703$	−74.6410	−2.81514
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.12436i	0.343157i
$708$	0	0
$709$	16.7846i	0.630359i	0.949032	+	0.315180i	$0.102065\pi$
$709$	16.7846i	0.630359i	−0.949032	+	0.315180i	$0.897935\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−43.0333	−1.61161
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	33.4641	1.24800	0.624000	−	0.781424i	$-0.285506\pi$
$719$	33.4641	1.24800	0.624000	+	0.781424i	$0.285506\pi$
$720$	0	0
$721$	8.92820	0.332504
$722$	0	0
$723$	0	0
$724$	0	0
$725$	15.4641i	0.574322i
$726$	0	0
$727$	41.7128	1.54704	0.773521	−	0.633770i	$-0.218494\pi$
$727$	41.7128	1.54704	0.773521	+	0.633770i	$0.218494\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.17691i	0.0435298i
$732$	0	0
$733$	10.9282i	0.403642i	0.979422	+	0.201821i	$0.0646860\pi$
$733$	10.9282i	0.403642i	−0.979422	+	0.201821i	$0.935314\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.3205	−1.15371
$738$	0	0
$739$	27.1769i	0.999719i	0.866107	+	0.499859i	$0.166615\pi$
$739$	27.1769i	0.999719i	−0.866107	+	0.499859i	$0.833385\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.48334	−0.0911049	−0.0455524	−	0.998962i	$-0.514505\pi$
$743$	−2.48334	−0.0911049	−0.0455524	+	0.998962i	$0.514505\pi$
$744$	0	0
$745$	9.17691	0.336216
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 6.73205i	− 0.245984i
$750$	0	0
$751$	26.9282	0.982624	0.491312	−	0.870984i	$-0.336517\pi$
$751$	26.9282	0.982624	0.491312	+	0.870984i	$0.336517\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10.9282i	0.397718i
$756$	0	0
$757$	− 40.0000i	− 1.45382i	−0.686730	−	0.726912i	$-0.740955\pi$
$757$	− 40.0000i	− 1.45382i	0.686730	−	0.726912i	$-0.259045\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.3397	0.664815	0.332408	−	0.943136i	$-0.392139\pi$
$761$	18.3397	0.664815	0.332408	+	0.943136i	$0.392139\pi$
$762$	0	0
$763$	2.92820i	0.106008i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−0.143594	−0.00517812	−0.00258906	−	0.999997i	$-0.500824\pi$
$769$	−0.143594	−0.00517812	−0.00258906	+	0.999997i	$0.500824\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 11.9474i	− 0.429720i	−0.976645	−	0.214860i	$-0.931071\pi$
$773$	− 11.9474i	− 0.429720i	0.976645	−	0.214860i	$-0.0689295\pi$
$774$	0	0
$775$	−22.0000	−0.790263
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 9.46410i	− 0.339087i
$780$	0	0
$781$	− 59.5692i	− 2.13155i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.07180	0.0382541
$786$	0	0
$787$	− 26.9282i	− 0.959887i	−0.877299	−	0.479943i	$-0.840657\pi$
$787$	− 26.9282i	− 0.959887i	0.877299	−	0.479943i	$-0.159343\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.53590	−0.303502
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 34.1962i	− 1.21129i	−0.795735	−	0.605645i	$-0.792915\pi$
$797$	− 34.1962i	− 1.21129i	0.795735	−	0.605645i	$-0.207085\pi$
$798$	0	0
$799$	−8.78461	−0.310777
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 43.6077i	− 1.53888i
$804$	0	0
$805$	− 6.39230i	− 0.225299i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	33.3205	1.17149	0.585743	−	0.810497i	$-0.300803\pi$
$809$	33.3205	1.17149	0.585743	+	0.810497i	$0.300803\pi$
$810$	0	0
$811$	42.6410i	1.49733i	0.662949	+	0.748664i	$0.269304\pi$
$811$	42.6410i	1.49733i	−0.662949	+	0.748664i	$0.730696\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.75129	0.0613450
$816$	0	0
$817$	−6.92820	−0.242387
$818$	0	0
$819$	0	0
$820$	0	0
$821$	53.3205i	1.86090i	0.366421	+	0.930449i	$0.380583\pi$
$821$	53.3205i	1.86090i	−0.366421	+	0.930449i	$0.619417\pi$
$822$	0	0
$823$	22.2487	0.775541	0.387771	−	0.921756i	$-0.373245\pi$
$823$	22.2487	0.775541	0.387771	+	0.921756i	$0.373245\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	53.6603i	1.86595i	0.359941	+	0.932975i	$0.382797\pi$
$827$	53.6603i	1.86595i	−0.359941	+	0.932975i	$0.617203\pi$
$828$	0	0
$829$	− 14.9282i	− 0.518478i	−0.965813	−	0.259239i	$-0.916528\pi$
$829$	− 14.9282i	− 0.518478i	0.965813	−	0.259239i	$-0.0834717\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.26795	0.0439318
$834$	0	0
$835$	4.00000i	0.138426i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	18.2487	0.630015	0.315008	−	0.949089i	$-0.397993\pi$
$839$	18.2487	0.630015	0.315008	+	0.949089i	$0.397993\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 9.51666i	− 0.327383i
$846$	0	0
$847$	−6.60770	−0.227043
$848$	0	0
$849$	0	0
$850$	0	0
$851$	87.3205i	2.99331i
$852$	0	0
$853$	16.3923i	0.561262i	0.959816	+	0.280631i	$0.0905437\pi$
$853$	16.3923i	0.561262i	−0.959816	+	0.280631i	$0.909456\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−51.5167	−1.75978	−0.879888	−	0.475182i	$-0.842382\pi$
$857$	−51.5167	−1.75978	−0.879888	+	0.475182i	$0.842382\pi$
$858$	0	0
$859$	− 24.2487i	− 0.827355i	−0.910423	−	0.413678i	$-0.864244\pi$
$859$	− 24.2487i	− 0.827355i	0.910423	−	0.413678i	$-0.135756\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.2679	−0.928212	−0.464106	−	0.885780i	$-0.653624\pi$
$863$	−27.2679	−0.928212	−0.464106	+	0.885780i	$0.653624\pi$
$864$	0	0
$865$	2.67949	0.0911055
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 39.7128i	− 1.34716i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 6.92820i	− 0.234216i
$876$	0	0
$877$	6.92820i	0.233949i	0.993135	+	0.116974i	$0.0373195\pi$
$877$	6.92820i	0.233949i	−0.993135	+	0.116974i	$0.962680\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.94744	0.200374	0.100187	−	0.994969i	$-0.468056\pi$
$881$	5.94744	0.200374	0.100187	+	0.994969i	$0.468056\pi$
$882$	0	0
$883$	50.4974i	1.69937i	0.527288	+	0.849687i	$0.323209\pi$
$883$	50.4974i	1.69937i	−0.527288	+	0.849687i	$0.676791\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.4641	−1.25792	−0.628961	−	0.777437i	$-0.716519\pi$
$887$	−37.4641	−1.25792	−0.628961	+	0.777437i	$0.716519\pi$
$888$	0	0
$889$	−0.392305	−0.0131575
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 51.7128i	− 1.73050i
$894$	0	0
$895$	5.71281	0.190958
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 17.0718i	− 0.569376i
$900$	0	0
$901$	− 4.39230i	− 0.146329i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.64102	−0.220755
$906$	0	0
$907$	− 19.8564i	− 0.659321i	−0.944100	−	0.329661i	$-0.893066\pi$
$907$	− 19.8564i	− 0.659321i	0.944100	−	0.329661i	$-0.106934\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.3397	0.408834	0.204417	−	0.978884i	$-0.434470\pi$
$911$	12.3397	0.408834	0.204417	+	0.978884i	$0.434470\pi$
$912$	0	0
$913$	−35.2154	−1.16546
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 20.3923i	− 0.673413i
$918$	0	0
$919$	13.0718	0.431199	0.215599	−	0.976482i	$-0.430829\pi$
$919$	13.0718	0.431199	0.215599	+	0.976482i	$0.430829\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	44.6410i	1.46779i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−27.8038	−0.912215	−0.456107	−	0.889925i	$-0.650757\pi$
$929$	−27.8038	−0.912215	−0.456107	+	0.889925i	$0.650757\pi$
$930$	0	0
$931$	7.46410i	0.244626i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.89488	−0.127376
$936$	0	0
$937$	32.1436	1.05009	0.525043	−	0.851076i	$-0.324049\pi$
$937$	32.1436	1.05009	0.525043	+	0.851076i	$0.324049\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	20.0526i	0.653695i	0.945077	+	0.326847i	$0.105986\pi$
$941$	20.0526i	0.653695i	−0.945077	+	0.326847i	$0.894014\pi$
$942$	0	0
$943$	−11.0718	−0.360547
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.0526i	0.456647i	0.973585	+	0.228323i	$0.0733244\pi$
$947$	14.0526i	0.456647i	−0.973585	+	0.228323i	$0.926676\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38.7846	−1.25636	−0.628178	−	0.778069i	$-0.716199\pi$
$953$	−38.7846	−1.25636	−0.628178	+	0.778069i	$0.716199\pi$
$954$	0	0
$955$	9.32051i	0.301605i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.3923	0.464752
$960$	0	0
$961$	−6.71281	−0.216542
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 9.85641i	− 0.317289i
$966$	0	0
$967$	38.2487	1.23000	0.614998	−	0.788529i	$-0.289157\pi$
$967$	38.2487	1.23000	0.614998	+	0.788529i	$0.289157\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	52.4974i	1.68472i	0.538913	+	0.842361i	$0.318835\pi$
$971$	52.4974i	1.68472i	−0.538913	+	0.842361i	$0.681165\pi$
$972$	0	0
$973$	− 2.92820i	− 0.0938739i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41.7128	−1.33451	−0.667256	−	0.744829i	$-0.732531\pi$
$977$	−41.7128	−1.33451	−0.667256	+	0.744829i	$0.732531\pi$
$978$	0	0
$979$	− 5.32051i	− 0.170044i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−56.4974	−1.80199	−0.900994	−	0.433832i	$-0.857161\pi$
$983$	−56.4974	−1.80199	−0.900994	+	0.433832i	$0.857161\pi$
$984$	0	0
$985$	18.2487	0.581452
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.10512i	0.257728i
$990$	0	0
$991$	−25.0718	−0.796432	−0.398216	−	0.917292i	$-0.630371\pi$
$991$	−25.0718	−0.796432	−0.398216	+	0.917292i	$0.630371\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.0000i	0.507234i
$996$	0	0
$997$	18.5359i	0.587038i	0.955953	+	0.293519i	$0.0948264\pi$
$997$	18.5359i	0.587038i	−0.955953	+	0.293519i	$0.905174\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.c.p.2017.2		4
3.2	odd	2		1344.2.c.g.673.2	yes	4
4.3	odd	2		4032.2.c.m.2017.2		4
8.3	odd	2		4032.2.c.m.2017.3		4
8.5	even	2	inner	4032.2.c.p.2017.3		4
12.11	even	2		1344.2.c.f.673.4	yes	4
24.5	odd	2		1344.2.c.g.673.3	yes	4
24.11	even	2		1344.2.c.f.673.1	✓	4
48.5	odd	4		5376.2.a.s.1.1		2
48.11	even	4		5376.2.a.be.1.1		2
48.29	odd	4		5376.2.a.x.1.2		2
48.35	even	4		5376.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.2.c.f.673.1	✓	4	24.11	even	2
1344.2.c.f.673.4	yes	4	12.11	even	2
1344.2.c.g.673.2	yes	4	3.2	odd	2
1344.2.c.g.673.3	yes	4	24.5	odd	2
4032.2.c.m.2017.2		4	4.3	odd	2
4032.2.c.m.2017.3		4	8.3	odd	2
4032.2.c.p.2017.2		4	1.1	even	1	trivial
4032.2.c.p.2017.3		4	8.5	even	2	inner
5376.2.a.p.1.2		2	48.35	even	4
5376.2.a.s.1.1		2	48.5	odd	4
5376.2.a.x.1.2		2	48.29	odd	4
5376.2.a.be.1.1		2	48.11	even	4

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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.732051i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-0.732051i q^{5} +1.00000 q^{7} -4.19615i q^{11} +1.26795 q^{17} +7.46410i q^{19} +8.73205 q^{23} +4.46410 q^{25} +3.46410i q^{29} -4.92820 q^{31} -0.732051i q^{35} +10.0000i q^{37} -1.26795 q^{41} +0.928203i q^{43} -6.92820 q^{47} +1.00000 q^{49} -3.46410i q^{53} -3.07180 q^{55} -10.9282i q^{59} -1.46410i q^{61} -7.46410i q^{67} +14.1962 q^{71} +10.3923 q^{73} -4.19615i q^{77} +9.46410 q^{79} -8.39230i q^{83} -0.928203i q^{85} +1.26795 q^{89} +5.46410 q^{95} +6.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} + 12 q^{17} + 28 q^{23} + 4 q^{25} + 8 q^{31} - 12 q^{41} + 4 q^{49} - 40 q^{55} + 36 q^{71} + 24 q^{79} + 12 q^{89} + 8 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 12 * q^17 + 28 * q^23 + 4 * q^25 + 8 * q^31 - 12 * q^41 + 4 * q^49 - 40 * q^55 + 36 * q^71 + 24 * q^79 + 12 * q^89 + 8 * q^95 - 16 * q^97

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