LMFDB - Embedded newform 4032.2.c.p.2017.4

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			2017.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	4032.2017
Dual form			4032.2.c.p.2017.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205i q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.73205i q^{5} +1.00000 q^{7} +6.19615i q^{11} +4.73205 q^{17} +0.535898i q^{19} +5.26795 q^{23} -2.46410 q^{25} -3.46410i q^{29} +8.92820 q^{31} +2.73205i q^{35} +10.0000i q^{37} -4.73205 q^{41} -12.9282i q^{43} +6.92820 q^{47} +1.00000 q^{49} +3.46410i q^{53} -16.9282 q^{55} +2.92820i q^{59} +5.46410i q^{61} -0.535898i q^{67} +3.80385 q^{71} -10.3923 q^{73} +6.19615i q^{77} +2.53590 q^{79} +12.3923i q^{83} +12.9282i q^{85} +4.73205 q^{89} -1.46410 q^{95} -14.3923 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.73205i	1.22181i	0.791704	+	0.610905i	$0.209194\pi$
$5$	2.73205i	1.22181i	−0.791704	+	0.610905i	$0.790806\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.19615i	1.86821i	0.356998	+	0.934105i	$0.383800\pi$
$11$	6.19615i	1.86821i	−0.356998	+	0.934105i	$0.616200\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.73205	1.14769	0.573845	−	0.818964i	$-0.305451\pi$
$17$	4.73205	1.14769	0.573845	+	0.818964i	$0.305451\pi$
$18$	0	0
$19$	0.535898i	0.122944i	0.998109	+	0.0614718i	$0.0195794\pi$
$19$	0.535898i	0.122944i	−0.998109	+	0.0614718i	$0.980421\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.26795	1.09844	0.549222	−	0.835677i	$-0.314924\pi$
$23$	5.26795	1.09844	0.549222	+	0.835677i	$0.314924\pi$
$24$	0	0
$25$	−2.46410	−0.492820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.46410i	− 0.643268i	−0.946864	−	0.321634i	$-0.895768\pi$
$29$	− 3.46410i	− 0.643268i	0.946864	−	0.321634i	$-0.104232\pi$
$30$	0	0
$31$	8.92820	1.60355	0.801776	−	0.597624i	$-0.203889\pi$
$31$	8.92820	1.60355	0.801776	+	0.597624i	$0.203889\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.73205i	0.461801i
$36$	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
$41$	−4.73205	−0.739022	−0.369511	−	0.929226i	$-0.620475\pi$
$41$	−4.73205	−0.739022	−0.369511	+	0.929226i	$0.620475\pi$
$42$	0	0
$43$	− 12.9282i	− 1.97153i	−0.168122	−	0.985766i	$-0.553770\pi$
$43$	− 12.9282i	− 1.97153i	0.168122	−	0.985766i	$-0.446230\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	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	$0.0764641\pi$
$53$	3.46410i	0.475831i	−0.971286	+	0.237915i	$0.923536\pi$
$54$	0	0
$55$	−16.9282	−2.28260
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.92820i	0.381220i	0.981666	+	0.190610i	$0.0610465\pi$
$59$	2.92820i	0.381220i	−0.981666	+	0.190610i	$0.938953\pi$
$60$	0	0
$61$	5.46410i	0.699607i	0.936823	+	0.349803i	$0.113752\pi$
$61$	5.46410i	0.699607i	−0.936823	+	0.349803i	$0.886248\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 0.535898i	− 0.0654704i	−0.999464	−	0.0327352i	$-0.989578\pi$
$67$	− 0.535898i	− 0.0654704i	0.999464	−	0.0327352i	$-0.0104218\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.80385	0.451434	0.225717	−	0.974193i	$-0.427528\pi$
$71$	3.80385	0.451434	0.225717	+	0.974193i	$0.427528\pi$
$72$	0	0
$73$	−10.3923	−1.21633	−0.608164	−	0.793812i	$-0.708094\pi$
$73$	−10.3923	−1.21633	−0.608164	+	0.793812i	$0.708094\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.19615i	0.706117i
$78$	0	0
$79$	2.53590	0.285311	0.142655	−	0.989772i	$-0.454436\pi$
$79$	2.53590	0.285311	0.142655	+	0.989772i	$0.454436\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.3923i	1.36023i	0.733104	+	0.680116i	$0.238071\pi$
$83$	12.3923i	1.36023i	−0.733104	+	0.680116i	$0.761929\pi$
$84$	0	0
$85$	12.9282i	1.40226i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.73205	0.501596	0.250798	−	0.968039i	$-0.419307\pi$
$89$	4.73205	0.501596	0.250798	+	0.968039i	$0.419307\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.46410	−0.150214
$96$	0	0
$97$	−14.3923	−1.46132	−0.730659	−	0.682743i	$-0.760787\pi$
$97$	−14.3923	−1.46132	−0.730659	+	0.682743i	$0.760787\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 15.1244i	− 1.50493i	−0.658632	−	0.752465i	$-0.728865\pi$
$101$	− 15.1244i	− 1.50493i	0.658632	−	0.752465i	$-0.271135\pi$
$102$	0	0
$103$	−4.92820	−0.485590	−0.242795	−	0.970078i	$-0.578064\pi$
$103$	−4.92820	−0.485590	−0.242795	+	0.970078i	$0.578064\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.26795i	− 0.315925i	−0.987445	−	0.157962i	$-0.949508\pi$
$107$	− 3.26795i	− 0.315925i	0.987445	−	0.157962i	$-0.0504924\pi$
$108$	0	0
$109$	− 10.9282i	− 1.04673i	−0.852108	−	0.523366i	$-0.824676\pi$
$109$	− 10.9282i	− 1.04673i	0.852108	−	0.523366i	$-0.175324\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−15.4641	−1.45474	−0.727370	−	0.686245i	$-0.759258\pi$
$113$	−15.4641	−1.45474	−0.727370	+	0.686245i	$0.759258\pi$
$114$	0	0
$115$	14.3923i	1.34209i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.73205	0.433786
$120$	0	0
$121$	−27.3923	−2.49021
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.92820i	0.619677i
$126$	0	0
$127$	20.3923	1.80952	0.904762	−	0.425917i	$-0.140048\pi$
$127$	20.3923	1.80952	0.904762	+	0.425917i	$0.140048\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.392305i	0.0342758i	0.999853	+	0.0171379i	$0.00545544\pi$
$131$	0.392305i	0.0342758i	−0.999853	+	0.0171379i	$0.994545\pi$
$132$	0	0
$133$	0.535898i	0.0464683i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.39230	−0.546131	−0.273066	−	0.961995i	$-0.588038\pi$
$137$	−6.39230	−0.546131	−0.273066	+	0.961995i	$0.588038\pi$
$138$	0	0
$139$	10.9282i	0.926918i	0.886118	+	0.463459i	$0.153392\pi$
$139$	10.9282i	0.926918i	−0.886118	+	0.463459i	$0.846608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	9.46410	0.785951
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.4641i	1.59456i	0.603609	+	0.797281i	$0.293729\pi$
$149$	19.4641i	1.59456i	−0.603609	+	0.797281i	$0.706271\pi$
$150$	0	0
$151$	−1.07180	−0.0872216	−0.0436108	−	0.999049i	$-0.513886\pi$
$151$	−1.07180	−0.0872216	−0.0436108	+	0.999049i	$0.513886\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	24.3923i	1.95924i
$156$	0	0
$157$	− 5.46410i	− 0.436083i	−0.975940	−	0.218041i	$-0.930033\pi$
$157$	− 5.46410i	− 0.436083i	0.975940	−	0.218041i	$-0.0699668\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.26795	0.415173
$162$	0	0
$163$	− 18.3923i	− 1.44060i	−0.693664	−	0.720298i	$-0.744005\pi$
$163$	− 18.3923i	− 1.44060i	0.693664	−	0.720298i	$-0.255995\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.46410	0.113296	0.0566478	−	0.998394i	$-0.481959\pi$
$167$	1.46410	0.113296	0.0566478	+	0.998394i	$0.481959\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 13.6603i	− 1.03857i	−0.854601	−	0.519285i	$-0.826198\pi$
$173$	− 13.6603i	− 1.03857i	0.854601	−	0.519285i	$-0.173802\pi$
$174$	0	0
$175$	−2.46410	−0.186269
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.1962i	1.36004i	0.733192	+	0.680022i	$0.238030\pi$
$179$	18.1962i	1.36004i	−0.733192	+	0.680022i	$0.761970\pi$
$180$	0	0
$181$	− 22.9282i	− 1.70424i	−0.523347	−	0.852120i	$-0.675317\pi$
$181$	− 22.9282i	− 1.70424i	0.523347	−	0.852120i	$-0.324683\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−27.3205	−2.00864
$186$	0	0
$187$	29.3205i	2.14413i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.26795	−0.670605	−0.335303	−	0.942110i	$-0.608839\pi$
$191$	−9.26795	−0.670605	−0.335303	+	0.942110i	$0.608839\pi$
$192$	0	0
$193$	6.53590	0.470464	0.235232	−	0.971939i	$-0.424415\pi$
$193$	6.53590	0.470464	0.235232	+	0.971939i	$0.424415\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.0718i	0.788833i	0.918932	+	0.394416i	$0.129053\pi$
$197$	11.0718i	0.788833i	−0.918932	+	0.394416i	$0.870947\pi$
$198$	0	0
$199$	5.85641	0.415150	0.207575	−	0.978219i	$-0.433443\pi$
$199$	5.85641	0.415150	0.207575	+	0.978219i	$0.433443\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.46410i	− 0.243132i
$204$	0	0
$205$	− 12.9282i	− 0.902945i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.32051	−0.229684
$210$	0	0
$211$	11.8564i	0.816229i	0.912931	+	0.408114i	$0.133814\pi$
$211$	11.8564i	0.816229i	−0.912931	+	0.408114i	$0.866186\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	35.3205	2.40884
$216$	0	0
$217$	8.92820	0.606086
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−16.7846	−1.12398	−0.561990	−	0.827144i	$-0.689964\pi$
$223$	−16.7846	−1.12398	−0.561990	+	0.827144i	$0.689964\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.53590i	0.433803i	0.976194	+	0.216901i	$0.0695950\pi$
$227$	6.53590i	0.433803i	−0.976194	+	0.216901i	$0.930405\pi$
$228$	0	0
$229$	− 12.7846i	− 0.844831i	−0.906402	−	0.422415i	$-0.861182\pi$
$229$	− 12.7846i	− 0.844831i	0.906402	−	0.422415i	$-0.138818\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.8564	−1.03879	−0.519394	−	0.854535i	$-0.673842\pi$
$233$	−15.8564	−1.03879	−0.519394	+	0.854535i	$0.673842\pi$
$234$	0	0
$235$	18.9282i	1.23474i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.0526	−1.68520	−0.842600	−	0.538540i	$-0.818976\pi$
$239$	−26.0526	−1.68520	−0.842600	+	0.538540i	$0.818976\pi$
$240$	0	0
$241$	24.2487	1.56200	0.780998	−	0.624533i	$-0.214711\pi$
$241$	24.2487	1.56200	0.780998	+	0.624533i	$0.214711\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.73205i	0.174544i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 21.4641i	− 1.35480i	−0.735614	−	0.677401i	$-0.763106\pi$
$251$	− 21.4641i	− 1.35480i	0.735614	−	0.677401i	$-0.236894\pi$
$252$	0	0
$253$	32.6410i	2.05212i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.5885	−0.660490	−0.330245	−	0.943895i	$-0.607131\pi$
$257$	−10.5885	−0.660490	−0.330245	+	0.943895i	$0.607131\pi$
$258$	0	0
$259$	10.0000i	0.621370i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.0526	−0.866518	−0.433259	−	0.901269i	$-0.642636\pi$
$263$	−14.0526	−0.866518	−0.433259	+	0.901269i	$0.642636\pi$
$264$	0	0
$265$	−9.46410	−0.581375
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 12.5885i	− 0.767532i	−0.923430	−	0.383766i	$-0.874627\pi$
$269$	− 12.5885i	− 0.767532i	0.923430	−	0.383766i	$-0.125373\pi$
$270$	0	0
$271$	0.143594	0.00872269	0.00436134	−	0.999990i	$-0.498612\pi$
$271$	0.143594	0.00872269	0.00436134	+	0.999990i	$0.498612\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 15.2679i	− 0.920692i
$276$	0	0
$277$	18.7846i	1.12866i	0.825550	+	0.564329i	$0.190865\pi$
$277$	18.7846i	1.12866i	−0.825550	+	0.564329i	$0.809135\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.8564	−0.707294	−0.353647	−	0.935379i	$-0.615059\pi$
$281$	−11.8564	−0.707294	−0.353647	+	0.935379i	$0.615059\pi$
$282$	0	0
$283$	− 8.53590i	− 0.507406i	−0.967282	−	0.253703i	$-0.918351\pi$
$283$	− 8.53590i	− 0.507406i	0.967282	−	0.253703i	$-0.0816487\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.73205	−0.279324
$288$	0	0
$289$	5.39230	0.317194
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.0526i	1.28832i	0.764889	+	0.644162i	$0.222794\pi$
$293$	22.0526i	1.28832i	−0.764889	+	0.644162i	$0.777206\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 12.9282i	− 0.745169i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−14.9282	−0.854786
$306$	0	0
$307$	− 0.535898i	− 0.0305853i	−0.999883	−	0.0152927i	$-0.995132\pi$
$307$	− 0.535898i	− 0.0305853i	0.999883	−	0.0152927i	$-0.00486800\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.9282	1.30014	0.650070	−	0.759875i	$-0.274740\pi$
$311$	22.9282	1.30014	0.650070	+	0.759875i	$0.274740\pi$
$312$	0	0
$313$	−16.9282	−0.956839	−0.478419	−	0.878132i	$-0.658790\pi$
$313$	−16.9282	−0.956839	−0.478419	+	0.878132i	$0.658790\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.32051i	0.523492i	0.965137	+	0.261746i	$0.0842983\pi$
$317$	9.32051i	0.523492i	−0.965137	+	0.261746i	$0.915702\pi$
$318$	0	0
$319$	21.4641	1.20176
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.53590i	0.141101i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.92820	0.381964
$330$	0	0
$331$	− 15.0718i	− 0.828421i	−0.910181	−	0.414210i	$-0.864058\pi$
$331$	− 15.0718i	− 0.828421i	0.910181	−	0.414210i	$-0.135942\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.46410	0.0799924
$336$	0	0
$337$	−11.8564	−0.645860	−0.322930	−	0.946423i	$-0.604668\pi$
$337$	−11.8564	−0.645860	−0.322930	+	0.946423i	$0.604668\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	55.3205i	2.99577i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.339746i	0.0182385i	0.999958	+	0.00911926i	$0.00290279\pi$
$347$	0.339746i	0.0182385i	−0.999958	+	0.00911926i	$0.997097\pi$
$348$	0	0
$349$	14.5359i	0.778089i	0.921219	+	0.389044i	$0.127195\pi$
$349$	14.5359i	0.778089i	−0.921219	+	0.389044i	$0.872805\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.41154	−0.0751288	−0.0375644	−	0.999294i	$-0.511960\pi$
$353$	−1.41154	−0.0751288	−0.0375644	+	0.999294i	$0.511960\pi$
$354$	0	0
$355$	10.3923i	0.551566i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.9090	1.68409	0.842045	−	0.539407i	$-0.181351\pi$
$359$	31.9090	1.68409	0.842045	+	0.539407i	$0.181351\pi$
$360$	0	0
$361$	18.7128	0.984885
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 28.3923i	− 1.48612i
$366$	0	0
$367$	37.8564	1.97609	0.988044	−	0.154171i	$-0.0492707\pi$
$367$	37.8564	1.97609	0.988044	+	0.154171i	$0.0492707\pi$
$368$	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
$379$	− 3.07180i	− 0.157788i	−0.996883	−	0.0788938i	$-0.974861\pi$
$379$	− 3.07180i	− 0.157788i	0.996883	−	0.0788938i	$-0.0251388\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7.60770	−0.388735	−0.194368	−	0.980929i	$-0.562265\pi$
$383$	−7.60770	−0.388735	−0.194368	+	0.980929i	$0.562265\pi$
$384$	0	0
$385$	−16.9282	−0.862741
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.53590i	0.432787i	0.976306	+	0.216394i	$0.0694294\pi$
$389$	8.53590i	0.432787i	−0.976306	+	0.216394i	$0.930571\pi$
$390$	0	0
$391$	24.9282	1.26067
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.92820i	0.348596i
$396$	0	0
$397$	− 31.3205i	− 1.57193i	−0.618270	−	0.785966i	$-0.712166\pi$
$397$	− 31.3205i	− 1.57193i	0.618270	−	0.785966i	$-0.287834\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.3923	−0.918468	−0.459234	−	0.888315i	$-0.651876\pi$
$401$	−18.3923	−0.918468	−0.459234	+	0.888315i	$0.651876\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−61.9615	−3.07132
$408$	0	0
$409$	17.6077	0.870644	0.435322	−	0.900275i	$-0.356634\pi$
$409$	17.6077	0.870644	0.435322	+	0.900275i	$0.356634\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.92820i	0.144087i
$414$	0	0
$415$	−33.8564	−1.66195
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 9.85641i	− 0.481517i	−0.970585	−	0.240758i	$-0.922604\pi$
$419$	− 9.85641i	− 0.481517i	0.970585	−	0.240758i	$-0.0773962\pi$
$420$	0	0
$421$	30.7846i	1.50035i	0.661239	+	0.750175i	$0.270031\pi$
$421$	30.7846i	1.50035i	−0.661239	+	0.750175i	$0.729969\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−11.6603	−0.565605
$426$	0	0
$427$	5.46410i	0.264426i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.0526	−1.44758	−0.723790	−	0.690020i	$-0.757602\pi$
$431$	−30.0526	−1.44758	−0.723790	+	0.690020i	$0.757602\pi$
$432$	0	0
$433$	−11.0718	−0.532077	−0.266038	−	0.963962i	$-0.585715\pi$
$433$	−11.0718	−0.532077	−0.266038	+	0.963962i	$0.585715\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.82309i	0.135046i
$438$	0	0
$439$	13.8564	0.661330	0.330665	−	0.943748i	$-0.392727\pi$
$439$	13.8564	0.661330	0.330665	+	0.943748i	$0.392727\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 8.73205i	− 0.414872i	−0.978249	−	0.207436i	$-0.933488\pi$
$443$	− 8.73205i	− 0.414872i	0.978249	−	0.207436i	$-0.0665119\pi$
$444$	0	0
$445$	12.9282i	0.612856i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15.4641	0.729796	0.364898	−	0.931047i	$-0.381104\pi$
$449$	15.4641	0.729796	0.364898	+	0.931047i	$0.381104\pi$
$450$	0	0
$451$	− 29.3205i	− 1.38065i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.9282	−0.604756	−0.302378	−	0.953188i	$-0.597780\pi$
$457$	−12.9282	−0.604756	−0.302378	+	0.953188i	$0.597780\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.5885i	0.586303i	0.956066	+	0.293151i	$0.0947040\pi$
$461$	12.5885i	0.586303i	−0.956066	+	0.293151i	$0.905296\pi$
$462$	0	0
$463$	22.2487	1.03399	0.516993	−	0.855990i	$-0.327051\pi$
$463$	22.2487	1.03399	0.516993	+	0.855990i	$0.327051\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 27.3205i	− 1.26424i	−0.774870	−	0.632121i	$-0.782184\pi$
$467$	− 27.3205i	− 1.26424i	0.774870	−	0.632121i	$-0.217816\pi$
$468$	0	0
$469$	− 0.535898i	− 0.0247455i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	80.1051	3.68324
$474$	0	0
$475$	− 1.32051i	− 0.0605891i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	19.7128	0.900701	0.450351	−	0.892852i	$-0.351299\pi$
$479$	19.7128	0.900701	0.450351	+	0.892852i	$0.351299\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 39.3205i	− 1.78545i
$486$	0	0
$487$	−5.85641	−0.265379	−0.132690	−	0.991158i	$-0.542361\pi$
$487$	−5.85641	−0.265379	−0.132690	+	0.991158i	$0.542361\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 4.05256i	− 0.182889i	−0.995810	−	0.0914447i	$-0.970852\pi$
$491$	− 4.05256i	− 0.182889i	0.995810	−	0.0914447i	$-0.0291485\pi$
$492$	0	0
$493$	− 16.3923i	− 0.738272i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.80385	0.170626
$498$	0	0
$499$	35.8564i	1.60515i	0.596549	+	0.802577i	$0.296538\pi$
$499$	35.8564i	1.60515i	−0.596549	+	0.802577i	$0.703462\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	36.3923	1.62265	0.811326	−	0.584594i	$-0.198746\pi$
$503$	36.3923	1.62265	0.811326	+	0.584594i	$0.198746\pi$
$504$	0	0
$505$	41.3205	1.83874
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 37.2679i	− 1.65187i	−0.563763	−	0.825936i	$-0.690647\pi$
$509$	− 37.2679i	− 1.65187i	0.563763	−	0.825936i	$-0.309353\pi$
$510$	0	0
$511$	−10.3923	−0.459728
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 13.4641i	− 0.593299i
$516$	0	0
$517$	42.9282i	1.88798i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.7321	−1.25877	−0.629387	−	0.777092i	$-0.716694\pi$
$521$	−28.7321	−1.25877	−0.629387	+	0.777092i	$0.716694\pi$
$522$	0	0
$523$	− 26.6410i	− 1.16493i	−0.812856	−	0.582465i	$-0.802088\pi$
$523$	− 26.6410i	− 1.16493i	0.812856	−	0.582465i	$-0.197912\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	42.2487	1.84038
$528$	0	0
$529$	4.75129	0.206578
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	8.92820	0.386000
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.19615i	0.266887i
$540$	0	0
$541$	− 27.8564i	− 1.19764i	−0.800883	−	0.598820i	$-0.795636\pi$
$541$	− 27.8564i	− 1.19764i	0.800883	−	0.598820i	$-0.204364\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	29.8564	1.27891
$546$	0	0
$547$	− 14.3923i	− 0.615371i	−0.951488	−	0.307685i	$-0.900446\pi$
$547$	− 14.3923i	− 0.615371i	0.951488	−	0.307685i	$-0.0995544\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.85641	0.0790856
$552$	0	0
$553$	2.53590	0.107837
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.32051i	0.225437i	0.993627	+	0.112719i	$0.0359559\pi$
$557$	5.32051i	0.225437i	−0.993627	+	0.112719i	$0.964044\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.85641i	0.246818i	0.992356	+	0.123409i	$0.0393827\pi$
$563$	5.85641i	0.246818i	−0.992356	+	0.123409i	$0.960617\pi$
$564$	0	0
$565$	− 42.2487i	− 1.77742i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.07180	−0.128776	−0.0643882	−	0.997925i	$-0.520510\pi$
$569$	−3.07180	−0.128776	−0.0643882	+	0.997925i	$0.520510\pi$
$570$	0	0
$571$	0.535898i	0.0224266i	0.999937	+	0.0112133i	$0.00356939\pi$
$571$	0.535898i	0.0224266i	−0.999937	+	0.0112133i	$0.996431\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.9808	−0.541335
$576$	0	0
$577$	22.7846	0.948536	0.474268	−	0.880381i	$-0.342713\pi$
$577$	22.7846	0.948536	0.474268	+	0.880381i	$0.342713\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.3923i	0.514119i
$582$	0	0
$583$	−21.4641	−0.888952
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.46410i	0.390625i	0.980741	+	0.195313i	$0.0625722\pi$
$587$	9.46410i	0.390625i	−0.980741	+	0.195313i	$0.937428\pi$
$588$	0	0
$589$	4.78461i	0.197146i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	44.8372	1.84124	0.920621	−	0.390458i	$-0.127683\pi$
$593$	44.8372	1.84124	0.920621	+	0.390458i	$0.127683\pi$
$594$	0	0
$595$	12.9282i	0.530005i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	25.6603	1.04845	0.524225	−	0.851580i	$-0.324355\pi$
$599$	25.6603	1.04845	0.524225	+	0.851580i	$0.324355\pi$
$600$	0	0
$601$	41.7128	1.70150	0.850751	−	0.525570i	$-0.176148\pi$
$601$	41.7128	1.70150	0.850751	+	0.525570i	$0.176148\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 74.8372i	− 3.04256i
$606$	0	0
$607$	−34.9282	−1.41769	−0.708846	−	0.705363i	$-0.750784\pi$
$607$	−34.9282	−1.41769	−0.708846	+	0.705363i	$0.750784\pi$
$608$	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
$617$	12.2487	0.493115	0.246557	−	0.969128i	$-0.420701\pi$
$617$	12.2487	0.493115	0.246557	+	0.969128i	$0.420701\pi$
$618$	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
$623$	4.73205	0.189586
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	0	0
$628$	0	0
$629$	47.3205i	1.88679i
$630$	0	0
$631$	−33.4641	−1.33218	−0.666092	−	0.745869i	$-0.732034\pi$
$631$	−33.4641	−1.33218	−0.666092	+	0.745869i	$0.732034\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	55.7128i	2.21090i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−46.3923	−1.83239	−0.916193	−	0.400737i	$-0.868754\pi$
$641$	−46.3923	−1.83239	−0.916193	+	0.400737i	$0.868754\pi$
$642$	0	0
$643$	− 14.3923i	− 0.567577i	−0.958887	−	0.283789i	$-0.908409\pi$
$643$	− 14.3923i	− 0.567577i	0.958887	−	0.283789i	$-0.0915914\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.6077	−0.456346	−0.228173	−	0.973621i	$-0.573275\pi$
$647$	−11.6077	−0.456346	−0.228173	+	0.973621i	$0.573275\pi$
$648$	0	0
$649$	−18.1436	−0.712198
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 22.7846i	− 0.891631i	−0.895125	−	0.445815i	$-0.852914\pi$
$653$	− 22.7846i	− 0.891631i	0.895125	−	0.445815i	$-0.147086\pi$
$654$	0	0
$655$	−1.07180	−0.0418786
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28.0526i	1.09277i	0.837533	+	0.546386i	$0.183997\pi$
$659$	28.0526i	1.09277i	−0.837533	+	0.546386i	$0.816003\pi$
$660$	0	0
$661$	14.5359i	0.565381i	0.959211	+	0.282690i	$0.0912269\pi$
$661$	14.5359i	0.565381i	−0.959211	+	0.282690i	$0.908773\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.46410	−0.0567754
$666$	0	0
$667$	− 18.2487i	− 0.706593i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33.8564	−1.30701
$672$	0	0
$673$	−31.3205	−1.20732	−0.603658	−	0.797243i	$-0.706291\pi$
$673$	−31.3205	−1.20732	−0.603658	+	0.797243i	$0.706291\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.4115i	− 0.592314i	−0.955139	−	0.296157i	$-0.904295\pi$
$677$	− 15.4115i	− 0.592314i	0.955139	−	0.296157i	$-0.0957051\pi$
$678$	0	0
$679$	−14.3923	−0.552326
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 2.87564i	− 0.110033i	−0.998485	−	0.0550167i	$-0.982479\pi$
$683$	− 2.87564i	− 0.110033i	0.998485	−	0.0550167i	$-0.0175212\pi$
$684$	0	0
$685$	− 17.4641i	− 0.667269i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 18.9282i	− 0.720063i	−0.932940	−	0.360031i	$-0.882766\pi$
$691$	− 18.9282i	− 0.720063i	0.932940	−	0.360031i	$-0.117234\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−29.8564	−1.13252
$696$	0	0
$697$	−22.3923	−0.848169
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 11.0718i	− 0.418176i	−0.977897	−	0.209088i	$-0.932950\pi$
$701$	− 11.0718i	− 0.418176i	0.977897	−	0.209088i	$-0.0670495\pi$
$702$	0	0
$703$	−5.35898	−0.202118
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.1244i	− 0.568810i
$708$	0	0
$709$	− 24.7846i	− 0.930806i	−0.885099	−	0.465403i	$-0.845909\pi$
$709$	− 24.7846i	− 0.930806i	0.885099	−	0.465403i	$-0.154091\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	47.0333	1.76141
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	26.5359	0.989622	0.494811	−	0.869001i	$-0.335237\pi$
$719$	26.5359	0.989622	0.494811	+	0.869001i	$0.335237\pi$
$720$	0	0
$721$	−4.92820	−0.183536
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.53590i	0.317015i
$726$	0	0
$727$	−13.7128	−0.508580	−0.254290	−	0.967128i	$-0.581842\pi$
$727$	−13.7128	−0.508580	−0.254290	+	0.967128i	$0.581842\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 61.1769i	− 2.26271i
$732$	0	0
$733$	− 2.92820i	− 0.108156i	−0.998537	−	0.0540778i	$-0.982778\pi$
$733$	− 2.92820i	− 0.108156i	0.998537	−	0.0540778i	$-0.0172219\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.32051	0.122312
$738$	0	0
$739$	− 35.1769i	− 1.29400i	−0.762488	−	0.647002i	$-0.776023\pi$
$739$	− 35.1769i	− 1.29400i	0.762488	−	0.647002i	$-0.223977\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−47.5167	−1.74322	−0.871609	−	0.490202i	$-0.836923\pi$
$743$	−47.5167	−1.74322	−0.871609	+	0.490202i	$0.836923\pi$
$744$	0	0
$745$	−53.1769	−1.94825
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 3.26795i	− 0.119408i
$750$	0	0
$751$	13.0718	0.476997	0.238498	−	0.971143i	$-0.423345\pi$
$751$	13.0718	0.476997	0.238498	+	0.971143i	$0.423345\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2.92820i	− 0.106568i
$756$	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
$761$	35.6603	1.29268	0.646342	−	0.763048i	$-0.276298\pi$
$761$	35.6603	1.29268	0.646342	+	0.763048i	$0.276298\pi$
$762$	0	0
$763$	− 10.9282i	− 0.395628i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−27.8564	−1.00453	−0.502264	−	0.864714i	$-0.667499\pi$
$769$	−27.8564	−1.00453	−0.502264	+	0.864714i	$0.667499\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 50.0526i	− 1.80027i	−0.435616	−	0.900133i	$-0.643469\pi$
$773$	− 50.0526i	− 1.80027i	0.435616	−	0.900133i	$-0.356531\pi$
$774$	0	0
$775$	−22.0000	−0.790263
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.53590i	− 0.0908580i
$780$	0	0
$781$	23.5692i	0.843373i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.9282	0.532810
$786$	0	0
$787$	− 13.0718i	− 0.465959i	−0.972482	−	0.232980i	$-0.925152\pi$
$787$	− 13.0718i	− 0.465959i	0.972482	−	0.232980i	$-0.0748475\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.4641	−0.549840
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 23.8038i	− 0.843176i	−0.906788	−	0.421588i	$-0.861473\pi$
$797$	− 23.8038i	− 0.843176i	0.906788	−	0.421588i	$-0.138527\pi$
$798$	0	0
$799$	32.7846	1.15984
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 64.3923i	− 2.27236i
$804$	0	0
$805$	14.3923i	0.507262i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.32051	−0.0464266	−0.0232133	−	0.999731i	$-0.507390\pi$
$809$	−1.32051	−0.0464266	−0.0232133	+	0.999731i	$0.507390\pi$
$810$	0	0
$811$	− 26.6410i	− 0.935493i	−0.883863	−	0.467746i	$-0.845066\pi$
$811$	− 26.6410i	− 0.935493i	0.883863	−	0.467746i	$-0.154934\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	50.2487	1.76014
$816$	0	0
$817$	6.92820	0.242387
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.6795i	0.651919i	0.945384	+	0.325959i	$0.105687\pi$
$821$	18.6795i	0.651919i	−0.945384	+	0.325959i	$0.894313\pi$
$822$	0	0
$823$	−26.2487	−0.914973	−0.457486	−	0.889217i	$-0.651250\pi$
$823$	−26.2487	−0.914973	−0.457486	+	0.889217i	$0.651250\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.3397i	1.26366i	0.775108	+	0.631828i	$0.217695\pi$
$827$	36.3397i	1.26366i	−0.775108	+	0.631828i	$0.782305\pi$
$828$	0	0
$829$	− 1.07180i	− 0.0372250i	−0.999827	−	0.0186125i	$-0.994075\pi$
$829$	− 1.07180i	− 0.0372250i	0.999827	−	0.0186125i	$-0.00592489\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.73205	0.163956
$834$	0	0
$835$	4.00000i	0.138426i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30.2487	−1.04430	−0.522151	−	0.852853i	$-0.674870\pi$
$839$	−30.2487	−1.04430	−0.522151	+	0.852853i	$0.674870\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	35.5167i	1.22181i
$846$	0	0
$847$	−27.3923	−0.941211
$848$	0	0
$849$	0	0
$850$	0	0
$851$	52.6795i	1.80583i
$852$	0	0
$853$	− 4.39230i	− 0.150390i	−0.997169	−	0.0751948i	$-0.976042\pi$
$853$	− 4.39230i	− 0.150390i	0.997169	−	0.0751948i	$-0.0239579\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.48334	−0.221467	−0.110733	−	0.993850i	$-0.535320\pi$
$857$	−6.48334	−0.221467	−0.110733	+	0.993850i	$0.535320\pi$
$858$	0	0
$859$	24.2487i	0.827355i	0.910423	+	0.413678i	$0.135756\pi$
$859$	24.2487i	0.827355i	−0.910423	+	0.413678i	$0.864244\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.7321	−1.04613	−0.523066	−	0.852292i	$-0.675212\pi$
$863$	−30.7321	−1.04613	−0.523066	+	0.852292i	$0.675212\pi$
$864$	0	0
$865$	37.3205	1.26894
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15.7128i	0.533021i
$870$	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	$-0.962680\pi$
$877$	− 6.92820i	− 0.233949i	0.993135	−	0.116974i	$-0.0373195\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	44.0526	1.48417	0.742084	−	0.670307i	$-0.233837\pi$
$881$	44.0526	1.48417	0.742084	+	0.670307i	$0.233837\pi$
$882$	0	0
$883$	− 46.4974i	− 1.56476i	−0.622800	−	0.782381i	$-0.714005\pi$
$883$	− 46.4974i	− 1.56476i	0.622800	−	0.782381i	$-0.285995\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.5359	−1.02530	−0.512648	−	0.858599i	$-0.671335\pi$
$887$	−30.5359	−1.02530	−0.512648	+	0.858599i	$0.671335\pi$
$888$	0	0
$889$	20.3923	0.683936
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.71281i	0.124245i
$894$	0	0
$895$	−49.7128	−1.66172
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 30.9282i	− 1.03151i
$900$	0	0
$901$	16.3923i	0.546107i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	62.6410	2.08226
$906$	0	0
$907$	7.85641i	0.260868i	0.991457	+	0.130434i	$0.0416370\pi$
$907$	7.85641i	0.260868i	−0.991457	+	0.130434i	$0.958363\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	29.6603	0.982688	0.491344	−	0.870966i	$-0.336506\pi$
$911$	29.6603	0.982688	0.491344	+	0.870966i	$0.336506\pi$
$912$	0	0
$913$	−76.7846	−2.54120
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.392305i	0.0129550i
$918$	0	0
$919$	26.9282	0.888279	0.444140	−	0.895958i	$-0.353509\pi$
$919$	26.9282	0.888279	0.444140	+	0.895958i	$0.353509\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 24.6410i	− 0.810192i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−38.1962	−1.25318	−0.626588	−	0.779351i	$-0.715549\pi$
$929$	−38.1962	−1.25318	−0.626588	+	0.779351i	$0.715549\pi$
$930$	0	0
$931$	0.535898i	0.0175634i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−80.1051	−2.61972
$936$	0	0
$937$	59.8564	1.95542	0.977712	−	0.209952i	$-0.0673306\pi$
$937$	59.8564	1.95542	0.977712	+	0.209952i	$0.0673306\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 18.0526i	− 0.588497i	−0.955729	−	0.294248i	$-0.904931\pi$
$941$	− 18.0526i	− 0.588497i	0.955729	−	0.294248i	$-0.0950693\pi$
$942$	0	0
$943$	−24.9282	−0.811774
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 24.0526i	− 0.781603i	−0.920475	−	0.390802i	$-0.872198\pi$
$947$	− 24.0526i	− 0.781603i	0.920475	−	0.390802i	$-0.127802\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.78461	0.0902024	0.0451012	−	0.998982i	$-0.485639\pi$
$953$	2.78461	0.0902024	0.0451012	+	0.998982i	$0.485639\pi$
$954$	0	0
$955$	− 25.3205i	− 0.819352i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.39230	−0.206418
$960$	0	0
$961$	48.7128	1.57138
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17.8564i	0.574818i
$966$	0	0
$967$	−10.2487	−0.329576	−0.164788	−	0.986329i	$-0.552694\pi$
$967$	−10.2487	−0.329576	−0.164788	+	0.986329i	$0.552694\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 44.4974i	− 1.42799i	−0.700151	−	0.713995i	$-0.746884\pi$
$971$	− 44.4974i	− 1.42799i	0.700151	−	0.713995i	$-0.253116\pi$
$972$	0	0
$973$	10.9282i	0.350342i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.7128	0.438712	0.219356	−	0.975645i	$-0.429604\pi$
$977$	13.7128	0.438712	0.219356	+	0.975645i	$0.429604\pi$
$978$	0	0
$979$	29.3205i	0.937088i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.4974	1.29167	0.645834	−	0.763478i	$-0.276510\pi$
$983$	40.4974	1.29167	0.645834	+	0.763478i	$0.276510\pi$
$984$	0	0
$985$	−30.2487	−0.963804
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 68.1051i	− 2.16562i
$990$	0	0
$991$	−38.9282	−1.23660	−0.618298	−	0.785944i	$-0.712177\pi$
$991$	−38.9282	−1.23660	−0.618298	+	0.785944i	$0.712177\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.0000i	0.507234i
$996$	0	0
$997$	25.4641i	0.806456i	0.915100	+	0.403228i	$0.132112\pi$
$997$	25.4641i	0.806456i	−0.915100	+	0.403228i	$0.867888\pi$
$998$	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.2017.4		4
3.2	odd	2		1344.2.c.g.673.1	yes	4
4.3	odd	2		4032.2.c.m.2017.4		4
8.3	odd	2		4032.2.c.m.2017.1		4
8.5	even	2	inner	4032.2.c.p.2017.1		4
12.11	even	2		1344.2.c.f.673.3	yes	4
24.5	odd	2		1344.2.c.g.673.4	yes	4
24.11	even	2		1344.2.c.f.673.2	✓	4
48.5	odd	4		5376.2.a.s.1.2		2
48.11	even	4		5376.2.a.be.1.2		2
48.29	odd	4		5376.2.a.x.1.1		2
48.35	even	4		5376.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.2.c.f.673.2	✓	4	24.11	even	2
1344.2.c.f.673.3	yes	4	12.11	even	2
1344.2.c.g.673.1	yes	4	3.2	odd	2
1344.2.c.g.673.4	yes	4	24.5	odd	2
4032.2.c.m.2017.1		4	8.3	odd	2
4032.2.c.m.2017.4		4	4.3	odd	2
4032.2.c.p.2017.1		4	8.5	even	2	inner
4032.2.c.p.2017.4		4	1.1	even	1	trivial
5376.2.a.p.1.1		2	48.35	even	4
5376.2.a.s.1.2		2	48.5	odd	4
5376.2.a.x.1.1		2	48.29	odd	4
5376.2.a.be.1.2		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)

\(f(q)\)	\(=\)	\(q+2.73205i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.73205i q^{5} +1.00000 q^{7} +6.19615i q^{11} +4.73205 q^{17} +0.535898i q^{19} +5.26795 q^{23} -2.46410 q^{25} -3.46410i q^{29} +8.92820 q^{31} +2.73205i q^{35} +10.0000i q^{37} -4.73205 q^{41} -12.9282i q^{43} +6.92820 q^{47} +1.00000 q^{49} +3.46410i q^{53} -16.9282 q^{55} +2.92820i q^{59} +5.46410i q^{61} -0.535898i q^{67} +3.80385 q^{71} -10.3923 q^{73} +6.19615i q^{77} +2.53590 q^{79} +12.3923i q^{83} +12.9282i q^{85} +4.73205 q^{89} -1.46410 q^{95} -14.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 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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