LMFDB - Embedded newform 4032.2.h.g.575.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(575,4032)]

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([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("4032.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.h (of order $2$, degree $1$, not 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:	$8$
Coefficient field:	8.0.5473632256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 49x^{4} + 256 $ x^8 + 49*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 2016)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.1
Root			$-1.10418 - 1.10418i$ of defining polynomial
Character	$\chi$	$=$	4032.575
Dual form			4032.2.h.g.575.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.62258i q^{5} -1.00000i q^{7} +O(q^{10})$	$q-3.62258i q^{5} -1.00000i q^{7} +5.03680 q^{11} -3.12311 q^{13} +6.45101i q^{17} +4.00000i q^{19} -5.03680 q^{23} -8.12311 q^{25} +8.65938i q^{29} +10.2462i q^{31} -3.62258 q^{35} +2.87689 q^{37} +9.27944i q^{41} +4.00000i q^{43} -1.24012 q^{47} -1.00000 q^{49} +8.65938i q^{53} -18.2462i q^{55} +10.0736 q^{59} +2.00000 q^{61} +11.3137i q^{65} -3.12311i q^{67} -9.45353 q^{71} -7.12311 q^{73} -5.03680i q^{77} -13.3693i q^{79} -8.83348 q^{83} +23.3693 q^{85} +3.62258i q^{89} +3.12311i q^{91} +14.4903 q^{95} -7.12311 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{13} - 32 q^{25} + 56 q^{37} - 8 q^{49} + 16 q^{61} - 24 q^{73} + 88 q^{85} - 24 q^{97}+O(q^{100}) $ 8 * q + 8 * q^13 - 32 * q^25 + 56 * q^37 - 8 * q^49 + 16 * q^61 - 24 * q^73 + 88 * q^85 - 24 * 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
$5$	− 3.62258i	− 1.62007i	−0.586383	−	0.810034i	$-0.699449\pi$
$5$	− 3.62258i	− 1.62007i	0.586383	−	0.810034i	$-0.300551\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.03680	1.51865	0.759326	−	0.650711i	$-0.225529\pi$
$11$	5.03680	1.51865	0.759326	+	0.650711i	$0.225529\pi$
$12$	0	0
$13$	−3.12311	−0.866194	−0.433097	−	0.901347i	$-0.642579\pi$
$13$	−3.12311	−0.866194	−0.433097	+	0.901347i	$0.642579\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.45101i	1.56460i	0.622902	+	0.782300i	$0.285954\pi$
$17$	6.45101i	1.56460i	−0.622902	+	0.782300i	$0.714046\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.03680	−1.05024	−0.525122	−	0.851027i	$-0.675980\pi$
$23$	−5.03680	−1.05024	−0.525122	+	0.851027i	$0.675980\pi$
$24$	0	0
$25$	−8.12311	−1.62462
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.65938i	1.60801i	0.594625	+	0.804003i	$0.297301\pi$
$29$	8.65938i	1.60801i	−0.594625	+	0.804003i	$0.702699\pi$
$30$	0	0
$31$	10.2462i	1.84027i	0.391597	+	0.920137i	$0.371923\pi$
$31$	10.2462i	1.84027i	−0.391597	+	0.920137i	$0.628077\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.62258	−0.612328
$36$	0	0
$37$	2.87689	0.472959	0.236479	−	0.971637i	$-0.424006\pi$
$37$	2.87689	0.472959	0.236479	+	0.971637i	$0.424006\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.27944i	1.44920i	0.689167	+	0.724602i	$0.257977\pi$
$41$	9.27944i	1.44920i	−0.689167	+	0.724602i	$0.742023\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.24012	−0.180889	−0.0904447	−	0.995901i	$-0.528829\pi$
$47$	−1.24012	−0.180889	−0.0904447	+	0.995901i	$0.528829\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.65938i	1.18946i	0.803927	+	0.594729i	$0.202740\pi$
$53$	8.65938i	1.18946i	−0.803927	+	0.594729i	$0.797260\pi$
$54$	0	0
$55$	− 18.2462i	− 2.46032i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.0736	1.31147	0.655735	−	0.754991i	$-0.272359\pi$
$59$	10.0736	1.31147	0.655735	+	0.754991i	$0.272359\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	11.3137i	1.40329i
$66$	0	0
$67$	− 3.12311i	− 0.381548i	−0.981634	−	0.190774i	$-0.938900\pi$
$67$	− 3.12311i	− 0.381548i	0.981634	−	0.190774i	$-0.0610998\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.45353	−1.12193	−0.560964	−	0.827840i	$-0.689569\pi$
$71$	−9.45353	−1.12193	−0.560964	+	0.827840i	$0.689569\pi$
$72$	0	0
$73$	−7.12311	−0.833696	−0.416848	−	0.908976i	$-0.636865\pi$
$73$	−7.12311	−0.833696	−0.416848	+	0.908976i	$0.636865\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.03680i	− 0.573996i
$78$	0	0
$79$	− 13.3693i	− 1.50417i	−0.659069	−	0.752083i	$-0.729049\pi$
$79$	− 13.3693i	− 1.50417i	0.659069	−	0.752083i	$-0.270951\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.83348	−0.969600	−0.484800	−	0.874625i	$-0.661108\pi$
$83$	−8.83348	−0.969600	−0.484800	+	0.874625i	$0.661108\pi$
$84$	0	0
$85$	23.3693	2.53476
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.62258i	0.383993i	0.981396	+	0.191997i	$0.0614962\pi$
$89$	3.62258i	0.383993i	−0.981396	+	0.191997i	$0.938504\pi$
$90$	0	0
$91$	3.12311i	0.327390i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	14.4903	1.48668
$96$	0	0
$97$	−7.12311	−0.723242	−0.361621	−	0.932325i	$-0.617777\pi$
$97$	−7.12311	−0.723242	−0.361621	+	0.932325i	$0.617777\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.794156i	0.0790214i	0.999219	+	0.0395107i	$0.0125799\pi$
$101$	0.794156i	0.0790214i	−0.999219	+	0.0395107i	$0.987420\pi$
$102$	0	0
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.1104	−1.46078	−0.730388	−	0.683032i	$-0.760661\pi$
$107$	−15.1104	−1.46078	−0.730388	+	0.683032i	$0.760661\pi$
$108$	0	0
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 14.6644i	− 1.37951i	−0.724041	−	0.689757i	$-0.757717\pi$
$113$	− 14.6644i	− 1.37951i	0.724041	−	0.689757i	$-0.242283\pi$
$114$	0	0
$115$	18.2462i	1.70147i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.45101	0.591363
$120$	0	0
$121$	14.3693	1.30630
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.3137i	1.01193i
$126$	0	0
$127$	5.36932i	0.476450i	0.971210	+	0.238225i	$0.0765655\pi$
$127$	5.36932i	0.476450i	−0.971210	+	0.238225i	$0.923434\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.3848i	1.57072i	0.619041	+	0.785359i	$0.287521\pi$
$137$	18.3848i	1.57072i	−0.619041	+	0.785359i	$0.712479\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−15.7304	−1.31545
$144$	0	0
$145$	31.3693	2.60508
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 10.2477i	− 0.839523i	−0.907634	−	0.419762i	$-0.862114\pi$
$149$	− 10.2477i	− 0.839523i	0.907634	−	0.419762i	$-0.137886\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	37.1177	2.98137
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.03680i	0.396955i
$162$	0	0
$163$	− 23.6155i	− 1.84971i	−0.380319	−	0.924855i	$-0.624186\pi$
$163$	− 23.6155i	− 1.84971i	0.380319	−	0.924855i	$-0.375814\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.41674	−0.341777	−0.170889	−	0.985290i	$-0.554664\pi$
$167$	−4.41674	−0.341777	−0.170889	+	0.985290i	$0.554664\pi$
$168$	0	0
$169$	−3.24621	−0.249709
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.27439i	0.248947i	0.992223	+	0.124474i	$0.0397242\pi$
$173$	3.27439i	0.248947i	−0.992223	+	0.124474i	$0.960276\pi$
$174$	0	0
$175$	8.12311i	0.614049i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.21342	0.613900	0.306950	−	0.951726i	$-0.400692\pi$
$179$	8.21342	0.613900	0.306950	+	0.951726i	$0.400692\pi$
$180$	0	0
$181$	−3.12311	−0.232139	−0.116069	−	0.993241i	$-0.537029\pi$
$181$	−3.12311	−0.232139	−0.116069	+	0.993241i	$0.537029\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 10.4218i	− 0.766225i
$186$	0	0
$187$	32.4924i	2.37608i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.6937	−0.773765	−0.386883	−	0.922129i	$-0.626448\pi$
$191$	−10.6937	−0.773765	−0.386883	+	0.922129i	$0.626448\pi$
$192$	0	0
$193$	1.12311	0.0808429	0.0404215	−	0.999183i	$-0.487130\pi$
$193$	1.12311	0.0808429	0.0404215	+	0.999183i	$0.487130\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.65433i	0.189113i	0.995519	+	0.0945566i	$0.0301433\pi$
$197$	2.65433i	0.189113i	−0.995519	+	0.0945566i	$0.969857\pi$
$198$	0	0
$199$	22.2462i	1.57699i	0.615040	+	0.788496i	$0.289140\pi$
$199$	22.2462i	1.57699i	−0.615040	+	0.788496i	$0.710860\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.65938	0.607769
$204$	0	0
$205$	33.6155	2.34781
$206$	0	0
$207$	0	0
$208$	0	0
$209$	20.1472i	1.39361i
$210$	0	0
$211$	− 16.4924i	− 1.13539i	−0.823241	−	0.567693i	$-0.807836\pi$
$211$	− 16.4924i	− 1.13539i	0.823241	−	0.567693i	$-0.192164\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14.4903	0.988232
$216$	0	0
$217$	10.2462	0.695558
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 20.1472i	− 1.35525i
$222$	0	0
$223$	14.2462i	0.953997i	0.878904	+	0.476998i	$0.158275\pi$
$223$	14.2462i	0.953997i	−0.878904	+	0.476998i	$0.841725\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.2107	−1.20869	−0.604343	−	0.796725i	$-0.706564\pi$
$227$	−18.2107	−1.20869	−0.604343	+	0.796725i	$0.706564\pi$
$228$	0	0
$229$	−19.1231	−1.26369	−0.631845	−	0.775095i	$-0.717702\pi$
$229$	−19.1231	−1.26369	−0.631845	+	0.775095i	$0.717702\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.83095i	0.381998i	0.981590	+	0.190999i	$0.0611728\pi$
$233$	5.83095i	0.381998i	−0.981590	+	0.190999i	$0.938827\pi$
$234$	0	0
$235$	4.49242i	0.293053i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.45353	−0.611498	−0.305749	−	0.952112i	$-0.598907\pi$
$239$	−9.45353	−0.611498	−0.305749	+	0.952112i	$0.598907\pi$
$240$	0	0
$241$	−0.876894	−0.0564857	−0.0282429	−	0.999601i	$-0.508991\pi$
$241$	−0.876894	−0.0564857	−0.0282429	+	0.999601i	$0.508991\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.62258i	0.231438i
$246$	0	0
$247$	− 12.4924i	− 0.794874i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.24012	0.0782754	0.0391377	−	0.999234i	$-0.487539\pi$
$251$	1.24012	0.0782754	0.0391377	+	0.999234i	$0.487539\pi$
$252$	0	0
$253$	−25.3693	−1.59496
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.8677i	0.677912i	0.940802	+	0.338956i	$0.110074\pi$
$257$	10.8677i	0.677912i	−0.940802	+	0.338956i	$0.889926\pi$
$258$	0	0
$259$	− 2.87689i	− 0.178762i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.55656	−0.157645	−0.0788223	−	0.996889i	$-0.525116\pi$
$263$	−2.55656	−0.157645	−0.0788223	+	0.996889i	$0.525116\pi$
$264$	0	0
$265$	31.3693	1.92700
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27.8383i	1.69733i	0.528929	+	0.848666i	$0.322594\pi$
$269$	27.8383i	1.69733i	−0.528929	+	0.848666i	$0.677406\pi$
$270$	0	0
$271$	− 4.00000i	− 0.242983i	−0.992592	−	0.121491i	$-0.961232\pi$
$271$	− 4.00000i	− 0.242983i	0.992592	−	0.121491i	$-0.0387677\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−40.9144	−2.46723
$276$	0	0
$277$	0.876894	0.0526875	0.0263437	−	0.999653i	$-0.491614\pi$
$277$	0.876894	0.0526875	0.0263437	+	0.999653i	$0.491614\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.41926i	0.442596i	0.975206	+	0.221298i	$0.0710294\pi$
$281$	7.41926i	0.442596i	−0.975206	+	0.221298i	$0.928971\pi$
$282$	0	0
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.27944	0.547748
$288$	0	0
$289$	−24.6155	−1.44797
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 0.794156i	− 0.0463951i	−0.999731	−	0.0231975i	$-0.992615\pi$
$293$	− 0.794156i	− 0.0463951i	0.999731	−	0.0231975i	$-0.00738467\pi$
$294$	0	0
$295$	− 36.4924i	− 2.12467i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	15.7304	0.909715
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 7.24517i	− 0.414857i
$306$	0	0
$307$	− 2.24621i	− 0.128198i	−0.997944	−	0.0640990i	$-0.979583\pi$
$307$	− 2.24621i	− 0.128198i	0.997944	−	0.0640990i	$-0.0204174\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.9071	1.07212	0.536061	−	0.844179i	$-0.319912\pi$
$311$	18.9071	1.07212	0.536061	+	0.844179i	$0.319912\pi$
$312$	0	0
$313$	3.75379	0.212177	0.106088	−	0.994357i	$-0.466167\pi$
$313$	3.75379	0.212177	0.106088	+	0.994357i	$0.466167\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.06602i	− 0.0598736i	−0.999552	−	0.0299368i	$-0.990469\pi$
$317$	− 1.06602i	− 0.0598736i	0.999552	−	0.0299368i	$-0.00953060\pi$
$318$	0	0
$319$	43.6155i	2.44200i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−25.8040	−1.43578
$324$	0	0
$325$	25.3693	1.40724
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.24012i	0.0683698i
$330$	0	0
$331$	− 32.4924i	− 1.78595i	−0.450111	−	0.892973i	$-0.648616\pi$
$331$	− 32.4924i	− 1.78595i	0.450111	−	0.892973i	$-0.351384\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	30.2462	1.64762	0.823808	−	0.566869i	$-0.191845\pi$
$337$	30.2462	1.64762	0.823808	+	0.566869i	$0.191845\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	51.6081i	2.79473i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.51703	0.403535	0.201768	−	0.979433i	$-0.435331\pi$
$347$	7.51703	0.403535	0.201768	+	0.979433i	$0.435331\pi$
$348$	0	0
$349$	−7.75379	−0.415051	−0.207525	−	0.978230i	$-0.566541\pi$
$349$	−7.75379	−0.415051	−0.207525	+	0.978230i	$0.566541\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.3530i	1.03006i	0.857173	+	0.515029i	$0.172219\pi$
$353$	19.3530i	1.03006i	−0.857173	+	0.515029i	$0.827781\pi$
$354$	0	0
$355$	34.2462i	1.81760i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.2475	1.22695	0.613477	−	0.789712i	$-0.289770\pi$
$359$	23.2475	1.22695	0.613477	+	0.789712i	$0.289770\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	25.8040i	1.35065i
$366$	0	0
$367$	3.50758i	0.183094i	0.995801	+	0.0915470i	$0.0291812\pi$
$367$	3.50758i	0.183094i	−0.995801	+	0.0915470i	$0.970819\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.65938	0.449573
$372$	0	0
$373$	−24.7386	−1.28092	−0.640459	−	0.767992i	$-0.721256\pi$
$373$	−24.7386	−1.28092	−0.640459	+	0.767992i	$0.721256\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 27.0442i	− 1.39284i
$378$	0	0
$379$	− 28.0000i	− 1.43826i	−0.694874	−	0.719132i	$-0.744540\pi$
$379$	− 28.0000i	− 1.43826i	0.694874	−	0.719132i	$-0.255460\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−28.2843	−1.44526	−0.722629	−	0.691236i	$-0.757067\pi$
$383$	−28.2843	−1.44526	−0.722629	+	0.691236i	$0.757067\pi$
$384$	0	0
$385$	−18.2462	−0.929913
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.3898i	1.23661i	0.785937	+	0.618307i	$0.212181\pi$
$389$	24.3898i	1.23661i	−0.785937	+	0.618307i	$0.787819\pi$
$390$	0	0
$391$	− 32.4924i	− 1.64321i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−48.4315	−2.43685
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 18.7330i	− 0.935480i	−0.883866	−	0.467740i	$-0.845068\pi$
$401$	− 18.7330i	− 0.935480i	0.883866	−	0.467740i	$-0.154932\pi$
$402$	0	0
$403$	− 32.0000i	− 1.59403i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.4903	0.718259
$408$	0	0
$409$	2.63068	0.130079	0.0650395	−	0.997883i	$-0.479283\pi$
$409$	2.63068	0.130079	0.0650395	+	0.997883i	$0.479283\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 10.0736i	− 0.495689i
$414$	0	0
$415$	32.0000i	1.57082i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13.2502	0.647315	0.323658	−	0.946174i	$-0.395087\pi$
$419$	13.2502	0.647315	0.323658	+	0.946174i	$0.395087\pi$
$420$	0	0
$421$	−24.8769	−1.21243	−0.606213	−	0.795302i	$-0.707312\pi$
$421$	−24.8769	−1.21243	−0.606213	+	0.795302i	$0.707312\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 52.4022i	− 2.54188i
$426$	0	0
$427$	− 2.00000i	− 0.0967868i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.9043	1.39227	0.696136	−	0.717910i	$-0.254901\pi$
$431$	28.9043	1.39227	0.696136	+	0.717910i	$0.254901\pi$
$432$	0	0
$433$	−20.7386	−0.996635	−0.498318	−	0.866995i	$-0.666049\pi$
$433$	−20.7386	−0.996635	−0.498318	+	0.866995i	$0.666049\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 20.1472i	− 0.963771i
$438$	0	0
$439$	32.9848i	1.57428i	0.616774	+	0.787140i	$0.288439\pi$
$439$	32.9848i	1.57428i	−0.616774	+	0.787140i	$0.711561\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.620058	−0.0294598	−0.0147299	−	0.999892i	$-0.504689\pi$
$443$	−0.620058	−0.0294598	−0.0147299	+	0.999892i	$0.504689\pi$
$444$	0	0
$445$	13.1231	0.622095
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.17915i	0.291612i	0.989313	+	0.145806i	$0.0465775\pi$
$449$	6.17915i	0.291612i	−0.989313	+	0.145806i	$0.953422\pi$
$450$	0	0
$451$	46.7386i	2.20084i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	11.3137	0.530395
$456$	0	0
$457$	−24.0000	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 12.1079i	− 0.563919i	−0.959426	−	0.281960i	$-0.909016\pi$
$461$	− 12.1079i	− 0.563919i	0.959426	−	0.281960i	$-0.0909845\pi$
$462$	0	0
$463$	− 33.8617i	− 1.57369i	−0.617152	−	0.786844i	$-0.711713\pi$
$463$	− 33.8617i	− 1.57369i	0.617152	−	0.786844i	$-0.288287\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.9071	0.874915	0.437457	−	0.899239i	$-0.355879\pi$
$467$	18.9071	0.874915	0.437457	+	0.899239i	$0.355879\pi$
$468$	0	0
$469$	−3.12311	−0.144212
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.1472i	0.926369i
$474$	0	0
$475$	− 32.4924i	− 1.49085i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.93651	0.0884812	0.0442406	−	0.999021i	$-0.485913\pi$
$479$	1.93651	0.0884812	0.0442406	+	0.999021i	$0.485913\pi$
$480$	0	0
$481$	−8.98485	−0.409674
$482$	0	0
$483$	0	0
$484$	0	0
$485$	25.8040i	1.17170i
$486$	0	0
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.55656	−0.115376	−0.0576881	−	0.998335i	$-0.518373\pi$
$491$	−2.55656	−0.115376	−0.0576881	+	0.998335i	$0.518373\pi$
$492$	0	0
$493$	−55.8617	−2.51589
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.45353i	0.424049i
$498$	0	0
$499$	36.9848i	1.65567i	0.560972	+	0.827835i	$0.310427\pi$
$499$	36.9848i	1.65567i	−0.560972	+	0.827835i	$0.689573\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33.9411	1.51336	0.756680	−	0.653785i	$-0.226820\pi$
$503$	33.9411	1.51336	0.756680	+	0.653785i	$0.226820\pi$
$504$	0	0
$505$	2.87689	0.128020
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17.4165i	0.771974i	0.922504	+	0.385987i	$0.126139\pi$
$509$	17.4165i	0.771974i	−0.922504	+	0.385987i	$0.873861\pi$
$510$	0	0
$511$	7.12311i	0.315108i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−43.4710	−1.91556
$516$	0	0
$517$	−6.24621	−0.274708
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.4611i	0.808796i	0.914583	+	0.404398i	$0.132519\pi$
$521$	18.4611i	0.808796i	−0.914583	+	0.404398i	$0.867481\pi$
$522$	0	0
$523$	− 4.49242i	− 0.196440i	−0.995165	−	0.0982200i	$-0.968685\pi$
$523$	− 4.49242i	− 0.196440i	0.995165	−	0.0982200i	$-0.0313149\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−66.0984	−2.87929
$528$	0	0
$529$	2.36932	0.103014
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 28.9807i	− 1.25529i
$534$	0	0
$535$	54.7386i	2.36656i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.03680	−0.216950
$540$	0	0
$541$	−15.1231	−0.650193	−0.325097	−	0.945681i	$-0.605397\pi$
$541$	−15.1231	−0.650193	−0.325097	+	0.945681i	$0.605397\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 43.4710i	− 1.86209i
$546$	0	0
$547$	19.1231i	0.817645i	0.912614	+	0.408822i	$0.134060\pi$
$547$	19.1231i	0.817645i	−0.912614	+	0.408822i	$0.865940\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−34.6375	−1.47561
$552$	0	0
$553$	−13.3693	−0.568521
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 18.3848i	− 0.778988i	−0.921029	−	0.389494i	$-0.872650\pi$
$557$	− 18.3848i	− 0.778988i	0.921029	−	0.389494i	$-0.127350\pi$
$558$	0	0
$559$	− 12.4924i	− 0.528373i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.4268	0.692309	0.346154	−	0.938178i	$-0.387487\pi$
$563$	16.4268	0.692309	0.346154	+	0.938178i	$0.387487\pi$
$564$	0	0
$565$	−53.1231	−2.23491
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.1152i	1.43018i	0.699030	+	0.715092i	$0.253615\pi$
$569$	34.1152i	1.43018i	−0.699030	+	0.715092i	$0.746385\pi$
$570$	0	0
$571$	− 43.1231i	− 1.80465i	−0.431061	−	0.902323i	$-0.641861\pi$
$571$	− 43.1231i	− 1.80465i	0.431061	−	0.902323i	$-0.358139\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	40.9144	1.70625
$576$	0	0
$577$	26.9848	1.12339	0.561697	−	0.827343i	$-0.310149\pi$
$577$	26.9848	1.12339	0.561697	+	0.827343i	$0.310149\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.83348i	0.366474i
$582$	0	0
$583$	43.6155i	1.80637i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	47.1913	1.94780	0.973898	−	0.226988i	$-0.0728878\pi$
$587$	47.1913	1.94780	0.973898	+	0.226988i	$0.0728878\pi$
$588$	0	0
$589$	−40.9848	−1.68875
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 12.9998i	− 0.533837i	−0.963719	−	0.266919i	$-0.913995\pi$
$593$	− 12.9998i	− 0.533837i	0.963719	−	0.266919i	$-0.0860055\pi$
$594$	0	0
$595$	− 23.3693i	− 0.958049i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	43.3947	1.77306	0.886529	−	0.462673i	$-0.153110\pi$
$599$	43.3947	1.77306	0.886529	+	0.462673i	$0.153110\pi$
$600$	0	0
$601$	30.9848	1.26390	0.631949	−	0.775010i	$-0.282255\pi$
$601$	30.9848	1.26390	0.631949	+	0.775010i	$0.282255\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 52.0540i	− 2.11630i
$606$	0	0
$607$	4.49242i	0.182342i	0.995835	+	0.0911709i	$0.0290610\pi$
$607$	4.49242i	0.182342i	−0.995835	+	0.0911709i	$0.970939\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.87301	0.156685
$612$	0	0
$613$	38.4924	1.55469	0.777347	−	0.629072i	$-0.216565\pi$
$613$	38.4924	1.55469	0.777347	+	0.629072i	$0.216565\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.3999i	1.46541i	0.680549	+	0.732703i	$0.261741\pi$
$617$	36.3999i	1.46541i	−0.680549	+	0.732703i	$0.738259\pi$
$618$	0	0
$619$	− 26.7386i	− 1.07472i	−0.843354	−	0.537358i	$-0.819422\pi$
$619$	− 26.7386i	− 1.07472i	0.843354	−	0.537358i	$-0.180578\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.62258	0.145136
$624$	0	0
$625$	0.369317	0.0147727
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.5589i	0.739991i
$630$	0	0
$631$	− 18.6307i	− 0.741676i	−0.928698	−	0.370838i	$-0.879071\pi$
$631$	− 18.6307i	− 0.741676i	0.928698	−	0.370838i	$-0.120929\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	19.4508	0.771881
$636$	0	0
$637$	3.12311	0.123742
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 5.48276i	− 0.216556i	−0.994121	−	0.108278i	$-0.965466\pi$
$641$	− 5.48276i	− 0.216556i	0.994121	−	0.108278i	$-0.0345336\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.89697	−0.271148	−0.135574	−	0.990767i	$-0.543288\pi$
$647$	−6.89697	−0.271148	−0.135574	+	0.990767i	$0.543288\pi$
$648$	0	0
$649$	50.7386	1.99167
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 21.2132i	− 0.830137i	−0.909790	−	0.415068i	$-0.863758\pi$
$653$	− 21.2132i	− 0.830137i	0.909790	−	0.415068i	$-0.136242\pi$
$654$	0	0
$655$	− 20.4924i	− 0.800705i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.75714	0.341130	0.170565	−	0.985346i	$-0.445441\pi$
$659$	8.75714	0.341130	0.170565	+	0.985346i	$0.445441\pi$
$660$	0	0
$661$	40.7386	1.58455	0.792275	−	0.610165i	$-0.208897\pi$
$661$	40.7386	1.58455	0.792275	+	0.610165i	$0.208897\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 14.4903i	− 0.561911i
$666$	0	0
$667$	− 43.6155i	− 1.68880i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.0736	0.388887
$672$	0	0
$673$	30.1080	1.16058	0.580288	−	0.814411i	$-0.302940\pi$
$673$	30.1080	1.16058	0.580288	+	0.814411i	$0.302940\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 38.8038i	− 1.49135i	−0.666309	−	0.745676i	$-0.732127\pi$
$677$	− 38.8038i	− 1.49135i	0.666309	−	0.745676i	$-0.267873\pi$
$678$	0	0
$679$	7.12311i	0.273360i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18.2870	0.699733	0.349867	−	0.936800i	$-0.386227\pi$
$683$	18.2870	0.699733	0.349867	+	0.936800i	$0.386227\pi$
$684$	0	0
$685$	66.6004	2.54467
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 27.0442i	− 1.03030i
$690$	0	0
$691$	30.2462i	1.15062i	0.817935	+	0.575310i	$0.195119\pi$
$691$	30.2462i	1.15062i	−0.817935	+	0.575310i	$0.804881\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−59.8617	−2.26743
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19.9731i	0.754373i	0.926137	+	0.377187i	$0.123108\pi$
$701$	19.9731i	0.754373i	−0.926137	+	0.377187i	$0.876892\pi$
$702$	0	0
$703$	11.5076i	0.434017i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.794156	0.0298673
$708$	0	0
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 51.6081i	− 1.93274i
$714$	0	0
$715$	56.9848i	2.13111i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−45.9512	−1.71369	−0.856846	−	0.515573i	$-0.827579\pi$
$719$	−45.9512	−1.71369	−0.856846	+	0.515573i	$0.827579\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 70.3411i	− 2.61240i
$726$	0	0
$727$	20.9848i	0.778285i	0.921178	+	0.389142i	$0.127229\pi$
$727$	20.9848i	0.778285i	−0.921178	+	0.389142i	$0.872771\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−25.8040	−0.954397
$732$	0	0
$733$	42.3542	1.56439	0.782193	−	0.623036i	$-0.214101\pi$
$733$	42.3542	1.56439	0.782193	+	0.623036i	$0.214101\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 15.7304i	− 0.579438i
$738$	0	0
$739$	31.6155i	1.16300i	0.813548	+	0.581498i	$0.197533\pi$
$739$	31.6155i	1.16300i	−0.813548	+	0.581498i	$0.802467\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.0810	1.17694	0.588468	−	0.808521i	$-0.299731\pi$
$743$	32.0810	1.17694	0.588468	+	0.808521i	$0.299731\pi$
$744$	0	0
$745$	−37.1231	−1.36009
$746$	0	0
$747$	0	0
$748$	0	0
$749$	15.1104i	0.552122i
$750$	0	0
$751$	3.50758i	0.127993i	0.997950	+	0.0639967i	$0.0203847\pi$
$751$	3.50758i	0.127993i	−0.997950	+	0.0639967i	$0.979615\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	16.4924	0.599427	0.299714	−	0.954029i	$-0.403109\pi$
$757$	16.4924	0.599427	0.299714	+	0.954029i	$0.403109\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 44.8089i	− 1.62432i	−0.583434	−	0.812160i	$-0.698292\pi$
$761$	− 44.8089i	− 1.62432i	0.583434	−	0.812160i	$-0.301708\pi$
$762$	0	0
$763$	− 12.0000i	− 0.434429i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31.4609	−1.13599
$768$	0	0
$769$	40.2462	1.45132	0.725658	−	0.688056i	$-0.241536\pi$
$769$	40.2462	1.45132	0.725658	+	0.688056i	$0.241536\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 20.9413i	− 0.753208i	−0.926374	−	0.376604i	$-0.877092\pi$
$773$	− 20.9413i	− 0.753208i	0.926374	−	0.376604i	$-0.122908\pi$
$774$	0	0
$775$	− 83.2311i	− 2.98975i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−37.1177	−1.32988
$780$	0	0
$781$	−47.6155	−1.70382
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 50.7162i	− 1.81014i
$786$	0	0
$787$	− 2.73863i	− 0.0976218i	−0.998808	−	0.0488109i	$-0.984457\pi$
$787$	− 2.73863i	− 0.0976218i	0.998808	−	0.0488109i	$-0.0155432\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14.6644	−0.521407
$792$	0	0
$793$	−6.24621	−0.221809
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.62763i	0.341028i	0.985355	+	0.170514i	$0.0545428\pi$
$797$	9.62763i	0.341028i	−0.985355	+	0.170514i	$0.945457\pi$
$798$	0	0
$799$	− 8.00000i	− 0.283020i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−35.8776	−1.26609
$804$	0	0
$805$	18.2462	0.643094
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 30.3949i	− 1.06863i	−0.845286	−	0.534313i	$-0.820570\pi$
$809$	− 30.3949i	− 1.06863i	0.845286	−	0.534313i	$-0.179430\pi$
$810$	0	0
$811$	22.2462i	0.781170i	0.920567	+	0.390585i	$0.127727\pi$
$811$	22.2462i	0.781170i	−0.920567	+	0.390585i	$0.872273\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−85.5492	−2.99666
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 11.1396i	− 0.388775i	−0.980925	−	0.194388i	$-0.937728\pi$
$821$	− 11.1396i	− 0.388775i	0.980925	−	0.194388i	$-0.0622719\pi$
$822$	0	0
$823$	− 29.3693i	− 1.02375i	−0.859060	−	0.511875i	$-0.828951\pi$
$823$	− 29.3693i	− 1.02375i	0.859060	−	0.511875i	$-0.171049\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.7277	0.894640	0.447320	−	0.894374i	$-0.352379\pi$
$827$	25.7277	0.894640	0.447320	+	0.894374i	$0.352379\pi$
$828$	0	0
$829$	−23.6155	−0.820201	−0.410101	−	0.912040i	$-0.634506\pi$
$829$	−23.6155	−0.820201	−0.410101	+	0.912040i	$0.634506\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 6.45101i	− 0.223514i
$834$	0	0
$835$	16.0000i	0.553703i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−21.3873	−0.738372	−0.369186	−	0.929356i	$-0.620363\pi$
$839$	−21.3873	−0.738372	−0.369186	+	0.929356i	$0.620363\pi$
$840$	0	0
$841$	−45.9848	−1.58568
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.7597i	0.404545i
$846$	0	0
$847$	− 14.3693i	− 0.493736i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14.4903	−0.496722
$852$	0	0
$853$	−32.2462	−1.10409	−0.552045	−	0.833815i	$-0.686152\pi$
$853$	−32.2462	−1.10409	−0.552045	+	0.833815i	$0.686152\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.2206i	1.47639i	0.674589	+	0.738193i	$0.264321\pi$
$857$	43.2206i	1.47639i	−0.674589	+	0.738193i	$0.735679\pi$
$858$	0	0
$859$	− 51.2311i	− 1.74798i	−0.485943	−	0.873991i	$-0.661524\pi$
$859$	− 51.2311i	− 1.74798i	0.485943	−	0.873991i	$-0.338476\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.1205	0.923192	0.461596	−	0.887090i	$-0.347277\pi$
$863$	27.1205	0.923192	0.461596	+	0.887090i	$0.347277\pi$
$864$	0	0
$865$	11.8617	0.403311
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 67.3385i	− 2.28430i
$870$	0	0
$871$	9.75379i	0.330495i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.3137	0.382473
$876$	0	0
$877$	40.2462	1.35902	0.679509	−	0.733667i	$-0.262193\pi$
$877$	40.2462	1.35902	0.679509	+	0.733667i	$0.262193\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38.6083i	1.30075i	0.759615	+	0.650373i	$0.225387\pi$
$881$	38.6083i	1.30075i	−0.759615	+	0.650373i	$0.774613\pi$
$882$	0	0
$883$	36.9848i	1.24464i	0.782763	+	0.622320i	$0.213810\pi$
$883$	36.9848i	1.24464i	−0.782763	+	0.622320i	$0.786190\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.93651	0.0650215	0.0325108	−	0.999471i	$-0.489650\pi$
$887$	1.93651	0.0650215	0.0325108	+	0.999471i	$0.489650\pi$
$888$	0	0
$889$	5.36932	0.180081
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 4.96046i	− 0.165996i
$894$	0	0
$895$	− 29.7538i	− 0.994559i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−88.7258	−2.95917
$900$	0	0
$901$	−55.8617	−1.86102
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.3137i	0.376080i
$906$	0	0
$907$	− 8.49242i	− 0.281986i	−0.990011	−	0.140993i	$-0.954970\pi$
$907$	− 8.49242i	− 0.281986i	0.990011	−	0.140993i	$-0.0450295\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	11.3900	0.377369	0.188684	−	0.982038i	$-0.439578\pi$
$911$	11.3900	0.377369	0.188684	+	0.982038i	$0.439578\pi$
$912$	0	0
$913$	−44.4924	−1.47248
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.65685i	− 0.186806i
$918$	0	0
$919$	32.9848i	1.08807i	0.839063	+	0.544035i	$0.183104\pi$
$919$	32.9848i	1.08807i	−0.839063	+	0.544035i	$0.816896\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	29.5244	0.971807
$924$	0	0
$925$	−23.3693	−0.768378
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.0048i	0.623528i	0.950160	+	0.311764i	$0.100920\pi$
$929$	19.0048i	0.623528i	−0.950160	+	0.311764i	$0.899080\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	117.706	3.84941
$936$	0	0
$937$	−60.7386	−1.98424	−0.992122	−	0.125273i	$-0.960019\pi$
$937$	−60.7386	−1.98424	−0.992122	+	0.125273i	$0.960019\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 24.6617i	− 0.803948i	−0.915651	−	0.401974i	$-0.868324\pi$
$941$	− 24.6617i	− 0.803948i	0.915651	−	0.401974i	$-0.131676\pi$
$942$	0	0
$943$	− 46.7386i	− 1.52202i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.620058	−0.0201492	−0.0100746	−	0.999949i	$-0.503207\pi$
$947$	−0.620058	−0.0201492	−0.0100746	+	0.999949i	$0.503207\pi$
$948$	0	0
$949$	22.2462	0.722143
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 57.0908i	− 1.84935i	−0.380753	−	0.924677i	$-0.624335\pi$
$953$	− 57.0908i	− 1.84935i	0.380753	−	0.924677i	$-0.375665\pi$
$954$	0	0
$955$	38.7386i	1.25355i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18.3848	0.593675
$960$	0	0
$961$	−73.9848	−2.38661
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 4.06854i	− 0.130971i
$966$	0	0
$967$	− 24.8769i	− 0.799987i	−0.916518	−	0.399993i	$-0.869012\pi$
$967$	− 24.8769i	− 0.799987i	0.916518	−	0.399993i	$-0.130988\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.17662	0.101943	0.0509713	−	0.998700i	$-0.483768\pi$
$971$	3.17662	0.101943	0.0509713	+	0.998700i	$0.483768\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.06602i	0.0341049i	0.999855	+	0.0170525i	$0.00542823\pi$
$977$	1.06602i	0.0341049i	−0.999855	+	0.0170525i	$0.994572\pi$
$978$	0	0
$979$	18.2462i	0.583151i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−23.3238	−0.743914	−0.371957	−	0.928250i	$-0.621313\pi$
$983$	−23.3238	−0.743914	−0.371957	+	0.928250i	$0.621313\pi$
$984$	0	0
$985$	9.61553	0.306376
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 20.1472i	− 0.640643i
$990$	0	0
$991$	24.0000i	0.762385i	0.924496	+	0.381193i	$0.124487\pi$
$991$	24.0000i	0.762385i	−0.924496	+	0.381193i	$0.875513\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	80.5887	2.55483
$996$	0	0
$997$	16.2462	0.514523	0.257261	−	0.966342i	$-0.417180\pi$
$997$	16.2462	0.514523	0.257261	+	0.966342i	$0.417180\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.h.g.575.1		8
3.2	odd	2	inner	4032.2.h.g.575.7		8
4.3	odd	2	inner	4032.2.h.g.575.2		8
8.3	odd	2		2016.2.h.e.575.8	yes	8
8.5	even	2		2016.2.h.e.575.7	yes	8
12.11	even	2	inner	4032.2.h.g.575.8		8
24.5	odd	2		2016.2.h.e.575.1	✓	8
24.11	even	2		2016.2.h.e.575.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.h.e.575.1	✓	8	24.5	odd	2
2016.2.h.e.575.2	yes	8	24.11	even	2
2016.2.h.e.575.7	yes	8	8.5	even	2
2016.2.h.e.575.8	yes	8	8.3	odd	2
4032.2.h.g.575.1		8	1.1	even	1	trivial
4032.2.h.g.575.2		8	4.3	odd	2	inner
4032.2.h.g.575.7		8	3.2	odd	2	inner
4032.2.h.g.575.8		8	12.11	even	2	inner

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.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.62258i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-3.62258i q^{5} -1.00000i q^{7} +5.03680 q^{11} -3.12311 q^{13} +6.45101i q^{17} +4.00000i q^{19} -5.03680 q^{23} -8.12311 q^{25} +8.65938i q^{29} +10.2462i q^{31} -3.62258 q^{35} +2.87689 q^{37} +9.27944i q^{41} +4.00000i q^{43} -1.24012 q^{47} -1.00000 q^{49} +8.65938i q^{53} -18.2462i q^{55} +10.0736 q^{59} +2.00000 q^{61} +11.3137i q^{65} -3.12311i q^{67} -9.45353 q^{71} -7.12311 q^{73} -5.03680i q^{77} -13.3693i q^{79} -8.83348 q^{83} +23.3693 q^{85} +3.62258i q^{89} +3.12311i q^{91} +14.4903 q^{95} -7.12311 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{13} - 32 q^{25} + 56 q^{37} - 8 q^{49} + 16 q^{61} - 24 q^{73} + 88 q^{85} - 24 q^{97}+O(q^{100}) \) 8 * q + 8 * q^13 - 32 * q^25 + 56 * q^37 - 8 * q^49 + 16 * q^61 - 24 * q^73 + 88 * q^85 - 24 * q^97

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