LMFDB - Embedded newform 4032.2.h.h.575.11

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:	$12$
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.11
Root			$-1.35489 - 0.405301i$ of defining polynomial
Character	$\chi$	$=$	4032.575
Dual form			4032.2.h.h.575.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.31339i q^{5} -1.00000i q^{7} +O(q^{10})$	$q+3.31339i q^{5} -1.00000i q^{7} +4.72761 q^{11} +4.97858 q^{13} -0.484966i q^{17} +2.29273i q^{19} +7.97002 q^{23} -5.97858 q^{25} +1.41421i q^{29} -7.66442i q^{31} +3.31339 q^{35} +2.39312 q^{37} -6.55580i q^{41} -5.37169i q^{43} +6.21280 q^{47} -1.00000 q^{49} -1.00023i q^{53} +15.6644i q^{55} -1.38392 q^{59} -13.6644 q^{61} +16.4960i q^{65} -3.27131i q^{67} +3.34369 q^{71} -2.10038 q^{73} -4.72761i q^{77} -12.0575i q^{79} +3.24241 q^{83} +1.60688 q^{85} -5.72784i q^{89} -4.97858i q^{91} -7.59672 q^{95} +12.0575 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 12 q^{25} - 8 q^{37} - 12 q^{49} - 56 q^{61} + 56 q^{85}+O(q^{100}) $ 12 * q - 12 * q^25 - 8 * q^37 - 12 * q^49 - 56 * q^61 + 56 * q^85

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
$5$	3.31339i	1.48179i	0.671618	+	0.740897i	$0.265600\pi$
$5$	3.31339i	1.48179i	−0.671618	+	0.740897i	$0.734400\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.72761	1.42543	0.712714	−	0.701455i	$-0.247466\pi$
$11$	4.72761	1.42543	0.712714	+	0.701455i	$0.247466\pi$
$12$	0	0
$13$	4.97858	1.38081	0.690404	−	0.723424i	$-0.257433\pi$
$13$	4.97858	1.38081	0.690404	+	0.723424i	$0.257433\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.484966i	− 0.117622i	−0.998269	−	0.0588108i	$-0.981269\pi$
$17$	− 0.484966i	− 0.117622i	0.998269	−	0.0588108i	$-0.0187309\pi$
$18$	0	0
$19$	2.29273i	0.525989i	0.964797	+	0.262994i	$0.0847100\pi$
$19$	2.29273i	0.525989i	−0.964797	+	0.262994i	$0.915290\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.97002	1.66186	0.830932	−	0.556374i	$-0.187808\pi$
$23$	7.97002	1.66186	0.830932	+	0.556374i	$0.187808\pi$
$24$	0	0
$25$	−5.97858	−1.19572
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.41421i	0.262613i	0.991342	+	0.131306i	$0.0419172\pi$
$29$	1.41421i	0.262613i	−0.991342	+	0.131306i	$0.958083\pi$
$30$	0	0
$31$	− 7.66442i	− 1.37657i	−0.725440	−	0.688286i	$-0.758364\pi$
$31$	− 7.66442i	− 1.37657i	0.725440	−	0.688286i	$-0.241636\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.31339	0.560066
$36$	0	0
$37$	2.39312	0.393426	0.196713	−	0.980461i	$-0.436973\pi$
$37$	2.39312	0.393426	0.196713	+	0.980461i	$0.436973\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.55580i	− 1.02384i	−0.859032	−	0.511922i	$-0.828934\pi$
$41$	− 6.55580i	− 1.02384i	0.859032	−	0.511922i	$-0.171066\pi$
$42$	0	0
$43$	− 5.37169i	− 0.819175i	−0.912271	−	0.409588i	$-0.865673\pi$
$43$	− 5.37169i	− 0.819175i	0.912271	−	0.409588i	$-0.134327\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.21280	0.906230	0.453115	−	0.891452i	$-0.350313\pi$
$47$	6.21280	0.906230	0.453115	+	0.891452i	$0.350313\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.00023i	− 0.137392i	−0.997638	−	0.0686960i	$-0.978116\pi$
$53$	− 1.00023i	− 0.137392i	0.997638	−	0.0686960i	$-0.0218839\pi$
$54$	0	0
$55$	15.6644i	2.11219i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.38392	−0.180171	−0.0900853	−	0.995934i	$-0.528714\pi$
$59$	−1.38392	−0.180171	−0.0900853	+	0.995934i	$0.528714\pi$
$60$	0	0
$61$	−13.6644	−1.74955	−0.874775	−	0.484529i	$-0.838991\pi$
$61$	−13.6644	−1.74955	−0.874775	+	0.484529i	$0.838991\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.4960i	2.04608i
$66$	0	0
$67$	− 3.27131i	− 0.399654i	−0.979831	−	0.199827i	$-0.935962\pi$
$67$	− 3.27131i	− 0.399654i	0.979831	−	0.199827i	$-0.0640380\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.34369	0.396823	0.198412	−	0.980119i	$-0.436422\pi$
$71$	3.34369	0.396823	0.198412	+	0.980119i	$0.436422\pi$
$72$	0	0
$73$	−2.10038	−0.245831	−0.122916	−	0.992417i	$-0.539224\pi$
$73$	−2.10038	−0.245831	−0.122916	+	0.992417i	$0.539224\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.72761i	− 0.538761i
$78$	0	0
$79$	− 12.0575i	− 1.35658i	−0.734795	−	0.678290i	$-0.762722\pi$
$79$	− 12.0575i	− 1.35658i	0.734795	−	0.678290i	$-0.237278\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.24241	0.355901	0.177950	−	0.984039i	$-0.443053\pi$
$83$	3.24241	0.355901	0.177950	+	0.984039i	$0.443053\pi$
$84$	0	0
$85$	1.60688	0.174291
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 5.72784i	− 0.607149i	−0.952808	−	0.303575i	$-0.901820\pi$
$89$	− 5.72784i	− 0.607149i	0.952808	−	0.303575i	$-0.0981802\pi$
$90$	0	0
$91$	− 4.97858i	− 0.521897i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.59672	−0.779407
$96$	0	0
$97$	12.0575	1.22426	0.612129	−	0.790758i	$-0.290313\pi$
$97$	12.0575	1.22426	0.612129	+	0.790758i	$0.290313\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 14.5665i	− 1.44942i	−0.689053	−	0.724711i	$-0.741973\pi$
$101$	− 14.5665i	− 1.44942i	0.689053	−	0.724711i	$-0.258027\pi$
$102$	0	0
$103$	5.70727i	0.562354i	0.959656	+	0.281177i	$0.0907248\pi$
$103$	5.70727i	0.562354i	−0.959656	+	0.281177i	$0.909275\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0413	1.55077	0.775386	−	0.631487i	$-0.217555\pi$
$107$	16.0413	1.55077	0.775386	+	0.631487i	$0.217555\pi$
$108$	0	0
$109$	−1.37169	−0.131384	−0.0656921	−	0.997840i	$-0.520926\pi$
$109$	−1.37169	−0.131384	−0.0656921	+	0.997840i	$0.520926\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 11.2834i	− 1.06145i	−0.847543	−	0.530727i	$-0.821919\pi$
$113$	− 11.2834i	− 1.06145i	0.847543	−	0.530727i	$-0.178081\pi$
$114$	0	0
$115$	26.4078i	2.46254i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.484966	−0.0444568
$120$	0	0
$121$	11.3503	1.03184
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 3.24241i	− 0.290010i
$126$	0	0
$127$	16.6430i	1.47683i	0.674348	+	0.738414i	$0.264425\pi$
$127$	16.6430i	1.47683i	−0.674348	+	0.738414i	$0.735575\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.59672	−0.663729	−0.331864	−	0.943327i	$-0.607678\pi$
$131$	−7.59672	−0.663729	−0.331864	+	0.943327i	$0.607678\pi$
$132$	0	0
$133$	2.29273	0.198805
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.42492i	0.805225i	0.915370	+	0.402613i	$0.131898\pi$
$137$	9.42492i	0.805225i	−0.915370	+	0.402613i	$0.868102\pi$
$138$	0	0
$139$	16.0000i	1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$	16.0000i	1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	23.5368	1.96824
$144$	0	0
$145$	−4.68585	−0.389138
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.4661i	1.51281i	0.654106	+	0.756403i	$0.273045\pi$
$149$	18.4661i	1.51281i	−0.654106	+	0.756403i	$0.726955\pi$
$150$	0	0
$151$	23.3288i	1.89847i	0.314561	+	0.949237i	$0.398143\pi$
$151$	23.3288i	1.89847i	−0.314561	+	0.949237i	$0.601857\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	25.3953	2.03980
$156$	0	0
$157$	−22.2499	−1.77573	−0.887867	−	0.460100i	$-0.847814\pi$
$157$	−22.2499	−1.77573	−0.887867	+	0.460100i	$0.847814\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 7.97002i	− 0.628125i
$162$	0	0
$163$	16.6430i	1.30358i	0.758399	+	0.651790i	$0.225982\pi$
$163$	16.6430i	1.30358i	−0.758399	+	0.651790i	$0.774018\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.98064	−0.694943	−0.347471	−	0.937691i	$-0.612960\pi$
$167$	−8.98064	−0.694943	−0.347471	+	0.937691i	$0.612960\pi$
$168$	0	0
$169$	11.7862	0.906633
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.75744i	0.209645i	0.994491	+	0.104822i	$0.0334274\pi$
$173$	2.75744i	0.209645i	−0.994491	+	0.104822i	$0.966573\pi$
$174$	0	0
$175$	5.97858i	0.451938i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.72761	0.353358	0.176679	−	0.984269i	$-0.443465\pi$
$179$	4.72761	0.353358	0.176679	+	0.984269i	$0.443465\pi$
$180$	0	0
$181$	8.39312	0.623855	0.311928	−	0.950106i	$-0.399025\pi$
$181$	8.39312	0.623855	0.311928	+	0.950106i	$0.399025\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.92933i	0.582976i
$186$	0	0
$187$	− 2.29273i	− 0.167661i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.48520	0.107465	0.0537325	−	0.998555i	$-0.482888\pi$
$191$	1.48520	0.107465	0.0537325	+	0.998555i	$0.482888\pi$
$192$	0	0
$193$	−8.35027	−0.601066	−0.300533	−	0.953771i	$-0.597164\pi$
$193$	−8.35027	−0.601066	−0.300533	+	0.953771i	$0.597164\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.82866i	− 0.272780i	−0.990655	−	0.136390i	$-0.956450\pi$
$197$	− 3.82866i	− 0.272780i	0.990655	−	0.136390i	$-0.0435501\pi$
$198$	0	0
$199$	6.04285i	0.428366i	0.976794	+	0.214183i	$0.0687089\pi$
$199$	6.04285i	0.428366i	−0.976794	+	0.214183i	$0.931291\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.41421	0.0992583
$204$	0	0
$205$	21.7220	1.51713
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.8391i	0.749758i
$210$	0	0
$211$	− 9.95715i	− 0.685479i	−0.939431	−	0.342739i	$-0.888645\pi$
$211$	− 9.95715i	− 0.685479i	0.939431	−	0.342739i	$-0.111355\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	17.7985	1.21385
$216$	0	0
$217$	−7.66442	−0.520295
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.41444i	− 0.162413i
$222$	0	0
$223$	0.786230i	0.0526499i	0.999653	+	0.0263249i	$0.00838046\pi$
$223$	0.786230i	0.0526499i	−0.999653	+	0.0263249i	$0.991620\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.2943	−1.34698	−0.673492	−	0.739195i	$-0.735206\pi$
$227$	−20.2943	−1.34698	−0.673492	+	0.739195i	$0.735206\pi$
$228$	0	0
$229$	4.97858	0.328994	0.164497	−	0.986378i	$-0.447400\pi$
$229$	4.97858	0.328994	0.164497	+	0.986378i	$0.447400\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 0.383688i	− 0.0251362i	−0.999921	−	0.0125681i	$-0.995999\pi$
$233$	− 0.383688i	− 0.0251362i	0.999921	−	0.0125681i	$-0.00400066\pi$
$234$	0	0
$235$	20.5855i	1.34285i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.48520	−0.0960693	−0.0480347	−	0.998846i	$-0.515296\pi$
$239$	−1.48520	−0.0960693	−0.0480347	+	0.998846i	$0.515296\pi$
$240$	0	0
$241$	−23.4721	−1.51197	−0.755985	−	0.654589i	$-0.772842\pi$
$241$	−23.4721	−1.51197	−0.755985	+	0.654589i	$0.772842\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3.31339i	− 0.211685i
$246$	0	0
$247$	11.4145i	0.726290i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.62633	−0.292011	−0.146006	−	0.989284i	$-0.546642\pi$
$251$	−4.62633	−0.292011	−0.146006	+	0.989284i	$0.546642\pi$
$252$	0	0
$253$	37.6791	2.36887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.78797i	0.236287i	0.992997	+	0.118144i	$0.0376943\pi$
$257$	3.78797i	0.236287i	−0.992997	+	0.118144i	$0.962306\pi$
$258$	0	0
$259$	− 2.39312i	− 0.148701i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.7989	0.789214	0.394607	−	0.918850i	$-0.370881\pi$
$263$	12.7989	0.789214	0.394607	+	0.918850i	$0.370881\pi$
$264$	0	0
$265$	3.31415	0.203587
$266$	0	0
$267$	0	0
$268$	0	0
$269$	25.4662i	1.55270i	0.630300	+	0.776352i	$0.282932\pi$
$269$	25.4662i	1.55270i	−0.630300	+	0.776352i	$0.717068\pi$
$270$	0	0
$271$	− 21.0361i	− 1.27785i	−0.769268	−	0.638927i	$-0.779379\pi$
$271$	− 21.0361i	− 1.27785i	0.769268	−	0.638927i	$-0.220621\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−28.2644	−1.70441
$276$	0	0
$277$	−8.39312	−0.504293	−0.252147	−	0.967689i	$-0.581137\pi$
$277$	−8.39312	−0.504293	−0.252147	+	0.967689i	$0.581137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0.111668i	0.00666157i	0.999994	+	0.00333078i	$0.00106022\pi$
$281$	0.111668i	0.00666157i	−0.999994	+	0.00333078i	$0.998940\pi$
$282$	0	0
$283$	11.4637i	0.681444i	0.940164	+	0.340722i	$0.110671\pi$
$283$	11.4637i	0.681444i	−0.940164	+	0.340722i	$0.889329\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.55580	−0.386977
$288$	0	0
$289$	16.7648	0.986165
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.8089i	1.04041i	0.854042	+	0.520204i	$0.174144\pi$
$293$	17.8089i	1.04041i	−0.854042	+	0.520204i	$0.825856\pi$
$294$	0	0
$295$	− 4.58546i	− 0.266976i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	39.6794	2.29472
$300$	0	0
$301$	−5.37169	−0.309619
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 45.2756i	− 2.59247i
$306$	0	0
$307$	− 20.2499i	− 1.15572i	−0.816135	−	0.577861i	$-0.803888\pi$
$307$	− 20.2499i	− 1.15572i	0.816135	−	0.577861i	$-0.196112\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.38392	−0.0784747	−0.0392374	−	0.999230i	$-0.512493\pi$
$311$	−1.38392	−0.0784747	−0.0392374	+	0.999230i	$0.512493\pi$
$312$	0	0
$313$	−30.1151	−1.70220	−0.851102	−	0.525000i	$-0.824065\pi$
$313$	−30.1151	−1.70220	−0.851102	+	0.525000i	$0.824065\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.27317i	0.296171i	0.988975	+	0.148085i	$0.0473110\pi$
$317$	5.27317i	0.296171i	−0.988975	+	0.148085i	$0.952689\pi$
$318$	0	0
$319$	6.68585i	0.374336i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.11190	0.0618676
$324$	0	0
$325$	−29.7648	−1.65105
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 6.21280i	− 0.342523i
$330$	0	0
$331$	− 9.95715i	− 0.547295i	−0.961830	−	0.273647i	$-0.911770\pi$
$331$	− 9.95715i	− 0.547295i	0.961830	−	0.273647i	$-0.0882301\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.8391	0.592205
$336$	0	0
$337$	−13.3717	−0.728402	−0.364201	−	0.931320i	$-0.618658\pi$
$337$	−13.3717	−0.728402	−0.364201	+	0.931320i	$0.618658\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 36.2344i	− 1.96220i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.2124	0.601915	0.300957	−	0.953638i	$-0.402694\pi$
$347$	11.2124	0.601915	0.300957	+	0.953638i	$0.402694\pi$
$348$	0	0
$349$	4.29273	0.229785	0.114892	−	0.993378i	$-0.463348\pi$
$349$	4.29273	0.229785	0.114892	+	0.993378i	$0.463348\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.78751i	0.0951397i	0.998868	+	0.0475698i	$0.0151477\pi$
$353$	1.78751i	0.0951397i	−0.998868	+	0.0475698i	$0.984852\pi$
$354$	0	0
$355$	11.0790i	0.588010i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.7989	0.675500	0.337750	−	0.941236i	$-0.390334\pi$
$359$	12.7989	0.675500	0.337750	+	0.941236i	$0.390334\pi$
$360$	0	0
$361$	13.7434	0.723336
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 6.95940i	− 0.364272i
$366$	0	0
$367$	− 30.1579i	− 1.57423i	−0.616806	−	0.787115i	$-0.711574\pi$
$367$	− 30.1579i	− 1.57423i	0.616806	−	0.787115i	$-0.288426\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.00023	−0.0519293
$372$	0	0
$373$	−5.21377	−0.269959	−0.134979	−	0.990848i	$-0.543097\pi$
$373$	−5.21377	−0.269959	−0.134979	+	0.990848i	$0.543097\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.04077i	0.362618i
$378$	0	0
$379$	16.7862i	0.862251i	0.902292	+	0.431125i	$0.141883\pi$
$379$	16.7862i	0.862251i	−0.902292	+	0.431125i	$0.858117\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−23.7393	−1.21302	−0.606511	−	0.795075i	$-0.707431\pi$
$383$	−23.7393	−1.21302	−0.606511	+	0.795075i	$0.707431\pi$
$384$	0	0
$385$	15.6644	0.798333
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 17.7682i	− 0.900885i	−0.892805	−	0.450443i	$-0.851266\pi$
$389$	− 17.7682i	− 0.900885i	0.892805	−	0.450443i	$-0.148734\pi$
$390$	0	0
$391$	− 3.86519i	− 0.195471i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	39.9514	2.01017
$396$	0	0
$397$	−24.4078	−1.22499	−0.612496	−	0.790473i	$-0.709835\pi$
$397$	−24.4078	−1.22499	−0.612496	+	0.790473i	$0.709835\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.1633i	1.45635i	0.685393	+	0.728173i	$0.259630\pi$
$401$	29.1633i	1.45635i	−0.685393	+	0.728173i	$0.740370\pi$
$402$	0	0
$403$	− 38.1579i	− 1.90078i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.3137	0.560800
$408$	0	0
$409$	−10.4851	−0.518454	−0.259227	−	0.965816i	$-0.583468\pi$
$409$	−10.4851	−0.518454	−0.259227	+	0.965816i	$0.583468\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.38392i	0.0680981i
$414$	0	0
$415$	10.7434i	0.527372i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	31.1335	1.52097	0.760485	−	0.649356i	$-0.224961\pi$
$419$	31.1335	1.52097	0.760485	+	0.649356i	$0.224961\pi$
$420$	0	0
$421$	−30.2646	−1.47501	−0.737503	−	0.675344i	$-0.763995\pi$
$421$	−30.2646	−1.47501	−0.737503	+	0.675344i	$0.763995\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.89941i	0.140642i
$426$	0	0
$427$	13.6644i	0.661268i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.23179	−0.107502	−0.0537508	−	0.998554i	$-0.517118\pi$
$431$	−2.23179	−0.107502	−0.0537508	+	0.998554i	$0.517118\pi$
$432$	0	0
$433$	−8.54262	−0.410532	−0.205266	−	0.978706i	$-0.565806\pi$
$433$	−8.54262	−0.410532	−0.205266	+	0.978706i	$0.565806\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.2731i	0.874121i
$438$	0	0
$439$	1.17092i	0.0558851i	0.999610	+	0.0279426i	$0.00889556\pi$
$439$	1.17092i	0.0558851i	−0.999610	+	0.0279426i	$0.991104\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.58610	−0.312915	−0.156458	−	0.987685i	$-0.550007\pi$
$443$	−6.58610	−0.312915	−0.156458	+	0.987685i	$0.550007\pi$
$444$	0	0
$445$	18.9786	0.899671
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11.2228i	0.529638i	0.964298	+	0.264819i	$0.0853121\pi$
$449$	11.2228i	0.529638i	−0.964298	+	0.264819i	$0.914688\pi$
$450$	0	0
$451$	− 30.9933i	− 1.45942i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	16.4960	0.773344
$456$	0	0
$457$	11.9143	0.557328	0.278664	−	0.960389i	$-0.410108\pi$
$457$	11.9143	0.557328	0.278664	+	0.960389i	$0.410108\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 18.4255i	− 0.858159i	−0.903267	−	0.429080i	$-0.858838\pi$
$461$	− 18.4255i	− 0.858159i	0.903267	−	0.429080i	$-0.141162\pi$
$462$	0	0
$463$	− 7.47208i	− 0.347257i	−0.984811	−	0.173628i	$-0.944451\pi$
$463$	− 7.47208i	− 0.347257i	0.984811	−	0.173628i	$-0.0555492\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.57548	−0.258003	−0.129001	−	0.991644i	$-0.541177\pi$
$467$	−5.57548	−0.258003	−0.129001	+	0.991644i	$0.541177\pi$
$468$	0	0
$469$	−3.27131	−0.151055
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 25.3953i	− 1.16767i
$474$	0	0
$475$	− 13.7073i	− 0.628933i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.9400	−0.728319	−0.364159	−	0.931337i	$-0.618644\pi$
$479$	−15.9400	−0.728319	−0.364159	+	0.931337i	$0.618644\pi$
$480$	0	0
$481$	11.9143	0.543246
$482$	0	0
$483$	0	0
$484$	0	0
$485$	39.9514i	1.81410i
$486$	0	0
$487$	− 34.0722i	− 1.54396i	−0.635647	−	0.771980i	$-0.719266\pi$
$487$	− 34.0722i	− 1.54396i	0.635647	−	0.771980i	$-0.280734\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−34.0026	−1.53452	−0.767258	−	0.641339i	$-0.778379\pi$
$491$	−34.0026	−1.53452	−0.767258	+	0.641339i	$0.778379\pi$
$492$	0	0
$493$	0.685846	0.0308890
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.34369i	− 0.149985i
$498$	0	0
$499$	− 5.37169i	− 0.240470i	−0.992745	−	0.120235i	$-0.961635\pi$
$499$	− 5.37169i	− 0.240470i	0.992745	−	0.120235i	$-0.0383648\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	37.3463	1.66519	0.832594	−	0.553884i	$-0.186855\pi$
$503$	37.3463	1.66519	0.832594	+	0.553884i	$0.186855\pi$
$504$	0	0
$505$	48.2646	2.14775
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.403595i	0.0178890i	0.999960	+	0.00894451i	$0.00284716\pi$
$509$	0.403595i	0.0178890i	−0.999960	+	0.00894451i	$0.997153\pi$
$510$	0	0
$511$	2.10038i	0.0929155i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−18.9104	−0.833293
$516$	0	0
$517$	29.3717	1.29177
$518$	0	0
$519$	0	0
$520$	0	0
$521$	42.9927i	1.88355i	0.336249	+	0.941773i	$0.390842\pi$
$521$	42.9927i	1.88355i	−0.336249	+	0.941773i	$0.609158\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.71699	−0.161915
$528$	0	0
$529$	40.5212	1.76179
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 32.6386i	− 1.41373i
$534$	0	0
$535$	53.1512i	2.29793i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.72761	−0.203632
$540$	0	0
$541$	−19.1365	−0.822742	−0.411371	−	0.911468i	$-0.634950\pi$
$541$	−19.1365	−0.822742	−0.411371	+	0.911468i	$0.634950\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 4.54496i	− 0.194685i
$546$	0	0
$547$	− 8.14323i	− 0.348179i	−0.984730	−	0.174090i	$-0.944302\pi$
$547$	− 8.14323i	− 0.348179i	0.984730	−	0.174090i	$-0.0556983\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.24241	−0.138131
$552$	0	0
$553$	−12.0575	−0.512739
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 39.0931i	− 1.65643i	−0.560412	−	0.828214i	$-0.689357\pi$
$557$	− 39.0931i	− 1.65643i	0.560412	−	0.828214i	$-0.310643\pi$
$558$	0	0
$559$	− 26.7434i	− 1.13112i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13.6468	−0.575143	−0.287572	−	0.957759i	$-0.592848\pi$
$563$	−13.6468	−0.575143	−0.287572	+	0.957759i	$0.592848\pi$
$564$	0	0
$565$	37.3864	1.57286
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 2.32355i	− 0.0974084i	−0.998813	−	0.0487042i	$-0.984491\pi$
$569$	− 2.32355i	− 0.0974084i	0.998813	−	0.0487042i	$-0.0155092\pi$
$570$	0	0
$571$	2.88661i	0.120801i	0.998174	+	0.0604005i	$0.0192378\pi$
$571$	2.88661i	0.120801i	−0.998174	+	0.0604005i	$0.980762\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−47.6494	−1.98712
$576$	0	0
$577$	−18.5855	−0.773723	−0.386861	−	0.922138i	$-0.626441\pi$
$577$	−18.5855	−0.773723	−0.386861	+	0.922138i	$0.626441\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 3.24241i	− 0.134518i
$582$	0	0
$583$	− 4.72869i	− 0.195842i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.92979	0.409846	0.204923	−	0.978778i	$-0.434306\pi$
$587$	9.92979	0.409846	0.204923	+	0.978778i	$0.434306\pi$
$588$	0	0
$589$	17.5725	0.724061
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.0823i	0.988942i	0.869194	+	0.494471i	$0.164638\pi$
$593$	24.0823i	0.988942i	−0.869194	+	0.494471i	$0.835362\pi$
$594$	0	0
$595$	− 1.60688i	− 0.0658759i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19.2837	0.787912	0.393956	−	0.919129i	$-0.371106\pi$
$599$	19.2837	0.787912	0.393956	+	0.919129i	$0.371106\pi$
$600$	0	0
$601$	10.5855	0.431790	0.215895	−	0.976417i	$-0.430733\pi$
$601$	10.5855	0.431790	0.215895	+	0.976417i	$0.430733\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	37.6079i	1.52898i
$606$	0	0
$607$	34.0722i	1.38295i	0.722401	+	0.691475i	$0.243039\pi$
$607$	34.0722i	1.38295i	−0.722401	+	0.691475i	$0.756961\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.9309	1.25133
$612$	0	0
$613$	17.3288	0.699906	0.349953	−	0.936767i	$-0.386198\pi$
$613$	17.3288	0.699906	0.349953	+	0.936767i	$0.386198\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.76760i	0.0711611i	0.999367	+	0.0355805i	$0.0113280\pi$
$617$	1.76760i	0.0711611i	−0.999367	+	0.0355805i	$0.988672\pi$
$618$	0	0
$619$	− 47.0424i	− 1.89079i	−0.325922	−	0.945397i	$-0.605675\pi$
$619$	− 47.0424i	− 1.89079i	0.325922	−	0.945397i	$-0.394325\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.72784	−0.229481
$624$	0	0
$625$	−19.1495	−0.765980
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.16058i	− 0.0462754i
$630$	0	0
$631$	− 14.0147i	− 0.557916i	−0.960303	−	0.278958i	$-0.910011\pi$
$631$	− 14.0147i	− 0.557916i	0.960303	−	0.278958i	$-0.0899890\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−55.1448	−2.18836
$636$	0	0
$637$	−4.97858	−0.197258
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 8.92956i	− 0.352696i	−0.984328	−	0.176348i	$-0.943572\pi$
$641$	− 8.92956i	− 0.352696i	0.984328	−	0.176348i	$-0.0564285\pi$
$642$	0	0
$643$	− 4.53635i	− 0.178896i	−0.995991	−	0.0894480i	$-0.971490\pi$
$643$	− 4.53635i	− 0.178896i	0.995991	−	0.0894480i	$-0.0285103\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.10091	0.200537	0.100269	−	0.994960i	$-0.468030\pi$
$647$	5.10091	0.200537	0.100269	+	0.994960i	$0.468030\pi$
$648$	0	0
$649$	−6.54262	−0.256820
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 26.2535i	− 1.02738i	−0.857976	−	0.513690i	$-0.828278\pi$
$653$	− 26.2535i	− 1.02738i	0.857976	−	0.513690i	$-0.171722\pi$
$654$	0	0
$655$	− 25.1709i	− 0.983509i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16.0413	−0.624881	−0.312440	−	0.949937i	$-0.601146\pi$
$659$	−16.0413	−0.624881	−0.312440	+	0.949937i	$0.601146\pi$
$660$	0	0
$661$	−29.8652	−1.16162	−0.580811	−	0.814039i	$-0.697264\pi$
$661$	−29.8652	−1.16162	−0.580811	+	0.814039i	$0.697264\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.59672i	0.294588i
$666$	0	0
$667$	11.2713i	0.436427i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−64.6000	−2.49386
$672$	0	0
$673$	−35.0080	−1.34946	−0.674729	−	0.738066i	$-0.735739\pi$
$673$	−35.0080	−1.34946	−0.674729	+	0.738066i	$0.735739\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.6893i	1.44852i	0.689529	+	0.724258i	$0.257818\pi$
$677$	37.6893i	1.44852i	−0.689529	+	0.724258i	$0.742182\pi$
$678$	0	0
$679$	− 12.0575i	− 0.462726i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21.5770	−0.825620	−0.412810	−	0.910817i	$-0.635453\pi$
$683$	−21.5770	−0.825620	−0.412810	+	0.910817i	$0.635453\pi$
$684$	0	0
$685$	−31.2285	−1.19318
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 4.97972i	− 0.189712i
$690$	0	0
$691$	− 14.0428i	− 0.534215i	−0.963667	−	0.267108i	$-0.913932\pi$
$691$	− 14.0428i	− 0.534215i	0.963667	−	0.267108i	$-0.0860679\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−53.0143	−2.01095
$696$	0	0
$697$	−3.17935	−0.120426
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27.9214i	1.05458i	0.849687	+	0.527288i	$0.176791\pi$
$701$	27.9214i	1.05458i	−0.849687	+	0.527288i	$0.823209\pi$
$702$	0	0
$703$	5.48677i	0.206937i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14.5665	−0.547830
$708$	0	0
$709$	10.5426	0.395936	0.197968	−	0.980208i	$-0.436566\pi$
$709$	10.5426	0.395936	0.197968	+	0.980208i	$0.436566\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 61.0856i	− 2.28767i
$714$	0	0
$715$	77.9865i	2.91653i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31.8801	−1.18893	−0.594463	−	0.804123i	$-0.702635\pi$
$719$	−31.8801	−1.18893	−0.594463	+	0.804123i	$0.702635\pi$
$720$	0	0
$721$	5.70727	0.212550
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 8.45498i	− 0.314010i
$726$	0	0
$727$	− 46.2070i	− 1.71372i	−0.515546	−	0.856862i	$-0.672411\pi$
$727$	− 46.2070i	− 1.71372i	0.515546	−	0.856862i	$-0.327589\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.60509	−0.0963528
$732$	0	0
$733$	−22.4360	−0.828691	−0.414346	−	0.910120i	$-0.635990\pi$
$733$	−22.4360	−0.828691	−0.414346	+	0.910120i	$0.635990\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 15.4655i	− 0.569678i
$738$	0	0
$739$	14.0147i	0.515539i	0.966206	+	0.257769i	$0.0829875\pi$
$739$	14.0147i	0.515539i	−0.966206	+	0.257769i	$0.917013\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25.9313	0.951327	0.475663	−	0.879627i	$-0.342208\pi$
$743$	25.9313	0.951327	0.475663	+	0.879627i	$0.342208\pi$
$744$	0	0
$745$	−61.1856	−2.24167
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 16.0413i	− 0.586137i
$750$	0	0
$751$	2.74338i	0.100108i	0.998747	+	0.0500538i	$0.0159393\pi$
$751$	2.74338i	0.100108i	−0.998747	+	0.0500538i	$0.984061\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−77.2977	−2.81315
$756$	0	0
$757$	15.7992	0.574233	0.287116	−	0.957896i	$-0.407303\pi$
$757$	15.7992	0.574233	0.287116	+	0.957896i	$0.407303\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 27.0527i	− 0.980660i	−0.871537	−	0.490330i	$-0.836876\pi$
$761$	− 27.0527i	− 0.980660i	0.871537	−	0.490330i	$-0.163124\pi$
$762$	0	0
$763$	1.37169i	0.0496586i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.88994	−0.248781
$768$	0	0
$769$	39.8715	1.43780	0.718901	−	0.695113i	$-0.244646\pi$
$769$	39.8715	1.43780	0.718901	+	0.695113i	$0.244646\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34.4261i	1.23822i	0.785304	+	0.619110i	$0.212507\pi$
$773$	34.4261i	1.23822i	−0.785304	+	0.619110i	$0.787493\pi$
$774$	0	0
$775$	45.8223i	1.64599i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15.0307	0.538531
$780$	0	0
$781$	15.8077	0.565642
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 73.7226i	− 2.63127i
$786$	0	0
$787$	− 24.1151i	− 0.859610i	−0.902922	−	0.429805i	$-0.858582\pi$
$787$	− 24.1151i	− 0.859610i	0.902922	−	0.429805i	$-0.141418\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11.2834	−0.401192
$792$	0	0
$793$	−68.0294	−2.41579
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 35.0634i	− 1.24201i	−0.783807	−	0.621005i	$-0.786725\pi$
$797$	− 35.0634i	− 1.24201i	0.783807	−	0.621005i	$-0.213275\pi$
$798$	0	0
$799$	− 3.01300i	− 0.106592i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.92979	−0.350415
$804$	0	0
$805$	26.4078	0.930753
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.9806i	0.772796i	0.922332	+	0.386398i	$0.126281\pi$
$809$	21.9806i	0.772796i	−0.922332	+	0.386398i	$0.873719\pi$
$810$	0	0
$811$	44.7005i	1.56965i	0.619719	+	0.784824i	$0.287247\pi$
$811$	44.7005i	1.56965i	−0.619719	+	0.784824i	$0.712753\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−55.1448	−1.93164
$816$	0	0
$817$	12.3158	0.430877
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.3651i	1.37385i	0.726727	+	0.686926i	$0.241040\pi$
$821$	39.3651i	1.37385i	−0.726727	+	0.686926i	$0.758960\pi$
$822$	0	0
$823$	− 31.9718i	− 1.11447i	−0.830355	−	0.557234i	$-0.811863\pi$
$823$	− 31.9718i	− 1.11447i	0.830355	−	0.557234i	$-0.188137\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.8702	−0.725728	−0.362864	−	0.931842i	$-0.618201\pi$
$827$	−20.8702	−0.725728	−0.362864	+	0.931842i	$0.618201\pi$
$828$	0	0
$829$	32.2217	1.11911	0.559554	−	0.828794i	$-0.310973\pi$
$829$	32.2217	1.11911	0.559554	+	0.828794i	$0.310973\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.484966i	0.0168031i
$834$	0	0
$835$	− 29.7564i	− 1.02976i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	48.9320	1.68932	0.844660	−	0.535303i	$-0.179802\pi$
$839$	48.9320	1.68932	0.844660	+	0.535303i	$0.179802\pi$
$840$	0	0
$841$	27.0000	0.931034
$842$	0	0
$843$	0	0
$844$	0	0
$845$	39.0524i	1.34344i
$846$	0	0
$847$	− 11.3503i	− 0.390000i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	19.0732	0.653820
$852$	0	0
$853$	−12.2927	−0.420895	−0.210448	−	0.977605i	$-0.567492\pi$
$853$	−12.2927	−0.420895	−0.210448	+	0.977605i	$0.567492\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	57.1349i	1.95169i	0.218463	+	0.975845i	$0.429896\pi$
$857$	57.1349i	1.95169i	−0.218463	+	0.975845i	$0.570104\pi$
$858$	0	0
$859$	− 21.8223i	− 0.744569i	−0.928119	−	0.372284i	$-0.878575\pi$
$859$	− 21.8223i	− 0.744569i	0.928119	−	0.372284i	$-0.121425\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−42.2764	−1.43911	−0.719553	−	0.694437i	$-0.755653\pi$
$863$	−42.2764	−1.43911	−0.719553	+	0.694437i	$0.755653\pi$
$864$	0	0
$865$	−9.13650	−0.310650
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 57.0033i	− 1.93370i
$870$	0	0
$871$	− 16.2865i	− 0.551846i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.24241	−0.109614
$876$	0	0
$877$	−26.2008	−0.884737	−0.442369	−	0.896833i	$-0.645862\pi$
$877$	−26.2008	−0.884737	−0.442369	+	0.896833i	$0.645862\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47.4165i	1.59750i	0.601661	+	0.798752i	$0.294506\pi$
$881$	47.4165i	1.59750i	−0.601661	+	0.798752i	$0.705494\pi$
$882$	0	0
$883$	− 26.6283i	− 0.896114i	−0.894005	−	0.448057i	$-0.852116\pi$
$883$	− 26.6283i	− 0.896114i	0.894005	−	0.448057i	$-0.147884\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.75684	−0.226873	−0.113436	−	0.993545i	$-0.536186\pi$
$887$	−6.75684	−0.226873	−0.113436	+	0.993545i	$0.536186\pi$
$888$	0	0
$889$	16.6430	0.558188
$890$	0	0
$891$	0	0
$892$	0	0
$893$	14.2443i	0.476667i
$894$	0	0
$895$	15.6644i	0.523604i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.8391	0.361505
$900$	0	0
$901$	−0.485078	−0.0161603
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27.8097i	0.924426i
$906$	0	0
$907$	25.2860i	0.839608i	0.907615	+	0.419804i	$0.137901\pi$
$907$	25.2860i	0.839608i	−0.907615	+	0.419804i	$0.862099\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	21.3050	0.705865	0.352932	−	0.935649i	$-0.385185\pi$
$911$	21.3050	0.705865	0.352932	+	0.935649i	$0.385185\pi$
$912$	0	0
$913$	15.3288	0.507311
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.59672i	0.250866i
$918$	0	0
$919$	48.4998i	1.59986i	0.600093	+	0.799930i	$0.295130\pi$
$919$	48.4998i	1.59986i	−0.600093	+	0.799930i	$0.704870\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	16.6468	0.547937
$924$	0	0
$925$	−14.3074	−0.470425
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.5567i	0.346355i	0.984891	+	0.173177i	$0.0554034\pi$
$929$	10.5567i	0.346355i	−0.984891	+	0.173177i	$0.944597\pi$
$930$	0	0
$931$	− 2.29273i	− 0.0751412i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.59672	0.248439
$936$	0	0
$937$	−29.5296	−0.964690	−0.482345	−	0.875981i	$-0.660215\pi$
$937$	−29.5296	−0.964690	−0.482345	+	0.875981i	$0.660215\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6.53503i	0.213036i	0.994311	+	0.106518i	$0.0339701\pi$
$941$	6.53503i	0.213036i	−0.994311	+	0.106518i	$0.966030\pi$
$942$	0	0
$943$	− 52.2499i	− 1.70149i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−43.2951	−1.40690	−0.703450	−	0.710745i	$-0.748358\pi$
$947$	−43.2951	−1.40690	−0.703450	+	0.710745i	$0.748358\pi$
$948$	0	0
$949$	−10.4569	−0.339446
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 19.4848i	− 0.631173i	−0.948897	−	0.315587i	$-0.897799\pi$
$953$	− 19.4848i	− 0.631173i	0.948897	−	0.315587i	$-0.102201\pi$
$954$	0	0
$955$	4.92104i	0.159241i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.42492	0.304346
$960$	0	0
$961$	−27.7434	−0.894948
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 27.6677i	− 0.890656i
$966$	0	0
$967$	28.4422i	0.914641i	0.889302	+	0.457320i	$0.151191\pi$
$967$	28.4422i	0.914641i	−0.889302	+	0.457320i	$0.848809\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.61534	0.276479	0.138240	−	0.990399i	$-0.455856\pi$
$971$	8.61534	0.276479	0.138240	+	0.990399i	$0.455856\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.8715i	1.14763i	0.818985	+	0.573815i	$0.194537\pi$
$977$	35.8715i	1.14763i	−0.818985	+	0.573815i	$0.805463\pi$
$978$	0	0
$979$	− 27.0790i	− 0.865447i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−41.0633	−1.30971	−0.654857	−	0.755752i	$-0.727271\pi$
$983$	−41.0633	−1.30971	−0.654857	+	0.755752i	$0.727271\pi$
$984$	0	0
$985$	12.6858	0.404205
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 42.8125i	− 1.36136i
$990$	0	0
$991$	25.6707i	0.815456i	0.913103	+	0.407728i	$0.133679\pi$
$991$	25.6707i	0.815456i	−0.913103	+	0.407728i	$0.866321\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−20.0223	−0.634750
$996$	0	0
$997$	2.45065	0.0776130	0.0388065	−	0.999247i	$-0.487644\pi$
$997$	2.45065	0.0776130	0.0388065	+	0.999247i	$0.487644\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.h.h.575.11		12
3.2	odd	2	inner	4032.2.h.h.575.1		12
4.3	odd	2	inner	4032.2.h.h.575.12		12
8.3	odd	2		252.2.e.a.71.12	yes	12
8.5	even	2		252.2.e.a.71.2	yes	12
12.11	even	2	inner	4032.2.h.h.575.2		12
24.5	odd	2		252.2.e.a.71.11	yes	12
24.11	even	2		252.2.e.a.71.1	✓	12
56.13	odd	2		1764.2.e.g.1079.2		12
56.27	even	2		1764.2.e.g.1079.12		12
168.83	odd	2		1764.2.e.g.1079.1		12
168.125	even	2		1764.2.e.g.1079.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.e.a.71.1	✓	12	24.11	even	2
252.2.e.a.71.2	yes	12	8.5	even	2
252.2.e.a.71.11	yes	12	24.5	odd	2
252.2.e.a.71.12	yes	12	8.3	odd	2
1764.2.e.g.1079.1		12	168.83	odd	2
1764.2.e.g.1079.2		12	56.13	odd	2
1764.2.e.g.1079.11		12	168.125	even	2
1764.2.e.g.1079.12		12	56.27	even	2
4032.2.h.h.575.1		12	3.2	odd	2	inner
4032.2.h.h.575.2		12	12.11	even	2	inner
4032.2.h.h.575.11		12	1.1	even	1	trivial
4032.2.h.h.575.12		12	4.3	odd	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)

\(f(q)\)	\(=\)	\(q+3.31339i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+3.31339i q^{5} -1.00000i q^{7} +4.72761 q^{11} +4.97858 q^{13} -0.484966i q^{17} +2.29273i q^{19} +7.97002 q^{23} -5.97858 q^{25} +1.41421i q^{29} -7.66442i q^{31} +3.31339 q^{35} +2.39312 q^{37} -6.55580i q^{41} -5.37169i q^{43} +6.21280 q^{47} -1.00000 q^{49} -1.00023i q^{53} +15.6644i q^{55} -1.38392 q^{59} -13.6644 q^{61} +16.4960i q^{65} -3.27131i q^{67} +3.34369 q^{71} -2.10038 q^{73} -4.72761i q^{77} -12.0575i q^{79} +3.24241 q^{83} +1.60688 q^{85} -5.72784i q^{89} -4.97858i q^{91} -7.59672 q^{95} +12.0575 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 12 q^{25} - 8 q^{37} - 12 q^{49} - 56 q^{61} + 56 q^{85}+O(q^{100}) \) 12 * q - 12 * q^25 - 8 * q^37 - 12 * q^49 - 56 * q^61 + 56 * q^85

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