LMFDB - Embedded newform 4032.2.h.h.575.7

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			575.7
Root			$1.16947 + 0.795191i$ of defining polynomial
Character	$\chi$	$=$	4032.575
Dual form			4032.2.h.h.575.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.665647i q^{5} -1.00000i q^{7} +O(q^{10})$	$q+0.665647i q^{5} -1.00000i q^{7} +2.07986 q^{11} -5.55691 q^{13} +2.16278i q^{17} -4.49828i q^{19} -4.28167 q^{23} +4.55691 q^{25} +1.41421i q^{29} +6.61555i q^{31} +0.665647 q^{35} +5.43965 q^{37} +5.69588i q^{41} +2.11727i q^{43} +10.5213 q^{47} -1.00000 q^{49} -10.6042i q^{53} +1.38445i q^{55} +13.5155 q^{59} +0.615547 q^{61} -3.69894i q^{65} +14.0552i q^{67} +15.5954 q^{71} -11.9379 q^{73} -2.07986i q^{77} -0.824101i q^{79} -6.36153 q^{83} -1.43965 q^{85} -12.6840i q^{89} +5.55691i q^{91} +2.99427 q^{95} +0.824101 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.665647i	0.297686i	0.988861	+	0.148843i	$0.0475550\pi$
$5$	0.665647i	0.297686i	−0.988861	+	0.148843i	$0.952445\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.07986	0.627102	0.313551	−	0.949571i	$-0.398481\pi$
$11$	2.07986	0.627102	0.313551	+	0.949571i	$0.398481\pi$
$12$	0	0
$13$	−5.55691	−1.54121	−0.770605	−	0.637313i	$-0.780046\pi$
$13$	−5.55691	−1.54121	−0.770605	+	0.637313i	$0.780046\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.16278i	0.524551i	0.964993	+	0.262276i	$0.0844730\pi$
$17$	2.16278i	0.524551i	−0.964993	+	0.262276i	$0.915527\pi$
$18$	0	0
$19$	− 4.49828i	− 1.03198i	−0.856596	−	0.515988i	$-0.827425\pi$
$19$	− 4.49828i	− 1.03198i	0.856596	−	0.515988i	$-0.172575\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.28167	−0.892790	−0.446395	−	0.894836i	$-0.647292\pi$
$23$	−4.28167	−0.892790	−0.446395	+	0.894836i	$0.647292\pi$
$24$	0	0
$25$	4.55691	0.911383
$26$	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
$31$	6.61555i	1.18819i	0.804396	+	0.594094i	$0.202489\pi$
$31$	6.61555i	1.18819i	−0.804396	+	0.594094i	$0.797511\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.665647	0.112515
$36$	0	0
$37$	5.43965	0.894273	0.447136	−	0.894466i	$-0.352444\pi$
$37$	5.43965	0.894273	0.447136	+	0.894466i	$0.352444\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.69588i	0.889548i	0.895643	+	0.444774i	$0.146716\pi$
$41$	5.69588i	0.889548i	−0.895643	+	0.444774i	$0.853284\pi$
$42$	0	0
$43$	2.11727i	0.322880i	0.986883	+	0.161440i	$0.0516138\pi$
$43$	2.11727i	0.322880i	−0.986883	+	0.161440i	$0.948386\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.5213	1.53468	0.767341	−	0.641239i	$-0.221579\pi$
$47$	10.5213	1.53468	0.767341	+	0.641239i	$0.221579\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.6042i	− 1.45659i	−0.685261	−	0.728297i	$-0.740312\pi$
$53$	− 10.6042i	− 1.45659i	0.685261	−	0.728297i	$-0.259688\pi$
$54$	0	0
$55$	1.38445i	0.186680i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.5155	1.75957	0.879785	−	0.475371i	$-0.157686\pi$
$59$	13.5155	1.75957	0.879785	+	0.475371i	$0.157686\pi$
$60$	0	0
$61$	0.615547	0.0788128	0.0394064	−	0.999223i	$-0.487453\pi$
$61$	0.615547	0.0788128	0.0394064	+	0.999223i	$0.487453\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 3.69894i	− 0.458797i
$66$	0	0
$67$	14.0552i	1.71712i	0.512717	+	0.858558i	$0.328639\pi$
$67$	14.0552i	1.71712i	−0.512717	+	0.858558i	$0.671361\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.5954	1.85083	0.925415	−	0.378954i	$-0.123716\pi$
$71$	15.5954	1.85083	0.925415	+	0.378954i	$0.123716\pi$
$72$	0	0
$73$	−11.9379	−1.39723	−0.698614	−	0.715498i	$-0.746200\pi$
$73$	−11.9379	−1.39723	−0.698614	+	0.715498i	$0.746200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.07986i	− 0.237022i
$78$	0	0
$79$	− 0.824101i	− 0.0927186i	−0.998925	−	0.0463593i	$-0.985238\pi$
$79$	− 0.824101i	− 0.0927186i	0.998925	−	0.0463593i	$-0.0147619\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.36153	−0.698269	−0.349134	−	0.937073i	$-0.613524\pi$
$83$	−6.36153	−0.698269	−0.349134	+	0.937073i	$0.613524\pi$
$84$	0	0
$85$	−1.43965	−0.156152
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.6840i	− 1.34450i	−0.740322	−	0.672252i	$-0.765327\pi$
$89$	− 12.6840i	− 1.34450i	0.740322	−	0.672252i	$-0.234673\pi$
$90$	0	0
$91$	5.55691i	0.582523i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.99427	0.307205
$96$	0	0
$97$	0.824101	0.0836747	0.0418374	−	0.999124i	$-0.486679\pi$
$97$	0.824101	0.0836747	0.0418374	+	0.999124i	$0.486679\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.8801i	1.77914i	0.456802	+	0.889569i	$0.348995\pi$
$101$	17.8801i	1.77914i	−0.456802	+	0.889569i	$0.651005\pi$
$102$	0	0
$103$	12.4983i	1.23149i	0.787945	+	0.615746i	$0.211145\pi$
$103$	12.4983i	1.23149i	−0.787945	+	0.615746i	$0.788855\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.3936	1.29481	0.647403	−	0.762148i	$-0.275855\pi$
$107$	13.3936	1.29481	0.647403	+	0.762148i	$0.275855\pi$
$108$	0	0
$109$	6.11727	0.585928	0.292964	−	0.956123i	$-0.405358\pi$
$109$	6.11727	0.585928	0.292964	+	0.956123i	$0.405358\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.61602i	0.340167i	0.985430	+	0.170083i	$0.0544037\pi$
$113$	3.61602i	0.340167i	−0.985430	+	0.170083i	$0.945596\pi$
$114$	0	0
$115$	− 2.85008i	− 0.265771i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.16278	0.198262
$120$	0	0
$121$	−6.67418	−0.606744
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.36153i	0.568993i
$126$	0	0
$127$	− 8.17246i	− 0.725189i	−0.931947	−	0.362594i	$-0.881891\pi$
$127$	− 8.17246i	− 0.725189i	0.931947	−	0.362594i	$-0.118109\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.99427	0.261610	0.130805	−	0.991408i	$-0.458244\pi$
$131$	2.99427	0.261610	0.130805	+	0.991408i	$0.458244\pi$
$132$	0	0
$133$	−4.49828	−0.390050
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.7700i	− 0.920144i	−0.887882	−	0.460072i	$-0.847824\pi$
$137$	− 10.7700i	− 0.920144i	0.887882	−	0.460072i	$-0.152176\pi$
$138$	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
$143$	−11.5576	−0.966496
$144$	0	0
$145$	−0.941367	−0.0781763
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.57967i	0.211335i	0.994402	+	0.105667i	$0.0336979\pi$
$149$	2.57967i	0.211335i	−0.994402	+	0.105667i	$0.966302\pi$
$150$	0	0
$151$	− 5.23109i	− 0.425700i	−0.977085	−	0.212850i	$-0.931725\pi$
$151$	− 5.23109i	− 0.425700i	0.977085	−	0.212850i	$-0.0682746\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.40362	−0.353707
$156$	0	0
$157$	5.61211	0.447895	0.223948	−	0.974601i	$-0.428106\pi$
$157$	5.61211	0.447895	0.223948	+	0.974601i	$0.428106\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.28167i	0.337443i
$162$	0	0
$163$	− 8.17246i	− 0.640117i	−0.947398	−	0.320058i	$-0.896297\pi$
$163$	− 8.17246i	− 0.640117i	0.947398	−	0.320058i	$-0.103703\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.5098	1.27757	0.638783	−	0.769387i	$-0.279438\pi$
$167$	16.5098	1.27757	0.638783	+	0.769387i	$0.279438\pi$
$168$	0	0
$169$	17.8793	1.37533
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4.19875i	− 0.319225i	−0.987180	−	0.159613i	$-0.948976\pi$
$173$	− 4.19875i	− 0.319225i	0.987180	−	0.159613i	$-0.0510245\pi$
$174$	0	0
$175$	− 4.55691i	− 0.344470i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.07986	0.155456	0.0777280	−	0.996975i	$-0.475233\pi$
$179$	2.07986	0.155456	0.0777280	+	0.996975i	$0.475233\pi$
$180$	0	0
$181$	11.4396	0.850302	0.425151	−	0.905122i	$-0.360221\pi$
$181$	11.4396	0.850302	0.425151	+	0.905122i	$0.360221\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.62088i	0.266213i
$186$	0	0
$187$	4.49828i	0.328947i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.44139	0.610798	0.305399	−	0.952225i	$-0.401210\pi$
$191$	8.44139	0.610798	0.305399	+	0.952225i	$0.401210\pi$
$192$	0	0
$193$	9.67418	0.696363	0.348181	−	0.937427i	$-0.386799\pi$
$193$	9.67418	0.696363	0.348181	+	0.937427i	$0.386799\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.4326i	− 0.957033i	−0.878079	−	0.478516i	$-0.841175\pi$
$197$	− 13.4326i	− 0.957033i	0.878079	−	0.478516i	$-0.158825\pi$
$198$	0	0
$199$	27.1138i	1.92205i	0.276464	+	0.961024i	$0.410837\pi$
$199$	27.1138i	1.92205i	−0.276464	+	0.961024i	$0.589163\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.41421	0.0992583
$204$	0	0
$205$	−3.79145	−0.264806
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 9.35580i	− 0.647154i
$210$	0	0
$211$	11.1138i	0.765108i	0.923933	+	0.382554i	$0.124955\pi$
$211$	11.1138i	0.765108i	−0.923933	+	0.382554i	$0.875045\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.40935	−0.0961170
$216$	0	0
$217$	6.61555	0.449093
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 12.0184i	− 0.808444i
$222$	0	0
$223$	6.87930i	0.460672i	0.973111	+	0.230336i	$0.0739825\pi$
$223$	6.87930i	0.460672i	−0.973111	+	0.230336i	$0.926018\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.19608	0.344876	0.172438	−	0.985020i	$-0.444836\pi$
$227$	5.19608	0.344876	0.172438	+	0.985020i	$0.444836\pi$
$228$	0	0
$229$	−5.55691	−0.367211	−0.183606	−	0.983000i	$-0.558777\pi$
$229$	−5.55691	−0.367211	−0.183606	+	0.983000i	$0.558777\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.1197i	1.58013i	0.613021	+	0.790067i	$0.289954\pi$
$233$	24.1197i	1.58013i	−0.613021	+	0.790067i	$0.710046\pi$
$234$	0	0
$235$	7.00344i	0.456854i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.44139	−0.546028	−0.273014	−	0.962010i	$-0.588021\pi$
$239$	−8.44139	−0.546028	−0.273014	+	0.962010i	$0.588021\pi$
$240$	0	0
$241$	−25.8207	−1.66326	−0.831628	−	0.555334i	$-0.812591\pi$
$241$	−25.8207	−1.66326	−0.831628	+	0.555334i	$0.812591\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 0.665647i	− 0.0425266i
$246$	0	0
$247$	24.9966i	1.59049i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.8770	1.25463	0.627314	−	0.778766i	$-0.284154\pi$
$251$	19.8770	1.25463	0.627314	+	0.778766i	$0.284154\pi$
$252$	0	0
$253$	−8.90528	−0.559870
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.3352i	1.33085i	0.746465	+	0.665425i	$0.231750\pi$
$257$	21.3352i	1.33085i	−0.746465	+	0.665425i	$0.768250\pi$
$258$	0	0
$259$	− 5.43965i	− 0.338003i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.7551	1.21815	0.609076	−	0.793112i	$-0.291541\pi$
$263$	19.7551	1.21815	0.609076	+	0.793112i	$0.291541\pi$
$264$	0	0
$265$	7.05863	0.433608
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.62356i	0.159961i	0.996796	+	0.0799806i	$0.0254858\pi$
$269$	2.62356i	0.159961i	−0.996796	+	0.0799806i	$0.974514\pi$
$270$	0	0
$271$	0.732814i	0.0445153i	0.999752	+	0.0222576i	$0.00708541\pi$
$271$	0.732814i	0.0445153i	−0.999752	+	0.0222576i	$0.992915\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.47775	0.571530
$276$	0	0
$277$	−11.4396	−0.687342	−0.343671	−	0.939090i	$-0.611670\pi$
$277$	−11.4396	−0.687342	−0.343671	+	0.939090i	$0.611670\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 0.875377i	− 0.0522206i	−0.999659	−	0.0261103i	$-0.991688\pi$
$281$	− 0.875377i	− 0.0522206i	0.999659	−	0.0261103i	$-0.00831212\pi$
$282$	0	0
$283$	− 22.4914i	− 1.33698i	−0.743723	−	0.668488i	$-0.766942\pi$
$283$	− 22.4914i	− 1.33698i	0.743723	−	0.668488i	$-0.233058\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.69588	0.336217
$288$	0	0
$289$	12.3224	0.724846
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 24.2416i	− 1.41621i	−0.706106	−	0.708106i	$-0.749550\pi$
$293$	− 24.2416i	− 1.41621i	0.706106	−	0.708106i	$-0.250450\pi$
$294$	0	0
$295$	8.99656i	0.523800i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	23.7929	1.37598
$300$	0	0
$301$	2.11727	0.122037
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.409737i	0.0234615i
$306$	0	0
$307$	7.61211i	0.434446i	0.976122	+	0.217223i	$0.0696999\pi$
$307$	7.61211i	0.434446i	−0.976122	+	0.217223i	$0.930300\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.5155	0.766395	0.383197	−	0.923666i	$-0.374823\pi$
$311$	13.5155	0.766395	0.383197	+	0.923666i	$0.374823\pi$
$312$	0	0
$313$	−7.64820	−0.432302	−0.216151	−	0.976360i	$-0.569350\pi$
$313$	−7.64820	−0.432302	−0.216151	+	0.976360i	$0.569350\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.7765i	1.67242i	0.548411	+	0.836209i	$0.315233\pi$
$317$	29.7765i	1.67242i	−0.548411	+	0.836209i	$0.684767\pi$
$318$	0	0
$319$	2.94137i	0.164685i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.72879	0.541325
$324$	0	0
$325$	−25.3224	−1.40463
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 10.5213i	− 0.580055i
$330$	0	0
$331$	11.1138i	0.610871i	0.952213	+	0.305436i	$0.0988021\pi$
$331$	11.1138i	0.610871i	−0.952213	+	0.305436i	$0.901198\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.35580	−0.511162
$336$	0	0
$337$	−5.88273	−0.320453	−0.160226	−	0.987080i	$-0.551222\pi$
$337$	−5.88273	−0.320453	−0.160226	+	0.987080i	$0.551222\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	13.7594i	0.745114i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.6432	−0.571357	−0.285678	−	0.958326i	$-0.592219\pi$
$347$	−10.6432	−0.571357	−0.285678	+	0.958326i	$0.592219\pi$
$348$	0	0
$349$	−2.49828	−0.133730	−0.0668650	−	0.997762i	$-0.521300\pi$
$349$	−2.49828	−0.133730	−0.0668650	+	0.997762i	$0.521300\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.126811i	0.00674945i	0.999994	+	0.00337472i	$0.00107421\pi$
$353$	0.126811i	0.00674945i	−0.999994	+	0.00337472i	$0.998926\pi$
$354$	0	0
$355$	10.3810i	0.550967i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.7551	1.04263	0.521317	−	0.853363i	$-0.325441\pi$
$359$	19.7551	1.04263	0.521317	+	0.853363i	$0.325441\pi$
$360$	0	0
$361$	−1.23453	−0.0649754
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 7.94645i	− 0.415936i
$366$	0	0
$367$	− 28.7620i	− 1.50137i	−0.660663	−	0.750683i	$-0.729725\pi$
$367$	− 28.7620i	− 1.50137i	0.660663	−	0.750683i	$-0.270275\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.6042	−0.550541
$372$	0	0
$373$	0.879296	0.0455282	0.0227641	−	0.999741i	$-0.492753\pi$
$373$	0.879296	0.0455282	0.0227641	+	0.999741i	$0.492753\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7.85866i	− 0.404742i
$378$	0	0
$379$	22.8793i	1.17523i	0.809140	+	0.587615i	$0.199933\pi$
$379$	22.8793i	1.17523i	−0.809140	+	0.587615i	$0.800067\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.3562	−1.65333	−0.826663	−	0.562698i	$-0.809763\pi$
$383$	−32.3562	−1.65333	−0.826663	+	0.562698i	$0.809763\pi$
$384$	0	0
$385$	1.38445	0.0705582
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.3391i	0.828424i	0.910180	+	0.414212i	$0.135943\pi$
$389$	16.3391i	0.828424i	−0.910180	+	0.414212i	$0.864057\pi$
$390$	0	0
$391$	− 9.26031i	− 0.468314i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.548560	0.0276010
$396$	0	0
$397$	4.85008	0.243419	0.121709	−	0.992566i	$-0.461162\pi$
$397$	4.85008	0.243419	0.121709	+	0.992566i	$0.461162\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 20.8305i	− 1.04022i	−0.854098	−	0.520112i	$-0.825890\pi$
$401$	− 20.8305i	− 1.04022i	0.854098	−	0.520112i	$-0.174110\pi$
$402$	0	0
$403$	− 36.7620i	− 1.83125i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.3137	0.560800
$408$	0	0
$409$	12.9345	0.639569	0.319785	−	0.947490i	$-0.396389\pi$
$409$	12.9345	0.639569	0.319785	+	0.947490i	$0.396389\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 13.5155i	− 0.665055i
$414$	0	0
$415$	− 4.23453i	− 0.207865i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14.5519	−0.710906	−0.355453	−	0.934694i	$-0.615673\pi$
$419$	−14.5519	−0.710906	−0.355453	+	0.934694i	$0.615673\pi$
$420$	0	0
$421$	29.9018	1.45733	0.728663	−	0.684872i	$-0.240142\pi$
$421$	29.9018	1.45733	0.728663	+	0.684872i	$0.240142\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.85560i	0.478067i
$426$	0	0
$427$	− 0.615547i	− 0.0297884i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.86658	−0.282583	−0.141292	−	0.989968i	$-0.545126\pi$
$431$	−5.86658	−0.282583	−0.141292	+	0.989968i	$0.545126\pi$
$432$	0	0
$433$	26.1104	1.25479	0.627393	−	0.778703i	$-0.284122\pi$
$433$	26.1104	1.25479	0.627393	+	0.778703i	$0.284122\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.2602i	0.921338i
$438$	0	0
$439$	− 25.9931i	− 1.24058i	−0.784371	−	0.620292i	$-0.787014\pi$
$439$	− 25.9931i	− 1.24058i	0.784371	−	0.620292i	$-0.212986\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.23385	−0.438713	−0.219357	−	0.975645i	$-0.570396\pi$
$443$	−9.23385	−0.438713	−0.219357	+	0.975645i	$0.570396\pi$
$444$	0	0
$445$	8.44309	0.400241
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 33.4755i	− 1.57981i	−0.613232	−	0.789903i	$-0.710131\pi$
$449$	− 33.4755i	− 1.57981i	0.613232	−	0.789903i	$-0.289869\pi$
$450$	0	0
$451$	11.8466i	0.557837i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.69894	−0.173409
$456$	0	0
$457$	−30.2277	−1.41399	−0.706995	−	0.707218i	$-0.749950\pi$
$457$	−30.2277	−1.41399	−0.706995	+	0.707218i	$0.749950\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 10.4822i	− 0.488206i	−0.969749	−	0.244103i	$-0.921507\pi$
$461$	− 10.4822i	− 0.488206i	0.969749	−	0.244103i	$-0.0784935\pi$
$462$	0	0
$463$	− 9.82066i	− 0.456405i	−0.973614	−	0.228202i	$-0.926715\pi$
$463$	− 9.82066i	− 0.456405i	0.973614	−	0.228202i	$-0.0732848\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.4620	−0.993141	−0.496571	−	0.867996i	$-0.665408\pi$
$467$	−21.4620	−0.993141	−0.496571	+	0.867996i	$0.665408\pi$
$468$	0	0
$469$	14.0552	0.649009
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.40362i	0.202479i
$474$	0	0
$475$	− 20.4983i	− 0.940526i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.56334	0.391269	0.195634	−	0.980677i	$-0.437323\pi$
$479$	8.56334	0.391269	0.195634	+	0.980677i	$0.437323\pi$
$480$	0	0
$481$	−30.2277	−1.37826
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.548560i	0.0249088i
$486$	0	0
$487$	9.46563i	0.428929i	0.976732	+	0.214464i	$0.0688005\pi$
$487$	9.46563i	0.428929i	−0.976732	+	0.214464i	$0.931199\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.6260	0.885709	0.442854	−	0.896593i	$-0.353966\pi$
$491$	19.6260	0.885709	0.442854	+	0.896593i	$0.353966\pi$
$492$	0	0
$493$	−3.05863	−0.137754
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 15.5954i	− 0.699548i
$498$	0	0
$499$	2.11727i	0.0947819i	0.998876	+	0.0473909i	$0.0150907\pi$
$499$	2.11727i	0.0947819i	−0.998876	+	0.0473909i	$0.984909\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.03062	−0.179717	−0.0898583	−	0.995955i	$-0.528641\pi$
$503$	−4.03062	−0.179717	−0.0898583	+	0.995955i	$0.528641\pi$
$504$	0	0
$505$	−11.9018	−0.529625
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.6423i	0.604686i	0.953199	+	0.302343i	$0.0977687\pi$
$509$	13.6423i	0.604686i	−0.953199	+	0.302343i	$0.902231\pi$
$510$	0	0
$511$	11.9379i	0.528103i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.31944	−0.366598
$516$	0	0
$517$	21.8827	0.962402
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.4585i	1.07155i	0.844362	+	0.535774i	$0.179980\pi$
$521$	24.4585i	1.07155i	−0.844362	+	0.535774i	$0.820020\pi$
$522$	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
$527$	−14.3080	−0.623265
$528$	0	0
$529$	−4.66730	−0.202926
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 31.6515i	− 1.37098i
$534$	0	0
$535$	8.91539i	0.385446i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.07986	−0.0895859
$540$	0	0
$541$	−7.20512	−0.309772	−0.154886	−	0.987932i	$-0.549501\pi$
$541$	−7.20512	−0.309772	−0.154886	+	0.987932i	$0.549501\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.07194i	0.174423i
$546$	0	0
$547$	− 39.0518i	− 1.66973i	−0.550453	−	0.834866i	$-0.685545\pi$
$547$	− 39.0518i	− 1.66973i	0.550453	−	0.834866i	$-0.314455\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.36153	0.271010
$552$	0	0
$553$	−0.824101	−0.0350443
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3.99874i	− 0.169432i	−0.996405	−	0.0847161i	$-0.973002\pi$
$557$	− 3.99874i	− 0.169432i	0.996405	−	0.0847161i	$-0.0269983\pi$
$558$	0	0
$559$	− 11.7655i	− 0.497626i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−39.1372	−1.64944	−0.824718	−	0.565544i	$-0.808666\pi$
$563$	−39.1372	−1.64944	−0.824718	+	0.565544i	$0.808666\pi$
$564$	0	0
$565$	−2.40699	−0.101263
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.7708i	1.37382i	0.726741	+	0.686912i	$0.241034\pi$
$569$	32.7708i	1.37382i	−0.726741	+	0.686912i	$0.758966\pi$
$570$	0	0
$571$	18.8172i	0.787476i	0.919223	+	0.393738i	$0.128818\pi$
$571$	18.8172i	0.787476i	−0.919223	+	0.393738i	$0.871182\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−19.5112	−0.813673
$576$	0	0
$577$	−5.00344	−0.208296	−0.104148	−	0.994562i	$-0.533212\pi$
$577$	−5.00344	−0.208296	−0.104148	+	0.994562i	$0.533212\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.36153i	0.263921i
$582$	0	0
$583$	− 22.0552i	− 0.913433i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.8292	1.02481	0.512406	−	0.858743i	$-0.328754\pi$
$587$	24.8292	1.02481	0.512406	+	0.858743i	$0.328754\pi$
$588$	0	0
$589$	29.7586	1.22618
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.1391i	0.662752i	0.943499	+	0.331376i	$0.107513\pi$
$593$	16.1391i	0.662752i	−0.943499	+	0.331376i	$0.892487\pi$
$594$	0	0
$595$	1.43965i	0.0590198i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.03204	0.287321	0.143661	−	0.989627i	$-0.454113\pi$
$599$	7.03204	0.287321	0.143661	+	0.989627i	$0.454113\pi$
$600$	0	0
$601$	−2.99656	−0.122232	−0.0611162	−	0.998131i	$-0.519466\pi$
$601$	−2.99656	−0.122232	−0.0611162	+	0.998131i	$0.519466\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.44265i	− 0.180619i
$606$	0	0
$607$	− 9.46563i	− 0.384198i	−0.981376	−	0.192099i	$-0.938471\pi$
$607$	− 9.46563i	− 0.384198i	0.981376	−	0.192099i	$-0.0615295\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−58.4657	−2.36527
$612$	0	0
$613$	−11.2311	−0.453620	−0.226810	−	0.973939i	$-0.572830\pi$
$613$	−11.2311	−0.453620	−0.226810	+	0.973939i	$0.572830\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 37.6352i	− 1.51514i	−0.652756	−	0.757568i	$-0.726387\pi$
$617$	− 37.6352i	− 1.51514i	0.652756	−	0.757568i	$-0.273613\pi$
$618$	0	0
$619$	43.3346i	1.74177i	0.491491	+	0.870883i	$0.336452\pi$
$619$	43.3346i	1.74177i	−0.491491	+	0.870883i	$0.663548\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.6840	−0.508175
$624$	0	0
$625$	18.5500	0.742002
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.7648i	0.469092i
$630$	0	0
$631$	18.2897i	0.728103i	0.931379	+	0.364051i	$0.118607\pi$
$631$	18.2897i	0.728103i	−0.931379	+	0.364051i	$0.881393\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.43997	0.215879
$636$	0	0
$637$	5.55691	0.220173
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 14.2251i	− 0.561856i	−0.959729	−	0.280928i	$-0.909358\pi$
$641$	− 14.2251i	− 0.561856i	0.959729	−	0.280928i	$-0.0906422\pi$
$642$	0	0
$643$	− 38.4914i	− 1.51795i	−0.651118	−	0.758976i	$-0.725700\pi$
$643$	− 38.4914i	− 1.51795i	0.651118	−	0.758976i	$-0.274300\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.792458	0.0311547	0.0155774	−	0.999879i	$-0.495041\pi$
$647$	0.792458	0.0311547	0.0155774	+	0.999879i	$0.495041\pi$
$648$	0	0
$649$	28.1104	1.10343
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.85380i	0.307343i	0.988122	+	0.153672i	$0.0491098\pi$
$653$	7.85380i	0.307343i	−0.988122	+	0.153672i	$0.950890\pi$
$654$	0	0
$655$	1.99312i	0.0778778i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13.3936	−0.521739	−0.260870	−	0.965374i	$-0.584009\pi$
$659$	−13.3936	−0.521739	−0.260870	+	0.965374i	$0.584009\pi$
$660$	0	0
$661$	−35.2603	−1.37147	−0.685734	−	0.727853i	$-0.740518\pi$
$661$	−35.2603	−1.37147	−0.685734	+	0.727853i	$0.740518\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 2.99427i	− 0.116113i
$666$	0	0
$667$	− 6.05520i	− 0.234458i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.28025	0.0494236
$672$	0	0
$673$	40.1364	1.54714	0.773572	−	0.633709i	$-0.218468\pi$
$673$	40.1364	1.54714	0.773572	+	0.633709i	$0.218468\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 20.2478i	− 0.778184i	−0.921199	−	0.389092i	$-0.872789\pi$
$677$	− 20.2478i	− 0.778184i	0.921199	−	0.389092i	$-0.127211\pi$
$678$	0	0
$679$	− 0.824101i	− 0.0316261i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	40.6685	1.55614	0.778068	−	0.628179i	$-0.216200\pi$
$683$	40.6685	1.55614	0.778068	+	0.628179i	$0.216200\pi$
$684$	0	0
$685$	7.16902	0.273914
$686$	0	0
$687$	0	0
$688$	0	0
$689$	58.9265i	2.24492i
$690$	0	0
$691$	− 35.1138i	− 1.33579i	−0.744254	−	0.667896i	$-0.767195\pi$
$691$	− 35.1138i	− 1.33579i	0.744254	−	0.667896i	$-0.232805\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.6504	−0.403991
$696$	0	0
$697$	−12.3189	−0.466613
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.73939i	0.254543i	0.991868	+	0.127272i	$0.0406220\pi$
$701$	6.73939i	0.254543i	−0.991868	+	0.127272i	$0.959378\pi$
$702$	0	0
$703$	− 24.4691i	− 0.922868i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.8801	0.672451
$708$	0	0
$709$	−24.1104	−0.905485	−0.452742	−	0.891641i	$-0.649554\pi$
$709$	−24.1104	−0.905485	−0.452742	+	0.891641i	$0.649554\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 28.3256i	− 1.06080i
$714$	0	0
$715$	− 7.69328i	− 0.287713i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	17.1267	0.638717	0.319359	−	0.947634i	$-0.396533\pi$
$719$	17.1267	0.638717	0.319359	+	0.947634i	$0.396533\pi$
$720$	0	0
$721$	12.4983	0.465460
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.44445i	0.239341i
$726$	0	0
$727$	2.72594i	0.101099i	0.998722	+	0.0505497i	$0.0160973\pi$
$727$	2.72594i	0.101099i	−0.998722	+	0.0505497i	$0.983903\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.57918	−0.169367
$732$	0	0
$733$	−46.5535	−1.71949	−0.859746	−	0.510722i	$-0.829378\pi$
$733$	−46.5535	−1.71949	−0.859746	+	0.510722i	$0.829378\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	29.2328i	1.07681i
$738$	0	0
$739$	− 18.2897i	− 0.672799i	−0.941719	−	0.336399i	$-0.890791\pi$
$739$	− 18.2897i	− 0.672799i	0.941719	−	0.336399i	$-0.109209\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37.3012	−1.36845	−0.684225	−	0.729271i	$-0.739859\pi$
$743$	−37.3012	−1.36845	−0.684225	+	0.729271i	$0.739859\pi$
$744$	0	0
$745$	−1.71715	−0.0629114
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 13.3936i	− 0.489390i
$750$	0	0
$751$	− 12.2345i	− 0.446444i	−0.974768	−	0.223222i	$-0.928342\pi$
$751$	− 12.2345i	− 0.446444i	0.974768	−	0.223222i	$-0.0716576\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.48206	0.126725
$756$	0	0
$757$	−3.87586	−0.140870	−0.0704352	−	0.997516i	$-0.522439\pi$
$757$	−3.87586	−0.140870	−0.0704352	+	0.997516i	$0.522439\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 33.0219i	− 1.19704i	−0.801107	−	0.598521i	$-0.795755\pi$
$761$	− 33.0219i	− 1.19704i	0.801107	−	0.598521i	$-0.204245\pi$
$762$	0	0
$763$	− 6.11727i	− 0.221460i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−75.1046	−2.71187
$768$	0	0
$769$	−23.3415	−0.841715	−0.420858	−	0.907127i	$-0.638271\pi$
$769$	−23.3415	−0.841715	−0.420858	+	0.907127i	$0.638271\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.7783i	1.14299i	0.820606	+	0.571494i	$0.193636\pi$
$773$	31.7783i	1.14299i	−0.820606	+	0.571494i	$0.806364\pi$
$774$	0	0
$775$	30.1465i	1.08289i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.6217	0.917992
$780$	0	0
$781$	32.4362	1.16066
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.73568i	0.133332i
$786$	0	0
$787$	− 1.64820i	− 0.0587520i	−0.999568	−	0.0293760i	$-0.990648\pi$
$787$	− 1.64820i	− 0.0587520i	0.999568	−	0.0293760i	$-0.00935202\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.61602	0.128571
$792$	0	0
$793$	−3.42054	−0.121467
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 20.8376i	− 0.738107i	−0.929408	−	0.369053i	$-0.879682\pi$
$797$	− 20.8376i	− 0.738107i	0.929408	−	0.369053i	$-0.120318\pi$
$798$	0	0
$799$	22.7552i	0.805019i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−24.8292	−0.876204
$804$	0	0
$805$	−2.85008	−0.100452
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 27.0262i	− 0.950190i	−0.879935	−	0.475095i	$-0.842414\pi$
$809$	− 27.0262i	− 0.950190i	0.879935	−	0.475095i	$-0.157586\pi$
$810$	0	0
$811$	8.65164i	0.303800i	0.988396	+	0.151900i	$0.0485392\pi$
$811$	8.65164i	0.303800i	−0.988396	+	0.151900i	$0.951461\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.43997	0.190554
$816$	0	0
$817$	9.52406	0.333205
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 19.2456i	− 0.671675i	−0.941920	−	0.335837i	$-0.890981\pi$
$821$	− 19.2456i	− 0.671675i	0.941920	−	0.335837i	$-0.109019\pi$
$822$	0	0
$823$	21.4036i	0.746081i	0.927815	+	0.373041i	$0.121685\pi$
$823$	21.4036i	0.746081i	−0.927815	+	0.373041i	$0.878315\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37.4303	−1.30158	−0.650790	−	0.759258i	$-0.725562\pi$
$827$	−37.4303	−1.30158	−0.650790	+	0.759258i	$0.725562\pi$
$828$	0	0
$829$	−49.0157	−1.70238	−0.851192	−	0.524854i	$-0.824120\pi$
$829$	−49.0157	−1.70238	−0.851192	+	0.524854i	$0.824120\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.16278i	− 0.0749359i
$834$	0	0
$835$	10.9897i	0.380314i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−15.9612	−0.551043	−0.275521	−	0.961295i	$-0.588850\pi$
$839$	−15.9612	−0.551043	−0.275521	+	0.961295i	$0.588850\pi$
$840$	0	0
$841$	27.0000	0.931034
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.9013i	0.409417i
$846$	0	0
$847$	6.67418i	0.229328i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−23.2908	−0.798398
$852$	0	0
$853$	−5.50172	−0.188375	−0.0941876	−	0.995554i	$-0.530025\pi$
$853$	−5.50172	−0.188375	−0.0941876	+	0.995554i	$0.530025\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.6007i	1.31857i	0.751892	+	0.659287i	$0.229142\pi$
$857$	38.6007i	1.31857i	−0.751892	+	0.659287i	$0.770858\pi$
$858$	0	0
$859$	− 6.14648i	− 0.209715i	−0.994487	−	0.104858i	$-0.966561\pi$
$859$	− 6.14648i	− 0.209715i	0.994487	−	0.104858i	$-0.0334387\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41.9631	−1.42844	−0.714220	−	0.699922i	$-0.753218\pi$
$863$	−41.9631	−1.42844	−0.714220	+	0.699922i	$0.753218\pi$
$864$	0	0
$865$	2.79488	0.0950289
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1.71401i	− 0.0581439i
$870$	0	0
$871$	− 78.1035i	− 2.64644i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.36153	0.215059
$876$	0	0
$877$	−45.8759	−1.54912	−0.774559	−	0.632502i	$-0.782028\pi$
$877$	−45.8759	−1.54912	−0.774559	+	0.632502i	$0.782028\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 39.3323i	− 1.32514i	−0.749000	−	0.662570i	$-0.769466\pi$
$881$	− 39.3323i	− 1.32514i	0.749000	−	0.662570i	$-0.230534\pi$
$882$	0	0
$883$	− 34.1173i	− 1.14814i	−0.818807	−	0.574069i	$-0.805364\pi$
$883$	− 34.1173i	− 1.14814i	0.818807	−	0.574069i	$-0.194636\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	35.9674	1.20767	0.603833	−	0.797111i	$-0.293639\pi$
$887$	35.9674	1.20767	0.603833	+	0.797111i	$0.293639\pi$
$888$	0	0
$889$	−8.17246	−0.274096
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 47.3275i	− 1.58376i
$894$	0	0
$895$	1.38445i	0.0462771i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9.35580	−0.312033
$900$	0	0
$901$	22.9345	0.764059
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.61477i	0.253123i
$906$	0	0
$907$	− 24.3449i	− 0.808360i	−0.914679	−	0.404180i	$-0.867557\pi$
$907$	− 24.3449i	− 0.808360i	0.914679	−	0.404180i	$-0.132443\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−17.4242	−0.577289	−0.288645	−	0.957436i	$-0.593205\pi$
$911$	−17.4242	−0.577289	−0.288645	+	0.957436i	$0.593205\pi$
$912$	0	0
$913$	−13.2311	−0.437885
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.99427i	− 0.0988794i
$918$	0	0
$919$	− 7.22422i	− 0.238305i	−0.992876	−	0.119152i	$-0.961982\pi$
$919$	− 7.22422i	− 0.238305i	0.992876	−	0.119152i	$-0.0380177\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−86.6622	−2.85252
$924$	0	0
$925$	24.7880	0.815025
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36.7208i	1.20477i	0.798206	+	0.602385i	$0.205783\pi$
$929$	36.7208i	1.20477i	−0.798206	+	0.602385i	$0.794217\pi$
$930$	0	0
$931$	4.49828i	0.147425i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.99427	−0.0979230
$936$	0	0
$937$	−20.6448	−0.674435	−0.337218	−	0.941427i	$-0.609486\pi$
$937$	−20.6448	−0.674435	−0.337218	+	0.941427i	$0.609486\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	39.9687i	1.30294i	0.758674	+	0.651471i	$0.225848\pi$
$941$	39.9687i	1.30294i	−0.758674	+	0.651471i	$0.774152\pi$
$942$	0	0
$943$	− 24.3879i	− 0.794179i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.1439	−0.524607	−0.262304	−	0.964985i	$-0.584482\pi$
$947$	−16.1439	−0.524607	−0.262304	+	0.964985i	$0.584482\pi$
$948$	0	0
$949$	66.3380	2.15342
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23.2394i	0.752800i	0.926457	+	0.376400i	$0.122838\pi$
$953$	23.2394i	0.752800i	−0.926457	+	0.376400i	$0.877162\pi$
$954$	0	0
$955$	5.61899i	0.181826i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.7700	−0.347782
$960$	0	0
$961$	−12.7655	−0.411789
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.43959i	0.207298i
$966$	0	0
$967$	− 16.0483i	− 0.516079i	−0.966134	−	0.258040i	$-0.916924\pi$
$967$	− 16.0483i	− 0.516079i	0.966134	−	0.258040i	$-0.0830765\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−28.8134	−0.924665	−0.462333	−	0.886707i	$-0.652987\pi$
$971$	−28.8134	−0.924665	−0.462333	+	0.886707i	$0.652987\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 35.3043i	− 1.12948i	−0.825267	−	0.564742i	$-0.808976\pi$
$977$	− 35.3043i	− 1.12948i	0.825267	−	0.564742i	$-0.191024\pi$
$978$	0	0
$979$	− 26.3810i	− 0.843141i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−10.2774	−0.327797	−0.163898	−	0.986477i	$-0.552407\pi$
$983$	−10.2774	−0.327797	−0.163898	+	0.986477i	$0.552407\pi$
$984$	0	0
$985$	8.94137	0.284896
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 9.06543i	− 0.288264i
$990$	0	0
$991$	− 57.2173i	− 1.81757i	−0.417266	−	0.908784i	$-0.637012\pi$
$991$	− 57.2173i	− 1.81757i	0.417266	−	0.908784i	$-0.362988\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−18.0482	−0.572168
$996$	0	0
$997$	−5.73625	−0.181669	−0.0908345	−	0.995866i	$-0.528953\pi$
$997$	−5.73625	−0.181669	−0.0908345	+	0.995866i	$0.528953\pi$
$998$	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.575.7		12
3.2	odd	2	inner	4032.2.h.h.575.5		12
4.3	odd	2	inner	4032.2.h.h.575.8		12
8.3	odd	2		252.2.e.a.71.3	✓	12
8.5	even	2		252.2.e.a.71.9	yes	12
12.11	even	2	inner	4032.2.h.h.575.6		12
24.5	odd	2		252.2.e.a.71.4	yes	12
24.11	even	2		252.2.e.a.71.10	yes	12
56.13	odd	2		1764.2.e.g.1079.9		12
56.27	even	2		1764.2.e.g.1079.3		12
168.83	odd	2		1764.2.e.g.1079.10		12
168.125	even	2		1764.2.e.g.1079.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.e.a.71.3	✓	12	8.3	odd	2
252.2.e.a.71.4	yes	12	24.5	odd	2
252.2.e.a.71.9	yes	12	8.5	even	2
252.2.e.a.71.10	yes	12	24.11	even	2
1764.2.e.g.1079.3		12	56.27	even	2
1764.2.e.g.1079.4		12	168.125	even	2
1764.2.e.g.1079.9		12	56.13	odd	2
1764.2.e.g.1079.10		12	168.83	odd	2
4032.2.h.h.575.5		12	3.2	odd	2	inner
4032.2.h.h.575.6		12	12.11	even	2	inner
4032.2.h.h.575.7		12	1.1	even	1	trivial
4032.2.h.h.575.8		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)

\(f(q)\)	\(=\)	\(q+0.665647i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+0.665647i q^{5} -1.00000i q^{7} +2.07986 q^{11} -5.55691 q^{13} +2.16278i q^{17} -4.49828i q^{19} -4.28167 q^{23} +4.55691 q^{25} +1.41421i q^{29} +6.61555i q^{31} +0.665647 q^{35} +5.43965 q^{37} +5.69588i q^{41} +2.11727i q^{43} +10.5213 q^{47} -1.00000 q^{49} -10.6042i q^{53} +1.38445i q^{55} +13.5155 q^{59} +0.615547 q^{61} -3.69894i q^{65} +14.0552i q^{67} +15.5954 q^{71} -11.9379 q^{73} -2.07986i q^{77} -0.824101i q^{79} -6.36153 q^{83} -1.43965 q^{85} -12.6840i q^{89} +5.55691i q^{91} +2.99427 q^{95} +0.824101 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 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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