LMFDB - Embedded newform 4032.2.i.b.1889.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1889,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))

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

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

chi := DirichletCharacter("4032.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.8
Root			$1.14412 - 1.14412i$ of defining polynomial
Character	$\chi$	$=$	4032.1889
Dual form			4032.2.i.b.1889.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{5} +(2.23607 + 1.41421i) q^{7} +O(q^{10})$	$q+2.00000i q^{5} +(2.23607 + 1.41421i) q^{7} -1.41421 q^{11} +4.47214 q^{17} +6.32456 q^{19} +3.16228i q^{23} +1.00000 q^{25} -3.16228 q^{29} -5.65685i q^{31} +(-2.82843 + 4.47214i) q^{35} +4.47214i q^{37} +4.47214 q^{41} -4.00000i q^{43} +8.00000 q^{47} +(3.00000 + 6.32456i) q^{49} -9.48683 q^{53} -2.82843i q^{55} -8.94427i q^{59} +8.48528 q^{61} +2.00000i q^{67} +9.48683i q^{71} -6.32456i q^{73} +(-3.16228 - 2.00000i) q^{77} +8.94427 q^{79} +8.94427i q^{85} -13.4164 q^{89} +12.6491i q^{95} +6.32456i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{25} + 64 q^{47} + 24 q^{49}+O(q^{100}) $ 8 * q + 8 * q^25 + 64 * q^47 + 24 * q^49

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$	2.23607	+	1.41421i	0.845154	+	0.534522i
$7$	2.23607	+	1.41421i	0.845154	+	0.534522i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.41421			−0.426401			−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421			−0.426401			−0.213201	+	0.977008i	$0.568389\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.47214			1.08465			0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214			1.08465			0.542326	+	0.840168i	$0.317544\pi$
$18$		0			0
$19$	6.32456			1.45095			0.725476	−	0.688247i	$-0.241620\pi$
$19$	6.32456			1.45095			0.725476	+	0.688247i	$0.241620\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.16228i			0.659380i	0.944089	+	0.329690i	$0.106944\pi$
$23$			3.16228i			0.659380i	−0.944089	+	0.329690i	$0.893056\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.16228			−0.587220			−0.293610	−	0.955925i	$-0.594857\pi$
$29$	−3.16228			−0.587220			−0.293610	+	0.955925i	$0.594857\pi$
$30$		0			0
$31$		−	5.65685i		−	1.01600i	−0.861357	−	0.508001i	$-0.830385\pi$
$31$		−	5.65685i		−	1.01600i	0.861357	−	0.508001i	$-0.169615\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.82843	+	4.47214i	−0.478091	+	0.755929i
$36$		0			0
$37$			4.47214i			0.735215i	0.929981	+	0.367607i	$0.119823\pi$
$37$			4.47214i			0.735215i	−0.929981	+	0.367607i	$0.880177\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.47214			0.698430			0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214			0.698430			0.349215	+	0.937043i	$0.386448\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	3.00000	+	6.32456i	0.428571	+	0.903508i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.48683			−1.30312			−0.651558	−	0.758599i	$-0.725884\pi$
$53$	−9.48683			−1.30312			−0.651558	+	0.758599i	$0.725884\pi$
$54$		0			0
$55$		−	2.82843i		−	0.381385i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	8.94427i		−	1.16445i	−0.813029	−	0.582223i	$-0.802183\pi$
$59$		−	8.94427i		−	1.16445i	0.813029	−	0.582223i	$-0.197817\pi$
$60$		0			0
$61$	8.48528			1.08643			0.543214	−	0.839594i	$-0.317207\pi$
$61$	8.48528			1.08643			0.543214	+	0.839594i	$0.317207\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			9.48683i			1.12588i	0.826498	+	0.562940i	$0.190330\pi$
$71$			9.48683i			1.12588i	−0.826498	+	0.562940i	$0.809670\pi$
$72$		0			0
$73$		−	6.32456i		−	0.740233i	−0.928985	−	0.370117i	$-0.879318\pi$
$73$		−	6.32456i		−	0.740233i	0.928985	−	0.370117i	$-0.120682\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.16228	−	2.00000i	−0.360375	−	0.227921i
$78$		0			0
$79$	8.94427			1.00631			0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427			1.00631			0.503155	+	0.864196i	$0.332173\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$			8.94427i			0.970143i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.4164			−1.42214			−0.711068	−	0.703123i	$-0.751788\pi$
$89$	−13.4164			−1.42214			−0.711068	+	0.703123i	$0.751788\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$			12.6491i			1.29777i
$96$		0			0
$97$			6.32456i			0.642161i	0.947052	+	0.321081i	$0.104046\pi$
$97$			6.32456i			0.642161i	−0.947052	+	0.321081i	$0.895954\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			18.0000i			1.79107i	0.444994	+	0.895533i	$0.353206\pi$
$101$			18.0000i			1.79107i	−0.444994	+	0.895533i	$0.646794\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.07107			0.683586			0.341793	−	0.939775i	$-0.388966\pi$
$107$	7.07107			0.683586			0.341793	+	0.939775i	$0.388966\pi$
$108$		0			0
$109$			4.47214i			0.428353i	0.976795	+	0.214176i	$0.0687068\pi$
$109$			4.47214i			0.428353i	−0.976795	+	0.214176i	$0.931293\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.7279i		−	1.19734i	−0.800995	−	0.598671i	$-0.795696\pi$
$113$		−	12.7279i		−	1.19734i	0.800995	−	0.598671i	$-0.204304\pi$
$114$		0			0
$115$	−6.32456			−0.589768
$116$		0			0
$117$		0			0
$118$		0			0
$119$	10.0000	+	6.32456i	0.916698	+	0.579771i
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$		0			0
$124$		0			0
$125$			12.0000i			1.07331i
$126$		0			0
$127$	8.94427			0.793676			0.396838	−	0.917889i	$-0.370108\pi$
$127$	8.94427			0.793676			0.396838	+	0.917889i	$0.370108\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	14.1421	+	8.94427i	1.22628	+	0.775567i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			9.89949i			0.845771i	0.906183	+	0.422885i	$0.138983\pi$
$137$			9.89949i			0.845771i	−0.906183	+	0.422885i	$0.861017\pi$
$138$		0			0
$139$	−12.6491			−1.07288			−0.536442	−	0.843937i	$-0.680232\pi$
$139$	−12.6491			−1.07288			−0.536442	+	0.843937i	$0.680232\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		−	6.32456i		−	0.525226i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−22.1359			−1.81345			−0.906724	−	0.421725i	$-0.861425\pi$
$149$	−22.1359			−1.81345			−0.906724	+	0.421725i	$0.861425\pi$
$150$		0			0
$151$	13.4164			1.09181			0.545906	−	0.837846i	$-0.316186\pi$
$151$	13.4164			1.09181			0.545906	+	0.837846i	$0.316186\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.3137			0.908739
$156$		0			0
$157$	−14.1421			−1.12867			−0.564333	−	0.825547i	$-0.690866\pi$
$157$	−14.1421			−1.12867			−0.564333	+	0.825547i	$0.690866\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.47214	+	7.07107i	−0.352454	+	0.557278i
$162$		0			0
$163$			6.00000i			0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$			6.00000i			0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$		0			0
$175$	2.23607	+	1.41421i	0.169031	+	0.106904i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.24264			−0.317110			−0.158555	−	0.987350i	$-0.550683\pi$
$179$	−4.24264			−0.317110			−0.158555	+	0.987350i	$0.550683\pi$
$180$		0			0
$181$	5.65685			0.420471			0.210235	−	0.977651i	$-0.432577\pi$
$181$	5.65685			0.420471			0.210235	+	0.977651i	$0.432577\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.94427			−0.657596
$186$		0			0
$187$	−6.32456			−0.462497
$188$		0			0
$189$		0			0
$190$		0			0
$191$			15.8114i			1.14407i	0.820229	+	0.572036i	$0.193846\pi$
$191$			15.8114i			1.14407i	−0.820229	+	0.572036i	$0.806154\pi$
$192$		0			0
$193$	4.00000			0.287926			0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000			0.287926			0.143963	+	0.989583i	$0.454015\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−22.1359			−1.57712			−0.788560	−	0.614957i	$-0.789173\pi$
$197$	−22.1359			−1.57712			−0.788560	+	0.614957i	$0.789173\pi$
$198$		0			0
$199$		−	25.4558i		−	1.80452i	−0.431196	−	0.902258i	$-0.641908\pi$
$199$		−	25.4558i		−	1.80452i	0.431196	−	0.902258i	$-0.358092\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.07107	−	4.47214i	−0.496292	−	0.313882i
$204$		0			0
$205$			8.94427i			0.624695i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.94427			−0.618688
$210$		0			0
$211$		−	12.0000i		−	0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$		−	12.0000i		−	0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.00000			0.545595
$216$		0			0
$217$	8.00000	−	12.6491i	0.543075	−	0.858678i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	19.7990i		−	1.32584i	−0.748691	−	0.662919i	$-0.769317\pi$
$223$		−	19.7990i		−	1.32584i	0.748691	−	0.662919i	$-0.230683\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			17.8885i			1.18730i	0.804722	+	0.593652i	$0.202314\pi$
$227$			17.8885i			1.18730i	−0.804722	+	0.593652i	$0.797686\pi$
$228$		0			0
$229$	16.9706			1.12145			0.560723	−	0.828003i	$-0.310523\pi$
$229$	16.9706			1.12145			0.560723	+	0.828003i	$0.310523\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1.41421i			0.0926482i	0.998926	+	0.0463241i	$0.0147507\pi$
$233$			1.41421i			0.0926482i	−0.998926	+	0.0463241i	$0.985249\pi$
$234$		0			0
$235$			16.0000i			1.04372i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			9.48683i			0.613652i	0.951766	+	0.306826i	$0.0992670\pi$
$239$			9.48683i			0.613652i	−0.951766	+	0.306826i	$0.900733\pi$
$240$		0			0
$241$			6.32456i			0.407400i	0.979033	+	0.203700i	$0.0652968\pi$
$241$			6.32456i			0.407400i	−0.979033	+	0.203700i	$0.934703\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−12.6491	+	6.00000i	−0.808122	+	0.383326i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			8.94427i			0.564557i	0.959332	+	0.282279i	$0.0910903\pi$
$251$			8.94427i			0.564557i	−0.959332	+	0.282279i	$0.908910\pi$
$252$		0			0
$253$		−	4.47214i		−	0.281161i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	31.3050			1.95275			0.976375	−	0.216085i	$-0.0693287\pi$
$257$	31.3050			1.95275			0.976375	+	0.216085i	$0.0693287\pi$
$258$		0			0
$259$	−6.32456	+	10.0000i	−0.392989	+	0.621370i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			9.48683i			0.584983i	0.956268	+	0.292492i	$0.0944843\pi$
$263$			9.48683i			0.584983i	−0.956268	+	0.292492i	$0.905516\pi$
$264$		0			0
$265$		−	18.9737i		−	1.16554i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			6.00000i			0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$			6.00000i			0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$		0			0
$271$			22.6274i			1.37452i	0.726413	+	0.687259i	$0.241186\pi$
$271$			22.6274i			1.37452i	−0.726413	+	0.687259i	$0.758814\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.41421			−0.0852803
$276$		0			0
$277$			26.8328i			1.61223i	0.591761	+	0.806114i	$0.298433\pi$
$277$			26.8328i			1.61223i	−0.591761	+	0.806114i	$0.701567\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	15.5563i		−	0.928014i	−0.885832	−	0.464007i	$-0.846411\pi$
$281$		−	15.5563i		−	0.928014i	0.885832	−	0.464007i	$-0.153589\pi$
$282$		0			0
$283$	18.9737			1.12787			0.563934	−	0.825820i	$-0.309287\pi$
$283$	18.9737			1.12787			0.563934	+	0.825820i	$0.309287\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.0000	+	6.32456i	0.590281	+	0.373327i
$288$		0			0
$289$	3.00000			0.176471
$290$		0			0
$291$		0			0
$292$		0			0
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$	17.8885			1.04151
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	5.65685	−	8.94427i	0.326056	−	0.515539i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			16.9706i			0.971732i
$306$		0			0
$307$	31.6228			1.80481			0.902404	−	0.430892i	$-0.141801\pi$
$307$	31.6228			1.80481			0.902404	+	0.430892i	$0.141801\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$			12.6491i			0.714970i	0.933919	+	0.357485i	$0.116366\pi$
$313$			12.6491i			0.714970i	−0.933919	+	0.357485i	$0.883634\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.48683			0.532834			0.266417	−	0.963858i	$-0.414160\pi$
$317$	9.48683			0.532834			0.266417	+	0.963858i	$0.414160\pi$
$318$		0			0
$319$	4.47214			0.250392
$320$		0			0
$321$		0			0
$322$		0			0
$323$	28.2843			1.57378
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	17.8885	+	11.3137i	0.986227	+	0.623745i
$330$		0			0
$331$		−	12.0000i		−	0.659580i	−0.944054	−	0.329790i	$-0.893022\pi$
$331$		−	12.0000i		−	0.659580i	0.944054	−	0.329790i	$-0.106978\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.00000			−0.218543
$336$		0			0
$337$	−8.00000			−0.435788			−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000			−0.435788			−0.217894	+	0.975972i	$0.569919\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			8.00000i			0.433224i
$342$		0			0
$343$	−2.23607	+	18.3848i	−0.120736	+	0.992685i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−35.3553			−1.89797			−0.948987	−	0.315315i	$-0.897890\pi$
$347$	−35.3553			−1.89797			−0.948987	+	0.315315i	$0.897890\pi$
$348$		0			0
$349$	−2.82843			−0.151402			−0.0757011	−	0.997131i	$-0.524119\pi$
$349$	−2.82843			−0.151402			−0.0757011	+	0.997131i	$0.524119\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.47214			−0.238028			−0.119014	−	0.992893i	$-0.537973\pi$
$353$	−4.47214			−0.238028			−0.119014	+	0.992893i	$0.537973\pi$
$354$		0			0
$355$	−18.9737			−1.00702
$356$		0			0
$357$		0			0
$358$		0			0
$359$			9.48683i			0.500696i	0.968156	+	0.250348i	$0.0805450\pi$
$359$			9.48683i			0.500696i	−0.968156	+	0.250348i	$0.919455\pi$
$360$		0			0
$361$	21.0000			1.10526
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.6491			0.662085
$366$		0			0
$367$		−	14.1421i		−	0.738213i	−0.929387	−	0.369107i	$-0.879664\pi$
$367$		−	14.1421i		−	0.738213i	0.929387	−	0.369107i	$-0.120336\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−21.2132	−	13.4164i	−1.10133	−	0.696545i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	20.0000i		−	1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$		−	20.0000i		−	1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	24.0000			1.22634			0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000			1.22634			0.613171	+	0.789950i	$0.289894\pi$
$384$		0			0
$385$	4.00000	−	6.32456i	0.203859	−	0.322329i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−22.1359			−1.12234			−0.561168	−	0.827702i	$-0.689648\pi$
$389$	−22.1359			−1.12234			−0.561168	+	0.827702i	$0.689648\pi$
$390$		0			0
$391$			14.1421i			0.715199i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			17.8885i			0.900070i
$396$		0			0
$397$	−2.82843			−0.141955			−0.0709773	−	0.997478i	$-0.522612\pi$
$397$	−2.82843			−0.141955			−0.0709773	+	0.997478i	$0.522612\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	15.5563i		−	0.776847i	−0.921481	−	0.388424i	$-0.873020\pi$
$401$		−	15.5563i		−	0.776847i	0.921481	−	0.388424i	$-0.126980\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	6.32456i		−	0.313497i
$408$		0			0
$409$			31.6228i			1.56365i	0.623501	+	0.781823i	$0.285710\pi$
$409$			31.6228i			1.56365i	−0.623501	+	0.781823i	$0.714290\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.6491	−	20.0000i	0.622422	−	0.984136i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	26.8328i		−	1.31087i	−0.755252	−	0.655434i	$-0.772486\pi$
$419$		−	26.8328i		−	1.31087i	0.755252	−	0.655434i	$-0.227514\pi$
$420$		0			0
$421$			26.8328i			1.30775i	0.756602	+	0.653876i	$0.226858\pi$
$421$			26.8328i			1.30775i	−0.756602	+	0.653876i	$0.773142\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.47214			0.216930
$426$		0			0
$427$	18.9737	+	12.0000i	0.918200	+	0.580721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	15.8114i		−	0.761608i	−0.924656	−	0.380804i	$-0.875647\pi$
$431$		−	15.8114i		−	0.761608i	0.924656	−	0.380804i	$-0.124353\pi$
$432$		0			0
$433$			12.6491i			0.607877i	0.952692	+	0.303939i	$0.0983018\pi$
$433$			12.6491i			0.607877i	−0.952692	+	0.303939i	$0.901698\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			20.0000i			0.956730i
$438$		0			0
$439$		−	25.4558i		−	1.21494i	−0.794342	−	0.607471i	$-0.792184\pi$
$439$		−	25.4558i		−	1.21494i	0.794342	−	0.607471i	$-0.207816\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.07107			0.335957			0.167978	−	0.985791i	$-0.446276\pi$
$443$	7.07107			0.335957			0.167978	+	0.985791i	$0.446276\pi$
$444$		0			0
$445$		−	26.8328i		−	1.27200i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	32.5269i		−	1.53504i	−0.641025	−	0.767520i	$-0.721491\pi$
$449$		−	32.5269i		−	1.53504i	0.641025	−	0.767520i	$-0.278509\pi$
$450$		0			0
$451$	−6.32456			−0.297812
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	8.00000			0.374224			0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000			0.374224			0.187112	+	0.982339i	$0.440087\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	30.0000i		−	1.39724i	−0.715493	−	0.698620i	$-0.753798\pi$
$461$		−	30.0000i		−	1.39724i	0.715493	−	0.698620i	$-0.246202\pi$
$462$		0			0
$463$	−26.8328			−1.24703			−0.623513	−	0.781813i	$-0.714295\pi$
$463$	−26.8328			−1.24703			−0.623513	+	0.781813i	$0.714295\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	8.94427i		−	0.413892i	−0.978352	−	0.206946i	$-0.933648\pi$
$467$		−	8.94427i		−	0.413892i	0.978352	−	0.206946i	$-0.0663524\pi$
$468$		0			0
$469$	−2.82843	+	4.47214i	−0.130605	+	0.206504i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5.65685i			0.260102i
$474$		0			0
$475$	6.32456			0.290191
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.00000			−0.182765			−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000			−0.182765			−0.0913823	+	0.995816i	$0.529129\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.6491			−0.574367
$486$		0			0
$487$	−13.4164			−0.607955			−0.303978	−	0.952679i	$-0.598315\pi$
$487$	−13.4164			−0.607955			−0.303978	+	0.952679i	$0.598315\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	26.8701			1.21263			0.606314	−	0.795225i	$-0.292647\pi$
$491$	26.8701			1.21263			0.606314	+	0.795225i	$0.292647\pi$
$492$		0			0
$493$	−14.1421			−0.636930
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−13.4164	+	21.2132i	−0.601808	+	0.951542i
$498$		0			0
$499$		−	20.0000i		−	0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$		−	20.0000i		−	0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16.0000			−0.713405			−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000			−0.713405			−0.356702	+	0.934218i	$0.616099\pi$
$504$		0			0
$505$	−36.0000			−1.60198
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	14.0000i		−	0.620539i	−0.950649	−	0.310270i	$-0.899581\pi$
$509$		−	14.0000i		−	0.620539i	0.950649	−	0.310270i	$-0.100419\pi$
$510$		0			0
$511$	8.94427	−	14.1421i	0.395671	−	0.625611i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−11.3137			−0.497576
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4.47214			0.195928			0.0979639	−	0.995190i	$-0.468767\pi$
$521$	4.47214			0.195928			0.0979639	+	0.995190i	$0.468767\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	25.2982i		−	1.10201i
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$			14.1421i			0.611418i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.24264	−	8.94427i	−0.182743	−	0.385257i
$540$		0			0
$541$		−	26.8328i		−	1.15363i	−0.816874	−	0.576816i	$-0.804295\pi$
$541$		−	26.8328i		−	1.15363i	0.816874	−	0.576816i	$-0.195705\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.94427			−0.383131
$546$		0			0
$547$		−	22.0000i		−	0.940652i	−0.882493	−	0.470326i	$-0.844136\pi$
$547$		−	22.0000i		−	0.940652i	0.882493	−	0.470326i	$-0.155864\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−20.0000			−0.852029
$552$		0			0
$553$	20.0000	+	12.6491i	0.850487	+	0.537895i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	34.7851			1.47389			0.736945	−	0.675953i	$-0.236268\pi$
$557$	34.7851			1.47389			0.736945	+	0.675953i	$0.236268\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	35.7771i		−	1.50782i	−0.656975	−	0.753912i	$-0.728164\pi$
$563$		−	35.7771i		−	1.50782i	0.656975	−	0.753912i	$-0.271836\pi$
$564$		0			0
$565$	25.4558			1.07094
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	24.0416i		−	1.00788i	−0.863739	−	0.503939i	$-0.831884\pi$
$569$		−	24.0416i		−	1.00788i	0.863739	−	0.503939i	$-0.168116\pi$
$570$		0			0
$571$			22.0000i			0.920671i	0.887745	+	0.460336i	$0.152271\pi$
$571$			22.0000i			0.920671i	−0.887745	+	0.460336i	$0.847729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			3.16228i			0.131876i
$576$		0			0
$577$		−	25.2982i		−	1.05318i	−0.850120	−	0.526589i	$-0.823471\pi$
$577$		−	25.2982i		−	1.05318i	0.850120	−	0.526589i	$-0.176529\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	13.4164			0.555651
$584$		0			0
$585$		0			0
$586$		0			0
$587$			8.94427i			0.369170i	0.982817	+	0.184585i	$0.0590940\pi$
$587$			8.94427i			0.369170i	−0.982817	+	0.184585i	$0.940906\pi$
$588$		0			0
$589$		−	35.7771i		−	1.47417i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−22.3607			−0.918243			−0.459122	−	0.888373i	$-0.651836\pi$
$593$	−22.3607			−0.918243			−0.459122	+	0.888373i	$0.651836\pi$
$594$		0			0
$595$	−12.6491	+	20.0000i	−0.518563	+	0.819920i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	9.48683i		−	0.387621i	−0.981039	−	0.193811i	$-0.937915\pi$
$599$		−	9.48683i		−	0.387621i	0.981039	−	0.193811i	$-0.0620848\pi$
$600$		0			0
$601$			12.6491i			0.515968i	0.966149	+	0.257984i	$0.0830582\pi$
$601$			12.6491i			0.515968i	−0.966149	+	0.257984i	$0.916942\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	18.0000i		−	0.731804i
$606$		0			0
$607$			25.4558i			1.03322i	0.856221	+	0.516610i	$0.172806\pi$
$607$			25.4558i			1.03322i	−0.856221	+	0.516610i	$0.827194\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	17.8885i		−	0.722511i	−0.932467	−	0.361256i	$-0.882348\pi$
$613$		−	17.8885i		−	0.722511i	0.932467	−	0.361256i	$-0.117652\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	35.3553i		−	1.42335i	−0.702508	−	0.711676i	$-0.747936\pi$
$617$		−	35.3553i		−	1.42335i	0.702508	−	0.711676i	$-0.252064\pi$
$618$		0			0
$619$	−12.6491			−0.508411			−0.254205	−	0.967150i	$-0.581814\pi$
$619$	−12.6491			−0.508411			−0.254205	+	0.967150i	$0.581814\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−30.0000	−	18.9737i	−1.20192	−	0.760164i
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			20.0000i			0.797452i
$630$		0			0
$631$	−26.8328			−1.06820			−0.534099	−	0.845422i	$-0.679349\pi$
$631$	−26.8328			−1.06820			−0.534099	+	0.845422i	$0.679349\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			17.8885i			0.709885i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			43.8406i			1.73160i	0.500390	+	0.865800i	$0.333190\pi$
$641$			43.8406i			1.73160i	−0.500390	+	0.865800i	$0.666810\pi$
$642$		0			0
$643$	−44.2719			−1.74591			−0.872956	−	0.487798i	$-0.837800\pi$
$643$	−44.2719			−1.74591			−0.872956	+	0.487798i	$0.837800\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.00000			0.314512			0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000			0.314512			0.157256	+	0.987558i	$0.449735\pi$
$648$		0			0
$649$			12.6491i			0.496521i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	47.4342			1.85624			0.928121	−	0.372278i	$-0.121423\pi$
$653$	47.4342			1.85624			0.928121	+	0.372278i	$0.121423\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	46.6690			1.81797			0.908984	−	0.416831i	$-0.136859\pi$
$659$	46.6690			1.81797			0.908984	+	0.416831i	$0.136859\pi$
$660$		0			0
$661$	−8.48528			−0.330039			−0.165020	−	0.986290i	$-0.552769\pi$
$661$	−8.48528			−0.330039			−0.165020	+	0.986290i	$0.552769\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−17.8885	+	28.2843i	−0.693688	+	1.09682i
$666$		0			0
$667$		−	10.0000i		−	0.387202i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.0000			−0.463255
$672$		0			0
$673$	−36.0000			−1.38770			−0.693849	−	0.720121i	$-0.744086\pi$
$673$	−36.0000			−1.38770			−0.693849	+	0.720121i	$0.744086\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			2.00000i			0.0768662i	0.999261	+	0.0384331i	$0.0122367\pi$
$677$			2.00000i			0.0768662i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$		0			0
$679$	−8.94427	+	14.1421i	−0.343250	+	0.542725i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−41.0122			−1.56929			−0.784644	−	0.619947i	$-0.787154\pi$
$683$	−41.0122			−1.56929			−0.784644	+	0.619947i	$0.787154\pi$
$684$		0			0
$685$	−19.7990			−0.756481
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	25.2982			0.962390			0.481195	−	0.876614i	$-0.340203\pi$
$691$	25.2982			0.962390			0.481195	+	0.876614i	$0.340203\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	25.2982i		−	0.959616i
$696$		0			0
$697$	20.0000			0.757554
$698$		0			0
$699$		0			0
$700$		0			0
$701$	41.1096			1.55269			0.776344	−	0.630309i	$-0.217072\pi$
$701$	41.1096			1.55269			0.776344	+	0.630309i	$0.217072\pi$
$702$		0			0
$703$			28.2843i			1.06676i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−25.4558	+	40.2492i	−0.957366	+	1.51373i
$708$		0			0
$709$			40.2492i			1.51159i	0.654808	+	0.755796i	$0.272750\pi$
$709$			40.2492i			1.51159i	−0.654808	+	0.755796i	$0.727250\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	17.8885			0.669931
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.16228			−0.117444
$726$		0			0
$727$			16.9706i			0.629403i	0.949191	+	0.314702i	$0.101904\pi$
$727$			16.9706i			0.629403i	−0.949191	+	0.314702i	$0.898096\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	17.8885i		−	0.661632i
$732$		0			0
$733$	−50.9117			−1.88047			−0.940233	−	0.340532i	$-0.889393\pi$
$733$	−50.9117			−1.88047			−0.940233	+	0.340532i	$0.889393\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	2.82843i		−	0.104186i
$738$		0			0
$739$			10.0000i			0.367856i	0.982940	+	0.183928i	$0.0588813\pi$
$739$			10.0000i			0.367856i	−0.982940	+	0.183928i	$0.941119\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	28.4605i		−	1.04411i	−0.852910	−	0.522057i	$-0.825165\pi$
$743$		−	28.4605i		−	1.04411i	0.852910	−	0.522057i	$-0.174835\pi$
$744$		0			0
$745$		−	44.2719i		−	1.62200i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15.8114	+	10.0000i	0.577736	+	0.365392i
$750$		0			0
$751$	−49.1935			−1.79510			−0.897548	−	0.440917i	$-0.854653\pi$
$751$	−49.1935			−1.79510			−0.897548	+	0.440917i	$0.854653\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			26.8328i			0.976546i
$756$		0			0
$757$		−	22.3607i		−	0.812713i	−0.913715	−	0.406356i	$-0.866799\pi$
$757$		−	22.3607i		−	0.812713i	0.913715	−	0.406356i	$-0.133201\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13.4164			−0.486344			−0.243172	−	0.969983i	$-0.578188\pi$
$761$	−13.4164			−0.486344			−0.243172	+	0.969983i	$0.578188\pi$
$762$		0			0
$763$	−6.32456	+	10.0000i	−0.228964	+	0.362024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	37.9473i		−	1.36842i	−0.729287	−	0.684208i	$-0.760148\pi$
$769$		−	37.9473i		−	1.36842i	0.729287	−	0.684208i	$-0.239852\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	46.0000i		−	1.65451i	−0.561830	−	0.827253i	$-0.689903\pi$
$773$		−	46.0000i		−	1.65451i	0.561830	−	0.827253i	$-0.310097\pi$
$774$		0			0
$775$		−	5.65685i		−	0.203200i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.2843			1.01339
$780$		0			0
$781$		−	13.4164i		−	0.480077i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	28.2843i		−	1.00951i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	18.0000	−	28.4605i	0.640006	−	1.01194i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	42.0000i		−	1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$		−	42.0000i		−	1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$		0			0
$799$	35.7771			1.26570
$800$		0			0
$801$		0			0
$802$		0			0
$803$			8.94427i			0.315637i
$804$		0			0
$805$	−14.1421	−	8.94427i	−0.498445	−	0.315244i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			24.0416i			0.845259i	0.906303	+	0.422629i	$0.138893\pi$
$809$			24.0416i			0.845259i	−0.906303	+	0.422629i	$0.861107\pi$
$810$		0			0
$811$	−12.6491			−0.444170			−0.222085	−	0.975027i	$-0.571286\pi$
$811$	−12.6491			−0.444170			−0.222085	+	0.975027i	$0.571286\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−12.0000			−0.420342
$816$		0			0
$817$		−	25.2982i		−	0.885073i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9.48683			0.331093			0.165546	−	0.986202i	$-0.447061\pi$
$821$	9.48683			0.331093			0.165546	+	0.986202i	$0.447061\pi$
$822$		0			0
$823$	−44.7214			−1.55889			−0.779444	−	0.626472i	$-0.784498\pi$
$823$	−44.7214			−1.55889			−0.779444	+	0.626472i	$0.784498\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	32.5269			1.13107			0.565536	−	0.824724i	$-0.308669\pi$
$827$	32.5269			1.13107			0.565536	+	0.824724i	$0.308669\pi$
$828$		0			0
$829$	11.3137			0.392941			0.196471	−	0.980510i	$-0.437052\pi$
$829$	11.3137			0.392941			0.196471	+	0.980510i	$0.437052\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	13.4164	+	28.2843i	0.464851	+	0.979992i
$834$		0			0
$835$			24.0000i			0.830554i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	36.0000			1.24286			0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000			1.24286			0.621429	+	0.783470i	$0.286552\pi$
$840$		0			0
$841$	−19.0000			−0.655172
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	26.0000i		−	0.894427i
$846$		0			0
$847$	−20.1246	−	12.7279i	−0.691490	−	0.437337i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−14.1421			−0.484786
$852$		0			0
$853$	48.0833			1.64634			0.823170	−	0.567795i	$-0.192204\pi$
$853$	48.0833			1.64634			0.823170	+	0.567795i	$0.192204\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	40.2492			1.37489			0.687444	−	0.726238i	$-0.258733\pi$
$857$	40.2492			1.37489			0.687444	+	0.726238i	$0.258733\pi$
$858$		0			0
$859$	31.6228			1.07896			0.539478	−	0.842000i	$-0.318622\pi$
$859$	31.6228			1.07896			0.539478	+	0.842000i	$0.318622\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	15.8114i		−	0.538226i	−0.963109	−	0.269113i	$-0.913270\pi$
$863$		−	15.8114i		−	0.538226i	0.963109	−	0.269113i	$-0.0867305\pi$
$864$		0			0
$865$	12.0000			0.408012
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.6491			−0.429092
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−16.9706	+	26.8328i	−0.573710	+	0.907115i
$876$		0			0
$877$		−	53.6656i		−	1.81216i	−0.423107	−	0.906080i	$-0.639061\pi$
$877$		−	53.6656i		−	1.81216i	0.423107	−	0.906080i	$-0.360939\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	22.3607			0.753350			0.376675	−	0.926345i	$-0.377067\pi$
$881$	22.3607			0.753350			0.376675	+	0.926345i	$0.377067\pi$
$882$		0			0
$883$		−	44.0000i		−	1.48072i	−0.672212	−	0.740359i	$-0.734656\pi$
$883$		−	44.0000i		−	1.48072i	0.672212	−	0.740359i	$-0.265344\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−52.0000			−1.74599			−0.872995	−	0.487730i	$-0.837825\pi$
$887$	−52.0000			−1.74599			−0.872995	+	0.487730i	$0.837825\pi$
$888$		0			0
$889$	20.0000	+	12.6491i	0.670778	+	0.424238i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	50.5964			1.69315
$894$		0			0
$895$		−	8.48528i		−	0.283632i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			17.8885i			0.596616i
$900$		0			0
$901$	−42.4264			−1.41343
$902$		0			0
$903$		0			0
$904$		0			0
$905$			11.3137i			0.376080i
$906$		0			0
$907$			52.0000i			1.72663i	0.504664	+	0.863316i	$0.331616\pi$
$907$			52.0000i			1.72663i	−0.504664	+	0.863316i	$0.668384\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	47.4342i		−	1.57156i	−0.618504	−	0.785782i	$-0.712261\pi$
$911$		−	47.4342i		−	1.57156i	0.618504	−	0.785782i	$-0.287739\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−22.3607			−0.737611			−0.368805	−	0.929507i	$-0.620233\pi$
$919$	−22.3607			−0.737611			−0.368805	+	0.929507i	$0.620233\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$			4.47214i			0.147043i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−49.1935			−1.61399			−0.806993	−	0.590561i	$-0.798907\pi$
$929$	−49.1935			−1.61399			−0.806993	+	0.590561i	$0.798907\pi$
$930$		0			0
$931$	18.9737	+	40.0000i	0.621837	+	1.31095i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	12.6491i		−	0.413670i
$936$		0			0
$937$			37.9473i			1.23969i	0.784726	+	0.619843i	$0.212804\pi$
$937$			37.9473i			1.23969i	−0.784726	+	0.619843i	$0.787196\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	30.0000i		−	0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$		−	30.0000i		−	0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$		0			0
$943$			14.1421i			0.460531i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.3848			−0.597425			−0.298712	−	0.954343i	$-0.596557\pi$
$947$	−18.3848			−0.597425			−0.298712	+	0.954343i	$0.596557\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	35.3553i		−	1.14527i	−0.819810	−	0.572636i	$-0.805921\pi$
$953$		−	35.3553i		−	1.14527i	0.819810	−	0.572636i	$-0.194079\pi$
$954$		0			0
$955$	−31.6228			−1.02329
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−14.0000	+	22.1359i	−0.452084	+	0.714807i
$960$		0			0
$961$	−1.00000			−0.0322581
$962$		0			0
$963$		0			0
$964$		0			0
$965$			8.00000i			0.257529i
$966$		0			0
$967$	−26.8328			−0.862885			−0.431443	−	0.902140i	$-0.641995\pi$
$967$	−26.8328			−0.862885			−0.431443	+	0.902140i	$0.641995\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	17.8885i		−	0.574071i	−0.957920	−	0.287035i	$-0.907330\pi$
$971$		−	17.8885i		−	0.574071i	0.957920	−	0.287035i	$-0.0926697\pi$
$972$		0			0
$973$	−28.2843	−	17.8885i	−0.906752	−	0.573480i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			24.0416i			0.769160i	0.923092	+	0.384580i	$0.125654\pi$
$977$			24.0416i			0.769160i	−0.923092	+	0.384580i	$0.874346\pi$
$978$		0			0
$979$	18.9737			0.606401
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−36.0000			−1.14822			−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000			−1.14822			−0.574111	+	0.818778i	$0.694652\pi$
$984$		0			0
$985$		−	44.2719i		−	1.41062i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.6491			0.402218
$990$		0			0
$991$	−13.4164			−0.426186			−0.213093	−	0.977032i	$-0.568354\pi$
$991$	−13.4164			−0.426186			−0.213093	+	0.977032i	$0.568354\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	50.9117			1.61401
$996$		0			0
$997$	53.7401			1.70197			0.850983	−	0.525193i	$-0.176007\pi$
$997$	53.7401			1.70197			0.850983	+	0.525193i	$0.176007\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.i.b.1889.8	yes	8
3.2	odd	2		4032.2.i.a.1889.4	yes	8
4.3	odd	2		4032.2.i.a.1889.5	yes	8
7.6	odd	2		4032.2.i.a.1889.3	yes	8
8.3	odd	2		4032.2.i.a.1889.1	✓	8
8.5	even	2	inner	4032.2.i.b.1889.4	yes	8
12.11	even	2	inner	4032.2.i.b.1889.1	yes	8
21.20	even	2	inner	4032.2.i.b.1889.7	yes	8
24.5	odd	2		4032.2.i.a.1889.8	yes	8
24.11	even	2	inner	4032.2.i.b.1889.5	yes	8
28.27	even	2	inner	4032.2.i.b.1889.2	yes	8
56.13	odd	2		4032.2.i.a.1889.7	yes	8
56.27	even	2	inner	4032.2.i.b.1889.6	yes	8
84.83	odd	2		4032.2.i.a.1889.6	yes	8
168.83	odd	2		4032.2.i.a.1889.2	yes	8
168.125	even	2	inner	4032.2.i.b.1889.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4032.2.i.a.1889.1	✓	8	8.3	odd	2
4032.2.i.a.1889.2	yes	8	168.83	odd	2
4032.2.i.a.1889.3	yes	8	7.6	odd	2
4032.2.i.a.1889.4	yes	8	3.2	odd	2
4032.2.i.a.1889.5	yes	8	4.3	odd	2
4032.2.i.a.1889.6	yes	8	84.83	odd	2
4032.2.i.a.1889.7	yes	8	56.13	odd	2
4032.2.i.a.1889.8	yes	8	24.5	odd	2
4032.2.i.b.1889.1	yes	8	12.11	even	2	inner
4032.2.i.b.1889.2	yes	8	28.27	even	2	inner
4032.2.i.b.1889.3	yes	8	168.125	even	2	inner
4032.2.i.b.1889.4	yes	8	8.5	even	2	inner
4032.2.i.b.1889.5	yes	8	24.11	even	2	inner
4032.2.i.b.1889.6	yes	8	56.27	even	2	inner
4032.2.i.b.1889.7	yes	8	21.20	even	2	inner
4032.2.i.b.1889.8	yes	8	1.1	even	1	trivial

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.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{5} +(2.23607 + 1.41421i) q^{7} +O(q^{10})\)	\(q+2.00000i q^{5} +(2.23607 + 1.41421i) q^{7} -1.41421 q^{11} +4.47214 q^{17} +6.32456 q^{19} +3.16228i q^{23} +1.00000 q^{25} -3.16228 q^{29} -5.65685i q^{31} +(-2.82843 + 4.47214i) q^{35} +4.47214i q^{37} +4.47214 q^{41} -4.00000i q^{43} +8.00000 q^{47} +(3.00000 + 6.32456i) q^{49} -9.48683 q^{53} -2.82843i q^{55} -8.94427i q^{59} +8.48528 q^{61} +2.00000i q^{67} +9.48683i q^{71} -6.32456i q^{73} +(-3.16228 - 2.00000i) q^{77} +8.94427 q^{79} +8.94427i q^{85} -13.4164 q^{89} +12.6491i q^{95} +6.32456i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{25} + 64 q^{47} + 24 q^{49}+O(q^{100}) \) 8 * q + 8 * q^25 + 64 * q^47 + 24 * q^49

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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