LMFDB - Embedded newform 4032.2.b.q.3583.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(3583,4032)]

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, 0, 1]))

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

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

chi := DirichletCharacter("4032.3583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.836829184.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 $ x^8 + 14x^6 + 61x^4 + 84*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3583.2
Root			$2.63640i$ of defining polynomial
Character	$\chi$	$=$	4032.3583
Dual form			4032.2.b.q.3583.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31423i q^{5} +(-0.222191 + 2.63640i) q^{7} +O(q^{10})$	$q-2.31423i q^{5} +(-0.222191 + 2.63640i) q^{7} -3.58704i q^{11} +2.82843i q^{13} +2.31423i q^{17} -7.90126 q^{19} -0.130157i q^{23} -0.355642 q^{25} +0.199975 q^{29} +3.45688 q^{31} +(6.10124 + 0.514201i) q^{35} +11.5581 q^{37} +7.97108i q^{41} -4.38404i q^{43} +6.54562 q^{47} +(-6.90126 - 1.17157i) q^{49} +10.3456 q^{53} -8.30121 q^{55} +9.65685 q^{59} +6.19998i q^{61} +6.54562 q^{65} -9.01250i q^{67} -4.75861i q^{71} +10.2853i q^{73} +(9.45688 + 0.797008i) q^{77} +4.24441i q^{79} -0.768089 q^{83} +5.35564 q^{85} -17.7486i q^{89} +(-7.45688 - 0.628452i) q^{91} +18.2853i q^{95} -9.02840i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7}+O(q^{10}) $ 8 * q + 4 * q^7	$ 8 q + 4 q^{7} - 8 q^{19} - 16 q^{25} - 16 q^{31} - 8 q^{35} - 8 q^{37} - 16 q^{47} + 16 q^{53} - 8 q^{55} + 32 q^{59} - 16 q^{65} + 32 q^{77} + 16 q^{83} + 56 q^{85} - 16 q^{91}+O(q^{100}) $ 8 * q + 4 * q^7 - 8 * q^19 - 16 * q^25 - 16 * q^31 - 8 * q^35 - 8 * q^37 - 16 * q^47 + 16 * q^53 - 8 * q^55 + 32 * q^59 - 16 * q^65 + 32 * q^77 + 16 * q^83 + 56 * q^85 - 16 * q^91

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
$5$		−	2.31423i		−	1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$5$		−	2.31423i		−	1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$6$		0			0
$7$	−0.222191	+	2.63640i	−0.0839804	+	0.996467i
$7$	−0.222191	+	2.63640i	−0.0839804	+	0.996467i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.58704i		−	1.08153i	−0.841173	−	0.540766i	$-0.818134\pi$
$11$		−	3.58704i		−	1.08153i	0.841173	−	0.540766i	$-0.181866\pi$
$12$		0			0
$13$			2.82843i			0.784465i	0.919866	+	0.392232i	$0.128297\pi$
$13$			2.82843i			0.784465i	−0.919866	+	0.392232i	$0.871703\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.31423i			0.561282i	0.959813	+	0.280641i	$0.0905471\pi$
$17$			2.31423i			0.561282i	−0.959813	+	0.280641i	$0.909453\pi$
$18$		0			0
$19$	−7.90126			−1.81267			−0.906337	−	0.422556i	$-0.861133\pi$
$19$	−7.90126			−1.81267			−0.906337	+	0.422556i	$0.861133\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.130157i		−	0.0271395i	−0.999908	−	0.0135698i	$-0.995680\pi$
$23$		−	0.130157i		−	0.0271395i	0.999908	−	0.0135698i	$-0.00431953\pi$
$24$		0			0
$25$	−0.355642			−0.0711284
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.199975			0.0371344			0.0185672	−	0.999828i	$-0.494090\pi$
$29$	0.199975			0.0371344			0.0185672	+	0.999828i	$0.494090\pi$
$30$		0			0
$31$	3.45688			0.620874			0.310437	−	0.950594i	$-0.399525\pi$
$31$	3.45688			0.620874			0.310437	+	0.950594i	$0.399525\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.10124	+	0.514201i	1.03130	+	0.0869158i
$36$		0			0
$37$	11.5581			1.90014			0.950071	−	0.312033i	$-0.101010\pi$
$37$	11.5581			1.90014			0.950071	+	0.312033i	$0.101010\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.97108i			1.24487i	0.782670	+	0.622437i	$0.213857\pi$
$41$			7.97108i			1.24487i	−0.782670	+	0.622437i	$0.786143\pi$
$42$		0			0
$43$		−	4.38404i		−	0.668560i	−0.942474	−	0.334280i	$-0.891507\pi$
$43$		−	4.38404i		−	0.668560i	0.942474	−	0.334280i	$-0.108493\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.54562			0.954777			0.477388	−	0.878692i	$-0.341583\pi$
$47$	6.54562			0.954777			0.477388	+	0.878692i	$0.341583\pi$
$48$		0			0
$49$	−6.90126	−	1.17157i	−0.985895	−	0.167368i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	10.3456			1.42108			0.710542	−	0.703655i	$-0.248450\pi$
$53$	10.3456			1.42108			0.710542	+	0.703655i	$0.248450\pi$
$54$		0			0
$55$	−8.30121			−1.11934
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.65685			1.25722			0.628608	−	0.777723i	$-0.283625\pi$
$59$	9.65685			1.25722			0.628608	+	0.777723i	$0.283625\pi$
$60$		0			0
$61$			6.19998i			0.793825i	0.917856	+	0.396913i	$0.129918\pi$
$61$			6.19998i			0.793825i	−0.917856	+	0.396913i	$0.870082\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.54562			0.811884
$66$		0			0
$67$		−	9.01250i		−	1.10105i	−0.834818	−	0.550526i	$-0.814427\pi$
$67$		−	9.01250i		−	1.10105i	0.834818	−	0.550526i	$-0.185573\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.75861i		−	0.564743i	−0.959305	−	0.282371i	$-0.908879\pi$
$71$		−	4.75861i		−	0.564743i	0.959305	−	0.282371i	$-0.0911211\pi$
$72$		0			0
$73$			10.2853i			1.20380i	0.798570	+	0.601902i	$0.205590\pi$
$73$			10.2853i			1.20380i	−0.798570	+	0.601902i	$0.794410\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.45688	+	0.797008i	1.07771	+	0.0908275i
$78$		0			0
$79$			4.24441i			0.477533i	0.971077	+	0.238767i	$0.0767431\pi$
$79$			4.24441i			0.477533i	−0.971077	+	0.238767i	$0.923257\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.768089			−0.0843087			−0.0421543	−	0.999111i	$-0.513422\pi$
$83$	−0.768089			−0.0843087			−0.0421543	+	0.999111i	$0.513422\pi$
$84$		0			0
$85$	5.35564			0.580901
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	17.7486i		−	1.88135i	−0.339310	−	0.940675i	$-0.610194\pi$
$89$		−	17.7486i		−	1.88135i	0.339310	−	0.940675i	$-0.389806\pi$
$90$		0			0
$91$	−7.45688	−	0.628452i	−0.781693	−	0.0658797i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			18.2853i			1.87603i
$96$		0			0
$97$		−	9.02840i		−	0.916695i	−0.888773	−	0.458348i	$-0.848441\pi$
$97$		−	9.02840i		−	0.916695i	0.888773	−	0.458348i	$-0.151559\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			13.6279i			1.35603i	0.735048	+	0.678015i	$0.237160\pi$
$101$			13.6279i			1.35603i	−0.735048	+	0.678015i	$0.762840\pi$
$102$		0			0
$103$	−11.4569			−1.12888			−0.564440	−	0.825474i	$-0.690908\pi$
$103$	−11.4569			−1.12888			−0.564440	+	0.825474i	$0.690908\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.95858i		−	0.672712i	−0.941735	−	0.336356i	$-0.890806\pi$
$107$		−	6.95858i		−	0.672712i	0.941735	−	0.336356i	$-0.109194\pi$
$108$		0			0
$109$	2.88877			0.276694			0.138347	−	0.990384i	$-0.455821\pi$
$109$	2.88877			0.276694			0.138347	+	0.990384i	$0.455821\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.36814			−0.222776			−0.111388	−	0.993777i	$-0.535530\pi$
$113$	−2.36814			−0.222776			−0.111388	+	0.993777i	$0.535530\pi$
$114$		0			0
$115$	−0.301212			−0.0280882
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.10124	−	0.514201i	−0.559299	−	0.0471367i
$120$		0			0
$121$	−1.86683			−0.169712
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	10.7481i		−	0.961339i
$126$		0			0
$127$			7.55812i			0.670674i	0.942098	+	0.335337i	$0.108850\pi$
$127$			7.55812i			0.670674i	−0.942098	+	0.335337i	$0.891150\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−17.4594			−1.52543			−0.762716	−	0.646733i	$-0.776135\pi$
$131$	−17.4594			−1.52543			−0.762716	+	0.646733i	$0.776135\pi$
$132$		0			0
$133$	1.75559	−	20.8309i	0.152229	−	1.80627i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	18.2025			1.55514			0.777571	−	0.628795i	$-0.216452\pi$
$137$	18.2025			1.55514			0.777571	+	0.628795i	$0.216452\pi$
$138$		0			0
$139$	15.3137			1.29889			0.649446	−	0.760408i	$-0.275001\pi$
$139$	15.3137			1.29889			0.649446	+	0.760408i	$0.275001\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	10.1457			0.848424
$144$		0			0
$145$		−	0.462787i		−	0.0384324i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	22.5481			1.84721			0.923607	−	0.383341i	$-0.125227\pi$
$149$	22.5481			1.84721			0.923607	+	0.383341i	$0.125227\pi$
$150$		0			0
$151$			17.8434i			1.45208i	0.687654	+	0.726039i	$0.258641\pi$
$151$			17.8434i			1.45208i	−0.687654	+	0.726039i	$0.741359\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	8.00000i		−	0.642575i
$156$		0			0
$157$		−	5.11373i		−	0.408120i	−0.978958	−	0.204060i	$-0.934586\pi$
$157$		−	5.11373i		−	0.408120i	0.978958	−	0.204060i	$-0.0654138\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.343146	+	0.0289197i	0.0270437	+	0.00227919i
$162$		0			0
$163$			10.3012i			0.806853i	0.915012	+	0.403427i	$0.132181\pi$
$163$			10.3012i			0.806853i	−0.915012	+	0.403427i	$0.867819\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	14.0250			1.08529			0.542643	−	0.839963i	$-0.317424\pi$
$167$	14.0250			1.08529			0.542643	+	0.839963i	$0.317424\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	8.02892i		−	0.610427i	−0.952284	−	0.305214i	$-0.901272\pi$
$173$		−	8.02892i		−	0.610427i	0.952284	−	0.305214i	$-0.0987279\pi$
$174$		0			0
$175$	0.0790206	−	0.937617i	0.00597340	−	0.0708772i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			21.1043i			1.57741i	0.614774	+	0.788703i	$0.289247\pi$
$179$			21.1043i			1.57741i	−0.614774	+	0.788703i	$0.710753\pi$
$180$		0			0
$181$		−	16.2060i		−	1.20458i	−0.798276	−	0.602292i	$-0.794254\pi$
$181$		−	16.2060i		−	1.20458i	0.798276	−	0.602292i	$-0.205746\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	26.7481i		−	1.96656i
$186$		0			0
$187$	8.30121			0.607045
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	7.60953i		−	0.550606i	−0.961357	−	0.275303i	$-0.911222\pi$
$191$		−	7.60953i		−	0.550606i	0.961357	−	0.275303i	$-0.0887782\pi$
$192$		0			0
$193$	10.6694			0.767997			0.383998	−	0.923334i	$-0.374547\pi$
$193$	10.6694			0.767997			0.383998	+	0.923334i	$0.374547\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.68879			0.334062			0.167031	−	0.985952i	$-0.446582\pi$
$197$	4.68879			0.334062			0.167031	+	0.985952i	$0.446582\pi$
$198$		0			0
$199$	22.4694			1.59281			0.796406	−	0.604762i	$-0.206732\pi$
$199$	22.4694			1.59281			0.796406	+	0.604762i	$0.206732\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.0444327	+	0.527215i	−0.00311857	+	0.0370032i
$204$		0			0
$205$	18.4469			1.28839
$206$		0			0
$207$		0			0
$208$		0			0
$209$			28.3421i			1.96046i
$210$		0			0
$211$		−	15.4185i		−	1.06145i	−0.847543	−	0.530726i	$-0.821919\pi$
$211$		−	15.4185i		−	1.06145i	0.847543	−	0.530726i	$-0.178081\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.1457			−0.691929
$216$		0			0
$217$	−0.768089	+	9.11373i	−0.0521413	+	0.618681i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.54562			−0.440306
$222$		0			0
$223$	−2.86933			−0.192144			−0.0960721	−	0.995374i	$-0.530628\pi$
$223$	−2.86933			−0.192144			−0.0960721	+	0.995374i	$0.530628\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.91376			0.193393			0.0966965	−	0.995314i	$-0.469172\pi$
$227$	2.91376			0.193393			0.0966965	+	0.995314i	$0.469172\pi$
$228$		0			0
$229$			20.0853i			1.32728i	0.748054	+	0.663638i	$0.230988\pi$
$229$			20.0853i			1.32728i	−0.748054	+	0.663638i	$0.769012\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.343146			0.0224802			0.0112401	−	0.999937i	$-0.496422\pi$
$233$	0.343146			0.0224802			0.0112401	+	0.999937i	$0.496422\pi$
$234$		0			0
$235$		−	15.1480i		−	0.988149i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			8.56113i			0.553774i	0.960903	+	0.276887i	$0.0893027\pi$
$239$			8.56113i			0.553774i	−0.960903	+	0.276887i	$0.910697\pi$
$240$		0			0
$241$		−	6.40598i		−	0.412646i	−0.978484	−	0.206323i	$-0.933850\pi$
$241$		−	6.40598i		−	0.412646i	0.978484	−	0.206323i	$-0.0661497\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.71128	+	15.9711i	−0.173218	+	1.02035i
$246$		0			0
$247$		−	22.3481i		−	1.42198i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−24.2025			−1.52765			−0.763823	−	0.645425i	$-0.776680\pi$
$251$	−24.2025			−1.52765			−0.763823	+	0.645425i	$0.776680\pi$
$252$		0			0
$253$	−0.466877			−0.0293523
$254$		0			0
$255$		0			0
$256$		0			0
$257$			1.22798i			0.0765996i	0.999266	+	0.0382998i	$0.0121942\pi$
$257$			1.22798i			0.0765996i	−0.999266	+	0.0382998i	$0.987806\pi$
$258$		0			0
$259$	−2.56811	+	30.4719i	−0.159575	+	1.89343i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	6.01551i		−	0.370932i	−0.982651	−	0.185466i	$-0.940620\pi$
$263$		−	6.01551i		−	0.370932i	0.982651	−	0.185466i	$-0.0593795\pi$
$264$		0			0
$265$		−	23.9422i		−	1.47076i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	22.3771i		−	1.36435i	−0.731187	−	0.682177i	$-0.761033\pi$
$269$		−	22.3771i		−	1.36435i	0.731187	−	0.682177i	$-0.238967\pi$
$270$		0			0
$271$	9.63436			0.585246			0.292623	−	0.956228i	$-0.405472\pi$
$271$	9.63436			0.585246			0.292623	+	0.956228i	$0.405472\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			1.27570i			0.0769277i
$276$		0			0
$277$	−11.9581			−0.718491			−0.359245	−	0.933243i	$-0.616966\pi$
$277$	−11.9581			−0.718491			−0.359245	+	0.933243i	$0.616966\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	17.6818			1.05481			0.527405	−	0.849614i	$-0.323165\pi$
$281$	17.6818			1.05481			0.527405	+	0.849614i	$0.323165\pi$
$282$		0			0
$283$	3.50131			0.208131			0.104066	−	0.994570i	$-0.466815\pi$
$283$	3.50131			0.208131			0.104066	+	0.994570i	$0.466815\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−21.0150	−	1.77111i	−1.24048	−	0.104545i
$288$		0			0
$289$	11.6444			0.684962
$290$		0			0
$291$		0			0
$292$		0			0
$293$			3.40047i			0.198657i	0.995055	+	0.0993287i	$0.0316695\pi$
$293$			3.40047i			0.198657i	−0.995055	+	0.0993287i	$0.968330\pi$
$294$		0			0
$295$		−	22.3481i		−	1.30116i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.368139			0.0212900
$300$		0			0
$301$	11.5581	+	0.974097i	0.666199	+	0.0561460i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	14.3481			0.821572
$306$		0			0
$307$	12.1037			0.690797			0.345398	−	0.938456i	$-0.387744\pi$
$307$	12.1037			0.690797			0.345398	+	0.938456i	$0.387744\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−26.7481			−1.51675			−0.758373	−	0.651821i	$-0.774005\pi$
$311$	−26.7481			−1.51675			−0.758373	+	0.651821i	$0.774005\pi$
$312$		0			0
$313$			27.8343i			1.57329i	0.617406	+	0.786645i	$0.288184\pi$
$313$			27.8343i			1.57329i	−0.617406	+	0.786645i	$0.711816\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.85683			0.328952			0.164476	−	0.986381i	$-0.447407\pi$
$317$	5.85683			0.328952			0.164476	+	0.986381i	$0.447407\pi$
$318$		0			0
$319$		−	0.717318i		−	0.0401621i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	18.2853i		−	1.01742i
$324$		0			0
$325$		−	1.00591i		−	0.0557977i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.45438	+	17.2569i	−0.0801826	+	0.951404i
$330$		0			0
$331$		−	2.84787i		−	0.156533i	−0.996932	−	0.0782665i	$-0.975062\pi$
$331$		−	2.84787i		−	0.156533i	0.996932	−	0.0782665i	$-0.0249385\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−20.8570			−1.13954
$336$		0			0
$337$	−10.4888			−0.571362			−0.285681	−	0.958325i	$-0.592220\pi$
$337$	−10.4888			−0.571362			−0.285681	+	0.958325i	$0.592220\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	12.4000i		−	0.671495i
$342$		0			0
$343$	4.62214	−	17.9342i	0.249572	−	0.968356i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	21.8145i		−	1.17106i	−0.810649	−	0.585532i	$-0.800886\pi$
$347$		−	21.8145i		−	1.17106i	0.810649	−	0.585532i	$-0.199114\pi$
$348$		0			0
$349$			28.7188i			1.53728i	0.639681	+	0.768641i	$0.279066\pi$
$349$			28.7188i			1.53728i	−0.639681	+	0.768641i	$0.720934\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	5.91428i		−	0.314785i	−0.987536	−	0.157393i	$-0.949691\pi$
$353$		−	5.91428i		−	0.314785i	0.987536	−	0.157393i	$-0.0503088\pi$
$354$		0			0
$355$	−11.0125			−0.584483
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	20.4723i		−	1.08048i	−0.841509	−	0.540242i	$-0.818333\pi$
$359$		−	20.4723i		−	1.08048i	0.841509	−	0.540242i	$-0.181667\pi$
$360$		0			0
$361$	43.4299			2.28579
$362$		0			0
$363$		0			0
$364$		0			0
$365$	23.8025			1.24588
$366$		0			0
$367$	28.4494			1.48505			0.742523	−	0.669821i	$-0.233629\pi$
$367$	28.4494			1.48505			0.742523	+	0.669821i	$0.233629\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.29871	+	27.2753i	−0.119343	+	1.41606i
$372$		0			0
$373$	−26.2913			−1.36131			−0.680657	−	0.732602i	$-0.738306\pi$
$373$	−26.2913			−1.36131			−0.680657	+	0.732602i	$0.738306\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			0.565615i			0.0291306i
$378$		0			0
$379$			9.44347i			0.485079i	0.970142	+	0.242539i	$0.0779803\pi$
$379$			9.44347i			0.485079i	−0.970142	+	0.242539i	$0.922020\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.14567			0.109639			0.0548193	−	0.998496i	$-0.482542\pi$
$383$	2.14567			0.109639			0.0548193	+	0.998496i	$0.482542\pi$
$384$		0			0
$385$	1.84446	−	21.8854i	0.0940023	−	1.11538i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−20.7388			−1.05150			−0.525749	−	0.850640i	$-0.676215\pi$
$389$	−20.7388			−1.05150			−0.525749	+	0.850640i	$0.676215\pi$
$390$		0			0
$391$	0.301212			0.0152329
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.82252			0.494225
$396$		0			0
$397$			19.8568i			0.996586i	0.867009	+	0.498293i	$0.166040\pi$
$397$			19.8568i			0.996586i	−0.867009	+	0.498293i	$0.833960\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.4249			−0.620472			−0.310236	−	0.950660i	$-0.600408\pi$
$401$	−12.4249			−0.620472			−0.310236	+	0.950660i	$0.600408\pi$
$402$		0			0
$403$			9.77753i			0.487054i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	41.4594i		−	2.05507i
$408$		0			0
$409$			9.54903i			0.472169i	0.971733	+	0.236085i	$0.0758642\pi$
$409$			9.54903i			0.472169i	−0.971733	+	0.236085i	$0.924136\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.14567	+	25.4594i	−0.105582	+	1.25277i
$414$		0			0
$415$			1.77753i			0.0872556i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	21.2569			1.03847			0.519234	−	0.854632i	$-0.326217\pi$
$419$	21.2569			1.03847			0.519234	+	0.854632i	$0.326217\pi$
$420$		0			0
$421$	−19.1582			−0.933712			−0.466856	−	0.884333i	$-0.654613\pi$
$421$	−19.1582			−0.933712			−0.466856	+	0.884333i	$0.654613\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	0.823037i		−	0.0399231i
$426$		0			0
$427$	−16.3456	−	1.37758i	−0.791021	−	0.0666658i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			23.3042i			1.12253i	0.827638	+	0.561263i	$0.189684\pi$
$431$			23.3042i			1.12253i	−0.827638	+	0.561263i	$0.810316\pi$
$432$		0			0
$433$		−	33.9482i		−	1.63145i	−0.578443	−	0.815723i	$-0.696339\pi$
$433$		−	33.9482i		−	1.63145i	0.578443	−	0.815723i	$-0.303661\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			1.02840i			0.0491951i
$438$		0			0
$439$	−9.13567			−0.436022			−0.218011	−	0.975946i	$-0.569957\pi$
$439$	−9.13567			−0.436022			−0.218011	+	0.975946i	$0.569957\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			28.5258i			1.35530i	0.735384	+	0.677651i	$0.237002\pi$
$443$			28.5258i			1.35530i	−0.735384	+	0.677651i	$0.762998\pi$
$444$		0			0
$445$	−41.0743			−1.94711
$446$		0			0
$447$		0			0
$448$		0			0
$449$	34.3731			1.62217			0.811084	−	0.584929i	$-0.198878\pi$
$449$	34.3731			1.62217			0.811084	+	0.584929i	$0.198878\pi$
$450$		0			0
$451$	28.5926			1.34637
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.45438	+	17.2569i	−0.0681824	+	0.809016i
$456$		0			0
$457$	−34.8938			−1.63226			−0.816131	−	0.577867i	$-0.803885\pi$
$457$	−34.8938			−1.63226			−0.816131	+	0.577867i	$0.803885\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			21.2908i			0.991612i	0.868433	+	0.495806i	$0.165127\pi$
$461$			21.2908i			0.991612i	−0.868433	+	0.495806i	$0.834873\pi$
$462$		0			0
$463$			1.33565i			0.0620728i	0.999518	+	0.0310364i	$0.00988078\pi$
$463$			1.33565i			0.0620728i	−0.999518	+	0.0310364i	$0.990119\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.98000			0.461820			0.230910	−	0.972975i	$-0.425830\pi$
$467$	9.98000			0.461820			0.230910	+	0.972975i	$0.425830\pi$
$468$		0			0
$469$	23.7606	+	2.00250i	1.09716	+	0.0924668i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−15.7257			−0.723070
$474$		0			0
$475$	2.81002			0.128933
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.42939			0.156693			0.0783464	−	0.996926i	$-0.475036\pi$
$479$	3.42939			0.156693			0.0783464	+	0.996926i	$0.475036\pi$
$480$		0			0
$481$			32.6913i			1.49059i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−20.8938			−0.948737
$486$		0			0
$487$			20.7522i			0.940371i	0.882568	+	0.470186i	$0.155813\pi$
$487$			20.7522i			0.940371i	−0.882568	+	0.470186i	$0.844187\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	36.6154i		−	1.65243i	−0.563354	−	0.826216i	$-0.690489\pi$
$491$		−	36.6154i		−	1.65243i	0.563354	−	0.826216i	$-0.309511\pi$
$492$		0			0
$493$			0.462787i			0.0208429i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.5456	+	1.05732i	0.562748	+	0.0474274i
$498$		0			0
$499$			34.7004i			1.55340i	0.629869	+	0.776701i	$0.283108\pi$
$499$			34.7004i			1.55340i	−0.629869	+	0.776701i	$0.716892\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	41.8593			1.86642			0.933208	−	0.359338i	$-0.116997\pi$
$503$	41.8593			1.86642			0.933208	+	0.359338i	$0.116997\pi$
$504$		0			0
$505$	31.5381			1.40343
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	37.2908i		−	1.65289i	−0.563020	−	0.826443i	$-0.690361\pi$
$509$		−	37.2908i		−	1.65289i	0.563020	−	0.826443i	$-0.309639\pi$
$510$		0			0
$511$	−27.1162	−	2.28531i	−1.19955	−	0.101096i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			26.5138i			1.16834i
$516$		0			0
$517$		−	23.4794i		−	1.03262i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			7.46330i			0.326973i	0.986546	+	0.163487i	$0.0522741\pi$
$521$			7.46330i			0.326973i	−0.986546	+	0.163487i	$0.947726\pi$
$522$		0			0
$523$	−2.17748			−0.0952146			−0.0476073	−	0.998866i	$-0.515160\pi$
$523$	−2.17748			−0.0952146			−0.0476073	+	0.998866i	$0.515160\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			8.00000i			0.348485i
$528$		0			0
$529$	22.9831			0.999263
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−22.5456			−0.976559
$534$		0			0
$535$	−16.1037			−0.696225
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.20247	+	24.7551i	−0.181013	+	1.06628i
$540$		0			0
$541$	16.1555			0.694581			0.347291	−	0.937758i	$-0.387102\pi$
$541$	16.1555			0.694581			0.347291	+	0.937758i	$0.387102\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	6.68526i		−	0.286365i
$546$		0			0
$547$		−	3.06930i		−	0.131234i	−0.997845	−	0.0656169i	$-0.979098\pi$
$547$		−	3.06930i		−	0.131234i	0.997845	−	0.0656169i	$-0.0209015\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.58005			−0.0673126
$552$		0			0
$553$	−11.1900	−	0.943071i	−0.475846	−	0.0401034i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.7656			−0.540895			−0.270448	−	0.962735i	$-0.587172\pi$
$557$	−12.7656			−0.540895			−0.270448	+	0.962735i	$0.587172\pi$
$558$		0			0
$559$	12.4000			0.524462
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17.5044			−0.737721			−0.368861	−	0.929485i	$-0.620252\pi$
$563$	−17.5044			−0.737721			−0.368861	+	0.929485i	$0.620252\pi$
$564$		0			0
$565$			5.48041i			0.230563i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.47937			0.229707			0.114854	−	0.993382i	$-0.463360\pi$
$569$	5.47937			0.229707			0.114854	+	0.993382i	$0.463360\pi$
$570$		0			0
$571$			3.67380i			0.153744i	0.997041	+	0.0768718i	$0.0244932\pi$
$571$			3.67380i			0.153744i	−0.997041	+	0.0768718i	$0.975507\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			0.0462892i			0.00193039i
$576$		0			0
$577$			36.0050i			1.49891i	0.662057	+	0.749454i	$0.269684\pi$
$577$			36.0050i			1.49891i	−0.662057	+	0.749454i	$0.730316\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.170663	−	2.02499i	0.00708028	−	0.0840109i
$582$		0			0
$583$		−	37.1102i		−	1.53695i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.2663			−0.753933			−0.376966	−	0.926227i	$-0.623033\pi$
$587$	−18.2663			−0.753933			−0.376966	+	0.926227i	$0.623033\pi$
$588$		0			0
$589$	−27.3137			−1.12544
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	24.4789i		−	1.00523i	−0.864511	−	0.502613i	$-0.832372\pi$
$593$		−	24.4789i		−	1.00523i	0.864511	−	0.502613i	$-0.167628\pi$
$594$		0			0
$595$	−1.18998	+	14.1196i	−0.0487843	+	0.578849i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			27.1257i			1.10833i	0.832408	+	0.554163i	$0.186962\pi$
$599$			27.1257i			1.10833i	−0.832408	+	0.554163i	$0.813038\pi$
$600$		0			0
$601$			46.5188i			1.89754i	0.315965	+	0.948771i	$0.397672\pi$
$601$			46.5188i			1.89754i	−0.315965	+	0.948771i	$0.602328\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			4.32026i			0.175644i
$606$		0			0
$607$	−15.9606			−0.647819			−0.323910	−	0.946088i	$-0.604997\pi$
$607$	−15.9606			−0.647819			−0.323910	+	0.946088i	$0.604997\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			18.5138i			0.748989i
$612$		0			0
$613$	−7.33870			−0.296407			−0.148204	−	0.988957i	$-0.547349\pi$
$613$	−7.33870			−0.296407			−0.148204	+	0.988957i	$0.547349\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−38.6524			−1.55609			−0.778044	−	0.628210i	$-0.783788\pi$
$617$	−38.6524			−1.55609			−0.778044	+	0.628210i	$0.783788\pi$
$618$		0			0
$619$	−26.8688			−1.07995			−0.539974	−	0.841682i	$-0.681566\pi$
$619$	−26.8688			−1.07995			−0.539974	+	0.841682i	$0.681566\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	46.7925	+	3.94359i	1.87470	+	0.157997i
$624$		0			0
$625$	−26.6517			−1.06607
$626$		0			0
$627$		0			0
$628$		0			0
$629$			26.7481i			1.06652i
$630$		0			0
$631$		−	22.7900i		−	0.907257i	−0.891191	−	0.453628i	$-0.850129\pi$
$631$		−	22.7900i		−	0.907257i	0.891191	−	0.453628i	$-0.149871\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17.4912			0.694117
$636$		0			0
$637$	3.31371	−	19.5197i	0.131294	−	0.773399i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.3776			−0.607378			−0.303689	−	0.952771i	$-0.598218\pi$
$641$	−15.3776			−0.607378			−0.303689	+	0.952771i	$0.598218\pi$
$642$		0			0
$643$	−29.1900			−1.15114			−0.575570	−	0.817752i	$-0.695220\pi$
$643$	−29.1900			−1.15114			−0.575570	+	0.817752i	$0.695220\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.2275			0.402083			0.201042	−	0.979583i	$-0.435567\pi$
$647$	10.2275			0.402083			0.201042	+	0.979583i	$0.435567\pi$
$648$		0			0
$649$		−	34.6395i		−	1.35972i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.48187			0.292788			0.146394	−	0.989226i	$-0.453233\pi$
$653$	7.48187			0.292788			0.146394	+	0.989226i	$0.453233\pi$
$654$		0			0
$655$			40.4049i			1.57875i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			27.9602i			1.08917i	0.838704	+	0.544587i	$0.183314\pi$
$659$			27.9602i			1.08917i	−0.838704	+	0.544587i	$0.816686\pi$
$660$		0			0
$661$			9.23441i			0.359177i	0.983742	+	0.179588i	$0.0574766\pi$
$661$			9.23441i			0.359177i	−0.983742	+	0.179588i	$0.942523\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−48.2075	−	4.06284i	−1.86941	−	0.157550i
$666$		0			0
$667$		−	0.0260281i		−	0.00100781i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	22.2395			0.858548
$672$		0			0
$673$	14.9607			0.576692			0.288346	−	0.957526i	$-0.406895\pi$
$673$	14.9607			0.576692			0.288346	+	0.957526i	$0.406895\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	5.45727i		−	0.209740i	−0.994486	−	0.104870i	$-0.966557\pi$
$677$		−	5.45727i		−	0.209740i	0.994486	−	0.104870i	$-0.0334426\pi$
$678$		0			0
$679$	23.8025	+	2.00603i	0.913457	+	0.0769845i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	25.4274i		−	0.972953i	−0.873694	−	0.486476i	$-0.838282\pi$
$683$		−	25.4274i		−	0.972953i	0.873694	−	0.486476i	$-0.161718\pi$
$684$		0			0
$685$		−	42.1246i		−	1.60950i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			29.2619i			1.11479i
$690$		0			0
$691$	29.0962			1.10687			0.553437	−	0.832891i	$-0.313316\pi$
$691$	29.0962			1.10687			0.553437	+	0.832891i	$0.313316\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	35.4394i		−	1.34429i
$696$		0			0
$697$	−18.4469			−0.698725
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−45.8618			−1.73218			−0.866089	−	0.499890i	$-0.833374\pi$
$701$	−45.8618			−1.73218			−0.866089	+	0.499890i	$0.833374\pi$
$702$		0			0
$703$	−91.3237			−3.44434
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−35.9288	−	3.02801i	−1.35124	−	0.113880i
$708$		0			0
$709$	−12.1775			−0.457335			−0.228667	−	0.973505i	$-0.573437\pi$
$709$	−12.1775			−0.457335			−0.228667	+	0.973505i	$0.573437\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	0.449936i		−	0.0168502i
$714$		0			0
$715$		−	23.4794i		−	0.878079i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12.5706			−0.468805			−0.234402	−	0.972140i	$-0.575313\pi$
$719$	−12.5706			−0.468805			−0.234402	+	0.972140i	$0.575313\pi$
$720$		0			0
$721$	2.54562	−	30.2050i	0.0948038	−	1.12489i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.0711196			−0.00264131
$726$		0			0
$727$	−9.22941			−0.342300			−0.171150	−	0.985245i	$-0.554748\pi$
$727$	−9.22941			−0.342300			−0.171150	+	0.985245i	$0.554748\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.1457			0.375251
$732$		0			0
$733$		−	25.2216i		−	0.931580i	−0.884895	−	0.465790i	$-0.845770\pi$
$733$		−	25.2216i		−	0.931580i	0.884895	−	0.465790i	$-0.154230\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−32.3281			−1.19082
$738$		0			0
$739$			4.93070i			0.181379i	0.995879	+	0.0906894i	$0.0289070\pi$
$739$			4.93070i			0.181379i	−0.995879	+	0.0906894i	$0.971093\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	50.3126i		−	1.84579i	−0.385050	−	0.922896i	$-0.625816\pi$
$743$		−	50.3126i		−	1.84579i	0.385050	−	0.922896i	$-0.374184\pi$
$744$		0			0
$745$		−	52.1814i		−	1.91178i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.3456	+	1.54614i	0.670335	+	0.0564946i
$750$		0			0
$751$		−	27.1321i		−	0.990066i	−0.868874	−	0.495033i	$-0.835156\pi$
$751$		−	27.1321i		−	0.990066i	0.868874	−	0.495033i	$-0.164844\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	41.2937			1.50283
$756$		0			0
$757$	34.6913			1.26088			0.630438	−	0.776239i	$-0.282875\pi$
$757$	34.6913			1.26088			0.630438	+	0.776239i	$0.282875\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	19.9082i		−	0.721673i	−0.932629	−	0.360837i	$-0.882491\pi$
$761$		−	19.9082i		−	0.721673i	0.932629	−	0.360837i	$-0.117509\pi$
$762$		0			0
$763$	−0.641859	+	7.61596i	−0.0232368	+	0.275716i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			27.3137i			0.986241i
$768$		0			0
$769$			3.83434i			0.138270i	0.997607	+	0.0691348i	$0.0220239\pi$
$769$			3.83434i			0.138270i	−0.997607	+	0.0691348i	$0.977976\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1.33660i		−	0.0480740i	−0.999711	−	0.0240370i	$-0.992348\pi$
$773$		−	1.33660i		−	0.0480740i	0.999711	−	0.0240370i	$-0.00765195\pi$
$774$		0			0
$775$	−1.22941			−0.0441618
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	62.9816i		−	2.25655i
$780$		0			0
$781$	−17.0693			−0.610788
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.8343			−0.422386
$786$		0			0
$787$	−2.11386			−0.0753509			−0.0376755	−	0.999290i	$-0.511995\pi$
$787$	−2.11386			−0.0753509			−0.0376755	+	0.999290i	$0.511995\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.526180	−	6.24337i	0.0187088	−	0.221989i
$792$		0			0
$793$	−17.5362			−0.622728
$794$		0			0
$795$		0			0
$796$		0			0
$797$			31.1770i			1.10434i	0.833730	+	0.552172i	$0.186201\pi$
$797$			31.1770i			1.10434i	−0.833730	+	0.552172i	$0.813799\pi$
$798$		0			0
$799$			15.1480i			0.535899i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	36.8938			1.30195
$804$		0			0
$805$	0.0669267	−	0.794117i	0.00235886	−	0.0279889i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	40.6395			1.42881			0.714404	−	0.699733i	$-0.246698\pi$
$809$	40.6395			1.42881			0.714404	+	0.699733i	$0.246698\pi$
$810$		0			0
$811$	−11.3187			−0.397454			−0.198727	−	0.980055i	$-0.563681\pi$
$811$	−11.3187			−0.397454			−0.198727	+	0.980055i	$0.563681\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	23.8393			0.835055
$816$		0			0
$817$			34.6395i			1.21188i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−37.2594			−1.30036			−0.650181	−	0.759779i	$-0.725307\pi$
$821$	−37.2594			−1.30036			−0.650181	+	0.759779i	$0.725307\pi$
$822$		0			0
$823$			50.9975i			1.77766i	0.458236	+	0.888831i	$0.348481\pi$
$823$			50.9975i			1.77766i	−0.458236	+	0.888831i	$0.651519\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	41.2439i		−	1.43419i	−0.696975	−	0.717095i	$-0.745471\pi$
$827$		−	41.2439i		−	1.43419i	0.696975	−	0.717095i	$-0.254529\pi$
$828$		0			0
$829$			10.8734i			0.377649i	0.982011	+	0.188825i	$0.0604678\pi$
$829$			10.8734i			0.377649i	−0.982011	+	0.188825i	$0.939532\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.71128	−	15.9711i	0.0939404	−	0.553365i
$834$		0			0
$835$		−	32.4570i		−	1.12322i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−36.5438			−1.26163			−0.630816	−	0.775932i	$-0.717280\pi$
$839$	−36.5438			−1.26163			−0.630816	+	0.775932i	$0.717280\pi$
$840$		0			0
$841$	−28.9600			−0.998621
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	11.5711i		−	0.398059i
$846$		0			0
$847$	0.414793	−	4.92171i	0.0142524	−	0.169112i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1.50437i		−	0.0515690i
$852$		0			0
$853$			4.09818i			0.140319i	0.997536	+	0.0701596i	$0.0223508\pi$
$853$			4.09818i			0.140319i	−0.997536	+	0.0701596i	$0.977649\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			7.40547i			0.252966i	0.991969	+	0.126483i	$0.0403689\pi$
$857$			7.40547i			0.252966i	−0.991969	+	0.126483i	$0.959631\pi$
$858$		0			0
$859$	−22.5287			−0.768669			−0.384334	−	0.923194i	$-0.625569\pi$
$859$	−22.5287			−0.768669			−0.384334	+	0.923194i	$0.625569\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			24.2748i			0.826324i	0.910658	+	0.413162i	$0.135576\pi$
$863$			24.2748i			0.826324i	−0.910658	+	0.413162i	$0.864424\pi$
$864$		0			0
$865$	−18.5807			−0.631764
$866$		0			0
$867$		0			0
$868$		0			0
$869$	15.2248			0.516467
$870$		0			0
$871$	25.4912			0.863736
$872$		0			0
$873$		0			0
$874$		0			0
$875$	28.3363	+	2.38813i	0.957943	+	0.0807337i
$876$		0			0
$877$	4.46993			0.150939			0.0754694	−	0.997148i	$-0.475954\pi$
$877$	4.46993			0.150939			0.0754694	+	0.997148i	$0.475954\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	19.8504i		−	0.668777i	−0.942435	−	0.334389i	$-0.891470\pi$
$881$		−	19.8504i		−	0.668777i	0.942435	−	0.334389i	$-0.108530\pi$
$882$		0			0
$883$		−	42.2116i		−	1.42053i	−0.703933	−	0.710266i	$-0.748575\pi$
$883$		−	42.2116i		−	1.42053i	0.703933	−	0.710266i	$-0.251425\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.22509			0.175441			0.0877207	−	0.996145i	$-0.472042\pi$
$887$	5.22509			0.175441			0.0877207	+	0.996145i	$0.472042\pi$
$888$		0			0
$889$	−19.9263	−	1.67935i	−0.668305	−	0.0563235i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−51.7187			−1.73070
$894$		0			0
$895$	48.8400			1.63254
$896$		0			0
$897$		0			0
$898$		0			0
$899$	0.691289			0.0230558
$900$		0			0
$901$			23.9422i			0.797629i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−37.5044			−1.24669
$906$		0			0
$907$		−	51.1371i		−	1.69798i	−0.528408	−	0.848990i	$-0.677211\pi$
$907$		−	51.1371i		−	1.69798i	0.528408	−	0.848990i	$-0.322789\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	18.9811i		−	0.628871i	−0.949279	−	0.314436i	$-0.898185\pi$
$911$		−	18.9811i		−	0.628871i	0.949279	−	0.314436i	$-0.101815\pi$
$912$		0			0
$913$			2.75516i			0.0911826i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.87932	−	46.0300i	0.128107	−	1.52004i
$918$		0			0
$919$		−	12.3890i		−	0.408677i	−0.978900	−	0.204338i	$-0.934496\pi$
$919$		−	12.3890i		−	0.408677i	0.978900	−	0.204338i	$-0.0655043\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13.4594			0.443021
$924$		0			0
$925$	−4.11055			−0.135154
$926$		0			0
$927$		0			0
$928$		0			0
$929$			2.87984i			0.0944845i	0.998883	+	0.0472423i	$0.0150433\pi$
$929$			2.87984i			0.0944845i	−0.998883	+	0.0472423i	$0.984957\pi$
$930$		0			0
$931$	54.5287	+	9.25690i	1.78711	+	0.303383i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	19.2109i		−	0.628263i
$936$		0			0
$937$		−	27.5551i		−	0.900185i	−0.892982	−	0.450092i	$-0.851391\pi$
$937$		−	27.5551i		−	0.900185i	0.892982	−	0.450092i	$-0.148609\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	0.707358i		−	0.0230592i	−0.999934	−	0.0115296i	$-0.996330\pi$
$941$		−	0.707358i		−	0.0230592i	0.999934	−	0.0115296i	$-0.00367007\pi$
$942$		0			0
$943$	1.03749			0.0337853
$944$		0			0
$945$		0			0
$946$		0			0
$947$			4.64725i			0.151015i	0.997145	+	0.0755077i	$0.0240577\pi$
$947$			4.64725i			0.151015i	−0.997145	+	0.0755077i	$0.975942\pi$
$948$		0			0
$949$	−29.0912			−0.944342
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−6.77516			−0.219469			−0.109734	−	0.993961i	$-0.535000\pi$
$953$	−6.77516			−0.219469			−0.109734	+	0.993961i	$0.535000\pi$
$954$		0			0
$955$	−17.6102			−0.569852
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.04443	+	47.9891i	−0.130602	+	1.54965i
$960$		0			0
$961$	−19.0500			−0.614516
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	24.6913i		−	0.794841i
$966$		0			0
$967$			15.0375i			0.483573i	0.970329	+	0.241787i	$0.0777334\pi$
$967$			15.0375i			0.483573i	−0.970329	+	0.241787i	$0.922267\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−35.5601			−1.14118			−0.570588	−	0.821236i	$-0.693285\pi$
$971$	−35.5601			−1.14118			−0.570588	+	0.821236i	$0.693285\pi$
$972$		0			0
$973$	−3.40257	+	40.3731i	−0.109082	+	1.29430i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−44.8370			−1.43446			−0.717231	−	0.696836i	$-0.754591\pi$
$977$	−44.8370			−1.43446			−0.717231	+	0.696836i	$0.754591\pi$
$978$		0			0
$979$	−63.6649			−2.03474
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.1386			0.578532			0.289266	−	0.957249i	$-0.406589\pi$
$983$	18.1386			0.578532			0.289266	+	0.957249i	$0.406589\pi$
$984$		0			0
$985$		−	10.8509i		−	0.345739i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.570613			−0.0181444
$990$		0			0
$991$			40.0659i			1.27273i	0.771386	+	0.636367i	$0.219564\pi$
$991$			40.0659i			1.27273i	−0.771386	+	0.636367i	$0.780436\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	51.9992i		−	1.64849i
$996$		0			0
$997$			44.7824i			1.41827i	0.705071	+	0.709136i	$0.250915\pi$
$997$			44.7824i			1.41827i	−0.705071	+	0.709136i	$0.749085\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.b.q.3583.2		8
3.2	odd	2		1344.2.b.g.895.7		8
4.3	odd	2		4032.2.b.o.3583.2		8
7.6	odd	2		4032.2.b.o.3583.7		8
8.3	odd	2		2016.2.b.a.1567.7		8
8.5	even	2		2016.2.b.c.1567.7		8
12.11	even	2		1344.2.b.h.895.7		8
21.20	even	2		1344.2.b.h.895.2		8
24.5	odd	2		672.2.b.b.223.2	yes	8
24.11	even	2		672.2.b.a.223.2	✓	8
28.27	even	2	inner	4032.2.b.q.3583.7		8
56.13	odd	2		2016.2.b.a.1567.2		8
56.27	even	2		2016.2.b.c.1567.2		8
84.83	odd	2		1344.2.b.g.895.2		8
168.83	odd	2		672.2.b.b.223.7	yes	8
168.125	even	2		672.2.b.a.223.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.b.a.223.2	✓	8	24.11	even	2
672.2.b.a.223.7	yes	8	168.125	even	2
672.2.b.b.223.2	yes	8	24.5	odd	2
672.2.b.b.223.7	yes	8	168.83	odd	2
1344.2.b.g.895.2		8	84.83	odd	2
1344.2.b.g.895.7		8	3.2	odd	2
1344.2.b.h.895.2		8	21.20	even	2
1344.2.b.h.895.7		8	12.11	even	2
2016.2.b.a.1567.2		8	56.13	odd	2
2016.2.b.a.1567.7		8	8.3	odd	2
2016.2.b.c.1567.2		8	56.27	even	2
2016.2.b.c.1567.7		8	8.5	even	2
4032.2.b.o.3583.2		8	4.3	odd	2
4032.2.b.o.3583.7		8	7.6	odd	2
4032.2.b.q.3583.2		8	1.1	even	1	trivial
4032.2.b.q.3583.7		8	28.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.31423i q^{5} +(-0.222191 + 2.63640i) q^{7} +O(q^{10})\)	\(q-2.31423i q^{5} +(-0.222191 + 2.63640i) q^{7} -3.58704i q^{11} +2.82843i q^{13} +2.31423i q^{17} -7.90126 q^{19} -0.130157i q^{23} -0.355642 q^{25} +0.199975 q^{29} +3.45688 q^{31} +(6.10124 + 0.514201i) q^{35} +11.5581 q^{37} +7.97108i q^{41} -4.38404i q^{43} +6.54562 q^{47} +(-6.90126 - 1.17157i) q^{49} +10.3456 q^{53} -8.30121 q^{55} +9.65685 q^{59} +6.19998i q^{61} +6.54562 q^{65} -9.01250i q^{67} -4.75861i q^{71} +10.2853i q^{73} +(9.45688 + 0.797008i) q^{77} +4.24441i q^{79} -0.768089 q^{83} +5.35564 q^{85} -17.7486i q^{89} +(-7.45688 - 0.628452i) q^{91} +18.2853i q^{95} -9.02840i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7}+O(q^{10}) \) 8 * q + 4 * q^7	\( 8 q + 4 q^{7} - 8 q^{19} - 16 q^{25} - 16 q^{31} - 8 q^{35} - 8 q^{37} - 16 q^{47} + 16 q^{53} - 8 q^{55} + 32 q^{59} - 16 q^{65} + 32 q^{77} + 16 q^{83} + 56 q^{85} - 16 q^{91}+O(q^{100}) \) 8 * q + 4 * q^7 - 8 * q^19 - 16 * q^25 - 16 * q^31 - 8 * q^35 - 8 * q^37 - 16 * q^47 + 16 * q^53 - 8 * q^55 + 32 * q^59 - 16 * q^65 + 32 * q^77 + 16 * q^83 + 56 * q^85 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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