LMFDB - Embedded newform 4032.2.v.d.3599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("4032.1583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (of order $4$, degree $2$, 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:	$36$
Relative dimension:	$18$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3599.2
Character	$\chi$	$=$	4032.3599
Dual form			4032.2.v.d.1583.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.53823 + 2.53823i) q^{5} +1.00000 q^{7} +O(q^{10})$	$q+(-2.53823 + 2.53823i) q^{5} +1.00000 q^{7} +(-0.490893 - 0.490893i) q^{11} +(-4.21725 + 4.21725i) q^{13} +6.71835i q^{17} +(5.38631 + 5.38631i) q^{19} -1.37152i q^{23} -7.88518i q^{25} +(1.45725 + 1.45725i) q^{29} -2.66033i q^{31} +(-2.53823 + 2.53823i) q^{35} +(2.41550 + 2.41550i) q^{37} +1.51556 q^{41} +(3.40983 - 3.40983i) q^{43} -13.2646 q^{47} +1.00000 q^{49} +(-9.42484 + 9.42484i) q^{53} +2.49199 q^{55} +(5.46141 + 5.46141i) q^{59} +(8.16928 - 8.16928i) q^{61} -21.4087i q^{65} +(2.47267 + 2.47267i) q^{67} -9.71605i q^{71} +2.38494i q^{73} +(-0.490893 - 0.490893i) q^{77} -4.70360i q^{79} +(-7.75521 + 7.75521i) q^{83} +(-17.0527 - 17.0527i) q^{85} +5.36941 q^{89} +(-4.21725 + 4.21725i) q^{91} -27.3433 q^{95} -3.44726 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 36 q^{7}+O(q^{10}) $ 36 * q + 36 * q^7	$ 36 q + 36 q^{7} - 16 q^{13} + 16 q^{19} + 20 q^{37} - 36 q^{43} + 36 q^{49} - 32 q^{55} + 112 q^{61} + 36 q^{67} - 96 q^{85} - 16 q^{91} + 56 q^{97}+O(q^{100}) $ 36 * q + 36 * q^7 - 16 * q^13 + 16 * q^19 + 20 * q^37 - 36 * q^43 + 36 * q^49 - 32 * q^55 + 112 * q^61 + 36 * q^67 - 96 * q^85 - 16 * q^91 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$5$	−2.53823	+	2.53823i	−1.13513	+	1.13513i	−0.145817	+	0.989312i	$0.546581\pi$
$5$	−2.53823	+	2.53823i	−1.13513	+	1.13513i	−0.989312	+	0.145817i	$0.953419\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.490893	−	0.490893i	−0.148010	−	0.148010i	0.629219	−	0.777228i	$-0.283375\pi$
$11$	−0.490893	−	0.490893i	−0.148010	−	0.148010i	−0.777228	+	0.629219i	$0.783375\pi$
$12$		0			0
$13$	−4.21725	+	4.21725i	−1.16966	+	1.16966i	−0.187366	+	0.982290i	$0.559995\pi$
$13$	−4.21725	+	4.21725i	−1.16966	+	1.16966i	−0.982290	+	0.187366i	$0.940005\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.71835i			1.62944i	0.579856	+	0.814719i	$0.303109\pi$
$17$			6.71835i			1.62944i	−0.579856	+	0.814719i	$0.696891\pi$
$18$		0			0
$19$	5.38631	+	5.38631i	1.23570	+	1.23570i	0.961740	+	0.273965i	$0.0883351\pi$
$19$	5.38631	+	5.38631i	1.23570	+	1.23570i	0.273965	+	0.961740i	$0.411665\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.37152i		−	0.285981i	−0.989724	−	0.142990i	$-0.954328\pi$
$23$		−	1.37152i		−	0.285981i	0.989724	−	0.142990i	$-0.0456718\pi$
$24$		0			0
$25$		−	7.88518i		−	1.57704i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.45725	+	1.45725i	0.270604	+	0.270604i	0.829343	−	0.558739i	$-0.188715\pi$
$29$	1.45725	+	1.45725i	0.270604	+	0.270604i	−0.558739	+	0.829343i	$0.688715\pi$
$30$		0			0
$31$		−	2.66033i		−	0.477809i	−0.971043	−	0.238905i	$-0.923212\pi$
$31$		−	2.66033i		−	0.477809i	0.971043	−	0.238905i	$-0.0767883\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.53823	+	2.53823i	−0.429038	+	0.429038i
$36$		0			0
$37$	2.41550	+	2.41550i	0.397106	+	0.397106i	0.877211	−	0.480105i	$-0.159401\pi$
$37$	2.41550	+	2.41550i	0.397106	+	0.397106i	−0.480105	+	0.877211i	$0.659401\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.51556			0.236690			0.118345	−	0.992973i	$-0.462241\pi$
$41$	1.51556			0.236690			0.118345	+	0.992973i	$0.462241\pi$
$42$		0			0
$43$	3.40983	−	3.40983i	0.519994	−	0.519994i	−0.397575	−	0.917570i	$-0.630148\pi$
$43$	3.40983	−	3.40983i	0.519994	−	0.519994i	0.917570	+	0.397575i	$0.130148\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−13.2646			−1.93484			−0.967418	−	0.253186i	$-0.918521\pi$
$47$	−13.2646			−1.93484			−0.967418	+	0.253186i	$0.918521\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.42484	+	9.42484i	−1.29460	+	1.29460i	−0.362693	+	0.931909i	$0.618143\pi$
$53$	−9.42484	+	9.42484i	−1.29460	+	1.29460i	−0.931909	+	0.362693i	$0.881857\pi$
$54$		0			0
$55$	2.49199			0.336020
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.46141	+	5.46141i	0.711015	+	0.711015i	0.966748	−	0.255732i	$-0.0823165\pi$
$59$	5.46141	+	5.46141i	0.711015	+	0.711015i	−0.255732	+	0.966748i	$0.582317\pi$
$60$		0			0
$61$	8.16928	−	8.16928i	1.04597	−	1.04597i	0.0470778	−	0.998891i	$-0.485009\pi$
$61$	8.16928	−	8.16928i	1.04597	−	1.04597i	0.998891	−	0.0470778i	$-0.0149909\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	21.4087i		−	2.65542i
$66$		0			0
$67$	2.47267	+	2.47267i	0.302084	+	0.302084i	0.841829	−	0.539745i	$-0.181479\pi$
$67$	2.47267	+	2.47267i	0.302084	+	0.302084i	−0.539745	+	0.841829i	$0.681479\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	9.71605i		−	1.15308i	−0.817068	−	0.576541i	$-0.804402\pi$
$71$		−	9.71605i		−	1.15308i	0.817068	−	0.576541i	$-0.195598\pi$
$72$		0			0
$73$			2.38494i			0.279136i	0.990212	+	0.139568i	$0.0445714\pi$
$73$			2.38494i			0.279136i	−0.990212	+	0.139568i	$0.955429\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.490893	−	0.490893i	−0.0559424	−	0.0559424i
$78$		0			0
$79$		−	4.70360i		−	0.529196i	−0.964359	−	0.264598i	$-0.914761\pi$
$79$		−	4.70360i		−	0.529196i	0.964359	−	0.264598i	$-0.0852393\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.75521	+	7.75521i	−0.851245	+	0.851245i	−0.990287	−	0.139042i	$-0.955598\pi$
$83$	−7.75521	+	7.75521i	−0.851245	+	0.851245i	0.139042	+	0.990287i	$0.455598\pi$
$84$		0			0
$85$	−17.0527	−	17.0527i	−1.84962	−	1.84962i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.36941			0.569157			0.284578	−	0.958653i	$-0.408146\pi$
$89$	5.36941			0.569157			0.284578	+	0.958653i	$0.408146\pi$
$90$		0			0
$91$	−4.21725	+	4.21725i	−0.442088	+	0.442088i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−27.3433			−2.80537
$96$		0			0
$97$	−3.44726			−0.350016			−0.175008	−	0.984567i	$-0.555995\pi$
$97$	−3.44726			−0.350016			−0.175008	+	0.984567i	$0.555995\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.49553	+	2.49553i	−0.248314	+	0.248314i	−0.820279	−	0.571964i	$-0.806182\pi$
$101$	−2.49553	+	2.49553i	−0.248314	+	0.248314i	0.571964	+	0.820279i	$0.306182\pi$
$102$		0			0
$103$	−7.88100			−0.776538			−0.388269	−	0.921546i	$-0.626927\pi$
$103$	−7.88100			−0.776538			−0.388269	+	0.921546i	$0.626927\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.26289	−	2.26289i	−0.218761	−	0.218761i	0.589215	−	0.807976i	$-0.299437\pi$
$107$	−2.26289	−	2.26289i	−0.218761	−	0.218761i	−0.807976	+	0.589215i	$0.799437\pi$
$108$		0			0
$109$	7.64611	−	7.64611i	0.732365	−	0.732365i	−0.238723	−	0.971088i	$-0.576729\pi$
$109$	7.64611	−	7.64611i	0.732365	−	0.732365i	0.971088	+	0.238723i	$0.0767288\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			3.74388i			0.352194i	0.984373	+	0.176097i	$0.0563473\pi$
$113$			3.74388i			0.352194i	−0.984373	+	0.176097i	$0.943653\pi$
$114$		0			0
$115$	3.48122	+	3.48122i	0.324625	+	0.324625i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			6.71835i			0.615870i
$120$		0			0
$121$		−	10.5180i		−	0.956186i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.32323	+	7.32323i	0.655010	+	0.655010i
$126$		0			0
$127$			12.6076i			1.11875i	0.828916	+	0.559373i	$0.188958\pi$
$127$			12.6076i			1.11875i	−0.828916	+	0.559373i	$0.811042\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.9113	+	10.9113i	−0.953325	+	0.953325i	−0.998958	−	0.0456330i	$-0.985470\pi$
$131$	−10.9113	+	10.9113i	−0.953325	+	0.953325i	0.0456330	+	0.998958i	$0.485470\pi$
$132$		0			0
$133$	5.38631	+	5.38631i	0.467052	+	0.467052i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.84548			0.755720			0.377860	−	0.925863i	$-0.376660\pi$
$137$	8.84548			0.755720			0.377860	+	0.925863i	$0.376660\pi$
$138$		0			0
$139$	−5.85221	+	5.85221i	−0.496378	+	0.496378i	−0.910309	−	0.413930i	$-0.864156\pi$
$139$	−5.85221	+	5.85221i	−0.496378	+	0.496378i	0.413930	+	0.910309i	$0.364156\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.14044			0.346241
$144$		0			0
$145$	−7.39764			−0.614340
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.71683	+	4.71683i	−0.386418	+	0.386418i	−0.873408	−	0.486990i	$-0.838095\pi$
$149$	−4.71683	+	4.71683i	−0.386418	+	0.386418i	0.486990	+	0.873408i	$0.338095\pi$
$150$		0			0
$151$	−10.6200			−0.864246			−0.432123	−	0.901815i	$-0.642235\pi$
$151$	−10.6200			−0.864246			−0.432123	+	0.901815i	$0.642235\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.75251	+	6.75251i	0.542375	+	0.542375i
$156$		0			0
$157$	−12.6976	+	12.6976i	−1.01338	+	1.01338i	−0.0134693	+	0.999909i	$0.504288\pi$
$157$	−12.6976	+	12.6976i	−1.01338	+	1.01338i	−0.999909	+	0.0134693i	$0.995712\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	1.37152i		−	0.108091i
$162$		0			0
$163$	−17.2947	−	17.2947i	−1.35462	−	1.35462i	−0.880409	−	0.474215i	$-0.842732\pi$
$163$	−17.2947	−	17.2947i	−1.35462	−	1.35462i	−0.474215	−	0.880409i	$-0.657268\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18.1120i		−	1.40155i	−0.713382	−	0.700776i	$-0.752837\pi$
$167$		−	18.1120i		−	1.40155i	0.713382	−	0.700776i	$-0.247163\pi$
$168$		0			0
$169$		−	22.5705i		−	1.73619i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.32043	−	7.32043i	−0.556562	−	0.556562i	0.371765	−	0.928327i	$-0.378753\pi$
$173$	−7.32043	−	7.32043i	−0.556562	−	0.556562i	−0.928327	+	0.371765i	$0.878753\pi$
$174$		0			0
$175$		−	7.88518i		−	0.596063i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.7625	−	10.7625i	0.804427	−	0.804427i	−0.179357	−	0.983784i	$-0.557402\pi$
$179$	10.7625	−	10.7625i	0.804427	−	0.804427i	0.983784	+	0.179357i	$0.0574018\pi$
$180$		0			0
$181$	−10.0479	−	10.0479i	−0.746855	−	0.746855i	0.227033	−	0.973887i	$-0.427098\pi$
$181$	−10.0479	−	10.0479i	−0.746855	−	0.746855i	−0.973887	+	0.227033i	$0.927098\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.2622			−0.901533
$186$		0			0
$187$	3.29799	−	3.29799i	0.241173	−	0.241173i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	19.9473			1.44334			0.721670	−	0.692238i	$-0.243375\pi$
$191$	19.9473			1.44334			0.721670	+	0.692238i	$0.243375\pi$
$192$		0			0
$193$	−14.6426			−1.05400			−0.527000	−	0.849865i	$-0.676683\pi$
$193$	−14.6426			−1.05400			−0.527000	+	0.849865i	$0.676683\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.16271	−	2.16271i	0.154087	−	0.154087i	−0.625854	−	0.779941i	$-0.715249\pi$
$197$	2.16271	−	2.16271i	0.154087	−	0.154087i	0.779941	+	0.625854i	$0.215249\pi$
$198$		0			0
$199$	8.68951			0.615983			0.307991	−	0.951389i	$-0.400343\pi$
$199$	8.68951			0.615983			0.307991	+	0.951389i	$0.400343\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.45725	+	1.45725i	0.102279	+	0.102279i
$204$		0			0
$205$	−3.84682	+	3.84682i	−0.268674	+	0.268674i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	5.28820i		−	0.365792i
$210$		0			0
$211$	−1.21024	−	1.21024i	−0.0833164	−	0.0833164i	0.664220	−	0.747537i	$-0.268764\pi$
$211$	−1.21024	−	1.21024i	−0.0833164	−	0.0833164i	−0.747537	+	0.664220i	$0.768764\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			17.3098i			1.18052i
$216$		0			0
$217$		−	2.66033i		−	0.180595i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−28.3330	−	28.3330i	−1.90588	−	1.90588i
$222$		0			0
$223$		−	0.184249i		−	0.0123382i	−0.999981	−	0.00616910i	$-0.998036\pi$
$223$		−	0.184249i		−	0.0123382i	0.999981	−	0.00616910i	$-0.00196370\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.79571	−	5.79571i	0.384675	−	0.384675i	−0.488108	−	0.872783i	$-0.662313\pi$
$227$	5.79571	−	5.79571i	0.384675	−	0.384675i	0.872783	+	0.488108i	$0.162313\pi$
$228$		0			0
$229$	15.7114	+	15.7114i	1.03824	+	1.03824i	0.999239	+	0.0390024i	$0.0124180\pi$
$229$	15.7114	+	15.7114i	1.03824	+	1.03824i	0.0390024	+	0.999239i	$0.487582\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.79337			0.641585			0.320793	−	0.947149i	$-0.396051\pi$
$233$	9.79337			0.641585			0.320793	+	0.947149i	$0.396051\pi$
$234$		0			0
$235$	33.6684	−	33.6684i	2.19629	−	2.19629i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.0153			−0.971257			−0.485629	−	0.874165i	$-0.661409\pi$
$239$	−15.0153			−0.971257			−0.485629	+	0.874165i	$0.661409\pi$
$240$		0			0
$241$	21.6047			1.39168			0.695841	−	0.718196i	$-0.255032\pi$
$241$	21.6047			1.39168			0.695841	+	0.718196i	$0.255032\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.53823	+	2.53823i	−0.162161	+	0.162161i
$246$		0			0
$247$	−45.4309			−2.89070
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.9759	−	14.9759i	−0.945269	−	0.945269i	0.0533095	−	0.998578i	$-0.483023\pi$
$251$	−14.9759	−	14.9759i	−0.945269	−	0.945269i	−0.998578	+	0.0533095i	$0.983023\pi$
$252$		0			0
$253$	−0.673267	+	0.673267i	−0.0423279	+	0.0423279i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	8.05949i		−	0.502737i	−0.967891	−	0.251369i	$-0.919119\pi$
$257$		−	8.05949i		−	0.502737i	0.967891	−	0.251369i	$-0.0808807\pi$
$258$		0			0
$259$	2.41550	+	2.41550i	0.150092	+	0.150092i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			15.9502i			0.983531i	0.870728	+	0.491766i	$0.163648\pi$
$263$			15.9502i			0.983531i	−0.870728	+	0.491766i	$0.836352\pi$
$264$		0			0
$265$		−	47.8448i		−	2.93908i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.75178	+	2.75178i	0.167779	+	0.167779i	0.786002	−	0.618223i	$-0.212147\pi$
$269$	2.75178	+	2.75178i	0.167779	+	0.167779i	−0.618223	+	0.786002i	$0.712147\pi$
$270$		0			0
$271$			2.32138i			0.141014i	0.997511	+	0.0705070i	$0.0224617\pi$
$271$			2.32138i			0.141014i	−0.997511	+	0.0705070i	$0.977538\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.87077	+	3.87077i	−0.233416	+	0.233416i
$276$		0			0
$277$	9.66939	+	9.66939i	0.580977	+	0.580977i	0.935172	−	0.354194i	$-0.115245\pi$
$277$	9.66939	+	9.66939i	0.580977	+	0.580977i	−0.354194	+	0.935172i	$0.615245\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−24.1266			−1.43927			−0.719637	−	0.694351i	$-0.755692\pi$
$281$	−24.1266			−1.43927			−0.719637	+	0.694351i	$0.755692\pi$
$282$		0			0
$283$	15.5365	−	15.5365i	0.923547	−	0.923547i	−0.0737311	−	0.997278i	$-0.523491\pi$
$283$	15.5365	−	15.5365i	0.923547	−	0.923547i	0.997278	+	0.0737311i	$0.0234907\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.51556			0.0894604
$288$		0			0
$289$	−28.1362			−1.65507
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.43691	+	3.43691i	−0.200787	+	0.200787i	−0.800337	−	0.599550i	$-0.795346\pi$
$293$	−3.43691	+	3.43691i	−0.200787	+	0.200787i	0.599550	+	0.800337i	$0.295346\pi$
$294$		0			0
$295$	−27.7246			−1.61419
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.78403	+	5.78403i	0.334499	+	0.334499i
$300$		0			0
$301$	3.40983	−	3.40983i	0.196539	−	0.196539i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			41.4709i			2.37462i
$306$		0			0
$307$	−21.6774	−	21.6774i	−1.23719	−	1.23719i	−0.961143	−	0.276050i	$-0.910974\pi$
$307$	−21.6774	−	21.6774i	−1.23719	−	1.23719i	−0.276050	−	0.961143i	$-0.589026\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	22.3827i		−	1.26920i	−0.772839	−	0.634602i	$-0.781164\pi$
$311$		−	22.3827i		−	1.26920i	0.772839	−	0.634602i	$-0.218836\pi$
$312$		0			0
$313$		−	18.7841i		−	1.06174i	−0.847454	−	0.530869i	$-0.821866\pi$
$313$		−	18.7841i		−	1.06174i	0.847454	−	0.530869i	$-0.178134\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.4581	−	10.4581i	−0.587388	−	0.587388i	0.349535	−	0.936923i	$-0.386340\pi$
$317$	−10.4581	−	10.4581i	−0.587388	−	0.587388i	−0.936923	+	0.349535i	$0.886340\pi$
$318$		0			0
$319$		−	1.43070i		−	0.0801040i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−36.1871	+	36.1871i	−2.01350	+	2.01350i
$324$		0			0
$325$	33.2538	+	33.2538i	1.84459	+	1.84459i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−13.2646			−0.731299
$330$		0			0
$331$	2.13347	−	2.13347i	0.117266	−	0.117266i	−0.646038	−	0.763305i	$-0.723575\pi$
$331$	2.13347	−	2.13347i	0.117266	−	0.117266i	0.763305	+	0.646038i	$0.223575\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.5524			−0.685809
$336$		0			0
$337$	8.08023			0.440158			0.220079	−	0.975482i	$-0.429368\pi$
$337$	8.08023			0.440158			0.220079	+	0.975482i	$0.429368\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.30594	+	1.30594i	−0.0707204	+	0.0707204i
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.45329	−	2.45329i	−0.131699	−	0.131699i	0.638184	−	0.769884i	$-0.279686\pi$
$347$	−2.45329	−	2.45329i	−0.131699	−	0.131699i	−0.769884	+	0.638184i	$0.779686\pi$
$348$		0			0
$349$	−18.5622	+	18.5622i	−0.993612	+	0.993612i	−0.999980	−	0.00636783i	$-0.997973\pi$
$349$	−18.5622	+	18.5622i	−0.993612	+	0.993612i	0.00636783	+	0.999980i	$0.497973\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			13.7220i			0.730348i	0.930939	+	0.365174i	$0.118991\pi$
$353$			13.7220i			0.730348i	−0.930939	+	0.365174i	$0.881009\pi$
$354$		0			0
$355$	24.6615	+	24.6615i	1.30890	+	1.30890i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	1.64418i		−	0.0867763i	−0.999058	−	0.0433881i	$-0.986185\pi$
$359$		−	1.64418i		−	0.0867763i	0.999058	−	0.0433881i	$-0.0138152\pi$
$360$		0			0
$361$			39.0247i			2.05393i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.05351	−	6.05351i	−0.316855	−	0.316855i
$366$		0			0
$367$			30.8382i			1.60974i	0.593449	+	0.804871i	$0.297766\pi$
$367$			30.8382i			1.60974i	−0.593449	+	0.804871i	$0.702234\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.42484	+	9.42484i	−0.489313	+	0.489313i
$372$		0			0
$373$	4.20792	+	4.20792i	0.217878	+	0.217878i	0.807604	−	0.589726i	$-0.200764\pi$
$373$	4.20792	+	4.20792i	0.217878	+	0.217878i	−0.589726	+	0.807604i	$0.700764\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.2912			−0.633027
$378$		0			0
$379$	1.62605	−	1.62605i	0.0835245	−	0.0835245i	−0.664110	−	0.747635i	$-0.731189\pi$
$379$	1.62605	−	1.62605i	0.0835245	−	0.0835245i	0.747635	+	0.664110i	$0.231189\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.9352			0.712053			0.356027	−	0.934476i	$-0.384131\pi$
$383$	13.9352			0.712053			0.356027	+	0.934476i	$0.384131\pi$
$384$		0			0
$385$	2.49199			0.127004
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−24.4433	+	24.4433i	−1.23933	+	1.23933i	−0.279048	+	0.960277i	$0.590019\pi$
$389$	−24.4433	+	24.4433i	−1.23933	+	1.23933i	−0.960277	+	0.279048i	$0.909981\pi$
$390$		0			0
$391$	9.21432			0.465988
$392$		0			0
$393$		0			0
$394$		0			0
$395$	11.9388	+	11.9388i	0.600706	+	0.600706i
$396$		0			0
$397$	19.0249	−	19.0249i	0.954832	−	0.954832i	−0.0441916	−	0.999023i	$-0.514071\pi$
$397$	19.0249	−	19.0249i	0.954832	−	0.954832i	0.999023	+	0.0441916i	$0.0140712\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			5.40410i			0.269868i	0.990855	+	0.134934i	$0.0430822\pi$
$401$			5.40410i			0.269868i	−0.990855	+	0.134934i	$0.956918\pi$
$402$		0			0
$403$	11.2193	+	11.2193i	0.558872	+	0.558872i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2.37150i		−	0.117551i
$408$		0			0
$409$			13.2695i			0.656134i	0.944655	+	0.328067i	$0.106397\pi$
$409$			13.2695i			0.656134i	−0.944655	+	0.328067i	$0.893603\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.46141	+	5.46141i	0.268739	+	0.268739i
$414$		0			0
$415$		−	39.3689i		−	1.93255i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.63116	+	3.63116i	−0.177394	+	0.177394i	−0.790219	−	0.612825i	$-0.790033\pi$
$419$	−3.63116	+	3.63116i	−0.177394	+	0.177394i	0.612825	+	0.790219i	$0.290033\pi$
$420$		0			0
$421$	23.9365	+	23.9365i	1.16659	+	1.16659i	0.983002	+	0.183592i	$0.0587726\pi$
$421$	23.9365	+	23.9365i	1.16659	+	1.16659i	0.183592	+	0.983002i	$0.441227\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	52.9753			2.56968
$426$		0			0
$427$	8.16928	−	8.16928i	0.395339	−	0.395339i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	21.6203			1.04141			0.520707	−	0.853736i	$-0.325669\pi$
$431$	21.6203			1.04141			0.520707	+	0.853736i	$0.325669\pi$
$432$		0			0
$433$	−5.44643			−0.261739			−0.130869	−	0.991400i	$-0.541777\pi$
$433$	−5.44643			−0.261739			−0.130869	+	0.991400i	$0.541777\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.38741	−	7.38741i	0.353388	−	0.353388i
$438$		0			0
$439$	8.74327			0.417294			0.208647	−	0.977991i	$-0.433094\pi$
$439$	8.74327			0.417294			0.208647	+	0.977991i	$0.433094\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.3728	+	16.3728i	0.777895	+	0.777895i	0.979473	−	0.201578i	$-0.0646068\pi$
$443$	16.3728	+	16.3728i	0.777895	+	0.777895i	−0.201578	+	0.979473i	$0.564607\pi$
$444$		0			0
$445$	−13.6288	+	13.6288i	−0.646066	+	0.646066i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	22.6779i		−	1.07024i	−0.844777	−	0.535118i	$-0.820267\pi$
$449$		−	22.6779i		−	1.07024i	0.844777	−	0.535118i	$-0.179733\pi$
$450$		0			0
$451$	−0.743975	−	0.743975i	−0.0350324	−	0.0350324i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	21.4087i		−	1.00365i
$456$		0			0
$457$			17.4655i			0.817001i	0.912758	+	0.408501i	$0.133948\pi$
$457$			17.4655i			0.817001i	−0.912758	+	0.408501i	$0.866052\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.90382	+	3.90382i	0.181819	+	0.181819i	0.792148	−	0.610329i	$-0.208963\pi$
$461$	3.90382	+	3.90382i	0.181819	+	0.181819i	−0.610329	+	0.792148i	$0.708963\pi$
$462$		0			0
$463$			1.25985i			0.0585503i	0.999571	+	0.0292752i	$0.00931990\pi$
$463$			1.25985i			0.0585503i	−0.999571	+	0.0292752i	$0.990680\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.48649	−	9.48649i	0.438982	−	0.438982i	−0.452687	−	0.891669i	$-0.649535\pi$
$467$	9.48649	−	9.48649i	0.438982	−	0.438982i	0.891669	+	0.452687i	$0.149535\pi$
$468$		0			0
$469$	2.47267	+	2.47267i	0.114177	+	0.114177i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.34772			−0.153928
$474$		0			0
$475$	42.4720	−	42.4720i	1.94875	−	1.94875i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.08989			0.278254			0.139127	−	0.990275i	$-0.455570\pi$
$479$	6.08989			0.278254			0.139127	+	0.990275i	$0.455570\pi$
$480$		0			0
$481$	−20.3736			−0.928955
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.74992	−	8.74992i	0.397313	−	0.397313i
$486$		0			0
$487$	−25.6331			−1.16155			−0.580773	−	0.814066i	$-0.697249\pi$
$487$	−25.6331			−1.16155			−0.580773	+	0.814066i	$0.697249\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−10.0392	−	10.0392i	−0.453062	−	0.453062i	0.443308	−	0.896370i	$-0.353805\pi$
$491$	−10.0392	−	10.0392i	−0.453062	−	0.453062i	−0.896370	+	0.443308i	$0.853805\pi$
$492$		0			0
$493$	−9.79028	+	9.79028i	−0.440932	+	0.440932i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	9.71605i		−	0.435824i
$498$		0			0
$499$	25.2497	+	25.2497i	1.13033	+	1.13033i	0.990122	+	0.140212i	$0.0447784\pi$
$499$	25.2497	+	25.2497i	1.13033	+	1.13033i	0.140212	+	0.990122i	$0.455222\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	26.9906i		−	1.20345i	−0.798702	−	0.601727i	$-0.794480\pi$
$503$		−	26.9906i		−	1.20345i	0.798702	−	0.601727i	$-0.205520\pi$
$504$		0			0
$505$		−	12.6684i		−	0.563738i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.51549	−	7.51549i	−0.333118	−	0.333118i	0.520651	−	0.853769i	$-0.325689\pi$
$509$	−7.51549	−	7.51549i	−0.333118	−	0.333118i	−0.853769	+	0.520651i	$0.825689\pi$
$510$		0			0
$511$			2.38494i			0.105503i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	20.0038	−	20.0038i	0.881471	−	0.881471i
$516$		0			0
$517$	6.51147	+	6.51147i	0.286374	+	0.286374i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	16.3971			0.718372			0.359186	−	0.933266i	$-0.383054\pi$
$521$	16.3971			0.718372			0.359186	+	0.933266i	$0.383054\pi$
$522$		0			0
$523$	−23.6070	+	23.6070i	−1.03226	+	1.03226i	−0.0328016	+	0.999462i	$0.510443\pi$
$523$	−23.6070	+	23.6070i	−1.03226	+	1.03226i	−0.999462	+	0.0328016i	$0.989557\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	17.8730			0.778560
$528$		0			0
$529$	21.1189			0.918215
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.39148	+	6.39148i	−0.276846	+	0.276846i
$534$		0			0
$535$	11.4874			0.496645
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.490893	−	0.490893i	−0.0211442	−	0.0211442i
$540$		0			0
$541$	3.00348	−	3.00348i	0.129130	−	0.129130i	−0.639588	−	0.768718i	$-0.720895\pi$
$541$	3.00348	−	3.00348i	0.129130	−	0.129130i	0.768718	+	0.639588i	$0.220895\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			38.8151i			1.66266i
$546$		0			0
$547$	−13.5039	−	13.5039i	−0.577384	−	0.577384i	0.356797	−	0.934182i	$-0.383869\pi$
$547$	−13.5039	−	13.5039i	−0.577384	−	0.577384i	−0.934182	+	0.356797i	$0.883869\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			15.6984i			0.668773i
$552$		0			0
$553$		−	4.70360i		−	0.200017i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.0607	+	15.0607i	0.638143	+	0.638143i	0.950097	−	0.311954i	$-0.100984\pi$
$557$	15.0607	+	15.0607i	0.638143	+	0.638143i	−0.311954	+	0.950097i	$0.600984\pi$
$558$		0			0
$559$			28.7602i			1.21643i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−11.0288	+	11.0288i	−0.464807	+	0.464807i	−0.900227	−	0.435420i	$-0.856600\pi$
$563$	−11.0288	+	11.0288i	−0.464807	+	0.464807i	0.435420	+	0.900227i	$0.356600\pi$
$564$		0			0
$565$	−9.50281	−	9.50281i	−0.399786	−	0.399786i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.00762			−0.126086			−0.0630430	−	0.998011i	$-0.520081\pi$
$569$	−3.00762			−0.126086			−0.0630430	+	0.998011i	$0.520081\pi$
$570$		0			0
$571$	−15.1905	+	15.1905i	−0.635703	+	0.635703i	−0.949493	−	0.313790i	$-0.898401\pi$
$571$	−15.1905	+	15.1905i	−0.635703	+	0.635703i	0.313790	+	0.949493i	$0.398401\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.8146			−0.451002
$576$		0			0
$577$	0.322228			0.0134145			0.00670727	−	0.999978i	$-0.497865\pi$
$577$	0.322228			0.0134145			0.00670727	+	0.999978i	$0.497865\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.75521	+	7.75521i	−0.321740	+	0.321740i
$582$		0			0
$583$	9.25317			0.383227
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21.1091	+	21.1091i	0.871266	+	0.871266i	0.992610	−	0.121344i	$-0.0387205\pi$
$587$	21.1091	+	21.1091i	0.871266	+	0.871266i	−0.121344	+	0.992610i	$0.538721\pi$
$588$		0			0
$589$	14.3294	−	14.3294i	0.590431	−	0.590431i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			7.41539i			0.304514i	0.988341	+	0.152257i	$0.0486541\pi$
$593$			7.41539i			0.304514i	−0.988341	+	0.152257i	$0.951346\pi$
$594$		0			0
$595$	−17.0527	−	17.0527i	−0.699092	−	0.699092i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			25.2627i			1.03221i	0.856526	+	0.516104i	$0.172618\pi$
$599$			25.2627i			1.03221i	−0.856526	+	0.516104i	$0.827382\pi$
$600$		0			0
$601$		−	5.95409i		−	0.242872i	−0.992599	−	0.121436i	$-0.961250\pi$
$601$		−	5.95409i		−	0.242872i	0.992599	−	0.121436i	$-0.0387500\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	26.6972	+	26.6972i	1.08539	+	1.08539i
$606$		0			0
$607$			39.7459i			1.61324i	0.591073	+	0.806618i	$0.298704\pi$
$607$			39.7459i			1.61324i	−0.591073	+	0.806618i	$0.701296\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	55.9400	−	55.9400i	2.26309	−	2.26309i
$612$		0			0
$613$	−21.8646	−	21.8646i	−0.883103	−	0.883103i	0.110746	−	0.993849i	$-0.464676\pi$
$613$	−21.8646	−	21.8646i	−0.883103	−	0.883103i	−0.993849	+	0.110746i	$0.964676\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14.1189			0.568405			0.284202	−	0.958764i	$-0.408271\pi$
$617$	14.1189			0.568405			0.284202	+	0.958764i	$0.408271\pi$
$618$		0			0
$619$	2.98080	−	2.98080i	0.119809	−	0.119809i	−0.644660	−	0.764469i	$-0.723001\pi$
$619$	2.98080	−	2.98080i	0.119809	−	0.119809i	0.764469	+	0.644660i	$0.223001\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.36941			0.215121
$624$		0			0
$625$	2.24986			0.0899946
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−16.2282	+	16.2282i	−0.647060	+	0.647060i
$630$		0			0
$631$	21.9208			0.872654			0.436327	−	0.899788i	$-0.356279\pi$
$631$	21.9208			0.872654			0.436327	+	0.899788i	$0.356279\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−32.0010	−	32.0010i	−1.26992	−	1.26992i
$636$		0			0
$637$	−4.21725	+	4.21725i	−0.167094	+	0.167094i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			6.74291i			0.266329i	0.991094	+	0.133164i	$0.0425139\pi$
$641$			6.74291i			0.266329i	−0.991094	+	0.133164i	$0.957486\pi$
$642$		0			0
$643$	−8.56927	−	8.56927i	−0.337939	−	0.337939i	0.517652	−	0.855591i	$-0.326806\pi$
$643$	−8.56927	−	8.56927i	−0.337939	−	0.337939i	−0.855591	+	0.517652i	$0.826806\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	39.6964i		−	1.56063i	−0.625389	−	0.780313i	$-0.715060\pi$
$647$		−	39.6964i		−	1.56063i	0.625389	−	0.780313i	$-0.284940\pi$
$648$		0			0
$649$		−	5.36193i		−	0.210474i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.80312	+	1.80312i	0.0705614	+	0.0705614i	0.741507	−	0.670945i	$-0.234112\pi$
$653$	1.80312	+	1.80312i	0.0705614	+	0.0705614i	−0.670945	+	0.741507i	$0.734112\pi$
$654$		0			0
$655$		−	55.3907i		−	2.16429i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.79636	+	2.79636i	−0.108931	+	0.108931i	−0.759471	−	0.650541i	$-0.774542\pi$
$659$	−2.79636	+	2.79636i	−0.108931	+	0.108931i	0.650541	+	0.759471i	$0.274542\pi$
$660$		0			0
$661$	−1.53027	−	1.53027i	−0.0595205	−	0.0595205i	0.676720	−	0.736241i	$-0.263401\pi$
$661$	−1.53027	−	1.53027i	−0.0595205	−	0.0595205i	−0.736241	+	0.676720i	$0.763401\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−27.3433			−1.06033
$666$		0			0
$667$	1.99864	−	1.99864i	0.0773875	−	0.0773875i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8.02048			−0.309627
$672$		0			0
$673$	−50.8286			−1.95930			−0.979649	−	0.200720i	$-0.935672\pi$
$673$	−50.8286			−1.95930			−0.979649	+	0.200720i	$0.935672\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.52373	+	1.52373i	−0.0585617	+	0.0585617i	−0.735781	−	0.677219i	$-0.763185\pi$
$677$	−1.52373	+	1.52373i	−0.0585617	+	0.0585617i	0.677219	+	0.735781i	$0.263185\pi$
$678$		0			0
$679$	−3.44726			−0.132294
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.480063	+	0.480063i	0.0183691	+	0.0183691i	0.716232	−	0.697863i	$-0.245865\pi$
$683$	0.480063	+	0.480063i	0.0183691	+	0.0183691i	−0.697863	+	0.716232i	$0.745865\pi$
$684$		0			0
$685$	−22.4518	+	22.4518i	−0.857840	+	0.857840i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	79.4939i		−	3.02848i
$690$		0			0
$691$	26.4280	+	26.4280i	1.00537	+	1.00537i	0.999986	+	0.00538425i	$0.00171387\pi$
$691$	26.4280	+	26.4280i	1.00537	+	1.00537i	0.00538425	+	0.999986i	$0.498286\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	29.7085i		−	1.12691i
$696$		0			0
$697$			10.1820i			0.385672i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.40089	−	2.40089i	−0.0906804	−	0.0906804i	0.660311	−	0.750992i	$-0.270424\pi$
$701$	−2.40089	−	2.40089i	−0.0906804	−	0.0906804i	−0.750992	+	0.660311i	$0.770424\pi$
$702$		0			0
$703$			26.0213i			0.981411i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.49553	+	2.49553i	−0.0938540	+	0.0938540i
$708$		0			0
$709$	27.3668	+	27.3668i	1.02778	+	1.02778i	0.999603	+	0.0281790i	$0.00897084\pi$
$709$	27.3668	+	27.3668i	1.02778	+	1.02778i	0.0281790	+	0.999603i	$0.491029\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.64868			−0.136644
$714$		0			0
$715$	−10.5094	+	10.5094i	−0.393028	+	0.393028i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.6007			−0.619101			−0.309551	−	0.950883i	$-0.600179\pi$
$719$	−16.6007			−0.619101			−0.309551	+	0.950883i	$0.600179\pi$
$720$		0			0
$721$	−7.88100			−0.293504
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11.4906	−	11.4906i	0.426752	−	0.426752i
$726$		0			0
$727$	11.7244			0.434835			0.217418	−	0.976079i	$-0.430237\pi$
$727$	11.7244			0.434835			0.217418	+	0.976079i	$0.430237\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	22.9084	+	22.9084i	0.847298	+	0.847298i
$732$		0			0
$733$	−15.3078	+	15.3078i	−0.565407	+	0.565407i	−0.930838	−	0.365431i	$-0.880921\pi$
$733$	−15.3078	+	15.3078i	−0.565407	+	0.565407i	0.365431	+	0.930838i	$0.380921\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	2.42763i		−	0.0894228i
$738$		0			0
$739$	−21.4937	−	21.4937i	−0.790659	−	0.790659i	0.190942	−	0.981601i	$-0.438846\pi$
$739$	−21.4937	−	21.4937i	−0.790659	−	0.790659i	−0.981601	+	0.190942i	$0.938846\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			31.6334i			1.16052i	0.814432	+	0.580259i	$0.197049\pi$
$743$			31.6334i			1.16052i	−0.814432	+	0.580259i	$0.802951\pi$
$744$		0			0
$745$		−	23.9448i		−	0.877269i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.26289	−	2.26289i	−0.0826840	−	0.0826840i
$750$		0			0
$751$			25.7604i			0.940011i	0.882663	+	0.470006i	$0.155748\pi$
$751$			25.7604i			0.940011i	−0.882663	+	0.470006i	$0.844252\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	26.9560	−	26.9560i	0.981031	−	0.981031i
$756$		0			0
$757$	18.8485	+	18.8485i	0.685061	+	0.685061i	0.961136	−	0.276075i	$-0.0890337\pi$
$757$	18.8485	+	18.8485i	0.685061	+	0.685061i	−0.276075	+	0.961136i	$0.589034\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.04604			−0.110419			−0.0552094	−	0.998475i	$-0.517583\pi$
$761$	−3.04604			−0.110419			−0.0552094	+	0.998475i	$0.517583\pi$
$762$		0			0
$763$	7.64611	−	7.64611i	0.276808	−	0.276808i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−46.0643			−1.66329
$768$		0			0
$769$	7.94309			0.286435			0.143218	−	0.989691i	$-0.454255\pi$
$769$	7.94309			0.286435			0.143218	+	0.989691i	$0.454255\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.3102	−	19.3102i	0.694541	−	0.694541i	−0.268687	−	0.963228i	$-0.586590\pi$
$773$	19.3102	−	19.3102i	0.694541	−	0.694541i	0.963228	+	0.268687i	$0.0865897\pi$
$774$		0			0
$775$	−20.9772			−0.753522
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.16325	+	8.16325i	0.292479	+	0.292479i
$780$		0			0
$781$	−4.76954	+	4.76954i	−0.170667	+	0.170667i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	64.4587i		−	2.30063i
$786$		0			0
$787$	−20.2277	−	20.2277i	−0.721041	−	0.721041i	0.247777	−	0.968817i	$-0.420300\pi$
$787$	−20.2277	−	20.2277i	−0.721041	−	0.721041i	−0.968817	+	0.247777i	$0.920300\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			3.74388i			0.133117i
$792$		0			0
$793$			68.9039i			2.44685i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−27.0767	−	27.0767i	−0.959105	−	0.959105i	0.0400908	−	0.999196i	$-0.487235\pi$
$797$	−27.0767	−	27.0767i	−0.959105	−	0.959105i	−0.999196	+	0.0400908i	$0.987235\pi$
$798$		0			0
$799$		−	89.1159i		−	3.15269i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.17075	−	1.17075i	0.0413148	−	0.0413148i
$804$		0			0
$805$	3.48122	+	3.48122i	0.122697	+	0.122697i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	51.2572			1.80211			0.901054	−	0.433706i	$-0.142794\pi$
$809$	51.2572			1.80211			0.901054	+	0.433706i	$0.142794\pi$
$810$		0			0
$811$	−15.6365	+	15.6365i	−0.549073	+	0.549073i	−0.926173	−	0.377099i	$-0.876921\pi$
$811$	−15.6365	+	15.6365i	−0.549073	+	0.549073i	0.377099	+	0.926173i	$0.376921\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	87.7956			3.07535
$816$		0			0
$817$	36.7328			1.28512
$818$		0			0
$819$		0			0
$820$		0			0
$821$	11.0162	−	11.0162i	0.384467	−	0.384467i	−0.488241	−	0.872709i	$-0.662361\pi$
$821$	11.0162	−	11.0162i	0.384467	−	0.384467i	0.872709	+	0.488241i	$0.162361\pi$
$822$		0			0
$823$	−34.8795			−1.21582			−0.607911	−	0.794005i	$-0.707992\pi$
$823$	−34.8795			−1.21582			−0.607911	+	0.794005i	$0.707992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.95741	+	9.95741i	0.346253	+	0.346253i	0.858712	−	0.512459i	$-0.171265\pi$
$827$	9.95741	+	9.95741i	0.346253	+	0.346253i	−0.512459	+	0.858712i	$0.671265\pi$
$828$		0			0
$829$	−19.7239	+	19.7239i	−0.685041	+	0.685041i	−0.961132	−	0.276091i	$-0.910961\pi$
$829$	−19.7239	+	19.7239i	−0.685041	+	0.685041i	0.276091	+	0.961132i	$0.410961\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			6.71835i			0.232777i
$834$		0			0
$835$	45.9724	+	45.9724i	1.59094	+	1.59094i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	17.7923i		−	0.614257i	−0.951668	−	0.307129i	$-0.900632\pi$
$839$		−	17.7923i		−	0.614257i	0.951668	−	0.307129i	$-0.0993682\pi$
$840$		0			0
$841$		−	24.7529i		−	0.853547i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	57.2890	+	57.2890i	1.97080	+	1.97080i
$846$		0			0
$847$		−	10.5180i		−	0.361404i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.31290	−	3.31290i	0.113565	−	0.113565i
$852$		0			0
$853$	−15.5555	−	15.5555i	−0.532610	−	0.532610i	0.388738	−	0.921348i	$-0.372911\pi$
$853$	−15.5555	−	15.5555i	−0.532610	−	0.532610i	−0.921348	+	0.388738i	$0.872911\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	55.7987			1.90605			0.953023	−	0.302898i	$-0.0979543\pi$
$857$	55.7987			1.90605			0.953023	+	0.302898i	$0.0979543\pi$
$858$		0			0
$859$	−5.54812	+	5.54812i	−0.189299	+	0.189299i	−0.795393	−	0.606094i	$-0.792736\pi$
$859$	−5.54812	+	5.54812i	−0.189299	+	0.189299i	0.606094	+	0.795393i	$0.292736\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.67668			−0.159196			−0.0795980	−	0.996827i	$-0.525364\pi$
$863$	−4.67668			−0.159196			−0.0795980	+	0.996827i	$0.525364\pi$
$864$		0			0
$865$	37.1618			1.26354
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.30896	+	2.30896i	−0.0783261	+	0.0783261i
$870$		0			0
$871$	−20.8557			−0.706670
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.32323	+	7.32323i	0.247570	+	0.247570i
$876$		0			0
$877$	30.6084	−	30.6084i	1.03357	−	1.03357i	0.0341550	−	0.999417i	$-0.489126\pi$
$877$	30.6084	−	30.6084i	1.03357	−	1.03357i	0.999417	−	0.0341550i	$-0.0108740\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	4.23931i		−	0.142826i	−0.997447	−	0.0714129i	$-0.977249\pi$
$881$		−	4.23931i		−	0.142826i	0.997447	−	0.0714129i	$-0.0227508\pi$
$882$		0			0
$883$	27.7473	+	27.7473i	0.933772	+	0.933772i	0.997939	−	0.0641668i	$-0.0204390\pi$
$883$	27.7473	+	27.7473i	0.933772	+	0.933772i	−0.0641668	+	0.997939i	$0.520439\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	46.9699i		−	1.57710i	−0.614973	−	0.788548i	$-0.710833\pi$
$887$		−	46.9699i		−	1.57710i	0.614973	−	0.788548i	$-0.289167\pi$
$888$		0			0
$889$			12.6076i			0.422846i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−71.4470	−	71.4470i	−2.39088	−	2.39088i
$894$		0			0
$895$			54.6353i			1.82626i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.87675	−	3.87675i	0.129297	−	0.129297i
$900$		0			0
$901$	−63.3193	−	63.3193i	−2.10947	−	2.10947i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	51.0077			1.69555
$906$		0			0
$907$	10.5685	−	10.5685i	0.350923	−	0.350923i	−0.509530	−	0.860453i	$-0.670181\pi$
$907$	10.5685	−	10.5685i	0.350923	−	0.350923i	0.860453	+	0.509530i	$0.170181\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	25.9324			0.859180			0.429590	−	0.903024i	$-0.358658\pi$
$911$	25.9324			0.859180			0.429590	+	0.903024i	$0.358658\pi$
$912$		0			0
$913$	7.61395			0.251985
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.9113	+	10.9113i	−0.360323	+	0.360323i
$918$		0			0
$919$	−1.79979			−0.0593697			−0.0296848	−	0.999559i	$-0.509450\pi$
$919$	−1.79979			−0.0593697			−0.0296848	+	0.999559i	$0.509450\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	40.9751	+	40.9751i	1.34871	+	1.34871i
$924$		0			0
$925$	19.0467	−	19.0467i	0.626250	−	0.626250i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	21.4683i		−	0.704353i	−0.935934	−	0.352176i	$-0.885442\pi$
$929$		−	21.4683i		−	0.704353i	0.935934	−	0.352176i	$-0.114558\pi$
$930$		0			0
$931$	5.38631	+	5.38631i	0.176529	+	0.176529i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			16.7421i			0.547524i
$936$		0			0
$937$		−	14.6323i		−	0.478016i	−0.971018	−	0.239008i	$-0.923178\pi$
$937$		−	14.6323i		−	0.478016i	0.971018	−	0.239008i	$-0.0768223\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	26.6247	+	26.6247i	0.867939	+	0.867939i	0.992244	−	0.124305i	$-0.0396702\pi$
$941$	26.6247	+	26.6247i	0.867939	+	0.867939i	−0.124305	+	0.992244i	$0.539670\pi$
$942$		0			0
$943$		−	2.07861i		−	0.0676888i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−39.4502	+	39.4502i	−1.28196	+	1.28196i	−0.342406	+	0.939552i	$0.611242\pi$
$947$	−39.4502	+	39.4502i	−1.28196	+	1.28196i	−0.939552	+	0.342406i	$0.888758\pi$
$948$		0			0
$949$	−10.0579	−	10.0579i	−0.326493	−	0.326493i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.54083			0.244271			0.122136	−	0.992513i	$-0.461026\pi$
$953$	7.54083			0.244271			0.122136	+	0.992513i	$0.461026\pi$
$954$		0			0
$955$	−50.6309	+	50.6309i	−1.63838	+	1.63838i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	8.84548			0.285635
$960$		0			0
$961$	23.9227			0.771698
$962$		0			0
$963$		0			0
$964$		0			0
$965$	37.1663	−	37.1663i	1.19643	−	1.19643i
$966$		0			0
$967$	−19.3259			−0.621480			−0.310740	−	0.950495i	$-0.600577\pi$
$967$	−19.3259			−0.621480			−0.310740	+	0.950495i	$0.600577\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	38.7441	+	38.7441i	1.24336	+	1.24336i	0.958599	+	0.284760i	$0.0919139\pi$
$971$	38.7441	+	38.7441i	1.24336	+	1.24336i	0.284760	+	0.958599i	$0.408086\pi$
$972$		0			0
$973$	−5.85221	+	5.85221i	−0.187613	+	0.187613i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.45784i		−	0.0466404i	−0.999728	−	0.0233202i	$-0.992576\pi$
$977$		−	1.45784i		−	0.0466404i	0.999728	−	0.0233202i	$-0.00742372\pi$
$978$		0			0
$979$	−2.63580	−	2.63580i	−0.0842407	−	0.0842407i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			25.9206i			0.826740i	0.910563	+	0.413370i	$0.135648\pi$
$983$			25.9206i			0.826740i	−0.910563	+	0.413370i	$0.864352\pi$
$984$		0			0
$985$			10.9789i			0.349817i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.67664	−	4.67664i	−0.148708	−	0.148708i
$990$		0			0
$991$			3.24733i			0.103155i	0.998669	+	0.0515775i	$0.0164249\pi$
$991$			3.24733i			0.103155i	−0.998669	+	0.0515775i	$0.983575\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−22.0559	+	22.0559i	−0.699220	+	0.699220i
$996$		0			0
$997$	16.0720	+	16.0720i	0.509006	+	0.509006i	0.914221	−	0.405215i	$-0.132803\pi$
$997$	16.0720	+	16.0720i	0.509006	+	0.509006i	−0.405215	+	0.914221i	$0.632803\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.d.3599.2		36
3.2	odd	2	inner	4032.2.v.d.3599.17		36
4.3	odd	2		1008.2.v.d.827.18	yes	36
12.11	even	2		1008.2.v.d.827.1	yes	36
16.3	odd	4	inner	4032.2.v.d.1583.17		36
16.13	even	4		1008.2.v.d.323.1	✓	36
48.29	odd	4		1008.2.v.d.323.18	yes	36
48.35	even	4	inner	4032.2.v.d.1583.2		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.d.323.1	✓	36	16.13	even	4
1008.2.v.d.323.18	yes	36	48.29	odd	4
1008.2.v.d.827.1	yes	36	12.11	even	2
1008.2.v.d.827.18	yes	36	4.3	odd	2
4032.2.v.d.1583.2		36	48.35	even	4	inner
4032.2.v.d.1583.17		36	16.3	odd	4	inner
4032.2.v.d.3599.2		36	1.1	even	1	trivial
4032.2.v.d.3599.17		36	3.2	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.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.53823 + 2.53823i) q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+(-2.53823 + 2.53823i) q^{5} +1.00000 q^{7} +(-0.490893 - 0.490893i) q^{11} +(-4.21725 + 4.21725i) q^{13} +6.71835i q^{17} +(5.38631 + 5.38631i) q^{19} -1.37152i q^{23} -7.88518i q^{25} +(1.45725 + 1.45725i) q^{29} -2.66033i q^{31} +(-2.53823 + 2.53823i) q^{35} +(2.41550 + 2.41550i) q^{37} +1.51556 q^{41} +(3.40983 - 3.40983i) q^{43} -13.2646 q^{47} +1.00000 q^{49} +(-9.42484 + 9.42484i) q^{53} +2.49199 q^{55} +(5.46141 + 5.46141i) q^{59} +(8.16928 - 8.16928i) q^{61} -21.4087i q^{65} +(2.47267 + 2.47267i) q^{67} -9.71605i q^{71} +2.38494i q^{73} +(-0.490893 - 0.490893i) q^{77} -4.70360i q^{79} +(-7.75521 + 7.75521i) q^{83} +(-17.0527 - 17.0527i) q^{85} +5.36941 q^{89} +(-4.21725 + 4.21725i) q^{91} -27.3433 q^{95} -3.44726 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 36 q^{7}+O(q^{10}) \) 36 * q + 36 * q^7	\( 36 q + 36 q^{7} - 16 q^{13} + 16 q^{19} + 20 q^{37} - 36 q^{43} + 36 q^{49} - 32 q^{55} + 112 q^{61} + 36 q^{67} - 96 q^{85} - 16 q^{91} + 56 q^{97}+O(q^{100}) \) 36 * q + 36 * q^7 - 16 * q^13 + 16 * q^19 + 20 * q^37 - 36 * q^43 + 36 * q^49 - 32 * q^55 + 112 * q^61 + 36 * q^67 - 96 * q^85 - 16 * q^91 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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