LMFDB - Embedded newform 4032.2.v.d.1583.11

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			1583.11
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.d.3599.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.495166 + 0.495166i) q^{5} +1.00000 q^{7} +O(q^{10})$	$q+(0.495166 + 0.495166i) q^{5} +1.00000 q^{7} +(0.675994 - 0.675994i) q^{11} +(-2.39551 - 2.39551i) q^{13} -1.03686i q^{17} +(-0.913823 + 0.913823i) q^{19} -2.92895i q^{23} -4.50962i q^{25} +(3.39091 - 3.39091i) q^{29} +9.43104i q^{31} +(0.495166 + 0.495166i) q^{35} +(5.47539 - 5.47539i) q^{37} -6.91226 q^{41} +(2.30822 + 2.30822i) q^{43} -9.68077 q^{47} +1.00000 q^{49} +(-8.02498 - 8.02498i) q^{53} +0.669458 q^{55} +(1.14297 - 1.14297i) q^{59} +(-7.05839 - 7.05839i) q^{61} -2.37235i q^{65} +(5.66980 - 5.66980i) q^{67} -7.26670i q^{71} -6.66570i q^{73} +(0.675994 - 0.675994i) q^{77} +2.26385i q^{79} +(0.619284 + 0.619284i) q^{83} +(0.513417 - 0.513417i) q^{85} +3.99233 q^{89} +(-2.39551 - 2.39551i) q^{91} -0.904988 q^{95} +6.82547 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$5$	0.495166	+	0.495166i	0.221445	+	0.221445i	0.809107	−	0.587662i	$-0.199951\pi$
$5$	0.495166	+	0.495166i	0.221445	+	0.221445i	−0.587662	+	0.809107i	$0.699951\pi$
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.675994	−	0.675994i	0.203820	−	0.203820i	−0.597815	−	0.801634i	$-0.703964\pi$
$11$	0.675994	−	0.675994i	0.203820	−	0.203820i	0.801634	+	0.597815i	$0.203964\pi$
$12$		0			0
$13$	−2.39551	−	2.39551i	−0.664394	−	0.664394i	0.292019	−	0.956413i	$-0.405673\pi$
$13$	−2.39551	−	2.39551i	−0.664394	−	0.664394i	−0.956413	+	0.292019i	$0.905673\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.03686i		−	0.251475i	−0.992064	−	0.125738i	$-0.959870\pi$
$17$		−	1.03686i		−	0.251475i	0.992064	−	0.125738i	$-0.0401297\pi$
$18$		0			0
$19$	−0.913823	+	0.913823i	−0.209645	+	0.209645i	−0.804117	−	0.594471i	$-0.797361\pi$
$19$	−0.913823	+	0.913823i	−0.209645	+	0.209645i	0.594471	+	0.804117i	$0.297361\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.92895i		−	0.610727i	−0.952236	−	0.305364i	$-0.901222\pi$
$23$		−	2.92895i		−	0.610727i	0.952236	−	0.305364i	$-0.0987780\pi$
$24$		0			0
$25$		−	4.50962i		−	0.901924i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.39091	−	3.39091i	0.629676	−	0.629676i	−0.318310	−	0.947987i	$-0.603115\pi$
$29$	3.39091	−	3.39091i	0.629676	−	0.629676i	0.947987	+	0.318310i	$0.103115\pi$
$30$		0			0
$31$			9.43104i			1.69386i	0.531701	+	0.846932i	$0.321553\pi$
$31$			9.43104i			1.69386i	−0.531701	+	0.846932i	$0.678447\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.495166	+	0.495166i	0.0836983	+	0.0836983i
$36$		0			0
$37$	5.47539	−	5.47539i	0.900148	−	0.900148i	−0.0953004	−	0.995449i	$-0.530381\pi$
$37$	5.47539	−	5.47539i	0.900148	−	0.900148i	0.995449	+	0.0953004i	$0.0303812\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.91226			−1.07951			−0.539757	−	0.841821i	$-0.681484\pi$
$41$	−6.91226			−1.07951			−0.539757	+	0.841821i	$0.681484\pi$
$42$		0			0
$43$	2.30822	+	2.30822i	0.352000	+	0.352000i	0.860853	−	0.508853i	$-0.169930\pi$
$43$	2.30822	+	2.30822i	0.352000	+	0.352000i	−0.508853	+	0.860853i	$0.669930\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.68077			−1.41209			−0.706043	−	0.708169i	$-0.749522\pi$
$47$	−9.68077			−1.41209			−0.706043	+	0.708169i	$0.749522\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.02498	−	8.02498i	−1.10232	−	1.10232i	−0.994131	−	0.108185i	$-0.965496\pi$
$53$	−8.02498	−	8.02498i	−1.10232	−	1.10232i	−0.108185	−	0.994131i	$-0.534504\pi$
$54$		0			0
$55$	0.669458			0.0902697
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.14297	−	1.14297i	0.148802	−	0.148802i	−0.628781	−	0.777583i	$-0.716446\pi$
$59$	1.14297	−	1.14297i	0.148802	−	0.148802i	0.777583	+	0.628781i	$0.216446\pi$
$60$		0			0
$61$	−7.05839	−	7.05839i	−0.903734	−	0.903734i	0.0920229	−	0.995757i	$-0.470667\pi$
$61$	−7.05839	−	7.05839i	−0.903734	−	0.903734i	−0.995757	+	0.0920229i	$0.970667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.37235i		−	0.294253i
$66$		0			0
$67$	5.66980	−	5.66980i	0.692677	−	0.692677i	−0.270143	−	0.962820i	$-0.587071\pi$
$67$	5.66980	−	5.66980i	0.692677	−	0.692677i	0.962820	+	0.270143i	$0.0870711\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.26670i		−	0.862399i	−0.902257	−	0.431199i	$-0.858091\pi$
$71$		−	7.26670i		−	0.862399i	0.902257	−	0.431199i	$-0.141909\pi$
$72$		0			0
$73$		−	6.66570i		−	0.780161i	−0.920781	−	0.390081i	$-0.872447\pi$
$73$		−	6.66570i		−	0.780161i	0.920781	−	0.390081i	$-0.127553\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.675994	−	0.675994i	0.0770366	−	0.0770366i
$78$		0			0
$79$			2.26385i			0.254704i	0.991858	+	0.127352i	$0.0406477\pi$
$79$			2.26385i			0.254704i	−0.991858	+	0.127352i	$0.959352\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.619284	+	0.619284i	0.0679753	+	0.0679753i	0.740277	−	0.672302i	$-0.234694\pi$
$83$	0.619284	+	0.619284i	0.0679753	+	0.0679753i	−0.672302	+	0.740277i	$0.734694\pi$
$84$		0			0
$85$	0.513417	−	0.513417i	0.0556879	−	0.0556879i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.99233			0.423186			0.211593	−	0.977358i	$-0.432135\pi$
$89$	3.99233			0.423186			0.211593	+	0.977358i	$0.432135\pi$
$90$		0			0
$91$	−2.39551	−	2.39551i	−0.251117	−	0.251117i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.904988			−0.0928498
$96$		0			0
$97$	6.82547			0.693022			0.346511	−	0.938046i	$-0.387366\pi$
$97$	6.82547			0.693022			0.346511	+	0.938046i	$0.387366\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.00990	+	2.00990i	0.199992	+	0.199992i	0.799997	−	0.600004i	$-0.204835\pi$
$101$	2.00990	+	2.00990i	0.199992	+	0.199992i	−0.600004	+	0.799997i	$0.704835\pi$
$102$		0			0
$103$	9.62061			0.947947			0.473973	−	0.880539i	$-0.342819\pi$
$103$	9.62061			0.947947			0.473973	+	0.880539i	$0.342819\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.18658	+	4.18658i	−0.404732	+	0.404732i	−0.879897	−	0.475165i	$-0.842388\pi$
$107$	−4.18658	+	4.18658i	−0.404732	+	0.404732i	0.475165	+	0.879897i	$0.342388\pi$
$108$		0			0
$109$	1.12051	+	1.12051i	0.107326	+	0.107326i	0.758730	−	0.651405i	$-0.225820\pi$
$109$	1.12051	+	1.12051i	0.107326	+	0.107326i	−0.651405	+	0.758730i	$0.725820\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	3.04200i		−	0.286167i	−0.989711	−	0.143084i	$-0.954298\pi$
$113$		−	3.04200i		−	0.286167i	0.989711	−	0.143084i	$-0.0457018\pi$
$114$		0			0
$115$	1.45031	−	1.45031i	0.135242	−	0.135242i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	1.03686i		−	0.0950486i
$120$		0			0
$121$			10.0861i			0.916915i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.70884	−	4.70884i	0.421171	−	0.421171i
$126$		0			0
$127$		−	2.15994i		−	0.191664i	−0.995398	−	0.0958318i	$-0.969449\pi$
$127$		−	2.15994i		−	0.191664i	0.995398	−	0.0958318i	$-0.0305511\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9.29314	−	9.29314i	−0.811945	−	0.811945i	0.172980	−	0.984925i	$-0.444660\pi$
$131$	−9.29314	−	9.29314i	−0.811945	−	0.811945i	−0.984925	+	0.172980i	$0.944660\pi$
$132$		0			0
$133$	−0.913823	+	0.913823i	−0.0792385	+	0.0792385i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−16.8459			−1.43924			−0.719620	−	0.694368i	$-0.755684\pi$
$137$	−16.8459			−1.43924			−0.719620	+	0.694368i	$0.755684\pi$
$138$		0			0
$139$	10.8283	+	10.8283i	0.918444	+	0.918444i	0.996916	−	0.0784726i	$-0.0250043\pi$
$139$	10.8283	+	10.8283i	0.918444	+	0.918444i	−0.0784726	+	0.996916i	$0.525004\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.23870			−0.270833
$144$		0			0
$145$	3.35812			0.278877
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.41586	+	5.41586i	0.443685	+	0.443685i	0.893248	−	0.449564i	$-0.148421\pi$
$149$	5.41586	+	5.41586i	0.443685	+	0.443685i	−0.449564	+	0.893248i	$0.648421\pi$
$150$		0			0
$151$	−1.86764			−0.151987			−0.0759933	−	0.997108i	$-0.524213\pi$
$151$	−1.86764			−0.151987			−0.0759933	+	0.997108i	$0.524213\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.66993	+	4.66993i	−0.375098	+	0.375098i
$156$		0			0
$157$	−6.27144	−	6.27144i	−0.500515	−	0.500515i	0.411083	−	0.911598i	$-0.365151\pi$
$157$	−6.27144	−	6.27144i	−0.500515	−	0.500515i	−0.911598	+	0.411083i	$0.865151\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.92895i		−	0.230833i
$162$		0			0
$163$	9.10671	−	9.10671i	0.713292	−	0.713292i	−0.253930	−	0.967223i	$-0.581723\pi$
$163$	9.10671	−	9.10671i	0.713292	−	0.713292i	0.967223	+	0.253930i	$0.0817233\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	9.44503i		−	0.730878i	−0.930835	−	0.365439i	$-0.880919\pi$
$167$		−	9.44503i		−	0.730878i	0.930835	−	0.365439i	$-0.119081\pi$
$168$		0			0
$169$		−	1.52309i		−	0.117161i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16.9743	−	16.9743i	1.29053	−	1.29053i	0.356078	−	0.934456i	$-0.384114\pi$
$173$	16.9743	−	16.9743i	1.29053	−	1.29053i	0.934456	−	0.356078i	$-0.115886\pi$
$174$		0			0
$175$		−	4.50962i		−	0.340895i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.830318	−	0.830318i	−0.0620609	−	0.0620609i	0.675395	−	0.737456i	$-0.263973\pi$
$179$	−0.830318	−	0.830318i	−0.0620609	−	0.0620609i	−0.737456	+	0.675395i	$0.763973\pi$
$180$		0			0
$181$	3.74015	−	3.74015i	0.278003	−	0.278003i	−0.554308	−	0.832311i	$-0.687017\pi$
$181$	3.74015	−	3.74015i	0.278003	−	0.278003i	0.832311	+	0.554308i	$0.187017\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.42245			0.398666
$186$		0			0
$187$	−0.700910	−	0.700910i	−0.0512556	−	0.0512556i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.3035			−0.745533			−0.372766	−	0.927925i	$-0.621591\pi$
$191$	−10.3035			−0.745533			−0.372766	+	0.927925i	$0.621591\pi$
$192$		0			0
$193$	12.1382			0.873726			0.436863	−	0.899528i	$-0.356089\pi$
$193$	12.1382			0.873726			0.436863	+	0.899528i	$0.356089\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.77271	−	8.77271i	−0.625029	−	0.625029i	0.321784	−	0.946813i	$-0.395718\pi$
$197$	−8.77271	−	8.77271i	−0.625029	−	0.625029i	−0.946813	+	0.321784i	$0.895718\pi$
$198$		0			0
$199$	−22.8194			−1.61762			−0.808812	−	0.588067i	$-0.799889\pi$
$199$	−22.8194			−1.61762			−0.808812	+	0.588067i	$0.799889\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.39091	−	3.39091i	0.237995	−	0.237995i
$204$		0			0
$205$	−3.42272	−	3.42272i	−0.239053	−	0.239053i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			1.23548i			0.0854598i
$210$		0			0
$211$	18.3550	−	18.3550i	1.26361	−	1.26361i	0.314283	−	0.949329i	$-0.398236\pi$
$211$	18.3550	−	18.3550i	1.26361	−	1.26361i	0.949329	−	0.314283i	$-0.101764\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			2.28590i			0.155897i
$216$		0			0
$217$			9.43104i			0.640221i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.48380	+	2.48380i	−0.167079	+	0.167079i
$222$		0			0
$223$			7.85008i			0.525680i	0.964839	+	0.262840i	$0.0846591\pi$
$223$			7.85008i			0.525680i	−0.964839	+	0.262840i	$0.915341\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.49085	+	8.49085i	0.563557	+	0.563557i	0.930316	−	0.366759i	$-0.119533\pi$
$227$	8.49085	+	8.49085i	0.563557	+	0.563557i	−0.366759	+	0.930316i	$0.619533\pi$
$228$		0			0
$229$	10.9830	−	10.9830i	0.725775	−	0.725775i	−0.244000	−	0.969775i	$-0.578460\pi$
$229$	10.9830	−	10.9830i	0.725775	−	0.725775i	0.969775	+	0.244000i	$0.0784598\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.0449847			−0.00294704			−0.00147352	−	0.999999i	$-0.500469\pi$
$233$	−0.0449847			−0.00294704			−0.00147352	+	0.999999i	$0.500469\pi$
$234$		0			0
$235$	−4.79359	−	4.79359i	−0.312699	−	0.312699i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−30.1340			−1.94921			−0.974605	−	0.223933i	$-0.928110\pi$
$239$	−30.1340			−1.94921			−0.974605	+	0.223933i	$0.928110\pi$
$240$		0			0
$241$	14.4042			0.927853			0.463927	−	0.885874i	$-0.346440\pi$
$241$	14.4042			0.927853			0.463927	+	0.885874i	$0.346440\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.495166	+	0.495166i	0.0316350	+	0.0316350i
$246$		0			0
$247$	4.37814			0.278574
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.14495	−	7.14495i	0.450986	−	0.450986i	−0.444696	−	0.895682i	$-0.646688\pi$
$251$	7.14495	−	7.14495i	0.450986	−	0.450986i	0.895682	+	0.444696i	$0.146688\pi$
$252$		0			0
$253$	−1.97995	−	1.97995i	−0.124478	−	0.124478i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			7.01111i			0.437341i	0.975799	+	0.218670i	$0.0701720\pi$
$257$			7.01111i			0.437341i	−0.975799	+	0.218670i	$0.929828\pi$
$258$		0			0
$259$	5.47539	−	5.47539i	0.340224	−	0.340224i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	28.2013i		−	1.73897i	−0.493964	−	0.869483i	$-0.664452\pi$
$263$		−	28.2013i		−	1.73897i	0.493964	−	0.869483i	$-0.335548\pi$
$264$		0			0
$265$		−	7.94739i		−	0.488204i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.65309	+	1.65309i	−0.100790	+	0.100790i	−0.755704	−	0.654913i	$-0.772705\pi$
$269$	−1.65309	+	1.65309i	−0.100790	+	0.100790i	0.654913	+	0.755704i	$0.272705\pi$
$270$		0			0
$271$			19.8214i			1.20406i	0.798472	+	0.602032i	$0.205642\pi$
$271$			19.8214i			1.20406i	−0.798472	+	0.602032i	$0.794358\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.04848	−	3.04848i	−0.183830	−	0.183830i
$276$		0			0
$277$	−3.17209	+	3.17209i	−0.190592	+	0.190592i	−0.795952	−	0.605360i	$-0.793029\pi$
$277$	−3.17209	+	3.17209i	−0.190592	+	0.190592i	0.605360	+	0.795952i	$0.293029\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.10515			0.0659280			0.0329640	−	0.999457i	$-0.489505\pi$
$281$	1.10515			0.0659280			0.0329640	+	0.999457i	$0.489505\pi$
$282$		0			0
$283$	7.08295	+	7.08295i	0.421038	+	0.421038i	0.885561	−	0.464523i	$-0.153774\pi$
$283$	7.08295	+	7.08295i	0.421038	+	0.421038i	−0.464523	+	0.885561i	$0.653774\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.91226			−0.408018
$288$		0			0
$289$	15.9249			0.936760
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−23.5296	−	23.5296i	−1.37461	−	1.37461i	−0.853470	−	0.521142i	$-0.825506\pi$
$293$	−23.5296	−	23.5296i	−1.37461	−	1.37461i	−0.521142	−	0.853470i	$-0.674494\pi$
$294$		0			0
$295$	1.13192			0.0659030
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.01631	+	7.01631i	−0.405764	+	0.405764i
$300$		0			0
$301$	2.30822	+	2.30822i	0.133043	+	0.133043i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	6.99014i		−	0.400254i
$306$		0			0
$307$	8.29187	−	8.29187i	0.473242	−	0.473242i	−0.429720	−	0.902962i	$-0.641388\pi$
$307$	8.29187	−	8.29187i	0.473242	−	0.473242i	0.902962	+	0.429720i	$0.141388\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	10.1040i		−	0.572945i	−0.958088	−	0.286472i	$-0.907517\pi$
$311$		−	10.1040i		−	0.572945i	0.958088	−	0.286472i	$-0.0924827\pi$
$312$		0			0
$313$		−	0.519053i		−	0.0293386i	−0.999892	−	0.0146693i	$-0.995330\pi$
$313$		−	0.519053i		−	0.0293386i	0.999892	−	0.0146693i	$-0.00466955\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.3517	−	10.3517i	0.581408	−	0.581408i	−0.353882	−	0.935290i	$-0.615139\pi$
$317$	10.3517	−	10.3517i	0.581408	−	0.581408i	0.935290	+	0.353882i	$0.115139\pi$
$318$		0			0
$319$		−	4.58447i		−	0.256681i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.947505	+	0.947505i	0.0527206	+	0.0527206i
$324$		0			0
$325$	−10.8028	+	10.8028i	−0.599233	+	0.599233i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9.68077			−0.533718
$330$		0			0
$331$	−23.4180	−	23.4180i	−1.28717	−	1.28717i	−0.936498	−	0.350672i	$-0.885953\pi$
$331$	−23.4180	−	23.4180i	−1.28717	−	1.28717i	−0.350672	−	0.936498i	$-0.614047\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.61498			0.306779
$336$		0			0
$337$	12.8612			0.700597			0.350298	−	0.936638i	$-0.386080\pi$
$337$	12.8612			0.700597			0.350298	+	0.936638i	$0.386080\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.37532	+	6.37532i	0.345243	+	0.345243i
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.07714	−	6.07714i	0.326238	−	0.326238i	−0.524916	−	0.851154i	$-0.675903\pi$
$347$	6.07714	−	6.07714i	0.326238	−	0.326238i	0.851154	+	0.524916i	$0.175903\pi$
$348$		0			0
$349$	−15.0938	−	15.0938i	−0.807952	−	0.807952i	0.176372	−	0.984324i	$-0.443564\pi$
$349$	−15.0938	−	15.0938i	−0.807952	−	0.807952i	−0.984324	+	0.176372i	$0.943564\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.0292i			0.959599i	0.877378	+	0.479799i	$0.159291\pi$
$353$			18.0292i			0.959599i	−0.877378	+	0.479799i	$0.840709\pi$
$354$		0			0
$355$	3.59822	−	3.59822i	0.190974	−	0.190974i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			14.7990i			0.781059i	0.920590	+	0.390529i	$0.127708\pi$
$359$			14.7990i			0.781059i	−0.920590	+	0.390529i	$0.872292\pi$
$360$		0			0
$361$			17.3299i			0.912098i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.30063	−	3.30063i	0.172763	−	0.172763i
$366$		0			0
$367$		−	15.4513i		−	0.806554i	−0.915078	−	0.403277i	$-0.867871\pi$
$367$		−	15.4513i		−	0.806554i	0.915078	−	0.403277i	$-0.132129\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.02498	−	8.02498i	−0.416636	−	0.416636i
$372$		0			0
$373$	−18.9447	+	18.9447i	−0.980918	+	0.980918i	−0.999821	−	0.0189036i	$-0.993982\pi$
$373$	−18.9447	+	18.9447i	−0.980918	+	0.980918i	0.0189036	+	0.999821i	$0.493982\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−16.2459			−0.836706
$378$		0			0
$379$	−3.79368	−	3.79368i	−0.194868	−	0.194868i	0.602928	−	0.797796i	$-0.294001\pi$
$379$	−3.79368	−	3.79368i	−0.194868	−	0.194868i	−0.797796	+	0.602928i	$0.794001\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.0654			−0.667608			−0.333804	−	0.942642i	$-0.608332\pi$
$383$	−13.0654			−0.667608			−0.333804	+	0.942642i	$0.608332\pi$
$384$		0			0
$385$	0.669458			0.0341187
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.7690	−	13.7690i	−0.698117	−	0.698117i	0.265887	−	0.964004i	$-0.414335\pi$
$389$	−13.7690	−	13.7690i	−0.698117	−	0.698117i	−0.964004	+	0.265887i	$0.914335\pi$
$390$		0			0
$391$	−3.03690			−0.153583
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.12098	+	1.12098i	−0.0564028	+	0.0564028i
$396$		0			0
$397$	−14.5406	−	14.5406i	−0.729772	−	0.729772i	0.240802	−	0.970574i	$-0.422589\pi$
$397$	−14.5406	−	14.5406i	−0.729772	−	0.729772i	−0.970574	+	0.240802i	$0.922589\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			31.9581i			1.59591i	0.602715	+	0.797956i	$0.294085\pi$
$401$			31.9581i			1.59591i	−0.602715	+	0.797956i	$0.705915\pi$
$402$		0			0
$403$	22.5921	−	22.5921i	1.12539	−	1.12539i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	7.40266i		−	0.366936i
$408$		0			0
$409$			32.5777i			1.61087i	0.592687	+	0.805433i	$0.298067\pi$
$409$			32.5777i			1.61087i	−0.592687	+	0.805433i	$0.701933\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.14297	−	1.14297i	0.0562420	−	0.0562420i
$414$		0			0
$415$			0.613296i			0.0301055i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11.3571	+	11.3571i	0.554829	+	0.554829i	0.927831	−	0.373002i	$-0.121671\pi$
$419$	11.3571	+	11.3571i	0.554829	+	0.554829i	−0.373002	+	0.927831i	$0.621671\pi$
$420$		0			0
$421$	12.6460	−	12.6460i	0.616330	−	0.616330i	−0.328258	−	0.944588i	$-0.606462\pi$
$421$	12.6460	−	12.6460i	0.616330	−	0.616330i	0.944588	+	0.328258i	$0.106462\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.67584			−0.226811
$426$		0			0
$427$	−7.05839	−	7.05839i	−0.341579	−	0.341579i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	31.2767			1.50655			0.753273	−	0.657708i	$-0.228474\pi$
$431$	31.2767			1.50655			0.753273	+	0.657708i	$0.228474\pi$
$432$		0			0
$433$	28.6869			1.37861			0.689303	−	0.724473i	$-0.257917\pi$
$433$	28.6869			1.37861			0.689303	+	0.724473i	$0.257917\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.67654	+	2.67654i	0.128036	+	0.128036i
$438$		0			0
$439$	−30.0241			−1.43297			−0.716487	−	0.697601i	$-0.754251\pi$
$439$	−30.0241			−1.43297			−0.716487	+	0.697601i	$0.754251\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.5402	−	16.5402i	0.785850	−	0.785850i	−0.194961	−	0.980811i	$-0.562458\pi$
$443$	16.5402	−	16.5402i	0.785850	−	0.785850i	0.980811	+	0.194961i	$0.0624580\pi$
$444$		0			0
$445$	1.97687	+	1.97687i	0.0937124	+	0.0937124i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	20.3748i		−	0.961546i	−0.876845	−	0.480773i	$-0.840356\pi$
$449$		−	20.3748i		−	0.961546i	0.876845	−	0.480773i	$-0.159644\pi$
$450$		0			0
$451$	−4.67265	+	4.67265i	−0.220026	+	0.220026i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	2.37235i		−	0.111217i
$456$		0			0
$457$			37.0265i			1.73203i	0.500020	+	0.866014i	$0.333326\pi$
$457$			37.0265i			1.73203i	−0.500020	+	0.866014i	$0.666674\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−22.3029	+	22.3029i	−1.03875	+	1.03875i	−0.0395325	+	0.999218i	$0.512587\pi$
$461$	−22.3029	+	22.3029i	−1.03875	+	1.03875i	−0.999218	+	0.0395325i	$0.987413\pi$
$462$		0			0
$463$			21.5480i			1.00142i	0.865616	+	0.500709i	$0.166927\pi$
$463$			21.5480i			1.00142i	−0.865616	+	0.500709i	$0.833073\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	21.7980	+	21.7980i	1.00869	+	1.00869i	0.999962	+	0.00873046i	$0.00277903\pi$
$467$	21.7980	+	21.7980i	1.00869	+	1.00869i	0.00873046	+	0.999962i	$0.497221\pi$
$468$		0			0
$469$	5.66980	−	5.66980i	0.261807	−	0.261807i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.12068			0.143489
$474$		0			0
$475$	4.12100	+	4.12100i	0.189084	+	0.189084i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	29.5744			1.35129			0.675644	−	0.737228i	$-0.263866\pi$
$479$	29.5744			1.35129			0.675644	+	0.737228i	$0.263866\pi$
$480$		0			0
$481$	−26.2327			−1.19611
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.37974	+	3.37974i	0.153466	+	0.153466i
$486$		0			0
$487$	−17.4819			−0.792179			−0.396089	−	0.918212i	$-0.629633\pi$
$487$	−17.4819			−0.792179			−0.396089	+	0.918212i	$0.629633\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	19.1434	−	19.1434i	0.863929	−	0.863929i	−0.127863	−	0.991792i	$-0.540812\pi$
$491$	19.1434	−	19.1434i	0.863929	−	0.863929i	0.991792	+	0.127863i	$0.0408118\pi$
$492$		0			0
$493$	−3.51589	−	3.51589i	−0.158348	−	0.158348i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	7.26670i		−	0.325956i
$498$		0			0
$499$	−1.34854	+	1.34854i	−0.0603691	+	0.0603691i	−0.736647	−	0.676278i	$-0.763592\pi$
$499$	−1.34854	+	1.34854i	−0.0603691	+	0.0603691i	0.676278	+	0.736647i	$0.263592\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9.93767i		−	0.443099i	−0.975149	−	0.221549i	$-0.928889\pi$
$503$		−	9.93767i		−	0.443099i	0.975149	−	0.221549i	$-0.0711114\pi$
$504$		0			0
$505$			1.99047i			0.0885745i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.46530	−	4.46530i	0.197921	−	0.197921i	−0.601187	−	0.799108i	$-0.705305\pi$
$509$	4.46530	−	4.46530i	0.197921	−	0.197921i	0.799108	+	0.601187i	$0.205305\pi$
$510$		0			0
$511$		−	6.66570i		−	0.294873i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.76380	+	4.76380i	0.209918	+	0.209918i
$516$		0			0
$517$	−6.54414	+	6.54414i	−0.287811	+	0.287811i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−20.2545			−0.887366			−0.443683	−	0.896184i	$-0.646328\pi$
$521$	−20.2545			−0.887366			−0.443683	+	0.896184i	$0.646328\pi$
$522$		0			0
$523$	16.2772	+	16.2772i	0.711754	+	0.711754i	0.966902	−	0.255148i	$-0.0821241\pi$
$523$	16.2772	+	16.2772i	0.711754	+	0.711754i	−0.255148	+	0.966902i	$0.582124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.77865			0.425965
$528$		0			0
$529$	14.4213			0.627012
$530$		0			0
$531$		0			0
$532$		0			0
$533$	16.5584	+	16.5584i	0.717223	+	0.717223i
$534$		0			0
$535$	−4.14610			−0.179251
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.675994	−	0.675994i	0.0291171	−	0.0291171i
$540$		0			0
$541$	19.1382	+	19.1382i	0.822815	+	0.822815i	0.986511	−	0.163696i	$-0.0523417\pi$
$541$	19.1382	+	19.1382i	0.822815	+	0.822815i	−0.163696	+	0.986511i	$0.552342\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1.10968i			0.0475334i
$546$		0			0
$547$	1.15918	−	1.15918i	0.0495629	−	0.0495629i	−0.681891	−	0.731454i	$-0.738842\pi$
$547$	1.15918	−	1.15918i	0.0495629	−	0.0495629i	0.731454	+	0.681891i	$0.238842\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			6.19739i			0.264018i
$552$		0			0
$553$			2.26385i			0.0962689i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.95406	−	3.95406i	0.167539	−	0.167539i	−0.618358	−	0.785897i	$-0.712202\pi$
$557$	3.95406	−	3.95406i	0.167539	−	0.167539i	0.785897	+	0.618358i	$0.212202\pi$
$558$		0			0
$559$		−	11.0587i		−	0.467733i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	32.5658	+	32.5658i	1.37248	+	1.37248i	0.856746	+	0.515739i	$0.172483\pi$
$563$	32.5658	+	32.5658i	1.37248	+	1.37248i	0.515739	+	0.856746i	$0.327517\pi$
$564$		0			0
$565$	1.50629	−	1.50629i	0.0633702	−	0.0633702i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.8687			−0.832940			−0.416470	−	0.909150i	$-0.636733\pi$
$569$	−19.8687			−0.832940			−0.416470	+	0.909150i	$0.636733\pi$
$570$		0			0
$571$	4.59303	+	4.59303i	0.192212	+	0.192212i	0.796651	−	0.604439i	$-0.206603\pi$
$571$	4.59303	+	4.59303i	0.192212	+	0.192212i	−0.604439	+	0.796651i	$0.706603\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−13.2084			−0.550830
$576$		0			0
$577$	−28.9231			−1.20408			−0.602041	−	0.798465i	$-0.705646\pi$
$577$	−28.9231			−1.20408			−0.602041	+	0.798465i	$0.705646\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.619284	+	0.619284i	0.0256922	+	0.0256922i
$582$		0			0
$583$	−10.8497			−0.449348
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.9197	+	14.9197i	−0.615803	+	0.615803i	−0.944452	−	0.328649i	$-0.893407\pi$
$587$	−14.9197	+	14.9197i	−0.615803	+	0.615803i	0.328649	+	0.944452i	$0.393407\pi$
$588$		0			0
$589$	−8.61830	−	8.61830i	−0.355111	−	0.355111i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	17.3114i		−	0.710894i	−0.934696	−	0.355447i	$-0.884329\pi$
$593$		−	17.3114i		−	0.710894i	0.934696	−	0.355447i	$-0.115671\pi$
$594$		0			0
$595$	0.513417	−	0.513417i	0.0210480	−	0.0210480i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	0.631879i		−	0.0258179i	−0.999917	−	0.0129089i	$-0.995891\pi$
$599$		−	0.631879i		−	0.0258179i	0.999917	−	0.0129089i	$-0.00410916\pi$
$600$		0			0
$601$		−	45.0956i		−	1.83949i	−0.392520	−	0.919744i	$-0.628397\pi$
$601$		−	45.0956i		−	1.83949i	0.392520	−	0.919744i	$-0.371603\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.99427	+	4.99427i	−0.203046	+	0.203046i
$606$		0			0
$607$			40.1367i			1.62910i	0.580094	+	0.814550i	$0.303016\pi$
$607$			40.1367i			1.62910i	−0.580094	+	0.814550i	$0.696984\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	23.1904	+	23.1904i	0.938182	+	0.938182i
$612$		0			0
$613$	7.38922	−	7.38922i	0.298448	−	0.298448i	−0.541958	−	0.840406i	$-0.682317\pi$
$613$	7.38922	−	7.38922i	0.298448	−	0.298448i	0.840406	+	0.541958i	$0.182317\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.8450			−0.597637			−0.298818	−	0.954310i	$-0.596592\pi$
$617$	−14.8450			−0.597637			−0.298818	+	0.954310i	$0.596592\pi$
$618$		0			0
$619$	−24.1413	−	24.1413i	−0.970322	−	0.970322i	0.0292499	−	0.999572i	$-0.490688\pi$
$619$	−24.1413	−	24.1413i	−0.970322	−	0.970322i	−0.999572	+	0.0292499i	$0.990688\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.99233			0.159949
$624$		0			0
$625$	−17.8848			−0.715392
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.67720	−	5.67720i	−0.226365	−	0.226365i
$630$		0			0
$631$	19.3942			0.772070			0.386035	−	0.922484i	$-0.373844\pi$
$631$	19.3942			0.772070			0.386035	+	0.922484i	$0.373844\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.06953	−	1.06953i	0.0424429	−	0.0424429i
$636$		0			0
$637$	−2.39551	−	2.39551i	−0.0949134	−	0.0949134i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			48.4253i			1.91268i	0.292250	+	0.956342i	$0.405596\pi$
$641$			48.4253i			1.91268i	−0.292250	+	0.956342i	$0.594404\pi$
$642$		0			0
$643$	−2.42503	+	2.42503i	−0.0956340	+	0.0956340i	−0.753305	−	0.657671i	$-0.771542\pi$
$643$	−2.42503	+	2.42503i	−0.0956340	+	0.0956340i	0.657671	+	0.753305i	$0.271542\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			32.9245i			1.29439i	0.762323	+	0.647197i	$0.224059\pi$
$647$			32.9245i			1.29439i	−0.762323	+	0.647197i	$0.775941\pi$
$648$		0			0
$649$		−	1.54528i		−	0.0606577i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−16.0855	+	16.0855i	−0.629476	+	0.629476i	−0.947936	−	0.318460i	$-0.896834\pi$
$653$	−16.0855	+	16.0855i	−0.629476	+	0.629476i	0.318460	+	0.947936i	$0.396834\pi$
$654$		0			0
$655$		−	9.20329i		−	0.359602i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−32.1819	−	32.1819i	−1.25363	−	1.25363i	−0.954082	−	0.299546i	$-0.903165\pi$
$659$	−32.1819	−	32.1819i	−1.25363	−	1.25363i	−0.299546	−	0.954082i	$-0.596835\pi$
$660$		0			0
$661$	−28.2861	+	28.2861i	−1.10020	+	1.10020i	−0.105818	+	0.994386i	$0.533746\pi$
$661$	−28.2861	+	28.2861i	−1.10020	+	1.10020i	−0.994386	+	0.105818i	$0.966254\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.904988			−0.0350939
$666$		0			0
$667$	−9.93179	−	9.93179i	−0.384561	−	0.384561i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.54285			−0.368398
$672$		0			0
$673$	−2.63038			−0.101394			−0.0506968	−	0.998714i	$-0.516144\pi$
$673$	−2.63038			−0.101394			−0.0506968	+	0.998714i	$0.516144\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.1362	+	15.1362i	0.581730	+	0.581730i	0.935378	−	0.353649i	$-0.115059\pi$
$677$	15.1362	+	15.1362i	0.581730	+	0.581730i	−0.353649	+	0.935378i	$0.615059\pi$
$678$		0			0
$679$	6.82547			0.261938
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−5.97609	+	5.97609i	−0.228669	+	0.228669i	−0.812136	−	0.583468i	$-0.801695\pi$
$683$	−5.97609	+	5.97609i	−0.228669	+	0.228669i	0.583468	+	0.812136i	$0.301695\pi$
$684$		0			0
$685$	−8.34150	−	8.34150i	−0.318712	−	0.318712i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			38.4478i			1.46474i
$690$		0			0
$691$	−13.6369	+	13.6369i	−0.518774	+	0.518774i	−0.917200	−	0.398426i	$-0.869556\pi$
$691$	−13.6369	+	13.6369i	−0.518774	+	0.518774i	0.398426	+	0.917200i	$0.369556\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			10.7236i			0.406769i
$696$		0			0
$697$			7.16704i			0.271471i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	23.6139	−	23.6139i	0.891886	−	0.891886i	−0.102815	−	0.994701i	$-0.532785\pi$
$701$	23.6139	−	23.6139i	0.891886	−	0.891886i	0.994701	+	0.102815i	$0.0327849\pi$
$702$		0			0
$703$			10.0071i			0.377424i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.00990	+	2.00990i	0.0755900	+	0.0755900i
$708$		0			0
$709$	13.4057	−	13.4057i	0.503460	−	0.503460i	−0.409051	−	0.912511i	$-0.634140\pi$
$709$	13.4057	−	13.4057i	0.503460	−	0.503460i	0.912511	+	0.409051i	$0.134140\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	27.6230			1.03449
$714$		0			0
$715$	−1.60369	−	1.60369i	−0.0599746	−	0.0599746i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.0030			−0.596810			−0.298405	−	0.954439i	$-0.596455\pi$
$719$	−16.0030			−0.596810			−0.298405	+	0.954439i	$0.596455\pi$
$720$		0			0
$721$	9.62061			0.358290
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.2917	−	15.2917i	−0.567920	−	0.567920i
$726$		0			0
$727$	−36.4754			−1.35280			−0.676399	−	0.736536i	$-0.736460\pi$
$727$	−36.4754			−1.35280			−0.676399	+	0.736536i	$0.736460\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.39329	−	2.39329i	0.0885192	−	0.0885192i
$732$		0			0
$733$	−2.37303	−	2.37303i	−0.0876497	−	0.0876497i	0.661923	−	0.749572i	$-0.269741\pi$
$733$	−2.37303	−	2.37303i	−0.0876497	−	0.0876497i	−0.749572	+	0.661923i	$0.769741\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	7.66550i		−	0.282362i
$738$		0			0
$739$	−10.6601	+	10.6601i	−0.392137	+	0.392137i	−0.875449	−	0.483311i	$-0.839434\pi$
$739$	−10.6601	+	10.6601i	−0.392137	+	0.392137i	0.483311	+	0.875449i	$0.339434\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			16.4432i			0.603244i	0.953428	+	0.301622i	$0.0975281\pi$
$743$			16.4432i			0.603244i	−0.953428	+	0.301622i	$0.902472\pi$
$744$		0			0
$745$			5.36350i			0.196503i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.18658	+	4.18658i	−0.152974	+	0.152974i
$750$		0			0
$751$			8.62234i			0.314634i	0.987548	+	0.157317i	$0.0502844\pi$
$751$			8.62234i			0.314634i	−0.987548	+	0.157317i	$0.949716\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.924793	−	0.924793i	−0.0336567	−	0.0336567i
$756$		0			0
$757$	14.7724	−	14.7724i	0.536910	−	0.536910i	−0.385710	−	0.922620i	$-0.626043\pi$
$757$	14.7724	−	14.7724i	0.536910	−	0.536910i	0.922620	+	0.385710i	$0.126043\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	49.3931			1.79050			0.895249	−	0.445566i	$-0.146998\pi$
$761$	49.3931			1.79050			0.895249	+	0.445566i	$0.146998\pi$
$762$		0			0
$763$	1.12051	+	1.12051i	0.0405653	+	0.0405653i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.47599			−0.197727
$768$		0			0
$769$	20.4566			0.737685			0.368843	−	0.929492i	$-0.379754\pi$
$769$	20.4566			0.737685			0.368843	+	0.929492i	$0.379754\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	30.7904	+	30.7904i	1.10745	+	1.10745i	0.993484	+	0.113967i	$0.0363560\pi$
$773$	30.7904	+	30.7904i	1.10745	+	1.10745i	0.113967	+	0.993484i	$0.463644\pi$
$774$		0			0
$775$	42.5304			1.52774
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.31659	−	6.31659i	0.226315	−	0.226315i
$780$		0			0
$781$	−4.91224	−	4.91224i	−0.175774	−	0.175774i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	6.21080i		−	0.221673i
$786$		0			0
$787$	−4.61526	+	4.61526i	−0.164516	+	0.164516i	−0.784564	−	0.620048i	$-0.787113\pi$
$787$	−4.61526	+	4.61526i	−0.164516	+	0.164516i	0.620048	+	0.784564i	$0.287113\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	3.04200i		−	0.108161i
$792$		0			0
$793$			33.8168i			1.20087i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	27.8979	−	27.8979i	0.988195	−	0.988195i	−0.0117360	−	0.999931i	$-0.503736\pi$
$797$	27.8979	−	27.8979i	0.988195	−	0.988195i	0.999931	+	0.0117360i	$0.00373577\pi$
$798$		0			0
$799$			10.0376i			0.355104i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4.50597	−	4.50597i	−0.159012	−	0.159012i
$804$		0			0
$805$	1.45031	−	1.45031i	0.0511168	−	0.0511168i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18.6104			−0.654308			−0.327154	−	0.944971i	$-0.606090\pi$
$809$	−18.6104			−0.654308			−0.327154	+	0.944971i	$0.606090\pi$
$810$		0			0
$811$	38.8480	+	38.8480i	1.36414	+	1.36414i	0.868568	+	0.495570i	$0.165041\pi$
$811$	38.8480	+	38.8480i	1.36414	+	1.36414i	0.495570	+	0.868568i	$0.334959\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9.01866			0.315910
$816$		0			0
$817$	−4.21861			−0.147590
$818$		0			0
$819$		0			0
$820$		0			0
$821$	37.4454	+	37.4454i	1.30685	+	1.30685i	0.923673	+	0.383181i	$0.125171\pi$
$821$	37.4454	+	37.4454i	1.30685	+	1.30685i	0.383181	+	0.923673i	$0.374829\pi$
$822$		0			0
$823$	−1.86401			−0.0649752			−0.0324876	−	0.999472i	$-0.510343\pi$
$823$	−1.86401			−0.0649752			−0.0324876	+	0.999472i	$0.510343\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.7774	+	31.7774i	−1.10501	+	1.10501i	−0.111211	+	0.993797i	$0.535473\pi$
$827$	−31.7774	+	31.7774i	−1.10501	+	1.10501i	−0.993797	+	0.111211i	$0.964527\pi$
$828$		0			0
$829$	14.5497	+	14.5497i	0.505333	+	0.505333i	0.913090	−	0.407757i	$-0.133689\pi$
$829$	14.5497	+	14.5497i	0.505333	+	0.505333i	−0.407757	+	0.913090i	$0.633689\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	1.03686i		−	0.0359250i
$834$		0			0
$835$	4.67686	−	4.67686i	0.161849	−	0.161849i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	48.2843i		−	1.66696i	−0.552549	−	0.833480i	$-0.686345\pi$
$839$		−	48.2843i		−	1.66696i	0.552549	−	0.833480i	$-0.313655\pi$
$840$		0			0
$841$			6.00345i			0.207016i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.754183	−	0.754183i	0.0259447	−	0.0259447i
$846$		0			0
$847$			10.0861i			0.346561i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−16.0371	−	16.0371i	−0.549745	−	0.549745i
$852$		0			0
$853$	−29.1174	+	29.1174i	−0.996962	+	0.996962i	−0.999995	−	0.00303388i	$-0.999034\pi$
$853$	−29.1174	+	29.1174i	−0.996962	+	0.996962i	0.00303388	+	0.999995i	$0.499034\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	30.4208			1.03915			0.519577	−	0.854424i	$-0.326090\pi$
$857$	30.4208			1.03915			0.519577	+	0.854424i	$0.326090\pi$
$858$		0			0
$859$	−12.4603	−	12.4603i	−0.425141	−	0.425141i	0.461828	−	0.886969i	$-0.347194\pi$
$859$	−12.4603	−	12.4603i	−0.425141	−	0.425141i	−0.886969	+	0.461828i	$0.847194\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	20.6468			0.702826			0.351413	−	0.936221i	$-0.385701\pi$
$863$	20.6468			0.702826			0.351413	+	0.936221i	$0.385701\pi$
$864$		0			0
$865$	16.8102			0.571564
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.53035	+	1.53035i	0.0519136	+	0.0519136i
$870$		0			0
$871$	−27.1641			−0.920421
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.70884	−	4.70884i	0.159188	−	0.159188i
$876$		0			0
$877$	−16.2158	−	16.2158i	−0.547570	−	0.547570i	0.378167	−	0.925737i	$-0.376554\pi$
$877$	−16.2158	−	16.2158i	−0.547570	−	0.547570i	−0.925737	+	0.378167i	$0.876554\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			7.21499i			0.243079i	0.992587	+	0.121540i	$0.0387831\pi$
$881$			7.21499i			0.243079i	−0.992587	+	0.121540i	$0.961217\pi$
$882$		0			0
$883$	−19.5756	+	19.5756i	−0.658771	+	0.658771i	−0.955089	−	0.296318i	$-0.904241\pi$
$883$	−19.5756	+	19.5756i	−0.658771	+	0.658771i	0.296318	+	0.955089i	$0.404241\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			22.3985i			0.752067i	0.926606	+	0.376034i	$0.122712\pi$
$887$			22.3985i			0.752067i	−0.926606	+	0.376034i	$0.877288\pi$
$888$		0			0
$889$		−	2.15994i		−	0.0724421i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.84652	−	8.84652i	0.296037	−	0.296037i
$894$		0			0
$895$		−	0.822290i		−	0.0274861i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	31.9798	+	31.9798i	1.06659	+	1.06659i
$900$		0			0
$901$	−8.32077	+	8.32077i	−0.277205	+	0.277205i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.70398			0.123125
$906$		0			0
$907$	2.55150	+	2.55150i	0.0847212	+	0.0847212i	0.748197	−	0.663476i	$-0.230920\pi$
$907$	2.55150	+	2.55150i	0.0847212	+	0.0847212i	−0.663476	+	0.748197i	$0.730920\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−28.3400			−0.938946			−0.469473	−	0.882947i	$-0.655556\pi$
$911$	−28.3400			−0.938946			−0.469473	+	0.882947i	$0.655556\pi$
$912$		0			0
$913$	0.837264			0.0277094
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.29314	−	9.29314i	−0.306886	−	0.306886i
$918$		0			0
$919$	−14.6615			−0.483639			−0.241819	−	0.970321i	$-0.577744\pi$
$919$	−14.6615			−0.483639			−0.241819	+	0.970321i	$0.577744\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−17.4074	+	17.4074i	−0.572973	+	0.572973i
$924$		0			0
$925$	−24.6919	−	24.6919i	−0.811866	−	0.811866i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			38.0843i			1.24950i	0.780823	+	0.624752i	$0.214800\pi$
$929$			38.0843i			1.24950i	−0.780823	+	0.624752i	$0.785200\pi$
$930$		0			0
$931$	−0.913823	+	0.913823i	−0.0299493	+	0.0299493i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	0.694133i		−	0.0227006i
$936$		0			0
$937$		−	31.6411i		−	1.03367i	−0.856085	−	0.516835i	$-0.827110\pi$
$937$		−	31.6411i		−	1.03367i	0.856085	−	0.516835i	$-0.172890\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.4691	+	12.4691i	−0.406480	+	0.406480i	−0.880509	−	0.474029i	$-0.842799\pi$
$941$	−12.4691	+	12.4691i	−0.406480	+	0.406480i	0.474029	+	0.880509i	$0.342799\pi$
$942$		0			0
$943$			20.2456i			0.659289i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	35.5835	+	35.5835i	1.15631	+	1.15631i	0.985263	+	0.171044i	$0.0547141\pi$
$947$	35.5835	+	35.5835i	1.15631	+	1.15631i	0.171044	+	0.985263i	$0.445286\pi$
$948$		0			0
$949$	−15.9677	+	15.9677i	−0.518335	+	0.518335i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.461925			0.0149632			0.00748160	−	0.999972i	$-0.497619\pi$
$953$	0.461925			0.0149632			0.00748160	+	0.999972i	$0.497619\pi$
$954$		0			0
$955$	−5.10192	−	5.10192i	−0.165094	−	0.165094i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−16.8459			−0.543982
$960$		0			0
$961$	−57.9445			−1.86918
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.01041	+	6.01041i	0.193482	+	0.193482i
$966$		0			0
$967$	31.2365			1.00450			0.502248	−	0.864723i	$-0.332506\pi$
$967$	31.2365			1.00450			0.502248	+	0.864723i	$0.332506\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−21.5210	+	21.5210i	−0.690642	+	0.690642i	−0.962373	−	0.271731i	$-0.912404\pi$
$971$	−21.5210	+	21.5210i	−0.690642	+	0.690642i	0.271731	+	0.962373i	$0.412404\pi$
$972$		0			0
$973$	10.8283	+	10.8283i	0.347139	+	0.347139i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	39.8191i		−	1.27393i	−0.770895	−	0.636963i	$-0.780191\pi$
$977$		−	39.8191i		−	1.27393i	0.770895	−	0.636963i	$-0.219809\pi$
$978$		0			0
$979$	2.69879	−	2.69879i	0.0862537	−	0.0862537i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			44.1551i			1.40833i	0.710037	+	0.704164i	$0.248678\pi$
$983$			44.1551i			1.40833i	−0.710037	+	0.704164i	$0.751322\pi$
$984$		0			0
$985$		−	8.68789i		−	0.276819i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.76065	−	6.76065i	0.214976	−	0.214976i
$990$		0			0
$991$		−	19.2245i		−	0.610687i	−0.952242	−	0.305343i	$-0.901229\pi$
$991$		−	19.2245i		−	0.610687i	0.952242	−	0.305343i	$-0.0987713\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−11.2994	−	11.2994i	−0.358214	−	0.358214i
$996$		0			0
$997$	−25.3176	+	25.3176i	−0.801818	+	0.801818i	−0.983380	−	0.181562i	$-0.941885\pi$
$997$	−25.3176	+	25.3176i	−0.801818	+	0.801818i	0.181562	+	0.983380i	$0.441885\pi$
$998$		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.1583.11		36
3.2	odd	2	inner	4032.2.v.d.1583.8		36
4.3	odd	2		1008.2.v.d.323.8	✓	36
12.11	even	2		1008.2.v.d.323.11	yes	36
16.5	even	4		1008.2.v.d.827.11	yes	36
16.11	odd	4	inner	4032.2.v.d.3599.8		36
48.5	odd	4		1008.2.v.d.827.8	yes	36
48.11	even	4	inner	4032.2.v.d.3599.11		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.d.323.8	✓	36	4.3	odd	2
1008.2.v.d.323.11	yes	36	12.11	even	2
1008.2.v.d.827.8	yes	36	48.5	odd	4
1008.2.v.d.827.11	yes	36	16.5	even	4
4032.2.v.d.1583.8		36	3.2	odd	2	inner
4032.2.v.d.1583.11		36	1.1	even	1	trivial
4032.2.v.d.3599.8		36	16.11	odd	4	inner
4032.2.v.d.3599.11		36	48.11	even	4	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)

\(f(q)\)	\(=\)	\(q+(0.495166 + 0.495166i) q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+(0.495166 + 0.495166i) q^{5} +1.00000 q^{7} +(0.675994 - 0.675994i) q^{11} +(-2.39551 - 2.39551i) q^{13} -1.03686i q^{17} +(-0.913823 + 0.913823i) q^{19} -2.92895i q^{23} -4.50962i q^{25} +(3.39091 - 3.39091i) q^{29} +9.43104i q^{31} +(0.495166 + 0.495166i) q^{35} +(5.47539 - 5.47539i) q^{37} -6.91226 q^{41} +(2.30822 + 2.30822i) q^{43} -9.68077 q^{47} +1.00000 q^{49} +(-8.02498 - 8.02498i) q^{53} +0.669458 q^{55} +(1.14297 - 1.14297i) q^{59} +(-7.05839 - 7.05839i) q^{61} -2.37235i q^{65} +(5.66980 - 5.66980i) q^{67} -7.26670i q^{71} -6.66570i q^{73} +(0.675994 - 0.675994i) q^{77} +2.26385i q^{79} +(0.619284 + 0.619284i) q^{83} +(0.513417 - 0.513417i) q^{85} +3.99233 q^{89} +(-2.39551 - 2.39551i) q^{91} -0.904988 q^{95} +6.82547 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 36 q^{7}+O(q^{10}) \) 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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