LMFDB - Embedded newform 1445.2.d.g.866.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1445,2,Mod(866,1445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1445, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1445.866");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1445 = 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1445.d (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:	$11.5383830921$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 $ x^12 + 18x^10 + 83x^8 + 152x^6 + 111x^4 + 22*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			866.12
Root			$-1.19804i$ of defining polynomial
Character	$\chi$	$=$	1445.866
Dual form			1445.2.d.g.866.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.24891 q^{2} +0.198035i q^{3} +3.05761 q^{4} -1.00000i q^{5} +0.445364i q^{6} -1.89231i q^{7} +2.37848 q^{8} +2.96078 q^{9} -2.24891i q^{10} -3.52898i q^{11} +0.605515i q^{12} -1.27324 q^{13} -4.25565i q^{14} +0.198035 q^{15} -0.766231 q^{16} +6.65854 q^{18} +4.69149 q^{19} -3.05761i q^{20} +0.374744 q^{21} -7.93637i q^{22} +0.574930i q^{23} +0.471022i q^{24} -1.00000 q^{25} -2.86340 q^{26} +1.18044i q^{27} -5.78596i q^{28} +5.39821i q^{29} +0.445364 q^{30} -6.21652i q^{31} -6.48015 q^{32} +0.698861 q^{33} -1.89231 q^{35} +9.05292 q^{36} -8.48642i q^{37} +10.5508 q^{38} -0.252146i q^{39} -2.37848i q^{40} +6.06547i q^{41} +0.842768 q^{42} -9.16040 q^{43} -10.7902i q^{44} -2.96078i q^{45} +1.29297i q^{46} +10.7932 q^{47} -0.151741i q^{48} +3.41915 q^{49} -2.24891 q^{50} -3.89307 q^{52} +9.90174 q^{53} +2.65472i q^{54} -3.52898 q^{55} -4.50083i q^{56} +0.929079i q^{57} +12.1401i q^{58} +3.15903 q^{59} +0.605515 q^{60} +5.14302i q^{61} -13.9804i q^{62} -5.60273i q^{63} -13.0408 q^{64} +1.27324i q^{65} +1.57168 q^{66} +0.281859 q^{67} -0.113856 q^{69} -4.25565 q^{70} +11.7434i q^{71} +7.04216 q^{72} +9.83451i q^{73} -19.0852i q^{74} -0.198035i q^{75} +14.3448 q^{76} -6.67793 q^{77} -0.567054i q^{78} +16.9407i q^{79} +0.766231i q^{80} +8.64858 q^{81} +13.6407i q^{82} -8.51139 q^{83} +1.14582 q^{84} -20.6010 q^{86} -1.06903 q^{87} -8.39360i q^{88} -16.7227 q^{89} -6.65854i q^{90} +2.40937i q^{91} +1.75791i q^{92} +1.23109 q^{93} +24.2729 q^{94} -4.69149i q^{95} -1.28330i q^{96} -5.91282i q^{97} +7.68938 q^{98} -10.4485i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} + 12 q^{4} - 12 q^{8} - 4 q^{9} - 8 q^{15} + 4 q^{16} + 28 q^{18} + 24 q^{19} + 16 q^{21} - 12 q^{25} + 24 q^{26} + 8 q^{30} + 12 q^{32} - 16 q^{33} + 16 q^{35} + 20 q^{36} - 24 q^{38} + 16 q^{42}+ \cdots - 44 q^{98}+O(q^{100}) $ 12 * q - 4 * q^2 + 12 * q^4 - 12 * q^8 - 4 * q^9 - 8 * q^15 + 4 * q^16 + 28 * q^18 + 24 * q^19 + 16 * q^21 - 12 * q^25 + 24 * q^26 + 8 * q^30 + 12 * q^32 - 16 * q^33 + 16 * q^35 + 20 * q^36 - 24 * q^38 + 16 * q^42 - 16 * q^43 + 48 * q^47 + 20 * q^49 + 4 * q^50 - 56 * q^52 + 32 * q^59 - 16 * q^60 - 28 * q^64 + 32 * q^66 - 8 * q^67 - 16 * q^70 - 20 * q^72 + 72 * q^76 - 48 * q^77 - 28 * q^81 - 40 * q^83 - 96 * q^86 - 48 * q^87 - 24 * q^89 - 72 * q^93 - 40 * q^94 - 44 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1445\mathbb{Z}\right)^\times$.

$n$	$581$	$1157$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.24891	1.59022	0.795111	−	0.606464i	$-0.207413\pi$
$2$	2.24891	1.59022	0.795111	+	0.606464i	$0.207413\pi$
$3$	0.198035i	0.114336i	0.998365	+	0.0571678i	$0.0182070\pi$
$3$	0.198035i	0.114336i	−0.998365	+	0.0571678i	$0.981793\pi$
$4$	3.05761	1.52881
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0.445364i	0.181819i
$7$	− 1.89231i	− 0.715227i	−0.933870	−	0.357613i	$-0.883591\pi$
$7$	− 1.89231i	− 0.715227i	0.933870	−	0.357613i	$-0.116409\pi$
$8$	2.37848	0.840919
$9$	2.96078	0.986927
$10$	− 2.24891i	− 0.711169i
$11$	− 3.52898i	− 1.06403i	−0.846736	−	0.532013i	$-0.821436\pi$
$11$	− 3.52898i	− 1.06403i	0.846736	−	0.532013i	$-0.178564\pi$
$12$	0.605515i	0.174797i
$13$	−1.27324	−0.353133	−0.176566	−	0.984289i	$-0.556499\pi$
$13$	−1.27324	−0.353133	−0.176566	+	0.984289i	$0.556499\pi$
$14$	− 4.25565i	− 1.13737i
$15$	0.198035	0.0511324
$16$	−0.766231	−0.191558
$17$	0	0
$17$	0	0
$18$	6.65854	1.56943
$19$	4.69149	1.07630	0.538151	−	0.842849i	$-0.319123\pi$
$19$	4.69149	1.07630	0.538151	+	0.842849i	$0.319123\pi$
$20$	− 3.05761i	− 0.683703i
$21$	0.374744	0.0817759
$22$	− 7.93637i	− 1.69204i
$23$	0.574930i	0.119881i	0.998202	+	0.0599405i	$0.0190911\pi$
$23$	0.574930i	0.119881i	−0.998202	+	0.0599405i	$0.980909\pi$
$24$	0.471022i	0.0961470i
$25$	−1.00000	−0.200000
$26$	−2.86340	−0.561560
$27$	1.18044i	0.227177i
$28$	− 5.78596i	− 1.09344i
$29$	5.39821i	1.00242i	0.865325	+	0.501211i	$0.167112\pi$
$29$	5.39821i	1.00242i	−0.865325	+	0.501211i	$0.832888\pi$
$30$	0.445364	0.0813119
$31$	− 6.21652i	− 1.11652i	−0.829666	−	0.558260i	$-0.811469\pi$
$31$	− 6.21652i	− 1.11652i	0.829666	−	0.558260i	$-0.188531\pi$
$32$	−6.48015	−1.14554
$33$	0.698861	0.121656
$34$	0	0
$35$	−1.89231	−0.319859
$36$	9.05292	1.50882
$37$	− 8.48642i	− 1.39516i	−0.716508	−	0.697579i	$-0.754261\pi$
$37$	− 8.48642i	− 1.39516i	0.716508	−	0.697579i	$-0.245739\pi$
$38$	10.5508	1.71156
$39$	− 0.252146i	− 0.0403757i
$40$	− 2.37848i	− 0.376071i
$41$	6.06547i	0.947267i	0.880722	+	0.473634i	$0.157058\pi$
$41$	6.06547i	0.947267i	−0.880722	+	0.473634i	$0.842942\pi$
$42$	0.842768	0.130042
$43$	−9.16040	−1.39695	−0.698474	−	0.715635i	$-0.746137\pi$
$43$	−9.16040	−1.39695	−0.698474	+	0.715635i	$0.746137\pi$
$44$	− 10.7902i	− 1.62669i
$45$	− 2.96078i	− 0.441367i
$46$	1.29297i	0.190638i
$47$	10.7932	1.57434	0.787172	−	0.616734i	$-0.211545\pi$
$47$	10.7932	1.57434	0.787172	+	0.616734i	$0.211545\pi$
$48$	− 0.151741i	− 0.0219019i
$49$	3.41915	0.488450
$50$	−2.24891	−0.318044
$51$	0	0
$52$	−3.89307	−0.539872
$53$	9.90174	1.36011	0.680054	−	0.733162i	$-0.261956\pi$
$53$	9.90174	1.36011	0.680054	+	0.733162i	$0.261956\pi$
$54$	2.65472i	0.361261i
$55$	−3.52898	−0.475847
$56$	− 4.50083i	− 0.601448i
$57$	0.929079i	0.123060i
$58$	12.1401i	1.59407i
$59$	3.15903	0.411270	0.205635	−	0.978629i	$-0.434074\pi$
$59$	3.15903	0.411270	0.205635	+	0.978629i	$0.434074\pi$
$60$	0.605515	0.0781716
$61$	5.14302i	0.658496i	0.944243	+	0.329248i	$0.106795\pi$
$61$	5.14302i	0.658496i	−0.944243	+	0.329248i	$0.893205\pi$
$62$	− 13.9804i	− 1.77551i
$63$	− 5.60273i	− 0.705877i
$64$	−13.0408	−1.63010
$65$	1.27324i	0.157926i
$66$	1.57168	0.193460
$67$	0.281859	0.0344345	0.0172173	−	0.999852i	$-0.494519\pi$
$67$	0.281859	0.0344345	0.0172173	+	0.999852i	$0.494519\pi$
$68$	0	0
$69$	−0.113856	−0.0137067
$70$	−4.25565	−0.508647
$71$	11.7434i	1.39369i	0.717223	+	0.696844i	$0.245413\pi$
$71$	11.7434i	1.39369i	−0.717223	+	0.696844i	$0.754587\pi$
$72$	7.04216	0.829926
$73$	9.83451i	1.15104i	0.817787	+	0.575521i	$0.195201\pi$
$73$	9.83451i	1.15104i	−0.817787	+	0.575521i	$0.804799\pi$
$74$	− 19.0852i	− 2.21861i
$75$	− 0.198035i	− 0.0228671i
$76$	14.3448	1.64546
$77$	−6.67793	−0.761021
$78$	− 0.567054i	− 0.0642063i
$79$	16.9407i	1.90597i	0.303014	+	0.952986i	$0.402007\pi$
$79$	16.9407i	1.90597i	−0.303014	+	0.952986i	$0.597993\pi$
$80$	0.766231i	0.0856672i
$81$	8.64858	0.960953
$82$	13.6407i	1.50637i
$83$	−8.51139	−0.934246	−0.467123	−	0.884192i	$-0.654709\pi$
$83$	−8.51139	−0.934246	−0.467123	+	0.884192i	$0.654709\pi$
$84$	1.14582	0.125020
$85$	0	0
$86$	−20.6010	−2.22146
$87$	−1.06903	−0.114613
$88$	− 8.39360i	− 0.894761i
$89$	−16.7227	−1.77260	−0.886300	−	0.463112i	$-0.846733\pi$
$89$	−16.7227	−1.77260	−0.886300	+	0.463112i	$0.846733\pi$
$90$	− 6.65854i	− 0.701872i
$91$	2.40937i	0.252570i
$92$	1.75791i	0.183275i
$93$	1.23109	0.127658
$94$	24.2729	2.50356
$95$	− 4.69149i	− 0.481337i
$96$	− 1.28330i	− 0.130976i
$97$	− 5.91282i	− 0.600356i	−0.953883	−	0.300178i	$-0.902954\pi$
$97$	− 5.91282i	− 0.600356i	0.953883	−	0.300178i	$-0.0970461\pi$
$98$	7.68938	0.776745
$99$	− 10.4485i	− 1.05012i
$100$	−3.05761	−0.305761
$101$	10.0971	1.00470	0.502348	−	0.864666i	$-0.332470\pi$
$101$	10.0971	1.00470	0.502348	+	0.864666i	$0.332470\pi$
$102$	0	0
$103$	−1.07322	−0.105747	−0.0528736	−	0.998601i	$-0.516838\pi$
$103$	−1.07322	−0.105747	−0.0528736	+	0.998601i	$0.516838\pi$
$104$	−3.02837	−0.296956
$105$	− 0.374744i	− 0.0365713i
$106$	22.2681	2.16287
$107$	− 6.34844i	− 0.613727i	−0.951754	−	0.306864i	$-0.900720\pi$
$107$	− 6.34844i	− 0.613727i	0.951754	−	0.306864i	$-0.0992795\pi$
$108$	3.60934i	0.347309i
$109$	10.0856i	0.966027i	0.875613	+	0.483014i	$0.160458\pi$
$109$	10.0856i	0.966027i	−0.875613	+	0.483014i	$0.839542\pi$
$110$	−7.93637	−0.756703
$111$	1.68061	0.159516
$112$	1.44995i	0.137007i
$113$	3.74391i	0.352197i	0.984373	+	0.176099i	$0.0563478\pi$
$113$	3.74391i	0.352197i	−0.984373	+	0.176099i	$0.943652\pi$
$114$	2.08942i	0.195692i
$115$	0.574930	0.0536125
$116$	16.5056i	1.53251i
$117$	−3.76978	−0.348516
$118$	7.10438	0.654011
$119$	0	0
$120$	0.471022	0.0429983
$121$	−1.45368	−0.132153
$122$	11.5662i	1.04716i
$123$	−1.20118	−0.108306
$124$	− 19.0077i	− 1.70694i
$125$	1.00000i	0.0894427i
$126$	− 12.6000i	− 1.12250i
$127$	−12.8352	−1.13894	−0.569470	−	0.822012i	$-0.692852\pi$
$127$	−12.8352	−1.13894	−0.569470	+	0.822012i	$0.692852\pi$
$128$	−16.3674	−1.44669
$129$	− 1.81408i	− 0.159721i
$130$	2.86340i	0.251137i
$131$	− 7.02467i	− 0.613748i	−0.951750	−	0.306874i	$-0.900717\pi$
$131$	− 7.02467i	− 0.613748i	0.951750	−	0.306874i	$-0.0992830\pi$
$132$	2.13685	0.185989
$133$	− 8.87776i	− 0.769800i
$134$	0.633876	0.0547585
$135$	1.18044	0.101596
$136$	0	0
$137$	−15.1475	−1.29414	−0.647071	−	0.762430i	$-0.724006\pi$
$137$	−15.1475	−1.29414	−0.647071	+	0.762430i	$0.724006\pi$
$138$	−0.256053	−0.0217967
$139$	− 12.4333i	− 1.05458i	−0.849686	−	0.527289i	$-0.823208\pi$
$139$	− 12.4333i	− 1.05458i	0.849686	−	0.527289i	$-0.176792\pi$
$140$	−5.78596	−0.489003
$141$	2.13742i	0.180004i
$142$	26.4099i	2.21627i
$143$	4.49323i	0.375743i
$144$	−2.26864	−0.189053
$145$	5.39821	0.448297
$146$	22.1170i	1.83041i
$147$	0.677112i	0.0558473i
$148$	− 25.9482i	− 2.13293i
$149$	−11.6094	−0.951076	−0.475538	−	0.879695i	$-0.657747\pi$
$149$	−11.6094	−0.951076	−0.475538	+	0.879695i	$0.657747\pi$
$150$	− 0.445364i	− 0.0363638i
$151$	1.31768	0.107231	0.0536156	−	0.998562i	$-0.482925\pi$
$151$	1.31768	0.107231	0.0536156	+	0.998562i	$0.482925\pi$
$152$	11.1586	0.905083
$153$	0	0
$154$	−15.0181	−1.21019
$155$	−6.21652	−0.499323
$156$	− 0.770965i	− 0.0617266i
$157$	0.183886	0.0146757	0.00733783	−	0.999973i	$-0.497664\pi$
$157$	0.183886	0.0146757	0.00733783	+	0.999973i	$0.497664\pi$
$158$	38.0981i	3.03092i
$159$	1.96089i	0.155509i
$160$	6.48015i	0.512300i
$161$	1.08795	0.0857422
$162$	19.4499	1.52813
$163$	18.0504i	1.41381i	0.707306	+	0.706907i	$0.249910\pi$
$163$	18.0504i	1.41381i	−0.707306	+	0.706907i	$0.750090\pi$
$164$	18.5459i	1.44819i
$165$	− 0.698861i	− 0.0544063i
$166$	−19.1414	−1.48566
$167$	− 7.57518i	− 0.586185i	−0.956084	−	0.293093i	$-0.905316\pi$
$167$	− 7.57518i	− 0.586185i	0.956084	−	0.293093i	$-0.0946844\pi$
$168$	0.891322	0.0687670
$169$	−11.3789	−0.875297
$170$	0	0
$171$	13.8905	1.06223
$172$	−28.0090	−2.13566
$173$	− 6.92236i	− 0.526298i	−0.964755	−	0.263149i	$-0.915239\pi$
$173$	− 6.92236i	− 0.526298i	0.964755	−	0.263149i	$-0.0847610\pi$
$174$	−2.40417	−0.182259
$175$	1.89231i	0.143045i
$176$	2.70401i	0.203822i
$177$	0.625598i	0.0470228i
$178$	−37.6078	−2.81883
$179$	10.2788	0.768274	0.384137	−	0.923276i	$-0.374499\pi$
$179$	10.2788	0.768274	0.384137	+	0.923276i	$0.374499\pi$
$180$	− 9.05292i	− 0.674765i
$181$	20.0601i	1.49105i	0.666475	+	0.745527i	$0.267802\pi$
$181$	20.0601i	1.49105i	−0.666475	+	0.745527i	$0.732198\pi$
$182$	5.41845i	0.401643i
$183$	−1.01850	−0.0752896
$184$	1.36746i	0.100810i
$185$	−8.48642	−0.623934
$186$	2.76861	0.203005
$187$	0	0
$188$	33.0013	2.40687
$189$	2.23377	0.162483
$190$	− 10.5508i	− 0.765432i
$191$	−4.11140	−0.297491	−0.148745	−	0.988876i	$-0.547523\pi$
$191$	−4.11140	−0.297491	−0.148745	+	0.988876i	$0.547523\pi$
$192$	− 2.58254i	− 0.186379i
$193$	14.2986i	1.02923i	0.857420	+	0.514617i	$0.172066\pi$
$193$	14.2986i	1.02923i	−0.857420	+	0.514617i	$0.827934\pi$
$194$	− 13.2974i	− 0.954700i
$195$	−0.252146	−0.0180565
$196$	10.4544	0.746746
$197$	5.35126i	0.381262i	0.981662	+	0.190631i	$0.0610533\pi$
$197$	5.35126i	0.381262i	−0.981662	+	0.190631i	$0.938947\pi$
$198$	− 23.4978i	− 1.66992i
$199$	− 3.37450i	− 0.239212i	−0.992821	−	0.119606i	$-0.961837\pi$
$199$	− 3.37450i	− 0.239212i	0.992821	−	0.119606i	$-0.0381631\pi$
$200$	−2.37848	−0.168184
$201$	0.0558179i	0.00393709i
$202$	22.7074	1.59769
$203$	10.2151	0.716959
$204$	0	0
$205$	6.06547	0.423631
$206$	−2.41357	−0.168162
$207$	1.70224i	0.118314i
$208$	0.975594	0.0676453
$209$	− 16.5562i	− 1.14521i
$210$	− 0.842768i	− 0.0581565i
$211$	− 2.35346i	− 0.162019i	−0.996713	−	0.0810094i	$-0.974186\pi$
$211$	− 2.35346i	− 0.162019i	0.996713	−	0.0810094i	$-0.0258144\pi$
$212$	30.2757	2.07934
$213$	−2.32561	−0.159348
$214$	− 14.2771i	− 0.975962i
$215$	9.16040i	0.624734i
$216$	2.80766i	0.191037i
$217$	−11.7636	−0.798565
$218$	22.6817i	1.53620i
$219$	−1.94758	−0.131605
$220$	−10.7902	−0.727478
$221$	0	0
$222$	3.77954	0.253666
$223$	−3.45702	−0.231499	−0.115750	−	0.993278i	$-0.536927\pi$
$223$	−3.45702	−0.231499	−0.115750	+	0.993278i	$0.536927\pi$
$224$	12.2625i	0.819320i
$225$	−2.96078	−0.197385
$226$	8.41973i	0.560072i
$227$	− 21.7164i	− 1.44137i	−0.693264	−	0.720684i	$-0.743828\pi$
$227$	− 21.7164i	− 1.44137i	0.693264	−	0.720684i	$-0.256172\pi$
$228$	2.84076i	0.188134i
$229$	−8.59173	−0.567757	−0.283879	−	0.958860i	$-0.591621\pi$
$229$	−8.59173	−0.567757	−0.283879	+	0.958860i	$0.591621\pi$
$230$	1.29297	0.0852557
$231$	− 1.32246i	− 0.0870118i
$232$	12.8395i	0.842956i
$233$	− 8.90817i	− 0.583594i	−0.956480	−	0.291797i	$-0.905747\pi$
$233$	− 8.90817i	− 0.583594i	0.956480	−	0.291797i	$-0.0942531\pi$
$234$	−8.47791	−0.554219
$235$	− 10.7932i	− 0.704068i
$236$	9.65908	0.628753
$237$	−3.35484	−0.217921
$238$	0	0
$239$	−12.3246	−0.797212	−0.398606	−	0.917122i	$-0.630506\pi$
$239$	−12.3246	−0.797212	−0.398606	+	0.917122i	$0.630506\pi$
$240$	−0.151741	−0.00979481
$241$	2.67257i	0.172155i	0.996288	+	0.0860777i	$0.0274333\pi$
$241$	2.67257i	0.172155i	−0.996288	+	0.0860777i	$0.972567\pi$
$242$	−3.26921	−0.210153
$243$	5.25405i	0.337048i
$244$	15.7254i	1.00671i
$245$	− 3.41915i	− 0.218442i
$246$	−2.70134	−0.172231
$247$	−5.97338	−0.380077
$248$	− 14.7859i	− 0.938903i
$249$	− 1.68555i	− 0.106818i
$250$	2.24891i	0.142234i
$251$	27.4415	1.73209	0.866046	−	0.499964i	$-0.166653\pi$
$251$	27.4415	1.73209	0.866046	+	0.499964i	$0.166653\pi$
$252$	− 17.1310i	− 1.07915i
$253$	2.02891	0.127557
$254$	−28.8653	−1.81117
$255$	0	0
$256$	−10.7272	−0.670451
$257$	22.8819	1.42734	0.713668	−	0.700484i	$-0.247032\pi$
$257$	22.8819	1.42734	0.713668	+	0.700484i	$0.247032\pi$
$258$	− 4.07971i	− 0.253992i
$259$	−16.0590	−0.997855
$260$	3.89307i	0.241438i
$261$	15.9829i	0.989318i
$262$	− 15.7979i	− 0.975996i
$263$	9.48703	0.584995	0.292498	−	0.956266i	$-0.405514\pi$
$263$	9.48703	0.584995	0.292498	+	0.956266i	$0.405514\pi$
$264$	1.66223	0.102303
$265$	− 9.90174i	− 0.608259i
$266$	− 19.9653i	− 1.22415i
$267$	− 3.31168i	− 0.202671i
$268$	0.861814	0.0526437
$269$	6.08811i	0.371198i	0.982626	+	0.185599i	$0.0594226\pi$
$269$	6.08811i	0.371198i	−0.982626	+	0.185599i	$0.940577\pi$
$270$	2.65472	0.161561
$271$	23.6580	1.43712	0.718562	−	0.695463i	$-0.244801\pi$
$271$	23.6580	1.43712	0.718562	+	0.695463i	$0.244801\pi$
$272$	0	0
$273$	−0.477139	−0.0288778
$274$	−34.0655	−2.05797
$275$	3.52898i	0.212805i
$276$	−0.348128	−0.0209549
$277$	8.23746i	0.494941i	0.968895	+	0.247471i	$0.0795994\pi$
$277$	8.23746i	0.494941i	−0.968895	+	0.247471i	$0.920401\pi$
$278$	− 27.9614i	− 1.67701i
$279$	− 18.4058i	− 1.10192i
$280$	−4.50083	−0.268976
$281$	−15.0835	−0.899808	−0.449904	−	0.893077i	$-0.648542\pi$
$281$	−15.0835	−0.899808	−0.449904	+	0.893077i	$0.648542\pi$
$282$	4.80688i	0.286246i
$283$	− 17.7506i	− 1.05517i	−0.849503	−	0.527583i	$-0.823098\pi$
$283$	− 17.7506i	− 1.05517i	0.849503	−	0.527583i	$-0.176902\pi$
$284$	35.9068i	2.13068i
$285$	0.929079	0.0550339
$286$	10.1049i	0.597515i
$287$	11.4778	0.677511
$288$	−19.1863	−1.13056
$289$	0	0
$290$	12.1401	0.712891
$291$	1.17095	0.0686421
$292$	30.0701i	1.75972i
$293$	−10.9192	−0.637907	−0.318953	−	0.947770i	$-0.603331\pi$
$293$	−10.9192	−0.637907	−0.318953	+	0.947770i	$0.603331\pi$
$294$	1.52277i	0.0888096i
$295$	− 3.15903i	− 0.183926i
$296$	− 20.1848i	− 1.17322i
$297$	4.16576	0.241722
$298$	−26.1084	−1.51242
$299$	− 0.732022i	− 0.0423340i
$300$	− 0.605515i	− 0.0349594i
$301$	17.3343i	0.999135i
$302$	2.96335	0.170522
$303$	1.99957i	0.114873i
$304$	−3.59476	−0.206174
$305$	5.14302	0.294489
$306$	0	0
$307$	6.82894	0.389748	0.194874	−	0.980828i	$-0.437570\pi$
$307$	6.82894	0.389748	0.194874	+	0.980828i	$0.437570\pi$
$308$	−20.4185	−1.16345
$309$	− 0.212535i	− 0.0120907i
$310$	−13.9804	−0.794034
$311$	− 0.661015i	− 0.0374827i	−0.999824	−	0.0187414i	$-0.994034\pi$
$311$	− 0.661015i	− 0.0374827i	0.999824	−	0.0187414i	$-0.00596591\pi$
$312$	− 0.599724i	− 0.0339527i
$313$	− 4.19329i	− 0.237019i	−0.992953	−	0.118510i	$-0.962188\pi$
$313$	− 4.19329i	− 0.237019i	0.992953	−	0.118510i	$-0.0378116\pi$
$314$	0.413543	0.0233376
$315$	−5.60273	−0.315678
$316$	51.7980i	2.91386i
$317$	21.2656i	1.19439i	0.802094	+	0.597197i	$0.203719\pi$
$317$	21.2656i	1.19439i	−0.802094	+	0.597197i	$0.796281\pi$
$318$	4.40988i	0.247294i
$319$	19.0502	1.06660
$320$	13.0408i	0.729004i
$321$	1.25721	0.0701709
$322$	2.44670	0.136349
$323$	0	0
$324$	26.4440	1.46911
$325$	1.27324	0.0706266
$326$	40.5937i	2.24828i
$327$	−1.99731	−0.110451
$328$	14.4266i	0.796575i
$329$	− 20.4240i	− 1.12601i
$330$	− 1.57168i	− 0.0865181i
$331$	27.7582	1.52573	0.762864	−	0.646559i	$-0.223793\pi$
$331$	27.7582	1.52573	0.762864	+	0.646559i	$0.223793\pi$
$332$	−26.0245	−1.42828
$333$	− 25.1264i	− 1.37692i
$334$	− 17.0359i	− 0.932164i
$335$	− 0.281859i	− 0.0153996i
$336$	−0.287141	−0.0156648
$337$	8.77964i	0.478257i	0.970988	+	0.239129i	$0.0768618\pi$
$337$	8.77964i	0.478257i	−0.970988	+	0.239129i	$0.923138\pi$
$338$	−25.5901	−1.39192
$339$	−0.741426	−0.0402687
$340$	0	0
$341$	−21.9380	−1.18801
$342$	31.2385	1.68918
$343$	− 19.7163i	− 1.06458i
$344$	−21.7878	−1.17472
$345$	0.113856i	0.00612981i
$346$	− 15.5678i	− 0.836930i
$347$	5.27026i	0.282922i	0.989944	+	0.141461i	$0.0451800\pi$
$347$	5.27026i	0.282922i	−0.989944	+	0.141461i	$0.954820\pi$
$348$	−3.26869	−0.175220
$349$	−10.8634	−0.581504	−0.290752	−	0.956799i	$-0.593905\pi$
$349$	−10.8634	−0.581504	−0.290752	+	0.956799i	$0.593905\pi$
$350$	4.25565i	0.227474i
$351$	− 1.50299i	− 0.0802235i
$352$	22.8683i	1.21888i
$353$	27.2304	1.44933	0.724664	−	0.689102i	$-0.241995\pi$
$353$	27.2304	1.44933	0.724664	+	0.689102i	$0.241995\pi$
$354$	1.40692i	0.0747768i
$355$	11.7434	0.623276
$356$	−51.1314	−2.70996
$357$	0	0
$358$	23.1161	1.22173
$359$	−13.0217	−0.687258	−0.343629	−	0.939105i	$-0.611656\pi$
$359$	−13.0217	−0.687258	−0.343629	+	0.939105i	$0.611656\pi$
$360$	− 7.04216i	− 0.371154i
$361$	3.01006	0.158424
$362$	45.1134i	2.37111i
$363$	− 0.287880i	− 0.0151098i
$364$	7.36691i	0.386131i
$365$	9.83451	0.514762
$366$	−2.29052	−0.119727
$367$	16.7636i	0.875051i	0.899206	+	0.437526i	$0.144145\pi$
$367$	16.7636i	0.875051i	−0.899206	+	0.437526i	$0.855855\pi$
$368$	− 0.440529i	− 0.0229641i
$369$	17.9585i	0.934884i
$370$	−19.0852	−0.992193
$371$	− 18.7372i	− 0.972786i
$372$	3.76419	0.195164
$373$	1.14057	0.0590567	0.0295283	−	0.999564i	$-0.490599\pi$
$373$	1.14057	0.0590567	0.0295283	+	0.999564i	$0.490599\pi$
$374$	0	0
$375$	−0.198035	−0.0102265
$376$	25.6713	1.32390
$377$	− 6.87321i	− 0.353988i
$378$	5.02355	0.258384
$379$	− 8.76900i	− 0.450433i	−0.974309	−	0.225217i	$-0.927691\pi$
$379$	− 8.76900i	− 0.450433i	0.974309	−	0.225217i	$-0.0723090\pi$
$380$	− 14.3448i	− 0.735870i
$381$	− 2.54182i	− 0.130221i
$382$	−9.24619	−0.473076
$383$	−11.9977	−0.613053	−0.306526	−	0.951862i	$-0.599167\pi$
$383$	−11.9977	−0.613053	−0.306526	+	0.951862i	$0.599167\pi$
$384$	− 3.24132i	− 0.165408i
$385$	6.67793i	0.340339i
$386$	32.1563i	1.63671i
$387$	−27.1220	−1.37869
$388$	− 18.0791i	− 0.917828i
$389$	0.165600	0.00839627	0.00419813	−	0.999991i	$-0.498664\pi$
$389$	0.165600	0.00839627	0.00419813	+	0.999991i	$0.498664\pi$
$390$	−0.567054	−0.0287139
$391$	0	0
$392$	8.13238	0.410747
$393$	1.39113	0.0701733
$394$	12.0345i	0.606291i
$395$	16.9407	0.852377
$396$	− 31.9476i	− 1.60543i
$397$	− 30.4455i	− 1.52802i	−0.645206	−	0.764009i	$-0.723228\pi$
$397$	− 30.4455i	− 1.52802i	0.645206	−	0.764009i	$-0.276772\pi$
$398$	− 7.58895i	− 0.380400i
$399$	1.75811	0.0880155
$400$	0.766231	0.0383115
$401$	− 36.7998i	− 1.83769i	−0.394613	−	0.918847i	$-0.629121\pi$
$401$	− 36.7998i	− 1.83769i	0.394613	−	0.918847i	$-0.370879\pi$
$402$	0.125530i	0.00626085i
$403$	7.91511i	0.394280i
$404$	30.8729	1.53599
$405$	− 8.64858i	− 0.429751i
$406$	22.9729	1.14012
$407$	−29.9484	−1.48449
$408$	0	0
$409$	7.05575	0.348885	0.174442	−	0.984667i	$-0.444188\pi$
$409$	7.05575	0.348885	0.174442	+	0.984667i	$0.444188\pi$
$410$	13.6407	0.673667
$411$	− 2.99975i	− 0.147967i
$412$	−3.28148	−0.161667
$413$	− 5.97787i	− 0.294152i
$414$	3.82819i	0.188145i
$415$	8.51139i	0.417808i
$416$	8.25077	0.404527
$417$	2.46223	0.120576
$418$	− 37.2334i	− 1.82114i
$419$	− 13.4854i	− 0.658803i	−0.944190	−	0.329402i	$-0.893153\pi$
$419$	− 13.4854i	− 0.658803i	0.944190	−	0.329402i	$-0.106847\pi$
$420$	− 1.14582i	− 0.0559104i
$421$	−11.9003	−0.579984	−0.289992	−	0.957029i	$-0.593653\pi$
$421$	−11.9003	−0.579984	−0.289992	+	0.957029i	$0.593653\pi$
$422$	− 5.29273i	− 0.257646i
$423$	31.9562	1.55376
$424$	23.5511	1.14374
$425$	0	0
$426$	−5.23010	−0.253399
$427$	9.73220	0.470974
$428$	− 19.4111i	− 0.938270i
$429$	−0.889817	−0.0429608
$430$	20.6010i	0.993466i
$431$	11.9863i	0.577362i	0.957425	+	0.288681i	$0.0932167\pi$
$431$	11.9863i	0.577362i	−0.957425	+	0.288681i	$0.906783\pi$
$432$	− 0.904492i	− 0.0435174i
$433$	−7.39430	−0.355347	−0.177674	−	0.984089i	$-0.556857\pi$
$433$	−7.39430	−0.355347	−0.177674	+	0.984089i	$0.556857\pi$
$434$	−26.4553	−1.26990
$435$	1.06903i	0.0512563i
$436$	30.8379i	1.47687i
$437$	2.69728i	0.129028i
$438$	−4.37994	−0.209281
$439$	31.6430i	1.51024i	0.655588	+	0.755119i	$0.272421\pi$
$439$	31.6430i	1.51024i	−0.655588	+	0.755119i	$0.727579\pi$
$440$	−8.39360	−0.400149
$441$	10.1234	0.482065
$442$	0	0
$443$	35.6597	1.69424	0.847121	−	0.531401i	$-0.178334\pi$
$443$	35.6597	1.69424	0.847121	+	0.531401i	$0.178334\pi$
$444$	5.13865	0.243870
$445$	16.7227i	0.792730i
$446$	−7.77454	−0.368135
$447$	− 2.29906i	− 0.108742i
$448$	24.6773i	1.16589i
$449$	− 1.16655i	− 0.0550529i	−0.999621	−	0.0275265i	$-0.991237\pi$
$449$	− 1.16655i	− 0.0550529i	0.999621	−	0.0275265i	$-0.00876305\pi$
$450$	−6.65854	−0.313887
$451$	21.4049	1.00792
$452$	11.4474i	0.538442i
$453$	0.260947i	0.0122604i
$454$	− 48.8383i	− 2.29209i
$455$	2.40937	0.112953
$456$	2.20980i	0.103483i
$457$	−37.6351	−1.76050	−0.880248	−	0.474514i	$-0.842624\pi$
$457$	−37.6351	−1.76050	−0.880248	+	0.474514i	$0.842624\pi$
$458$	−19.3221	−0.902860
$459$	0	0
$460$	1.75791	0.0819631
$461$	−22.0741	−1.02809	−0.514046	−	0.857763i	$-0.671854\pi$
$461$	−22.0741	−1.02809	−0.514046	+	0.857763i	$0.671854\pi$
$462$	− 2.97411i	− 0.138368i
$463$	−11.0361	−0.512893	−0.256447	−	0.966558i	$-0.582552\pi$
$463$	−11.0361	−0.512893	−0.256447	+	0.966558i	$0.582552\pi$
$464$	− 4.13627i	− 0.192022i
$465$	− 1.23109i	− 0.0570904i
$466$	− 20.0337i	− 0.928044i
$467$	22.0407	1.01992	0.509960	−	0.860198i	$-0.329660\pi$
$467$	22.0407	1.01992	0.509960	+	0.860198i	$0.329660\pi$
$468$	−11.5265	−0.532814
$469$	− 0.533365i	− 0.0246285i
$470$	− 24.2729i	− 1.11962i
$471$	0.0364158i	0.00167795i
$472$	7.51368	0.345845
$473$	32.3269i	1.48639i
$474$	−7.54475	−0.346542
$475$	−4.69149	−0.215260
$476$	0	0
$477$	29.3169	1.34233
$478$	−27.7170	−1.26774
$479$	− 31.3474i	− 1.43230i	−0.697947	−	0.716149i	$-0.745903\pi$
$479$	− 31.3474i	− 1.43230i	0.697947	−	0.716149i	$-0.254097\pi$
$480$	−1.28330	−0.0585742
$481$	10.8052i	0.492676i
$482$	6.01038i	0.273765i
$483$	0.215452i	0.00980339i
$484$	−4.44480	−0.202036
$485$	−5.91282	−0.268487
$486$	11.8159i	0.535981i
$487$	2.67826i	0.121364i	0.998157	+	0.0606818i	$0.0193275\pi$
$487$	2.67826i	0.121364i	−0.998157	+	0.0606818i	$0.980673\pi$
$488$	12.2326i	0.553742i
$489$	−3.57461	−0.161649
$490$	− 7.68938i	− 0.347371i
$491$	19.7985	0.893496	0.446748	−	0.894660i	$-0.352582\pi$
$491$	19.7985	0.893496	0.446748	+	0.894660i	$0.352582\pi$
$492$	−3.67273	−0.165580
$493$	0	0
$494$	−13.4336	−0.604407
$495$	−10.4485	−0.469627
$496$	4.76329i	0.213878i
$497$	22.2222	0.996803
$498$	− 3.79066i	− 0.169864i
$499$	27.7102i	1.24048i	0.784412	+	0.620240i	$0.212965\pi$
$499$	27.7102i	1.24048i	−0.784412	+	0.620240i	$0.787035\pi$
$500$	3.05761i	0.136741i
$501$	1.50015	0.0670218
$502$	61.7136	2.75441
$503$	4.11047i	0.183277i	0.995792	+	0.0916384i	$0.0292104\pi$
$503$	4.11047i	0.183277i	−0.995792	+	0.0916384i	$0.970790\pi$
$504$	− 13.3260i	− 0.593586i
$505$	− 10.0971i	− 0.449314i
$506$	4.56285	0.202843
$507$	− 2.25341i	− 0.100078i
$508$	−39.2451	−1.74122
$509$	41.9347	1.85872	0.929362	−	0.369169i	$-0.120358\pi$
$509$	41.9347	1.85872	0.929362	+	0.369169i	$0.120358\pi$
$510$	0	0
$511$	18.6100	0.823256
$512$	8.61021	0.380521
$513$	5.53804i	0.244510i
$514$	51.4595	2.26978
$515$	1.07322i	0.0472916i
$516$	− 5.54676i	− 0.244182i
$517$	− 38.0888i	− 1.67514i
$518$	−36.1152	−1.58681
$519$	1.37087	0.0601746
$520$	3.02837i	0.132803i
$521$	− 35.4127i	− 1.55146i	−0.631066	−	0.775729i	$-0.717382\pi$
$521$	− 35.4127i	− 1.55146i	0.631066	−	0.775729i	$-0.282618\pi$
$522$	35.9442i	1.57323i
$523$	−6.37043	−0.278560	−0.139280	−	0.990253i	$-0.544479\pi$
$523$	−6.37043	−0.278560	−0.139280	+	0.990253i	$0.544479\pi$
$524$	− 21.4787i	− 0.938302i
$525$	−0.374744	−0.0163552
$526$	21.3355	0.930273
$527$	0	0
$528$	−0.535489	−0.0233042
$529$	22.6695	0.985629
$530$	− 22.2681i	− 0.967267i
$531$	9.35319	0.405894
$532$	− 27.1448i	− 1.17687i
$533$	− 7.72279i	− 0.334511i
$534$	− 7.44767i	− 0.322292i
$535$	−6.34844	−0.274467
$536$	0.670395	0.0289566
$537$	2.03556i	0.0878411i
$538$	13.6916i	0.590288i
$539$	− 12.0661i	− 0.519724i
$540$	3.60934	0.155321
$541$	− 16.3452i	− 0.702735i	−0.936238	−	0.351367i	$-0.885717\pi$
$541$	− 16.3452i	− 0.702735i	0.936238	−	0.351367i	$-0.114283\pi$
$542$	53.2049	2.28535
$543$	−3.97260	−0.170481
$544$	0	0
$545$	10.0856	0.432021
$546$	−1.07304	−0.0459221
$547$	− 1.28671i	− 0.0550156i	−0.999622	−	0.0275078i	$-0.991243\pi$
$547$	− 1.28671i	− 0.0550156i	0.999622	−	0.0275078i	$-0.00875711\pi$
$548$	−46.3153	−1.97849
$549$	15.2274i	0.649888i
$550$	7.93637i	0.338408i
$551$	25.3256i	1.07891i
$552$	−0.270805	−0.0115262
$553$	32.0570	1.36320
$554$	18.5253i	0.787066i
$555$	− 1.68061i	− 0.0713379i
$556$	− 38.0162i	− 1.61225i
$557$	−35.2859	−1.49511	−0.747555	−	0.664200i	$-0.768773\pi$
$557$	−35.2859	−1.49511	−0.747555	+	0.664200i	$0.768773\pi$
$558$	− 41.3930i	− 1.75230i
$559$	11.6634	0.493308
$560$	1.44995	0.0612715
$561$	0	0
$562$	−33.9216	−1.43089
$563$	−0.931153	−0.0392434	−0.0196217	−	0.999807i	$-0.506246\pi$
$563$	−0.931153	−0.0392434	−0.0196217	+	0.999807i	$0.506246\pi$
$564$	6.53541i	0.275191i
$565$	3.74391	0.157507
$566$	− 39.9197i	− 1.67795i
$567$	− 16.3658i	− 0.687299i
$568$	27.9315i	1.17198i
$569$	17.5921	0.737498	0.368749	−	0.929529i	$-0.379786\pi$
$569$	17.5921	0.737498	0.368749	+	0.929529i	$0.379786\pi$
$570$	2.08942	0.0875161
$571$	− 17.2534i	− 0.722032i	−0.932560	−	0.361016i	$-0.882430\pi$
$571$	− 17.2534i	− 0.722032i	0.932560	−	0.361016i	$-0.117570\pi$
$572$	13.7386i	0.574438i
$573$	− 0.814202i	− 0.0340138i
$574$	25.8125	1.07739
$575$	− 0.574930i	− 0.0239762i
$576$	−38.6110	−1.60879
$577$	8.50534	0.354082	0.177041	−	0.984203i	$-0.443347\pi$
$577$	8.50534	0.354082	0.177041	+	0.984203i	$0.443347\pi$
$578$	0	0
$579$	−2.83162	−0.117678
$580$	16.5056	0.685359
$581$	16.1062i	0.668198i
$582$	2.63336	0.109156
$583$	− 34.9430i	− 1.44719i
$584$	23.3912i	0.967934i
$585$	3.76978i	0.155861i
$586$	−24.5563	−1.01441
$587$	−1.55649	−0.0642433	−0.0321217	−	0.999484i	$-0.510226\pi$
$587$	−1.55649	−0.0642433	−0.0321217	+	0.999484i	$0.510226\pi$
$588$	2.07035i	0.0853797i
$589$	− 29.1647i	− 1.20171i
$590$	− 7.10438i	− 0.292483i
$591$	−1.05974	−0.0435918
$592$	6.50255i	0.267253i
$593$	−19.6862	−0.808415	−0.404207	−	0.914667i	$-0.632453\pi$
$593$	−19.6862	−0.808415	−0.404207	+	0.914667i	$0.632453\pi$
$594$	9.36844	0.384392
$595$	0	0
$596$	−35.4969	−1.45401
$597$	0.668269	0.0273504
$598$	− 1.64626i	− 0.0673204i
$599$	24.7552	1.01147	0.505735	−	0.862689i	$-0.331221\pi$
$599$	24.7552	1.01147	0.505735	+	0.862689i	$0.331221\pi$
$600$	− 0.471022i	− 0.0192294i
$601$	28.3317i	1.15568i	0.816152	+	0.577838i	$0.196103\pi$
$601$	28.3317i	1.15568i	−0.816152	+	0.577838i	$0.803897\pi$
$602$	38.9834i	1.58885i
$603$	0.834522	0.0339844
$604$	4.02895	0.163936
$605$	1.45368i	0.0591006i
$606$	4.49687i	0.182673i
$607$	− 46.5969i	− 1.89131i	−0.325169	−	0.945656i	$-0.605421\pi$
$607$	− 46.5969i	− 1.89131i	0.325169	−	0.945656i	$-0.394579\pi$
$608$	−30.4015	−1.23294
$609$	2.02295i	0.0819740i
$610$	11.5662	0.468302
$611$	−13.7423	−0.555952
$612$	0	0
$613$	−42.0358	−1.69781	−0.848905	−	0.528546i	$-0.822737\pi$
$613$	−42.0358	−1.69781	−0.848905	+	0.528546i	$0.822737\pi$
$614$	15.3577	0.619786
$615$	1.20118i	0.0484361i
$616$	−15.8833	−0.639957
$617$	− 18.2061i	− 0.732951i	−0.930428	−	0.366476i	$-0.880564\pi$
$617$	− 18.2061i	− 0.732951i	0.930428	−	0.366476i	$-0.119436\pi$
$618$	− 0.477972i	− 0.0192269i
$619$	− 13.7770i	− 0.553746i	−0.960907	−	0.276873i	$-0.910702\pi$
$619$	− 13.7770i	− 0.553746i	0.960907	−	0.276873i	$-0.0892981\pi$
$620$	−19.0077	−0.763368
$621$	−0.678672	−0.0272342
$622$	− 1.48657i	− 0.0596058i
$623$	31.6445i	1.26781i
$624$	0.193202i	0.00773427i
$625$	1.00000	0.0400000
$626$	− 9.43036i	− 0.376913i
$627$	3.27870	0.130939
$628$	0.562251	0.0224362
$629$	0	0
$630$	−12.6000	−0.501998
$631$	−20.2307	−0.805372	−0.402686	−	0.915338i	$-0.631923\pi$
$631$	−20.2307	−0.805372	−0.402686	+	0.915338i	$0.631923\pi$
$632$	40.2930i	1.60277i
$633$	0.466068	0.0185245
$634$	47.8245i	1.89935i
$635$	12.8352i	0.509350i
$636$	5.99565i	0.237743i
$637$	−4.35340	−0.172488
$638$	42.8421	1.69614
$639$	34.7697i	1.37547i
$640$	16.3674i	0.646978i
$641$	4.77563i	0.188626i	0.995543	+	0.0943131i	$0.0300655\pi$
$641$	4.77563i	0.188626i	−0.995543	+	0.0943131i	$0.969935\pi$
$642$	2.82737	0.111587
$643$	− 9.85730i	− 0.388734i	−0.980929	−	0.194367i	$-0.937735\pi$
$643$	− 9.85730i	− 0.388734i	0.980929	−	0.194367i	$-0.0622652\pi$
$644$	3.32652	0.131083
$645$	−1.81408	−0.0714294
$646$	0	0
$647$	−36.4555	−1.43321	−0.716607	−	0.697477i	$-0.754306\pi$
$647$	−36.4555	−1.43321	−0.716607	+	0.697477i	$0.754306\pi$
$648$	20.5705	0.808084
$649$	− 11.1481i	− 0.437603i
$650$	2.86340	0.112312
$651$	− 2.32961i	− 0.0913045i
$652$	55.1910i	2.16145i
$653$	18.0074i	0.704686i	0.935871	+	0.352343i	$0.114615\pi$
$653$	18.0074i	0.704686i	−0.935871	+	0.352343i	$0.885385\pi$
$654$	−4.49177	−0.175642
$655$	−7.02467	−0.274476
$656$	− 4.64755i	− 0.181456i
$657$	29.1178i	1.13600i
$658$	− 45.9319i	− 1.79061i
$659$	0.175710	0.00684470	0.00342235	−	0.999994i	$-0.498911\pi$
$659$	0.175710	0.00684470	0.00342235	+	0.999994i	$0.498911\pi$
$660$	− 2.13685i	− 0.0831767i
$661$	−4.14751	−0.161319	−0.0806597	−	0.996742i	$-0.525703\pi$
$661$	−4.14751	−0.161319	−0.0806597	+	0.996742i	$0.525703\pi$
$662$	62.4258	2.42625
$663$	0	0
$664$	−20.2442	−0.785626
$665$	−8.87776	−0.344265
$666$	− 56.5072i	− 2.18961i
$667$	−3.10359	−0.120171
$668$	− 23.1620i	− 0.896163i
$669$	− 0.684612i	− 0.0264686i
$670$	− 0.633876i	− 0.0244888i
$671$	18.1496	0.700658
$672$	−2.42840	−0.0936775
$673$	− 8.74027i	− 0.336913i	−0.985709	−	0.168456i	$-0.946122\pi$
$673$	− 8.74027i	− 0.336913i	0.985709	−	0.168456i	$-0.0538782\pi$
$674$	19.7446i	0.760535i
$675$	− 1.18044i	− 0.0454353i
$676$	−34.7922	−1.33816
$677$	9.37205i	0.360197i	0.983649	+	0.180099i	$0.0576417\pi$
$677$	9.37205i	0.360197i	−0.983649	+	0.180099i	$0.942358\pi$
$678$	−1.66740	−0.0640362
$679$	−11.1889	−0.429391
$680$	0	0
$681$	4.30061	0.164800
$682$	−49.3366	−1.88920
$683$	− 37.4610i	− 1.43340i	−0.697379	−	0.716702i	$-0.745651\pi$
$683$	− 37.4610i	− 1.43340i	0.697379	−	0.716702i	$-0.254349\pi$
$684$	42.4717	1.62395
$685$	15.1475i	0.578758i
$686$	− 44.3402i	− 1.69292i
$687$	− 1.70146i	− 0.0649149i
$688$	7.01898	0.267596
$689$	−12.6073	−0.480299
$690$	0.256053i	0.00974776i
$691$	44.6752i	1.69952i	0.527166	+	0.849762i	$0.323255\pi$
$691$	44.6752i	1.69952i	−0.527166	+	0.849762i	$0.676745\pi$
$692$	− 21.1659i	− 0.804607i
$693$	−19.7719	−0.751072
$694$	11.8524i	0.449909i
$695$	−12.4333	−0.471622
$696$	−2.54268	−0.0963799
$697$	0	0
$698$	−24.4308	−0.924720
$699$	1.76413	0.0667256
$700$	5.78596i	0.218689i
$701$	18.7355	0.707630	0.353815	−	0.935315i	$-0.384884\pi$
$701$	18.7355	0.707630	0.353815	+	0.935315i	$0.384884\pi$
$702$	− 3.38009i	− 0.127573i
$703$	− 39.8139i	− 1.50161i
$704$	46.0208i	1.73447i
$705$	2.13742	0.0805000
$706$	61.2388	2.30475
$707$	− 19.1068i	− 0.718586i
$708$	1.91284i	0.0718888i
$709$	− 5.72933i	− 0.215169i	−0.994196	−	0.107585i	$-0.965688\pi$
$709$	− 5.72933i	− 0.215169i	0.994196	−	0.107585i	$-0.0343117\pi$
$710$	26.4099	0.991147
$711$	50.1576i	1.88106i
$712$	−39.7745	−1.49061
$713$	3.57406	0.133850
$714$	0	0
$715$	4.49323	0.168037
$716$	31.4286	1.17454
$717$	− 2.44070i	− 0.0911498i
$718$	−29.2846	−1.09289
$719$	29.3658i	1.09516i	0.836754	+	0.547579i	$0.184450\pi$
$719$	29.3658i	1.09516i	−0.836754	+	0.547579i	$0.815550\pi$
$720$	2.26864i	0.0845473i
$721$	2.03086i	0.0756333i
$722$	6.76937	0.251930
$723$	−0.529263	−0.0196835
$724$	61.3360i	2.27953i
$725$	− 5.39821i	− 0.200484i
$726$	− 0.647418i	− 0.0240279i
$727$	48.1662	1.78639	0.893193	−	0.449674i	$-0.148460\pi$
$727$	48.1662	1.78639	0.893193	+	0.449674i	$0.148460\pi$
$728$	5.73063i	0.212391i
$729$	24.9052	0.922416
$730$	22.1170	0.818585
$731$	0	0
$732$	−3.11417	−0.115103
$733$	−14.3419	−0.529731	−0.264866	−	0.964285i	$-0.585328\pi$
$733$	−14.3419	−0.529731	−0.264866	+	0.964285i	$0.585328\pi$
$734$	37.6998i	1.39153i
$735$	0.677112	0.0249757
$736$	− 3.72563i	− 0.137328i
$737$	− 0.994673i	− 0.0366392i
$738$	40.3872i	1.48667i
$739$	−9.73758	−0.358203	−0.179101	−	0.983831i	$-0.557319\pi$
$739$	−9.73758	−0.358203	−0.179101	+	0.983831i	$0.557319\pi$
$740$	−25.9482	−0.953874
$741$	− 1.18294i	− 0.0434564i
$742$	− 42.1383i	− 1.54695i
$743$	24.0030i	0.880585i	0.897854	+	0.440292i	$0.145125\pi$
$743$	24.0030i	0.880585i	−0.897854	+	0.440292i	$0.854875\pi$
$744$	2.92812	0.107350
$745$	11.6094i	0.425334i
$746$	2.56505	0.0939133
$747$	−25.2004	−0.922033
$748$	0	0
$749$	−12.0132	−0.438954
$750$	−0.445364	−0.0162624
$751$	47.9548i	1.74990i	0.484217	+	0.874948i	$0.339105\pi$
$751$	47.9548i	1.74990i	−0.484217	+	0.874948i	$0.660895\pi$
$752$	−8.27004	−0.301578
$753$	5.43438i	0.198040i
$754$	− 15.4572i	− 0.562920i
$755$	− 1.31768i	− 0.0479553i
$756$	6.83000	0.248405
$757$	−18.5200	−0.673119	−0.336560	−	0.941662i	$-0.609263\pi$
$757$	−18.5200	−0.673119	−0.336560	+	0.941662i	$0.609263\pi$
$758$	− 19.7207i	− 0.716289i
$759$	0.401796i	0.0145843i
$760$	− 11.1586i	− 0.404765i
$761$	−35.9145	−1.30190	−0.650950	−	0.759120i	$-0.725629\pi$
$761$	−35.9145	−1.30190	−0.650950	+	0.759120i	$0.725629\pi$
$762$	− 5.71634i	− 0.207081i
$763$	19.0851	0.690929
$764$	−12.5711	−0.454806
$765$	0	0
$766$	−26.9818	−0.974890
$767$	−4.02220	−0.145233
$768$	− 2.12437i	− 0.0766564i
$769$	−17.7831	−0.641276	−0.320638	−	0.947202i	$-0.603897\pi$
$769$	−17.7831	−0.641276	−0.320638	+	0.947202i	$0.603897\pi$
$770$	15.0181i	0.541214i
$771$	4.53143i	0.163195i
$772$	43.7195i	1.57350i
$773$	−7.33018	−0.263648	−0.131824	−	0.991273i	$-0.542083\pi$
$773$	−7.33018	−0.263648	−0.131824	+	0.991273i	$0.542083\pi$
$774$	−60.9949	−2.19242
$775$	6.21652i	0.223304i
$776$	− 14.0635i	− 0.504851i
$777$	− 3.18024i	− 0.114090i
$778$	0.372421	0.0133519
$779$	28.4561i	1.01954i
$780$	−0.770965	−0.0276050
$781$	41.4423	1.48292
$782$	0	0
$783$	−6.37228	−0.227727
$784$	−2.61986	−0.0935664
$785$	− 0.183886i	− 0.00656316i
$786$	3.12853	0.111591
$787$	− 38.7949i	− 1.38289i	−0.722429	−	0.691445i	$-0.756975\pi$
$787$	− 38.7949i	− 1.38289i	0.722429	−	0.691445i	$-0.243025\pi$
$788$	16.3621i	0.582875i
$789$	1.87877i	0.0668858i
$790$	38.0981	1.35547
$791$	7.08465	0.251901
$792$	− 24.8516i	− 0.883064i
$793$	− 6.54829i	− 0.232537i
$794$	− 68.4694i	− 2.42989i
$795$	1.96089	0.0695457
$796$	− 10.3179i	− 0.365708i
$797$	−36.6386	−1.29780	−0.648902	−	0.760872i	$-0.724772\pi$
$797$	−36.6386	−1.29780	−0.648902	+	0.760872i	$0.724772\pi$
$798$	3.95383	0.139964
$799$	0	0
$800$	6.48015	0.229108
$801$	−49.5122	−1.74943
$802$	− 82.7596i	− 2.92234i
$803$	34.7058	1.22474
$804$	0.170670i	0.00601905i
$805$	− 1.08795i	− 0.0383451i
$806$	17.8004i	0.626993i
$807$	−1.20566	−0.0424412
$808$	24.0157	0.844868
$809$	− 15.6748i	− 0.551098i	−0.961287	−	0.275549i	$-0.911140\pi$
$809$	− 15.6748i	− 0.551098i	0.961287	−	0.275549i	$-0.0888596\pi$
$810$	− 19.4499i	− 0.683400i
$811$	− 7.89645i	− 0.277282i	−0.990343	−	0.138641i	$-0.955727\pi$
$811$	− 7.89645i	− 0.277282i	0.990343	−	0.138641i	$-0.0442734\pi$
$812$	31.2338	1.09609
$813$	4.68512i	0.164314i
$814$	−67.3513	−2.36066
$815$	18.0504	0.632277
$816$	0	0
$817$	−42.9759	−1.50354
$818$	15.8678	0.554804
$819$	7.13361i	0.249268i
$820$	18.5459	0.647649
$821$	− 23.6001i	− 0.823649i	−0.911263	−	0.411824i	$-0.864892\pi$
$821$	− 23.6001i	− 0.823649i	0.911263	−	0.411824i	$-0.135108\pi$
$822$	− 6.74617i	− 0.235300i
$823$	− 14.1914i	− 0.494680i	−0.968929	−	0.247340i	$-0.920444\pi$
$823$	− 14.1914i	− 0.494680i	0.968929	−	0.247340i	$-0.0795565\pi$
$824$	−2.55263	−0.0889250
$825$	−0.698861	−0.0243312
$826$	− 13.4437i	− 0.467766i
$827$	− 37.9599i	− 1.32000i	−0.751268	−	0.659998i	$-0.770557\pi$
$827$	− 37.9599i	− 1.32000i	0.751268	−	0.659998i	$-0.229443\pi$
$828$	5.20479i	0.180879i
$829$	−4.91163	−0.170588	−0.0852939	−	0.996356i	$-0.527183\pi$
$829$	−4.91163	−0.170588	−0.0852939	+	0.996356i	$0.527183\pi$
$830$	19.1414i	0.664407i
$831$	−1.63131	−0.0565894
$832$	16.6041	0.575643
$833$	0	0
$834$	5.53734	0.191743
$835$	−7.57518	−0.262150
$836$	− 50.6223i	− 1.75081i
$837$	7.33826	0.253647
$838$	− 30.3274i	− 1.04764i
$839$	− 11.0386i	− 0.381094i	−0.981678	−	0.190547i	$-0.938974\pi$
$839$	− 11.0386i	− 0.381094i	0.981678	−	0.190547i	$-0.0610262\pi$
$840$	− 0.891322i	− 0.0307535i
$841$	−0.140644	−0.00484980
$842$	−26.7627	−0.922303
$843$	− 2.98707i	− 0.102880i
$844$	− 7.19597i	− 0.247695i
$845$	11.3789i	0.391445i
$846$	71.8667	2.47083
$847$	2.75082i	0.0945194i
$848$	−7.58701	−0.260539
$849$	3.51525	0.120643
$850$	0	0
$851$	4.87909	0.167253
$852$	−7.11081	−0.243612
$853$	40.7976i	1.39688i	0.715667	+	0.698442i	$0.246123\pi$
$853$	40.7976i	1.39688i	−0.715667	+	0.698442i	$0.753877\pi$
$854$	21.8869	0.748954
$855$	− 13.8905i	− 0.475044i
$856$	− 15.0996i	− 0.516095i
$857$	− 4.12864i	− 0.141032i	−0.997511	−	0.0705158i	$-0.977535\pi$
$857$	− 4.12864i	− 0.141032i	0.997511	−	0.0705158i	$-0.0224645\pi$
$858$	−2.00112	−0.0683172
$859$	8.53800	0.291313	0.145656	−	0.989335i	$-0.453471\pi$
$859$	8.53800	0.291313	0.145656	+	0.989335i	$0.453471\pi$
$860$	28.0090i	0.955098i
$861$	2.27300i	0.0774637i
$862$	26.9563i	0.918134i
$863$	−3.08921	−0.105158	−0.0525790	−	0.998617i	$-0.516744\pi$
$863$	−3.08921	−0.105158	−0.0525790	+	0.998617i	$0.516744\pi$
$864$	− 7.64945i	− 0.260240i
$865$	−6.92236	−0.235367
$866$	−16.6291	−0.565081
$867$	0	0
$868$	−35.9685	−1.22085
$869$	59.7832	2.02801
$870$	2.40417i	0.0815089i
$871$	−0.358873	−0.0121600
$872$	23.9884i	0.812351i
$873$	− 17.5066i	− 0.592508i
$874$	6.06594i	0.205183i
$875$	1.89231	0.0639718
$876$	−5.95494	−0.201199
$877$	− 30.7103i	− 1.03701i	−0.855074	−	0.518506i	$-0.826488\pi$
$877$	− 30.7103i	− 1.03701i	0.855074	−	0.518506i	$-0.173512\pi$
$878$	71.1623i	2.40161i
$879$	− 2.16239i	− 0.0729354i
$880$	2.70401	0.0911522
$881$	− 8.75043i	− 0.294809i	−0.989076	−	0.147405i	$-0.952908\pi$
$881$	− 8.75043i	− 0.294809i	0.989076	−	0.147405i	$-0.0470920\pi$
$882$	22.7666	0.766591
$883$	−29.5875	−0.995698	−0.497849	−	0.867264i	$-0.665877\pi$
$883$	−29.5875	−0.995698	−0.497849	+	0.867264i	$0.665877\pi$
$884$	0	0
$885$	0.625598	0.0210293
$886$	80.1955	2.69422
$887$	8.37625i	0.281247i	0.990063	+	0.140623i	$0.0449107\pi$
$887$	8.37625i	0.281247i	−0.990063	+	0.140623i	$0.955089\pi$
$888$	3.99729	0.134140
$889$	24.2882i	0.814601i
$890$	37.6078i	1.26062i
$891$	− 30.5206i	− 1.02248i
$892$	−10.5702	−0.353917
$893$	50.6360	1.69447
$894$	− 5.17039i	− 0.172924i
$895$	− 10.2788i	− 0.343583i
$896$	30.9722i	1.03471i
$897$	0.144966	0.00484028
$898$	− 2.62347i	− 0.0875464i
$899$	33.5581	1.11922
$900$	−9.05292	−0.301764
$901$	0	0
$902$	48.1378	1.60281
$903$	−3.43281	−0.114237
$904$	8.90481i	0.296170i
$905$	20.0601	0.666820
$906$	0.586847i	0.0194967i
$907$	− 20.1257i	− 0.668264i	−0.942526	−	0.334132i	$-0.891557\pi$
$907$	− 20.1257i	− 0.668264i	0.942526	−	0.334132i	$-0.108443\pi$
$908$	− 66.4003i	− 2.20357i
$909$	29.8952	0.991562
$910$	5.41845	0.179620
$911$	8.02227i	0.265790i	0.991130	+	0.132895i	$0.0424273\pi$
$911$	8.02227i	0.265790i	−0.991130	+	0.132895i	$0.957573\pi$
$912$	− 0.711889i	− 0.0235730i
$913$	30.0365i	0.994063i
$914$	−84.6381	−2.79958
$915$	1.01850i	0.0336705i
$916$	−26.2702	−0.867991
$917$	−13.2929	−0.438969
$918$	0	0
$919$	30.7678	1.01494	0.507468	−	0.861671i	$-0.330582\pi$
$919$	30.7678	1.01494	0.507468	+	0.861671i	$0.330582\pi$
$920$	1.36746	0.0450838
$921$	1.35237i	0.0445621i
$922$	−49.6427	−1.63489
$923$	− 14.9522i	− 0.492157i
$924$	− 4.04358i	− 0.133024i
$925$	8.48642i	0.279032i
$926$	−24.8193	−0.815614
$927$	−3.17756	−0.104365
$928$	− 34.9812i	− 1.14831i
$929$	− 36.1110i	− 1.18476i	−0.805657	−	0.592382i	$-0.798188\pi$
$929$	− 36.1110i	− 1.18476i	0.805657	−	0.592382i	$-0.201812\pi$
$930$	− 2.76861i	− 0.0907864i
$931$	16.0409	0.525720
$932$	− 27.2377i	− 0.892202i
$933$	0.130904	0.00428561
$934$	49.5675	1.62190
$935$	0	0
$936$	−8.96635	−0.293074
$937$	−36.7492	−1.20054	−0.600272	−	0.799796i	$-0.704941\pi$
$937$	−36.7492	−1.20054	−0.600272	+	0.799796i	$0.704941\pi$
$938$	− 1.19949i	− 0.0391648i
$939$	0.830420	0.0270997
$940$	− 33.0013i	− 1.07638i
$941$	11.6655i	0.380283i	0.981757	+	0.190142i	$0.0608947\pi$
$941$	11.6655i	0.380283i	−0.981757	+	0.190142i	$0.939105\pi$
$942$	0.0818960i	0.00266832i
$943$	−3.48722	−0.113559
$944$	−2.42054	−0.0787820
$945$	− 2.23377i	− 0.0726645i
$946$	72.7003i	2.36369i
$947$	− 46.6305i	− 1.51529i	−0.652669	−	0.757644i	$-0.726351\pi$
$947$	− 46.6305i	− 1.51529i	0.652669	−	0.757644i	$-0.273649\pi$
$948$	−10.2578	−0.333158
$949$	− 12.5217i	− 0.406471i
$950$	−10.5508	−0.342312
$951$	−4.21133	−0.136562
$952$	0	0
$953$	15.8298	0.512778	0.256389	−	0.966574i	$-0.417467\pi$
$953$	15.8298	0.512778	0.256389	+	0.966574i	$0.417467\pi$
$954$	65.9311	2.13460
$955$	4.11140i	0.133042i
$956$	−37.6839	−1.21878
$957$	3.77260i	0.121951i
$958$	− 70.4975i	− 2.27767i
$959$	28.6639i	0.925605i
$960$	−2.58254	−0.0833512
$961$	−7.64513	−0.246617
$962$	24.3000i	0.783465i
$963$	− 18.7964i	− 0.605704i
$964$	8.17169i	0.263192i
$965$	14.2986	0.460288
$966$	0.484532i	0.0155896i
$967$	−18.4139	−0.592152	−0.296076	−	0.955164i	$-0.595678\pi$
$967$	−18.4139	−0.592152	−0.296076	+	0.955164i	$0.595678\pi$
$968$	−3.45755	−0.111130
$969$	0	0
$970$	−13.2974	−0.426955
$971$	−16.3118	−0.523470	−0.261735	−	0.965140i	$-0.584295\pi$
$971$	−16.3118	−0.523470	−0.261735	+	0.965140i	$0.584295\pi$
$972$	16.0649i	0.515281i
$973$	−23.5277	−0.754263
$974$	6.02318i	0.192995i
$975$	0.252146i	0.00807513i
$976$	− 3.94074i	− 0.126140i
$977$	−52.7849	−1.68874	−0.844369	−	0.535762i	$-0.820024\pi$
$977$	−52.7849	−1.68874	−0.844369	+	0.535762i	$0.820024\pi$
$978$	−8.03898	−0.257058
$979$	59.0139i	1.88609i
$980$	− 10.4544i	− 0.333955i
$981$	29.8613i	0.953399i
$982$	44.5252	1.42086
$983$	− 33.3291i	− 1.06303i	−0.847048	−	0.531516i	$-0.821622\pi$
$983$	− 33.3291i	− 1.06303i	0.847048	−	0.531516i	$-0.178378\pi$
$984$	−2.85697	−0.0910770
$985$	5.35126	0.170505
$986$	0	0
$987$	4.04467	0.128743
$988$	−18.2643	−0.581065
$989$	− 5.26659i	− 0.167468i
$990$	−23.4978	−0.746811
$991$	39.4944i	1.25458i	0.778786	+	0.627290i	$0.215836\pi$
$991$	39.4944i	1.25458i	−0.778786	+	0.627290i	$0.784164\pi$
$992$	40.2840i	1.27902i
$993$	5.49710i	0.174445i
$994$	49.9759	1.58514
$995$	−3.37450	−0.106979
$996$	− 5.15377i	− 0.163303i
$997$	− 26.6781i	− 0.844904i	−0.906385	−	0.422452i	$-0.861170\pi$
$997$	− 26.6781i	− 0.844904i	0.906385	−	0.422452i	$-0.138830\pi$
$998$	62.3179i	1.97264i
$999$	10.0177	0.316947

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1445.2.d.g.866.12		12
17.2	even	8		85.2.e.a.81.6	yes	12
17.4	even	4		1445.2.a.n.1.1		6
17.8	even	8		85.2.e.a.21.1	✓	12
17.13	even	4		1445.2.a.o.1.1		6
17.16	even	2	inner	1445.2.d.g.866.11		12
51.2	odd	8		765.2.k.b.676.1		12
51.8	odd	8		765.2.k.b.361.6		12
68.19	odd	8		1360.2.bt.d.81.3		12
68.59	odd	8		1360.2.bt.d.1041.3		12
85.2	odd	8		425.2.j.c.149.1		12
85.4	even	4		7225.2.a.bb.1.6		6
85.8	odd	8		425.2.j.c.174.1		12
85.19	even	8		425.2.e.f.251.1		12
85.42	odd	8		425.2.j.b.174.6		12
85.53	odd	8		425.2.j.b.149.6		12
85.59	even	8		425.2.e.f.276.6		12
85.64	even	4		7225.2.a.z.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.e.a.21.1	✓	12	17.8	even	8
85.2.e.a.81.6	yes	12	17.2	even	8
425.2.e.f.251.1		12	85.19	even	8
425.2.e.f.276.6		12	85.59	even	8
425.2.j.b.149.6		12	85.53	odd	8
425.2.j.b.174.6		12	85.42	odd	8
425.2.j.c.149.1		12	85.2	odd	8
425.2.j.c.174.1		12	85.8	odd	8
765.2.k.b.361.6		12	51.8	odd	8
765.2.k.b.676.1		12	51.2	odd	8
1360.2.bt.d.81.3		12	68.19	odd	8
1360.2.bt.d.1041.3		12	68.59	odd	8
1445.2.a.n.1.1		6	17.4	even	4
1445.2.a.o.1.1		6	17.13	even	4
1445.2.d.g.866.11		12	17.16	even	2	inner
1445.2.d.g.866.12		12	1.1	even	1	trivial
7225.2.a.z.1.6		6	85.64	even	4
7225.2.a.bb.1.6		6	85.4	even	4

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:	\( N \)	\(=\)	\( 1445 = 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1445.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.24891 q^{2} +0.198035i q^{3} +3.05761 q^{4} -1.00000i q^{5} +0.445364i q^{6} -1.89231i q^{7} +2.37848 q^{8} +2.96078 q^{9} -2.24891i q^{10} -3.52898i q^{11} +0.605515i q^{12} -1.27324 q^{13} -4.25565i q^{14} +0.198035 q^{15} -0.766231 q^{16} +6.65854 q^{18} +4.69149 q^{19} -3.05761i q^{20} +0.374744 q^{21} -7.93637i q^{22} +0.574930i q^{23} +0.471022i q^{24} -1.00000 q^{25} -2.86340 q^{26} +1.18044i q^{27} -5.78596i q^{28} +5.39821i q^{29} +0.445364 q^{30} -6.21652i q^{31} -6.48015 q^{32} +0.698861 q^{33} -1.89231 q^{35} +9.05292 q^{36} -8.48642i q^{37} +10.5508 q^{38} -0.252146i q^{39} -2.37848i q^{40} +6.06547i q^{41} +0.842768 q^{42} -9.16040 q^{43} -10.7902i q^{44} -2.96078i q^{45} +1.29297i q^{46} +10.7932 q^{47} -0.151741i q^{48} +3.41915 q^{49} -2.24891 q^{50} -3.89307 q^{52} +9.90174 q^{53} +2.65472i q^{54} -3.52898 q^{55} -4.50083i q^{56} +0.929079i q^{57} +12.1401i q^{58} +3.15903 q^{59} +0.605515 q^{60} +5.14302i q^{61} -13.9804i q^{62} -5.60273i q^{63} -13.0408 q^{64} +1.27324i q^{65} +1.57168 q^{66} +0.281859 q^{67} -0.113856 q^{69} -4.25565 q^{70} +11.7434i q^{71} +7.04216 q^{72} +9.83451i q^{73} -19.0852i q^{74} -0.198035i q^{75} +14.3448 q^{76} -6.67793 q^{77} -0.567054i q^{78} +16.9407i q^{79} +0.766231i q^{80} +8.64858 q^{81} +13.6407i q^{82} -8.51139 q^{83} +1.14582 q^{84} -20.6010 q^{86} -1.06903 q^{87} -8.39360i q^{88} -16.7227 q^{89} -6.65854i q^{90} +2.40937i q^{91} +1.75791i q^{92} +1.23109 q^{93} +24.2729 q^{94} -4.69149i q^{95} -1.28330i q^{96} -5.91282i q^{97} +7.68938 q^{98} -10.4485i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 12 q^{4} - 12 q^{8} - 4 q^{9} - 8 q^{15} + 4 q^{16} + 28 q^{18} + 24 q^{19} + 16 q^{21} - 12 q^{25} + 24 q^{26} + 8 q^{30} + 12 q^{32} - 16 q^{33} + 16 q^{35} + 20 q^{36} - 24 q^{38} + 16 q^{42}+ \cdots - 44 q^{98}+O(q^{100}) \) 12 * q - 4 * q^2 + 12 * q^4 - 12 * q^8 - 4 * q^9 - 8 * q^15 + 4 * q^16 + 28 * q^18 + 24 * q^19 + 16 * q^21 - 12 * q^25 + 24 * q^26 + 8 * q^30 + 12 * q^32 - 16 * q^33 + 16 * q^35 + 20 * q^36 - 24 * q^38 + 16 * q^42 - 16 * q^43 + 48 * q^47 + 20 * q^49 + 4 * q^50 - 56 * q^52 + 32 * q^59 - 16 * q^60 - 28 * q^64 + 32 * q^66 - 8 * q^67 - 16 * q^70 - 20 * q^72 + 72 * q^76 - 48 * q^77 - 28 * q^81 - 40 * q^83 - 96 * q^86 - 48 * q^87 - 24 * q^89 - 72 * q^93 - 40 * q^94 - 44 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more