LMFDB - Embedded newform 3380.2.f.j.3041.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.f (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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 39x^{16} + 605x^{14} + 4764x^{12} + 20080x^{10} + 43783x^{8} + 43791x^{6} + 14071x^{4} + 378x^{2} + 1 $ x^18 + 39x^16 + 605x^14 + 4764x^12 + 20080x^10 + 43783x^8 + 43791x^6 + 14071x^4 + 378x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.18
Root			$-2.81781i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.j.3041.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.81781 q^{3} +1.00000i q^{5} +0.974186i q^{7} +4.94004 q^{9} +O(q^{10})$	$q+2.81781 q^{3} +1.00000i q^{5} +0.974186i q^{7} +4.94004 q^{9} +3.42555i q^{11} +2.81781i q^{15} -0.283829 q^{17} +6.03105i q^{19} +2.74507i q^{21} -8.91511 q^{23} -1.00000 q^{25} +5.46666 q^{27} +0.461096 q^{29} +7.96959i q^{31} +9.65254i q^{33} -0.974186 q^{35} +2.95339i q^{37} -8.76466i q^{41} -8.81154 q^{43} +4.94004i q^{45} +10.3775i q^{47} +6.05096 q^{49} -0.799776 q^{51} +4.96945 q^{53} -3.42555 q^{55} +16.9943i q^{57} -7.88601i q^{59} +11.4869 q^{61} +4.81252i q^{63} -3.96340i q^{67} -25.1211 q^{69} -6.22524i q^{71} -9.94784i q^{73} -2.81781 q^{75} -3.33712 q^{77} +6.63735 q^{79} +0.583871 q^{81} +7.05049i q^{83} -0.283829i q^{85} +1.29928 q^{87} -8.66990i q^{89} +22.4568i q^{93} -6.03105 q^{95} -0.332881i q^{97} +16.9224i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 2 q^{3} + 24 q^{9}+O(q^{10}) $ 18 * q - 2 * q^3 + 24 * q^9	$ 18 q - 2 q^{3} + 24 q^{9} - 26 q^{17} - 24 q^{23} - 18 q^{25} - 8 q^{27} + 32 q^{29} + 2 q^{35} - 2 q^{43} - 40 q^{49} - 22 q^{51} + 60 q^{53} - 14 q^{55} + 42 q^{61} - 30 q^{69} + 2 q^{75} - 92 q^{77} + 62 q^{79} + 82 q^{81} + 56 q^{87} + 8 q^{95}+O(q^{100}) $ 18 * q - 2 * q^3 + 24 * q^9 - 26 * q^17 - 24 * q^23 - 18 * q^25 - 8 * q^27 + 32 * q^29 + 2 * q^35 - 2 * q^43 - 40 * q^49 - 22 * q^51 + 60 * q^53 - 14 * q^55 + 42 * q^61 - 30 * q^69 + 2 * q^75 - 92 * q^77 + 62 * q^79 + 82 * q^81 + 56 * q^87 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.81781	1.62686	0.813431	−	0.581661i	$-0.197597\pi$
$3$	2.81781	1.62686	0.813431	+	0.581661i	$0.197597\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	0.974186i	0.368208i	0.982907	+	0.184104i	$0.0589383\pi$
$7$	0.974186i	0.368208i	−0.982907	+	0.184104i	$0.941062\pi$
$8$	0	0
$9$	4.94004	1.64668
$10$	0	0
$11$	3.42555i	1.03284i	0.856335	+	0.516421i	$0.172736\pi$
$11$	3.42555i	1.03284i	−0.856335	+	0.516421i	$0.827264\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.81781i	0.727555i
$16$	0	0
$17$	−0.283829	−0.0688387	−0.0344193	−	0.999407i	$-0.510958\pi$
$17$	−0.283829	−0.0688387	−0.0344193	+	0.999407i	$0.510958\pi$
$18$	0	0
$19$	6.03105i	1.38362i	0.722081	+	0.691809i	$0.243186\pi$
$19$	6.03105i	1.38362i	−0.722081	+	0.691809i	$0.756814\pi$
$20$	0	0
$21$	2.74507i	0.599023i
$22$	0	0
$23$	−8.91511	−1.85893	−0.929465	−	0.368911i	$-0.879731\pi$
$23$	−8.91511	−1.85893	−0.929465	+	0.368911i	$0.879731\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.46666	1.05206
$28$	0	0
$29$	0.461096	0.0856233	0.0428117	−	0.999083i	$-0.486368\pi$
$29$	0.461096	0.0856233	0.0428117	+	0.999083i	$0.486368\pi$
$30$	0	0
$31$	7.96959i	1.43138i	0.698418	+	0.715690i	$0.253888\pi$
$31$	7.96959i	1.43138i	−0.698418	+	0.715690i	$0.746112\pi$
$32$	0	0
$33$	9.65254i	1.68029i
$34$	0	0
$35$	−0.974186	−0.164668
$36$	0	0
$37$	2.95339i	0.485535i	0.970085	+	0.242767i	$0.0780552\pi$
$37$	2.95339i	0.485535i	−0.970085	+	0.242767i	$0.921945\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.76466i	− 1.36881i	−0.729102	−	0.684405i	$-0.760062\pi$
$41$	− 8.76466i	− 1.36881i	0.729102	−	0.684405i	$-0.239938\pi$
$42$	0	0
$43$	−8.81154	−1.34375	−0.671873	−	0.740666i	$-0.734510\pi$
$43$	−8.81154	−1.34375	−0.671873	+	0.740666i	$0.734510\pi$
$44$	0	0
$45$	4.94004i	0.736418i
$46$	0	0
$47$	10.3775i	1.51372i	0.653578	+	0.756859i	$0.273267\pi$
$47$	10.3775i	1.51372i	−0.653578	+	0.756859i	$0.726733\pi$
$48$	0	0
$49$	6.05096	0.864423
$50$	0	0
$51$	−0.799776	−0.111991
$52$	0	0
$53$	4.96945	0.682606	0.341303	−	0.939953i	$-0.389132\pi$
$53$	4.96945	0.682606	0.341303	+	0.939953i	$0.389132\pi$
$54$	0	0
$55$	−3.42555	−0.461901
$56$	0	0
$57$	16.9943i	2.25096i
$58$	0	0
$59$	− 7.88601i	− 1.02667i	−0.858188	−	0.513336i	$-0.828410\pi$
$59$	− 7.88601i	− 1.02667i	0.858188	−	0.513336i	$-0.171590\pi$
$60$	0	0
$61$	11.4869	1.47075	0.735375	−	0.677661i	$-0.237006\pi$
$61$	11.4869	1.47075	0.735375	+	0.677661i	$0.237006\pi$
$62$	0	0
$63$	4.81252i	0.606320i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.96340i	− 0.484206i	−0.970251	−	0.242103i	$-0.922163\pi$
$67$	− 3.96340i	− 0.484206i	0.970251	−	0.242103i	$-0.0778372\pi$
$68$	0	0
$69$	−25.1211	−3.02422
$70$	0	0
$71$	− 6.22524i	− 0.738801i	−0.929270	−	0.369400i	$-0.879563\pi$
$71$	− 6.22524i	− 0.738801i	0.929270	−	0.369400i	$-0.120437\pi$
$72$	0	0
$73$	− 9.94784i	− 1.16431i	−0.813079	−	0.582153i	$-0.802210\pi$
$73$	− 9.94784i	− 1.16431i	0.813079	−	0.582153i	$-0.197790\pi$
$74$	0	0
$75$	−2.81781	−0.325372
$76$	0	0
$77$	−3.33712	−0.380301
$78$	0	0
$79$	6.63735	0.746760	0.373380	−	0.927679i	$-0.378199\pi$
$79$	6.63735	0.746760	0.373380	+	0.927679i	$0.378199\pi$
$80$	0	0
$81$	0.583871	0.0648745
$82$	0	0
$83$	7.05049i	0.773891i	0.922103	+	0.386946i	$0.126470\pi$
$83$	7.05049i	0.773891i	−0.922103	+	0.386946i	$0.873530\pi$
$84$	0	0
$85$	− 0.283829i	− 0.0307856i
$86$	0	0
$87$	1.29928	0.139297
$88$	0	0
$89$	− 8.66990i	− 0.919007i	−0.888176	−	0.459504i	$-0.848027\pi$
$89$	− 8.66990i	− 0.919007i	0.888176	−	0.459504i	$-0.151973\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	22.4568i	2.32866i
$94$	0	0
$95$	−6.03105	−0.618773
$96$	0	0
$97$	− 0.332881i	− 0.0337990i	−0.999857	−	0.0168995i	$-0.994620\pi$
$97$	− 0.332881i	− 0.0337990i	0.999857	−	0.0168995i	$-0.00537953\pi$
$98$	0	0
$99$	16.9224i	1.70076i
$100$	0	0
$101$	14.0197	1.39501	0.697506	−	0.716579i	$-0.254293\pi$
$101$	14.0197	1.39501	0.697506	+	0.716579i	$0.254293\pi$
$102$	0	0
$103$	18.6039	1.83310	0.916549	−	0.399923i	$-0.130963\pi$
$103$	18.6039	1.83310	0.916549	+	0.399923i	$0.130963\pi$
$104$	0	0
$105$	−2.74507	−0.267891
$106$	0	0
$107$	−10.8007	−1.04414	−0.522071	−	0.852902i	$-0.674840\pi$
$107$	−10.8007	−1.04414	−0.522071	+	0.852902i	$0.674840\pi$
$108$	0	0
$109$	2.87817i	0.275679i	0.990455	+	0.137839i	$0.0440158\pi$
$109$	2.87817i	0.275679i	−0.990455	+	0.137839i	$0.955984\pi$
$110$	0	0
$111$	8.32209i	0.789898i
$112$	0	0
$113$	9.77628	0.919675	0.459838	−	0.888003i	$-0.347908\pi$
$113$	9.77628	0.919675	0.459838	+	0.888003i	$0.347908\pi$
$114$	0	0
$115$	− 8.91511i	− 0.831338i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 0.276502i	− 0.0253469i
$120$	0	0
$121$	−0.734394	−0.0667631
$122$	0	0
$123$	− 24.6971i	− 2.22686i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−14.3824	−1.27623	−0.638114	−	0.769942i	$-0.720285\pi$
$127$	−14.3824	−1.27623	−0.638114	+	0.769942i	$0.720285\pi$
$128$	0	0
$129$	−24.8292	−2.18609
$130$	0	0
$131$	16.9106	1.47749	0.738745	−	0.673985i	$-0.235419\pi$
$131$	16.9106	1.47749	0.738745	+	0.673985i	$0.235419\pi$
$132$	0	0
$133$	−5.87537	−0.509459
$134$	0	0
$135$	5.46666i	0.470495i
$136$	0	0
$137$	− 7.71889i	− 0.659469i	−0.944074	−	0.329735i	$-0.893041\pi$
$137$	− 7.71889i	− 0.659469i	0.944074	−	0.329735i	$-0.106959\pi$
$138$	0	0
$139$	−18.7958	−1.59424	−0.797120	−	0.603821i	$-0.793644\pi$
$139$	−18.7958	−1.59424	−0.797120	+	0.603821i	$0.793644\pi$
$140$	0	0
$141$	29.2419i	2.46261i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.461096i	0.0382919i
$146$	0	0
$147$	17.0504	1.40630
$148$	0	0
$149$	3.52113i	0.288462i	0.989544	+	0.144231i	$0.0460708\pi$
$149$	3.52113i	0.288462i	−0.989544	+	0.144231i	$0.953929\pi$
$150$	0	0
$151$	11.0498i	0.899217i	0.893226	+	0.449608i	$0.148436\pi$
$151$	11.0498i	0.899217i	−0.893226	+	0.449608i	$0.851564\pi$
$152$	0	0
$153$	−1.40213	−0.113355
$154$	0	0
$155$	−7.96959	−0.640133
$156$	0	0
$157$	4.47826	0.357404	0.178702	−	0.983903i	$-0.442810\pi$
$157$	4.47826	0.357404	0.178702	+	0.983903i	$0.442810\pi$
$158$	0	0
$159$	14.0029	1.11051
$160$	0	0
$161$	− 8.68498i	− 0.684472i
$162$	0	0
$163$	11.3490i	0.888923i	0.895798	+	0.444462i	$0.146605\pi$
$163$	11.3490i	0.888923i	−0.895798	+	0.444462i	$0.853395\pi$
$164$	0	0
$165$	−9.65254	−0.751449
$166$	0	0
$167$	8.82022i	0.682529i	0.939967	+	0.341265i	$0.110855\pi$
$167$	8.82022i	0.682529i	−0.939967	+	0.341265i	$0.889145\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	29.7936i	2.27838i
$172$	0	0
$173$	4.64718	0.353319	0.176659	−	0.984272i	$-0.443471\pi$
$173$	4.64718	0.353319	0.176659	+	0.984272i	$0.443471\pi$
$174$	0	0
$175$	− 0.974186i	− 0.0736416i
$176$	0	0
$177$	− 22.2213i	− 1.67025i
$178$	0	0
$179$	25.3186	1.89240	0.946200	−	0.323581i	$-0.104887\pi$
$179$	25.3186	1.89240	0.946200	+	0.323581i	$0.104887\pi$
$180$	0	0
$181$	−4.32715	−0.321635	−0.160817	−	0.986984i	$-0.551413\pi$
$181$	−4.32715	−0.321635	−0.160817	+	0.986984i	$0.551413\pi$
$182$	0	0
$183$	32.3679	2.39271
$184$	0	0
$185$	−2.95339	−0.217138
$186$	0	0
$187$	− 0.972271i	− 0.0710995i
$188$	0	0
$189$	5.32554i	0.387376i
$190$	0	0
$191$	−2.62340	−0.189823	−0.0949114	−	0.995486i	$-0.530257\pi$
$191$	−2.62340	−0.189823	−0.0949114	+	0.995486i	$0.530257\pi$
$192$	0	0
$193$	21.6639i	1.55940i	0.626153	+	0.779700i	$0.284629\pi$
$193$	21.6639i	1.55940i	−0.626153	+	0.779700i	$0.715371\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.8601i	1.48622i	0.669169	+	0.743110i	$0.266650\pi$
$197$	20.8601i	1.48622i	−0.669169	+	0.743110i	$0.733350\pi$
$198$	0	0
$199$	−22.7819	−1.61497	−0.807485	−	0.589888i	$-0.799172\pi$
$199$	−22.7819	−1.61497	−0.807485	+	0.589888i	$0.799172\pi$
$200$	0	0
$201$	− 11.1681i	− 0.787736i
$202$	0	0
$203$	0.449193i	0.0315272i
$204$	0	0
$205$	8.76466	0.612150
$206$	0	0
$207$	−44.0410	−3.06106
$208$	0	0
$209$	−20.6597	−1.42906
$210$	0	0
$211$	15.9891	1.10073	0.550366	−	0.834923i	$-0.314488\pi$
$211$	15.9891	1.10073	0.550366	+	0.834923i	$0.314488\pi$
$212$	0	0
$213$	− 17.5415i	− 1.20193i
$214$	0	0
$215$	− 8.81154i	− 0.600942i
$216$	0	0
$217$	−7.76387	−0.527046
$218$	0	0
$219$	− 28.0311i	− 1.89417i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 7.54613i	− 0.505326i	−0.967554	−	0.252663i	$-0.918694\pi$
$223$	− 7.54613i	− 0.505326i	0.967554	−	0.252663i	$-0.0813064\pi$
$224$	0	0
$225$	−4.94004	−0.329336
$226$	0	0
$227$	− 28.4988i	− 1.89153i	−0.324853	−	0.945765i	$-0.605315\pi$
$227$	− 28.4988i	− 1.89153i	0.324853	−	0.945765i	$-0.394685\pi$
$228$	0	0
$229$	19.3902i	1.28134i	0.767815	+	0.640671i	$0.221344\pi$
$229$	19.3902i	1.28134i	−0.767815	+	0.640671i	$0.778656\pi$
$230$	0	0
$231$	−9.40337	−0.618696
$232$	0	0
$233$	12.7622	0.836080	0.418040	−	0.908429i	$-0.362717\pi$
$233$	12.7622	0.836080	0.418040	+	0.908429i	$0.362717\pi$
$234$	0	0
$235$	−10.3775	−0.676955
$236$	0	0
$237$	18.7028	1.21488
$238$	0	0
$239$	− 11.1424i	− 0.720741i	−0.932809	−	0.360371i	$-0.882650\pi$
$239$	− 11.1424i	− 0.720741i	0.932809	−	0.360371i	$-0.117350\pi$
$240$	0	0
$241$	5.48277i	0.353176i	0.984285	+	0.176588i	$0.0565061\pi$
$241$	5.48277i	0.353176i	−0.984285	+	0.176588i	$0.943494\pi$
$242$	0	0
$243$	−14.7547	−0.946517
$244$	0	0
$245$	6.05096i	0.386582i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	19.8669i	1.25901i
$250$	0	0
$251$	24.4904	1.54582	0.772910	−	0.634516i	$-0.218801\pi$
$251$	24.4904	1.54582	0.772910	+	0.634516i	$0.218801\pi$
$252$	0	0
$253$	− 30.5392i	− 1.91998i
$254$	0	0
$255$	− 0.799776i	− 0.0500839i
$256$	0	0
$257$	14.7828	0.922128	0.461064	−	0.887367i	$-0.347468\pi$
$257$	14.7828	0.922128	0.461064	+	0.887367i	$0.347468\pi$
$258$	0	0
$259$	−2.87715	−0.178778
$260$	0	0
$261$	2.27783	0.140994
$262$	0	0
$263$	3.01732	0.186056	0.0930279	−	0.995664i	$-0.470345\pi$
$263$	3.01732	0.186056	0.0930279	+	0.995664i	$0.470345\pi$
$264$	0	0
$265$	4.96945i	0.305271i
$266$	0	0
$267$	− 24.4301i	− 1.49510i
$268$	0	0
$269$	−8.46709	−0.516248	−0.258124	−	0.966112i	$-0.583104\pi$
$269$	−8.46709	−0.516248	−0.258124	+	0.966112i	$0.583104\pi$
$270$	0	0
$271$	− 25.7542i	− 1.56445i	−0.622994	−	0.782227i	$-0.714084\pi$
$271$	− 25.7542i	− 1.56445i	0.622994	−	0.782227i	$-0.285916\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.42555i	− 0.206568i
$276$	0	0
$277$	−14.6848	−0.882322	−0.441161	−	0.897428i	$-0.645433\pi$
$277$	−14.6848	−0.882322	−0.441161	+	0.897428i	$0.645433\pi$
$278$	0	0
$279$	39.3701i	2.35703i
$280$	0	0
$281$	− 25.1933i	− 1.50290i	−0.659788	−	0.751452i	$-0.729354\pi$
$281$	− 25.1933i	− 1.50290i	0.659788	−	0.751452i	$-0.270646\pi$
$282$	0	0
$283$	19.3058	1.14761	0.573807	−	0.818991i	$-0.305466\pi$
$283$	19.3058	1.14761	0.573807	+	0.818991i	$0.305466\pi$
$284$	0	0
$285$	−16.9943	−1.00666
$286$	0	0
$287$	8.53841	0.504006
$288$	0	0
$289$	−16.9194	−0.995261
$290$	0	0
$291$	− 0.937995i	− 0.0549862i
$292$	0	0
$293$	− 1.23340i	− 0.0720561i	−0.999351	−	0.0360281i	$-0.988529\pi$
$293$	− 1.23340i	− 0.0720561i	0.999351	−	0.0360281i	$-0.0114706\pi$
$294$	0	0
$295$	7.88601	0.459141
$296$	0	0
$297$	18.7263i	1.08661i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 8.58408i	− 0.494778i
$302$	0	0
$303$	39.5048	2.26949
$304$	0	0
$305$	11.4869i	0.657739i
$306$	0	0
$307$	− 7.36503i	− 0.420344i	−0.977664	−	0.210172i	$-0.932598\pi$
$307$	− 7.36503i	− 0.420344i	0.977664	−	0.210172i	$-0.0674025\pi$
$308$	0	0
$309$	52.4222	2.98220
$310$	0	0
$311$	7.77672	0.440977	0.220489	−	0.975390i	$-0.429235\pi$
$311$	7.77672	0.440977	0.220489	+	0.975390i	$0.429235\pi$
$312$	0	0
$313$	0.104459	0.00590438	0.00295219	−	0.999996i	$-0.499060\pi$
$313$	0.104459	0.00590438	0.00295219	+	0.999996i	$0.499060\pi$
$314$	0	0
$315$	−4.81252	−0.271155
$316$	0	0
$317$	32.3972i	1.81961i	0.415040	+	0.909803i	$0.363768\pi$
$317$	32.3972i	1.81961i	−0.415040	+	0.909803i	$0.636232\pi$
$318$	0	0
$319$	1.57951i	0.0884354i
$320$	0	0
$321$	−30.4342	−1.69867
$322$	0	0
$323$	− 1.71179i	− 0.0952464i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.11013i	0.448491i
$328$	0	0
$329$	−10.1096	−0.557363
$330$	0	0
$331$	− 19.1973i	− 1.05518i	−0.849500	−	0.527589i	$-0.823096\pi$
$331$	− 19.1973i	− 1.05518i	0.849500	−	0.527589i	$-0.176904\pi$
$332$	0	0
$333$	14.5899i	0.799520i
$334$	0	0
$335$	3.96340	0.216543
$336$	0	0
$337$	21.6042	1.17686	0.588429	−	0.808549i	$-0.299747\pi$
$337$	21.6042	1.17686	0.588429	+	0.808549i	$0.299747\pi$
$338$	0	0
$339$	27.5477	1.49618
$340$	0	0
$341$	−27.3002	−1.47839
$342$	0	0
$343$	12.7141i	0.686495i
$344$	0	0
$345$	− 25.1211i	− 1.35247i
$346$	0	0
$347$	−12.4830	−0.670120	−0.335060	−	0.942197i	$-0.608757\pi$
$347$	−12.4830	−0.670120	−0.335060	+	0.942197i	$0.608757\pi$
$348$	0	0
$349$	− 26.1232i	− 1.39834i	−0.714954	−	0.699171i	$-0.753552\pi$
$349$	− 26.1232i	− 1.39834i	0.714954	−	0.699171i	$-0.246448\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 19.7057i	− 1.04883i	−0.851464	−	0.524413i	$-0.824285\pi$
$353$	− 19.7057i	− 1.04883i	0.851464	−	0.524413i	$-0.175715\pi$
$354$	0	0
$355$	6.22524	0.330402
$356$	0	0
$357$	− 0.779130i	− 0.0412360i
$358$	0	0
$359$	6.56325i	0.346395i	0.984887	+	0.173198i	$0.0554099\pi$
$359$	6.56325i	0.346395i	−0.984887	+	0.173198i	$0.944590\pi$
$360$	0	0
$361$	−17.3736	−0.914399
$362$	0	0
$363$	−2.06938	−0.108614
$364$	0	0
$365$	9.94784	0.520694
$366$	0	0
$367$	−33.1532	−1.73058	−0.865291	−	0.501270i	$-0.832866\pi$
$367$	−33.1532	−1.73058	−0.865291	+	0.501270i	$0.832866\pi$
$368$	0	0
$369$	− 43.2978i	− 2.25399i
$370$	0	0
$371$	4.84117i	0.251341i
$372$	0	0
$373$	30.0243	1.55460	0.777300	−	0.629130i	$-0.216589\pi$
$373$	30.0243	1.55460	0.777300	+	0.629130i	$0.216589\pi$
$374$	0	0
$375$	− 2.81781i	− 0.145511i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 11.1570i	− 0.573097i	−0.958066	−	0.286549i	$-0.907492\pi$
$379$	− 11.1570i	− 0.573097i	0.958066	−	0.286549i	$-0.0925080\pi$
$380$	0	0
$381$	−40.5267	−2.07625
$382$	0	0
$383$	− 11.5707i	− 0.591233i	−0.955307	−	0.295616i	$-0.904475\pi$
$383$	− 11.5707i	− 0.591233i	0.955307	−	0.295616i	$-0.0955250\pi$
$384$	0	0
$385$	− 3.33712i	− 0.170076i
$386$	0	0
$387$	−43.5293	−2.21272
$388$	0	0
$389$	18.6724	0.946727	0.473363	−	0.880867i	$-0.343040\pi$
$389$	18.6724	0.946727	0.473363	+	0.880867i	$0.343040\pi$
$390$	0	0
$391$	2.53037	0.127966
$392$	0	0
$393$	47.6509	2.40367
$394$	0	0
$395$	6.63735i	0.333961i
$396$	0	0
$397$	− 14.7838i	− 0.741977i	−0.928638	−	0.370988i	$-0.879019\pi$
$397$	− 14.7838i	− 0.741977i	0.928638	−	0.370988i	$-0.120981\pi$
$398$	0	0
$399$	−16.5557	−0.828819
$400$	0	0
$401$	13.5374i	0.676025i	0.941142	+	0.338012i	$0.109755\pi$
$401$	13.5374i	0.676025i	−0.941142	+	0.338012i	$0.890245\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.583871i	0.0290128i
$406$	0	0
$407$	−10.1170	−0.501481
$408$	0	0
$409$	9.79211i	0.484189i	0.970253	+	0.242094i	$0.0778343\pi$
$409$	9.79211i	0.484189i	−0.970253	+	0.242094i	$0.922166\pi$
$410$	0	0
$411$	− 21.7504i	− 1.07287i
$412$	0	0
$413$	7.68244	0.378028
$414$	0	0
$415$	−7.05049	−0.346095
$416$	0	0
$417$	−52.9630	−2.59361
$418$	0	0
$419$	−19.3450	−0.945065	−0.472533	−	0.881313i	$-0.656660\pi$
$419$	−19.3450	−0.945065	−0.472533	+	0.881313i	$0.656660\pi$
$420$	0	0
$421$	10.8582i	0.529199i	0.964358	+	0.264599i	$0.0852397\pi$
$421$	10.8582i	0.529199i	−0.964358	+	0.264599i	$0.914760\pi$
$422$	0	0
$423$	51.2654i	2.49261i
$424$	0	0
$425$	0.283829	0.0137677
$426$	0	0
$427$	11.1904i	0.541541i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.47538i	0.408245i	0.978945	+	0.204122i	$0.0654341\pi$
$431$	8.47538i	0.408245i	−0.978945	+	0.204122i	$0.934566\pi$
$432$	0	0
$433$	14.6325	0.703193	0.351596	−	0.936152i	$-0.385639\pi$
$433$	14.6325	0.703193	0.351596	+	0.936152i	$0.385639\pi$
$434$	0	0
$435$	1.29928i	0.0622957i
$436$	0	0
$437$	− 53.7675i	− 2.57205i
$438$	0	0
$439$	−10.2299	−0.488245	−0.244123	−	0.969744i	$-0.578500\pi$
$439$	−10.2299	−0.488245	−0.244123	+	0.969744i	$0.578500\pi$
$440$	0	0
$441$	29.8920	1.42343
$442$	0	0
$443$	1.89017	0.0898045	0.0449023	−	0.998991i	$-0.485702\pi$
$443$	1.89017	0.0898045	0.0449023	+	0.998991i	$0.485702\pi$
$444$	0	0
$445$	8.66990	0.410993
$446$	0	0
$447$	9.92185i	0.469288i
$448$	0	0
$449$	− 10.6031i	− 0.500393i	−0.968195	−	0.250196i	$-0.919505\pi$
$449$	− 10.6031i	− 0.500393i	0.968195	−	0.250196i	$-0.0804952\pi$
$450$	0	0
$451$	30.0238	1.41376
$452$	0	0
$453$	31.1361i	1.46290i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 28.2158i	− 1.31988i	−0.751319	−	0.659939i	$-0.770582\pi$
$457$	− 28.2158i	− 1.31988i	0.751319	−	0.659939i	$-0.229418\pi$
$458$	0	0
$459$	−1.55160	−0.0724223
$460$	0	0
$461$	− 14.1918i	− 0.660976i	−0.943810	−	0.330488i	$-0.892787\pi$
$461$	− 14.1918i	− 0.660976i	0.943810	−	0.330488i	$-0.107213\pi$
$462$	0	0
$463$	18.9826i	0.882195i	0.897459	+	0.441098i	$0.145411\pi$
$463$	18.9826i	0.882195i	−0.897459	+	0.441098i	$0.854589\pi$
$464$	0	0
$465$	−22.4568	−1.04141
$466$	0	0
$467$	9.88823	0.457573	0.228786	−	0.973477i	$-0.426524\pi$
$467$	9.88823	0.457573	0.228786	+	0.973477i	$0.426524\pi$
$468$	0	0
$469$	3.86109	0.178288
$470$	0	0
$471$	12.6189	0.581447
$472$	0	0
$473$	− 30.1844i	− 1.38788i
$474$	0	0
$475$	− 6.03105i	− 0.276724i
$476$	0	0
$477$	24.5493	1.12403
$478$	0	0
$479$	− 26.4914i	− 1.21042i	−0.796065	−	0.605211i	$-0.793089\pi$
$479$	− 26.4914i	− 1.21042i	0.796065	−	0.605211i	$-0.206911\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 24.4726i	− 1.11354i
$484$	0	0
$485$	0.332881	0.0151154
$486$	0	0
$487$	34.8349i	1.57852i	0.614058	+	0.789261i	$0.289536\pi$
$487$	34.8349i	1.57852i	−0.614058	+	0.789261i	$0.710464\pi$
$488$	0	0
$489$	31.9793i	1.44616i
$490$	0	0
$491$	0.113245	0.00511068	0.00255534	−	0.999997i	$-0.499187\pi$
$491$	0.113245	0.00511068	0.00255534	+	0.999997i	$0.499187\pi$
$492$	0	0
$493$	−0.130872	−0.00589420
$494$	0	0
$495$	−16.9224	−0.760603
$496$	0	0
$497$	6.06455	0.272032
$498$	0	0
$499$	7.69924i	0.344665i	0.985039	+	0.172333i	$0.0551304\pi$
$499$	7.69924i	0.344665i	−0.985039	+	0.172333i	$0.944870\pi$
$500$	0	0
$501$	24.8537i	1.11038i
$502$	0	0
$503$	−1.09165	−0.0486742	−0.0243371	−	0.999704i	$-0.507748\pi$
$503$	−1.09165	−0.0486742	−0.0243371	+	0.999704i	$0.507748\pi$
$504$	0	0
$505$	14.0197i	0.623869i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.78054i	0.167569i	0.996484	+	0.0837847i	$0.0267008\pi$
$509$	3.78054i	0.167569i	−0.996484	+	0.0837847i	$0.973299\pi$
$510$	0	0
$511$	9.69105	0.428707
$512$	0	0
$513$	32.9697i	1.45565i
$514$	0	0
$515$	18.6039i	0.819786i
$516$	0	0
$517$	−35.5487	−1.56343
$518$	0	0
$519$	13.0949	0.574800
$520$	0	0
$521$	−8.70631	−0.381430	−0.190715	−	0.981645i	$-0.561081\pi$
$521$	−8.70631	−0.381430	−0.190715	+	0.981645i	$0.561081\pi$
$522$	0	0
$523$	−11.8023	−0.516077	−0.258038	−	0.966135i	$-0.583076\pi$
$523$	−11.8023	−0.516077	−0.258038	+	0.966135i	$0.583076\pi$
$524$	0	0
$525$	− 2.74507i	− 0.119805i
$526$	0	0
$527$	− 2.26200i	− 0.0985343i
$528$	0	0
$529$	56.4792	2.45562
$530$	0	0
$531$	− 38.9572i	− 1.69060i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 10.8007i	− 0.466954i
$536$	0	0
$537$	71.3429	3.07867
$538$	0	0
$539$	20.7279i	0.892813i
$540$	0	0
$541$	21.0010i	0.902902i	0.892296	+	0.451451i	$0.149093\pi$
$541$	21.0010i	0.902902i	−0.892296	+	0.451451i	$0.850907\pi$
$542$	0	0
$543$	−12.1931	−0.523255
$544$	0	0
$545$	−2.87817	−0.123287
$546$	0	0
$547$	−19.6759	−0.841282	−0.420641	−	0.907227i	$-0.638195\pi$
$547$	−19.6759	−0.841282	−0.420641	+	0.907227i	$0.638195\pi$
$548$	0	0
$549$	56.7458	2.42185
$550$	0	0
$551$	2.78089i	0.118470i
$552$	0	0
$553$	6.46601i	0.274963i
$554$	0	0
$555$	−8.32209	−0.353253
$556$	0	0
$557$	− 21.5516i	− 0.913172i	−0.889679	−	0.456586i	$-0.849072\pi$
$557$	− 21.5516i	− 0.913172i	0.889679	−	0.456586i	$-0.150928\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 2.73967i	− 0.115669i
$562$	0	0
$563$	−10.6586	−0.449205	−0.224602	−	0.974450i	$-0.572108\pi$
$563$	−10.6586	−0.449205	−0.224602	+	0.974450i	$0.572108\pi$
$564$	0	0
$565$	9.77628i	0.411291i
$566$	0	0
$567$	0.568799i	0.0238873i
$568$	0	0
$569$	−27.5445	−1.15473	−0.577363	−	0.816488i	$-0.695918\pi$
$569$	−27.5445	−1.15473	−0.577363	+	0.816488i	$0.695918\pi$
$570$	0	0
$571$	42.1038	1.76199	0.880994	−	0.473127i	$-0.156875\pi$
$571$	42.1038	1.76199	0.880994	+	0.473127i	$0.156875\pi$
$572$	0	0
$573$	−7.39224	−0.308815
$574$	0	0
$575$	8.91511	0.371786
$576$	0	0
$577$	33.7205i	1.40380i	0.712274	+	0.701901i	$0.247665\pi$
$577$	33.7205i	1.40380i	−0.712274	+	0.701901i	$0.752335\pi$
$578$	0	0
$579$	61.0447i	2.53693i
$580$	0	0
$581$	−6.86849	−0.284953
$582$	0	0
$583$	17.0231i	0.705024i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.3282i	0.550114i	0.961428	+	0.275057i	$0.0886968\pi$
$587$	13.3282i	0.550114i	−0.961428	+	0.275057i	$0.911303\pi$
$588$	0	0
$589$	−48.0650	−1.98048
$590$	0	0
$591$	58.7797i	2.41788i
$592$	0	0
$593$	1.07378i	0.0440949i	0.999757	+	0.0220474i	$0.00701848\pi$
$593$	1.07378i	0.0440949i	−0.999757	+	0.0220474i	$0.992982\pi$
$594$	0	0
$595$	0.276502	0.0113355
$596$	0	0
$597$	−64.1951	−2.62733
$598$	0	0
$599$	−18.3111	−0.748169	−0.374085	−	0.927395i	$-0.622043\pi$
$599$	−18.3111	−0.748169	−0.374085	+	0.927395i	$0.622043\pi$
$600$	0	0
$601$	−1.14497	−0.0467043	−0.0233521	−	0.999727i	$-0.507434\pi$
$601$	−1.14497	−0.0467043	−0.0233521	+	0.999727i	$0.507434\pi$
$602$	0	0
$603$	− 19.5793i	− 0.797332i
$604$	0	0
$605$	− 0.734394i	− 0.0298573i
$606$	0	0
$607$	−11.3920	−0.462385	−0.231193	−	0.972908i	$-0.574263\pi$
$607$	−11.3920	−0.462385	−0.231193	+	0.972908i	$0.574263\pi$
$608$	0	0
$609$	1.26574i	0.0512904i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	6.51518i	0.263146i	0.991306	+	0.131573i	$0.0420027\pi$
$613$	6.51518i	0.263146i	−0.991306	+	0.131573i	$0.957997\pi$
$614$	0	0
$615$	24.6971	0.995884
$616$	0	0
$617$	− 4.43338i	− 0.178481i	−0.996010	−	0.0892405i	$-0.971556\pi$
$617$	− 4.43338i	− 0.178481i	0.996010	−	0.0892405i	$-0.0284440\pi$
$618$	0	0
$619$	− 2.32341i	− 0.0933857i	−0.998909	−	0.0466929i	$-0.985132\pi$
$619$	− 2.32341i	− 0.0933857i	0.998909	−	0.0466929i	$-0.0148682\pi$
$620$	0	0
$621$	−48.7359	−1.95570
$622$	0	0
$623$	8.44610	0.338386
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−58.2150	−2.32488
$628$	0	0
$629$	− 0.838259i	− 0.0334236i
$630$	0	0
$631$	− 25.4463i	− 1.01300i	−0.862239	−	0.506501i	$-0.830939\pi$
$631$	− 25.4463i	− 1.01300i	0.862239	−	0.506501i	$-0.169061\pi$
$632$	0	0
$633$	45.0541	1.79074
$634$	0	0
$635$	− 14.3824i	− 0.570746i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 30.7530i	− 1.21657i
$640$	0	0
$641$	−9.30285	−0.367440	−0.183720	−	0.982979i	$-0.558814\pi$
$641$	−9.30285	−0.367440	−0.183720	+	0.982979i	$0.558814\pi$
$642$	0	0
$643$	49.3771i	1.94724i	0.228172	+	0.973621i	$0.426725\pi$
$643$	49.3771i	1.94724i	−0.228172	+	0.973621i	$0.573275\pi$
$644$	0	0
$645$	− 24.8292i	− 0.977649i
$646$	0	0
$647$	−26.1654	−1.02867	−0.514334	−	0.857590i	$-0.671961\pi$
$647$	−26.1654	−1.02867	−0.514334	+	0.857590i	$0.671961\pi$
$648$	0	0
$649$	27.0139	1.06039
$650$	0	0
$651$	−21.8771	−0.857430
$652$	0	0
$653$	−12.5177	−0.489857	−0.244929	−	0.969541i	$-0.578765\pi$
$653$	−12.5177	−0.489857	−0.244929	+	0.969541i	$0.578765\pi$
$654$	0	0
$655$	16.9106i	0.660753i
$656$	0	0
$657$	− 49.1427i	− 1.91724i
$658$	0	0
$659$	−9.30544	−0.362488	−0.181244	−	0.983438i	$-0.558012\pi$
$659$	−9.30544	−0.362488	−0.181244	+	0.983438i	$0.558012\pi$
$660$	0	0
$661$	0.778487i	0.0302796i	0.999885	+	0.0151398i	$0.00481934\pi$
$661$	0.778487i	0.0302796i	−0.999885	+	0.0151398i	$0.995181\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 5.87537i	− 0.227837i
$666$	0	0
$667$	−4.11072	−0.159168
$668$	0	0
$669$	− 21.2635i	− 0.822096i
$670$	0	0
$671$	39.3490i	1.51905i
$672$	0	0
$673$	−18.6069	−0.717243	−0.358621	−	0.933483i	$-0.616753\pi$
$673$	−18.6069	−0.717243	−0.358621	+	0.933483i	$0.616753\pi$
$674$	0	0
$675$	−5.46666	−0.210412
$676$	0	0
$677$	0.451238	0.0173425	0.00867125	−	0.999962i	$-0.497240\pi$
$677$	0.451238	0.0173425	0.00867125	+	0.999962i	$0.497240\pi$
$678$	0	0
$679$	0.324288	0.0124450
$680$	0	0
$681$	− 80.3040i	− 3.07726i
$682$	0	0
$683$	− 35.3929i	− 1.35427i	−0.735858	−	0.677136i	$-0.763221\pi$
$683$	− 35.3929i	− 1.35427i	0.735858	−	0.677136i	$-0.236779\pi$
$684$	0	0
$685$	7.71889	0.294924
$686$	0	0
$687$	54.6379i	2.08457i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 39.0369i	− 1.48503i	−0.669828	−	0.742516i	$-0.733632\pi$
$691$	− 39.0369i	− 1.48503i	0.669828	−	0.742516i	$-0.266368\pi$
$692$	0	0
$693$	−16.4855	−0.626233
$694$	0	0
$695$	− 18.7958i	− 0.712966i
$696$	0	0
$697$	2.48767i	0.0942270i
$698$	0	0
$699$	35.9614	1.36019
$700$	0	0
$701$	−18.0395	−0.681344	−0.340672	−	0.940182i	$-0.610655\pi$
$701$	−18.0395	−0.681344	−0.340672	+	0.940182i	$0.610655\pi$
$702$	0	0
$703$	−17.8121	−0.671795
$704$	0	0
$705$	−29.2419	−1.10131
$706$	0	0
$707$	13.6578i	0.513654i
$708$	0	0
$709$	19.7805i	0.742871i	0.928459	+	0.371436i	$0.121134\pi$
$709$	19.7805i	0.742871i	−0.928459	+	0.371436i	$0.878866\pi$
$710$	0	0
$711$	32.7888	1.22967
$712$	0	0
$713$	− 71.0498i	− 2.66084i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 31.3971i	− 1.17255i
$718$	0	0
$719$	10.2849	0.383562	0.191781	−	0.981438i	$-0.438574\pi$
$719$	10.2849	0.383562	0.191781	+	0.981438i	$0.438574\pi$
$720$	0	0
$721$	18.1237i	0.674961i
$722$	0	0
$723$	15.4494i	0.574569i
$724$	0	0
$725$	−0.461096	−0.0171247
$726$	0	0
$727$	7.08016	0.262589	0.131294	−	0.991343i	$-0.458087\pi$
$727$	7.08016	0.262589	0.131294	+	0.991343i	$0.458087\pi$
$728$	0	0
$729$	−43.3276	−1.60473
$730$	0	0
$731$	2.50097	0.0925017
$732$	0	0
$733$	6.29532i	0.232523i	0.993219	+	0.116261i	$0.0370911\pi$
$733$	6.29532i	0.232523i	−0.993219	+	0.116261i	$0.962909\pi$
$734$	0	0
$735$	17.0504i	0.628915i
$736$	0	0
$737$	13.5768	0.500108
$738$	0	0
$739$	33.4301i	1.22975i	0.788626	+	0.614873i	$0.210793\pi$
$739$	33.4301i	1.22975i	−0.788626	+	0.614873i	$0.789207\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 3.84274i	− 0.140976i	−0.997513	−	0.0704882i	$-0.977544\pi$
$743$	− 3.84274i	− 0.140976i	0.997513	−	0.0704882i	$-0.0224557\pi$
$744$	0	0
$745$	−3.52113	−0.129004
$746$	0	0
$747$	34.8297i	1.27435i
$748$	0	0
$749$	− 10.5219i	− 0.384461i
$750$	0	0
$751$	34.7285	1.26726	0.633630	−	0.773636i	$-0.281564\pi$
$751$	34.7285	1.26726	0.633630	+	0.773636i	$0.281564\pi$
$752$	0	0
$753$	69.0092	2.51483
$754$	0	0
$755$	−11.0498	−0.402142
$756$	0	0
$757$	8.10935	0.294739	0.147370	−	0.989081i	$-0.452919\pi$
$757$	8.10935	0.294739	0.147370	+	0.989081i	$0.452919\pi$
$758$	0	0
$759$	− 86.0535i	− 3.12354i
$760$	0	0
$761$	− 10.4717i	− 0.379598i	−0.981823	−	0.189799i	$-0.939216\pi$
$761$	− 10.4717i	− 0.379598i	0.981823	−	0.189799i	$-0.0607837\pi$
$762$	0	0
$763$	−2.80388	−0.101507
$764$	0	0
$765$	− 1.40213i	− 0.0506940i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 44.2939i	− 1.59728i	−0.601810	−	0.798639i	$-0.705554\pi$
$769$	− 44.2939i	− 1.59728i	0.601810	−	0.798639i	$-0.294446\pi$
$770$	0	0
$771$	41.6552	1.50017
$772$	0	0
$773$	− 12.2760i	− 0.441537i	−0.975326	−	0.220769i	$-0.929143\pi$
$773$	− 12.2760i	− 0.441537i	0.975326	−	0.220769i	$-0.0708566\pi$
$774$	0	0
$775$	− 7.96959i	− 0.286276i
$776$	0	0
$777$	−8.10727	−0.290847
$778$	0	0
$779$	52.8601	1.89391
$780$	0	0
$781$	21.3249	0.763064
$782$	0	0
$783$	2.52065	0.0900808
$784$	0	0
$785$	4.47826i	0.159836i
$786$	0	0
$787$	7.08241i	0.252461i	0.992001	+	0.126230i	$0.0402878\pi$
$787$	7.08241i	0.252461i	−0.992001	+	0.126230i	$0.959712\pi$
$788$	0	0
$789$	8.50222	0.302687
$790$	0	0
$791$	9.52392i	0.338632i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	14.0029i	0.496633i
$796$	0	0
$797$	19.6799	0.697099	0.348550	−	0.937290i	$-0.386674\pi$
$797$	19.6799	0.697099	0.348550	+	0.937290i	$0.386674\pi$
$798$	0	0
$799$	− 2.94544i	− 0.104202i
$800$	0	0
$801$	− 42.8296i	− 1.51331i
$802$	0	0
$803$	34.0768	1.20254
$804$	0	0
$805$	8.68498	0.306105
$806$	0	0
$807$	−23.8586	−0.839864
$808$	0	0
$809$	26.3066	0.924890	0.462445	−	0.886648i	$-0.346972\pi$
$809$	26.3066	0.924890	0.462445	+	0.886648i	$0.346972\pi$
$810$	0	0
$811$	52.2096i	1.83333i	0.399662	+	0.916663i	$0.369128\pi$
$811$	52.2096i	1.83333i	−0.399662	+	0.916663i	$0.630872\pi$
$812$	0	0
$813$	− 72.5702i	− 2.54515i
$814$	0	0
$815$	−11.3490	−0.397539
$816$	0	0
$817$	− 53.1428i	− 1.85923i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 48.9551i	− 1.70855i	−0.519825	−	0.854273i	$-0.674003\pi$
$821$	− 48.9551i	− 1.70855i	0.519825	−	0.854273i	$-0.325997\pi$
$822$	0	0
$823$	−28.3073	−0.986731	−0.493366	−	0.869822i	$-0.664233\pi$
$823$	−28.3073	−0.986731	−0.493366	+	0.869822i	$0.664233\pi$
$824$	0	0
$825$	− 9.65254i	− 0.336058i
$826$	0	0
$827$	− 11.1223i	− 0.386759i	−0.981124	−	0.193380i	$-0.938055\pi$
$827$	− 11.1223i	− 0.386759i	0.981124	−	0.193380i	$-0.0619449\pi$
$828$	0	0
$829$	1.14566	0.0397905	0.0198952	−	0.999802i	$-0.493667\pi$
$829$	1.14566	0.0397905	0.0198952	+	0.999802i	$0.493667\pi$
$830$	0	0
$831$	−41.3788	−1.43542
$832$	0	0
$833$	−1.71744	−0.0595057
$834$	0	0
$835$	−8.82022	−0.305236
$836$	0	0
$837$	43.5670i	1.50590i
$838$	0	0
$839$	28.3167i	0.977602i	0.872395	+	0.488801i	$0.162566\pi$
$839$	28.3167i	0.977602i	−0.872395	+	0.488801i	$0.837434\pi$
$840$	0	0
$841$	−28.7874	−0.992669
$842$	0	0
$843$	− 70.9898i	− 2.44502i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.715436i	− 0.0245827i
$848$	0	0
$849$	54.4002	1.86701
$850$	0	0
$851$	− 26.3298i	− 0.902574i
$852$	0	0
$853$	− 29.6180i	− 1.01410i	−0.861916	−	0.507051i	$-0.830736\pi$
$853$	− 29.6180i	− 1.01410i	0.861916	−	0.507051i	$-0.169264\pi$
$854$	0	0
$855$	−29.7936	−1.01892
$856$	0	0
$857$	−14.3690	−0.490835	−0.245417	−	0.969418i	$-0.578925\pi$
$857$	−14.3690	−0.490835	−0.245417	+	0.969418i	$0.578925\pi$
$858$	0	0
$859$	−9.17850	−0.313166	−0.156583	−	0.987665i	$-0.550048\pi$
$859$	−9.17850	−0.313166	−0.156583	+	0.987665i	$0.550048\pi$
$860$	0	0
$861$	24.0596	0.819949
$862$	0	0
$863$	− 32.1458i	− 1.09426i	−0.837049	−	0.547128i	$-0.815721\pi$
$863$	− 32.1458i	− 1.09426i	0.837049	−	0.547128i	$-0.184279\pi$
$864$	0	0
$865$	4.64718i	0.158009i
$866$	0	0
$867$	−47.6757	−1.61915
$868$	0	0
$869$	22.7366i	0.771285i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 1.64445i	− 0.0556561i
$874$	0	0
$875$	0.974186	0.0329335
$876$	0	0
$877$	− 25.9633i	− 0.876720i	−0.898800	−	0.438360i	$-0.855560\pi$
$877$	− 25.9633i	− 0.876720i	0.898800	−	0.438360i	$-0.144440\pi$
$878$	0	0
$879$	− 3.47549i	− 0.117225i
$880$	0	0
$881$	17.1165	0.576671	0.288335	−	0.957529i	$-0.406898\pi$
$881$	17.1165	0.576671	0.288335	+	0.957529i	$0.406898\pi$
$882$	0	0
$883$	18.3273	0.616762	0.308381	−	0.951263i	$-0.400213\pi$
$883$	18.3273	0.616762	0.308381	+	0.951263i	$0.400213\pi$
$884$	0	0
$885$	22.2213	0.746960
$886$	0	0
$887$	−36.2866	−1.21838	−0.609192	−	0.793023i	$-0.708506\pi$
$887$	−36.2866	−1.21838	−0.609192	+	0.793023i	$0.708506\pi$
$888$	0	0
$889$	− 14.0111i	− 0.469917i
$890$	0	0
$891$	2.00008i	0.0670051i
$892$	0	0
$893$	−62.5874	−2.09441
$894$	0	0
$895$	25.3186i	0.846307i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.67475i	0.122560i
$900$	0	0
$901$	−1.41047	−0.0469897
$902$	0	0
$903$	− 24.1883i	− 0.804935i
$904$	0	0
$905$	− 4.32715i	− 0.143839i
$906$	0	0
$907$	−10.7409	−0.356645	−0.178323	−	0.983972i	$-0.557067\pi$
$907$	−10.7409	−0.356645	−0.178323	+	0.983972i	$0.557067\pi$
$908$	0	0
$909$	69.2579	2.29714
$910$	0	0
$911$	−31.8649	−1.05573	−0.527866	−	0.849327i	$-0.677008\pi$
$911$	−31.8649	−1.05573	−0.527866	+	0.849327i	$0.677008\pi$
$912$	0	0
$913$	−24.1518	−0.799308
$914$	0	0
$915$	32.3679i	1.07005i
$916$	0	0
$917$	16.4741i	0.544023i
$918$	0	0
$919$	14.6467	0.483149	0.241574	−	0.970382i	$-0.422336\pi$
$919$	14.6467	0.483149	0.241574	+	0.970382i	$0.422336\pi$
$920$	0	0
$921$	− 20.7532i	− 0.683842i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 2.95339i	− 0.0971069i
$926$	0	0
$927$	91.9040	3.01852
$928$	0	0
$929$	− 56.9334i	− 1.86792i	−0.357373	−	0.933962i	$-0.616328\pi$
$929$	− 56.9334i	− 1.86792i	0.357373	−	0.933962i	$-0.383672\pi$
$930$	0	0
$931$	36.4937i	1.19603i
$932$	0	0
$933$	21.9133	0.717409
$934$	0	0
$935$	0.972271	0.0317967
$936$	0	0
$937$	3.76516	0.123002	0.0615012	−	0.998107i	$-0.480411\pi$
$937$	3.76516	0.123002	0.0615012	+	0.998107i	$0.480411\pi$
$938$	0	0
$939$	0.294346	0.00960562
$940$	0	0
$941$	− 28.9802i	− 0.944728i	−0.881404	−	0.472364i	$-0.843401\pi$
$941$	− 28.9802i	− 0.944728i	0.881404	−	0.472364i	$-0.156599\pi$
$942$	0	0
$943$	78.1379i	2.54452i
$944$	0	0
$945$	−5.32554	−0.173240
$946$	0	0
$947$	9.77047i	0.317498i	0.987319	+	0.158749i	$0.0507460\pi$
$947$	9.77047i	0.317498i	−0.987319	+	0.158749i	$0.949254\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	91.2890i	2.96025i
$952$	0	0
$953$	−42.3749	−1.37266	−0.686330	−	0.727291i	$-0.740779\pi$
$953$	−42.3749	−1.37266	−0.686330	+	0.727291i	$0.740779\pi$
$954$	0	0
$955$	− 2.62340i	− 0.0848913i
$956$	0	0
$957$	4.45075i	0.143872i
$958$	0	0
$959$	7.51964	0.242822
$960$	0	0
$961$	−32.5144	−1.04885
$962$	0	0
$963$	−53.3558	−1.71937
$964$	0	0
$965$	−21.6639	−0.697385
$966$	0	0
$967$	− 5.65301i	− 0.181788i	−0.995861	−	0.0908942i	$-0.971027\pi$
$967$	− 5.65301i	− 0.181788i	0.995861	−	0.0908942i	$-0.0289725\pi$
$968$	0	0
$969$	− 4.82349i	− 0.154953i
$970$	0	0
$971$	40.0547	1.28542	0.642709	−	0.766111i	$-0.277811\pi$
$971$	40.0547	1.28542	0.642709	+	0.766111i	$0.277811\pi$
$972$	0	0
$973$	− 18.3106i	− 0.587012i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.7119i	0.502668i	0.967900	+	0.251334i	$0.0808692\pi$
$977$	15.7119i	0.502668i	−0.967900	+	0.251334i	$0.919131\pi$
$978$	0	0
$979$	29.6992	0.949190
$980$	0	0
$981$	14.2183i	0.453955i
$982$	0	0
$983$	− 46.6671i	− 1.48845i	−0.667929	−	0.744225i	$-0.732819\pi$
$983$	− 46.6671i	− 1.48845i	0.667929	−	0.744225i	$-0.267181\pi$
$984$	0	0
$985$	−20.8601	−0.664658
$986$	0	0
$987$	−28.4870	−0.906752
$988$	0	0
$989$	78.5558	2.49793
$990$	0	0
$991$	1.23358	0.0391859	0.0195930	−	0.999808i	$-0.493763\pi$
$991$	1.23358	0.0391859	0.0195930	+	0.999808i	$0.493763\pi$
$992$	0	0
$993$	− 54.0942i	− 1.71663i
$994$	0	0
$995$	− 22.7819i	− 0.722236i
$996$	0	0
$997$	32.9285	1.04286	0.521428	−	0.853295i	$-0.325400\pi$
$997$	32.9285	1.04286	0.521428	+	0.853295i	$0.325400\pi$
$998$	0	0
$999$	16.1452i	0.510811i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.j.3041.18		18
13.5	odd	4		3380.2.a.s.1.9	yes	9
13.8	odd	4		3380.2.a.r.1.9	✓	9
13.12	even	2	inner	3380.2.f.j.3041.17		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.r.1.9	✓	9	13.8	odd	4
3380.2.a.s.1.9	yes	9	13.5	odd	4
3380.2.f.j.3041.17		18	13.12	even	2	inner
3380.2.f.j.3041.18		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.81781 q^{3} +1.00000i q^{5} +0.974186i q^{7} +4.94004 q^{9} +O(q^{10})\)	\(q+2.81781 q^{3} +1.00000i q^{5} +0.974186i q^{7} +4.94004 q^{9} +3.42555i q^{11} +2.81781i q^{15} -0.283829 q^{17} +6.03105i q^{19} +2.74507i q^{21} -8.91511 q^{23} -1.00000 q^{25} +5.46666 q^{27} +0.461096 q^{29} +7.96959i q^{31} +9.65254i q^{33} -0.974186 q^{35} +2.95339i q^{37} -8.76466i q^{41} -8.81154 q^{43} +4.94004i q^{45} +10.3775i q^{47} +6.05096 q^{49} -0.799776 q^{51} +4.96945 q^{53} -3.42555 q^{55} +16.9943i q^{57} -7.88601i q^{59} +11.4869 q^{61} +4.81252i q^{63} -3.96340i q^{67} -25.1211 q^{69} -6.22524i q^{71} -9.94784i q^{73} -2.81781 q^{75} -3.33712 q^{77} +6.63735 q^{79} +0.583871 q^{81} +7.05049i q^{83} -0.283829i q^{85} +1.29928 q^{87} -8.66990i q^{89} +22.4568i q^{93} -6.03105 q^{95} -0.332881i q^{97} +16.9224i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 2 q^{3} + 24 q^{9}+O(q^{10}) \) 18 * q - 2 * q^3 + 24 * q^9	\( 18 q - 2 q^{3} + 24 q^{9} - 26 q^{17} - 24 q^{23} - 18 q^{25} - 8 q^{27} + 32 q^{29} + 2 q^{35} - 2 q^{43} - 40 q^{49} - 22 q^{51} + 60 q^{53} - 14 q^{55} + 42 q^{61} - 30 q^{69} + 2 q^{75} - 92 q^{77} + 62 q^{79} + 82 q^{81} + 56 q^{87} + 8 q^{95}+O(q^{100}) \) 18 * q - 2 * q^3 + 24 * q^9 - 26 * q^17 - 24 * q^23 - 18 * q^25 - 8 * q^27 + 32 * q^29 + 2 * q^35 - 2 * q^43 - 40 * q^49 - 22 * q^51 + 60 * q^53 - 14 * q^55 + 42 * q^61 - 30 * q^69 + 2 * q^75 - 92 * q^77 + 62 * q^79 + 82 * q^81 + 56 * q^87 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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