LMFDB - Embedded newform 4680.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(1,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.a (trivial)

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:	yes
Analytic conductor:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1849.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 8 $ x^3 - x^2 - 14*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.615072$ of defining polynomial
Character	$\chi$	$=$	4680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +2.61507 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +2.61507 q^{7} +6.39154 q^{11} +1.00000 q^{13} +0.615072 q^{17} +7.77647 q^{19} -2.61507 q^{23} +1.00000 q^{25} -7.00662 q^{29} +5.23014 q^{31} +2.61507 q^{35} +7.16140 q^{37} -7.16140 q^{41} +4.00000 q^{43} +1.23014 q^{47} -0.161400 q^{49} +13.6217 q^{53} +6.39154 q^{55} -0.391544 q^{61} +1.00000 q^{65} -14.0132 q^{67} -5.16140 q^{71} +0.993385 q^{73} +16.7143 q^{77} -14.3915 q^{79} -18.0132 q^{83} +0.615072 q^{85} -14.8518 q^{89} +2.61507 q^{91} +7.77647 q^{95} +7.38493 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 5 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 + 5 * q^7	$ 3 q + 3 q^{5} + 5 q^{7} - 3 q^{11} + 3 q^{13} - q^{17} + 4 q^{19} - 5 q^{23} + 3 q^{25} + 4 q^{29} + 10 q^{31} + 5 q^{35} + 5 q^{37} - 5 q^{41} + 12 q^{43} - 2 q^{47} + 16 q^{49} + 13 q^{53} - 3 q^{55} + 21 q^{61} + 3 q^{65} + 8 q^{67} + q^{71} + 28 q^{73} - 5 q^{77} - 21 q^{79} - 4 q^{83} - q^{85} - 11 q^{89} + 5 q^{91} + 4 q^{95} + 25 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 + 5 * q^7 - 3 * q^11 + 3 * q^13 - q^17 + 4 * q^19 - 5 * q^23 + 3 * q^25 + 4 * q^29 + 10 * q^31 + 5 * q^35 + 5 * q^37 - 5 * q^41 + 12 * q^43 - 2 * q^47 + 16 * q^49 + 13 * q^53 - 3 * q^55 + 21 * q^61 + 3 * q^65 + 8 * q^67 + q^71 + 28 * q^73 - 5 * q^77 - 21 * q^79 - 4 * q^83 - q^85 - 11 * q^89 + 5 * q^91 + 4 * q^95 + 25 * q^97

Display 100 coefficients 10 coefficients

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.61507	0.988404	0.494202	−	0.869347i	$-0.335460\pi$
$7$	2.61507	0.988404	0.494202	+	0.869347i	$0.335460\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.39154	1.92712	0.963561	−	0.267487i	$-0.0861933\pi$
$11$	6.39154	1.92712	0.963561	+	0.267487i	$0.0861933\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.615072	0.149177	0.0745884	−	0.997214i	$-0.476236\pi$
$17$	0.615072	0.149177	0.0745884	+	0.997214i	$0.476236\pi$
$18$	0	0
$19$	7.77647	1.78405	0.892023	−	0.451991i	$-0.149286\pi$
$19$	7.77647	1.78405	0.892023	+	0.451991i	$0.149286\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.61507	−0.545280	−0.272640	−	0.962116i	$-0.587897\pi$
$23$	−2.61507	−0.545280	−0.272640	+	0.962116i	$0.587897\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.00662	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$29$	−7.00662	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$30$	0	0
$31$	5.23014	0.939361	0.469681	−	0.882836i	$-0.344369\pi$
$31$	5.23014	0.939361	0.469681	+	0.882836i	$0.344369\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.61507	0.442028
$36$	0	0
$37$	7.16140	1.17733	0.588663	−	0.808378i	$-0.299654\pi$
$37$	7.16140	1.17733	0.588663	+	0.808378i	$0.299654\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.16140	−1.11842	−0.559211	−	0.829025i	$-0.688896\pi$
$41$	−7.16140	−1.11842	−0.559211	+	0.829025i	$0.688896\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.23014	0.179435	0.0897174	−	0.995967i	$-0.471404\pi$
$47$	1.23014	0.179435	0.0897174	+	0.995967i	$0.471404\pi$
$48$	0	0
$49$	−0.161400	−0.0230572
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.6217	1.87108	0.935541	−	0.353217i	$-0.114912\pi$
$53$	13.6217	1.87108	0.935541	+	0.353217i	$0.114912\pi$
$54$	0	0
$55$	6.39154	0.861836
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−0.391544	−0.0501320	−0.0250660	−	0.999686i	$-0.507980\pi$
$61$	−0.391544	−0.0501320	−0.0250660	+	0.999686i	$0.507980\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−14.0132	−1.71199	−0.855994	−	0.516985i	$-0.827054\pi$
$67$	−14.0132	−1.71199	−0.855994	+	0.516985i	$0.827054\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.16140	−0.612546	−0.306273	−	0.951944i	$-0.599082\pi$
$71$	−5.16140	−0.612546	−0.306273	+	0.951944i	$0.599082\pi$
$72$	0	0
$73$	0.993385	0.116267	0.0581334	−	0.998309i	$-0.481485\pi$
$73$	0.993385	0.116267	0.0581334	+	0.998309i	$0.481485\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.7143	1.90478
$78$	0	0
$79$	−14.3915	−1.61918	−0.809588	−	0.586999i	$-0.800309\pi$
$79$	−14.3915	−1.61918	−0.809588	+	0.586999i	$0.800309\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−18.0132	−1.97721	−0.988604	−	0.150536i	$-0.951900\pi$
$83$	−18.0132	−1.97721	−0.988604	+	0.150536i	$0.951900\pi$
$84$	0	0
$85$	0.615072	0.0667139
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.8518	−1.57429	−0.787145	−	0.616767i	$-0.788442\pi$
$89$	−14.8518	−1.57429	−0.787145	+	0.616767i	$0.788442\pi$
$90$	0	0
$91$	2.61507	0.274134
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.77647	0.797849
$96$	0	0
$97$	7.38493	0.749826	0.374913	−	0.927060i	$-0.377673\pi$
$97$	7.38493	0.749826	0.374913	+	0.927060i	$0.377673\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.77647	−0.972795	−0.486398	−	0.873738i	$-0.661689\pi$
$101$	−9.77647	−0.972795	−0.486398	+	0.873738i	$0.661689\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.3915	1.00459	0.502294	−	0.864697i	$-0.332489\pi$
$107$	10.3915	1.00459	0.502294	+	0.864697i	$0.332489\pi$
$108$	0	0
$109$	−8.23676	−0.788938	−0.394469	−	0.918909i	$-0.629072\pi$
$109$	−8.23676	−0.788938	−0.394469	+	0.918909i	$0.629072\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11.0066	−1.03542	−0.517708	−	0.855558i	$-0.673214\pi$
$113$	−11.0066	−1.03542	−0.517708	+	0.855558i	$0.673214\pi$
$114$	0	0
$115$	−2.61507	−0.243857
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.60846	0.147447
$120$	0	0
$121$	29.8518	2.71380
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.7831	1.13432	0.567158	−	0.823609i	$-0.308043\pi$
$127$	12.7831	1.13432	0.567158	+	0.823609i	$0.308043\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.7765	1.37840	0.689198	−	0.724573i	$-0.257963\pi$
$131$	15.7765	1.37840	0.689198	+	0.724573i	$0.257963\pi$
$132$	0	0
$133$	20.3360	1.76336
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.0132	−1.36810	−0.684051	−	0.729434i	$-0.739784\pi$
$137$	−16.0132	−1.36810	−0.684051	+	0.729434i	$0.739784\pi$
$138$	0	0
$139$	−21.1614	−1.79489	−0.897443	−	0.441130i	$-0.854578\pi$
$139$	−21.1614	−1.79489	−0.897443	+	0.441130i	$0.854578\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.39154	0.534488
$144$	0	0
$145$	−7.00662	−0.581868
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.6217	1.44362	0.721812	−	0.692089i	$-0.243309\pi$
$149$	17.6217	1.44362	0.721812	+	0.692089i	$0.243309\pi$
$150$	0	0
$151$	−15.5529	−1.26568	−0.632840	−	0.774282i	$-0.718111\pi$
$151$	−15.5529	−1.26568	−0.632840	+	0.774282i	$0.718111\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.23014	0.420095
$156$	0	0
$157$	5.55294	0.443173	0.221587	−	0.975141i	$-0.428876\pi$
$157$	5.55294	0.443173	0.221587	+	0.975141i	$0.428876\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.83860	−0.538957
$162$	0	0
$163$	−7.93126	−0.621224	−0.310612	−	0.950537i	$-0.600534\pi$
$163$	−7.93126	−0.621224	−0.310612	+	0.950537i	$0.600534\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.5529	1.33453	0.667263	−	0.744822i	$-0.267466\pi$
$173$	17.5529	1.33453	0.667263	+	0.744822i	$0.267466\pi$
$174$	0	0
$175$	2.61507	0.197681
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.223528	0.0167073	0.00835363	−	0.999965i	$-0.497341\pi$
$179$	0.223528	0.0167073	0.00835363	+	0.999965i	$0.497341\pi$
$180$	0	0
$181$	15.1614	1.12694	0.563469	−	0.826137i	$-0.309466\pi$
$181$	15.1614	1.12694	0.563469	+	0.826137i	$0.309466\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.16140	0.526517
$186$	0	0
$187$	3.93126	0.287482
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.53971	0.400840	0.200420	−	0.979710i	$-0.435769\pi$
$191$	5.53971	0.400840	0.200420	+	0.979710i	$0.435769\pi$
$192$	0	0
$193$	4.61507	0.332200	0.166100	−	0.986109i	$-0.446883\pi$
$193$	4.61507	0.332200	0.166100	+	0.986109i	$0.446883\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.769857	0.0548500	0.0274250	−	0.999624i	$-0.491269\pi$
$197$	0.769857	0.0548500	0.0274250	+	0.999624i	$0.491269\pi$
$198$	0	0
$199$	−18.4603	−1.30862	−0.654308	−	0.756229i	$-0.727040\pi$
$199$	−18.4603	−1.30862	−0.654308	+	0.756229i	$0.727040\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−18.3228	−1.28601
$204$	0	0
$205$	−7.16140	−0.500174
$206$	0	0
$207$	0	0
$208$	0	0
$209$	49.7037	3.43807
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	13.6772	0.928469
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.615072	0.0413742
$222$	0	0
$223$	−11.4669	−0.767881	−0.383940	−	0.923358i	$-0.625433\pi$
$223$	−11.4669	−0.767881	−0.383940	+	0.923358i	$0.625433\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.69043	−0.510432	−0.255216	−	0.966884i	$-0.582147\pi$
$227$	−7.69043	−0.510432	−0.255216	+	0.966884i	$0.582147\pi$
$228$	0	0
$229$	7.00662	0.463010	0.231505	−	0.972834i	$-0.425635\pi$
$229$	7.00662	0.463010	0.231505	+	0.972834i	$0.425635\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.7077	−1.16007	−0.580036	−	0.814591i	$-0.696962\pi$
$233$	−17.7077	−1.16007	−0.580036	+	0.814591i	$0.696962\pi$
$234$	0	0
$235$	1.23014	0.0802457
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.9445	1.16073	0.580366	−	0.814356i	$-0.302909\pi$
$239$	17.9445	1.16073	0.580366	+	0.814356i	$0.302909\pi$
$240$	0	0
$241$	−0.460287	−0.0296497	−0.0148248	−	0.999890i	$-0.504719\pi$
$241$	−0.460287	−0.0296497	−0.0148248	+	0.999890i	$0.504719\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.161400	−0.0103115
$246$	0	0
$247$	7.77647	0.494805
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.5596	−0.792752	−0.396376	−	0.918088i	$-0.629732\pi$
$251$	−12.5596	−0.792752	−0.396376	+	0.918088i	$0.629732\pi$
$252$	0	0
$253$	−16.7143	−1.05082
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25.0198	1.56070	0.780348	−	0.625346i	$-0.215042\pi$
$257$	25.0198	1.56070	0.780348	+	0.625346i	$0.215042\pi$
$258$	0	0
$259$	18.7276	1.16367
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.54633	0.403664	0.201832	−	0.979420i	$-0.435311\pi$
$263$	6.54633	0.403664	0.201832	+	0.979420i	$0.435311\pi$
$264$	0	0
$265$	13.6217	0.836774
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.77647	−0.108313	−0.0541567	−	0.998532i	$-0.517247\pi$
$269$	−1.77647	−0.108313	−0.0541567	+	0.998532i	$0.517247\pi$
$270$	0	0
$271$	28.3360	1.72129	0.860646	−	0.509204i	$-0.170060\pi$
$271$	28.3360	1.72129	0.860646	+	0.509204i	$0.170060\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.39154	0.385425
$276$	0	0
$277$	−12.0132	−0.721805	−0.360903	−	0.932604i	$-0.617531\pi$
$277$	−12.0132	−0.721805	−0.360903	+	0.932604i	$0.617531\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.53971	−0.211162	−0.105581	−	0.994411i	$-0.533670\pi$
$281$	−3.53971	−0.211162	−0.105581	+	0.994411i	$0.533670\pi$
$282$	0	0
$283$	1.53971	0.0915265	0.0457632	−	0.998952i	$-0.485428\pi$
$283$	1.53971	0.0915265	0.0457632	+	0.998952i	$0.485428\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−18.7276	−1.10545
$288$	0	0
$289$	−16.6217	−0.977746
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.2301	1.12344	0.561718	−	0.827328i	$-0.310140\pi$
$293$	19.2301	1.12344	0.561718	+	0.827328i	$0.310140\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.61507	−0.151233
$300$	0	0
$301$	10.4603	0.602921
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.391544	−0.0224197
$306$	0	0
$307$	18.0820	1.03199	0.515996	−	0.856591i	$-0.327422\pi$
$307$	18.0820	1.03199	0.515996	+	0.856591i	$0.327422\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	12.0132	0.679028	0.339514	−	0.940601i	$-0.389737\pi$
$313$	12.0132	0.679028	0.339514	+	0.940601i	$0.389737\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.7831	−0.830301	−0.415150	−	0.909753i	$-0.636271\pi$
$317$	−14.7831	−0.830301	−0.415150	+	0.909753i	$0.636271\pi$
$318$	0	0
$319$	−44.7831	−2.50737
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.78309	0.266138
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.21691	0.177354
$330$	0	0
$331$	−15.4669	−0.850138	−0.425069	−	0.905161i	$-0.639750\pi$
$331$	−15.4669	−0.850138	−0.425069	+	0.905161i	$0.639750\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−14.0132	−0.765625
$336$	0	0
$337$	17.6904	0.963659	0.481830	−	0.876265i	$-0.339972\pi$
$337$	17.6904	0.963659	0.481830	+	0.876265i	$0.339972\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.4287	1.81026
$342$	0	0
$343$	−18.7276	−1.01119
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.9445	1.39277	0.696387	−	0.717667i	$-0.254790\pi$
$347$	25.9445	1.39277	0.696387	+	0.717667i	$0.254790\pi$
$348$	0	0
$349$	−21.7765	−1.16567	−0.582834	−	0.812591i	$-0.698056\pi$
$349$	−21.7765	−1.16567	−0.582834	+	0.812591i	$0.698056\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.4735	−0.983246	−0.491623	−	0.870808i	$-0.663596\pi$
$353$	−18.4735	−0.983246	−0.491623	+	0.870808i	$0.663596\pi$
$354$	0	0
$355$	−5.16140	−0.273939
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.5529	0.609741	0.304871	−	0.952394i	$-0.401387\pi$
$359$	11.5529	0.609741	0.304871	+	0.952394i	$0.401387\pi$
$360$	0	0
$361$	41.4735	2.18282
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.993385	0.0519961
$366$	0	0
$367$	−33.5662	−1.75214	−0.876070	−	0.482184i	$-0.839844\pi$
$367$	−33.5662	−1.75214	−0.876070	+	0.482184i	$0.839844\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	35.6217	1.84939
$372$	0	0
$373$	0.769857	0.0398617	0.0199308	−	0.999801i	$-0.493655\pi$
$373$	0.769857	0.0398617	0.0199308	+	0.999801i	$0.493655\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.00662	−0.360859
$378$	0	0
$379$	28.2500	1.45110	0.725552	−	0.688167i	$-0.241584\pi$
$379$	28.2500	1.45110	0.725552	+	0.688167i	$0.241584\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0.783087	0.0400139	0.0200069	−	0.999800i	$-0.493631\pi$
$383$	0.783087	0.0400139	0.0200069	+	0.999800i	$0.493631\pi$
$384$	0	0
$385$	16.7143	0.851842
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.2235	0.721161	0.360581	−	0.932728i	$-0.382579\pi$
$389$	14.2235	0.721161	0.360581	+	0.932728i	$0.382579\pi$
$390$	0	0
$391$	−1.60846	−0.0813431
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14.3915	−0.724117
$396$	0	0
$397$	−0.838600	−0.0420881	−0.0210441	−	0.999779i	$-0.506699\pi$
$397$	−0.838600	−0.0420881	−0.0210441	+	0.999779i	$0.506699\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.4735	0.722773	0.361386	−	0.932416i	$-0.382304\pi$
$401$	14.4735	0.722773	0.361386	+	0.932416i	$0.382304\pi$
$402$	0	0
$403$	5.23014	0.260532
$404$	0	0
$405$	0	0
$406$	0	0
$407$	45.7724	2.26885
$408$	0	0
$409$	9.55294	0.472363	0.236181	−	0.971709i	$-0.424104\pi$
$409$	9.55294	0.472363	0.236181	+	0.971709i	$0.424104\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−18.0132	−0.884235
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−25.7897	−1.25991	−0.629955	−	0.776632i	$-0.716927\pi$
$419$	−25.7897	−1.25991	−0.629955	+	0.776632i	$0.716927\pi$
$420$	0	0
$421$	−23.7897	−1.15944	−0.579720	−	0.814816i	$-0.696838\pi$
$421$	−23.7897	−1.15944	−0.579720	+	0.814816i	$0.696838\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.615072	0.0298354
$426$	0	0
$427$	−1.02391	−0.0495507
$428$	0	0
$429$	0	0
$430$	0	0
$431$	32.4735	1.56419	0.782097	−	0.623157i	$-0.214150\pi$
$431$	32.4735	1.56419	0.782097	+	0.623157i	$0.214150\pi$
$432$	0	0
$433$	−3.23014	−0.155231	−0.0776154	−	0.996983i	$-0.524731\pi$
$433$	−3.23014	−0.155231	−0.0776154	+	0.996983i	$0.524731\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−20.3360	−0.972804
$438$	0	0
$439$	−12.0687	−0.576010	−0.288005	−	0.957629i	$-0.592992\pi$
$439$	−12.0687	−0.576010	−0.288005	+	0.957629i	$0.592992\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.4048	0.589369	0.294684	−	0.955595i	$-0.404786\pi$
$443$	12.4048	0.589369	0.294684	+	0.955595i	$0.404786\pi$
$444$	0	0
$445$	−14.8518	−0.704044
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.9313	1.03500	0.517500	−	0.855683i	$-0.326863\pi$
$449$	21.9313	1.03500	0.517500	+	0.855683i	$0.326863\pi$
$450$	0	0
$451$	−45.7724	−2.15534
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.61507	0.122596
$456$	0	0
$457$	−10.1812	−0.476259	−0.238129	−	0.971233i	$-0.576534\pi$
$457$	−10.1812	−0.476259	−0.238129	+	0.971233i	$0.576534\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.63492	−0.355594	−0.177797	−	0.984067i	$-0.556897\pi$
$461$	−7.63492	−0.355594	−0.177797	+	0.984067i	$0.556897\pi$
$462$	0	0
$463$	28.6283	1.33047	0.665235	−	0.746634i	$-0.268331\pi$
$463$	28.6283	1.33047	0.665235	+	0.746634i	$0.268331\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.3121	1.07876	0.539378	−	0.842064i	$-0.318659\pi$
$467$	23.3121	1.07876	0.539378	+	0.842064i	$0.318659\pi$
$468$	0	0
$469$	−36.6456	−1.69214
$470$	0	0
$471$	0	0
$472$	0	0
$473$	25.5662	1.17553
$474$	0	0
$475$	7.77647	0.356809
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.37831	−0.382815	−0.191407	−	0.981511i	$-0.561305\pi$
$479$	−8.37831	−0.382815	−0.191407	+	0.981511i	$0.561305\pi$
$480$	0	0
$481$	7.16140	0.326532
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.38493	0.335332
$486$	0	0
$487$	28.9379	1.31130	0.655650	−	0.755065i	$-0.272395\pi$
$487$	28.9379	1.31130	0.655650	+	0.755065i	$0.272395\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21.0066	−0.948015	−0.474008	−	0.880521i	$-0.657193\pi$
$491$	−21.0066	−0.948015	−0.474008	+	0.880521i	$0.657193\pi$
$492$	0	0
$493$	−4.30957	−0.194093
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.4974	−0.605443
$498$	0	0
$499$	−23.7765	−1.06438	−0.532191	−	0.846625i	$-0.678631\pi$
$499$	−23.7765	−1.06438	−0.532191	+	0.846625i	$0.678631\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14.0993	−0.628656	−0.314328	−	0.949315i	$-0.601779\pi$
$503$	−14.0993	−0.628656	−0.314328	+	0.949315i	$0.601779\pi$
$504$	0	0
$505$	−9.77647	−0.435047
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.0687	0.446289	0.223145	−	0.974785i	$-0.428368\pi$
$509$	10.0687	0.446289	0.223145	+	0.974785i	$0.428368\pi$
$510$	0	0
$511$	2.59777	0.114919
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.86251	0.345793
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.4735	−0.634096	−0.317048	−	0.948409i	$-0.602692\pi$
$521$	−14.4735	−0.634096	−0.317048	+	0.948409i	$0.602692\pi$
$522$	0	0
$523$	−6.46029	−0.282489	−0.141244	−	0.989975i	$-0.545110\pi$
$523$	−6.46029	−0.282489	−0.141244	+	0.989975i	$0.545110\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.21691	0.140131
$528$	0	0
$529$	−16.1614	−0.702670
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.16140	−0.310195
$534$	0	0
$535$	10.3915	0.449266
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.03160	−0.0444341
$540$	0	0
$541$	−3.00662	−0.129264	−0.0646322	−	0.997909i	$-0.520587\pi$
$541$	−3.00662	−0.129264	−0.0646322	+	0.997909i	$0.520587\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.23676	−0.352824
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−54.4867	−2.32121
$552$	0	0
$553$	−37.6349	−1.60040
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.07943	0.0457368	0.0228684	−	0.999738i	$-0.492720\pi$
$557$	1.07943	0.0457368	0.0228684	+	0.999738i	$0.492720\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.1879	−1.39870	−0.699351	−	0.714779i	$-0.746527\pi$
$563$	−33.1879	−1.39870	−0.699351	+	0.714779i	$0.746527\pi$
$564$	0	0
$565$	−11.0066	−0.463052
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.7963	0.536450	0.268225	−	0.963356i	$-0.413563\pi$
$569$	12.7963	0.536450	0.268225	+	0.963356i	$0.413563\pi$
$570$	0	0
$571$	−41.9445	−1.75532	−0.877661	−	0.479282i	$-0.840897\pi$
$571$	−41.9445	−1.75532	−0.877661	+	0.479282i	$0.840897\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.61507	−0.109056
$576$	0	0
$577$	38.9379	1.62100	0.810502	−	0.585735i	$-0.199194\pi$
$577$	38.9379	1.62100	0.810502	+	0.585735i	$0.199194\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−47.1059	−1.95428
$582$	0	0
$583$	87.0636	3.60581
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.2301	−1.20646	−0.603229	−	0.797568i	$-0.706119\pi$
$587$	−29.2301	−1.20646	−0.603229	+	0.797568i	$0.706119\pi$
$588$	0	0
$589$	40.6721	1.67586
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11.6772	−0.479525	−0.239763	−	0.970832i	$-0.577070\pi$
$593$	−11.6772	−0.479525	−0.239763	+	0.970832i	$0.577070\pi$
$594$	0	0
$595$	1.60846	0.0659403
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.0265	−1.47200	−0.736001	−	0.676981i	$-0.763288\pi$
$599$	−36.0265	−1.47200	−0.736001	+	0.676981i	$0.763288\pi$
$600$	0	0
$601$	−0.0555123	−0.00226439	−0.00113220	−	0.999999i	$-0.500360\pi$
$601$	−0.0555123	−0.00226439	−0.00113220	+	0.999999i	$0.500360\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	29.8518	1.21365
$606$	0	0
$607$	33.5662	1.36241	0.681204	−	0.732093i	$-0.261456\pi$
$607$	33.5662	1.36241	0.681204	+	0.732093i	$0.261456\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.23014	0.0497663
$612$	0	0
$613$	25.4842	1.02930	0.514649	−	0.857401i	$-0.327922\pi$
$613$	25.4842	1.02930	0.514649	+	0.857401i	$0.327922\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.55294	0.384587	0.192294	−	0.981337i	$-0.438407\pi$
$617$	9.55294	0.384587	0.192294	+	0.981337i	$0.438407\pi$
$618$	0	0
$619$	−26.2368	−1.05454	−0.527272	−	0.849696i	$-0.676785\pi$
$619$	−26.2368	−1.05454	−0.527272	+	0.849696i	$0.676785\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−38.8386	−1.55604
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.40477	0.175630
$630$	0	0
$631$	−18.4603	−0.734892	−0.367446	−	0.930045i	$-0.619768\pi$
$631$	−18.4603	−0.734892	−0.367446	+	0.930045i	$0.619768\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.7831	0.507281
$636$	0	0
$637$	−0.161400	−0.00639492
$638$	0	0
$639$	0	0
$640$	0	0
$641$	41.8890	1.65452	0.827258	−	0.561823i	$-0.189900\pi$
$641$	41.8890	1.65452	0.827258	+	0.561823i	$0.189900\pi$
$642$	0	0
$643$	23.6217	0.931548	0.465774	−	0.884904i	$-0.345776\pi$
$643$	23.6217	0.931548	0.465774	+	0.884904i	$0.345776\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.2607	0.599959	0.299979	−	0.953946i	$-0.403020\pi$
$647$	15.2607	0.599959	0.299979	+	0.953946i	$0.403020\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	0	0
$655$	15.7765	0.616438
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.0198	0.585090	0.292545	−	0.956252i	$-0.405498\pi$
$659$	15.0198	0.585090	0.292545	+	0.956252i	$0.405498\pi$
$660$	0	0
$661$	−31.7897	−1.23648	−0.618238	−	0.785991i	$-0.712153\pi$
$661$	−31.7897	−1.23648	−0.618238	+	0.785991i	$0.712153\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	20.3360	0.788597
$666$	0	0
$667$	18.3228	0.709462
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.50257	−0.0966106
$672$	0	0
$673$	38.0265	1.46581	0.732906	−	0.680330i	$-0.238163\pi$
$673$	38.0265	1.46581	0.732906	+	0.680330i	$0.238163\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.7143	1.64165	0.820823	−	0.571183i	$-0.193515\pi$
$677$	42.7143	1.64165	0.820823	+	0.571183i	$0.193515\pi$
$678$	0	0
$679$	19.3121	0.741131
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21.0927	0.807088	0.403544	−	0.914960i	$-0.367778\pi$
$683$	21.0927	0.807088	0.403544	+	0.914960i	$0.367778\pi$
$684$	0	0
$685$	−16.0132	−0.611834
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13.6217	0.518945
$690$	0	0
$691$	31.1573	1.18528	0.592640	−	0.805467i	$-0.298086\pi$
$691$	31.1573	1.18528	0.592640	+	0.805467i	$0.298086\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21.1614	−0.802698
$696$	0	0
$697$	−4.40477	−0.166843
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−40.8824	−1.54411	−0.772053	−	0.635558i	$-0.780770\pi$
$701$	−40.8824	−1.54411	−0.772053	+	0.635558i	$0.780770\pi$
$702$	0	0
$703$	55.6904	2.10040
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−25.5662	−0.961515
$708$	0	0
$709$	4.68381	0.175904	0.0879522	−	0.996125i	$-0.471968\pi$
$709$	4.68381	0.175904	0.0879522	+	0.996125i	$0.471968\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−13.6772	−0.512215
$714$	0	0
$715$	6.39154	0.239030
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0.447056	0.0166724	0.00833619	−	0.999965i	$-0.497346\pi$
$719$	0.447056	0.0166724	0.00833619	+	0.999965i	$0.497346\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.00662	−0.260219
$726$	0	0
$727$	−33.2566	−1.23342	−0.616710	−	0.787191i	$-0.711535\pi$
$727$	−33.2566	−1.23342	−0.616710	+	0.787191i	$0.711535\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.46029	0.0909970
$732$	0	0
$733$	−14.3783	−0.531075	−0.265538	−	0.964101i	$-0.585549\pi$
$733$	−14.3783	−0.531075	−0.265538	+	0.964101i	$0.585549\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−89.5662	−3.29921
$738$	0	0
$739$	−17.3426	−0.637960	−0.318980	−	0.947762i	$-0.603340\pi$
$739$	−17.3426	−0.637960	−0.318980	+	0.947762i	$0.603340\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−35.2434	−1.29295	−0.646477	−	0.762933i	$-0.723758\pi$
$743$	−35.2434	−1.29295	−0.646477	+	0.762933i	$0.723758\pi$
$744$	0	0
$745$	17.6217	0.645609
$746$	0	0
$747$	0	0
$748$	0	0
$749$	27.1746	0.992939
$750$	0	0
$751$	−13.6349	−0.497545	−0.248773	−	0.968562i	$-0.580027\pi$
$751$	−13.6349	−0.497545	−0.248773	+	0.968562i	$0.580027\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.5529	−0.566030
$756$	0	0
$757$	−28.3228	−1.02941	−0.514705	−	0.857367i	$-0.672098\pi$
$757$	−28.3228	−1.02941	−0.514705	+	0.857367i	$0.672098\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.0927	−0.402109	−0.201054	−	0.979580i	$-0.564437\pi$
$761$	−11.0927	−0.402109	−0.201054	+	0.979580i	$0.564437\pi$
$762$	0	0
$763$	−21.5397	−0.779790
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−32.3228	−1.16559	−0.582795	−	0.812619i	$-0.698041\pi$
$769$	−32.3228	−1.16559	−0.582795	+	0.812619i	$0.698041\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.33603	−0.227891	−0.113946	−	0.993487i	$-0.536349\pi$
$773$	−6.33603	−0.227891	−0.113946	+	0.993487i	$0.536349\pi$
$774$	0	0
$775$	5.23014	0.187872
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−55.6904	−1.99532
$780$	0	0
$781$	−32.9893	−1.18045
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.55294	0.198193
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−28.7831	−1.02341
$792$	0	0
$793$	−0.391544	−0.0139041
$794$	0	0
$795$	0	0
$796$	0	0
$797$	50.4048	1.78543	0.892714	−	0.450623i	$-0.148798\pi$
$797$	50.4048	1.78543	0.892714	+	0.450623i	$0.148798\pi$
$798$	0	0
$799$	0.756626	0.0267675
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.34926	0.224061
$804$	0	0
$805$	−6.83860	−0.241029
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	28.5596	1.00286	0.501431	−	0.865198i	$-0.332807\pi$
$811$	28.5596	1.00286	0.501431	+	0.865198i	$0.332807\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.93126	−0.277820
$816$	0	0
$817$	31.1059	1.08826
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.94449	0.137664	0.0688318	−	0.997628i	$-0.478073\pi$
$821$	3.94449	0.137664	0.0688318	+	0.997628i	$0.478073\pi$
$822$	0	0
$823$	−20.9206	−0.729245	−0.364623	−	0.931155i	$-0.618802\pi$
$823$	−20.9206	−0.729245	−0.364623	+	0.931155i	$0.618802\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−49.8757	−1.73435	−0.867175	−	0.498004i	$-0.834067\pi$
$827$	−49.8757	−1.73435	−0.867175	+	0.498004i	$0.834067\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.0992728	−0.00343960
$834$	0	0
$835$	4.00000	0.138426
$836$	0	0
$837$	0	0
$838$	0	0
$839$	54.8651	1.89415	0.947076	−	0.321009i	$-0.104022\pi$
$839$	54.8651	1.89415	0.947076	+	0.321009i	$0.104022\pi$
$840$	0	0
$841$	20.0927	0.692850
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	78.0647	2.68233
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−18.7276	−0.641973
$852$	0	0
$853$	−31.8070	−1.08905	−0.544526	−	0.838744i	$-0.683290\pi$
$853$	−31.8070	−1.08905	−0.544526	+	0.838744i	$0.683290\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.4114	1.07299	0.536496	−	0.843903i	$-0.319748\pi$
$857$	31.4114	1.07299	0.536496	+	0.843903i	$0.319748\pi$
$858$	0	0
$859$	−57.6349	−1.96648	−0.983239	−	0.182321i	$-0.941639\pi$
$859$	−57.6349	−1.96648	−0.983239	+	0.182321i	$0.941639\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9.40223	−0.320056	−0.160028	−	0.987113i	$-0.551158\pi$
$863$	−9.40223	−0.320056	−0.160028	+	0.987113i	$0.551158\pi$
$864$	0	0
$865$	17.5529	0.596818
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−91.9842	−3.12035
$870$	0	0
$871$	−14.0132	−0.474820
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.61507	0.0884056
$876$	0	0
$877$	46.9470	1.58529	0.792644	−	0.609684i	$-0.208704\pi$
$877$	46.9470	1.58529	0.792644	+	0.609684i	$0.208704\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−44.0132	−1.48284	−0.741422	−	0.671039i	$-0.765848\pi$
$881$	−44.0132	−1.48284	−0.741422	+	0.671039i	$0.765848\pi$
$882$	0	0
$883$	−22.4603	−0.755849	−0.377924	−	0.925836i	$-0.623362\pi$
$883$	−22.4603	−0.755849	−0.377924	+	0.925836i	$0.623362\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12.3187	−0.413623	−0.206811	−	0.978381i	$-0.566309\pi$
$887$	−12.3187	−0.413623	−0.206811	+	0.978381i	$0.566309\pi$
$888$	0	0
$889$	33.4287	1.12116
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.56617	0.320120
$894$	0	0
$895$	0.223528	0.00747172
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−36.6456	−1.22220
$900$	0	0
$901$	8.37831	0.279122
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.1614	0.503982
$906$	0	0
$907$	−6.32280	−0.209945	−0.104973	−	0.994475i	$-0.533475\pi$
$907$	−6.32280	−0.209945	−0.104973	+	0.994475i	$0.533475\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	11.5794	0.383643	0.191821	−	0.981430i	$-0.438561\pi$
$911$	11.5794	0.383643	0.191821	+	0.981430i	$0.438561\pi$
$912$	0	0
$913$	−115.132	−3.81032
$914$	0	0
$915$	0	0
$916$	0	0
$917$	41.2566	1.36241
$918$	0	0
$919$	−17.4710	−0.576314	−0.288157	−	0.957583i	$-0.593043\pi$
$919$	−17.4710	−0.576314	−0.288157	+	0.957583i	$0.593043\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.16140	−0.169890
$924$	0	0
$925$	7.16140	0.235465
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−34.8783	−1.14432	−0.572160	−	0.820142i	$-0.693894\pi$
$929$	−34.8783	−1.14432	−0.572160	+	0.820142i	$0.693894\pi$
$930$	0	0
$931$	−1.25513	−0.0411351
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.93126	0.128566
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−47.3253	−1.54276	−0.771381	−	0.636373i	$-0.780434\pi$
$941$	−47.3253	−1.54276	−0.771381	+	0.636373i	$0.780434\pi$
$942$	0	0
$943$	18.7276	0.609854
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.1191	−1.33619	−0.668096	−	0.744075i	$-0.732890\pi$
$947$	−41.1191	−1.33619	−0.668096	+	0.744075i	$0.732890\pi$
$948$	0	0
$949$	0.993385	0.0322466
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.53564	0.179317	0.0896586	−	0.995973i	$-0.471422\pi$
$953$	5.53564	0.179317	0.0896586	+	0.995973i	$0.471422\pi$
$954$	0	0
$955$	5.53971	0.179261
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−41.8757	−1.35224
$960$	0	0
$961$	−3.64560	−0.117600
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.61507	0.148564
$966$	0	0
$967$	−21.4801	−0.690754	−0.345377	−	0.938464i	$-0.612249\pi$
$967$	−21.4801	−0.690754	−0.345377	+	0.938464i	$0.612249\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.4669	1.00982	0.504910	−	0.863172i	$-0.331526\pi$
$971$	31.4669	1.00982	0.504910	+	0.863172i	$0.331526\pi$
$972$	0	0
$973$	−55.3386	−1.77407
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.3228	−0.522213	−0.261106	−	0.965310i	$-0.584087\pi$
$977$	−16.3228	−0.522213	−0.261106	+	0.965310i	$0.584087\pi$
$978$	0	0
$979$	−94.9261	−3.03385
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−24.9206	−0.794843	−0.397421	−	0.917636i	$-0.630095\pi$
$983$	−24.9206	−0.794843	−0.397421	+	0.917636i	$0.630095\pi$
$984$	0	0
$985$	0.769857	0.0245297
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.4603	−0.332618
$990$	0	0
$991$	52.8783	1.67973	0.839867	−	0.542792i	$-0.182633\pi$
$991$	52.8783	1.67973	0.839867	+	0.542792i	$0.182633\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−18.4603	−0.585230
$996$	0	0
$997$	32.1507	1.01822	0.509112	−	0.860700i	$-0.329974\pi$
$997$	32.1507	1.01822	0.509112	+	0.860700i	$0.329974\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.bl.1.2		3
3.2	odd	2		1560.2.a.p.1.2	✓	3
4.3	odd	2		9360.2.a.db.1.2		3
12.11	even	2		3120.2.a.bh.1.2		3
15.14	odd	2		7800.2.a.bf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.p.1.2	✓	3	3.2	odd	2
3120.2.a.bh.1.2		3	12.11	even	2
4680.2.a.bl.1.2		3	1.1	even	1	trivial
7800.2.a.bf.1.2		3	15.14	odd	2
9360.2.a.db.1.2		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +2.61507 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +2.61507 q^{7} +6.39154 q^{11} +1.00000 q^{13} +0.615072 q^{17} +7.77647 q^{19} -2.61507 q^{23} +1.00000 q^{25} -7.00662 q^{29} +5.23014 q^{31} +2.61507 q^{35} +7.16140 q^{37} -7.16140 q^{41} +4.00000 q^{43} +1.23014 q^{47} -0.161400 q^{49} +13.6217 q^{53} +6.39154 q^{55} -0.391544 q^{61} +1.00000 q^{65} -14.0132 q^{67} -5.16140 q^{71} +0.993385 q^{73} +16.7143 q^{77} -14.3915 q^{79} -18.0132 q^{83} +0.615072 q^{85} -14.8518 q^{89} +2.61507 q^{91} +7.77647 q^{95} +7.38493 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 5 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 + 5 * q^7	\( 3 q + 3 q^{5} + 5 q^{7} - 3 q^{11} + 3 q^{13} - q^{17} + 4 q^{19} - 5 q^{23} + 3 q^{25} + 4 q^{29} + 10 q^{31} + 5 q^{35} + 5 q^{37} - 5 q^{41} + 12 q^{43} - 2 q^{47} + 16 q^{49} + 13 q^{53} - 3 q^{55} + 21 q^{61} + 3 q^{65} + 8 q^{67} + q^{71} + 28 q^{73} - 5 q^{77} - 21 q^{79} - 4 q^{83} - q^{85} - 11 q^{89} + 5 q^{91} + 4 q^{95} + 25 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 + 5 * q^7 - 3 * q^11 + 3 * q^13 - q^17 + 4 * q^19 - 5 * q^23 + 3 * q^25 + 4 * q^29 + 10 * q^31 + 5 * q^35 + 5 * q^37 - 5 * q^41 + 12 * q^43 - 2 * q^47 + 16 * q^49 + 13 * q^53 - 3 * q^55 + 21 * q^61 + 3 * q^65 + 8 * q^67 + q^71 + 28 * q^73 - 5 * q^77 - 21 * q^79 - 4 * q^83 - q^85 - 11 * q^89 + 5 * q^91 + 4 * q^95 + 25 * q^97

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