LMFDB - Embedded newform 4680.2.a.bi.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:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.66908$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} +1.45490 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +1.45490 q^{7} +4.79306 q^{11} +1.00000 q^{13} -0.545096 q^{17} +7.33816 q^{19} -0.545096 q^{23} +1.00000 q^{25} +5.33816 q^{29} +2.00000 q^{31} -1.45490 q^{35} -3.88325 q^{37} -5.70287 q^{41} +2.90981 q^{43} -1.09019 q^{47} -4.88325 q^{49} -4.79306 q^{53} -4.79306 q^{55} -10.6763 q^{59} +7.88325 q^{61} -1.00000 q^{65} +3.09019 q^{67} +2.97345 q^{71} -4.42835 q^{73} +6.97345 q^{77} +11.7029 q^{79} +0.545096 q^{85} +9.70287 q^{89} +1.45490 q^{91} -7.33816 q^{95} -7.04103 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + 3 * q^7	$ 3 q - 3 q^{5} + 3 q^{7} - 3 q^{11} + 3 q^{13} - 3 q^{17} + 6 q^{19} - 3 q^{23} + 3 q^{25} + 6 q^{31} - 3 q^{35} + 3 q^{37} + 3 q^{41} + 6 q^{43} - 6 q^{47} + 3 q^{53} + 3 q^{55} + 9 q^{61} - 3 q^{65} + 12 q^{67} - 3 q^{71} + 9 q^{77} + 15 q^{79} + 3 q^{85} + 9 q^{89} + 3 q^{91} - 6 q^{95} + 15 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 - 3 * q^11 + 3 * q^13 - 3 * q^17 + 6 * q^19 - 3 * q^23 + 3 * q^25 + 6 * q^31 - 3 * q^35 + 3 * q^37 + 3 * q^41 + 6 * q^43 - 6 * q^47 + 3 * q^53 + 3 * q^55 + 9 * q^61 - 3 * q^65 + 12 * q^67 - 3 * q^71 + 9 * q^77 + 15 * q^79 + 3 * q^85 + 9 * q^89 + 3 * q^91 - 6 * q^95 + 15 * 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
$7$	1.45490	0.549902	0.274951	−	0.961458i	$-0.411338\pi$
$7$	1.45490	0.549902	0.274951	+	0.961458i	$0.411338\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.79306	1.44516	0.722581	−	0.691286i	$-0.242955\pi$
$11$	4.79306	1.44516	0.722581	+	0.691286i	$0.242955\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.545096	−0.132205	−0.0661026	−	0.997813i	$-0.521056\pi$
$17$	−0.545096	−0.132205	−0.0661026	+	0.997813i	$0.521056\pi$
$18$	0	0
$19$	7.33816	1.68349	0.841744	−	0.539876i	$-0.181529\pi$
$19$	7.33816	1.68349	0.841744	+	0.539876i	$0.181529\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.545096	−0.113660	−0.0568302	−	0.998384i	$-0.518099\pi$
$23$	−0.545096	−0.113660	−0.0568302	+	0.998384i	$0.518099\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.33816	0.991271	0.495636	−	0.868531i	$-0.334935\pi$
$29$	5.33816	0.991271	0.495636	+	0.868531i	$0.334935\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.45490	−0.245924
$36$	0	0
$37$	−3.88325	−0.638403	−0.319202	−	0.947687i	$-0.603415\pi$
$37$	−3.88325	−0.638403	−0.319202	+	0.947687i	$0.603415\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.70287	−0.890639	−0.445319	−	0.895372i	$-0.646910\pi$
$41$	−5.70287	−0.890639	−0.445319	+	0.895372i	$0.646910\pi$
$42$	0	0
$43$	2.90981	0.443742	0.221871	−	0.975076i	$-0.428784\pi$
$43$	2.90981	0.443742	0.221871	+	0.975076i	$0.428784\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.09019	−0.159021	−0.0795104	−	0.996834i	$-0.525336\pi$
$47$	−1.09019	−0.159021	−0.0795104	+	0.996834i	$0.525336\pi$
$48$	0	0
$49$	−4.88325	−0.697608
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.79306	−0.658378	−0.329189	−	0.944264i	$-0.606775\pi$
$53$	−4.79306	−0.658378	−0.329189	+	0.944264i	$0.606775\pi$
$54$	0	0
$55$	−4.79306	−0.646296
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.6763	−1.38994	−0.694969	−	0.719040i	$-0.744582\pi$
$59$	−10.6763	−1.38994	−0.694969	+	0.719040i	$0.744582\pi$
$60$	0	0
$61$	7.88325	1.00935	0.504674	−	0.863310i	$-0.331613\pi$
$61$	7.88325	1.00935	0.504674	+	0.863310i	$0.331613\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	3.09019	0.377527	0.188764	−	0.982023i	$-0.439552\pi$
$67$	3.09019	0.377527	0.188764	+	0.982023i	$0.439552\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.97345	0.352883	0.176442	−	0.984311i	$-0.443541\pi$
$71$	2.97345	0.352883	0.176442	+	0.984311i	$0.443541\pi$
$72$	0	0
$73$	−4.42835	−0.518299	−0.259150	−	0.965837i	$-0.583442\pi$
$73$	−4.42835	−0.518299	−0.259150	+	0.965837i	$0.583442\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.97345	0.794698
$78$	0	0
$79$	11.7029	1.31668	0.658338	−	0.752723i	$-0.271260\pi$
$79$	11.7029	1.31668	0.658338	+	0.752723i	$0.271260\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0.545096	0.0591240
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.70287	1.02850	0.514251	−	0.857640i	$-0.328070\pi$
$89$	9.70287	1.02850	0.514251	+	0.857640i	$0.328070\pi$
$90$	0	0
$91$	1.45490	0.152515
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.33816	−0.752879
$96$	0	0
$97$	−7.04103	−0.714908	−0.357454	−	0.933931i	$-0.616355\pi$
$97$	−7.04103	−0.714908	−0.357454	+	0.933931i	$0.616355\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.42835	0.639645	0.319822	−	0.947478i	$-0.396377\pi$
$101$	6.42835	0.639645	0.319822	+	0.947478i	$0.396377\pi$
$102$	0	0
$103$	15.7665	1.55352	0.776760	−	0.629797i	$-0.216862\pi$
$103$	15.7665	1.55352	0.776760	+	0.629797i	$0.216862\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.9734	−1.06084	−0.530422	−	0.847734i	$-0.677966\pi$
$107$	−10.9734	−1.06084	−0.530422	+	0.847734i	$0.677966\pi$
$108$	0	0
$109$	15.1047	1.44676	0.723382	−	0.690448i	$-0.242586\pi$
$109$	15.1047	1.44676	0.723382	+	0.690448i	$0.242586\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.0145	−1.13023	−0.565113	−	0.825013i	$-0.691168\pi$
$113$	−12.0145	−1.13023	−0.565113	+	0.825013i	$0.691168\pi$
$114$	0	0
$115$	0.545096	0.0508305
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.793062	−0.0726999
$120$	0	0
$121$	11.9734	1.08850
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.49593	−0.753892	−0.376946	−	0.926235i	$-0.623026\pi$
$127$	−8.49593	−0.753892	−0.376946	+	0.926235i	$0.623026\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.4283	−0.911129	−0.455565	−	0.890203i	$-0.650563\pi$
$131$	−10.4283	−0.911129	−0.455565	+	0.890203i	$0.650563\pi$
$132$	0	0
$133$	10.6763	0.925754
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.90981	0.419473	0.209737	−	0.977758i	$-0.432739\pi$
$137$	4.90981	0.419473	0.209737	+	0.977758i	$0.432739\pi$
$138$	0	0
$139$	8.79306	0.745818	0.372909	−	0.927868i	$-0.378360\pi$
$139$	8.79306	0.745818	0.372909	+	0.927868i	$0.378360\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.79306	0.400816
$144$	0	0
$145$	−5.33816	−0.443310
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.973446	−0.0797478	−0.0398739	−	0.999205i	$-0.512696\pi$
$149$	−0.973446	−0.0797478	−0.0398739	+	0.999205i	$0.512696\pi$
$150$	0	0
$151$	−5.76651	−0.469272	−0.234636	−	0.972083i	$-0.575390\pi$
$151$	−5.76651	−0.469272	−0.234636	+	0.972083i	$0.575390\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	8.90981	0.711080	0.355540	−	0.934661i	$-0.384297\pi$
$157$	8.90981	0.711080	0.355540	+	0.934661i	$0.384297\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.793062	−0.0625021
$162$	0	0
$163$	−2.79306	−0.218770	−0.109385	−	0.993999i	$-0.534888\pi$
$163$	−2.79306	−0.218770	−0.109385	+	0.993999i	$0.534888\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.58612	0.741797	0.370898	−	0.928673i	$-0.379050\pi$
$167$	9.58612	0.741797	0.370898	+	0.928673i	$0.379050\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.7665	1.19871	0.599353	−	0.800485i	$-0.295425\pi$
$173$	15.7665	1.19871	0.599353	+	0.800485i	$0.295425\pi$
$174$	0	0
$175$	1.45490	0.109980
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.1949	1.06097	0.530487	−	0.847693i	$-0.322009\pi$
$179$	14.1949	1.06097	0.530487	+	0.847693i	$0.322009\pi$
$180$	0	0
$181$	7.52249	0.559142	0.279571	−	0.960125i	$-0.409808\pi$
$181$	7.52249	0.559142	0.279571	+	0.960125i	$0.409808\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.88325	0.285503
$186$	0	0
$187$	−2.61268	−0.191058
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.5861	−0.983057	−0.491529	−	0.870861i	$-0.663562\pi$
$191$	−13.5861	−0.983057	−0.491529	+	0.870861i	$0.663562\pi$
$192$	0	0
$193$	19.4018	1.39657	0.698286	−	0.715819i	$-0.253946\pi$
$193$	19.4018	1.39657	0.698286	+	0.715819i	$0.253946\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.5861	−0.825477	−0.412739	−	0.910850i	$-0.635428\pi$
$197$	−11.5861	−0.825477	−0.412739	+	0.910850i	$0.635428\pi$
$198$	0	0
$199$	1.32368	0.0938334	0.0469167	−	0.998899i	$-0.485060\pi$
$199$	1.32368	0.0938334	0.0469167	+	0.998899i	$0.485060\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.76651	0.545102
$204$	0	0
$205$	5.70287	0.398306
$206$	0	0
$207$	0	0
$208$	0	0
$209$	35.1722	2.43292
$210$	0	0
$211$	−14.3155	−0.985523	−0.492762	−	0.870164i	$-0.664013\pi$
$211$	−14.3155	−0.985523	−0.492762	+	0.870164i	$0.664013\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.90981	−0.198447
$216$	0	0
$217$	2.90981	0.197531
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.545096	−0.0366671
$222$	0	0
$223$	13.1578	0.881110	0.440555	−	0.897726i	$-0.354782\pi$
$223$	13.1578	0.881110	0.440555	+	0.897726i	$0.354782\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	3.33816	0.220592	0.110296	−	0.993899i	$-0.464820\pi$
$229$	3.33816	0.220592	0.110296	+	0.993899i	$0.464820\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.8606	1.75970	0.879850	−	0.475252i	$-0.157643\pi$
$233$	26.8606	1.75970	0.879850	+	0.475252i	$0.157643\pi$
$234$	0	0
$235$	1.09019	0.0711163
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.55957	−0.553673	−0.276836	−	0.960917i	$-0.589286\pi$
$239$	−8.55957	−0.553673	−0.276836	+	0.960917i	$0.589286\pi$
$240$	0	0
$241$	−13.1722	−0.848499	−0.424250	−	0.905545i	$-0.639462\pi$
$241$	−13.1722	−0.848499	−0.424250	+	0.905545i	$0.639462\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.88325	0.311980
$246$	0	0
$247$	7.33816	0.466916
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.57165	−0.604157	−0.302079	−	0.953283i	$-0.597681\pi$
$251$	−9.57165	−0.604157	−0.302079	+	0.953283i	$0.597681\pi$
$252$	0	0
$253$	−2.61268	−0.164258
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.97739	0.310481	0.155241	−	0.987877i	$-0.450385\pi$
$257$	4.97739	0.310481	0.155241	+	0.987877i	$0.450385\pi$
$258$	0	0
$259$	−5.64976	−0.351059
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.9243	1.16692	0.583461	−	0.812141i	$-0.301698\pi$
$263$	18.9243	1.16692	0.583461	+	0.812141i	$0.301698\pi$
$264$	0	0
$265$	4.79306	0.294435
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.84223	0.295236	0.147618	−	0.989044i	$-0.452839\pi$
$269$	4.84223	0.295236	0.147618	+	0.989044i	$0.452839\pi$
$270$	0	0
$271$	11.5861	0.703807	0.351903	−	0.936036i	$-0.385535\pi$
$271$	11.5861	0.703807	0.351903	+	0.936036i	$0.385535\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.79306	0.289033
$276$	0	0
$277$	24.3155	1.46098	0.730490	−	0.682924i	$-0.239292\pi$
$277$	24.3155	1.46098	0.730490	+	0.682924i	$0.239292\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−0.676316	−0.0403457	−0.0201728	−	0.999797i	$-0.506422\pi$
$281$	−0.676316	−0.0403457	−0.0201728	+	0.999797i	$0.506422\pi$
$282$	0	0
$283$	−12.8567	−0.764251	−0.382126	−	0.924110i	$-0.624808\pi$
$283$	−12.8567	−0.764251	−0.382126	+	0.924110i	$0.624808\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.29713	−0.489764
$288$	0	0
$289$	−16.7029	−0.982522
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.1804	0.711586	0.355793	−	0.934565i	$-0.384211\pi$
$293$	12.1804	0.711586	0.355793	+	0.934565i	$0.384211\pi$
$294$	0	0
$295$	10.6763	0.621599
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.545096	−0.0315237
$300$	0	0
$301$	4.23349	0.244014
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7.88325	−0.451394
$306$	0	0
$307$	1.70287	0.0971879	0.0485940	−	0.998819i	$-0.484526\pi$
$307$	1.70287	0.0971879	0.0485940	+	0.998819i	$0.484526\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.94689	0.564036	0.282018	−	0.959409i	$-0.408996\pi$
$311$	9.94689	0.564036	0.282018	+	0.959409i	$0.408996\pi$
$312$	0	0
$313$	14.7294	0.832556	0.416278	−	0.909237i	$-0.363334\pi$
$313$	14.7294	0.832556	0.416278	+	0.909237i	$0.363334\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.3526	−1.53628	−0.768138	−	0.640284i	$-0.778817\pi$
$317$	−27.3526	−1.53628	−0.768138	+	0.640284i	$0.778817\pi$
$318$	0	0
$319$	25.5861	1.43255
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.58612	−0.0874459
$330$	0	0
$331$	30.5104	1.67700	0.838502	−	0.544899i	$-0.183432\pi$
$331$	30.5104	1.67700	0.838502	+	0.544899i	$0.183432\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.09019	−0.168835
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.58612	0.519118
$342$	0	0
$343$	−17.2890	−0.933518
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−35.1086	−1.88473	−0.942365	−	0.334588i	$-0.891403\pi$
$347$	−35.1086	−1.88473	−0.942365	+	0.334588i	$0.891403\pi$
$348$	0	0
$349$	−11.8341	−0.633464	−0.316732	−	0.948515i	$-0.602586\pi$
$349$	−11.8341	−0.633464	−0.316732	+	0.948515i	$0.602586\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.67632	−0.248895	−0.124448	−	0.992226i	$-0.539716\pi$
$353$	−4.67632	−0.248895	−0.124448	+	0.992226i	$0.539716\pi$
$354$	0	0
$355$	−2.97345	−0.157814
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.3526	−1.33806	−0.669030	−	0.743235i	$-0.733290\pi$
$359$	−25.3526	−1.33806	−0.669030	+	0.743235i	$0.733290\pi$
$360$	0	0
$361$	34.8486	1.83414
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.42835	0.231790
$366$	0	0
$367$	24.0289	1.25430	0.627150	−	0.778898i	$-0.284221\pi$
$367$	24.0289	1.25430	0.627150	+	0.778898i	$0.284221\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.97345	−0.362043
$372$	0	0
$373$	−31.8486	−1.64906	−0.824528	−	0.565821i	$-0.808559\pi$
$373$	−31.8486	−1.64906	−0.824528	+	0.565821i	$0.808559\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.33816	0.274929
$378$	0	0
$379$	14.2480	0.731869	0.365934	−	0.930641i	$-0.380749\pi$
$379$	14.2480	0.731869	0.365934	+	0.930641i	$0.380749\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−6.97345	−0.355400
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.4815	0.632835	0.316418	−	0.948620i	$-0.397520\pi$
$389$	12.4815	0.632835	0.316418	+	0.948620i	$0.397520\pi$
$390$	0	0
$391$	0.297130	0.0150265
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.7029	−0.588835
$396$	0	0
$397$	23.2890	1.16884	0.584421	−	0.811451i	$-0.301322\pi$
$397$	23.2890	1.16884	0.584421	+	0.811451i	$0.301322\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.49593	−0.324391	−0.162196	−	0.986759i	$-0.551858\pi$
$401$	−6.49593	−0.324391	−0.162196	+	0.986759i	$0.551858\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.6127	−0.922596
$408$	0	0
$409$	−20.0821	−0.992994	−0.496497	−	0.868038i	$-0.665381\pi$
$409$	−20.0821	−0.992994	−0.496497	+	0.868038i	$0.665381\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−15.5330	−0.764330
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.15777	0.349680	0.174840	−	0.984597i	$-0.444059\pi$
$419$	7.15777	0.349680	0.174840	+	0.984597i	$0.444059\pi$
$420$	0	0
$421$	−7.33816	−0.357640	−0.178820	−	0.983882i	$-0.557228\pi$
$421$	−7.33816	−0.357640	−0.178820	+	0.983882i	$0.557228\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.545096	−0.0264410
$426$	0	0
$427$	11.4694	0.555042
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.3526	−1.02852	−0.514260	−	0.857634i	$-0.671933\pi$
$431$	−21.3526	−1.02852	−0.514260	+	0.857634i	$0.671933\pi$
$432$	0	0
$433$	23.5861	1.13348	0.566738	−	0.823898i	$-0.308205\pi$
$433$	23.5861	1.13348	0.566738	+	0.823898i	$0.308205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	−8.19880	−0.391308	−0.195654	−	0.980673i	$-0.562683\pi$
$439$	−8.19880	−0.391308	−0.195654	+	0.980673i	$0.562683\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−33.2890	−1.58161	−0.790804	−	0.612070i	$-0.790337\pi$
$443$	−33.2890	−1.58161	−0.790804	+	0.612070i	$0.790337\pi$
$444$	0	0
$445$	−9.70287	−0.459960
$446$	0	0
$447$	0	0
$448$	0	0
$449$	31.6498	1.49364	0.746822	−	0.665024i	$-0.231579\pi$
$449$	31.6498	1.49364	0.746822	+	0.665024i	$0.231579\pi$
$450$	0	0
$451$	−27.3342	−1.28712
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.45490	−0.0682070
$456$	0	0
$457$	20.1312	0.941699	0.470849	−	0.882214i	$-0.343948\pi$
$457$	20.1312	0.941699	0.470849	+	0.882214i	$0.343948\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.2890	1.08468	0.542338	−	0.840160i	$-0.317539\pi$
$461$	23.2890	1.08468	0.542338	+	0.840160i	$0.317539\pi$
$462$	0	0
$463$	1.95084	0.0906631	0.0453315	−	0.998972i	$-0.485566\pi$
$463$	1.95084	0.0906631	0.0453315	+	0.998972i	$0.485566\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.62081	−0.445198	−0.222599	−	0.974910i	$-0.571454\pi$
$467$	−9.62081	−0.445198	−0.222599	+	0.974910i	$0.571454\pi$
$468$	0	0
$469$	4.49593	0.207603
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13.9469	0.641279
$474$	0	0
$475$	7.33816	0.336698
$476$	0	0
$477$	0	0
$478$	0	0
$479$	40.5596	1.85321	0.926607	−	0.376031i	$-0.122711\pi$
$479$	40.5596	1.85321	0.926607	+	0.376031i	$0.122711\pi$
$480$	0	0
$481$	−3.88325	−0.177061
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.04103	0.319717
$486$	0	0
$487$	−13.9508	−0.632173	−0.316086	−	0.948730i	$-0.602369\pi$
$487$	−13.9508	−0.632173	−0.316086	+	0.948730i	$0.602369\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.8422	−0.579562	−0.289781	−	0.957093i	$-0.593582\pi$
$491$	−12.8422	−0.579562	−0.289781	+	0.957093i	$0.593582\pi$
$492$	0	0
$493$	−2.90981	−0.131051
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.32608	0.194051
$498$	0	0
$499$	−14.7439	−0.660028	−0.330014	−	0.943976i	$-0.607053\pi$
$499$	−14.7439	−0.660028	−0.330014	+	0.943976i	$0.607053\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.4283	−0.464977	−0.232489	−	0.972599i	$-0.574687\pi$
$503$	−10.4283	−0.464977	−0.232489	+	0.972599i	$0.574687\pi$
$504$	0	0
$505$	−6.42835	−0.286058
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.0636	0.446063	0.223031	−	0.974811i	$-0.428405\pi$
$509$	10.0636	0.446063	0.223031	+	0.974811i	$0.428405\pi$
$510$	0	0
$511$	−6.44282	−0.285014
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15.7665	−0.694755
$516$	0	0
$517$	−5.22536	−0.229811
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.7294	1.08342	0.541708	−	0.840567i	$-0.317778\pi$
$521$	24.7294	1.08342	0.541708	+	0.840567i	$0.317778\pi$
$522$	0	0
$523$	−25.8196	−1.12901	−0.564506	−	0.825429i	$-0.690933\pi$
$523$	−25.8196	−1.12901	−0.564506	+	0.825429i	$0.690933\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.09019	−0.0474895
$528$	0	0
$529$	−22.7029	−0.987081
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.70287	−0.247019
$534$	0	0
$535$	10.9734	0.474423
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−23.4057	−1.00816
$540$	0	0
$541$	−18.9774	−0.815902	−0.407951	−	0.913004i	$-0.633757\pi$
$541$	−18.9774	−0.815902	−0.407951	+	0.913004i	$0.633757\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−15.1047	−0.647013
$546$	0	0
$547$	−9.94689	−0.425298	−0.212649	−	0.977129i	$-0.568209\pi$
$547$	−9.94689	−0.425298	−0.212649	+	0.977129i	$0.568209\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	39.1722	1.66879
$552$	0	0
$553$	17.0266	0.724043
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−37.1722	−1.57504	−0.787519	−	0.616290i	$-0.788635\pi$
$557$	−37.1722	−1.57504	−0.787519	+	0.616290i	$0.788635\pi$
$558$	0	0
$559$	2.90981	0.123072
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18.0105	−0.759053	−0.379527	−	0.925181i	$-0.623913\pi$
$563$	−18.0105	−0.759053	−0.379527	+	0.925181i	$0.623913\pi$
$564$	0	0
$565$	12.0145	0.505453
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.9098	0.792740	0.396370	−	0.918091i	$-0.370270\pi$
$569$	18.9098	0.792740	0.396370	+	0.918091i	$0.370270\pi$
$570$	0	0
$571$	31.7318	1.32794	0.663968	−	0.747761i	$-0.268871\pi$
$571$	31.7318	1.32794	0.663968	+	0.747761i	$0.268871\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.545096	−0.0227321
$576$	0	0
$577$	2.80754	0.116879	0.0584396	−	0.998291i	$-0.481387\pi$
$577$	2.80754	0.116879	0.0584396	+	0.998291i	$0.481387\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−22.9734	−0.951463
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	0	0
$589$	14.6763	0.604727
$590$	0	0
$591$	0	0
$592$	0	0
$593$	38.3897	1.57648	0.788238	−	0.615370i	$-0.210993\pi$
$593$	38.3897	1.57648	0.788238	+	0.615370i	$0.210993\pi$
$594$	0	0
$595$	0.793062	0.0325124
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.9469	0.569855	0.284927	−	0.958549i	$-0.408031\pi$
$599$	13.9469	0.569855	0.284927	+	0.958549i	$0.408031\pi$
$600$	0	0
$601$	−38.6947	−1.57839	−0.789196	−	0.614142i	$-0.789502\pi$
$601$	−38.6947	−1.57839	−0.789196	+	0.614142i	$0.789502\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.9734	−0.486790
$606$	0	0
$607$	−34.6763	−1.40747	−0.703734	−	0.710463i	$-0.748485\pi$
$607$	−34.6763	−1.40747	−0.703734	+	0.710463i	$0.748485\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.09019	−0.0441044
$612$	0	0
$613$	−19.2890	−0.779075	−0.389538	−	0.921011i	$-0.627365\pi$
$613$	−19.2890	−0.779075	−0.389538	+	0.921011i	$0.627365\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.35263	0.134972	0.0674860	−	0.997720i	$-0.478502\pi$
$617$	3.35263	0.134972	0.0674860	+	0.997720i	$0.478502\pi$
$618$	0	0
$619$	15.8341	0.636426	0.318213	−	0.948019i	$-0.396917\pi$
$619$	15.8341	0.636426	0.318213	+	0.948019i	$0.396917\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.1167	0.565575
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.11675	0.0844002
$630$	0	0
$631$	13.0371	0.518998	0.259499	−	0.965743i	$-0.416443\pi$
$631$	13.0371	0.518998	0.259499	+	0.965743i	$0.416443\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.49593	0.337151
$636$	0	0
$637$	−4.88325	−0.193482
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27.1722	−1.07324	−0.536620	−	0.843824i	$-0.680299\pi$
$641$	−27.1722	−1.07324	−0.536620	+	0.843824i	$0.680299\pi$
$642$	0	0
$643$	7.88325	0.310885	0.155443	−	0.987845i	$-0.450320\pi$
$643$	7.88325	0.310885	0.155443	+	0.987845i	$0.450320\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.9138	−0.507692	−0.253846	−	0.967245i	$-0.581696\pi$
$647$	−12.9138	−0.507692	−0.253846	+	0.967245i	$0.581696\pi$
$648$	0	0
$649$	−51.1722	−2.00869
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.90981	−0.113870	−0.0569348	−	0.998378i	$-0.518133\pi$
$653$	−2.90981	−0.113870	−0.0569348	+	0.998378i	$0.518133\pi$
$654$	0	0
$655$	10.4283	0.407469
$656$	0	0
$657$	0	0
$658$	0	0
$659$	36.1496	1.40819	0.704095	−	0.710106i	$-0.251353\pi$
$659$	36.1496	1.40819	0.704095	+	0.710106i	$0.251353\pi$
$660$	0	0
$661$	−32.0965	−1.24841	−0.624205	−	0.781260i	$-0.714577\pi$
$661$	−32.0965	−1.24841	−0.624205	+	0.781260i	$0.714577\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−10.6763	−0.414010
$666$	0	0
$667$	−2.90981	−0.112668
$668$	0	0
$669$	0	0
$670$	0	0
$671$	37.7849	1.45867
$672$	0	0
$673$	33.6682	1.29781	0.648907	−	0.760868i	$-0.275227\pi$
$673$	33.6682	1.29781	0.648907	+	0.760868i	$0.275227\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.1086	−1.04187	−0.520934	−	0.853597i	$-0.674416\pi$
$677$	−27.1086	−1.04187	−0.520934	+	0.853597i	$0.674416\pi$
$678$	0	0
$679$	−10.2440	−0.393129
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.81962	−0.222681	−0.111341	−	0.993782i	$-0.535514\pi$
$683$	−5.81962	−0.222681	−0.111341	+	0.993782i	$0.535514\pi$
$684$	0	0
$685$	−4.90981	−0.187594
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.79306	−0.182601
$690$	0	0
$691$	−40.8260	−1.55309	−0.776546	−	0.630060i	$-0.783030\pi$
$691$	−40.8260	−1.55309	−0.776546	+	0.630060i	$0.783030\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.79306	−0.333540
$696$	0	0
$697$	3.10861	0.117747
$698$	0	0
$699$	0	0
$700$	0	0
$701$	43.4202	1.63996	0.819979	−	0.572393i	$-0.193985\pi$
$701$	43.4202	1.63996	0.819979	+	0.572393i	$0.193985\pi$
$702$	0	0
$703$	−28.4959	−1.07474
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.35263	0.351742
$708$	0	0
$709$	−2.48146	−0.0931931	−0.0465966	−	0.998914i	$-0.514838\pi$
$709$	−2.48146	−0.0931931	−0.0465966	+	0.998914i	$0.514838\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.09019	−0.0408280
$714$	0	0
$715$	−4.79306	−0.179250
$716$	0	0
$717$	0	0
$718$	0	0
$719$	31.6682	1.18102	0.590512	−	0.807029i	$-0.298926\pi$
$719$	31.6682	1.18102	0.590512	+	0.807029i	$0.298926\pi$
$720$	0	0
$721$	22.9388	0.854284
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.33816	0.198254
$726$	0	0
$727$	−51.7955	−1.92099	−0.960494	−	0.278302i	$-0.910228\pi$
$727$	−51.7955	−1.92099	−0.960494	+	0.278302i	$0.910228\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.58612	−0.0586649
$732$	0	0
$733$	−5.70287	−0.210640	−0.105320	−	0.994438i	$-0.533587\pi$
$733$	−5.70287	−0.210640	−0.105320	+	0.994438i	$0.533587\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.8115	0.545588
$738$	0	0
$739$	2.60873	0.0959638	0.0479819	−	0.998848i	$-0.484721\pi$
$739$	2.60873	0.0959638	0.0479819	+	0.998848i	$0.484721\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.1352	1.32567	0.662835	−	0.748765i	$-0.269353\pi$
$743$	36.1352	1.32567	0.662835	+	0.748765i	$0.269353\pi$
$744$	0	0
$745$	0.973446	0.0356643
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−15.9653	−0.583360
$750$	0	0
$751$	43.4983	1.58728	0.793638	−	0.608390i	$-0.208184\pi$
$751$	43.4983	1.58728	0.793638	+	0.608390i	$0.208184\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.76651	0.209865
$756$	0	0
$757$	−10.8278	−0.393541	−0.196771	−	0.980450i	$-0.563045\pi$
$757$	−10.8278	−0.393541	−0.196771	+	0.980450i	$0.563045\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.4959	0.525477	0.262739	−	0.964867i	$-0.415374\pi$
$761$	14.4959	0.525477	0.262739	+	0.964867i	$0.415374\pi$
$762$	0	0
$763$	21.9758	0.795579
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.6763	−0.385499
$768$	0	0
$769$	40.3445	1.45486	0.727430	−	0.686182i	$-0.240715\pi$
$769$	40.3445	1.45486	0.727430	+	0.686182i	$0.240715\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4.05311	0.145780	0.0728901	−	0.997340i	$-0.476778\pi$
$773$	4.05311	0.145780	0.0728901	+	0.997340i	$0.476778\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−41.8486	−1.49938
$780$	0	0
$781$	14.2519	0.509973
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8.90981	−0.318005
$786$	0	0
$787$	33.7665	1.20365	0.601823	−	0.798629i	$-0.294441\pi$
$787$	33.7665	1.20365	0.601823	+	0.798629i	$0.294441\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−17.4799	−0.621514
$792$	0	0
$793$	7.88325	0.279943
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.1457	0.642753	0.321377	−	0.946951i	$-0.395854\pi$
$797$	18.1457	0.642753	0.321377	+	0.946951i	$0.395854\pi$
$798$	0	0
$799$	0.594259	0.0210234
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−21.2254	−0.749027
$804$	0	0
$805$	0.793062	0.0279518
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−40.1641	−1.41210	−0.706048	−	0.708164i	$-0.749524\pi$
$809$	−40.1641	−1.41210	−0.706048	+	0.708164i	$0.749524\pi$
$810$	0	0
$811$	−47.6006	−1.67148	−0.835742	−	0.549123i	$-0.814962\pi$
$811$	−47.6006	−1.67148	−0.835742	+	0.549123i	$0.814962\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.79306	0.0978367
$816$	0	0
$817$	21.3526	0.747034
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.5065	0.994882	0.497441	−	0.867498i	$-0.334273\pi$
$821$	28.5065	0.994882	0.497441	+	0.867498i	$0.334273\pi$
$822$	0	0
$823$	−12.9919	−0.452868	−0.226434	−	0.974027i	$-0.572707\pi$
$823$	−12.9919	−0.452868	−0.226434	+	0.974027i	$0.572707\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22.4139	−0.779407	−0.389703	−	0.920940i	$-0.627422\pi$
$827$	−22.4139	−0.779407	−0.389703	+	0.920940i	$0.627422\pi$
$828$	0	0
$829$	−41.5330	−1.44250	−0.721251	−	0.692674i	$-0.756432\pi$
$829$	−41.5330	−1.44250	−0.721251	+	0.692674i	$0.756432\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.66184	0.0922274
$834$	0	0
$835$	−9.58612	−0.331742
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38.9734	1.34551	0.672756	−	0.739864i	$-0.265110\pi$
$839$	38.9734	1.34551	0.672756	+	0.739864i	$0.265110\pi$
$840$	0	0
$841$	−0.504067	−0.0173816
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	17.4202	0.598566
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.11675	0.0725611
$852$	0	0
$853$	−18.2971	−0.626482	−0.313241	−	0.949674i	$-0.601415\pi$
$853$	−18.2971	−0.626482	−0.313241	+	0.949674i	$0.601415\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−49.7994	−1.70112	−0.850558	−	0.525882i	$-0.823735\pi$
$857$	−49.7994	−1.70112	−0.850558	+	0.525882i	$0.823735\pi$
$858$	0	0
$859$	−21.4163	−0.730714	−0.365357	−	0.930868i	$-0.619053\pi$
$859$	−21.4163	−0.730714	−0.365357	+	0.930868i	$0.619053\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.77464	0.0944499	0.0472250	−	0.998884i	$-0.484962\pi$
$863$	2.77464	0.0944499	0.0472250	+	0.998884i	$0.484962\pi$
$864$	0	0
$865$	−15.7665	−0.536077
$866$	0	0
$867$	0	0
$868$	0	0
$869$	56.0926	1.90281
$870$	0	0
$871$	3.09019	0.104707
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.45490	−0.0491847
$876$	0	0
$877$	−32.7053	−1.10438	−0.552189	−	0.833719i	$-0.686207\pi$
$877$	−32.7053	−1.10438	−0.552189	+	0.833719i	$0.686207\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−10.4428	−0.351828	−0.175914	−	0.984406i	$-0.556288\pi$
$881$	−10.4428	−0.351828	−0.175914	+	0.984406i	$0.556288\pi$
$882$	0	0
$883$	−55.2995	−1.86098	−0.930489	−	0.366321i	$-0.880617\pi$
$883$	−55.2995	−1.86098	−0.930489	+	0.366321i	$0.880617\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29.3035	−0.983914	−0.491957	−	0.870619i	$-0.663718\pi$
$887$	−29.3035	−0.983914	−0.491957	+	0.870619i	$0.663718\pi$
$888$	0	0
$889$	−12.3608	−0.414567
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	−14.1949	−0.474482
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.6763	0.356075
$900$	0	0
$901$	2.61268	0.0870409
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.52249	−0.250056
$906$	0	0
$907$	−18.0821	−0.600405	−0.300202	−	0.953876i	$-0.597054\pi$
$907$	−18.0821	−0.600405	−0.300202	+	0.953876i	$0.597054\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19.5041	0.646199	0.323099	−	0.946365i	$-0.395275\pi$
$911$	19.5041	0.646199	0.323099	+	0.946365i	$0.395275\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.1722	−0.501032
$918$	0	0
$919$	−11.4404	−0.377385	−0.188692	−	0.982036i	$-0.560425\pi$
$919$	−11.4404	−0.377385	−0.188692	+	0.982036i	$0.560425\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.97345	0.0978722
$924$	0	0
$925$	−3.88325	−0.127681
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−23.6498	−0.775924	−0.387962	−	0.921675i	$-0.626821\pi$
$929$	−23.6498	−0.775924	−0.387962	+	0.921675i	$0.626821\pi$
$930$	0	0
$931$	−35.8341	−1.17441
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.61268	0.0854437
$936$	0	0
$937$	−12.5490	−0.409959	−0.204980	−	0.978766i	$-0.565713\pi$
$937$	−12.5490	−0.409959	−0.204980	+	0.978766i	$0.565713\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.1909	−0.593007	−0.296503	−	0.955032i	$-0.595821\pi$
$941$	−18.1909	−0.593007	−0.296503	+	0.955032i	$0.595821\pi$
$942$	0	0
$943$	3.10861	0.101230
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.7665	−0.642325	−0.321163	−	0.947024i	$-0.604074\pi$
$947$	−19.7665	−0.642325	−0.321163	+	0.947024i	$0.604074\pi$
$948$	0	0
$949$	−4.42835	−0.143750
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15.0941	0.488947	0.244474	−	0.969656i	$-0.421385\pi$
$953$	15.0941	0.488947	0.244474	+	0.969656i	$0.421385\pi$
$954$	0	0
$955$	13.5861	0.439637
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.14330	0.230669
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19.4018	−0.624566
$966$	0	0
$967$	−21.0516	−0.676972	−0.338486	−	0.940971i	$-0.609915\pi$
$967$	−21.0516	−0.676972	−0.338486	+	0.940971i	$0.609915\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.5717	−0.563901	−0.281951	−	0.959429i	$-0.590981\pi$
$971$	−17.5717	−0.563901	−0.281951	+	0.959429i	$0.590981\pi$
$972$	0	0
$973$	12.7931	0.410127
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−33.1433	−1.06035	−0.530174	−	0.847889i	$-0.677873\pi$
$977$	−33.1433	−1.06035	−0.530174	+	0.847889i	$0.677873\pi$
$978$	0	0
$979$	46.5065	1.48635
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−6.31555	−0.201435	−0.100717	−	0.994915i	$-0.532114\pi$
$983$	−6.31555	−0.201435	−0.100717	+	0.994915i	$0.532114\pi$
$984$	0	0
$985$	11.5861	0.369165
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.58612	−0.0504358
$990$	0	0
$991$	−47.2359	−1.50050	−0.750249	−	0.661156i	$-0.770066\pi$
$991$	−47.2359	−1.50050	−0.750249	+	0.661156i	$0.770066\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.32368	−0.0419636
$996$	0	0
$997$	−53.6682	−1.69969	−0.849844	−	0.527034i	$-0.823304\pi$
$997$	−53.6682	−1.69969	−0.849844	+	0.527034i	$0.823304\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.bi.1.2	✓	3
3.2	odd	2		4680.2.a.bk.1.2	yes	3
4.3	odd	2		9360.2.a.cx.1.2		3
12.11	even	2		9360.2.a.dc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4680.2.a.bi.1.2	✓	3	1.1	even	1	trivial
4680.2.a.bk.1.2	yes	3	3.2	odd	2
9360.2.a.cx.1.2		3	4.3	odd	2
9360.2.a.dc.1.2		3	12.11	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +1.45490 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +1.45490 q^{7} +4.79306 q^{11} +1.00000 q^{13} -0.545096 q^{17} +7.33816 q^{19} -0.545096 q^{23} +1.00000 q^{25} +5.33816 q^{29} +2.00000 q^{31} -1.45490 q^{35} -3.88325 q^{37} -5.70287 q^{41} +2.90981 q^{43} -1.09019 q^{47} -4.88325 q^{49} -4.79306 q^{53} -4.79306 q^{55} -10.6763 q^{59} +7.88325 q^{61} -1.00000 q^{65} +3.09019 q^{67} +2.97345 q^{71} -4.42835 q^{73} +6.97345 q^{77} +11.7029 q^{79} +0.545096 q^{85} +9.70287 q^{89} +1.45490 q^{91} -7.33816 q^{95} -7.04103 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 3 * q^7	\( 3 q - 3 q^{5} + 3 q^{7} - 3 q^{11} + 3 q^{13} - 3 q^{17} + 6 q^{19} - 3 q^{23} + 3 q^{25} + 6 q^{31} - 3 q^{35} + 3 q^{37} + 3 q^{41} + 6 q^{43} - 6 q^{47} + 3 q^{53} + 3 q^{55} + 9 q^{61} - 3 q^{65} + 12 q^{67} - 3 q^{71} + 9 q^{77} + 15 q^{79} + 3 q^{85} + 9 q^{89} + 3 q^{91} - 6 q^{95} + 15 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 - 3 * q^11 + 3 * q^13 - 3 * q^17 + 6 * q^19 - 3 * q^23 + 3 * q^25 + 6 * q^31 - 3 * q^35 + 3 * q^37 + 3 * q^41 + 6 * q^43 - 6 * q^47 + 3 * q^53 + 3 * q^55 + 9 * q^61 - 3 * q^65 + 12 * q^67 - 3 * q^71 + 9 * q^77 + 15 * q^79 + 3 * q^85 + 9 * q^89 + 3 * q^91 - 6 * q^95 + 15 * q^97

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