LMFDB - Embedded newform 1287.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1287.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.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:	$10.2767467401$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	1287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.311108 q^{2} -1.90321 q^{4} -2.21432 q^{5} +1.52543 q^{7} +1.21432 q^{8} +O(q^{10})$	\(q-0.311108 q^{2} -1.90321 q^{4} -2.21432 q^{5} +1.52543 q^{7} +1.21432 q^{8} +0.688892 q^{10} -1.00000 q^{11} +1.00000 q^{13} -0.474572 q^{14} +3.42864 q^{16} +1.73975 q^{17} +4.28100 q^{19} +4.21432 q^{20} +0.311108 q^{22} -0.474572 q^{23} -0.0967881 q^{25} -0.311108 q^{26} -2.90321 q^{28} -5.11753 q^{29} -4.21432 q^{31} -3.49532 q^{32} -0.541249 q^{34} -3.37778 q^{35} +0.428639 q^{37} -1.33185 q^{38} -2.68889 q^{40} -6.28100 q^{41} -8.16839 q^{43} +1.90321 q^{44} +0.147643 q^{46} -5.05086 q^{47} -4.67307 q^{49} +0.0301115 q^{50} -1.90321 q^{52} +14.0415 q^{53} +2.21432 q^{55} +1.85236 q^{56} +1.59210 q^{58} +1.67307 q^{59} -5.80642 q^{61} +1.31111 q^{62} -5.76986 q^{64} -2.21432 q^{65} -6.02074 q^{67} -3.31111 q^{68} +1.05086 q^{70} -5.18421 q^{71} -8.08742 q^{73} -0.133353 q^{74} -8.14764 q^{76} -1.52543 q^{77} +0.260253 q^{79} -7.59210 q^{80} +1.95407 q^{82} +1.37778 q^{83} -3.85236 q^{85} +2.54125 q^{86} -1.21432 q^{88} +10.1541 q^{89} +1.52543 q^{91} +0.903212 q^{92} +1.57136 q^{94} -9.47949 q^{95} +0.428639 q^{97} +1.45383 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + q^4 - 2 * q^7 - 3 * q^8	$ 3 q - q^{2} + q^{4} - 2 q^{7} - 3 q^{8} + 2 q^{10} - 3 q^{11} + 3 q^{13} - 8 q^{14} - 3 q^{16} - 8 q^{17} + 6 q^{19} + 6 q^{20} + q^{22} - 8 q^{23} - 7 q^{25} - q^{26} - 2 q^{28} - 2 q^{29} - 6 q^{31} + 3 q^{32} - 8 q^{34} - 10 q^{35} - 12 q^{37} + 16 q^{38} - 8 q^{40} - 12 q^{41} + 2 q^{43} - q^{44} - 6 q^{46} - 2 q^{47} - q^{49} + 7 q^{50} + q^{52} + 2 q^{53} + 12 q^{56} - 2 q^{58} - 8 q^{59} - 4 q^{61} + 4 q^{62} - 11 q^{64} + 2 q^{67} - 10 q^{68} - 10 q^{70} - 2 q^{71} - 4 q^{73} - 18 q^{76} + 2 q^{77} + 14 q^{79} - 16 q^{80} - 14 q^{82} + 4 q^{83} - 18 q^{85} + 14 q^{86} + 3 q^{88} + 10 q^{89} - 2 q^{91} - 4 q^{92} + 18 q^{94} - 2 q^{95} - 12 q^{97} + 31 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 - 2 * q^7 - 3 * q^8 + 2 * q^10 - 3 * q^11 + 3 * q^13 - 8 * q^14 - 3 * q^16 - 8 * q^17 + 6 * q^19 + 6 * q^20 + q^22 - 8 * q^23 - 7 * q^25 - q^26 - 2 * q^28 - 2 * q^29 - 6 * q^31 + 3 * q^32 - 8 * q^34 - 10 * q^35 - 12 * q^37 + 16 * q^38 - 8 * q^40 - 12 * q^41 + 2 * q^43 - q^44 - 6 * q^46 - 2 * q^47 - q^49 + 7 * q^50 + q^52 + 2 * q^53 + 12 * q^56 - 2 * q^58 - 8 * q^59 - 4 * q^61 + 4 * q^62 - 11 * q^64 + 2 * q^67 - 10 * q^68 - 10 * q^70 - 2 * q^71 - 4 * q^73 - 18 * q^76 + 2 * q^77 + 14 * q^79 - 16 * q^80 - 14 * q^82 + 4 * q^83 - 18 * q^85 + 14 * q^86 + 3 * q^88 + 10 * q^89 - 2 * q^91 - 4 * q^92 + 18 * q^94 - 2 * q^95 - 12 * q^97 + 31 * q^98

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 $

$2$	−0.311108	−0.219986	−0.109993	−	0.993932i	$-0.535083\pi$
$2$	−0.311108	−0.219986	−0.109993	+	0.993932i	$0.535083\pi$
$3$	0	0
$3$	0	0
$4$	−1.90321	−0.951606
$5$	−2.21432	−0.990274	−0.495137	−	0.868815i	$-0.664882\pi$
$5$	−2.21432	−0.990274	−0.495137	+	0.868815i	$0.664882\pi$
$6$	0	0
$7$	1.52543	0.576557	0.288279	−	0.957547i	$-0.406917\pi$
$7$	1.52543	0.576557	0.288279	+	0.957547i	$0.406917\pi$
$8$	1.21432	0.429327
$9$	0	0
$10$	0.688892	0.217847
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.474572	−0.126835
$15$	0	0
$16$	3.42864	0.857160
$17$	1.73975	0.421951	0.210975	−	0.977491i	$-0.432336\pi$
$17$	1.73975	0.421951	0.210975	+	0.977491i	$0.432336\pi$
$18$	0	0
$19$	4.28100	0.982128	0.491064	−	0.871124i	$-0.336608\pi$
$19$	4.28100	0.982128	0.491064	+	0.871124i	$0.336608\pi$
$20$	4.21432	0.942351
$21$	0	0
$22$	0.311108	0.0663284
$23$	−0.474572	−0.0989552	−0.0494776	−	0.998775i	$-0.515756\pi$
$23$	−0.474572	−0.0989552	−0.0494776	+	0.998775i	$0.515756\pi$
$24$	0	0
$25$	−0.0967881	−0.0193576
$26$	−0.311108	−0.0610133
$27$	0	0
$28$	−2.90321	−0.548655
$29$	−5.11753	−0.950302	−0.475151	−	0.879904i	$-0.657607\pi$
$29$	−5.11753	−0.950302	−0.475151	+	0.879904i	$0.657607\pi$
$30$	0	0
$31$	−4.21432	−0.756914	−0.378457	−	0.925619i	$-0.623545\pi$
$31$	−4.21432	−0.756914	−0.378457	+	0.925619i	$0.623545\pi$
$32$	−3.49532	−0.617890
$33$	0	0
$34$	−0.541249	−0.0928234
$35$	−3.37778	−0.570950
$36$	0	0
$37$	0.428639	0.0704679	0.0352339	−	0.999379i	$-0.488782\pi$
$37$	0.428639	0.0704679	0.0352339	+	0.999379i	$0.488782\pi$
$38$	−1.33185	−0.216055
$39$	0	0
$40$	−2.68889	−0.425151
$41$	−6.28100	−0.980927	−0.490463	−	0.871462i	$-0.663172\pi$
$41$	−6.28100	−0.980927	−0.490463	+	0.871462i	$0.663172\pi$
$42$	0	0
$43$	−8.16839	−1.24567	−0.622834	−	0.782354i	$-0.714019\pi$
$43$	−8.16839	−1.24567	−0.622834	+	0.782354i	$0.714019\pi$
$44$	1.90321	0.286920
$45$	0	0
$46$	0.147643	0.0217688
$47$	−5.05086	−0.736743	−0.368371	−	0.929679i	$-0.620085\pi$
$47$	−5.05086	−0.736743	−0.368371	+	0.929679i	$0.620085\pi$
$48$	0	0
$49$	−4.67307	−0.667582
$50$	0.0301115	0.00425841
$51$	0	0
$52$	−1.90321	−0.263928
$53$	14.0415	1.92875	0.964373	−	0.264545i	$-0.0852218\pi$
$53$	14.0415	1.92875	0.964373	+	0.264545i	$0.0852218\pi$
$54$	0	0
$55$	2.21432	0.298579
$56$	1.85236	0.247532
$57$	0	0
$58$	1.59210	0.209054
$59$	1.67307	0.217815	0.108908	−	0.994052i	$-0.465265\pi$
$59$	1.67307	0.217815	0.108908	+	0.994052i	$0.465265\pi$
$60$	0	0
$61$	−5.80642	−0.743436	−0.371718	−	0.928346i	$-0.621231\pi$
$61$	−5.80642	−0.743436	−0.371718	+	0.928346i	$0.621231\pi$
$62$	1.31111	0.166511
$63$	0	0
$64$	−5.76986	−0.721232
$65$	−2.21432	−0.274653
$66$	0	0
$67$	−6.02074	−0.735551	−0.367775	−	0.929915i	$-0.619880\pi$
$67$	−6.02074	−0.735551	−0.367775	+	0.929915i	$0.619880\pi$
$68$	−3.31111	−0.401531
$69$	0	0
$70$	1.05086	0.125601
$71$	−5.18421	−0.615252	−0.307626	−	0.951507i	$-0.599535\pi$
$71$	−5.18421	−0.615252	−0.307626	+	0.951507i	$0.599535\pi$
$72$	0	0
$73$	−8.08742	−0.946561	−0.473280	−	0.880912i	$-0.656930\pi$
$73$	−8.08742	−0.946561	−0.473280	+	0.880912i	$0.656930\pi$
$74$	−0.133353	−0.0155020
$75$	0	0
$76$	−8.14764	−0.934599
$77$	−1.52543	−0.173839
$78$	0	0
$79$	0.260253	0.0292807	0.0146404	−	0.999893i	$-0.495340\pi$
$79$	0.260253	0.0292807	0.0146404	+	0.999893i	$0.495340\pi$
$80$	−7.59210	−0.848823
$81$	0	0
$82$	1.95407	0.215791
$83$	1.37778	0.151231	0.0756157	−	0.997137i	$-0.475908\pi$
$83$	1.37778	0.151231	0.0756157	+	0.997137i	$0.475908\pi$
$84$	0	0
$85$	−3.85236	−0.417847
$86$	2.54125	0.274030
$87$	0	0
$88$	−1.21432	−0.129447
$89$	10.1541	1.07633	0.538166	−	0.842839i	$-0.319117\pi$
$89$	10.1541	1.07633	0.538166	+	0.842839i	$0.319117\pi$
$90$	0	0
$91$	1.52543	0.159908
$92$	0.903212	0.0941664
$93$	0	0
$94$	1.57136	0.162073
$95$	−9.47949	−0.972576
$96$	0	0
$97$	0.428639	0.0435217	0.0217609	−	0.999763i	$-0.493073\pi$
$97$	0.428639	0.0435217	0.0217609	+	0.999763i	$0.493073\pi$
$98$	1.45383	0.146859
$99$	0	0
$100$	0.184208	0.0184208
$101$	−3.93332	−0.391380	−0.195690	−	0.980666i	$-0.562695\pi$
$101$	−3.93332	−0.391380	−0.195690	+	0.980666i	$0.562695\pi$
$102$	0	0
$103$	−13.8064	−1.36039	−0.680194	−	0.733032i	$-0.738104\pi$
$103$	−13.8064	−1.36039	−0.680194	+	0.733032i	$0.738104\pi$
$104$	1.21432	0.119074
$105$	0	0
$106$	−4.36842	−0.424298
$107$	−9.67307	−0.935131	−0.467566	−	0.883958i	$-0.654869\pi$
$107$	−9.67307	−0.935131	−0.467566	+	0.883958i	$0.654869\pi$
$108$	0	0
$109$	−17.6271	−1.68837	−0.844187	−	0.536049i	$-0.819916\pi$
$109$	−17.6271	−1.68837	−0.844187	+	0.536049i	$0.819916\pi$
$110$	−0.688892	−0.0656833
$111$	0	0
$112$	5.23014	0.494202
$113$	−13.4795	−1.26804	−0.634022	−	0.773315i	$-0.718597\pi$
$113$	−13.4795	−1.26804	−0.634022	+	0.773315i	$0.718597\pi$
$114$	0	0
$115$	1.05086	0.0979927
$116$	9.73975	0.904313
$117$	0	0
$118$	−0.520505	−0.0479164
$119$	2.65386	0.243279
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.80642	0.163546
$123$	0	0
$124$	8.02074	0.720284
$125$	11.2859	1.00944
$126$	0	0
$127$	20.2415	1.79614	0.898072	−	0.439848i	$-0.144968\pi$
$127$	20.2415	1.79614	0.898072	+	0.439848i	$0.144968\pi$
$128$	8.78568	0.776552
$129$	0	0
$130$	0.688892	0.0604198
$131$	−8.29529	−0.724763	−0.362381	−	0.932030i	$-0.618036\pi$
$131$	−8.29529	−0.724763	−0.362381	+	0.932030i	$0.618036\pi$
$132$	0	0
$133$	6.53035	0.566253
$134$	1.87310	0.161811
$135$	0	0
$136$	2.11261	0.181155
$137$	14.7447	1.25972	0.629861	−	0.776708i	$-0.283112\pi$
$137$	14.7447	1.25972	0.629861	+	0.776708i	$0.283112\pi$
$138$	0	0
$139$	−9.05731	−0.768231	−0.384115	−	0.923285i	$-0.625494\pi$
$139$	−9.05731	−0.768231	−0.384115	+	0.923285i	$0.625494\pi$
$140$	6.42864	0.543319
$141$	0	0
$142$	1.61285	0.135347
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	11.3319	0.941059
$146$	2.51606	0.208231
$147$	0	0
$148$	−0.815792	−0.0670577
$149$	2.01429	0.165017	0.0825085	−	0.996590i	$-0.473707\pi$
$149$	2.01429	0.165017	0.0825085	+	0.996590i	$0.473707\pi$
$150$	0	0
$151$	0.668149	0.0543732	0.0271866	−	0.999630i	$-0.491345\pi$
$151$	0.668149	0.0543732	0.0271866	+	0.999630i	$0.491345\pi$
$152$	5.19850	0.421654
$153$	0	0
$154$	0.474572	0.0382421
$155$	9.33185	0.749552
$156$	0	0
$157$	4.70964	0.375870	0.187935	−	0.982181i	$-0.439821\pi$
$157$	4.70964	0.375870	0.187935	+	0.982181i	$0.439821\pi$
$158$	−0.0809666	−0.00644136
$159$	0	0
$160$	7.73975	0.611881
$161$	−0.723926	−0.0570534
$162$	0	0
$163$	−2.73483	−0.214208	−0.107104	−	0.994248i	$-0.534158\pi$
$163$	−2.73483	−0.214208	−0.107104	+	0.994248i	$0.534158\pi$
$164$	11.9541	0.933456
$165$	0	0
$166$	−0.428639	−0.0332689
$167$	0.561993	0.0434883	0.0217441	−	0.999764i	$-0.493078\pi$
$167$	0.561993	0.0434883	0.0217441	+	0.999764i	$0.493078\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	1.19850	0.0919206
$171$	0	0
$172$	15.5462	1.18538
$173$	−8.88247	−0.675322	−0.337661	−	0.941268i	$-0.609636\pi$
$173$	−8.88247	−0.675322	−0.337661	+	0.941268i	$0.609636\pi$
$174$	0	0
$175$	−0.147643	−0.0111608
$176$	−3.42864	−0.258443
$177$	0	0
$178$	−3.15902	−0.236778
$179$	4.51606	0.337546	0.168773	−	0.985655i	$-0.446019\pi$
$179$	4.51606	0.337546	0.168773	+	0.985655i	$0.446019\pi$
$180$	0	0
$181$	−24.7096	−1.83665	−0.918326	−	0.395824i	$-0.870459\pi$
$181$	−24.7096	−1.83665	−0.918326	+	0.395824i	$0.870459\pi$
$182$	−0.474572	−0.0351776
$183$	0	0
$184$	−0.576283	−0.0424841
$185$	−0.949145	−0.0697825
$186$	0	0
$187$	−1.73975	−0.127223
$188$	9.61285	0.701089
$189$	0	0
$190$	2.94914	0.213953
$191$	−12.9906	−0.939969	−0.469985	−	0.882675i	$-0.655741\pi$
$191$	−12.9906	−0.939969	−0.469985	+	0.882675i	$0.655741\pi$
$192$	0	0
$193$	15.0464	1.08306	0.541532	−	0.840680i	$-0.317844\pi$
$193$	15.0464	1.08306	0.541532	+	0.840680i	$0.317844\pi$
$194$	−0.133353	−0.00957419
$195$	0	0
$196$	8.89384	0.635275
$197$	−15.8622	−1.13014	−0.565068	−	0.825045i	$-0.691150\pi$
$197$	−15.8622	−1.13014	−0.565068	+	0.825045i	$0.691150\pi$
$198$	0	0
$199$	13.4193	0.951267	0.475633	−	0.879644i	$-0.342219\pi$
$199$	13.4193	0.951267	0.475633	+	0.879644i	$0.342219\pi$
$200$	−0.117532	−0.00831074
$201$	0	0
$202$	1.22369	0.0860984
$203$	−7.80642	−0.547904
$204$	0	0
$205$	13.9081	0.971386
$206$	4.29529	0.299267
$207$	0	0
$208$	3.42864	0.237733
$209$	−4.28100	−0.296123
$210$	0	0
$211$	24.1684	1.66382	0.831910	−	0.554910i	$-0.187247\pi$
$211$	24.1684	1.66382	0.831910	+	0.554910i	$0.187247\pi$
$212$	−26.7239	−1.83541
$213$	0	0
$214$	3.00937	0.205716
$215$	18.0874	1.23355
$216$	0	0
$217$	−6.42864	−0.436404
$218$	5.48394	0.371419
$219$	0	0
$220$	−4.21432	−0.284129
$221$	1.73975	0.117028
$222$	0	0
$223$	−9.50024	−0.636183	−0.318092	−	0.948060i	$-0.603042\pi$
$223$	−9.50024	−0.636183	−0.318092	+	0.948060i	$0.603042\pi$
$224$	−5.33185	−0.356249
$225$	0	0
$226$	4.19358	0.278953
$227$	−8.41435	−0.558480	−0.279240	−	0.960221i	$-0.590083\pi$
$227$	−8.41435	−0.558480	−0.279240	+	0.960221i	$0.590083\pi$
$228$	0	0
$229$	29.0321	1.91850	0.959248	−	0.282565i	$-0.0911852\pi$
$229$	29.0321	1.91850	0.959248	+	0.282565i	$0.0911852\pi$
$230$	−0.326929	−0.0215571
$231$	0	0
$232$	−6.21432	−0.407990
$233$	−12.3017	−0.805914	−0.402957	−	0.915219i	$-0.632018\pi$
$233$	−12.3017	−0.805914	−0.402957	+	0.915219i	$0.632018\pi$
$234$	0	0
$235$	11.1842	0.729577
$236$	−3.18421	−0.207274
$237$	0	0
$238$	−0.825636	−0.0535180
$239$	−21.5669	−1.39505	−0.697524	−	0.716562i	$-0.745715\pi$
$239$	−21.5669	−1.39505	−0.697524	+	0.716562i	$0.745715\pi$
$240$	0	0
$241$	−25.0005	−1.61042	−0.805211	−	0.592988i	$-0.797948\pi$
$241$	−25.0005	−1.61042	−0.805211	+	0.592988i	$0.797948\pi$
$242$	−0.311108	−0.0199988
$243$	0	0
$244$	11.0509	0.707459
$245$	10.3477	0.661089
$246$	0	0
$247$	4.28100	0.272393
$248$	−5.11753	−0.324964
$249$	0	0
$250$	−3.51114	−0.222064
$251$	−11.4795	−0.724579	−0.362290	−	0.932066i	$-0.618005\pi$
$251$	−11.4795	−0.724579	−0.362290	+	0.932066i	$0.618005\pi$
$252$	0	0
$253$	0.474572	0.0298361
$254$	−6.29729	−0.395127
$255$	0	0
$256$	8.80642	0.550401
$257$	−12.1017	−0.754884	−0.377442	−	0.926033i	$-0.623196\pi$
$257$	−12.1017	−0.754884	−0.377442	+	0.926033i	$0.623196\pi$
$258$	0	0
$259$	0.653858	0.0406288
$260$	4.21432	0.261361
$261$	0	0
$262$	2.58073	0.159438
$263$	19.6543	1.21194	0.605969	−	0.795488i	$-0.292785\pi$
$263$	19.6543	1.21194	0.605969	+	0.795488i	$0.292785\pi$
$264$	0	0
$265$	−31.0923	−1.90999
$266$	−2.03164	−0.124568
$267$	0	0
$268$	11.4588	0.699955
$269$	4.62222	0.281821	0.140911	−	0.990022i	$-0.454997\pi$
$269$	4.62222	0.281821	0.140911	+	0.990022i	$0.454997\pi$
$270$	0	0
$271$	−7.37334	−0.447898	−0.223949	−	0.974601i	$-0.571895\pi$
$271$	−7.37334	−0.447898	−0.223949	+	0.974601i	$0.571895\pi$
$272$	5.96497	0.361679
$273$	0	0
$274$	−4.58718	−0.277122
$275$	0.0967881	0.00583654
$276$	0	0
$277$	24.0098	1.44261	0.721306	−	0.692617i	$-0.243542\pi$
$277$	24.0098	1.44261	0.721306	+	0.692617i	$0.243542\pi$
$278$	2.81780	0.169000
$279$	0	0
$280$	−4.10171	−0.245124
$281$	−10.6178	−0.633403	−0.316702	−	0.948525i	$-0.602575\pi$
$281$	−10.6178	−0.633403	−0.316702	+	0.948525i	$0.602575\pi$
$282$	0	0
$283$	8.88247	0.528008	0.264004	−	0.964522i	$-0.414957\pi$
$283$	8.88247	0.528008	0.264004	+	0.964522i	$0.414957\pi$
$284$	9.86665	0.585478
$285$	0	0
$286$	0.311108	0.0183962
$287$	−9.58120	−0.565561
$288$	0	0
$289$	−13.9733	−0.821958
$290$	−3.52543	−0.207020
$291$	0	0
$292$	15.3921	0.900753
$293$	21.1985	1.23843	0.619215	−	0.785222i	$-0.287451\pi$
$293$	21.1985	1.23843	0.619215	+	0.785222i	$0.287451\pi$
$294$	0	0
$295$	−3.70471	−0.215697
$296$	0.520505	0.0302538
$297$	0	0
$298$	−0.626661	−0.0363015
$299$	−0.474572	−0.0274452
$300$	0	0
$301$	−12.4603	−0.718199
$302$	−0.207866	−0.0119614
$303$	0	0
$304$	14.6780	0.841841
$305$	12.8573	0.736206
$306$	0	0
$307$	10.0874	0.575719	0.287860	−	0.957673i	$-0.407056\pi$
$307$	10.0874	0.575719	0.287860	+	0.957673i	$0.407056\pi$
$308$	2.90321	0.165426
$309$	0	0
$310$	−2.90321	−0.164891
$311$	−17.0366	−0.966055	−0.483027	−	0.875605i	$-0.660463\pi$
$311$	−17.0366	−0.966055	−0.483027	+	0.875605i	$0.660463\pi$
$312$	0	0
$313$	−19.5986	−1.10778	−0.553888	−	0.832591i	$-0.686856\pi$
$313$	−19.5986	−1.10778	−0.553888	+	0.832591i	$0.686856\pi$
$314$	−1.46520	−0.0826863
$315$	0	0
$316$	−0.495316	−0.0278637
$317$	31.8687	1.78992	0.894961	−	0.446144i	$-0.147203\pi$
$317$	31.8687	1.78992	0.894961	+	0.446144i	$0.147203\pi$
$318$	0	0
$319$	5.11753	0.286527
$320$	12.7763	0.714218
$321$	0	0
$322$	0.225219	0.0125510
$323$	7.44785	0.414410
$324$	0	0
$325$	−0.0967881	−0.00536884
$326$	0.850825	0.0471229
$327$	0	0
$328$	−7.62714	−0.421138
$329$	−7.70471	−0.424775
$330$	0	0
$331$	1.09033	0.0599302	0.0299651	−	0.999551i	$-0.490460\pi$
$331$	1.09033	0.0599302	0.0299651	+	0.999551i	$0.490460\pi$
$332$	−2.62222	−0.143913
$333$	0	0
$334$	−0.174840	−0.00956683
$335$	13.3319	0.728397
$336$	0	0
$337$	−11.0825	−0.603702	−0.301851	−	0.953355i	$-0.597605\pi$
$337$	−11.0825	−0.603702	−0.301851	+	0.953355i	$0.597605\pi$
$338$	−0.311108	−0.0169220
$339$	0	0
$340$	7.33185	0.397625
$341$	4.21432	0.228218
$342$	0	0
$343$	−17.8064	−0.961457
$344$	−9.91903	−0.534798
$345$	0	0
$346$	2.76341	0.148562
$347$	0.326929	0.0175505	0.00877524	−	0.999961i	$-0.497207\pi$
$347$	0.326929	0.0175505	0.00877524	+	0.999961i	$0.497207\pi$
$348$	0	0
$349$	−3.93978	−0.210891	−0.105446	−	0.994425i	$-0.533627\pi$
$349$	−3.93978	−0.210891	−0.105446	+	0.994425i	$0.533627\pi$
$350$	0.0459330	0.00245522
$351$	0	0
$352$	3.49532	0.186301
$353$	36.7862	1.95793	0.978965	−	0.204029i	$-0.0654038\pi$
$353$	36.7862	1.95793	0.978965	+	0.204029i	$0.0654038\pi$
$354$	0	0
$355$	11.4795	0.609268
$356$	−19.3254	−1.02424
$357$	0	0
$358$	−1.40498	−0.0742556
$359$	5.48394	0.289431	0.144716	−	0.989473i	$-0.453773\pi$
$359$	5.48394	0.289431	0.144716	+	0.989473i	$0.453773\pi$
$360$	0	0
$361$	−0.673071	−0.0354248
$362$	7.68736	0.404039
$363$	0	0
$364$	−2.90321	−0.152170
$365$	17.9081	0.937355
$366$	0	0
$367$	5.28592	0.275923	0.137961	−	0.990438i	$-0.455945\pi$
$367$	5.28592	0.275923	0.137961	+	0.990438i	$0.455945\pi$
$368$	−1.62714	−0.0848204
$369$	0	0
$370$	0.295286	0.0153512
$371$	21.4193	1.11203
$372$	0	0
$373$	−13.5714	−0.702698	−0.351349	−	0.936244i	$-0.614277\pi$
$373$	−13.5714	−0.702698	−0.351349	+	0.936244i	$0.614277\pi$
$374$	0.541249	0.0279873
$375$	0	0
$376$	−6.13335	−0.316304
$377$	−5.11753	−0.263566
$378$	0	0
$379$	0.836535	0.0429699	0.0214850	−	0.999769i	$-0.493161\pi$
$379$	0.836535	0.0429699	0.0214850	+	0.999769i	$0.493161\pi$
$380$	18.0415	0.925509
$381$	0	0
$382$	4.04149	0.206780
$383$	12.1334	0.619985	0.309993	−	0.950739i	$-0.399673\pi$
$383$	12.1334	0.619985	0.309993	+	0.950739i	$0.399673\pi$
$384$	0	0
$385$	3.37778	0.172148
$386$	−4.68106	−0.238259
$387$	0	0
$388$	−0.815792	−0.0414156
$389$	20.9175	1.06056	0.530280	−	0.847823i	$-0.322087\pi$
$389$	20.9175	1.06056	0.530280	+	0.847823i	$0.322087\pi$
$390$	0	0
$391$	−0.825636	−0.0417542
$392$	−5.67460	−0.286611
$393$	0	0
$394$	4.93485	0.248614
$395$	−0.576283	−0.0289959
$396$	0	0
$397$	3.45091	0.173196	0.0865982	−	0.996243i	$-0.472400\pi$
$397$	3.45091	0.173196	0.0865982	+	0.996243i	$0.472400\pi$
$398$	−4.17484	−0.209266
$399$	0	0
$400$	−0.331851	−0.0165926
$401$	−38.1639	−1.90582	−0.952908	−	0.303259i	$-0.901925\pi$
$401$	−38.1639	−1.90582	−0.952908	+	0.303259i	$0.901925\pi$
$402$	0	0
$403$	−4.21432	−0.209930
$404$	7.48595	0.372440
$405$	0	0
$406$	2.42864	0.120531
$407$	−0.428639	−0.0212469
$408$	0	0
$409$	38.0973	1.88379	0.941894	−	0.335910i	$-0.109044\pi$
$409$	38.0973	1.88379	0.941894	+	0.335910i	$0.109044\pi$
$410$	−4.32693	−0.213692
$411$	0	0
$412$	26.2766	1.29455
$413$	2.55215	0.125583
$414$	0	0
$415$	−3.05086	−0.149761
$416$	−3.49532	−0.171372
$417$	0	0
$418$	1.33185	0.0651430
$419$	−9.73191	−0.475435	−0.237717	−	0.971334i	$-0.576399\pi$
$419$	−9.73191	−0.475435	−0.237717	+	0.971334i	$0.576399\pi$
$420$	0	0
$421$	−25.0509	−1.22090	−0.610452	−	0.792053i	$-0.709012\pi$
$421$	−25.0509	−1.22090	−0.610452	+	0.792053i	$0.709012\pi$
$422$	−7.51897	−0.366018
$423$	0	0
$424$	17.0509	0.828063
$425$	−0.168387	−0.00816796
$426$	0	0
$427$	−8.85728	−0.428634
$428$	18.4099	0.889876
$429$	0	0
$430$	−5.62714	−0.271365
$431$	23.2543	1.12012	0.560060	−	0.828452i	$-0.310778\pi$
$431$	23.2543	1.12012	0.560060	+	0.828452i	$0.310778\pi$
$432$	0	0
$433$	−20.1017	−0.966027	−0.483013	−	0.875613i	$-0.660458\pi$
$433$	−20.1017	−0.966027	−0.483013	+	0.875613i	$0.660458\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	33.5482	1.60667
$437$	−2.03164	−0.0971867
$438$	0	0
$439$	32.1180	1.53291	0.766454	−	0.642299i	$-0.222019\pi$
$439$	32.1180	1.53291	0.766454	+	0.642299i	$0.222019\pi$
$440$	2.68889	0.128188
$441$	0	0
$442$	−0.541249	−0.0257446
$443$	−30.8385	−1.46518	−0.732592	−	0.680668i	$-0.761689\pi$
$443$	−30.8385	−1.46518	−0.732592	+	0.680668i	$0.761689\pi$
$444$	0	0
$445$	−22.4844	−1.06586
$446$	2.95560	0.139952
$447$	0	0
$448$	−8.80150	−0.415832
$449$	19.3254	0.912022	0.456011	−	0.889974i	$-0.349278\pi$
$449$	19.3254	0.912022	0.456011	+	0.889974i	$0.349278\pi$
$450$	0	0
$451$	6.28100	0.295761
$452$	25.6543	1.20668
$453$	0	0
$454$	2.61777	0.122858
$455$	−3.37778	−0.158353
$456$	0	0
$457$	−8.58073	−0.401390	−0.200695	−	0.979654i	$-0.564320\pi$
$457$	−8.58073	−0.401390	−0.200695	+	0.979654i	$0.564320\pi$
$458$	−9.03212	−0.422043
$459$	0	0
$460$	−2.00000	−0.0932505
$461$	4.94470	0.230298	0.115149	−	0.993348i	$-0.463266\pi$
$461$	4.94470	0.230298	0.115149	+	0.993348i	$0.463266\pi$
$462$	0	0
$463$	19.9289	0.926173	0.463087	−	0.886313i	$-0.346742\pi$
$463$	19.9289	0.926173	0.463087	+	0.886313i	$0.346742\pi$
$464$	−17.5462	−0.814561
$465$	0	0
$466$	3.82717	0.177290
$467$	−18.5763	−0.859608	−0.429804	−	0.902922i	$-0.641417\pi$
$467$	−18.5763	−0.859608	−0.429804	+	0.902922i	$0.641417\pi$
$468$	0	0
$469$	−9.18421	−0.424087
$470$	−3.47949	−0.160497
$471$	0	0
$472$	2.03164	0.0935139
$473$	8.16839	0.375583
$474$	0	0
$475$	−0.414349	−0.0190117
$476$	−5.05086	−0.231506
$477$	0	0
$478$	6.70964	0.306892
$479$	1.09679	0.0501135	0.0250568	−	0.999686i	$-0.492023\pi$
$479$	1.09679	0.0501135	0.0250568	+	0.999686i	$0.492023\pi$
$480$	0	0
$481$	0.428639	0.0195443
$482$	7.77784	0.354271
$483$	0	0
$484$	−1.90321	−0.0865096
$485$	−0.949145	−0.0430984
$486$	0	0
$487$	−28.6113	−1.29650	−0.648251	−	0.761427i	$-0.724499\pi$
$487$	−28.6113	−1.29650	−0.648251	+	0.761427i	$0.724499\pi$
$488$	−7.05086	−0.319177
$489$	0	0
$490$	−3.21924	−0.145431
$491$	19.4608	0.878252	0.439126	−	0.898426i	$-0.355288\pi$
$491$	19.4608	0.878252	0.439126	+	0.898426i	$0.355288\pi$
$492$	0	0
$493$	−8.90321	−0.400980
$494$	−1.33185	−0.0599228
$495$	0	0
$496$	−14.4494	−0.648796
$497$	−7.90813	−0.354728
$498$	0	0
$499$	−0.601472	−0.0269256	−0.0134628	−	0.999909i	$-0.504285\pi$
$499$	−0.601472	−0.0269256	−0.0134628	+	0.999909i	$0.504285\pi$
$500$	−21.4795	−0.960592
$501$	0	0
$502$	3.57136	0.159398
$503$	−8.69535	−0.387706	−0.193853	−	0.981031i	$-0.562099\pi$
$503$	−8.69535	−0.387706	−0.193853	+	0.981031i	$0.562099\pi$
$504$	0	0
$505$	8.70964	0.387574
$506$	−0.147643	−0.00656354
$507$	0	0
$508$	−38.5239	−1.70922
$509$	3.94761	0.174975	0.0874874	−	0.996166i	$-0.472116\pi$
$509$	3.94761	0.174975	0.0874874	+	0.996166i	$0.472116\pi$
$510$	0	0
$511$	−12.3368	−0.545747
$512$	−20.3111	−0.897633
$513$	0	0
$514$	3.76494	0.166064
$515$	30.5718	1.34716
$516$	0	0
$517$	5.05086	0.222136
$518$	−0.203420	−0.00893778
$519$	0	0
$520$	−2.68889	−0.117916
$521$	−22.4701	−0.984434	−0.492217	−	0.870472i	$-0.663813\pi$
$521$	−22.4701	−0.984434	−0.492217	+	0.870472i	$0.663813\pi$
$522$	0	0
$523$	−21.4257	−0.936882	−0.468441	−	0.883495i	$-0.655184\pi$
$523$	−21.4257	−0.936882	−0.468441	+	0.883495i	$0.655184\pi$
$524$	15.7877	0.689688
$525$	0	0
$526$	−6.11462	−0.266610
$527$	−7.33185	−0.319380
$528$	0	0
$529$	−22.7748	−0.990208
$530$	9.67307	0.420171
$531$	0	0
$532$	−12.4286	−0.538850
$533$	−6.28100	−0.272060
$534$	0	0
$535$	21.4193	0.926036
$536$	−7.31111	−0.315792
$537$	0	0
$538$	−1.43801	−0.0619969
$539$	4.67307	0.201283
$540$	0	0
$541$	−26.6735	−1.14679	−0.573393	−	0.819281i	$-0.694373\pi$
$541$	−26.6735	−1.14679	−0.573393	+	0.819281i	$0.694373\pi$
$542$	2.29390	0.0985316
$543$	0	0
$544$	−6.08097	−0.260719
$545$	39.0321	1.67195
$546$	0	0
$547$	−16.1497	−0.690509	−0.345255	−	0.938509i	$-0.612207\pi$
$547$	−16.1497	−0.690509	−0.345255	+	0.938509i	$0.612207\pi$
$548$	−28.0622	−1.19876
$549$	0	0
$550$	−0.0301115	−0.00128396
$551$	−21.9081	−0.933318
$552$	0	0
$553$	0.396997	0.0168820
$554$	−7.46965	−0.317355
$555$	0	0
$556$	17.2380	0.731053
$557$	25.2400	1.06945	0.534726	−	0.845025i	$-0.320415\pi$
$557$	25.2400	1.06945	0.534726	+	0.845025i	$0.320415\pi$
$558$	0	0
$559$	−8.16839	−0.345486
$560$	−11.5812	−0.489395
$561$	0	0
$562$	3.30327	0.139340
$563$	8.37826	0.353102	0.176551	−	0.984292i	$-0.443506\pi$
$563$	8.37826	0.353102	0.176551	+	0.984292i	$0.443506\pi$
$564$	0	0
$565$	29.8479	1.25571
$566$	−2.76341	−0.116155
$567$	0	0
$568$	−6.29529	−0.264144
$569$	−37.9432	−1.59066	−0.795330	−	0.606176i	$-0.792703\pi$
$569$	−37.9432	−1.59066	−0.795330	+	0.606176i	$0.792703\pi$
$570$	0	0
$571$	30.1782	1.26292	0.631460	−	0.775409i	$-0.282456\pi$
$571$	30.1782	1.26292	0.631460	+	0.775409i	$0.282456\pi$
$572$	1.90321	0.0795773
$573$	0	0
$574$	2.98079	0.124416
$575$	0.0459330	0.00191554
$576$	0	0
$577$	−24.1936	−1.00719	−0.503596	−	0.863939i	$-0.667990\pi$
$577$	−24.1936	−1.00719	−0.503596	+	0.863939i	$0.667990\pi$
$578$	4.34720	0.180820
$579$	0	0
$580$	−21.5669	−0.895517
$581$	2.10171	0.0871936
$582$	0	0
$583$	−14.0415	−0.581539
$584$	−9.82071	−0.406384
$585$	0	0
$586$	−6.59502	−0.272438
$587$	30.7971	1.27113	0.635565	−	0.772047i	$-0.280767\pi$
$587$	30.7971	1.27113	0.635565	+	0.772047i	$0.280767\pi$
$588$	0	0
$589$	−18.0415	−0.743387
$590$	1.15257	0.0474504
$591$	0	0
$592$	1.46965	0.0604023
$593$	−14.7382	−0.605226	−0.302613	−	0.953114i	$-0.597859\pi$
$593$	−14.7382	−0.605226	−0.302613	+	0.953114i	$0.597859\pi$
$594$	0	0
$595$	−5.87649	−0.240913
$596$	−3.83362	−0.157031
$597$	0	0
$598$	0.147643	0.00603758
$599$	−11.5999	−0.473961	−0.236980	−	0.971514i	$-0.576158\pi$
$599$	−11.5999	−0.473961	−0.236980	+	0.971514i	$0.576158\pi$
$600$	0	0
$601$	27.3274	1.11471	0.557354	−	0.830275i	$-0.311817\pi$
$601$	27.3274	1.11471	0.557354	+	0.830275i	$0.311817\pi$
$602$	3.87649	0.157994
$603$	0	0
$604$	−1.27163	−0.0517418
$605$	−2.21432	−0.0900249
$606$	0	0
$607$	0.0952567	0.00386635	0.00193318	−	0.999998i	$-0.499385\pi$
$607$	0.0952567	0.00386635	0.00193318	+	0.999998i	$0.499385\pi$
$608$	−14.9634	−0.606847
$609$	0	0
$610$	−4.00000	−0.161955
$611$	−5.05086	−0.204336
$612$	0	0
$613$	33.3230	1.34590	0.672951	−	0.739687i	$-0.265027\pi$
$613$	33.3230	1.34590	0.672951	+	0.739687i	$0.265027\pi$
$614$	−3.13828	−0.126650
$615$	0	0
$616$	−1.85236	−0.0746336
$617$	−41.7353	−1.68020	−0.840100	−	0.542432i	$-0.817504\pi$
$617$	−41.7353	−1.68020	−0.840100	+	0.542432i	$0.817504\pi$
$618$	0	0
$619$	29.8687	1.20052	0.600261	−	0.799804i	$-0.295063\pi$
$619$	29.8687	1.20052	0.600261	+	0.799804i	$0.295063\pi$
$620$	−17.7605	−0.713278
$621$	0	0
$622$	5.30021	0.212519
$623$	15.4893	0.620567
$624$	0	0
$625$	−24.5067	−0.980268
$626$	6.09726	0.243696
$627$	0	0
$628$	−8.96343	−0.357680
$629$	0.745724	0.0297340
$630$	0	0
$631$	12.3160	0.490293	0.245147	−	0.969486i	$-0.421164\pi$
$631$	12.3160	0.490293	0.245147	+	0.969486i	$0.421164\pi$
$632$	0.316030	0.0125710
$633$	0	0
$634$	−9.91459	−0.393759
$635$	−44.8212	−1.77867
$636$	0	0
$637$	−4.67307	−0.185154
$638$	−1.59210	−0.0630320
$639$	0	0
$640$	−19.4543	−0.768999
$641$	−44.0928	−1.74156	−0.870781	−	0.491671i	$-0.836386\pi$
$641$	−44.0928	−1.74156	−0.870781	+	0.491671i	$0.836386\pi$
$642$	0	0
$643$	32.1225	1.26679	0.633393	−	0.773830i	$-0.281662\pi$
$643$	32.1225	1.26679	0.633393	+	0.773830i	$0.281662\pi$
$644$	1.37778	0.0542923
$645$	0	0
$646$	−2.31708	−0.0911645
$647$	−7.47949	−0.294049	−0.147025	−	0.989133i	$-0.546970\pi$
$647$	−7.47949	−0.294049	−0.147025	+	0.989133i	$0.546970\pi$
$648$	0	0
$649$	−1.67307	−0.0656738
$650$	0.0301115	0.00118107
$651$	0	0
$652$	5.20495	0.203842
$653$	−41.3087	−1.61653	−0.808267	−	0.588817i	$-0.799594\pi$
$653$	−41.3087	−1.61653	−0.808267	+	0.588817i	$0.799594\pi$
$654$	0	0
$655$	18.3684	0.717713
$656$	−21.5353	−0.840811
$657$	0	0
$658$	2.39700	0.0934447
$659$	12.6637	0.493308	0.246654	−	0.969104i	$-0.420669\pi$
$659$	12.6637	0.493308	0.246654	+	0.969104i	$0.420669\pi$
$660$	0	0
$661$	13.5999	0.528976	0.264488	−	0.964389i	$-0.414797\pi$
$661$	13.5999	0.528976	0.264488	+	0.964389i	$0.414797\pi$
$662$	−0.339212	−0.0131838
$663$	0	0
$664$	1.67307	0.0649277
$665$	−14.4603	−0.560746
$666$	0	0
$667$	2.42864	0.0940373
$668$	−1.06959	−0.0413837
$669$	0	0
$670$	−4.14764	−0.160237
$671$	5.80642	0.224155
$672$	0	0
$673$	−12.9621	−0.499650	−0.249825	−	0.968291i	$-0.580373\pi$
$673$	−12.9621	−0.499650	−0.249825	+	0.968291i	$0.580373\pi$
$674$	3.44785	0.132806
$675$	0	0
$676$	−1.90321	−0.0732005
$677$	19.0988	0.734026	0.367013	−	0.930216i	$-0.380380\pi$
$677$	19.0988	0.734026	0.367013	+	0.930216i	$0.380380\pi$
$678$	0	0
$679$	0.653858	0.0250928
$680$	−4.67799	−0.179393
$681$	0	0
$682$	−1.31111	−0.0502049
$683$	−47.0450	−1.80013	−0.900064	−	0.435758i	$-0.856480\pi$
$683$	−47.0450	−1.80013	−0.900064	+	0.435758i	$0.856480\pi$
$684$	0	0
$685$	−32.6494	−1.24747
$686$	5.53972	0.211507
$687$	0	0
$688$	−28.0065	−1.06774
$689$	14.0415	0.534938
$690$	0	0
$691$	44.5827	1.69601	0.848004	−	0.529990i	$-0.177804\pi$
$691$	44.5827	1.69601	0.848004	+	0.529990i	$0.177804\pi$
$692$	16.9052	0.642640
$693$	0	0
$694$	−0.101710	−0.00386087
$695$	20.0558	0.760759
$696$	0	0
$697$	−10.9273	−0.413903
$698$	1.22570	0.0463933
$699$	0	0
$700$	0.280996	0.0106207
$701$	−43.3624	−1.63778	−0.818888	−	0.573953i	$-0.805409\pi$
$701$	−43.3624	−1.63778	−0.818888	+	0.573953i	$0.805409\pi$
$702$	0	0
$703$	1.83500	0.0692085
$704$	5.76986	0.217460
$705$	0	0
$706$	−11.4445	−0.430718
$707$	−6.00000	−0.225653
$708$	0	0
$709$	−3.34614	−0.125667	−0.0628335	−	0.998024i	$-0.520014\pi$
$709$	−3.34614	−0.125667	−0.0628335	+	0.998024i	$0.520014\pi$
$710$	−3.57136	−0.134031
$711$	0	0
$712$	12.3303	0.462098
$713$	2.00000	0.0749006
$714$	0	0
$715$	2.21432	0.0828109
$716$	−8.59502	−0.321211
$717$	0	0
$718$	−1.70610	−0.0636710
$719$	45.3274	1.69043	0.845213	−	0.534429i	$-0.179473\pi$
$719$	45.3274	1.69043	0.845213	+	0.534429i	$0.179473\pi$
$720$	0	0
$721$	−21.0607	−0.784341
$722$	0.209398	0.00779297
$723$	0	0
$724$	47.0277	1.74777
$725$	0.495316	0.0183956
$726$	0	0
$727$	−24.2163	−0.898134	−0.449067	−	0.893498i	$-0.648244\pi$
$727$	−24.2163	−0.898134	−0.449067	+	0.893498i	$0.648244\pi$
$728$	1.85236	0.0686529
$729$	0	0
$730$	−5.57136	−0.206205
$731$	−14.2109	−0.525610
$732$	0	0
$733$	−31.9037	−1.17839	−0.589195	−	0.807991i	$-0.700555\pi$
$733$	−31.9037	−1.17839	−0.589195	+	0.807991i	$0.700555\pi$
$734$	−1.64449	−0.0606993
$735$	0	0
$736$	1.65878	0.0611435
$737$	6.02074	0.221777
$738$	0	0
$739$	44.3926	1.63301	0.816503	−	0.577341i	$-0.195910\pi$
$739$	44.3926	1.63301	0.816503	+	0.577341i	$0.195910\pi$
$740$	1.80642	0.0664055
$741$	0	0
$742$	−6.66370	−0.244632
$743$	52.2449	1.91668	0.958340	−	0.285630i	$-0.0922029\pi$
$743$	52.2449	1.91668	0.958340	+	0.285630i	$0.0922029\pi$
$744$	0	0
$745$	−4.46028	−0.163412
$746$	4.22216	0.154584
$747$	0	0
$748$	3.31111	0.121066
$749$	−14.7556	−0.539157
$750$	0	0
$751$	−9.81933	−0.358312	−0.179156	−	0.983821i	$-0.557337\pi$
$751$	−9.81933	−0.358312	−0.179156	+	0.983821i	$0.557337\pi$
$752$	−17.3176	−0.631506
$753$	0	0
$754$	1.59210	0.0579810
$755$	−1.47949	−0.0538443
$756$	0	0
$757$	−22.5990	−0.821376	−0.410688	−	0.911776i	$-0.634711\pi$
$757$	−22.5990	−0.821376	−0.410688	+	0.911776i	$0.634711\pi$
$758$	−0.260253	−0.00945280
$759$	0	0
$760$	−11.5111	−0.417553
$761$	−0.0745132	−0.00270110	−0.00135055	−	0.999999i	$-0.500430\pi$
$761$	−0.0745132	−0.00270110	−0.00135055	+	0.999999i	$0.500430\pi$
$762$	0	0
$763$	−26.8889	−0.973444
$764$	24.7239	0.894480
$765$	0	0
$766$	−3.77478	−0.136388
$767$	1.67307	0.0604111
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	−1.05086	−0.0378702
$771$	0	0
$772$	−28.6365	−1.03065
$773$	2.66170	0.0957345	0.0478673	−	0.998854i	$-0.484758\pi$
$773$	2.66170	0.0957345	0.0478673	+	0.998854i	$0.484758\pi$
$774$	0	0
$775$	0.407896	0.0146521
$776$	0.520505	0.0186851
$777$	0	0
$778$	−6.50760	−0.233309
$779$	−26.8889	−0.963396
$780$	0	0
$781$	5.18421	0.185506
$782$	0.256862	0.00918536
$783$	0	0
$784$	−16.0223	−0.572224
$785$	−10.4286	−0.372214
$786$	0	0
$787$	28.5763	1.01863	0.509317	−	0.860579i	$-0.329898\pi$
$787$	28.5763	1.01863	0.509317	+	0.860579i	$0.329898\pi$
$788$	30.1891	1.07544
$789$	0	0
$790$	0.179286	0.00637871
$791$	−20.5620	−0.731100
$792$	0	0
$793$	−5.80642	−0.206192
$794$	−1.07361	−0.0381009
$795$	0	0
$796$	−25.5397	−0.905231
$797$	27.6860	0.980688	0.490344	−	0.871529i	$-0.336871\pi$
$797$	27.6860	0.980688	0.490344	+	0.871529i	$0.336871\pi$
$798$	0	0
$799$	−8.78721	−0.310869
$800$	0.338305	0.0119609
$801$	0	0
$802$	11.8731	0.419254
$803$	8.08742	0.285399
$804$	0	0
$805$	1.60300	0.0564984
$806$	1.31111	0.0461818
$807$	0	0
$808$	−4.77631	−0.168030
$809$	43.7911	1.53961	0.769806	−	0.638278i	$-0.220353\pi$
$809$	43.7911	1.53961	0.769806	+	0.638278i	$0.220353\pi$
$810$	0	0
$811$	42.4371	1.49017	0.745084	−	0.666971i	$-0.232409\pi$
$811$	42.4371	1.49017	0.745084	+	0.666971i	$0.232409\pi$
$812$	14.8573	0.521388
$813$	0	0
$814$	0.133353	0.00467402
$815$	6.05578	0.212125
$816$	0	0
$817$	−34.9688	−1.22340
$818$	−11.8524	−0.414408
$819$	0	0
$820$	−26.4701	−0.924377
$821$	44.5705	1.55552	0.777760	−	0.628562i	$-0.216356\pi$
$821$	44.5705	1.55552	0.777760	+	0.628562i	$0.216356\pi$
$822$	0	0
$823$	−19.8666	−0.692508	−0.346254	−	0.938141i	$-0.612546\pi$
$823$	−19.8666	−0.692508	−0.346254	+	0.938141i	$0.612546\pi$
$824$	−16.7654	−0.584051
$825$	0	0
$826$	−0.793993	−0.0276266
$827$	23.3822	0.813080	0.406540	−	0.913633i	$-0.366735\pi$
$827$	23.3822	0.813080	0.406540	+	0.913633i	$0.366735\pi$
$828$	0	0
$829$	9.52987	0.330986	0.165493	−	0.986211i	$-0.447078\pi$
$829$	9.52987	0.330986	0.165493	+	0.986211i	$0.447078\pi$
$830$	0.949145	0.0329453
$831$	0	0
$832$	−5.76986	−0.200034
$833$	−8.12996	−0.281686
$834$	0	0
$835$	−1.24443	−0.0430653
$836$	8.14764	0.281792
$837$	0	0
$838$	3.02767	0.104589
$839$	6.58073	0.227192	0.113596	−	0.993527i	$-0.463763\pi$
$839$	6.58073	0.227192	0.113596	+	0.993527i	$0.463763\pi$
$840$	0	0
$841$	−2.81087	−0.0969265
$842$	7.79352	0.268582
$843$	0	0
$844$	−45.9976	−1.58330
$845$	−2.21432	−0.0761749
$846$	0	0
$847$	1.52543	0.0524143
$848$	48.1432	1.65324
$849$	0	0
$850$	0.0523864	0.00179684
$851$	−0.203420	−0.00697316
$852$	0	0
$853$	0.726989	0.0248916	0.0124458	−	0.999923i	$-0.496038\pi$
$853$	0.726989	0.0248916	0.0124458	+	0.999923i	$0.496038\pi$
$854$	2.75557	0.0942936
$855$	0	0
$856$	−11.7462	−0.401477
$857$	−28.7304	−0.981411	−0.490706	−	0.871325i	$-0.663261\pi$
$857$	−28.7304	−0.981411	−0.490706	+	0.871325i	$0.663261\pi$
$858$	0	0
$859$	56.9273	1.94234	0.971168	−	0.238396i	$-0.0766217\pi$
$859$	56.9273	1.94234	0.971168	+	0.238396i	$0.0766217\pi$
$860$	−34.4242	−1.17386
$861$	0	0
$862$	−7.23459	−0.246411
$863$	−7.86665	−0.267784	−0.133892	−	0.990996i	$-0.542747\pi$
$863$	−7.86665	−0.267784	−0.133892	+	0.990996i	$0.542747\pi$
$864$	0	0
$865$	19.6686	0.668753
$866$	6.25380	0.212513
$867$	0	0
$868$	12.2351	0.415285
$869$	−0.260253	−0.00882847
$870$	0	0
$871$	−6.02074	−0.204005
$872$	−21.4050	−0.724864
$873$	0	0
$874$	0.632060	0.0213797
$875$	17.2159	0.582002
$876$	0	0
$877$	−5.18421	−0.175058	−0.0875291	−	0.996162i	$-0.527897\pi$
$877$	−5.18421	−0.175058	−0.0875291	+	0.996162i	$0.527897\pi$
$878$	−9.99216	−0.337219
$879$	0	0
$880$	7.59210	0.255930
$881$	−23.5111	−0.792110	−0.396055	−	0.918227i	$-0.629621\pi$
$881$	−23.5111	−0.792110	−0.396055	+	0.918227i	$0.629621\pi$
$882$	0	0
$883$	−6.07313	−0.204377	−0.102189	−	0.994765i	$-0.532585\pi$
$883$	−6.07313	−0.204377	−0.102189	+	0.994765i	$0.532585\pi$
$884$	−3.31111	−0.111365
$885$	0	0
$886$	9.59411	0.322320
$887$	−46.4612	−1.56002	−0.780008	−	0.625770i	$-0.784785\pi$
$887$	−46.4612	−1.56002	−0.780008	+	0.625770i	$0.784785\pi$
$888$	0	0
$889$	30.8770	1.03558
$890$	6.99508	0.234476
$891$	0	0
$892$	18.0810	0.605396
$893$	−21.6227	−0.723576
$894$	0	0
$895$	−10.0000	−0.334263
$896$	13.4019	0.447727
$897$	0	0
$898$	−6.01228	−0.200632
$899$	21.5669	0.719297
$900$	0	0
$901$	24.4286	0.813836
$902$	−1.95407	−0.0650633
$903$	0	0
$904$	−16.3684	−0.544405
$905$	54.7150	1.81879
$906$	0	0
$907$	44.3912	1.47398	0.736992	−	0.675901i	$-0.236245\pi$
$907$	44.3912	1.47398	0.736992	+	0.675901i	$0.236245\pi$
$908$	16.0143	0.531453
$909$	0	0
$910$	1.05086	0.0348355
$911$	−10.4385	−0.345842	−0.172921	−	0.984936i	$-0.555321\pi$
$911$	−10.4385	−0.345842	−0.172921	+	0.984936i	$0.555321\pi$
$912$	0	0
$913$	−1.37778	−0.0455980
$914$	2.66953	0.0883003
$915$	0	0
$916$	−55.2543	−1.82565
$917$	−12.6539	−0.417867
$918$	0	0
$919$	−20.4953	−0.676078	−0.338039	−	0.941132i	$-0.609764\pi$
$919$	−20.4953	−0.676078	−0.338039	+	0.941132i	$0.609764\pi$
$920$	1.27607	0.0420709
$921$	0	0
$922$	−1.53833	−0.0506624
$923$	−5.18421	−0.170640
$924$	0	0
$925$	−0.0414872	−0.00136409
$926$	−6.20003	−0.203746
$927$	0	0
$928$	17.8874	0.587182
$929$	12.8178	0.420538	0.210269	−	0.977644i	$-0.432566\pi$
$929$	12.8178	0.420538	0.210269	+	0.977644i	$0.432566\pi$
$930$	0	0
$931$	−20.0054	−0.655650
$932$	23.4128	0.766912
$933$	0	0
$934$	5.77923	0.189102
$935$	3.85236	0.125986
$936$	0	0
$937$	−33.3145	−1.08834	−0.544169	−	0.838976i	$-0.683155\pi$
$937$	−33.3145	−1.08834	−0.544169	+	0.838976i	$0.683155\pi$
$938$	2.85728	0.0932935
$939$	0	0
$940$	−21.2859	−0.694270
$941$	−55.1008	−1.79623	−0.898117	−	0.439756i	$-0.855065\pi$
$941$	−55.1008	−1.79623	−0.898117	+	0.439756i	$0.855065\pi$
$942$	0	0
$943$	2.98079	0.0970678
$944$	5.73636	0.186703
$945$	0	0
$946$	−2.54125	−0.0826231
$947$	13.6958	0.445054	0.222527	−	0.974926i	$-0.428569\pi$
$947$	13.6958	0.445054	0.222527	+	0.974926i	$0.428569\pi$
$948$	0	0
$949$	−8.08742	−0.262529
$950$	0.128907	0.00418231
$951$	0	0
$952$	3.22263	0.104446
$953$	12.0350	0.389853	0.194926	−	0.980818i	$-0.437553\pi$
$953$	12.0350	0.389853	0.194926	+	0.980818i	$0.437553\pi$
$954$	0	0
$955$	28.7654	0.930827
$956$	41.0464	1.32754
$957$	0	0
$958$	−0.341219	−0.0110243
$959$	22.4919	0.726302
$960$	0	0
$961$	−13.2395	−0.427081
$962$	−0.133353	−0.00429948
$963$	0	0
$964$	47.5812	1.53249
$965$	−33.3176	−1.07253
$966$	0	0
$967$	−59.5768	−1.91586	−0.957930	−	0.287003i	$-0.907341\pi$
$967$	−59.5768	−1.91586	−0.957930	+	0.287003i	$0.907341\pi$
$968$	1.21432	0.0390297
$969$	0	0
$970$	0.295286	0.00948107
$971$	53.2944	1.71030	0.855149	−	0.518382i	$-0.173466\pi$
$971$	53.2944	1.71030	0.855149	+	0.518382i	$0.173466\pi$
$972$	0	0
$973$	−13.8163	−0.442929
$974$	8.90120	0.285213
$975$	0	0
$976$	−19.9081	−0.637244
$977$	0.481026	0.0153894	0.00769469	−	0.999970i	$-0.497551\pi$
$977$	0.481026	0.0153894	0.00769469	+	0.999970i	$0.497551\pi$
$978$	0	0
$979$	−10.1541	−0.324526
$980$	−19.6938	−0.629096
$981$	0	0
$982$	−6.05439	−0.193203
$983$	−32.4612	−1.03535	−0.517676	−	0.855577i	$-0.673203\pi$
$983$	−32.4612	−1.03535	−0.517676	+	0.855577i	$0.673203\pi$
$984$	0	0
$985$	35.1240	1.11914
$986$	2.76986	0.0882103
$987$	0	0
$988$	−8.14764	−0.259211
$989$	3.87649	0.123265
$990$	0	0
$991$	61.3056	1.94744	0.973718	−	0.227755i	$-0.0731386\pi$
$991$	61.3056	1.94744	0.973718	+	0.227755i	$0.0731386\pi$
$992$	14.7304	0.467690
$993$	0	0
$994$	2.46028	0.0780354
$995$	−29.7146	−0.942015
$996$	0	0
$997$	−3.20294	−0.101438	−0.0507191	−	0.998713i	$-0.516151\pi$
$997$	−3.20294	−0.101438	−0.0507191	+	0.998713i	$0.516151\pi$
$998$	0.187123	0.00592326
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.i.1.2		3
3.2	odd	2		429.2.a.f.1.2	✓	3
12.11	even	2		6864.2.a.bp.1.3		3
33.32	even	2		4719.2.a.t.1.2		3
39.38	odd	2		5577.2.a.k.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.f.1.2	✓	3	3.2	odd	2
1287.2.a.i.1.2		3	1.1	even	1	trivial
4719.2.a.t.1.2		3	33.32	even	2
5577.2.a.k.1.2		3	39.38	odd	2
6864.2.a.bp.1.3		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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q-0.311108 q^{2} -1.90321 q^{4} -2.21432 q^{5} +1.52543 q^{7} +1.21432 q^{8} +O(q^{10})\)	\(q-0.311108 q^{2} -1.90321 q^{4} -2.21432 q^{5} +1.52543 q^{7} +1.21432 q^{8} +0.688892 q^{10} -1.00000 q^{11} +1.00000 q^{13} -0.474572 q^{14} +3.42864 q^{16} +1.73975 q^{17} +4.28100 q^{19} +4.21432 q^{20} +0.311108 q^{22} -0.474572 q^{23} -0.0967881 q^{25} -0.311108 q^{26} -2.90321 q^{28} -5.11753 q^{29} -4.21432 q^{31} -3.49532 q^{32} -0.541249 q^{34} -3.37778 q^{35} +0.428639 q^{37} -1.33185 q^{38} -2.68889 q^{40} -6.28100 q^{41} -8.16839 q^{43} +1.90321 q^{44} +0.147643 q^{46} -5.05086 q^{47} -4.67307 q^{49} +0.0301115 q^{50} -1.90321 q^{52} +14.0415 q^{53} +2.21432 q^{55} +1.85236 q^{56} +1.59210 q^{58} +1.67307 q^{59} -5.80642 q^{61} +1.31111 q^{62} -5.76986 q^{64} -2.21432 q^{65} -6.02074 q^{67} -3.31111 q^{68} +1.05086 q^{70} -5.18421 q^{71} -8.08742 q^{73} -0.133353 q^{74} -8.14764 q^{76} -1.52543 q^{77} +0.260253 q^{79} -7.59210 q^{80} +1.95407 q^{82} +1.37778 q^{83} -3.85236 q^{85} +2.54125 q^{86} -1.21432 q^{88} +10.1541 q^{89} +1.52543 q^{91} +0.903212 q^{92} +1.57136 q^{94} -9.47949 q^{95} +0.428639 q^{97} +1.45383 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + q^4 - 2 * q^7 - 3 * q^8	\( 3 q - q^{2} + q^{4} - 2 q^{7} - 3 q^{8} + 2 q^{10} - 3 q^{11} + 3 q^{13} - 8 q^{14} - 3 q^{16} - 8 q^{17} + 6 q^{19} + 6 q^{20} + q^{22} - 8 q^{23} - 7 q^{25} - q^{26} - 2 q^{28} - 2 q^{29} - 6 q^{31} + 3 q^{32} - 8 q^{34} - 10 q^{35} - 12 q^{37} + 16 q^{38} - 8 q^{40} - 12 q^{41} + 2 q^{43} - q^{44} - 6 q^{46} - 2 q^{47} - q^{49} + 7 q^{50} + q^{52} + 2 q^{53} + 12 q^{56} - 2 q^{58} - 8 q^{59} - 4 q^{61} + 4 q^{62} - 11 q^{64} + 2 q^{67} - 10 q^{68} - 10 q^{70} - 2 q^{71} - 4 q^{73} - 18 q^{76} + 2 q^{77} + 14 q^{79} - 16 q^{80} - 14 q^{82} + 4 q^{83} - 18 q^{85} + 14 q^{86} + 3 q^{88} + 10 q^{89} - 2 q^{91} - 4 q^{92} + 18 q^{94} - 2 q^{95} - 12 q^{97} + 31 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 - 2 * q^7 - 3 * q^8 + 2 * q^10 - 3 * q^11 + 3 * q^13 - 8 * q^14 - 3 * q^16 - 8 * q^17 + 6 * q^19 + 6 * q^20 + q^22 - 8 * q^23 - 7 * q^25 - q^26 - 2 * q^28 - 2 * q^29 - 6 * q^31 + 3 * q^32 - 8 * q^34 - 10 * q^35 - 12 * q^37 + 16 * q^38 - 8 * q^40 - 12 * q^41 + 2 * q^43 - q^44 - 6 * q^46 - 2 * q^47 - q^49 + 7 * q^50 + q^52 + 2 * q^53 + 12 * q^56 - 2 * q^58 - 8 * q^59 - 4 * q^61 + 4 * q^62 - 11 * q^64 + 2 * q^67 - 10 * q^68 - 10 * q^70 - 2 * q^71 - 4 * q^73 - 18 * q^76 + 2 * q^77 + 14 * q^79 - 16 * q^80 - 14 * q^82 + 4 * q^83 - 18 * q^85 + 14 * q^86 + 3 * q^88 + 10 * q^89 - 2 * q^91 - 4 * q^92 + 18 * q^94 - 2 * q^95 - 12 * q^97 + 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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