LMFDB - Embedded newform 1287.2.a.h.1.3

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			1.3
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)

$f(q)$	$=$	$q+1.21432 q^{2} -0.525428 q^{4} -1.31111 q^{5} +1.52543 q^{7} -3.06668 q^{8} +O(q^{10})$	\(q+1.21432 q^{2} -0.525428 q^{4} -1.31111 q^{5} +1.52543 q^{7} -3.06668 q^{8} -1.59210 q^{10} +1.00000 q^{11} -1.00000 q^{13} +1.85236 q^{14} -2.67307 q^{16} -2.21432 q^{17} -1.52543 q^{19} +0.688892 q^{20} +1.21432 q^{22} -7.95407 q^{23} -3.28100 q^{25} -1.21432 q^{26} -0.801502 q^{28} -7.39853 q^{29} +4.68889 q^{31} +2.88739 q^{32} -2.68889 q^{34} -2.00000 q^{35} +8.85728 q^{37} -1.85236 q^{38} +4.02074 q^{40} +3.52543 q^{41} -8.77631 q^{43} -0.525428 q^{44} -9.65878 q^{46} -9.18421 q^{47} -4.67307 q^{49} -3.98418 q^{50} +0.525428 q^{52} -3.67307 q^{53} -1.31111 q^{55} -4.67799 q^{56} -8.98418 q^{58} -9.37778 q^{59} -11.4795 q^{61} +5.69381 q^{62} +8.85236 q^{64} +1.31111 q^{65} +5.25088 q^{67} +1.16346 q^{68} -2.42864 q^{70} +14.4701 q^{71} +3.13828 q^{73} +10.7556 q^{74} +0.801502 q^{76} +1.52543 q^{77} -5.03011 q^{79} +3.50468 q^{80} +4.28100 q^{82} -5.37778 q^{83} +2.90321 q^{85} -10.6572 q^{86} -3.06668 q^{88} -0.688892 q^{89} -1.52543 q^{91} +4.17929 q^{92} -11.1526 q^{94} +2.00000 q^{95} -12.8573 q^{97} -5.67460 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 5 q^{4} - 4 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 5 * q^4 - 4 * q^5 - 2 * q^7 - 9 * q^8	$ 3 q - 3 q^{2} + 5 q^{4} - 4 q^{5} - 2 q^{7} - 9 q^{8} + 2 q^{10} + 3 q^{11} - 3 q^{13} + 12 q^{14} + 5 q^{16} + 2 q^{19} + 2 q^{20} - 3 q^{22} - 4 q^{23} - 3 q^{25} + 3 q^{26} - 22 q^{28} - 2 q^{29} + 14 q^{31} - 11 q^{32} - 8 q^{34} - 6 q^{35} - 12 q^{38} - 8 q^{40} + 4 q^{41} - 6 q^{43} + 5 q^{44} - 22 q^{46} - 14 q^{47} - q^{49} + q^{50} - 5 q^{52} + 2 q^{53} - 4 q^{55} + 32 q^{56} - 14 q^{58} - 28 q^{59} - 8 q^{61} - 16 q^{62} + 33 q^{64} + 4 q^{65} + 2 q^{67} + 10 q^{68} + 6 q^{70} - 10 q^{71} - 24 q^{73} + 32 q^{74} + 22 q^{76} - 2 q^{77} - 22 q^{79} + 24 q^{80} + 6 q^{82} - 16 q^{83} + 2 q^{85} - 6 q^{86} - 9 q^{88} - 2 q^{89} + 2 q^{91} + 32 q^{92} + 6 q^{94} + 6 q^{95} - 12 q^{97} - 23 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 5 * q^4 - 4 * q^5 - 2 * q^7 - 9 * q^8 + 2 * q^10 + 3 * q^11 - 3 * q^13 + 12 * q^14 + 5 * q^16 + 2 * q^19 + 2 * q^20 - 3 * q^22 - 4 * q^23 - 3 * q^25 + 3 * q^26 - 22 * q^28 - 2 * q^29 + 14 * q^31 - 11 * q^32 - 8 * q^34 - 6 * q^35 - 12 * q^38 - 8 * q^40 + 4 * q^41 - 6 * q^43 + 5 * q^44 - 22 * q^46 - 14 * q^47 - q^49 + q^50 - 5 * q^52 + 2 * q^53 - 4 * q^55 + 32 * q^56 - 14 * q^58 - 28 * q^59 - 8 * q^61 - 16 * q^62 + 33 * q^64 + 4 * q^65 + 2 * q^67 + 10 * q^68 + 6 * q^70 - 10 * q^71 - 24 * q^73 + 32 * q^74 + 22 * q^76 - 2 * q^77 - 22 * q^79 + 24 * q^80 + 6 * q^82 - 16 * q^83 + 2 * q^85 - 6 * q^86 - 9 * q^88 - 2 * q^89 + 2 * q^91 + 32 * q^92 + 6 * q^94 + 6 * q^95 - 12 * q^97 - 23 * 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 $

$2$	1.21432	0.858654	0.429327	−	0.903149i	$-0.358751\pi$
$2$	1.21432	0.858654	0.429327	+	0.903149i	$0.358751\pi$
$3$	0	0
$3$	0	0
$4$	−0.525428	−0.262714
$5$	−1.31111	−0.586345	−0.293173	−	0.956060i	$-0.594711\pi$
$5$	−1.31111	−0.586345	−0.293173	+	0.956060i	$0.594711\pi$
$6$	0	0
$7$	1.52543	0.576557	0.288279	−	0.957547i	$-0.406917\pi$
$7$	1.52543	0.576557	0.288279	+	0.957547i	$0.406917\pi$
$8$	−3.06668	−1.08423
$9$	0	0
$10$	−1.59210	−0.503468
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.85236	0.495063
$15$	0	0
$16$	−2.67307	−0.668268
$17$	−2.21432	−0.537051	−0.268526	−	0.963273i	$-0.586536\pi$
$17$	−2.21432	−0.537051	−0.268526	+	0.963273i	$0.586536\pi$
$18$	0	0
$19$	−1.52543	−0.349957	−0.174979	−	0.984572i	$-0.555986\pi$
$19$	−1.52543	−0.349957	−0.174979	+	0.984572i	$0.555986\pi$
$20$	0.688892	0.154041
$21$	0	0
$22$	1.21432	0.258894
$23$	−7.95407	−1.65854	−0.829269	−	0.558850i	$-0.811243\pi$
$23$	−7.95407	−1.65854	−0.829269	+	0.558850i	$0.811243\pi$
$24$	0	0
$25$	−3.28100	−0.656199
$26$	−1.21432	−0.238148
$27$	0	0
$28$	−0.801502	−0.151470
$29$	−7.39853	−1.37387	−0.686936	−	0.726718i	$-0.741045\pi$
$29$	−7.39853	−1.37387	−0.686936	+	0.726718i	$0.741045\pi$
$30$	0	0
$31$	4.68889	0.842150	0.421075	−	0.907026i	$-0.361653\pi$
$31$	4.68889	0.842150	0.421075	+	0.907026i	$0.361653\pi$
$32$	2.88739	0.510423
$33$	0	0
$34$	−2.68889	−0.461141
$35$	−2.00000	−0.338062
$36$	0	0
$37$	8.85728	1.45613	0.728064	−	0.685509i	$-0.240420\pi$
$37$	8.85728	1.45613	0.728064	+	0.685509i	$0.240420\pi$
$38$	−1.85236	−0.300492
$39$	0	0
$40$	4.02074	0.635735
$41$	3.52543	0.550579	0.275290	−	0.961361i	$-0.411226\pi$
$41$	3.52543	0.550579	0.275290	+	0.961361i	$0.411226\pi$
$42$	0	0
$43$	−8.77631	−1.33838	−0.669188	−	0.743094i	$-0.733358\pi$
$43$	−8.77631	−1.33838	−0.669188	+	0.743094i	$0.733358\pi$
$44$	−0.525428	−0.0792112
$45$	0	0
$46$	−9.65878	−1.42411
$47$	−9.18421	−1.33965	−0.669827	−	0.742517i	$-0.733632\pi$
$47$	−9.18421	−1.33965	−0.669827	+	0.742517i	$0.733632\pi$
$48$	0	0
$49$	−4.67307	−0.667582
$50$	−3.98418	−0.563448
$51$	0	0
$52$	0.525428	0.0728637
$53$	−3.67307	−0.504535	−0.252268	−	0.967658i	$-0.581176\pi$
$53$	−3.67307	−0.504535	−0.252268	+	0.967658i	$0.581176\pi$
$54$	0	0
$55$	−1.31111	−0.176790
$56$	−4.67799	−0.625123
$57$	0	0
$58$	−8.98418	−1.17968
$59$	−9.37778	−1.22088	−0.610442	−	0.792061i	$-0.709008\pi$
$59$	−9.37778	−1.22088	−0.610442	+	0.792061i	$0.709008\pi$
$60$	0	0
$61$	−11.4795	−1.46980	−0.734899	−	0.678176i	$-0.762771\pi$
$61$	−11.4795	−1.46980	−0.734899	+	0.678176i	$0.762771\pi$
$62$	5.69381	0.723115
$63$	0	0
$64$	8.85236	1.10654
$65$	1.31111	0.162623
$66$	0	0
$67$	5.25088	0.641498	0.320749	−	0.947164i	$-0.396065\pi$
$67$	5.25088	0.641498	0.320749	+	0.947164i	$0.396065\pi$
$68$	1.16346	0.141091
$69$	0	0
$70$	−2.42864	−0.290278
$71$	14.4701	1.71729	0.858644	−	0.512572i	$-0.171307\pi$
$71$	14.4701	1.71729	0.858644	+	0.512572i	$0.171307\pi$
$72$	0	0
$73$	3.13828	0.367307	0.183654	−	0.982991i	$-0.441208\pi$
$73$	3.13828	0.367307	0.183654	+	0.982991i	$0.441208\pi$
$74$	10.7556	1.25031
$75$	0	0
$76$	0.801502	0.0919385
$77$	1.52543	0.173839
$78$	0	0
$79$	−5.03011	−0.565932	−0.282966	−	0.959130i	$-0.591318\pi$
$79$	−5.03011	−0.565932	−0.282966	+	0.959130i	$0.591318\pi$
$80$	3.50468	0.391836
$81$	0	0
$82$	4.28100	0.472757
$83$	−5.37778	−0.590289	−0.295144	−	0.955453i	$-0.595368\pi$
$83$	−5.37778	−0.590289	−0.295144	+	0.955453i	$0.595368\pi$
$84$	0	0
$85$	2.90321	0.314898
$86$	−10.6572	−1.14920
$87$	0	0
$88$	−3.06668	−0.326909
$89$	−0.688892	−0.0730224	−0.0365112	−	0.999333i	$-0.511624\pi$
$89$	−0.688892	−0.0730224	−0.0365112	+	0.999333i	$0.511624\pi$
$90$	0	0
$91$	−1.52543	−0.159908
$92$	4.17929	0.435721
$93$	0	0
$94$	−11.1526	−1.15030
$95$	2.00000	0.205196
$96$	0	0
$97$	−12.8573	−1.30546	−0.652729	−	0.757591i	$-0.726376\pi$
$97$	−12.8573	−1.30546	−0.652729	+	0.757591i	$0.726376\pi$
$98$	−5.67460	−0.573221
$99$	0	0
$100$	1.72393	0.172393
$101$	16.0207	1.59412	0.797062	−	0.603898i	$-0.206386\pi$
$101$	16.0207	1.59412	0.797062	+	0.603898i	$0.206386\pi$
$102$	0	0
$103$	−7.61285	−0.750116	−0.375058	−	0.927001i	$-0.622377\pi$
$103$	−7.61285	−0.750116	−0.375058	+	0.927001i	$0.622377\pi$
$104$	3.06668	0.300712
$105$	0	0
$106$	−4.46028	−0.433221
$107$	12.9906	1.25585	0.627926	−	0.778273i	$-0.283904\pi$
$107$	12.9906	1.25585	0.627926	+	0.778273i	$0.283904\pi$
$108$	0	0
$109$	3.82071	0.365958	0.182979	−	0.983117i	$-0.441426\pi$
$109$	3.82071	0.365958	0.182979	+	0.983117i	$0.441426\pi$
$110$	−1.59210	−0.151801
$111$	0	0
$112$	−4.07758	−0.385295
$113$	18.3368	1.72498	0.862489	−	0.506075i	$-0.168904\pi$
$113$	18.3368	1.72498	0.862489	+	0.506075i	$0.168904\pi$
$114$	0	0
$115$	10.4286	0.972476
$116$	3.88739	0.360935
$117$	0	0
$118$	−11.3876	−1.04832
$119$	−3.37778	−0.309641
$120$	0	0
$121$	1.00000	0.0909091
$122$	−13.9398	−1.26205
$123$	0	0
$124$	−2.46367	−0.221244
$125$	10.8573	0.971105
$126$	0	0
$127$	−20.5827	−1.82642	−0.913211	−	0.407486i	$-0.866405\pi$
$127$	−20.5827	−1.82642	−0.913211	+	0.407486i	$0.866405\pi$
$128$	4.97481	0.439715
$129$	0	0
$130$	1.59210	0.139637
$131$	0.561993	0.0491015	0.0245508	−	0.999699i	$-0.492184\pi$
$131$	0.561993	0.0491015	0.0245508	+	0.999699i	$0.492184\pi$
$132$	0	0
$133$	−2.32693	−0.201770
$134$	6.37625	0.550824
$135$	0	0
$136$	6.79060	0.582289
$137$	−5.07604	−0.433676	−0.216838	−	0.976208i	$-0.569574\pi$
$137$	−5.07604	−0.433676	−0.216838	+	0.976208i	$0.569574\pi$
$138$	0	0
$139$	18.5511	1.57348	0.786742	−	0.617282i	$-0.211766\pi$
$139$	18.5511	1.57348	0.786742	+	0.617282i	$0.211766\pi$
$140$	1.05086	0.0888135
$141$	0	0
$142$	17.5714	1.47456
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	9.70027	0.805563
$146$	3.81087	0.315390
$147$	0	0
$148$	−4.65386	−0.382545
$149$	4.38271	0.359045	0.179523	−	0.983754i	$-0.442545\pi$
$149$	4.38271	0.359045	0.179523	+	0.983754i	$0.442545\pi$
$150$	0	0
$151$	1.23014	0.100107	0.0500537	−	0.998747i	$-0.484061\pi$
$151$	1.23014	0.100107	0.0500537	+	0.998747i	$0.484061\pi$
$152$	4.67799	0.379435
$153$	0	0
$154$	1.85236	0.149267
$155$	−6.14764	−0.493791
$156$	0	0
$157$	7.62714	0.608712	0.304356	−	0.952558i	$-0.401559\pi$
$157$	7.62714	0.608712	0.304356	+	0.952558i	$0.401559\pi$
$158$	−6.10816	−0.485939
$159$	0	0
$160$	−3.78568	−0.299284
$161$	−12.1334	−0.956242
$162$	0	0
$163$	7.25088	0.567933	0.283967	−	0.958834i	$-0.408350\pi$
$163$	7.25088	0.567933	0.283967	+	0.958834i	$0.408350\pi$
$164$	−1.85236	−0.144645
$165$	0	0
$166$	−6.53035	−0.506853
$167$	−0.295286	−0.0228499	−0.0114250	−	0.999935i	$-0.503637\pi$
$167$	−0.295286	−0.0228499	−0.0114250	+	0.999935i	$0.503637\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	3.52543	0.270388
$171$	0	0
$172$	4.61132	0.351610
$173$	17.5605	1.33510	0.667549	−	0.744566i	$-0.267344\pi$
$173$	17.5605	1.33510	0.667549	+	0.744566i	$0.267344\pi$
$174$	0	0
$175$	−5.00492	−0.378337
$176$	−2.67307	−0.201490
$177$	0	0
$178$	−0.836535	−0.0627010
$179$	−12.0874	−0.903456	−0.451728	−	0.892156i	$-0.649192\pi$
$179$	−12.0874	−0.903456	−0.451728	+	0.892156i	$0.649192\pi$
$180$	0	0
$181$	7.71900	0.573749	0.286875	−	0.957968i	$-0.407384\pi$
$181$	7.71900	0.573749	0.286875	+	0.957968i	$0.407384\pi$
$182$	−1.85236	−0.137306
$183$	0	0
$184$	24.3926	1.79824
$185$	−11.6128	−0.853794
$186$	0	0
$187$	−2.21432	−0.161927
$188$	4.82564	0.351946
$189$	0	0
$190$	2.42864	0.176192
$191$	16.6035	1.20139	0.600693	−	0.799480i	$-0.294891\pi$
$191$	16.6035	1.20139	0.600693	+	0.799480i	$0.294891\pi$
$192$	0	0
$193$	−7.52543	−0.541692	−0.270846	−	0.962623i	$-0.587303\pi$
$193$	−7.52543	−0.541692	−0.270846	+	0.962623i	$0.587303\pi$
$194$	−15.6128	−1.12094
$195$	0	0
$196$	2.45536	0.175383
$197$	−12.2494	−0.872730	−0.436365	−	0.899770i	$-0.643734\pi$
$197$	−12.2494	−0.872730	−0.436365	+	0.899770i	$0.643734\pi$
$198$	0	0
$199$	11.6128	0.823213	0.411606	−	0.911362i	$-0.364968\pi$
$199$	11.6128	0.823213	0.411606	+	0.911362i	$0.364968\pi$
$200$	10.0618	0.711473
$201$	0	0
$202$	19.4543	1.36880
$203$	−11.2859	−0.792116
$204$	0	0
$205$	−4.62222	−0.322830
$206$	−9.24443	−0.644090
$207$	0	0
$208$	2.67307	0.185344
$209$	−1.52543	−0.105516
$210$	0	0
$211$	5.16346	0.355468	0.177734	−	0.984079i	$-0.443123\pi$
$211$	5.16346	0.355468	0.177734	+	0.984079i	$0.443123\pi$
$212$	1.92993	0.132548
$213$	0	0
$214$	15.7748	1.07834
$215$	11.5067	0.784750
$216$	0	0
$217$	7.15257	0.485548
$218$	4.63957	0.314231
$219$	0	0
$220$	0.688892	0.0464451
$221$	2.21432	0.148951
$222$	0	0
$223$	−8.30174	−0.555926	−0.277963	−	0.960592i	$-0.589659\pi$
$223$	−8.30174	−0.555926	−0.277963	+	0.960592i	$0.589659\pi$
$224$	4.40451	0.294288
$225$	0	0
$226$	22.2667	1.48116
$227$	5.27163	0.349890	0.174945	−	0.984578i	$-0.444025\pi$
$227$	5.27163	0.349890	0.174945	+	0.984578i	$0.444025\pi$
$228$	0	0
$229$	−25.5526	−1.68856	−0.844282	−	0.535898i	$-0.819973\pi$
$229$	−25.5526	−1.68856	−0.844282	+	0.535898i	$0.819973\pi$
$230$	12.6637	0.835020
$231$	0	0
$232$	22.6889	1.48960
$233$	−2.40790	−0.157747	−0.0788733	−	0.996885i	$-0.525132\pi$
$233$	−2.40790	−0.157747	−0.0788733	+	0.996885i	$0.525132\pi$
$234$	0	0
$235$	12.0415	0.785500
$236$	4.92735	0.320743
$237$	0	0
$238$	−4.10171	−0.265874
$239$	−17.8622	−1.15541	−0.577705	−	0.816246i	$-0.696052\pi$
$239$	−17.8622	−1.15541	−0.577705	+	0.816246i	$0.696052\pi$
$240$	0	0
$241$	−3.93978	−0.253783	−0.126892	−	0.991917i	$-0.540500\pi$
$241$	−3.93978	−0.253783	−0.126892	+	0.991917i	$0.540500\pi$
$242$	1.21432	0.0780594
$243$	0	0
$244$	6.03164	0.386136
$245$	6.12690	0.391433
$246$	0	0
$247$	1.52543	0.0970606
$248$	−14.3793	−0.913087
$249$	0	0
$250$	13.1842	0.833843
$251$	−23.3590	−1.47441	−0.737205	−	0.675669i	$-0.763855\pi$
$251$	−23.3590	−1.47441	−0.737205	+	0.675669i	$0.763855\pi$
$252$	0	0
$253$	−7.95407	−0.500068
$254$	−24.9940	−1.56826
$255$	0	0
$256$	−11.6637	−0.728981
$257$	−1.70471	−0.106337	−0.0531686	−	0.998586i	$-0.516932\pi$
$257$	−1.70471	−0.106337	−0.0531686	+	0.998586i	$0.516932\pi$
$258$	0	0
$259$	13.5111	0.839541
$260$	−0.688892	−0.0427233
$261$	0	0
$262$	0.682439	0.0421612
$263$	1.11108	0.0685120	0.0342560	−	0.999413i	$-0.489094\pi$
$263$	1.11108	0.0685120	0.0342560	+	0.999413i	$0.489094\pi$
$264$	0	0
$265$	4.81579	0.295832
$266$	−2.82564	−0.173251
$267$	0	0
$268$	−2.75896	−0.168530
$269$	−8.79706	−0.536366	−0.268183	−	0.963368i	$-0.586423\pi$
$269$	−8.79706	−0.536366	−0.268183	+	0.963368i	$0.586423\pi$
$270$	0	0
$271$	27.3733	1.66281	0.831406	−	0.555665i	$-0.187536\pi$
$271$	27.3733	1.66281	0.831406	+	0.555665i	$0.187536\pi$
$272$	5.91903	0.358894
$273$	0	0
$274$	−6.16394	−0.372377
$275$	−3.28100	−0.197852
$276$	0	0
$277$	−25.9081	−1.55667	−0.778334	−	0.627850i	$-0.783935\pi$
$277$	−25.9081	−1.55667	−0.778334	+	0.627850i	$0.783935\pi$
$278$	22.5270	1.35108
$279$	0	0
$280$	6.13335	0.366538
$281$	5.76049	0.343642	0.171821	−	0.985128i	$-0.445035\pi$
$281$	5.76049	0.343642	0.171821	+	0.985128i	$0.445035\pi$
$282$	0	0
$283$	−3.03011	−0.180121	−0.0900607	−	0.995936i	$-0.528706\pi$
$283$	−3.03011	−0.180121	−0.0900607	+	0.995936i	$0.528706\pi$
$284$	−7.60300	−0.451155
$285$	0	0
$286$	−1.21432	−0.0718042
$287$	5.37778	0.317441
$288$	0	0
$289$	−12.0968	−0.711576
$290$	11.7792	0.691700
$291$	0	0
$292$	−1.64894	−0.0964967
$293$	9.07805	0.530345	0.265173	−	0.964201i	$-0.414571\pi$
$293$	9.07805	0.530345	0.265173	+	0.964201i	$0.414571\pi$
$294$	0	0
$295$	12.2953	0.715859
$296$	−27.1624	−1.57878
$297$	0	0
$298$	5.32201	0.308296
$299$	7.95407	0.459996
$300$	0	0
$301$	−13.3876	−0.771650
$302$	1.49378	0.0859577
$303$	0	0
$304$	4.07758	0.233865
$305$	15.0509	0.861809
$306$	0	0
$307$	−5.43356	−0.310110	−0.155055	−	0.987906i	$-0.549555\pi$
$307$	−5.43356	−0.310110	−0.155055	+	0.987906i	$0.549555\pi$
$308$	−0.801502	−0.0456698
$309$	0	0
$310$	−7.46520	−0.423995
$311$	9.46520	0.536723	0.268361	−	0.963318i	$-0.413518\pi$
$311$	9.46520	0.536723	0.268361	+	0.963318i	$0.413518\pi$
$312$	0	0
$313$	−23.6400	−1.33621	−0.668107	−	0.744065i	$-0.732895\pi$
$313$	−23.6400	−1.33621	−0.668107	+	0.744065i	$0.732895\pi$
$314$	9.26178	0.522673
$315$	0	0
$316$	2.64296	0.148678
$317$	−15.6479	−0.878873	−0.439436	−	0.898274i	$-0.644822\pi$
$317$	−15.6479	−0.878873	−0.439436	+	0.898274i	$0.644822\pi$
$318$	0	0
$319$	−7.39853	−0.414238
$320$	−11.6064	−0.648817
$321$	0	0
$322$	−14.7338	−0.821081
$323$	3.37778	0.187945
$324$	0	0
$325$	3.28100	0.181997
$326$	8.80489	0.487658
$327$	0	0
$328$	−10.8113	−0.596957
$329$	−14.0098	−0.772388
$330$	0	0
$331$	−19.6795	−1.08168	−0.540842	−	0.841124i	$-0.681894\pi$
$331$	−19.6795	−1.08168	−0.540842	+	0.841124i	$0.681894\pi$
$332$	2.82564	0.155077
$333$	0	0
$334$	−0.358572	−0.0196202
$335$	−6.88448	−0.376139
$336$	0	0
$337$	10.9906	0.598698	0.299349	−	0.954144i	$-0.403231\pi$
$337$	10.9906	0.598698	0.299349	+	0.954144i	$0.403231\pi$
$338$	1.21432	0.0660503
$339$	0	0
$340$	−1.52543	−0.0827279
$341$	4.68889	0.253918
$342$	0	0
$343$	−17.8064	−0.961457
$344$	26.9141	1.45111
$345$	0	0
$346$	21.3240	1.14639
$347$	−17.3176	−0.929655	−0.464828	−	0.885401i	$-0.653884\pi$
$347$	−17.3176	−0.929655	−0.464828	+	0.885401i	$0.653884\pi$
$348$	0	0
$349$	−17.1842	−0.919850	−0.459925	−	0.887958i	$-0.652124\pi$
$349$	−17.1842	−0.919850	−0.459925	+	0.887958i	$0.652124\pi$
$350$	−6.07758	−0.324860
$351$	0	0
$352$	2.88739	0.153898
$353$	14.4351	0.768302	0.384151	−	0.923270i	$-0.374494\pi$
$353$	14.4351	0.768302	0.384151	+	0.923270i	$0.374494\pi$
$354$	0	0
$355$	−18.9719	−1.00692
$356$	0.361963	0.0191840
$357$	0	0
$358$	−14.6780	−0.775756
$359$	16.1476	0.852240	0.426120	−	0.904667i	$-0.359880\pi$
$359$	16.1476	0.852240	0.426120	+	0.904667i	$0.359880\pi$
$360$	0	0
$361$	−16.6731	−0.877530
$362$	9.37334	0.492652
$363$	0	0
$364$	0.801502	0.0420101
$365$	−4.11462	−0.215369
$366$	0	0
$367$	26.7971	1.39879	0.699397	−	0.714733i	$-0.253452\pi$
$367$	26.7971	1.39879	0.699397	+	0.714733i	$0.253452\pi$
$368$	21.2618	1.10835
$369$	0	0
$370$	−14.1017	−0.733113
$371$	−5.60300	−0.290893
$372$	0	0
$373$	13.3590	0.691705	0.345853	−	0.938289i	$-0.387590\pi$
$373$	13.3590	0.691705	0.345853	+	0.938289i	$0.387590\pi$
$374$	−2.68889	−0.139039
$375$	0	0
$376$	28.1650	1.45250
$377$	7.39853	0.381044
$378$	0	0
$379$	6.85083	0.351903	0.175952	−	0.984399i	$-0.443700\pi$
$379$	6.85083	0.351903	0.175952	+	0.984399i	$0.443700\pi$
$380$	−1.05086	−0.0539077
$381$	0	0
$382$	20.1619	1.03157
$383$	−3.57136	−0.182488	−0.0912440	−	0.995829i	$-0.529084\pi$
$383$	−3.57136	−0.182488	−0.0912440	+	0.995829i	$0.529084\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	−9.13828	−0.465126
$387$	0	0
$388$	6.75557	0.342962
$389$	8.35551	0.423641	0.211821	−	0.977309i	$-0.432061\pi$
$389$	8.35551	0.423641	0.211821	+	0.977309i	$0.432061\pi$
$390$	0	0
$391$	17.6128	0.890720
$392$	14.3308	0.723815
$393$	0	0
$394$	−14.8746	−0.749373
$395$	6.59502	0.331831
$396$	0	0
$397$	22.1847	1.11342	0.556709	−	0.830708i	$-0.312064\pi$
$397$	22.1847	1.11342	0.556709	+	0.830708i	$0.312064\pi$
$398$	14.1017	0.706855
$399$	0	0
$400$	8.77034	0.438517
$401$	−20.6987	−1.03365	−0.516823	−	0.856092i	$-0.672885\pi$
$401$	−20.6987	−1.03365	−0.516823	+	0.856092i	$0.672885\pi$
$402$	0	0
$403$	−4.68889	−0.233570
$404$	−8.41774	−0.418798
$405$	0	0
$406$	−13.7047	−0.680154
$407$	8.85728	0.439039
$408$	0	0
$409$	20.8528	1.03111	0.515553	−	0.856858i	$-0.327586\pi$
$409$	20.8528	1.03111	0.515553	+	0.856858i	$0.327586\pi$
$410$	−5.61285	−0.277199
$411$	0	0
$412$	4.00000	0.197066
$413$	−14.3051	−0.703909
$414$	0	0
$415$	7.05086	0.346113
$416$	−2.88739	−0.141566
$417$	0	0
$418$	−1.85236	−0.0906017
$419$	21.2400	1.03764	0.518821	−	0.854883i	$-0.326371\pi$
$419$	21.2400	1.03764	0.518821	+	0.854883i	$0.326371\pi$
$420$	0	0
$421$	−3.15257	−0.153647	−0.0768233	−	0.997045i	$-0.524478\pi$
$421$	−3.15257	−0.153647	−0.0768233	+	0.997045i	$0.524478\pi$
$422$	6.27010	0.305224
$423$	0	0
$424$	11.2641	0.547034
$425$	7.26517	0.352413
$426$	0	0
$427$	−17.5111	−0.847423
$428$	−6.82564	−0.329930
$429$	0	0
$430$	13.9728	0.673828
$431$	−40.1116	−1.93211	−0.966053	−	0.258345i	$-0.916823\pi$
$431$	−40.1116	−1.93211	−0.966053	+	0.258345i	$0.916823\pi$
$432$	0	0
$433$	−36.1017	−1.73494	−0.867469	−	0.497492i	$-0.834254\pi$
$433$	−36.1017	−1.73494	−0.867469	+	0.497492i	$0.834254\pi$
$434$	8.68550	0.416917
$435$	0	0
$436$	−2.00751	−0.0961422
$437$	12.1334	0.580417
$438$	0	0
$439$	13.2050	0.630238	0.315119	−	0.949052i	$-0.397956\pi$
$439$	13.2050	0.630238	0.315119	+	0.949052i	$0.397956\pi$
$440$	4.02074	0.191681
$441$	0	0
$442$	2.68889	0.127898
$443$	−13.9813	−0.664270	−0.332135	−	0.943232i	$-0.607769\pi$
$443$	−13.9813	−0.664270	−0.332135	+	0.943232i	$0.607769\pi$
$444$	0	0
$445$	0.903212	0.0428164
$446$	−10.0810	−0.477348
$447$	0	0
$448$	13.5036	0.637987
$449$	−19.3210	−0.911812	−0.455906	−	0.890028i	$-0.650685\pi$
$449$	−19.3210	−0.911812	−0.455906	+	0.890028i	$0.650685\pi$
$450$	0	0
$451$	3.52543	0.166006
$452$	−9.63465	−0.453176
$453$	0	0
$454$	6.40144	0.300435
$455$	2.00000	0.0937614
$456$	0	0
$457$	−36.3970	−1.70258	−0.851290	−	0.524696i	$-0.824179\pi$
$457$	−36.3970	−1.70258	−0.851290	+	0.524696i	$0.824179\pi$
$458$	−31.0291	−1.44989
$459$	0	0
$460$	−5.47949	−0.255483
$461$	23.5254	1.09569	0.547844	−	0.836580i	$-0.315449\pi$
$461$	23.5254	1.09569	0.547844	+	0.836580i	$0.315449\pi$
$462$	0	0
$463$	−36.3531	−1.68947	−0.844735	−	0.535184i	$-0.820242\pi$
$463$	−36.3531	−1.68947	−0.844735	+	0.535184i	$0.820242\pi$
$464$	19.7768	0.918114
$465$	0	0
$466$	−2.92396	−0.135450
$467$	32.5161	1.50466	0.752332	−	0.658784i	$-0.228929\pi$
$467$	32.5161	1.50466	0.752332	+	0.658784i	$0.228929\pi$
$468$	0	0
$469$	8.00984	0.369860
$470$	14.6222	0.674473
$471$	0	0
$472$	28.7586	1.32372
$473$	−8.77631	−0.403535
$474$	0	0
$475$	5.00492	0.229642
$476$	1.77478	0.0813470
$477$	0	0
$478$	−21.6904	−0.992097
$479$	−33.6499	−1.53750	−0.768751	−	0.639548i	$-0.779122\pi$
$479$	−33.6499	−1.53750	−0.768751	+	0.639548i	$0.779122\pi$
$480$	0	0
$481$	−8.85728	−0.403857
$482$	−4.78415	−0.217912
$483$	0	0
$484$	−0.525428	−0.0238831
$485$	16.8573	0.765450
$486$	0	0
$487$	17.0988	0.774820	0.387410	−	0.921907i	$-0.373370\pi$
$487$	17.0988	0.774820	0.387410	+	0.921907i	$0.373370\pi$
$488$	35.2039	1.59361
$489$	0	0
$490$	7.44002	0.336106
$491$	−30.2034	−1.36306	−0.681531	−	0.731790i	$-0.738685\pi$
$491$	−30.2034	−1.36306	−0.681531	+	0.731790i	$0.738685\pi$
$492$	0	0
$493$	16.3827	0.737840
$494$	1.85236	0.0833415
$495$	0	0
$496$	−12.5337	−0.562782
$497$	22.0731	0.990115
$498$	0	0
$499$	38.1367	1.70724	0.853618	−	0.520900i	$-0.174404\pi$
$499$	38.1367	1.70724	0.853618	+	0.520900i	$0.174404\pi$
$500$	−5.70471	−0.255123
$501$	0	0
$502$	−28.3654	−1.26601
$503$	23.8666	1.06416	0.532081	−	0.846694i	$-0.321410\pi$
$503$	23.8666	1.06416	0.532081	+	0.846694i	$0.321410\pi$
$504$	0	0
$505$	−21.0049	−0.934707
$506$	−9.65878	−0.429385
$507$	0	0
$508$	10.8147	0.479826
$509$	−12.8222	−0.568336	−0.284168	−	0.958774i	$-0.591717\pi$
$509$	−12.8222	−0.568336	−0.284168	+	0.958774i	$0.591717\pi$
$510$	0	0
$511$	4.78721	0.211774
$512$	−24.1131	−1.06566
$513$	0	0
$514$	−2.07007	−0.0913068
$515$	9.98126	0.439827
$516$	0	0
$517$	−9.18421	−0.403921
$518$	16.4068	0.720875
$519$	0	0
$520$	−4.02074	−0.176321
$521$	−3.63158	−0.159103	−0.0795513	−	0.996831i	$-0.525349\pi$
$521$	−3.63158	−0.159103	−0.0795513	+	0.996831i	$0.525349\pi$
$522$	0	0
$523$	−9.40837	−0.411399	−0.205700	−	0.978615i	$-0.565947\pi$
$523$	−9.40837	−0.411399	−0.205700	+	0.978615i	$0.565947\pi$
$524$	−0.295286	−0.0128996
$525$	0	0
$526$	1.34920	0.0588281
$527$	−10.3827	−0.452278
$528$	0	0
$529$	40.2672	1.75075
$530$	5.84791	0.254017
$531$	0	0
$532$	1.22263	0.0530079
$533$	−3.52543	−0.152703
$534$	0	0
$535$	−17.0321	−0.736363
$536$	−16.1028	−0.695534
$537$	0	0
$538$	−10.6824	−0.460553
$539$	−4.67307	−0.201283
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	33.2400	1.42778
$543$	0	0
$544$	−6.39361	−0.274124
$545$	−5.00937	−0.214578
$546$	0	0
$547$	1.73530	0.0741961	0.0370981	−	0.999312i	$-0.488189\pi$
$547$	1.73530	0.0741961	0.0370981	+	0.999312i	$0.488189\pi$
$548$	2.66709	0.113933
$549$	0	0
$550$	−3.98418	−0.169886
$551$	11.2859	0.480796
$552$	0	0
$553$	−7.67307	−0.326292
$554$	−31.4608	−1.33664
$555$	0	0
$556$	−9.74726	−0.413376
$557$	21.2400	0.899967	0.449984	−	0.893037i	$-0.351430\pi$
$557$	21.2400	0.899967	0.449984	+	0.893037i	$0.351430\pi$
$558$	0	0
$559$	8.77631	0.371198
$560$	5.34614	0.225916
$561$	0	0
$562$	6.99508	0.295070
$563$	−22.7841	−0.960237	−0.480119	−	0.877204i	$-0.659406\pi$
$563$	−22.7841	−0.960237	−0.480119	+	0.877204i	$0.659406\pi$
$564$	0	0
$565$	−24.0415	−1.01143
$566$	−3.67952	−0.154662
$567$	0	0
$568$	−44.3752	−1.86194
$569$	−31.9704	−1.34027	−0.670134	−	0.742240i	$-0.733763\pi$
$569$	−31.9704	−1.34027	−0.670134	+	0.742240i	$0.733763\pi$
$570$	0	0
$571$	14.7032	0.615309	0.307655	−	0.951498i	$-0.400456\pi$
$571$	14.7032	0.615309	0.307655	+	0.951498i	$0.400456\pi$
$572$	0.525428	0.0219692
$573$	0	0
$574$	6.53035	0.272572
$575$	26.0973	1.08833
$576$	0	0
$577$	18.9491	0.788863	0.394432	−	0.918925i	$-0.370942\pi$
$577$	18.9491	0.788863	0.394432	+	0.918925i	$0.370942\pi$
$578$	−14.6894	−0.610997
$579$	0	0
$580$	−5.09679	−0.211633
$581$	−8.20342	−0.340335
$582$	0	0
$583$	−3.67307	−0.152123
$584$	−9.62408	−0.398247
$585$	0	0
$586$	11.0237	0.455383
$587$	29.0736	1.20000	0.599998	−	0.800001i	$-0.295168\pi$
$587$	29.0736	1.20000	0.599998	+	0.800001i	$0.295168\pi$
$588$	0	0
$589$	−7.15257	−0.294716
$590$	14.9304	0.614675
$591$	0	0
$592$	−23.6761	−0.973083
$593$	−0.312639	−0.0128386	−0.00641928	−	0.999979i	$-0.502043\pi$
$593$	−0.312639	−0.0128386	−0.00641928	+	0.999979i	$0.502043\pi$
$594$	0	0
$595$	4.42864	0.181557
$596$	−2.30279	−0.0943262
$597$	0	0
$598$	9.65878	0.394977
$599$	19.8666	0.811729	0.405865	−	0.913933i	$-0.366970\pi$
$599$	19.8666	0.811729	0.405865	+	0.913933i	$0.366970\pi$
$600$	0	0
$601$	−23.9813	−0.978216	−0.489108	−	0.872223i	$-0.662678\pi$
$601$	−23.9813	−0.978216	−0.489108	+	0.872223i	$0.662678\pi$
$602$	−16.2569	−0.662580
$603$	0	0
$604$	−0.646350	−0.0262996
$605$	−1.31111	−0.0533041
$606$	0	0
$607$	12.5412	0.509034	0.254517	−	0.967068i	$-0.418084\pi$
$607$	12.5412	0.509034	0.254517	+	0.967068i	$0.418084\pi$
$608$	−4.40451	−0.178626
$609$	0	0
$610$	18.2766	0.739996
$611$	9.18421	0.371553
$612$	0	0
$613$	3.79213	0.153163	0.0765814	−	0.997063i	$-0.475599\pi$
$613$	3.79213	0.153163	0.0765814	+	0.997063i	$0.475599\pi$
$614$	−6.59808	−0.266277
$615$	0	0
$616$	−4.67799	−0.188482
$617$	−0.278989	−0.0112317	−0.00561583	−	0.999984i	$-0.501788\pi$
$617$	−0.278989	−0.0112317	−0.00561583	+	0.999984i	$0.501788\pi$
$618$	0	0
$619$	9.69826	0.389806	0.194903	−	0.980823i	$-0.437561\pi$
$619$	9.69826	0.389806	0.194903	+	0.980823i	$0.437561\pi$
$620$	3.23014	0.129726
$621$	0	0
$622$	11.4938	0.460859
$623$	−1.05086	−0.0421016
$624$	0	0
$625$	2.16992	0.0867967
$626$	−28.7066	−1.14735
$627$	0	0
$628$	−4.00751	−0.159917
$629$	−19.6128	−0.782015
$630$	0	0
$631$	10.6889	0.425518	0.212759	−	0.977105i	$-0.431755\pi$
$631$	10.6889	0.425518	0.212759	+	0.977105i	$0.431755\pi$
$632$	15.4257	0.613602
$633$	0	0
$634$	−19.0015	−0.754647
$635$	26.9862	1.07091
$636$	0	0
$637$	4.67307	0.185154
$638$	−8.98418	−0.355687
$639$	0	0
$640$	−6.52251	−0.257825
$641$	−31.0321	−1.22570	−0.612848	−	0.790201i	$-0.709976\pi$
$641$	−31.0321	−1.22570	−0.612848	+	0.790201i	$0.709976\pi$
$642$	0	0
$643$	24.6099	0.970521	0.485261	−	0.874370i	$-0.338725\pi$
$643$	24.6099	0.970521	0.485261	+	0.874370i	$0.338725\pi$
$644$	6.37520	0.251218
$645$	0	0
$646$	4.10171	0.161380
$647$	−24.3368	−0.956777	−0.478389	−	0.878148i	$-0.658779\pi$
$647$	−24.3368	−0.956777	−0.478389	+	0.878148i	$0.658779\pi$
$648$	0	0
$649$	−9.37778	−0.368110
$650$	3.98418	0.156272
$651$	0	0
$652$	−3.80981	−0.149204
$653$	17.7877	0.696086	0.348043	−	0.937479i	$-0.386846\pi$
$653$	17.7877	0.696086	0.348043	+	0.937479i	$0.386846\pi$
$654$	0	0
$655$	−0.736833	−0.0287904
$656$	−9.42372	−0.367934
$657$	0	0
$658$	−17.0124	−0.663214
$659$	−28.8256	−1.12289	−0.561444	−	0.827515i	$-0.689754\pi$
$659$	−28.8256	−1.12289	−0.561444	+	0.827515i	$0.689754\pi$
$660$	0	0
$661$	−39.1655	−1.52336	−0.761680	−	0.647953i	$-0.775625\pi$
$661$	−39.1655	−1.52336	−0.761680	+	0.647953i	$0.775625\pi$
$662$	−23.8972	−0.928792
$663$	0	0
$664$	16.4919	0.640011
$665$	3.05086	0.118307
$666$	0	0
$667$	58.8484	2.27862
$668$	0.155152	0.00600300
$669$	0	0
$670$	−8.35996	−0.322973
$671$	−11.4795	−0.443161
$672$	0	0
$673$	−46.8671	−1.80659	−0.903297	−	0.429015i	$-0.858861\pi$
$673$	−46.8671	−1.80659	−0.903297	+	0.429015i	$0.858861\pi$
$674$	13.3461	0.514074
$675$	0	0
$676$	−0.525428	−0.0202088
$677$	−22.9699	−0.882805	−0.441402	−	0.897309i	$-0.645519\pi$
$677$	−22.9699	−0.882805	−0.441402	+	0.897309i	$0.645519\pi$
$678$	0	0
$679$	−19.6128	−0.752672
$680$	−8.90321	−0.341423
$681$	0	0
$682$	5.69381	0.218027
$683$	21.7333	0.831601	0.415801	−	0.909456i	$-0.363501\pi$
$683$	21.7333	0.831601	0.415801	+	0.909456i	$0.363501\pi$
$684$	0	0
$685$	6.65524	0.254284
$686$	−21.6227	−0.825558
$687$	0	0
$688$	23.4597	0.894393
$689$	3.67307	0.139933
$690$	0	0
$691$	33.4445	1.27229	0.636144	−	0.771571i	$-0.280529\pi$
$691$	33.4445	1.27229	0.636144	+	0.771571i	$0.280529\pi$
$692$	−9.22675	−0.350748
$693$	0	0
$694$	−21.0291	−0.798252
$695$	−24.3225	−0.922604
$696$	0	0
$697$	−7.80642	−0.295689
$698$	−20.8671	−0.789832
$699$	0	0
$700$	2.62972	0.0993942
$701$	31.7540	1.19933	0.599667	−	0.800250i	$-0.295300\pi$
$701$	31.7540	1.19933	0.599667	+	0.800250i	$0.295300\pi$
$702$	0	0
$703$	−13.5111	−0.509582
$704$	8.85236	0.333636
$705$	0	0
$706$	17.5288	0.659706
$707$	24.4385	0.919104
$708$	0	0
$709$	25.3778	0.953083	0.476541	−	0.879152i	$-0.341890\pi$
$709$	25.3778	0.953083	0.476541	+	0.879152i	$0.341890\pi$
$710$	−23.0379	−0.864599
$711$	0	0
$712$	2.11261	0.0791734
$713$	−37.2958	−1.39674
$714$	0	0
$715$	1.31111	0.0490327
$716$	6.35106	0.237350
$717$	0	0
$718$	19.6084	0.731779
$719$	11.6128	0.433086	0.216543	−	0.976273i	$-0.430522\pi$
$719$	11.6128	0.433086	0.216543	+	0.976273i	$0.430522\pi$
$720$	0	0
$721$	−11.6128	−0.432485
$722$	−20.2464	−0.753494
$723$	0	0
$724$	−4.05578	−0.150732
$725$	24.2745	0.901534
$726$	0	0
$727$	−42.5303	−1.57736	−0.788682	−	0.614802i	$-0.789236\pi$
$727$	−42.5303	−1.57736	−0.788682	+	0.614802i	$0.789236\pi$
$728$	4.67799	0.173378
$729$	0	0
$730$	−4.99646	−0.184927
$731$	19.4336	0.718776
$732$	0	0
$733$	16.2997	0.602044	0.301022	−	0.953617i	$-0.402672\pi$
$733$	16.2997	0.602044	0.301022	+	0.953617i	$0.402672\pi$
$734$	32.5402	1.20108
$735$	0	0
$736$	−22.9665	−0.846556
$737$	5.25088	0.193419
$738$	0	0
$739$	27.5353	1.01290	0.506451	−	0.862269i	$-0.330957\pi$
$739$	27.5353	1.01290	0.506451	+	0.862269i	$0.330957\pi$
$740$	6.10171	0.224303
$741$	0	0
$742$	−6.80384	−0.249777
$743$	−34.1432	−1.25259	−0.626296	−	0.779585i	$-0.715430\pi$
$743$	−34.1432	−1.25259	−0.626296	+	0.779585i	$0.715430\pi$
$744$	0	0
$745$	−5.74620	−0.210525
$746$	16.2222	0.593935
$747$	0	0
$748$	1.16346	0.0425405
$749$	19.8163	0.724071
$750$	0	0
$751$	5.08250	0.185463	0.0927315	−	0.995691i	$-0.470440\pi$
$751$	5.08250	0.185463	0.0927315	+	0.995691i	$0.470440\pi$
$752$	24.5500	0.895248
$753$	0	0
$754$	8.98418	0.327184
$755$	−1.61285	−0.0586975
$756$	0	0
$757$	−20.1062	−0.730771	−0.365385	−	0.930856i	$-0.619063\pi$
$757$	−20.1062	−0.730771	−0.365385	+	0.930856i	$0.619063\pi$
$758$	8.31909	0.302163
$759$	0	0
$760$	−6.13335	−0.222480
$761$	7.47505	0.270970	0.135485	−	0.990779i	$-0.456741\pi$
$761$	7.47505	0.270970	0.135485	+	0.990779i	$0.456741\pi$
$762$	0	0
$763$	5.82822	0.210996
$764$	−8.72393	−0.315621
$765$	0	0
$766$	−4.33677	−0.156694
$767$	9.37778	0.338612
$768$	0	0
$769$	−50.9403	−1.83695	−0.918476	−	0.395476i	$-0.870580\pi$
$769$	−50.9403	−1.83695	−0.918476	+	0.395476i	$0.870580\pi$
$770$	−2.42864	−0.0875221
$771$	0	0
$772$	3.95407	0.142310
$773$	−47.7309	−1.71676	−0.858380	−	0.513015i	$-0.828529\pi$
$773$	−47.7309	−1.71676	−0.858380	+	0.513015i	$0.828529\pi$
$774$	0	0
$775$	−15.3842	−0.552618
$776$	39.4291	1.41542
$777$	0	0
$778$	10.1463	0.363761
$779$	−5.37778	−0.192679
$780$	0	0
$781$	14.4701	0.517782
$782$	21.3876	0.764820
$783$	0	0
$784$	12.4914	0.446123
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−17.6084	−0.627672	−0.313836	−	0.949477i	$-0.601614\pi$
$787$	−17.6084	−0.627672	−0.313836	+	0.949477i	$0.601614\pi$
$788$	6.43615	0.229278
$789$	0	0
$790$	8.00846	0.284928
$791$	27.9714	0.994549
$792$	0	0
$793$	11.4795	0.407649
$794$	26.9393	0.956040
$795$	0	0
$796$	−6.10171	−0.216269
$797$	−7.32741	−0.259550	−0.129775	−	0.991543i	$-0.541426\pi$
$797$	−7.32741	−0.259550	−0.129775	+	0.991543i	$0.541426\pi$
$798$	0	0
$799$	20.3368	0.719463
$800$	−9.47352	−0.334939
$801$	0	0
$802$	−25.1349	−0.887544
$803$	3.13828	0.110747
$804$	0	0
$805$	15.9081	0.560688
$806$	−5.69381	−0.200556
$807$	0	0
$808$	−49.1304	−1.72840
$809$	−19.2968	−0.678440	−0.339220	−	0.940707i	$-0.610163\pi$
$809$	−19.2968	−0.678440	−0.339220	+	0.940707i	$0.610163\pi$
$810$	0	0
$811$	31.6271	1.11058	0.555290	−	0.831657i	$-0.312607\pi$
$811$	31.6271	1.11058	0.555290	+	0.831657i	$0.312607\pi$
$812$	5.92993	0.208100
$813$	0	0
$814$	10.7556	0.376982
$815$	−9.50669	−0.333005
$816$	0	0
$817$	13.3876	0.468374
$818$	25.3220	0.885363
$819$	0	0
$820$	2.42864	0.0848118
$821$	−37.4652	−1.30754	−0.653772	−	0.756691i	$-0.726815\pi$
$821$	−37.4652	−1.30754	−0.653772	+	0.756691i	$0.726815\pi$
$822$	0	0
$823$	28.8988	1.00735	0.503674	−	0.863894i	$-0.331981\pi$
$823$	28.8988	1.00735	0.503674	+	0.863894i	$0.331981\pi$
$824$	23.3461	0.813301
$825$	0	0
$826$	−17.3710	−0.604414
$827$	−19.7319	−0.686146	−0.343073	−	0.939309i	$-0.611468\pi$
$827$	−19.7319	−0.686146	−0.343073	+	0.939309i	$0.611468\pi$
$828$	0	0
$829$	−3.44785	−0.119749	−0.0598744	−	0.998206i	$-0.519070\pi$
$829$	−3.44785	−0.119749	−0.0598744	+	0.998206i	$0.519070\pi$
$830$	8.56199	0.297191
$831$	0	0
$832$	−8.85236	−0.306900
$833$	10.3477	0.358526
$834$	0	0
$835$	0.387152	0.0133980
$836$	0.801502	0.0277205
$837$	0	0
$838$	25.7921	0.890974
$839$	52.2578	1.80414	0.902070	−	0.431590i	$-0.142047\pi$
$839$	52.2578	1.80414	0.902070	+	0.431590i	$0.142047\pi$
$840$	0	0
$841$	25.7382	0.887525
$842$	−3.82822	−0.131929
$843$	0	0
$844$	−2.71303	−0.0933862
$845$	−1.31111	−0.0451035
$846$	0	0
$847$	1.52543	0.0524143
$848$	9.81838	0.337164
$849$	0	0
$850$	8.82225	0.302601
$851$	−70.4514	−2.41504
$852$	0	0
$853$	31.2988	1.07165	0.535826	−	0.844329i	$-0.320000\pi$
$853$	31.2988	1.07165	0.535826	+	0.844329i	$0.320000\pi$
$854$	−21.2641	−0.727643
$855$	0	0
$856$	−39.8381	−1.36164
$857$	−37.1131	−1.26776	−0.633879	−	0.773432i	$-0.718538\pi$
$857$	−37.1131	−1.26776	−0.633879	+	0.773432i	$0.718538\pi$
$858$	0	0
$859$	−16.6953	−0.569638	−0.284819	−	0.958581i	$-0.591933\pi$
$859$	−16.6953	−0.569638	−0.284819	+	0.958581i	$0.591933\pi$
$860$	−6.04593	−0.206165
$861$	0	0
$862$	−48.7083	−1.65901
$863$	41.8765	1.42549	0.712746	−	0.701422i	$-0.247451\pi$
$863$	41.8765	1.42549	0.712746	+	0.701422i	$0.247451\pi$
$864$	0	0
$865$	−23.0237	−0.782828
$866$	−43.8390	−1.48971
$867$	0	0
$868$	−3.75815	−0.127560
$869$	−5.03011	−0.170635
$870$	0	0
$871$	−5.25088	−0.177919
$872$	−11.7169	−0.396784
$873$	0	0
$874$	14.7338	0.498377
$875$	16.5620	0.559898
$876$	0	0
$877$	−6.54909	−0.221147	−0.110573	−	0.993868i	$-0.535269\pi$
$877$	−6.54909	−0.221147	−0.110573	+	0.993868i	$0.535269\pi$
$878$	16.0350	0.541156
$879$	0	0
$880$	3.50468	0.118143
$881$	15.6316	0.526641	0.263321	−	0.964708i	$-0.415182\pi$
$881$	15.6316	0.526641	0.263321	+	0.964708i	$0.415182\pi$
$882$	0	0
$883$	19.2257	0.646996	0.323498	−	0.946229i	$-0.395141\pi$
$883$	19.2257	0.646996	0.323498	+	0.946229i	$0.395141\pi$
$884$	−1.16346	−0.0391315
$885$	0	0
$886$	−16.9777	−0.570378
$887$	−8.88892	−0.298461	−0.149230	−	0.988802i	$-0.547680\pi$
$887$	−8.88892	−0.298461	−0.149230	+	0.988802i	$0.547680\pi$
$888$	0	0
$889$	−31.3975	−1.05304
$890$	1.09679	0.0367644
$891$	0	0
$892$	4.36196	0.146049
$893$	14.0098	0.468822
$894$	0	0
$895$	15.8479	0.529737
$896$	7.58871	0.253521
$897$	0	0
$898$	−23.4618	−0.782931
$899$	−34.6909	−1.15701
$900$	0	0
$901$	8.13335	0.270961
$902$	4.28100	0.142542
$903$	0	0
$904$	−56.2330	−1.87028
$905$	−10.1204	−0.336415
$906$	0	0
$907$	−15.5714	−0.517039	−0.258519	−	0.966006i	$-0.583235\pi$
$907$	−15.5714	−0.517039	−0.258519	+	0.966006i	$0.583235\pi$
$908$	−2.76986	−0.0919210
$909$	0	0
$910$	2.42864	0.0805086
$911$	42.5215	1.40880	0.704399	−	0.709804i	$-0.251216\pi$
$911$	42.5215	1.40880	0.704399	+	0.709804i	$0.251216\pi$
$912$	0	0
$913$	−5.37778	−0.177979
$914$	−44.1976	−1.46193
$915$	0	0
$916$	13.4261	0.443609
$917$	0.857279	0.0283098
$918$	0	0
$919$	−14.5640	−0.480422	−0.240211	−	0.970721i	$-0.577217\pi$
$919$	−14.5640	−0.480422	−0.240211	+	0.970721i	$0.577217\pi$
$920$	−31.9813	−1.05439
$921$	0	0
$922$	28.5674	0.940817
$923$	−14.4701	−0.476290
$924$	0	0
$925$	−29.0607	−0.955510
$926$	−44.1443	−1.45067
$927$	0	0
$928$	−21.3624	−0.701256
$929$	39.5689	1.29821	0.649107	−	0.760697i	$-0.275143\pi$
$929$	39.5689	1.29821	0.649107	+	0.760697i	$0.275143\pi$
$930$	0	0
$931$	7.12843	0.233625
$932$	1.26517	0.0414422
$933$	0	0
$934$	39.4849	1.29199
$935$	2.90321	0.0949452
$936$	0	0
$937$	31.9625	1.04417	0.522085	−	0.852893i	$-0.325154\pi$
$937$	31.9625	1.04417	0.522085	+	0.852893i	$0.325154\pi$
$938$	9.72651	0.317582
$939$	0	0
$940$	−6.32693	−0.206362
$941$	−50.1891	−1.63612	−0.818059	−	0.575134i	$-0.804950\pi$
$941$	−50.1891	−1.63612	−0.818059	+	0.575134i	$0.804950\pi$
$942$	0	0
$943$	−28.0415	−0.913156
$944$	25.0675	0.815877
$945$	0	0
$946$	−10.6572	−0.346497
$947$	0.742662	0.0241333	0.0120666	−	0.999927i	$-0.496159\pi$
$947$	0.742662	0.0241333	0.0120666	+	0.999927i	$0.496159\pi$
$948$	0	0
$949$	−3.13828	−0.101873
$950$	6.07758	0.197183
$951$	0	0
$952$	10.3586	0.335723
$953$	−42.3861	−1.37302	−0.686510	−	0.727120i	$-0.740858\pi$
$953$	−42.3861	−1.37302	−0.686510	+	0.727120i	$0.740858\pi$
$954$	0	0
$955$	−21.7690	−0.704427
$956$	9.38529	0.303542
$957$	0	0
$958$	−40.8617	−1.32018
$959$	−7.74314	−0.250039
$960$	0	0
$961$	−9.01429	−0.290784
$962$	−10.7556	−0.346773
$963$	0	0
$964$	2.07007	0.0666724
$965$	9.86665	0.317619
$966$	0	0
$967$	−33.2083	−1.06791	−0.533954	−	0.845513i	$-0.679295\pi$
$967$	−33.2083	−1.06791	−0.533954	+	0.845513i	$0.679295\pi$
$968$	−3.06668	−0.0985667
$969$	0	0
$970$	20.4701	0.657256
$971$	−29.2904	−0.939973	−0.469986	−	0.882674i	$-0.655741\pi$
$971$	−29.2904	−0.939973	−0.469986	+	0.882674i	$0.655741\pi$
$972$	0	0
$973$	28.2983	0.907203
$974$	20.7634	0.665302
$975$	0	0
$976$	30.6855	0.982219
$977$	28.3847	0.908107	0.454054	−	0.890974i	$-0.349977\pi$
$977$	28.3847	0.908107	0.454054	+	0.890974i	$0.349977\pi$
$978$	0	0
$979$	−0.688892	−0.0220171
$980$	−3.21924	−0.102835
$981$	0	0
$982$	−36.6766	−1.17040
$983$	−44.2578	−1.41161	−0.705803	−	0.708409i	$-0.749413\pi$
$983$	−44.2578	−1.41161	−0.705803	+	0.708409i	$0.749413\pi$
$984$	0	0
$985$	16.0602	0.511721
$986$	19.8938	0.633549
$987$	0	0
$988$	−0.801502	−0.0254992
$989$	69.8074	2.21975
$990$	0	0
$991$	32.4197	1.02985	0.514924	−	0.857236i	$-0.327820\pi$
$991$	32.4197	1.02985	0.514924	+	0.857236i	$0.327820\pi$
$992$	13.5387	0.429853
$993$	0	0
$994$	26.8038	0.850166
$995$	−15.2257	−0.482687
$996$	0	0
$997$	42.5205	1.34664	0.673319	−	0.739352i	$-0.264868\pi$
$997$	42.5205	1.34664	0.673319	+	0.739352i	$0.264868\pi$
$998$	46.3102	1.46592
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.h.1.3		3
3.2	odd	2		429.2.a.g.1.1	✓	3
12.11	even	2		6864.2.a.bs.1.2		3
33.32	even	2		4719.2.a.q.1.3		3
39.38	odd	2		5577.2.a.j.1.3		3

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

\(f(q)\)	\(=\)	\(q+1.21432 q^{2} -0.525428 q^{4} -1.31111 q^{5} +1.52543 q^{7} -3.06668 q^{8} +O(q^{10})\)	\(q+1.21432 q^{2} -0.525428 q^{4} -1.31111 q^{5} +1.52543 q^{7} -3.06668 q^{8} -1.59210 q^{10} +1.00000 q^{11} -1.00000 q^{13} +1.85236 q^{14} -2.67307 q^{16} -2.21432 q^{17} -1.52543 q^{19} +0.688892 q^{20} +1.21432 q^{22} -7.95407 q^{23} -3.28100 q^{25} -1.21432 q^{26} -0.801502 q^{28} -7.39853 q^{29} +4.68889 q^{31} +2.88739 q^{32} -2.68889 q^{34} -2.00000 q^{35} +8.85728 q^{37} -1.85236 q^{38} +4.02074 q^{40} +3.52543 q^{41} -8.77631 q^{43} -0.525428 q^{44} -9.65878 q^{46} -9.18421 q^{47} -4.67307 q^{49} -3.98418 q^{50} +0.525428 q^{52} -3.67307 q^{53} -1.31111 q^{55} -4.67799 q^{56} -8.98418 q^{58} -9.37778 q^{59} -11.4795 q^{61} +5.69381 q^{62} +8.85236 q^{64} +1.31111 q^{65} +5.25088 q^{67} +1.16346 q^{68} -2.42864 q^{70} +14.4701 q^{71} +3.13828 q^{73} +10.7556 q^{74} +0.801502 q^{76} +1.52543 q^{77} -5.03011 q^{79} +3.50468 q^{80} +4.28100 q^{82} -5.37778 q^{83} +2.90321 q^{85} -10.6572 q^{86} -3.06668 q^{88} -0.688892 q^{89} -1.52543 q^{91} +4.17929 q^{92} -11.1526 q^{94} +2.00000 q^{95} -12.8573 q^{97} -5.67460 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 5 q^{4} - 4 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 5 * q^4 - 4 * q^5 - 2 * q^7 - 9 * q^8	\( 3 q - 3 q^{2} + 5 q^{4} - 4 q^{5} - 2 q^{7} - 9 q^{8} + 2 q^{10} + 3 q^{11} - 3 q^{13} + 12 q^{14} + 5 q^{16} + 2 q^{19} + 2 q^{20} - 3 q^{22} - 4 q^{23} - 3 q^{25} + 3 q^{26} - 22 q^{28} - 2 q^{29} + 14 q^{31} - 11 q^{32} - 8 q^{34} - 6 q^{35} - 12 q^{38} - 8 q^{40} + 4 q^{41} - 6 q^{43} + 5 q^{44} - 22 q^{46} - 14 q^{47} - q^{49} + q^{50} - 5 q^{52} + 2 q^{53} - 4 q^{55} + 32 q^{56} - 14 q^{58} - 28 q^{59} - 8 q^{61} - 16 q^{62} + 33 q^{64} + 4 q^{65} + 2 q^{67} + 10 q^{68} + 6 q^{70} - 10 q^{71} - 24 q^{73} + 32 q^{74} + 22 q^{76} - 2 q^{77} - 22 q^{79} + 24 q^{80} + 6 q^{82} - 16 q^{83} + 2 q^{85} - 6 q^{86} - 9 q^{88} - 2 q^{89} + 2 q^{91} + 32 q^{92} + 6 q^{94} + 6 q^{95} - 12 q^{97} - 23 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 5 * q^4 - 4 * q^5 - 2 * q^7 - 9 * q^8 + 2 * q^10 + 3 * q^11 - 3 * q^13 + 12 * q^14 + 5 * q^16 + 2 * q^19 + 2 * q^20 - 3 * q^22 - 4 * q^23 - 3 * q^25 + 3 * q^26 - 22 * q^28 - 2 * q^29 + 14 * q^31 - 11 * q^32 - 8 * q^34 - 6 * q^35 - 12 * q^38 - 8 * q^40 + 4 * q^41 - 6 * q^43 + 5 * q^44 - 22 * q^46 - 14 * q^47 - q^49 + q^50 - 5 * q^52 + 2 * q^53 - 4 * q^55 + 32 * q^56 - 14 * q^58 - 28 * q^59 - 8 * q^61 - 16 * q^62 + 33 * q^64 + 4 * q^65 + 2 * q^67 + 10 * q^68 + 6 * q^70 - 10 * q^71 - 24 * q^73 + 32 * q^74 + 22 * q^76 - 2 * q^77 - 22 * q^79 + 24 * q^80 + 6 * q^82 - 16 * q^83 + 2 * q^85 - 6 * q^86 - 9 * q^88 - 2 * q^89 + 2 * q^91 + 32 * q^92 + 6 * q^94 + 6 * q^95 - 12 * q^97 - 23 * q^98

Introduction

L-functions

Modular forms

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