LMFDB - Embedded newform 2151.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 256 x^{7} - 134 x^{6} + 571 x^{5} + 23 x^{4} + \cdots - 31 $ x^12 - 3x^11 - 15x^10 + 47x^9 + 75x^8 - 256x^7 - 134x^6 + 571x^5 + 23x^4 - 479x^3 + 129x^2 + 88*x - 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.78161$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78161 q^{2} +5.73734 q^{4} +1.06781 q^{5} +2.18960 q^{7} -10.3958 q^{8} +O(q^{10})$	\(q-2.78161 q^{2} +5.73734 q^{4} +1.06781 q^{5} +2.18960 q^{7} -10.3958 q^{8} -2.97024 q^{10} +1.68112 q^{11} +0.385556 q^{13} -6.09060 q^{14} +17.4424 q^{16} +3.21887 q^{17} -2.35719 q^{19} +6.12640 q^{20} -4.67622 q^{22} -1.83491 q^{23} -3.85978 q^{25} -1.07246 q^{26} +12.5625 q^{28} -1.30265 q^{29} +2.03380 q^{31} -27.7262 q^{32} -8.95364 q^{34} +2.33808 q^{35} +11.7158 q^{37} +6.55679 q^{38} -11.1008 q^{40} +4.50764 q^{41} -4.09379 q^{43} +9.64516 q^{44} +5.10401 q^{46} +3.20852 q^{47} -2.20566 q^{49} +10.7364 q^{50} +2.21206 q^{52} +12.3901 q^{53} +1.79512 q^{55} -22.7626 q^{56} +3.62346 q^{58} +6.97090 q^{59} +13.2806 q^{61} -5.65724 q^{62} +42.2386 q^{64} +0.411701 q^{65} -7.74686 q^{67} +18.4678 q^{68} -6.50362 q^{70} +11.2837 q^{71} -14.7720 q^{73} -32.5889 q^{74} -13.5240 q^{76} +3.68098 q^{77} -0.872260 q^{79} +18.6252 q^{80} -12.5385 q^{82} -3.55591 q^{83} +3.43715 q^{85} +11.3873 q^{86} -17.4766 q^{88} +2.26911 q^{89} +0.844212 q^{91} -10.5275 q^{92} -8.92485 q^{94} -2.51704 q^{95} -17.6862 q^{97} +6.13529 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8} - 15 q^{11} + 7 q^{13} + 6 q^{14} + 21 q^{16} + 3 q^{17} + 10 q^{19} + 4 q^{20} + 23 q^{22} - 20 q^{23} + 19 q^{25} + 10 q^{26} + 34 q^{28} - 2 q^{29} + 10 q^{31} - 26 q^{32} + 12 q^{34} - 7 q^{35} + 30 q^{37} + 3 q^{38} + 25 q^{40} + 28 q^{41} + 48 q^{43} - 25 q^{44} + 22 q^{46} - 13 q^{47} + 19 q^{49} - 12 q^{50} + 24 q^{52} + 2 q^{53} + 8 q^{55} + 7 q^{56} + 42 q^{58} + 14 q^{59} + 14 q^{61} - 8 q^{62} + 9 q^{64} + 35 q^{65} + 52 q^{67} - 3 q^{68} - 33 q^{70} + 7 q^{71} + 14 q^{73} + 13 q^{74} - 12 q^{76} + 6 q^{77} + 15 q^{79} + 8 q^{80} - 61 q^{82} - 29 q^{83} + 8 q^{85} + 9 q^{86} + 11 q^{88} + 71 q^{89} + 13 q^{91} - 2 q^{92} - 22 q^{94} - 2 q^{95} - 2 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8 - 15 * q^11 + 7 * q^13 + 6 * q^14 + 21 * q^16 + 3 * q^17 + 10 * q^19 + 4 * q^20 + 23 * q^22 - 20 * q^23 + 19 * q^25 + 10 * q^26 + 34 * q^28 - 2 * q^29 + 10 * q^31 - 26 * q^32 + 12 * q^34 - 7 * q^35 + 30 * q^37 + 3 * q^38 + 25 * q^40 + 28 * q^41 + 48 * q^43 - 25 * q^44 + 22 * q^46 - 13 * q^47 + 19 * q^49 - 12 * q^50 + 24 * q^52 + 2 * q^53 + 8 * q^55 + 7 * q^56 + 42 * q^58 + 14 * q^59 + 14 * q^61 - 8 * q^62 + 9 * q^64 + 35 * q^65 + 52 * q^67 - 3 * q^68 - 33 * q^70 + 7 * q^71 + 14 * q^73 + 13 * q^74 - 12 * q^76 + 6 * q^77 + 15 * q^79 + 8 * q^80 - 61 * q^82 - 29 * q^83 + 8 * q^85 + 9 * q^86 + 11 * q^88 + 71 * q^89 + 13 * q^91 - 2 * q^92 - 22 * q^94 - 2 * q^95 - 2 * 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 $

$2$	−2.78161	−1.96689	−0.983447	−	0.181198i	$-0.942002\pi$
$2$	−2.78161	−1.96689	−0.983447	+	0.181198i	$0.942002\pi$
$3$	0	0
$3$	0	0
$4$	5.73734	2.86867
$5$	1.06781	0.477540	0.238770	−	0.971076i	$-0.423256\pi$
$5$	1.06781	0.477540	0.238770	+	0.971076i	$0.423256\pi$
$6$	0	0
$7$	2.18960	0.827590	0.413795	−	0.910370i	$-0.364203\pi$
$7$	2.18960	0.827590	0.413795	+	0.910370i	$0.364203\pi$
$8$	−10.3958	−3.67547
$9$	0	0
$10$	−2.97024	−0.939271
$11$	1.68112	0.506877	0.253439	−	0.967351i	$-0.418438\pi$
$11$	1.68112	0.506877	0.253439	+	0.967351i	$0.418438\pi$
$12$	0	0
$13$	0.385556	0.106934	0.0534670	−	0.998570i	$-0.482973\pi$
$13$	0.385556	0.106934	0.0534670	+	0.998570i	$0.482973\pi$
$14$	−6.09060	−1.62778
$15$	0	0
$16$	17.4424	4.36059
$17$	3.21887	0.780692	0.390346	−	0.920668i	$-0.372355\pi$
$17$	3.21887	0.780692	0.390346	+	0.920668i	$0.372355\pi$
$18$	0	0
$19$	−2.35719	−0.540777	−0.270389	−	0.962751i	$-0.587152\pi$
$19$	−2.35719	−0.540777	−0.270389	+	0.962751i	$0.587152\pi$
$20$	6.12640	1.36991
$21$	0	0
$22$	−4.67622	−0.996974
$23$	−1.83491	−0.382606	−0.191303	−	0.981531i	$-0.561271\pi$
$23$	−1.83491	−0.382606	−0.191303	+	0.981531i	$0.561271\pi$
$24$	0	0
$25$	−3.85978	−0.771955
$26$	−1.07246	−0.210328
$27$	0	0
$28$	12.5625	2.37408
$29$	−1.30265	−0.241896	−0.120948	−	0.992659i	$-0.538593\pi$
$29$	−1.30265	−0.241896	−0.120948	+	0.992659i	$0.538593\pi$
$30$	0	0
$31$	2.03380	0.365282	0.182641	−	0.983180i	$-0.441535\pi$
$31$	2.03380	0.365282	0.182641	+	0.983180i	$0.441535\pi$
$32$	−27.7262	−4.90134
$33$	0	0
$34$	−8.95364	−1.53554
$35$	2.33808	0.395208
$36$	0	0
$37$	11.7158	1.92607	0.963037	−	0.269370i	$-0.0868156\pi$
$37$	11.7158	1.92607	0.963037	+	0.269370i	$0.0868156\pi$
$38$	6.55679	1.06365
$39$	0	0
$40$	−11.1008	−1.75519
$41$	4.50764	0.703975	0.351987	−	0.936005i	$-0.385506\pi$
$41$	4.50764	0.703975	0.351987	+	0.936005i	$0.385506\pi$
$42$	0	0
$43$	−4.09379	−0.624298	−0.312149	−	0.950033i	$-0.601049\pi$
$43$	−4.09379	−0.624298	−0.312149	+	0.950033i	$0.601049\pi$
$44$	9.64516	1.45406
$45$	0	0
$46$	5.10401	0.752545
$47$	3.20852	0.468011	0.234006	−	0.972235i	$-0.424817\pi$
$47$	3.20852	0.468011	0.234006	+	0.972235i	$0.424817\pi$
$48$	0	0
$49$	−2.20566	−0.315095
$50$	10.7364	1.51835
$51$	0	0
$52$	2.21206	0.306758
$53$	12.3901	1.70191	0.850957	−	0.525236i	$-0.176023\pi$
$53$	12.3901	1.70191	0.850957	+	0.525236i	$0.176023\pi$
$54$	0	0
$55$	1.79512	0.242054
$56$	−22.7626	−3.04178
$57$	0	0
$58$	3.62346	0.475783
$59$	6.97090	0.907533	0.453767	−	0.891121i	$-0.350080\pi$
$59$	6.97090	0.907533	0.453767	+	0.891121i	$0.350080\pi$
$60$	0	0
$61$	13.2806	1.70040	0.850201	−	0.526458i	$-0.176480\pi$
$61$	13.2806	1.70040	0.850201	+	0.526458i	$0.176480\pi$
$62$	−5.65724	−0.718470
$63$	0	0
$64$	42.2386	5.27983
$65$	0.411701	0.0510653
$66$	0	0
$67$	−7.74686	−0.946429	−0.473215	−	0.880947i	$-0.656906\pi$
$67$	−7.74686	−0.946429	−0.473215	+	0.880947i	$0.656906\pi$
$68$	18.4678	2.23955
$69$	0	0
$70$	−6.50362	−0.777331
$71$	11.2837	1.33913	0.669565	−	0.742753i	$-0.266480\pi$
$71$	11.2837	1.33913	0.669565	+	0.742753i	$0.266480\pi$
$72$	0	0
$73$	−14.7720	−1.72893	−0.864466	−	0.502692i	$-0.832343\pi$
$73$	−14.7720	−1.72893	−0.864466	+	0.502692i	$0.832343\pi$
$74$	−32.5889	−3.78838
$75$	0	0
$76$	−13.5240	−1.55131
$77$	3.68098	0.419487
$78$	0	0
$79$	−0.872260	−0.0981369	−0.0490684	−	0.998795i	$-0.515625\pi$
$79$	−0.872260	−0.0981369	−0.0490684	+	0.998795i	$0.515625\pi$
$80$	18.6252	2.08236
$81$	0	0
$82$	−12.5385	−1.38464
$83$	−3.55591	−0.390312	−0.195156	−	0.980772i	$-0.562521\pi$
$83$	−3.55591	−0.390312	−0.195156	+	0.980772i	$0.562521\pi$
$84$	0	0
$85$	3.43715	0.372812
$86$	11.3873	1.22793
$87$	0	0
$88$	−17.4766	−1.86301
$89$	2.26911	0.240526	0.120263	−	0.992742i	$-0.461626\pi$
$89$	2.26911	0.240526	0.120263	+	0.992742i	$0.461626\pi$
$90$	0	0
$91$	0.844212	0.0884974
$92$	−10.5275	−1.09757
$93$	0	0
$94$	−8.92485	−0.920528
$95$	−2.51704	−0.258243
$96$	0	0
$97$	−17.6862	−1.79576	−0.897882	−	0.440235i	$-0.854895\pi$
$97$	−17.6862	−1.79576	−0.897882	+	0.440235i	$0.854895\pi$
$98$	6.13529	0.619758
$99$	0	0
$100$	−22.1448	−2.21448
$101$	1.63843	0.163030	0.0815151	−	0.996672i	$-0.474024\pi$
$101$	1.63843	0.163030	0.0815151	+	0.996672i	$0.474024\pi$
$102$	0	0
$103$	11.0366	1.08747	0.543734	−	0.839258i	$-0.317010\pi$
$103$	11.0366	1.08747	0.543734	+	0.839258i	$0.317010\pi$
$104$	−4.00816	−0.393033
$105$	0	0
$106$	−34.4644	−3.34748
$107$	−20.4261	−1.97467	−0.987333	−	0.158664i	$-0.949281\pi$
$107$	−20.4261	−1.97467	−0.987333	+	0.158664i	$0.949281\pi$
$108$	0	0
$109$	16.3055	1.56178	0.780891	−	0.624667i	$-0.214765\pi$
$109$	16.3055	1.56178	0.780891	+	0.624667i	$0.214765\pi$
$110$	−4.99333	−0.476095
$111$	0	0
$112$	38.1917	3.60878
$113$	11.4231	1.07460	0.537298	−	0.843392i	$-0.319445\pi$
$113$	11.4231	1.07460	0.537298	+	0.843392i	$0.319445\pi$
$114$	0	0
$115$	−1.95934	−0.182710
$116$	−7.47374	−0.693919
$117$	0	0
$118$	−19.3903	−1.78502
$119$	7.04804	0.646092
$120$	0	0
$121$	−8.17383	−0.743075
$122$	−36.9413	−3.34451
$123$	0	0
$124$	11.6686	1.04787
$125$	−9.46058	−0.846180
$126$	0	0
$127$	5.54806	0.492311	0.246155	−	0.969230i	$-0.420833\pi$
$127$	5.54806	0.492311	0.246155	+	0.969230i	$0.420833\pi$
$128$	−62.0389	−5.48352
$129$	0	0
$130$	−1.14519	−0.100440
$131$	−3.57696	−0.312521	−0.156260	−	0.987716i	$-0.549944\pi$
$131$	−3.57696	−0.312521	−0.156260	+	0.987716i	$0.549944\pi$
$132$	0	0
$133$	−5.16131	−0.447542
$134$	21.5487	1.86153
$135$	0	0
$136$	−33.4628	−2.86941
$137$	−17.0126	−1.45348	−0.726741	−	0.686911i	$-0.758966\pi$
$137$	−17.0126	−1.45348	−0.726741	+	0.686911i	$0.758966\pi$
$138$	0	0
$139$	15.9062	1.34915	0.674574	−	0.738207i	$-0.264327\pi$
$139$	15.9062	1.34915	0.674574	+	0.738207i	$0.264327\pi$
$140$	13.4144	1.13372
$141$	0	0
$142$	−31.3869	−2.63393
$143$	0.648166	0.0542024
$144$	0	0
$145$	−1.39099	−0.115515
$146$	41.0899	3.40062
$147$	0	0
$148$	67.2178	5.52527
$149$	−0.869889	−0.0712641	−0.0356320	−	0.999365i	$-0.511344\pi$
$149$	−0.869889	−0.0712641	−0.0356320	+	0.999365i	$0.511344\pi$
$150$	0	0
$151$	13.3917	1.08980	0.544900	−	0.838501i	$-0.316568\pi$
$151$	13.3917	1.08980	0.544900	+	0.838501i	$0.316568\pi$
$152$	24.5049	1.98761
$153$	0	0
$154$	−10.2390	−0.825085
$155$	2.17172	0.174437
$156$	0	0
$157$	−3.80986	−0.304060	−0.152030	−	0.988376i	$-0.548581\pi$
$157$	−3.80986	−0.304060	−0.152030	+	0.988376i	$0.548581\pi$
$158$	2.42628	0.193025
$159$	0	0
$160$	−29.6064	−2.34059
$161$	−4.01772	−0.316641
$162$	0	0
$163$	5.64768	0.442361	0.221180	−	0.975233i	$-0.429009\pi$
$163$	5.64768	0.442361	0.221180	+	0.975233i	$0.429009\pi$
$164$	25.8618	2.01947
$165$	0	0
$166$	9.89115	0.767702
$167$	−13.7433	−1.06349	−0.531743	−	0.846905i	$-0.678463\pi$
$167$	−13.7433	−1.06349	−0.531743	+	0.846905i	$0.678463\pi$
$168$	0	0
$169$	−12.8513	−0.988565
$170$	−9.56081	−0.733281
$171$	0	0
$172$	−23.4875	−1.79090
$173$	10.5190	0.799743	0.399872	−	0.916571i	$-0.369055\pi$
$173$	10.5190	0.799743	0.399872	+	0.916571i	$0.369055\pi$
$174$	0	0
$175$	−8.45135	−0.638862
$176$	29.3227	2.21028
$177$	0	0
$178$	−6.31178	−0.473088
$179$	0.756985	0.0565797	0.0282899	−	0.999600i	$-0.490994\pi$
$179$	0.756985	0.0565797	0.0282899	+	0.999600i	$0.490994\pi$
$180$	0	0
$181$	13.3393	0.991504	0.495752	−	0.868464i	$-0.334892\pi$
$181$	13.3393	0.991504	0.495752	+	0.868464i	$0.334892\pi$
$182$	−2.34827	−0.174065
$183$	0	0
$184$	19.0754	1.40626
$185$	12.5103	0.919778
$186$	0	0
$187$	5.41132	0.395715
$188$	18.4084	1.34257
$189$	0	0
$190$	7.00142	0.507936
$191$	3.92533	0.284027	0.142014	−	0.989865i	$-0.454642\pi$
$191$	3.92533	0.284027	0.142014	+	0.989865i	$0.454642\pi$
$192$	0	0
$193$	11.5817	0.833667	0.416834	−	0.908983i	$-0.363140\pi$
$193$	11.5817	0.833667	0.416834	+	0.908983i	$0.363140\pi$
$194$	49.1961	3.53208
$195$	0	0
$196$	−12.6546	−0.903903
$197$	−5.60687	−0.399473	−0.199736	−	0.979850i	$-0.564009\pi$
$197$	−5.60687	−0.399473	−0.199736	+	0.979850i	$0.564009\pi$
$198$	0	0
$199$	−2.89593	−0.205287	−0.102643	−	0.994718i	$-0.532730\pi$
$199$	−2.89593	−0.205287	−0.102643	+	0.994718i	$0.532730\pi$
$200$	40.1255	2.83730
$201$	0	0
$202$	−4.55747	−0.320663
$203$	−2.85228	−0.200191
$204$	0	0
$205$	4.81331	0.336176
$206$	−30.6994	−2.13893
$207$	0	0
$208$	6.72500	0.466295
$209$	−3.96273	−0.274108
$210$	0	0
$211$	9.13389	0.628803	0.314401	−	0.949290i	$-0.398196\pi$
$211$	9.13389	0.628803	0.314401	+	0.949290i	$0.398196\pi$
$212$	71.0863	4.88223
$213$	0	0
$214$	56.8174	3.88396
$215$	−4.37140	−0.298127
$216$	0	0
$217$	4.45321	0.302303
$218$	−45.3555	−3.07186
$219$	0	0
$220$	10.2992	0.694374
$221$	1.24106	0.0834824
$222$	0	0
$223$	20.9949	1.40592	0.702960	−	0.711229i	$-0.251861\pi$
$223$	20.9949	1.40592	0.702960	+	0.711229i	$0.251861\pi$
$224$	−60.7092	−4.05630
$225$	0	0
$226$	−31.7746	−2.11362
$227$	−16.3350	−1.08419	−0.542097	−	0.840316i	$-0.682369\pi$
$227$	−16.3350	−1.08419	−0.542097	+	0.840316i	$0.682369\pi$
$228$	0	0
$229$	3.59578	0.237616	0.118808	−	0.992917i	$-0.462093\pi$
$229$	3.59578	0.237616	0.118808	+	0.992917i	$0.462093\pi$
$230$	5.45013	0.359371
$231$	0	0
$232$	13.5421	0.889081
$233$	19.7418	1.29333	0.646665	−	0.762774i	$-0.276163\pi$
$233$	19.7418	1.29333	0.646665	+	0.762774i	$0.276163\pi$
$234$	0	0
$235$	3.42610	0.223494
$236$	39.9944	2.60341
$237$	0	0
$238$	−19.6049	−1.27079
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	22.6222	1.45722	0.728611	−	0.684927i	$-0.240166\pi$
$241$	22.6222	1.45722	0.728611	+	0.684927i	$0.240166\pi$
$242$	22.7364	1.46155
$243$	0	0
$244$	76.1951	4.87789
$245$	−2.35524	−0.150471
$246$	0	0
$247$	−0.908830	−0.0578274
$248$	−21.1430	−1.34258
$249$	0	0
$250$	26.3156	1.66435
$251$	24.5135	1.54728	0.773638	−	0.633628i	$-0.218435\pi$
$251$	24.5135	1.54728	0.773638	+	0.633628i	$0.218435\pi$
$252$	0	0
$253$	−3.08471	−0.193934
$254$	−15.4325	−0.968322
$255$	0	0
$256$	88.0906	5.50566
$257$	5.37597	0.335344	0.167672	−	0.985843i	$-0.446375\pi$
$257$	5.37597	0.335344	0.167672	+	0.985843i	$0.446375\pi$
$258$	0	0
$259$	25.6530	1.59400
$260$	2.36207	0.146489
$261$	0	0
$262$	9.94970	0.614694
$263$	−20.8527	−1.28583	−0.642917	−	0.765936i	$-0.722276\pi$
$263$	−20.8527	−1.28583	−0.642917	+	0.765936i	$0.722276\pi$
$264$	0	0
$265$	13.2303	0.812732
$266$	14.3567	0.880267
$267$	0	0
$268$	−44.4463	−2.71499
$269$	−23.0236	−1.40377	−0.701887	−	0.712289i	$-0.747659\pi$
$269$	−23.0236	−1.40377	−0.701887	+	0.712289i	$0.747659\pi$
$270$	0	0
$271$	−13.4391	−0.816365	−0.408182	−	0.912900i	$-0.633837\pi$
$271$	−13.4391	−0.816365	−0.408182	+	0.912900i	$0.633837\pi$
$272$	56.1448	3.40428
$273$	0	0
$274$	47.3223	2.85885
$275$	−6.48876	−0.391287
$276$	0	0
$277$	−24.2084	−1.45454	−0.727272	−	0.686349i	$-0.759212\pi$
$277$	−24.2084	−1.45454	−0.727272	+	0.686349i	$0.759212\pi$
$278$	−44.2449	−2.65363
$279$	0	0
$280$	−24.3062	−1.45257
$281$	12.1982	0.727684	0.363842	−	0.931461i	$-0.381465\pi$
$281$	12.1982	0.727684	0.363842	+	0.931461i	$0.381465\pi$
$282$	0	0
$283$	−5.86861	−0.348852	−0.174426	−	0.984670i	$-0.555807\pi$
$283$	−5.86861	−0.348852	−0.174426	+	0.984670i	$0.555807\pi$
$284$	64.7385	3.84152
$285$	0	0
$286$	−1.80294	−0.106610
$287$	9.86991	0.582602
$288$	0	0
$289$	−6.63885	−0.390521
$290$	3.86917	0.227206
$291$	0	0
$292$	−84.7519	−4.95973
$293$	0.238352	0.0139247	0.00696235	−	0.999976i	$-0.497784\pi$
$293$	0.238352	0.0139247	0.00696235	+	0.999976i	$0.497784\pi$
$294$	0	0
$295$	7.44361	0.433384
$296$	−121.796	−7.07923
$297$	0	0
$298$	2.41969	0.140169
$299$	−0.707462	−0.0409136
$300$	0	0
$301$	−8.96376	−0.516662
$302$	−37.2504	−2.14352
$303$	0	0
$304$	−41.1150	−2.35811
$305$	14.1812	0.812011
$306$	0	0
$307$	−14.6984	−0.838881	−0.419440	−	0.907783i	$-0.637774\pi$
$307$	−14.6984	−0.838881	−0.419440	+	0.907783i	$0.637774\pi$
$308$	21.1190	1.20337
$309$	0	0
$310$	−6.04087	−0.343098
$311$	15.7771	0.894638	0.447319	−	0.894374i	$-0.352379\pi$
$311$	15.7771	0.894638	0.447319	+	0.894374i	$0.352379\pi$
$312$	0	0
$313$	19.0878	1.07891	0.539454	−	0.842015i	$-0.318631\pi$
$313$	19.0878	1.07891	0.539454	+	0.842015i	$0.318631\pi$
$314$	10.5975	0.598054
$315$	0	0
$316$	−5.00445	−0.281522
$317$	18.4886	1.03842	0.519210	−	0.854647i	$-0.326226\pi$
$317$	18.4886	1.03842	0.519210	+	0.854647i	$0.326226\pi$
$318$	0	0
$319$	−2.18991	−0.122612
$320$	45.1030	2.52133
$321$	0	0
$322$	11.1757	0.622799
$323$	−7.58751	−0.422180
$324$	0	0
$325$	−1.48816	−0.0825482
$326$	−15.7096	−0.870076
$327$	0	0
$328$	−46.8605	−2.58744
$329$	7.02537	0.387321
$330$	0	0
$331$	9.27884	0.510011	0.255006	−	0.966940i	$-0.417923\pi$
$331$	9.27884	0.510011	0.255006	+	0.966940i	$0.417923\pi$
$332$	−20.4015	−1.11968
$333$	0	0
$334$	38.2284	2.09176
$335$	−8.27219	−0.451958
$336$	0	0
$337$	20.9531	1.14139	0.570693	−	0.821164i	$-0.306675\pi$
$337$	20.9531	1.14139	0.570693	+	0.821164i	$0.306675\pi$
$338$	35.7474	1.94440
$339$	0	0
$340$	19.7201	1.06947
$341$	3.41907	0.185153
$342$	0	0
$343$	−20.1567	−1.08836
$344$	42.5583	2.29459
$345$	0	0
$346$	−29.2597	−1.57301
$347$	19.6522	1.05499	0.527493	−	0.849559i	$-0.323132\pi$
$347$	19.6522	1.05499	0.527493	+	0.849559i	$0.323132\pi$
$348$	0	0
$349$	15.0391	0.805024	0.402512	−	0.915415i	$-0.368137\pi$
$349$	15.0391	0.805024	0.402512	+	0.915415i	$0.368137\pi$
$350$	23.5083	1.25657
$351$	0	0
$352$	−46.6111	−2.48438
$353$	−0.925452	−0.0492568	−0.0246284	−	0.999697i	$-0.507840\pi$
$353$	−0.925452	−0.0492568	−0.0246284	+	0.999697i	$0.507840\pi$
$354$	0	0
$355$	12.0489	0.639489
$356$	13.0187	0.689988
$357$	0	0
$358$	−2.10563	−0.111286
$359$	3.93664	0.207768	0.103884	−	0.994589i	$-0.466873\pi$
$359$	3.93664	0.207768	0.103884	+	0.994589i	$0.466873\pi$
$360$	0	0
$361$	−13.4436	−0.707560
$362$	−37.1048	−1.95018
$363$	0	0
$364$	4.84353	0.253870
$365$	−15.7737	−0.825635
$366$	0	0
$367$	25.9022	1.35208	0.676042	−	0.736863i	$-0.263694\pi$
$367$	25.9022	1.35208	0.676042	+	0.736863i	$0.263694\pi$
$368$	−32.0052	−1.66839
$369$	0	0
$370$	−34.7988	−1.80910
$371$	27.1294	1.40849
$372$	0	0
$373$	19.4536	1.00727	0.503634	−	0.863917i	$-0.331996\pi$
$373$	19.4536	1.00727	0.503634	+	0.863917i	$0.331996\pi$
$374$	−15.0522	−0.778329
$375$	0	0
$376$	−33.3552	−1.72016
$377$	−0.502244	−0.0258669
$378$	0	0
$379$	29.9718	1.53955	0.769775	−	0.638315i	$-0.220368\pi$
$379$	29.9718	1.53955	0.769775	+	0.638315i	$0.220368\pi$
$380$	−14.4411	−0.740814
$381$	0	0
$382$	−10.9187	−0.558651
$383$	4.44865	0.227315	0.113658	−	0.993520i	$-0.463743\pi$
$383$	4.44865	0.227315	0.113658	+	0.993520i	$0.463743\pi$
$384$	0	0
$385$	3.93060	0.200322
$386$	−32.2157	−1.63973
$387$	0	0
$388$	−101.472	−5.15145
$389$	8.23153	0.417355	0.208678	−	0.977984i	$-0.433084\pi$
$389$	8.23153	0.417355	0.208678	+	0.977984i	$0.433084\pi$
$390$	0	0
$391$	−5.90636	−0.298697
$392$	22.9296	1.15812
$393$	0	0
$394$	15.5961	0.785720
$395$	−0.931410	−0.0468643
$396$	0	0
$397$	10.2238	0.513119	0.256560	−	0.966528i	$-0.417411\pi$
$397$	10.2238	0.513119	0.256560	+	0.966528i	$0.417411\pi$
$398$	8.05533	0.403777
$399$	0	0
$400$	−67.3236	−3.36618
$401$	32.1298	1.60449	0.802244	−	0.596997i	$-0.203639\pi$
$401$	32.1298	1.60449	0.802244	+	0.596997i	$0.203639\pi$
$402$	0	0
$403$	0.784144	0.0390610
$404$	9.40024	0.467679
$405$	0	0
$406$	7.93391	0.393753
$407$	19.6958	0.976283
$408$	0	0
$409$	8.80377	0.435319	0.217659	−	0.976025i	$-0.430158\pi$
$409$	8.80377	0.435319	0.217659	+	0.976025i	$0.430158\pi$
$410$	−13.3887	−0.661223
$411$	0	0
$412$	63.3206	3.11958
$413$	15.2635	0.751065
$414$	0	0
$415$	−3.79705	−0.186390
$416$	−10.6900	−0.524120
$417$	0	0
$418$	11.0228	0.539141
$419$	−33.1065	−1.61736	−0.808680	−	0.588248i	$-0.799818\pi$
$419$	−33.1065	−1.61736	−0.808680	+	0.588248i	$0.799818\pi$
$420$	0	0
$421$	−22.4473	−1.09401	−0.547006	−	0.837128i	$-0.684233\pi$
$421$	−22.4473	−1.09401	−0.547006	+	0.837128i	$0.684233\pi$
$422$	−25.4069	−1.23679
$423$	0	0
$424$	−128.805	−6.25533
$425$	−12.4241	−0.602659
$426$	0	0
$427$	29.0791	1.40724
$428$	−117.191	−5.66466
$429$	0	0
$430$	12.1595	0.586385
$431$	−12.6186	−0.607816	−0.303908	−	0.952701i	$-0.598292\pi$
$431$	−12.6186	−0.607816	−0.303908	+	0.952701i	$0.598292\pi$
$432$	0	0
$433$	−33.7256	−1.62075	−0.810374	−	0.585913i	$-0.800736\pi$
$433$	−33.7256	−1.62075	−0.810374	+	0.585913i	$0.800736\pi$
$434$	−12.3871	−0.594599
$435$	0	0
$436$	93.5501	4.48024
$437$	4.32525	0.206905
$438$	0	0
$439$	−32.4633	−1.54939	−0.774695	−	0.632335i	$-0.782097\pi$
$439$	−32.4633	−1.54939	−0.774695	+	0.632335i	$0.782097\pi$
$440$	−18.6618	−0.889664
$441$	0	0
$442$	−3.45213	−0.164201
$443$	−40.8241	−1.93961	−0.969807	−	0.243875i	$-0.921581\pi$
$443$	−40.8241	−1.93961	−0.969807	+	0.243875i	$0.921581\pi$
$444$	0	0
$445$	2.42299	0.114861
$446$	−58.3995	−2.76530
$447$	0	0
$448$	92.4856	4.36953
$449$	28.2281	1.33217	0.666083	−	0.745877i	$-0.267970\pi$
$449$	28.2281	1.33217	0.666083	+	0.745877i	$0.267970\pi$
$450$	0	0
$451$	7.57789	0.356829
$452$	65.5383	3.08266
$453$	0	0
$454$	45.4376	2.13249
$455$	0.901460	0.0422611
$456$	0	0
$457$	0.0427512	0.00199982	0.000999909	−	1.00000i	$-0.499682\pi$
$457$	0.0427512	0.00199982	0.000999909		1.00000i	$0.499682\pi$
$458$	−10.0021	−0.467366
$459$	0	0
$460$	−11.2414	−0.524134
$461$	4.92326	0.229299	0.114649	−	0.993406i	$-0.463426\pi$
$461$	4.92326	0.229299	0.114649	+	0.993406i	$0.463426\pi$
$462$	0	0
$463$	9.56366	0.444461	0.222231	−	0.974994i	$-0.428666\pi$
$463$	9.56366	0.444461	0.222231	+	0.974994i	$0.428666\pi$
$464$	−22.7213	−1.05481
$465$	0	0
$466$	−54.9140	−2.54384
$467$	−30.1697	−1.39609	−0.698044	−	0.716055i	$-0.745946\pi$
$467$	−30.1697	−1.39609	−0.698044	+	0.716055i	$0.745946\pi$
$468$	0	0
$469$	−16.9625	−0.783255
$470$	−9.53007	−0.439589
$471$	0	0
$472$	−72.4680	−3.33561
$473$	−6.88217	−0.316442
$474$	0	0
$475$	9.09824	0.417456
$476$	40.4370	1.85343
$477$	0	0
$478$	2.78161	0.127228
$479$	−28.2462	−1.29060	−0.645301	−	0.763929i	$-0.723268\pi$
$479$	−28.2462	−1.29060	−0.645301	+	0.763929i	$0.723268\pi$
$480$	0	0
$481$	4.51711	0.205963
$482$	−62.9260	−2.86620
$483$	0	0
$484$	−46.8960	−2.13164
$485$	−18.8856	−0.857550
$486$	0	0
$487$	23.3990	1.06031	0.530155	−	0.847901i	$-0.322134\pi$
$487$	23.3990	1.06031	0.530155	+	0.847901i	$0.322134\pi$
$488$	−138.062	−6.24978
$489$	0	0
$490$	6.55134	0.295959
$491$	−15.2278	−0.687221	−0.343610	−	0.939112i	$-0.611650\pi$
$491$	−15.2278	−0.687221	−0.343610	+	0.939112i	$0.611650\pi$
$492$	0	0
$493$	−4.19306	−0.188846
$494$	2.52801	0.113740
$495$	0	0
$496$	35.4743	1.59284
$497$	24.7068	1.10825
$498$	0	0
$499$	−8.07973	−0.361698	−0.180849	−	0.983511i	$-0.557885\pi$
$499$	−8.07973	−0.361698	−0.180849	+	0.983511i	$0.557885\pi$
$500$	−54.2785	−2.42741
$501$	0	0
$502$	−68.1868	−3.04333
$503$	18.4869	0.824290	0.412145	−	0.911118i	$-0.364780\pi$
$503$	18.4869	0.824290	0.412145	+	0.911118i	$0.364780\pi$
$504$	0	0
$505$	1.74954	0.0778535
$506$	8.58046	0.381448
$507$	0	0
$508$	31.8311	1.41228
$509$	17.0307	0.754873	0.377436	−	0.926036i	$-0.376806\pi$
$509$	17.0307	0.754873	0.377436	+	0.926036i	$0.376806\pi$
$510$	0	0
$511$	−32.3447	−1.43085
$512$	−120.956	−5.34553
$513$	0	0
$514$	−14.9538	−0.659585
$515$	11.7850	0.519310
$516$	0	0
$517$	5.39392	0.237224
$518$	−71.3565	−3.13523
$519$	0	0
$520$	−4.27997	−0.187689
$521$	29.1053	1.27512	0.637562	−	0.770399i	$-0.279943\pi$
$521$	29.1053	1.27512	0.637562	+	0.770399i	$0.279943\pi$
$522$	0	0
$523$	−33.4022	−1.46058	−0.730288	−	0.683139i	$-0.760614\pi$
$523$	−33.4022	−1.46058	−0.730288	+	0.683139i	$0.760614\pi$
$524$	−20.5222	−0.896518
$525$	0	0
$526$	58.0040	2.52910
$527$	6.54655	0.285172
$528$	0	0
$529$	−19.6331	−0.853613
$530$	−36.8016	−1.59856
$531$	0	0
$532$	−29.6121	−1.28385
$533$	1.73795	0.0752788
$534$	0	0
$535$	−21.8112	−0.942982
$536$	80.5348	3.47857
$537$	0	0
$538$	64.0426	2.76107
$539$	−3.70799	−0.159714
$540$	0	0
$541$	−31.8722	−1.37029	−0.685147	−	0.728405i	$-0.740262\pi$
$541$	−31.8722	−1.37029	−0.685147	+	0.728405i	$0.740262\pi$
$542$	37.3822	1.60570
$543$	0	0
$544$	−89.2471	−3.82644
$545$	17.4112	0.745814
$546$	0	0
$547$	−38.2048	−1.63352	−0.816759	−	0.576979i	$-0.804231\pi$
$547$	−38.2048	−1.63352	−0.816759	+	0.576979i	$0.804231\pi$
$548$	−97.6069	−4.16956
$549$	0	0
$550$	18.0492	0.769619
$551$	3.07060	0.130812
$552$	0	0
$553$	−1.90990	−0.0812171
$554$	67.3384	2.86093
$555$	0	0
$556$	91.2594	3.87026
$557$	37.1090	1.57236	0.786179	−	0.617999i	$-0.212056\pi$
$557$	37.1090	1.57236	0.786179	+	0.617999i	$0.212056\pi$
$558$	0	0
$559$	−1.57839	−0.0667586
$560$	40.7816	1.72334
$561$	0	0
$562$	−33.9306	−1.43128
$563$	−28.7136	−1.21014	−0.605068	−	0.796174i	$-0.706854\pi$
$563$	−28.7136	−1.21014	−0.605068	+	0.796174i	$0.706854\pi$
$564$	0	0
$565$	12.1978	0.513163
$566$	16.3242	0.686155
$567$	0	0
$568$	−117.303	−4.92193
$569$	−28.2036	−1.18236	−0.591178	−	0.806541i	$-0.701337\pi$
$569$	−28.2036	−1.18236	−0.591178	+	0.806541i	$0.701337\pi$
$570$	0	0
$571$	42.3934	1.77411	0.887055	−	0.461664i	$-0.152747\pi$
$571$	42.3934	1.77411	0.887055	+	0.461664i	$0.152747\pi$
$572$	3.71875	0.155489
$573$	0	0
$574$	−27.4542	−1.14592
$575$	7.08236	0.295355
$576$	0	0
$577$	38.9229	1.62038	0.810190	−	0.586167i	$-0.199364\pi$
$577$	38.9229	1.62038	0.810190	+	0.586167i	$0.199364\pi$
$578$	18.4667	0.768113
$579$	0	0
$580$	−7.98055	−0.331374
$581$	−7.78602	−0.323018
$582$	0	0
$583$	20.8293	0.862661
$584$	153.567	6.35464
$585$	0	0
$586$	−0.663003	−0.0273884
$587$	9.00364	0.371620	0.185810	−	0.982586i	$-0.440509\pi$
$587$	9.00364	0.371620	0.185810	+	0.982586i	$0.440509\pi$
$588$	0	0
$589$	−4.79407	−0.197536
$590$	−20.7052	−0.852420
$591$	0	0
$592$	204.352	8.39882
$593$	−6.63569	−0.272495	−0.136247	−	0.990675i	$-0.543504\pi$
$593$	−6.63569	−0.272495	−0.136247	+	0.990675i	$0.543504\pi$
$594$	0	0
$595$	7.52598	0.308535
$596$	−4.99084	−0.204433
$597$	0	0
$598$	1.96788	0.0804726
$599$	−39.9522	−1.63240	−0.816202	−	0.577766i	$-0.803925\pi$
$599$	−39.9522	−1.63240	−0.816202	+	0.577766i	$0.803925\pi$
$600$	0	0
$601$	−32.8969	−1.34189	−0.670947	−	0.741505i	$-0.734112\pi$
$601$	−32.8969	−1.34189	−0.670947	+	0.741505i	$0.734112\pi$
$602$	24.9337	1.01622
$603$	0	0
$604$	76.8326	3.12627
$605$	−8.72812	−0.354848
$606$	0	0
$607$	20.8635	0.846824	0.423412	−	0.905937i	$-0.360832\pi$
$607$	20.8635	0.846824	0.423412	+	0.905937i	$0.360832\pi$
$608$	65.3560	2.65054
$609$	0	0
$610$	−39.4464	−1.59714
$611$	1.23706	0.0500463
$612$	0	0
$613$	18.2393	0.736680	0.368340	−	0.929691i	$-0.379926\pi$
$613$	18.2393	0.736680	0.368340	+	0.929691i	$0.379926\pi$
$614$	40.8851	1.64999
$615$	0	0
$616$	−38.2667	−1.54181
$617$	15.1256	0.608934	0.304467	−	0.952523i	$-0.401522\pi$
$617$	15.1256	0.608934	0.304467	+	0.952523i	$0.401522\pi$
$618$	0	0
$619$	−28.4122	−1.14198	−0.570991	−	0.820956i	$-0.693441\pi$
$619$	−28.4122	−1.14198	−0.570991	+	0.820956i	$0.693441\pi$
$620$	12.4599	0.500401
$621$	0	0
$622$	−43.8857	−1.75966
$623$	4.96845	0.199057
$624$	0	0
$625$	9.19675	0.367870
$626$	−53.0949	−2.12210
$627$	0	0
$628$	−21.8585	−0.872248
$629$	37.7118	1.50367
$630$	0	0
$631$	27.4940	1.09452	0.547260	−	0.836963i	$-0.315671\pi$
$631$	27.4940	1.09452	0.547260	+	0.836963i	$0.315671\pi$
$632$	9.06784	0.360699
$633$	0	0
$634$	−51.4279	−2.04246
$635$	5.92429	0.235098
$636$	0	0
$637$	−0.850406	−0.0336943
$638$	6.09147	0.241164
$639$	0	0
$640$	−66.2459	−2.61860
$641$	−34.2161	−1.35145	−0.675727	−	0.737152i	$-0.736170\pi$
$641$	−34.2161	−1.35145	−0.675727	+	0.737152i	$0.736170\pi$
$642$	0	0
$643$	37.9108	1.49506	0.747529	−	0.664230i	$-0.231240\pi$
$643$	37.9108	1.49506	0.747529	+	0.664230i	$0.231240\pi$
$644$	−23.0510	−0.908338
$645$	0	0
$646$	21.1055	0.830384
$647$	−19.6787	−0.773649	−0.386824	−	0.922153i	$-0.626428\pi$
$647$	−19.6787	−0.773649	−0.386824	+	0.922153i	$0.626428\pi$
$648$	0	0
$649$	11.7189	0.460008
$650$	4.13947	0.162363
$651$	0	0
$652$	32.4027	1.26899
$653$	−28.3671	−1.11009	−0.555046	−	0.831820i	$-0.687299\pi$
$653$	−28.3671	−1.11009	−0.555046	+	0.831820i	$0.687299\pi$
$654$	0	0
$655$	−3.81952	−0.149241
$656$	78.6238	3.06975
$657$	0	0
$658$	−19.5418	−0.761820
$659$	20.9453	0.815914	0.407957	−	0.913001i	$-0.366241\pi$
$659$	20.9453	0.815914	0.407957	+	0.913001i	$0.366241\pi$
$660$	0	0
$661$	−17.7746	−0.691351	−0.345676	−	0.938354i	$-0.612350\pi$
$661$	−17.7746	−0.691351	−0.345676	+	0.938354i	$0.612350\pi$
$662$	−25.8101	−1.00314
$663$	0	0
$664$	36.9666	1.43458
$665$	−5.51131	−0.213719
$666$	0	0
$667$	2.39025	0.0925508
$668$	−78.8498	−3.05079
$669$	0	0
$670$	23.0100	0.888953
$671$	22.3263	0.861896
$672$	0	0
$673$	−6.51133	−0.250993	−0.125497	−	0.992094i	$-0.540052\pi$
$673$	−6.51133	−0.250993	−0.125497	+	0.992094i	$0.540052\pi$
$674$	−58.2832	−2.24498
$675$	0	0
$676$	−73.7325	−2.83587
$677$	−44.8472	−1.72362	−0.861809	−	0.507233i	$-0.830668\pi$
$677$	−44.8472	−1.72362	−0.861809	+	0.507233i	$0.830668\pi$
$678$	0	0
$679$	−38.7257	−1.48616
$680$	−35.7320	−1.37026
$681$	0	0
$682$	−9.51051	−0.364176
$683$	27.0247	1.03407	0.517035	−	0.855964i	$-0.327036\pi$
$683$	27.0247	1.03407	0.517035	+	0.855964i	$0.327036\pi$
$684$	0	0
$685$	−18.1663	−0.694097
$686$	56.0680	2.14069
$687$	0	0
$688$	−71.4054	−2.72231
$689$	4.77708	0.181992
$690$	0	0
$691$	−49.4834	−1.88244	−0.941218	−	0.337798i	$-0.890318\pi$
$691$	−49.4834	−1.88244	−0.941218	+	0.337798i	$0.890318\pi$
$692$	60.3509	2.29420
$693$	0	0
$694$	−54.6647	−2.07505
$695$	16.9849	0.644273
$696$	0	0
$697$	14.5095	0.549587
$698$	−41.8328	−1.58340
$699$	0	0
$700$	−48.4883	−1.83268
$701$	11.7075	0.442186	0.221093	−	0.975253i	$-0.429037\pi$
$701$	11.7075	0.442186	0.221093	+	0.975253i	$0.429037\pi$
$702$	0	0
$703$	−27.6165	−1.04158
$704$	71.0083	2.67623
$705$	0	0
$706$	2.57424	0.0968829
$707$	3.58751	0.134922
$708$	0	0
$709$	−4.91434	−0.184562	−0.0922809	−	0.995733i	$-0.529416\pi$
$709$	−4.91434	−0.184562	−0.0922809	+	0.995733i	$0.529416\pi$
$710$	−33.5153	−1.25781
$711$	0	0
$712$	−23.5893	−0.884045
$713$	−3.73185	−0.139759
$714$	0	0
$715$	0.692120	0.0258838
$716$	4.34308	0.162308
$717$	0	0
$718$	−10.9502	−0.408657
$719$	16.4213	0.612410	0.306205	−	0.951966i	$-0.400941\pi$
$719$	16.4213	0.612410	0.306205	+	0.951966i	$0.400941\pi$
$720$	0	0
$721$	24.1657	0.899977
$722$	37.3949	1.39169
$723$	0	0
$724$	76.5322	2.84430
$725$	5.02793	0.186733
$726$	0	0
$727$	42.8239	1.58825	0.794125	−	0.607754i	$-0.207930\pi$
$727$	42.8239	1.58825	0.794125	+	0.607754i	$0.207930\pi$
$728$	−8.77626	−0.325270
$729$	0	0
$730$	43.8763	1.62394
$731$	−13.1774	−0.487384
$732$	0	0
$733$	5.32382	0.196640	0.0983198	−	0.995155i	$-0.468653\pi$
$733$	5.32382	0.196640	0.0983198	+	0.995155i	$0.468653\pi$
$734$	−72.0497	−2.65940
$735$	0	0
$736$	50.8752	1.87528
$737$	−13.0234	−0.479724
$738$	0	0
$739$	−21.7444	−0.799880	−0.399940	−	0.916541i	$-0.630969\pi$
$739$	−21.7444	−0.799880	−0.399940	+	0.916541i	$0.630969\pi$
$740$	71.7760	2.63854
$741$	0	0
$742$	−75.4632	−2.77034
$743$	16.2285	0.595364	0.297682	−	0.954665i	$-0.403786\pi$
$743$	16.2285	0.595364	0.297682	+	0.954665i	$0.403786\pi$
$744$	0	0
$745$	−0.928878	−0.0340315
$746$	−54.1122	−1.98119
$747$	0	0
$748$	31.0466	1.13517
$749$	−44.7249	−1.63421
$750$	0	0
$751$	−1.58740	−0.0579249	−0.0289625	−	0.999581i	$-0.509220\pi$
$751$	−1.58740	−0.0579249	−0.0289625	+	0.999581i	$0.509220\pi$
$752$	55.9642	2.04081
$753$	0	0
$754$	1.39704	0.0508774
$755$	14.2998	0.520423
$756$	0	0
$757$	−14.7480	−0.536025	−0.268012	−	0.963415i	$-0.586367\pi$
$757$	−14.7480	−0.536025	−0.268012	+	0.963415i	$0.586367\pi$
$758$	−83.3699	−3.02813
$759$	0	0
$760$	26.1667	0.949165
$761$	36.6411	1.32824	0.664120	−	0.747626i	$-0.268806\pi$
$761$	36.6411	1.32824	0.664120	+	0.747626i	$0.268806\pi$
$762$	0	0
$763$	35.7025	1.29252
$764$	22.5210	0.814780
$765$	0	0
$766$	−12.3744	−0.447105
$767$	2.68767	0.0970461
$768$	0	0
$769$	11.6303	0.419398	0.209699	−	0.977766i	$-0.432752\pi$
$769$	11.6303	0.419398	0.209699	+	0.977766i	$0.432752\pi$
$770$	−10.9334	−0.394012
$771$	0	0
$772$	66.4480	2.39152
$773$	11.4126	0.410484	0.205242	−	0.978711i	$-0.434202\pi$
$773$	11.4126	0.410484	0.205242	+	0.978711i	$0.434202\pi$
$774$	0	0
$775$	−7.85002	−0.281981
$776$	183.863	6.60028
$777$	0	0
$778$	−22.8969	−0.820893
$779$	−10.6254	−0.380694
$780$	0	0
$781$	18.9693	0.678775
$782$	16.4292	0.587506
$783$	0	0
$784$	−38.4720	−1.37400
$785$	−4.06822	−0.145201
$786$	0	0
$787$	−34.6084	−1.23365	−0.616827	−	0.787098i	$-0.711582\pi$
$787$	−34.6084	−1.23365	−0.616827	+	0.787098i	$0.711582\pi$
$788$	−32.1685	−1.14595
$789$	0	0
$790$	2.59082	0.0921771
$791$	25.0120	0.889326
$792$	0	0
$793$	5.12040	0.181831
$794$	−28.4387	−1.00925
$795$	0	0
$796$	−16.6149	−0.588900
$797$	25.9268	0.918376	0.459188	−	0.888339i	$-0.348141\pi$
$797$	25.9268	0.918376	0.459188	+	0.888339i	$0.348141\pi$
$798$	0	0
$799$	10.3278	0.365372
$800$	107.017	3.78362
$801$	0	0
$802$	−89.3726	−3.15586
$803$	−24.8335	−0.876356
$804$	0	0
$805$	−4.29018	−0.151209
$806$	−2.18118	−0.0768288
$807$	0	0
$808$	−17.0328	−0.599212
$809$	6.66361	0.234280	0.117140	−	0.993115i	$-0.462627\pi$
$809$	6.66361	0.234280	0.117140	+	0.993115i	$0.462627\pi$
$810$	0	0
$811$	−12.8667	−0.451809	−0.225905	−	0.974149i	$-0.572534\pi$
$811$	−12.8667	−0.451809	−0.225905	+	0.974149i	$0.572534\pi$
$812$	−16.3645	−0.574280
$813$	0	0
$814$	−54.7859	−1.92024
$815$	6.03067	0.211245
$816$	0	0
$817$	9.64987	0.337606
$818$	−24.4886	−0.856225
$819$	0	0
$820$	27.6156	0.964379
$821$	11.8457	0.413418	0.206709	−	0.978402i	$-0.433725\pi$
$821$	11.8457	0.413418	0.206709	+	0.978402i	$0.433725\pi$
$822$	0	0
$823$	−2.64071	−0.0920494	−0.0460247	−	0.998940i	$-0.514655\pi$
$823$	−2.64071	−0.0920494	−0.0460247	+	0.998940i	$0.514655\pi$
$824$	−114.734	−3.99695
$825$	0	0
$826$	−42.4569	−1.47727
$827$	−36.8806	−1.28247	−0.641233	−	0.767346i	$-0.721577\pi$
$827$	−36.8806	−1.28247	−0.641233	+	0.767346i	$0.721577\pi$
$828$	0	0
$829$	−29.3699	−1.02006	−0.510029	−	0.860157i	$-0.670365\pi$
$829$	−29.3699	−1.02006	−0.510029	+	0.860157i	$0.670365\pi$
$830$	10.5619	0.366609
$831$	0	0
$832$	16.2853	0.564593
$833$	−7.09975	−0.245992
$834$	0	0
$835$	−14.6752	−0.507858
$836$	−22.7355	−0.786325
$837$	0	0
$838$	92.0894	3.18118
$839$	−3.48288	−0.120242	−0.0601211	−	0.998191i	$-0.519149\pi$
$839$	−3.48288	−0.120242	−0.0601211	+	0.998191i	$0.519149\pi$
$840$	0	0
$841$	−27.3031	−0.941486
$842$	62.4395	2.15181
$843$	0	0
$844$	52.4042	1.80383
$845$	−13.7228	−0.472080
$846$	0	0
$847$	−17.8974	−0.614962
$848$	216.113	7.42135
$849$	0	0
$850$	34.5590	1.18537
$851$	−21.4976	−0.736927
$852$	0	0
$853$	25.8074	0.883628	0.441814	−	0.897107i	$-0.354335\pi$
$853$	25.8074	0.883628	0.441814	+	0.897107i	$0.354335\pi$
$854$	−80.8866	−2.76788
$855$	0	0
$856$	212.346	7.25783
$857$	24.6920	0.843463	0.421732	−	0.906721i	$-0.361422\pi$
$857$	24.6920	0.843463	0.421732	+	0.906721i	$0.361422\pi$
$858$	0	0
$859$	−19.2402	−0.656466	−0.328233	−	0.944597i	$-0.606453\pi$
$859$	−19.2402	−0.656466	−0.328233	+	0.944597i	$0.606453\pi$
$860$	−25.0802	−0.855228
$861$	0	0
$862$	35.0999	1.19551
$863$	35.9640	1.22423	0.612114	−	0.790769i	$-0.290319\pi$
$863$	35.9640	1.22423	0.612114	+	0.790769i	$0.290319\pi$
$864$	0	0
$865$	11.2323	0.381910
$866$	93.8113	3.18784
$867$	0	0
$868$	25.5496	0.867209
$869$	−1.46637	−0.0497434
$870$	0	0
$871$	−2.98684	−0.101205
$872$	−169.509	−5.74029
$873$	0	0
$874$	−12.0311	−0.406959
$875$	−20.7149	−0.700290
$876$	0	0
$877$	−8.27736	−0.279506	−0.139753	−	0.990186i	$-0.544631\pi$
$877$	−8.27736	−0.279506	−0.139753	+	0.990186i	$0.544631\pi$
$878$	90.3002	3.04748
$879$	0	0
$880$	31.3112	1.05550
$881$	48.7508	1.64246	0.821228	−	0.570599i	$-0.193289\pi$
$881$	48.7508	1.64246	0.821228	+	0.570599i	$0.193289\pi$
$882$	0	0
$883$	−33.0923	−1.11365	−0.556823	−	0.830631i	$-0.687980\pi$
$883$	−33.0923	−1.11365	−0.556823	+	0.830631i	$0.687980\pi$
$884$	7.12035	0.239483
$885$	0	0
$886$	113.557	3.81501
$887$	−55.6500	−1.86855	−0.934273	−	0.356559i	$-0.883950\pi$
$887$	−55.6500	−1.86855	−0.934273	+	0.356559i	$0.883950\pi$
$888$	0	0
$889$	12.1480	0.407431
$890$	−6.73980	−0.225919
$891$	0	0
$892$	120.455	4.03312
$893$	−7.56311	−0.253090
$894$	0	0
$895$	0.808318	0.0270191
$896$	−135.840	−4.53810
$897$	0	0
$898$	−78.5195	−2.62023
$899$	−2.64933	−0.0883601
$900$	0	0
$901$	39.8822	1.32867
$902$	−21.0787	−0.701844
$903$	0	0
$904$	−118.752	−3.94965
$905$	14.2439	0.473483
$906$	0	0
$907$	−13.1898	−0.437960	−0.218980	−	0.975729i	$-0.570273\pi$
$907$	−13.1898	−0.437960	−0.218980	+	0.975729i	$0.570273\pi$
$908$	−93.7195	−3.11019
$909$	0	0
$910$	−2.50751	−0.0831231
$911$	−1.46739	−0.0486169	−0.0243084	−	0.999705i	$-0.507738\pi$
$911$	−1.46739	−0.0486169	−0.0243084	+	0.999705i	$0.507738\pi$
$912$	0	0
$913$	−5.97792	−0.197840
$914$	−0.118917	−0.00393343
$915$	0	0
$916$	20.6302	0.681642
$917$	−7.83210	−0.258639
$918$	0	0
$919$	−46.3333	−1.52839	−0.764197	−	0.644982i	$-0.776865\pi$
$919$	−46.3333	−1.52839	−0.764197	+	0.644982i	$0.776865\pi$
$920$	20.3690	0.671545
$921$	0	0
$922$	−13.6946	−0.451006
$923$	4.35050	0.143198
$924$	0	0
$925$	−45.2205	−1.48684
$926$	−26.6024	−0.874208
$927$	0	0
$928$	36.1175	1.18561
$929$	−46.4479	−1.52391	−0.761953	−	0.647632i	$-0.775759\pi$
$929$	−46.4479	−1.52391	−0.761953	+	0.647632i	$0.775759\pi$
$930$	0	0
$931$	5.19918	0.170396
$932$	113.265	3.71013
$933$	0	0
$934$	83.9202	2.74595
$935$	5.77828	0.188970
$936$	0	0
$937$	3.04807	0.0995760	0.0497880	−	0.998760i	$-0.484145\pi$
$937$	3.04807	0.0995760	0.0497880	+	0.998760i	$0.484145\pi$
$938$	47.1830	1.54058
$939$	0	0
$940$	19.6567	0.641131
$941$	−14.9295	−0.486686	−0.243343	−	0.969940i	$-0.578244\pi$
$941$	−14.9295	−0.486686	−0.243343	+	0.969940i	$0.578244\pi$
$942$	0	0
$943$	−8.27113	−0.269345
$944$	121.589	3.95738
$945$	0	0
$946$	19.1435	0.622408
$947$	−7.11410	−0.231177	−0.115589	−	0.993297i	$-0.536875\pi$
$947$	−7.11410	−0.231177	−0.115589	+	0.993297i	$0.536875\pi$
$948$	0	0
$949$	−5.69543	−0.184881
$950$	−25.3077	−0.821091
$951$	0	0
$952$	−73.2700	−2.37469
$953$	0.848663	0.0274909	0.0137454	−	0.999906i	$-0.495625\pi$
$953$	0.848663	0.0274909	0.0137454	+	0.999906i	$0.495625\pi$
$954$	0	0
$955$	4.19152	0.135634
$956$	−5.73734	−0.185559
$957$	0	0
$958$	78.5698	2.53847
$959$	−37.2507	−1.20289
$960$	0	0
$961$	−26.8636	−0.866569
$962$	−12.5648	−0.405106
$963$	0	0
$964$	129.791	4.18029
$965$	12.3671	0.398110
$966$	0	0
$967$	−50.7160	−1.63092	−0.815458	−	0.578816i	$-0.803515\pi$
$967$	−50.7160	−1.63092	−0.815458	+	0.578816i	$0.803515\pi$
$968$	84.9735	2.73115
$969$	0	0
$970$	52.5323	1.68671
$971$	−38.4397	−1.23359	−0.616794	−	0.787125i	$-0.711569\pi$
$971$	−38.4397	−1.23359	−0.616794	+	0.787125i	$0.711569\pi$
$972$	0	0
$973$	34.8282	1.11654
$974$	−65.0868	−2.08551
$975$	0	0
$976$	231.644	7.41476
$977$	8.88196	0.284159	0.142080	−	0.989855i	$-0.454621\pi$
$977$	8.88196	0.284159	0.142080	+	0.989855i	$0.454621\pi$
$978$	0	0
$979$	3.81466	0.121917
$980$	−13.5128	−0.431650
$981$	0	0
$982$	42.3577	1.35169
$983$	30.1665	0.962162	0.481081	−	0.876676i	$-0.340244\pi$
$983$	30.1665	0.962162	0.481081	+	0.876676i	$0.340244\pi$
$984$	0	0
$985$	−5.98708	−0.190764
$986$	11.6635	0.371440
$987$	0	0
$988$	−5.21426	−0.165888
$989$	7.51176	0.238860
$990$	0	0
$991$	−18.3992	−0.584470	−0.292235	−	0.956347i	$-0.594399\pi$
$991$	−18.3992	−0.584470	−0.292235	+	0.956347i	$0.594399\pi$
$992$	−56.3896	−1.79037
$993$	0	0
$994$	−68.7246	−2.17981
$995$	−3.09231	−0.0980327
$996$	0	0
$997$	−17.2306	−0.545700	−0.272850	−	0.962057i	$-0.587966\pi$
$997$	−17.2306	−0.545700	−0.272850	+	0.962057i	$0.587966\pi$
$998$	22.4746	0.711422
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.h.1.1		12
3.2	odd	2		717.2.a.g.1.12	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.g.1.12	✓	12	3.2	odd	2
2151.2.a.h.1.1		12	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78161 q^{2} +5.73734 q^{4} +1.06781 q^{5} +2.18960 q^{7} -10.3958 q^{8} +O(q^{10})\)	\(q-2.78161 q^{2} +5.73734 q^{4} +1.06781 q^{5} +2.18960 q^{7} -10.3958 q^{8} -2.97024 q^{10} +1.68112 q^{11} +0.385556 q^{13} -6.09060 q^{14} +17.4424 q^{16} +3.21887 q^{17} -2.35719 q^{19} +6.12640 q^{20} -4.67622 q^{22} -1.83491 q^{23} -3.85978 q^{25} -1.07246 q^{26} +12.5625 q^{28} -1.30265 q^{29} +2.03380 q^{31} -27.7262 q^{32} -8.95364 q^{34} +2.33808 q^{35} +11.7158 q^{37} +6.55679 q^{38} -11.1008 q^{40} +4.50764 q^{41} -4.09379 q^{43} +9.64516 q^{44} +5.10401 q^{46} +3.20852 q^{47} -2.20566 q^{49} +10.7364 q^{50} +2.21206 q^{52} +12.3901 q^{53} +1.79512 q^{55} -22.7626 q^{56} +3.62346 q^{58} +6.97090 q^{59} +13.2806 q^{61} -5.65724 q^{62} +42.2386 q^{64} +0.411701 q^{65} -7.74686 q^{67} +18.4678 q^{68} -6.50362 q^{70} +11.2837 q^{71} -14.7720 q^{73} -32.5889 q^{74} -13.5240 q^{76} +3.68098 q^{77} -0.872260 q^{79} +18.6252 q^{80} -12.5385 q^{82} -3.55591 q^{83} +3.43715 q^{85} +11.3873 q^{86} -17.4766 q^{88} +2.26911 q^{89} +0.844212 q^{91} -10.5275 q^{92} -8.92485 q^{94} -2.51704 q^{95} -17.6862 q^{97} +6.13529 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8} - 15 q^{11} + 7 q^{13} + 6 q^{14} + 21 q^{16} + 3 q^{17} + 10 q^{19} + 4 q^{20} + 23 q^{22} - 20 q^{23} + 19 q^{25} + 10 q^{26} + 34 q^{28} - 2 q^{29} + 10 q^{31} - 26 q^{32} + 12 q^{34} - 7 q^{35} + 30 q^{37} + 3 q^{38} + 25 q^{40} + 28 q^{41} + 48 q^{43} - 25 q^{44} + 22 q^{46} - 13 q^{47} + 19 q^{49} - 12 q^{50} + 24 q^{52} + 2 q^{53} + 8 q^{55} + 7 q^{56} + 42 q^{58} + 14 q^{59} + 14 q^{61} - 8 q^{62} + 9 q^{64} + 35 q^{65} + 52 q^{67} - 3 q^{68} - 33 q^{70} + 7 q^{71} + 14 q^{73} + 13 q^{74} - 12 q^{76} + 6 q^{77} + 15 q^{79} + 8 q^{80} - 61 q^{82} - 29 q^{83} + 8 q^{85} + 9 q^{86} + 11 q^{88} + 71 q^{89} + 13 q^{91} - 2 q^{92} - 22 q^{94} - 2 q^{95} - 2 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8 - 15 * q^11 + 7 * q^13 + 6 * q^14 + 21 * q^16 + 3 * q^17 + 10 * q^19 + 4 * q^20 + 23 * q^22 - 20 * q^23 + 19 * q^25 + 10 * q^26 + 34 * q^28 - 2 * q^29 + 10 * q^31 - 26 * q^32 + 12 * q^34 - 7 * q^35 + 30 * q^37 + 3 * q^38 + 25 * q^40 + 28 * q^41 + 48 * q^43 - 25 * q^44 + 22 * q^46 - 13 * q^47 + 19 * q^49 - 12 * q^50 + 24 * q^52 + 2 * q^53 + 8 * q^55 + 7 * q^56 + 42 * q^58 + 14 * q^59 + 14 * q^61 - 8 * q^62 + 9 * q^64 + 35 * q^65 + 52 * q^67 - 3 * q^68 - 33 * q^70 + 7 * q^71 + 14 * q^73 + 13 * q^74 - 12 * q^76 + 6 * q^77 + 15 * q^79 + 8 * q^80 - 61 * q^82 - 29 * q^83 + 8 * q^85 + 9 * q^86 + 11 * q^88 + 71 * q^89 + 13 * q^91 - 2 * q^92 - 22 * q^94 - 2 * q^95 - 2 * q^98

Introduction

L-functions

Modular forms

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Database

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