LMFDB - Embedded newform 1331.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1331,2,Mod(1,1331)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1331, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1331.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1331 = 11^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1331.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:	$10.6280885090$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	10.10.43681195334656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 16x^{8} + 98x^{6} - 285x^{4} + 390x^{2} - 199 $ x^10 - 16x^8 + 98x^6 - 285x^4 + 390x^2 - 199
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.07599$ of defining polynomial
Character	$\chi$	$=$	1331.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.07599 q^{2} -1.30972 q^{3} +2.30972 q^{4} -2.91899 q^{5} +2.71896 q^{6} -3.98379 q^{7} -0.642978 q^{8} -1.28463 q^{9} +O(q^{10})$	\(q-2.07599 q^{2} -1.30972 q^{3} +2.30972 q^{4} -2.91899 q^{5} +2.71896 q^{6} -3.98379 q^{7} -0.642978 q^{8} -1.28463 q^{9} +6.05978 q^{10} -3.02509 q^{12} +1.61602 q^{13} +8.27029 q^{14} +3.82306 q^{15} -3.28463 q^{16} +3.20989 q^{17} +2.66687 q^{18} +5.52557 q^{19} -6.74204 q^{20} +5.21765 q^{21} +5.04159 q^{23} +0.842122 q^{24} +3.52048 q^{25} -3.35484 q^{26} +5.61167 q^{27} -9.20144 q^{28} +1.47629 q^{29} -7.93662 q^{30} -8.91121 q^{31} +8.10480 q^{32} -6.66369 q^{34} +11.6286 q^{35} -2.96714 q^{36} -4.70335 q^{37} -11.4710 q^{38} -2.11654 q^{39} +1.87684 q^{40} -3.21591 q^{41} -10.8318 q^{42} +11.3728 q^{43} +3.74982 q^{45} -10.4663 q^{46} -6.60660 q^{47} +4.30195 q^{48} +8.87058 q^{49} -7.30847 q^{50} -4.20406 q^{51} +3.73256 q^{52} +11.5459 q^{53} -11.6498 q^{54} +2.56149 q^{56} -7.23696 q^{57} -3.06475 q^{58} -2.01695 q^{59} +8.83020 q^{60} +5.07060 q^{61} +18.4996 q^{62} +5.11769 q^{63} -10.2562 q^{64} -4.71714 q^{65} -8.57509 q^{67} +7.41396 q^{68} -6.60308 q^{69} -24.1409 q^{70} +3.62279 q^{71} +0.825988 q^{72} -9.08386 q^{73} +9.76410 q^{74} -4.61085 q^{75} +12.7625 q^{76} +4.39390 q^{78} -13.3907 q^{79} +9.58779 q^{80} -3.49584 q^{81} +6.67618 q^{82} +12.8907 q^{83} +12.0513 q^{84} -9.36963 q^{85} -23.6098 q^{86} -1.93353 q^{87} +2.51455 q^{89} -7.78457 q^{90} -6.43788 q^{91} +11.6447 q^{92} +11.6712 q^{93} +13.7152 q^{94} -16.1291 q^{95} -10.6150 q^{96} -3.61907 q^{97} -18.4152 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} + 12 q^{4} - 12 q^{5} - 12 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 + 12 * q^4 - 12 * q^5 - 12 * q^9	$ 10 q - 2 q^{3} + 12 q^{4} - 12 q^{5} - 12 q^{9} - 20 q^{12} + 2 q^{14} - 2 q^{15} - 32 q^{16} - 10 q^{20} - 2 q^{23} - 18 q^{25} - 12 q^{26} + 4 q^{27} - 42 q^{31} - 28 q^{34} - 10 q^{36} - 30 q^{37} - 28 q^{38} + 4 q^{42} + 10 q^{45} - 24 q^{47} + 2 q^{48} - 8 q^{49} + 24 q^{53} - 6 q^{56} - 42 q^{58} - 8 q^{59} + 24 q^{60} - 18 q^{64} - 24 q^{67} + 18 q^{69} - 64 q^{70} + 32 q^{71} + 8 q^{75} - 46 q^{78} + 34 q^{80} - 22 q^{81} - 50 q^{82} - 22 q^{86} - 24 q^{89} - 42 q^{91} - 20 q^{92} + 26 q^{93} - 42 q^{97}+O(q^{100}) $ 10 * q - 2 * q^3 + 12 * q^4 - 12 * q^5 - 12 * q^9 - 20 * q^12 + 2 * q^14 - 2 * q^15 - 32 * q^16 - 10 * q^20 - 2 * q^23 - 18 * q^25 - 12 * q^26 + 4 * q^27 - 42 * q^31 - 28 * q^34 - 10 * q^36 - 30 * q^37 - 28 * q^38 + 4 * q^42 + 10 * q^45 - 24 * q^47 + 2 * q^48 - 8 * q^49 + 24 * q^53 - 6 * q^56 - 42 * q^58 - 8 * q^59 + 24 * q^60 - 18 * q^64 - 24 * q^67 + 18 * q^69 - 64 * q^70 + 32 * q^71 + 8 * q^75 - 46 * q^78 + 34 * q^80 - 22 * q^81 - 50 * q^82 - 22 * q^86 - 24 * q^89 - 42 * q^91 - 20 * q^92 + 26 * q^93 - 42 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.07599	−1.46794	−0.733972	−	0.679180i	$-0.762336\pi$
$2$	−2.07599	−1.46794	−0.733972	+	0.679180i	$0.762336\pi$
$3$	−1.30972	−0.756168	−0.378084	−	0.925771i	$-0.623417\pi$
$3$	−1.30972	−0.756168	−0.378084	+	0.925771i	$0.623417\pi$
$4$	2.30972	1.15486
$5$	−2.91899	−1.30541	−0.652705	−	0.757612i	$-0.726366\pi$
$5$	−2.91899	−1.30541	−0.652705	+	0.757612i	$0.726366\pi$
$6$	2.71896	1.11001
$7$	−3.98379	−1.50573	−0.752865	−	0.658174i	$-0.771329\pi$
$7$	−3.98379	−1.50573	−0.752865	+	0.658174i	$0.771329\pi$
$8$	−0.642978	−0.227327
$9$	−1.28463	−0.428210
$10$	6.05978	1.91627
$11$	0	0
$11$	0	0
$12$	−3.02509	−0.873269
$13$	1.61602	0.448203	0.224102	−	0.974566i	$-0.428055\pi$
$13$	1.61602	0.448203	0.224102	+	0.974566i	$0.428055\pi$
$14$	8.27029	2.21033
$15$	3.82306	0.987109
$16$	−3.28463	−0.821157
$17$	3.20989	0.778513	0.389257	−	0.921129i	$-0.372732\pi$
$17$	3.20989	0.778513	0.389257	+	0.921129i	$0.372732\pi$
$18$	2.66687	0.628588
$19$	5.52557	1.26765	0.633826	−	0.773475i	$-0.281483\pi$
$19$	5.52557	1.26765	0.633826	+	0.773475i	$0.281483\pi$
$20$	−6.74204	−1.50757
$21$	5.21765	1.13859
$22$	0	0
$23$	5.04159	1.05124	0.525622	−	0.850718i	$-0.323833\pi$
$23$	5.04159	1.05124	0.525622	+	0.850718i	$0.323833\pi$
$24$	0.842122	0.171897
$25$	3.52048	0.704096
$26$	−3.35484	−0.657937
$27$	5.61167	1.07997
$28$	−9.20144	−1.73891
$29$	1.47629	0.274140	0.137070	−	0.990561i	$-0.456232\pi$
$29$	1.47629	0.274140	0.137070	+	0.990561i	$0.456232\pi$
$30$	−7.93662	−1.44902
$31$	−8.91121	−1.60050	−0.800251	−	0.599666i	$-0.795300\pi$
$31$	−8.91121	−1.60050	−0.800251	+	0.599666i	$0.795300\pi$
$32$	8.10480	1.43274
$33$	0	0
$34$	−6.66369	−1.14281
$35$	11.6286	1.96560
$36$	−2.96714	−0.494523
$37$	−4.70335	−0.773227	−0.386613	−	0.922242i	$-0.626355\pi$
$37$	−4.70335	−0.773227	−0.386613	+	0.922242i	$0.626355\pi$
$38$	−11.4710	−1.86084
$39$	−2.11654	−0.338917
$40$	1.87684	0.296755
$41$	−3.21591	−0.502240	−0.251120	−	0.967956i	$-0.580799\pi$
$41$	−3.21591	−0.502240	−0.251120	+	0.967956i	$0.580799\pi$
$42$	−10.8318	−1.67138
$43$	11.3728	1.73434	0.867168	−	0.498016i	$-0.165938\pi$
$43$	11.3728	1.73434	0.867168	+	0.498016i	$0.165938\pi$
$44$	0	0
$45$	3.74982	0.558990
$46$	−10.4663	−1.54317
$47$	−6.60660	−0.963672	−0.481836	−	0.876261i	$-0.660030\pi$
$47$	−6.60660	−0.963672	−0.481836	+	0.876261i	$0.660030\pi$
$48$	4.30195	0.620933
$49$	8.87058	1.26723
$50$	−7.30847	−1.03357
$51$	−4.20406	−0.588687
$52$	3.73256	0.517612
$53$	11.5459	1.58596	0.792979	−	0.609249i	$-0.208529\pi$
$53$	11.5459	1.58596	0.792979	+	0.609249i	$0.208529\pi$
$54$	−11.6498	−1.58533
$55$	0	0
$56$	2.56149	0.342293
$57$	−7.23696	−0.958559
$58$	−3.06475	−0.402422
$59$	−2.01695	−0.262584	−0.131292	−	0.991344i	$-0.541913\pi$
$59$	−2.01695	−0.262584	−0.131292	+	0.991344i	$0.541913\pi$
$60$	8.83020	1.13997
$61$	5.07060	0.649224	0.324612	−	0.945847i	$-0.394766\pi$
$61$	5.07060	0.649224	0.324612	+	0.945847i	$0.394766\pi$
$62$	18.4996	2.34945
$63$	5.11769	0.644769
$64$	−10.2562	−1.28203
$65$	−4.71714	−0.585089
$66$	0	0
$67$	−8.57509	−1.04761	−0.523807	−	0.851837i	$-0.675489\pi$
$67$	−8.57509	−1.04761	−0.523807	+	0.851837i	$0.675489\pi$
$68$	7.41396	0.899074
$69$	−6.60308	−0.794917
$70$	−24.1409	−2.88539
$71$	3.62279	0.429946	0.214973	−	0.976620i	$-0.431034\pi$
$71$	3.62279	0.429946	0.214973	+	0.976620i	$0.431034\pi$
$72$	0.825988	0.0973436
$73$	−9.08386	−1.06319	−0.531593	−	0.847000i	$-0.678406\pi$
$73$	−9.08386	−1.06319	−0.531593	+	0.847000i	$0.678406\pi$
$74$	9.76410	1.13505
$75$	−4.61085	−0.532415
$76$	12.7625	1.46396
$77$	0	0
$78$	4.39390	0.497511
$79$	−13.3907	−1.50657	−0.753285	−	0.657695i	$-0.771532\pi$
$79$	−13.3907	−1.50657	−0.753285	+	0.657695i	$0.771532\pi$
$80$	9.58779	1.07195
$81$	−3.49584	−0.388426
$82$	6.67618	0.737261
$83$	12.8907	1.41494	0.707470	−	0.706743i	$-0.249836\pi$
$83$	12.8907	1.41494	0.707470	+	0.706743i	$0.249836\pi$
$84$	12.0513	1.31491
$85$	−9.36963	−1.01628
$86$	−23.6098	−2.54591
$87$	−1.93353	−0.207296
$88$	0	0
$89$	2.51455	0.266541	0.133271	−	0.991080i	$-0.457452\pi$
$89$	2.51455	0.266541	0.133271	+	0.991080i	$0.457452\pi$
$90$	−7.78457	−0.820566
$91$	−6.43788	−0.674873
$92$	11.6447	1.21404
$93$	11.6712	1.21025
$94$	13.7152	1.41462
$95$	−16.1291	−1.65481
$96$	−10.6150	−1.08339
$97$	−3.61907	−0.367461	−0.183730	−	0.982977i	$-0.558817\pi$
$97$	−3.61907	−0.367461	−0.183730	+	0.982977i	$0.558817\pi$
$98$	−18.4152	−1.86022
$99$	0	0
$100$	8.13133	0.813133
$101$	9.90253	0.985338	0.492669	−	0.870217i	$-0.336021\pi$
$101$	9.90253	0.985338	0.492669	+	0.870217i	$0.336021\pi$
$102$	8.72758	0.864159
$103$	12.9246	1.27349	0.636747	−	0.771073i	$-0.280279\pi$
$103$	12.9246	1.27349	0.636747	+	0.771073i	$0.280279\pi$
$104$	−1.03906	−0.101889
$105$	−15.2303	−1.48632
$106$	−23.9692	−2.32810
$107$	4.07035	0.393495	0.196748	−	0.980454i	$-0.436962\pi$
$107$	4.07035	0.393495	0.196748	+	0.980454i	$0.436962\pi$
$108$	12.9614	1.24721
$109$	−15.6733	−1.50123	−0.750616	−	0.660739i	$-0.770243\pi$
$109$	−15.6733	−1.50123	−0.750616	+	0.660739i	$0.770243\pi$
$110$	0	0
$111$	6.16008	0.584689
$112$	13.0853	1.23644
$113$	−0.222569	−0.0209376	−0.0104688	−	0.999945i	$-0.503332\pi$
$113$	−0.222569	−0.0209376	−0.0104688	+	0.999945i	$0.503332\pi$
$114$	15.0238	1.40711
$115$	−14.7163	−1.37230
$116$	3.40981	0.316593
$117$	−2.07599	−0.191925
$118$	4.18715	0.385459
$119$	−12.7875	−1.17223
$120$	−2.45814	−0.224397
$121$	0	0
$122$	−10.5265	−0.953025
$123$	4.21194	0.379778
$124$	−20.5824	−1.84836
$125$	4.31870	0.386276
$126$	−10.6243	−0.946485
$127$	5.94368	0.527417	0.263708	−	0.964602i	$-0.415054\pi$
$127$	5.94368	0.527417	0.263708	+	0.964602i	$0.415054\pi$
$128$	5.08214	0.449202
$129$	−14.8952	−1.31145
$130$	9.79272	0.858878
$131$	−15.2260	−1.33031	−0.665153	−	0.746707i	$-0.731634\pi$
$131$	−15.2260	−1.33031	−0.665153	+	0.746707i	$0.731634\pi$
$132$	0	0
$133$	−22.0127	−1.90874
$134$	17.8018	1.53784
$135$	−16.3804	−1.40980
$136$	−2.06389	−0.176977
$137$	0.321738	0.0274879	0.0137440	−	0.999906i	$-0.495625\pi$
$137$	0.321738	0.0274879	0.0137440	+	0.999906i	$0.495625\pi$
$138$	13.7079	1.16689
$139$	6.53839	0.554579	0.277289	−	0.960786i	$-0.410564\pi$
$139$	6.53839	0.554579	0.277289	+	0.960786i	$0.410564\pi$
$140$	26.8589	2.26999
$141$	8.65281	0.728698
$142$	−7.52087	−0.631137
$143$	0	0
$144$	4.21953	0.351628
$145$	−4.30926	−0.357865
$146$	18.8580	1.56070
$147$	−11.6180	−0.958236
$148$	−10.8634	−0.892969
$149$	−18.4169	−1.50877	−0.754386	−	0.656431i	$-0.772065\pi$
$149$	−18.4169	−1.50877	−0.754386	+	0.656431i	$0.772065\pi$
$150$	9.57206	0.781555
$151$	−5.44174	−0.442843	−0.221421	−	0.975178i	$-0.571070\pi$
$151$	−5.44174	−0.442843	−0.221421	+	0.975178i	$0.571070\pi$
$152$	−3.55282	−0.288172
$153$	−4.12352	−0.333367
$154$	0	0
$155$	26.0117	2.08931
$156$	−4.88861	−0.391402
$157$	−22.6050	−1.80408	−0.902038	−	0.431657i	$-0.857929\pi$
$157$	−22.6050	−1.80408	−0.902038	+	0.431657i	$0.857929\pi$
$158$	27.7989	2.21156
$159$	−15.1220	−1.19925
$160$	−23.6578	−1.87031
$161$	−20.0846	−1.58289
$162$	7.25731	0.570188
$163$	3.19187	0.250007	0.125003	−	0.992156i	$-0.460106\pi$
$163$	3.19187	0.250007	0.125003	+	0.992156i	$0.460106\pi$
$164$	−7.42785	−0.580017
$165$	0	0
$166$	−26.7610	−2.07705
$167$	−1.65721	−0.128239	−0.0641195	−	0.997942i	$-0.520424\pi$
$167$	−1.65721	−0.128239	−0.0641195	+	0.997942i	$0.520424\pi$
$168$	−3.35484	−0.258831
$169$	−10.3885	−0.799114
$170$	19.4512	1.49184
$171$	−7.09831	−0.542822
$172$	26.2680	2.00292
$173$	−19.3125	−1.46830	−0.734149	−	0.678988i	$-0.762419\pi$
$173$	−19.3125	−1.46830	−0.734149	+	0.678988i	$0.762419\pi$
$174$	4.01397	0.304298
$175$	−14.0248	−1.06018
$176$	0	0
$177$	2.64164	0.198558
$178$	−5.22016	−0.391268
$179$	14.2614	1.06595	0.532973	−	0.846132i	$-0.321075\pi$
$179$	14.2614	1.06595	0.532973	+	0.846132i	$0.321075\pi$
$180$	8.66103	0.645555
$181$	−22.0550	−1.63933	−0.819666	−	0.572841i	$-0.805841\pi$
$181$	−22.0550	−1.63933	−0.819666	+	0.572841i	$0.805841\pi$
$182$	13.3650	0.990677
$183$	−6.64108	−0.490922
$184$	−3.24163	−0.238976
$185$	13.7290	1.00938
$186$	−24.2293	−1.77658
$187$	0	0
$188$	−15.2594	−1.11291
$189$	−22.3557	−1.62614
$190$	33.4837	2.42916
$191$	0.788848	0.0570790	0.0285395	−	0.999593i	$-0.490914\pi$
$191$	0.788848	0.0570790	0.0285395	+	0.999593i	$0.490914\pi$
$192$	13.4328	0.969427
$193$	7.44439	0.535859	0.267929	−	0.963439i	$-0.413661\pi$
$193$	7.44439	0.535859	0.267929	+	0.963439i	$0.413661\pi$
$194$	7.51314	0.539412
$195$	6.17814	0.442426
$196$	20.4886	1.46347
$197$	13.0384	0.928948	0.464474	−	0.885587i	$-0.346243\pi$
$197$	13.0384	0.928948	0.464474	+	0.885587i	$0.346243\pi$
$198$	0	0
$199$	21.2005	1.50286	0.751432	−	0.659811i	$-0.229364\pi$
$199$	21.2005	1.50286	0.751432	+	0.659811i	$0.229364\pi$
$200$	−2.26359	−0.160060
$201$	11.2310	0.792172
$202$	−20.5575	−1.44642
$203$	−5.88122	−0.412781
$204$	−9.71022	−0.679851
$205$	9.38718	0.655629
$206$	−26.8312	−1.86942
$207$	−6.47657	−0.450153
$208$	−5.30803	−0.368045
$209$	0	0
$210$	31.6178	2.18184
$211$	−21.4203	−1.47464	−0.737318	−	0.675546i	$-0.763908\pi$
$211$	−21.4203	−1.47464	−0.737318	+	0.675546i	$0.763908\pi$
$212$	26.6679	1.83156
$213$	−4.74485	−0.325112
$214$	−8.44998	−0.577629
$215$	−33.1970	−2.26402
$216$	−3.60818	−0.245506
$217$	35.5004	2.40992
$218$	32.5376	2.20372
$219$	11.8973	0.803947
$220$	0	0
$221$	5.18725	0.348932
$222$	−12.7883	−0.858291
$223$	−0.676884	−0.0453275	−0.0226638	−	0.999743i	$-0.507215\pi$
$223$	−0.676884	−0.0453275	−0.0226638	+	0.999743i	$0.507215\pi$
$224$	−32.2878	−2.15732
$225$	−4.52251	−0.301501
$226$	0.462051	0.0307352
$227$	0.615833	0.0408743	0.0204371	−	0.999791i	$-0.493494\pi$
$227$	0.615833	0.0408743	0.0204371	+	0.999791i	$0.493494\pi$
$228$	−16.7154	−1.10700
$229$	−9.11509	−0.602342	−0.301171	−	0.953570i	$-0.597377\pi$
$229$	−9.11509	−0.602342	−0.301171	+	0.953570i	$0.597377\pi$
$230$	30.5509	2.01447
$231$	0	0
$232$	−0.949220	−0.0623193
$233$	15.2194	0.997054	0.498527	−	0.866874i	$-0.333875\pi$
$233$	15.2194	0.997054	0.498527	+	0.866874i	$0.333875\pi$
$234$	4.30972	0.281735
$235$	19.2846	1.25799
$236$	−4.65858	−0.303248
$237$	17.5381	1.13922
$238$	26.5468	1.72077
$239$	24.4782	1.58336	0.791680	−	0.610936i	$-0.209207\pi$
$239$	24.4782	1.58336	0.791680	+	0.610936i	$0.209207\pi$
$240$	−12.5573	−0.810572
$241$	9.23935	0.595159	0.297580	−	0.954697i	$-0.403821\pi$
$241$	9.23935	0.595159	0.297580	+	0.954697i	$0.403821\pi$
$242$	0	0
$243$	−12.2564	−0.786251
$244$	11.7117	0.749763
$245$	−25.8931	−1.65425
$246$	−8.74393	−0.557493
$247$	8.92943	0.568166
$248$	5.72971	0.363837
$249$	−16.8833	−1.06993
$250$	−8.96557	−0.567032
$251$	4.48310	0.282971	0.141485	−	0.989940i	$-0.454812\pi$
$251$	4.48310	0.282971	0.141485	+	0.989940i	$0.454812\pi$
$252$	11.8204	0.744618
$253$	0	0
$254$	−12.3390	−0.774218
$255$	12.2716	0.768478
$256$	9.96195	0.622622
$257$	7.33009	0.457238	0.228619	−	0.973516i	$-0.426579\pi$
$257$	7.33009	0.457238	0.228619	+	0.973516i	$0.426579\pi$
$258$	30.9222	1.92513
$259$	18.7372	1.16427
$260$	−10.8953	−0.675696
$261$	−1.89648	−0.117389
$262$	31.6091	1.95281
$263$	−30.0844	−1.85508	−0.927541	−	0.373721i	$-0.878082\pi$
$263$	−30.0844	−1.85508	−0.927541	+	0.373721i	$0.878082\pi$
$264$	0	0
$265$	−33.7025	−2.07033
$266$	45.6981	2.80193
$267$	−3.29335	−0.201550
$268$	−19.8061	−1.20985
$269$	−3.03664	−0.185147	−0.0925736	−	0.995706i	$-0.529509\pi$
$269$	−3.03664	−0.185147	−0.0925736	+	0.995706i	$0.529509\pi$
$270$	34.0055	2.06951
$271$	23.1183	1.40434	0.702168	−	0.712011i	$-0.252215\pi$
$271$	23.1183	1.40434	0.702168	+	0.712011i	$0.252215\pi$
$272$	−10.5433	−0.639282
$273$	8.43183	0.510318
$274$	−0.667923	−0.0403507
$275$	0	0
$276$	−15.2513	−0.918018
$277$	11.6041	0.697224	0.348612	−	0.937267i	$-0.386653\pi$
$277$	11.6041	0.697224	0.348612	+	0.937267i	$0.386653\pi$
$278$	−13.5736	−0.814091
$279$	11.4476	0.685351
$280$	−7.47695	−0.446833
$281$	6.84916	0.408587	0.204293	−	0.978910i	$-0.434510\pi$
$281$	6.84916	0.408587	0.204293	+	0.978910i	$0.434510\pi$
$282$	−17.9631	−1.06969
$283$	10.6065	0.630493	0.315246	−	0.949010i	$-0.397913\pi$
$283$	10.6065	0.630493	0.315246	+	0.949010i	$0.397913\pi$
$284$	8.36764	0.496528
$285$	21.1246	1.25131
$286$	0	0
$287$	12.8115	0.756239
$288$	−10.4117	−0.613514
$289$	−6.69660	−0.393917
$290$	8.94597	0.525326
$291$	4.73997	0.277862
$292$	−20.9812	−1.22783
$293$	−10.0908	−0.589512	−0.294756	−	0.955573i	$-0.595238\pi$
$293$	−10.0908	−0.589512	−0.294756	+	0.955573i	$0.595238\pi$
$294$	24.1188	1.40664
$295$	5.88744	0.342780
$296$	3.02415	0.175775
$297$	0	0
$298$	38.2333	2.21479
$299$	8.14730	0.471171
$300$	−10.6498	−0.614865
$301$	−45.3068	−2.61144
$302$	11.2970	0.650069
$303$	−12.9696	−0.745081
$304$	−18.1495	−1.04094
$305$	−14.8010	−0.847504
$306$	8.56038	0.489364
$307$	28.6523	1.63527	0.817636	−	0.575736i	$-0.195284\pi$
$307$	28.6523	1.63527	0.817636	+	0.575736i	$0.195284\pi$
$308$	0	0
$309$	−16.9276	−0.962976
$310$	−54.0000	−3.06699
$311$	−22.0072	−1.24791	−0.623957	−	0.781459i	$-0.714476\pi$
$311$	−22.0072	−1.24791	−0.623957	+	0.781459i	$0.714476\pi$
$312$	1.36089	0.0770450
$313$	2.82645	0.159760	0.0798801	−	0.996804i	$-0.474546\pi$
$313$	2.82645	0.159760	0.0798801	+	0.996804i	$0.474546\pi$
$314$	46.9277	2.64828
$315$	−14.9385	−0.841688
$316$	−30.9287	−1.73988
$317$	24.8797	1.39739	0.698693	−	0.715422i	$-0.253766\pi$
$317$	24.8797	1.39739	0.698693	+	0.715422i	$0.253766\pi$
$318$	31.3930	1.76043
$319$	0	0
$320$	29.9377	1.67357
$321$	−5.33102	−0.297548
$322$	41.6954	2.32359
$323$	17.7365	0.986884
$324$	−8.07441	−0.448578
$325$	5.68916	0.315578
$326$	−6.62628	−0.366996
$327$	20.5277	1.13518
$328$	2.06776	0.114173
$329$	26.3193	1.45103
$330$	0	0
$331$	−6.74841	−0.370926	−0.185463	−	0.982651i	$-0.559379\pi$
$331$	−6.74841	−0.370926	−0.185463	+	0.982651i	$0.559379\pi$
$332$	29.7740	1.63406
$333$	6.04207	0.331103
$334$	3.44035	0.188248
$335$	25.0306	1.36757
$336$	−17.1381	−0.934958
$337$	27.5205	1.49914	0.749568	−	0.661927i	$-0.230261\pi$
$337$	27.5205	1.49914	0.749568	+	0.661927i	$0.230261\pi$
$338$	21.5663	1.17305
$339$	0.291504	0.0158323
$340$	−21.6412	−1.17366
$341$	0	0
$342$	14.7360	0.796832
$343$	−7.45200	−0.402370
$344$	−7.31246	−0.394261
$345$	19.2743	1.03769
$346$	40.0924	2.15538
$347$	−4.99965	−0.268395	−0.134198	−	0.990955i	$-0.542846\pi$
$347$	−4.99965	−0.268395	−0.134198	+	0.990955i	$0.542846\pi$
$348$	−4.46590	−0.239398
$349$	−14.2950	−0.765192	−0.382596	−	0.923916i	$-0.624970\pi$
$349$	−14.2950	−0.765192	−0.382596	+	0.923916i	$0.624970\pi$
$350$	29.1154	1.55628
$351$	9.06857	0.484045
$352$	0	0
$353$	−12.0233	−0.639938	−0.319969	−	0.947428i	$-0.603672\pi$
$353$	−12.0233	−0.639938	−0.319969	+	0.947428i	$0.603672\pi$
$354$	−5.48401	−0.291472
$355$	−10.5749	−0.561256
$356$	5.80790	0.307818
$357$	16.7481	0.886404
$358$	−29.6065	−1.56475
$359$	19.5131	1.02986	0.514931	−	0.857232i	$-0.327818\pi$
$359$	19.5131	1.02986	0.514931	+	0.857232i	$0.327818\pi$
$360$	−2.41105	−0.127073
$361$	11.5319	0.606944
$362$	45.7858	2.40645
$363$	0	0
$364$	−14.8697	−0.779385
$365$	26.5157	1.38789
$366$	13.7868	0.720647
$367$	−14.2673	−0.744747	−0.372374	−	0.928083i	$-0.621456\pi$
$367$	−14.2673	−0.744747	−0.372374	+	0.928083i	$0.621456\pi$
$368$	−16.5597	−0.863237
$369$	4.13125	0.215064
$370$	−28.5013	−1.48171
$371$	−45.9966	−2.38803
$372$	26.9572	1.39767
$373$	−21.5605	−1.11636	−0.558181	−	0.829719i	$-0.688501\pi$
$373$	−21.5605	−1.11636	−0.558181	+	0.829719i	$0.688501\pi$
$374$	0	0
$375$	−5.65630	−0.292090
$376$	4.24790	0.219069
$377$	2.38571	0.122870
$378$	46.4102	2.38708
$379$	−11.9542	−0.614048	−0.307024	−	0.951702i	$-0.599333\pi$
$379$	−11.9542	−0.614048	−0.307024	+	0.951702i	$0.599333\pi$
$380$	−37.2536	−1.91107
$381$	−7.78457	−0.398816
$382$	−1.63764	−0.0837888
$383$	−22.8560	−1.16789	−0.583943	−	0.811794i	$-0.698491\pi$
$383$	−22.8560	−1.16789	−0.583943	+	0.811794i	$0.698491\pi$
$384$	−6.65619	−0.339672
$385$	0	0
$386$	−15.4544	−0.786611
$387$	−14.6098	−0.742660
$388$	−8.35904	−0.424366
$389$	0.507191	0.0257156	0.0128578	−	0.999917i	$-0.495907\pi$
$389$	0.507191	0.0257156	0.0128578	+	0.999917i	$0.495907\pi$
$390$	−12.8257	−0.649456
$391$	16.1829	0.818407
$392$	−5.70358	−0.288075
$393$	19.9419	1.00593
$394$	−27.0676	−1.36364
$395$	39.0872	1.96669
$396$	0	0
$397$	−19.1018	−0.958694	−0.479347	−	0.877625i	$-0.659126\pi$
$397$	−19.1018	−0.958694	−0.479347	+	0.877625i	$0.659126\pi$
$398$	−44.0120	−2.20612
$399$	28.8305	1.44333
$400$	−11.5635	−0.578173
$401$	−15.1098	−0.754545	−0.377273	−	0.926102i	$-0.623138\pi$
$401$	−15.1098	−0.754545	−0.377273	+	0.926102i	$0.623138\pi$
$402$	−23.3154	−1.16286
$403$	−14.4007	−0.717350
$404$	22.8721	1.13793
$405$	10.2043	0.507056
$406$	12.2093	0.605939
$407$	0	0
$408$	2.70312	0.133824
$409$	2.92674	0.144718	0.0723589	−	0.997379i	$-0.476947\pi$
$409$	2.92674	0.144718	0.0723589	+	0.997379i	$0.476947\pi$
$410$	−19.4877	−0.962427
$411$	−0.421387	−0.0207855
$412$	29.8521	1.47071
$413$	8.03509	0.395381
$414$	13.4453	0.660799
$415$	−37.6278	−1.84708
$416$	13.0975	0.642159
$417$	−8.56346	−0.419355
$418$	0	0
$419$	−13.2738	−0.648467	−0.324233	−	0.945977i	$-0.605106\pi$
$419$	−13.2738	−0.648467	−0.324233	+	0.945977i	$0.605106\pi$
$420$	−35.1777	−1.71649
$421$	−25.1717	−1.22679	−0.613397	−	0.789775i	$-0.710197\pi$
$421$	−25.1717	−1.22679	−0.613397	+	0.789775i	$0.710197\pi$
$422$	44.4683	2.16468
$423$	8.48704	0.412654
$424$	−7.42379	−0.360531
$425$	11.3004	0.548148
$426$	9.85024	0.477246
$427$	−20.2002	−0.977557
$428$	9.40137	0.454432
$429$	0	0
$430$	68.9166	3.32345
$431$	17.6775	0.851495	0.425747	−	0.904842i	$-0.360011\pi$
$431$	17.6775	0.851495	0.425747	+	0.904842i	$0.360011\pi$
$432$	−18.4323	−0.886823
$433$	4.07049	0.195615	0.0978076	−	0.995205i	$-0.468817\pi$
$433$	4.07049	0.195615	0.0978076	+	0.995205i	$0.468817\pi$
$434$	−73.6984	−3.53764
$435$	5.64393	0.270606
$436$	−36.2010	−1.73371
$437$	27.8577	1.33261
$438$	−24.6987	−1.18015
$439$	−19.4195	−0.926843	−0.463422	−	0.886138i	$-0.653378\pi$
$439$	−19.4195	−0.926843	−0.463422	+	0.886138i	$0.653378\pi$
$440$	0	0
$441$	−11.3954	−0.542639
$442$	−10.7687	−0.512213
$443$	−21.3894	−1.01624	−0.508120	−	0.861286i	$-0.669659\pi$
$443$	−21.3894	−1.01624	−0.508120	+	0.861286i	$0.669659\pi$
$444$	14.2281	0.675235
$445$	−7.33992	−0.347946
$446$	1.40520	0.0665383
$447$	24.1210	1.14089
$448$	40.8586	1.93039
$449$	5.04186	0.237940	0.118970	−	0.992898i	$-0.462041\pi$
$449$	5.04186	0.237940	0.118970	+	0.992898i	$0.462041\pi$
$450$	9.38867	0.442586
$451$	0	0
$452$	−0.514073	−0.0241800
$453$	7.12717	0.334864
$454$	−1.27846	−0.0600012
$455$	18.7921	0.880987
$456$	4.65320	0.217906
$457$	−22.4307	−1.04927	−0.524633	−	0.851329i	$-0.675797\pi$
$457$	−22.4307	−1.04927	−0.524633	+	0.851329i	$0.675797\pi$
$458$	18.9228	0.884204
$459$	18.0129	0.840768
$460$	−33.9906	−1.58482
$461$	−15.4392	−0.719076	−0.359538	−	0.933130i	$-0.617066\pi$
$461$	−15.4392	−0.719076	−0.359538	+	0.933130i	$0.617066\pi$
$462$	0	0
$463$	27.7529	1.28979	0.644894	−	0.764272i	$-0.276902\pi$
$463$	27.7529	1.28979	0.644894	+	0.764272i	$0.276902\pi$
$464$	−4.84906	−0.225112
$465$	−34.0681	−1.57987
$466$	−31.5952	−1.46362
$467$	−12.8341	−0.593891	−0.296946	−	0.954894i	$-0.595968\pi$
$467$	−12.8341	−0.593891	−0.296946	+	0.954894i	$0.595968\pi$
$468$	−4.79495	−0.221647
$469$	34.1613	1.57742
$470$	−40.0345	−1.84666
$471$	29.6063	1.36418
$472$	1.29685	0.0596924
$473$	0	0
$474$	−36.4088	−1.67231
$475$	19.4527	0.892549
$476$	−29.5356	−1.35376
$477$	−14.8323	−0.679123
$478$	−50.8163	−2.32428
$479$	−4.90117	−0.223940	−0.111970	−	0.993712i	$-0.535716\pi$
$479$	−4.90117	−0.223940	−0.111970	+	0.993712i	$0.535716\pi$
$480$	30.9851	1.41427
$481$	−7.60071	−0.346563
$482$	−19.1808	−0.873660
$483$	26.3053	1.19693
$484$	0	0
$485$	10.5640	0.479687
$486$	25.4442	1.15417
$487$	6.81660	0.308890	0.154445	−	0.988001i	$-0.450641\pi$
$487$	6.81660	0.308890	0.154445	+	0.988001i	$0.450641\pi$
$488$	−3.26028	−0.147586
$489$	−4.18046	−0.189047
$490$	53.7537	2.42835
$491$	28.1797	1.27173	0.635867	−	0.771799i	$-0.280643\pi$
$491$	28.1797	1.27173	0.635867	+	0.771799i	$0.280643\pi$
$492$	9.72841	0.438591
$493$	4.73872	0.213421
$494$	−18.5374	−0.834036
$495$	0	0
$496$	29.2700	1.31426
$497$	−14.4324	−0.647383
$498$	35.0494	1.57060
$499$	−2.15205	−0.0963389	−0.0481694	−	0.998839i	$-0.515339\pi$
$499$	−2.15205	−0.0963389	−0.0481694	+	0.998839i	$0.515339\pi$
$500$	9.97500	0.446095
$501$	2.17049	0.0969703
$502$	−9.30686	−0.415386
$503$	−33.1043	−1.47605	−0.738024	−	0.674774i	$-0.764241\pi$
$503$	−33.1043	−1.47605	−0.738024	+	0.674774i	$0.764241\pi$
$504$	−3.29056	−0.146573
$505$	−28.9053	−1.28627
$506$	0	0
$507$	13.6060	0.604264
$508$	13.7283	0.609093
$509$	11.0759	0.490932	0.245466	−	0.969405i	$-0.421059\pi$
$509$	11.0759	0.490932	0.245466	+	0.969405i	$0.421059\pi$
$510$	−25.4757	−1.12808
$511$	36.1882	1.60087
$512$	−30.8452	−1.36318
$513$	31.0077	1.36902
$514$	−15.2172	−0.671200
$515$	−37.7266	−1.66243
$516$	−34.4038	−1.51454
$517$	0	0
$518$	−38.8981	−1.70909
$519$	25.2939	1.11028
$520$	3.03302	0.133007
$521$	−15.6816	−0.687023	−0.343512	−	0.939148i	$-0.611617\pi$
$521$	−15.6816	−0.687023	−0.343512	+	0.939148i	$0.611617\pi$
$522$	3.93707	0.172321
$523$	4.40351	0.192552	0.0962761	−	0.995355i	$-0.469307\pi$
$523$	4.40351	0.192552	0.0962761	+	0.995355i	$0.469307\pi$
$524$	−35.1679	−1.53632
$525$	18.3686	0.801673
$526$	62.4548	2.72316
$527$	−28.6040	−1.24601
$528$	0	0
$529$	2.41760	0.105113
$530$	69.9658	3.03912
$531$	2.59103	0.112441
$532$	−50.8432	−2.20433
$533$	−5.19697	−0.225106
$534$	6.83696	0.295864
$535$	−11.8813	−0.513673
$536$	5.51359	0.238151
$537$	−18.6784	−0.806034
$538$	6.30403	0.271786
$539$	0	0
$540$	−37.8341	−1.62812
$541$	−18.2167	−0.783196	−0.391598	−	0.920136i	$-0.628078\pi$
$541$	−18.2167	−0.783196	−0.391598	+	0.920136i	$0.628078\pi$
$542$	−47.9933	−2.06149
$543$	28.8859	1.23961
$544$	26.0155	1.11541
$545$	45.7502	1.95972
$546$	−17.5044	−0.749118
$547$	7.03378	0.300743	0.150371	−	0.988630i	$-0.451953\pi$
$547$	7.03378	0.300743	0.150371	+	0.988630i	$0.451953\pi$
$548$	0.743125	0.0317447
$549$	−6.51384	−0.278004
$550$	0	0
$551$	8.15733	0.347514
$552$	4.24563	0.180706
$553$	53.3457	2.26849
$554$	−24.0900	−1.02349
$555$	−17.9812	−0.763259
$556$	15.1018	0.640461
$557$	−10.0059	−0.423963	−0.211981	−	0.977274i	$-0.567992\pi$
$557$	−10.0059	−0.423963	−0.211981	+	0.977274i	$0.567992\pi$
$558$	−23.7651	−1.00606
$559$	18.3787	0.777335
$560$	−38.1957	−1.61406
$561$	0	0
$562$	−14.2188	−0.599783
$563$	−14.4334	−0.608294	−0.304147	−	0.952625i	$-0.598371\pi$
$563$	−14.4334	−0.608294	−0.304147	+	0.952625i	$0.598371\pi$
$564$	19.9856	0.841545
$565$	0.649677	0.0273321
$566$	−22.0190	−0.925528
$567$	13.9267	0.584866
$568$	−2.32937	−0.0977383
$569$	−23.6057	−0.989603	−0.494802	−	0.869006i	$-0.664759\pi$
$569$	−23.6057	−0.989603	−0.494802	+	0.869006i	$0.664759\pi$
$570$	−43.8544	−1.83686
$571$	−1.36283	−0.0570328	−0.0285164	−	0.999593i	$-0.509078\pi$
$571$	−1.36283	−0.0570328	−0.0285164	+	0.999593i	$0.509078\pi$
$572$	0	0
$573$	−1.03317	−0.0431613
$574$	−26.5965	−1.11012
$575$	17.7488	0.740176
$576$	13.1754	0.548976
$577$	−17.3274	−0.721350	−0.360675	−	0.932692i	$-0.617454\pi$
$577$	−17.3274	−0.721350	−0.360675	+	0.932692i	$0.617454\pi$
$578$	13.9020	0.578249
$579$	−9.75007	−0.405199
$580$	−9.95319	−0.413284
$581$	−51.3539	−2.13052
$582$	−9.84012	−0.407886
$583$	0	0
$584$	5.84072	0.241691
$585$	6.05978	0.250541
$586$	20.9484	0.865371
$587$	−26.7265	−1.10312	−0.551560	−	0.834135i	$-0.685967\pi$
$587$	−26.7265	−1.10312	−0.551560	+	0.834135i	$0.685967\pi$
$588$	−26.8343	−1.10663
$589$	−49.2396	−2.02888
$590$	−12.2222	−0.503182
$591$	−17.0767	−0.702441
$592$	15.4488	0.634941
$593$	−18.3552	−0.753759	−0.376880	−	0.926262i	$-0.623003\pi$
$593$	−18.3552	−0.753759	−0.376880	+	0.926262i	$0.623003\pi$
$594$	0	0
$595$	37.3266	1.53024
$596$	−42.5380	−1.74242
$597$	−27.7668	−1.13642
$598$	−16.9137	−0.691653
$599$	20.2006	0.825375	0.412688	−	0.910873i	$-0.364590\pi$
$599$	20.2006	0.825375	0.412688	+	0.910873i	$0.364590\pi$
$600$	2.96467	0.121032
$601$	−1.44518	−0.0589500	−0.0294750	−	0.999566i	$-0.509384\pi$
$601$	−1.44518	−0.0589500	−0.0294750	+	0.999566i	$0.509384\pi$
$602$	94.0564	3.83345
$603$	11.0158	0.448598
$604$	−12.5689	−0.511422
$605$	0	0
$606$	26.9246	1.09374
$607$	4.46020	0.181034	0.0905170	−	0.995895i	$-0.471148\pi$
$607$	4.46020	0.181034	0.0905170	+	0.995895i	$0.471148\pi$
$608$	44.7837	1.81622
$609$	7.70276	0.312131
$610$	30.7267	1.24409
$611$	−10.6764	−0.431921
$612$	−9.52419	−0.384992
$613$	34.4575	1.39173	0.695863	−	0.718175i	$-0.255022\pi$
$613$	34.4575	1.39173	0.695863	+	0.718175i	$0.255022\pi$
$614$	−59.4817	−2.40049
$615$	−12.2946	−0.495766
$616$	0	0
$617$	−32.5312	−1.30966	−0.654829	−	0.755777i	$-0.727259\pi$
$617$	−32.5312	−1.30966	−0.654829	+	0.755777i	$0.727259\pi$
$618$	35.1414	1.41360
$619$	1.45388	0.0584364	0.0292182	−	0.999573i	$-0.490698\pi$
$619$	1.45388	0.0584364	0.0292182	+	0.999573i	$0.490698\pi$
$620$	60.0798	2.41286
$621$	28.2917	1.13531
$622$	45.6867	1.83187
$623$	−10.0174	−0.401339
$624$	6.95204	0.278304
$625$	−30.2086	−1.20834
$626$	−5.86766	−0.234519
$627$	0	0
$628$	−52.2113	−2.08346
$629$	−15.0973	−0.601967
$630$	31.0121	1.23555
$631$	−28.9615	−1.15294	−0.576469	−	0.817119i	$-0.695570\pi$
$631$	−28.9615	−1.15294	−0.576469	+	0.817119i	$0.695570\pi$
$632$	8.60991	0.342484
$633$	28.0546	1.11507
$634$	−51.6500	−2.05128
$635$	−17.3495	−0.688495
$636$	−34.9275	−1.38497
$637$	14.3350	0.567975
$638$	0	0
$639$	−4.65394	−0.184107
$640$	−14.8347	−0.586393
$641$	35.0645	1.38496	0.692482	−	0.721436i	$-0.256517\pi$
$641$	35.0645	1.38496	0.692482	+	0.721436i	$0.256517\pi$
$642$	11.0671	0.436785
$643$	−26.2747	−1.03617	−0.518087	−	0.855328i	$-0.673356\pi$
$643$	−26.2747	−1.03617	−0.518087	+	0.855328i	$0.673356\pi$
$644$	−46.3899	−1.82802
$645$	43.4789	1.71198
$646$	−36.8207	−1.44869
$647$	24.0786	0.946626	0.473313	−	0.880894i	$-0.343058\pi$
$647$	24.0786	0.946626	0.473313	+	0.880894i	$0.343058\pi$
$648$	2.24775	0.0882998
$649$	0	0
$650$	−11.8106	−0.463251
$651$	−46.4956	−1.82231
$652$	7.37233	0.288723
$653$	−23.5863	−0.923005	−0.461502	−	0.887139i	$-0.652689\pi$
$653$	−23.5863	−0.923005	−0.461502	+	0.887139i	$0.652689\pi$
$654$	−42.6152	−1.66639
$655$	44.4446	1.73659
$656$	10.5631	0.412418
$657$	11.6694	0.455266
$658$	−54.6385	−2.13003
$659$	2.26037	0.0880516	0.0440258	−	0.999030i	$-0.485982\pi$
$659$	2.26037	0.0880516	0.0440258	+	0.999030i	$0.485982\pi$
$660$	0	0
$661$	−5.92950	−0.230631	−0.115315	−	0.993329i	$-0.536788\pi$
$661$	−5.92950	−0.230631	−0.115315	+	0.993329i	$0.536788\pi$
$662$	14.0096	0.544499
$663$	−6.79385	−0.263851
$664$	−8.28845	−0.321654
$665$	64.2548	2.49169
$666$	−12.5433	−0.486041
$667$	7.44283	0.288188
$668$	−3.82770	−0.148098
$669$	0.886530	0.0342752
$670$	−51.9631	−2.00751
$671$	0	0
$672$	42.2881	1.63130
$673$	−25.8596	−0.996813	−0.498407	−	0.866943i	$-0.666081\pi$
$673$	−25.8596	−0.996813	−0.498407	+	0.866943i	$0.666081\pi$
$674$	−57.1322	−2.20065
$675$	19.7558	0.760400
$676$	−23.9945	−0.922865
$677$	−16.6186	−0.638704	−0.319352	−	0.947636i	$-0.603465\pi$
$677$	−16.6186	−0.638704	−0.319352	+	0.947636i	$0.603465\pi$
$678$	−0.605158	−0.0232410
$679$	14.4176	0.553297
$680$	6.02446	0.231028
$681$	−0.806570	−0.0309078
$682$	0	0
$683$	33.9910	1.30063	0.650314	−	0.759666i	$-0.274637\pi$
$683$	33.9910	1.30063	0.650314	+	0.759666i	$0.274637\pi$
$684$	−16.3951	−0.626883
$685$	−0.939148	−0.0358830
$686$	15.4702	0.590657
$687$	11.9382	0.455472
$688$	−37.3554	−1.42416
$689$	18.6585	0.710831
$690$	−40.0132	−1.52328
$691$	27.9749	1.06422	0.532108	−	0.846676i	$-0.321400\pi$
$691$	27.9749	1.06422	0.532108	+	0.846676i	$0.321400\pi$
$692$	−44.6064	−1.69568
$693$	0	0
$694$	10.3792	0.393989
$695$	−19.0855	−0.723953
$696$	1.24321	0.0471239
$697$	−10.3227	−0.391001
$698$	29.6762	1.12326
$699$	−19.9331	−0.753940
$700$	−32.3935	−1.22436
$701$	−49.1565	−1.85662	−0.928308	−	0.371813i	$-0.878736\pi$
$701$	−49.1565	−1.85662	−0.928308	+	0.371813i	$0.878736\pi$
$702$	−18.8262	−0.710550
$703$	−25.9887	−0.980183
$704$	0	0
$705$	−25.2574	−0.951250
$706$	24.9603	0.939393
$707$	−39.4496	−1.48365
$708$	6.10145	0.229306
$709$	−9.35694	−0.351407	−0.175704	−	0.984443i	$-0.556220\pi$
$709$	−9.35694	−0.351407	−0.175704	+	0.984443i	$0.556220\pi$
$710$	21.9533	0.823893
$711$	17.2021	0.645128
$712$	−1.61680	−0.0605920
$713$	−44.9267	−1.68252
$714$	−34.7688	−1.30119
$715$	0	0
$716$	32.9398	1.23102
$717$	−32.0596	−1.19729
$718$	−40.5089	−1.51178
$719$	−15.0207	−0.560179	−0.280090	−	0.959974i	$-0.590364\pi$
$719$	−15.0207	−0.560179	−0.280090	+	0.959974i	$0.590364\pi$
$720$	−12.3168	−0.459018
$721$	−51.4887	−1.91754
$722$	−23.9402	−0.890960
$723$	−12.1010	−0.450040
$724$	−50.9408	−1.89320
$725$	5.19724	0.193021
$726$	0	0
$727$	28.9167	1.07246	0.536231	−	0.844071i	$-0.319848\pi$
$727$	28.9167	1.07246	0.536231	+	0.844071i	$0.319848\pi$
$728$	4.13941	0.153417
$729$	26.5400	0.982964
$730$	−55.0462	−2.03735
$731$	36.5055	1.35020
$732$	−15.3390	−0.566947
$733$	−40.3514	−1.49041	−0.745206	−	0.666834i	$-0.767649\pi$
$733$	−40.3514	−1.49041	−0.745206	+	0.666834i	$0.767649\pi$
$734$	29.6187	1.09325
$735$	33.9127	1.25089
$736$	40.8611	1.50616
$737$	0	0
$738$	−8.57642	−0.315702
$739$	26.0543	0.958421	0.479211	−	0.877700i	$-0.340923\pi$
$739$	26.0543	0.958421	0.479211	+	0.877700i	$0.340923\pi$
$740$	31.7102	1.16569
$741$	−11.6951	−0.429629
$742$	95.4884	3.50549
$743$	7.26190	0.266413	0.133207	−	0.991088i	$-0.457473\pi$
$743$	7.26190	0.266413	0.133207	+	0.991088i	$0.457473\pi$
$744$	−7.50433	−0.275122
$745$	53.7587	1.96957
$746$	44.7594	1.63876
$747$	−16.5598	−0.605892
$748$	0	0
$749$	−16.2154	−0.592498
$750$	11.7424	0.428772
$751$	−10.4520	−0.381400	−0.190700	−	0.981648i	$-0.561076\pi$
$751$	−10.4520	−0.381400	−0.190700	+	0.981648i	$0.561076\pi$
$752$	21.7002	0.791326
$753$	−5.87162	−0.213974
$754$	−4.95270	−0.180367
$755$	15.8844	0.578092
$756$	−51.6355	−1.87796
$757$	−16.0120	−0.581968	−0.290984	−	0.956728i	$-0.593983\pi$
$757$	−16.0120	−0.581968	−0.290984	+	0.956728i	$0.593983\pi$
$758$	24.8168	0.901388
$759$	0	0
$760$	10.3706	0.376182
$761$	11.2103	0.406372	0.203186	−	0.979140i	$-0.434870\pi$
$761$	11.2103	0.406372	0.203186	+	0.979140i	$0.434870\pi$
$762$	16.1607	0.585439
$763$	62.4392	2.26045
$764$	1.82202	0.0659183
$765$	12.0365	0.435181
$766$	47.4488	1.71439
$767$	−3.25942	−0.117691
$768$	−13.0474	−0.470807
$769$	25.4455	0.917589	0.458794	−	0.888542i	$-0.348281\pi$
$769$	25.4455	0.917589	0.458794	+	0.888542i	$0.348281\pi$
$770$	0	0
$771$	−9.60037	−0.345749
$772$	17.1945	0.618842
$773$	−30.8291	−1.10885	−0.554423	−	0.832235i	$-0.687061\pi$
$773$	−30.8291	−1.10885	−0.554423	+	0.832235i	$0.687061\pi$
$774$	30.3298	1.09018
$775$	−31.3717	−1.12691
$776$	2.32698	0.0835337
$777$	−24.5405	−0.880385
$778$	−1.05292	−0.0377491
$779$	−17.7697	−0.636666
$780$	14.2698	0.510940
$781$	0	0
$782$	−33.5956	−1.20138
$783$	8.28444	0.296062
$784$	−29.1366	−1.04059
$785$	65.9837	2.35506
$786$	−41.3991	−1.47666
$787$	−17.3470	−0.618352	−0.309176	−	0.951005i	$-0.600053\pi$
$787$	−17.3470	−0.618352	−0.309176	+	0.951005i	$0.600053\pi$
$788$	30.1151	1.07281
$789$	39.4022	1.40275
$790$	−81.1445	−2.88699
$791$	0.886669	0.0315263
$792$	0	0
$793$	8.19419	0.290984
$794$	39.6552	1.40731
$795$	44.1408	1.56551
$796$	48.9673	1.73560
$797$	26.5509	0.940480	0.470240	−	0.882538i	$-0.344167\pi$
$797$	26.5509	0.940480	0.470240	+	0.882538i	$0.344167\pi$
$798$	−59.8518	−2.11873
$799$	−21.2065	−0.750231
$800$	28.5328	1.00879
$801$	−3.23026	−0.114136
$802$	31.3677	1.10763
$803$	0	0
$804$	25.9404	0.914848
$805$	58.6267	2.06632
$806$	29.8957	1.05303
$807$	3.97715	0.140002
$808$	−6.36710	−0.223994
$809$	−13.4805	−0.473951	−0.236975	−	0.971516i	$-0.576156\pi$
$809$	−13.4805	−0.473951	−0.236975	+	0.971516i	$0.576156\pi$
$810$	−21.1840	−0.744330
$811$	−10.5470	−0.370355	−0.185178	−	0.982705i	$-0.559286\pi$
$811$	−10.5470	−0.370355	−0.185178	+	0.982705i	$0.559286\pi$
$812$	−13.5840	−0.476704
$813$	−30.2785	−1.06191
$814$	0	0
$815$	−9.31703	−0.326361
$816$	13.8088	0.483404
$817$	62.8412	2.19854
$818$	−6.07587	−0.212438
$819$	8.27029	0.288987
$820$	21.6818	0.757161
$821$	37.8334	1.32039	0.660197	−	0.751092i	$-0.270473\pi$
$821$	37.8334	1.32039	0.660197	+	0.751092i	$0.270473\pi$
$822$	0.874793	0.0305119
$823$	25.2067	0.878650	0.439325	−	0.898328i	$-0.355218\pi$
$823$	25.2067	0.878650	0.439325	+	0.898328i	$0.355218\pi$
$824$	−8.31020	−0.289500
$825$	0	0
$826$	−16.6807	−0.580397
$827$	4.69694	0.163329	0.0816644	−	0.996660i	$-0.473976\pi$
$827$	4.69694	0.163329	0.0816644	+	0.996660i	$0.473976\pi$
$828$	−14.9591	−0.519864
$829$	−19.2590	−0.668891	−0.334446	−	0.942415i	$-0.608549\pi$
$829$	−19.2590	−0.668891	−0.334446	+	0.942415i	$0.608549\pi$
$830$	78.1149	2.71141
$831$	−15.1982	−0.527219
$832$	−16.5742	−0.574608
$833$	28.4736	0.986552
$834$	17.7776	0.615589
$835$	4.83738	0.167405
$836$	0	0
$837$	−50.0068	−1.72849
$838$	27.5562	0.951913
$839$	20.4215	0.705029	0.352515	−	0.935806i	$-0.385327\pi$
$839$	20.4215	0.705029	0.352515	+	0.935806i	$0.385327\pi$
$840$	9.79272	0.337881
$841$	−26.8206	−0.924847
$842$	52.2561	1.80087
$843$	−8.97050	−0.308960
$844$	−49.4750	−1.70300
$845$	30.3238	1.04317
$846$	−17.6190	−0.605753
$847$	0	0
$848$	−37.9242	−1.30232
$849$	−13.8916	−0.476759
$850$	−23.4594	−0.804650
$851$	−23.7124	−0.812850
$852$	−10.9593	−0.375459
$853$	11.9691	0.409815	0.204907	−	0.978781i	$-0.434311\pi$
$853$	11.9691	0.409815	0.204907	+	0.978781i	$0.434311\pi$
$854$	41.9354	1.43500
$855$	20.7199	0.708605
$856$	−2.61714	−0.0894521
$857$	−27.8709	−0.952053	−0.476026	−	0.879431i	$-0.657923\pi$
$857$	−27.8709	−0.952053	−0.476026	+	0.879431i	$0.657923\pi$
$858$	0	0
$859$	49.5376	1.69020	0.845101	−	0.534606i	$-0.179540\pi$
$859$	49.5376	1.69020	0.845101	+	0.534606i	$0.179540\pi$
$860$	−76.6759	−2.61463
$861$	−16.7795	−0.571843
$862$	−36.6982	−1.24995
$863$	5.17762	0.176248	0.0881242	−	0.996109i	$-0.471913\pi$
$863$	5.17762	0.176248	0.0881242	+	0.996109i	$0.471913\pi$
$864$	45.4815	1.54731
$865$	56.3728	1.91673
$866$	−8.45028	−0.287152
$867$	8.77068	0.297868
$868$	81.9960	2.78313
$869$	0	0
$870$	−11.7167	−0.397234
$871$	−13.8575	−0.469544
$872$	10.0776	0.341270
$873$	4.64916	0.157350
$874$	−57.8321	−1.95620
$875$	−17.2048	−0.581628
$876$	27.4795	0.928447
$877$	55.4553	1.87259	0.936296	−	0.351212i	$-0.114231\pi$
$877$	55.4553	1.87259	0.936296	+	0.351212i	$0.114231\pi$
$878$	40.3147	1.36055
$879$	13.2162	0.445770
$880$	0	0
$881$	−22.1859	−0.747461	−0.373730	−	0.927537i	$-0.621921\pi$
$881$	−22.1859	−0.747461	−0.373730	+	0.927537i	$0.621921\pi$
$882$	23.6567	0.796563
$883$	3.35918	0.113045	0.0565226	−	0.998401i	$-0.481999\pi$
$883$	3.35918	0.113045	0.0565226	+	0.998401i	$0.481999\pi$
$884$	11.9811	0.402968
$885$	−7.71090	−0.259199
$886$	44.4041	1.49178
$887$	8.11226	0.272383	0.136191	−	0.990683i	$-0.456514\pi$
$887$	8.11226	0.272383	0.136191	+	0.990683i	$0.456514\pi$
$888$	−3.96080	−0.132916
$889$	−23.6784	−0.794147
$890$	15.2376	0.510765
$891$	0	0
$892$	−1.56341	−0.0523470
$893$	−36.5053	−1.22160
$894$	−50.0749	−1.67476
$895$	−41.6288	−1.39150
$896$	−20.2462	−0.676378
$897$	−10.6707	−0.356284
$898$	−10.4668	−0.349282
$899$	−13.1555	−0.438761
$900$	−10.4457	−0.348191
$901$	37.0612	1.23469
$902$	0	0
$903$	59.3393	1.97469
$904$	0.143107	0.00475967
$905$	64.3781	2.14000
$906$	−14.7959	−0.491561
$907$	−14.0669	−0.467084	−0.233542	−	0.972347i	$-0.575032\pi$
$907$	−14.0669	−0.467084	−0.233542	+	0.972347i	$0.575032\pi$
$908$	1.42240	0.0472041
$909$	−12.7211	−0.421932
$910$	−39.0121	−1.29324
$911$	−16.3728	−0.542456	−0.271228	−	0.962515i	$-0.587430\pi$
$911$	−16.3728	−0.542456	−0.271228	+	0.962515i	$0.587430\pi$
$912$	23.7707	0.787128
$913$	0	0
$914$	46.5659	1.54026
$915$	19.3852	0.640855
$916$	−21.0533	−0.695621
$917$	60.6573	2.00308
$918$	−37.3945	−1.23420
$919$	−31.5515	−1.04079	−0.520394	−	0.853926i	$-0.674215\pi$
$919$	−31.5515	−1.04079	−0.520394	+	0.853926i	$0.674215\pi$
$920$	9.46227	0.311962
$921$	−37.5265	−1.23654
$922$	32.0516	1.05556
$923$	5.85450	0.192703
$924$	0	0
$925$	−16.5581	−0.544426
$926$	−57.6147	−1.89334
$927$	−16.6033	−0.545323
$928$	11.9650	0.392771
$929$	−44.0641	−1.44570	−0.722848	−	0.691007i	$-0.757167\pi$
$929$	−44.0641	−1.44570	−0.722848	+	0.691007i	$0.757167\pi$
$930$	70.7249	2.31916
$931$	49.0150	1.60640
$932$	35.1525	1.15146
$933$	28.8233	0.943633
$934$	26.6434	0.871800
$935$	0	0
$936$	1.33481	0.0436297
$937$	0.807693	0.0263862	0.0131931	−	0.999913i	$-0.495800\pi$
$937$	0.807693	0.0263862	0.0131931	+	0.999913i	$0.495800\pi$
$938$	−70.9185	−2.31557
$939$	−3.70186	−0.120806
$940$	44.5420	1.45280
$941$	54.9603	1.79166	0.895828	−	0.444402i	$-0.146584\pi$
$941$	54.9603	1.79166	0.895828	+	0.444402i	$0.146584\pi$
$942$	−61.4622	−2.00255
$943$	−16.2133	−0.527977
$944$	6.62492	0.215623
$945$	65.2560	2.12278
$946$	0	0
$947$	32.8742	1.06827	0.534135	−	0.845399i	$-0.320637\pi$
$947$	32.8742	1.06827	0.534135	+	0.845399i	$0.320637\pi$
$948$	40.5080	1.31564
$949$	−14.6797	−0.476523
$950$	−40.3835	−1.31021
$951$	−32.5855	−1.05666
$952$	8.22210	0.266480
$953$	−35.0917	−1.13673	−0.568367	−	0.822775i	$-0.692424\pi$
$953$	−35.0917	−1.13673	−0.568367	+	0.822775i	$0.692424\pi$
$954$	30.7916	0.996914
$955$	−2.30264	−0.0745115
$956$	56.5377	1.82856
$957$	0	0
$958$	10.1748	0.328732
$959$	−1.28174	−0.0413894
$960$	−39.2101	−1.26550
$961$	48.4097	1.56160
$962$	15.7790	0.508735
$963$	−5.22889	−0.168499
$964$	21.3403	0.687326
$965$	−21.7301	−0.699515
$966$	−54.6094	−1.75703
$967$	−17.8513	−0.574060	−0.287030	−	0.957922i	$-0.592668\pi$
$967$	−17.8513	−0.574060	−0.287030	+	0.957922i	$0.592668\pi$
$968$	0	0
$969$	−23.2299	−0.746250
$970$	−21.9307	−0.704154
$971$	−25.0431	−0.803672	−0.401836	−	0.915712i	$-0.631628\pi$
$971$	−25.0431	−0.803672	−0.401836	+	0.915712i	$0.631628\pi$
$972$	−28.3090	−0.908010
$973$	−26.0476	−0.835046
$974$	−14.1512	−0.453433
$975$	−7.45122	−0.238630
$976$	−16.6550	−0.533115
$977$	−38.2959	−1.22520	−0.612598	−	0.790395i	$-0.709876\pi$
$977$	−38.2959	−1.22520	−0.612598	+	0.790395i	$0.709876\pi$
$978$	8.67858	0.277510
$979$	0	0
$980$	−59.8058	−1.91043
$981$	20.1344	0.642842
$982$	−58.5008	−1.86683
$983$	−53.8415	−1.71728	−0.858638	−	0.512582i	$-0.828689\pi$
$983$	−53.8415	−1.71728	−0.858638	+	0.512582i	$0.828689\pi$
$984$	−2.70818	−0.0863338
$985$	−38.0589	−1.21266
$986$	−9.83753	−0.313291
$987$	−34.4710	−1.09722
$988$	20.6245	0.656153
$989$	57.3370	1.82321
$990$	0	0
$991$	−22.2411	−0.706511	−0.353255	−	0.935527i	$-0.614925\pi$
$991$	−22.2411	−0.706511	−0.353255	+	0.935527i	$0.614925\pi$
$992$	−72.2236	−2.29310
$993$	8.83854	0.280483
$994$	29.9615	0.950323
$995$	−61.8840	−1.96185
$996$	−38.9956	−1.23562
$997$	−34.6543	−1.09751	−0.548756	−	0.835982i	$-0.684899\pi$
$997$	−34.6543	−1.09751	−0.548756	+	0.835982i	$0.684899\pi$
$998$	4.46762	0.141420
$999$	−26.3937	−0.835059

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1331.2.a.c.1.2	✓	10
11.10	odd	2	inner	1331.2.a.c.1.9	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1331.2.a.c.1.2	✓	10	1.1	even	1	trivial
1331.2.a.c.1.9	yes	10	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 1331 = 11^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1331.a (trivial)

\(f(q)\)	\(=\)	\(q-2.07599 q^{2} -1.30972 q^{3} +2.30972 q^{4} -2.91899 q^{5} +2.71896 q^{6} -3.98379 q^{7} -0.642978 q^{8} -1.28463 q^{9} +O(q^{10})\)	\(q-2.07599 q^{2} -1.30972 q^{3} +2.30972 q^{4} -2.91899 q^{5} +2.71896 q^{6} -3.98379 q^{7} -0.642978 q^{8} -1.28463 q^{9} +6.05978 q^{10} -3.02509 q^{12} +1.61602 q^{13} +8.27029 q^{14} +3.82306 q^{15} -3.28463 q^{16} +3.20989 q^{17} +2.66687 q^{18} +5.52557 q^{19} -6.74204 q^{20} +5.21765 q^{21} +5.04159 q^{23} +0.842122 q^{24} +3.52048 q^{25} -3.35484 q^{26} +5.61167 q^{27} -9.20144 q^{28} +1.47629 q^{29} -7.93662 q^{30} -8.91121 q^{31} +8.10480 q^{32} -6.66369 q^{34} +11.6286 q^{35} -2.96714 q^{36} -4.70335 q^{37} -11.4710 q^{38} -2.11654 q^{39} +1.87684 q^{40} -3.21591 q^{41} -10.8318 q^{42} +11.3728 q^{43} +3.74982 q^{45} -10.4663 q^{46} -6.60660 q^{47} +4.30195 q^{48} +8.87058 q^{49} -7.30847 q^{50} -4.20406 q^{51} +3.73256 q^{52} +11.5459 q^{53} -11.6498 q^{54} +2.56149 q^{56} -7.23696 q^{57} -3.06475 q^{58} -2.01695 q^{59} +8.83020 q^{60} +5.07060 q^{61} +18.4996 q^{62} +5.11769 q^{63} -10.2562 q^{64} -4.71714 q^{65} -8.57509 q^{67} +7.41396 q^{68} -6.60308 q^{69} -24.1409 q^{70} +3.62279 q^{71} +0.825988 q^{72} -9.08386 q^{73} +9.76410 q^{74} -4.61085 q^{75} +12.7625 q^{76} +4.39390 q^{78} -13.3907 q^{79} +9.58779 q^{80} -3.49584 q^{81} +6.67618 q^{82} +12.8907 q^{83} +12.0513 q^{84} -9.36963 q^{85} -23.6098 q^{86} -1.93353 q^{87} +2.51455 q^{89} -7.78457 q^{90} -6.43788 q^{91} +11.6447 q^{92} +11.6712 q^{93} +13.7152 q^{94} -16.1291 q^{95} -10.6150 q^{96} -3.61907 q^{97} -18.4152 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} + 12 q^{4} - 12 q^{5} - 12 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 + 12 * q^4 - 12 * q^5 - 12 * q^9	\( 10 q - 2 q^{3} + 12 q^{4} - 12 q^{5} - 12 q^{9} - 20 q^{12} + 2 q^{14} - 2 q^{15} - 32 q^{16} - 10 q^{20} - 2 q^{23} - 18 q^{25} - 12 q^{26} + 4 q^{27} - 42 q^{31} - 28 q^{34} - 10 q^{36} - 30 q^{37} - 28 q^{38} + 4 q^{42} + 10 q^{45} - 24 q^{47} + 2 q^{48} - 8 q^{49} + 24 q^{53} - 6 q^{56} - 42 q^{58} - 8 q^{59} + 24 q^{60} - 18 q^{64} - 24 q^{67} + 18 q^{69} - 64 q^{70} + 32 q^{71} + 8 q^{75} - 46 q^{78} + 34 q^{80} - 22 q^{81} - 50 q^{82} - 22 q^{86} - 24 q^{89} - 42 q^{91} - 20 q^{92} + 26 q^{93} - 42 q^{97}+O(q^{100}) \) 10 * q - 2 * q^3 + 12 * q^4 - 12 * q^5 - 12 * q^9 - 20 * q^12 + 2 * q^14 - 2 * q^15 - 32 * q^16 - 10 * q^20 - 2 * q^23 - 18 * q^25 - 12 * q^26 + 4 * q^27 - 42 * q^31 - 28 * q^34 - 10 * q^36 - 30 * q^37 - 28 * q^38 + 4 * q^42 + 10 * q^45 - 24 * q^47 + 2 * q^48 - 8 * q^49 + 24 * q^53 - 6 * q^56 - 42 * q^58 - 8 * q^59 + 24 * q^60 - 18 * q^64 - 24 * q^67 + 18 * q^69 - 64 * q^70 + 32 * q^71 + 8 * q^75 - 46 * q^78 + 34 * q^80 - 22 * q^81 - 50 * q^82 - 22 * q^86 - 24 * q^89 - 42 * q^91 - 20 * q^92 + 26 * q^93 - 42 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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