LMFDB - Embedded newform 1331.2.a.e.1.5

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:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	\(q-1.62967 q^{2} -3.23376 q^{3} +0.655813 q^{4} -0.934058 q^{5} +5.26996 q^{6} -3.14288 q^{7} +2.19058 q^{8} +7.45722 q^{9} +1.52220 q^{10} -2.12074 q^{12} +3.95136 q^{13} +5.12185 q^{14} +3.02052 q^{15} -4.88154 q^{16} -5.78643 q^{17} -12.1528 q^{18} -1.48329 q^{19} -0.612568 q^{20} +10.1633 q^{21} -1.92737 q^{23} -7.08380 q^{24} -4.12754 q^{25} -6.43940 q^{26} -14.4136 q^{27} -2.06115 q^{28} -4.13485 q^{29} -4.92245 q^{30} -3.20660 q^{31} +3.57412 q^{32} +9.42995 q^{34} +2.93564 q^{35} +4.89055 q^{36} -5.71858 q^{37} +2.41727 q^{38} -12.7778 q^{39} -2.04613 q^{40} +11.7813 q^{41} -16.5629 q^{42} -2.34870 q^{43} -6.96548 q^{45} +3.14097 q^{46} -3.54657 q^{47} +15.7857 q^{48} +2.87772 q^{49} +6.72651 q^{50} +18.7119 q^{51} +2.59135 q^{52} -9.44683 q^{53} +23.4894 q^{54} -6.88473 q^{56} +4.79662 q^{57} +6.73842 q^{58} -14.0004 q^{59} +1.98090 q^{60} +8.02865 q^{61} +5.22569 q^{62} -23.4372 q^{63} +3.93844 q^{64} -3.69080 q^{65} +1.19184 q^{67} -3.79482 q^{68} +6.23265 q^{69} -4.78411 q^{70} +5.59466 q^{71} +16.3356 q^{72} -0.0557016 q^{73} +9.31938 q^{74} +13.3475 q^{75} -0.972763 q^{76} +20.8235 q^{78} -5.92689 q^{79} +4.55964 q^{80} +24.2385 q^{81} -19.1996 q^{82} -0.787284 q^{83} +6.66526 q^{84} +5.40486 q^{85} +3.82760 q^{86} +13.3711 q^{87} -14.8017 q^{89} +11.3514 q^{90} -12.4187 q^{91} -1.26399 q^{92} +10.3694 q^{93} +5.77973 q^{94} +1.38548 q^{95} -11.5579 q^{96} +10.4524 q^{97} -4.68973 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} - 3 q^{3} + 25 q^{4} - 3 q^{5} + 15 q^{6} + 19 q^{7} + 9 q^{8} + 22 q^{9} + 25 q^{10} - 12 q^{12} + 46 q^{13} - 12 q^{14} - 15 q^{15} + 13 q^{16} + 14 q^{17} + 6 q^{18} + 45 q^{19} - 24 q^{20}+ \cdots - 39 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 - 3 * q^3 + 25 * q^4 - 3 * q^5 + 15 * q^6 + 19 * q^7 + 9 * q^8 + 22 * q^9 + 25 * q^10 - 12 * q^12 + 46 * q^13 - 12 * q^14 - 15 * q^15 + 13 * q^16 + 14 * q^17 + 6 * q^18 + 45 * q^19 - 24 * q^20 + 49 * q^21 + 2 * q^23 + 36 * q^24 + 16 * q^25 - 7 * q^26 - 21 * q^27 + 40 * q^28 + 40 * q^29 - 15 * q^30 - 4 * q^31 + 3 * q^32 - 24 * q^34 - q^35 + 38 * q^36 - 21 * q^37 + 30 * q^38 + 18 * q^39 + 54 * q^40 + 49 * q^41 + 13 * q^42 + 14 * q^43 - 42 * q^45 + 40 * q^46 - 2 * q^47 - 5 * q^48 + 18 * q^49 - 12 * q^50 + 7 * q^51 + 57 * q^52 - 25 * q^53 + 33 * q^54 - 11 * q^56 + 4 * q^57 + 23 * q^58 - 22 * q^59 - 12 * q^60 + 128 * q^61 - 8 * q^62 + 12 * q^63 + 31 * q^64 + 16 * q^67 - 6 * q^68 - 57 * q^69 + 38 * q^70 - 34 * q^71 - 15 * q^72 + 71 * q^73 - 21 * q^74 - 10 * q^75 + 95 * q^76 + 75 * q^78 + 53 * q^79 - 29 * q^80 + 21 * q^81 + 33 * q^82 - 6 * q^83 + 18 * q^84 + 101 * q^85 + 38 * q^86 - 16 * q^87 + 5 * q^89 - 20 * q^90 + 16 * q^91 + 53 * q^92 + 7 * q^93 + 57 * q^94 - 16 * q^95 + 32 * q^96 + 14 * q^97 - 39 * q^98

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 $

$2$	−1.62967	−1.15235	−0.576174	−	0.817327i	$-0.695455\pi$
$2$	−1.62967	−1.15235	−0.576174	+	0.817327i	$0.695455\pi$
$3$	−3.23376	−1.86701	−0.933507	−	0.358559i	$-0.883268\pi$
$3$	−3.23376	−1.86701	−0.933507	+	0.358559i	$0.883268\pi$
$4$	0.655813	0.327907
$5$	−0.934058	−0.417724	−0.208862	−	0.977945i	$-0.566976\pi$
$5$	−0.934058	−0.417724	−0.208862	+	0.977945i	$0.566976\pi$
$6$	5.26996	2.15145
$7$	−3.14288	−1.18790	−0.593949	−	0.804502i	$-0.702432\pi$
$7$	−3.14288	−1.18790	−0.593949	+	0.804502i	$0.702432\pi$
$8$	2.19058	0.774486
$9$	7.45722	2.48574
$10$	1.52220	0.481363
$11$	0	0
$11$	0	0
$12$	−2.12074	−0.612206
$13$	3.95136	1.09591	0.547955	−	0.836508i	$-0.315407\pi$
$13$	3.95136	1.09591	0.547955	+	0.836508i	$0.315407\pi$
$14$	5.12185	1.36887
$15$	3.02052	0.779896
$16$	−4.88154	−1.22038
$17$	−5.78643	−1.40342	−0.701708	−	0.712465i	$-0.747579\pi$
$17$	−5.78643	−1.40342	−0.701708	+	0.712465i	$0.747579\pi$
$18$	−12.1528	−2.86444
$19$	−1.48329	−0.340291	−0.170145	−	0.985419i	$-0.554424\pi$
$19$	−1.48329	−0.340291	−0.170145	+	0.985419i	$0.554424\pi$
$20$	−0.612568	−0.136974
$21$	10.1633	2.21782
$22$	0	0
$23$	−1.92737	−0.401884	−0.200942	−	0.979603i	$-0.564400\pi$
$23$	−1.92737	−0.401884	−0.200942	+	0.979603i	$0.564400\pi$
$24$	−7.08380	−1.44598
$25$	−4.12754	−0.825507
$26$	−6.43940	−1.26287
$27$	−14.4136	−2.77390
$28$	−2.06115	−0.389520
$29$	−4.13485	−0.767822	−0.383911	−	0.923370i	$-0.625423\pi$
$29$	−4.13485	−0.767822	−0.383911	+	0.923370i	$0.625423\pi$
$30$	−4.92245	−0.898711
$31$	−3.20660	−0.575922	−0.287961	−	0.957642i	$-0.592977\pi$
$31$	−3.20660	−0.575922	−0.287961	+	0.957642i	$0.592977\pi$
$32$	3.57412	0.631822
$33$	0	0
$34$	9.42995	1.61722
$35$	2.93564	0.496213
$36$	4.89055	0.815091
$37$	−5.71858	−0.940129	−0.470064	−	0.882632i	$-0.655769\pi$
$37$	−5.71858	−0.940129	−0.470064	+	0.882632i	$0.655769\pi$
$38$	2.41727	0.392133
$39$	−12.7778	−2.04608
$40$	−2.04613	−0.323521
$41$	11.7813	1.83993	0.919964	−	0.392002i	$-0.128217\pi$
$41$	11.7813	1.83993	0.919964	+	0.392002i	$0.128217\pi$
$42$	−16.5629	−2.55570
$43$	−2.34870	−0.358173	−0.179087	−	0.983833i	$-0.557314\pi$
$43$	−2.34870	−0.358173	−0.179087	+	0.983833i	$0.557314\pi$
$44$	0	0
$45$	−6.96548	−1.03835
$46$	3.14097	0.463110
$47$	−3.54657	−0.517320	−0.258660	−	0.965968i	$-0.583281\pi$
$47$	−3.54657	−0.517320	−0.258660	+	0.965968i	$0.583281\pi$
$48$	15.7857	2.27847
$49$	2.87772	0.411103
$50$	6.72651	0.951272
$51$	18.7119	2.62020
$52$	2.59135	0.359356
$53$	−9.44683	−1.29762	−0.648811	−	0.760950i	$-0.724733\pi$
$53$	−9.44683	−1.29762	−0.648811	+	0.760950i	$0.724733\pi$
$54$	23.4894	3.19650
$55$	0	0
$56$	−6.88473	−0.920010
$57$	4.79662	0.635327
$58$	6.73842	0.884798
$59$	−14.0004	−1.82269	−0.911347	−	0.411638i	$-0.864957\pi$
$59$	−14.0004	−1.82269	−0.911347	+	0.411638i	$0.864957\pi$
$60$	1.98090	0.255733
$61$	8.02865	1.02796	0.513982	−	0.857801i	$-0.328170\pi$
$61$	8.02865	1.02796	0.513982	+	0.857801i	$0.328170\pi$
$62$	5.22569	0.663663
$63$	−23.4372	−2.95281
$64$	3.93844	0.492305
$65$	−3.69080	−0.457787
$66$	0	0
$67$	1.19184	0.145606	0.0728030	−	0.997346i	$-0.476806\pi$
$67$	1.19184	0.145606	0.0728030	+	0.997346i	$0.476806\pi$
$68$	−3.79482	−0.460189
$69$	6.23265	0.750323
$70$	−4.78411	−0.571810
$71$	5.59466	0.663964	0.331982	−	0.943286i	$-0.392283\pi$
$71$	5.59466	0.663964	0.331982	+	0.943286i	$0.392283\pi$
$72$	16.3356	1.92517
$73$	−0.0557016	−0.00651938	−0.00325969	−	0.999995i	$-0.501038\pi$
$73$	−0.0557016	−0.00651938	−0.00325969	+	0.999995i	$0.501038\pi$
$74$	9.31938	1.08336
$75$	13.3475	1.54123
$76$	−0.972763	−0.111584
$77$	0	0
$78$	20.8235	2.35779
$79$	−5.92689	−0.666827	−0.333414	−	0.942781i	$-0.608201\pi$
$79$	−5.92689	−0.666827	−0.333414	+	0.942781i	$0.608201\pi$
$80$	4.55964	0.509783
$81$	24.2385	2.69317
$82$	−19.1996	−2.12024
$83$	−0.787284	−0.0864157	−0.0432078	−	0.999066i	$-0.513758\pi$
$83$	−0.787284	−0.0864157	−0.0432078	+	0.999066i	$0.513758\pi$
$84$	6.66526	0.727239
$85$	5.40486	0.586240
$86$	3.82760	0.412740
$87$	13.3711	1.43353
$88$	0	0
$89$	−14.8017	−1.56898	−0.784488	−	0.620144i	$-0.787074\pi$
$89$	−14.8017	−1.56898	−0.784488	+	0.620144i	$0.787074\pi$
$90$	11.3514	1.19654
$91$	−12.4187	−1.30183
$92$	−1.26399	−0.131780
$93$	10.3694	1.07526
$94$	5.77973	0.596133
$95$	1.38548	0.142147
$96$	−11.5579	−1.17962
$97$	10.4524	1.06128	0.530638	−	0.847599i	$-0.321952\pi$
$97$	10.4524	1.06128	0.530638	+	0.847599i	$0.321952\pi$
$98$	−4.68973	−0.473734
$99$	0	0
$100$	−2.70689	−0.270689
$101$	1.16663	0.116084	0.0580421	−	0.998314i	$-0.481514\pi$
$101$	1.16663	0.116084	0.0580421	+	0.998314i	$0.481514\pi$
$102$	−30.4942	−3.01938
$103$	−3.85120	−0.379470	−0.189735	−	0.981835i	$-0.560763\pi$
$103$	−3.85120	−0.379470	−0.189735	+	0.981835i	$0.560763\pi$
$104$	8.65575	0.848766
$105$	−9.49315	−0.926437
$106$	15.3952	1.49531
$107$	−12.0786	−1.16768	−0.583840	−	0.811869i	$-0.698450\pi$
$107$	−12.0786	−1.16768	−0.583840	+	0.811869i	$0.698450\pi$
$108$	−9.45263	−0.909580
$109$	12.8601	1.23178	0.615888	−	0.787834i	$-0.288797\pi$
$109$	12.8601	1.23178	0.615888	+	0.787834i	$0.288797\pi$
$110$	0	0
$111$	18.4925	1.75523
$112$	15.3421	1.44969
$113$	1.06284	0.0999840	0.0499920	−	0.998750i	$-0.484080\pi$
$113$	1.06284	0.0999840	0.0499920	+	0.998750i	$0.484080\pi$
$114$	−7.81689	−0.732119
$115$	1.80027	0.167876
$116$	−2.71169	−0.251774
$117$	29.4661	2.72415
$118$	22.8160	2.10038
$119$	18.1861	1.66712
$120$	6.61669	0.604018
$121$	0	0
$122$	−13.0840	−1.18457
$123$	−38.0979	−3.43517
$124$	−2.10293	−0.188849
$125$	8.52565	0.762557
$126$	38.1948	3.40266
$127$	−14.9779	−1.32908	−0.664538	−	0.747254i	$-0.731372\pi$
$127$	−14.9779	−1.32908	−0.664538	+	0.747254i	$0.731372\pi$
$128$	−13.5666	−1.19913
$129$	7.59513	0.668714
$130$	6.01477	0.527530
$131$	7.33497	0.640859	0.320430	−	0.947272i	$-0.396173\pi$
$131$	7.33497	0.640859	0.320430	+	0.947272i	$0.396173\pi$
$132$	0	0
$133$	4.66182	0.404231
$134$	−1.94230	−0.167789
$135$	13.4631	1.15872
$136$	−12.6756	−1.08693
$137$	−5.22649	−0.446530	−0.223265	−	0.974758i	$-0.571671\pi$
$137$	−5.22649	−0.446530	−0.223265	+	0.974758i	$0.571671\pi$
$138$	−10.1571	−0.864634
$139$	15.1116	1.28175	0.640877	−	0.767644i	$-0.278571\pi$
$139$	15.1116	1.28175	0.640877	+	0.767644i	$0.278571\pi$
$140$	1.92523	0.162712
$141$	11.4688	0.965844
$142$	−9.11744	−0.765118
$143$	0	0
$144$	−36.4027	−3.03356
$145$	3.86219	0.320737
$146$	0.0907750	0.00751259
$147$	−9.30587	−0.767535
$148$	−3.75032	−0.308274
$149$	−9.59596	−0.786132	−0.393066	−	0.919510i	$-0.628586\pi$
$149$	−9.59596	−0.786132	−0.393066	+	0.919510i	$0.628586\pi$
$150$	−21.7519	−1.77604
$151$	12.6043	1.02572	0.512862	−	0.858471i	$-0.328585\pi$
$151$	12.6043	1.02572	0.512862	+	0.858471i	$0.328585\pi$
$152$	−3.24927	−0.263550
$153$	−43.1507	−3.48853
$154$	0	0
$155$	2.99515	0.240576
$156$	−8.37982	−0.670923
$157$	18.1108	1.44540	0.722700	−	0.691162i	$-0.242901\pi$
$157$	18.1108	1.44540	0.722700	+	0.691162i	$0.242901\pi$
$158$	9.65886	0.768417
$159$	30.5488	2.42268
$160$	−3.33844	−0.263927
$161$	6.05750	0.477398
$162$	−39.5007	−3.10346
$163$	−10.0441	−0.786714	−0.393357	−	0.919386i	$-0.628686\pi$
$163$	−10.0441	−0.786714	−0.393357	+	0.919386i	$0.628686\pi$
$164$	7.72633	0.603325
$165$	0	0
$166$	1.28301	0.0995810
$167$	−14.4023	−1.11449	−0.557243	−	0.830350i	$-0.688141\pi$
$167$	−14.4023	−1.11449	−0.557243	+	0.830350i	$0.688141\pi$
$168$	22.2636	1.71767
$169$	2.61323	0.201017
$170$	−8.80813	−0.675552
$171$	−11.0612	−0.845874
$172$	−1.54031	−0.117447
$173$	14.2248	1.08149	0.540744	−	0.841187i	$-0.318143\pi$
$173$	14.2248	1.08149	0.540744	+	0.841187i	$0.318143\pi$
$174$	−21.7905	−1.65193
$175$	12.9724	0.980619
$176$	0	0
$177$	45.2739	3.40300
$178$	24.1218	1.80801
$179$	9.70539	0.725415	0.362707	−	0.931903i	$-0.381852\pi$
$179$	9.70539	0.725415	0.362707	+	0.931903i	$0.381852\pi$
$180$	−4.56805	−0.340483
$181$	6.61148	0.491427	0.245714	−	0.969342i	$-0.420978\pi$
$181$	6.61148	0.491427	0.245714	+	0.969342i	$0.420978\pi$
$182$	20.2383	1.50016
$183$	−25.9627	−1.91922
$184$	−4.22205	−0.311253
$185$	5.34149	0.392714
$186$	−16.8986	−1.23907
$187$	0	0
$188$	−2.32589	−0.169633
$189$	45.3003	3.29511
$190$	−2.25787	−0.163803
$191$	7.25228	0.524757	0.262378	−	0.964965i	$-0.415493\pi$
$191$	7.25228	0.524757	0.262378	+	0.964965i	$0.415493\pi$
$192$	−12.7360	−0.919141
$193$	24.7374	1.78064	0.890318	−	0.455339i	$-0.150482\pi$
$193$	24.7374	1.78064	0.890318	+	0.455339i	$0.150482\pi$
$194$	−17.0339	−1.22296
$195$	11.9352	0.854695
$196$	1.88725	0.134803
$197$	1.08099	0.0770177	0.0385088	−	0.999258i	$-0.487739\pi$
$197$	1.08099	0.0770177	0.0385088	+	0.999258i	$0.487739\pi$
$198$	0	0
$199$	0.658437	0.0466754	0.0233377	−	0.999728i	$-0.492571\pi$
$199$	0.658437	0.0466754	0.0233377	+	0.999728i	$0.492571\pi$
$200$	−9.04168	−0.639343
$201$	−3.85411	−0.271848
$202$	−1.90122	−0.133769
$203$	12.9953	0.912095
$204$	12.2715	0.859180
$205$	−11.0044	−0.768582
$206$	6.27618	0.437282
$207$	−14.3728	−0.998980
$208$	−19.2887	−1.33743
$209$	0	0
$210$	15.4707	1.06758
$211$	0.404094	0.0278190	0.0139095	−	0.999903i	$-0.495572\pi$
$211$	0.404094	0.0278190	0.0139095	+	0.999903i	$0.495572\pi$
$212$	−6.19535	−0.425499
$213$	−18.0918	−1.23963
$214$	19.6840	1.34557
$215$	2.19382	0.149617
$216$	−31.5741	−2.14834
$217$	10.0780	0.684137
$218$	−20.9577	−1.41943
$219$	0.180126	0.0121718
$220$	0	0
$221$	−22.8643	−1.53802
$222$	−30.1367	−2.02264
$223$	−15.1521	−1.01466	−0.507331	−	0.861751i	$-0.669368\pi$
$223$	−15.1521	−1.01466	−0.507331	+	0.861751i	$0.669368\pi$
$224$	−11.2331	−0.750540
$225$	−30.7799	−2.05200
$226$	−1.73208	−0.115216
$227$	−0.629848	−0.0418045	−0.0209022	−	0.999782i	$-0.506654\pi$
$227$	−0.629848	−0.0418045	−0.0209022	+	0.999782i	$0.506654\pi$
$228$	3.14569	0.208328
$229$	−27.6597	−1.82781	−0.913903	−	0.405934i	$-0.866946\pi$
$229$	−27.6597	−1.82781	−0.913903	+	0.405934i	$0.866946\pi$
$230$	−2.93385	−0.193452
$231$	0	0
$232$	−9.05770	−0.594667
$233$	−6.98602	−0.457669	−0.228835	−	0.973465i	$-0.573491\pi$
$233$	−6.98602	−0.457669	−0.228835	+	0.973465i	$0.573491\pi$
$234$	−48.0200	−3.13917
$235$	3.31270	0.216097
$236$	−9.18164	−0.597674
$237$	19.1662	1.24498
$238$	−29.6372	−1.92110
$239$	−16.7634	−1.08433	−0.542167	−	0.840270i	$-0.682396\pi$
$239$	−16.7634	−1.08433	−0.542167	+	0.840270i	$0.682396\pi$
$240$	−14.7448	−0.951772
$241$	−16.4072	−1.05688	−0.528439	−	0.848971i	$-0.677223\pi$
$241$	−16.4072	−1.05688	−0.528439	+	0.848971i	$0.677223\pi$
$242$	0	0
$243$	−35.1407	−2.25428
$244$	5.26529	0.337076
$245$	−2.68796	−0.171727
$246$	62.0869	3.95851
$247$	−5.86102	−0.372928
$248$	−7.02430	−0.446044
$249$	2.54589	0.161339
$250$	−13.8940	−0.878732
$251$	6.76681	0.427117	0.213558	−	0.976930i	$-0.431495\pi$
$251$	6.76681	0.427117	0.213558	+	0.976930i	$0.431495\pi$
$252$	−15.3704	−0.968245
$253$	0	0
$254$	24.4090	1.53156
$255$	−17.4780	−1.09452
$256$	14.2321	0.889509
$257$	−0.136839	−0.00853576	−0.00426788	−	0.999991i	$-0.501359\pi$
$257$	−0.136839	−0.00853576	−0.00426788	+	0.999991i	$0.501359\pi$
$258$	−12.3775	−0.770592
$259$	17.9728	1.11678
$260$	−2.42047	−0.150111
$261$	−30.8345	−1.90861
$262$	−11.9536	−0.738493
$263$	−16.6239	−1.02507	−0.512537	−	0.858665i	$-0.671294\pi$
$263$	−16.6239	−1.02507	−0.512537	+	0.858665i	$0.671294\pi$
$264$	0	0
$265$	8.82389	0.542047
$266$	−7.59721	−0.465815
$267$	47.8651	2.92930
$268$	0.781622	0.0477452
$269$	−1.36202	−0.0830441	−0.0415220	−	0.999138i	$-0.513221\pi$
$269$	−1.36202	−0.0830441	−0.0415220	+	0.999138i	$0.513221\pi$
$270$	−21.9404	−1.33525
$271$	20.6770	1.25604	0.628018	−	0.778199i	$-0.283866\pi$
$271$	20.6770	1.25604	0.628018	+	0.778199i	$0.283866\pi$
$272$	28.2467	1.71271
$273$	40.1590	2.43053
$274$	8.51744	0.514558
$275$	0	0
$276$	4.08746	0.246036
$277$	26.4680	1.59031	0.795154	−	0.606407i	$-0.207390\pi$
$277$	26.4680	1.59031	0.795154	+	0.606407i	$0.207390\pi$
$278$	−24.6269	−1.47703
$279$	−23.9123	−1.43159
$280$	6.43074	0.384310
$281$	−4.65579	−0.277741	−0.138871	−	0.990311i	$-0.544347\pi$
$281$	−4.65579	−0.277741	−0.138871	+	0.990311i	$0.544347\pi$
$282$	−18.6903	−1.11299
$283$	22.0637	1.31155	0.655776	−	0.754955i	$-0.272342\pi$
$283$	22.0637	1.31155	0.655776	+	0.754955i	$0.272342\pi$
$284$	3.66906	0.217718
$285$	−4.48032	−0.265391
$286$	0	0
$287$	−37.0272	−2.18565
$288$	26.6530	1.57054
$289$	16.4828	0.969575
$290$	−6.29408	−0.369601
$291$	−33.8004	−1.98142
$292$	−0.0365298	−0.00213775
$293$	0.308905	0.0180464	0.00902321	−	0.999959i	$-0.497128\pi$
$293$	0.308905	0.0180464	0.00902321	+	0.999959i	$0.497128\pi$
$294$	15.1655	0.884468
$295$	13.0772	0.761383
$296$	−12.5270	−0.728116
$297$	0	0
$298$	15.6382	0.905898
$299$	−7.61572	−0.440429
$300$	8.75345	0.505381
$301$	7.38169	0.425473
$302$	−20.5408	−1.18199
$303$	−3.77261	−0.216731
$304$	7.24075	0.415285
$305$	−7.49923	−0.429404
$306$	70.3212	4.02000
$307$	−2.54346	−0.145163	−0.0725814	−	0.997362i	$-0.523124\pi$
$307$	−2.54346	−0.145163	−0.0725814	+	0.997362i	$0.523124\pi$
$308$	0	0
$309$	12.4539	0.708476
$310$	−4.88110	−0.277228
$311$	−7.84600	−0.444906	−0.222453	−	0.974943i	$-0.571406\pi$
$311$	−7.84600	−0.444906	−0.222453	+	0.974943i	$0.571406\pi$
$312$	−27.9906	−1.58466
$313$	28.3651	1.60329	0.801644	−	0.597801i	$-0.203959\pi$
$313$	28.3651	1.60329	0.801644	+	0.597801i	$0.203959\pi$
$314$	−29.5146	−1.66560
$315$	21.8917	1.23346
$316$	−3.88693	−0.218657
$317$	−5.04616	−0.283420	−0.141710	−	0.989908i	$-0.545260\pi$
$317$	−5.04616	−0.283420	−0.141710	+	0.989908i	$0.545260\pi$
$318$	−49.7843	−2.79177
$319$	0	0
$320$	−3.67873	−0.205647
$321$	39.0592	2.18007
$322$	−9.87170	−0.550128
$323$	8.58297	0.477569
$324$	15.8959	0.883107
$325$	−16.3094	−0.904681
$326$	16.3685	0.906568
$327$	−41.5866	−2.29974
$328$	25.8078	1.42500
$329$	11.1465	0.614524
$330$	0	0
$331$	32.9653	1.81194	0.905969	−	0.423343i	$-0.139144\pi$
$331$	32.9653	1.81194	0.905969	+	0.423343i	$0.139144\pi$
$332$	−0.516312	−0.0283363
$333$	−42.6447	−2.33692
$334$	23.4710	1.28428
$335$	−1.11324	−0.0608230
$336$	−49.6127	−2.70660
$337$	−7.92651	−0.431784	−0.215892	−	0.976417i	$-0.569266\pi$
$337$	−7.92651	−0.431784	−0.215892	+	0.976417i	$0.569266\pi$
$338$	−4.25869	−0.231642
$339$	−3.43699	−0.186672
$340$	3.54458	0.192232
$341$	0	0
$342$	18.0261	0.974742
$343$	12.9558	0.699550
$344$	−5.14500	−0.277400
$345$	−5.82166	−0.313428
$346$	−23.1816	−1.24625
$347$	−12.3791	−0.664544	−0.332272	−	0.943184i	$-0.607815\pi$
$347$	−12.3791	−0.664544	−0.332272	+	0.943184i	$0.607815\pi$
$348$	8.76896	0.470065
$349$	12.1559	0.650693	0.325346	−	0.945595i	$-0.394519\pi$
$349$	12.1559	0.650693	0.325346	+	0.945595i	$0.394519\pi$
$350$	−21.1406	−1.13001
$351$	−56.9533	−3.03994
$352$	0	0
$353$	−13.9022	−0.739942	−0.369971	−	0.929043i	$-0.620632\pi$
$353$	−13.9022	−0.739942	−0.369971	+	0.929043i	$0.620632\pi$
$354$	−73.7814	−3.92144
$355$	−5.22574	−0.277354
$356$	−9.70714	−0.514478
$357$	−58.8095	−3.11253
$358$	−15.8165	−0.835931
$359$	−1.94064	−0.102423	−0.0512116	−	0.998688i	$-0.516308\pi$
$359$	−1.94064	−0.102423	−0.0512116	+	0.998688i	$0.516308\pi$
$360$	−15.2584	−0.804189
$361$	−16.7998	−0.884202
$362$	−10.7745	−0.566296
$363$	0	0
$364$	−8.14432	−0.426878
$365$	0.0520285	0.00272330
$366$	42.3106	2.21161
$367$	5.98445	0.312386	0.156193	−	0.987727i	$-0.450078\pi$
$367$	5.98445	0.312386	0.156193	+	0.987727i	$0.450078\pi$
$368$	9.40852	0.490453
$369$	87.8557	4.57358
$370$	−8.70484	−0.452543
$371$	29.6903	1.54144
$372$	6.80038	0.352583
$373$	33.5193	1.73556	0.867782	−	0.496944i	$-0.165545\pi$
$373$	33.5193	1.73556	0.867782	+	0.496944i	$0.165545\pi$
$374$	0	0
$375$	−27.5699	−1.42371
$376$	−7.76903	−0.400657
$377$	−16.3383	−0.841463
$378$	−73.8243	−3.79711
$379$	4.59681	0.236122	0.118061	−	0.993006i	$-0.462332\pi$
$379$	4.59681	0.236122	0.118061	+	0.993006i	$0.462332\pi$
$380$	0.908618	0.0466111
$381$	48.4351	2.48140
$382$	−11.8188	−0.604702
$383$	19.3485	0.988663	0.494331	−	0.869274i	$-0.335413\pi$
$383$	19.3485	0.988663	0.494331	+	0.869274i	$0.335413\pi$
$384$	43.8711	2.23879
$385$	0	0
$386$	−40.3137	−2.05191
$387$	−17.5148	−0.890326
$388$	6.85479	0.347999
$389$	18.9514	0.960873	0.480437	−	0.877029i	$-0.340478\pi$
$389$	18.9514	0.960873	0.480437	+	0.877029i	$0.340478\pi$
$390$	−19.4503	−0.984906
$391$	11.1526	0.564010
$392$	6.30387	0.318393
$393$	−23.7195	−1.19649
$394$	−1.76166	−0.0887512
$395$	5.53606	0.278550
$396$	0	0
$397$	−11.3455	−0.569412	−0.284706	−	0.958615i	$-0.591896\pi$
$397$	−11.3455	−0.569412	−0.284706	+	0.958615i	$0.591896\pi$
$398$	−1.07303	−0.0537863
$399$	−15.0752	−0.754705
$400$	20.1487	1.00744
$401$	−10.8271	−0.540679	−0.270339	−	0.962765i	$-0.587136\pi$
$401$	−10.8271	−0.540679	−0.270339	+	0.962765i	$0.587136\pi$
$402$	6.28092	0.313264
$403$	−12.6704	−0.631159
$404$	0.765092	0.0380648
$405$	−22.6402	−1.12500
$406$	−21.1781	−1.05105
$407$	0	0
$408$	40.9899	2.02930
$409$	32.7178	1.61779	0.808896	−	0.587952i	$-0.200066\pi$
$409$	32.7178	1.61779	0.808896	+	0.587952i	$0.200066\pi$
$410$	17.9335	0.885674
$411$	16.9012	0.833677
$412$	−2.52567	−0.124431
$413$	44.0016	2.16518
$414$	23.4229	1.15117
$415$	0.735369	0.0360979
$416$	14.1226	0.692419
$417$	−48.8675	−2.39305
$418$	0	0
$419$	16.5244	0.807271	0.403636	−	0.914920i	$-0.367746\pi$
$419$	16.5244	0.807271	0.403636	+	0.914920i	$0.367746\pi$
$420$	−6.22574	−0.303785
$421$	−11.1452	−0.543184	−0.271592	−	0.962412i	$-0.587550\pi$
$421$	−11.1452	−0.543184	−0.271592	+	0.962412i	$0.587550\pi$
$422$	−0.658539	−0.0320572
$423$	−26.4476	−1.28592
$424$	−20.6940	−1.00499
$425$	23.8837	1.15853
$426$	29.4836	1.42849
$427$	−25.2331	−1.22112
$428$	−7.92129	−0.382890
$429$	0	0
$430$	−3.57520	−0.172411
$431$	−26.2629	−1.26504	−0.632520	−	0.774544i	$-0.717979\pi$
$431$	−26.2629	−1.26504	−0.632520	+	0.774544i	$0.717979\pi$
$432$	70.3605	3.38522
$433$	9.99011	0.480094	0.240047	−	0.970761i	$-0.422837\pi$
$433$	9.99011	0.480094	0.240047	+	0.970761i	$0.422837\pi$
$434$	−16.4237	−0.788365
$435$	−12.4894	−0.598821
$436$	8.43384	0.403908
$437$	2.85885	0.136757
$438$	−0.293545	−0.0140261
$439$	22.2044	1.05976	0.529880	−	0.848072i	$-0.322237\pi$
$439$	22.2044	1.05976	0.529880	+	0.848072i	$0.322237\pi$
$440$	0	0
$441$	21.4598	1.02190
$442$	37.2611	1.77233
$443$	29.3758	1.39569	0.697844	−	0.716250i	$-0.254143\pi$
$443$	29.3758	1.39569	0.697844	+	0.716250i	$0.254143\pi$
$444$	12.1276	0.575553
$445$	13.8256	0.655398
$446$	24.6929	1.16924
$447$	31.0311	1.46772
$448$	−12.3781	−0.584809
$449$	−33.7859	−1.59445	−0.797227	−	0.603680i	$-0.793700\pi$
$449$	−33.7859	−1.59445	−0.797227	+	0.603680i	$0.793700\pi$
$450$	50.1610	2.36461
$451$	0	0
$452$	0.697028	0.0327854
$453$	−40.7594	−1.91504
$454$	1.02644	0.0481733
$455$	11.5998	0.543805
$456$	10.5074	0.492052
$457$	−17.5898	−0.822815	−0.411408	−	0.911451i	$-0.634963\pi$
$457$	−17.5898	−0.822815	−0.411408	+	0.911451i	$0.634963\pi$
$458$	45.0761	2.10627
$459$	83.4033	3.89293
$460$	1.18064	0.0550478
$461$	31.0570	1.44647	0.723234	−	0.690603i	$-0.242655\pi$
$461$	31.0570	1.44647	0.723234	+	0.690603i	$0.242655\pi$
$462$	0	0
$463$	37.3368	1.73519	0.867594	−	0.497273i	$-0.165665\pi$
$463$	37.3368	1.73519	0.867594	+	0.497273i	$0.165665\pi$
$464$	20.1844	0.937038
$465$	−9.68561	−0.449159
$466$	11.3849	0.527394
$467$	4.37372	0.202392	0.101196	−	0.994867i	$-0.467733\pi$
$467$	4.37372	0.202392	0.101196	+	0.994867i	$0.467733\pi$
$468$	19.3243	0.893266
$469$	−3.74580	−0.172965
$470$	−5.39860	−0.249019
$471$	−58.5660	−2.69858
$472$	−30.6689	−1.41165
$473$	0	0
$474$	−31.2345	−1.43465
$475$	6.12234	0.280912
$476$	11.9267	0.546658
$477$	−70.4471	−3.22555
$478$	27.3188	1.24953
$479$	10.4683	0.478309	0.239154	−	0.970982i	$-0.423130\pi$
$479$	10.4683	0.478309	0.239154	+	0.970982i	$0.423130\pi$
$480$	10.7957	0.492755
$481$	−22.5962	−1.03030
$482$	26.7382	1.21789
$483$	−19.5885	−0.891308
$484$	0	0
$485$	−9.76311	−0.443320
$486$	57.2677	2.59771
$487$	−5.80738	−0.263157	−0.131579	−	0.991306i	$-0.542005\pi$
$487$	−5.80738	−0.263157	−0.131579	+	0.991306i	$0.542005\pi$
$488$	17.5874	0.796143
$489$	32.4802	1.46881
$490$	4.38048	0.197890
$491$	−7.07576	−0.319325	−0.159662	−	0.987172i	$-0.551041\pi$
$491$	−7.07576	−0.319325	−0.159662	+	0.987172i	$0.551041\pi$
$492$	−24.9851	−1.12642
$493$	23.9260	1.07757
$494$	9.55151	0.429743
$495$	0	0
$496$	15.6531	0.702847
$497$	−17.5834	−0.788722
$498$	−4.14895	−0.185919
$499$	6.49078	0.290567	0.145284	−	0.989390i	$-0.453591\pi$
$499$	6.49078	0.290567	0.145284	+	0.989390i	$0.453591\pi$
$500$	5.59124	0.250048
$501$	46.5737	2.08076
$502$	−11.0276	−0.492188
$503$	5.90604	0.263338	0.131669	−	0.991294i	$-0.457966\pi$
$503$	5.90604	0.263338	0.131669	+	0.991294i	$0.457966\pi$
$504$	−51.3409	−2.28691
$505$	−1.08970	−0.0484911
$506$	0	0
$507$	−8.45056	−0.375302
$508$	−9.82273	−0.435813
$509$	−9.33224	−0.413644	−0.206822	−	0.978379i	$-0.566312\pi$
$509$	−9.33224	−0.413644	−0.206822	+	0.978379i	$0.566312\pi$
$510$	28.4834	1.26127
$511$	0.175064	0.00774436
$512$	3.93954	0.174105
$513$	21.3796	0.943932
$514$	0.223001	0.00983617
$515$	3.59725	0.158514
$516$	4.98099	0.219276
$517$	0	0
$518$	−29.2897	−1.28692
$519$	−45.9995	−2.01915
$520$	−8.08498	−0.354550
$521$	31.1924	1.36656	0.683282	−	0.730155i	$-0.260552\pi$
$521$	31.1924	1.36656	0.683282	+	0.730155i	$0.260552\pi$
$522$	50.2499	2.19938
$523$	−17.4309	−0.762198	−0.381099	−	0.924534i	$-0.624454\pi$
$523$	−17.4309	−0.762198	−0.381099	+	0.924534i	$0.624454\pi$
$524$	4.81037	0.210142
$525$	−41.9495	−1.83083
$526$	27.0914	1.18124
$527$	18.5548	0.808259
$528$	0	0
$529$	−19.2853	−0.838489
$530$	−14.3800	−0.624627
$531$	−104.404	−4.53075
$532$	3.05728	0.132550
$533$	46.5521	2.01639
$534$	−78.0042	−3.37557
$535$	11.2821	0.487767
$536$	2.61081	0.112770
$537$	−31.3849	−1.35436
$538$	2.21965	0.0956957
$539$	0	0
$540$	8.82931	0.379953
$541$	23.5369	1.01193	0.505965	−	0.862554i	$-0.331137\pi$
$541$	23.5369	1.01193	0.505965	+	0.862554i	$0.331137\pi$
$542$	−33.6966	−1.44739
$543$	−21.3800	−0.917502
$544$	−20.6814	−0.886708
$545$	−12.0121	−0.514542
$546$	−65.4458	−2.80082
$547$	4.70986	0.201379	0.100690	−	0.994918i	$-0.467895\pi$
$547$	4.70986	0.201379	0.100690	+	0.994918i	$0.467895\pi$
$548$	−3.42760	−0.146420
$549$	59.8714	2.55525
$550$	0	0
$551$	6.13319	0.261283
$552$	13.6531	0.581115
$553$	18.6275	0.792123
$554$	−43.1340	−1.83259
$555$	−17.2731	−0.733202
$556$	9.91042	0.420295
$557$	−33.1180	−1.40326	−0.701628	−	0.712543i	$-0.747543\pi$
$557$	−33.1180	−1.40326	−0.701628	+	0.712543i	$0.747543\pi$
$558$	38.9691	1.64969
$559$	−9.28055	−0.392525
$560$	−14.3304	−0.605571
$561$	0	0
$562$	7.58739	0.320055
$563$	7.48682	0.315532	0.157766	−	0.987477i	$-0.449571\pi$
$563$	7.48682	0.315532	0.157766	+	0.987477i	$0.449571\pi$
$564$	7.52137	0.316707
$565$	−0.992759	−0.0417657
$566$	−35.9565	−1.51136
$567$	−76.1788	−3.19921
$568$	12.2555	0.514231
$569$	−29.8028	−1.24940	−0.624698	−	0.780866i	$-0.714778\pi$
$569$	−29.8028	−1.24940	−0.624698	+	0.780866i	$0.714778\pi$
$570$	7.30143	0.305823
$571$	−13.7647	−0.576034	−0.288017	−	0.957625i	$-0.592996\pi$
$571$	−13.7647	−0.576034	−0.288017	+	0.957625i	$0.592996\pi$
$572$	0	0
$573$	−23.4522	−0.979728
$574$	60.3420	2.51863
$575$	7.95528	0.331758
$576$	29.3698	1.22374
$577$	29.0765	1.21047	0.605234	−	0.796047i	$-0.293079\pi$
$577$	29.0765	1.21047	0.605234	+	0.796047i	$0.293079\pi$
$578$	−26.8614	−1.11729
$579$	−79.9948	−3.32447
$580$	2.53288	0.105172
$581$	2.47434	0.102653
$582$	55.0834	2.28328
$583$	0	0
$584$	−0.122019	−0.00504916
$585$	−27.5231	−1.13794
$586$	−0.503412	−0.0207958
$587$	28.7574	1.18694	0.593472	−	0.804854i	$-0.297757\pi$
$587$	28.7574	1.18694	0.593472	+	0.804854i	$0.297757\pi$
$588$	−6.10291	−0.251680
$589$	4.75633	0.195981
$590$	−21.3114	−0.877378
$591$	−3.49568	−0.143793
$592$	27.9155	1.14732
$593$	−26.5838	−1.09166	−0.545832	−	0.837895i	$-0.683786\pi$
$593$	−26.5838	−1.09166	−0.545832	+	0.837895i	$0.683786\pi$
$594$	0	0
$595$	−16.9869	−0.696393
$596$	−6.29316	−0.257778
$597$	−2.12923	−0.0871436
$598$	12.4111	0.507527
$599$	11.6062	0.474215	0.237108	−	0.971483i	$-0.423801\pi$
$599$	11.6062	0.474215	0.237108	+	0.971483i	$0.423801\pi$
$600$	29.2386	1.19366
$601$	−30.2074	−1.23219	−0.616094	−	0.787673i	$-0.711286\pi$
$601$	−30.2074	−1.23219	−0.616094	+	0.787673i	$0.711286\pi$
$602$	−12.0297	−0.490294
$603$	8.88778	0.361939
$604$	8.26608	0.336342
$605$	0	0
$606$	6.14810	0.249749
$607$	15.9646	0.647983	0.323991	−	0.946060i	$-0.394975\pi$
$607$	15.9646	0.647983	0.323991	+	0.946060i	$0.394975\pi$
$608$	−5.30147	−0.215003
$609$	−42.0239	−1.70289
$610$	12.2212	0.494823
$611$	−14.0138	−0.566936
$612$	−28.2988	−1.14391
$613$	0.239609	0.00967770	0.00483885	−	0.999988i	$-0.498460\pi$
$613$	0.239609	0.00967770	0.00483885	+	0.999988i	$0.498460\pi$
$614$	4.14499	0.167278
$615$	35.5857	1.43495
$616$	0	0
$617$	−6.81566	−0.274388	−0.137194	−	0.990544i	$-0.543808\pi$
$617$	−6.81566	−0.274388	−0.137194	+	0.990544i	$0.543808\pi$
$618$	−20.2957	−0.816411
$619$	24.0332	0.965978	0.482989	−	0.875627i	$-0.339551\pi$
$619$	24.0332	0.965978	0.482989	+	0.875627i	$0.339551\pi$
$620$	1.96426	0.0788866
$621$	27.7803	1.11479
$622$	12.7864	0.512687
$623$	46.5200	1.86378
$624$	62.3751	2.49700
$625$	12.6742	0.506969
$626$	−46.2256	−1.84755
$627$	0	0
$628$	11.8773	0.473956
$629$	33.0902	1.31939
$630$	−35.6762	−1.42137
$631$	−15.8884	−0.632508	−0.316254	−	0.948674i	$-0.602425\pi$
$631$	−15.8884	−0.632508	−0.316254	+	0.948674i	$0.602425\pi$
$632$	−12.9833	−0.516448
$633$	−1.30675	−0.0519385
$634$	8.22356	0.326599
$635$	13.9903	0.555187
$636$	20.0343	0.794412
$637$	11.3709	0.450532
$638$	0	0
$639$	41.7206	1.65044
$640$	12.6720	0.500904
$641$	14.0783	0.556059	0.278029	−	0.960573i	$-0.410319\pi$
$641$	14.0783	0.556059	0.278029	+	0.960573i	$0.410319\pi$
$642$	−63.6535	−2.51220
$643$	−28.2102	−1.11250	−0.556252	−	0.831014i	$-0.687761\pi$
$643$	−28.2102	−1.11250	−0.556252	+	0.831014i	$0.687761\pi$
$644$	3.97259	0.156542
$645$	−7.09430	−0.279338
$646$	−13.9874	−0.550326
$647$	20.4001	0.802012	0.401006	−	0.916075i	$-0.368661\pi$
$647$	20.4001	0.802012	0.401006	+	0.916075i	$0.368661\pi$
$648$	53.0962	2.08582
$649$	0	0
$650$	26.5788	1.04251
$651$	−32.5898	−1.27729
$652$	−6.58705	−0.257969
$653$	16.5223	0.646566	0.323283	−	0.946302i	$-0.395213\pi$
$653$	16.5223	0.646566	0.323283	+	0.946302i	$0.395213\pi$
$654$	67.7722	2.65010
$655$	−6.85129	−0.267702
$656$	−57.5108	−2.24542
$657$	−0.415379	−0.0162055
$658$	−18.1650	−0.708146
$659$	42.0970	1.63987	0.819933	−	0.572460i	$-0.194011\pi$
$659$	42.0970	1.63987	0.819933	+	0.572460i	$0.194011\pi$
$660$	0	0
$661$	−26.6792	−1.03770	−0.518850	−	0.854865i	$-0.673640\pi$
$661$	−26.6792	−1.03770	−0.518850	+	0.854865i	$0.673640\pi$
$662$	−53.7225	−2.08798
$663$	73.9376	2.87150
$664$	−1.72461	−0.0669277
$665$	−4.35441	−0.168857
$666$	69.4967	2.69294
$667$	7.96938	0.308575
$668$	−9.44524	−0.365447
$669$	48.9984	1.89439
$670$	1.81422	0.0700893
$671$	0	0
$672$	36.3250	1.40127
$673$	−15.1914	−0.585586	−0.292793	−	0.956176i	$-0.594585\pi$
$673$	−15.1914	−0.585586	−0.292793	+	0.956176i	$0.594585\pi$
$674$	12.9176	0.497566
$675$	59.4926	2.28987
$676$	1.71379	0.0659150
$677$	−26.7863	−1.02948	−0.514740	−	0.857346i	$-0.672112\pi$
$677$	−26.7863	−1.02948	−0.514740	+	0.857346i	$0.672112\pi$
$678$	5.60114	0.215111
$679$	−32.8505	−1.26069
$680$	11.8398	0.454034
$681$	2.03678	0.0780495
$682$	0	0
$683$	17.7240	0.678192	0.339096	−	0.940752i	$-0.389879\pi$
$683$	17.7240	0.678192	0.339096	+	0.940752i	$0.389879\pi$
$684$	−7.25411	−0.277368
$685$	4.88185	0.186526
$686$	−21.1137	−0.806125
$687$	89.4449	3.41254
$688$	11.4653	0.437109
$689$	−37.3278	−1.42208
$690$	9.48737	0.361178
$691$	15.0790	0.573632	0.286816	−	0.957986i	$-0.407403\pi$
$691$	15.0790	0.573632	0.286816	+	0.957986i	$0.407403\pi$
$692$	9.32878	0.354627
$693$	0	0
$694$	20.1738	0.765786
$695$	−14.1152	−0.535419
$696$	29.2905	1.11025
$697$	−68.1716	−2.58218
$698$	−19.8101	−0.749825
$699$	22.5911	0.854475
$700$	8.50745	0.321551
$701$	−39.8913	−1.50667	−0.753337	−	0.657634i	$-0.771557\pi$
$701$	−39.8913	−1.50667	−0.753337	+	0.657634i	$0.771557\pi$
$702$	92.8148	3.50307
$703$	8.48233	0.319917
$704$	0	0
$705$	−10.7125	−0.403456
$706$	22.6560	0.852670
$707$	−3.66659	−0.137896
$708$	29.6912	1.11587
$709$	−28.1244	−1.05623	−0.528116	−	0.849172i	$-0.677101\pi$
$709$	−28.1244	−1.05623	−0.528116	+	0.849172i	$0.677101\pi$
$710$	8.51622	0.319608
$711$	−44.1981	−1.65756
$712$	−32.4242	−1.21515
$713$	6.18030	0.231454
$714$	95.8398	3.58672
$715$	0	0
$716$	6.36492	0.237868
$717$	54.2089	2.02447
$718$	3.16260	0.118027
$719$	39.5088	1.47343	0.736715	−	0.676203i	$-0.236376\pi$
$719$	39.5088	1.47343	0.736715	+	0.676203i	$0.236376\pi$
$720$	34.0022	1.26719
$721$	12.1039	0.450772
$722$	27.3781	1.01891
$723$	53.0569	1.97321
$724$	4.33590	0.161142
$725$	17.0667	0.633842
$726$	0	0
$727$	−4.95971	−0.183946	−0.0919728	−	0.995762i	$-0.529317\pi$
$727$	−4.95971	−0.183946	−0.0919728	+	0.995762i	$0.529317\pi$
$728$	−27.2040	−1.00825
$729$	40.9213	1.51560
$730$	−0.0847891	−0.00313819
$731$	13.5906	0.502666
$732$	−17.0267	−0.629325
$733$	12.1731	0.449624	0.224812	−	0.974402i	$-0.427823\pi$
$733$	12.1731	0.449624	0.224812	+	0.974402i	$0.427823\pi$
$734$	−9.75266	−0.359977
$735$	8.69222	0.320617
$736$	−6.88865	−0.253919
$737$	0	0
$738$	−143.175	−5.27036
$739$	−29.8936	−1.09966	−0.549828	−	0.835278i	$-0.685307\pi$
$739$	−29.8936	−1.09966	−0.549828	+	0.835278i	$0.685307\pi$
$740$	3.50302	0.128774
$741$	18.9532	0.696261
$742$	−48.3853	−1.77628
$743$	13.1288	0.481649	0.240824	−	0.970569i	$-0.422582\pi$
$743$	13.1288	0.481649	0.240824	+	0.970569i	$0.422582\pi$
$744$	22.7149	0.832770
$745$	8.96319	0.328386
$746$	−54.6253	−1.99998
$747$	−5.87095	−0.214807
$748$	0	0
$749$	37.9615	1.38708
$750$	44.9298	1.64060
$751$	−17.2193	−0.628341	−0.314171	−	0.949367i	$-0.601726\pi$
$751$	−17.2193	−0.628341	−0.314171	+	0.949367i	$0.601726\pi$
$752$	17.3127	0.631330
$753$	−21.8822	−0.797433
$754$	26.6259	0.969659
$755$	−11.7732	−0.428469
$756$	29.7085	1.08049
$757$	−17.0776	−0.620697	−0.310349	−	0.950623i	$-0.600446\pi$
$757$	−17.0776	−0.620697	−0.310349	+	0.950623i	$0.600446\pi$
$758$	−7.49127	−0.272095
$759$	0	0
$760$	3.03500	0.110091
$761$	−3.30126	−0.119670	−0.0598352	−	0.998208i	$-0.519058\pi$
$761$	−3.30126	−0.119670	−0.0598352	+	0.998208i	$0.519058\pi$
$762$	−78.9330	−2.85944
$763$	−40.4179	−1.46322
$764$	4.75614	0.172071
$765$	40.3053	1.45724
$766$	−31.5316	−1.13928
$767$	−55.3205	−1.99751
$768$	−46.0234	−1.66073
$769$	−26.7422	−0.964348	−0.482174	−	0.876075i	$-0.660153\pi$
$769$	−26.7422	−0.964348	−0.482174	+	0.876075i	$0.660153\pi$
$770$	0	0
$771$	0.442504	0.0159364
$772$	16.2231	0.583882
$773$	−13.8206	−0.497093	−0.248547	−	0.968620i	$-0.579953\pi$
$773$	−13.8206	−0.497093	−0.248547	+	0.968620i	$0.579953\pi$
$774$	28.5432	1.02597
$775$	13.2354	0.475428
$776$	22.8967	0.821943
$777$	−58.1199	−2.08504
$778$	−30.8844	−1.10726
$779$	−17.4751	−0.626111
$780$	7.82724	0.280260
$781$	0	0
$782$	−18.1750	−0.649936
$783$	59.5980	2.12986
$784$	−14.0477	−0.501703
$785$	−16.9165	−0.603777
$786$	38.6549	1.37878
$787$	−10.9843	−0.391548	−0.195774	−	0.980649i	$-0.562722\pi$
$787$	−10.9843	−0.391548	−0.195774	+	0.980649i	$0.562722\pi$
$788$	0.708931	0.0252546
$789$	53.7578	1.91383
$790$	−9.02194	−0.320986
$791$	−3.34040	−0.118771
$792$	0	0
$793$	31.7241	1.12655
$794$	18.4893	0.656161
$795$	−28.5344	−1.01201
$796$	0.431812	0.0153052
$797$	18.4437	0.653308	0.326654	−	0.945144i	$-0.394079\pi$
$797$	18.4437	0.653308	0.326654	+	0.945144i	$0.394079\pi$
$798$	24.5676	0.869683
$799$	20.5220	0.726016
$800$	−14.7523	−0.521573
$801$	−110.379	−3.90007
$802$	17.6445	0.623050
$803$	0	0
$804$	−2.52758	−0.0891409
$805$	−5.65805	−0.199420
$806$	20.6486	0.727315
$807$	4.40446	0.155044
$808$	2.55560	0.0899055
$809$	12.5013	0.439521	0.219760	−	0.975554i	$-0.429472\pi$
$809$	12.5013	0.439521	0.219760	+	0.975554i	$0.429472\pi$
$810$	36.8959	1.29639
$811$	21.3578	0.749973	0.374987	−	0.927030i	$-0.377647\pi$
$811$	21.3578	0.749973	0.374987	+	0.927030i	$0.377647\pi$
$812$	8.52252	0.299082
$813$	−66.8644	−2.34504
$814$	0	0
$815$	9.38176	0.328629
$816$	−91.3430	−3.19765
$817$	3.48381	0.121883
$818$	−53.3191	−1.86426
$819$	−92.6087	−3.23601
$820$	−7.21684	−0.252023
$821$	8.71755	0.304245	0.152122	−	0.988362i	$-0.451389\pi$
$821$	8.71755	0.304245	0.152122	+	0.988362i	$0.451389\pi$
$822$	−27.5434	−0.960686
$823$	−26.2385	−0.914616	−0.457308	−	0.889308i	$-0.651186\pi$
$823$	−26.2385	−0.914616	−0.457308	+	0.889308i	$0.651186\pi$
$824$	−8.43635	−0.293894
$825$	0	0
$826$	−71.7079	−2.49504
$827$	28.6887	0.997605	0.498802	−	0.866716i	$-0.333773\pi$
$827$	28.6887	0.997605	0.498802	+	0.866716i	$0.333773\pi$
$828$	−9.42588	−0.327572
$829$	33.2537	1.15495	0.577474	−	0.816409i	$-0.304039\pi$
$829$	33.2537	1.15495	0.577474	+	0.816409i	$0.304039\pi$
$830$	−1.19841	−0.0415973
$831$	−85.5913	−2.96913
$832$	15.5622	0.539522
$833$	−16.6517	−0.576948
$834$	79.6377	2.75763
$835$	13.4526	0.465547
$836$	0	0
$837$	46.2186	1.59755
$838$	−26.9293	−0.930258
$839$	42.3483	1.46203	0.731013	−	0.682364i	$-0.239048\pi$
$839$	42.3483	1.46203	0.731013	+	0.682364i	$0.239048\pi$
$840$	−20.7955	−0.717512
$841$	−11.9030	−0.410449
$842$	18.1630	0.625938
$843$	15.0557	0.518547
$844$	0.265010	0.00912204
$845$	−2.44091	−0.0839697
$846$	43.1007	1.48183
$847$	0	0
$848$	46.1150	1.58360
$849$	−71.3488	−2.44869
$850$	−38.9225	−1.33503
$851$	11.0218	0.377823
$852$	−11.8649	−0.406483
$853$	−34.2986	−1.17436	−0.587181	−	0.809455i	$-0.699763\pi$
$853$	−34.2986	−1.17436	−0.587181	+	0.809455i	$0.699763\pi$
$854$	41.1216	1.40715
$855$	10.3318	0.353342
$856$	−26.4590	−0.904351
$857$	32.6414	1.11501	0.557505	−	0.830174i	$-0.311759\pi$
$857$	32.6414	1.11501	0.557505	+	0.830174i	$0.311759\pi$
$858$	0	0
$859$	−24.8732	−0.848663	−0.424331	−	0.905507i	$-0.639491\pi$
$859$	−24.8732	−0.848663	−0.424331	+	0.905507i	$0.639491\pi$
$860$	1.43874	0.0490605
$861$	119.737	4.08064
$862$	42.7998	1.45777
$863$	34.2263	1.16508	0.582538	−	0.812804i	$-0.302060\pi$
$863$	34.2263	1.16508	0.582538	+	0.812804i	$0.302060\pi$
$864$	−51.5160	−1.75261
$865$	−13.2867	−0.451763
$866$	−16.2806	−0.553236
$867$	−53.3014	−1.81021
$868$	6.60927	0.224333
$869$	0	0
$870$	20.3536	0.690051
$871$	4.70937	0.159571
$872$	28.1711	0.953993
$873$	77.9455	2.63806
$874$	−4.65898	−0.157592
$875$	−26.7951	−0.905841
$876$	0.118129	0.00399120
$877$	17.7942	0.600869	0.300435	−	0.953802i	$-0.402868\pi$
$877$	17.7942	0.600869	0.300435	+	0.953802i	$0.402868\pi$
$878$	−36.1858	−1.22121
$879$	−0.998925	−0.0336929
$880$	0	0
$881$	−16.3604	−0.551197	−0.275599	−	0.961273i	$-0.588876\pi$
$881$	−16.3604	−0.551197	−0.275599	+	0.961273i	$0.588876\pi$
$882$	−34.9723	−1.17758
$883$	−12.1775	−0.409806	−0.204903	−	0.978782i	$-0.565688\pi$
$883$	−12.1775	−0.409806	−0.204903	+	0.978782i	$0.565688\pi$
$884$	−14.9947	−0.504326
$885$	−42.2885	−1.42151
$886$	−47.8728	−1.60832
$887$	−7.70306	−0.258643	−0.129322	−	0.991603i	$-0.541280\pi$
$887$	−7.70306	−0.258643	−0.129322	+	0.991603i	$0.541280\pi$
$888$	40.5093	1.35940
$889$	47.0739	1.57881
$890$	−22.5312	−0.755247
$891$	0	0
$892$	−9.93697	−0.332714
$893$	5.26060	0.176039
$894$	−50.5703	−1.69132
$895$	−9.06540	−0.303023
$896$	42.6382	1.42444
$897$	24.6274	0.822286
$898$	55.0597	1.83737
$899$	13.2588	0.442206
$900$	−20.1859	−0.672863
$901$	54.6634	1.82110
$902$	0	0
$903$	−23.8706	−0.794365
$904$	2.32824	0.0774362
$905$	−6.17551	−0.205281
$906$	66.4242	2.20680
$907$	−1.16828	−0.0387921	−0.0193961	−	0.999812i	$-0.506174\pi$
$907$	−1.16828	−0.0387921	−0.0193961	+	0.999812i	$0.506174\pi$
$908$	−0.413063	−0.0137080
$909$	8.69983	0.288555
$910$	−18.9037	−0.626653
$911$	−57.1088	−1.89210	−0.946049	−	0.324024i	$-0.894964\pi$
$911$	−57.1088	−1.89210	−0.946049	+	0.324024i	$0.894964\pi$
$912$	−23.4149	−0.775343
$913$	0	0
$914$	28.6655	0.948170
$915$	24.2507	0.801704
$916$	−18.1396	−0.599350
$917$	−23.0530	−0.761276
$918$	−135.920	−4.48601
$919$	−10.6905	−0.352647	−0.176324	−	0.984332i	$-0.556421\pi$
$919$	−10.6905	−0.352647	−0.176324	+	0.984332i	$0.556421\pi$
$920$	3.94364	0.130018
$921$	8.22494	0.271021
$922$	−50.6125	−1.66684
$923$	22.1065	0.727645
$924$	0	0
$925$	23.6036	0.776083
$926$	−60.8465	−1.99954
$927$	−28.7193	−0.943265
$928$	−14.7785	−0.485127
$929$	−33.8631	−1.11101	−0.555506	−	0.831513i	$-0.687475\pi$
$929$	−33.8631	−1.11101	−0.555506	+	0.831513i	$0.687475\pi$
$930$	15.7843	0.517588
$931$	−4.26850	−0.139895
$932$	−4.58152	−0.150073
$933$	25.3721	0.830646
$934$	−7.12771	−0.233226
$935$	0	0
$936$	64.5478	2.10981
$937$	40.7999	1.33287	0.666437	−	0.745561i	$-0.267818\pi$
$937$	40.7999	1.33287	0.666437	+	0.745561i	$0.267818\pi$
$938$	6.10441	0.199316
$939$	−91.7259	−2.99336
$940$	2.17252	0.0708596
$941$	−0.0192262	−0.000626757 0	−0.000313378	−	1.00000i	$-0.500100\pi$
$941$	−0.0192262	−0.000626757 0	−0.000313378		1.00000i	$0.500100\pi$
$942$	95.4431	3.10970
$943$	−22.7069	−0.739438
$944$	68.3434	2.22439
$945$	−42.3131	−1.37644
$946$	0	0
$947$	32.4815	1.05551	0.527753	−	0.849398i	$-0.323035\pi$
$947$	32.4815	1.05551	0.527753	+	0.849398i	$0.323035\pi$
$948$	12.5694	0.408236
$949$	−0.220097	−0.00714465
$950$	−9.97738	−0.323709
$951$	16.3181	0.529150
$952$	39.8380	1.29116
$953$	7.92375	0.256675	0.128338	−	0.991731i	$-0.459036\pi$
$953$	7.92375	0.256675	0.128338	+	0.991731i	$0.459036\pi$
$954$	114.805	3.71696
$955$	−6.77405	−0.219203
$956$	−10.9937	−0.355561
$957$	0	0
$958$	−17.0598	−0.551178
$959$	16.4263	0.530432
$960$	11.8962	0.383947
$961$	−20.7177	−0.668313
$962$	36.8242	1.18726
$963$	−90.0726	−2.90255
$964$	−10.7600	−0.346558
$965$	−23.1062	−0.743814
$966$	31.9227	1.02710
$967$	−34.1966	−1.09969	−0.549844	−	0.835268i	$-0.685313\pi$
$967$	−34.1966	−1.09969	−0.549844	+	0.835268i	$0.685313\pi$
$968$	0	0
$969$	−27.7553	−0.891629
$970$	15.9106	0.510859
$971$	8.68724	0.278787	0.139393	−	0.990237i	$-0.455485\pi$
$971$	8.68724	0.278787	0.139393	+	0.990237i	$0.455485\pi$
$972$	−23.0458	−0.739193
$973$	−47.4942	−1.52259
$974$	9.46409	0.303249
$975$	52.7406	1.68905
$976$	−39.1921	−1.25451
$977$	−8.48837	−0.271567	−0.135783	−	0.990739i	$-0.543355\pi$
$977$	−8.48837	−0.271567	−0.135783	+	0.990739i	$0.543355\pi$
$978$	−52.9319	−1.69258
$979$	0	0
$980$	−1.76280	−0.0563106
$981$	95.9008	3.06188
$982$	11.5311	0.367973
$983$	−19.2677	−0.614544	−0.307272	−	0.951622i	$-0.599416\pi$
$983$	−19.2677	−0.614544	−0.307272	+	0.951622i	$0.599416\pi$
$984$	−83.4563	−2.66049
$985$	−1.00971	−0.0321721
$986$	−38.9914	−1.24174
$987$	−36.0450	−1.14733
$988$	−3.84374	−0.122286
$989$	4.52681	0.143944
$990$	0	0
$991$	−42.3749	−1.34608	−0.673041	−	0.739605i	$-0.735012\pi$
$991$	−42.3749	−1.34608	−0.673041	+	0.739605i	$0.735012\pi$
$992$	−11.4608	−0.363880
$993$	−106.602	−3.38291
$994$	28.6550	0.908883
$995$	−0.615019	−0.0194974
$996$	1.66963	0.0529042
$997$	−0.832594	−0.0263685	−0.0131843	−	0.999913i	$-0.504197\pi$
$997$	−0.832594	−0.0263685	−0.0131843	+	0.999913i	$0.504197\pi$
$998$	−10.5778	−0.334835
$999$	82.4253	2.60782

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1331.2.a.e.1.5	yes	25
11.10	odd	2		1331.2.a.d.1.21	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1331.2.a.d.1.21	✓	25	11.10	odd	2
1331.2.a.e.1.5	yes	25	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.62967 q^{2} -3.23376 q^{3} +0.655813 q^{4} -0.934058 q^{5} +5.26996 q^{6} -3.14288 q^{7} +2.19058 q^{8} +7.45722 q^{9} +1.52220 q^{10} -2.12074 q^{12} +3.95136 q^{13} +5.12185 q^{14} +3.02052 q^{15} -4.88154 q^{16} -5.78643 q^{17} -12.1528 q^{18} -1.48329 q^{19} -0.612568 q^{20} +10.1633 q^{21} -1.92737 q^{23} -7.08380 q^{24} -4.12754 q^{25} -6.43940 q^{26} -14.4136 q^{27} -2.06115 q^{28} -4.13485 q^{29} -4.92245 q^{30} -3.20660 q^{31} +3.57412 q^{32} +9.42995 q^{34} +2.93564 q^{35} +4.89055 q^{36} -5.71858 q^{37} +2.41727 q^{38} -12.7778 q^{39} -2.04613 q^{40} +11.7813 q^{41} -16.5629 q^{42} -2.34870 q^{43} -6.96548 q^{45} +3.14097 q^{46} -3.54657 q^{47} +15.7857 q^{48} +2.87772 q^{49} +6.72651 q^{50} +18.7119 q^{51} +2.59135 q^{52} -9.44683 q^{53} +23.4894 q^{54} -6.88473 q^{56} +4.79662 q^{57} +6.73842 q^{58} -14.0004 q^{59} +1.98090 q^{60} +8.02865 q^{61} +5.22569 q^{62} -23.4372 q^{63} +3.93844 q^{64} -3.69080 q^{65} +1.19184 q^{67} -3.79482 q^{68} +6.23265 q^{69} -4.78411 q^{70} +5.59466 q^{71} +16.3356 q^{72} -0.0557016 q^{73} +9.31938 q^{74} +13.3475 q^{75} -0.972763 q^{76} +20.8235 q^{78} -5.92689 q^{79} +4.55964 q^{80} +24.2385 q^{81} -19.1996 q^{82} -0.787284 q^{83} +6.66526 q^{84} +5.40486 q^{85} +3.82760 q^{86} +13.3711 q^{87} -14.8017 q^{89} +11.3514 q^{90} -12.4187 q^{91} -1.26399 q^{92} +10.3694 q^{93} +5.77973 q^{94} +1.38548 q^{95} -11.5579 q^{96} +10.4524 q^{97} -4.68973 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} - 3 q^{3} + 25 q^{4} - 3 q^{5} + 15 q^{6} + 19 q^{7} + 9 q^{8} + 22 q^{9} + 25 q^{10} - 12 q^{12} + 46 q^{13} - 12 q^{14} - 15 q^{15} + 13 q^{16} + 14 q^{17} + 6 q^{18} + 45 q^{19} - 24 q^{20}+ \cdots - 39 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 - 3 * q^3 + 25 * q^4 - 3 * q^5 + 15 * q^6 + 19 * q^7 + 9 * q^8 + 22 * q^9 + 25 * q^10 - 12 * q^12 + 46 * q^13 - 12 * q^14 - 15 * q^15 + 13 * q^16 + 14 * q^17 + 6 * q^18 + 45 * q^19 - 24 * q^20 + 49 * q^21 + 2 * q^23 + 36 * q^24 + 16 * q^25 - 7 * q^26 - 21 * q^27 + 40 * q^28 + 40 * q^29 - 15 * q^30 - 4 * q^31 + 3 * q^32 - 24 * q^34 - q^35 + 38 * q^36 - 21 * q^37 + 30 * q^38 + 18 * q^39 + 54 * q^40 + 49 * q^41 + 13 * q^42 + 14 * q^43 - 42 * q^45 + 40 * q^46 - 2 * q^47 - 5 * q^48 + 18 * q^49 - 12 * q^50 + 7 * q^51 + 57 * q^52 - 25 * q^53 + 33 * q^54 - 11 * q^56 + 4 * q^57 + 23 * q^58 - 22 * q^59 - 12 * q^60 + 128 * q^61 - 8 * q^62 + 12 * q^63 + 31 * q^64 + 16 * q^67 - 6 * q^68 - 57 * q^69 + 38 * q^70 - 34 * q^71 - 15 * q^72 + 71 * q^73 - 21 * q^74 - 10 * q^75 + 95 * q^76 + 75 * q^78 + 53 * q^79 - 29 * q^80 + 21 * q^81 + 33 * q^82 - 6 * q^83 + 18 * q^84 + 101 * q^85 + 38 * q^86 - 16 * q^87 + 5 * q^89 - 20 * q^90 + 16 * q^91 + 53 * q^92 + 7 * q^93 + 57 * q^94 - 16 * q^95 + 32 * q^96 + 14 * q^97 - 39 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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