LMFDB - Embedded newform 8001.2.a.o.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 $ x^13 - 4x^12 - 10x^11 + 53x^10 + 19x^9 - 242x^8 + 61x^7 + 467x^6 - 211x^5 - 372x^4 + 146x^3 + 116x^2 - 12x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-2.45160$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.45160 q^{2} +4.01034 q^{4} -1.23909 q^{5} +1.00000 q^{7} +4.92856 q^{8} +O(q^{10})$	\(q+2.45160 q^{2} +4.01034 q^{4} -1.23909 q^{5} +1.00000 q^{7} +4.92856 q^{8} -3.03775 q^{10} -2.19586 q^{11} +1.52432 q^{13} +2.45160 q^{14} +4.06217 q^{16} -7.06712 q^{17} -1.45237 q^{19} -4.96917 q^{20} -5.38336 q^{22} -2.60977 q^{23} -3.46466 q^{25} +3.73703 q^{26} +4.01034 q^{28} -6.11799 q^{29} +1.83292 q^{31} +0.101697 q^{32} -17.3258 q^{34} -1.23909 q^{35} +0.914123 q^{37} -3.56064 q^{38} -6.10692 q^{40} +1.13505 q^{41} +2.48757 q^{43} -8.80614 q^{44} -6.39812 q^{46} -0.620901 q^{47} +1.00000 q^{49} -8.49396 q^{50} +6.11305 q^{52} -6.86259 q^{53} +2.72086 q^{55} +4.92856 q^{56} -14.9989 q^{58} -8.13041 q^{59} +5.09932 q^{61} +4.49360 q^{62} -7.87502 q^{64} -1.88877 q^{65} -5.28798 q^{67} -28.3416 q^{68} -3.03775 q^{70} -4.94368 q^{71} +3.24571 q^{73} +2.24106 q^{74} -5.82451 q^{76} -2.19586 q^{77} -8.62522 q^{79} -5.03339 q^{80} +2.78269 q^{82} +14.6973 q^{83} +8.75679 q^{85} +6.09853 q^{86} -10.8224 q^{88} -2.62512 q^{89} +1.52432 q^{91} -10.4661 q^{92} -1.52220 q^{94} +1.79962 q^{95} +4.23555 q^{97} +2.45160 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.45160	1.73354	0.866772	−	0.498705i	$-0.166191\pi$
$2$	2.45160	1.73354	0.866772	+	0.498705i	$0.166191\pi$
$3$	0	0
$3$	0	0
$4$	4.01034	2.00517
$5$	−1.23909	−0.554137	−0.277069	−	0.960850i	$-0.589363\pi$
$5$	−1.23909	−0.554137	−0.277069	+	0.960850i	$0.589363\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	4.92856	1.74251
$9$	0	0
$10$	−3.03775	−0.960621
$11$	−2.19586	−0.662076	−0.331038	−	0.943618i	$-0.607399\pi$
$11$	−2.19586	−0.662076	−0.331038	+	0.943618i	$0.607399\pi$
$12$	0	0
$13$	1.52432	0.422771	0.211385	−	0.977403i	$-0.432203\pi$
$13$	1.52432	0.422771	0.211385	+	0.977403i	$0.432203\pi$
$14$	2.45160	0.655218
$15$	0	0
$16$	4.06217	1.01554
$17$	−7.06712	−1.71403	−0.857014	−	0.515292i	$-0.827683\pi$
$17$	−7.06712	−1.71403	−0.857014	+	0.515292i	$0.827683\pi$
$18$	0	0
$19$	−1.45237	−0.333197	−0.166599	−	0.986025i	$-0.553278\pi$
$19$	−1.45237	−0.333197	−0.166599	+	0.986025i	$0.553278\pi$
$20$	−4.96917	−1.11114
$21$	0	0
$22$	−5.38336	−1.14774
$23$	−2.60977	−0.544175	−0.272088	−	0.962272i	$-0.587714\pi$
$23$	−2.60977	−0.544175	−0.272088	+	0.962272i	$0.587714\pi$
$24$	0	0
$25$	−3.46466	−0.692932
$26$	3.73703	0.732891
$27$	0	0
$28$	4.01034	0.757884
$29$	−6.11799	−1.13608	−0.568041	−	0.823000i	$-0.692299\pi$
$29$	−6.11799	−1.13608	−0.568041	+	0.823000i	$0.692299\pi$
$30$	0	0
$31$	1.83292	0.329203	0.164601	−	0.986360i	$-0.447366\pi$
$31$	1.83292	0.329203	0.164601	+	0.986360i	$0.447366\pi$
$32$	0.101697	0.0179777
$33$	0	0
$34$	−17.3258	−2.97134
$35$	−1.23909	−0.209444
$36$	0	0
$37$	0.914123	0.150281	0.0751404	−	0.997173i	$-0.476059\pi$
$37$	0.914123	0.150281	0.0751404	+	0.997173i	$0.476059\pi$
$38$	−3.56064	−0.577611
$39$	0	0
$40$	−6.10692	−0.965589
$41$	1.13505	0.177265	0.0886325	−	0.996064i	$-0.471750\pi$
$41$	1.13505	0.177265	0.0886325	+	0.996064i	$0.471750\pi$
$42$	0	0
$43$	2.48757	0.379351	0.189675	−	0.981847i	$-0.439256\pi$
$43$	2.48757	0.379351	0.189675	+	0.981847i	$0.439256\pi$
$44$	−8.80614	−1.32758
$45$	0	0
$46$	−6.39812	−0.943352
$47$	−0.620901	−0.0905677	−0.0452839	−	0.998974i	$-0.514419\pi$
$47$	−0.620901	−0.0905677	−0.0452839	+	0.998974i	$0.514419\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−8.49396	−1.20123
$51$	0	0
$52$	6.11305	0.847728
$53$	−6.86259	−0.942649	−0.471325	−	0.881960i	$-0.656224\pi$
$53$	−6.86259	−0.942649	−0.471325	+	0.881960i	$0.656224\pi$
$54$	0	0
$55$	2.72086	0.366881
$56$	4.92856	0.658606
$57$	0	0
$58$	−14.9989	−1.96945
$59$	−8.13041	−1.05849	−0.529245	−	0.848469i	$-0.677525\pi$
$59$	−8.13041	−1.05849	−0.529245	+	0.848469i	$0.677525\pi$
$60$	0	0
$61$	5.09932	0.652901	0.326451	−	0.945214i	$-0.394147\pi$
$61$	5.09932	0.652901	0.326451	+	0.945214i	$0.394147\pi$
$62$	4.49360	0.570687
$63$	0	0
$64$	−7.87502	−0.984377
$65$	−1.88877	−0.234273
$66$	0	0
$67$	−5.28798	−0.646029	−0.323015	−	0.946394i	$-0.604696\pi$
$67$	−5.28798	−0.646029	−0.323015	+	0.946394i	$0.604696\pi$
$68$	−28.3416	−3.43692
$69$	0	0
$70$	−3.03775	−0.363081
$71$	−4.94368	−0.586707	−0.293353	−	0.956004i	$-0.594771\pi$
$71$	−4.94368	−0.586707	−0.293353	+	0.956004i	$0.594771\pi$
$72$	0	0
$73$	3.24571	0.379882	0.189941	−	0.981796i	$-0.439170\pi$
$73$	3.24571	0.379882	0.189941	+	0.981796i	$0.439170\pi$
$74$	2.24106	0.260518
$75$	0	0
$76$	−5.82451	−0.668117
$77$	−2.19586	−0.250241
$78$	0	0
$79$	−8.62522	−0.970413	−0.485206	−	0.874400i	$-0.661255\pi$
$79$	−8.62522	−0.970413	−0.485206	+	0.874400i	$0.661255\pi$
$80$	−5.03339	−0.562750
$81$	0	0
$82$	2.78269	0.307296
$83$	14.6973	1.61324	0.806621	−	0.591069i	$-0.201294\pi$
$83$	14.6973	1.61324	0.806621	+	0.591069i	$0.201294\pi$
$84$	0	0
$85$	8.75679	0.949807
$86$	6.09853	0.657621
$87$	0	0
$88$	−10.8224	−1.15367
$89$	−2.62512	−0.278262	−0.139131	−	0.990274i	$-0.544431\pi$
$89$	−2.62512	−0.278262	−0.139131	+	0.990274i	$0.544431\pi$
$90$	0	0
$91$	1.52432	0.159792
$92$	−10.4661	−1.09117
$93$	0	0
$94$	−1.52220	−0.157003
$95$	1.79962	0.184637
$96$	0	0
$97$	4.23555	0.430055	0.215028	−	0.976608i	$-0.431016\pi$
$97$	4.23555	0.430055	0.215028	+	0.976608i	$0.431016\pi$
$98$	2.45160	0.247649
$99$	0	0
$100$	−13.8945	−1.38945
$101$	8.53644	0.849407	0.424704	−	0.905332i	$-0.360378\pi$
$101$	8.53644	0.849407	0.424704	+	0.905332i	$0.360378\pi$
$102$	0	0
$103$	−0.764721	−0.0753502	−0.0376751	−	0.999290i	$-0.511995\pi$
$103$	−0.764721	−0.0753502	−0.0376751	+	0.999290i	$0.511995\pi$
$104$	7.51271	0.736681
$105$	0	0
$106$	−16.8243	−1.63412
$107$	−5.26387	−0.508878	−0.254439	−	0.967089i	$-0.581891\pi$
$107$	−5.26387	−0.508878	−0.254439	+	0.967089i	$0.581891\pi$
$108$	0	0
$109$	−7.88983	−0.755709	−0.377854	−	0.925865i	$-0.623338\pi$
$109$	−7.88983	−0.755709	−0.377854	+	0.925865i	$0.623338\pi$
$110$	6.67046	0.636004
$111$	0	0
$112$	4.06217	0.383839
$113$	3.38966	0.318872	0.159436	−	0.987208i	$-0.449032\pi$
$113$	3.38966	0.318872	0.159436	+	0.987208i	$0.449032\pi$
$114$	0	0
$115$	3.23374	0.301548
$116$	−24.5352	−2.27804
$117$	0	0
$118$	−19.9325	−1.83494
$119$	−7.06712	−0.647842
$120$	0	0
$121$	−6.17822	−0.561656
$122$	12.5015	1.13183
$123$	0	0
$124$	7.35066	0.660108
$125$	10.4885	0.938117
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−19.5098	−1.72444
$129$	0	0
$130$	−4.63051	−0.406122
$131$	15.2901	1.33590	0.667951	−	0.744205i	$-0.267172\pi$
$131$	15.2901	1.33590	0.667951	+	0.744205i	$0.267172\pi$
$132$	0	0
$133$	−1.45237	−0.125937
$134$	−12.9640	−1.11992
$135$	0	0
$136$	−34.8307	−2.98671
$137$	−11.5984	−0.990920	−0.495460	−	0.868631i	$-0.665001\pi$
$137$	−11.5984	−0.990920	−0.495460	+	0.868631i	$0.665001\pi$
$138$	0	0
$139$	−13.6844	−1.16069	−0.580347	−	0.814369i	$-0.697083\pi$
$139$	−13.6844	−1.16069	−0.580347	+	0.814369i	$0.697083\pi$
$140$	−4.96917	−0.419972
$141$	0	0
$142$	−12.1199	−1.01708
$143$	−3.34719	−0.279906
$144$	0	0
$145$	7.58073	0.629545
$146$	7.95719	0.658541
$147$	0	0
$148$	3.66595	0.301339
$149$	17.3140	1.41842	0.709210	−	0.704997i	$-0.249052\pi$
$149$	17.3140	1.41842	0.709210	+	0.704997i	$0.249052\pi$
$150$	0	0
$151$	1.01569	0.0826559	0.0413280	−	0.999146i	$-0.486841\pi$
$151$	1.01569	0.0826559	0.0413280	+	0.999146i	$0.486841\pi$
$152$	−7.15810	−0.580599
$153$	0	0
$154$	−5.38336	−0.433804
$155$	−2.27115	−0.182424
$156$	0	0
$157$	−7.65737	−0.611125	−0.305562	−	0.952172i	$-0.598844\pi$
$157$	−7.65737	−0.611125	−0.305562	+	0.952172i	$0.598844\pi$
$158$	−21.1456	−1.68225
$159$	0	0
$160$	−0.126012	−0.00996212
$161$	−2.60977	−0.205679
$162$	0	0
$163$	21.5168	1.68533	0.842664	−	0.538439i	$-0.180986\pi$
$163$	21.5168	1.68533	0.842664	+	0.538439i	$0.180986\pi$
$164$	4.55194	0.355447
$165$	0	0
$166$	36.0320	2.79662
$167$	15.3106	1.18477	0.592387	−	0.805654i	$-0.298186\pi$
$167$	15.3106	1.18477	0.592387	+	0.805654i	$0.298186\pi$
$168$	0	0
$169$	−10.6764	−0.821265
$170$	21.4681	1.64653
$171$	0	0
$172$	9.97601	0.760664
$173$	−25.8227	−1.96326	−0.981632	−	0.190784i	$-0.938897\pi$
$173$	−25.8227	−1.96326	−0.981632	+	0.190784i	$0.938897\pi$
$174$	0	0
$175$	−3.46466	−0.261904
$176$	−8.91994	−0.672366
$177$	0	0
$178$	−6.43573	−0.482379
$179$	−5.00100	−0.373792	−0.186896	−	0.982380i	$-0.559843\pi$
$179$	−5.00100	−0.373792	−0.186896	+	0.982380i	$0.559843\pi$
$180$	0	0
$181$	7.03422	0.522850	0.261425	−	0.965224i	$-0.415808\pi$
$181$	7.03422	0.522850	0.261425	+	0.965224i	$0.415808\pi$
$182$	3.73703	0.277007
$183$	0	0
$184$	−12.8624	−0.948231
$185$	−1.13268	−0.0832762
$186$	0	0
$187$	15.5184	1.13482
$188$	−2.49003	−0.181604
$189$	0	0
$190$	4.41194	0.320076
$191$	9.51278	0.688321	0.344160	−	0.938911i	$-0.388164\pi$
$191$	9.51278	0.688321	0.344160	+	0.938911i	$0.388164\pi$
$192$	0	0
$193$	5.64395	0.406261	0.203130	−	0.979152i	$-0.434888\pi$
$193$	5.64395	0.406261	0.203130	+	0.979152i	$0.434888\pi$
$194$	10.3839	0.745519
$195$	0	0
$196$	4.01034	0.286453
$197$	0.962282	0.0685597	0.0342799	−	0.999412i	$-0.489086\pi$
$197$	0.962282	0.0685597	0.0342799	+	0.999412i	$0.489086\pi$
$198$	0	0
$199$	−8.51386	−0.603532	−0.301766	−	0.953382i	$-0.597576\pi$
$199$	−8.51386	−0.603532	−0.301766	+	0.953382i	$0.597576\pi$
$200$	−17.0758	−1.20744
$201$	0	0
$202$	20.9279	1.47248
$203$	−6.11799	−0.429399
$204$	0	0
$205$	−1.40643	−0.0982291
$206$	−1.87479	−0.130623
$207$	0	0
$208$	6.19205	0.429341
$209$	3.18920	0.220602
$210$	0	0
$211$	4.62207	0.318196	0.159098	−	0.987263i	$-0.449141\pi$
$211$	4.62207	0.318196	0.159098	+	0.987263i	$0.449141\pi$
$212$	−27.5213	−1.89017
$213$	0	0
$214$	−12.9049	−0.882161
$215$	−3.08232	−0.210212
$216$	0	0
$217$	1.83292	0.124427
$218$	−19.3427	−1.31005
$219$	0	0
$220$	10.9116	0.735659
$221$	−10.7726	−0.724641
$222$	0	0
$223$	−2.37627	−0.159127	−0.0795635	−	0.996830i	$-0.525353\pi$
$223$	−2.37627	−0.159127	−0.0795635	+	0.996830i	$0.525353\pi$
$224$	0.101697	0.00679494
$225$	0	0
$226$	8.31009	0.552779
$227$	−23.8414	−1.58241	−0.791205	−	0.611550i	$-0.790546\pi$
$227$	−23.8414	−1.58241	−0.791205	+	0.611550i	$0.790546\pi$
$228$	0	0
$229$	1.15914	0.0765981	0.0382991	−	0.999266i	$-0.487806\pi$
$229$	1.15914	0.0765981	0.0382991	+	0.999266i	$0.487806\pi$
$230$	7.92784	0.522746
$231$	0	0
$232$	−30.1529	−1.97963
$233$	−0.949682	−0.0622157	−0.0311079	−	0.999516i	$-0.509904\pi$
$233$	−0.949682	−0.0622157	−0.0311079	+	0.999516i	$0.509904\pi$
$234$	0	0
$235$	0.769351	0.0501869
$236$	−32.6057	−2.12245
$237$	0	0
$238$	−17.3258	−1.12306
$239$	5.71424	0.369623	0.184812	−	0.982774i	$-0.440833\pi$
$239$	5.71424	0.369623	0.184812	+	0.982774i	$0.440833\pi$
$240$	0	0
$241$	22.3910	1.44233	0.721166	−	0.692762i	$-0.243607\pi$
$241$	22.3910	1.44233	0.721166	+	0.692762i	$0.243607\pi$
$242$	−15.1465	−0.973655
$243$	0	0
$244$	20.4500	1.30918
$245$	−1.23909	−0.0791625
$246$	0	0
$247$	−2.21388	−0.140866
$248$	9.03367	0.573639
$249$	0	0
$250$	25.7135	1.62627
$251$	1.54081	0.0972550	0.0486275	−	0.998817i	$-0.484515\pi$
$251$	1.54081	0.0972550	0.0486275	+	0.998817i	$0.484515\pi$
$252$	0	0
$253$	5.73069	0.360285
$254$	−2.45160	−0.153827
$255$	0	0
$256$	−32.0802	−2.00501
$257$	3.86350	0.240999	0.120499	−	0.992713i	$-0.461550\pi$
$257$	3.86350	0.240999	0.120499	+	0.992713i	$0.461550\pi$
$258$	0	0
$259$	0.914123	0.0568008
$260$	−7.57461	−0.469757
$261$	0	0
$262$	37.4852	2.31584
$263$	3.85898	0.237955	0.118977	−	0.992897i	$-0.462038\pi$
$263$	3.85898	0.237955	0.118977	+	0.992897i	$0.462038\pi$
$264$	0	0
$265$	8.50336	0.522357
$266$	−3.56064	−0.218317
$267$	0	0
$268$	−21.2066	−1.29540
$269$	−29.8634	−1.82080	−0.910401	−	0.413727i	$-0.864227\pi$
$269$	−29.8634	−1.82080	−0.910401	+	0.413727i	$0.864227\pi$
$270$	0	0
$271$	31.4327	1.90940	0.954700	−	0.297571i	$-0.0961767\pi$
$271$	31.4327	1.90940	0.954700	+	0.297571i	$0.0961767\pi$
$272$	−28.7078	−1.74067
$273$	0	0
$274$	−28.4347	−1.71780
$275$	7.60789	0.458773
$276$	0	0
$277$	6.75055	0.405601	0.202801	−	0.979220i	$-0.434996\pi$
$277$	6.75055	0.405601	0.202801	+	0.979220i	$0.434996\pi$
$278$	−33.5486	−2.01211
$279$	0	0
$280$	−6.10692	−0.364958
$281$	−7.23858	−0.431818	−0.215909	−	0.976414i	$-0.569271\pi$
$281$	−7.23858	−0.431818	−0.215909	+	0.976414i	$0.569271\pi$
$282$	0	0
$283$	12.0636	0.717108	0.358554	−	0.933509i	$-0.383270\pi$
$283$	12.0636	0.717108	0.358554	+	0.933509i	$0.383270\pi$
$284$	−19.8259	−1.17645
$285$	0	0
$286$	−8.20597	−0.485229
$287$	1.13505	0.0669998
$288$	0	0
$289$	32.9442	1.93790
$290$	18.5849	1.09134
$291$	0	0
$292$	13.0164	0.761728
$293$	1.86633	0.109032	0.0545162	−	0.998513i	$-0.482638\pi$
$293$	1.86633	0.109032	0.0545162	+	0.998513i	$0.482638\pi$
$294$	0	0
$295$	10.0743	0.586548
$296$	4.50531	0.261866
$297$	0	0
$298$	42.4471	2.45889
$299$	−3.97813	−0.230061
$300$	0	0
$301$	2.48757	0.143381
$302$	2.49007	0.143288
$303$	0	0
$304$	−5.89978	−0.338376
$305$	−6.31851	−0.361797
$306$	0	0
$307$	−3.17131	−0.180996	−0.0904981	−	0.995897i	$-0.528846\pi$
$307$	−3.17131	−0.180996	−0.0904981	+	0.995897i	$0.528846\pi$
$308$	−8.80614	−0.501776
$309$	0	0
$310$	−5.56796	−0.316239
$311$	6.97271	0.395386	0.197693	−	0.980264i	$-0.436655\pi$
$311$	6.97271	0.395386	0.197693	+	0.980264i	$0.436655\pi$
$312$	0	0
$313$	−10.0961	−0.570668	−0.285334	−	0.958428i	$-0.592105\pi$
$313$	−10.0961	−0.570668	−0.285334	+	0.958428i	$0.592105\pi$
$314$	−18.7728	−1.05941
$315$	0	0
$316$	−34.5901	−1.94584
$317$	−32.1544	−1.80597	−0.902986	−	0.429670i	$-0.858630\pi$
$317$	−32.1544	−1.80597	−0.902986	+	0.429670i	$0.858630\pi$
$318$	0	0
$319$	13.4342	0.752172
$320$	9.75784	0.545480
$321$	0	0
$322$	−6.39812	−0.356553
$323$	10.2641	0.571109
$324$	0	0
$325$	−5.28125	−0.292951
$326$	52.7507	2.92159
$327$	0	0
$328$	5.59416	0.308886
$329$	−0.620901	−0.0342314
$330$	0	0
$331$	−24.3663	−1.33930	−0.669648	−	0.742679i	$-0.733555\pi$
$331$	−24.3663	−1.33930	−0.669648	+	0.742679i	$0.733555\pi$
$332$	58.9413	3.23483
$333$	0	0
$334$	37.5356	2.05386
$335$	6.55227	0.357989
$336$	0	0
$337$	7.06781	0.385008	0.192504	−	0.981296i	$-0.438339\pi$
$337$	7.06781	0.385008	0.192504	+	0.981296i	$0.438339\pi$
$338$	−26.1744	−1.42370
$339$	0	0
$340$	35.1177	1.90453
$341$	−4.02484	−0.217957
$342$	0	0
$343$	1.00000	0.0539949
$344$	12.2601	0.661022
$345$	0	0
$346$	−63.3070	−3.40340
$347$	−26.8575	−1.44178	−0.720892	−	0.693047i	$-0.756268\pi$
$347$	−26.8575	−1.44178	−0.720892	+	0.693047i	$0.756268\pi$
$348$	0	0
$349$	−25.3862	−1.35889	−0.679445	−	0.733726i	$-0.737779\pi$
$349$	−25.3862	−1.35889	−0.679445	+	0.733726i	$0.737779\pi$
$350$	−8.49396	−0.454021
$351$	0	0
$352$	−0.223313	−0.0119026
$353$	−25.8412	−1.37539	−0.687695	−	0.726000i	$-0.741377\pi$
$353$	−25.8412	−1.37539	−0.687695	+	0.726000i	$0.741377\pi$
$354$	0	0
$355$	6.12566	0.325116
$356$	−10.5276	−0.557963
$357$	0	0
$358$	−12.2604	−0.647985
$359$	24.9557	1.31711	0.658556	−	0.752531i	$-0.271167\pi$
$359$	24.9557	1.31711	0.658556	+	0.752531i	$0.271167\pi$
$360$	0	0
$361$	−16.8906	−0.888980
$362$	17.2451	0.906383
$363$	0	0
$364$	6.11305	0.320411
$365$	−4.02172	−0.210507
$366$	0	0
$367$	−11.8416	−0.618126	−0.309063	−	0.951042i	$-0.600015\pi$
$367$	−11.8416	−0.618126	−0.309063	+	0.951042i	$0.600015\pi$
$368$	−10.6013	−0.552633
$369$	0	0
$370$	−2.77688	−0.144363
$371$	−6.86259	−0.356288
$372$	0	0
$373$	−0.196072	−0.0101522	−0.00507611	−	0.999987i	$-0.501616\pi$
$373$	−0.196072	−0.0101522	−0.00507611	+	0.999987i	$0.501616\pi$
$374$	38.0449	1.96725
$375$	0	0
$376$	−3.06015	−0.157815
$377$	−9.32578	−0.480302
$378$	0	0
$379$	−24.0964	−1.23775	−0.618874	−	0.785490i	$-0.712411\pi$
$379$	−24.0964	−1.23775	−0.618874	+	0.785490i	$0.712411\pi$
$380$	7.21709	0.370229
$381$	0	0
$382$	23.3215	1.19323
$383$	−20.1432	−1.02927	−0.514635	−	0.857409i	$-0.672073\pi$
$383$	−20.1432	−1.02927	−0.514635	+	0.857409i	$0.672073\pi$
$384$	0	0
$385$	2.72086	0.138668
$386$	13.8367	0.704270
$387$	0	0
$388$	16.9860	0.862334
$389$	21.1063	1.07013	0.535067	−	0.844810i	$-0.320286\pi$
$389$	21.1063	1.07013	0.535067	+	0.844810i	$0.320286\pi$
$390$	0	0
$391$	18.4436	0.932733
$392$	4.92856	0.248930
$393$	0	0
$394$	2.35913	0.118851
$395$	10.6874	0.537742
$396$	0	0
$397$	11.3506	0.569673	0.284836	−	0.958576i	$-0.408061\pi$
$397$	11.3506	0.569673	0.284836	+	0.958576i	$0.408061\pi$
$398$	−20.8726	−1.04625
$399$	0	0
$400$	−14.0740	−0.703702
$401$	−21.6751	−1.08240	−0.541200	−	0.840894i	$-0.682030\pi$
$401$	−21.6751	−1.08240	−0.541200	+	0.840894i	$0.682030\pi$
$402$	0	0
$403$	2.79396	0.139177
$404$	34.2340	1.70321
$405$	0	0
$406$	−14.9989	−0.744381
$407$	−2.00728	−0.0994973
$408$	0	0
$409$	24.7253	1.22259	0.611294	−	0.791404i	$-0.290649\pi$
$409$	24.7253	1.22259	0.611294	+	0.791404i	$0.290649\pi$
$410$	−3.44800	−0.170284
$411$	0	0
$412$	−3.06679	−0.151090
$413$	−8.13041	−0.400071
$414$	0	0
$415$	−18.2113	−0.893957
$416$	0.155019	0.00760045
$417$	0	0
$418$	7.81864	0.382422
$419$	0.0686235	0.00335248	0.00167624	−	0.999999i	$-0.499466\pi$
$419$	0.0686235	0.00335248	0.00167624	+	0.999999i	$0.499466\pi$
$420$	0	0
$421$	−11.2289	−0.547261	−0.273630	−	0.961835i	$-0.588224\pi$
$421$	−11.2289	−0.547261	−0.273630	+	0.961835i	$0.588224\pi$
$422$	11.3315	0.551607
$423$	0	0
$424$	−33.8227	−1.64257
$425$	24.4852	1.18771
$426$	0	0
$427$	5.09932	0.246773
$428$	−21.1099	−1.02039
$429$	0	0
$430$	−7.55661	−0.364412
$431$	−3.76067	−0.181145	−0.0905725	−	0.995890i	$-0.528870\pi$
$431$	−3.76067	−0.181145	−0.0905725	+	0.995890i	$0.528870\pi$
$432$	0	0
$433$	1.15019	0.0552748	0.0276374	−	0.999618i	$-0.491202\pi$
$433$	1.15019	0.0552748	0.0276374	+	0.999618i	$0.491202\pi$
$434$	4.49360	0.215700
$435$	0	0
$436$	−31.6409	−1.51533
$437$	3.79036	0.181318
$438$	0	0
$439$	10.1838	0.486046	0.243023	−	0.970021i	$-0.421861\pi$
$439$	10.1838	0.486046	0.243023	+	0.970021i	$0.421861\pi$
$440$	13.4099	0.639293
$441$	0	0
$442$	−26.4100	−1.25620
$443$	−3.18877	−0.151503	−0.0757514	−	0.997127i	$-0.524136\pi$
$443$	−3.18877	−0.151503	−0.0757514	+	0.997127i	$0.524136\pi$
$444$	0	0
$445$	3.25275	0.154195
$446$	−5.82567	−0.275854
$447$	0	0
$448$	−7.87502	−0.372060
$449$	−19.2378	−0.907889	−0.453945	−	0.891030i	$-0.649984\pi$
$449$	−19.2378	−0.907889	−0.453945	+	0.891030i	$0.649984\pi$
$450$	0	0
$451$	−2.49240	−0.117363
$452$	13.5937	0.639394
$453$	0	0
$454$	−58.4496	−2.74318
$455$	−1.88877	−0.0885468
$456$	0	0
$457$	9.07754	0.424629	0.212315	−	0.977201i	$-0.431900\pi$
$457$	9.07754	0.424629	0.212315	+	0.977201i	$0.431900\pi$
$458$	2.84175	0.132786
$459$	0	0
$460$	12.9684	0.604655
$461$	22.2184	1.03481	0.517407	−	0.855739i	$-0.326897\pi$
$461$	22.2184	1.03481	0.517407	+	0.855739i	$0.326897\pi$
$462$	0	0
$463$	−10.3963	−0.483156	−0.241578	−	0.970381i	$-0.577665\pi$
$463$	−10.3963	−0.483156	−0.241578	+	0.970381i	$0.577665\pi$
$464$	−24.8523	−1.15374
$465$	0	0
$466$	−2.32824	−0.107854
$467$	−10.0389	−0.464547	−0.232273	−	0.972651i	$-0.574616\pi$
$467$	−10.0389	−0.464547	−0.232273	+	0.972651i	$0.574616\pi$
$468$	0	0
$469$	−5.28798	−0.244176
$470$	1.88614	0.0870012
$471$	0	0
$472$	−40.0712	−1.84443
$473$	−5.46235	−0.251159
$474$	0	0
$475$	5.03198	0.230883
$476$	−28.3416	−1.29903
$477$	0	0
$478$	14.0090	0.640758
$479$	8.89671	0.406501	0.203251	−	0.979127i	$-0.434849\pi$
$479$	8.89671	0.406501	0.203251	+	0.979127i	$0.434849\pi$
$480$	0	0
$481$	1.39342	0.0635343
$482$	54.8938	2.50034
$483$	0	0
$484$	−24.7768	−1.12622
$485$	−5.24822	−0.238310
$486$	0	0
$487$	28.1080	1.27369	0.636847	−	0.770990i	$-0.280238\pi$
$487$	28.1080	1.27369	0.636847	+	0.770990i	$0.280238\pi$
$488$	25.1323	1.13769
$489$	0	0
$490$	−3.03775	−0.137232
$491$	3.08069	0.139030	0.0695149	−	0.997581i	$-0.477855\pi$
$491$	3.08069	0.139030	0.0695149	+	0.997581i	$0.477855\pi$
$492$	0	0
$493$	43.2366	1.94728
$494$	−5.42755	−0.244197
$495$	0	0
$496$	7.44565	0.334319
$497$	−4.94368	−0.221754
$498$	0	0
$499$	−3.72255	−0.166644	−0.0833221	−	0.996523i	$-0.526553\pi$
$499$	−3.72255	−0.166644	−0.0833221	+	0.996523i	$0.526553\pi$
$500$	42.0623	1.88108
$501$	0	0
$502$	3.77745	0.168596
$503$	−4.44735	−0.198298	−0.0991488	−	0.995073i	$-0.531612\pi$
$503$	−4.44735	−0.198298	−0.0991488	+	0.995073i	$0.531612\pi$
$504$	0	0
$505$	−10.5774	−0.470688
$506$	14.0494	0.624570
$507$	0	0
$508$	−4.01034	−0.177930
$509$	−6.22906	−0.276098	−0.138049	−	0.990425i	$-0.544083\pi$
$509$	−6.22906	−0.276098	−0.138049	+	0.990425i	$0.544083\pi$
$510$	0	0
$511$	3.24571	0.143582
$512$	−39.6282	−1.75133
$513$	0	0
$514$	9.47177	0.417782
$515$	0.947557	0.0417544
$516$	0	0
$517$	1.36341	0.0599627
$518$	2.24106	0.0984667
$519$	0	0
$520$	−9.30891	−0.408223
$521$	−11.1532	−0.488631	−0.244316	−	0.969696i	$-0.578563\pi$
$521$	−11.1532	−0.488631	−0.244316	+	0.969696i	$0.578563\pi$
$522$	0	0
$523$	10.4772	0.458134	0.229067	−	0.973411i	$-0.426432\pi$
$523$	10.4772	0.458134	0.229067	+	0.973411i	$0.426432\pi$
$524$	61.3186	2.67871
$525$	0	0
$526$	9.46067	0.412505
$527$	−12.9535	−0.564263
$528$	0	0
$529$	−16.1891	−0.703873
$530$	20.8468	0.905528
$531$	0	0
$532$	−5.82451	−0.252525
$533$	1.73018	0.0749424
$534$	0	0
$535$	6.52240	0.281988
$536$	−26.0621	−1.12571
$537$	0	0
$538$	−73.2131	−3.15644
$539$	−2.19586	−0.0945822
$540$	0	0
$541$	6.36150	0.273502	0.136751	−	0.990605i	$-0.456334\pi$
$541$	6.36150	0.273502	0.136751	+	0.990605i	$0.456334\pi$
$542$	77.0604	3.31003
$543$	0	0
$544$	−0.718708	−0.0308143
$545$	9.77620	0.418766
$546$	0	0
$547$	4.58101	0.195870	0.0979349	−	0.995193i	$-0.468776\pi$
$547$	4.58101	0.195870	0.0979349	+	0.995193i	$0.468776\pi$
$548$	−46.5137	−1.98697
$549$	0	0
$550$	18.6515	0.795303
$551$	8.88560	0.378539
$552$	0	0
$553$	−8.62522	−0.366781
$554$	16.5497	0.703127
$555$	0	0
$556$	−54.8791	−2.32739
$557$	−3.52897	−0.149527	−0.0747637	−	0.997201i	$-0.523820\pi$
$557$	−3.52897	−0.149527	−0.0747637	+	0.997201i	$0.523820\pi$
$558$	0	0
$559$	3.79185	0.160378
$560$	−5.03339	−0.212699
$561$	0	0
$562$	−17.7461	−0.748575
$563$	9.67953	0.407944	0.203972	−	0.978977i	$-0.434615\pi$
$563$	9.67953	0.407944	0.203972	+	0.978977i	$0.434615\pi$
$564$	0	0
$565$	−4.20009	−0.176699
$566$	29.5752	1.24314
$567$	0	0
$568$	−24.3652	−1.02234
$569$	26.8002	1.12352	0.561762	−	0.827299i	$-0.310124\pi$
$569$	26.8002	1.12352	0.561762	+	0.827299i	$0.310124\pi$
$570$	0	0
$571$	−1.99922	−0.0836649	−0.0418325	−	0.999125i	$-0.513320\pi$
$571$	−1.99922	−0.0836649	−0.0418325	+	0.999125i	$0.513320\pi$
$572$	−13.4234	−0.561260
$573$	0	0
$574$	2.78269	0.116147
$575$	9.04198	0.377077
$576$	0	0
$577$	−6.27902	−0.261399	−0.130699	−	0.991422i	$-0.541722\pi$
$577$	−6.27902	−0.261399	−0.130699	+	0.991422i	$0.541722\pi$
$578$	80.7661	3.35943
$579$	0	0
$580$	30.4013	1.26235
$581$	14.6973	0.609748
$582$	0	0
$583$	15.0693	0.624105
$584$	15.9967	0.661947
$585$	0	0
$586$	4.57551	0.189012
$587$	24.0066	0.990859	0.495429	−	0.868648i	$-0.335011\pi$
$587$	24.0066	0.990859	0.495429	+	0.868648i	$0.335011\pi$
$588$	0	0
$589$	−2.66209	−0.109689
$590$	24.6981	1.01681
$591$	0	0
$592$	3.71332	0.152617
$593$	−12.2147	−0.501599	−0.250799	−	0.968039i	$-0.580693\pi$
$593$	−12.2147	−0.501599	−0.250799	+	0.968039i	$0.580693\pi$
$594$	0	0
$595$	8.75679	0.358993
$596$	69.4352	2.84418
$597$	0	0
$598$	−9.75279	−0.398821
$599$	0.687692	0.0280983	0.0140492	−	0.999901i	$-0.495528\pi$
$599$	0.687692	0.0280983	0.0140492	+	0.999901i	$0.495528\pi$
$600$	0	0
$601$	1.55551	0.0634504	0.0317252	−	0.999497i	$-0.489900\pi$
$601$	1.55551	0.0634504	0.0317252	+	0.999497i	$0.489900\pi$
$602$	6.09853	0.248557
$603$	0	0
$604$	4.07328	0.165739
$605$	7.65536	0.311234
$606$	0	0
$607$	24.4768	0.993485	0.496742	−	0.867898i	$-0.334529\pi$
$607$	24.4768	0.993485	0.496742	+	0.867898i	$0.334529\pi$
$608$	−0.147702	−0.00599012
$609$	0	0
$610$	−15.4905	−0.627190
$611$	−0.946453	−0.0382894
$612$	0	0
$613$	−5.97120	−0.241174	−0.120587	−	0.992703i	$-0.538478\pi$
$613$	−5.97120	−0.241174	−0.120587	+	0.992703i	$0.538478\pi$
$614$	−7.77478	−0.313765
$615$	0	0
$616$	−10.8224	−0.436047
$617$	46.9374	1.88963	0.944814	−	0.327607i	$-0.106242\pi$
$617$	46.9374	1.88963	0.944814	+	0.327607i	$0.106242\pi$
$618$	0	0
$619$	11.9186	0.479047	0.239524	−	0.970891i	$-0.423009\pi$
$619$	11.9186	0.479047	0.239524	+	0.970891i	$0.423009\pi$
$620$	−9.10811	−0.365791
$621$	0	0
$622$	17.0943	0.685419
$623$	−2.62512	−0.105173
$624$	0	0
$625$	4.32717	0.173087
$626$	−24.7517	−0.989278
$627$	0	0
$628$	−30.7087	−1.22541
$629$	−6.46022	−0.257586
$630$	0	0
$631$	−30.4351	−1.21160	−0.605802	−	0.795615i	$-0.707148\pi$
$631$	−30.4351	−1.21160	−0.605802	+	0.795615i	$0.707148\pi$
$632$	−42.5099	−1.69095
$633$	0	0
$634$	−78.8298	−3.13073
$635$	1.23909	0.0491717
$636$	0	0
$637$	1.52432	0.0603958
$638$	32.9354	1.30392
$639$	0	0
$640$	24.1744	0.955575
$641$	40.1834	1.58715	0.793574	−	0.608474i	$-0.208218\pi$
$641$	40.1834	1.58715	0.793574	+	0.608474i	$0.208218\pi$
$642$	0	0
$643$	−22.8069	−0.899415	−0.449708	−	0.893176i	$-0.648472\pi$
$643$	−22.8069	−0.899415	−0.449708	+	0.893176i	$0.648472\pi$
$644$	−10.4661	−0.412422
$645$	0	0
$646$	25.1635	0.990043
$647$	8.81515	0.346559	0.173280	−	0.984873i	$-0.444564\pi$
$647$	8.81515	0.346559	0.173280	+	0.984873i	$0.444564\pi$
$648$	0	0
$649$	17.8532	0.700800
$650$	−12.9475	−0.507844
$651$	0	0
$652$	86.2899	3.37937
$653$	−12.8758	−0.503871	−0.251935	−	0.967744i	$-0.581067\pi$
$653$	−12.8758	−0.503871	−0.251935	+	0.967744i	$0.581067\pi$
$654$	0	0
$655$	−18.9458	−0.740273
$656$	4.61076	0.180020
$657$	0	0
$658$	−1.52220	−0.0593416
$659$	19.2643	0.750430	0.375215	−	0.926938i	$-0.377569\pi$
$659$	19.2643	0.750430	0.375215	+	0.926938i	$0.377569\pi$
$660$	0	0
$661$	30.2257	1.17565	0.587823	−	0.808990i	$-0.299985\pi$
$661$	30.2257	1.17565	0.587823	+	0.808990i	$0.299985\pi$
$662$	−59.7365	−2.32173
$663$	0	0
$664$	72.4366	2.81109
$665$	1.79962	0.0697862
$666$	0	0
$667$	15.9666	0.618228
$668$	61.4009	2.37567
$669$	0	0
$670$	16.0636	0.620589
$671$	−11.1974	−0.432270
$672$	0	0
$673$	−42.8167	−1.65046	−0.825232	−	0.564794i	$-0.808956\pi$
$673$	−42.8167	−1.65046	−0.825232	+	0.564794i	$0.808956\pi$
$674$	17.3274	0.667428
$675$	0	0
$676$	−42.8162	−1.64678
$677$	8.01207	0.307929	0.153964	−	0.988076i	$-0.450796\pi$
$677$	8.01207	0.307929	0.153964	+	0.988076i	$0.450796\pi$
$678$	0	0
$679$	4.23555	0.162546
$680$	43.1584	1.65505
$681$	0	0
$682$	−9.86729	−0.377838
$683$	−12.8698	−0.492451	−0.246225	−	0.969213i	$-0.579190\pi$
$683$	−12.8698	−0.492451	−0.246225	+	0.969213i	$0.579190\pi$
$684$	0	0
$685$	14.3715	0.549106
$686$	2.45160	0.0936025
$687$	0	0
$688$	10.1049	0.385247
$689$	−10.4608	−0.398524
$690$	0	0
$691$	−9.04215	−0.343980	−0.171990	−	0.985099i	$-0.555020\pi$
$691$	−9.04215	−0.343980	−0.171990	+	0.985099i	$0.555020\pi$
$692$	−103.558	−3.93668
$693$	0	0
$694$	−65.8438	−2.49940
$695$	16.9562	0.643184
$696$	0	0
$697$	−8.02153	−0.303837
$698$	−62.2367	−2.35569
$699$	0	0
$700$	−13.8945	−0.525162
$701$	−43.2487	−1.63348	−0.816741	−	0.577004i	$-0.804222\pi$
$701$	−43.2487	−1.63348	−0.816741	+	0.577004i	$0.804222\pi$
$702$	0	0
$703$	−1.32765	−0.0500731
$704$	17.2924	0.651732
$705$	0	0
$706$	−63.3523	−2.38430
$707$	8.53644	0.321046
$708$	0	0
$709$	21.8179	0.819388	0.409694	−	0.912223i	$-0.365635\pi$
$709$	21.8179	0.819388	0.409694	+	0.912223i	$0.365635\pi$
$710$	15.0177	0.563603
$711$	0	0
$712$	−12.9380	−0.484874
$713$	−4.78352	−0.179144
$714$	0	0
$715$	4.14746	0.155106
$716$	−20.0557	−0.749517
$717$	0	0
$718$	61.1815	2.28327
$719$	−2.71418	−0.101222	−0.0506109	−	0.998718i	$-0.516117\pi$
$719$	−2.71418	−0.101222	−0.0506109	+	0.998718i	$0.516117\pi$
$720$	0	0
$721$	−0.764721	−0.0284797
$722$	−41.4090	−1.54108
$723$	0	0
$724$	28.2097	1.04840
$725$	21.1968	0.787228
$726$	0	0
$727$	−3.24994	−0.120533	−0.0602667	−	0.998182i	$-0.519195\pi$
$727$	−3.24994	−0.120533	−0.0602667	+	0.998182i	$0.519195\pi$
$728$	7.51271	0.278439
$729$	0	0
$730$	−9.85966	−0.364922
$731$	−17.5800	−0.650218
$732$	0	0
$733$	22.8484	0.843925	0.421962	−	0.906613i	$-0.361341\pi$
$733$	22.8484	0.843925	0.421962	+	0.906613i	$0.361341\pi$
$734$	−29.0308	−1.07155
$735$	0	0
$736$	−0.265407	−0.00978303
$737$	11.6116	0.427720
$738$	0	0
$739$	37.8282	1.39153	0.695766	−	0.718269i	$-0.255065\pi$
$739$	37.8282	1.39153	0.695766	+	0.718269i	$0.255065\pi$
$740$	−4.54243	−0.166983
$741$	0	0
$742$	−16.8243	−0.617641
$743$	22.5284	0.826487	0.413244	−	0.910620i	$-0.364396\pi$
$743$	22.5284	0.826487	0.413244	+	0.910620i	$0.364396\pi$
$744$	0	0
$745$	−21.4536	−0.785999
$746$	−0.480690	−0.0175993
$747$	0	0
$748$	62.2341	2.27550
$749$	−5.26387	−0.192338
$750$	0	0
$751$	−38.7581	−1.41430	−0.707151	−	0.707063i	$-0.750020\pi$
$751$	−38.7581	−1.41430	−0.707151	+	0.707063i	$0.750020\pi$
$752$	−2.52221	−0.0919754
$753$	0	0
$754$	−22.8631	−0.832625
$755$	−1.25853	−0.0458027
$756$	0	0
$757$	0.582273	0.0211631	0.0105815	−	0.999944i	$-0.496632\pi$
$757$	0.582273	0.0211631	0.0105815	+	0.999944i	$0.496632\pi$
$758$	−59.0747	−2.14569
$759$	0	0
$760$	8.86952	0.321731
$761$	−33.7263	−1.22258	−0.611288	−	0.791408i	$-0.709348\pi$
$761$	−33.7263	−1.22258	−0.611288	+	0.791408i	$0.709348\pi$
$762$	0	0
$763$	−7.88983	−0.285631
$764$	38.1495	1.38020
$765$	0	0
$766$	−49.3831	−1.78428
$767$	−12.3934	−0.447498
$768$	0	0
$769$	31.0918	1.12120	0.560599	−	0.828087i	$-0.310571\pi$
$769$	31.0918	1.12120	0.560599	+	0.828087i	$0.310571\pi$
$770$	6.67046	0.240387
$771$	0	0
$772$	22.6342	0.814623
$773$	46.5433	1.67405	0.837023	−	0.547168i	$-0.184294\pi$
$773$	46.5433	1.67405	0.837023	+	0.547168i	$0.184294\pi$
$774$	0	0
$775$	−6.35046	−0.228115
$776$	20.8752	0.749375
$777$	0	0
$778$	51.7443	1.85512
$779$	−1.64851	−0.0590641
$780$	0	0
$781$	10.8556	0.388444
$782$	45.2163	1.61693
$783$	0	0
$784$	4.06217	0.145077
$785$	9.48816	0.338647
$786$	0	0
$787$	−37.7996	−1.34741	−0.673706	−	0.739000i	$-0.735299\pi$
$787$	−37.7996	−1.34741	−0.673706	+	0.739000i	$0.735299\pi$
$788$	3.85908	0.137474
$789$	0	0
$790$	26.2012	0.932198
$791$	3.38966	0.120522
$792$	0	0
$793$	7.77300	0.276027
$794$	27.8273	0.987552
$795$	0	0
$796$	−34.1435	−1.21018
$797$	−20.3689	−0.721504	−0.360752	−	0.932662i	$-0.617480\pi$
$797$	−20.3689	−0.721504	−0.360752	+	0.932662i	$0.617480\pi$
$798$	0	0
$799$	4.38798	0.155236
$800$	−0.352347	−0.0124573
$801$	0	0
$802$	−53.1386	−1.87639
$803$	−7.12711	−0.251510
$804$	0	0
$805$	3.23374	0.113974
$806$	6.84968	0.241270
$807$	0	0
$808$	42.0723	1.48010
$809$	−8.20215	−0.288372	−0.144186	−	0.989551i	$-0.546056\pi$
$809$	−8.20215	−0.288372	−0.144186	+	0.989551i	$0.546056\pi$
$810$	0	0
$811$	26.9018	0.944648	0.472324	−	0.881425i	$-0.343415\pi$
$811$	26.9018	0.944648	0.472324	+	0.881425i	$0.343415\pi$
$812$	−24.5352	−0.861018
$813$	0	0
$814$	−4.92105	−0.172483
$815$	−26.6613	−0.933903
$816$	0	0
$817$	−3.61288	−0.126399
$818$	60.6166	2.11941
$819$	0	0
$820$	−5.64025	−0.196966
$821$	20.6788	0.721694	0.360847	−	0.932625i	$-0.382488\pi$
$821$	20.6788	0.721694	0.360847	+	0.932625i	$0.382488\pi$
$822$	0	0
$823$	−16.4881	−0.574740	−0.287370	−	0.957820i	$-0.592781\pi$
$823$	−16.4881	−0.574740	−0.287370	+	0.957820i	$0.592781\pi$
$824$	−3.76897	−0.131298
$825$	0	0
$826$	−19.9325	−0.693541
$827$	−45.9300	−1.59714	−0.798571	−	0.601900i	$-0.794411\pi$
$827$	−45.9300	−1.59714	−0.798571	+	0.601900i	$0.794411\pi$
$828$	0	0
$829$	45.1979	1.56979	0.784894	−	0.619630i	$-0.212717\pi$
$829$	45.1979	1.56979	0.784894	+	0.619630i	$0.212717\pi$
$830$	−44.6468	−1.54971
$831$	0	0
$832$	−12.0041	−0.416166
$833$	−7.06712	−0.244861
$834$	0	0
$835$	−18.9712	−0.656527
$836$	12.7898	0.442344
$837$	0	0
$838$	0.168237	0.00581167
$839$	34.0233	1.17462	0.587308	−	0.809363i	$-0.300188\pi$
$839$	34.0233	1.17462	0.587308	+	0.809363i	$0.300188\pi$
$840$	0	0
$841$	8.42981	0.290683
$842$	−27.5287	−0.948700
$843$	0	0
$844$	18.5361	0.638038
$845$	13.2291	0.455094
$846$	0	0
$847$	−6.17822	−0.212286
$848$	−27.8770	−0.957300
$849$	0	0
$850$	60.0279	2.05894
$851$	−2.38565	−0.0817791
$852$	0	0
$853$	31.0563	1.06335	0.531674	−	0.846949i	$-0.321563\pi$
$853$	31.0563	1.06335	0.531674	+	0.846949i	$0.321563\pi$
$854$	12.5015	0.427792
$855$	0	0
$856$	−25.9433	−0.886724
$857$	−9.85200	−0.336538	−0.168269	−	0.985741i	$-0.553818\pi$
$857$	−9.85200	−0.336538	−0.168269	+	0.985741i	$0.553818\pi$
$858$	0	0
$859$	−34.6137	−1.18101	−0.590503	−	0.807036i	$-0.701070\pi$
$859$	−34.6137	−1.18101	−0.590503	+	0.807036i	$0.701070\pi$
$860$	−12.3612	−0.421512
$861$	0	0
$862$	−9.21965	−0.314023
$863$	26.6628	0.907613	0.453806	−	0.891100i	$-0.350066\pi$
$863$	26.6628	0.907613	0.453806	+	0.891100i	$0.350066\pi$
$864$	0	0
$865$	31.9966	1.08792
$866$	2.81982	0.0958212
$867$	0	0
$868$	7.35066	0.249497
$869$	18.9397	0.642486
$870$	0	0
$871$	−8.06058	−0.273122
$872$	−38.8855	−1.31683
$873$	0	0
$874$	9.29246	0.314322
$875$	10.4885	0.354575
$876$	0	0
$877$	20.7876	0.701948	0.350974	−	0.936385i	$-0.385851\pi$
$877$	20.7876	0.701948	0.350974	+	0.936385i	$0.385851\pi$
$878$	24.9666	0.842581
$879$	0	0
$880$	11.0526	0.372583
$881$	−27.9289	−0.940949	−0.470474	−	0.882414i	$-0.655917\pi$
$881$	−27.9289	−0.940949	−0.470474	+	0.882414i	$0.655917\pi$
$882$	0	0
$883$	38.9829	1.31188	0.655939	−	0.754814i	$-0.272273\pi$
$883$	38.9829	1.31188	0.655939	+	0.754814i	$0.272273\pi$
$884$	−43.2017	−1.45303
$885$	0	0
$886$	−7.81758	−0.262637
$887$	−3.83646	−0.128816	−0.0644079	−	0.997924i	$-0.520516\pi$
$887$	−3.83646	−0.128816	−0.0644079	+	0.997924i	$0.520516\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	7.97444	0.267304
$891$	0	0
$892$	−9.52967	−0.319077
$893$	0.901780	0.0301769
$894$	0	0
$895$	6.19668	0.207132
$896$	−19.5098	−0.651776
$897$	0	0
$898$	−47.1635	−1.57387
$899$	−11.2138	−0.374002
$900$	0	0
$901$	48.4988	1.61573
$902$	−6.11038	−0.203453
$903$	0	0
$904$	16.7061	0.555638
$905$	−8.71602	−0.289730
$906$	0	0
$907$	1.86310	0.0618631	0.0309315	−	0.999522i	$-0.490153\pi$
$907$	1.86310	0.0618631	0.0309315	+	0.999522i	$0.490153\pi$
$908$	−95.6123	−3.17301
$909$	0	0
$910$	−4.63051	−0.153500
$911$	−31.7046	−1.05042	−0.525210	−	0.850973i	$-0.676013\pi$
$911$	−31.7046	−1.05042	−0.525210	+	0.850973i	$0.676013\pi$
$912$	0	0
$913$	−32.2732	−1.06809
$914$	22.2545	0.736114
$915$	0	0
$916$	4.64855	0.153592
$917$	15.2901	0.504924
$918$	0	0
$919$	−6.20652	−0.204734	−0.102367	−	0.994747i	$-0.532642\pi$
$919$	−6.20652	−0.204734	−0.102367	+	0.994747i	$0.532642\pi$
$920$	15.9377	0.525450
$921$	0	0
$922$	54.4707	1.79390
$923$	−7.53575	−0.248042
$924$	0	0
$925$	−3.16712	−0.104134
$926$	−25.4875	−0.837572
$927$	0	0
$928$	−0.622183	−0.0204242
$929$	−11.9844	−0.393196	−0.196598	−	0.980484i	$-0.562989\pi$
$929$	−11.9844	−0.393196	−0.196598	+	0.980484i	$0.562989\pi$
$930$	0	0
$931$	−1.45237	−0.0475996
$932$	−3.80855	−0.124753
$933$	0	0
$934$	−24.6115	−0.805312
$935$	−19.2287	−0.628844
$936$	0	0
$937$	18.5854	0.607159	0.303580	−	0.952806i	$-0.401818\pi$
$937$	18.5854	0.607159	0.303580	+	0.952806i	$0.401818\pi$
$938$	−12.9640	−0.423290
$939$	0	0
$940$	3.08536	0.100633
$941$	−29.4475	−0.959962	−0.479981	−	0.877279i	$-0.659357\pi$
$941$	−29.4475	−0.959962	−0.479981	+	0.877279i	$0.659357\pi$
$942$	0	0
$943$	−2.96222	−0.0964632
$944$	−33.0271	−1.07494
$945$	0	0
$946$	−13.3915	−0.435395
$947$	−1.05890	−0.0344096	−0.0172048	−	0.999852i	$-0.505477\pi$
$947$	−1.05890	−0.0344096	−0.0172048	+	0.999852i	$0.505477\pi$
$948$	0	0
$949$	4.94751	0.160603
$950$	12.3364	0.400245
$951$	0	0
$952$	−34.8307	−1.12887
$953$	57.6761	1.86831	0.934156	−	0.356864i	$-0.116154\pi$
$953$	57.6761	1.86831	0.934156	+	0.356864i	$0.116154\pi$
$954$	0	0
$955$	−11.7872	−0.381424
$956$	22.9161	0.741158
$957$	0	0
$958$	21.8112	0.704687
$959$	−11.5984	−0.374533
$960$	0	0
$961$	−27.6404	−0.891625
$962$	3.41610	0.110139
$963$	0	0
$964$	89.7957	2.89212
$965$	−6.99336	−0.225124
$966$	0	0
$967$	17.5642	0.564827	0.282414	−	0.959293i	$-0.408865\pi$
$967$	17.5642	0.564827	0.282414	+	0.959293i	$0.408865\pi$
$968$	−30.4497	−0.978690
$969$	0	0
$970$	−12.8665	−0.413120
$971$	43.8415	1.40694	0.703470	−	0.710725i	$-0.251633\pi$
$971$	43.8415	1.40694	0.703470	+	0.710725i	$0.251633\pi$
$972$	0	0
$973$	−13.6844	−0.438701
$974$	68.9095	2.20800
$975$	0	0
$976$	20.7143	0.663049
$977$	13.1600	0.421027	0.210513	−	0.977591i	$-0.432486\pi$
$977$	13.1600	0.421027	0.210513	+	0.977591i	$0.432486\pi$
$978$	0	0
$979$	5.76438	0.184230
$980$	−4.96917	−0.158734
$981$	0	0
$982$	7.55263	0.241014
$983$	49.4485	1.57716	0.788581	−	0.614931i	$-0.210816\pi$
$983$	49.4485	1.57716	0.788581	+	0.614931i	$0.210816\pi$
$984$	0	0
$985$	−1.19235	−0.0379915
$986$	105.999	3.37569
$987$	0	0
$988$	−8.87843	−0.282460
$989$	−6.49199	−0.206433
$990$	0	0
$991$	−1.17880	−0.0374459	−0.0187229	−	0.999825i	$-0.505960\pi$
$991$	−1.17880	−0.0374459	−0.0187229	+	0.999825i	$0.505960\pi$
$992$	0.186403	0.00591832
$993$	0	0
$994$	−12.1199	−0.384421
$995$	10.5494	0.334439
$996$	0	0
$997$	−6.51368	−0.206290	−0.103145	−	0.994666i	$-0.532891\pi$
$997$	−6.51368	−0.206290	−0.103145	+	0.994666i	$0.532891\pi$
$998$	−9.12621	−0.288885
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.o.1.13		13
3.2	odd	2		2667.2.a.l.1.1	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.1	✓	13	3.2	odd	2
8001.2.a.o.1.13		13	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+2.45160 q^{2} +4.01034 q^{4} -1.23909 q^{5} +1.00000 q^{7} +4.92856 q^{8} +O(q^{10})\)	\(q+2.45160 q^{2} +4.01034 q^{4} -1.23909 q^{5} +1.00000 q^{7} +4.92856 q^{8} -3.03775 q^{10} -2.19586 q^{11} +1.52432 q^{13} +2.45160 q^{14} +4.06217 q^{16} -7.06712 q^{17} -1.45237 q^{19} -4.96917 q^{20} -5.38336 q^{22} -2.60977 q^{23} -3.46466 q^{25} +3.73703 q^{26} +4.01034 q^{28} -6.11799 q^{29} +1.83292 q^{31} +0.101697 q^{32} -17.3258 q^{34} -1.23909 q^{35} +0.914123 q^{37} -3.56064 q^{38} -6.10692 q^{40} +1.13505 q^{41} +2.48757 q^{43} -8.80614 q^{44} -6.39812 q^{46} -0.620901 q^{47} +1.00000 q^{49} -8.49396 q^{50} +6.11305 q^{52} -6.86259 q^{53} +2.72086 q^{55} +4.92856 q^{56} -14.9989 q^{58} -8.13041 q^{59} +5.09932 q^{61} +4.49360 q^{62} -7.87502 q^{64} -1.88877 q^{65} -5.28798 q^{67} -28.3416 q^{68} -3.03775 q^{70} -4.94368 q^{71} +3.24571 q^{73} +2.24106 q^{74} -5.82451 q^{76} -2.19586 q^{77} -8.62522 q^{79} -5.03339 q^{80} +2.78269 q^{82} +14.6973 q^{83} +8.75679 q^{85} +6.09853 q^{86} -10.8224 q^{88} -2.62512 q^{89} +1.52432 q^{91} -10.4661 q^{92} -1.52220 q^{94} +1.79962 q^{95} +4.23555 q^{97} +2.45160 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more