LMFDB - Embedded newform 8001.2.a.q.1.12

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.32572$ 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)

$f(q)$	$=$	$q+1.32572 q^{2} -0.242472 q^{4} +1.07970 q^{5} -1.00000 q^{7} -2.97289 q^{8} +O(q^{10})$	\(q+1.32572 q^{2} -0.242472 q^{4} +1.07970 q^{5} -1.00000 q^{7} -2.97289 q^{8} +1.43138 q^{10} +4.78334 q^{11} -3.48769 q^{13} -1.32572 q^{14} -3.45626 q^{16} +2.77181 q^{17} -2.57728 q^{19} -0.261797 q^{20} +6.34136 q^{22} +0.614993 q^{23} -3.83424 q^{25} -4.62370 q^{26} +0.242472 q^{28} +7.90799 q^{29} -3.61669 q^{31} +1.36374 q^{32} +3.67463 q^{34} -1.07970 q^{35} -6.08123 q^{37} -3.41675 q^{38} -3.20983 q^{40} -4.86701 q^{41} -1.75805 q^{43} -1.15983 q^{44} +0.815308 q^{46} -8.64126 q^{47} +1.00000 q^{49} -5.08313 q^{50} +0.845667 q^{52} +8.77331 q^{53} +5.16458 q^{55} +2.97289 q^{56} +10.4838 q^{58} -14.8356 q^{59} +3.37035 q^{61} -4.79471 q^{62} +8.72046 q^{64} -3.76567 q^{65} -0.747683 q^{67} -0.672085 q^{68} -1.43138 q^{70} -9.58491 q^{71} +0.421112 q^{73} -8.06200 q^{74} +0.624918 q^{76} -4.78334 q^{77} +10.1734 q^{79} -3.73173 q^{80} -6.45228 q^{82} +10.5240 q^{83} +2.99273 q^{85} -2.33068 q^{86} -14.2203 q^{88} +7.71591 q^{89} +3.48769 q^{91} -0.149119 q^{92} -11.4559 q^{94} -2.78270 q^{95} -12.5382 q^{97} +1.32572 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * 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 $

$2$	1.32572	0.937424	0.468712	−	0.883351i	$-0.344718\pi$
$2$	1.32572	0.937424	0.468712	+	0.883351i	$0.344718\pi$
$3$	0	0
$3$	0	0
$4$	−0.242472	−0.121236
$5$	1.07970	0.482857	0.241429	−	0.970419i	$-0.422384\pi$
$5$	1.07970	0.482857	0.241429	+	0.970419i	$0.422384\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.97289	−1.05107
$9$	0	0
$10$	1.43138	0.452642
$11$	4.78334	1.44223	0.721115	−	0.692815i	$-0.243630\pi$
$11$	4.78334	1.44223	0.721115	+	0.692815i	$0.243630\pi$
$12$	0	0
$13$	−3.48769	−0.967312	−0.483656	−	0.875258i	$-0.660691\pi$
$13$	−3.48769	−0.967312	−0.483656	+	0.875258i	$0.660691\pi$
$14$	−1.32572	−0.354313
$15$	0	0
$16$	−3.45626	−0.864066
$17$	2.77181	0.672262	0.336131	−	0.941815i	$-0.390882\pi$
$17$	2.77181	0.672262	0.336131	+	0.941815i	$0.390882\pi$
$18$	0	0
$19$	−2.57728	−0.591269	−0.295634	−	0.955301i	$-0.595531\pi$
$19$	−2.57728	−0.591269	−0.295634	+	0.955301i	$0.595531\pi$
$20$	−0.261797	−0.0585397
$21$	0	0
$22$	6.34136	1.35198
$23$	0.614993	0.128235	0.0641175	−	0.997942i	$-0.479577\pi$
$23$	0.614993	0.128235	0.0641175	+	0.997942i	$0.479577\pi$
$24$	0	0
$25$	−3.83424	−0.766849
$26$	−4.62370	−0.906781
$27$	0	0
$28$	0.242472	0.0458229
$29$	7.90799	1.46848	0.734238	−	0.678892i	$-0.237539\pi$
$29$	7.90799	1.46848	0.734238	+	0.678892i	$0.237539\pi$
$30$	0	0
$31$	−3.61669	−0.649576	−0.324788	−	0.945787i	$-0.605293\pi$
$31$	−3.61669	−0.649576	−0.324788	+	0.945787i	$0.605293\pi$
$32$	1.36374	0.241077
$33$	0	0
$34$	3.67463	0.630195
$35$	−1.07970	−0.182503
$36$	0	0
$37$	−6.08123	−0.999749	−0.499874	−	0.866098i	$-0.666620\pi$
$37$	−6.08123	−0.999749	−0.499874	+	0.866098i	$0.666620\pi$
$38$	−3.41675	−0.554270
$39$	0	0
$40$	−3.20983	−0.507519
$41$	−4.86701	−0.760099	−0.380050	−	0.924966i	$-0.624093\pi$
$41$	−4.86701	−0.760099	−0.380050	+	0.924966i	$0.624093\pi$
$42$	0	0
$43$	−1.75805	−0.268100	−0.134050	−	0.990975i	$-0.542798\pi$
$43$	−1.75805	−0.268100	−0.134050	+	0.990975i	$0.542798\pi$
$44$	−1.15983	−0.174850
$45$	0	0
$46$	0.815308	0.120211
$47$	−8.64126	−1.26046	−0.630229	−	0.776409i	$-0.717039\pi$
$47$	−8.64126	−1.26046	−0.630229	+	0.776409i	$0.717039\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.08313	−0.718862
$51$	0	0
$52$	0.845667	0.117273
$53$	8.77331	1.20511	0.602553	−	0.798079i	$-0.294150\pi$
$53$	8.77331	1.20511	0.602553	+	0.798079i	$0.294150\pi$
$54$	0	0
$55$	5.16458	0.696392
$56$	2.97289	0.397269
$57$	0	0
$58$	10.4838	1.37659
$59$	−14.8356	−1.93142	−0.965712	−	0.259614i	$-0.916405\pi$
$59$	−14.8356	−1.93142	−0.965712	+	0.259614i	$0.916405\pi$
$60$	0	0
$61$	3.37035	0.431529	0.215764	−	0.976445i	$-0.430776\pi$
$61$	3.37035	0.431529	0.215764	+	0.976445i	$0.430776\pi$
$62$	−4.79471	−0.608928
$63$	0	0
$64$	8.72046	1.09006
$65$	−3.76567	−0.467074
$66$	0	0
$67$	−0.747683	−0.0913441	−0.0456720	−	0.998956i	$-0.514543\pi$
$67$	−0.747683	−0.0913441	−0.0456720	+	0.998956i	$0.514543\pi$
$68$	−0.672085	−0.0815023
$69$	0	0
$70$	−1.43138	−0.171083
$71$	−9.58491	−1.13752	−0.568760	−	0.822504i	$-0.692577\pi$
$71$	−9.58491	−1.13752	−0.568760	+	0.822504i	$0.692577\pi$
$72$	0	0
$73$	0.421112	0.0492874	0.0246437	−	0.999696i	$-0.492155\pi$
$73$	0.421112	0.0492874	0.0246437	+	0.999696i	$0.492155\pi$
$74$	−8.06200	−0.937189
$75$	0	0
$76$	0.624918	0.0716831
$77$	−4.78334	−0.545112
$78$	0	0
$79$	10.1734	1.14460	0.572298	−	0.820046i	$-0.306052\pi$
$79$	10.1734	1.14460	0.572298	+	0.820046i	$0.306052\pi$
$80$	−3.73173	−0.417221
$81$	0	0
$82$	−6.45228	−0.712535
$83$	10.5240	1.15516	0.577579	−	0.816335i	$-0.303998\pi$
$83$	10.5240	1.15516	0.577579	+	0.816335i	$0.303998\pi$
$84$	0	0
$85$	2.99273	0.324607
$86$	−2.33068	−0.251323
$87$	0	0
$88$	−14.2203	−1.51589
$89$	7.71591	0.817885	0.408943	−	0.912560i	$-0.365898\pi$
$89$	7.71591	0.817885	0.408943	+	0.912560i	$0.365898\pi$
$90$	0	0
$91$	3.48769	0.365609
$92$	−0.149119	−0.0155467
$93$	0	0
$94$	−11.4559	−1.18158
$95$	−2.78270	−0.285499
$96$	0	0
$97$	−12.5382	−1.27306	−0.636532	−	0.771251i	$-0.719632\pi$
$97$	−12.5382	−1.27306	−0.636532	+	0.771251i	$0.719632\pi$
$98$	1.32572	0.133918
$99$	0	0
$100$	0.929696	0.0929696
$101$	−12.9232	−1.28591	−0.642955	−	0.765904i	$-0.722292\pi$
$101$	−12.9232	−1.28591	−0.642955	+	0.765904i	$0.722292\pi$
$102$	0	0
$103$	−0.717482	−0.0706956	−0.0353478	−	0.999375i	$-0.511254\pi$
$103$	−0.717482	−0.0706956	−0.0353478	+	0.999375i	$0.511254\pi$
$104$	10.3685	1.01672
$105$	0	0
$106$	11.6309	1.12970
$107$	10.4070	1.00608	0.503042	−	0.864262i	$-0.332214\pi$
$107$	10.4070	1.00608	0.503042	+	0.864262i	$0.332214\pi$
$108$	0	0
$109$	−7.63503	−0.731303	−0.365651	−	0.930752i	$-0.619154\pi$
$109$	−7.63503	−0.731303	−0.365651	+	0.930752i	$0.619154\pi$
$110$	6.84677	0.652814
$111$	0	0
$112$	3.45626	0.326586
$113$	−12.8912	−1.21270	−0.606352	−	0.795196i	$-0.707368\pi$
$113$	−12.8912	−1.21270	−0.606352	+	0.795196i	$0.707368\pi$
$114$	0	0
$115$	0.664009	0.0619192
$116$	−1.91747	−0.178032
$117$	0	0
$118$	−19.6678	−1.81056
$119$	−2.77181	−0.254091
$120$	0	0
$121$	11.8803	1.08003
$122$	4.46813	0.404525
$123$	0	0
$124$	0.876945	0.0787520
$125$	−9.53835	−0.853136
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	8.83339	0.780769
$129$	0	0
$130$	−4.99221	−0.437846
$131$	0.311413	0.0272083	0.0136041	−	0.999907i	$-0.495670\pi$
$131$	0.311413	0.0272083	0.0136041	+	0.999907i	$0.495670\pi$
$132$	0	0
$133$	2.57728	0.223479
$134$	−0.991217	−0.0856281
$135$	0	0
$136$	−8.24026	−0.706597
$137$	8.75520	0.748007	0.374004	−	0.927427i	$-0.377985\pi$
$137$	8.75520	0.748007	0.374004	+	0.927427i	$0.377985\pi$
$138$	0	0
$139$	9.08402	0.770497	0.385248	−	0.922813i	$-0.374116\pi$
$139$	9.08402	0.770497	0.385248	+	0.922813i	$0.374116\pi$
$140$	0.261797	0.0221259
$141$	0	0
$142$	−12.7069	−1.06634
$143$	−16.6828	−1.39509
$144$	0	0
$145$	8.53827	0.709065
$146$	0.558275	0.0462032
$147$	0	0
$148$	1.47453	0.121205
$149$	−12.2412	−1.00284	−0.501420	−	0.865204i	$-0.667189\pi$
$149$	−12.2412	−1.00284	−0.501420	+	0.865204i	$0.667189\pi$
$150$	0	0
$151$	−20.4028	−1.66036	−0.830178	−	0.557498i	$-0.811761\pi$
$151$	−20.4028	−1.66036	−0.830178	+	0.557498i	$0.811761\pi$
$152$	7.66196	0.621467
$153$	0	0
$154$	−6.34136	−0.511001
$155$	−3.90494	−0.313653
$156$	0	0
$157$	−16.6195	−1.32638	−0.663191	−	0.748450i	$-0.730798\pi$
$157$	−16.6195	−1.32638	−0.663191	+	0.748450i	$0.730798\pi$
$158$	13.4871	1.07297
$159$	0	0
$160$	1.47243	0.116406
$161$	−0.614993	−0.0484683
$162$	0	0
$163$	−17.5880	−1.37760	−0.688800	−	0.724951i	$-0.741862\pi$
$163$	−17.5880	−1.37760	−0.688800	+	0.724951i	$0.741862\pi$
$164$	1.18011	0.0921513
$165$	0	0
$166$	13.9518	1.08287
$167$	−13.0374	−1.00886	−0.504432	−	0.863452i	$-0.668298\pi$
$167$	−13.0374	−1.00886	−0.504432	+	0.863452i	$0.668298\pi$
$168$	0	0
$169$	−0.836009	−0.0643084
$170$	3.96751	0.304294
$171$	0	0
$172$	0.426277	0.0325034
$173$	2.62626	0.199671	0.0998355	−	0.995004i	$-0.468168\pi$
$173$	2.62626	0.199671	0.0998355	+	0.995004i	$0.468168\pi$
$174$	0	0
$175$	3.83424	0.289842
$176$	−16.5325	−1.24618
$177$	0	0
$178$	10.2291	0.766705
$179$	−18.5152	−1.38389	−0.691945	−	0.721950i	$-0.743246\pi$
$179$	−18.5152	−1.38389	−0.691945	+	0.721950i	$0.743246\pi$
$180$	0	0
$181$	2.58461	0.192112	0.0960561	−	0.995376i	$-0.469377\pi$
$181$	2.58461	0.192112	0.0960561	+	0.995376i	$0.469377\pi$
$182$	4.62370	0.342731
$183$	0	0
$184$	−1.82830	−0.134784
$185$	−6.56592	−0.482736
$186$	0	0
$187$	13.2585	0.969557
$188$	2.09526	0.152813
$189$	0	0
$190$	−3.68907	−0.267633
$191$	−15.0100	−1.08608	−0.543041	−	0.839706i	$-0.682727\pi$
$191$	−15.0100	−1.08608	−0.543041	+	0.839706i	$0.682727\pi$
$192$	0	0
$193$	6.82279	0.491115	0.245558	−	0.969382i	$-0.421029\pi$
$193$	6.82279	0.491115	0.245558	+	0.969382i	$0.421029\pi$
$194$	−16.6221	−1.19340
$195$	0	0
$196$	−0.242472	−0.0173194
$197$	4.95055	0.352712	0.176356	−	0.984326i	$-0.443569\pi$
$197$	4.95055	0.352712	0.176356	+	0.984326i	$0.443569\pi$
$198$	0	0
$199$	−4.98488	−0.353368	−0.176684	−	0.984268i	$-0.556537\pi$
$199$	−4.98488	−0.353368	−0.176684	+	0.984268i	$0.556537\pi$
$200$	11.3988	0.806014
$201$	0	0
$202$	−17.1326	−1.20544
$203$	−7.90799	−0.555032
$204$	0	0
$205$	−5.25492	−0.367019
$206$	−0.951178	−0.0662717
$207$	0	0
$208$	12.0544	0.835821
$209$	−12.3280	−0.852746
$210$	0	0
$211$	−10.6537	−0.733432	−0.366716	−	0.930333i	$-0.619518\pi$
$211$	−10.6537	−0.733432	−0.366716	+	0.930333i	$0.619518\pi$
$212$	−2.12728	−0.146102
$213$	0	0
$214$	13.7968	0.943128
$215$	−1.89817	−0.129454
$216$	0	0
$217$	3.61669	0.245517
$218$	−10.1219	−0.685541
$219$	0	0
$220$	−1.25227	−0.0844277
$221$	−9.66721	−0.650287
$222$	0	0
$223$	−18.4982	−1.23873	−0.619367	−	0.785102i	$-0.712611\pi$
$223$	−18.4982	−1.23873	−0.619367	+	0.785102i	$0.712611\pi$
$224$	−1.36374	−0.0911187
$225$	0	0
$226$	−17.0901	−1.13682
$227$	−13.5842	−0.901618	−0.450809	−	0.892620i	$-0.648864\pi$
$227$	−13.5842	−0.901618	−0.450809	+	0.892620i	$0.648864\pi$
$228$	0	0
$229$	−27.7830	−1.83595	−0.917976	−	0.396635i	$-0.870178\pi$
$229$	−27.7830	−1.83595	−0.917976	+	0.396635i	$0.870178\pi$
$230$	0.880289	0.0580446
$231$	0	0
$232$	−23.5095	−1.54348
$233$	20.0906	1.31618	0.658090	−	0.752939i	$-0.271365\pi$
$233$	20.0906	1.31618	0.658090	+	0.752939i	$0.271365\pi$
$234$	0	0
$235$	−9.32999	−0.608621
$236$	3.59721	0.234158
$237$	0	0
$238$	−3.67463	−0.238191
$239$	0.00392957	0.000254183 0	0.000127091	−	1.00000i	$-0.499960\pi$
$239$	0.00392957	0.000254183 0	0.000127091		1.00000i	$0.499960\pi$
$240$	0	0
$241$	22.2894	1.43579	0.717895	−	0.696152i	$-0.245106\pi$
$241$	22.2894	1.43579	0.717895	+	0.696152i	$0.245106\pi$
$242$	15.7499	1.01244
$243$	0	0
$244$	−0.817214	−0.0523168
$245$	1.07970	0.0689796
$246$	0	0
$247$	8.98876	0.571941
$248$	10.7520	0.682752
$249$	0	0
$250$	−12.6452	−0.799750
$251$	27.9133	1.76187	0.880937	−	0.473234i	$-0.156913\pi$
$251$	27.9133	1.76187	0.880937	+	0.473234i	$0.156913\pi$
$252$	0	0
$253$	2.94172	0.184944
$254$	1.32572	0.0831829
$255$	0	0
$256$	−5.73034	−0.358146
$257$	−26.5246	−1.65456	−0.827281	−	0.561788i	$-0.810114\pi$
$257$	−26.5246	−1.65456	−0.827281	+	0.561788i	$0.810114\pi$
$258$	0	0
$259$	6.08123	0.377869
$260$	0.913069	0.0566261
$261$	0	0
$262$	0.412846	0.0255057
$263$	22.8853	1.41117	0.705584	−	0.708626i	$-0.250685\pi$
$263$	22.8853	1.41117	0.705584	+	0.708626i	$0.250685\pi$
$264$	0	0
$265$	9.47256	0.581895
$266$	3.41675	0.209494
$267$	0	0
$268$	0.181292	0.0110742
$269$	−5.61271	−0.342213	−0.171107	−	0.985253i	$-0.554734\pi$
$269$	−5.61271	−0.342213	−0.171107	+	0.985253i	$0.554734\pi$
$270$	0	0
$271$	24.5734	1.49273	0.746363	−	0.665539i	$-0.231798\pi$
$271$	24.5734	1.49273	0.746363	+	0.665539i	$0.231798\pi$
$272$	−9.58009	−0.580879
$273$	0	0
$274$	11.6069	0.701200
$275$	−18.3405	−1.10597
$276$	0	0
$277$	13.3210	0.800379	0.400190	−	0.916432i	$-0.368944\pi$
$277$	13.3210	0.800379	0.400190	+	0.916432i	$0.368944\pi$
$278$	12.0429	0.722282
$279$	0	0
$280$	3.20983	0.191824
$281$	−9.81012	−0.585223	−0.292611	−	0.956231i	$-0.594524\pi$
$281$	−9.81012	−0.585223	−0.292611	+	0.956231i	$0.594524\pi$
$282$	0	0
$283$	−0.859898	−0.0511156	−0.0255578	−	0.999673i	$-0.508136\pi$
$283$	−0.859898	−0.0511156	−0.0255578	+	0.999673i	$0.508136\pi$
$284$	2.32407	0.137908
$285$	0	0
$286$	−22.1167	−1.30779
$287$	4.86701	0.287290
$288$	0	0
$289$	−9.31709	−0.548064
$290$	11.3193	0.664695
$291$	0	0
$292$	−0.102108	−0.00597541
$293$	8.97464	0.524304	0.262152	−	0.965027i	$-0.415568\pi$
$293$	8.97464	0.524304	0.262152	+	0.965027i	$0.415568\pi$
$294$	0	0
$295$	−16.0180	−0.932603
$296$	18.0788	1.05081
$297$	0	0
$298$	−16.2284	−0.940087
$299$	−2.14491	−0.124043
$300$	0	0
$301$	1.75805	0.101332
$302$	−27.0484	−1.55646
$303$	0	0
$304$	8.90776	0.510895
$305$	3.63897	0.208367
$306$	0	0
$307$	16.7241	0.954494	0.477247	−	0.878769i	$-0.341635\pi$
$307$	16.7241	0.954494	0.477247	+	0.878769i	$0.341635\pi$
$308$	1.15983	0.0660872
$309$	0	0
$310$	−5.17685	−0.294025
$311$	−30.0754	−1.70542	−0.852709	−	0.522386i	$-0.825042\pi$
$311$	−30.0754	−1.70542	−0.852709	+	0.522386i	$0.825042\pi$
$312$	0	0
$313$	−14.7111	−0.831520	−0.415760	−	0.909474i	$-0.636484\pi$
$313$	−14.7111	−0.831520	−0.415760	+	0.909474i	$0.636484\pi$
$314$	−22.0328	−1.24338
$315$	0	0
$316$	−2.46676	−0.138766
$317$	0.868539	0.0487820	0.0243910	−	0.999702i	$-0.492235\pi$
$317$	0.868539	0.0487820	0.0243910	+	0.999702i	$0.492235\pi$
$318$	0	0
$319$	37.8266	2.11788
$320$	9.41550	0.526342
$321$	0	0
$322$	−0.815308	−0.0454353
$323$	−7.14373	−0.397488
$324$	0	0
$325$	13.3727	0.741782
$326$	−23.3168	−1.29140
$327$	0	0
$328$	14.4691	0.798920
$329$	8.64126	0.476408
$330$	0	0
$331$	0.458447	0.0251985	0.0125993	−	0.999921i	$-0.495989\pi$
$331$	0.458447	0.0251985	0.0125993	+	0.999921i	$0.495989\pi$
$332$	−2.55177	−0.140047
$333$	0	0
$334$	−17.2839	−0.945733
$335$	−0.807275	−0.0441062
$336$	0	0
$337$	20.3840	1.11039	0.555194	−	0.831721i	$-0.312644\pi$
$337$	20.3840	1.11039	0.555194	+	0.831721i	$0.312644\pi$
$338$	−1.10831	−0.0602842
$339$	0	0
$340$	−0.725652	−0.0393540
$341$	−17.2998	−0.936838
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	5.22648	0.281793
$345$	0	0
$346$	3.48168	0.187176
$347$	−25.4635	−1.36695	−0.683476	−	0.729973i	$-0.739533\pi$
$347$	−25.4635	−1.36695	−0.683476	+	0.729973i	$0.739533\pi$
$348$	0	0
$349$	−14.1799	−0.759031	−0.379516	−	0.925185i	$-0.623909\pi$
$349$	−14.1799	−0.759031	−0.379516	+	0.925185i	$0.623909\pi$
$350$	5.08313	0.271704
$351$	0	0
$352$	6.52323	0.347689
$353$	21.3784	1.13786	0.568928	−	0.822388i	$-0.307358\pi$
$353$	21.3784	1.13786	0.568928	+	0.822388i	$0.307358\pi$
$354$	0	0
$355$	−10.3488	−0.549260
$356$	−1.87089	−0.0991571
$357$	0	0
$358$	−24.5459	−1.29729
$359$	20.7681	1.09610	0.548049	−	0.836446i	$-0.315371\pi$
$359$	20.7681	1.09610	0.548049	+	0.836446i	$0.315371\pi$
$360$	0	0
$361$	−12.3576	−0.650401
$362$	3.42646	0.180091
$363$	0	0
$364$	−0.845667	−0.0443250
$365$	0.454675	0.0237988
$366$	0	0
$367$	−4.93228	−0.257463	−0.128731	−	0.991679i	$-0.541091\pi$
$367$	−4.93228	−0.257463	−0.128731	+	0.991679i	$0.541091\pi$
$368$	−2.12558	−0.110803
$369$	0	0
$370$	−8.70456	−0.452528
$371$	−8.77331	−0.455487
$372$	0	0
$373$	−13.0488	−0.675644	−0.337822	−	0.941210i	$-0.609690\pi$
$373$	−13.0488	−0.675644	−0.337822	+	0.941210i	$0.609690\pi$
$374$	17.5770	0.908886
$375$	0	0
$376$	25.6895	1.32483
$377$	−27.5806	−1.42047
$378$	0	0
$379$	−18.7555	−0.963404	−0.481702	−	0.876335i	$-0.659981\pi$
$379$	−18.7555	−0.963404	−0.481702	+	0.876335i	$0.659981\pi$
$380$	0.674726	0.0346127
$381$	0	0
$382$	−19.8990	−1.01812
$383$	−6.00562	−0.306873	−0.153436	−	0.988159i	$-0.549034\pi$
$383$	−6.00562	−0.306873	−0.153436	+	0.988159i	$0.549034\pi$
$384$	0	0
$385$	−5.16458	−0.263211
$386$	9.04510	0.460384
$387$	0	0
$388$	3.04017	0.154341
$389$	16.9632	0.860069	0.430034	−	0.902812i	$-0.358501\pi$
$389$	16.9632	0.860069	0.430034	+	0.902812i	$0.358501\pi$
$390$	0	0
$391$	1.70464	0.0862075
$392$	−2.97289	−0.150153
$393$	0	0
$394$	6.56304	0.330641
$395$	10.9842	0.552677
$396$	0	0
$397$	2.47989	0.124462	0.0622310	−	0.998062i	$-0.480178\pi$
$397$	2.47989	0.124462	0.0622310	+	0.998062i	$0.480178\pi$
$398$	−6.60854	−0.331256
$399$	0	0
$400$	13.2522	0.662608
$401$	23.0060	1.14886	0.574432	−	0.818552i	$-0.305223\pi$
$401$	23.0060	1.14886	0.574432	+	0.818552i	$0.305223\pi$
$402$	0	0
$403$	12.6139	0.628342
$404$	3.13352	0.155899
$405$	0	0
$406$	−10.4838	−0.520301
$407$	−29.0886	−1.44187
$408$	0	0
$409$	10.9306	0.540482	0.270241	−	0.962793i	$-0.412897\pi$
$409$	10.9306	0.540482	0.270241	+	0.962793i	$0.412897\pi$
$410$	−6.96654	−0.344053
$411$	0	0
$412$	0.173969	0.00857084
$413$	14.8356	0.730010
$414$	0	0
$415$	11.3628	0.557776
$416$	−4.75630	−0.233197
$417$	0	0
$418$	−16.3435	−0.799385
$419$	−33.2351	−1.62364	−0.811821	−	0.583906i	$-0.801524\pi$
$419$	−33.2351	−1.62364	−0.811821	+	0.583906i	$0.801524\pi$
$420$	0	0
$421$	11.0309	0.537611	0.268806	−	0.963194i	$-0.413371\pi$
$421$	11.0309	0.537611	0.268806	+	0.963194i	$0.413371\pi$
$422$	−14.1238	−0.687537
$423$	0	0
$424$	−26.0820	−1.26666
$425$	−10.6278	−0.515523
$426$	0	0
$427$	−3.37035	−0.163102
$428$	−2.52341	−0.121974
$429$	0	0
$430$	−2.51644	−0.121353
$431$	4.18755	0.201707	0.100854	−	0.994901i	$-0.467843\pi$
$431$	4.18755	0.201707	0.100854	+	0.994901i	$0.467843\pi$
$432$	0	0
$433$	3.63646	0.174757	0.0873785	−	0.996175i	$-0.472151\pi$
$433$	3.63646	0.174757	0.0873785	+	0.996175i	$0.472151\pi$
$434$	4.79471	0.230153
$435$	0	0
$436$	1.85128	0.0886602
$437$	−1.58501	−0.0758213
$438$	0	0
$439$	−8.53268	−0.407243	−0.203621	−	0.979050i	$-0.565271\pi$
$439$	−8.53268	−0.407243	−0.203621	+	0.979050i	$0.565271\pi$
$440$	−15.3537	−0.731959
$441$	0	0
$442$	−12.8160	−0.609594
$443$	−26.5914	−1.26339	−0.631697	−	0.775216i	$-0.717641\pi$
$443$	−26.5914	−1.26339	−0.631697	+	0.775216i	$0.717641\pi$
$444$	0	0
$445$	8.33089	0.394922
$446$	−24.5234	−1.16122
$447$	0	0
$448$	−8.72046	−0.412003
$449$	−20.8074	−0.981964	−0.490982	−	0.871170i	$-0.663362\pi$
$449$	−20.8074	−0.981964	−0.490982	+	0.871170i	$0.663362\pi$
$450$	0	0
$451$	−23.2805	−1.09624
$452$	3.12576	0.147023
$453$	0	0
$454$	−18.0089	−0.845199
$455$	3.76567	0.176537
$456$	0	0
$457$	−2.39350	−0.111963	−0.0559817	−	0.998432i	$-0.517829\pi$
$457$	−2.39350	−0.111963	−0.0559817	+	0.998432i	$0.517829\pi$
$458$	−36.8324	−1.72107
$459$	0	0
$460$	−0.161004	−0.00750683
$461$	−8.02854	−0.373926	−0.186963	−	0.982367i	$-0.559864\pi$
$461$	−8.02854	−0.373926	−0.186963	+	0.982367i	$0.559864\pi$
$462$	0	0
$463$	−20.8286	−0.967989	−0.483994	−	0.875071i	$-0.660815\pi$
$463$	−20.8286	−0.967989	−0.483994	+	0.875071i	$0.660815\pi$
$464$	−27.3321	−1.26886
$465$	0	0
$466$	26.6345	1.23382
$467$	26.6270	1.23215	0.616076	−	0.787687i	$-0.288721\pi$
$467$	26.6270	1.23215	0.616076	+	0.787687i	$0.288721\pi$
$468$	0	0
$469$	0.747683	0.0345248
$470$	−12.3689	−0.570536
$471$	0	0
$472$	44.1044	2.03007
$473$	−8.40934	−0.386662
$474$	0	0
$475$	9.88193	0.453414
$476$	0.672085	0.0308050
$477$	0	0
$478$	0.00520950	0.000238277 0
$479$	19.1425	0.874643	0.437322	−	0.899305i	$-0.355927\pi$
$479$	19.1425	0.874643	0.437322	+	0.899305i	$0.355927\pi$
$480$	0	0
$481$	21.2095	0.967068
$482$	29.5495	1.34594
$483$	0	0
$484$	−2.88064	−0.130938
$485$	−13.5375	−0.614708
$486$	0	0
$487$	9.87224	0.447354	0.223677	−	0.974663i	$-0.428194\pi$
$487$	9.87224	0.447354	0.223677	+	0.974663i	$0.428194\pi$
$488$	−10.0197	−0.453568
$489$	0	0
$490$	1.43138	0.0646632
$491$	31.7645	1.43351	0.716756	−	0.697324i	$-0.245626\pi$
$491$	31.7645	1.43351	0.716756	+	0.697324i	$0.245626\pi$
$492$	0	0
$493$	21.9194	0.987201
$494$	11.9166	0.536152
$495$	0	0
$496$	12.5002	0.561276
$497$	9.58491	0.429942
$498$	0	0
$499$	16.0776	0.719734	0.359867	−	0.933004i	$-0.382822\pi$
$499$	16.0776	0.719734	0.359867	+	0.933004i	$0.382822\pi$
$500$	2.31278	0.103431
$501$	0	0
$502$	37.0052	1.65162
$503$	8.24213	0.367498	0.183749	−	0.982973i	$-0.441177\pi$
$503$	8.24213	0.367498	0.183749	+	0.982973i	$0.441177\pi$
$504$	0	0
$505$	−13.9532	−0.620911
$506$	3.89989	0.173371
$507$	0	0
$508$	−0.242472	−0.0107580
$509$	−10.2126	−0.452664	−0.226332	−	0.974050i	$-0.572673\pi$
$509$	−10.2126	−0.452664	−0.226332	+	0.974050i	$0.572673\pi$
$510$	0	0
$511$	−0.421112	−0.0186289
$512$	−25.2636	−1.11650
$513$	0	0
$514$	−35.1642	−1.55103
$515$	−0.774666	−0.0341359
$516$	0	0
$517$	−41.3341	−1.81787
$518$	8.06200	0.354224
$519$	0	0
$520$	11.1949	0.490929
$521$	3.26353	0.142978	0.0714890	−	0.997441i	$-0.477225\pi$
$521$	3.26353	0.142978	0.0714890	+	0.997441i	$0.477225\pi$
$522$	0	0
$523$	−28.4322	−1.24325	−0.621626	−	0.783314i	$-0.713528\pi$
$523$	−28.4322	−1.24325	−0.621626	+	0.783314i	$0.713528\pi$
$524$	−0.0755090	−0.00329862
$525$	0	0
$526$	30.3395	1.32286
$527$	−10.0248	−0.436685
$528$	0	0
$529$	−22.6218	−0.983556
$530$	12.5579	0.545482
$531$	0	0
$532$	−0.624918	−0.0270937
$533$	16.9746	0.735253
$534$	0	0
$535$	11.2365	0.485795
$536$	2.22278	0.0960093
$537$	0	0
$538$	−7.44087	−0.320799
$539$	4.78334	0.206033
$540$	0	0
$541$	−11.7662	−0.505869	−0.252935	−	0.967483i	$-0.581396\pi$
$541$	−11.7662	−0.505869	−0.252935	+	0.967483i	$0.581396\pi$
$542$	32.5774	1.39932
$543$	0	0
$544$	3.78002	0.162067
$545$	−8.24355	−0.353115
$546$	0	0
$547$	9.47909	0.405297	0.202648	−	0.979252i	$-0.435045\pi$
$547$	9.47909	0.405297	0.202648	+	0.979252i	$0.435045\pi$
$548$	−2.12289	−0.0906854
$549$	0	0
$550$	−24.3143	−1.03677
$551$	−20.3811	−0.868265
$552$	0	0
$553$	−10.1734	−0.432617
$554$	17.6598	0.750295
$555$	0	0
$556$	−2.20262	−0.0934119
$557$	−44.8884	−1.90198	−0.950991	−	0.309219i	$-0.899932\pi$
$557$	−44.8884	−1.90198	−0.950991	+	0.309219i	$0.899932\pi$
$558$	0	0
$559$	6.13153	0.259336
$560$	3.73173	0.157695
$561$	0	0
$562$	−13.0055	−0.548602
$563$	34.1789	1.44047	0.720234	−	0.693731i	$-0.244034\pi$
$563$	34.1789	1.44047	0.720234	+	0.693731i	$0.244034\pi$
$564$	0	0
$565$	−13.9187	−0.585563
$566$	−1.13998	−0.0479170
$567$	0	0
$568$	28.4948	1.19562
$569$	21.6376	0.907097	0.453548	−	0.891232i	$-0.350158\pi$
$569$	21.6376	0.907097	0.453548	+	0.891232i	$0.350158\pi$
$570$	0	0
$571$	27.3427	1.14426	0.572128	−	0.820165i	$-0.306118\pi$
$571$	27.3427	1.14426	0.572128	+	0.820165i	$0.306118\pi$
$572$	4.04511	0.169135
$573$	0	0
$574$	6.45228	0.269313
$575$	−2.35803	−0.0983368
$576$	0	0
$577$	43.6578	1.81750	0.908749	−	0.417343i	$-0.137039\pi$
$577$	43.6578	1.81750	0.908749	+	0.417343i	$0.137039\pi$
$578$	−12.3518	−0.513768
$579$	0	0
$580$	−2.07029	−0.0859642
$581$	−10.5240	−0.436608
$582$	0	0
$583$	41.9657	1.73804
$584$	−1.25192	−0.0518047
$585$	0	0
$586$	11.8978	0.491495
$587$	1.64579	0.0679290	0.0339645	−	0.999423i	$-0.489187\pi$
$587$	1.64579	0.0679290	0.0339645	+	0.999423i	$0.489187\pi$
$588$	0	0
$589$	9.32122	0.384074
$590$	−21.2353	−0.874244
$591$	0	0
$592$	21.0183	0.863849
$593$	20.9349	0.859694	0.429847	−	0.902902i	$-0.358567\pi$
$593$	20.9349	0.859694	0.429847	+	0.902902i	$0.358567\pi$
$594$	0	0
$595$	−2.99273	−0.122690
$596$	2.96815	0.121580
$597$	0	0
$598$	−2.84354	−0.116281
$599$	0.150271	0.00613991	0.00306996	−	0.999995i	$-0.499023\pi$
$599$	0.150271	0.00613991	0.00306996	+	0.999995i	$0.499023\pi$
$600$	0	0
$601$	17.0191	0.694225	0.347113	−	0.937823i	$-0.387162\pi$
$601$	17.0191	0.694225	0.347113	+	0.937823i	$0.387162\pi$
$602$	2.33068	0.0949913
$603$	0	0
$604$	4.94711	0.201295
$605$	12.8272	0.521500
$606$	0	0
$607$	1.14397	0.0464325	0.0232162	−	0.999730i	$-0.492609\pi$
$607$	1.14397	0.0464325	0.0232162	+	0.999730i	$0.492609\pi$
$608$	−3.51474	−0.142542
$609$	0	0
$610$	4.82425	0.195328
$611$	30.1381	1.21926
$612$	0	0
$613$	−18.4635	−0.745735	−0.372868	−	0.927885i	$-0.621625\pi$
$613$	−18.4635	−0.745735	−0.372868	+	0.927885i	$0.621625\pi$
$614$	22.1714	0.894765
$615$	0	0
$616$	14.2203	0.572953
$617$	−26.8607	−1.08137	−0.540685	−	0.841225i	$-0.681835\pi$
$617$	−26.8607	−1.08137	−0.540685	+	0.841225i	$0.681835\pi$
$618$	0	0
$619$	−8.70634	−0.349937	−0.174969	−	0.984574i	$-0.555982\pi$
$619$	−8.70634	−0.349937	−0.174969	+	0.984574i	$0.555982\pi$
$620$	0.946839	0.0380260
$621$	0	0
$622$	−39.8715	−1.59870
$623$	−7.71591	−0.309132
$624$	0	0
$625$	8.87264	0.354906
$626$	−19.5027	−0.779487
$627$	0	0
$628$	4.02977	0.160805
$629$	−16.8560	−0.672093
$630$	0	0
$631$	32.9059	1.30996	0.654982	−	0.755644i	$-0.272676\pi$
$631$	32.9059	1.30996	0.654982	+	0.755644i	$0.272676\pi$
$632$	−30.2443	−1.20306
$633$	0	0
$634$	1.15144	0.0457294
$635$	1.07970	0.0428467
$636$	0	0
$637$	−3.48769	−0.138187
$638$	50.1474	1.98535
$639$	0	0
$640$	9.53743	0.377000
$641$	36.6692	1.44834	0.724172	−	0.689619i	$-0.242222\pi$
$641$	36.6692	1.44834	0.724172	+	0.689619i	$0.242222\pi$
$642$	0	0
$643$	−34.2787	−1.35182	−0.675909	−	0.736985i	$-0.736249\pi$
$643$	−34.2787	−1.35182	−0.675909	+	0.736985i	$0.736249\pi$
$644$	0.149119	0.00587610
$645$	0	0
$646$	−9.47057	−0.372614
$647$	−9.00511	−0.354028	−0.177014	−	0.984208i	$-0.556644\pi$
$647$	−9.00511	−0.354028	−0.177014	+	0.984208i	$0.556644\pi$
$648$	0	0
$649$	−70.9635	−2.78556
$650$	17.7284	0.695364
$651$	0	0
$652$	4.26460	0.167015
$653$	22.0431	0.862613	0.431307	−	0.902205i	$-0.358053\pi$
$653$	22.0431	0.862613	0.431307	+	0.902205i	$0.358053\pi$
$654$	0	0
$655$	0.336233	0.0131377
$656$	16.8217	0.656776
$657$	0	0
$658$	11.4559	0.446597
$659$	−37.2955	−1.45282	−0.726412	−	0.687259i	$-0.758814\pi$
$659$	−37.2955	−1.45282	−0.726412	+	0.687259i	$0.758814\pi$
$660$	0	0
$661$	15.3434	0.596788	0.298394	−	0.954443i	$-0.403549\pi$
$661$	15.3434	0.596788	0.298394	+	0.954443i	$0.403549\pi$
$662$	0.607771	0.0236217
$663$	0	0
$664$	−31.2866	−1.21416
$665$	2.78270	0.107908
$666$	0	0
$667$	4.86336	0.188310
$668$	3.16120	0.122311
$669$	0	0
$670$	−1.07022	−0.0413462
$671$	16.1215	0.622364
$672$	0	0
$673$	−31.8567	−1.22799	−0.613993	−	0.789311i	$-0.710438\pi$
$673$	−31.8567	−1.22799	−0.613993	+	0.789311i	$0.710438\pi$
$674$	27.0234	1.04090
$675$	0	0
$676$	0.202709	0.00779649
$677$	22.2441	0.854911	0.427455	−	0.904036i	$-0.359410\pi$
$677$	22.2441	0.854911	0.427455	+	0.904036i	$0.359410\pi$
$678$	0	0
$679$	12.5382	0.481173
$680$	−8.89703	−0.341185
$681$	0	0
$682$	−22.9347	−0.878215
$683$	38.5950	1.47680	0.738398	−	0.674365i	$-0.235583\pi$
$683$	38.5950	1.47680	0.738398	+	0.674365i	$0.235583\pi$
$684$	0	0
$685$	9.45301	0.361181
$686$	−1.32572	−0.0506161
$687$	0	0
$688$	6.07628	0.231656
$689$	−30.5986	−1.16571
$690$	0	0
$691$	−6.59695	−0.250960	−0.125480	−	0.992096i	$-0.540047\pi$
$691$	−6.59695	−0.250960	−0.125480	+	0.992096i	$0.540047\pi$
$692$	−0.636795	−0.0242073
$693$	0	0
$694$	−33.7574	−1.28141
$695$	9.80804	0.372040
$696$	0	0
$697$	−13.4904	−0.510986
$698$	−18.7985	−0.711534
$699$	0	0
$700$	−0.929696	−0.0351392
$701$	49.0148	1.85126	0.925632	−	0.378426i	$-0.123535\pi$
$701$	49.0148	1.85126	0.925632	+	0.378426i	$0.123535\pi$
$702$	0	0
$703$	15.6731	0.591120
$704$	41.7129	1.57211
$705$	0	0
$706$	28.3417	1.06665
$707$	12.9232	0.486028
$708$	0	0
$709$	−27.8304	−1.04519	−0.522596	−	0.852580i	$-0.675036\pi$
$709$	−27.8304	−1.04519	−0.522596	+	0.852580i	$0.675036\pi$
$710$	−13.7197	−0.514889
$711$	0	0
$712$	−22.9385	−0.859658
$713$	−2.22424	−0.0832983
$714$	0	0
$715$	−18.0125	−0.673628
$716$	4.48941	0.167777
$717$	0	0
$718$	27.5326	1.02751
$719$	−33.0772	−1.23357	−0.616785	−	0.787131i	$-0.711565\pi$
$719$	−33.0772	−1.23357	−0.616785	+	0.787131i	$0.711565\pi$
$720$	0	0
$721$	0.717482	0.0267204
$722$	−16.3827	−0.609702
$723$	0	0
$724$	−0.626694	−0.0232909
$725$	−30.3212	−1.12610
$726$	0	0
$727$	40.0650	1.48593	0.742965	−	0.669330i	$-0.233419\pi$
$727$	40.0650	1.48593	0.742965	+	0.669330i	$0.233419\pi$
$728$	−10.3685	−0.384282
$729$	0	0
$730$	0.602771	0.0223096
$731$	−4.87297	−0.180233
$732$	0	0
$733$	−14.8400	−0.548128	−0.274064	−	0.961711i	$-0.588368\pi$
$733$	−14.8400	−0.548128	−0.274064	+	0.961711i	$0.588368\pi$
$734$	−6.53881	−0.241352
$735$	0	0
$736$	0.838691	0.0309146
$737$	−3.57642	−0.131739
$738$	0	0
$739$	29.9549	1.10191	0.550955	−	0.834535i	$-0.314264\pi$
$739$	29.9549	1.10191	0.550955	+	0.834535i	$0.314264\pi$
$740$	1.59205	0.0585250
$741$	0	0
$742$	−11.6309	−0.426985
$743$	17.7684	0.651859	0.325930	−	0.945394i	$-0.394323\pi$
$743$	17.7684	0.651859	0.325930	+	0.945394i	$0.394323\pi$
$744$	0	0
$745$	−13.2169	−0.484229
$746$	−17.2991	−0.633365
$747$	0	0
$748$	−3.21481	−0.117545
$749$	−10.4070	−0.380264
$750$	0	0
$751$	41.9211	1.52972	0.764861	−	0.644195i	$-0.222807\pi$
$751$	41.9211	1.52972	0.764861	+	0.644195i	$0.222807\pi$
$752$	29.8665	1.08912
$753$	0	0
$754$	−36.5641	−1.33159
$755$	−22.0289	−0.801715
$756$	0	0
$757$	−20.5645	−0.747429	−0.373714	−	0.927544i	$-0.621916\pi$
$757$	−20.5645	−0.747429	−0.373714	+	0.927544i	$0.621916\pi$
$758$	−24.8645	−0.903118
$759$	0	0
$760$	8.27264	0.300080
$761$	−17.6310	−0.639124	−0.319562	−	0.947565i	$-0.603536\pi$
$761$	−17.6310	−0.639124	−0.319562	+	0.947565i	$0.603536\pi$
$762$	0	0
$763$	7.63503	0.276406
$764$	3.63950	0.131672
$765$	0	0
$766$	−7.96176	−0.287670
$767$	51.7418	1.86829
$768$	0	0
$769$	−2.17432	−0.0784078	−0.0392039	−	0.999231i	$-0.512482\pi$
$769$	−2.17432	−0.0784078	−0.0392039	+	0.999231i	$0.512482\pi$
$770$	−6.84677	−0.246741
$771$	0	0
$772$	−1.65434	−0.0595409
$773$	0.757770	0.0272551	0.0136275	−	0.999907i	$-0.495662\pi$
$773$	0.757770	0.0272551	0.0136275	+	0.999907i	$0.495662\pi$
$774$	0	0
$775$	13.8673	0.498126
$776$	37.2747	1.33808
$777$	0	0
$778$	22.4884	0.806249
$779$	12.5437	0.449423
$780$	0	0
$781$	−45.8479	−1.64057
$782$	2.25987	0.0808130
$783$	0	0
$784$	−3.45626	−0.123438
$785$	−17.9441	−0.640454
$786$	0	0
$787$	−50.7033	−1.80738	−0.903689	−	0.428189i	$-0.859152\pi$
$787$	−50.7033	−1.80738	−0.903689	+	0.428189i	$0.859152\pi$
$788$	−1.20037	−0.0427614
$789$	0	0
$790$	14.5620	0.518093
$791$	12.8912	0.458359
$792$	0	0
$793$	−11.7547	−0.417423
$794$	3.28763	0.116674
$795$	0	0
$796$	1.20869	0.0428410
$797$	53.4300	1.89259	0.946294	−	0.323306i	$-0.104794\pi$
$797$	53.4300	1.89259	0.946294	+	0.323306i	$0.104794\pi$
$798$	0	0
$799$	−23.9519	−0.847358
$800$	−5.22891	−0.184870
$801$	0	0
$802$	30.4994	1.07697
$803$	2.01432	0.0710838
$804$	0	0
$805$	−0.664009	−0.0234033
$806$	16.7225	0.589023
$807$	0	0
$808$	38.4193	1.35159
$809$	−24.8179	−0.872552	−0.436276	−	0.899813i	$-0.643703\pi$
$809$	−24.8179	−0.872552	−0.436276	+	0.899813i	$0.643703\pi$
$810$	0	0
$811$	30.0721	1.05597	0.527987	−	0.849252i	$-0.322947\pi$
$811$	30.0721	1.05597	0.527987	+	0.849252i	$0.322947\pi$
$812$	1.91747	0.0672899
$813$	0	0
$814$	−38.5633	−1.35164
$815$	−18.9898	−0.665185
$816$	0	0
$817$	4.53099	0.158519
$818$	14.4909	0.506661
$819$	0	0
$820$	1.27417	0.0444960
$821$	26.3441	0.919416	0.459708	−	0.888070i	$-0.347954\pi$
$821$	26.3441	0.919416	0.459708	+	0.888070i	$0.347954\pi$
$822$	0	0
$823$	39.6141	1.38086	0.690429	−	0.723400i	$-0.257422\pi$
$823$	39.6141	1.38086	0.690429	+	0.723400i	$0.257422\pi$
$824$	2.13299	0.0743062
$825$	0	0
$826$	19.6678	0.684329
$827$	8.01733	0.278790	0.139395	−	0.990237i	$-0.455484\pi$
$827$	8.01733	0.278790	0.139395	+	0.990237i	$0.455484\pi$
$828$	0	0
$829$	18.8203	0.653656	0.326828	−	0.945084i	$-0.394020\pi$
$829$	18.8203	0.653656	0.326828	+	0.945084i	$0.394020\pi$
$830$	15.0638	0.522873
$831$	0	0
$832$	−30.4143	−1.05443
$833$	2.77181	0.0960374
$834$	0	0
$835$	−14.0765	−0.487137
$836$	2.98920	0.103383
$837$	0	0
$838$	−44.0604	−1.52204
$839$	−38.3113	−1.32265	−0.661326	−	0.750098i	$-0.730006\pi$
$839$	−38.3113	−1.32265	−0.661326	+	0.750098i	$0.730006\pi$
$840$	0	0
$841$	33.5363	1.15642
$842$	14.6238	0.503970
$843$	0	0
$844$	2.58323	0.0889183
$845$	−0.902640	−0.0310518
$846$	0	0
$847$	−11.8803	−0.408212
$848$	−30.3229	−1.04129
$849$	0	0
$850$	−14.0894	−0.483264
$851$	−3.73992	−0.128203
$852$	0	0
$853$	50.2220	1.71957	0.859785	−	0.510657i	$-0.170598\pi$
$853$	50.2220	1.71957	0.859785	+	0.510657i	$0.170598\pi$
$854$	−4.46813	−0.152896
$855$	0	0
$856$	−30.9389	−1.05747
$857$	17.2975	0.590871	0.295436	−	0.955363i	$-0.404535\pi$
$857$	17.2975	0.590871	0.295436	+	0.955363i	$0.404535\pi$
$858$	0	0
$859$	−29.6158	−1.01048	−0.505240	−	0.862979i	$-0.668596\pi$
$859$	−29.6158	−1.01048	−0.505240	+	0.862979i	$0.668596\pi$
$860$	0.460253	0.0156945
$861$	0	0
$862$	5.55151	0.189085
$863$	50.9049	1.73282	0.866411	−	0.499332i	$-0.166421\pi$
$863$	50.9049	1.73282	0.866411	+	0.499332i	$0.166421\pi$
$864$	0	0
$865$	2.83558	0.0964126
$866$	4.82092	0.163821
$867$	0	0
$868$	−0.876945	−0.0297654
$869$	48.6628	1.65077
$870$	0	0
$871$	2.60769	0.0883582
$872$	22.6981	0.768653
$873$	0	0
$874$	−2.10128	−0.0710768
$875$	9.53835	0.322455
$876$	0	0
$877$	−11.0160	−0.371982	−0.185991	−	0.982551i	$-0.559550\pi$
$877$	−11.0160	−0.371982	−0.185991	+	0.982551i	$0.559550\pi$
$878$	−11.3119	−0.381759
$879$	0	0
$880$	−17.8501	−0.601728
$881$	2.18598	0.0736474	0.0368237	−	0.999322i	$-0.488276\pi$
$881$	2.18598	0.0736474	0.0368237	+	0.999322i	$0.488276\pi$
$882$	0	0
$883$	−16.7582	−0.563960	−0.281980	−	0.959420i	$-0.590991\pi$
$883$	−16.7582	−0.563960	−0.281980	+	0.959420i	$0.590991\pi$
$884$	2.34403	0.0788381
$885$	0	0
$886$	−35.2526	−1.18434
$887$	51.8568	1.74118	0.870590	−	0.492008i	$-0.163737\pi$
$887$	51.8568	1.74118	0.870590	+	0.492008i	$0.163737\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	11.0444	0.370209
$891$	0	0
$892$	4.48530	0.150179
$893$	22.2710	0.745269
$894$	0	0
$895$	−19.9909	−0.668222
$896$	−8.83339	−0.295103
$897$	0	0
$898$	−27.5848	−0.920517
$899$	−28.6007	−0.953887
$900$	0	0
$901$	24.3179	0.810147
$902$	−30.8634	−1.02764
$903$	0	0
$904$	38.3241	1.27464
$905$	2.79060	0.0927628
$906$	0	0
$907$	49.4221	1.64104	0.820518	−	0.571621i	$-0.193685\pi$
$907$	49.4221	1.64104	0.820518	+	0.571621i	$0.193685\pi$
$908$	3.29380	0.109309
$909$	0	0
$910$	4.99221	0.165490
$911$	36.2697	1.20167	0.600834	−	0.799374i	$-0.294835\pi$
$911$	36.2697	1.20167	0.600834	+	0.799374i	$0.294835\pi$
$912$	0	0
$913$	50.3398	1.66600
$914$	−3.17311	−0.104957
$915$	0	0
$916$	6.73660	0.222583
$917$	−0.311413	−0.0102838
$918$	0	0
$919$	−56.8577	−1.87556	−0.937781	−	0.347226i	$-0.887124\pi$
$919$	−56.8577	−1.87556	−0.937781	+	0.347226i	$0.887124\pi$
$920$	−1.97402	−0.0650816
$921$	0	0
$922$	−10.6436	−0.350528
$923$	33.4292	1.10034
$924$	0	0
$925$	23.3169	0.766656
$926$	−27.6129	−0.907416
$927$	0	0
$928$	10.7844	0.354017
$929$	−9.37481	−0.307578	−0.153789	−	0.988104i	$-0.549148\pi$
$929$	−9.37481	−0.307578	−0.153789	+	0.988104i	$0.549148\pi$
$930$	0	0
$931$	−2.57728	−0.0844670
$932$	−4.87141	−0.159568
$933$	0	0
$934$	35.2999	1.15505
$935$	14.3152	0.468158
$936$	0	0
$937$	1.57816	0.0515561	0.0257781	−	0.999668i	$-0.491794\pi$
$937$	1.57816	0.0515561	0.0257781	+	0.999668i	$0.491794\pi$
$938$	0.991217	0.0323644
$939$	0	0
$940$	2.26226	0.0737868
$941$	−8.80232	−0.286948	−0.143474	−	0.989654i	$-0.545827\pi$
$941$	−8.80232	−0.286948	−0.143474	+	0.989654i	$0.545827\pi$
$942$	0	0
$943$	−2.99318	−0.0974713
$944$	51.2756	1.66888
$945$	0	0
$946$	−11.1484	−0.362466
$947$	−1.92069	−0.0624142	−0.0312071	−	0.999513i	$-0.509935\pi$
$947$	−1.92069	−0.0624142	−0.0312071	+	0.999513i	$0.509935\pi$
$948$	0	0
$949$	−1.46871	−0.0476763
$950$	13.1006	0.425041
$951$	0	0
$952$	8.24026	0.267068
$953$	14.8677	0.481613	0.240807	−	0.970573i	$-0.422588\pi$
$953$	14.8677	0.481613	0.240807	+	0.970573i	$0.422588\pi$
$954$	0	0
$955$	−16.2063	−0.524423
$956$	−0.000952810 0	−3.08161e−5 0
$957$	0	0
$958$	25.3776	0.819912
$959$	−8.75520	−0.282720
$960$	0	0
$961$	−17.9196	−0.578051
$962$	28.1178	0.906553
$963$	0	0
$964$	−5.40456	−0.174069
$965$	7.36658	0.237139
$966$	0	0
$967$	−11.4470	−0.368109	−0.184055	−	0.982916i	$-0.558922\pi$
$967$	−11.4470	−0.368109	−0.184055	+	0.982916i	$0.558922\pi$
$968$	−35.3188	−1.13519
$969$	0	0
$970$	−17.9470	−0.576242
$971$	−49.0052	−1.57265	−0.786326	−	0.617812i	$-0.788019\pi$
$971$	−49.0052	−1.57265	−0.786326	+	0.617812i	$0.788019\pi$
$972$	0	0
$973$	−9.08402	−0.291220
$974$	13.0878	0.419360
$975$	0	0
$976$	−11.6488	−0.372869
$977$	−13.8494	−0.443082	−0.221541	−	0.975151i	$-0.571109\pi$
$977$	−13.8494	−0.443082	−0.221541	+	0.975151i	$0.571109\pi$
$978$	0	0
$979$	36.9078	1.17958
$980$	−0.261797	−0.00836281
$981$	0	0
$982$	42.1108	1.34381
$983$	−50.3518	−1.60597	−0.802987	−	0.595997i	$-0.796757\pi$
$983$	−50.3518	−1.60597	−0.802987	+	0.595997i	$0.796757\pi$
$984$	0	0
$985$	5.34512	0.170310
$986$	29.0590	0.925426
$987$	0	0
$988$	−2.17952	−0.0693399
$989$	−1.08119	−0.0343798
$990$	0	0
$991$	28.2028	0.895892	0.447946	−	0.894061i	$-0.352156\pi$
$991$	28.2028	0.895892	0.447946	+	0.894061i	$0.352156\pi$
$992$	−4.93222	−0.156598
$993$	0	0
$994$	12.7069	0.403038
$995$	−5.38218	−0.170627
$996$	0	0
$997$	18.9945	0.601561	0.300781	−	0.953693i	$-0.402753\pi$
$997$	18.9945	0.601561	0.300781	+	0.953693i	$0.402753\pi$
$998$	21.3144	0.674696
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.q.1.12		15
3.2	odd	2		889.2.a.b.1.4	✓	15
21.20	even	2		6223.2.a.j.1.4		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.4	✓	15	3.2	odd	2
6223.2.a.j.1.4		15	21.20	even	2
8001.2.a.q.1.12		15	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)

\(f(q)\)	\(=\)	\(q+1.32572 q^{2} -0.242472 q^{4} +1.07970 q^{5} -1.00000 q^{7} -2.97289 q^{8} +O(q^{10})\)	\(q+1.32572 q^{2} -0.242472 q^{4} +1.07970 q^{5} -1.00000 q^{7} -2.97289 q^{8} +1.43138 q^{10} +4.78334 q^{11} -3.48769 q^{13} -1.32572 q^{14} -3.45626 q^{16} +2.77181 q^{17} -2.57728 q^{19} -0.261797 q^{20} +6.34136 q^{22} +0.614993 q^{23} -3.83424 q^{25} -4.62370 q^{26} +0.242472 q^{28} +7.90799 q^{29} -3.61669 q^{31} +1.36374 q^{32} +3.67463 q^{34} -1.07970 q^{35} -6.08123 q^{37} -3.41675 q^{38} -3.20983 q^{40} -4.86701 q^{41} -1.75805 q^{43} -1.15983 q^{44} +0.815308 q^{46} -8.64126 q^{47} +1.00000 q^{49} -5.08313 q^{50} +0.845667 q^{52} +8.77331 q^{53} +5.16458 q^{55} +2.97289 q^{56} +10.4838 q^{58} -14.8356 q^{59} +3.37035 q^{61} -4.79471 q^{62} +8.72046 q^{64} -3.76567 q^{65} -0.747683 q^{67} -0.672085 q^{68} -1.43138 q^{70} -9.58491 q^{71} +0.421112 q^{73} -8.06200 q^{74} +0.624918 q^{76} -4.78334 q^{77} +10.1734 q^{79} -3.73173 q^{80} -6.45228 q^{82} +10.5240 q^{83} +2.99273 q^{85} -2.33068 q^{86} -14.2203 q^{88} +7.71591 q^{89} +3.48769 q^{91} -0.149119 q^{92} -11.4559 q^{94} -2.78270 q^{95} -12.5382 q^{97} +1.32572 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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