LMFDB - Embedded newform 8001.2.a.t.1.4

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
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			1.4
Root			$-1.48078$ 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)

$f(q)$	$=$	$q-1.48078 q^{2} +0.192706 q^{4} -1.52223 q^{5} -1.00000 q^{7} +2.67620 q^{8} +O(q^{10})$	\(q-1.48078 q^{2} +0.192706 q^{4} -1.52223 q^{5} -1.00000 q^{7} +2.67620 q^{8} +2.25409 q^{10} +0.865340 q^{11} -1.04410 q^{13} +1.48078 q^{14} -4.34828 q^{16} +4.97921 q^{17} +0.815782 q^{19} -0.293343 q^{20} -1.28138 q^{22} -1.49743 q^{23} -2.68282 q^{25} +1.54608 q^{26} -0.192706 q^{28} +3.83213 q^{29} -10.0365 q^{31} +1.08643 q^{32} -7.37310 q^{34} +1.52223 q^{35} +10.7792 q^{37} -1.20799 q^{38} -4.07380 q^{40} -6.50981 q^{41} -2.33357 q^{43} +0.166756 q^{44} +2.21736 q^{46} +10.1623 q^{47} +1.00000 q^{49} +3.97266 q^{50} -0.201204 q^{52} +8.29563 q^{53} -1.31725 q^{55} -2.67620 q^{56} -5.67454 q^{58} +5.53895 q^{59} -8.99905 q^{61} +14.8619 q^{62} +7.08779 q^{64} +1.58936 q^{65} -4.35668 q^{67} +0.959522 q^{68} -2.25409 q^{70} -14.1352 q^{71} +10.9769 q^{73} -15.9616 q^{74} +0.157206 q^{76} -0.865340 q^{77} +12.9973 q^{79} +6.61908 q^{80} +9.63959 q^{82} +15.9319 q^{83} -7.57950 q^{85} +3.45550 q^{86} +2.31583 q^{88} -12.5926 q^{89} +1.04410 q^{91} -0.288563 q^{92} -15.0482 q^{94} -1.24181 q^{95} +4.12134 q^{97} -1.48078 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.48078	−1.04707	−0.523534	−	0.852005i	$-0.675387\pi$
$2$	−1.48078	−1.04707	−0.523534	+	0.852005i	$0.675387\pi$
$3$	0	0
$3$	0	0
$4$	0.192706	0.0963529
$5$	−1.52223	−0.680762	−0.340381	−	0.940288i	$-0.610556\pi$
$5$	−1.52223	−0.680762	−0.340381	+	0.940288i	$0.610556\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.67620	0.946181
$9$	0	0
$10$	2.25409	0.712805
$11$	0.865340	0.260910	0.130455	−	0.991454i	$-0.458356\pi$
$11$	0.865340	0.260910	0.130455	+	0.991454i	$0.458356\pi$
$12$	0	0
$13$	−1.04410	−0.289581	−0.144791	−	0.989462i	$-0.546251\pi$
$13$	−1.04410	−0.289581	−0.144791	+	0.989462i	$0.546251\pi$
$14$	1.48078	0.395755
$15$	0	0
$16$	−4.34828	−1.08707
$17$	4.97921	1.20764	0.603818	−	0.797123i	$-0.293646\pi$
$17$	4.97921	1.20764	0.603818	+	0.797123i	$0.293646\pi$
$18$	0	0
$19$	0.815782	0.187153	0.0935766	−	0.995612i	$-0.470170\pi$
$19$	0.815782	0.187153	0.0935766	+	0.995612i	$0.470170\pi$
$20$	−0.293343	−0.0655934
$21$	0	0
$22$	−1.28138	−0.273191
$23$	−1.49743	−0.312235	−0.156118	−	0.987738i	$-0.549898\pi$
$23$	−1.49743	−0.312235	−0.156118	+	0.987738i	$0.549898\pi$
$24$	0	0
$25$	−2.68282	−0.536563
$26$	1.54608	0.303212
$27$	0	0
$28$	−0.192706	−0.0364180
$29$	3.83213	0.711609	0.355804	−	0.934560i	$-0.384207\pi$
$29$	3.83213	0.711609	0.355804	+	0.934560i	$0.384207\pi$
$30$	0	0
$31$	−10.0365	−1.80261	−0.901306	−	0.433182i	$-0.857391\pi$
$31$	−10.0365	−1.80261	−0.901306	+	0.433182i	$0.857391\pi$
$32$	1.08643	0.192055
$33$	0	0
$34$	−7.37310	−1.26448
$35$	1.52223	0.257304
$36$	0	0
$37$	10.7792	1.77209	0.886045	−	0.463600i	$-0.153442\pi$
$37$	10.7792	1.77209	0.886045	+	0.463600i	$0.153442\pi$
$38$	−1.20799	−0.195962
$39$	0	0
$40$	−4.07380	−0.644124
$41$	−6.50981	−1.01666	−0.508331	−	0.861162i	$-0.669737\pi$
$41$	−6.50981	−1.01666	−0.508331	+	0.861162i	$0.669737\pi$
$42$	0	0
$43$	−2.33357	−0.355866	−0.177933	−	0.984043i	$-0.556941\pi$
$43$	−2.33357	−0.355866	−0.177933	+	0.984043i	$0.556941\pi$
$44$	0.166756	0.0251394
$45$	0	0
$46$	2.21736	0.326932
$47$	10.1623	1.48233	0.741165	−	0.671322i	$-0.234273\pi$
$47$	10.1623	1.48233	0.741165	+	0.671322i	$0.234273\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.97266	0.561819
$51$	0	0
$52$	−0.201204	−0.0279020
$53$	8.29563	1.13949	0.569746	−	0.821821i	$-0.307041\pi$
$53$	8.29563	1.13949	0.569746	+	0.821821i	$0.307041\pi$
$54$	0	0
$55$	−1.31725	−0.177618
$56$	−2.67620	−0.357623
$57$	0	0
$58$	−5.67454	−0.745103
$59$	5.53895	0.721110	0.360555	−	0.932738i	$-0.382587\pi$
$59$	5.53895	0.721110	0.360555	+	0.932738i	$0.382587\pi$
$60$	0	0
$61$	−8.99905	−1.15221	−0.576105	−	0.817376i	$-0.695428\pi$
$61$	−8.99905	−1.15221	−0.576105	+	0.817376i	$0.695428\pi$
$62$	14.8619	1.88746
$63$	0	0
$64$	7.08779	0.885974
$65$	1.58936	0.197136
$66$	0	0
$67$	−4.35668	−0.532254	−0.266127	−	0.963938i	$-0.585744\pi$
$67$	−4.35668	−0.532254	−0.266127	+	0.963938i	$0.585744\pi$
$68$	0.959522	0.116359
$69$	0	0
$70$	−2.25409	−0.269415
$71$	−14.1352	−1.67753	−0.838767	−	0.544490i	$-0.816723\pi$
$71$	−14.1352	−1.67753	−0.838767	+	0.544490i	$0.816723\pi$
$72$	0	0
$73$	10.9769	1.28475	0.642373	−	0.766392i	$-0.277950\pi$
$73$	10.9769	1.28475	0.642373	+	0.766392i	$0.277950\pi$
$74$	−15.9616	−1.85550
$75$	0	0
$76$	0.157206	0.0180328
$77$	−0.865340	−0.0986147
$78$	0	0
$79$	12.9973	1.46231	0.731155	−	0.682211i	$-0.238982\pi$
$79$	12.9973	1.46231	0.731155	+	0.682211i	$0.238982\pi$
$80$	6.61908	0.740035
$81$	0	0
$82$	9.63959	1.06451
$83$	15.9319	1.74876	0.874379	−	0.485244i	$-0.161269\pi$
$83$	15.9319	1.74876	0.874379	+	0.485244i	$0.161269\pi$
$84$	0	0
$85$	−7.57950	−0.822112
$86$	3.45550	0.372616
$87$	0	0
$88$	2.31583	0.246868
$89$	−12.5926	−1.33481	−0.667407	−	0.744693i	$-0.732596\pi$
$89$	−12.5926	−1.33481	−0.667407	+	0.744693i	$0.732596\pi$
$90$	0	0
$91$	1.04410	0.109451
$92$	−0.288563	−0.0300848
$93$	0	0
$94$	−15.0482	−1.55210
$95$	−1.24181	−0.127407
$96$	0	0
$97$	4.12134	0.418459	0.209229	−	0.977867i	$-0.432904\pi$
$97$	4.12134	0.418459	0.209229	+	0.977867i	$0.432904\pi$
$98$	−1.48078	−0.149581
$99$	0	0
$100$	−0.516994	−0.0516994
$101$	2.55649	0.254380	0.127190	−	0.991878i	$-0.459404\pi$
$101$	2.55649	0.254380	0.127190	+	0.991878i	$0.459404\pi$
$102$	0	0
$103$	−13.2780	−1.30832	−0.654158	−	0.756358i	$-0.726977\pi$
$103$	−13.2780	−1.30832	−0.654158	+	0.756358i	$0.726977\pi$
$104$	−2.79422	−0.273996
$105$	0	0
$106$	−12.2840	−1.19313
$107$	−6.06579	−0.586402	−0.293201	−	0.956051i	$-0.594721\pi$
$107$	−6.06579	−0.586402	−0.293201	+	0.956051i	$0.594721\pi$
$108$	0	0
$109$	−5.10177	−0.488661	−0.244330	−	0.969692i	$-0.578568\pi$
$109$	−5.10177	−0.488661	−0.244330	+	0.969692i	$0.578568\pi$
$110$	1.95055	0.185978
$111$	0	0
$112$	4.34828	0.410873
$113$	−4.08229	−0.384029	−0.192015	−	0.981392i	$-0.561502\pi$
$113$	−4.08229	−0.384029	−0.192015	+	0.981392i	$0.561502\pi$
$114$	0	0
$115$	2.27943	0.212558
$116$	0.738474	0.0685656
$117$	0	0
$118$	−8.20196	−0.755052
$119$	−4.97921	−0.456443
$120$	0	0
$121$	−10.2512	−0.931926
$122$	13.3256	1.20644
$123$	0	0
$124$	−1.93410	−0.173687
$125$	11.6950	1.04603
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−12.6683	−1.11973
$129$	0	0
$130$	−2.35349	−0.206415
$131$	2.37886	0.207842	0.103921	−	0.994586i	$-0.466861\pi$
$131$	2.37886	0.207842	0.103921	+	0.994586i	$0.466861\pi$
$132$	0	0
$133$	−0.815782	−0.0707373
$134$	6.45129	0.557306
$135$	0	0
$136$	13.3254	1.14264
$137$	−22.3714	−1.91132	−0.955659	−	0.294477i	$-0.904855\pi$
$137$	−22.3714	−1.91132	−0.955659	+	0.294477i	$0.904855\pi$
$138$	0	0
$139$	−8.66152	−0.734661	−0.367330	−	0.930091i	$-0.619728\pi$
$139$	−8.66152	−0.734661	−0.367330	+	0.930091i	$0.619728\pi$
$140$	0.293343	0.0247920
$141$	0	0
$142$	20.9310	1.75649
$143$	−0.903502	−0.0755546
$144$	0	0
$145$	−5.83338	−0.484436
$146$	−16.2543	−1.34522
$147$	0	0
$148$	2.07721	0.170746
$149$	−17.4223	−1.42729	−0.713647	−	0.700506i	$-0.752958\pi$
$149$	−17.4223	−1.42729	−0.713647	+	0.700506i	$0.752958\pi$
$150$	0	0
$151$	7.73444	0.629420	0.314710	−	0.949188i	$-0.398093\pi$
$151$	7.73444	0.629420	0.314710	+	0.949188i	$0.398093\pi$
$152$	2.18320	0.177081
$153$	0	0
$154$	1.28138	0.103256
$155$	15.2779	1.22715
$156$	0	0
$157$	20.6035	1.64434	0.822169	−	0.569244i	$-0.192764\pi$
$157$	20.6035	1.64434	0.822169	+	0.569244i	$0.192764\pi$
$158$	−19.2461	−1.53114
$159$	0	0
$160$	−1.65380	−0.130744
$161$	1.49743	0.118014
$162$	0	0
$163$	−3.80843	−0.298299	−0.149150	−	0.988815i	$-0.547654\pi$
$163$	−3.80843	−0.298299	−0.149150	+	0.988815i	$0.547654\pi$
$164$	−1.25448	−0.0979583
$165$	0	0
$166$	−23.5917	−1.83107
$167$	13.4087	1.03760	0.518800	−	0.854896i	$-0.326379\pi$
$167$	13.4087	1.03760	0.518800	+	0.854896i	$0.326379\pi$
$168$	0	0
$169$	−11.9099	−0.916143
$170$	11.2236	0.860808
$171$	0	0
$172$	−0.449692	−0.0342887
$173$	−17.6553	−1.34231	−0.671155	−	0.741317i	$-0.734201\pi$
$173$	−17.6553	−1.34231	−0.671155	+	0.741317i	$0.734201\pi$
$174$	0	0
$175$	2.68282	0.202802
$176$	−3.76274	−0.283627
$177$	0	0
$178$	18.6469	1.39764
$179$	23.0982	1.72644	0.863221	−	0.504825i	$-0.168443\pi$
$179$	23.0982	1.72644	0.863221	+	0.504825i	$0.168443\pi$
$180$	0	0
$181$	−16.5259	−1.22836	−0.614181	−	0.789165i	$-0.710514\pi$
$181$	−16.5259	−1.22836	−0.614181	+	0.789165i	$0.710514\pi$
$182$	−1.54608	−0.114603
$183$	0	0
$184$	−4.00742	−0.295431
$185$	−16.4084	−1.20637
$186$	0	0
$187$	4.30871	0.315084
$188$	1.95834	0.142827
$189$	0	0
$190$	1.83884	0.133404
$191$	7.27254	0.526223	0.263111	−	0.964765i	$-0.415251\pi$
$191$	7.27254	0.526223	0.263111	+	0.964765i	$0.415251\pi$
$192$	0	0
$193$	10.9223	0.786204	0.393102	−	0.919495i	$-0.371402\pi$
$193$	10.9223	0.786204	0.393102	+	0.919495i	$0.371402\pi$
$194$	−6.10279	−0.438155
$195$	0	0
$196$	0.192706	0.0137647
$197$	−17.9350	−1.27781	−0.638907	−	0.769284i	$-0.720613\pi$
$197$	−17.9350	−1.27781	−0.638907	+	0.769284i	$0.720613\pi$
$198$	0	0
$199$	13.3884	0.949078	0.474539	−	0.880235i	$-0.342615\pi$
$199$	13.3884	0.949078	0.474539	+	0.880235i	$0.342615\pi$
$200$	−7.17976	−0.507686
$201$	0	0
$202$	−3.78560	−0.266354
$203$	−3.83213	−0.268963
$204$	0	0
$205$	9.90942	0.692104
$206$	19.6617	1.36990
$207$	0	0
$208$	4.54004	0.314795
$209$	0.705929	0.0488301
$210$	0	0
$211$	−12.8278	−0.883102	−0.441551	−	0.897236i	$-0.645572\pi$
$211$	−12.8278	−0.883102	−0.441551	+	0.897236i	$0.645572\pi$
$212$	1.59862	0.109793
$213$	0	0
$214$	8.98210	0.614003
$215$	3.55223	0.242260
$216$	0	0
$217$	10.0365	0.681324
$218$	7.55459	0.511662
$219$	0	0
$220$	−0.253841	−0.0171140
$221$	−5.19879	−0.349709
$222$	0	0
$223$	−12.4662	−0.834797	−0.417399	−	0.908724i	$-0.637058\pi$
$223$	−12.4662	−0.834797	−0.417399	+	0.908724i	$0.637058\pi$
$224$	−1.08643	−0.0725901
$225$	0	0
$226$	6.04496	0.402105
$227$	12.9737	0.861092	0.430546	−	0.902569i	$-0.358321\pi$
$227$	12.9737	0.861092	0.430546	+	0.902569i	$0.358321\pi$
$228$	0	0
$229$	−2.53558	−0.167556	−0.0837781	−	0.996484i	$-0.526699\pi$
$229$	−2.53558	−0.167556	−0.0837781	+	0.996484i	$0.526699\pi$
$230$	−3.37533	−0.222563
$231$	0	0
$232$	10.2556	0.673311
$233$	−5.62162	−0.368285	−0.184142	−	0.982900i	$-0.558951\pi$
$233$	−5.62162	−0.368285	−0.184142	+	0.982900i	$0.558951\pi$
$234$	0	0
$235$	−15.4694	−1.00911
$236$	1.06739	0.0694811
$237$	0	0
$238$	7.37310	0.477927
$239$	1.99554	0.129081	0.0645404	−	0.997915i	$-0.479442\pi$
$239$	1.99554	0.129081	0.0645404	+	0.997915i	$0.479442\pi$
$240$	0	0
$241$	−12.1610	−0.783360	−0.391680	−	0.920102i	$-0.628106\pi$
$241$	−12.1610	−0.783360	−0.391680	+	0.920102i	$0.628106\pi$
$242$	15.1797	0.975791
$243$	0	0
$244$	−1.73417	−0.111019
$245$	−1.52223	−0.0972517
$246$	0	0
$247$	−0.851759	−0.0541961
$248$	−26.8598	−1.70560
$249$	0	0
$250$	−17.3177	−1.09527
$251$	−4.69171	−0.296138	−0.148069	−	0.988977i	$-0.547306\pi$
$251$	−4.69171	−0.296138	−0.148069	+	0.988977i	$0.547306\pi$
$252$	0	0
$253$	−1.29578	−0.0814653
$254$	1.48078	0.0929123
$255$	0	0
$256$	4.58338	0.286461
$257$	19.0546	1.18859	0.594296	−	0.804246i	$-0.297431\pi$
$257$	19.0546	1.18859	0.594296	+	0.804246i	$0.297431\pi$
$258$	0	0
$259$	−10.7792	−0.669787
$260$	0.306279	0.0189946
$261$	0	0
$262$	−3.52257	−0.217625
$263$	29.9375	1.84603	0.923014	−	0.384767i	$-0.125718\pi$
$263$	29.9375	1.84603	0.923014	+	0.384767i	$0.125718\pi$
$264$	0	0
$265$	−12.6279	−0.775723
$266$	1.20799	0.0740668
$267$	0	0
$268$	−0.839558	−0.0512842
$269$	9.98159	0.608588	0.304294	−	0.952578i	$-0.401579\pi$
$269$	9.98159	0.608588	0.304294	+	0.952578i	$0.401579\pi$
$270$	0	0
$271$	−17.8738	−1.08575	−0.542877	−	0.839812i	$-0.682665\pi$
$271$	−17.8738	−1.08575	−0.542877	+	0.839812i	$0.682665\pi$
$272$	−21.6510	−1.31278
$273$	0	0
$274$	33.1271	2.00128
$275$	−2.32155	−0.139995
$276$	0	0
$277$	6.86173	0.412281	0.206141	−	0.978522i	$-0.433910\pi$
$277$	6.86173	0.412281	0.206141	+	0.978522i	$0.433910\pi$
$278$	12.8258	0.769240
$279$	0	0
$280$	4.07380	0.243456
$281$	25.3082	1.50976	0.754880	−	0.655863i	$-0.227695\pi$
$281$	25.3082	1.50976	0.754880	+	0.655863i	$0.227695\pi$
$282$	0	0
$283$	32.8700	1.95392	0.976958	−	0.213430i	$-0.0684634\pi$
$283$	32.8700	1.95392	0.976958	+	0.213430i	$0.0684634\pi$
$284$	−2.72393	−0.161635
$285$	0	0
$286$	1.33789	0.0791109
$287$	6.50981	0.384262
$288$	0	0
$289$	7.79250	0.458382
$290$	8.63795	0.507238
$291$	0	0
$292$	2.11531	0.123789
$293$	25.0230	1.46186	0.730929	−	0.682453i	$-0.239087\pi$
$293$	25.0230	1.46186	0.730929	+	0.682453i	$0.239087\pi$
$294$	0	0
$295$	−8.43156	−0.490904
$296$	28.8473	1.67672
$297$	0	0
$298$	25.7986	1.49447
$299$	1.56346	0.0904175
$300$	0	0
$301$	2.33357	0.134505
$302$	−11.4530	−0.659045
$303$	0	0
$304$	−3.54725	−0.203449
$305$	13.6986	0.784381
$306$	0	0
$307$	−19.6929	−1.12393	−0.561967	−	0.827160i	$-0.689955\pi$
$307$	−19.6929	−1.12393	−0.561967	+	0.827160i	$0.689955\pi$
$308$	−0.166756	−0.00950181
$309$	0	0
$310$	−22.6232	−1.28491
$311$	11.2125	0.635805	0.317902	−	0.948123i	$-0.397022\pi$
$311$	11.2125	0.635805	0.317902	+	0.948123i	$0.397022\pi$
$312$	0	0
$313$	−16.2306	−0.917410	−0.458705	−	0.888589i	$-0.651686\pi$
$313$	−16.2306	−0.917410	−0.458705	+	0.888589i	$0.651686\pi$
$314$	−30.5092	−1.72173
$315$	0	0
$316$	2.50465	0.140898
$317$	−2.01720	−0.113297	−0.0566487	−	0.998394i	$-0.518042\pi$
$317$	−2.01720	−0.113297	−0.0566487	+	0.998394i	$0.518042\pi$
$318$	0	0
$319$	3.31610	0.185666
$320$	−10.7892	−0.603137
$321$	0	0
$322$	−2.21736	−0.123569
$323$	4.06195	0.226013
$324$	0	0
$325$	2.80113	0.155379
$326$	5.63944	0.312340
$327$	0	0
$328$	−17.4216	−0.961945
$329$	−10.1623	−0.560268
$330$	0	0
$331$	10.4074	0.572044	0.286022	−	0.958223i	$-0.407667\pi$
$331$	10.4074	0.572044	0.286022	+	0.958223i	$0.407667\pi$
$332$	3.07018	0.168498
$333$	0	0
$334$	−19.8554	−1.08644
$335$	6.63187	0.362338
$336$	0	0
$337$	−20.7231	−1.12886	−0.564431	−	0.825480i	$-0.690904\pi$
$337$	−20.7231	−1.12886	−0.564431	+	0.825480i	$0.690904\pi$
$338$	17.6359	0.959264
$339$	0	0
$340$	−1.46061	−0.0792129
$341$	−8.68501	−0.470319
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−6.24510	−0.336713
$345$	0	0
$346$	26.1436	1.40549
$347$	0.446661	0.0239780	0.0119890	−	0.999928i	$-0.496184\pi$
$347$	0.446661	0.0239780	0.0119890	+	0.999928i	$0.496184\pi$
$348$	0	0
$349$	36.6960	1.96429	0.982147	−	0.188115i	$-0.0602377\pi$
$349$	36.6960	1.96429	0.982147	+	0.188115i	$0.0602377\pi$
$350$	−3.97266	−0.212347
$351$	0	0
$352$	0.940131	0.0501091
$353$	−0.310894	−0.0165472	−0.00827362	−	0.999966i	$-0.502634\pi$
$353$	−0.310894	−0.0165472	−0.00827362	+	0.999966i	$0.502634\pi$
$354$	0	0
$355$	21.5170	1.14200
$356$	−2.42667	−0.128613
$357$	0	0
$358$	−34.2034	−1.80770
$359$	−27.3736	−1.44472	−0.722361	−	0.691516i	$-0.756943\pi$
$359$	−27.3736	−1.44472	−0.722361	+	0.691516i	$0.756943\pi$
$360$	0	0
$361$	−18.3345	−0.964974
$362$	24.4712	1.28618
$363$	0	0
$364$	0.201204	0.0105460
$365$	−16.7093	−0.874606
$366$	0	0
$367$	5.74303	0.299784	0.149892	−	0.988702i	$-0.452107\pi$
$367$	5.74303	0.299784	0.149892	+	0.988702i	$0.452107\pi$
$368$	6.51123	0.339421
$369$	0	0
$370$	24.2972	1.26315
$371$	−8.29563	−0.430688
$372$	0	0
$373$	−30.9957	−1.60489	−0.802447	−	0.596723i	$-0.796469\pi$
$373$	−30.9957	−1.60489	−0.802447	+	0.596723i	$0.796469\pi$
$374$	−6.38024	−0.329914
$375$	0	0
$376$	27.1965	1.40255
$377$	−4.00113	−0.206069
$378$	0	0
$379$	3.85939	0.198243	0.0991217	−	0.995075i	$-0.468397\pi$
$379$	3.85939	0.198243	0.0991217	+	0.995075i	$0.468397\pi$
$380$	−0.239304	−0.0122760
$381$	0	0
$382$	−10.7690	−0.550991
$383$	12.4738	0.637380	0.318690	−	0.947859i	$-0.396757\pi$
$383$	12.4738	0.637380	0.318690	+	0.947859i	$0.396757\pi$
$384$	0	0
$385$	1.31725	0.0671331
$386$	−16.1735	−0.823210
$387$	0	0
$388$	0.794206	0.0403197
$389$	29.1443	1.47767	0.738836	−	0.673885i	$-0.235376\pi$
$389$	29.1443	1.47767	0.738836	+	0.673885i	$0.235376\pi$
$390$	0	0
$391$	−7.45600	−0.377066
$392$	2.67620	0.135169
$393$	0	0
$394$	26.5577	1.33796
$395$	−19.7849	−0.995485
$396$	0	0
$397$	24.0043	1.20474	0.602371	−	0.798216i	$-0.294223\pi$
$397$	24.0043	1.20474	0.602371	+	0.798216i	$0.294223\pi$
$398$	−19.8252	−0.993750
$399$	0	0
$400$	11.6656	0.583281
$401$	32.2760	1.61178	0.805892	−	0.592062i	$-0.201686\pi$
$401$	32.2760	1.61178	0.805892	+	0.592062i	$0.201686\pi$
$402$	0	0
$403$	10.4791	0.522003
$404$	0.492651	0.0245103
$405$	0	0
$406$	5.67454	0.281623
$407$	9.32767	0.462356
$408$	0	0
$409$	−2.60665	−0.128891	−0.0644453	−	0.997921i	$-0.520528\pi$
$409$	−2.60665	−0.128891	−0.0644453	+	0.997921i	$0.520528\pi$
$410$	−14.6737	−0.724681
$411$	0	0
$412$	−2.55874	−0.126060
$413$	−5.53895	−0.272554
$414$	0	0
$415$	−24.2521	−1.19049
$416$	−1.13434	−0.0556157
$417$	0	0
$418$	−1.04532	−0.0511285
$419$	−19.3070	−0.943210	−0.471605	−	0.881810i	$-0.656325\pi$
$419$	−19.3070	−0.943210	−0.471605	+	0.881810i	$0.656325\pi$
$420$	0	0
$421$	33.2581	1.62090	0.810451	−	0.585806i	$-0.199222\pi$
$421$	33.2581	1.62090	0.810451	+	0.585806i	$0.199222\pi$
$422$	18.9951	0.924669
$423$	0	0
$424$	22.2008	1.07817
$425$	−13.3583	−0.647973
$426$	0	0
$427$	8.99905	0.435494
$428$	−1.16891	−0.0565016
$429$	0	0
$430$	−5.26006	−0.253663
$431$	33.8535	1.63066	0.815332	−	0.578993i	$-0.196554\pi$
$431$	33.8535	1.63066	0.815332	+	0.578993i	$0.196554\pi$
$432$	0	0
$433$	9.97618	0.479424	0.239712	−	0.970844i	$-0.422947\pi$
$433$	9.97618	0.479424	0.239712	+	0.970844i	$0.422947\pi$
$434$	−14.8619	−0.713393
$435$	0	0
$436$	−0.983141	−0.0470839
$437$	−1.22157	−0.0584358
$438$	0	0
$439$	−26.2024	−1.25057	−0.625287	−	0.780395i	$-0.715018\pi$
$439$	−26.2024	−1.25057	−0.625287	+	0.780395i	$0.715018\pi$
$440$	−3.52522	−0.168058
$441$	0	0
$442$	7.69826	0.366169
$443$	−2.04449	−0.0971366	−0.0485683	−	0.998820i	$-0.515466\pi$
$443$	−2.04449	−0.0971366	−0.0485683	+	0.998820i	$0.515466\pi$
$444$	0	0
$445$	19.1689	0.908691
$446$	18.4597	0.874090
$447$	0	0
$448$	−7.08779	−0.334867
$449$	13.2581	0.625689	0.312844	−	0.949804i	$-0.398718\pi$
$449$	13.2581	0.625689	0.312844	+	0.949804i	$0.398718\pi$
$450$	0	0
$451$	−5.63320	−0.265257
$452$	−0.786680	−0.0370023
$453$	0	0
$454$	−19.2111	−0.901623
$455$	−1.58936	−0.0745104
$456$	0	0
$457$	9.60418	0.449265	0.224632	−	0.974444i	$-0.427882\pi$
$457$	9.60418	0.449265	0.224632	+	0.974444i	$0.427882\pi$
$458$	3.75464	0.175443
$459$	0	0
$460$	0.439259	0.0204806
$461$	−10.6527	−0.496144	−0.248072	−	0.968742i	$-0.579797\pi$
$461$	−10.6527	−0.496144	−0.248072	+	0.968742i	$0.579797\pi$
$462$	0	0
$463$	5.33069	0.247738	0.123869	−	0.992299i	$-0.460470\pi$
$463$	5.33069	0.247738	0.123869	+	0.992299i	$0.460470\pi$
$464$	−16.6632	−0.773568
$465$	0	0
$466$	8.32438	0.385619
$467$	5.61910	0.260021	0.130010	−	0.991513i	$-0.458499\pi$
$467$	5.61910	0.260021	0.130010	+	0.991513i	$0.458499\pi$
$468$	0	0
$469$	4.35668	0.201173
$470$	22.9068	1.05661
$471$	0	0
$472$	14.8234	0.682300
$473$	−2.01933	−0.0928489
$474$	0	0
$475$	−2.18859	−0.100420
$476$	−0.959522	−0.0439796
$477$	0	0
$478$	−2.95496	−0.135157
$479$	24.6499	1.12628	0.563141	−	0.826361i	$-0.309593\pi$
$479$	24.6499	1.12628	0.563141	+	0.826361i	$0.309593\pi$
$480$	0	0
$481$	−11.2546	−0.513164
$482$	18.0078	0.820231
$483$	0	0
$484$	−1.97546	−0.0897938
$485$	−6.27363	−0.284871
$486$	0	0
$487$	19.6466	0.890271	0.445135	−	0.895463i	$-0.353156\pi$
$487$	19.6466	0.890271	0.445135	+	0.895463i	$0.353156\pi$
$488$	−24.0833	−1.09020
$489$	0	0
$490$	2.25409	0.101829
$491$	37.5098	1.69279	0.846397	−	0.532552i	$-0.178767\pi$
$491$	37.5098	1.69279	0.846397	+	0.532552i	$0.178767\pi$
$492$	0	0
$493$	19.0810	0.859364
$494$	1.26127	0.0567470
$495$	0	0
$496$	43.6416	1.95956
$497$	14.1352	0.634048
$498$	0	0
$499$	41.1745	1.84322	0.921612	−	0.388113i	$-0.126873\pi$
$499$	41.1745	1.84322	0.921612	+	0.388113i	$0.126873\pi$
$500$	2.25370	0.100788
$501$	0	0
$502$	6.94739	0.310077
$503$	−16.9741	−0.756837	−0.378419	−	0.925635i	$-0.623532\pi$
$503$	−16.9741	−0.756837	−0.378419	+	0.925635i	$0.623532\pi$
$504$	0	0
$505$	−3.89157	−0.173173
$506$	1.91877	0.0852997
$507$	0	0
$508$	−0.192706	−0.00854994
$509$	1.99436	0.0883984	0.0441992	−	0.999023i	$-0.485926\pi$
$509$	1.99436	0.0883984	0.0441992	+	0.999023i	$0.485926\pi$
$510$	0	0
$511$	−10.9769	−0.485588
$512$	18.5496	0.819786
$513$	0	0
$514$	−28.2156	−1.24454
$515$	20.2121	0.890652
$516$	0	0
$517$	8.79389	0.386755
$518$	15.9616	0.701313
$519$	0	0
$520$	4.25345	0.186526
$521$	41.4305	1.81510	0.907552	−	0.419940i	$-0.137949\pi$
$521$	41.4305	1.81510	0.907552	+	0.419940i	$0.137949\pi$
$522$	0	0
$523$	−4.08834	−0.178771	−0.0893853	−	0.995997i	$-0.528490\pi$
$523$	−4.08834	−0.178771	−0.0893853	+	0.995997i	$0.528490\pi$
$524$	0.458420	0.0200262
$525$	0	0
$526$	−44.3309	−1.93292
$527$	−49.9739	−2.17690
$528$	0	0
$529$	−20.7577	−0.902509
$530$	18.6991	0.812235
$531$	0	0
$532$	−0.157206	−0.00681574
$533$	6.79689	0.294406
$534$	0	0
$535$	9.23353	0.399200
$536$	−11.6594	−0.503608
$537$	0	0
$538$	−14.7805	−0.637234
$539$	0.865340	0.0372728
$540$	0	0
$541$	42.1070	1.81032	0.905160	−	0.425072i	$-0.139751\pi$
$541$	42.1070	1.81032	0.905160	+	0.425072i	$0.139751\pi$
$542$	26.4671	1.13686
$543$	0	0
$544$	5.40956	0.231933
$545$	7.76607	0.332662
$546$	0	0
$547$	−15.8876	−0.679304	−0.339652	−	0.940551i	$-0.610309\pi$
$547$	−15.8876	−0.679304	−0.339652	+	0.940551i	$0.610309\pi$
$548$	−4.31110	−0.184161
$549$	0	0
$550$	3.43770	0.146584
$551$	3.12618	0.133180
$552$	0	0
$553$	−12.9973	−0.552701
$554$	−10.1607	−0.431687
$555$	0	0
$556$	−1.66913	−0.0707867
$557$	3.60503	0.152750	0.0763751	−	0.997079i	$-0.475665\pi$
$557$	3.60503	0.152750	0.0763751	+	0.997079i	$0.475665\pi$
$558$	0	0
$559$	2.43648	0.103052
$560$	−6.61908	−0.279707
$561$	0	0
$562$	−37.4758	−1.58082
$563$	34.6910	1.46205	0.731026	−	0.682350i	$-0.239042\pi$
$563$	34.6910	1.46205	0.731026	+	0.682350i	$0.239042\pi$
$564$	0	0
$565$	6.21418	0.261432
$566$	−48.6732	−2.04589
$567$	0	0
$568$	−37.8285	−1.58725
$569$	−20.6266	−0.864711	−0.432355	−	0.901703i	$-0.642317\pi$
$569$	−20.6266	−0.864711	−0.432355	+	0.901703i	$0.642317\pi$
$570$	0	0
$571$	17.3911	0.727796	0.363898	−	0.931439i	$-0.381446\pi$
$571$	17.3911	0.727796	0.363898	+	0.931439i	$0.381446\pi$
$572$	−0.174110	−0.00727991
$573$	0	0
$574$	−9.63959	−0.402349
$575$	4.01732	0.167534
$576$	0	0
$577$	24.2037	1.00761	0.503806	−	0.863817i	$-0.331932\pi$
$577$	24.2037	1.00761	0.503806	+	0.863817i	$0.331932\pi$
$578$	−11.5390	−0.479958
$579$	0	0
$580$	−1.12413	−0.0466768
$581$	−15.9319	−0.660968
$582$	0	0
$583$	7.17854	0.297305
$584$	29.3763	1.21560
$585$	0	0
$586$	−37.0535	−1.53067
$587$	−1.63945	−0.0676675	−0.0338338	−	0.999427i	$-0.510772\pi$
$587$	−1.63945	−0.0676675	−0.0338338	+	0.999427i	$0.510772\pi$
$588$	0	0
$589$	−8.18762	−0.337365
$590$	12.4853	0.514011
$591$	0	0
$592$	−46.8709	−1.92638
$593$	−11.1286	−0.456998	−0.228499	−	0.973544i	$-0.573382\pi$
$593$	−11.1286	−0.456998	−0.228499	+	0.973544i	$0.573382\pi$
$594$	0	0
$595$	7.57950	0.310729
$596$	−3.35739	−0.137524
$597$	0	0
$598$	−2.31515	−0.0946734
$599$	−13.0414	−0.532859	−0.266429	−	0.963854i	$-0.585844\pi$
$599$	−13.0414	−0.532859	−0.266429	+	0.963854i	$0.585844\pi$
$600$	0	0
$601$	2.80138	0.114271	0.0571353	−	0.998366i	$-0.481803\pi$
$601$	2.80138	0.114271	0.0571353	+	0.998366i	$0.481803\pi$
$602$	−3.45550	−0.140836
$603$	0	0
$604$	1.49047	0.0606464
$605$	15.6047	0.634420
$606$	0	0
$607$	31.1419	1.26401	0.632004	−	0.774965i	$-0.282232\pi$
$607$	31.1419	1.26401	0.632004	+	0.774965i	$0.282232\pi$
$608$	0.886290	0.0359438
$609$	0	0
$610$	−20.2846	−0.821301
$611$	−10.6105	−0.429255
$612$	0	0
$613$	−9.49033	−0.383311	−0.191655	−	0.981462i	$-0.561386\pi$
$613$	−9.49033	−0.383311	−0.191655	+	0.981462i	$0.561386\pi$
$614$	29.1608	1.17684
$615$	0	0
$616$	−2.31583	−0.0933073
$617$	−30.4102	−1.22427	−0.612133	−	0.790755i	$-0.709688\pi$
$617$	−30.4102	−1.22427	−0.612133	+	0.790755i	$0.709688\pi$
$618$	0	0
$619$	33.6359	1.35194	0.675970	−	0.736929i	$-0.263725\pi$
$619$	33.6359	1.35194	0.675970	+	0.736929i	$0.263725\pi$
$620$	2.94414	0.118239
$621$	0	0
$622$	−16.6033	−0.665731
$623$	12.5926	0.504513
$624$	0	0
$625$	−4.38842	−0.175537
$626$	24.0340	0.960591
$627$	0	0
$628$	3.97041	0.158437
$629$	53.6719	2.14004
$630$	0	0
$631$	−35.2319	−1.40256	−0.701281	−	0.712885i	$-0.747388\pi$
$631$	−35.2319	−1.40256	−0.701281	+	0.712885i	$0.747388\pi$
$632$	34.7834	1.38361
$633$	0	0
$634$	2.98703	0.118630
$635$	1.52223	0.0604079
$636$	0	0
$637$	−1.04410	−0.0413688
$638$	−4.91041	−0.194405
$639$	0	0
$640$	19.2841	0.762270
$641$	−48.2278	−1.90488	−0.952442	−	0.304719i	$-0.901437\pi$
$641$	−48.2278	−1.90488	−0.952442	+	0.304719i	$0.901437\pi$
$642$	0	0
$643$	16.5052	0.650903	0.325451	−	0.945559i	$-0.394484\pi$
$643$	16.5052	0.650903	0.325451	+	0.945559i	$0.394484\pi$
$644$	0.288563	0.0113710
$645$	0	0
$646$	−6.01485	−0.236651
$647$	−3.26517	−0.128367	−0.0641836	−	0.997938i	$-0.520444\pi$
$647$	−3.26517	−0.128367	−0.0641836	+	0.997938i	$0.520444\pi$
$648$	0	0
$649$	4.79308	0.188145
$650$	−4.14785	−0.162692
$651$	0	0
$652$	−0.733907	−0.0287420
$653$	40.7841	1.59601	0.798003	−	0.602653i	$-0.205890\pi$
$653$	40.7841	1.59601	0.798003	+	0.602653i	$0.205890\pi$
$654$	0	0
$655$	−3.62117	−0.141491
$656$	28.3064	1.10518
$657$	0	0
$658$	15.0482	0.586639
$659$	−9.97990	−0.388762	−0.194381	−	0.980926i	$-0.562270\pi$
$659$	−9.97990	−0.388762	−0.194381	+	0.980926i	$0.562270\pi$
$660$	0	0
$661$	−18.6517	−0.725466	−0.362733	−	0.931893i	$-0.618156\pi$
$661$	−18.6517	−0.725466	−0.362733	+	0.931893i	$0.618156\pi$
$662$	−15.4111	−0.598969
$663$	0	0
$664$	42.6371	1.65464
$665$	1.24181	0.0481552
$666$	0	0
$667$	−5.73834	−0.222189
$668$	2.58394	0.0999757
$669$	0	0
$670$	−9.82034	−0.379393
$671$	−7.78724	−0.300623
$672$	0	0
$673$	5.65105	0.217832	0.108916	−	0.994051i	$-0.465262\pi$
$673$	5.65105	0.217832	0.108916	+	0.994051i	$0.465262\pi$
$674$	30.6864	1.18200
$675$	0	0
$676$	−2.29510	−0.0882730
$677$	−12.6562	−0.486417	−0.243209	−	0.969974i	$-0.578200\pi$
$677$	−12.6562	−0.486417	−0.243209	+	0.969974i	$0.578200\pi$
$678$	0	0
$679$	−4.12134	−0.158162
$680$	−20.2843	−0.777866
$681$	0	0
$682$	12.8606	0.492457
$683$	−32.4257	−1.24073	−0.620367	−	0.784312i	$-0.713016\pi$
$683$	−32.4257	−1.24073	−0.620367	+	0.784312i	$0.713016\pi$
$684$	0	0
$685$	34.0544	1.30115
$686$	1.48078	0.0565364
$687$	0	0
$688$	10.1470	0.386851
$689$	−8.66147	−0.329976
$690$	0	0
$691$	−24.6758	−0.938711	−0.469355	−	0.883009i	$-0.655514\pi$
$691$	−24.6758	−0.938711	−0.469355	+	0.883009i	$0.655514\pi$
$692$	−3.40228	−0.129335
$693$	0	0
$694$	−0.661406	−0.0251066
$695$	13.1848	0.500129
$696$	0	0
$697$	−32.4137	−1.22776
$698$	−54.3387	−2.05675
$699$	0	0
$700$	0.516994	0.0195405
$701$	16.6489	0.628818	0.314409	−	0.949288i	$-0.398194\pi$
$701$	16.6489	0.628818	0.314409	+	0.949288i	$0.398194\pi$
$702$	0	0
$703$	8.79348	0.331652
$704$	6.13335	0.231159
$705$	0	0
$706$	0.460366	0.0173261
$707$	−2.55649	−0.0961468
$708$	0	0
$709$	−9.56092	−0.359068	−0.179534	−	0.983752i	$-0.557459\pi$
$709$	−9.56092	−0.359068	−0.179534	+	0.983752i	$0.557459\pi$
$710$	−31.8619	−1.19575
$711$	0	0
$712$	−33.7004	−1.26298
$713$	15.0290	0.562839
$714$	0	0
$715$	1.37534	0.0514347
$716$	4.45116	0.166348
$717$	0	0
$718$	40.5342	1.51272
$719$	47.0868	1.75604	0.878021	−	0.478622i	$-0.158863\pi$
$719$	47.0868	1.75604	0.878021	+	0.478622i	$0.158863\pi$
$720$	0	0
$721$	13.2780	0.494497
$722$	27.1493	1.01039
$723$	0	0
$724$	−3.18464	−0.118356
$725$	−10.2809	−0.381823
$726$	0	0
$727$	9.81644	0.364072	0.182036	−	0.983292i	$-0.441731\pi$
$727$	9.81644	0.364072	0.182036	+	0.983292i	$0.441731\pi$
$728$	2.79422	0.103561
$729$	0	0
$730$	24.7428	0.915773
$731$	−11.6193	−0.429756
$732$	0	0
$733$	−16.7000	−0.616829	−0.308414	−	0.951252i	$-0.599798\pi$
$733$	−16.7000	−0.616829	−0.308414	+	0.951252i	$0.599798\pi$
$734$	−8.50415	−0.313894
$735$	0	0
$736$	−1.62685	−0.0599665
$737$	−3.77001	−0.138870
$738$	0	0
$739$	18.5712	0.683152	0.341576	−	0.939854i	$-0.389039\pi$
$739$	18.5712	0.683152	0.341576	+	0.939854i	$0.389039\pi$
$740$	−3.16200	−0.116237
$741$	0	0
$742$	12.2840	0.450960
$743$	−17.4217	−0.639140	−0.319570	−	0.947563i	$-0.603539\pi$
$743$	−17.4217	−0.639140	−0.319570	+	0.947563i	$0.603539\pi$
$744$	0	0
$745$	26.5208	0.971647
$746$	45.8977	1.68043
$747$	0	0
$748$	0.830313	0.0303593
$749$	6.06579	0.221639
$750$	0	0
$751$	30.9453	1.12921	0.564606	−	0.825361i	$-0.309028\pi$
$751$	30.9453	1.12921	0.564606	+	0.825361i	$0.309028\pi$
$752$	−44.1887	−1.61140
$753$	0	0
$754$	5.92479	0.215768
$755$	−11.7736	−0.428485
$756$	0	0
$757$	−11.7251	−0.426154	−0.213077	−	0.977035i	$-0.568349\pi$
$757$	−11.7251	−0.426154	−0.213077	+	0.977035i	$0.568349\pi$
$758$	−5.71490	−0.207574
$759$	0	0
$760$	−3.32333	−0.120550
$761$	31.9501	1.15819	0.579095	−	0.815260i	$-0.303406\pi$
$761$	31.9501	1.15819	0.579095	+	0.815260i	$0.303406\pi$
$762$	0	0
$763$	5.10177	0.184696
$764$	1.40146	0.0507031
$765$	0	0
$766$	−18.4709	−0.667381
$767$	−5.78322	−0.208820
$768$	0	0
$769$	40.5215	1.46124	0.730622	−	0.682782i	$-0.239230\pi$
$769$	40.5215	1.46124	0.730622	+	0.682782i	$0.239230\pi$
$770$	−1.95055	−0.0702930
$771$	0	0
$772$	2.10479	0.0757531
$773$	−2.23455	−0.0803712	−0.0401856	−	0.999192i	$-0.512795\pi$
$773$	−2.23455	−0.0803712	−0.0401856	+	0.999192i	$0.512795\pi$
$774$	0	0
$775$	26.9261	0.967216
$776$	11.0295	0.395937
$777$	0	0
$778$	−43.1562	−1.54723
$779$	−5.31059	−0.190271
$780$	0	0
$781$	−12.2317	−0.437685
$782$	11.0407	0.394814
$783$	0	0
$784$	−4.34828	−0.155296
$785$	−31.3632	−1.11940
$786$	0	0
$787$	1.61067	0.0574143	0.0287071	−	0.999588i	$-0.490861\pi$
$787$	1.61067	0.0574143	0.0287071	+	0.999588i	$0.490861\pi$
$788$	−3.45618	−0.123121
$789$	0	0
$790$	29.2970	1.04234
$791$	4.08229	0.145149
$792$	0	0
$793$	9.39591	0.333659
$794$	−35.5451	−1.26145
$795$	0	0
$796$	2.58002	0.0914464
$797$	−32.9082	−1.16567	−0.582835	−	0.812591i	$-0.698056\pi$
$797$	−32.9082	−1.16567	−0.582835	+	0.812591i	$0.698056\pi$
$798$	0	0
$799$	50.6004	1.79011
$800$	−2.91469	−0.103050
$801$	0	0
$802$	−47.7936	−1.68765
$803$	9.49873	0.335203
$804$	0	0
$805$	−2.27943	−0.0803393
$806$	−15.5173	−0.546573
$807$	0	0
$808$	6.84169	0.240690
$809$	−26.1660	−0.919948	−0.459974	−	0.887932i	$-0.652141\pi$
$809$	−26.1660	−0.919948	−0.459974	+	0.887932i	$0.652141\pi$
$810$	0	0
$811$	−11.6615	−0.409492	−0.204746	−	0.978815i	$-0.565637\pi$
$811$	−11.6615	−0.409492	−0.204746	+	0.978815i	$0.565637\pi$
$812$	−0.738474	−0.0259154
$813$	0	0
$814$	−13.8122	−0.484118
$815$	5.79731	0.203071
$816$	0	0
$817$	−1.90368	−0.0666015
$818$	3.85987	0.134957
$819$	0	0
$820$	1.90960	0.0666863
$821$	1.77911	0.0620912	0.0310456	−	0.999518i	$-0.490116\pi$
$821$	1.77911	0.0620912	0.0310456	+	0.999518i	$0.490116\pi$
$822$	0	0
$823$	−2.09679	−0.0730894	−0.0365447	−	0.999332i	$-0.511635\pi$
$823$	−2.09679	−0.0730894	−0.0365447	+	0.999332i	$0.511635\pi$
$824$	−35.5345	−1.23790
$825$	0	0
$826$	8.20196	0.285383
$827$	2.83267	0.0985015	0.0492508	−	0.998786i	$-0.484317\pi$
$827$	2.83267	0.0985015	0.0492508	+	0.998786i	$0.484317\pi$
$828$	0	0
$829$	−24.7826	−0.860735	−0.430367	−	0.902654i	$-0.641616\pi$
$829$	−24.7826	−0.860735	−0.430367	+	0.902654i	$0.641616\pi$
$830$	35.9120	1.24652
$831$	0	0
$832$	−7.40037	−0.256562
$833$	4.97921	0.172519
$834$	0	0
$835$	−20.4112	−0.706358
$836$	0.136037	0.00470493
$837$	0	0
$838$	28.5894	0.987606
$839$	22.9922	0.793778	0.396889	−	0.917867i	$-0.370090\pi$
$839$	22.9922	0.793778	0.396889	+	0.917867i	$0.370090\pi$
$840$	0	0
$841$	−14.3148	−0.493613
$842$	−49.2479	−1.69720
$843$	0	0
$844$	−2.47199	−0.0850895
$845$	18.1295	0.623675
$846$	0	0
$847$	10.2512	0.352235
$848$	−36.0717	−1.23871
$849$	0	0
$850$	19.7807	0.678472
$851$	−16.1411	−0.553309
$852$	0	0
$853$	12.0576	0.412844	0.206422	−	0.978463i	$-0.433818\pi$
$853$	12.0576	0.412844	0.206422	+	0.978463i	$0.433818\pi$
$854$	−13.3256	−0.455993
$855$	0	0
$856$	−16.2333	−0.554842
$857$	−19.8401	−0.677725	−0.338863	−	0.940836i	$-0.610042\pi$
$857$	−19.8401	−0.677725	−0.338863	+	0.940836i	$0.610042\pi$
$858$	0	0
$859$	13.8046	0.471007	0.235503	−	0.971874i	$-0.424326\pi$
$859$	13.8046	0.471007	0.235503	+	0.971874i	$0.424326\pi$
$860$	0.684535	0.0233425
$861$	0	0
$862$	−50.1295	−1.70742
$863$	−51.8268	−1.76420	−0.882102	−	0.471058i	$-0.843872\pi$
$863$	−51.8268	−1.76420	−0.882102	+	0.471058i	$0.843872\pi$
$864$	0	0
$865$	26.8755	0.913793
$866$	−14.7725	−0.501990
$867$	0	0
$868$	1.93410	0.0656475
$869$	11.2471	0.381531
$870$	0	0
$871$	4.54882	0.154131
$872$	−13.6534	−0.462362
$873$	0	0
$874$	1.80888	0.0611863
$875$	−11.6950	−0.395364
$876$	0	0
$877$	10.1392	0.342375	0.171188	−	0.985238i	$-0.445240\pi$
$877$	10.1392	0.342375	0.171188	+	0.985238i	$0.445240\pi$
$878$	38.8000	1.30944
$879$	0	0
$880$	5.72775	0.193082
$881$	21.3641	0.719774	0.359887	−	0.932996i	$-0.382815\pi$
$881$	21.3641	0.719774	0.359887	+	0.932996i	$0.382815\pi$
$882$	0	0
$883$	40.9216	1.37712	0.688560	−	0.725179i	$-0.258243\pi$
$883$	40.9216	1.37712	0.688560	+	0.725179i	$0.258243\pi$
$884$	−1.00184	−0.0336954
$885$	0	0
$886$	3.02744	0.101709
$887$	8.78762	0.295059	0.147530	−	0.989058i	$-0.452868\pi$
$887$	8.78762	0.295059	0.147530	+	0.989058i	$0.452868\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−28.3848	−0.951462
$891$	0	0
$892$	−2.40231	−0.0804351
$893$	8.29026	0.277423
$894$	0	0
$895$	−35.1608	−1.17530
$896$	12.6683	0.423218
$897$	0	0
$898$	−19.6323	−0.655139
$899$	−38.4613	−1.28276
$900$	0	0
$901$	41.3057	1.37609
$902$	8.34152	0.277742
$903$	0	0
$904$	−10.9250	−0.363361
$905$	25.1563	0.836223
$906$	0	0
$907$	10.8368	0.359832	0.179916	−	0.983682i	$-0.442417\pi$
$907$	10.8368	0.359832	0.179916	+	0.983682i	$0.442417\pi$
$908$	2.50010	0.0829688
$909$	0	0
$910$	2.35349	0.0780175
$911$	9.36241	0.310190	0.155095	−	0.987900i	$-0.450432\pi$
$911$	9.36241	0.310190	0.155095	+	0.987900i	$0.450432\pi$
$912$	0	0
$913$	13.7865	0.456268
$914$	−14.2217	−0.470411
$915$	0	0
$916$	−0.488622	−0.0161445
$917$	−2.37886	−0.0785569
$918$	0	0
$919$	−23.6594	−0.780451	−0.390225	−	0.920719i	$-0.627603\pi$
$919$	−23.6594	−0.780451	−0.390225	+	0.920719i	$0.627603\pi$
$920$	6.10021	0.201118
$921$	0	0
$922$	15.7742	0.519497
$923$	14.7585	0.485783
$924$	0	0
$925$	−28.9186	−0.950838
$926$	−7.89358	−0.259399
$927$	0	0
$928$	4.16334	0.136668
$929$	−9.91065	−0.325158	−0.162579	−	0.986696i	$-0.551981\pi$
$929$	−9.91065	−0.325158	−0.162579	+	0.986696i	$0.551981\pi$
$930$	0	0
$931$	0.815782	0.0267362
$932$	−1.08332	−0.0354853
$933$	0	0
$934$	−8.32065	−0.272260
$935$	−6.55884	−0.214497
$936$	0	0
$937$	28.5637	0.933137	0.466568	−	0.884485i	$-0.345490\pi$
$937$	28.5637	0.933137	0.466568	+	0.884485i	$0.345490\pi$
$938$	−6.45129	−0.210642
$939$	0	0
$940$	−2.98105	−0.0972311
$941$	30.6305	0.998525	0.499262	−	0.866451i	$-0.333604\pi$
$941$	30.6305	0.998525	0.499262	+	0.866451i	$0.333604\pi$
$942$	0	0
$943$	9.74797	0.317437
$944$	−24.0849	−0.783896
$945$	0	0
$946$	2.99018	0.0972192
$947$	31.3812	1.01975	0.509876	−	0.860248i	$-0.329691\pi$
$947$	31.3812	1.01975	0.509876	+	0.860248i	$0.329691\pi$
$948$	0	0
$949$	−11.4610	−0.372039
$950$	3.24082	0.105146
$951$	0	0
$952$	−13.3254	−0.431878
$953$	33.0380	1.07021	0.535103	−	0.844787i	$-0.320273\pi$
$953$	33.0380	1.07021	0.535103	+	0.844787i	$0.320273\pi$
$954$	0	0
$955$	−11.0705	−0.358232
$956$	0.384552	0.0124373
$957$	0	0
$958$	−36.5010	−1.17929
$959$	22.3714	0.722410
$960$	0	0
$961$	69.7318	2.24941
$962$	16.6655	0.537318
$963$	0	0
$964$	−2.34350	−0.0754790
$965$	−16.6263	−0.535218
$966$	0	0
$967$	−3.84001	−0.123486	−0.0617432	−	0.998092i	$-0.519666\pi$
$967$	−3.84001	−0.123486	−0.0617432	+	0.998092i	$0.519666\pi$
$968$	−27.4343	−0.881770
$969$	0	0
$970$	9.28985	0.298279
$971$	11.5868	0.371837	0.185918	−	0.982565i	$-0.440474\pi$
$971$	11.5868	0.371837	0.185918	+	0.982565i	$0.440474\pi$
$972$	0	0
$973$	8.66152	0.277676
$974$	−29.0922	−0.932175
$975$	0	0
$976$	39.1303	1.25253
$977$	−17.8845	−0.572177	−0.286088	−	0.958203i	$-0.592355\pi$
$977$	−17.8845	−0.572177	−0.286088	+	0.958203i	$0.592355\pi$
$978$	0	0
$979$	−10.8969	−0.348266
$980$	−0.293343	−0.00937049
$981$	0	0
$982$	−55.5437	−1.77247
$983$	−20.2259	−0.645106	−0.322553	−	0.946551i	$-0.604541\pi$
$983$	−20.2259	−0.645106	−0.322553	+	0.946551i	$0.604541\pi$
$984$	0	0
$985$	27.3012	0.869888
$986$	−28.2547	−0.899813
$987$	0	0
$988$	−0.164139	−0.00522195
$989$	3.49435	0.111114
$990$	0	0
$991$	15.8972	0.504990	0.252495	−	0.967598i	$-0.418749\pi$
$991$	15.8972	0.504990	0.252495	+	0.967598i	$0.418749\pi$
$992$	−10.9040	−0.346201
$993$	0	0
$994$	−20.9310	−0.663892
$995$	−20.3802	−0.646096
$996$	0	0
$997$	−10.2702	−0.325261	−0.162631	−	0.986687i	$-0.551998\pi$
$997$	−10.2702	−0.325261	−0.162631	+	0.986687i	$0.551998\pi$
$998$	−60.9703	−1.92998
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.t.1.4		16
3.2	odd	2		889.2.a.c.1.13	✓	16
21.20	even	2		6223.2.a.k.1.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.13	✓	16	3.2	odd	2
6223.2.a.k.1.13		16	21.20	even	2
8001.2.a.t.1.4		16	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)

\(f(q)\)	\(=\)	\(q-1.48078 q^{2} +0.192706 q^{4} -1.52223 q^{5} -1.00000 q^{7} +2.67620 q^{8} +O(q^{10})\)	\(q-1.48078 q^{2} +0.192706 q^{4} -1.52223 q^{5} -1.00000 q^{7} +2.67620 q^{8} +2.25409 q^{10} +0.865340 q^{11} -1.04410 q^{13} +1.48078 q^{14} -4.34828 q^{16} +4.97921 q^{17} +0.815782 q^{19} -0.293343 q^{20} -1.28138 q^{22} -1.49743 q^{23} -2.68282 q^{25} +1.54608 q^{26} -0.192706 q^{28} +3.83213 q^{29} -10.0365 q^{31} +1.08643 q^{32} -7.37310 q^{34} +1.52223 q^{35} +10.7792 q^{37} -1.20799 q^{38} -4.07380 q^{40} -6.50981 q^{41} -2.33357 q^{43} +0.166756 q^{44} +2.21736 q^{46} +10.1623 q^{47} +1.00000 q^{49} +3.97266 q^{50} -0.201204 q^{52} +8.29563 q^{53} -1.31725 q^{55} -2.67620 q^{56} -5.67454 q^{58} +5.53895 q^{59} -8.99905 q^{61} +14.8619 q^{62} +7.08779 q^{64} +1.58936 q^{65} -4.35668 q^{67} +0.959522 q^{68} -2.25409 q^{70} -14.1352 q^{71} +10.9769 q^{73} -15.9616 q^{74} +0.157206 q^{76} -0.865340 q^{77} +12.9973 q^{79} +6.61908 q^{80} +9.63959 q^{82} +15.9319 q^{83} -7.57950 q^{85} +3.45550 q^{86} +2.31583 q^{88} -12.5926 q^{89} +1.04410 q^{91} -0.288563 q^{92} -15.0482 q^{94} -1.24181 q^{95} +4.12134 q^{97} -1.48078 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more