LMFDB - Embedded newform 8001.2.a.l.1.7

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:	$7$
Coefficient field:	7.7.118870813.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 9x^{5} - 3x^{4} + 20x^{3} + 7x^{2} - 13x - 4 $ x^7 - 9x^5 - 3x^4 + 20x^3 + 7x^2 - 13*x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.13462$ 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)

$f(q)$	$=$	$q+2.48001 q^{2} +4.15043 q^{4} +0.783950 q^{5} +1.00000 q^{7} +5.33307 q^{8} +O(q^{10})$	\(q+2.48001 q^{2} +4.15043 q^{4} +0.783950 q^{5} +1.00000 q^{7} +5.33307 q^{8} +1.94420 q^{10} -4.88599 q^{11} -6.65543 q^{13} +2.48001 q^{14} +4.92520 q^{16} -1.25567 q^{17} -2.07830 q^{19} +3.25373 q^{20} -12.1173 q^{22} -6.86976 q^{23} -4.38542 q^{25} -16.5055 q^{26} +4.15043 q^{28} +9.79804 q^{29} -3.45591 q^{31} +1.54837 q^{32} -3.11406 q^{34} +0.783950 q^{35} -2.67789 q^{37} -5.15419 q^{38} +4.18086 q^{40} +4.19977 q^{41} -4.11683 q^{43} -20.2789 q^{44} -17.0370 q^{46} +6.51041 q^{47} +1.00000 q^{49} -10.8759 q^{50} -27.6229 q^{52} -3.48731 q^{53} -3.83037 q^{55} +5.33307 q^{56} +24.2992 q^{58} -12.2268 q^{59} +6.13863 q^{61} -8.57067 q^{62} -6.01043 q^{64} -5.21753 q^{65} -15.0576 q^{67} -5.21155 q^{68} +1.94420 q^{70} +7.18924 q^{71} +2.24951 q^{73} -6.64118 q^{74} -8.62583 q^{76} -4.88599 q^{77} -7.38716 q^{79} +3.86111 q^{80} +10.4154 q^{82} +3.82920 q^{83} -0.984380 q^{85} -10.2098 q^{86} -26.0573 q^{88} +16.4372 q^{89} -6.65543 q^{91} -28.5124 q^{92} +16.1458 q^{94} -1.62928 q^{95} +16.8967 q^{97} +2.48001 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * 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 $

$2$	2.48001	1.75363	0.876814	−	0.480829i	$-0.159664\pi$
$2$	2.48001	1.75363	0.876814	+	0.480829i	$0.159664\pi$
$3$	0	0
$3$	0	0
$4$	4.15043	2.07521
$5$	0.783950	0.350593	0.175297	−	0.984516i	$-0.443912\pi$
$5$	0.783950	0.350593	0.175297	+	0.984516i	$0.443912\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	5.33307	1.88553
$9$	0	0
$10$	1.94420	0.614810
$11$	−4.88599	−1.47318	−0.736590	−	0.676340i	$-0.763565\pi$
$11$	−4.88599	−1.47318	−0.736590	+	0.676340i	$0.763565\pi$
$12$	0	0
$13$	−6.65543	−1.84589	−0.922943	−	0.384937i	$-0.874223\pi$
$13$	−6.65543	−1.84589	−0.922943	+	0.384937i	$0.874223\pi$
$14$	2.48001	0.662809
$15$	0	0
$16$	4.92520	1.23130
$17$	−1.25567	−0.304544	−0.152272	−	0.988339i	$-0.548659\pi$
$17$	−1.25567	−0.304544	−0.152272	+	0.988339i	$0.548659\pi$
$18$	0	0
$19$	−2.07830	−0.476795	−0.238397	−	0.971168i	$-0.576622\pi$
$19$	−2.07830	−0.476795	−0.238397	+	0.971168i	$0.576622\pi$
$20$	3.25373	0.727556
$21$	0	0
$22$	−12.1173	−2.58341
$23$	−6.86976	−1.43244	−0.716222	−	0.697872i	$-0.754130\pi$
$23$	−6.86976	−1.43244	−0.716222	+	0.697872i	$0.754130\pi$
$24$	0	0
$25$	−4.38542	−0.877084
$26$	−16.5055	−3.23700
$27$	0	0
$28$	4.15043	0.784357
$29$	9.79804	1.81945	0.909726	−	0.415210i	$-0.136292\pi$
$29$	9.79804	1.81945	0.909726	+	0.415210i	$0.136292\pi$
$30$	0	0
$31$	−3.45591	−0.620699	−0.310349	−	0.950623i	$-0.600446\pi$
$31$	−3.45591	−0.620699	−0.310349	+	0.950623i	$0.600446\pi$
$32$	1.54837	0.273715
$33$	0	0
$34$	−3.11406	−0.534057
$35$	0.783950	0.132512
$36$	0	0
$37$	−2.67789	−0.440242	−0.220121	−	0.975473i	$-0.570645\pi$
$37$	−2.67789	−0.440242	−0.220121	+	0.975473i	$0.570645\pi$
$38$	−5.15419	−0.836121
$39$	0	0
$40$	4.18086	0.661052
$41$	4.19977	0.655893	0.327947	−	0.944696i	$-0.393643\pi$
$41$	4.19977	0.655893	0.327947	+	0.944696i	$0.393643\pi$
$42$	0	0
$43$	−4.11683	−0.627811	−0.313906	−	0.949454i	$-0.601638\pi$
$43$	−4.11683	−0.627811	−0.313906	+	0.949454i	$0.601638\pi$
$44$	−20.2789	−3.05716
$45$	0	0
$46$	−17.0370	−2.51198
$47$	6.51041	0.949640	0.474820	−	0.880083i	$-0.342513\pi$
$47$	6.51041	0.949640	0.474820	+	0.880083i	$0.342513\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−10.8759	−1.53808
$51$	0	0
$52$	−27.6229	−3.83061
$53$	−3.48731	−0.479019	−0.239510	−	0.970894i	$-0.576987\pi$
$53$	−3.48731	−0.479019	−0.239510	+	0.970894i	$0.576987\pi$
$54$	0	0
$55$	−3.83037	−0.516487
$56$	5.33307	0.712662
$57$	0	0
$58$	24.2992	3.19064
$59$	−12.2268	−1.59180	−0.795899	−	0.605430i	$-0.793001\pi$
$59$	−12.2268	−1.59180	−0.795899	+	0.605430i	$0.793001\pi$
$60$	0	0
$61$	6.13863	0.785971	0.392986	−	0.919545i	$-0.371442\pi$
$61$	6.13863	0.785971	0.392986	+	0.919545i	$0.371442\pi$
$62$	−8.57067	−1.08848
$63$	0	0
$64$	−6.01043	−0.751304
$65$	−5.21753	−0.647155
$66$	0	0
$67$	−15.0576	−1.83958	−0.919790	−	0.392412i	$-0.871641\pi$
$67$	−15.0576	−1.83958	−0.919790	+	0.392412i	$0.871641\pi$
$68$	−5.21155	−0.631994
$69$	0	0
$70$	1.94420	0.232376
$71$	7.18924	0.853206	0.426603	−	0.904439i	$-0.359710\pi$
$71$	7.18924	0.853206	0.426603	+	0.904439i	$0.359710\pi$
$72$	0	0
$73$	2.24951	0.263285	0.131643	−	0.991297i	$-0.457975\pi$
$73$	2.24951	0.263285	0.131643	+	0.991297i	$0.457975\pi$
$74$	−6.64118	−0.772021
$75$	0	0
$76$	−8.62583	−0.989451
$77$	−4.88599	−0.556810
$78$	0	0
$79$	−7.38716	−0.831120	−0.415560	−	0.909566i	$-0.636414\pi$
$79$	−7.38716	−0.831120	−0.415560	+	0.909566i	$0.636414\pi$
$80$	3.86111	0.431685
$81$	0	0
$82$	10.4154	1.15019
$83$	3.82920	0.420310	0.210155	−	0.977668i	$-0.432603\pi$
$83$	3.82920	0.420310	0.210155	+	0.977668i	$0.432603\pi$
$84$	0	0
$85$	−0.984380	−0.106771
$86$	−10.2098	−1.10095
$87$	0	0
$88$	−26.0573	−2.77772
$89$	16.4372	1.74234	0.871171	−	0.490980i	$-0.163361\pi$
$89$	16.4372	1.74234	0.871171	+	0.490980i	$0.163361\pi$
$90$	0	0
$91$	−6.65543	−0.697679
$92$	−28.5124	−2.97263
$93$	0	0
$94$	16.1458	1.66532
$95$	−1.62928	−0.167161
$96$	0	0
$97$	16.8967	1.71560	0.857798	−	0.513987i	$-0.171832\pi$
$97$	16.8967	1.71560	0.857798	+	0.513987i	$0.171832\pi$
$98$	2.48001	0.250518
$99$	0	0
$100$	−18.2014	−1.82014
$101$	−0.691702	−0.0688269	−0.0344134	−	0.999408i	$-0.510956\pi$
$101$	−0.691702	−0.0688269	−0.0344134	+	0.999408i	$0.510956\pi$
$102$	0	0
$103$	−16.5225	−1.62801	−0.814003	−	0.580861i	$-0.802716\pi$
$103$	−16.5225	−1.62801	−0.814003	+	0.580861i	$0.802716\pi$
$104$	−35.4939	−3.48047
$105$	0	0
$106$	−8.64855	−0.840022
$107$	19.1211	1.84851	0.924255	−	0.381776i	$-0.124687\pi$
$107$	19.1211	1.84851	0.924255	+	0.381776i	$0.124687\pi$
$108$	0	0
$109$	14.7383	1.41167	0.705837	−	0.708374i	$-0.250571\pi$
$109$	14.7383	1.41167	0.705837	+	0.708374i	$0.250571\pi$
$110$	−9.49933	−0.905726
$111$	0	0
$112$	4.92520	0.465387
$113$	−12.5353	−1.17922	−0.589612	−	0.807686i	$-0.700719\pi$
$113$	−12.5353	−1.17922	−0.589612	+	0.807686i	$0.700719\pi$
$114$	0	0
$115$	−5.38555	−0.502205
$116$	40.6661	3.77575
$117$	0	0
$118$	−30.3226	−2.79142
$119$	−1.25567	−0.115107
$120$	0	0
$121$	12.8729	1.17026
$122$	15.2238	1.37830
$123$	0	0
$124$	−14.3435	−1.28808
$125$	−7.35770	−0.658093
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−18.0026	−1.59122
$129$	0	0
$130$	−12.9395	−1.13487
$131$	0.687639	0.0600793	0.0300397	−	0.999549i	$-0.490437\pi$
$131$	0.687639	0.0600793	0.0300397	+	0.999549i	$0.490437\pi$
$132$	0	0
$133$	−2.07830	−0.180211
$134$	−37.3429	−3.22594
$135$	0	0
$136$	−6.69656	−0.574225
$137$	7.96226	0.680262	0.340131	−	0.940378i	$-0.389529\pi$
$137$	7.96226	0.680262	0.340131	+	0.940378i	$0.389529\pi$
$138$	0	0
$139$	−0.410908	−0.0348528	−0.0174264	−	0.999848i	$-0.505547\pi$
$139$	−0.410908	−0.0348528	−0.0174264	+	0.999848i	$0.505547\pi$
$140$	3.25373	0.274990
$141$	0	0
$142$	17.8294	1.49621
$143$	32.5184	2.71932
$144$	0	0
$145$	7.68118	0.637887
$146$	5.57880	0.461705
$147$	0	0
$148$	−11.1144	−0.913597
$149$	−23.6085	−1.93408	−0.967041	−	0.254620i	$-0.918050\pi$
$149$	−23.6085	−1.93408	−0.967041	+	0.254620i	$0.918050\pi$
$150$	0	0
$151$	−6.43593	−0.523748	−0.261874	−	0.965102i	$-0.584341\pi$
$151$	−6.43593	−0.523748	−0.261874	+	0.965102i	$0.584341\pi$
$152$	−11.0837	−0.899009
$153$	0	0
$154$	−12.1173	−0.976438
$155$	−2.70926	−0.217613
$156$	0	0
$157$	−14.4987	−1.15712	−0.578561	−	0.815639i	$-0.696385\pi$
$157$	−14.4987	−1.15712	−0.578561	+	0.815639i	$0.696385\pi$
$158$	−18.3202	−1.45748
$159$	0	0
$160$	1.21384	0.0959627
$161$	−6.86976	−0.541413
$162$	0	0
$163$	8.06383	0.631608	0.315804	−	0.948824i	$-0.397726\pi$
$163$	8.06383	0.631608	0.315804	+	0.948824i	$0.397726\pi$
$164$	17.4308	1.36112
$165$	0	0
$166$	9.49645	0.737067
$167$	−4.60533	−0.356371	−0.178186	−	0.983997i	$-0.557023\pi$
$167$	−4.60533	−0.356371	−0.178186	+	0.983997i	$0.557023\pi$
$168$	0	0
$169$	31.2948	2.40729
$170$	−2.44127	−0.187237
$171$	0	0
$172$	−17.0866	−1.30284
$173$	11.5019	0.874475	0.437237	−	0.899346i	$-0.355957\pi$
$173$	11.5019	0.874475	0.437237	+	0.899346i	$0.355957\pi$
$174$	0	0
$175$	−4.38542	−0.331507
$176$	−24.0644	−1.81393
$177$	0	0
$178$	40.7644	3.05542
$179$	−22.8385	−1.70703	−0.853515	−	0.521068i	$-0.825534\pi$
$179$	−22.8385	−1.70703	−0.853515	+	0.521068i	$0.825534\pi$
$180$	0	0
$181$	4.40434	0.327372	0.163686	−	0.986512i	$-0.447662\pi$
$181$	4.40434	0.327372	0.163686	+	0.986512i	$0.447662\pi$
$182$	−16.5055	−1.22347
$183$	0	0
$184$	−36.6369	−2.70091
$185$	−2.09933	−0.154346
$186$	0	0
$187$	6.13517	0.448648
$188$	27.0210	1.97071
$189$	0	0
$190$	−4.04063	−0.293138
$191$	−20.5561	−1.48739	−0.743695	−	0.668520i	$-0.766928\pi$
$191$	−20.5561	−1.48739	−0.743695	+	0.668520i	$0.766928\pi$
$192$	0	0
$193$	−18.0715	−1.30082	−0.650409	−	0.759584i	$-0.725402\pi$
$193$	−18.0715	−1.30082	−0.650409	+	0.759584i	$0.725402\pi$
$194$	41.9038	3.00852
$195$	0	0
$196$	4.15043	0.296459
$197$	24.7109	1.76058	0.880291	−	0.474434i	$-0.157347\pi$
$197$	24.7109	1.76058	0.880291	+	0.474434i	$0.157347\pi$
$198$	0	0
$199$	16.6604	1.18102	0.590511	−	0.807030i	$-0.298926\pi$
$199$	16.6604	1.18102	0.590511	+	0.807030i	$0.298926\pi$
$200$	−23.3878	−1.65377
$201$	0	0
$202$	−1.71542	−0.120697
$203$	9.79804	0.687688
$204$	0	0
$205$	3.29241	0.229952
$206$	−40.9758	−2.85492
$207$	0	0
$208$	−32.7793	−2.27284
$209$	10.1545	0.702404
$210$	0	0
$211$	14.5672	1.00284	0.501422	−	0.865203i	$-0.332810\pi$
$211$	14.5672	1.00284	0.501422	+	0.865203i	$0.332810\pi$
$212$	−14.4738	−0.994067
$213$	0	0
$214$	47.4205	3.24160
$215$	−3.22739	−0.220106
$216$	0	0
$217$	−3.45591	−0.234602
$218$	36.5511	2.47555
$219$	0	0
$220$	−15.8977	−1.07182
$221$	8.35701	0.562153
$222$	0	0
$223$	−9.76267	−0.653757	−0.326878	−	0.945066i	$-0.605997\pi$
$223$	−9.76267	−0.653757	−0.326878	+	0.945066i	$0.605997\pi$
$224$	1.54837	0.103455
$225$	0	0
$226$	−31.0877	−2.06792
$227$	26.0747	1.73064	0.865321	−	0.501219i	$-0.167115\pi$
$227$	26.0747	1.73064	0.865321	+	0.501219i	$0.167115\pi$
$228$	0	0
$229$	6.24716	0.412824	0.206412	−	0.978465i	$-0.433821\pi$
$229$	6.24716	0.412824	0.206412	+	0.978465i	$0.433821\pi$
$230$	−13.3562	−0.880681
$231$	0	0
$232$	52.2537	3.43062
$233$	−6.77809	−0.444047	−0.222024	−	0.975041i	$-0.571266\pi$
$233$	−6.77809	−0.444047	−0.222024	+	0.975041i	$0.571266\pi$
$234$	0	0
$235$	5.10383	0.332937
$236$	−50.7466	−3.30332
$237$	0	0
$238$	−3.11406	−0.201855
$239$	23.7065	1.53345	0.766724	−	0.641977i	$-0.221885\pi$
$239$	23.7065	1.53345	0.766724	+	0.641977i	$0.221885\pi$
$240$	0	0
$241$	−9.28434	−0.598057	−0.299028	−	0.954244i	$-0.596663\pi$
$241$	−9.28434	−0.598057	−0.299028	+	0.954244i	$0.596663\pi$
$242$	31.9247	2.05220
$243$	0	0
$244$	25.4779	1.63106
$245$	0.783950	0.0500847
$246$	0	0
$247$	13.8320	0.880108
$248$	−18.4306	−1.17034
$249$	0	0
$250$	−18.2471	−1.15405
$251$	−0.539366	−0.0340445	−0.0170222	−	0.999855i	$-0.505419\pi$
$251$	−0.539366	−0.0340445	−0.0170222	+	0.999855i	$0.505419\pi$
$252$	0	0
$253$	33.5656	2.11025
$254$	−2.48001	−0.155609
$255$	0	0
$256$	−32.6258	−2.03911
$257$	4.16847	0.260022	0.130011	−	0.991513i	$-0.458499\pi$
$257$	4.16847	0.260022	0.130011	+	0.991513i	$0.458499\pi$
$258$	0	0
$259$	−2.67789	−0.166396
$260$	−21.6550	−1.34298
$261$	0	0
$262$	1.70535	0.105357
$263$	23.9281	1.47547	0.737734	−	0.675092i	$-0.235896\pi$
$263$	23.9281	1.47547	0.737734	+	0.675092i	$0.235896\pi$
$264$	0	0
$265$	−2.73388	−0.167941
$266$	−5.15419	−0.316024
$267$	0	0
$268$	−62.4955	−3.81752
$269$	30.9708	1.88832	0.944160	−	0.329487i	$-0.106876\pi$
$269$	30.9708	1.88832	0.944160	+	0.329487i	$0.106876\pi$
$270$	0	0
$271$	−23.8214	−1.44705	−0.723523	−	0.690300i	$-0.757478\pi$
$271$	−23.8214	−1.44705	−0.723523	+	0.690300i	$0.757478\pi$
$272$	−6.18440	−0.374985
$273$	0	0
$274$	19.7465	1.19293
$275$	21.4271	1.29210
$276$	0	0
$277$	−14.3357	−0.861351	−0.430675	−	0.902507i	$-0.641725\pi$
$277$	−14.3357	−0.861351	−0.430675	+	0.902507i	$0.641725\pi$
$278$	−1.01905	−0.0611188
$279$	0	0
$280$	4.18086	0.249854
$281$	−20.6212	−1.23016	−0.615079	−	0.788466i	$-0.710876\pi$
$281$	−20.6212	−1.23016	−0.615079	+	0.788466i	$0.710876\pi$
$282$	0	0
$283$	3.77609	0.224465	0.112233	−	0.993682i	$-0.464200\pi$
$283$	3.77609	0.224465	0.112233	+	0.993682i	$0.464200\pi$
$284$	29.8384	1.77058
$285$	0	0
$286$	80.6457	4.76868
$287$	4.19977	0.247904
$288$	0	0
$289$	−15.4233	−0.907253
$290$	19.0494	1.11862
$291$	0	0
$292$	9.33644	0.546374
$293$	24.2919	1.41915	0.709573	−	0.704632i	$-0.248888\pi$
$293$	24.2919	1.41915	0.709573	+	0.704632i	$0.248888\pi$
$294$	0	0
$295$	−9.58522	−0.558073
$296$	−14.2814	−0.830088
$297$	0	0
$298$	−58.5492	−3.39166
$299$	45.7212	2.64413
$300$	0	0
$301$	−4.11683	−0.237290
$302$	−15.9611	−0.918460
$303$	0	0
$304$	−10.2360	−0.587077
$305$	4.81238	0.275556
$306$	0	0
$307$	−18.6482	−1.06431	−0.532153	−	0.846648i	$-0.678617\pi$
$307$	−18.6482	−1.06431	−0.532153	+	0.846648i	$0.678617\pi$
$308$	−20.2789	−1.15550
$309$	0	0
$310$	−6.71897	−0.381612
$311$	−20.8747	−1.18370	−0.591848	−	0.806050i	$-0.701601\pi$
$311$	−20.8747	−1.18370	−0.591848	+	0.806050i	$0.701601\pi$
$312$	0	0
$313$	−7.93347	−0.448426	−0.224213	−	0.974540i	$-0.571981\pi$
$313$	−7.93347	−0.448426	−0.224213	+	0.974540i	$0.571981\pi$
$314$	−35.9568	−2.02916
$315$	0	0
$316$	−30.6599	−1.72475
$317$	−11.3644	−0.638290	−0.319145	−	0.947706i	$-0.603396\pi$
$317$	−11.3644	−0.638290	−0.319145	+	0.947706i	$0.603396\pi$
$318$	0	0
$319$	−47.8731	−2.68038
$320$	−4.71188	−0.263402
$321$	0	0
$322$	−17.0370	−0.949437
$323$	2.60965	0.145205
$324$	0	0
$325$	29.1869	1.61900
$326$	19.9983	1.10761
$327$	0	0
$328$	22.3977	1.23670
$329$	6.51041	0.358930
$330$	0	0
$331$	−19.1983	−1.05523	−0.527616	−	0.849483i	$-0.676914\pi$
$331$	−19.1983	−1.05523	−0.527616	+	0.849483i	$0.676914\pi$
$332$	15.8928	0.872233
$333$	0	0
$334$	−11.4213	−0.624943
$335$	−11.8044	−0.644944
$336$	0	0
$337$	8.90770	0.485234	0.242617	−	0.970122i	$-0.421994\pi$
$337$	8.90770	0.485234	0.242617	+	0.970122i	$0.421994\pi$
$338$	77.6113	4.22150
$339$	0	0
$340$	−4.08560	−0.221573
$341$	16.8855	0.914401
$342$	0	0
$343$	1.00000	0.0539949
$344$	−21.9554	−1.18375
$345$	0	0
$346$	28.5248	1.53350
$347$	1.17952	0.0633201	0.0316601	−	0.999499i	$-0.489921\pi$
$347$	1.17952	0.0633201	0.0316601	+	0.999499i	$0.489921\pi$
$348$	0	0
$349$	−29.5422	−1.58136	−0.790679	−	0.612231i	$-0.790272\pi$
$349$	−29.5422	−1.58136	−0.790679	+	0.612231i	$0.790272\pi$
$350$	−10.8759	−0.581340
$351$	0	0
$352$	−7.56530	−0.403232
$353$	−10.1955	−0.542653	−0.271327	−	0.962487i	$-0.587462\pi$
$353$	−10.1955	−0.542653	−0.271327	+	0.962487i	$0.587462\pi$
$354$	0	0
$355$	5.63601	0.299128
$356$	68.2215	3.61573
$357$	0	0
$358$	−56.6396	−2.99350
$359$	−30.6151	−1.61580	−0.807900	−	0.589319i	$-0.799396\pi$
$359$	−30.6151	−1.61580	−0.807900	+	0.589319i	$0.799396\pi$
$360$	0	0
$361$	−14.6807	−0.772667
$362$	10.9228	0.574089
$363$	0	0
$364$	−27.6229	−1.44783
$365$	1.76350	0.0923061
$366$	0	0
$367$	−10.8181	−0.564702	−0.282351	−	0.959311i	$-0.591114\pi$
$367$	−10.8181	−0.564702	−0.282351	+	0.959311i	$0.591114\pi$
$368$	−33.8349	−1.76377
$369$	0	0
$370$	−5.20635	−0.270665
$371$	−3.48731	−0.181052
$372$	0	0
$373$	10.2006	0.528169	0.264085	−	0.964499i	$-0.414930\pi$
$373$	10.2006	0.528169	0.264085	+	0.964499i	$0.414930\pi$
$374$	15.2153	0.786762
$375$	0	0
$376$	34.7205	1.79057
$377$	−65.2102	−3.35850
$378$	0	0
$379$	6.58231	0.338110	0.169055	−	0.985607i	$-0.445928\pi$
$379$	6.58231	0.338110	0.169055	+	0.985607i	$0.445928\pi$
$380$	−6.76222	−0.346895
$381$	0	0
$382$	−50.9793	−2.60833
$383$	20.2431	1.03438	0.517188	−	0.855872i	$-0.326979\pi$
$383$	20.2431	1.03438	0.517188	+	0.855872i	$0.326979\pi$
$384$	0	0
$385$	−3.83037	−0.195214
$386$	−44.8175	−2.28115
$387$	0	0
$388$	70.1283	3.56023
$389$	−8.10612	−0.410997	−0.205498	−	0.978657i	$-0.565882\pi$
$389$	−8.10612	−0.410997	−0.205498	+	0.978657i	$0.565882\pi$
$390$	0	0
$391$	8.62613	0.436242
$392$	5.33307	0.269361
$393$	0	0
$394$	61.2833	3.08741
$395$	−5.79116	−0.291385
$396$	0	0
$397$	−15.9348	−0.799745	−0.399872	−	0.916571i	$-0.630946\pi$
$397$	−15.9348	−0.799745	−0.399872	+	0.916571i	$0.630946\pi$
$398$	41.3178	2.07107
$399$	0	0
$400$	−21.5991	−1.07995
$401$	−35.9972	−1.79761	−0.898807	−	0.438344i	$-0.855565\pi$
$401$	−35.9972	−1.79761	−0.898807	+	0.438344i	$0.855565\pi$
$402$	0	0
$403$	23.0006	1.14574
$404$	−2.87086	−0.142831
$405$	0	0
$406$	24.2992	1.20595
$407$	13.0841	0.648556
$408$	0	0
$409$	−21.3091	−1.05367	−0.526833	−	0.849969i	$-0.676621\pi$
$409$	−21.3091	−1.05367	−0.526833	+	0.849969i	$0.676621\pi$
$410$	8.16519	0.403250
$411$	0	0
$412$	−68.5752	−3.37846
$413$	−12.2268	−0.601643
$414$	0	0
$415$	3.00190	0.147358
$416$	−10.3051	−0.505247
$417$	0	0
$418$	25.1833	1.23176
$419$	26.8399	1.31122	0.655608	−	0.755101i	$-0.272412\pi$
$419$	26.8399	1.31122	0.655608	+	0.755101i	$0.272412\pi$
$420$	0	0
$421$	1.71312	0.0834923	0.0417462	−	0.999128i	$-0.486708\pi$
$421$	1.71312	0.0834923	0.0417462	+	0.999128i	$0.486708\pi$
$422$	36.1266	1.75862
$423$	0	0
$424$	−18.5981	−0.903203
$425$	5.50663	0.267111
$426$	0	0
$427$	6.13863	0.297069
$428$	79.3609	3.83605
$429$	0	0
$430$	−8.00395	−0.385985
$431$	−29.8518	−1.43791	−0.718954	−	0.695057i	$-0.755379\pi$
$431$	−29.8518	−1.43791	−0.718954	+	0.695057i	$0.755379\pi$
$432$	0	0
$433$	−6.53077	−0.313849	−0.156924	−	0.987611i	$-0.550158\pi$
$433$	−6.53077	−0.313849	−0.156924	+	0.987611i	$0.550158\pi$
$434$	−8.57067	−0.411405
$435$	0	0
$436$	61.1703	2.92953
$437$	14.2774	0.682982
$438$	0	0
$439$	−14.9276	−0.712457	−0.356228	−	0.934399i	$-0.615938\pi$
$439$	−14.9276	−0.712457	−0.356228	+	0.934399i	$0.615938\pi$
$440$	−20.4276	−0.973849
$441$	0	0
$442$	20.7254	0.985808
$443$	4.63421	0.220178	0.110089	−	0.993922i	$-0.464886\pi$
$443$	4.63421	0.220178	0.110089	+	0.993922i	$0.464886\pi$
$444$	0	0
$445$	12.8860	0.610853
$446$	−24.2115	−1.14645
$447$	0	0
$448$	−6.01043	−0.283966
$449$	6.63579	0.313162	0.156581	−	0.987665i	$-0.449953\pi$
$449$	6.63579	0.313162	0.156581	+	0.987665i	$0.449953\pi$
$450$	0	0
$451$	−20.5200	−0.966249
$452$	−52.0270	−2.44714
$453$	0	0
$454$	64.6655	3.03490
$455$	−5.21753	−0.244601
$456$	0	0
$457$	−17.6826	−0.827157	−0.413578	−	0.910468i	$-0.635721\pi$
$457$	−17.6826	−0.827157	−0.413578	+	0.910468i	$0.635721\pi$
$458$	15.4930	0.723940
$459$	0	0
$460$	−22.3523	−1.04218
$461$	40.2668	1.87541	0.937706	−	0.347431i	$-0.112946\pi$
$461$	40.2668	1.87541	0.937706	+	0.347431i	$0.112946\pi$
$462$	0	0
$463$	−3.88536	−0.180568	−0.0902841	−	0.995916i	$-0.528777\pi$
$463$	−3.88536	−0.180568	−0.0902841	+	0.995916i	$0.528777\pi$
$464$	48.2573	2.24029
$465$	0	0
$466$	−16.8097	−0.778694
$467$	1.32929	0.0615123	0.0307561	−	0.999527i	$-0.490208\pi$
$467$	1.32929	0.0615123	0.0307561	+	0.999527i	$0.490208\pi$
$468$	0	0
$469$	−15.0576	−0.695296
$470$	12.6575	0.583849
$471$	0	0
$472$	−65.2066	−3.00138
$473$	20.1148	0.924879
$474$	0	0
$475$	9.11422	0.418189
$476$	−5.21155	−0.238871
$477$	0	0
$478$	58.7923	2.68910
$479$	10.9654	0.501022	0.250511	−	0.968114i	$-0.419401\pi$
$479$	10.9654	0.501022	0.250511	+	0.968114i	$0.419401\pi$
$480$	0	0
$481$	17.8225	0.812637
$482$	−23.0252	−1.04877
$483$	0	0
$484$	53.4278	2.42854
$485$	13.2461	0.601476
$486$	0	0
$487$	7.42363	0.336397	0.168198	−	0.985753i	$-0.446205\pi$
$487$	7.42363	0.336397	0.168198	+	0.985753i	$0.446205\pi$
$488$	32.7378	1.48197
$489$	0	0
$490$	1.94420	0.0878300
$491$	−10.4742	−0.472694	−0.236347	−	0.971669i	$-0.575950\pi$
$491$	−10.4742	−0.472694	−0.236347	+	0.971669i	$0.575950\pi$
$492$	0	0
$493$	−12.3031	−0.554103
$494$	34.3034	1.54338
$495$	0	0
$496$	−17.0210	−0.764266
$497$	7.18924	0.322482
$498$	0	0
$499$	26.6794	1.19433	0.597167	−	0.802117i	$-0.296293\pi$
$499$	26.6794	1.19433	0.597167	+	0.802117i	$0.296293\pi$
$500$	−30.5376	−1.36568
$501$	0	0
$502$	−1.33763	−0.0597013
$503$	−27.1665	−1.21129	−0.605646	−	0.795734i	$-0.707085\pi$
$503$	−27.1665	−1.21129	−0.605646	+	0.795734i	$0.707085\pi$
$504$	0	0
$505$	−0.542259	−0.0241302
$506$	83.2428	3.70059
$507$	0	0
$508$	−4.15043	−0.184145
$509$	5.67100	0.251363	0.125681	−	0.992071i	$-0.459888\pi$
$509$	5.67100	0.251363	0.125681	+	0.992071i	$0.459888\pi$
$510$	0	0
$511$	2.24951	0.0995126
$512$	−44.9069	−1.98462
$513$	0	0
$514$	10.3378	0.455982
$515$	−12.9528	−0.570767
$516$	0	0
$517$	−31.8098	−1.39899
$518$	−6.64118	−0.291797
$519$	0	0
$520$	−27.8255	−1.22023
$521$	−32.7163	−1.43333	−0.716664	−	0.697419i	$-0.754332\pi$
$521$	−32.7163	−1.43333	−0.716664	+	0.697419i	$0.754332\pi$
$522$	0	0
$523$	20.8450	0.911486	0.455743	−	0.890111i	$-0.349374\pi$
$523$	20.8450	0.911486	0.455743	+	0.890111i	$0.349374\pi$
$524$	2.85400	0.124677
$525$	0	0
$526$	59.3417	2.58742
$527$	4.33946	0.189030
$528$	0	0
$529$	24.1936	1.05190
$530$	−6.78003	−0.294506
$531$	0	0
$532$	−8.62583	−0.373977
$533$	−27.9513	−1.21070
$534$	0	0
$535$	14.9900	0.648075
$536$	−80.3033	−3.46857
$537$	0	0
$538$	76.8077	3.31141
$539$	−4.88599	−0.210454
$540$	0	0
$541$	−21.1270	−0.908319	−0.454159	−	0.890920i	$-0.650060\pi$
$541$	−21.1270	−0.908319	−0.454159	+	0.890920i	$0.650060\pi$
$542$	−59.0772	−2.53758
$543$	0	0
$544$	−1.94423	−0.0833583
$545$	11.5541	0.494923
$546$	0	0
$547$	10.1713	0.434895	0.217448	−	0.976072i	$-0.430227\pi$
$547$	10.1713	0.434895	0.217448	+	0.976072i	$0.430227\pi$
$548$	33.0468	1.41169
$549$	0	0
$550$	53.1394	2.26587
$551$	−20.3633	−0.867504
$552$	0	0
$553$	−7.38716	−0.314134
$554$	−35.5527	−1.51049
$555$	0	0
$556$	−1.70544	−0.0723270
$557$	29.5187	1.25075	0.625373	−	0.780326i	$-0.284947\pi$
$557$	29.5187	1.25075	0.625373	+	0.780326i	$0.284947\pi$
$558$	0	0
$559$	27.3993	1.15887
$560$	3.86111	0.163162
$561$	0	0
$562$	−51.1407	−2.15724
$563$	−29.4720	−1.24210	−0.621049	−	0.783772i	$-0.713293\pi$
$563$	−29.4720	−1.24210	−0.621049	+	0.783772i	$0.713293\pi$
$564$	0	0
$565$	−9.82707	−0.413428
$566$	9.36473	0.393629
$567$	0	0
$568$	38.3408	1.60874
$569$	−18.1639	−0.761471	−0.380735	−	0.924684i	$-0.624329\pi$
$569$	−18.1639	−0.761471	−0.380735	+	0.924684i	$0.624329\pi$
$570$	0	0
$571$	−5.20524	−0.217832	−0.108916	−	0.994051i	$-0.534738\pi$
$571$	−5.20524	−0.217832	−0.108916	+	0.994051i	$0.534738\pi$
$572$	134.965	5.64317
$573$	0	0
$574$	10.4154	0.434732
$575$	30.1268	1.25637
$576$	0	0
$577$	22.5036	0.936835	0.468418	−	0.883507i	$-0.344824\pi$
$577$	22.5036	0.936835	0.468418	+	0.883507i	$0.344824\pi$
$578$	−38.2499	−1.59099
$579$	0	0
$580$	31.8802	1.32375
$581$	3.82920	0.158862
$582$	0	0
$583$	17.0390	0.705681
$584$	11.9968	0.496432
$585$	0	0
$586$	60.2440	2.48866
$587$	11.1742	0.461208	0.230604	−	0.973048i	$-0.425930\pi$
$587$	11.1742	0.461208	0.230604	+	0.973048i	$0.425930\pi$
$588$	0	0
$589$	7.18241	0.295946
$590$	−23.7714	−0.978653
$591$	0	0
$592$	−13.1891	−0.542070
$593$	−10.7553	−0.441669	−0.220835	−	0.975311i	$-0.570878\pi$
$593$	−10.7553	−0.441669	−0.220835	+	0.975311i	$0.570878\pi$
$594$	0	0
$595$	−0.984380	−0.0403556
$596$	−97.9853	−4.01364
$597$	0	0
$598$	113.389	4.63682
$599$	−16.4493	−0.672099	−0.336049	−	0.941844i	$-0.609091\pi$
$599$	−16.4493	−0.672099	−0.336049	+	0.941844i	$0.609091\pi$
$600$	0	0
$601$	24.6614	1.00596	0.502979	−	0.864298i	$-0.332237\pi$
$601$	24.6614	1.00596	0.502979	+	0.864298i	$0.332237\pi$
$602$	−10.2098	−0.416119
$603$	0	0
$604$	−26.7118	−1.08689
$605$	10.0917	0.410285
$606$	0	0
$607$	−6.40510	−0.259975	−0.129988	−	0.991516i	$-0.541494\pi$
$607$	−6.40510	−0.259975	−0.129988	+	0.991516i	$0.541494\pi$
$608$	−3.21797	−0.130506
$609$	0	0
$610$	11.9347	0.483223
$611$	−43.3296	−1.75293
$612$	0	0
$613$	9.77647	0.394868	0.197434	−	0.980316i	$-0.436739\pi$
$613$	9.77647	0.394868	0.197434	+	0.980316i	$0.436739\pi$
$614$	−46.2475	−1.86640
$615$	0	0
$616$	−26.0573	−1.04988
$617$	−36.0074	−1.44960	−0.724802	−	0.688958i	$-0.758069\pi$
$617$	−36.0074	−1.44960	−0.724802	+	0.688958i	$0.758069\pi$
$618$	0	0
$619$	−18.4220	−0.740441	−0.370220	−	0.928944i	$-0.620718\pi$
$619$	−18.4220	−0.740441	−0.370220	+	0.928944i	$0.620718\pi$
$620$	−11.2446	−0.451593
$621$	0	0
$622$	−51.7694	−2.07576
$623$	16.4372	0.658543
$624$	0	0
$625$	16.1590	0.646362
$626$	−19.6750	−0.786373
$627$	0	0
$628$	−60.1758	−2.40127
$629$	3.36254	0.134073
$630$	0	0
$631$	−21.8713	−0.870682	−0.435341	−	0.900266i	$-0.643372\pi$
$631$	−21.8713	−0.870682	−0.435341	+	0.900266i	$0.643372\pi$
$632$	−39.3962	−1.56710
$633$	0	0
$634$	−28.1838	−1.11932
$635$	−0.783950	−0.0311101
$636$	0	0
$637$	−6.65543	−0.263698
$638$	−118.726	−4.70039
$639$	0	0
$640$	−14.1132	−0.557872
$641$	5.35545	0.211527	0.105764	−	0.994391i	$-0.466271\pi$
$641$	5.35545	0.211527	0.105764	+	0.994391i	$0.466271\pi$
$642$	0	0
$643$	12.1586	0.479488	0.239744	−	0.970836i	$-0.422936\pi$
$643$	12.1586	0.479488	0.239744	+	0.970836i	$0.422936\pi$
$644$	−28.5124	−1.12355
$645$	0	0
$646$	6.47195	0.254635
$647$	3.92615	0.154353	0.0771764	−	0.997017i	$-0.475410\pi$
$647$	3.92615	0.154353	0.0771764	+	0.997017i	$0.475410\pi$
$648$	0	0
$649$	59.7401	2.34500
$650$	72.3837	2.83912
$651$	0	0
$652$	33.4683	1.31072
$653$	−3.49414	−0.136736	−0.0683681	−	0.997660i	$-0.521779\pi$
$653$	−3.49414	−0.136736	−0.0683681	+	0.997660i	$0.521779\pi$
$654$	0	0
$655$	0.539075	0.0210634
$656$	20.6847	0.807601
$657$	0	0
$658$	16.1458	0.629431
$659$	−26.1167	−1.01736	−0.508682	−	0.860955i	$-0.669867\pi$
$659$	−26.1167	−1.01736	−0.508682	+	0.860955i	$0.669867\pi$
$660$	0	0
$661$	−34.1023	−1.32643	−0.663213	−	0.748431i	$-0.730808\pi$
$661$	−34.1023	−1.32643	−0.663213	+	0.748431i	$0.730808\pi$
$662$	−47.6118	−1.85049
$663$	0	0
$664$	20.4214	0.792505
$665$	−1.62928	−0.0631809
$666$	0	0
$667$	−67.3102	−2.60626
$668$	−19.1141	−0.739547
$669$	0	0
$670$	−29.2750	−1.13099
$671$	−29.9933	−1.15788
$672$	0	0
$673$	−11.3642	−0.438058	−0.219029	−	0.975718i	$-0.570289\pi$
$673$	−11.3642	−0.438058	−0.219029	+	0.975718i	$0.570289\pi$
$674$	22.0912	0.850920
$675$	0	0
$676$	129.887	4.99565
$677$	8.47890	0.325870	0.162935	−	0.986637i	$-0.447904\pi$
$677$	8.47890	0.325870	0.162935	+	0.986637i	$0.447904\pi$
$678$	0	0
$679$	16.8967	0.648434
$680$	−5.24977	−0.201319
$681$	0	0
$682$	41.8761	1.60352
$683$	−13.5516	−0.518537	−0.259269	−	0.965805i	$-0.583482\pi$
$683$	−13.5516	−0.518537	−0.259269	+	0.965805i	$0.583482\pi$
$684$	0	0
$685$	6.24201	0.238495
$686$	2.48001	0.0946871
$687$	0	0
$688$	−20.2762	−0.773023
$689$	23.2096	0.884214
$690$	0	0
$691$	−43.4677	−1.65359	−0.826794	−	0.562505i	$-0.809838\pi$
$691$	−43.4677	−1.65359	−0.826794	+	0.562505i	$0.809838\pi$
$692$	47.7379	1.81472
$693$	0	0
$694$	2.92522	0.111040
$695$	−0.322131	−0.0122191
$696$	0	0
$697$	−5.27351	−0.199748
$698$	−73.2649	−2.77312
$699$	0	0
$700$	−18.2014	−0.687947
$701$	−28.8144	−1.08831	−0.544153	−	0.838986i	$-0.683149\pi$
$701$	−28.8144	−1.08831	−0.544153	+	0.838986i	$0.683149\pi$
$702$	0	0
$703$	5.56545	0.209905
$704$	29.3669	1.10681
$705$	0	0
$706$	−25.2850	−0.951612
$707$	−0.691702	−0.0260141
$708$	0	0
$709$	−20.9862	−0.788154	−0.394077	−	0.919077i	$-0.628936\pi$
$709$	−20.9862	−0.788154	−0.394077	+	0.919077i	$0.628936\pi$
$710$	13.9773	0.524560
$711$	0	0
$712$	87.6609	3.28523
$713$	23.7412	0.889117
$714$	0	0
$715$	25.4928	0.953375
$716$	−94.7896	−3.54245
$717$	0	0
$718$	−75.9255	−2.83351
$719$	4.40085	0.164124	0.0820620	−	0.996627i	$-0.473849\pi$
$719$	4.40085	0.164124	0.0820620	+	0.996627i	$0.473849\pi$
$720$	0	0
$721$	−16.5225	−0.615328
$722$	−36.4081	−1.35497
$723$	0	0
$724$	18.2799	0.679368
$725$	−42.9686	−1.59581
$726$	0	0
$727$	21.6396	0.802568	0.401284	−	0.915954i	$-0.368564\pi$
$727$	21.6396	0.802568	0.401284	+	0.915954i	$0.368564\pi$
$728$	−35.4939	−1.31549
$729$	0	0
$730$	4.37350	0.161871
$731$	5.16937	0.191196
$732$	0	0
$733$	30.9010	1.14136	0.570678	−	0.821174i	$-0.306680\pi$
$733$	30.9010	1.14136	0.570678	+	0.821174i	$0.306680\pi$
$734$	−26.8291	−0.990279
$735$	0	0
$736$	−10.6369	−0.392082
$737$	73.5712	2.71003
$738$	0	0
$739$	−32.2746	−1.18724	−0.593620	−	0.804745i	$-0.702302\pi$
$739$	−32.2746	−1.18724	−0.593620	+	0.804745i	$0.702302\pi$
$740$	−8.71312	−0.320301
$741$	0	0
$742$	−8.64855	−0.317498
$743$	−28.5122	−1.04601	−0.523006	−	0.852329i	$-0.675190\pi$
$743$	−28.5122	−1.04601	−0.523006	+	0.852329i	$0.675190\pi$
$744$	0	0
$745$	−18.5079	−0.678076
$746$	25.2977	0.926213
$747$	0	0
$748$	25.4636	0.931040
$749$	19.1211	0.698671
$750$	0	0
$751$	−24.0517	−0.877659	−0.438829	−	0.898570i	$-0.644607\pi$
$751$	−24.0517	−0.877659	−0.438829	+	0.898570i	$0.644607\pi$
$752$	32.0650	1.16929
$753$	0	0
$754$	−161.722	−5.88956
$755$	−5.04544	−0.183623
$756$	0	0
$757$	−33.1952	−1.20650	−0.603249	−	0.797553i	$-0.706128\pi$
$757$	−33.1952	−1.20650	−0.603249	+	0.797553i	$0.706128\pi$
$758$	16.3242	0.592920
$759$	0	0
$760$	−8.68908	−0.315186
$761$	−27.5176	−0.997511	−0.498755	−	0.866743i	$-0.666209\pi$
$761$	−27.5176	−0.997511	−0.498755	+	0.866743i	$0.666209\pi$
$762$	0	0
$763$	14.7383	0.533563
$764$	−85.3167	−3.08665
$765$	0	0
$766$	50.2031	1.81391
$767$	81.3748	2.93828
$768$	0	0
$769$	−14.6074	−0.526757	−0.263378	−	0.964693i	$-0.584837\pi$
$769$	−14.6074	−0.526757	−0.263378	+	0.964693i	$0.584837\pi$
$770$	−9.49933	−0.342332
$771$	0	0
$772$	−75.0046	−2.69948
$773$	−4.97622	−0.178982	−0.0894912	−	0.995988i	$-0.528524\pi$
$773$	−4.97622	−0.178982	−0.0894912	+	0.995988i	$0.528524\pi$
$774$	0	0
$775$	15.1556	0.544405
$776$	90.1111	3.23480
$777$	0	0
$778$	−20.1032	−0.720736
$779$	−8.72837	−0.312726
$780$	0	0
$781$	−35.1265	−1.25693
$782$	21.3928	0.765007
$783$	0	0
$784$	4.92520	0.175900
$785$	−11.3662	−0.405679
$786$	0	0
$787$	20.1321	0.717633	0.358816	−	0.933408i	$-0.383180\pi$
$787$	20.1321	0.717633	0.358816	+	0.933408i	$0.383180\pi$
$788$	102.561	3.65358
$789$	0	0
$790$	−14.3621	−0.510981
$791$	−12.5353	−0.445705
$792$	0	0
$793$	−40.8553	−1.45081
$794$	−39.5184	−1.40246
$795$	0	0
$796$	69.1476	2.45087
$797$	35.3654	1.25271	0.626354	−	0.779539i	$-0.284546\pi$
$797$	35.3654	1.25271	0.626354	+	0.779539i	$0.284546\pi$
$798$	0	0
$799$	−8.17490	−0.289207
$800$	−6.79025	−0.240071
$801$	0	0
$802$	−89.2733	−3.15235
$803$	−10.9911	−0.387867
$804$	0	0
$805$	−5.38555	−0.189816
$806$	57.0415	2.00920
$807$	0	0
$808$	−3.68890	−0.129775
$809$	−9.88279	−0.347460	−0.173730	−	0.984793i	$-0.555582\pi$
$809$	−9.88279	−0.347460	−0.173730	+	0.984793i	$0.555582\pi$
$810$	0	0
$811$	29.3646	1.03113	0.515566	−	0.856850i	$-0.327582\pi$
$811$	29.3646	1.03113	0.515566	+	0.856850i	$0.327582\pi$
$812$	40.6661	1.42710
$813$	0	0
$814$	32.4487	1.13733
$815$	6.32164	0.221437
$816$	0	0
$817$	8.55601	0.299337
$818$	−52.8466	−1.84774
$819$	0	0
$820$	13.6649	0.477199
$821$	11.5855	0.404338	0.202169	−	0.979351i	$-0.435201\pi$
$821$	11.5855	0.404338	0.202169	+	0.979351i	$0.435201\pi$
$822$	0	0
$823$	32.4979	1.13281	0.566404	−	0.824128i	$-0.308334\pi$
$823$	32.4979	1.13281	0.566404	+	0.824128i	$0.308334\pi$
$824$	−88.1154	−3.06965
$825$	0	0
$826$	−30.3226	−1.05506
$827$	34.8327	1.21125	0.605626	−	0.795749i	$-0.292923\pi$
$827$	34.8327	1.21125	0.605626	+	0.795749i	$0.292923\pi$
$828$	0	0
$829$	25.0563	0.870241	0.435120	−	0.900372i	$-0.356706\pi$
$829$	25.0563	0.870241	0.435120	+	0.900372i	$0.356706\pi$
$830$	7.44474	0.258411
$831$	0	0
$832$	40.0020	1.38682
$833$	−1.25567	−0.0435063
$834$	0	0
$835$	−3.61035	−0.124941
$836$	42.1457	1.45764
$837$	0	0
$838$	66.5632	2.29939
$839$	16.0782	0.555082	0.277541	−	0.960714i	$-0.410481\pi$
$839$	16.0782	0.555082	0.277541	+	0.960714i	$0.410481\pi$
$840$	0	0
$841$	67.0017	2.31040
$842$	4.24855	0.146415
$843$	0	0
$844$	60.4599	2.08112
$845$	24.5336	0.843980
$846$	0	0
$847$	12.8729	0.442316
$848$	−17.1757	−0.589816
$849$	0	0
$850$	13.6565	0.468413
$851$	18.3965	0.630622
$852$	0	0
$853$	9.71995	0.332805	0.166402	−	0.986058i	$-0.446785\pi$
$853$	9.71995	0.332805	0.166402	+	0.986058i	$0.446785\pi$
$854$	15.2238	0.520949
$855$	0	0
$856$	101.974	3.48541
$857$	−41.2846	−1.41025	−0.705127	−	0.709081i	$-0.749110\pi$
$857$	−41.2846	−1.41025	−0.705127	+	0.709081i	$0.749110\pi$
$858$	0	0
$859$	−1.05236	−0.0359061	−0.0179531	−	0.999839i	$-0.505715\pi$
$859$	−1.05236	−0.0359061	−0.0179531	+	0.999839i	$0.505715\pi$
$860$	−13.3951	−0.456768
$861$	0	0
$862$	−74.0325	−2.52156
$863$	−34.4783	−1.17365	−0.586827	−	0.809712i	$-0.699623\pi$
$863$	−34.4783	−1.17365	−0.586827	+	0.809712i	$0.699623\pi$
$864$	0	0
$865$	9.01693	0.306585
$866$	−16.1963	−0.550374
$867$	0	0
$868$	−14.3435	−0.486850
$869$	36.0935	1.22439
$870$	0	0
$871$	100.215	3.39565
$872$	78.6005	2.66175
$873$	0	0
$874$	35.4081	1.19770
$875$	−7.35770	−0.248736
$876$	0	0
$877$	41.2483	1.39285	0.696427	−	0.717627i	$-0.254772\pi$
$877$	41.2483	1.39285	0.696427	+	0.717627i	$0.254772\pi$
$878$	−37.0206	−1.24939
$879$	0	0
$880$	−18.8653	−0.635950
$881$	−16.3701	−0.551523	−0.275761	−	0.961226i	$-0.588930\pi$
$881$	−16.3701	−0.551523	−0.275761	+	0.961226i	$0.588930\pi$
$882$	0	0
$883$	39.4843	1.32875	0.664376	−	0.747398i	$-0.268697\pi$
$883$	39.4843	1.32875	0.664376	+	0.747398i	$0.268697\pi$
$884$	34.6851	1.16659
$885$	0	0
$886$	11.4929	0.386110
$887$	14.0226	0.470831	0.235416	−	0.971895i	$-0.424355\pi$
$887$	14.0226	0.470831	0.235416	+	0.971895i	$0.424355\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	31.9572	1.07121
$891$	0	0
$892$	−40.5193	−1.35669
$893$	−13.5306	−0.452783
$894$	0	0
$895$	−17.9042	−0.598473
$896$	−18.0026	−0.601426
$897$	0	0
$898$	16.4568	0.549171
$899$	−33.8611	−1.12933
$900$	0	0
$901$	4.37890	0.145882
$902$	−50.8897	−1.69444
$903$	0	0
$904$	−66.8518	−2.22346
$905$	3.45278	0.114774
$906$	0	0
$907$	−35.6241	−1.18288	−0.591440	−	0.806349i	$-0.701440\pi$
$907$	−35.6241	−1.18288	−0.591440	+	0.806349i	$0.701440\pi$
$908$	108.221	3.59145
$909$	0	0
$910$	−12.9395	−0.428940
$911$	−4.45318	−0.147541	−0.0737703	−	0.997275i	$-0.523503\pi$
$911$	−4.45318	−0.147541	−0.0737703	+	0.997275i	$0.523503\pi$
$912$	0	0
$913$	−18.7094	−0.619192
$914$	−43.8529	−1.45053
$915$	0	0
$916$	25.9284	0.856698
$917$	0.687639	0.0227078
$918$	0	0
$919$	38.4633	1.26879	0.634394	−	0.773010i	$-0.281250\pi$
$919$	38.4633	1.26879	0.634394	+	0.773010i	$0.281250\pi$
$920$	−28.7215	−0.946921
$921$	0	0
$922$	99.8618	3.28877
$923$	−47.8475	−1.57492
$924$	0	0
$925$	11.7437	0.386130
$926$	−9.63572	−0.316649
$927$	0	0
$928$	15.1710	0.498012
$929$	−41.8359	−1.37259	−0.686295	−	0.727323i	$-0.740764\pi$
$929$	−41.8359	−1.37259	−0.686295	+	0.727323i	$0.740764\pi$
$930$	0	0
$931$	−2.07830	−0.0681135
$932$	−28.1320	−0.921493
$933$	0	0
$934$	3.29665	0.107870
$935$	4.80966	0.157293
$936$	0	0
$937$	27.1190	0.885938	0.442969	−	0.896537i	$-0.353925\pi$
$937$	27.1190	0.885938	0.442969	+	0.896537i	$0.353925\pi$
$938$	−37.3429	−1.21929
$939$	0	0
$940$	21.1831	0.690916
$941$	28.1483	0.917608	0.458804	−	0.888538i	$-0.348278\pi$
$941$	28.1483	0.917608	0.458804	+	0.888538i	$0.348278\pi$
$942$	0	0
$943$	−28.8514	−0.939531
$944$	−60.2195	−1.95998
$945$	0	0
$946$	49.8848	1.62189
$947$	−26.7498	−0.869251	−0.434625	−	0.900611i	$-0.643119\pi$
$947$	−26.7498	−0.869251	−0.434625	+	0.900611i	$0.643119\pi$
$948$	0	0
$949$	−14.9715	−0.485995
$950$	22.6033	0.733349
$951$	0	0
$952$	−6.69656	−0.217037
$953$	59.4329	1.92522	0.962611	−	0.270888i	$-0.0873174\pi$
$953$	59.4329	1.92522	0.962611	+	0.270888i	$0.0873174\pi$
$954$	0	0
$955$	−16.1150	−0.521468
$956$	98.3923	3.18223
$957$	0	0
$958$	27.1943	0.878607
$959$	7.96226	0.257115
$960$	0	0
$961$	−19.0567	−0.614733
$962$	44.1999	1.42506
$963$	0	0
$964$	−38.5340	−1.24110
$965$	−14.1672	−0.456058
$966$	0	0
$967$	−4.26909	−0.137285	−0.0686424	−	0.997641i	$-0.521867\pi$
$967$	−4.26909	−0.137285	−0.0686424	+	0.997641i	$0.521867\pi$
$968$	68.6519	2.20655
$969$	0	0
$970$	32.8505	1.05477
$971$	−53.2969	−1.71038	−0.855190	−	0.518315i	$-0.826559\pi$
$971$	−53.2969	−1.71038	−0.855190	+	0.518315i	$0.826559\pi$
$972$	0	0
$973$	−0.410908	−0.0131731
$974$	18.4106	0.589915
$975$	0	0
$976$	30.2340	0.967766
$977$	−19.3371	−0.618647	−0.309324	−	0.950957i	$-0.600103\pi$
$977$	−19.3371	−0.618647	−0.309324	+	0.950957i	$0.600103\pi$
$978$	0	0
$979$	−80.3120	−2.56678
$980$	3.25373	0.103937
$981$	0	0
$982$	−25.9761	−0.828930
$983$	38.4570	1.22659	0.613294	−	0.789855i	$-0.289844\pi$
$983$	38.4570	1.22659	0.613294	+	0.789855i	$0.289844\pi$
$984$	0	0
$985$	19.3721	0.617248
$986$	−30.5117	−0.971690
$987$	0	0
$988$	57.4087	1.82641
$989$	28.2817	0.899305
$990$	0	0
$991$	25.2038	0.800626	0.400313	−	0.916379i	$-0.368901\pi$
$991$	25.2038	0.800626	0.400313	+	0.916379i	$0.368901\pi$
$992$	−5.35101	−0.169895
$993$	0	0
$994$	17.8294	0.565513
$995$	13.0609	0.414058
$996$	0	0
$997$	42.0562	1.33193	0.665966	−	0.745982i	$-0.268019\pi$
$997$	42.0562	1.33193	0.665966	+	0.745982i	$0.268019\pi$
$998$	66.1651	2.09442
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.l.1.7		7
3.2	odd	2		2667.2.a.j.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.j.1.1	✓	7	3.2	odd	2
8001.2.a.l.1.7		7	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)

\(f(q)\)	\(=\)	\(q+2.48001 q^{2} +4.15043 q^{4} +0.783950 q^{5} +1.00000 q^{7} +5.33307 q^{8} +O(q^{10})\)	\(q+2.48001 q^{2} +4.15043 q^{4} +0.783950 q^{5} +1.00000 q^{7} +5.33307 q^{8} +1.94420 q^{10} -4.88599 q^{11} -6.65543 q^{13} +2.48001 q^{14} +4.92520 q^{16} -1.25567 q^{17} -2.07830 q^{19} +3.25373 q^{20} -12.1173 q^{22} -6.86976 q^{23} -4.38542 q^{25} -16.5055 q^{26} +4.15043 q^{28} +9.79804 q^{29} -3.45591 q^{31} +1.54837 q^{32} -3.11406 q^{34} +0.783950 q^{35} -2.67789 q^{37} -5.15419 q^{38} +4.18086 q^{40} +4.19977 q^{41} -4.11683 q^{43} -20.2789 q^{44} -17.0370 q^{46} +6.51041 q^{47} +1.00000 q^{49} -10.8759 q^{50} -27.6229 q^{52} -3.48731 q^{53} -3.83037 q^{55} +5.33307 q^{56} +24.2992 q^{58} -12.2268 q^{59} +6.13863 q^{61} -8.57067 q^{62} -6.01043 q^{64} -5.21753 q^{65} -15.0576 q^{67} -5.21155 q^{68} +1.94420 q^{70} +7.18924 q^{71} +2.24951 q^{73} -6.64118 q^{74} -8.62583 q^{76} -4.88599 q^{77} -7.38716 q^{79} +3.86111 q^{80} +10.4154 q^{82} +3.82920 q^{83} -0.984380 q^{85} -10.2098 q^{86} -26.0573 q^{88} +16.4372 q^{89} -6.65543 q^{91} -28.5124 q^{92} +16.1458 q^{94} -1.62928 q^{95} +16.8967 q^{97} +2.48001 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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