LMFDB - Embedded newform 8001.2.a.v.1.19

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:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Root			$-2.69590$ 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)

$f(q)$	$=$	$q+2.69590 q^{2} +5.26787 q^{4} -1.47463 q^{5} +1.00000 q^{7} +8.80983 q^{8} +O(q^{10})$	\(q+2.69590 q^{2} +5.26787 q^{4} -1.47463 q^{5} +1.00000 q^{7} +8.80983 q^{8} -3.97544 q^{10} +3.56252 q^{11} +0.225914 q^{13} +2.69590 q^{14} +13.2147 q^{16} -1.78744 q^{17} +4.11236 q^{19} -7.76814 q^{20} +9.60418 q^{22} +2.13512 q^{23} -2.82548 q^{25} +0.609042 q^{26} +5.26787 q^{28} +3.51167 q^{29} -2.58402 q^{31} +18.0058 q^{32} -4.81876 q^{34} -1.47463 q^{35} +1.05351 q^{37} +11.0865 q^{38} -12.9912 q^{40} -1.46417 q^{41} +7.65218 q^{43} +18.7669 q^{44} +5.75605 q^{46} -6.34127 q^{47} +1.00000 q^{49} -7.61719 q^{50} +1.19009 q^{52} -0.703666 q^{53} -5.25338 q^{55} +8.80983 q^{56} +9.46711 q^{58} -4.82613 q^{59} +7.39852 q^{61} -6.96626 q^{62} +22.1123 q^{64} -0.333140 q^{65} +9.46650 q^{67} -9.41600 q^{68} -3.97544 q^{70} -11.6949 q^{71} +10.0871 q^{73} +2.84016 q^{74} +21.6634 q^{76} +3.56252 q^{77} +4.56427 q^{79} -19.4867 q^{80} -3.94726 q^{82} +3.07538 q^{83} +2.63581 q^{85} +20.6295 q^{86} +31.3852 q^{88} +6.29747 q^{89} +0.225914 q^{91} +11.2475 q^{92} -17.0954 q^{94} -6.06420 q^{95} -11.5065 q^{97} +2.69590 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.69590	1.90629	0.953144	−	0.302517i	$-0.0978270\pi$
$2$	2.69590	1.90629	0.953144	+	0.302517i	$0.0978270\pi$
$3$	0	0
$3$	0	0
$4$	5.26787	2.63393
$5$	−1.47463	−0.659473	−0.329737	−	0.944073i	$-0.606960\pi$
$5$	−1.47463	−0.659473	−0.329737	+	0.944073i	$0.606960\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	8.80983	3.11475
$9$	0	0
$10$	−3.97544	−1.25715
$11$	3.56252	1.07414	0.537070	−	0.843538i	$-0.319531\pi$
$11$	3.56252	1.07414	0.537070	+	0.843538i	$0.319531\pi$
$12$	0	0
$13$	0.225914	0.0626574	0.0313287	−	0.999509i	$-0.490026\pi$
$13$	0.225914	0.0626574	0.0313287	+	0.999509i	$0.490026\pi$
$14$	2.69590	0.720509
$15$	0	0
$16$	13.2147	3.30367
$17$	−1.78744	−0.433518	−0.216759	−	0.976225i	$-0.569549\pi$
$17$	−1.78744	−0.433518	−0.216759	+	0.976225i	$0.569549\pi$
$18$	0	0
$19$	4.11236	0.943441	0.471720	−	0.881748i	$-0.343633\pi$
$19$	4.11236	0.943441	0.471720	+	0.881748i	$0.343633\pi$
$20$	−7.76814	−1.73701
$21$	0	0
$22$	9.60418	2.04762
$23$	2.13512	0.445202	0.222601	−	0.974910i	$-0.428545\pi$
$23$	2.13512	0.445202	0.222601	+	0.974910i	$0.428545\pi$
$24$	0	0
$25$	−2.82548	−0.565095
$26$	0.609042	0.119443
$27$	0	0
$28$	5.26787	0.995533
$29$	3.51167	0.652101	0.326051	−	0.945352i	$-0.394282\pi$
$29$	3.51167	0.652101	0.326051	+	0.945352i	$0.394282\pi$
$30$	0	0
$31$	−2.58402	−0.464104	−0.232052	−	0.972703i	$-0.574544\pi$
$31$	−2.58402	−0.464104	−0.232052	+	0.972703i	$0.574544\pi$
$32$	18.0058	3.18300
$33$	0	0
$34$	−4.81876	−0.826411
$35$	−1.47463	−0.249257
$36$	0	0
$37$	1.05351	0.173196	0.0865980	−	0.996243i	$-0.472400\pi$
$37$	1.05351	0.173196	0.0865980	+	0.996243i	$0.472400\pi$
$38$	11.0865	1.79847
$39$	0	0
$40$	−12.9912	−2.05409
$41$	−1.46417	−0.228665	−0.114333	−	0.993443i	$-0.536473\pi$
$41$	−1.46417	−0.228665	−0.114333	+	0.993443i	$0.536473\pi$
$42$	0	0
$43$	7.65218	1.16695	0.583473	−	0.812133i	$-0.301693\pi$
$43$	7.65218	1.16695	0.583473	+	0.812133i	$0.301693\pi$
$44$	18.7669	2.82921
$45$	0	0
$46$	5.75605	0.848684
$47$	−6.34127	−0.924969	−0.462485	−	0.886627i	$-0.653042\pi$
$47$	−6.34127	−0.924969	−0.462485	+	0.886627i	$0.653042\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−7.61719	−1.07723
$51$	0	0
$52$	1.19009	0.165035
$53$	−0.703666	−0.0966560	−0.0483280	−	0.998832i	$-0.515389\pi$
$53$	−0.703666	−0.0966560	−0.0483280	+	0.998832i	$0.515389\pi$
$54$	0	0
$55$	−5.25338	−0.708366
$56$	8.80983	1.17726
$57$	0	0
$58$	9.46711	1.24309
$59$	−4.82613	−0.628309	−0.314154	−	0.949372i	$-0.601721\pi$
$59$	−4.82613	−0.628309	−0.314154	+	0.949372i	$0.601721\pi$
$60$	0	0
$61$	7.39852	0.947283	0.473641	−	0.880718i	$-0.342939\pi$
$61$	7.39852	0.947283	0.473641	+	0.880718i	$0.342939\pi$
$62$	−6.96626	−0.884716
$63$	0	0
$64$	22.1123	2.76404
$65$	−0.333140	−0.0413209
$66$	0	0
$67$	9.46650	1.15652	0.578258	−	0.815854i	$-0.303733\pi$
$67$	9.46650	1.15652	0.578258	+	0.815854i	$0.303733\pi$
$68$	−9.41600	−1.14186
$69$	0	0
$70$	−3.97544	−0.475156
$71$	−11.6949	−1.38793	−0.693966	−	0.720008i	$-0.744138\pi$
$71$	−11.6949	−1.38793	−0.693966	+	0.720008i	$0.744138\pi$
$72$	0	0
$73$	10.0871	1.18061	0.590305	−	0.807181i	$-0.299008\pi$
$73$	10.0871	1.18061	0.590305	+	0.807181i	$0.299008\pi$
$74$	2.84016	0.330162
$75$	0	0
$76$	21.6634	2.48496
$77$	3.56252	0.405986
$78$	0	0
$79$	4.56427	0.513520	0.256760	−	0.966475i	$-0.417345\pi$
$79$	4.56427	0.513520	0.256760	+	0.966475i	$0.417345\pi$
$80$	−19.4867	−2.17868
$81$	0	0
$82$	−3.94726	−0.435902
$83$	3.07538	0.337566	0.168783	−	0.985653i	$-0.446016\pi$
$83$	3.07538	0.337566	0.168783	+	0.985653i	$0.446016\pi$
$84$	0	0
$85$	2.63581	0.285894
$86$	20.6295	2.22454
$87$	0	0
$88$	31.3852	3.34567
$89$	6.29747	0.667530	0.333765	−	0.942656i	$-0.391681\pi$
$89$	6.29747	0.667530	0.333765	+	0.942656i	$0.391681\pi$
$90$	0	0
$91$	0.225914	0.0236823
$92$	11.2475	1.17263
$93$	0	0
$94$	−17.0954	−1.76326
$95$	−6.06420	−0.622174
$96$	0	0
$97$	−11.5065	−1.16831	−0.584156	−	0.811641i	$-0.698574\pi$
$97$	−11.5065	−1.16831	−0.584156	+	0.811641i	$0.698574\pi$
$98$	2.69590	0.272327
$99$	0	0
$100$	−14.8842	−1.48842
$101$	−13.2282	−1.31626	−0.658129	−	0.752906i	$-0.728652\pi$
$101$	−13.2282	−1.31626	−0.658129	+	0.752906i	$0.728652\pi$
$102$	0	0
$103$	−0.287794	−0.0283572	−0.0141786	−	0.999899i	$-0.504513\pi$
$103$	−0.287794	−0.0283572	−0.0141786	+	0.999899i	$0.504513\pi$
$104$	1.99027	0.195162
$105$	0	0
$106$	−1.89701	−0.184254
$107$	2.14910	0.207761	0.103881	−	0.994590i	$-0.466874\pi$
$107$	2.14910	0.207761	0.103881	+	0.994590i	$0.466874\pi$
$108$	0	0
$109$	15.3817	1.47330	0.736648	−	0.676276i	$-0.236407\pi$
$109$	15.3817	1.47330	0.736648	+	0.676276i	$0.236407\pi$
$110$	−14.1626	−1.35035
$111$	0	0
$112$	13.2147	1.24867
$113$	−15.0444	−1.41525	−0.707627	−	0.706586i	$-0.750234\pi$
$113$	−15.0444	−1.41525	−0.707627	+	0.706586i	$0.750234\pi$
$114$	0	0
$115$	−3.14850	−0.293599
$116$	18.4990	1.71759
$117$	0	0
$118$	−13.0108	−1.19774
$119$	−1.78744	−0.163855
$120$	0	0
$121$	1.69152	0.153775
$122$	19.9456	1.80579
$123$	0	0
$124$	−13.6123	−1.22242
$125$	11.5397	1.03214
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	23.6011	2.08606
$129$	0	0
$130$	−0.898110	−0.0787695
$131$	8.23090	0.719137	0.359568	−	0.933119i	$-0.382924\pi$
$131$	8.23090	0.719137	0.359568	+	0.933119i	$0.382924\pi$
$132$	0	0
$133$	4.11236	0.356587
$134$	25.5207	2.20465
$135$	0	0
$136$	−15.7471	−1.35030
$137$	−12.9097	−1.10295	−0.551476	−	0.834191i	$-0.685935\pi$
$137$	−12.9097	−1.10295	−0.551476	+	0.834191i	$0.685935\pi$
$138$	0	0
$139$	1.60973	0.136536	0.0682678	−	0.997667i	$-0.478253\pi$
$139$	1.60973	0.136536	0.0682678	+	0.997667i	$0.478253\pi$
$140$	−7.76814	−0.656527
$141$	0	0
$142$	−31.5283	−2.64580
$143$	0.804824	0.0673028
$144$	0	0
$145$	−5.17841	−0.430043
$146$	27.1939	2.25058
$147$	0	0
$148$	5.54975	0.456187
$149$	7.63434	0.625430	0.312715	−	0.949847i	$-0.398762\pi$
$149$	7.63434	0.625430	0.312715	+	0.949847i	$0.398762\pi$
$150$	0	0
$151$	13.6790	1.11318	0.556592	−	0.830786i	$-0.312109\pi$
$151$	13.6790	1.11318	0.556592	+	0.830786i	$0.312109\pi$
$152$	36.2292	2.93858
$153$	0	0
$154$	9.60418	0.773927
$155$	3.81047	0.306064
$156$	0	0
$157$	1.96314	0.156676	0.0783378	−	0.996927i	$-0.475039\pi$
$157$	1.96314	0.156676	0.0783378	+	0.996927i	$0.475039\pi$
$158$	12.3048	0.978918
$159$	0	0
$160$	−26.5518	−2.09910
$161$	2.13512	0.168271
$162$	0	0
$163$	−13.5412	−1.06063	−0.530313	−	0.847802i	$-0.677926\pi$
$163$	−13.5412	−1.06063	−0.530313	+	0.847802i	$0.677926\pi$
$164$	−7.71306	−0.602289
$165$	0	0
$166$	8.29090	0.643499
$167$	−6.38113	−0.493787	−0.246893	−	0.969043i	$-0.579410\pi$
$167$	−6.38113	−0.493787	−0.246893	+	0.969043i	$0.579410\pi$
$168$	0	0
$169$	−12.9490	−0.996074
$170$	7.10587	0.544996
$171$	0	0
$172$	40.3106	3.07366
$173$	−4.05461	−0.308266	−0.154133	−	0.988050i	$-0.549258\pi$
$173$	−4.05461	−0.308266	−0.154133	+	0.988050i	$0.549258\pi$
$174$	0	0
$175$	−2.82548	−0.213586
$176$	47.0775	3.54860
$177$	0	0
$178$	16.9773	1.27250
$179$	2.72091	0.203370	0.101685	−	0.994817i	$-0.467577\pi$
$179$	2.72091	0.203370	0.101685	+	0.994817i	$0.467577\pi$
$180$	0	0
$181$	−9.49771	−0.705959	−0.352980	−	0.935631i	$-0.614832\pi$
$181$	−9.49771	−0.705959	−0.352980	+	0.935631i	$0.614832\pi$
$182$	0.609042	0.0451452
$183$	0	0
$184$	18.8100	1.38669
$185$	−1.55353	−0.114218
$186$	0	0
$187$	−6.36779	−0.465659
$188$	−33.4050	−2.43631
$189$	0	0
$190$	−16.3485	−1.18604
$191$	25.9208	1.87556	0.937782	−	0.347225i	$-0.112876\pi$
$191$	25.9208	1.87556	0.937782	+	0.347225i	$0.112876\pi$
$192$	0	0
$193$	21.7161	1.56316	0.781581	−	0.623804i	$-0.214414\pi$
$193$	21.7161	1.56316	0.781581	+	0.623804i	$0.214414\pi$
$194$	−31.0205	−2.22714
$195$	0	0
$196$	5.26787	0.376276
$197$	2.08379	0.148464	0.0742319	−	0.997241i	$-0.476349\pi$
$197$	2.08379	0.148464	0.0742319	+	0.997241i	$0.476349\pi$
$198$	0	0
$199$	−27.7717	−1.96868	−0.984340	−	0.176278i	$-0.943594\pi$
$199$	−27.7717	−1.96868	−0.984340	+	0.176278i	$0.943594\pi$
$200$	−24.8920	−1.76013
$201$	0	0
$202$	−35.6619	−2.50917
$203$	3.51167	0.246471
$204$	0	0
$205$	2.15911	0.150799
$206$	−0.775865	−0.0540570
$207$	0	0
$208$	2.98539	0.206999
$209$	14.6504	1.01339
$210$	0	0
$211$	9.29244	0.639718	0.319859	−	0.947465i	$-0.396364\pi$
$211$	9.29244	0.639718	0.319859	+	0.947465i	$0.396364\pi$
$212$	−3.70682	−0.254585
$213$	0	0
$214$	5.79376	0.396053
$215$	−11.2841	−0.769570
$216$	0	0
$217$	−2.58402	−0.175415
$218$	41.4674	2.80853
$219$	0	0
$220$	−27.6741	−1.86579
$221$	−0.403809	−0.0271631
$222$	0	0
$223$	7.34565	0.491901	0.245950	−	0.969282i	$-0.420900\pi$
$223$	7.34565	0.491901	0.245950	+	0.969282i	$0.420900\pi$
$224$	18.0058	1.20306
$225$	0	0
$226$	−40.5581	−2.69788
$227$	4.97373	0.330118	0.165059	−	0.986284i	$-0.447219\pi$
$227$	4.97373	0.330118	0.165059	+	0.986284i	$0.447219\pi$
$228$	0	0
$229$	15.3181	1.01225	0.506125	−	0.862460i	$-0.331078\pi$
$229$	15.3181	1.01225	0.506125	+	0.862460i	$0.331078\pi$
$230$	−8.48803	−0.559684
$231$	0	0
$232$	30.9372	2.03113
$233$	−13.1332	−0.860386	−0.430193	−	0.902737i	$-0.641554\pi$
$233$	−13.1332	−0.860386	−0.430193	+	0.902737i	$0.641554\pi$
$234$	0	0
$235$	9.35101	0.609993
$236$	−25.4234	−1.65492
$237$	0	0
$238$	−4.81876	−0.312354
$239$	−1.96196	−0.126909	−0.0634543	−	0.997985i	$-0.520212\pi$
$239$	−1.96196	−0.126909	−0.0634543	+	0.997985i	$0.520212\pi$
$240$	0	0
$241$	5.36551	0.345623	0.172811	−	0.984955i	$-0.444715\pi$
$241$	5.36551	0.345623	0.172811	+	0.984955i	$0.444715\pi$
$242$	4.56018	0.293139
$243$	0	0
$244$	38.9744	2.49508
$245$	−1.47463	−0.0942105
$246$	0	0
$247$	0.929042	0.0591135
$248$	−22.7648	−1.44557
$249$	0	0
$250$	31.1097	1.96755
$251$	8.17749	0.516159	0.258079	−	0.966124i	$-0.416910\pi$
$251$	8.17749	0.516159	0.258079	+	0.966124i	$0.416910\pi$
$252$	0	0
$253$	7.60639	0.478209
$254$	2.69590	0.169156
$255$	0	0
$256$	19.4015	1.21259
$257$	−16.4034	−1.02322	−0.511609	−	0.859218i	$-0.670951\pi$
$257$	−16.4034	−1.02322	−0.511609	+	0.859218i	$0.670951\pi$
$258$	0	0
$259$	1.05351	0.0654620
$260$	−1.75493	−0.108836
$261$	0	0
$262$	22.1897	1.37088
$263$	−22.1474	−1.36567	−0.682835	−	0.730573i	$-0.739253\pi$
$263$	−22.1474	−1.36567	−0.682835	+	0.730573i	$0.739253\pi$
$264$	0	0
$265$	1.03765	0.0637420
$266$	11.0865	0.679758
$267$	0	0
$268$	49.8683	3.04619
$269$	20.8780	1.27296	0.636479	−	0.771294i	$-0.280390\pi$
$269$	20.8780	1.27296	0.636479	+	0.771294i	$0.280390\pi$
$270$	0	0
$271$	−31.9527	−1.94099	−0.970496	−	0.241119i	$-0.922486\pi$
$271$	−31.9527	−1.94099	−0.970496	+	0.241119i	$0.922486\pi$
$272$	−23.6205	−1.43220
$273$	0	0
$274$	−34.8033	−2.10254
$275$	−10.0658	−0.606991
$276$	0	0
$277$	0.338014	0.0203093	0.0101546	−	0.999948i	$-0.496768\pi$
$277$	0.338014	0.0203093	0.0101546	+	0.999948i	$0.496768\pi$
$278$	4.33967	0.260276
$279$	0	0
$280$	−12.9912	−0.776374
$281$	−9.50219	−0.566853	−0.283427	−	0.958994i	$-0.591471\pi$
$281$	−9.50219	−0.566853	−0.283427	+	0.958994i	$0.591471\pi$
$282$	0	0
$283$	8.92113	0.530306	0.265153	−	0.964206i	$-0.414578\pi$
$283$	8.92113	0.530306	0.265153	+	0.964206i	$0.414578\pi$
$284$	−61.6073	−3.65572
$285$	0	0
$286$	2.16972	0.128298
$287$	−1.46417	−0.0864273
$288$	0	0
$289$	−13.8051	−0.812062
$290$	−13.9605	−0.819786
$291$	0	0
$292$	53.1376	3.10965
$293$	9.87457	0.576879	0.288439	−	0.957498i	$-0.406864\pi$
$293$	9.87457	0.576879	0.288439	+	0.957498i	$0.406864\pi$
$294$	0	0
$295$	7.11674	0.414353
$296$	9.28125	0.539462
$297$	0	0
$298$	20.5814	1.19225
$299$	0.482354	0.0278952
$300$	0	0
$301$	7.65218	0.441064
$302$	36.8773	2.12205
$303$	0	0
$304$	54.3436	3.11682
$305$	−10.9101	−0.624708
$306$	0	0
$307$	−27.4317	−1.56561	−0.782804	−	0.622269i	$-0.786211\pi$
$307$	−27.4317	−1.56561	−0.782804	+	0.622269i	$0.786211\pi$
$308$	18.7669	1.06934
$309$	0	0
$310$	10.2726	0.583447
$311$	−0.352596	−0.0199939	−0.00999693	−	0.999950i	$-0.503182\pi$
$311$	−0.352596	−0.0199939	−0.00999693	+	0.999950i	$0.503182\pi$
$312$	0	0
$313$	7.26435	0.410605	0.205303	−	0.978699i	$-0.434182\pi$
$313$	7.26435	0.410605	0.205303	+	0.978699i	$0.434182\pi$
$314$	5.29242	0.298669
$315$	0	0
$316$	24.0440	1.35258
$317$	−12.1028	−0.679764	−0.339882	−	0.940468i	$-0.610387\pi$
$317$	−12.1028	−0.679764	−0.339882	+	0.940468i	$0.610387\pi$
$318$	0	0
$319$	12.5104	0.700447
$320$	−32.6075	−1.82281
$321$	0	0
$322$	5.75605	0.320772
$323$	−7.35061	−0.408999
$324$	0	0
$325$	−0.638316	−0.0354074
$326$	−36.5056	−2.02186
$327$	0	0
$328$	−12.8991	−0.712234
$329$	−6.34127	−0.349606
$330$	0	0
$331$	−25.5738	−1.40566	−0.702832	−	0.711356i	$-0.748081\pi$
$331$	−25.5738	−1.40566	−0.702832	+	0.711356i	$0.748081\pi$
$332$	16.2007	0.889127
$333$	0	0
$334$	−17.2029	−0.941300
$335$	−13.9596	−0.762692
$336$	0	0
$337$	11.3205	0.616664	0.308332	−	0.951279i	$-0.400229\pi$
$337$	11.3205	0.616664	0.308332	+	0.951279i	$0.400229\pi$
$338$	−34.9091	−1.89880
$339$	0	0
$340$	13.8851	0.753025
$341$	−9.20562	−0.498513
$342$	0	0
$343$	1.00000	0.0539949
$344$	67.4144	3.63474
$345$	0	0
$346$	−10.9308	−0.587644
$347$	0.0553482	0.00297125	0.00148562	−	0.999999i	$-0.499527\pi$
$347$	0.0553482	0.00297125	0.00148562	+	0.999999i	$0.499527\pi$
$348$	0	0
$349$	25.8360	1.38297	0.691484	−	0.722391i	$-0.256957\pi$
$349$	25.8360	1.38297	0.691484	+	0.722391i	$0.256957\pi$
$350$	−7.61719	−0.407156
$351$	0	0
$352$	64.1458	3.41898
$353$	−7.72286	−0.411046	−0.205523	−	0.978652i	$-0.565890\pi$
$353$	−7.72286	−0.411046	−0.205523	+	0.978652i	$0.565890\pi$
$354$	0	0
$355$	17.2456	0.915304
$356$	33.1742	1.75823
$357$	0	0
$358$	7.33529	0.387682
$359$	−9.97746	−0.526590	−0.263295	−	0.964715i	$-0.584809\pi$
$359$	−9.97746	−0.526590	−0.263295	+	0.964715i	$0.584809\pi$
$360$	0	0
$361$	−2.08847	−0.109919
$362$	−25.6049	−1.34576
$363$	0	0
$364$	1.19009	0.0623775
$365$	−14.8748	−0.778580
$366$	0	0
$367$	10.8593	0.566849	0.283424	−	0.958995i	$-0.408529\pi$
$367$	10.8593	0.566849	0.283424	+	0.958995i	$0.408529\pi$
$368$	28.2149	1.47080
$369$	0	0
$370$	−4.18817	−0.217733
$371$	−0.703666	−0.0365325
$372$	0	0
$373$	3.59117	0.185944	0.0929718	−	0.995669i	$-0.470363\pi$
$373$	3.59117	0.185944	0.0929718	+	0.995669i	$0.470363\pi$
$374$	−17.1669	−0.887680
$375$	0	0
$376$	−55.8655	−2.88105
$377$	0.793337	0.0408590
$378$	0	0
$379$	−27.5754	−1.41645	−0.708226	−	0.705986i	$-0.750504\pi$
$379$	−27.5754	−1.41645	−0.708226	+	0.705986i	$0.750504\pi$
$380$	−31.9454	−1.63876
$381$	0	0
$382$	69.8799	3.57536
$383$	−19.8678	−1.01520	−0.507598	−	0.861594i	$-0.669466\pi$
$383$	−19.8678	−1.01520	−0.507598	+	0.861594i	$0.669466\pi$
$384$	0	0
$385$	−5.25338	−0.267737
$386$	58.5445	2.97984
$387$	0	0
$388$	−60.6149	−3.07726
$389$	15.1334	0.767296	0.383648	−	0.923479i	$-0.374668\pi$
$389$	15.1334	0.767296	0.383648	+	0.923479i	$0.374668\pi$
$390$	0	0
$391$	−3.81640	−0.193003
$392$	8.80983	0.444964
$393$	0	0
$394$	5.61768	0.283015
$395$	−6.73059	−0.338653
$396$	0	0
$397$	1.31022	0.0657578	0.0328789	−	0.999459i	$-0.489532\pi$
$397$	1.31022	0.0657578	0.0328789	+	0.999459i	$0.489532\pi$
$398$	−74.8696	−3.75287
$399$	0	0
$400$	−37.3378	−1.86689
$401$	−6.97286	−0.348208	−0.174104	−	0.984727i	$-0.555703\pi$
$401$	−6.97286	−0.348208	−0.174104	+	0.984727i	$0.555703\pi$
$402$	0	0
$403$	−0.583768	−0.0290796
$404$	−69.6845	−3.46693
$405$	0	0
$406$	9.46711	0.469845
$407$	3.75315	0.186037
$408$	0	0
$409$	7.41016	0.366409	0.183204	−	0.983075i	$-0.441353\pi$
$409$	7.41016	0.366409	0.183204	+	0.983075i	$0.441353\pi$
$410$	5.82073	0.287465
$411$	0	0
$412$	−1.51606	−0.0746911
$413$	−4.82613	−0.237478
$414$	0	0
$415$	−4.53503	−0.222616
$416$	4.06776	0.199438
$417$	0	0
$418$	39.4959	1.93181
$419$	−20.9835	−1.02511	−0.512555	−	0.858655i	$-0.671301\pi$
$419$	−20.9835	−1.02511	−0.512555	+	0.858655i	$0.671301\pi$
$420$	0	0
$421$	31.6075	1.54045	0.770227	−	0.637770i	$-0.220143\pi$
$421$	31.6075	1.54045	0.770227	+	0.637770i	$0.220143\pi$
$422$	25.0515	1.21949
$423$	0	0
$424$	−6.19918	−0.301059
$425$	5.05037	0.244979
$426$	0	0
$427$	7.39852	0.358039
$428$	11.3212	0.547230
$429$	0	0
$430$	−30.4208	−1.46702
$431$	−17.7328	−0.854158	−0.427079	−	0.904214i	$-0.640457\pi$
$431$	−17.7328	−0.854158	−0.427079	+	0.904214i	$0.640457\pi$
$432$	0	0
$433$	16.7334	0.804156	0.402078	−	0.915605i	$-0.368288\pi$
$433$	16.7334	0.804156	0.402078	+	0.915605i	$0.368288\pi$
$434$	−6.96626	−0.334391
$435$	0	0
$436$	81.0285	3.88056
$437$	8.78037	0.420022
$438$	0	0
$439$	−10.0050	−0.477511	−0.238756	−	0.971080i	$-0.576739\pi$
$439$	−10.0050	−0.477511	−0.238756	+	0.971080i	$0.576739\pi$
$440$	−46.2814	−2.20638
$441$	0	0
$442$	−1.08863	−0.0517807
$443$	−1.99760	−0.0949089	−0.0474544	−	0.998873i	$-0.515111\pi$
$443$	−1.99760	−0.0949089	−0.0474544	+	0.998873i	$0.515111\pi$
$444$	0	0
$445$	−9.28641	−0.440218
$446$	19.8031	0.937705
$447$	0	0
$448$	22.1123	1.04471
$449$	16.0463	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$449$	16.0463	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$450$	0	0
$451$	−5.21613	−0.245618
$452$	−79.2517	−3.72769
$453$	0	0
$454$	13.4087	0.629300
$455$	−0.333140	−0.0156178
$456$	0	0
$457$	−7.98051	−0.373313	−0.186656	−	0.982425i	$-0.559765\pi$
$457$	−7.98051	−0.373313	−0.186656	+	0.982425i	$0.559765\pi$
$458$	41.2961	1.92964
$459$	0	0
$460$	−16.5859	−0.773320
$461$	−0.274598	−0.0127893	−0.00639466	−	0.999980i	$-0.502035\pi$
$461$	−0.274598	−0.0127893	−0.00639466	+	0.999980i	$0.502035\pi$
$462$	0	0
$463$	−13.9300	−0.647384	−0.323692	−	0.946163i	$-0.604924\pi$
$463$	−13.9300	−0.647384	−0.323692	+	0.946163i	$0.604924\pi$
$464$	46.4056	2.15433
$465$	0	0
$466$	−35.4058	−1.64014
$467$	8.85623	0.409818	0.204909	−	0.978781i	$-0.434310\pi$
$467$	8.85623	0.409818	0.204909	+	0.978781i	$0.434310\pi$
$468$	0	0
$469$	9.46650	0.437122
$470$	25.2094	1.16282
$471$	0	0
$472$	−42.5174	−1.95702
$473$	27.2610	1.25346
$474$	0	0
$475$	−11.6194	−0.533134
$476$	−9.41600	−0.431582
$477$	0	0
$478$	−5.28924	−0.241924
$479$	−30.7285	−1.40402	−0.702010	−	0.712168i	$-0.747714\pi$
$479$	−30.7285	−1.40402	−0.702010	+	0.712168i	$0.747714\pi$
$480$	0	0
$481$	0.238003	0.0108520
$482$	14.4649	0.658856
$483$	0	0
$484$	8.91072	0.405033
$485$	16.9679	0.770471
$486$	0	0
$487$	−8.27967	−0.375188	−0.187594	−	0.982247i	$-0.560069\pi$
$487$	−8.27967	−0.375188	−0.187594	+	0.982247i	$0.560069\pi$
$488$	65.1797	2.95055
$489$	0	0
$490$	−3.97544	−0.179592
$491$	13.2902	0.599778	0.299889	−	0.953974i	$-0.403050\pi$
$491$	13.2902	0.599778	0.299889	+	0.953974i	$0.403050\pi$
$492$	0	0
$493$	−6.27691	−0.282698
$494$	2.50460	0.112687
$495$	0	0
$496$	−34.1470	−1.53325
$497$	−11.6949	−0.524589
$498$	0	0
$499$	11.9744	0.536048	0.268024	−	0.963412i	$-0.413629\pi$
$499$	11.9744	0.536048	0.268024	+	0.963412i	$0.413629\pi$
$500$	60.7894	2.71858
$501$	0	0
$502$	22.0457	0.983947
$503$	10.3059	0.459519	0.229759	−	0.973247i	$-0.426206\pi$
$503$	10.3059	0.459519	0.229759	+	0.973247i	$0.426206\pi$
$504$	0	0
$505$	19.5067	0.868036
$506$	20.5060	0.911605
$507$	0	0
$508$	5.26787	0.233724
$509$	−36.4192	−1.61425	−0.807126	−	0.590379i	$-0.798978\pi$
$509$	−36.4192	−1.61425	−0.807126	+	0.590379i	$0.798978\pi$
$510$	0	0
$511$	10.0871	0.446228
$512$	5.10215	0.225485
$513$	0	0
$514$	−44.2220	−1.95055
$515$	0.424389	0.0187008
$516$	0	0
$517$	−22.5909	−0.993546
$518$	2.84016	0.124789
$519$	0	0
$520$	−2.93490	−0.128704
$521$	−21.2597	−0.931404	−0.465702	−	0.884942i	$-0.654198\pi$
$521$	−21.2597	−0.931404	−0.465702	+	0.884942i	$0.654198\pi$
$522$	0	0
$523$	−37.1814	−1.62583	−0.812914	−	0.582384i	$-0.802120\pi$
$523$	−37.1814	−1.62583	−0.812914	+	0.582384i	$0.802120\pi$
$524$	43.3593	1.89416
$525$	0	0
$526$	−59.7072	−2.60336
$527$	4.61879	0.201198
$528$	0	0
$529$	−18.4413	−0.801795
$530$	2.79739	0.121511
$531$	0	0
$532$	21.6634	0.939227
$533$	−0.330777	−0.0143276
$534$	0	0
$535$	−3.16912	−0.137013
$536$	83.3983	3.60226
$537$	0	0
$538$	56.2851	2.42662
$539$	3.56252	0.153448
$540$	0	0
$541$	−44.0566	−1.89414	−0.947071	−	0.321024i	$-0.895973\pi$
$541$	−44.0566	−1.89414	−0.947071	+	0.321024i	$0.895973\pi$
$542$	−86.1414	−3.70009
$543$	0	0
$544$	−32.1843	−1.37989
$545$	−22.6822	−0.971599
$546$	0	0
$547$	−4.36781	−0.186754	−0.0933770	−	0.995631i	$-0.529766\pi$
$547$	−4.36781	−0.186754	−0.0933770	+	0.995631i	$0.529766\pi$
$548$	−68.0067	−2.90510
$549$	0	0
$550$	−27.1364	−1.15710
$551$	14.4413	0.615219
$552$	0	0
$553$	4.56427	0.194092
$554$	0.911251	0.0387154
$555$	0	0
$556$	8.47984	0.359626
$557$	−4.46834	−0.189330	−0.0946648	−	0.995509i	$-0.530178\pi$
$557$	−4.46834	−0.189330	−0.0946648	+	0.995509i	$0.530178\pi$
$558$	0	0
$559$	1.72874	0.0731178
$560$	−19.4867	−0.823464
$561$	0	0
$562$	−25.6169	−1.08058
$563$	−20.1736	−0.850214	−0.425107	−	0.905143i	$-0.639764\pi$
$563$	−20.1736	−0.850214	−0.425107	+	0.905143i	$0.639764\pi$
$564$	0	0
$565$	22.1848	0.933323
$566$	24.0505	1.01092
$567$	0	0
$568$	−103.030	−4.32306
$569$	−19.9292	−0.835475	−0.417738	−	0.908568i	$-0.637177\pi$
$569$	−19.9292	−0.835475	−0.417738	+	0.908568i	$0.637177\pi$
$570$	0	0
$571$	−23.3154	−0.975720	−0.487860	−	0.872922i	$-0.662222\pi$
$571$	−23.3154	−0.975720	−0.487860	+	0.872922i	$0.662222\pi$
$572$	4.23971	0.177271
$573$	0	0
$574$	−3.94726	−0.164755
$575$	−6.03272	−0.251582
$576$	0	0
$577$	3.13955	0.130701	0.0653507	−	0.997862i	$-0.479183\pi$
$577$	3.13955	0.130701	0.0653507	+	0.997862i	$0.479183\pi$
$578$	−37.2170	−1.54802
$579$	0	0
$580$	−27.2791	−1.13270
$581$	3.07538	0.127588
$582$	0	0
$583$	−2.50682	−0.103822
$584$	88.8659	3.67730
$585$	0	0
$586$	26.6208	1.09970
$587$	−24.9262	−1.02881	−0.514407	−	0.857546i	$-0.671988\pi$
$587$	−24.9262	−1.02881	−0.514407	+	0.857546i	$0.671988\pi$
$588$	0	0
$589$	−10.6264	−0.437855
$590$	19.1860	0.789876
$591$	0	0
$592$	13.9218	0.572183
$593$	41.7620	1.71496	0.857480	−	0.514517i	$-0.172029\pi$
$593$	41.7620	1.71496	0.857480	+	0.514517i	$0.172029\pi$
$594$	0	0
$595$	2.63581	0.108058
$596$	40.2167	1.64734
$597$	0	0
$598$	1.30038	0.0531763
$599$	31.5948	1.29093	0.645465	−	0.763790i	$-0.276664\pi$
$599$	31.5948	1.29093	0.645465	+	0.763790i	$0.276664\pi$
$600$	0	0
$601$	−43.7000	−1.78256	−0.891279	−	0.453454i	$-0.850191\pi$
$601$	−43.7000	−1.78256	−0.891279	+	0.453454i	$0.850191\pi$
$602$	20.6295	0.840795
$603$	0	0
$604$	72.0593	2.93205
$605$	−2.49437	−0.101410
$606$	0	0
$607$	−40.7124	−1.65246	−0.826232	−	0.563329i	$-0.809520\pi$
$607$	−40.7124	−1.65246	−0.826232	+	0.563329i	$0.809520\pi$
$608$	74.0462	3.00297
$609$	0	0
$610$	−29.4124	−1.19087
$611$	−1.43258	−0.0579562
$612$	0	0
$613$	33.8182	1.36590	0.682952	−	0.730464i	$-0.260696\pi$
$613$	33.8182	1.36590	0.682952	+	0.730464i	$0.260696\pi$
$614$	−73.9529	−2.98450
$615$	0	0
$616$	31.3852	1.26454
$617$	−10.0308	−0.403824	−0.201912	−	0.979404i	$-0.564715\pi$
$617$	−10.0308	−0.403824	−0.201912	+	0.979404i	$0.564715\pi$
$618$	0	0
$619$	−17.0587	−0.685647	−0.342824	−	0.939400i	$-0.611383\pi$
$619$	−17.0587	−0.685647	−0.342824	+	0.939400i	$0.611383\pi$
$620$	20.0730	0.806153
$621$	0	0
$622$	−0.950562	−0.0381141
$623$	6.29747	0.252303
$624$	0	0
$625$	−2.88931	−0.115572
$626$	19.5839	0.782731
$627$	0	0
$628$	10.3416	0.412673
$629$	−1.88309	−0.0750837
$630$	0	0
$631$	−27.9445	−1.11245	−0.556226	−	0.831031i	$-0.687751\pi$
$631$	−27.9445	−1.11245	−0.556226	+	0.831031i	$0.687751\pi$
$632$	40.2105	1.59949
$633$	0	0
$634$	−32.6280	−1.29582
$635$	−1.47463	−0.0585188
$636$	0	0
$637$	0.225914	0.00895106
$638$	33.7267	1.33525
$639$	0	0
$640$	−34.8028	−1.37570
$641$	−11.6463	−0.460002	−0.230001	−	0.973190i	$-0.573873\pi$
$641$	−11.6463	−0.460002	−0.230001	+	0.973190i	$0.573873\pi$
$642$	0	0
$643$	1.33579	0.0526785	0.0263393	−	0.999653i	$-0.491615\pi$
$643$	1.33579	0.0526785	0.0263393	+	0.999653i	$0.491615\pi$
$644$	11.2475	0.443214
$645$	0	0
$646$	−19.8165	−0.779670
$647$	18.7574	0.737428	0.368714	−	0.929543i	$-0.379798\pi$
$647$	18.7574	0.737428	0.368714	+	0.929543i	$0.379798\pi$
$648$	0	0
$649$	−17.1932	−0.674891
$650$	−1.72083	−0.0674967
$651$	0	0
$652$	−71.3331	−2.79362
$653$	22.5480	0.882370	0.441185	−	0.897416i	$-0.354558\pi$
$653$	22.5480	0.882370	0.441185	+	0.897416i	$0.354558\pi$
$654$	0	0
$655$	−12.1375	−0.474251
$656$	−19.3486	−0.755434
$657$	0	0
$658$	−17.0954	−0.666449
$659$	6.84788	0.266756	0.133378	−	0.991065i	$-0.457418\pi$
$659$	6.84788	0.266756	0.133378	+	0.991065i	$0.457418\pi$
$660$	0	0
$661$	29.5627	1.14985	0.574927	−	0.818205i	$-0.305030\pi$
$661$	29.5627	1.14985	0.574927	+	0.818205i	$0.305030\pi$
$662$	−68.9444	−2.67960
$663$	0	0
$664$	27.0936	1.05143
$665$	−6.06420	−0.235160
$666$	0	0
$667$	7.49783	0.290317
$668$	−33.6149	−1.30060
$669$	0	0
$670$	−37.6335	−1.45391
$671$	26.3573	1.01751
$672$	0	0
$673$	14.7263	0.567656	0.283828	−	0.958875i	$-0.408396\pi$
$673$	14.7263	0.567656	0.283828	+	0.958875i	$0.408396\pi$
$674$	30.5188	1.17554
$675$	0	0
$676$	−68.2134	−2.62359
$677$	24.8508	0.955095	0.477548	−	0.878606i	$-0.341526\pi$
$677$	24.8508	0.955095	0.477548	+	0.878606i	$0.341526\pi$
$678$	0	0
$679$	−11.5065	−0.441581
$680$	23.2210	0.890487
$681$	0	0
$682$	−24.8174	−0.950308
$683$	−19.4189	−0.743043	−0.371521	−	0.928424i	$-0.621164\pi$
$683$	−19.4189	−0.743043	−0.371521	+	0.928424i	$0.621164\pi$
$684$	0	0
$685$	19.0370	0.727367
$686$	2.69590	0.102930
$687$	0	0
$688$	101.121	3.85521
$689$	−0.158968	−0.00605621
$690$	0	0
$691$	14.1538	0.538435	0.269218	−	0.963079i	$-0.413235\pi$
$691$	14.1538	0.538435	0.269218	+	0.963079i	$0.413235\pi$
$692$	−21.3591	−0.811952
$693$	0	0
$694$	0.149213	0.00566405
$695$	−2.37375	−0.0900415
$696$	0	0
$697$	2.61712	0.0991305
$698$	69.6512	2.63634
$699$	0	0
$700$	−14.8842	−0.562571
$701$	32.1195	1.21314	0.606568	−	0.795032i	$-0.292546\pi$
$701$	32.1195	1.21314	0.606568	+	0.795032i	$0.292546\pi$
$702$	0	0
$703$	4.33242	0.163400
$704$	78.7756	2.96897
$705$	0	0
$706$	−20.8200	−0.783572
$707$	−13.2282	−0.497498
$708$	0	0
$709$	41.5911	1.56199	0.780993	−	0.624540i	$-0.214713\pi$
$709$	41.5911	1.56199	0.780993	+	0.624540i	$0.214713\pi$
$710$	46.4925	1.74483
$711$	0	0
$712$	55.4796	2.07919
$713$	−5.51719	−0.206620
$714$	0	0
$715$	−1.18682	−0.0443844
$716$	14.3334	0.535664
$717$	0	0
$718$	−26.8982	−1.00383
$719$	3.50030	0.130539	0.0652697	−	0.997868i	$-0.479209\pi$
$719$	3.50030	0.130539	0.0652697	+	0.997868i	$0.479209\pi$
$720$	0	0
$721$	−0.287794	−0.0107180
$722$	−5.63030	−0.209538
$723$	0	0
$724$	−50.0327	−1.85945
$725$	−9.92214	−0.368499
$726$	0	0
$727$	4.18230	0.155113	0.0775565	−	0.996988i	$-0.475288\pi$
$727$	4.18230	0.155113	0.0775565	+	0.996988i	$0.475288\pi$
$728$	1.99027	0.0737643
$729$	0	0
$730$	−40.1008	−1.48420
$731$	−13.6778	−0.505893
$732$	0	0
$733$	−40.3111	−1.48892	−0.744462	−	0.667664i	$-0.767294\pi$
$733$	−40.3111	−1.48892	−0.744462	+	0.667664i	$0.767294\pi$
$734$	29.2755	1.08058
$735$	0	0
$736$	38.4444	1.41708
$737$	33.7246	1.24226
$738$	0	0
$739$	0.374850	0.0137891	0.00689454	−	0.999976i	$-0.497805\pi$
$739$	0.374850	0.0137891	0.00689454	+	0.999976i	$0.497805\pi$
$740$	−8.18381	−0.300843
$741$	0	0
$742$	−1.89701	−0.0696415
$743$	−30.0132	−1.10108	−0.550539	−	0.834809i	$-0.685578\pi$
$743$	−30.0132	−1.10108	−0.550539	+	0.834809i	$0.685578\pi$
$744$	0	0
$745$	−11.2578	−0.412454
$746$	9.68142	0.354462
$747$	0	0
$748$	−33.5447	−1.22651
$749$	2.14910	0.0785264
$750$	0	0
$751$	6.64282	0.242400	0.121200	−	0.992628i	$-0.461326\pi$
$751$	6.64282	0.242400	0.121200	+	0.992628i	$0.461326\pi$
$752$	−83.7979	−3.05579
$753$	0	0
$754$	2.13876	0.0778889
$755$	−20.1715	−0.734115
$756$	0	0
$757$	45.9262	1.66922	0.834608	−	0.550844i	$-0.185694\pi$
$757$	45.9262	1.66922	0.834608	+	0.550844i	$0.185694\pi$
$758$	−74.3404	−2.70016
$759$	0	0
$760$	−53.4246	−1.93791
$761$	16.9692	0.615134	0.307567	−	0.951526i	$-0.400485\pi$
$761$	16.9692	0.615134	0.307567	+	0.951526i	$0.400485\pi$
$762$	0	0
$763$	15.3817	0.556854
$764$	136.547	4.94011
$765$	0	0
$766$	−53.5615	−1.93526
$767$	−1.09029	−0.0393682
$768$	0	0
$769$	−17.4485	−0.629211	−0.314605	−	0.949223i	$-0.601872\pi$
$769$	−17.4485	−0.629211	−0.314605	+	0.949223i	$0.601872\pi$
$770$	−14.1626	−0.510384
$771$	0	0
$772$	114.398	4.11726
$773$	24.3032	0.874127	0.437064	−	0.899431i	$-0.356018\pi$
$773$	24.3032	0.874127	0.437064	+	0.899431i	$0.356018\pi$
$774$	0	0
$775$	7.30109	0.262263
$776$	−101.371	−3.63900
$777$	0	0
$778$	40.7982	1.46269
$779$	−6.02120	−0.215732
$780$	0	0
$781$	−41.6634	−1.49083
$782$	−10.2886	−0.367920
$783$	0	0
$784$	13.2147	0.471953
$785$	−2.89490	−0.103323
$786$	0	0
$787$	13.9741	0.498124	0.249062	−	0.968488i	$-0.419878\pi$
$787$	13.9741	0.498124	0.249062	+	0.968488i	$0.419878\pi$
$788$	10.9771	0.391044
$789$	0	0
$790$	−18.1450	−0.645570
$791$	−15.0444	−0.534916
$792$	0	0
$793$	1.67143	0.0593543
$794$	3.53221	0.125353
$795$	0	0
$796$	−146.297	−5.18537
$797$	−14.3507	−0.508328	−0.254164	−	0.967161i	$-0.581800\pi$
$797$	−14.3507	−0.508328	−0.254164	+	0.967161i	$0.581800\pi$
$798$	0	0
$799$	11.3347	0.400991
$800$	−50.8748	−1.79870
$801$	0	0
$802$	−18.7981	−0.663785
$803$	35.9356	1.26814
$804$	0	0
$805$	−3.14850	−0.110970
$806$	−1.57378	−0.0554340
$807$	0	0
$808$	−116.538	−4.09981
$809$	−40.9336	−1.43915	−0.719575	−	0.694415i	$-0.755663\pi$
$809$	−40.9336	−1.43915	−0.719575	+	0.694415i	$0.755663\pi$
$810$	0	0
$811$	36.9592	1.29781	0.648906	−	0.760868i	$-0.275227\pi$
$811$	36.9592	1.29781	0.648906	+	0.760868i	$0.275227\pi$
$812$	18.4990	0.649188
$813$	0	0
$814$	10.1181	0.354639
$815$	19.9682	0.699455
$816$	0	0
$817$	31.4685	1.10094
$818$	19.9770	0.698480
$819$	0	0
$820$	11.3739	0.397193
$821$	−36.5291	−1.27487	−0.637436	−	0.770503i	$-0.720005\pi$
$821$	−36.5291	−1.27487	−0.637436	+	0.770503i	$0.720005\pi$
$822$	0	0
$823$	−4.79059	−0.166989	−0.0834947	−	0.996508i	$-0.526608\pi$
$823$	−4.79059	−0.166989	−0.0834947	+	0.996508i	$0.526608\pi$
$824$	−2.53542	−0.0883256
$825$	0	0
$826$	−13.0108	−0.452702
$827$	−25.5392	−0.888086	−0.444043	−	0.896005i	$-0.646456\pi$
$827$	−25.5392	−0.888086	−0.444043	+	0.896005i	$0.646456\pi$
$828$	0	0
$829$	−46.5382	−1.61634	−0.808169	−	0.588950i	$-0.799541\pi$
$829$	−46.5382	−1.61634	−0.808169	+	0.588950i	$0.799541\pi$
$830$	−12.2260	−0.424370
$831$	0	0
$832$	4.99550	0.173188
$833$	−1.78744	−0.0619312
$834$	0	0
$835$	9.40979	0.325639
$836$	77.1761	2.66919
$837$	0	0
$838$	−56.5693	−1.95415
$839$	−9.11328	−0.314625	−0.157313	−	0.987549i	$-0.550283\pi$
$839$	−9.11328	−0.314625	−0.157313	+	0.987549i	$0.550283\pi$
$840$	0	0
$841$	−16.6682	−0.574764
$842$	85.2105	2.93655
$843$	0	0
$844$	48.9513	1.68497
$845$	19.0949	0.656884
$846$	0	0
$847$	1.69152	0.0581215
$848$	−9.29872	−0.319319
$849$	0	0
$850$	13.6153	0.467001
$851$	2.24937	0.0771073
$852$	0	0
$853$	−10.1833	−0.348668	−0.174334	−	0.984687i	$-0.555777\pi$
$853$	−10.1833	−0.348668	−0.174334	+	0.984687i	$0.555777\pi$
$854$	19.9456	0.682526
$855$	0	0
$856$	18.9332	0.647124
$857$	−26.7522	−0.913838	−0.456919	−	0.889508i	$-0.651047\pi$
$857$	−26.7522	−0.913838	−0.456919	+	0.889508i	$0.651047\pi$
$858$	0	0
$859$	49.2129	1.67912	0.839561	−	0.543266i	$-0.182812\pi$
$859$	49.2129	1.67912	0.839561	+	0.543266i	$0.182812\pi$
$860$	−59.4432	−2.02700
$861$	0	0
$862$	−47.8058	−1.62827
$863$	42.3482	1.44155	0.720775	−	0.693169i	$-0.243786\pi$
$863$	42.3482	1.44155	0.720775	+	0.693169i	$0.243786\pi$
$864$	0	0
$865$	5.97903	0.203293
$866$	45.1116	1.53295
$867$	0	0
$868$	−13.6123	−0.462031
$869$	16.2603	0.551592
$870$	0	0
$871$	2.13862	0.0724643
$872$	135.510	4.58894
$873$	0	0
$874$	23.6710	0.800683
$875$	11.5397	0.390112
$876$	0	0
$877$	−15.5136	−0.523857	−0.261929	−	0.965087i	$-0.584359\pi$
$877$	−15.5136	−0.523857	−0.261929	+	0.965087i	$0.584359\pi$
$878$	−26.9724	−0.910273
$879$	0	0
$880$	−69.4218	−2.34021
$881$	8.51738	0.286958	0.143479	−	0.989653i	$-0.454171\pi$
$881$	8.51738	0.286958	0.143479	+	0.989653i	$0.454171\pi$
$882$	0	0
$883$	−7.04199	−0.236982	−0.118491	−	0.992955i	$-0.537806\pi$
$883$	−7.04199	−0.236982	−0.118491	+	0.992955i	$0.537806\pi$
$884$	−2.12721	−0.0715459
$885$	0	0
$886$	−5.38533	−0.180924
$887$	−12.0385	−0.404213	−0.202106	−	0.979364i	$-0.564779\pi$
$887$	−12.0385	−0.404213	−0.202106	+	0.979364i	$0.564779\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−25.0352	−0.839182
$891$	0	0
$892$	38.6959	1.29563
$893$	−26.0776	−0.872654
$894$	0	0
$895$	−4.01233	−0.134117
$896$	23.6011	0.788457
$897$	0	0
$898$	43.2592	1.44358
$899$	−9.07424	−0.302643
$900$	0	0
$901$	1.25776	0.0419021
$902$	−14.0622	−0.468219
$903$	0	0
$904$	−132.538	−4.40816
$905$	14.0056	0.465561
$906$	0	0
$907$	−34.0032	−1.12906	−0.564529	−	0.825413i	$-0.690942\pi$
$907$	−34.0032	−1.12906	−0.564529	+	0.825413i	$0.690942\pi$
$908$	26.2009	0.869509
$909$	0	0
$910$	−0.898110	−0.0297721
$911$	6.98648	0.231472	0.115736	−	0.993280i	$-0.463077\pi$
$911$	6.98648	0.231472	0.115736	+	0.993280i	$0.463077\pi$
$912$	0	0
$913$	10.9561	0.362593
$914$	−21.5146	−0.711641
$915$	0	0
$916$	80.6938	2.66620
$917$	8.23090	0.271808
$918$	0	0
$919$	−57.1332	−1.88465	−0.942325	−	0.334699i	$-0.891365\pi$
$919$	−57.1332	−1.88465	−0.942325	+	0.334699i	$0.891365\pi$
$920$	−27.7378	−0.914487
$921$	0	0
$922$	−0.740290	−0.0243801
$923$	−2.64205	−0.0869642
$924$	0	0
$925$	−2.97667	−0.0978722
$926$	−37.5540	−1.23410
$927$	0	0
$928$	63.2303	2.07564
$929$	13.2551	0.434885	0.217443	−	0.976073i	$-0.430229\pi$
$929$	13.2551	0.434885	0.217443	+	0.976073i	$0.430229\pi$
$930$	0	0
$931$	4.11236	0.134777
$932$	−69.1840	−2.26620
$933$	0	0
$934$	23.8755	0.781230
$935$	9.39012	0.307090
$936$	0	0
$937$	−37.0553	−1.21054	−0.605272	−	0.796018i	$-0.706936\pi$
$937$	−37.0553	−1.21054	−0.605272	+	0.796018i	$0.706936\pi$
$938$	25.5207	0.833281
$939$	0	0
$940$	49.2599	1.60668
$941$	−37.0926	−1.20919	−0.604593	−	0.796535i	$-0.706664\pi$
$941$	−37.0926	−1.20919	−0.604593	+	0.796535i	$0.706664\pi$
$942$	0	0
$943$	−3.12618	−0.101802
$944$	−63.7758	−2.07573
$945$	0	0
$946$	73.4929	2.38946
$947$	−34.5410	−1.12243	−0.561216	−	0.827669i	$-0.689666\pi$
$947$	−34.5410	−1.12243	−0.561216	+	0.827669i	$0.689666\pi$
$948$	0	0
$949$	2.27883	0.0739739
$950$	−31.3247	−1.01631
$951$	0	0
$952$	−15.7471	−0.510365
$953$	−59.7161	−1.93439	−0.967196	−	0.254030i	$-0.918244\pi$
$953$	−59.7161	−1.93439	−0.967196	+	0.254030i	$0.918244\pi$
$954$	0	0
$955$	−38.2235	−1.23688
$956$	−10.3353	−0.334269
$957$	0	0
$958$	−82.8408	−2.67646
$959$	−12.9097	−0.416877
$960$	0	0
$961$	−24.3228	−0.784607
$962$	0.641633	0.0206871
$963$	0	0
$964$	28.2648	0.910347
$965$	−32.0232	−1.03086
$966$	0	0
$967$	53.0691	1.70659	0.853293	−	0.521432i	$-0.174602\pi$
$967$	53.0691	1.70659	0.853293	+	0.521432i	$0.174602\pi$
$968$	14.9020	0.478970
$969$	0	0
$970$	45.7436	1.46874
$971$	−1.31432	−0.0421784	−0.0210892	−	0.999778i	$-0.506713\pi$
$971$	−1.31432	−0.0421784	−0.0210892	+	0.999778i	$0.506713\pi$
$972$	0	0
$973$	1.60973	0.0516056
$974$	−22.3212	−0.715216
$975$	0	0
$976$	97.7690	3.12951
$977$	−31.6311	−1.01197	−0.505985	−	0.862542i	$-0.668871\pi$
$977$	−31.6311	−1.01197	−0.505985	+	0.862542i	$0.668871\pi$
$978$	0	0
$979$	22.4348	0.717020
$980$	−7.76814	−0.248144
$981$	0	0
$982$	35.8290	1.14335
$983$	50.3689	1.60652	0.803260	−	0.595629i	$-0.203097\pi$
$983$	50.3689	1.60652	0.803260	+	0.595629i	$0.203097\pi$
$984$	0	0
$985$	−3.07281	−0.0979079
$986$	−16.9219	−0.538903
$987$	0	0
$988$	4.89407	0.155701
$989$	16.3383	0.519527
$990$	0	0
$991$	11.2592	0.357662	0.178831	−	0.983880i	$-0.442769\pi$
$991$	11.2592	0.357662	0.178831	+	0.983880i	$0.442769\pi$
$992$	−46.5273	−1.47724
$993$	0	0
$994$	−31.5283	−1.00002
$995$	40.9528	1.29829
$996$	0	0
$997$	20.9813	0.664485	0.332242	−	0.943194i	$-0.392195\pi$
$997$	20.9813	0.664485	0.332242	+	0.943194i	$0.392195\pi$
$998$	32.2818	1.02186
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.19		19
3.2	odd	2		2667.2.a.q.1.1	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.1	✓	19	3.2	odd	2
8001.2.a.v.1.19		19	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)

\(f(q)\)	\(=\)	\(q+2.69590 q^{2} +5.26787 q^{4} -1.47463 q^{5} +1.00000 q^{7} +8.80983 q^{8} +O(q^{10})\)	\(q+2.69590 q^{2} +5.26787 q^{4} -1.47463 q^{5} +1.00000 q^{7} +8.80983 q^{8} -3.97544 q^{10} +3.56252 q^{11} +0.225914 q^{13} +2.69590 q^{14} +13.2147 q^{16} -1.78744 q^{17} +4.11236 q^{19} -7.76814 q^{20} +9.60418 q^{22} +2.13512 q^{23} -2.82548 q^{25} +0.609042 q^{26} +5.26787 q^{28} +3.51167 q^{29} -2.58402 q^{31} +18.0058 q^{32} -4.81876 q^{34} -1.47463 q^{35} +1.05351 q^{37} +11.0865 q^{38} -12.9912 q^{40} -1.46417 q^{41} +7.65218 q^{43} +18.7669 q^{44} +5.75605 q^{46} -6.34127 q^{47} +1.00000 q^{49} -7.61719 q^{50} +1.19009 q^{52} -0.703666 q^{53} -5.25338 q^{55} +8.80983 q^{56} +9.46711 q^{58} -4.82613 q^{59} +7.39852 q^{61} -6.96626 q^{62} +22.1123 q^{64} -0.333140 q^{65} +9.46650 q^{67} -9.41600 q^{68} -3.97544 q^{70} -11.6949 q^{71} +10.0871 q^{73} +2.84016 q^{74} +21.6634 q^{76} +3.56252 q^{77} +4.56427 q^{79} -19.4867 q^{80} -3.94726 q^{82} +3.07538 q^{83} +2.63581 q^{85} +20.6295 q^{86} +31.3852 q^{88} +6.29747 q^{89} +0.225914 q^{91} +11.2475 q^{92} -17.0954 q^{94} -6.06420 q^{95} -11.5065 q^{97} +2.69590 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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