LMFDB - Embedded newform 8001.2.a.s.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
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} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 $ x^16 - 4x^15 - 18x^14 + 83x^13 + 112x^12 - 668x^11 - 235x^10 + 2648x^9 - 298x^8 - 5422x^7 + 2075x^6 + 5385x^5 - 3163x^4 - 1882x^3 + 1614x^2 - 200*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-1.35902$ 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)

$f(q)$	$=$	$q+1.35902 q^{2} -0.153057 q^{4} +2.54404 q^{5} -1.00000 q^{7} -2.92605 q^{8} +O(q^{10})$	\(q+1.35902 q^{2} -0.153057 q^{4} +2.54404 q^{5} -1.00000 q^{7} -2.92605 q^{8} +3.45741 q^{10} +2.34338 q^{11} +6.63119 q^{13} -1.35902 q^{14} -3.67046 q^{16} +0.289541 q^{17} +5.41849 q^{19} -0.389383 q^{20} +3.18471 q^{22} +4.52880 q^{23} +1.47214 q^{25} +9.01194 q^{26} +0.153057 q^{28} -1.87262 q^{29} -6.77646 q^{31} +0.863869 q^{32} +0.393493 q^{34} -2.54404 q^{35} -4.93282 q^{37} +7.36385 q^{38} -7.44400 q^{40} +8.23587 q^{41} +4.70763 q^{43} -0.358671 q^{44} +6.15474 q^{46} -7.01449 q^{47} +1.00000 q^{49} +2.00067 q^{50} -1.01495 q^{52} +5.73553 q^{53} +5.96166 q^{55} +2.92605 q^{56} -2.54493 q^{58} -0.869543 q^{59} -4.57288 q^{61} -9.20936 q^{62} +8.51494 q^{64} +16.8700 q^{65} -10.2654 q^{67} -0.0443163 q^{68} -3.45741 q^{70} -4.28238 q^{71} +8.87861 q^{73} -6.70382 q^{74} -0.829338 q^{76} -2.34338 q^{77} +13.1403 q^{79} -9.33780 q^{80} +11.1927 q^{82} -2.55190 q^{83} +0.736605 q^{85} +6.39777 q^{86} -6.85686 q^{88} -14.5511 q^{89} -6.63119 q^{91} -0.693164 q^{92} -9.53285 q^{94} +13.7849 q^{95} +9.24773 q^{97} +1.35902 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.35902	0.960974	0.480487	−	0.877002i	$-0.340460\pi$
$2$	1.35902	0.960974	0.480487	+	0.877002i	$0.340460\pi$
$3$	0	0
$3$	0	0
$4$	−0.153057	−0.0765285
$5$	2.54404	1.13773	0.568865	−	0.822431i	$-0.307383\pi$
$5$	2.54404	1.13773	0.568865	+	0.822431i	$0.307383\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.92605	−1.03452
$9$	0	0
$10$	3.45741	1.09333
$11$	2.34338	0.706556	0.353278	−	0.935518i	$-0.385067\pi$
$11$	2.34338	0.706556	0.353278	+	0.935518i	$0.385067\pi$
$12$	0	0
$13$	6.63119	1.83916	0.919581	−	0.392901i	$-0.128529\pi$
$13$	6.63119	1.83916	0.919581	+	0.392901i	$0.128529\pi$
$14$	−1.35902	−0.363214
$15$	0	0
$16$	−3.67046	−0.917615
$17$	0.289541	0.0702241	0.0351120	−	0.999383i	$-0.488821\pi$
$17$	0.289541	0.0702241	0.0351120	+	0.999383i	$0.488821\pi$
$18$	0	0
$19$	5.41849	1.24309	0.621543	−	0.783380i	$-0.286506\pi$
$19$	5.41849	1.24309	0.621543	+	0.783380i	$0.286506\pi$
$20$	−0.389383	−0.0870687
$21$	0	0
$22$	3.18471	0.678983
$23$	4.52880	0.944319	0.472160	−	0.881513i	$-0.343475\pi$
$23$	4.52880	0.944319	0.472160	+	0.881513i	$0.343475\pi$
$24$	0	0
$25$	1.47214	0.294428
$26$	9.01194	1.76739
$27$	0	0
$28$	0.153057	0.0289251
$29$	−1.87262	−0.347737	−0.173868	−	0.984769i	$-0.555627\pi$
$29$	−1.87262	−0.347737	−0.173868	+	0.984769i	$0.555627\pi$
$30$	0	0
$31$	−6.77646	−1.21709	−0.608544	−	0.793520i	$-0.708246\pi$
$31$	−6.77646	−1.21709	−0.608544	+	0.793520i	$0.708246\pi$
$32$	0.863869	0.152712
$33$	0	0
$34$	0.393493	0.0674835
$35$	−2.54404	−0.430021
$36$	0	0
$37$	−4.93282	−0.810951	−0.405475	−	0.914106i	$-0.632894\pi$
$37$	−4.93282	−0.810951	−0.405475	+	0.914106i	$0.632894\pi$
$38$	7.36385	1.19457
$39$	0	0
$40$	−7.44400	−1.17700
$41$	8.23587	1.28623	0.643113	−	0.765771i	$-0.277642\pi$
$41$	8.23587	1.28623	0.643113	+	0.765771i	$0.277642\pi$
$42$	0	0
$43$	4.70763	0.717906	0.358953	−	0.933356i	$-0.383134\pi$
$43$	4.70763	0.717906	0.358953	+	0.933356i	$0.383134\pi$
$44$	−0.358671	−0.0540717
$45$	0	0
$46$	6.15474	0.907466
$47$	−7.01449	−1.02317	−0.511584	−	0.859233i	$-0.670941\pi$
$47$	−7.01449	−1.02317	−0.511584	+	0.859233i	$0.670941\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.00067	0.282938
$51$	0	0
$52$	−1.01495	−0.140748
$53$	5.73553	0.787836	0.393918	−	0.919146i	$-0.371119\pi$
$53$	5.73553	0.787836	0.393918	+	0.919146i	$0.371119\pi$
$54$	0	0
$55$	5.96166	0.803870
$56$	2.92605	0.391010
$57$	0	0
$58$	−2.54493	−0.334166
$59$	−0.869543	−0.113205	−0.0566025	−	0.998397i	$-0.518027\pi$
$59$	−0.869543	−0.113205	−0.0566025	+	0.998397i	$0.518027\pi$
$60$	0	0
$61$	−4.57288	−0.585498	−0.292749	−	0.956189i	$-0.594570\pi$
$61$	−4.57288	−0.585498	−0.292749	+	0.956189i	$0.594570\pi$
$62$	−9.20936	−1.16959
$63$	0	0
$64$	8.51494	1.06437
$65$	16.8700	2.09247
$66$	0	0
$67$	−10.2654	−1.25412	−0.627059	−	0.778972i	$-0.715741\pi$
$67$	−10.2654	−1.25412	−0.627059	+	0.778972i	$0.715741\pi$
$68$	−0.0443163	−0.00537414
$69$	0	0
$70$	−3.45741	−0.413239
$71$	−4.28238	−0.508225	−0.254112	−	0.967175i	$-0.581783\pi$
$71$	−4.28238	−0.508225	−0.254112	+	0.967175i	$0.581783\pi$
$72$	0	0
$73$	8.87861	1.03916	0.519581	−	0.854421i	$-0.326088\pi$
$73$	8.87861	1.03916	0.519581	+	0.854421i	$0.326088\pi$
$74$	−6.70382	−0.779303
$75$	0	0
$76$	−0.829338	−0.0951316
$77$	−2.34338	−0.267053
$78$	0	0
$79$	13.1403	1.47839	0.739197	−	0.673489i	$-0.235205\pi$
$79$	13.1403	1.47839	0.739197	+	0.673489i	$0.235205\pi$
$80$	−9.33780	−1.04400
$81$	0	0
$82$	11.1927	1.23603
$83$	−2.55190	−0.280107	−0.140053	−	0.990144i	$-0.544727\pi$
$83$	−2.55190	−0.280107	−0.140053	+	0.990144i	$0.544727\pi$
$84$	0	0
$85$	0.736605	0.0798960
$86$	6.39777	0.689889
$87$	0	0
$88$	−6.85686	−0.730944
$89$	−14.5511	−1.54242	−0.771208	−	0.636584i	$-0.780347\pi$
$89$	−14.5511	−1.54242	−0.771208	+	0.636584i	$0.780347\pi$
$90$	0	0
$91$	−6.63119	−0.695138
$92$	−0.693164	−0.0722673
$93$	0	0
$94$	−9.53285	−0.983239
$95$	13.7849	1.41430
$96$	0	0
$97$	9.24773	0.938965	0.469482	−	0.882942i	$-0.344441\pi$
$97$	9.24773	0.938965	0.469482	+	0.882942i	$0.344441\pi$
$98$	1.35902	0.137282
$99$	0	0
$100$	−0.225321	−0.0225321
$101$	−9.46262	−0.941566	−0.470783	−	0.882249i	$-0.656029\pi$
$101$	−9.46262	−0.941566	−0.470783	+	0.882249i	$0.656029\pi$
$102$	0	0
$103$	17.7645	1.75039	0.875195	−	0.483770i	$-0.160733\pi$
$103$	17.7645	1.75039	0.875195	+	0.483770i	$0.160733\pi$
$104$	−19.4032	−1.90264
$105$	0	0
$106$	7.79472	0.757090
$107$	13.0844	1.26492	0.632458	−	0.774595i	$-0.282046\pi$
$107$	13.0844	1.26492	0.632458	+	0.774595i	$0.282046\pi$
$108$	0	0
$109$	8.79184	0.842106	0.421053	−	0.907036i	$-0.361661\pi$
$109$	8.79184	0.842106	0.421053	+	0.907036i	$0.361661\pi$
$110$	8.10203	0.772498
$111$	0	0
$112$	3.67046	0.346826
$113$	−5.04132	−0.474248	−0.237124	−	0.971479i	$-0.576205\pi$
$113$	−5.04132	−0.474248	−0.237124	+	0.971479i	$0.576205\pi$
$114$	0	0
$115$	11.5214	1.07438
$116$	0.286618	0.0266118
$117$	0	0
$118$	−1.18173	−0.108787
$119$	−0.289541	−0.0265422
$120$	0	0
$121$	−5.50856	−0.500778
$122$	−6.21465	−0.562648
$123$	0	0
$124$	1.03718	0.0931419
$125$	−8.97502	−0.802750
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	9.84426	0.870117
$129$	0	0
$130$	22.9267	2.01081
$131$	−1.83427	−0.160261	−0.0801303	−	0.996784i	$-0.525534\pi$
$131$	−1.83427	−0.160261	−0.0801303	+	0.996784i	$0.525534\pi$
$132$	0	0
$133$	−5.41849	−0.469843
$134$	−13.9509	−1.20517
$135$	0	0
$136$	−0.847213	−0.0726480
$137$	0.295919	0.0252820	0.0126410	−	0.999920i	$-0.495976\pi$
$137$	0.295919	0.0252820	0.0126410	+	0.999920i	$0.495976\pi$
$138$	0	0
$139$	−9.64612	−0.818173	−0.409087	−	0.912496i	$-0.634153\pi$
$139$	−9.64612	−0.818173	−0.409087	+	0.912496i	$0.634153\pi$
$140$	0.389383	0.0329089
$141$	0	0
$142$	−5.81985	−0.488391
$143$	15.5394	1.29947
$144$	0	0
$145$	−4.76402	−0.395630
$146$	12.0662	0.998609
$147$	0	0
$148$	0.755003	0.0620609
$149$	−22.3896	−1.83423	−0.917114	−	0.398624i	$-0.869488\pi$
$149$	−22.3896	−1.83423	−0.917114	+	0.398624i	$0.869488\pi$
$150$	0	0
$151$	19.4752	1.58487	0.792435	−	0.609956i	$-0.208813\pi$
$151$	19.4752	1.58487	0.792435	+	0.609956i	$0.208813\pi$
$152$	−15.8548	−1.28599
$153$	0	0
$154$	−3.18471	−0.256631
$155$	−17.2396	−1.38472
$156$	0	0
$157$	4.65498	0.371508	0.185754	−	0.982596i	$-0.440527\pi$
$157$	4.65498	0.371508	0.185754	+	0.982596i	$0.440527\pi$
$158$	17.8579	1.42070
$159$	0	0
$160$	2.19772	0.173745
$161$	−4.52880	−0.356919
$162$	0	0
$163$	18.9795	1.48659	0.743296	−	0.668963i	$-0.233262\pi$
$163$	18.9795	1.48659	0.743296	+	0.668963i	$0.233262\pi$
$164$	−1.26056	−0.0984330
$165$	0	0
$166$	−3.46809	−0.269176
$167$	15.6163	1.20842	0.604211	−	0.796824i	$-0.293488\pi$
$167$	15.6163	1.20842	0.604211	+	0.796824i	$0.293488\pi$
$168$	0	0
$169$	30.9727	2.38251
$170$	1.00106	0.0767780
$171$	0	0
$172$	−0.720535	−0.0549403
$173$	−11.2062	−0.851993	−0.425996	−	0.904725i	$-0.640076\pi$
$173$	−11.2062	−0.851993	−0.425996	+	0.904725i	$0.640076\pi$
$174$	0	0
$175$	−1.47214	−0.111283
$176$	−8.60129	−0.648347
$177$	0	0
$178$	−19.7753	−1.48222
$179$	15.0538	1.12517	0.562586	−	0.826739i	$-0.309807\pi$
$179$	15.0538	1.12517	0.562586	+	0.826739i	$0.309807\pi$
$180$	0	0
$181$	25.6108	1.90364	0.951819	−	0.306660i	$-0.0992115\pi$
$181$	25.6108	1.90364	0.951819	+	0.306660i	$0.0992115\pi$
$182$	−9.01194	−0.668009
$183$	0	0
$184$	−13.2515	−0.976914
$185$	−12.5493	−0.922643
$186$	0	0
$187$	0.678506	0.0496173
$188$	1.07362	0.0783016
$189$	0	0
$190$	18.7339	1.35910
$191$	19.2059	1.38969	0.694846	−	0.719159i	$-0.255473\pi$
$191$	19.2059	1.38969	0.694846	+	0.719159i	$0.255473\pi$
$192$	0	0
$193$	−8.77981	−0.631985	−0.315992	−	0.948762i	$-0.602337\pi$
$193$	−8.77981	−0.631985	−0.315992	+	0.948762i	$0.602337\pi$
$194$	12.5679	0.902321
$195$	0	0
$196$	−0.153057	−0.0109326
$197$	5.78587	0.412226	0.206113	−	0.978528i	$-0.433919\pi$
$197$	5.78587	0.412226	0.206113	+	0.978528i	$0.433919\pi$
$198$	0	0
$199$	−1.53549	−0.108848	−0.0544241	−	0.998518i	$-0.517332\pi$
$199$	−1.53549	−0.108848	−0.0544241	+	0.998518i	$0.517332\pi$
$200$	−4.30756	−0.304590
$201$	0	0
$202$	−12.8599	−0.904821
$203$	1.87262	0.131432
$204$	0	0
$205$	20.9524	1.46338
$206$	24.1424	1.68208
$207$	0	0
$208$	−24.3395	−1.68764
$209$	12.6976	0.878311
$210$	0	0
$211$	12.1099	0.833677	0.416839	−	0.908980i	$-0.363138\pi$
$211$	12.1099	0.833677	0.416839	+	0.908980i	$0.363138\pi$
$212$	−0.877864	−0.0602919
$213$	0	0
$214$	17.7820	1.21555
$215$	11.9764	0.816783
$216$	0	0
$217$	6.77646	0.460016
$218$	11.9483	0.809242
$219$	0	0
$220$	−0.912474	−0.0615190
$221$	1.92000	0.129153
$222$	0	0
$223$	13.4345	0.899643	0.449821	−	0.893119i	$-0.351488\pi$
$223$	13.4345	0.899643	0.449821	+	0.893119i	$0.351488\pi$
$224$	−0.863869	−0.0577197
$225$	0	0
$226$	−6.85127	−0.455740
$227$	−1.67653	−0.111275	−0.0556376	−	0.998451i	$-0.517719\pi$
$227$	−1.67653	−0.111275	−0.0556376	+	0.998451i	$0.517719\pi$
$228$	0	0
$229$	−19.2166	−1.26987	−0.634933	−	0.772567i	$-0.718972\pi$
$229$	−19.2166	−1.26987	−0.634933	+	0.772567i	$0.718972\pi$
$230$	15.6579	1.03245
$231$	0	0
$232$	5.47939	0.359739
$233$	11.9964	0.785912	0.392956	−	0.919557i	$-0.371452\pi$
$233$	11.9964	0.785912	0.392956	+	0.919557i	$0.371452\pi$
$234$	0	0
$235$	−17.8451	−1.16409
$236$	0.133090	0.00866340
$237$	0	0
$238$	−0.393493	−0.0255064
$239$	−0.719854	−0.0465635	−0.0232818	−	0.999729i	$-0.507411\pi$
$239$	−0.719854	−0.0465635	−0.0232818	+	0.999729i	$0.507411\pi$
$240$	0	0
$241$	−15.9444	−1.02707	−0.513534	−	0.858069i	$-0.671664\pi$
$241$	−15.9444	−1.02707	−0.513534	+	0.858069i	$0.671664\pi$
$242$	−7.48625	−0.481235
$243$	0	0
$244$	0.699912	0.0448073
$245$	2.54404	0.162533
$246$	0	0
$247$	35.9310	2.28624
$248$	19.8283	1.25910
$249$	0	0
$250$	−12.1973	−0.771422
$251$	17.9860	1.13527	0.567633	−	0.823282i	$-0.307859\pi$
$251$	17.9860	1.13527	0.567633	+	0.823282i	$0.307859\pi$
$252$	0	0
$253$	10.6127	0.667215
$254$	−1.35902	−0.0852727
$255$	0	0
$256$	−3.65131	−0.228207
$257$	−3.28178	−0.204712	−0.102356	−	0.994748i	$-0.532638\pi$
$257$	−3.28178	−0.204712	−0.102356	+	0.994748i	$0.532638\pi$
$258$	0	0
$259$	4.93282	0.306511
$260$	−2.58207	−0.160133
$261$	0	0
$262$	−2.49281	−0.154006
$263$	8.43804	0.520312	0.260156	−	0.965567i	$-0.416226\pi$
$263$	8.43804	0.520312	0.260156	+	0.965567i	$0.416226\pi$
$264$	0	0
$265$	14.5914	0.896344
$266$	−7.36385	−0.451507
$267$	0	0
$268$	1.57119	0.0959758
$269$	−6.07405	−0.370341	−0.185171	−	0.982706i	$-0.559284\pi$
$269$	−6.07405	−0.370341	−0.185171	+	0.982706i	$0.559284\pi$
$270$	0	0
$271$	−2.06286	−0.125310	−0.0626550	−	0.998035i	$-0.519957\pi$
$271$	−2.06286	−0.125310	−0.0626550	+	0.998035i	$0.519957\pi$
$272$	−1.06275	−0.0644387
$273$	0	0
$274$	0.402160	0.0242954
$275$	3.44979	0.208030
$276$	0	0
$277$	−7.63544	−0.458769	−0.229385	−	0.973336i	$-0.573671\pi$
$277$	−7.63544	−0.458769	−0.229385	+	0.973336i	$0.573671\pi$
$278$	−13.1093	−0.786243
$279$	0	0
$280$	7.44400	0.444864
$281$	−3.35625	−0.200217	−0.100108	−	0.994977i	$-0.531919\pi$
$281$	−3.35625	−0.200217	−0.100108	+	0.994977i	$0.531919\pi$
$282$	0	0
$283$	−10.1276	−0.602024	−0.301012	−	0.953620i	$-0.597324\pi$
$283$	−10.1276	−0.602024	−0.301012	+	0.953620i	$0.597324\pi$
$284$	0.655448	0.0388937
$285$	0	0
$286$	21.1184	1.24876
$287$	−8.23587	−0.486148
$288$	0	0
$289$	−16.9162	−0.995069
$290$	−6.47441	−0.380191
$291$	0	0
$292$	−1.35893	−0.0795256
$293$	5.53542	0.323383	0.161691	−	0.986841i	$-0.448305\pi$
$293$	5.53542	0.323383	0.161691	+	0.986841i	$0.448305\pi$
$294$	0	0
$295$	−2.21215	−0.128797
$296$	14.4337	0.838942
$297$	0	0
$298$	−30.4280	−1.76265
$299$	30.0313	1.73676
$300$	0	0
$301$	−4.70763	−0.271343
$302$	26.4673	1.52302
$303$	0	0
$304$	−19.8883	−1.14068
$305$	−11.6336	−0.666138
$306$	0	0
$307$	−5.34774	−0.305212	−0.152606	−	0.988287i	$-0.548767\pi$
$307$	−5.34774	−0.305212	−0.152606	+	0.988287i	$0.548767\pi$
$308$	0.358671	0.0204372
$309$	0	0
$310$	−23.4290	−1.33068
$311$	−31.1835	−1.76825	−0.884127	−	0.467246i	$-0.845246\pi$
$311$	−31.1835	−1.76825	−0.884127	+	0.467246i	$0.845246\pi$
$312$	0	0
$313$	27.2020	1.53755	0.768775	−	0.639520i	$-0.220867\pi$
$313$	27.2020	1.53755	0.768775	+	0.639520i	$0.220867\pi$
$314$	6.32623	0.357010
$315$	0	0
$316$	−2.01121	−0.113139
$317$	−16.8413	−0.945903	−0.472952	−	0.881088i	$-0.656811\pi$
$317$	−16.8413	−0.945903	−0.472952	+	0.881088i	$0.656811\pi$
$318$	0	0
$319$	−4.38827	−0.245696
$320$	21.6623	1.21096
$321$	0	0
$322$	−6.15474	−0.342990
$323$	1.56888	0.0872946
$324$	0	0
$325$	9.76204	0.541500
$326$	25.7936	1.42858
$327$	0	0
$328$	−24.0986	−1.33062
$329$	7.01449	0.386721
$330$	0	0
$331$	33.7745	1.85642	0.928208	−	0.372063i	$-0.121349\pi$
$331$	33.7745	1.85642	0.928208	+	0.372063i	$0.121349\pi$
$332$	0.390586	0.0214362
$333$	0	0
$334$	21.2228	1.16126
$335$	−26.1156	−1.42685
$336$	0	0
$337$	22.3423	1.21707	0.608533	−	0.793529i	$-0.291758\pi$
$337$	22.3423	1.21707	0.608533	+	0.793529i	$0.291758\pi$
$338$	42.0926	2.28954
$339$	0	0
$340$	−0.112743	−0.00611432
$341$	−15.8798	−0.859941
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−13.7748	−0.742686
$345$	0	0
$346$	−15.2295	−0.818743
$347$	−19.0034	−1.02016	−0.510078	−	0.860128i	$-0.670383\pi$
$347$	−19.0034	−1.02016	−0.510078	+	0.860128i	$0.670383\pi$
$348$	0	0
$349$	2.46216	0.131796	0.0658981	−	0.997826i	$-0.479009\pi$
$349$	2.46216	0.131796	0.0658981	+	0.997826i	$0.479009\pi$
$350$	−2.00067	−0.106940
$351$	0	0
$352$	2.02438	0.107900
$353$	−13.9188	−0.740824	−0.370412	−	0.928868i	$-0.620784\pi$
$353$	−13.9188	−0.740824	−0.370412	+	0.928868i	$0.620784\pi$
$354$	0	0
$355$	−10.8945	−0.578222
$356$	2.22715	0.118039
$357$	0	0
$358$	20.4584	1.08126
$359$	−31.8327	−1.68007	−0.840033	−	0.542535i	$-0.817465\pi$
$359$	−31.8327	−1.68007	−0.840033	+	0.542535i	$0.817465\pi$
$360$	0	0
$361$	10.3600	0.545265
$362$	34.8057	1.82935
$363$	0	0
$364$	1.01495	0.0531979
$365$	22.5875	1.18229
$366$	0	0
$367$	11.3031	0.590019	0.295009	−	0.955494i	$-0.404677\pi$
$367$	11.3031	0.590019	0.295009	+	0.955494i	$0.404677\pi$
$368$	−16.6228	−0.866521
$369$	0	0
$370$	−17.0548	−0.886636
$371$	−5.73553	−0.297774
$372$	0	0
$373$	−23.3621	−1.20965	−0.604823	−	0.796360i	$-0.706756\pi$
$373$	−23.3621	−1.20965	−0.604823	+	0.796360i	$0.706756\pi$
$374$	0.922105	0.0476809
$375$	0	0
$376$	20.5248	1.05848
$377$	−12.4177	−0.639544
$378$	0	0
$379$	21.0432	1.08092	0.540458	−	0.841371i	$-0.318251\pi$
$379$	21.0432	1.08092	0.540458	+	0.841371i	$0.318251\pi$
$380$	−2.10987	−0.108234
$381$	0	0
$382$	26.1013	1.33546
$383$	9.06375	0.463136	0.231568	−	0.972819i	$-0.425614\pi$
$383$	9.06375	0.463136	0.231568	+	0.972819i	$0.425614\pi$
$384$	0	0
$385$	−5.96166	−0.303834
$386$	−11.9320	−0.607321
$387$	0	0
$388$	−1.41543	−0.0718576
$389$	−5.29041	−0.268234	−0.134117	−	0.990965i	$-0.542820\pi$
$389$	−5.29041	−0.268234	−0.134117	+	0.990965i	$0.542820\pi$
$390$	0	0
$391$	1.31127	0.0663140
$392$	−2.92605	−0.147788
$393$	0	0
$394$	7.86313	0.396139
$395$	33.4293	1.68201
$396$	0	0
$397$	−27.3903	−1.37468	−0.687341	−	0.726335i	$-0.741222\pi$
$397$	−27.3903	−1.37468	−0.687341	+	0.726335i	$0.741222\pi$
$398$	−2.08677	−0.104600
$399$	0	0
$400$	−5.40343	−0.270171
$401$	−9.17289	−0.458072	−0.229036	−	0.973418i	$-0.573557\pi$
$401$	−9.17289	−0.458072	−0.229036	+	0.973418i	$0.573557\pi$
$402$	0	0
$403$	−44.9360	−2.23842
$404$	1.44832	0.0720567
$405$	0	0
$406$	2.54493	0.126303
$407$	−11.5595	−0.572983
$408$	0	0
$409$	15.3027	0.756667	0.378334	−	0.925669i	$-0.376497\pi$
$409$	15.3027	0.756667	0.378334	+	0.925669i	$0.376497\pi$
$410$	28.4748	1.40627
$411$	0	0
$412$	−2.71898	−0.133955
$413$	0.869543	0.0427874
$414$	0	0
$415$	−6.49213	−0.318686
$416$	5.72848	0.280862
$417$	0	0
$418$	17.2563	0.844034
$419$	−21.6363	−1.05700	−0.528501	−	0.848933i	$-0.677246\pi$
$419$	−21.6363	−1.05700	−0.528501	+	0.848933i	$0.677246\pi$
$420$	0	0
$421$	20.6838	1.00807	0.504034	−	0.863684i	$-0.331849\pi$
$421$	20.6838	1.00807	0.504034	+	0.863684i	$0.331849\pi$
$422$	16.4576	0.801143
$423$	0	0
$424$	−16.7825	−0.815029
$425$	0.426245	0.0206759
$426$	0	0
$427$	4.57288	0.221297
$428$	−2.00266	−0.0968021
$429$	0	0
$430$	16.2762	0.784907
$431$	−30.4476	−1.46661	−0.733304	−	0.679900i	$-0.762023\pi$
$431$	−30.4476	−1.46661	−0.733304	+	0.679900i	$0.762023\pi$
$432$	0	0
$433$	23.9773	1.15228	0.576139	−	0.817352i	$-0.304559\pi$
$433$	23.9773	1.15228	0.576139	+	0.817352i	$0.304559\pi$
$434$	9.20936	0.442063
$435$	0	0
$436$	−1.34565	−0.0644451
$437$	24.5392	1.17387
$438$	0	0
$439$	23.1661	1.10566	0.552828	−	0.833295i	$-0.313549\pi$
$439$	23.1661	1.10566	0.552828	+	0.833295i	$0.313549\pi$
$440$	−17.4441	−0.831617
$441$	0	0
$442$	2.60933	0.124113
$443$	−10.7635	−0.511391	−0.255695	−	0.966757i	$-0.582304\pi$
$443$	−10.7635	−0.511391	−0.255695	+	0.966757i	$0.582304\pi$
$444$	0	0
$445$	−37.0186	−1.75485
$446$	18.2578	0.864534
$447$	0	0
$448$	−8.51494	−0.402293
$449$	13.6259	0.643045	0.321522	−	0.946902i	$-0.395805\pi$
$449$	13.6259	0.643045	0.321522	+	0.946902i	$0.395805\pi$
$450$	0	0
$451$	19.2998	0.908792
$452$	0.771609	0.0362935
$453$	0	0
$454$	−2.27844	−0.106933
$455$	−16.8700	−0.790878
$456$	0	0
$457$	−11.6167	−0.543405	−0.271703	−	0.962381i	$-0.587587\pi$
$457$	−11.6167	−0.543405	−0.271703	+	0.962381i	$0.587587\pi$
$458$	−26.1157	−1.22031
$459$	0	0
$460$	−1.76344	−0.0822207
$461$	−19.3170	−0.899681	−0.449841	−	0.893109i	$-0.648519\pi$
$461$	−19.3170	−0.899681	−0.449841	+	0.893109i	$0.648519\pi$
$462$	0	0
$463$	3.54701	0.164844	0.0824218	−	0.996598i	$-0.473735\pi$
$463$	3.54701	0.164844	0.0824218	+	0.996598i	$0.473735\pi$
$464$	6.87338	0.319089
$465$	0	0
$466$	16.3034	0.755241
$467$	−9.00052	−0.416494	−0.208247	−	0.978076i	$-0.566776\pi$
$467$	−9.00052	−0.416494	−0.208247	+	0.978076i	$0.566776\pi$
$468$	0	0
$469$	10.2654	0.474012
$470$	−24.2520	−1.11866
$471$	0	0
$472$	2.54433	0.117112
$473$	11.0318	0.507241
$474$	0	0
$475$	7.97677	0.366000
$476$	0.0443163	0.00203124
$477$	0	0
$478$	−0.978299	−0.0447463
$479$	0.286076	0.0130711	0.00653556	−	0.999979i	$-0.497920\pi$
$479$	0.286076	0.0130711	0.00653556	+	0.999979i	$0.497920\pi$
$480$	0	0
$481$	−32.7105	−1.49147
$482$	−21.6688	−0.986986
$483$	0	0
$484$	0.843123	0.0383238
$485$	23.5266	1.06829
$486$	0	0
$487$	5.72000	0.259198	0.129599	−	0.991566i	$-0.458631\pi$
$487$	5.72000	0.259198	0.129599	+	0.991566i	$0.458631\pi$
$488$	13.3805	0.605707
$489$	0	0
$490$	3.45741	0.156190
$491$	7.65675	0.345544	0.172772	−	0.984962i	$-0.444728\pi$
$491$	7.65675	0.345544	0.172772	+	0.984962i	$0.444728\pi$
$492$	0	0
$493$	−0.542201	−0.0244195
$494$	48.8311	2.19702
$495$	0	0
$496$	24.8727	1.11682
$497$	4.28238	0.192091
$498$	0	0
$499$	−40.6026	−1.81762	−0.908811	−	0.417208i	$-0.863009\pi$
$499$	−40.6026	−1.81762	−0.908811	+	0.417208i	$0.863009\pi$
$500$	1.37369	0.0614333
$501$	0	0
$502$	24.4434	1.09096
$503$	31.4879	1.40398	0.701988	−	0.712189i	$-0.252296\pi$
$503$	31.4879	1.40398	0.701988	+	0.712189i	$0.252296\pi$
$504$	0	0
$505$	−24.0733	−1.07125
$506$	14.4229	0.641176
$507$	0	0
$508$	0.153057	0.00679081
$509$	34.1250	1.51257	0.756283	−	0.654245i	$-0.227013\pi$
$509$	34.1250	1.51257	0.756283	+	0.654245i	$0.227013\pi$
$510$	0	0
$511$	−8.87861	−0.392767
$512$	−24.6507	−1.08942
$513$	0	0
$514$	−4.46002	−0.196723
$515$	45.1937	1.99147
$516$	0	0
$517$	−16.4376	−0.722926
$518$	6.70382	0.294549
$519$	0	0
$520$	−49.3626	−2.16469
$521$	−13.1479	−0.576021	−0.288010	−	0.957627i	$-0.592994\pi$
$521$	−13.1479	−0.576021	−0.288010	+	0.957627i	$0.592994\pi$
$522$	0	0
$523$	−16.0852	−0.703356	−0.351678	−	0.936121i	$-0.614389\pi$
$523$	−16.0852	−0.703356	−0.351678	+	0.936121i	$0.614389\pi$
$524$	0.280747	0.0122645
$525$	0	0
$526$	11.4675	0.500006
$527$	−1.96206	−0.0854689
$528$	0	0
$529$	−2.49001	−0.108261
$530$	19.8301	0.861364
$531$	0	0
$532$	0.829338	0.0359564
$533$	54.6136	2.36558
$534$	0	0
$535$	33.2872	1.43913
$536$	30.0371	1.29741
$537$	0	0
$538$	−8.25478	−0.355889
$539$	2.34338	0.100937
$540$	0	0
$541$	38.9689	1.67540	0.837702	−	0.546128i	$-0.183899\pi$
$541$	38.9689	1.67540	0.837702	+	0.546128i	$0.183899\pi$
$542$	−2.80348	−0.120420
$543$	0	0
$544$	0.250126	0.0107241
$545$	22.3668	0.958088
$546$	0	0
$547$	−37.8253	−1.61729	−0.808646	−	0.588295i	$-0.799799\pi$
$547$	−37.8253	−1.61729	−0.808646	+	0.588295i	$0.799799\pi$
$548$	−0.0452924	−0.00193480
$549$	0	0
$550$	4.68834	0.199911
$551$	−10.1468	−0.432267
$552$	0	0
$553$	−13.1403	−0.558780
$554$	−10.3767	−0.440865
$555$	0	0
$556$	1.47641	0.0626136
$557$	−44.7160	−1.89468	−0.947339	−	0.320233i	$-0.896239\pi$
$557$	−44.7160	−1.89468	−0.947339	+	0.320233i	$0.896239\pi$
$558$	0	0
$559$	31.2172	1.32035
$560$	9.33780	0.394594
$561$	0	0
$562$	−4.56122	−0.192403
$563$	−5.35554	−0.225709	−0.112855	−	0.993612i	$-0.535999\pi$
$563$	−5.35554	−0.225709	−0.112855	+	0.993612i	$0.535999\pi$
$564$	0	0
$565$	−12.8253	−0.539565
$566$	−13.7637	−0.578530
$567$	0	0
$568$	12.5305	0.525767
$569$	1.76437	0.0739663	0.0369831	−	0.999316i	$-0.488225\pi$
$569$	1.76437	0.0739663	0.0369831	+	0.999316i	$0.488225\pi$
$570$	0	0
$571$	−18.1733	−0.760528	−0.380264	−	0.924878i	$-0.624167\pi$
$571$	−18.1733	−0.760528	−0.380264	+	0.924878i	$0.624167\pi$
$572$	−2.37842	−0.0994466
$573$	0	0
$574$	−11.1927	−0.467176
$575$	6.66702	0.278034
$576$	0	0
$577$	22.2207	0.925060	0.462530	−	0.886604i	$-0.346942\pi$
$577$	22.2207	0.925060	0.462530	+	0.886604i	$0.346942\pi$
$578$	−22.9895	−0.956235
$579$	0	0
$580$	0.729167	0.0302770
$581$	2.55190	0.105870
$582$	0	0
$583$	13.4406	0.556651
$584$	−25.9793	−1.07503
$585$	0	0
$586$	7.52276	0.310763
$587$	0.439685	0.0181478	0.00907388	−	0.999959i	$-0.497112\pi$
$587$	0.439685	0.0181478	0.00907388	+	0.999959i	$0.497112\pi$
$588$	0	0
$589$	−36.7182	−1.51295
$590$	−3.00637	−0.123770
$591$	0	0
$592$	18.1057	0.744141
$593$	−19.3335	−0.793931	−0.396966	−	0.917833i	$-0.629937\pi$
$593$	−19.3335	−0.793931	−0.396966	+	0.917833i	$0.629937\pi$
$594$	0	0
$595$	−0.736605	−0.0301978
$596$	3.42689	0.140371
$597$	0	0
$598$	40.8132	1.66898
$599$	12.7982	0.522920	0.261460	−	0.965214i	$-0.415796\pi$
$599$	12.7982	0.522920	0.261460	+	0.965214i	$0.415796\pi$
$600$	0	0
$601$	−25.4863	−1.03961	−0.519804	−	0.854286i	$-0.673995\pi$
$601$	−25.4863	−1.03961	−0.519804	+	0.854286i	$0.673995\pi$
$602$	−6.39777	−0.260754
$603$	0	0
$604$	−2.98082	−0.121288
$605$	−14.0140	−0.569750
$606$	0	0
$607$	−19.4915	−0.791135	−0.395568	−	0.918437i	$-0.629452\pi$
$607$	−19.4915	−0.791135	−0.395568	+	0.918437i	$0.629452\pi$
$608$	4.68087	0.189834
$609$	0	0
$610$	−15.8103	−0.640142
$611$	−46.5144	−1.88177
$612$	0	0
$613$	18.1506	0.733096	0.366548	−	0.930399i	$-0.380539\pi$
$613$	18.1506	0.733096	0.366548	+	0.930399i	$0.380539\pi$
$614$	−7.26770	−0.293301
$615$	0	0
$616$	6.85686	0.276271
$617$	−0.313028	−0.0126020	−0.00630101	−	0.999980i	$-0.502006\pi$
$617$	−0.313028	−0.0126020	−0.00630101	+	0.999980i	$0.502006\pi$
$618$	0	0
$619$	−41.2474	−1.65787	−0.828936	−	0.559344i	$-0.811053\pi$
$619$	−41.2474	−1.65787	−0.828936	+	0.559344i	$0.811053\pi$
$620$	2.63864	0.105970
$621$	0	0
$622$	−42.3791	−1.69925
$623$	14.5511	0.582978
$624$	0	0
$625$	−30.1935	−1.20774
$626$	36.9682	1.47754
$627$	0	0
$628$	−0.712478	−0.0284310
$629$	−1.42826	−0.0569483
$630$	0	0
$631$	11.4993	0.457779	0.228889	−	0.973452i	$-0.426491\pi$
$631$	11.4993	0.457779	0.228889	+	0.973452i	$0.426491\pi$
$632$	−38.4491	−1.52942
$633$	0	0
$634$	−22.8877	−0.908988
$635$	−2.54404	−0.100957
$636$	0	0
$637$	6.63119	0.262737
$638$	−5.96376	−0.236107
$639$	0	0
$640$	25.0442	0.989958
$641$	2.89721	0.114433	0.0572165	−	0.998362i	$-0.481777\pi$
$641$	2.89721	0.114433	0.0572165	+	0.998362i	$0.481777\pi$
$642$	0	0
$643$	−13.7118	−0.540740	−0.270370	−	0.962756i	$-0.587146\pi$
$643$	−13.7118	−0.540740	−0.270370	+	0.962756i	$0.587146\pi$
$644$	0.693164	0.0273145
$645$	0	0
$646$	2.13214	0.0838879
$647$	−4.64392	−0.182571	−0.0912856	−	0.995825i	$-0.529098\pi$
$647$	−4.64392	−0.182571	−0.0912856	+	0.995825i	$0.529098\pi$
$648$	0	0
$649$	−2.03767	−0.0799857
$650$	13.2668	0.520368
$651$	0	0
$652$	−2.90495	−0.113767
$653$	−45.5919	−1.78415	−0.892075	−	0.451888i	$-0.850751\pi$
$653$	−45.5919	−1.78415	−0.892075	+	0.451888i	$0.850751\pi$
$654$	0	0
$655$	−4.66645	−0.182333
$656$	−30.2294	−1.18026
$657$	0	0
$658$	9.53285	0.371629
$659$	4.44895	0.173307	0.0866533	−	0.996239i	$-0.472383\pi$
$659$	4.44895	0.173307	0.0866533	+	0.996239i	$0.472383\pi$
$660$	0	0
$661$	20.8881	0.812453	0.406227	−	0.913772i	$-0.366844\pi$
$661$	20.8881	0.812453	0.406227	+	0.913772i	$0.366844\pi$
$662$	45.9003	1.78397
$663$	0	0
$664$	7.46698	0.289775
$665$	−13.7849	−0.534554
$666$	0	0
$667$	−8.48072	−0.328375
$668$	−2.39018	−0.0924788
$669$	0	0
$670$	−35.4917	−1.37116
$671$	−10.7160	−0.413687
$672$	0	0
$673$	−6.95247	−0.267998	−0.133999	−	0.990981i	$-0.542782\pi$
$673$	−6.95247	−0.267998	−0.133999	+	0.990981i	$0.542782\pi$
$674$	30.3638	1.16957
$675$	0	0
$676$	−4.74059	−0.182330
$677$	28.3240	1.08858	0.544289	−	0.838898i	$-0.316800\pi$
$677$	28.3240	1.08858	0.544289	+	0.838898i	$0.316800\pi$
$678$	0	0
$679$	−9.24773	−0.354895
$680$	−2.15534	−0.0826537
$681$	0	0
$682$	−21.5811	−0.826381
$683$	−13.5458	−0.518317	−0.259159	−	0.965835i	$-0.583445\pi$
$683$	−13.5458	−0.518317	−0.259159	+	0.965835i	$0.583445\pi$
$684$	0	0
$685$	0.752829	0.0287641
$686$	−1.35902	−0.0518877
$687$	0	0
$688$	−17.2792	−0.658761
$689$	38.0334	1.44896
$690$	0	0
$691$	−48.4837	−1.84441	−0.922204	−	0.386705i	$-0.873613\pi$
$691$	−48.4837	−1.84441	−0.922204	+	0.386705i	$0.873613\pi$
$692$	1.71519	0.0652017
$693$	0	0
$694$	−25.8260	−0.980343
$695$	−24.5401	−0.930860
$696$	0	0
$697$	2.38462	0.0903241
$698$	3.34613	0.126653
$699$	0	0
$700$	0.225321	0.00851635
$701$	21.4838	0.811433	0.405716	−	0.913999i	$-0.367022\pi$
$701$	21.4838	0.811433	0.405716	+	0.913999i	$0.367022\pi$
$702$	0	0
$703$	−26.7284	−1.00808
$704$	19.9538	0.752035
$705$	0	0
$706$	−18.9160	−0.711913
$707$	9.46262	0.355879
$708$	0	0
$709$	13.1243	0.492894	0.246447	−	0.969156i	$-0.420737\pi$
$709$	13.1243	0.492894	0.246447	+	0.969156i	$0.420737\pi$
$710$	−14.8059	−0.555657
$711$	0	0
$712$	42.5774	1.59565
$713$	−30.6892	−1.14932
$714$	0	0
$715$	39.5329	1.47845
$716$	−2.30409	−0.0861077
$717$	0	0
$718$	−43.2614	−1.61450
$719$	32.5608	1.21431	0.607157	−	0.794582i	$-0.292310\pi$
$719$	32.5608	1.21431	0.607157	+	0.794582i	$0.292310\pi$
$720$	0	0
$721$	−17.7645	−0.661585
$722$	14.0795	0.523986
$723$	0	0
$724$	−3.91992	−0.145683
$725$	−2.75676	−0.102383
$726$	0	0
$727$	−17.2007	−0.637939	−0.318970	−	0.947765i	$-0.603337\pi$
$727$	−17.2007	−0.637939	−0.318970	+	0.947765i	$0.603337\pi$
$728$	19.4032	0.719131
$729$	0	0
$730$	30.6970	1.13615
$731$	1.36305	0.0504143
$732$	0	0
$733$	−3.41623	−0.126181	−0.0630907	−	0.998008i	$-0.520096\pi$
$733$	−3.41623	−0.126181	−0.0630907	+	0.998008i	$0.520096\pi$
$734$	15.3612	0.566993
$735$	0	0
$736$	3.91229	0.144209
$737$	−24.0558	−0.886105
$738$	0	0
$739$	−22.0238	−0.810158	−0.405079	−	0.914282i	$-0.632756\pi$
$739$	−22.0238	−0.810158	−0.405079	+	0.914282i	$0.632756\pi$
$740$	1.92076	0.0706085
$741$	0	0
$742$	−7.79472	−0.286153
$743$	−14.0090	−0.513939	−0.256969	−	0.966420i	$-0.582724\pi$
$743$	−14.0090	−0.513939	−0.256969	+	0.966420i	$0.582724\pi$
$744$	0	0
$745$	−56.9601	−2.08686
$746$	−31.7497	−1.16244
$747$	0	0
$748$	−0.103850	−0.00379714
$749$	−13.0844	−0.478093
$750$	0	0
$751$	−37.6992	−1.37566	−0.687832	−	0.725870i	$-0.741437\pi$
$751$	−37.6992	−1.37566	−0.687832	+	0.725870i	$0.741437\pi$
$752$	25.7464	0.938875
$753$	0	0
$754$	−16.8759	−0.614586
$755$	49.5457	1.80315
$756$	0	0
$757$	−21.5129	−0.781900	−0.390950	−	0.920412i	$-0.627854\pi$
$757$	−21.5129	−0.781900	−0.390950	+	0.920412i	$0.627854\pi$
$758$	28.5982	1.03873
$759$	0	0
$760$	−40.3352	−1.46311
$761$	−2.47769	−0.0898162	−0.0449081	−	0.998991i	$-0.514299\pi$
$761$	−2.47769	−0.0898162	−0.0449081	+	0.998991i	$0.514299\pi$
$762$	0	0
$763$	−8.79184	−0.318286
$764$	−2.93960	−0.106351
$765$	0	0
$766$	12.3178	0.445062
$767$	−5.76611	−0.208202
$768$	0	0
$769$	−25.5173	−0.920178	−0.460089	−	0.887873i	$-0.652183\pi$
$769$	−25.5173	−0.920178	−0.460089	+	0.887873i	$0.652183\pi$
$770$	−8.10203	−0.291977
$771$	0	0
$772$	1.34381	0.0483649
$773$	8.14480	0.292948	0.146474	−	0.989215i	$-0.453208\pi$
$773$	8.14480	0.292948	0.146474	+	0.989215i	$0.453208\pi$
$774$	0	0
$775$	−9.97589	−0.358345
$776$	−27.0594	−0.971374
$777$	0	0
$778$	−7.18979	−0.257766
$779$	44.6260	1.59889
$780$	0	0
$781$	−10.0353	−0.359090
$782$	1.78205	0.0637260
$783$	0	0
$784$	−3.67046	−0.131088
$785$	11.8425	0.422675
$786$	0	0
$787$	43.5440	1.55217	0.776087	−	0.630625i	$-0.217202\pi$
$787$	43.5440	1.55217	0.776087	+	0.630625i	$0.217202\pi$
$788$	−0.885568	−0.0315470
$789$	0	0
$790$	45.4312	1.61637
$791$	5.04132	0.179249
$792$	0	0
$793$	−30.3237	−1.07683
$794$	−37.2241	−1.32103
$795$	0	0
$796$	0.235018	0.00832999
$797$	30.4581	1.07888	0.539441	−	0.842023i	$-0.318636\pi$
$797$	30.4581	1.07888	0.539441	+	0.842023i	$0.318636\pi$
$798$	0	0
$799$	−2.03099	−0.0718511
$800$	1.27174	0.0449627
$801$	0	0
$802$	−12.4662	−0.440196
$803$	20.8060	0.734227
$804$	0	0
$805$	−11.5214	−0.406077
$806$	−61.0690	−2.15106
$807$	0	0
$808$	27.6881	0.974066
$809$	−50.0946	−1.76123	−0.880617	−	0.473829i	$-0.842871\pi$
$809$	−50.0946	−1.76123	−0.880617	+	0.473829i	$0.842871\pi$
$810$	0	0
$811$	−44.1877	−1.55164	−0.775820	−	0.630954i	$-0.782664\pi$
$811$	−44.1877	−1.55164	−0.775820	+	0.630954i	$0.782664\pi$
$812$	−0.286618	−0.0100583
$813$	0	0
$814$	−15.7096	−0.550622
$815$	48.2847	1.69134
$816$	0	0
$817$	25.5082	0.892420
$818$	20.7967	0.727138
$819$	0	0
$820$	−3.20691	−0.111990
$821$	54.6550	1.90747	0.953737	−	0.300643i	$-0.0972011\pi$
$821$	54.6550	1.90747	0.953737	+	0.300643i	$0.0972011\pi$
$822$	0	0
$823$	6.20626	0.216336	0.108168	−	0.994133i	$-0.465501\pi$
$823$	6.20626	0.216336	0.108168	+	0.994133i	$0.465501\pi$
$824$	−51.9799	−1.81081
$825$	0	0
$826$	1.18173	0.0411176
$827$	−11.0456	−0.384093	−0.192046	−	0.981386i	$-0.561512\pi$
$827$	−11.0456	−0.384093	−0.192046	+	0.981386i	$0.561512\pi$
$828$	0	0
$829$	13.0527	0.453338	0.226669	−	0.973972i	$-0.427216\pi$
$829$	13.0527	0.453338	0.226669	+	0.973972i	$0.427216\pi$
$830$	−8.82295	−0.306249
$831$	0	0
$832$	56.4642	1.95754
$833$	0.289541	0.0100320
$834$	0	0
$835$	39.7284	1.37486
$836$	−1.94346	−0.0672158
$837$	0	0
$838$	−29.4042	−1.01575
$839$	43.0772	1.48719	0.743596	−	0.668629i	$-0.233119\pi$
$839$	43.0772	1.48719	0.743596	+	0.668629i	$0.233119\pi$
$840$	0	0
$841$	−25.4933	−0.879079
$842$	28.1098	0.968727
$843$	0	0
$844$	−1.85350	−0.0638001
$845$	78.7958	2.71066
$846$	0	0
$847$	5.50856	0.189276
$848$	−21.0520	−0.722930
$849$	0	0
$850$	0.579277	0.0198690
$851$	−22.3397	−0.765797
$852$	0	0
$853$	37.7048	1.29099	0.645494	−	0.763765i	$-0.276651\pi$
$853$	37.7048	1.29099	0.645494	+	0.763765i	$0.276651\pi$
$854$	6.21465	0.212661
$855$	0	0
$856$	−38.2856	−1.30858
$857$	−41.1966	−1.40725	−0.703624	−	0.710572i	$-0.748436\pi$
$857$	−41.1966	−1.40725	−0.703624	+	0.710572i	$0.748436\pi$
$858$	0	0
$859$	43.0888	1.47017	0.735086	−	0.677974i	$-0.237142\pi$
$859$	43.0888	1.47017	0.735086	+	0.677974i	$0.237142\pi$
$860$	−1.83307	−0.0625072
$861$	0	0
$862$	−41.3790	−1.40937
$863$	9.10932	0.310085	0.155042	−	0.987908i	$-0.450449\pi$
$863$	9.10932	0.310085	0.155042	+	0.987908i	$0.450449\pi$
$864$	0	0
$865$	−28.5091	−0.969337
$866$	32.5857	1.10731
$867$	0	0
$868$	−1.03718	−0.0352043
$869$	30.7926	1.04457
$870$	0	0
$871$	−68.0718	−2.30653
$872$	−25.7254	−0.871172
$873$	0	0
$874$	33.3494	1.12806
$875$	8.97502	0.303411
$876$	0	0
$877$	28.0344	0.946654	0.473327	−	0.880887i	$-0.343053\pi$
$877$	28.0344	0.946654	0.473327	+	0.880887i	$0.343053\pi$
$878$	31.4832	1.06251
$879$	0	0
$880$	−21.8820	−0.737643
$881$	−38.7353	−1.30503	−0.652513	−	0.757778i	$-0.726285\pi$
$881$	−38.7353	−1.30503	−0.652513	+	0.757778i	$0.726285\pi$
$882$	0	0
$883$	53.9225	1.81464	0.907319	−	0.420443i	$-0.138125\pi$
$883$	53.9225	1.81464	0.907319	+	0.420443i	$0.138125\pi$
$884$	−0.293870	−0.00988392
$885$	0	0
$886$	−14.6279	−0.491433
$887$	6.53629	0.219467	0.109733	−	0.993961i	$-0.465000\pi$
$887$	6.53629	0.219467	0.109733	+	0.993961i	$0.465000\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−50.3092	−1.68637
$891$	0	0
$892$	−2.05625	−0.0688483
$893$	−38.0080	−1.27189
$894$	0	0
$895$	38.2974	1.28014
$896$	−9.84426	−0.328873
$897$	0	0
$898$	18.5179	0.617950
$899$	12.6897	0.423226
$900$	0	0
$901$	1.66067	0.0553251
$902$	26.2289	0.873326
$903$	0	0
$904$	14.7512	0.490617
$905$	65.1550	2.16582
$906$	0	0
$907$	41.9634	1.39337	0.696685	−	0.717377i	$-0.254657\pi$
$907$	41.9634	1.39337	0.696685	+	0.717377i	$0.254657\pi$
$908$	0.256605	0.00851572
$909$	0	0
$910$	−22.9267	−0.760014
$911$	−40.0565	−1.32713	−0.663565	−	0.748119i	$-0.730957\pi$
$911$	−40.0565	−1.32713	−0.663565	+	0.748119i	$0.730957\pi$
$912$	0	0
$913$	−5.98007	−0.197911
$914$	−15.7873	−0.522199
$915$	0	0
$916$	2.94123	0.0971809
$917$	1.83427	0.0605728
$918$	0	0
$919$	−19.0015	−0.626801	−0.313400	−	0.949621i	$-0.601468\pi$
$919$	−19.0015	−0.626801	−0.313400	+	0.949621i	$0.601468\pi$
$920$	−33.7123	−1.11146
$921$	0	0
$922$	−26.2522	−0.864571
$923$	−28.3973	−0.934707
$924$	0	0
$925$	−7.26180	−0.238767
$926$	4.82047	0.158410
$927$	0	0
$928$	−1.61770	−0.0531036
$929$	15.5617	0.510562	0.255281	−	0.966867i	$-0.417832\pi$
$929$	15.5617	0.510562	0.255281	+	0.966867i	$0.417832\pi$
$930$	0	0
$931$	5.41849	0.177584
$932$	−1.83614	−0.0601447
$933$	0	0
$934$	−12.2319	−0.400240
$935$	1.72615	0.0564510
$936$	0	0
$937$	−53.7849	−1.75707	−0.878537	−	0.477674i	$-0.841480\pi$
$937$	−53.7849	−1.75707	−0.878537	+	0.477674i	$0.841480\pi$
$938$	13.9509	0.455513
$939$	0	0
$940$	2.73133	0.0890860
$941$	−29.5597	−0.963618	−0.481809	−	0.876276i	$-0.660020\pi$
$941$	−29.5597	−0.963618	−0.481809	+	0.876276i	$0.660020\pi$
$942$	0	0
$943$	37.2986	1.21461
$944$	3.19162	0.103879
$945$	0	0
$946$	14.9924	0.487446
$947$	14.7088	0.477973	0.238987	−	0.971023i	$-0.423185\pi$
$947$	14.7088	0.477973	0.238987	+	0.971023i	$0.423185\pi$
$948$	0	0
$949$	58.8758	1.91119
$950$	10.8406	0.351716
$951$	0	0
$952$	0.847213	0.0274583
$953$	36.3926	1.17887	0.589436	−	0.807815i	$-0.299350\pi$
$953$	36.3926	1.17887	0.589436	+	0.807815i	$0.299350\pi$
$954$	0	0
$955$	48.8606	1.58109
$956$	0.110179	0.00356344
$957$	0	0
$958$	0.388783	0.0125610
$959$	−0.295919	−0.00955571
$960$	0	0
$961$	14.9204	0.481302
$962$	−44.4543	−1.43326
$963$	0	0
$964$	2.44040	0.0786000
$965$	−22.3362	−0.719028
$966$	0	0
$967$	36.5601	1.17569	0.587847	−	0.808972i	$-0.299976\pi$
$967$	36.5601	1.17569	0.587847	+	0.808972i	$0.299976\pi$
$968$	16.1183	0.518063
$969$	0	0
$970$	31.9732	1.02660
$971$	35.9938	1.15509	0.577547	−	0.816357i	$-0.304010\pi$
$971$	35.9938	1.15509	0.577547	+	0.816357i	$0.304010\pi$
$972$	0	0
$973$	9.64612	0.309240
$974$	7.77362	0.249083
$975$	0	0
$976$	16.7846	0.537262
$977$	−60.4714	−1.93465	−0.967325	−	0.253539i	$-0.918406\pi$
$977$	−60.4714	−1.93465	−0.967325	+	0.253539i	$0.918406\pi$
$978$	0	0
$979$	−34.0988	−1.08980
$980$	−0.389383	−0.0124384
$981$	0	0
$982$	10.4057	0.332059
$983$	4.00509	0.127743	0.0638713	−	0.997958i	$-0.479655\pi$
$983$	4.00509	0.127743	0.0638713	+	0.997958i	$0.479655\pi$
$984$	0	0
$985$	14.7195	0.469002
$986$	−0.736864	−0.0234665
$987$	0	0
$988$	−5.49950	−0.174962
$989$	21.3199	0.677933
$990$	0	0
$991$	−6.32986	−0.201075	−0.100537	−	0.994933i	$-0.532056\pi$
$991$	−6.32986	−0.201075	−0.100537	+	0.994933i	$0.532056\pi$
$992$	−5.85397	−0.185864
$993$	0	0
$994$	5.81985	0.184594
$995$	−3.90636	−0.123840
$996$	0	0
$997$	0.224764	0.00711835	0.00355918	−	0.999994i	$-0.498867\pi$
$997$	0.224764	0.00711835	0.00355918	+	0.999994i	$0.498867\pi$
$998$	−55.1799	−1.74669
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.12		16
3.2	odd	2		2667.2.a.n.1.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.5	✓	16	3.2	odd	2
8001.2.a.s.1.12		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)

\(f(q)\)	\(=\)	\(q+1.35902 q^{2} -0.153057 q^{4} +2.54404 q^{5} -1.00000 q^{7} -2.92605 q^{8} +O(q^{10})\)	\(q+1.35902 q^{2} -0.153057 q^{4} +2.54404 q^{5} -1.00000 q^{7} -2.92605 q^{8} +3.45741 q^{10} +2.34338 q^{11} +6.63119 q^{13} -1.35902 q^{14} -3.67046 q^{16} +0.289541 q^{17} +5.41849 q^{19} -0.389383 q^{20} +3.18471 q^{22} +4.52880 q^{23} +1.47214 q^{25} +9.01194 q^{26} +0.153057 q^{28} -1.87262 q^{29} -6.77646 q^{31} +0.863869 q^{32} +0.393493 q^{34} -2.54404 q^{35} -4.93282 q^{37} +7.36385 q^{38} -7.44400 q^{40} +8.23587 q^{41} +4.70763 q^{43} -0.358671 q^{44} +6.15474 q^{46} -7.01449 q^{47} +1.00000 q^{49} +2.00067 q^{50} -1.01495 q^{52} +5.73553 q^{53} +5.96166 q^{55} +2.92605 q^{56} -2.54493 q^{58} -0.869543 q^{59} -4.57288 q^{61} -9.20936 q^{62} +8.51494 q^{64} +16.8700 q^{65} -10.2654 q^{67} -0.0443163 q^{68} -3.45741 q^{70} -4.28238 q^{71} +8.87861 q^{73} -6.70382 q^{74} -0.829338 q^{76} -2.34338 q^{77} +13.1403 q^{79} -9.33780 q^{80} +11.1927 q^{82} -2.55190 q^{83} +0.736605 q^{85} +6.39777 q^{86} -6.85686 q^{88} -14.5511 q^{89} -6.63119 q^{91} -0.693164 q^{92} -9.53285 q^{94} +13.7849 q^{95} +9.24773 q^{97} +1.35902 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

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Varieties

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Database

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