LMFDB - Embedded newform 8001.2.a.u.1.16

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 $ x^18 - 6x^17 - 11x^16 + 123x^15 - 35x^14 - 982x^13 + 988x^12 + 3872x^11 - 5421x^10 - 7882x^9 + 13376x^8 + 7948x^7 - 15795x^6 - 3858x^5 + 8199x^4 + 1453x^3 - 1610x^2 - 380*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.59597$ 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)

$f(q)$	$=$	$q+2.59597 q^{2} +4.73905 q^{4} +1.63234 q^{5} +1.00000 q^{7} +7.11048 q^{8} +O(q^{10})$	\(q+2.59597 q^{2} +4.73905 q^{4} +1.63234 q^{5} +1.00000 q^{7} +7.11048 q^{8} +4.23750 q^{10} -4.79599 q^{11} +0.263712 q^{13} +2.59597 q^{14} +8.98048 q^{16} +5.12058 q^{17} +5.93565 q^{19} +7.73574 q^{20} -12.4502 q^{22} +4.10470 q^{23} -2.33547 q^{25} +0.684587 q^{26} +4.73905 q^{28} -0.459771 q^{29} -9.24495 q^{31} +9.09207 q^{32} +13.2929 q^{34} +1.63234 q^{35} +6.64452 q^{37} +15.4088 q^{38} +11.6067 q^{40} +3.43282 q^{41} +2.33770 q^{43} -22.7284 q^{44} +10.6557 q^{46} +5.69134 q^{47} +1.00000 q^{49} -6.06280 q^{50} +1.24974 q^{52} -2.55281 q^{53} -7.82868 q^{55} +7.11048 q^{56} -1.19355 q^{58} +13.3394 q^{59} +10.9442 q^{61} -23.9996 q^{62} +5.64176 q^{64} +0.430467 q^{65} +1.34491 q^{67} +24.2667 q^{68} +4.23750 q^{70} -5.05046 q^{71} -16.4455 q^{73} +17.2490 q^{74} +28.1293 q^{76} -4.79599 q^{77} -0.111124 q^{79} +14.6592 q^{80} +8.91149 q^{82} -6.65678 q^{83} +8.35852 q^{85} +6.06861 q^{86} -34.1018 q^{88} -11.2897 q^{89} +0.263712 q^{91} +19.4524 q^{92} +14.7745 q^{94} +9.68900 q^{95} -5.51451 q^{97} +2.59597 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * 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 $

$2$	2.59597	1.83563	0.917813	−	0.397013i	$-0.129953\pi$
$2$	2.59597	1.83563	0.917813	+	0.397013i	$0.129953\pi$
$3$	0	0
$3$	0	0
$4$	4.73905	2.36952
$5$	1.63234	0.730005	0.365002	−	0.931007i	$-0.381068\pi$
$5$	1.63234	0.730005	0.365002	+	0.931007i	$0.381068\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	7.11048	2.51393
$9$	0	0
$10$	4.23750	1.34002
$11$	−4.79599	−1.44604	−0.723022	−	0.690825i	$-0.757248\pi$
$11$	−4.79599	−1.44604	−0.723022	+	0.690825i	$0.757248\pi$
$12$	0	0
$13$	0.263712	0.0731405	0.0365702	−	0.999331i	$-0.488357\pi$
$13$	0.263712	0.0731405	0.0365702	+	0.999331i	$0.488357\pi$
$14$	2.59597	0.693802
$15$	0	0
$16$	8.98048	2.24512
$17$	5.12058	1.24192	0.620961	−	0.783841i	$-0.286742\pi$
$17$	5.12058	1.24192	0.620961	+	0.783841i	$0.286742\pi$
$18$	0	0
$19$	5.93565	1.36173	0.680866	−	0.732408i	$-0.261604\pi$
$19$	5.93565	1.36173	0.680866	+	0.732408i	$0.261604\pi$
$20$	7.73574	1.72976
$21$	0	0
$22$	−12.4502	−2.65440
$23$	4.10470	0.855889	0.427944	−	0.903805i	$-0.359238\pi$
$23$	4.10470	0.855889	0.427944	+	0.903805i	$0.359238\pi$
$24$	0	0
$25$	−2.33547	−0.467093
$26$	0.684587	0.134259
$27$	0	0
$28$	4.73905	0.895596
$29$	−0.459771	−0.0853773	−0.0426886	−	0.999088i	$-0.513592\pi$
$29$	−0.459771	−0.0853773	−0.0426886	+	0.999088i	$0.513592\pi$
$30$	0	0
$31$	−9.24495	−1.66044	−0.830221	−	0.557434i	$-0.811786\pi$
$31$	−9.24495	−1.66044	−0.830221	+	0.557434i	$0.811786\pi$
$32$	9.09207	1.60727
$33$	0	0
$34$	13.2929	2.27971
$35$	1.63234	0.275916
$36$	0	0
$37$	6.64452	1.09235	0.546176	−	0.837670i	$-0.316083\pi$
$37$	6.64452	1.09235	0.546176	+	0.837670i	$0.316083\pi$
$38$	15.4088	2.49963
$39$	0	0
$40$	11.6067	1.83518
$41$	3.43282	0.536116	0.268058	−	0.963403i	$-0.413618\pi$
$41$	3.43282	0.536116	0.268058	+	0.963403i	$0.413618\pi$
$42$	0	0
$43$	2.33770	0.356497	0.178248	−	0.983986i	$-0.442957\pi$
$43$	2.33770	0.356497	0.178248	+	0.983986i	$0.442957\pi$
$44$	−22.7284	−3.42644
$45$	0	0
$46$	10.6557	1.57109
$47$	5.69134	0.830167	0.415083	−	0.909783i	$-0.363752\pi$
$47$	5.69134	0.830167	0.415083	+	0.909783i	$0.363752\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−6.06280	−0.857409
$51$	0	0
$52$	1.24974	0.173308
$53$	−2.55281	−0.350656	−0.175328	−	0.984510i	$-0.556099\pi$
$53$	−2.55281	−0.350656	−0.175328	+	0.984510i	$0.556099\pi$
$54$	0	0
$55$	−7.82868	−1.05562
$56$	7.11048	0.950178
$57$	0	0
$58$	−1.19355	−0.156721
$59$	13.3394	1.73664	0.868320	−	0.496005i	$-0.165200\pi$
$59$	13.3394	1.73664	0.868320	+	0.496005i	$0.165200\pi$
$60$	0	0
$61$	10.9442	1.40126	0.700631	−	0.713524i	$-0.252902\pi$
$61$	10.9442	1.40126	0.700631	+	0.713524i	$0.252902\pi$
$62$	−23.9996	−3.04795
$63$	0	0
$64$	5.64176	0.705220
$65$	0.430467	0.0533929
$66$	0	0
$67$	1.34491	0.164307	0.0821535	−	0.996620i	$-0.473820\pi$
$67$	1.34491	0.164307	0.0821535	+	0.996620i	$0.473820\pi$
$68$	24.2667	2.94276
$69$	0	0
$70$	4.23750	0.506478
$71$	−5.05046	−0.599379	−0.299690	−	0.954037i	$-0.596883\pi$
$71$	−5.05046	−0.599379	−0.299690	+	0.954037i	$0.596883\pi$
$72$	0	0
$73$	−16.4455	−1.92480	−0.962401	−	0.271634i	$-0.912436\pi$
$73$	−16.4455	−1.92480	−0.962401	+	0.271634i	$0.912436\pi$
$74$	17.2490	2.00515
$75$	0	0
$76$	28.1293	3.22666
$77$	−4.79599	−0.546554
$78$	0	0
$79$	−0.111124	−0.0125025	−0.00625124	−	0.999980i	$-0.501990\pi$
$79$	−0.111124	−0.0125025	−0.00625124	+	0.999980i	$0.501990\pi$
$80$	14.6592	1.63895
$81$	0	0
$82$	8.91149	0.984109
$83$	−6.65678	−0.730676	−0.365338	−	0.930875i	$-0.619047\pi$
$83$	−6.65678	−0.730676	−0.365338	+	0.930875i	$0.619047\pi$
$84$	0	0
$85$	8.35852	0.906609
$86$	6.06861	0.654395
$87$	0	0
$88$	−34.1018	−3.63526
$89$	−11.2897	−1.19671	−0.598353	−	0.801233i	$-0.704178\pi$
$89$	−11.2897	−1.19671	−0.598353	+	0.801233i	$0.704178\pi$
$90$	0	0
$91$	0.263712	0.0276445
$92$	19.4524	2.02805
$93$	0	0
$94$	14.7745	1.52388
$95$	9.68900	0.994070
$96$	0	0
$97$	−5.51451	−0.559913	−0.279957	−	0.960013i	$-0.590320\pi$
$97$	−5.51451	−0.559913	−0.279957	+	0.960013i	$0.590320\pi$
$98$	2.59597	0.262232
$99$	0	0
$100$	−11.0679	−1.10679
$101$	−0.809406	−0.0805389	−0.0402694	−	0.999189i	$-0.512822\pi$
$101$	−0.809406	−0.0805389	−0.0402694	+	0.999189i	$0.512822\pi$
$102$	0	0
$103$	−5.13235	−0.505705	−0.252853	−	0.967505i	$-0.581369\pi$
$103$	−5.13235	−0.505705	−0.252853	+	0.967505i	$0.581369\pi$
$104$	1.87512	0.183870
$105$	0	0
$106$	−6.62702	−0.643673
$107$	17.1879	1.66161	0.830807	−	0.556560i	$-0.187879\pi$
$107$	17.1879	1.66161	0.830807	+	0.556560i	$0.187879\pi$
$108$	0	0
$109$	8.44424	0.808812	0.404406	−	0.914580i	$-0.367478\pi$
$109$	8.44424	0.808812	0.404406	+	0.914580i	$0.367478\pi$
$110$	−20.3230	−1.93772
$111$	0	0
$112$	8.98048	0.848575
$113$	2.27777	0.214275	0.107137	−	0.994244i	$-0.465832\pi$
$113$	2.27777	0.214275	0.107137	+	0.994244i	$0.465832\pi$
$114$	0	0
$115$	6.70026	0.624803
$116$	−2.17888	−0.202303
$117$	0	0
$118$	34.6286	3.18782
$119$	5.12058	0.469403
$120$	0	0
$121$	12.0015	1.09105
$122$	28.4108	2.57219
$123$	0	0
$124$	−43.8123	−3.93446
$125$	−11.9740	−1.07098
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−3.53831	−0.312745
$129$	0	0
$130$	1.11748	0.0980094
$131$	−7.70860	−0.673503	−0.336752	−	0.941593i	$-0.609328\pi$
$131$	−7.70860	−0.673503	−0.336752	+	0.941593i	$0.609328\pi$
$132$	0	0
$133$	5.93565	0.514686
$134$	3.49134	0.301606
$135$	0	0
$136$	36.4098	3.12211
$137$	2.92001	0.249473	0.124737	−	0.992190i	$-0.460191\pi$
$137$	2.92001	0.249473	0.124737	+	0.992190i	$0.460191\pi$
$138$	0	0
$139$	1.52119	0.129026	0.0645130	−	0.997917i	$-0.479451\pi$
$139$	1.52119	0.129026	0.0645130	+	0.997917i	$0.479451\pi$
$140$	7.73574	0.653789
$141$	0	0
$142$	−13.1108	−1.10024
$143$	−1.26476	−0.105764
$144$	0	0
$145$	−0.750502	−0.0623258
$146$	−42.6920	−3.53322
$147$	0	0
$148$	31.4887	2.58835
$149$	−12.0258	−0.985195	−0.492598	−	0.870257i	$-0.663953\pi$
$149$	−12.0258	−0.985195	−0.492598	+	0.870257i	$0.663953\pi$
$150$	0	0
$151$	4.51556	0.367471	0.183736	−	0.982976i	$-0.441181\pi$
$151$	4.51556	0.367471	0.183736	+	0.982976i	$0.441181\pi$
$152$	42.2053	3.42330
$153$	0	0
$154$	−12.4502	−1.00327
$155$	−15.0909	−1.21213
$156$	0	0
$157$	2.59506	0.207109	0.103554	−	0.994624i	$-0.466978\pi$
$157$	2.59506	0.207109	0.103554	+	0.994624i	$0.466978\pi$
$158$	−0.288475	−0.0229499
$159$	0	0
$160$	14.8413	1.17331
$161$	4.10470	0.323495
$162$	0	0
$163$	20.3837	1.59658	0.798288	−	0.602275i	$-0.205739\pi$
$163$	20.3837	1.59658	0.798288	+	0.602275i	$0.205739\pi$
$164$	16.2683	1.27034
$165$	0	0
$166$	−17.2808	−1.34125
$167$	−7.15571	−0.553726	−0.276863	−	0.960909i	$-0.589295\pi$
$167$	−7.15571	−0.553726	−0.276863	+	0.960909i	$0.589295\pi$
$168$	0	0
$169$	−12.9305	−0.994650
$170$	21.6985	1.66420
$171$	0	0
$172$	11.0785	0.844727
$173$	9.08892	0.691018	0.345509	−	0.938415i	$-0.387706\pi$
$173$	9.08892	0.691018	0.345509	+	0.938415i	$0.387706\pi$
$174$	0	0
$175$	−2.33547	−0.176545
$176$	−43.0703	−3.24654
$177$	0	0
$178$	−29.3077	−2.19670
$179$	19.6649	1.46982	0.734912	−	0.678162i	$-0.237223\pi$
$179$	19.6649	1.46982	0.734912	+	0.678162i	$0.237223\pi$
$180$	0	0
$181$	5.64241	0.419397	0.209699	−	0.977766i	$-0.432752\pi$
$181$	5.64241	0.419397	0.209699	+	0.977766i	$0.432752\pi$
$182$	0.684587	0.0507450
$183$	0	0
$184$	29.1864	2.15165
$185$	10.8461	0.797422
$186$	0	0
$187$	−24.5582	−1.79588
$188$	26.9715	1.96710
$189$	0	0
$190$	25.1523	1.82474
$191$	14.5073	1.04971	0.524855	−	0.851192i	$-0.324119\pi$
$191$	14.5073	1.04971	0.524855	+	0.851192i	$0.324119\pi$
$192$	0	0
$193$	−13.6112	−0.979753	−0.489877	−	0.871792i	$-0.662958\pi$
$193$	−13.6112	−0.979753	−0.489877	+	0.871792i	$0.662958\pi$
$194$	−14.3155	−1.02779
$195$	0	0
$196$	4.73905	0.338503
$197$	−6.88652	−0.490644	−0.245322	−	0.969442i	$-0.578894\pi$
$197$	−6.88652	−0.490644	−0.245322	+	0.969442i	$0.578894\pi$
$198$	0	0
$199$	−25.0663	−1.77690	−0.888451	−	0.458972i	$-0.848218\pi$
$199$	−25.0663	−1.77690	−0.888451	+	0.458972i	$0.848218\pi$
$200$	−16.6063	−1.17424
$201$	0	0
$202$	−2.10119	−0.147839
$203$	−0.459771	−0.0322696
$204$	0	0
$205$	5.60353	0.391367
$206$	−13.3234	−0.928286
$207$	0	0
$208$	2.36826	0.164209
$209$	−28.4673	−1.96913
$210$	0	0
$211$	−22.2975	−1.53502	−0.767512	−	0.641035i	$-0.778505\pi$
$211$	−22.2975	−1.53502	−0.767512	+	0.641035i	$0.778505\pi$
$212$	−12.0979	−0.830887
$213$	0	0
$214$	44.6192	3.05010
$215$	3.81593	0.260244
$216$	0	0
$217$	−9.24495	−0.627588
$218$	21.9210	1.48468
$219$	0	0
$220$	−37.1005	−2.50131
$221$	1.35036	0.0908348
$222$	0	0
$223$	−9.15329	−0.612950	−0.306475	−	0.951879i	$-0.599150\pi$
$223$	−9.15329	−0.612950	−0.306475	+	0.951879i	$0.599150\pi$
$224$	9.09207	0.607489
$225$	0	0
$226$	5.91303	0.393329
$227$	−18.4647	−1.22554	−0.612771	−	0.790260i	$-0.709945\pi$
$227$	−18.4647	−1.22554	−0.612771	+	0.790260i	$0.709945\pi$
$228$	0	0
$229$	−21.5708	−1.42544	−0.712721	−	0.701448i	$-0.752537\pi$
$229$	−21.5708	−1.42544	−0.712721	+	0.701448i	$0.752537\pi$
$230$	17.3937	1.14690
$231$	0	0
$232$	−3.26919	−0.214633
$233$	8.86474	0.580749	0.290374	−	0.956913i	$-0.406220\pi$
$233$	8.86474	0.580749	0.290374	+	0.956913i	$0.406220\pi$
$234$	0	0
$235$	9.29020	0.606026
$236$	63.2160	4.11501
$237$	0	0
$238$	13.2929	0.861648
$239$	−20.1987	−1.30654	−0.653272	−	0.757124i	$-0.726604\pi$
$239$	−20.1987	−1.30654	−0.653272	+	0.757124i	$0.726604\pi$
$240$	0	0
$241$	0.276725	0.0178254	0.00891272	−	0.999960i	$-0.497163\pi$
$241$	0.276725	0.0178254	0.00891272	+	0.999960i	$0.497163\pi$
$242$	31.1555	2.00275
$243$	0	0
$244$	51.8651	3.32032
$245$	1.63234	0.104286
$246$	0	0
$247$	1.56530	0.0995977
$248$	−65.7360	−4.17424
$249$	0	0
$250$	−31.0840	−1.96593
$251$	23.2801	1.46942	0.734712	−	0.678379i	$-0.237317\pi$
$251$	23.2801	1.46942	0.734712	+	0.678379i	$0.237317\pi$
$252$	0	0
$253$	−19.6861	−1.23765
$254$	2.59597	0.162885
$255$	0	0
$256$	−20.4689	−1.27930
$257$	−23.4788	−1.46457	−0.732285	−	0.680998i	$-0.761546\pi$
$257$	−23.4788	−1.46457	−0.732285	+	0.680998i	$0.761546\pi$
$258$	0	0
$259$	6.64452	0.412870
$260$	2.04000	0.126516
$261$	0	0
$262$	−20.0113	−1.23630
$263$	2.69710	0.166310	0.0831551	−	0.996537i	$-0.473500\pi$
$263$	2.69710	0.166310	0.0831551	+	0.996537i	$0.473500\pi$
$264$	0	0
$265$	−4.16706	−0.255980
$266$	15.4088	0.944772
$267$	0	0
$268$	6.37360	0.389329
$269$	4.37198	0.266565	0.133282	−	0.991078i	$-0.457448\pi$
$269$	4.37198	0.266565	0.133282	+	0.991078i	$0.457448\pi$
$270$	0	0
$271$	−8.85747	−0.538053	−0.269026	−	0.963133i	$-0.586702\pi$
$271$	−8.85747	−0.538053	−0.269026	+	0.963133i	$0.586702\pi$
$272$	45.9852	2.78826
$273$	0	0
$274$	7.58024	0.457939
$275$	11.2009	0.675438
$276$	0	0
$277$	−19.4438	−1.16826	−0.584131	−	0.811659i	$-0.698565\pi$
$277$	−19.4438	−1.16826	−0.584131	+	0.811659i	$0.698565\pi$
$278$	3.94897	0.236843
$279$	0	0
$280$	11.6067	0.693634
$281$	0.140909	0.00840591	0.00420295	−	0.999991i	$-0.498662\pi$
$281$	0.140909	0.00840591	0.00420295	+	0.999991i	$0.498662\pi$
$282$	0	0
$283$	22.7659	1.35329	0.676645	−	0.736309i	$-0.263433\pi$
$283$	22.7659	1.35329	0.676645	+	0.736309i	$0.263433\pi$
$284$	−23.9344	−1.42024
$285$	0	0
$286$	−3.28327	−0.194144
$287$	3.43282	0.202633
$288$	0	0
$289$	9.22031	0.542371
$290$	−1.94828	−0.114407
$291$	0	0
$292$	−77.9361	−4.56086
$293$	29.5536	1.72654	0.863271	−	0.504741i	$-0.168412\pi$
$293$	29.5536	1.72654	0.863271	+	0.504741i	$0.168412\pi$
$294$	0	0
$295$	21.7744	1.26775
$296$	47.2457	2.74610
$297$	0	0
$298$	−31.2187	−1.80845
$299$	1.08246	0.0626001
$300$	0	0
$301$	2.33770	0.134743
$302$	11.7223	0.674540
$303$	0	0
$304$	53.3050	3.05725
$305$	17.8647	1.02293
$306$	0	0
$307$	−17.4005	−0.993097	−0.496549	−	0.868009i	$-0.665400\pi$
$307$	−17.4005	−0.993097	−0.496549	+	0.868009i	$0.665400\pi$
$308$	−22.7284	−1.29507
$309$	0	0
$310$	−39.1755	−2.22502
$311$	17.8934	1.01464	0.507322	−	0.861757i	$-0.330635\pi$
$311$	17.8934	1.01464	0.507322	+	0.861757i	$0.330635\pi$
$312$	0	0
$313$	−13.0691	−0.738712	−0.369356	−	0.929288i	$-0.620422\pi$
$313$	−13.0691	−0.738712	−0.369356	+	0.929288i	$0.620422\pi$
$314$	6.73670	0.380174
$315$	0	0
$316$	−0.526624	−0.0296249
$317$	24.7771	1.39162	0.695809	−	0.718226i	$-0.255046\pi$
$317$	24.7771	1.39162	0.695809	+	0.718226i	$0.255046\pi$
$318$	0	0
$319$	2.20505	0.123459
$320$	9.20927	0.514814
$321$	0	0
$322$	10.6557	0.593817
$323$	30.3940	1.69117
$324$	0	0
$325$	−0.615890	−0.0341634
$326$	52.9155	2.93072
$327$	0	0
$328$	24.4090	1.34776
$329$	5.69134	0.313774
$330$	0	0
$331$	−14.3948	−0.791210	−0.395605	−	0.918421i	$-0.629465\pi$
$331$	−14.3948	−0.791210	−0.395605	+	0.918421i	$0.629465\pi$
$332$	−31.5468	−1.73136
$333$	0	0
$334$	−18.5760	−1.01643
$335$	2.19535	0.119945
$336$	0	0
$337$	−8.13837	−0.443325	−0.221663	−	0.975123i	$-0.571148\pi$
$337$	−8.13837	−0.443325	−0.221663	+	0.975123i	$0.571148\pi$
$338$	−33.5670	−1.82581
$339$	0	0
$340$	39.6114	2.14823
$341$	44.3387	2.40107
$342$	0	0
$343$	1.00000	0.0539949
$344$	16.6222	0.896209
$345$	0	0
$346$	23.5946	1.26845
$347$	23.6307	1.26856	0.634280	−	0.773103i	$-0.281297\pi$
$347$	23.6307	1.26856	0.634280	+	0.773103i	$0.281297\pi$
$348$	0	0
$349$	−27.0018	−1.44537	−0.722686	−	0.691176i	$-0.757093\pi$
$349$	−27.0018	−1.44537	−0.722686	+	0.691176i	$0.757093\pi$
$350$	−6.06280	−0.324070
$351$	0	0
$352$	−43.6055	−2.32418
$353$	−20.7847	−1.10626	−0.553130	−	0.833095i	$-0.686567\pi$
$353$	−20.7847	−1.10626	−0.553130	+	0.833095i	$0.686567\pi$
$354$	0	0
$355$	−8.24406	−0.437549
$356$	−53.5024	−2.83562
$357$	0	0
$358$	51.0495	2.69805
$359$	7.34347	0.387574	0.193787	−	0.981044i	$-0.437923\pi$
$359$	7.34347	0.387574	0.193787	+	0.981044i	$0.437923\pi$
$360$	0	0
$361$	16.2320	0.854314
$362$	14.6475	0.769856
$363$	0	0
$364$	1.24974	0.0655043
$365$	−26.8447	−1.40511
$366$	0	0
$367$	3.27104	0.170747	0.0853735	−	0.996349i	$-0.472792\pi$
$367$	3.27104	0.170747	0.0853735	+	0.996349i	$0.472792\pi$
$368$	36.8621	1.92157
$369$	0	0
$370$	28.1562	1.46377
$371$	−2.55281	−0.132535
$372$	0	0
$373$	−30.7576	−1.59257	−0.796284	−	0.604923i	$-0.793204\pi$
$373$	−30.7576	−1.59257	−0.796284	+	0.604923i	$0.793204\pi$
$374$	−63.7524	−3.29656
$375$	0	0
$376$	40.4681	2.08698
$377$	−0.121247	−0.00624453
$378$	0	0
$379$	16.1652	0.830353	0.415176	−	0.909741i	$-0.363720\pi$
$379$	16.1652	0.830353	0.415176	+	0.909741i	$0.363720\pi$
$380$	45.9166	2.35547
$381$	0	0
$382$	37.6604	1.92688
$383$	−10.0448	−0.513263	−0.256632	−	0.966509i	$-0.582613\pi$
$383$	−10.0448	−0.513263	−0.256632	+	0.966509i	$0.582613\pi$
$384$	0	0
$385$	−7.82868	−0.398987
$386$	−35.3341	−1.79846
$387$	0	0
$388$	−26.1335	−1.32673
$389$	−29.6168	−1.50163	−0.750816	−	0.660511i	$-0.770340\pi$
$389$	−29.6168	−1.50163	−0.750816	+	0.660511i	$0.770340\pi$
$390$	0	0
$391$	21.0184	1.06295
$392$	7.11048	0.359133
$393$	0	0
$394$	−17.8772	−0.900640
$395$	−0.181393	−0.00912687
$396$	0	0
$397$	0.0564815	0.00283472	0.00141736	−	0.999999i	$-0.499549\pi$
$397$	0.0564815	0.00283472	0.00141736	+	0.999999i	$0.499549\pi$
$398$	−65.0713	−3.26173
$399$	0	0
$400$	−20.9736	−1.04868
$401$	13.6688	0.682587	0.341294	−	0.939957i	$-0.389135\pi$
$401$	13.6688	0.682587	0.341294	+	0.939957i	$0.389135\pi$
$402$	0	0
$403$	−2.43800	−0.121446
$404$	−3.83581	−0.190839
$405$	0	0
$406$	−1.19355	−0.0592349
$407$	−31.8670	−1.57959
$408$	0	0
$409$	−37.5240	−1.85544	−0.927722	−	0.373272i	$-0.878236\pi$
$409$	−37.5240	−1.85544	−0.927722	+	0.373272i	$0.878236\pi$
$410$	14.5466	0.718404
$411$	0	0
$412$	−24.3224	−1.19828
$413$	13.3394	0.656388
$414$	0	0
$415$	−10.8661	−0.533397
$416$	2.39769	0.117556
$417$	0	0
$418$	−73.9002	−3.61458
$419$	−9.96006	−0.486581	−0.243290	−	0.969954i	$-0.578227\pi$
$419$	−9.96006	−0.486581	−0.243290	+	0.969954i	$0.578227\pi$
$420$	0	0
$421$	−0.917111	−0.0446973	−0.0223486	−	0.999750i	$-0.507114\pi$
$421$	−0.917111	−0.0446973	−0.0223486	+	0.999750i	$0.507114\pi$
$422$	−57.8836	−2.81773
$423$	0	0
$424$	−18.1517	−0.881526
$425$	−11.9589	−0.580094
$426$	0	0
$427$	10.9442	0.529627
$428$	81.4542	3.93724
$429$	0	0
$430$	9.90603	0.477711
$431$	28.8257	1.38848	0.694242	−	0.719741i	$-0.255740\pi$
$431$	28.8257	1.38848	0.694242	+	0.719741i	$0.255740\pi$
$432$	0	0
$433$	−29.4309	−1.41436	−0.707178	−	0.707035i	$-0.750032\pi$
$433$	−29.4309	−1.41436	−0.707178	+	0.707035i	$0.750032\pi$
$434$	−23.9996	−1.15202
$435$	0	0
$436$	40.0177	1.91650
$437$	24.3641	1.16549
$438$	0	0
$439$	−21.0634	−1.00530	−0.502650	−	0.864490i	$-0.667642\pi$
$439$	−21.0634	−1.00530	−0.502650	+	0.864490i	$0.667642\pi$
$440$	−55.6657	−2.65376
$441$	0	0
$442$	3.50548	0.166739
$443$	−37.1991	−1.76738	−0.883691	−	0.468071i	$-0.844949\pi$
$443$	−37.1991	−1.76738	−0.883691	+	0.468071i	$0.844949\pi$
$444$	0	0
$445$	−18.4286	−0.873600
$446$	−23.7617	−1.12515
$447$	0	0
$448$	5.64176	0.266548
$449$	−20.6942	−0.976621	−0.488310	−	0.872670i	$-0.662387\pi$
$449$	−20.6942	−0.976621	−0.488310	+	0.872670i	$0.662387\pi$
$450$	0	0
$451$	−16.4638	−0.775248
$452$	10.7945	0.507729
$453$	0	0
$454$	−47.9337	−2.24964
$455$	0.430467	0.0201806
$456$	0	0
$457$	29.9339	1.40025	0.700124	−	0.714021i	$-0.253128\pi$
$457$	29.9339	1.40025	0.700124	+	0.714021i	$0.253128\pi$
$458$	−55.9972	−2.61658
$459$	0	0
$460$	31.7529	1.48048
$461$	−22.9494	−1.06886	−0.534429	−	0.845213i	$-0.679473\pi$
$461$	−22.9494	−1.06886	−0.534429	+	0.845213i	$0.679473\pi$
$462$	0	0
$463$	−14.6390	−0.680333	−0.340167	−	0.940365i	$-0.610484\pi$
$463$	−14.6390	−0.680333	−0.340167	+	0.940365i	$0.610484\pi$
$464$	−4.12896	−0.191682
$465$	0	0
$466$	23.0126	1.06604
$467$	−2.88366	−0.133440	−0.0667200	−	0.997772i	$-0.521253\pi$
$467$	−2.88366	−0.133440	−0.0667200	+	0.997772i	$0.521253\pi$
$468$	0	0
$469$	1.34491	0.0621022
$470$	24.1171	1.11244
$471$	0	0
$472$	94.8494	4.36580
$473$	−11.2116	−0.515510
$474$	0	0
$475$	−13.8625	−0.636056
$476$	24.2667	1.11226
$477$	0	0
$478$	−52.4351	−2.39833
$479$	−9.17227	−0.419092	−0.209546	−	0.977799i	$-0.567199\pi$
$479$	−9.17227	−0.419092	−0.209546	+	0.977799i	$0.567199\pi$
$480$	0	0
$481$	1.75224	0.0798952
$482$	0.718370	0.0327209
$483$	0	0
$484$	56.8757	2.58526
$485$	−9.00155	−0.408739
$486$	0	0
$487$	−26.2962	−1.19160	−0.595798	−	0.803134i	$-0.703164\pi$
$487$	−26.2962	−1.19160	−0.595798	+	0.803134i	$0.703164\pi$
$488$	77.8185	3.52268
$489$	0	0
$490$	4.23750	0.191431
$491$	−14.2675	−0.643882	−0.321941	−	0.946760i	$-0.604335\pi$
$491$	−14.2675	−0.643882	−0.321941	+	0.946760i	$0.604335\pi$
$492$	0	0
$493$	−2.35429	−0.106032
$494$	4.06347	0.182824
$495$	0	0
$496$	−83.0241	−3.72789
$497$	−5.05046	−0.226544
$498$	0	0
$499$	31.6638	1.41747	0.708734	−	0.705476i	$-0.249267\pi$
$499$	31.6638	1.41747	0.708734	+	0.705476i	$0.249267\pi$
$500$	−56.7452	−2.53772
$501$	0	0
$502$	60.4343	2.69731
$503$	38.6248	1.72219	0.861097	−	0.508441i	$-0.169778\pi$
$503$	38.6248	1.72219	0.861097	+	0.508441i	$0.169778\pi$
$504$	0	0
$505$	−1.32123	−0.0587937
$506$	−51.1044	−2.27187
$507$	0	0
$508$	4.73905	0.210261
$509$	15.2747	0.677040	0.338520	−	0.940959i	$-0.390074\pi$
$509$	15.2747	0.677040	0.338520	+	0.940959i	$0.390074\pi$
$510$	0	0
$511$	−16.4455	−0.727507
$512$	−46.0599	−2.03558
$513$	0	0
$514$	−60.9503	−2.68840
$515$	−8.37773	−0.369167
$516$	0	0
$517$	−27.2956	−1.20046
$518$	17.2490	0.757876
$519$	0	0
$520$	3.06083	0.134226
$521$	−10.6429	−0.466273	−0.233136	−	0.972444i	$-0.574899\pi$
$521$	−10.6429	−0.466273	−0.233136	+	0.972444i	$0.574899\pi$
$522$	0	0
$523$	28.1697	1.23178	0.615888	−	0.787834i	$-0.288797\pi$
$523$	28.1697	1.23178	0.615888	+	0.787834i	$0.288797\pi$
$524$	−36.5314	−1.59588
$525$	0	0
$526$	7.00158	0.305283
$527$	−47.3395	−2.06214
$528$	0	0
$529$	−6.15146	−0.267455
$530$	−10.8175	−0.469884
$531$	0	0
$532$	28.1293	1.21956
$533$	0.905275	0.0392118
$534$	0	0
$535$	28.0565	1.21299
$536$	9.56296	0.413057
$537$	0	0
$538$	11.3495	0.489313
$539$	−4.79599	−0.206578
$540$	0	0
$541$	11.9718	0.514708	0.257354	−	0.966317i	$-0.417149\pi$
$541$	11.9718	0.514708	0.257354	+	0.966317i	$0.417149\pi$
$542$	−22.9937	−0.987664
$543$	0	0
$544$	46.5566	1.99610
$545$	13.7839	0.590436
$546$	0	0
$547$	38.6453	1.65235	0.826177	−	0.563410i	$-0.190511\pi$
$547$	38.6453	1.65235	0.826177	+	0.563410i	$0.190511\pi$
$548$	13.8381	0.591132
$549$	0	0
$550$	29.0771	1.23985
$551$	−2.72904	−0.116261
$552$	0	0
$553$	−0.111124	−0.00472549
$554$	−50.4754	−2.14449
$555$	0	0
$556$	7.20901	0.305730
$557$	27.9185	1.18295	0.591473	−	0.806325i	$-0.298547\pi$
$557$	27.9185	1.18295	0.591473	+	0.806325i	$0.298547\pi$
$558$	0	0
$559$	0.616480	0.0260743
$560$	14.6592	0.619464
$561$	0	0
$562$	0.365794	0.0154301
$563$	−17.5654	−0.740292	−0.370146	−	0.928974i	$-0.620692\pi$
$563$	−17.5654	−0.740292	−0.370146	+	0.928974i	$0.620692\pi$
$564$	0	0
$565$	3.71810	0.156422
$566$	59.0994	2.48413
$567$	0	0
$568$	−35.9112	−1.50680
$569$	6.23445	0.261362	0.130681	−	0.991424i	$-0.458284\pi$
$569$	6.23445	0.261362	0.130681	+	0.991424i	$0.458284\pi$
$570$	0	0
$571$	10.2350	0.428321	0.214161	−	0.976798i	$-0.431298\pi$
$571$	10.2350	0.428321	0.214161	+	0.976798i	$0.431298\pi$
$572$	−5.99375	−0.250611
$573$	0	0
$574$	8.91149	0.371958
$575$	−9.58639	−0.399780
$576$	0	0
$577$	−6.56718	−0.273395	−0.136698	−	0.990613i	$-0.543649\pi$
$577$	−6.56718	−0.273395	−0.136698	+	0.990613i	$0.543649\pi$
$578$	23.9356	0.995591
$579$	0	0
$580$	−3.55666	−0.147682
$581$	−6.65678	−0.276170
$582$	0	0
$583$	12.2433	0.507064
$584$	−116.935	−4.83882
$585$	0	0
$586$	76.7202	3.16928
$587$	3.47722	0.143520	0.0717602	−	0.997422i	$-0.477138\pi$
$587$	3.47722	0.143520	0.0717602	+	0.997422i	$0.477138\pi$
$588$	0	0
$589$	−54.8748	−2.26108
$590$	56.5256	2.32712
$591$	0	0
$592$	59.6709	2.45246
$593$	18.0049	0.739372	0.369686	−	0.929157i	$-0.379465\pi$
$593$	18.0049	0.739372	0.369686	+	0.929157i	$0.379465\pi$
$594$	0	0
$595$	8.35852	0.342666
$596$	−56.9910	−2.33444
$597$	0	0
$598$	2.81002	0.114910
$599$	−5.91219	−0.241565	−0.120783	−	0.992679i	$-0.538540\pi$
$599$	−5.91219	−0.241565	−0.120783	+	0.992679i	$0.538540\pi$
$600$	0	0
$601$	−8.32532	−0.339597	−0.169799	−	0.985479i	$-0.554312\pi$
$601$	−8.32532	−0.339597	−0.169799	+	0.985479i	$0.554312\pi$
$602$	6.06861	0.247338
$603$	0	0
$604$	21.3995	0.870732
$605$	19.5905	0.796468
$606$	0	0
$607$	12.2488	0.497163	0.248581	−	0.968611i	$-0.420036\pi$
$607$	12.2488	0.497163	0.248581	+	0.968611i	$0.420036\pi$
$608$	53.9673	2.18867
$609$	0	0
$610$	46.3761	1.87771
$611$	1.50087	0.0607188
$612$	0	0
$613$	5.93637	0.239768	0.119884	−	0.992788i	$-0.461748\pi$
$613$	5.93637	0.239768	0.119884	+	0.992788i	$0.461748\pi$
$614$	−45.1711	−1.82296
$615$	0	0
$616$	−34.1018	−1.37400
$617$	44.2173	1.78012	0.890061	−	0.455841i	$-0.150662\pi$
$617$	44.2173	1.78012	0.890061	+	0.455841i	$0.150662\pi$
$618$	0	0
$619$	−30.2653	−1.21647	−0.608233	−	0.793759i	$-0.708121\pi$
$619$	−30.2653	−1.21647	−0.608233	+	0.793759i	$0.708121\pi$
$620$	−71.5165	−2.87217
$621$	0	0
$622$	46.4508	1.86251
$623$	−11.2897	−0.452312
$624$	0	0
$625$	−7.86826	−0.314730
$626$	−33.9271	−1.35600
$627$	0	0
$628$	12.2981	0.490749
$629$	34.0238	1.35662
$630$	0	0
$631$	37.4959	1.49269	0.746344	−	0.665561i	$-0.231808\pi$
$631$	37.4959	1.49269	0.746344	+	0.665561i	$0.231808\pi$
$632$	−0.790148	−0.0314304
$633$	0	0
$634$	64.3205	2.55449
$635$	1.63234	0.0647774
$636$	0	0
$637$	0.263712	0.0104486
$638$	5.72425	0.226625
$639$	0	0
$640$	−5.77572	−0.228305
$641$	−20.9319	−0.826762	−0.413381	−	0.910558i	$-0.635652\pi$
$641$	−20.9319	−0.826762	−0.413381	+	0.910558i	$0.635652\pi$
$642$	0	0
$643$	42.0230	1.65723	0.828613	−	0.559821i	$-0.189130\pi$
$643$	42.0230	1.65723	0.828613	+	0.559821i	$0.189130\pi$
$644$	19.4524	0.766530
$645$	0	0
$646$	78.9017	3.10435
$647$	−7.99073	−0.314148	−0.157074	−	0.987587i	$-0.550206\pi$
$647$	−7.99073	−0.314148	−0.157074	+	0.987587i	$0.550206\pi$
$648$	0	0
$649$	−63.9755	−2.51126
$650$	−1.59883	−0.0627113
$651$	0	0
$652$	96.5995	3.78313
$653$	6.91364	0.270552	0.135276	−	0.990808i	$-0.456808\pi$
$653$	6.91364	0.270552	0.135276	+	0.990808i	$0.456808\pi$
$654$	0	0
$655$	−12.5831	−0.491661
$656$	30.8284	1.20364
$657$	0	0
$658$	14.7745	0.575971
$659$	−29.0812	−1.13284	−0.566421	−	0.824116i	$-0.691672\pi$
$659$	−29.0812	−1.13284	−0.566421	+	0.824116i	$0.691672\pi$
$660$	0	0
$661$	−30.8242	−1.19892	−0.599461	−	0.800404i	$-0.704618\pi$
$661$	−30.8242	−1.19892	−0.599461	+	0.800404i	$0.704618\pi$
$662$	−37.3685	−1.45237
$663$	0	0
$664$	−47.3329	−1.83687
$665$	9.68900	0.375723
$666$	0	0
$667$	−1.88722	−0.0730734
$668$	−33.9113	−1.31207
$669$	0	0
$670$	5.69906	0.220174
$671$	−52.4883	−2.02629
$672$	0	0
$673$	25.7018	0.990733	0.495366	−	0.868684i	$-0.335034\pi$
$673$	25.7018	0.990733	0.495366	+	0.868684i	$0.335034\pi$
$674$	−21.1270	−0.813780
$675$	0	0
$676$	−61.2780	−2.35685
$677$	21.9171	0.842342	0.421171	−	0.906981i	$-0.361619\pi$
$677$	21.9171	0.842342	0.421171	+	0.906981i	$0.361619\pi$
$678$	0	0
$679$	−5.51451	−0.211627
$680$	59.4331	2.27916
$681$	0	0
$682$	115.102	4.40747
$683$	−16.0446	−0.613931	−0.306966	−	0.951721i	$-0.599314\pi$
$683$	−16.0446	−0.613931	−0.306966	+	0.951721i	$0.599314\pi$
$684$	0	0
$685$	4.76644	0.182116
$686$	2.59597	0.0991145
$687$	0	0
$688$	20.9937	0.800377
$689$	−0.673207	−0.0256471
$690$	0	0
$691$	−11.0457	−0.420199	−0.210099	−	0.977680i	$-0.567379\pi$
$691$	−11.0457	−0.420199	−0.210099	+	0.977680i	$0.567379\pi$
$692$	43.0728	1.63738
$693$	0	0
$694$	61.3444	2.32860
$695$	2.48310	0.0941895
$696$	0	0
$697$	17.5780	0.665815
$698$	−70.0958	−2.65316
$699$	0	0
$700$	−11.0679	−0.418327
$701$	7.99153	0.301836	0.150918	−	0.988546i	$-0.451777\pi$
$701$	7.99153	0.301836	0.150918	+	0.988546i	$0.451777\pi$
$702$	0	0
$703$	39.4395	1.48749
$704$	−27.0578	−1.01978
$705$	0	0
$706$	−53.9565	−2.03068
$707$	−0.809406	−0.0304408
$708$	0	0
$709$	10.5416	0.395900	0.197950	−	0.980212i	$-0.436572\pi$
$709$	10.5416	0.395900	0.197950	+	0.980212i	$0.436572\pi$
$710$	−21.4013	−0.803177
$711$	0	0
$712$	−80.2751	−3.00844
$713$	−37.9477	−1.42115
$714$	0	0
$715$	−2.06452	−0.0772085
$716$	93.1929	3.48278
$717$	0	0
$718$	19.0634	0.711440
$719$	25.0664	0.934820	0.467410	−	0.884041i	$-0.345187\pi$
$719$	25.0664	0.934820	0.467410	+	0.884041i	$0.345187\pi$
$720$	0	0
$721$	−5.13235	−0.191139
$722$	42.1376	1.56820
$723$	0	0
$724$	26.7397	0.993771
$725$	1.07378	0.0398792
$726$	0	0
$727$	−21.6940	−0.804588	−0.402294	−	0.915511i	$-0.631787\pi$
$727$	−21.6940	−0.804588	−0.402294	+	0.915511i	$0.631787\pi$
$728$	1.87512	0.0694964
$729$	0	0
$730$	−69.6879	−2.57926
$731$	11.9704	0.442741
$732$	0	0
$733$	21.1204	0.780099	0.390050	−	0.920794i	$-0.372458\pi$
$733$	21.1204	0.780099	0.390050	+	0.920794i	$0.372458\pi$
$734$	8.49152	0.313428
$735$	0	0
$736$	37.3202	1.37564
$737$	−6.45018	−0.237595
$738$	0	0
$739$	−30.2759	−1.11372	−0.556858	−	0.830608i	$-0.687993\pi$
$739$	−30.2759	−1.11372	−0.556858	+	0.830608i	$0.687993\pi$
$740$	51.4002	1.88951
$741$	0	0
$742$	−6.62702	−0.243286
$743$	46.3530	1.70053	0.850264	−	0.526356i	$-0.176442\pi$
$743$	46.3530	1.70053	0.850264	+	0.526356i	$0.176442\pi$
$744$	0	0
$745$	−19.6303	−0.719197
$746$	−79.8457	−2.92336
$747$	0	0
$748$	−116.383	−4.25537
$749$	17.1879	0.628031
$750$	0	0
$751$	28.3709	1.03527	0.517634	−	0.855602i	$-0.326813\pi$
$751$	28.3709	1.03527	0.517634	+	0.855602i	$0.326813\pi$
$752$	51.1109	1.86382
$753$	0	0
$754$	−0.314753	−0.0114626
$755$	7.37094	0.268256
$756$	0	0
$757$	−29.1551	−1.05966	−0.529831	−	0.848103i	$-0.677745\pi$
$757$	−29.1551	−1.05966	−0.529831	+	0.848103i	$0.677745\pi$
$758$	41.9644	1.52422
$759$	0	0
$760$	68.8934	2.49903
$761$	51.7911	1.87743	0.938713	−	0.344700i	$-0.112019\pi$
$761$	51.7911	1.87743	0.938713	+	0.344700i	$0.112019\pi$
$762$	0	0
$763$	8.44424	0.305702
$764$	68.7507	2.48731
$765$	0	0
$766$	−26.0759	−0.942160
$767$	3.51775	0.127019
$768$	0	0
$769$	44.8361	1.61683	0.808415	−	0.588613i	$-0.200326\pi$
$769$	44.8361	1.61683	0.808415	+	0.588613i	$0.200326\pi$
$770$	−20.3230	−0.732390
$771$	0	0
$772$	−64.5040	−2.32155
$773$	45.1243	1.62301	0.811504	−	0.584347i	$-0.198649\pi$
$773$	45.1243	1.62301	0.811504	+	0.584347i	$0.198649\pi$
$774$	0	0
$775$	21.5913	0.775582
$776$	−39.2108	−1.40758
$777$	0	0
$778$	−76.8843	−2.75644
$779$	20.3760	0.730047
$780$	0	0
$781$	24.2219	0.866729
$782$	54.5631	1.95117
$783$	0	0
$784$	8.98048	0.320731
$785$	4.23603	0.151190
$786$	0	0
$787$	−31.6116	−1.12683	−0.563416	−	0.826173i	$-0.690513\pi$
$787$	−31.6116	−1.12683	−0.563416	+	0.826173i	$0.690513\pi$
$788$	−32.6355	−1.16259
$789$	0	0
$790$	−0.470890	−0.0167535
$791$	2.27777	0.0809883
$792$	0	0
$793$	2.88611	0.102489
$794$	0.146624	0.00520349
$795$	0	0
$796$	−118.790	−4.21041
$797$	−9.37863	−0.332208	−0.166104	−	0.986108i	$-0.553119\pi$
$797$	−9.37863	−0.332208	−0.166104	+	0.986108i	$0.553119\pi$
$798$	0	0
$799$	29.1429	1.03100
$800$	−21.2342	−0.750743
$801$	0	0
$802$	35.4838	1.25297
$803$	78.8725	2.78335
$804$	0	0
$805$	6.70026	0.236153
$806$	−6.32897	−0.222929
$807$	0	0
$808$	−5.75526	−0.202469
$809$	−30.7716	−1.08187	−0.540935	−	0.841064i	$-0.681930\pi$
$809$	−30.7716	−1.08187	−0.540935	+	0.841064i	$0.681930\pi$
$810$	0	0
$811$	5.29510	0.185936	0.0929681	−	0.995669i	$-0.470365\pi$
$811$	5.29510	0.185936	0.0929681	+	0.995669i	$0.470365\pi$
$812$	−2.17888	−0.0764635
$813$	0	0
$814$	−82.7258	−2.89954
$815$	33.2732	1.16551
$816$	0	0
$817$	13.8758	0.485453
$818$	−97.4111	−3.40590
$819$	0	0
$820$	26.5554	0.927354
$821$	46.7930	1.63309	0.816543	−	0.577284i	$-0.195888\pi$
$821$	46.7930	1.63309	0.816543	+	0.577284i	$0.195888\pi$
$822$	0	0
$823$	−9.35491	−0.326092	−0.163046	−	0.986618i	$-0.552132\pi$
$823$	−9.35491	−0.326092	−0.163046	+	0.986618i	$0.552132\pi$
$824$	−36.4934	−1.27131
$825$	0	0
$826$	34.6286	1.20488
$827$	−40.5945	−1.41161	−0.705805	−	0.708406i	$-0.749414\pi$
$827$	−40.5945	−1.41161	−0.705805	+	0.708406i	$0.749414\pi$
$828$	0	0
$829$	−11.2936	−0.392243	−0.196121	−	0.980580i	$-0.562835\pi$
$829$	−11.2936	−0.392243	−0.196121	+	0.980580i	$0.562835\pi$
$830$	−28.2081	−0.979118
$831$	0	0
$832$	1.48780	0.0515801
$833$	5.12058	0.177417
$834$	0	0
$835$	−11.6806	−0.404222
$836$	−134.908	−4.66589
$837$	0	0
$838$	−25.8560	−0.893180
$839$	−6.00322	−0.207254	−0.103627	−	0.994616i	$-0.533045\pi$
$839$	−6.00322	−0.207254	−0.103627	+	0.994616i	$0.533045\pi$
$840$	0	0
$841$	−28.7886	−0.992711
$842$	−2.38079	−0.0820475
$843$	0	0
$844$	−105.669	−3.63727
$845$	−21.1069	−0.726099
$846$	0	0
$847$	12.0015	0.412377
$848$	−22.9255	−0.787264
$849$	0	0
$850$	−31.0450	−1.06484
$851$	27.2737	0.934932
$852$	0	0
$853$	43.1151	1.47623	0.738116	−	0.674674i	$-0.235716\pi$
$853$	43.1151	1.47623	0.738116	+	0.674674i	$0.235716\pi$
$854$	28.4108	0.972197
$855$	0	0
$856$	122.214	4.17719
$857$	−14.7372	−0.503414	−0.251707	−	0.967803i	$-0.580992\pi$
$857$	−14.7372	−0.503414	−0.251707	+	0.967803i	$0.580992\pi$
$858$	0	0
$859$	−31.1822	−1.06392	−0.531961	−	0.846769i	$-0.678545\pi$
$859$	−31.1822	−1.06392	−0.531961	+	0.846769i	$0.678545\pi$
$860$	18.0839	0.616655
$861$	0	0
$862$	74.8305	2.54874
$863$	26.3519	0.897028	0.448514	−	0.893776i	$-0.351953\pi$
$863$	26.3519	0.897028	0.448514	+	0.893776i	$0.351953\pi$
$864$	0	0
$865$	14.8362	0.504446
$866$	−76.4016	−2.59623
$867$	0	0
$868$	−43.8123	−1.48709
$869$	0.532952	0.0180791
$870$	0	0
$871$	0.354669	0.0120175
$872$	60.0426	2.03330
$873$	0	0
$874$	63.2483	2.13941
$875$	−11.9740	−0.404794
$876$	0	0
$877$	−44.9642	−1.51833	−0.759167	−	0.650896i	$-0.774393\pi$
$877$	−44.9642	−1.51833	−0.759167	+	0.650896i	$0.774393\pi$
$878$	−54.6798	−1.84536
$879$	0	0
$880$	−70.3053	−2.36999
$881$	−11.3353	−0.381897	−0.190949	−	0.981600i	$-0.561156\pi$
$881$	−11.3353	−0.381897	−0.190949	+	0.981600i	$0.561156\pi$
$882$	0	0
$883$	37.7221	1.26945	0.634724	−	0.772739i	$-0.281114\pi$
$883$	37.7221	1.26945	0.634724	+	0.772739i	$0.281114\pi$
$884$	6.39940	0.215235
$885$	0	0
$886$	−96.5676	−3.24425
$887$	−5.00523	−0.168059	−0.0840295	−	0.996463i	$-0.526779\pi$
$887$	−5.00523	−0.168059	−0.0840295	+	0.996463i	$0.526779\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−47.8401	−1.60360
$891$	0	0
$892$	−43.3779	−1.45240
$893$	33.7818	1.13046
$894$	0	0
$895$	32.0998	1.07298
$896$	−3.53831	−0.118207
$897$	0	0
$898$	−53.7215	−1.79271
$899$	4.25056	0.141764
$900$	0	0
$901$	−13.0719	−0.435487
$902$	−42.7394	−1.42307
$903$	0	0
$904$	16.1961	0.538673
$905$	9.21033	0.306162
$906$	0	0
$907$	−1.70559	−0.0566333	−0.0283166	−	0.999599i	$-0.509015\pi$
$907$	−1.70559	−0.0566333	−0.0283166	+	0.999599i	$0.509015\pi$
$908$	−87.5049	−2.90395
$909$	0	0
$910$	1.11748	0.0370441
$911$	49.3674	1.63561	0.817807	−	0.575493i	$-0.195190\pi$
$911$	49.3674	1.63561	0.817807	+	0.575493i	$0.195190\pi$
$912$	0	0
$913$	31.9258	1.05659
$914$	77.7074	2.57033
$915$	0	0
$916$	−102.225	−3.37762
$917$	−7.70860	−0.254560
$918$	0	0
$919$	−40.1970	−1.32598	−0.662989	−	0.748629i	$-0.730712\pi$
$919$	−40.1970	−1.32598	−0.662989	+	0.748629i	$0.730712\pi$
$920$	47.6421	1.57071
$921$	0	0
$922$	−59.5758	−1.96202
$923$	−1.33186	−0.0438389
$924$	0	0
$925$	−15.5181	−0.510230
$926$	−38.0024	−1.24884
$927$	0	0
$928$	−4.18027	−0.137224
$929$	−23.2396	−0.762465	−0.381233	−	0.924479i	$-0.624500\pi$
$929$	−23.2396	−0.762465	−0.381233	+	0.924479i	$0.624500\pi$
$930$	0	0
$931$	5.93565	0.194533
$932$	42.0104	1.37610
$933$	0	0
$934$	−7.48590	−0.244946
$935$	−40.0874	−1.31100
$936$	0	0
$937$	32.6176	1.06557	0.532785	−	0.846251i	$-0.321146\pi$
$937$	32.6176	1.06557	0.532785	+	0.846251i	$0.321146\pi$
$938$	3.49134	0.113996
$939$	0	0
$940$	44.0267	1.43599
$941$	−0.939169	−0.0306160	−0.0153080	−	0.999883i	$-0.504873\pi$
$941$	−0.939169	−0.0306160	−0.0153080	+	0.999883i	$0.504873\pi$
$942$	0	0
$943$	14.0907	0.458856
$944$	119.794	3.89896
$945$	0	0
$946$	−29.1050	−0.946284
$947$	3.25233	0.105686	0.0528432	−	0.998603i	$-0.483172\pi$
$947$	3.25233	0.105686	0.0528432	+	0.998603i	$0.483172\pi$
$948$	0	0
$949$	−4.33687	−0.140781
$950$	−35.9866	−1.16756
$951$	0	0
$952$	36.4098	1.18005
$953$	−20.1286	−0.652029	−0.326015	−	0.945365i	$-0.605706\pi$
$953$	−20.1286	−0.652029	−0.326015	+	0.945365i	$0.605706\pi$
$954$	0	0
$955$	23.6808	0.766293
$956$	−95.7225	−3.09589
$957$	0	0
$958$	−23.8109	−0.769296
$959$	2.92001	0.0942919
$960$	0	0
$961$	54.4691	1.75707
$962$	4.54875	0.146658
$963$	0	0
$964$	1.31141	0.0422378
$965$	−22.2180	−0.715224
$966$	0	0
$967$	−46.0528	−1.48096	−0.740479	−	0.672080i	$-0.765401\pi$
$967$	−46.0528	−1.48096	−0.740479	+	0.672080i	$0.765401\pi$
$968$	85.3364	2.74282
$969$	0	0
$970$	−23.3677	−0.750292
$971$	−15.1771	−0.487055	−0.243527	−	0.969894i	$-0.578305\pi$
$971$	−15.1771	−0.487055	−0.243527	+	0.969894i	$0.578305\pi$
$972$	0	0
$973$	1.52119	0.0487672
$974$	−68.2641	−2.18732
$975$	0	0
$976$	98.2841	3.14600
$977$	36.8954	1.18039	0.590194	−	0.807261i	$-0.299051\pi$
$977$	36.8954	1.18039	0.590194	+	0.807261i	$0.299051\pi$
$978$	0	0
$979$	54.1452	1.73049
$980$	7.73574	0.247109
$981$	0	0
$982$	−37.0379	−1.18193
$983$	−13.2244	−0.421794	−0.210897	−	0.977508i	$-0.567638\pi$
$983$	−13.2244	−0.421794	−0.210897	+	0.977508i	$0.567638\pi$
$984$	0	0
$985$	−11.2411	−0.358173
$986$	−6.11166	−0.194635
$987$	0	0
$988$	7.41804	0.235999
$989$	9.59557	0.305121
$990$	0	0
$991$	23.1367	0.734960	0.367480	−	0.930031i	$-0.380221\pi$
$991$	23.1367	0.734960	0.367480	+	0.930031i	$0.380221\pi$
$992$	−84.0557	−2.66877
$993$	0	0
$994$	−13.1108	−0.415850
$995$	−40.9167	−1.29715
$996$	0	0
$997$	−39.4462	−1.24927	−0.624637	−	0.780915i	$-0.714753\pi$
$997$	−39.4462	−1.24927	−0.624637	+	0.780915i	$0.714753\pi$
$998$	82.1982	2.60194
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.16		18
3.2	odd	2		2667.2.a.p.1.3	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.3	✓	18	3.2	odd	2
8001.2.a.u.1.16		18	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)

\(f(q)\)	\(=\)	\(q+2.59597 q^{2} +4.73905 q^{4} +1.63234 q^{5} +1.00000 q^{7} +7.11048 q^{8} +O(q^{10})\)	\(q+2.59597 q^{2} +4.73905 q^{4} +1.63234 q^{5} +1.00000 q^{7} +7.11048 q^{8} +4.23750 q^{10} -4.79599 q^{11} +0.263712 q^{13} +2.59597 q^{14} +8.98048 q^{16} +5.12058 q^{17} +5.93565 q^{19} +7.73574 q^{20} -12.4502 q^{22} +4.10470 q^{23} -2.33547 q^{25} +0.684587 q^{26} +4.73905 q^{28} -0.459771 q^{29} -9.24495 q^{31} +9.09207 q^{32} +13.2929 q^{34} +1.63234 q^{35} +6.64452 q^{37} +15.4088 q^{38} +11.6067 q^{40} +3.43282 q^{41} +2.33770 q^{43} -22.7284 q^{44} +10.6557 q^{46} +5.69134 q^{47} +1.00000 q^{49} -6.06280 q^{50} +1.24974 q^{52} -2.55281 q^{53} -7.82868 q^{55} +7.11048 q^{56} -1.19355 q^{58} +13.3394 q^{59} +10.9442 q^{61} -23.9996 q^{62} +5.64176 q^{64} +0.430467 q^{65} +1.34491 q^{67} +24.2667 q^{68} +4.23750 q^{70} -5.05046 q^{71} -16.4455 q^{73} +17.2490 q^{74} +28.1293 q^{76} -4.79599 q^{77} -0.111124 q^{79} +14.6592 q^{80} +8.91149 q^{82} -6.65678 q^{83} +8.35852 q^{85} +6.06861 q^{86} -34.1018 q^{88} -11.2897 q^{89} +0.263712 q^{91} +19.4524 q^{92} +14.7745 q^{94} +9.68900 q^{95} -5.51451 q^{97} +2.59597 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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