LMFDB - Embedded newform 8001.2.a.u.1.17

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			1.17
Root			$2.73355$ 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)

$f(q)$	$=$	$q+2.73355 q^{2} +5.47227 q^{4} -4.25416 q^{5} +1.00000 q^{7} +9.49162 q^{8} +O(q^{10})$	\(q+2.73355 q^{2} +5.47227 q^{4} -4.25416 q^{5} +1.00000 q^{7} +9.49162 q^{8} -11.6289 q^{10} -0.824446 q^{11} -6.07408 q^{13} +2.73355 q^{14} +15.0012 q^{16} +0.158117 q^{17} +2.99692 q^{19} -23.2799 q^{20} -2.25366 q^{22} +7.25197 q^{23} +13.0978 q^{25} -16.6038 q^{26} +5.47227 q^{28} +8.45102 q^{29} +3.64198 q^{31} +22.0233 q^{32} +0.432220 q^{34} -4.25416 q^{35} -7.96478 q^{37} +8.19223 q^{38} -40.3788 q^{40} +2.33165 q^{41} -7.65441 q^{43} -4.51159 q^{44} +19.8236 q^{46} +1.87334 q^{47} +1.00000 q^{49} +35.8035 q^{50} -33.2390 q^{52} +4.08642 q^{53} +3.50732 q^{55} +9.49162 q^{56} +23.1013 q^{58} +5.07072 q^{59} -10.9691 q^{61} +9.95552 q^{62} +30.1993 q^{64} +25.8401 q^{65} -8.96906 q^{67} +0.865260 q^{68} -11.6289 q^{70} +13.5264 q^{71} +3.98307 q^{73} -21.7721 q^{74} +16.4000 q^{76} -0.824446 q^{77} +10.1633 q^{79} -63.8175 q^{80} +6.37367 q^{82} +14.9750 q^{83} -0.672655 q^{85} -20.9237 q^{86} -7.82533 q^{88} +2.30581 q^{89} -6.07408 q^{91} +39.6847 q^{92} +5.12087 q^{94} -12.7494 q^{95} +12.0013 q^{97} +2.73355 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.73355	1.93291	0.966454	−	0.256838i	$-0.0826808\pi$
$2$	2.73355	1.93291	0.966454	+	0.256838i	$0.0826808\pi$
$3$	0	0
$3$	0	0
$4$	5.47227	2.73614
$5$	−4.25416	−1.90252	−0.951258	−	0.308396i	$-0.900208\pi$
$5$	−4.25416	−1.90252	−0.951258	+	0.308396i	$0.900208\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	9.49162	3.35579
$9$	0	0
$10$	−11.6289	−3.67739
$11$	−0.824446	−0.248580	−0.124290	−	0.992246i	$-0.539665\pi$
$11$	−0.824446	−0.248580	−0.124290	+	0.992246i	$0.539665\pi$
$12$	0	0
$13$	−6.07408	−1.68465	−0.842324	−	0.538972i	$-0.818813\pi$
$13$	−6.07408	−1.68465	−0.842324	+	0.538972i	$0.818813\pi$
$14$	2.73355	0.730571
$15$	0	0
$16$	15.0012	3.75031
$17$	0.158117	0.0383490	0.0191745	−	0.999816i	$-0.493896\pi$
$17$	0.158117	0.0383490	0.0191745	+	0.999816i	$0.493896\pi$
$18$	0	0
$19$	2.99692	0.687542	0.343771	−	0.939054i	$-0.388296\pi$
$19$	2.99692	0.687542	0.343771	+	0.939054i	$0.388296\pi$
$20$	−23.2799	−5.20554
$21$	0	0
$22$	−2.25366	−0.480482
$23$	7.25197	1.51214	0.756070	−	0.654491i	$-0.227117\pi$
$23$	7.25197	1.51214	0.756070	+	0.654491i	$0.227117\pi$
$24$	0	0
$25$	13.0978	2.61957
$26$	−16.6038	−3.25627
$27$	0	0
$28$	5.47227	1.03416
$29$	8.45102	1.56932	0.784658	−	0.619929i	$-0.212839\pi$
$29$	8.45102	1.56932	0.784658	+	0.619929i	$0.212839\pi$
$30$	0	0
$31$	3.64198	0.654119	0.327059	−	0.945004i	$-0.393942\pi$
$31$	3.64198	0.654119	0.327059	+	0.945004i	$0.393942\pi$
$32$	22.0233	3.89321
$33$	0	0
$34$	0.432220	0.0741252
$35$	−4.25416	−0.719084
$36$	0	0
$37$	−7.96478	−1.30940	−0.654701	−	0.755888i	$-0.727206\pi$
$37$	−7.96478	−1.30940	−0.654701	+	0.755888i	$0.727206\pi$
$38$	8.19223	1.32896
$39$	0	0
$40$	−40.3788	−6.38445
$41$	2.33165	0.364142	0.182071	−	0.983285i	$-0.441720\pi$
$41$	2.33165	0.364142	0.182071	+	0.983285i	$0.441720\pi$
$42$	0	0
$43$	−7.65441	−1.16729	−0.583643	−	0.812010i	$-0.698373\pi$
$43$	−7.65441	−1.16729	−0.583643	+	0.812010i	$0.698373\pi$
$44$	−4.51159	−0.680148
$45$	0	0
$46$	19.8236	2.92283
$47$	1.87334	0.273255	0.136628	−	0.990622i	$-0.456374\pi$
$47$	1.87334	0.273255	0.136628	+	0.990622i	$0.456374\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	35.8035	5.06339
$51$	0	0
$52$	−33.2390	−4.60942
$53$	4.08642	0.561312	0.280656	−	0.959808i	$-0.409448\pi$
$53$	4.08642	0.561312	0.280656	+	0.959808i	$0.409448\pi$
$54$	0	0
$55$	3.50732	0.472927
$56$	9.49162	1.26837
$57$	0	0
$58$	23.1013	3.03334
$59$	5.07072	0.660152	0.330076	−	0.943954i	$-0.392926\pi$
$59$	5.07072	0.660152	0.330076	+	0.943954i	$0.392926\pi$
$60$	0	0
$61$	−10.9691	−1.40445	−0.702224	−	0.711956i	$-0.747809\pi$
$61$	−10.9691	−1.40445	−0.702224	+	0.711956i	$0.747809\pi$
$62$	9.95552	1.26435
$63$	0	0
$64$	30.1993	3.77491
$65$	25.8401	3.20507
$66$	0	0
$67$	−8.96906	−1.09574	−0.547872	−	0.836562i	$-0.684562\pi$
$67$	−8.96906	−1.09574	−0.547872	+	0.836562i	$0.684562\pi$
$68$	0.865260	0.104928
$69$	0	0
$70$	−11.6289	−1.38992
$71$	13.5264	1.60529	0.802643	−	0.596459i	$-0.203426\pi$
$71$	13.5264	1.60529	0.802643	+	0.596459i	$0.203426\pi$
$72$	0	0
$73$	3.98307	0.466183	0.233091	−	0.972455i	$-0.425116\pi$
$73$	3.98307	0.466183	0.233091	+	0.972455i	$0.425116\pi$
$74$	−21.7721	−2.53096
$75$	0	0
$76$	16.4000	1.88121
$77$	−0.824446	−0.0939544
$78$	0	0
$79$	10.1633	1.14346	0.571732	−	0.820441i	$-0.306272\pi$
$79$	10.1633	1.14346	0.571732	+	0.820441i	$0.306272\pi$
$80$	−63.8175	−7.13502
$81$	0	0
$82$	6.37367	0.703854
$83$	14.9750	1.64372	0.821859	−	0.569691i	$-0.192937\pi$
$83$	14.9750	1.64372	0.821859	+	0.569691i	$0.192937\pi$
$84$	0	0
$85$	−0.672655	−0.0729596
$86$	−20.9237	−2.25626
$87$	0	0
$88$	−7.82533	−0.834183
$89$	2.30581	0.244415	0.122208	−	0.992505i	$-0.461003\pi$
$89$	2.30581	0.244415	0.122208	+	0.992505i	$0.461003\pi$
$90$	0	0
$91$	−6.07408	−0.636737
$92$	39.6847	4.13742
$93$	0	0
$94$	5.12087	0.528177
$95$	−12.7494	−1.30806
$96$	0	0
$97$	12.0013	1.21854	0.609272	−	0.792961i	$-0.291462\pi$
$97$	12.0013	1.21854	0.609272	+	0.792961i	$0.291462\pi$
$98$	2.73355	0.276130
$99$	0	0
$100$	71.6750	7.16750
$101$	14.7534	1.46802	0.734008	−	0.679140i	$-0.237647\pi$
$101$	14.7534	1.46802	0.734008	+	0.679140i	$0.237647\pi$
$102$	0	0
$103$	20.0095	1.97159	0.985797	−	0.167943i	$-0.0537124\pi$
$103$	20.0095	1.97159	0.985797	+	0.167943i	$0.0537124\pi$
$104$	−57.6529	−5.65333
$105$	0	0
$106$	11.1704	1.08497
$107$	−3.53204	−0.341455	−0.170728	−	0.985318i	$-0.554612\pi$
$107$	−3.53204	−0.341455	−0.170728	+	0.985318i	$0.554612\pi$
$108$	0	0
$109$	−7.28193	−0.697483	−0.348741	−	0.937219i	$-0.613391\pi$
$109$	−7.28193	−0.697483	−0.348741	+	0.937219i	$0.613391\pi$
$110$	9.58743	0.914125
$111$	0	0
$112$	15.0012	1.41748
$113$	0.617948	0.0581316	0.0290658	−	0.999578i	$-0.490747\pi$
$113$	0.617948	0.0581316	0.0290658	+	0.999578i	$0.490747\pi$
$114$	0	0
$115$	−30.8510	−2.87687
$116$	46.2463	4.29386
$117$	0	0
$118$	13.8611	1.27601
$119$	0.158117	0.0144946
$120$	0	0
$121$	−10.3203	−0.938208
$122$	−29.9845	−2.71467
$123$	0	0
$124$	19.9299	1.78976
$125$	−34.4495	−3.08125
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	38.5045	3.40334
$129$	0	0
$130$	70.6351	6.19510
$131$	−5.77893	−0.504907	−0.252454	−	0.967609i	$-0.581237\pi$
$131$	−5.77893	−0.504907	−0.252454	+	0.967609i	$0.581237\pi$
$132$	0	0
$133$	2.99692	0.259866
$134$	−24.5173	−2.11797
$135$	0	0
$136$	1.50079	0.128691
$137$	−10.6326	−0.908402	−0.454201	−	0.890899i	$-0.650075\pi$
$137$	−10.6326	−0.908402	−0.454201	+	0.890899i	$0.650075\pi$
$138$	0	0
$139$	11.1927	0.949357	0.474678	−	0.880159i	$-0.342564\pi$
$139$	11.1927	0.949357	0.474678	+	0.880159i	$0.342564\pi$
$140$	−23.2799	−1.96751
$141$	0	0
$142$	36.9750	3.10287
$143$	5.00775	0.418769
$144$	0	0
$145$	−35.9520	−2.98565
$146$	10.8879	0.901088
$147$	0	0
$148$	−43.5855	−3.58270
$149$	5.51178	0.451543	0.225771	−	0.974180i	$-0.427510\pi$
$149$	5.51178	0.451543	0.225771	+	0.974180i	$0.427510\pi$
$150$	0	0
$151$	−9.28026	−0.755217	−0.377608	−	0.925965i	$-0.623253\pi$
$151$	−9.28026	−0.755217	−0.377608	+	0.925965i	$0.623253\pi$
$152$	28.4457	2.30725
$153$	0	0
$154$	−2.25366	−0.181605
$155$	−15.4936	−1.24447
$156$	0	0
$157$	6.26218	0.499777	0.249888	−	0.968275i	$-0.419606\pi$
$157$	6.26218	0.499777	0.249888	+	0.968275i	$0.419606\pi$
$158$	27.7819	2.21021
$159$	0	0
$160$	−93.6906	−7.40689
$161$	7.25197	0.571535
$162$	0	0
$163$	9.87167	0.773209	0.386605	−	0.922246i	$-0.373648\pi$
$163$	9.87167	0.773209	0.386605	+	0.922246i	$0.373648\pi$
$164$	12.7594	0.996343
$165$	0	0
$166$	40.9348	3.17716
$167$	3.42627	0.265132	0.132566	−	0.991174i	$-0.457678\pi$
$167$	3.42627	0.265132	0.132566	+	0.991174i	$0.457678\pi$
$168$	0	0
$169$	23.8945	1.83804
$170$	−1.83873	−0.141024
$171$	0	0
$172$	−41.8870	−3.19385
$173$	24.0689	1.82992	0.914961	−	0.403542i	$-0.132221\pi$
$173$	24.0689	1.82992	0.914961	+	0.403542i	$0.132221\pi$
$174$	0	0
$175$	13.0978	0.990104
$176$	−12.3677	−0.932251
$177$	0	0
$178$	6.30304	0.472433
$179$	−1.62717	−0.121620	−0.0608101	−	0.998149i	$-0.519368\pi$
$179$	−1.62717	−0.121620	−0.0608101	+	0.998149i	$0.519368\pi$
$180$	0	0
$181$	−21.3974	−1.59046	−0.795229	−	0.606309i	$-0.792650\pi$
$181$	−21.3974	−1.59046	−0.795229	+	0.606309i	$0.792650\pi$
$182$	−16.6038	−1.23075
$183$	0	0
$184$	68.8329	5.07443
$185$	33.8834	2.49116
$186$	0	0
$187$	−0.130359	−0.00953280
$188$	10.2514	0.747663
$189$	0	0
$190$	−34.8510	−2.52836
$191$	−3.16587	−0.229074	−0.114537	−	0.993419i	$-0.536538\pi$
$191$	−3.16587	−0.229074	−0.114537	+	0.993419i	$0.536538\pi$
$192$	0	0
$193$	17.9622	1.29295	0.646474	−	0.762936i	$-0.276243\pi$
$193$	17.9622	1.29295	0.646474	+	0.762936i	$0.276243\pi$
$194$	32.8060	2.35534
$195$	0	0
$196$	5.47227	0.390877
$197$	−3.38471	−0.241150	−0.120575	−	0.992704i	$-0.538474\pi$
$197$	−3.38471	−0.241150	−0.120575	+	0.992704i	$0.538474\pi$
$198$	0	0
$199$	−8.58653	−0.608683	−0.304342	−	0.952563i	$-0.598436\pi$
$199$	−8.58653	−0.608683	−0.304342	+	0.952563i	$0.598436\pi$
$200$	124.320	8.79073
$201$	0	0
$202$	40.3291	2.83754
$203$	8.45102	0.593146
$204$	0	0
$205$	−9.91919	−0.692787
$206$	54.6969	3.81091
$207$	0	0
$208$	−91.1187	−6.31794
$209$	−2.47080	−0.170909
$210$	0	0
$211$	−1.57255	−0.108259	−0.0541293	−	0.998534i	$-0.517238\pi$
$211$	−1.57255	−0.108259	−0.0541293	+	0.998534i	$0.517238\pi$
$212$	22.3620	1.53583
$213$	0	0
$214$	−9.65500	−0.660002
$215$	32.5630	2.22078
$216$	0	0
$217$	3.64198	0.247234
$218$	−19.9055	−1.34817
$219$	0	0
$220$	19.1930	1.29399
$221$	−0.960416	−0.0646046
$222$	0	0
$223$	20.6473	1.38264	0.691322	−	0.722546i	$-0.257028\pi$
$223$	20.6473	1.38264	0.691322	+	0.722546i	$0.257028\pi$
$224$	22.0233	1.47149
$225$	0	0
$226$	1.68919	0.112363
$227$	−16.1218	−1.07004	−0.535021	−	0.844839i	$-0.679696\pi$
$227$	−16.1218	−1.07004	−0.535021	+	0.844839i	$0.679696\pi$
$228$	0	0
$229$	0.466165	0.0308051	0.0154025	−	0.999881i	$-0.495097\pi$
$229$	0.466165	0.0308051	0.0154025	+	0.999881i	$0.495097\pi$
$230$	−84.3326	−5.56073
$231$	0	0
$232$	80.2139	5.26630
$233$	−15.7869	−1.03423	−0.517117	−	0.855914i	$-0.672995\pi$
$233$	−15.7869	−1.03423	−0.517117	+	0.855914i	$0.672995\pi$
$234$	0	0
$235$	−7.96949	−0.519872
$236$	27.7484	1.80627
$237$	0	0
$238$	0.432220	0.0280167
$239$	−21.2755	−1.37620	−0.688100	−	0.725616i	$-0.741555\pi$
$239$	−21.2755	−1.37620	−0.688100	+	0.725616i	$0.741555\pi$
$240$	0	0
$241$	0.979158	0.0630731	0.0315366	−	0.999503i	$-0.489960\pi$
$241$	0.979158	0.0630731	0.0315366	+	0.999503i	$0.489960\pi$
$242$	−28.2110	−1.81347
$243$	0	0
$244$	−60.0258	−3.84276
$245$	−4.25416	−0.271788
$246$	0	0
$247$	−18.2036	−1.15827
$248$	34.5683	2.19509
$249$	0	0
$250$	−94.1692	−5.95578
$251$	−22.4410	−1.41646	−0.708232	−	0.705980i	$-0.750507\pi$
$251$	−22.4410	−1.41646	−0.708232	+	0.705980i	$0.750507\pi$
$252$	0	0
$253$	−5.97886	−0.375887
$254$	2.73355	0.171518
$255$	0	0
$256$	44.8552	2.80345
$257$	−1.84487	−0.115080	−0.0575400	−	0.998343i	$-0.518326\pi$
$257$	−1.84487	−0.115080	−0.0575400	+	0.998343i	$0.518326\pi$
$258$	0	0
$259$	−7.96478	−0.494908
$260$	141.404	8.76950
$261$	0	0
$262$	−15.7970	−0.975939
$263$	−10.0151	−0.617555	−0.308778	−	0.951134i	$-0.599920\pi$
$263$	−10.0151	−0.617555	−0.308778	+	0.951134i	$0.599920\pi$
$264$	0	0
$265$	−17.3843	−1.06791
$266$	8.19223	0.502298
$267$	0	0
$268$	−49.0811	−2.99811
$269$	−5.91408	−0.360588	−0.180294	−	0.983613i	$-0.557705\pi$
$269$	−5.91408	−0.360588	−0.180294	+	0.983613i	$0.557705\pi$
$270$	0	0
$271$	5.40067	0.328067	0.164034	−	0.986455i	$-0.447549\pi$
$271$	5.40067	0.328067	0.164034	+	0.986455i	$0.447549\pi$
$272$	2.37195	0.143821
$273$	0	0
$274$	−29.0646	−1.75586
$275$	−10.7985	−0.651172
$276$	0	0
$277$	−29.8821	−1.79544	−0.897719	−	0.440568i	$-0.854777\pi$
$277$	−29.8821	−1.79544	−0.897719	+	0.440568i	$0.854777\pi$
$278$	30.5959	1.83502
$279$	0	0
$280$	−40.3788	−2.41310
$281$	−9.16406	−0.546682	−0.273341	−	0.961917i	$-0.588129\pi$
$281$	−9.16406	−0.546682	−0.273341	+	0.961917i	$0.588129\pi$
$282$	0	0
$283$	−17.9317	−1.06593	−0.532966	−	0.846137i	$-0.678923\pi$
$283$	−17.9317	−1.06593	−0.532966	+	0.846137i	$0.678923\pi$
$284$	74.0201	4.39228
$285$	0	0
$286$	13.6889	0.809443
$287$	2.33165	0.137633
$288$	0	0
$289$	−16.9750	−0.998529
$290$	−98.2764	−5.77099
$291$	0	0
$292$	21.7964	1.27554
$293$	10.8736	0.635241	0.317621	−	0.948218i	$-0.397116\pi$
$293$	10.8736	0.635241	0.317621	+	0.948218i	$0.397116\pi$
$294$	0	0
$295$	−21.5716	−1.25595
$296$	−75.5987	−4.39408
$297$	0	0
$298$	15.0667	0.872791
$299$	−44.0490	−2.54742
$300$	0	0
$301$	−7.65441	−0.441193
$302$	−25.3680	−1.45976
$303$	0	0
$304$	44.9575	2.57849
$305$	46.6642	2.67198
$306$	0	0
$307$	15.4755	0.883234	0.441617	−	0.897204i	$-0.354405\pi$
$307$	15.4755	0.883234	0.441617	+	0.897204i	$0.354405\pi$
$308$	−4.51159	−0.257072
$309$	0	0
$310$	−42.3523	−2.40545
$311$	6.68109	0.378850	0.189425	−	0.981895i	$-0.439338\pi$
$311$	6.68109	0.378850	0.189425	+	0.981895i	$0.439338\pi$
$312$	0	0
$313$	−2.10828	−0.119167	−0.0595837	−	0.998223i	$-0.518977\pi$
$313$	−2.10828	−0.119167	−0.0595837	+	0.998223i	$0.518977\pi$
$314$	17.1180	0.966022
$315$	0	0
$316$	55.6165	3.12867
$317$	14.5298	0.816073	0.408037	−	0.912966i	$-0.366214\pi$
$317$	14.5298	0.816073	0.408037	+	0.912966i	$0.366214\pi$
$318$	0	0
$319$	−6.96741	−0.390100
$320$	−128.472	−7.18182
$321$	0	0
$322$	19.8236	1.10473
$323$	0.473865	0.0263666
$324$	0	0
$325$	−79.5573	−4.41305
$326$	26.9847	1.49454
$327$	0	0
$328$	22.1311	1.22199
$329$	1.87334	0.103281
$330$	0	0
$331$	3.02729	0.166395	0.0831976	−	0.996533i	$-0.473487\pi$
$331$	3.02729	0.166395	0.0831976	+	0.996533i	$0.473487\pi$
$332$	81.9472	4.49744
$333$	0	0
$334$	9.36585	0.512477
$335$	38.1558	2.08467
$336$	0	0
$337$	23.2809	1.26819	0.634095	−	0.773255i	$-0.281373\pi$
$337$	23.2809	1.26819	0.634095	+	0.773255i	$0.281373\pi$
$338$	65.3166	3.55276
$339$	0	0
$340$	−3.68095	−0.199628
$341$	−3.00262	−0.162601
$342$	0	0
$343$	1.00000	0.0539949
$344$	−72.6527	−3.91717
$345$	0	0
$346$	65.7934	3.53707
$347$	−12.7330	−0.683545	−0.341772	−	0.939783i	$-0.611027\pi$
$347$	−12.7330	−0.683545	−0.341772	+	0.939783i	$0.611027\pi$
$348$	0	0
$349$	−4.43634	−0.237472	−0.118736	−	0.992926i	$-0.537884\pi$
$349$	−4.43634	−0.237472	−0.118736	+	0.992926i	$0.537884\pi$
$350$	35.8035	1.91378
$351$	0	0
$352$	−18.1570	−0.967773
$353$	−30.1229	−1.60328	−0.801640	−	0.597808i	$-0.796039\pi$
$353$	−30.1229	−1.60328	−0.801640	+	0.597808i	$0.796039\pi$
$354$	0	0
$355$	−57.5433	−3.05408
$356$	12.6180	0.668754
$357$	0	0
$358$	−4.44794	−0.235081
$359$	−16.9749	−0.895899	−0.447950	−	0.894059i	$-0.647846\pi$
$359$	−16.9749	−0.895899	−0.447950	+	0.894059i	$0.647846\pi$
$360$	0	0
$361$	−10.0184	−0.527286
$362$	−58.4909	−3.07421
$363$	0	0
$364$	−33.2390	−1.74220
$365$	−16.9446	−0.886920
$366$	0	0
$367$	−30.7822	−1.60682	−0.803409	−	0.595428i	$-0.796983\pi$
$367$	−30.7822	−1.60682	−0.803409	+	0.595428i	$0.796983\pi$
$368$	108.788	5.67099
$369$	0	0
$370$	92.6219	4.81518
$371$	4.08642	0.212156
$372$	0	0
$373$	−18.9019	−0.978701	−0.489350	−	0.872087i	$-0.662766\pi$
$373$	−18.9019	−0.978701	−0.489350	+	0.872087i	$0.662766\pi$
$374$	−0.356342	−0.0184260
$375$	0	0
$376$	17.7810	0.916987
$377$	−51.3322	−2.64374
$378$	0	0
$379$	27.2163	1.39801	0.699004	−	0.715117i	$-0.253627\pi$
$379$	27.2163	1.39801	0.699004	+	0.715117i	$0.253627\pi$
$380$	−69.7681	−3.57903
$381$	0	0
$382$	−8.65404	−0.442779
$383$	−22.7566	−1.16281	−0.581403	−	0.813616i	$-0.697496\pi$
$383$	−22.7566	−1.16281	−0.581403	+	0.813616i	$0.697496\pi$
$384$	0	0
$385$	3.50732	0.178750
$386$	49.1005	2.49915
$387$	0	0
$388$	65.6742	3.33410
$389$	29.3879	1.49003	0.745014	−	0.667049i	$-0.232443\pi$
$389$	29.3879	1.49003	0.745014	+	0.667049i	$0.232443\pi$
$390$	0	0
$391$	1.14666	0.0579891
$392$	9.49162	0.479399
$393$	0	0
$394$	−9.25226	−0.466122
$395$	−43.2364	−2.17546
$396$	0	0
$397$	13.9738	0.701326	0.350663	−	0.936502i	$-0.385956\pi$
$397$	13.9738	0.701326	0.350663	+	0.936502i	$0.385956\pi$
$398$	−23.4717	−1.17653
$399$	0	0
$400$	196.484	9.82418
$401$	13.0010	0.649240	0.324620	−	0.945844i	$-0.394764\pi$
$401$	13.0010	0.649240	0.324620	+	0.945844i	$0.394764\pi$
$402$	0	0
$403$	−22.1217	−1.10196
$404$	80.7346	4.01669
$405$	0	0
$406$	23.1013	1.14650
$407$	6.56654	0.325491
$408$	0	0
$409$	20.1740	0.997539	0.498769	−	0.866735i	$-0.333785\pi$
$409$	20.1740	0.997539	0.498769	+	0.866735i	$0.333785\pi$
$410$	−27.1146	−1.33909
$411$	0	0
$412$	109.497	5.39455
$413$	5.07072	0.249514
$414$	0	0
$415$	−63.7059	−3.12720
$416$	−133.771	−6.55868
$417$	0	0
$418$	−6.75405	−0.330352
$419$	10.2203	0.499296	0.249648	−	0.968337i	$-0.419685\pi$
$419$	10.2203	0.499296	0.249648	+	0.968337i	$0.419685\pi$
$420$	0	0
$421$	17.7217	0.863705	0.431852	−	0.901944i	$-0.357860\pi$
$421$	17.7217	0.863705	0.431852	+	0.901944i	$0.357860\pi$
$422$	−4.29863	−0.209254
$423$	0	0
$424$	38.7867	1.88365
$425$	2.07099	0.100458
$426$	0	0
$427$	−10.9691	−0.530831
$428$	−19.3283	−0.934269
$429$	0	0
$430$	89.0125	4.29257
$431$	−9.52244	−0.458680	−0.229340	−	0.973346i	$-0.573657\pi$
$431$	−9.52244	−0.458680	−0.229340	+	0.973346i	$0.573657\pi$
$432$	0	0
$433$	−4.07745	−0.195950	−0.0979748	−	0.995189i	$-0.531236\pi$
$433$	−4.07745	−0.195950	−0.0979748	+	0.995189i	$0.531236\pi$
$434$	9.95552	0.477880
$435$	0	0
$436$	−39.8487	−1.90841
$437$	21.7336	1.03966
$438$	0	0
$439$	17.8790	0.853320	0.426660	−	0.904412i	$-0.359690\pi$
$439$	17.8790	0.853320	0.426660	+	0.904412i	$0.359690\pi$
$440$	33.2902	1.58705
$441$	0	0
$442$	−2.62534	−0.124875
$443$	−13.3161	−0.632666	−0.316333	−	0.948648i	$-0.602452\pi$
$443$	−13.3161	−0.632666	−0.316333	+	0.948648i	$0.602452\pi$
$444$	0	0
$445$	−9.80927	−0.465004
$446$	56.4403	2.67253
$447$	0	0
$448$	30.1993	1.42678
$449$	9.52844	0.449675	0.224837	−	0.974396i	$-0.427815\pi$
$449$	9.52844	0.449675	0.224837	+	0.974396i	$0.427815\pi$
$450$	0	0
$451$	−1.92232	−0.0905184
$452$	3.38158	0.159056
$453$	0	0
$454$	−44.0697	−2.06829
$455$	25.8401	1.21140
$456$	0	0
$457$	−16.6666	−0.779630	−0.389815	−	0.920893i	$-0.627461\pi$
$457$	−16.6666	−0.779630	−0.389815	+	0.920893i	$0.627461\pi$
$458$	1.27428	0.0595434
$459$	0	0
$460$	−168.825	−7.87151
$461$	−6.78637	−0.316073	−0.158036	−	0.987433i	$-0.550516\pi$
$461$	−6.78637	−0.316073	−0.158036	+	0.987433i	$0.550516\pi$
$462$	0	0
$463$	−33.8866	−1.57484	−0.787422	−	0.616415i	$-0.788585\pi$
$463$	−33.8866	−1.57484	−0.787422	+	0.616415i	$0.788585\pi$
$464$	126.776	5.88541
$465$	0	0
$466$	−43.1542	−1.99908
$467$	−13.2514	−0.613200	−0.306600	−	0.951838i	$-0.599191\pi$
$467$	−13.2514	−0.613200	−0.306600	+	0.951838i	$0.599191\pi$
$468$	0	0
$469$	−8.96906	−0.414153
$470$	−21.7850	−1.00487
$471$	0	0
$472$	48.1294	2.21533
$473$	6.31064	0.290164
$474$	0	0
$475$	39.2532	1.80106
$476$	0.865260	0.0396591
$477$	0	0
$478$	−58.1576	−2.66007
$479$	31.4248	1.43584	0.717919	−	0.696127i	$-0.245095\pi$
$479$	31.4248	1.43584	0.717919	+	0.696127i	$0.245095\pi$
$480$	0	0
$481$	48.3787	2.20588
$482$	2.67657	0.121915
$483$	0	0
$484$	−56.4754	−2.56707
$485$	−51.0553	−2.31830
$486$	0	0
$487$	−6.66403	−0.301976	−0.150988	−	0.988536i	$-0.548245\pi$
$487$	−6.66403	−0.301976	−0.150988	+	0.988536i	$0.548245\pi$
$488$	−104.114	−4.71304
$489$	0	0
$490$	−11.6289	−0.525341
$491$	−2.68603	−0.121219	−0.0606095	−	0.998162i	$-0.519304\pi$
$491$	−2.68603	−0.121219	−0.0606095	+	0.998162i	$0.519304\pi$
$492$	0	0
$493$	1.33625	0.0601817
$494$	−49.7603	−2.23882
$495$	0	0
$496$	54.6342	2.45315
$497$	13.5264	0.606741
$498$	0	0
$499$	9.17753	0.410843	0.205421	−	0.978674i	$-0.434144\pi$
$499$	9.17753	0.410843	0.205421	+	0.978674i	$0.434144\pi$
$500$	−188.517	−8.43073
$501$	0	0
$502$	−61.3435	−2.73789
$503$	−7.82917	−0.349085	−0.174543	−	0.984650i	$-0.555845\pi$
$503$	−7.82917	−0.349085	−0.174543	+	0.984650i	$0.555845\pi$
$504$	0	0
$505$	−62.7632	−2.79293
$506$	−16.3435	−0.726556
$507$	0	0
$508$	5.47227	0.242793
$509$	−13.5889	−0.602318	−0.301159	−	0.953574i	$-0.597374\pi$
$509$	−13.5889	−0.602318	−0.301159	+	0.953574i	$0.597374\pi$
$510$	0	0
$511$	3.98307	0.176200
$512$	45.6048	2.01546
$513$	0	0
$514$	−5.04304	−0.222439
$515$	−85.1235	−3.75099
$516$	0	0
$517$	−1.54447	−0.0679257
$518$	−21.7721	−0.956611
$519$	0	0
$520$	245.264	10.7555
$521$	35.6930	1.56374	0.781869	−	0.623443i	$-0.214266\pi$
$521$	35.6930	1.56374	0.781869	+	0.623443i	$0.214266\pi$
$522$	0	0
$523$	−38.0116	−1.66213	−0.831066	−	0.556173i	$-0.812269\pi$
$523$	−38.0116	−1.66213	−0.831066	+	0.556173i	$0.812269\pi$
$524$	−31.6239	−1.38149
$525$	0	0
$526$	−27.3766	−1.19368
$527$	0.575859	0.0250848
$528$	0	0
$529$	29.5910	1.28657
$530$	−47.5206	−2.06417
$531$	0	0
$532$	16.4000	0.711030
$533$	−14.1626	−0.613451
$534$	0	0
$535$	15.0259	0.649624
$536$	−85.1309	−3.67709
$537$	0	0
$538$	−16.1664	−0.696983
$539$	−0.824446	−0.0355114
$540$	0	0
$541$	−18.2336	−0.783922	−0.391961	−	0.919982i	$-0.628203\pi$
$541$	−18.2336	−0.783922	−0.391961	+	0.919982i	$0.628203\pi$
$542$	14.7630	0.634124
$543$	0	0
$544$	3.48226	0.149301
$545$	30.9785	1.32697
$546$	0	0
$547$	20.9218	0.894550	0.447275	−	0.894396i	$-0.352394\pi$
$547$	20.9218	0.894550	0.447275	+	0.894396i	$0.352394\pi$
$548$	−58.1844	−2.48551
$549$	0	0
$550$	−29.5181	−1.25866
$551$	25.3271	1.07897
$552$	0	0
$553$	10.1633	0.432188
$554$	−81.6840	−3.47042
$555$	0	0
$556$	61.2498	2.59757
$557$	25.8713	1.09620	0.548101	−	0.836412i	$-0.315351\pi$
$557$	25.8713	1.09620	0.548101	+	0.836412i	$0.315351\pi$
$558$	0	0
$559$	46.4935	1.96646
$560$	−63.8175	−2.69678
$561$	0	0
$562$	−25.0504	−1.05669
$563$	5.55729	0.234212	0.117106	−	0.993119i	$-0.462638\pi$
$563$	5.55729	0.234212	0.117106	+	0.993119i	$0.462638\pi$
$564$	0	0
$565$	−2.62885	−0.110596
$566$	−49.0173	−2.06035
$567$	0	0
$568$	128.387	5.38701
$569$	46.5246	1.95041	0.975207	−	0.221295i	$-0.0710283\pi$
$569$	46.5246	1.95041	0.975207	+	0.221295i	$0.0710283\pi$
$570$	0	0
$571$	20.3185	0.850303	0.425151	−	0.905122i	$-0.360221\pi$
$571$	20.3185	0.850303	0.425151	+	0.905122i	$0.360221\pi$
$572$	27.4038	1.14581
$573$	0	0
$574$	6.37367	0.266032
$575$	94.9851	3.96115
$576$	0	0
$577$	−25.7829	−1.07336	−0.536679	−	0.843787i	$-0.680321\pi$
$577$	−25.7829	−1.07336	−0.536679	+	0.843787i	$0.680321\pi$
$578$	−46.4019	−1.93007
$579$	0	0
$580$	−196.739	−8.16914
$581$	14.9750	0.621267
$582$	0	0
$583$	−3.36903	−0.139531
$584$	37.8057	1.56441
$585$	0	0
$586$	29.7234	1.22786
$587$	−10.4729	−0.432264	−0.216132	−	0.976364i	$-0.569344\pi$
$587$	−10.4729	−0.432264	−0.216132	+	0.976364i	$0.569344\pi$
$588$	0	0
$589$	10.9147	0.449734
$590$	−58.9671	−2.42764
$591$	0	0
$592$	−119.482	−4.91066
$593$	0.853843	0.0350631	0.0175316	−	0.999846i	$-0.494419\pi$
$593$	0.853843	0.0350631	0.0175316	+	0.999846i	$0.494419\pi$
$594$	0	0
$595$	−0.672655	−0.0275762
$596$	30.1620	1.23548
$597$	0	0
$598$	−120.410	−4.92393
$599$	−25.2120	−1.03013	−0.515066	−	0.857150i	$-0.672233\pi$
$599$	−25.2120	−1.03013	−0.515066	+	0.857150i	$0.672233\pi$
$600$	0	0
$601$	4.40447	0.179662	0.0898310	−	0.995957i	$-0.471367\pi$
$601$	4.40447	0.179662	0.0898310	+	0.995957i	$0.471367\pi$
$602$	−20.9237	−0.852785
$603$	0	0
$604$	−50.7841	−2.06638
$605$	43.9041	1.78496
$606$	0	0
$607$	−30.6486	−1.24399	−0.621995	−	0.783021i	$-0.713678\pi$
$607$	−30.6486	−1.24399	−0.621995	+	0.783021i	$0.713678\pi$
$608$	66.0022	2.67674
$609$	0	0
$610$	127.559	5.16470
$611$	−11.3788	−0.460338
$612$	0	0
$613$	18.9715	0.766253	0.383127	−	0.923696i	$-0.374847\pi$
$613$	18.9715	0.766253	0.383127	+	0.923696i	$0.374847\pi$
$614$	42.3030	1.70721
$615$	0	0
$616$	−7.82533	−0.315291
$617$	32.5842	1.31179	0.655896	−	0.754851i	$-0.272291\pi$
$617$	32.5842	1.31179	0.655896	+	0.754851i	$0.272291\pi$
$618$	0	0
$619$	11.4269	0.459286	0.229643	−	0.973275i	$-0.426244\pi$
$619$	11.4269	0.459286	0.229643	+	0.973275i	$0.426244\pi$
$620$	−84.7849	−3.40504
$621$	0	0
$622$	18.2631	0.732282
$623$	2.30581	0.0923803
$624$	0	0
$625$	81.0642	3.24257
$626$	−5.76309	−0.230340
$627$	0	0
$628$	34.2684	1.36746
$629$	−1.25937	−0.0502143
$630$	0	0
$631$	−8.98000	−0.357488	−0.178744	−	0.983896i	$-0.557203\pi$
$631$	−8.98000	−0.357488	−0.178744	+	0.983896i	$0.557203\pi$
$632$	96.4664	3.83723
$633$	0	0
$634$	39.7178	1.57739
$635$	−4.25416	−0.168821
$636$	0	0
$637$	−6.07408	−0.240664
$638$	−19.0457	−0.754028
$639$	0	0
$640$	−163.804	−6.47492
$641$	−47.3990	−1.87215	−0.936075	−	0.351801i	$-0.885569\pi$
$641$	−47.3990	−1.87215	−0.936075	+	0.351801i	$0.885569\pi$
$642$	0	0
$643$	−31.5756	−1.24522	−0.622610	−	0.782532i	$-0.713928\pi$
$643$	−31.5756	−1.24522	−0.622610	+	0.782532i	$0.713928\pi$
$644$	39.6847	1.56380
$645$	0	0
$646$	1.29533	0.0509641
$647$	25.6927	1.01009	0.505043	−	0.863094i	$-0.331477\pi$
$647$	25.6927	1.01009	0.505043	+	0.863094i	$0.331477\pi$
$648$	0	0
$649$	−4.18054	−0.164100
$650$	−217.474	−8.53002
$651$	0	0
$652$	54.0205	2.11561
$653$	44.9646	1.75960	0.879801	−	0.475343i	$-0.157676\pi$
$653$	44.9646	1.75960	0.879801	+	0.475343i	$0.157676\pi$
$654$	0	0
$655$	24.5845	0.960594
$656$	34.9776	1.36564
$657$	0	0
$658$	5.12087	0.199632
$659$	−22.2509	−0.866774	−0.433387	−	0.901208i	$-0.642682\pi$
$659$	−22.2509	−0.866774	−0.433387	+	0.901208i	$0.642682\pi$
$660$	0	0
$661$	1.25962	0.0489934	0.0244967	−	0.999700i	$-0.492202\pi$
$661$	1.25962	0.0489934	0.0244967	+	0.999700i	$0.492202\pi$
$662$	8.27525	0.321627
$663$	0	0
$664$	142.137	5.51598
$665$	−12.7494	−0.494400
$666$	0	0
$667$	61.2865	2.37302
$668$	18.7495	0.725438
$669$	0	0
$670$	104.301	4.02948
$671$	9.04342	0.349117
$672$	0	0
$673$	−7.14715	−0.275503	−0.137751	−	0.990467i	$-0.543987\pi$
$673$	−7.14715	−0.275503	−0.137751	+	0.990467i	$0.543987\pi$
$674$	63.6393	2.45129
$675$	0	0
$676$	130.757	5.02912
$677$	−43.3438	−1.66584	−0.832919	−	0.553394i	$-0.813332\pi$
$677$	−43.3438	−1.66584	−0.832919	+	0.553394i	$0.813332\pi$
$678$	0	0
$679$	12.0013	0.460566
$680$	−6.38458	−0.244838
$681$	0	0
$682$	−8.20779	−0.314293
$683$	−10.7838	−0.412630	−0.206315	−	0.978486i	$-0.566147\pi$
$683$	−10.7838	−0.412630	−0.206315	+	0.978486i	$0.566147\pi$
$684$	0	0
$685$	45.2326	1.72825
$686$	2.73355	0.104367
$687$	0	0
$688$	−114.825	−4.37768
$689$	−24.8212	−0.945613
$690$	0	0
$691$	−27.4142	−1.04289	−0.521443	−	0.853286i	$-0.674606\pi$
$691$	−27.4142	−1.04289	−0.521443	+	0.853286i	$0.674606\pi$
$692$	131.711	5.00692
$693$	0	0
$694$	−34.8063	−1.32123
$695$	−47.6157	−1.80617
$696$	0	0
$697$	0.368673	0.0139645
$698$	−12.1269	−0.459011
$699$	0	0
$700$	71.6750	2.70906
$701$	40.6532	1.53545	0.767726	−	0.640778i	$-0.221388\pi$
$701$	40.6532	1.53545	0.767726	+	0.640778i	$0.221388\pi$
$702$	0	0
$703$	−23.8699	−0.900269
$704$	−24.8977	−0.938366
$705$	0	0
$706$	−82.3423	−3.09899
$707$	14.7534	0.554858
$708$	0	0
$709$	−1.15285	−0.0432962	−0.0216481	−	0.999766i	$-0.506891\pi$
$709$	−1.15285	−0.0432962	−0.0216481	+	0.999766i	$0.506891\pi$
$710$	−157.297	−5.90326
$711$	0	0
$712$	21.8859	0.820208
$713$	26.4115	0.989119
$714$	0	0
$715$	−21.3038	−0.796715
$716$	−8.90431	−0.332770
$717$	0	0
$718$	−46.4016	−1.73169
$719$	3.46292	0.129145	0.0645726	−	0.997913i	$-0.479432\pi$
$719$	3.46292	0.129145	0.0645726	+	0.997913i	$0.479432\pi$
$720$	0	0
$721$	20.0095	0.745192
$722$	−27.3859	−1.01920
$723$	0	0
$724$	−117.093	−4.35171
$725$	110.690	4.11093
$726$	0	0
$727$	−17.7957	−0.660005	−0.330002	−	0.943980i	$-0.607050\pi$
$727$	−17.7957	−0.660005	−0.330002	+	0.943980i	$0.607050\pi$
$728$	−57.6529	−2.13676
$729$	0	0
$730$	−46.3188	−1.71434
$731$	−1.21029	−0.0447643
$732$	0	0
$733$	28.8583	1.06591	0.532953	−	0.846145i	$-0.321082\pi$
$733$	28.8583	1.06591	0.532953	+	0.846145i	$0.321082\pi$
$734$	−84.1445	−3.10583
$735$	0	0
$736$	159.712	5.88707
$737$	7.39451	0.272380
$738$	0	0
$739$	−54.0990	−1.99006	−0.995031	−	0.0995614i	$-0.968256\pi$
$739$	−54.0990	−1.99006	−0.995031	+	0.0995614i	$0.968256\pi$
$740$	185.419	6.81615
$741$	0	0
$742$	11.1704	0.410079
$743$	11.4467	0.419938	0.209969	−	0.977708i	$-0.432664\pi$
$743$	11.4467	0.419938	0.209969	+	0.977708i	$0.432664\pi$
$744$	0	0
$745$	−23.4480	−0.859067
$746$	−51.6691	−1.89174
$747$	0	0
$748$	−0.713360	−0.0260830
$749$	−3.53204	−0.129058
$750$	0	0
$751$	27.0757	0.988005	0.494003	−	0.869460i	$-0.335533\pi$
$751$	27.0757	0.988005	0.494003	+	0.869460i	$0.335533\pi$
$752$	28.1024	1.02479
$753$	0	0
$754$	−140.319	−5.11011
$755$	39.4797	1.43681
$756$	0	0
$757$	−8.33274	−0.302859	−0.151429	−	0.988468i	$-0.548388\pi$
$757$	−8.33274	−0.302859	−0.151429	+	0.988468i	$0.548388\pi$
$758$	74.3971	2.70222
$759$	0	0
$760$	−121.012	−4.38958
$761$	−29.5344	−1.07062	−0.535310	−	0.844656i	$-0.679805\pi$
$761$	−29.5344	−1.07062	−0.535310	+	0.844656i	$0.679805\pi$
$762$	0	0
$763$	−7.28193	−0.263624
$764$	−17.3245	−0.626778
$765$	0	0
$766$	−62.2061	−2.24760
$767$	−30.8000	−1.11212
$768$	0	0
$769$	−16.5765	−0.597763	−0.298881	−	0.954290i	$-0.596613\pi$
$769$	−16.5765	−0.597763	−0.298881	+	0.954290i	$0.596613\pi$
$770$	9.58743	0.345507
$771$	0	0
$772$	98.2942	3.53768
$773$	14.6432	0.526678	0.263339	−	0.964703i	$-0.415176\pi$
$773$	14.6432	0.526678	0.263339	+	0.964703i	$0.415176\pi$
$774$	0	0
$775$	47.7021	1.71351
$776$	113.911	4.08918
$777$	0	0
$778$	80.3333	2.88009
$779$	6.98777	0.250363
$780$	0	0
$781$	−11.1518	−0.399042
$782$	3.13445	0.112088
$783$	0	0
$784$	15.0012	0.535758
$785$	−26.6403	−0.950833
$786$	0	0
$787$	16.1508	0.575713	0.287857	−	0.957674i	$-0.407057\pi$
$787$	16.1508	0.575713	0.287857	+	0.957674i	$0.407057\pi$
$788$	−18.5220	−0.659821
$789$	0	0
$790$	−118.189	−4.20496
$791$	0.617948	0.0219717
$792$	0	0
$793$	66.6271	2.36600
$794$	38.1981	1.35560
$795$	0	0
$796$	−46.9878	−1.66544
$797$	−17.4408	−0.617783	−0.308892	−	0.951097i	$-0.599958\pi$
$797$	−17.4408	−0.617783	−0.308892	+	0.951097i	$0.599958\pi$
$798$	0	0
$799$	0.296207	0.0104791
$800$	288.458	10.1985
$801$	0	0
$802$	35.5389	1.25492
$803$	−3.28382	−0.115884
$804$	0	0
$805$	−30.8510	−1.08735
$806$	−60.4706	−2.12999
$807$	0	0
$808$	140.034	4.92636
$809$	16.1442	0.567600	0.283800	−	0.958883i	$-0.408405\pi$
$809$	16.1442	0.567600	0.283800	+	0.958883i	$0.408405\pi$
$810$	0	0
$811$	−20.7568	−0.728870	−0.364435	−	0.931229i	$-0.618738\pi$
$811$	−20.7568	−0.728870	−0.364435	+	0.931229i	$0.618738\pi$
$812$	46.2463	1.62293
$813$	0	0
$814$	17.9499	0.629145
$815$	−41.9956	−1.47104
$816$	0	0
$817$	−22.9397	−0.802558
$818$	55.1465	1.92815
$819$	0	0
$820$	−54.2805	−1.89556
$821$	34.1101	1.19045	0.595225	−	0.803559i	$-0.297063\pi$
$821$	34.1101	1.19045	0.595225	+	0.803559i	$0.297063\pi$
$822$	0	0
$823$	40.0748	1.39692	0.698460	−	0.715649i	$-0.253869\pi$
$823$	40.0748	1.39692	0.698460	+	0.715649i	$0.253869\pi$
$824$	189.922	6.61626
$825$	0	0
$826$	13.8611	0.482288
$827$	−17.2128	−0.598547	−0.299274	−	0.954167i	$-0.596744\pi$
$827$	−17.2128	−0.598547	−0.299274	+	0.954167i	$0.596744\pi$
$828$	0	0
$829$	1.98879	0.0690734	0.0345367	−	0.999403i	$-0.489004\pi$
$829$	1.98879	0.0690734	0.0345367	+	0.999403i	$0.489004\pi$
$830$	−174.143	−6.04459
$831$	0	0
$832$	−183.433	−6.35939
$833$	0.158117	0.00547843
$834$	0	0
$835$	−14.5759	−0.504419
$836$	−13.5209	−0.467630
$837$	0	0
$838$	27.9377	0.965093
$839$	40.5028	1.39831	0.699157	−	0.714969i	$-0.253559\pi$
$839$	40.5028	1.39831	0.699157	+	0.714969i	$0.253559\pi$
$840$	0	0
$841$	42.4198	1.46275
$842$	48.4432	1.66946
$843$	0	0
$844$	−8.60541	−0.296210
$845$	−101.651	−3.49689
$846$	0	0
$847$	−10.3203	−0.354609
$848$	61.3013	2.10509
$849$	0	0
$850$	5.66115	0.194176
$851$	−57.7604	−1.98000
$852$	0	0
$853$	15.5137	0.531179	0.265590	−	0.964086i	$-0.414433\pi$
$853$	15.5137	0.531179	0.265590	+	0.964086i	$0.414433\pi$
$854$	−29.9845	−1.02605
$855$	0	0
$856$	−33.5248	−1.14585
$857$	35.7100	1.21983	0.609914	−	0.792467i	$-0.291204\pi$
$857$	35.7100	1.21983	0.609914	+	0.792467i	$0.291204\pi$
$858$	0	0
$859$	−5.81711	−0.198477	−0.0992386	−	0.995064i	$-0.531641\pi$
$859$	−5.81711	−0.198477	−0.0992386	+	0.995064i	$0.531641\pi$
$860$	178.194	6.07636
$861$	0	0
$862$	−26.0300	−0.886587
$863$	32.7541	1.11496	0.557481	−	0.830190i	$-0.311768\pi$
$863$	32.7541	1.11496	0.557481	+	0.830190i	$0.311768\pi$
$864$	0	0
$865$	−102.393	−3.48146
$866$	−11.1459	−0.378753
$867$	0	0
$868$	19.9299	0.676465
$869$	−8.37911	−0.284242
$870$	0	0
$871$	54.4788	1.84594
$872$	−69.1173	−2.34061
$873$	0	0
$874$	59.4098	2.00957
$875$	−34.4495	−1.16460
$876$	0	0
$877$	−17.7749	−0.600216	−0.300108	−	0.953905i	$-0.597023\pi$
$877$	−17.7749	−0.600216	−0.300108	+	0.953905i	$0.597023\pi$
$878$	48.8732	1.64939
$879$	0	0
$880$	52.6141	1.77362
$881$	25.2583	0.850974	0.425487	−	0.904965i	$-0.360103\pi$
$881$	25.2583	0.850974	0.425487	+	0.904965i	$0.360103\pi$
$882$	0	0
$883$	−11.7438	−0.395212	−0.197606	−	0.980282i	$-0.563317\pi$
$883$	−11.7438	−0.395212	−0.197606	+	0.980282i	$0.563317\pi$
$884$	−5.25566	−0.176767
$885$	0	0
$886$	−36.4001	−1.22288
$887$	14.7378	0.494848	0.247424	−	0.968907i	$-0.420416\pi$
$887$	14.7378	0.494848	0.247424	+	0.968907i	$0.420416\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−26.8141	−0.898811
$891$	0	0
$892$	112.988	3.78311
$893$	5.61427	0.187874
$894$	0	0
$895$	6.92222	0.231384
$896$	38.5045	1.28634
$897$	0	0
$898$	26.0464	0.869180
$899$	30.7785	1.02652
$900$	0	0
$901$	0.646132	0.0215258
$902$	−5.25474	−0.174964
$903$	0	0
$904$	5.86532	0.195078
$905$	91.0280	3.02587
$906$	0	0
$907$	30.1984	1.00272	0.501361	−	0.865238i	$-0.332833\pi$
$907$	30.1984	1.00272	0.501361	+	0.865238i	$0.332833\pi$
$908$	−88.2229	−2.92778
$909$	0	0
$910$	70.6351	2.34153
$911$	−25.3123	−0.838636	−0.419318	−	0.907840i	$-0.637731\pi$
$911$	−25.3123	−0.838636	−0.419318	+	0.907840i	$0.637731\pi$
$912$	0	0
$913$	−12.3461	−0.408595
$914$	−45.5589	−1.50695
$915$	0	0
$916$	2.55098	0.0842869
$917$	−5.77893	−0.190837
$918$	0	0
$919$	−11.7299	−0.386933	−0.193467	−	0.981107i	$-0.561973\pi$
$919$	−11.7299	−0.386933	−0.193467	+	0.981107i	$0.561973\pi$
$920$	−292.826	−9.65418
$921$	0	0
$922$	−18.5509	−0.610940
$923$	−82.1604	−2.70434
$924$	0	0
$925$	−104.321	−3.43007
$926$	−92.6306	−3.04403
$927$	0	0
$928$	186.119	6.10967
$929$	−29.3813	−0.963971	−0.481985	−	0.876179i	$-0.660084\pi$
$929$	−29.3813	−0.963971	−0.481985	+	0.876179i	$0.660084\pi$
$930$	0	0
$931$	2.99692	0.0982202
$932$	−86.3903	−2.82981
$933$	0	0
$934$	−36.2232	−1.18526
$935$	0.554568	0.0181363
$936$	0	0
$937$	−44.3816	−1.44988	−0.724941	−	0.688811i	$-0.758133\pi$
$937$	−44.3816	−1.44988	−0.724941	+	0.688811i	$0.758133\pi$
$938$	−24.5173	−0.800519
$939$	0	0
$940$	−43.6112	−1.42244
$941$	55.8015	1.81908	0.909538	−	0.415620i	$-0.136435\pi$
$941$	55.8015	1.81908	0.909538	+	0.415620i	$0.136435\pi$
$942$	0	0
$943$	16.9090	0.550634
$944$	76.0670	2.47577
$945$	0	0
$946$	17.2504	0.560860
$947$	−33.7324	−1.09615	−0.548077	−	0.836428i	$-0.684640\pi$
$947$	−33.7324	−1.09615	−0.548077	+	0.836428i	$0.684640\pi$
$948$	0	0
$949$	−24.1935	−0.785353
$950$	107.301	3.48129
$951$	0	0
$952$	1.50079	0.0486408
$953$	47.4188	1.53605	0.768023	−	0.640423i	$-0.221241\pi$
$953$	47.4188	1.53605	0.768023	+	0.640423i	$0.221241\pi$
$954$	0	0
$955$	13.4681	0.435817
$956$	−116.425	−3.76547
$957$	0	0
$958$	85.9012	2.77534
$959$	−10.6326	−0.343344
$960$	0	0
$961$	−17.7360	−0.572128
$962$	132.246	4.26377
$963$	0	0
$964$	5.35822	0.172577
$965$	−76.4141	−2.45986
$966$	0	0
$967$	−1.40282	−0.0451117	−0.0225558	−	0.999746i	$-0.507180\pi$
$967$	−1.40282	−0.0451117	−0.0225558	+	0.999746i	$0.507180\pi$
$968$	−97.9562	−3.14843
$969$	0	0
$970$	−139.562	−4.48106
$971$	5.29054	0.169781	0.0848907	−	0.996390i	$-0.472946\pi$
$971$	5.29054	0.169781	0.0848907	+	0.996390i	$0.472946\pi$
$972$	0	0
$973$	11.1927	0.358823
$974$	−18.2164	−0.583692
$975$	0	0
$976$	−164.550	−5.26711
$977$	−41.5812	−1.33030	−0.665149	−	0.746710i	$-0.731632\pi$
$977$	−41.5812	−1.33030	−0.665149	+	0.746710i	$0.731632\pi$
$978$	0	0
$979$	−1.90102	−0.0607567
$980$	−23.2799	−0.743649
$981$	0	0
$982$	−7.34240	−0.234305
$983$	−42.2822	−1.34859	−0.674296	−	0.738461i	$-0.735553\pi$
$983$	−42.2822	−1.34859	−0.674296	+	0.738461i	$0.735553\pi$
$984$	0	0
$985$	14.3991	0.458793
$986$	3.65270	0.116326
$987$	0	0
$988$	−99.6149	−3.16917
$989$	−55.5095	−1.76510
$990$	0	0
$991$	38.2496	1.21504	0.607519	−	0.794305i	$-0.292165\pi$
$991$	38.2496	1.21504	0.607519	+	0.794305i	$0.292165\pi$
$992$	80.2084	2.54662
$993$	0	0
$994$	36.9750	1.17278
$995$	36.5284	1.15803
$996$	0	0
$997$	11.6674	0.369512	0.184756	−	0.982784i	$-0.440851\pi$
$997$	11.6674	0.369512	0.184756	+	0.982784i	$0.440851\pi$
$998$	25.0872	0.794121
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.17		18
3.2	odd	2		2667.2.a.p.1.2	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.2	✓	18	3.2	odd	2
8001.2.a.u.1.17		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)

\(f(q)\)	\(=\)	\(q+2.73355 q^{2} +5.47227 q^{4} -4.25416 q^{5} +1.00000 q^{7} +9.49162 q^{8} +O(q^{10})\)	\(q+2.73355 q^{2} +5.47227 q^{4} -4.25416 q^{5} +1.00000 q^{7} +9.49162 q^{8} -11.6289 q^{10} -0.824446 q^{11} -6.07408 q^{13} +2.73355 q^{14} +15.0012 q^{16} +0.158117 q^{17} +2.99692 q^{19} -23.2799 q^{20} -2.25366 q^{22} +7.25197 q^{23} +13.0978 q^{25} -16.6038 q^{26} +5.47227 q^{28} +8.45102 q^{29} +3.64198 q^{31} +22.0233 q^{32} +0.432220 q^{34} -4.25416 q^{35} -7.96478 q^{37} +8.19223 q^{38} -40.3788 q^{40} +2.33165 q^{41} -7.65441 q^{43} -4.51159 q^{44} +19.8236 q^{46} +1.87334 q^{47} +1.00000 q^{49} +35.8035 q^{50} -33.2390 q^{52} +4.08642 q^{53} +3.50732 q^{55} +9.49162 q^{56} +23.1013 q^{58} +5.07072 q^{59} -10.9691 q^{61} +9.95552 q^{62} +30.1993 q^{64} +25.8401 q^{65} -8.96906 q^{67} +0.865260 q^{68} -11.6289 q^{70} +13.5264 q^{71} +3.98307 q^{73} -21.7721 q^{74} +16.4000 q^{76} -0.824446 q^{77} +10.1633 q^{79} -63.8175 q^{80} +6.37367 q^{82} +14.9750 q^{83} -0.672655 q^{85} -20.9237 q^{86} -7.82533 q^{88} +2.30581 q^{89} -6.07408 q^{91} +39.6847 q^{92} +5.12087 q^{94} -12.7494 q^{95} +12.0013 q^{97} +2.73355 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 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

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Database

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