LMFDB - Embedded newform 8001.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.59170$ 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)

$f(q)$	$=$	$q-2.59170 q^{2} +4.71692 q^{4} -3.45466 q^{5} -1.00000 q^{7} -7.04143 q^{8} +O(q^{10})$	\(q-2.59170 q^{2} +4.71692 q^{4} -3.45466 q^{5} -1.00000 q^{7} -7.04143 q^{8} +8.95345 q^{10} +0.590039 q^{11} -0.545545 q^{13} +2.59170 q^{14} +8.81546 q^{16} +5.73286 q^{17} +0.427552 q^{19} -16.2953 q^{20} -1.52920 q^{22} +2.92076 q^{23} +6.93468 q^{25} +1.41389 q^{26} -4.71692 q^{28} -1.28121 q^{29} +1.26471 q^{31} -8.76418 q^{32} -14.8579 q^{34} +3.45466 q^{35} -9.04182 q^{37} -1.10809 q^{38} +24.3258 q^{40} -2.37966 q^{41} -5.15076 q^{43} +2.78316 q^{44} -7.56975 q^{46} -12.5952 q^{47} +1.00000 q^{49} -17.9726 q^{50} -2.57329 q^{52} +5.31416 q^{53} -2.03838 q^{55} +7.04143 q^{56} +3.32052 q^{58} -9.87573 q^{59} +5.02047 q^{61} -3.27775 q^{62} +5.08321 q^{64} +1.88467 q^{65} +2.64992 q^{67} +27.0414 q^{68} -8.95345 q^{70} -13.1288 q^{71} +15.5717 q^{73} +23.4337 q^{74} +2.01673 q^{76} -0.590039 q^{77} -7.77587 q^{79} -30.4544 q^{80} +6.16738 q^{82} +3.24532 q^{83} -19.8051 q^{85} +13.3492 q^{86} -4.15472 q^{88} +6.62676 q^{89} +0.545545 q^{91} +13.7770 q^{92} +32.6430 q^{94} -1.47705 q^{95} -3.67364 q^{97} -2.59170 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

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 $

$2$	−2.59170	−1.83261	−0.916305	−	0.400482i	$-0.868843\pi$
$2$	−2.59170	−1.83261	−0.916305	+	0.400482i	$0.868843\pi$
$3$	0	0
$3$	0	0
$4$	4.71692	2.35846
$5$	−3.45466	−1.54497	−0.772486	−	0.635032i	$-0.780987\pi$
$5$	−3.45466	−1.54497	−0.772486	+	0.635032i	$0.780987\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−7.04143	−2.48952
$9$	0	0
$10$	8.95345	2.83133
$11$	0.590039	0.177903	0.0889517	−	0.996036i	$-0.471648\pi$
$11$	0.590039	0.177903	0.0889517	+	0.996036i	$0.471648\pi$
$12$	0	0
$13$	−0.545545	−0.151307	−0.0756535	−	0.997134i	$-0.524104\pi$
$13$	−0.545545	−0.151307	−0.0756535	+	0.997134i	$0.524104\pi$
$14$	2.59170	0.692661
$15$	0	0
$16$	8.81546	2.20387
$17$	5.73286	1.39042	0.695211	−	0.718806i	$-0.255311\pi$
$17$	5.73286	1.39042	0.695211	+	0.718806i	$0.255311\pi$
$18$	0	0
$19$	0.427552	0.0980871	0.0490436	−	0.998797i	$-0.484383\pi$
$19$	0.427552	0.0980871	0.0490436	+	0.998797i	$0.484383\pi$
$20$	−16.2953	−3.64375
$21$	0	0
$22$	−1.52920	−0.326027
$23$	2.92076	0.609021	0.304511	−	0.952509i	$-0.401507\pi$
$23$	2.92076	0.609021	0.304511	+	0.952509i	$0.401507\pi$
$24$	0	0
$25$	6.93468	1.38694
$26$	1.41389	0.277287
$27$	0	0
$28$	−4.71692	−0.891413
$29$	−1.28121	−0.237915	−0.118958	−	0.992899i	$-0.537955\pi$
$29$	−1.28121	−0.237915	−0.118958	+	0.992899i	$0.537955\pi$
$30$	0	0
$31$	1.26471	0.227149	0.113574	−	0.993530i	$-0.463770\pi$
$31$	1.26471	0.227149	0.113574	+	0.993530i	$0.463770\pi$
$32$	−8.76418	−1.54930
$33$	0	0
$34$	−14.8579	−2.54810
$35$	3.45466	0.583944
$36$	0	0
$37$	−9.04182	−1.48647	−0.743233	−	0.669033i	$-0.766709\pi$
$37$	−9.04182	−1.48647	−0.743233	+	0.669033i	$0.766709\pi$
$38$	−1.10809	−0.179755
$39$	0	0
$40$	24.3258	3.84624
$41$	−2.37966	−0.371641	−0.185821	−	0.982584i	$-0.559494\pi$
$41$	−2.37966	−0.371641	−0.185821	+	0.982584i	$0.559494\pi$
$42$	0	0
$43$	−5.15076	−0.785484	−0.392742	−	0.919649i	$-0.628473\pi$
$43$	−5.15076	−0.785484	−0.392742	+	0.919649i	$0.628473\pi$
$44$	2.78316	0.419578
$45$	0	0
$46$	−7.56975	−1.11610
$47$	−12.5952	−1.83720	−0.918600	−	0.395188i	$-0.870679\pi$
$47$	−12.5952	−1.83720	−0.918600	+	0.395188i	$0.870679\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−17.9726	−2.54171
$51$	0	0
$52$	−2.57329	−0.356851
$53$	5.31416	0.729956	0.364978	−	0.931016i	$-0.381077\pi$
$53$	5.31416	0.729956	0.364978	+	0.931016i	$0.381077\pi$
$54$	0	0
$55$	−2.03838	−0.274856
$56$	7.04143	0.940951
$57$	0	0
$58$	3.32052	0.436005
$59$	−9.87573	−1.28571	−0.642855	−	0.765988i	$-0.722250\pi$
$59$	−9.87573	−1.28571	−0.642855	+	0.765988i	$0.722250\pi$
$60$	0	0
$61$	5.02047	0.642805	0.321403	−	0.946943i	$-0.395846\pi$
$61$	5.02047	0.642805	0.321403	+	0.946943i	$0.395846\pi$
$62$	−3.27775	−0.416275
$63$	0	0
$64$	5.08321	0.635401
$65$	1.88467	0.233765
$66$	0	0
$67$	2.64992	0.323739	0.161870	−	0.986812i	$-0.448248\pi$
$67$	2.64992	0.323739	0.161870	+	0.986812i	$0.448248\pi$
$68$	27.0414	3.27925
$69$	0	0
$70$	−8.95345	−1.07014
$71$	−13.1288	−1.55811	−0.779054	−	0.626957i	$-0.784300\pi$
$71$	−13.1288	−1.55811	−0.779054	+	0.626957i	$0.784300\pi$
$72$	0	0
$73$	15.5717	1.82253	0.911266	−	0.411819i	$-0.135107\pi$
$73$	15.5717	1.82253	0.911266	+	0.411819i	$0.135107\pi$
$74$	23.4337	2.72411
$75$	0	0
$76$	2.01673	0.231334
$77$	−0.590039	−0.0672412
$78$	0	0
$79$	−7.77587	−0.874854	−0.437427	−	0.899254i	$-0.644110\pi$
$79$	−7.77587	−0.874854	−0.437427	+	0.899254i	$0.644110\pi$
$80$	−30.4544	−3.40491
$81$	0	0
$82$	6.16738	0.681073
$83$	3.24532	0.356220	0.178110	−	0.984011i	$-0.443002\pi$
$83$	3.24532	0.356220	0.178110	+	0.984011i	$0.443002\pi$
$84$	0	0
$85$	−19.8051	−2.14816
$86$	13.3492	1.43949
$87$	0	0
$88$	−4.15472	−0.442895
$89$	6.62676	0.702435	0.351217	−	0.936294i	$-0.385768\pi$
$89$	6.62676	0.702435	0.351217	+	0.936294i	$0.385768\pi$
$90$	0	0
$91$	0.545545	0.0571886
$92$	13.7770	1.43635
$93$	0	0
$94$	32.6430	3.36687
$95$	−1.47705	−0.151542
$96$	0	0
$97$	−3.67364	−0.373002	−0.186501	−	0.982455i	$-0.559715\pi$
$97$	−3.67364	−0.373002	−0.186501	+	0.982455i	$0.559715\pi$
$98$	−2.59170	−0.261801
$99$	0	0
$100$	32.7103	3.27103
$101$	−3.42151	−0.340452	−0.170226	−	0.985405i	$-0.554450\pi$
$101$	−3.42151	−0.340452	−0.170226	+	0.985405i	$0.554450\pi$
$102$	0	0
$103$	0.396896	0.0391073	0.0195536	−	0.999809i	$-0.493775\pi$
$103$	0.396896	0.0391073	0.0195536	+	0.999809i	$0.493775\pi$
$104$	3.84142	0.376682
$105$	0	0
$106$	−13.7727	−1.33772
$107$	3.28344	0.317422	0.158711	−	0.987325i	$-0.449266\pi$
$107$	3.28344	0.317422	0.158711	+	0.987325i	$0.449266\pi$
$108$	0	0
$109$	16.5859	1.58865	0.794323	−	0.607496i	$-0.207826\pi$
$109$	16.5859	1.58865	0.794323	+	0.607496i	$0.207826\pi$
$110$	5.28288	0.503703
$111$	0	0
$112$	−8.81546	−0.832983
$113$	17.2066	1.61866	0.809331	−	0.587352i	$-0.199830\pi$
$113$	17.2066	1.61866	0.809331	+	0.587352i	$0.199830\pi$
$114$	0	0
$115$	−10.0902	−0.940920
$116$	−6.04337	−0.561113
$117$	0	0
$118$	25.5949	2.35620
$119$	−5.73286	−0.525530
$120$	0	0
$121$	−10.6519	−0.968350
$122$	−13.0116	−1.17801
$123$	0	0
$124$	5.96553	0.535720
$125$	−6.68366	−0.597805
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	4.35421	0.384861
$129$	0	0
$130$	−4.88451	−0.428400
$131$	7.59333	0.663432	0.331716	−	0.943379i	$-0.392372\pi$
$131$	7.59333	0.663432	0.331716	+	0.943379i	$0.392372\pi$
$132$	0	0
$133$	−0.427552	−0.0370735
$134$	−6.86780	−0.593288
$135$	0	0
$136$	−40.3675	−3.46149
$137$	16.0871	1.37441	0.687207	−	0.726462i	$-0.258837\pi$
$137$	16.0871	1.37441	0.687207	+	0.726462i	$0.258837\pi$
$138$	0	0
$139$	20.2511	1.71768	0.858839	−	0.512245i	$-0.171186\pi$
$139$	20.2511	1.71768	0.858839	+	0.512245i	$0.171186\pi$
$140$	16.2953	1.37721
$141$	0	0
$142$	34.0260	2.85540
$143$	−0.321893	−0.0269180
$144$	0	0
$145$	4.42615	0.367572
$146$	−40.3572	−3.33999
$147$	0	0
$148$	−42.6495	−3.50577
$149$	−2.47695	−0.202920	−0.101460	−	0.994840i	$-0.532351\pi$
$149$	−2.47695	−0.202920	−0.101460	+	0.994840i	$0.532351\pi$
$150$	0	0
$151$	1.63845	0.133335	0.0666675	−	0.997775i	$-0.478763\pi$
$151$	1.63845	0.133335	0.0666675	+	0.997775i	$0.478763\pi$
$152$	−3.01058	−0.244190
$153$	0	0
$154$	1.52920	0.123227
$155$	−4.36914	−0.350938
$156$	0	0
$157$	12.5782	1.00385	0.501924	−	0.864912i	$-0.332626\pi$
$157$	12.5782	1.00385	0.501924	+	0.864912i	$0.332626\pi$
$158$	20.1527	1.60327
$159$	0	0
$160$	30.2773	2.39363
$161$	−2.92076	−0.230188
$162$	0	0
$163$	12.2616	0.960401	0.480200	−	0.877159i	$-0.340564\pi$
$163$	12.2616	0.960401	0.480200	+	0.877159i	$0.340564\pi$
$164$	−11.2247	−0.876500
$165$	0	0
$166$	−8.41089	−0.652812
$167$	9.98217	0.772443	0.386222	−	0.922406i	$-0.373780\pi$
$167$	9.98217	0.772443	0.386222	+	0.922406i	$0.373780\pi$
$168$	0	0
$169$	−12.7024	−0.977106
$170$	51.3288	3.93674
$171$	0	0
$172$	−24.2957	−1.85253
$173$	16.2376	1.23452	0.617260	−	0.786759i	$-0.288242\pi$
$173$	16.2376	1.23452	0.617260	+	0.786759i	$0.288242\pi$
$174$	0	0
$175$	−6.93468	−0.524213
$176$	5.20147	0.392075
$177$	0	0
$178$	−17.1746	−1.28729
$179$	−17.1608	−1.28266	−0.641328	−	0.767267i	$-0.721616\pi$
$179$	−17.1608	−1.28266	−0.641328	+	0.767267i	$0.721616\pi$
$180$	0	0
$181$	−2.70008	−0.200695	−0.100348	−	0.994952i	$-0.531996\pi$
$181$	−2.70008	−0.200695	−0.100348	+	0.994952i	$0.531996\pi$
$182$	−1.41389	−0.104804
$183$	0	0
$184$	−20.5664	−1.51617
$185$	31.2364	2.29655
$186$	0	0
$187$	3.38261	0.247361
$188$	−59.4106	−4.33296
$189$	0	0
$190$	3.82806	0.277717
$191$	21.5596	1.56000	0.780001	−	0.625779i	$-0.215219\pi$
$191$	21.5596	1.56000	0.780001	+	0.625779i	$0.215219\pi$
$192$	0	0
$193$	0.817434	0.0588402	0.0294201	−	0.999567i	$-0.490634\pi$
$193$	0.817434	0.0588402	0.0294201	+	0.999567i	$0.490634\pi$
$194$	9.52098	0.683566
$195$	0	0
$196$	4.71692	0.336923
$197$	14.7570	1.05139	0.525696	−	0.850672i	$-0.323805\pi$
$197$	14.7570	1.05139	0.525696	+	0.850672i	$0.323805\pi$
$198$	0	0
$199$	−10.8319	−0.767853	−0.383926	−	0.923364i	$-0.625428\pi$
$199$	−10.8319	−0.767853	−0.383926	+	0.923364i	$0.625428\pi$
$200$	−48.8301	−3.45281
$201$	0	0
$202$	8.86752	0.623916
$203$	1.28121	0.0899235
$204$	0	0
$205$	8.22093	0.574175
$206$	−1.02864	−0.0716684
$207$	0	0
$208$	−4.80923	−0.333460
$209$	0.252272	0.0174500
$210$	0	0
$211$	24.8434	1.71029	0.855147	−	0.518386i	$-0.173467\pi$
$211$	24.8434	1.71029	0.855147	+	0.518386i	$0.173467\pi$
$212$	25.0664	1.72157
$213$	0	0
$214$	−8.50970	−0.581711
$215$	17.7941	1.21355
$216$	0	0
$217$	−1.26471	−0.0858541
$218$	−42.9858	−2.91137
$219$	0	0
$220$	−9.61489	−0.648235
$221$	−3.12753	−0.210381
$222$	0	0
$223$	−16.6863	−1.11740	−0.558700	−	0.829370i	$-0.688700\pi$
$223$	−16.6863	−1.11740	−0.558700	+	0.829370i	$0.688700\pi$
$224$	8.76418	0.585581
$225$	0	0
$226$	−44.5944	−2.96638
$227$	−5.03157	−0.333957	−0.166978	−	0.985961i	$-0.553401\pi$
$227$	−5.03157	−0.333957	−0.166978	+	0.985961i	$0.553401\pi$
$228$	0	0
$229$	−27.9865	−1.84940	−0.924699	−	0.380698i	$-0.875684\pi$
$229$	−27.9865	−1.84940	−0.924699	+	0.380698i	$0.875684\pi$
$230$	26.1509	1.72434
$231$	0	0
$232$	9.02157	0.592295
$233$	−19.8978	−1.30355	−0.651776	−	0.758412i	$-0.725976\pi$
$233$	−19.8978	−1.30355	−0.651776	+	0.758412i	$0.725976\pi$
$234$	0	0
$235$	43.5122	2.83842
$236$	−46.5830	−3.03229
$237$	0	0
$238$	14.8579	0.963092
$239$	−1.54763	−0.100108	−0.0500539	−	0.998747i	$-0.515939\pi$
$239$	−1.54763	−0.100108	−0.0500539	+	0.998747i	$0.515939\pi$
$240$	0	0
$241$	−12.9633	−0.835042	−0.417521	−	0.908667i	$-0.637101\pi$
$241$	−12.9633	−0.835042	−0.417521	+	0.908667i	$0.637101\pi$
$242$	27.6064	1.77461
$243$	0	0
$244$	23.6811	1.51603
$245$	−3.45466	−0.220710
$246$	0	0
$247$	−0.233249	−0.0148413
$248$	−8.90537	−0.565492
$249$	0	0
$250$	17.3221	1.09554
$251$	−30.7618	−1.94167	−0.970834	−	0.239751i	$-0.922934\pi$
$251$	−30.7618	−1.94167	−0.970834	+	0.239751i	$0.922934\pi$
$252$	0	0
$253$	1.72336	0.108347
$254$	−2.59170	−0.162618
$255$	0	0
$256$	−21.4512	−1.34070
$257$	−9.94280	−0.620215	−0.310107	−	0.950702i	$-0.600365\pi$
$257$	−9.94280	−0.620215	−0.310107	+	0.950702i	$0.600365\pi$
$258$	0	0
$259$	9.04182	0.561831
$260$	8.88984	0.551325
$261$	0	0
$262$	−19.6796	−1.21581
$263$	−16.1247	−0.994291	−0.497146	−	0.867667i	$-0.665619\pi$
$263$	−16.1247	−0.994291	−0.497146	+	0.867667i	$0.665619\pi$
$264$	0	0
$265$	−18.3586	−1.12776
$266$	1.10809	0.0679412
$267$	0	0
$268$	12.4994	0.763525
$269$	24.8233	1.51350	0.756751	−	0.653703i	$-0.226786\pi$
$269$	24.8233	1.51350	0.756751	+	0.653703i	$0.226786\pi$
$270$	0	0
$271$	−4.56220	−0.277134	−0.138567	−	0.990353i	$-0.544250\pi$
$271$	−4.56220	−0.277134	−0.138567	+	0.990353i	$0.544250\pi$
$272$	50.5378	3.06430
$273$	0	0
$274$	−41.6930	−2.51876
$275$	4.09173	0.246741
$276$	0	0
$277$	8.66916	0.520879	0.260440	−	0.965490i	$-0.416132\pi$
$277$	8.66916	0.520879	0.260440	+	0.965490i	$0.416132\pi$
$278$	−52.4849	−3.14783
$279$	0	0
$280$	−24.3258	−1.45374
$281$	−5.67698	−0.338660	−0.169330	−	0.985559i	$-0.554160\pi$
$281$	−5.67698	−0.338660	−0.169330	+	0.985559i	$0.554160\pi$
$282$	0	0
$283$	11.2309	0.667610	0.333805	−	0.942642i	$-0.391667\pi$
$283$	11.2309	0.667610	0.333805	+	0.942642i	$0.391667\pi$
$284$	−61.9276	−3.67473
$285$	0	0
$286$	0.834250	0.0493302
$287$	2.37966	0.140467
$288$	0	0
$289$	15.8657	0.933274
$290$	−11.4713	−0.673616
$291$	0	0
$292$	73.4505	4.29836
$293$	−21.7083	−1.26822	−0.634108	−	0.773245i	$-0.718632\pi$
$293$	−21.7083	−1.26822	−0.634108	+	0.773245i	$0.718632\pi$
$294$	0	0
$295$	34.1173	1.98638
$296$	63.6674	3.70059
$297$	0	0
$298$	6.41951	0.371872
$299$	−1.59341	−0.0921491
$300$	0	0
$301$	5.15076	0.296885
$302$	−4.24637	−0.244351
$303$	0	0
$304$	3.76907	0.216171
$305$	−17.3440	−0.993116
$306$	0	0
$307$	−0.768564	−0.0438643	−0.0219321	−	0.999759i	$-0.506982\pi$
$307$	−0.768564	−0.0438643	−0.0219321	+	0.999759i	$0.506982\pi$
$308$	−2.78316	−0.158585
$309$	0	0
$310$	11.3235	0.643132
$311$	−26.5965	−1.50815	−0.754073	−	0.656790i	$-0.771914\pi$
$311$	−26.5965	−1.50815	−0.754073	+	0.656790i	$0.771914\pi$
$312$	0	0
$313$	12.1099	0.684491	0.342246	−	0.939611i	$-0.388813\pi$
$313$	12.1099	0.684491	0.342246	+	0.939611i	$0.388813\pi$
$314$	−32.5989	−1.83966
$315$	0	0
$316$	−36.6781	−2.06331
$317$	20.1474	1.13159	0.565795	−	0.824546i	$-0.308569\pi$
$317$	20.1474	1.13159	0.565795	+	0.824546i	$0.308569\pi$
$318$	0	0
$319$	−0.755965	−0.0423259
$320$	−17.5607	−0.981676
$321$	0	0
$322$	7.56975	0.421846
$323$	2.45109	0.136383
$324$	0	0
$325$	−3.78318	−0.209853
$326$	−31.7783	−1.76004
$327$	0	0
$328$	16.7562	0.925209
$329$	12.5952	0.694396
$330$	0	0
$331$	−1.97584	−0.108602	−0.0543009	−	0.998525i	$-0.517293\pi$
$331$	−1.97584	−0.108602	−0.0543009	+	0.998525i	$0.517293\pi$
$332$	15.3079	0.840130
$333$	0	0
$334$	−25.8708	−1.41559
$335$	−9.15457	−0.500168
$336$	0	0
$337$	−0.623977	−0.0339902	−0.0169951	−	0.999856i	$-0.505410\pi$
$337$	−0.623977	−0.0339902	−0.0169951	+	0.999856i	$0.505410\pi$
$338$	32.9208	1.79065
$339$	0	0
$340$	−93.4189	−5.06635
$341$	0.746228	0.0404105
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	36.2687	1.95548
$345$	0	0
$346$	−42.0830	−2.26239
$347$	−22.9149	−1.23014	−0.615069	−	0.788473i	$-0.710872\pi$
$347$	−22.9149	−1.23014	−0.615069	+	0.788473i	$0.710872\pi$
$348$	0	0
$349$	−24.0670	−1.28828	−0.644138	−	0.764909i	$-0.722784\pi$
$349$	−24.0670	−1.28828	−0.644138	+	0.764909i	$0.722784\pi$
$350$	17.9726	0.960677
$351$	0	0
$352$	−5.17120	−0.275626
$353$	−9.85326	−0.524436	−0.262218	−	0.965009i	$-0.584454\pi$
$353$	−9.85326	−0.524436	−0.262218	+	0.965009i	$0.584454\pi$
$354$	0	0
$355$	45.3557	2.40723
$356$	31.2578	1.65666
$357$	0	0
$358$	44.4756	2.35061
$359$	1.22312	0.0645537	0.0322768	−	0.999479i	$-0.489724\pi$
$359$	1.22312	0.0645537	0.0322768	+	0.999479i	$0.489724\pi$
$360$	0	0
$361$	−18.8172	−0.990379
$362$	6.99780	0.367796
$363$	0	0
$364$	2.57329	0.134877
$365$	−53.7950	−2.81576
$366$	0	0
$367$	−33.3549	−1.74111	−0.870556	−	0.492069i	$-0.836241\pi$
$367$	−33.3549	−1.74111	−0.870556	+	0.492069i	$0.836241\pi$
$368$	25.7479	1.34220
$369$	0	0
$370$	−80.9555	−4.20867
$371$	−5.31416	−0.275897
$372$	0	0
$373$	13.7021	0.709467	0.354734	−	0.934967i	$-0.384572\pi$
$373$	13.7021	0.709467	0.354734	+	0.934967i	$0.384572\pi$
$374$	−8.76671	−0.453316
$375$	0	0
$376$	88.6883	4.57375
$377$	0.698959	0.0359982
$378$	0	0
$379$	7.57636	0.389171	0.194586	−	0.980886i	$-0.437664\pi$
$379$	7.57636	0.389171	0.194586	+	0.980886i	$0.437664\pi$
$380$	−6.96711	−0.357405
$381$	0	0
$382$	−55.8762	−2.85887
$383$	−23.7384	−1.21297	−0.606487	−	0.795094i	$-0.707422\pi$
$383$	−23.7384	−1.21297	−0.606487	+	0.795094i	$0.707422\pi$
$384$	0	0
$385$	2.03838	0.103886
$386$	−2.11855	−0.107831
$387$	0	0
$388$	−17.3283	−0.879709
$389$	−13.7807	−0.698708	−0.349354	−	0.936991i	$-0.613599\pi$
$389$	−13.7807	−0.698708	−0.349354	+	0.936991i	$0.613599\pi$
$390$	0	0
$391$	16.7443	0.846797
$392$	−7.04143	−0.355646
$393$	0	0
$394$	−38.2457	−1.92679
$395$	26.8630	1.35162
$396$	0	0
$397$	−23.6664	−1.18778	−0.593891	−	0.804546i	$-0.702409\pi$
$397$	−23.6664	−1.18778	−0.593891	+	0.804546i	$0.702409\pi$
$398$	28.0730	1.40717
$399$	0	0
$400$	61.1324	3.05662
$401$	−26.4614	−1.32142	−0.660711	−	0.750641i	$-0.729745\pi$
$401$	−26.4614	−1.32142	−0.660711	+	0.750641i	$0.729745\pi$
$402$	0	0
$403$	−0.689956	−0.0343691
$404$	−16.1390	−0.802943
$405$	0	0
$406$	−3.32052	−0.164795
$407$	−5.33502	−0.264447
$408$	0	0
$409$	2.36060	0.116724	0.0583620	−	0.998295i	$-0.481412\pi$
$409$	2.36060	0.116724	0.0583620	+	0.998295i	$0.481412\pi$
$410$	−21.3062	−1.05224
$411$	0	0
$412$	1.87212	0.0922329
$413$	9.87573	0.485953
$414$	0	0
$415$	−11.2115	−0.550350
$416$	4.78125	0.234420
$417$	0	0
$418$	−0.653814	−0.0319791
$419$	19.3453	0.945082	0.472541	−	0.881309i	$-0.343337\pi$
$419$	19.3453	0.945082	0.472541	+	0.881309i	$0.343337\pi$
$420$	0	0
$421$	34.7259	1.69244	0.846219	−	0.532836i	$-0.178874\pi$
$421$	34.7259	1.69244	0.846219	+	0.532836i	$0.178874\pi$
$422$	−64.3868	−3.13430
$423$	0	0
$424$	−37.4193	−1.81724
$425$	39.7555	1.92843
$426$	0	0
$427$	−5.02047	−0.242958
$428$	15.4877	0.748627
$429$	0	0
$430$	−46.1171	−2.22396
$431$	23.7336	1.14321	0.571604	−	0.820529i	$-0.306321\pi$
$431$	23.7336	1.14321	0.571604	+	0.820529i	$0.306321\pi$
$432$	0	0
$433$	30.3046	1.45635	0.728174	−	0.685392i	$-0.240369\pi$
$433$	30.3046	1.45635	0.728174	+	0.685392i	$0.240369\pi$
$434$	3.27775	0.157337
$435$	0	0
$436$	78.2345	3.74675
$437$	1.24878	0.0597372
$438$	0	0
$439$	−11.6799	−0.557451	−0.278725	−	0.960371i	$-0.589912\pi$
$439$	−11.6799	−0.557451	−0.278725	+	0.960371i	$0.589912\pi$
$440$	14.3531	0.684259
$441$	0	0
$442$	8.10563	0.385545
$443$	28.8707	1.37169	0.685843	−	0.727750i	$-0.259434\pi$
$443$	28.8707	1.37169	0.685843	+	0.727750i	$0.259434\pi$
$444$	0	0
$445$	−22.8932	−1.08524
$446$	43.2460	2.04776
$447$	0	0
$448$	−5.08321	−0.240159
$449$	27.4480	1.29535	0.647675	−	0.761916i	$-0.275741\pi$
$449$	27.4480	1.29535	0.647675	+	0.761916i	$0.275741\pi$
$450$	0	0
$451$	−1.40409	−0.0661162
$452$	81.1622	3.81755
$453$	0	0
$454$	13.0403	0.612012
$455$	−1.88467	−0.0883548
$456$	0	0
$457$	−17.4358	−0.815612	−0.407806	−	0.913069i	$-0.633706\pi$
$457$	−17.4358	−0.815612	−0.407806	+	0.913069i	$0.633706\pi$
$458$	72.5326	3.38923
$459$	0	0
$460$	−47.5948	−2.21912
$461$	−32.2238	−1.50081	−0.750406	−	0.660977i	$-0.770142\pi$
$461$	−32.2238	−1.50081	−0.750406	+	0.660977i	$0.770142\pi$
$462$	0	0
$463$	29.4985	1.37091	0.685456	−	0.728114i	$-0.259603\pi$
$463$	29.4985	1.37091	0.685456	+	0.728114i	$0.259603\pi$
$464$	−11.2945	−0.524333
$465$	0	0
$466$	51.5693	2.38890
$467$	23.2936	1.07790	0.538951	−	0.842337i	$-0.318821\pi$
$467$	23.2936	1.07790	0.538951	+	0.842337i	$0.318821\pi$
$468$	0	0
$469$	−2.64992	−0.122362
$470$	−112.771	−5.20172
$471$	0	0
$472$	69.5393	3.20080
$473$	−3.03915	−0.139740
$474$	0	0
$475$	2.96494	0.136041
$476$	−27.0414	−1.23944
$477$	0	0
$478$	4.01099	0.183459
$479$	−21.7631	−0.994381	−0.497191	−	0.867641i	$-0.665635\pi$
$479$	−21.7631	−0.994381	−0.497191	+	0.867641i	$0.665635\pi$
$480$	0	0
$481$	4.93272	0.224913
$482$	33.5971	1.53031
$483$	0	0
$484$	−50.2439	−2.28381
$485$	12.6912	0.576277
$486$	0	0
$487$	−25.6867	−1.16397	−0.581987	−	0.813198i	$-0.697725\pi$
$487$	−25.6867	−1.16397	−0.581987	+	0.813198i	$0.697725\pi$
$488$	−35.3513	−1.60028
$489$	0	0
$490$	8.95345	0.404476
$491$	12.8321	0.579104	0.289552	−	0.957162i	$-0.406494\pi$
$491$	12.8321	0.579104	0.289552	+	0.957162i	$0.406494\pi$
$492$	0	0
$493$	−7.34501	−0.330802
$494$	0.604511	0.0271982
$495$	0	0
$496$	11.1490	0.500605
$497$	13.1288	0.588909
$498$	0	0
$499$	2.06676	0.0925208	0.0462604	−	0.998929i	$-0.485270\pi$
$499$	2.06676	0.0925208	0.0462604	+	0.998929i	$0.485270\pi$
$500$	−31.5263	−1.40990
$501$	0	0
$502$	79.7255	3.55832
$503$	−17.6546	−0.787181	−0.393590	−	0.919286i	$-0.628767\pi$
$503$	−17.6546	−0.787181	−0.393590	+	0.919286i	$0.628767\pi$
$504$	0	0
$505$	11.8201	0.525989
$506$	−4.46644	−0.198558
$507$	0	0
$508$	4.71692	0.209279
$509$	5.98594	0.265322	0.132661	−	0.991161i	$-0.457648\pi$
$509$	5.98594	0.265322	0.132661	+	0.991161i	$0.457648\pi$
$510$	0	0
$511$	−15.5717	−0.688852
$512$	46.8867	2.07212
$513$	0	0
$514$	25.7688	1.13661
$515$	−1.37114	−0.0604197
$516$	0	0
$517$	−7.43166	−0.326844
$518$	−23.4337	−1.02962
$519$	0	0
$520$	−13.2708	−0.581963
$521$	37.4421	1.64037	0.820183	−	0.572101i	$-0.193871\pi$
$521$	37.4421	1.64037	0.820183	+	0.572101i	$0.193871\pi$
$522$	0	0
$523$	−6.54065	−0.286003	−0.143001	−	0.989723i	$-0.545675\pi$
$523$	−6.54065	−0.286003	−0.143001	+	0.989723i	$0.545675\pi$
$524$	35.8171	1.56468
$525$	0	0
$526$	41.7904	1.82215
$527$	7.25040	0.315832
$528$	0	0
$529$	−14.4691	−0.629093
$530$	47.5800	2.06674
$531$	0	0
$532$	−2.01673	−0.0874362
$533$	1.29821	0.0562319
$534$	0	0
$535$	−11.3432	−0.490408
$536$	−18.6592	−0.805956
$537$	0	0
$538$	−64.3345	−2.77366
$539$	0.590039	0.0254148
$540$	0	0
$541$	34.1271	1.46724	0.733618	−	0.679562i	$-0.237830\pi$
$541$	34.1271	1.46724	0.733618	+	0.679562i	$0.237830\pi$
$542$	11.8239	0.507879
$543$	0	0
$544$	−50.2438	−2.15418
$545$	−57.2988	−2.45441
$546$	0	0
$547$	−35.3609	−1.51192	−0.755962	−	0.654615i	$-0.772831\pi$
$547$	−35.3609	−1.51192	−0.755962	+	0.654615i	$0.772831\pi$
$548$	75.8815	3.24150
$549$	0	0
$550$	−10.6045	−0.452179
$551$	−0.547785	−0.0233364
$552$	0	0
$553$	7.77587	0.330664
$554$	−22.4679	−0.954569
$555$	0	0
$556$	95.5229	4.05107
$557$	14.3335	0.607330	0.303665	−	0.952779i	$-0.401790\pi$
$557$	14.3335	0.607330	0.303665	+	0.952779i	$0.401790\pi$
$558$	0	0
$559$	2.80997	0.118849
$560$	30.4544	1.28693
$561$	0	0
$562$	14.7130	0.620632
$563$	2.57618	0.108573	0.0542866	−	0.998525i	$-0.482712\pi$
$563$	2.57618	0.108573	0.0542866	+	0.998525i	$0.482712\pi$
$564$	0	0
$565$	−59.4430	−2.50079
$566$	−29.1073	−1.22347
$567$	0	0
$568$	92.4459	3.87894
$569$	−40.4285	−1.69485	−0.847425	−	0.530915i	$-0.821848\pi$
$569$	−40.4285	−1.69485	−0.847425	+	0.530915i	$0.821848\pi$
$570$	0	0
$571$	−0.0295823	−0.00123798	−0.000618991	−	1.00000i	$-0.500197\pi$
$571$	−0.0295823	−0.00123798	−0.000618991		1.00000i	$0.500197\pi$
$572$	−1.51834	−0.0634850
$573$	0	0
$574$	−6.16738	−0.257421
$575$	20.2546	0.844674
$576$	0	0
$577$	−16.0505	−0.668189	−0.334095	−	0.942540i	$-0.608430\pi$
$577$	−16.0505	−0.668189	−0.334095	+	0.942540i	$0.608430\pi$
$578$	−41.1191	−1.71033
$579$	0	0
$580$	20.8778	0.866903
$581$	−3.24532	−0.134638
$582$	0	0
$583$	3.13556	0.129862
$584$	−109.647	−4.53723
$585$	0	0
$586$	56.2616	2.32414
$587$	−17.8635	−0.737304	−0.368652	−	0.929567i	$-0.620181\pi$
$587$	−17.8635	−0.737304	−0.368652	+	0.929567i	$0.620181\pi$
$588$	0	0
$589$	0.540729	0.0222804
$590$	−88.4218	−3.64027
$591$	0	0
$592$	−79.7078	−3.27597
$593$	−26.0207	−1.06854	−0.534270	−	0.845314i	$-0.679414\pi$
$593$	−26.0207	−1.06854	−0.534270	+	0.845314i	$0.679414\pi$
$594$	0	0
$595$	19.8051	0.811929
$596$	−11.6836	−0.478577
$597$	0	0
$598$	4.12964	0.168873
$599$	−32.9219	−1.34515	−0.672577	−	0.740027i	$-0.734813\pi$
$599$	−32.9219	−1.34515	−0.672577	+	0.740027i	$0.734813\pi$
$600$	0	0
$601$	22.7518	0.928067	0.464033	−	0.885818i	$-0.346402\pi$
$601$	22.7518	0.928067	0.464033	+	0.885818i	$0.346402\pi$
$602$	−13.3492	−0.544074
$603$	0	0
$604$	7.72842	0.314465
$605$	36.7985	1.49607
$606$	0	0
$607$	8.12806	0.329908	0.164954	−	0.986301i	$-0.447252\pi$
$607$	8.12806	0.329908	0.164954	+	0.986301i	$0.447252\pi$
$608$	−3.74714	−0.151967
$609$	0	0
$610$	44.9505	1.81999
$611$	6.87125	0.277981
$612$	0	0
$613$	−10.8964	−0.440100	−0.220050	−	0.975489i	$-0.570622\pi$
$613$	−10.8964	−0.440100	−0.220050	+	0.975489i	$0.570622\pi$
$614$	1.99189	0.0803861
$615$	0	0
$616$	4.15472	0.167398
$617$	11.3094	0.455301	0.227650	−	0.973743i	$-0.426896\pi$
$617$	11.3094	0.455301	0.227650	+	0.973743i	$0.426896\pi$
$618$	0	0
$619$	32.4228	1.30318	0.651592	−	0.758570i	$-0.274102\pi$
$619$	32.4228	1.30318	0.651592	+	0.758570i	$0.274102\pi$
$620$	−20.6089	−0.827672
$621$	0	0
$622$	68.9301	2.76384
$623$	−6.62676	−0.265495
$624$	0	0
$625$	−11.5836	−0.463344
$626$	−31.3852	−1.25440
$627$	0	0
$628$	59.3302	2.36753
$629$	−51.8355	−2.06682
$630$	0	0
$631$	−24.6671	−0.981980	−0.490990	−	0.871165i	$-0.663365\pi$
$631$	−24.6671	−0.981980	−0.490990	+	0.871165i	$0.663365\pi$
$632$	54.7533	2.17797
$633$	0	0
$634$	−52.2160	−2.07376
$635$	−3.45466	−0.137094
$636$	0	0
$637$	−0.545545	−0.0216153
$638$	1.95924	0.0775669
$639$	0	0
$640$	−15.0423	−0.594599
$641$	−5.72518	−0.226131	−0.113066	−	0.993588i	$-0.536067\pi$
$641$	−5.72518	−0.226131	−0.113066	+	0.993588i	$0.536067\pi$
$642$	0	0
$643$	21.3308	0.841206	0.420603	−	0.907245i	$-0.361819\pi$
$643$	21.3308	0.841206	0.420603	+	0.907245i	$0.361819\pi$
$644$	−13.7770	−0.542890
$645$	0	0
$646$	−6.35251	−0.249936
$647$	−8.81203	−0.346437	−0.173218	−	0.984883i	$-0.555417\pi$
$647$	−8.81203	−0.346437	−0.173218	+	0.984883i	$0.555417\pi$
$648$	0	0
$649$	−5.82706	−0.228732
$650$	9.80487	0.384579
$651$	0	0
$652$	57.8368	2.26506
$653$	26.9468	1.05451	0.527254	−	0.849708i	$-0.323222\pi$
$653$	26.9468	1.05451	0.527254	+	0.849708i	$0.323222\pi$
$654$	0	0
$655$	−26.2324	−1.02498
$656$	−20.9778	−0.819047
$657$	0	0
$658$	−32.6430	−1.27256
$659$	28.6932	1.11773	0.558864	−	0.829259i	$-0.311237\pi$
$659$	28.6932	1.11773	0.558864	+	0.829259i	$0.311237\pi$
$660$	0	0
$661$	9.02952	0.351208	0.175604	−	0.984461i	$-0.443812\pi$
$661$	9.02952	0.351208	0.175604	+	0.984461i	$0.443812\pi$
$662$	5.12078	0.199025
$663$	0	0
$664$	−22.8517	−0.886818
$665$	1.47705	0.0572774
$666$	0	0
$667$	−3.74212	−0.144895
$668$	47.0850	1.82177
$669$	0	0
$670$	23.7259	0.916612
$671$	2.96227	0.114357
$672$	0	0
$673$	−4.18116	−0.161172	−0.0805858	−	0.996748i	$-0.525679\pi$
$673$	−4.18116	−0.161172	−0.0805858	+	0.996748i	$0.525679\pi$
$674$	1.61716	0.0622907
$675$	0	0
$676$	−59.9161	−2.30446
$677$	−12.8698	−0.494628	−0.247314	−	0.968935i	$-0.579548\pi$
$677$	−12.8698	−0.494628	−0.247314	+	0.968935i	$0.579548\pi$
$678$	0	0
$679$	3.67364	0.140981
$680$	139.456	5.34790
$681$	0	0
$682$	−1.93400	−0.0740567
$683$	29.6643	1.13507	0.567537	−	0.823348i	$-0.307896\pi$
$683$	29.6643	1.13507	0.567537	+	0.823348i	$0.307896\pi$
$684$	0	0
$685$	−55.5755	−2.12343
$686$	2.59170	0.0989516
$687$	0	0
$688$	−45.4063	−1.73110
$689$	−2.89911	−0.110447
$690$	0	0
$691$	12.0335	0.457776	0.228888	−	0.973453i	$-0.426491\pi$
$691$	12.0335	0.457776	0.228888	+	0.973453i	$0.426491\pi$
$692$	76.5913	2.91157
$693$	0	0
$694$	59.3887	2.25436
$695$	−69.9608	−2.65376
$696$	0	0
$697$	−13.6423	−0.516738
$698$	62.3744	2.36091
$699$	0	0
$700$	−32.7103	−1.23633
$701$	−0.127438	−0.00481327	−0.00240663	−	0.999997i	$-0.500766\pi$
$701$	−0.127438	−0.00481327	−0.00240663	+	0.999997i	$0.500766\pi$
$702$	0	0
$703$	−3.86585	−0.145803
$704$	2.99929	0.113040
$705$	0	0
$706$	25.5367	0.961086
$707$	3.42151	0.128679
$708$	0	0
$709$	−46.4111	−1.74301	−0.871503	−	0.490391i	$-0.836854\pi$
$709$	−46.4111	−1.74301	−0.871503	+	0.490391i	$0.836854\pi$
$710$	−117.548	−4.41151
$711$	0	0
$712$	−46.6619	−1.74873
$713$	3.69392	0.138338
$714$	0	0
$715$	1.11203	0.0415876
$716$	−80.9459	−3.02509
$717$	0	0
$718$	−3.16995	−0.118302
$719$	7.24602	0.270231	0.135116	−	0.990830i	$-0.456859\pi$
$719$	7.24602	0.270231	0.135116	+	0.990830i	$0.456859\pi$
$720$	0	0
$721$	−0.396896	−0.0147812
$722$	48.7686	1.81498
$723$	0	0
$724$	−12.7360	−0.473331
$725$	−8.88480	−0.329973
$726$	0	0
$727$	−19.1956	−0.711926	−0.355963	−	0.934500i	$-0.615847\pi$
$727$	−19.1956	−0.711926	−0.355963	+	0.934500i	$0.615847\pi$
$728$	−3.84142	−0.142372
$729$	0	0
$730$	139.421	5.16019
$731$	−29.5286	−1.09215
$732$	0	0
$733$	3.99021	0.147382	0.0736908	−	0.997281i	$-0.476522\pi$
$733$	3.99021	0.147382	0.0736908	+	0.997281i	$0.476522\pi$
$734$	86.4460	3.19078
$735$	0	0
$736$	−25.5981	−0.943558
$737$	1.56356	0.0575943
$738$	0	0
$739$	10.5965	0.389799	0.194899	−	0.980823i	$-0.437562\pi$
$739$	10.5965	0.389799	0.194899	+	0.980823i	$0.437562\pi$
$740$	147.340	5.41631
$741$	0	0
$742$	13.7727	0.505612
$743$	−23.4001	−0.858466	−0.429233	−	0.903194i	$-0.641216\pi$
$743$	−23.4001	−0.858466	−0.429233	+	0.903194i	$0.641216\pi$
$744$	0	0
$745$	8.55702	0.313505
$746$	−35.5117	−1.30018
$747$	0	0
$748$	15.9555	0.583390
$749$	−3.28344	−0.119974
$750$	0	0
$751$	−18.0578	−0.658940	−0.329470	−	0.944166i	$-0.606870\pi$
$751$	−18.0578	−0.658940	−0.329470	+	0.944166i	$0.606870\pi$
$752$	−111.033	−4.04894
$753$	0	0
$754$	−1.81149	−0.0659706
$755$	−5.66028	−0.205999
$756$	0	0
$757$	36.0339	1.30968	0.654838	−	0.755770i	$-0.272737\pi$
$757$	36.0339	1.30968	0.654838	+	0.755770i	$0.272737\pi$
$758$	−19.6357	−0.713199
$759$	0	0
$760$	10.4005	0.377267
$761$	13.6580	0.495103	0.247551	−	0.968875i	$-0.420374\pi$
$761$	13.6580	0.495103	0.247551	+	0.968875i	$0.420374\pi$
$762$	0	0
$763$	−16.5859	−0.600451
$764$	101.695	3.67920
$765$	0	0
$766$	61.5227	2.22291
$767$	5.38765	0.194537
$768$	0	0
$769$	−26.6626	−0.961477	−0.480738	−	0.876864i	$-0.659631\pi$
$769$	−26.6626	−0.961477	−0.480738	+	0.876864i	$0.659631\pi$
$770$	−5.28288	−0.190382
$771$	0	0
$772$	3.85577	0.138772
$773$	−5.82941	−0.209669	−0.104835	−	0.994490i	$-0.533431\pi$
$773$	−5.82941	−0.209669	−0.104835	+	0.994490i	$0.533431\pi$
$774$	0	0
$775$	8.77036	0.315041
$776$	25.8677	0.928596
$777$	0	0
$778$	35.7154	1.28046
$779$	−1.01743	−0.0364532
$780$	0	0
$781$	−7.74653	−0.277193
$782$	−43.3963	−1.55185
$783$	0	0
$784$	8.81546	0.314838
$785$	−43.4533	−1.55091
$786$	0	0
$787$	−19.1050	−0.681018	−0.340509	−	0.940241i	$-0.610599\pi$
$787$	−19.1050	−0.681018	−0.340509	+	0.940241i	$0.610599\pi$
$788$	69.6075	2.47967
$789$	0	0
$790$	−69.6208	−2.47700
$791$	−17.2066	−0.611797
$792$	0	0
$793$	−2.73889	−0.0972609
$794$	61.3362	2.17674
$795$	0	0
$796$	−51.0931	−1.81095
$797$	38.3406	1.35810	0.679048	−	0.734094i	$-0.262393\pi$
$797$	38.3406	1.35810	0.679048	+	0.734094i	$0.262393\pi$
$798$	0	0
$799$	−72.2066	−2.55448
$800$	−60.7768	−2.14878
$801$	0	0
$802$	68.5801	2.42165
$803$	9.18791	0.324234
$804$	0	0
$805$	10.0902	0.355634
$806$	1.78816	0.0629852
$807$	0	0
$808$	24.0923	0.847564
$809$	5.83655	0.205202	0.102601	−	0.994723i	$-0.467283\pi$
$809$	5.83655	0.205202	0.102601	+	0.994723i	$0.467283\pi$
$810$	0	0
$811$	−36.9840	−1.29868	−0.649342	−	0.760497i	$-0.724955\pi$
$811$	−36.9840	−1.29868	−0.649342	+	0.760497i	$0.724955\pi$
$812$	6.04337	0.212081
$813$	0	0
$814$	13.8268	0.484629
$815$	−42.3596	−1.48379
$816$	0	0
$817$	−2.20222	−0.0770459
$818$	−6.11796	−0.213909
$819$	0	0
$820$	38.7774	1.35417
$821$	−21.5315	−0.751455	−0.375727	−	0.926730i	$-0.622607\pi$
$821$	−21.5315	−0.751455	−0.375727	+	0.926730i	$0.622607\pi$
$822$	0	0
$823$	−13.9567	−0.486499	−0.243249	−	0.969964i	$-0.578213\pi$
$823$	−13.9567	−0.486499	−0.243249	+	0.969964i	$0.578213\pi$
$824$	−2.79472	−0.0973585
$825$	0	0
$826$	−25.5949	−0.890562
$827$	−54.6420	−1.90009	−0.950044	−	0.312117i	$-0.898962\pi$
$827$	−54.6420	−1.90009	−0.950044	+	0.312117i	$0.898962\pi$
$828$	0	0
$829$	−39.4091	−1.36874	−0.684368	−	0.729137i	$-0.739922\pi$
$829$	−39.4091	−1.36874	−0.684368	+	0.729137i	$0.739922\pi$
$830$	29.0568	1.00858
$831$	0	0
$832$	−2.77312	−0.0961405
$833$	5.73286	0.198632
$834$	0	0
$835$	−34.4850	−1.19340
$836$	1.18995	0.0411552
$837$	0	0
$838$	−50.1374	−1.73197
$839$	−15.7620	−0.544165	−0.272083	−	0.962274i	$-0.587712\pi$
$839$	−15.7620	−0.544165	−0.272083	+	0.962274i	$0.587712\pi$
$840$	0	0
$841$	−27.3585	−0.943396
$842$	−89.9992	−3.10158
$843$	0	0
$844$	117.184	4.03365
$845$	43.8824	1.50960
$846$	0	0
$847$	10.6519	0.366002
$848$	46.8467	1.60872
$849$	0	0
$850$	−103.034	−3.53405
$851$	−26.4090	−0.905289
$852$	0	0
$853$	−45.1146	−1.54470	−0.772348	−	0.635200i	$-0.780918\pi$
$853$	−45.1146	−1.54470	−0.772348	+	0.635200i	$0.780918\pi$
$854$	13.0116	0.445246
$855$	0	0
$856$	−23.1201	−0.790230
$857$	−56.8501	−1.94196	−0.970981	−	0.239156i	$-0.923129\pi$
$857$	−56.8501	−1.94196	−0.970981	+	0.239156i	$0.923129\pi$
$858$	0	0
$859$	−31.0874	−1.06069	−0.530343	−	0.847783i	$-0.677937\pi$
$859$	−31.0874	−1.06069	−0.530343	+	0.847783i	$0.677937\pi$
$860$	83.9334	2.86211
$861$	0	0
$862$	−61.5105	−2.09506
$863$	12.8286	0.436691	0.218345	−	0.975872i	$-0.429934\pi$
$863$	12.8286	0.436691	0.218345	+	0.975872i	$0.429934\pi$
$864$	0	0
$865$	−56.0954	−1.90730
$866$	−78.5406	−2.66892
$867$	0	0
$868$	−5.96553	−0.202483
$869$	−4.58806	−0.155639
$870$	0	0
$871$	−1.44565	−0.0489840
$872$	−116.789	−3.95497
$873$	0	0
$874$	−3.23646	−0.109475
$875$	6.68366	0.225949
$876$	0	0
$877$	−21.5829	−0.728803	−0.364401	−	0.931242i	$-0.618726\pi$
$877$	−21.5829	−0.728803	−0.364401	+	0.931242i	$0.618726\pi$
$878$	30.2708	1.02159
$879$	0	0
$880$	−17.9693	−0.605745
$881$	−9.39054	−0.316375	−0.158188	−	0.987409i	$-0.550565\pi$
$881$	−9.39054	−0.316375	−0.158188	+	0.987409i	$0.550565\pi$
$882$	0	0
$883$	7.52141	0.253115	0.126558	−	0.991959i	$-0.459607\pi$
$883$	7.52141	0.253115	0.126558	+	0.991959i	$0.459607\pi$
$884$	−14.7523	−0.496174
$885$	0	0
$886$	−74.8241	−2.51376
$887$	11.2171	0.376635	0.188317	−	0.982108i	$-0.439697\pi$
$887$	11.2171	0.376635	0.188317	+	0.982108i	$0.439697\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	59.3323	1.98882
$891$	0	0
$892$	−78.7081	−2.63534
$893$	−5.38511	−0.180206
$894$	0	0
$895$	59.2847	1.98167
$896$	−4.35421	−0.145464
$897$	0	0
$898$	−71.1370	−2.37387
$899$	−1.62036	−0.0540421
$900$	0	0
$901$	30.4653	1.01495
$902$	3.63899	0.121165
$903$	0	0
$904$	−121.159	−4.02970
$905$	9.32786	0.310069
$906$	0	0
$907$	0.525349	0.0174439	0.00872197	−	0.999962i	$-0.497224\pi$
$907$	0.525349	0.0174439	0.00872197	+	0.999962i	$0.497224\pi$
$908$	−23.7335	−0.787623
$909$	0	0
$910$	4.88451	0.161920
$911$	−40.5231	−1.34259	−0.671295	−	0.741190i	$-0.734262\pi$
$911$	−40.5231	−1.34259	−0.671295	+	0.741190i	$0.734262\pi$
$912$	0	0
$913$	1.91486	0.0633727
$914$	45.1884	1.49470
$915$	0	0
$916$	−132.010	−4.36173
$917$	−7.59333	−0.250754
$918$	0	0
$919$	−42.6123	−1.40565	−0.702825	−	0.711362i	$-0.748079\pi$
$919$	−42.6123	−1.40565	−0.702825	+	0.711362i	$0.748079\pi$
$920$	71.0498	2.34244
$921$	0	0
$922$	83.5145	2.75040
$923$	7.16237	0.235752
$924$	0	0
$925$	−62.7021	−2.06163
$926$	−76.4514	−2.51235
$927$	0	0
$928$	11.2288	0.368602
$929$	29.7018	0.974484	0.487242	−	0.873267i	$-0.338003\pi$
$929$	29.7018	0.974484	0.487242	+	0.873267i	$0.338003\pi$
$930$	0	0
$931$	0.427552	0.0140124
$932$	−93.8565	−3.07437
$933$	0	0
$934$	−60.3702	−1.97537
$935$	−11.6858	−0.382165
$936$	0	0
$937$	−18.4474	−0.602651	−0.301326	−	0.953521i	$-0.597429\pi$
$937$	−18.4474	−0.602651	−0.301326	+	0.953521i	$0.597429\pi$
$938$	6.86780	0.224242
$939$	0	0
$940$	205.243	6.69430
$941$	41.9960	1.36903	0.684516	−	0.728998i	$-0.260014\pi$
$941$	41.9960	1.36903	0.684516	+	0.728998i	$0.260014\pi$
$942$	0	0
$943$	−6.95044	−0.226337
$944$	−87.0591	−2.83353
$945$	0	0
$946$	7.87657	0.256089
$947$	4.73041	0.153718	0.0768589	−	0.997042i	$-0.475511\pi$
$947$	4.73041	0.153718	0.0768589	+	0.997042i	$0.475511\pi$
$948$	0	0
$949$	−8.49507	−0.275762
$950$	−7.68423	−0.249309
$951$	0	0
$952$	40.3675	1.30832
$953$	21.9136	0.709851	0.354925	−	0.934895i	$-0.384506\pi$
$953$	21.9136	0.709851	0.354925	+	0.934895i	$0.384506\pi$
$954$	0	0
$955$	−74.4813	−2.41016
$956$	−7.30004	−0.236100
$957$	0	0
$958$	56.4034	1.82231
$959$	−16.0871	−0.519480
$960$	0	0
$961$	−29.4005	−0.948404
$962$	−12.7841	−0.412177
$963$	0	0
$964$	−61.1470	−1.96941
$965$	−2.82396	−0.0909064
$966$	0	0
$967$	55.7878	1.79401	0.897007	−	0.442016i	$-0.145737\pi$
$967$	55.7878	1.79401	0.897007	+	0.442016i	$0.145737\pi$
$968$	75.0043	2.41073
$969$	0	0
$970$	−32.8917	−1.05609
$971$	17.1268	0.549626	0.274813	−	0.961498i	$-0.411384\pi$
$971$	17.1268	0.549626	0.274813	+	0.961498i	$0.411384\pi$
$972$	0	0
$973$	−20.2511	−0.649221
$974$	66.5722	2.13311
$975$	0	0
$976$	44.2578	1.41666
$977$	52.0777	1.66611	0.833056	−	0.553189i	$-0.186589\pi$
$977$	52.0777	1.66611	0.833056	+	0.553189i	$0.186589\pi$
$978$	0	0
$979$	3.91004	0.124966
$980$	−16.2953	−0.520536
$981$	0	0
$982$	−33.2569	−1.06127
$983$	−7.80268	−0.248867	−0.124433	−	0.992228i	$-0.539711\pi$
$983$	−7.80268	−0.248867	−0.124433	+	0.992228i	$0.539711\pi$
$984$	0	0
$985$	−50.9804	−1.62437
$986$	19.0361	0.606232
$987$	0	0
$988$	−1.10021	−0.0350025
$989$	−15.0442	−0.478376
$990$	0	0
$991$	47.1419	1.49751	0.748757	−	0.662845i	$-0.230651\pi$
$991$	47.1419	1.49751	0.748757	+	0.662845i	$0.230651\pi$
$992$	−11.0841	−0.351922
$993$	0	0
$994$	−34.0260	−1.07924
$995$	37.4205	1.18631
$996$	0	0
$997$	−12.5330	−0.396923	−0.198461	−	0.980109i	$-0.563594\pi$
$997$	−12.5330	−0.396923	−0.198461	+	0.980109i	$0.563594\pi$
$998$	−5.35642	−0.169555
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.q.1.1		15
3.2	odd	2		889.2.a.b.1.15	✓	15
21.20	even	2		6223.2.a.j.1.15		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.15	✓	15	3.2	odd	2
6223.2.a.j.1.15		15	21.20	even	2
8001.2.a.q.1.1		15	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)

\(f(q)\)	\(=\)	\(q-2.59170 q^{2} +4.71692 q^{4} -3.45466 q^{5} -1.00000 q^{7} -7.04143 q^{8} +O(q^{10})\)	\(q-2.59170 q^{2} +4.71692 q^{4} -3.45466 q^{5} -1.00000 q^{7} -7.04143 q^{8} +8.95345 q^{10} +0.590039 q^{11} -0.545545 q^{13} +2.59170 q^{14} +8.81546 q^{16} +5.73286 q^{17} +0.427552 q^{19} -16.2953 q^{20} -1.52920 q^{22} +2.92076 q^{23} +6.93468 q^{25} +1.41389 q^{26} -4.71692 q^{28} -1.28121 q^{29} +1.26471 q^{31} -8.76418 q^{32} -14.8579 q^{34} +3.45466 q^{35} -9.04182 q^{37} -1.10809 q^{38} +24.3258 q^{40} -2.37966 q^{41} -5.15076 q^{43} +2.78316 q^{44} -7.56975 q^{46} -12.5952 q^{47} +1.00000 q^{49} -17.9726 q^{50} -2.57329 q^{52} +5.31416 q^{53} -2.03838 q^{55} +7.04143 q^{56} +3.32052 q^{58} -9.87573 q^{59} +5.02047 q^{61} -3.27775 q^{62} +5.08321 q^{64} +1.88467 q^{65} +2.64992 q^{67} +27.0414 q^{68} -8.95345 q^{70} -13.1288 q^{71} +15.5717 q^{73} +23.4337 q^{74} +2.01673 q^{76} -0.590039 q^{77} -7.77587 q^{79} -30.4544 q^{80} +6.16738 q^{82} +3.24532 q^{83} -19.8051 q^{85} +13.3492 q^{86} -4.15472 q^{88} +6.62676 q^{89} +0.545545 q^{91} +13.7770 q^{92} +32.6430 q^{94} -1.47705 q^{95} -3.67364 q^{97} -2.59170 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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