LMFDB - Embedded newform 8037.2.a.n.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 13 x^{14} + 65 x^{13} + 47 x^{12} - 390 x^{11} + 4 x^{10} + 1115 x^{9} - 320 x^{8} + \cdots - 25 $ x^16 - 4x^15 - 13x^14 + 65x^13 + 47x^12 - 390x^11 + 4x^10 + 1115x^9 - 320x^8 - 1639x^7 + 618x^6 + 1250x^5 - 487x^4 - 456x^3 + 179x^2 + 62*x - 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.469922$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.469922 q^{2} -1.77917 q^{4} -0.709772 q^{5} -4.48221 q^{7} -1.77592 q^{8} +O(q^{10})$	\(q+0.469922 q^{2} -1.77917 q^{4} -0.709772 q^{5} -4.48221 q^{7} -1.77592 q^{8} -0.333538 q^{10} -1.95928 q^{11} +5.21511 q^{13} -2.10629 q^{14} +2.72380 q^{16} -2.18280 q^{17} -1.00000 q^{19} +1.26281 q^{20} -0.920708 q^{22} +5.80559 q^{23} -4.49622 q^{25} +2.45069 q^{26} +7.97463 q^{28} -3.22466 q^{29} +5.70086 q^{31} +4.83181 q^{32} -1.02574 q^{34} +3.18135 q^{35} -2.09562 q^{37} -0.469922 q^{38} +1.26050 q^{40} +3.55511 q^{41} +3.30509 q^{43} +3.48589 q^{44} +2.72817 q^{46} +1.00000 q^{47} +13.0902 q^{49} -2.11287 q^{50} -9.27858 q^{52} +10.0140 q^{53} +1.39064 q^{55} +7.96004 q^{56} -1.51534 q^{58} +7.99904 q^{59} +9.24255 q^{61} +2.67896 q^{62} -3.17703 q^{64} -3.70154 q^{65} -1.93632 q^{67} +3.88357 q^{68} +1.49499 q^{70} -12.1911 q^{71} -7.52728 q^{73} -0.984777 q^{74} +1.77917 q^{76} +8.78189 q^{77} -13.3009 q^{79} -1.93328 q^{80} +1.67062 q^{82} -12.3503 q^{83} +1.54929 q^{85} +1.55313 q^{86} +3.47951 q^{88} +15.7300 q^{89} -23.3752 q^{91} -10.3291 q^{92} +0.469922 q^{94} +0.709772 q^{95} +0.573560 q^{97} +6.15139 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) $ 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8	$ 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8} - 15 q^{10} - 19 q^{13} + 6 q^{14} + 10 q^{16} + 8 q^{17} - 16 q^{19} + 11 q^{20} - 12 q^{22} + 5 q^{23} - 3 q^{25} - 9 q^{26} - 17 q^{28} + 2 q^{29} - 18 q^{31} - 3 q^{32} - 14 q^{34} + 11 q^{35} - 24 q^{37} - 4 q^{38} - 50 q^{40} + 6 q^{41} - 34 q^{43} + 4 q^{44} - 3 q^{46} + 16 q^{47} + 5 q^{49} - 26 q^{50} - 44 q^{52} + 23 q^{53} - 48 q^{55} + 3 q^{56} - 26 q^{58} + 32 q^{59} - 16 q^{61} - 32 q^{62} + 7 q^{64} + 18 q^{65} - 67 q^{67} + 19 q^{68} + 24 q^{70} - 19 q^{71} - 2 q^{73} + 29 q^{74} - 10 q^{76} - 14 q^{77} - 27 q^{79} - 15 q^{80} - 56 q^{82} + 17 q^{83} + 15 q^{85} + q^{86} - 13 q^{88} - 20 q^{89} - 42 q^{91} - 45 q^{92} + 4 q^{94} - q^{95} - 50 q^{97} - 13 q^{98}+O(q^{100}) $ 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8 - 15 * q^10 - 19 * q^13 + 6 * q^14 + 10 * q^16 + 8 * q^17 - 16 * q^19 + 11 * q^20 - 12 * q^22 + 5 * q^23 - 3 * q^25 - 9 * q^26 - 17 * q^28 + 2 * q^29 - 18 * q^31 - 3 * q^32 - 14 * q^34 + 11 * q^35 - 24 * q^37 - 4 * q^38 - 50 * q^40 + 6 * q^41 - 34 * q^43 + 4 * q^44 - 3 * q^46 + 16 * q^47 + 5 * q^49 - 26 * q^50 - 44 * q^52 + 23 * q^53 - 48 * q^55 + 3 * q^56 - 26 * q^58 + 32 * q^59 - 16 * q^61 - 32 * q^62 + 7 * q^64 + 18 * q^65 - 67 * q^67 + 19 * q^68 + 24 * q^70 - 19 * q^71 - 2 * q^73 + 29 * q^74 - 10 * q^76 - 14 * q^77 - 27 * q^79 - 15 * q^80 - 56 * q^82 + 17 * q^83 + 15 * q^85 + q^86 - 13 * q^88 - 20 * q^89 - 42 * q^91 - 45 * q^92 + 4 * q^94 - q^95 - 50 * q^97 - 13 * 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 $

$2$	0.469922	0.332285	0.166143	−	0.986102i	$-0.446869\pi$
$2$	0.469922	0.332285	0.166143	+	0.986102i	$0.446869\pi$
$3$	0	0
$3$	0	0
$4$	−1.77917	−0.889587
$5$	−0.709772	−0.317420	−0.158710	−	0.987325i	$-0.550733\pi$
$5$	−0.709772	−0.317420	−0.158710	+	0.987325i	$0.550733\pi$
$6$	0	0
$7$	−4.48221	−1.69412	−0.847058	−	0.531500i	$-0.821629\pi$
$7$	−4.48221	−1.69412	−0.847058	+	0.531500i	$0.821629\pi$
$8$	−1.77592	−0.627882
$9$	0	0
$10$	−0.333538	−0.105474
$11$	−1.95928	−0.590744	−0.295372	−	0.955382i	$-0.595444\pi$
$11$	−1.95928	−0.590744	−0.295372	+	0.955382i	$0.595444\pi$
$12$	0	0
$13$	5.21511	1.44641	0.723205	−	0.690633i	$-0.242668\pi$
$13$	5.21511	1.44641	0.723205	+	0.690633i	$0.242668\pi$
$14$	−2.10629	−0.562930
$15$	0	0
$16$	2.72380	0.680951
$17$	−2.18280	−0.529406	−0.264703	−	0.964330i	$-0.585274\pi$
$17$	−2.18280	−0.529406	−0.264703	+	0.964330i	$0.585274\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.26281	0.282372
$21$	0	0
$22$	−0.920708	−0.196296
$23$	5.80559	1.21055	0.605274	−	0.796017i	$-0.293063\pi$
$23$	5.80559	1.21055	0.605274	+	0.796017i	$0.293063\pi$
$24$	0	0
$25$	−4.49622	−0.899245
$26$	2.45069	0.480621
$27$	0	0
$28$	7.97463	1.50706
$29$	−3.22466	−0.598804	−0.299402	−	0.954127i	$-0.596787\pi$
$29$	−3.22466	−0.598804	−0.299402	+	0.954127i	$0.596787\pi$
$30$	0	0
$31$	5.70086	1.02390	0.511952	−	0.859014i	$-0.328923\pi$
$31$	5.70086	1.02390	0.511952	+	0.859014i	$0.328923\pi$
$32$	4.83181	0.854151
$33$	0	0
$34$	−1.02574	−0.175914
$35$	3.18135	0.537746
$36$	0	0
$37$	−2.09562	−0.344518	−0.172259	−	0.985052i	$-0.555107\pi$
$37$	−2.09562	−0.344518	−0.172259	+	0.985052i	$0.555107\pi$
$38$	−0.469922	−0.0762314
$39$	0	0
$40$	1.26050	0.199302
$41$	3.55511	0.555215	0.277607	−	0.960695i	$-0.410459\pi$
$41$	3.55511	0.555215	0.277607	+	0.960695i	$0.410459\pi$
$42$	0	0
$43$	3.30509	0.504021	0.252011	−	0.967724i	$-0.418908\pi$
$43$	3.30509	0.504021	0.252011	+	0.967724i	$0.418908\pi$
$44$	3.48589	0.525518
$45$	0	0
$46$	2.72817	0.402247
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	13.0902	1.87003
$50$	−2.11287	−0.298806
$51$	0	0
$52$	−9.27858	−1.28671
$53$	10.0140	1.37553	0.687767	−	0.725932i	$-0.258591\pi$
$53$	10.0140	1.37553	0.687767	+	0.725932i	$0.258591\pi$
$54$	0	0
$55$	1.39064	0.187514
$56$	7.96004	1.06370
$57$	0	0
$58$	−1.51534	−0.198974
$59$	7.99904	1.04139	0.520693	−	0.853744i	$-0.325673\pi$
$59$	7.99904	1.04139	0.520693	+	0.853744i	$0.325673\pi$
$60$	0	0
$61$	9.24255	1.18339	0.591693	−	0.806163i	$-0.298460\pi$
$61$	9.24255	1.18339	0.591693	+	0.806163i	$0.298460\pi$
$62$	2.67896	0.340228
$63$	0	0
$64$	−3.17703	−0.397129
$65$	−3.70154	−0.459120
$66$	0	0
$67$	−1.93632	−0.236559	−0.118279	−	0.992980i	$-0.537738\pi$
$67$	−1.93632	−0.236559	−0.118279	+	0.992980i	$0.537738\pi$
$68$	3.88357	0.470952
$69$	0	0
$70$	1.49499	0.178685
$71$	−12.1911	−1.44682	−0.723410	−	0.690419i	$-0.757426\pi$
$71$	−12.1911	−1.44682	−0.723410	+	0.690419i	$0.757426\pi$
$72$	0	0
$73$	−7.52728	−0.881001	−0.440501	−	0.897752i	$-0.645199\pi$
$73$	−7.52728	−0.881001	−0.440501	+	0.897752i	$0.645199\pi$
$74$	−0.984777	−0.114478
$75$	0	0
$76$	1.77917	0.204085
$77$	8.78189	1.00079
$78$	0	0
$79$	−13.3009	−1.49647	−0.748237	−	0.663432i	$-0.769099\pi$
$79$	−13.3009	−1.49647	−0.748237	+	0.663432i	$0.769099\pi$
$80$	−1.93328	−0.216147
$81$	0	0
$82$	1.67062	0.184490
$83$	−12.3503	−1.35562	−0.677809	−	0.735238i	$-0.737070\pi$
$83$	−12.3503	−1.35562	−0.677809	+	0.735238i	$0.737070\pi$
$84$	0	0
$85$	1.54929	0.168044
$86$	1.55313	0.167479
$87$	0	0
$88$	3.47951	0.370917
$89$	15.7300	1.66738	0.833690	−	0.552233i	$-0.186224\pi$
$89$	15.7300	1.66738	0.833690	+	0.552233i	$0.186224\pi$
$90$	0	0
$91$	−23.3752	−2.45039
$92$	−10.3291	−1.07689
$93$	0	0
$94$	0.469922	0.0484688
$95$	0.709772	0.0728211
$96$	0	0
$97$	0.573560	0.0582361	0.0291181	−	0.999576i	$-0.490730\pi$
$97$	0.573560	0.0582361	0.0291181	+	0.999576i	$0.490730\pi$
$98$	6.15139	0.621384
$99$	0	0
$100$	7.99956	0.799956
$101$	1.01612	0.101108	0.0505540	−	0.998721i	$-0.483901\pi$
$101$	1.01612	0.101108	0.0505540	+	0.998721i	$0.483901\pi$
$102$	0	0
$103$	−11.1391	−1.09757	−0.548785	−	0.835963i	$-0.684910\pi$
$103$	−11.1391	−1.09757	−0.548785	+	0.835963i	$0.684910\pi$
$104$	−9.26160	−0.908175
$105$	0	0
$106$	4.70582	0.457069
$107$	5.62930	0.544205	0.272103	−	0.962268i	$-0.412281\pi$
$107$	5.62930	0.544205	0.272103	+	0.962268i	$0.412281\pi$
$108$	0	0
$109$	9.81460	0.940068	0.470034	−	0.882648i	$-0.344242\pi$
$109$	9.81460	0.940068	0.470034	+	0.882648i	$0.344242\pi$
$110$	0.653493	0.0623081
$111$	0	0
$112$	−12.2087	−1.15361
$113$	−8.04550	−0.756857	−0.378428	−	0.925631i	$-0.623535\pi$
$113$	−8.04550	−0.756857	−0.378428	+	0.925631i	$0.623535\pi$
$114$	0	0
$115$	−4.12065	−0.384252
$116$	5.73722	0.532688
$117$	0	0
$118$	3.75893	0.346037
$119$	9.78376	0.896876
$120$	0	0
$121$	−7.16123	−0.651021
$122$	4.34328	0.393222
$123$	0	0
$124$	−10.1428	−0.910851
$125$	6.74016	0.602858
$126$	0	0
$127$	−10.6703	−0.946839	−0.473420	−	0.880837i	$-0.656981\pi$
$127$	−10.6703	−0.946839	−0.473420	+	0.880837i	$0.656981\pi$
$128$	−11.1566	−0.986111
$129$	0	0
$130$	−1.73944	−0.152559
$131$	12.5829	1.09937	0.549687	−	0.835371i	$-0.314747\pi$
$131$	12.5829	1.09937	0.549687	+	0.835371i	$0.314747\pi$
$132$	0	0
$133$	4.48221	0.388657
$134$	−0.909918	−0.0786049
$135$	0	0
$136$	3.87647	0.332404
$137$	−5.93575	−0.507125	−0.253562	−	0.967319i	$-0.581602\pi$
$137$	−5.93575	−0.507125	−0.253562	+	0.967319i	$0.581602\pi$
$138$	0	0
$139$	17.9729	1.52444	0.762221	−	0.647317i	$-0.224109\pi$
$139$	17.9729	1.52444	0.762221	+	0.647317i	$0.224109\pi$
$140$	−5.66017	−0.478372
$141$	0	0
$142$	−5.72888	−0.480757
$143$	−10.2178	−0.854459
$144$	0	0
$145$	2.28877	0.190072
$146$	−3.53723	−0.292744
$147$	0	0
$148$	3.72847	0.306478
$149$	15.8903	1.30178	0.650892	−	0.759170i	$-0.274395\pi$
$149$	15.8903	1.30178	0.650892	+	0.759170i	$0.274395\pi$
$150$	0	0
$151$	−10.5160	−0.855776	−0.427888	−	0.903832i	$-0.640742\pi$
$151$	−10.5160	−0.855776	−0.427888	+	0.903832i	$0.640742\pi$
$152$	1.77592	0.144046
$153$	0	0
$154$	4.12681	0.332548
$155$	−4.04631	−0.325007
$156$	0	0
$157$	−17.4693	−1.39420	−0.697100	−	0.716974i	$-0.745527\pi$
$157$	−17.4693	−1.39420	−0.697100	+	0.716974i	$0.745527\pi$
$158$	−6.25041	−0.497256
$159$	0	0
$160$	−3.42949	−0.271125
$161$	−26.0219	−2.05081
$162$	0	0
$163$	16.2148	1.27004	0.635021	−	0.772495i	$-0.280992\pi$
$163$	16.2148	1.27004	0.635021	+	0.772495i	$0.280992\pi$
$164$	−6.32516	−0.493912
$165$	0	0
$166$	−5.80366	−0.450452
$167$	6.58649	0.509678	0.254839	−	0.966983i	$-0.417978\pi$
$167$	6.58649	0.509678	0.254839	+	0.966983i	$0.417978\pi$
$168$	0	0
$169$	14.1974	1.09210
$170$	0.728045	0.0558385
$171$	0	0
$172$	−5.88033	−0.448371
$173$	10.8140	0.822170	0.411085	−	0.911597i	$-0.365150\pi$
$173$	10.8140	0.822170	0.411085	+	0.911597i	$0.365150\pi$
$174$	0	0
$175$	20.1530	1.52343
$176$	−5.33668	−0.402268
$177$	0	0
$178$	7.39189	0.554046
$179$	−22.1513	−1.65566	−0.827832	−	0.560976i	$-0.810426\pi$
$179$	−22.1513	−1.65566	−0.827832	+	0.560976i	$0.810426\pi$
$180$	0	0
$181$	6.72204	0.499646	0.249823	−	0.968292i	$-0.419628\pi$
$181$	6.72204	0.499646	0.249823	+	0.968292i	$0.419628\pi$
$182$	−10.9845	−0.814228
$183$	0	0
$184$	−10.3102	−0.760081
$185$	1.48741	0.109357
$186$	0	0
$187$	4.27670	0.312743
$188$	−1.77917	−0.129760
$189$	0	0
$190$	0.333538	0.0241974
$191$	−25.5176	−1.84639	−0.923194	−	0.384334i	$-0.874431\pi$
$191$	−25.5176	−1.84639	−0.923194	+	0.384334i	$0.874431\pi$
$192$	0	0
$193$	−19.8020	−1.42538	−0.712691	−	0.701478i	$-0.752524\pi$
$193$	−19.8020	−1.42538	−0.712691	+	0.701478i	$0.752524\pi$
$194$	0.269528	0.0193510
$195$	0	0
$196$	−23.2898	−1.66356
$197$	−2.13533	−0.152136	−0.0760680	−	0.997103i	$-0.524237\pi$
$197$	−2.13533	−0.152136	−0.0760680	+	0.997103i	$0.524237\pi$
$198$	0	0
$199$	14.0001	0.992440	0.496220	−	0.868197i	$-0.334721\pi$
$199$	14.0001	0.992440	0.496220	+	0.868197i	$0.334721\pi$
$200$	7.98492	0.564619
$201$	0	0
$202$	0.477498	0.0335967
$203$	14.4536	1.01444
$204$	0	0
$205$	−2.52332	−0.176236
$206$	−5.23452	−0.364706
$207$	0	0
$208$	14.2049	0.984935
$209$	1.95928	0.135526
$210$	0	0
$211$	−16.4188	−1.13031	−0.565157	−	0.824983i	$-0.691185\pi$
$211$	−16.4188	−1.13031	−0.565157	+	0.824983i	$0.691185\pi$
$212$	−17.8167	−1.22366
$213$	0	0
$214$	2.64533	0.180831
$215$	−2.34586	−0.159986
$216$	0	0
$217$	−25.5524	−1.73461
$218$	4.61210	0.312371
$219$	0	0
$220$	−2.47419	−0.166810
$221$	−11.3835	−0.765738
$222$	0	0
$223$	−16.3014	−1.09162	−0.545812	−	0.837908i	$-0.683779\pi$
$223$	−16.3014	−1.09162	−0.545812	+	0.837908i	$0.683779\pi$
$224$	−21.6572	−1.44703
$225$	0	0
$226$	−3.78076	−0.251492
$227$	16.0849	1.06759	0.533795	−	0.845614i	$-0.320765\pi$
$227$	16.0849	1.06759	0.533795	+	0.845614i	$0.320765\pi$
$228$	0	0
$229$	−25.7727	−1.70311	−0.851553	−	0.524268i	$-0.824339\pi$
$229$	−25.7727	−1.70311	−0.851553	+	0.524268i	$0.824339\pi$
$230$	−1.93638	−0.127681
$231$	0	0
$232$	5.72672	0.375978
$233$	−19.1626	−1.25538	−0.627692	−	0.778462i	$-0.716000\pi$
$233$	−19.1626	−1.25538	−0.627692	+	0.778462i	$0.716000\pi$
$234$	0	0
$235$	−0.709772	−0.0463005
$236$	−14.2317	−0.926404
$237$	0	0
$238$	4.59760	0.298018
$239$	10.5338	0.681374	0.340687	−	0.940177i	$-0.389340\pi$
$239$	10.5338	0.681374	0.340687	+	0.940177i	$0.389340\pi$
$240$	0	0
$241$	0.186996	0.0120455	0.00602273	−	0.999982i	$-0.498083\pi$
$241$	0.186996	0.0120455	0.00602273	+	0.999982i	$0.498083\pi$
$242$	−3.36522	−0.216325
$243$	0	0
$244$	−16.4441	−1.05273
$245$	−9.29108	−0.593585
$246$	0	0
$247$	−5.21511	−0.331829
$248$	−10.1242	−0.642890
$249$	0	0
$250$	3.16735	0.200321
$251$	8.11113	0.511970	0.255985	−	0.966681i	$-0.417600\pi$
$251$	8.11113	0.511970	0.255985	+	0.966681i	$0.417600\pi$
$252$	0	0
$253$	−11.3748	−0.715125
$254$	−5.01423	−0.314621
$255$	0	0
$256$	1.11134	0.0694589
$257$	31.1915	1.94567	0.972835	−	0.231499i	$-0.0743629\pi$
$257$	31.1915	1.94567	0.972835	+	0.231499i	$0.0743629\pi$
$258$	0	0
$259$	9.39301	0.583653
$260$	6.58568	0.408427
$261$	0	0
$262$	5.91298	0.365305
$263$	−3.91399	−0.241347	−0.120674	−	0.992692i	$-0.538505\pi$
$263$	−3.91399	−0.241347	−0.120674	+	0.992692i	$0.538505\pi$
$264$	0	0
$265$	−7.10769	−0.436622
$266$	2.10629	0.129145
$267$	0	0
$268$	3.44504	0.210439
$269$	3.40439	0.207569	0.103785	−	0.994600i	$-0.466905\pi$
$269$	3.40439	0.207569	0.103785	+	0.994600i	$0.466905\pi$
$270$	0	0
$271$	−14.8420	−0.901586	−0.450793	−	0.892629i	$-0.648859\pi$
$271$	−14.8420	−0.901586	−0.450793	+	0.892629i	$0.648859\pi$
$272$	−5.94551	−0.360499
$273$	0	0
$274$	−2.78934	−0.168510
$275$	8.80935	0.531223
$276$	0	0
$277$	−27.6969	−1.66414	−0.832072	−	0.554667i	$-0.812846\pi$
$277$	−27.6969	−1.66414	−0.832072	+	0.554667i	$0.812846\pi$
$278$	8.44587	0.506550
$279$	0	0
$280$	−5.64982	−0.337641
$281$	21.1240	1.26015	0.630076	−	0.776533i	$-0.283024\pi$
$281$	21.1240	1.26015	0.630076	+	0.776533i	$0.283024\pi$
$282$	0	0
$283$	−30.2193	−1.79635	−0.898174	−	0.439639i	$-0.855106\pi$
$283$	−30.2193	−1.79635	−0.898174	+	0.439639i	$0.855106\pi$
$284$	21.6901	1.28707
$285$	0	0
$286$	−4.80159	−0.283924
$287$	−15.9348	−0.940599
$288$	0	0
$289$	−12.2354	−0.719729
$290$	1.07554	0.0631582
$291$	0	0
$292$	13.3923	0.783727
$293$	3.67315	0.214587	0.107294	−	0.994227i	$-0.465781\pi$
$293$	3.67315	0.214587	0.107294	+	0.994227i	$0.465781\pi$
$294$	0	0
$295$	−5.67750	−0.330557
$296$	3.72164	0.216316
$297$	0	0
$298$	7.46721	0.432564
$299$	30.2768	1.75095
$300$	0	0
$301$	−14.8141	−0.853871
$302$	−4.94168	−0.284362
$303$	0	0
$304$	−2.72380	−0.156221
$305$	−6.56011	−0.375631
$306$	0	0
$307$	−24.8444	−1.41794	−0.708972	−	0.705237i	$-0.750841\pi$
$307$	−24.8444	−1.41794	−0.708972	+	0.705237i	$0.750841\pi$
$308$	−15.6245	−0.890289
$309$	0	0
$310$	−1.90145	−0.107995
$311$	−23.1931	−1.31516	−0.657580	−	0.753385i	$-0.728420\pi$
$311$	−23.1931	−1.31516	−0.657580	+	0.753385i	$0.728420\pi$
$312$	0	0
$313$	10.5096	0.594036	0.297018	−	0.954872i	$-0.404008\pi$
$313$	10.5096	0.594036	0.297018	+	0.954872i	$0.404008\pi$
$314$	−8.20920	−0.463272
$315$	0	0
$316$	23.6647	1.33124
$317$	−12.2705	−0.689181	−0.344590	−	0.938753i	$-0.611982\pi$
$317$	−12.2705	−0.689181	−0.344590	+	0.938753i	$0.611982\pi$
$318$	0	0
$319$	6.31799	0.353740
$320$	2.25497	0.126057
$321$	0	0
$322$	−12.2283	−0.681454
$323$	2.18280	0.121454
$324$	0	0
$325$	−23.4483	−1.30068
$326$	7.61970	0.422016
$327$	0	0
$328$	−6.31358	−0.348609
$329$	−4.48221	−0.247112
$330$	0	0
$331$	−0.434345	−0.0238738	−0.0119369	−	0.999929i	$-0.503800\pi$
$331$	−0.434345	−0.0238738	−0.0119369	+	0.999929i	$0.503800\pi$
$332$	21.9733	1.20594
$333$	0	0
$334$	3.09514	0.169358
$335$	1.37434	0.0750884
$336$	0	0
$337$	7.68321	0.418531	0.209266	−	0.977859i	$-0.432893\pi$
$337$	7.68321	0.418531	0.209266	+	0.977859i	$0.432893\pi$
$338$	6.67165	0.362890
$339$	0	0
$340$	−2.75645	−0.149490
$341$	−11.1696	−0.604865
$342$	0	0
$343$	−27.2977	−1.47394
$344$	−5.86957	−0.316466
$345$	0	0
$346$	5.08172	0.273195
$347$	12.3453	0.662728	0.331364	−	0.943503i	$-0.392491\pi$
$347$	12.3453	0.662728	0.331364	+	0.943503i	$0.392491\pi$
$348$	0	0
$349$	4.14497	0.221875	0.110938	−	0.993827i	$-0.464615\pi$
$349$	4.14497	0.221875	0.110938	+	0.993827i	$0.464615\pi$
$350$	9.47035	0.506212
$351$	0	0
$352$	−9.46685	−0.504585
$353$	−13.8317	−0.736185	−0.368093	−	0.929789i	$-0.619989\pi$
$353$	−13.8317	−0.736185	−0.368093	+	0.929789i	$0.619989\pi$
$354$	0	0
$355$	8.65292	0.459249
$356$	−27.9864	−1.48328
$357$	0	0
$358$	−10.4094	−0.550153
$359$	−3.64626	−0.192442	−0.0962211	−	0.995360i	$-0.530676\pi$
$359$	−3.64626	−0.192442	−0.0962211	+	0.995360i	$0.530676\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	3.15884	0.166025
$363$	0	0
$364$	41.5886	2.17983
$365$	5.34265	0.279647
$366$	0	0
$367$	−5.46714	−0.285383	−0.142691	−	0.989767i	$-0.545576\pi$
$367$	−5.46714	−0.285383	−0.142691	+	0.989767i	$0.545576\pi$
$368$	15.8133	0.824324
$369$	0	0
$370$	0.698968	0.0363376
$371$	−44.8850	−2.33032
$372$	0	0
$373$	24.1704	1.25150	0.625748	−	0.780025i	$-0.284794\pi$
$373$	24.1704	1.25150	0.625748	+	0.780025i	$0.284794\pi$
$374$	2.00972	0.103920
$375$	0	0
$376$	−1.77592	−0.0915859
$377$	−16.8169	−0.866116
$378$	0	0
$379$	18.0348	0.926385	0.463192	−	0.886258i	$-0.346704\pi$
$379$	18.0348	0.926385	0.463192	+	0.886258i	$0.346704\pi$
$380$	−1.26281	−0.0647807
$381$	0	0
$382$	−11.9913	−0.613527
$383$	1.33092	0.0680066	0.0340033	−	0.999422i	$-0.489174\pi$
$383$	1.33092	0.0680066	0.0340033	+	0.999422i	$0.489174\pi$
$384$	0	0
$385$	−6.23315	−0.317671
$386$	−9.30542	−0.473633
$387$	0	0
$388$	−1.02046	−0.0518061
$389$	−15.8422	−0.803231	−0.401615	−	0.915808i	$-0.631551\pi$
$389$	−15.8422	−0.803231	−0.401615	+	0.915808i	$0.631551\pi$
$390$	0	0
$391$	−12.6724	−0.640872
$392$	−23.2472	−1.17416
$393$	0	0
$394$	−1.00344	−0.0505526
$395$	9.44065	0.475010
$396$	0	0
$397$	0.794816	0.0398907	0.0199453	−	0.999801i	$-0.493651\pi$
$397$	0.794816	0.0398907	0.0199453	+	0.999801i	$0.493651\pi$
$398$	6.57895	0.329773
$399$	0	0
$400$	−12.2468	−0.612341
$401$	−24.4027	−1.21861	−0.609307	−	0.792935i	$-0.708552\pi$
$401$	−24.4027	−1.21861	−0.609307	+	0.792935i	$0.708552\pi$
$402$	0	0
$403$	29.7306	1.48099
$404$	−1.80786	−0.0899443
$405$	0	0
$406$	6.79206	0.337084
$407$	4.10590	0.203522
$408$	0	0
$409$	−21.8314	−1.07949	−0.539747	−	0.841828i	$-0.681480\pi$
$409$	−21.8314	−1.07949	−0.539747	+	0.841828i	$0.681480\pi$
$410$	−1.18576	−0.0585607
$411$	0	0
$412$	19.8184	0.976384
$413$	−35.8534	−1.76423
$414$	0	0
$415$	8.76588	0.430300
$416$	25.1984	1.23545
$417$	0	0
$418$	0.920708	0.0450333
$419$	33.7746	1.64999	0.824997	−	0.565136i	$-0.191176\pi$
$419$	33.7746	1.64999	0.824997	+	0.565136i	$0.191176\pi$
$420$	0	0
$421$	−25.4662	−1.24115	−0.620574	−	0.784148i	$-0.713100\pi$
$421$	−25.4662	−1.24115	−0.620574	+	0.784148i	$0.713100\pi$
$422$	−7.71554	−0.375587
$423$	0	0
$424$	−17.7841	−0.863672
$425$	9.81434	0.476065
$426$	0	0
$427$	−41.4271	−2.00480
$428$	−10.0155	−0.484118
$429$	0	0
$430$	−1.10237	−0.0531611
$431$	−23.5143	−1.13264	−0.566322	−	0.824184i	$-0.691634\pi$
$431$	−23.5143	−1.13264	−0.566322	+	0.824184i	$0.691634\pi$
$432$	0	0
$433$	−23.3811	−1.12363	−0.561813	−	0.827265i	$-0.689896\pi$
$433$	−23.3811	−1.12363	−0.561813	+	0.827265i	$0.689896\pi$
$434$	−12.0077	−0.576386
$435$	0	0
$436$	−17.4619	−0.836272
$437$	−5.80559	−0.277719
$438$	0	0
$439$	3.68654	0.175949	0.0879745	−	0.996123i	$-0.471961\pi$
$439$	3.68654	0.175949	0.0879745	+	0.996123i	$0.471961\pi$
$440$	−2.46966	−0.117737
$441$	0	0
$442$	−5.34937	−0.254443
$443$	29.4842	1.40084	0.700418	−	0.713733i	$-0.252997\pi$
$443$	29.4842	1.40084	0.700418	+	0.713733i	$0.252997\pi$
$444$	0	0
$445$	−11.1647	−0.529260
$446$	−7.66040	−0.362730
$447$	0	0
$448$	14.2401	0.672783
$449$	32.4581	1.53179	0.765895	−	0.642965i	$-0.222296\pi$
$449$	32.4581	1.53179	0.765895	+	0.642965i	$0.222296\pi$
$450$	0	0
$451$	−6.96544	−0.327990
$452$	14.3143	0.673290
$453$	0	0
$454$	7.55863	0.354744
$455$	16.5911	0.777802
$456$	0	0
$457$	9.89985	0.463095	0.231548	−	0.972824i	$-0.425621\pi$
$457$	9.89985	0.463095	0.231548	+	0.972824i	$0.425621\pi$
$458$	−12.1112	−0.565917
$459$	0	0
$460$	7.33134	0.341826
$461$	−8.07888	−0.376271	−0.188135	−	0.982143i	$-0.560244\pi$
$461$	−8.07888	−0.376271	−0.188135	+	0.982143i	$0.560244\pi$
$462$	0	0
$463$	35.4136	1.64581	0.822905	−	0.568178i	$-0.192352\pi$
$463$	35.4136	1.64581	0.822905	+	0.568178i	$0.192352\pi$
$464$	−8.78333	−0.407756
$465$	0	0
$466$	−9.00494	−0.417146
$467$	3.03307	0.140354	0.0701770	−	0.997535i	$-0.477644\pi$
$467$	3.03307	0.140354	0.0701770	+	0.997535i	$0.477644\pi$
$468$	0	0
$469$	8.67898	0.400758
$470$	−0.333538	−0.0153850
$471$	0	0
$472$	−14.2056	−0.653868
$473$	−6.47559	−0.297748
$474$	0	0
$475$	4.49622	0.206301
$476$	−17.4070	−0.797848
$477$	0	0
$478$	4.95006	0.226410
$479$	−14.5937	−0.666802	−0.333401	−	0.942785i	$-0.608196\pi$
$479$	−14.5937	−0.666802	−0.333401	+	0.942785i	$0.608196\pi$
$480$	0	0
$481$	−10.9289	−0.498314
$482$	0.0878735	0.00400253
$483$	0	0
$484$	12.7411	0.579140
$485$	−0.407097	−0.0184853
$486$	0	0
$487$	−8.25333	−0.373994	−0.186997	−	0.982360i	$-0.559875\pi$
$487$	−8.25333	−0.373994	−0.186997	+	0.982360i	$0.559875\pi$
$488$	−16.4140	−0.743027
$489$	0	0
$490$	−4.36609	−0.197240
$491$	−17.1302	−0.773075	−0.386537	−	0.922274i	$-0.626329\pi$
$491$	−17.1302	−0.773075	−0.386537	+	0.922274i	$0.626329\pi$
$492$	0	0
$493$	7.03877	0.317010
$494$	−2.45069	−0.110262
$495$	0	0
$496$	15.5280	0.697228
$497$	54.6432	2.45108
$498$	0	0
$499$	−10.2948	−0.460858	−0.230429	−	0.973089i	$-0.574013\pi$
$499$	−10.2948	−0.460858	−0.230429	+	0.973089i	$0.574013\pi$
$500$	−11.9919	−0.536294
$501$	0	0
$502$	3.81160	0.170120
$503$	3.52902	0.157351	0.0786756	−	0.996900i	$-0.474931\pi$
$503$	3.52902	0.157351	0.0786756	+	0.996900i	$0.474931\pi$
$504$	0	0
$505$	−0.721216	−0.0320937
$506$	−5.34525	−0.237625
$507$	0	0
$508$	18.9844	0.842296
$509$	1.99736	0.0885313	0.0442656	−	0.999020i	$-0.485905\pi$
$509$	1.99736	0.0885313	0.0442656	+	0.999020i	$0.485905\pi$
$510$	0	0
$511$	33.7389	1.49252
$512$	22.8354	1.00919
$513$	0	0
$514$	14.6576	0.646517
$515$	7.90625	0.348391
$516$	0	0
$517$	−1.95928	−0.0861689
$518$	4.41398	0.193939
$519$	0	0
$520$	6.57363	0.288273
$521$	1.46175	0.0640402	0.0320201	−	0.999487i	$-0.489806\pi$
$521$	1.46175	0.0640402	0.0320201	+	0.999487i	$0.489806\pi$
$522$	0	0
$523$	20.3916	0.891665	0.445832	−	0.895117i	$-0.352908\pi$
$523$	20.3916	0.891665	0.445832	+	0.895117i	$0.352908\pi$
$524$	−22.3872	−0.977988
$525$	0	0
$526$	−1.83927	−0.0801961
$527$	−12.4438	−0.542061
$528$	0	0
$529$	10.7048	0.465428
$530$	−3.34006	−0.145083
$531$	0	0
$532$	−7.97463	−0.345744
$533$	18.5403	0.803069
$534$	0	0
$535$	−3.99552	−0.172742
$536$	3.43874	0.148531
$537$	0	0
$538$	1.59980	0.0689722
$539$	−25.6474	−1.10471
$540$	0	0
$541$	−8.35787	−0.359333	−0.179667	−	0.983728i	$-0.557502\pi$
$541$	−8.35787	−0.359333	−0.179667	+	0.983728i	$0.557502\pi$
$542$	−6.97457	−0.299584
$543$	0	0
$544$	−10.5469	−0.452193
$545$	−6.96613	−0.298396
$546$	0	0
$547$	−33.2771	−1.42282	−0.711412	−	0.702775i	$-0.751944\pi$
$547$	−33.2771	−1.42282	−0.711412	+	0.702775i	$0.751944\pi$
$548$	10.5607	0.451132
$549$	0	0
$550$	4.13971	0.176518
$551$	3.22466	0.137375
$552$	0	0
$553$	59.6177	2.53520
$554$	−13.0154	−0.552971
$555$	0	0
$556$	−31.9769	−1.35612
$557$	31.3565	1.32862	0.664308	−	0.747459i	$-0.268726\pi$
$557$	31.3565	1.32862	0.664308	+	0.747459i	$0.268726\pi$
$558$	0	0
$559$	17.2364	0.729022
$560$	8.66537	0.366179
$561$	0	0
$562$	9.92664	0.418730
$563$	2.22957	0.0939651	0.0469825	−	0.998896i	$-0.485039\pi$
$563$	2.22957	0.0939651	0.0469825	+	0.998896i	$0.485039\pi$
$564$	0	0
$565$	5.71047	0.240241
$566$	−14.2007	−0.596900
$567$	0	0
$568$	21.6504	0.908432
$569$	2.88177	0.120810	0.0604049	−	0.998174i	$-0.480761\pi$
$569$	2.88177	0.120810	0.0604049	+	0.998174i	$0.480761\pi$
$570$	0	0
$571$	−18.1157	−0.758117	−0.379059	−	0.925373i	$-0.623752\pi$
$571$	−18.1157	−0.758117	−0.379059	+	0.925373i	$0.623752\pi$
$572$	18.1793	0.760115
$573$	0	0
$574$	−7.48810	−0.312547
$575$	−26.1032	−1.08858
$576$	0	0
$577$	29.1220	1.21236	0.606182	−	0.795326i	$-0.292700\pi$
$577$	29.1220	1.21236	0.606182	+	0.795326i	$0.292700\pi$
$578$	−5.74969	−0.239155
$579$	0	0
$580$	−4.07212	−0.169086
$581$	55.3565	2.29657
$582$	0	0
$583$	−19.6203	−0.812589
$584$	13.3678	0.553164
$585$	0	0
$586$	1.72609	0.0713042
$587$	22.0669	0.910800	0.455400	−	0.890287i	$-0.349496\pi$
$587$	22.0669	0.910800	0.455400	+	0.890287i	$0.349496\pi$
$588$	0	0
$589$	−5.70086	−0.234900
$590$	−2.66798	−0.109839
$591$	0	0
$592$	−5.70805	−0.234600
$593$	−8.27458	−0.339796	−0.169898	−	0.985462i	$-0.554344\pi$
$593$	−8.27458	−0.339796	−0.169898	+	0.985462i	$0.554344\pi$
$594$	0	0
$595$	−6.94424	−0.284686
$596$	−28.2716	−1.15805
$597$	0	0
$598$	14.2277	0.581815
$599$	26.8802	1.09829	0.549147	−	0.835726i	$-0.314953\pi$
$599$	26.8802	1.09829	0.549147	+	0.835726i	$0.314953\pi$
$600$	0	0
$601$	17.1116	0.697997	0.348998	−	0.937123i	$-0.386522\pi$
$601$	17.1116	0.697997	0.348998	+	0.937123i	$0.386522\pi$
$602$	−6.96148	−0.283729
$603$	0	0
$604$	18.7097	0.761287
$605$	5.08285	0.206647
$606$	0	0
$607$	9.36332	0.380045	0.190023	−	0.981780i	$-0.439144\pi$
$607$	9.36332	0.380045	0.190023	+	0.981780i	$0.439144\pi$
$608$	−4.83181	−0.195956
$609$	0	0
$610$	−3.08274	−0.124816
$611$	5.21511	0.210981
$612$	0	0
$613$	16.4011	0.662432	0.331216	−	0.943555i	$-0.392541\pi$
$613$	16.4011	0.662432	0.331216	+	0.943555i	$0.392541\pi$
$614$	−11.6749	−0.471162
$615$	0	0
$616$	−15.5959	−0.628377
$617$	−44.5400	−1.79311	−0.896556	−	0.442930i	$-0.853939\pi$
$617$	−44.5400	−1.79311	−0.896556	+	0.442930i	$0.853939\pi$
$618$	0	0
$619$	−9.66339	−0.388404	−0.194202	−	0.980962i	$-0.562212\pi$
$619$	−9.66339	−0.388404	−0.194202	+	0.980962i	$0.562212\pi$
$620$	7.19909	0.289122
$621$	0	0
$622$	−10.8989	−0.437008
$623$	−70.5053	−2.82474
$624$	0	0
$625$	17.6971	0.707885
$626$	4.93868	0.197389
$627$	0	0
$628$	31.0809	1.24026
$629$	4.57431	0.182390
$630$	0	0
$631$	27.4290	1.09193	0.545966	−	0.837807i	$-0.316163\pi$
$631$	27.4290	1.09193	0.545966	+	0.837807i	$0.316163\pi$
$632$	23.6214	0.939608
$633$	0	0
$634$	−5.76619	−0.229005
$635$	7.57351	0.300546
$636$	0	0
$637$	68.2669	2.70483
$638$	2.96897	0.117542
$639$	0	0
$640$	7.91863	0.313011
$641$	−15.0405	−0.594064	−0.297032	−	0.954868i	$-0.595997\pi$
$641$	−15.0405	−0.594064	−0.297032	+	0.954868i	$0.595997\pi$
$642$	0	0
$643$	−11.8167	−0.466006	−0.233003	−	0.972476i	$-0.574855\pi$
$643$	−11.8167	−0.466006	−0.233003	+	0.972476i	$0.574855\pi$
$644$	46.2974	1.82437
$645$	0	0
$646$	1.02574	0.0403574
$647$	−11.6027	−0.456150	−0.228075	−	0.973644i	$-0.573243\pi$
$647$	−11.6027	−0.456150	−0.228075	+	0.973644i	$0.573243\pi$
$648$	0	0
$649$	−15.6723	−0.615193
$650$	−11.0189	−0.432196
$651$	0	0
$652$	−28.8489	−1.12981
$653$	3.68993	0.144398	0.0721991	−	0.997390i	$-0.476998\pi$
$653$	3.68993	0.144398	0.0721991	+	0.997390i	$0.476998\pi$
$654$	0	0
$655$	−8.93100	−0.348963
$656$	9.68342	0.378074
$657$	0	0
$658$	−2.10629	−0.0821118
$659$	1.32549	0.0516339	0.0258170	−	0.999667i	$-0.491781\pi$
$659$	1.32549	0.0516339	0.0258170	+	0.999667i	$0.491781\pi$
$660$	0	0
$661$	3.25095	0.126447	0.0632236	−	0.997999i	$-0.479862\pi$
$661$	3.25095	0.126447	0.0632236	+	0.997999i	$0.479862\pi$
$662$	−0.204108	−0.00793290
$663$	0	0
$664$	21.9330	0.851167
$665$	−3.18135	−0.123367
$666$	0	0
$667$	−18.7210	−0.724881
$668$	−11.7185	−0.453403
$669$	0	0
$670$	0.645835	0.0249508
$671$	−18.1087	−0.699079
$672$	0	0
$673$	49.4867	1.90757	0.953787	−	0.300484i	$-0.0971481\pi$
$673$	49.4867	1.90757	0.953787	+	0.300484i	$0.0971481\pi$
$674$	3.61051	0.139072
$675$	0	0
$676$	−25.2596	−0.971521
$677$	20.9648	0.805744	0.402872	−	0.915256i	$-0.368012\pi$
$677$	20.9648	0.805744	0.402872	+	0.915256i	$0.368012\pi$
$678$	0	0
$679$	−2.57082	−0.0986588
$680$	−2.75141	−0.105512
$681$	0	0
$682$	−5.24882	−0.200988
$683$	−39.8494	−1.52479	−0.762397	−	0.647109i	$-0.775978\pi$
$683$	−39.8494	−1.52479	−0.762397	+	0.647109i	$0.775978\pi$
$684$	0	0
$685$	4.21303	0.160972
$686$	−12.8278	−0.489767
$687$	0	0
$688$	9.00242	0.343214
$689$	52.2243	1.98959
$690$	0	0
$691$	−16.5947	−0.631292	−0.315646	−	0.948877i	$-0.602221\pi$
$691$	−16.5947	−0.631292	−0.315646	+	0.948877i	$0.602221\pi$
$692$	−19.2399	−0.731392
$693$	0	0
$694$	5.80131	0.220215
$695$	−12.7567	−0.483888
$696$	0	0
$697$	−7.76008	−0.293934
$698$	1.94781	0.0737259
$699$	0	0
$700$	−35.8557	−1.35522
$701$	−17.6615	−0.667067	−0.333534	−	0.942738i	$-0.608241\pi$
$701$	−17.6615	−0.667067	−0.333534	+	0.942738i	$0.608241\pi$
$702$	0	0
$703$	2.09562	0.0790377
$704$	6.22469	0.234602
$705$	0	0
$706$	−6.49981	−0.244623
$707$	−4.55448	−0.171289
$708$	0	0
$709$	13.2910	0.499155	0.249577	−	0.968355i	$-0.419708\pi$
$709$	13.2910	0.499155	0.249577	+	0.968355i	$0.419708\pi$
$710$	4.06620	0.152602
$711$	0	0
$712$	−27.9352	−1.04692
$713$	33.0968	1.23949
$714$	0	0
$715$	7.25234	0.271222
$716$	39.4110	1.47286
$717$	0	0
$718$	−1.71346	−0.0639457
$719$	−46.3193	−1.72742	−0.863708	−	0.503992i	$-0.831864\pi$
$719$	−46.3193	−1.72742	−0.863708	+	0.503992i	$0.831864\pi$
$720$	0	0
$721$	49.9279	1.85941
$722$	0.469922	0.0174887
$723$	0	0
$724$	−11.9597	−0.444478
$725$	14.4988	0.538471
$726$	0	0
$727$	−23.6805	−0.878262	−0.439131	−	0.898423i	$-0.644714\pi$
$727$	−23.6805	−0.878262	−0.439131	+	0.898423i	$0.644714\pi$
$728$	41.5125	1.53855
$729$	0	0
$730$	2.51063	0.0929226
$731$	−7.21434	−0.266832
$732$	0	0
$733$	51.4763	1.90132	0.950661	−	0.310233i	$-0.100407\pi$
$733$	51.4763	1.90132	0.950661	+	0.310233i	$0.100407\pi$
$734$	−2.56913	−0.0948284
$735$	0	0
$736$	28.0515	1.03399
$737$	3.79378	0.139746
$738$	0	0
$739$	−15.2674	−0.561619	−0.280809	−	0.959764i	$-0.590603\pi$
$739$	−15.2674	−0.561619	−0.280809	+	0.959764i	$0.590603\pi$
$740$	−2.64636	−0.0972823
$741$	0	0
$742$	−21.0925	−0.774329
$743$	10.8460	0.397902	0.198951	−	0.980009i	$-0.436247\pi$
$743$	10.8460	0.397902	0.198951	+	0.980009i	$0.436247\pi$
$744$	0	0
$745$	−11.2785	−0.413212
$746$	11.3582	0.415854
$747$	0	0
$748$	−7.60899	−0.278212
$749$	−25.2317	−0.921947
$750$	0	0
$751$	−9.27745	−0.338539	−0.169269	−	0.985570i	$-0.554141\pi$
$751$	−9.27745	−0.338539	−0.169269	+	0.985570i	$0.554141\pi$
$752$	2.72380	0.0993269
$753$	0	0
$754$	−7.90265	−0.287798
$755$	7.46394	0.271640
$756$	0	0
$757$	−48.8267	−1.77464	−0.887318	−	0.461159i	$-0.847434\pi$
$757$	−48.8267	−1.77464	−0.887318	+	0.461159i	$0.847434\pi$
$758$	8.47494	0.307824
$759$	0	0
$760$	−1.26050	−0.0457230
$761$	−38.8583	−1.40861	−0.704306	−	0.709896i	$-0.748742\pi$
$761$	−38.8583	−1.40861	−0.704306	+	0.709896i	$0.748742\pi$
$762$	0	0
$763$	−43.9911	−1.59259
$764$	45.4002	1.64252
$765$	0	0
$766$	0.625427	0.0225976
$767$	41.7159	1.50627
$768$	0	0
$769$	−49.1463	−1.77226	−0.886130	−	0.463437i	$-0.846616\pi$
$769$	−49.1463	−1.77226	−0.886130	+	0.463437i	$0.846616\pi$
$770$	−2.92909	−0.105557
$771$	0	0
$772$	35.2313	1.26800
$773$	43.5775	1.56737	0.783686	−	0.621157i	$-0.213337\pi$
$773$	43.5775	1.56737	0.783686	+	0.621157i	$0.213337\pi$
$774$	0	0
$775$	−25.6323	−0.920740
$776$	−1.01859	−0.0365654
$777$	0	0
$778$	−7.44460	−0.266902
$779$	−3.55511	−0.127375
$780$	0	0
$781$	23.8858	0.854701
$782$	−5.95505	−0.212952
$783$	0	0
$784$	35.6552	1.27340
$785$	12.3992	0.442547
$786$	0	0
$787$	19.2967	0.687854	0.343927	−	0.938996i	$-0.388243\pi$
$787$	19.2967	0.687854	0.343927	+	0.938996i	$0.388243\pi$
$788$	3.79912	0.135338
$789$	0	0
$790$	4.43637	0.157839
$791$	36.0616	1.28220
$792$	0	0
$793$	48.2009	1.71166
$794$	0.373501	0.0132551
$795$	0	0
$796$	−24.9086	−0.882861
$797$	−39.5008	−1.39919	−0.699595	−	0.714540i	$-0.746636\pi$
$797$	−39.5008	−1.39919	−0.699595	+	0.714540i	$0.746636\pi$
$798$	0	0
$799$	−2.18280	−0.0772218
$800$	−21.7249	−0.768091
$801$	0	0
$802$	−11.4674	−0.404927
$803$	14.7480	0.520446
$804$	0	0
$805$	18.4696	0.650968
$806$	13.9711	0.492109
$807$	0	0
$808$	−1.80455	−0.0634838
$809$	−43.3647	−1.52462	−0.762310	−	0.647212i	$-0.775935\pi$
$809$	−43.3647	−1.52462	−0.762310	+	0.647212i	$0.775935\pi$
$810$	0	0
$811$	−5.63347	−0.197818	−0.0989090	−	0.995096i	$-0.531535\pi$
$811$	−5.63347	−0.197818	−0.0989090	+	0.995096i	$0.531535\pi$
$812$	−25.7154	−0.902435
$813$	0	0
$814$	1.92945	0.0676272
$815$	−11.5088	−0.403136
$816$	0	0
$817$	−3.30509	−0.115630
$818$	−10.2591	−0.358700
$819$	0	0
$820$	4.48942	0.156777
$821$	−24.1990	−0.844552	−0.422276	−	0.906467i	$-0.638769\pi$
$821$	−24.1990	−0.844552	−0.422276	+	0.906467i	$0.638769\pi$
$822$	0	0
$823$	4.21460	0.146912	0.0734558	−	0.997298i	$-0.476597\pi$
$823$	4.21460	0.146912	0.0734558	+	0.997298i	$0.476597\pi$
$824$	19.7822	0.689144
$825$	0	0
$826$	−16.8483	−0.586228
$827$	−42.9502	−1.49352	−0.746762	−	0.665092i	$-0.768392\pi$
$827$	−42.9502	−1.49352	−0.746762	+	0.665092i	$0.768392\pi$
$828$	0	0
$829$	−0.684238	−0.0237646	−0.0118823	−	0.999929i	$-0.503782\pi$
$829$	−0.684238	−0.0237646	−0.0118823	+	0.999929i	$0.503782\pi$
$830$	4.11928	0.142982
$831$	0	0
$832$	−16.5686	−0.574412
$833$	−28.5733	−0.990006
$834$	0	0
$835$	−4.67491	−0.161782
$836$	−3.48589	−0.120562
$837$	0	0
$838$	15.8714	0.548269
$839$	35.9931	1.24262	0.621309	−	0.783565i	$-0.286601\pi$
$839$	35.9931	1.24262	0.621309	+	0.783565i	$0.286601\pi$
$840$	0	0
$841$	−18.6016	−0.641434
$842$	−11.9671	−0.412415
$843$	0	0
$844$	29.2118	1.00551
$845$	−10.0769	−0.346656
$846$	0	0
$847$	32.0982	1.10291
$848$	27.2763	0.936671
$849$	0	0
$850$	4.61198	0.158189
$851$	−12.1663	−0.417055
$852$	0	0
$853$	−19.1053	−0.654154	−0.327077	−	0.944998i	$-0.606064\pi$
$853$	−19.1053	−0.654154	−0.327077	+	0.944998i	$0.606064\pi$
$854$	−19.4675	−0.666164
$855$	0	0
$856$	−9.99717	−0.341696
$857$	35.9259	1.22720	0.613602	−	0.789616i	$-0.289720\pi$
$857$	35.9259	1.22720	0.613602	+	0.789616i	$0.289720\pi$
$858$	0	0
$859$	−17.9356	−0.611954	−0.305977	−	0.952039i	$-0.598983\pi$
$859$	−17.9356	−0.611954	−0.305977	+	0.952039i	$0.598983\pi$
$860$	4.17369	0.142322
$861$	0	0
$862$	−11.0499	−0.376361
$863$	−54.8002	−1.86542	−0.932710	−	0.360627i	$-0.882563\pi$
$863$	−54.8002	−1.86542	−0.932710	+	0.360627i	$0.882563\pi$
$864$	0	0
$865$	−7.67545	−0.260973
$866$	−10.9873	−0.373364
$867$	0	0
$868$	45.4622	1.54309
$869$	26.0602	0.884033
$870$	0	0
$871$	−10.0981	−0.342161
$872$	−17.4299	−0.590251
$873$	0	0
$874$	−2.72817	−0.0922819
$875$	−30.2108	−1.02131
$876$	0	0
$877$	−12.8858	−0.435121	−0.217561	−	0.976047i	$-0.569810\pi$
$877$	−12.8858	−0.435121	−0.217561	+	0.976047i	$0.569810\pi$
$878$	1.73239	0.0584652
$879$	0	0
$880$	3.78783	0.127688
$881$	7.83989	0.264133	0.132066	−	0.991241i	$-0.457839\pi$
$881$	7.83989	0.264133	0.132066	+	0.991241i	$0.457839\pi$
$882$	0	0
$883$	−50.6172	−1.70341	−0.851703	−	0.524025i	$-0.824430\pi$
$883$	−50.6172	−1.70341	−0.851703	+	0.524025i	$0.824430\pi$
$884$	20.2533	0.681191
$885$	0	0
$886$	13.8553	0.465477
$887$	23.9589	0.804460	0.402230	−	0.915539i	$-0.368235\pi$
$887$	23.9589	0.804460	0.402230	+	0.915539i	$0.368235\pi$
$888$	0	0
$889$	47.8267	1.60406
$890$	−5.24656	−0.175865
$891$	0	0
$892$	29.0030	0.971094
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	15.7224	0.525541
$896$	50.0061	1.67059
$897$	0	0
$898$	15.2528	0.508991
$899$	−18.3833	−0.613117
$900$	0	0
$901$	−21.8586	−0.728216
$902$	−3.27322	−0.108986
$903$	0	0
$904$	14.2881	0.475216
$905$	−4.77112	−0.158597
$906$	0	0
$907$	−10.7709	−0.357642	−0.178821	−	0.983882i	$-0.557228\pi$
$907$	−10.7709	−0.357642	−0.178821	+	0.983882i	$0.557228\pi$
$908$	−28.6178	−0.949714
$909$	0	0
$910$	7.79652	0.258452
$911$	−5.48993	−0.181889	−0.0909447	−	0.995856i	$-0.528989\pi$
$911$	−5.48993	−0.181889	−0.0909447	+	0.995856i	$0.528989\pi$
$912$	0	0
$913$	24.1976	0.800823
$914$	4.65216	0.153880
$915$	0	0
$916$	45.8540	1.51506
$917$	−56.3992	−1.86247
$918$	0	0
$919$	−36.1357	−1.19201	−0.596004	−	0.802982i	$-0.703246\pi$
$919$	−36.1357	−1.19201	−0.596004	+	0.802982i	$0.703246\pi$
$920$	7.31793	0.241265
$921$	0	0
$922$	−3.79644	−0.125029
$923$	−63.5780	−2.09270
$924$	0	0
$925$	9.42237	0.309806
$926$	16.6416	0.546879
$927$	0	0
$928$	−15.5809	−0.511469
$929$	3.94818	0.129535	0.0647677	−	0.997900i	$-0.479369\pi$
$929$	3.94818	0.129535	0.0647677	+	0.997900i	$0.479369\pi$
$930$	0	0
$931$	−13.0902	−0.429015
$932$	34.0936	1.11677
$933$	0	0
$934$	1.42531	0.0466375
$935$	−3.03549	−0.0992710
$936$	0	0
$937$	38.9919	1.27381	0.636905	−	0.770942i	$-0.280214\pi$
$937$	38.9919	1.27381	0.636905	+	0.770942i	$0.280214\pi$
$938$	4.07845	0.133166
$939$	0	0
$940$	1.26281	0.0411883
$941$	46.9294	1.52985	0.764927	−	0.644116i	$-0.222775\pi$
$941$	46.9294	1.52985	0.764927	+	0.644116i	$0.222775\pi$
$942$	0	0
$943$	20.6395	0.672115
$944$	21.7878	0.709133
$945$	0	0
$946$	−3.04302	−0.0989372
$947$	−42.8452	−1.39228	−0.696140	−	0.717906i	$-0.745101\pi$
$947$	−42.8452	−1.39228	−0.696140	+	0.717906i	$0.745101\pi$
$948$	0	0
$949$	−39.2556	−1.27429
$950$	2.11287	0.0685507
$951$	0	0
$952$	−17.3751	−0.563132
$953$	−26.9769	−0.873869	−0.436934	−	0.899493i	$-0.643936\pi$
$953$	−26.9769	−0.873869	−0.436934	+	0.899493i	$0.643936\pi$
$954$	0	0
$955$	18.1117	0.586080
$956$	−18.7414	−0.606141
$957$	0	0
$958$	−6.85789	−0.221568
$959$	26.6053	0.859129
$960$	0	0
$961$	1.49975	0.0483791
$962$	−5.13572	−0.165582
$963$	0	0
$964$	−0.332698	−0.0107155
$965$	14.0549	0.452445
$966$	0	0
$967$	21.7193	0.698445	0.349222	−	0.937040i	$-0.386446\pi$
$967$	21.7193	0.698445	0.349222	+	0.937040i	$0.386446\pi$
$968$	12.7178	0.408764
$969$	0	0
$970$	−0.191304	−0.00614239
$971$	48.9419	1.57062	0.785310	−	0.619103i	$-0.212504\pi$
$971$	48.9419	1.57062	0.785310	+	0.619103i	$0.212504\pi$
$972$	0	0
$973$	−80.5584	−2.58258
$974$	−3.87842	−0.124273
$975$	0	0
$976$	25.1749	0.805828
$977$	24.5468	0.785322	0.392661	−	0.919683i	$-0.371555\pi$
$977$	24.5468	0.785322	0.392661	+	0.919683i	$0.371555\pi$
$978$	0	0
$979$	−30.8195	−0.984995
$980$	16.5304	0.528046
$981$	0	0
$982$	−8.04986	−0.256881
$983$	−33.7068	−1.07508	−0.537539	−	0.843239i	$-0.680646\pi$
$983$	−33.7068	−1.07508	−0.537539	+	0.843239i	$0.680646\pi$
$984$	0	0
$985$	1.51560	0.0482910
$986$	3.30767	0.105338
$987$	0	0
$988$	9.27858	0.295191
$989$	19.1880	0.610143
$990$	0	0
$991$	−47.0026	−1.49309	−0.746544	−	0.665336i	$-0.768288\pi$
$991$	−47.0026	−1.49309	−0.746544	+	0.665336i	$0.768288\pi$
$992$	27.5454	0.874569
$993$	0	0
$994$	25.6780	0.814458
$995$	−9.93688	−0.315020
$996$	0	0
$997$	54.1301	1.71432	0.857159	−	0.515052i	$-0.172227\pi$
$997$	54.1301	1.71432	0.857159	+	0.515052i	$0.172227\pi$
$998$	−4.83775	−0.153136
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.n.1.8		16
3.2	odd	2		893.2.a.b.1.9	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.b.1.9	✓	16	3.2	odd	2
8037.2.a.n.1.8		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.469922 q^{2} -1.77917 q^{4} -0.709772 q^{5} -4.48221 q^{7} -1.77592 q^{8} +O(q^{10})\)	\(q+0.469922 q^{2} -1.77917 q^{4} -0.709772 q^{5} -4.48221 q^{7} -1.77592 q^{8} -0.333538 q^{10} -1.95928 q^{11} +5.21511 q^{13} -2.10629 q^{14} +2.72380 q^{16} -2.18280 q^{17} -1.00000 q^{19} +1.26281 q^{20} -0.920708 q^{22} +5.80559 q^{23} -4.49622 q^{25} +2.45069 q^{26} +7.97463 q^{28} -3.22466 q^{29} +5.70086 q^{31} +4.83181 q^{32} -1.02574 q^{34} +3.18135 q^{35} -2.09562 q^{37} -0.469922 q^{38} +1.26050 q^{40} +3.55511 q^{41} +3.30509 q^{43} +3.48589 q^{44} +2.72817 q^{46} +1.00000 q^{47} +13.0902 q^{49} -2.11287 q^{50} -9.27858 q^{52} +10.0140 q^{53} +1.39064 q^{55} +7.96004 q^{56} -1.51534 q^{58} +7.99904 q^{59} +9.24255 q^{61} +2.67896 q^{62} -3.17703 q^{64} -3.70154 q^{65} -1.93632 q^{67} +3.88357 q^{68} +1.49499 q^{70} -12.1911 q^{71} -7.52728 q^{73} -0.984777 q^{74} +1.77917 q^{76} +8.78189 q^{77} -13.3009 q^{79} -1.93328 q^{80} +1.67062 q^{82} -12.3503 q^{83} +1.54929 q^{85} +1.55313 q^{86} +3.47951 q^{88} +15.7300 q^{89} -23.3752 q^{91} -10.3291 q^{92} +0.469922 q^{94} +0.709772 q^{95} +0.573560 q^{97} +6.15139 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) \) 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8	\( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8} - 15 q^{10} - 19 q^{13} + 6 q^{14} + 10 q^{16} + 8 q^{17} - 16 q^{19} + 11 q^{20} - 12 q^{22} + 5 q^{23} - 3 q^{25} - 9 q^{26} - 17 q^{28} + 2 q^{29} - 18 q^{31} - 3 q^{32} - 14 q^{34} + 11 q^{35} - 24 q^{37} - 4 q^{38} - 50 q^{40} + 6 q^{41} - 34 q^{43} + 4 q^{44} - 3 q^{46} + 16 q^{47} + 5 q^{49} - 26 q^{50} - 44 q^{52} + 23 q^{53} - 48 q^{55} + 3 q^{56} - 26 q^{58} + 32 q^{59} - 16 q^{61} - 32 q^{62} + 7 q^{64} + 18 q^{65} - 67 q^{67} + 19 q^{68} + 24 q^{70} - 19 q^{71} - 2 q^{73} + 29 q^{74} - 10 q^{76} - 14 q^{77} - 27 q^{79} - 15 q^{80} - 56 q^{82} + 17 q^{83} + 15 q^{85} + q^{86} - 13 q^{88} - 20 q^{89} - 42 q^{91} - 45 q^{92} + 4 q^{94} - q^{95} - 50 q^{97} - 13 q^{98}+O(q^{100}) \) 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8 - 15 * q^10 - 19 * q^13 + 6 * q^14 + 10 * q^16 + 8 * q^17 - 16 * q^19 + 11 * q^20 - 12 * q^22 + 5 * q^23 - 3 * q^25 - 9 * q^26 - 17 * q^28 + 2 * q^29 - 18 * q^31 - 3 * q^32 - 14 * q^34 + 11 * q^35 - 24 * q^37 - 4 * q^38 - 50 * q^40 + 6 * q^41 - 34 * q^43 + 4 * q^44 - 3 * q^46 + 16 * q^47 + 5 * q^49 - 26 * q^50 - 44 * q^52 + 23 * q^53 - 48 * q^55 + 3 * q^56 - 26 * q^58 + 32 * q^59 - 16 * q^61 - 32 * q^62 + 7 * q^64 + 18 * q^65 - 67 * q^67 + 19 * q^68 + 24 * q^70 - 19 * q^71 - 2 * q^73 + 29 * q^74 - 10 * q^76 - 14 * q^77 - 27 * q^79 - 15 * q^80 - 56 * q^82 + 17 * q^83 + 15 * q^85 + q^86 - 13 * q^88 - 20 * q^89 - 42 * q^91 - 45 * q^92 + 4 * q^94 - q^95 - 50 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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