LMFDB - Embedded newform 6039.2.a.k.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$-2.21976$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21976 q^{2} +2.92734 q^{4} +2.17512 q^{5} +5.05675 q^{7} +2.05847 q^{8} +O(q^{10})$	\(q+2.21976 q^{2} +2.92734 q^{4} +2.17512 q^{5} +5.05675 q^{7} +2.05847 q^{8} +4.82825 q^{10} -1.00000 q^{11} -3.20178 q^{13} +11.2248 q^{14} -1.28537 q^{16} +5.61523 q^{17} +1.25758 q^{19} +6.36732 q^{20} -2.21976 q^{22} -0.955523 q^{23} -0.268843 q^{25} -7.10719 q^{26} +14.8028 q^{28} -2.64307 q^{29} +5.28519 q^{31} -6.97015 q^{32} +12.4645 q^{34} +10.9990 q^{35} +7.64117 q^{37} +2.79153 q^{38} +4.47742 q^{40} +3.21692 q^{41} -8.70811 q^{43} -2.92734 q^{44} -2.12103 q^{46} +6.21668 q^{47} +18.5707 q^{49} -0.596767 q^{50} -9.37269 q^{52} -2.85705 q^{53} -2.17512 q^{55} +10.4091 q^{56} -5.86699 q^{58} +6.34839 q^{59} -1.00000 q^{61} +11.7319 q^{62} -12.9013 q^{64} -6.96426 q^{65} -14.1507 q^{67} +16.4377 q^{68} +24.4152 q^{70} -10.5413 q^{71} +4.23786 q^{73} +16.9616 q^{74} +3.68136 q^{76} -5.05675 q^{77} +17.1827 q^{79} -2.79584 q^{80} +7.14080 q^{82} +3.89639 q^{83} +12.2138 q^{85} -19.3299 q^{86} -2.05847 q^{88} -9.09472 q^{89} -16.1906 q^{91} -2.79714 q^{92} +13.7995 q^{94} +2.73539 q^{95} -6.63181 q^{97} +41.2225 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * 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 $

$2$	2.21976	1.56961	0.784804	−	0.619744i	$-0.212764\pi$
$2$	2.21976	1.56961	0.784804	+	0.619744i	$0.212764\pi$
$3$	0	0
$3$	0	0
$4$	2.92734	1.46367
$5$	2.17512	0.972744	0.486372	−	0.873752i	$-0.338320\pi$
$5$	2.17512	0.972744	0.486372	+	0.873752i	$0.338320\pi$
$6$	0	0
$7$	5.05675	1.91127	0.955635	−	0.294552i	$-0.0951706\pi$
$7$	5.05675	1.91127	0.955635	+	0.294552i	$0.0951706\pi$
$8$	2.05847	0.727778
$9$	0	0
$10$	4.82825	1.52683
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.20178	−0.888014	−0.444007	−	0.896023i	$-0.646443\pi$
$13$	−3.20178	−0.888014	−0.444007	+	0.896023i	$0.646443\pi$
$14$	11.2248	2.99995
$15$	0	0
$16$	−1.28537	−0.321343
$17$	5.61523	1.36189	0.680947	−	0.732333i	$-0.261568\pi$
$17$	5.61523	1.36189	0.680947	+	0.732333i	$0.261568\pi$
$18$	0	0
$19$	1.25758	0.288509	0.144254	−	0.989541i	$-0.453922\pi$
$19$	1.25758	0.288509	0.144254	+	0.989541i	$0.453922\pi$
$20$	6.36732	1.42378
$21$	0	0
$22$	−2.21976	−0.473255
$23$	−0.955523	−0.199240	−0.0996201	−	0.995026i	$-0.531763\pi$
$23$	−0.955523	−0.199240	−0.0996201	+	0.995026i	$0.531763\pi$
$24$	0	0
$25$	−0.268843	−0.0537686
$26$	−7.10719	−1.39383
$27$	0	0
$28$	14.8028	2.79747
$29$	−2.64307	−0.490806	−0.245403	−	0.969421i	$-0.578920\pi$
$29$	−2.64307	−0.490806	−0.245403	+	0.969421i	$0.578920\pi$
$30$	0	0
$31$	5.28519	0.949249	0.474624	−	0.880188i	$-0.342584\pi$
$31$	5.28519	0.949249	0.474624	+	0.880188i	$0.342584\pi$
$32$	−6.97015	−1.23216
$33$	0	0
$34$	12.4645	2.13764
$35$	10.9990	1.85918
$36$	0	0
$37$	7.64117	1.25620	0.628100	−	0.778132i	$-0.283833\pi$
$37$	7.64117	1.25620	0.628100	+	0.778132i	$0.283833\pi$
$38$	2.79153	0.452846
$39$	0	0
$40$	4.47742	0.707942
$41$	3.21692	0.502399	0.251200	−	0.967935i	$-0.419175\pi$
$41$	3.21692	0.502399	0.251200	+	0.967935i	$0.419175\pi$
$42$	0	0
$43$	−8.70811	−1.32797	−0.663987	−	0.747744i	$-0.731137\pi$
$43$	−8.70811	−1.32797	−0.663987	+	0.747744i	$0.731137\pi$
$44$	−2.92734	−0.441313
$45$	0	0
$46$	−2.12103	−0.312729
$47$	6.21668	0.906796	0.453398	−	0.891308i	$-0.350212\pi$
$47$	6.21668	0.906796	0.453398	+	0.891308i	$0.350212\pi$
$48$	0	0
$49$	18.5707	2.65296
$50$	−0.596767	−0.0843957
$51$	0	0
$52$	−9.37269	−1.29976
$53$	−2.85705	−0.392445	−0.196223	−	0.980559i	$-0.562868\pi$
$53$	−2.85705	−0.392445	−0.196223	+	0.980559i	$0.562868\pi$
$54$	0	0
$55$	−2.17512	−0.293293
$56$	10.4091	1.39098
$57$	0	0
$58$	−5.86699	−0.770373
$59$	6.34839	0.826491	0.413245	−	0.910620i	$-0.364395\pi$
$59$	6.34839	0.826491	0.413245	+	0.910620i	$0.364395\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	11.7319	1.48995
$63$	0	0
$64$	−12.9013	−1.61267
$65$	−6.96426	−0.863811
$66$	0	0
$67$	−14.1507	−1.72878	−0.864391	−	0.502821i	$-0.832296\pi$
$67$	−14.1507	−1.72878	−0.864391	+	0.502821i	$0.832296\pi$
$68$	16.4377	1.99336
$69$	0	0
$70$	24.4152	2.91818
$71$	−10.5413	−1.25102	−0.625511	−	0.780215i	$-0.715110\pi$
$71$	−10.5413	−1.25102	−0.625511	+	0.780215i	$0.715110\pi$
$72$	0	0
$73$	4.23786	0.496004	0.248002	−	0.968760i	$-0.420226\pi$
$73$	4.23786	0.496004	0.248002	+	0.968760i	$0.420226\pi$
$74$	16.9616	1.97174
$75$	0	0
$76$	3.68136	0.422281
$77$	−5.05675	−0.576270
$78$	0	0
$79$	17.1827	1.93320	0.966601	−	0.256288i	$-0.0824994\pi$
$79$	17.1827	1.93320	0.966601	+	0.256288i	$0.0824994\pi$
$80$	−2.79584	−0.312584
$81$	0	0
$82$	7.14080	0.788570
$83$	3.89639	0.427685	0.213842	−	0.976868i	$-0.431402\pi$
$83$	3.89639	0.427685	0.213842	+	0.976868i	$0.431402\pi$
$84$	0	0
$85$	12.2138	1.32477
$86$	−19.3299	−2.08440
$87$	0	0
$88$	−2.05847	−0.219433
$89$	−9.09472	−0.964039	−0.482019	−	0.876161i	$-0.660096\pi$
$89$	−9.09472	−0.964039	−0.482019	+	0.876161i	$0.660096\pi$
$90$	0	0
$91$	−16.1906	−1.69724
$92$	−2.79714	−0.291622
$93$	0	0
$94$	13.7995	1.42331
$95$	2.73539	0.280645
$96$	0	0
$97$	−6.63181	−0.673359	−0.336679	−	0.941619i	$-0.609304\pi$
$97$	−6.63181	−0.673359	−0.336679	+	0.941619i	$0.609304\pi$
$98$	41.2225	4.16410
$99$	0	0
$100$	−0.786995	−0.0786995
$101$	4.51837	0.449594	0.224797	−	0.974406i	$-0.427828\pi$
$101$	4.51837	0.449594	0.224797	+	0.974406i	$0.427828\pi$
$102$	0	0
$103$	16.3610	1.61210	0.806051	−	0.591846i	$-0.201601\pi$
$103$	16.3610	1.61210	0.806051	+	0.591846i	$0.201601\pi$
$104$	−6.59076	−0.646277
$105$	0	0
$106$	−6.34196	−0.615985
$107$	−8.60049	−0.831441	−0.415721	−	0.909492i	$-0.636471\pi$
$107$	−8.60049	−0.831441	−0.415721	+	0.909492i	$0.636471\pi$
$108$	0	0
$109$	18.4620	1.76834	0.884170	−	0.467165i	$-0.154724\pi$
$109$	18.4620	1.76834	0.884170	+	0.467165i	$0.154724\pi$
$110$	−4.82825	−0.460356
$111$	0	0
$112$	−6.49980	−0.614173
$113$	−15.5049	−1.45858	−0.729291	−	0.684203i	$-0.760150\pi$
$113$	−15.5049	−1.45858	−0.729291	+	0.684203i	$0.760150\pi$
$114$	0	0
$115$	−2.07838	−0.193810
$116$	−7.73717	−0.718378
$117$	0	0
$118$	14.0919	1.29727
$119$	28.3948	2.60295
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.21976	−0.200968
$123$	0	0
$124$	15.4715	1.38939
$125$	−11.4604	−1.02505
$126$	0	0
$127$	5.45540	0.484089	0.242044	−	0.970265i	$-0.422182\pi$
$127$	5.45540	0.484089	0.242044	+	0.970265i	$0.422182\pi$
$128$	−14.6975	−1.29909
$129$	0	0
$130$	−15.4590	−1.35584
$131$	−15.8906	−1.38836	−0.694182	−	0.719799i	$-0.744234\pi$
$131$	−15.8906	−1.38836	−0.694182	+	0.719799i	$0.744234\pi$
$132$	0	0
$133$	6.35927	0.551418
$134$	−31.4111	−2.71351
$135$	0	0
$136$	11.5588	0.991156
$137$	2.94993	0.252029	0.126015	−	0.992028i	$-0.459781\pi$
$137$	2.94993	0.252029	0.126015	+	0.992028i	$0.459781\pi$
$138$	0	0
$139$	0.637252	0.0540510	0.0270255	−	0.999635i	$-0.491396\pi$
$139$	0.637252	0.0540510	0.0270255	+	0.999635i	$0.491396\pi$
$140$	32.1979	2.72122
$141$	0	0
$142$	−23.3992	−1.96361
$143$	3.20178	0.267746
$144$	0	0
$145$	−5.74901	−0.477429
$146$	9.40703	0.778531
$147$	0	0
$148$	22.3683	1.83866
$149$	−13.8076	−1.13116	−0.565581	−	0.824693i	$-0.691348\pi$
$149$	−13.8076	−1.13116	−0.565581	+	0.824693i	$0.691348\pi$
$150$	0	0
$151$	−20.4624	−1.66520	−0.832602	−	0.553872i	$-0.813150\pi$
$151$	−20.4624	−1.66520	−0.832602	+	0.553872i	$0.813150\pi$
$152$	2.58869	0.209970
$153$	0	0
$154$	−11.2248	−0.904518
$155$	11.4959	0.923376
$156$	0	0
$157$	−0.115826	−0.00924395	−0.00462198	−	0.999989i	$-0.501471\pi$
$157$	−0.115826	−0.00924395	−0.00462198	+	0.999989i	$0.501471\pi$
$158$	38.1414	3.03437
$159$	0	0
$160$	−15.1609	−1.19858
$161$	−4.83184	−0.380802
$162$	0	0
$163$	−21.4977	−1.68383	−0.841914	−	0.539612i	$-0.818571\pi$
$163$	−21.4977	−1.68383	−0.841914	+	0.539612i	$0.818571\pi$
$164$	9.41702	0.735346
$165$	0	0
$166$	8.64906	0.671297
$167$	−3.66249	−0.283412	−0.141706	−	0.989909i	$-0.545259\pi$
$167$	−3.66249	−0.283412	−0.141706	+	0.989909i	$0.545259\pi$
$168$	0	0
$169$	−2.74860	−0.211431
$170$	27.1117	2.07938
$171$	0	0
$172$	−25.4916	−1.94371
$173$	−16.7593	−1.27419	−0.637095	−	0.770785i	$-0.719864\pi$
$173$	−16.7593	−1.27419	−0.637095	+	0.770785i	$0.719864\pi$
$174$	0	0
$175$	−1.35947	−0.102766
$176$	1.28537	0.0968885
$177$	0	0
$178$	−20.1881	−1.51316
$179$	−16.8397	−1.25866	−0.629329	−	0.777139i	$-0.716670\pi$
$179$	−16.8397	−1.25866	−0.629329	+	0.777139i	$0.716670\pi$
$180$	0	0
$181$	13.3009	0.988645	0.494323	−	0.869279i	$-0.335416\pi$
$181$	13.3009	0.988645	0.494323	+	0.869279i	$0.335416\pi$
$182$	−35.9392	−2.66399
$183$	0	0
$184$	−1.96691	−0.145003
$185$	16.6205	1.22196
$186$	0	0
$187$	−5.61523	−0.410626
$188$	18.1983	1.32725
$189$	0	0
$190$	6.07191	0.440503
$191$	20.8693	1.51005	0.755024	−	0.655698i	$-0.227625\pi$
$191$	20.8693	1.51005	0.755024	+	0.655698i	$0.227625\pi$
$192$	0	0
$193$	−2.99802	−0.215802	−0.107901	−	0.994162i	$-0.534413\pi$
$193$	−2.99802	−0.215802	−0.107901	+	0.994162i	$0.534413\pi$
$194$	−14.7210	−1.05691
$195$	0	0
$196$	54.3627	3.88305
$197$	−16.0388	−1.14272	−0.571360	−	0.820699i	$-0.693584\pi$
$197$	−16.0388	−1.14272	−0.571360	+	0.820699i	$0.693584\pi$
$198$	0	0
$199$	0.533497	0.0378186	0.0189093	−	0.999821i	$-0.493981\pi$
$199$	0.533497	0.0378186	0.0189093	+	0.999821i	$0.493981\pi$
$200$	−0.553405	−0.0391316
$201$	0	0
$202$	10.0297	0.705687
$203$	−13.3654	−0.938064
$204$	0	0
$205$	6.99720	0.488706
$206$	36.3176	2.53037
$207$	0	0
$208$	4.11548	0.285357
$209$	−1.25758	−0.0869887
$210$	0	0
$211$	−5.67278	−0.390530	−0.195265	−	0.980750i	$-0.562557\pi$
$211$	−5.67278	−0.390530	−0.195265	+	0.980750i	$0.562557\pi$
$212$	−8.36354	−0.574410
$213$	0	0
$214$	−19.0910	−1.30504
$215$	−18.9412	−1.29178
$216$	0	0
$217$	26.7259	1.81427
$218$	40.9813	2.77560
$219$	0	0
$220$	−6.36732	−0.429284
$221$	−17.9787	−1.20938
$222$	0	0
$223$	14.2351	0.953255	0.476627	−	0.879105i	$-0.341859\pi$
$223$	14.2351	0.953255	0.476627	+	0.879105i	$0.341859\pi$
$224$	−35.2463	−2.35499
$225$	0	0
$226$	−34.4173	−2.28940
$227$	3.30880	0.219613	0.109806	−	0.993953i	$-0.464977\pi$
$227$	3.30880	0.219613	0.109806	+	0.993953i	$0.464977\pi$
$228$	0	0
$229$	−3.72809	−0.246359	−0.123180	−	0.992384i	$-0.539309\pi$
$229$	−3.72809	−0.246359	−0.123180	+	0.992384i	$0.539309\pi$
$230$	−4.61350	−0.304205
$231$	0	0
$232$	−5.44068	−0.357198
$233$	−11.3017	−0.740402	−0.370201	−	0.928952i	$-0.620711\pi$
$233$	−11.3017	−0.740402	−0.370201	+	0.928952i	$0.620711\pi$
$234$	0	0
$235$	13.5220	0.882081
$236$	18.5839	1.20971
$237$	0	0
$238$	63.0297	4.08561
$239$	−2.35225	−0.152155	−0.0760773	−	0.997102i	$-0.524240\pi$
$239$	−2.35225	−0.152155	−0.0760773	+	0.997102i	$0.524240\pi$
$240$	0	0
$241$	7.75854	0.499771	0.249886	−	0.968275i	$-0.419607\pi$
$241$	7.75854	0.499771	0.249886	+	0.968275i	$0.419607\pi$
$242$	2.21976	0.142692
$243$	0	0
$244$	−2.92734	−0.187404
$245$	40.3935	2.58065
$246$	0	0
$247$	−4.02650	−0.256200
$248$	10.8794	0.690842
$249$	0	0
$250$	−25.4393	−1.60892
$251$	4.26673	0.269314	0.134657	−	0.990892i	$-0.457007\pi$
$251$	4.26673	0.269314	0.134657	+	0.990892i	$0.457007\pi$
$252$	0	0
$253$	0.955523	0.0600732
$254$	12.1097	0.759829
$255$	0	0
$256$	−6.82239	−0.426399
$257$	−4.14948	−0.258838	−0.129419	−	0.991590i	$-0.541311\pi$
$257$	−4.14948	−0.258838	−0.129419	+	0.991590i	$0.541311\pi$
$258$	0	0
$259$	38.6395	2.40094
$260$	−20.3867	−1.26433
$261$	0	0
$262$	−35.2732	−2.17919
$263$	−11.8060	−0.727990	−0.363995	−	0.931401i	$-0.618588\pi$
$263$	−11.8060	−0.727990	−0.363995	+	0.931401i	$0.618588\pi$
$264$	0	0
$265$	−6.21442	−0.381749
$266$	14.1160	0.865511
$267$	0	0
$268$	−41.4238	−2.53036
$269$	27.5004	1.67673	0.838363	−	0.545112i	$-0.183513\pi$
$269$	27.5004	1.67673	0.838363	+	0.545112i	$0.183513\pi$
$270$	0	0
$271$	19.8308	1.20463	0.602317	−	0.798257i	$-0.294244\pi$
$271$	19.8308	1.20463	0.602317	+	0.798257i	$0.294244\pi$
$272$	−7.21766	−0.437635
$273$	0	0
$274$	6.54814	0.395587
$275$	0.268843	0.0162119
$276$	0	0
$277$	9.78672	0.588027	0.294013	−	0.955801i	$-0.405009\pi$
$277$	9.78672	0.588027	0.294013	+	0.955801i	$0.405009\pi$
$278$	1.41455	0.0848389
$279$	0	0
$280$	22.6412	1.35307
$281$	−7.31594	−0.436432	−0.218216	−	0.975900i	$-0.570024\pi$
$281$	−7.31594	−0.436432	−0.218216	+	0.975900i	$0.570024\pi$
$282$	0	0
$283$	14.1525	0.841279	0.420640	−	0.907228i	$-0.361806\pi$
$283$	14.1525	0.841279	0.420640	+	0.907228i	$0.361806\pi$
$284$	−30.8579	−1.83108
$285$	0	0
$286$	7.10719	0.420257
$287$	16.2672	0.960221
$288$	0	0
$289$	14.5308	0.854755
$290$	−12.7614	−0.749376
$291$	0	0
$292$	12.4056	0.725985
$293$	15.8736	0.927347	0.463673	−	0.886006i	$-0.346531\pi$
$293$	15.8736	0.927347	0.463673	+	0.886006i	$0.346531\pi$
$294$	0	0
$295$	13.8085	0.803964
$296$	15.7291	0.914235
$297$	0	0
$298$	−30.6496	−1.77548
$299$	3.05937	0.176928
$300$	0	0
$301$	−44.0347	−2.53812
$302$	−45.4216	−2.61372
$303$	0	0
$304$	−1.61646	−0.0927103
$305$	−2.17512	−0.124547
$306$	0	0
$307$	−22.3977	−1.27830	−0.639151	−	0.769081i	$-0.720714\pi$
$307$	−22.3977	−1.27830	−0.639151	+	0.769081i	$0.720714\pi$
$308$	−14.8028	−0.843468
$309$	0	0
$310$	25.5182	1.44934
$311$	11.6603	0.661194	0.330597	−	0.943772i	$-0.392750\pi$
$311$	11.6603	0.661194	0.330597	+	0.943772i	$0.392750\pi$
$312$	0	0
$313$	1.19268	0.0674142	0.0337071	−	0.999432i	$-0.489269\pi$
$313$	1.19268	0.0674142	0.0337071	+	0.999432i	$0.489269\pi$
$314$	−0.257107	−0.0145094
$315$	0	0
$316$	50.2995	2.82957
$317$	−13.3231	−0.748300	−0.374150	−	0.927368i	$-0.622065\pi$
$317$	−13.3231	−0.748300	−0.374150	+	0.927368i	$0.622065\pi$
$318$	0	0
$319$	2.64307	0.147984
$320$	−28.0620	−1.56871
$321$	0	0
$322$	−10.7255	−0.597710
$323$	7.06161	0.392918
$324$	0	0
$325$	0.860777	0.0477473
$326$	−47.7197	−2.64295
$327$	0	0
$328$	6.62193	0.365635
$329$	31.4362	1.73313
$330$	0	0
$331$	23.8884	1.31303	0.656514	−	0.754314i	$-0.272030\pi$
$331$	23.8884	1.31303	0.656514	+	0.754314i	$0.272030\pi$
$332$	11.4061	0.625988
$333$	0	0
$334$	−8.12985	−0.444845
$335$	−30.7795	−1.68166
$336$	0	0
$337$	7.52823	0.410089	0.205044	−	0.978753i	$-0.434266\pi$
$337$	7.52823	0.410089	0.205044	+	0.978753i	$0.434266\pi$
$338$	−6.10124	−0.331864
$339$	0	0
$340$	35.7540	1.93903
$341$	−5.28519	−0.286209
$342$	0	0
$343$	58.5100	3.15925
$344$	−17.9253	−0.966470
$345$	0	0
$346$	−37.2017	−1.99998
$347$	−12.0728	−0.648102	−0.324051	−	0.946040i	$-0.605045\pi$
$347$	−12.0728	−0.648102	−0.324051	+	0.946040i	$0.605045\pi$
$348$	0	0
$349$	−13.4052	−0.717564	−0.358782	−	0.933421i	$-0.616808\pi$
$349$	−13.4052	−0.717564	−0.358782	+	0.933421i	$0.616808\pi$
$350$	−3.01770	−0.161303
$351$	0	0
$352$	6.97015	0.371510
$353$	−26.2962	−1.39961	−0.699804	−	0.714335i	$-0.746729\pi$
$353$	−26.2962	−1.39961	−0.699804	+	0.714335i	$0.746729\pi$
$354$	0	0
$355$	−22.9286	−1.21692
$356$	−26.6233	−1.41103
$357$	0	0
$358$	−37.3801	−1.97560
$359$	−4.55740	−0.240530	−0.120265	−	0.992742i	$-0.538374\pi$
$359$	−4.55740	−0.240530	−0.120265	+	0.992742i	$0.538374\pi$
$360$	0	0
$361$	−17.4185	−0.916763
$362$	29.5247	1.55179
$363$	0	0
$364$	−47.3953	−2.48419
$365$	9.21786	0.482485
$366$	0	0
$367$	−14.5909	−0.761638	−0.380819	−	0.924650i	$-0.624358\pi$
$367$	−14.5909	−0.761638	−0.380819	+	0.924650i	$0.624358\pi$
$368$	1.22820	0.0640244
$369$	0	0
$370$	36.8935	1.91800
$371$	−14.4474	−0.750069
$372$	0	0
$373$	−27.5057	−1.42419	−0.712097	−	0.702081i	$-0.752254\pi$
$373$	−27.5057	−1.42419	−0.712097	+	0.702081i	$0.752254\pi$
$374$	−12.4645	−0.644522
$375$	0	0
$376$	12.7968	0.659946
$377$	8.46254	0.435843
$378$	0	0
$379$	−9.32525	−0.479006	−0.239503	−	0.970896i	$-0.576984\pi$
$379$	−9.32525	−0.479006	−0.239503	+	0.970896i	$0.576984\pi$
$380$	8.00741	0.410772
$381$	0	0
$382$	46.3248	2.37018
$383$	−26.5579	−1.35704	−0.678522	−	0.734581i	$-0.737379\pi$
$383$	−26.5579	−1.35704	−0.678522	+	0.734581i	$0.737379\pi$
$384$	0	0
$385$	−10.9990	−0.560563
$386$	−6.65488	−0.338724
$387$	0	0
$388$	−19.4136	−0.985574
$389$	−22.2104	−1.12611	−0.563055	−	0.826419i	$-0.690374\pi$
$389$	−22.2104	−1.12611	−0.563055	+	0.826419i	$0.690374\pi$
$390$	0	0
$391$	−5.36548	−0.271344
$392$	38.2271	1.93076
$393$	0	0
$394$	−35.6024	−1.79362
$395$	37.3744	1.88051
$396$	0	0
$397$	19.9200	0.999754	0.499877	−	0.866096i	$-0.333378\pi$
$397$	19.9200	0.999754	0.499877	+	0.866096i	$0.333378\pi$
$398$	1.18424	0.0593604
$399$	0	0
$400$	0.345563	0.0172782
$401$	37.7336	1.88433	0.942163	−	0.335154i	$-0.108788\pi$
$401$	37.7336	1.88433	0.942163	+	0.335154i	$0.108788\pi$
$402$	0	0
$403$	−16.9220	−0.842946
$404$	13.2268	0.658057
$405$	0	0
$406$	−29.6679	−1.47239
$407$	−7.64117	−0.378759
$408$	0	0
$409$	6.44752	0.318809	0.159405	−	0.987213i	$-0.449043\pi$
$409$	6.44752	0.318809	0.159405	+	0.987213i	$0.449043\pi$
$410$	15.5321	0.767076
$411$	0	0
$412$	47.8943	2.35958
$413$	32.1022	1.57965
$414$	0	0
$415$	8.47513	0.416028
$416$	22.3169	1.09418
$417$	0	0
$418$	−2.79153	−0.136538
$419$	17.0964	0.835212	0.417606	−	0.908628i	$-0.362869\pi$
$419$	17.0964	0.835212	0.417606	+	0.908628i	$0.362869\pi$
$420$	0	0
$421$	19.2338	0.937400	0.468700	−	0.883357i	$-0.344723\pi$
$421$	19.2338	0.937400	0.468700	+	0.883357i	$0.344723\pi$
$422$	−12.5922	−0.612980
$423$	0	0
$424$	−5.88113	−0.285613
$425$	−1.50962	−0.0732272
$426$	0	0
$427$	−5.05675	−0.244713
$428$	−25.1765	−1.21695
$429$	0	0
$430$	−42.0449	−2.02759
$431$	15.0711	0.725951	0.362976	−	0.931799i	$-0.381761\pi$
$431$	15.0711	0.725951	0.362976	+	0.931799i	$0.381761\pi$
$432$	0	0
$433$	−1.16749	−0.0561058	−0.0280529	−	0.999606i	$-0.508931\pi$
$433$	−1.16749	−0.0561058	−0.0280529	+	0.999606i	$0.508931\pi$
$434$	59.3251	2.84769
$435$	0	0
$436$	54.0446	2.58826
$437$	−1.20165	−0.0574826
$438$	0	0
$439$	16.3226	0.779035	0.389518	−	0.921019i	$-0.372642\pi$
$439$	16.3226	0.779035	0.389518	+	0.921019i	$0.372642\pi$
$440$	−4.47742	−0.213452
$441$	0	0
$442$	−39.9085	−1.89825
$443$	−4.47718	−0.212717	−0.106359	−	0.994328i	$-0.533919\pi$
$443$	−4.47718	−0.212717	−0.106359	+	0.994328i	$0.533919\pi$
$444$	0	0
$445$	−19.7821	−0.937763
$446$	31.5986	1.49624
$447$	0	0
$448$	−65.2387	−3.08224
$449$	26.7590	1.26284	0.631418	−	0.775442i	$-0.282473\pi$
$449$	26.7590	1.26284	0.631418	+	0.775442i	$0.282473\pi$
$450$	0	0
$451$	−3.21692	−0.151479
$452$	−45.3882	−2.13488
$453$	0	0
$454$	7.34475	0.344706
$455$	−35.2165	−1.65098
$456$	0	0
$457$	6.98309	0.326655	0.163328	−	0.986572i	$-0.447777\pi$
$457$	6.98309	0.326655	0.163328	+	0.986572i	$0.447777\pi$
$458$	−8.27547	−0.386687
$459$	0	0
$460$	−6.08411	−0.283673
$461$	0.637167	0.0296758	0.0148379	−	0.999890i	$-0.495277\pi$
$461$	0.637167	0.0296758	0.0148379	+	0.999890i	$0.495277\pi$
$462$	0	0
$463$	18.7279	0.870361	0.435180	−	0.900343i	$-0.356685\pi$
$463$	18.7279	0.870361	0.435180	+	0.900343i	$0.356685\pi$
$464$	3.39733	0.157717
$465$	0	0
$466$	−25.0872	−1.16214
$467$	10.7986	0.499698	0.249849	−	0.968285i	$-0.419619\pi$
$467$	10.7986	0.499698	0.249849	+	0.968285i	$0.419619\pi$
$468$	0	0
$469$	−71.5564	−3.30417
$470$	30.0157	1.38452
$471$	0	0
$472$	13.0680	0.601501
$473$	8.70811	0.400399
$474$	0	0
$475$	−0.338092	−0.0155127
$476$	83.1212	3.80985
$477$	0	0
$478$	−5.22144	−0.238823
$479$	37.6783	1.72157	0.860783	−	0.508972i	$-0.169974\pi$
$479$	37.6783	1.72157	0.860783	+	0.508972i	$0.169974\pi$
$480$	0	0
$481$	−24.4654	−1.11552
$482$	17.2221	0.784445
$483$	0	0
$484$	2.92734	0.133061
$485$	−14.4250	−0.655006
$486$	0	0
$487$	9.34251	0.423349	0.211675	−	0.977340i	$-0.432108\pi$
$487$	9.34251	0.423349	0.211675	+	0.977340i	$0.432108\pi$
$488$	−2.05847	−0.0931824
$489$	0	0
$490$	89.6639	4.05060
$491$	−41.6005	−1.87741	−0.938703	−	0.344728i	$-0.887971\pi$
$491$	−41.6005	−1.87741	−0.938703	+	0.344728i	$0.887971\pi$
$492$	0	0
$493$	−14.8415	−0.668426
$494$	−8.93786	−0.402133
$495$	0	0
$496$	−6.79344	−0.305034
$497$	−53.3047	−2.39104
$498$	0	0
$499$	−9.80103	−0.438755	−0.219377	−	0.975640i	$-0.570403\pi$
$499$	−9.80103	−0.438755	−0.219377	+	0.975640i	$0.570403\pi$
$500$	−33.5484	−1.50033
$501$	0	0
$502$	9.47113	0.422717
$503$	−36.9840	−1.64904	−0.824518	−	0.565836i	$-0.808554\pi$
$503$	−36.9840	−1.64904	−0.824518	+	0.565836i	$0.808554\pi$
$504$	0	0
$505$	9.82800	0.437340
$506$	2.12103	0.0942913
$507$	0	0
$508$	15.9698	0.708545
$509$	−27.0976	−1.20108	−0.600540	−	0.799594i	$-0.705048\pi$
$509$	−27.0976	−1.20108	−0.600540	+	0.799594i	$0.705048\pi$
$510$	0	0
$511$	21.4298	0.947997
$512$	14.2510	0.629812
$513$	0	0
$514$	−9.21086	−0.406274
$515$	35.5873	1.56816
$516$	0	0
$517$	−6.21668	−0.273409
$518$	85.7704	3.76853
$519$	0	0
$520$	−14.3357	−0.628662
$521$	−29.9679	−1.31292	−0.656459	−	0.754361i	$-0.727947\pi$
$521$	−29.9679	−1.31292	−0.656459	+	0.754361i	$0.727947\pi$
$522$	0	0
$523$	29.1056	1.27270	0.636348	−	0.771402i	$-0.280444\pi$
$523$	29.1056	1.27270	0.636348	+	0.771402i	$0.280444\pi$
$524$	−46.5170	−2.03211
$525$	0	0
$526$	−26.2065	−1.14266
$527$	29.6776	1.29278
$528$	0	0
$529$	−22.0870	−0.960303
$530$	−13.7945	−0.599196
$531$	0	0
$532$	18.6157	0.807094
$533$	−10.2999	−0.446138
$534$	0	0
$535$	−18.7071	−0.808780
$536$	−29.1287	−1.25817
$537$	0	0
$538$	61.0442	2.63180
$539$	−18.5707	−0.799896
$540$	0	0
$541$	42.6909	1.83543	0.917713	−	0.397245i	$-0.130034\pi$
$541$	42.6909	1.83543	0.917713	+	0.397245i	$0.130034\pi$
$542$	44.0196	1.89080
$543$	0	0
$544$	−39.1390	−1.67807
$545$	40.1571	1.72014
$546$	0	0
$547$	−27.5597	−1.17837	−0.589184	−	0.807999i	$-0.700551\pi$
$547$	−27.5597	−1.17837	−0.589184	+	0.807999i	$0.700551\pi$
$548$	8.63544	0.368888
$549$	0	0
$550$	0.596767	0.0254462
$551$	−3.32388	−0.141602
$552$	0	0
$553$	86.8884	3.69487
$554$	21.7242	0.922971
$555$	0	0
$556$	1.86545	0.0791127
$557$	−0.987301	−0.0418333	−0.0209166	−	0.999781i	$-0.506658\pi$
$557$	−0.987301	−0.0418333	−0.0209166	+	0.999781i	$0.506658\pi$
$558$	0	0
$559$	27.8815	1.17926
$560$	−14.1379	−0.597434
$561$	0	0
$562$	−16.2396	−0.685028
$563$	42.4725	1.79000	0.895001	−	0.446064i	$-0.147175\pi$
$563$	42.4725	1.79000	0.895001	+	0.446064i	$0.147175\pi$
$564$	0	0
$565$	−33.7252	−1.41883
$566$	31.4152	1.32048
$567$	0	0
$568$	−21.6989	−0.910466
$569$	−25.8404	−1.08329	−0.541643	−	0.840609i	$-0.682198\pi$
$569$	−25.8404	−1.08329	−0.541643	+	0.840609i	$0.682198\pi$
$570$	0	0
$571$	−7.75510	−0.324541	−0.162270	−	0.986746i	$-0.551882\pi$
$571$	−7.75510	−0.324541	−0.162270	+	0.986746i	$0.551882\pi$
$572$	9.37269	0.391892
$573$	0	0
$574$	36.1092	1.50717
$575$	0.256886	0.0107129
$576$	0	0
$577$	−17.0522	−0.709893	−0.354946	−	0.934887i	$-0.615501\pi$
$577$	−17.0522	−0.709893	−0.354946	+	0.934887i	$0.615501\pi$
$578$	32.2550	1.34163
$579$	0	0
$580$	−16.8293	−0.698798
$581$	19.7031	0.817421
$582$	0	0
$583$	2.85705	0.118327
$584$	8.72349	0.360980
$585$	0	0
$586$	35.2356	1.45557
$587$	−18.5551	−0.765852	−0.382926	−	0.923779i	$-0.625084\pi$
$587$	−18.5551	−0.765852	−0.382926	+	0.923779i	$0.625084\pi$
$588$	0	0
$589$	6.64656	0.273867
$590$	30.6516	1.26191
$591$	0	0
$592$	−9.82174	−0.403671
$593$	15.3999	0.632399	0.316199	−	0.948693i	$-0.397593\pi$
$593$	15.3999	0.632399	0.316199	+	0.948693i	$0.397593\pi$
$594$	0	0
$595$	61.7622	2.53200
$596$	−40.4195	−1.65565
$597$	0	0
$598$	6.79108	0.277708
$599$	30.4897	1.24578	0.622888	−	0.782311i	$-0.285959\pi$
$599$	30.4897	1.24578	0.622888	+	0.782311i	$0.285959\pi$
$600$	0	0
$601$	−8.97417	−0.366064	−0.183032	−	0.983107i	$-0.558591\pi$
$601$	−8.97417	−0.366064	−0.183032	+	0.983107i	$0.558591\pi$
$602$	−97.7465	−3.98385
$603$	0	0
$604$	−59.9003	−2.43731
$605$	2.17512	0.0884313
$606$	0	0
$607$	−4.13601	−0.167875	−0.0839376	−	0.996471i	$-0.526750\pi$
$607$	−4.13601	−0.167875	−0.0839376	+	0.996471i	$0.526750\pi$
$608$	−8.76553	−0.355489
$609$	0	0
$610$	−4.82825	−0.195490
$611$	−19.9044	−0.805248
$612$	0	0
$613$	36.6287	1.47942	0.739710	−	0.672926i	$-0.234963\pi$
$613$	36.6287	1.47942	0.739710	+	0.672926i	$0.234963\pi$
$614$	−49.7175	−2.00643
$615$	0	0
$616$	−10.4091	−0.419396
$617$	−8.24763	−0.332037	−0.166019	−	0.986123i	$-0.553091\pi$
$617$	−8.24763	−0.332037	−0.166019	+	0.986123i	$0.553091\pi$
$618$	0	0
$619$	11.4400	0.459811	0.229906	−	0.973213i	$-0.426158\pi$
$619$	11.4400	0.459811	0.229906	+	0.973213i	$0.426158\pi$
$620$	33.6525	1.35152
$621$	0	0
$622$	25.8830	1.03781
$623$	−45.9897	−1.84254
$624$	0	0
$625$	−23.5835	−0.943340
$626$	2.64746	0.105814
$627$	0	0
$628$	−0.339063	−0.0135301
$629$	42.9069	1.71081
$630$	0	0
$631$	−37.8435	−1.50653	−0.753263	−	0.657719i	$-0.771521\pi$
$631$	−37.8435	−1.50653	−0.753263	+	0.657719i	$0.771521\pi$
$632$	35.3699	1.40694
$633$	0	0
$634$	−29.5741	−1.17454
$635$	11.8662	0.470895
$636$	0	0
$637$	−59.4593	−2.35586
$638$	5.86699	0.232276
$639$	0	0
$640$	−31.9690	−1.26368
$641$	26.7866	1.05801	0.529003	−	0.848620i	$-0.322566\pi$
$641$	26.7866	1.05801	0.529003	+	0.848620i	$0.322566\pi$
$642$	0	0
$643$	−25.8395	−1.01901	−0.509505	−	0.860468i	$-0.670172\pi$
$643$	−25.8395	−1.01901	−0.509505	+	0.860468i	$0.670172\pi$
$644$	−14.1444	−0.557368
$645$	0	0
$646$	15.6751	0.616728
$647$	−30.5070	−1.19935	−0.599677	−	0.800242i	$-0.704704\pi$
$647$	−30.5070	−1.19935	−0.599677	+	0.800242i	$0.704704\pi$
$648$	0	0
$649$	−6.34839	−0.249196
$650$	1.91072	0.0749445
$651$	0	0
$652$	−62.9309	−2.46457
$653$	−1.47873	−0.0578672	−0.0289336	−	0.999581i	$-0.509211\pi$
$653$	−1.47873	−0.0578672	−0.0289336	+	0.999581i	$0.509211\pi$
$654$	0	0
$655$	−34.5639	−1.35052
$656$	−4.13494	−0.161442
$657$	0	0
$658$	69.7808	2.72034
$659$	4.68280	0.182416	0.0912080	−	0.995832i	$-0.470927\pi$
$659$	4.68280	0.182416	0.0912080	+	0.995832i	$0.470927\pi$
$660$	0	0
$661$	14.5913	0.567536	0.283768	−	0.958893i	$-0.408415\pi$
$661$	14.5913	0.567536	0.283768	+	0.958893i	$0.408415\pi$
$662$	53.0266	2.06094
$663$	0	0
$664$	8.02059	0.311259
$665$	13.8322	0.536389
$666$	0	0
$667$	2.52552	0.0977884
$668$	−10.7213	−0.414821
$669$	0	0
$670$	−68.3231	−2.63955
$671$	1.00000	0.0386046
$672$	0	0
$673$	−2.33640	−0.0900616	−0.0450308	−	0.998986i	$-0.514339\pi$
$673$	−2.33640	−0.0900616	−0.0450308	+	0.998986i	$0.514339\pi$
$674$	16.7109	0.643679
$675$	0	0
$676$	−8.04608	−0.309465
$677$	−0.867708	−0.0333487	−0.0166744	−	0.999861i	$-0.505308\pi$
$677$	−0.867708	−0.0333487	−0.0166744	+	0.999861i	$0.505308\pi$
$678$	0	0
$679$	−33.5354	−1.28697
$680$	25.1417	0.964141
$681$	0	0
$682$	−11.7319	−0.449236
$683$	2.82033	0.107917	0.0539585	−	0.998543i	$-0.482816\pi$
$683$	2.82033	0.107917	0.0539585	+	0.998543i	$0.482816\pi$
$684$	0	0
$685$	6.41646	0.245160
$686$	129.878	4.95878
$687$	0	0
$688$	11.1932	0.426735
$689$	9.14763	0.348497
$690$	0	0
$691$	−35.0428	−1.33309	−0.666545	−	0.745465i	$-0.732227\pi$
$691$	−35.0428	−1.33309	−0.666545	+	0.745465i	$0.732227\pi$
$692$	−49.0603	−1.86499
$693$	0	0
$694$	−26.7987	−1.01727
$695$	1.38610	0.0525778
$696$	0	0
$697$	18.0638	0.684214
$698$	−29.7563	−1.12629
$699$	0	0
$700$	−3.97963	−0.150416
$701$	−46.6126	−1.76053	−0.880267	−	0.474478i	$-0.842637\pi$
$701$	−46.6126	−1.76053	−0.880267	+	0.474478i	$0.842637\pi$
$702$	0	0
$703$	9.60939	0.362425
$704$	12.9013	0.486237
$705$	0	0
$706$	−58.3714	−2.19684
$707$	22.8482	0.859296
$708$	0	0
$709$	14.6994	0.552049	0.276025	−	0.961151i	$-0.410983\pi$
$709$	14.6994	0.552049	0.276025	+	0.961151i	$0.410983\pi$
$710$	−50.8960	−1.91009
$711$	0	0
$712$	−18.7212	−0.701606
$713$	−5.05012	−0.189129
$714$	0	0
$715$	6.96426	0.260449
$716$	−49.2954	−1.84226
$717$	0	0
$718$	−10.1163	−0.377538
$719$	−7.93574	−0.295953	−0.147977	−	0.988991i	$-0.547276\pi$
$719$	−7.93574	−0.295953	−0.147977	+	0.988991i	$0.547276\pi$
$720$	0	0
$721$	82.7337	3.08116
$722$	−38.6649	−1.43896
$723$	0	0
$724$	38.9361	1.44705
$725$	0.710572	0.0263900
$726$	0	0
$727$	−44.4043	−1.64686	−0.823432	−	0.567415i	$-0.807944\pi$
$727$	−44.4043	−1.64686	−0.823432	+	0.567415i	$0.807944\pi$
$728$	−33.3278	−1.23521
$729$	0	0
$730$	20.4614	0.757312
$731$	−48.8980	−1.80856
$732$	0	0
$733$	−27.4350	−1.01333	−0.506667	−	0.862142i	$-0.669122\pi$
$733$	−27.4350	−1.01333	−0.506667	+	0.862142i	$0.669122\pi$
$734$	−32.3883	−1.19547
$735$	0	0
$736$	6.66014	0.245496
$737$	14.1507	0.521247
$738$	0	0
$739$	−4.73622	−0.174225	−0.0871124	−	0.996198i	$-0.527764\pi$
$739$	−4.73622	−0.174225	−0.0871124	+	0.996198i	$0.527764\pi$
$740$	48.6538	1.78855
$741$	0	0
$742$	−32.0697	−1.17731
$743$	23.6033	0.865920	0.432960	−	0.901413i	$-0.357469\pi$
$743$	23.6033	0.865920	0.432960	+	0.901413i	$0.357469\pi$
$744$	0	0
$745$	−30.0332	−1.10033
$746$	−61.0562	−2.23543
$747$	0	0
$748$	−16.4377	−0.601021
$749$	−43.4905	−1.58911
$750$	0	0
$751$	−41.3633	−1.50937	−0.754684	−	0.656089i	$-0.772210\pi$
$751$	−41.3633	−1.50937	−0.754684	+	0.656089i	$0.772210\pi$
$752$	−7.99075	−0.291393
$753$	0	0
$754$	18.7848	0.684102
$755$	−44.5082	−1.61982
$756$	0	0
$757$	−9.70134	−0.352601	−0.176301	−	0.984336i	$-0.556413\pi$
$757$	−9.70134	−0.352601	−0.176301	+	0.984336i	$0.556413\pi$
$758$	−20.6998	−0.751851
$759$	0	0
$760$	5.63071	0.204247
$761$	−32.7697	−1.18790	−0.593951	−	0.804501i	$-0.702433\pi$
$761$	−32.7697	−1.18790	−0.593951	+	0.804501i	$0.702433\pi$
$762$	0	0
$763$	93.3578	3.37978
$764$	61.0914	2.21021
$765$	0	0
$766$	−58.9521	−2.13003
$767$	−20.3262	−0.733935
$768$	0	0
$769$	−13.2509	−0.477840	−0.238920	−	0.971039i	$-0.576793\pi$
$769$	−13.2509	−0.477840	−0.238920	+	0.971039i	$0.576793\pi$
$770$	−24.4152	−0.879864
$771$	0	0
$772$	−8.77620	−0.315862
$773$	24.4785	0.880430	0.440215	−	0.897892i	$-0.354902\pi$
$773$	24.4785	0.880430	0.440215	+	0.897892i	$0.354902\pi$
$774$	0	0
$775$	−1.42089	−0.0510398
$776$	−13.6514	−0.490055
$777$	0	0
$778$	−49.3017	−1.76755
$779$	4.04554	0.144947
$780$	0	0
$781$	10.5413	0.377197
$782$	−11.9101	−0.425904
$783$	0	0
$784$	−23.8702	−0.852508
$785$	−0.251937	−0.00899200
$786$	0	0
$787$	−7.47104	−0.266314	−0.133157	−	0.991095i	$-0.542511\pi$
$787$	−7.47104	−0.266314	−0.133157	+	0.991095i	$0.542511\pi$
$788$	−46.9511	−1.67256
$789$	0	0
$790$	82.9622	2.95166
$791$	−78.4046	−2.78775
$792$	0	0
$793$	3.20178	0.113699
$794$	44.2176	1.56922
$795$	0	0
$796$	1.56173	0.0553539
$797$	−4.02337	−0.142515	−0.0712576	−	0.997458i	$-0.522701\pi$
$797$	−4.02337	−0.142515	−0.0712576	+	0.997458i	$0.522701\pi$
$798$	0	0
$799$	34.9081	1.23496
$800$	1.87388	0.0662516
$801$	0	0
$802$	83.7596	2.95765
$803$	−4.23786	−0.149551
$804$	0	0
$805$	−10.5098	−0.370423
$806$	−37.5629	−1.32310
$807$	0	0
$808$	9.30091	0.327205
$809$	35.2530	1.23943	0.619715	−	0.784827i	$-0.287248\pi$
$809$	35.2530	1.23943	0.619715	+	0.784827i	$0.287248\pi$
$810$	0	0
$811$	41.0721	1.44224	0.721118	−	0.692812i	$-0.243628\pi$
$811$	41.0721	1.44224	0.721118	+	0.692812i	$0.243628\pi$
$812$	−39.1249	−1.37301
$813$	0	0
$814$	−16.9616	−0.594503
$815$	−46.7601	−1.63793
$816$	0	0
$817$	−10.9511	−0.383132
$818$	14.3119	0.500405
$819$	0	0
$820$	20.4832	0.715303
$821$	−21.1227	−0.737189	−0.368594	−	0.929590i	$-0.620161\pi$
$821$	−21.1227	−0.737189	−0.368594	+	0.929590i	$0.620161\pi$
$822$	0	0
$823$	39.4675	1.37575	0.687875	−	0.725829i	$-0.258544\pi$
$823$	39.4675	1.37575	0.687875	+	0.725829i	$0.258544\pi$
$824$	33.6787	1.17325
$825$	0	0
$826$	71.2592	2.47943
$827$	−35.0703	−1.21951	−0.609757	−	0.792588i	$-0.708733\pi$
$827$	−35.0703	−1.21951	−0.609757	+	0.792588i	$0.708733\pi$
$828$	0	0
$829$	16.4013	0.569642	0.284821	−	0.958581i	$-0.408066\pi$
$829$	16.4013	0.569642	0.284821	+	0.958581i	$0.408066\pi$
$830$	18.8128	0.653000
$831$	0	0
$832$	41.3072	1.43207
$833$	104.279	3.61304
$834$	0	0
$835$	−7.96636	−0.275687
$836$	−3.68136	−0.127323
$837$	0	0
$838$	37.9498	1.31095
$839$	12.1840	0.420639	0.210319	−	0.977633i	$-0.432550\pi$
$839$	12.1840	0.420639	0.210319	+	0.977633i	$0.432550\pi$
$840$	0	0
$841$	−22.0142	−0.759109
$842$	42.6945	1.47135
$843$	0	0
$844$	−16.6061	−0.571607
$845$	−5.97854	−0.205668
$846$	0	0
$847$	5.05675	0.173752
$848$	3.67237	0.126110
$849$	0	0
$850$	−3.35099	−0.114938
$851$	−7.30131	−0.250286
$852$	0	0
$853$	49.9463	1.71013	0.855065	−	0.518521i	$-0.173517\pi$
$853$	49.9463	1.71013	0.855065	+	0.518521i	$0.173517\pi$
$854$	−11.2248	−0.384104
$855$	0	0
$856$	−17.7038	−0.605104
$857$	27.5472	0.940994	0.470497	−	0.882402i	$-0.344075\pi$
$857$	27.5472	0.940994	0.470497	+	0.882402i	$0.344075\pi$
$858$	0	0
$859$	−23.6340	−0.806383	−0.403192	−	0.915116i	$-0.632099\pi$
$859$	−23.6340	−0.806383	−0.403192	+	0.915116i	$0.632099\pi$
$860$	−55.4473	−1.89074
$861$	0	0
$862$	33.4543	1.13946
$863$	−19.1966	−0.653461	−0.326731	−	0.945118i	$-0.605947\pi$
$863$	−19.1966	−0.653461	−0.326731	+	0.945118i	$0.605947\pi$
$864$	0	0
$865$	−36.4536	−1.23946
$866$	−2.59154	−0.0880641
$867$	0	0
$868$	78.2357	2.65549
$869$	−17.1827	−0.582882
$870$	0	0
$871$	45.3074	1.53518
$872$	38.0034	1.28696
$873$	0	0
$874$	−2.66737	−0.0902251
$875$	−57.9522	−1.95914
$876$	0	0
$877$	−33.6570	−1.13652	−0.568259	−	0.822850i	$-0.692383\pi$
$877$	−33.6570	−1.13652	−0.568259	+	0.822850i	$0.692383\pi$
$878$	36.2323	1.22278
$879$	0	0
$880$	2.79584	0.0942478
$881$	9.88462	0.333021	0.166511	−	0.986040i	$-0.446750\pi$
$881$	9.88462	0.333021	0.166511	+	0.986040i	$0.446750\pi$
$882$	0	0
$883$	−30.6737	−1.03225	−0.516126	−	0.856513i	$-0.672626\pi$
$883$	−30.6737	−1.03225	−0.516126	+	0.856513i	$0.672626\pi$
$884$	−52.6298	−1.77013
$885$	0	0
$886$	−9.93827	−0.333883
$887$	5.63849	0.189322	0.0946610	−	0.995510i	$-0.469823\pi$
$887$	5.63849	0.189322	0.0946610	+	0.995510i	$0.469823\pi$
$888$	0	0
$889$	27.5866	0.925225
$890$	−43.9116	−1.47192
$891$	0	0
$892$	41.6710	1.39525
$893$	7.81798	0.261619
$894$	0	0
$895$	−36.6284	−1.22435
$896$	−74.3218	−2.48292
$897$	0	0
$898$	59.3986	1.98216
$899$	−13.9692	−0.465897
$900$	0	0
$901$	−16.0430	−0.534469
$902$	−7.14080	−0.237763
$903$	0	0
$904$	−31.9164	−1.06152
$905$	28.9310	0.961699
$906$	0	0
$907$	6.27594	0.208389	0.104195	−	0.994557i	$-0.466774\pi$
$907$	6.27594	0.208389	0.104195	+	0.994557i	$0.466774\pi$
$908$	9.68598	0.321440
$909$	0	0
$910$	−78.1722	−2.59138
$911$	−15.6434	−0.518290	−0.259145	−	0.965838i	$-0.583441\pi$
$911$	−15.6434	−0.518290	−0.259145	+	0.965838i	$0.583441\pi$
$912$	0	0
$913$	−3.89639	−0.128952
$914$	15.5008	0.512720
$915$	0	0
$916$	−10.9134	−0.360588
$917$	−80.3545	−2.65354
$918$	0	0
$919$	−37.8167	−1.24746	−0.623728	−	0.781641i	$-0.714383\pi$
$919$	−37.8167	−1.24746	−0.623728	+	0.781641i	$0.714383\pi$
$920$	−4.27827	−0.141050
$921$	0	0
$922$	1.41436	0.0465794
$923$	33.7509	1.11093
$924$	0	0
$925$	−2.05428	−0.0675442
$926$	41.5715	1.36613
$927$	0	0
$928$	18.4226	0.604752
$929$	−28.9770	−0.950705	−0.475352	−	0.879796i	$-0.657679\pi$
$929$	−28.9770	−0.950705	−0.475352	+	0.879796i	$0.657679\pi$
$930$	0	0
$931$	23.3541	0.765401
$932$	−33.0840	−1.08370
$933$	0	0
$934$	23.9702	0.784331
$935$	−12.2138	−0.399434
$936$	0	0
$937$	53.8625	1.75961	0.879805	−	0.475334i	$-0.157673\pi$
$937$	53.8625	1.75961	0.879805	+	0.475334i	$0.157673\pi$
$938$	−158.838	−5.18625
$939$	0	0
$940$	39.5836	1.29107
$941$	−17.7491	−0.578603	−0.289301	−	0.957238i	$-0.593423\pi$
$941$	−17.7491	−0.578603	−0.289301	+	0.957238i	$0.593423\pi$
$942$	0	0
$943$	−3.07384	−0.100098
$944$	−8.16005	−0.265587
$945$	0	0
$946$	19.3299	0.628470
$947$	36.6133	1.18977	0.594886	−	0.803810i	$-0.297197\pi$
$947$	36.6133	1.18977	0.594886	+	0.803810i	$0.297197\pi$
$948$	0	0
$949$	−13.5687	−0.440458
$950$	−0.750483	−0.0243489
$951$	0	0
$952$	58.4498	1.89437
$953$	−4.57686	−0.148259	−0.0741295	−	0.997249i	$-0.523618\pi$
$953$	−4.57686	−0.148259	−0.0741295	+	0.997249i	$0.523618\pi$
$954$	0	0
$955$	45.3932	1.46889
$956$	−6.88584	−0.222704
$957$	0	0
$958$	83.6368	2.70218
$959$	14.9170	0.481697
$960$	0	0
$961$	−3.06672	−0.0989265
$962$	−54.3072	−1.75094
$963$	0	0
$964$	22.7119	0.731500
$965$	−6.52105	−0.209920
$966$	0	0
$967$	22.2757	0.716340	0.358170	−	0.933656i	$-0.383401\pi$
$967$	22.2757	0.716340	0.358170	+	0.933656i	$0.383401\pi$
$968$	2.05847	0.0661616
$969$	0	0
$970$	−32.0201	−1.02810
$971$	25.4408	0.816435	0.408217	−	0.912885i	$-0.366151\pi$
$971$	25.4408	0.816435	0.408217	+	0.912885i	$0.366151\pi$
$972$	0	0
$973$	3.22242	0.103306
$974$	20.7381	0.664493
$975$	0	0
$976$	1.28537	0.0411437
$977$	−31.8193	−1.01799	−0.508995	−	0.860770i	$-0.669983\pi$
$977$	−31.8193	−1.01799	−0.508995	+	0.860770i	$0.669983\pi$
$978$	0	0
$979$	9.09472	0.290669
$980$	118.245	3.77721
$981$	0	0
$982$	−92.3432	−2.94679
$983$	−2.49700	−0.0796418	−0.0398209	−	0.999207i	$-0.512679\pi$
$983$	−2.49700	−0.0796418	−0.0398209	+	0.999207i	$0.512679\pi$
$984$	0	0
$985$	−34.8864	−1.11157
$986$	−32.9445	−1.04917
$987$	0	0
$988$	−11.7869	−0.374992
$989$	8.32079	0.264586
$990$	0	0
$991$	37.7292	1.19851	0.599253	−	0.800560i	$-0.295464\pi$
$991$	37.7292	1.19851	0.599253	+	0.800560i	$0.295464\pi$
$992$	−36.8386	−1.16963
$993$	0	0
$994$	−118.324	−3.75300
$995$	1.16042	0.0367878
$996$	0	0
$997$	−33.8874	−1.07322	−0.536612	−	0.843829i	$-0.680296\pi$
$997$	−33.8874	−1.07322	−0.536612	+	0.843829i	$0.680296\pi$
$998$	−21.7559	−0.688672
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.18		19
3.2	odd	2		671.2.a.c.1.2	✓	19
33.32	even	2		7381.2.a.i.1.18		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.2	✓	19	3.2	odd	2
6039.2.a.k.1.18		19	1.1	even	1	trivial
7381.2.a.i.1.18		19	33.32	even	2

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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+2.21976 q^{2} +2.92734 q^{4} +2.17512 q^{5} +5.05675 q^{7} +2.05847 q^{8} +O(q^{10})\)	\(q+2.21976 q^{2} +2.92734 q^{4} +2.17512 q^{5} +5.05675 q^{7} +2.05847 q^{8} +4.82825 q^{10} -1.00000 q^{11} -3.20178 q^{13} +11.2248 q^{14} -1.28537 q^{16} +5.61523 q^{17} +1.25758 q^{19} +6.36732 q^{20} -2.21976 q^{22} -0.955523 q^{23} -0.268843 q^{25} -7.10719 q^{26} +14.8028 q^{28} -2.64307 q^{29} +5.28519 q^{31} -6.97015 q^{32} +12.4645 q^{34} +10.9990 q^{35} +7.64117 q^{37} +2.79153 q^{38} +4.47742 q^{40} +3.21692 q^{41} -8.70811 q^{43} -2.92734 q^{44} -2.12103 q^{46} +6.21668 q^{47} +18.5707 q^{49} -0.596767 q^{50} -9.37269 q^{52} -2.85705 q^{53} -2.17512 q^{55} +10.4091 q^{56} -5.86699 q^{58} +6.34839 q^{59} -1.00000 q^{61} +11.7319 q^{62} -12.9013 q^{64} -6.96426 q^{65} -14.1507 q^{67} +16.4377 q^{68} +24.4152 q^{70} -10.5413 q^{71} +4.23786 q^{73} +16.9616 q^{74} +3.68136 q^{76} -5.05675 q^{77} +17.1827 q^{79} -2.79584 q^{80} +7.14080 q^{82} +3.89639 q^{83} +12.2138 q^{85} -19.3299 q^{86} -2.05847 q^{88} -9.09472 q^{89} -16.1906 q^{91} -2.79714 q^{92} +13.7995 q^{94} +2.73539 q^{95} -6.63181 q^{97} +41.2225 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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