LMFDB - Embedded newform 6039.2.a.j.1.5

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} + \cdots - 1 $ x^14 - x^13 - 21x^12 + 20x^11 + 167x^10 - 148x^9 - 627x^8 + 497x^7 + 1123x^6 - 745x^5 - 802x^4 + 386x^3 + 74x^2 - 15x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.18140$ 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)

$f(q)$	$=$	$q-1.18140 q^{2} -0.604292 q^{4} +2.90081 q^{5} -0.528840 q^{7} +3.07671 q^{8} +O(q^{10})$	\(q-1.18140 q^{2} -0.604292 q^{4} +2.90081 q^{5} -0.528840 q^{7} +3.07671 q^{8} -3.42702 q^{10} +1.00000 q^{11} -0.150802 q^{13} +0.624772 q^{14} -2.42625 q^{16} -0.0673519 q^{17} -0.482246 q^{19} -1.75293 q^{20} -1.18140 q^{22} +4.10835 q^{23} +3.41469 q^{25} +0.178158 q^{26} +0.319574 q^{28} +5.83447 q^{29} -2.84419 q^{31} -3.28705 q^{32} +0.0795696 q^{34} -1.53406 q^{35} +6.02828 q^{37} +0.569726 q^{38} +8.92495 q^{40} -11.4101 q^{41} +11.3555 q^{43} -0.604292 q^{44} -4.85361 q^{46} +3.90970 q^{47} -6.72033 q^{49} -4.03411 q^{50} +0.0911284 q^{52} +0.588357 q^{53} +2.90081 q^{55} -1.62709 q^{56} -6.89285 q^{58} -11.9392 q^{59} +1.00000 q^{61} +3.36013 q^{62} +8.73582 q^{64} -0.437448 q^{65} +9.38126 q^{67} +0.0407002 q^{68} +1.81234 q^{70} -12.6773 q^{71} +13.0690 q^{73} -7.12182 q^{74} +0.291417 q^{76} -0.528840 q^{77} +9.65447 q^{79} -7.03808 q^{80} +13.4799 q^{82} +11.6417 q^{83} -0.195375 q^{85} -13.4154 q^{86} +3.07671 q^{88} +12.7098 q^{89} +0.0797502 q^{91} -2.48264 q^{92} -4.61892 q^{94} -1.39890 q^{95} -1.66875 q^{97} +7.93940 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * 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 $

$2$	−1.18140	−0.835377	−0.417688	−	0.908590i	$-0.637160\pi$
$2$	−1.18140	−0.835377	−0.417688	+	0.908590i	$0.637160\pi$
$3$	0	0
$3$	0	0
$4$	−0.604292	−0.302146
$5$	2.90081	1.29728	0.648640	−	0.761095i	$-0.275338\pi$
$5$	2.90081	1.29728	0.648640	+	0.761095i	$0.275338\pi$
$6$	0	0
$7$	−0.528840	−0.199883	−0.0999414	−	0.994993i	$-0.531866\pi$
$7$	−0.528840	−0.199883	−0.0999414	+	0.994993i	$0.531866\pi$
$8$	3.07671	1.08778
$9$	0	0
$10$	−3.42702	−1.08372
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.150802	−0.0418250	−0.0209125	−	0.999781i	$-0.506657\pi$
$13$	−0.150802	−0.0418250	−0.0209125	+	0.999781i	$0.506657\pi$
$14$	0.624772	0.166977
$15$	0	0
$16$	−2.42625	−0.606562
$17$	−0.0673519	−0.0163352	−0.00816762	−	0.999967i	$-0.502600\pi$
$17$	−0.0673519	−0.0163352	−0.00816762	+	0.999967i	$0.502600\pi$
$18$	0	0
$19$	−0.482246	−0.110635	−0.0553174	−	0.998469i	$-0.517617\pi$
$19$	−0.482246	−0.110635	−0.0553174	+	0.998469i	$0.517617\pi$
$20$	−1.75293	−0.391968
$21$	0	0
$22$	−1.18140	−0.251876
$23$	4.10835	0.856651	0.428325	−	0.903625i	$-0.359104\pi$
$23$	4.10835	0.856651	0.428325	+	0.903625i	$0.359104\pi$
$24$	0	0
$25$	3.41469	0.682937
$26$	0.178158	0.0349396
$27$	0	0
$28$	0.319574	0.0603937
$29$	5.83447	1.08343	0.541717	−	0.840561i	$-0.317774\pi$
$29$	5.83447	1.08343	0.541717	+	0.840561i	$0.317774\pi$
$30$	0	0
$31$	−2.84419	−0.510832	−0.255416	−	0.966831i	$-0.582213\pi$
$31$	−2.84419	−0.510832	−0.255416	+	0.966831i	$0.582213\pi$
$32$	−3.28705	−0.581074
$33$	0	0
$34$	0.0795696	0.0136461
$35$	−1.53406	−0.259304
$36$	0	0
$37$	6.02828	0.991043	0.495522	−	0.868596i	$-0.334977\pi$
$37$	6.02828	0.991043	0.495522	+	0.868596i	$0.334977\pi$
$38$	0.569726	0.0924217
$39$	0	0
$40$	8.92495	1.41116
$41$	−11.4101	−1.78196	−0.890982	−	0.454038i	$-0.849983\pi$
$41$	−11.4101	−1.78196	−0.890982	+	0.454038i	$0.849983\pi$
$42$	0	0
$43$	11.3555	1.73170	0.865849	−	0.500305i	$-0.166779\pi$
$43$	11.3555	1.73170	0.865849	+	0.500305i	$0.166779\pi$
$44$	−0.604292	−0.0911004
$45$	0	0
$46$	−4.85361	−0.715626
$47$	3.90970	0.570288	0.285144	−	0.958485i	$-0.407959\pi$
$47$	3.90970	0.570288	0.285144	+	0.958485i	$0.407959\pi$
$48$	0	0
$49$	−6.72033	−0.960047
$50$	−4.03411	−0.570510
$51$	0	0
$52$	0.0911284	0.0126372
$53$	0.588357	0.0808171	0.0404085	−	0.999183i	$-0.487134\pi$
$53$	0.588357	0.0808171	0.0404085	+	0.999183i	$0.487134\pi$
$54$	0	0
$55$	2.90081	0.391145
$56$	−1.62709	−0.217429
$57$	0	0
$58$	−6.89285	−0.905076
$59$	−11.9392	−1.55435	−0.777176	−	0.629284i	$-0.783348\pi$
$59$	−11.9392	−1.55435	−0.777176	+	0.629284i	$0.783348\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	3.36013	0.426738
$63$	0	0
$64$	8.73582	1.09198
$65$	−0.437448	−0.0542587
$66$	0	0
$67$	9.38126	1.14610	0.573052	−	0.819519i	$-0.305759\pi$
$67$	9.38126	1.14610	0.573052	+	0.819519i	$0.305759\pi$
$68$	0.0407002	0.00493562
$69$	0	0
$70$	1.81234	0.216617
$71$	−12.6773	−1.50452	−0.752258	−	0.658869i	$-0.771035\pi$
$71$	−12.6773	−1.50452	−0.752258	+	0.658869i	$0.771035\pi$
$72$	0	0
$73$	13.0690	1.52961	0.764806	−	0.644261i	$-0.222835\pi$
$73$	13.0690	1.52961	0.764806	+	0.644261i	$0.222835\pi$
$74$	−7.12182	−0.827894
$75$	0	0
$76$	0.291417	0.0334278
$77$	−0.528840	−0.0602669
$78$	0	0
$79$	9.65447	1.08621	0.543107	−	0.839664i	$-0.317248\pi$
$79$	9.65447	1.08621	0.543107	+	0.839664i	$0.317248\pi$
$80$	−7.03808	−0.786881
$81$	0	0
$82$	13.4799	1.48861
$83$	11.6417	1.27784	0.638919	−	0.769274i	$-0.279382\pi$
$83$	11.6417	1.27784	0.638919	+	0.769274i	$0.279382\pi$
$84$	0	0
$85$	−0.195375	−0.0211914
$86$	−13.4154	−1.44662
$87$	0	0
$88$	3.07671	0.327979
$89$	12.7098	1.34724	0.673620	−	0.739078i	$-0.264738\pi$
$89$	12.7098	1.34724	0.673620	+	0.739078i	$0.264738\pi$
$90$	0	0
$91$	0.0797502	0.00836009
$92$	−2.48264	−0.258833
$93$	0	0
$94$	−4.61892	−0.476406
$95$	−1.39890	−0.143524
$96$	0	0
$97$	−1.66875	−0.169436	−0.0847178	−	0.996405i	$-0.526999\pi$
$97$	−1.66875	−0.169436	−0.0847178	+	0.996405i	$0.526999\pi$
$98$	7.93940	0.802001
$99$	0	0
$100$	−2.06347	−0.206347
$101$	−7.21215	−0.717636	−0.358818	−	0.933408i	$-0.616820\pi$
$101$	−7.21215	−0.717636	−0.358818	+	0.933408i	$0.616820\pi$
$102$	0	0
$103$	10.7724	1.06143	0.530717	−	0.847549i	$-0.321923\pi$
$103$	10.7724	1.06143	0.530717	+	0.847549i	$0.321923\pi$
$104$	−0.463975	−0.0454965
$105$	0	0
$106$	−0.695086	−0.0675127
$107$	2.91453	0.281759	0.140879	−	0.990027i	$-0.455007\pi$
$107$	2.91453	0.281759	0.140879	+	0.990027i	$0.455007\pi$
$108$	0	0
$109$	−15.6562	−1.49960	−0.749798	−	0.661667i	$-0.769849\pi$
$109$	−15.6562	−1.49960	−0.749798	+	0.661667i	$0.769849\pi$
$110$	−3.42702	−0.326753
$111$	0	0
$112$	1.28310	0.121241
$113$	17.6863	1.66379	0.831895	−	0.554933i	$-0.187256\pi$
$113$	17.6863	1.66379	0.831895	+	0.554933i	$0.187256\pi$
$114$	0	0
$115$	11.9175	1.11132
$116$	−3.52572	−0.327355
$117$	0	0
$118$	14.1050	1.29847
$119$	0.0356184	0.00326513
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.18140	−0.106959
$123$	0	0
$124$	1.71872	0.154346
$125$	−4.59869	−0.411319
$126$	0	0
$127$	−20.8464	−1.84981	−0.924907	−	0.380192i	$-0.875858\pi$
$127$	−20.8464	−1.84981	−0.924907	+	0.380192i	$0.875858\pi$
$128$	−3.74641	−0.331139
$129$	0	0
$130$	0.516801	0.0453265
$131$	8.61884	0.753031	0.376516	−	0.926410i	$-0.377122\pi$
$131$	8.61884	0.753031	0.376516	+	0.926410i	$0.377122\pi$
$132$	0	0
$133$	0.255031	0.0221140
$134$	−11.0830	−0.957428
$135$	0	0
$136$	−0.207222	−0.0177692
$137$	−3.31289	−0.283039	−0.141520	−	0.989935i	$-0.545199\pi$
$137$	−3.31289	−0.283039	−0.141520	+	0.989935i	$0.545199\pi$
$138$	0	0
$139$	16.9686	1.43926	0.719628	−	0.694360i	$-0.244312\pi$
$139$	16.9686	1.43926	0.719628	+	0.694360i	$0.244312\pi$
$140$	0.927022	0.0783476
$141$	0	0
$142$	14.9769	1.25684
$143$	−0.150802	−0.0126107
$144$	0	0
$145$	16.9247	1.40552
$146$	−15.4397	−1.27780
$147$	0	0
$148$	−3.64284	−0.299440
$149$	10.5728	0.866156	0.433078	−	0.901356i	$-0.357427\pi$
$149$	10.5728	0.866156	0.433078	+	0.901356i	$0.357427\pi$
$150$	0	0
$151$	−11.7619	−0.957168	−0.478584	−	0.878042i	$-0.658850\pi$
$151$	−11.7619	−0.957168	−0.478584	+	0.878042i	$0.658850\pi$
$152$	−1.48373	−0.120347
$153$	0	0
$154$	0.624772	0.0503456
$155$	−8.25046	−0.662693
$156$	0	0
$157$	14.7885	1.18025	0.590126	−	0.807311i	$-0.299078\pi$
$157$	14.7885	1.18025	0.590126	+	0.807311i	$0.299078\pi$
$158$	−11.4058	−0.907397
$159$	0	0
$160$	−9.53511	−0.753817
$161$	−2.17266	−0.171230
$162$	0	0
$163$	−20.8054	−1.62961	−0.814803	−	0.579739i	$-0.803155\pi$
$163$	−20.8054	−1.62961	−0.814803	+	0.579739i	$0.803155\pi$
$164$	6.89505	0.538413
$165$	0	0
$166$	−13.7535	−1.06748
$167$	13.2699	1.02686	0.513430	−	0.858132i	$-0.328375\pi$
$167$	13.2699	1.02686	0.513430	+	0.858132i	$0.328375\pi$
$168$	0	0
$169$	−12.9773	−0.998251
$170$	0.230816	0.0177028
$171$	0	0
$172$	−6.86204	−0.523226
$173$	−12.4101	−0.943521	−0.471760	−	0.881727i	$-0.656381\pi$
$173$	−12.4101	−0.943521	−0.471760	+	0.881727i	$0.656381\pi$
$174$	0	0
$175$	−1.80582	−0.136507
$176$	−2.42625	−0.182885
$177$	0	0
$178$	−15.0154	−1.12545
$179$	−8.62506	−0.644668	−0.322334	−	0.946626i	$-0.604467\pi$
$179$	−8.62506	−0.644668	−0.322334	+	0.946626i	$0.604467\pi$
$180$	0	0
$181$	−1.29192	−0.0960277	−0.0480139	−	0.998847i	$-0.515289\pi$
$181$	−1.29192	−0.0960277	−0.0480139	+	0.998847i	$0.515289\pi$
$182$	−0.0942169	−0.00698382
$183$	0	0
$184$	12.6402	0.931849
$185$	17.4869	1.28566
$186$	0	0
$187$	−0.0673519	−0.00492526
$188$	−2.36260	−0.172310
$189$	0	0
$190$	1.65266	0.119897
$191$	−9.07558	−0.656686	−0.328343	−	0.944559i	$-0.606490\pi$
$191$	−9.07558	−0.656686	−0.328343	+	0.944559i	$0.606490\pi$
$192$	0	0
$193$	21.0767	1.51713	0.758566	−	0.651596i	$-0.225900\pi$
$193$	21.0767	1.51713	0.758566	+	0.651596i	$0.225900\pi$
$194$	1.97146	0.141543
$195$	0	0
$196$	4.06104	0.290074
$197$	−6.18993	−0.441014	−0.220507	−	0.975385i	$-0.570771\pi$
$197$	−6.18993	−0.441014	−0.220507	+	0.975385i	$0.570771\pi$
$198$	0	0
$199$	−20.3389	−1.44179	−0.720894	−	0.693045i	$-0.756269\pi$
$199$	−20.3389	−1.44179	−0.720894	+	0.693045i	$0.756269\pi$
$200$	10.5060	0.742887
$201$	0	0
$202$	8.52045	0.599496
$203$	−3.08550	−0.216560
$204$	0	0
$205$	−33.0986	−2.31171
$206$	−12.7265	−0.886697
$207$	0	0
$208$	0.365883	0.0253694
$209$	−0.482246	−0.0333576
$210$	0	0
$211$	28.7585	1.97981	0.989907	−	0.141717i	$-0.0452623\pi$
$211$	28.7585	1.97981	0.989907	+	0.141717i	$0.0452623\pi$
$212$	−0.355539	−0.0244185
$213$	0	0
$214$	−3.44323	−0.235374
$215$	32.9401	2.24650
$216$	0	0
$217$	1.50412	0.102107
$218$	18.4963	1.25273
$219$	0	0
$220$	−1.75293	−0.118183
$221$	0.0101568	0.000683220 0
$222$	0	0
$223$	−21.4498	−1.43639	−0.718194	−	0.695843i	$-0.755031\pi$
$223$	−21.4498	−1.43639	−0.718194	+	0.695843i	$0.755031\pi$
$224$	1.73833	0.116147
$225$	0	0
$226$	−20.8946	−1.38989
$227$	−20.1374	−1.33657	−0.668283	−	0.743908i	$-0.732970\pi$
$227$	−20.1374	−1.33657	−0.668283	+	0.743908i	$0.732970\pi$
$228$	0	0
$229$	23.4322	1.54844	0.774221	−	0.632915i	$-0.218142\pi$
$229$	23.4322	1.54844	0.774221	+	0.632915i	$0.218142\pi$
$230$	−14.0794	−0.928368
$231$	0	0
$232$	17.9510	1.17854
$233$	1.49700	0.0980719	0.0490360	−	0.998797i	$-0.484385\pi$
$233$	1.49700	0.0980719	0.0490360	+	0.998797i	$0.484385\pi$
$234$	0	0
$235$	11.3413	0.739824
$236$	7.21476	0.469641
$237$	0	0
$238$	−0.0420796	−0.00272761
$239$	−11.3579	−0.734680	−0.367340	−	0.930087i	$-0.619731\pi$
$239$	−11.3579	−0.734680	−0.367340	+	0.930087i	$0.619731\pi$
$240$	0	0
$241$	19.2590	1.24058	0.620290	−	0.784373i	$-0.287015\pi$
$241$	19.2590	1.24058	0.620290	+	0.784373i	$0.287015\pi$
$242$	−1.18140	−0.0759433
$243$	0	0
$244$	−0.604292	−0.0386858
$245$	−19.4944	−1.24545
$246$	0	0
$247$	0.0727237	0.00462730
$248$	−8.75077	−0.555675
$249$	0	0
$250$	5.43290	0.343607
$251$	9.41617	0.594344	0.297172	−	0.954824i	$-0.403957\pi$
$251$	9.41617	0.594344	0.297172	+	0.954824i	$0.403957\pi$
$252$	0	0
$253$	4.10835	0.258290
$254$	24.6279	1.54529
$255$	0	0
$256$	−13.0456	−0.815353
$257$	−15.6633	−0.977049	−0.488524	−	0.872550i	$-0.662465\pi$
$257$	−15.6633	−0.977049	−0.488524	+	0.872550i	$0.662465\pi$
$258$	0	0
$259$	−3.18800	−0.198092
$260$	0.264346	0.0163940
$261$	0	0
$262$	−10.1823	−0.629065
$263$	7.59913	0.468582	0.234291	−	0.972166i	$-0.424723\pi$
$263$	7.59913	0.468582	0.234291	+	0.972166i	$0.424723\pi$
$264$	0	0
$265$	1.70671	0.104842
$266$	−0.301294	−0.0184735
$267$	0	0
$268$	−5.66902	−0.346290
$269$	24.2461	1.47831	0.739154	−	0.673536i	$-0.235226\pi$
$269$	24.2461	1.47831	0.739154	+	0.673536i	$0.235226\pi$
$270$	0	0
$271$	3.99382	0.242607	0.121304	−	0.992615i	$-0.461293\pi$
$271$	3.99382	0.242607	0.121304	+	0.992615i	$0.461293\pi$
$272$	0.163412	0.00990833
$273$	0	0
$274$	3.91385	0.236445
$275$	3.41469	0.205913
$276$	0	0
$277$	7.93878	0.476995	0.238497	−	0.971143i	$-0.423345\pi$
$277$	7.93878	0.476995	0.238497	+	0.971143i	$0.423345\pi$
$278$	−20.0467	−1.20232
$279$	0	0
$280$	−4.71987	−0.282066
$281$	−3.12368	−0.186343	−0.0931717	−	0.995650i	$-0.529701\pi$
$281$	−3.12368	−0.186343	−0.0931717	+	0.995650i	$0.529701\pi$
$282$	0	0
$283$	5.18624	0.308290	0.154145	−	0.988048i	$-0.450738\pi$
$283$	5.18624	0.308290	0.154145	+	0.988048i	$0.450738\pi$
$284$	7.66077	0.454583
$285$	0	0
$286$	0.178158	0.0105347
$287$	6.03414	0.356184
$288$	0	0
$289$	−16.9955	−0.999733
$290$	−19.9948	−1.17414
$291$	0	0
$292$	−7.89749	−0.462166
$293$	22.5194	1.31560	0.657799	−	0.753194i	$-0.271488\pi$
$293$	22.5194	1.31560	0.657799	+	0.753194i	$0.271488\pi$
$294$	0	0
$295$	−34.6333	−2.01643
$296$	18.5473	1.07804
$297$	0	0
$298$	−12.4907	−0.723567
$299$	−0.619548	−0.0358294
$300$	0	0
$301$	−6.00525	−0.346137
$302$	13.8955	0.799596
$303$	0	0
$304$	1.17005	0.0671069
$305$	2.90081	0.166100
$306$	0	0
$307$	−6.51791	−0.371997	−0.185998	−	0.982550i	$-0.559552\pi$
$307$	−6.51791	−0.371997	−0.185998	+	0.982550i	$0.559552\pi$
$308$	0.319574	0.0182094
$309$	0	0
$310$	9.74711	0.553598
$311$	−17.4530	−0.989667	−0.494834	−	0.868988i	$-0.664771\pi$
$311$	−17.4530	−0.989667	−0.494834	+	0.868988i	$0.664771\pi$
$312$	0	0
$313$	20.6548	1.16748	0.583739	−	0.811941i	$-0.301589\pi$
$313$	20.6548	1.16748	0.583739	+	0.811941i	$0.301589\pi$
$314$	−17.4712	−0.985955
$315$	0	0
$316$	−5.83412	−0.328195
$317$	13.7550	0.772558	0.386279	−	0.922382i	$-0.373760\pi$
$317$	13.7550	0.772558	0.386279	+	0.922382i	$0.373760\pi$
$318$	0	0
$319$	5.83447	0.326668
$320$	25.3409	1.41660
$321$	0	0
$322$	2.56678	0.143041
$323$	0.0324802	0.00180724
$324$	0	0
$325$	−0.514942	−0.0285638
$326$	24.5795	1.36133
$327$	0	0
$328$	−35.1057	−1.93839
$329$	−2.06761	−0.113991
$330$	0	0
$331$	26.0426	1.43143	0.715716	−	0.698392i	$-0.246101\pi$
$331$	26.0426	1.43143	0.715716	+	0.698392i	$0.246101\pi$
$332$	−7.03495	−0.386093
$333$	0	0
$334$	−15.6771	−0.857814
$335$	27.2132	1.48682
$336$	0	0
$337$	32.9988	1.79756	0.898781	−	0.438399i	$-0.144454\pi$
$337$	32.9988	1.79756	0.898781	+	0.438399i	$0.144454\pi$
$338$	15.3313	0.833915
$339$	0	0
$340$	0.118063	0.00640289
$341$	−2.84419	−0.154022
$342$	0	0
$343$	7.25586	0.391780
$344$	34.9376	1.88371
$345$	0	0
$346$	14.6613	0.788195
$347$	−26.3834	−1.41633	−0.708166	−	0.706045i	$-0.750477\pi$
$347$	−26.3834	−1.41633	−0.708166	+	0.706045i	$0.750477\pi$
$348$	0	0
$349$	−14.5133	−0.776881	−0.388441	−	0.921474i	$-0.626986\pi$
$349$	−14.5133	−0.776881	−0.388441	+	0.921474i	$0.626986\pi$
$350$	2.13340	0.114035
$351$	0	0
$352$	−3.28705	−0.175201
$353$	−34.6089	−1.84205	−0.921024	−	0.389507i	$-0.872645\pi$
$353$	−34.6089	−1.84205	−0.921024	+	0.389507i	$0.872645\pi$
$354$	0	0
$355$	−36.7743	−1.95178
$356$	−7.68045	−0.407063
$357$	0	0
$358$	10.1897	0.538540
$359$	−19.3153	−1.01942	−0.509712	−	0.860345i	$-0.670248\pi$
$359$	−19.3153	−1.01942	−0.509712	+	0.860345i	$0.670248\pi$
$360$	0	0
$361$	−18.7674	−0.987760
$362$	1.52628	0.0802193
$363$	0	0
$364$	−0.0481924	−0.00252597
$365$	37.9107	1.98434
$366$	0	0
$367$	14.0802	0.734979	0.367489	−	0.930028i	$-0.380217\pi$
$367$	14.0802	0.734979	0.367489	+	0.930028i	$0.380217\pi$
$368$	−9.96788	−0.519612
$369$	0	0
$370$	−20.6590	−1.07401
$371$	−0.311147	−0.0161539
$372$	0	0
$373$	26.0491	1.34877	0.674386	−	0.738379i	$-0.264409\pi$
$373$	26.0491	1.34877	0.674386	+	0.738379i	$0.264409\pi$
$374$	0.0795696	0.00411445
$375$	0	0
$376$	12.0290	0.620349
$377$	−0.879850	−0.0453146
$378$	0	0
$379$	14.8469	0.762635	0.381317	−	0.924444i	$-0.375470\pi$
$379$	14.8469	0.762635	0.381317	+	0.924444i	$0.375470\pi$
$380$	0.845345	0.0433653
$381$	0	0
$382$	10.7219	0.548580
$383$	−11.6961	−0.597643	−0.298821	−	0.954309i	$-0.596593\pi$
$383$	−11.6961	−0.597643	−0.298821	+	0.954309i	$0.596593\pi$
$384$	0	0
$385$	−1.53406	−0.0781831
$386$	−24.9000	−1.26738
$387$	0	0
$388$	1.00841	0.0511943
$389$	14.7146	0.746059	0.373029	−	0.927820i	$-0.378319\pi$
$389$	14.7146	0.746059	0.373029	+	0.927820i	$0.378319\pi$
$390$	0	0
$391$	−0.276705	−0.0139936
$392$	−20.6765	−1.04432
$393$	0	0
$394$	7.31279	0.368413
$395$	28.0058	1.40912
$396$	0	0
$397$	29.4067	1.47588	0.737940	−	0.674867i	$-0.235799\pi$
$397$	29.4067	1.47588	0.737940	+	0.674867i	$0.235799\pi$
$398$	24.0284	1.20444
$399$	0	0
$400$	−8.28488	−0.414244
$401$	17.6578	0.881786	0.440893	−	0.897560i	$-0.354662\pi$
$401$	17.6578	0.881786	0.440893	+	0.897560i	$0.354662\pi$
$402$	0	0
$403$	0.428910	0.0213656
$404$	4.35824	0.216831
$405$	0	0
$406$	3.64522	0.180909
$407$	6.02828	0.298811
$408$	0	0
$409$	−11.0602	−0.546893	−0.273447	−	0.961887i	$-0.588164\pi$
$409$	−11.0602	−0.546893	−0.273447	+	0.961887i	$0.588164\pi$
$410$	39.1027	1.93115
$411$	0	0
$412$	−6.50965	−0.320708
$413$	6.31393	0.310688
$414$	0	0
$415$	33.7702	1.65771
$416$	0.495694	0.0243034
$417$	0	0
$418$	0.569726	0.0278662
$419$	−9.64189	−0.471037	−0.235519	−	0.971870i	$-0.575679\pi$
$419$	−9.64189	−0.471037	−0.235519	+	0.971870i	$0.575679\pi$
$420$	0	0
$421$	31.6981	1.54487	0.772435	−	0.635094i	$-0.219039\pi$
$421$	31.6981	1.54487	0.772435	+	0.635094i	$0.219039\pi$
$422$	−33.9753	−1.65389
$423$	0	0
$424$	1.81021	0.0879114
$425$	−0.229986	−0.0111559
$426$	0	0
$427$	−0.528840	−0.0255924
$428$	−1.76123	−0.0851322
$429$	0	0
$430$	−38.9155	−1.87667
$431$	19.2048	0.925061	0.462530	−	0.886603i	$-0.346942\pi$
$431$	19.2048	0.925061	0.462530	+	0.886603i	$0.346942\pi$
$432$	0	0
$433$	−13.3020	−0.639254	−0.319627	−	0.947544i	$-0.603558\pi$
$433$	−13.3020	−0.639254	−0.319627	+	0.947544i	$0.603558\pi$
$434$	−1.77697	−0.0852975
$435$	0	0
$436$	9.46094	0.453097
$437$	−1.98124	−0.0947753
$438$	0	0
$439$	37.7505	1.80174	0.900868	−	0.434094i	$-0.142931\pi$
$439$	37.7505	1.80174	0.900868	+	0.434094i	$0.142931\pi$
$440$	8.92495	0.425480
$441$	0	0
$442$	−0.0119993	−0.000570746 0
$443$	−12.5575	−0.596623	−0.298311	−	0.954469i	$-0.596423\pi$
$443$	−12.5575	−0.596623	−0.298311	+	0.954469i	$0.596423\pi$
$444$	0	0
$445$	36.8688	1.74775
$446$	25.3409	1.19992
$447$	0	0
$448$	−4.61985	−0.218268
$449$	27.8448	1.31408	0.657040	−	0.753856i	$-0.271808\pi$
$449$	27.8448	1.31408	0.657040	+	0.753856i	$0.271808\pi$
$450$	0	0
$451$	−11.4101	−0.537282
$452$	−10.6877	−0.502707
$453$	0	0
$454$	23.7903	1.11654
$455$	0.231340	0.0108454
$456$	0	0
$457$	−15.9549	−0.746338	−0.373169	−	0.927763i	$-0.621729\pi$
$457$	−15.9549	−0.746338	−0.373169	+	0.927763i	$0.621729\pi$
$458$	−27.6828	−1.29353
$459$	0	0
$460$	−7.20167	−0.335780
$461$	21.5790	1.00503	0.502517	−	0.864568i	$-0.332408\pi$
$461$	21.5790	1.00503	0.502517	+	0.864568i	$0.332408\pi$
$462$	0	0
$463$	−1.41157	−0.0656011	−0.0328006	−	0.999462i	$-0.510443\pi$
$463$	−1.41157	−0.0656011	−0.0328006	+	0.999462i	$0.510443\pi$
$464$	−14.1559	−0.657170
$465$	0	0
$466$	−1.76856	−0.0819270
$467$	1.05698	0.0489112	0.0244556	−	0.999701i	$-0.492215\pi$
$467$	1.05698	0.0489112	0.0244556	+	0.999701i	$0.492215\pi$
$468$	0	0
$469$	−4.96119	−0.229086
$470$	−13.3986	−0.618032
$471$	0	0
$472$	−36.7335	−1.69080
$473$	11.3555	0.522127
$474$	0	0
$475$	−1.64672	−0.0755566
$476$	−0.0215239	−0.000986546 0
$477$	0	0
$478$	13.4182	0.613734
$479$	24.6689	1.12715	0.563575	−	0.826065i	$-0.309426\pi$
$479$	24.6689	1.12715	0.563575	+	0.826065i	$0.309426\pi$
$480$	0	0
$481$	−0.909077	−0.0414503
$482$	−22.7526	−1.03635
$483$	0	0
$484$	−0.604292	−0.0274678
$485$	−4.84072	−0.219806
$486$	0	0
$487$	8.86822	0.401857	0.200929	−	0.979606i	$-0.435604\pi$
$487$	8.86822	0.401857	0.200929	+	0.979606i	$0.435604\pi$
$488$	3.07671	0.139276
$489$	0	0
$490$	23.0307	1.04042
$491$	−14.8167	−0.668670	−0.334335	−	0.942454i	$-0.608512\pi$
$491$	−14.8167	−0.668670	−0.334335	+	0.942454i	$0.608512\pi$
$492$	0	0
$493$	−0.392963	−0.0176981
$494$	−0.0859158	−0.00386553
$495$	0	0
$496$	6.90072	0.309852
$497$	6.70425	0.300727
$498$	0	0
$499$	−1.89914	−0.0850174	−0.0425087	−	0.999096i	$-0.513535\pi$
$499$	−1.89914	−0.0850174	−0.0425087	+	0.999096i	$0.513535\pi$
$500$	2.77895	0.124278
$501$	0	0
$502$	−11.1243	−0.496501
$503$	−22.0264	−0.982109	−0.491055	−	0.871129i	$-0.663388\pi$
$503$	−22.0264	−0.982109	−0.491055	+	0.871129i	$0.663388\pi$
$504$	0	0
$505$	−20.9211	−0.930976
$506$	−4.85361	−0.215769
$507$	0	0
$508$	12.5973	0.558914
$509$	32.0102	1.41882	0.709412	−	0.704794i	$-0.248960\pi$
$509$	32.0102	1.41882	0.709412	+	0.704794i	$0.248960\pi$
$510$	0	0
$511$	−6.91142	−0.305743
$512$	22.9049	1.01227
$513$	0	0
$514$	18.5046	0.816204
$515$	31.2486	1.37698
$516$	0	0
$517$	3.90970	0.171948
$518$	3.76630	0.165482
$519$	0	0
$520$	−1.34590	−0.0590217
$521$	−28.1925	−1.23514	−0.617569	−	0.786517i	$-0.711882\pi$
$521$	−28.1925	−1.23514	−0.617569	+	0.786517i	$0.711882\pi$
$522$	0	0
$523$	−8.41440	−0.367936	−0.183968	−	0.982932i	$-0.558894\pi$
$523$	−8.41440	−0.367936	−0.183968	+	0.982932i	$0.558894\pi$
$524$	−5.20829	−0.227525
$525$	0	0
$526$	−8.97762	−0.391443
$527$	0.191562	0.00834457
$528$	0	0
$529$	−6.12144	−0.266150
$530$	−2.01631	−0.0875829
$531$	0	0
$532$	−0.154113	−0.00668165
$533$	1.72067	0.0745306
$534$	0	0
$535$	8.45450	0.365520
$536$	28.8634	1.24671
$537$	0	0
$538$	−28.6443	−1.23494
$539$	−6.72033	−0.289465
$540$	0	0
$541$	27.0427	1.16266	0.581329	−	0.813669i	$-0.302533\pi$
$541$	27.0427	1.16266	0.581329	+	0.813669i	$0.302533\pi$
$542$	−4.71830	−0.202668
$543$	0	0
$544$	0.221389	0.00949199
$545$	−45.4158	−1.94540
$546$	0	0
$547$	16.9856	0.726253	0.363126	−	0.931740i	$-0.381709\pi$
$547$	16.9856	0.726253	0.363126	+	0.931740i	$0.381709\pi$
$548$	2.00195	0.0855192
$549$	0	0
$550$	−4.03411	−0.172015
$551$	−2.81365	−0.119865
$552$	0	0
$553$	−5.10567	−0.217115
$554$	−9.37888	−0.398470
$555$	0	0
$556$	−10.2540	−0.434865
$557$	29.1185	1.23379	0.616896	−	0.787045i	$-0.288390\pi$
$557$	29.1185	1.23379	0.616896	+	0.787045i	$0.288390\pi$
$558$	0	0
$559$	−1.71243	−0.0724282
$560$	3.72202	0.157284
$561$	0	0
$562$	3.69032	0.155667
$563$	−33.1163	−1.39569	−0.697843	−	0.716250i	$-0.745857\pi$
$563$	−33.1163	−1.39569	−0.697843	+	0.716250i	$0.745857\pi$
$564$	0	0
$565$	51.3046	2.15840
$566$	−6.12703	−0.257538
$567$	0	0
$568$	−39.0043	−1.63659
$569$	9.13425	0.382928	0.191464	−	0.981500i	$-0.438677\pi$
$569$	9.13425	0.382928	0.191464	+	0.981500i	$0.438677\pi$
$570$	0	0
$571$	31.6452	1.32431	0.662156	−	0.749366i	$-0.269642\pi$
$571$	31.6452	1.32431	0.662156	+	0.749366i	$0.269642\pi$
$572$	0.0911284	0.00381027
$573$	0	0
$574$	−7.12874	−0.297548
$575$	14.0287	0.585039
$576$	0	0
$577$	−5.78481	−0.240825	−0.120412	−	0.992724i	$-0.538422\pi$
$577$	−5.78481	−0.240825	−0.120412	+	0.992724i	$0.538422\pi$
$578$	20.0785	0.835154
$579$	0	0
$580$	−10.2274	−0.424671
$581$	−6.15657	−0.255418
$582$	0	0
$583$	0.588357	0.0243673
$584$	40.2096	1.66388
$585$	0	0
$586$	−26.6044	−1.09902
$587$	8.04522	0.332062	0.166031	−	0.986121i	$-0.446905\pi$
$587$	8.04522	0.332062	0.166031	+	0.986121i	$0.446905\pi$
$588$	0	0
$589$	1.37160	0.0565158
$590$	40.9158	1.68448
$591$	0	0
$592$	−14.6261	−0.601129
$593$	7.77065	0.319103	0.159551	−	0.987190i	$-0.448995\pi$
$593$	7.77065	0.319103	0.159551	+	0.987190i	$0.448995\pi$
$594$	0	0
$595$	0.103322	0.00423579
$596$	−6.38905	−0.261706
$597$	0	0
$598$	0.731935	0.0299310
$599$	−5.13824	−0.209943	−0.104971	−	0.994475i	$-0.533475\pi$
$599$	−5.13824	−0.209943	−0.104971	+	0.994475i	$0.533475\pi$
$600$	0	0
$601$	−18.9195	−0.771744	−0.385872	−	0.922552i	$-0.626099\pi$
$601$	−18.9195	−0.771744	−0.385872	+	0.922552i	$0.626099\pi$
$602$	7.09461	0.289155
$603$	0	0
$604$	7.10760	0.289204
$605$	2.90081	0.117935
$606$	0	0
$607$	31.8620	1.29324	0.646620	−	0.762812i	$-0.276182\pi$
$607$	31.8620	1.29324	0.646620	+	0.762812i	$0.276182\pi$
$608$	1.58517	0.0642870
$609$	0	0
$610$	−3.42702	−0.138756
$611$	−0.589591	−0.0238523
$612$	0	0
$613$	−14.7696	−0.596539	−0.298269	−	0.954482i	$-0.596409\pi$
$613$	−14.7696	−0.596539	−0.298269	+	0.954482i	$0.596409\pi$
$614$	7.70027	0.310758
$615$	0	0
$616$	−1.62709	−0.0655573
$617$	45.2708	1.82253	0.911266	−	0.411817i	$-0.135106\pi$
$617$	45.2708	1.82253	0.911266	+	0.411817i	$0.135106\pi$
$618$	0	0
$619$	39.0064	1.56780	0.783900	−	0.620887i	$-0.213227\pi$
$619$	39.0064	1.56780	0.783900	+	0.620887i	$0.213227\pi$
$620$	4.98569	0.200230
$621$	0	0
$622$	20.6190	0.826745
$623$	−6.72147	−0.269290
$624$	0	0
$625$	−30.4133	−1.21653
$626$	−24.4016	−0.975284
$627$	0	0
$628$	−8.93657	−0.356608
$629$	−0.406016	−0.0161889
$630$	0	0
$631$	−37.8923	−1.50847	−0.754235	−	0.656605i	$-0.771992\pi$
$631$	−37.8923	−1.50847	−0.754235	+	0.656605i	$0.771992\pi$
$632$	29.7040	1.18156
$633$	0	0
$634$	−16.2502	−0.645377
$635$	−60.4713	−2.39973
$636$	0	0
$637$	1.01344	0.0401539
$638$	−6.89285	−0.272891
$639$	0	0
$640$	−10.8676	−0.429580
$641$	15.2355	0.601767	0.300883	−	0.953661i	$-0.402719\pi$
$641$	15.2355	0.601767	0.300883	+	0.953661i	$0.402719\pi$
$642$	0	0
$643$	0.685252	0.0270237	0.0135118	−	0.999909i	$-0.495699\pi$
$643$	0.685252	0.0270237	0.0135118	+	0.999909i	$0.495699\pi$
$644$	1.31292	0.0517363
$645$	0	0
$646$	−0.0383721	−0.00150973
$647$	−32.3211	−1.27067	−0.635337	−	0.772235i	$-0.719139\pi$
$647$	−32.3211	−1.27067	−0.635337	+	0.772235i	$0.719139\pi$
$648$	0	0
$649$	−11.9392	−0.468655
$650$	0.608353	0.0238616
$651$	0	0
$652$	12.5725	0.492378
$653$	19.3514	0.757281	0.378640	−	0.925544i	$-0.376392\pi$
$653$	19.3514	0.757281	0.378640	+	0.925544i	$0.376392\pi$
$654$	0	0
$655$	25.0016	0.976893
$656$	27.6838	1.08087
$657$	0	0
$658$	2.44267	0.0952253
$659$	−35.8402	−1.39613	−0.698067	−	0.716032i	$-0.745956\pi$
$659$	−35.8402	−1.39613	−0.698067	+	0.716032i	$0.745956\pi$
$660$	0	0
$661$	22.2012	0.863526	0.431763	−	0.901987i	$-0.357892\pi$
$661$	22.2012	0.863526	0.431763	+	0.901987i	$0.357892\pi$
$662$	−30.7668	−1.19578
$663$	0	0
$664$	35.8180	1.39001
$665$	0.739796	0.0286880
$666$	0	0
$667$	23.9701	0.928125
$668$	−8.01892	−0.310261
$669$	0	0
$670$	−32.1497	−1.24205
$671$	1.00000	0.0386046
$672$	0	0
$673$	−1.26748	−0.0488576	−0.0244288	−	0.999702i	$-0.507777\pi$
$673$	−1.26748	−0.0488576	−0.0244288	+	0.999702i	$0.507777\pi$
$674$	−38.9849	−1.50164
$675$	0	0
$676$	7.84205	0.301617
$677$	−22.7065	−0.872681	−0.436340	−	0.899782i	$-0.643726\pi$
$677$	−22.7065	−0.872681	−0.436340	+	0.899782i	$0.643726\pi$
$678$	0	0
$679$	0.882501	0.0338673
$680$	−0.601112	−0.0230516
$681$	0	0
$682$	3.36013	0.128666
$683$	34.5294	1.32123	0.660616	−	0.750724i	$-0.270295\pi$
$683$	34.5294	1.32123	0.660616	+	0.750724i	$0.270295\pi$
$684$	0	0
$685$	−9.61006	−0.367182
$686$	−8.57208	−0.327284
$687$	0	0
$688$	−27.5513	−1.05038
$689$	−0.0887255	−0.00338017
$690$	0	0
$691$	−22.4560	−0.854265	−0.427132	−	0.904189i	$-0.640476\pi$
$691$	−22.4560	−0.854265	−0.427132	+	0.904189i	$0.640476\pi$
$692$	7.49931	0.285081
$693$	0	0
$694$	31.1693	1.18317
$695$	49.2226	1.86712
$696$	0	0
$697$	0.768494	0.0291088
$698$	17.1461	0.648988
$699$	0	0
$700$	1.09124	0.0412451
$701$	31.5776	1.19267	0.596335	−	0.802736i	$-0.296623\pi$
$701$	31.5776	1.19267	0.596335	+	0.802736i	$0.296623\pi$
$702$	0	0
$703$	−2.90711	−0.109644
$704$	8.73582	0.329244
$705$	0	0
$706$	40.8870	1.53880
$707$	3.81408	0.143443
$708$	0	0
$709$	−9.16530	−0.344210	−0.172105	−	0.985079i	$-0.555057\pi$
$709$	−9.16530	−0.344210	−0.172105	+	0.985079i	$0.555057\pi$
$710$	43.4453	1.63047
$711$	0	0
$712$	39.1045	1.46550
$713$	−11.6850	−0.437605
$714$	0	0
$715$	−0.437448	−0.0163596
$716$	5.21205	0.194784
$717$	0	0
$718$	22.8191	0.851603
$719$	3.15397	0.117623	0.0588116	−	0.998269i	$-0.481269\pi$
$719$	3.15397	0.117623	0.0588116	+	0.998269i	$0.481269\pi$
$720$	0	0
$721$	−5.69686	−0.212162
$722$	22.1719	0.825152
$723$	0	0
$724$	0.780697	0.0290144
$725$	19.9229	0.739917
$726$	0	0
$727$	39.5017	1.46504	0.732519	−	0.680747i	$-0.238345\pi$
$727$	39.5017	1.46504	0.732519	+	0.680747i	$0.238345\pi$
$728$	0.245368	0.00909396
$729$	0	0
$730$	−44.7877	−1.65767
$731$	−0.764815	−0.0282877
$732$	0	0
$733$	23.5163	0.868595	0.434297	−	0.900770i	$-0.356997\pi$
$733$	23.5163	0.868595	0.434297	+	0.900770i	$0.356997\pi$
$734$	−16.6343	−0.613984
$735$	0	0
$736$	−13.5044	−0.497778
$737$	9.38126	0.345563
$738$	0	0
$739$	28.9368	1.06446	0.532228	−	0.846601i	$-0.321355\pi$
$739$	28.9368	1.06446	0.532228	+	0.846601i	$0.321355\pi$
$740$	−10.5672	−0.388457
$741$	0	0
$742$	0.367589	0.0134946
$743$	36.1642	1.32674	0.663369	−	0.748293i	$-0.269126\pi$
$743$	36.1642	1.32674	0.663369	+	0.748293i	$0.269126\pi$
$744$	0	0
$745$	30.6696	1.12365
$746$	−30.7744	−1.12673
$747$	0	0
$748$	0.0407002	0.00148815
$749$	−1.54132	−0.0563187
$750$	0	0
$751$	−37.9916	−1.38633	−0.693167	−	0.720777i	$-0.743785\pi$
$751$	−37.9916	−1.38633	−0.693167	+	0.720777i	$0.743785\pi$
$752$	−9.48590	−0.345915
$753$	0	0
$754$	1.03946	0.0378548
$755$	−34.1189	−1.24172
$756$	0	0
$757$	−6.48633	−0.235750	−0.117875	−	0.993028i	$-0.537608\pi$
$757$	−6.48633	−0.235750	−0.117875	+	0.993028i	$0.537608\pi$
$758$	−17.5402	−0.637087
$759$	0	0
$760$	−4.30402	−0.156123
$761$	0.142118	0.00515178	0.00257589	−	0.999997i	$-0.499180\pi$
$761$	0.142118	0.00515178	0.00257589	+	0.999997i	$0.499180\pi$
$762$	0	0
$763$	8.27965	0.299743
$764$	5.48430	0.198415
$765$	0	0
$766$	13.8178	0.499257
$767$	1.80046	0.0650107
$768$	0	0
$769$	37.9256	1.36763	0.683816	−	0.729655i	$-0.260319\pi$
$769$	37.9256	1.36763	0.683816	+	0.729655i	$0.260319\pi$
$770$	1.81234	0.0653124
$771$	0	0
$772$	−12.7365	−0.458395
$773$	−7.81766	−0.281182	−0.140591	−	0.990068i	$-0.544900\pi$
$773$	−7.81766	−0.281182	−0.140591	+	0.990068i	$0.544900\pi$
$774$	0	0
$775$	−9.71203	−0.348867
$776$	−5.13426	−0.184309
$777$	0	0
$778$	−17.3838	−0.623240
$779$	5.50249	0.197147
$780$	0	0
$781$	−12.6773	−0.453629
$782$	0.326900	0.0116899
$783$	0	0
$784$	16.3052	0.582328
$785$	42.8986	1.53112
$786$	0	0
$787$	20.2740	0.722692	0.361346	−	0.932432i	$-0.382317\pi$
$787$	20.2740	0.722692	0.361346	+	0.932432i	$0.382317\pi$
$788$	3.74052	0.133251
$789$	0	0
$790$	−33.0860	−1.17715
$791$	−9.35324	−0.332563
$792$	0	0
$793$	−0.150802	−0.00535514
$794$	−34.7411	−1.23292
$795$	0	0
$796$	12.2906	0.435630
$797$	−5.99639	−0.212403	−0.106201	−	0.994345i	$-0.533869\pi$
$797$	−5.99639	−0.212403	−0.106201	+	0.994345i	$0.533869\pi$
$798$	0	0
$799$	−0.263326	−0.00931579
$800$	−11.2243	−0.396837
$801$	0	0
$802$	−20.8609	−0.736624
$803$	13.0690	0.461195
$804$	0	0
$805$	−6.30247	−0.222133
$806$	−0.506715	−0.0178483
$807$	0	0
$808$	−22.1897	−0.780632
$809$	38.7682	1.36302	0.681508	−	0.731811i	$-0.261324\pi$
$809$	38.7682	1.36302	0.681508	+	0.731811i	$0.261324\pi$
$810$	0	0
$811$	−20.8789	−0.733159	−0.366579	−	0.930387i	$-0.619471\pi$
$811$	−20.8789	−0.733159	−0.366579	+	0.930387i	$0.619471\pi$
$812$	1.86454	0.0654326
$813$	0	0
$814$	−7.12182	−0.249620
$815$	−60.3525	−2.11406
$816$	0	0
$817$	−5.47615	−0.191586
$818$	13.0666	0.456862
$819$	0	0
$820$	20.0012	0.698473
$821$	−33.6755	−1.17528	−0.587642	−	0.809121i	$-0.699944\pi$
$821$	−33.6755	−1.17528	−0.587642	+	0.809121i	$0.699944\pi$
$822$	0	0
$823$	−48.6794	−1.69686	−0.848428	−	0.529311i	$-0.822450\pi$
$823$	−48.6794	−1.69686	−0.848428	+	0.529311i	$0.822450\pi$
$824$	33.1435	1.15461
$825$	0	0
$826$	−7.45928	−0.259542
$827$	−2.49408	−0.0867276	−0.0433638	−	0.999059i	$-0.513807\pi$
$827$	−2.49408	−0.0867276	−0.0433638	+	0.999059i	$0.513807\pi$
$828$	0	0
$829$	20.9559	0.727829	0.363915	−	0.931432i	$-0.381440\pi$
$829$	20.9559	0.727829	0.363915	+	0.931432i	$0.381440\pi$
$830$	−39.8962	−1.38482
$831$	0	0
$832$	−1.31738	−0.0456719
$833$	0.452627	0.0156826
$834$	0	0
$835$	38.4936	1.33212
$836$	0.291417	0.0100789
$837$	0	0
$838$	11.3909	0.393493
$839$	−46.2064	−1.59522	−0.797611	−	0.603172i	$-0.793903\pi$
$839$	−46.2064	−1.59522	−0.797611	+	0.603172i	$0.793903\pi$
$840$	0	0
$841$	5.04106	0.173830
$842$	−37.4481	−1.29055
$843$	0	0
$844$	−17.3785	−0.598193
$845$	−37.6445	−1.29501
$846$	0	0
$847$	−0.528840	−0.0181712
$848$	−1.42750	−0.0490206
$849$	0	0
$850$	0.271705	0.00931941
$851$	24.7663	0.848978
$852$	0	0
$853$	−12.1396	−0.415653	−0.207826	−	0.978166i	$-0.566639\pi$
$853$	−12.1396	−0.415653	−0.207826	+	0.978166i	$0.566639\pi$
$854$	0.624772	0.0213793
$855$	0	0
$856$	8.96718	0.306492
$857$	39.4393	1.34722	0.673610	−	0.739087i	$-0.264743\pi$
$857$	39.4393	1.34722	0.673610	+	0.739087i	$0.264743\pi$
$858$	0	0
$859$	−25.7414	−0.878285	−0.439143	−	0.898417i	$-0.644718\pi$
$859$	−25.7414	−0.878285	−0.439143	+	0.898417i	$0.644718\pi$
$860$	−19.9055	−0.678770
$861$	0	0
$862$	−22.6885	−0.772774
$863$	−15.9889	−0.544269	−0.272134	−	0.962259i	$-0.587730\pi$
$863$	−15.9889	−0.544269	−0.272134	+	0.962259i	$0.587730\pi$
$864$	0	0
$865$	−35.9992	−1.22401
$866$	15.7150	0.534018
$867$	0	0
$868$	−0.908930	−0.0308511
$869$	9.65447	0.327506
$870$	0	0
$871$	−1.41471	−0.0479357
$872$	−48.1698	−1.63123
$873$	0	0
$874$	2.34063	0.0791731
$875$	2.43197	0.0822157
$876$	0	0
$877$	49.8159	1.68216	0.841082	−	0.540908i	$-0.181919\pi$
$877$	49.8159	1.68216	0.841082	+	0.540908i	$0.181919\pi$
$878$	−44.5985	−1.50513
$879$	0	0
$880$	−7.03808	−0.237254
$881$	−26.3085	−0.886355	−0.443178	−	0.896434i	$-0.646149\pi$
$881$	−26.3085	−0.886355	−0.443178	+	0.896434i	$0.646149\pi$
$882$	0	0
$883$	−27.4250	−0.922925	−0.461462	−	0.887160i	$-0.652675\pi$
$883$	−27.4250	−0.922925	−0.461462	+	0.887160i	$0.652675\pi$
$884$	−0.00613767	−0.000206432 0
$885$	0	0
$886$	14.8354	0.498405
$887$	32.1898	1.08083	0.540413	−	0.841400i	$-0.318268\pi$
$887$	32.1898	1.08083	0.540413	+	0.841400i	$0.318268\pi$
$888$	0	0
$889$	11.0244	0.369746
$890$	−43.5568	−1.46003
$891$	0	0
$892$	12.9620	0.433998
$893$	−1.88544	−0.0630937
$894$	0	0
$895$	−25.0197	−0.836315
$896$	1.98125	0.0661889
$897$	0	0
$898$	−32.8959	−1.09775
$899$	−16.5944	−0.553453
$900$	0	0
$901$	−0.0396270	−0.00132017
$902$	13.4799	0.448833
$903$	0	0
$904$	54.4158	1.80984
$905$	−3.74762	−0.124575
$906$	0	0
$907$	−1.28331	−0.0426114	−0.0213057	−	0.999773i	$-0.506782\pi$
$907$	−1.28331	−0.0426114	−0.0213057	+	0.999773i	$0.506782\pi$
$908$	12.1689	0.403838
$909$	0	0
$910$	−0.273305	−0.00905998
$911$	49.4716	1.63907	0.819534	−	0.573030i	$-0.194232\pi$
$911$	49.4716	1.63907	0.819534	+	0.573030i	$0.194232\pi$
$912$	0	0
$913$	11.6417	0.385283
$914$	18.8491	0.623473
$915$	0	0
$916$	−14.1599	−0.467856
$917$	−4.55799	−0.150518
$918$	0	0
$919$	−29.7366	−0.980921	−0.490461	−	0.871463i	$-0.663171\pi$
$919$	−29.7366	−0.980921	−0.490461	+	0.871463i	$0.663171\pi$
$920$	36.6668	1.20887
$921$	0	0
$922$	−25.4934	−0.839581
$923$	1.91176	0.0629263
$924$	0	0
$925$	20.5847	0.676820
$926$	1.66763	0.0548017
$927$	0	0
$928$	−19.1782	−0.629556
$929$	−9.03584	−0.296456	−0.148228	−	0.988953i	$-0.547357\pi$
$929$	−9.03584	−0.296456	−0.148228	+	0.988953i	$0.547357\pi$
$930$	0	0
$931$	3.24085	0.106215
$932$	−0.904626	−0.0296320
$933$	0	0
$934$	−1.24872	−0.0408593
$935$	−0.195375	−0.00638944
$936$	0	0
$937$	−16.0261	−0.523551	−0.261776	−	0.965129i	$-0.584308\pi$
$937$	−16.0261	−0.523551	−0.261776	+	0.965129i	$0.584308\pi$
$938$	5.86115	0.191373
$939$	0	0
$940$	−6.85344	−0.223535
$941$	3.42319	0.111593	0.0557963	−	0.998442i	$-0.482230\pi$
$941$	3.42319	0.111593	0.0557963	+	0.998442i	$0.482230\pi$
$942$	0	0
$943$	−46.8769	−1.52652
$944$	28.9675	0.942811
$945$	0	0
$946$	−13.4154	−0.436173
$947$	−26.8391	−0.872153	−0.436076	−	0.899910i	$-0.643632\pi$
$947$	−26.8391	−0.872153	−0.436076	+	0.899910i	$0.643632\pi$
$948$	0	0
$949$	−1.97083	−0.0639760
$950$	1.94543	0.0631182
$951$	0	0
$952$	0.109588	0.00355175
$953$	−5.87575	−0.190334	−0.0951671	−	0.995461i	$-0.530339\pi$
$953$	−5.87575	−0.190334	−0.0951671	+	0.995461i	$0.530339\pi$
$954$	0	0
$955$	−26.3265	−0.851906
$956$	6.86347	0.221980
$957$	0	0
$958$	−29.1438	−0.941595
$959$	1.75199	0.0565747
$960$	0	0
$961$	−22.9106	−0.739050
$962$	1.07398	0.0346267
$963$	0	0
$964$	−11.6380	−0.374836
$965$	61.1394	1.96815
$966$	0	0
$967$	41.7810	1.34359	0.671793	−	0.740739i	$-0.265525\pi$
$967$	41.7810	1.34359	0.671793	+	0.740739i	$0.265525\pi$
$968$	3.07671	0.0988893
$969$	0	0
$970$	5.71883	0.183620
$971$	−30.2221	−0.969873	−0.484937	−	0.874549i	$-0.661157\pi$
$971$	−30.2221	−0.969873	−0.484937	+	0.874549i	$0.661157\pi$
$972$	0	0
$973$	−8.97366	−0.287682
$974$	−10.4769	−0.335702
$975$	0	0
$976$	−2.42625	−0.0776623
$977$	−12.5432	−0.401292	−0.200646	−	0.979664i	$-0.564304\pi$
$977$	−12.5432	−0.401292	−0.200646	+	0.979664i	$0.564304\pi$
$978$	0	0
$979$	12.7098	0.406208
$980$	11.7803	0.376308
$981$	0	0
$982$	17.5045	0.558591
$983$	−35.7613	−1.14061	−0.570304	−	0.821434i	$-0.693175\pi$
$983$	−35.7613	−1.14061	−0.570304	+	0.821434i	$0.693175\pi$
$984$	0	0
$985$	−17.9558	−0.572119
$986$	0.464246	0.0147846
$987$	0	0
$988$	−0.0439463	−0.00139812
$989$	46.6524	1.48346
$990$	0	0
$991$	26.5390	0.843039	0.421519	−	0.906819i	$-0.361497\pi$
$991$	26.5390	0.843039	0.421519	+	0.906819i	$0.361497\pi$
$992$	9.34902	0.296832
$993$	0	0
$994$	−7.92041	−0.251220
$995$	−58.9993	−1.87040
$996$	0	0
$997$	−0.481491	−0.0152490	−0.00762448	−	0.999971i	$-0.502427\pi$
$997$	−0.481491	−0.0152490	−0.00762448	+	0.999971i	$0.502427\pi$
$998$	2.24365	0.0710216
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.j.1.5		14
3.2	odd	2		2013.2.a.h.1.10	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.h.1.10	✓	14	3.2	odd	2
6039.2.a.j.1.5		14	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-1.18140 q^{2} -0.604292 q^{4} +2.90081 q^{5} -0.528840 q^{7} +3.07671 q^{8} +O(q^{10})\)	\(q-1.18140 q^{2} -0.604292 q^{4} +2.90081 q^{5} -0.528840 q^{7} +3.07671 q^{8} -3.42702 q^{10} +1.00000 q^{11} -0.150802 q^{13} +0.624772 q^{14} -2.42625 q^{16} -0.0673519 q^{17} -0.482246 q^{19} -1.75293 q^{20} -1.18140 q^{22} +4.10835 q^{23} +3.41469 q^{25} +0.178158 q^{26} +0.319574 q^{28} +5.83447 q^{29} -2.84419 q^{31} -3.28705 q^{32} +0.0795696 q^{34} -1.53406 q^{35} +6.02828 q^{37} +0.569726 q^{38} +8.92495 q^{40} -11.4101 q^{41} +11.3555 q^{43} -0.604292 q^{44} -4.85361 q^{46} +3.90970 q^{47} -6.72033 q^{49} -4.03411 q^{50} +0.0911284 q^{52} +0.588357 q^{53} +2.90081 q^{55} -1.62709 q^{56} -6.89285 q^{58} -11.9392 q^{59} +1.00000 q^{61} +3.36013 q^{62} +8.73582 q^{64} -0.437448 q^{65} +9.38126 q^{67} +0.0407002 q^{68} +1.81234 q^{70} -12.6773 q^{71} +13.0690 q^{73} -7.12182 q^{74} +0.291417 q^{76} -0.528840 q^{77} +9.65447 q^{79} -7.03808 q^{80} +13.4799 q^{82} +11.6417 q^{83} -0.195375 q^{85} -13.4154 q^{86} +3.07671 q^{88} +12.7098 q^{89} +0.0797502 q^{91} -2.48264 q^{92} -4.61892 q^{94} -1.39890 q^{95} -1.66875 q^{97} +7.93940 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more