LMFDB - Embedded newform 6039.2.a.g.1.4

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:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 55 x^{10} + 32 x^{9} - 266 x^{8} + 13 x^{7} + 534 x^{6} - 141 x^{5} + \cdots - 11 $ x^13 - 4x^12 - 11x^11 + 55x^10 + 32x^9 - 266x^8 + 13x^7 + 534x^6 - 141x^5 - 404x^4 + 98x^3 + 118x^2 - 16x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.88022$ 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)

$f(q)$	$=$	$q-1.88022 q^{2} +1.53524 q^{4} -1.27827 q^{5} +0.0298223 q^{7} +0.873847 q^{8} +O(q^{10})$	\(q-1.88022 q^{2} +1.53524 q^{4} -1.27827 q^{5} +0.0298223 q^{7} +0.873847 q^{8} +2.40344 q^{10} +1.00000 q^{11} +5.58822 q^{13} -0.0560727 q^{14} -4.71351 q^{16} -1.33955 q^{17} +6.70962 q^{19} -1.96246 q^{20} -1.88022 q^{22} -3.41259 q^{23} -3.36602 q^{25} -10.5071 q^{26} +0.0457846 q^{28} -1.60973 q^{29} -6.25509 q^{31} +7.11477 q^{32} +2.51865 q^{34} -0.0381211 q^{35} +1.71832 q^{37} -12.6156 q^{38} -1.11701 q^{40} -0.668829 q^{41} +2.57228 q^{43} +1.53524 q^{44} +6.41643 q^{46} +0.587690 q^{47} -6.99911 q^{49} +6.32887 q^{50} +8.57927 q^{52} -7.62049 q^{53} -1.27827 q^{55} +0.0260602 q^{56} +3.02664 q^{58} -10.5438 q^{59} -1.00000 q^{61} +11.7610 q^{62} -3.95034 q^{64} -7.14327 q^{65} -8.58956 q^{67} -2.05653 q^{68} +0.0716763 q^{70} -2.03577 q^{71} -3.80905 q^{73} -3.23083 q^{74} +10.3009 q^{76} +0.0298223 q^{77} +7.69951 q^{79} +6.02516 q^{80} +1.25755 q^{82} +0.742344 q^{83} +1.71231 q^{85} -4.83647 q^{86} +0.873847 q^{88} +13.4255 q^{89} +0.166654 q^{91} -5.23915 q^{92} -1.10499 q^{94} -8.57673 q^{95} +1.56491 q^{97} +13.1599 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8} + 8 q^{10} + 13 q^{11} - 9 q^{13} - 19 q^{14} + 18 q^{16} - 7 q^{17} + 2 q^{19} - 15 q^{20} - 4 q^{22} - 23 q^{23} + 10 q^{25} - 8 q^{26} + 9 q^{28} - 16 q^{29} + 9 q^{31} - 29 q^{32} + 2 q^{34} - 16 q^{35} + 14 q^{37} - 8 q^{38} + 16 q^{40} - 19 q^{41} + 7 q^{43} + 12 q^{44} + 4 q^{46} - 26 q^{47} + 8 q^{49} + 15 q^{50} - 17 q^{52} - 18 q^{53} - 7 q^{55} - 44 q^{56} - q^{58} - 31 q^{59} - 13 q^{61} + 5 q^{62} - 17 q^{64} - 31 q^{65} + 14 q^{67} + 32 q^{68} - 20 q^{70} - 37 q^{71} - 16 q^{73} + 6 q^{74} - 7 q^{76} + 5 q^{77} - 17 q^{79} + 2 q^{80} - 2 q^{82} - 30 q^{83} - 16 q^{85} + 22 q^{86} - 15 q^{88} - 35 q^{89} - q^{91} - 24 q^{92} - 11 q^{94} - 13 q^{95} - q^{97} + q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8 + 8 * q^10 + 13 * q^11 - 9 * q^13 - 19 * q^14 + 18 * q^16 - 7 * q^17 + 2 * q^19 - 15 * q^20 - 4 * q^22 - 23 * q^23 + 10 * q^25 - 8 * q^26 + 9 * q^28 - 16 * q^29 + 9 * q^31 - 29 * q^32 + 2 * q^34 - 16 * q^35 + 14 * q^37 - 8 * q^38 + 16 * q^40 - 19 * q^41 + 7 * q^43 + 12 * q^44 + 4 * q^46 - 26 * q^47 + 8 * q^49 + 15 * q^50 - 17 * q^52 - 18 * q^53 - 7 * q^55 - 44 * q^56 - q^58 - 31 * q^59 - 13 * q^61 + 5 * q^62 - 17 * q^64 - 31 * q^65 + 14 * q^67 + 32 * q^68 - 20 * q^70 - 37 * q^71 - 16 * q^73 + 6 * q^74 - 7 * q^76 + 5 * q^77 - 17 * q^79 + 2 * q^80 - 2 * q^82 - 30 * q^83 - 16 * q^85 + 22 * q^86 - 15 * q^88 - 35 * q^89 - q^91 - 24 * q^92 - 11 * q^94 - 13 * q^95 - q^97 + 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 $

$2$	−1.88022	−1.32952	−0.664760	−	0.747057i	$-0.731466\pi$
$2$	−1.88022	−1.32952	−0.664760	+	0.747057i	$0.731466\pi$
$3$	0	0
$3$	0	0
$4$	1.53524	0.767622
$5$	−1.27827	−0.571661	−0.285831	−	0.958280i	$-0.592269\pi$
$5$	−1.27827	−0.571661	−0.285831	+	0.958280i	$0.592269\pi$
$6$	0	0
$7$	0.0298223	0.0112718	0.00563589	−	0.999984i	$-0.498206\pi$
$7$	0.0298223	0.0112718	0.00563589	+	0.999984i	$0.498206\pi$
$8$	0.873847	0.308951
$9$	0	0
$10$	2.40344	0.760035
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.58822	1.54989	0.774946	−	0.632027i	$-0.217777\pi$
$13$	5.58822	1.54989	0.774946	+	0.632027i	$0.217777\pi$
$14$	−0.0560727	−0.0149861
$15$	0	0
$16$	−4.71351	−1.17838
$17$	−1.33955	−0.324888	−0.162444	−	0.986718i	$-0.551938\pi$
$17$	−1.33955	−0.324888	−0.162444	+	0.986718i	$0.551938\pi$
$18$	0	0
$19$	6.70962	1.53929	0.769647	−	0.638470i	$-0.220432\pi$
$19$	6.70962	1.53929	0.769647	+	0.638470i	$0.220432\pi$
$20$	−1.96246	−0.438820
$21$	0	0
$22$	−1.88022	−0.400865
$23$	−3.41259	−0.711574	−0.355787	−	0.934567i	$-0.615787\pi$
$23$	−3.41259	−0.711574	−0.355787	+	0.934567i	$0.615787\pi$
$24$	0	0
$25$	−3.36602	−0.673203
$26$	−10.5071	−2.06061
$27$	0	0
$28$	0.0457846	0.00865247
$29$	−1.60973	−0.298918	−0.149459	−	0.988768i	$-0.547753\pi$
$29$	−1.60973	−0.298918	−0.149459	+	0.988768i	$0.547753\pi$
$30$	0	0
$31$	−6.25509	−1.12345	−0.561724	−	0.827325i	$-0.689862\pi$
$31$	−6.25509	−1.12345	−0.561724	+	0.827325i	$0.689862\pi$
$32$	7.11477	1.25773
$33$	0	0
$34$	2.51865	0.431944
$35$	−0.0381211	−0.00644365
$36$	0	0
$37$	1.71832	0.282490	0.141245	−	0.989975i	$-0.454889\pi$
$37$	1.71832	0.282490	0.141245	+	0.989975i	$0.454889\pi$
$38$	−12.6156	−2.04652
$39$	0	0
$40$	−1.11701	−0.176616
$41$	−0.668829	−0.104454	−0.0522268	−	0.998635i	$-0.516632\pi$
$41$	−0.668829	−0.104454	−0.0522268	+	0.998635i	$0.516632\pi$
$42$	0	0
$43$	2.57228	0.392270	0.196135	−	0.980577i	$-0.437161\pi$
$43$	2.57228	0.392270	0.196135	+	0.980577i	$0.437161\pi$
$44$	1.53524	0.231447
$45$	0	0
$46$	6.41643	0.946051
$47$	0.587690	0.0857233	0.0428617	−	0.999081i	$-0.486353\pi$
$47$	0.587690	0.0857233	0.0428617	+	0.999081i	$0.486353\pi$
$48$	0	0
$49$	−6.99911	−0.999873
$50$	6.32887	0.895037
$51$	0	0
$52$	8.57927	1.18973
$53$	−7.62049	−1.04675	−0.523377	−	0.852101i	$-0.675328\pi$
$53$	−7.62049	−1.04675	−0.523377	+	0.852101i	$0.675328\pi$
$54$	0	0
$55$	−1.27827	−0.172362
$56$	0.0260602	0.00348243
$57$	0	0
$58$	3.02664	0.397418
$59$	−10.5438	−1.37269	−0.686345	−	0.727276i	$-0.740786\pi$
$59$	−10.5438	−1.37269	−0.686345	+	0.727276i	$0.740786\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	11.7610	1.49365
$63$	0	0
$64$	−3.95034	−0.493792
$65$	−7.14327	−0.886014
$66$	0	0
$67$	−8.58956	−1.04938	−0.524691	−	0.851293i	$-0.675819\pi$
$67$	−8.58956	−1.04938	−0.524691	+	0.851293i	$0.675819\pi$
$68$	−2.05653	−0.249391
$69$	0	0
$70$	0.0716763	0.00856695
$71$	−2.03577	−0.241601	−0.120801	−	0.992677i	$-0.538546\pi$
$71$	−2.03577	−0.241601	−0.120801	+	0.992677i	$0.538546\pi$
$72$	0	0
$73$	−3.80905	−0.445816	−0.222908	−	0.974840i	$-0.571555\pi$
$73$	−3.80905	−0.445816	−0.222908	+	0.974840i	$0.571555\pi$
$74$	−3.23083	−0.375576
$75$	0	0
$76$	10.3009	1.18159
$77$	0.0298223	0.00339857
$78$	0	0
$79$	7.69951	0.866262	0.433131	−	0.901331i	$-0.357409\pi$
$79$	7.69951	0.866262	0.433131	+	0.901331i	$0.357409\pi$
$80$	6.02516	0.673633
$81$	0	0
$82$	1.25755	0.138873
$83$	0.742344	0.0814828	0.0407414	−	0.999170i	$-0.487028\pi$
$83$	0.742344	0.0814828	0.0407414	+	0.999170i	$0.487028\pi$
$84$	0	0
$85$	1.71231	0.185726
$86$	−4.83647	−0.521530
$87$	0	0
$88$	0.873847	0.0931524
$89$	13.4255	1.42310	0.711552	−	0.702634i	$-0.247993\pi$
$89$	13.4255	1.42310	0.711552	+	0.702634i	$0.247993\pi$
$90$	0	0
$91$	0.166654	0.0174701
$92$	−5.23915	−0.546219
$93$	0	0
$94$	−1.10499	−0.113971
$95$	−8.57673	−0.879954
$96$	0	0
$97$	1.56491	0.158893	0.0794464	−	0.996839i	$-0.474685\pi$
$97$	1.56491	0.158893	0.0794464	+	0.996839i	$0.474685\pi$
$98$	13.1599	1.32935
$99$	0	0
$100$	−5.16766	−0.516766
$101$	13.7243	1.36561	0.682807	−	0.730598i	$-0.260759\pi$
$101$	13.7243	1.36561	0.682807	+	0.730598i	$0.260759\pi$
$102$	0	0
$103$	−8.90102	−0.877043	−0.438522	−	0.898721i	$-0.644498\pi$
$103$	−8.90102	−0.877043	−0.438522	+	0.898721i	$0.644498\pi$
$104$	4.88324	0.478842
$105$	0	0
$106$	14.3282	1.39168
$107$	8.38029	0.810154	0.405077	−	0.914283i	$-0.367245\pi$
$107$	8.38029	0.810154	0.405077	+	0.914283i	$0.367245\pi$
$108$	0	0
$109$	−11.9870	−1.14814	−0.574072	−	0.818805i	$-0.694637\pi$
$109$	−11.9870	−1.14814	−0.574072	+	0.818805i	$0.694637\pi$
$110$	2.40344	0.229159
$111$	0	0
$112$	−0.140568	−0.0132824
$113$	−1.22669	−0.115398	−0.0576988	−	0.998334i	$-0.518376\pi$
$113$	−1.22669	−0.115398	−0.0576988	+	0.998334i	$0.518376\pi$
$114$	0	0
$115$	4.36222	0.406779
$116$	−2.47132	−0.229456
$117$	0	0
$118$	19.8248	1.82502
$119$	−0.0399484	−0.00366206
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.88022	0.170228
$123$	0	0
$124$	−9.60309	−0.862383
$125$	10.6941	0.956506
$126$	0	0
$127$	−6.73121	−0.597298	−0.298649	−	0.954363i	$-0.596536\pi$
$127$	−6.73121	−0.597298	−0.298649	+	0.954363i	$0.596536\pi$
$128$	−6.80202	−0.601220
$129$	0	0
$130$	13.4310	1.17797
$131$	1.11554	0.0974650	0.0487325	−	0.998812i	$-0.484482\pi$
$131$	1.11554	0.0974650	0.0487325	+	0.998812i	$0.484482\pi$
$132$	0	0
$133$	0.200097	0.0173506
$134$	16.1503	1.39517
$135$	0	0
$136$	−1.17056	−0.100374
$137$	−4.67046	−0.399025	−0.199512	−	0.979895i	$-0.563936\pi$
$137$	−4.67046	−0.399025	−0.199512	+	0.979895i	$0.563936\pi$
$138$	0	0
$139$	−3.77266	−0.319993	−0.159996	−	0.987118i	$-0.551148\pi$
$139$	−3.77266	−0.319993	−0.159996	+	0.987118i	$0.551148\pi$
$140$	−0.0585252	−0.00494628
$141$	0	0
$142$	3.82770	0.321214
$143$	5.58822	0.467310
$144$	0	0
$145$	2.05767	0.170880
$146$	7.16187	0.592720
$147$	0	0
$148$	2.63804	0.216845
$149$	7.87763	0.645360	0.322680	−	0.946508i	$-0.395416\pi$
$149$	7.87763	0.645360	0.322680	+	0.946508i	$0.395416\pi$
$150$	0	0
$151$	18.2796	1.48757	0.743785	−	0.668419i	$-0.233029\pi$
$151$	18.2796	1.48757	0.743785	+	0.668419i	$0.233029\pi$
$152$	5.86318	0.475567
$153$	0	0
$154$	−0.0560727	−0.00451847
$155$	7.99572	0.642232
$156$	0	0
$157$	−21.7284	−1.73412	−0.867059	−	0.498206i	$-0.833992\pi$
$157$	−21.7284	−1.73412	−0.867059	+	0.498206i	$0.833992\pi$
$158$	−14.4768	−1.15171
$159$	0	0
$160$	−9.09462	−0.718993
$161$	−0.101771	−0.00802071
$162$	0	0
$163$	−10.1629	−0.796023	−0.398012	−	0.917380i	$-0.630300\pi$
$163$	−10.1629	−0.796023	−0.398012	+	0.917380i	$0.630300\pi$
$164$	−1.02682	−0.0801809
$165$	0	0
$166$	−1.39577	−0.108333
$167$	11.2296	0.868970	0.434485	−	0.900679i	$-0.356930\pi$
$167$	11.2296	0.868970	0.434485	+	0.900679i	$0.356930\pi$
$168$	0	0
$169$	18.2282	1.40217
$170$	−3.21952	−0.246926
$171$	0	0
$172$	3.94908	0.301115
$173$	−11.3300	−0.861404	−0.430702	−	0.902494i	$-0.641734\pi$
$173$	−11.3300	−0.861404	−0.430702	+	0.902494i	$0.641734\pi$
$174$	0	0
$175$	−0.100383	−0.00758821
$176$	−4.71351	−0.355295
$177$	0	0
$178$	−25.2430	−1.89204
$179$	9.37041	0.700377	0.350189	−	0.936679i	$-0.386117\pi$
$179$	9.37041	0.700377	0.350189	+	0.936679i	$0.386117\pi$
$180$	0	0
$181$	−17.0680	−1.26865	−0.634327	−	0.773065i	$-0.718723\pi$
$181$	−17.0680	−1.26865	−0.634327	+	0.773065i	$0.718723\pi$
$182$	−0.313346	−0.0232268
$183$	0	0
$184$	−2.98208	−0.219842
$185$	−2.19648	−0.161489
$186$	0	0
$187$	−1.33955	−0.0979573
$188$	0.902247	0.0658031
$189$	0	0
$190$	16.1262	1.16992
$191$	−21.8340	−1.57985	−0.789927	−	0.613201i	$-0.789882\pi$
$191$	−21.8340	−1.57985	−0.789927	+	0.613201i	$0.789882\pi$
$192$	0	0
$193$	10.7246	0.771975	0.385988	−	0.922504i	$-0.373861\pi$
$193$	10.7246	0.771975	0.385988	+	0.922504i	$0.373861\pi$
$194$	−2.94239	−0.211251
$195$	0	0
$196$	−10.7453	−0.767524
$197$	14.5895	1.03946	0.519730	−	0.854331i	$-0.326033\pi$
$197$	14.5895	1.03946	0.519730	+	0.854331i	$0.326033\pi$
$198$	0	0
$199$	19.4483	1.37865	0.689326	−	0.724451i	$-0.257907\pi$
$199$	19.4483	1.37865	0.689326	+	0.724451i	$0.257907\pi$
$200$	−2.94138	−0.207987
$201$	0	0
$202$	−25.8047	−1.81561
$203$	−0.0480058	−0.00336935
$204$	0	0
$205$	0.854947	0.0597121
$206$	16.7359	1.16605
$207$	0	0
$208$	−26.3401	−1.82636
$209$	6.70962	0.464114
$210$	0	0
$211$	−21.2613	−1.46369	−0.731843	−	0.681473i	$-0.761340\pi$
$211$	−21.2613	−1.46369	−0.731843	+	0.681473i	$0.761340\pi$
$212$	−11.6993	−0.803511
$213$	0	0
$214$	−15.7568	−1.07711
$215$	−3.28808	−0.224245
$216$	0	0
$217$	−0.186542	−0.0126633
$218$	22.5382	1.52648
$219$	0	0
$220$	−1.96246	−0.132309
$221$	−7.48568	−0.503541
$222$	0	0
$223$	−9.20342	−0.616306	−0.308153	−	0.951337i	$-0.599711\pi$
$223$	−9.20342	−0.616306	−0.308153	+	0.951337i	$0.599711\pi$
$224$	0.212179	0.0141768
$225$	0	0
$226$	2.30646	0.153423
$227$	−9.75094	−0.647193	−0.323596	−	0.946195i	$-0.604892\pi$
$227$	−9.75094	−0.647193	−0.323596	+	0.946195i	$0.604892\pi$
$228$	0	0
$229$	8.00584	0.529041	0.264521	−	0.964380i	$-0.414786\pi$
$229$	8.00584	0.529041	0.264521	+	0.964380i	$0.414786\pi$
$230$	−8.20195	−0.540821
$231$	0	0
$232$	−1.40665	−0.0923513
$233$	−6.15569	−0.403273	−0.201636	−	0.979460i	$-0.564626\pi$
$233$	−6.15569	−0.403273	−0.201636	+	0.979460i	$0.564626\pi$
$234$	0	0
$235$	−0.751228	−0.0490047
$236$	−16.1873	−1.05371
$237$	0	0
$238$	0.0751120	0.00486879
$239$	−23.9973	−1.55225	−0.776127	−	0.630576i	$-0.782819\pi$
$239$	−23.9973	−1.55225	−0.776127	+	0.630576i	$0.782819\pi$
$240$	0	0
$241$	−17.6264	−1.13542	−0.567708	−	0.823230i	$-0.692170\pi$
$241$	−17.6264	−1.13542	−0.567708	+	0.823230i	$0.692170\pi$
$242$	−1.88022	−0.120865
$243$	0	0
$244$	−1.53524	−0.0982839
$245$	8.94678	0.571589
$246$	0	0
$247$	37.4948	2.38574
$248$	−5.46599	−0.347091
$249$	0	0
$250$	−20.1072	−1.27169
$251$	−7.69236	−0.485538	−0.242769	−	0.970084i	$-0.578056\pi$
$251$	−7.69236	−0.485538	−0.242769	+	0.970084i	$0.578056\pi$
$252$	0	0
$253$	−3.41259	−0.214548
$254$	12.6562	0.794120
$255$	0	0
$256$	20.6900	1.29313
$257$	4.61272	0.287734	0.143867	−	0.989597i	$-0.454046\pi$
$257$	4.61272	0.287734	0.143867	+	0.989597i	$0.454046\pi$
$258$	0	0
$259$	0.0512443	0.00318417
$260$	−10.9667	−0.680123
$261$	0	0
$262$	−2.09746	−0.129582
$263$	9.97272	0.614944	0.307472	−	0.951557i	$-0.400517\pi$
$263$	9.97272	0.614944	0.307472	+	0.951557i	$0.400517\pi$
$264$	0	0
$265$	9.74107	0.598389
$266$	−0.376227	−0.0230679
$267$	0	0
$268$	−13.1871	−0.805528
$269$	−21.8938	−1.33489	−0.667444	−	0.744660i	$-0.732612\pi$
$269$	−21.8938	−1.33489	−0.667444	+	0.744660i	$0.732612\pi$
$270$	0	0
$271$	18.1929	1.10514	0.552571	−	0.833466i	$-0.313647\pi$
$271$	18.1929	1.10514	0.552571	+	0.833466i	$0.313647\pi$
$272$	6.31397	0.382841
$273$	0	0
$274$	8.78152	0.530511
$275$	−3.36602	−0.202978
$276$	0	0
$277$	24.8770	1.49471	0.747357	−	0.664423i	$-0.231323\pi$
$277$	24.8770	1.49471	0.747357	+	0.664423i	$0.231323\pi$
$278$	7.09344	0.425436
$279$	0	0
$280$	−0.0333120	−0.00199077
$281$	16.7545	0.999487	0.499744	−	0.866173i	$-0.333428\pi$
$281$	16.7545	0.999487	0.499744	+	0.866173i	$0.333428\pi$
$282$	0	0
$283$	−10.4893	−0.623527	−0.311763	−	0.950160i	$-0.600920\pi$
$283$	−10.4893	−0.623527	−0.311763	+	0.950160i	$0.600920\pi$
$284$	−3.12540	−0.185458
$285$	0	0
$286$	−10.5071	−0.621298
$287$	−0.0199461	−0.00117738
$288$	0	0
$289$	−15.2056	−0.894448
$290$	−3.86888	−0.227188
$291$	0	0
$292$	−5.84782	−0.342218
$293$	23.2899	1.36061	0.680305	−	0.732930i	$-0.261848\pi$
$293$	23.2899	1.36061	0.680305	+	0.732930i	$0.261848\pi$
$294$	0	0
$295$	13.4779	0.784714
$296$	1.50155	0.0872757
$297$	0	0
$298$	−14.8117	−0.858019
$299$	−19.0703	−1.10286
$300$	0	0
$301$	0.0767116	0.00442158
$302$	−34.3697	−1.97775
$303$	0	0
$304$	−31.6259	−1.81387
$305$	1.27827	0.0731937
$306$	0	0
$307$	−5.37282	−0.306643	−0.153321	−	0.988176i	$-0.548997\pi$
$307$	−5.37282	−0.306643	−0.153321	+	0.988176i	$0.548997\pi$
$308$	0.0457846	0.00260882
$309$	0	0
$310$	−15.0337	−0.853859
$311$	−0.901161	−0.0511001	−0.0255501	−	0.999674i	$-0.508134\pi$
$311$	−0.901161	−0.0511001	−0.0255501	+	0.999674i	$0.508134\pi$
$312$	0	0
$313$	4.66099	0.263455	0.131727	−	0.991286i	$-0.457948\pi$
$313$	4.66099	0.263455	0.131727	+	0.991286i	$0.457948\pi$
$314$	40.8543	2.30554
$315$	0	0
$316$	11.8206	0.664962
$317$	7.15317	0.401762	0.200881	−	0.979616i	$-0.435620\pi$
$317$	7.15317	0.401762	0.200881	+	0.979616i	$0.435620\pi$
$318$	0	0
$319$	−1.60973	−0.0901273
$320$	5.04961	0.282282
$321$	0	0
$322$	0.191353	0.0106637
$323$	−8.98785	−0.500097
$324$	0	0
$325$	−18.8100	−1.04339
$326$	19.1086	1.05833
$327$	0	0
$328$	−0.584454	−0.0322711
$329$	0.0175263	0.000966255 0
$330$	0	0
$331$	3.97759	0.218628	0.109314	−	0.994007i	$-0.465135\pi$
$331$	3.97759	0.218628	0.109314	+	0.994007i	$0.465135\pi$
$332$	1.13968	0.0625480
$333$	0	0
$334$	−21.1141	−1.15531
$335$	10.9798	0.599891
$336$	0	0
$337$	−19.6675	−1.07135	−0.535677	−	0.844423i	$-0.679944\pi$
$337$	−19.6675	−1.07135	−0.535677	+	0.844423i	$0.679944\pi$
$338$	−34.2731	−1.86421
$339$	0	0
$340$	2.62881	0.142567
$341$	−6.25509	−0.338732
$342$	0	0
$343$	−0.417486	−0.0225421
$344$	2.24778	0.121192
$345$	0	0
$346$	21.3029	1.14525
$347$	−14.8057	−0.794812	−0.397406	−	0.917643i	$-0.630090\pi$
$347$	−14.8057	−0.794812	−0.397406	+	0.917643i	$0.630090\pi$
$348$	0	0
$349$	−32.1526	−1.72109	−0.860546	−	0.509373i	$-0.829877\pi$
$349$	−32.1526	−1.72109	−0.860546	+	0.509373i	$0.829877\pi$
$350$	0.188742	0.0100887
$351$	0	0
$352$	7.11477	0.379219
$353$	2.07749	0.110574	0.0552868	−	0.998471i	$-0.482393\pi$
$353$	2.07749	0.110574	0.0552868	+	0.998471i	$0.482393\pi$
$354$	0	0
$355$	2.60227	0.138114
$356$	20.6115	1.09241
$357$	0	0
$358$	−17.6185	−0.931165
$359$	8.95752	0.472760	0.236380	−	0.971661i	$-0.424039\pi$
$359$	8.95752	0.472760	0.236380	+	0.971661i	$0.424039\pi$
$360$	0	0
$361$	26.0190	1.36942
$362$	32.0917	1.68670
$363$	0	0
$364$	0.255854	0.0134104
$365$	4.86901	0.254856
$366$	0	0
$367$	6.69870	0.349669	0.174835	−	0.984598i	$-0.444061\pi$
$367$	6.69870	0.349669	0.174835	+	0.984598i	$0.444061\pi$
$368$	16.0853	0.838503
$369$	0	0
$370$	4.12988	0.214702
$371$	−0.227261	−0.0117988
$372$	0	0
$373$	−34.7971	−1.80172	−0.900862	−	0.434107i	$-0.857064\pi$
$373$	−34.7971	−1.80172	−0.900862	+	0.434107i	$0.857064\pi$
$374$	2.51865	0.130236
$375$	0	0
$376$	0.513550	0.0264843
$377$	−8.99550	−0.463292
$378$	0	0
$379$	7.77701	0.399478	0.199739	−	0.979849i	$-0.435991\pi$
$379$	7.77701	0.399478	0.199739	+	0.979849i	$0.435991\pi$
$380$	−13.1674	−0.675472
$381$	0	0
$382$	41.0528	2.10045
$383$	−24.8333	−1.26892	−0.634462	−	0.772954i	$-0.718778\pi$
$383$	−24.8333	−1.26892	−0.634462	+	0.772954i	$0.718778\pi$
$384$	0	0
$385$	−0.0381211	−0.00194283
$386$	−20.1647	−1.02636
$387$	0	0
$388$	2.40252	0.121970
$389$	−26.7412	−1.35583	−0.677916	−	0.735140i	$-0.737117\pi$
$389$	−26.7412	−1.35583	−0.677916	+	0.735140i	$0.737117\pi$
$390$	0	0
$391$	4.57132	0.231181
$392$	−6.11615	−0.308912
$393$	0	0
$394$	−27.4316	−1.38198
$395$	−9.84208	−0.495209
$396$	0	0
$397$	16.7943	0.842880	0.421440	−	0.906856i	$-0.361525\pi$
$397$	16.7943	0.842880	0.421440	+	0.906856i	$0.361525\pi$
$398$	−36.5672	−1.83295
$399$	0	0
$400$	15.8658	0.793288
$401$	14.3276	0.715486	0.357743	−	0.933820i	$-0.383546\pi$
$401$	14.3276	0.715486	0.357743	+	0.933820i	$0.383546\pi$
$402$	0	0
$403$	−34.9548	−1.74122
$404$	21.0701	1.04828
$405$	0	0
$406$	0.0902617	0.00447961
$407$	1.71832	0.0851740
$408$	0	0
$409$	4.28656	0.211957	0.105978	−	0.994368i	$-0.466203\pi$
$409$	4.28656	0.211957	0.105978	+	0.994368i	$0.466203\pi$
$410$	−1.60749	−0.0793884
$411$	0	0
$412$	−13.6652	−0.673238
$413$	−0.314442	−0.0154727
$414$	0	0
$415$	−0.948919	−0.0465806
$416$	39.7589	1.94934
$417$	0	0
$418$	−12.6156	−0.617049
$419$	−10.7345	−0.524414	−0.262207	−	0.965012i	$-0.584450\pi$
$419$	−10.7345	−0.524414	−0.262207	+	0.965012i	$0.584450\pi$
$420$	0	0
$421$	−7.07807	−0.344964	−0.172482	−	0.985013i	$-0.555179\pi$
$421$	−7.07807	−0.344964	−0.172482	+	0.985013i	$0.555179\pi$
$422$	39.9760	1.94600
$423$	0	0
$424$	−6.65913	−0.323396
$425$	4.50893	0.218715
$426$	0	0
$427$	−0.0298223	−0.00144320
$428$	12.8658	0.621891
$429$	0	0
$430$	6.18233	0.298139
$431$	18.2462	0.878890	0.439445	−	0.898269i	$-0.355175\pi$
$431$	18.2462	0.878890	0.439445	+	0.898269i	$0.355175\pi$
$432$	0	0
$433$	−5.19879	−0.249838	−0.124919	−	0.992167i	$-0.539867\pi$
$433$	−5.19879	−0.249838	−0.124919	+	0.992167i	$0.539867\pi$
$434$	0.350740	0.0168361
$435$	0	0
$436$	−18.4029	−0.881340
$437$	−22.8972	−1.09532
$438$	0	0
$439$	−35.1802	−1.67906	−0.839529	−	0.543315i	$-0.817169\pi$
$439$	−35.1802	−1.67906	−0.839529	+	0.543315i	$0.817169\pi$
$440$	−1.11701	−0.0532516
$441$	0	0
$442$	14.0747	0.669468
$443$	13.3949	0.636409	0.318204	−	0.948022i	$-0.396920\pi$
$443$	13.3949	0.636409	0.318204	+	0.948022i	$0.396920\pi$
$444$	0	0
$445$	−17.1615	−0.813533
$446$	17.3045	0.819391
$447$	0	0
$448$	−0.117808	−0.00556592
$449$	−17.7582	−0.838061	−0.419030	−	0.907972i	$-0.637630\pi$
$449$	−17.7582	−0.838061	−0.419030	+	0.907972i	$0.637630\pi$
$450$	0	0
$451$	−0.668829	−0.0314939
$452$	−1.88327	−0.0885817
$453$	0	0
$454$	18.3340	0.860455
$455$	−0.213029	−0.00998696
$456$	0	0
$457$	21.6364	1.01211	0.506054	−	0.862502i	$-0.331104\pi$
$457$	21.6364	1.01211	0.506054	+	0.862502i	$0.331104\pi$
$458$	−15.0528	−0.703370
$459$	0	0
$460$	6.69707	0.312252
$461$	−9.15352	−0.426322	−0.213161	−	0.977017i	$-0.568376\pi$
$461$	−9.15352	−0.426322	−0.213161	+	0.977017i	$0.568376\pi$
$462$	0	0
$463$	−7.70387	−0.358029	−0.179015	−	0.983846i	$-0.557291\pi$
$463$	−7.70387	−0.358029	−0.179015	+	0.983846i	$0.557291\pi$
$464$	7.58746	0.352239
$465$	0	0
$466$	11.5741	0.536159
$467$	10.5122	0.486448	0.243224	−	0.969970i	$-0.421795\pi$
$467$	10.5122	0.486448	0.243224	+	0.969970i	$0.421795\pi$
$468$	0	0
$469$	−0.256161	−0.0118284
$470$	1.41248	0.0651527
$471$	0	0
$472$	−9.21369	−0.424094
$473$	2.57228	0.118274
$474$	0	0
$475$	−22.5847	−1.03626
$476$	−0.0613305	−0.00281108
$477$	0	0
$478$	45.1203	2.06375
$479$	−22.2273	−1.01559	−0.507795	−	0.861478i	$-0.669539\pi$
$479$	−22.2273	−1.01559	−0.507795	+	0.861478i	$0.669539\pi$
$480$	0	0
$481$	9.60235	0.437829
$482$	33.1416	1.50956
$483$	0	0
$484$	1.53524	0.0697838
$485$	−2.00039	−0.0908328
$486$	0	0
$487$	6.57806	0.298081	0.149040	−	0.988831i	$-0.452382\pi$
$487$	6.57806	0.298081	0.149040	+	0.988831i	$0.452382\pi$
$488$	−0.873847	−0.0395572
$489$	0	0
$490$	−16.8219	−0.759938
$491$	−12.9133	−0.582769	−0.291384	−	0.956606i	$-0.594116\pi$
$491$	−12.9133	−0.582769	−0.291384	+	0.956606i	$0.594116\pi$
$492$	0	0
$493$	2.15630	0.0971149
$494$	−70.4987	−3.17189
$495$	0	0
$496$	29.4835	1.32385
$497$	−0.0607114	−0.00272328
$498$	0	0
$499$	34.3660	1.53843	0.769217	−	0.638988i	$-0.220647\pi$
$499$	34.3660	1.53843	0.769217	+	0.638988i	$0.220647\pi$
$500$	16.4180	0.734235
$501$	0	0
$502$	14.4634	0.645532
$503$	21.4268	0.955373	0.477687	−	0.878530i	$-0.341475\pi$
$503$	21.4268	0.955373	0.477687	+	0.878530i	$0.341475\pi$
$504$	0	0
$505$	−17.5434	−0.780669
$506$	6.41643	0.285245
$507$	0	0
$508$	−10.3340	−0.458499
$509$	11.0074	0.487896	0.243948	−	0.969788i	$-0.421557\pi$
$509$	11.0074	0.487896	0.243948	+	0.969788i	$0.421557\pi$
$510$	0	0
$511$	−0.113595	−0.00502514
$512$	−25.2978	−1.11802
$513$	0	0
$514$	−8.67295	−0.382547
$515$	11.3779	0.501372
$516$	0	0
$517$	0.587690	0.0258466
$518$	−0.0963508	−0.00423341
$519$	0	0
$520$	−6.24212	−0.273735
$521$	−42.8741	−1.87835	−0.939175	−	0.343440i	$-0.888408\pi$
$521$	−42.8741	−1.87835	−0.939175	+	0.343440i	$0.888408\pi$
$522$	0	0
$523$	−15.8294	−0.692172	−0.346086	−	0.938203i	$-0.612489\pi$
$523$	−15.8294	−0.692172	−0.346086	+	0.938203i	$0.612489\pi$
$524$	1.71262	0.0748163
$525$	0	0
$526$	−18.7510	−0.817581
$527$	8.37899	0.364994
$528$	0	0
$529$	−11.3542	−0.493663
$530$	−18.3154	−0.795570
$531$	0	0
$532$	0.307197	0.0133187
$533$	−3.73756	−0.161892
$534$	0	0
$535$	−10.7123	−0.463133
$536$	−7.50596	−0.324208
$537$	0	0
$538$	41.1653	1.77476
$539$	−6.99911	−0.301473
$540$	0	0
$541$	12.6615	0.544360	0.272180	−	0.962246i	$-0.412255\pi$
$541$	12.6615	0.544360	0.272180	+	0.962246i	$0.412255\pi$
$542$	−34.2068	−1.46931
$543$	0	0
$544$	−9.53056	−0.408620
$545$	15.3226	0.656349
$546$	0	0
$547$	38.1275	1.63021	0.815106	−	0.579312i	$-0.196679\pi$
$547$	38.1275	1.63021	0.815106	+	0.579312i	$0.196679\pi$
$548$	−7.17030	−0.306300
$549$	0	0
$550$	6.32887	0.269864
$551$	−10.8007	−0.460123
$552$	0	0
$553$	0.229617	0.00976432
$554$	−46.7743	−1.98725
$555$	0	0
$556$	−5.79195	−0.245633
$557$	−14.8321	−0.628455	−0.314228	−	0.949348i	$-0.601746\pi$
$557$	−14.8321	−0.628455	−0.314228	+	0.949348i	$0.601746\pi$
$558$	0	0
$559$	14.3745	0.607976
$560$	0.179684	0.00759305
$561$	0	0
$562$	−31.5021	−1.32884
$563$	2.52615	0.106464	0.0532322	−	0.998582i	$-0.483048\pi$
$563$	2.52615	0.106464	0.0532322	+	0.998582i	$0.483048\pi$
$564$	0	0
$565$	1.56805	0.0659683
$566$	19.7223	0.828991
$567$	0	0
$568$	−1.77895	−0.0746431
$569$	−10.0494	−0.421291	−0.210645	−	0.977563i	$-0.567557\pi$
$569$	−10.0494	−0.421291	−0.210645	+	0.977563i	$0.567557\pi$
$570$	0	0
$571$	−19.7352	−0.825891	−0.412946	−	0.910756i	$-0.635500\pi$
$571$	−19.7352	−0.825891	−0.412946	+	0.910756i	$0.635500\pi$
$572$	8.57927	0.358717
$573$	0	0
$574$	0.0375031	0.00156535
$575$	11.4868	0.479034
$576$	0	0
$577$	−1.47224	−0.0612900	−0.0306450	−	0.999530i	$-0.509756\pi$
$577$	−1.47224	−0.0612900	−0.0306450	+	0.999530i	$0.509756\pi$
$578$	28.5900	1.18919
$579$	0	0
$580$	3.15902	0.131171
$581$	0.0221384	0.000918457 0
$582$	0	0
$583$	−7.62049	−0.315608
$584$	−3.32852	−0.137735
$585$	0	0
$586$	−43.7902	−1.80896
$587$	−37.3090	−1.53991	−0.769953	−	0.638101i	$-0.779720\pi$
$587$	−37.3090	−1.53991	−0.769953	+	0.638101i	$0.779720\pi$
$588$	0	0
$589$	−41.9693	−1.72932
$590$	−25.3415	−1.04329
$591$	0	0
$592$	−8.09933	−0.332880
$593$	−23.1904	−0.952314	−0.476157	−	0.879360i	$-0.657971\pi$
$593$	−23.1904	−0.952314	−0.476157	+	0.879360i	$0.657971\pi$
$594$	0	0
$595$	0.0510650	0.00209346
$596$	12.0941	0.495393
$597$	0	0
$598$	35.8564	1.46628
$599$	−10.9690	−0.448183	−0.224091	−	0.974568i	$-0.571941\pi$
$599$	−10.9690	−0.448183	−0.224091	+	0.974568i	$0.571941\pi$
$600$	0	0
$601$	8.97680	0.366171	0.183086	−	0.983097i	$-0.441391\pi$
$601$	8.97680	0.366171	0.183086	+	0.983097i	$0.441391\pi$
$602$	−0.144235	−0.00587858
$603$	0	0
$604$	28.0636	1.14189
$605$	−1.27827	−0.0519692
$606$	0	0
$607$	21.8511	0.886907	0.443453	−	0.896297i	$-0.353753\pi$
$607$	21.8511	0.886907	0.443453	+	0.896297i	$0.353753\pi$
$608$	47.7374	1.93601
$609$	0	0
$610$	−2.40344	−0.0973125
$611$	3.28414	0.132862
$612$	0	0
$613$	6.69910	0.270574	0.135287	−	0.990806i	$-0.456804\pi$
$613$	6.69910	0.270574	0.135287	+	0.990806i	$0.456804\pi$
$614$	10.1021	0.407688
$615$	0	0
$616$	0.0260602	0.00104999
$617$	−14.6447	−0.589572	−0.294786	−	0.955563i	$-0.595248\pi$
$617$	−14.6447	−0.589572	−0.294786	+	0.955563i	$0.595248\pi$
$618$	0	0
$619$	4.61819	0.185621	0.0928103	−	0.995684i	$-0.470415\pi$
$619$	4.61819	0.185621	0.0928103	+	0.995684i	$0.470415\pi$
$620$	12.2754	0.492991
$621$	0	0
$622$	1.69438	0.0679386
$623$	0.400381	0.0160409
$624$	0	0
$625$	3.16015	0.126406
$626$	−8.76370	−0.350268
$627$	0	0
$628$	−33.3584	−1.33115
$629$	−2.30177	−0.0917775
$630$	0	0
$631$	−7.21368	−0.287172	−0.143586	−	0.989638i	$-0.545863\pi$
$631$	−7.21368	−0.287172	−0.143586	+	0.989638i	$0.545863\pi$
$632$	6.72819	0.267633
$633$	0	0
$634$	−13.4496	−0.534151
$635$	8.60433	0.341452
$636$	0	0
$637$	−39.1126	−1.54970
$638$	3.02664	0.119826
$639$	0	0
$640$	8.69485	0.343694
$641$	−42.0066	−1.65916	−0.829580	−	0.558387i	$-0.811420\pi$
$641$	−42.0066	−1.65916	−0.829580	+	0.558387i	$0.811420\pi$
$642$	0	0
$643$	3.82823	0.150971	0.0754853	−	0.997147i	$-0.475949\pi$
$643$	3.82823	0.150971	0.0754853	+	0.997147i	$0.475949\pi$
$644$	−0.156244	−0.00615687
$645$	0	0
$646$	16.8992	0.664889
$647$	26.9638	1.06006	0.530028	−	0.847980i	$-0.322181\pi$
$647$	26.9638	1.06006	0.530028	+	0.847980i	$0.322181\pi$
$648$	0	0
$649$	−10.5438	−0.413882
$650$	35.3671	1.38721
$651$	0	0
$652$	−15.6026	−0.611045
$653$	−3.40054	−0.133073	−0.0665366	−	0.997784i	$-0.521195\pi$
$653$	−3.40054	−0.133073	−0.0665366	+	0.997784i	$0.521195\pi$
$654$	0	0
$655$	−1.42596	−0.0557170
$656$	3.15254	0.123086
$657$	0	0
$658$	−0.0329533	−0.00128466
$659$	−32.0857	−1.24988	−0.624941	−	0.780672i	$-0.714877\pi$
$659$	−32.0857	−1.24988	−0.624941	+	0.780672i	$0.714877\pi$
$660$	0	0
$661$	1.07251	0.0417157	0.0208578	−	0.999782i	$-0.493360\pi$
$661$	1.07251	0.0417157	0.0208578	+	0.999782i	$0.493360\pi$
$662$	−7.47876	−0.290670
$663$	0	0
$664$	0.648695	0.0251742
$665$	−0.255778	−0.00991866
$666$	0	0
$667$	5.49333	0.212703
$668$	17.2401	0.667040
$669$	0	0
$670$	−20.6445	−0.797567
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	10.2363	0.394578	0.197289	−	0.980345i	$-0.436786\pi$
$673$	10.2363	0.394578	0.197289	+	0.980345i	$0.436786\pi$
$674$	36.9792	1.42439
$675$	0	0
$676$	27.9847	1.07633
$677$	35.6445	1.36993	0.684965	−	0.728576i	$-0.259818\pi$
$677$	35.6445	1.36993	0.684965	+	0.728576i	$0.259818\pi$
$678$	0	0
$679$	0.0466694	0.00179101
$680$	1.49629	0.0573802
$681$	0	0
$682$	11.7610	0.450351
$683$	26.9142	1.02984	0.514921	−	0.857238i	$-0.327821\pi$
$683$	26.9142	1.02984	0.514921	+	0.857238i	$0.327821\pi$
$684$	0	0
$685$	5.97013	0.228107
$686$	0.784968	0.0299702
$687$	0	0
$688$	−12.1245	−0.462242
$689$	−42.5849	−1.62236
$690$	0	0
$691$	−2.50922	−0.0954551	−0.0477276	−	0.998860i	$-0.515198\pi$
$691$	−2.50922	−0.0954551	−0.0477276	+	0.998860i	$0.515198\pi$
$692$	−17.3943	−0.661233
$693$	0	0
$694$	27.8380	1.05672
$695$	4.82249	0.182927
$696$	0	0
$697$	0.895928	0.0339357
$698$	60.4542	2.28822
$699$	0	0
$700$	−0.154112	−0.00582487
$701$	1.91591	0.0723628	0.0361814	−	0.999345i	$-0.488481\pi$
$701$	1.91591	0.0723628	0.0361814	+	0.999345i	$0.488481\pi$
$702$	0	0
$703$	11.5293	0.434835
$704$	−3.95034	−0.148884
$705$	0	0
$706$	−3.90615	−0.147010
$707$	0.409290	0.0153929
$708$	0	0
$709$	−2.61979	−0.0983884	−0.0491942	−	0.998789i	$-0.515665\pi$
$709$	−2.61979	−0.0983884	−0.0491942	+	0.998789i	$0.515665\pi$
$710$	−4.89285	−0.183625
$711$	0	0
$712$	11.7319	0.439670
$713$	21.3460	0.799416
$714$	0	0
$715$	−7.14327	−0.267143
$716$	14.3859	0.537625
$717$	0	0
$718$	−16.8422	−0.628544
$719$	−36.1632	−1.34866	−0.674331	−	0.738429i	$-0.735568\pi$
$719$	−36.1632	−1.34866	−0.674331	+	0.738429i	$0.735568\pi$
$720$	0	0
$721$	−0.265449	−0.00988585
$722$	−48.9216	−1.82067
$723$	0	0
$724$	−26.2035	−0.973847
$725$	5.41836	0.201233
$726$	0	0
$727$	36.6906	1.36078	0.680389	−	0.732851i	$-0.261811\pi$
$727$	36.6906	1.36078	0.680389	+	0.732851i	$0.261811\pi$
$728$	0.145630	0.00539740
$729$	0	0
$730$	−9.15483	−0.338835
$731$	−3.44569	−0.127444
$732$	0	0
$733$	30.2021	1.11554	0.557770	−	0.829996i	$-0.311657\pi$
$733$	30.2021	1.11554	0.557770	+	0.829996i	$0.311657\pi$
$734$	−12.5951	−0.464892
$735$	0	0
$736$	−24.2798	−0.894964
$737$	−8.58956	−0.316401
$738$	0	0
$739$	−46.7194	−1.71860	−0.859301	−	0.511471i	$-0.829101\pi$
$739$	−46.7194	−1.71860	−0.859301	+	0.511471i	$0.829101\pi$
$740$	−3.37214	−0.123962
$741$	0	0
$742$	0.427301	0.0156867
$743$	−12.0904	−0.443554	−0.221777	−	0.975097i	$-0.571186\pi$
$743$	−12.0904	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$744$	0	0
$745$	−10.0698	−0.368928
$746$	65.4263	2.39543
$747$	0	0
$748$	−2.05653	−0.0751942
$749$	0.249920	0.00913188
$750$	0	0
$751$	−27.1495	−0.990701	−0.495350	−	0.868693i	$-0.664960\pi$
$751$	−27.1495	−0.990701	−0.495350	+	0.868693i	$0.664960\pi$
$752$	−2.77008	−0.101015
$753$	0	0
$754$	16.9135	0.615955
$755$	−23.3663	−0.850386
$756$	0	0
$757$	−30.1523	−1.09590	−0.547952	−	0.836510i	$-0.684592\pi$
$757$	−30.1523	−1.09590	−0.547952	+	0.836510i	$0.684592\pi$
$758$	−14.6225	−0.531114
$759$	0	0
$760$	−7.49475	−0.271863
$761$	22.1105	0.801504	0.400752	−	0.916186i	$-0.368749\pi$
$761$	22.1105	0.801504	0.400752	+	0.916186i	$0.368749\pi$
$762$	0	0
$763$	−0.357480	−0.0129416
$764$	−33.5205	−1.21273
$765$	0	0
$766$	46.6922	1.68706
$767$	−58.9212	−2.12752
$768$	0	0
$769$	−23.3843	−0.843259	−0.421630	−	0.906768i	$-0.638542\pi$
$769$	−23.3843	−0.843259	−0.421630	+	0.906768i	$0.638542\pi$
$770$	0.0716763	0.00258303
$771$	0	0
$772$	16.4649	0.592585
$773$	−49.4566	−1.77883	−0.889415	−	0.457100i	$-0.848888\pi$
$773$	−49.4566	−1.77883	−0.889415	+	0.457100i	$0.848888\pi$
$774$	0	0
$775$	21.0547	0.756309
$776$	1.36749	0.0490901
$777$	0	0
$778$	50.2794	1.80260
$779$	−4.48759	−0.160785
$780$	0	0
$781$	−2.03577	−0.0728455
$782$	−8.59510	−0.307360
$783$	0	0
$784$	32.9904	1.17823
$785$	27.7749	0.991328
$786$	0	0
$787$	−14.0106	−0.499425	−0.249713	−	0.968320i	$-0.580336\pi$
$787$	−14.0106	−0.499425	−0.249713	+	0.968320i	$0.580336\pi$
$788$	22.3985	0.797912
$789$	0	0
$790$	18.5053	0.658389
$791$	−0.0365829	−0.00130074
$792$	0	0
$793$	−5.58822	−0.198443
$794$	−31.5770	−1.12063
$795$	0	0
$796$	29.8579	1.05828
$797$	−29.6569	−1.05050	−0.525252	−	0.850947i	$-0.676029\pi$
$797$	−29.6569	−1.05050	−0.525252	+	0.850947i	$0.676029\pi$
$798$	0	0
$799$	−0.787237	−0.0278505
$800$	−23.9484	−0.846705
$801$	0	0
$802$	−26.9391	−0.951253
$803$	−3.80905	−0.134418
$804$	0	0
$805$	0.130092	0.00458513
$806$	65.7229	2.31499
$807$	0	0
$808$	11.9929	0.421909
$809$	28.7985	1.01250	0.506250	−	0.862387i	$-0.331031\pi$
$809$	28.7985	1.01250	0.506250	+	0.862387i	$0.331031\pi$
$810$	0	0
$811$	34.4307	1.20902	0.604512	−	0.796596i	$-0.293368\pi$
$811$	34.4307	1.20902	0.604512	+	0.796596i	$0.293368\pi$
$812$	−0.0737006	−0.00258638
$813$	0	0
$814$	−3.23083	−0.113240
$815$	12.9910	0.455056
$816$	0	0
$817$	17.2591	0.603818
$818$	−8.05970	−0.281801
$819$	0	0
$820$	1.31255	0.0458363
$821$	−39.8355	−1.39027	−0.695134	−	0.718880i	$-0.744655\pi$
$821$	−39.8355	−1.39027	−0.695134	+	0.718880i	$0.744655\pi$
$822$	0	0
$823$	−37.9114	−1.32151	−0.660754	−	0.750602i	$-0.729763\pi$
$823$	−37.9114	−1.32151	−0.660754	+	0.750602i	$0.729763\pi$
$824$	−7.77812	−0.270964
$825$	0	0
$826$	0.591221	0.0205712
$827$	10.8584	0.377584	0.188792	−	0.982017i	$-0.439543\pi$
$827$	10.8584	0.377584	0.188792	+	0.982017i	$0.439543\pi$
$828$	0	0
$829$	−36.4132	−1.26468	−0.632342	−	0.774690i	$-0.717906\pi$
$829$	−36.4132	−1.26468	−0.632342	+	0.774690i	$0.717906\pi$
$830$	1.78418	0.0619298
$831$	0	0
$832$	−22.0753	−0.765325
$833$	9.37563	0.324846
$834$	0	0
$835$	−14.3545	−0.496757
$836$	10.3009	0.356264
$837$	0	0
$838$	20.1832	0.697218
$839$	−28.0543	−0.968542	−0.484271	−	0.874918i	$-0.660915\pi$
$839$	−28.0543	−0.968542	−0.484271	+	0.874918i	$0.660915\pi$
$840$	0	0
$841$	−26.4088	−0.910648
$842$	13.3084	0.458636
$843$	0	0
$844$	−32.6412	−1.12356
$845$	−23.3006	−0.801565
$846$	0	0
$847$	0.0298223	0.00102471
$848$	35.9193	1.23347
$849$	0	0
$850$	−8.47781	−0.290786
$851$	−5.86392	−0.201012
$852$	0	0
$853$	−47.5752	−1.62894	−0.814471	−	0.580204i	$-0.802973\pi$
$853$	−47.5752	−1.62894	−0.814471	+	0.580204i	$0.802973\pi$
$854$	0.0560727	0.00191877
$855$	0	0
$856$	7.32309	0.250298
$857$	−40.4027	−1.38013	−0.690065	−	0.723748i	$-0.742418\pi$
$857$	−40.4027	−1.38013	−0.690065	+	0.723748i	$0.742418\pi$
$858$	0	0
$859$	−35.8636	−1.22365	−0.611825	−	0.790993i	$-0.709564\pi$
$859$	−35.8636	−1.22365	−0.611825	+	0.790993i	$0.709564\pi$
$860$	−5.04801	−0.172136
$861$	0	0
$862$	−34.3070	−1.16850
$863$	−35.3569	−1.20356	−0.601782	−	0.798661i	$-0.705542\pi$
$863$	−35.3569	−1.20356	−0.601782	+	0.798661i	$0.705542\pi$
$864$	0	0
$865$	14.4828	0.492431
$866$	9.77488	0.332164
$867$	0	0
$868$	−0.286387	−0.00972060
$869$	7.69951	0.261188
$870$	0	0
$871$	−48.0003	−1.62643
$872$	−10.4748	−0.354721
$873$	0	0
$874$	43.0518	1.45625
$875$	0.318922	0.0107815
$876$	0	0
$877$	27.6647	0.934170	0.467085	−	0.884212i	$-0.345304\pi$
$877$	27.6647	0.934170	0.467085	+	0.884212i	$0.345304\pi$
$878$	66.1466	2.23234
$879$	0	0
$880$	6.02516	0.203108
$881$	6.81227	0.229511	0.114756	−	0.993394i	$-0.463392\pi$
$881$	6.81227	0.229511	0.114756	+	0.993394i	$0.463392\pi$
$882$	0	0
$883$	−16.0376	−0.539707	−0.269853	−	0.962901i	$-0.586975\pi$
$883$	−16.0376	−0.539707	−0.269853	+	0.962901i	$0.586975\pi$
$884$	−11.4923	−0.386529
$885$	0	0
$886$	−25.1853	−0.846118
$887$	−43.1339	−1.44829	−0.724147	−	0.689646i	$-0.757766\pi$
$887$	−43.1339	−1.44829	−0.724147	+	0.689646i	$0.757766\pi$
$888$	0	0
$889$	−0.200741	−0.00673262
$890$	32.2675	1.08161
$891$	0	0
$892$	−14.1295	−0.473090
$893$	3.94318	0.131953
$894$	0	0
$895$	−11.9779	−0.400379
$896$	−0.202852	−0.00677682
$897$	0	0
$898$	33.3894	1.11422
$899$	10.0690	0.335819
$900$	0	0
$901$	10.2080	0.340078
$902$	1.25755	0.0418718
$903$	0	0
$904$	−1.07194	−0.0356522
$905$	21.8176	0.725241
$906$	0	0
$907$	4.97879	0.165318	0.0826590	−	0.996578i	$-0.473659\pi$
$907$	4.97879	0.165318	0.0826590	+	0.996578i	$0.473659\pi$
$908$	−14.9701	−0.496799
$909$	0	0
$910$	0.400543	0.0132779
$911$	46.5597	1.54259	0.771296	−	0.636476i	$-0.219609\pi$
$911$	46.5597	1.54259	0.771296	+	0.636476i	$0.219609\pi$
$912$	0	0
$913$	0.742344	0.0245680
$914$	−40.6812	−1.34562
$915$	0	0
$916$	12.2909	0.406103
$917$	0.0332680	0.00109861
$918$	0	0
$919$	−2.86125	−0.0943839	−0.0471920	−	0.998886i	$-0.515027\pi$
$919$	−2.86125	−0.0943839	−0.0471920	+	0.998886i	$0.515027\pi$
$920$	3.81191	0.125675
$921$	0	0
$922$	17.2107	0.566803
$923$	−11.3763	−0.374456
$924$	0	0
$925$	−5.78389	−0.190173
$926$	14.4850	0.476007
$927$	0	0
$928$	−11.4528	−0.375957
$929$	12.6025	0.413476	0.206738	−	0.978396i	$-0.433715\pi$
$929$	12.6025	0.413476	0.206738	+	0.978396i	$0.433715\pi$
$930$	0	0
$931$	−46.9614	−1.53910
$932$	−9.45048	−0.309561
$933$	0	0
$934$	−19.7654	−0.646742
$935$	1.71231	0.0559984
$936$	0	0
$937$	−13.0474	−0.426240	−0.213120	−	0.977026i	$-0.568363\pi$
$937$	−13.0474	−0.426240	−0.213120	+	0.977026i	$0.568363\pi$
$938$	0.481640	0.0157261
$939$	0	0
$940$	−1.15332	−0.0376171
$941$	−3.70818	−0.120883	−0.0604416	−	0.998172i	$-0.519251\pi$
$941$	−3.70818	−0.120883	−0.0604416	+	0.998172i	$0.519251\pi$
$942$	0	0
$943$	2.28244	0.0743264
$944$	49.6985	1.61755
$945$	0	0
$946$	−4.83647	−0.157247
$947$	−47.5677	−1.54574	−0.772871	−	0.634564i	$-0.781180\pi$
$947$	−47.5677	−1.54574	−0.772871	+	0.634564i	$0.781180\pi$
$948$	0	0
$949$	−21.2858	−0.690966
$950$	42.4643	1.37772
$951$	0	0
$952$	−0.0349088	−0.00113140
$953$	7.37757	0.238983	0.119491	−	0.992835i	$-0.461874\pi$
$953$	7.37757	0.238983	0.119491	+	0.992835i	$0.461874\pi$
$954$	0	0
$955$	27.9098	0.903141
$956$	−36.8417	−1.19154
$957$	0	0
$958$	41.7922	1.35025
$959$	−0.139284	−0.00449772
$960$	0	0
$961$	8.12618	0.262135
$962$	−18.0546	−0.582102
$963$	0	0
$964$	−27.0608	−0.871570
$965$	−13.7090	−0.441308
$966$	0	0
$967$	−36.9737	−1.18899	−0.594497	−	0.804098i	$-0.702649\pi$
$967$	−36.9737	−1.18899	−0.594497	+	0.804098i	$0.702649\pi$
$968$	0.873847	0.0280865
$969$	0	0
$970$	3.76117	0.120764
$971$	−40.9475	−1.31407	−0.657034	−	0.753861i	$-0.728189\pi$
$971$	−40.9475	−1.31407	−0.657034	+	0.753861i	$0.728189\pi$
$972$	0	0
$973$	−0.112510	−0.00360689
$974$	−12.3682	−0.396304
$975$	0	0
$976$	4.71351	0.150876
$977$	−2.35066	−0.0752044	−0.0376022	−	0.999293i	$-0.511972\pi$
$977$	−2.35066	−0.0752044	−0.0376022	+	0.999293i	$0.511972\pi$
$978$	0	0
$979$	13.4255	0.429082
$980$	13.7355	0.438764
$981$	0	0
$982$	24.2799	0.774802
$983$	42.1964	1.34586	0.672928	−	0.739708i	$-0.265036\pi$
$983$	42.1964	1.34586	0.672928	+	0.739708i	$0.265036\pi$
$984$	0	0
$985$	−18.6494	−0.594219
$986$	−4.05433	−0.129116
$987$	0	0
$988$	57.5637	1.83135
$989$	−8.77814	−0.279129
$990$	0	0
$991$	31.5627	1.00262	0.501312	−	0.865267i	$-0.332851\pi$
$991$	31.5627	1.00262	0.501312	+	0.865267i	$0.332851\pi$
$992$	−44.5036	−1.41299
$993$	0	0
$994$	0.114151	0.00362065
$995$	−24.8602	−0.788123
$996$	0	0
$997$	23.6223	0.748126	0.374063	−	0.927403i	$-0.377964\pi$
$997$	23.6223	0.748126	0.374063	+	0.927403i	$0.377964\pi$
$998$	−64.6158	−2.04538
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.g.1.4		13
3.2	odd	2		2013.2.a.f.1.10	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.f.1.10	✓	13	3.2	odd	2
6039.2.a.g.1.4		13	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)

\(f(q)\)	\(=\)	\(q-1.88022 q^{2} +1.53524 q^{4} -1.27827 q^{5} +0.0298223 q^{7} +0.873847 q^{8} +O(q^{10})\)	\(q-1.88022 q^{2} +1.53524 q^{4} -1.27827 q^{5} +0.0298223 q^{7} +0.873847 q^{8} +2.40344 q^{10} +1.00000 q^{11} +5.58822 q^{13} -0.0560727 q^{14} -4.71351 q^{16} -1.33955 q^{17} +6.70962 q^{19} -1.96246 q^{20} -1.88022 q^{22} -3.41259 q^{23} -3.36602 q^{25} -10.5071 q^{26} +0.0457846 q^{28} -1.60973 q^{29} -6.25509 q^{31} +7.11477 q^{32} +2.51865 q^{34} -0.0381211 q^{35} +1.71832 q^{37} -12.6156 q^{38} -1.11701 q^{40} -0.668829 q^{41} +2.57228 q^{43} +1.53524 q^{44} +6.41643 q^{46} +0.587690 q^{47} -6.99911 q^{49} +6.32887 q^{50} +8.57927 q^{52} -7.62049 q^{53} -1.27827 q^{55} +0.0260602 q^{56} +3.02664 q^{58} -10.5438 q^{59} -1.00000 q^{61} +11.7610 q^{62} -3.95034 q^{64} -7.14327 q^{65} -8.58956 q^{67} -2.05653 q^{68} +0.0716763 q^{70} -2.03577 q^{71} -3.80905 q^{73} -3.23083 q^{74} +10.3009 q^{76} +0.0298223 q^{77} +7.69951 q^{79} +6.02516 q^{80} +1.25755 q^{82} +0.742344 q^{83} +1.71231 q^{85} -4.83647 q^{86} +0.873847 q^{88} +13.4255 q^{89} +0.166654 q^{91} -5.23915 q^{92} -1.10499 q^{94} -8.57673 q^{95} +1.56491 q^{97} +13.1599 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8} + 8 q^{10} + 13 q^{11} - 9 q^{13} - 19 q^{14} + 18 q^{16} - 7 q^{17} + 2 q^{19} - 15 q^{20} - 4 q^{22} - 23 q^{23} + 10 q^{25} - 8 q^{26} + 9 q^{28} - 16 q^{29} + 9 q^{31} - 29 q^{32} + 2 q^{34} - 16 q^{35} + 14 q^{37} - 8 q^{38} + 16 q^{40} - 19 q^{41} + 7 q^{43} + 12 q^{44} + 4 q^{46} - 26 q^{47} + 8 q^{49} + 15 q^{50} - 17 q^{52} - 18 q^{53} - 7 q^{55} - 44 q^{56} - q^{58} - 31 q^{59} - 13 q^{61} + 5 q^{62} - 17 q^{64} - 31 q^{65} + 14 q^{67} + 32 q^{68} - 20 q^{70} - 37 q^{71} - 16 q^{73} + 6 q^{74} - 7 q^{76} + 5 q^{77} - 17 q^{79} + 2 q^{80} - 2 q^{82} - 30 q^{83} - 16 q^{85} + 22 q^{86} - 15 q^{88} - 35 q^{89} - q^{91} - 24 q^{92} - 11 q^{94} - 13 q^{95} - q^{97} + q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8 + 8 * q^10 + 13 * q^11 - 9 * q^13 - 19 * q^14 + 18 * q^16 - 7 * q^17 + 2 * q^19 - 15 * q^20 - 4 * q^22 - 23 * q^23 + 10 * q^25 - 8 * q^26 + 9 * q^28 - 16 * q^29 + 9 * q^31 - 29 * q^32 + 2 * q^34 - 16 * q^35 + 14 * q^37 - 8 * q^38 + 16 * q^40 - 19 * q^41 + 7 * q^43 + 12 * q^44 + 4 * q^46 - 26 * q^47 + 8 * q^49 + 15 * q^50 - 17 * q^52 - 18 * q^53 - 7 * q^55 - 44 * q^56 - q^58 - 31 * q^59 - 13 * q^61 + 5 * q^62 - 17 * q^64 - 31 * q^65 + 14 * q^67 + 32 * q^68 - 20 * q^70 - 37 * q^71 - 16 * q^73 + 6 * q^74 - 7 * q^76 + 5 * q^77 - 17 * q^79 + 2 * q^80 - 2 * q^82 - 30 * q^83 - 16 * q^85 + 22 * q^86 - 15 * q^88 - 35 * q^89 - q^91 - 24 * q^92 - 11 * q^94 - 13 * q^95 - q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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