LMFDB - Embedded newform 6039.2.a.d.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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4x^{10} - 6x^{9} + 37x^{8} - 2x^{7} - 109x^{6} + 55x^{5} + 115x^{4} - 76x^{3} - 29x^{2} + 14x + 3 $ x^11 - 4x^10 - 6x^9 + 37x^8 - 2x^7 - 109x^6 + 55x^5 + 115x^4 - 76x^3 - 29x^2 + 14x + 3
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			$-0.423080$ 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)

$f(q)$	$=$	$q-0.423080 q^{2} -1.82100 q^{4} +1.36483 q^{5} +0.865577 q^{7} +1.61659 q^{8} +O(q^{10})$	\(q-0.423080 q^{2} -1.82100 q^{4} +1.36483 q^{5} +0.865577 q^{7} +1.61659 q^{8} -0.577432 q^{10} +1.00000 q^{11} +2.63744 q^{13} -0.366209 q^{14} +2.95806 q^{16} +3.12547 q^{17} -0.281623 q^{19} -2.48536 q^{20} -0.423080 q^{22} -1.73752 q^{23} -3.13724 q^{25} -1.11585 q^{26} -1.57622 q^{28} +7.07603 q^{29} -4.28304 q^{31} -4.48468 q^{32} -1.32232 q^{34} +1.18136 q^{35} +3.94276 q^{37} +0.119149 q^{38} +2.20637 q^{40} +7.38287 q^{41} +5.77261 q^{43} -1.82100 q^{44} +0.735109 q^{46} +3.73936 q^{47} -6.25078 q^{49} +1.32731 q^{50} -4.80278 q^{52} +5.86335 q^{53} +1.36483 q^{55} +1.39928 q^{56} -2.99373 q^{58} +7.96157 q^{59} +1.00000 q^{61} +1.81207 q^{62} -4.01874 q^{64} +3.59965 q^{65} +4.80971 q^{67} -5.69149 q^{68} -0.499812 q^{70} +7.35945 q^{71} -16.3671 q^{73} -1.66811 q^{74} +0.512837 q^{76} +0.865577 q^{77} -6.03539 q^{79} +4.03724 q^{80} -3.12355 q^{82} -10.7520 q^{83} +4.26573 q^{85} -2.44228 q^{86} +1.61659 q^{88} +13.8901 q^{89} +2.28290 q^{91} +3.16402 q^{92} -1.58205 q^{94} -0.384368 q^{95} +12.3781 q^{97} +2.64458 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) $ 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8	$ 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8} + 6 q^{10} + 11 q^{11} - 3 q^{13} + 9 q^{14} + 4 q^{16} + 7 q^{17} - 8 q^{19} + 25 q^{20} + 4 q^{22} + 15 q^{23} + 4 q^{25} + 2 q^{26} + 13 q^{28} + 8 q^{29} - 17 q^{31} + 27 q^{32} - 18 q^{34} + 2 q^{35} - 10 q^{37} + 30 q^{38} + 10 q^{40} + 25 q^{41} - 7 q^{43} + 6 q^{44} + 32 q^{46} + 30 q^{47} - 2 q^{49} - 11 q^{50} - 7 q^{52} + 18 q^{53} + 13 q^{55} + 20 q^{56} - 13 q^{58} + 43 q^{59} + 11 q^{61} - 7 q^{62} + 25 q^{64} + 27 q^{65} - 30 q^{67} - 10 q^{68} - 4 q^{70} + 7 q^{71} + 6 q^{73} + 44 q^{74} - 19 q^{76} - 5 q^{77} + 17 q^{79} + 22 q^{80} + 8 q^{82} + 34 q^{83} + 10 q^{85} - 2 q^{86} + 9 q^{88} + 41 q^{89} - 39 q^{91} + 32 q^{92} + 55 q^{94} + 9 q^{95} - 41 q^{97} + 29 q^{98}+O(q^{100}) $ 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8 + 6 * q^10 + 11 * q^11 - 3 * q^13 + 9 * q^14 + 4 * q^16 + 7 * q^17 - 8 * q^19 + 25 * q^20 + 4 * q^22 + 15 * q^23 + 4 * q^25 + 2 * q^26 + 13 * q^28 + 8 * q^29 - 17 * q^31 + 27 * q^32 - 18 * q^34 + 2 * q^35 - 10 * q^37 + 30 * q^38 + 10 * q^40 + 25 * q^41 - 7 * q^43 + 6 * q^44 + 32 * q^46 + 30 * q^47 - 2 * q^49 - 11 * q^50 - 7 * q^52 + 18 * q^53 + 13 * q^55 + 20 * q^56 - 13 * q^58 + 43 * q^59 + 11 * q^61 - 7 * q^62 + 25 * q^64 + 27 * q^65 - 30 * q^67 - 10 * q^68 - 4 * q^70 + 7 * q^71 + 6 * q^73 + 44 * q^74 - 19 * q^76 - 5 * q^77 + 17 * q^79 + 22 * q^80 + 8 * q^82 + 34 * q^83 + 10 * q^85 - 2 * q^86 + 9 * q^88 + 41 * q^89 - 39 * q^91 + 32 * q^92 + 55 * q^94 + 9 * q^95 - 41 * q^97 + 29 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.423080	−0.299163	−0.149582	−	0.988749i	$-0.547793\pi$
$2$	−0.423080	−0.299163	−0.149582	+	0.988749i	$0.547793\pi$
$3$	0	0
$3$	0	0
$4$	−1.82100	−0.910501
$5$	1.36483	0.610370	0.305185	−	0.952293i	$-0.401282\pi$
$5$	1.36483	0.610370	0.305185	+	0.952293i	$0.401282\pi$
$6$	0	0
$7$	0.865577	0.327157	0.163579	−	0.986530i	$-0.447696\pi$
$7$	0.865577	0.327157	0.163579	+	0.986530i	$0.447696\pi$
$8$	1.61659	0.571551
$9$	0	0
$10$	−0.577432	−0.182600
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.63744	0.731493	0.365747	−	0.930714i	$-0.380814\pi$
$13$	2.63744	0.731493	0.365747	+	0.930714i	$0.380814\pi$
$14$	−0.366209	−0.0978733
$15$	0	0
$16$	2.95806	0.739514
$17$	3.12547	0.758037	0.379019	−	0.925389i	$-0.376262\pi$
$17$	3.12547	0.758037	0.379019	+	0.925389i	$0.376262\pi$
$18$	0	0
$19$	−0.281623	−0.0646088	−0.0323044	−	0.999478i	$-0.510285\pi$
$19$	−0.281623	−0.0646088	−0.0323044	+	0.999478i	$0.510285\pi$
$20$	−2.48536	−0.555742
$21$	0	0
$22$	−0.423080	−0.0902010
$23$	−1.73752	−0.362297	−0.181149	−	0.983456i	$-0.557982\pi$
$23$	−1.73752	−0.362297	−0.181149	+	0.983456i	$0.557982\pi$
$24$	0	0
$25$	−3.13724	−0.627449
$26$	−1.11585	−0.218836
$27$	0	0
$28$	−1.57622	−0.297877
$29$	7.07603	1.31399	0.656993	−	0.753897i	$-0.271828\pi$
$29$	7.07603	1.31399	0.656993	+	0.753897i	$0.271828\pi$
$30$	0	0
$31$	−4.28304	−0.769257	−0.384628	−	0.923071i	$-0.625670\pi$
$31$	−4.28304	−0.769257	−0.384628	+	0.923071i	$0.625670\pi$
$32$	−4.48468	−0.792787
$33$	0	0
$34$	−1.32232	−0.226777
$35$	1.18136	0.199687
$36$	0	0
$37$	3.94276	0.648186	0.324093	−	0.946025i	$-0.394941\pi$
$37$	3.94276	0.648186	0.324093	+	0.946025i	$0.394941\pi$
$38$	0.119149	0.0193286
$39$	0	0
$40$	2.20637	0.348858
$41$	7.38287	1.15301	0.576505	−	0.817094i	$-0.304416\pi$
$41$	7.38287	1.15301	0.576505	+	0.817094i	$0.304416\pi$
$42$	0	0
$43$	5.77261	0.880315	0.440157	−	0.897921i	$-0.354923\pi$
$43$	5.77261	0.880315	0.440157	+	0.897921i	$0.354923\pi$
$44$	−1.82100	−0.274527
$45$	0	0
$46$	0.735109	0.108386
$47$	3.73936	0.545441	0.272721	−	0.962093i	$-0.412076\pi$
$47$	3.73936	0.545441	0.272721	+	0.962093i	$0.412076\pi$
$48$	0	0
$49$	−6.25078	−0.892968
$50$	1.32731	0.187709
$51$	0	0
$52$	−4.80278	−0.666026
$53$	5.86335	0.805393	0.402696	−	0.915334i	$-0.368073\pi$
$53$	5.86335	0.805393	0.402696	+	0.915334i	$0.368073\pi$
$54$	0	0
$55$	1.36483	0.184033
$56$	1.39928	0.186987
$57$	0	0
$58$	−2.99373	−0.393096
$59$	7.96157	1.03651	0.518254	−	0.855227i	$-0.326582\pi$
$59$	7.96157	1.03651	0.518254	+	0.855227i	$0.326582\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	1.81207	0.230133
$63$	0	0
$64$	−4.01874	−0.502342
$65$	3.59965	0.446481
$66$	0	0
$67$	4.80971	0.587599	0.293800	−	0.955867i	$-0.405080\pi$
$67$	4.80971	0.587599	0.293800	+	0.955867i	$0.405080\pi$
$68$	−5.69149	−0.690194
$69$	0	0
$70$	−0.499812	−0.0597389
$71$	7.35945	0.873406	0.436703	−	0.899606i	$-0.356146\pi$
$71$	7.35945	0.873406	0.436703	+	0.899606i	$0.356146\pi$
$72$	0	0
$73$	−16.3671	−1.91563	−0.957814	−	0.287390i	$-0.907212\pi$
$73$	−16.3671	−1.91563	−0.957814	+	0.287390i	$0.907212\pi$
$74$	−1.66811	−0.193913
$75$	0	0
$76$	0.512837	0.0588265
$77$	0.865577	0.0986416
$78$	0	0
$79$	−6.03539	−0.679035	−0.339517	−	0.940600i	$-0.610264\pi$
$79$	−6.03539	−0.679035	−0.339517	+	0.940600i	$0.610264\pi$
$80$	4.03724	0.451377
$81$	0	0
$82$	−3.12355	−0.344938
$83$	−10.7520	−1.18019	−0.590093	−	0.807336i	$-0.700909\pi$
$83$	−10.7520	−1.18019	−0.590093	+	0.807336i	$0.700909\pi$
$84$	0	0
$85$	4.26573	0.462683
$86$	−2.44228	−0.263358
$87$	0	0
$88$	1.61659	0.172329
$89$	13.8901	1.47235	0.736175	−	0.676791i	$-0.236630\pi$
$89$	13.8901	1.47235	0.736175	+	0.676791i	$0.236630\pi$
$90$	0	0
$91$	2.28290	0.239313
$92$	3.16402	0.329872
$93$	0	0
$94$	−1.58205	−0.163176
$95$	−0.384368	−0.0394353
$96$	0	0
$97$	12.3781	1.25680	0.628402	−	0.777888i	$-0.283709\pi$
$97$	12.3781	1.25680	0.628402	+	0.777888i	$0.283709\pi$
$98$	2.64458	0.267143
$99$	0	0
$100$	5.71293	0.571293
$101$	−0.0427311	−0.00425191	−0.00212595	−	0.999998i	$-0.500677\pi$
$101$	−0.0427311	−0.00425191	−0.00212595	+	0.999998i	$0.500677\pi$
$102$	0	0
$103$	−14.7060	−1.44902	−0.724511	−	0.689263i	$-0.757935\pi$
$103$	−14.7060	−1.44902	−0.724511	+	0.689263i	$0.757935\pi$
$104$	4.26366	0.418086
$105$	0	0
$106$	−2.48067	−0.240944
$107$	−14.8784	−1.43835	−0.719175	−	0.694829i	$-0.755480\pi$
$107$	−14.8784	−1.43835	−0.719175	+	0.694829i	$0.755480\pi$
$108$	0	0
$109$	−9.34470	−0.895060	−0.447530	−	0.894269i	$-0.647696\pi$
$109$	−9.34470	−0.895060	−0.447530	+	0.894269i	$0.647696\pi$
$110$	−0.577432	−0.0550560
$111$	0	0
$112$	2.56043	0.241938
$113$	−15.9919	−1.50440	−0.752198	−	0.658938i	$-0.771006\pi$
$113$	−15.9919	−1.50440	−0.752198	+	0.658938i	$0.771006\pi$
$114$	0	0
$115$	−2.37141	−0.221135
$116$	−12.8855	−1.19639
$117$	0	0
$118$	−3.36839	−0.310085
$119$	2.70533	0.247997
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.423080	−0.0383039
$123$	0	0
$124$	7.79943	0.700410
$125$	−11.1059	−0.993345
$126$	0	0
$127$	−17.6871	−1.56948	−0.784740	−	0.619825i	$-0.787204\pi$
$127$	−17.6871	−1.56948	−0.784740	+	0.619825i	$0.787204\pi$
$128$	10.6696	0.943069
$129$	0	0
$130$	−1.52294	−0.133571
$131$	20.0215	1.74929	0.874643	−	0.484768i	$-0.161096\pi$
$131$	20.0215	1.74929	0.874643	+	0.484768i	$0.161096\pi$
$132$	0	0
$133$	−0.243767	−0.0211373
$134$	−2.03489	−0.175788
$135$	0	0
$136$	5.05260	0.433257
$137$	16.4175	1.40264	0.701322	−	0.712844i	$-0.252593\pi$
$137$	16.4175	1.40264	0.701322	+	0.712844i	$0.252593\pi$
$138$	0	0
$139$	−6.82389	−0.578795	−0.289397	−	0.957209i	$-0.593455\pi$
$139$	−6.82389	−0.578795	−0.289397	+	0.957209i	$0.593455\pi$
$140$	−2.15127	−0.181815
$141$	0	0
$142$	−3.11364	−0.261291
$143$	2.63744	0.220554
$144$	0	0
$145$	9.65757	0.802017
$146$	6.92461	0.573085
$147$	0	0
$148$	−7.17979	−0.590175
$149$	−15.0883	−1.23608	−0.618039	−	0.786148i	$-0.712073\pi$
$149$	−15.0883	−1.23608	−0.618039	+	0.786148i	$0.712073\pi$
$150$	0	0
$151$	−8.59674	−0.699593	−0.349797	−	0.936826i	$-0.613749\pi$
$151$	−8.59674	−0.699593	−0.349797	+	0.936826i	$0.613749\pi$
$152$	−0.455270	−0.0369273
$153$	0	0
$154$	−0.366209	−0.0295099
$155$	−5.84562	−0.469531
$156$	0	0
$157$	9.95066	0.794149	0.397074	−	0.917786i	$-0.370025\pi$
$157$	9.95066	0.794149	0.397074	+	0.917786i	$0.370025\pi$
$158$	2.55346	0.203142
$159$	0	0
$160$	−6.12082	−0.483893
$161$	−1.50395	−0.118528
$162$	0	0
$163$	21.9852	1.72201	0.861005	−	0.508596i	$-0.169835\pi$
$163$	21.9852	1.72201	0.861005	+	0.508596i	$0.169835\pi$
$164$	−13.4442	−1.04982
$165$	0	0
$166$	4.54896	0.353068
$167$	−11.4954	−0.889539	−0.444770	−	0.895645i	$-0.646714\pi$
$167$	−11.4954	−0.889539	−0.444770	+	0.895645i	$0.646714\pi$
$168$	0	0
$169$	−6.04393	−0.464918
$170$	−1.80474	−0.138418
$171$	0	0
$172$	−10.5119	−0.801528
$173$	20.5992	1.56613	0.783065	−	0.621940i	$-0.213655\pi$
$173$	20.5992	1.56613	0.783065	+	0.621940i	$0.213655\pi$
$174$	0	0
$175$	−2.71553	−0.205274
$176$	2.95806	0.222972
$177$	0	0
$178$	−5.87664	−0.440473
$179$	9.33212	0.697515	0.348758	−	0.937213i	$-0.386604\pi$
$179$	9.33212	0.697515	0.348758	+	0.937213i	$0.386604\pi$
$180$	0	0
$181$	10.1819	0.756814	0.378407	−	0.925639i	$-0.376472\pi$
$181$	10.1819	0.756814	0.378407	+	0.925639i	$0.376472\pi$
$182$	−0.965852	−0.0715937
$183$	0	0
$184$	−2.80886	−0.207072
$185$	5.38119	0.395633
$186$	0	0
$187$	3.12547	0.228557
$188$	−6.80938	−0.496625
$189$	0	0
$190$	0.162618	0.0117976
$191$	−7.53307	−0.545074	−0.272537	−	0.962145i	$-0.587863\pi$
$191$	−7.53307	−0.545074	−0.272537	+	0.962145i	$0.587863\pi$
$192$	0	0
$193$	27.2060	1.95833	0.979165	−	0.203067i	$-0.0650909\pi$
$193$	27.2060	1.95833	0.979165	+	0.203067i	$0.0650909\pi$
$194$	−5.23693	−0.375990
$195$	0	0
$196$	11.3827	0.813049
$197$	−5.30033	−0.377633	−0.188817	−	0.982012i	$-0.560465\pi$
$197$	−5.30033	−0.377633	−0.188817	+	0.982012i	$0.560465\pi$
$198$	0	0
$199$	26.0687	1.84796	0.923981	−	0.382438i	$-0.124916\pi$
$199$	26.0687	1.84796	0.923981	+	0.382438i	$0.124916\pi$
$200$	−5.07164	−0.358619
$201$	0	0
$202$	0.0180787	0.00127201
$203$	6.12485	0.429880
$204$	0	0
$205$	10.0763	0.703763
$206$	6.22181	0.433494
$207$	0	0
$208$	7.80169	0.540950
$209$	−0.281623	−0.0194803
$210$	0	0
$211$	21.3309	1.46848	0.734238	−	0.678892i	$-0.237539\pi$
$211$	21.3309	1.46848	0.734238	+	0.678892i	$0.237539\pi$
$212$	−10.6772	−0.733311
$213$	0	0
$214$	6.29476	0.430301
$215$	7.87862	0.537317
$216$	0	0
$217$	−3.70730	−0.251668
$218$	3.95356	0.267769
$219$	0	0
$220$	−2.48536	−0.167563
$221$	8.24322	0.554499
$222$	0	0
$223$	10.5749	0.708148	0.354074	−	0.935217i	$-0.384796\pi$
$223$	10.5749	0.708148	0.354074	+	0.935217i	$0.384796\pi$
$224$	−3.88183	−0.259366
$225$	0	0
$226$	6.76588	0.450059
$227$	10.0676	0.668210	0.334105	−	0.942536i	$-0.391566\pi$
$227$	10.0676	0.668210	0.334105	+	0.942536i	$0.391566\pi$
$228$	0	0
$229$	−18.2971	−1.20910	−0.604552	−	0.796566i	$-0.706648\pi$
$229$	−18.2971	−1.20910	−0.604552	+	0.796566i	$0.706648\pi$
$230$	1.00330	0.0661555
$231$	0	0
$232$	11.4391	0.751011
$233$	20.1119	1.31758	0.658788	−	0.752328i	$-0.271069\pi$
$233$	20.1119	1.31758	0.658788	+	0.752328i	$0.271069\pi$
$234$	0	0
$235$	5.10358	0.332921
$236$	−14.4981	−0.943743
$237$	0	0
$238$	−1.14457	−0.0741916
$239$	9.24058	0.597723	0.298862	−	0.954296i	$-0.403393\pi$
$239$	9.24058	0.597723	0.298862	+	0.954296i	$0.403393\pi$
$240$	0	0
$241$	−4.46915	−0.287884	−0.143942	−	0.989586i	$-0.545978\pi$
$241$	−4.46915	−0.287884	−0.143942	+	0.989586i	$0.545978\pi$
$242$	−0.423080	−0.0271966
$243$	0	0
$244$	−1.82100	−0.116578
$245$	−8.53124	−0.545041
$246$	0	0
$247$	−0.742764	−0.0472609
$248$	−6.92393	−0.439670
$249$	0	0
$250$	4.69871	0.297172
$251$	−25.3784	−1.60187	−0.800935	−	0.598752i	$-0.795664\pi$
$251$	−25.3784	−1.60187	−0.800935	+	0.598752i	$0.795664\pi$
$252$	0	0
$253$	−1.73752	−0.109237
$254$	7.48308	0.469530
$255$	0	0
$256$	3.52337	0.220211
$257$	−11.7192	−0.731022	−0.365511	−	0.930807i	$-0.619106\pi$
$257$	−11.7192	−0.731022	−0.365511	+	0.930807i	$0.619106\pi$
$258$	0	0
$259$	3.41276	0.212059
$260$	−6.55497	−0.406522
$261$	0	0
$262$	−8.47070	−0.523321
$263$	−12.2142	−0.753163	−0.376581	−	0.926384i	$-0.622900\pi$
$263$	−12.2142	−0.753163	−0.376581	+	0.926384i	$0.622900\pi$
$264$	0	0
$265$	8.00246	0.491587
$266$	0.103133	0.00632348
$267$	0	0
$268$	−8.75849	−0.535010
$269$	22.0290	1.34313	0.671567	−	0.740944i	$-0.265622\pi$
$269$	22.0290	1.34313	0.671567	+	0.740944i	$0.265622\pi$
$270$	0	0
$271$	5.27206	0.320255	0.160127	−	0.987096i	$-0.448810\pi$
$271$	5.27206	0.320255	0.160127	+	0.987096i	$0.448810\pi$
$272$	9.24531	0.560579
$273$	0	0
$274$	−6.94594	−0.419619
$275$	−3.13724	−0.189183
$276$	0	0
$277$	25.0372	1.50434	0.752171	−	0.658969i	$-0.229007\pi$
$277$	25.0372	1.50434	0.752171	+	0.658969i	$0.229007\pi$
$278$	2.88705	0.173154
$279$	0	0
$280$	1.90978	0.114131
$281$	−19.6180	−1.17031	−0.585157	−	0.810920i	$-0.698967\pi$
$281$	−19.6180	−1.17031	−0.585157	+	0.810920i	$0.698967\pi$
$282$	0	0
$283$	−3.64470	−0.216655	−0.108327	−	0.994115i	$-0.534549\pi$
$283$	−3.64470	−0.216655	−0.108327	+	0.994115i	$0.534549\pi$
$284$	−13.4016	−0.795238
$285$	0	0
$286$	−1.11585	−0.0659815
$287$	6.39044	0.377216
$288$	0	0
$289$	−7.23145	−0.425380
$290$	−4.08593	−0.239934
$291$	0	0
$292$	29.8046	1.74418
$293$	5.22920	0.305493	0.152747	−	0.988265i	$-0.451188\pi$
$293$	5.22920	0.305493	0.152747	+	0.988265i	$0.451188\pi$
$294$	0	0
$295$	10.8662	0.632654
$296$	6.37384	0.370472
$297$	0	0
$298$	6.38354	0.369789
$299$	−4.58259	−0.265018
$300$	0	0
$301$	4.99664	0.288001
$302$	3.63711	0.209292
$303$	0	0
$304$	−0.833058	−0.0477792
$305$	1.36483	0.0781498
$306$	0	0
$307$	−24.2309	−1.38293	−0.691467	−	0.722408i	$-0.743035\pi$
$307$	−24.2309	−1.38293	−0.691467	+	0.722408i	$0.743035\pi$
$308$	−1.57622	−0.0898133
$309$	0	0
$310$	2.47317	0.140466
$311$	−1.91505	−0.108593	−0.0542964	−	0.998525i	$-0.517292\pi$
$311$	−1.91505	−0.108593	−0.0542964	+	0.998525i	$0.517292\pi$
$312$	0	0
$313$	18.8745	1.06685	0.533425	−	0.845847i	$-0.320905\pi$
$313$	18.8745	1.06685	0.533425	+	0.845847i	$0.320905\pi$
$314$	−4.20993	−0.237580
$315$	0	0
$316$	10.9905	0.618262
$317$	29.4492	1.65403	0.827017	−	0.562176i	$-0.190036\pi$
$317$	29.4492	1.65403	0.827017	+	0.562176i	$0.190036\pi$
$318$	0	0
$319$	7.07603	0.396182
$320$	−5.48488	−0.306614
$321$	0	0
$322$	0.636294	0.0354593
$323$	−0.880205	−0.0489759
$324$	0	0
$325$	−8.27428	−0.458975
$326$	−9.30149	−0.515162
$327$	0	0
$328$	11.9351	0.659005
$329$	3.23670	0.178445
$330$	0	0
$331$	−12.7781	−0.702347	−0.351173	−	0.936310i	$-0.614217\pi$
$331$	−12.7781	−0.702347	−0.351173	+	0.936310i	$0.614217\pi$
$332$	19.5794	1.07456
$333$	0	0
$334$	4.86347	0.266117
$335$	6.56442	0.358653
$336$	0	0
$337$	28.0453	1.52773	0.763863	−	0.645378i	$-0.223300\pi$
$337$	28.0453	1.52773	0.763863	+	0.645378i	$0.223300\pi$
$338$	2.55707	0.139086
$339$	0	0
$340$	−7.76790	−0.421273
$341$	−4.28304	−0.231940
$342$	0	0
$343$	−11.4696	−0.619298
$344$	9.33195	0.503145
$345$	0	0
$346$	−8.71513	−0.468528
$347$	20.4356	1.09704	0.548520	−	0.836137i	$-0.315191\pi$
$347$	20.4356	1.09704	0.548520	+	0.836137i	$0.315191\pi$
$348$	0	0
$349$	17.9641	0.961598	0.480799	−	0.876831i	$-0.340347\pi$
$349$	17.9641	0.961598	0.480799	+	0.876831i	$0.340347\pi$
$350$	1.14889	0.0614105
$351$	0	0
$352$	−4.48468	−0.239034
$353$	−20.5387	−1.09317	−0.546583	−	0.837405i	$-0.684072\pi$
$353$	−20.5387	−1.09317	−0.546583	+	0.837405i	$0.684072\pi$
$354$	0	0
$355$	10.0444	0.533101
$356$	−25.2940	−1.34058
$357$	0	0
$358$	−3.94824	−0.208671
$359$	19.6886	1.03913	0.519564	−	0.854432i	$-0.326095\pi$
$359$	19.6886	1.03913	0.519564	+	0.854432i	$0.326095\pi$
$360$	0	0
$361$	−18.9207	−0.995826
$362$	−4.30776	−0.226411
$363$	0	0
$364$	−4.15717	−0.217895
$365$	−22.3383	−1.16924
$366$	0	0
$367$	−13.8596	−0.723465	−0.361733	−	0.932282i	$-0.617815\pi$
$367$	−13.8596	−0.723465	−0.361733	+	0.932282i	$0.617815\pi$
$368$	−5.13968	−0.267924
$369$	0	0
$370$	−2.27668	−0.118359
$371$	5.07518	0.263490
$372$	0	0
$373$	15.4187	0.798351	0.399175	−	0.916875i	$-0.369296\pi$
$373$	15.4187	0.798351	0.399175	+	0.916875i	$0.369296\pi$
$374$	−1.32232	−0.0683757
$375$	0	0
$376$	6.04501	0.311748
$377$	18.6626	0.961172
$378$	0	0
$379$	0.426414	0.0219034	0.0109517	−	0.999940i	$-0.496514\pi$
$379$	0.426414	0.0219034	0.0109517	+	0.999940i	$0.496514\pi$
$380$	0.699935	0.0359059
$381$	0	0
$382$	3.18709	0.163066
$383$	−35.5235	−1.81517	−0.907584	−	0.419870i	$-0.862076\pi$
$383$	−35.5235	−1.81517	−0.907584	+	0.419870i	$0.862076\pi$
$384$	0	0
$385$	1.18136	0.0602079
$386$	−11.5103	−0.585860
$387$	0	0
$388$	−22.5405	−1.14432
$389$	13.9775	0.708686	0.354343	−	0.935115i	$-0.384704\pi$
$389$	13.9775	0.708686	0.354343	+	0.935115i	$0.384704\pi$
$390$	0	0
$391$	−5.43055	−0.274635
$392$	−10.1050	−0.510377
$393$	0	0
$394$	2.24247	0.112974
$395$	−8.23727	−0.414462
$396$	0	0
$397$	24.6805	1.23868	0.619338	−	0.785124i	$-0.287401\pi$
$397$	24.6805	1.23868	0.619338	+	0.785124i	$0.287401\pi$
$398$	−11.0292	−0.552842
$399$	0	0
$400$	−9.28015	−0.464008
$401$	38.1654	1.90589	0.952944	−	0.303148i	$-0.0980375\pi$
$401$	38.1654	1.90589	0.952944	+	0.303148i	$0.0980375\pi$
$402$	0	0
$403$	−11.2963	−0.562706
$404$	0.0778135	0.00387137
$405$	0	0
$406$	−2.59130	−0.128604
$407$	3.94276	0.195436
$408$	0	0
$409$	−6.08708	−0.300987	−0.150493	−	0.988611i	$-0.548086\pi$
$409$	−6.08708	−0.300987	−0.150493	+	0.988611i	$0.548086\pi$
$410$	−4.26310	−0.210540
$411$	0	0
$412$	26.7796	1.31934
$413$	6.89135	0.339101
$414$	0	0
$415$	−14.6746	−0.720349
$416$	−11.8281	−0.579918
$417$	0	0
$418$	0.119149	0.00582779
$419$	14.7329	0.719752	0.359876	−	0.933000i	$-0.382819\pi$
$419$	14.7329	0.719752	0.359876	+	0.933000i	$0.382819\pi$
$420$	0	0
$421$	−12.0408	−0.586834	−0.293417	−	0.955985i	$-0.594792\pi$
$421$	−12.0408	−0.586834	−0.293417	+	0.955985i	$0.594792\pi$
$422$	−9.02467	−0.439314
$423$	0	0
$424$	9.47864	0.460323
$425$	−9.80536	−0.475630
$426$	0	0
$427$	0.865577	0.0418882
$428$	27.0936	1.30962
$429$	0	0
$430$	−3.33329	−0.160745
$431$	−26.7332	−1.28769	−0.643846	−	0.765155i	$-0.722662\pi$
$431$	−26.7332	−1.28769	−0.643846	+	0.765155i	$0.722662\pi$
$432$	0	0
$433$	−33.0285	−1.58725	−0.793624	−	0.608409i	$-0.791808\pi$
$433$	−33.0285	−1.58725	−0.793624	+	0.608409i	$0.791808\pi$
$434$	1.56849	0.0752898
$435$	0	0
$436$	17.0167	0.814954
$437$	0.489326	0.0234076
$438$	0	0
$439$	32.4991	1.55110	0.775548	−	0.631288i	$-0.217474\pi$
$439$	32.4991	1.55110	0.775548	+	0.631288i	$0.217474\pi$
$440$	2.20637	0.105185
$441$	0	0
$442$	−3.48755	−0.165886
$443$	23.0625	1.09573	0.547865	−	0.836567i	$-0.315441\pi$
$443$	23.0625	1.09573	0.547865	+	0.836567i	$0.315441\pi$
$444$	0	0
$445$	18.9576	0.898678
$446$	−4.47404	−0.211852
$447$	0	0
$448$	−3.47852	−0.164345
$449$	−27.4385	−1.29490	−0.647450	−	0.762108i	$-0.724165\pi$
$449$	−27.4385	−1.29490	−0.647450	+	0.762108i	$0.724165\pi$
$450$	0	0
$451$	7.38287	0.347646
$452$	29.1214	1.36975
$453$	0	0
$454$	−4.25940	−0.199904
$455$	3.11577	0.146070
$456$	0	0
$457$	32.4706	1.51891	0.759456	−	0.650559i	$-0.225465\pi$
$457$	32.4706	1.51891	0.759456	+	0.650559i	$0.225465\pi$
$458$	7.74113	0.361719
$459$	0	0
$460$	4.31835	0.201344
$461$	26.3178	1.22574	0.612871	−	0.790183i	$-0.290015\pi$
$461$	26.3178	1.22574	0.612871	+	0.790183i	$0.290015\pi$
$462$	0	0
$463$	−31.9939	−1.48688	−0.743441	−	0.668801i	$-0.766808\pi$
$463$	−31.9939	−1.48688	−0.743441	+	0.668801i	$0.766808\pi$
$464$	20.9313	0.971712
$465$	0	0
$466$	−8.50897	−0.394170
$467$	18.9096	0.875032	0.437516	−	0.899211i	$-0.355858\pi$
$467$	18.9096	0.875032	0.437516	+	0.899211i	$0.355858\pi$
$468$	0	0
$469$	4.16317	0.192237
$470$	−2.15923	−0.0995976
$471$	0	0
$472$	12.8706	0.592418
$473$	5.77261	0.265425
$474$	0	0
$475$	0.883522	0.0405388
$476$	−4.92642	−0.225802
$477$	0	0
$478$	−3.90951	−0.178817
$479$	37.8765	1.73062	0.865311	−	0.501235i	$-0.167121\pi$
$479$	37.8765	1.73062	0.865311	+	0.501235i	$0.167121\pi$
$480$	0	0
$481$	10.3988	0.474144
$482$	1.89081	0.0861241
$483$	0	0
$484$	−1.82100	−0.0827729
$485$	16.8940	0.767116
$486$	0	0
$487$	9.67080	0.438226	0.219113	−	0.975699i	$-0.429684\pi$
$487$	9.67080	0.438226	0.219113	+	0.975699i	$0.429684\pi$
$488$	1.61659	0.0731797
$489$	0	0
$490$	3.60940	0.163056
$491$	−9.63253	−0.434710	−0.217355	−	0.976093i	$-0.569743\pi$
$491$	−9.63253	−0.434710	−0.217355	+	0.976093i	$0.569743\pi$
$492$	0	0
$493$	22.1159	0.996050
$494$	0.314249	0.0141387
$495$	0	0
$496$	−12.6695	−0.568877
$497$	6.37017	0.285741
$498$	0	0
$499$	28.9699	1.29687	0.648436	−	0.761269i	$-0.275423\pi$
$499$	28.9699	1.29687	0.648436	+	0.761269i	$0.275423\pi$
$500$	20.2239	0.904442
$501$	0	0
$502$	10.7371	0.479220
$503$	−18.9084	−0.843084	−0.421542	−	0.906809i	$-0.638511\pi$
$503$	−18.9084	−0.843084	−0.421542	+	0.906809i	$0.638511\pi$
$504$	0	0
$505$	−0.0583206	−0.00259523
$506$	0.735109	0.0326796
$507$	0	0
$508$	32.2083	1.42901
$509$	8.63556	0.382765	0.191382	−	0.981516i	$-0.438703\pi$
$509$	8.63556	0.382765	0.191382	+	0.981516i	$0.438703\pi$
$510$	0	0
$511$	−14.1670	−0.626711
$512$	−22.8299	−1.00895
$513$	0	0
$514$	4.95816	0.218695
$515$	−20.0711	−0.884439
$516$	0	0
$517$	3.73936	0.164457
$518$	−1.44387	−0.0634402
$519$	0	0
$520$	5.81916	0.255187
$521$	5.59504	0.245123	0.122562	−	0.992461i	$-0.460889\pi$
$521$	5.59504	0.245123	0.122562	+	0.992461i	$0.460889\pi$
$522$	0	0
$523$	38.6887	1.69174	0.845870	−	0.533390i	$-0.179082\pi$
$523$	38.6887	1.69174	0.845870	+	0.533390i	$0.179082\pi$
$524$	−36.4592	−1.59273
$525$	0	0
$526$	5.16761	0.225318
$527$	−13.3865	−0.583125
$528$	0	0
$529$	−19.9810	−0.868741
$530$	−3.38569	−0.147065
$531$	0	0
$532$	0.443900	0.0192455
$533$	19.4718	0.843419
$534$	0	0
$535$	−20.3065	−0.877925
$536$	7.77533	0.335843
$537$	0	0
$538$	−9.32005	−0.401816
$539$	−6.25078	−0.269240
$540$	0	0
$541$	3.22362	0.138594	0.0692971	−	0.997596i	$-0.477924\pi$
$541$	3.22362	0.138594	0.0692971	+	0.997596i	$0.477924\pi$
$542$	−2.23050	−0.0958083
$543$	0	0
$544$	−14.0167	−0.600962
$545$	−12.7539	−0.546318
$546$	0	0
$547$	−42.0810	−1.79925	−0.899627	−	0.436658i	$-0.856162\pi$
$547$	−42.0810	−1.79925	−0.899627	+	0.436658i	$0.856162\pi$
$548$	−29.8964	−1.27711
$549$	0	0
$550$	1.32731	0.0565965
$551$	−1.99278	−0.0848951
$552$	0	0
$553$	−5.22410	−0.222151
$554$	−10.5928	−0.450043
$555$	0	0
$556$	12.4263	0.526994
$557$	39.2686	1.66386	0.831932	−	0.554878i	$-0.187235\pi$
$557$	39.2686	1.66386	0.831932	+	0.554878i	$0.187235\pi$
$558$	0	0
$559$	15.2249	0.643944
$560$	3.49454	0.147671
$561$	0	0
$562$	8.30000	0.350115
$563$	−18.1501	−0.764935	−0.382468	−	0.923969i	$-0.624926\pi$
$563$	−18.1501	−0.764935	−0.382468	+	0.923969i	$0.624926\pi$
$564$	0	0
$565$	−21.8263	−0.918237
$566$	1.54200	0.0648151
$567$	0	0
$568$	11.8972	0.499197
$569$	−10.8482	−0.454782	−0.227391	−	0.973804i	$-0.573020\pi$
$569$	−10.8482	−0.454782	−0.227391	+	0.973804i	$0.573020\pi$
$570$	0	0
$571$	20.0265	0.838081	0.419041	−	0.907967i	$-0.362366\pi$
$571$	20.0265	0.838081	0.419041	+	0.907967i	$0.362366\pi$
$572$	−4.80278	−0.200814
$573$	0	0
$574$	−2.70367	−0.112849
$575$	5.45102	0.227323
$576$	0	0
$577$	−34.8606	−1.45127	−0.725634	−	0.688081i	$-0.758453\pi$
$577$	−34.8606	−1.45127	−0.725634	+	0.688081i	$0.758453\pi$
$578$	3.05949	0.127258
$579$	0	0
$580$	−17.5865	−0.730238
$581$	−9.30668	−0.386106
$582$	0	0
$583$	5.86335	0.242835
$584$	−26.4590	−1.09488
$585$	0	0
$586$	−2.21237	−0.0913922
$587$	14.8050	0.611066	0.305533	−	0.952181i	$-0.401165\pi$
$587$	14.8050	0.611066	0.305533	+	0.952181i	$0.401165\pi$
$588$	0	0
$589$	1.20620	0.0497008
$590$	−4.59727	−0.189267
$591$	0	0
$592$	11.6629	0.479343
$593$	−15.7941	−0.648586	−0.324293	−	0.945957i	$-0.605126\pi$
$593$	−15.7941	−0.648586	−0.324293	+	0.945957i	$0.605126\pi$
$594$	0	0
$595$	3.69231	0.151370
$596$	27.4757	1.12545
$597$	0	0
$598$	1.93880	0.0792836
$599$	−21.0304	−0.859279	−0.429640	−	0.903000i	$-0.641359\pi$
$599$	−21.0304	−0.859279	−0.429640	+	0.903000i	$0.641359\pi$
$600$	0	0
$601$	−26.1583	−1.06702	−0.533510	−	0.845794i	$-0.679127\pi$
$601$	−26.1583	−1.06702	−0.533510	+	0.845794i	$0.679127\pi$
$602$	−2.11398	−0.0861593
$603$	0	0
$604$	15.6547	0.636980
$605$	1.36483	0.0554882
$606$	0	0
$607$	37.7328	1.53153	0.765764	−	0.643122i	$-0.222361\pi$
$607$	37.7328	1.53153	0.765764	+	0.643122i	$0.222361\pi$
$608$	1.26299	0.0512210
$609$	0	0
$610$	−0.577432	−0.0233795
$611$	9.86232	0.398987
$612$	0	0
$613$	34.1803	1.38053	0.690264	−	0.723558i	$-0.257494\pi$
$613$	34.1803	1.38053	0.690264	+	0.723558i	$0.257494\pi$
$614$	10.2516	0.413722
$615$	0	0
$616$	1.39928	0.0563788
$617$	−19.5940	−0.788826	−0.394413	−	0.918933i	$-0.629052\pi$
$617$	−19.5940	−0.788826	−0.394413	+	0.918933i	$0.629052\pi$
$618$	0	0
$619$	−27.3391	−1.09885	−0.549425	−	0.835543i	$-0.685153\pi$
$619$	−27.3391	−1.09885	−0.549425	+	0.835543i	$0.685153\pi$
$620$	10.6449	0.427509
$621$	0	0
$622$	0.810222	0.0324869
$623$	12.0230	0.481690
$624$	0	0
$625$	0.528524	0.0211410
$626$	−7.98543	−0.319162
$627$	0	0
$628$	−18.1202	−0.723074
$629$	12.3230	0.491349
$630$	0	0
$631$	29.4107	1.17082	0.585411	−	0.810737i	$-0.300933\pi$
$631$	29.4107	1.17082	0.585411	+	0.810737i	$0.300933\pi$
$632$	−9.75676	−0.388103
$633$	0	0
$634$	−12.4594	−0.494826
$635$	−24.1399	−0.957963
$636$	0	0
$637$	−16.4860	−0.653200
$638$	−2.99373	−0.118523
$639$	0	0
$640$	14.5622	0.575621
$641$	36.2587	1.43213	0.716066	−	0.698032i	$-0.245941\pi$
$641$	36.2587	1.43213	0.716066	+	0.698032i	$0.245941\pi$
$642$	0	0
$643$	5.46621	0.215566	0.107783	−	0.994174i	$-0.465625\pi$
$643$	5.46621	0.215566	0.107783	+	0.994174i	$0.465625\pi$
$644$	2.73871	0.107920
$645$	0	0
$646$	0.372397	0.0146518
$647$	−8.88569	−0.349333	−0.174666	−	0.984628i	$-0.555885\pi$
$647$	−8.88569	−0.349333	−0.174666	+	0.984628i	$0.555885\pi$
$648$	0	0
$649$	7.96157	0.312519
$650$	3.50069	0.137308
$651$	0	0
$652$	−40.0350	−1.56789
$653$	−42.2586	−1.65371	−0.826853	−	0.562418i	$-0.809871\pi$
$653$	−42.2586	−1.65371	−0.826853	+	0.562418i	$0.809871\pi$
$654$	0	0
$655$	27.3259	1.06771
$656$	21.8390	0.852668
$657$	0	0
$658$	−1.36938	−0.0533842
$659$	32.6335	1.27122	0.635610	−	0.772010i	$-0.280749\pi$
$659$	32.6335	1.27122	0.635610	+	0.772010i	$0.280749\pi$
$660$	0	0
$661$	10.9510	0.425943	0.212971	−	0.977058i	$-0.431686\pi$
$661$	10.9510	0.425943	0.212971	+	0.977058i	$0.431686\pi$
$662$	5.40615	0.210116
$663$	0	0
$664$	−17.3816	−0.674537
$665$	−0.332700	−0.0129015
$666$	0	0
$667$	−12.2947	−0.476054
$668$	20.9331	0.809927
$669$	0	0
$670$	−2.77728	−0.107296
$671$	1.00000	0.0386046
$672$	0	0
$673$	11.8740	0.457709	0.228854	−	0.973461i	$-0.426502\pi$
$673$	11.8740	0.457709	0.228854	+	0.973461i	$0.426502\pi$
$674$	−11.8654	−0.457039
$675$	0	0
$676$	11.0060	0.423308
$677$	−4.07190	−0.156496	−0.0782478	−	0.996934i	$-0.524933\pi$
$677$	−4.07190	−0.156496	−0.0782478	+	0.996934i	$0.524933\pi$
$678$	0	0
$679$	10.7142	0.411173
$680$	6.89594	0.264447
$681$	0	0
$682$	1.81207	0.0693878
$683$	35.2353	1.34824	0.674120	−	0.738622i	$-0.264523\pi$
$683$	35.2353	1.34824	0.674120	+	0.738622i	$0.264523\pi$
$684$	0	0
$685$	22.4071	0.856132
$686$	4.85255	0.185271
$687$	0	0
$688$	17.0757	0.651005
$689$	15.4642	0.589140
$690$	0	0
$691$	−29.7507	−1.13177	−0.565885	−	0.824484i	$-0.691465\pi$
$691$	−29.7507	−1.13177	−0.565885	+	0.824484i	$0.691465\pi$
$692$	−37.5112	−1.42596
$693$	0	0
$694$	−8.64590	−0.328194
$695$	−9.31344	−0.353279
$696$	0	0
$697$	23.0749	0.874025
$698$	−7.60027	−0.287675
$699$	0	0
$700$	4.94498	0.186903
$701$	−36.1389	−1.36495	−0.682475	−	0.730909i	$-0.739096\pi$
$701$	−36.1389	−1.36495	−0.682475	+	0.730909i	$0.739096\pi$
$702$	0	0
$703$	−1.11037	−0.0418786
$704$	−4.01874	−0.151462
$705$	0	0
$706$	8.68954	0.327035
$707$	−0.0369871	−0.00139104
$708$	0	0
$709$	−5.65735	−0.212466	−0.106233	−	0.994341i	$-0.533879\pi$
$709$	−5.65735	−0.212466	−0.106233	+	0.994341i	$0.533879\pi$
$710$	−4.24958	−0.159484
$711$	0	0
$712$	22.4547	0.841524
$713$	7.44186	0.278700
$714$	0	0
$715$	3.59965	0.134619
$716$	−16.9938	−0.635089
$717$	0	0
$718$	−8.32988	−0.310868
$719$	19.1968	0.715920	0.357960	−	0.933737i	$-0.383472\pi$
$719$	19.1968	0.715920	0.357960	+	0.933737i	$0.383472\pi$
$720$	0	0
$721$	−12.7291	−0.474058
$722$	8.00497	0.297914
$723$	0	0
$724$	−18.5412	−0.689080
$725$	−22.1992	−0.824459
$726$	0	0
$727$	27.1989	1.00875	0.504375	−	0.863485i	$-0.331723\pi$
$727$	27.1989	1.00875	0.504375	+	0.863485i	$0.331723\pi$
$728$	3.69052	0.136780
$729$	0	0
$730$	9.45090	0.349794
$731$	18.0421	0.667311
$732$	0	0
$733$	12.3262	0.455277	0.227638	−	0.973746i	$-0.426900\pi$
$733$	12.3262	0.455277	0.227638	+	0.973746i	$0.426900\pi$
$734$	5.86372	0.216434
$735$	0	0
$736$	7.79221	0.287225
$737$	4.80971	0.177168
$738$	0	0
$739$	−45.4141	−1.67059	−0.835293	−	0.549806i	$-0.814702\pi$
$739$	−45.4141	−1.67059	−0.835293	+	0.549806i	$0.814702\pi$
$740$	−9.79917	−0.360225
$741$	0	0
$742$	−2.14721	−0.0788265
$743$	−38.0717	−1.39672	−0.698358	−	0.715749i	$-0.746086\pi$
$743$	−38.0717	−1.39672	−0.698358	+	0.715749i	$0.746086\pi$
$744$	0	0
$745$	−20.5929	−0.754464
$746$	−6.52336	−0.238837
$747$	0	0
$748$	−5.69149	−0.208101
$749$	−12.8784	−0.470567
$750$	0	0
$751$	12.2499	0.447006	0.223503	−	0.974703i	$-0.428251\pi$
$751$	12.2499	0.447006	0.223503	+	0.974703i	$0.428251\pi$
$752$	11.0612	0.403362
$753$	0	0
$754$	−7.89577	−0.287547
$755$	−11.7331	−0.427010
$756$	0	0
$757$	1.84173	0.0669390	0.0334695	−	0.999440i	$-0.489344\pi$
$757$	1.84173	0.0669390	0.0334695	+	0.999440i	$0.489344\pi$
$758$	−0.180408	−0.00655270
$759$	0	0
$760$	−0.621365	−0.0225393
$761$	−36.8876	−1.33717	−0.668587	−	0.743634i	$-0.733100\pi$
$761$	−36.8876	−1.33717	−0.668587	+	0.743634i	$0.733100\pi$
$762$	0	0
$763$	−8.08856	−0.292825
$764$	13.7177	0.496291
$765$	0	0
$766$	15.0293	0.543031
$767$	20.9982	0.758199
$768$	0	0
$769$	−18.9650	−0.683894	−0.341947	−	0.939719i	$-0.611086\pi$
$769$	−18.9650	−0.683894	−0.341947	+	0.939719i	$0.611086\pi$
$770$	−0.499812	−0.0180120
$771$	0	0
$772$	−49.5422	−1.78306
$773$	−34.9730	−1.25789	−0.628946	−	0.777449i	$-0.716513\pi$
$773$	−34.9730	−1.25789	−0.628946	+	0.777449i	$0.716513\pi$
$774$	0	0
$775$	13.4369	0.482669
$776$	20.0103	0.718329
$777$	0	0
$778$	−5.91360	−0.212013
$779$	−2.07919	−0.0744947
$780$	0	0
$781$	7.35945	0.263342
$782$	2.29756	0.0821606
$783$	0	0
$784$	−18.4902	−0.660363
$785$	13.5809	0.484724
$786$	0	0
$787$	−3.25717	−0.116106	−0.0580529	−	0.998314i	$-0.518489\pi$
$787$	−3.25717	−0.116106	−0.0580529	+	0.998314i	$0.518489\pi$
$788$	9.65192	0.343835
$789$	0	0
$790$	3.48503	0.123992
$791$	−13.8423	−0.492174
$792$	0	0
$793$	2.63744	0.0936581
$794$	−10.4418	−0.370566
$795$	0	0
$796$	−47.4712	−1.68257
$797$	23.1563	0.820238	0.410119	−	0.912032i	$-0.365487\pi$
$797$	23.1563	0.820238	0.410119	+	0.912032i	$0.365487\pi$
$798$	0	0
$799$	11.6872	0.413465
$800$	14.0695	0.497433
$801$	0	0
$802$	−16.1470	−0.570171
$803$	−16.3671	−0.577583
$804$	0	0
$805$	−2.05264	−0.0723460
$806$	4.77922	0.168341
$807$	0	0
$808$	−0.0690788	−0.00243018
$809$	−1.91012	−0.0671564	−0.0335782	−	0.999436i	$-0.510690\pi$
$809$	−1.91012	−0.0671564	−0.0335782	+	0.999436i	$0.510690\pi$
$810$	0	0
$811$	−7.87730	−0.276609	−0.138305	−	0.990390i	$-0.544165\pi$
$811$	−7.87730	−0.276609	−0.138305	+	0.990390i	$0.544165\pi$
$812$	−11.1534	−0.391406
$813$	0	0
$814$	−1.66811	−0.0584671
$815$	30.0060	1.05106
$816$	0	0
$817$	−1.62570	−0.0568761
$818$	2.57532	0.0900440
$819$	0	0
$820$	−18.3491	−0.640777
$821$	10.6394	0.371318	0.185659	−	0.982614i	$-0.440558\pi$
$821$	10.6394	0.371318	0.185659	+	0.982614i	$0.440558\pi$
$822$	0	0
$823$	−2.50922	−0.0874658	−0.0437329	−	0.999043i	$-0.513925\pi$
$823$	−2.50922	−0.0874658	−0.0437329	+	0.999043i	$0.513925\pi$
$824$	−23.7736	−0.828191
$825$	0	0
$826$	−2.91560	−0.101447
$827$	5.89475	0.204980	0.102490	−	0.994734i	$-0.467319\pi$
$827$	5.89475	0.204980	0.102490	+	0.994734i	$0.467319\pi$
$828$	0	0
$829$	−15.3165	−0.531964	−0.265982	−	0.963978i	$-0.585696\pi$
$829$	−15.3165	−0.531964	−0.265982	+	0.963978i	$0.585696\pi$
$830$	6.20855	0.215502
$831$	0	0
$832$	−10.5992	−0.367460
$833$	−19.5366	−0.676903
$834$	0	0
$835$	−15.6892	−0.542948
$836$	0.512837	0.0177368
$837$	0	0
$838$	−6.23322	−0.215323
$839$	35.8612	1.23807	0.619033	−	0.785365i	$-0.287525\pi$
$839$	35.8612	1.23807	0.619033	+	0.785365i	$0.287525\pi$
$840$	0	0
$841$	21.0702	0.726560
$842$	5.09424	0.175559
$843$	0	0
$844$	−38.8435	−1.33705
$845$	−8.24892	−0.283772
$846$	0	0
$847$	0.865577	0.0297416
$848$	17.3441	0.595600
$849$	0	0
$850$	4.14845	0.142291
$851$	−6.85062	−0.234836
$852$	0	0
$853$	1.26182	0.0432041	0.0216020	−	0.999767i	$-0.493123\pi$
$853$	1.26182	0.0432041	0.0216020	+	0.999767i	$0.493123\pi$
$854$	−0.366209	−0.0125314
$855$	0	0
$856$	−24.0523	−0.822091
$857$	−30.6772	−1.04791	−0.523956	−	0.851745i	$-0.675545\pi$
$857$	−30.6772	−1.04791	−0.523956	+	0.851745i	$0.675545\pi$
$858$	0	0
$859$	5.82456	0.198731	0.0993657	−	0.995051i	$-0.468319\pi$
$859$	5.82456	0.198731	0.0993657	+	0.995051i	$0.468319\pi$
$860$	−14.3470	−0.489228
$861$	0	0
$862$	11.3103	0.385230
$863$	−47.2448	−1.60823	−0.804116	−	0.594472i	$-0.797361\pi$
$863$	−47.2448	−1.60823	−0.804116	+	0.594472i	$0.797361\pi$
$864$	0	0
$865$	28.1144	0.955918
$866$	13.9737	0.474846
$867$	0	0
$868$	6.75101	0.229144
$869$	−6.03539	−0.204737
$870$	0	0
$871$	12.6853	0.429825
$872$	−15.1066	−0.511573
$873$	0	0
$874$	−0.207024	−0.00700269
$875$	−9.61304	−0.324980
$876$	0	0
$877$	−26.3489	−0.889740	−0.444870	−	0.895595i	$-0.646750\pi$
$877$	−26.3489	−0.889740	−0.444870	+	0.895595i	$0.646750\pi$
$878$	−13.7497	−0.464031
$879$	0	0
$880$	4.03724	0.136095
$881$	3.59627	0.121162	0.0605808	−	0.998163i	$-0.480705\pi$
$881$	3.59627	0.121162	0.0605808	+	0.998163i	$0.480705\pi$
$882$	0	0
$883$	−9.91315	−0.333604	−0.166802	−	0.985990i	$-0.553344\pi$
$883$	−9.91315	−0.333604	−0.166802	+	0.985990i	$0.553344\pi$
$884$	−15.0109	−0.504872
$885$	0	0
$886$	−9.75728	−0.327802
$887$	15.1609	0.509052	0.254526	−	0.967066i	$-0.418081\pi$
$887$	15.1609	0.509052	0.254526	+	0.967066i	$0.418081\pi$
$888$	0	0
$889$	−15.3096	−0.513467
$890$	−8.02061	−0.268851
$891$	0	0
$892$	−19.2569	−0.644770
$893$	−1.05309	−0.0352403
$894$	0	0
$895$	12.7367	0.425742
$896$	9.23536	0.308532
$897$	0	0
$898$	11.6087	0.387386
$899$	−30.3069	−1.01079
$900$	0	0
$901$	18.3257	0.610518
$902$	−3.12355	−0.104003
$903$	0	0
$904$	−25.8524	−0.859839
$905$	13.8965	0.461936
$906$	0	0
$907$	−37.2166	−1.23576	−0.617879	−	0.786273i	$-0.712008\pi$
$907$	−37.2166	−1.23576	−0.617879	+	0.786273i	$0.712008\pi$
$908$	−18.3331	−0.608406
$909$	0	0
$910$	−1.31822	−0.0436986
$911$	9.44668	0.312982	0.156491	−	0.987679i	$-0.449982\pi$
$911$	9.44668	0.312982	0.156491	+	0.987679i	$0.449982\pi$
$912$	0	0
$913$	−10.7520	−0.355839
$914$	−13.7377	−0.454402
$915$	0	0
$916$	33.3190	1.10089
$917$	17.3301	0.572291
$918$	0	0
$919$	−32.5338	−1.07319	−0.536596	−	0.843839i	$-0.680290\pi$
$919$	−32.5338	−1.07319	−0.536596	+	0.843839i	$0.680290\pi$
$920$	−3.83360	−0.126390
$921$	0	0
$922$	−11.1345	−0.366697
$923$	19.4101	0.638891
$924$	0	0
$925$	−12.3694	−0.406704
$926$	13.5360	0.444820
$927$	0	0
$928$	−31.7337	−1.04171
$929$	28.2779	0.927767	0.463883	−	0.885896i	$-0.346456\pi$
$929$	28.2779	0.927767	0.463883	+	0.885896i	$0.346456\pi$
$930$	0	0
$931$	1.76037	0.0576936
$932$	−36.6239	−1.19966
$933$	0	0
$934$	−8.00029	−0.261777
$935$	4.26573	0.139504
$936$	0	0
$937$	10.7114	0.349927	0.174964	−	0.984575i	$-0.444019\pi$
$937$	10.7114	0.349927	0.174964	+	0.984575i	$0.444019\pi$
$938$	−1.76136	−0.0575103
$939$	0	0
$940$	−9.29364	−0.303125
$941$	−1.24454	−0.0405708	−0.0202854	−	0.999794i	$-0.506457\pi$
$941$	−1.24454	−0.0405708	−0.0202854	+	0.999794i	$0.506457\pi$
$942$	0	0
$943$	−12.8279	−0.417733
$944$	23.5508	0.766513
$945$	0	0
$946$	−2.44228	−0.0794053
$947$	31.5849	1.02637	0.513186	−	0.858277i	$-0.328465\pi$
$947$	31.5849	1.02637	0.513186	+	0.858277i	$0.328465\pi$
$948$	0	0
$949$	−43.1673	−1.40127
$950$	−0.373801	−0.0121277
$951$	0	0
$952$	4.37342	0.141743
$953$	−8.16979	−0.264645	−0.132323	−	0.991207i	$-0.542244\pi$
$953$	−8.16979	−0.264645	−0.132323	+	0.991207i	$0.542244\pi$
$954$	0	0
$955$	−10.2813	−0.332697
$956$	−16.8271	−0.544228
$957$	0	0
$958$	−16.0248	−0.517738
$959$	14.2106	0.458885
$960$	0	0
$961$	−12.6556	−0.408244
$962$	−4.39952	−0.141846
$963$	0	0
$964$	8.13834	0.262118
$965$	37.1315	1.19531
$966$	0	0
$967$	−8.87766	−0.285486	−0.142743	−	0.989760i	$-0.545592\pi$
$967$	−8.87766	−0.285486	−0.142743	+	0.989760i	$0.545592\pi$
$968$	1.61659	0.0519592
$969$	0	0
$970$	−7.14751	−0.229493
$971$	3.89090	0.124865	0.0624324	−	0.998049i	$-0.480114\pi$
$971$	3.89090	0.124865	0.0624324	+	0.998049i	$0.480114\pi$
$972$	0	0
$973$	−5.90660	−0.189357
$974$	−4.09153	−0.131101
$975$	0	0
$976$	2.95806	0.0946851
$977$	−52.3391	−1.67447	−0.837237	−	0.546840i	$-0.815831\pi$
$977$	−52.3391	−1.67447	−0.837237	+	0.546840i	$0.815831\pi$
$978$	0	0
$979$	13.8901	0.443930
$980$	15.5354	0.496260
$981$	0	0
$982$	4.07533	0.130049
$983$	−3.51579	−0.112136	−0.0560682	−	0.998427i	$-0.517856\pi$
$983$	−3.51579	−0.112136	−0.0560682	+	0.998427i	$0.517856\pi$
$984$	0	0
$985$	−7.23404	−0.230496
$986$	−9.35681	−0.297981
$987$	0	0
$988$	1.35258	0.0430312
$989$	−10.0300	−0.318936
$990$	0	0
$991$	−59.0461	−1.87566	−0.937831	−	0.347092i	$-0.887169\pi$
$991$	−59.0461	−1.87566	−0.937831	+	0.347092i	$0.887169\pi$
$992$	19.2081	0.609857
$993$	0	0
$994$	−2.69509	−0.0854832
$995$	35.5793	1.12794
$996$	0	0
$997$	2.67568	0.0847396	0.0423698	−	0.999102i	$-0.486509\pi$
$997$	2.67568	0.0847396	0.0423698	+	0.999102i	$0.486509\pi$
$998$	−12.2566	−0.387976
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.d.1.4		11
3.2	odd	2		2013.2.a.a.1.8	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.a.1.8	✓	11	3.2	odd	2
6039.2.a.d.1.4		11	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)

\(f(q)\)	\(=\)	\(q-0.423080 q^{2} -1.82100 q^{4} +1.36483 q^{5} +0.865577 q^{7} +1.61659 q^{8} +O(q^{10})\)	\(q-0.423080 q^{2} -1.82100 q^{4} +1.36483 q^{5} +0.865577 q^{7} +1.61659 q^{8} -0.577432 q^{10} +1.00000 q^{11} +2.63744 q^{13} -0.366209 q^{14} +2.95806 q^{16} +3.12547 q^{17} -0.281623 q^{19} -2.48536 q^{20} -0.423080 q^{22} -1.73752 q^{23} -3.13724 q^{25} -1.11585 q^{26} -1.57622 q^{28} +7.07603 q^{29} -4.28304 q^{31} -4.48468 q^{32} -1.32232 q^{34} +1.18136 q^{35} +3.94276 q^{37} +0.119149 q^{38} +2.20637 q^{40} +7.38287 q^{41} +5.77261 q^{43} -1.82100 q^{44} +0.735109 q^{46} +3.73936 q^{47} -6.25078 q^{49} +1.32731 q^{50} -4.80278 q^{52} +5.86335 q^{53} +1.36483 q^{55} +1.39928 q^{56} -2.99373 q^{58} +7.96157 q^{59} +1.00000 q^{61} +1.81207 q^{62} -4.01874 q^{64} +3.59965 q^{65} +4.80971 q^{67} -5.69149 q^{68} -0.499812 q^{70} +7.35945 q^{71} -16.3671 q^{73} -1.66811 q^{74} +0.512837 q^{76} +0.865577 q^{77} -6.03539 q^{79} +4.03724 q^{80} -3.12355 q^{82} -10.7520 q^{83} +4.26573 q^{85} -2.44228 q^{86} +1.61659 q^{88} +13.8901 q^{89} +2.28290 q^{91} +3.16402 q^{92} -1.58205 q^{94} -0.384368 q^{95} +12.3781 q^{97} +2.64458 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) \) 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8	\( 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8} + 6 q^{10} + 11 q^{11} - 3 q^{13} + 9 q^{14} + 4 q^{16} + 7 q^{17} - 8 q^{19} + 25 q^{20} + 4 q^{22} + 15 q^{23} + 4 q^{25} + 2 q^{26} + 13 q^{28} + 8 q^{29} - 17 q^{31} + 27 q^{32} - 18 q^{34} + 2 q^{35} - 10 q^{37} + 30 q^{38} + 10 q^{40} + 25 q^{41} - 7 q^{43} + 6 q^{44} + 32 q^{46} + 30 q^{47} - 2 q^{49} - 11 q^{50} - 7 q^{52} + 18 q^{53} + 13 q^{55} + 20 q^{56} - 13 q^{58} + 43 q^{59} + 11 q^{61} - 7 q^{62} + 25 q^{64} + 27 q^{65} - 30 q^{67} - 10 q^{68} - 4 q^{70} + 7 q^{71} + 6 q^{73} + 44 q^{74} - 19 q^{76} - 5 q^{77} + 17 q^{79} + 22 q^{80} + 8 q^{82} + 34 q^{83} + 10 q^{85} - 2 q^{86} + 9 q^{88} + 41 q^{89} - 39 q^{91} + 32 q^{92} + 55 q^{94} + 9 q^{95} - 41 q^{97} + 29 q^{98}+O(q^{100}) \) 11 * q + 4 * q^2 + 6 * q^4 + 13 * q^5 - 5 * q^7 + 9 * q^8 + 6 * q^10 + 11 * q^11 - 3 * q^13 + 9 * q^14 + 4 * q^16 + 7 * q^17 - 8 * q^19 + 25 * q^20 + 4 * q^22 + 15 * q^23 + 4 * q^25 + 2 * q^26 + 13 * q^28 + 8 * q^29 - 17 * q^31 + 27 * q^32 - 18 * q^34 + 2 * q^35 - 10 * q^37 + 30 * q^38 + 10 * q^40 + 25 * q^41 - 7 * q^43 + 6 * q^44 + 32 * q^46 + 30 * q^47 - 2 * q^49 - 11 * q^50 - 7 * q^52 + 18 * q^53 + 13 * q^55 + 20 * q^56 - 13 * q^58 + 43 * q^59 + 11 * q^61 - 7 * q^62 + 25 * q^64 + 27 * q^65 - 30 * q^67 - 10 * q^68 - 4 * q^70 + 7 * q^71 + 6 * q^73 + 44 * q^74 - 19 * q^76 - 5 * q^77 + 17 * q^79 + 22 * q^80 + 8 * q^82 + 34 * q^83 + 10 * q^85 - 2 * q^86 + 9 * q^88 + 41 * q^89 - 39 * q^91 + 32 * q^92 + 55 * q^94 + 9 * q^95 - 41 * q^97 + 29 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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