LMFDB - Embedded newform 6039.2.a.h.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:	$1$
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} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} + \cdots - 35 $ x^13 - 4x^12 - 11x^11 + 57x^10 + 28x^9 - 290x^8 + 51x^7 + 644x^6 - 259x^5 - 640x^4 + 274x^3 + 256x^2 - 74x - 35
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.31287$ 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)

$f(q)$	$=$	$q-1.31287 q^{2} -0.276371 q^{4} +3.61438 q^{5} -0.837447 q^{7} +2.98858 q^{8} +O(q^{10})$	\(q-1.31287 q^{2} -0.276371 q^{4} +3.61438 q^{5} -0.837447 q^{7} +2.98858 q^{8} -4.74521 q^{10} -1.00000 q^{11} +1.11084 q^{13} +1.09946 q^{14} -3.37088 q^{16} -4.37994 q^{17} +3.54367 q^{19} -0.998910 q^{20} +1.31287 q^{22} -4.06422 q^{23} +8.06373 q^{25} -1.45840 q^{26} +0.231446 q^{28} +1.67338 q^{29} -2.81104 q^{31} -1.55164 q^{32} +5.75030 q^{34} -3.02685 q^{35} +1.86448 q^{37} -4.65237 q^{38} +10.8019 q^{40} -8.79562 q^{41} -0.247282 q^{43} +0.276371 q^{44} +5.33580 q^{46} -7.35580 q^{47} -6.29868 q^{49} -10.5866 q^{50} -0.307006 q^{52} -2.90217 q^{53} -3.61438 q^{55} -2.50278 q^{56} -2.19694 q^{58} +0.446773 q^{59} +1.00000 q^{61} +3.69054 q^{62} +8.77885 q^{64} +4.01501 q^{65} +13.5843 q^{67} +1.21049 q^{68} +3.97386 q^{70} -2.92885 q^{71} -3.22502 q^{73} -2.44782 q^{74} -0.979367 q^{76} +0.837447 q^{77} -1.75929 q^{79} -12.1836 q^{80} +11.5475 q^{82} -0.319558 q^{83} -15.8308 q^{85} +0.324650 q^{86} -2.98858 q^{88} -15.7237 q^{89} -0.930274 q^{91} +1.12323 q^{92} +9.65721 q^{94} +12.8081 q^{95} -7.75619 q^{97} +8.26935 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + 2 q^{10} - 13 q^{11} + 9 q^{13} - 7 q^{14} + 2 q^{16} - 19 q^{17} + 14 q^{19} - 19 q^{20} + 4 q^{22} - 5 q^{23} + 2 q^{25} + 4 q^{26} + 7 q^{28} - 10 q^{29} - q^{31} - 7 q^{32} - 2 q^{34} - 16 q^{35} - 8 q^{37} + 10 q^{38} + 14 q^{40} - 21 q^{41} + 11 q^{43} - 12 q^{44} - 8 q^{46} - 22 q^{47} - 19 q^{50} - q^{52} - 16 q^{53} + 7 q^{55} - 13 q^{58} - 19 q^{59} + 13 q^{61} - 3 q^{62} - 13 q^{64} - 13 q^{65} + 12 q^{67} - 36 q^{68} - 20 q^{70} - 5 q^{71} + 18 q^{73} - 6 q^{74} - 5 q^{76} - 7 q^{77} - q^{79} - 6 q^{80} - 22 q^{82} - 48 q^{83} - 2 q^{85} - 26 q^{86} + 9 q^{88} - 15 q^{89} - 11 q^{91} + 24 q^{92} - 23 q^{94} - 17 q^{95} - 17 q^{97} + 15 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + 2 * q^10 - 13 * q^11 + 9 * q^13 - 7 * q^14 + 2 * q^16 - 19 * q^17 + 14 * q^19 - 19 * q^20 + 4 * q^22 - 5 * q^23 + 2 * q^25 + 4 * q^26 + 7 * q^28 - 10 * q^29 - q^31 - 7 * q^32 - 2 * q^34 - 16 * q^35 - 8 * q^37 + 10 * q^38 + 14 * q^40 - 21 * q^41 + 11 * q^43 - 12 * q^44 - 8 * q^46 - 22 * q^47 - 19 * q^50 - q^52 - 16 * q^53 + 7 * q^55 - 13 * q^58 - 19 * q^59 + 13 * q^61 - 3 * q^62 - 13 * q^64 - 13 * q^65 + 12 * q^67 - 36 * q^68 - 20 * q^70 - 5 * q^71 + 18 * q^73 - 6 * q^74 - 5 * q^76 - 7 * q^77 - q^79 - 6 * q^80 - 22 * q^82 - 48 * q^83 - 2 * q^85 - 26 * q^86 + 9 * q^88 - 15 * q^89 - 11 * q^91 + 24 * q^92 - 23 * q^94 - 17 * q^95 - 17 * q^97 + 15 * 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 $

$2$	−1.31287	−0.928340	−0.464170	−	0.885746i	$-0.653647\pi$
$2$	−1.31287	−0.928340	−0.464170	+	0.885746i	$0.653647\pi$
$3$	0	0
$3$	0	0
$4$	−0.276371	−0.138186
$5$	3.61438	1.61640	0.808200	−	0.588909i	$-0.200442\pi$
$5$	3.61438	1.61640	0.808200	+	0.588909i	$0.200442\pi$
$6$	0	0
$7$	−0.837447	−0.316525	−0.158263	−	0.987397i	$-0.550589\pi$
$7$	−0.837447	−0.316525	−0.158263	+	0.987397i	$0.550589\pi$
$8$	2.98858	1.05662
$9$	0	0
$10$	−4.74521	−1.50057
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.11084	0.308093	0.154046	−	0.988064i	$-0.450769\pi$
$13$	1.11084	0.308093	0.154046	+	0.988064i	$0.450769\pi$
$14$	1.09946	0.293843
$15$	0	0
$16$	−3.37088	−0.842719
$17$	−4.37994	−1.06229	−0.531146	−	0.847280i	$-0.678238\pi$
$17$	−4.37994	−1.06229	−0.531146	+	0.847280i	$0.678238\pi$
$18$	0	0
$19$	3.54367	0.812973	0.406486	−	0.913657i	$-0.366754\pi$
$19$	3.54367	0.812973	0.406486	+	0.913657i	$0.366754\pi$
$20$	−0.998910	−0.223363
$21$	0	0
$22$	1.31287	0.279905
$23$	−4.06422	−0.847449	−0.423724	−	0.905791i	$-0.639278\pi$
$23$	−4.06422	−0.847449	−0.423724	+	0.905791i	$0.639278\pi$
$24$	0	0
$25$	8.06373	1.61275
$26$	−1.45840	−0.286015
$27$	0	0
$28$	0.231446	0.0437392
$29$	1.67338	0.310740	0.155370	−	0.987856i	$-0.450343\pi$
$29$	1.67338	0.310740	0.155370	+	0.987856i	$0.450343\pi$
$30$	0	0
$31$	−2.81104	−0.504879	−0.252439	−	0.967613i	$-0.581233\pi$
$31$	−2.81104	−0.504879	−0.252439	+	0.967613i	$0.581233\pi$
$32$	−1.55164	−0.274293
$33$	0	0
$34$	5.75030	0.986168
$35$	−3.02685	−0.511631
$36$	0	0
$37$	1.86448	0.306519	0.153259	−	0.988186i	$-0.451023\pi$
$37$	1.86448	0.306519	0.153259	+	0.988186i	$0.451023\pi$
$38$	−4.65237	−0.754715
$39$	0	0
$40$	10.8019	1.70792
$41$	−8.79562	−1.37365	−0.686823	−	0.726825i	$-0.740995\pi$
$41$	−8.79562	−1.37365	−0.686823	+	0.726825i	$0.740995\pi$
$42$	0	0
$43$	−0.247282	−0.0377102	−0.0188551	−	0.999822i	$-0.506002\pi$
$43$	−0.247282	−0.0377102	−0.0188551	+	0.999822i	$0.506002\pi$
$44$	0.276371	0.0416645
$45$	0	0
$46$	5.33580	0.786720
$47$	−7.35580	−1.07295	−0.536477	−	0.843915i	$-0.680245\pi$
$47$	−7.35580	−1.07295	−0.536477	+	0.843915i	$0.680245\pi$
$48$	0	0
$49$	−6.29868	−0.899812
$50$	−10.5866	−1.49718
$51$	0	0
$52$	−0.307006	−0.0425740
$53$	−2.90217	−0.398644	−0.199322	−	0.979934i	$-0.563874\pi$
$53$	−2.90217	−0.398644	−0.199322	+	0.979934i	$0.563874\pi$
$54$	0	0
$55$	−3.61438	−0.487363
$56$	−2.50278	−0.334448
$57$	0	0
$58$	−2.19694	−0.288472
$59$	0.446773	0.0581649	0.0290825	−	0.999577i	$-0.490741\pi$
$59$	0.446773	0.0581649	0.0290825	+	0.999577i	$0.490741\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	3.69054	0.468699
$63$	0	0
$64$	8.77885	1.09736
$65$	4.01501	0.498001
$66$	0	0
$67$	13.5843	1.65959	0.829795	−	0.558069i	$-0.188457\pi$
$67$	13.5843	1.65959	0.829795	+	0.558069i	$0.188457\pi$
$68$	1.21049	0.146794
$69$	0	0
$70$	3.97386	0.474967
$71$	−2.92885	−0.347591	−0.173795	−	0.984782i	$-0.555603\pi$
$71$	−2.92885	−0.347591	−0.173795	+	0.984782i	$0.555603\pi$
$72$	0	0
$73$	−3.22502	−0.377460	−0.188730	−	0.982029i	$-0.560437\pi$
$73$	−3.22502	−0.377460	−0.188730	+	0.982029i	$0.560437\pi$
$74$	−2.44782	−0.284554
$75$	0	0
$76$	−0.979367	−0.112341
$77$	0.837447	0.0954359
$78$	0	0
$79$	−1.75929	−0.197936	−0.0989679	−	0.995091i	$-0.531554\pi$
$79$	−1.75929	−0.197936	−0.0989679	+	0.995091i	$0.531554\pi$
$80$	−12.1836	−1.36217
$81$	0	0
$82$	11.5475	1.27521
$83$	−0.319558	−0.0350761	−0.0175380	−	0.999846i	$-0.505583\pi$
$83$	−0.319558	−0.0350761	−0.0175380	+	0.999846i	$0.505583\pi$
$84$	0	0
$85$	−15.8308	−1.71709
$86$	0.324650	0.0350079
$87$	0	0
$88$	−2.98858	−0.318584
$89$	−15.7237	−1.66671	−0.833356	−	0.552737i	$-0.813583\pi$
$89$	−15.7237	−1.66671	−0.833356	+	0.552737i	$0.813583\pi$
$90$	0	0
$91$	−0.930274	−0.0975192
$92$	1.12323	0.117105
$93$	0	0
$94$	9.65721	0.996065
$95$	12.8081	1.31409
$96$	0	0
$97$	−7.75619	−0.787521	−0.393761	−	0.919213i	$-0.628826\pi$
$97$	−7.75619	−0.787521	−0.393761	+	0.919213i	$0.628826\pi$
$98$	8.26935	0.835331
$99$	0	0
$100$	−2.22858	−0.222858
$101$	−0.542833	−0.0540139	−0.0270069	−	0.999635i	$-0.508598\pi$
$101$	−0.542833	−0.0540139	−0.0270069	+	0.999635i	$0.508598\pi$
$102$	0	0
$103$	−18.9963	−1.87176	−0.935878	−	0.352323i	$-0.885392\pi$
$103$	−18.9963	−1.87176	−0.935878	+	0.352323i	$0.885392\pi$
$104$	3.31985	0.325538
$105$	0	0
$106$	3.81017	0.370077
$107$	−1.11430	−0.107724	−0.0538619	−	0.998548i	$-0.517153\pi$
$107$	−1.11430	−0.107724	−0.0538619	+	0.998548i	$0.517153\pi$
$108$	0	0
$109$	−7.68359	−0.735954	−0.367977	−	0.929835i	$-0.619950\pi$
$109$	−7.68359	−0.735954	−0.367977	+	0.929835i	$0.619950\pi$
$110$	4.74521	0.452438
$111$	0	0
$112$	2.82293	0.266742
$113$	−13.4812	−1.26821	−0.634103	−	0.773249i	$-0.718630\pi$
$113$	−13.4812	−1.26821	−0.634103	+	0.773249i	$0.718630\pi$
$114$	0	0
$115$	−14.6896	−1.36982
$116$	−0.462475	−0.0429398
$117$	0	0
$118$	−0.586555	−0.0539968
$119$	3.66797	0.336242
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.31287	−0.118862
$123$	0	0
$124$	0.776892	0.0697670
$125$	11.0735	0.990443
$126$	0	0
$127$	2.50614	0.222384	0.111192	−	0.993799i	$-0.464533\pi$
$127$	2.50614	0.222384	0.111192	+	0.993799i	$0.464533\pi$
$128$	−8.42222	−0.744426
$129$	0	0
$130$	−5.27119	−0.462314
$131$	−7.59263	−0.663371	−0.331686	−	0.943390i	$-0.607617\pi$
$131$	−7.59263	−0.663371	−0.331686	+	0.943390i	$0.607617\pi$
$132$	0	0
$133$	−2.96763	−0.257326
$134$	−17.8345	−1.54066
$135$	0	0
$136$	−13.0898	−1.12244
$137$	10.8126	0.923782	0.461891	−	0.886937i	$-0.347171\pi$
$137$	10.8126	0.923782	0.461891	+	0.886937i	$0.347171\pi$
$138$	0	0
$139$	9.10264	0.772076	0.386038	−	0.922483i	$-0.373843\pi$
$139$	9.10264	0.772076	0.386038	+	0.922483i	$0.373843\pi$
$140$	0.836534	0.0707001
$141$	0	0
$142$	3.84520	0.322682
$143$	−1.11084	−0.0928935
$144$	0	0
$145$	6.04825	0.502279
$146$	4.23404	0.350411
$147$	0	0
$148$	−0.515289	−0.0423565
$149$	−6.88180	−0.563779	−0.281890	−	0.959447i	$-0.590961\pi$
$149$	−6.88180	−0.563779	−0.281890	+	0.959447i	$0.590961\pi$
$150$	0	0
$151$	5.97340	0.486108	0.243054	−	0.970013i	$-0.421851\pi$
$151$	5.97340	0.486108	0.243054	+	0.970013i	$0.421851\pi$
$152$	10.5905	0.859005
$153$	0	0
$154$	−1.09946	−0.0885970
$155$	−10.1602	−0.816085
$156$	0	0
$157$	−13.7456	−1.09702	−0.548511	−	0.836144i	$-0.684805\pi$
$157$	−13.7456	−1.09702	−0.548511	+	0.836144i	$0.684805\pi$
$158$	2.30972	0.183752
$159$	0	0
$160$	−5.60820	−0.443367
$161$	3.40357	0.268239
$162$	0	0
$163$	13.1473	1.02978	0.514888	−	0.857257i	$-0.327833\pi$
$163$	13.1473	1.02978	0.514888	+	0.857257i	$0.327833\pi$
$164$	2.43086	0.189818
$165$	0	0
$166$	0.419539	0.0325625
$167$	23.2428	1.79858	0.899292	−	0.437349i	$-0.144082\pi$
$167$	23.2428	1.79858	0.899292	+	0.437349i	$0.144082\pi$
$168$	0	0
$169$	−11.7660	−0.905079
$170$	20.7838	1.59404
$171$	0	0
$172$	0.0683417	0.00521101
$173$	−5.59970	−0.425737	−0.212869	−	0.977081i	$-0.568281\pi$
$173$	−5.59970	−0.425737	−0.212869	+	0.977081i	$0.568281\pi$
$174$	0	0
$175$	−6.75295	−0.510475
$176$	3.37088	0.254089
$177$	0	0
$178$	20.6432	1.54727
$179$	−6.95344	−0.519724	−0.259862	−	0.965646i	$-0.583677\pi$
$179$	−6.95344	−0.519724	−0.259862	+	0.965646i	$0.583677\pi$
$180$	0	0
$181$	12.9447	0.962174	0.481087	−	0.876673i	$-0.340242\pi$
$181$	12.9447	0.962174	0.481087	+	0.876673i	$0.340242\pi$
$182$	1.22133	0.0905309
$183$	0	0
$184$	−12.1463	−0.895434
$185$	6.73894	0.495457
$186$	0	0
$187$	4.37994	0.320293
$188$	2.03293	0.148267
$189$	0	0
$190$	−16.8154	−1.21992
$191$	18.5614	1.34306	0.671529	−	0.740978i	$-0.265638\pi$
$191$	18.5614	1.34306	0.671529	+	0.740978i	$0.265638\pi$
$192$	0	0
$193$	−10.6201	−0.764450	−0.382225	−	0.924069i	$-0.624842\pi$
$193$	−10.6201	−0.764450	−0.382225	+	0.924069i	$0.624842\pi$
$194$	10.1829	0.731087
$195$	0	0
$196$	1.74077	0.124341
$197$	−8.69998	−0.619848	−0.309924	−	0.950761i	$-0.600304\pi$
$197$	−8.69998	−0.619848	−0.309924	+	0.950761i	$0.600304\pi$
$198$	0	0
$199$	13.2999	0.942803	0.471402	−	0.881919i	$-0.343748\pi$
$199$	13.2999	0.942803	0.471402	+	0.881919i	$0.343748\pi$
$200$	24.0991	1.70406
$201$	0	0
$202$	0.712669	0.0501432
$203$	−1.40137	−0.0983570
$204$	0	0
$205$	−31.7907	−2.22036
$206$	24.9396	1.73763
$207$	0	0
$208$	−3.74452	−0.259636
$209$	−3.54367	−0.245120
$210$	0	0
$211$	−4.35272	−0.299654	−0.149827	−	0.988712i	$-0.547872\pi$
$211$	−4.35272	−0.299654	−0.149827	+	0.988712i	$0.547872\pi$
$212$	0.802076	0.0550868
$213$	0	0
$214$	1.46294	0.100004
$215$	−0.893772	−0.0609548
$216$	0	0
$217$	2.35410	0.159807
$218$	10.0876	0.683216
$219$	0	0
$220$	0.998910	0.0673465
$221$	−4.86544	−0.327285
$222$	0	0
$223$	9.37001	0.627462	0.313731	−	0.949512i	$-0.398421\pi$
$223$	9.37001	0.627462	0.313731	+	0.949512i	$0.398421\pi$
$224$	1.29941	0.0868207
$225$	0	0
$226$	17.6991	1.17733
$227$	20.2223	1.34220	0.671100	−	0.741367i	$-0.265822\pi$
$227$	20.2223	1.34220	0.671100	+	0.741367i	$0.265822\pi$
$228$	0	0
$229$	−12.0588	−0.796866	−0.398433	−	0.917197i	$-0.630446\pi$
$229$	−12.0588	−0.796866	−0.398433	+	0.917197i	$0.630446\pi$
$230$	19.2856	1.27165
$231$	0	0
$232$	5.00104	0.328335
$233$	22.8322	1.49579	0.747895	−	0.663817i	$-0.231065\pi$
$233$	22.8322	1.49579	0.747895	+	0.663817i	$0.231065\pi$
$234$	0	0
$235$	−26.5866	−1.73432
$236$	−0.123475	−0.00803756
$237$	0	0
$238$	−4.81557	−0.312147
$239$	−12.9994	−0.840864	−0.420432	−	0.907324i	$-0.638121\pi$
$239$	−12.9994	−0.840864	−0.420432	+	0.907324i	$0.638121\pi$
$240$	0	0
$241$	25.7692	1.65994	0.829971	−	0.557807i	$-0.188357\pi$
$241$	25.7692	1.65994	0.829971	+	0.557807i	$0.188357\pi$
$242$	−1.31287	−0.0843945
$243$	0	0
$244$	−0.276371	−0.0176929
$245$	−22.7658	−1.45446
$246$	0	0
$247$	3.93646	0.250471
$248$	−8.40103	−0.533466
$249$	0	0
$250$	−14.5381	−0.919467
$251$	2.41536	0.152456	0.0762281	−	0.997090i	$-0.475712\pi$
$251$	2.41536	0.152456	0.0762281	+	0.997090i	$0.475712\pi$
$252$	0	0
$253$	4.06422	0.255515
$254$	−3.29024	−0.206448
$255$	0	0
$256$	−6.50042	−0.406276
$257$	−20.1663	−1.25794	−0.628969	−	0.777431i	$-0.716523\pi$
$257$	−20.1663	−1.25794	−0.628969	+	0.777431i	$0.716523\pi$
$258$	0	0
$259$	−1.56140	−0.0970209
$260$	−1.10963	−0.0688166
$261$	0	0
$262$	9.96814	0.615834
$263$	15.8182	0.975392	0.487696	−	0.873013i	$-0.337837\pi$
$263$	15.8182	0.975392	0.487696	+	0.873013i	$0.337837\pi$
$264$	0	0
$265$	−10.4895	−0.644367
$266$	3.89612	0.238886
$267$	0	0
$268$	−3.75432	−0.229331
$269$	−18.4287	−1.12362	−0.561808	−	0.827267i	$-0.689894\pi$
$269$	−18.4287	−1.12362	−0.561808	+	0.827267i	$0.689894\pi$
$270$	0	0
$271$	−11.2933	−0.686017	−0.343009	−	0.939332i	$-0.611446\pi$
$271$	−11.2933	−0.686017	−0.343009	+	0.939332i	$0.611446\pi$
$272$	14.7642	0.895214
$273$	0	0
$274$	−14.1955	−0.857584
$275$	−8.06373	−0.486261
$276$	0	0
$277$	25.1075	1.50856	0.754282	−	0.656550i	$-0.227985\pi$
$277$	25.1075	1.50856	0.754282	+	0.656550i	$0.227985\pi$
$278$	−11.9506	−0.716749
$279$	0	0
$280$	−9.04599	−0.540601
$281$	−9.59612	−0.572457	−0.286228	−	0.958161i	$-0.592402\pi$
$281$	−9.59612	−0.572457	−0.286228	+	0.958161i	$0.592402\pi$
$282$	0	0
$283$	−28.2833	−1.68126	−0.840632	−	0.541606i	$-0.817816\pi$
$283$	−28.2833	−1.68126	−0.840632	+	0.541606i	$0.817816\pi$
$284$	0.809450	0.0480320
$285$	0	0
$286$	1.45840	0.0862367
$287$	7.36587	0.434793
$288$	0	0
$289$	2.18390	0.128465
$290$	−7.94056	−0.466286
$291$	0	0
$292$	0.891303	0.0521596
$293$	−9.34876	−0.546160	−0.273080	−	0.961991i	$-0.588042\pi$
$293$	−9.34876	−0.546160	−0.273080	+	0.961991i	$0.588042\pi$
$294$	0	0
$295$	1.61481	0.0940177
$296$	5.57215	0.323875
$297$	0	0
$298$	9.03491	0.523379
$299$	−4.51472	−0.261093
$300$	0	0
$301$	0.207086	0.0119362
$302$	−7.84230	−0.451274
$303$	0	0
$304$	−11.9453	−0.685108
$305$	3.61438	0.206959
$306$	0	0
$307$	16.0672	0.917005	0.458503	−	0.888693i	$-0.348386\pi$
$307$	16.0672	0.917005	0.458503	+	0.888693i	$0.348386\pi$
$308$	−0.231446	−0.0131879
$309$	0	0
$310$	13.3390	0.757604
$311$	5.53945	0.314113	0.157057	−	0.987590i	$-0.449799\pi$
$311$	5.53945	0.314113	0.157057	+	0.987590i	$0.449799\pi$
$312$	0	0
$313$	6.72316	0.380016	0.190008	−	0.981783i	$-0.439149\pi$
$313$	6.72316	0.380016	0.190008	+	0.981783i	$0.439149\pi$
$314$	18.0462	1.01841
$315$	0	0
$316$	0.486218	0.0273519
$317$	−25.5153	−1.43308	−0.716541	−	0.697545i	$-0.754276\pi$
$317$	−25.5153	−1.43308	−0.716541	+	0.697545i	$0.754276\pi$
$318$	0	0
$319$	−1.67338	−0.0936916
$320$	31.7301	1.77377
$321$	0	0
$322$	−4.46845	−0.249017
$323$	−15.5211	−0.863615
$324$	0	0
$325$	8.95755	0.496876
$326$	−17.2607	−0.955982
$327$	0	0
$328$	−26.2864	−1.45142
$329$	6.16009	0.339617
$330$	0	0
$331$	−3.18003	−0.174790	−0.0873950	−	0.996174i	$-0.527854\pi$
$331$	−3.18003	−0.174790	−0.0873950	+	0.996174i	$0.527854\pi$
$332$	0.0883167	0.00484701
$333$	0	0
$334$	−30.5148	−1.66970
$335$	49.0989	2.68256
$336$	0	0
$337$	−3.13033	−0.170520	−0.0852599	−	0.996359i	$-0.527172\pi$
$337$	−3.13033	−0.170520	−0.0852599	+	0.996359i	$0.527172\pi$
$338$	15.4473	0.840220
$339$	0	0
$340$	4.37517	0.237277
$341$	2.81104	0.152227
$342$	0	0
$343$	11.1369	0.601338
$344$	−0.739023	−0.0398455
$345$	0	0
$346$	7.35168	0.395229
$347$	−21.9508	−1.17838	−0.589190	−	0.807994i	$-0.700553\pi$
$347$	−21.9508	−1.17838	−0.589190	+	0.807994i	$0.700553\pi$
$348$	0	0
$349$	7.14342	0.382378	0.191189	−	0.981553i	$-0.438766\pi$
$349$	7.14342	0.382378	0.191189	+	0.981553i	$0.438766\pi$
$350$	8.86575	0.473894
$351$	0	0
$352$	1.55164	0.0827025
$353$	7.76027	0.413037	0.206519	−	0.978443i	$-0.433787\pi$
$353$	7.76027	0.413037	0.206519	+	0.978443i	$0.433787\pi$
$354$	0	0
$355$	−10.5860	−0.561845
$356$	4.34559	0.230316
$357$	0	0
$358$	9.12896	0.482480
$359$	5.97613	0.315408	0.157704	−	0.987486i	$-0.449591\pi$
$359$	5.97613	0.315408	0.157704	+	0.987486i	$0.449591\pi$
$360$	0	0
$361$	−6.44243	−0.339075
$362$	−16.9947	−0.893224
$363$	0	0
$364$	0.257101	0.0134757
$365$	−11.6565	−0.610126
$366$	0	0
$367$	25.5116	1.33170	0.665848	−	0.746088i	$-0.268070\pi$
$367$	25.5116	1.33170	0.665848	+	0.746088i	$0.268070\pi$
$368$	13.7000	0.714161
$369$	0	0
$370$	−8.84735	−0.459952
$371$	2.43041	0.126181
$372$	0	0
$373$	3.08998	0.159993	0.0799967	−	0.996795i	$-0.474509\pi$
$373$	3.08998	0.159993	0.0799967	+	0.996795i	$0.474509\pi$
$374$	−5.75030	−0.297341
$375$	0	0
$376$	−21.9834	−1.13371
$377$	1.85887	0.0957367
$378$	0	0
$379$	23.8740	1.22633	0.613163	−	0.789956i	$-0.289897\pi$
$379$	23.8740	1.22633	0.613163	+	0.789956i	$0.289897\pi$
$380$	−3.53980	−0.181588
$381$	0	0
$382$	−24.3688	−1.24681
$383$	−4.74359	−0.242386	−0.121193	−	0.992629i	$-0.538672\pi$
$383$	−4.74359	−0.242386	−0.121193	+	0.992629i	$0.538672\pi$
$384$	0	0
$385$	3.02685	0.154263
$386$	13.9428	0.709669
$387$	0	0
$388$	2.14359	0.108824
$389$	−10.1189	−0.513048	−0.256524	−	0.966538i	$-0.582577\pi$
$389$	−10.1189	−0.513048	−0.256524	+	0.966538i	$0.582577\pi$
$390$	0	0
$391$	17.8011	0.900238
$392$	−18.8241	−0.950762
$393$	0	0
$394$	11.4220	0.575430
$395$	−6.35875	−0.319943
$396$	0	0
$397$	−15.0570	−0.755688	−0.377844	−	0.925869i	$-0.623334\pi$
$397$	−15.0570	−0.755688	−0.377844	+	0.925869i	$0.623334\pi$
$398$	−17.4610	−0.875242
$399$	0	0
$400$	−27.1818	−1.35909
$401$	−10.5678	−0.527730	−0.263865	−	0.964560i	$-0.584997\pi$
$401$	−10.5678	−0.527730	−0.263865	+	0.964560i	$0.584997\pi$
$402$	0	0
$403$	−3.12263	−0.155550
$404$	0.150023	0.00746394
$405$	0	0
$406$	1.83982	0.0913087
$407$	−1.86448	−0.0924189
$408$	0	0
$409$	5.26303	0.260240	0.130120	−	0.991498i	$-0.458464\pi$
$409$	5.26303	0.260240	0.130120	+	0.991498i	$0.458464\pi$
$410$	41.7371	2.06125
$411$	0	0
$412$	5.25002	0.258650
$413$	−0.374149	−0.0184107
$414$	0	0
$415$	−1.15500	−0.0566970
$416$	−1.72363	−0.0845078
$417$	0	0
$418$	4.65237	0.227555
$419$	−19.6551	−0.960216	−0.480108	−	0.877209i	$-0.659402\pi$
$419$	−19.6551	−0.960216	−0.480108	+	0.877209i	$0.659402\pi$
$420$	0	0
$421$	−3.75600	−0.183056	−0.0915280	−	0.995803i	$-0.529175\pi$
$421$	−3.75600	−0.183056	−0.0915280	+	0.995803i	$0.529175\pi$
$422$	5.71456	0.278180
$423$	0	0
$424$	−8.67337	−0.421216
$425$	−35.3187	−1.71321
$426$	0	0
$427$	−0.837447	−0.0405269
$428$	0.307962	0.0148859
$429$	0	0
$430$	1.17341	0.0565867
$431$	−6.44918	−0.310646	−0.155323	−	0.987864i	$-0.549642\pi$
$431$	−6.44918	−0.310646	−0.155323	+	0.987864i	$0.549642\pi$
$432$	0	0
$433$	36.7936	1.76819	0.884094	−	0.467310i	$-0.154777\pi$
$433$	36.7936	1.76819	0.884094	+	0.467310i	$0.154777\pi$
$434$	−3.09063	−0.148355
$435$	0	0
$436$	2.12352	0.101698
$437$	−14.4022	−0.688953
$438$	0	0
$439$	−9.74500	−0.465104	−0.232552	−	0.972584i	$-0.574708\pi$
$439$	−9.74500	−0.465104	−0.232552	+	0.972584i	$0.574708\pi$
$440$	−10.8019	−0.514959
$441$	0	0
$442$	6.38769	0.303831
$443$	−6.37596	−0.302931	−0.151466	−	0.988463i	$-0.548399\pi$
$443$	−6.37596	−0.302931	−0.151466	+	0.988463i	$0.548399\pi$
$444$	0	0
$445$	−56.8315	−2.69407
$446$	−12.3016	−0.582498
$447$	0	0
$448$	−7.35182	−0.347341
$449$	−17.4495	−0.823495	−0.411747	−	0.911298i	$-0.635081\pi$
$449$	−17.4495	−0.823495	−0.411747	+	0.911298i	$0.635081\pi$
$450$	0	0
$451$	8.79562	0.414170
$452$	3.72582	0.175248
$453$	0	0
$454$	−26.5492	−1.24602
$455$	−3.36236	−0.157630
$456$	0	0
$457$	−41.4643	−1.93962	−0.969808	−	0.243869i	$-0.921583\pi$
$457$	−41.4643	−1.93962	−0.969808	+	0.243869i	$0.921583\pi$
$458$	15.8316	0.739763
$459$	0	0
$460$	4.05979	0.189289
$461$	23.3164	1.08595	0.542976	−	0.839748i	$-0.317298\pi$
$461$	23.3164	1.08595	0.542976	+	0.839748i	$0.317298\pi$
$462$	0	0
$463$	−38.3434	−1.78197	−0.890984	−	0.454036i	$-0.849984\pi$
$463$	−38.3434	−1.78197	−0.890984	+	0.454036i	$0.849984\pi$
$464$	−5.64077	−0.261866
$465$	0	0
$466$	−29.9758	−1.38860
$467$	−26.9505	−1.24712	−0.623561	−	0.781775i	$-0.714315\pi$
$467$	−26.9505	−1.24712	−0.623561	+	0.781775i	$0.714315\pi$
$468$	0	0
$469$	−11.3762	−0.525302
$470$	34.9048	1.61004
$471$	0	0
$472$	1.33522	0.0614584
$473$	0.247282	0.0113701
$474$	0	0
$475$	28.5752	1.31112
$476$	−1.01372	−0.0464638
$477$	0	0
$478$	17.0666	0.780607
$479$	−32.4812	−1.48411	−0.742053	−	0.670342i	$-0.766148\pi$
$479$	−32.4812	−1.48411	−0.742053	+	0.670342i	$0.766148\pi$
$480$	0	0
$481$	2.07115	0.0944363
$482$	−33.8317	−1.54099
$483$	0	0
$484$	−0.276371	−0.0125623
$485$	−28.0338	−1.27295
$486$	0	0
$487$	−21.1728	−0.959432	−0.479716	−	0.877424i	$-0.659260\pi$
$487$	−21.1728	−0.959432	−0.479716	+	0.877424i	$0.659260\pi$
$488$	2.98858	0.135287
$489$	0	0
$490$	29.8886	1.35023
$491$	17.4594	0.787930	0.393965	−	0.919125i	$-0.371103\pi$
$491$	17.4594	0.787930	0.393965	+	0.919125i	$0.371103\pi$
$492$	0	0
$493$	−7.32933	−0.330096
$494$	−5.16807	−0.232522
$495$	0	0
$496$	9.47568	0.425471
$497$	2.45276	0.110021
$498$	0	0
$499$	−28.8978	−1.29364	−0.646822	−	0.762641i	$-0.723902\pi$
$499$	−28.8978	−1.29364	−0.646822	+	0.762641i	$0.723902\pi$
$500$	−3.06039	−0.136865
$501$	0	0
$502$	−3.17105	−0.141531
$503$	14.6656	0.653905	0.326953	−	0.945041i	$-0.393978\pi$
$503$	14.6656	0.653905	0.326953	+	0.945041i	$0.393978\pi$
$504$	0	0
$505$	−1.96200	−0.0873080
$506$	−5.33580	−0.237205
$507$	0	0
$508$	−0.692626	−0.0307303
$509$	−11.3607	−0.503556	−0.251778	−	0.967785i	$-0.581015\pi$
$509$	−11.3607	−0.503556	−0.251778	+	0.967785i	$0.581015\pi$
$510$	0	0
$511$	2.70078	0.119476
$512$	25.3786	1.12159
$513$	0	0
$514$	26.4757	1.16779
$515$	−68.6597	−3.02551
$516$	0	0
$517$	7.35580	0.323508
$518$	2.04992	0.0900684
$519$	0	0
$520$	11.9992	0.526199
$521$	−4.14492	−0.181592	−0.0907961	−	0.995870i	$-0.528941\pi$
$521$	−4.14492	−0.181592	−0.0907961	+	0.995870i	$0.528941\pi$
$522$	0	0
$523$	5.81064	0.254082	0.127041	−	0.991897i	$-0.459452\pi$
$523$	5.81064	0.254082	0.127041	+	0.991897i	$0.459452\pi$
$524$	2.09838	0.0916683
$525$	0	0
$526$	−20.7673	−0.905495
$527$	12.3122	0.536329
$528$	0	0
$529$	−6.48210	−0.281830
$530$	13.7714	0.598192
$531$	0	0
$532$	0.820168	0.0355588
$533$	−9.77057	−0.423210
$534$	0	0
$535$	−4.02752	−0.174125
$536$	40.5978	1.75356
$537$	0	0
$538$	24.1945	1.04310
$539$	6.29868	0.271303
$540$	0	0
$541$	20.0635	0.862598	0.431299	−	0.902209i	$-0.358055\pi$
$541$	20.0635	0.862598	0.431299	+	0.902209i	$0.358055\pi$
$542$	14.8266	0.636857
$543$	0	0
$544$	6.79608	0.291380
$545$	−27.7714	−1.18960
$546$	0	0
$547$	23.1230	0.988669	0.494334	−	0.869272i	$-0.335412\pi$
$547$	23.1230	0.988669	0.494334	+	0.869272i	$0.335412\pi$
$548$	−2.98829	−0.127653
$549$	0	0
$550$	10.5866	0.451416
$551$	5.92992	0.252623
$552$	0	0
$553$	1.47331	0.0626517
$554$	−32.9629	−1.40046
$555$	0	0
$556$	−2.51571	−0.106690
$557$	−10.3408	−0.438154	−0.219077	−	0.975708i	$-0.570305\pi$
$557$	−10.3408	−0.438154	−0.219077	+	0.975708i	$0.570305\pi$
$558$	0	0
$559$	−0.274692	−0.0116182
$560$	10.2031	0.431161
$561$	0	0
$562$	12.5985	0.531434
$563$	−8.42500	−0.355071	−0.177536	−	0.984114i	$-0.556813\pi$
$563$	−8.42500	−0.355071	−0.177536	+	0.984114i	$0.556813\pi$
$564$	0	0
$565$	−48.7262	−2.04993
$566$	37.1322	1.56078
$567$	0	0
$568$	−8.75311	−0.367272
$569$	−6.71812	−0.281638	−0.140819	−	0.990035i	$-0.544974\pi$
$569$	−6.71812	−0.281638	−0.140819	+	0.990035i	$0.544974\pi$
$570$	0	0
$571$	6.05227	0.253279	0.126640	−	0.991949i	$-0.459581\pi$
$571$	6.05227	0.253279	0.126640	+	0.991949i	$0.459581\pi$
$572$	0.307006	0.0128365
$573$	0	0
$574$	−9.67043	−0.403636
$575$	−32.7728	−1.36672
$576$	0	0
$577$	39.9435	1.66287	0.831435	−	0.555622i	$-0.187520\pi$
$577$	39.9435	1.66287	0.831435	+	0.555622i	$0.187520\pi$
$578$	−2.86718	−0.119259
$579$	0	0
$580$	−1.67156	−0.0694078
$581$	0.267613	0.0111025
$582$	0	0
$583$	2.90217	0.120196
$584$	−9.63824	−0.398833
$585$	0	0
$586$	12.2737	0.507022
$587$	−1.60472	−0.0662339	−0.0331169	−	0.999451i	$-0.510543\pi$
$587$	−1.60472	−0.0662339	−0.0331169	+	0.999451i	$0.510543\pi$
$588$	0	0
$589$	−9.96140	−0.410452
$590$	−2.12003	−0.0872804
$591$	0	0
$592$	−6.28493	−0.258309
$593$	12.8886	0.529273	0.264636	−	0.964348i	$-0.414748\pi$
$593$	12.8886	0.529273	0.264636	+	0.964348i	$0.414748\pi$
$594$	0	0
$595$	13.2574	0.543502
$596$	1.90193	0.0779062
$597$	0	0
$598$	5.92724	0.242383
$599$	46.2862	1.89120	0.945601	−	0.325329i	$-0.105475\pi$
$599$	46.2862	1.89120	0.945601	+	0.325329i	$0.105475\pi$
$600$	0	0
$601$	−33.8954	−1.38262	−0.691311	−	0.722557i	$-0.742967\pi$
$601$	−33.8954	−1.38262	−0.691311	+	0.722557i	$0.742967\pi$
$602$	−0.271877	−0.0110809
$603$	0	0
$604$	−1.65088	−0.0671732
$605$	3.61438	0.146945
$606$	0	0
$607$	20.3795	0.827179	0.413589	−	0.910464i	$-0.364275\pi$
$607$	20.3795	0.827179	0.413589	+	0.910464i	$0.364275\pi$
$608$	−5.49848	−0.222993
$609$	0	0
$610$	−4.74521	−0.192128
$611$	−8.17115	−0.330569
$612$	0	0
$613$	−16.8249	−0.679553	−0.339776	−	0.940506i	$-0.610351\pi$
$613$	−16.8249	−0.679553	−0.339776	+	0.940506i	$0.610351\pi$
$614$	−21.0942	−0.851292
$615$	0	0
$616$	2.50278	0.100840
$617$	1.36945	0.0551318	0.0275659	−	0.999620i	$-0.491224\pi$
$617$	1.36945	0.0551318	0.0275659	+	0.999620i	$0.491224\pi$
$618$	0	0
$619$	−42.6806	−1.71548	−0.857740	−	0.514084i	$-0.828132\pi$
$619$	−42.6806	−1.71548	−0.857740	+	0.514084i	$0.828132\pi$
$620$	2.80798	0.112771
$621$	0	0
$622$	−7.27258	−0.291604
$623$	13.1678	0.527556
$624$	0	0
$625$	−0.294885	−0.0117954
$626$	−8.82664	−0.352784
$627$	0	0
$628$	3.79890	0.151593
$629$	−8.16632	−0.325612
$630$	0	0
$631$	−31.4686	−1.25275	−0.626373	−	0.779524i	$-0.715461\pi$
$631$	−31.4686	−1.25275	−0.626373	+	0.779524i	$0.715461\pi$
$632$	−5.25779	−0.209143
$633$	0	0
$634$	33.4983	1.33039
$635$	9.05815	0.359462
$636$	0	0
$637$	−6.99686	−0.277226
$638$	2.19694	0.0869776
$639$	0	0
$640$	−30.4411	−1.20329
$641$	15.2489	0.602294	0.301147	−	0.953578i	$-0.402630\pi$
$641$	15.2489	0.602294	0.301147	+	0.953578i	$0.402630\pi$
$642$	0	0
$643$	−29.6334	−1.16863	−0.584313	−	0.811529i	$-0.698636\pi$
$643$	−29.6334	−1.16863	−0.584313	+	0.811529i	$0.698636\pi$
$644$	−0.940649	−0.0370668
$645$	0	0
$646$	20.3771	0.801728
$647$	−31.1055	−1.22288	−0.611442	−	0.791289i	$-0.709410\pi$
$647$	−31.1055	−1.22288	−0.611442	+	0.791289i	$0.709410\pi$
$648$	0	0
$649$	−0.446773	−0.0175374
$650$	−11.7601	−0.461269
$651$	0	0
$652$	−3.63354	−0.142300
$653$	24.3750	0.953867	0.476934	−	0.878939i	$-0.341748\pi$
$653$	24.3750	0.953867	0.476934	+	0.878939i	$0.341748\pi$
$654$	0	0
$655$	−27.4426	−1.07227
$656$	29.6489	1.15760
$657$	0	0
$658$	−8.08740	−0.315280
$659$	20.4867	0.798049	0.399025	−	0.916940i	$-0.369349\pi$
$659$	20.4867	0.798049	0.399025	+	0.916940i	$0.369349\pi$
$660$	0	0
$661$	−36.1634	−1.40659	−0.703297	−	0.710896i	$-0.748290\pi$
$661$	−36.1634	−1.40659	−0.703297	+	0.710896i	$0.748290\pi$
$662$	4.17496	0.162265
$663$	0	0
$664$	−0.955026	−0.0370622
$665$	−10.7261	−0.415942
$666$	0	0
$667$	−6.80101	−0.263336
$668$	−6.42365	−0.248538
$669$	0	0
$670$	−64.4605	−2.49033
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−19.7440	−0.761076	−0.380538	−	0.924765i	$-0.624261\pi$
$673$	−19.7440	−0.761076	−0.380538	+	0.924765i	$0.624261\pi$
$674$	4.10971	0.158300
$675$	0	0
$676$	3.25179	0.125069
$677$	−16.9323	−0.650761	−0.325380	−	0.945583i	$-0.605492\pi$
$677$	−16.9323	−0.650761	−0.325380	+	0.945583i	$0.605492\pi$
$678$	0	0
$679$	6.49539	0.249270
$680$	−47.3115	−1.81431
$681$	0	0
$682$	−3.69054	−0.141318
$683$	−22.3016	−0.853346	−0.426673	−	0.904406i	$-0.640315\pi$
$683$	−22.3016	−0.853346	−0.426673	+	0.904406i	$0.640315\pi$
$684$	0	0
$685$	39.0808	1.49320
$686$	−14.6214	−0.558246
$687$	0	0
$688$	0.833558	0.0317791
$689$	−3.22386	−0.122819
$690$	0	0
$691$	−13.5228	−0.514431	−0.257216	−	0.966354i	$-0.582805\pi$
$691$	−13.5228	−0.514431	−0.257216	+	0.966354i	$0.582805\pi$
$692$	1.54760	0.0588307
$693$	0	0
$694$	28.8185	1.09394
$695$	32.9004	1.24798
$696$	0	0
$697$	38.5243	1.45921
$698$	−9.37838	−0.354977
$699$	0	0
$700$	1.86632	0.0705403
$701$	−0.111492	−0.00421098	−0.00210549	−	0.999998i	$-0.500670\pi$
$701$	−0.111492	−0.00421098	−0.00210549	+	0.999998i	$0.500670\pi$
$702$	0	0
$703$	6.60710	0.249191
$704$	−8.77885	−0.330865
$705$	0	0
$706$	−10.1882	−0.383439
$707$	0.454594	0.0170967
$708$	0	0
$709$	−16.1234	−0.605526	−0.302763	−	0.953066i	$-0.597909\pi$
$709$	−16.1234	−0.605526	−0.302763	+	0.953066i	$0.597909\pi$
$710$	13.8980	0.521583
$711$	0	0
$712$	−46.9916	−1.76109
$713$	11.4247	0.427859
$714$	0	0
$715$	−4.01501	−0.150153
$716$	1.92173	0.0718184
$717$	0	0
$718$	−7.84589	−0.292806
$719$	1.62281	0.0605205	0.0302602	−	0.999542i	$-0.490366\pi$
$719$	1.62281	0.0605205	0.0302602	+	0.999542i	$0.490366\pi$
$720$	0	0
$721$	15.9084	0.592458
$722$	8.45808	0.314777
$723$	0	0
$724$	−3.57755	−0.132959
$725$	13.4937	0.501144
$726$	0	0
$727$	−21.1068	−0.782807	−0.391404	−	0.920219i	$-0.628010\pi$
$727$	−21.1068	−0.782807	−0.391404	+	0.920219i	$0.628010\pi$
$728$	−2.78020	−0.103041
$729$	0	0
$730$	15.3034	0.566405
$731$	1.08308	0.0400593
$732$	0	0
$733$	−32.6890	−1.20739	−0.603697	−	0.797213i	$-0.706307\pi$
$733$	−32.6890	−1.20739	−0.603697	+	0.797213i	$0.706307\pi$
$734$	−33.4934	−1.23627
$735$	0	0
$736$	6.30620	0.232450
$737$	−13.5843	−0.500385
$738$	0	0
$739$	−46.9356	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$739$	−46.9356	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$740$	−1.86245	−0.0684650
$741$	0	0
$742$	−3.19082	−0.117139
$743$	19.1077	0.700993	0.350497	−	0.936564i	$-0.386013\pi$
$743$	19.1077	0.700993	0.350497	+	0.936564i	$0.386013\pi$
$744$	0	0
$745$	−24.8734	−0.911292
$746$	−4.05675	−0.148528
$747$	0	0
$748$	−1.21049	−0.0442599
$749$	0.933171	0.0340973
$750$	0	0
$751$	−30.0661	−1.09713	−0.548565	−	0.836108i	$-0.684826\pi$
$751$	−30.0661	−1.09713	−0.548565	+	0.836108i	$0.684826\pi$
$752$	24.7955	0.904199
$753$	0	0
$754$	−2.44046	−0.0888762
$755$	21.5901	0.785745
$756$	0	0
$757$	25.3394	0.920976	0.460488	−	0.887666i	$-0.347674\pi$
$757$	25.3394	0.920976	0.460488	+	0.887666i	$0.347674\pi$
$758$	−31.3435	−1.13845
$759$	0	0
$760$	38.2782	1.38850
$761$	−1.16161	−0.0421084	−0.0210542	−	0.999778i	$-0.506702\pi$
$761$	−1.16161	−0.0421084	−0.0210542	+	0.999778i	$0.506702\pi$
$762$	0	0
$763$	6.43460	0.232948
$764$	−5.12985	−0.185591
$765$	0	0
$766$	6.22772	0.225016
$767$	0.496296	0.0179202
$768$	0	0
$769$	43.2961	1.56130	0.780648	−	0.624971i	$-0.214889\pi$
$769$	43.2961	1.56130	0.780648	+	0.624971i	$0.214889\pi$
$770$	−3.97386	−0.143208
$771$	0	0
$772$	2.93509	0.105636
$773$	45.6382	1.64149	0.820747	−	0.571292i	$-0.193558\pi$
$773$	45.6382	1.64149	0.820747	+	0.571292i	$0.193558\pi$
$774$	0	0
$775$	−22.6675	−0.814241
$776$	−23.1800	−0.832113
$777$	0	0
$778$	13.2848	0.476283
$779$	−31.1687	−1.11674
$780$	0	0
$781$	2.92885	0.104803
$782$	−23.3705	−0.835727
$783$	0	0
$784$	21.2321	0.758289
$785$	−49.6819	−1.77322
$786$	0	0
$787$	20.1357	0.717760	0.358880	−	0.933384i	$-0.383159\pi$
$787$	20.1357	0.717760	0.358880	+	0.933384i	$0.383159\pi$
$788$	2.40443	0.0856541
$789$	0	0
$790$	8.34821	0.297016
$791$	11.2898	0.401419
$792$	0	0
$793$	1.11084	0.0394473
$794$	19.7679	0.701535
$795$	0	0
$796$	−3.67570	−0.130282
$797$	32.4143	1.14817	0.574087	−	0.818794i	$-0.305357\pi$
$797$	32.4143	1.14817	0.574087	+	0.818794i	$0.305357\pi$
$798$	0	0
$799$	32.2180	1.13979
$800$	−12.5120	−0.442365
$801$	0	0
$802$	13.8741	0.489913
$803$	3.22502	0.113809
$804$	0	0
$805$	12.3018	0.433581
$806$	4.09961	0.144403
$807$	0	0
$808$	−1.62230	−0.0570723
$809$	15.1263	0.531811	0.265905	−	0.963999i	$-0.414329\pi$
$809$	15.1263	0.531811	0.265905	+	0.963999i	$0.414329\pi$
$810$	0	0
$811$	40.8120	1.43310	0.716552	−	0.697534i	$-0.245719\pi$
$811$	40.8120	1.43310	0.716552	+	0.697534i	$0.245719\pi$
$812$	0.387299	0.0135915
$813$	0	0
$814$	2.44782	0.0857961
$815$	47.5194	1.66453
$816$	0	0
$817$	−0.876286	−0.0306574
$818$	−6.90968	−0.241591
$819$	0	0
$820$	8.78604	0.306822
$821$	22.2786	0.777528	0.388764	−	0.921337i	$-0.372902\pi$
$821$	22.2786	0.777528	0.388764	+	0.921337i	$0.372902\pi$
$822$	0	0
$823$	−38.1070	−1.32833	−0.664163	−	0.747588i	$-0.731212\pi$
$823$	−38.1070	−1.32833	−0.664163	+	0.747588i	$0.731212\pi$
$824$	−56.7718	−1.97774
$825$	0	0
$826$	0.491209	0.0170913
$827$	−20.0408	−0.696887	−0.348443	−	0.937330i	$-0.613290\pi$
$827$	−20.0408	−0.696887	−0.348443	+	0.937330i	$0.613290\pi$
$828$	0	0
$829$	10.5019	0.364746	0.182373	−	0.983229i	$-0.441622\pi$
$829$	10.5019	0.364746	0.182373	+	0.983229i	$0.441622\pi$
$830$	1.51637	0.0526340
$831$	0	0
$832$	9.75194	0.338088
$833$	27.5879	0.955863
$834$	0	0
$835$	84.0084	2.90723
$836$	0.979367	0.0338721
$837$	0	0
$838$	25.8046	0.891406
$839$	−23.5773	−0.813979	−0.406990	−	0.913433i	$-0.633421\pi$
$839$	−23.5773	−0.813979	−0.406990	+	0.913433i	$0.633421\pi$
$840$	0	0
$841$	−26.1998	−0.903441
$842$	4.93114	0.169938
$843$	0	0
$844$	1.20297	0.0414078
$845$	−42.5269	−1.46297
$846$	0	0
$847$	−0.837447	−0.0287750
$848$	9.78285	0.335945
$849$	0	0
$850$	46.3689	1.59044
$851$	−7.57766	−0.259759
$852$	0	0
$853$	43.7066	1.49649	0.748243	−	0.663425i	$-0.230898\pi$
$853$	43.7066	1.49649	0.748243	+	0.663425i	$0.230898\pi$
$854$	1.09946	0.0376227
$855$	0	0
$856$	−3.33019	−0.113823
$857$	−50.7072	−1.73213	−0.866063	−	0.499935i	$-0.833357\pi$
$857$	−50.7072	−1.73213	−0.866063	+	0.499935i	$0.833357\pi$
$858$	0	0
$859$	28.6892	0.978861	0.489431	−	0.872042i	$-0.337205\pi$
$859$	28.6892	0.978861	0.489431	+	0.872042i	$0.337205\pi$
$860$	0.247013	0.00842307
$861$	0	0
$862$	8.46694	0.288385
$863$	−30.4241	−1.03565	−0.517824	−	0.855487i	$-0.673258\pi$
$863$	−30.4241	−1.03565	−0.517824	+	0.855487i	$0.673258\pi$
$864$	0	0
$865$	−20.2394	−0.688161
$866$	−48.3052	−1.64148
$867$	0	0
$868$	−0.650606	−0.0220830
$869$	1.75929	0.0596799
$870$	0	0
$871$	15.0901	0.511308
$872$	−22.9630	−0.777626
$873$	0	0
$874$	18.9083	0.639582
$875$	−9.27346	−0.313500
$876$	0	0
$877$	−35.6625	−1.20424	−0.602118	−	0.798407i	$-0.705676\pi$
$877$	−35.6625	−1.20424	−0.602118	+	0.798407i	$0.705676\pi$
$878$	12.7939	0.431774
$879$	0	0
$880$	12.1836	0.410710
$881$	21.9830	0.740627	0.370314	−	0.928907i	$-0.379250\pi$
$881$	21.9830	0.740627	0.370314	+	0.928907i	$0.379250\pi$
$882$	0	0
$883$	−24.6476	−0.829458	−0.414729	−	0.909945i	$-0.636124\pi$
$883$	−24.6476	−0.829458	−0.414729	+	0.909945i	$0.636124\pi$
$884$	1.34467	0.0452260
$885$	0	0
$886$	8.37081	0.281223
$887$	21.9777	0.737938	0.368969	−	0.929442i	$-0.379711\pi$
$887$	21.9777	0.737938	0.368969	+	0.929442i	$0.379711\pi$
$888$	0	0
$889$	−2.09876	−0.0703902
$890$	74.6124	2.50101
$891$	0	0
$892$	−2.58960	−0.0867063
$893$	−26.0665	−0.872282
$894$	0	0
$895$	−25.1323	−0.840082
$896$	7.05316	0.235630
$897$	0	0
$898$	22.9090	0.764483
$899$	−4.70396	−0.156886
$900$	0	0
$901$	12.7113	0.423476
$902$	−11.5475	−0.384490
$903$	0	0
$904$	−40.2897	−1.34002
$905$	46.7871	1.55526
$906$	0	0
$907$	19.5697	0.649801	0.324900	−	0.945748i	$-0.394669\pi$
$907$	19.5697	0.649801	0.324900	+	0.945748i	$0.394669\pi$
$908$	−5.58885	−0.185473
$909$	0	0
$910$	4.41434	0.146334
$911$	−29.9244	−0.991440	−0.495720	−	0.868482i	$-0.665096\pi$
$911$	−29.9244	−0.991440	−0.495720	+	0.868482i	$0.665096\pi$
$912$	0	0
$913$	0.319558	0.0105758
$914$	54.4372	1.80062
$915$	0	0
$916$	3.33270	0.110115
$917$	6.35842	0.209974
$918$	0	0
$919$	13.0486	0.430433	0.215217	−	0.976566i	$-0.430954\pi$
$919$	13.0486	0.430433	0.215217	+	0.976566i	$0.430954\pi$
$920$	−43.9012	−1.44738
$921$	0	0
$922$	−30.6114	−1.00813
$923$	−3.25350	−0.107090
$924$	0	0
$925$	15.0347	0.494337
$926$	50.3399	1.65427
$927$	0	0
$928$	−2.59649	−0.0852338
$929$	48.1672	1.58032	0.790158	−	0.612904i	$-0.209999\pi$
$929$	48.1672	1.58032	0.790158	+	0.612904i	$0.209999\pi$
$930$	0	0
$931$	−22.3204	−0.731522
$932$	−6.31018	−0.206697
$933$	0	0
$934$	35.3826	1.15775
$935$	15.8308	0.517722
$936$	0	0
$937$	−44.7933	−1.46333	−0.731667	−	0.681663i	$-0.761257\pi$
$937$	−44.7933	−1.46333	−0.731667	+	0.681663i	$0.761257\pi$
$938$	14.9354	0.487659
$939$	0	0
$940$	7.34778	0.239658
$941$	−24.4400	−0.796722	−0.398361	−	0.917229i	$-0.630421\pi$
$941$	−24.4400	−0.796722	−0.398361	+	0.917229i	$0.630421\pi$
$942$	0	0
$943$	35.7474	1.16409
$944$	−1.50602	−0.0490167
$945$	0	0
$946$	−0.324650	−0.0105553
$947$	11.3436	0.368617	0.184309	−	0.982868i	$-0.440995\pi$
$947$	11.3436	0.368617	0.184309	+	0.982868i	$0.440995\pi$
$948$	0	0
$949$	−3.58250	−0.116293
$950$	−37.5155	−1.21716
$951$	0	0
$952$	10.9620	0.355281
$953$	13.2016	0.427641	0.213821	−	0.976873i	$-0.431409\pi$
$953$	13.2016	0.427641	0.213821	+	0.976873i	$0.431409\pi$
$954$	0	0
$955$	67.0881	2.17092
$956$	3.59267	0.116195
$957$	0	0
$958$	42.6436	1.37775
$959$	−9.05497	−0.292400
$960$	0	0
$961$	−23.0980	−0.745098
$962$	−2.71915	−0.0876689
$963$	0	0
$964$	−7.12187	−0.229380
$965$	−38.3850	−1.23566
$966$	0	0
$967$	16.6355	0.534961	0.267480	−	0.963563i	$-0.413809\pi$
$967$	16.6355	0.534961	0.267480	+	0.963563i	$0.413809\pi$
$968$	2.98858	0.0960566
$969$	0	0
$970$	36.8047	1.18173
$971$	10.0363	0.322080	0.161040	−	0.986948i	$-0.448515\pi$
$971$	10.0363	0.322080	0.161040	+	0.986948i	$0.448515\pi$
$972$	0	0
$973$	−7.62298	−0.244382
$974$	27.7972	0.890679
$975$	0	0
$976$	−3.37088	−0.107899
$977$	−11.7119	−0.374695	−0.187348	−	0.982294i	$-0.559989\pi$
$977$	−11.7119	−0.374695	−0.187348	+	0.982294i	$0.559989\pi$
$978$	0	0
$979$	15.7237	0.502533
$980$	6.29182	0.200985
$981$	0	0
$982$	−22.9219	−0.731466
$983$	−13.8777	−0.442631	−0.221316	−	0.975202i	$-0.571035\pi$
$983$	−13.8777	−0.442631	−0.221316	+	0.975202i	$0.571035\pi$
$984$	0	0
$985$	−31.4450	−1.00192
$986$	9.62246	0.306442
$987$	0	0
$988$	−1.08793	−0.0346115
$989$	1.00501	0.0319575
$990$	0	0
$991$	13.9982	0.444666	0.222333	−	0.974971i	$-0.428633\pi$
$991$	13.9982	0.444666	0.222333	+	0.974971i	$0.428633\pi$
$992$	4.36172	0.138485
$993$	0	0
$994$	−3.22015	−0.102137
$995$	48.0708	1.52395
$996$	0	0
$997$	−42.5735	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$997$	−42.5735	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$998$	37.9391	1.20094
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.h.1.5		13
3.2	odd	2		2013.2.a.g.1.9	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.g.1.9	✓	13	3.2	odd	2
6039.2.a.h.1.5		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)

\(f(q)\)	\(=\)	\(q-1.31287 q^{2} -0.276371 q^{4} +3.61438 q^{5} -0.837447 q^{7} +2.98858 q^{8} +O(q^{10})\)	\(q-1.31287 q^{2} -0.276371 q^{4} +3.61438 q^{5} -0.837447 q^{7} +2.98858 q^{8} -4.74521 q^{10} -1.00000 q^{11} +1.11084 q^{13} +1.09946 q^{14} -3.37088 q^{16} -4.37994 q^{17} +3.54367 q^{19} -0.998910 q^{20} +1.31287 q^{22} -4.06422 q^{23} +8.06373 q^{25} -1.45840 q^{26} +0.231446 q^{28} +1.67338 q^{29} -2.81104 q^{31} -1.55164 q^{32} +5.75030 q^{34} -3.02685 q^{35} +1.86448 q^{37} -4.65237 q^{38} +10.8019 q^{40} -8.79562 q^{41} -0.247282 q^{43} +0.276371 q^{44} +5.33580 q^{46} -7.35580 q^{47} -6.29868 q^{49} -10.5866 q^{50} -0.307006 q^{52} -2.90217 q^{53} -3.61438 q^{55} -2.50278 q^{56} -2.19694 q^{58} +0.446773 q^{59} +1.00000 q^{61} +3.69054 q^{62} +8.77885 q^{64} +4.01501 q^{65} +13.5843 q^{67} +1.21049 q^{68} +3.97386 q^{70} -2.92885 q^{71} -3.22502 q^{73} -2.44782 q^{74} -0.979367 q^{76} +0.837447 q^{77} -1.75929 q^{79} -12.1836 q^{80} +11.5475 q^{82} -0.319558 q^{83} -15.8308 q^{85} +0.324650 q^{86} -2.98858 q^{88} -15.7237 q^{89} -0.930274 q^{91} +1.12323 q^{92} +9.65721 q^{94} +12.8081 q^{95} -7.75619 q^{97} +8.26935 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + 2 q^{10} - 13 q^{11} + 9 q^{13} - 7 q^{14} + 2 q^{16} - 19 q^{17} + 14 q^{19} - 19 q^{20} + 4 q^{22} - 5 q^{23} + 2 q^{25} + 4 q^{26} + 7 q^{28} - 10 q^{29} - q^{31} - 7 q^{32} - 2 q^{34} - 16 q^{35} - 8 q^{37} + 10 q^{38} + 14 q^{40} - 21 q^{41} + 11 q^{43} - 12 q^{44} - 8 q^{46} - 22 q^{47} - 19 q^{50} - q^{52} - 16 q^{53} + 7 q^{55} - 13 q^{58} - 19 q^{59} + 13 q^{61} - 3 q^{62} - 13 q^{64} - 13 q^{65} + 12 q^{67} - 36 q^{68} - 20 q^{70} - 5 q^{71} + 18 q^{73} - 6 q^{74} - 5 q^{76} - 7 q^{77} - q^{79} - 6 q^{80} - 22 q^{82} - 48 q^{83} - 2 q^{85} - 26 q^{86} + 9 q^{88} - 15 q^{89} - 11 q^{91} + 24 q^{92} - 23 q^{94} - 17 q^{95} - 17 q^{97} + 15 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + 2 * q^10 - 13 * q^11 + 9 * q^13 - 7 * q^14 + 2 * q^16 - 19 * q^17 + 14 * q^19 - 19 * q^20 + 4 * q^22 - 5 * q^23 + 2 * q^25 + 4 * q^26 + 7 * q^28 - 10 * q^29 - q^31 - 7 * q^32 - 2 * q^34 - 16 * q^35 - 8 * q^37 + 10 * q^38 + 14 * q^40 - 21 * q^41 + 11 * q^43 - 12 * q^44 - 8 * q^46 - 22 * q^47 - 19 * q^50 - q^52 - 16 * q^53 + 7 * q^55 - 13 * q^58 - 19 * q^59 + 13 * q^61 - 3 * q^62 - 13 * q^64 - 13 * q^65 + 12 * q^67 - 36 * q^68 - 20 * q^70 - 5 * q^71 + 18 * q^73 - 6 * q^74 - 5 * q^76 - 7 * q^77 - q^79 - 6 * q^80 - 22 * q^82 - 48 * q^83 - 2 * q^85 - 26 * q^86 + 9 * q^88 - 15 * q^89 - 11 * q^91 + 24 * q^92 - 23 * q^94 - 17 * q^95 - 17 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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