LMFDB - Embedded newform 6039.2.a.k.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.7
Root			$1.31423$ 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)

$f(q)$	$=$	$q-1.31423 q^{2} -0.272811 q^{4} +1.38757 q^{5} +2.03012 q^{7} +2.98699 q^{8} +O(q^{10})$	\(q-1.31423 q^{2} -0.272811 q^{4} +1.38757 q^{5} +2.03012 q^{7} +2.98699 q^{8} -1.82358 q^{10} -1.00000 q^{11} +5.73746 q^{13} -2.66804 q^{14} -3.37995 q^{16} -6.94204 q^{17} -3.55211 q^{19} -0.378545 q^{20} +1.31423 q^{22} -0.806289 q^{23} -3.07464 q^{25} -7.54032 q^{26} -0.553841 q^{28} -8.32766 q^{29} +4.54221 q^{31} -1.53195 q^{32} +9.12341 q^{34} +2.81694 q^{35} +8.85665 q^{37} +4.66828 q^{38} +4.14466 q^{40} +4.04638 q^{41} +0.832544 q^{43} +0.272811 q^{44} +1.05965 q^{46} -4.66753 q^{47} -2.87860 q^{49} +4.04077 q^{50} -1.56524 q^{52} +9.49286 q^{53} -1.38757 q^{55} +6.06395 q^{56} +10.9444 q^{58} +12.6362 q^{59} -1.00000 q^{61} -5.96948 q^{62} +8.77324 q^{64} +7.96114 q^{65} +6.13660 q^{67} +1.89387 q^{68} -3.70210 q^{70} -12.3352 q^{71} +8.32305 q^{73} -11.6396 q^{74} +0.969057 q^{76} -2.03012 q^{77} +3.78450 q^{79} -4.68993 q^{80} -5.31786 q^{82} +12.6260 q^{83} -9.63258 q^{85} -1.09415 q^{86} -2.98699 q^{88} +14.8592 q^{89} +11.6478 q^{91} +0.219965 q^{92} +6.13418 q^{94} -4.92881 q^{95} +7.09938 q^{97} +3.78312 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.31423	−0.929298	−0.464649	−	0.885495i	$-0.653819\pi$
$2$	−1.31423	−0.929298	−0.464649	+	0.885495i	$0.653819\pi$
$3$	0	0
$3$	0	0
$4$	−0.272811	−0.136406
$5$	1.38757	0.620541	0.310271	−	0.950648i	$-0.399580\pi$
$5$	1.38757	0.620541	0.310271	+	0.950648i	$0.399580\pi$
$6$	0	0
$7$	2.03012	0.767315	0.383657	−	0.923476i	$-0.374664\pi$
$7$	2.03012	0.767315	0.383657	+	0.923476i	$0.374664\pi$
$8$	2.98699	1.05606
$9$	0	0
$10$	−1.82358	−0.576668
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.73746	1.59129	0.795643	−	0.605766i	$-0.207133\pi$
$13$	5.73746	1.59129	0.795643	+	0.605766i	$0.207133\pi$
$14$	−2.66804	−0.713064
$15$	0	0
$16$	−3.37995	−0.844988
$17$	−6.94204	−1.68369	−0.841846	−	0.539718i	$-0.818531\pi$
$17$	−6.94204	−1.68369	−0.841846	+	0.539718i	$0.818531\pi$
$18$	0	0
$19$	−3.55211	−0.814911	−0.407455	−	0.913225i	$-0.633584\pi$
$19$	−3.55211	−0.814911	−0.407455	+	0.913225i	$0.633584\pi$
$20$	−0.378545	−0.0846453
$21$	0	0
$22$	1.31423	0.280194
$23$	−0.806289	−0.168123	−0.0840615	−	0.996461i	$-0.526789\pi$
$23$	−0.806289	−0.168123	−0.0840615	+	0.996461i	$0.526789\pi$
$24$	0	0
$25$	−3.07464	−0.614929
$26$	−7.54032	−1.47878
$27$	0	0
$28$	−0.553841	−0.104666
$29$	−8.32766	−1.54641	−0.773204	−	0.634158i	$-0.781347\pi$
$29$	−8.32766	−1.54641	−0.773204	+	0.634158i	$0.781347\pi$
$30$	0	0
$31$	4.54221	0.815804	0.407902	−	0.913026i	$-0.366260\pi$
$31$	4.54221	0.815804	0.407902	+	0.913026i	$0.366260\pi$
$32$	−1.53195	−0.270814
$33$	0	0
$34$	9.12341	1.56465
$35$	2.81694	0.476150
$36$	0	0
$37$	8.85665	1.45602	0.728012	−	0.685565i	$-0.240445\pi$
$37$	8.85665	1.45602	0.728012	+	0.685565i	$0.240445\pi$
$38$	4.66828	0.757295
$39$	0	0
$40$	4.14466	0.655328
$41$	4.04638	0.631939	0.315969	−	0.948769i	$-0.397670\pi$
$41$	4.04638	0.631939	0.315969	+	0.948769i	$0.397670\pi$
$42$	0	0
$43$	0.832544	0.126962	0.0634809	−	0.997983i	$-0.479780\pi$
$43$	0.832544	0.126962	0.0634809	+	0.997983i	$0.479780\pi$
$44$	0.272811	0.0411279
$45$	0	0
$46$	1.05965	0.156236
$47$	−4.66753	−0.680829	−0.340414	−	0.940276i	$-0.610567\pi$
$47$	−4.66753	−0.680829	−0.340414	+	0.940276i	$0.610567\pi$
$48$	0	0
$49$	−2.87860	−0.411228
$50$	4.04077	0.571452
$51$	0	0
$52$	−1.56524	−0.217060
$53$	9.49286	1.30394	0.651972	−	0.758243i	$-0.273942\pi$
$53$	9.49286	1.30394	0.651972	+	0.758243i	$0.273942\pi$
$54$	0	0
$55$	−1.38757	−0.187100
$56$	6.06395	0.810330
$57$	0	0
$58$	10.9444	1.43707
$59$	12.6362	1.64510	0.822549	−	0.568694i	$-0.192551\pi$
$59$	12.6362	1.64510	0.822549	+	0.568694i	$0.192551\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−5.96948	−0.758125
$63$	0	0
$64$	8.77324	1.09665
$65$	7.96114	0.987458
$66$	0	0
$67$	6.13660	0.749705	0.374852	−	0.927085i	$-0.377693\pi$
$67$	6.13660	0.749705	0.374852	+	0.927085i	$0.377693\pi$
$68$	1.89387	0.229665
$69$	0	0
$70$	−3.70210	−0.442486
$71$	−12.3352	−1.46392	−0.731958	−	0.681350i	$-0.761393\pi$
$71$	−12.3352	−1.46392	−0.731958	+	0.681350i	$0.761393\pi$
$72$	0	0
$73$	8.32305	0.974139	0.487070	−	0.873363i	$-0.338066\pi$
$73$	8.32305	0.974139	0.487070	+	0.873363i	$0.338066\pi$
$74$	−11.6396	−1.35308
$75$	0	0
$76$	0.969057	0.111158
$77$	−2.03012	−0.231354
$78$	0	0
$79$	3.78450	0.425789	0.212895	−	0.977075i	$-0.431711\pi$
$79$	3.78450	0.425789	0.212895	+	0.977075i	$0.431711\pi$
$80$	−4.68993	−0.524350
$81$	0	0
$82$	−5.31786	−0.587259
$83$	12.6260	1.38588	0.692939	−	0.720996i	$-0.256315\pi$
$83$	12.6260	1.38588	0.692939	+	0.720996i	$0.256315\pi$
$84$	0	0
$85$	−9.63258	−1.04480
$86$	−1.09415	−0.117985
$87$	0	0
$88$	−2.98699	−0.318414
$89$	14.8592	1.57507	0.787534	−	0.616271i	$-0.211357\pi$
$89$	14.8592	1.57507	0.787534	+	0.616271i	$0.211357\pi$
$90$	0	0
$91$	11.6478	1.22102
$92$	0.219965	0.0229329
$93$	0	0
$94$	6.13418	0.632693
$95$	−4.92881	−0.505686
$96$	0	0
$97$	7.09938	0.720833	0.360417	−	0.932791i	$-0.382635\pi$
$97$	7.09938	0.720833	0.360417	+	0.932791i	$0.382635\pi$
$98$	3.78312	0.382153
$99$	0	0
$100$	0.838797	0.0838797
$101$	−14.2585	−1.41877	−0.709387	−	0.704819i	$-0.751028\pi$
$101$	−14.2585	−1.41877	−0.709387	+	0.704819i	$0.751028\pi$
$102$	0	0
$103$	1.69876	0.167384	0.0836920	−	0.996492i	$-0.473329\pi$
$103$	1.69876	0.167384	0.0836920	+	0.996492i	$0.473329\pi$
$104$	17.1377	1.68049
$105$	0	0
$106$	−12.4758	−1.21175
$107$	14.6034	1.41176	0.705881	−	0.708331i	$-0.250551\pi$
$107$	14.6034	1.41176	0.705881	+	0.708331i	$0.250551\pi$
$108$	0	0
$109$	7.22627	0.692151	0.346076	−	0.938207i	$-0.387514\pi$
$109$	7.22627	0.692151	0.346076	+	0.938207i	$0.387514\pi$
$110$	1.82358	0.173872
$111$	0	0
$112$	−6.86172	−0.648372
$113$	−17.4970	−1.64598	−0.822990	−	0.568056i	$-0.807696\pi$
$113$	−17.4970	−1.64598	−0.822990	+	0.568056i	$0.807696\pi$
$114$	0	0
$115$	−1.11878	−0.104327
$116$	2.27188	0.210939
$117$	0	0
$118$	−16.6069	−1.52879
$119$	−14.0932	−1.29192
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.31423	0.118984
$123$	0	0
$124$	−1.23917	−0.111280
$125$	−11.2042	−1.00213
$126$	0	0
$127$	2.20928	0.196042	0.0980209	−	0.995184i	$-0.468749\pi$
$127$	2.20928	0.196042	0.0980209	+	0.995184i	$0.468749\pi$
$128$	−8.46610	−0.748305
$129$	0	0
$130$	−10.4627	−0.917643
$131$	2.07784	0.181541	0.0907707	−	0.995872i	$-0.471067\pi$
$131$	2.07784	0.181541	0.0907707	+	0.995872i	$0.471067\pi$
$132$	0	0
$133$	−7.21123	−0.625293
$134$	−8.06487	−0.696699
$135$	0	0
$136$	−20.7358	−1.77808
$137$	−10.7795	−0.920951	−0.460476	−	0.887672i	$-0.652321\pi$
$137$	−10.7795	−0.920951	−0.460476	+	0.887672i	$0.652321\pi$
$138$	0	0
$139$	−11.4790	−0.973636	−0.486818	−	0.873504i	$-0.661842\pi$
$139$	−11.4790	−0.973636	−0.486818	+	0.873504i	$0.661842\pi$
$140$	−0.768494	−0.0649496
$141$	0	0
$142$	16.2112	1.36041
$143$	−5.73746	−0.479791
$144$	0	0
$145$	−11.5552	−0.959609
$146$	−10.9384	−0.905266
$147$	0	0
$148$	−2.41619	−0.198610
$149$	11.4894	0.941246	0.470623	−	0.882334i	$-0.344029\pi$
$149$	11.4894	0.941246	0.470623	+	0.882334i	$0.344029\pi$
$150$	0	0
$151$	−2.05463	−0.167204	−0.0836018	−	0.996499i	$-0.526642\pi$
$151$	−2.05463	−0.167204	−0.0836018	+	0.996499i	$0.526642\pi$
$152$	−10.6101	−0.860594
$153$	0	0
$154$	2.66804	0.214997
$155$	6.30264	0.506240
$156$	0	0
$157$	24.4022	1.94751	0.973755	−	0.227600i	$-0.0730879\pi$
$157$	24.4022	1.94751	0.973755	+	0.227600i	$0.0730879\pi$
$158$	−4.97368	−0.395685
$159$	0	0
$160$	−2.12570	−0.168051
$161$	−1.63687	−0.129003
$162$	0	0
$163$	−9.62670	−0.754021	−0.377011	−	0.926209i	$-0.623048\pi$
$163$	−9.62670	−0.754021	−0.377011	+	0.926209i	$0.623048\pi$
$164$	−1.10390	−0.0862000
$165$	0	0
$166$	−16.5933	−1.28789
$167$	16.8772	1.30600	0.652999	−	0.757359i	$-0.273511\pi$
$167$	16.8772	1.30600	0.652999	+	0.757359i	$0.273511\pi$
$168$	0	0
$169$	19.9185	1.53219
$170$	12.6594	0.970930
$171$	0	0
$172$	−0.227127	−0.0173183
$173$	5.79912	0.440899	0.220450	−	0.975398i	$-0.429248\pi$
$173$	5.79912	0.440899	0.220450	+	0.975398i	$0.429248\pi$
$174$	0	0
$175$	−6.24191	−0.471844
$176$	3.37995	0.254773
$177$	0	0
$178$	−19.5283	−1.46371
$179$	−4.27998	−0.319900	−0.159950	−	0.987125i	$-0.551133\pi$
$179$	−4.27998	−0.319900	−0.159950	+	0.987125i	$0.551133\pi$
$180$	0	0
$181$	−20.1052	−1.49441	−0.747205	−	0.664594i	$-0.768605\pi$
$181$	−20.1052	−1.49441	−0.747205	+	0.664594i	$0.768605\pi$
$182$	−15.3078	−1.13469
$183$	0	0
$184$	−2.40837	−0.177548
$185$	12.2892	0.903523
$186$	0	0
$187$	6.94204	0.507652
$188$	1.27335	0.0928689
$189$	0	0
$190$	6.47757	0.469933
$191$	−9.14553	−0.661748	−0.330874	−	0.943675i	$-0.607343\pi$
$191$	−9.14553	−0.661748	−0.330874	+	0.943675i	$0.607343\pi$
$192$	0	0
$193$	19.5388	1.40644	0.703218	−	0.710974i	$-0.251746\pi$
$193$	19.5388	1.40644	0.703218	+	0.710974i	$0.251746\pi$
$194$	−9.33019	−0.669869
$195$	0	0
$196$	0.785314	0.0560938
$197$	21.7664	1.55079	0.775395	−	0.631476i	$-0.217551\pi$
$197$	21.7664	1.55079	0.775395	+	0.631476i	$0.217551\pi$
$198$	0	0
$199$	21.7232	1.53992	0.769958	−	0.638094i	$-0.220277\pi$
$199$	21.7232	1.53992	0.769958	+	0.638094i	$0.220277\pi$
$200$	−9.18392	−0.649401
$201$	0	0
$202$	18.7389	1.31846
$203$	−16.9062	−1.18658
$204$	0	0
$205$	5.61465	0.392144
$206$	−2.23256	−0.155550
$207$	0	0
$208$	−19.3923	−1.34462
$209$	3.55211	0.245705
$210$	0	0
$211$	3.03388	0.208861	0.104430	−	0.994532i	$-0.466698\pi$
$211$	3.03388	0.208861	0.104430	+	0.994532i	$0.466698\pi$
$212$	−2.58976	−0.177865
$213$	0	0
$214$	−19.1921	−1.31195
$215$	1.15522	0.0787850
$216$	0	0
$217$	9.22124	0.625979
$218$	−9.49695	−0.643215
$219$	0	0
$220$	0.378545	0.0255215
$221$	−39.8297	−2.67924
$222$	0	0
$223$	4.76902	0.319357	0.159678	−	0.987169i	$-0.448954\pi$
$223$	4.76902	0.319357	0.159678	+	0.987169i	$0.448954\pi$
$224$	−3.11006	−0.207800
$225$	0	0
$226$	22.9950	1.52961
$227$	26.6405	1.76819	0.884095	−	0.467307i	$-0.154776\pi$
$227$	26.6405	1.76819	0.884095	+	0.467307i	$0.154776\pi$
$228$	0	0
$229$	11.6882	0.772377	0.386188	−	0.922420i	$-0.373792\pi$
$229$	11.6882	0.772377	0.386188	+	0.922420i	$0.373792\pi$
$230$	1.47034	0.0969510
$231$	0	0
$232$	−24.8746	−1.63310
$233$	3.48827	0.228524	0.114262	−	0.993451i	$-0.463550\pi$
$233$	3.48827	0.228524	0.114262	+	0.993451i	$0.463550\pi$
$234$	0	0
$235$	−6.47653	−0.422482
$236$	−3.44731	−0.224401
$237$	0	0
$238$	18.5216	1.20058
$239$	−18.1468	−1.17382	−0.586911	−	0.809652i	$-0.699656\pi$
$239$	−18.1468	−1.17382	−0.586911	+	0.809652i	$0.699656\pi$
$240$	0	0
$241$	−4.01737	−0.258782	−0.129391	−	0.991594i	$-0.541302\pi$
$241$	−4.01737	−0.258782	−0.129391	+	0.991594i	$0.541302\pi$
$242$	−1.31423	−0.0844816
$243$	0	0
$244$	0.272811	0.0174650
$245$	−3.99426	−0.255184
$246$	0	0
$247$	−20.3801	−1.29676
$248$	13.5675	0.861538
$249$	0	0
$250$	14.7248	0.931277
$251$	7.05879	0.445547	0.222774	−	0.974870i	$-0.428489\pi$
$251$	7.05879	0.445547	0.222774	+	0.974870i	$0.428489\pi$
$252$	0	0
$253$	0.806289	0.0506910
$254$	−2.90349	−0.182181
$255$	0	0
$256$	−6.42011	−0.401257
$257$	6.03995	0.376762	0.188381	−	0.982096i	$-0.439676\pi$
$257$	6.03995	0.376762	0.188381	+	0.982096i	$0.439676\pi$
$258$	0	0
$259$	17.9801	1.11723
$260$	−2.17189	−0.134695
$261$	0	0
$262$	−2.73075	−0.168706
$263$	−11.0908	−0.683890	−0.341945	−	0.939720i	$-0.611086\pi$
$263$	−11.0908	−0.683890	−0.341945	+	0.939720i	$0.611086\pi$
$264$	0	0
$265$	13.1720	0.809151
$266$	9.47718	0.581083
$267$	0	0
$268$	−1.67413	−0.102264
$269$	−5.22740	−0.318720	−0.159360	−	0.987221i	$-0.550943\pi$
$269$	−5.22740	−0.318720	−0.159360	+	0.987221i	$0.550943\pi$
$270$	0	0
$271$	−13.4323	−0.815956	−0.407978	−	0.912992i	$-0.633766\pi$
$271$	−13.4323	−0.815956	−0.407978	+	0.912992i	$0.633766\pi$
$272$	23.4638	1.42270
$273$	0	0
$274$	14.1666	0.855838
$275$	3.07464	0.185408
$276$	0	0
$277$	31.7919	1.91019	0.955096	−	0.296297i	$-0.0957518\pi$
$277$	31.7919	1.91019	0.955096	+	0.296297i	$0.0957518\pi$
$278$	15.0860	0.904797
$279$	0	0
$280$	8.41417	0.502843
$281$	−22.8024	−1.36028	−0.680140	−	0.733083i	$-0.738081\pi$
$281$	−22.8024	−1.36028	−0.680140	+	0.733083i	$0.738081\pi$
$282$	0	0
$283$	−19.0479	−1.13228	−0.566140	−	0.824309i	$-0.691564\pi$
$283$	−19.0479	−1.13228	−0.566140	+	0.824309i	$0.691564\pi$
$284$	3.36517	0.199686
$285$	0	0
$286$	7.54032	0.445868
$287$	8.21466	0.484896
$288$	0	0
$289$	31.1919	1.83482
$290$	15.1862	0.891763
$291$	0	0
$292$	−2.27062	−0.132878
$293$	4.36266	0.254869	0.127435	−	0.991847i	$-0.459326\pi$
$293$	4.36266	0.254869	0.127435	+	0.991847i	$0.459326\pi$
$294$	0	0
$295$	17.5337	1.02085
$296$	26.4547	1.53765
$297$	0	0
$298$	−15.0996	−0.874698
$299$	−4.62605	−0.267532
$300$	0	0
$301$	1.69017	0.0974197
$302$	2.70025	0.155382
$303$	0	0
$304$	12.0060	0.688590
$305$	−1.38757	−0.0794522
$306$	0	0
$307$	−11.4166	−0.651580	−0.325790	−	0.945442i	$-0.605630\pi$
$307$	−11.4166	−0.651580	−0.325790	+	0.945442i	$0.605630\pi$
$308$	0.553841	0.0315580
$309$	0	0
$310$	−8.28309	−0.470448
$311$	30.5731	1.73364	0.866820	−	0.498621i	$-0.166160\pi$
$311$	30.5731	1.73364	0.866820	+	0.498621i	$0.166160\pi$
$312$	0	0
$313$	9.04130	0.511044	0.255522	−	0.966803i	$-0.417753\pi$
$313$	9.04130	0.511044	0.255522	+	0.966803i	$0.417753\pi$
$314$	−32.0700	−1.80982
$315$	0	0
$316$	−1.03245	−0.0580801
$317$	21.5081	1.20802	0.604009	−	0.796978i	$-0.293569\pi$
$317$	21.5081	1.20802	0.604009	+	0.796978i	$0.293569\pi$
$318$	0	0
$319$	8.32766	0.466259
$320$	12.1735	0.680519
$321$	0	0
$322$	2.15121	0.119882
$323$	24.6589	1.37206
$324$	0	0
$325$	−17.6406	−0.978527
$326$	12.6517	0.700710
$327$	0	0
$328$	12.0865	0.667365
$329$	−9.47566	−0.522410
$330$	0	0
$331$	−23.2813	−1.27966	−0.639829	−	0.768518i	$-0.720995\pi$
$331$	−23.2813	−1.27966	−0.639829	+	0.768518i	$0.720995\pi$
$332$	−3.44450	−0.189042
$333$	0	0
$334$	−22.1804	−1.21366
$335$	8.51497	0.465223
$336$	0	0
$337$	−13.9809	−0.761586	−0.380793	−	0.924660i	$-0.624349\pi$
$337$	−13.9809	−0.761586	−0.380793	+	0.924660i	$0.624349\pi$
$338$	−26.1774	−1.42386
$339$	0	0
$340$	2.62788	0.142517
$341$	−4.54221	−0.245974
$342$	0	0
$343$	−20.0548	−1.08286
$344$	2.48680	0.134079
$345$	0	0
$346$	−7.62135	−0.409726
$347$	21.9732	1.17959	0.589793	−	0.807555i	$-0.299209\pi$
$347$	21.9732	1.17959	0.589793	+	0.807555i	$0.299209\pi$
$348$	0	0
$349$	12.0629	0.645710	0.322855	−	0.946448i	$-0.395357\pi$
$349$	12.0629	0.645710	0.322855	+	0.946448i	$0.395357\pi$
$350$	8.20327	0.438483
$351$	0	0
$352$	1.53195	0.0816535
$353$	1.86971	0.0995146	0.0497573	−	0.998761i	$-0.484155\pi$
$353$	1.86971	0.0995146	0.0497573	+	0.998761i	$0.484155\pi$
$354$	0	0
$355$	−17.1159	−0.908420
$356$	−4.05375	−0.214848
$357$	0	0
$358$	5.62485	0.297283
$359$	−32.2819	−1.70377	−0.851886	−	0.523727i	$-0.824541\pi$
$359$	−32.2819	−1.70377	−0.851886	+	0.523727i	$0.824541\pi$
$360$	0	0
$361$	−6.38249	−0.335921
$362$	26.4228	1.38875
$363$	0	0
$364$	−3.17764	−0.166554
$365$	11.5488	0.604494
$366$	0	0
$367$	15.1555	0.791112	0.395556	−	0.918442i	$-0.370552\pi$
$367$	15.1555	0.791112	0.395556	+	0.918442i	$0.370552\pi$
$368$	2.72522	0.142062
$369$	0	0
$370$	−16.1508	−0.839642
$371$	19.2717	1.00054
$372$	0	0
$373$	−4.47779	−0.231851	−0.115926	−	0.993258i	$-0.536983\pi$
$373$	−4.47779	−0.231851	−0.115926	+	0.993258i	$0.536983\pi$
$374$	−9.12341	−0.471760
$375$	0	0
$376$	−13.9418	−0.718996
$377$	−47.7796	−2.46078
$378$	0	0
$379$	12.5805	0.646215	0.323107	−	0.946362i	$-0.395273\pi$
$379$	12.5805	0.646215	0.323107	+	0.946362i	$0.395273\pi$
$380$	1.34464	0.0689784
$381$	0	0
$382$	12.0193	0.614961
$383$	4.18886	0.214041	0.107020	−	0.994257i	$-0.465869\pi$
$383$	4.18886	0.214041	0.107020	+	0.994257i	$0.465869\pi$
$384$	0	0
$385$	−2.81694	−0.143565
$386$	−25.6784	−1.30700
$387$	0	0
$388$	−1.93679	−0.0983257
$389$	21.2015	1.07496	0.537480	−	0.843277i	$-0.319377\pi$
$389$	21.2015	1.07496	0.537480	+	0.843277i	$0.319377\pi$
$390$	0	0
$391$	5.59729	0.283067
$392$	−8.59833	−0.434281
$393$	0	0
$394$	−28.6059	−1.44115
$395$	5.25127	0.264220
$396$	0	0
$397$	6.72041	0.337288	0.168644	−	0.985677i	$-0.446061\pi$
$397$	6.72041	0.337288	0.168644	+	0.985677i	$0.446061\pi$
$398$	−28.5492	−1.43104
$399$	0	0
$400$	10.3921	0.519607
$401$	−2.60439	−0.130057	−0.0650286	−	0.997883i	$-0.520714\pi$
$401$	−2.60439	−0.130057	−0.0650286	+	0.997883i	$0.520714\pi$
$402$	0	0
$403$	26.0607	1.29818
$404$	3.88988	0.193529
$405$	0	0
$406$	22.2185	1.10269
$407$	−8.85665	−0.439008
$408$	0	0
$409$	2.08400	0.103047	0.0515237	−	0.998672i	$-0.483592\pi$
$409$	2.08400	0.103047	0.0515237	+	0.998672i	$0.483592\pi$
$410$	−7.37891	−0.364418
$411$	0	0
$412$	−0.463441	−0.0228321
$413$	25.6531	1.26231
$414$	0	0
$415$	17.5194	0.859995
$416$	−8.78953	−0.430942
$417$	0	0
$418$	−4.66828	−0.228333
$419$	24.0304	1.17396	0.586982	−	0.809600i	$-0.300316\pi$
$419$	24.0304	1.17396	0.586982	+	0.809600i	$0.300316\pi$
$420$	0	0
$421$	1.01272	0.0493568	0.0246784	−	0.999695i	$-0.492144\pi$
$421$	1.01272	0.0493568	0.0246784	+	0.999695i	$0.492144\pi$
$422$	−3.98720	−0.194094
$423$	0	0
$424$	28.3550	1.37704
$425$	21.3443	1.03535
$426$	0	0
$427$	−2.03012	−0.0982446
$428$	−3.98397	−0.192572
$429$	0	0
$430$	−1.51821	−0.0732147
$431$	16.4442	0.792088	0.396044	−	0.918232i	$-0.370383\pi$
$431$	16.4442	0.792088	0.396044	+	0.918232i	$0.370383\pi$
$432$	0	0
$433$	−22.5857	−1.08540	−0.542700	−	0.839926i	$-0.682598\pi$
$433$	−22.5857	−1.08540	−0.542700	+	0.839926i	$0.682598\pi$
$434$	−12.1188	−0.581721
$435$	0	0
$436$	−1.97141	−0.0944134
$437$	2.86403	0.137005
$438$	0	0
$439$	−34.8689	−1.66420	−0.832101	−	0.554625i	$-0.812862\pi$
$439$	−34.8689	−1.66420	−0.832101	+	0.554625i	$0.812862\pi$
$440$	−4.14466	−0.197589
$441$	0	0
$442$	52.3452	2.48981
$443$	31.1718	1.48102	0.740508	−	0.672048i	$-0.234585\pi$
$443$	31.1718	1.48102	0.740508	+	0.672048i	$0.234585\pi$
$444$	0	0
$445$	20.6182	0.977395
$446$	−6.26756	−0.296778
$447$	0	0
$448$	17.8108	0.841479
$449$	−25.8963	−1.22212	−0.611060	−	0.791584i	$-0.709257\pi$
$449$	−25.8963	−1.22212	−0.611060	+	0.791584i	$0.709257\pi$
$450$	0	0
$451$	−4.04638	−0.190537
$452$	4.77338	0.224521
$453$	0	0
$454$	−35.0116	−1.64317
$455$	16.1621	0.757691
$456$	0	0
$457$	6.73065	0.314847	0.157423	−	0.987531i	$-0.449681\pi$
$457$	6.73065	0.314847	0.157423	+	0.987531i	$0.449681\pi$
$458$	−15.3609	−0.717768
$459$	0	0
$460$	0.305217	0.0142308
$461$	−38.5957	−1.79758	−0.898791	−	0.438378i	$-0.855553\pi$
$461$	−38.5957	−1.79758	−0.898791	+	0.438378i	$0.855553\pi$
$462$	0	0
$463$	28.2081	1.31094	0.655472	−	0.755220i	$-0.272470\pi$
$463$	28.2081	1.31094	0.655472	+	0.755220i	$0.272470\pi$
$464$	28.1471	1.30670
$465$	0	0
$466$	−4.58437	−0.212367
$467$	16.3664	0.757346	0.378673	−	0.925531i	$-0.376381\pi$
$467$	16.3664	0.757346	0.378673	+	0.925531i	$0.376381\pi$
$468$	0	0
$469$	12.4581	0.575260
$470$	8.51162	0.392612
$471$	0	0
$472$	37.7443	1.73732
$473$	−0.832544	−0.0382804
$474$	0	0
$475$	10.9215	0.501112
$476$	3.84478	0.176225
$477$	0	0
$478$	23.8490	1.09083
$479$	16.7773	0.766572	0.383286	−	0.923630i	$-0.374792\pi$
$479$	16.7773	0.766572	0.383286	+	0.923630i	$0.374792\pi$
$480$	0	0
$481$	50.8147	2.31695
$482$	5.27974	0.240485
$483$	0	0
$484$	−0.272811	−0.0124005
$485$	9.85091	0.447307
$486$	0	0
$487$	−28.7178	−1.30133	−0.650665	−	0.759365i	$-0.725510\pi$
$487$	−28.7178	−1.30133	−0.650665	+	0.759365i	$0.725510\pi$
$488$	−2.98699	−0.135215
$489$	0	0
$490$	5.24936	0.237142
$491$	−32.6549	−1.47369	−0.736847	−	0.676060i	$-0.763686\pi$
$491$	−32.6549	−1.47369	−0.736847	+	0.676060i	$0.763686\pi$
$492$	0	0
$493$	57.8109	2.60367
$494$	26.7841	1.20507
$495$	0	0
$496$	−15.3524	−0.689345
$497$	−25.0419	−1.12328
$498$	0	0
$499$	−7.11998	−0.318734	−0.159367	−	0.987219i	$-0.550945\pi$
$499$	−7.11998	−0.318734	−0.159367	+	0.987219i	$0.550945\pi$
$500$	3.05662	0.136696
$501$	0	0
$502$	−9.27685	−0.414046
$503$	12.3496	0.550641	0.275321	−	0.961352i	$-0.411216\pi$
$503$	12.3496	0.550641	0.275321	+	0.961352i	$0.411216\pi$
$504$	0	0
$505$	−19.7847	−0.880408
$506$	−1.05965	−0.0471070
$507$	0	0
$508$	−0.602716	−0.0267412
$509$	11.4262	0.506459	0.253229	−	0.967406i	$-0.418507\pi$
$509$	11.4262	0.506459	0.253229	+	0.967406i	$0.418507\pi$
$510$	0	0
$511$	16.8968	0.747472
$512$	25.3697	1.12119
$513$	0	0
$514$	−7.93786	−0.350124
$515$	2.35715	0.103869
$516$	0	0
$517$	4.66753	0.205278
$518$	−23.6299	−1.03824
$519$	0	0
$520$	23.7798	1.04281
$521$	30.2067	1.32338	0.661690	−	0.749777i	$-0.269839\pi$
$521$	30.2067	1.32338	0.661690	+	0.749777i	$0.269839\pi$
$522$	0	0
$523$	−2.64676	−0.115735	−0.0578674	−	0.998324i	$-0.518430\pi$
$523$	−2.64676	−0.115735	−0.0578674	+	0.998324i	$0.518430\pi$
$524$	−0.566857	−0.0247633
$525$	0	0
$526$	14.5758	0.635537
$527$	−31.5322	−1.37356
$528$	0	0
$529$	−22.3499	−0.971735
$530$	−17.3110	−0.751943
$531$	0	0
$532$	1.96731	0.0852935
$533$	23.2160	1.00559
$534$	0	0
$535$	20.2632	0.876056
$536$	18.3299	0.791733
$537$	0	0
$538$	6.86998	0.296186
$539$	2.87860	0.123990
$540$	0	0
$541$	16.9705	0.729617	0.364809	−	0.931082i	$-0.381134\pi$
$541$	16.9705	0.729617	0.364809	+	0.931082i	$0.381134\pi$
$542$	17.6531	0.758266
$543$	0	0
$544$	10.6349	0.455967
$545$	10.0270	0.429509
$546$	0	0
$547$	−12.3888	−0.529707	−0.264853	−	0.964289i	$-0.585324\pi$
$547$	−12.3888	−0.529707	−0.264853	+	0.964289i	$0.585324\pi$
$548$	2.94076	0.125623
$549$	0	0
$550$	−4.04077	−0.172299
$551$	29.5808	1.26018
$552$	0	0
$553$	7.68300	0.326714
$554$	−41.7818	−1.77514
$555$	0	0
$556$	3.13160	0.132809
$557$	28.0597	1.18893	0.594463	−	0.804123i	$-0.297365\pi$
$557$	28.0597	1.18893	0.594463	+	0.804123i	$0.297365\pi$
$558$	0	0
$559$	4.77669	0.202033
$560$	−9.52113	−0.402341
$561$	0	0
$562$	29.9675	1.26410
$563$	−34.0521	−1.43513	−0.717563	−	0.696493i	$-0.754743\pi$
$563$	−34.0521	−1.43513	−0.717563	+	0.696493i	$0.754743\pi$
$564$	0	0
$565$	−24.2784	−1.02140
$566$	25.0332	1.05223
$567$	0	0
$568$	−36.8450	−1.54598
$569$	4.18354	0.175383	0.0876916	−	0.996148i	$-0.472051\pi$
$569$	4.18354	0.175383	0.0876916	+	0.996148i	$0.472051\pi$
$570$	0	0
$571$	−19.9054	−0.833013	−0.416507	−	0.909133i	$-0.636746\pi$
$571$	−19.9054	−0.833013	−0.416507	+	0.909133i	$0.636746\pi$
$572$	1.56524	0.0654462
$573$	0	0
$574$	−10.7959	−0.450613
$575$	2.47905	0.103384
$576$	0	0
$577$	−26.0834	−1.08587	−0.542933	−	0.839776i	$-0.682686\pi$
$577$	−26.0834	−1.08587	−0.542933	+	0.839776i	$0.682686\pi$
$578$	−40.9932	−1.70509
$579$	0	0
$580$	3.15240	0.130896
$581$	25.6322	1.06340
$582$	0	0
$583$	−9.49286	−0.393154
$584$	24.8608	1.02875
$585$	0	0
$586$	−5.73352	−0.236849
$587$	16.9706	0.700451	0.350225	−	0.936665i	$-0.386105\pi$
$587$	16.9706	0.700451	0.350225	+	0.936665i	$0.386105\pi$
$588$	0	0
$589$	−16.1344	−0.664808
$590$	−23.0432	−0.948675
$591$	0	0
$592$	−29.9350	−1.23032
$593$	28.1953	1.15784	0.578921	−	0.815383i	$-0.303474\pi$
$593$	28.1953	1.15784	0.578921	+	0.815383i	$0.303474\pi$
$594$	0	0
$595$	−19.5553	−0.801691
$596$	−3.13443	−0.128391
$597$	0	0
$598$	6.07968	0.248617
$599$	−16.6875	−0.681834	−0.340917	−	0.940093i	$-0.610737\pi$
$599$	−16.6875	−0.681834	−0.340917	+	0.940093i	$0.610737\pi$
$600$	0	0
$601$	−34.9576	−1.42595	−0.712975	−	0.701189i	$-0.752653\pi$
$601$	−34.9576	−1.42595	−0.712975	+	0.701189i	$0.752653\pi$
$602$	−2.22126	−0.0905319
$603$	0	0
$604$	0.560527	0.0228075
$605$	1.38757	0.0564128
$606$	0	0
$607$	16.5940	0.673528	0.336764	−	0.941589i	$-0.390668\pi$
$607$	16.5940	0.673528	0.336764	+	0.941589i	$0.390668\pi$
$608$	5.44168	0.220689
$609$	0	0
$610$	1.82358	0.0738347
$611$	−26.7798	−1.08339
$612$	0	0
$613$	8.29426	0.335002	0.167501	−	0.985872i	$-0.446430\pi$
$613$	8.29426	0.335002	0.167501	+	0.985872i	$0.446430\pi$
$614$	15.0040	0.605512
$615$	0	0
$616$	−6.06395	−0.244324
$617$	−39.9235	−1.60726	−0.803630	−	0.595129i	$-0.797101\pi$
$617$	−39.9235	−1.60726	−0.803630	+	0.595129i	$0.797101\pi$
$618$	0	0
$619$	6.86944	0.276106	0.138053	−	0.990425i	$-0.455916\pi$
$619$	6.86944	0.276106	0.138053	+	0.990425i	$0.455916\pi$
$620$	−1.71943	−0.0690540
$621$	0	0
$622$	−40.1799	−1.61107
$623$	30.1660	1.20857
$624$	0	0
$625$	−0.173355	−0.00693420
$626$	−11.8823	−0.474912
$627$	0	0
$628$	−6.65720	−0.265651
$629$	−61.4832	−2.45150
$630$	0	0
$631$	−37.2753	−1.48390	−0.741952	−	0.670453i	$-0.766100\pi$
$631$	−37.2753	−1.48390	−0.741952	+	0.670453i	$0.766100\pi$
$632$	11.3042	0.449659
$633$	0	0
$634$	−28.2666	−1.12261
$635$	3.06553	0.121652
$636$	0	0
$637$	−16.5158	−0.654381
$638$	−10.9444	−0.433294
$639$	0	0
$640$	−11.7473	−0.464354
$641$	28.5638	1.12820	0.564101	−	0.825706i	$-0.309223\pi$
$641$	28.5638	1.12820	0.564101	+	0.825706i	$0.309223\pi$
$642$	0	0
$643$	13.8862	0.547618	0.273809	−	0.961784i	$-0.411716\pi$
$643$	13.8862	0.547618	0.273809	+	0.961784i	$0.411716\pi$
$644$	0.446556	0.0175968
$645$	0	0
$646$	−32.4074	−1.27505
$647$	2.36581	0.0930097	0.0465048	−	0.998918i	$-0.485192\pi$
$647$	2.36581	0.0930097	0.0465048	+	0.998918i	$0.485192\pi$
$648$	0	0
$649$	−12.6362	−0.496016
$650$	23.1838	0.909343
$651$	0	0
$652$	2.62627	0.102853
$653$	44.6721	1.74815	0.874077	−	0.485788i	$-0.161467\pi$
$653$	44.6721	1.74815	0.874077	+	0.485788i	$0.161467\pi$
$654$	0	0
$655$	2.88315	0.112654
$656$	−13.6766	−0.533980
$657$	0	0
$658$	12.4532	0.485474
$659$	3.04987	0.118806	0.0594031	−	0.998234i	$-0.481080\pi$
$659$	3.04987	0.118806	0.0594031	+	0.998234i	$0.481080\pi$
$660$	0	0
$661$	15.0839	0.586694	0.293347	−	0.956006i	$-0.405231\pi$
$661$	15.0839	0.586694	0.293347	+	0.956006i	$0.405231\pi$
$662$	30.5969	1.18918
$663$	0	0
$664$	37.7135	1.46357
$665$	−10.0061	−0.388020
$666$	0	0
$667$	6.71450	0.259986
$668$	−4.60429	−0.178145
$669$	0	0
$670$	−11.1906	−0.432330
$671$	1.00000	0.0386046
$672$	0	0
$673$	0.744519	0.0286991	0.0143495	−	0.999897i	$-0.495432\pi$
$673$	0.744519	0.0286991	0.0143495	+	0.999897i	$0.495432\pi$
$674$	18.3740	0.707740
$675$	0	0
$676$	−5.43399	−0.208999
$677$	21.5042	0.826475	0.413237	−	0.910623i	$-0.364398\pi$
$677$	21.5042	0.826475	0.413237	+	0.910623i	$0.364398\pi$
$678$	0	0
$679$	14.4126	0.553106
$680$	−28.7724	−1.10337
$681$	0	0
$682$	5.96948	0.228583
$683$	−32.8678	−1.25765	−0.628825	−	0.777547i	$-0.716464\pi$
$683$	−32.8678	−1.25765	−0.628825	+	0.777547i	$0.716464\pi$
$684$	0	0
$685$	−14.9573	−0.571488
$686$	26.3565	1.00630
$687$	0	0
$688$	−2.81396	−0.107281
$689$	54.4649	2.07495
$690$	0	0
$691$	40.7898	1.55172	0.775859	−	0.630906i	$-0.217317\pi$
$691$	40.7898	1.55172	0.775859	+	0.630906i	$0.217317\pi$
$692$	−1.58207	−0.0601411
$693$	0	0
$694$	−28.8778	−1.09619
$695$	−15.9279	−0.604181
$696$	0	0
$697$	−28.0901	−1.06399
$698$	−15.8533	−0.600057
$699$	0	0
$700$	1.70286	0.0643622
$701$	−16.5302	−0.624339	−0.312169	−	0.950026i	$-0.601056\pi$
$701$	−16.5302	−0.624339	−0.312169	+	0.950026i	$0.601056\pi$
$702$	0	0
$703$	−31.4598	−1.18653
$704$	−8.77324	−0.330654
$705$	0	0
$706$	−2.45722	−0.0924787
$707$	−28.9465	−1.08865
$708$	0	0
$709$	9.67562	0.363375	0.181688	−	0.983356i	$-0.441844\pi$
$709$	9.67562	0.363375	0.181688	+	0.983356i	$0.441844\pi$
$710$	22.4942	0.844192
$711$	0	0
$712$	44.3841	1.66337
$713$	−3.66233	−0.137155
$714$	0	0
$715$	−7.96114	−0.297730
$716$	1.16763	0.0436362
$717$	0	0
$718$	42.4257	1.58331
$719$	−0.412582	−0.0153867	−0.00769336	−	0.999970i	$-0.502449\pi$
$719$	−0.412582	−0.0153867	−0.00769336	+	0.999970i	$0.502449\pi$
$720$	0	0
$721$	3.44870	0.128436
$722$	8.38803	0.312170
$723$	0	0
$724$	5.48493	0.203846
$725$	25.6046	0.950930
$726$	0	0
$727$	−3.71500	−0.137782	−0.0688908	−	0.997624i	$-0.521946\pi$
$727$	−3.71500	−0.137782	−0.0688908	+	0.997624i	$0.521946\pi$
$728$	34.7917	1.28947
$729$	0	0
$730$	−15.1778	−0.561755
$731$	−5.77955	−0.213765
$732$	0	0
$733$	−20.3180	−0.750464	−0.375232	−	0.926931i	$-0.622437\pi$
$733$	−20.3180	−0.750464	−0.375232	+	0.926931i	$0.622437\pi$
$734$	−19.9178	−0.735179
$735$	0	0
$736$	1.23520	0.0455300
$737$	−6.13660	−0.226044
$738$	0	0
$739$	−13.1912	−0.485248	−0.242624	−	0.970120i	$-0.578008\pi$
$739$	−13.1912	−0.485248	−0.242624	+	0.970120i	$0.578008\pi$
$740$	−3.35264	−0.123246
$741$	0	0
$742$	−25.3273	−0.929796
$743$	−36.2465	−1.32975	−0.664877	−	0.746953i	$-0.731516\pi$
$743$	−36.2465	−1.32975	−0.664877	+	0.746953i	$0.731516\pi$
$744$	0	0
$745$	15.9423	0.584082
$746$	5.88483	0.215459
$747$	0	0
$748$	−1.89387	−0.0692466
$749$	29.6467	1.08327
$750$	0	0
$751$	25.4285	0.927901	0.463950	−	0.885861i	$-0.346432\pi$
$751$	25.4285	0.927901	0.463950	+	0.885861i	$0.346432\pi$
$752$	15.7760	0.575292
$753$	0	0
$754$	62.7932	2.28679
$755$	−2.85095	−0.103757
$756$	0	0
$757$	−1.51402	−0.0550280	−0.0275140	−	0.999621i	$-0.508759\pi$
$757$	−1.51402	−0.0550280	−0.0275140	+	0.999621i	$0.508759\pi$
$758$	−16.5336	−0.600526
$759$	0	0
$760$	−14.7223	−0.534034
$761$	22.5979	0.819172	0.409586	−	0.912272i	$-0.365673\pi$
$761$	22.5979	0.819172	0.409586	+	0.912272i	$0.365673\pi$
$762$	0	0
$763$	14.6702	0.531098
$764$	2.49501	0.0902661
$765$	0	0
$766$	−5.50510	−0.198907
$767$	72.4999	2.61782
$768$	0	0
$769$	−11.8829	−0.428509	−0.214254	−	0.976778i	$-0.568732\pi$
$769$	−11.8829	−0.428509	−0.214254	+	0.976778i	$0.568732\pi$
$770$	3.70210	0.133414
$771$	0	0
$772$	−5.33042	−0.191846
$773$	14.3931	0.517683	0.258841	−	0.965920i	$-0.416659\pi$
$773$	14.3931	0.517683	0.258841	+	0.965920i	$0.416659\pi$
$774$	0	0
$775$	−13.9657	−0.501661
$776$	21.2058	0.761243
$777$	0	0
$778$	−27.8636	−0.998957
$779$	−14.3732	−0.514973
$780$	0	0
$781$	12.3352	0.441387
$782$	−7.35610	−0.263054
$783$	0	0
$784$	9.72952	0.347483
$785$	33.8598	1.20851
$786$	0	0
$787$	−4.50354	−0.160534	−0.0802668	−	0.996773i	$-0.525577\pi$
$787$	−4.50354	−0.160534	−0.0802668	+	0.996773i	$0.525577\pi$
$788$	−5.93812	−0.211537
$789$	0	0
$790$	−6.90135	−0.245539
$791$	−35.5211	−1.26298
$792$	0	0
$793$	−5.73746	−0.203743
$794$	−8.83213	−0.313441
$795$	0	0
$796$	−5.92633	−0.210053
$797$	−0.483472	−0.0171255	−0.00856273	−	0.999963i	$-0.502726\pi$
$797$	−0.483472	−0.0171255	−0.00856273	+	0.999963i	$0.502726\pi$
$798$	0	0
$799$	32.4022	1.14631
$800$	4.71021	0.166531
$801$	0	0
$802$	3.42276	0.120862
$803$	−8.32305	−0.293714
$804$	0	0
$805$	−2.27127	−0.0800518
$806$	−34.2497	−1.20639
$807$	0	0
$808$	−42.5899	−1.49831
$809$	13.2547	0.466010	0.233005	−	0.972476i	$-0.425144\pi$
$809$	13.2547	0.466010	0.233005	+	0.972476i	$0.425144\pi$
$810$	0	0
$811$	−25.8912	−0.909162	−0.454581	−	0.890705i	$-0.650211\pi$
$811$	−25.8912	−0.909162	−0.454581	+	0.890705i	$0.650211\pi$
$812$	4.61220	0.161856
$813$	0	0
$814$	11.6396	0.407969
$815$	−13.3577	−0.467901
$816$	0	0
$817$	−2.95729	−0.103463
$818$	−2.73885	−0.0957617
$819$	0	0
$820$	−1.53174	−0.0534906
$821$	29.0101	1.01246	0.506230	−	0.862399i	$-0.331039\pi$
$821$	29.0101	1.01246	0.506230	+	0.862399i	$0.331039\pi$
$822$	0	0
$823$	51.9525	1.81095	0.905476	−	0.424398i	$-0.139514\pi$
$823$	51.9525	1.81095	0.905476	+	0.424398i	$0.139514\pi$
$824$	5.07418	0.176767
$825$	0	0
$826$	−33.7140	−1.17306
$827$	7.79500	0.271059	0.135529	−	0.990773i	$-0.456727\pi$
$827$	7.79500	0.271059	0.135529	+	0.990773i	$0.456727\pi$
$828$	0	0
$829$	21.1894	0.735940	0.367970	−	0.929838i	$-0.380053\pi$
$829$	21.1894	0.735940	0.367970	+	0.929838i	$0.380053\pi$
$830$	−23.0245	−0.799191
$831$	0	0
$832$	50.3361	1.74509
$833$	19.9833	0.692381
$834$	0	0
$835$	23.4183	0.810425
$836$	−0.969057	−0.0335155
$837$	0	0
$838$	−31.5814	−1.09096
$839$	16.8696	0.582402	0.291201	−	0.956662i	$-0.405945\pi$
$839$	16.8696	0.582402	0.291201	+	0.956662i	$0.405945\pi$
$840$	0	0
$841$	40.3499	1.39138
$842$	−1.33094	−0.0458672
$843$	0	0
$844$	−0.827675	−0.0284898
$845$	27.6383	0.950787
$846$	0	0
$847$	2.03012	0.0697559
$848$	−32.0854	−1.10182
$849$	0	0
$850$	−28.0512	−0.962149
$851$	−7.14102	−0.244791
$852$	0	0
$853$	40.2710	1.37885	0.689426	−	0.724356i	$-0.257863\pi$
$853$	40.2710	1.37885	0.689426	+	0.724356i	$0.257863\pi$
$854$	2.66804	0.0912985
$855$	0	0
$856$	43.6201	1.49090
$857$	−54.7718	−1.87097	−0.935485	−	0.353366i	$-0.885037\pi$
$857$	−54.7718	−1.87097	−0.935485	+	0.353366i	$0.885037\pi$
$858$	0	0
$859$	25.8871	0.883257	0.441629	−	0.897198i	$-0.354401\pi$
$859$	25.8871	0.883257	0.441629	+	0.897198i	$0.354401\pi$
$860$	−0.315156	−0.0107467
$861$	0	0
$862$	−21.6114	−0.736086
$863$	13.5251	0.460399	0.230199	−	0.973143i	$-0.426062\pi$
$863$	13.5251	0.460399	0.230199	+	0.973143i	$0.426062\pi$
$864$	0	0
$865$	8.04670	0.273596
$866$	29.6827	1.00866
$867$	0	0
$868$	−2.51566	−0.0853870
$869$	−3.78450	−0.128380
$870$	0	0
$871$	35.2085	1.19299
$872$	21.5848	0.730953
$873$	0	0
$874$	−3.76398	−0.127319
$875$	−22.7458	−0.768949
$876$	0	0
$877$	50.5645	1.70744	0.853721	−	0.520730i	$-0.174340\pi$
$877$	50.5645	1.70744	0.853721	+	0.520730i	$0.174340\pi$
$878$	45.8256	1.54654
$879$	0	0
$880$	4.68993	0.158097
$881$	−24.1533	−0.813744	−0.406872	−	0.913485i	$-0.633381\pi$
$881$	−24.1533	−0.813744	−0.406872	+	0.913485i	$0.633381\pi$
$882$	0	0
$883$	−52.5172	−1.76735	−0.883673	−	0.468105i	$-0.844937\pi$
$883$	−52.5172	−1.76735	−0.883673	+	0.468105i	$0.844937\pi$
$884$	10.8660	0.365463
$885$	0	0
$886$	−40.9667	−1.37630
$887$	−24.0349	−0.807013	−0.403507	−	0.914977i	$-0.632209\pi$
$887$	−24.0349	−0.807013	−0.403507	+	0.914977i	$0.632209\pi$
$888$	0	0
$889$	4.48511	0.150426
$890$	−27.0969	−0.908291
$891$	0	0
$892$	−1.30104	−0.0435621
$893$	16.5796	0.554815
$894$	0	0
$895$	−5.93878	−0.198511
$896$	−17.1872	−0.574185
$897$	0	0
$898$	34.0335	1.13571
$899$	−37.8259	−1.26157
$900$	0	0
$901$	−65.8998	−2.19544
$902$	5.31786	0.177065
$903$	0	0
$904$	−52.2633	−1.73825
$905$	−27.8975	−0.927343
$906$	0	0
$907$	−27.3556	−0.908328	−0.454164	−	0.890918i	$-0.650062\pi$
$907$	−27.3556	−0.908328	−0.454164	+	0.890918i	$0.650062\pi$
$908$	−7.26782	−0.241191
$909$	0	0
$910$	−21.2407	−0.704121
$911$	−3.27520	−0.108512	−0.0542562	−	0.998527i	$-0.517279\pi$
$911$	−3.27520	−0.108512	−0.0542562	+	0.998527i	$0.517279\pi$
$912$	0	0
$913$	−12.6260	−0.417858
$914$	−8.84559	−0.292586
$915$	0	0
$916$	−3.18867	−0.105357
$917$	4.21827	0.139299
$918$	0	0
$919$	−9.33515	−0.307938	−0.153969	−	0.988076i	$-0.549206\pi$
$919$	−9.33515	−0.307938	−0.153969	+	0.988076i	$0.549206\pi$
$920$	−3.34179	−0.110176
$921$	0	0
$922$	50.7235	1.67049
$923$	−70.7726	−2.32951
$924$	0	0
$925$	−27.2310	−0.895351
$926$	−37.0718	−1.21826
$927$	0	0
$928$	12.7576	0.418789
$929$	41.7443	1.36958	0.684792	−	0.728738i	$-0.259893\pi$
$929$	41.7443	1.36958	0.684792	+	0.728738i	$0.259893\pi$
$930$	0	0
$931$	10.2251	0.335114
$932$	−0.951639	−0.0311720
$933$	0	0
$934$	−21.5091	−0.703800
$935$	9.63258	0.315019
$936$	0	0
$937$	17.4250	0.569250	0.284625	−	0.958639i	$-0.408131\pi$
$937$	17.4250	0.569250	0.284625	+	0.958639i	$0.408131\pi$
$938$	−16.3727	−0.534587
$939$	0	0
$940$	1.76687	0.0576290
$941$	−29.2922	−0.954899	−0.477450	−	0.878659i	$-0.658439\pi$
$941$	−29.2922	−0.954899	−0.477450	+	0.878659i	$0.658439\pi$
$942$	0	0
$943$	−3.26255	−0.106243
$944$	−42.7099	−1.39009
$945$	0	0
$946$	1.09415	0.0355739
$947$	−8.76532	−0.284835	−0.142417	−	0.989807i	$-0.545487\pi$
$947$	−8.76532	−0.284835	−0.142417	+	0.989807i	$0.545487\pi$
$948$	0	0
$949$	47.7532	1.55013
$950$	−14.3533	−0.465682
$951$	0	0
$952$	−42.0962	−1.36435
$953$	32.0962	1.03970	0.519849	−	0.854258i	$-0.325988\pi$
$953$	32.0962	1.03970	0.519849	+	0.854258i	$0.325988\pi$
$954$	0	0
$955$	−12.6901	−0.410642
$956$	4.95066	0.160116
$957$	0	0
$958$	−22.0491	−0.712374
$959$	−21.8836	−0.706659
$960$	0	0
$961$	−10.3684	−0.334463
$962$	−66.7819	−2.15314
$963$	0	0
$964$	1.09599	0.0352993
$965$	27.1116	0.872752
$966$	0	0
$967$	6.77178	0.217766	0.108883	−	0.994055i	$-0.465273\pi$
$967$	6.77178	0.217766	0.108883	+	0.994055i	$0.465273\pi$
$968$	2.98699	0.0960054
$969$	0	0
$970$	−12.9463	−0.415681
$971$	−24.4938	−0.786045	−0.393022	−	0.919529i	$-0.628571\pi$
$971$	−24.4938	−0.786045	−0.393022	+	0.919529i	$0.628571\pi$
$972$	0	0
$973$	−23.3038	−0.747085
$974$	37.7417	1.20932
$975$	0	0
$976$	3.37995	0.108190
$977$	34.4114	1.10092	0.550459	−	0.834862i	$-0.314453\pi$
$977$	34.4114	1.10092	0.550459	+	0.834862i	$0.314453\pi$
$978$	0	0
$979$	−14.8592	−0.474901
$980$	1.08968	0.0348085
$981$	0	0
$982$	42.9158	1.36950
$983$	24.8714	0.793275	0.396638	−	0.917975i	$-0.370177\pi$
$983$	24.8714	0.793275	0.396638	+	0.917975i	$0.370177\pi$
$984$	0	0
$985$	30.2024	0.962329
$986$	−75.9766	−2.41959
$987$	0	0
$988$	5.55993	0.176885
$989$	−0.671271	−0.0213452
$990$	0	0
$991$	30.8026	0.978475	0.489238	−	0.872151i	$-0.337275\pi$
$991$	30.8026	0.978475	0.489238	+	0.872151i	$0.337275\pi$
$992$	−6.95845	−0.220931
$993$	0	0
$994$	32.9107	1.04387
$995$	30.1425	0.955582
$996$	0	0
$997$	23.9102	0.757244	0.378622	−	0.925551i	$-0.376398\pi$
$997$	23.9102	0.757244	0.378622	+	0.925551i	$0.376398\pi$
$998$	9.35726	0.296199
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.7		19
3.2	odd	2		671.2.a.c.1.13	✓	19
33.32	even	2		7381.2.a.i.1.7		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.13	✓	19	3.2	odd	2
6039.2.a.k.1.7		19	1.1	even	1	trivial
7381.2.a.i.1.7		19	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.31423 q^{2} -0.272811 q^{4} +1.38757 q^{5} +2.03012 q^{7} +2.98699 q^{8} +O(q^{10})\)	\(q-1.31423 q^{2} -0.272811 q^{4} +1.38757 q^{5} +2.03012 q^{7} +2.98699 q^{8} -1.82358 q^{10} -1.00000 q^{11} +5.73746 q^{13} -2.66804 q^{14} -3.37995 q^{16} -6.94204 q^{17} -3.55211 q^{19} -0.378545 q^{20} +1.31423 q^{22} -0.806289 q^{23} -3.07464 q^{25} -7.54032 q^{26} -0.553841 q^{28} -8.32766 q^{29} +4.54221 q^{31} -1.53195 q^{32} +9.12341 q^{34} +2.81694 q^{35} +8.85665 q^{37} +4.66828 q^{38} +4.14466 q^{40} +4.04638 q^{41} +0.832544 q^{43} +0.272811 q^{44} +1.05965 q^{46} -4.66753 q^{47} -2.87860 q^{49} +4.04077 q^{50} -1.56524 q^{52} +9.49286 q^{53} -1.38757 q^{55} +6.06395 q^{56} +10.9444 q^{58} +12.6362 q^{59} -1.00000 q^{61} -5.96948 q^{62} +8.77324 q^{64} +7.96114 q^{65} +6.13660 q^{67} +1.89387 q^{68} -3.70210 q^{70} -12.3352 q^{71} +8.32305 q^{73} -11.6396 q^{74} +0.969057 q^{76} -2.03012 q^{77} +3.78450 q^{79} -4.68993 q^{80} -5.31786 q^{82} +12.6260 q^{83} -9.63258 q^{85} -1.09415 q^{86} -2.98699 q^{88} +14.8592 q^{89} +11.6478 q^{91} +0.219965 q^{92} +6.13418 q^{94} -4.92881 q^{95} +7.09938 q^{97} +3.78312 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more