LMFDB - Embedded newform 6039.2.a.i.1.10

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:	$13$
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} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 $ x^13 - 2x^12 - 19x^11 + 35x^10 + 136x^9 - 220x^8 - 469x^7 + 610x^6 + 841x^5 - 760x^4 - 742x^3 + 366x^2 + 236x - 47
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			1.10
Root			$-1.33092$ 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)

$f(q)$	$=$	$q+1.33092 q^{2} -0.228660 q^{4} +2.36819 q^{5} -2.92169 q^{7} -2.96616 q^{8} +O(q^{10})$	\(q+1.33092 q^{2} -0.228660 q^{4} +2.36819 q^{5} -2.92169 q^{7} -2.96616 q^{8} +3.15187 q^{10} -1.00000 q^{11} +4.67063 q^{13} -3.88853 q^{14} -3.49039 q^{16} +3.08651 q^{17} -5.01988 q^{19} -0.541511 q^{20} -1.33092 q^{22} +4.73072 q^{23} +0.608331 q^{25} +6.21622 q^{26} +0.668075 q^{28} +2.14373 q^{29} +1.22908 q^{31} +1.28690 q^{32} +4.10789 q^{34} -6.91913 q^{35} -5.71888 q^{37} -6.68104 q^{38} -7.02444 q^{40} +9.30697 q^{41} -3.00386 q^{43} +0.228660 q^{44} +6.29619 q^{46} +1.19144 q^{47} +1.53629 q^{49} +0.809639 q^{50} -1.06799 q^{52} +4.66301 q^{53} -2.36819 q^{55} +8.66621 q^{56} +2.85313 q^{58} +8.94746 q^{59} -1.00000 q^{61} +1.63581 q^{62} +8.69354 q^{64} +11.0609 q^{65} +8.99807 q^{67} -0.705762 q^{68} -9.20879 q^{70} -14.8610 q^{71} +9.79980 q^{73} -7.61136 q^{74} +1.14785 q^{76} +2.92169 q^{77} +11.9696 q^{79} -8.26592 q^{80} +12.3868 q^{82} -12.9510 q^{83} +7.30945 q^{85} -3.99788 q^{86} +2.96616 q^{88} +0.425469 q^{89} -13.6462 q^{91} -1.08173 q^{92} +1.58570 q^{94} -11.8880 q^{95} +5.28118 q^{97} +2.04467 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * 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 $

$2$	1.33092	0.941100	0.470550	−	0.882373i	$-0.344055\pi$
$2$	1.33092	0.941100	0.470550	+	0.882373i	$0.344055\pi$
$3$	0	0
$3$	0	0
$4$	−0.228660	−0.114330
$5$	2.36819	1.05909	0.529544	−	0.848283i	$-0.322363\pi$
$5$	2.36819	1.05909	0.529544	+	0.848283i	$0.322363\pi$
$6$	0	0
$7$	−2.92169	−1.10430	−0.552148	−	0.833746i	$-0.686192\pi$
$7$	−2.92169	−1.10430	−0.552148	+	0.833746i	$0.686192\pi$
$8$	−2.96616	−1.04870
$9$	0	0
$10$	3.15187	0.996708
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.67063	1.29540	0.647700	−	0.761895i	$-0.275731\pi$
$13$	4.67063	1.29540	0.647700	+	0.761895i	$0.275731\pi$
$14$	−3.88853	−1.03925
$15$	0	0
$16$	−3.49039	−0.872599
$17$	3.08651	0.748589	0.374294	−	0.927310i	$-0.377885\pi$
$17$	3.08651	0.748589	0.374294	+	0.927310i	$0.377885\pi$
$18$	0	0
$19$	−5.01988	−1.15164	−0.575820	−	0.817577i	$-0.695317\pi$
$19$	−5.01988	−1.15164	−0.575820	+	0.817577i	$0.695317\pi$
$20$	−0.541511	−0.121086
$21$	0	0
$22$	−1.33092	−0.283752
$23$	4.73072	0.986423	0.493212	−	0.869909i	$-0.335823\pi$
$23$	4.73072	0.986423	0.493212	+	0.869909i	$0.335823\pi$
$24$	0	0
$25$	0.608331	0.121666
$26$	6.21622	1.21910
$27$	0	0
$28$	0.668075	0.126254
$29$	2.14373	0.398081	0.199040	−	0.979991i	$-0.436218\pi$
$29$	2.14373	0.398081	0.199040	+	0.979991i	$0.436218\pi$
$30$	0	0
$31$	1.22908	0.220750	0.110375	−	0.993890i	$-0.464795\pi$
$31$	1.22908	0.220750	0.110375	+	0.993890i	$0.464795\pi$
$32$	1.28690	0.227494
$33$	0	0
$34$	4.10789	0.704497
$35$	−6.91913	−1.16955
$36$	0	0
$37$	−5.71888	−0.940178	−0.470089	−	0.882619i	$-0.655778\pi$
$37$	−5.71888	−0.940178	−0.470089	+	0.882619i	$0.655778\pi$
$38$	−6.68104	−1.08381
$39$	0	0
$40$	−7.02444	−1.11066
$41$	9.30697	1.45350	0.726752	−	0.686900i	$-0.241029\pi$
$41$	9.30697	1.45350	0.726752	+	0.686900i	$0.241029\pi$
$42$	0	0
$43$	−3.00386	−0.458084	−0.229042	−	0.973417i	$-0.573559\pi$
$43$	−3.00386	−0.458084	−0.229042	+	0.973417i	$0.573559\pi$
$44$	0.228660	0.0344718
$45$	0	0
$46$	6.29619	0.928323
$47$	1.19144	0.173789	0.0868944	−	0.996218i	$-0.472306\pi$
$47$	1.19144	0.173789	0.0868944	+	0.996218i	$0.472306\pi$
$48$	0	0
$49$	1.53629	0.219470
$50$	0.809639	0.114500
$51$	0	0
$52$	−1.06799	−0.148103
$53$	4.66301	0.640514	0.320257	−	0.947331i	$-0.396231\pi$
$53$	4.66301	0.640514	0.320257	+	0.947331i	$0.396231\pi$
$54$	0	0
$55$	−2.36819	−0.319327
$56$	8.66621	1.15807
$57$	0	0
$58$	2.85313	0.374634
$59$	8.94746	1.16486	0.582430	−	0.812881i	$-0.302102\pi$
$59$	8.94746	1.16486	0.582430	+	0.812881i	$0.302102\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	1.63581	0.207747
$63$	0	0
$64$	8.69354	1.08669
$65$	11.0609	1.37194
$66$	0	0
$67$	8.99807	1.09929	0.549645	−	0.835398i	$-0.314763\pi$
$67$	8.99807	1.09929	0.549645	+	0.835398i	$0.314763\pi$
$68$	−0.705762	−0.0855862
$69$	0	0
$70$	−9.20879	−1.10066
$71$	−14.8610	−1.76368	−0.881838	−	0.471553i	$-0.843694\pi$
$71$	−14.8610	−1.76368	−0.881838	+	0.471553i	$0.843694\pi$
$72$	0	0
$73$	9.79980	1.14698	0.573490	−	0.819213i	$-0.305589\pi$
$73$	9.79980	1.14698	0.573490	+	0.819213i	$0.305589\pi$
$74$	−7.61136	−0.884802
$75$	0	0
$76$	1.14785	0.131667
$77$	2.92169	0.332958
$78$	0	0
$79$	11.9696	1.34669	0.673343	−	0.739331i	$-0.264858\pi$
$79$	11.9696	1.34669	0.673343	+	0.739331i	$0.264858\pi$
$80$	−8.26592	−0.924158
$81$	0	0
$82$	12.3868	1.36789
$83$	−12.9510	−1.42156	−0.710781	−	0.703414i	$-0.751658\pi$
$83$	−12.9510	−1.42156	−0.710781	+	0.703414i	$0.751658\pi$
$84$	0	0
$85$	7.30945	0.792821
$86$	−3.99788	−0.431103
$87$	0	0
$88$	2.96616	0.316194
$89$	0.425469	0.0450997	0.0225498	−	0.999746i	$-0.492822\pi$
$89$	0.425469	0.0450997	0.0225498	+	0.999746i	$0.492822\pi$
$90$	0	0
$91$	−13.6462	−1.43051
$92$	−1.08173	−0.112778
$93$	0	0
$94$	1.58570	0.163553
$95$	−11.8880	−1.21969
$96$	0	0
$97$	5.28118	0.536222	0.268111	−	0.963388i	$-0.413601\pi$
$97$	5.28118	0.536222	0.268111	+	0.963388i	$0.413601\pi$
$98$	2.04467	0.206543
$99$	0	0
$100$	−0.139101	−0.0139101
$101$	18.0972	1.80074	0.900368	−	0.435130i	$-0.143297\pi$
$101$	18.0972	1.80074	0.900368	+	0.435130i	$0.143297\pi$
$102$	0	0
$103$	5.40800	0.532867	0.266433	−	0.963853i	$-0.414155\pi$
$103$	5.40800	0.532867	0.266433	+	0.963853i	$0.414155\pi$
$104$	−13.8538	−1.35848
$105$	0	0
$106$	6.20608	0.602788
$107$	5.18077	0.500844	0.250422	−	0.968137i	$-0.419431\pi$
$107$	5.18077	0.500844	0.250422	+	0.968137i	$0.419431\pi$
$108$	0	0
$109$	−11.1822	−1.07106	−0.535530	−	0.844516i	$-0.679888\pi$
$109$	−11.1822	−1.07106	−0.535530	+	0.844516i	$0.679888\pi$
$110$	−3.15187	−0.300519
$111$	0	0
$112$	10.1979	0.963607
$113$	8.56972	0.806172	0.403086	−	0.915162i	$-0.367938\pi$
$113$	8.56972	0.806172	0.403086	+	0.915162i	$0.367938\pi$
$114$	0	0
$115$	11.2033	1.04471
$116$	−0.490186	−0.0455126
$117$	0	0
$118$	11.9083	1.09625
$119$	−9.01784	−0.826664
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.33092	−0.120496
$123$	0	0
$124$	−0.281042	−0.0252383
$125$	−10.4003	−0.930232
$126$	0	0
$127$	0.489906	0.0434721	0.0217361	−	0.999764i	$-0.493081\pi$
$127$	0.489906	0.0434721	0.0217361	+	0.999764i	$0.493081\pi$
$128$	8.99659	0.795193
$129$	0	0
$130$	14.7212	1.29114
$131$	4.25776	0.372002	0.186001	−	0.982550i	$-0.440447\pi$
$131$	4.25776	0.372002	0.186001	+	0.982550i	$0.440447\pi$
$132$	0	0
$133$	14.6665	1.27175
$134$	11.9757	1.03454
$135$	0	0
$136$	−9.15509	−0.785042
$137$	20.9037	1.78592	0.892961	−	0.450134i	$-0.148624\pi$
$137$	20.9037	1.78592	0.892961	+	0.450134i	$0.148624\pi$
$138$	0	0
$139$	10.2136	0.866307	0.433153	−	0.901320i	$-0.357401\pi$
$139$	10.2136	0.866307	0.433153	+	0.901320i	$0.357401\pi$
$140$	1.58213	0.133714
$141$	0	0
$142$	−19.7788	−1.65980
$143$	−4.67063	−0.390578
$144$	0	0
$145$	5.07676	0.421602
$146$	13.0427	1.07942
$147$	0	0
$148$	1.30768	0.107491
$149$	13.3117	1.09054	0.545268	−	0.838262i	$-0.316428\pi$
$149$	13.3117	1.09054	0.545268	+	0.838262i	$0.316428\pi$
$150$	0	0
$151$	−14.1469	−1.15126	−0.575628	−	0.817711i	$-0.695243\pi$
$151$	−14.1469	−1.15126	−0.575628	+	0.817711i	$0.695243\pi$
$152$	14.8898	1.20772
$153$	0	0
$154$	3.88853	0.313347
$155$	2.91070	0.233793
$156$	0	0
$157$	15.2651	1.21829	0.609144	−	0.793059i	$-0.291513\pi$
$157$	15.2651	1.21829	0.609144	+	0.793059i	$0.291513\pi$
$158$	15.9305	1.26737
$159$	0	0
$160$	3.04762	0.240936
$161$	−13.8217	−1.08930
$162$	0	0
$163$	18.0609	1.41464	0.707321	−	0.706892i	$-0.249903\pi$
$163$	18.0609	1.41464	0.707321	+	0.706892i	$0.249903\pi$
$164$	−2.12813	−0.166179
$165$	0	0
$166$	−17.2368	−1.33783
$167$	18.8759	1.46066	0.730329	−	0.683096i	$-0.239367\pi$
$167$	18.8759	1.46066	0.730329	+	0.683096i	$0.239367\pi$
$168$	0	0
$169$	8.81480	0.678061
$170$	9.72827	0.746124
$171$	0	0
$172$	0.686863	0.0523728
$173$	−20.7395	−1.57680	−0.788398	−	0.615165i	$-0.789089\pi$
$173$	−20.7395	−1.57680	−0.788398	+	0.615165i	$0.789089\pi$
$174$	0	0
$175$	−1.77736	−0.134356
$176$	3.49039	0.263098
$177$	0	0
$178$	0.566264	0.0424433
$179$	−10.9017	−0.814834	−0.407417	−	0.913242i	$-0.633570\pi$
$179$	−10.9017	−0.814834	−0.407417	+	0.913242i	$0.633570\pi$
$180$	0	0
$181$	9.68426	0.719825	0.359913	−	0.932986i	$-0.382806\pi$
$181$	9.68426	0.719825	0.359913	+	0.932986i	$0.382806\pi$
$182$	−18.1619	−1.34625
$183$	0	0
$184$	−14.0321	−1.03446
$185$	−13.5434	−0.995731
$186$	0	0
$187$	−3.08651	−0.225708
$188$	−0.272434	−0.0198693
$189$	0	0
$190$	−15.8220	−1.14785
$191$	10.8157	0.782597	0.391298	−	0.920264i	$-0.372026\pi$
$191$	10.8157	0.782597	0.391298	+	0.920264i	$0.372026\pi$
$192$	0	0
$193$	−6.19562	−0.445970	−0.222985	−	0.974822i	$-0.571580\pi$
$193$	−6.19562	−0.445970	−0.222985	+	0.974822i	$0.571580\pi$
$194$	7.02881	0.504639
$195$	0	0
$196$	−0.351288	−0.0250920
$197$	1.14995	0.0819308	0.0409654	−	0.999161i	$-0.486957\pi$
$197$	1.14995	0.0819308	0.0409654	+	0.999161i	$0.486957\pi$
$198$	0	0
$199$	13.8061	0.978687	0.489344	−	0.872091i	$-0.337236\pi$
$199$	13.8061	0.978687	0.489344	+	0.872091i	$0.337236\pi$
$200$	−1.80441	−0.127591
$201$	0	0
$202$	24.0858	1.69467
$203$	−6.26332	−0.439599
$204$	0	0
$205$	22.0407	1.53939
$206$	7.19760	0.501481
$207$	0	0
$208$	−16.3023	−1.13036
$209$	5.01988	0.347232
$210$	0	0
$211$	8.67180	0.596991	0.298496	−	0.954411i	$-0.403515\pi$
$211$	8.67180	0.596991	0.298496	+	0.954411i	$0.403515\pi$
$212$	−1.06624	−0.0732300
$213$	0	0
$214$	6.89518	0.471345
$215$	−7.11371	−0.485151
$216$	0	0
$217$	−3.59100	−0.243773
$218$	−14.8826	−1.00797
$219$	0	0
$220$	0.541511	0.0365087
$221$	14.4160	0.969722
$222$	0	0
$223$	0.540497	0.0361944	0.0180972	−	0.999836i	$-0.494239\pi$
$223$	0.540497	0.0361944	0.0180972	+	0.999836i	$0.494239\pi$
$224$	−3.75992	−0.251220
$225$	0	0
$226$	11.4056	0.758688
$227$	−5.60176	−0.371802	−0.185901	−	0.982568i	$-0.559520\pi$
$227$	−5.60176	−0.371802	−0.185901	+	0.982568i	$0.559520\pi$
$228$	0	0
$229$	−12.8413	−0.848578	−0.424289	−	0.905527i	$-0.639476\pi$
$229$	−12.8413	−0.848578	−0.424289	+	0.905527i	$0.639476\pi$
$230$	14.9106	0.983176
$231$	0	0
$232$	−6.35865	−0.417466
$233$	−26.1667	−1.71423	−0.857117	−	0.515121i	$-0.827747\pi$
$233$	−26.1667	−1.71423	−0.857117	+	0.515121i	$0.827747\pi$
$234$	0	0
$235$	2.82155	0.184058
$236$	−2.04593	−0.133179
$237$	0	0
$238$	−12.0020	−0.777974
$239$	−17.1822	−1.11142	−0.555711	−	0.831376i	$-0.687554\pi$
$239$	−17.1822	−1.11142	−0.555711	+	0.831376i	$0.687554\pi$
$240$	0	0
$241$	20.5527	1.32392	0.661959	−	0.749540i	$-0.269725\pi$
$241$	20.5527	1.32392	0.661959	+	0.749540i	$0.269725\pi$
$242$	1.33092	0.0855546
$243$	0	0
$244$	0.228660	0.0146385
$245$	3.63823	0.232438
$246$	0	0
$247$	−23.4460	−1.49183
$248$	−3.64565	−0.231499
$249$	0	0
$250$	−13.8420	−0.875442
$251$	−22.6512	−1.42973	−0.714865	−	0.699262i	$-0.753512\pi$
$251$	−22.6512	−1.42973	−0.714865	+	0.699262i	$0.753512\pi$
$252$	0	0
$253$	−4.73072	−0.297418
$254$	0.652024	0.0409116
$255$	0	0
$256$	−5.41338	−0.338336
$257$	12.2924	0.766778	0.383389	−	0.923587i	$-0.374757\pi$
$257$	12.2924	0.766778	0.383389	+	0.923587i	$0.374757\pi$
$258$	0	0
$259$	16.7088	1.03824
$260$	−2.52920	−0.156854
$261$	0	0
$262$	5.66672	0.350091
$263$	−11.9232	−0.735219	−0.367609	−	0.929980i	$-0.619824\pi$
$263$	−11.9232	−0.735219	−0.367609	+	0.929980i	$0.619824\pi$
$264$	0	0
$265$	11.0429	0.678360
$266$	19.5200	1.19684
$267$	0	0
$268$	−2.05750	−0.125682
$269$	8.16149	0.497615	0.248807	−	0.968553i	$-0.419961\pi$
$269$	8.16149	0.497615	0.248807	+	0.968553i	$0.419961\pi$
$270$	0	0
$271$	27.6984	1.68256	0.841278	−	0.540602i	$-0.181804\pi$
$271$	27.6984	1.68256	0.841278	+	0.540602i	$0.181804\pi$
$272$	−10.7731	−0.653218
$273$	0	0
$274$	27.8211	1.68073
$275$	−0.608331	−0.0366838
$276$	0	0
$277$	−1.78976	−0.107536	−0.0537681	−	0.998553i	$-0.517123\pi$
$277$	−1.78976	−0.107536	−0.0537681	+	0.998553i	$0.517123\pi$
$278$	13.5935	0.815281
$279$	0	0
$280$	20.5233	1.22650
$281$	11.9883	0.715164	0.357582	−	0.933882i	$-0.383601\pi$
$281$	11.9883	0.715164	0.357582	+	0.933882i	$0.383601\pi$
$282$	0	0
$283$	−18.0687	−1.07407	−0.537035	−	0.843560i	$-0.680456\pi$
$283$	−18.0687	−1.07407	−0.537035	+	0.843560i	$0.680456\pi$
$284$	3.39812	0.201641
$285$	0	0
$286$	−6.21622	−0.367573
$287$	−27.1921	−1.60510
$288$	0	0
$289$	−7.47345	−0.439615
$290$	6.75675	0.396770
$291$	0	0
$292$	−2.24082	−0.131134
$293$	−23.4090	−1.36757	−0.683784	−	0.729684i	$-0.739667\pi$
$293$	−23.4090	−1.36757	−0.683784	+	0.729684i	$0.739667\pi$
$294$	0	0
$295$	21.1893	1.23369
$296$	16.9631	0.985962
$297$	0	0
$298$	17.7168	1.02630
$299$	22.0954	1.27781
$300$	0	0
$301$	8.77635	0.505860
$302$	−18.8283	−1.08345
$303$	0	0
$304$	17.5214	1.00492
$305$	−2.36819	−0.135602
$306$	0	0
$307$	8.48031	0.483997	0.241998	−	0.970277i	$-0.422197\pi$
$307$	8.48031	0.483997	0.241998	+	0.970277i	$0.422197\pi$
$308$	−0.668075	−0.0380671
$309$	0	0
$310$	3.87390	0.220023
$311$	16.1743	0.917160	0.458580	−	0.888653i	$-0.348358\pi$
$311$	16.1743	0.917160	0.458580	+	0.888653i	$0.348358\pi$
$312$	0	0
$313$	0.716595	0.0405044	0.0202522	−	0.999795i	$-0.493553\pi$
$313$	0.716595	0.0405044	0.0202522	+	0.999795i	$0.493553\pi$
$314$	20.3166	1.14653
$315$	0	0
$316$	−2.73697	−0.153967
$317$	2.26108	0.126995	0.0634974	−	0.997982i	$-0.479775\pi$
$317$	2.26108	0.126995	0.0634974	+	0.997982i	$0.479775\pi$
$318$	0	0
$319$	−2.14373	−0.120026
$320$	20.5880	1.15090
$321$	0	0
$322$	−18.3955	−1.02514
$323$	−15.4939	−0.862104
$324$	0	0
$325$	2.84129	0.157607
$326$	24.0376	1.33132
$327$	0	0
$328$	−27.6060	−1.52429
$329$	−3.48101	−0.191914
$330$	0	0
$331$	−11.8409	−0.650836	−0.325418	−	0.945570i	$-0.605505\pi$
$331$	−11.8409	−0.650836	−0.325418	+	0.945570i	$0.605505\pi$
$332$	2.96139	0.162527
$333$	0	0
$334$	25.1222	1.37463
$335$	21.3092	1.16424
$336$	0	0
$337$	30.4451	1.65845	0.829225	−	0.558915i	$-0.188782\pi$
$337$	30.4451	1.65845	0.829225	+	0.558915i	$0.188782\pi$
$338$	11.7318	0.638124
$339$	0	0
$340$	−1.67138	−0.0906433
$341$	−1.22908	−0.0665585
$342$	0	0
$343$	15.9633	0.861936
$344$	8.90993	0.480391
$345$	0	0
$346$	−27.6026	−1.48392
$347$	−12.9041	−0.692728	−0.346364	−	0.938100i	$-0.612584\pi$
$347$	−12.9041	−0.692728	−0.346364	+	0.938100i	$0.612584\pi$
$348$	0	0
$349$	−18.0493	−0.966159	−0.483080	−	0.875576i	$-0.660482\pi$
$349$	−18.0493	−0.966159	−0.483080	+	0.875576i	$0.660482\pi$
$350$	−2.36552	−0.126442
$351$	0	0
$352$	−1.28690	−0.0685919
$353$	−11.3839	−0.605903	−0.302951	−	0.953006i	$-0.597972\pi$
$353$	−11.3839	−0.605903	−0.302951	+	0.953006i	$0.597972\pi$
$354$	0	0
$355$	−35.1937	−1.86789
$356$	−0.0972879	−0.00515625
$357$	0	0
$358$	−14.5093	−0.766841
$359$	−3.51661	−0.185599	−0.0927997	−	0.995685i	$-0.529582\pi$
$359$	−3.51661	−0.185599	−0.0927997	+	0.995685i	$0.529582\pi$
$360$	0	0
$361$	6.19918	0.326273
$362$	12.8889	0.677428
$363$	0	0
$364$	3.12033	0.163550
$365$	23.2078	1.21475
$366$	0	0
$367$	−6.46554	−0.337498	−0.168749	−	0.985659i	$-0.553973\pi$
$367$	−6.46554	−0.337498	−0.168749	+	0.985659i	$0.553973\pi$
$368$	−16.5121	−0.860751
$369$	0	0
$370$	−18.0252	−0.937083
$371$	−13.6239	−0.707317
$372$	0	0
$373$	4.51638	0.233849	0.116925	−	0.993141i	$-0.462696\pi$
$373$	4.51638	0.233849	0.116925	+	0.993141i	$0.462696\pi$
$374$	−4.10789	−0.212414
$375$	0	0
$376$	−3.53399	−0.182252
$377$	10.0126	0.515674
$378$	0	0
$379$	−26.1477	−1.34312	−0.671558	−	0.740952i	$-0.734375\pi$
$379$	−26.1477	−1.34312	−0.671558	+	0.740952i	$0.734375\pi$
$380$	2.71832	0.139447
$381$	0	0
$382$	14.3948	0.736502
$383$	23.3062	1.19089	0.595445	−	0.803396i	$-0.296976\pi$
$383$	23.3062	1.19089	0.595445	+	0.803396i	$0.296976\pi$
$384$	0	0
$385$	6.91913	0.352631
$386$	−8.24585	−0.419703
$387$	0	0
$388$	−1.20760	−0.0613064
$389$	−19.3335	−0.980249	−0.490125	−	0.871652i	$-0.663049\pi$
$389$	−19.3335	−0.980249	−0.490125	+	0.871652i	$0.663049\pi$
$390$	0	0
$391$	14.6014	0.738425
$392$	−4.55688	−0.230157
$393$	0	0
$394$	1.53049	0.0771051
$395$	28.3463	1.42626
$396$	0	0
$397$	8.98364	0.450876	0.225438	−	0.974258i	$-0.427619\pi$
$397$	8.98364	0.450876	0.225438	+	0.974258i	$0.427619\pi$
$398$	18.3748	0.921043
$399$	0	0
$400$	−2.12332	−0.106166
$401$	6.20996	0.310110	0.155055	−	0.987906i	$-0.450444\pi$
$401$	6.20996	0.310110	0.155055	+	0.987906i	$0.450444\pi$
$402$	0	0
$403$	5.74059	0.285959
$404$	−4.13810	−0.205878
$405$	0	0
$406$	−8.33596	−0.413707
$407$	5.71888	0.283474
$408$	0	0
$409$	27.1966	1.34478	0.672392	−	0.740196i	$-0.265267\pi$
$409$	27.1966	1.34478	0.672392	+	0.740196i	$0.265267\pi$
$410$	29.3343	1.44872
$411$	0	0
$412$	−1.23660	−0.0609227
$413$	−26.1417	−1.28635
$414$	0	0
$415$	−30.6705	−1.50556
$416$	6.01063	0.294695
$417$	0	0
$418$	6.68104	0.326780
$419$	−17.1176	−0.836250	−0.418125	−	0.908389i	$-0.637313\pi$
$419$	−17.1176	−0.836250	−0.418125	+	0.908389i	$0.637313\pi$
$420$	0	0
$421$	−16.4220	−0.800357	−0.400179	−	0.916437i	$-0.631052\pi$
$421$	−16.4220	−0.800357	−0.400179	+	0.916437i	$0.631052\pi$
$422$	11.5414	0.561829
$423$	0	0
$424$	−13.8312	−0.671705
$425$	1.87762	0.0910780
$426$	0	0
$427$	2.92169	0.141391
$428$	−1.18464	−0.0572615
$429$	0	0
$430$	−9.46776	−0.456576
$431$	−39.0693	−1.88190	−0.940951	−	0.338542i	$-0.890066\pi$
$431$	−39.0693	−1.88190	−0.940951	+	0.338542i	$0.890066\pi$
$432$	0	0
$433$	20.6507	0.992409	0.496205	−	0.868206i	$-0.334727\pi$
$433$	20.6507	0.992409	0.496205	+	0.868206i	$0.334727\pi$
$434$	−4.77932	−0.229415
$435$	0	0
$436$	2.55692	0.122454
$437$	−23.7476	−1.13600
$438$	0	0
$439$	−36.3101	−1.73299	−0.866493	−	0.499189i	$-0.833631\pi$
$439$	−36.3101	−1.73299	−0.866493	+	0.499189i	$0.833631\pi$
$440$	7.02444	0.334877
$441$	0	0
$442$	19.1864	0.912606
$443$	−3.64725	−0.173286	−0.0866430	−	0.996239i	$-0.527614\pi$
$443$	−3.64725	−0.173286	−0.0866430	+	0.996239i	$0.527614\pi$
$444$	0	0
$445$	1.00759	0.0477645
$446$	0.719357	0.0340625
$447$	0	0
$448$	−25.3999	−1.20003
$449$	−38.3189	−1.80838	−0.904191	−	0.427129i	$-0.859525\pi$
$449$	−38.3189	−1.80838	−0.904191	+	0.427129i	$0.859525\pi$
$450$	0	0
$451$	−9.30697	−0.438248
$452$	−1.95955	−0.0921697
$453$	0	0
$454$	−7.45548	−0.349903
$455$	−32.3167	−1.51503
$456$	0	0
$457$	10.6266	0.497093	0.248547	−	0.968620i	$-0.420047\pi$
$457$	10.6266	0.497093	0.248547	+	0.968620i	$0.420047\pi$
$458$	−17.0907	−0.798597
$459$	0	0
$460$	−2.56174	−0.119442
$461$	24.2727	1.13049	0.565246	−	0.824923i	$-0.308781\pi$
$461$	24.2727	1.13049	0.565246	+	0.824923i	$0.308781\pi$
$462$	0	0
$463$	5.27536	0.245167	0.122583	−	0.992458i	$-0.460882\pi$
$463$	5.27536	0.245167	0.122583	+	0.992458i	$0.460882\pi$
$464$	−7.48246	−0.347365
$465$	0	0
$466$	−34.8256	−1.61327
$467$	−5.97949	−0.276698	−0.138349	−	0.990384i	$-0.544180\pi$
$467$	−5.97949	−0.276698	−0.138349	+	0.990384i	$0.544180\pi$
$468$	0	0
$469$	−26.2896	−1.21394
$470$	3.75525	0.173217
$471$	0	0
$472$	−26.5396	−1.22158
$473$	3.00386	0.138118
$474$	0	0
$475$	−3.05375	−0.140116
$476$	2.06202	0.0945125
$477$	0	0
$478$	−22.8680	−1.04596
$479$	10.7587	0.491579	0.245790	−	0.969323i	$-0.420953\pi$
$479$	10.7587	0.491579	0.245790	+	0.969323i	$0.420953\pi$
$480$	0	0
$481$	−26.7108	−1.21791
$482$	27.3540	1.24594
$483$	0	0
$484$	−0.228660	−0.0103936
$485$	12.5068	0.567907
$486$	0	0
$487$	27.3853	1.24095	0.620474	−	0.784227i	$-0.286940\pi$
$487$	27.3853	1.24095	0.620474	+	0.784227i	$0.286940\pi$
$488$	2.96616	0.134272
$489$	0	0
$490$	4.84218	0.218747
$491$	12.9998	0.586672	0.293336	−	0.956009i	$-0.405235\pi$
$491$	12.9998	0.586672	0.293336	+	0.956009i	$0.405235\pi$
$492$	0	0
$493$	6.61665	0.297999
$494$	−31.2047	−1.40396
$495$	0	0
$496$	−4.28998	−0.192626
$497$	43.4193	1.94762
$498$	0	0
$499$	−18.5160	−0.828889	−0.414444	−	0.910075i	$-0.636024\pi$
$499$	−18.5160	−0.828889	−0.414444	+	0.910075i	$0.636024\pi$
$500$	2.37814	0.106354
$501$	0	0
$502$	−30.1469	−1.34552
$503$	−19.8385	−0.884556	−0.442278	−	0.896878i	$-0.645830\pi$
$503$	−19.8385	−0.884556	−0.442278	+	0.896878i	$0.645830\pi$
$504$	0	0
$505$	42.8576	1.90714
$506$	−6.29619	−0.279900
$507$	0	0
$508$	−0.112022	−0.00497017
$509$	9.34304	0.414123	0.207061	−	0.978328i	$-0.433610\pi$
$509$	9.34304	0.414123	0.207061	+	0.978328i	$0.433610\pi$
$510$	0	0
$511$	−28.6320	−1.26661
$512$	−25.1979	−1.11360
$513$	0	0
$514$	16.3601	0.721615
$515$	12.8072	0.564352
$516$	0	0
$517$	−1.19144	−0.0523993
$518$	22.2380	0.977084
$519$	0	0
$520$	−32.8086	−1.43875
$521$	18.6247	0.815961	0.407980	−	0.912991i	$-0.366233\pi$
$521$	18.6247	0.815961	0.407980	+	0.912991i	$0.366233\pi$
$522$	0	0
$523$	7.84831	0.343183	0.171591	−	0.985168i	$-0.445109\pi$
$523$	7.84831	0.343183	0.171591	+	0.985168i	$0.445109\pi$
$524$	−0.973579	−0.0425310
$525$	0	0
$526$	−15.8688	−0.691915
$527$	3.79357	0.165251
$528$	0	0
$529$	−0.620292	−0.0269692
$530$	14.6972	0.638405
$531$	0	0
$532$	−3.35365	−0.145399
$533$	43.4694	1.88287
$534$	0	0
$535$	12.2691	0.530438
$536$	−26.6897	−1.15282
$537$	0	0
$538$	10.8623	0.468305
$539$	−1.53629	−0.0661727
$540$	0	0
$541$	−28.4103	−1.22145	−0.610726	−	0.791842i	$-0.709122\pi$
$541$	−28.4103	−1.22145	−0.610726	+	0.791842i	$0.709122\pi$
$542$	36.8642	1.58345
$543$	0	0
$544$	3.97203	0.170299
$545$	−26.4816	−1.13435
$546$	0	0
$547$	−30.5691	−1.30704	−0.653520	−	0.756909i	$-0.726708\pi$
$547$	−30.5691	−1.30704	−0.653520	+	0.756909i	$0.726708\pi$
$548$	−4.77984	−0.204185
$549$	0	0
$550$	−0.809639	−0.0345231
$551$	−10.7613	−0.458445
$552$	0	0
$553$	−34.9715	−1.48714
$554$	−2.38202	−0.101202
$555$	0	0
$556$	−2.33544	−0.0990449
$557$	−28.3132	−1.19967	−0.599835	−	0.800124i	$-0.704767\pi$
$557$	−28.3132	−1.19967	−0.599835	+	0.800124i	$0.704767\pi$
$558$	0	0
$559$	−14.0299	−0.593402
$560$	24.1505	1.02054
$561$	0	0
$562$	15.9555	0.673041
$563$	−35.6071	−1.50066	−0.750331	−	0.661063i	$-0.770106\pi$
$563$	−35.6071	−1.50066	−0.750331	+	0.661063i	$0.770106\pi$
$564$	0	0
$565$	20.2947	0.853806
$566$	−24.0479	−1.01081
$567$	0	0
$568$	44.0801	1.84956
$569$	14.0979	0.591015	0.295507	−	0.955340i	$-0.404511\pi$
$569$	14.0979	0.591015	0.295507	+	0.955340i	$0.404511\pi$
$570$	0	0
$571$	−21.3065	−0.891648	−0.445824	−	0.895121i	$-0.647089\pi$
$571$	−21.3065	−0.891648	−0.445824	+	0.895121i	$0.647089\pi$
$572$	1.06799	0.0446548
$573$	0	0
$574$	−36.1904	−1.51056
$575$	2.87785	0.120014
$576$	0	0
$577$	33.3841	1.38980	0.694900	−	0.719107i	$-0.255449\pi$
$577$	33.3841	1.38980	0.694900	+	0.719107i	$0.255449\pi$
$578$	−9.94654	−0.413722
$579$	0	0
$580$	−1.16085	−0.0482018
$581$	37.8390	1.56982
$582$	0	0
$583$	−4.66301	−0.193122
$584$	−29.0678	−1.20283
$585$	0	0
$586$	−31.1554	−1.28702
$587$	−7.08280	−0.292338	−0.146169	−	0.989260i	$-0.546694\pi$
$587$	−7.08280	−0.292338	−0.146169	+	0.989260i	$0.546694\pi$
$588$	0	0
$589$	−6.16984	−0.254224
$590$	28.2012	1.16103
$591$	0	0
$592$	19.9612	0.820398
$593$	24.1694	0.992519	0.496260	−	0.868174i	$-0.334706\pi$
$593$	24.1694	0.992519	0.496260	+	0.868174i	$0.334706\pi$
$594$	0	0
$595$	−21.3560	−0.875509
$596$	−3.04385	−0.124681
$597$	0	0
$598$	29.4072	1.20255
$599$	−15.8654	−0.648242	−0.324121	−	0.946016i	$-0.605068\pi$
$599$	−15.8654	−0.648242	−0.324121	+	0.946016i	$0.605068\pi$
$600$	0	0
$601$	−21.4050	−0.873128	−0.436564	−	0.899673i	$-0.643805\pi$
$601$	−21.4050	−0.873128	−0.436564	+	0.899673i	$0.643805\pi$
$602$	11.6806	0.476065
$603$	0	0
$604$	3.23483	0.131623
$605$	2.36819	0.0962807
$606$	0	0
$607$	37.9016	1.53838	0.769189	−	0.639021i	$-0.220660\pi$
$607$	37.9016	1.53838	0.769189	+	0.639021i	$0.220660\pi$
$608$	−6.46007	−0.261991
$609$	0	0
$610$	−3.15187	−0.127615
$611$	5.56476	0.225126
$612$	0	0
$613$	−5.28282	−0.213371	−0.106686	−	0.994293i	$-0.534024\pi$
$613$	−5.28282	−0.213371	−0.106686	+	0.994293i	$0.534024\pi$
$614$	11.2866	0.455489
$615$	0	0
$616$	−8.66621	−0.349172
$617$	44.2406	1.78106	0.890529	−	0.454926i	$-0.150334\pi$
$617$	44.2406	1.78106	0.890529	+	0.454926i	$0.150334\pi$
$618$	0	0
$619$	23.8962	0.960468	0.480234	−	0.877140i	$-0.340552\pi$
$619$	23.8962	0.960468	0.480234	+	0.877140i	$0.340552\pi$
$620$	−0.665561	−0.0267296
$621$	0	0
$622$	21.5266	0.863139
$623$	−1.24309	−0.0498034
$624$	0	0
$625$	−27.6716	−1.10686
$626$	0.953729	0.0381187
$627$	0	0
$628$	−3.49052	−0.139287
$629$	−17.6514	−0.703807
$630$	0	0
$631$	−4.59242	−0.182821	−0.0914106	−	0.995813i	$-0.529138\pi$
$631$	−4.59242	−0.182821	−0.0914106	+	0.995813i	$0.529138\pi$
$632$	−35.5038	−1.41226
$633$	0	0
$634$	3.00931	0.119515
$635$	1.16019	0.0460408
$636$	0	0
$637$	7.17544	0.284301
$638$	−2.85313	−0.112956
$639$	0	0
$640$	21.3056	0.842179
$641$	20.0957	0.793731	0.396865	−	0.917877i	$-0.370098\pi$
$641$	20.0957	0.793731	0.396865	+	0.917877i	$0.370098\pi$
$642$	0	0
$643$	25.7989	1.01741	0.508705	−	0.860941i	$-0.330124\pi$
$643$	25.7989	1.01741	0.508705	+	0.860941i	$0.330124\pi$
$644$	3.16047	0.124540
$645$	0	0
$646$	−20.6211	−0.811327
$647$	44.0280	1.73092	0.865460	−	0.500977i	$-0.167026\pi$
$647$	44.0280	1.73092	0.865460	+	0.500977i	$0.167026\pi$
$648$	0	0
$649$	−8.94746	−0.351219
$650$	3.78152	0.148324
$651$	0	0
$652$	−4.12982	−0.161736
$653$	5.99511	0.234607	0.117303	−	0.993096i	$-0.462575\pi$
$653$	5.99511	0.234607	0.117303	+	0.993096i	$0.462575\pi$
$654$	0	0
$655$	10.0832	0.393982
$656$	−32.4850	−1.26833
$657$	0	0
$658$	−4.63294	−0.180611
$659$	−36.8221	−1.43439	−0.717193	−	0.696874i	$-0.754574\pi$
$659$	−36.8221	−1.43439	−0.717193	+	0.696874i	$0.754574\pi$
$660$	0	0
$661$	34.6404	1.34736	0.673678	−	0.739025i	$-0.264714\pi$
$661$	34.6404	1.34736	0.673678	+	0.739025i	$0.264714\pi$
$662$	−15.7593	−0.612502
$663$	0	0
$664$	38.4149	1.49079
$665$	34.7332	1.34690
$666$	0	0
$667$	10.1414	0.392676
$668$	−4.31616	−0.166997
$669$	0	0
$670$	28.3607	1.09567
$671$	1.00000	0.0386046
$672$	0	0
$673$	−35.1235	−1.35391	−0.676955	−	0.736024i	$-0.736701\pi$
$673$	−35.1235	−1.35391	−0.676955	+	0.736024i	$0.736701\pi$
$674$	40.5199	1.56077
$675$	0	0
$676$	−2.01559	−0.0775228
$677$	−46.6369	−1.79240	−0.896200	−	0.443650i	$-0.853683\pi$
$677$	−46.6369	−1.79240	−0.896200	+	0.443650i	$0.853683\pi$
$678$	0	0
$679$	−15.4300	−0.592148
$680$	−21.6810	−0.831429
$681$	0	0
$682$	−1.63581	−0.0626382
$683$	26.3591	1.00860	0.504301	−	0.863528i	$-0.331750\pi$
$683$	26.3591	1.00860	0.504301	+	0.863528i	$0.331750\pi$
$684$	0	0
$685$	49.5039	1.89145
$686$	21.2458	0.811169
$687$	0	0
$688$	10.4846	0.399723
$689$	21.7792	0.829722
$690$	0	0
$691$	−43.4335	−1.65229	−0.826145	−	0.563458i	$-0.809471\pi$
$691$	−43.4335	−1.65229	−0.826145	+	0.563458i	$0.809471\pi$
$692$	4.74230	0.180275
$693$	0	0
$694$	−17.1743	−0.651926
$695$	24.1878	0.917494
$696$	0	0
$697$	28.7261	1.08808
$698$	−24.0222	−0.909253
$699$	0	0
$700$	0.406411	0.0153609
$701$	29.6826	1.12110	0.560548	−	0.828122i	$-0.310591\pi$
$701$	29.6826	1.12110	0.560548	+	0.828122i	$0.310591\pi$
$702$	0	0
$703$	28.7081	1.08275
$704$	−8.69354	−0.327650
$705$	0	0
$706$	−15.1510	−0.570215
$707$	−52.8744	−1.98855
$708$	0	0
$709$	32.3827	1.21616	0.608079	−	0.793877i	$-0.291940\pi$
$709$	32.3827	1.21616	0.608079	+	0.793877i	$0.291940\pi$
$710$	−46.8399	−1.75787
$711$	0	0
$712$	−1.26201	−0.0472959
$713$	5.81444	0.217752
$714$	0	0
$715$	−11.0609	−0.413656
$716$	2.49279	0.0931600
$717$	0	0
$718$	−4.68031	−0.174668
$719$	38.1158	1.42148	0.710741	−	0.703454i	$-0.248360\pi$
$719$	38.1158	1.42148	0.710741	+	0.703454i	$0.248360\pi$
$720$	0	0
$721$	−15.8005	−0.588442
$722$	8.25059	0.307055
$723$	0	0
$724$	−2.21440	−0.0822977
$725$	1.30410	0.0484330
$726$	0	0
$727$	−32.9791	−1.22313	−0.611564	−	0.791195i	$-0.709459\pi$
$727$	−32.9791	−1.22313	−0.611564	+	0.791195i	$0.709459\pi$
$728$	40.4767	1.50017
$729$	0	0
$730$	30.8877	1.14320
$731$	−9.27144	−0.342917
$732$	0	0
$733$	22.0044	0.812751	0.406376	−	0.913706i	$-0.366792\pi$
$733$	22.0044	0.812751	0.406376	+	0.913706i	$0.366792\pi$
$734$	−8.60510	−0.317620
$735$	0	0
$736$	6.08796	0.224405
$737$	−8.99807	−0.331448
$738$	0	0
$739$	2.36992	0.0871789	0.0435894	−	0.999050i	$-0.486121\pi$
$739$	2.36992	0.0871789	0.0435894	+	0.999050i	$0.486121\pi$
$740$	3.09684	0.113842
$741$	0	0
$742$	−18.1323	−0.665656
$743$	−22.5722	−0.828094	−0.414047	−	0.910255i	$-0.635885\pi$
$743$	−22.5722	−0.828094	−0.414047	+	0.910255i	$0.635885\pi$
$744$	0	0
$745$	31.5246	1.15497
$746$	6.01092	0.220076
$747$	0	0
$748$	0.705762	0.0258052
$749$	−15.1366	−0.553080
$750$	0	0
$751$	4.61181	0.168287	0.0841437	−	0.996454i	$-0.473185\pi$
$751$	4.61181	0.168287	0.0841437	+	0.996454i	$0.473185\pi$
$752$	−4.15858	−0.151648
$753$	0	0
$754$	13.3259	0.485301
$755$	−33.5025	−1.21928
$756$	0	0
$757$	−2.35032	−0.0854237	−0.0427119	−	0.999087i	$-0.513600\pi$
$757$	−2.35032	−0.0854237	−0.0427119	+	0.999087i	$0.513600\pi$
$758$	−34.8004	−1.26401
$759$	0	0
$760$	35.2618	1.27908
$761$	−45.2527	−1.64041	−0.820204	−	0.572071i	$-0.806140\pi$
$761$	−45.2527	−1.64041	−0.820204	+	0.572071i	$0.806140\pi$
$762$	0	0
$763$	32.6709	1.18277
$764$	−2.47312	−0.0894744
$765$	0	0
$766$	31.0186	1.12075
$767$	41.7903	1.50896
$768$	0	0
$769$	−19.5824	−0.706158	−0.353079	−	0.935594i	$-0.614865\pi$
$769$	−19.5824	−0.706158	−0.353079	+	0.935594i	$0.614865\pi$
$770$	9.20879	0.331862
$771$	0	0
$772$	1.41669	0.0509878
$773$	20.2345	0.727785	0.363892	−	0.931441i	$-0.381448\pi$
$773$	20.2345	0.727785	0.363892	+	0.931441i	$0.381448\pi$
$774$	0	0
$775$	0.747689	0.0268578
$776$	−15.6648	−0.562335
$777$	0	0
$778$	−25.7313	−0.922513
$779$	−46.7199	−1.67391
$780$	0	0
$781$	14.8610	0.531768
$782$	19.4333	0.694932
$783$	0	0
$784$	−5.36226	−0.191509
$785$	36.1507	1.29027
$786$	0	0
$787$	19.3105	0.688344	0.344172	−	0.938907i	$-0.388160\pi$
$787$	19.3105	0.688344	0.344172	+	0.938907i	$0.388160\pi$
$788$	−0.262949	−0.00936716
$789$	0	0
$790$	37.7266	1.34225
$791$	−25.0381	−0.890252
$792$	0	0
$793$	−4.67063	−0.165859
$794$	11.9565	0.424320
$795$	0	0
$796$	−3.15690	−0.111893
$797$	−12.0619	−0.427253	−0.213627	−	0.976915i	$-0.568528\pi$
$797$	−12.0619	−0.427253	−0.213627	+	0.976915i	$0.568528\pi$
$798$	0	0
$799$	3.67738	0.130096
$800$	0.782861	0.0276783
$801$	0	0
$802$	8.26494	0.291845
$803$	−9.79980	−0.345827
$804$	0	0
$805$	−32.7325	−1.15367
$806$	7.64024	0.269116
$807$	0	0
$808$	−53.6791	−1.88843
$809$	−35.8603	−1.26078	−0.630390	−	0.776278i	$-0.717105\pi$
$809$	−35.8603	−1.26078	−0.630390	+	0.776278i	$0.717105\pi$
$810$	0	0
$811$	43.1282	1.51444	0.757218	−	0.653163i	$-0.226558\pi$
$811$	43.1282	1.51444	0.757218	+	0.653163i	$0.226558\pi$
$812$	1.43217	0.0502594
$813$	0	0
$814$	7.61136	0.266778
$815$	42.7718	1.49823
$816$	0	0
$817$	15.0790	0.527547
$818$	36.1964	1.26558
$819$	0	0
$820$	−5.03983	−0.175998
$821$	21.6892	0.756957	0.378479	−	0.925610i	$-0.376447\pi$
$821$	21.6892	0.756957	0.378479	+	0.925610i	$0.376447\pi$
$822$	0	0
$823$	6.12406	0.213471	0.106736	−	0.994287i	$-0.465960\pi$
$823$	6.12406	0.213471	0.106736	+	0.994287i	$0.465960\pi$
$824$	−16.0410	−0.558815
$825$	0	0
$826$	−34.7925	−1.21059
$827$	−50.6057	−1.75973	−0.879866	−	0.475221i	$-0.842368\pi$
$827$	−50.6057	−1.75973	−0.879866	+	0.475221i	$0.842368\pi$
$828$	0	0
$829$	−15.7599	−0.547365	−0.273682	−	0.961820i	$-0.588242\pi$
$829$	−15.7599	−0.547365	−0.273682	+	0.961820i	$0.588242\pi$
$830$	−40.8199	−1.41688
$831$	0	0
$832$	40.6043	1.40770
$833$	4.74178	0.164293
$834$	0	0
$835$	44.7016	1.54696
$836$	−1.14785	−0.0396991
$837$	0	0
$838$	−22.7821	−0.786996
$839$	20.8938	0.721334	0.360667	−	0.932695i	$-0.382549\pi$
$839$	20.8938	0.721334	0.360667	+	0.932695i	$0.382549\pi$
$840$	0	0
$841$	−24.4044	−0.841532
$842$	−21.8563	−0.753217
$843$	0	0
$844$	−1.98290	−0.0682541
$845$	20.8751	0.718126
$846$	0	0
$847$	−2.92169	−0.100391
$848$	−16.2757	−0.558911
$849$	0	0
$850$	2.49896	0.0857136
$851$	−27.0544	−0.927414
$852$	0	0
$853$	19.2873	0.660383	0.330192	−	0.943914i	$-0.392887\pi$
$853$	19.2873	0.660383	0.330192	+	0.943914i	$0.392887\pi$
$854$	3.88853	0.133063
$855$	0	0
$856$	−15.3670	−0.525233
$857$	−4.94606	−0.168954	−0.0844771	−	0.996425i	$-0.526922\pi$
$857$	−4.94606	−0.168954	−0.0844771	+	0.996425i	$0.526922\pi$
$858$	0	0
$859$	−28.0355	−0.956558	−0.478279	−	0.878208i	$-0.658739\pi$
$859$	−28.0355	−0.956558	−0.478279	+	0.878208i	$0.658739\pi$
$860$	1.62662	0.0554674
$861$	0	0
$862$	−51.9980	−1.77106
$863$	52.8568	1.79927	0.899634	−	0.436645i	$-0.143833\pi$
$863$	52.8568	1.79927	0.899634	+	0.436645i	$0.143833\pi$
$864$	0	0
$865$	−49.1152	−1.66997
$866$	27.4844	0.933957
$867$	0	0
$868$	0.821118	0.0278706
$869$	−11.9696	−0.406041
$870$	0	0
$871$	42.0267	1.42402
$872$	33.1682	1.12322
$873$	0	0
$874$	−31.6061	−1.06909
$875$	30.3865	1.02725
$876$	0	0
$877$	2.79315	0.0943180	0.0471590	−	0.998887i	$-0.484983\pi$
$877$	2.79315	0.0943180	0.0471590	+	0.998887i	$0.484983\pi$
$878$	−48.3257	−1.63091
$879$	0	0
$880$	8.26592	0.278644
$881$	−21.8113	−0.734843	−0.367421	−	0.930055i	$-0.619759\pi$
$881$	−21.8113	−0.734843	−0.367421	+	0.930055i	$0.619759\pi$
$882$	0	0
$883$	−6.55226	−0.220501	−0.110251	−	0.993904i	$-0.535165\pi$
$883$	−6.55226	−0.220501	−0.110251	+	0.993904i	$0.535165\pi$
$884$	−3.29635	−0.110868
$885$	0	0
$886$	−4.85418	−0.163080
$887$	−17.6112	−0.591325	−0.295662	−	0.955292i	$-0.595540\pi$
$887$	−17.6112	−0.591325	−0.295662	+	0.955292i	$0.595540\pi$
$888$	0	0
$889$	−1.43135	−0.0480061
$890$	1.34102	0.0449512
$891$	0	0
$892$	−0.123590	−0.00413811
$893$	−5.98087	−0.200142
$894$	0	0
$895$	−25.8174	−0.862980
$896$	−26.2853	−0.878129
$897$	0	0
$898$	−50.9993	−1.70187
$899$	2.63482	0.0878761
$900$	0	0
$901$	14.3924	0.479481
$902$	−12.3868	−0.412435
$903$	0	0
$904$	−25.4192	−0.845429
$905$	22.9342	0.762358
$906$	0	0
$907$	−31.0718	−1.03172	−0.515861	−	0.856672i	$-0.672528\pi$
$907$	−31.0718	−1.03172	−0.515861	+	0.856672i	$0.672528\pi$
$908$	1.28090	0.0425082
$909$	0	0
$910$	−43.0108	−1.42580
$911$	3.53312	0.117058	0.0585288	−	0.998286i	$-0.481359\pi$
$911$	3.53312	0.117058	0.0585288	+	0.998286i	$0.481359\pi$
$912$	0	0
$913$	12.9510	0.428617
$914$	14.1432	0.467815
$915$	0	0
$916$	2.93630	0.0970180
$917$	−12.4399	−0.410800
$918$	0	0
$919$	39.8129	1.31331	0.656653	−	0.754193i	$-0.271972\pi$
$919$	39.8129	1.31331	0.656653	+	0.754193i	$0.271972\pi$
$920$	−33.2307	−1.09558
$921$	0	0
$922$	32.3049	1.06391
$923$	−69.4102	−2.28467
$924$	0	0
$925$	−3.47898	−0.114388
$926$	7.02106	0.230726
$927$	0	0
$928$	2.75876	0.0905608
$929$	−26.4621	−0.868193	−0.434097	−	0.900866i	$-0.642932\pi$
$929$	−26.4621	−0.868193	−0.434097	+	0.900866i	$0.642932\pi$
$930$	0	0
$931$	−7.71199	−0.252750
$932$	5.98327	0.195989
$933$	0	0
$934$	−7.95821	−0.260401
$935$	−7.30945	−0.239045
$936$	0	0
$937$	−5.91019	−0.193077	−0.0965387	−	0.995329i	$-0.530777\pi$
$937$	−5.91019	−0.193077	−0.0965387	+	0.995329i	$0.530777\pi$
$938$	−34.9893	−1.14244
$939$	0	0
$940$	−0.645176	−0.0210433
$941$	41.2125	1.34349	0.671744	−	0.740784i	$-0.265546\pi$
$941$	41.2125	1.34349	0.671744	+	0.740784i	$0.265546\pi$
$942$	0	0
$943$	44.0287	1.43377
$944$	−31.2302	−1.01646
$945$	0	0
$946$	3.99788	0.129982
$947$	−3.35032	−0.108871	−0.0544354	−	0.998517i	$-0.517336\pi$
$947$	−3.35032	−0.108871	−0.0544354	+	0.998517i	$0.517336\pi$
$948$	0	0
$949$	45.7712	1.48580
$950$	−4.06429	−0.131863
$951$	0	0
$952$	26.7484	0.866919
$953$	7.53957	0.244231	0.122115	−	0.992516i	$-0.461032\pi$
$953$	7.53957	0.244231	0.122115	+	0.992516i	$0.461032\pi$
$954$	0	0
$955$	25.6137	0.828838
$956$	3.92888	0.127069
$957$	0	0
$958$	14.3190	0.462626
$959$	−61.0741	−1.97219
$960$	0	0
$961$	−29.4894	−0.951270
$962$	−35.5498	−1.14617
$963$	0	0
$964$	−4.69959	−0.151364
$965$	−14.6724	−0.472322
$966$	0	0
$967$	−36.1572	−1.16274	−0.581369	−	0.813640i	$-0.697483\pi$
$967$	−36.1572	−1.16274	−0.581369	+	0.813640i	$0.697483\pi$
$968$	−2.96616	−0.0953360
$969$	0	0
$970$	16.6456	0.534457
$971$	20.8300	0.668468	0.334234	−	0.942490i	$-0.391522\pi$
$971$	20.8300	0.668468	0.334234	+	0.942490i	$0.391522\pi$
$972$	0	0
$973$	−29.8410	−0.956659
$974$	36.4476	1.16786
$975$	0	0
$976$	3.49039	0.111725
$977$	−23.9880	−0.767444	−0.383722	−	0.923449i	$-0.625358\pi$
$977$	−23.9880	−0.767444	−0.383722	+	0.923449i	$0.625358\pi$
$978$	0	0
$979$	−0.425469	−0.0135981
$980$	−0.831918	−0.0265746
$981$	0	0
$982$	17.3016	0.552117
$983$	1.23884	0.0395129	0.0197565	−	0.999805i	$-0.493711\pi$
$983$	1.23884	0.0395129	0.0197565	+	0.999805i	$0.493711\pi$
$984$	0	0
$985$	2.72331	0.0867719
$986$	8.80621	0.280447
$987$	0	0
$988$	5.36117	0.170561
$989$	−14.2104	−0.451865
$990$	0	0
$991$	2.76799	0.0879281	0.0439641	−	0.999033i	$-0.486001\pi$
$991$	2.76799	0.0879281	0.0439641	+	0.999033i	$0.486001\pi$
$992$	1.58170	0.0502191
$993$	0	0
$994$	57.7874	1.83291
$995$	32.6955	1.03652
$996$	0	0
$997$	27.1886	0.861071	0.430536	−	0.902574i	$-0.358325\pi$
$997$	27.1886	0.861071	0.430536	+	0.902574i	$0.358325\pi$
$998$	−24.6432	−0.780068
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.33092 q^{2} -0.228660 q^{4} +2.36819 q^{5} -2.92169 q^{7} -2.96616 q^{8} +O(q^{10})\)	\(q+1.33092 q^{2} -0.228660 q^{4} +2.36819 q^{5} -2.92169 q^{7} -2.96616 q^{8} +3.15187 q^{10} -1.00000 q^{11} +4.67063 q^{13} -3.88853 q^{14} -3.49039 q^{16} +3.08651 q^{17} -5.01988 q^{19} -0.541511 q^{20} -1.33092 q^{22} +4.73072 q^{23} +0.608331 q^{25} +6.21622 q^{26} +0.668075 q^{28} +2.14373 q^{29} +1.22908 q^{31} +1.28690 q^{32} +4.10789 q^{34} -6.91913 q^{35} -5.71888 q^{37} -6.68104 q^{38} -7.02444 q^{40} +9.30697 q^{41} -3.00386 q^{43} +0.228660 q^{44} +6.29619 q^{46} +1.19144 q^{47} +1.53629 q^{49} +0.809639 q^{50} -1.06799 q^{52} +4.66301 q^{53} -2.36819 q^{55} +8.66621 q^{56} +2.85313 q^{58} +8.94746 q^{59} -1.00000 q^{61} +1.63581 q^{62} +8.69354 q^{64} +11.0609 q^{65} +8.99807 q^{67} -0.705762 q^{68} -9.20879 q^{70} -14.8610 q^{71} +9.79980 q^{73} -7.61136 q^{74} +1.14785 q^{76} +2.92169 q^{77} +11.9696 q^{79} -8.26592 q^{80} +12.3868 q^{82} -12.9510 q^{83} +7.30945 q^{85} -3.99788 q^{86} +2.96616 q^{88} +0.425469 q^{89} -13.6462 q^{91} -1.08173 q^{92} +1.58570 q^{94} -11.8880 q^{95} +5.28118 q^{97} +2.04467 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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