LMFDB - Embedded newform 6027.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1197392.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 14x - 2 $ x^5 - 2x^4 - 8x^3 + 6x^2 + 14x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 861)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.136381$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.863619 q^{2} +1.00000 q^{3} -1.25416 q^{4} +3.66459 q^{5} +0.863619 q^{6} -2.81036 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.863619 q^{2} +1.00000 q^{3} -1.25416 q^{4} +3.66459 q^{5} +0.863619 q^{6} -2.81036 q^{8} +1.00000 q^{9} +3.16481 q^{10} -1.58958 q^{11} -1.25416 q^{12} -6.92942 q^{13} +3.66459 q^{15} +0.0812501 q^{16} +7.35419 q^{17} +0.863619 q^{18} -1.24478 q^{19} -4.59599 q^{20} -1.37279 q^{22} +3.92003 q^{23} -2.81036 q^{24} +8.42920 q^{25} -5.98437 q^{26} +1.00000 q^{27} +2.06137 q^{29} +3.16481 q^{30} -1.93438 q^{31} +5.69088 q^{32} -1.58958 q^{33} +6.35121 q^{34} -1.25416 q^{36} +10.5833 q^{37} -1.07501 q^{38} -6.92942 q^{39} -10.2988 q^{40} +1.00000 q^{41} -0.664587 q^{43} +1.99359 q^{44} +3.66459 q^{45} +3.38541 q^{46} -11.5991 q^{47} +0.0812501 q^{48} +7.27961 q^{50} +7.35419 q^{51} +8.69062 q^{52} +7.59768 q^{53} +0.863619 q^{54} -5.82514 q^{55} -1.24478 q^{57} +1.78024 q^{58} +0.881376 q^{59} -4.59599 q^{60} +11.8368 q^{61} -1.67056 q^{62} +4.75225 q^{64} -25.3935 q^{65} -1.37279 q^{66} -5.51360 q^{67} -9.22335 q^{68} +3.92003 q^{69} +7.03440 q^{71} -2.81036 q^{72} +8.31996 q^{73} +9.13997 q^{74} +8.42920 q^{75} +1.56115 q^{76} -5.98437 q^{78} -1.50124 q^{79} +0.297748 q^{80} +1.00000 q^{81} +0.863619 q^{82} +10.6524 q^{83} +26.9501 q^{85} -0.573950 q^{86} +2.06137 q^{87} +4.46727 q^{88} +0.808487 q^{89} +3.16481 q^{90} -4.91636 q^{92} -1.93438 q^{93} -10.0172 q^{94} -4.56159 q^{95} +5.69088 q^{96} -1.66587 q^{97} -1.58958 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 5 q^{3} + 11 q^{4} + 9 q^{5} + 3 q^{6} + 9 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q + 3 * q^2 + 5 * q^3 + 11 * q^4 + 9 * q^5 + 3 * q^6 + 9 * q^8 + 5 * q^9	$ 5 q + 3 q^{2} + 5 q^{3} + 11 q^{4} + 9 q^{5} + 3 q^{6} + 9 q^{8} + 5 q^{9} - 9 q^{10} + 11 q^{12} + 3 q^{13} + 9 q^{15} + 27 q^{16} + 16 q^{17} + 3 q^{18} - 4 q^{19} + 7 q^{20} - 6 q^{22} - 3 q^{23} + 9 q^{24} + 20 q^{25} - 17 q^{26} + 5 q^{27} + 13 q^{29} - 9 q^{30} + 4 q^{31} + 21 q^{32} + 4 q^{34} + 11 q^{36} + 17 q^{37} - 4 q^{38} + 3 q^{39} - 37 q^{40} + 5 q^{41} + 6 q^{43} + 32 q^{44} + 9 q^{45} + 27 q^{46} + 15 q^{47} + 27 q^{48} - 14 q^{50} + 16 q^{51} + 17 q^{52} + 11 q^{53} + 3 q^{54} + 16 q^{55} - 4 q^{57} + 9 q^{58} - 12 q^{59} + 7 q^{60} - 12 q^{61} - 8 q^{62} + 19 q^{64} - 19 q^{65} - 6 q^{66} + 11 q^{67} + 28 q^{68} - 3 q^{69} + 18 q^{71} + 9 q^{72} - 12 q^{73} - 27 q^{74} + 20 q^{75} + 26 q^{76} - 17 q^{78} + 23 q^{79} + 7 q^{80} + 5 q^{81} + 3 q^{82} + 2 q^{83} + 20 q^{85} + 18 q^{86} + 13 q^{87} - 22 q^{88} + 28 q^{89} - 9 q^{90} - 7 q^{92} + 4 q^{93} + 3 q^{94} - 10 q^{95} + 21 q^{96} - 3 q^{97}+O(q^{100}) $ 5 * q + 3 * q^2 + 5 * q^3 + 11 * q^4 + 9 * q^5 + 3 * q^6 + 9 * q^8 + 5 * q^9 - 9 * q^10 + 11 * q^12 + 3 * q^13 + 9 * q^15 + 27 * q^16 + 16 * q^17 + 3 * q^18 - 4 * q^19 + 7 * q^20 - 6 * q^22 - 3 * q^23 + 9 * q^24 + 20 * q^25 - 17 * q^26 + 5 * q^27 + 13 * q^29 - 9 * q^30 + 4 * q^31 + 21 * q^32 + 4 * q^34 + 11 * q^36 + 17 * q^37 - 4 * q^38 + 3 * q^39 - 37 * q^40 + 5 * q^41 + 6 * q^43 + 32 * q^44 + 9 * q^45 + 27 * q^46 + 15 * q^47 + 27 * q^48 - 14 * q^50 + 16 * q^51 + 17 * q^52 + 11 * q^53 + 3 * q^54 + 16 * q^55 - 4 * q^57 + 9 * q^58 - 12 * q^59 + 7 * q^60 - 12 * q^61 - 8 * q^62 + 19 * q^64 - 19 * q^65 - 6 * q^66 + 11 * q^67 + 28 * q^68 - 3 * q^69 + 18 * q^71 + 9 * q^72 - 12 * q^73 - 27 * q^74 + 20 * q^75 + 26 * q^76 - 17 * q^78 + 23 * q^79 + 7 * q^80 + 5 * q^81 + 3 * q^82 + 2 * q^83 + 20 * q^85 + 18 * q^86 + 13 * q^87 - 22 * q^88 + 28 * q^89 - 9 * q^90 - 7 * q^92 + 4 * q^93 + 3 * q^94 - 10 * q^95 + 21 * q^96 - 3 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.863619	0.610671	0.305335	−	0.952245i	$-0.401231\pi$
$2$	0.863619	0.610671	0.305335	+	0.952245i	$0.401231\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.25416	−0.627081
$5$	3.66459	1.63885	0.819427	−	0.573184i	$-0.194292\pi$
$5$	3.66459	1.63885	0.819427	+	0.573184i	$0.194292\pi$
$6$	0.863619	0.352571
$7$	0	0
$7$	0	0
$8$	−2.81036	−0.993611
$9$	1.00000	0.333333
$10$	3.16481	1.00080
$11$	−1.58958	−0.479275	−0.239638	−	0.970862i	$-0.577029\pi$
$11$	−1.58958	−0.479275	−0.239638	+	0.970862i	$0.577029\pi$
$12$	−1.25416	−0.362046
$13$	−6.92942	−1.92187	−0.960937	−	0.276766i	$-0.910737\pi$
$13$	−6.92942	−1.92187	−0.960937	+	0.276766i	$0.910737\pi$
$14$	0	0
$15$	3.66459	0.946192
$16$	0.0812501	0.0203125
$17$	7.35419	1.78365	0.891826	−	0.452378i	$-0.149424\pi$
$17$	7.35419	1.78365	0.891826	+	0.452378i	$0.149424\pi$
$18$	0.863619	0.203557
$19$	−1.24478	−0.285571	−0.142786	−	0.989754i	$-0.545606\pi$
$19$	−1.24478	−0.285571	−0.142786	+	0.989754i	$0.545606\pi$
$20$	−4.59599	−1.02769
$21$	0	0
$22$	−1.37279	−0.292679
$23$	3.92003	0.817383	0.408691	−	0.912673i	$-0.365985\pi$
$23$	3.92003	0.817383	0.408691	+	0.912673i	$0.365985\pi$
$24$	−2.81036	−0.573661
$25$	8.42920	1.68584
$26$	−5.98437	−1.17363
$27$	1.00000	0.192450
$28$	0	0
$29$	2.06137	0.382787	0.191393	−	0.981513i	$-0.438699\pi$
$29$	2.06137	0.382787	0.191393	+	0.981513i	$0.438699\pi$
$30$	3.16481	0.577812
$31$	−1.93438	−0.347424	−0.173712	−	0.984796i	$-0.555576\pi$
$31$	−1.93438	−0.347424	−0.173712	+	0.984796i	$0.555576\pi$
$32$	5.69088	1.00602
$33$	−1.58958	−0.276710
$34$	6.35121	1.08922
$35$	0	0
$36$	−1.25416	−0.209027
$37$	10.5833	1.73989	0.869945	−	0.493149i	$-0.164154\pi$
$37$	10.5833	1.73989	0.869945	+	0.493149i	$0.164154\pi$
$38$	−1.07501	−0.174390
$39$	−6.92942	−1.10959
$40$	−10.2988	−1.62838
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−0.664587	−0.101349	−0.0506743	−	0.998715i	$-0.516137\pi$
$43$	−0.664587	−0.101349	−0.0506743	+	0.998715i	$0.516137\pi$
$44$	1.99359	0.300545
$45$	3.66459	0.546284
$46$	3.38541	0.499152
$47$	−11.5991	−1.69191	−0.845954	−	0.533256i	$-0.820968\pi$
$47$	−11.5991	−1.69191	−0.845954	+	0.533256i	$0.820968\pi$
$48$	0.0812501	0.0117274
$49$	0	0
$50$	7.27961	1.02949
$51$	7.35419	1.02979
$52$	8.69062	1.20517
$53$	7.59768	1.04362	0.521811	−	0.853061i	$-0.325257\pi$
$53$	7.59768	1.04362	0.521811	+	0.853061i	$0.325257\pi$
$54$	0.863619	0.117524
$55$	−5.82514	−0.785462
$56$	0	0
$57$	−1.24478	−0.164875
$58$	1.78024	0.233757
$59$	0.881376	0.114745	0.0573727	−	0.998353i	$-0.481728\pi$
$59$	0.881376	0.114745	0.0573727	+	0.998353i	$0.481728\pi$
$60$	−4.59599	−0.593340
$61$	11.8368	1.51555	0.757775	−	0.652516i	$-0.226286\pi$
$61$	11.8368	1.51555	0.757775	+	0.652516i	$0.226286\pi$
$62$	−1.67056	−0.212162
$63$	0	0
$64$	4.75225	0.594031
$65$	−25.3935	−3.14967
$66$	−1.37279	−0.168978
$67$	−5.51360	−0.673593	−0.336797	−	0.941577i	$-0.609343\pi$
$67$	−5.51360	−0.673593	−0.336797	+	0.941577i	$0.609343\pi$
$68$	−9.22335	−1.11850
$69$	3.92003	0.471916
$70$	0	0
$71$	7.03440	0.834830	0.417415	−	0.908716i	$-0.362936\pi$
$71$	7.03440	0.834830	0.417415	+	0.908716i	$0.362936\pi$
$72$	−2.81036	−0.331204
$73$	8.31996	0.973778	0.486889	−	0.873464i	$-0.338132\pi$
$73$	8.31996	0.973778	0.486889	+	0.873464i	$0.338132\pi$
$74$	9.13997	1.06250
$75$	8.42920	0.973320
$76$	1.56115	0.179076
$77$	0	0
$78$	−5.98437	−0.677597
$79$	−1.50124	−0.168902	−0.0844511	−	0.996428i	$-0.526914\pi$
$79$	−1.50124	−0.168902	−0.0844511	+	0.996428i	$0.526914\pi$
$80$	0.297748	0.0332892
$81$	1.00000	0.111111
$82$	0.863619	0.0953707
$83$	10.6524	1.16925	0.584626	−	0.811303i	$-0.301241\pi$
$83$	10.6524	1.16925	0.584626	+	0.811303i	$0.301241\pi$
$84$	0	0
$85$	26.9501	2.92314
$86$	−0.573950	−0.0618906
$87$	2.06137	0.221002
$88$	4.46727	0.476213
$89$	0.808487	0.0856995	0.0428497	−	0.999082i	$-0.486356\pi$
$89$	0.808487	0.0856995	0.0428497	+	0.999082i	$0.486356\pi$
$90$	3.16481	0.333600
$91$	0	0
$92$	−4.91636	−0.512566
$93$	−1.93438	−0.200585
$94$	−10.0172	−1.03320
$95$	−4.56159	−0.468009
$96$	5.69088	0.580823
$97$	−1.66587	−0.169143	−0.0845716	−	0.996417i	$-0.526952\pi$
$97$	−1.66587	−0.169143	−0.0845716	+	0.996417i	$0.526952\pi$
$98$	0	0
$99$	−1.58958	−0.159758
$100$	−10.5716	−1.05716
$101$	−2.61047	−0.259752	−0.129876	−	0.991530i	$-0.541458\pi$
$101$	−2.61047	−0.259752	−0.129876	+	0.991530i	$0.541458\pi$
$102$	6.35121	0.628864
$103$	16.2276	1.59896	0.799478	−	0.600695i	$-0.205110\pi$
$103$	16.2276	1.59896	0.799478	+	0.600695i	$0.205110\pi$
$104$	19.4741	1.90960
$105$	0	0
$106$	6.56150	0.637309
$107$	10.5687	1.02172	0.510859	−	0.859665i	$-0.329328\pi$
$107$	10.5687	1.02172	0.510859	+	0.859665i	$0.329328\pi$
$108$	−1.25416	−0.120682
$109$	16.8026	1.60940	0.804698	−	0.593684i	$-0.202327\pi$
$109$	16.8026	1.60940	0.804698	+	0.593684i	$0.202327\pi$
$110$	−5.03070	−0.479658
$111$	10.5833	1.00453
$112$	0	0
$113$	11.4856	1.08048	0.540238	−	0.841512i	$-0.318334\pi$
$113$	11.4856	1.08048	0.540238	+	0.841512i	$0.318334\pi$
$114$	−1.07501	−0.100684
$115$	14.3653	1.33957
$116$	−2.58529	−0.240038
$117$	−6.92942	−0.640625
$118$	0.761172	0.0700716
$119$	0	0
$120$	−10.2988	−0.940147
$121$	−8.47325	−0.770295
$122$	10.2225	0.925502
$123$	1.00000	0.0901670
$124$	2.42602	0.217863
$125$	12.5666	1.12399
$126$	0	0
$127$	−4.70196	−0.417232	−0.208616	−	0.977998i	$-0.566896\pi$
$127$	−4.70196	−0.417232	−0.208616	+	0.977998i	$0.566896\pi$
$128$	−7.27763	−0.643258
$129$	−0.664587	−0.0585136
$130$	−21.9303	−1.92341
$131$	−12.1982	−1.06576	−0.532880	−	0.846191i	$-0.678890\pi$
$131$	−12.1982	−1.06576	−0.532880	+	0.846191i	$0.678890\pi$
$132$	1.99359	0.173519
$133$	0	0
$134$	−4.76165	−0.411344
$135$	3.66459	0.315397
$136$	−20.6679	−1.77226
$137$	−9.07417	−0.775259	−0.387629	−	0.921815i	$-0.626706\pi$
$137$	−9.07417	−0.775259	−0.387629	+	0.921815i	$0.626706\pi$
$138$	3.38541	0.288185
$139$	−10.2364	−0.868241	−0.434120	−	0.900855i	$-0.642941\pi$
$139$	−10.2364	−0.868241	−0.434120	+	0.900855i	$0.642941\pi$
$140$	0	0
$141$	−11.5991	−0.976824
$142$	6.07504	0.509806
$143$	11.0148	0.921107
$144$	0.0812501	0.00677084
$145$	7.55407	0.627331
$146$	7.18527	0.594658
$147$	0	0
$148$	−13.2732	−1.09105
$149$	11.5863	0.949190	0.474595	−	0.880204i	$-0.342595\pi$
$149$	11.5863	0.949190	0.474595	+	0.880204i	$0.342595\pi$
$150$	7.27961	0.594378
$151$	−11.9752	−0.974525	−0.487262	−	0.873256i	$-0.662004\pi$
$151$	−11.9752	−0.974525	−0.487262	+	0.873256i	$0.662004\pi$
$152$	3.49826	0.283747
$153$	7.35419	0.594551
$154$	0	0
$155$	−7.08869	−0.569377
$156$	8.69062	0.695806
$157$	3.17206	0.253158	0.126579	−	0.991957i	$-0.459600\pi$
$157$	3.17206	0.253158	0.126579	+	0.991957i	$0.459600\pi$
$158$	−1.29650	−0.103144
$159$	7.59768	0.602535
$160$	20.8547	1.64871
$161$	0	0
$162$	0.863619	0.0678523
$163$	19.8001	1.55087	0.775433	−	0.631430i	$-0.217532\pi$
$163$	19.8001	1.55087	0.775433	+	0.631430i	$0.217532\pi$
$164$	−1.25416	−0.0979337
$165$	−5.82514	−0.453486
$166$	9.19961	0.714028
$167$	−24.4163	−1.88939	−0.944695	−	0.327949i	$-0.893642\pi$
$167$	−24.4163	−1.88939	−0.944695	+	0.327949i	$0.893642\pi$
$168$	0	0
$169$	35.0168	2.69360
$170$	23.2746	1.78508
$171$	−1.24478	−0.0951904
$172$	0.833500	0.0635538
$173$	−19.1568	−1.45646	−0.728232	−	0.685331i	$-0.759658\pi$
$173$	−19.1568	−1.45646	−0.728232	+	0.685331i	$0.759658\pi$
$174$	1.78024	0.134959
$175$	0	0
$176$	−0.129153	−0.00973528
$177$	0.881376	0.0662482
$178$	0.698225	0.0523342
$179$	−3.86166	−0.288634	−0.144317	−	0.989531i	$-0.546099\pi$
$179$	−3.86166	−0.288634	−0.144317	+	0.989531i	$0.546099\pi$
$180$	−4.59599	−0.342565
$181$	−0.813759	−0.0604862	−0.0302431	−	0.999543i	$-0.509628\pi$
$181$	−0.813759	−0.0604862	−0.0302431	+	0.999543i	$0.509628\pi$
$182$	0	0
$183$	11.8368	0.875003
$184$	−11.0167	−0.812160
$185$	38.7836	2.85142
$186$	−1.67056	−0.122492
$187$	−11.6900	−0.854860
$188$	14.5472	1.06096
$189$	0	0
$190$	−3.93947	−0.285799
$191$	9.00956	0.651909	0.325955	−	0.945385i	$-0.394314\pi$
$191$	9.00956	0.651909	0.325955	+	0.945385i	$0.394314\pi$
$192$	4.75225	0.342964
$193$	13.6085	0.979563	0.489782	−	0.871845i	$-0.337077\pi$
$193$	13.6085	0.979563	0.489782	+	0.871845i	$0.337077\pi$
$194$	−1.43867	−0.103291
$195$	−25.3935	−1.81846
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−1.37279	−0.0975597
$199$	−22.7992	−1.61619	−0.808096	−	0.589051i	$-0.799502\pi$
$199$	−22.7992	−1.61619	−0.808096	+	0.589051i	$0.799502\pi$
$200$	−23.6890	−1.67507
$201$	−5.51360	−0.388899
$202$	−2.25445	−0.158623
$203$	0	0
$204$	−9.22335	−0.645764
$205$	3.66459	0.255946
$206$	14.0145	0.976435
$207$	3.92003	0.272461
$208$	−0.563016	−0.0390381
$209$	1.97867	0.136867
$210$	0	0
$211$	15.4799	1.06568	0.532840	−	0.846216i	$-0.321125\pi$
$211$	15.4799	1.06568	0.532840	+	0.846216i	$0.321125\pi$
$212$	−9.52873	−0.654436
$213$	7.03440	0.481989
$214$	9.12735	0.623933
$215$	−2.43544	−0.166095
$216$	−2.81036	−0.191220
$217$	0	0
$218$	14.5110	0.982811
$219$	8.31996	0.562211
$220$	7.30567	0.492548
$221$	−50.9602	−3.42796
$222$	9.13997	0.613434
$223$	−11.2155	−0.751046	−0.375523	−	0.926813i	$-0.622537\pi$
$223$	−11.2155	−0.751046	−0.375523	+	0.926813i	$0.622537\pi$
$224$	0	0
$225$	8.42920	0.561947
$226$	9.91919	0.659815
$227$	−24.7615	−1.64348	−0.821741	−	0.569861i	$-0.806997\pi$
$227$	−24.7615	−1.64348	−0.821741	+	0.569861i	$0.806997\pi$
$228$	1.56115	0.103390
$229$	−2.98268	−0.197101	−0.0985505	−	0.995132i	$-0.531421\pi$
$229$	−2.98268	−0.197101	−0.0985505	+	0.995132i	$0.531421\pi$
$230$	12.4061	0.818036
$231$	0	0
$232$	−5.79318	−0.380341
$233$	−15.1079	−0.989749	−0.494874	−	0.868965i	$-0.664786\pi$
$233$	−15.1079	−0.989749	−0.494874	+	0.868965i	$0.664786\pi$
$234$	−5.98437	−0.391211
$235$	−42.5061	−2.77279
$236$	−1.10539	−0.0719547
$237$	−1.50124	−0.0975158
$238$	0	0
$239$	−12.0311	−0.778229	−0.389115	−	0.921189i	$-0.627219\pi$
$239$	−12.0311	−0.778229	−0.389115	+	0.921189i	$0.627219\pi$
$240$	0.297748	0.0192195
$241$	11.4900	0.740135	0.370068	−	0.929005i	$-0.379335\pi$
$241$	11.4900	0.740135	0.370068	+	0.929005i	$0.379335\pi$
$242$	−7.31766	−0.470397
$243$	1.00000	0.0641500
$244$	−14.8453	−0.950373
$245$	0	0
$246$	0.863619	0.0550623
$247$	8.62557	0.548832
$248$	5.43628	0.345204
$249$	10.6524	0.675068
$250$	10.8527	0.686388
$251$	21.8840	1.38131	0.690654	−	0.723186i	$-0.257323\pi$
$251$	21.8840	1.38131	0.690654	+	0.723186i	$0.257323\pi$
$252$	0	0
$253$	−6.23118	−0.391751
$254$	−4.06070	−0.254791
$255$	26.9501	1.68768
$256$	−15.7896	−0.986850
$257$	−1.87982	−0.117260	−0.0586300	−	0.998280i	$-0.518673\pi$
$257$	−1.87982	−0.117260	−0.0586300	+	0.998280i	$0.518673\pi$
$258$	−0.573950	−0.0357326
$259$	0	0
$260$	31.8475	1.97510
$261$	2.06137	0.127596
$262$	−10.5346	−0.650829
$263$	16.2855	1.00421	0.502103	−	0.864808i	$-0.332560\pi$
$263$	16.2855	1.00421	0.502103	+	0.864808i	$0.332560\pi$
$264$	4.46727	0.274942
$265$	27.8424	1.71034
$266$	0	0
$267$	0.808487	0.0494786
$268$	6.91495	0.422398
$269$	−11.6964	−0.713143	−0.356572	−	0.934268i	$-0.616054\pi$
$269$	−11.6964	−0.713143	−0.356572	+	0.934268i	$0.616054\pi$
$270$	3.16481	0.192604
$271$	19.7391	1.19907	0.599533	−	0.800350i	$-0.295353\pi$
$271$	19.7391	1.19907	0.599533	+	0.800350i	$0.295353\pi$
$272$	0.597528	0.0362305
$273$	0	0
$274$	−7.83662	−0.473428
$275$	−13.3988	−0.807981
$276$	−4.91636	−0.295930
$277$	−17.6810	−1.06235	−0.531173	−	0.847263i	$-0.678249\pi$
$277$	−17.6810	−1.06235	−0.531173	+	0.847263i	$0.678249\pi$
$278$	−8.84035	−0.530209
$279$	−1.93438	−0.115808
$280$	0	0
$281$	−22.9566	−1.36948	−0.684739	−	0.728788i	$-0.740084\pi$
$281$	−22.9566	−1.36948	−0.684739	+	0.728788i	$0.740084\pi$
$282$	−10.0172	−0.596517
$283$	−18.0695	−1.07412	−0.537059	−	0.843544i	$-0.680465\pi$
$283$	−18.0695	−1.07412	−0.537059	+	0.843544i	$0.680465\pi$
$284$	−8.82228	−0.523506
$285$	−4.56159	−0.270205
$286$	9.51262	0.562493
$287$	0	0
$288$	5.69088	0.335338
$289$	37.0841	2.18142
$290$	6.52384	0.383093
$291$	−1.66587	−0.0976549
$292$	−10.4346	−0.610638
$293$	9.30077	0.543357	0.271678	−	0.962388i	$-0.412421\pi$
$293$	9.30077	0.543357	0.271678	+	0.962388i	$0.412421\pi$
$294$	0	0
$295$	3.22988	0.188051
$296$	−29.7429	−1.72877
$297$	−1.58958	−0.0922365
$298$	10.0062	0.579642
$299$	−27.1635	−1.57091
$300$	−10.5716	−0.610351
$301$	0	0
$302$	−10.3420	−0.595114
$303$	−2.61047	−0.149968
$304$	−0.101138	−0.00580067
$305$	43.3771	2.48376
$306$	6.35121	0.363075
$307$	−10.5466	−0.601924	−0.300962	−	0.953636i	$-0.597308\pi$
$307$	−10.5466	−0.601924	−0.300962	+	0.953636i	$0.597308\pi$
$308$	0	0
$309$	16.2276	0.923158
$310$	−6.12192	−0.347702
$311$	19.8621	1.12628	0.563138	−	0.826363i	$-0.309594\pi$
$311$	19.8621	1.12628	0.563138	+	0.826363i	$0.309594\pi$
$312$	19.4741	1.10251
$313$	−3.78537	−0.213962	−0.106981	−	0.994261i	$-0.534118\pi$
$313$	−3.78537	−0.213962	−0.106981	+	0.994261i	$0.534118\pi$
$314$	2.73945	0.154596
$315$	0	0
$316$	1.88279	0.105915
$317$	−11.9422	−0.670742	−0.335371	−	0.942086i	$-0.608862\pi$
$317$	−11.9422	−0.670742	−0.335371	+	0.942086i	$0.608862\pi$
$318$	6.56150	0.367951
$319$	−3.27670	−0.183460
$320$	17.4150	0.973530
$321$	10.5687	0.589889
$322$	0	0
$323$	−9.15431	−0.509360
$324$	−1.25416	−0.0696757
$325$	−58.4094	−3.23997
$326$	17.0998	0.947068
$327$	16.8026	0.929186
$328$	−2.81036	−0.155176
$329$	0	0
$330$	−5.03070	−0.276931
$331$	−4.40745	−0.242255	−0.121128	−	0.992637i	$-0.538651\pi$
$331$	−4.40745	−0.242255	−0.121128	+	0.992637i	$0.538651\pi$
$332$	−13.3598	−0.733217
$333$	10.5833	0.579963
$334$	−21.0864	−1.15380
$335$	−20.2051	−1.10392
$336$	0	0
$337$	−5.69619	−0.310291	−0.155146	−	0.987892i	$-0.549585\pi$
$337$	−5.69619	−0.310291	−0.155146	+	0.987892i	$0.549585\pi$
$338$	30.2412	1.64490
$339$	11.4856	0.623813
$340$	−33.7998	−1.83305
$341$	3.07484	0.166512
$342$	−1.07501	−0.0581300
$343$	0	0
$344$	1.86773	0.100701
$345$	14.3653	0.773401
$346$	−16.5442	−0.889419
$347$	−16.0847	−0.863475	−0.431737	−	0.901999i	$-0.642099\pi$
$347$	−16.0847	−0.863475	−0.431737	+	0.901999i	$0.642099\pi$
$348$	−2.58529	−0.138586
$349$	−4.51115	−0.241476	−0.120738	−	0.992684i	$-0.538526\pi$
$349$	−4.51115	−0.241476	−0.120738	+	0.992684i	$0.538526\pi$
$350$	0	0
$351$	−6.92942	−0.369865
$352$	−9.04609	−0.482158
$353$	11.7751	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$353$	11.7751	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$354$	0.761172	0.0404559
$355$	25.7782	1.36816
$356$	−1.01397	−0.0537406
$357$	0	0
$358$	−3.33500	−0.176260
$359$	11.4107	0.602233	0.301117	−	0.953587i	$-0.402641\pi$
$359$	11.4107	0.602233	0.301117	+	0.953587i	$0.402641\pi$
$360$	−10.2988	−0.542794
$361$	−17.4505	−0.918449
$362$	−0.702778	−0.0369372
$363$	−8.47325	−0.444730
$364$	0	0
$365$	30.4892	1.59588
$366$	10.2225	0.534339
$367$	24.2729	1.26704	0.633519	−	0.773727i	$-0.281610\pi$
$367$	24.2729	1.26704	0.633519	+	0.773727i	$0.281610\pi$
$368$	0.318503	0.0166031
$369$	1.00000	0.0520579
$370$	33.4942	1.74128
$371$	0	0
$372$	2.42602	0.125783
$373$	−31.4251	−1.62713	−0.813566	−	0.581473i	$-0.802477\pi$
$373$	−31.4251	−1.62713	−0.813566	+	0.581473i	$0.802477\pi$
$374$	−10.0957	−0.522038
$375$	12.5666	0.648936
$376$	32.5977	1.68110
$377$	−14.2841	−0.735668
$378$	0	0
$379$	30.5217	1.56780	0.783898	−	0.620889i	$-0.213228\pi$
$379$	30.5217	1.56780	0.783898	+	0.620889i	$0.213228\pi$
$380$	5.72097	0.293480
$381$	−4.70196	−0.240889
$382$	7.78083	0.398102
$383$	6.77356	0.346113	0.173056	−	0.984912i	$-0.444636\pi$
$383$	6.77356	0.346113	0.173056	+	0.984912i	$0.444636\pi$
$384$	−7.27763	−0.371385
$385$	0	0
$386$	11.7526	0.598190
$387$	−0.664587	−0.0337829
$388$	2.08927	0.106067
$389$	12.2961	0.623435	0.311718	−	0.950175i	$-0.399096\pi$
$389$	12.2961	0.623435	0.311718	+	0.950175i	$0.399096\pi$
$390$	−21.9303	−1.11048
$391$	28.8286	1.45793
$392$	0	0
$393$	−12.1982	−0.615317
$394$	−1.72724	−0.0870170
$395$	−5.50141	−0.276806
$396$	1.99359	0.100182
$397$	16.5927	0.832765	0.416382	−	0.909190i	$-0.363298\pi$
$397$	16.5927	0.832765	0.416382	+	0.909190i	$0.363298\pi$
$398$	−19.6898	−0.986961
$399$	0	0
$400$	0.684873	0.0342436
$401$	−24.8302	−1.23996	−0.619980	−	0.784618i	$-0.712859\pi$
$401$	−24.8302	−1.23996	−0.619980	+	0.784618i	$0.712859\pi$
$402$	−4.76165	−0.237489
$403$	13.4041	0.667706
$404$	3.27396	0.162886
$405$	3.66459	0.182095
$406$	0	0
$407$	−16.8230	−0.833886
$408$	−20.6679	−1.02321
$409$	27.4612	1.35787	0.678935	−	0.734198i	$-0.262442\pi$
$409$	27.4612	1.35787	0.678935	+	0.734198i	$0.262442\pi$
$410$	3.16481	0.156299
$411$	−9.07417	−0.447596
$412$	−20.3521	−1.00268
$413$	0	0
$414$	3.38541	0.166384
$415$	39.0366	1.91623
$416$	−39.4345	−1.93343
$417$	−10.2364	−0.501279
$418$	1.70881	0.0835807
$419$	−0.860041	−0.0420157	−0.0210079	−	0.999779i	$-0.506688\pi$
$419$	−0.860041	−0.0420157	−0.0210079	+	0.999779i	$0.506688\pi$
$420$	0	0
$421$	0.777061	0.0378716	0.0189358	−	0.999821i	$-0.493972\pi$
$421$	0.777061	0.0378716	0.0189358	+	0.999821i	$0.493972\pi$
$422$	13.3687	0.650780
$423$	−11.5991	−0.563969
$424$	−21.3522	−1.03695
$425$	61.9899	3.00695
$426$	6.07504	0.294337
$427$	0	0
$428$	−13.2549	−0.640700
$429$	11.0148	0.531801
$430$	−2.10329	−0.101430
$431$	32.5308	1.56695	0.783477	−	0.621421i	$-0.213444\pi$
$431$	32.5308	1.56695	0.783477	+	0.621421i	$0.213444\pi$
$432$	0.0812501	0.00390914
$433$	22.9152	1.10124	0.550618	−	0.834757i	$-0.314392\pi$
$433$	22.9152	1.10124	0.550618	+	0.834757i	$0.314392\pi$
$434$	0	0
$435$	7.55407	0.362190
$436$	−21.0732	−1.00922
$437$	−4.87956	−0.233421
$438$	7.18527	0.343326
$439$	−20.3246	−0.970038	−0.485019	−	0.874504i	$-0.661187\pi$
$439$	−20.3246	−0.970038	−0.485019	+	0.874504i	$0.661187\pi$
$440$	16.3707	0.780443
$441$	0	0
$442$	−44.0102	−2.09335
$443$	−18.7016	−0.888541	−0.444271	−	0.895893i	$-0.646537\pi$
$443$	−18.7016	−0.888541	−0.444271	+	0.895893i	$0.646537\pi$
$444$	−13.2732	−0.629920
$445$	2.96277	0.140449
$446$	−9.68592	−0.458642
$447$	11.5863	0.548015
$448$	0	0
$449$	23.6765	1.11736	0.558682	−	0.829382i	$-0.311307\pi$
$449$	23.6765	1.11736	0.558682	+	0.829382i	$0.311307\pi$
$450$	7.27961	0.343164
$451$	−1.58958	−0.0748502
$452$	−14.4048	−0.677546
$453$	−11.9752	−0.562642
$454$	−21.3845	−1.00363
$455$	0	0
$456$	3.49826	0.163821
$457$	−27.4821	−1.28556	−0.642779	−	0.766051i	$-0.722219\pi$
$457$	−27.4821	−1.28556	−0.642779	+	0.766051i	$0.722219\pi$
$458$	−2.57590	−0.120364
$459$	7.35419	0.343264
$460$	−18.0164	−0.840020
$461$	−42.2145	−1.96612	−0.983062	−	0.183271i	$-0.941331\pi$
$461$	−42.2145	−1.96612	−0.983062	+	0.183271i	$0.941331\pi$
$462$	0	0
$463$	28.2010	1.31061	0.655305	−	0.755364i	$-0.272540\pi$
$463$	28.2010	1.31061	0.655305	+	0.755364i	$0.272540\pi$
$464$	0.167486	0.00777536
$465$	−7.08869	−0.328730
$466$	−13.0474	−0.604410
$467$	−39.4229	−1.82427	−0.912136	−	0.409887i	$-0.865568\pi$
$467$	−39.4229	−1.82427	−0.912136	+	0.409887i	$0.865568\pi$
$468$	8.69062	0.401724
$469$	0	0
$470$	−36.7090	−1.69326
$471$	3.17206	0.146161
$472$	−2.47698	−0.114012
$473$	1.05641	0.0485739
$474$	−1.29650	−0.0595500
$475$	−10.4925	−0.481427
$476$	0	0
$477$	7.59768	0.347874
$478$	−10.3903	−0.475242
$479$	−17.0884	−0.780790	−0.390395	−	0.920647i	$-0.627662\pi$
$479$	−17.0884	−0.780790	−0.390395	+	0.920647i	$0.627662\pi$
$480$	20.8547	0.951884
$481$	−73.3364	−3.34385
$482$	9.92297	0.451979
$483$	0	0
$484$	10.6268	0.483038
$485$	−6.10472	−0.277201
$486$	0.863619	0.0391745
$487$	−8.54990	−0.387433	−0.193717	−	0.981058i	$-0.562054\pi$
$487$	−8.54990	−0.387433	−0.193717	+	0.981058i	$0.562054\pi$
$488$	−33.2657	−1.50587
$489$	19.8001	0.895393
$490$	0	0
$491$	16.6790	0.752713	0.376357	−	0.926475i	$-0.377177\pi$
$491$	16.6790	0.752713	0.376357	+	0.926475i	$0.377177\pi$
$492$	−1.25416	−0.0565420
$493$	15.1597	0.682759
$494$	7.44920	0.335155
$495$	−5.82514	−0.261821
$496$	−0.157168	−0.00705706
$497$	0	0
$498$	9.19961	0.412244
$499$	4.44850	0.199142	0.0995711	−	0.995030i	$-0.468253\pi$
$499$	4.44850	0.199142	0.0995711	+	0.995030i	$0.468253\pi$
$500$	−15.7606	−0.704834
$501$	−24.4163	−1.09084
$502$	18.8995	0.843524
$503$	24.2876	1.08293	0.541466	−	0.840723i	$-0.317870\pi$
$503$	24.2876	1.08293	0.541466	+	0.840723i	$0.317870\pi$
$504$	0	0
$505$	−9.56631	−0.425695
$506$	−5.38137	−0.239231
$507$	35.0168	1.55515
$508$	5.89703	0.261638
$509$	29.8110	1.32135	0.660675	−	0.750672i	$-0.270270\pi$
$509$	29.8110	1.32135	0.660675	+	0.750672i	$0.270270\pi$
$510$	23.2746	1.03062
$511$	0	0
$512$	0.919067	0.0406174
$513$	−1.24478	−0.0549582
$514$	−1.62345	−0.0716072
$515$	59.4676	2.62045
$516$	0.833500	0.0366928
$517$	18.4377	0.810889
$518$	0	0
$519$	−19.1568	−0.840890
$520$	71.3646	3.12955
$521$	−16.0363	−0.702561	−0.351281	−	0.936270i	$-0.614254\pi$
$521$	−16.0363	−0.702561	−0.351281	+	0.936270i	$0.614254\pi$
$522$	1.78024	0.0779189
$523$	−12.6325	−0.552380	−0.276190	−	0.961103i	$-0.589072\pi$
$523$	−12.6325	−0.552380	−0.276190	+	0.961103i	$0.589072\pi$
$524$	15.2985	0.668319
$525$	0	0
$526$	14.0644	0.613239
$527$	−14.2258	−0.619684
$528$	−0.129153	−0.00562067
$529$	−7.63336	−0.331885
$530$	24.0452	1.04446
$531$	0.881376	0.0382484
$532$	0	0
$533$	−6.92942	−0.300146
$534$	0.698225	0.0302151
$535$	38.7300	1.67445
$536$	15.4952	0.669289
$537$	−3.86166	−0.166643
$538$	−10.1012	−0.435496
$539$	0	0
$540$	−4.59599	−0.197780
$541$	8.44217	0.362957	0.181479	−	0.983395i	$-0.441912\pi$
$541$	8.44217	0.362957	0.181479	+	0.983395i	$0.441912\pi$
$542$	17.0471	0.732234
$543$	−0.813759	−0.0349217
$544$	41.8518	1.79438
$545$	61.5746	2.63757
$546$	0	0
$547$	20.2500	0.865826	0.432913	−	0.901436i	$-0.357486\pi$
$547$	20.2500	0.865826	0.432913	+	0.901436i	$0.357486\pi$
$548$	11.3805	0.486150
$549$	11.8368	0.505183
$550$	−11.5715	−0.493410
$551$	−2.56594	−0.109313
$552$	−11.0167	−0.468901
$553$	0	0
$554$	−15.2696	−0.648744
$555$	38.7836	1.64627
$556$	12.8381	0.544458
$557$	−7.95706	−0.337152	−0.168576	−	0.985689i	$-0.553917\pi$
$557$	−7.95706	−0.337152	−0.168576	+	0.985689i	$0.553917\pi$
$558$	−1.67056	−0.0707206
$559$	4.60520	0.194779
$560$	0	0
$561$	−11.6900	−0.493554
$562$	−19.8258	−0.836300
$563$	19.8237	0.835471	0.417735	−	0.908569i	$-0.362824\pi$
$563$	19.8237	0.835471	0.417735	+	0.908569i	$0.362824\pi$
$564$	14.5472	0.612548
$565$	42.0900	1.77074
$566$	−15.6051	−0.655933
$567$	0	0
$568$	−19.7692	−0.829496
$569$	−5.94470	−0.249215	−0.124607	−	0.992206i	$-0.539767\pi$
$569$	−5.94470	−0.249215	−0.124607	+	0.992206i	$0.539767\pi$
$570$	−3.93947	−0.165006
$571$	12.0248	0.503224	0.251612	−	0.967828i	$-0.419039\pi$
$571$	12.0248	0.503224	0.251612	+	0.967828i	$0.419039\pi$
$572$	−13.8144	−0.577609
$573$	9.00956	0.376380
$574$	0	0
$575$	33.0427	1.37798
$576$	4.75225	0.198010
$577$	14.2745	0.594256	0.297128	−	0.954838i	$-0.403971\pi$
$577$	14.2745	0.594256	0.297128	+	0.954838i	$0.403971\pi$
$578$	32.0265	1.33213
$579$	13.6085	0.565551
$580$	−9.47403	−0.393388
$581$	0	0
$582$	−1.43867	−0.0596350
$583$	−12.0771	−0.500182
$584$	−23.3821	−0.967556
$585$	−25.3935	−1.04989
$586$	8.03232	0.331812
$587$	−12.8109	−0.528764	−0.264382	−	0.964418i	$-0.585168\pi$
$587$	−12.8109	−0.528764	−0.264382	+	0.964418i	$0.585168\pi$
$588$	0	0
$589$	2.40786	0.0992143
$590$	2.78938	0.114837
$591$	−2.00000	−0.0822690
$592$	0.859897	0.0353415
$593$	−43.2844	−1.77748	−0.888739	−	0.458413i	$-0.848418\pi$
$593$	−43.2844	−1.77748	−0.888739	+	0.458413i	$0.848418\pi$
$594$	−1.37279	−0.0563261
$595$	0	0
$596$	−14.5312	−0.595219
$597$	−22.7992	−0.933109
$598$	−23.4589	−0.959307
$599$	−7.99513	−0.326672	−0.163336	−	0.986570i	$-0.552225\pi$
$599$	−7.99513	−0.326672	−0.163336	+	0.986570i	$0.552225\pi$
$600$	−23.6890	−0.967101
$601$	29.3636	1.19777	0.598883	−	0.800836i	$-0.295611\pi$
$601$	29.3636	1.19777	0.598883	+	0.800836i	$0.295611\pi$
$602$	0	0
$603$	−5.51360	−0.224531
$604$	15.0188	0.611106
$605$	−31.0510	−1.26240
$606$	−2.25445	−0.0915809
$607$	−27.0644	−1.09851	−0.549255	−	0.835655i	$-0.685088\pi$
$607$	−27.0644	−1.09851	−0.549255	+	0.835655i	$0.685088\pi$
$608$	−7.08387	−0.287289
$609$	0	0
$610$	37.4612	1.51676
$611$	80.3753	3.25164
$612$	−9.22335	−0.372832
$613$	38.6762	1.56212	0.781059	−	0.624458i	$-0.214680\pi$
$613$	38.6762	1.56212	0.781059	+	0.624458i	$0.214680\pi$
$614$	−9.10821	−0.367578
$615$	3.66459	0.147770
$616$	0	0
$617$	−15.0624	−0.606388	−0.303194	−	0.952929i	$-0.598053\pi$
$617$	−15.0624	−0.606388	−0.303194	+	0.952929i	$0.598053\pi$
$618$	14.0145	0.563745
$619$	−6.23499	−0.250605	−0.125303	−	0.992119i	$-0.539990\pi$
$619$	−6.23499	−0.250605	−0.125303	+	0.992119i	$0.539990\pi$
$620$	8.89037	0.357046
$621$	3.92003	0.157305
$622$	17.1533	0.687784
$623$	0	0
$624$	−0.563016	−0.0225387
$625$	3.90540	0.156216
$626$	−3.26912	−0.130660
$627$	1.97867	0.0790203
$628$	−3.97828	−0.158751
$629$	77.8318	3.10336
$630$	0	0
$631$	−11.9336	−0.475068	−0.237534	−	0.971379i	$-0.576339\pi$
$631$	−11.9336	−0.475068	−0.237534	+	0.971379i	$0.576339\pi$
$632$	4.21901	0.167823
$633$	15.4799	0.615271
$634$	−10.3135	−0.409602
$635$	−17.2307	−0.683781
$636$	−9.52873	−0.377839
$637$	0	0
$638$	−2.82982	−0.112034
$639$	7.03440	0.278277
$640$	−26.6695	−1.05420
$641$	19.9403	0.787595	0.393798	−	0.919197i	$-0.371161\pi$
$641$	19.9403	0.787595	0.393798	+	0.919197i	$0.371161\pi$
$642$	9.12735	0.360228
$643$	27.6581	1.09073	0.545365	−	0.838199i	$-0.316391\pi$
$643$	27.6581	1.09073	0.545365	+	0.838199i	$0.316391\pi$
$644$	0	0
$645$	−2.43544	−0.0958953
$646$	−7.90584	−0.311051
$647$	−38.7029	−1.52157	−0.760784	−	0.649005i	$-0.775185\pi$
$647$	−38.7029	−1.52157	−0.760784	+	0.649005i	$0.775185\pi$
$648$	−2.81036	−0.110401
$649$	−1.40101	−0.0549946
$650$	−50.4435	−1.97856
$651$	0	0
$652$	−24.8326	−0.972519
$653$	−8.67098	−0.339322	−0.169661	−	0.985503i	$-0.554267\pi$
$653$	−8.67098	−0.339322	−0.169661	+	0.985503i	$0.554267\pi$
$654$	14.5110	0.567426
$655$	−44.7013	−1.74663
$656$	0.0812501	0.00317228
$657$	8.31996	0.324593
$658$	0	0
$659$	41.4899	1.61622	0.808108	−	0.589034i	$-0.200492\pi$
$659$	41.4899	1.61622	0.808108	+	0.589034i	$0.200492\pi$
$660$	7.30567	0.284373
$661$	−19.8732	−0.772979	−0.386490	−	0.922294i	$-0.626312\pi$
$661$	−19.8732	−0.772979	−0.386490	+	0.922294i	$0.626312\pi$
$662$	−3.80636	−0.147938
$663$	−50.9602	−1.97913
$664$	−29.9370	−1.16178
$665$	0	0
$666$	9.13997	0.354167
$667$	8.08063	0.312883
$668$	30.6220	1.18480
$669$	−11.2155	−0.433617
$670$	−17.4495	−0.674132
$671$	−18.8155	−0.726365
$672$	0	0
$673$	−12.0656	−0.465097	−0.232548	−	0.972585i	$-0.574706\pi$
$673$	−12.0656	−0.465097	−0.232548	+	0.972585i	$0.574706\pi$
$674$	−4.91933	−0.189486
$675$	8.42920	0.324440
$676$	−43.9168	−1.68911
$677$	12.8812	0.495065	0.247532	−	0.968880i	$-0.420380\pi$
$677$	12.8812	0.495065	0.247532	+	0.968880i	$0.420380\pi$
$678$	9.91919	0.380944
$679$	0	0
$680$	−75.7393	−2.90447
$681$	−24.7615	−0.948865
$682$	2.65549	0.101684
$683$	−40.7908	−1.56082	−0.780408	−	0.625271i	$-0.784988\pi$
$683$	−40.7908	−1.56082	−0.780408	+	0.625271i	$0.784988\pi$
$684$	1.56115	0.0596921
$685$	−33.2531	−1.27053
$686$	0	0
$687$	−2.98268	−0.113796
$688$	−0.0539977	−0.00205864
$689$	−52.6475	−2.00571
$690$	12.4061	0.472293
$691$	−17.6626	−0.671918	−0.335959	−	0.941877i	$-0.609060\pi$
$691$	−17.6626	−0.671918	−0.335959	+	0.941877i	$0.609060\pi$
$692$	24.0257	0.913321
$693$	0	0
$694$	−13.8911	−0.527298
$695$	−37.5122	−1.42292
$696$	−5.79318	−0.219590
$697$	7.35419	0.278560
$698$	−3.89592	−0.147463
$699$	−15.1079	−0.571432
$700$	0	0
$701$	−14.6878	−0.554750	−0.277375	−	0.960762i	$-0.589464\pi$
$701$	−14.6878	−0.554750	−0.277375	+	0.960762i	$0.589464\pi$
$702$	−5.98437	−0.225866
$703$	−13.1739	−0.496862
$704$	−7.55406	−0.284704
$705$	−42.5061	−1.60087
$706$	10.1692	0.382723
$707$	0	0
$708$	−1.10539	−0.0415430
$709$	−36.2747	−1.36233	−0.681163	−	0.732132i	$-0.738526\pi$
$709$	−36.2747	−1.36233	−0.681163	+	0.732132i	$0.738526\pi$
$710$	22.2625	0.835497
$711$	−1.50124	−0.0563008
$712$	−2.27214	−0.0851519
$713$	−7.58281	−0.283979
$714$	0	0
$715$	40.3648	1.50956
$716$	4.84315	0.180997
$717$	−12.0311	−0.449311
$718$	9.85449	0.367766
$719$	18.7250	0.698325	0.349162	−	0.937062i	$-0.386466\pi$
$719$	18.7250	0.698325	0.349162	+	0.937062i	$0.386466\pi$
$720$	0.297748	0.0110964
$721$	0	0
$722$	−15.0706	−0.560870
$723$	11.4900	0.427317
$724$	1.02059	0.0379298
$725$	17.3757	0.645317
$726$	−7.31766	−0.271584
$727$	42.1365	1.56276	0.781379	−	0.624057i	$-0.214517\pi$
$727$	42.1365	1.56276	0.781379	+	0.624057i	$0.214517\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	26.3311	0.974556
$731$	−4.88750	−0.180771
$732$	−14.8453	−0.548698
$733$	42.4902	1.56941	0.784707	−	0.619867i	$-0.212814\pi$
$733$	42.4902	1.56941	0.784707	+	0.619867i	$0.212814\pi$
$734$	20.9626	0.773743
$735$	0	0
$736$	22.3084	0.822299
$737$	8.76428	0.322836
$738$	0.863619	0.0317902
$739$	7.70222	0.283331	0.141665	−	0.989915i	$-0.454754\pi$
$739$	7.70222	0.283331	0.141665	+	0.989915i	$0.454754\pi$
$740$	−48.6409	−1.78808
$741$	8.62557	0.316868
$742$	0	0
$743$	−8.34182	−0.306032	−0.153016	−	0.988224i	$-0.548899\pi$
$743$	−8.34182	−0.306032	−0.153016	+	0.988224i	$0.548899\pi$
$744$	5.43628	0.199304
$745$	42.4591	1.55558
$746$	−27.1393	−0.993641
$747$	10.6524	0.389751
$748$	14.6612	0.536067
$749$	0	0
$750$	10.8527	0.396286
$751$	−30.7499	−1.12208	−0.561041	−	0.827788i	$-0.689599\pi$
$751$	−30.7499	−1.12208	−0.561041	+	0.827788i	$0.689599\pi$
$752$	−0.942431	−0.0343669
$753$	21.8840	0.797498
$754$	−12.3360	−0.449251
$755$	−43.8840	−1.59710
$756$	0	0
$757$	−14.0937	−0.512243	−0.256121	−	0.966645i	$-0.582445\pi$
$757$	−14.0937	−0.512243	−0.256121	+	0.966645i	$0.582445\pi$
$758$	26.3591	0.957407
$759$	−6.23118	−0.226178
$760$	12.8197	0.465019
$761$	−4.93684	−0.178960	−0.0894802	−	0.995989i	$-0.528521\pi$
$761$	−4.93684	−0.178960	−0.0894802	+	0.995989i	$0.528521\pi$
$762$	−4.06070	−0.147104
$763$	0	0
$764$	−11.2995	−0.408800
$765$	26.9501	0.974381
$766$	5.84977	0.211361
$767$	−6.10742	−0.220526
$768$	−15.7896	−0.569758
$769$	−40.9298	−1.47597	−0.737983	−	0.674820i	$-0.764221\pi$
$769$	−40.9298	−1.47597	−0.737983	+	0.674820i	$0.764221\pi$
$770$	0	0
$771$	−1.87982	−0.0677001
$772$	−17.0673	−0.614266
$773$	45.6176	1.64075	0.820375	−	0.571826i	$-0.193765\pi$
$773$	45.6176	1.64075	0.820375	+	0.571826i	$0.193765\pi$
$774$	−0.573950	−0.0206302
$775$	−16.3052	−0.585701
$776$	4.68168	0.168063
$777$	0	0
$778$	10.6191	0.380714
$779$	−1.24478	−0.0445987
$780$	31.8475	1.14032
$781$	−11.1817	−0.400113
$782$	24.8969	0.890313
$783$	2.06137	0.0736674
$784$	0	0
$785$	11.6243	0.414889
$786$	−10.5346	−0.375756
$787$	−16.2377	−0.578811	−0.289406	−	0.957207i	$-0.593458\pi$
$787$	−16.2377	−0.578811	−0.289406	+	0.957207i	$0.593458\pi$
$788$	2.50833	0.0893554
$789$	16.2855	0.579778
$790$	−4.75112	−0.169037
$791$	0	0
$792$	4.46727	0.158738
$793$	−82.0223	−2.91270
$794$	14.3298	0.508545
$795$	27.8424	0.987467
$796$	28.5939	1.01348
$797$	24.5104	0.868202	0.434101	−	0.900864i	$-0.357066\pi$
$797$	24.5104	0.868202	0.434101	+	0.900864i	$0.357066\pi$
$798$	0	0
$799$	−85.3022	−3.01778
$800$	47.9696	1.69598
$801$	0.808487	0.0285665
$802$	−21.4438	−0.757207
$803$	−13.2252	−0.466707
$804$	6.91495	0.243871
$805$	0	0
$806$	11.5760	0.407748
$807$	−11.6964	−0.411734
$808$	7.33636	0.258092
$809$	5.19577	0.182673	0.0913367	−	0.995820i	$-0.470886\pi$
$809$	5.19577	0.182673	0.0913367	+	0.995820i	$0.470886\pi$
$810$	3.16481	0.111200
$811$	−41.7718	−1.46681	−0.733403	−	0.679794i	$-0.762069\pi$
$811$	−41.7718	−1.46681	−0.733403	+	0.679794i	$0.762069\pi$
$812$	0	0
$813$	19.7391	0.692281
$814$	−14.5287	−0.509230
$815$	72.5593	2.54164
$816$	0.597528	0.0209177
$817$	0.827262	0.0289422
$818$	23.7160	0.829212
$819$	0	0
$820$	−4.59599	−0.160499
$821$	−31.8757	−1.11247	−0.556235	−	0.831025i	$-0.687755\pi$
$821$	−31.8757	−1.11247	−0.556235	+	0.831025i	$0.687755\pi$
$822$	−7.83662	−0.273334
$823$	26.1901	0.912931	0.456465	−	0.889741i	$-0.349115\pi$
$823$	26.1901	0.912931	0.456465	+	0.889741i	$0.349115\pi$
$824$	−45.6054	−1.58874
$825$	−13.3988	−0.466488
$826$	0	0
$827$	−40.6996	−1.41526	−0.707632	−	0.706581i	$-0.750237\pi$
$827$	−40.6996	−1.41526	−0.707632	+	0.706581i	$0.750237\pi$
$828$	−4.91636	−0.170855
$829$	−28.0820	−0.975329	−0.487665	−	0.873031i	$-0.662151\pi$
$829$	−28.0820	−0.975329	−0.487665	+	0.873031i	$0.662151\pi$
$830$	33.7128	1.17019
$831$	−17.6810	−0.613346
$832$	−32.9303	−1.14165
$833$	0	0
$834$	−8.84035	−0.306116
$835$	−89.4757	−3.09643
$836$	−2.48157	−0.0858268
$837$	−1.93438	−0.0668618
$838$	−0.742747	−0.0256578
$839$	27.1910	0.938737	0.469368	−	0.883002i	$-0.344482\pi$
$839$	27.1910	0.938737	0.469368	+	0.883002i	$0.344482\pi$
$840$	0	0
$841$	−24.7508	−0.853474
$842$	0.671084	0.0231271
$843$	−22.9566	−0.790669
$844$	−19.4143	−0.668268
$845$	128.322	4.41442
$846$	−10.0172	−0.344400
$847$	0	0
$848$	0.617312	0.0211986
$849$	−18.0695	−0.620143
$850$	53.5356	1.83626
$851$	41.4870	1.42216
$852$	−8.82228	−0.302246
$853$	−41.0367	−1.40507	−0.702535	−	0.711649i	$-0.747949\pi$
$853$	−41.0367	−1.40507	−0.702535	+	0.711649i	$0.747949\pi$
$854$	0	0
$855$	−4.56159	−0.156003
$856$	−29.7019	−1.01519
$857$	1.75256	0.0598662	0.0299331	−	0.999552i	$-0.490471\pi$
$857$	1.75256	0.0598662	0.0299331	+	0.999552i	$0.490471\pi$
$858$	9.51262	0.324755
$859$	−40.2035	−1.37172	−0.685862	−	0.727731i	$-0.740575\pi$
$859$	−40.2035	−1.37172	−0.685862	+	0.727731i	$0.740575\pi$
$860$	3.05444	0.104155
$861$	0	0
$862$	28.0942	0.956893
$863$	−19.8459	−0.675562	−0.337781	−	0.941225i	$-0.609676\pi$
$863$	−19.8459	−0.675562	−0.337781	+	0.941225i	$0.609676\pi$
$864$	5.69088	0.193608
$865$	−70.2017	−2.38693
$866$	19.7900	0.672493
$867$	37.0841	1.25944
$868$	0	0
$869$	2.38633	0.0809507
$870$	6.52384	0.221179
$871$	38.2060	1.29456
$872$	−47.2213	−1.59911
$873$	−1.66587	−0.0563811
$874$	−4.21408	−0.142543
$875$	0	0
$876$	−10.4346	−0.352552
$877$	52.1507	1.76100	0.880502	−	0.474042i	$-0.157205\pi$
$877$	52.1507	1.76100	0.880502	+	0.474042i	$0.157205\pi$
$878$	−17.5527	−0.592374
$879$	9.30077	0.313707
$880$	−0.473293	−0.0159547
$881$	0.719181	0.0242298	0.0121149	−	0.999927i	$-0.496144\pi$
$881$	0.719181	0.0242298	0.0121149	+	0.999927i	$0.496144\pi$
$882$	0	0
$883$	−34.4345	−1.15881	−0.579406	−	0.815039i	$-0.696715\pi$
$883$	−34.4345	−1.15881	−0.579406	+	0.815039i	$0.696715\pi$
$884$	63.9124	2.14961
$885$	3.22988	0.108571
$886$	−16.1511	−0.542606
$887$	13.2949	0.446401	0.223200	−	0.974773i	$-0.428350\pi$
$887$	13.2949	0.446401	0.223200	+	0.974773i	$0.428350\pi$
$888$	−29.7429	−0.998108
$889$	0	0
$890$	2.55871	0.0857680
$891$	−1.58958	−0.0532528
$892$	14.0661	0.470967
$893$	14.4383	0.483160
$894$	10.0062	0.334657
$895$	−14.1514	−0.473029
$896$	0	0
$897$	−27.1635	−0.906964
$898$	20.4475	0.682342
$899$	−3.98746	−0.132989
$900$	−10.5716	−0.352386
$901$	55.8748	1.86146
$902$	−1.37279	−0.0457088
$903$	0	0
$904$	−32.2787	−1.07357
$905$	−2.98209	−0.0991281
$906$	−10.3420	−0.343589
$907$	−6.32989	−0.210181	−0.105090	−	0.994463i	$-0.533513\pi$
$907$	−6.32989	−0.210181	−0.105090	+	0.994463i	$0.533513\pi$
$908$	31.0550	1.03060
$909$	−2.61047	−0.0865839
$910$	0	0
$911$	30.8451	1.02195	0.510973	−	0.859597i	$-0.329285\pi$
$911$	30.8451	1.02195	0.510973	+	0.859597i	$0.329285\pi$
$912$	−0.101138	−0.00334902
$913$	−16.9328	−0.560394
$914$	−23.7341	−0.785053
$915$	43.3771	1.43400
$916$	3.74077	0.123598
$917$	0	0
$918$	6.35121	0.209621
$919$	−5.89697	−0.194523	−0.0972616	−	0.995259i	$-0.531008\pi$
$919$	−5.89697	−0.194523	−0.0972616	+	0.995259i	$0.531008\pi$
$920$	−40.3716	−1.33101
$921$	−10.5466	−0.347521
$922$	−36.4572	−1.20065
$923$	−48.7443	−1.60444
$924$	0	0
$925$	89.2091	2.93318
$926$	24.3549	0.800352
$927$	16.2276	0.532985
$928$	11.7310	0.385089
$929$	10.1233	0.332135	0.166067	−	0.986114i	$-0.446893\pi$
$929$	10.1233	0.332135	0.166067	+	0.986114i	$0.446893\pi$
$930$	−6.12192	−0.200746
$931$	0	0
$932$	18.9477	0.620653
$933$	19.8621	0.650256
$934$	−34.0463	−1.11403
$935$	−42.8392	−1.40099
$936$	19.4741	0.636532
$937$	−32.6682	−1.06722	−0.533611	−	0.845730i	$-0.679165\pi$
$937$	−32.6682	−1.06722	−0.533611	+	0.845730i	$0.679165\pi$
$938$	0	0
$939$	−3.78537	−0.123531
$940$	53.3095	1.73876
$941$	46.6087	1.51940	0.759700	−	0.650274i	$-0.225346\pi$
$941$	46.6087	1.51940	0.759700	+	0.650274i	$0.225346\pi$
$942$	2.73945	0.0892562
$943$	3.92003	0.127654
$944$	0.0716118	0.00233077
$945$	0	0
$946$	0.912337	0.0296626
$947$	−16.0484	−0.521503	−0.260751	−	0.965406i	$-0.583970\pi$
$947$	−16.0484	−0.521503	−0.260751	+	0.965406i	$0.583970\pi$
$948$	1.88279	0.0611503
$949$	−57.6525	−1.87148
$950$	−9.06148	−0.293993
$951$	−11.9422	−0.387253
$952$	0	0
$953$	−18.5606	−0.601238	−0.300619	−	0.953744i	$-0.597193\pi$
$953$	−18.5606	−0.601238	−0.300619	+	0.953744i	$0.597193\pi$
$954$	6.56150	0.212436
$955$	33.0163	1.06838
$956$	15.0890	0.488013
$957$	−3.27670	−0.105921
$958$	−14.7579	−0.476805
$959$	0	0
$960$	17.4150	0.562068
$961$	−27.2582	−0.879296
$962$	−63.3346	−2.04199
$963$	10.5687	0.340573
$964$	−14.4103	−0.464125
$965$	49.8696	1.60536
$966$	0	0
$967$	24.0948	0.774837	0.387418	−	0.921904i	$-0.373367\pi$
$967$	24.0948	0.774837	0.387418	+	0.921904i	$0.373367\pi$
$968$	23.8128	0.765374
$969$	−9.15431	−0.294079
$970$	−5.27215	−0.169278
$971$	−47.7350	−1.53189	−0.765944	−	0.642907i	$-0.777728\pi$
$971$	−47.7350	−1.53189	−0.765944	+	0.642907i	$0.777728\pi$
$972$	−1.25416	−0.0402273
$973$	0	0
$974$	−7.38386	−0.236594
$975$	−58.4094	−1.87060
$976$	0.961743	0.0307846
$977$	26.1960	0.838085	0.419042	−	0.907967i	$-0.362366\pi$
$977$	26.1960	0.838085	0.419042	+	0.907967i	$0.362366\pi$
$978$	17.0998	0.546790
$979$	−1.28515	−0.0410736
$980$	0	0
$981$	16.8026	0.536466
$982$	14.4043	0.459660
$983$	−7.31517	−0.233318	−0.116659	−	0.993172i	$-0.537218\pi$
$983$	−7.31517	−0.233318	−0.116659	+	0.993172i	$0.537218\pi$
$984$	−2.81036	−0.0895909
$985$	−7.32917	−0.233527
$986$	13.0922	0.416941
$987$	0	0
$988$	−10.8179	−0.344162
$989$	−2.60520	−0.0828406
$990$	−5.03070	−0.159886
$991$	2.75793	0.0876086	0.0438043	−	0.999040i	$-0.486052\pi$
$991$	2.75793	0.0876086	0.0438043	+	0.999040i	$0.486052\pi$
$992$	−11.0083	−0.349514
$993$	−4.40745	−0.139866
$994$	0	0
$995$	−83.5496	−2.64870
$996$	−13.3598	−0.423323
$997$	−32.7683	−1.03778	−0.518891	−	0.854840i	$-0.673655\pi$
$997$	−32.7683	−1.03778	−0.518891	+	0.854840i	$0.673655\pi$
$998$	3.84181	0.121610
$999$	10.5833	0.334842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.x.1.3		5
7.6	odd	2		861.2.a.k.1.3	✓	5
21.20	even	2		2583.2.a.q.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.a.k.1.3	✓	5	7.6	odd	2
2583.2.a.q.1.3		5	21.20	even	2
6027.2.a.x.1.3		5	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q+0.863619 q^{2} +1.00000 q^{3} -1.25416 q^{4} +3.66459 q^{5} +0.863619 q^{6} -2.81036 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.863619 q^{2} +1.00000 q^{3} -1.25416 q^{4} +3.66459 q^{5} +0.863619 q^{6} -2.81036 q^{8} +1.00000 q^{9} +3.16481 q^{10} -1.58958 q^{11} -1.25416 q^{12} -6.92942 q^{13} +3.66459 q^{15} +0.0812501 q^{16} +7.35419 q^{17} +0.863619 q^{18} -1.24478 q^{19} -4.59599 q^{20} -1.37279 q^{22} +3.92003 q^{23} -2.81036 q^{24} +8.42920 q^{25} -5.98437 q^{26} +1.00000 q^{27} +2.06137 q^{29} +3.16481 q^{30} -1.93438 q^{31} +5.69088 q^{32} -1.58958 q^{33} +6.35121 q^{34} -1.25416 q^{36} +10.5833 q^{37} -1.07501 q^{38} -6.92942 q^{39} -10.2988 q^{40} +1.00000 q^{41} -0.664587 q^{43} +1.99359 q^{44} +3.66459 q^{45} +3.38541 q^{46} -11.5991 q^{47} +0.0812501 q^{48} +7.27961 q^{50} +7.35419 q^{51} +8.69062 q^{52} +7.59768 q^{53} +0.863619 q^{54} -5.82514 q^{55} -1.24478 q^{57} +1.78024 q^{58} +0.881376 q^{59} -4.59599 q^{60} +11.8368 q^{61} -1.67056 q^{62} +4.75225 q^{64} -25.3935 q^{65} -1.37279 q^{66} -5.51360 q^{67} -9.22335 q^{68} +3.92003 q^{69} +7.03440 q^{71} -2.81036 q^{72} +8.31996 q^{73} +9.13997 q^{74} +8.42920 q^{75} +1.56115 q^{76} -5.98437 q^{78} -1.50124 q^{79} +0.297748 q^{80} +1.00000 q^{81} +0.863619 q^{82} +10.6524 q^{83} +26.9501 q^{85} -0.573950 q^{86} +2.06137 q^{87} +4.46727 q^{88} +0.808487 q^{89} +3.16481 q^{90} -4.91636 q^{92} -1.93438 q^{93} -10.0172 q^{94} -4.56159 q^{95} +5.69088 q^{96} -1.66587 q^{97} -1.58958 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 5 q^{3} + 11 q^{4} + 9 q^{5} + 3 q^{6} + 9 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q + 3 * q^2 + 5 * q^3 + 11 * q^4 + 9 * q^5 + 3 * q^6 + 9 * q^8 + 5 * q^9	\( 5 q + 3 q^{2} + 5 q^{3} + 11 q^{4} + 9 q^{5} + 3 q^{6} + 9 q^{8} + 5 q^{9} - 9 q^{10} + 11 q^{12} + 3 q^{13} + 9 q^{15} + 27 q^{16} + 16 q^{17} + 3 q^{18} - 4 q^{19} + 7 q^{20} - 6 q^{22} - 3 q^{23} + 9 q^{24} + 20 q^{25} - 17 q^{26} + 5 q^{27} + 13 q^{29} - 9 q^{30} + 4 q^{31} + 21 q^{32} + 4 q^{34} + 11 q^{36} + 17 q^{37} - 4 q^{38} + 3 q^{39} - 37 q^{40} + 5 q^{41} + 6 q^{43} + 32 q^{44} + 9 q^{45} + 27 q^{46} + 15 q^{47} + 27 q^{48} - 14 q^{50} + 16 q^{51} + 17 q^{52} + 11 q^{53} + 3 q^{54} + 16 q^{55} - 4 q^{57} + 9 q^{58} - 12 q^{59} + 7 q^{60} - 12 q^{61} - 8 q^{62} + 19 q^{64} - 19 q^{65} - 6 q^{66} + 11 q^{67} + 28 q^{68} - 3 q^{69} + 18 q^{71} + 9 q^{72} - 12 q^{73} - 27 q^{74} + 20 q^{75} + 26 q^{76} - 17 q^{78} + 23 q^{79} + 7 q^{80} + 5 q^{81} + 3 q^{82} + 2 q^{83} + 20 q^{85} + 18 q^{86} + 13 q^{87} - 22 q^{88} + 28 q^{89} - 9 q^{90} - 7 q^{92} + 4 q^{93} + 3 q^{94} - 10 q^{95} + 21 q^{96} - 3 q^{97}+O(q^{100}) \) 5 * q + 3 * q^2 + 5 * q^3 + 11 * q^4 + 9 * q^5 + 3 * q^6 + 9 * q^8 + 5 * q^9 - 9 * q^10 + 11 * q^12 + 3 * q^13 + 9 * q^15 + 27 * q^16 + 16 * q^17 + 3 * q^18 - 4 * q^19 + 7 * q^20 - 6 * q^22 - 3 * q^23 + 9 * q^24 + 20 * q^25 - 17 * q^26 + 5 * q^27 + 13 * q^29 - 9 * q^30 + 4 * q^31 + 21 * q^32 + 4 * q^34 + 11 * q^36 + 17 * q^37 - 4 * q^38 + 3 * q^39 - 37 * q^40 + 5 * q^41 + 6 * q^43 + 32 * q^44 + 9 * q^45 + 27 * q^46 + 15 * q^47 + 27 * q^48 - 14 * q^50 + 16 * q^51 + 17 * q^52 + 11 * q^53 + 3 * q^54 + 16 * q^55 - 4 * q^57 + 9 * q^58 - 12 * q^59 + 7 * q^60 - 12 * q^61 - 8 * q^62 + 19 * q^64 - 19 * q^65 - 6 * q^66 + 11 * q^67 + 28 * q^68 - 3 * q^69 + 18 * q^71 + 9 * q^72 - 12 * q^73 - 27 * q^74 + 20 * q^75 + 26 * q^76 - 17 * q^78 + 23 * q^79 + 7 * q^80 + 5 * q^81 + 3 * q^82 + 2 * q^83 + 20 * q^85 + 18 * q^86 + 13 * q^87 - 22 * q^88 + 28 * q^89 - 9 * q^90 - 7 * q^92 + 4 * q^93 + 3 * q^94 - 10 * q^95 + 21 * q^96 - 3 * q^97

Introduction

L-functions

Modular forms

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Database

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