LMFDB - Embedded newform 4029.2.a.k.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4029,2,Mod(1,4029)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4029, 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("4029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4029 = 3 \cdot 17 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4029.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:	$32.1717269744$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	4029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0488355 q^{2} +1.00000 q^{3} -1.99762 q^{4} -0.188126 q^{5} +0.0488355 q^{6} +3.46836 q^{7} -0.195225 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.0488355 q^{2} +1.00000 q^{3} -1.99762 q^{4} -0.188126 q^{5} +0.0488355 q^{6} +3.46836 q^{7} -0.195225 q^{8} +1.00000 q^{9} -0.00918720 q^{10} +3.08722 q^{11} -1.99762 q^{12} -6.69012 q^{13} +0.169379 q^{14} -0.188126 q^{15} +3.98570 q^{16} +1.00000 q^{17} +0.0488355 q^{18} -1.82042 q^{19} +0.375802 q^{20} +3.46836 q^{21} +0.150766 q^{22} +3.48683 q^{23} -0.195225 q^{24} -4.96461 q^{25} -0.326715 q^{26} +1.00000 q^{27} -6.92845 q^{28} -4.01796 q^{29} -0.00918720 q^{30} +10.0379 q^{31} +0.585094 q^{32} +3.08722 q^{33} +0.0488355 q^{34} -0.652488 q^{35} -1.99762 q^{36} +8.47735 q^{37} -0.0889012 q^{38} -6.69012 q^{39} +0.0367269 q^{40} +5.10831 q^{41} +0.169379 q^{42} +10.8185 q^{43} -6.16709 q^{44} -0.188126 q^{45} +0.170281 q^{46} -8.85864 q^{47} +3.98570 q^{48} +5.02954 q^{49} -0.242449 q^{50} +1.00000 q^{51} +13.3643 q^{52} -8.03495 q^{53} +0.0488355 q^{54} -0.580786 q^{55} -0.677113 q^{56} -1.82042 q^{57} -0.196219 q^{58} +12.4261 q^{59} +0.375802 q^{60} +6.94654 q^{61} +0.490206 q^{62} +3.46836 q^{63} -7.94282 q^{64} +1.25858 q^{65} +0.150766 q^{66} -1.50745 q^{67} -1.99762 q^{68} +3.48683 q^{69} -0.0318645 q^{70} -9.99510 q^{71} -0.195225 q^{72} -2.67757 q^{73} +0.413996 q^{74} -4.96461 q^{75} +3.63650 q^{76} +10.7076 q^{77} -0.326715 q^{78} +1.00000 q^{79} -0.749811 q^{80} +1.00000 q^{81} +0.249467 q^{82} -13.1230 q^{83} -6.92845 q^{84} -0.188126 q^{85} +0.528329 q^{86} -4.01796 q^{87} -0.602705 q^{88} +4.05252 q^{89} -0.00918720 q^{90} -23.2038 q^{91} -6.96534 q^{92} +10.0379 q^{93} -0.432616 q^{94} +0.342468 q^{95} +0.585094 q^{96} +8.47463 q^{97} +0.245620 q^{98} +3.08722 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) $ 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9} + 5 q^{10} + 26 q^{11} + 34 q^{12} + 7 q^{13} + 19 q^{14} + 11 q^{15} + 40 q^{16} + 31 q^{17} + 4 q^{18} + 32 q^{19} + 23 q^{20} + 4 q^{21} + 2 q^{22} + 29 q^{23} + 12 q^{24} + 32 q^{25} + 13 q^{26} + 31 q^{27} - 13 q^{28} + 25 q^{29} + 5 q^{30} + 22 q^{31} + 28 q^{32} + 26 q^{33} + 4 q^{34} + 20 q^{35} + 34 q^{36} - 4 q^{37} + 19 q^{38} + 7 q^{39} - 3 q^{40} + 33 q^{41} + 19 q^{42} + 6 q^{43} + 30 q^{44} + 11 q^{45} - 11 q^{46} + 23 q^{47} + 40 q^{48} + 31 q^{49} + 6 q^{50} + 31 q^{51} - 7 q^{52} + 12 q^{53} + 4 q^{54} + 40 q^{56} + 32 q^{57} + 9 q^{58} + 27 q^{59} + 23 q^{60} - 4 q^{61} + 25 q^{62} + 4 q^{63} + 10 q^{64} + 54 q^{65} + 2 q^{66} + 34 q^{68} + 29 q^{69} - 59 q^{70} + 35 q^{71} + 12 q^{72} + 5 q^{73} + 48 q^{74} + 32 q^{75} + 32 q^{76} + 42 q^{77} + 13 q^{78} + 31 q^{79} + 24 q^{80} + 31 q^{81} + 5 q^{82} + 67 q^{83} - 13 q^{84} + 11 q^{85} - 20 q^{86} + 25 q^{87} - 7 q^{88} + 22 q^{89} + 5 q^{90} + 16 q^{91} + 57 q^{92} + 22 q^{93} + 45 q^{94} + 73 q^{95} + 28 q^{96} - 13 q^{97} - 19 q^{98} + 26 q^{99}+O(q^{100}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9 + 5 * q^10 + 26 * q^11 + 34 * q^12 + 7 * q^13 + 19 * q^14 + 11 * q^15 + 40 * q^16 + 31 * q^17 + 4 * q^18 + 32 * q^19 + 23 * q^20 + 4 * q^21 + 2 * q^22 + 29 * q^23 + 12 * q^24 + 32 * q^25 + 13 * q^26 + 31 * q^27 - 13 * q^28 + 25 * q^29 + 5 * q^30 + 22 * q^31 + 28 * q^32 + 26 * q^33 + 4 * q^34 + 20 * q^35 + 34 * q^36 - 4 * q^37 + 19 * q^38 + 7 * q^39 - 3 * q^40 + 33 * q^41 + 19 * q^42 + 6 * q^43 + 30 * q^44 + 11 * q^45 - 11 * q^46 + 23 * q^47 + 40 * q^48 + 31 * q^49 + 6 * q^50 + 31 * q^51 - 7 * q^52 + 12 * q^53 + 4 * q^54 + 40 * q^56 + 32 * q^57 + 9 * q^58 + 27 * q^59 + 23 * q^60 - 4 * q^61 + 25 * q^62 + 4 * q^63 + 10 * q^64 + 54 * q^65 + 2 * q^66 + 34 * q^68 + 29 * q^69 - 59 * q^70 + 35 * q^71 + 12 * q^72 + 5 * q^73 + 48 * q^74 + 32 * q^75 + 32 * q^76 + 42 * q^77 + 13 * q^78 + 31 * q^79 + 24 * q^80 + 31 * q^81 + 5 * q^82 + 67 * q^83 - 13 * q^84 + 11 * q^85 - 20 * q^86 + 25 * q^87 - 7 * q^88 + 22 * q^89 + 5 * q^90 + 16 * q^91 + 57 * q^92 + 22 * q^93 + 45 * q^94 + 73 * q^95 + 28 * q^96 - 13 * q^97 - 19 * q^98 + 26 * q^99

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 $

$2$	0.0488355	0.0345319	0.0172660	−	0.999851i	$-0.494504\pi$
$2$	0.0488355	0.0345319	0.0172660	+	0.999851i	$0.494504\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.99762	−0.998808
$5$	−0.188126	−0.0841323	−0.0420661	−	0.999115i	$-0.513394\pi$
$5$	−0.188126	−0.0841323	−0.0420661	+	0.999115i	$0.513394\pi$
$6$	0.0488355	0.0199370
$7$	3.46836	1.31092	0.655459	−	0.755231i	$-0.272475\pi$
$7$	3.46836	1.31092	0.655459	+	0.755231i	$0.272475\pi$
$8$	−0.195225	−0.0690226
$9$	1.00000	0.333333
$10$	−0.00918720	−0.00290525
$11$	3.08722	0.930833	0.465417	−	0.885092i	$-0.345905\pi$
$11$	3.08722	0.930833	0.465417	+	0.885092i	$0.345905\pi$
$12$	−1.99762	−0.576662
$13$	−6.69012	−1.85551	−0.927753	−	0.373195i	$-0.878262\pi$
$13$	−6.69012	−1.85551	−0.927753	+	0.373195i	$0.878262\pi$
$14$	0.169379	0.0452685
$15$	−0.188126	−0.0485738
$16$	3.98570	0.996424
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0.0488355	0.0115106
$19$	−1.82042	−0.417633	−0.208817	−	0.977955i	$-0.566961\pi$
$19$	−1.82042	−0.417633	−0.208817	+	0.977955i	$0.566961\pi$
$20$	0.375802	0.0840320
$21$	3.46836	0.756859
$22$	0.150766	0.0321434
$23$	3.48683	0.727053	0.363527	−	0.931584i	$-0.381573\pi$
$23$	3.48683	0.727053	0.363527	+	0.931584i	$0.381573\pi$
$24$	−0.195225	−0.0398502
$25$	−4.96461	−0.992922
$26$	−0.326715	−0.0640742
$27$	1.00000	0.192450
$28$	−6.92845	−1.30935
$29$	−4.01796	−0.746116	−0.373058	−	0.927808i	$-0.621691\pi$
$29$	−4.01796	−0.746116	−0.373058	+	0.927808i	$0.621691\pi$
$30$	−0.00918720	−0.00167735
$31$	10.0379	1.80286	0.901431	−	0.432923i	$-0.142518\pi$
$31$	10.0379	1.80286	0.901431	+	0.432923i	$0.142518\pi$
$32$	0.585094	0.103431
$33$	3.08722	0.537417
$34$	0.0488355	0.00837522
$35$	−0.652488	−0.110291
$36$	−1.99762	−0.332936
$37$	8.47735	1.39367	0.696834	−	0.717232i	$-0.254591\pi$
$37$	8.47735	1.39367	0.696834	+	0.717232i	$0.254591\pi$
$38$	−0.0889012	−0.0144217
$39$	−6.69012	−1.07128
$40$	0.0367269	0.00580703
$41$	5.10831	0.797783	0.398892	−	0.916998i	$-0.369395\pi$
$41$	5.10831	0.797783	0.398892	+	0.916998i	$0.369395\pi$
$42$	0.169379	0.0261358
$43$	10.8185	1.64981	0.824906	−	0.565270i	$-0.191228\pi$
$43$	10.8185	1.64981	0.824906	+	0.565270i	$0.191228\pi$
$44$	−6.16709	−0.929723
$45$	−0.188126	−0.0280441
$46$	0.170281	0.0251065
$47$	−8.85864	−1.29217	−0.646083	−	0.763267i	$-0.723594\pi$
$47$	−8.85864	−1.29217	−0.646083	+	0.763267i	$0.723594\pi$
$48$	3.98570	0.575286
$49$	5.02954	0.718506
$50$	−0.242449	−0.0342875
$51$	1.00000	0.140028
$52$	13.3643	1.85329
$53$	−8.03495	−1.10369	−0.551843	−	0.833948i	$-0.686075\pi$
$53$	−8.03495	−1.10369	−0.551843	+	0.833948i	$0.686075\pi$
$54$	0.0488355	0.00664567
$55$	−0.580786	−0.0783131
$56$	−0.677113	−0.0904830
$57$	−1.82042	−0.241121
$58$	−0.196219	−0.0257648
$59$	12.4261	1.61774	0.808871	−	0.587986i	$-0.200079\pi$
$59$	12.4261	1.61774	0.808871	+	0.587986i	$0.200079\pi$
$60$	0.375802	0.0485159
$61$	6.94654	0.889413	0.444706	−	0.895676i	$-0.353308\pi$
$61$	6.94654	0.889413	0.444706	+	0.895676i	$0.353308\pi$
$62$	0.490206	0.0622563
$63$	3.46836	0.436973
$64$	−7.94282	−0.992852
$65$	1.25858	0.156108
$66$	0.150766	0.0185580
$67$	−1.50745	−0.184164	−0.0920820	−	0.995751i	$-0.529352\pi$
$67$	−1.50745	−0.184164	−0.0920820	+	0.995751i	$0.529352\pi$
$68$	−1.99762	−0.242246
$69$	3.48683	0.419764
$70$	−0.0318645	−0.00380854
$71$	−9.99510	−1.18620	−0.593100	−	0.805129i	$-0.702096\pi$
$71$	−9.99510	−1.18620	−0.593100	+	0.805129i	$0.702096\pi$
$72$	−0.195225	−0.0230075
$73$	−2.67757	−0.313386	−0.156693	−	0.987647i	$-0.550083\pi$
$73$	−2.67757	−0.313386	−0.156693	+	0.987647i	$0.550083\pi$
$74$	0.413996	0.0481260
$75$	−4.96461	−0.573264
$76$	3.63650	0.417135
$77$	10.7076	1.22025
$78$	−0.326715	−0.0369932
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	−0.749811	−0.0838314
$81$	1.00000	0.111111
$82$	0.249467	0.0275490
$83$	−13.1230	−1.44044	−0.720221	−	0.693745i	$-0.755960\pi$
$83$	−13.1230	−1.44044	−0.720221	+	0.693745i	$0.755960\pi$
$84$	−6.92845	−0.755956
$85$	−0.188126	−0.0204051
$86$	0.528329	0.0569711
$87$	−4.01796	−0.430770
$88$	−0.602705	−0.0642486
$89$	4.05252	0.429567	0.214783	−	0.976662i	$-0.431095\pi$
$89$	4.05252	0.429567	0.214783	+	0.976662i	$0.431095\pi$
$90$	−0.00918720	−0.000968416 0
$91$	−23.2038	−2.43242
$92$	−6.96534	−0.726186
$93$	10.0379	1.04088
$94$	−0.432616	−0.0446209
$95$	0.342468	0.0351364
$96$	0.585094	0.0597159
$97$	8.47463	0.860468	0.430234	−	0.902717i	$-0.358431\pi$
$97$	8.47463	0.860468	0.430234	+	0.902717i	$0.358431\pi$
$98$	0.245620	0.0248114
$99$	3.08722	0.310278
$100$	9.91738	0.991738
$101$	16.5028	1.64209	0.821045	−	0.570864i	$-0.193391\pi$
$101$	16.5028	1.64209	0.821045	+	0.570864i	$0.193391\pi$
$102$	0.0488355	0.00483543
$103$	−3.65982	−0.360612	−0.180306	−	0.983611i	$-0.557709\pi$
$103$	−3.65982	−0.360612	−0.180306	+	0.983611i	$0.557709\pi$
$104$	1.30608	0.128072
$105$	−0.652488	−0.0636763
$106$	−0.392391	−0.0381123
$107$	19.0585	1.84246	0.921228	−	0.389023i	$-0.127187\pi$
$107$	19.0585	1.84246	0.921228	+	0.389023i	$0.127187\pi$
$108$	−1.99762	−0.192221
$109$	−12.7995	−1.22597	−0.612987	−	0.790093i	$-0.710032\pi$
$109$	−12.7995	−1.22597	−0.612987	+	0.790093i	$0.710032\pi$
$110$	−0.0283630	−0.00270430
$111$	8.47735	0.804635
$112$	13.8238	1.30623
$113$	10.3696	0.975487	0.487744	−	0.872987i	$-0.337820\pi$
$113$	10.3696	0.975487	0.487744	+	0.872987i	$0.337820\pi$
$114$	−0.0889012	−0.00832636
$115$	−0.655961	−0.0611687
$116$	8.02633	0.745226
$117$	−6.69012	−0.618502
$118$	0.606835	0.0558637
$119$	3.46836	0.317944
$120$	0.0367269	0.00335269
$121$	−1.46904	−0.133549
$122$	0.339237	0.0307131
$123$	5.10831	0.460600
$124$	−20.0519	−1.80071
$125$	1.87460	0.167669
$126$	0.169379	0.0150895
$127$	3.38064	0.299983	0.149992	−	0.988687i	$-0.452075\pi$
$127$	3.38064	0.299983	0.149992	+	0.988687i	$0.452075\pi$
$128$	−1.55808	−0.137716
$129$	10.8185	0.952519
$130$	0.0614635	0.00539071
$131$	14.9439	1.30565	0.652827	−	0.757507i	$-0.273583\pi$
$131$	14.9439	1.30565	0.652827	+	0.757507i	$0.273583\pi$
$132$	−6.16709	−0.536776
$133$	−6.31388	−0.547483
$134$	−0.0736170	−0.00635954
$135$	−0.188126	−0.0161913
$136$	−0.195225	−0.0167404
$137$	−3.55003	−0.303300	−0.151650	−	0.988434i	$-0.548459\pi$
$137$	−3.55003	−0.303300	−0.151650	+	0.988434i	$0.548459\pi$
$138$	0.170281	0.0144953
$139$	−20.5032	−1.73906	−0.869530	−	0.493880i	$-0.835578\pi$
$139$	−20.5032	−1.73906	−0.869530	+	0.493880i	$0.835578\pi$
$140$	1.30342	0.110159
$141$	−8.85864	−0.746032
$142$	−0.488116	−0.0409617
$143$	−20.6539	−1.72717
$144$	3.98570	0.332141
$145$	0.755880	0.0627724
$146$	−0.130760	−0.0108218
$147$	5.02954	0.414830
$148$	−16.9345	−1.39201
$149$	5.79237	0.474529	0.237265	−	0.971445i	$-0.423749\pi$
$149$	5.79237	0.474529	0.237265	+	0.971445i	$0.423749\pi$
$150$	−0.242449	−0.0197959
$151$	−7.38646	−0.601102	−0.300551	−	0.953766i	$-0.597170\pi$
$151$	−7.38646	−0.601102	−0.300551	+	0.953766i	$0.597170\pi$
$152$	0.355393	0.0288261
$153$	1.00000	0.0808452
$154$	0.522912	0.0421374
$155$	−1.88839	−0.151679
$156$	13.3643	1.07000
$157$	12.3260	0.983718	0.491859	−	0.870675i	$-0.336317\pi$
$157$	12.3260	0.983718	0.491859	+	0.870675i	$0.336317\pi$
$158$	0.0488355	0.00388514
$159$	−8.03495	−0.637213
$160$	−0.110071	−0.00870189
$161$	12.0936	0.953107
$162$	0.0488355	0.00383688
$163$	13.4096	1.05032	0.525161	−	0.851003i	$-0.324005\pi$
$163$	13.4096	1.05032	0.525161	+	0.851003i	$0.324005\pi$
$164$	−10.2044	−0.796832
$165$	−0.580786	−0.0452141
$166$	−0.640871	−0.0497412
$167$	−5.38383	−0.416613	−0.208307	−	0.978064i	$-0.566795\pi$
$167$	−5.38383	−0.416613	−0.208307	+	0.978064i	$0.566795\pi$
$168$	−0.677113	−0.0522404
$169$	31.7577	2.44290
$170$	−0.00918720	−0.000704626 0
$171$	−1.82042	−0.139211
$172$	−21.6113	−1.64784
$173$	12.1312	0.922320	0.461160	−	0.887317i	$-0.347433\pi$
$173$	12.1312	0.922320	0.461160	+	0.887317i	$0.347433\pi$
$174$	−0.196219	−0.0148753
$175$	−17.2191	−1.30164
$176$	12.3047	0.927505
$177$	12.4261	0.934004
$178$	0.197907	0.0148338
$179$	5.40371	0.403892	0.201946	−	0.979397i	$-0.435273\pi$
$179$	5.40371	0.403892	0.201946	+	0.979397i	$0.435273\pi$
$180$	0.375802	0.0280107
$181$	21.0681	1.56598	0.782989	−	0.622035i	$-0.213694\pi$
$181$	21.0681	1.56598	0.782989	+	0.622035i	$0.213694\pi$
$182$	−1.13317	−0.0839960
$183$	6.94654	0.513503
$184$	−0.680717	−0.0501831
$185$	−1.59481	−0.117253
$186$	0.490206	0.0359437
$187$	3.08722	0.225760
$188$	17.6962	1.29062
$189$	3.46836	0.252286
$190$	0.0167246	0.00121333
$191$	27.1557	1.96492	0.982458	−	0.186482i	$-0.0597085\pi$
$191$	27.1557	1.96492	0.982458	+	0.186482i	$0.0597085\pi$
$192$	−7.94282	−0.573224
$193$	9.27823	0.667861	0.333931	−	0.942598i	$-0.391625\pi$
$193$	9.27823	0.667861	0.333931	+	0.942598i	$0.391625\pi$
$194$	0.413863	0.0297136
$195$	1.25858	0.0901290
$196$	−10.0471	−0.717649
$197$	−20.3967	−1.45321	−0.726603	−	0.687058i	$-0.758902\pi$
$197$	−20.3967	−1.45321	−0.726603	+	0.687058i	$0.758902\pi$
$198$	0.150766	0.0107145
$199$	−10.4301	−0.739370	−0.369685	−	0.929157i	$-0.620534\pi$
$199$	−10.4301	−0.739370	−0.369685	+	0.929157i	$0.620534\pi$
$200$	0.969218	0.0685341
$201$	−1.50745	−0.106327
$202$	0.805922	0.0567045
$203$	−13.9357	−0.978097
$204$	−1.99762	−0.139861
$205$	−0.961003	−0.0671193
$206$	−0.178729	−0.0124526
$207$	3.48683	0.242351
$208$	−26.6648	−1.84887
$209$	−5.62005	−0.388747
$210$	−0.0318645	−0.00219886
$211$	4.41022	0.303612	0.151806	−	0.988410i	$-0.451491\pi$
$211$	4.41022	0.303612	0.151806	+	0.988410i	$0.451491\pi$
$212$	16.0507	1.10237
$213$	−9.99510	−0.684853
$214$	0.930732	0.0636235
$215$	−2.03524	−0.138802
$216$	−0.195225	−0.0132834
$217$	34.8151	2.36340
$218$	−0.625072	−0.0423352
$219$	−2.67757	−0.180933
$220$	1.16019	0.0782198
$221$	−6.69012	−0.450026
$222$	0.413996	0.0277856
$223$	−25.4398	−1.70357	−0.851786	−	0.523890i	$-0.824480\pi$
$223$	−25.4398	−1.70357	−0.851786	+	0.523890i	$0.824480\pi$
$224$	2.02932	0.135590
$225$	−4.96461	−0.330974
$226$	0.506403	0.0336854
$227$	0.0735588	0.00488227	0.00244113	−	0.999997i	$-0.499223\pi$
$227$	0.0735588	0.00488227	0.00244113	+	0.999997i	$0.499223\pi$
$228$	3.63650	0.240833
$229$	10.5527	0.697343	0.348672	−	0.937245i	$-0.386633\pi$
$229$	10.5527	0.697343	0.348672	+	0.937245i	$0.386633\pi$
$230$	−0.0320342	−0.00211227
$231$	10.7076	0.704509
$232$	0.784408	0.0514989
$233$	18.3410	1.20156	0.600779	−	0.799415i	$-0.294857\pi$
$233$	18.3410	1.20156	0.600779	+	0.799415i	$0.294857\pi$
$234$	−0.326715	−0.0213581
$235$	1.66654	0.108713
$236$	−24.8226	−1.61581
$237$	1.00000	0.0649570
$238$	0.169379	0.0109792
$239$	27.2352	1.76170	0.880849	−	0.473397i	$-0.156972\pi$
$239$	27.2352	1.76170	0.880849	+	0.473397i	$0.156972\pi$
$240$	−0.749811	−0.0484001
$241$	5.06644	0.326358	0.163179	−	0.986596i	$-0.447825\pi$
$241$	5.06644	0.326358	0.163179	+	0.986596i	$0.447825\pi$
$242$	−0.0717415	−0.00461172
$243$	1.00000	0.0641500
$244$	−13.8765	−0.888352
$245$	−0.946185	−0.0604496
$246$	0.249467	0.0159054
$247$	12.1788	0.774921
$248$	−1.95966	−0.124438
$249$	−13.1230	−0.831639
$250$	0.0915469	0.00578993
$251$	12.4395	0.785173	0.392587	−	0.919715i	$-0.371580\pi$
$251$	12.4395	0.785173	0.392587	+	0.919715i	$0.371580\pi$
$252$	−6.92845	−0.436452
$253$	10.7646	0.676766
$254$	0.165095	0.0103590
$255$	−0.188126	−0.0117809
$256$	15.8095	0.988097
$257$	2.31761	0.144569	0.0722843	−	0.997384i	$-0.476971\pi$
$257$	2.31761	0.144569	0.0722843	+	0.997384i	$0.476971\pi$
$258$	0.528329	0.0328923
$259$	29.4025	1.82699
$260$	−2.51416	−0.155922
$261$	−4.01796	−0.248705
$262$	0.729793	0.0450867
$263$	−14.3401	−0.884250	−0.442125	−	0.896953i	$-0.645775\pi$
$263$	−14.3401	−0.884250	−0.442125	+	0.896953i	$0.645775\pi$
$264$	−0.602705	−0.0370939
$265$	1.51158	0.0928555
$266$	−0.308342	−0.0189056
$267$	4.05252	0.248010
$268$	3.01130	0.183944
$269$	−5.00886	−0.305395	−0.152698	−	0.988273i	$-0.548796\pi$
$269$	−5.00886	−0.305395	−0.152698	+	0.988273i	$0.548796\pi$
$270$	−0.00918720	−0.000559115 0
$271$	−4.90415	−0.297906	−0.148953	−	0.988844i	$-0.547590\pi$
$271$	−4.90415	−0.297906	−0.148953	+	0.988844i	$0.547590\pi$
$272$	3.98570	0.241668
$273$	−23.2038	−1.40436
$274$	−0.173367	−0.0104735
$275$	−15.3269	−0.924245
$276$	−6.96534	−0.419264
$277$	10.9342	0.656970	0.328485	−	0.944509i	$-0.393462\pi$
$277$	10.9342	0.656970	0.328485	+	0.944509i	$0.393462\pi$
$278$	−1.00128	−0.0600531
$279$	10.0379	0.600954
$280$	0.127382	0.00761254
$281$	19.8441	1.18380	0.591899	−	0.806012i	$-0.298378\pi$
$281$	19.8441	1.18380	0.591899	+	0.806012i	$0.298378\pi$
$282$	−0.432616	−0.0257619
$283$	−5.54080	−0.329366	−0.164683	−	0.986347i	$-0.552660\pi$
$283$	−5.54080	−0.329366	−0.164683	+	0.986347i	$0.552660\pi$
$284$	19.9664	1.18479
$285$	0.342468	0.0202860
$286$	−1.00864	−0.0596424
$287$	17.7175	1.04583
$288$	0.585094	0.0344770
$289$	1.00000	0.0588235
$290$	0.0369138	0.00216765
$291$	8.47463	0.496792
$292$	5.34875	0.313012
$293$	14.1163	0.824681	0.412341	−	0.911030i	$-0.364711\pi$
$293$	14.1163	0.824681	0.412341	+	0.911030i	$0.364711\pi$
$294$	0.245620	0.0143249
$295$	−2.33767	−0.136104
$296$	−1.65500	−0.0961947
$297$	3.08722	0.179139
$298$	0.282873	0.0163864
$299$	−23.3273	−1.34905
$300$	9.91738	0.572580
$301$	37.5226	2.16277
$302$	−0.360721	−0.0207572
$303$	16.5028	0.948061
$304$	−7.25565	−0.416140
$305$	−1.30682	−0.0748283
$306$	0.0488355	0.00279174
$307$	16.7907	0.958295	0.479147	−	0.877734i	$-0.340946\pi$
$307$	16.7907	0.958295	0.479147	+	0.877734i	$0.340946\pi$
$308$	−21.3897	−1.21879
$309$	−3.65982	−0.208200
$310$	−0.0922203	−0.00523776
$311$	17.2105	0.975920	0.487960	−	0.872866i	$-0.337741\pi$
$311$	17.2105	0.975920	0.487960	+	0.872866i	$0.337741\pi$
$312$	1.30608	0.0739424
$313$	11.0304	0.623474	0.311737	−	0.950168i	$-0.399089\pi$
$313$	11.0304	0.623474	0.311737	+	0.950168i	$0.399089\pi$
$314$	0.601944	0.0339697
$315$	−0.652488	−0.0367635
$316$	−1.99762	−0.112375
$317$	−13.9115	−0.781345	−0.390673	−	0.920530i	$-0.627758\pi$
$317$	−13.9115	−0.781345	−0.390673	+	0.920530i	$0.627758\pi$
$318$	−0.392391	−0.0220042
$319$	−12.4043	−0.694510
$320$	1.49425	0.0835309
$321$	19.0585	1.06374
$322$	0.590596	0.0329126
$323$	−1.82042	−0.101291
$324$	−1.99762	−0.110979
$325$	33.2138	1.84237
$326$	0.654864	0.0362696
$327$	−12.7995	−0.707817
$328$	−0.997272	−0.0550651
$329$	−30.7250	−1.69392
$330$	−0.0283630	−0.00156133
$331$	−17.3979	−0.956275	−0.478137	−	0.878285i	$-0.658688\pi$
$331$	−17.3979	−0.956275	−0.478137	+	0.878285i	$0.658688\pi$
$332$	26.2148	1.43872
$333$	8.47735	0.464556
$334$	−0.262922	−0.0143864
$335$	0.283589	0.0154941
$336$	13.8238	0.754152
$337$	−16.4679	−0.897064	−0.448532	−	0.893767i	$-0.648053\pi$
$337$	−16.4679	−0.897064	−0.448532	+	0.893767i	$0.648053\pi$
$338$	1.55090	0.0843581
$339$	10.3696	0.563198
$340$	0.375802	0.0203807
$341$	30.9893	1.67816
$342$	−0.0889012	−0.00480722
$343$	−6.83426	−0.369016
$344$	−2.11205	−0.113874
$345$	−0.655961	−0.0353157
$346$	0.592434	0.0318495
$347$	−12.1439	−0.651921	−0.325961	−	0.945383i	$-0.605688\pi$
$347$	−12.1439	−0.651921	−0.325961	+	0.945383i	$0.605688\pi$
$348$	8.02633	0.430257
$349$	−24.7722	−1.32603	−0.663013	−	0.748607i	$-0.730723\pi$
$349$	−24.7722	−1.32603	−0.663013	+	0.748607i	$0.730723\pi$
$350$	−0.840901	−0.0449481
$351$	−6.69012	−0.357092
$352$	1.80632	0.0962771
$353$	−18.9307	−1.00758	−0.503790	−	0.863826i	$-0.668061\pi$
$353$	−18.9307	−1.00758	−0.503790	+	0.863826i	$0.668061\pi$
$354$	0.606835	0.0322529
$355$	1.88033	0.0997977
$356$	−8.09538	−0.429054
$357$	3.46836	0.183565
$358$	0.263893	0.0139472
$359$	26.0487	1.37480	0.687398	−	0.726281i	$-0.258753\pi$
$359$	26.0487	1.37480	0.687398	+	0.726281i	$0.258753\pi$
$360$	0.0367269	0.00193568
$361$	−15.6861	−0.825582
$362$	1.02887	0.0540762
$363$	−1.46904	−0.0771048
$364$	46.3522	2.42952
$365$	0.503719	0.0263659
$366$	0.339237	0.0177322
$367$	−17.6229	−0.919908	−0.459954	−	0.887943i	$-0.652134\pi$
$367$	−17.6229	−0.919908	−0.459954	+	0.887943i	$0.652134\pi$
$368$	13.8974	0.724454
$369$	5.10831	0.265928
$370$	−0.0778832	−0.00404895
$371$	−27.8681	−1.44684
$372$	−20.0519	−1.03964
$373$	−27.0562	−1.40091	−0.700457	−	0.713694i	$-0.747021\pi$
$373$	−27.0562	−1.40091	−0.700457	+	0.713694i	$0.747021\pi$
$374$	0.150766	0.00779593
$375$	1.87460	0.0968038
$376$	1.72943	0.0891887
$377$	26.8806	1.38442
$378$	0.169379	0.00871193
$379$	14.4095	0.740165	0.370083	−	0.928999i	$-0.379329\pi$
$379$	14.4095	0.740165	0.370083	+	0.928999i	$0.379329\pi$
$380$	−0.684119	−0.0350945
$381$	3.38064	0.173195
$382$	1.32616	0.0678523
$383$	3.52687	0.180215	0.0901074	−	0.995932i	$-0.471279\pi$
$383$	3.52687	0.180215	0.0901074	+	0.995932i	$0.471279\pi$
$384$	−1.55808	−0.0795104
$385$	−2.01438	−0.102662
$386$	0.453107	0.0230625
$387$	10.8185	0.549937
$388$	−16.9290	−0.859442
$389$	19.4993	0.988656	0.494328	−	0.869275i	$-0.335414\pi$
$389$	19.4993	0.988656	0.494328	+	0.869275i	$0.335414\pi$
$390$	0.0614635	0.00311233
$391$	3.48683	0.176336
$392$	−0.981895	−0.0495932
$393$	14.9439	0.753820
$394$	−0.996084	−0.0501820
$395$	−0.188126	−0.00946562
$396$	−6.16709	−0.309908
$397$	−10.3140	−0.517646	−0.258823	−	0.965925i	$-0.583335\pi$
$397$	−10.3140	−0.517646	−0.258823	+	0.965925i	$0.583335\pi$
$398$	−0.509359	−0.0255318
$399$	−6.31388	−0.316089
$400$	−19.7874	−0.989371
$401$	13.9178	0.695024	0.347512	−	0.937676i	$-0.387027\pi$
$401$	13.9178	0.695024	0.347512	+	0.937676i	$0.387027\pi$
$402$	−0.0736170	−0.00367168
$403$	−67.1549	−3.34522
$404$	−32.9662	−1.64013
$405$	−0.188126	−0.00934803
$406$	−0.680558	−0.0337755
$407$	26.1715	1.29727
$408$	−0.195225	−0.00966510
$409$	−8.13110	−0.402057	−0.201029	−	0.979585i	$-0.564428\pi$
$409$	−8.13110	−0.402057	−0.201029	+	0.979585i	$0.564428\pi$
$410$	−0.0469310	−0.00231776
$411$	−3.55003	−0.175110
$412$	7.31090	0.360182
$413$	43.0983	2.12073
$414$	0.170281	0.00836885
$415$	2.46878	0.121188
$416$	−3.91435	−0.191917
$417$	−20.5032	−1.00405
$418$	−0.274458	−0.0134242
$419$	−28.7896	−1.40646	−0.703232	−	0.710961i	$-0.748260\pi$
$419$	−28.7896	−1.40646	−0.703232	+	0.710961i	$0.748260\pi$
$420$	1.30342	0.0636003
$421$	−6.66404	−0.324786	−0.162393	−	0.986726i	$-0.551921\pi$
$421$	−6.66404	−0.324786	−0.162393	+	0.986726i	$0.551921\pi$
$422$	0.215375	0.0104843
$423$	−8.85864	−0.430722
$424$	1.56863	0.0761792
$425$	−4.96461	−0.240819
$426$	−0.488116	−0.0236493
$427$	24.0931	1.16595
$428$	−38.0716	−1.84026
$429$	−20.6539	−0.997180
$430$	−0.0993921	−0.00479311
$431$	1.08855	0.0524336	0.0262168	−	0.999656i	$-0.491654\pi$
$431$	1.08855	0.0524336	0.0262168	+	0.999656i	$0.491654\pi$
$432$	3.98570	0.191762
$433$	−20.9678	−1.00765	−0.503823	−	0.863807i	$-0.668074\pi$
$433$	−20.9678	−1.00765	−0.503823	+	0.863807i	$0.668074\pi$
$434$	1.70021	0.0816129
$435$	0.755880	0.0362417
$436$	25.5686	1.22451
$437$	−6.34749	−0.303642
$438$	−0.130760	−0.00624797
$439$	−2.44992	−0.116928	−0.0584641	−	0.998290i	$-0.518620\pi$
$439$	−2.44992	−0.116928	−0.0584641	+	0.998290i	$0.518620\pi$
$440$	0.113384	0.00540538
$441$	5.02954	0.239502
$442$	−0.326715	−0.0155403
$443$	−6.16931	−0.293113	−0.146556	−	0.989202i	$-0.546819\pi$
$443$	−6.16931	−0.293113	−0.146556	+	0.989202i	$0.546819\pi$
$444$	−16.9345	−0.803675
$445$	−0.762383	−0.0361404
$446$	−1.24236	−0.0588276
$447$	5.79237	0.273970
$448$	−27.5486	−1.30155
$449$	−41.4632	−1.95677	−0.978384	−	0.206795i	$-0.933697\pi$
$449$	−41.4632	−1.95677	−0.978384	+	0.206795i	$0.933697\pi$
$450$	−0.242449	−0.0114292
$451$	15.7705	0.742603
$452$	−20.7144	−0.974324
$453$	−7.38646	−0.347046
$454$	0.00359228	0.000168594 0
$455$	4.36522	0.204645
$456$	0.355393	0.0166428
$457$	−37.6028	−1.75898	−0.879492	−	0.475913i	$-0.842118\pi$
$457$	−37.6028	−1.75898	−0.879492	+	0.475913i	$0.842118\pi$
$458$	0.515347	0.0240806
$459$	1.00000	0.0466760
$460$	1.31036	0.0610957
$461$	22.0115	1.02518	0.512588	−	0.858635i	$-0.328687\pi$
$461$	22.0115	1.02518	0.512588	+	0.858635i	$0.328687\pi$
$462$	0.522912	0.0243281
$463$	−29.7303	−1.38168	−0.690841	−	0.723006i	$-0.742760\pi$
$463$	−29.7303	−1.38168	−0.690841	+	0.723006i	$0.742760\pi$
$464$	−16.0144	−0.743448
$465$	−1.88839	−0.0875719
$466$	0.895691	0.0414921
$467$	−20.0102	−0.925961	−0.462981	−	0.886368i	$-0.653220\pi$
$467$	−20.0102	−0.925961	−0.462981	+	0.886368i	$0.653220\pi$
$468$	13.3643	0.617765
$469$	−5.22838	−0.241424
$470$	0.0813861	0.00375406
$471$	12.3260	0.567950
$472$	−2.42589	−0.111661
$473$	33.3993	1.53570
$474$	0.0488355	0.00224309
$475$	9.03768	0.414677
$476$	−6.92845	−0.317565
$477$	−8.03495	−0.367895
$478$	1.33004	0.0608348
$479$	−4.05162	−0.185123	−0.0925616	−	0.995707i	$-0.529506\pi$
$479$	−4.05162	−0.185123	−0.0925616	+	0.995707i	$0.529506\pi$
$480$	−0.110071	−0.00502404
$481$	−56.7145	−2.58596
$482$	0.247422	0.0112698
$483$	12.0936	0.550277
$484$	2.93458	0.133390
$485$	−1.59429	−0.0723932
$486$	0.0488355	0.00221522
$487$	−16.7295	−0.758084	−0.379042	−	0.925379i	$-0.623746\pi$
$487$	−16.7295	−0.758084	−0.379042	+	0.925379i	$0.623746\pi$
$488$	−1.35614	−0.0613896
$489$	13.4096	0.606403
$490$	−0.0462074	−0.00208744
$491$	−42.0166	−1.89618	−0.948092	−	0.317996i	$-0.896990\pi$
$491$	−42.0166	−1.89618	−0.948092	+	0.317996i	$0.896990\pi$
$492$	−10.2044	−0.460051
$493$	−4.01796	−0.180960
$494$	0.594760	0.0267595
$495$	−0.580786	−0.0261044
$496$	40.0081	1.79642
$497$	−34.6666	−1.55501
$498$	−0.640871	−0.0287181
$499$	−29.6144	−1.32572	−0.662861	−	0.748742i	$-0.730658\pi$
$499$	−29.6144	−1.32572	−0.662861	+	0.748742i	$0.730658\pi$
$500$	−3.74472	−0.167469
$501$	−5.38383	−0.240532
$502$	0.607488	0.0271135
$503$	14.8661	0.662846	0.331423	−	0.943482i	$-0.392471\pi$
$503$	14.8661	0.662846	0.331423	+	0.943482i	$0.392471\pi$
$504$	−0.677113	−0.0301610
$505$	−3.10460	−0.138153
$506$	0.525695	0.0233700
$507$	31.7577	1.41041
$508$	−6.75321	−0.299625
$509$	37.4954	1.66195	0.830977	−	0.556306i	$-0.187782\pi$
$509$	37.4954	1.66195	0.830977	+	0.556306i	$0.187782\pi$
$510$	−0.00918720	−0.000406816 0
$511$	−9.28678	−0.410823
$512$	3.88823	0.171837
$513$	−1.82042	−0.0803736
$514$	0.113182	0.00499223
$515$	0.688505	0.0303391
$516$	−21.6113	−0.951384
$517$	−27.3486	−1.20279
$518$	1.43589	0.0630893
$519$	12.1312	0.532502
$520$	−0.245707	−0.0107750
$521$	−23.8055	−1.04294	−0.521468	−	0.853271i	$-0.674616\pi$
$521$	−23.8055	−1.04294	−0.521468	+	0.853271i	$0.674616\pi$
$522$	−0.196219	−0.00858827
$523$	40.3639	1.76499	0.882494	−	0.470323i	$-0.155863\pi$
$523$	40.3639	1.76499	0.882494	+	0.470323i	$0.155863\pi$
$524$	−29.8522	−1.30410
$525$	−17.2191	−0.751502
$526$	−0.700307	−0.0305348
$527$	10.0379	0.437258
$528$	12.3047	0.535495
$529$	−10.8420	−0.471393
$530$	0.0738187	0.00320648
$531$	12.4261	0.539248
$532$	12.6127	0.546830
$533$	−34.1752	−1.48029
$534$	0.197907	0.00856427
$535$	−3.58539	−0.155010
$536$	0.294292	0.0127115
$537$	5.40371	0.233187
$538$	−0.244610	−0.0105459
$539$	15.5273	0.668809
$540$	0.375802	0.0161720
$541$	−10.8843	−0.467954	−0.233977	−	0.972242i	$-0.575174\pi$
$541$	−10.8843	−0.467954	−0.233977	+	0.972242i	$0.575174\pi$
$542$	−0.239497	−0.0102873
$543$	21.0681	0.904118
$544$	0.585094	0.0250857
$545$	2.40792	0.103144
$546$	−1.13317	−0.0484951
$547$	12.0756	0.516314	0.258157	−	0.966103i	$-0.416885\pi$
$547$	12.0756	0.516314	0.258157	+	0.966103i	$0.416885\pi$
$548$	7.09159	0.302938
$549$	6.94654	0.296471
$550$	−0.748495	−0.0319159
$551$	7.31438	0.311603
$552$	−0.680717	−0.0289732
$553$	3.46836	0.147490
$554$	0.533975	0.0226864
$555$	−1.59481	−0.0676958
$556$	40.9575	1.73699
$557$	−18.2844	−0.774737	−0.387368	−	0.921925i	$-0.626616\pi$
$557$	−18.2844	−0.774737	−0.387368	+	0.921925i	$0.626616\pi$
$558$	0.490206	0.0207521
$559$	−72.3774	−3.06124
$560$	−2.60062	−0.109896
$561$	3.08722	0.130343
$562$	0.969096	0.0408788
$563$	18.2598	0.769558	0.384779	−	0.923009i	$-0.374278\pi$
$563$	18.2598	0.769558	0.384779	+	0.923009i	$0.374278\pi$
$564$	17.6962	0.745143
$565$	−1.95078	−0.0820700
$566$	−0.270588	−0.0113737
$567$	3.46836	0.145658
$568$	1.95130	0.0818747
$569$	−0.690518	−0.0289480	−0.0144740	−	0.999895i	$-0.504607\pi$
$569$	−0.690518	−0.0289480	−0.0144740	+	0.999895i	$0.504607\pi$
$570$	0.0167246	0.000700515 0
$571$	22.6322	0.947128	0.473564	−	0.880759i	$-0.342967\pi$
$571$	22.6322	0.947128	0.473564	+	0.880759i	$0.342967\pi$
$572$	41.2586	1.72511
$573$	27.1557	1.13445
$574$	0.865241	0.0361145
$575$	−17.3107	−0.721907
$576$	−7.94282	−0.330951
$577$	41.8894	1.74388	0.871939	−	0.489615i	$-0.162863\pi$
$577$	41.8894	1.74388	0.871939	+	0.489615i	$0.162863\pi$
$578$	0.0488355	0.00203129
$579$	9.27823	0.385590
$580$	−1.50996	−0.0626976
$581$	−45.5155	−1.88830
$582$	0.413863	0.0171552
$583$	−24.8057	−1.02735
$584$	0.522730	0.0216307
$585$	1.25858	0.0520360
$586$	0.689375	0.0284778
$587$	−39.0445	−1.61154	−0.805769	−	0.592230i	$-0.798248\pi$
$587$	−39.0445	−1.61154	−0.805769	+	0.592230i	$0.798248\pi$
$588$	−10.0471	−0.414335
$589$	−18.2732	−0.752935
$590$	−0.114161	−0.00469994
$591$	−20.3967	−0.839009
$592$	33.7882	1.38868
$593$	40.1747	1.64978	0.824888	−	0.565296i	$-0.191238\pi$
$593$	40.1747	1.64978	0.824888	+	0.565296i	$0.191238\pi$
$594$	0.150766	0.00618601
$595$	−0.652488	−0.0267494
$596$	−11.5709	−0.473963
$597$	−10.4301	−0.426875
$598$	−1.13920	−0.0465853
$599$	−4.51721	−0.184568	−0.0922841	−	0.995733i	$-0.529417\pi$
$599$	−4.51721	−0.184568	−0.0922841	+	0.995733i	$0.529417\pi$
$600$	0.969218	0.0395682
$601$	5.24339	0.213882	0.106941	−	0.994265i	$-0.465894\pi$
$601$	5.24339	0.213882	0.106941	+	0.994265i	$0.465894\pi$
$602$	1.83244	0.0746845
$603$	−1.50745	−0.0613880
$604$	14.7553	0.600385
$605$	0.276365	0.0112358
$606$	0.805922	0.0327384
$607$	−32.4421	−1.31679	−0.658393	−	0.752674i	$-0.728763\pi$
$607$	−32.4421	−1.31679	−0.658393	+	0.752674i	$0.728763\pi$
$608$	−1.06512	−0.0431963
$609$	−13.9357	−0.564704
$610$	−0.0638192	−0.00258396
$611$	59.2654	2.39762
$612$	−1.99762	−0.0807488
$613$	−19.4452	−0.785383	−0.392692	−	0.919670i	$-0.628456\pi$
$613$	−19.4452	−0.785383	−0.392692	+	0.919670i	$0.628456\pi$
$614$	0.819981	0.0330917
$615$	−0.961003	−0.0387514
$616$	−2.09040	−0.0842246
$617$	30.8778	1.24309	0.621546	−	0.783377i	$-0.286505\pi$
$617$	30.8778	1.24309	0.621546	+	0.783377i	$0.286505\pi$
$618$	−0.178729	−0.00718953
$619$	1.50085	0.0603244	0.0301622	−	0.999545i	$-0.490398\pi$
$619$	1.50085	0.0603244	0.0301622	+	0.999545i	$0.490398\pi$
$620$	3.77227	0.151498
$621$	3.48683	0.139921
$622$	0.840485	0.0337004
$623$	14.0556	0.563127
$624$	−26.6648	−1.06745
$625$	24.4704	0.978815
$626$	0.538674	0.0215298
$627$	−5.62005	−0.224443
$628$	−24.6225	−0.982545
$629$	8.47735	0.338014
$630$	−0.0318645	−0.00126951
$631$	−20.0363	−0.797634	−0.398817	−	0.917031i	$-0.630579\pi$
$631$	−20.0363	−0.797634	−0.398817	+	0.917031i	$0.630579\pi$
$632$	−0.195225	−0.00776565
$633$	4.41022	0.175290
$634$	−0.679373	−0.0269813
$635$	−0.635984	−0.0252383
$636$	16.0507	0.636453
$637$	−33.6483	−1.33319
$638$	−0.605772	−0.0239827
$639$	−9.99510	−0.395400
$640$	0.293115	0.0115864
$641$	−46.6685	−1.84329	−0.921646	−	0.388031i	$-0.873155\pi$
$641$	−46.6685	−1.84329	−0.921646	+	0.388031i	$0.873155\pi$
$642$	0.930732	0.0367331
$643$	18.7704	0.740231	0.370115	−	0.928986i	$-0.379318\pi$
$643$	18.7704	0.740231	0.370115	+	0.928986i	$0.379318\pi$
$644$	−24.1583	−0.951971
$645$	−2.03524	−0.0801376
$646$	−0.0889012	−0.00349777
$647$	−36.9010	−1.45073	−0.725364	−	0.688366i	$-0.758328\pi$
$647$	−36.9010	−1.45073	−0.725364	+	0.688366i	$0.758328\pi$
$648$	−0.195225	−0.00766918
$649$	38.3622	1.50585
$650$	1.62201	0.0636206
$651$	34.8151	1.36451
$652$	−26.7872	−1.04907
$653$	−19.9016	−0.778808	−0.389404	−	0.921067i	$-0.627319\pi$
$653$	−19.9016	−0.778808	−0.389404	+	0.921067i	$0.627319\pi$
$654$	−0.625072	−0.0244423
$655$	−2.81133	−0.109848
$656$	20.3602	0.794931
$657$	−2.67757	−0.104462
$658$	−1.50047	−0.0584944
$659$	2.98589	0.116314	0.0581568	−	0.998307i	$-0.481478\pi$
$659$	2.98589	0.116314	0.0581568	+	0.998307i	$0.481478\pi$
$660$	1.16019	0.0451602
$661$	25.4871	0.991334	0.495667	−	0.868513i	$-0.334924\pi$
$661$	25.4871	0.991334	0.495667	+	0.868513i	$0.334924\pi$
$662$	−0.849635	−0.0330220
$663$	−6.69012	−0.259823
$664$	2.56195	0.0994231
$665$	1.18780	0.0460610
$666$	0.413996	0.0160420
$667$	−14.0099	−0.542466
$668$	10.7548	0.416116
$669$	−25.4398	−0.983558
$670$	0.0138492	0.000535042 0
$671$	21.4455	0.827895
$672$	2.02932	0.0782827
$673$	−5.43037	−0.209325	−0.104663	−	0.994508i	$-0.533376\pi$
$673$	−5.43037	−0.209325	−0.104663	+	0.994508i	$0.533376\pi$
$674$	−0.804218	−0.0309773
$675$	−4.96461	−0.191088
$676$	−63.4397	−2.43999
$677$	−10.6352	−0.408742	−0.204371	−	0.978893i	$-0.565515\pi$
$677$	−10.6352	−0.408742	−0.204371	+	0.978893i	$0.565515\pi$
$678$	0.506403	0.0194483
$679$	29.3931	1.12800
$680$	0.0367269	0.00140841
$681$	0.0735588	0.00281878
$682$	1.51338	0.0579502
$683$	4.50761	0.172479	0.0862395	−	0.996274i	$-0.472515\pi$
$683$	4.50761	0.172479	0.0862395	+	0.996274i	$0.472515\pi$
$684$	3.63650	0.139045
$685$	0.667851	0.0255173
$686$	−0.333755	−0.0127428
$687$	10.5527	0.402611
$688$	43.1194	1.64391
$689$	53.7548	2.04789
$690$	−0.0320342	−0.00121952
$691$	7.65022	0.291028	0.145514	−	0.989356i	$-0.453516\pi$
$691$	7.65022	0.291028	0.145514	+	0.989356i	$0.453516\pi$
$692$	−24.2335	−0.921220
$693$	10.7076	0.406749
$694$	−0.593055	−0.0225121
$695$	3.85718	0.146311
$696$	0.784408	0.0297329
$697$	5.10831	0.193491
$698$	−1.20976	−0.0457902
$699$	18.3410	0.693720
$700$	34.3971	1.30009
$701$	22.7818	0.860458	0.430229	−	0.902720i	$-0.358433\pi$
$701$	22.7818	0.860458	0.430229	+	0.902720i	$0.358433\pi$
$702$	−0.326715	−0.0123311
$703$	−15.4324	−0.582042
$704$	−24.5213	−0.924180
$705$	1.66654	0.0627654
$706$	−0.924490	−0.0347937
$707$	57.2377	2.15265
$708$	−24.8226	−0.932890
$709$	−4.36980	−0.164111	−0.0820557	−	0.996628i	$-0.526149\pi$
$709$	−4.36980	−0.164111	−0.0820557	+	0.996628i	$0.526149\pi$
$710$	0.0918270	0.00344621
$711$	1.00000	0.0375029
$712$	−0.791156	−0.0296498
$713$	35.0005	1.31078
$714$	0.169379	0.00633886
$715$	3.88553	0.145311
$716$	−10.7945	−0.403411
$717$	27.2352	1.01712
$718$	1.27210	0.0474743
$719$	15.3235	0.571471	0.285736	−	0.958308i	$-0.407762\pi$
$719$	15.3235	0.571471	0.285736	+	0.958308i	$0.407762\pi$
$720$	−0.749811	−0.0279438
$721$	−12.6936	−0.472733
$722$	−0.766037	−0.0285089
$723$	5.06644	0.188423
$724$	−42.0859	−1.56411
$725$	19.9476	0.740835
$726$	−0.0717415	−0.00266258
$727$	−45.1511	−1.67456	−0.837280	−	0.546774i	$-0.815856\pi$
$727$	−45.1511	−1.67456	−0.837280	+	0.546774i	$0.815856\pi$
$728$	4.52997	0.167892
$729$	1.00000	0.0370370
$730$	0.0245994	0.000910463 0
$731$	10.8185	0.400138
$732$	−13.8765	−0.512890
$733$	36.4108	1.34486	0.672431	−	0.740160i	$-0.265250\pi$
$733$	36.4108	1.34486	0.672431	+	0.740160i	$0.265250\pi$
$734$	−0.860623	−0.0317662
$735$	−0.946185	−0.0349006
$736$	2.04012	0.0751999
$737$	−4.65383	−0.171426
$738$	0.249467	0.00918299
$739$	26.4373	0.972511	0.486256	−	0.873817i	$-0.338362\pi$
$739$	26.4373	0.972511	0.486256	+	0.873817i	$0.338362\pi$
$740$	3.18581	0.117113
$741$	12.1788	0.447401
$742$	−1.36095	−0.0499622
$743$	25.0440	0.918777	0.459388	−	0.888235i	$-0.348069\pi$
$743$	25.0440	0.918777	0.459388	+	0.888235i	$0.348069\pi$
$744$	−1.95966	−0.0718445
$745$	−1.08969	−0.0399232
$746$	−1.32130	−0.0483763
$747$	−13.1230	−0.480147
$748$	−6.16709	−0.225491
$749$	66.1018	2.41531
$750$	0.0915469	0.00334282
$751$	−33.2486	−1.21326	−0.606630	−	0.794985i	$-0.707479\pi$
$751$	−33.2486	−1.21326	−0.606630	+	0.794985i	$0.707479\pi$
$752$	−35.3079	−1.28755
$753$	12.4395	0.453320
$754$	1.31273	0.0478068
$755$	1.38958	0.0505721
$756$	−6.92845	−0.251985
$757$	−0.252350	−0.00917182	−0.00458591	−	0.999989i	$-0.501460\pi$
$757$	−0.252350	−0.00917182	−0.00458591	+	0.999989i	$0.501460\pi$
$758$	0.703694	0.0255593
$759$	10.7646	0.390731
$760$	−0.0668584	−0.00242521
$761$	−25.2599	−0.915671	−0.457835	−	0.889037i	$-0.651375\pi$
$761$	−25.2599	−0.915671	−0.457835	+	0.889037i	$0.651375\pi$
$762$	0.165095	0.00598077
$763$	−44.3935	−1.60715
$764$	−54.2466	−1.96257
$765$	−0.188126	−0.00680169
$766$	0.172236	0.00622316
$767$	−83.1322	−3.00173
$768$	15.8095	0.570478
$769$	−13.6542	−0.492385	−0.246192	−	0.969221i	$-0.579179\pi$
$769$	−13.6542	−0.492385	−0.246192	+	0.969221i	$0.579179\pi$
$770$	−0.0983730	−0.00354512
$771$	2.31761	0.0834667
$772$	−18.5343	−0.667065
$773$	−23.8219	−0.856813	−0.428406	−	0.903586i	$-0.640925\pi$
$773$	−23.8219	−0.856813	−0.428406	+	0.903586i	$0.640925\pi$
$774$	0.528329	0.0189904
$775$	−49.8343	−1.79010
$776$	−1.65446	−0.0593918
$777$	29.4025	1.05481
$778$	0.952260	0.0341402
$779$	−9.29927	−0.333181
$780$	−2.51416	−0.0900215
$781$	−30.8571	−1.10415
$782$	0.170281	0.00608923
$783$	−4.01796	−0.143590
$784$	20.0462	0.715937
$785$	−2.31883	−0.0827625
$786$	0.729793	0.0260308
$787$	−5.27091	−0.187888	−0.0939438	−	0.995577i	$-0.529947\pi$
$787$	−5.27091	−0.187888	−0.0939438	+	0.995577i	$0.529947\pi$
$788$	40.7448	1.45147
$789$	−14.3401	−0.510522
$790$	−0.00918720	−0.000326866 0
$791$	35.9654	1.27878
$792$	−0.602705	−0.0214162
$793$	−46.4732	−1.65031
$794$	−0.503690	−0.0178753
$795$	1.51158	0.0536102
$796$	20.8353	0.738488
$797$	19.8387	0.702722	0.351361	−	0.936240i	$-0.385719\pi$
$797$	19.8387	0.702722	0.351361	+	0.936240i	$0.385719\pi$
$798$	−0.308342	−0.0109152
$799$	−8.85864	−0.313396
$800$	−2.90476	−0.102699
$801$	4.05252	0.143189
$802$	0.679684	0.0240005
$803$	−8.26626	−0.291710
$804$	3.01130	0.106200
$805$	−2.27511	−0.0801871
$806$	−3.27954	−0.115517
$807$	−5.00886	−0.176320
$808$	−3.22177	−0.113341
$809$	31.8772	1.12074	0.560371	−	0.828242i	$-0.310659\pi$
$809$	31.8772	1.12074	0.560371	+	0.828242i	$0.310659\pi$
$810$	−0.00918720	−0.000322805 0
$811$	−25.3291	−0.889425	−0.444713	−	0.895673i	$-0.646694\pi$
$811$	−25.3291	−0.889425	−0.444713	+	0.895673i	$0.646694\pi$
$812$	27.8382	0.976930
$813$	−4.90415	−0.171996
$814$	1.27810	0.0447973
$815$	−2.52269	−0.0883659
$816$	3.98570	0.139527
$817$	−19.6943	−0.689016
$818$	−0.397086	−0.0138838
$819$	−23.2038	−0.810805
$820$	1.91971	0.0670393
$821$	−18.9958	−0.662958	−0.331479	−	0.943463i	$-0.607548\pi$
$821$	−18.9958	−0.662958	−0.331479	+	0.943463i	$0.607548\pi$
$822$	−0.173367	−0.00604688
$823$	9.08785	0.316782	0.158391	−	0.987376i	$-0.449369\pi$
$823$	9.08785	0.316782	0.158391	+	0.987376i	$0.449369\pi$
$824$	0.714489	0.0248904
$825$	−15.3269	−0.533613
$826$	2.10473	0.0732328
$827$	27.4456	0.954376	0.477188	−	0.878801i	$-0.341656\pi$
$827$	27.4456	0.954376	0.477188	+	0.878801i	$0.341656\pi$
$828$	−6.96534	−0.242062
$829$	53.2375	1.84901	0.924507	−	0.381164i	$-0.124477\pi$
$829$	53.2375	1.84901	0.924507	+	0.381164i	$0.124477\pi$
$830$	0.120564	0.00418484
$831$	10.9342	0.379302
$832$	53.1384	1.84224
$833$	5.02954	0.174263
$834$	−1.00128	−0.0346716
$835$	1.01284	0.0350506
$836$	11.2267	0.388283
$837$	10.0379	0.346961
$838$	−1.40595	−0.0485679
$839$	6.57324	0.226934	0.113467	−	0.993542i	$-0.463804\pi$
$839$	6.57324	0.226934	0.113467	+	0.993542i	$0.463804\pi$
$840$	0.127382	0.00439510
$841$	−12.8560	−0.443311
$842$	−0.325442	−0.0112155
$843$	19.8441	0.683467
$844$	−8.80991	−0.303250
$845$	−5.97444	−0.205527
$846$	−0.432616	−0.0148736
$847$	−5.09518	−0.175072
$848$	−32.0249	−1.09974
$849$	−5.54080	−0.190160
$850$	−0.242449	−0.00831594
$851$	29.5591	1.01327
$852$	19.9664	0.684036
$853$	−25.0968	−0.859297	−0.429648	−	0.902996i	$-0.641363\pi$
$853$	−25.0968	−0.859297	−0.429648	+	0.902996i	$0.641363\pi$
$854$	1.17660	0.0402624
$855$	0.342468	0.0117121
$856$	−3.72071	−0.127171
$857$	−31.7374	−1.08413	−0.542065	−	0.840336i	$-0.682357\pi$
$857$	−31.7374	−1.08413	−0.542065	+	0.840336i	$0.682357\pi$
$858$	−1.00864	−0.0344345
$859$	−43.1041	−1.47069	−0.735347	−	0.677691i	$-0.762981\pi$
$859$	−43.1041	−1.47069	−0.735347	+	0.677691i	$0.762981\pi$
$860$	4.06563	0.138637
$861$	17.7175	0.603809
$862$	0.0531599	0.00181063
$863$	35.8401	1.22001	0.610006	−	0.792397i	$-0.291167\pi$
$863$	35.8401	1.22001	0.610006	+	0.792397i	$0.291167\pi$
$864$	0.585094	0.0199053
$865$	−2.28219	−0.0775969
$866$	−1.02397	−0.0347960
$867$	1.00000	0.0339618
$868$	−69.5472	−2.36059
$869$	3.08722	0.104727
$870$	0.0369138	0.00125149
$871$	10.0850	0.341718
$872$	2.49880	0.0846200
$873$	8.47463	0.286823
$874$	−0.309983	−0.0104853
$875$	6.50178	0.219800
$876$	5.34875	0.180718
$877$	36.3503	1.22746	0.613730	−	0.789516i	$-0.289668\pi$
$877$	36.3503	1.22746	0.613730	+	0.789516i	$0.289668\pi$
$878$	−0.119643	−0.00403775
$879$	14.1163	0.476130
$880$	−2.31484	−0.0780331
$881$	−49.9058	−1.68137	−0.840684	−	0.541526i	$-0.817847\pi$
$881$	−49.9058	−1.68137	−0.840684	+	0.541526i	$0.817847\pi$
$882$	0.245620	0.00827046
$883$	1.84497	0.0620882	0.0310441	−	0.999518i	$-0.490117\pi$
$883$	1.84497	0.0620882	0.0310441	+	0.999518i	$0.490117\pi$
$884$	13.3643	0.449490
$885$	−2.33767	−0.0785799
$886$	−0.301281	−0.0101217
$887$	7.83251	0.262990	0.131495	−	0.991317i	$-0.458022\pi$
$887$	7.83251	0.262990	0.131495	+	0.991317i	$0.458022\pi$
$888$	−1.65500	−0.0555380
$889$	11.7253	0.393253
$890$	−0.0372314	−0.00124800
$891$	3.08722	0.103426
$892$	50.8188	1.70154
$893$	16.1265	0.539651
$894$	0.282873	0.00946069
$895$	−1.01658	−0.0339804
$896$	−5.40399	−0.180535
$897$	−23.3273	−0.778876
$898$	−2.02487	−0.0675710
$899$	−40.3319	−1.34514
$900$	9.91738	0.330579
$901$	−8.03495	−0.267683
$902$	0.770160	0.0256435
$903$	37.5226	1.24867
$904$	−2.02440	−0.0673307
$905$	−3.96344	−0.131749
$906$	−0.360721	−0.0119842
$907$	9.38676	0.311682	0.155841	−	0.987782i	$-0.450191\pi$
$907$	9.38676	0.311682	0.155841	+	0.987782i	$0.450191\pi$
$908$	−0.146942	−0.00487645
$909$	16.5028	0.547363
$910$	0.213178	0.00706677
$911$	−33.2686	−1.10224	−0.551118	−	0.834427i	$-0.685799\pi$
$911$	−33.2686	−1.10224	−0.551118	+	0.834427i	$0.685799\pi$
$912$	−7.25565	−0.240258
$913$	−40.5138	−1.34081
$914$	−1.83635	−0.0607411
$915$	−1.30682	−0.0432022
$916$	−21.0803	−0.696512
$917$	51.8309	1.71161
$918$	0.0488355	0.00161181
$919$	4.42083	0.145830	0.0729148	−	0.997338i	$-0.476770\pi$
$919$	4.42083	0.145830	0.0729148	+	0.997338i	$0.476770\pi$
$920$	0.128060	0.00422202
$921$	16.7907	0.553272
$922$	1.07494	0.0354013
$923$	66.8684	2.20100
$924$	−21.3897	−0.703669
$925$	−42.0868	−1.38380
$926$	−1.45189	−0.0477121
$927$	−3.65982	−0.120204
$928$	−2.35088	−0.0771716
$929$	13.9434	0.457468	0.228734	−	0.973489i	$-0.426541\pi$
$929$	13.9434	0.457468	0.228734	+	0.973489i	$0.426541\pi$
$930$	−0.0922203	−0.00302402
$931$	−9.15589	−0.300072
$932$	−36.6382	−1.20013
$933$	17.2105	0.563448
$934$	−0.977207	−0.0319752
$935$	−0.580786	−0.0189937
$936$	1.30608	0.0426906
$937$	9.60780	0.313873	0.156937	−	0.987609i	$-0.449838\pi$
$937$	9.60780	0.313873	0.156937	+	0.987609i	$0.449838\pi$
$938$	−0.255330	−0.00833683
$939$	11.0304	0.359963
$940$	−3.32910	−0.108583
$941$	−27.7472	−0.904532	−0.452266	−	0.891883i	$-0.649384\pi$
$941$	−27.7472	−0.904532	−0.452266	+	0.891883i	$0.649384\pi$
$942$	0.601944	0.0196124
$943$	17.8118	0.580031
$944$	49.5267	1.61196
$945$	−0.652488	−0.0212254
$946$	1.63107	0.0530306
$947$	−46.9156	−1.52455	−0.762276	−	0.647252i	$-0.775918\pi$
$947$	−46.9156	−1.52455	−0.762276	+	0.647252i	$0.775918\pi$
$948$	−1.99762	−0.0648795
$949$	17.9133	0.581489
$950$	0.441360	0.0143196
$951$	−13.9115	−0.451110
$952$	−0.677113	−0.0219454
$953$	23.8600	0.772902	0.386451	−	0.922310i	$-0.373701\pi$
$953$	23.8600	0.772902	0.386451	+	0.922310i	$0.373701\pi$
$954$	−0.392391	−0.0127041
$955$	−5.10868	−0.165313
$956$	−54.4054	−1.75960
$957$	−12.4043	−0.400975
$958$	−0.197863	−0.00639266
$959$	−12.3128	−0.397601
$960$	1.49425	0.0482266
$961$	69.7597	2.25031
$962$	−2.76968	−0.0892981
$963$	19.0585	0.614152
$964$	−10.1208	−0.325969
$965$	−1.74547	−0.0561887
$966$	0.590596	0.0190021
$967$	11.8915	0.382405	0.191203	−	0.981551i	$-0.438761\pi$
$967$	11.8915	0.382405	0.191203	+	0.981551i	$0.438761\pi$
$968$	0.286795	0.00921793
$969$	−1.82042	−0.0584804
$970$	−0.0778581	−0.00249987
$971$	56.1500	1.80194	0.900969	−	0.433883i	$-0.142857\pi$
$971$	56.1500	1.80194	0.900969	+	0.433883i	$0.142857\pi$
$972$	−1.99762	−0.0640735
$973$	−71.1126	−2.27977
$974$	−0.816992	−0.0261781
$975$	33.2138	1.06369
$976$	27.6868	0.886232
$977$	37.7889	1.20897	0.604486	−	0.796615i	$-0.293378\pi$
$977$	37.7889	1.20897	0.604486	+	0.796615i	$0.293378\pi$
$978$	0.654864	0.0209403
$979$	12.5111	0.399855
$980$	1.89011	0.0603775
$981$	−12.7995	−0.408658
$982$	−2.05190	−0.0654789
$983$	−45.3739	−1.44720	−0.723601	−	0.690218i	$-0.757514\pi$
$983$	−45.3739	−1.44720	−0.723601	+	0.690218i	$0.757514\pi$
$984$	−0.997272	−0.0317919
$985$	3.83714	0.122262
$986$	−0.196219	−0.00624888
$987$	−30.7250	−0.977987
$988$	−24.3286	−0.773997
$989$	37.7224	1.19950
$990$	−0.0283630	−0.000901434 0
$991$	1.53256	0.0486834	0.0243417	−	0.999704i	$-0.492251\pi$
$991$	1.53256	0.0486834	0.0243417	+	0.999704i	$0.492251\pi$
$992$	5.87313	0.186472
$993$	−17.3979	−0.552105
$994$	−1.69296	−0.0536975
$995$	1.96217	0.0622049
$996$	26.2148	0.830648
$997$	28.8584	0.913953	0.456977	−	0.889479i	$-0.348932\pi$
$997$	28.8584	0.913953	0.456977	+	0.889479i	$0.348932\pi$
$998$	−1.44623	−0.0457797
$999$	8.47735	0.268212

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.k.1.15	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.k.1.15	✓	31	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:	\( N \)	\(=\)	\( 4029 = 3 \cdot 17 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4029.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0488355 q^{2} +1.00000 q^{3} -1.99762 q^{4} -0.188126 q^{5} +0.0488355 q^{6} +3.46836 q^{7} -0.195225 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.0488355 q^{2} +1.00000 q^{3} -1.99762 q^{4} -0.188126 q^{5} +0.0488355 q^{6} +3.46836 q^{7} -0.195225 q^{8} +1.00000 q^{9} -0.00918720 q^{10} +3.08722 q^{11} -1.99762 q^{12} -6.69012 q^{13} +0.169379 q^{14} -0.188126 q^{15} +3.98570 q^{16} +1.00000 q^{17} +0.0488355 q^{18} -1.82042 q^{19} +0.375802 q^{20} +3.46836 q^{21} +0.150766 q^{22} +3.48683 q^{23} -0.195225 q^{24} -4.96461 q^{25} -0.326715 q^{26} +1.00000 q^{27} -6.92845 q^{28} -4.01796 q^{29} -0.00918720 q^{30} +10.0379 q^{31} +0.585094 q^{32} +3.08722 q^{33} +0.0488355 q^{34} -0.652488 q^{35} -1.99762 q^{36} +8.47735 q^{37} -0.0889012 q^{38} -6.69012 q^{39} +0.0367269 q^{40} +5.10831 q^{41} +0.169379 q^{42} +10.8185 q^{43} -6.16709 q^{44} -0.188126 q^{45} +0.170281 q^{46} -8.85864 q^{47} +3.98570 q^{48} +5.02954 q^{49} -0.242449 q^{50} +1.00000 q^{51} +13.3643 q^{52} -8.03495 q^{53} +0.0488355 q^{54} -0.580786 q^{55} -0.677113 q^{56} -1.82042 q^{57} -0.196219 q^{58} +12.4261 q^{59} +0.375802 q^{60} +6.94654 q^{61} +0.490206 q^{62} +3.46836 q^{63} -7.94282 q^{64} +1.25858 q^{65} +0.150766 q^{66} -1.50745 q^{67} -1.99762 q^{68} +3.48683 q^{69} -0.0318645 q^{70} -9.99510 q^{71} -0.195225 q^{72} -2.67757 q^{73} +0.413996 q^{74} -4.96461 q^{75} +3.63650 q^{76} +10.7076 q^{77} -0.326715 q^{78} +1.00000 q^{79} -0.749811 q^{80} +1.00000 q^{81} +0.249467 q^{82} -13.1230 q^{83} -6.92845 q^{84} -0.188126 q^{85} +0.528329 q^{86} -4.01796 q^{87} -0.602705 q^{88} +4.05252 q^{89} -0.00918720 q^{90} -23.2038 q^{91} -6.96534 q^{92} +10.0379 q^{93} -0.432616 q^{94} +0.342468 q^{95} +0.585094 q^{96} +8.47463 q^{97} +0.245620 q^{98} +3.08722 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9} + 5 q^{10} + 26 q^{11} + 34 q^{12} + 7 q^{13} + 19 q^{14} + 11 q^{15} + 40 q^{16} + 31 q^{17} + 4 q^{18} + 32 q^{19} + 23 q^{20} + 4 q^{21} + 2 q^{22} + 29 q^{23} + 12 q^{24} + 32 q^{25} + 13 q^{26} + 31 q^{27} - 13 q^{28} + 25 q^{29} + 5 q^{30} + 22 q^{31} + 28 q^{32} + 26 q^{33} + 4 q^{34} + 20 q^{35} + 34 q^{36} - 4 q^{37} + 19 q^{38} + 7 q^{39} - 3 q^{40} + 33 q^{41} + 19 q^{42} + 6 q^{43} + 30 q^{44} + 11 q^{45} - 11 q^{46} + 23 q^{47} + 40 q^{48} + 31 q^{49} + 6 q^{50} + 31 q^{51} - 7 q^{52} + 12 q^{53} + 4 q^{54} + 40 q^{56} + 32 q^{57} + 9 q^{58} + 27 q^{59} + 23 q^{60} - 4 q^{61} + 25 q^{62} + 4 q^{63} + 10 q^{64} + 54 q^{65} + 2 q^{66} + 34 q^{68} + 29 q^{69} - 59 q^{70} + 35 q^{71} + 12 q^{72} + 5 q^{73} + 48 q^{74} + 32 q^{75} + 32 q^{76} + 42 q^{77} + 13 q^{78} + 31 q^{79} + 24 q^{80} + 31 q^{81} + 5 q^{82} + 67 q^{83} - 13 q^{84} + 11 q^{85} - 20 q^{86} + 25 q^{87} - 7 q^{88} + 22 q^{89} + 5 q^{90} + 16 q^{91} + 57 q^{92} + 22 q^{93} + 45 q^{94} + 73 q^{95} + 28 q^{96} - 13 q^{97} - 19 q^{98} + 26 q^{99}+O(q^{100}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9 + 5 * q^10 + 26 * q^11 + 34 * q^12 + 7 * q^13 + 19 * q^14 + 11 * q^15 + 40 * q^16 + 31 * q^17 + 4 * q^18 + 32 * q^19 + 23 * q^20 + 4 * q^21 + 2 * q^22 + 29 * q^23 + 12 * q^24 + 32 * q^25 + 13 * q^26 + 31 * q^27 - 13 * q^28 + 25 * q^29 + 5 * q^30 + 22 * q^31 + 28 * q^32 + 26 * q^33 + 4 * q^34 + 20 * q^35 + 34 * q^36 - 4 * q^37 + 19 * q^38 + 7 * q^39 - 3 * q^40 + 33 * q^41 + 19 * q^42 + 6 * q^43 + 30 * q^44 + 11 * q^45 - 11 * q^46 + 23 * q^47 + 40 * q^48 + 31 * q^49 + 6 * q^50 + 31 * q^51 - 7 * q^52 + 12 * q^53 + 4 * q^54 + 40 * q^56 + 32 * q^57 + 9 * q^58 + 27 * q^59 + 23 * q^60 - 4 * q^61 + 25 * q^62 + 4 * q^63 + 10 * q^64 + 54 * q^65 + 2 * q^66 + 34 * q^68 + 29 * q^69 - 59 * q^70 + 35 * q^71 + 12 * q^72 + 5 * q^73 + 48 * q^74 + 32 * q^75 + 32 * q^76 + 42 * q^77 + 13 * q^78 + 31 * q^79 + 24 * q^80 + 31 * q^81 + 5 * q^82 + 67 * q^83 - 13 * q^84 + 11 * q^85 - 20 * q^86 + 25 * q^87 - 7 * q^88 + 22 * q^89 + 5 * q^90 + 16 * q^91 + 57 * q^92 + 22 * q^93 + 45 * q^94 + 73 * q^95 + 28 * q^96 - 13 * q^97 - 19 * q^98 + 26 * q^99

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