LMFDB - Embedded newform 4029.2.a.k.1.14

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			1.14
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)

$f(q)$	$=$	$q-0.103363 q^{2} +1.00000 q^{3} -1.98932 q^{4} -2.25663 q^{5} -0.103363 q^{6} +4.11147 q^{7} +0.412347 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.103363 q^{2} +1.00000 q^{3} -1.98932 q^{4} -2.25663 q^{5} -0.103363 q^{6} +4.11147 q^{7} +0.412347 q^{8} +1.00000 q^{9} +0.233252 q^{10} +3.41549 q^{11} -1.98932 q^{12} +3.88047 q^{13} -0.424973 q^{14} -2.25663 q^{15} +3.93601 q^{16} +1.00000 q^{17} -0.103363 q^{18} -2.55629 q^{19} +4.48916 q^{20} +4.11147 q^{21} -0.353034 q^{22} +2.82500 q^{23} +0.412347 q^{24} +0.0923998 q^{25} -0.401096 q^{26} +1.00000 q^{27} -8.17901 q^{28} +9.03694 q^{29} +0.233252 q^{30} +2.07562 q^{31} -1.23153 q^{32} +3.41549 q^{33} -0.103363 q^{34} -9.27808 q^{35} -1.98932 q^{36} +0.222616 q^{37} +0.264225 q^{38} +3.88047 q^{39} -0.930516 q^{40} -9.66686 q^{41} -0.424973 q^{42} -11.3497 q^{43} -6.79448 q^{44} -2.25663 q^{45} -0.291999 q^{46} +0.936384 q^{47} +3.93601 q^{48} +9.90418 q^{49} -0.00955069 q^{50} +1.00000 q^{51} -7.71948 q^{52} -1.47935 q^{53} -0.103363 q^{54} -7.70751 q^{55} +1.69535 q^{56} -2.55629 q^{57} -0.934083 q^{58} +8.03969 q^{59} +4.48916 q^{60} +1.23827 q^{61} -0.214542 q^{62} +4.11147 q^{63} -7.74473 q^{64} -8.75680 q^{65} -0.353034 q^{66} -5.75544 q^{67} -1.98932 q^{68} +2.82500 q^{69} +0.959008 q^{70} +14.6544 q^{71} +0.412347 q^{72} -9.54271 q^{73} -0.0230102 q^{74} +0.0923998 q^{75} +5.08527 q^{76} +14.0427 q^{77} -0.401096 q^{78} +1.00000 q^{79} -8.88214 q^{80} +1.00000 q^{81} +0.999194 q^{82} +7.96281 q^{83} -8.17901 q^{84} -2.25663 q^{85} +1.17313 q^{86} +9.03694 q^{87} +1.40836 q^{88} +0.331886 q^{89} +0.233252 q^{90} +15.9544 q^{91} -5.61981 q^{92} +2.07562 q^{93} -0.0967872 q^{94} +5.76862 q^{95} -1.23153 q^{96} -13.1352 q^{97} -1.02372 q^{98} +3.41549 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.103363	−0.0730885	−0.0365443	−	0.999332i	$-0.511635\pi$
$2$	−0.103363	−0.0730885	−0.0365443	+	0.999332i	$0.511635\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.98932	−0.994658
$5$	−2.25663	−1.00920	−0.504599	−	0.863354i	$-0.668360\pi$
$5$	−2.25663	−1.00920	−0.504599	+	0.863354i	$0.668360\pi$
$6$	−0.103363	−0.0421977
$7$	4.11147	1.55399	0.776995	−	0.629507i	$-0.216743\pi$
$7$	4.11147	1.55399	0.776995	+	0.629507i	$0.216743\pi$
$8$	0.412347	0.145787
$9$	1.00000	0.333333
$10$	0.233252	0.0737608
$11$	3.41549	1.02981	0.514904	−	0.857248i	$-0.327828\pi$
$11$	3.41549	1.02981	0.514904	+	0.857248i	$0.327828\pi$
$12$	−1.98932	−0.574266
$13$	3.88047	1.07625	0.538124	−	0.842865i	$-0.319133\pi$
$13$	3.88047	1.07625	0.538124	+	0.842865i	$0.319133\pi$
$14$	−0.424973	−0.113579
$15$	−2.25663	−0.582661
$16$	3.93601	0.984003
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−0.103363	−0.0243628
$19$	−2.55629	−0.586454	−0.293227	−	0.956043i	$-0.594729\pi$
$19$	−2.55629	−0.586454	−0.293227	+	0.956043i	$0.594729\pi$
$20$	4.48916	1.00381
$21$	4.11147	0.897196
$22$	−0.353034	−0.0752671
$23$	2.82500	0.589052	0.294526	−	0.955643i	$-0.404838\pi$
$23$	2.82500	0.589052	0.294526	+	0.955643i	$0.404838\pi$
$24$	0.412347	0.0841699
$25$	0.0923998	0.0184800
$26$	−0.401096	−0.0786614
$27$	1.00000	0.192450
$28$	−8.17901	−1.54569
$29$	9.03694	1.67812	0.839059	−	0.544041i	$-0.183106\pi$
$29$	9.03694	1.67812	0.839059	+	0.544041i	$0.183106\pi$
$30$	0.233252	0.0425858
$31$	2.07562	0.372792	0.186396	−	0.982475i	$-0.440319\pi$
$31$	2.07562	0.372792	0.186396	+	0.982475i	$0.440319\pi$
$32$	−1.23153	−0.217706
$33$	3.41549	0.594560
$34$	−0.103363	−0.0177266
$35$	−9.27808	−1.56828
$36$	−1.98932	−0.331553
$37$	0.222616	0.0365978	0.0182989	−	0.999833i	$-0.494175\pi$
$37$	0.222616	0.0365978	0.0182989	+	0.999833i	$0.494175\pi$
$38$	0.264225	0.0428630
$39$	3.88047	0.621372
$40$	−0.930516	−0.147127
$41$	−9.66686	−1.50971	−0.754855	−	0.655891i	$-0.772293\pi$
$41$	−9.66686	−1.50971	−0.754855	+	0.655891i	$0.772293\pi$
$42$	−0.424973	−0.0655747
$43$	−11.3497	−1.73081	−0.865405	−	0.501073i	$-0.832939\pi$
$43$	−11.3497	−1.73081	−0.865405	+	0.501073i	$0.832939\pi$
$44$	−6.79448	−1.02431
$45$	−2.25663	−0.336399
$46$	−0.291999	−0.0430530
$47$	0.936384	0.136586	0.0682928	−	0.997665i	$-0.478245\pi$
$47$	0.936384	0.136586	0.0682928	+	0.997665i	$0.478245\pi$
$48$	3.93601	0.568114
$49$	9.90418	1.41488
$50$	−0.00955069	−0.00135067
$51$	1.00000	0.140028
$52$	−7.71948	−1.07050
$53$	−1.47935	−0.203204	−0.101602	−	0.994825i	$-0.532397\pi$
$53$	−1.47935	−0.203204	−0.101602	+	0.994825i	$0.532397\pi$
$54$	−0.103363	−0.0140659
$55$	−7.70751	−1.03928
$56$	1.69535	0.226551
$57$	−2.55629	−0.338589
$58$	−0.934083	−0.122651
$59$	8.03969	1.04668	0.523339	−	0.852125i	$-0.324686\pi$
$59$	8.03969	1.04668	0.523339	+	0.852125i	$0.324686\pi$
$60$	4.48916	0.579548
$61$	1.23827	0.158544	0.0792719	−	0.996853i	$-0.474740\pi$
$61$	1.23827	0.158544	0.0792719	+	0.996853i	$0.474740\pi$
$62$	−0.214542	−0.0272468
$63$	4.11147	0.517996
$64$	−7.74473	−0.968091
$65$	−8.75680	−1.08615
$66$	−0.353034	−0.0434555
$67$	−5.75544	−0.703139	−0.351570	−	0.936162i	$-0.614352\pi$
$67$	−5.75544	−0.703139	−0.351570	+	0.936162i	$0.614352\pi$
$68$	−1.98932	−0.241240
$69$	2.82500	0.340090
$70$	0.959008	0.114623
$71$	14.6544	1.73916	0.869579	−	0.493794i	$-0.164390\pi$
$71$	14.6544	1.73916	0.869579	+	0.493794i	$0.164390\pi$
$72$	0.412347	0.0485955
$73$	−9.54271	−1.11689	−0.558445	−	0.829541i	$-0.688602\pi$
$73$	−9.54271	−1.11689	−0.558445	+	0.829541i	$0.688602\pi$
$74$	−0.0230102	−0.00267488
$75$	0.0923998	0.0106694
$76$	5.08527	0.583321
$77$	14.0427	1.60031
$78$	−0.401096	−0.0454152
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	−8.88214	−0.993053
$81$	1.00000	0.111111
$82$	0.999194	0.110342
$83$	7.96281	0.874031	0.437016	−	0.899454i	$-0.356035\pi$
$83$	7.96281	0.874031	0.437016	+	0.899454i	$0.356035\pi$
$84$	−8.17901	−0.892403
$85$	−2.25663	−0.244766
$86$	1.17313	0.126502
$87$	9.03694	0.968862
$88$	1.40836	0.150132
$89$	0.331886	0.0351799	0.0175899	−	0.999845i	$-0.494401\pi$
$89$	0.331886	0.0351799	0.0175899	+	0.999845i	$0.494401\pi$
$90$	0.233252	0.0245869
$91$	15.9544	1.67248
$92$	−5.61981	−0.585906
$93$	2.07562	0.215232
$94$	−0.0967872	−0.00998284
$95$	5.76862	0.591848
$96$	−1.23153	−0.125693
$97$	−13.1352	−1.33368	−0.666838	−	0.745203i	$-0.732353\pi$
$97$	−13.1352	−1.33368	−0.666838	+	0.745203i	$0.732353\pi$
$98$	−1.02372	−0.103412
$99$	3.41549	0.343269
$100$	−0.183812	−0.0183812
$101$	13.8201	1.37515	0.687573	−	0.726115i	$-0.258676\pi$
$101$	13.8201	1.37515	0.687573	+	0.726115i	$0.258676\pi$
$102$	−0.103363	−0.0102344
$103$	7.92873	0.781241	0.390621	−	0.920552i	$-0.372260\pi$
$103$	7.92873	0.781241	0.390621	+	0.920552i	$0.372260\pi$
$104$	1.60010	0.156903
$105$	−9.27808	−0.905448
$106$	0.152910	0.0148519
$107$	−7.91500	−0.765172	−0.382586	−	0.923920i	$-0.624966\pi$
$107$	−7.91500	−0.765172	−0.382586	+	0.923920i	$0.624966\pi$
$108$	−1.98932	−0.191422
$109$	−1.10428	−0.105771	−0.0528853	−	0.998601i	$-0.516842\pi$
$109$	−1.10428	−0.105771	−0.0528853	+	0.998601i	$0.516842\pi$
$110$	0.796669	0.0759594
$111$	0.222616	0.0211298
$112$	16.1828	1.52913
$113$	4.66181	0.438546	0.219273	−	0.975664i	$-0.429631\pi$
$113$	4.66181	0.438546	0.219273	+	0.975664i	$0.429631\pi$
$114$	0.264225	0.0247470
$115$	−6.37498	−0.594470
$116$	−17.9773	−1.66915
$117$	3.88047	0.358750
$118$	−0.831004	−0.0765002
$119$	4.11147	0.376898
$120$	−0.930516	−0.0849441
$121$	0.665550	0.0605046
$122$	−0.127991	−0.0115877
$123$	−9.66686	−0.871632
$124$	−4.12906	−0.370801
$125$	11.0747	0.990548
$126$	−0.424973	−0.0378596
$127$	13.9060	1.23396	0.616978	−	0.786980i	$-0.288357\pi$
$127$	13.9060	1.23396	0.616978	+	0.786980i	$0.288357\pi$
$128$	3.26358	0.288462
$129$	−11.3497	−0.999284
$130$	0.905127	0.0793849
$131$	−2.67978	−0.234134	−0.117067	−	0.993124i	$-0.537349\pi$
$131$	−2.67978	−0.234134	−0.117067	+	0.993124i	$0.537349\pi$
$132$	−6.79448	−0.591384
$133$	−10.5101	−0.911343
$134$	0.594899	0.0513914
$135$	−2.25663	−0.194220
$136$	0.412347	0.0353584
$137$	12.1779	1.04042	0.520212	−	0.854037i	$-0.325853\pi$
$137$	12.1779	1.04042	0.520212	+	0.854037i	$0.325853\pi$
$138$	−0.291999	−0.0248566
$139$	21.5354	1.82661	0.913303	−	0.407282i	$-0.133523\pi$
$139$	21.5354	1.82661	0.913303	+	0.407282i	$0.133523\pi$
$140$	18.4570	1.55990
$141$	0.936384	0.0788577
$142$	−1.51472	−0.127112
$143$	13.2537	1.10833
$144$	3.93601	0.328001
$145$	−20.3931	−1.69355
$146$	0.986361	0.0816318
$147$	9.90418	0.816883
$148$	−0.442853	−0.0364023
$149$	14.9571	1.22533	0.612665	−	0.790342i	$-0.290097\pi$
$149$	14.9571	1.22533	0.612665	+	0.790342i	$0.290097\pi$
$150$	−0.00955069	−0.000779811 0
$151$	−7.46476	−0.607474	−0.303737	−	0.952756i	$-0.598234\pi$
$151$	−7.46476	−0.607474	−0.303737	+	0.952756i	$0.598234\pi$
$152$	−1.05408	−0.0854971
$153$	1.00000	0.0808452
$154$	−1.45149	−0.116964
$155$	−4.68391	−0.376221
$156$	−7.71948	−0.618053
$157$	−4.39671	−0.350896	−0.175448	−	0.984489i	$-0.556137\pi$
$157$	−4.39671	−0.350896	−0.175448	+	0.984489i	$0.556137\pi$
$158$	−0.103363	−0.00822310
$159$	−1.47935	−0.117320
$160$	2.77911	0.219708
$161$	11.6149	0.915381
$162$	−0.103363	−0.00812095
$163$	−22.5135	−1.76340	−0.881698	−	0.471815i	$-0.843599\pi$
$163$	−22.5135	−1.76340	−0.881698	+	0.471815i	$0.843599\pi$
$164$	19.2304	1.50165
$165$	−7.70751	−0.600029
$166$	−0.823058	−0.0638817
$167$	−2.64140	−0.204398	−0.102199	−	0.994764i	$-0.532588\pi$
$167$	−2.64140	−0.204398	−0.102199	+	0.994764i	$0.532588\pi$
$168$	1.69535	0.130799
$169$	2.05804	0.158311
$170$	0.233252	0.0178896
$171$	−2.55629	−0.195485
$172$	22.5781	1.72156
$173$	−22.4015	−1.70315	−0.851577	−	0.524229i	$-0.824353\pi$
$173$	−22.4015	−1.70315	−0.851577	+	0.524229i	$0.824353\pi$
$174$	−0.934083	−0.0708127
$175$	0.379899	0.0287176
$176$	13.4434	1.01333
$177$	8.03969	0.604300
$178$	−0.0343047	−0.00257124
$179$	−15.5245	−1.16035	−0.580176	−	0.814491i	$-0.697016\pi$
$179$	−15.5245	−1.16035	−0.580176	+	0.814491i	$0.697016\pi$
$180$	4.48916	0.334602
$181$	−7.46336	−0.554747	−0.277373	−	0.960762i	$-0.589464\pi$
$181$	−7.46336	−0.554747	−0.277373	+	0.960762i	$0.589464\pi$
$182$	−1.64909	−0.122239
$183$	1.23827	0.0915353
$184$	1.16488	0.0858759
$185$	−0.502362	−0.0369344
$186$	−0.214542	−0.0157310
$187$	3.41549	0.249765
$188$	−1.86276	−0.135856
$189$	4.11147	0.299065
$190$	−0.596260	−0.0432573
$191$	16.0425	1.16079	0.580396	−	0.814334i	$-0.302898\pi$
$191$	16.0425	1.16079	0.580396	+	0.814334i	$0.302898\pi$
$192$	−7.74473	−0.558928
$193$	−6.59109	−0.474437	−0.237219	−	0.971456i	$-0.576236\pi$
$193$	−6.59109	−0.474437	−0.237219	+	0.971456i	$0.576236\pi$
$194$	1.35769	0.0974763
$195$	−8.75680	−0.627088
$196$	−19.7025	−1.40732
$197$	13.5393	0.964634	0.482317	−	0.875997i	$-0.339795\pi$
$197$	13.5393	0.964634	0.482317	+	0.875997i	$0.339795\pi$
$198$	−0.353034	−0.0250890
$199$	27.5511	1.95305	0.976523	−	0.215412i	$-0.0691093\pi$
$199$	27.5511	1.95305	0.976523	+	0.215412i	$0.0691093\pi$
$200$	0.0381007	0.00269413
$201$	−5.75544	−0.405958
$202$	−1.42848	−0.100507
$203$	37.1551	2.60778
$204$	−1.98932	−0.139280
$205$	21.8146	1.52360
$206$	−0.819536	−0.0570998
$207$	2.82500	0.196351
$208$	15.2736	1.05903
$209$	−8.73098	−0.603935
$210$	0.959008	0.0661779
$211$	24.3779	1.67825	0.839123	−	0.543942i	$-0.183069\pi$
$211$	24.3779	1.67825	0.839123	+	0.543942i	$0.183069\pi$
$212$	2.94290	0.202119
$213$	14.6544	1.00410
$214$	0.818116	0.0559253
$215$	25.6121	1.74673
$216$	0.412347	0.0280566
$217$	8.53384	0.579315
$218$	0.114141	0.00773062
$219$	−9.54271	−0.644837
$220$	15.3327	1.03373
$221$	3.88047	0.261029
$222$	−0.0230102	−0.00154434
$223$	−15.1922	−1.01735	−0.508673	−	0.860960i	$-0.669864\pi$
$223$	−15.1922	−1.01735	−0.508673	+	0.860960i	$0.669864\pi$
$224$	−5.06340	−0.338313
$225$	0.0923998	0.00615998
$226$	−0.481858	−0.0320527
$227$	−21.5081	−1.42755	−0.713773	−	0.700377i	$-0.753015\pi$
$227$	−21.5081	−1.42755	−0.713773	+	0.700377i	$0.753015\pi$
$228$	5.08527	0.336780
$229$	12.9452	0.855442	0.427721	−	0.903911i	$-0.359317\pi$
$229$	12.9452	0.855442	0.427721	+	0.903911i	$0.359317\pi$
$230$	0.658936	0.0434490
$231$	14.0427	0.923940
$232$	3.72635	0.244647
$233$	7.29501	0.477912	0.238956	−	0.971030i	$-0.423195\pi$
$233$	7.29501	0.477912	0.238956	+	0.971030i	$0.423195\pi$
$234$	−0.401096	−0.0262205
$235$	−2.11308	−0.137842
$236$	−15.9935	−1.04109
$237$	1.00000	0.0649570
$238$	−0.424973	−0.0275469
$239$	−24.1130	−1.55974	−0.779869	−	0.625942i	$-0.784715\pi$
$239$	−24.1130	−1.55974	−0.779869	+	0.625942i	$0.784715\pi$
$240$	−8.88214	−0.573340
$241$	13.5990	0.875992	0.437996	−	0.898977i	$-0.355688\pi$
$241$	13.5990	0.875992	0.437996	+	0.898977i	$0.355688\pi$
$242$	−0.0687931	−0.00442219
$243$	1.00000	0.0641500
$244$	−2.46330	−0.157697
$245$	−22.3501	−1.42790
$246$	0.999194	0.0637063
$247$	−9.91961	−0.631170
$248$	0.855874	0.0543481
$249$	7.96281	0.504622
$250$	−1.14471	−0.0723977
$251$	−8.69606	−0.548890	−0.274445	−	0.961603i	$-0.588494\pi$
$251$	−8.69606	−0.548890	−0.274445	+	0.961603i	$0.588494\pi$
$252$	−8.17901	−0.515229
$253$	9.64874	0.606611
$254$	−1.43736	−0.0901881
$255$	−2.25663	−0.141316
$256$	15.1521	0.947008
$257$	26.4319	1.64878	0.824388	−	0.566025i	$-0.191520\pi$
$257$	26.4319	1.64878	0.824388	+	0.566025i	$0.191520\pi$
$258$	1.17313	0.0730362
$259$	0.915278	0.0568726
$260$	17.4200	1.08035
$261$	9.03694	0.559373
$262$	0.276990	0.0171125
$263$	−9.73137	−0.600062	−0.300031	−	0.953930i	$-0.596997\pi$
$263$	−9.73137	−0.600062	−0.300031	+	0.953930i	$0.596997\pi$
$264$	1.40836	0.0866789
$265$	3.33835	0.205073
$266$	1.08635	0.0666087
$267$	0.331886	0.0203111
$268$	11.4494	0.699383
$269$	8.36111	0.509786	0.254893	−	0.966969i	$-0.417960\pi$
$269$	8.36111	0.509786	0.254893	+	0.966969i	$0.417960\pi$
$270$	0.233252	0.0141953
$271$	24.9853	1.51775	0.758873	−	0.651239i	$-0.225750\pi$
$271$	24.9853	1.51775	0.758873	+	0.651239i	$0.225750\pi$
$272$	3.93601	0.238656
$273$	15.9544	0.965606
$274$	−1.25874	−0.0760431
$275$	0.315590	0.0190308
$276$	−5.61981	−0.338273
$277$	4.74679	0.285207	0.142604	−	0.989780i	$-0.454453\pi$
$277$	4.74679	0.285207	0.142604	+	0.989780i	$0.454453\pi$
$278$	−2.22595	−0.133504
$279$	2.07562	0.124264
$280$	−3.82579	−0.228635
$281$	20.0142	1.19395	0.596973	−	0.802261i	$-0.296370\pi$
$281$	20.0142	1.19395	0.596973	+	0.802261i	$0.296370\pi$
$282$	−0.0967872	−0.00576359
$283$	0.611321	0.0363393	0.0181696	−	0.999835i	$-0.494216\pi$
$283$	0.611321	0.0363393	0.0181696	+	0.999835i	$0.494216\pi$
$284$	−29.1522	−1.72987
$285$	5.76862	0.341703
$286$	−1.36994	−0.0810062
$287$	−39.7450	−2.34607
$288$	−1.23153	−0.0725686
$289$	1.00000	0.0588235
$290$	2.10788	0.123779
$291$	−13.1352	−0.769998
$292$	18.9835	1.11092
$293$	−7.19357	−0.420253	−0.210127	−	0.977674i	$-0.567388\pi$
$293$	−7.19357	−0.420253	−0.210127	+	0.977674i	$0.567388\pi$
$294$	−1.02372	−0.0597048
$295$	−18.1426	−1.05631
$296$	0.0917949	0.00533547
$297$	3.41549	0.198187
$298$	−1.54600	−0.0895576
$299$	10.9623	0.633967
$300$	−0.183812	−0.0106124
$301$	−46.6639	−2.68966
$302$	0.771579	0.0443994
$303$	13.8201	0.793941
$304$	−10.0616	−0.577072
$305$	−2.79432	−0.160002
$306$	−0.103363	−0.00590886
$307$	23.8255	1.35980	0.679898	−	0.733307i	$-0.262024\pi$
$307$	23.8255	1.35980	0.679898	+	0.733307i	$0.262024\pi$
$308$	−27.9353	−1.59176
$309$	7.92873	0.451050
$310$	0.484142	0.0274974
$311$	1.82437	0.103451	0.0517253	−	0.998661i	$-0.483528\pi$
$311$	1.82437	0.103451	0.0517253	+	0.998661i	$0.483528\pi$
$312$	1.60010	0.0905878
$313$	−28.2597	−1.59733	−0.798667	−	0.601773i	$-0.794461\pi$
$313$	−28.2597	−1.59733	−0.798667	+	0.601773i	$0.794461\pi$
$314$	0.454456	0.0256465
$315$	−9.27808	−0.522761
$316$	−1.98932	−0.111908
$317$	22.8240	1.28192	0.640961	−	0.767574i	$-0.278536\pi$
$317$	22.8240	1.28192	0.640961	+	0.767574i	$0.278536\pi$
$318$	0.152910	0.00857476
$319$	30.8655	1.72814
$320$	17.4770	0.976995
$321$	−7.91500	−0.441772
$322$	−1.20055	−0.0669039
$323$	−2.55629	−0.142236
$324$	−1.98932	−0.110518
$325$	0.358554	0.0198890
$326$	2.32706	0.128884
$327$	−1.10428	−0.0610667
$328$	−3.98610	−0.220096
$329$	3.84991	0.212253
$330$	0.796669	0.0438552
$331$	−20.8797	−1.14765	−0.573827	−	0.818977i	$-0.694542\pi$
$331$	−20.8797	−1.14765	−0.573827	+	0.818977i	$0.694542\pi$
$332$	−15.8405	−0.869362
$333$	0.222616	0.0121993
$334$	0.273023	0.0149391
$335$	12.9879	0.709607
$336$	16.1828	0.882843
$337$	11.7891	0.642195	0.321098	−	0.947046i	$-0.395948\pi$
$337$	11.7891	0.642195	0.321098	+	0.947046i	$0.395948\pi$
$338$	−0.212725	−0.0115707
$339$	4.66181	0.253195
$340$	4.48916	0.243459
$341$	7.08925	0.383904
$342$	0.264225	0.0142877
$343$	11.9404	0.644723
$344$	−4.68001	−0.252329
$345$	−6.37498	−0.343218
$346$	2.31548	0.124481
$347$	1.22964	0.0660106	0.0330053	−	0.999455i	$-0.489492\pi$
$347$	1.22964	0.0660106	0.0330053	+	0.999455i	$0.489492\pi$
$348$	−17.9773	−0.963686
$349$	28.7056	1.53658	0.768289	−	0.640103i	$-0.221108\pi$
$349$	28.7056	1.53658	0.768289	+	0.640103i	$0.221108\pi$
$350$	−0.0392674	−0.00209893
$351$	3.88047	0.207124
$352$	−4.20628	−0.224195
$353$	−2.01481	−0.107238	−0.0536188	−	0.998561i	$-0.517076\pi$
$353$	−2.01481	−0.107238	−0.0536188	+	0.998561i	$0.517076\pi$
$354$	−0.831004	−0.0441674
$355$	−33.0696	−1.75515
$356$	−0.660227	−0.0349919
$357$	4.11147	0.217602
$358$	1.60465	0.0848085
$359$	−23.8578	−1.25917	−0.629583	−	0.776934i	$-0.716774\pi$
$359$	−23.8578	−1.25917	−0.629583	+	0.776934i	$0.716774\pi$
$360$	−0.930516	−0.0490425
$361$	−12.4654	−0.656072
$362$	0.771433	0.0405456
$363$	0.665550	0.0349323
$364$	−31.7384	−1.66354
$365$	21.5344	1.12716
$366$	−0.127991	−0.00669018
$367$	−0.797399	−0.0416239	−0.0208120	−	0.999783i	$-0.506625\pi$
$367$	−0.797399	−0.0416239	−0.0208120	+	0.999783i	$0.506625\pi$
$368$	11.1192	0.579629
$369$	−9.66686	−0.503237
$370$	0.0519256	0.00269948
$371$	−6.08231	−0.315778
$372$	−4.12906	−0.214082
$373$	14.2509	0.737884	0.368942	−	0.929452i	$-0.379720\pi$
$373$	14.2509	0.737884	0.368942	+	0.929452i	$0.379720\pi$
$374$	−0.353034	−0.0182550
$375$	11.0747	0.571893
$376$	0.386115	0.0199123
$377$	35.0676	1.80607
$378$	−0.424973	−0.0218582
$379$	−30.7021	−1.57706	−0.788530	−	0.614997i	$-0.789157\pi$
$379$	−30.7021	−1.57706	−0.788530	+	0.614997i	$0.789157\pi$
$380$	−11.4756	−0.588686
$381$	13.9060	0.712425
$382$	−1.65819	−0.0848406
$383$	13.8220	0.706273	0.353136	−	0.935572i	$-0.385115\pi$
$383$	13.8220	0.706273	0.353136	+	0.935572i	$0.385115\pi$
$384$	3.26358	0.166544
$385$	−31.6892	−1.61503
$386$	0.681273	0.0346759
$387$	−11.3497	−0.576937
$388$	26.1300	1.32655
$389$	−31.6403	−1.60423	−0.802114	−	0.597171i	$-0.796292\pi$
$389$	−31.6403	−1.60423	−0.802114	+	0.597171i	$0.796292\pi$
$390$	0.905127	0.0458329
$391$	2.82500	0.142866
$392$	4.08396	0.206271
$393$	−2.67978	−0.135177
$394$	−1.39946	−0.0705036
$395$	−2.25663	−0.113544
$396$	−6.79448	−0.341436
$397$	−15.0934	−0.757514	−0.378757	−	0.925496i	$-0.623648\pi$
$397$	−15.0934	−0.757514	−0.378757	+	0.925496i	$0.623648\pi$
$398$	−2.84776	−0.142745
$399$	−10.5101	−0.526164
$400$	0.363686	0.0181843
$401$	15.7747	0.787750	0.393875	−	0.919164i	$-0.371134\pi$
$401$	15.7747	0.787750	0.393875	+	0.919164i	$0.371134\pi$
$402$	0.594899	0.0296708
$403$	8.05437	0.401217
$404$	−27.4925	−1.36780
$405$	−2.25663	−0.112133
$406$	−3.84045	−0.190599
$407$	0.760341	0.0376887
$408$	0.412347	0.0204142
$409$	27.4829	1.35894	0.679471	−	0.733702i	$-0.262209\pi$
$409$	27.4829	1.35894	0.679471	+	0.733702i	$0.262209\pi$
$410$	−2.25482	−0.111357
$411$	12.1779	0.600689
$412$	−15.7728	−0.777068
$413$	33.0549	1.62653
$414$	−0.291999	−0.0143510
$415$	−17.9691	−0.882070
$416$	−4.77892	−0.234306
$417$	21.5354	1.05459
$418$	0.902459	0.0441407
$419$	−0.342752	−0.0167446	−0.00837228	−	0.999965i	$-0.502665\pi$
$419$	−0.342752	−0.0167446	−0.00837228	+	0.999965i	$0.502665\pi$
$420$	18.4570	0.900611
$421$	19.8389	0.966889	0.483445	−	0.875375i	$-0.339385\pi$
$421$	19.8389	0.966889	0.483445	+	0.875375i	$0.339385\pi$
$422$	−2.51977	−0.122661
$423$	0.936384	0.0455285
$424$	−0.610006	−0.0296245
$425$	0.0923998	0.00448205
$426$	−1.51472	−0.0733884
$427$	5.09110	0.246375
$428$	15.7454	0.761084
$429$	13.2537	0.639894
$430$	−2.64734	−0.127666
$431$	−27.0817	−1.30448	−0.652239	−	0.758013i	$-0.726170\pi$
$431$	−27.0817	−1.30448	−0.652239	+	0.758013i	$0.726170\pi$
$432$	3.93601	0.189371
$433$	−35.0547	−1.68462	−0.842310	−	0.538993i	$-0.818805\pi$
$433$	−35.0547	−1.68462	−0.842310	+	0.538993i	$0.818805\pi$
$434$	−0.882081	−0.0423413
$435$	−20.3931	−0.977773
$436$	2.19676	0.105206
$437$	−7.22152	−0.345452
$438$	0.986361	0.0471302
$439$	−22.1696	−1.05810	−0.529049	−	0.848591i	$-0.677451\pi$
$439$	−22.1696	−1.05810	−0.529049	+	0.848591i	$0.677451\pi$
$440$	−3.17817	−0.151513
$441$	9.90418	0.471628
$442$	−0.401096	−0.0190782
$443$	−1.58164	−0.0751462	−0.0375731	−	0.999294i	$-0.511963\pi$
$443$	−1.58164	−0.0751462	−0.0375731	+	0.999294i	$0.511963\pi$
$444$	−0.442853	−0.0210169
$445$	−0.748946	−0.0355034
$446$	1.57031	0.0743563
$447$	14.9571	0.707445
$448$	−31.8422	−1.50440
$449$	−4.16269	−0.196449	−0.0982247	−	0.995164i	$-0.531316\pi$
$449$	−4.16269	−0.196449	−0.0982247	+	0.995164i	$0.531316\pi$
$450$	−0.00955069	−0.000450224 0
$451$	−33.0170	−1.55471
$452$	−9.27381	−0.436204
$453$	−7.46476	−0.350725
$454$	2.22314	0.104337
$455$	−36.0033	−1.68786
$456$	−1.05408	−0.0493618
$457$	−26.7430	−1.25099	−0.625493	−	0.780230i	$-0.715102\pi$
$457$	−26.7430	−1.25099	−0.625493	+	0.780230i	$0.715102\pi$
$458$	−1.33805	−0.0625230
$459$	1.00000	0.0466760
$460$	12.6819	0.591295
$461$	−27.5308	−1.28224	−0.641118	−	0.767442i	$-0.721529\pi$
$461$	−27.5308	−1.28224	−0.641118	+	0.767442i	$0.721529\pi$
$462$	−1.45149	−0.0675294
$463$	−40.1196	−1.86452	−0.932258	−	0.361795i	$-0.882164\pi$
$463$	−40.1196	−1.86452	−0.932258	+	0.361795i	$0.882164\pi$
$464$	35.5695	1.65127
$465$	−4.68391	−0.217211
$466$	−0.754033	−0.0349299
$467$	5.85171	0.270785	0.135392	−	0.990792i	$-0.456771\pi$
$467$	5.85171	0.270785	0.135392	+	0.990792i	$0.456771\pi$
$468$	−7.71948	−0.356833
$469$	−23.6633	−1.09267
$470$	0.218413	0.0100747
$471$	−4.39671	−0.202590
$472$	3.31514	0.152592
$473$	−38.7647	−1.78240
$474$	−0.103363	−0.00474761
$475$	−0.236201	−0.0108376
$476$	−8.17901	−0.374884
$477$	−1.47935	−0.0677348
$478$	2.49238	0.113999
$479$	−25.8669	−1.18189	−0.590944	−	0.806713i	$-0.701245\pi$
$479$	−25.8669	−1.18189	−0.590944	+	0.806713i	$0.701245\pi$
$480$	2.77911	0.126849
$481$	0.863854	0.0393883
$482$	−1.40564	−0.0640249
$483$	11.6149	0.528496
$484$	−1.32399	−0.0601814
$485$	29.6413	1.34594
$486$	−0.103363	−0.00468863
$487$	14.7433	0.668081	0.334040	−	0.942559i	$-0.391588\pi$
$487$	14.7433	0.668081	0.334040	+	0.942559i	$0.391588\pi$
$488$	0.510595	0.0231136
$489$	−22.5135	−1.01810
$490$	2.31017	0.104363
$491$	16.8884	0.762162	0.381081	−	0.924542i	$-0.375552\pi$
$491$	16.8884	0.762162	0.381081	+	0.924542i	$0.375552\pi$
$492$	19.2304	0.866975
$493$	9.03694	0.407003
$494$	1.02532	0.0461313
$495$	−7.70751	−0.346427
$496$	8.16966	0.366828
$497$	60.2511	2.70263
$498$	−0.823058	−0.0368821
$499$	−18.3125	−0.819779	−0.409890	−	0.912135i	$-0.634433\pi$
$499$	−18.3125	−0.819779	−0.409890	+	0.912135i	$0.634433\pi$
$500$	−22.0310	−0.985256
$501$	−2.64140	−0.118009
$502$	0.898849	0.0401176
$503$	13.6167	0.607140	0.303570	−	0.952809i	$-0.401821\pi$
$503$	13.6167	0.607140	0.303570	+	0.952809i	$0.401821\pi$
$504$	1.69535	0.0755169
$505$	−31.1868	−1.38779
$506$	−0.997320	−0.0443363
$507$	2.05804	0.0914009
$508$	−27.6634	−1.22737
$509$	13.2859	0.588888	0.294444	−	0.955669i	$-0.404865\pi$
$509$	13.2859	0.588888	0.294444	+	0.955669i	$0.404865\pi$
$510$	0.233252	0.0103286
$511$	−39.2346	−1.73564
$512$	−8.09332	−0.357678
$513$	−2.55629	−0.112863
$514$	−2.73207	−0.120507
$515$	−17.8923	−0.788427
$516$	22.5781	0.993946
$517$	3.19821	0.140657
$518$	−0.0946057	−0.00415673
$519$	−22.4015	−0.983317
$520$	−3.61084	−0.158346
$521$	41.0750	1.79953	0.899765	−	0.436375i	$-0.143738\pi$
$521$	41.0750	1.79953	0.899765	+	0.436375i	$0.143738\pi$
$522$	−0.934083	−0.0408837
$523$	−0.662665	−0.0289763	−0.0144882	−	0.999895i	$-0.504612\pi$
$523$	−0.662665	−0.0289763	−0.0144882	+	0.999895i	$0.504612\pi$
$524$	5.33094	0.232883
$525$	0.379899	0.0165801
$526$	1.00586	0.0438576
$527$	2.07562	0.0904154
$528$	13.4434	0.585049
$529$	−15.0194	−0.653017
$530$	−0.345062	−0.0149885
$531$	8.03969	0.348893
$532$	20.9079	0.906474
$533$	−37.5120	−1.62482
$534$	−0.0343047	−0.00148451
$535$	17.8613	0.772210
$536$	−2.37324	−0.102508
$537$	−15.5245	−0.669930
$538$	−0.864228	−0.0372595
$539$	33.8276	1.45706
$540$	4.48916	0.193183
$541$	20.7675	0.892866	0.446433	−	0.894817i	$-0.352694\pi$
$541$	20.7675	0.892866	0.446433	+	0.894817i	$0.352694\pi$
$542$	−2.58254	−0.110930
$543$	−7.46336	−0.320283
$544$	−1.23153	−0.0528014
$545$	2.49195	0.106743
$546$	−1.64909	−0.0705747
$547$	11.2688	0.481818	0.240909	−	0.970548i	$-0.422554\pi$
$547$	11.2688	0.481818	0.240909	+	0.970548i	$0.422554\pi$
$548$	−24.2256	−1.03487
$549$	1.23827	0.0528479
$550$	−0.0326203	−0.00139093
$551$	−23.1011	−0.984138
$552$	1.16488	0.0495805
$553$	4.11147	0.174837
$554$	−0.490641	−0.0208454
$555$	−0.502362	−0.0213241
$556$	−42.8406	−1.81685
$557$	1.38234	0.0585716	0.0292858	−	0.999571i	$-0.490677\pi$
$557$	1.38234	0.0585716	0.0292858	+	0.999571i	$0.490677\pi$
$558$	−0.214542	−0.00908227
$559$	−44.0421	−1.86278
$560$	−36.5186	−1.54319
$561$	3.41549	0.144202
$562$	−2.06872	−0.0872638
$563$	16.2735	0.685844	0.342922	−	0.939364i	$-0.388583\pi$
$563$	16.2735	0.685844	0.342922	+	0.939364i	$0.388583\pi$
$564$	−1.86276	−0.0784365
$565$	−10.5200	−0.442580
$566$	−0.0631879	−0.00265598
$567$	4.11147	0.172665
$568$	6.04270	0.253546
$569$	30.0681	1.26052	0.630260	−	0.776384i	$-0.282948\pi$
$569$	30.0681	1.26052	0.630260	+	0.776384i	$0.282948\pi$
$570$	−0.596260	−0.0249746
$571$	18.3594	0.768315	0.384158	−	0.923267i	$-0.374492\pi$
$571$	18.3594	0.768315	0.384158	+	0.923267i	$0.374492\pi$
$572$	−26.3658	−1.10241
$573$	16.0425	0.670184
$574$	4.10815	0.171471
$575$	0.261029	0.0108857
$576$	−7.74473	−0.322697
$577$	45.5266	1.89530	0.947648	−	0.319317i	$-0.103454\pi$
$577$	45.5266	1.89530	0.947648	+	0.319317i	$0.103454\pi$
$578$	−0.103363	−0.00429932
$579$	−6.59109	−0.273916
$580$	40.5683	1.68451
$581$	32.7388	1.35824
$582$	1.35769	0.0562780
$583$	−5.05270	−0.209262
$584$	−3.93491	−0.162828
$585$	−8.75680	−0.362049
$586$	0.743548	0.0307157
$587$	21.5288	0.888587	0.444294	−	0.895881i	$-0.353455\pi$
$587$	21.5288	0.888587	0.444294	+	0.895881i	$0.353455\pi$
$588$	−19.7025	−0.812519
$589$	−5.30589	−0.218625
$590$	1.87527	0.0772038
$591$	13.5393	0.556931
$592$	0.876218	0.0360123
$593$	−18.5086	−0.760058	−0.380029	−	0.924974i	$-0.624086\pi$
$593$	−18.5086	−0.760058	−0.380029	+	0.924974i	$0.624086\pi$
$594$	−0.353034	−0.0144852
$595$	−9.27808	−0.380364
$596$	−29.7543	−1.21878
$597$	27.5511	1.12759
$598$	−1.13309	−0.0463357
$599$	−38.7741	−1.58427	−0.792134	−	0.610347i	$-0.791030\pi$
$599$	−38.7741	−1.58427	−0.792134	+	0.610347i	$0.791030\pi$
$600$	0.0381007	0.00155546
$601$	−9.51671	−0.388195	−0.194097	−	0.980982i	$-0.562178\pi$
$601$	−9.51671	−0.388195	−0.194097	+	0.980982i	$0.562178\pi$
$602$	4.82331	0.196583
$603$	−5.75544	−0.234380
$604$	14.8498	0.604229
$605$	−1.50190	−0.0610611
$606$	−1.42848	−0.0580280
$607$	−37.6589	−1.52853	−0.764263	−	0.644905i	$-0.776897\pi$
$607$	−37.6589	−1.52853	−0.764263	+	0.644905i	$0.776897\pi$
$608$	3.14815	0.127674
$609$	37.1551	1.50560
$610$	0.288828	0.0116943
$611$	3.63361	0.147000
$612$	−1.98932	−0.0804133
$613$	−17.7438	−0.716665	−0.358333	−	0.933594i	$-0.616655\pi$
$613$	−17.7438	−0.716665	−0.358333	+	0.933594i	$0.616655\pi$
$614$	−2.46267	−0.0993855
$615$	21.8146	0.879649
$616$	5.79045	0.233304
$617$	7.26489	0.292473	0.146237	−	0.989250i	$-0.453284\pi$
$617$	7.26489	0.292473	0.146237	+	0.989250i	$0.453284\pi$
$618$	−0.819536	−0.0329666
$619$	3.82609	0.153784	0.0768919	−	0.997039i	$-0.475500\pi$
$619$	3.82609	0.153784	0.0768919	+	0.997039i	$0.475500\pi$
$620$	9.31778	0.374211
$621$	2.82500	0.113363
$622$	−0.188572	−0.00756105
$623$	1.36454	0.0546691
$624$	15.2736	0.611432
$625$	−25.4535	−1.01814
$626$	2.92100	0.116747
$627$	−8.73098	−0.348682
$628$	8.74645	0.349021
$629$	0.222616	0.00887627
$630$	0.959008	0.0382078
$631$	8.87289	0.353224	0.176612	−	0.984281i	$-0.443486\pi$
$631$	8.87289	0.353224	0.176612	+	0.984281i	$0.443486\pi$
$632$	0.412347	0.0164023
$633$	24.3779	0.968936
$634$	−2.35915	−0.0936937
$635$	−31.3807	−1.24531
$636$	2.94290	0.116693
$637$	38.4329	1.52277
$638$	−3.19035	−0.126307
$639$	14.6544	0.579719
$640$	−7.36470	−0.291115
$641$	−9.93501	−0.392409	−0.196205	−	0.980563i	$-0.562862\pi$
$641$	−9.93501	−0.392409	−0.196205	+	0.980563i	$0.562862\pi$
$642$	0.818116	0.0322885
$643$	−23.7309	−0.935854	−0.467927	−	0.883767i	$-0.654999\pi$
$643$	−23.7309	−0.935854	−0.467927	+	0.883767i	$0.654999\pi$
$644$	−23.1057	−0.910491
$645$	25.6121	1.00847
$646$	0.264225	0.0103958
$647$	37.2200	1.46327	0.731634	−	0.681698i	$-0.238758\pi$
$647$	37.2200	1.46327	0.731634	+	0.681698i	$0.238758\pi$
$648$	0.412347	0.0161985
$649$	27.4594	1.07788
$650$	−0.0370612	−0.00145366
$651$	8.53384	0.334468
$652$	44.7865	1.75398
$653$	−36.4939	−1.42812	−0.714058	−	0.700087i	$-0.753145\pi$
$653$	−36.4939	−1.42812	−0.714058	+	0.700087i	$0.753145\pi$
$654$	0.114141	0.00446328
$655$	6.04729	0.236287
$656$	−38.0489	−1.48556
$657$	−9.54271	−0.372297
$658$	−0.397938	−0.0155132
$659$	12.0293	0.468595	0.234297	−	0.972165i	$-0.424721\pi$
$659$	12.0293	0.468595	0.234297	+	0.972165i	$0.424721\pi$
$660$	15.3327	0.596823
$661$	−5.99406	−0.233142	−0.116571	−	0.993182i	$-0.537190\pi$
$661$	−5.99406	−0.233142	−0.116571	+	0.993182i	$0.537190\pi$
$662$	2.15819	0.0838803
$663$	3.88047	0.150705
$664$	3.28344	0.127422
$665$	23.7175	0.919725
$666$	−0.0230102	−0.000891627 0
$667$	25.5293	0.988499
$668$	5.25459	0.203306
$669$	−15.1922	−0.587365
$670$	−1.34247	−0.0518641
$671$	4.22928	0.163270
$672$	−5.06340	−0.195325
$673$	−19.0368	−0.733814	−0.366907	−	0.930258i	$-0.619583\pi$
$673$	−19.0368	−0.733814	−0.366907	+	0.930258i	$0.619583\pi$
$674$	−1.21856	−0.0469371
$675$	0.0923998	0.00355647
$676$	−4.09410	−0.157465
$677$	−22.7123	−0.872906	−0.436453	−	0.899727i	$-0.643766\pi$
$677$	−22.7123	−0.872906	−0.436453	+	0.899727i	$0.643766\pi$
$678$	−0.481858	−0.0185056
$679$	−54.0049	−2.07252
$680$	−0.930516	−0.0356837
$681$	−21.5081	−0.824194
$682$	−0.732764	−0.0280590
$683$	−48.9665	−1.87365	−0.936826	−	0.349796i	$-0.886251\pi$
$683$	−48.9665	−1.87365	−0.936826	+	0.349796i	$0.886251\pi$
$684$	5.08527	0.194440
$685$	−27.4810	−1.04999
$686$	−1.23420	−0.0471219
$687$	12.9452	0.493889
$688$	−44.6725	−1.70312
$689$	−5.74058	−0.218699
$690$	0.658936	0.0250853
$691$	10.3956	0.395469	0.197734	−	0.980256i	$-0.436642\pi$
$691$	10.3956	0.395469	0.197734	+	0.980256i	$0.436642\pi$
$692$	44.5637	1.69406
$693$	14.0427	0.533437
$694$	−0.127099	−0.00482461
$695$	−48.5974	−1.84341
$696$	3.72635	0.141247
$697$	−9.66686	−0.366159
$698$	−2.96709	−0.112306
$699$	7.29501	0.275923
$700$	−0.755739	−0.0285642
$701$	28.3636	1.07128	0.535640	−	0.844447i	$-0.320071\pi$
$701$	28.3636	1.07128	0.535640	+	0.844447i	$0.320071\pi$
$702$	−0.401096	−0.0151384
$703$	−0.569071	−0.0214629
$704$	−26.4520	−0.996948
$705$	−2.11308	−0.0795830
$706$	0.208257	0.00783784
$707$	56.8207	2.13696
$708$	−15.9935	−0.601072
$709$	−4.85767	−0.182434	−0.0912169	−	0.995831i	$-0.529076\pi$
$709$	−4.85767	−0.182434	−0.0912169	+	0.995831i	$0.529076\pi$
$710$	3.41817	0.128282
$711$	1.00000	0.0375029
$712$	0.136852	0.00512875
$713$	5.86361	0.219594
$714$	−0.424973	−0.0159042
$715$	−29.9087	−1.11852
$716$	30.8831	1.15415
$717$	−24.1130	−0.900516
$718$	2.46601	0.0920305
$719$	14.7157	0.548804	0.274402	−	0.961615i	$-0.411520\pi$
$719$	14.7157	0.548804	0.274402	+	0.961615i	$0.411520\pi$
$720$	−8.88214	−0.331018
$721$	32.5987	1.21404
$722$	1.28846	0.0479513
$723$	13.5990	0.505754
$724$	14.8470	0.551783
$725$	0.835011	0.0310115
$726$	−0.0687931	−0.00255315
$727$	45.5142	1.68803	0.844015	−	0.536320i	$-0.180186\pi$
$727$	45.5142	1.68803	0.844015	+	0.536320i	$0.180186\pi$
$728$	6.57876	0.243825
$729$	1.00000	0.0370370
$730$	−2.22586	−0.0823827
$731$	−11.3497	−0.419783
$732$	−2.46330	−0.0910463
$733$	−26.9213	−0.994362	−0.497181	−	0.867647i	$-0.665631\pi$
$733$	−26.9213	−0.994362	−0.497181	+	0.867647i	$0.665631\pi$
$734$	0.0824214	0.00304223
$735$	−22.3501	−0.824396
$736$	−3.47907	−0.128240
$737$	−19.6576	−0.724099
$738$	0.999194	0.0367808
$739$	−13.5725	−0.499272	−0.249636	−	0.968340i	$-0.580311\pi$
$739$	−13.5725	−0.499272	−0.249636	+	0.968340i	$0.580311\pi$
$740$	0.999358	0.0367371
$741$	−9.91961	−0.364406
$742$	0.628684	0.0230797
$743$	45.7597	1.67876	0.839380	−	0.543545i	$-0.182918\pi$
$743$	45.7597	1.67876	0.839380	+	0.543545i	$0.182918\pi$
$744$	0.855874	0.0313779
$745$	−33.7526	−1.23660
$746$	−1.47301	−0.0539308
$747$	7.96281	0.291344
$748$	−6.79448	−0.248431
$749$	−32.5423	−1.18907
$750$	−1.14471	−0.0417988
$751$	−29.6084	−1.08043	−0.540213	−	0.841528i	$-0.681656\pi$
$751$	−29.6084	−1.08043	−0.540213	+	0.841528i	$0.681656\pi$
$752$	3.68562	0.134401
$753$	−8.69606	−0.316902
$754$	−3.62468	−0.132003
$755$	16.8452	0.613061
$756$	−8.17901	−0.297468
$757$	25.2492	0.917698	0.458849	−	0.888514i	$-0.348262\pi$
$757$	25.2492	0.917698	0.458849	+	0.888514i	$0.348262\pi$
$758$	3.17345	0.115265
$759$	9.64874	0.350227
$760$	2.37867	0.0862835
$761$	−24.0148	−0.870537	−0.435268	−	0.900301i	$-0.643347\pi$
$761$	−24.0148	−0.870537	−0.435268	+	0.900301i	$0.643347\pi$
$762$	−1.43736	−0.0520701
$763$	−4.54020	−0.164366
$764$	−31.9135	−1.15459
$765$	−2.25663	−0.0815888
$766$	−1.42868	−0.0516204
$767$	31.1978	1.12649
$768$	15.1521	0.546755
$769$	−18.4740	−0.666188	−0.333094	−	0.942894i	$-0.608093\pi$
$769$	−18.4740	−0.666188	−0.333094	+	0.942894i	$0.608093\pi$
$770$	3.27548	0.118040
$771$	26.4319	0.951921
$772$	13.1118	0.471903
$773$	19.4386	0.699157	0.349579	−	0.936907i	$-0.386325\pi$
$773$	19.4386	0.699157	0.349579	+	0.936907i	$0.386325\pi$
$774$	1.17313	0.0421675
$775$	0.191787	0.00688918
$776$	−5.41625	−0.194432
$777$	0.915278	0.0328354
$778$	3.27043	0.117251
$779$	24.7113	0.885375
$780$	17.4200	0.623738
$781$	50.0519	1.79100
$782$	−0.291999	−0.0104419
$783$	9.03694	0.322954
$784$	38.9830	1.39225
$785$	9.92177	0.354123
$786$	0.276990	0.00987990
$787$	20.0273	0.713895	0.356947	−	0.934124i	$-0.383818\pi$
$787$	20.0273	0.713895	0.356947	+	0.934124i	$0.383818\pi$
$788$	−26.9339	−0.959481
$789$	−9.73137	−0.346446
$790$	0.233252	0.00829873
$791$	19.1669	0.681496
$792$	1.40836	0.0500441
$793$	4.80506	0.170633
$794$	1.56009	0.0553656
$795$	3.33835	0.118399
$796$	−54.8079	−1.94261
$797$	13.8042	0.488970	0.244485	−	0.969653i	$-0.421381\pi$
$797$	13.8042	0.488970	0.244485	+	0.969653i	$0.421381\pi$
$798$	1.08635	0.0384565
$799$	0.936384	0.0331269
$800$	−0.113793	−0.00402319
$801$	0.331886	0.0117266
$802$	−1.63051	−0.0575754
$803$	−32.5930	−1.15018
$804$	11.4494	0.403789
$805$	−26.2106	−0.923801
$806$	−0.832522	−0.0293243
$807$	8.36111	0.294325
$808$	5.69865	0.200478
$809$	24.0426	0.845293	0.422647	−	0.906295i	$-0.361101\pi$
$809$	24.0426	0.845293	0.422647	+	0.906295i	$0.361101\pi$
$810$	0.233252	0.00819564
$811$	−14.1894	−0.498256	−0.249128	−	0.968471i	$-0.580144\pi$
$811$	−14.1894	−0.498256	−0.249128	+	0.968471i	$0.580144\pi$
$812$	−73.9132	−2.59385
$813$	24.9853	0.876271
$814$	−0.0785910	−0.00275461
$815$	50.8048	1.77961
$816$	3.93601	0.137788
$817$	29.0131	1.01504
$818$	−2.84071	−0.0993230
$819$	15.9544	0.557493
$820$	−43.3961	−1.51546
$821$	−25.4942	−0.889753	−0.444877	−	0.895592i	$-0.646752\pi$
$821$	−25.4942	−0.889753	−0.444877	+	0.895592i	$0.646752\pi$
$822$	−1.25874	−0.0439035
$823$	−25.5858	−0.891866	−0.445933	−	0.895066i	$-0.647128\pi$
$823$	−25.5858	−0.891866	−0.445933	+	0.895066i	$0.647128\pi$
$824$	3.26939	0.113895
$825$	0.315590	0.0109874
$826$	−3.41665	−0.118880
$827$	−20.1308	−0.700015	−0.350007	−	0.936747i	$-0.613821\pi$
$827$	−20.1308	−0.700015	−0.350007	+	0.936747i	$0.613821\pi$
$828$	−5.61981	−0.195302
$829$	−5.42513	−0.188422	−0.0942112	−	0.995552i	$-0.530033\pi$
$829$	−5.42513	−0.188422	−0.0942112	+	0.995552i	$0.530033\pi$
$830$	1.85734	0.0644692
$831$	4.74679	0.164664
$832$	−30.0532	−1.04191
$833$	9.90418	0.343159
$834$	−2.22595	−0.0770785
$835$	5.96068	0.206278
$836$	17.3687	0.600709
$837$	2.07562	0.0717439
$838$	0.0354278	0.00122383
$839$	28.9248	0.998596	0.499298	−	0.866430i	$-0.333591\pi$
$839$	28.9248	0.998596	0.499298	+	0.866430i	$0.333591\pi$
$840$	−3.82579	−0.132002
$841$	52.6663	1.81608
$842$	−2.05061	−0.0706685
$843$	20.0142	0.689325
$844$	−48.4954	−1.66928
$845$	−4.64425	−0.159767
$846$	−0.0967872	−0.00332761
$847$	2.73639	0.0940235
$848$	−5.82274	−0.199954
$849$	0.611321	0.0209805
$850$	−0.00955069	−0.000327586 0
$851$	0.628889	0.0215580
$852$	−29.1522	−0.998740
$853$	29.9618	1.02587	0.512936	−	0.858427i	$-0.328558\pi$
$853$	29.9618	1.02587	0.512936	+	0.858427i	$0.328558\pi$
$854$	−0.526230	−0.0180072
$855$	5.76862	0.197283
$856$	−3.26372	−0.111552
$857$	8.18399	0.279560	0.139780	−	0.990183i	$-0.455360\pi$
$857$	8.18399	0.279560	0.139780	+	0.990183i	$0.455360\pi$
$858$	−1.36994	−0.0467689
$859$	19.6111	0.669123	0.334561	−	0.942374i	$-0.391412\pi$
$859$	19.6111	0.669123	0.334561	+	0.942374i	$0.391412\pi$
$860$	−50.9505	−1.73740
$861$	−39.7450	−1.35451
$862$	2.79924	0.0953424
$863$	21.1317	0.719333	0.359666	−	0.933081i	$-0.382890\pi$
$863$	21.1317	0.719333	0.359666	+	0.933081i	$0.382890\pi$
$864$	−1.23153	−0.0418975
$865$	50.5520	1.71882
$866$	3.62335	0.123126
$867$	1.00000	0.0339618
$868$	−16.9765	−0.576220
$869$	3.41549	0.115862
$870$	2.10788	0.0714640
$871$	−22.3338	−0.756753
$872$	−0.455345	−0.0154199
$873$	−13.1352	−0.444558
$874$	0.746436	0.0252486
$875$	45.5331	1.53930
$876$	18.9835	0.641392
$877$	−17.4488	−0.589203	−0.294601	−	0.955620i	$-0.595187\pi$
$877$	−17.4488	−0.589203	−0.294601	+	0.955620i	$0.595187\pi$
$878$	2.29151	0.0773349
$879$	−7.19357	−0.242633
$880$	−30.3368	−1.02265
$881$	−6.42031	−0.216306	−0.108153	−	0.994134i	$-0.534494\pi$
$881$	−6.42031	−0.216306	−0.108153	+	0.994134i	$0.534494\pi$
$882$	−1.02372	−0.0344706
$883$	32.2212	1.08433	0.542164	−	0.840272i	$-0.317605\pi$
$883$	32.2212	1.08433	0.542164	+	0.840272i	$0.317605\pi$
$884$	−7.71948	−0.259634
$885$	−18.1426	−0.609858
$886$	0.163483	0.00549232
$887$	−43.8386	−1.47196	−0.735979	−	0.677005i	$-0.763278\pi$
$887$	−43.8386	−1.47196	−0.735979	+	0.677005i	$0.763278\pi$
$888$	0.0917949	0.00308044
$889$	57.1740	1.91756
$890$	0.0774131	0.00259489
$891$	3.41549	0.114423
$892$	30.2221	1.01191
$893$	−2.39367	−0.0801011
$894$	−1.54600	−0.0517061
$895$	35.0330	1.17103
$896$	13.4181	0.448267
$897$	10.9623	0.366021
$898$	0.430267	0.0143582
$899$	18.7572	0.625589
$900$	−0.183812	−0.00612708
$901$	−1.47935	−0.0492843
$902$	3.41273	0.113632
$903$	−46.6639	−1.55288
$904$	1.92228	0.0639342
$905$	16.8421	0.559849
$906$	0.771579	0.0256340
$907$	43.9748	1.46016	0.730080	−	0.683361i	$-0.239483\pi$
$907$	43.9748	1.46016	0.730080	+	0.683361i	$0.239483\pi$
$908$	42.7865	1.41992
$909$	13.8201	0.458382
$910$	3.72140	0.123363
$911$	38.5233	1.27633	0.638167	−	0.769898i	$-0.279693\pi$
$911$	38.5233	1.27633	0.638167	+	0.769898i	$0.279693\pi$
$912$	−10.0616	−0.333173
$913$	27.1969	0.900085
$914$	2.76423	0.0914327
$915$	−2.79432	−0.0923772
$916$	−25.7521	−0.850872
$917$	−11.0178	−0.363841
$918$	−0.103363	−0.00341148
$919$	−18.6949	−0.616687	−0.308343	−	0.951275i	$-0.599775\pi$
$919$	−18.6949	−0.616687	−0.308343	+	0.951275i	$0.599775\pi$
$920$	−2.62870	−0.0866658
$921$	23.8255	0.785078
$922$	2.84566	0.0937167
$923$	56.8660	1.87177
$924$	−27.9353	−0.919004
$925$	0.0205696	0.000676326 0
$926$	4.14687	0.136275
$927$	7.92873	0.260414
$928$	−11.1293	−0.365336
$929$	−41.5483	−1.36315	−0.681577	−	0.731746i	$-0.738706\pi$
$929$	−41.5483	−1.36315	−0.681577	+	0.731746i	$0.738706\pi$
$930$	0.484142	0.0158756
$931$	−25.3180	−0.829763
$932$	−14.5121	−0.475359
$933$	1.82437	0.0597273
$934$	−0.604849	−0.0197913
$935$	−7.70751	−0.252062
$936$	1.60010	0.0523009
$937$	−50.2457	−1.64146	−0.820729	−	0.571318i	$-0.806432\pi$
$937$	−50.2457	−1.64146	−0.820729	+	0.571318i	$0.806432\pi$
$938$	2.44591	0.0798617
$939$	−28.2597	−0.922221
$940$	4.20358	0.137106
$941$	55.5146	1.80972	0.904862	−	0.425706i	$-0.139974\pi$
$941$	55.5146	1.80972	0.904862	+	0.425706i	$0.139974\pi$
$942$	0.454456	0.0148070
$943$	−27.3089	−0.889299
$944$	31.6443	1.02993
$945$	−9.27808	−0.301816
$946$	4.00683	0.130273
$947$	−30.7473	−0.999152	−0.499576	−	0.866270i	$-0.666511\pi$
$947$	−30.7473	−0.999152	−0.499576	+	0.866270i	$0.666511\pi$
$948$	−1.98932	−0.0646100
$949$	−37.0302	−1.20205
$950$	0.0244144	0.000792107 0
$951$	22.8240	0.740118
$952$	1.69535	0.0549466
$953$	32.6368	1.05721	0.528605	−	0.848868i	$-0.322715\pi$
$953$	32.6368	1.05721	0.528605	+	0.848868i	$0.322715\pi$
$954$	0.152910	0.00495064
$955$	−36.2020	−1.17147
$956$	47.9683	1.55141
$957$	30.8655	0.997741
$958$	2.67367	0.0863825
$959$	50.0689	1.61681
$960$	17.4770	0.564068
$961$	−26.6918	−0.861026
$962$	−0.0892903	−0.00287884
$963$	−7.91500	−0.255057
$964$	−27.0528	−0.871312
$965$	14.8737	0.478801
$966$	−1.20055	−0.0386270
$967$	−46.0847	−1.48198	−0.740992	−	0.671514i	$-0.765644\pi$
$967$	−46.0847	−1.48198	−0.740992	+	0.671514i	$0.765644\pi$
$968$	0.274437	0.00882076
$969$	−2.55629	−0.0821199
$970$	−3.06381	−0.0983729
$971$	−21.4748	−0.689159	−0.344579	−	0.938757i	$-0.611978\pi$
$971$	−21.4748	−0.689159	−0.344579	+	0.938757i	$0.611978\pi$
$972$	−1.98932	−0.0638073
$973$	88.5420	2.83852
$974$	−1.52390	−0.0488290
$975$	0.358554	0.0114829
$976$	4.87383	0.156008
$977$	−34.2612	−1.09611	−0.548057	−	0.836441i	$-0.684632\pi$
$977$	−34.2612	−1.09611	−0.548057	+	0.836441i	$0.684632\pi$
$978$	2.32706	0.0744112
$979$	1.13355	0.0362285
$980$	44.4614	1.42027
$981$	−1.10428	−0.0352569
$982$	−1.74563	−0.0557053
$983$	−18.8464	−0.601106	−0.300553	−	0.953765i	$-0.597171\pi$
$983$	−18.8464	−0.601106	−0.300553	+	0.953765i	$0.597171\pi$
$984$	−3.98610	−0.127072
$985$	−30.5532	−0.973506
$986$	−0.934083	−0.0297473
$987$	3.84991	0.122544
$988$	19.7332	0.627798
$989$	−32.0628	−1.01954
$990$	0.796669	0.0253198
$991$	−56.1745	−1.78444	−0.892221	−	0.451599i	$-0.850854\pi$
$991$	−56.1745	−1.78444	−0.892221	+	0.451599i	$0.850854\pi$
$992$	−2.55619	−0.0811590
$993$	−20.8797	−0.662598
$994$	−6.22772	−0.197531
$995$	−62.1728	−1.97101
$996$	−15.8405	−0.501927
$997$	−11.6470	−0.368865	−0.184432	−	0.982845i	$-0.559045\pi$
$997$	−11.6470	−0.368865	−0.184432	+	0.982845i	$0.559045\pi$
$998$	1.89283	0.0599164
$999$	0.222616	0.00704325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.k.1.14	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.k.1.14	✓	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)

\(f(q)\)	\(=\)	\(q-0.103363 q^{2} +1.00000 q^{3} -1.98932 q^{4} -2.25663 q^{5} -0.103363 q^{6} +4.11147 q^{7} +0.412347 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.103363 q^{2} +1.00000 q^{3} -1.98932 q^{4} -2.25663 q^{5} -0.103363 q^{6} +4.11147 q^{7} +0.412347 q^{8} +1.00000 q^{9} +0.233252 q^{10} +3.41549 q^{11} -1.98932 q^{12} +3.88047 q^{13} -0.424973 q^{14} -2.25663 q^{15} +3.93601 q^{16} +1.00000 q^{17} -0.103363 q^{18} -2.55629 q^{19} +4.48916 q^{20} +4.11147 q^{21} -0.353034 q^{22} +2.82500 q^{23} +0.412347 q^{24} +0.0923998 q^{25} -0.401096 q^{26} +1.00000 q^{27} -8.17901 q^{28} +9.03694 q^{29} +0.233252 q^{30} +2.07562 q^{31} -1.23153 q^{32} +3.41549 q^{33} -0.103363 q^{34} -9.27808 q^{35} -1.98932 q^{36} +0.222616 q^{37} +0.264225 q^{38} +3.88047 q^{39} -0.930516 q^{40} -9.66686 q^{41} -0.424973 q^{42} -11.3497 q^{43} -6.79448 q^{44} -2.25663 q^{45} -0.291999 q^{46} +0.936384 q^{47} +3.93601 q^{48} +9.90418 q^{49} -0.00955069 q^{50} +1.00000 q^{51} -7.71948 q^{52} -1.47935 q^{53} -0.103363 q^{54} -7.70751 q^{55} +1.69535 q^{56} -2.55629 q^{57} -0.934083 q^{58} +8.03969 q^{59} +4.48916 q^{60} +1.23827 q^{61} -0.214542 q^{62} +4.11147 q^{63} -7.74473 q^{64} -8.75680 q^{65} -0.353034 q^{66} -5.75544 q^{67} -1.98932 q^{68} +2.82500 q^{69} +0.959008 q^{70} +14.6544 q^{71} +0.412347 q^{72} -9.54271 q^{73} -0.0230102 q^{74} +0.0923998 q^{75} +5.08527 q^{76} +14.0427 q^{77} -0.401096 q^{78} +1.00000 q^{79} -8.88214 q^{80} +1.00000 q^{81} +0.999194 q^{82} +7.96281 q^{83} -8.17901 q^{84} -2.25663 q^{85} +1.17313 q^{86} +9.03694 q^{87} +1.40836 q^{88} +0.331886 q^{89} +0.233252 q^{90} +15.9544 q^{91} -5.61981 q^{92} +2.07562 q^{93} -0.0967872 q^{94} +5.76862 q^{95} -1.23153 q^{96} -13.1352 q^{97} -1.02372 q^{98} +3.41549 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) \) 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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