LMFDB - Embedded newform 4030.2.a.n.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, 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("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 16x^{6} + 18x^{5} + 64x^{4} - 84x^{3} - 19x^{2} + 22x + 4 $ x^8 - x^7 - 16x^6 + 18x^5 + 64x^4 - 84x^3 - 19x^2 + 22x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.447568$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.447568 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.447568 q^{6} +1.86515 q^{7} +1.00000 q^{8} -2.79968 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.447568 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.447568 q^{6} +1.86515 q^{7} +1.00000 q^{8} -2.79968 q^{9} -1.00000 q^{10} -4.77230 q^{11} +0.447568 q^{12} -1.00000 q^{13} +1.86515 q^{14} -0.447568 q^{15} +1.00000 q^{16} +0.778658 q^{17} -2.79968 q^{18} +1.72026 q^{19} -1.00000 q^{20} +0.834782 q^{21} -4.77230 q^{22} +7.35604 q^{23} +0.447568 q^{24} +1.00000 q^{25} -1.00000 q^{26} -2.59575 q^{27} +1.86515 q^{28} +4.16592 q^{29} -0.447568 q^{30} -1.00000 q^{31} +1.00000 q^{32} -2.13593 q^{33} +0.778658 q^{34} -1.86515 q^{35} -2.79968 q^{36} +3.34010 q^{37} +1.72026 q^{38} -0.447568 q^{39} -1.00000 q^{40} +9.99536 q^{41} +0.834782 q^{42} +3.66184 q^{43} -4.77230 q^{44} +2.79968 q^{45} +7.35604 q^{46} +13.1874 q^{47} +0.447568 q^{48} -3.52121 q^{49} +1.00000 q^{50} +0.348503 q^{51} -1.00000 q^{52} +4.43178 q^{53} -2.59575 q^{54} +4.77230 q^{55} +1.86515 q^{56} +0.769933 q^{57} +4.16592 q^{58} +0.881231 q^{59} -0.447568 q^{60} +10.9344 q^{61} -1.00000 q^{62} -5.22183 q^{63} +1.00000 q^{64} +1.00000 q^{65} -2.13593 q^{66} +12.5130 q^{67} +0.778658 q^{68} +3.29233 q^{69} -1.86515 q^{70} -11.6243 q^{71} -2.79968 q^{72} -3.07944 q^{73} +3.34010 q^{74} +0.447568 q^{75} +1.72026 q^{76} -8.90107 q^{77} -0.447568 q^{78} -6.67559 q^{79} -1.00000 q^{80} +7.23727 q^{81} +9.99536 q^{82} +9.02853 q^{83} +0.834782 q^{84} -0.778658 q^{85} +3.66184 q^{86} +1.86453 q^{87} -4.77230 q^{88} +1.00767 q^{89} +2.79968 q^{90} -1.86515 q^{91} +7.35604 q^{92} -0.447568 q^{93} +13.1874 q^{94} -1.72026 q^{95} +0.447568 q^{96} -0.347181 q^{97} -3.52121 q^{98} +13.3609 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9} - 8 q^{10} + 4 q^{11} - q^{12} - 8 q^{13} + q^{14} + q^{15} + 8 q^{16} - 5 q^{17} + 9 q^{18} + 2 q^{19} - 8 q^{20} + 17 q^{21} + 4 q^{22} + 4 q^{23} - q^{24} + 8 q^{25} - 8 q^{26} + 11 q^{27} + q^{28} + 11 q^{29} + q^{30} - 8 q^{31} + 8 q^{32} + 10 q^{33} - 5 q^{34} - q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} + q^{39} - 8 q^{40} + 10 q^{41} + 17 q^{42} + 19 q^{43} + 4 q^{44} - 9 q^{45} + 4 q^{46} + 11 q^{47} - q^{48} + 11 q^{49} + 8 q^{50} + 7 q^{51} - 8 q^{52} + 8 q^{53} + 11 q^{54} - 4 q^{55} + q^{56} - 11 q^{57} + 11 q^{58} + 28 q^{59} + q^{60} - 12 q^{61} - 8 q^{62} + 20 q^{63} + 8 q^{64} + 8 q^{65} + 10 q^{66} + 24 q^{67} - 5 q^{68} + 30 q^{69} - q^{70} + 18 q^{71} + 9 q^{72} - 3 q^{73} + 19 q^{74} - q^{75} + 2 q^{76} - 7 q^{77} + q^{78} + 22 q^{79} - 8 q^{80} + 24 q^{81} + 10 q^{82} + 17 q^{83} + 17 q^{84} + 5 q^{85} + 19 q^{86} + 11 q^{87} + 4 q^{88} + 17 q^{89} - 9 q^{90} - q^{91} + 4 q^{92} + q^{93} + 11 q^{94} - 2 q^{95} - q^{96} - 24 q^{97} + 11 q^{98} + 23 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9 - 8 * q^10 + 4 * q^11 - q^12 - 8 * q^13 + q^14 + q^15 + 8 * q^16 - 5 * q^17 + 9 * q^18 + 2 * q^19 - 8 * q^20 + 17 * q^21 + 4 * q^22 + 4 * q^23 - q^24 + 8 * q^25 - 8 * q^26 + 11 * q^27 + q^28 + 11 * q^29 + q^30 - 8 * q^31 + 8 * q^32 + 10 * q^33 - 5 * q^34 - q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 + q^39 - 8 * q^40 + 10 * q^41 + 17 * q^42 + 19 * q^43 + 4 * q^44 - 9 * q^45 + 4 * q^46 + 11 * q^47 - q^48 + 11 * q^49 + 8 * q^50 + 7 * q^51 - 8 * q^52 + 8 * q^53 + 11 * q^54 - 4 * q^55 + q^56 - 11 * q^57 + 11 * q^58 + 28 * q^59 + q^60 - 12 * q^61 - 8 * q^62 + 20 * q^63 + 8 * q^64 + 8 * q^65 + 10 * q^66 + 24 * q^67 - 5 * q^68 + 30 * q^69 - q^70 + 18 * q^71 + 9 * q^72 - 3 * q^73 + 19 * q^74 - q^75 + 2 * q^76 - 7 * q^77 + q^78 + 22 * q^79 - 8 * q^80 + 24 * q^81 + 10 * q^82 + 17 * q^83 + 17 * q^84 + 5 * q^85 + 19 * q^86 + 11 * q^87 + 4 * q^88 + 17 * q^89 - 9 * q^90 - q^91 + 4 * q^92 + q^93 + 11 * q^94 - 2 * q^95 - q^96 - 24 * q^97 + 11 * q^98 + 23 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.447568	0.258404	0.129202	−	0.991618i	$-0.458758\pi$
$3$	0.447568	0.258404	0.129202	+	0.991618i	$0.458758\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.447568	0.182719
$7$	1.86515	0.704961	0.352480	−	0.935819i	$-0.385338\pi$
$7$	1.86515	0.704961	0.352480	+	0.935819i	$0.385338\pi$
$8$	1.00000	0.353553
$9$	−2.79968	−0.933228
$10$	−1.00000	−0.316228
$11$	−4.77230	−1.43890	−0.719452	−	0.694542i	$-0.755607\pi$
$11$	−4.77230	−1.43890	−0.719452	+	0.694542i	$0.755607\pi$
$12$	0.447568	0.129202
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.86515	0.498482
$15$	−0.447568	−0.115562
$16$	1.00000	0.250000
$17$	0.778658	0.188852	0.0944262	−	0.995532i	$-0.469898\pi$
$17$	0.778658	0.188852	0.0944262	+	0.995532i	$0.469898\pi$
$18$	−2.79968	−0.659892
$19$	1.72026	0.394654	0.197327	−	0.980338i	$-0.436774\pi$
$19$	1.72026	0.394654	0.197327	+	0.980338i	$0.436774\pi$
$20$	−1.00000	−0.223607
$21$	0.834782	0.182164
$22$	−4.77230	−1.01746
$23$	7.35604	1.53384	0.766920	−	0.641742i	$-0.221788\pi$
$23$	7.35604	1.53384	0.766920	+	0.641742i	$0.221788\pi$
$24$	0.447568	0.0913595
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	−2.59575	−0.499553
$28$	1.86515	0.352480
$29$	4.16592	0.773591	0.386796	−	0.922165i	$-0.373582\pi$
$29$	4.16592	0.773591	0.386796	+	0.922165i	$0.373582\pi$
$30$	−0.447568	−0.0817144
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	−2.13593	−0.371818
$34$	0.778658	0.133539
$35$	−1.86515	−0.315268
$36$	−2.79968	−0.466614
$37$	3.34010	0.549109	0.274554	−	0.961572i	$-0.411470\pi$
$37$	3.34010	0.549109	0.274554	+	0.961572i	$0.411470\pi$
$38$	1.72026	0.279063
$39$	−0.447568	−0.0716683
$40$	−1.00000	−0.158114
$41$	9.99536	1.56101	0.780507	−	0.625147i	$-0.214961\pi$
$41$	9.99536	1.56101	0.780507	+	0.625147i	$0.214961\pi$
$42$	0.834782	0.128810
$43$	3.66184	0.558426	0.279213	−	0.960229i	$-0.409926\pi$
$43$	3.66184	0.558426	0.279213	+	0.960229i	$0.409926\pi$
$44$	−4.77230	−0.719452
$45$	2.79968	0.417352
$46$	7.35604	1.08459
$47$	13.1874	1.92358	0.961789	−	0.273793i	$-0.0882783\pi$
$47$	13.1874	1.92358	0.961789	+	0.273793i	$0.0882783\pi$
$48$	0.447568	0.0646009
$49$	−3.52121	−0.503030
$50$	1.00000	0.141421
$51$	0.348503	0.0488001
$52$	−1.00000	−0.138675
$53$	4.43178	0.608751	0.304376	−	0.952552i	$-0.401552\pi$
$53$	4.43178	0.608751	0.304376	+	0.952552i	$0.401552\pi$
$54$	−2.59575	−0.353237
$55$	4.77230	0.643497
$56$	1.86515	0.249241
$57$	0.769933	0.101980
$58$	4.16592	0.547012
$59$	0.881231	0.114726	0.0573632	−	0.998353i	$-0.481731\pi$
$59$	0.881231	0.114726	0.0573632	+	0.998353i	$0.481731\pi$
$60$	−0.447568	−0.0577808
$61$	10.9344	1.40001	0.700003	−	0.714140i	$-0.253182\pi$
$61$	10.9344	1.40001	0.700003	+	0.714140i	$0.253182\pi$
$62$	−1.00000	−0.127000
$63$	−5.22183	−0.657889
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	−2.13593	−0.262915
$67$	12.5130	1.52870	0.764350	−	0.644801i	$-0.223060\pi$
$67$	12.5130	1.52870	0.764350	+	0.644801i	$0.223060\pi$
$68$	0.778658	0.0944262
$69$	3.29233	0.396350
$70$	−1.86515	−0.222928
$71$	−11.6243	−1.37955	−0.689776	−	0.724023i	$-0.742291\pi$
$71$	−11.6243	−1.37955	−0.689776	+	0.724023i	$0.742291\pi$
$72$	−2.79968	−0.329946
$73$	−3.07944	−0.360421	−0.180211	−	0.983628i	$-0.557678\pi$
$73$	−3.07944	−0.360421	−0.180211	+	0.983628i	$0.557678\pi$
$74$	3.34010	0.388278
$75$	0.447568	0.0516807
$76$	1.72026	0.197327
$77$	−8.90107	−1.01437
$78$	−0.447568	−0.0506771
$79$	−6.67559	−0.751063	−0.375531	−	0.926810i	$-0.622540\pi$
$79$	−6.67559	−0.751063	−0.375531	+	0.926810i	$0.622540\pi$
$80$	−1.00000	−0.111803
$81$	7.23727	0.804141
$82$	9.99536	1.10380
$83$	9.02853	0.991010	0.495505	−	0.868605i	$-0.334983\pi$
$83$	9.02853	0.991010	0.495505	+	0.868605i	$0.334983\pi$
$84$	0.834782	0.0910822
$85$	−0.778658	−0.0844573
$86$	3.66184	0.394867
$87$	1.86453	0.199899
$88$	−4.77230	−0.508729
$89$	1.00767	0.106813	0.0534065	−	0.998573i	$-0.482992\pi$
$89$	1.00767	0.106813	0.0534065	+	0.998573i	$0.482992\pi$
$90$	2.79968	0.295112
$91$	−1.86515	−0.195521
$92$	7.35604	0.766920
$93$	−0.447568	−0.0464107
$94$	13.1874	1.36017
$95$	−1.72026	−0.176495
$96$	0.447568	0.0456797
$97$	−0.347181	−0.0352509	−0.0176255	−	0.999845i	$-0.505611\pi$
$97$	−0.347181	−0.0352509	−0.0176255	+	0.999845i	$0.505611\pi$
$98$	−3.52121	−0.355696
$99$	13.3609	1.34282
$100$	1.00000	0.100000
$101$	−13.2052	−1.31396	−0.656982	−	0.753906i	$-0.728167\pi$
$101$	−13.2052	−1.31396	−0.656982	+	0.753906i	$0.728167\pi$
$102$	0.348503	0.0345069
$103$	−4.60964	−0.454202	−0.227101	−	0.973871i	$-0.572925\pi$
$103$	−4.60964	−0.454202	−0.227101	+	0.973871i	$0.572925\pi$
$104$	−1.00000	−0.0980581
$105$	−0.834782	−0.0814664
$106$	4.43178	0.430452
$107$	9.36385	0.905237	0.452619	−	0.891704i	$-0.350490\pi$
$107$	9.36385	0.905237	0.452619	+	0.891704i	$0.350490\pi$
$108$	−2.59575	−0.249777
$109$	0.494017	0.0473182	0.0236591	−	0.999720i	$-0.492468\pi$
$109$	0.494017	0.0473182	0.0236591	+	0.999720i	$0.492468\pi$
$110$	4.77230	0.455021
$111$	1.49492	0.141892
$112$	1.86515	0.176240
$113$	5.70682	0.536852	0.268426	−	0.963300i	$-0.413496\pi$
$113$	5.70682	0.536852	0.268426	+	0.963300i	$0.413496\pi$
$114$	0.769933	0.0721108
$115$	−7.35604	−0.685954
$116$	4.16592	0.386796
$117$	2.79968	0.258831
$118$	0.881231	0.0811239
$119$	1.45231	0.133133
$120$	−0.447568	−0.0408572
$121$	11.7749	1.07044
$122$	10.9344	0.989954
$123$	4.47361	0.403372
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	−5.22183	−0.465198
$127$	13.5704	1.20418	0.602088	−	0.798430i	$-0.294336\pi$
$127$	13.5704	1.20418	0.602088	+	0.798430i	$0.294336\pi$
$128$	1.00000	0.0883883
$129$	1.63892	0.144299
$130$	1.00000	0.0877058
$131$	−11.4293	−0.998587	−0.499293	−	0.866433i	$-0.666407\pi$
$131$	−11.4293	−0.998587	−0.499293	+	0.866433i	$0.666407\pi$
$132$	−2.13593	−0.185909
$133$	3.20854	0.278216
$134$	12.5130	1.08095
$135$	2.59575	0.223407
$136$	0.778658	0.0667694
$137$	−3.83202	−0.327392	−0.163696	−	0.986511i	$-0.552342\pi$
$137$	−3.83202	−0.327392	−0.163696	+	0.986511i	$0.552342\pi$
$138$	3.29233	0.280262
$139$	−5.95360	−0.504977	−0.252489	−	0.967600i	$-0.581249\pi$
$139$	−5.95360	−0.504977	−0.252489	+	0.967600i	$0.581249\pi$
$140$	−1.86515	−0.157634
$141$	5.90225	0.497059
$142$	−11.6243	−0.975491
$143$	4.77230	0.399080
$144$	−2.79968	−0.233307
$145$	−4.16592	−0.345961
$146$	−3.07944	−0.254856
$147$	−1.57598	−0.129985
$148$	3.34010	0.274554
$149$	−1.40807	−0.115354	−0.0576769	−	0.998335i	$-0.518369\pi$
$149$	−1.40807	−0.115354	−0.0576769	+	0.998335i	$0.518369\pi$
$150$	0.447568	0.0365438
$151$	−17.8726	−1.45445	−0.727227	−	0.686397i	$-0.759191\pi$
$151$	−17.8726	−1.45445	−0.727227	+	0.686397i	$0.759191\pi$
$152$	1.72026	0.139531
$153$	−2.18000	−0.176242
$154$	−8.90107	−0.717268
$155$	1.00000	0.0803219
$156$	−0.447568	−0.0358341
$157$	3.24145	0.258696	0.129348	−	0.991599i	$-0.458712\pi$
$157$	3.24145	0.258696	0.129348	+	0.991599i	$0.458712\pi$
$158$	−6.67559	−0.531082
$159$	1.98352	0.157303
$160$	−1.00000	−0.0790569
$161$	13.7201	1.08130
$162$	7.23727	0.568614
$163$	−8.73810	−0.684420	−0.342210	−	0.939623i	$-0.611176\pi$
$163$	−8.73810	−0.684420	−0.342210	+	0.939623i	$0.611176\pi$
$164$	9.99536	0.780507
$165$	2.13593	0.166282
$166$	9.02853	0.700750
$167$	−19.0952	−1.47763	−0.738817	−	0.673907i	$-0.764615\pi$
$167$	−19.0952	−1.47763	−0.738817	+	0.673907i	$0.764615\pi$
$168$	0.834782	0.0644048
$169$	1.00000	0.0769231
$170$	−0.778658	−0.0597204
$171$	−4.81618	−0.368302
$172$	3.66184	0.279213
$173$	4.53336	0.344665	0.172333	−	0.985039i	$-0.444870\pi$
$173$	4.53336	0.344665	0.172333	+	0.985039i	$0.444870\pi$
$174$	1.86453	0.141350
$175$	1.86515	0.140992
$176$	−4.77230	−0.359726
$177$	0.394411	0.0296457
$178$	1.00767	0.0755282
$179$	0.428265	0.0320100	0.0160050	−	0.999872i	$-0.494905\pi$
$179$	0.428265	0.0320100	0.0160050	+	0.999872i	$0.494905\pi$
$180$	2.79968	0.208676
$181$	−25.2013	−1.87320	−0.936598	−	0.350407i	$-0.886043\pi$
$181$	−25.2013	−1.87320	−0.936598	+	0.350407i	$0.886043\pi$
$182$	−1.86515	−0.138254
$183$	4.89389	0.361767
$184$	7.35604	0.542295
$185$	−3.34010	−0.245569
$186$	−0.447568	−0.0328173
$187$	−3.71599	−0.271740
$188$	13.1874	0.961789
$189$	−4.84147	−0.352165
$190$	−1.72026	−0.124801
$191$	−22.3986	−1.62071	−0.810353	−	0.585942i	$-0.800724\pi$
$191$	−22.3986	−1.62071	−0.810353	+	0.585942i	$0.800724\pi$
$192$	0.447568	0.0323005
$193$	7.83137	0.563714	0.281857	−	0.959456i	$-0.409050\pi$
$193$	7.83137	0.563714	0.281857	+	0.959456i	$0.409050\pi$
$194$	−0.347181	−0.0249262
$195$	0.447568	0.0320510
$196$	−3.52121	−0.251515
$197$	−0.917816	−0.0653917	−0.0326959	−	0.999465i	$-0.510409\pi$
$197$	−0.917816	−0.0653917	−0.0326959	+	0.999465i	$0.510409\pi$
$198$	13.3609	0.949521
$199$	−1.53767	−0.109003	−0.0545014	−	0.998514i	$-0.517357\pi$
$199$	−1.53767	−0.109003	−0.0545014	+	0.998514i	$0.517357\pi$
$200$	1.00000	0.0707107
$201$	5.60040	0.395022
$202$	−13.2052	−0.929113
$203$	7.77006	0.545352
$204$	0.348503	0.0244001
$205$	−9.99536	−0.698107
$206$	−4.60964	−0.321169
$207$	−20.5946	−1.43142
$208$	−1.00000	−0.0693375
$209$	−8.20959	−0.567869
$210$	−0.834782	−0.0576054
$211$	−11.8224	−0.813889	−0.406944	−	0.913453i	$-0.633406\pi$
$211$	−11.8224	−0.813889	−0.406944	+	0.913453i	$0.633406\pi$
$212$	4.43178	0.304376
$213$	−5.20267	−0.356481
$214$	9.36385	0.640099
$215$	−3.66184	−0.249736
$216$	−2.59575	−0.176619
$217$	−1.86515	−0.126615
$218$	0.494017	0.0334590
$219$	−1.37826	−0.0931341
$220$	4.77230	0.321749
$221$	−0.778658	−0.0523782
$222$	1.49492	0.100333
$223$	−0.987764	−0.0661456	−0.0330728	−	0.999453i	$-0.510529\pi$
$223$	−0.987764	−0.0661456	−0.0330728	+	0.999453i	$0.510529\pi$
$224$	1.86515	0.124621
$225$	−2.79968	−0.186646
$226$	5.70682	0.379612
$227$	29.3711	1.94943	0.974713	−	0.223462i	$-0.0717358\pi$
$227$	29.3711	1.94943	0.974713	+	0.223462i	$0.0717358\pi$
$228$	0.769933	0.0509900
$229$	22.0436	1.45668	0.728342	−	0.685214i	$-0.240291\pi$
$229$	22.0436	1.45668	0.728342	+	0.685214i	$0.240291\pi$
$230$	−7.35604	−0.485043
$231$	−3.98383	−0.262117
$232$	4.16592	0.273506
$233$	29.9428	1.96162	0.980808	−	0.194977i	$-0.0624632\pi$
$233$	29.9428	1.96162	0.980808	+	0.194977i	$0.0624632\pi$
$234$	2.79968	0.183021
$235$	−13.1874	−0.860250
$236$	0.881231	0.0573632
$237$	−2.98778	−0.194077
$238$	1.45231	0.0941396
$239$	−2.20402	−0.142566	−0.0712830	−	0.997456i	$-0.522709\pi$
$239$	−2.20402	−0.142566	−0.0712830	+	0.997456i	$0.522709\pi$
$240$	−0.447568	−0.0288904
$241$	−2.96453	−0.190962	−0.0954812	−	0.995431i	$-0.530439\pi$
$241$	−2.96453	−0.190962	−0.0954812	+	0.995431i	$0.530439\pi$
$242$	11.7749	0.756919
$243$	11.0264	0.707346
$244$	10.9344	0.700003
$245$	3.52121	0.224962
$246$	4.47361	0.285227
$247$	−1.72026	−0.109457
$248$	−1.00000	−0.0635001
$249$	4.04088	0.256081
$250$	−1.00000	−0.0632456
$251$	13.3033	0.839695	0.419848	−	0.907595i	$-0.362084\pi$
$251$	13.3033	0.839695	0.419848	+	0.907595i	$0.362084\pi$
$252$	−5.22183	−0.328944
$253$	−35.1053	−2.20705
$254$	13.5704	0.851481
$255$	−0.348503	−0.0218241
$256$	1.00000	0.0625000
$257$	−19.2191	−1.19885	−0.599426	−	0.800430i	$-0.704605\pi$
$257$	−19.2191	−1.19885	−0.599426	+	0.800430i	$0.704605\pi$
$258$	1.63892	0.102035
$259$	6.22978	0.387100
$260$	1.00000	0.0620174
$261$	−11.6632	−0.721937
$262$	−11.4293	−0.706107
$263$	4.55763	0.281036	0.140518	−	0.990078i	$-0.455123\pi$
$263$	4.55763	0.281036	0.140518	+	0.990078i	$0.455123\pi$
$264$	−2.13593	−0.131458
$265$	−4.43178	−0.272242
$266$	3.20854	0.196728
$267$	0.451002	0.0276009
$268$	12.5130	0.764350
$269$	20.1656	1.22952	0.614759	−	0.788715i	$-0.289253\pi$
$269$	20.1656	1.22952	0.614759	+	0.788715i	$0.289253\pi$
$270$	2.59575	0.157973
$271$	−7.81439	−0.474690	−0.237345	−	0.971425i	$-0.576277\pi$
$271$	−7.81439	−0.474690	−0.237345	+	0.971425i	$0.576277\pi$
$272$	0.778658	0.0472131
$273$	−0.834782	−0.0505233
$274$	−3.83202	−0.231501
$275$	−4.77230	−0.287781
$276$	3.29233	0.198175
$277$	−20.2372	−1.21593	−0.607967	−	0.793962i	$-0.708015\pi$
$277$	−20.2372	−1.21593	−0.607967	+	0.793962i	$0.708015\pi$
$278$	−5.95360	−0.357073
$279$	2.79968	0.167613
$280$	−1.86515	−0.111464
$281$	−4.54675	−0.271236	−0.135618	−	0.990761i	$-0.543302\pi$
$281$	−4.54675	−0.271236	−0.135618	+	0.990761i	$0.543302\pi$
$282$	5.90225	0.351474
$283$	−7.74070	−0.460137	−0.230068	−	0.973174i	$-0.573895\pi$
$283$	−7.74070	−0.460137	−0.230068	+	0.973174i	$0.573895\pi$
$284$	−11.6243	−0.689776
$285$	−0.769933	−0.0456069
$286$	4.77230	0.282192
$287$	18.6429	1.10045
$288$	−2.79968	−0.164973
$289$	−16.3937	−0.964335
$290$	−4.16592	−0.244631
$291$	−0.155387	−0.00910896
$292$	−3.07944	−0.180211
$293$	27.0373	1.57954	0.789769	−	0.613405i	$-0.210201\pi$
$293$	27.0373	1.57954	0.789769	+	0.613405i	$0.210201\pi$
$294$	−1.57598	−0.0919132
$295$	−0.881231	−0.0513072
$296$	3.34010	0.194139
$297$	12.3877	0.718809
$298$	−1.40807	−0.0815675
$299$	−7.35604	−0.425411
$300$	0.447568	0.0258404
$301$	6.82989	0.393668
$302$	−17.8726	−1.02845
$303$	−5.91022	−0.339533
$304$	1.72026	0.0986635
$305$	−10.9344	−0.626102
$306$	−2.18000	−0.124622
$307$	21.7107	1.23909	0.619546	−	0.784960i	$-0.287317\pi$
$307$	21.7107	1.23909	0.619546	+	0.784960i	$0.287317\pi$
$308$	−8.90107	−0.507185
$309$	−2.06313	−0.117367
$310$	1.00000	0.0567962
$311$	12.3391	0.699686	0.349843	−	0.936808i	$-0.386235\pi$
$311$	12.3391	0.699686	0.349843	+	0.936808i	$0.386235\pi$
$312$	−0.447568	−0.0253386
$313$	15.3649	0.868476	0.434238	−	0.900798i	$-0.357018\pi$
$313$	15.3649	0.868476	0.434238	+	0.900798i	$0.357018\pi$
$314$	3.24145	0.182926
$315$	5.22183	0.294217
$316$	−6.67559	−0.375531
$317$	15.9180	0.894043	0.447022	−	0.894523i	$-0.352485\pi$
$317$	15.9180	0.894043	0.447022	+	0.894523i	$0.352485\pi$
$318$	1.98352	0.111230
$319$	−19.8810	−1.11312
$320$	−1.00000	−0.0559017
$321$	4.19096	0.233917
$322$	13.7201	0.764593
$323$	1.33949	0.0745314
$324$	7.23727	0.402071
$325$	−1.00000	−0.0554700
$326$	−8.73810	−0.483958
$327$	0.221106	0.0122272
$328$	9.99536	0.551902
$329$	24.5965	1.35605
$330$	2.13593	0.117579
$331$	18.4853	1.01604	0.508021	−	0.861345i	$-0.330377\pi$
$331$	18.4853	1.01604	0.508021	+	0.861345i	$0.330377\pi$
$332$	9.02853	0.495505
$333$	−9.35121	−0.512443
$334$	−19.0952	−1.04484
$335$	−12.5130	−0.683656
$336$	0.834782	0.0455411
$337$	−16.4309	−0.895050	−0.447525	−	0.894271i	$-0.647695\pi$
$337$	−16.4309	−0.895050	−0.447525	+	0.894271i	$0.647695\pi$
$338$	1.00000	0.0543928
$339$	2.55419	0.138725
$340$	−0.778658	−0.0422287
$341$	4.77230	0.258435
$342$	−4.81618	−0.260429
$343$	−19.6236	−1.05958
$344$	3.66184	0.197433
$345$	−3.29233	−0.177253
$346$	4.53336	0.243715
$347$	7.61718	0.408912	0.204456	−	0.978876i	$-0.434458\pi$
$347$	7.61718	0.408912	0.204456	+	0.978876i	$0.434458\pi$
$348$	1.86453	0.0999494
$349$	22.4423	1.20131	0.600655	−	0.799508i	$-0.294906\pi$
$349$	22.4423	1.20131	0.600655	+	0.799508i	$0.294906\pi$
$350$	1.86515	0.0996965
$351$	2.59575	0.138551
$352$	−4.77230	−0.254365
$353$	−3.91489	−0.208369	−0.104184	−	0.994558i	$-0.533223\pi$
$353$	−3.91489	−0.208369	−0.104184	+	0.994558i	$0.533223\pi$
$354$	0.394411	0.0209627
$355$	11.6243	0.616955
$356$	1.00767	0.0534065
$357$	0.650010	0.0344022
$358$	0.428265	0.0226345
$359$	27.3518	1.44357	0.721787	−	0.692116i	$-0.243321\pi$
$359$	27.3518	1.44357	0.721787	+	0.692116i	$0.243321\pi$
$360$	2.79968	0.147556
$361$	−16.0407	−0.844248
$362$	−25.2013	−1.32455
$363$	5.27007	0.276607
$364$	−1.86515	−0.0977605
$365$	3.07944	0.161185
$366$	4.89389	0.255808
$367$	−16.1452	−0.842771	−0.421386	−	0.906881i	$-0.638456\pi$
$367$	−16.1452	−0.842771	−0.421386	+	0.906881i	$0.638456\pi$
$368$	7.35604	0.383460
$369$	−27.9839	−1.45678
$370$	−3.34010	−0.173643
$371$	8.26593	0.429146
$372$	−0.447568	−0.0232053
$373$	−21.2717	−1.10141	−0.550703	−	0.834701i	$-0.685640\pi$
$373$	−21.2717	−1.10141	−0.550703	+	0.834701i	$0.685640\pi$
$374$	−3.71599	−0.192149
$375$	−0.447568	−0.0231123
$376$	13.1874	0.680087
$377$	−4.16592	−0.214556
$378$	−4.84147	−0.249018
$379$	26.2908	1.35047	0.675234	−	0.737604i	$-0.264043\pi$
$379$	26.2908	1.35047	0.675234	+	0.737604i	$0.264043\pi$
$380$	−1.72026	−0.0882474
$381$	6.07367	0.311163
$382$	−22.3986	−1.14601
$383$	−11.4722	−0.586204	−0.293102	−	0.956081i	$-0.594687\pi$
$383$	−11.4722	−0.586204	−0.293102	+	0.956081i	$0.594687\pi$
$384$	0.447568	0.0228399
$385$	8.90107	0.453640
$386$	7.83137	0.398606
$387$	−10.2520	−0.521138
$388$	−0.347181	−0.0176255
$389$	−2.58562	−0.131096	−0.0655481	−	0.997849i	$-0.520880\pi$
$389$	−2.58562	−0.131096	−0.0655481	+	0.997849i	$0.520880\pi$
$390$	0.447568	0.0226635
$391$	5.72784	0.289669
$392$	−3.52121	−0.177848
$393$	−5.11541	−0.258038
$394$	−0.917816	−0.0462389
$395$	6.67559	0.335885
$396$	13.3609	0.671412
$397$	7.28233	0.365490	0.182745	−	0.983160i	$-0.441502\pi$
$397$	7.28233	0.365490	0.182745	+	0.983160i	$0.441502\pi$
$398$	−1.53767	−0.0770766
$399$	1.43604	0.0718919
$400$	1.00000	0.0500000
$401$	9.87129	0.492948	0.246474	−	0.969149i	$-0.420728\pi$
$401$	9.87129	0.492948	0.246474	+	0.969149i	$0.420728\pi$
$402$	5.60040	0.279323
$403$	1.00000	0.0498135
$404$	−13.2052	−0.656982
$405$	−7.23727	−0.359623
$406$	7.77006	0.385622
$407$	−15.9400	−0.790114
$408$	0.348503	0.0172535
$409$	−7.54987	−0.373317	−0.186658	−	0.982425i	$-0.559766\pi$
$409$	−7.54987	−0.373317	−0.186658	+	0.982425i	$0.559766\pi$
$410$	−9.99536	−0.493636
$411$	−1.71509	−0.0845992
$412$	−4.60964	−0.227101
$413$	1.64363	0.0808777
$414$	−20.5946	−1.01217
$415$	−9.02853	−0.443193
$416$	−1.00000	−0.0490290
$417$	−2.66464	−0.130488
$418$	−8.20959	−0.401544
$419$	3.67874	0.179718	0.0898591	−	0.995954i	$-0.471358\pi$
$419$	3.67874	0.179718	0.0898591	+	0.995954i	$0.471358\pi$
$420$	−0.834782	−0.0407332
$421$	25.5642	1.24592	0.622961	−	0.782253i	$-0.285930\pi$
$421$	25.5642	1.24592	0.622961	+	0.782253i	$0.285930\pi$
$422$	−11.8224	−0.575506
$423$	−36.9205	−1.79514
$424$	4.43178	0.215226
$425$	0.778658	0.0377705
$426$	−5.20267	−0.252070
$427$	20.3943	0.986949
$428$	9.36385	0.452619
$429$	2.13593	0.103124
$430$	−3.66184	−0.176590
$431$	6.61648	0.318705	0.159352	−	0.987222i	$-0.449059\pi$
$431$	6.61648	0.318705	0.159352	+	0.987222i	$0.449059\pi$
$432$	−2.59575	−0.124888
$433$	34.7456	1.66977	0.834883	−	0.550428i	$-0.185535\pi$
$433$	34.7456	1.66977	0.834883	+	0.550428i	$0.185535\pi$
$434$	−1.86515	−0.0895301
$435$	−1.86453	−0.0893975
$436$	0.494017	0.0236591
$437$	12.6543	0.605337
$438$	−1.37826	−0.0658558
$439$	2.41999	0.115500	0.0577499	−	0.998331i	$-0.481607\pi$
$439$	2.41999	0.115500	0.0577499	+	0.998331i	$0.481607\pi$
$440$	4.77230	0.227511
$441$	9.85828	0.469442
$442$	−0.778658	−0.0370370
$443$	−1.74933	−0.0831130	−0.0415565	−	0.999136i	$-0.513232\pi$
$443$	−1.74933	−0.0831130	−0.0415565	+	0.999136i	$0.513232\pi$
$444$	1.49492	0.0709458
$445$	−1.00767	−0.0477682
$446$	−0.987764	−0.0467720
$447$	−0.630209	−0.0298079
$448$	1.86515	0.0881201
$449$	−10.9051	−0.514644	−0.257322	−	0.966326i	$-0.582840\pi$
$449$	−10.9051	−0.514644	−0.257322	+	0.966326i	$0.582840\pi$
$450$	−2.79968	−0.131978
$451$	−47.7009	−2.24615
$452$	5.70682	0.268426
$453$	−7.99922	−0.375836
$454$	29.3711	1.37845
$455$	1.86515	0.0874396
$456$	0.769933	0.0360554
$457$	−10.9185	−0.510744	−0.255372	−	0.966843i	$-0.582198\pi$
$457$	−10.9185	−0.510744	−0.255372	+	0.966843i	$0.582198\pi$
$458$	22.0436	1.03003
$459$	−2.02120	−0.0943418
$460$	−7.35604	−0.342977
$461$	8.49891	0.395834	0.197917	−	0.980219i	$-0.436582\pi$
$461$	8.49891	0.395834	0.197917	+	0.980219i	$0.436582\pi$
$462$	−3.98383	−0.185345
$463$	−4.70088	−0.218469	−0.109234	−	0.994016i	$-0.534840\pi$
$463$	−4.70088	−0.218469	−0.109234	+	0.994016i	$0.534840\pi$
$464$	4.16592	0.193398
$465$	0.447568	0.0207555
$466$	29.9428	1.38707
$467$	35.0427	1.62158	0.810791	−	0.585336i	$-0.199038\pi$
$467$	35.0427	1.62158	0.810791	+	0.585336i	$0.199038\pi$
$468$	2.79968	0.129415
$469$	23.3385	1.07767
$470$	−13.1874	−0.608289
$471$	1.45077	0.0668479
$472$	0.881231	0.0405619
$473$	−17.4754	−0.803521
$474$	−2.98778	−0.137233
$475$	1.72026	0.0789308
$476$	1.45231	0.0665667
$477$	−12.4076	−0.568103
$478$	−2.20402	−0.100809
$479$	−7.61684	−0.348022	−0.174011	−	0.984744i	$-0.555673\pi$
$479$	−7.61684	−0.348022	−0.174011	+	0.984744i	$0.555673\pi$
$480$	−0.447568	−0.0204286
$481$	−3.34010	−0.152295
$482$	−2.96453	−0.135031
$483$	6.14069	0.279411
$484$	11.7749	0.535222
$485$	0.347181	0.0157647
$486$	11.0264	0.500169
$487$	−18.2794	−0.828319	−0.414160	−	0.910204i	$-0.635924\pi$
$487$	−18.2794	−0.828319	−0.414160	+	0.910204i	$0.635924\pi$
$488$	10.9344	0.494977
$489$	−3.91089	−0.176857
$490$	3.52121	0.159072
$491$	16.4804	0.743748	0.371874	−	0.928283i	$-0.378715\pi$
$491$	16.4804	0.743748	0.371874	+	0.928283i	$0.378715\pi$
$492$	4.47361	0.201686
$493$	3.24383	0.146095
$494$	−1.72026	−0.0773980
$495$	−13.3609	−0.600530
$496$	−1.00000	−0.0449013
$497$	−21.6811	−0.972530
$498$	4.04088	0.181076
$499$	−15.8554	−0.709787	−0.354894	−	0.934907i	$-0.615483\pi$
$499$	−15.8554	−0.709787	−0.354894	+	0.934907i	$0.615483\pi$
$500$	−1.00000	−0.0447214
$501$	−8.54642	−0.381826
$502$	13.3033	0.593754
$503$	30.0903	1.34166	0.670830	−	0.741611i	$-0.265938\pi$
$503$	30.0903	1.34166	0.670830	+	0.741611i	$0.265938\pi$
$504$	−5.22183	−0.232599
$505$	13.2052	0.587623
$506$	−35.1053	−1.56062
$507$	0.447568	0.0198772
$508$	13.5704	0.602088
$509$	−14.0717	−0.623716	−0.311858	−	0.950129i	$-0.600951\pi$
$509$	−14.0717	−0.623716	−0.311858	+	0.950129i	$0.600951\pi$
$510$	−0.348503	−0.0154320
$511$	−5.74362	−0.254083
$512$	1.00000	0.0441942
$513$	−4.46536	−0.197151
$514$	−19.2191	−0.847717
$515$	4.60964	0.203125
$516$	1.63892	0.0721496
$517$	−62.9342	−2.76784
$518$	6.22978	0.273721
$519$	2.02899	0.0890627
$520$	1.00000	0.0438529
$521$	−39.8181	−1.74446	−0.872230	−	0.489095i	$-0.837327\pi$
$521$	−39.8181	−1.74446	−0.872230	+	0.489095i	$0.837327\pi$
$522$	−11.6632	−0.510486
$523$	−30.3377	−1.32658	−0.663288	−	0.748364i	$-0.730840\pi$
$523$	−30.3377	−1.32658	−0.663288	+	0.748364i	$0.730840\pi$
$524$	−11.4293	−0.499293
$525$	0.834782	0.0364329
$526$	4.55763	0.198722
$527$	−0.778658	−0.0339189
$528$	−2.13593	−0.0929545
$529$	31.1113	1.35267
$530$	−4.43178	−0.192504
$531$	−2.46717	−0.107066
$532$	3.20854	0.139108
$533$	−9.99536	−0.432947
$534$	0.451002	0.0195168
$535$	−9.36385	−0.404834
$536$	12.5130	0.540477
$537$	0.191678	0.00827151
$538$	20.1656	0.869401
$539$	16.8043	0.723813
$540$	2.59575	0.111703
$541$	3.36965	0.144873	0.0724363	−	0.997373i	$-0.476923\pi$
$541$	3.36965	0.144873	0.0724363	+	0.997373i	$0.476923\pi$
$542$	−7.81439	−0.335657
$543$	−11.2793	−0.484040
$544$	0.778658	0.0333847
$545$	−0.494017	−0.0211614
$546$	−0.834782	−0.0357254
$547$	−7.31261	−0.312664	−0.156332	−	0.987705i	$-0.549967\pi$
$547$	−7.31261	−0.312664	−0.156332	+	0.987705i	$0.549967\pi$
$548$	−3.83202	−0.163696
$549$	−30.6128	−1.30652
$550$	−4.77230	−0.203492
$551$	7.16645	0.305301
$552$	3.29233	0.140131
$553$	−12.4510	−0.529470
$554$	−20.2372	−0.859795
$555$	−1.49492	−0.0634559
$556$	−5.95360	−0.252489
$557$	−16.7686	−0.710509	−0.355255	−	0.934770i	$-0.615606\pi$
$557$	−16.7686	−0.710509	−0.355255	+	0.934770i	$0.615606\pi$
$558$	2.79968	0.118520
$559$	−3.66184	−0.154879
$560$	−1.86515	−0.0788170
$561$	−1.66316	−0.0702187
$562$	−4.54675	−0.191793
$563$	−27.3655	−1.15332	−0.576659	−	0.816985i	$-0.695644\pi$
$563$	−27.3655	−1.15332	−0.576659	+	0.816985i	$0.695644\pi$
$564$	5.90225	0.248530
$565$	−5.70682	−0.240088
$566$	−7.74070	−0.325366
$567$	13.4986	0.566888
$568$	−11.6243	−0.487745
$569$	8.59225	0.360206	0.180103	−	0.983648i	$-0.442357\pi$
$569$	8.59225	0.360206	0.180103	+	0.983648i	$0.442357\pi$
$570$	−0.769933	−0.0322489
$571$	−1.01112	−0.0423142	−0.0211571	−	0.999776i	$-0.506735\pi$
$571$	−1.01112	−0.0423142	−0.0211571	+	0.999776i	$0.506735\pi$
$572$	4.77230	0.199540
$573$	−10.0249	−0.418796
$574$	18.6429	0.778138
$575$	7.35604	0.306768
$576$	−2.79968	−0.116653
$577$	−31.4920	−1.31103	−0.655514	−	0.755183i	$-0.727548\pi$
$577$	−31.4920	−1.31103	−0.655514	+	0.755183i	$0.727548\pi$
$578$	−16.3937	−0.681888
$579$	3.50507	0.145666
$580$	−4.16592	−0.172980
$581$	16.8396	0.698623
$582$	−0.155387	−0.00644101
$583$	−21.1498	−0.875934
$584$	−3.07944	−0.127428
$585$	−2.79968	−0.115753
$586$	27.0373	1.11690
$587$	−13.0002	−0.536577	−0.268289	−	0.963339i	$-0.586458\pi$
$587$	−13.0002	−0.536577	−0.268289	+	0.963339i	$0.586458\pi$
$588$	−1.57598	−0.0649924
$589$	−1.72026	−0.0708820
$590$	−0.881231	−0.0362797
$591$	−0.410785	−0.0168975
$592$	3.34010	0.137277
$593$	8.86766	0.364151	0.182076	−	0.983285i	$-0.441718\pi$
$593$	8.86766	0.364151	0.182076	+	0.983285i	$0.441718\pi$
$594$	12.3877	0.508275
$595$	−1.45231	−0.0595391
$596$	−1.40807	−0.0576769
$597$	−0.688213	−0.0281667
$598$	−7.35604	−0.300811
$599$	9.02508	0.368755	0.184377	−	0.982856i	$-0.440973\pi$
$599$	9.02508	0.368755	0.184377	+	0.982856i	$0.440973\pi$
$600$	0.447568	0.0182719
$601$	−20.2693	−0.826803	−0.413401	−	0.910549i	$-0.635659\pi$
$601$	−20.2693	−0.826803	−0.413401	+	0.910549i	$0.635659\pi$
$602$	6.82989	0.278365
$603$	−35.0323	−1.42663
$604$	−17.8726	−0.727227
$605$	−11.7749	−0.478718
$606$	−5.91022	−0.240086
$607$	−28.9615	−1.17551	−0.587756	−	0.809038i	$-0.699988\pi$
$607$	−28.9615	−1.17551	−0.587756	+	0.809038i	$0.699988\pi$
$608$	1.72026	0.0697657
$609$	3.47763	0.140921
$610$	−10.9344	−0.442721
$611$	−13.1874	−0.533504
$612$	−2.18000	−0.0881211
$613$	19.1521	0.773545	0.386773	−	0.922175i	$-0.373590\pi$
$613$	19.1521	0.773545	0.386773	+	0.922175i	$0.373590\pi$
$614$	21.7107	0.876171
$615$	−4.47361	−0.180393
$616$	−8.90107	−0.358634
$617$	−48.6447	−1.95836	−0.979180	−	0.202992i	$-0.934933\pi$
$617$	−48.6447	−1.95836	−0.979180	+	0.202992i	$0.934933\pi$
$618$	−2.06313	−0.0829913
$619$	12.5591	0.504794	0.252397	−	0.967624i	$-0.418781\pi$
$619$	12.5591	0.504794	0.252397	+	0.967624i	$0.418781\pi$
$620$	1.00000	0.0401610
$621$	−19.0945	−0.766235
$622$	12.3391	0.494753
$623$	1.87946	0.0752989
$624$	−0.447568	−0.0179171
$625$	1.00000	0.0400000
$626$	15.3649	0.614106
$627$	−3.67435	−0.146740
$628$	3.24145	0.129348
$629$	2.60079	0.103700
$630$	5.22183	0.208043
$631$	−11.0498	−0.439886	−0.219943	−	0.975513i	$-0.570587\pi$
$631$	−11.0498	−0.439886	−0.219943	+	0.975513i	$0.570587\pi$
$632$	−6.67559	−0.265541
$633$	−5.29134	−0.210312
$634$	15.9180	0.632184
$635$	−13.5704	−0.538524
$636$	1.98352	0.0786517
$637$	3.52121	0.139516
$638$	−19.8810	−0.787097
$639$	32.5444	1.28744
$640$	−1.00000	−0.0395285
$641$	−10.0605	−0.397367	−0.198683	−	0.980064i	$-0.563667\pi$
$641$	−10.0605	−0.397367	−0.198683	+	0.980064i	$0.563667\pi$
$642$	4.19096	0.165404
$643$	−4.39066	−0.173151	−0.0865754	−	0.996245i	$-0.527592\pi$
$643$	−4.39066	−0.173151	−0.0865754	+	0.996245i	$0.527592\pi$
$644$	13.7201	0.540649
$645$	−1.63892	−0.0645326
$646$	1.33949	0.0527016
$647$	−26.0147	−1.02274	−0.511372	−	0.859359i	$-0.670863\pi$
$647$	−26.0147	−1.02274	−0.511372	+	0.859359i	$0.670863\pi$
$648$	7.23727	0.284307
$649$	−4.20550	−0.165080
$650$	−1.00000	−0.0392232
$651$	−0.834782	−0.0327177
$652$	−8.73810	−0.342210
$653$	33.9554	1.32878	0.664388	−	0.747388i	$-0.268692\pi$
$653$	33.9554	1.32878	0.664388	+	0.747388i	$0.268692\pi$
$654$	0.221106	0.00864594
$655$	11.4293	0.446582
$656$	9.99536	0.390253
$657$	8.62145	0.336355
$658$	24.5965	0.958870
$659$	35.9335	1.39977	0.699886	−	0.714255i	$-0.253234\pi$
$659$	35.9335	1.39977	0.699886	+	0.714255i	$0.253234\pi$
$660$	2.13593	0.0831410
$661$	−24.1467	−0.939197	−0.469598	−	0.882880i	$-0.655601\pi$
$661$	−24.1467	−0.939197	−0.469598	+	0.882880i	$0.655601\pi$
$662$	18.4853	0.718450
$663$	−0.348503	−0.0135347
$664$	9.02853	0.350375
$665$	−3.20854	−0.124422
$666$	−9.35121	−0.362352
$667$	30.6447	1.18657
$668$	−19.0952	−0.738817
$669$	−0.442092	−0.0170923
$670$	−12.5130	−0.483417
$671$	−52.1823	−2.01447
$672$	0.834782	0.0322024
$673$	−4.55474	−0.175572	−0.0877861	−	0.996139i	$-0.527979\pi$
$673$	−4.55474	−0.175572	−0.0877861	+	0.996139i	$0.527979\pi$
$674$	−16.4309	−0.632896
$675$	−2.59575	−0.0999106
$676$	1.00000	0.0384615
$677$	24.0091	0.922744	0.461372	−	0.887207i	$-0.347357\pi$
$677$	24.0091	0.922744	0.461372	+	0.887207i	$0.347357\pi$
$678$	2.55419	0.0980931
$679$	−0.647545	−0.0248505
$680$	−0.778658	−0.0298602
$681$	13.1456	0.503739
$682$	4.77230	0.182741
$683$	31.4528	1.20351	0.601754	−	0.798682i	$-0.294469\pi$
$683$	31.4528	1.20351	0.601754	+	0.798682i	$0.294469\pi$
$684$	−4.81618	−0.184151
$685$	3.83202	0.146414
$686$	−19.6236	−0.749234
$687$	9.86602	0.376412
$688$	3.66184	0.139606
$689$	−4.43178	−0.168837
$690$	−3.29233	−0.125337
$691$	−6.27565	−0.238737	−0.119369	−	0.992850i	$-0.538087\pi$
$691$	−6.27565	−0.238737	−0.119369	+	0.992850i	$0.538087\pi$
$692$	4.53336	0.172333
$693$	24.9202	0.946639
$694$	7.61718	0.289144
$695$	5.95360	0.225833
$696$	1.86453	0.0706749
$697$	7.78297	0.294801
$698$	22.4423	0.849455
$699$	13.4014	0.506889
$700$	1.86515	0.0704961
$701$	−0.574070	−0.0216823	−0.0108412	−	0.999941i	$-0.503451\pi$
$701$	−0.574070	−0.0216823	−0.0108412	+	0.999941i	$0.503451\pi$
$702$	2.59575	0.0979704
$703$	5.74583	0.216708
$704$	−4.77230	−0.179863
$705$	−5.90225	−0.222292
$706$	−3.91489	−0.147339
$707$	−24.6296	−0.926293
$708$	0.394411	0.0148229
$709$	−27.9485	−1.04963	−0.524815	−	0.851216i	$-0.675865\pi$
$709$	−27.9485	−1.04963	−0.524815	+	0.851216i	$0.675865\pi$
$710$	11.6243	0.436253
$711$	18.6895	0.700912
$712$	1.00767	0.0377641
$713$	−7.35604	−0.275486
$714$	0.650010	0.0243260
$715$	−4.77230	−0.178474
$716$	0.428265	0.0160050
$717$	−0.986448	−0.0368396
$718$	27.3518	1.02076
$719$	41.4300	1.54508	0.772540	−	0.634966i	$-0.218986\pi$
$719$	41.4300	1.54508	0.772540	+	0.634966i	$0.218986\pi$
$720$	2.79968	0.104338
$721$	−8.59768	−0.320194
$722$	−16.0407	−0.596974
$723$	−1.32683	−0.0493453
$724$	−25.2013	−0.936598
$725$	4.16592	0.154718
$726$	5.27007	0.195591
$727$	−34.8225	−1.29150	−0.645748	−	0.763551i	$-0.723454\pi$
$727$	−34.8225	−1.29150	−0.645748	+	0.763551i	$0.723454\pi$
$728$	−1.86515	−0.0691271
$729$	−16.7767	−0.621360
$730$	3.07944	0.113975
$731$	2.85132	0.105460
$732$	4.89389	0.180883
$733$	−18.3137	−0.676433	−0.338216	−	0.941068i	$-0.609824\pi$
$733$	−18.3137	−0.676433	−0.338216	+	0.941068i	$0.609824\pi$
$734$	−16.1452	−0.595929
$735$	1.57598	0.0581310
$736$	7.35604	0.271147
$737$	−59.7156	−2.19965
$738$	−27.9839	−1.03010
$739$	21.5968	0.794450	0.397225	−	0.917721i	$-0.369973\pi$
$739$	21.5968	0.794450	0.397225	+	0.917721i	$0.369973\pi$
$740$	−3.34010	−0.122784
$741$	−0.769933	−0.0282842
$742$	8.26593	0.303452
$743$	12.1199	0.444636	0.222318	−	0.974974i	$-0.428638\pi$
$743$	12.1199	0.444636	0.222318	+	0.974974i	$0.428638\pi$
$744$	−0.447568	−0.0164086
$745$	1.40807	0.0515878
$746$	−21.2717	−0.778812
$747$	−25.2770	−0.924838
$748$	−3.71599	−0.135870
$749$	17.4650	0.638157
$750$	−0.447568	−0.0163429
$751$	53.1822	1.94065	0.970324	−	0.241810i	$-0.0777412\pi$
$751$	53.1822	1.94065	0.970324	+	0.241810i	$0.0777412\pi$
$752$	13.1874	0.480894
$753$	5.95412	0.216980
$754$	−4.16592	−0.151714
$755$	17.8726	0.650452
$756$	−4.84147	−0.176083
$757$	−23.9118	−0.869091	−0.434545	−	0.900650i	$-0.643091\pi$
$757$	−23.9118	−0.869091	−0.434545	+	0.900650i	$0.643091\pi$
$758$	26.2908	0.954925
$759$	−15.7120	−0.570310
$760$	−1.72026	−0.0624003
$761$	−20.9029	−0.757729	−0.378864	−	0.925452i	$-0.623685\pi$
$761$	−20.9029	−0.757729	−0.378864	+	0.925452i	$0.623685\pi$
$762$	6.07367	0.220026
$763$	0.921416	0.0333575
$764$	−22.3986	−0.810353
$765$	2.18000	0.0788179
$766$	−11.4722	−0.414509
$767$	−0.881231	−0.0318194
$768$	0.447568	0.0161502
$769$	34.3679	1.23934	0.619670	−	0.784863i	$-0.287267\pi$
$769$	34.3679	1.23934	0.619670	+	0.784863i	$0.287267\pi$
$770$	8.90107	0.320772
$771$	−8.60184	−0.309788
$772$	7.83137	0.281857
$773$	−19.7225	−0.709368	−0.354684	−	0.934986i	$-0.615412\pi$
$773$	−19.7225	−0.709368	−0.354684	+	0.934986i	$0.615412\pi$
$774$	−10.2520	−0.368500
$775$	−1.00000	−0.0359211
$776$	−0.347181	−0.0124631
$777$	2.78825	0.100028
$778$	−2.58562	−0.0926990
$779$	17.1946	0.616061
$780$	0.447568	0.0160255
$781$	55.4748	1.98504
$782$	5.72784	0.204827
$783$	−10.8137	−0.386450
$784$	−3.52121	−0.125758
$785$	−3.24145	−0.115692
$786$	−5.11541	−0.182461
$787$	41.5231	1.48014	0.740069	−	0.672531i	$-0.234793\pi$
$787$	41.5231	1.48014	0.740069	+	0.672531i	$0.234793\pi$
$788$	−0.917816	−0.0326959
$789$	2.03985	0.0726206
$790$	6.67559	0.237507
$791$	10.6441	0.378460
$792$	13.3609	0.474760
$793$	−10.9344	−0.388292
$794$	7.28233	0.258440
$795$	−1.98352	−0.0703483
$796$	−1.53767	−0.0545014
$797$	46.4075	1.64384	0.821919	−	0.569605i	$-0.192904\pi$
$797$	46.4075	1.64384	0.821919	+	0.569605i	$0.192904\pi$
$798$	1.43604	0.0508353
$799$	10.2685	0.363272
$800$	1.00000	0.0353553
$801$	−2.82116	−0.0996808
$802$	9.87129	0.348567
$803$	14.6960	0.518611
$804$	5.60040	0.197511
$805$	−13.7201	−0.483571
$806$	1.00000	0.0352235
$807$	9.02549	0.317712
$808$	−13.2052	−0.464557
$809$	4.22007	0.148370	0.0741848	−	0.997245i	$-0.476365\pi$
$809$	4.22007	0.148370	0.0741848	+	0.997245i	$0.476365\pi$
$810$	−7.23727	−0.254292
$811$	−35.4098	−1.24341	−0.621704	−	0.783252i	$-0.713559\pi$
$811$	−35.4098	−1.24341	−0.621704	+	0.783252i	$0.713559\pi$
$812$	7.77006	0.272676
$813$	−3.49747	−0.122662
$814$	−15.9400	−0.558695
$815$	8.73810	0.306082
$816$	0.348503	0.0122000
$817$	6.29931	0.220385
$818$	−7.54987	−0.263975
$819$	5.22183	0.182466
$820$	−9.99536	−0.349053
$821$	−30.9267	−1.07935	−0.539675	−	0.841873i	$-0.681453\pi$
$821$	−30.9267	−1.07935	−0.539675	+	0.841873i	$0.681453\pi$
$822$	−1.71509	−0.0598207
$823$	52.9075	1.84424	0.922120	−	0.386904i	$-0.126456\pi$
$823$	52.9075	1.84424	0.922120	+	0.386904i	$0.126456\pi$
$824$	−4.60964	−0.160585
$825$	−2.13593	−0.0743636
$826$	1.64363	0.0571891
$827$	−37.9031	−1.31802	−0.659009	−	0.752135i	$-0.729024\pi$
$827$	−37.9031	−1.31802	−0.659009	+	0.752135i	$0.729024\pi$
$828$	−20.5946	−0.715711
$829$	22.5811	0.784273	0.392136	−	0.919907i	$-0.371736\pi$
$829$	22.5811	0.784273	0.392136	+	0.919907i	$0.371736\pi$
$830$	−9.02853	−0.313385
$831$	−9.05752	−0.314202
$832$	−1.00000	−0.0346688
$833$	−2.74182	−0.0949985
$834$	−2.66464	−0.0922690
$835$	19.0952	0.660818
$836$	−8.20959	−0.283935
$837$	2.59575	0.0897224
$838$	3.67874	0.127080
$839$	26.2954	0.907818	0.453909	−	0.891048i	$-0.350029\pi$
$839$	26.2954	0.907818	0.453909	+	0.891048i	$0.350029\pi$
$840$	−0.834782	−0.0288027
$841$	−11.6451	−0.401556
$842$	25.5642	0.881000
$843$	−2.03498	−0.0700884
$844$	−11.8224	−0.406944
$845$	−1.00000	−0.0344010
$846$	−36.9205	−1.26935
$847$	21.9620	0.754622
$848$	4.43178	0.152188
$849$	−3.46449	−0.118901
$850$	0.778658	0.0267078
$851$	24.5699	0.842245
$852$	−5.20267	−0.178241
$853$	−43.2843	−1.48203	−0.741014	−	0.671490i	$-0.765655\pi$
$853$	−43.2843	−1.48203	−0.741014	+	0.671490i	$0.765655\pi$
$854$	20.3943	0.697878
$855$	4.81618	0.164710
$856$	9.36385	0.320050
$857$	6.22499	0.212642	0.106321	−	0.994332i	$-0.466093\pi$
$857$	6.22499	0.212642	0.106321	+	0.994332i	$0.466093\pi$
$858$	2.13593	0.0729195
$859$	−56.3879	−1.92393	−0.961965	−	0.273173i	$-0.911927\pi$
$859$	−56.3879	−1.92393	−0.961965	+	0.273173i	$0.911927\pi$
$860$	−3.66184	−0.124868
$861$	8.34395	0.284361
$862$	6.61648	0.225358
$863$	31.7152	1.07960	0.539799	−	0.841794i	$-0.318500\pi$
$863$	31.7152	1.07960	0.539799	+	0.841794i	$0.318500\pi$
$864$	−2.59575	−0.0883093
$865$	−4.53336	−0.154139
$866$	34.7456	1.18070
$867$	−7.33729	−0.249188
$868$	−1.86515	−0.0633073
$869$	31.8580	1.08071
$870$	−1.86453	−0.0632136
$871$	−12.5130	−0.423985
$872$	0.494017	0.0167295
$873$	0.971997	0.0328971
$874$	12.6543	0.428038
$875$	−1.86515	−0.0630536
$876$	−1.37826	−0.0465671
$877$	49.8873	1.68457	0.842287	−	0.539029i	$-0.181209\pi$
$877$	49.8873	1.68457	0.842287	+	0.539029i	$0.181209\pi$
$878$	2.41999	0.0816706
$879$	12.1010	0.408158
$880$	4.77230	0.160874
$881$	44.7380	1.50726	0.753631	−	0.657297i	$-0.228300\pi$
$881$	44.7380	1.50726	0.753631	+	0.657297i	$0.228300\pi$
$882$	9.85828	0.331946
$883$	−53.3699	−1.79604	−0.898020	−	0.439954i	$-0.854995\pi$
$883$	−53.3699	−1.79604	−0.898020	+	0.439954i	$0.854995\pi$
$884$	−0.778658	−0.0261891
$885$	−0.394411	−0.0132580
$886$	−1.74933	−0.0587698
$887$	−21.8610	−0.734020	−0.367010	−	0.930217i	$-0.619618\pi$
$887$	−21.8610	−0.734020	−0.367010	+	0.930217i	$0.619618\pi$
$888$	1.49492	0.0501663
$889$	25.3108	0.848897
$890$	−1.00767	−0.0337772
$891$	−34.5385	−1.15708
$892$	−0.987764	−0.0330728
$893$	22.6857	0.759148
$894$	−0.630209	−0.0210773
$895$	−0.428265	−0.0143153
$896$	1.86515	0.0623103
$897$	−3.29233	−0.109928
$898$	−10.9051	−0.363908
$899$	−4.16592	−0.138941
$900$	−2.79968	−0.0933228
$901$	3.45084	0.114964
$902$	−47.7009	−1.58827
$903$	3.05684	0.101725
$904$	5.70682	0.189806
$905$	25.2013	0.837718
$906$	−7.99922	−0.265756
$907$	−50.0962	−1.66342	−0.831709	−	0.555212i	$-0.812637\pi$
$907$	−50.0962	−1.66342	−0.831709	+	0.555212i	$0.812637\pi$
$908$	29.3711	0.974713
$909$	36.9703	1.22623
$910$	1.86515	0.0618291
$911$	−46.3166	−1.53454	−0.767269	−	0.641326i	$-0.778385\pi$
$911$	−46.3166	−1.53454	−0.767269	+	0.641326i	$0.778385\pi$
$912$	0.769933	0.0254950
$913$	−43.0869	−1.42597
$914$	−10.9185	−0.361151
$915$	−4.89389	−0.161787
$916$	22.0436	0.728342
$917$	−21.3175	−0.703964
$918$	−2.02120	−0.0667097
$919$	2.41549	0.0796798	0.0398399	−	0.999206i	$-0.487315\pi$
$919$	2.41549	0.0796798	0.0398399	+	0.999206i	$0.487315\pi$
$920$	−7.35604	−0.242521
$921$	9.71700	0.320186
$922$	8.49891	0.279897
$923$	11.6243	0.382619
$924$	−3.98383	−0.131059
$925$	3.34010	0.109822
$926$	−4.70088	−0.154481
$927$	12.9055	0.423874
$928$	4.16592	0.136753
$929$	−2.38825	−0.0783561	−0.0391780	−	0.999232i	$-0.512474\pi$
$929$	−2.38825	−0.0783561	−0.0391780	+	0.999232i	$0.512474\pi$
$930$	0.447568	0.0146763
$931$	−6.05739	−0.198523
$932$	29.9428	0.980808
$933$	5.52259	0.180801
$934$	35.0427	1.14663
$935$	3.71599	0.121526
$936$	2.79968	0.0915105
$937$	10.4567	0.341605	0.170802	−	0.985305i	$-0.445364\pi$
$937$	10.4567	0.341605	0.170802	+	0.985305i	$0.445364\pi$
$938$	23.3385	0.762030
$939$	6.87685	0.224417
$940$	−13.1874	−0.430125
$941$	−39.2641	−1.27997	−0.639987	−	0.768386i	$-0.721060\pi$
$941$	−39.2641	−1.27997	−0.639987	+	0.768386i	$0.721060\pi$
$942$	1.45077	0.0472686
$943$	73.5263	2.39435
$944$	0.881231	0.0286816
$945$	4.84147	0.157493
$946$	−17.4754	−0.568175
$947$	45.8543	1.49006	0.745032	−	0.667029i	$-0.232434\pi$
$947$	45.8543	1.49006	0.745032	+	0.667029i	$0.232434\pi$
$948$	−2.98778	−0.0970387
$949$	3.07944	0.0999628
$950$	1.72026	0.0558125
$951$	7.12438	0.231024
$952$	1.45231	0.0470698
$953$	−45.6183	−1.47772	−0.738860	−	0.673859i	$-0.764636\pi$
$953$	−45.6183	−1.47772	−0.738860	+	0.673859i	$0.764636\pi$
$954$	−12.4076	−0.401710
$955$	22.3986	0.724801
$956$	−2.20402	−0.0712830
$957$	−8.89812	−0.287635
$958$	−7.61684	−0.246089
$959$	−7.14730	−0.230798
$960$	−0.447568	−0.0144452
$961$	1.00000	0.0322581
$962$	−3.34010	−0.107689
$963$	−26.2158	−0.844792
$964$	−2.96453	−0.0954812
$965$	−7.83137	−0.252101
$966$	6.14069	0.197574
$967$	−3.28378	−0.105599	−0.0527996	−	0.998605i	$-0.516814\pi$
$967$	−3.28378	−0.105599	−0.0527996	+	0.998605i	$0.516814\pi$
$968$	11.7749	0.378459
$969$	0.599514	0.0192592
$970$	0.347181	0.0111473
$971$	−36.7536	−1.17948	−0.589740	−	0.807593i	$-0.700770\pi$
$971$	−36.7536	−1.17948	−0.589740	+	0.807593i	$0.700770\pi$
$972$	11.0264	0.353673
$973$	−11.1044	−0.355989
$974$	−18.2794	−0.585710
$975$	−0.447568	−0.0143337
$976$	10.9344	0.350001
$977$	3.93943	0.126033	0.0630167	−	0.998012i	$-0.479928\pi$
$977$	3.93943	0.126033	0.0630167	+	0.998012i	$0.479928\pi$
$978$	−3.91089	−0.125057
$979$	−4.80892	−0.153694
$980$	3.52121	0.112481
$981$	−1.38309	−0.0441587
$982$	16.4804	0.525909
$983$	−39.5430	−1.26123	−0.630613	−	0.776097i	$-0.717197\pi$
$983$	−39.5430	−1.26123	−0.630613	+	0.776097i	$0.717197\pi$
$984$	4.47361	0.142613
$985$	0.917816	0.0292441
$986$	3.24383	0.103304
$987$	11.0086	0.350407
$988$	−1.72026	−0.0547287
$989$	26.9367	0.856536
$990$	−13.3609	−0.424639
$991$	10.2319	0.325027	0.162513	−	0.986706i	$-0.448040\pi$
$991$	10.2319	0.325027	0.162513	+	0.986706i	$0.448040\pi$
$992$	−1.00000	−0.0317500
$993$	8.27342	0.262549
$994$	−21.6811	−0.687683
$995$	1.53767	0.0487475
$996$	4.04088	0.128040
$997$	−27.3618	−0.866556	−0.433278	−	0.901260i	$-0.642643\pi$
$997$	−27.3618	−0.866556	−0.433278	+	0.901260i	$0.642643\pi$
$998$	−15.8554	−0.501895
$999$	−8.67007	−0.274309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.n.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.n.1.6	✓	8	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:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.447568 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.447568 q^{6} +1.86515 q^{7} +1.00000 q^{8} -2.79968 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.447568 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.447568 q^{6} +1.86515 q^{7} +1.00000 q^{8} -2.79968 q^{9} -1.00000 q^{10} -4.77230 q^{11} +0.447568 q^{12} -1.00000 q^{13} +1.86515 q^{14} -0.447568 q^{15} +1.00000 q^{16} +0.778658 q^{17} -2.79968 q^{18} +1.72026 q^{19} -1.00000 q^{20} +0.834782 q^{21} -4.77230 q^{22} +7.35604 q^{23} +0.447568 q^{24} +1.00000 q^{25} -1.00000 q^{26} -2.59575 q^{27} +1.86515 q^{28} +4.16592 q^{29} -0.447568 q^{30} -1.00000 q^{31} +1.00000 q^{32} -2.13593 q^{33} +0.778658 q^{34} -1.86515 q^{35} -2.79968 q^{36} +3.34010 q^{37} +1.72026 q^{38} -0.447568 q^{39} -1.00000 q^{40} +9.99536 q^{41} +0.834782 q^{42} +3.66184 q^{43} -4.77230 q^{44} +2.79968 q^{45} +7.35604 q^{46} +13.1874 q^{47} +0.447568 q^{48} -3.52121 q^{49} +1.00000 q^{50} +0.348503 q^{51} -1.00000 q^{52} +4.43178 q^{53} -2.59575 q^{54} +4.77230 q^{55} +1.86515 q^{56} +0.769933 q^{57} +4.16592 q^{58} +0.881231 q^{59} -0.447568 q^{60} +10.9344 q^{61} -1.00000 q^{62} -5.22183 q^{63} +1.00000 q^{64} +1.00000 q^{65} -2.13593 q^{66} +12.5130 q^{67} +0.778658 q^{68} +3.29233 q^{69} -1.86515 q^{70} -11.6243 q^{71} -2.79968 q^{72} -3.07944 q^{73} +3.34010 q^{74} +0.447568 q^{75} +1.72026 q^{76} -8.90107 q^{77} -0.447568 q^{78} -6.67559 q^{79} -1.00000 q^{80} +7.23727 q^{81} +9.99536 q^{82} +9.02853 q^{83} +0.834782 q^{84} -0.778658 q^{85} +3.66184 q^{86} +1.86453 q^{87} -4.77230 q^{88} +1.00767 q^{89} +2.79968 q^{90} -1.86515 q^{91} +7.35604 q^{92} -0.447568 q^{93} +13.1874 q^{94} -1.72026 q^{95} +0.447568 q^{96} -0.347181 q^{97} -3.52121 q^{98} +13.3609 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9} - 8 q^{10} + 4 q^{11} - q^{12} - 8 q^{13} + q^{14} + q^{15} + 8 q^{16} - 5 q^{17} + 9 q^{18} + 2 q^{19} - 8 q^{20} + 17 q^{21} + 4 q^{22} + 4 q^{23} - q^{24} + 8 q^{25} - 8 q^{26} + 11 q^{27} + q^{28} + 11 q^{29} + q^{30} - 8 q^{31} + 8 q^{32} + 10 q^{33} - 5 q^{34} - q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} + q^{39} - 8 q^{40} + 10 q^{41} + 17 q^{42} + 19 q^{43} + 4 q^{44} - 9 q^{45} + 4 q^{46} + 11 q^{47} - q^{48} + 11 q^{49} + 8 q^{50} + 7 q^{51} - 8 q^{52} + 8 q^{53} + 11 q^{54} - 4 q^{55} + q^{56} - 11 q^{57} + 11 q^{58} + 28 q^{59} + q^{60} - 12 q^{61} - 8 q^{62} + 20 q^{63} + 8 q^{64} + 8 q^{65} + 10 q^{66} + 24 q^{67} - 5 q^{68} + 30 q^{69} - q^{70} + 18 q^{71} + 9 q^{72} - 3 q^{73} + 19 q^{74} - q^{75} + 2 q^{76} - 7 q^{77} + q^{78} + 22 q^{79} - 8 q^{80} + 24 q^{81} + 10 q^{82} + 17 q^{83} + 17 q^{84} + 5 q^{85} + 19 q^{86} + 11 q^{87} + 4 q^{88} + 17 q^{89} - 9 q^{90} - q^{91} + 4 q^{92} + q^{93} + 11 q^{94} - 2 q^{95} - q^{96} - 24 q^{97} + 11 q^{98} + 23 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9 - 8 * q^10 + 4 * q^11 - q^12 - 8 * q^13 + q^14 + q^15 + 8 * q^16 - 5 * q^17 + 9 * q^18 + 2 * q^19 - 8 * q^20 + 17 * q^21 + 4 * q^22 + 4 * q^23 - q^24 + 8 * q^25 - 8 * q^26 + 11 * q^27 + q^28 + 11 * q^29 + q^30 - 8 * q^31 + 8 * q^32 + 10 * q^33 - 5 * q^34 - q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 + q^39 - 8 * q^40 + 10 * q^41 + 17 * q^42 + 19 * q^43 + 4 * q^44 - 9 * q^45 + 4 * q^46 + 11 * q^47 - q^48 + 11 * q^49 + 8 * q^50 + 7 * q^51 - 8 * q^52 + 8 * q^53 + 11 * q^54 - 4 * q^55 + q^56 - 11 * q^57 + 11 * q^58 + 28 * q^59 + q^60 - 12 * q^61 - 8 * q^62 + 20 * q^63 + 8 * q^64 + 8 * q^65 + 10 * q^66 + 24 * q^67 - 5 * q^68 + 30 * q^69 - q^70 + 18 * q^71 + 9 * q^72 - 3 * q^73 + 19 * q^74 - q^75 + 2 * q^76 - 7 * q^77 + q^78 + 22 * q^79 - 8 * q^80 + 24 * q^81 + 10 * q^82 + 17 * q^83 + 17 * q^84 + 5 * q^85 + 19 * q^86 + 11 * q^87 + 4 * q^88 + 17 * q^89 - 9 * q^90 - q^91 + 4 * q^92 + q^93 + 11 * q^94 - 2 * q^95 - q^96 - 24 * q^97 + 11 * q^98 + 23 * q^99

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