LMFDB - Embedded newform 4030.2.a.r.1.4

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 8x^{7} + 39x^{6} + 13x^{5} - 106x^{4} + 9x^{3} + 74x^{2} - 3x - 7 $ x^9 - 4x^8 - 8x^7 + 39x^6 + 13x^5 - 106x^4 + 9x^3 + 74x^2 - 3x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.351548$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.748457 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.748457 q^{6} +4.59004 q^{7} +1.00000 q^{8} -2.43981 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.748457 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.748457 q^{6} +4.59004 q^{7} +1.00000 q^{8} -2.43981 q^{9} +1.00000 q^{10} -2.22796 q^{11} -0.748457 q^{12} +1.00000 q^{13} +4.59004 q^{14} -0.748457 q^{15} +1.00000 q^{16} -3.52524 q^{17} -2.43981 q^{18} -0.371373 q^{19} +1.00000 q^{20} -3.43544 q^{21} -2.22796 q^{22} +4.53717 q^{23} -0.748457 q^{24} +1.00000 q^{25} +1.00000 q^{26} +4.07146 q^{27} +4.59004 q^{28} +3.98426 q^{29} -0.748457 q^{30} -1.00000 q^{31} +1.00000 q^{32} +1.66753 q^{33} -3.52524 q^{34} +4.59004 q^{35} -2.43981 q^{36} +2.32928 q^{37} -0.371373 q^{38} -0.748457 q^{39} +1.00000 q^{40} +6.05311 q^{41} -3.43544 q^{42} +1.20748 q^{43} -2.22796 q^{44} -2.43981 q^{45} +4.53717 q^{46} -3.11300 q^{47} -0.748457 q^{48} +14.0684 q^{49} +1.00000 q^{50} +2.63849 q^{51} +1.00000 q^{52} +13.5626 q^{53} +4.07146 q^{54} -2.22796 q^{55} +4.59004 q^{56} +0.277957 q^{57} +3.98426 q^{58} +9.99269 q^{59} -0.748457 q^{60} +5.68755 q^{61} -1.00000 q^{62} -11.1988 q^{63} +1.00000 q^{64} +1.00000 q^{65} +1.66753 q^{66} -9.75709 q^{67} -3.52524 q^{68} -3.39588 q^{69} +4.59004 q^{70} +9.98191 q^{71} -2.43981 q^{72} -11.4843 q^{73} +2.32928 q^{74} -0.748457 q^{75} -0.371373 q^{76} -10.2264 q^{77} -0.748457 q^{78} +6.97071 q^{79} +1.00000 q^{80} +4.27212 q^{81} +6.05311 q^{82} -13.6130 q^{83} -3.43544 q^{84} -3.52524 q^{85} +1.20748 q^{86} -2.98205 q^{87} -2.22796 q^{88} -8.00674 q^{89} -2.43981 q^{90} +4.59004 q^{91} +4.53717 q^{92} +0.748457 q^{93} -3.11300 q^{94} -0.371373 q^{95} -0.748457 q^{96} -1.68726 q^{97} +14.0684 q^{98} +5.43581 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + 3 q^{3} + 9 q^{4} + 9 q^{5} + 3 q^{6} + 9 q^{7} + 9 q^{8} + 14 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 + 3 * q^3 + 9 * q^4 + 9 * q^5 + 3 * q^6 + 9 * q^7 + 9 * q^8 + 14 * q^9	\( 9 q + 9 q^{2} + 3 q^{3} + 9 q^{4} + 9 q^{5} + 3 q^{6} + 9 q^{7} + 9 q^{8} + 14 q^{9} + 9 q^{10} + 10 q^{11} + 3 q^{12} + 9 q^{13} + 9 q^{14} + 3 q^{15} + 9 q^{16} + q^{17} + 14 q^{18} + 10 q^{19} + 9 q^{20} + 3 q^{21} + 10 q^{22} + 8 q^{23} + 3 q^{24} + 9 q^{25} + 9 q^{26} + 15 q^{27} + 9 q^{28} + 9 q^{29} + 3 q^{30} - 9 q^{31} + 9 q^{32} + 4 q^{33} + q^{34} + 9 q^{35} + 14 q^{36} - 3 q^{37} + 10 q^{38} + 3 q^{39} + 9 q^{40} + 3 q^{42} + 7 q^{43} + 10 q^{44} + 14 q^{45} + 8 q^{46} + 7 q^{47} + 3 q^{48} + 8 q^{49} + 9 q^{50} - 3 q^{51} + 9 q^{52} + 8 q^{53} + 15 q^{54} + 10 q^{55} + 9 q^{56} - 7 q^{57} + 9 q^{58} + 2 q^{59} + 3 q^{60} - 2 q^{61} - 9 q^{62} + 10 q^{63} + 9 q^{64} + 9 q^{65} + 4 q^{66} + 18 q^{67} + q^{68} - 16 q^{69} + 9 q^{70} + 14 q^{71} + 14 q^{72} + q^{73} - 3 q^{74} + 3 q^{75} + 10 q^{76} - 5 q^{77} + 3 q^{78} + 6 q^{79} + 9 q^{80} + q^{81} + 7 q^{83} + 3 q^{84} + q^{85} + 7 q^{86} + 11 q^{87} + 10 q^{88} - 19 q^{89} + 14 q^{90} + 9 q^{91} + 8 q^{92} - 3 q^{93} + 7 q^{94} + 10 q^{95} + 3 q^{96} - 6 q^{97} + 8 q^{98} - 5 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 + 3 * q^3 + 9 * q^4 + 9 * q^5 + 3 * q^6 + 9 * q^7 + 9 * q^8 + 14 * q^9 + 9 * q^10 + 10 * q^11 + 3 * q^12 + 9 * q^13 + 9 * q^14 + 3 * q^15 + 9 * q^16 + q^17 + 14 * q^18 + 10 * q^19 + 9 * q^20 + 3 * q^21 + 10 * q^22 + 8 * q^23 + 3 * q^24 + 9 * q^25 + 9 * q^26 + 15 * q^27 + 9 * q^28 + 9 * q^29 + 3 * q^30 - 9 * q^31 + 9 * q^32 + 4 * q^33 + q^34 + 9 * q^35 + 14 * q^36 - 3 * q^37 + 10 * q^38 + 3 * q^39 + 9 * q^40 + 3 * q^42 + 7 * q^43 + 10 * q^44 + 14 * q^45 + 8 * q^46 + 7 * q^47 + 3 * q^48 + 8 * q^49 + 9 * q^50 - 3 * q^51 + 9 * q^52 + 8 * q^53 + 15 * q^54 + 10 * q^55 + 9 * q^56 - 7 * q^57 + 9 * q^58 + 2 * q^59 + 3 * q^60 - 2 * q^61 - 9 * q^62 + 10 * q^63 + 9 * q^64 + 9 * q^65 + 4 * q^66 + 18 * q^67 + q^68 - 16 * q^69 + 9 * q^70 + 14 * q^71 + 14 * q^72 + q^73 - 3 * q^74 + 3 * q^75 + 10 * q^76 - 5 * q^77 + 3 * q^78 + 6 * q^79 + 9 * q^80 + q^81 + 7 * q^83 + 3 * q^84 + q^85 + 7 * q^86 + 11 * q^87 + 10 * q^88 - 19 * q^89 + 14 * q^90 + 9 * q^91 + 8 * q^92 - 3 * q^93 + 7 * q^94 + 10 * q^95 + 3 * q^96 - 6 * q^97 + 8 * q^98 - 5 * 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
$3$	−0.748457	−0.432122	−0.216061	−	0.976380i	$-0.569321\pi$
$3$	−0.748457	−0.432122	−0.216061	+	0.976380i	$0.569321\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.748457	−0.305556
$7$	4.59004	1.73487	0.867435	−	0.497550i	$-0.165767\pi$
$7$	4.59004	1.73487	0.867435	+	0.497550i	$0.165767\pi$
$8$	1.00000	0.353553
$9$	−2.43981	−0.813271
$10$	1.00000	0.316228
$11$	−2.22796	−0.671756	−0.335878	−	0.941906i	$-0.609033\pi$
$11$	−2.22796	−0.671756	−0.335878	+	0.941906i	$0.609033\pi$
$12$	−0.748457	−0.216061
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	4.59004	1.22674
$15$	−0.748457	−0.193251
$16$	1.00000	0.250000
$17$	−3.52524	−0.854996	−0.427498	−	0.904016i	$-0.640605\pi$
$17$	−3.52524	−0.854996	−0.427498	+	0.904016i	$0.640605\pi$
$18$	−2.43981	−0.575069
$19$	−0.371373	−0.0851989	−0.0425994	−	0.999092i	$-0.513564\pi$
$19$	−0.371373	−0.0851989	−0.0425994	+	0.999092i	$0.513564\pi$
$20$	1.00000	0.223607
$21$	−3.43544	−0.749675
$22$	−2.22796	−0.475003
$23$	4.53717	0.946065	0.473033	−	0.881045i	$-0.343159\pi$
$23$	4.53717	0.946065	0.473033	+	0.881045i	$0.343159\pi$
$24$	−0.748457	−0.152778
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	4.07146	0.783554
$28$	4.59004	0.867435
$29$	3.98426	0.739859	0.369930	−	0.929060i	$-0.379382\pi$
$29$	3.98426	0.739859	0.369930	+	0.929060i	$0.379382\pi$
$30$	−0.748457	−0.136649
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	1.66753	0.290280
$34$	−3.52524	−0.604573
$35$	4.59004	0.775858
$36$	−2.43981	−0.406635
$37$	2.32928	0.382931	0.191466	−	0.981499i	$-0.438676\pi$
$37$	2.32928	0.382931	0.191466	+	0.981499i	$0.438676\pi$
$38$	−0.371373	−0.0602447
$39$	−0.748457	−0.119849
$40$	1.00000	0.158114
$41$	6.05311	0.945337	0.472668	−	0.881240i	$-0.343291\pi$
$41$	6.05311	0.945337	0.472668	+	0.881240i	$0.343291\pi$
$42$	−3.43544	−0.530101
$43$	1.20748	0.184139	0.0920697	−	0.995753i	$-0.470652\pi$
$43$	1.20748	0.184139	0.0920697	+	0.995753i	$0.470652\pi$
$44$	−2.22796	−0.335878
$45$	−2.43981	−0.363706
$46$	4.53717	0.668969
$47$	−3.11300	−0.454078	−0.227039	−	0.973886i	$-0.572905\pi$
$47$	−3.11300	−0.454078	−0.227039	+	0.973886i	$0.572905\pi$
$48$	−0.748457	−0.108030
$49$	14.0684	2.00978
$50$	1.00000	0.141421
$51$	2.63849	0.369462
$52$	1.00000	0.138675
$53$	13.5626	1.86297	0.931484	−	0.363783i	$-0.118515\pi$
$53$	13.5626	1.86297	0.931484	+	0.363783i	$0.118515\pi$
$54$	4.07146	0.554056
$55$	−2.22796	−0.300418
$56$	4.59004	0.613369
$57$	0.277957	0.0368163
$58$	3.98426	0.523159
$59$	9.99269	1.30094	0.650469	−	0.759533i	$-0.274573\pi$
$59$	9.99269	1.30094	0.650469	+	0.759533i	$0.274573\pi$
$60$	−0.748457	−0.0966254
$61$	5.68755	0.728216	0.364108	−	0.931357i	$-0.381374\pi$
$61$	5.68755	0.728216	0.364108	+	0.931357i	$0.381374\pi$
$62$	−1.00000	−0.127000
$63$	−11.1988	−1.41092
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	1.66753	0.205259
$67$	−9.75709	−1.19202	−0.596009	−	0.802978i	$-0.703248\pi$
$67$	−9.75709	−1.19202	−0.596009	+	0.802978i	$0.703248\pi$
$68$	−3.52524	−0.427498
$69$	−3.39588	−0.408815
$70$	4.59004	0.548614
$71$	9.98191	1.18463	0.592317	−	0.805705i	$-0.298213\pi$
$71$	9.98191	1.18463	0.592317	+	0.805705i	$0.298213\pi$
$72$	−2.43981	−0.287535
$73$	−11.4843	−1.34414	−0.672069	−	0.740489i	$-0.734594\pi$
$73$	−11.4843	−1.34414	−0.672069	+	0.740489i	$0.734594\pi$
$74$	2.32928	0.270773
$75$	−0.748457	−0.0864244
$76$	−0.371373	−0.0425994
$77$	−10.2264	−1.16541
$78$	−0.748457	−0.0847461
$79$	6.97071	0.784266	0.392133	−	0.919909i	$-0.371737\pi$
$79$	6.97071	0.784266	0.392133	+	0.919909i	$0.371737\pi$
$80$	1.00000	0.111803
$81$	4.27212	0.474680
$82$	6.05311	0.668454
$83$	−13.6130	−1.49422	−0.747111	−	0.664699i	$-0.768560\pi$
$83$	−13.6130	−1.49422	−0.747111	+	0.664699i	$0.768560\pi$
$84$	−3.43544	−0.374838
$85$	−3.52524	−0.382366
$86$	1.20748	0.130206
$87$	−2.98205	−0.319709
$88$	−2.22796	−0.237502
$89$	−8.00674	−0.848712	−0.424356	−	0.905495i	$-0.639500\pi$
$89$	−8.00674	−0.848712	−0.424356	+	0.905495i	$0.639500\pi$
$90$	−2.43981	−0.257179
$91$	4.59004	0.481167
$92$	4.53717	0.473033
$93$	0.748457	0.0776114
$94$	−3.11300	−0.321082
$95$	−0.371373	−0.0381021
$96$	−0.748457	−0.0763891
$97$	−1.68726	−0.171315	−0.0856576	−	0.996325i	$-0.527299\pi$
$97$	−1.68726	−0.171315	−0.0856576	+	0.996325i	$0.527299\pi$
$98$	14.0684	1.42113
$99$	5.43581	0.546319
$100$	1.00000	0.100000
$101$	0.293435	0.0291979	0.0145989	−	0.999893i	$-0.495353\pi$
$101$	0.293435	0.0291979	0.0145989	+	0.999893i	$0.495353\pi$
$102$	2.63849	0.261249
$103$	13.6160	1.34162	0.670810	−	0.741629i	$-0.265946\pi$
$103$	13.6160	1.34162	0.670810	+	0.741629i	$0.265946\pi$
$104$	1.00000	0.0980581
$105$	−3.43544	−0.335265
$106$	13.5626	1.31732
$107$	−4.47912	−0.433013	−0.216507	−	0.976281i	$-0.569466\pi$
$107$	−4.47912	−0.433013	−0.216507	+	0.976281i	$0.569466\pi$
$108$	4.07146	0.391777
$109$	8.62440	0.826068	0.413034	−	0.910716i	$-0.364469\pi$
$109$	8.62440	0.826068	0.413034	+	0.910716i	$0.364469\pi$
$110$	−2.22796	−0.212428
$111$	−1.74337	−0.165473
$112$	4.59004	0.433718
$113$	6.49554	0.611049	0.305525	−	0.952184i	$-0.401168\pi$
$113$	6.49554	0.611049	0.305525	+	0.952184i	$0.401168\pi$
$114$	0.277957	0.0260330
$115$	4.53717	0.423093
$116$	3.98426	0.369930
$117$	−2.43981	−0.225561
$118$	9.99269	0.919901
$119$	−16.1810	−1.48331
$120$	−0.748457	−0.0683245
$121$	−6.03619	−0.548744
$122$	5.68755	0.514927
$123$	−4.53049	−0.408501
$124$	−1.00000	−0.0898027
$125$	1.00000	0.0894427
$126$	−11.1988	−0.997671
$127$	−6.93317	−0.615219	−0.307610	−	0.951513i	$-0.599529\pi$
$127$	−6.93317	−0.615219	−0.307610	+	0.951513i	$0.599529\pi$
$128$	1.00000	0.0883883
$129$	−0.903748	−0.0795706
$130$	1.00000	0.0877058
$131$	16.5844	1.44898	0.724491	−	0.689285i	$-0.242075\pi$
$131$	16.5844	1.44898	0.724491	+	0.689285i	$0.242075\pi$
$132$	1.66753	0.145140
$133$	−1.70462	−0.147809
$134$	−9.75709	−0.842884
$135$	4.07146	0.350416
$136$	−3.52524	−0.302287
$137$	8.23564	0.703618	0.351809	−	0.936072i	$-0.385567\pi$
$137$	8.23564	0.703618	0.351809	+	0.936072i	$0.385567\pi$
$138$	−3.39588	−0.289076
$139$	−21.8427	−1.85267	−0.926335	−	0.376701i	$-0.877058\pi$
$139$	−21.8427	−1.85267	−0.926335	+	0.376701i	$0.877058\pi$
$140$	4.59004	0.387929
$141$	2.32995	0.196217
$142$	9.98191	0.837663
$143$	−2.22796	−0.186312
$144$	−2.43981	−0.203318
$145$	3.98426	0.330875
$146$	−11.4843	−0.950449
$147$	−10.5296	−0.868468
$148$	2.32928	0.191466
$149$	−10.6426	−0.871878	−0.435939	−	0.899976i	$-0.643584\pi$
$149$	−10.6426	−0.871878	−0.435939	+	0.899976i	$0.643584\pi$
$150$	−0.748457	−0.0611112
$151$	−10.6052	−0.863035	−0.431518	−	0.902105i	$-0.642022\pi$
$151$	−10.6052	−0.863035	−0.431518	+	0.902105i	$0.642022\pi$
$152$	−0.371373	−0.0301223
$153$	8.60092	0.695343
$154$	−10.2264	−0.824069
$155$	−1.00000	−0.0803219
$156$	−0.748457	−0.0599245
$157$	−15.1867	−1.21203	−0.606017	−	0.795452i	$-0.707233\pi$
$157$	−15.1867	−1.21203	−0.606017	+	0.795452i	$0.707233\pi$
$158$	6.97071	0.554560
$159$	−10.1510	−0.805029
$160$	1.00000	0.0790569
$161$	20.8258	1.64130
$162$	4.27212	0.335650
$163$	−9.80160	−0.767720	−0.383860	−	0.923391i	$-0.625406\pi$
$163$	−9.80160	−0.767720	−0.383860	+	0.923391i	$0.625406\pi$
$164$	6.05311	0.472668
$165$	1.66753	0.129817
$166$	−13.6130	−1.05658
$167$	14.0417	1.08658	0.543290	−	0.839545i	$-0.317178\pi$
$167$	14.0417	1.08658	0.543290	+	0.839545i	$0.317178\pi$
$168$	−3.43544	−0.265050
$169$	1.00000	0.0769231
$170$	−3.52524	−0.270373
$171$	0.906081	0.0692897
$172$	1.20748	0.0920697
$173$	17.1015	1.30020	0.650101	−	0.759848i	$-0.274727\pi$
$173$	17.1015	1.30020	0.650101	+	0.759848i	$0.274727\pi$
$174$	−2.98205	−0.226069
$175$	4.59004	0.346974
$176$	−2.22796	−0.167939
$177$	−7.47910	−0.562163
$178$	−8.00674	−0.600130
$179$	−1.34014	−0.100167	−0.0500835	−	0.998745i	$-0.515949\pi$
$179$	−1.34014	−0.100167	−0.0500835	+	0.998745i	$0.515949\pi$
$180$	−2.43981	−0.181853
$181$	2.21505	0.164643	0.0823217	−	0.996606i	$-0.473766\pi$
$181$	2.21505	0.164643	0.0823217	+	0.996606i	$0.473766\pi$
$182$	4.59004	0.340236
$183$	−4.25689	−0.314678
$184$	4.53717	0.334485
$185$	2.32928	0.171252
$186$	0.748457	0.0548795
$187$	7.85410	0.574348
$188$	−3.11300	−0.227039
$189$	18.6882	1.35936
$190$	−0.371373	−0.0269422
$191$	−0.880684	−0.0637241	−0.0318620	−	0.999492i	$-0.510144\pi$
$191$	−0.880684	−0.0637241	−0.0318620	+	0.999492i	$0.510144\pi$
$192$	−0.748457	−0.0540152
$193$	3.62453	0.260899	0.130450	−	0.991455i	$-0.458358\pi$
$193$	3.62453	0.260899	0.130450	+	0.991455i	$0.458358\pi$
$194$	−1.68726	−0.121138
$195$	−0.748457	−0.0535981
$196$	14.0684	1.00489
$197$	7.82180	0.557281	0.278640	−	0.960396i	$-0.410116\pi$
$197$	7.82180	0.557281	0.278640	+	0.960396i	$0.410116\pi$
$198$	5.43581	0.386306
$199$	−16.3425	−1.15849	−0.579246	−	0.815153i	$-0.696653\pi$
$199$	−16.3425	−1.15849	−0.579246	+	0.815153i	$0.696653\pi$
$200$	1.00000	0.0707107
$201$	7.30276	0.515097
$202$	0.293435	0.0206460
$203$	18.2879	1.28356
$204$	2.63849	0.184731
$205$	6.05311	0.422767
$206$	13.6160	0.948669
$207$	−11.0698	−0.769407
$208$	1.00000	0.0693375
$209$	0.827405	0.0572328
$210$	−3.43544	−0.237068
$211$	17.9147	1.23330	0.616649	−	0.787238i	$-0.288490\pi$
$211$	17.9147	1.23330	0.616649	+	0.787238i	$0.288490\pi$
$212$	13.5626	0.931484
$213$	−7.47103	−0.511907
$214$	−4.47912	−0.306187
$215$	1.20748	0.0823496
$216$	4.07146	0.277028
$217$	−4.59004	−0.311592
$218$	8.62440	0.584118
$219$	8.59552	0.580831
$220$	−2.22796	−0.150209
$221$	−3.52524	−0.237133
$222$	−1.74337	−0.117007
$223$	11.4277	0.765258	0.382629	−	0.923902i	$-0.375019\pi$
$223$	11.4277	0.765258	0.382629	+	0.923902i	$0.375019\pi$
$224$	4.59004	0.306685
$225$	−2.43981	−0.162654
$226$	6.49554	0.432077
$227$	−5.92640	−0.393349	−0.196674	−	0.980469i	$-0.563014\pi$
$227$	−5.92640	−0.393349	−0.196674	+	0.980469i	$0.563014\pi$
$228$	0.277957	0.0184081
$229$	−11.2993	−0.746680	−0.373340	−	0.927695i	$-0.621788\pi$
$229$	−11.2993	−0.746680	−0.373340	+	0.927695i	$0.621788\pi$
$230$	4.53717	0.299172
$231$	7.65404	0.503599
$232$	3.98426	0.261580
$233$	5.58511	0.365892	0.182946	−	0.983123i	$-0.441437\pi$
$233$	5.58511	0.365892	0.182946	+	0.983123i	$0.441437\pi$
$234$	−2.43981	−0.159496
$235$	−3.11300	−0.203070
$236$	9.99269	0.650469
$237$	−5.21727	−0.338898
$238$	−16.1810	−1.04886
$239$	1.73687	0.112349	0.0561745	−	0.998421i	$-0.482110\pi$
$239$	1.73687	0.112349	0.0561745	+	0.998421i	$0.482110\pi$
$240$	−0.748457	−0.0483127
$241$	1.53690	0.0990005	0.0495002	−	0.998774i	$-0.484237\pi$
$241$	1.53690	0.0990005	0.0495002	+	0.998774i	$0.484237\pi$
$242$	−6.03619	−0.388021
$243$	−15.4119	−0.988673
$244$	5.68755	0.364108
$245$	14.0684	0.898799
$246$	−4.53049	−0.288854
$247$	−0.371373	−0.0236299
$248$	−1.00000	−0.0635001
$249$	10.1888	0.645686
$250$	1.00000	0.0632456
$251$	19.0083	1.19980	0.599898	−	0.800076i	$-0.295208\pi$
$251$	19.0083	1.19980	0.599898	+	0.800076i	$0.295208\pi$
$252$	−11.1988	−0.705460
$253$	−10.1086	−0.635525
$254$	−6.93317	−0.435026
$255$	2.63849	0.165229
$256$	1.00000	0.0625000
$257$	10.7678	0.671674	0.335837	−	0.941920i	$-0.390981\pi$
$257$	10.7678	0.671674	0.335837	+	0.941920i	$0.390981\pi$
$258$	−0.903748	−0.0562649
$259$	10.6915	0.664337
$260$	1.00000	0.0620174
$261$	−9.72085	−0.601706
$262$	16.5844	1.02458
$263$	−1.21715	−0.0750526	−0.0375263	−	0.999296i	$-0.511948\pi$
$263$	−1.21715	−0.0750526	−0.0375263	+	0.999296i	$0.511948\pi$
$264$	1.66753	0.102630
$265$	13.5626	0.833144
$266$	−1.70462	−0.104517
$267$	5.99270	0.366747
$268$	−9.75709	−0.596009
$269$	−16.5409	−1.00852	−0.504259	−	0.863552i	$-0.668234\pi$
$269$	−16.5409	−1.00852	−0.504259	+	0.863552i	$0.668234\pi$
$270$	4.07146	0.247781
$271$	−5.60514	−0.340488	−0.170244	−	0.985402i	$-0.554456\pi$
$271$	−5.60514	−0.340488	−0.170244	+	0.985402i	$0.554456\pi$
$272$	−3.52524	−0.213749
$273$	−3.43544	−0.207923
$274$	8.23564	0.497533
$275$	−2.22796	−0.134351
$276$	−3.39588	−0.204408
$277$	−16.1894	−0.972726	−0.486363	−	0.873757i	$-0.661677\pi$
$277$	−16.1894	−0.972726	−0.486363	+	0.873757i	$0.661677\pi$
$278$	−21.8427	−1.31004
$279$	2.43981	0.146068
$280$	4.59004	0.274307
$281$	−23.8583	−1.42327	−0.711634	−	0.702551i	$-0.752044\pi$
$281$	−23.8583	−1.42327	−0.711634	+	0.702551i	$0.752044\pi$
$282$	2.32995	0.138746
$283$	12.6373	0.751209	0.375604	−	0.926780i	$-0.377435\pi$
$283$	12.6373	0.751209	0.375604	+	0.926780i	$0.377435\pi$
$284$	9.98191	0.592317
$285$	0.277957	0.0164647
$286$	−2.22796	−0.131742
$287$	27.7840	1.64004
$288$	−2.43981	−0.143767
$289$	−4.57270	−0.268982
$290$	3.98426	0.233964
$291$	1.26284	0.0740290
$292$	−11.4843	−0.672069
$293$	5.31797	0.310679	0.155340	−	0.987861i	$-0.450353\pi$
$293$	5.31797	0.310679	0.155340	+	0.987861i	$0.450353\pi$
$294$	−10.5296	−0.614100
$295$	9.99269	0.581797
$296$	2.32928	0.135387
$297$	−9.07107	−0.526357
$298$	−10.6426	−0.616511
$299$	4.53717	0.262391
$300$	−0.748457	−0.0432122
$301$	5.54239	0.319458
$302$	−10.6052	−0.610258
$303$	−0.219623	−0.0126170
$304$	−0.371373	−0.0212997
$305$	5.68755	0.325668
$306$	8.60092	0.491682
$307$	−8.89821	−0.507848	−0.253924	−	0.967224i	$-0.581721\pi$
$307$	−8.89821	−0.507848	−0.253924	+	0.967224i	$0.581721\pi$
$308$	−10.2264	−0.582705
$309$	−10.1910	−0.579744
$310$	−1.00000	−0.0567962
$311$	24.9009	1.41200	0.706001	−	0.708211i	$-0.250497\pi$
$311$	24.9009	1.41200	0.706001	+	0.708211i	$0.250497\pi$
$312$	−0.748457	−0.0423730
$313$	−14.8308	−0.838287	−0.419144	−	0.907920i	$-0.637670\pi$
$313$	−14.8308	−0.838287	−0.419144	+	0.907920i	$0.637670\pi$
$314$	−15.1867	−0.857037
$315$	−11.1988	−0.630982
$316$	6.97071	0.392133
$317$	14.6519	0.822935	0.411467	−	0.911424i	$-0.365016\pi$
$317$	14.6519	0.822935	0.411467	+	0.911424i	$0.365016\pi$
$318$	−10.1510	−0.569241
$319$	−8.87679	−0.497005
$320$	1.00000	0.0559017
$321$	3.35243	0.187114
$322$	20.8258	1.16057
$323$	1.30918	0.0728447
$324$	4.27212	0.237340
$325$	1.00000	0.0554700
$326$	−9.80160	−0.542860
$327$	−6.45499	−0.356962
$328$	6.05311	0.334227
$329$	−14.2888	−0.787767
$330$	1.66753	0.0917947
$331$	6.28024	0.345193	0.172597	−	0.984993i	$-0.444784\pi$
$331$	6.28024	0.345193	0.172597	+	0.984993i	$0.444784\pi$
$332$	−13.6130	−0.747111
$333$	−5.68301	−0.311427
$334$	14.0417	0.768328
$335$	−9.75709	−0.533087
$336$	−3.43544	−0.187419
$337$	−20.6968	−1.12743	−0.563714	−	0.825970i	$-0.690628\pi$
$337$	−20.6968	−1.12743	−0.563714	+	0.825970i	$0.690628\pi$
$338$	1.00000	0.0543928
$339$	−4.86163	−0.264048
$340$	−3.52524	−0.191183
$341$	2.22796	0.120651
$342$	0.906081	0.0489952
$343$	32.4444	1.75183
$344$	1.20748	0.0651031
$345$	−3.39588	−0.182828
$346$	17.1015	0.919381
$347$	5.91487	0.317527	0.158763	−	0.987317i	$-0.449249\pi$
$347$	5.91487	0.317527	0.158763	+	0.987317i	$0.449249\pi$
$348$	−2.98205	−0.159855
$349$	−18.6387	−0.997707	−0.498853	−	0.866686i	$-0.666245\pi$
$349$	−18.6387	−0.997707	−0.498853	+	0.866686i	$0.666245\pi$
$350$	4.59004	0.245348
$351$	4.07146	0.217319
$352$	−2.22796	−0.118751
$353$	8.85408	0.471255	0.235628	−	0.971843i	$-0.424285\pi$
$353$	8.85408	0.471255	0.235628	+	0.971843i	$0.424285\pi$
$354$	−7.47910	−0.397509
$355$	9.98191	0.529785
$356$	−8.00674	−0.424356
$357$	12.1108	0.640969
$358$	−1.34014	−0.0708288
$359$	7.21216	0.380643	0.190322	−	0.981722i	$-0.439047\pi$
$359$	7.21216	0.380643	0.190322	+	0.981722i	$0.439047\pi$
$360$	−2.43981	−0.128589
$361$	−18.8621	−0.992741
$362$	2.21505	0.116421
$363$	4.51782	0.237124
$364$	4.59004	0.240583
$365$	−11.4843	−0.601117
$366$	−4.25689	−0.222511
$367$	−3.09747	−0.161687	−0.0808433	−	0.996727i	$-0.525761\pi$
$367$	−3.09747	−0.161687	−0.0808433	+	0.996727i	$0.525761\pi$
$368$	4.53717	0.236516
$369$	−14.7684	−0.768815
$370$	2.32928	0.121094
$371$	62.2529	3.23201
$372$	0.748457	0.0388057
$373$	30.1133	1.55921	0.779603	−	0.626274i	$-0.215421\pi$
$373$	30.1133	1.55921	0.779603	+	0.626274i	$0.215421\pi$
$374$	7.85410	0.406126
$375$	−0.748457	−0.0386501
$376$	−3.11300	−0.160541
$377$	3.98426	0.205200
$378$	18.6882	0.961216
$379$	23.7167	1.21825	0.609123	−	0.793076i	$-0.291522\pi$
$379$	23.7167	1.21825	0.609123	+	0.793076i	$0.291522\pi$
$380$	−0.371373	−0.0190510
$381$	5.18918	0.265850
$382$	−0.880684	−0.0450597
$383$	−24.2779	−1.24054	−0.620271	−	0.784387i	$-0.712977\pi$
$383$	−24.2779	−1.24054	−0.620271	+	0.784387i	$0.712977\pi$
$384$	−0.748457	−0.0381945
$385$	−10.2264	−0.521187
$386$	3.62453	0.184484
$387$	−2.94603	−0.149755
$388$	−1.68726	−0.0856576
$389$	−30.5758	−1.55025	−0.775126	−	0.631806i	$-0.782314\pi$
$389$	−30.5758	−1.55025	−0.775126	+	0.631806i	$0.782314\pi$
$390$	−0.748457	−0.0378996
$391$	−15.9946	−0.808882
$392$	14.0684	0.710563
$393$	−12.4127	−0.626136
$394$	7.82180	0.394057
$395$	6.97071	0.350734
$396$	5.43581	0.273160
$397$	−1.65554	−0.0830893	−0.0415446	−	0.999137i	$-0.513228\pi$
$397$	−1.65554	−0.0830893	−0.0415446	+	0.999137i	$0.513228\pi$
$398$	−16.3425	−0.819178
$399$	1.27583	0.0638715
$400$	1.00000	0.0500000
$401$	14.4225	0.720226	0.360113	−	0.932909i	$-0.382738\pi$
$401$	14.4225	0.720226	0.360113	+	0.932909i	$0.382738\pi$
$402$	7.30276	0.364229
$403$	−1.00000	−0.0498135
$404$	0.293435	0.0145989
$405$	4.27212	0.212283
$406$	18.2879	0.907614
$407$	−5.18955	−0.257236
$408$	2.63849	0.130625
$409$	11.7521	0.581104	0.290552	−	0.956859i	$-0.406161\pi$
$409$	11.7521	0.581104	0.290552	+	0.956859i	$0.406161\pi$
$410$	6.05311	0.298942
$411$	−6.16402	−0.304049
$412$	13.6160	0.670810
$413$	45.8668	2.25696
$414$	−11.0698	−0.544053
$415$	−13.6130	−0.668237
$416$	1.00000	0.0490290
$417$	16.3483	0.800579
$418$	0.827405	0.0404697
$419$	−20.9600	−1.02396	−0.511980	−	0.858997i	$-0.671088\pi$
$419$	−20.9600	−1.02396	−0.511980	+	0.858997i	$0.671088\pi$
$420$	−3.43544	−0.167633
$421$	−28.5335	−1.39064	−0.695319	−	0.718702i	$-0.744737\pi$
$421$	−28.5335	−1.39064	−0.695319	+	0.718702i	$0.744737\pi$
$422$	17.9147	0.872074
$423$	7.59515	0.369289
$424$	13.5626	0.658659
$425$	−3.52524	−0.170999
$426$	−7.47103	−0.361973
$427$	26.1061	1.26336
$428$	−4.47912	−0.216507
$429$	1.66753	0.0805093
$430$	1.20748	0.0582300
$431$	24.8710	1.19799	0.598996	−	0.800752i	$-0.295567\pi$
$431$	24.8710	1.19799	0.598996	+	0.800752i	$0.295567\pi$
$432$	4.07146	0.195888
$433$	11.0528	0.531163	0.265581	−	0.964088i	$-0.414436\pi$
$433$	11.0528	0.531163	0.265581	+	0.964088i	$0.414436\pi$
$434$	−4.59004	−0.220329
$435$	−2.98205	−0.142978
$436$	8.62440	0.413034
$437$	−1.68498	−0.0806037
$438$	8.59552	0.410710
$439$	−10.3481	−0.493889	−0.246945	−	0.969030i	$-0.579427\pi$
$439$	−10.3481	−0.493889	−0.246945	+	0.969030i	$0.579427\pi$
$440$	−2.22796	−0.106214
$441$	−34.3243	−1.63449
$442$	−3.52524	−0.167678
$443$	−15.2443	−0.724281	−0.362140	−	0.932124i	$-0.617954\pi$
$443$	−15.2443	−0.724281	−0.362140	+	0.932124i	$0.617954\pi$
$444$	−1.74337	−0.0827365
$445$	−8.00674	−0.379556
$446$	11.4277	0.541119
$447$	7.96555	0.376758
$448$	4.59004	0.216859
$449$	−13.6284	−0.643165	−0.321583	−	0.946881i	$-0.604215\pi$
$449$	−13.6284	−0.643165	−0.321583	+	0.946881i	$0.604215\pi$
$450$	−2.43981	−0.115014
$451$	−13.4861	−0.635035
$452$	6.49554	0.305525
$453$	7.93750	0.372936
$454$	−5.92640	−0.278140
$455$	4.59004	0.215184
$456$	0.277957	0.0130165
$457$	−7.81492	−0.365567	−0.182783	−	0.983153i	$-0.558511\pi$
$457$	−7.81492	−0.365567	−0.182783	+	0.983153i	$0.558511\pi$
$458$	−11.2993	−0.527982
$459$	−14.3529	−0.669935
$460$	4.53717	0.211547
$461$	−18.7847	−0.874892	−0.437446	−	0.899245i	$-0.644117\pi$
$461$	−18.7847	−0.874892	−0.437446	+	0.899245i	$0.644117\pi$
$462$	7.65404	0.356098
$463$	−24.8780	−1.15618	−0.578090	−	0.815973i	$-0.696202\pi$
$463$	−24.8780	−1.15618	−0.578090	+	0.815973i	$0.696202\pi$
$464$	3.98426	0.184965
$465$	0.748457	0.0347089
$466$	5.58511	0.258725
$467$	12.4676	0.576932	0.288466	−	0.957490i	$-0.406855\pi$
$467$	12.4676	0.576932	0.288466	+	0.957490i	$0.406855\pi$
$468$	−2.43981	−0.112780
$469$	−44.7854	−2.06800
$470$	−3.11300	−0.143592
$471$	11.3666	0.523746
$472$	9.99269	0.459951
$473$	−2.69022	−0.123697
$474$	−5.21727	−0.239637
$475$	−0.371373	−0.0170398
$476$	−16.1810	−0.741653
$477$	−33.0902	−1.51510
$478$	1.73687	0.0794427
$479$	17.4551	0.797543	0.398771	−	0.917050i	$-0.369437\pi$
$479$	17.4551	0.797543	0.398771	+	0.917050i	$0.369437\pi$
$480$	−0.748457	−0.0341622
$481$	2.32928	0.106206
$482$	1.53690	0.0700039
$483$	−15.5872	−0.709242
$484$	−6.03619	−0.274372
$485$	−1.68726	−0.0766144
$486$	−15.4119	−0.699098
$487$	−39.8230	−1.80455	−0.902277	−	0.431156i	$-0.858106\pi$
$487$	−39.8230	−1.80455	−0.902277	+	0.431156i	$0.858106\pi$
$488$	5.68755	0.257463
$489$	7.33607	0.331749
$490$	14.0684	0.635547
$491$	8.68419	0.391912	0.195956	−	0.980613i	$-0.437219\pi$
$491$	8.68419	0.391912	0.195956	+	0.980613i	$0.437219\pi$
$492$	−4.53049	−0.204250
$493$	−14.0455	−0.632576
$494$	−0.371373	−0.0167089
$495$	5.43581	0.244321
$496$	−1.00000	−0.0449013
$497$	45.8173	2.05519
$498$	10.1888	0.456569
$499$	9.60305	0.429892	0.214946	−	0.976626i	$-0.431043\pi$
$499$	9.60305	0.429892	0.214946	+	0.976626i	$0.431043\pi$
$500$	1.00000	0.0447214
$501$	−10.5096	−0.469535
$502$	19.0083	0.848384
$503$	−24.9903	−1.11426	−0.557131	−	0.830425i	$-0.688098\pi$
$503$	−24.9903	−1.11426	−0.557131	+	0.830425i	$0.688098\pi$
$504$	−11.1988	−0.498835
$505$	0.293435	0.0130577
$506$	−10.1086	−0.449384
$507$	−0.748457	−0.0332401
$508$	−6.93317	−0.307610
$509$	25.8204	1.14447	0.572234	−	0.820090i	$-0.306077\pi$
$509$	25.8204	1.14447	0.572234	+	0.820090i	$0.306077\pi$
$510$	2.63849	0.116834
$511$	−52.7134	−2.33191
$512$	1.00000	0.0441942
$513$	−1.51203	−0.0667579
$514$	10.7678	0.474946
$515$	13.6160	0.599991
$516$	−0.903748	−0.0397853
$517$	6.93565	0.305030
$518$	10.6915	0.469757
$519$	−12.7997	−0.561845
$520$	1.00000	0.0438529
$521$	−4.80582	−0.210547	−0.105273	−	0.994443i	$-0.533572\pi$
$521$	−4.80582	−0.210547	−0.105273	+	0.994443i	$0.533572\pi$
$522$	−9.72085	−0.425470
$523$	34.6045	1.51315	0.756574	−	0.653908i	$-0.226871\pi$
$523$	34.6045	1.51315	0.756574	+	0.653908i	$0.226871\pi$
$524$	16.5844	0.724491
$525$	−3.43544	−0.149935
$526$	−1.21715	−0.0530702
$527$	3.52524	0.153562
$528$	1.66753	0.0725701
$529$	−2.41410	−0.104961
$530$	13.5626	0.589122
$531$	−24.3803	−1.05801
$532$	−1.70462	−0.0739045
$533$	6.05311	0.262189
$534$	5.99270	0.259329
$535$	−4.47912	−0.193649
$536$	−9.75709	−0.421442
$537$	1.00304	0.0432843
$538$	−16.5409	−0.713131
$539$	−31.3439	−1.35008
$540$	4.07146	0.175208
$541$	1.83799	0.0790212	0.0395106	−	0.999219i	$-0.487420\pi$
$541$	1.83799	0.0790212	0.0395106	+	0.999219i	$0.487420\pi$
$542$	−5.60514	−0.240762
$543$	−1.65787	−0.0711460
$544$	−3.52524	−0.151143
$545$	8.62440	0.369429
$546$	−3.43544	−0.147023
$547$	36.2299	1.54908	0.774538	−	0.632527i	$-0.217982\pi$
$547$	36.2299	1.54908	0.774538	+	0.632527i	$0.217982\pi$
$548$	8.23564	0.351809
$549$	−13.8766	−0.592237
$550$	−2.22796	−0.0950006
$551$	−1.47965	−0.0630352
$552$	−3.39588	−0.144538
$553$	31.9958	1.36060
$554$	−16.1894	−0.687821
$555$	−1.74337	−0.0740018
$556$	−21.8427	−0.926335
$557$	−26.6497	−1.12918	−0.564592	−	0.825370i	$-0.690966\pi$
$557$	−26.6497	−1.12918	−0.564592	+	0.825370i	$0.690966\pi$
$558$	2.43981	0.103285
$559$	1.20748	0.0510711
$560$	4.59004	0.193964
$561$	−5.87845	−0.248188
$562$	−23.8583	−1.00640
$563$	12.7088	0.535611	0.267806	−	0.963473i	$-0.413701\pi$
$563$	12.7088	0.535611	0.267806	+	0.963473i	$0.413701\pi$
$564$	2.32995	0.0981086
$565$	6.49554	0.273270
$566$	12.6373	0.531185
$567$	19.6092	0.823509
$568$	9.98191	0.418832
$569$	0.650347	0.0272640	0.0136320	−	0.999907i	$-0.495661\pi$
$569$	0.650347	0.0272640	0.0136320	+	0.999907i	$0.495661\pi$
$570$	0.277957	0.0116423
$571$	−3.41866	−0.143066	−0.0715332	−	0.997438i	$-0.522789\pi$
$571$	−3.41866	−0.143066	−0.0715332	+	0.997438i	$0.522789\pi$
$572$	−2.22796	−0.0931558
$573$	0.659154	0.0275366
$574$	27.7840	1.15968
$575$	4.53717	0.189213
$576$	−2.43981	−0.101659
$577$	−39.2770	−1.63512	−0.817560	−	0.575843i	$-0.804674\pi$
$577$	−39.2770	−1.63512	−0.817560	+	0.575843i	$0.804674\pi$
$578$	−4.57270	−0.190199
$579$	−2.71280	−0.112740
$580$	3.98426	0.165438
$581$	−62.4843	−2.59228
$582$	1.26284	0.0523464
$583$	−30.2170	−1.25146
$584$	−11.4843	−0.475224
$585$	−2.43981	−0.100874
$586$	5.31797	0.219683
$587$	−35.5987	−1.46932	−0.734658	−	0.678437i	$-0.762658\pi$
$587$	−35.5987	−1.46932	−0.734658	+	0.678437i	$0.762658\pi$
$588$	−10.5296	−0.434234
$589$	0.371373	0.0153022
$590$	9.99269	0.411392
$591$	−5.85428	−0.240813
$592$	2.32928	0.0957329
$593$	−1.42320	−0.0584438	−0.0292219	−	0.999573i	$-0.509303\pi$
$593$	−1.42320	−0.0584438	−0.0292219	+	0.999573i	$0.509303\pi$
$594$	−9.07107	−0.372190
$595$	−16.1810	−0.663355
$596$	−10.6426	−0.435939
$597$	12.2317	0.500610
$598$	4.53717	0.185539
$599$	−29.9870	−1.22523	−0.612617	−	0.790380i	$-0.709883\pi$
$599$	−29.9870	−1.22523	−0.612617	+	0.790380i	$0.709883\pi$
$600$	−0.748457	−0.0305556
$601$	−27.7258	−1.13096	−0.565480	−	0.824762i	$-0.691309\pi$
$601$	−27.7258	−1.13096	−0.565480	+	0.824762i	$0.691309\pi$
$602$	5.54239	0.225891
$603$	23.8055	0.969434
$604$	−10.6052	−0.431518
$605$	−6.03619	−0.245406
$606$	−0.219623	−0.00892159
$607$	−38.0903	−1.54604	−0.773018	−	0.634384i	$-0.781254\pi$
$607$	−38.0903	−1.54604	−0.773018	+	0.634384i	$0.781254\pi$
$608$	−0.371373	−0.0150612
$609$	−13.6877	−0.554654
$610$	5.68755	0.230282
$611$	−3.11300	−0.125939
$612$	8.60092	0.347672
$613$	−23.2562	−0.939311	−0.469655	−	0.882850i	$-0.655622\pi$
$613$	−23.2562	−0.939311	−0.469655	+	0.882850i	$0.655622\pi$
$614$	−8.89821	−0.359103
$615$	−4.53049	−0.182687
$616$	−10.2264	−0.412034
$617$	28.7014	1.15547	0.577737	−	0.816223i	$-0.303936\pi$
$617$	28.7014	1.15547	0.577737	+	0.816223i	$0.303936\pi$
$618$	−10.1910	−0.409941
$619$	29.3850	1.18108	0.590542	−	0.807007i	$-0.298914\pi$
$619$	29.3850	1.18108	0.590542	+	0.807007i	$0.298914\pi$
$620$	−1.00000	−0.0401610
$621$	18.4729	0.741293
$622$	24.9009	0.998436
$623$	−36.7512	−1.47241
$624$	−0.748457	−0.0299623
$625$	1.00000	0.0400000
$626$	−14.8308	−0.592759
$627$	−0.619277	−0.0247316
$628$	−15.1867	−0.606017
$629$	−8.21127	−0.327405
$630$	−11.1988	−0.446172
$631$	−23.4233	−0.932467	−0.466234	−	0.884662i	$-0.654389\pi$
$631$	−23.4233	−0.932467	−0.466234	+	0.884662i	$0.654389\pi$
$632$	6.97071	0.277280
$633$	−13.4084	−0.532935
$634$	14.6519	0.581903
$635$	−6.93317	−0.275134
$636$	−10.1510	−0.402514
$637$	14.0684	0.557412
$638$	−8.87679	−0.351435
$639$	−24.3540	−0.963429
$640$	1.00000	0.0395285
$641$	−46.8042	−1.84865	−0.924327	−	0.381600i	$-0.875373\pi$
$641$	−46.8042	−1.84865	−0.924327	+	0.381600i	$0.875373\pi$
$642$	3.35243	0.132310
$643$	−32.6679	−1.28830	−0.644149	−	0.764900i	$-0.722788\pi$
$643$	−32.6679	−1.28830	−0.644149	+	0.764900i	$0.722788\pi$
$644$	20.8258	0.820650
$645$	−0.903748	−0.0355851
$646$	1.30918	0.0515090
$647$	27.3030	1.07339	0.536696	−	0.843776i	$-0.319672\pi$
$647$	27.3030	1.07339	0.536696	+	0.843776i	$0.319672\pi$
$648$	4.27212	0.167825
$649$	−22.2633	−0.873912
$650$	1.00000	0.0392232
$651$	3.43544	0.134646
$652$	−9.80160	−0.383860
$653$	−11.2383	−0.439790	−0.219895	−	0.975524i	$-0.570572\pi$
$653$	−11.2383	−0.439790	−0.219895	+	0.975524i	$0.570572\pi$
$654$	−6.45499	−0.252410
$655$	16.5844	0.648004
$656$	6.05311	0.236334
$657$	28.0196	1.09315
$658$	−14.2888	−0.557036
$659$	−2.81261	−0.109564	−0.0547819	−	0.998498i	$-0.517446\pi$
$659$	−2.81261	−0.109564	−0.0547819	+	0.998498i	$0.517446\pi$
$660$	1.66753	0.0649087
$661$	−15.8004	−0.614566	−0.307283	−	0.951618i	$-0.599420\pi$
$661$	−15.8004	−0.614566	−0.307283	+	0.951618i	$0.599420\pi$
$662$	6.28024	0.244089
$663$	2.63849	0.102470
$664$	−13.6130	−0.528288
$665$	−1.70462	−0.0661022
$666$	−5.68301	−0.220212
$667$	18.0773	0.699955
$668$	14.0417	0.543290
$669$	−8.55317	−0.330685
$670$	−9.75709	−0.376949
$671$	−12.6716	−0.489183
$672$	−3.43544	−0.132525
$673$	−1.62729	−0.0627275	−0.0313637	−	0.999508i	$-0.509985\pi$
$673$	−1.62729	−0.0627275	−0.0313637	+	0.999508i	$0.509985\pi$
$674$	−20.6968	−0.797212
$675$	4.07146	0.156711
$676$	1.00000	0.0384615
$677$	−36.3245	−1.39606	−0.698032	−	0.716067i	$-0.745941\pi$
$677$	−36.3245	−1.39606	−0.698032	+	0.716067i	$0.745941\pi$
$678$	−4.86163	−0.186710
$679$	−7.74458	−0.297210
$680$	−3.52524	−0.135187
$681$	4.43565	0.169975
$682$	2.22796	0.0853131
$683$	19.9259	0.762442	0.381221	−	0.924484i	$-0.375504\pi$
$683$	19.9259	0.762442	0.381221	+	0.924484i	$0.375504\pi$
$684$	0.906081	0.0346449
$685$	8.23564	0.314668
$686$	32.4444	1.23873
$687$	8.45705	0.322657
$688$	1.20748	0.0460348
$689$	13.5626	0.516694
$690$	−3.39588	−0.129279
$691$	44.5720	1.69560	0.847799	−	0.530318i	$-0.177928\pi$
$691$	44.5720	1.69560	0.847799	+	0.530318i	$0.177928\pi$
$692$	17.1015	0.650101
$693$	24.9506	0.947793
$694$	5.91487	0.224525
$695$	−21.8427	−0.828539
$696$	−2.98205	−0.113034
$697$	−21.3386	−0.808259
$698$	−18.6387	−0.705485
$699$	−4.18021	−0.158110
$700$	4.59004	0.173487
$701$	−31.2843	−1.18159	−0.590797	−	0.806820i	$-0.701186\pi$
$701$	−31.2843	−1.18159	−0.590797	+	0.806820i	$0.701186\pi$
$702$	4.07146	0.153668
$703$	−0.865033	−0.0326253
$704$	−2.22796	−0.0839695
$705$	2.32995	0.0877510
$706$	8.85408	0.333228
$707$	1.34688	0.0506545
$708$	−7.47910	−0.281082
$709$	−24.2610	−0.911139	−0.455570	−	0.890200i	$-0.650564\pi$
$709$	−24.2610	−0.911139	−0.455570	+	0.890200i	$0.650564\pi$
$710$	9.98191	0.374614
$711$	−17.0072	−0.637821
$712$	−8.00674	−0.300065
$713$	−4.53717	−0.169918
$714$	12.1108	0.453234
$715$	−2.22796	−0.0833211
$716$	−1.34014	−0.0500835
$717$	−1.29997	−0.0485484
$718$	7.21216	0.269155
$719$	−45.3139	−1.68992	−0.844962	−	0.534826i	$-0.820377\pi$
$719$	−45.3139	−1.68992	−0.844962	+	0.534826i	$0.820377\pi$
$720$	−2.43981	−0.0909264
$721$	62.4978	2.32754
$722$	−18.8621	−0.701974
$723$	−1.15030	−0.0427803
$724$	2.21505	0.0823217
$725$	3.98426	0.147972
$726$	4.51782	0.167672
$727$	−6.10103	−0.226275	−0.113137	−	0.993579i	$-0.536090\pi$
$727$	−6.10103	−0.226275	−0.113137	+	0.993579i	$0.536090\pi$
$728$	4.59004	0.170118
$729$	−1.28123	−0.0474528
$730$	−11.4843	−0.425054
$731$	−4.25666	−0.157438
$732$	−4.25689	−0.157339
$733$	39.6578	1.46479	0.732397	−	0.680877i	$-0.238401\pi$
$733$	39.6578	1.46479	0.732397	+	0.680877i	$0.238401\pi$
$734$	−3.09747	−0.114330
$735$	−10.5296	−0.388391
$736$	4.53717	0.167242
$737$	21.7384	0.800745
$738$	−14.7684	−0.543634
$739$	−4.68469	−0.172329	−0.0861646	−	0.996281i	$-0.527461\pi$
$739$	−4.68469	−0.172329	−0.0861646	+	0.996281i	$0.527461\pi$
$740$	2.32928	0.0856261
$741$	0.277957	0.0102110
$742$	62.2529	2.28537
$743$	−22.6511	−0.830990	−0.415495	−	0.909595i	$-0.636392\pi$
$743$	−22.6511	−0.830990	−0.415495	+	0.909595i	$0.636392\pi$
$744$	0.748457	0.0274398
$745$	−10.6426	−0.389916
$746$	30.1133	1.10253
$747$	33.2132	1.21521
$748$	7.85410	0.287174
$749$	−20.5593	−0.751222
$750$	−0.748457	−0.0273298
$751$	1.19792	0.0437127	0.0218564	−	0.999761i	$-0.493042\pi$
$751$	1.19792	0.0437127	0.0218564	+	0.999761i	$0.493042\pi$
$752$	−3.11300	−0.113520
$753$	−14.2269	−0.518458
$754$	3.98426	0.145098
$755$	−10.6052	−0.385961
$756$	18.6882	0.679682
$757$	26.4252	0.960442	0.480221	−	0.877148i	$-0.340557\pi$
$757$	26.4252	0.960442	0.480221	+	0.877148i	$0.340557\pi$
$758$	23.7167	0.861430
$759$	7.56588	0.274624
$760$	−0.371373	−0.0134711
$761$	−34.7446	−1.25949	−0.629745	−	0.776802i	$-0.716841\pi$
$761$	−34.7446	−1.25949	−0.629745	+	0.776802i	$0.716841\pi$
$762$	5.18918	0.187984
$763$	39.5863	1.43312
$764$	−0.880684	−0.0318620
$765$	8.60092	0.310967
$766$	−24.2779	−0.877196
$767$	9.99269	0.360815
$768$	−0.748457	−0.0270076
$769$	−30.7095	−1.10741	−0.553707	−	0.832712i	$-0.686787\pi$
$769$	−30.7095	−1.10741	−0.553707	+	0.832712i	$0.686787\pi$
$770$	−10.2264	−0.368535
$771$	−8.05920	−0.290245
$772$	3.62453	0.130450
$773$	−18.6275	−0.669984	−0.334992	−	0.942221i	$-0.608734\pi$
$773$	−18.6275	−0.669984	−0.334992	+	0.942221i	$0.608734\pi$
$774$	−2.94603	−0.105893
$775$	−1.00000	−0.0359211
$776$	−1.68726	−0.0605690
$777$	−8.00212	−0.287074
$778$	−30.5758	−1.09619
$779$	−2.24796	−0.0805416
$780$	−0.748457	−0.0267991
$781$	−22.2393	−0.795785
$782$	−15.9946	−0.571966
$783$	16.2218	0.579719
$784$	14.0684	0.502444
$785$	−15.1867	−0.542038
$786$	−12.4127	−0.442745
$787$	10.3997	0.370708	0.185354	−	0.982672i	$-0.440657\pi$
$787$	10.3997	0.370708	0.185354	+	0.982672i	$0.440657\pi$
$788$	7.82180	0.278640
$789$	0.910983	0.0324319
$790$	6.97071	0.248007
$791$	29.8148	1.06009
$792$	5.43581	0.193153
$793$	5.68755	0.201971
$794$	−1.65554	−0.0587530
$795$	−10.1510	−0.360020
$796$	−16.3425	−0.579246
$797$	−0.351550	−0.0124525	−0.00622627	−	0.999981i	$-0.501982\pi$
$797$	−0.351550	−0.0124525	−0.00622627	+	0.999981i	$0.501982\pi$
$798$	1.27583	0.0451640
$799$	10.9741	0.388235
$800$	1.00000	0.0353553
$801$	19.5349	0.690233
$802$	14.4225	0.509277
$803$	25.5866	0.902932
$804$	7.30276	0.257549
$805$	20.8258	0.734012
$806$	−1.00000	−0.0352235
$807$	12.3802	0.435803
$808$	0.293435	0.0103230
$809$	22.0770	0.776186	0.388093	−	0.921620i	$-0.373134\pi$
$809$	22.0770	0.776186	0.388093	+	0.921620i	$0.373134\pi$
$810$	4.27212	0.150107
$811$	37.4020	1.31336	0.656681	−	0.754169i	$-0.271960\pi$
$811$	37.4020	1.31336	0.656681	+	0.754169i	$0.271960\pi$
$812$	18.2879	0.641780
$813$	4.19521	0.147132
$814$	−5.18955	−0.181894
$815$	−9.80160	−0.343335
$816$	2.63849	0.0923656
$817$	−0.448427	−0.0156885
$818$	11.7521	0.410902
$819$	−11.1988	−0.391319
$820$	6.05311	0.211384
$821$	−3.70244	−0.129216	−0.0646081	−	0.997911i	$-0.520580\pi$
$821$	−3.70244	−0.129216	−0.0646081	+	0.997911i	$0.520580\pi$
$822$	−6.16402	−0.214995
$823$	23.8720	0.832125	0.416062	−	0.909336i	$-0.363410\pi$
$823$	23.8720	0.832125	0.416062	+	0.909336i	$0.363410\pi$
$824$	13.6160	0.474335
$825$	1.66753	0.0580561
$826$	45.8668	1.59591
$827$	38.0667	1.32371	0.661854	−	0.749633i	$-0.269770\pi$
$827$	38.0667	1.32371	0.661854	+	0.749633i	$0.269770\pi$
$828$	−11.0698	−0.384704
$829$	−17.2784	−0.600104	−0.300052	−	0.953923i	$-0.597004\pi$
$829$	−17.2784	−0.600104	−0.300052	+	0.953923i	$0.597004\pi$
$830$	−13.6130	−0.472515
$831$	12.1171	0.420336
$832$	1.00000	0.0346688
$833$	−49.5946	−1.71835
$834$	16.3483	0.566095
$835$	14.0417	0.485933
$836$	0.827405	0.0286164
$837$	−4.07146	−0.140730
$838$	−20.9600	−0.724050
$839$	−41.0346	−1.41667	−0.708336	−	0.705875i	$-0.750554\pi$
$839$	−41.0346	−1.41667	−0.708336	+	0.705875i	$0.750554\pi$
$840$	−3.43544	−0.118534
$841$	−13.1256	−0.452609
$842$	−28.5335	−0.983329
$843$	17.8569	0.615025
$844$	17.9147	0.616649
$845$	1.00000	0.0344010
$846$	7.59515	0.261126
$847$	−27.7063	−0.952000
$848$	13.5626	0.465742
$849$	−9.45847	−0.324614
$850$	−3.52524	−0.120915
$851$	10.5683	0.362278
$852$	−7.47103	−0.255953
$853$	6.35266	0.217511	0.108755	−	0.994069i	$-0.465313\pi$
$853$	6.35266	0.217511	0.108755	+	0.994069i	$0.465313\pi$
$854$	26.1061	0.893331
$855$	0.906081	0.0309873
$856$	−4.47912	−0.153093
$857$	−51.1485	−1.74720	−0.873600	−	0.486644i	$-0.838221\pi$
$857$	−51.1485	−1.74720	−0.873600	+	0.486644i	$0.838221\pi$
$858$	1.66753	0.0569287
$859$	−52.2973	−1.78436	−0.892180	−	0.451680i	$-0.850825\pi$
$859$	−52.2973	−1.78436	−0.892180	+	0.451680i	$0.850825\pi$
$860$	1.20748	0.0411748
$861$	−20.7951	−0.708696
$862$	24.8710	0.847108
$863$	22.6187	0.769951	0.384976	−	0.922927i	$-0.374210\pi$
$863$	22.6187	0.769951	0.384976	+	0.922927i	$0.374210\pi$
$864$	4.07146	0.138514
$865$	17.1015	0.581468
$866$	11.0528	0.375589
$867$	3.42247	0.116233
$868$	−4.59004	−0.155796
$869$	−15.5305	−0.526835
$870$	−2.98205	−0.101101
$871$	−9.75709	−0.330606
$872$	8.62440	0.292059
$873$	4.11659	0.139326
$874$	−1.68498	−0.0569954
$875$	4.59004	0.155172
$876$	8.59552	0.290416
$877$	12.2136	0.412425	0.206212	−	0.978507i	$-0.433886\pi$
$877$	12.2136	0.412425	0.206212	+	0.978507i	$0.433886\pi$
$878$	−10.3481	−0.349232
$879$	−3.98027	−0.134251
$880$	−2.22796	−0.0751046
$881$	16.0047	0.539213	0.269606	−	0.962971i	$-0.413106\pi$
$881$	16.0047	0.539213	0.269606	+	0.962971i	$0.413106\pi$
$882$	−34.3243	−1.15576
$883$	−4.37834	−0.147343	−0.0736714	−	0.997283i	$-0.523472\pi$
$883$	−4.37834	−0.147343	−0.0736714	+	0.997283i	$0.523472\pi$
$884$	−3.52524	−0.118567
$885$	−7.47910	−0.251407
$886$	−15.2443	−0.512144
$887$	25.4300	0.853857	0.426929	−	0.904285i	$-0.359596\pi$
$887$	25.4300	0.853857	0.426929	+	0.904285i	$0.359596\pi$
$888$	−1.74337	−0.0585035
$889$	−31.8235	−1.06733
$890$	−8.00674	−0.268386
$891$	−9.51812	−0.318869
$892$	11.4277	0.382629
$893$	1.15609	0.0386870
$894$	7.96555	0.266408
$895$	−1.34014	−0.0447960
$896$	4.59004	0.153342
$897$	−3.39588	−0.113385
$898$	−13.6284	−0.454787
$899$	−3.98426	−0.132883
$900$	−2.43981	−0.0813271
$901$	−47.8114	−1.59283
$902$	−13.4861	−0.449038
$903$	−4.14824	−0.138045
$904$	6.49554	0.216039
$905$	2.21505	0.0736308
$906$	7.93750	0.263706
$907$	3.56461	0.118361	0.0591805	−	0.998247i	$-0.481151\pi$
$907$	3.56461	0.118361	0.0591805	+	0.998247i	$0.481151\pi$
$908$	−5.92640	−0.196674
$909$	−0.715926	−0.0237458
$910$	4.59004	0.152158
$911$	11.6929	0.387404	0.193702	−	0.981060i	$-0.437950\pi$
$911$	11.6929	0.387404	0.193702	+	0.981060i	$0.437950\pi$
$912$	0.277957	0.00920407
$913$	30.3293	1.00375
$914$	−7.81492	−0.258495
$915$	−4.25689	−0.140728
$916$	−11.2993	−0.373340
$917$	76.1228	2.51380
$918$	−14.3529	−0.473716
$919$	24.6073	0.811718	0.405859	−	0.913936i	$-0.366972\pi$
$919$	24.6073	0.811718	0.405859	+	0.913936i	$0.366972\pi$
$920$	4.53717	0.149586
$921$	6.65993	0.219452
$922$	−18.7847	−0.618642
$923$	9.98191	0.328559
$924$	7.65404	0.251799
$925$	2.32928	0.0765863
$926$	−24.8780	−0.817542
$927$	−33.2204	−1.09110
$928$	3.98426	0.130790
$929$	13.4896	0.442581	0.221290	−	0.975208i	$-0.428973\pi$
$929$	13.4896	0.442581	0.221290	+	0.975208i	$0.428973\pi$
$930$	0.748457	0.0245429
$931$	−5.22464	−0.171231
$932$	5.58511	0.182946
$933$	−18.6373	−0.610157
$934$	12.4676	0.407952
$935$	7.85410	0.256856
$936$	−2.43981	−0.0797478
$937$	53.0082	1.73170	0.865851	−	0.500302i	$-0.166778\pi$
$937$	53.0082	1.73170	0.865851	+	0.500302i	$0.166778\pi$
$938$	−44.7854	−1.46230
$939$	11.1002	0.362242
$940$	−3.11300	−0.101535
$941$	−0.940849	−0.0306708	−0.0153354	−	0.999882i	$-0.504882\pi$
$941$	−0.940849	−0.0306708	−0.0153354	+	0.999882i	$0.504882\pi$
$942$	11.3666	0.370344
$943$	27.4640	0.894350
$944$	9.99269	0.325234
$945$	18.6882	0.607926
$946$	−2.69022	−0.0874667
$947$	38.7540	1.25933	0.629667	−	0.776865i	$-0.283191\pi$
$947$	38.7540	1.25933	0.629667	+	0.776865i	$0.283191\pi$
$948$	−5.21727	−0.169449
$949$	−11.4843	−0.372797
$950$	−0.371373	−0.0120489
$951$	−10.9663	−0.355608
$952$	−16.1810	−0.524428
$953$	−33.0033	−1.06908	−0.534541	−	0.845143i	$-0.679515\pi$
$953$	−33.0033	−1.06908	−0.534541	+	0.845143i	$0.679515\pi$
$954$	−33.0902	−1.07134
$955$	−0.880684	−0.0284983
$956$	1.73687	0.0561745
$957$	6.64389	0.214767
$958$	17.4551	0.563948
$959$	37.8019	1.22069
$960$	−0.748457	−0.0241563
$961$	1.00000	0.0322581
$962$	2.32928	0.0750990
$963$	10.9282	0.352157
$964$	1.53690	0.0495002
$965$	3.62453	0.116678
$966$	−15.5872	−0.501510
$967$	−0.407423	−0.0131018	−0.00655092	−	0.999979i	$-0.502085\pi$
$967$	−0.407423	−0.0131018	−0.00655092	+	0.999979i	$0.502085\pi$
$968$	−6.03619	−0.194010
$969$	−0.979864	−0.0314778
$970$	−1.68726	−0.0541746
$971$	50.9665	1.63559	0.817797	−	0.575507i	$-0.195195\pi$
$971$	50.9665	1.63559	0.817797	+	0.575507i	$0.195195\pi$
$972$	−15.4119	−0.494337
$973$	−100.259	−3.21414
$974$	−39.8230	−1.27601
$975$	−0.748457	−0.0239698
$976$	5.68755	0.182054
$977$	−2.93496	−0.0938977	−0.0469488	−	0.998897i	$-0.514950\pi$
$977$	−2.93496	−0.0938977	−0.0469488	+	0.998897i	$0.514950\pi$
$978$	7.33607	0.234582
$979$	17.8387	0.570127
$980$	14.0684	0.449400
$981$	−21.0419	−0.671817
$982$	8.68419	0.277123
$983$	−43.5117	−1.38781	−0.693904	−	0.720068i	$-0.744111\pi$
$983$	−43.5117	−1.38781	−0.693904	+	0.720068i	$0.744111\pi$
$984$	−4.53049	−0.144427
$985$	7.82180	0.249223
$986$	−14.0455	−0.447299
$987$	10.6946	0.340411
$988$	−0.371373	−0.0118150
$989$	5.47855	0.174208
$990$	5.43581	0.172761
$991$	36.4023	1.15636	0.578178	−	0.815911i	$-0.303764\pi$
$991$	36.4023	1.15636	0.578178	+	0.815911i	$0.303764\pi$
$992$	−1.00000	−0.0317500
$993$	−4.70049	−0.149166
$994$	45.8173	1.45324
$995$	−16.3425	−0.518094
$996$	10.1888	0.322843
$997$	−48.6017	−1.53923	−0.769616	−	0.638507i	$-0.779552\pi$
$997$	−48.6017	−1.53923	−0.769616	+	0.638507i	$0.779552\pi$
$998$	9.60305	0.303979
$999$	9.48359	0.300047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.r.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.r.1.4	✓	9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.748457 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.748457 q^{6} +4.59004 q^{7} +1.00000 q^{8} -2.43981 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.748457 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.748457 q^{6} +4.59004 q^{7} +1.00000 q^{8} -2.43981 q^{9} +1.00000 q^{10} -2.22796 q^{11} -0.748457 q^{12} +1.00000 q^{13} +4.59004 q^{14} -0.748457 q^{15} +1.00000 q^{16} -3.52524 q^{17} -2.43981 q^{18} -0.371373 q^{19} +1.00000 q^{20} -3.43544 q^{21} -2.22796 q^{22} +4.53717 q^{23} -0.748457 q^{24} +1.00000 q^{25} +1.00000 q^{26} +4.07146 q^{27} +4.59004 q^{28} +3.98426 q^{29} -0.748457 q^{30} -1.00000 q^{31} +1.00000 q^{32} +1.66753 q^{33} -3.52524 q^{34} +4.59004 q^{35} -2.43981 q^{36} +2.32928 q^{37} -0.371373 q^{38} -0.748457 q^{39} +1.00000 q^{40} +6.05311 q^{41} -3.43544 q^{42} +1.20748 q^{43} -2.22796 q^{44} -2.43981 q^{45} +4.53717 q^{46} -3.11300 q^{47} -0.748457 q^{48} +14.0684 q^{49} +1.00000 q^{50} +2.63849 q^{51} +1.00000 q^{52} +13.5626 q^{53} +4.07146 q^{54} -2.22796 q^{55} +4.59004 q^{56} +0.277957 q^{57} +3.98426 q^{58} +9.99269 q^{59} -0.748457 q^{60} +5.68755 q^{61} -1.00000 q^{62} -11.1988 q^{63} +1.00000 q^{64} +1.00000 q^{65} +1.66753 q^{66} -9.75709 q^{67} -3.52524 q^{68} -3.39588 q^{69} +4.59004 q^{70} +9.98191 q^{71} -2.43981 q^{72} -11.4843 q^{73} +2.32928 q^{74} -0.748457 q^{75} -0.371373 q^{76} -10.2264 q^{77} -0.748457 q^{78} +6.97071 q^{79} +1.00000 q^{80} +4.27212 q^{81} +6.05311 q^{82} -13.6130 q^{83} -3.43544 q^{84} -3.52524 q^{85} +1.20748 q^{86} -2.98205 q^{87} -2.22796 q^{88} -8.00674 q^{89} -2.43981 q^{90} +4.59004 q^{91} +4.53717 q^{92} +0.748457 q^{93} -3.11300 q^{94} -0.371373 q^{95} -0.748457 q^{96} -1.68726 q^{97} +14.0684 q^{98} +5.43581 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + 3 q^{3} + 9 q^{4} + 9 q^{5} + 3 q^{6} + 9 q^{7} + 9 q^{8} + 14 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 + 3 * q^3 + 9 * q^4 + 9 * q^5 + 3 * q^6 + 9 * q^7 + 9 * q^8 + 14 * q^9	\( 9 q + 9 q^{2} + 3 q^{3} + 9 q^{4} + 9 q^{5} + 3 q^{6} + 9 q^{7} + 9 q^{8} + 14 q^{9} + 9 q^{10} + 10 q^{11} + 3 q^{12} + 9 q^{13} + 9 q^{14} + 3 q^{15} + 9 q^{16} + q^{17} + 14 q^{18} + 10 q^{19} + 9 q^{20} + 3 q^{21} + 10 q^{22} + 8 q^{23} + 3 q^{24} + 9 q^{25} + 9 q^{26} + 15 q^{27} + 9 q^{28} + 9 q^{29} + 3 q^{30} - 9 q^{31} + 9 q^{32} + 4 q^{33} + q^{34} + 9 q^{35} + 14 q^{36} - 3 q^{37} + 10 q^{38} + 3 q^{39} + 9 q^{40} + 3 q^{42} + 7 q^{43} + 10 q^{44} + 14 q^{45} + 8 q^{46} + 7 q^{47} + 3 q^{48} + 8 q^{49} + 9 q^{50} - 3 q^{51} + 9 q^{52} + 8 q^{53} + 15 q^{54} + 10 q^{55} + 9 q^{56} - 7 q^{57} + 9 q^{58} + 2 q^{59} + 3 q^{60} - 2 q^{61} - 9 q^{62} + 10 q^{63} + 9 q^{64} + 9 q^{65} + 4 q^{66} + 18 q^{67} + q^{68} - 16 q^{69} + 9 q^{70} + 14 q^{71} + 14 q^{72} + q^{73} - 3 q^{74} + 3 q^{75} + 10 q^{76} - 5 q^{77} + 3 q^{78} + 6 q^{79} + 9 q^{80} + q^{81} + 7 q^{83} + 3 q^{84} + q^{85} + 7 q^{86} + 11 q^{87} + 10 q^{88} - 19 q^{89} + 14 q^{90} + 9 q^{91} + 8 q^{92} - 3 q^{93} + 7 q^{94} + 10 q^{95} + 3 q^{96} - 6 q^{97} + 8 q^{98} - 5 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 + 3 * q^3 + 9 * q^4 + 9 * q^5 + 3 * q^6 + 9 * q^7 + 9 * q^8 + 14 * q^9 + 9 * q^10 + 10 * q^11 + 3 * q^12 + 9 * q^13 + 9 * q^14 + 3 * q^15 + 9 * q^16 + q^17 + 14 * q^18 + 10 * q^19 + 9 * q^20 + 3 * q^21 + 10 * q^22 + 8 * q^23 + 3 * q^24 + 9 * q^25 + 9 * q^26 + 15 * q^27 + 9 * q^28 + 9 * q^29 + 3 * q^30 - 9 * q^31 + 9 * q^32 + 4 * q^33 + q^34 + 9 * q^35 + 14 * q^36 - 3 * q^37 + 10 * q^38 + 3 * q^39 + 9 * q^40 + 3 * q^42 + 7 * q^43 + 10 * q^44 + 14 * q^45 + 8 * q^46 + 7 * q^47 + 3 * q^48 + 8 * q^49 + 9 * q^50 - 3 * q^51 + 9 * q^52 + 8 * q^53 + 15 * q^54 + 10 * q^55 + 9 * q^56 - 7 * q^57 + 9 * q^58 + 2 * q^59 + 3 * q^60 - 2 * q^61 - 9 * q^62 + 10 * q^63 + 9 * q^64 + 9 * q^65 + 4 * q^66 + 18 * q^67 + q^68 - 16 * q^69 + 9 * q^70 + 14 * q^71 + 14 * q^72 + q^73 - 3 * q^74 + 3 * q^75 + 10 * q^76 - 5 * q^77 + 3 * q^78 + 6 * q^79 + 9 * q^80 + q^81 + 7 * q^83 + 3 * q^84 + q^85 + 7 * q^86 + 11 * q^87 + 10 * q^88 - 19 * q^89 + 14 * q^90 + 9 * q^91 + 8 * q^92 - 3 * q^93 + 7 * q^94 + 10 * q^95 + 3 * q^96 - 6 * q^97 + 8 * q^98 - 5 * q^99

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