LMFDB - Embedded newform 4730.2.a.x.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4730,2,Mod(1,4730)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4730, 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("4730.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4730 = 2 \cdot 5 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4730.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:	$37.7692401561$
Analytic rank:	$1$
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} - 17x^{6} - 3x^{5} + 86x^{4} + 27x^{3} - 136x^{2} - 24x + 64 $ x^8 - 17x^6 - 3x^5 + 86x^4 + 27x^3 - 136x^2 - 24x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.970143$ of defining polynomial
Character	$\chi$	$=$	4730.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.970143 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.970143 q^{6} -0.827277 q^{7} -1.00000 q^{8} -2.05882 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.970143 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.970143 q^{6} -0.827277 q^{7} -1.00000 q^{8} -2.05882 q^{9} +1.00000 q^{10} -1.00000 q^{11} -0.970143 q^{12} -0.399865 q^{13} +0.827277 q^{14} +0.970143 q^{15} +1.00000 q^{16} -2.38237 q^{17} +2.05882 q^{18} -3.72626 q^{19} -1.00000 q^{20} +0.802578 q^{21} +1.00000 q^{22} +4.45353 q^{23} +0.970143 q^{24} +1.00000 q^{25} +0.399865 q^{26} +4.90778 q^{27} -0.827277 q^{28} +3.28953 q^{29} -0.970143 q^{30} +8.18303 q^{31} -1.00000 q^{32} +0.970143 q^{33} +2.38237 q^{34} +0.827277 q^{35} -2.05882 q^{36} +10.2719 q^{37} +3.72626 q^{38} +0.387926 q^{39} +1.00000 q^{40} +3.28709 q^{41} -0.802578 q^{42} -1.00000 q^{43} -1.00000 q^{44} +2.05882 q^{45} -4.45353 q^{46} +11.2371 q^{47} -0.970143 q^{48} -6.31561 q^{49} -1.00000 q^{50} +2.31124 q^{51} -0.399865 q^{52} -12.0181 q^{53} -4.90778 q^{54} +1.00000 q^{55} +0.827277 q^{56} +3.61501 q^{57} -3.28953 q^{58} +8.41377 q^{59} +0.970143 q^{60} -10.3312 q^{61} -8.18303 q^{62} +1.70322 q^{63} +1.00000 q^{64} +0.399865 q^{65} -0.970143 q^{66} -4.03666 q^{67} -2.38237 q^{68} -4.32056 q^{69} -0.827277 q^{70} -10.9219 q^{71} +2.05882 q^{72} -4.77844 q^{73} -10.2719 q^{74} -0.970143 q^{75} -3.72626 q^{76} +0.827277 q^{77} -0.387926 q^{78} -0.730224 q^{79} -1.00000 q^{80} +1.41521 q^{81} -3.28709 q^{82} +6.10174 q^{83} +0.802578 q^{84} +2.38237 q^{85} +1.00000 q^{86} -3.19131 q^{87} +1.00000 q^{88} +11.4701 q^{89} -2.05882 q^{90} +0.330799 q^{91} +4.45353 q^{92} -7.93872 q^{93} -11.2371 q^{94} +3.72626 q^{95} +0.970143 q^{96} -5.35966 q^{97} +6.31561 q^{98} +2.05882 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} - 5 q^{7} - 8 q^{8} + 10 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 - 5 * q^7 - 8 * q^8 + 10 * q^9	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} - 5 q^{7} - 8 q^{8} + 10 q^{9} + 8 q^{10} - 8 q^{11} - 3 q^{13} + 5 q^{14} + 8 q^{16} + 10 q^{17} - 10 q^{18} - 15 q^{19} - 8 q^{20} - 6 q^{21} + 8 q^{22} + 12 q^{23} + 8 q^{25} + 3 q^{26} + 9 q^{27} - 5 q^{28} - 6 q^{29} - 4 q^{31} - 8 q^{32} - 10 q^{34} + 5 q^{35} + 10 q^{36} - q^{37} + 15 q^{38} - 12 q^{39} + 8 q^{40} + 3 q^{41} + 6 q^{42} - 8 q^{43} - 8 q^{44} - 10 q^{45} - 12 q^{46} + 11 q^{47} + 7 q^{49} - 8 q^{50} - 13 q^{51} - 3 q^{52} + 24 q^{53} - 9 q^{54} + 8 q^{55} + 5 q^{56} - 7 q^{57} + 6 q^{58} - 5 q^{59} - 26 q^{61} + 4 q^{62} + 12 q^{63} + 8 q^{64} + 3 q^{65} - 5 q^{67} + 10 q^{68} - 22 q^{69} - 5 q^{70} + 10 q^{71} - 10 q^{72} - q^{73} + q^{74} - 15 q^{76} + 5 q^{77} + 12 q^{78} - 24 q^{79} - 8 q^{80} - 3 q^{82} - 15 q^{83} - 6 q^{84} - 10 q^{85} + 8 q^{86} + 15 q^{87} + 8 q^{88} - q^{89} + 10 q^{90} - 35 q^{91} + 12 q^{92} + 23 q^{93} - 11 q^{94} + 15 q^{95} - 30 q^{97} - 7 q^{98} - 10 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 - 5 * q^7 - 8 * q^8 + 10 * q^9 + 8 * q^10 - 8 * q^11 - 3 * q^13 + 5 * q^14 + 8 * q^16 + 10 * q^17 - 10 * q^18 - 15 * q^19 - 8 * q^20 - 6 * q^21 + 8 * q^22 + 12 * q^23 + 8 * q^25 + 3 * q^26 + 9 * q^27 - 5 * q^28 - 6 * q^29 - 4 * q^31 - 8 * q^32 - 10 * q^34 + 5 * q^35 + 10 * q^36 - q^37 + 15 * q^38 - 12 * q^39 + 8 * q^40 + 3 * q^41 + 6 * q^42 - 8 * q^43 - 8 * q^44 - 10 * q^45 - 12 * q^46 + 11 * q^47 + 7 * q^49 - 8 * q^50 - 13 * q^51 - 3 * q^52 + 24 * q^53 - 9 * q^54 + 8 * q^55 + 5 * q^56 - 7 * q^57 + 6 * q^58 - 5 * q^59 - 26 * q^61 + 4 * q^62 + 12 * q^63 + 8 * q^64 + 3 * q^65 - 5 * q^67 + 10 * q^68 - 22 * q^69 - 5 * q^70 + 10 * q^71 - 10 * q^72 - q^73 + q^74 - 15 * q^76 + 5 * q^77 + 12 * q^78 - 24 * q^79 - 8 * q^80 - 3 * q^82 - 15 * q^83 - 6 * q^84 - 10 * q^85 + 8 * q^86 + 15 * q^87 + 8 * q^88 - q^89 + 10 * q^90 - 35 * q^91 + 12 * q^92 + 23 * q^93 - 11 * q^94 + 15 * q^95 - 30 * q^97 - 7 * q^98 - 10 * 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
$3$	−0.970143	−0.560113	−0.280056	−	0.959984i	$-0.590353\pi$
$3$	−0.970143	−0.560113	−0.280056	+	0.959984i	$0.590353\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.970143	0.396059
$7$	−0.827277	−0.312681	−0.156341	−	0.987703i	$-0.549970\pi$
$7$	−0.827277	−0.312681	−0.156341	+	0.987703i	$0.549970\pi$
$8$	−1.00000	−0.353553
$9$	−2.05882	−0.686274
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.970143	−0.280056
$13$	−0.399865	−0.110902	−0.0554512	−	0.998461i	$-0.517660\pi$
$13$	−0.399865	−0.110902	−0.0554512	+	0.998461i	$0.517660\pi$
$14$	0.827277	0.221099
$15$	0.970143	0.250490
$16$	1.00000	0.250000
$17$	−2.38237	−0.577809	−0.288905	−	0.957358i	$-0.593291\pi$
$17$	−2.38237	−0.577809	−0.288905	+	0.957358i	$0.593291\pi$
$18$	2.05882	0.485269
$19$	−3.72626	−0.854863	−0.427431	−	0.904048i	$-0.640581\pi$
$19$	−3.72626	−0.854863	−0.427431	+	0.904048i	$0.640581\pi$
$20$	−1.00000	−0.223607
$21$	0.802578	0.175137
$22$	1.00000	0.213201
$23$	4.45353	0.928625	0.464313	−	0.885671i	$-0.346301\pi$
$23$	4.45353	0.928625	0.464313	+	0.885671i	$0.346301\pi$
$24$	0.970143	0.198030
$25$	1.00000	0.200000
$26$	0.399865	0.0784199
$27$	4.90778	0.944503
$28$	−0.827277	−0.156341
$29$	3.28953	0.610850	0.305425	−	0.952216i	$-0.401202\pi$
$29$	3.28953	0.610850	0.305425	+	0.952216i	$0.401202\pi$
$30$	−0.970143	−0.177123
$31$	8.18303	1.46972	0.734858	−	0.678221i	$-0.237249\pi$
$31$	8.18303	1.46972	0.734858	+	0.678221i	$0.237249\pi$
$32$	−1.00000	−0.176777
$33$	0.970143	0.168880
$34$	2.38237	0.408573
$35$	0.827277	0.139835
$36$	−2.05882	−0.343137
$37$	10.2719	1.68869	0.844345	−	0.535801i	$-0.179990\pi$
$37$	10.2719	1.68869	0.844345	+	0.535801i	$0.179990\pi$
$38$	3.72626	0.604479
$39$	0.387926	0.0621179
$40$	1.00000	0.158114
$41$	3.28709	0.513358	0.256679	−	0.966497i	$-0.417372\pi$
$41$	3.28709	0.513358	0.256679	+	0.966497i	$0.417372\pi$
$42$	−0.802578	−0.123840
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−1.00000	−0.150756
$45$	2.05882	0.306911
$46$	−4.45353	−0.656637
$47$	11.2371	1.63910	0.819549	−	0.573010i	$-0.194224\pi$
$47$	11.2371	1.63910	0.819549	+	0.573010i	$0.194224\pi$
$48$	−0.970143	−0.140028
$49$	−6.31561	−0.902230
$50$	−1.00000	−0.141421
$51$	2.31124	0.323638
$52$	−0.399865	−0.0554512
$53$	−12.0181	−1.65082	−0.825409	−	0.564535i	$-0.809056\pi$
$53$	−12.0181	−1.65082	−0.825409	+	0.564535i	$0.809056\pi$
$54$	−4.90778	−0.667865
$55$	1.00000	0.134840
$56$	0.827277	0.110550
$57$	3.61501	0.478819
$58$	−3.28953	−0.431936
$59$	8.41377	1.09538	0.547690	−	0.836682i	$-0.315507\pi$
$59$	8.41377	1.09538	0.547690	+	0.836682i	$0.315507\pi$
$60$	0.970143	0.125245
$61$	−10.3312	−1.32277	−0.661387	−	0.750045i	$-0.730032\pi$
$61$	−10.3312	−1.32277	−0.661387	+	0.750045i	$0.730032\pi$
$62$	−8.18303	−1.03925
$63$	1.70322	0.214585
$64$	1.00000	0.125000
$65$	0.399865	0.0495971
$66$	−0.970143	−0.119416
$67$	−4.03666	−0.493157	−0.246578	−	0.969123i	$-0.579306\pi$
$67$	−4.03666	−0.493157	−0.246578	+	0.969123i	$0.579306\pi$
$68$	−2.38237	−0.288905
$69$	−4.32056	−0.520135
$70$	−0.827277	−0.0988786
$71$	−10.9219	−1.29619	−0.648094	−	0.761561i	$-0.724433\pi$
$71$	−10.9219	−1.29619	−0.648094	+	0.761561i	$0.724433\pi$
$72$	2.05882	0.242634
$73$	−4.77844	−0.559274	−0.279637	−	0.960106i	$-0.590214\pi$
$73$	−4.77844	−0.559274	−0.279637	+	0.960106i	$0.590214\pi$
$74$	−10.2719	−1.19408
$75$	−0.970143	−0.112023
$76$	−3.72626	−0.427431
$77$	0.827277	0.0942770
$78$	−0.387926	−0.0439240
$79$	−0.730224	−0.0821567	−0.0410783	−	0.999156i	$-0.513079\pi$
$79$	−0.730224	−0.0821567	−0.0410783	+	0.999156i	$0.513079\pi$
$80$	−1.00000	−0.111803
$81$	1.41521	0.157246
$82$	−3.28709	−0.362999
$83$	6.10174	0.669753	0.334877	−	0.942262i	$-0.391305\pi$
$83$	6.10174	0.669753	0.334877	+	0.942262i	$0.391305\pi$
$84$	0.802578	0.0875684
$85$	2.38237	0.258404
$86$	1.00000	0.107833
$87$	−3.19131	−0.342145
$88$	1.00000	0.106600
$89$	11.4701	1.21583	0.607915	−	0.794002i	$-0.292006\pi$
$89$	11.4701	1.21583	0.607915	+	0.794002i	$0.292006\pi$
$90$	−2.05882	−0.217019
$91$	0.330799	0.0346771
$92$	4.45353	0.464313
$93$	−7.93872	−0.823207
$94$	−11.2371	−1.15902
$95$	3.72626	0.382306
$96$	0.970143	0.0990149
$97$	−5.35966	−0.544191	−0.272096	−	0.962270i	$-0.587717\pi$
$97$	−5.35966	−0.544191	−0.272096	+	0.962270i	$0.587717\pi$
$98$	6.31561	0.637973
$99$	2.05882	0.206919
$100$	1.00000	0.100000
$101$	−7.32711	−0.729075	−0.364537	−	0.931189i	$-0.618773\pi$
$101$	−7.32711	−0.729075	−0.364537	+	0.931189i	$0.618773\pi$
$102$	−2.31124	−0.228847
$103$	−11.1941	−1.10299	−0.551493	−	0.834180i	$-0.685942\pi$
$103$	−11.1941	−1.10299	−0.551493	+	0.834180i	$0.685942\pi$
$104$	0.399865	0.0392099
$105$	−0.802578	−0.0783236
$106$	12.0181	1.16730
$107$	6.86646	0.663806	0.331903	−	0.943314i	$-0.392309\pi$
$107$	6.86646	0.663806	0.331903	+	0.943314i	$0.392309\pi$
$108$	4.90778	0.472252
$109$	9.05352	0.867170	0.433585	−	0.901113i	$-0.357248\pi$
$109$	9.05352	0.867170	0.433585	+	0.901113i	$0.357248\pi$
$110$	−1.00000	−0.0953463
$111$	−9.96521	−0.945856
$112$	−0.827277	−0.0781704
$113$	10.8083	1.01676	0.508378	−	0.861134i	$-0.330245\pi$
$113$	10.8083	1.01676	0.508378	+	0.861134i	$0.330245\pi$
$114$	−3.61501	−0.338577
$115$	−4.45353	−0.415294
$116$	3.28953	0.305425
$117$	0.823250	0.0761095
$118$	−8.41377	−0.774550
$119$	1.97088	0.180670
$120$	−0.970143	−0.0885616
$121$	1.00000	0.0909091
$122$	10.3312	0.935342
$123$	−3.18895	−0.287538
$124$	8.18303	0.734858
$125$	−1.00000	−0.0894427
$126$	−1.70322	−0.151735
$127$	−0.168832	−0.0149814	−0.00749069	−	0.999972i	$-0.502384\pi$
$127$	−0.168832	−0.0149814	−0.00749069	+	0.999972i	$0.502384\pi$
$128$	−1.00000	−0.0883883
$129$	0.970143	0.0854164
$130$	−0.399865	−0.0350704
$131$	−16.9996	−1.48527	−0.742633	−	0.669699i	$-0.766423\pi$
$131$	−16.9996	−1.48527	−0.742633	+	0.669699i	$0.766423\pi$
$132$	0.970143	0.0844402
$133$	3.08265	0.267300
$134$	4.03666	0.348714
$135$	−4.90778	−0.422395
$136$	2.38237	0.204287
$137$	19.3953	1.65705	0.828526	−	0.559951i	$-0.189180\pi$
$137$	19.3953	1.65705	0.828526	+	0.559951i	$0.189180\pi$
$138$	4.32056	0.367791
$139$	−21.4091	−1.81590	−0.907949	−	0.419081i	$-0.862352\pi$
$139$	−21.4091	−1.81590	−0.907949	+	0.419081i	$0.862352\pi$
$140$	0.827277	0.0699177
$141$	−10.9016	−0.918079
$142$	10.9219	0.916543
$143$	0.399865	0.0334383
$144$	−2.05882	−0.171568
$145$	−3.28953	−0.273180
$146$	4.77844	0.395466
$147$	6.12705	0.505351
$148$	10.2719	0.844345
$149$	−14.5840	−1.19477	−0.597383	−	0.801956i	$-0.703793\pi$
$149$	−14.5840	−1.19477	−0.597383	+	0.801956i	$0.703793\pi$
$150$	0.970143	0.0792119
$151$	−11.8233	−0.962168	−0.481084	−	0.876675i	$-0.659757\pi$
$151$	−11.8233	−0.962168	−0.481084	+	0.876675i	$0.659757\pi$
$152$	3.72626	0.302240
$153$	4.90487	0.396536
$154$	−0.827277	−0.0666639
$155$	−8.18303	−0.657277
$156$	0.387926	0.0310589
$157$	11.7861	0.940632	0.470316	−	0.882498i	$-0.344140\pi$
$157$	11.7861	0.940632	0.470316	+	0.882498i	$0.344140\pi$
$158$	0.730224	0.0580935
$159$	11.6593	0.924644
$160$	1.00000	0.0790569
$161$	−3.68430	−0.290364
$162$	−1.41521	−0.111190
$163$	−2.36233	−0.185032	−0.0925161	−	0.995711i	$-0.529491\pi$
$163$	−2.36233	−0.185032	−0.0925161	+	0.995711i	$0.529491\pi$
$164$	3.28709	0.256679
$165$	−0.970143	−0.0755256
$166$	−6.10174	−0.473587
$167$	−14.7432	−1.14087	−0.570433	−	0.821344i	$-0.693225\pi$
$167$	−14.7432	−1.14087	−0.570433	+	0.821344i	$0.693225\pi$
$168$	−0.802578	−0.0619202
$169$	−12.8401	−0.987701
$170$	−2.38237	−0.182719
$171$	7.67171	0.586670
$172$	−1.00000	−0.0762493
$173$	11.6610	0.886569	0.443284	−	0.896381i	$-0.353813\pi$
$173$	11.6610	0.886569	0.443284	+	0.896381i	$0.353813\pi$
$174$	3.19131	0.241933
$175$	−0.827277	−0.0625363
$176$	−1.00000	−0.0753778
$177$	−8.16256	−0.613536
$178$	−11.4701	−0.859722
$179$	−10.7548	−0.803854	−0.401927	−	0.915672i	$-0.631660\pi$
$179$	−10.7548	−0.803854	−0.401927	+	0.915672i	$0.631660\pi$
$180$	2.05882	0.153456
$181$	−7.47049	−0.555277	−0.277638	−	0.960686i	$-0.589552\pi$
$181$	−7.47049	−0.555277	−0.277638	+	0.960686i	$0.589552\pi$
$182$	−0.330799	−0.0245204
$183$	10.0227	0.740902
$184$	−4.45353	−0.328319
$185$	−10.2719	−0.755205
$186$	7.93872	0.582095
$187$	2.38237	0.174216
$188$	11.2371	0.819549
$189$	−4.06010	−0.295329
$190$	−3.72626	−0.270331
$191$	−17.9787	−1.30090	−0.650448	−	0.759551i	$-0.725419\pi$
$191$	−17.9787	−1.30090	−0.650448	+	0.759551i	$0.725419\pi$
$192$	−0.970143	−0.0700141
$193$	0.681617	0.0490639	0.0245319	−	0.999699i	$-0.492190\pi$
$193$	0.681617	0.0490639	0.0245319	+	0.999699i	$0.492190\pi$
$194$	5.35966	0.384801
$195$	−0.387926	−0.0277800
$196$	−6.31561	−0.451115
$197$	14.1094	1.00525	0.502625	−	0.864504i	$-0.332368\pi$
$197$	14.1094	1.00525	0.502625	+	0.864504i	$0.332368\pi$
$198$	−2.05882	−0.146314
$199$	13.0486	0.924993	0.462496	−	0.886621i	$-0.346954\pi$
$199$	13.0486	0.924993	0.462496	+	0.886621i	$0.346954\pi$
$200$	−1.00000	−0.0707107
$201$	3.91614	0.276223
$202$	7.32711	0.515534
$203$	−2.72135	−0.191001
$204$	2.31124	0.161819
$205$	−3.28709	−0.229581
$206$	11.1941	0.779929
$207$	−9.16902	−0.637291
$208$	−0.399865	−0.0277256
$209$	3.72626	0.257751
$210$	0.802578	0.0553831
$211$	10.3390	0.711767	0.355884	−	0.934530i	$-0.384180\pi$
$211$	10.3390	0.711767	0.355884	+	0.934530i	$0.384180\pi$
$212$	−12.0181	−0.825409
$213$	10.5958	0.726011
$214$	−6.86646	−0.469381
$215$	1.00000	0.0681994
$216$	−4.90778	−0.333932
$217$	−6.76964	−0.459553
$218$	−9.05352	−0.613182
$219$	4.63577	0.313256
$220$	1.00000	0.0674200
$221$	0.952625	0.0640805
$222$	9.96521	0.668821
$223$	−3.67944	−0.246393	−0.123197	−	0.992382i	$-0.539315\pi$
$223$	−3.67944	−0.246393	−0.123197	+	0.992382i	$0.539315\pi$
$224$	0.827277	0.0552748
$225$	−2.05882	−0.137255
$226$	−10.8083	−0.718954
$227$	−19.7649	−1.31184	−0.655921	−	0.754829i	$-0.727720\pi$
$227$	−19.7649	−1.31184	−0.655921	+	0.754829i	$0.727720\pi$
$228$	3.61501	0.239410
$229$	−12.3531	−0.816316	−0.408158	−	0.912911i	$-0.633829\pi$
$229$	−12.3531	−0.816316	−0.408158	+	0.912911i	$0.633829\pi$
$230$	4.45353	0.293657
$231$	−0.802578	−0.0528057
$232$	−3.28953	−0.215968
$233$	2.58763	0.169522	0.0847608	−	0.996401i	$-0.472987\pi$
$233$	2.58763	0.169522	0.0847608	+	0.996401i	$0.472987\pi$
$234$	−0.823250	−0.0538175
$235$	−11.2371	−0.733027
$236$	8.41377	0.547690
$237$	0.708423	0.0460170
$238$	−1.97088	−0.127753
$239$	4.28284	0.277034	0.138517	−	0.990360i	$-0.455766\pi$
$239$	4.28284	0.277034	0.138517	+	0.990360i	$0.455766\pi$
$240$	0.970143	0.0626225
$241$	−4.35843	−0.280751	−0.140376	−	0.990098i	$-0.544831\pi$
$241$	−4.35843	−0.280751	−0.140376	+	0.990098i	$0.544831\pi$
$242$	−1.00000	−0.0642824
$243$	−16.0963	−1.03258
$244$	−10.3312	−0.661387
$245$	6.31561	0.403490
$246$	3.18895	0.203320
$247$	1.49000	0.0948064
$248$	−8.18303	−0.519623
$249$	−5.91957	−0.375137
$250$	1.00000	0.0632456
$251$	21.2399	1.34065	0.670325	−	0.742068i	$-0.266155\pi$
$251$	21.2399	1.34065	0.670325	+	0.742068i	$0.266155\pi$
$252$	1.70322	0.107293
$253$	−4.45353	−0.279991
$254$	0.168832	0.0105934
$255$	−2.31124	−0.144735
$256$	1.00000	0.0625000
$257$	10.0422	0.626416	0.313208	−	0.949684i	$-0.398596\pi$
$257$	10.0422	0.626416	0.313208	+	0.949684i	$0.398596\pi$
$258$	−0.970143	−0.0603985
$259$	−8.49771	−0.528022
$260$	0.399865	0.0247985
$261$	−6.77255	−0.419210
$262$	16.9996	1.05024
$263$	−15.6335	−0.964004	−0.482002	−	0.876170i	$-0.660090\pi$
$263$	−15.6335	−0.964004	−0.482002	+	0.876170i	$0.660090\pi$
$264$	−0.970143	−0.0597082
$265$	12.0181	0.738268
$266$	−3.08265	−0.189010
$267$	−11.1277	−0.681002
$268$	−4.03666	−0.246578
$269$	−10.6882	−0.651671	−0.325836	−	0.945426i	$-0.605646\pi$
$269$	−10.6882	−0.651671	−0.325836	+	0.945426i	$0.605646\pi$
$270$	4.90778	0.298678
$271$	31.0544	1.88642	0.943210	−	0.332196i	$-0.107790\pi$
$271$	31.0544	1.88642	0.943210	+	0.332196i	$0.107790\pi$
$272$	−2.38237	−0.144452
$273$	−0.320922	−0.0194231
$274$	−19.3953	−1.17171
$275$	−1.00000	−0.0603023
$276$	−4.32056	−0.260067
$277$	−24.1974	−1.45388	−0.726940	−	0.686701i	$-0.759058\pi$
$277$	−24.1974	−1.45388	−0.726940	+	0.686701i	$0.759058\pi$
$278$	21.4091	1.28403
$279$	−16.8474	−1.00863
$280$	−0.827277	−0.0494393
$281$	14.8121	0.883619	0.441809	−	0.897109i	$-0.354337\pi$
$281$	14.8121	0.883619	0.441809	+	0.897109i	$0.354337\pi$
$282$	10.9016	0.649180
$283$	25.9713	1.54383	0.771916	−	0.635724i	$-0.219298\pi$
$283$	25.9713	1.54383	0.771916	+	0.635724i	$0.219298\pi$
$284$	−10.9219	−0.648094
$285$	−3.61501	−0.214135
$286$	−0.399865	−0.0236445
$287$	−2.71934	−0.160517
$288$	2.05882	0.121317
$289$	−11.3243	−0.666136
$290$	3.28953	0.193168
$291$	5.19964	0.304808
$292$	−4.77844	−0.279637
$293$	−2.66216	−0.155525	−0.0777626	−	0.996972i	$-0.524778\pi$
$293$	−2.66216	−0.155525	−0.0777626	+	0.996972i	$0.524778\pi$
$294$	−6.12705	−0.357337
$295$	−8.41377	−0.489869
$296$	−10.2719	−0.597042
$297$	−4.90778	−0.284778
$298$	14.5840	0.844827
$299$	−1.78081	−0.102987
$300$	−0.970143	−0.0560113
$301$	0.827277	0.0476835
$302$	11.8233	0.680355
$303$	7.10835	0.408364
$304$	−3.72626	−0.213716
$305$	10.3312	0.591562
$306$	−4.90487	−0.280393
$307$	26.9058	1.53560	0.767799	−	0.640691i	$-0.221352\pi$
$307$	26.9058	1.53560	0.767799	+	0.640691i	$0.221352\pi$
$308$	0.827277	0.0471385
$309$	10.8599	0.617796
$310$	8.18303	0.464765
$311$	−31.7828	−1.80224	−0.901118	−	0.433573i	$-0.857253\pi$
$311$	−31.7828	−1.80224	−0.901118	+	0.433573i	$0.857253\pi$
$312$	−0.387926	−0.0219620
$313$	1.71903	0.0971650	0.0485825	−	0.998819i	$-0.484530\pi$
$313$	1.71903	0.0971650	0.0485825	+	0.998819i	$0.484530\pi$
$314$	−11.7861	−0.665127
$315$	−1.70322	−0.0959654
$316$	−0.730224	−0.0410783
$317$	−8.69339	−0.488269	−0.244135	−	0.969741i	$-0.578504\pi$
$317$	−8.69339	−0.488269	−0.244135	+	0.969741i	$0.578504\pi$
$318$	−11.6593	−0.653822
$319$	−3.28953	−0.184178
$320$	−1.00000	−0.0559017
$321$	−6.66145	−0.371806
$322$	3.68430	0.205318
$323$	8.87733	0.493948
$324$	1.41521	0.0786229
$325$	−0.399865	−0.0221805
$326$	2.36233	0.130838
$327$	−8.78321	−0.485713
$328$	−3.28709	−0.181499
$329$	−9.29619	−0.512515
$330$	0.970143	0.0534046
$331$	−20.5876	−1.13160	−0.565798	−	0.824544i	$-0.691432\pi$
$331$	−20.5876	−1.13160	−0.565798	+	0.824544i	$0.691432\pi$
$332$	6.10174	0.334877
$333$	−21.1480	−1.15890
$334$	14.7432	0.806714
$335$	4.03666	0.220546
$336$	0.802578	0.0437842
$337$	−0.235074	−0.0128053	−0.00640264	−	0.999980i	$-0.502038\pi$
$337$	−0.235074	−0.0128053	−0.00640264	+	0.999980i	$0.502038\pi$
$338$	12.8401	0.698410
$339$	−10.4856	−0.569497
$340$	2.38237	0.129202
$341$	−8.18303	−0.443136
$342$	−7.67171	−0.414838
$343$	11.0157	0.594792
$344$	1.00000	0.0539164
$345$	4.32056	0.232611
$346$	−11.6610	−0.626899
$347$	31.7096	1.70226	0.851130	−	0.524955i	$-0.175918\pi$
$347$	31.7096	1.70226	0.851130	+	0.524955i	$0.175918\pi$
$348$	−3.19131	−0.171072
$349$	−22.3713	−1.19751	−0.598753	−	0.800933i	$-0.704337\pi$
$349$	−22.3713	−1.19751	−0.598753	+	0.800933i	$0.704337\pi$
$350$	0.827277	0.0442198
$351$	−1.96245	−0.104748
$352$	1.00000	0.0533002
$353$	20.4334	1.08756	0.543781	−	0.839227i	$-0.316992\pi$
$353$	20.4334	1.08756	0.543781	+	0.839227i	$0.316992\pi$
$354$	8.16256	0.433835
$355$	10.9219	0.579673
$356$	11.4701	0.607915
$357$	−1.91204	−0.101196
$358$	10.7548	0.568411
$359$	17.7230	0.935383	0.467692	−	0.883892i	$-0.345086\pi$
$359$	17.7230	0.935383	0.467692	+	0.883892i	$0.345086\pi$
$360$	−2.05882	−0.108509
$361$	−5.11498	−0.269209
$362$	7.47049	0.392640
$363$	−0.970143	−0.0509193
$364$	0.330799	0.0173386
$365$	4.77844	0.250115
$366$	−10.0227	−0.523897
$367$	−24.3595	−1.27156	−0.635779	−	0.771871i	$-0.719321\pi$
$367$	−24.3595	−1.27156	−0.635779	+	0.771871i	$0.719321\pi$
$368$	4.45353	0.232156
$369$	−6.76754	−0.352304
$370$	10.2719	0.534010
$371$	9.94234	0.516180
$372$	−7.93872	−0.411603
$373$	−32.4034	−1.67778	−0.838892	−	0.544298i	$-0.816796\pi$
$373$	−32.4034	−1.67778	−0.838892	+	0.544298i	$0.816796\pi$
$374$	−2.38237	−0.123189
$375$	0.970143	0.0500980
$376$	−11.2371	−0.579508
$377$	−1.31536	−0.0677447
$378$	4.06010	0.208829
$379$	−21.7671	−1.11810	−0.559052	−	0.829133i	$-0.688835\pi$
$379$	−21.7671	−1.11810	−0.559052	+	0.829133i	$0.688835\pi$
$380$	3.72626	0.191153
$381$	0.163791	0.00839126
$382$	17.9787	0.919872
$383$	−30.9067	−1.57926	−0.789630	−	0.613584i	$-0.789727\pi$
$383$	−30.9067	−1.57926	−0.789630	+	0.613584i	$0.789727\pi$
$384$	0.970143	0.0495074
$385$	−0.827277	−0.0421620
$386$	−0.681617	−0.0346934
$387$	2.05882	0.104656
$388$	−5.35966	−0.272096
$389$	5.66489	0.287221	0.143611	−	0.989634i	$-0.454129\pi$
$389$	5.66489	0.287221	0.143611	+	0.989634i	$0.454129\pi$
$390$	0.387926	0.0196434
$391$	−10.6100	−0.536568
$392$	6.31561	0.318987
$393$	16.4921	0.831916
$394$	−14.1094	−0.710819
$395$	0.730224	0.0367416
$396$	2.05882	0.103460
$397$	13.9769	0.701481	0.350740	−	0.936473i	$-0.385930\pi$
$397$	13.9769	0.701481	0.350740	+	0.936473i	$0.385930\pi$
$398$	−13.0486	−0.654069
$399$	−2.99061	−0.149718
$400$	1.00000	0.0500000
$401$	16.5879	0.828362	0.414181	−	0.910194i	$-0.364068\pi$
$401$	16.5879	0.828362	0.414181	+	0.910194i	$0.364068\pi$
$402$	−3.91614	−0.195319
$403$	−3.27210	−0.162995
$404$	−7.32711	−0.364537
$405$	−1.41521	−0.0703224
$406$	2.72135	0.135058
$407$	−10.2719	−0.509159
$408$	−2.31124	−0.114423
$409$	9.10903	0.450413	0.225206	−	0.974311i	$-0.427694\pi$
$409$	9.10903	0.450413	0.225206	+	0.974311i	$0.427694\pi$
$410$	3.28709	0.162338
$411$	−18.8162	−0.928136
$412$	−11.1941	−0.551493
$413$	−6.96052	−0.342505
$414$	9.16902	0.450633
$415$	−6.10174	−0.299523
$416$	0.399865	0.0196050
$417$	20.7699	1.01711
$418$	−3.72626	−0.182257
$419$	15.9208	0.777782	0.388891	−	0.921284i	$-0.372858\pi$
$419$	15.9208	0.777782	0.388891	+	0.921284i	$0.372858\pi$
$420$	−0.802578	−0.0391618
$421$	−33.1714	−1.61667	−0.808336	−	0.588721i	$-0.799632\pi$
$421$	−33.1714	−1.61667	−0.808336	+	0.588721i	$0.799632\pi$
$422$	−10.3390	−0.503296
$423$	−23.1352	−1.12487
$424$	12.0181	0.583652
$425$	−2.38237	−0.115562
$426$	−10.5958	−0.513367
$427$	8.54676	0.413607
$428$	6.86646	0.331903
$429$	−0.387926	−0.0187292
$430$	−1.00000	−0.0482243
$431$	−22.6573	−1.09136	−0.545681	−	0.837993i	$-0.683729\pi$
$431$	−22.6573	−1.09136	−0.545681	+	0.837993i	$0.683729\pi$
$432$	4.90778	0.236126
$433$	−16.4851	−0.792223	−0.396112	−	0.918202i	$-0.629641\pi$
$433$	−16.4851	−0.792223	−0.396112	+	0.918202i	$0.629641\pi$
$434$	6.76964	0.324953
$435$	3.19131	0.153012
$436$	9.05352	0.433585
$437$	−16.5950	−0.793847
$438$	−4.63577	−0.221506
$439$	−38.8406	−1.85376	−0.926881	−	0.375355i	$-0.877521\pi$
$439$	−38.8406	−1.85376	−0.926881	+	0.375355i	$0.877521\pi$
$440$	−1.00000	−0.0476731
$441$	13.0027	0.619177
$442$	−0.952625	−0.0453118
$443$	−16.0614	−0.763099	−0.381549	−	0.924349i	$-0.624609\pi$
$443$	−16.0614	−0.763099	−0.381549	+	0.924349i	$0.624609\pi$
$444$	−9.96521	−0.472928
$445$	−11.4701	−0.543736
$446$	3.67944	0.174226
$447$	14.1485	0.669203
$448$	−0.827277	−0.0390852
$449$	−8.83587	−0.416990	−0.208495	−	0.978023i	$-0.566857\pi$
$449$	−8.83587	−0.416990	−0.208495	+	0.978023i	$0.566857\pi$
$450$	2.05882	0.0970538
$451$	−3.28709	−0.154783
$452$	10.8083	0.508378
$453$	11.4703	0.538922
$454$	19.7649	0.927613
$455$	−0.330799	−0.0155081
$456$	−3.61501	−0.169288
$457$	4.30297	0.201285	0.100642	−	0.994923i	$-0.467910\pi$
$457$	4.30297	0.201285	0.100642	+	0.994923i	$0.467910\pi$
$458$	12.3531	0.577222
$459$	−11.6922	−0.545743
$460$	−4.45353	−0.207647
$461$	−10.0599	−0.468536	−0.234268	−	0.972172i	$-0.575269\pi$
$461$	−10.0599	−0.468536	−0.234268	+	0.972172i	$0.575269\pi$
$462$	0.802578	0.0373393
$463$	28.0482	1.30351	0.651755	−	0.758430i	$-0.274033\pi$
$463$	28.0482	1.30351	0.651755	+	0.758430i	$0.274033\pi$
$464$	3.28953	0.152712
$465$	7.93872	0.368149
$466$	−2.58763	−0.119870
$467$	12.4846	0.577718	0.288859	−	0.957372i	$-0.406724\pi$
$467$	12.4846	0.577718	0.288859	+	0.957372i	$0.406724\pi$
$468$	0.823250	0.0380547
$469$	3.33944	0.154201
$470$	11.2371	0.518328
$471$	−11.4342	−0.526860
$472$	−8.41377	−0.387275
$473$	1.00000	0.0459800
$474$	−0.708423	−0.0325389
$475$	−3.72626	−0.170973
$476$	1.97088	0.0903352
$477$	24.7432	1.13291
$478$	−4.28284	−0.195893
$479$	5.80107	0.265058	0.132529	−	0.991179i	$-0.457690\pi$
$479$	5.80107	0.265058	0.132529	+	0.991179i	$0.457690\pi$
$480$	−0.970143	−0.0442808
$481$	−4.10737	−0.187280
$482$	4.35843	0.198521
$483$	3.57430	0.162636
$484$	1.00000	0.0454545
$485$	5.35966	0.243370
$486$	16.0963	0.730143
$487$	6.16836	0.279515	0.139758	−	0.990186i	$-0.455368\pi$
$487$	6.16836	0.279515	0.139758	+	0.990186i	$0.455368\pi$
$488$	10.3312	0.467671
$489$	2.29180	0.103639
$490$	−6.31561	−0.285310
$491$	2.78822	0.125831	0.0629153	−	0.998019i	$-0.479960\pi$
$491$	2.78822	0.125831	0.0629153	+	0.998019i	$0.479960\pi$
$492$	−3.18895	−0.143769
$493$	−7.83687	−0.352955
$494$	−1.49000	−0.0670383
$495$	−2.05882	−0.0925372
$496$	8.18303	0.367429
$497$	9.03542	0.405294
$498$	5.91957	0.265262
$499$	−36.9207	−1.65280	−0.826398	−	0.563087i	$-0.809614\pi$
$499$	−36.9207	−1.65280	−0.826398	+	0.563087i	$0.809614\pi$
$500$	−1.00000	−0.0447214
$501$	14.3031	0.639014
$502$	−21.2399	−0.947983
$503$	−26.7003	−1.19051	−0.595254	−	0.803537i	$-0.702949\pi$
$503$	−26.7003	−1.19051	−0.595254	+	0.803537i	$0.702949\pi$
$504$	−1.70322	−0.0758673
$505$	7.32711	0.326052
$506$	4.45353	0.197984
$507$	12.4567	0.553224
$508$	−0.168832	−0.00749069
$509$	21.7185	0.962655	0.481327	−	0.876541i	$-0.340155\pi$
$509$	21.7185	0.962655	0.481327	+	0.876541i	$0.340155\pi$
$510$	2.31124	0.102343
$511$	3.95309	0.174875
$512$	−1.00000	−0.0441942
$513$	−18.2877	−0.807421
$514$	−10.0422	−0.442943
$515$	11.1941	0.493270
$516$	0.970143	0.0427082
$517$	−11.2371	−0.494206
$518$	8.49771	0.373368
$519$	−11.3128	−0.496578
$520$	−0.399865	−0.0175352
$521$	−28.5151	−1.24927	−0.624634	−	0.780917i	$-0.714752\pi$
$521$	−28.5151	−1.24927	−0.624634	+	0.780917i	$0.714752\pi$
$522$	6.77255	0.296426
$523$	−34.0106	−1.48718	−0.743591	−	0.668635i	$-0.766879\pi$
$523$	−34.0106	−1.48718	−0.743591	+	0.668635i	$0.766879\pi$
$524$	−16.9996	−0.742633
$525$	0.802578	0.0350274
$526$	15.6335	0.681653
$527$	−19.4950	−0.849216
$528$	0.970143	0.0422201
$529$	−3.16608	−0.137656
$530$	−12.0181	−0.522035
$531$	−17.3224	−0.751730
$532$	3.08265	0.133650
$533$	−1.31439	−0.0569326
$534$	11.1277	0.481541
$535$	−6.86646	−0.296863
$536$	4.03666	0.174357
$537$	10.4337	0.450249
$538$	10.6882	0.460801
$539$	6.31561	0.272033
$540$	−4.90778	−0.211197
$541$	3.07396	0.132160	0.0660799	−	0.997814i	$-0.478951\pi$
$541$	3.07396	0.132160	0.0660799	+	0.997814i	$0.478951\pi$
$542$	−31.0544	−1.33390
$543$	7.24744	0.311018
$544$	2.38237	0.102143
$545$	−9.05352	−0.387810
$546$	0.320922	0.0137342
$547$	−23.9147	−1.02252	−0.511259	−	0.859427i	$-0.670821\pi$
$547$	−23.9147	−1.02252	−0.511259	+	0.859427i	$0.670821\pi$
$548$	19.3953	0.828526
$549$	21.2701	0.907785
$550$	1.00000	0.0426401
$551$	−12.2576	−0.522193
$552$	4.32056	0.183895
$553$	0.604098	0.0256889
$554$	24.1974	1.02805
$555$	9.96521	0.423000
$556$	−21.4091	−0.907949
$557$	−25.1506	−1.06566	−0.532832	−	0.846221i	$-0.678872\pi$
$557$	−25.1506	−1.06566	−0.532832	+	0.846221i	$0.678872\pi$
$558$	16.8474	0.713208
$559$	0.399865	0.0169125
$560$	0.827277	0.0349589
$561$	−2.31124	−0.0975806
$562$	−14.8121	−0.624813
$563$	−30.2622	−1.27540	−0.637700	−	0.770284i	$-0.720114\pi$
$563$	−30.2622	−1.27540	−0.637700	+	0.770284i	$0.720114\pi$
$564$	−10.9016	−0.459039
$565$	−10.8083	−0.454707
$566$	−25.9713	−1.09165
$567$	−1.17077	−0.0491678
$568$	10.9219	0.458271
$569$	32.3895	1.35784	0.678918	−	0.734214i	$-0.262449\pi$
$569$	32.3895	1.35784	0.678918	+	0.734214i	$0.262449\pi$
$570$	3.61501	0.151416
$571$	−1.44238	−0.0603617	−0.0301808	−	0.999544i	$-0.509608\pi$
$571$	−1.44238	−0.0603617	−0.0301808	+	0.999544i	$0.509608\pi$
$572$	0.399865	0.0167192
$573$	17.4419	0.728648
$574$	2.71934	0.113503
$575$	4.45353	0.185725
$576$	−2.05882	−0.0857842
$577$	13.3593	0.556155	0.278077	−	0.960559i	$-0.410303\pi$
$577$	13.3593	0.556155	0.278077	+	0.960559i	$0.410303\pi$
$578$	11.3243	0.471029
$579$	−0.661267	−0.0274813
$580$	−3.28953	−0.136590
$581$	−5.04783	−0.209419
$582$	−5.19964	−0.215532
$583$	12.0181	0.497740
$584$	4.77844	0.197733
$585$	−0.823250	−0.0340372
$586$	2.66216	0.109973
$587$	11.1815	0.461509	0.230755	−	0.973012i	$-0.425881\pi$
$587$	11.1815	0.461509	0.230755	+	0.973012i	$0.425881\pi$
$588$	6.12705	0.252675
$589$	−30.4921	−1.25641
$590$	8.41377	0.346389
$591$	−13.6881	−0.563053
$592$	10.2719	0.422172
$593$	−38.6909	−1.58884	−0.794422	−	0.607366i	$-0.792226\pi$
$593$	−38.6909	−1.58884	−0.794422	+	0.607366i	$0.792226\pi$
$594$	4.90778	0.201369
$595$	−1.97088	−0.0807982
$596$	−14.5840	−0.597383
$597$	−12.6590	−0.518100
$598$	1.78081	0.0728227
$599$	−28.2948	−1.15609	−0.578046	−	0.816004i	$-0.696185\pi$
$599$	−28.2948	−1.15609	−0.578046	+	0.816004i	$0.696185\pi$
$600$	0.970143	0.0396059
$601$	−25.7532	−1.05050	−0.525248	−	0.850949i	$-0.676028\pi$
$601$	−25.7532	−1.05050	−0.525248	+	0.850949i	$0.676028\pi$
$602$	−0.827277	−0.0337173
$603$	8.31077	0.338441
$604$	−11.8233	−0.481084
$605$	−1.00000	−0.0406558
$606$	−7.10835	−0.288757
$607$	16.6181	0.674509	0.337255	−	0.941413i	$-0.390502\pi$
$607$	16.6181	0.674509	0.337255	+	0.941413i	$0.390502\pi$
$608$	3.72626	0.151120
$609$	2.64010	0.106982
$610$	−10.3312	−0.418298
$611$	−4.49331	−0.181780
$612$	4.90487	0.198268
$613$	−18.8135	−0.759870	−0.379935	−	0.925013i	$-0.624054\pi$
$613$	−18.8135	−0.759870	−0.379935	+	0.925013i	$0.624054\pi$
$614$	−26.9058	−1.08583
$615$	3.18895	0.128591
$616$	−0.827277	−0.0333320
$617$	36.1064	1.45359	0.726794	−	0.686856i	$-0.241009\pi$
$617$	36.1064	1.45359	0.726794	+	0.686856i	$0.241009\pi$
$618$	−10.8599	−0.436848
$619$	1.86112	0.0748048	0.0374024	−	0.999300i	$-0.488092\pi$
$619$	1.86112	0.0748048	0.0374024	+	0.999300i	$0.488092\pi$
$620$	−8.18303	−0.328639
$621$	21.8570	0.877089
$622$	31.7828	1.27437
$623$	−9.48898	−0.380168
$624$	0.387926	0.0155295
$625$	1.00000	0.0400000
$626$	−1.71903	−0.0687061
$627$	−3.61501	−0.144370
$628$	11.7861	0.470316
$629$	−24.4714	−0.975741
$630$	1.70322	0.0678578
$631$	0.454748	0.0181032	0.00905161	−	0.999959i	$-0.497119\pi$
$631$	0.454748	0.0181032	0.00905161	+	0.999959i	$0.497119\pi$
$632$	0.730224	0.0290468
$633$	−10.0303	−0.398670
$634$	8.69339	0.345258
$635$	0.168832	0.00669988
$636$	11.6593	0.462322
$637$	2.52539	0.100060
$638$	3.28953	0.130234
$639$	22.4862	0.889540
$640$	1.00000	0.0395285
$641$	−6.90667	−0.272797	−0.136399	−	0.990654i	$-0.543553\pi$
$641$	−6.90667	−0.272797	−0.136399	+	0.990654i	$0.543553\pi$
$642$	6.66145	0.262906
$643$	−15.5936	−0.614953	−0.307476	−	0.951556i	$-0.599485\pi$
$643$	−15.5936	−0.614953	−0.307476	+	0.951556i	$0.599485\pi$
$644$	−3.68430	−0.145182
$645$	−0.970143	−0.0381994
$646$	−8.87733	−0.349274
$647$	5.49745	0.216127	0.108063	−	0.994144i	$-0.465535\pi$
$647$	5.49745	0.216127	0.108063	+	0.994144i	$0.465535\pi$
$648$	−1.41521	−0.0555948
$649$	−8.41377	−0.330269
$650$	0.399865	0.0156840
$651$	6.56752	0.257401
$652$	−2.36233	−0.0925161
$653$	47.7941	1.87033	0.935164	−	0.354215i	$-0.115252\pi$
$653$	47.7941	1.87033	0.935164	+	0.354215i	$0.115252\pi$
$654$	8.78321	0.343451
$655$	16.9996	0.664231
$656$	3.28709	0.128339
$657$	9.83795	0.383815
$658$	9.29619	0.362403
$659$	−1.40130	−0.0545868	−0.0272934	−	0.999627i	$-0.508689\pi$
$659$	−1.40130	−0.0545868	−0.0272934	+	0.999627i	$0.508689\pi$
$660$	−0.970143	−0.0377628
$661$	23.9984	0.933430	0.466715	−	0.884408i	$-0.345437\pi$
$661$	23.9984	0.933430	0.466715	+	0.884408i	$0.345437\pi$
$662$	20.5876	0.800159
$663$	−0.924183	−0.0358923
$664$	−6.10174	−0.236794
$665$	−3.08265	−0.119540
$666$	21.1480	0.819468
$667$	14.6500	0.567250
$668$	−14.7432	−0.570433
$669$	3.56958	0.138008
$670$	−4.03666	−0.155950
$671$	10.3312	0.398831
$672$	−0.802578	−0.0309601
$673$	5.81218	0.224043	0.112022	−	0.993706i	$-0.464267\pi$
$673$	5.81218	0.224043	0.112022	+	0.993706i	$0.464267\pi$
$674$	0.235074	0.00905470
$675$	4.90778	0.188901
$676$	−12.8401	−0.493850
$677$	35.9242	1.38068	0.690339	−	0.723486i	$-0.257461\pi$
$677$	35.9242	1.38068	0.690339	+	0.723486i	$0.257461\pi$
$678$	10.4856	0.402695
$679$	4.43393	0.170159
$680$	−2.38237	−0.0913597
$681$	19.1748	0.734780
$682$	8.18303	0.313345
$683$	−23.2517	−0.889701	−0.444850	−	0.895605i	$-0.646743\pi$
$683$	−23.2517	−0.889701	−0.444850	+	0.895605i	$0.646743\pi$
$684$	7.67171	0.293335
$685$	−19.3953	−0.741056
$686$	−11.0157	−0.420582
$687$	11.9843	0.457229
$688$	−1.00000	−0.0381246
$689$	4.80563	0.183080
$690$	−4.32056	−0.164481
$691$	−18.5181	−0.704462	−0.352231	−	0.935913i	$-0.614577\pi$
$691$	−18.5181	−0.704462	−0.352231	+	0.935913i	$0.614577\pi$
$692$	11.6610	0.443284
$693$	−1.70322	−0.0646999
$694$	−31.7096	−1.20368
$695$	21.4091	0.812094
$696$	3.19131	0.120966
$697$	−7.83107	−0.296623
$698$	22.3713	0.846765
$699$	−2.51038	−0.0949511
$700$	−0.827277	−0.0312681
$701$	47.0385	1.77662	0.888310	−	0.459245i	$-0.151880\pi$
$701$	47.0385	1.77662	0.888310	+	0.459245i	$0.151880\pi$
$702$	1.96245	0.0740678
$703$	−38.2758	−1.44360
$704$	−1.00000	−0.0376889
$705$	10.9016	0.410577
$706$	−20.4334	−0.769023
$707$	6.06155	0.227968
$708$	−8.16256	−0.306768
$709$	18.9139	0.710327	0.355163	−	0.934804i	$-0.384425\pi$
$709$	18.9139	0.710327	0.355163	+	0.934804i	$0.384425\pi$
$710$	−10.9219	−0.409890
$711$	1.50340	0.0563820
$712$	−11.4701	−0.429861
$713$	36.4434	1.36482
$714$	1.91204	0.0715562
$715$	−0.399865	−0.0149541
$716$	−10.7548	−0.401927
$717$	−4.15497	−0.155170
$718$	−17.7230	−0.661416
$719$	40.4176	1.50732	0.753660	−	0.657264i	$-0.228286\pi$
$719$	40.4176	1.50732	0.753660	+	0.657264i	$0.228286\pi$
$720$	2.05882	0.0767278
$721$	9.26061	0.344883
$722$	5.11498	0.190360
$723$	4.22830	0.157252
$724$	−7.47049	−0.277638
$725$	3.28953	0.122170
$726$	0.970143	0.0360054
$727$	36.0019	1.33524	0.667618	−	0.744504i	$-0.267314\pi$
$727$	36.0019	1.33524	0.667618	+	0.744504i	$0.267314\pi$
$728$	−0.330799	−0.0122602
$729$	11.3701	0.421115
$730$	−4.77844	−0.176858
$731$	2.38237	0.0881151
$732$	10.0227	0.370451
$733$	11.0214	0.407085	0.203543	−	0.979066i	$-0.434754\pi$
$733$	11.0214	0.407085	0.203543	+	0.979066i	$0.434754\pi$
$734$	24.3595	0.899127
$735$	−6.12705	−0.226000
$736$	−4.45353	−0.164159
$737$	4.03666	0.148692
$738$	6.76754	0.249117
$739$	−47.8650	−1.76074	−0.880371	−	0.474287i	$-0.842706\pi$
$739$	−47.8650	−1.76074	−0.880371	+	0.474287i	$0.842706\pi$
$740$	−10.2719	−0.377602
$741$	−1.44551	−0.0531023
$742$	−9.94234	−0.364995
$743$	9.94719	0.364927	0.182464	−	0.983213i	$-0.441593\pi$
$743$	9.94719	0.364927	0.182464	+	0.983213i	$0.441593\pi$
$744$	7.93872	0.291047
$745$	14.5840	0.534315
$746$	32.4034	1.18637
$747$	−12.5624	−0.459634
$748$	2.38237	0.0871081
$749$	−5.68047	−0.207560
$750$	−0.970143	−0.0354246
$751$	13.2378	0.483053	0.241526	−	0.970394i	$-0.422352\pi$
$751$	13.2378	0.483053	0.241526	+	0.970394i	$0.422352\pi$
$752$	11.2371	0.409774
$753$	−20.6057	−0.750915
$754$	1.31536	0.0479028
$755$	11.8233	0.430294
$756$	−4.06010	−0.147664
$757$	3.88339	0.141144	0.0705722	−	0.997507i	$-0.477517\pi$
$757$	3.88339	0.141144	0.0705722	+	0.997507i	$0.477517\pi$
$758$	21.7671	0.790618
$759$	4.32056	0.156826
$760$	−3.72626	−0.135166
$761$	−44.6321	−1.61791	−0.808957	−	0.587868i	$-0.799968\pi$
$761$	−44.6321	−1.61791	−0.808957	+	0.587868i	$0.799968\pi$
$762$	−0.163791	−0.00593352
$763$	−7.48977	−0.271148
$764$	−17.9787	−0.650448
$765$	−4.90487	−0.177336
$766$	30.9067	1.11670
$767$	−3.36437	−0.121480
$768$	−0.970143	−0.0350070
$769$	48.0453	1.73256	0.866278	−	0.499562i	$-0.166506\pi$
$769$	48.0453	1.73256	0.866278	+	0.499562i	$0.166506\pi$
$770$	0.827277	0.0298130
$771$	−9.74239	−0.350864
$772$	0.681617	0.0245319
$773$	3.79206	0.136391	0.0681954	−	0.997672i	$-0.478276\pi$
$773$	3.79206	0.136391	0.0681954	+	0.997672i	$0.478276\pi$
$774$	−2.05882	−0.0740028
$775$	8.18303	0.293943
$776$	5.35966	0.192401
$777$	8.24399	0.295752
$778$	−5.66489	−0.203096
$779$	−12.2486	−0.438850
$780$	−0.387926	−0.0138900
$781$	10.9219	0.390815
$782$	10.6100	0.379411
$783$	16.1443	0.576949
$784$	−6.31561	−0.225558
$785$	−11.7861	−0.420663
$786$	−16.4921	−0.588253
$787$	−52.9764	−1.88841	−0.944203	−	0.329365i	$-0.893165\pi$
$787$	−52.9764	−1.88841	−0.944203	+	0.329365i	$0.893165\pi$
$788$	14.1094	0.502625
$789$	15.1667	0.539951
$790$	−0.730224	−0.0259802
$791$	−8.94143	−0.317920
$792$	−2.05882	−0.0731570
$793$	4.13108	0.146699
$794$	−13.9769	−0.496022
$795$	−11.6593	−0.413513
$796$	13.0486	0.462496
$797$	8.12439	0.287781	0.143890	−	0.989594i	$-0.454039\pi$
$797$	8.12439	0.287781	0.143890	+	0.989594i	$0.454039\pi$
$798$	2.99061	0.105867
$799$	−26.7709	−0.947086
$800$	−1.00000	−0.0353553
$801$	−23.6149	−0.834393
$802$	−16.5879	−0.585741
$803$	4.77844	0.168627
$804$	3.91614	0.138112
$805$	3.68430	0.129855
$806$	3.27210	0.115255
$807$	10.3691	0.365009
$808$	7.32711	0.257767
$809$	28.0801	0.987245	0.493622	−	0.869676i	$-0.335673\pi$
$809$	28.0801	0.987245	0.493622	+	0.869676i	$0.335673\pi$
$810$	1.41521	0.0497255
$811$	36.0818	1.26700	0.633502	−	0.773741i	$-0.281617\pi$
$811$	36.0818	1.26700	0.633502	+	0.773741i	$0.281617\pi$
$812$	−2.72135	−0.0955007
$813$	−30.1272	−1.05661
$814$	10.2719	0.360030
$815$	2.36233	0.0827489
$816$	2.31124	0.0809096
$817$	3.72626	0.130365
$818$	−9.10903	−0.318490
$819$	−0.681056	−0.0237980
$820$	−3.28709	−0.114790
$821$	−18.6946	−0.652446	−0.326223	−	0.945293i	$-0.605776\pi$
$821$	−18.6946	−0.652446	−0.326223	+	0.945293i	$0.605776\pi$
$822$	18.8162	0.656291
$823$	−36.5256	−1.27320	−0.636601	−	0.771194i	$-0.719660\pi$
$823$	−36.5256	−1.27320	−0.636601	+	0.771194i	$0.719660\pi$
$824$	11.1941	0.389964
$825$	0.970143	0.0337761
$826$	6.96052	0.242187
$827$	5.06096	0.175987	0.0879934	−	0.996121i	$-0.471955\pi$
$827$	5.06096	0.175987	0.0879934	+	0.996121i	$0.471955\pi$
$828$	−9.16902	−0.318646
$829$	8.32879	0.289271	0.144635	−	0.989485i	$-0.453799\pi$
$829$	8.32879	0.289271	0.144635	+	0.989485i	$0.453799\pi$
$830$	6.10174	0.211795
$831$	23.4749	0.814337
$832$	−0.399865	−0.0138628
$833$	15.0461	0.521317
$834$	−20.7699	−0.719203
$835$	14.7432	0.510211
$836$	3.72626	0.128875
$837$	40.1605	1.38815
$838$	−15.9208	−0.549975
$839$	1.66254	0.0573971	0.0286986	−	0.999588i	$-0.490864\pi$
$839$	1.66254	0.0573971	0.0286986	+	0.999588i	$0.490864\pi$
$840$	0.802578	0.0276916
$841$	−18.1790	−0.626863
$842$	33.1714	1.14316
$843$	−14.3699	−0.494926
$844$	10.3390	0.355884
$845$	12.8401	0.441713
$846$	23.1352	0.795403
$847$	−0.827277	−0.0284256
$848$	−12.0181	−0.412705
$849$	−25.1959	−0.864720
$850$	2.38237	0.0817146
$851$	45.7462	1.56816
$852$	10.5958	0.363005
$853$	17.8988	0.612844	0.306422	−	0.951896i	$-0.400868\pi$
$853$	17.8988	0.612844	0.306422	+	0.951896i	$0.400868\pi$
$854$	−8.54676	−0.292464
$855$	−7.67171	−0.262367
$856$	−6.86646	−0.234691
$857$	−27.7115	−0.946606	−0.473303	−	0.880900i	$-0.656938\pi$
$857$	−27.7115	−0.946606	−0.473303	+	0.880900i	$0.656938\pi$
$858$	0.387926	0.0132436
$859$	32.3540	1.10390	0.551952	−	0.833876i	$-0.313883\pi$
$859$	32.3540	1.10390	0.551952	+	0.833876i	$0.313883\pi$
$860$	1.00000	0.0340997
$861$	2.63815	0.0899078
$862$	22.6573	0.771710
$863$	−17.2369	−0.586750	−0.293375	−	0.955997i	$-0.594779\pi$
$863$	−17.2369	−0.586750	−0.293375	+	0.955997i	$0.594779\pi$
$864$	−4.90778	−0.166966
$865$	−11.6610	−0.396486
$866$	16.4851	0.560186
$867$	10.9862	0.373111
$868$	−6.76964	−0.229777
$869$	0.730224	0.0247712
$870$	−3.19131	−0.108196
$871$	1.61412	0.0546923
$872$	−9.05352	−0.306591
$873$	11.0346	0.373464
$874$	16.5950	0.561335
$875$	0.827277	0.0279671
$876$	4.63577	0.156628
$877$	49.0813	1.65736	0.828678	−	0.559725i	$-0.189093\pi$
$877$	49.0813	1.65736	0.828678	+	0.559725i	$0.189093\pi$
$878$	38.8406	1.31081
$879$	2.58268	0.0871116
$880$	1.00000	0.0337100
$881$	0.140773	0.00474277	0.00237139	−	0.999997i	$-0.499245\pi$
$881$	0.140773	0.00474277	0.00237139	+	0.999997i	$0.499245\pi$
$882$	−13.0027	−0.437824
$883$	−41.5781	−1.39921	−0.699607	−	0.714528i	$-0.746641\pi$
$883$	−41.5781	−1.39921	−0.699607	+	0.714528i	$0.746641\pi$
$884$	0.952625	0.0320402
$885$	8.16256	0.274382
$886$	16.0614	0.539592
$887$	−24.7901	−0.832370	−0.416185	−	0.909280i	$-0.636633\pi$
$887$	−24.7901	−0.832370	−0.416185	+	0.909280i	$0.636633\pi$
$888$	9.96521	0.334411
$889$	0.139671	0.00468440
$890$	11.4701	0.384479
$891$	−1.41521	−0.0474114
$892$	−3.67944	−0.123197
$893$	−41.8723	−1.40120
$894$	−14.1485	−0.473198
$895$	10.7548	0.359495
$896$	0.827277	0.0276374
$897$	1.72764	0.0576842
$898$	8.83587	0.294857
$899$	26.9183	0.897776
$900$	−2.05882	−0.0686274
$901$	28.6316	0.953858
$902$	3.28709	0.109448
$903$	−0.802578	−0.0267081
$904$	−10.8083	−0.359477
$905$	7.47049	0.248327
$906$	−11.4703	−0.381076
$907$	−48.1550	−1.59896	−0.799480	−	0.600693i	$-0.794891\pi$
$907$	−48.1550	−1.59896	−0.799480	+	0.600693i	$0.794891\pi$
$908$	−19.7649	−0.655921
$909$	15.0852	0.500345
$910$	0.330799	0.0109659
$911$	21.3214	0.706408	0.353204	−	0.935546i	$-0.385092\pi$
$911$	21.3214	0.706408	0.353204	+	0.935546i	$0.385092\pi$
$912$	3.61501	0.119705
$913$	−6.10174	−0.201938
$914$	−4.30297	−0.142330
$915$	−10.0227	−0.331342
$916$	−12.3531	−0.408158
$917$	14.0634	0.464415
$918$	11.6922	0.385899
$919$	34.2705	1.13048	0.565240	−	0.824926i	$-0.308783\pi$
$919$	34.2705	1.13048	0.565240	+	0.824926i	$0.308783\pi$
$920$	4.45353	0.146829
$921$	−26.1025	−0.860107
$922$	10.0599	0.331305
$923$	4.36727	0.143750
$924$	−0.802578	−0.0264029
$925$	10.2719	0.337738
$926$	−28.0482	−0.921721
$927$	23.0466	0.756950
$928$	−3.28953	−0.107984
$929$	32.2941	1.05953	0.529767	−	0.848143i	$-0.322279\pi$
$929$	32.2941	1.05953	0.529767	+	0.848143i	$0.322279\pi$
$930$	−7.93872	−0.260321
$931$	23.5336	0.771283
$932$	2.58763	0.0847608
$933$	30.8339	1.00946
$934$	−12.4846	−0.408508
$935$	−2.38237	−0.0779118
$936$	−0.823250	−0.0269088
$937$	−26.2531	−0.857652	−0.428826	−	0.903387i	$-0.641073\pi$
$937$	−26.2531	−0.857652	−0.428826	+	0.903387i	$0.641073\pi$
$938$	−3.33944	−0.109037
$939$	−1.66770	−0.0544234
$940$	−11.2371	−0.366513
$941$	−40.1571	−1.30908	−0.654542	−	0.756026i	$-0.727138\pi$
$941$	−40.1571	−1.30908	−0.654542	+	0.756026i	$0.727138\pi$
$942$	11.4342	0.372546
$943$	14.6392	0.476717
$944$	8.41377	0.273845
$945$	4.06010	0.132075
$946$	−1.00000	−0.0325128
$947$	23.3582	0.759040	0.379520	−	0.925183i	$-0.376089\pi$
$947$	23.3582	0.759040	0.379520	+	0.925183i	$0.376089\pi$
$948$	0.708423	0.0230085
$949$	1.91073	0.0620248
$950$	3.72626	0.120896
$951$	8.43383	0.273486
$952$	−1.97088	−0.0638766
$953$	26.1544	0.847225	0.423613	−	0.905843i	$-0.360762\pi$
$953$	26.1544	0.847225	0.423613	+	0.905843i	$0.360762\pi$
$954$	−24.7432	−0.801091
$955$	17.9787	0.581778
$956$	4.28284	0.138517
$957$	3.19131	0.103160
$958$	−5.80107	−0.187424
$959$	−16.0453	−0.518129
$960$	0.970143	0.0313112
$961$	35.9620	1.16007
$962$	4.10737	0.132427
$963$	−14.1368	−0.455552
$964$	−4.35843	−0.140376
$965$	−0.681617	−0.0219420
$966$	−3.57430	−0.115001
$967$	−7.40757	−0.238211	−0.119106	−	0.992882i	$-0.538003\pi$
$967$	−7.40757	−0.238211	−0.119106	+	0.992882i	$0.538003\pi$
$968$	−1.00000	−0.0321412
$969$	−8.61228	−0.276666
$970$	−5.35966	−0.172088
$971$	10.5570	0.338791	0.169396	−	0.985548i	$-0.445818\pi$
$971$	10.5570	0.338791	0.169396	+	0.985548i	$0.445818\pi$
$972$	−16.0963	−0.516289
$973$	17.7113	0.567798
$974$	−6.16836	−0.197647
$975$	0.387926	0.0124236
$976$	−10.3312	−0.330693
$977$	46.4746	1.48685	0.743427	−	0.668817i	$-0.233199\pi$
$977$	46.4746	1.48685	0.743427	+	0.668817i	$0.233199\pi$
$978$	−2.29180	−0.0732837
$979$	−11.4701	−0.366587
$980$	6.31561	0.201745
$981$	−18.6396	−0.595116
$982$	−2.78822	−0.0889756
$983$	−11.9035	−0.379663	−0.189832	−	0.981817i	$-0.560794\pi$
$983$	−11.9035	−0.379663	−0.189832	+	0.981817i	$0.560794\pi$
$984$	3.18895	0.101660
$985$	−14.1094	−0.449562
$986$	7.83687	0.249577
$987$	9.01863	0.287066
$988$	1.49000	0.0474032
$989$	−4.45353	−0.141614
$990$	2.05882	0.0654336
$991$	−62.2185	−1.97643	−0.988217	−	0.153058i	$-0.951088\pi$
$991$	−62.2185	−1.97643	−0.988217	+	0.153058i	$0.951088\pi$
$992$	−8.18303	−0.259812
$993$	19.9729	0.633821
$994$	−9.03542	−0.286586
$995$	−13.0486	−0.413669
$996$	−5.91957	−0.187569
$997$	−26.6020	−0.842493	−0.421246	−	0.906946i	$-0.638407\pi$
$997$	−26.6020	−0.842493	−0.421246	+	0.906946i	$0.638407\pi$
$998$	36.9207	1.16870
$999$	50.4122	1.59497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4730.2.a.x.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4730.2.a.x.1.4	✓	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:	\( N \)	\(=\)	\( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4730.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.970143 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.970143 q^{6} -0.827277 q^{7} -1.00000 q^{8} -2.05882 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.970143 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.970143 q^{6} -0.827277 q^{7} -1.00000 q^{8} -2.05882 q^{9} +1.00000 q^{10} -1.00000 q^{11} -0.970143 q^{12} -0.399865 q^{13} +0.827277 q^{14} +0.970143 q^{15} +1.00000 q^{16} -2.38237 q^{17} +2.05882 q^{18} -3.72626 q^{19} -1.00000 q^{20} +0.802578 q^{21} +1.00000 q^{22} +4.45353 q^{23} +0.970143 q^{24} +1.00000 q^{25} +0.399865 q^{26} +4.90778 q^{27} -0.827277 q^{28} +3.28953 q^{29} -0.970143 q^{30} +8.18303 q^{31} -1.00000 q^{32} +0.970143 q^{33} +2.38237 q^{34} +0.827277 q^{35} -2.05882 q^{36} +10.2719 q^{37} +3.72626 q^{38} +0.387926 q^{39} +1.00000 q^{40} +3.28709 q^{41} -0.802578 q^{42} -1.00000 q^{43} -1.00000 q^{44} +2.05882 q^{45} -4.45353 q^{46} +11.2371 q^{47} -0.970143 q^{48} -6.31561 q^{49} -1.00000 q^{50} +2.31124 q^{51} -0.399865 q^{52} -12.0181 q^{53} -4.90778 q^{54} +1.00000 q^{55} +0.827277 q^{56} +3.61501 q^{57} -3.28953 q^{58} +8.41377 q^{59} +0.970143 q^{60} -10.3312 q^{61} -8.18303 q^{62} +1.70322 q^{63} +1.00000 q^{64} +0.399865 q^{65} -0.970143 q^{66} -4.03666 q^{67} -2.38237 q^{68} -4.32056 q^{69} -0.827277 q^{70} -10.9219 q^{71} +2.05882 q^{72} -4.77844 q^{73} -10.2719 q^{74} -0.970143 q^{75} -3.72626 q^{76} +0.827277 q^{77} -0.387926 q^{78} -0.730224 q^{79} -1.00000 q^{80} +1.41521 q^{81} -3.28709 q^{82} +6.10174 q^{83} +0.802578 q^{84} +2.38237 q^{85} +1.00000 q^{86} -3.19131 q^{87} +1.00000 q^{88} +11.4701 q^{89} -2.05882 q^{90} +0.330799 q^{91} +4.45353 q^{92} -7.93872 q^{93} -11.2371 q^{94} +3.72626 q^{95} +0.970143 q^{96} -5.35966 q^{97} +6.31561 q^{98} +2.05882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} - 5 q^{7} - 8 q^{8} + 10 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 - 5 * q^7 - 8 * q^8 + 10 * q^9	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} - 5 q^{7} - 8 q^{8} + 10 q^{9} + 8 q^{10} - 8 q^{11} - 3 q^{13} + 5 q^{14} + 8 q^{16} + 10 q^{17} - 10 q^{18} - 15 q^{19} - 8 q^{20} - 6 q^{21} + 8 q^{22} + 12 q^{23} + 8 q^{25} + 3 q^{26} + 9 q^{27} - 5 q^{28} - 6 q^{29} - 4 q^{31} - 8 q^{32} - 10 q^{34} + 5 q^{35} + 10 q^{36} - q^{37} + 15 q^{38} - 12 q^{39} + 8 q^{40} + 3 q^{41} + 6 q^{42} - 8 q^{43} - 8 q^{44} - 10 q^{45} - 12 q^{46} + 11 q^{47} + 7 q^{49} - 8 q^{50} - 13 q^{51} - 3 q^{52} + 24 q^{53} - 9 q^{54} + 8 q^{55} + 5 q^{56} - 7 q^{57} + 6 q^{58} - 5 q^{59} - 26 q^{61} + 4 q^{62} + 12 q^{63} + 8 q^{64} + 3 q^{65} - 5 q^{67} + 10 q^{68} - 22 q^{69} - 5 q^{70} + 10 q^{71} - 10 q^{72} - q^{73} + q^{74} - 15 q^{76} + 5 q^{77} + 12 q^{78} - 24 q^{79} - 8 q^{80} - 3 q^{82} - 15 q^{83} - 6 q^{84} - 10 q^{85} + 8 q^{86} + 15 q^{87} + 8 q^{88} - q^{89} + 10 q^{90} - 35 q^{91} + 12 q^{92} + 23 q^{93} - 11 q^{94} + 15 q^{95} - 30 q^{97} - 7 q^{98} - 10 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 - 5 * q^7 - 8 * q^8 + 10 * q^9 + 8 * q^10 - 8 * q^11 - 3 * q^13 + 5 * q^14 + 8 * q^16 + 10 * q^17 - 10 * q^18 - 15 * q^19 - 8 * q^20 - 6 * q^21 + 8 * q^22 + 12 * q^23 + 8 * q^25 + 3 * q^26 + 9 * q^27 - 5 * q^28 - 6 * q^29 - 4 * q^31 - 8 * q^32 - 10 * q^34 + 5 * q^35 + 10 * q^36 - q^37 + 15 * q^38 - 12 * q^39 + 8 * q^40 + 3 * q^41 + 6 * q^42 - 8 * q^43 - 8 * q^44 - 10 * q^45 - 12 * q^46 + 11 * q^47 + 7 * q^49 - 8 * q^50 - 13 * q^51 - 3 * q^52 + 24 * q^53 - 9 * q^54 + 8 * q^55 + 5 * q^56 - 7 * q^57 + 6 * q^58 - 5 * q^59 - 26 * q^61 + 4 * q^62 + 12 * q^63 + 8 * q^64 + 3 * q^65 - 5 * q^67 + 10 * q^68 - 22 * q^69 - 5 * q^70 + 10 * q^71 - 10 * q^72 - q^73 + q^74 - 15 * q^76 + 5 * q^77 + 12 * q^78 - 24 * q^79 - 8 * q^80 - 3 * q^82 - 15 * q^83 - 6 * q^84 - 10 * q^85 + 8 * q^86 + 15 * q^87 + 8 * q^88 - q^89 + 10 * q^90 - 35 * q^91 + 12 * q^92 + 23 * q^93 - 11 * q^94 + 15 * q^95 - 30 * q^97 - 7 * q^98 - 10 * q^99

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