LMFDB - Embedded newform 4730.2.a.x.1.8

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			1.8
Root			$3.09237$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +3.09237 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.09237 q^{6} -2.30961 q^{7} -1.00000 q^{8} +6.56278 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.09237 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.09237 q^{6} -2.30961 q^{7} -1.00000 q^{8} +6.56278 q^{9} +1.00000 q^{10} -1.00000 q^{11} +3.09237 q^{12} +1.84222 q^{13} +2.30961 q^{14} -3.09237 q^{15} +1.00000 q^{16} -1.97995 q^{17} -6.56278 q^{18} -6.24277 q^{19} -1.00000 q^{20} -7.14218 q^{21} +1.00000 q^{22} -1.04558 q^{23} -3.09237 q^{24} +1.00000 q^{25} -1.84222 q^{26} +11.0174 q^{27} -2.30961 q^{28} -5.26513 q^{29} +3.09237 q^{30} +4.56461 q^{31} -1.00000 q^{32} -3.09237 q^{33} +1.97995 q^{34} +2.30961 q^{35} +6.56278 q^{36} -9.21013 q^{37} +6.24277 q^{38} +5.69682 q^{39} +1.00000 q^{40} -1.67436 q^{41} +7.14218 q^{42} -1.00000 q^{43} -1.00000 q^{44} -6.56278 q^{45} +1.04558 q^{46} -6.13413 q^{47} +3.09237 q^{48} -1.66570 q^{49} -1.00000 q^{50} -6.12275 q^{51} +1.84222 q^{52} +11.4025 q^{53} -11.0174 q^{54} +1.00000 q^{55} +2.30961 q^{56} -19.3050 q^{57} +5.26513 q^{58} -8.26598 q^{59} -3.09237 q^{60} -8.57037 q^{61} -4.56461 q^{62} -15.1575 q^{63} +1.00000 q^{64} -1.84222 q^{65} +3.09237 q^{66} -8.29005 q^{67} -1.97995 q^{68} -3.23332 q^{69} -2.30961 q^{70} +8.71065 q^{71} -6.56278 q^{72} -13.6268 q^{73} +9.21013 q^{74} +3.09237 q^{75} -6.24277 q^{76} +2.30961 q^{77} -5.69682 q^{78} +8.60734 q^{79} -1.00000 q^{80} +14.3817 q^{81} +1.67436 q^{82} +7.00464 q^{83} -7.14218 q^{84} +1.97995 q^{85} +1.00000 q^{86} -16.2818 q^{87} +1.00000 q^{88} +2.89026 q^{89} +6.56278 q^{90} -4.25480 q^{91} -1.04558 q^{92} +14.1155 q^{93} +6.13413 q^{94} +6.24277 q^{95} -3.09237 q^{96} -13.1837 q^{97} +1.66570 q^{98} -6.56278 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.09237	1.78538	0.892692	−	0.450668i	$-0.148814\pi$
$3$	3.09237	1.78538	0.892692	+	0.450668i	$0.148814\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.09237	−1.26246
$7$	−2.30961	−0.872951	−0.436475	−	0.899716i	$-0.643773\pi$
$7$	−2.30961	−0.872951	−0.436475	+	0.899716i	$0.643773\pi$
$8$	−1.00000	−0.353553
$9$	6.56278	2.18759
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	3.09237	0.892692
$13$	1.84222	0.510939	0.255469	−	0.966817i	$-0.417770\pi$
$13$	1.84222	0.510939	0.255469	+	0.966817i	$0.417770\pi$
$14$	2.30961	0.617269
$15$	−3.09237	−0.798448
$16$	1.00000	0.250000
$17$	−1.97995	−0.480209	−0.240105	−	0.970747i	$-0.577182\pi$
$17$	−1.97995	−0.480209	−0.240105	+	0.970747i	$0.577182\pi$
$18$	−6.56278	−1.54686
$19$	−6.24277	−1.43219	−0.716095	−	0.698003i	$-0.754072\pi$
$19$	−6.24277	−1.43219	−0.716095	+	0.698003i	$0.754072\pi$
$20$	−1.00000	−0.223607
$21$	−7.14218	−1.55855
$22$	1.00000	0.213201
$23$	−1.04558	−0.218018	−0.109009	−	0.994041i	$-0.534768\pi$
$23$	−1.04558	−0.218018	−0.109009	+	0.994041i	$0.534768\pi$
$24$	−3.09237	−0.631228
$25$	1.00000	0.200000
$26$	−1.84222	−0.361288
$27$	11.0174	2.12031
$28$	−2.30961	−0.436475
$29$	−5.26513	−0.977711	−0.488855	−	0.872365i	$-0.662585\pi$
$29$	−5.26513	−0.977711	−0.488855	+	0.872365i	$0.662585\pi$
$30$	3.09237	0.564588
$31$	4.56461	0.819829	0.409914	−	0.912124i	$-0.365559\pi$
$31$	4.56461	0.819829	0.409914	+	0.912124i	$0.365559\pi$
$32$	−1.00000	−0.176777
$33$	−3.09237	−0.538313
$34$	1.97995	0.339559
$35$	2.30961	0.390395
$36$	6.56278	1.09380
$37$	−9.21013	−1.51414	−0.757068	−	0.653336i	$-0.773369\pi$
$37$	−9.21013	−1.51414	−0.757068	+	0.653336i	$0.773369\pi$
$38$	6.24277	1.01271
$39$	5.69682	0.912222
$40$	1.00000	0.158114
$41$	−1.67436	−0.261490	−0.130745	−	0.991416i	$-0.541737\pi$
$41$	−1.67436	−0.261490	−0.130745	+	0.991416i	$0.541737\pi$
$42$	7.14218	1.10206
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−1.00000	−0.150756
$45$	−6.56278	−0.978321
$46$	1.04558	0.154162
$47$	−6.13413	−0.894755	−0.447377	−	0.894345i	$-0.647642\pi$
$47$	−6.13413	−0.894755	−0.447377	+	0.894345i	$0.647642\pi$
$48$	3.09237	0.446346
$49$	−1.66570	−0.237957
$50$	−1.00000	−0.141421
$51$	−6.12275	−0.857357
$52$	1.84222	0.255469
$53$	11.4025	1.56625	0.783125	−	0.621864i	$-0.213624\pi$
$53$	11.4025	1.56625	0.783125	+	0.621864i	$0.213624\pi$
$54$	−11.0174	−1.49928
$55$	1.00000	0.134840
$56$	2.30961	0.308635
$57$	−19.3050	−2.55701
$58$	5.26513	0.691346
$59$	−8.26598	−1.07614	−0.538070	−	0.842900i	$-0.680846\pi$
$59$	−8.26598	−1.07614	−0.538070	+	0.842900i	$0.680846\pi$
$60$	−3.09237	−0.399224
$61$	−8.57037	−1.09732	−0.548662	−	0.836044i	$-0.684863\pi$
$61$	−8.57037	−1.09732	−0.548662	+	0.836044i	$0.684863\pi$
$62$	−4.56461	−0.579706
$63$	−15.1575	−1.90966
$64$	1.00000	0.125000
$65$	−1.84222	−0.228499
$66$	3.09237	0.380645
$67$	−8.29005	−1.01279	−0.506395	−	0.862301i	$-0.669022\pi$
$67$	−8.29005	−1.01279	−0.506395	+	0.862301i	$0.669022\pi$
$68$	−1.97995	−0.240105
$69$	−3.23332	−0.389246
$70$	−2.30961	−0.276051
$71$	8.71065	1.03376	0.516882	−	0.856057i	$-0.327093\pi$
$71$	8.71065	1.03376	0.516882	+	0.856057i	$0.327093\pi$
$72$	−6.56278	−0.773431
$73$	−13.6268	−1.59489	−0.797446	−	0.603390i	$-0.793816\pi$
$73$	−13.6268	−1.59489	−0.797446	+	0.603390i	$0.793816\pi$
$74$	9.21013	1.07066
$75$	3.09237	0.357077
$76$	−6.24277	−0.716095
$77$	2.30961	0.263205
$78$	−5.69682	−0.645038
$79$	8.60734	0.968402	0.484201	−	0.874957i	$-0.339110\pi$
$79$	8.60734	0.968402	0.484201	+	0.874957i	$0.339110\pi$
$80$	−1.00000	−0.111803
$81$	14.3817	1.59797
$82$	1.67436	0.184902
$83$	7.00464	0.768860	0.384430	−	0.923154i	$-0.374398\pi$
$83$	7.00464	0.768860	0.384430	+	0.923154i	$0.374398\pi$
$84$	−7.14218	−0.779276
$85$	1.97995	0.214756
$86$	1.00000	0.107833
$87$	−16.2818	−1.74559
$88$	1.00000	0.106600
$89$	2.89026	0.306367	0.153183	−	0.988198i	$-0.451048\pi$
$89$	2.89026	0.306367	0.153183	+	0.988198i	$0.451048\pi$
$90$	6.56278	0.691778
$91$	−4.25480	−0.446024
$92$	−1.04558	−0.109009
$93$	14.1155	1.46371
$94$	6.13413	0.632687
$95$	6.24277	0.640494
$96$	−3.09237	−0.315614
$97$	−13.1837	−1.33860	−0.669301	−	0.742991i	$-0.733406\pi$
$97$	−13.1837	−1.33860	−0.669301	+	0.742991i	$0.733406\pi$
$98$	1.66570	0.168261
$99$	−6.56278	−0.659584
$100$	1.00000	0.100000
$101$	−17.6319	−1.75444	−0.877221	−	0.480086i	$-0.840605\pi$
$101$	−17.6319	−1.75444	−0.877221	+	0.480086i	$0.840605\pi$
$102$	6.12275	0.606243
$103$	16.7220	1.64766	0.823832	−	0.566835i	$-0.191832\pi$
$103$	16.7220	1.64766	0.823832	+	0.566835i	$0.191832\pi$
$104$	−1.84222	−0.180644
$105$	7.14218	0.697005
$106$	−11.4025	−1.10751
$107$	4.58725	0.443467	0.221733	−	0.975107i	$-0.428829\pi$
$107$	4.58725	0.443467	0.221733	+	0.975107i	$0.428829\pi$
$108$	11.0174	1.06015
$109$	8.53501	0.817506	0.408753	−	0.912645i	$-0.365964\pi$
$109$	8.53501	0.817506	0.408753	+	0.912645i	$0.365964\pi$
$110$	−1.00000	−0.0953463
$111$	−28.4812	−2.70331
$112$	−2.30961	−0.218238
$113$	8.36324	0.786748	0.393374	−	0.919379i	$-0.371308\pi$
$113$	8.36324	0.786748	0.393374	+	0.919379i	$0.371308\pi$
$114$	19.3050	1.80808
$115$	1.04558	0.0975007
$116$	−5.26513	−0.488855
$117$	12.0901	1.11773
$118$	8.26598	0.760945
$119$	4.57292	0.419199
$120$	3.09237	0.282294
$121$	1.00000	0.0909091
$122$	8.57037	0.775925
$123$	−5.17773	−0.466860
$124$	4.56461	0.409914
$125$	−1.00000	−0.0894427
$126$	15.1575	1.35033
$127$	1.25971	0.111781	0.0558904	−	0.998437i	$-0.482200\pi$
$127$	1.25971	0.111781	0.0558904	+	0.998437i	$0.482200\pi$
$128$	−1.00000	−0.0883883
$129$	−3.09237	−0.272268
$130$	1.84222	0.161573
$131$	9.55472	0.834800	0.417400	−	0.908723i	$-0.362941\pi$
$131$	9.55472	0.834800	0.417400	+	0.908723i	$0.362941\pi$
$132$	−3.09237	−0.269157
$133$	14.4184	1.25023
$134$	8.29005	0.716151
$135$	−11.0174	−0.948231
$136$	1.97995	0.169780
$137$	1.95906	0.167374	0.0836870	−	0.996492i	$-0.473330\pi$
$137$	1.95906	0.167374	0.0836870	+	0.996492i	$0.473330\pi$
$138$	3.23332	0.275239
$139$	−16.3091	−1.38332	−0.691659	−	0.722224i	$-0.743120\pi$
$139$	−16.3091	−1.38332	−0.691659	+	0.722224i	$0.743120\pi$
$140$	2.30961	0.195198
$141$	−18.9690	−1.59748
$142$	−8.71065	−0.730981
$143$	−1.84222	−0.154054
$144$	6.56278	0.546898
$145$	5.26513	0.437246
$146$	13.6268	1.12776
$147$	−5.15096	−0.424844
$148$	−9.21013	−0.757068
$149$	1.00521	0.0823504	0.0411752	−	0.999152i	$-0.486890\pi$
$149$	1.00521	0.0823504	0.0411752	+	0.999152i	$0.486890\pi$
$150$	−3.09237	−0.252491
$151$	−15.4309	−1.25575	−0.627876	−	0.778313i	$-0.716076\pi$
$151$	−15.4309	−1.25575	−0.627876	+	0.778313i	$0.716076\pi$
$152$	6.24277	0.506355
$153$	−12.9940	−1.05050
$154$	−2.30961	−0.186114
$155$	−4.56461	−0.366638
$156$	5.69682	0.456111
$157$	10.6544	0.850314	0.425157	−	0.905120i	$-0.360219\pi$
$157$	10.6544	0.850314	0.425157	+	0.905120i	$0.360219\pi$
$158$	−8.60734	−0.684763
$159$	35.2607	2.79636
$160$	1.00000	0.0790569
$161$	2.41488	0.190319
$162$	−14.3817	−1.12994
$163$	−19.1660	−1.50120	−0.750599	−	0.660758i	$-0.770235\pi$
$163$	−19.1660	−1.50120	−0.750599	+	0.660758i	$0.770235\pi$
$164$	−1.67436	−0.130745
$165$	3.09237	0.240741
$166$	−7.00464	−0.543666
$167$	−2.54750	−0.197132	−0.0985659	−	0.995131i	$-0.531426\pi$
$167$	−2.54750	−0.197132	−0.0985659	+	0.995131i	$0.531426\pi$
$168$	7.14218	0.551031
$169$	−9.60624	−0.738942
$170$	−1.97995	−0.151855
$171$	−40.9699	−3.13305
$172$	−1.00000	−0.0762493
$173$	0.341149	0.0259371	0.0129685	−	0.999916i	$-0.495872\pi$
$173$	0.341149	0.0259371	0.0129685	+	0.999916i	$0.495872\pi$
$174$	16.2818	1.23432
$175$	−2.30961	−0.174590
$176$	−1.00000	−0.0753778
$177$	−25.5615	−1.92132
$178$	−2.89026	−0.216634
$179$	−11.0830	−0.828382	−0.414191	−	0.910190i	$-0.635935\pi$
$179$	−11.0830	−0.828382	−0.414191	+	0.910190i	$0.635935\pi$
$180$	−6.56278	−0.489161
$181$	−0.0454526	−0.00337847	−0.00168923	−	0.999999i	$-0.500538\pi$
$181$	−0.0454526	−0.00337847	−0.00168923	+	0.999999i	$0.500538\pi$
$182$	4.25480	0.315387
$183$	−26.5028	−1.95914
$184$	1.04558	0.0770811
$185$	9.21013	0.677142
$186$	−14.1155	−1.03500
$187$	1.97995	0.144788
$188$	−6.13413	−0.447377
$189$	−25.4460	−1.85093
$190$	−6.24277	−0.452898
$191$	−18.5249	−1.34042	−0.670208	−	0.742174i	$-0.733795\pi$
$191$	−18.5249	−1.34042	−0.670208	+	0.742174i	$0.733795\pi$
$192$	3.09237	0.223173
$193$	10.1472	0.730410	0.365205	−	0.930927i	$-0.380999\pi$
$193$	10.1472	0.730410	0.365205	+	0.930927i	$0.380999\pi$
$194$	13.1837	0.946535
$195$	−5.69682	−0.407958
$196$	−1.66570	−0.118978
$197$	3.91393	0.278856	0.139428	−	0.990232i	$-0.455474\pi$
$197$	3.91393	0.278856	0.139428	+	0.990232i	$0.455474\pi$
$198$	6.56278	0.466396
$199$	5.21583	0.369740	0.184870	−	0.982763i	$-0.440814\pi$
$199$	5.21583	0.369740	0.184870	+	0.982763i	$0.440814\pi$
$200$	−1.00000	−0.0707107
$201$	−25.6359	−1.80822
$202$	17.6319	1.24058
$203$	12.1604	0.853494
$204$	−6.12275	−0.428679
$205$	1.67436	0.116942
$206$	−16.7220	−1.16507
$207$	−6.86190	−0.476935
$208$	1.84222	0.127735
$209$	6.24277	0.431821
$210$	−7.14218	−0.492857
$211$	−11.4474	−0.788070	−0.394035	−	0.919095i	$-0.628921\pi$
$211$	−11.4474	−0.788070	−0.394035	+	0.919095i	$0.628921\pi$
$212$	11.4025	0.783125
$213$	26.9366	1.84566
$214$	−4.58725	−0.313578
$215$	1.00000	0.0681994
$216$	−11.0174	−0.749642
$217$	−10.5425	−0.715670
$218$	−8.53501	−0.578064
$219$	−42.1391	−2.84749
$220$	1.00000	0.0674200
$221$	−3.64750	−0.245357
$222$	28.4812	1.91153
$223$	−4.76668	−0.319200	−0.159600	−	0.987182i	$-0.551021\pi$
$223$	−4.76668	−0.319200	−0.159600	+	0.987182i	$0.551021\pi$
$224$	2.30961	0.154317
$225$	6.56278	0.437519
$226$	−8.36324	−0.556315
$227$	−1.79236	−0.118963	−0.0594814	−	0.998229i	$-0.518945\pi$
$227$	−1.79236	−0.118963	−0.0594814	+	0.998229i	$0.518945\pi$
$228$	−19.3050	−1.27850
$229$	−7.65953	−0.506156	−0.253078	−	0.967446i	$-0.581443\pi$
$229$	−7.65953	−0.506156	−0.253078	+	0.967446i	$0.581443\pi$
$230$	−1.04558	−0.0689434
$231$	7.14218	0.469921
$232$	5.26513	0.345673
$233$	−5.24643	−0.343705	−0.171853	−	0.985123i	$-0.554975\pi$
$233$	−5.24643	−0.343705	−0.171853	+	0.985123i	$0.554975\pi$
$234$	−12.0901	−0.790352
$235$	6.13413	0.400146
$236$	−8.26598	−0.538070
$237$	26.6171	1.72897
$238$	−4.57292	−0.296418
$239$	10.4893	0.678499	0.339249	−	0.940696i	$-0.389827\pi$
$239$	10.4893	0.678499	0.339249	+	0.940696i	$0.389827\pi$
$240$	−3.09237	−0.199612
$241$	16.9970	1.09487	0.547437	−	0.836847i	$-0.315604\pi$
$241$	16.9970	1.09487	0.547437	+	0.836847i	$0.315604\pi$
$242$	−1.00000	−0.0642824
$243$	11.4214	0.732682
$244$	−8.57037	−0.548662
$245$	1.66570	0.106418
$246$	5.17773	0.330120
$247$	−11.5005	−0.731761
$248$	−4.56461	−0.289853
$249$	21.6610	1.37271
$250$	1.00000	0.0632456
$251$	−16.0124	−1.01069	−0.505347	−	0.862916i	$-0.668636\pi$
$251$	−16.0124	−1.01069	−0.505347	+	0.862916i	$0.668636\pi$
$252$	−15.1575	−0.954831
$253$	1.04558	0.0657350
$254$	−1.25971	−0.0790409
$255$	6.12275	0.383422
$256$	1.00000	0.0625000
$257$	7.23799	0.451494	0.225747	−	0.974186i	$-0.427518\pi$
$257$	7.23799	0.451494	0.225747	+	0.974186i	$0.427518\pi$
$258$	3.09237	0.192523
$259$	21.2718	1.32177
$260$	−1.84222	−0.114249
$261$	−34.5539	−2.13883
$262$	−9.55472	−0.590293
$263$	−18.8966	−1.16521	−0.582607	−	0.812754i	$-0.697967\pi$
$263$	−18.8966	−1.16521	−0.582607	+	0.812754i	$0.697967\pi$
$264$	3.09237	0.190322
$265$	−11.4025	−0.700448
$266$	−14.4184	−0.884046
$267$	8.93776	0.546982
$268$	−8.29005	−0.506395
$269$	7.47058	0.455489	0.227745	−	0.973721i	$-0.426865\pi$
$269$	7.47058	0.455489	0.227745	+	0.973721i	$0.426865\pi$
$270$	11.0174	0.670501
$271$	19.6848	1.19576	0.597882	−	0.801584i	$-0.296009\pi$
$271$	19.6848	1.19576	0.597882	+	0.801584i	$0.296009\pi$
$272$	−1.97995	−0.120052
$273$	−13.1574	−0.796325
$274$	−1.95906	−0.118351
$275$	−1.00000	−0.0603023
$276$	−3.23332	−0.194623
$277$	−10.3385	−0.621181	−0.310591	−	0.950544i	$-0.600527\pi$
$277$	−10.3385	−0.621181	−0.310591	+	0.950544i	$0.600527\pi$
$278$	16.3091	0.978154
$279$	29.9565	1.79345
$280$	−2.30961	−0.138026
$281$	−12.5862	−0.750833	−0.375416	−	0.926856i	$-0.622500\pi$
$281$	−12.5862	−0.750833	−0.375416	+	0.926856i	$0.622500\pi$
$282$	18.9690	1.12959
$283$	−24.6778	−1.46694	−0.733471	−	0.679721i	$-0.762101\pi$
$283$	−24.6778	−1.46694	−0.733471	+	0.679721i	$0.762101\pi$
$284$	8.71065	0.516882
$285$	19.3050	1.14353
$286$	1.84222	0.108933
$287$	3.86711	0.228268
$288$	−6.56278	−0.386716
$289$	−13.0798	−0.769399
$290$	−5.26513	−0.309179
$291$	−40.7689	−2.38992
$292$	−13.6268	−0.797446
$293$	25.4075	1.48432	0.742161	−	0.670222i	$-0.233801\pi$
$293$	25.4075	1.48432	0.742161	+	0.670222i	$0.233801\pi$
$294$	5.15096	0.300410
$295$	8.26598	0.481264
$296$	9.21013	0.535328
$297$	−11.0174	−0.639297
$298$	−1.00521	−0.0582305
$299$	−1.92618	−0.111394
$300$	3.09237	0.178538
$301$	2.30961	0.133124
$302$	15.4309	0.887951
$303$	−54.5245	−3.13235
$304$	−6.24277	−0.358047
$305$	8.57037	0.490738
$306$	12.9940	0.742817
$307$	−25.0161	−1.42775	−0.713873	−	0.700276i	$-0.753060\pi$
$307$	−25.0161	−1.42775	−0.713873	+	0.700276i	$0.753060\pi$
$308$	2.30961	0.131602
$309$	51.7105	2.94171
$310$	4.56461	0.259253
$311$	15.9607	0.905047	0.452523	−	0.891753i	$-0.350524\pi$
$311$	15.9607	0.905047	0.452523	+	0.891753i	$0.350524\pi$
$312$	−5.69682	−0.322519
$313$	21.7606	1.22998	0.614990	−	0.788535i	$-0.289160\pi$
$313$	21.7606	1.22998	0.614990	+	0.788535i	$0.289160\pi$
$314$	−10.6544	−0.601263
$315$	15.1575	0.854026
$316$	8.60734	0.484201
$317$	−19.5623	−1.09873	−0.549363	−	0.835584i	$-0.685130\pi$
$317$	−19.5623	−1.09873	−0.549363	+	0.835584i	$0.685130\pi$
$318$	−35.2607	−1.97732
$319$	5.26513	0.294791
$320$	−1.00000	−0.0559017
$321$	14.1855	0.791758
$322$	−2.41488	−0.134576
$323$	12.3604	0.687750
$324$	14.3817	0.798985
$325$	1.84222	0.102188
$326$	19.1660	1.06151
$327$	26.3934	1.45956
$328$	1.67436	0.0924508
$329$	14.1674	0.781077
$330$	−3.09237	−0.170230
$331$	−8.41201	−0.462366	−0.231183	−	0.972910i	$-0.574260\pi$
$331$	−8.41201	−0.462366	−0.231183	+	0.972910i	$0.574260\pi$
$332$	7.00464	0.384430
$333$	−60.4440	−3.31231
$334$	2.54750	0.139393
$335$	8.29005	0.452934
$336$	−7.14218	−0.389638
$337$	0.0897684	0.00489000	0.00244500	−	0.999997i	$-0.499222\pi$
$337$	0.0897684	0.00489000	0.00244500	+	0.999997i	$0.499222\pi$
$338$	9.60624	0.522511
$339$	25.8623	1.40465
$340$	1.97995	0.107378
$341$	−4.56461	−0.247188
$342$	40.9699	2.21540
$343$	20.0144	1.08068
$344$	1.00000	0.0539164
$345$	3.23332	0.174076
$346$	−0.341149	−0.0183403
$347$	−18.9755	−1.01866	−0.509328	−	0.860572i	$-0.670106\pi$
$347$	−18.9755	−1.01866	−0.509328	+	0.860572i	$0.670106\pi$
$348$	−16.2818	−0.872794
$349$	22.4731	1.20296	0.601479	−	0.798889i	$-0.294578\pi$
$349$	22.4731	1.20296	0.601479	+	0.798889i	$0.294578\pi$
$350$	2.30961	0.123454
$351$	20.2965	1.08335
$352$	1.00000	0.0533002
$353$	24.3202	1.29443	0.647217	−	0.762306i	$-0.275933\pi$
$353$	24.3202	1.29443	0.647217	+	0.762306i	$0.275933\pi$
$354$	25.5615	1.35858
$355$	−8.71065	−0.462313
$356$	2.89026	0.153183
$357$	14.1412	0.748431
$358$	11.0830	0.585754
$359$	0.852884	0.0450135	0.0225067	−	0.999747i	$-0.492835\pi$
$359$	0.852884	0.0450135	0.0225067	+	0.999747i	$0.492835\pi$
$360$	6.56278	0.345889
$361$	19.9721	1.05117
$362$	0.0454526	0.00238894
$363$	3.09237	0.162308
$364$	−4.25480	−0.223012
$365$	13.6268	0.713258
$366$	26.5028	1.38532
$367$	18.9665	0.990043	0.495022	−	0.868881i	$-0.335160\pi$
$367$	18.9665	0.990043	0.495022	+	0.868881i	$0.335160\pi$
$368$	−1.04558	−0.0545046
$369$	−10.9884	−0.572034
$370$	−9.21013	−0.478812
$371$	−26.3353	−1.36726
$372$	14.1155	0.731854
$373$	12.7360	0.659445	0.329723	−	0.944078i	$-0.393045\pi$
$373$	12.7360	0.659445	0.329723	+	0.944078i	$0.393045\pi$
$374$	−1.97995	−0.102381
$375$	−3.09237	−0.159690
$376$	6.13413	0.316344
$377$	−9.69952	−0.499550
$378$	25.4460	1.30880
$379$	36.0044	1.84942	0.924712	−	0.380667i	$-0.124306\pi$
$379$	36.0044	1.84942	0.924712	+	0.380667i	$0.124306\pi$
$380$	6.24277	0.320247
$381$	3.89548	0.199572
$382$	18.5249	0.947817
$383$	−0.811006	−0.0414405	−0.0207202	−	0.999785i	$-0.506596\pi$
$383$	−0.811006	−0.0414405	−0.0207202	+	0.999785i	$0.506596\pi$
$384$	−3.09237	−0.157807
$385$	−2.30961	−0.117709
$386$	−10.1472	−0.516478
$387$	−6.56278	−0.333605
$388$	−13.1837	−0.669301
$389$	−9.45110	−0.479190	−0.239595	−	0.970873i	$-0.577015\pi$
$389$	−9.45110	−0.479190	−0.239595	+	0.970873i	$0.577015\pi$
$390$	5.69682	0.288470
$391$	2.07020	0.104694
$392$	1.66570	0.0841305
$393$	29.5468	1.49044
$394$	−3.91393	−0.197181
$395$	−8.60734	−0.433082
$396$	−6.56278	−0.329792
$397$	−3.99166	−0.200336	−0.100168	−	0.994971i	$-0.531938\pi$
$397$	−3.99166	−0.200336	−0.100168	+	0.994971i	$0.531938\pi$
$398$	−5.21583	−0.261446
$399$	44.5870	2.23214
$400$	1.00000	0.0500000
$401$	24.5862	1.22778	0.613888	−	0.789393i	$-0.289605\pi$
$401$	24.5862	1.22778	0.613888	+	0.789393i	$0.289605\pi$
$402$	25.6359	1.27860
$403$	8.40900	0.418882
$404$	−17.6319	−0.877221
$405$	−14.3817	−0.714634
$406$	−12.1604	−0.603511
$407$	9.21013	0.456529
$408$	6.12275	0.303122
$409$	−32.2209	−1.59322	−0.796610	−	0.604493i	$-0.793376\pi$
$409$	−32.2209	−1.59322	−0.796610	+	0.604493i	$0.793376\pi$
$410$	−1.67436	−0.0826905
$411$	6.05815	0.298827
$412$	16.7220	0.823832
$413$	19.0912	0.939416
$414$	6.86190	0.337244
$415$	−7.00464	−0.343844
$416$	−1.84222	−0.0903221
$417$	−50.4338	−2.46975
$418$	−6.24277	−0.305344
$419$	−9.15384	−0.447195	−0.223597	−	0.974682i	$-0.571780\pi$
$419$	−9.15384	−0.447195	−0.223597	+	0.974682i	$0.571780\pi$
$420$	7.14218	0.348503
$421$	4.86053	0.236888	0.118444	−	0.992961i	$-0.462209\pi$
$421$	4.86053	0.236888	0.118444	+	0.992961i	$0.462209\pi$
$422$	11.4474	0.557250
$423$	−40.2569	−1.95736
$424$	−11.4025	−0.553753
$425$	−1.97995	−0.0960418
$426$	−26.9366	−1.30508
$427$	19.7942	0.957909
$428$	4.58725	0.221733
$429$	−5.69682	−0.275045
$430$	−1.00000	−0.0482243
$431$	1.05622	0.0508762	0.0254381	−	0.999676i	$-0.491902\pi$
$431$	1.05622	0.0508762	0.0254381	+	0.999676i	$0.491902\pi$
$432$	11.0174	0.530077
$433$	−2.44052	−0.117284	−0.0586420	−	0.998279i	$-0.518677\pi$
$433$	−2.44052	−0.117284	−0.0586420	+	0.998279i	$0.518677\pi$
$434$	10.5425	0.506055
$435$	16.2818	0.780651
$436$	8.53501	0.408753
$437$	6.52730	0.312243
$438$	42.1391	2.01348
$439$	−10.9702	−0.523578	−0.261789	−	0.965125i	$-0.584312\pi$
$439$	−10.9702	−0.523578	−0.261789	+	0.965125i	$0.584312\pi$
$440$	−1.00000	−0.0476731
$441$	−10.9316	−0.520553
$442$	3.64750	0.173494
$443$	24.8491	1.18062	0.590309	−	0.807177i	$-0.299006\pi$
$443$	24.8491	1.18062	0.590309	+	0.807177i	$0.299006\pi$
$444$	−28.4812	−1.35166
$445$	−2.89026	−0.137011
$446$	4.76668	0.225709
$447$	3.10850	0.147027
$448$	−2.30961	−0.109119
$449$	−0.899022	−0.0424275	−0.0212137	−	0.999775i	$-0.506753\pi$
$449$	−0.899022	−0.0424275	−0.0212137	+	0.999775i	$0.506753\pi$
$450$	−6.56278	−0.309372
$451$	1.67436	0.0788423
$452$	8.36324	0.393374
$453$	−47.7183	−2.24200
$454$	1.79236	0.0841195
$455$	4.25480	0.199468
$456$	19.3050	0.904038
$457$	32.5366	1.52200	0.760999	−	0.648753i	$-0.224709\pi$
$457$	32.5366	1.52200	0.760999	+	0.648753i	$0.224709\pi$
$458$	7.65953	0.357906
$459$	−21.8140	−1.01819
$460$	1.04558	0.0487504
$461$	−7.22236	−0.336379	−0.168189	−	0.985755i	$-0.553792\pi$
$461$	−7.22236	−0.336379	−0.168189	+	0.985755i	$0.553792\pi$
$462$	−7.14218	−0.332284
$463$	−42.6331	−1.98133	−0.990665	−	0.136319i	$-0.956473\pi$
$463$	−42.6331	−1.98133	−0.990665	+	0.136319i	$0.956473\pi$
$464$	−5.26513	−0.244428
$465$	−14.1155	−0.654590
$466$	5.24643	0.243036
$467$	36.2928	1.67943	0.839716	−	0.543025i	$-0.182721\pi$
$467$	36.2928	1.67943	0.839716	+	0.543025i	$0.182721\pi$
$468$	12.0901	0.558863
$469$	19.1468	0.884116
$470$	−6.13413	−0.282946
$471$	32.9474	1.51814
$472$	8.26598	0.380473
$473$	1.00000	0.0459800
$474$	−26.6171	−1.22257
$475$	−6.24277	−0.286438
$476$	4.57292	0.209599
$477$	74.8319	3.42632
$478$	−10.4893	−0.479771
$479$	−3.89776	−0.178093	−0.0890466	−	0.996027i	$-0.528382\pi$
$479$	−3.89776	−0.178093	−0.0890466	+	0.996027i	$0.528382\pi$
$480$	3.09237	0.141147
$481$	−16.9670	−0.773631
$482$	−16.9970	−0.774192
$483$	7.46771	0.339793
$484$	1.00000	0.0454545
$485$	13.1837	0.598641
$486$	−11.4214	−0.518084
$487$	14.2153	0.644155	0.322078	−	0.946713i	$-0.395619\pi$
$487$	14.2153	0.644155	0.322078	+	0.946713i	$0.395619\pi$
$488$	8.57037	0.387962
$489$	−59.2685	−2.68021
$490$	−1.66570	−0.0752486
$491$	16.4064	0.740409	0.370204	−	0.928950i	$-0.379288\pi$
$491$	16.4064	0.740409	0.370204	+	0.928950i	$0.379288\pi$
$492$	−5.17773	−0.233430
$493$	10.4247	0.469506
$494$	11.5005	0.517433
$495$	6.56278	0.294975
$496$	4.56461	0.204957
$497$	−20.1182	−0.902425
$498$	−21.6610	−0.970652
$499$	−24.3133	−1.08841	−0.544207	−	0.838951i	$-0.683169\pi$
$499$	−24.3133	−1.08841	−0.544207	+	0.838951i	$0.683169\pi$
$500$	−1.00000	−0.0447214
$501$	−7.87783	−0.351956
$502$	16.0124	0.714669
$503$	33.9546	1.51396	0.756980	−	0.653438i	$-0.226674\pi$
$503$	33.9546	1.51396	0.756980	+	0.653438i	$0.226674\pi$
$504$	15.1575	0.675167
$505$	17.6319	0.784610
$506$	−1.04558	−0.0464816
$507$	−29.7061	−1.31929
$508$	1.25971	0.0558904
$509$	−34.2860	−1.51970	−0.759849	−	0.650099i	$-0.774727\pi$
$509$	−34.2860	−1.51970	−0.759849	+	0.650099i	$0.774727\pi$
$510$	−6.12275	−0.271120
$511$	31.4725	1.39226
$512$	−1.00000	−0.0441942
$513$	−68.7794	−3.03668
$514$	−7.23799	−0.319254
$515$	−16.7220	−0.736857
$516$	−3.09237	−0.136134
$517$	6.13413	0.269779
$518$	−21.2718	−0.934630
$519$	1.05496	0.0463077
$520$	1.84222	0.0807865
$521$	30.9193	1.35460	0.677300	−	0.735707i	$-0.263150\pi$
$521$	30.9193	1.35460	0.677300	+	0.735707i	$0.263150\pi$
$522$	34.5539	1.51238
$523$	−38.9509	−1.70320	−0.851602	−	0.524189i	$-0.824369\pi$
$523$	−38.9509	−1.70320	−0.851602	+	0.524189i	$0.824369\pi$
$524$	9.55472	0.417400
$525$	−7.14218	−0.311710
$526$	18.8966	0.823931
$527$	−9.03772	−0.393689
$528$	−3.09237	−0.134578
$529$	−21.9068	−0.952468
$530$	11.4025	0.495292
$531$	−54.2478	−2.35415
$532$	14.4184	0.625115
$533$	−3.08452	−0.133606
$534$	−8.93776	−0.386775
$535$	−4.58725	−0.198324
$536$	8.29005	0.358076
$537$	−34.2728	−1.47898
$538$	−7.47058	−0.322079
$539$	1.66570	0.0717467
$540$	−11.0174	−0.474116
$541$	10.3299	0.444116	0.222058	−	0.975033i	$-0.428722\pi$
$541$	10.3299	0.444116	0.222058	+	0.975033i	$0.428722\pi$
$542$	−19.6848	−0.845532
$543$	−0.140557	−0.00603186
$544$	1.97995	0.0848898
$545$	−8.53501	−0.365600
$546$	13.1574	0.563086
$547$	−9.42055	−0.402794	−0.201397	−	0.979510i	$-0.564548\pi$
$547$	−9.42055	−0.402794	−0.201397	+	0.979510i	$0.564548\pi$
$548$	1.95906	0.0836870
$549$	−56.2455	−2.40050
$550$	1.00000	0.0426401
$551$	32.8690	1.40027
$552$	3.23332	0.137619
$553$	−19.8796	−0.845367
$554$	10.3385	0.439242
$555$	28.4812	1.20896
$556$	−16.3091	−0.691659
$557$	5.82569	0.246842	0.123421	−	0.992354i	$-0.460613\pi$
$557$	5.82569	0.246842	0.123421	+	0.992354i	$0.460613\pi$
$558$	−29.9565	−1.26816
$559$	−1.84222	−0.0779174
$560$	2.30961	0.0975989
$561$	6.12275	0.258503
$562$	12.5862	0.530919
$563$	23.8355	1.00454	0.502272	−	0.864709i	$-0.332497\pi$
$563$	23.8355	1.00454	0.502272	+	0.864709i	$0.332497\pi$
$564$	−18.9690	−0.798740
$565$	−8.36324	−0.351844
$566$	24.6778	1.03728
$567$	−33.2162	−1.39495
$568$	−8.71065	−0.365491
$569$	25.8647	1.08430	0.542151	−	0.840281i	$-0.317610\pi$
$569$	25.8647	1.08430	0.542151	+	0.840281i	$0.317610\pi$
$570$	−19.3050	−0.808596
$571$	14.2475	0.596238	0.298119	−	0.954529i	$-0.403641\pi$
$571$	14.2475	0.596238	0.298119	+	0.954529i	$0.403641\pi$
$572$	−1.84222	−0.0770269
$573$	−57.2859	−2.39315
$574$	−3.86711	−0.161410
$575$	−1.04558	−0.0436036
$576$	6.56278	0.273449
$577$	−10.0202	−0.417147	−0.208573	−	0.978007i	$-0.566882\pi$
$577$	−10.0202	−0.417147	−0.208573	+	0.978007i	$0.566882\pi$
$578$	13.0798	0.544047
$579$	31.3789	1.30406
$580$	5.26513	0.218623
$581$	−16.1780	−0.671177
$582$	40.7689	1.68993
$583$	−11.4025	−0.472242
$584$	13.6268	0.563880
$585$	−12.0901	−0.499862
$586$	−25.4075	−1.04957
$587$	3.01055	0.124259	0.0621294	−	0.998068i	$-0.480211\pi$
$587$	3.01055	0.124259	0.0621294	+	0.998068i	$0.480211\pi$
$588$	−5.15096	−0.212422
$589$	−28.4958	−1.17415
$590$	−8.26598	−0.340305
$591$	12.1033	0.497865
$592$	−9.21013	−0.378534
$593$	33.0047	1.35534	0.677670	−	0.735366i	$-0.262990\pi$
$593$	33.0047	1.35534	0.677670	+	0.735366i	$0.262990\pi$
$594$	11.0174	0.452051
$595$	−4.57292	−0.187471
$596$	1.00521	0.0411752
$597$	16.1293	0.660128
$598$	1.92618	0.0787674
$599$	47.3375	1.93416	0.967078	−	0.254480i	$-0.0819043\pi$
$599$	47.3375	1.93416	0.967078	+	0.254480i	$0.0819043\pi$
$600$	−3.09237	−0.126246
$601$	−39.3084	−1.60342	−0.801711	−	0.597712i	$-0.796077\pi$
$601$	−39.3084	−1.60342	−0.801711	+	0.597712i	$0.796077\pi$
$602$	−2.30961	−0.0941327
$603$	−54.4058	−2.21557
$604$	−15.4309	−0.627876
$605$	−1.00000	−0.0406558
$606$	54.5245	2.21491
$607$	5.17029	0.209855	0.104928	−	0.994480i	$-0.466539\pi$
$607$	5.17029	0.209855	0.104928	+	0.994480i	$0.466539\pi$
$608$	6.24277	0.253178
$609$	37.6045	1.52381
$610$	−8.57037	−0.347004
$611$	−11.3004	−0.457165
$612$	−12.9940	−0.525251
$613$	−0.810567	−0.0327385	−0.0163693	−	0.999866i	$-0.505211\pi$
$613$	−0.810567	−0.0327385	−0.0163693	+	0.999866i	$0.505211\pi$
$614$	25.0161	1.00957
$615$	5.17773	0.208786
$616$	−2.30961	−0.0930569
$617$	−15.4105	−0.620402	−0.310201	−	0.950671i	$-0.600396\pi$
$617$	−15.4105	−0.620402	−0.310201	+	0.950671i	$0.600396\pi$
$618$	−51.7105	−2.08010
$619$	−44.8088	−1.80102	−0.900508	−	0.434839i	$-0.856805\pi$
$619$	−44.8088	−1.80102	−0.900508	+	0.434839i	$0.856805\pi$
$620$	−4.56461	−0.183319
$621$	−11.5196	−0.462266
$622$	−15.9607	−0.639965
$623$	−6.67537	−0.267443
$624$	5.69682	0.228055
$625$	1.00000	0.0400000
$626$	−21.7606	−0.869728
$627$	19.3050	0.770966
$628$	10.6544	0.425157
$629$	18.2356	0.727102
$630$	−15.1575	−0.603888
$631$	−39.0315	−1.55382	−0.776911	−	0.629611i	$-0.783214\pi$
$631$	−39.0315	−1.55382	−0.776911	+	0.629611i	$0.783214\pi$
$632$	−8.60734	−0.342382
$633$	−35.3996	−1.40701
$634$	19.5623	0.776917
$635$	−1.25971	−0.0499899
$636$	35.2607	1.39818
$637$	−3.06858	−0.121581
$638$	−5.26513	−0.208449
$639$	57.1661	2.26145
$640$	1.00000	0.0395285
$641$	−46.9673	−1.85510	−0.927549	−	0.373701i	$-0.878089\pi$
$641$	−46.9673	−1.85510	−0.927549	+	0.373701i	$0.878089\pi$
$642$	−14.1855	−0.559857
$643$	33.5690	1.32383	0.661917	−	0.749577i	$-0.269743\pi$
$643$	33.5690	1.32383	0.661917	+	0.749577i	$0.269743\pi$
$644$	2.41488	0.0951596
$645$	3.09237	0.121762
$646$	−12.3604	−0.486313
$647$	−1.08896	−0.0428115	−0.0214058	−	0.999771i	$-0.506814\pi$
$647$	−1.08896	−0.0428115	−0.0214058	+	0.999771i	$0.506814\pi$
$648$	−14.3817	−0.564968
$649$	8.26598	0.324468
$650$	−1.84222	−0.0722577
$651$	−32.6013	−1.27775
$652$	−19.1660	−0.750599
$653$	−7.34075	−0.287266	−0.143633	−	0.989631i	$-0.545878\pi$
$653$	−7.34075	−0.287266	−0.143633	+	0.989631i	$0.545878\pi$
$654$	−26.3934	−1.03207
$655$	−9.55472	−0.373334
$656$	−1.67436	−0.0653726
$657$	−89.4295	−3.48898
$658$	−14.1674	−0.552305
$659$	−3.36151	−0.130946	−0.0654729	−	0.997854i	$-0.520856\pi$
$659$	−3.36151	−0.130946	−0.0654729	+	0.997854i	$0.520856\pi$
$660$	3.09237	0.120371
$661$	13.3832	0.520546	0.260273	−	0.965535i	$-0.416187\pi$
$661$	13.3832	0.520546	0.260273	+	0.965535i	$0.416187\pi$
$662$	8.41201	0.326942
$663$	−11.2794	−0.438057
$664$	−7.00464	−0.271833
$665$	−14.4184	−0.559120
$666$	60.4440	2.34216
$667$	5.50511	0.213159
$668$	−2.54750	−0.0985659
$669$	−14.7404	−0.569895
$670$	−8.29005	−0.320273
$671$	8.57037	0.330855
$672$	7.14218	0.275516
$673$	10.4181	0.401589	0.200795	−	0.979633i	$-0.435648\pi$
$673$	10.4181	0.401589	0.200795	+	0.979633i	$0.435648\pi$
$674$	−0.0897684	−0.00345775
$675$	11.0174	0.424062
$676$	−9.60624	−0.369471
$677$	15.2740	0.587029	0.293514	−	0.955955i	$-0.405175\pi$
$677$	15.2740	0.587029	0.293514	+	0.955955i	$0.405175\pi$
$678$	−25.8623	−0.993235
$679$	30.4492	1.16853
$680$	−1.97995	−0.0759277
$681$	−5.54264	−0.212394
$682$	4.56461	0.174788
$683$	−30.3503	−1.16132	−0.580661	−	0.814145i	$-0.697206\pi$
$683$	−30.3503	−1.16132	−0.580661	+	0.814145i	$0.697206\pi$
$684$	−40.9699	−1.56652
$685$	−1.95906	−0.0748519
$686$	−20.0144	−0.764153
$687$	−23.6861	−0.903683
$688$	−1.00000	−0.0381246
$689$	21.0058	0.800258
$690$	−3.23332	−0.123090
$691$	−8.43589	−0.320916	−0.160458	−	0.987043i	$-0.551297\pi$
$691$	−8.43589	−0.320916	−0.160458	+	0.987043i	$0.551297\pi$
$692$	0.341149	0.0129685
$693$	15.1575	0.575785
$694$	18.9755	0.720299
$695$	16.3091	0.618639
$696$	16.2818	0.617159
$697$	3.31514	0.125570
$698$	−22.4731	−0.850620
$699$	−16.2239	−0.613645
$700$	−2.30961	−0.0872951
$701$	−11.1304	−0.420391	−0.210195	−	0.977659i	$-0.567410\pi$
$701$	−11.1304	−0.420391	−0.210195	+	0.977659i	$0.567410\pi$
$702$	−20.2965	−0.766043
$703$	57.4967	2.16853
$704$	−1.00000	−0.0376889
$705$	18.9690	0.714415
$706$	−24.3202	−0.915303
$707$	40.7229	1.53154
$708$	−25.5615	−0.960660
$709$	5.94863	0.223405	0.111703	−	0.993742i	$-0.464370\pi$
$709$	5.94863	0.223405	0.111703	+	0.993742i	$0.464370\pi$
$710$	8.71065	0.326905
$711$	56.4881	2.11847
$712$	−2.89026	−0.108317
$713$	−4.77266	−0.178738
$714$	−14.1412	−0.529220
$715$	1.84222	0.0688950
$716$	−11.0830	−0.414191
$717$	32.4370	1.21138
$718$	−0.852884	−0.0318293
$719$	31.3351	1.16860	0.584301	−	0.811537i	$-0.301369\pi$
$719$	31.3351	1.16860	0.584301	+	0.811537i	$0.301369\pi$
$720$	−6.56278	−0.244580
$721$	−38.6212	−1.43833
$722$	−19.9721	−0.743286
$723$	52.5611	1.95477
$724$	−0.0454526	−0.00168923
$725$	−5.26513	−0.195542
$726$	−3.09237	−0.114769
$727$	−6.47165	−0.240020	−0.120010	−	0.992773i	$-0.538293\pi$
$727$	−6.47165	−0.240020	−0.120010	+	0.992773i	$0.538293\pi$
$728$	4.25480	0.157693
$729$	−7.82605	−0.289854
$730$	−13.6268	−0.504349
$731$	1.97995	0.0732312
$732$	−26.5028	−0.979571
$733$	−23.7758	−0.878181	−0.439090	−	0.898443i	$-0.644699\pi$
$733$	−23.7758	−0.878181	−0.439090	+	0.898443i	$0.644699\pi$
$734$	−18.9665	−0.700066
$735$	5.15096	0.189996
$736$	1.04558	0.0385405
$737$	8.29005	0.305368
$738$	10.9884	0.404489
$739$	41.7341	1.53521	0.767606	−	0.640922i	$-0.221448\pi$
$739$	41.7341	1.53521	0.767606	+	0.640922i	$0.221448\pi$
$740$	9.21013	0.338571
$741$	−35.5639	−1.30647
$742$	26.3353	0.966798
$743$	23.7397	0.870924	0.435462	−	0.900207i	$-0.356585\pi$
$743$	23.7397	0.870924	0.435462	+	0.900207i	$0.356585\pi$
$744$	−14.1155	−0.517499
$745$	−1.00521	−0.0368282
$746$	−12.7360	−0.466298
$747$	45.9699	1.68195
$748$	1.97995	0.0723942
$749$	−10.5948	−0.387125
$750$	3.09237	0.112918
$751$	46.7820	1.70710	0.853549	−	0.521012i	$-0.174445\pi$
$751$	46.7820	1.70710	0.853549	+	0.521012i	$0.174445\pi$
$752$	−6.13413	−0.223689
$753$	−49.5164	−1.80448
$754$	9.69952	0.353236
$755$	15.4309	0.561589
$756$	−25.4460	−0.925463
$757$	23.3977	0.850403	0.425202	−	0.905099i	$-0.360203\pi$
$757$	23.3977	0.850403	0.425202	+	0.905099i	$0.360203\pi$
$758$	−36.0044	−1.30774
$759$	3.23332	0.117362
$760$	−6.24277	−0.226449
$761$	33.5825	1.21736	0.608682	−	0.793414i	$-0.291698\pi$
$761$	33.5825	1.21736	0.608682	+	0.793414i	$0.291698\pi$
$762$	−3.89548	−0.141118
$763$	−19.7125	−0.713642
$764$	−18.5249	−0.670208
$765$	12.9940	0.469799
$766$	0.811006	0.0293028
$767$	−15.2277	−0.549841
$768$	3.09237	0.111586
$769$	−39.1699	−1.41250	−0.706251	−	0.707962i	$-0.749615\pi$
$769$	−39.1699	−1.41250	−0.706251	+	0.707962i	$0.749615\pi$
$770$	2.30961	0.0832326
$771$	22.3826	0.806089
$772$	10.1472	0.365205
$773$	−26.2401	−0.943789	−0.471895	−	0.881655i	$-0.656430\pi$
$773$	−26.2401	−0.943789	−0.471895	+	0.881655i	$0.656430\pi$
$774$	6.56278	0.235894
$775$	4.56461	0.163966
$776$	13.1837	0.473267
$777$	65.7804	2.35986
$778$	9.45110	0.338838
$779$	10.4526	0.374504
$780$	−5.69682	−0.203979
$781$	−8.71065	−0.311692
$782$	−2.07020	−0.0740301
$783$	−58.0084	−2.07305
$784$	−1.66570	−0.0594892
$785$	−10.6544	−0.380272
$786$	−29.5468	−1.05390
$787$	38.0556	1.35653	0.678267	−	0.734816i	$-0.262731\pi$
$787$	38.0556	1.35653	0.678267	+	0.734816i	$0.262731\pi$
$788$	3.91393	0.139428
$789$	−58.4354	−2.08035
$790$	8.60734	0.306236
$791$	−19.3158	−0.686792
$792$	6.56278	0.233198
$793$	−15.7885	−0.560665
$794$	3.99166	0.141659
$795$	−35.2607	−1.25057
$796$	5.21583	0.184870
$797$	33.8380	1.19860	0.599302	−	0.800523i	$-0.295445\pi$
$797$	33.8380	1.19860	0.599302	+	0.800523i	$0.295445\pi$
$798$	−44.5870	−1.57836
$799$	12.1453	0.429669
$800$	−1.00000	−0.0353553
$801$	18.9681	0.670206
$802$	−24.5862	−0.868169
$803$	13.6268	0.480878
$804$	−25.6359	−0.904110
$805$	−2.41488	−0.0851133
$806$	−8.40900	−0.296194
$807$	23.1018	0.813223
$808$	17.6319	0.620289
$809$	39.4361	1.38650	0.693249	−	0.720698i	$-0.256179\pi$
$809$	39.4361	1.38650	0.693249	+	0.720698i	$0.256179\pi$
$810$	14.3817	0.505323
$811$	−27.1427	−0.953108	−0.476554	−	0.879145i	$-0.658114\pi$
$811$	−27.1427	−0.953108	−0.476554	+	0.879145i	$0.658114\pi$
$812$	12.1604	0.426747
$813$	60.8726	2.13490
$814$	−9.21013	−0.322815
$815$	19.1660	0.671356
$816$	−6.12275	−0.214339
$817$	6.24277	0.218407
$818$	32.2209	1.12658
$819$	−27.9233	−0.975720
$820$	1.67436	0.0584710
$821$	−13.4147	−0.468175	−0.234088	−	0.972215i	$-0.575210\pi$
$821$	−13.4147	−0.468175	−0.234088	+	0.972215i	$0.575210\pi$
$822$	−6.05815	−0.211302
$823$	−2.23835	−0.0780238	−0.0390119	−	0.999239i	$-0.512421\pi$
$823$	−2.23835	−0.0780238	−0.0390119	+	0.999239i	$0.512421\pi$
$824$	−16.7220	−0.582537
$825$	−3.09237	−0.107663
$826$	−19.0912	−0.664268
$827$	−41.7661	−1.45235	−0.726176	−	0.687509i	$-0.758704\pi$
$827$	−41.7661	−1.45235	−0.726176	+	0.687509i	$0.758704\pi$
$828$	−6.86190	−0.238468
$829$	24.8074	0.861597	0.430799	−	0.902448i	$-0.358232\pi$
$829$	24.8074	0.861597	0.430799	+	0.902448i	$0.358232\pi$
$830$	7.00464	0.243135
$831$	−31.9706	−1.10905
$832$	1.84222	0.0638673
$833$	3.29800	0.114269
$834$	50.4338	1.74638
$835$	2.54750	0.0881600
$836$	6.24277	0.215911
$837$	50.2904	1.73829
$838$	9.15384	0.316214
$839$	−44.0250	−1.51991	−0.759956	−	0.649974i	$-0.774780\pi$
$839$	−44.0250	−1.51991	−0.759956	+	0.649974i	$0.774780\pi$
$840$	−7.14218	−0.246429
$841$	−1.27836	−0.0440813
$842$	−4.86053	−0.167505
$843$	−38.9214	−1.34052
$844$	−11.4474	−0.394035
$845$	9.60624	0.330465
$846$	40.2569	1.38406
$847$	−2.30961	−0.0793592
$848$	11.4025	0.391563
$849$	−76.3129	−2.61905
$850$	1.97995	0.0679118
$851$	9.62991	0.330109
$852$	26.9366	0.922832
$853$	−12.5178	−0.428602	−0.214301	−	0.976768i	$-0.568747\pi$
$853$	−12.5178	−0.428602	−0.214301	+	0.976768i	$0.568747\pi$
$854$	−19.7942	−0.677344
$855$	40.9699	1.40114
$856$	−4.58725	−0.156789
$857$	40.5911	1.38656	0.693282	−	0.720666i	$-0.256164\pi$
$857$	40.5911	1.38656	0.693282	+	0.720666i	$0.256164\pi$
$858$	5.69682	0.194486
$859$	23.0072	0.784996	0.392498	−	0.919753i	$-0.371611\pi$
$859$	23.0072	0.784996	0.392498	+	0.919753i	$0.371611\pi$
$860$	1.00000	0.0340997
$861$	11.9585	0.407546
$862$	−1.05622	−0.0359749
$863$	6.38511	0.217352	0.108676	−	0.994077i	$-0.465339\pi$
$863$	6.38511	0.217352	0.108676	+	0.994077i	$0.465339\pi$
$864$	−11.0174	−0.374821
$865$	−0.341149	−0.0115994
$866$	2.44052	0.0829324
$867$	−40.4476	−1.37367
$868$	−10.5425	−0.357835
$869$	−8.60734	−0.291984
$870$	−16.2818	−0.552004
$871$	−15.2721	−0.517474
$872$	−8.53501	−0.289032
$873$	−86.5217	−2.92832
$874$	−6.52730	−0.220789
$875$	2.30961	0.0780791
$876$	−42.1391	−1.42375
$877$	2.41900	0.0816838	0.0408419	−	0.999166i	$-0.486996\pi$
$877$	2.41900	0.0816838	0.0408419	+	0.999166i	$0.486996\pi$
$878$	10.9702	0.370226
$879$	78.5695	2.65008
$880$	1.00000	0.0337100
$881$	45.5331	1.53405	0.767024	−	0.641618i	$-0.221737\pi$
$881$	45.5331	1.53405	0.767024	+	0.641618i	$0.221737\pi$
$882$	10.9316	0.368087
$883$	−49.9667	−1.68151	−0.840757	−	0.541412i	$-0.817890\pi$
$883$	−49.9667	−1.68151	−0.840757	+	0.541412i	$0.817890\pi$
$884$	−3.64750	−0.122679
$885$	25.5615	0.859241
$886$	−24.8491	−0.834823
$887$	−9.63919	−0.323652	−0.161826	−	0.986819i	$-0.551738\pi$
$887$	−9.63919	−0.323652	−0.161826	+	0.986819i	$0.551738\pi$
$888$	28.4812	0.955765
$889$	−2.90943	−0.0975791
$890$	2.89026	0.0968816
$891$	−14.3817	−0.481806
$892$	−4.76668	−0.159600
$893$	38.2939	1.28146
$894$	−3.10850	−0.103964
$895$	11.0830	0.370464
$896$	2.30961	0.0771587
$897$	−5.95647	−0.198881
$898$	0.899022	0.0300008
$899$	−24.0333	−0.801555
$900$	6.56278	0.218759
$901$	−22.5764	−0.752127
$902$	−1.67436	−0.0557499
$903$	7.14218	0.237677
$904$	−8.36324	−0.278157
$905$	0.0454526	0.00151090
$906$	47.7183	1.58533
$907$	−26.0322	−0.864385	−0.432193	−	0.901781i	$-0.642260\pi$
$907$	−26.0322	−0.864385	−0.432193	+	0.901781i	$0.642260\pi$
$908$	−1.79236	−0.0594814
$909$	−115.714	−3.83801
$910$	−4.25480	−0.141045
$911$	−37.9547	−1.25750	−0.628748	−	0.777609i	$-0.716432\pi$
$911$	−37.9547	−1.25750	−0.628748	+	0.777609i	$0.716432\pi$
$912$	−19.3050	−0.639252
$913$	−7.00464	−0.231820
$914$	−32.5366	−1.07622
$915$	26.5028	0.876155
$916$	−7.65953	−0.253078
$917$	−22.0677	−0.728739
$918$	21.8140	0.719970
$919$	−14.9876	−0.494395	−0.247198	−	0.968965i	$-0.579510\pi$
$919$	−14.9876	−0.494395	−0.247198	+	0.968965i	$0.579510\pi$
$920$	−1.04558	−0.0344717
$921$	−77.3592	−2.54907
$922$	7.22236	0.237856
$923$	16.0469	0.528190
$924$	7.14218	0.234961
$925$	−9.21013	−0.302827
$926$	42.6331	1.40101
$927$	109.743	3.60442
$928$	5.26513	0.172837
$929$	−9.77185	−0.320604	−0.160302	−	0.987068i	$-0.551247\pi$
$929$	−9.77185	−0.320604	−0.160302	+	0.987068i	$0.551247\pi$
$930$	14.1155	0.462865
$931$	10.3986	0.340799
$932$	−5.24643	−0.171853
$933$	49.3564	1.61586
$934$	−36.2928	−1.18754
$935$	−1.97995	−0.0647514
$936$	−12.0901	−0.395176
$937$	35.0382	1.14465	0.572324	−	0.820027i	$-0.306042\pi$
$937$	35.0382	1.14465	0.572324	+	0.820027i	$0.306042\pi$
$938$	−19.1468	−0.625165
$939$	67.2919	2.19599
$940$	6.13413	0.200073
$941$	14.1601	0.461607	0.230804	−	0.973000i	$-0.425864\pi$
$941$	14.1601	0.461607	0.230804	+	0.973000i	$0.425864\pi$
$942$	−32.9474	−1.07348
$943$	1.75067	0.0570097
$944$	−8.26598	−0.269035
$945$	25.4460	0.827759
$946$	−1.00000	−0.0325128
$947$	−19.3333	−0.628247	−0.314123	−	0.949382i	$-0.601711\pi$
$947$	−19.3333	−0.628247	−0.314123	+	0.949382i	$0.601711\pi$
$948$	26.6171	0.864484
$949$	−25.1035	−0.814893
$950$	6.24277	0.202542
$951$	−60.4939	−1.96165
$952$	−4.57292	−0.148209
$953$	−4.70628	−0.152451	−0.0762257	−	0.997091i	$-0.524287\pi$
$953$	−4.70628	−0.152451	−0.0762257	+	0.997091i	$0.524287\pi$
$954$	−74.8319	−2.42277
$955$	18.5249	0.599452
$956$	10.4893	0.339249
$957$	16.2818	0.526315
$958$	3.89776	0.125931
$959$	−4.52467	−0.146109
$960$	−3.09237	−0.0998060
$961$	−10.1643	−0.327881
$962$	16.9670	0.547039
$963$	30.1051	0.970125
$964$	16.9970	0.547437
$965$	−10.1472	−0.326649
$966$	−7.46771	−0.240270
$967$	24.2369	0.779406	0.389703	−	0.920941i	$-0.372578\pi$
$967$	24.2369	0.779406	0.389703	+	0.920941i	$0.372578\pi$
$968$	−1.00000	−0.0321412
$969$	38.2229	1.22790
$970$	−13.1837	−0.423303
$971$	22.6764	0.727719	0.363860	−	0.931454i	$-0.381459\pi$
$971$	22.6764	0.727719	0.363860	+	0.931454i	$0.381459\pi$
$972$	11.4214	0.366341
$973$	37.6676	1.20757
$974$	−14.2153	−0.455487
$975$	5.69682	0.182444
$976$	−8.57037	−0.274331
$977$	20.5188	0.656453	0.328227	−	0.944599i	$-0.393549\pi$
$977$	20.5188	0.656453	0.328227	+	0.944599i	$0.393549\pi$
$978$	59.2685	1.89520
$979$	−2.89026	−0.0923730
$980$	1.66570	0.0532088
$981$	56.0134	1.78837
$982$	−16.4064	−0.523548
$983$	−24.7956	−0.790856	−0.395428	−	0.918497i	$-0.629404\pi$
$983$	−24.7956	−0.790856	−0.395428	+	0.918497i	$0.629404\pi$
$984$	5.17773	0.165060
$985$	−3.91393	−0.124708
$986$	−10.4247	−0.331991
$987$	43.8111	1.39452
$988$	−11.5005	−0.365880
$989$	1.04558	0.0332475
$990$	−6.56278	−0.208579
$991$	−24.7895	−0.787465	−0.393733	−	0.919225i	$-0.628816\pi$
$991$	−24.7895	−0.787465	−0.393733	+	0.919225i	$0.628816\pi$
$992$	−4.56461	−0.144927
$993$	−26.0131	−0.825501
$994$	20.1182	0.638111
$995$	−5.21583	−0.165353
$996$	21.6610	0.686354
$997$	35.9662	1.13906	0.569530	−	0.821971i	$-0.307125\pi$
$997$	35.9662	1.13906	0.569530	+	0.821971i	$0.307125\pi$
$998$	24.3133	0.769624
$999$	−101.472	−3.21044

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4730.2.a.x.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4730.2.a.x.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.09237 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.09237 q^{6} -2.30961 q^{7} -1.00000 q^{8} +6.56278 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.09237 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.09237 q^{6} -2.30961 q^{7} -1.00000 q^{8} +6.56278 q^{9} +1.00000 q^{10} -1.00000 q^{11} +3.09237 q^{12} +1.84222 q^{13} +2.30961 q^{14} -3.09237 q^{15} +1.00000 q^{16} -1.97995 q^{17} -6.56278 q^{18} -6.24277 q^{19} -1.00000 q^{20} -7.14218 q^{21} +1.00000 q^{22} -1.04558 q^{23} -3.09237 q^{24} +1.00000 q^{25} -1.84222 q^{26} +11.0174 q^{27} -2.30961 q^{28} -5.26513 q^{29} +3.09237 q^{30} +4.56461 q^{31} -1.00000 q^{32} -3.09237 q^{33} +1.97995 q^{34} +2.30961 q^{35} +6.56278 q^{36} -9.21013 q^{37} +6.24277 q^{38} +5.69682 q^{39} +1.00000 q^{40} -1.67436 q^{41} +7.14218 q^{42} -1.00000 q^{43} -1.00000 q^{44} -6.56278 q^{45} +1.04558 q^{46} -6.13413 q^{47} +3.09237 q^{48} -1.66570 q^{49} -1.00000 q^{50} -6.12275 q^{51} +1.84222 q^{52} +11.4025 q^{53} -11.0174 q^{54} +1.00000 q^{55} +2.30961 q^{56} -19.3050 q^{57} +5.26513 q^{58} -8.26598 q^{59} -3.09237 q^{60} -8.57037 q^{61} -4.56461 q^{62} -15.1575 q^{63} +1.00000 q^{64} -1.84222 q^{65} +3.09237 q^{66} -8.29005 q^{67} -1.97995 q^{68} -3.23332 q^{69} -2.30961 q^{70} +8.71065 q^{71} -6.56278 q^{72} -13.6268 q^{73} +9.21013 q^{74} +3.09237 q^{75} -6.24277 q^{76} +2.30961 q^{77} -5.69682 q^{78} +8.60734 q^{79} -1.00000 q^{80} +14.3817 q^{81} +1.67436 q^{82} +7.00464 q^{83} -7.14218 q^{84} +1.97995 q^{85} +1.00000 q^{86} -16.2818 q^{87} +1.00000 q^{88} +2.89026 q^{89} +6.56278 q^{90} -4.25480 q^{91} -1.04558 q^{92} +14.1155 q^{93} +6.13413 q^{94} +6.24277 q^{95} -3.09237 q^{96} -13.1837 q^{97} +1.66570 q^{98} -6.56278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} - 5 q^{7} - 8 q^{8} + 10 q^{9}+O(q^{10}) \) 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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