LMFDB - Embedded newform 4730.2.a.p.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.414214 q^{6} -2.41421 q^{7} +1.00000 q^{8} -2.82843 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.414214 q^{6} -2.41421 q^{7} +1.00000 q^{8} -2.82843 q^{9} -1.00000 q^{10} -1.00000 q^{11} +0.414214 q^{12} +3.41421 q^{13} -2.41421 q^{14} -0.414214 q^{15} +1.00000 q^{16} +5.82843 q^{17} -2.82843 q^{18} +0.414214 q^{19} -1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -3.41421 q^{23} +0.414214 q^{24} +1.00000 q^{25} +3.41421 q^{26} -2.41421 q^{27} -2.41421 q^{28} +2.41421 q^{29} -0.414214 q^{30} -1.82843 q^{31} +1.00000 q^{32} -0.414214 q^{33} +5.82843 q^{34} +2.41421 q^{35} -2.82843 q^{36} -9.24264 q^{37} +0.414214 q^{38} +1.41421 q^{39} -1.00000 q^{40} -6.24264 q^{41} -1.00000 q^{42} +1.00000 q^{43} -1.00000 q^{44} +2.82843 q^{45} -3.41421 q^{46} +3.65685 q^{47} +0.414214 q^{48} -1.17157 q^{49} +1.00000 q^{50} +2.41421 q^{51} +3.41421 q^{52} -9.82843 q^{53} -2.41421 q^{54} +1.00000 q^{55} -2.41421 q^{56} +0.171573 q^{57} +2.41421 q^{58} -10.7279 q^{59} -0.414214 q^{60} +6.07107 q^{61} -1.82843 q^{62} +6.82843 q^{63} +1.00000 q^{64} -3.41421 q^{65} -0.414214 q^{66} -3.65685 q^{67} +5.82843 q^{68} -1.41421 q^{69} +2.41421 q^{70} +1.24264 q^{71} -2.82843 q^{72} -13.3137 q^{73} -9.24264 q^{74} +0.414214 q^{75} +0.414214 q^{76} +2.41421 q^{77} +1.41421 q^{78} +6.24264 q^{79} -1.00000 q^{80} +7.48528 q^{81} -6.24264 q^{82} -3.89949 q^{83} -1.00000 q^{84} -5.82843 q^{85} +1.00000 q^{86} +1.00000 q^{87} -1.00000 q^{88} -4.41421 q^{89} +2.82843 q^{90} -8.24264 q^{91} -3.41421 q^{92} -0.757359 q^{93} +3.65685 q^{94} -0.414214 q^{95} +0.414214 q^{96} -10.8284 q^{97} -1.17157 q^{98} +2.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} + 6 q^{17} - 2 q^{19} - 2 q^{20} - 2 q^{21} - 2 q^{22} - 4 q^{23} - 2 q^{24} + 2 q^{25} + 4 q^{26} - 2 q^{27} - 2 q^{28} + 2 q^{29} + 2 q^{30} + 2 q^{31} + 2 q^{32} + 2 q^{33} + 6 q^{34} + 2 q^{35} - 10 q^{37} - 2 q^{38} - 2 q^{40} - 4 q^{41} - 2 q^{42} + 2 q^{43} - 2 q^{44} - 4 q^{46} - 4 q^{47} - 2 q^{48} - 8 q^{49} + 2 q^{50} + 2 q^{51} + 4 q^{52} - 14 q^{53} - 2 q^{54} + 2 q^{55} - 2 q^{56} + 6 q^{57} + 2 q^{58} + 4 q^{59} + 2 q^{60} - 2 q^{61} + 2 q^{62} + 8 q^{63} + 2 q^{64} - 4 q^{65} + 2 q^{66} + 4 q^{67} + 6 q^{68} + 2 q^{70} - 6 q^{71} - 4 q^{73} - 10 q^{74} - 2 q^{75} - 2 q^{76} + 2 q^{77} + 4 q^{79} - 2 q^{80} - 2 q^{81} - 4 q^{82} + 12 q^{83} - 2 q^{84} - 6 q^{85} + 2 q^{86} + 2 q^{87} - 2 q^{88} - 6 q^{89} - 8 q^{91} - 4 q^{92} - 10 q^{93} - 4 q^{94} + 2 q^{95} - 2 q^{96} - 16 q^{97} - 8 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^11 - 2 * q^12 + 4 * q^13 - 2 * q^14 + 2 * q^15 + 2 * q^16 + 6 * q^17 - 2 * q^19 - 2 * q^20 - 2 * q^21 - 2 * q^22 - 4 * q^23 - 2 * q^24 + 2 * q^25 + 4 * q^26 - 2 * q^27 - 2 * q^28 + 2 * q^29 + 2 * q^30 + 2 * q^31 + 2 * q^32 + 2 * q^33 + 6 * q^34 + 2 * q^35 - 10 * q^37 - 2 * q^38 - 2 * q^40 - 4 * q^41 - 2 * q^42 + 2 * q^43 - 2 * q^44 - 4 * q^46 - 4 * q^47 - 2 * q^48 - 8 * q^49 + 2 * q^50 + 2 * q^51 + 4 * q^52 - 14 * q^53 - 2 * q^54 + 2 * q^55 - 2 * q^56 + 6 * q^57 + 2 * q^58 + 4 * q^59 + 2 * q^60 - 2 * q^61 + 2 * q^62 + 8 * q^63 + 2 * q^64 - 4 * q^65 + 2 * q^66 + 4 * q^67 + 6 * q^68 + 2 * q^70 - 6 * q^71 - 4 * q^73 - 10 * q^74 - 2 * q^75 - 2 * q^76 + 2 * q^77 + 4 * q^79 - 2 * q^80 - 2 * q^81 - 4 * q^82 + 12 * q^83 - 2 * q^84 - 6 * q^85 + 2 * q^86 + 2 * q^87 - 2 * q^88 - 6 * q^89 - 8 * q^91 - 4 * q^92 - 10 * q^93 - 4 * q^94 + 2 * q^95 - 2 * q^96 - 16 * q^97 - 8 * q^98

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
$3$	0.414214	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.414214	0.169102
$7$	−2.41421	−0.912487	−0.456243	−	0.889855i	$-0.650805\pi$
$7$	−2.41421	−0.912487	−0.456243	+	0.889855i	$0.650805\pi$
$8$	1.00000	0.353553
$9$	−2.82843	−0.942809
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.414214	0.119573
$13$	3.41421	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$14$	−2.41421	−0.645226
$15$	−0.414214	−0.106949
$16$	1.00000	0.250000
$17$	5.82843	1.41360	0.706801	−	0.707413i	$-0.250138\pi$
$17$	5.82843	1.41360	0.706801	+	0.707413i	$0.250138\pi$
$18$	−2.82843	−0.666667
$19$	0.414214	0.0950271	0.0475136	−	0.998871i	$-0.484870\pi$
$19$	0.414214	0.0950271	0.0475136	+	0.998871i	$0.484870\pi$
$20$	−1.00000	−0.223607
$21$	−1.00000	−0.218218
$22$	−1.00000	−0.213201
$23$	−3.41421	−0.711913	−0.355956	−	0.934503i	$-0.615845\pi$
$23$	−3.41421	−0.711913	−0.355956	+	0.934503i	$0.615845\pi$
$24$	0.414214	0.0845510
$25$	1.00000	0.200000
$26$	3.41421	0.669582
$27$	−2.41421	−0.464616
$28$	−2.41421	−0.456243
$29$	2.41421	0.448308	0.224154	−	0.974554i	$-0.428038\pi$
$29$	2.41421	0.448308	0.224154	+	0.974554i	$0.428038\pi$
$30$	−0.414214	−0.0756247
$31$	−1.82843	−0.328395	−0.164198	−	0.986427i	$-0.552503\pi$
$31$	−1.82843	−0.328395	−0.164198	+	0.986427i	$0.552503\pi$
$32$	1.00000	0.176777
$33$	−0.414214	−0.0721053
$34$	5.82843	0.999567
$35$	2.41421	0.408077
$36$	−2.82843	−0.471405
$37$	−9.24264	−1.51948	−0.759740	−	0.650227i	$-0.774674\pi$
$37$	−9.24264	−1.51948	−0.759740	+	0.650227i	$0.774674\pi$
$38$	0.414214	0.0671943
$39$	1.41421	0.226455
$40$	−1.00000	−0.158114
$41$	−6.24264	−0.974937	−0.487468	−	0.873141i	$-0.662080\pi$
$41$	−6.24264	−0.974937	−0.487468	+	0.873141i	$0.662080\pi$
$42$	−1.00000	−0.154303
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−1.00000	−0.150756
$45$	2.82843	0.421637
$46$	−3.41421	−0.503398
$47$	3.65685	0.533407	0.266704	−	0.963779i	$-0.414066\pi$
$47$	3.65685	0.533407	0.266704	+	0.963779i	$0.414066\pi$
$48$	0.414214	0.0597866
$49$	−1.17157	−0.167368
$50$	1.00000	0.141421
$51$	2.41421	0.338058
$52$	3.41421	0.473466
$53$	−9.82843	−1.35004	−0.675019	−	0.737800i	$-0.735865\pi$
$53$	−9.82843	−1.35004	−0.675019	+	0.737800i	$0.735865\pi$
$54$	−2.41421	−0.328533
$55$	1.00000	0.134840
$56$	−2.41421	−0.322613
$57$	0.171573	0.0227254
$58$	2.41421	0.317002
$59$	−10.7279	−1.39666	−0.698328	−	0.715778i	$-0.746072\pi$
$59$	−10.7279	−1.39666	−0.698328	+	0.715778i	$0.746072\pi$
$60$	−0.414214	−0.0534747
$61$	6.07107	0.777321	0.388660	−	0.921381i	$-0.372938\pi$
$61$	6.07107	0.777321	0.388660	+	0.921381i	$0.372938\pi$
$62$	−1.82843	−0.232210
$63$	6.82843	0.860301
$64$	1.00000	0.125000
$65$	−3.41421	−0.423481
$66$	−0.414214	−0.0509862
$67$	−3.65685	−0.446756	−0.223378	−	0.974732i	$-0.571708\pi$
$67$	−3.65685	−0.446756	−0.223378	+	0.974732i	$0.571708\pi$
$68$	5.82843	0.706801
$69$	−1.41421	−0.170251
$70$	2.41421	0.288554
$71$	1.24264	0.147474	0.0737372	−	0.997278i	$-0.476507\pi$
$71$	1.24264	0.147474	0.0737372	+	0.997278i	$0.476507\pi$
$72$	−2.82843	−0.333333
$73$	−13.3137	−1.55825	−0.779126	−	0.626868i	$-0.784337\pi$
$73$	−13.3137	−1.55825	−0.779126	+	0.626868i	$0.784337\pi$
$74$	−9.24264	−1.07444
$75$	0.414214	0.0478293
$76$	0.414214	0.0475136
$77$	2.41421	0.275125
$78$	1.41421	0.160128
$79$	6.24264	0.702352	0.351176	−	0.936309i	$-0.385782\pi$
$79$	6.24264	0.702352	0.351176	+	0.936309i	$0.385782\pi$
$80$	−1.00000	−0.111803
$81$	7.48528	0.831698
$82$	−6.24264	−0.689384
$83$	−3.89949	−0.428025	−0.214013	−	0.976831i	$-0.568653\pi$
$83$	−3.89949	−0.428025	−0.214013	+	0.976831i	$0.568653\pi$
$84$	−1.00000	−0.109109
$85$	−5.82843	−0.632182
$86$	1.00000	0.107833
$87$	1.00000	0.107211
$88$	−1.00000	−0.106600
$89$	−4.41421	−0.467906	−0.233953	−	0.972248i	$-0.575166\pi$
$89$	−4.41421	−0.467906	−0.233953	+	0.972248i	$0.575166\pi$
$90$	2.82843	0.298142
$91$	−8.24264	−0.864064
$92$	−3.41421	−0.355956
$93$	−0.757359	−0.0785345
$94$	3.65685	0.377176
$95$	−0.414214	−0.0424974
$96$	0.414214	0.0422755
$97$	−10.8284	−1.09946	−0.549730	−	0.835342i	$-0.685269\pi$
$97$	−10.8284	−1.09946	−0.549730	+	0.835342i	$0.685269\pi$
$98$	−1.17157	−0.118347
$99$	2.82843	0.284268
$100$	1.00000	0.100000
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	2.41421	0.239043
$103$	−13.1716	−1.29783	−0.648917	−	0.760859i	$-0.724778\pi$
$103$	−13.1716	−1.29783	−0.648917	+	0.760859i	$0.724778\pi$
$104$	3.41421	0.334791
$105$	1.00000	0.0975900
$106$	−9.82843	−0.954621
$107$	14.1421	1.36717	0.683586	−	0.729870i	$-0.260419\pi$
$107$	14.1421	1.36717	0.683586	+	0.729870i	$0.260419\pi$
$108$	−2.41421	−0.232308
$109$	−13.1716	−1.26161	−0.630804	−	0.775942i	$-0.717275\pi$
$109$	−13.1716	−1.26161	−0.630804	+	0.775942i	$0.717275\pi$
$110$	1.00000	0.0953463
$111$	−3.82843	−0.363378
$112$	−2.41421	−0.228122
$113$	−0.485281	−0.0456514	−0.0228257	−	0.999739i	$-0.507266\pi$
$113$	−0.485281	−0.0456514	−0.0228257	+	0.999739i	$0.507266\pi$
$114$	0.171573	0.0160693
$115$	3.41421	0.318377
$116$	2.41421	0.224154
$117$	−9.65685	−0.892776
$118$	−10.7279	−0.987585
$119$	−14.0711	−1.28989
$120$	−0.414214	−0.0378124
$121$	1.00000	0.0909091
$122$	6.07107	0.549649
$123$	−2.58579	−0.233153
$124$	−1.82843	−0.164198
$125$	−1.00000	−0.0894427
$126$	6.82843	0.608325
$127$	3.65685	0.324493	0.162247	−	0.986750i	$-0.448126\pi$
$127$	3.65685	0.324493	0.162247	+	0.986750i	$0.448126\pi$
$128$	1.00000	0.0883883
$129$	0.414214	0.0364695
$130$	−3.41421	−0.299446
$131$	−6.07107	−0.530432	−0.265216	−	0.964189i	$-0.585443\pi$
$131$	−6.07107	−0.530432	−0.265216	+	0.964189i	$0.585443\pi$
$132$	−0.414214	−0.0360527
$133$	−1.00000	−0.0867110
$134$	−3.65685	−0.315904
$135$	2.41421	0.207782
$136$	5.82843	0.499784
$137$	−18.2426	−1.55857	−0.779287	−	0.626667i	$-0.784419\pi$
$137$	−18.2426	−1.55857	−0.779287	+	0.626667i	$0.784419\pi$
$138$	−1.41421	−0.120386
$139$	19.6569	1.66727	0.833636	−	0.552314i	$-0.186255\pi$
$139$	19.6569	1.66727	0.833636	+	0.552314i	$0.186255\pi$
$140$	2.41421	0.204038
$141$	1.51472	0.127562
$142$	1.24264	0.104280
$143$	−3.41421	−0.285511
$144$	−2.82843	−0.235702
$145$	−2.41421	−0.200490
$146$	−13.3137	−1.10185
$147$	−0.485281	−0.0400253
$148$	−9.24264	−0.759740
$149$	−7.92893	−0.649563	−0.324782	−	0.945789i	$-0.605291\pi$
$149$	−7.92893	−0.649563	−0.324782	+	0.945789i	$0.605291\pi$
$150$	0.414214	0.0338204
$151$	4.24264	0.345261	0.172631	−	0.984987i	$-0.444773\pi$
$151$	4.24264	0.345261	0.172631	+	0.984987i	$0.444773\pi$
$152$	0.414214	0.0335972
$153$	−16.4853	−1.33276
$154$	2.41421	0.194543
$155$	1.82843	0.146863
$156$	1.41421	0.113228
$157$	15.7279	1.25522	0.627612	−	0.778526i	$-0.284032\pi$
$157$	15.7279	1.25522	0.627612	+	0.778526i	$0.284032\pi$
$158$	6.24264	0.496638
$159$	−4.07107	−0.322857
$160$	−1.00000	−0.0790569
$161$	8.24264	0.649611
$162$	7.48528	0.588099
$163$	−17.3848	−1.36168	−0.680840	−	0.732432i	$-0.738385\pi$
$163$	−17.3848	−1.36168	−0.680840	+	0.732432i	$0.738385\pi$
$164$	−6.24264	−0.487468
$165$	0.414214	0.0322465
$166$	−3.89949	−0.302660
$167$	0.313708	0.0242755	0.0121377	−	0.999926i	$-0.496136\pi$
$167$	0.313708	0.0242755	0.0121377	+	0.999926i	$0.496136\pi$
$168$	−1.00000	−0.0771517
$169$	−1.34315	−0.103319
$170$	−5.82843	−0.447020
$171$	−1.17157	−0.0895924
$172$	1.00000	0.0762493
$173$	−14.9706	−1.13819	−0.569095	−	0.822272i	$-0.692706\pi$
$173$	−14.9706	−1.13819	−0.569095	+	0.822272i	$0.692706\pi$
$174$	1.00000	0.0758098
$175$	−2.41421	−0.182497
$176$	−1.00000	−0.0753778
$177$	−4.44365	−0.334005
$178$	−4.41421	−0.330859
$179$	2.34315	0.175135	0.0875675	−	0.996159i	$-0.472091\pi$
$179$	2.34315	0.175135	0.0875675	+	0.996159i	$0.472091\pi$
$180$	2.82843	0.210819
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	−8.24264	−0.610985
$183$	2.51472	0.185893
$184$	−3.41421	−0.251699
$185$	9.24264	0.679532
$186$	−0.757359	−0.0555323
$187$	−5.82843	−0.426217
$188$	3.65685	0.266704
$189$	5.82843	0.423956
$190$	−0.414214	−0.0300502
$191$	−2.00000	−0.144715	−0.0723575	−	0.997379i	$-0.523052\pi$
$191$	−2.00000	−0.144715	−0.0723575	+	0.997379i	$0.523052\pi$
$192$	0.414214	0.0298933
$193$	1.48528	0.106913	0.0534564	−	0.998570i	$-0.482976\pi$
$193$	1.48528	0.106913	0.0534564	+	0.998570i	$0.482976\pi$
$194$	−10.8284	−0.777436
$195$	−1.41421	−0.101274
$196$	−1.17157	−0.0836838
$197$	14.8284	1.05648	0.528241	−	0.849095i	$-0.322852\pi$
$197$	14.8284	1.05648	0.528241	+	0.849095i	$0.322852\pi$
$198$	2.82843	0.201008
$199$	6.55635	0.464767	0.232384	−	0.972624i	$-0.425348\pi$
$199$	6.55635	0.464767	0.232384	+	0.972624i	$0.425348\pi$
$200$	1.00000	0.0707107
$201$	−1.51472	−0.106840
$202$	−8.00000	−0.562878
$203$	−5.82843	−0.409075
$204$	2.41421	0.169029
$205$	6.24264	0.436005
$206$	−13.1716	−0.917707
$207$	9.65685	0.671198
$208$	3.41421	0.236733
$209$	−0.414214	−0.0286518
$210$	1.00000	0.0690066
$211$	24.8995	1.71415	0.857076	−	0.515190i	$-0.172279\pi$
$211$	24.8995	1.71415	0.857076	+	0.515190i	$0.172279\pi$
$212$	−9.82843	−0.675019
$213$	0.514719	0.0352679
$214$	14.1421	0.966736
$215$	−1.00000	−0.0681994
$216$	−2.41421	−0.164266
$217$	4.41421	0.299656
$218$	−13.1716	−0.892091
$219$	−5.51472	−0.372650
$220$	1.00000	0.0674200
$221$	19.8995	1.33858
$222$	−3.82843	−0.256947
$223$	−13.4142	−0.898282	−0.449141	−	0.893461i	$-0.648270\pi$
$223$	−13.4142	−0.898282	−0.449141	+	0.893461i	$0.648270\pi$
$224$	−2.41421	−0.161306
$225$	−2.82843	−0.188562
$226$	−0.485281	−0.0322804
$227$	5.31371	0.352683	0.176342	−	0.984329i	$-0.443574\pi$
$227$	5.31371	0.352683	0.176342	+	0.984329i	$0.443574\pi$
$228$	0.171573	0.0113627
$229$	−1.31371	−0.0868123	−0.0434062	−	0.999058i	$-0.513821\pi$
$229$	−1.31371	−0.0868123	−0.0434062	+	0.999058i	$0.513821\pi$
$230$	3.41421	0.225127
$231$	1.00000	0.0657952
$232$	2.41421	0.158501
$233$	8.89949	0.583025	0.291513	−	0.956567i	$-0.405841\pi$
$233$	8.89949	0.583025	0.291513	+	0.956567i	$0.405841\pi$
$234$	−9.65685	−0.631288
$235$	−3.65685	−0.238547
$236$	−10.7279	−0.698328
$237$	2.58579	0.167965
$238$	−14.0711	−0.912092
$239$	−16.3431	−1.05715	−0.528575	−	0.848887i	$-0.677274\pi$
$239$	−16.3431	−1.05715	−0.528575	+	0.848887i	$0.677274\pi$
$240$	−0.414214	−0.0267374
$241$	−27.0711	−1.74380	−0.871900	−	0.489683i	$-0.837112\pi$
$241$	−27.0711	−1.74380	−0.871900	+	0.489683i	$0.837112\pi$
$242$	1.00000	0.0642824
$243$	10.3431	0.663513
$244$	6.07107	0.388660
$245$	1.17157	0.0748490
$246$	−2.58579	−0.164864
$247$	1.41421	0.0899843
$248$	−1.82843	−0.116105
$249$	−1.61522	−0.102361
$250$	−1.00000	−0.0632456
$251$	18.7279	1.18210	0.591048	−	0.806636i	$-0.298714\pi$
$251$	18.7279	1.18210	0.591048	+	0.806636i	$0.298714\pi$
$252$	6.82843	0.430150
$253$	3.41421	0.214650
$254$	3.65685	0.229451
$255$	−2.41421	−0.151184
$256$	1.00000	0.0625000
$257$	−17.3137	−1.08000	−0.540000	−	0.841665i	$-0.681576\pi$
$257$	−17.3137	−1.08000	−0.540000	+	0.841665i	$0.681576\pi$
$258$	0.414214	0.0257878
$259$	22.3137	1.38651
$260$	−3.41421	−0.211741
$261$	−6.82843	−0.422669
$262$	−6.07107	−0.375072
$263$	11.5858	0.714410	0.357205	−	0.934026i	$-0.383730\pi$
$263$	11.5858	0.714410	0.357205	+	0.934026i	$0.383730\pi$
$264$	−0.414214	−0.0254931
$265$	9.82843	0.603755
$266$	−1.00000	−0.0613139
$267$	−1.82843	−0.111898
$268$	−3.65685	−0.223378
$269$	7.55635	0.460719	0.230359	−	0.973106i	$-0.426010\pi$
$269$	7.55635	0.460719	0.230359	+	0.973106i	$0.426010\pi$
$270$	2.41421	0.146924
$271$	−2.58579	−0.157075	−0.0785377	−	0.996911i	$-0.525025\pi$
$271$	−2.58579	−0.157075	−0.0785377	+	0.996911i	$0.525025\pi$
$272$	5.82843	0.353400
$273$	−3.41421	−0.206638
$274$	−18.2426	−1.10208
$275$	−1.00000	−0.0603023
$276$	−1.41421	−0.0851257
$277$	7.89949	0.474635	0.237317	−	0.971432i	$-0.423732\pi$
$277$	7.89949	0.474635	0.237317	+	0.971432i	$0.423732\pi$
$278$	19.6569	1.17894
$279$	5.17157	0.309614
$280$	2.41421	0.144277
$281$	20.3848	1.21605	0.608027	−	0.793916i	$-0.291961\pi$
$281$	20.3848	1.21605	0.608027	+	0.793916i	$0.291961\pi$
$282$	1.51472	0.0902002
$283$	12.8284	0.762571	0.381285	−	0.924457i	$-0.375482\pi$
$283$	12.8284	0.762571	0.381285	+	0.924457i	$0.375482\pi$
$284$	1.24264	0.0737372
$285$	−0.171573	−0.0101631
$286$	−3.41421	−0.201887
$287$	15.0711	0.889617
$288$	−2.82843	−0.166667
$289$	16.9706	0.998268
$290$	−2.41421	−0.141768
$291$	−4.48528	−0.262932
$292$	−13.3137	−0.779126
$293$	−9.75736	−0.570031	−0.285016	−	0.958523i	$-0.591999\pi$
$293$	−9.75736	−0.570031	−0.285016	+	0.958523i	$0.591999\pi$
$294$	−0.485281	−0.0283022
$295$	10.7279	0.624604
$296$	−9.24264	−0.537218
$297$	2.41421	0.140087
$298$	−7.92893	−0.459311
$299$	−11.6569	−0.674133
$300$	0.414214	0.0239146
$301$	−2.41421	−0.139153
$302$	4.24264	0.244137
$303$	−3.31371	−0.190368
$304$	0.414214	0.0237568
$305$	−6.07107	−0.347628
$306$	−16.4853	−0.942401
$307$	−4.34315	−0.247876	−0.123938	−	0.992290i	$-0.539552\pi$
$307$	−4.34315	−0.247876	−0.123938	+	0.992290i	$0.539552\pi$
$308$	2.41421	0.137563
$309$	−5.45584	−0.310372
$310$	1.82843	0.103848
$311$	23.0000	1.30421	0.652105	−	0.758129i	$-0.273886\pi$
$311$	23.0000	1.30421	0.652105	+	0.758129i	$0.273886\pi$
$312$	1.41421	0.0800641
$313$	23.4558	1.32580	0.662901	−	0.748707i	$-0.269325\pi$
$313$	23.4558	1.32580	0.662901	+	0.748707i	$0.269325\pi$
$314$	15.7279	0.887578
$315$	−6.82843	−0.384738
$316$	6.24264	0.351176
$317$	−6.65685	−0.373886	−0.186943	−	0.982371i	$-0.559858\pi$
$317$	−6.65685	−0.373886	−0.186943	+	0.982371i	$0.559858\pi$
$318$	−4.07107	−0.228294
$319$	−2.41421	−0.135170
$320$	−1.00000	−0.0559017
$321$	5.85786	0.326954
$322$	8.24264	0.459344
$323$	2.41421	0.134330
$324$	7.48528	0.415849
$325$	3.41421	0.189386
$326$	−17.3848	−0.962854
$327$	−5.45584	−0.301709
$328$	−6.24264	−0.344692
$329$	−8.82843	−0.486727
$330$	0.414214	0.0228017
$331$	−9.17157	−0.504115	−0.252058	−	0.967712i	$-0.581107\pi$
$331$	−9.17157	−0.504115	−0.252058	+	0.967712i	$0.581107\pi$
$332$	−3.89949	−0.214013
$333$	26.1421	1.43258
$334$	0.313708	0.0171654
$335$	3.65685	0.199795
$336$	−1.00000	−0.0545545
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	−1.34315	−0.0730575
$339$	−0.201010	−0.0109174
$340$	−5.82843	−0.316091
$341$	1.82843	0.0990149
$342$	−1.17157	−0.0633514
$343$	19.7279	1.06521
$344$	1.00000	0.0539164
$345$	1.41421	0.0761387
$346$	−14.9706	−0.804822
$347$	11.6569	0.625773	0.312886	−	0.949791i	$-0.398704\pi$
$347$	11.6569	0.625773	0.312886	+	0.949791i	$0.398704\pi$
$348$	1.00000	0.0536056
$349$	1.51472	0.0810810	0.0405405	−	0.999178i	$-0.487092\pi$
$349$	1.51472	0.0810810	0.0405405	+	0.999178i	$0.487092\pi$
$350$	−2.41421	−0.129045
$351$	−8.24264	−0.439960
$352$	−1.00000	−0.0533002
$353$	34.2426	1.82255	0.911276	−	0.411796i	$-0.135098\pi$
$353$	34.2426	1.82255	0.911276	+	0.411796i	$0.135098\pi$
$354$	−4.44365	−0.236177
$355$	−1.24264	−0.0659525
$356$	−4.41421	−0.233953
$357$	−5.82843	−0.308473
$358$	2.34315	0.123839
$359$	19.7990	1.04495	0.522475	−	0.852654i	$-0.325009\pi$
$359$	19.7990	1.04495	0.522475	+	0.852654i	$0.325009\pi$
$360$	2.82843	0.149071
$361$	−18.8284	−0.990970
$362$	−18.0000	−0.946059
$363$	0.414214	0.0217406
$364$	−8.24264	−0.432032
$365$	13.3137	0.696871
$366$	2.51472	0.131446
$367$	−11.0711	−0.577905	−0.288953	−	0.957343i	$-0.593307\pi$
$367$	−11.0711	−0.577905	−0.288953	+	0.957343i	$0.593307\pi$
$368$	−3.41421	−0.177978
$369$	17.6569	0.919179
$370$	9.24264	0.480502
$371$	23.7279	1.23189
$372$	−0.757359	−0.0392673
$373$	1.41421	0.0732252	0.0366126	−	0.999330i	$-0.488343\pi$
$373$	1.41421	0.0732252	0.0366126	+	0.999330i	$0.488343\pi$
$374$	−5.82843	−0.301381
$375$	−0.414214	−0.0213899
$376$	3.65685	0.188588
$377$	8.24264	0.424518
$378$	5.82843	0.299782
$379$	−9.75736	−0.501202	−0.250601	−	0.968090i	$-0.580628\pi$
$379$	−9.75736	−0.501202	−0.250601	+	0.968090i	$0.580628\pi$
$380$	−0.414214	−0.0212487
$381$	1.51472	0.0776014
$382$	−2.00000	−0.102329
$383$	8.92893	0.456247	0.228124	−	0.973632i	$-0.426741\pi$
$383$	8.92893	0.456247	0.228124	+	0.973632i	$0.426741\pi$
$384$	0.414214	0.0211377
$385$	−2.41421	−0.123040
$386$	1.48528	0.0755988
$387$	−2.82843	−0.143777
$388$	−10.8284	−0.549730
$389$	−21.8995	−1.11035	−0.555174	−	0.831734i	$-0.687348\pi$
$389$	−21.8995	−1.11035	−0.555174	+	0.831734i	$0.687348\pi$
$390$	−1.41421	−0.0716115
$391$	−19.8995	−1.00636
$392$	−1.17157	−0.0591734
$393$	−2.51472	−0.126851
$394$	14.8284	0.747045
$395$	−6.24264	−0.314101
$396$	2.82843	0.142134
$397$	−17.1716	−0.861817	−0.430908	−	0.902396i	$-0.641807\pi$
$397$	−17.1716	−0.861817	−0.430908	+	0.902396i	$0.641807\pi$
$398$	6.55635	0.328640
$399$	−0.414214	−0.0207366
$400$	1.00000	0.0500000
$401$	28.4558	1.42102	0.710509	−	0.703689i	$-0.248465\pi$
$401$	28.4558	1.42102	0.710509	+	0.703689i	$0.248465\pi$
$402$	−1.51472	−0.0755473
$403$	−6.24264	−0.310968
$404$	−8.00000	−0.398015
$405$	−7.48528	−0.371947
$406$	−5.82843	−0.289260
$407$	9.24264	0.458141
$408$	2.41421	0.119521
$409$	−26.1421	−1.29265	−0.646323	−	0.763064i	$-0.723694\pi$
$409$	−26.1421	−1.29265	−0.646323	+	0.763064i	$0.723694\pi$
$410$	6.24264	0.308302
$411$	−7.55635	−0.372727
$412$	−13.1716	−0.648917
$413$	25.8995	1.27443
$414$	9.65685	0.474608
$415$	3.89949	0.191419
$416$	3.41421	0.167396
$417$	8.14214	0.398722
$418$	−0.414214	−0.0202598
$419$	−17.0711	−0.833976	−0.416988	−	0.908912i	$-0.636914\pi$
$419$	−17.0711	−0.833976	−0.416988	+	0.908912i	$0.636914\pi$
$420$	1.00000	0.0487950
$421$	−40.2843	−1.96334	−0.981668	−	0.190601i	$-0.938956\pi$
$421$	−40.2843	−1.96334	−0.981668	+	0.190601i	$0.938956\pi$
$422$	24.8995	1.21209
$423$	−10.3431	−0.502901
$424$	−9.82843	−0.477311
$425$	5.82843	0.282720
$426$	0.514719	0.0249382
$427$	−14.6569	−0.709295
$428$	14.1421	0.683586
$429$	−1.41421	−0.0682789
$430$	−1.00000	−0.0482243
$431$	5.41421	0.260793	0.130397	−	0.991462i	$-0.458375\pi$
$431$	5.41421	0.260793	0.130397	+	0.991462i	$0.458375\pi$
$432$	−2.41421	−0.116154
$433$	37.6569	1.80967	0.904836	−	0.425759i	$-0.139993\pi$
$433$	37.6569	1.80967	0.904836	+	0.425759i	$0.139993\pi$
$434$	4.41421	0.211889
$435$	−1.00000	−0.0479463
$436$	−13.1716	−0.630804
$437$	−1.41421	−0.0676510
$438$	−5.51472	−0.263503
$439$	−20.0416	−0.956535	−0.478267	−	0.878214i	$-0.658735\pi$
$439$	−20.0416	−0.956535	−0.478267	+	0.878214i	$0.658735\pi$
$440$	1.00000	0.0476731
$441$	3.31371	0.157796
$442$	19.8995	0.946522
$443$	−0.828427	−0.0393598	−0.0196799	−	0.999806i	$-0.506265\pi$
$443$	−0.828427	−0.0393598	−0.0196799	+	0.999806i	$0.506265\pi$
$444$	−3.82843	−0.181689
$445$	4.41421	0.209254
$446$	−13.4142	−0.635181
$447$	−3.28427	−0.155341
$448$	−2.41421	−0.114061
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	−2.82843	−0.133333
$451$	6.24264	0.293954
$452$	−0.485281	−0.0228257
$453$	1.75736	0.0825679
$454$	5.31371	0.249385
$455$	8.24264	0.386421
$456$	0.171573	0.00803464
$457$	35.7279	1.67128	0.835641	−	0.549276i	$-0.185097\pi$
$457$	35.7279	1.67128	0.835641	+	0.549276i	$0.185097\pi$
$458$	−1.31371	−0.0613856
$459$	−14.0711	−0.656781
$460$	3.41421	0.159189
$461$	5.97056	0.278077	0.139038	−	0.990287i	$-0.455599\pi$
$461$	5.97056	0.278077	0.139038	+	0.990287i	$0.455599\pi$
$462$	1.00000	0.0465242
$463$	−25.5147	−1.18577	−0.592885	−	0.805287i	$-0.702011\pi$
$463$	−25.5147	−1.18577	−0.592885	+	0.805287i	$0.702011\pi$
$464$	2.41421	0.112077
$465$	0.757359	0.0351217
$466$	8.89949	0.412261
$467$	15.5858	0.721224	0.360612	−	0.932716i	$-0.382568\pi$
$467$	15.5858	0.721224	0.360612	+	0.932716i	$0.382568\pi$
$468$	−9.65685	−0.446388
$469$	8.82843	0.407659
$470$	−3.65685	−0.168678
$471$	6.51472	0.300182
$472$	−10.7279	−0.493793
$473$	−1.00000	−0.0459800
$474$	2.58579	0.118769
$475$	0.414214	0.0190054
$476$	−14.0711	−0.644946
$477$	27.7990	1.27283
$478$	−16.3431	−0.747518
$479$	−37.4558	−1.71140	−0.855701	−	0.517471i	$-0.826873\pi$
$479$	−37.4558	−1.71140	−0.855701	+	0.517471i	$0.826873\pi$
$480$	−0.414214	−0.0189062
$481$	−31.5563	−1.43885
$482$	−27.0711	−1.23305
$483$	3.41421	0.155352
$484$	1.00000	0.0454545
$485$	10.8284	0.491694
$486$	10.3431	0.469175
$487$	2.10051	0.0951830	0.0475915	−	0.998867i	$-0.484845\pi$
$487$	2.10051	0.0951830	0.0475915	+	0.998867i	$0.484845\pi$
$488$	6.07107	0.274824
$489$	−7.20101	−0.325641
$490$	1.17157	0.0529263
$491$	−34.3553	−1.55043	−0.775217	−	0.631695i	$-0.782360\pi$
$491$	−34.3553	−1.55043	−0.775217	+	0.631695i	$0.782360\pi$
$492$	−2.58579	−0.116576
$493$	14.0711	0.633729
$494$	1.41421	0.0636285
$495$	−2.82843	−0.127128
$496$	−1.82843	−0.0820988
$497$	−3.00000	−0.134568
$498$	−1.61522	−0.0723799
$499$	−6.24264	−0.279459	−0.139729	−	0.990190i	$-0.544623\pi$
$499$	−6.24264	−0.279459	−0.139729	+	0.990190i	$0.544623\pi$
$500$	−1.00000	−0.0447214
$501$	0.129942	0.00580539
$502$	18.7279	0.835868
$503$	3.02944	0.135076	0.0675380	−	0.997717i	$-0.478486\pi$
$503$	3.02944	0.135076	0.0675380	+	0.997717i	$0.478486\pi$
$504$	6.82843	0.304162
$505$	8.00000	0.355995
$506$	3.41421	0.151780
$507$	−0.556349	−0.0247083
$508$	3.65685	0.162247
$509$	−39.7990	−1.76406	−0.882030	−	0.471194i	$-0.843823\pi$
$509$	−39.7990	−1.76406	−0.882030	+	0.471194i	$0.843823\pi$
$510$	−2.41421	−0.106903
$511$	32.1421	1.42188
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−17.3137	−0.763675
$515$	13.1716	0.580409
$516$	0.414214	0.0182347
$517$	−3.65685	−0.160828
$518$	22.3137	0.980408
$519$	−6.20101	−0.272194
$520$	−3.41421	−0.149723
$521$	5.02944	0.220344	0.110172	−	0.993913i	$-0.464860\pi$
$521$	5.02944	0.220344	0.110172	+	0.993913i	$0.464860\pi$
$522$	−6.82843	−0.298872
$523$	−15.6985	−0.686447	−0.343223	−	0.939254i	$-0.611519\pi$
$523$	−15.6985	−0.686447	−0.343223	+	0.939254i	$0.611519\pi$
$524$	−6.07107	−0.265216
$525$	−1.00000	−0.0436436
$526$	11.5858	0.505164
$527$	−10.6569	−0.464220
$528$	−0.414214	−0.0180263
$529$	−11.3431	−0.493180
$530$	9.82843	0.426920
$531$	30.3431	1.31678
$532$	−1.00000	−0.0433555
$533$	−21.3137	−0.923199
$534$	−1.82843	−0.0791238
$535$	−14.1421	−0.611418
$536$	−3.65685	−0.157952
$537$	0.970563	0.0418829
$538$	7.55635	0.325777
$539$	1.17157	0.0504632
$540$	2.41421	0.103891
$541$	14.4558	0.621505	0.310753	−	0.950491i	$-0.399419\pi$
$541$	14.4558	0.621505	0.310753	+	0.950491i	$0.399419\pi$
$542$	−2.58579	−0.111069
$543$	−7.45584	−0.319961
$544$	5.82843	0.249892
$545$	13.1716	0.564208
$546$	−3.41421	−0.146115
$547$	14.6274	0.625423	0.312712	−	0.949848i	$-0.398763\pi$
$547$	14.6274	0.625423	0.312712	+	0.949848i	$0.398763\pi$
$548$	−18.2426	−0.779287
$549$	−17.1716	−0.732865
$550$	−1.00000	−0.0426401
$551$	1.00000	0.0426014
$552$	−1.41421	−0.0601929
$553$	−15.0711	−0.640887
$554$	7.89949	0.335617
$555$	3.82843	0.162508
$556$	19.6569	0.833636
$557$	−5.31371	−0.225149	−0.112575	−	0.993643i	$-0.535910\pi$
$557$	−5.31371	−0.225149	−0.112575	+	0.993643i	$0.535910\pi$
$558$	5.17157	0.218930
$559$	3.41421	0.144406
$560$	2.41421	0.102019
$561$	−2.41421	−0.101928
$562$	20.3848	0.859880
$563$	−27.6985	−1.16735	−0.583676	−	0.811987i	$-0.698386\pi$
$563$	−27.6985	−1.16735	−0.583676	+	0.811987i	$0.698386\pi$
$564$	1.51472	0.0637812
$565$	0.485281	0.0204159
$566$	12.8284	0.539219
$567$	−18.0711	−0.758914
$568$	1.24264	0.0521400
$569$	−3.27208	−0.137173	−0.0685863	−	0.997645i	$-0.521849\pi$
$569$	−3.27208	−0.137173	−0.0685863	+	0.997645i	$0.521849\pi$
$570$	−0.171573	−0.00718640
$571$	8.21320	0.343712	0.171856	−	0.985122i	$-0.445024\pi$
$571$	8.21320	0.343712	0.171856	+	0.985122i	$0.445024\pi$
$572$	−3.41421	−0.142755
$573$	−0.828427	−0.0346080
$574$	15.0711	0.629054
$575$	−3.41421	−0.142383
$576$	−2.82843	−0.117851
$577$	18.4853	0.769552	0.384776	−	0.923010i	$-0.374279\pi$
$577$	18.4853	0.769552	0.384776	+	0.923010i	$0.374279\pi$
$578$	16.9706	0.705882
$579$	0.615224	0.0255678
$580$	−2.41421	−0.100245
$581$	9.41421	0.390567
$582$	−4.48528	−0.185921
$583$	9.82843	0.407052
$584$	−13.3137	−0.550925
$585$	9.65685	0.399262
$586$	−9.75736	−0.403073
$587$	42.3553	1.74819	0.874096	−	0.485753i	$-0.161455\pi$
$587$	42.3553	1.74819	0.874096	+	0.485753i	$0.161455\pi$
$588$	−0.485281	−0.0200127
$589$	−0.757359	−0.0312064
$590$	10.7279	0.441662
$591$	6.14214	0.252654
$592$	−9.24264	−0.379870
$593$	5.79899	0.238136	0.119068	−	0.992886i	$-0.462009\pi$
$593$	5.79899	0.238136	0.119068	+	0.992886i	$0.462009\pi$
$594$	2.41421	0.0990564
$595$	14.0711	0.576858
$596$	−7.92893	−0.324782
$597$	2.71573	0.111147
$598$	−11.6569	−0.476684
$599$	−11.0000	−0.449448	−0.224724	−	0.974422i	$-0.572148\pi$
$599$	−11.0000	−0.449448	−0.224724	+	0.974422i	$0.572148\pi$
$600$	0.414214	0.0169102
$601$	29.4142	1.19983	0.599915	−	0.800063i	$-0.295201\pi$
$601$	29.4142	1.19983	0.599915	+	0.800063i	$0.295201\pi$
$602$	−2.41421	−0.0983960
$603$	10.3431	0.421205
$604$	4.24264	0.172631
$605$	−1.00000	−0.0406558
$606$	−3.31371	−0.134610
$607$	−5.24264	−0.212792	−0.106396	−	0.994324i	$-0.533931\pi$
$607$	−5.24264	−0.212792	−0.106396	+	0.994324i	$0.533931\pi$
$608$	0.414214	0.0167986
$609$	−2.41421	−0.0978289
$610$	−6.07107	−0.245810
$611$	12.4853	0.505100
$612$	−16.4853	−0.666378
$613$	−27.6985	−1.11873	−0.559366	−	0.828921i	$-0.688955\pi$
$613$	−27.6985	−1.11873	−0.559366	+	0.828921i	$0.688955\pi$
$614$	−4.34315	−0.175275
$615$	2.58579	0.104269
$616$	2.41421	0.0972714
$617$	19.7574	0.795401	0.397701	−	0.917515i	$-0.369808\pi$
$617$	19.7574	0.795401	0.397701	+	0.917515i	$0.369808\pi$
$618$	−5.45584	−0.219466
$619$	−8.04163	−0.323220	−0.161610	−	0.986855i	$-0.551669\pi$
$619$	−8.04163	−0.323220	−0.161610	+	0.986855i	$0.551669\pi$
$620$	1.82843	0.0734314
$621$	8.24264	0.330766
$622$	23.0000	0.922216
$623$	10.6569	0.426958
$624$	1.41421	0.0566139
$625$	1.00000	0.0400000
$626$	23.4558	0.937484
$627$	−0.171573	−0.00685196
$628$	15.7279	0.627612
$629$	−53.8701	−2.14794
$630$	−6.82843	−0.272051
$631$	45.2426	1.80108	0.900541	−	0.434772i	$-0.143171\pi$
$631$	45.2426	1.80108	0.900541	+	0.434772i	$0.143171\pi$
$632$	6.24264	0.248319
$633$	10.3137	0.409933
$634$	−6.65685	−0.264377
$635$	−3.65685	−0.145118
$636$	−4.07107	−0.161428
$637$	−4.00000	−0.158486
$638$	−2.41421	−0.0955796
$639$	−3.51472	−0.139040
$640$	−1.00000	−0.0395285
$641$	−21.2426	−0.839034	−0.419517	−	0.907748i	$-0.637801\pi$
$641$	−21.2426	−0.839034	−0.419517	+	0.907748i	$0.637801\pi$
$642$	5.85786	0.231191
$643$	−5.14214	−0.202786	−0.101393	−	0.994846i	$-0.532330\pi$
$643$	−5.14214	−0.202786	−0.101393	+	0.994846i	$0.532330\pi$
$644$	8.24264	0.324806
$645$	−0.414214	−0.0163096
$646$	2.41421	0.0949860
$647$	−27.0711	−1.06427	−0.532137	−	0.846658i	$-0.678611\pi$
$647$	−27.0711	−1.06427	−0.532137	+	0.846658i	$0.678611\pi$
$648$	7.48528	0.294050
$649$	10.7279	0.421108
$650$	3.41421	0.133916
$651$	1.82843	0.0716617
$652$	−17.3848	−0.680840
$653$	14.7574	0.577500	0.288750	−	0.957405i	$-0.406760\pi$
$653$	14.7574	0.577500	0.288750	+	0.957405i	$0.406760\pi$
$654$	−5.45584	−0.213340
$655$	6.07107	0.237216
$656$	−6.24264	−0.243734
$657$	37.6569	1.46913
$658$	−8.82843	−0.344168
$659$	20.7990	0.810214	0.405107	−	0.914269i	$-0.367234\pi$
$659$	20.7990	0.810214	0.405107	+	0.914269i	$0.367234\pi$
$660$	0.414214	0.0161232
$661$	5.02944	0.195622	0.0978112	−	0.995205i	$-0.468816\pi$
$661$	5.02944	0.195622	0.0978112	+	0.995205i	$0.468816\pi$
$662$	−9.17157	−0.356463
$663$	8.24264	0.320118
$664$	−3.89949	−0.151330
$665$	1.00000	0.0387783
$666$	26.1421	1.01299
$667$	−8.24264	−0.319156
$668$	0.313708	0.0121377
$669$	−5.55635	−0.214821
$670$	3.65685	0.141277
$671$	−6.07107	−0.234371
$672$	−1.00000	−0.0385758
$673$	−9.44365	−0.364026	−0.182013	−	0.983296i	$-0.558261\pi$
$673$	−9.44365	−0.364026	−0.182013	+	0.983296i	$0.558261\pi$
$674$	31.0000	1.19408
$675$	−2.41421	−0.0929231
$676$	−1.34315	−0.0516595
$677$	−21.3553	−0.820752	−0.410376	−	0.911916i	$-0.634603\pi$
$677$	−21.3553	−0.820752	−0.410376	+	0.911916i	$0.634603\pi$
$678$	−0.201010	−0.00771975
$679$	26.1421	1.00324
$680$	−5.82843	−0.223510
$681$	2.20101	0.0843429
$682$	1.82843	0.0700141
$683$	−36.4558	−1.39494	−0.697472	−	0.716612i	$-0.745692\pi$
$683$	−36.4558	−1.39494	−0.697472	+	0.716612i	$0.745692\pi$
$684$	−1.17157	−0.0447962
$685$	18.2426	0.697015
$686$	19.7279	0.753216
$687$	−0.544156	−0.0207608
$688$	1.00000	0.0381246
$689$	−33.5563	−1.27840
$690$	1.41421	0.0538382
$691$	28.7279	1.09286	0.546431	−	0.837504i	$-0.315986\pi$
$691$	28.7279	1.09286	0.546431	+	0.837504i	$0.315986\pi$
$692$	−14.9706	−0.569095
$693$	−6.82843	−0.259390
$694$	11.6569	0.442488
$695$	−19.6569	−0.745627
$696$	1.00000	0.0379049
$697$	−36.3848	−1.37817
$698$	1.51472	0.0573329
$699$	3.68629	0.139428
$700$	−2.41421	−0.0912487
$701$	39.4853	1.49134	0.745669	−	0.666316i	$-0.232130\pi$
$701$	39.4853	1.49134	0.745669	+	0.666316i	$0.232130\pi$
$702$	−8.24264	−0.311098
$703$	−3.82843	−0.144392
$704$	−1.00000	−0.0376889
$705$	−1.51472	−0.0570476
$706$	34.2426	1.28874
$707$	19.3137	0.726367
$708$	−4.44365	−0.167003
$709$	6.24264	0.234447	0.117224	−	0.993106i	$-0.462601\pi$
$709$	6.24264	0.234447	0.117224	+	0.993106i	$0.462601\pi$
$710$	−1.24264	−0.0466355
$711$	−17.6569	−0.662184
$712$	−4.41421	−0.165430
$713$	6.24264	0.233789
$714$	−5.82843	−0.218123
$715$	3.41421	0.127684
$716$	2.34315	0.0875675
$717$	−6.76955	−0.252814
$718$	19.7990	0.738892
$719$	49.4264	1.84329	0.921647	−	0.388029i	$-0.126844\pi$
$719$	49.4264	1.84329	0.921647	+	0.388029i	$0.126844\pi$
$720$	2.82843	0.105409
$721$	31.7990	1.18426
$722$	−18.8284	−0.700721
$723$	−11.2132	−0.417024
$724$	−18.0000	−0.668965
$725$	2.41421	0.0896616
$726$	0.414214	0.0153729
$727$	−35.4558	−1.31498	−0.657492	−	0.753461i	$-0.728383\pi$
$727$	−35.4558	−1.31498	−0.657492	+	0.753461i	$0.728383\pi$
$728$	−8.24264	−0.305493
$729$	−18.1716	−0.673021
$730$	13.3137	0.492762
$731$	5.82843	0.215572
$732$	2.51472	0.0929467
$733$	23.6985	0.875324	0.437662	−	0.899140i	$-0.355807\pi$
$733$	23.6985	0.875324	0.437662	+	0.899140i	$0.355807\pi$
$734$	−11.0711	−0.408641
$735$	0.485281	0.0178999
$736$	−3.41421	−0.125850
$737$	3.65685	0.134702
$738$	17.6569	0.649958
$739$	−20.3431	−0.748335	−0.374167	−	0.927361i	$-0.622071\pi$
$739$	−20.3431	−0.748335	−0.374167	+	0.927361i	$0.622071\pi$
$740$	9.24264	0.339766
$741$	0.585786	0.0215194
$742$	23.7279	0.871079
$743$	−4.21320	−0.154567	−0.0772837	−	0.997009i	$-0.524625\pi$
$743$	−4.21320	−0.154567	−0.0772837	+	0.997009i	$0.524625\pi$
$744$	−0.757359	−0.0277661
$745$	7.92893	0.290494
$746$	1.41421	0.0517780
$747$	11.0294	0.403546
$748$	−5.82843	−0.213108
$749$	−34.1421	−1.24753
$750$	−0.414214	−0.0151249
$751$	−10.2721	−0.374833	−0.187417	−	0.982280i	$-0.560011\pi$
$751$	−10.2721	−0.374833	−0.187417	+	0.982280i	$0.560011\pi$
$752$	3.65685	0.133352
$753$	7.75736	0.282694
$754$	8.24264	0.300179
$755$	−4.24264	−0.154406
$756$	5.82843	0.211978
$757$	−19.1716	−0.696803	−0.348401	−	0.937345i	$-0.613275\pi$
$757$	−19.1716	−0.696803	−0.348401	+	0.937345i	$0.613275\pi$
$758$	−9.75736	−0.354403
$759$	1.41421	0.0513327
$760$	−0.414214	−0.0150251
$761$	15.0711	0.546326	0.273163	−	0.961968i	$-0.411930\pi$
$761$	15.0711	0.546326	0.273163	+	0.961968i	$0.411930\pi$
$762$	1.51472	0.0548725
$763$	31.7990	1.15120
$764$	−2.00000	−0.0723575
$765$	16.4853	0.596027
$766$	8.92893	0.322615
$767$	−36.6274	−1.32254
$768$	0.414214	0.0149466
$769$	−49.7990	−1.79580	−0.897899	−	0.440202i	$-0.854907\pi$
$769$	−49.7990	−1.79580	−0.897899	+	0.440202i	$0.854907\pi$
$770$	−2.41421	−0.0870022
$771$	−7.17157	−0.258278
$772$	1.48528	0.0534564
$773$	−19.1005	−0.686997	−0.343499	−	0.939153i	$-0.611612\pi$
$773$	−19.1005	−0.686997	−0.343499	+	0.939153i	$0.611612\pi$
$774$	−2.82843	−0.101666
$775$	−1.82843	−0.0656790
$776$	−10.8284	−0.388718
$777$	9.24264	0.331578
$778$	−21.8995	−0.785135
$779$	−2.58579	−0.0926454
$780$	−1.41421	−0.0506370
$781$	−1.24264	−0.0444652
$782$	−19.8995	−0.711604
$783$	−5.82843	−0.208291
$784$	−1.17157	−0.0418419
$785$	−15.7279	−0.561354
$786$	−2.51472	−0.0896970
$787$	23.5147	0.838209	0.419105	−	0.907938i	$-0.362344\pi$
$787$	23.5147	0.838209	0.419105	+	0.907938i	$0.362344\pi$
$788$	14.8284	0.528241
$789$	4.79899	0.170849
$790$	−6.24264	−0.222103
$791$	1.17157	0.0416563
$792$	2.82843	0.100504
$793$	20.7279	0.736070
$794$	−17.1716	−0.609396
$795$	4.07107	0.144386
$796$	6.55635	0.232384
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	−0.414214	−0.0146630
$799$	21.3137	0.754025
$800$	1.00000	0.0353553
$801$	12.4853	0.441146
$802$	28.4558	1.00481
$803$	13.3137	0.469831
$804$	−1.51472	−0.0534200
$805$	−8.24264	−0.290515
$806$	−6.24264	−0.219888
$807$	3.12994	0.110179
$808$	−8.00000	−0.281439
$809$	−42.0416	−1.47811	−0.739053	−	0.673648i	$-0.764726\pi$
$809$	−42.0416	−1.47811	−0.739053	+	0.673648i	$0.764726\pi$
$810$	−7.48528	−0.263006
$811$	−45.5269	−1.59867	−0.799333	−	0.600888i	$-0.794814\pi$
$811$	−45.5269	−1.59867	−0.799333	+	0.600888i	$0.794814\pi$
$812$	−5.82843	−0.204538
$813$	−1.07107	−0.0375640
$814$	9.24264	0.323954
$815$	17.3848	0.608962
$816$	2.41421	0.0845144
$817$	0.414214	0.0144915
$818$	−26.1421	−0.914038
$819$	23.3137	0.814647
$820$	6.24264	0.218002
$821$	−14.8284	−0.517516	−0.258758	−	0.965942i	$-0.583313\pi$
$821$	−14.8284	−0.517516	−0.258758	+	0.965942i	$0.583313\pi$
$822$	−7.55635	−0.263558
$823$	−32.7279	−1.14082	−0.570412	−	0.821359i	$-0.693216\pi$
$823$	−32.7279	−1.14082	−0.570412	+	0.821359i	$0.693216\pi$
$824$	−13.1716	−0.458853
$825$	−0.414214	−0.0144211
$826$	25.8995	0.901159
$827$	31.5147	1.09587	0.547937	−	0.836520i	$-0.315413\pi$
$827$	31.5147	1.09587	0.547937	+	0.836520i	$0.315413\pi$
$828$	9.65685	0.335599
$829$	6.72792	0.233670	0.116835	−	0.993151i	$-0.462725\pi$
$829$	6.72792	0.233670	0.116835	+	0.993151i	$0.462725\pi$
$830$	3.89949	0.135353
$831$	3.27208	0.113507
$832$	3.41421	0.118367
$833$	−6.82843	−0.236591
$834$	8.14214	0.281939
$835$	−0.313708	−0.0108563
$836$	−0.414214	−0.0143259
$837$	4.41421	0.152578
$838$	−17.0711	−0.589710
$839$	48.9706	1.69065	0.845326	−	0.534251i	$-0.179406\pi$
$839$	48.9706	1.69065	0.845326	+	0.534251i	$0.179406\pi$
$840$	1.00000	0.0345033
$841$	−23.1716	−0.799020
$842$	−40.2843	−1.38829
$843$	8.44365	0.290815
$844$	24.8995	0.857076
$845$	1.34315	0.0462056
$846$	−10.3431	−0.355605
$847$	−2.41421	−0.0829534
$848$	−9.82843	−0.337510
$849$	5.31371	0.182366
$850$	5.82843	0.199913
$851$	31.5563	1.08174
$852$	0.514719	0.0176340
$853$	−36.3848	−1.24579	−0.622896	−	0.782305i	$-0.714044\pi$
$853$	−36.3848	−1.24579	−0.622896	+	0.782305i	$0.714044\pi$
$854$	−14.6569	−0.501547
$855$	1.17157	0.0400669
$856$	14.1421	0.483368
$857$	22.3137	0.762222	0.381111	−	0.924529i	$-0.375542\pi$
$857$	22.3137	0.762222	0.381111	+	0.924529i	$0.375542\pi$
$858$	−1.41421	−0.0482805
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	−1.00000	−0.0340997
$861$	6.24264	0.212749
$862$	5.41421	0.184409
$863$	39.9411	1.35961	0.679806	−	0.733392i	$-0.262064\pi$
$863$	39.9411	1.35961	0.679806	+	0.733392i	$0.262064\pi$
$864$	−2.41421	−0.0821332
$865$	14.9706	0.509014
$866$	37.6569	1.27963
$867$	7.02944	0.238732
$868$	4.41421	0.149828
$869$	−6.24264	−0.211767
$870$	−1.00000	−0.0339032
$871$	−12.4853	−0.423048
$872$	−13.1716	−0.446046
$873$	30.6274	1.03658
$874$	−1.41421	−0.0478365
$875$	2.41421	0.0816153
$876$	−5.51472	−0.186325
$877$	−33.5147	−1.13171	−0.565856	−	0.824504i	$-0.691454\pi$
$877$	−33.5147	−1.13171	−0.565856	+	0.824504i	$0.691454\pi$
$878$	−20.0416	−0.676372
$879$	−4.04163	−0.136321
$880$	1.00000	0.0337100
$881$	12.2843	0.413868	0.206934	−	0.978355i	$-0.433652\pi$
$881$	12.2843	0.413868	0.206934	+	0.978355i	$0.433652\pi$
$882$	3.31371	0.111578
$883$	22.5980	0.760483	0.380241	−	0.924887i	$-0.375841\pi$
$883$	22.5980	0.760483	0.380241	+	0.924887i	$0.375841\pi$
$884$	19.8995	0.669292
$885$	4.44365	0.149372
$886$	−0.828427	−0.0278316
$887$	57.9411	1.94547	0.972736	−	0.231914i	$-0.0744988\pi$
$887$	57.9411	1.94547	0.972736	+	0.231914i	$0.0744988\pi$
$888$	−3.82843	−0.128474
$889$	−8.82843	−0.296096
$890$	4.41421	0.147965
$891$	−7.48528	−0.250766
$892$	−13.4142	−0.449141
$893$	1.51472	0.0506881
$894$	−3.28427	−0.109842
$895$	−2.34315	−0.0783227
$896$	−2.41421	−0.0806532
$897$	−4.82843	−0.161216
$898$	−10.0000	−0.333704
$899$	−4.41421	−0.147222
$900$	−2.82843	−0.0942809
$901$	−57.2843	−1.90842
$902$	6.24264	0.207857
$903$	−1.00000	−0.0332779
$904$	−0.485281	−0.0161402
$905$	18.0000	0.598340
$906$	1.75736	0.0583844
$907$	53.7696	1.78539	0.892694	−	0.450663i	$-0.148812\pi$
$907$	53.7696	1.78539	0.892694	+	0.450663i	$0.148812\pi$
$908$	5.31371	0.176342
$909$	22.6274	0.750504
$910$	8.24264	0.273241
$911$	18.0711	0.598721	0.299361	−	0.954140i	$-0.403227\pi$
$911$	18.0711	0.598721	0.299361	+	0.954140i	$0.403227\pi$
$912$	0.171573	0.00568135
$913$	3.89949	0.129054
$914$	35.7279	1.18177
$915$	−2.51472	−0.0831340
$916$	−1.31371	−0.0434062
$917$	14.6569	0.484012
$918$	−14.0711	−0.464414
$919$	15.7990	0.521161	0.260580	−	0.965452i	$-0.416086\pi$
$919$	15.7990	0.521161	0.260580	+	0.965452i	$0.416086\pi$
$920$	3.41421	0.112563
$921$	−1.79899	−0.0592787
$922$	5.97056	0.196630
$923$	4.24264	0.139648
$924$	1.00000	0.0328976
$925$	−9.24264	−0.303896
$926$	−25.5147	−0.838466
$927$	37.2548	1.22361
$928$	2.41421	0.0792504
$929$	16.0711	0.527275	0.263637	−	0.964622i	$-0.415078\pi$
$929$	16.0711	0.527275	0.263637	+	0.964622i	$0.415078\pi$
$930$	0.757359	0.0248348
$931$	−0.485281	−0.0159045
$932$	8.89949	0.291513
$933$	9.52691	0.311897
$934$	15.5858	0.509983
$935$	5.82843	0.190610
$936$	−9.65685	−0.315644
$937$	−13.8579	−0.452717	−0.226358	−	0.974044i	$-0.572682\pi$
$937$	−13.8579	−0.452717	−0.226358	+	0.974044i	$0.572682\pi$
$938$	8.82843	0.288258
$939$	9.71573	0.317061
$940$	−3.65685	−0.119273
$941$	22.8579	0.745145	0.372573	−	0.928003i	$-0.378476\pi$
$941$	22.8579	0.745145	0.372573	+	0.928003i	$0.378476\pi$
$942$	6.51472	0.212261
$943$	21.3137	0.694070
$944$	−10.7279	−0.349164
$945$	−5.82843	−0.189599
$946$	−1.00000	−0.0325128
$947$	−10.7990	−0.350920	−0.175460	−	0.984487i	$-0.556141\pi$
$947$	−10.7990	−0.350920	−0.175460	+	0.984487i	$0.556141\pi$
$948$	2.58579	0.0839824
$949$	−45.4558	−1.47556
$950$	0.414214	0.0134389
$951$	−2.75736	−0.0894135
$952$	−14.0711	−0.456046
$953$	44.2132	1.43221	0.716103	−	0.697995i	$-0.245924\pi$
$953$	44.2132	1.43221	0.716103	+	0.697995i	$0.245924\pi$
$954$	27.7990	0.900026
$955$	2.00000	0.0647185
$956$	−16.3431	−0.528575
$957$	−1.00000	−0.0323254
$958$	−37.4558	−1.21014
$959$	44.0416	1.42218
$960$	−0.414214	−0.0133687
$961$	−27.6569	−0.892157
$962$	−31.5563	−1.01742
$963$	−40.0000	−1.28898
$964$	−27.0711	−0.871900
$965$	−1.48528	−0.0478129
$966$	3.41421	0.109851
$967$	−36.4558	−1.17234	−0.586170	−	0.810188i	$-0.699365\pi$
$967$	−36.4558	−1.17234	−0.586170	+	0.810188i	$0.699365\pi$
$968$	1.00000	0.0321412
$969$	1.00000	0.0321246
$970$	10.8284	0.347680
$971$	−32.2843	−1.03605	−0.518026	−	0.855365i	$-0.673333\pi$
$971$	−32.2843	−1.03605	−0.518026	+	0.855365i	$0.673333\pi$
$972$	10.3431	0.331757
$973$	−47.4558	−1.52136
$974$	2.10051	0.0673045
$975$	1.41421	0.0452911
$976$	6.07107	0.194330
$977$	1.17157	0.0374819	0.0187410	−	0.999824i	$-0.494034\pi$
$977$	1.17157	0.0374819	0.0187410	+	0.999824i	$0.494034\pi$
$978$	−7.20101	−0.230263
$979$	4.41421	0.141079
$980$	1.17157	0.0374245
$981$	37.2548	1.18946
$982$	−34.3553	−1.09632
$983$	20.9289	0.667529	0.333765	−	0.942656i	$-0.391681\pi$
$983$	20.9289	0.667529	0.333765	+	0.942656i	$0.391681\pi$
$984$	−2.58579	−0.0824319
$985$	−14.8284	−0.472473
$986$	14.0711	0.448114
$987$	−3.65685	−0.116399
$988$	1.41421	0.0449921
$989$	−3.41421	−0.108566
$990$	−2.82843	−0.0898933
$991$	12.6863	0.402993	0.201497	−	0.979489i	$-0.435419\pi$
$991$	12.6863	0.402993	0.201497	+	0.979489i	$0.435419\pi$
$992$	−1.82843	−0.0580526
$993$	−3.79899	−0.120557
$994$	−3.00000	−0.0951542
$995$	−6.55635	−0.207850
$996$	−1.61522	−0.0511803
$997$	−8.48528	−0.268732	−0.134366	−	0.990932i	$-0.542900\pi$
$997$	−8.48528	−0.268732	−0.134366	+	0.990932i	$0.542900\pi$
$998$	−6.24264	−0.197607
$999$	22.3137	0.705974

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4730.2.a.p.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4730.2.a.p.1.2	✓	2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.414214 q^{6} -2.41421 q^{7} +1.00000 q^{8} -2.82843 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.414214 q^{6} -2.41421 q^{7} +1.00000 q^{8} -2.82843 q^{9} -1.00000 q^{10} -1.00000 q^{11} +0.414214 q^{12} +3.41421 q^{13} -2.41421 q^{14} -0.414214 q^{15} +1.00000 q^{16} +5.82843 q^{17} -2.82843 q^{18} +0.414214 q^{19} -1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -3.41421 q^{23} +0.414214 q^{24} +1.00000 q^{25} +3.41421 q^{26} -2.41421 q^{27} -2.41421 q^{28} +2.41421 q^{29} -0.414214 q^{30} -1.82843 q^{31} +1.00000 q^{32} -0.414214 q^{33} +5.82843 q^{34} +2.41421 q^{35} -2.82843 q^{36} -9.24264 q^{37} +0.414214 q^{38} +1.41421 q^{39} -1.00000 q^{40} -6.24264 q^{41} -1.00000 q^{42} +1.00000 q^{43} -1.00000 q^{44} +2.82843 q^{45} -3.41421 q^{46} +3.65685 q^{47} +0.414214 q^{48} -1.17157 q^{49} +1.00000 q^{50} +2.41421 q^{51} +3.41421 q^{52} -9.82843 q^{53} -2.41421 q^{54} +1.00000 q^{55} -2.41421 q^{56} +0.171573 q^{57} +2.41421 q^{58} -10.7279 q^{59} -0.414214 q^{60} +6.07107 q^{61} -1.82843 q^{62} +6.82843 q^{63} +1.00000 q^{64} -3.41421 q^{65} -0.414214 q^{66} -3.65685 q^{67} +5.82843 q^{68} -1.41421 q^{69} +2.41421 q^{70} +1.24264 q^{71} -2.82843 q^{72} -13.3137 q^{73} -9.24264 q^{74} +0.414214 q^{75} +0.414214 q^{76} +2.41421 q^{77} +1.41421 q^{78} +6.24264 q^{79} -1.00000 q^{80} +7.48528 q^{81} -6.24264 q^{82} -3.89949 q^{83} -1.00000 q^{84} -5.82843 q^{85} +1.00000 q^{86} +1.00000 q^{87} -1.00000 q^{88} -4.41421 q^{89} +2.82843 q^{90} -8.24264 q^{91} -3.41421 q^{92} -0.757359 q^{93} +3.65685 q^{94} -0.414214 q^{95} +0.414214 q^{96} -10.8284 q^{97} -1.17157 q^{98} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} + 6 q^{17} - 2 q^{19} - 2 q^{20} - 2 q^{21} - 2 q^{22} - 4 q^{23} - 2 q^{24} + 2 q^{25} + 4 q^{26} - 2 q^{27} - 2 q^{28} + 2 q^{29} + 2 q^{30} + 2 q^{31} + 2 q^{32} + 2 q^{33} + 6 q^{34} + 2 q^{35} - 10 q^{37} - 2 q^{38} - 2 q^{40} - 4 q^{41} - 2 q^{42} + 2 q^{43} - 2 q^{44} - 4 q^{46} - 4 q^{47} - 2 q^{48} - 8 q^{49} + 2 q^{50} + 2 q^{51} + 4 q^{52} - 14 q^{53} - 2 q^{54} + 2 q^{55} - 2 q^{56} + 6 q^{57} + 2 q^{58} + 4 q^{59} + 2 q^{60} - 2 q^{61} + 2 q^{62} + 8 q^{63} + 2 q^{64} - 4 q^{65} + 2 q^{66} + 4 q^{67} + 6 q^{68} + 2 q^{70} - 6 q^{71} - 4 q^{73} - 10 q^{74} - 2 q^{75} - 2 q^{76} + 2 q^{77} + 4 q^{79} - 2 q^{80} - 2 q^{81} - 4 q^{82} + 12 q^{83} - 2 q^{84} - 6 q^{85} + 2 q^{86} + 2 q^{87} - 2 q^{88} - 6 q^{89} - 8 q^{91} - 4 q^{92} - 10 q^{93} - 4 q^{94} + 2 q^{95} - 2 q^{96} - 16 q^{97} - 8 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^11 - 2 * q^12 + 4 * q^13 - 2 * q^14 + 2 * q^15 + 2 * q^16 + 6 * q^17 - 2 * q^19 - 2 * q^20 - 2 * q^21 - 2 * q^22 - 4 * q^23 - 2 * q^24 + 2 * q^25 + 4 * q^26 - 2 * q^27 - 2 * q^28 + 2 * q^29 + 2 * q^30 + 2 * q^31 + 2 * q^32 + 2 * q^33 + 6 * q^34 + 2 * q^35 - 10 * q^37 - 2 * q^38 - 2 * q^40 - 4 * q^41 - 2 * q^42 + 2 * q^43 - 2 * q^44 - 4 * q^46 - 4 * q^47 - 2 * q^48 - 8 * q^49 + 2 * q^50 + 2 * q^51 + 4 * q^52 - 14 * q^53 - 2 * q^54 + 2 * q^55 - 2 * q^56 + 6 * q^57 + 2 * q^58 + 4 * q^59 + 2 * q^60 - 2 * q^61 + 2 * q^62 + 8 * q^63 + 2 * q^64 - 4 * q^65 + 2 * q^66 + 4 * q^67 + 6 * q^68 + 2 * q^70 - 6 * q^71 - 4 * q^73 - 10 * q^74 - 2 * q^75 - 2 * q^76 + 2 * q^77 + 4 * q^79 - 2 * q^80 - 2 * q^81 - 4 * q^82 + 12 * q^83 - 2 * q^84 - 6 * q^85 + 2 * q^86 + 2 * q^87 - 2 * q^88 - 6 * q^89 - 8 * q^91 - 4 * q^92 - 10 * q^93 - 4 * q^94 + 2 * q^95 - 2 * q^96 - 16 * q^97 - 8 * q^98

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