LMFDB - Embedded newform 4030.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.1797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 18x^{6} + 12x^{5} + 98x^{4} - 18x^{3} - 173x^{2} - 48x + 32 $ x^8 - x^7 - 18x^6 + 12x^5 + 98x^4 - 18x^3 - 173x^2 - 48x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.89749$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.89749 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.89749 q^{6} +0.730984 q^{7} -1.00000 q^{8} +0.600463 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.89749 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.89749 q^{6} +0.730984 q^{7} -1.00000 q^{8} +0.600463 q^{9} +1.00000 q^{10} +2.37058 q^{11} -1.89749 q^{12} +1.00000 q^{13} -0.730984 q^{14} +1.89749 q^{15} +1.00000 q^{16} -6.53685 q^{17} -0.600463 q^{18} +7.54171 q^{19} -1.00000 q^{20} -1.38703 q^{21} -2.37058 q^{22} +6.56109 q^{23} +1.89749 q^{24} +1.00000 q^{25} -1.00000 q^{26} +4.55309 q^{27} +0.730984 q^{28} -3.52646 q^{29} -1.89749 q^{30} -1.00000 q^{31} -1.00000 q^{32} -4.49815 q^{33} +6.53685 q^{34} -0.730984 q^{35} +0.600463 q^{36} -8.37632 q^{37} -7.54171 q^{38} -1.89749 q^{39} +1.00000 q^{40} +10.4911 q^{41} +1.38703 q^{42} -11.2196 q^{43} +2.37058 q^{44} -0.600463 q^{45} -6.56109 q^{46} +3.79386 q^{47} -1.89749 q^{48} -6.46566 q^{49} -1.00000 q^{50} +12.4036 q^{51} +1.00000 q^{52} +9.62823 q^{53} -4.55309 q^{54} -2.37058 q^{55} -0.730984 q^{56} -14.3103 q^{57} +3.52646 q^{58} -0.122502 q^{59} +1.89749 q^{60} -9.26121 q^{61} +1.00000 q^{62} +0.438929 q^{63} +1.00000 q^{64} -1.00000 q^{65} +4.49815 q^{66} +7.43808 q^{67} -6.53685 q^{68} -12.4496 q^{69} +0.730984 q^{70} +13.4786 q^{71} -0.600463 q^{72} -1.92325 q^{73} +8.37632 q^{74} -1.89749 q^{75} +7.54171 q^{76} +1.73286 q^{77} +1.89749 q^{78} -0.500415 q^{79} -1.00000 q^{80} -10.4408 q^{81} -10.4911 q^{82} -15.2331 q^{83} -1.38703 q^{84} +6.53685 q^{85} +11.2196 q^{86} +6.69142 q^{87} -2.37058 q^{88} +5.13489 q^{89} +0.600463 q^{90} +0.730984 q^{91} +6.56109 q^{92} +1.89749 q^{93} -3.79386 q^{94} -7.54171 q^{95} +1.89749 q^{96} -13.8506 q^{97} +6.46566 q^{98} +1.42345 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9} + 8 q^{10} + 12 q^{11} - q^{12} + 8 q^{13} + 8 q^{14} + q^{15} + 8 q^{16} - 13 q^{18} - 3 q^{19} - 8 q^{20} + 27 q^{21} - 12 q^{22} + 17 q^{23} + q^{24} + 8 q^{25} - 8 q^{26} - 13 q^{27} - 8 q^{28} - 10 q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 4 q^{33} + 8 q^{35} + 13 q^{36} + 2 q^{37} + 3 q^{38} - q^{39} + 8 q^{40} + 14 q^{41} - 27 q^{42} - 19 q^{43} + 12 q^{44} - 13 q^{45} - 17 q^{46} + 8 q^{47} - q^{48} + 16 q^{49} - 8 q^{50} + 35 q^{51} + 8 q^{52} + 4 q^{53} + 13 q^{54} - 12 q^{55} + 8 q^{56} - 33 q^{57} + 10 q^{58} - 13 q^{59} + q^{60} + 11 q^{61} + 8 q^{62} - 27 q^{63} + 8 q^{64} - 8 q^{65} - 4 q^{66} - 34 q^{67} + 12 q^{69} - 8 q^{70} + 8 q^{71} - 13 q^{72} - 21 q^{73} - 2 q^{74} - q^{75} - 3 q^{76} + 19 q^{77} + q^{78} - 36 q^{79} - 8 q^{80} + 20 q^{81} - 14 q^{82} + 27 q^{84} + 19 q^{86} - 29 q^{87} - 12 q^{88} - 16 q^{89} + 13 q^{90} - 8 q^{91} + 17 q^{92} + q^{93} - 8 q^{94} + 3 q^{95} + q^{96} + q^{97} - 16 q^{98} + 65 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9 + 8 * q^10 + 12 * q^11 - q^12 + 8 * q^13 + 8 * q^14 + q^15 + 8 * q^16 - 13 * q^18 - 3 * q^19 - 8 * q^20 + 27 * q^21 - 12 * q^22 + 17 * q^23 + q^24 + 8 * q^25 - 8 * q^26 - 13 * q^27 - 8 * q^28 - 10 * q^29 - q^30 - 8 * q^31 - 8 * q^32 + 4 * q^33 + 8 * q^35 + 13 * q^36 + 2 * q^37 + 3 * q^38 - q^39 + 8 * q^40 + 14 * q^41 - 27 * q^42 - 19 * q^43 + 12 * q^44 - 13 * q^45 - 17 * q^46 + 8 * q^47 - q^48 + 16 * q^49 - 8 * q^50 + 35 * q^51 + 8 * q^52 + 4 * q^53 + 13 * q^54 - 12 * q^55 + 8 * q^56 - 33 * q^57 + 10 * q^58 - 13 * q^59 + q^60 + 11 * q^61 + 8 * q^62 - 27 * q^63 + 8 * q^64 - 8 * q^65 - 4 * q^66 - 34 * q^67 + 12 * q^69 - 8 * q^70 + 8 * q^71 - 13 * q^72 - 21 * q^73 - 2 * q^74 - q^75 - 3 * q^76 + 19 * q^77 + q^78 - 36 * q^79 - 8 * q^80 + 20 * q^81 - 14 * q^82 + 27 * q^84 + 19 * q^86 - 29 * q^87 - 12 * q^88 - 16 * q^89 + 13 * q^90 - 8 * q^91 + 17 * q^92 + q^93 - 8 * q^94 + 3 * q^95 + q^96 + q^97 - 16 * q^98 + 65 * 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
$3$	−1.89749	−1.09552	−0.547758	−	0.836637i	$-0.684518\pi$
$3$	−1.89749	−1.09552	−0.547758	+	0.836637i	$0.684518\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.89749	0.774646
$7$	0.730984	0.276286	0.138143	−	0.990412i	$-0.455887\pi$
$7$	0.730984	0.276286	0.138143	+	0.990412i	$0.455887\pi$
$8$	−1.00000	−0.353553
$9$	0.600463	0.200154
$10$	1.00000	0.316228
$11$	2.37058	0.714758	0.357379	−	0.933959i	$-0.383670\pi$
$11$	2.37058	0.714758	0.357379	+	0.933959i	$0.383670\pi$
$12$	−1.89749	−0.547758
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.730984	−0.195364
$15$	1.89749	0.489929
$16$	1.00000	0.250000
$17$	−6.53685	−1.58542	−0.792709	−	0.609600i	$-0.791330\pi$
$17$	−6.53685	−1.58542	−0.792709	+	0.609600i	$0.791330\pi$
$18$	−0.600463	−0.141531
$19$	7.54171	1.73019	0.865094	−	0.501610i	$-0.167259\pi$
$19$	7.54171	1.73019	0.865094	+	0.501610i	$0.167259\pi$
$20$	−1.00000	−0.223607
$21$	−1.38703	−0.302676
$22$	−2.37058	−0.505410
$23$	6.56109	1.36808	0.684040	−	0.729444i	$-0.260221\pi$
$23$	6.56109	1.36808	0.684040	+	0.729444i	$0.260221\pi$
$24$	1.89749	0.387323
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	4.55309	0.876243
$28$	0.730984	0.138143
$29$	−3.52646	−0.654847	−0.327424	−	0.944878i	$-0.606180\pi$
$29$	−3.52646	−0.654847	−0.327424	+	0.944878i	$0.606180\pi$
$30$	−1.89749	−0.346432
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	−4.49815	−0.783028
$34$	6.53685	1.12106
$35$	−0.730984	−0.123559
$36$	0.600463	0.100077
$37$	−8.37632	−1.37706	−0.688530	−	0.725208i	$-0.741743\pi$
$37$	−8.37632	−1.37706	−0.688530	+	0.725208i	$0.741743\pi$
$38$	−7.54171	−1.22343
$39$	−1.89749	−0.303841
$40$	1.00000	0.158114
$41$	10.4911	1.63843	0.819217	−	0.573484i	$-0.194408\pi$
$41$	10.4911	1.63843	0.819217	+	0.573484i	$0.194408\pi$
$42$	1.38703	0.214024
$43$	−11.2196	−1.71098	−0.855489	−	0.517822i	$-0.826743\pi$
$43$	−11.2196	−1.71098	−0.855489	+	0.517822i	$0.826743\pi$
$44$	2.37058	0.357379
$45$	−0.600463	−0.0895118
$46$	−6.56109	−0.967379
$47$	3.79386	0.553392	0.276696	−	0.960958i	$-0.410761\pi$
$47$	3.79386	0.553392	0.276696	+	0.960958i	$0.410761\pi$
$48$	−1.89749	−0.273879
$49$	−6.46566	−0.923666
$50$	−1.00000	−0.141421
$51$	12.4036	1.73685
$52$	1.00000	0.138675
$53$	9.62823	1.32254	0.661270	−	0.750148i	$-0.270018\pi$
$53$	9.62823	1.32254	0.661270	+	0.750148i	$0.270018\pi$
$54$	−4.55309	−0.619598
$55$	−2.37058	−0.319649
$56$	−0.730984	−0.0976819
$57$	−14.3103	−1.89545
$58$	3.52646	0.463047
$59$	−0.122502	−0.0159485	−0.00797423	−	0.999968i	$-0.502538\pi$
$59$	−0.122502	−0.0159485	−0.00797423	+	0.999968i	$0.502538\pi$
$60$	1.89749	0.244965
$61$	−9.26121	−1.18578	−0.592889	−	0.805285i	$-0.702013\pi$
$61$	−9.26121	−1.18578	−0.592889	+	0.805285i	$0.702013\pi$
$62$	1.00000	0.127000
$63$	0.438929	0.0552999
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	4.49815	0.553685
$67$	7.43808	0.908706	0.454353	−	0.890822i	$-0.349870\pi$
$67$	7.43808	0.908706	0.454353	+	0.890822i	$0.349870\pi$
$68$	−6.53685	−0.792709
$69$	−12.4496	−1.49875
$70$	0.730984	0.0873694
$71$	13.4786	1.59961	0.799807	−	0.600257i	$-0.204935\pi$
$71$	13.4786	1.59961	0.799807	+	0.600257i	$0.204935\pi$
$72$	−0.600463	−0.0707653
$73$	−1.92325	−0.225099	−0.112550	−	0.993646i	$-0.535902\pi$
$73$	−1.92325	−0.225099	−0.112550	+	0.993646i	$0.535902\pi$
$74$	8.37632	0.973728
$75$	−1.89749	−0.219103
$76$	7.54171	0.865094
$77$	1.73286	0.197478
$78$	1.89749	0.214848
$79$	−0.500415	−0.0563011	−0.0281505	−	0.999604i	$-0.508962\pi$
$79$	−0.500415	−0.0563011	−0.0281505	+	0.999604i	$0.508962\pi$
$80$	−1.00000	−0.111803
$81$	−10.4408	−1.16009
$82$	−10.4911	−1.15855
$83$	−15.2331	−1.67205	−0.836023	−	0.548695i	$-0.815125\pi$
$83$	−15.2331	−1.67205	−0.836023	+	0.548695i	$0.815125\pi$
$84$	−1.38703	−0.151338
$85$	6.53685	0.709020
$86$	11.2196	1.20984
$87$	6.69142	0.717395
$88$	−2.37058	−0.252705
$89$	5.13489	0.544297	0.272149	−	0.962255i	$-0.412266\pi$
$89$	5.13489	0.544297	0.272149	+	0.962255i	$0.412266\pi$
$90$	0.600463	0.0632944
$91$	0.730984	0.0766280
$92$	6.56109	0.684040
$93$	1.89749	0.196760
$94$	−3.79386	−0.391307
$95$	−7.54171	−0.773763
$96$	1.89749	0.193662
$97$	−13.8506	−1.40631	−0.703155	−	0.711036i	$-0.748226\pi$
$97$	−13.8506	−1.40631	−0.703155	+	0.711036i	$0.748226\pi$
$98$	6.46566	0.653130
$99$	1.42345	0.143062
$100$	1.00000	0.100000
$101$	2.97421	0.295945	0.147973	−	0.988991i	$-0.452725\pi$
$101$	2.97421	0.295945	0.147973	+	0.988991i	$0.452725\pi$
$102$	−12.4036	−1.22814
$103$	−6.52262	−0.642693	−0.321346	−	0.946962i	$-0.604135\pi$
$103$	−6.52262	−0.642693	−0.321346	+	0.946962i	$0.604135\pi$
$104$	−1.00000	−0.0980581
$105$	1.38703	0.135361
$106$	−9.62823	−0.935176
$107$	16.0520	1.55181	0.775903	−	0.630852i	$-0.217295\pi$
$107$	16.0520	1.55181	0.775903	+	0.630852i	$0.217295\pi$
$108$	4.55309	0.438122
$109$	12.2158	1.17006	0.585032	−	0.811010i	$-0.301082\pi$
$109$	12.2158	1.17006	0.585032	+	0.811010i	$0.301082\pi$
$110$	2.37058	0.226026
$111$	15.8940	1.50859
$112$	0.730984	0.0690715
$113$	9.05045	0.851394	0.425697	−	0.904866i	$-0.360029\pi$
$113$	9.05045	0.851394	0.425697	+	0.904866i	$0.360029\pi$
$114$	14.3103	1.34028
$115$	−6.56109	−0.611824
$116$	−3.52646	−0.327424
$117$	0.600463	0.0555128
$118$	0.122502	0.0112773
$119$	−4.77833	−0.438029
$120$	−1.89749	−0.173216
$121$	−5.38034	−0.489121
$122$	9.26121	0.838471
$123$	−19.9067	−1.79493
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	−0.438929	−0.0391029
$127$	−20.3085	−1.80208	−0.901042	−	0.433732i	$-0.857196\pi$
$127$	−20.3085	−1.80208	−0.901042	+	0.433732i	$0.857196\pi$
$128$	−1.00000	−0.0883883
$129$	21.2891	1.87440
$130$	1.00000	0.0877058
$131$	−16.1499	−1.41102	−0.705510	−	0.708700i	$-0.749282\pi$
$131$	−16.1499	−1.41102	−0.705510	+	0.708700i	$0.749282\pi$
$132$	−4.49815	−0.391514
$133$	5.51287	0.478027
$134$	−7.43808	−0.642552
$135$	−4.55309	−0.391868
$136$	6.53685	0.560530
$137$	−6.96419	−0.594991	−0.297495	−	0.954723i	$-0.596151\pi$
$137$	−6.96419	−0.594991	−0.297495	+	0.954723i	$0.596151\pi$
$138$	12.4496	1.05978
$139$	11.2302	0.952534	0.476267	−	0.879301i	$-0.341990\pi$
$139$	11.2302	0.952534	0.476267	+	0.879301i	$0.341990\pi$
$140$	−0.730984	−0.0617795
$141$	−7.19881	−0.606249
$142$	−13.4786	−1.13110
$143$	2.37058	0.198238
$144$	0.600463	0.0500386
$145$	3.52646	0.292857
$146$	1.92325	0.159169
$147$	12.2685	1.01189
$148$	−8.37632	−0.688530
$149$	19.7656	1.61926	0.809632	−	0.586938i	$-0.199667\pi$
$149$	19.7656	1.61926	0.809632	+	0.586938i	$0.199667\pi$
$150$	1.89749	0.154929
$151$	−19.9669	−1.62489	−0.812443	−	0.583040i	$-0.801863\pi$
$151$	−19.9669	−1.62489	−0.812443	+	0.583040i	$0.801863\pi$
$152$	−7.54171	−0.611714
$153$	−3.92513	−0.317328
$154$	−1.73286	−0.139638
$155$	1.00000	0.0803219
$156$	−1.89749	−0.151921
$157$	−19.4424	−1.55168	−0.775838	−	0.630933i	$-0.782672\pi$
$157$	−19.4424	−1.55168	−0.775838	+	0.630933i	$0.782672\pi$
$158$	0.500415	0.0398109
$159$	−18.2695	−1.44886
$160$	1.00000	0.0790569
$161$	4.79605	0.377982
$162$	10.4408	0.820309
$163$	10.5751	0.828305	0.414153	−	0.910207i	$-0.364078\pi$
$163$	10.5751	0.828305	0.414153	+	0.910207i	$0.364078\pi$
$164$	10.4911	0.819217
$165$	4.49815	0.350181
$166$	15.2331	1.18231
$167$	3.35904	0.259931	0.129965	−	0.991519i	$-0.458513\pi$
$167$	3.35904	0.259931	0.129965	+	0.991519i	$0.458513\pi$
$168$	1.38703	0.107012
$169$	1.00000	0.0769231
$170$	−6.53685	−0.501353
$171$	4.52852	0.346305
$172$	−11.2196	−0.855489
$173$	17.6888	1.34485	0.672427	−	0.740164i	$-0.265252\pi$
$173$	17.6888	1.34485	0.672427	+	0.740164i	$0.265252\pi$
$174$	−6.69142	−0.507275
$175$	0.730984	0.0552572
$176$	2.37058	0.178689
$177$	0.232447	0.0174718
$178$	−5.13489	−0.384876
$179$	−10.5497	−0.788521	−0.394260	−	0.918999i	$-0.628999\pi$
$179$	−10.5497	−0.788521	−0.394260	+	0.918999i	$0.628999\pi$
$180$	−0.600463	−0.0447559
$181$	−5.93296	−0.440993	−0.220497	−	0.975388i	$-0.570768\pi$
$181$	−5.93296	−0.440993	−0.220497	+	0.975388i	$0.570768\pi$
$182$	−0.730984	−0.0541842
$183$	17.5730	1.29904
$184$	−6.56109	−0.483690
$185$	8.37632	0.615840
$186$	−1.89749	−0.139131
$187$	−15.4961	−1.13319
$188$	3.79386	0.276696
$189$	3.32824	0.242094
$190$	7.54171	0.547133
$191$	−0.975874	−0.0706117	−0.0353059	−	0.999377i	$-0.511241\pi$
$191$	−0.975874	−0.0706117	−0.0353059	+	0.999377i	$0.511241\pi$
$192$	−1.89749	−0.136939
$193$	4.67059	0.336196	0.168098	−	0.985770i	$-0.446237\pi$
$193$	4.67059	0.336196	0.168098	+	0.985770i	$0.446237\pi$
$194$	13.8506	0.994412
$195$	1.89749	0.135882
$196$	−6.46566	−0.461833
$197$	9.32542	0.664408	0.332204	−	0.943207i	$-0.392208\pi$
$197$	9.32542	0.664408	0.332204	+	0.943207i	$0.392208\pi$
$198$	−1.42345	−0.101160
$199$	25.7320	1.82409	0.912045	−	0.410090i	$-0.134503\pi$
$199$	25.7320	1.82409	0.912045	+	0.410090i	$0.134503\pi$
$200$	−1.00000	−0.0707107
$201$	−14.1137	−0.995502
$202$	−2.97421	−0.209265
$203$	−2.57779	−0.180925
$204$	12.4036	0.868425
$205$	−10.4911	−0.732730
$206$	6.52262	0.454453
$207$	3.93969	0.273827
$208$	1.00000	0.0693375
$209$	17.8783	1.23666
$210$	−1.38703	−0.0957145
$211$	15.8187	1.08901	0.544504	−	0.838758i	$-0.316718\pi$
$211$	15.8187	1.08901	0.544504	+	0.838758i	$0.316718\pi$
$212$	9.62823	0.661270
$213$	−25.5755	−1.75240
$214$	−16.0520	−1.09729
$215$	11.2196	0.765172
$216$	−4.55309	−0.309799
$217$	−0.730984	−0.0496225
$218$	−12.2158	−0.827361
$219$	3.64934	0.246600
$220$	−2.37058	−0.159825
$221$	−6.53685	−0.439716
$222$	−15.8940	−1.06673
$223$	23.7847	1.59274	0.796372	−	0.604807i	$-0.206750\pi$
$223$	23.7847	1.59274	0.796372	+	0.604807i	$0.206750\pi$
$224$	−0.730984	−0.0488410
$225$	0.600463	0.0400309
$226$	−9.05045	−0.602027
$227$	8.15837	0.541490	0.270745	−	0.962651i	$-0.412730\pi$
$227$	8.15837	0.541490	0.270745	+	0.962651i	$0.412730\pi$
$228$	−14.3103	−0.947724
$229$	13.9300	0.920523	0.460261	−	0.887783i	$-0.347756\pi$
$229$	13.9300	0.920523	0.460261	+	0.887783i	$0.347756\pi$
$230$	6.56109	0.432625
$231$	−3.28808	−0.216340
$232$	3.52646	0.231523
$233$	−4.82542	−0.316124	−0.158062	−	0.987429i	$-0.550525\pi$
$233$	−4.82542	−0.316124	−0.158062	+	0.987429i	$0.550525\pi$
$234$	−0.600463	−0.0392535
$235$	−3.79386	−0.247484
$236$	−0.122502	−0.00797423
$237$	0.949532	0.0616787
$238$	4.77833	0.309733
$239$	24.0466	1.55544	0.777721	−	0.628610i	$-0.216376\pi$
$239$	24.0466	1.55544	0.777721	+	0.628610i	$0.216376\pi$
$240$	1.89749	0.122482
$241$	23.7872	1.53227	0.766134	−	0.642680i	$-0.222178\pi$
$241$	23.7872	1.53227	0.766134	+	0.642680i	$0.222178\pi$
$242$	5.38034	0.345861
$243$	6.15208	0.394656
$244$	−9.26121	−0.592889
$245$	6.46566	0.413076
$246$	19.9067	1.26921
$247$	7.54171	0.479868
$248$	1.00000	0.0635001
$249$	28.9046	1.83175
$250$	1.00000	0.0632456
$251$	14.1520	0.893267	0.446634	−	0.894717i	$-0.352623\pi$
$251$	14.1520	0.893267	0.446634	+	0.894717i	$0.352623\pi$
$252$	0.438929	0.0276499
$253$	15.5536	0.977846
$254$	20.3085	1.27427
$255$	−12.4036	−0.776743
$256$	1.00000	0.0625000
$257$	−2.38838	−0.148983	−0.0744916	−	0.997222i	$-0.523733\pi$
$257$	−2.38838	−0.148983	−0.0744916	+	0.997222i	$0.523733\pi$
$258$	−21.2891	−1.32540
$259$	−6.12296	−0.380462
$260$	−1.00000	−0.0620174
$261$	−2.11751	−0.131071
$262$	16.1499	0.997741
$263$	−5.84166	−0.360212	−0.180106	−	0.983647i	$-0.557644\pi$
$263$	−5.84166	−0.360212	−0.180106	+	0.983647i	$0.557644\pi$
$264$	4.49815	0.276842
$265$	−9.62823	−0.591458
$266$	−5.51287	−0.338016
$267$	−9.74339	−0.596286
$268$	7.43808	0.454353
$269$	−5.55795	−0.338874	−0.169437	−	0.985541i	$-0.554195\pi$
$269$	−5.55795	−0.338874	−0.169437	+	0.985541i	$0.554195\pi$
$270$	4.55309	0.277092
$271$	−21.2403	−1.29026	−0.645130	−	0.764073i	$-0.723197\pi$
$271$	−21.2403	−1.29026	−0.645130	+	0.764073i	$0.723197\pi$
$272$	−6.53685	−0.396354
$273$	−1.38703	−0.0839472
$274$	6.96419	0.420722
$275$	2.37058	0.142952
$276$	−12.4496	−0.749377
$277$	9.40187	0.564904	0.282452	−	0.959281i	$-0.408852\pi$
$277$	9.40187	0.564904	0.282452	+	0.959281i	$0.408852\pi$
$278$	−11.2302	−0.673543
$279$	−0.600463	−0.0359488
$280$	0.730984	0.0436847
$281$	16.0252	0.955984	0.477992	−	0.878364i	$-0.341365\pi$
$281$	16.0252	0.955984	0.477992	+	0.878364i	$0.341365\pi$
$282$	7.19881	0.428683
$283$	20.6081	1.22503	0.612513	−	0.790461i	$-0.290159\pi$
$283$	20.6081	1.22503	0.612513	+	0.790461i	$0.290159\pi$
$284$	13.4786	0.799807
$285$	14.3103	0.847670
$286$	−2.37058	−0.140176
$287$	7.66883	0.452677
$288$	−0.600463	−0.0353826
$289$	25.7303	1.51355
$290$	−3.52646	−0.207081
$291$	26.2813	1.54064
$292$	−1.92325	−0.112550
$293$	4.48881	0.262239	0.131119	−	0.991367i	$-0.458143\pi$
$293$	4.48881	0.262239	0.131119	+	0.991367i	$0.458143\pi$
$294$	−12.2685	−0.715515
$295$	0.122502	0.00713236
$296$	8.37632	0.486864
$297$	10.7935	0.626302
$298$	−19.7656	−1.14499
$299$	6.56109	0.379437
$300$	−1.89749	−0.109552
$301$	−8.20137	−0.472719
$302$	19.9669	1.14897
$303$	−5.64353	−0.324212
$304$	7.54171	0.432547
$305$	9.26121	0.530296
$306$	3.92513	0.224385
$307$	−23.8447	−1.36089	−0.680444	−	0.732800i	$-0.738213\pi$
$307$	−23.8447	−1.36089	−0.680444	+	0.732800i	$0.738213\pi$
$308$	1.73286	0.0987388
$309$	12.3766	0.704080
$310$	−1.00000	−0.0567962
$311$	−11.5989	−0.657711	−0.328856	−	0.944380i	$-0.606663\pi$
$311$	−11.5989	−0.657711	−0.328856	+	0.944380i	$0.606663\pi$
$312$	1.89749	0.107424
$313$	16.5923	0.937855	0.468927	−	0.883237i	$-0.344641\pi$
$313$	16.5923	0.937855	0.468927	+	0.883237i	$0.344641\pi$
$314$	19.4424	1.09720
$315$	−0.438929	−0.0247309
$316$	−0.500415	−0.0281505
$317$	−13.5684	−0.762077	−0.381039	−	0.924559i	$-0.624434\pi$
$317$	−13.5684	−0.762077	−0.381039	+	0.924559i	$0.624434\pi$
$318$	18.2695	1.02450
$319$	−8.35977	−0.468057
$320$	−1.00000	−0.0559017
$321$	−30.4585	−1.70003
$322$	−4.79605	−0.267274
$323$	−49.2990	−2.74307
$324$	−10.4408	−0.580046
$325$	1.00000	0.0554700
$326$	−10.5751	−0.585700
$327$	−23.1794	−1.28182
$328$	−10.4911	−0.579274
$329$	2.77325	0.152894
$330$	−4.49815	−0.247615
$331$	25.2961	1.39040	0.695199	−	0.718817i	$-0.255316\pi$
$331$	25.2961	1.39040	0.695199	+	0.718817i	$0.255316\pi$
$332$	−15.2331	−0.836023
$333$	−5.02967	−0.275624
$334$	−3.35904	−0.183799
$335$	−7.43808	−0.406386
$336$	−1.38703	−0.0756690
$337$	32.7052	1.78157	0.890783	−	0.454429i	$-0.150157\pi$
$337$	32.7052	1.78157	0.890783	+	0.454429i	$0.150157\pi$
$338$	−1.00000	−0.0543928
$339$	−17.1731	−0.932716
$340$	6.53685	0.354510
$341$	−2.37058	−0.128374
$342$	−4.52852	−0.244874
$343$	−9.84319	−0.531482
$344$	11.2196	0.604922
$345$	12.4496	0.670263
$346$	−17.6888	−0.950955
$347$	−11.6421	−0.624981	−0.312491	−	0.949921i	$-0.601163\pi$
$347$	−11.6421	−0.624981	−0.312491	+	0.949921i	$0.601163\pi$
$348$	6.69142	0.358698
$349$	−5.53711	−0.296395	−0.148197	−	0.988958i	$-0.547347\pi$
$349$	−5.53711	−0.296395	−0.148197	+	0.988958i	$0.547347\pi$
$350$	−0.730984	−0.0390728
$351$	4.55309	0.243026
$352$	−2.37058	−0.126353
$353$	0.288700	0.0153659	0.00768297	−	0.999970i	$-0.497554\pi$
$353$	0.288700	0.0153659	0.00768297	+	0.999970i	$0.497554\pi$
$354$	−0.232447	−0.0123544
$355$	−13.4786	−0.715369
$356$	5.13489	0.272149
$357$	9.06683	0.479868
$358$	10.5497	0.557568
$359$	25.9539	1.36980	0.684898	−	0.728639i	$-0.259847\pi$
$359$	25.9539	1.36980	0.684898	+	0.728639i	$0.259847\pi$
$360$	0.600463	0.0316472
$361$	37.8774	1.99355
$362$	5.93296	0.311829
$363$	10.2091	0.535840
$364$	0.730984	0.0383140
$365$	1.92325	0.100667
$366$	−17.5730	−0.918558
$367$	−4.42528	−0.230998	−0.115499	−	0.993308i	$-0.536847\pi$
$367$	−4.42528	−0.230998	−0.115499	+	0.993308i	$0.536847\pi$
$368$	6.56109	0.342020
$369$	6.29952	0.327940
$370$	−8.37632	−0.435464
$371$	7.03809	0.365399
$372$	1.89749	0.0983802
$373$	19.3156	1.00013	0.500063	−	0.865989i	$-0.333310\pi$
$373$	19.3156	1.00013	0.500063	+	0.865989i	$0.333310\pi$
$374$	15.4961	0.801286
$375$	1.89749	0.0979859
$376$	−3.79386	−0.195653
$377$	−3.52646	−0.181622
$378$	−3.32824	−0.171186
$379$	−9.39084	−0.482375	−0.241188	−	0.970479i	$-0.577537\pi$
$379$	−9.39084	−0.482375	−0.241188	+	0.970479i	$0.577537\pi$
$380$	−7.54171	−0.386882
$381$	38.5351	1.97421
$382$	0.975874	0.0499300
$383$	25.3022	1.29288	0.646441	−	0.762964i	$-0.276257\pi$
$383$	25.3022	1.29288	0.646441	+	0.762964i	$0.276257\pi$
$384$	1.89749	0.0968308
$385$	−1.73286	−0.0883147
$386$	−4.67059	−0.237727
$387$	−6.73697	−0.342460
$388$	−13.8506	−0.703155
$389$	−18.2693	−0.926291	−0.463145	−	0.886282i	$-0.653279\pi$
$389$	−18.2693	−0.926291	−0.463145	+	0.886282i	$0.653279\pi$
$390$	−1.89749	−0.0960831
$391$	−42.8888	−2.16898
$392$	6.46566	0.326565
$393$	30.6442	1.54579
$394$	−9.32542	−0.469808
$395$	0.500415	0.0251786
$396$	1.42345	0.0715309
$397$	25.2156	1.26554	0.632768	−	0.774342i	$-0.281919\pi$
$397$	25.2156	1.26554	0.632768	+	0.774342i	$0.281919\pi$
$398$	−25.7320	−1.28983
$399$	−10.4606	−0.523686
$400$	1.00000	0.0500000
$401$	0.360931	0.0180240	0.00901202	−	0.999959i	$-0.497131\pi$
$401$	0.360931	0.0180240	0.00901202	+	0.999959i	$0.497131\pi$
$402$	14.1137	0.703926
$403$	−1.00000	−0.0498135
$404$	2.97421	0.147973
$405$	10.4408	0.518809
$406$	2.57779	0.127933
$407$	−19.8568	−0.984264
$408$	−12.4036	−0.614069
$409$	16.2130	0.801680	0.400840	−	0.916148i	$-0.368718\pi$
$409$	16.2130	0.801680	0.400840	+	0.916148i	$0.368718\pi$
$410$	10.4911	0.518118
$411$	13.2145	0.651822
$412$	−6.52262	−0.321346
$413$	−0.0895473	−0.00440634
$414$	−3.93969	−0.193625
$415$	15.2331	0.747761
$416$	−1.00000	−0.0490290
$417$	−21.3092	−1.04352
$418$	−17.8783	−0.874454
$419$	−18.6449	−0.910865	−0.455433	−	0.890270i	$-0.650515\pi$
$419$	−18.6449	−0.910865	−0.455433	+	0.890270i	$0.650515\pi$
$420$	1.38703	0.0676804
$421$	3.29229	0.160457	0.0802283	−	0.996777i	$-0.474435\pi$
$421$	3.29229	0.160457	0.0802283	+	0.996777i	$0.474435\pi$
$422$	−15.8187	−0.770045
$423$	2.27807	0.110764
$424$	−9.62823	−0.467588
$425$	−6.53685	−0.317084
$426$	25.5755	1.23914
$427$	−6.76980	−0.327614
$428$	16.0520	0.775903
$429$	−4.49815	−0.217173
$430$	−11.2196	−0.541059
$431$	−0.316054	−0.0152238	−0.00761189	−	0.999971i	$-0.502423\pi$
$431$	−0.316054	−0.0152238	−0.00761189	+	0.999971i	$0.502423\pi$
$432$	4.55309	0.219061
$433$	−7.57496	−0.364030	−0.182015	−	0.983296i	$-0.558262\pi$
$433$	−7.57496	−0.364030	−0.182015	+	0.983296i	$0.558262\pi$
$434$	0.730984	0.0350884
$435$	−6.69142	−0.320829
$436$	12.2158	0.585032
$437$	49.4818	2.36704
$438$	−3.64934	−0.174372
$439$	37.4454	1.78717	0.893585	−	0.448895i	$-0.148182\pi$
$439$	37.4454	1.78717	0.893585	+	0.448895i	$0.148182\pi$
$440$	2.37058	0.113013
$441$	−3.88239	−0.184876
$442$	6.53685	0.310926
$443$	−10.2772	−0.488283	−0.244141	−	0.969740i	$-0.578506\pi$
$443$	−10.2772	−0.488283	−0.244141	+	0.969740i	$0.578506\pi$
$444$	15.8940	0.754295
$445$	−5.13489	−0.243417
$446$	−23.7847	−1.12624
$447$	−37.5051	−1.77393
$448$	0.730984	0.0345358
$449$	5.48302	0.258760	0.129380	−	0.991595i	$-0.458701\pi$
$449$	5.48302	0.258760	0.129380	+	0.991595i	$0.458701\pi$
$450$	−0.600463	−0.0283061
$451$	24.8700	1.17108
$452$	9.05045	0.425697
$453$	37.8870	1.78009
$454$	−8.15837	−0.382891
$455$	−0.730984	−0.0342691
$456$	14.3103	0.670142
$457$	24.3973	1.14126	0.570628	−	0.821209i	$-0.306700\pi$
$457$	24.3973	1.14126	0.570628	+	0.821209i	$0.306700\pi$
$458$	−13.9300	−0.650908
$459$	−29.7629	−1.38921
$460$	−6.56109	−0.305912
$461$	1.52224	0.0708976	0.0354488	−	0.999371i	$-0.488714\pi$
$461$	1.52224	0.0708976	0.0354488	+	0.999371i	$0.488714\pi$
$462$	3.28808	0.152975
$463$	−21.9797	−1.02148	−0.510741	−	0.859735i	$-0.670629\pi$
$463$	−21.9797	−1.02148	−0.510741	+	0.859735i	$0.670629\pi$
$464$	−3.52646	−0.163712
$465$	−1.89749	−0.0879939
$466$	4.82542	0.223533
$467$	30.1686	1.39604	0.698018	−	0.716080i	$-0.254066\pi$
$467$	30.1686	1.39604	0.698018	+	0.716080i	$0.254066\pi$
$468$	0.600463	0.0277564
$469$	5.43712	0.251063
$470$	3.79386	0.174998
$471$	36.8918	1.69988
$472$	0.122502	0.00563863
$473$	−26.5971	−1.22293
$474$	−0.949532	−0.0436134
$475$	7.54171	0.346037
$476$	−4.77833	−0.219015
$477$	5.78140	0.264712
$478$	−24.0466	−1.09986
$479$	−7.76219	−0.354663	−0.177332	−	0.984151i	$-0.556747\pi$
$479$	−7.76219	−0.354663	−0.177332	+	0.984151i	$0.556747\pi$
$480$	−1.89749	−0.0866081
$481$	−8.37632	−0.381927
$482$	−23.7872	−1.08348
$483$	−9.10045	−0.414085
$484$	−5.38034	−0.244561
$485$	13.8506	0.628921
$486$	−6.15208	−0.279064
$487$	−16.9679	−0.768891	−0.384445	−	0.923148i	$-0.625607\pi$
$487$	−16.9679	−0.768891	−0.384445	+	0.923148i	$0.625607\pi$
$488$	9.26121	0.419235
$489$	−20.0661	−0.907421
$490$	−6.46566	−0.292089
$491$	7.39757	0.333848	0.166924	−	0.985970i	$-0.446617\pi$
$491$	7.39757	0.333848	0.166924	+	0.985970i	$0.446617\pi$
$492$	−19.9067	−0.897465
$493$	23.0519	1.03821
$494$	−7.54171	−0.339318
$495$	−1.42345	−0.0639792
$496$	−1.00000	−0.0449013
$497$	9.85264	0.441951
$498$	−28.9046	−1.29524
$499$	18.6118	0.833179	0.416590	−	0.909095i	$-0.363225\pi$
$499$	18.6118	0.833179	0.416590	+	0.909095i	$0.363225\pi$
$500$	−1.00000	−0.0447214
$501$	−6.37375	−0.284758
$502$	−14.1520	−0.631635
$503$	29.8171	1.32948	0.664739	−	0.747076i	$-0.268543\pi$
$503$	29.8171	1.32948	0.664739	+	0.747076i	$0.268543\pi$
$504$	−0.438929	−0.0195515
$505$	−2.97421	−0.132351
$506$	−15.5536	−0.691442
$507$	−1.89749	−0.0842704
$508$	−20.3085	−0.901042
$509$	−30.5308	−1.35325	−0.676627	−	0.736326i	$-0.736559\pi$
$509$	−30.5308	−1.35325	−0.676627	+	0.736326i	$0.736559\pi$
$510$	12.4036	0.549240
$511$	−1.40586	−0.0621918
$512$	−1.00000	−0.0441942
$513$	34.3381	1.51607
$514$	2.38838	0.105347
$515$	6.52262	0.287421
$516$	21.2891	0.937201
$517$	8.99366	0.395541
$518$	6.12296	0.269028
$519$	−33.5643	−1.47331
$520$	1.00000	0.0438529
$521$	42.9717	1.88263	0.941313	−	0.337536i	$-0.109594\pi$
$521$	42.9717	1.88263	0.941313	+	0.337536i	$0.109594\pi$
$522$	2.11751	0.0926809
$523$	42.7865	1.87092	0.935462	−	0.353427i	$-0.114984\pi$
$523$	42.7865	1.87092	0.935462	+	0.353427i	$0.114984\pi$
$524$	−16.1499	−0.705510
$525$	−1.38703	−0.0605352
$526$	5.84166	0.254708
$527$	6.53685	0.284749
$528$	−4.49815	−0.195757
$529$	20.0478	0.871645
$530$	9.62823	0.418224
$531$	−0.0735582	−0.00319215
$532$	5.51287	0.239013
$533$	10.4911	0.454420
$534$	9.74339	0.421638
$535$	−16.0520	−0.693989
$536$	−7.43808	−0.321276
$537$	20.0179	0.863837
$538$	5.55795	0.239620
$539$	−15.3274	−0.660197
$540$	−4.55309	−0.195934
$541$	−4.33977	−0.186581	−0.0932906	−	0.995639i	$-0.529739\pi$
$541$	−4.33977	−0.186581	−0.0932906	+	0.995639i	$0.529739\pi$
$542$	21.2403	0.912351
$543$	11.2577	0.483115
$544$	6.53685	0.280265
$545$	−12.2158	−0.523269
$546$	1.38703	0.0593596
$547$	−35.0703	−1.49950	−0.749749	−	0.661722i	$-0.769826\pi$
$547$	−35.0703	−1.49950	−0.749749	+	0.661722i	$0.769826\pi$
$548$	−6.96419	−0.297495
$549$	−5.56102	−0.237338
$550$	−2.37058	−0.101082
$551$	−26.5955	−1.13301
$552$	12.4496	0.529890
$553$	−0.365796	−0.0155552
$554$	−9.40187	−0.399447
$555$	−15.8940	−0.674662
$556$	11.2302	0.476267
$557$	−0.514860	−0.0218153	−0.0109077	−	0.999941i	$-0.503472\pi$
$557$	−0.514860	−0.0218153	−0.0109077	+	0.999941i	$0.503472\pi$
$558$	0.600463	0.0254196
$559$	−11.2196	−0.474540
$560$	−0.730984	−0.0308897
$561$	29.4037	1.24143
$562$	−16.0252	−0.675983
$563$	15.2882	0.644322	0.322161	−	0.946685i	$-0.395591\pi$
$563$	15.2882	0.644322	0.322161	+	0.946685i	$0.395591\pi$
$564$	−7.19881	−0.303125
$565$	−9.05045	−0.380755
$566$	−20.6081	−0.866224
$567$	−7.63209	−0.320518
$568$	−13.4786	−0.565549
$569$	−15.1152	−0.633664	−0.316832	−	0.948482i	$-0.602619\pi$
$569$	−15.1152	−0.633664	−0.316832	+	0.948482i	$0.602619\pi$
$570$	−14.3103	−0.599393
$571$	25.2815	1.05800	0.528998	−	0.848623i	$-0.322568\pi$
$571$	25.2815	1.05800	0.528998	+	0.848623i	$0.322568\pi$
$572$	2.37058	0.0991191
$573$	1.85171	0.0773563
$574$	−7.66883	−0.320091
$575$	6.56109	0.273616
$576$	0.600463	0.0250193
$577$	−19.1211	−0.796020	−0.398010	−	0.917381i	$-0.630299\pi$
$577$	−19.1211	−0.796020	−0.398010	+	0.917381i	$0.630299\pi$
$578$	−25.7303	−1.07024
$579$	−8.86239	−0.368308
$580$	3.52646	0.146428
$581$	−11.1351	−0.461963
$582$	−26.2813	−1.08939
$583$	22.8245	0.945295
$584$	1.92325	0.0795846
$585$	−0.600463	−0.0248261
$586$	−4.48881	−0.185431
$587$	1.13631	0.0469005	0.0234503	−	0.999725i	$-0.492535\pi$
$587$	1.13631	0.0469005	0.0234503	+	0.999725i	$0.492535\pi$
$588$	12.2685	0.505945
$589$	−7.54171	−0.310751
$590$	−0.122502	−0.00504334
$591$	−17.6949	−0.727870
$592$	−8.37632	−0.344265
$593$	27.1289	1.11405	0.557026	−	0.830495i	$-0.311942\pi$
$593$	27.1289	1.11405	0.557026	+	0.830495i	$0.311942\pi$
$594$	−10.7935	−0.442862
$595$	4.77833	0.195893
$596$	19.7656	0.809632
$597$	−48.8261	−1.99832
$598$	−6.56109	−0.268303
$599$	14.4179	0.589099	0.294550	−	0.955636i	$-0.404830\pi$
$599$	14.4179	0.589099	0.294550	+	0.955636i	$0.404830\pi$
$600$	1.89749	0.0774646
$601$	−3.62914	−0.148036	−0.0740178	−	0.997257i	$-0.523582\pi$
$601$	−3.62914	−0.148036	−0.0740178	+	0.997257i	$0.523582\pi$
$602$	8.20137	0.334263
$603$	4.46629	0.181882
$604$	−19.9669	−0.812443
$605$	5.38034	0.218742
$606$	5.64353	0.229253
$607$	−39.8239	−1.61640	−0.808200	−	0.588908i	$-0.799558\pi$
$607$	−39.8239	−1.61640	−0.808200	+	0.588908i	$0.799558\pi$
$608$	−7.54171	−0.305857
$609$	4.89132	0.198206
$610$	−9.26121	−0.374976
$611$	3.79386	0.153483
$612$	−3.92513	−0.158664
$613$	38.6606	1.56149	0.780743	−	0.624852i	$-0.214841\pi$
$613$	38.6606	1.56149	0.780743	+	0.624852i	$0.214841\pi$
$614$	23.8447	0.962293
$615$	19.9067	0.802717
$616$	−1.73286	−0.0698189
$617$	−36.0210	−1.45015	−0.725076	−	0.688669i	$-0.758195\pi$
$617$	−36.0210	−1.45015	−0.725076	+	0.688669i	$0.758195\pi$
$618$	−12.3766	−0.497860
$619$	−8.72389	−0.350643	−0.175321	−	0.984511i	$-0.556096\pi$
$619$	−8.72389	−0.350643	−0.175321	+	0.984511i	$0.556096\pi$
$620$	1.00000	0.0401610
$621$	29.8732	1.19877
$622$	11.5989	0.465072
$623$	3.75352	0.150382
$624$	−1.89749	−0.0759603
$625$	1.00000	0.0400000
$626$	−16.5923	−0.663163
$627$	−33.9238	−1.35479
$628$	−19.4424	−0.775838
$629$	54.7547	2.18321
$630$	0.438929	0.0174874
$631$	11.6964	0.465626	0.232813	−	0.972522i	$-0.425207\pi$
$631$	11.6964	0.465626	0.232813	+	0.972522i	$0.425207\pi$
$632$	0.500415	0.0199054
$633$	−30.0159	−1.19302
$634$	13.5684	0.538870
$635$	20.3085	0.805916
$636$	−18.2695	−0.724431
$637$	−6.46566	−0.256179
$638$	8.35977	0.330966
$639$	8.09340	0.320170
$640$	1.00000	0.0395285
$641$	−2.59338	−0.102432	−0.0512161	−	0.998688i	$-0.516310\pi$
$641$	−2.59338	−0.102432	−0.0512161	+	0.998688i	$0.516310\pi$
$642$	30.4585	1.20210
$643$	−40.8858	−1.61238	−0.806189	−	0.591658i	$-0.798474\pi$
$643$	−40.8858	−1.61238	−0.806189	+	0.591658i	$0.798474\pi$
$644$	4.79605	0.188991
$645$	−21.2891	−0.838258
$646$	49.2990	1.93964
$647$	44.4977	1.74938	0.874692	−	0.484679i	$-0.161064\pi$
$647$	44.4977	1.74938	0.874692	+	0.484679i	$0.161064\pi$
$648$	10.4408	0.410155
$649$	−0.290402	−0.0113993
$650$	−1.00000	−0.0392232
$651$	1.38703	0.0543622
$652$	10.5751	0.414153
$653$	−9.78880	−0.383065	−0.191533	−	0.981486i	$-0.561346\pi$
$653$	−9.78880	−0.383065	−0.191533	+	0.981486i	$0.561346\pi$
$654$	23.1794	0.906386
$655$	16.1499	0.631027
$656$	10.4911	0.409608
$657$	−1.15484	−0.0450546
$658$	−2.77325	−0.108113
$659$	−27.5202	−1.07204	−0.536018	−	0.844207i	$-0.680072\pi$
$659$	−27.5202	−1.07204	−0.536018	+	0.844207i	$0.680072\pi$
$660$	4.49815	0.175090
$661$	−11.1566	−0.433943	−0.216972	−	0.976178i	$-0.569618\pi$
$661$	−11.1566	−0.433943	−0.216972	+	0.976178i	$0.569618\pi$
$662$	−25.2961	−0.983160
$663$	12.4036	0.481716
$664$	15.2331	0.591157
$665$	−5.51287	−0.213780
$666$	5.02967	0.194896
$667$	−23.1374	−0.895884
$668$	3.35904	0.129965
$669$	−45.1313	−1.74488
$670$	7.43808	0.287358
$671$	−21.9545	−0.847543
$672$	1.38703	0.0535060
$673$	−5.36542	−0.206822	−0.103411	−	0.994639i	$-0.532976\pi$
$673$	−5.36542	−0.206822	−0.103411	+	0.994639i	$0.532976\pi$
$674$	−32.7052	−1.25976
$675$	4.55309	0.175249
$676$	1.00000	0.0384615
$677$	−29.8509	−1.14726	−0.573631	−	0.819114i	$-0.694466\pi$
$677$	−29.8509	−1.14726	−0.573631	+	0.819114i	$0.694466\pi$
$678$	17.1731	0.659530
$679$	−10.1245	−0.388544
$680$	−6.53685	−0.250677
$681$	−15.4804	−0.593211
$682$	2.37058	0.0907743
$683$	−27.8555	−1.06586	−0.532931	−	0.846159i	$-0.678909\pi$
$683$	−27.8555	−1.06586	−0.532931	+	0.846159i	$0.678909\pi$
$684$	4.52852	0.173152
$685$	6.96419	0.266088
$686$	9.84319	0.375815
$687$	−26.4321	−1.00845
$688$	−11.2196	−0.427744
$689$	9.62823	0.366806
$690$	−12.4496	−0.473948
$691$	−47.2608	−1.79788	−0.898942	−	0.438067i	$-0.855663\pi$
$691$	−47.2608	−1.79788	−0.898942	+	0.438067i	$0.855663\pi$
$692$	17.6888	0.672427
$693$	1.04052	0.0395260
$694$	11.6421	0.441928
$695$	−11.2302	−0.425986
$696$	−6.69142	−0.253638
$697$	−68.5787	−2.59760
$698$	5.53711	0.209583
$699$	9.15617	0.346318
$700$	0.730984	0.0276286
$701$	−18.8352	−0.711396	−0.355698	−	0.934601i	$-0.615757\pi$
$701$	−18.8352	−0.711396	−0.355698	+	0.934601i	$0.615757\pi$
$702$	−4.55309	−0.171845
$703$	−63.1718	−2.38257
$704$	2.37058	0.0893447
$705$	7.19881	0.271123
$706$	−0.288700	−0.0108654
$707$	2.17410	0.0817655
$708$	0.232447	0.00873589
$709$	−23.5315	−0.883744	−0.441872	−	0.897078i	$-0.645685\pi$
$709$	−23.5315	−0.883744	−0.441872	+	0.897078i	$0.645685\pi$
$710$	13.4786	0.505843
$711$	−0.300481	−0.0112689
$712$	−5.13489	−0.192438
$713$	−6.56109	−0.245715
$714$	−9.06683	−0.339318
$715$	−2.37058	−0.0886548
$716$	−10.5497	−0.394260
$717$	−45.6281	−1.70401
$718$	−25.9539	−0.968592
$719$	29.7653	1.11006	0.555029	−	0.831831i	$-0.312707\pi$
$719$	29.7653	1.11006	0.555029	+	0.831831i	$0.312707\pi$
$720$	−0.600463	−0.0223779
$721$	−4.76793	−0.177567
$722$	−37.8774	−1.40965
$723$	−45.1360	−1.67862
$724$	−5.93296	−0.220497
$725$	−3.52646	−0.130969
$726$	−10.2091	−0.378896
$727$	5.84982	0.216958	0.108479	−	0.994099i	$-0.465402\pi$
$727$	5.84982	0.216958	0.108479	+	0.994099i	$0.465402\pi$
$728$	−0.730984	−0.0270921
$729$	19.6490	0.727741
$730$	−1.92325	−0.0711826
$731$	73.3410	2.71261
$732$	17.5730	0.649519
$733$	−21.6145	−0.798351	−0.399176	−	0.916875i	$-0.630704\pi$
$733$	−21.6145	−0.798351	−0.399176	+	0.916875i	$0.630704\pi$
$734$	4.42528	0.163340
$735$	−12.2685	−0.452531
$736$	−6.56109	−0.241845
$737$	17.6326	0.649505
$738$	−6.29952	−0.231888
$739$	6.19071	0.227729	0.113864	−	0.993496i	$-0.463677\pi$
$739$	6.19071	0.227729	0.113864	+	0.993496i	$0.463677\pi$
$740$	8.37632	0.307920
$741$	−14.3103	−0.525702
$742$	−7.03809	−0.258376
$743$	33.5358	1.23031	0.615155	−	0.788406i	$-0.289094\pi$
$743$	33.5358	1.23031	0.615155	+	0.788406i	$0.289094\pi$
$744$	−1.89749	−0.0695653
$745$	−19.7656	−0.724157
$746$	−19.3156	−0.707196
$747$	−9.14689	−0.334667
$748$	−15.4961	−0.566595
$749$	11.7338	0.428742
$750$	−1.89749	−0.0692865
$751$	31.7577	1.15886	0.579428	−	0.815023i	$-0.303276\pi$
$751$	31.7577	1.15886	0.579428	+	0.815023i	$0.303276\pi$
$752$	3.79386	0.138348
$753$	−26.8533	−0.978588
$754$	3.52646	0.128426
$755$	19.9669	0.726671
$756$	3.32824	0.121047
$757$	−16.1093	−0.585504	−0.292752	−	0.956188i	$-0.594571\pi$
$757$	−16.1093	−0.585504	−0.292752	+	0.956188i	$0.594571\pi$
$758$	9.39084	0.341091
$759$	−29.5128	−1.07125
$760$	7.54171	0.273567
$761$	−16.0028	−0.580099	−0.290050	−	0.957012i	$-0.593672\pi$
$761$	−16.0028	−0.580099	−0.290050	+	0.957012i	$0.593672\pi$
$762$	−38.5351	−1.39598
$763$	8.92958	0.323273
$764$	−0.975874	−0.0353059
$765$	3.92513	0.141914
$766$	−25.3022	−0.914206
$767$	−0.122502	−0.00442330
$768$	−1.89749	−0.0684697
$769$	20.4951	0.739072	0.369536	−	0.929216i	$-0.379517\pi$
$769$	20.4951	0.739072	0.369536	+	0.929216i	$0.379517\pi$
$770$	1.73286	0.0624479
$771$	4.53193	0.163213
$772$	4.67059	0.168098
$773$	−24.8713	−0.894559	−0.447279	−	0.894394i	$-0.647607\pi$
$773$	−24.8713	−0.894559	−0.447279	+	0.894394i	$0.647607\pi$
$774$	6.73697	0.242155
$775$	−1.00000	−0.0359211
$776$	13.8506	0.497206
$777$	11.6183	0.416802
$778$	18.2693	0.654986
$779$	79.1208	2.83480
$780$	1.89749	0.0679410
$781$	31.9521	1.14334
$782$	42.8888	1.53370
$783$	−16.0563	−0.573806
$784$	−6.46566	−0.230916
$785$	19.4424	0.693930
$786$	−30.6442	−1.09304
$787$	11.0934	0.395438	0.197719	−	0.980259i	$-0.436647\pi$
$787$	11.0934	0.395438	0.197719	+	0.980259i	$0.436647\pi$
$788$	9.32542	0.332204
$789$	11.0845	0.394618
$790$	−0.500415	−0.0178040
$791$	6.61574	0.235229
$792$	−1.42345	−0.0505800
$793$	−9.26121	−0.328875
$794$	−25.2156	−0.894869
$795$	18.2695	0.647951
$796$	25.7320	0.912045
$797$	5.37055	0.190235	0.0951174	−	0.995466i	$-0.469677\pi$
$797$	5.37055	0.190235	0.0951174	+	0.995466i	$0.469677\pi$
$798$	10.4606	0.370302
$799$	−24.7999	−0.877357
$800$	−1.00000	−0.0353553
$801$	3.08331	0.108943
$802$	−0.360931	−0.0127449
$803$	−4.55922	−0.160891
$804$	−14.1137	−0.497751
$805$	−4.79605	−0.169039
$806$	1.00000	0.0352235
$807$	10.5461	0.371242
$808$	−2.97421	−0.104632
$809$	5.88261	0.206822	0.103411	−	0.994639i	$-0.467024\pi$
$809$	5.88261	0.206822	0.103411	+	0.994639i	$0.467024\pi$
$810$	−10.4408	−0.366853
$811$	−43.0462	−1.51155	−0.755777	−	0.654829i	$-0.772741\pi$
$811$	−43.0462	−1.51155	−0.755777	+	0.654829i	$0.772741\pi$
$812$	−2.57779	−0.0904626
$813$	40.3033	1.41350
$814$	19.8568	0.695979
$815$	−10.5751	−0.370429
$816$	12.4036	0.434213
$817$	−84.6152	−2.96031
$818$	−16.2130	−0.566873
$819$	0.438929	0.0153374
$820$	−10.4911	−0.366365
$821$	26.1700	0.913341	0.456670	−	0.889636i	$-0.349042\pi$
$821$	26.1700	0.913341	0.456670	+	0.889636i	$0.349042\pi$
$822$	−13.2145	−0.460908
$823$	27.2675	0.950487	0.475243	−	0.879854i	$-0.342360\pi$
$823$	27.2675	0.950487	0.475243	+	0.879854i	$0.342360\pi$
$824$	6.52262	0.227226
$825$	−4.49815	−0.156606
$826$	0.0895473	0.00311575
$827$	4.33624	0.150786	0.0753929	−	0.997154i	$-0.475979\pi$
$827$	4.33624	0.150786	0.0753929	+	0.997154i	$0.475979\pi$
$828$	3.93969	0.136914
$829$	27.9667	0.971325	0.485662	−	0.874147i	$-0.338578\pi$
$829$	27.9667	0.971325	0.485662	+	0.874147i	$0.338578\pi$
$830$	−15.2331	−0.528747
$831$	−17.8399	−0.618861
$832$	1.00000	0.0346688
$833$	42.2650	1.46440
$834$	21.3092	0.737877
$835$	−3.35904	−0.116244
$836$	17.8783	0.618332
$837$	−4.55309	−0.157378
$838$	18.6449	0.644079
$839$	47.7589	1.64882	0.824409	−	0.565994i	$-0.191507\pi$
$839$	47.7589	1.64882	0.824409	+	0.565994i	$0.191507\pi$
$840$	−1.38703	−0.0478572
$841$	−16.5641	−0.571175
$842$	−3.29229	−0.113460
$843$	−30.4077	−1.04730
$844$	15.8187	0.544504
$845$	−1.00000	−0.0344010
$846$	−2.27807	−0.0783218
$847$	−3.93294	−0.135137
$848$	9.62823	0.330635
$849$	−39.1037	−1.34203
$850$	6.53685	0.224212
$851$	−54.9578	−1.88393
$852$	−25.5755	−0.876201
$853$	−8.01342	−0.274374	−0.137187	−	0.990545i	$-0.543806\pi$
$853$	−8.01342	−0.274374	−0.137187	+	0.990545i	$0.543806\pi$
$854$	6.76980	0.231658
$855$	−4.52852	−0.154872
$856$	−16.0520	−0.548646
$857$	−4.47914	−0.153005	−0.0765023	−	0.997069i	$-0.524375\pi$
$857$	−4.47914	−0.153005	−0.0765023	+	0.997069i	$0.524375\pi$
$858$	4.49815	0.153564
$859$	1.79072	0.0610986	0.0305493	−	0.999533i	$-0.490274\pi$
$859$	1.79072	0.0610986	0.0305493	+	0.999533i	$0.490274\pi$
$860$	11.2196	0.382586
$861$	−14.5515	−0.495914
$862$	0.316054	0.0107648
$863$	15.0291	0.511597	0.255799	−	0.966730i	$-0.417662\pi$
$863$	15.0291	0.511597	0.255799	+	0.966730i	$0.417662\pi$
$864$	−4.55309	−0.154899
$865$	−17.6888	−0.601437
$866$	7.57496	0.257408
$867$	−48.8230	−1.65812
$868$	−0.730984	−0.0248112
$869$	−1.18628	−0.0402416
$870$	6.69142	0.226860
$871$	7.43808	0.252030
$872$	−12.2158	−0.413680
$873$	−8.31675	−0.281479
$874$	−49.4818	−1.67375
$875$	−0.730984	−0.0247118
$876$	3.64934	0.123300
$877$	5.63667	0.190337	0.0951684	−	0.995461i	$-0.469661\pi$
$877$	5.63667	0.190337	0.0951684	+	0.995461i	$0.469661\pi$
$878$	−37.4454	−1.26372
$879$	−8.51746	−0.287287
$880$	−2.37058	−0.0799123
$881$	13.9448	0.469811	0.234906	−	0.972018i	$-0.424522\pi$
$881$	13.9448	0.469811	0.234906	+	0.972018i	$0.424522\pi$
$882$	3.88239	0.130727
$883$	−54.6621	−1.83952	−0.919762	−	0.392476i	$-0.871619\pi$
$883$	−54.6621	−1.83952	−0.919762	+	0.392476i	$0.871619\pi$
$884$	−6.53685	−0.219858
$885$	−0.232447	−0.00781362
$886$	10.2772	0.345268
$887$	−34.9893	−1.17483	−0.587413	−	0.809287i	$-0.699854\pi$
$887$	−34.9893	−1.17483	−0.587413	+	0.809287i	$0.699854\pi$
$888$	−15.8940	−0.533367
$889$	−14.8452	−0.497891
$890$	5.13489	0.172122
$891$	−24.7509	−0.829185
$892$	23.7847	0.796372
$893$	28.6122	0.957471
$894$	37.5051	1.25436
$895$	10.5497	0.352637
$896$	−0.730984	−0.0244205
$897$	−12.4496	−0.415680
$898$	−5.48302	−0.182971
$899$	3.52646	0.117614
$900$	0.600463	0.0200154
$901$	−62.9383	−2.09678
$902$	−24.8700	−0.828081
$903$	15.5620	0.517871
$904$	−9.05045	−0.301013
$905$	5.93296	0.197218
$906$	−37.8870	−1.25871
$907$	42.9172	1.42504	0.712522	−	0.701650i	$-0.247553\pi$
$907$	42.9172	1.42504	0.712522	+	0.701650i	$0.247553\pi$
$908$	8.15837	0.270745
$909$	1.78590	0.0592347
$910$	0.730984	0.0242319
$911$	−28.4113	−0.941307	−0.470654	−	0.882318i	$-0.655982\pi$
$911$	−28.4113	−0.941307	−0.470654	+	0.882318i	$0.655982\pi$
$912$	−14.3103	−0.473862
$913$	−36.1112	−1.19511
$914$	−24.3973	−0.806990
$915$	−17.5730	−0.580947
$916$	13.9300	0.460261
$917$	−11.8053	−0.389845
$918$	29.7629	0.982321
$919$	−13.5579	−0.447235	−0.223617	−	0.974677i	$-0.571787\pi$
$919$	−13.5579	−0.447235	−0.223617	+	0.974677i	$0.571787\pi$
$920$	6.56109	0.216313
$921$	45.2450	1.49087
$922$	−1.52224	−0.0501322
$923$	13.4786	0.443653
$924$	−3.28808	−0.108170
$925$	−8.37632	−0.275412
$926$	21.9797	0.722297
$927$	−3.91659	−0.128638
$928$	3.52646	0.115762
$929$	−9.70553	−0.318428	−0.159214	−	0.987244i	$-0.550896\pi$
$929$	−9.70553	−0.318428	−0.159214	+	0.987244i	$0.550896\pi$
$930$	1.89749	0.0622211
$931$	−48.7622	−1.59812
$932$	−4.82542	−0.158062
$933$	22.0087	0.720533
$934$	−30.1686	−0.987147
$935$	15.4961	0.506778
$936$	−0.600463	−0.0196268
$937$	−9.63317	−0.314702	−0.157351	−	0.987543i	$-0.550295\pi$
$937$	−9.63317	−0.314702	−0.157351	+	0.987543i	$0.550295\pi$
$938$	−5.43712	−0.177528
$939$	−31.4838	−1.02743
$940$	−3.79386	−0.123742
$941$	−26.2228	−0.854840	−0.427420	−	0.904053i	$-0.640577\pi$
$941$	−26.2228	−0.854840	−0.427420	+	0.904053i	$0.640577\pi$
$942$	−36.8918	−1.20200
$943$	68.8330	2.24151
$944$	−0.122502	−0.00398711
$945$	−3.32824	−0.108268
$946$	26.5971	0.864745
$947$	−33.2890	−1.08175	−0.540874	−	0.841104i	$-0.681906\pi$
$947$	−33.2890	−1.08175	−0.540874	+	0.841104i	$0.681906\pi$
$948$	0.949532	0.0308394
$949$	−1.92325	−0.0624313
$950$	−7.54171	−0.244685
$951$	25.7459	0.834868
$952$	4.77833	0.154867
$953$	8.34005	0.270161	0.135080	−	0.990835i	$-0.456871\pi$
$953$	8.34005	0.270161	0.135080	+	0.990835i	$0.456871\pi$
$954$	−5.78140	−0.187180
$955$	0.975874	0.0315785
$956$	24.0466	0.777721
$957$	15.8626	0.512764
$958$	7.76219	0.250785
$959$	−5.09071	−0.164388
$960$	1.89749	0.0612412
$961$	1.00000	0.0322581
$962$	8.37632	0.270063
$963$	9.63864	0.310601
$964$	23.7872	0.766134
$965$	−4.67059	−0.150352
$966$	9.10045	0.292802
$967$	22.4200	0.720979	0.360489	−	0.932763i	$-0.382610\pi$
$967$	22.4200	0.720979	0.360489	+	0.932763i	$0.382610\pi$
$968$	5.38034	0.172931
$969$	93.5443	3.00508
$970$	−13.8506	−0.444714
$971$	13.3573	0.428657	0.214329	−	0.976762i	$-0.431244\pi$
$971$	13.3573	0.428657	0.214329	+	0.976762i	$0.431244\pi$
$972$	6.15208	0.197328
$973$	8.20911	0.263172
$974$	16.9679	0.543688
$975$	−1.89749	−0.0607683
$976$	−9.26121	−0.296444
$977$	−13.4439	−0.430110	−0.215055	−	0.976602i	$-0.568993\pi$
$977$	−13.4439	−0.430110	−0.215055	+	0.976602i	$0.568993\pi$
$978$	20.0661	0.641644
$979$	12.1727	0.389041
$980$	6.46566	0.206538
$981$	7.33516	0.234194
$982$	−7.39757	−0.236066
$983$	19.5278	0.622840	0.311420	−	0.950272i	$-0.399195\pi$
$983$	19.5278	0.622840	0.311420	+	0.950272i	$0.399195\pi$
$984$	19.9067	0.634603
$985$	−9.32542	−0.297133
$986$	−23.0519	−0.734123
$987$	−5.26222	−0.167498
$988$	7.54171	0.239934
$989$	−73.6129	−2.34076
$990$	1.42345	0.0452401
$991$	−13.0584	−0.414814	−0.207407	−	0.978255i	$-0.566502\pi$
$991$	−13.0584	−0.414814	−0.207407	+	0.978255i	$0.566502\pi$
$992$	1.00000	0.0317500
$993$	−47.9990	−1.52320
$994$	−9.85264	−0.312507
$995$	−25.7320	−0.815758
$996$	28.9046	0.915876
$997$	45.3703	1.43689	0.718445	−	0.695583i	$-0.244854\pi$
$997$	45.3703	1.43689	0.718445	+	0.695583i	$0.244854\pi$
$998$	−18.6118	−0.589147
$999$	−38.1382	−1.20664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.k.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.k.1.3	✓	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.89749 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.89749 q^{6} +0.730984 q^{7} -1.00000 q^{8} +0.600463 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.89749 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.89749 q^{6} +0.730984 q^{7} -1.00000 q^{8} +0.600463 q^{9} +1.00000 q^{10} +2.37058 q^{11} -1.89749 q^{12} +1.00000 q^{13} -0.730984 q^{14} +1.89749 q^{15} +1.00000 q^{16} -6.53685 q^{17} -0.600463 q^{18} +7.54171 q^{19} -1.00000 q^{20} -1.38703 q^{21} -2.37058 q^{22} +6.56109 q^{23} +1.89749 q^{24} +1.00000 q^{25} -1.00000 q^{26} +4.55309 q^{27} +0.730984 q^{28} -3.52646 q^{29} -1.89749 q^{30} -1.00000 q^{31} -1.00000 q^{32} -4.49815 q^{33} +6.53685 q^{34} -0.730984 q^{35} +0.600463 q^{36} -8.37632 q^{37} -7.54171 q^{38} -1.89749 q^{39} +1.00000 q^{40} +10.4911 q^{41} +1.38703 q^{42} -11.2196 q^{43} +2.37058 q^{44} -0.600463 q^{45} -6.56109 q^{46} +3.79386 q^{47} -1.89749 q^{48} -6.46566 q^{49} -1.00000 q^{50} +12.4036 q^{51} +1.00000 q^{52} +9.62823 q^{53} -4.55309 q^{54} -2.37058 q^{55} -0.730984 q^{56} -14.3103 q^{57} +3.52646 q^{58} -0.122502 q^{59} +1.89749 q^{60} -9.26121 q^{61} +1.00000 q^{62} +0.438929 q^{63} +1.00000 q^{64} -1.00000 q^{65} +4.49815 q^{66} +7.43808 q^{67} -6.53685 q^{68} -12.4496 q^{69} +0.730984 q^{70} +13.4786 q^{71} -0.600463 q^{72} -1.92325 q^{73} +8.37632 q^{74} -1.89749 q^{75} +7.54171 q^{76} +1.73286 q^{77} +1.89749 q^{78} -0.500415 q^{79} -1.00000 q^{80} -10.4408 q^{81} -10.4911 q^{82} -15.2331 q^{83} -1.38703 q^{84} +6.53685 q^{85} +11.2196 q^{86} +6.69142 q^{87} -2.37058 q^{88} +5.13489 q^{89} +0.600463 q^{90} +0.730984 q^{91} +6.56109 q^{92} +1.89749 q^{93} -3.79386 q^{94} -7.54171 q^{95} +1.89749 q^{96} -13.8506 q^{97} +6.46566 q^{98} +1.42345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9} + 8 q^{10} + 12 q^{11} - q^{12} + 8 q^{13} + 8 q^{14} + q^{15} + 8 q^{16} - 13 q^{18} - 3 q^{19} - 8 q^{20} + 27 q^{21} - 12 q^{22} + 17 q^{23} + q^{24} + 8 q^{25} - 8 q^{26} - 13 q^{27} - 8 q^{28} - 10 q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 4 q^{33} + 8 q^{35} + 13 q^{36} + 2 q^{37} + 3 q^{38} - q^{39} + 8 q^{40} + 14 q^{41} - 27 q^{42} - 19 q^{43} + 12 q^{44} - 13 q^{45} - 17 q^{46} + 8 q^{47} - q^{48} + 16 q^{49} - 8 q^{50} + 35 q^{51} + 8 q^{52} + 4 q^{53} + 13 q^{54} - 12 q^{55} + 8 q^{56} - 33 q^{57} + 10 q^{58} - 13 q^{59} + q^{60} + 11 q^{61} + 8 q^{62} - 27 q^{63} + 8 q^{64} - 8 q^{65} - 4 q^{66} - 34 q^{67} + 12 q^{69} - 8 q^{70} + 8 q^{71} - 13 q^{72} - 21 q^{73} - 2 q^{74} - q^{75} - 3 q^{76} + 19 q^{77} + q^{78} - 36 q^{79} - 8 q^{80} + 20 q^{81} - 14 q^{82} + 27 q^{84} + 19 q^{86} - 29 q^{87} - 12 q^{88} - 16 q^{89} + 13 q^{90} - 8 q^{91} + 17 q^{92} + q^{93} - 8 q^{94} + 3 q^{95} + q^{96} + q^{97} - 16 q^{98} + 65 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9 + 8 * q^10 + 12 * q^11 - q^12 + 8 * q^13 + 8 * q^14 + q^15 + 8 * q^16 - 13 * q^18 - 3 * q^19 - 8 * q^20 + 27 * q^21 - 12 * q^22 + 17 * q^23 + q^24 + 8 * q^25 - 8 * q^26 - 13 * q^27 - 8 * q^28 - 10 * q^29 - q^30 - 8 * q^31 - 8 * q^32 + 4 * q^33 + 8 * q^35 + 13 * q^36 + 2 * q^37 + 3 * q^38 - q^39 + 8 * q^40 + 14 * q^41 - 27 * q^42 - 19 * q^43 + 12 * q^44 - 13 * q^45 - 17 * q^46 + 8 * q^47 - q^48 + 16 * q^49 - 8 * q^50 + 35 * q^51 + 8 * q^52 + 4 * q^53 + 13 * q^54 - 12 * q^55 + 8 * q^56 - 33 * q^57 + 10 * q^58 - 13 * q^59 + q^60 + 11 * q^61 + 8 * q^62 - 27 * q^63 + 8 * q^64 - 8 * q^65 - 4 * q^66 - 34 * q^67 + 12 * q^69 - 8 * q^70 + 8 * q^71 - 13 * q^72 - 21 * q^73 - 2 * q^74 - q^75 - 3 * q^76 + 19 * q^77 + q^78 - 36 * q^79 - 8 * q^80 + 20 * q^81 - 14 * q^82 + 27 * q^84 + 19 * q^86 - 29 * q^87 - 12 * q^88 - 16 * q^89 + 13 * q^90 - 8 * q^91 + 17 * q^92 + q^93 - 8 * q^94 + 3 * q^95 + q^96 + q^97 - 16 * q^98 + 65 * q^99

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