LMFDB - Embedded newform 6038.2.a.d.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$0$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.79695 q^{3} +1.00000 q^{4} -3.22250 q^{5} +2.79695 q^{6} +3.03239 q^{7} -1.00000 q^{8} +4.82293 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.79695 q^{3} +1.00000 q^{4} -3.22250 q^{5} +2.79695 q^{6} +3.03239 q^{7} -1.00000 q^{8} +4.82293 q^{9} +3.22250 q^{10} -1.74517 q^{11} -2.79695 q^{12} +2.14019 q^{13} -3.03239 q^{14} +9.01318 q^{15} +1.00000 q^{16} +0.190161 q^{17} -4.82293 q^{18} +2.73132 q^{19} -3.22250 q^{20} -8.48146 q^{21} +1.74517 q^{22} -6.25778 q^{23} +2.79695 q^{24} +5.38453 q^{25} -2.14019 q^{26} -5.09864 q^{27} +3.03239 q^{28} +7.43690 q^{29} -9.01318 q^{30} -3.12082 q^{31} -1.00000 q^{32} +4.88115 q^{33} -0.190161 q^{34} -9.77190 q^{35} +4.82293 q^{36} +5.85047 q^{37} -2.73132 q^{38} -5.98600 q^{39} +3.22250 q^{40} +7.17262 q^{41} +8.48146 q^{42} +10.0352 q^{43} -1.74517 q^{44} -15.5419 q^{45} +6.25778 q^{46} -0.0406837 q^{47} -2.79695 q^{48} +2.19542 q^{49} -5.38453 q^{50} -0.531870 q^{51} +2.14019 q^{52} +8.58100 q^{53} +5.09864 q^{54} +5.62381 q^{55} -3.03239 q^{56} -7.63936 q^{57} -7.43690 q^{58} +7.78387 q^{59} +9.01318 q^{60} +2.43396 q^{61} +3.12082 q^{62} +14.6250 q^{63} +1.00000 q^{64} -6.89677 q^{65} -4.88115 q^{66} -9.57563 q^{67} +0.190161 q^{68} +17.5027 q^{69} +9.77190 q^{70} +2.86144 q^{71} -4.82293 q^{72} +16.0283 q^{73} -5.85047 q^{74} -15.0602 q^{75} +2.73132 q^{76} -5.29204 q^{77} +5.98600 q^{78} -10.7863 q^{79} -3.22250 q^{80} -0.208142 q^{81} -7.17262 q^{82} -8.45607 q^{83} -8.48146 q^{84} -0.612794 q^{85} -10.0352 q^{86} -20.8006 q^{87} +1.74517 q^{88} -15.0458 q^{89} +15.5419 q^{90} +6.48990 q^{91} -6.25778 q^{92} +8.72878 q^{93} +0.0406837 q^{94} -8.80168 q^{95} +2.79695 q^{96} -12.0926 q^{97} -2.19542 q^{98} -8.41682 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) $ 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * 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
$3$	−2.79695	−1.61482	−0.807410	−	0.589991i	$-0.799131\pi$
$3$	−2.79695	−1.61482	−0.807410	+	0.589991i	$0.799131\pi$
$4$	1.00000	0.500000
$5$	−3.22250	−1.44115	−0.720574	−	0.693378i	$-0.756121\pi$
$5$	−3.22250	−1.44115	−0.720574	+	0.693378i	$0.756121\pi$
$6$	2.79695	1.14185
$7$	3.03239	1.14614	0.573069	−	0.819507i	$-0.305753\pi$
$7$	3.03239	1.14614	0.573069	+	0.819507i	$0.305753\pi$
$8$	−1.00000	−0.353553
$9$	4.82293	1.60764
$10$	3.22250	1.01904
$11$	−1.74517	−0.526188	−0.263094	−	0.964770i	$-0.584743\pi$
$11$	−1.74517	−0.526188	−0.263094	+	0.964770i	$0.584743\pi$
$12$	−2.79695	−0.807410
$13$	2.14019	0.593582	0.296791	−	0.954942i	$-0.404084\pi$
$13$	2.14019	0.593582	0.296791	+	0.954942i	$0.404084\pi$
$14$	−3.03239	−0.810442
$15$	9.01318	2.32719
$16$	1.00000	0.250000
$17$	0.190161	0.0461208	0.0230604	−	0.999734i	$-0.492659\pi$
$17$	0.190161	0.0461208	0.0230604	+	0.999734i	$0.492659\pi$
$18$	−4.82293	−1.13678
$19$	2.73132	0.626607	0.313304	−	0.949653i	$-0.398564\pi$
$19$	2.73132	0.626607	0.313304	+	0.949653i	$0.398564\pi$
$20$	−3.22250	−0.720574
$21$	−8.48146	−1.85081
$22$	1.74517	0.372071
$23$	−6.25778	−1.30484	−0.652419	−	0.757858i	$-0.726246\pi$
$23$	−6.25778	−1.30484	−0.652419	+	0.757858i	$0.726246\pi$
$24$	2.79695	0.570925
$25$	5.38453	1.07691
$26$	−2.14019	−0.419726
$27$	−5.09864	−0.981234
$28$	3.03239	0.573069
$29$	7.43690	1.38100	0.690499	−	0.723333i	$-0.257391\pi$
$29$	7.43690	1.38100	0.690499	+	0.723333i	$0.257391\pi$
$30$	−9.01318	−1.64557
$31$	−3.12082	−0.560516	−0.280258	−	0.959925i	$-0.590420\pi$
$31$	−3.12082	−0.560516	−0.280258	+	0.959925i	$0.590420\pi$
$32$	−1.00000	−0.176777
$33$	4.88115	0.849698
$34$	−0.190161	−0.0326123
$35$	−9.77190	−1.65175
$36$	4.82293	0.803822
$37$	5.85047	0.961811	0.480906	−	0.876772i	$-0.340308\pi$
$37$	5.85047	0.961811	0.480906	+	0.876772i	$0.340308\pi$
$38$	−2.73132	−0.443078
$39$	−5.98600	−0.958528
$40$	3.22250	0.509522
$41$	7.17262	1.12018	0.560088	−	0.828433i	$-0.310767\pi$
$41$	7.17262	1.12018	0.560088	+	0.828433i	$0.310767\pi$
$42$	8.48146	1.30872
$43$	10.0352	1.53035	0.765174	−	0.643824i	$-0.222653\pi$
$43$	10.0352	1.53035	0.765174	+	0.643824i	$0.222653\pi$
$44$	−1.74517	−0.263094
$45$	−15.5419	−2.31685
$46$	6.25778	0.922660
$47$	−0.0406837	−0.00593432	−0.00296716	−	0.999996i	$-0.500944\pi$
$47$	−0.0406837	−0.00593432	−0.00296716	+	0.999996i	$0.500944\pi$
$48$	−2.79695	−0.403705
$49$	2.19542	0.313631
$50$	−5.38453	−0.761487
$51$	−0.531870	−0.0744768
$52$	2.14019	0.296791
$53$	8.58100	1.17869	0.589345	−	0.807881i	$-0.299386\pi$
$53$	8.58100	1.17869	0.589345	+	0.807881i	$0.299386\pi$
$54$	5.09864	0.693837
$55$	5.62381	0.758314
$56$	−3.03239	−0.405221
$57$	−7.63936	−1.01186
$58$	−7.43690	−0.976513
$59$	7.78387	1.01337	0.506687	−	0.862130i	$-0.330870\pi$
$59$	7.78387	1.01337	0.506687	+	0.862130i	$0.330870\pi$
$60$	9.01318	1.16360
$61$	2.43396	0.311637	0.155819	−	0.987786i	$-0.450198\pi$
$61$	2.43396	0.311637	0.155819	+	0.987786i	$0.450198\pi$
$62$	3.12082	0.396345
$63$	14.6250	1.84258
$64$	1.00000	0.125000
$65$	−6.89677	−0.855439
$66$	−4.88115	−0.600828
$67$	−9.57563	−1.16985	−0.584925	−	0.811088i	$-0.698876\pi$
$67$	−9.57563	−1.16985	−0.584925	+	0.811088i	$0.698876\pi$
$68$	0.190161	0.0230604
$69$	17.5027	2.10708
$70$	9.77190	1.16797
$71$	2.86144	0.339591	0.169795	−	0.985479i	$-0.445689\pi$
$71$	2.86144	0.339591	0.169795	+	0.985479i	$0.445689\pi$
$72$	−4.82293	−0.568388
$73$	16.0283	1.87597	0.937983	−	0.346680i	$-0.112691\pi$
$73$	16.0283	1.87597	0.937983	+	0.346680i	$0.112691\pi$
$74$	−5.85047	−0.680103
$75$	−15.0602	−1.73901
$76$	2.73132	0.313304
$77$	−5.29204	−0.603084
$78$	5.98600	0.677782
$79$	−10.7863	−1.21355	−0.606776	−	0.794873i	$-0.707537\pi$
$79$	−10.7863	−1.21355	−0.606776	+	0.794873i	$0.707537\pi$
$80$	−3.22250	−0.360287
$81$	−0.208142	−0.0231269
$82$	−7.17262	−0.792084
$83$	−8.45607	−0.928175	−0.464087	−	0.885789i	$-0.653618\pi$
$83$	−8.45607	−0.928175	−0.464087	+	0.885789i	$0.653618\pi$
$84$	−8.48146	−0.925403
$85$	−0.612794	−0.0664668
$86$	−10.0352	−1.08212
$87$	−20.8006	−2.23006
$88$	1.74517	0.186035
$89$	−15.0458	−1.59485	−0.797427	−	0.603415i	$-0.793806\pi$
$89$	−15.0458	−1.59485	−0.797427	+	0.603415i	$0.793806\pi$
$90$	15.5419	1.63826
$91$	6.48990	0.680326
$92$	−6.25778	−0.652419
$93$	8.72878	0.905133
$94$	0.0406837	0.00419620
$95$	−8.80168	−0.903033
$96$	2.79695	0.285463
$97$	−12.0926	−1.22781	−0.613907	−	0.789378i	$-0.710403\pi$
$97$	−12.0926	−1.22781	−0.613907	+	0.789378i	$0.710403\pi$
$98$	−2.19542	−0.221771
$99$	−8.41682	−0.845922
$100$	5.38453	0.538453
$101$	−9.53811	−0.949078	−0.474539	−	0.880235i	$-0.657385\pi$
$101$	−9.53811	−0.949078	−0.474539	+	0.880235i	$0.657385\pi$
$102$	0.531870	0.0526630
$103$	5.18064	0.510463	0.255232	−	0.966880i	$-0.417848\pi$
$103$	5.18064	0.510463	0.255232	+	0.966880i	$0.417848\pi$
$104$	−2.14019	−0.209863
$105$	27.3315	2.66728
$106$	−8.58100	−0.833460
$107$	−1.77409	−0.171508	−0.0857538	−	0.996316i	$-0.527330\pi$
$107$	−1.77409	−0.171508	−0.0857538	+	0.996316i	$0.527330\pi$
$108$	−5.09864	−0.490617
$109$	−11.9318	−1.14286	−0.571431	−	0.820650i	$-0.693612\pi$
$109$	−11.9318	−1.14286	−0.571431	+	0.820650i	$0.693612\pi$
$110$	−5.62381	−0.536209
$111$	−16.3635	−1.55315
$112$	3.03239	0.286534
$113$	−4.55944	−0.428916	−0.214458	−	0.976733i	$-0.568799\pi$
$113$	−4.55944	−0.428916	−0.214458	+	0.976733i	$0.568799\pi$
$114$	7.63936	0.715492
$115$	20.1657	1.88046
$116$	7.43690	0.690499
$117$	10.3220	0.954268
$118$	−7.78387	−0.716563
$119$	0.576643	0.0528608
$120$	−9.01318	−0.822787
$121$	−7.95439	−0.723126
$122$	−2.43396	−0.220361
$123$	−20.0615	−1.80888
$124$	−3.12082	−0.280258
$125$	−1.23914	−0.110832
$126$	−14.6250	−1.30290
$127$	−4.46269	−0.396000	−0.198000	−	0.980202i	$-0.563445\pi$
$127$	−4.46269	−0.396000	−0.198000	+	0.980202i	$0.563445\pi$
$128$	−1.00000	−0.0883883
$129$	−28.0678	−2.47124
$130$	6.89677	0.604887
$131$	2.86878	0.250646	0.125323	−	0.992116i	$-0.460003\pi$
$131$	2.86878	0.250646	0.125323	+	0.992116i	$0.460003\pi$
$132$	4.88115	0.424849
$133$	8.28243	0.718178
$134$	9.57563	0.827208
$135$	16.4304	1.41410
$136$	−0.190161	−0.0163062
$137$	11.1458	0.952254	0.476127	−	0.879377i	$-0.342040\pi$
$137$	11.1458	0.952254	0.476127	+	0.879377i	$0.342040\pi$
$138$	−17.5027	−1.48993
$139$	0.345699	0.0293218	0.0146609	−	0.999893i	$-0.495333\pi$
$139$	0.345699	0.0293218	0.0146609	+	0.999893i	$0.495333\pi$
$140$	−9.77190	−0.825876
$141$	0.113790	0.00958286
$142$	−2.86144	−0.240127
$143$	−3.73499	−0.312336
$144$	4.82293	0.401911
$145$	−23.9654	−1.99022
$146$	−16.0283	−1.32651
$147$	−6.14048	−0.506458
$148$	5.85047	0.480906
$149$	−21.0841	−1.72728	−0.863640	−	0.504109i	$-0.831821\pi$
$149$	−21.0841	−1.72728	−0.863640	+	0.504109i	$0.831821\pi$
$150$	15.0602	1.22966
$151$	23.0801	1.87824	0.939118	−	0.343596i	$-0.111645\pi$
$151$	23.0801	1.87824	0.939118	+	0.343596i	$0.111645\pi$
$152$	−2.73132	−0.221539
$153$	0.917132	0.0741458
$154$	5.29204	0.426444
$155$	10.0569	0.807786
$156$	−5.98600	−0.479264
$157$	8.62529	0.688373	0.344187	−	0.938901i	$-0.388155\pi$
$157$	8.62529	0.688373	0.344187	+	0.938901i	$0.388155\pi$
$158$	10.7863	0.858110
$159$	−24.0006	−1.90337
$160$	3.22250	0.254761
$161$	−18.9761	−1.49552
$162$	0.208142	0.0163532
$163$	19.7683	1.54837	0.774186	−	0.632959i	$-0.218160\pi$
$163$	19.7683	1.54837	0.774186	+	0.632959i	$0.218160\pi$
$164$	7.17262	0.560088
$165$	−15.7295	−1.22454
$166$	8.45607	0.656319
$167$	−3.87698	−0.300010	−0.150005	−	0.988685i	$-0.547929\pi$
$167$	−3.87698	−0.300010	−0.150005	+	0.988685i	$0.547929\pi$
$168$	8.48146	0.654359
$169$	−8.41959	−0.647661
$170$	0.612794	0.0469991
$171$	13.1730	1.00736
$172$	10.0352	0.765174
$173$	−6.47775	−0.492494	−0.246247	−	0.969207i	$-0.579197\pi$
$173$	−6.47775	−0.492494	−0.246247	+	0.969207i	$0.579197\pi$
$174$	20.8006	1.57689
$175$	16.3280	1.23428
$176$	−1.74517	−0.131547
$177$	−21.7711	−1.63642
$178$	15.0458	1.12773
$179$	13.8771	1.03722	0.518612	−	0.855009i	$-0.326449\pi$
$179$	13.8771	1.03722	0.518612	+	0.855009i	$0.326449\pi$
$180$	−15.5419	−1.15843
$181$	21.0384	1.56377	0.781887	−	0.623420i	$-0.214257\pi$
$181$	21.0384	1.56377	0.781887	+	0.623420i	$0.214257\pi$
$182$	−6.48990	−0.481063
$183$	−6.80768	−0.503238
$184$	6.25778	0.461330
$185$	−18.8532	−1.38611
$186$	−8.72878	−0.640026
$187$	−0.331863	−0.0242682
$188$	−0.0406837	−0.00296716
$189$	−15.4611	−1.12463
$190$	8.80168	0.638541
$191$	−21.7058	−1.57058	−0.785288	−	0.619130i	$-0.787485\pi$
$191$	−21.7058	−1.57058	−0.785288	+	0.619130i	$0.787485\pi$
$192$	−2.79695	−0.201852
$193$	−2.75513	−0.198319	−0.0991593	−	0.995072i	$-0.531615\pi$
$193$	−2.75513	−0.198319	−0.0991593	+	0.995072i	$0.531615\pi$
$194$	12.0926	0.868195
$195$	19.2899	1.38138
$196$	2.19542	0.156816
$197$	−0.764855	−0.0544936	−0.0272468	−	0.999629i	$-0.508674\pi$
$197$	−0.764855	−0.0544936	−0.0272468	+	0.999629i	$0.508674\pi$
$198$	8.41682	0.598157
$199$	−4.78566	−0.339247	−0.169623	−	0.985509i	$-0.554255\pi$
$199$	−4.78566	−0.339247	−0.169623	+	0.985509i	$0.554255\pi$
$200$	−5.38453	−0.380743
$201$	26.7826	1.88910
$202$	9.53811	0.671099
$203$	22.5516	1.58281
$204$	−0.531870	−0.0372384
$205$	−23.1138	−1.61434
$206$	−5.18064	−0.360952
$207$	−30.1809	−2.09771
$208$	2.14019	0.148395
$209$	−4.76661	−0.329713
$210$	−27.3315	−1.88605
$211$	−1.83031	−0.126004	−0.0630020	−	0.998013i	$-0.520067\pi$
$211$	−1.83031	−0.126004	−0.0630020	+	0.998013i	$0.520067\pi$
$212$	8.58100	0.589345
$213$	−8.00332	−0.548378
$214$	1.77409	0.121274
$215$	−32.3383	−2.20546
$216$	5.09864	0.346919
$217$	−9.46357	−0.642429
$218$	11.9318	0.808126
$219$	−44.8303	−3.02935
$220$	5.62381	0.379157
$221$	0.406980	0.0273765
$222$	16.3635	1.09824
$223$	−14.8028	−0.991268	−0.495634	−	0.868532i	$-0.665064\pi$
$223$	−14.8028	−0.991268	−0.495634	+	0.868532i	$0.665064\pi$
$224$	−3.03239	−0.202610
$225$	25.9692	1.73128
$226$	4.55944	0.303290
$227$	19.9626	1.32496	0.662482	−	0.749077i	$-0.269503\pi$
$227$	19.9626	1.32496	0.662482	+	0.749077i	$0.269503\pi$
$228$	−7.63936	−0.505929
$229$	25.5576	1.68889	0.844446	−	0.535641i	$-0.179930\pi$
$229$	25.5576	1.68889	0.844446	+	0.535641i	$0.179930\pi$
$230$	−20.1657	−1.32969
$231$	14.8016	0.973871
$232$	−7.43690	−0.488257
$233$	−28.8657	−1.89105	−0.945526	−	0.325547i	$-0.894452\pi$
$233$	−28.8657	−1.89105	−0.945526	+	0.325547i	$0.894452\pi$
$234$	−10.3220	−0.674769
$235$	0.131103	0.00855223
$236$	7.78387	0.506687
$237$	30.1687	1.95967
$238$	−0.576643	−0.0373782
$239$	−20.3663	−1.31739	−0.658694	−	0.752411i	$-0.728891\pi$
$239$	−20.3663	−1.31739	−0.658694	+	0.752411i	$0.728891\pi$
$240$	9.01318	0.581798
$241$	15.4077	0.992495	0.496247	−	0.868181i	$-0.334711\pi$
$241$	15.4077	0.992495	0.496247	+	0.868181i	$0.334711\pi$
$242$	7.95439	0.511328
$243$	15.8781	1.01858
$244$	2.43396	0.155819
$245$	−7.07474	−0.451989
$246$	20.0615	1.27907
$247$	5.84554	0.371943
$248$	3.12082	0.198172
$249$	23.6512	1.49883
$250$	1.23914	0.0783698
$251$	−3.81649	−0.240895	−0.120447	−	0.992720i	$-0.538433\pi$
$251$	−3.81649	−0.240895	−0.120447	+	0.992720i	$0.538433\pi$
$252$	14.6250	0.921290
$253$	10.9209	0.686590
$254$	4.46269	0.280014
$255$	1.71395	0.107332
$256$	1.00000	0.0625000
$257$	8.23600	0.513748	0.256874	−	0.966445i	$-0.417307\pi$
$257$	8.23600	0.513748	0.256874	+	0.966445i	$0.417307\pi$
$258$	28.0678	1.74743
$259$	17.7409	1.10237
$260$	−6.89677	−0.427719
$261$	35.8677	2.22015
$262$	−2.86878	−0.177234
$263$	11.3349	0.698938	0.349469	−	0.936948i	$-0.386362\pi$
$263$	11.3349	0.698938	0.349469	+	0.936948i	$0.386362\pi$
$264$	−4.88115	−0.300414
$265$	−27.6523	−1.69867
$266$	−8.28243	−0.507829
$267$	42.0824	2.57540
$268$	−9.57563	−0.584925
$269$	26.4845	1.61479	0.807394	−	0.590012i	$-0.200877\pi$
$269$	26.4845	1.61479	0.807394	+	0.590012i	$0.200877\pi$
$270$	−16.4304	−0.999922
$271$	−11.3642	−0.690328	−0.345164	−	0.938542i	$-0.612177\pi$
$271$	−11.3642	−0.690328	−0.345164	+	0.938542i	$0.612177\pi$
$272$	0.190161	0.0115302
$273$	−18.1519	−1.09860
$274$	−11.1458	−0.673345
$275$	−9.39690	−0.566654
$276$	17.5027	1.05354
$277$	13.3312	0.800993	0.400496	−	0.916298i	$-0.368838\pi$
$277$	13.3312	0.800993	0.400496	+	0.916298i	$0.368838\pi$
$278$	−0.345699	−0.0207336
$279$	−15.0515	−0.901110
$280$	9.77190	0.583983
$281$	1.21676	0.0725860	0.0362930	−	0.999341i	$-0.488445\pi$
$281$	1.21676	0.0725860	0.0362930	+	0.999341i	$0.488445\pi$
$282$	−0.113790	−0.00677611
$283$	−32.4820	−1.93085	−0.965426	−	0.260676i	$-0.916055\pi$
$283$	−32.4820	−1.93085	−0.965426	+	0.260676i	$0.916055\pi$
$284$	2.86144	0.169795
$285$	24.6179	1.45824
$286$	3.73499	0.220855
$287$	21.7502	1.28387
$288$	−4.82293	−0.284194
$289$	−16.9638	−0.997873
$290$	23.9654	1.40730
$291$	33.8223	1.98270
$292$	16.0283	0.937983
$293$	5.40608	0.315827	0.157913	−	0.987453i	$-0.449523\pi$
$293$	5.40608	0.315827	0.157913	+	0.987453i	$0.449523\pi$
$294$	6.14048	0.358120
$295$	−25.0835	−1.46042
$296$	−5.85047	−0.340052
$297$	8.89798	0.516313
$298$	21.0841	1.22137
$299$	−13.3928	−0.774528
$300$	−15.0602	−0.869504
$301$	30.4306	1.75399
$302$	−23.0801	−1.32811
$303$	26.6776	1.53259
$304$	2.73132	0.156652
$305$	−7.84346	−0.449115
$306$	−0.917132	−0.0524290
$307$	13.0663	0.745732	0.372866	−	0.927885i	$-0.378375\pi$
$307$	13.0663	0.745732	0.372866	+	0.927885i	$0.378375\pi$
$308$	−5.29204	−0.301542
$309$	−14.4900	−0.824306
$310$	−10.0569	−0.571191
$311$	16.6846	0.946095	0.473048	−	0.881037i	$-0.343154\pi$
$311$	16.6846	0.946095	0.473048	+	0.881037i	$0.343154\pi$
$312$	5.98600	0.338891
$313$	14.4685	0.817808	0.408904	−	0.912577i	$-0.365911\pi$
$313$	14.4685	0.817808	0.408904	+	0.912577i	$0.365911\pi$
$314$	−8.62529	−0.486753
$315$	−47.1292	−2.65543
$316$	−10.7863	−0.606776
$317$	−1.13796	−0.0639143	−0.0319571	−	0.999489i	$-0.510174\pi$
$317$	−1.13796	−0.0639143	−0.0319571	+	0.999489i	$0.510174\pi$
$318$	24.0006	1.34589
$319$	−12.9786	−0.726664
$320$	−3.22250	−0.180143
$321$	4.96204	0.276954
$322$	18.9761	1.05750
$323$	0.519390	0.0288996
$324$	−0.208142	−0.0115634
$325$	11.5239	0.639231
$326$	−19.7683	−1.09486
$327$	33.3727	1.84552
$328$	−7.17262	−0.396042
$329$	−0.123369	−0.00680155
$330$	15.7295	0.865881
$331$	−9.24366	−0.508078	−0.254039	−	0.967194i	$-0.581759\pi$
$331$	−9.24366	−0.508078	−0.254039	+	0.967194i	$0.581759\pi$
$332$	−8.45607	−0.464087
$333$	28.2164	1.54625
$334$	3.87698	0.212139
$335$	30.8575	1.68592
$336$	−8.48146	−0.462701
$337$	34.7442	1.89264	0.946320	−	0.323231i	$-0.104769\pi$
$337$	34.7442	1.89264	0.946320	+	0.323231i	$0.104769\pi$
$338$	8.41959	0.457965
$339$	12.7525	0.692623
$340$	−0.612794	−0.0332334
$341$	5.44636	0.294937
$342$	−13.1730	−0.712312
$343$	−14.5694	−0.786673
$344$	−10.0352	−0.541060
$345$	−56.4025	−3.03661
$346$	6.47775	0.348246
$347$	−5.25304	−0.281998	−0.140999	−	0.990010i	$-0.545031\pi$
$347$	−5.25304	−0.281998	−0.140999	+	0.990010i	$0.545031\pi$
$348$	−20.8006	−1.11503
$349$	12.6015	0.674543	0.337272	−	0.941407i	$-0.390496\pi$
$349$	12.6015	0.674543	0.337272	+	0.941407i	$0.390496\pi$
$350$	−16.3280	−0.872769
$351$	−10.9121	−0.582443
$352$	1.74517	0.0930177
$353$	3.06593	0.163183	0.0815914	−	0.996666i	$-0.474000\pi$
$353$	3.06593	0.163183	0.0815914	+	0.996666i	$0.474000\pi$
$354$	21.7711	1.15712
$355$	−9.22101	−0.489401
$356$	−15.0458	−0.797427
$357$	−1.61284	−0.0853606
$358$	−13.8771	−0.733429
$359$	15.6304	0.824943	0.412472	−	0.910970i	$-0.364666\pi$
$359$	15.6304	0.824943	0.412472	+	0.910970i	$0.364666\pi$
$360$	15.5419	0.819130
$361$	−11.5399	−0.607363
$362$	−21.0384	−1.10576
$363$	22.2480	1.16772
$364$	6.48990	0.340163
$365$	−51.6511	−2.70354
$366$	6.80768	0.355843
$367$	7.52525	0.392815	0.196407	−	0.980522i	$-0.437072\pi$
$367$	7.52525	0.392815	0.196407	+	0.980522i	$0.437072\pi$
$368$	−6.25778	−0.326210
$369$	34.5930	1.80084
$370$	18.8532	0.980129
$371$	26.0210	1.35094
$372$	8.72878	0.452566
$373$	7.77597	0.402624	0.201312	−	0.979527i	$-0.435479\pi$
$373$	7.77597	0.402624	0.201312	+	0.979527i	$0.435479\pi$
$374$	0.331863	0.0171602
$375$	3.46580	0.178973
$376$	0.0406837	0.00209810
$377$	15.9164	0.819735
$378$	15.4611	0.795233
$379$	−12.2017	−0.626758	−0.313379	−	0.949628i	$-0.601461\pi$
$379$	−12.2017	−0.626758	−0.313379	+	0.949628i	$0.601461\pi$
$380$	−8.80168	−0.451517
$381$	12.4819	0.639468
$382$	21.7058	1.11057
$383$	−29.4551	−1.50509	−0.752543	−	0.658543i	$-0.771173\pi$
$383$	−29.4551	−1.50509	−0.752543	+	0.658543i	$0.771173\pi$
$384$	2.79695	0.142731
$385$	17.0536	0.869132
$386$	2.75513	0.140232
$387$	48.3989	2.46025
$388$	−12.0926	−0.613907
$389$	−9.92720	−0.503329	−0.251664	−	0.967815i	$-0.580978\pi$
$389$	−9.92720	−0.503329	−0.251664	+	0.967815i	$0.580978\pi$
$390$	−19.2899	−0.976783
$391$	−1.18999	−0.0601802
$392$	−2.19542	−0.110885
$393$	−8.02383	−0.404748
$394$	0.764855	0.0385328
$395$	34.7588	1.74891
$396$	−8.41682	−0.422961
$397$	−20.8355	−1.04570	−0.522852	−	0.852424i	$-0.675132\pi$
$397$	−20.8355	−1.04570	−0.522852	+	0.852424i	$0.675132\pi$
$398$	4.78566	0.239884
$399$	−23.1656	−1.15973
$400$	5.38453	0.269226
$401$	−27.4961	−1.37309	−0.686544	−	0.727088i	$-0.740873\pi$
$401$	−27.4961	−1.37309	−0.686544	+	0.727088i	$0.740873\pi$
$402$	−26.7826	−1.33579
$403$	−6.67915	−0.332712
$404$	−9.53811	−0.474539
$405$	0.670739	0.0333293
$406$	−22.5516	−1.11922
$407$	−10.2101	−0.506093
$408$	0.531870	0.0263315
$409$	−10.4490	−0.516670	−0.258335	−	0.966055i	$-0.583174\pi$
$409$	−10.4490	−0.516670	−0.258335	+	0.966055i	$0.583174\pi$
$410$	23.1138	1.14151
$411$	−31.1744	−1.53772
$412$	5.18064	0.255232
$413$	23.6038	1.16147
$414$	30.1809	1.48331
$415$	27.2497	1.33764
$416$	−2.14019	−0.104931
$417$	−0.966902	−0.0473494
$418$	4.76661	0.233142
$419$	8.51918	0.416189	0.208095	−	0.978109i	$-0.433274\pi$
$419$	8.51918	0.416189	0.208095	+	0.978109i	$0.433274\pi$
$420$	27.3315	1.33364
$421$	1.82370	0.0888815	0.0444407	−	0.999012i	$-0.485849\pi$
$421$	1.82370	0.0888815	0.0444407	+	0.999012i	$0.485849\pi$
$422$	1.83031	0.0890982
$423$	−0.196214	−0.00954027
$424$	−8.58100	−0.416730
$425$	1.02393	0.0496677
$426$	8.00332	0.387762
$427$	7.38074	0.357179
$428$	−1.77409	−0.0857538
$429$	10.4466	0.504366
$430$	32.3383	1.55949
$431$	12.3387	0.594333	0.297167	−	0.954826i	$-0.403958\pi$
$431$	12.3387	0.594333	0.297167	+	0.954826i	$0.403958\pi$
$432$	−5.09864	−0.245309
$433$	30.8715	1.48359	0.741795	−	0.670627i	$-0.233975\pi$
$433$	30.8715	1.48359	0.741795	+	0.670627i	$0.233975\pi$
$434$	9.46357	0.454266
$435$	67.0301	3.21385
$436$	−11.9318	−0.571431
$437$	−17.0920	−0.817621
$438$	44.8303	2.14207
$439$	23.6064	1.12667	0.563335	−	0.826228i	$-0.309518\pi$
$439$	23.6064	1.12667	0.563335	+	0.826228i	$0.309518\pi$
$440$	−5.62381	−0.268105
$441$	10.5883	0.504207
$442$	−0.406980	−0.0193581
$443$	−5.10605	−0.242596	−0.121298	−	0.992616i	$-0.538706\pi$
$443$	−5.10605	−0.242596	−0.121298	+	0.992616i	$0.538706\pi$
$444$	−16.3635	−0.776576
$445$	48.4852	2.29842
$446$	14.8028	0.700932
$447$	58.9713	2.78925
$448$	3.03239	0.143267
$449$	30.3891	1.43415	0.717075	−	0.696996i	$-0.245481\pi$
$449$	30.3891	1.43415	0.717075	+	0.696996i	$0.245481\pi$
$450$	−25.9692	−1.22420
$451$	−12.5174	−0.589423
$452$	−4.55944	−0.214458
$453$	−64.5540	−3.03301
$454$	−19.9626	−0.936892
$455$	−20.9137	−0.980451
$456$	7.63936	0.357746
$457$	18.6741	0.873537	0.436769	−	0.899574i	$-0.356123\pi$
$457$	18.6741	0.873537	0.436769	+	0.899574i	$0.356123\pi$
$458$	−25.5576	−1.19423
$459$	−0.969562	−0.0452553
$460$	20.1657	0.940232
$461$	−10.7485	−0.500609	−0.250304	−	0.968167i	$-0.580531\pi$
$461$	−10.7485	−0.500609	−0.250304	+	0.968167i	$0.580531\pi$
$462$	−14.8016	−0.688631
$463$	−0.134754	−0.00626253	−0.00313127	−	0.999995i	$-0.500997\pi$
$463$	−0.134754	−0.00626253	−0.00313127	+	0.999995i	$0.500997\pi$
$464$	7.43690	0.345250
$465$	−28.1285	−1.30443
$466$	28.8657	1.33718
$467$	40.8006	1.88803	0.944013	−	0.329908i	$-0.107018\pi$
$467$	40.8006	1.88803	0.944013	+	0.329908i	$0.107018\pi$
$468$	10.3220	0.477134
$469$	−29.0371	−1.34081
$470$	−0.131103	−0.00604734
$471$	−24.1245	−1.11160
$472$	−7.78387	−0.358282
$473$	−17.5130	−0.805250
$474$	−30.1687	−1.38569
$475$	14.7069	0.674797
$476$	0.576643	0.0264304
$477$	41.3855	1.89491
$478$	20.3663	0.931533
$479$	−19.0310	−0.869548	−0.434774	−	0.900540i	$-0.643172\pi$
$479$	−19.0310	−0.869548	−0.434774	+	0.900540i	$0.643172\pi$
$480$	−9.01318	−0.411393
$481$	12.5211	0.570914
$482$	−15.4077	−0.701800
$483$	53.0751	2.41500
$484$	−7.95439	−0.361563
$485$	38.9683	1.76946
$486$	−15.8781	−0.720245
$487$	31.9457	1.44760	0.723800	−	0.690010i	$-0.242394\pi$
$487$	31.9457	1.44760	0.723800	+	0.690010i	$0.242394\pi$
$488$	−2.43396	−0.110180
$489$	−55.2909	−2.50034
$490$	7.07474	0.319604
$491$	−7.12100	−0.321366	−0.160683	−	0.987006i	$-0.551370\pi$
$491$	−7.12100	−0.321366	−0.160683	+	0.987006i	$0.551370\pi$
$492$	−20.0615	−0.904441
$493$	1.41421	0.0636927
$494$	−5.84554	−0.263003
$495$	27.1232	1.21910
$496$	−3.12082	−0.140129
$497$	8.67703	0.389218
$498$	−23.6512	−1.05984
$499$	−34.7435	−1.55533	−0.777666	−	0.628677i	$-0.783597\pi$
$499$	−34.7435	−1.55533	−0.777666	+	0.628677i	$0.783597\pi$
$500$	−1.23914	−0.0554158
$501$	10.8437	0.484462
$502$	3.81649	0.170338
$503$	−3.59623	−0.160348	−0.0801741	−	0.996781i	$-0.525548\pi$
$503$	−3.59623	−0.160348	−0.0801741	+	0.996781i	$0.525548\pi$
$504$	−14.6250	−0.651450
$505$	30.7366	1.36776
$506$	−10.9209	−0.485492
$507$	23.5492	1.04586
$508$	−4.46269	−0.198000
$509$	16.1438	0.715560	0.357780	−	0.933806i	$-0.383534\pi$
$509$	16.1438	0.715560	0.357780	+	0.933806i	$0.383534\pi$
$510$	−1.71395	−0.0758952
$511$	48.6040	2.15012
$512$	−1.00000	−0.0441942
$513$	−13.9260	−0.614848
$514$	−8.23600	−0.363275
$515$	−16.6946	−0.735653
$516$	−28.0678	−1.23562
$517$	0.0709998	0.00312257
$518$	−17.7409	−0.779492
$519$	18.1179	0.795289
$520$	6.89677	0.302443
$521$	26.6083	1.16573	0.582866	−	0.812568i	$-0.301931\pi$
$521$	26.6083	1.16573	0.582866	+	0.812568i	$0.301931\pi$
$522$	−35.8677	−1.56988
$523$	−23.3074	−1.01916	−0.509582	−	0.860422i	$-0.670200\pi$
$523$	−23.3074	−1.01916	−0.509582	+	0.860422i	$0.670200\pi$
$524$	2.86878	0.125323
$525$	−45.6686	−1.99314
$526$	−11.3349	−0.494224
$527$	−0.593458	−0.0258514
$528$	4.88115	0.212425
$529$	16.1599	0.702603
$530$	27.6523	1.20114
$531$	37.5411	1.62914
$532$	8.28243	0.359089
$533$	15.3508	0.664916
$534$	−42.0824	−1.82108
$535$	5.71701	0.247168
$536$	9.57563	0.413604
$537$	−38.8136	−1.67493
$538$	−26.4845	−1.14183
$539$	−3.83137	−0.165029
$540$	16.4304	0.707051
$541$	30.9248	1.32956	0.664782	−	0.747038i	$-0.268525\pi$
$541$	30.9248	1.32956	0.664782	+	0.747038i	$0.268525\pi$
$542$	11.3642	0.488136
$543$	−58.8434	−2.52521
$544$	−0.190161	−0.00815308
$545$	38.4504	1.64703
$546$	18.1519	0.776831
$547$	−24.3192	−1.03982	−0.519908	−	0.854222i	$-0.674034\pi$
$547$	−24.3192	−1.03982	−0.519908	+	0.854222i	$0.674034\pi$
$548$	11.1458	0.476127
$549$	11.7388	0.501001
$550$	9.39690	0.400685
$551$	20.3125	0.865343
$552$	−17.5027	−0.744965
$553$	−32.7083	−1.39090
$554$	−13.3312	−0.566387
$555$	52.7313	2.23832
$556$	0.345699	0.0146609
$557$	33.7609	1.43049	0.715247	−	0.698871i	$-0.246314\pi$
$557$	33.7609	1.43049	0.715247	+	0.698871i	$0.246314\pi$
$558$	15.0515	0.637181
$559$	21.4772	0.908387
$560$	−9.77190	−0.412938
$561$	0.928203	0.0391888
$562$	−1.21676	−0.0513260
$563$	−25.4393	−1.07214	−0.536070	−	0.844174i	$-0.680092\pi$
$563$	−25.4393	−1.07214	−0.536070	+	0.844174i	$0.680092\pi$
$564$	0.113790	0.00479143
$565$	14.6928	0.618132
$566$	32.4820	1.36532
$567$	−0.631169	−0.0265066
$568$	−2.86144	−0.120064
$569$	1.56406	0.0655686	0.0327843	−	0.999462i	$-0.489563\pi$
$569$	1.56406	0.0655686	0.0327843	+	0.999462i	$0.489563\pi$
$570$	−24.6179	−1.03113
$571$	17.2805	0.723167	0.361583	−	0.932340i	$-0.382236\pi$
$571$	17.2805	0.723167	0.361583	+	0.932340i	$0.382236\pi$
$572$	−3.73499	−0.156168
$573$	60.7100	2.53620
$574$	−21.7502	−0.907837
$575$	−33.6952	−1.40519
$576$	4.82293	0.200955
$577$	−4.68685	−0.195116	−0.0975580	−	0.995230i	$-0.531103\pi$
$577$	−4.68685	−0.195116	−0.0975580	+	0.995230i	$0.531103\pi$
$578$	16.9638	0.705603
$579$	7.70596	0.320249
$580$	−23.9654	−0.995111
$581$	−25.6422	−1.06382
$582$	−33.8223	−1.40198
$583$	−14.9753	−0.620213
$584$	−16.0283	−0.663254
$585$	−33.2626	−1.37524
$586$	−5.40608	−0.223323
$587$	−42.5914	−1.75793	−0.878967	−	0.476883i	$-0.841767\pi$
$587$	−42.5914	−1.75793	−0.878967	+	0.476883i	$0.841767\pi$
$588$	−6.14048	−0.253229
$589$	−8.52396	−0.351224
$590$	25.0835	1.03267
$591$	2.13926	0.0879974
$592$	5.85047	0.240453
$593$	41.3362	1.69747	0.848736	−	0.528817i	$-0.177364\pi$
$593$	41.3362	1.69747	0.848736	+	0.528817i	$0.177364\pi$
$594$	−8.89798	−0.365089
$595$	−1.85823	−0.0761801
$596$	−21.0841	−0.863640
$597$	13.3853	0.547822
$598$	13.3928	0.547674
$599$	7.46599	0.305052	0.152526	−	0.988299i	$-0.451259\pi$
$599$	7.46599	0.305052	0.152526	+	0.988299i	$0.451259\pi$
$600$	15.0602	0.614832
$601$	17.8899	0.729745	0.364872	−	0.931058i	$-0.381113\pi$
$601$	17.8899	0.729745	0.364872	+	0.931058i	$0.381113\pi$
$602$	−30.4306	−1.24026
$603$	−46.1826	−1.88070
$604$	23.0801	0.939118
$605$	25.6330	1.04213
$606$	−26.6776	−1.08370
$607$	−16.6096	−0.674164	−0.337082	−	0.941475i	$-0.609440\pi$
$607$	−16.6096	−0.674164	−0.337082	+	0.941475i	$0.609440\pi$
$608$	−2.73132	−0.110770
$609$	−63.0758	−2.55596
$610$	7.84346	0.317572
$611$	−0.0870708	−0.00352251
$612$	0.917132	0.0370729
$613$	38.2903	1.54653	0.773266	−	0.634082i	$-0.218622\pi$
$613$	38.2903	1.54653	0.773266	+	0.634082i	$0.218622\pi$
$614$	−13.0663	−0.527312
$615$	64.6481	2.60686
$616$	5.29204	0.213222
$617$	26.8228	1.07985	0.539923	−	0.841714i	$-0.318453\pi$
$617$	26.8228	1.07985	0.539923	+	0.841714i	$0.318453\pi$
$618$	14.4900	0.582873
$619$	−19.7224	−0.792710	−0.396355	−	0.918097i	$-0.629725\pi$
$619$	−19.7224	−0.792710	−0.396355	+	0.918097i	$0.629725\pi$
$620$	10.0569	0.403893
$621$	31.9062	1.28035
$622$	−16.6846	−0.668990
$623$	−45.6249	−1.82792
$624$	−5.98600	−0.239632
$625$	−22.9295	−0.917180
$626$	−14.4685	−0.578278
$627$	13.3320	0.532427
$628$	8.62529	0.344187
$629$	1.11253	0.0443595
$630$	47.1292	1.87767
$631$	3.06465	0.122002	0.0610009	−	0.998138i	$-0.480571\pi$
$631$	3.06465	0.122002	0.0610009	+	0.998138i	$0.480571\pi$
$632$	10.7863	0.429055
$633$	5.11929	0.203474
$634$	1.13796	0.0451942
$635$	14.3810	0.570694
$636$	−24.0006	−0.951686
$637$	4.69861	0.186166
$638$	12.9786	0.513829
$639$	13.8005	0.545941
$640$	3.22250	0.127381
$641$	17.6330	0.696461	0.348230	−	0.937409i	$-0.386783\pi$
$641$	17.6330	0.696461	0.348230	+	0.937409i	$0.386783\pi$
$642$	−4.96204	−0.195836
$643$	27.9863	1.10367	0.551837	−	0.833952i	$-0.313927\pi$
$643$	27.9863	1.10367	0.551837	+	0.833952i	$0.313927\pi$
$644$	−18.9761	−0.747762
$645$	90.4487	3.56141
$646$	−0.519390	−0.0204351
$647$	17.9188	0.704462	0.352231	−	0.935913i	$-0.385423\pi$
$647$	17.9188	0.704462	0.352231	+	0.935913i	$0.385423\pi$
$648$	0.208142	0.00817659
$649$	−13.5842	−0.533225
$650$	−11.5239	−0.452005
$651$	26.4691	1.03741
$652$	19.7683	0.774186
$653$	31.0312	1.21435	0.607173	−	0.794570i	$-0.292303\pi$
$653$	31.0312	1.21435	0.607173	+	0.794570i	$0.292303\pi$
$654$	−33.3727	−1.30498
$655$	−9.24464	−0.361218
$656$	7.17262	0.280044
$657$	77.3032	3.01589
$658$	0.123369	0.00480942
$659$	−15.4371	−0.601343	−0.300671	−	0.953728i	$-0.597211\pi$
$659$	−15.4371	−0.601343	−0.300671	+	0.953728i	$0.597211\pi$
$660$	−15.7295	−0.612270
$661$	−12.7160	−0.494594	−0.247297	−	0.968940i	$-0.579542\pi$
$661$	−12.7160	−0.494594	−0.247297	+	0.968940i	$0.579542\pi$
$662$	9.24366	0.359265
$663$	−1.13830	−0.0442081
$664$	8.45607	0.328159
$665$	−26.6902	−1.03500
$666$	−28.2164	−1.09336
$667$	−46.5385	−1.80198
$668$	−3.87698	−0.150005
$669$	41.4026	1.60072
$670$	−30.8575	−1.19213
$671$	−4.24768	−0.163980
$672$	8.48146	0.327179
$673$	−1.29189	−0.0497987	−0.0248993	−	0.999690i	$-0.507927\pi$
$673$	−1.29189	−0.0497987	−0.0248993	+	0.999690i	$0.507927\pi$
$674$	−34.7442	−1.33830
$675$	−27.4538	−1.05670
$676$	−8.41959	−0.323830
$677$	15.1559	0.582489	0.291245	−	0.956649i	$-0.405931\pi$
$677$	15.1559	0.582489	0.291245	+	0.956649i	$0.405931\pi$
$678$	−12.7525	−0.489758
$679$	−36.6694	−1.40724
$680$	0.612794	0.0234996
$681$	−55.8344	−2.13958
$682$	−5.44636	−0.208552
$683$	−11.7651	−0.450181	−0.225090	−	0.974338i	$-0.572268\pi$
$683$	−11.7651	−0.450181	−0.225090	+	0.974338i	$0.572268\pi$
$684$	13.1730	0.503680
$685$	−35.9175	−1.37234
$686$	14.5694	0.556262
$687$	−71.4832	−2.72726
$688$	10.0352	0.382587
$689$	18.3650	0.699649
$690$	56.4025	2.14721
$691$	−8.51173	−0.323801	−0.161901	−	0.986807i	$-0.551762\pi$
$691$	−8.51173	−0.323801	−0.161901	+	0.986807i	$0.551762\pi$
$692$	−6.47775	−0.246247
$693$	−25.5231	−0.969543
$694$	5.25304	0.199403
$695$	−1.11402	−0.0422570
$696$	20.8006	0.788446
$697$	1.36395	0.0516634
$698$	−12.6015	−0.476974
$699$	80.7358	3.05371
$700$	16.3280	0.617141
$701$	2.09347	0.0790691	0.0395346	−	0.999218i	$-0.487412\pi$
$701$	2.09347	0.0790691	0.0395346	+	0.999218i	$0.487412\pi$
$702$	10.9121	0.411849
$703$	15.9795	0.602678
$704$	−1.74517	−0.0657735
$705$	−0.366689	−0.0138103
$706$	−3.06593	−0.115388
$707$	−28.9233	−1.08777
$708$	−21.7711	−0.818208
$709$	−8.08647	−0.303694	−0.151847	−	0.988404i	$-0.548522\pi$
$709$	−8.08647	−0.303694	−0.151847	+	0.988404i	$0.548522\pi$
$710$	9.22101	0.346058
$711$	−52.0215	−1.95096
$712$	15.0458	0.563866
$713$	19.5294	0.731383
$714$	1.61284	0.0603591
$715$	12.0360	0.450121
$716$	13.8771	0.518612
$717$	56.9636	2.12734
$718$	−15.6304	−0.583323
$719$	16.3847	0.611047	0.305523	−	0.952185i	$-0.401169\pi$
$719$	16.3847	0.611047	0.305523	+	0.952185i	$0.401169\pi$
$720$	−15.5419	−0.579213
$721$	15.7097	0.585061
$722$	11.5399	0.429471
$723$	−43.0945	−1.60270
$724$	21.0384	0.781887
$725$	40.0442	1.48720
$726$	−22.2480	−0.825702
$727$	24.7958	0.919627	0.459813	−	0.888016i	$-0.347916\pi$
$727$	24.7958	0.919627	0.459813	+	0.888016i	$0.347916\pi$
$728$	−6.48990	−0.240532
$729$	−43.7858	−1.62170
$730$	51.6511	1.91169
$731$	1.90829	0.0705808
$732$	−6.80768	−0.251619
$733$	−19.9232	−0.735881	−0.367941	−	0.929849i	$-0.619937\pi$
$733$	−19.9232	−0.735881	−0.367941	+	0.929849i	$0.619937\pi$
$734$	−7.52525	−0.277762
$735$	19.7877	0.729880
$736$	6.25778	0.230665
$737$	16.7111	0.615560
$738$	−34.5930	−1.27339
$739$	−7.46898	−0.274751	−0.137375	−	0.990519i	$-0.543867\pi$
$739$	−7.46898	−0.274751	−0.137375	+	0.990519i	$0.543867\pi$
$740$	−18.8532	−0.693056
$741$	−16.3497	−0.600621
$742$	−26.0210	−0.955260
$743$	17.8822	0.656033	0.328016	−	0.944672i	$-0.393620\pi$
$743$	17.8822	0.656033	0.328016	+	0.944672i	$0.393620\pi$
$744$	−8.72878	−0.320013
$745$	67.9437	2.48926
$746$	−7.77597	−0.284698
$747$	−40.7830	−1.49217
$748$	−0.331863	−0.0121341
$749$	−5.37974	−0.196571
$750$	−3.46580	−0.126553
$751$	−36.4346	−1.32952	−0.664758	−	0.747058i	$-0.731465\pi$
$751$	−36.4346	−1.32952	−0.664758	+	0.747058i	$0.731465\pi$
$752$	−0.0406837	−0.00148358
$753$	10.6745	0.389002
$754$	−15.9164	−0.579640
$755$	−74.3758	−2.70681
$756$	−15.4611	−0.562315
$757$	19.6455	0.714027	0.357013	−	0.934099i	$-0.383795\pi$
$757$	19.6455	0.714027	0.357013	+	0.934099i	$0.383795\pi$
$758$	12.2017	0.443185
$759$	−30.5452	−1.10872
$760$	8.80168	0.319271
$761$	12.8759	0.466750	0.233375	−	0.972387i	$-0.425023\pi$
$761$	12.8759	0.466750	0.233375	+	0.972387i	$0.425023\pi$
$762$	−12.4819	−0.452172
$763$	−36.1820	−1.30988
$764$	−21.7058	−0.785288
$765$	−2.95546	−0.106855
$766$	29.4551	1.06426
$767$	16.6590	0.601520
$768$	−2.79695	−0.100926
$769$	−31.1578	−1.12358	−0.561790	−	0.827280i	$-0.689887\pi$
$769$	−31.1578	−1.12358	−0.561790	+	0.827280i	$0.689887\pi$
$770$	−17.0536	−0.614569
$771$	−23.0357	−0.829610
$772$	−2.75513	−0.0991593
$773$	−9.79963	−0.352468	−0.176234	−	0.984348i	$-0.556392\pi$
$773$	−9.79963	−0.352468	−0.176234	+	0.984348i	$0.556392\pi$
$774$	−48.3989	−1.73966
$775$	−16.8041	−0.603623
$776$	12.0926	0.434098
$777$	−49.6205	−1.78013
$778$	9.92720	0.355907
$779$	19.5907	0.701910
$780$	19.2899	0.690690
$781$	−4.99370	−0.178689
$782$	1.18999	0.0425538
$783$	−37.9181	−1.35508
$784$	2.19542	0.0784078
$785$	−27.7950	−0.992047
$786$	8.02383	0.286200
$787$	41.1550	1.46702	0.733508	−	0.679681i	$-0.237882\pi$
$787$	41.1550	1.46702	0.733508	+	0.679681i	$0.237882\pi$
$788$	−0.764855	−0.0272468
$789$	−31.7031	−1.12866
$790$	−34.7588	−1.23666
$791$	−13.8260	−0.491597
$792$	8.41682	0.299079
$793$	5.20915	0.184982
$794$	20.8355	0.739424
$795$	77.3421	2.74304
$796$	−4.78566	−0.169623
$797$	−14.0289	−0.496930	−0.248465	−	0.968641i	$-0.579926\pi$
$797$	−14.0289	−0.496930	−0.248465	+	0.968641i	$0.579926\pi$
$798$	23.1656	0.820052
$799$	−0.00773644	−0.000273696 0
$800$	−5.38453	−0.190372
$801$	−72.5650	−2.56396
$802$	27.4961	0.970920
$803$	−27.9720	−0.987111
$804$	26.7826	0.944548
$805$	61.1505	2.15527
$806$	6.67915	0.235263
$807$	−74.0758	−2.60759
$808$	9.53811	0.335550
$809$	9.61411	0.338014	0.169007	−	0.985615i	$-0.445944\pi$
$809$	9.61411	0.338014	0.169007	+	0.985615i	$0.445944\pi$
$810$	−0.670739	−0.0235673
$811$	44.5202	1.56332	0.781658	−	0.623707i	$-0.214374\pi$
$811$	44.5202	1.56332	0.781658	+	0.623707i	$0.214374\pi$
$812$	22.5516	0.791407
$813$	31.7852	1.11476
$814$	10.2101	0.357862
$815$	−63.7034	−2.23143
$816$	−0.531870	−0.0186192
$817$	27.4092	0.958927
$818$	10.4490	0.365341
$819$	31.3003	1.09372
$820$	−23.1138	−0.807169
$821$	27.0311	0.943393	0.471697	−	0.881761i	$-0.343642\pi$
$821$	27.0311	0.943393	0.471697	+	0.881761i	$0.343642\pi$
$822$	31.1744	1.08733
$823$	−0.540074	−0.0188258	−0.00941290	−	0.999956i	$-0.502996\pi$
$823$	−0.540074	−0.0188258	−0.00941290	+	0.999956i	$0.502996\pi$
$824$	−5.18064	−0.180476
$825$	26.2827	0.915045
$826$	−23.6038	−0.821280
$827$	−14.3895	−0.500371	−0.250185	−	0.968198i	$-0.580492\pi$
$827$	−14.3895	−0.500371	−0.250185	+	0.968198i	$0.580492\pi$
$828$	−30.1809	−1.04886
$829$	−32.6802	−1.13503	−0.567515	−	0.823363i	$-0.692095\pi$
$829$	−32.6802	−1.13503	−0.567515	+	0.823363i	$0.692095\pi$
$830$	−27.2497	−0.945852
$831$	−37.2866	−1.29346
$832$	2.14019	0.0741977
$833$	0.417483	0.0144649
$834$	0.966902	0.0334811
$835$	12.4936	0.432358
$836$	−4.76661	−0.164857
$837$	15.9120	0.549998
$838$	−8.51918	−0.294290
$839$	49.7199	1.71652	0.858262	−	0.513213i	$-0.171545\pi$
$839$	49.7199	1.71652	0.858262	+	0.513213i	$0.171545\pi$
$840$	−27.3315	−0.943027
$841$	26.3075	0.907155
$842$	−1.82370	−0.0628487
$843$	−3.40322	−0.117213
$844$	−1.83031	−0.0630020
$845$	27.1321	0.933374
$846$	0.196214	0.00674599
$847$	−24.1209	−0.828802
$848$	8.58100	0.294673
$849$	90.8505	3.11798
$850$	−1.02393	−0.0351204
$851$	−36.6110	−1.25501
$852$	−8.00332	−0.274189
$853$	−34.7092	−1.18842	−0.594209	−	0.804310i	$-0.702535\pi$
$853$	−34.7092	−1.18842	−0.594209	+	0.804310i	$0.702535\pi$
$854$	−7.38074	−0.252564
$855$	−42.4499	−1.45176
$856$	1.77409	0.0606371
$857$	−38.4877	−1.31471	−0.657357	−	0.753579i	$-0.728326\pi$
$857$	−38.4877	−1.31471	−0.657357	+	0.753579i	$0.728326\pi$
$858$	−10.4466	−0.356640
$859$	−1.48503	−0.0506687	−0.0253344	−	0.999679i	$-0.508065\pi$
$859$	−1.48503	−0.0506687	−0.0253344	+	0.999679i	$0.508065\pi$
$860$	−32.3383	−1.10273
$861$	−60.8343	−2.07323
$862$	−12.3387	−0.420257
$863$	24.1472	0.821981	0.410990	−	0.911640i	$-0.365183\pi$
$863$	24.1472	0.821981	0.410990	+	0.911640i	$0.365183\pi$
$864$	5.09864	0.173459
$865$	20.8746	0.709756
$866$	−30.8715	−1.04906
$867$	47.4470	1.61138
$868$	−9.46357	−0.321214
$869$	18.8239	0.638556
$870$	−67.0301	−2.27253
$871$	−20.4937	−0.694401
$872$	11.9318	0.404063
$873$	−58.3216	−1.97389
$874$	17.0920	0.578146
$875$	−3.75755	−0.127028
$876$	−44.8303	−1.51467
$877$	12.4362	0.419939	0.209970	−	0.977708i	$-0.432663\pi$
$877$	12.4362	0.419939	0.209970	+	0.977708i	$0.432663\pi$
$878$	−23.6064	−0.796677
$879$	−15.1205	−0.510003
$880$	5.62381	0.189579
$881$	−41.9167	−1.41221	−0.706105	−	0.708108i	$-0.749549\pi$
$881$	−41.9167	−1.41221	−0.706105	+	0.708108i	$0.749549\pi$
$882$	−10.5883	−0.356528
$883$	31.4498	1.05837	0.529186	−	0.848506i	$-0.322498\pi$
$883$	31.4498	1.05837	0.529186	+	0.848506i	$0.322498\pi$
$884$	0.406980	0.0136882
$885$	70.1574	2.35832
$886$	5.10605	0.171541
$887$	−25.8278	−0.867212	−0.433606	−	0.901103i	$-0.642759\pi$
$887$	−25.8278	−0.867212	−0.433606	+	0.901103i	$0.642759\pi$
$888$	16.3635	0.549122
$889$	−13.5326	−0.453870
$890$	−48.4852	−1.62523
$891$	0.363243	0.0121691
$892$	−14.8028	−0.495634
$893$	−0.111120	−0.00371849
$894$	−58.9713	−1.97229
$895$	−44.7191	−1.49479
$896$	−3.03239	−0.101305
$897$	37.4591	1.25072
$898$	−30.3891	−1.01410
$899$	−23.2093	−0.774072
$900$	25.9692	0.865640
$901$	1.63177	0.0543621
$902$	12.5174	0.416785
$903$	−85.1128	−2.83238
$904$	4.55944	0.151645
$905$	−67.7964	−2.25363
$906$	64.5540	2.14466
$907$	−17.5642	−0.583210	−0.291605	−	0.956539i	$-0.594189\pi$
$907$	−17.5642	−0.583210	−0.291605	+	0.956539i	$0.594189\pi$
$908$	19.9626	0.662482
$909$	−46.0016	−1.52578
$910$	20.9137	0.693283
$911$	−24.0483	−0.796756	−0.398378	−	0.917221i	$-0.630427\pi$
$911$	−24.0483	−0.796756	−0.398378	+	0.917221i	$0.630427\pi$
$912$	−7.63936	−0.252964
$913$	14.7573	0.488394
$914$	−18.6741	−0.617684
$915$	21.9378	0.725240
$916$	25.5576	0.844446
$917$	8.69926	0.287275
$918$	0.969562	0.0320003
$919$	−24.9884	−0.824290	−0.412145	−	0.911118i	$-0.635220\pi$
$919$	−24.9884	−0.824290	−0.412145	+	0.911118i	$0.635220\pi$
$920$	−20.1657	−0.664844
$921$	−36.5458	−1.20422
$922$	10.7485	0.353984
$923$	6.12403	0.201575
$924$	14.8016	0.486936
$925$	31.5020	1.03578
$926$	0.134754	0.00442828
$927$	24.9858	0.820643
$928$	−7.43690	−0.244128
$929$	42.6450	1.39914	0.699568	−	0.714566i	$-0.253376\pi$
$929$	42.6450	1.39914	0.699568	+	0.714566i	$0.253376\pi$
$930$	28.1285	0.922371
$931$	5.99639	0.196524
$932$	−28.8657	−0.945526
$933$	−46.6659	−1.52777
$934$	−40.8006	−1.33504
$935$	1.06943	0.0349740
$936$	−10.3220	−0.337385
$937$	28.4158	0.928305	0.464153	−	0.885755i	$-0.346359\pi$
$937$	28.4158	0.928305	0.464153	+	0.885755i	$0.346359\pi$
$938$	29.0371	0.948095
$939$	−40.4677	−1.32061
$940$	0.131103	0.00427612
$941$	−24.3562	−0.793989	−0.396994	−	0.917821i	$-0.629947\pi$
$941$	−24.3562	−0.793989	−0.396994	+	0.917821i	$0.629947\pi$
$942$	24.1245	0.786019
$943$	−44.8847	−1.46165
$944$	7.78387	0.253343
$945$	49.8234	1.62076
$946$	17.5130	0.569398
$947$	39.1118	1.27096	0.635481	−	0.772117i	$-0.280802\pi$
$947$	39.1118	1.27096	0.635481	+	0.772117i	$0.280802\pi$
$948$	30.1687	0.979833
$949$	34.3035	1.11354
$950$	−14.7069	−0.477153
$951$	3.18282	0.103210
$952$	−0.576643	−0.0186891
$953$	42.9414	1.39101	0.695504	−	0.718523i	$-0.255181\pi$
$953$	42.9414	1.39101	0.695504	+	0.718523i	$0.255181\pi$
$954$	−41.3855	−1.33991
$955$	69.9470	2.26343
$956$	−20.3663	−0.658694
$957$	36.3006	1.17343
$958$	19.0310	0.614863
$959$	33.7986	1.09141
$960$	9.01318	0.290899
$961$	−21.2605	−0.685822
$962$	−12.5211	−0.403697
$963$	−8.55630	−0.275723
$964$	15.4077	0.496247
$965$	8.87842	0.285806
$966$	−53.0751	−1.70766
$967$	−18.7846	−0.604073	−0.302037	−	0.953296i	$-0.597667\pi$
$967$	−18.7846	−0.604073	−0.302037	+	0.953296i	$0.597667\pi$
$968$	7.95439	0.255664
$969$	−1.45271	−0.0466677
$970$	−38.9683	−1.25120
$971$	11.6044	0.372404	0.186202	−	0.982511i	$-0.440382\pi$
$971$	11.6044	0.372404	0.186202	+	0.982511i	$0.440382\pi$
$972$	15.8781	0.509290
$973$	1.04829	0.0336068
$974$	−31.9457	−1.02361
$975$	−32.2318	−1.03224
$976$	2.43396	0.0779093
$977$	−22.3336	−0.714515	−0.357257	−	0.934006i	$-0.616288\pi$
$977$	−22.3336	−0.714515	−0.357257	+	0.934006i	$0.616288\pi$
$978$	55.2909	1.76801
$979$	26.2575	0.839193
$980$	−7.07474	−0.225994
$981$	−57.5464	−1.83731
$982$	7.12100	0.227240
$983$	33.6364	1.07283	0.536417	−	0.843953i	$-0.319778\pi$
$983$	33.6364	1.07283	0.536417	+	0.843953i	$0.319778\pi$
$984$	20.0615	0.639536
$985$	2.46475	0.0785334
$986$	−1.41421	−0.0450375
$987$	0.345057	0.0109833
$988$	5.84554	0.185971
$989$	−62.7979	−1.99686
$990$	−27.1232	−0.862033
$991$	−19.3583	−0.614936	−0.307468	−	0.951558i	$-0.599482\pi$
$991$	−19.3583	−0.614936	−0.307468	+	0.951558i	$0.599482\pi$
$992$	3.12082	0.0990862
$993$	25.8541	0.820454
$994$	−8.67703	−0.275219
$995$	15.4218	0.488904
$996$	23.6512	0.749417
$997$	48.6129	1.53959	0.769793	−	0.638293i	$-0.220359\pi$
$997$	48.6129	1.53959	0.769793	+	0.638293i	$0.220359\pi$
$998$	34.7435	1.09979
$999$	−29.8294	−0.943762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.7	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.d.1.7	✓	69	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.79695 q^{3} +1.00000 q^{4} -3.22250 q^{5} +2.79695 q^{6} +3.03239 q^{7} -1.00000 q^{8} +4.82293 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.79695 q^{3} +1.00000 q^{4} -3.22250 q^{5} +2.79695 q^{6} +3.03239 q^{7} -1.00000 q^{8} +4.82293 q^{9} +3.22250 q^{10} -1.74517 q^{11} -2.79695 q^{12} +2.14019 q^{13} -3.03239 q^{14} +9.01318 q^{15} +1.00000 q^{16} +0.190161 q^{17} -4.82293 q^{18} +2.73132 q^{19} -3.22250 q^{20} -8.48146 q^{21} +1.74517 q^{22} -6.25778 q^{23} +2.79695 q^{24} +5.38453 q^{25} -2.14019 q^{26} -5.09864 q^{27} +3.03239 q^{28} +7.43690 q^{29} -9.01318 q^{30} -3.12082 q^{31} -1.00000 q^{32} +4.88115 q^{33} -0.190161 q^{34} -9.77190 q^{35} +4.82293 q^{36} +5.85047 q^{37} -2.73132 q^{38} -5.98600 q^{39} +3.22250 q^{40} +7.17262 q^{41} +8.48146 q^{42} +10.0352 q^{43} -1.74517 q^{44} -15.5419 q^{45} +6.25778 q^{46} -0.0406837 q^{47} -2.79695 q^{48} +2.19542 q^{49} -5.38453 q^{50} -0.531870 q^{51} +2.14019 q^{52} +8.58100 q^{53} +5.09864 q^{54} +5.62381 q^{55} -3.03239 q^{56} -7.63936 q^{57} -7.43690 q^{58} +7.78387 q^{59} +9.01318 q^{60} +2.43396 q^{61} +3.12082 q^{62} +14.6250 q^{63} +1.00000 q^{64} -6.89677 q^{65} -4.88115 q^{66} -9.57563 q^{67} +0.190161 q^{68} +17.5027 q^{69} +9.77190 q^{70} +2.86144 q^{71} -4.82293 q^{72} +16.0283 q^{73} -5.85047 q^{74} -15.0602 q^{75} +2.73132 q^{76} -5.29204 q^{77} +5.98600 q^{78} -10.7863 q^{79} -3.22250 q^{80} -0.208142 q^{81} -7.17262 q^{82} -8.45607 q^{83} -8.48146 q^{84} -0.612794 q^{85} -10.0352 q^{86} -20.8006 q^{87} +1.74517 q^{88} -15.0458 q^{89} +15.5419 q^{90} +6.48990 q^{91} -6.25778 q^{92} +8.72878 q^{93} +0.0406837 q^{94} -8.80168 q^{95} +2.79695 q^{96} -12.0926 q^{97} -2.19542 q^{98} -8.41682 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * q^99

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