LMFDB - Embedded newform 8038.2.a.a.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.85626 q^{3} +1.00000 q^{4} +0.772797 q^{5} -2.85626 q^{6} -4.64843 q^{7} +1.00000 q^{8} +5.15824 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.85626 q^{3} +1.00000 q^{4} +0.772797 q^{5} -2.85626 q^{6} -4.64843 q^{7} +1.00000 q^{8} +5.15824 q^{9} +0.772797 q^{10} -0.325948 q^{11} -2.85626 q^{12} -2.15313 q^{13} -4.64843 q^{14} -2.20731 q^{15} +1.00000 q^{16} +1.42043 q^{17} +5.15824 q^{18} +0.0380090 q^{19} +0.772797 q^{20} +13.2772 q^{21} -0.325948 q^{22} +0.908911 q^{23} -2.85626 q^{24} -4.40278 q^{25} -2.15313 q^{26} -6.16451 q^{27} -4.64843 q^{28} +1.71174 q^{29} -2.20731 q^{30} +5.16581 q^{31} +1.00000 q^{32} +0.930993 q^{33} +1.42043 q^{34} -3.59230 q^{35} +5.15824 q^{36} +3.23019 q^{37} +0.0380090 q^{38} +6.14991 q^{39} +0.772797 q^{40} +4.81159 q^{41} +13.2772 q^{42} +9.53926 q^{43} -0.325948 q^{44} +3.98627 q^{45} +0.908911 q^{46} -12.1263 q^{47} -2.85626 q^{48} +14.6079 q^{49} -4.40278 q^{50} -4.05712 q^{51} -2.15313 q^{52} +8.34916 q^{53} -6.16451 q^{54} -0.251891 q^{55} -4.64843 q^{56} -0.108564 q^{57} +1.71174 q^{58} -2.65101 q^{59} -2.20731 q^{60} -8.89428 q^{61} +5.16581 q^{62} -23.9777 q^{63} +1.00000 q^{64} -1.66393 q^{65} +0.930993 q^{66} -1.63119 q^{67} +1.42043 q^{68} -2.59609 q^{69} -3.59230 q^{70} -2.78220 q^{71} +5.15824 q^{72} -2.74375 q^{73} +3.23019 q^{74} +12.5755 q^{75} +0.0380090 q^{76} +1.51515 q^{77} +6.14991 q^{78} +8.41118 q^{79} +0.772797 q^{80} +2.13274 q^{81} +4.81159 q^{82} -11.8073 q^{83} +13.2772 q^{84} +1.09770 q^{85} +9.53926 q^{86} -4.88917 q^{87} -0.325948 q^{88} +15.4163 q^{89} +3.98627 q^{90} +10.0087 q^{91} +0.908911 q^{92} -14.7549 q^{93} -12.1263 q^{94} +0.0293732 q^{95} -2.85626 q^{96} +9.85488 q^{97} +14.6079 q^{98} -1.68132 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) $ 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * 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
$3$	−2.85626	−1.64906	−0.824532	−	0.565815i	$-0.808562\pi$
$3$	−2.85626	−1.64906	−0.824532	+	0.565815i	$0.808562\pi$
$4$	1.00000	0.500000
$5$	0.772797	0.345605	0.172803	−	0.984956i	$-0.444718\pi$
$5$	0.772797	0.345605	0.172803	+	0.984956i	$0.444718\pi$
$6$	−2.85626	−1.16606
$7$	−4.64843	−1.75694	−0.878471	−	0.477795i	$-0.841436\pi$
$7$	−4.64843	−1.75694	−0.878471	+	0.477795i	$0.841436\pi$
$8$	1.00000	0.353553
$9$	5.15824	1.71941
$10$	0.772797	0.244380
$11$	−0.325948	−0.0982770	−0.0491385	−	0.998792i	$-0.515648\pi$
$11$	−0.325948	−0.0982770	−0.0491385	+	0.998792i	$0.515648\pi$
$12$	−2.85626	−0.824532
$13$	−2.15313	−0.597171	−0.298585	−	0.954383i	$-0.596515\pi$
$13$	−2.15313	−0.597171	−0.298585	+	0.954383i	$0.596515\pi$
$14$	−4.64843	−1.24235
$15$	−2.20731	−0.569925
$16$	1.00000	0.250000
$17$	1.42043	0.344505	0.172252	−	0.985053i	$-0.444896\pi$
$17$	1.42043	0.344505	0.172252	+	0.985053i	$0.444896\pi$
$18$	5.15824	1.21581
$19$	0.0380090	0.00871986	0.00435993	−	0.999990i	$-0.498612\pi$
$19$	0.0380090	0.00871986	0.00435993	+	0.999990i	$0.498612\pi$
$20$	0.772797	0.172803
$21$	13.2772	2.89731
$22$	−0.325948	−0.0694923
$23$	0.908911	0.189521	0.0947605	−	0.995500i	$-0.469791\pi$
$23$	0.908911	0.189521	0.0947605	+	0.995500i	$0.469791\pi$
$24$	−2.85626	−0.583032
$25$	−4.40278	−0.880557
$26$	−2.15313	−0.422264
$27$	−6.16451	−1.18636
$28$	−4.64843	−0.878471
$29$	1.71174	0.317861	0.158931	−	0.987290i	$-0.449195\pi$
$29$	1.71174	0.317861	0.158931	+	0.987290i	$0.449195\pi$
$30$	−2.20731	−0.402998
$31$	5.16581	0.927807	0.463904	−	0.885886i	$-0.346448\pi$
$31$	5.16581	0.927807	0.463904	+	0.885886i	$0.346448\pi$
$32$	1.00000	0.176777
$33$	0.930993	0.162065
$34$	1.42043	0.243602
$35$	−3.59230	−0.607209
$36$	5.15824	0.859707
$37$	3.23019	0.531039	0.265520	−	0.964105i	$-0.414456\pi$
$37$	3.23019	0.531039	0.265520	+	0.964105i	$0.414456\pi$
$38$	0.0380090	0.00616587
$39$	6.14991	0.984773
$40$	0.772797	0.122190
$41$	4.81159	0.751444	0.375722	−	0.926732i	$-0.377395\pi$
$41$	4.81159	0.751444	0.375722	+	0.926732i	$0.377395\pi$
$42$	13.2772	2.04871
$43$	9.53926	1.45472	0.727361	−	0.686255i	$-0.240746\pi$
$43$	9.53926	1.45472	0.727361	+	0.686255i	$0.240746\pi$
$44$	−0.325948	−0.0491385
$45$	3.98627	0.594239
$46$	0.908911	0.134012
$47$	−12.1263	−1.76881	−0.884403	−	0.466723i	$-0.845434\pi$
$47$	−12.1263	−1.76881	−0.884403	+	0.466723i	$0.845434\pi$
$48$	−2.85626	−0.412266
$49$	14.6079	2.08685
$50$	−4.40278	−0.622648
$51$	−4.05712	−0.568111
$52$	−2.15313	−0.298585
$53$	8.34916	1.14685	0.573423	−	0.819260i	$-0.305615\pi$
$53$	8.34916	1.14685	0.573423	+	0.819260i	$0.305615\pi$
$54$	−6.16451	−0.838883
$55$	−0.251891	−0.0339650
$56$	−4.64843	−0.621173
$57$	−0.108564	−0.0143796
$58$	1.71174	0.224762
$59$	−2.65101	−0.345132	−0.172566	−	0.984998i	$-0.555206\pi$
$59$	−2.65101	−0.345132	−0.172566	+	0.984998i	$0.555206\pi$
$60$	−2.20731	−0.284963
$61$	−8.89428	−1.13880	−0.569398	−	0.822062i	$-0.692824\pi$
$61$	−8.89428	−1.13880	−0.569398	+	0.822062i	$0.692824\pi$
$62$	5.16581	0.656059
$63$	−23.9777	−3.02091
$64$	1.00000	0.125000
$65$	−1.66393	−0.206385
$66$	0.930993	0.114597
$67$	−1.63119	−0.199281	−0.0996406	−	0.995023i	$-0.531769\pi$
$67$	−1.63119	−0.199281	−0.0996406	+	0.995023i	$0.531769\pi$
$68$	1.42043	0.172252
$69$	−2.59609	−0.312532
$70$	−3.59230	−0.429361
$71$	−2.78220	−0.330187	−0.165093	−	0.986278i	$-0.552793\pi$
$71$	−2.78220	−0.330187	−0.165093	+	0.986278i	$0.552793\pi$
$72$	5.15824	0.607905
$73$	−2.74375	−0.321131	−0.160566	−	0.987025i	$-0.551332\pi$
$73$	−2.74375	−0.321131	−0.160566	+	0.987025i	$0.551332\pi$
$74$	3.23019	0.375501
$75$	12.5755	1.45210
$76$	0.0380090	0.00435993
$77$	1.51515	0.172667
$78$	6.14991	0.696340
$79$	8.41118	0.946332	0.473166	−	0.880973i	$-0.343111\pi$
$79$	8.41118	0.946332	0.473166	+	0.880973i	$0.343111\pi$
$80$	0.772797	0.0864013
$81$	2.13274	0.236971
$82$	4.81159	0.531351
$83$	−11.8073	−1.29602	−0.648010	−	0.761631i	$-0.724399\pi$
$83$	−11.8073	−1.29602	−0.648010	+	0.761631i	$0.724399\pi$
$84$	13.2772	1.44866
$85$	1.09770	0.119063
$86$	9.53926	1.02864
$87$	−4.88917	−0.524174
$88$	−0.325948	−0.0347462
$89$	15.4163	1.63413	0.817064	−	0.576547i	$-0.195600\pi$
$89$	15.4163	1.63413	0.817064	+	0.576547i	$0.195600\pi$
$90$	3.98627	0.420190
$91$	10.0087	1.04920
$92$	0.908911	0.0947605
$93$	−14.7549	−1.53001
$94$	−12.1263	−1.25074
$95$	0.0293732	0.00301363
$96$	−2.85626	−0.291516
$97$	9.85488	1.00061	0.500306	−	0.865849i	$-0.333221\pi$
$97$	9.85488	1.00061	0.500306	+	0.865849i	$0.333221\pi$
$98$	14.6079	1.47562
$99$	−1.68132	−0.168979
$100$	−4.40278	−0.440278
$101$	13.7298	1.36617	0.683084	−	0.730339i	$-0.260638\pi$
$101$	13.7298	1.36617	0.683084	+	0.730339i	$0.260638\pi$
$102$	−4.05712	−0.401715
$103$	−13.7670	−1.35650	−0.678252	−	0.734830i	$-0.737262\pi$
$103$	−13.7670	−1.35650	−0.678252	+	0.734830i	$0.737262\pi$
$104$	−2.15313	−0.211132
$105$	10.2605	1.00133
$106$	8.34916	0.810942
$107$	−0.771043	−0.0745395	−0.0372698	−	0.999305i	$-0.511866\pi$
$107$	−0.771043	−0.0745395	−0.0372698	+	0.999305i	$0.511866\pi$
$108$	−6.16451	−0.593180
$109$	−4.04142	−0.387098	−0.193549	−	0.981091i	$-0.562000\pi$
$109$	−4.04142	−0.387098	−0.193549	+	0.981091i	$0.562000\pi$
$110$	−0.251891	−0.0240169
$111$	−9.22626	−0.875718
$112$	−4.64843	−0.439236
$113$	−2.48631	−0.233893	−0.116946	−	0.993138i	$-0.537311\pi$
$113$	−2.48631	−0.233893	−0.116946	+	0.993138i	$0.537311\pi$
$114$	−0.108564	−0.0101679
$115$	0.702404	0.0654995
$116$	1.71174	0.158931
$117$	−11.1064	−1.02678
$118$	−2.65101	−0.244045
$119$	−6.60278	−0.605276
$120$	−2.20731	−0.201499
$121$	−10.8938	−0.990342
$122$	−8.89428	−0.805250
$123$	−13.7432	−1.23918
$124$	5.16581	0.463904
$125$	−7.26644	−0.649930
$126$	−23.9777	−2.13611
$127$	4.26977	0.378880	0.189440	−	0.981892i	$-0.439333\pi$
$127$	4.26977	0.378880	0.189440	+	0.981892i	$0.439333\pi$
$128$	1.00000	0.0883883
$129$	−27.2466	−2.39893
$130$	−1.66393	−0.145937
$131$	−1.40132	−0.122434	−0.0612171	−	0.998124i	$-0.519498\pi$
$131$	−1.40132	−0.122434	−0.0612171	+	0.998124i	$0.519498\pi$
$132$	0.930993	0.0810325
$133$	−0.176682	−0.0153203
$134$	−1.63119	−0.140913
$135$	−4.76391	−0.410012
$136$	1.42043	0.121801
$137$	3.58185	0.306018	0.153009	−	0.988225i	$-0.451104\pi$
$137$	3.58185	0.306018	0.153009	+	0.988225i	$0.451104\pi$
$138$	−2.59609	−0.220994
$139$	4.99709	0.423847	0.211924	−	0.977286i	$-0.432027\pi$
$139$	4.99709	0.423847	0.211924	+	0.977286i	$0.432027\pi$
$140$	−3.59230	−0.303604
$141$	34.6360	2.91688
$142$	−2.78220	−0.233477
$143$	0.701808	0.0586881
$144$	5.15824	0.429854
$145$	1.32282	0.109855
$146$	−2.74375	−0.227074
$147$	−41.7241	−3.44135
$148$	3.23019	0.265520
$149$	−3.40948	−0.279316	−0.139658	−	0.990200i	$-0.544600\pi$
$149$	−3.40948	−0.279316	−0.139658	+	0.990200i	$0.544600\pi$
$150$	12.5755	1.02679
$151$	6.65757	0.541785	0.270892	−	0.962610i	$-0.412681\pi$
$151$	6.65757	0.541785	0.270892	+	0.962610i	$0.412681\pi$
$152$	0.0380090	0.00308294
$153$	7.32692	0.592347
$154$	1.51515	0.122094
$155$	3.99212	0.320655
$156$	6.14991	0.492387
$157$	−13.2512	−1.05756	−0.528779	−	0.848759i	$-0.677350\pi$
$157$	−13.2512	−1.05756	−0.528779	+	0.848759i	$0.677350\pi$
$158$	8.41118	0.669158
$159$	−23.8474	−1.89122
$160$	0.772797	0.0610950
$161$	−4.22501	−0.332978
$162$	2.13274	0.167564
$163$	−20.1827	−1.58083	−0.790414	−	0.612573i	$-0.790134\pi$
$163$	−20.1827	−1.58083	−0.790414	+	0.612573i	$0.790134\pi$
$164$	4.81159	0.375722
$165$	0.719468	0.0560105
$166$	−11.8073	−0.916425
$167$	−13.0844	−1.01250	−0.506249	−	0.862387i	$-0.668968\pi$
$167$	−13.0844	−1.01250	−0.506249	+	0.862387i	$0.668968\pi$
$168$	13.2772	1.02435
$169$	−8.36403	−0.643387
$170$	1.09770	0.0841901
$171$	0.196060	0.0149930
$172$	9.53926	0.727361
$173$	−0.248483	−0.0188918	−0.00944590	−	0.999955i	$-0.503007\pi$
$173$	−0.248483	−0.0188918	−0.00944590	+	0.999955i	$0.503007\pi$
$174$	−4.88917	−0.370647
$175$	20.4661	1.54709
$176$	−0.325948	−0.0245692
$177$	7.57197	0.569144
$178$	15.4163	1.15550
$179$	−4.17865	−0.312327	−0.156164	−	0.987731i	$-0.549913\pi$
$179$	−4.17865	−0.312327	−0.156164	+	0.987731i	$0.549913\pi$
$180$	3.98627	0.297119
$181$	22.4539	1.66898	0.834492	−	0.551020i	$-0.185761\pi$
$181$	22.4539	1.66898	0.834492	+	0.551020i	$0.185761\pi$
$182$	10.0087	0.741893
$183$	25.4044	1.87795
$184$	0.908911	0.0670058
$185$	2.49628	0.183530
$186$	−14.7549	−1.08188
$187$	−0.462986	−0.0338569
$188$	−12.1263	−0.884403
$189$	28.6553	2.08437
$190$	0.0293732	0.00213096
$191$	7.55821	0.546893	0.273446	−	0.961887i	$-0.411836\pi$
$191$	7.55821	0.546893	0.273446	+	0.961887i	$0.411836\pi$
$192$	−2.85626	−0.206133
$193$	9.11114	0.655834	0.327917	−	0.944706i	$-0.393653\pi$
$193$	9.11114	0.655834	0.327917	+	0.944706i	$0.393653\pi$
$194$	9.85488	0.707539
$195$	4.75263	0.340343
$196$	14.6079	1.04342
$197$	3.81717	0.271962	0.135981	−	0.990711i	$-0.456581\pi$
$197$	3.81717	0.271962	0.135981	+	0.990711i	$0.456581\pi$
$198$	−1.68132	−0.119486
$199$	−10.4497	−0.740760	−0.370380	−	0.928880i	$-0.620772\pi$
$199$	−10.4497	−0.740760	−0.370380	+	0.928880i	$0.620772\pi$
$200$	−4.40278	−0.311324
$201$	4.65910	0.328628
$202$	13.7298	0.966027
$203$	−7.95689	−0.558464
$204$	−4.05712	−0.284055
$205$	3.71838	0.259703
$206$	−13.7670	−0.959193
$207$	4.68838	0.325865
$208$	−2.15313	−0.149293
$209$	−0.0123889	−0.000856961 0
$210$	10.2605	0.708045
$211$	−24.1823	−1.66478	−0.832389	−	0.554192i	$-0.813027\pi$
$211$	−24.1823	−1.66478	−0.832389	+	0.554192i	$0.813027\pi$
$212$	8.34916	0.573423
$213$	7.94671	0.544499
$214$	−0.771043	−0.0527074
$215$	7.37191	0.502760
$216$	−6.16451	−0.419442
$217$	−24.0129	−1.63010
$218$	−4.04142	−0.273720
$219$	7.83687	0.529566
$220$	−0.251891	−0.0169825
$221$	−3.05837	−0.205728
$222$	−9.22626	−0.619226
$223$	9.47379	0.634412	0.317206	−	0.948357i	$-0.397255\pi$
$223$	9.47379	0.634412	0.317206	+	0.948357i	$0.397255\pi$
$224$	−4.64843	−0.310587
$225$	−22.7106	−1.51404
$226$	−2.48631	−0.165387
$227$	−19.6010	−1.30096	−0.650481	−	0.759523i	$-0.725432\pi$
$227$	−19.6010	−1.30096	−0.650481	+	0.759523i	$0.725432\pi$
$228$	−0.108564	−0.00718981
$229$	−0.799017	−0.0528005	−0.0264003	−	0.999651i	$-0.508404\pi$
$229$	−0.799017	−0.0528005	−0.0264003	+	0.999651i	$0.508404\pi$
$230$	0.702404	0.0463151
$231$	−4.32766	−0.284739
$232$	1.71174	0.112381
$233$	2.37314	0.155470	0.0777349	−	0.996974i	$-0.475231\pi$
$233$	2.37314	0.155470	0.0777349	+	0.996974i	$0.475231\pi$
$234$	−11.1064	−0.726046
$235$	−9.37119	−0.611309
$236$	−2.65101	−0.172566
$237$	−24.0246	−1.56056
$238$	−6.60278	−0.427994
$239$	−11.5667	−0.748191	−0.374095	−	0.927390i	$-0.622047\pi$
$239$	−11.5667	−0.748191	−0.374095	+	0.927390i	$0.622047\pi$
$240$	−2.20731	−0.142481
$241$	2.57079	0.165599	0.0827997	−	0.996566i	$-0.473614\pi$
$241$	2.57079	0.165599	0.0827997	+	0.996566i	$0.473614\pi$
$242$	−10.8938	−0.700277
$243$	12.4019	0.795580
$244$	−8.89428	−0.569398
$245$	11.2890	0.721226
$246$	−13.7432	−0.876232
$247$	−0.0818383	−0.00520725
$248$	5.16581	0.328029
$249$	33.7248	2.13722
$250$	−7.26644	−0.459570
$251$	1.02856	0.0649224	0.0324612	−	0.999473i	$-0.489665\pi$
$251$	1.02856	0.0649224	0.0324612	+	0.999473i	$0.489665\pi$
$252$	−23.9777	−1.51046
$253$	−0.296258	−0.0186256
$254$	4.26977	0.267909
$255$	−3.13533	−0.196342
$256$	1.00000	0.0625000
$257$	5.86654	0.365945	0.182972	−	0.983118i	$-0.441428\pi$
$257$	5.86654	0.365945	0.182972	+	0.983118i	$0.441428\pi$
$258$	−27.2466	−1.69630
$259$	−15.0153	−0.933006
$260$	−1.66393	−0.103193
$261$	8.82955	0.546535
$262$	−1.40132	−0.0865740
$263$	7.33848	0.452510	0.226255	−	0.974068i	$-0.427352\pi$
$263$	7.33848	0.452510	0.226255	+	0.974068i	$0.427352\pi$
$264$	0.930993	0.0572986
$265$	6.45220	0.396356
$266$	−0.176682	−0.0108331
$267$	−44.0331	−2.69478
$268$	−1.63119	−0.0996406
$269$	−1.56158	−0.0952113	−0.0476056	−	0.998866i	$-0.515159\pi$
$269$	−1.56158	−0.0952113	−0.0476056	+	0.998866i	$0.515159\pi$
$270$	−4.76391	−0.289923
$271$	12.6243	0.766875	0.383437	−	0.923567i	$-0.374740\pi$
$271$	12.6243	0.766875	0.383437	+	0.923567i	$0.374740\pi$
$272$	1.42043	0.0861262
$273$	−28.5874	−1.73019
$274$	3.58185	0.216388
$275$	1.43508	0.0865385
$276$	−2.59609	−0.156266
$277$	−14.7948	−0.888931	−0.444465	−	0.895796i	$-0.646606\pi$
$277$	−14.7948	−0.888931	−0.444465	+	0.895796i	$0.646606\pi$
$278$	4.99709	0.299705
$279$	26.6465	1.59528
$280$	−3.59230	−0.214681
$281$	−1.96400	−0.117162	−0.0585812	−	0.998283i	$-0.518658\pi$
$281$	−1.96400	−0.117162	−0.0585812	+	0.998283i	$0.518658\pi$
$282$	34.6360	2.06254
$283$	−23.1223	−1.37448	−0.687240	−	0.726431i	$-0.741178\pi$
$283$	−23.1223	−1.37448	−0.687240	+	0.726431i	$0.741178\pi$
$284$	−2.78220	−0.165093
$285$	−0.0838977	−0.00496967
$286$	0.701808	0.0414988
$287$	−22.3663	−1.32024
$288$	5.15824	0.303952
$289$	−14.9824	−0.881316
$290$	1.32282	0.0776789
$291$	−28.1481	−1.65007
$292$	−2.74375	−0.160566
$293$	−29.2409	−1.70827	−0.854137	−	0.520049i	$-0.825914\pi$
$293$	−29.2409	−1.70827	−0.854137	+	0.520049i	$0.825914\pi$
$294$	−41.7241	−2.43340
$295$	−2.04869	−0.119279
$296$	3.23019	0.187751
$297$	2.00931	0.116592
$298$	−3.40948	−0.197506
$299$	−1.95700	−0.113176
$300$	12.5755	0.726048
$301$	−44.3426	−2.55586
$302$	6.65757	0.383100
$303$	−39.2160	−2.25290
$304$	0.0380090	0.00217996
$305$	−6.87347	−0.393574
$306$	7.32692	0.418852
$307$	−12.8603	−0.733974	−0.366987	−	0.930226i	$-0.619611\pi$
$307$	−12.8603	−0.733974	−0.366987	+	0.930226i	$0.619611\pi$
$308$	1.51515	0.0863335
$309$	39.3222	2.23696
$310$	3.99212	0.226737
$311$	−0.890715	−0.0505078	−0.0252539	−	0.999681i	$-0.508039\pi$
$311$	−0.890715	−0.0505078	−0.0252539	+	0.999681i	$0.508039\pi$
$312$	6.14991	0.348170
$313$	−22.1991	−1.25477	−0.627384	−	0.778710i	$-0.715874\pi$
$313$	−22.1991	−1.25477	−0.627384	+	0.778710i	$0.715874\pi$
$314$	−13.2512	−0.747807
$315$	−18.5299	−1.04404
$316$	8.41118	0.473166
$317$	9.56953	0.537478	0.268739	−	0.963213i	$-0.413393\pi$
$317$	9.56953	0.537478	0.268739	+	0.963213i	$0.413393\pi$
$318$	−23.8474	−1.33730
$319$	−0.557936	−0.0312384
$320$	0.772797	0.0432007
$321$	2.20230	0.122921
$322$	−4.22501	−0.235451
$323$	0.0539891	0.00300403
$324$	2.13274	0.118485
$325$	9.47977	0.525843
$326$	−20.1827	−1.11781
$327$	11.5434	0.638350
$328$	4.81159	0.265675
$329$	56.3684	3.10769
$330$	0.719468	0.0396054
$331$	1.69426	0.0931250	0.0465625	−	0.998915i	$-0.485173\pi$
$331$	1.69426	0.0931250	0.0465625	+	0.998915i	$0.485173\pi$
$332$	−11.8073	−0.648010
$333$	16.6621	0.913076
$334$	−13.0844	−0.715945
$335$	−1.26058	−0.0688727
$336$	13.2772	0.724328
$337$	−2.35496	−0.128283	−0.0641414	−	0.997941i	$-0.520431\pi$
$337$	−2.35496	−0.128283	−0.0641414	+	0.997941i	$0.520431\pi$
$338$	−8.36403	−0.454943
$339$	7.10157	0.385704
$340$	1.09770	0.0595314
$341$	−1.68378	−0.0911821
$342$	0.196060	0.0106017
$343$	−35.3650	−1.90953
$344$	9.53926	0.514322
$345$	−2.00625	−0.108013
$346$	−0.248483	−0.0133585
$347$	27.1868	1.45946	0.729731	−	0.683734i	$-0.239645\pi$
$347$	27.1868	1.45946	0.729731	+	0.683734i	$0.239645\pi$
$348$	−4.88917	−0.262087
$349$	−22.9581	−1.22892	−0.614459	−	0.788949i	$-0.710626\pi$
$349$	−22.9581	−1.22892	−0.614459	+	0.788949i	$0.710626\pi$
$350$	20.4661	1.09396
$351$	13.2730	0.708460
$352$	−0.325948	−0.0173731
$353$	−6.77308	−0.360495	−0.180247	−	0.983621i	$-0.557690\pi$
$353$	−6.77308	−0.360495	−0.180247	+	0.983621i	$0.557690\pi$
$354$	7.57197	0.402446
$355$	−2.15008	−0.114114
$356$	15.4163	0.817064
$357$	18.8593	0.998138
$358$	−4.17865	−0.220849
$359$	1.42520	0.0752192	0.0376096	−	0.999293i	$-0.488026\pi$
$359$	1.42520	0.0752192	0.0376096	+	0.999293i	$0.488026\pi$
$360$	3.98627	0.210095
$361$	−18.9986	−0.999924
$362$	22.4539	1.18015
$363$	31.1154	1.63314
$364$	10.0087	0.524598
$365$	−2.12036	−0.110985
$366$	25.4044	1.32791
$367$	−23.4780	−1.22554	−0.612771	−	0.790261i	$-0.709945\pi$
$367$	−23.4780	−1.22554	−0.612771	+	0.790261i	$0.709945\pi$
$368$	0.908911	0.0473803
$369$	24.8193	1.29204
$370$	2.49628	0.129775
$371$	−38.8105	−2.01494
$372$	−14.7549	−0.765007
$373$	26.5795	1.37623	0.688117	−	0.725600i	$-0.258438\pi$
$373$	26.5795	1.37623	0.688117	+	0.725600i	$0.258438\pi$
$374$	−0.462986	−0.0239404
$375$	20.7549	1.07178
$376$	−12.1263	−0.625368
$377$	−3.68559	−0.189818
$378$	28.6553	1.47387
$379$	−5.54304	−0.284727	−0.142363	−	0.989814i	$-0.545470\pi$
$379$	−5.54304	−0.284727	−0.142363	+	0.989814i	$0.545470\pi$
$380$	0.0293732	0.00150681
$381$	−12.1956	−0.624798
$382$	7.55821	0.386712
$383$	13.1404	0.671441	0.335721	−	0.941962i	$-0.391020\pi$
$383$	13.1404	0.671441	0.335721	+	0.941962i	$0.391020\pi$
$384$	−2.85626	−0.145758
$385$	1.17090	0.0596746
$386$	9.11114	0.463745
$387$	49.2058	2.50127
$388$	9.85488	0.500306
$389$	5.43808	0.275722	0.137861	−	0.990452i	$-0.455977\pi$
$389$	5.43808	0.275722	0.137861	+	0.990452i	$0.455977\pi$
$390$	4.75263	0.240659
$391$	1.29104	0.0652909
$392$	14.6079	0.737812
$393$	4.00255	0.201902
$394$	3.81717	0.192306
$395$	6.50014	0.327057
$396$	−1.68132	−0.0844894
$397$	−30.1906	−1.51522	−0.757611	−	0.652707i	$-0.773633\pi$
$397$	−30.1906	−1.51522	−0.757611	+	0.652707i	$0.773633\pi$
$398$	−10.4497	−0.523796
$399$	0.504651	0.0252642
$400$	−4.40278	−0.220139
$401$	28.8556	1.44098	0.720490	−	0.693466i	$-0.243917\pi$
$401$	28.8556	1.44098	0.720490	+	0.693466i	$0.243917\pi$
$402$	4.65910	0.232375
$403$	−11.1227	−0.554059
$404$	13.7298	0.683084
$405$	1.64817	0.0818983
$406$	−7.95689	−0.394894
$407$	−1.05287	−0.0521889
$408$	−4.05712	−0.200858
$409$	−2.16453	−0.107029	−0.0535145	−	0.998567i	$-0.517042\pi$
$409$	−2.16453	−0.107029	−0.0535145	+	0.998567i	$0.517042\pi$
$410$	3.71838	0.183638
$411$	−10.2307	−0.504644
$412$	−13.7670	−0.678252
$413$	12.3230	0.606376
$414$	4.68838	0.230421
$415$	−9.12465	−0.447912
$416$	−2.15313	−0.105566
$417$	−14.2730	−0.698952
$418$	−0.0123889	−0.000605963 0
$419$	5.35225	0.261474	0.130737	−	0.991417i	$-0.458266\pi$
$419$	5.35225	0.261474	0.130737	+	0.991417i	$0.458266\pi$
$420$	10.2605	0.500663
$421$	3.13782	0.152928	0.0764641	−	0.997072i	$-0.475637\pi$
$421$	3.13782	0.152928	0.0764641	+	0.997072i	$0.475637\pi$
$422$	−24.1823	−1.17718
$423$	−62.5505	−3.04131
$424$	8.34916	0.405471
$425$	−6.25385	−0.303356
$426$	7.94671	0.385019
$427$	41.3445	2.00080
$428$	−0.771043	−0.0372698
$429$	−2.00455	−0.0967805
$430$	7.37191	0.355505
$431$	4.95228	0.238543	0.119271	−	0.992862i	$-0.461944\pi$
$431$	4.95228	0.238543	0.119271	+	0.992862i	$0.461944\pi$
$432$	−6.16451	−0.296590
$433$	−12.2375	−0.588099	−0.294049	−	0.955790i	$-0.595003\pi$
$433$	−12.2375	−0.588099	−0.294049	+	0.955790i	$0.595003\pi$
$434$	−24.0129	−1.15266
$435$	−3.77833	−0.181157
$436$	−4.04142	−0.193549
$437$	0.0345468	0.00165260
$438$	7.83687	0.374460
$439$	−30.2706	−1.44474	−0.722368	−	0.691509i	$-0.756946\pi$
$439$	−30.2706	−1.44474	−0.722368	+	0.691509i	$0.756946\pi$
$440$	−0.251891	−0.0120085
$441$	75.3513	3.58816
$442$	−3.05837	−0.145472
$443$	−14.3311	−0.680892	−0.340446	−	0.940264i	$-0.610578\pi$
$443$	−14.3311	−0.680892	−0.340446	+	0.940264i	$0.610578\pi$
$444$	−9.22626	−0.437859
$445$	11.9137	0.564763
$446$	9.47379	0.448597
$447$	9.73838	0.460609
$448$	−4.64843	−0.219618
$449$	−25.9061	−1.22258	−0.611291	−	0.791406i	$-0.709350\pi$
$449$	−25.9061	−1.22258	−0.611291	+	0.791406i	$0.709350\pi$
$450$	−22.7106	−1.07059
$451$	−1.56833	−0.0738496
$452$	−2.48631	−0.116946
$453$	−19.0158	−0.893438
$454$	−19.6010	−0.919918
$455$	7.73468	0.362607
$456$	−0.108564	−0.00508396
$457$	−22.2930	−1.04282	−0.521410	−	0.853306i	$-0.674594\pi$
$457$	−22.2930	−1.04282	−0.521410	+	0.853306i	$0.674594\pi$
$458$	−0.799017	−0.0373356
$459$	−8.75626	−0.408707
$460$	0.702404	0.0327497
$461$	20.6565	0.962067	0.481034	−	0.876702i	$-0.340261\pi$
$461$	20.6565	0.962067	0.481034	+	0.876702i	$0.340261\pi$
$462$	−4.32766	−0.201341
$463$	22.7977	1.05950	0.529750	−	0.848154i	$-0.322286\pi$
$463$	22.7977	1.05950	0.529750	+	0.848154i	$0.322286\pi$
$464$	1.71174	0.0794653
$465$	−11.4026	−0.528781
$466$	2.37314	0.109934
$467$	−24.8504	−1.14994	−0.574969	−	0.818175i	$-0.694986\pi$
$467$	−24.8504	−1.14994	−0.574969	+	0.818175i	$0.694986\pi$
$468$	−11.1064	−0.513392
$469$	7.58247	0.350126
$470$	−9.37119	−0.432261
$471$	37.8489	1.74398
$472$	−2.65101	−0.122022
$473$	−3.10930	−0.142966
$474$	−24.0246	−1.10348
$475$	−0.167345	−0.00767833
$476$	−6.60278	−0.302638
$477$	43.0670	1.97190
$478$	−11.5667	−0.529051
$479$	−38.3607	−1.75275	−0.876373	−	0.481634i	$-0.840044\pi$
$479$	−38.3607	−1.75275	−0.876373	+	0.481634i	$0.840044\pi$
$480$	−2.20731	−0.100750
$481$	−6.95501	−0.317121
$482$	2.57079	0.117096
$483$	12.0678	0.549102
$484$	−10.8938	−0.495171
$485$	7.61582	0.345817
$486$	12.4019	0.562560
$487$	−1.88064	−0.0852200	−0.0426100	−	0.999092i	$-0.513567\pi$
$487$	−1.88064	−0.0852200	−0.0426100	+	0.999092i	$0.513567\pi$
$488$	−8.89428	−0.402625
$489$	57.6470	2.60689
$490$	11.2890	0.509984
$491$	−13.1935	−0.595416	−0.297708	−	0.954657i	$-0.596222\pi$
$491$	−13.1935	−0.595416	−0.297708	+	0.954657i	$0.596222\pi$
$492$	−13.7432	−0.619590
$493$	2.43140	0.109505
$494$	−0.0818383	−0.00368208
$495$	−1.29932	−0.0584000
$496$	5.16581	0.231952
$497$	12.9329	0.580119
$498$	33.7248	1.51124
$499$	19.2300	0.860853	0.430427	−	0.902626i	$-0.358363\pi$
$499$	19.2300	0.860853	0.430427	+	0.902626i	$0.358363\pi$
$500$	−7.26644	−0.324965
$501$	37.3724	1.66968
$502$	1.02856	0.0459071
$503$	−39.5007	−1.76125	−0.880624	−	0.473816i	$-0.842876\pi$
$503$	−39.5007	−1.76125	−0.880624	+	0.473816i	$0.842876\pi$
$504$	−23.9777	−1.06805
$505$	10.6104	0.472155
$506$	−0.296258	−0.0131703
$507$	23.8899	1.06099
$508$	4.26977	0.189440
$509$	1.25265	0.0555225	0.0277613	−	0.999615i	$-0.491162\pi$
$509$	1.25265	0.0555225	0.0277613	+	0.999615i	$0.491162\pi$
$510$	−3.13533	−0.138835
$511$	12.7541	0.564209
$512$	1.00000	0.0441942
$513$	−0.234307	−0.0103449
$514$	5.86654	0.258762
$515$	−10.6391	−0.468815
$516$	−27.2466	−1.19947
$517$	3.95255	0.173833
$518$	−15.0153	−0.659735
$519$	0.709732	0.0311538
$520$	−1.66393	−0.0729683
$521$	−5.70712	−0.250033	−0.125017	−	0.992155i	$-0.539898\pi$
$521$	−5.70712	−0.250033	−0.125017	+	0.992155i	$0.539898\pi$
$522$	8.82955	0.386459
$523$	−15.9667	−0.698174	−0.349087	−	0.937090i	$-0.613508\pi$
$523$	−15.9667	−0.698174	−0.349087	+	0.937090i	$0.613508\pi$
$524$	−1.40132	−0.0612171
$525$	−58.4564	−2.55125
$526$	7.33848	0.319973
$527$	7.33767	0.319634
$528$	0.930993	0.0405163
$529$	−22.1739	−0.964082
$530$	6.45220	0.280266
$531$	−13.6745	−0.593424
$532$	−0.176682	−0.00766015
$533$	−10.3600	−0.448740
$534$	−44.0331	−1.90550
$535$	−0.595860	−0.0257613
$536$	−1.63119	−0.0704566
$537$	11.9353	0.515048
$538$	−1.56158	−0.0673246
$539$	−4.76143	−0.205089
$540$	−4.76391	−0.205006
$541$	−31.4989	−1.35424	−0.677122	−	0.735871i	$-0.736773\pi$
$541$	−31.4989	−1.35424	−0.677122	+	0.735871i	$0.736773\pi$
$542$	12.6243	0.542262
$543$	−64.1342	−2.75226
$544$	1.42043	0.0609004
$545$	−3.12320	−0.133783
$546$	−28.5874	−1.22343
$547$	29.5438	1.26320	0.631600	−	0.775295i	$-0.282399\pi$
$547$	29.5438	1.26320	0.631600	+	0.775295i	$0.282399\pi$
$548$	3.58185	0.153009
$549$	−45.8788	−1.95806
$550$	1.43508	0.0611919
$551$	0.0650613	0.00277171
$552$	−2.59609	−0.110497
$553$	−39.0988	−1.66265
$554$	−14.7948	−0.628569
$555$	−7.13003	−0.302653
$556$	4.99709	0.211924
$557$	42.1183	1.78461	0.892306	−	0.451432i	$-0.149087\pi$
$557$	42.1183	1.78461	0.892306	+	0.451432i	$0.149087\pi$
$558$	26.6465	1.12804
$559$	−20.5393	−0.868718
$560$	−3.59230	−0.151802
$561$	1.32241	0.0558322
$562$	−1.96400	−0.0828464
$563$	22.4862	0.947681	0.473840	−	0.880611i	$-0.342867\pi$
$563$	22.4862	0.947681	0.473840	+	0.880611i	$0.342867\pi$
$564$	34.6360	1.45844
$565$	−1.92142	−0.0808346
$566$	−23.1223	−0.971904
$567$	−9.91389	−0.416344
$568$	−2.78220	−0.116739
$569$	−11.9765	−0.502082	−0.251041	−	0.967976i	$-0.580773\pi$
$569$	−11.9765	−0.502082	−0.251041	+	0.967976i	$0.580773\pi$
$570$	−0.0838977	−0.00351409
$571$	−39.9149	−1.67039	−0.835194	−	0.549956i	$-0.814645\pi$
$571$	−39.9149	−1.67039	−0.835194	+	0.549956i	$0.814645\pi$
$572$	0.701808	0.0293441
$573$	−21.5882	−0.901862
$574$	−22.3663	−0.933553
$575$	−4.00174	−0.166884
$576$	5.15824	0.214927
$577$	−1.72368	−0.0717578	−0.0358789	−	0.999356i	$-0.511423\pi$
$577$	−1.72368	−0.0717578	−0.0358789	+	0.999356i	$0.511423\pi$
$578$	−14.9824	−0.623185
$579$	−26.0238	−1.08151
$580$	1.32282	0.0549273
$581$	54.8855	2.27703
$582$	−28.1481	−1.16678
$583$	−2.72139	−0.112708
$584$	−2.74375	−0.113537
$585$	−8.58297	−0.354862
$586$	−29.2409	−1.20793
$587$	−15.2242	−0.628370	−0.314185	−	0.949362i	$-0.601731\pi$
$587$	−15.2242	−0.628370	−0.314185	+	0.949362i	$0.601731\pi$
$588$	−41.7241	−1.72067
$589$	0.196347	0.00809035
$590$	−2.04869	−0.0843432
$591$	−10.9028	−0.448483
$592$	3.23019	0.132760
$593$	16.2534	0.667449	0.333724	−	0.942671i	$-0.391695\pi$
$593$	16.2534	0.667449	0.333724	+	0.942671i	$0.391695\pi$
$594$	2.00931	0.0824429
$595$	−5.10261	−0.209186
$596$	−3.40948	−0.139658
$597$	29.8471	1.22156
$598$	−1.95700	−0.0800278
$599$	14.3763	0.587400	0.293700	−	0.955898i	$-0.405113\pi$
$599$	14.3763	0.587400	0.293700	+	0.955898i	$0.405113\pi$
$600$	12.5755	0.513393
$601$	4.13386	0.168624	0.0843119	−	0.996439i	$-0.473131\pi$
$601$	4.13386	0.168624	0.0843119	+	0.996439i	$0.473131\pi$
$602$	−44.3426	−1.80727
$603$	−8.41406	−0.342647
$604$	6.65757	0.270892
$605$	−8.41866	−0.342267
$606$	−39.2160	−1.59304
$607$	−40.9939	−1.66389	−0.831946	−	0.554857i	$-0.812773\pi$
$607$	−40.9939	−1.66389	−0.831946	+	0.554857i	$0.812773\pi$
$608$	0.0380090	0.00154147
$609$	22.7270	0.920944
$610$	−6.87347	−0.278299
$611$	26.1096	1.05628
$612$	7.32692	0.296173
$613$	38.6900	1.56267	0.781337	−	0.624110i	$-0.214538\pi$
$613$	38.6900	1.56267	0.781337	+	0.624110i	$0.214538\pi$
$614$	−12.8603	−0.518998
$615$	−10.6207	−0.428267
$616$	1.51515	0.0610470
$617$	−21.7593	−0.875998	−0.437999	−	0.898975i	$-0.644313\pi$
$617$	−21.7593	−0.875998	−0.437999	+	0.898975i	$0.644313\pi$
$618$	39.3222	1.58177
$619$	0.411694	0.0165474	0.00827368	−	0.999966i	$-0.497366\pi$
$619$	0.411694	0.0165474	0.00827368	+	0.999966i	$0.497366\pi$
$620$	3.99212	0.160328
$621$	−5.60299	−0.224840
$622$	−0.890715	−0.0357144
$623$	−71.6618	−2.87107
$624$	6.14991	0.246193
$625$	16.3984	0.655938
$626$	−22.1991	−0.887255
$627$	0.0353861	0.00141318
$628$	−13.2512	−0.528779
$629$	4.58825	0.182946
$630$	−18.5299	−0.738250
$631$	−24.0999	−0.959401	−0.479701	−	0.877432i	$-0.659255\pi$
$631$	−24.0999	−0.959401	−0.479701	+	0.877432i	$0.659255\pi$
$632$	8.41118	0.334579
$633$	69.0710	2.74533
$634$	9.56953	0.380055
$635$	3.29966	0.130943
$636$	−23.8474	−0.945611
$637$	−31.4528	−1.24621
$638$	−0.557936	−0.0220889
$639$	−14.3513	−0.567728
$640$	0.772797	0.0305475
$641$	21.9632	0.867493	0.433746	−	0.901035i	$-0.357191\pi$
$641$	21.9632	0.867493	0.433746	+	0.901035i	$0.357191\pi$
$642$	2.20230	0.0869179
$643$	−16.0315	−0.632219	−0.316110	−	0.948723i	$-0.602377\pi$
$643$	−16.0315	−0.632219	−0.316110	+	0.948723i	$0.602377\pi$
$644$	−4.22501	−0.166489
$645$	−21.0561	−0.829084
$646$	0.0539891	0.00212417
$647$	30.8458	1.21267	0.606336	−	0.795208i	$-0.292639\pi$
$647$	30.8458	1.21267	0.606336	+	0.795208i	$0.292639\pi$
$648$	2.13274	0.0837818
$649$	0.864089	0.0339185
$650$	9.47977	0.371827
$651$	68.5873	2.68815
$652$	−20.1827	−0.790414
$653$	−7.87323	−0.308103	−0.154052	−	0.988063i	$-0.549232\pi$
$653$	−7.87323	−0.308103	−0.154052	+	0.988063i	$0.549232\pi$
$654$	11.5434	0.451382
$655$	−1.08294	−0.0423139
$656$	4.81159	0.187861
$657$	−14.1529	−0.552158
$658$	56.3684	2.19747
$659$	10.1254	0.394428	0.197214	−	0.980360i	$-0.436811\pi$
$659$	10.1254	0.394428	0.197214	+	0.980360i	$0.436811\pi$
$660$	0.719468	0.0280053
$661$	−2.24967	−0.0875018	−0.0437509	−	0.999042i	$-0.513931\pi$
$661$	−2.24967	−0.0875018	−0.0437509	+	0.999042i	$0.513931\pi$
$662$	1.69426	0.0658493
$663$	8.73552	0.339259
$664$	−11.8073	−0.458213
$665$	−0.136540	−0.00529477
$666$	16.6621	0.645642
$667$	1.55582	0.0602414
$668$	−13.0844	−0.506249
$669$	−27.0596	−1.04619
$670$	−1.26058	−0.0487003
$671$	2.89907	0.111917
$672$	13.2772	0.512177
$673$	9.34042	0.360047	0.180023	−	0.983662i	$-0.442383\pi$
$673$	9.34042	0.360047	0.180023	+	0.983662i	$0.442383\pi$
$674$	−2.35496	−0.0907096
$675$	27.1410	1.04466
$676$	−8.36403	−0.321693
$677$	12.2725	0.471669	0.235834	−	0.971793i	$-0.424218\pi$
$677$	12.2725	0.471669	0.235834	+	0.971793i	$0.424218\pi$
$678$	7.10157	0.272734
$679$	−45.8098	−1.75802
$680$	1.09770	0.0420950
$681$	55.9855	2.14537
$682$	−1.68378	−0.0644754
$683$	−27.8718	−1.06648	−0.533242	−	0.845963i	$-0.679026\pi$
$683$	−27.8718	−1.06648	−0.533242	+	0.845963i	$0.679026\pi$
$684$	0.196060	0.00749652
$685$	2.76804	0.105762
$686$	−35.3650	−1.35024
$687$	2.28220	0.0870715
$688$	9.53926	0.363681
$689$	−17.9768	−0.684862
$690$	−2.00625	−0.0763766
$691$	50.4000	1.91731	0.958653	−	0.284578i	$-0.0918535\pi$
$691$	50.4000	1.91731	0.958653	+	0.284578i	$0.0918535\pi$
$692$	−0.248483	−0.00944590
$693$	7.81549	0.296886
$694$	27.1868	1.03200
$695$	3.86173	0.146484
$696$	−4.88917	−0.185323
$697$	6.83452	0.258876
$698$	−22.9581	−0.868976
$699$	−6.77833	−0.256380
$700$	20.4661	0.773544
$701$	−19.7450	−0.745759	−0.372879	−	0.927880i	$-0.621629\pi$
$701$	−19.7450	−0.745759	−0.372879	+	0.927880i	$0.621629\pi$
$702$	13.2730	0.500957
$703$	0.122776	0.00463059
$704$	−0.325948	−0.0122846
$705$	26.7666	1.00809
$706$	−6.77308	−0.254908
$707$	−63.8222	−2.40028
$708$	7.57197	0.284572
$709$	−21.6909	−0.814619	−0.407309	−	0.913290i	$-0.633533\pi$
$709$	−21.6909	−0.814619	−0.407309	+	0.913290i	$0.633533\pi$
$710$	−2.15008	−0.0806910
$711$	43.3869	1.62714
$712$	15.4163	0.577752
$713$	4.69526	0.175839
$714$	18.8593	0.705790
$715$	0.542355	0.0202829
$716$	−4.17865	−0.156164
$717$	33.0377	1.23382
$718$	1.42520	0.0531880
$719$	19.0651	0.711008	0.355504	−	0.934675i	$-0.384309\pi$
$719$	19.0651	0.711008	0.355504	+	0.934675i	$0.384309\pi$
$720$	3.98627	0.148560
$721$	63.9950	2.38330
$722$	−18.9986	−0.707053
$723$	−7.34286	−0.273084
$724$	22.4539	0.834492
$725$	−7.53640	−0.279895
$726$	31.1154	1.15480
$727$	−16.3554	−0.606587	−0.303294	−	0.952897i	$-0.598086\pi$
$727$	−16.3554	−0.606587	−0.303294	+	0.952897i	$0.598086\pi$
$728$	10.0087	0.370946
$729$	−41.8212	−1.54893
$730$	−2.12036	−0.0784780
$731$	13.5498	0.501159
$732$	25.4044	0.938974
$733$	−10.9863	−0.405789	−0.202894	−	0.979201i	$-0.565035\pi$
$733$	−10.9863	−0.405789	−0.202894	+	0.979201i	$0.565035\pi$
$734$	−23.4780	−0.866589
$735$	−32.2443	−1.18935
$736$	0.908911	0.0335029
$737$	0.531682	0.0195848
$738$	24.8193	0.913612
$739$	47.4845	1.74674	0.873372	−	0.487053i	$-0.161928\pi$
$739$	47.4845	1.74674	0.873372	+	0.487053i	$0.161928\pi$
$740$	2.49628	0.0917650
$741$	0.233752	0.00858708
$742$	−38.8105	−1.42478
$743$	−17.4182	−0.639010	−0.319505	−	0.947585i	$-0.603517\pi$
$743$	−17.4182	−0.639010	−0.319505	+	0.947585i	$0.603517\pi$
$744$	−14.7549	−0.540942
$745$	−2.63484	−0.0965330
$746$	26.5795	0.973144
$747$	−60.9050	−2.22840
$748$	−0.462986	−0.0169284
$749$	3.58414	0.130962
$750$	20.7549	0.757861
$751$	−18.5791	−0.677960	−0.338980	−	0.940794i	$-0.610082\pi$
$751$	−18.5791	−0.677960	−0.338980	+	0.940794i	$0.610082\pi$
$752$	−12.1263	−0.442202
$753$	−2.93785	−0.107061
$754$	−3.68559	−0.134221
$755$	5.14495	0.187244
$756$	28.6553	1.04218
$757$	3.75875	0.136614	0.0683070	−	0.997664i	$-0.478240\pi$
$757$	3.75875	0.136614	0.0683070	+	0.997664i	$0.478240\pi$
$758$	−5.54304	−0.201332
$759$	0.846190	0.0307147
$760$	0.0293732	0.00106548
$761$	−38.5253	−1.39654	−0.698271	−	0.715833i	$-0.746047\pi$
$761$	−38.5253	−1.39654	−0.698271	+	0.715833i	$0.746047\pi$
$762$	−12.1956	−0.441799
$763$	18.7863	0.680110
$764$	7.55821	0.273446
$765$	5.66222	0.204718
$766$	13.1404	0.474781
$767$	5.70796	0.206102
$768$	−2.85626	−0.103067
$769$	−26.7584	−0.964933	−0.482467	−	0.875914i	$-0.660259\pi$
$769$	−26.7584	−0.964933	−0.482467	+	0.875914i	$0.660259\pi$
$770$	1.17090	0.0421963
$771$	−16.7564	−0.603467
$772$	9.11114	0.327917
$773$	10.8667	0.390847	0.195423	−	0.980719i	$-0.437392\pi$
$773$	10.8667	0.390847	0.195423	+	0.980719i	$0.437392\pi$
$774$	49.2058	1.76867
$775$	−22.7440	−0.816987
$776$	9.85488	0.353770
$777$	42.8877	1.53859
$778$	5.43808	0.194965
$779$	0.182884	0.00655248
$780$	4.75263	0.170171
$781$	0.906853	0.0324498
$782$	1.29104	0.0461677
$783$	−10.5520	−0.377098
$784$	14.6079	0.521712
$785$	−10.2405	−0.365498
$786$	4.00255	0.142766
$787$	−9.76333	−0.348025	−0.174013	−	0.984743i	$-0.555673\pi$
$787$	−9.76333	−0.348025	−0.174013	+	0.984743i	$0.555673\pi$
$788$	3.81717	0.135981
$789$	−20.9606	−0.746218
$790$	6.50014	0.231264
$791$	11.5575	0.410936
$792$	−1.68132	−0.0597430
$793$	19.1505	0.680055
$794$	−30.1906	−1.07142
$795$	−18.4292	−0.653616
$796$	−10.4497	−0.370380
$797$	14.0491	0.497645	0.248822	−	0.968549i	$-0.419956\pi$
$797$	14.0491	0.497645	0.248822	+	0.968549i	$0.419956\pi$
$798$	0.504651	0.0178645
$799$	−17.2246	−0.609363
$800$	−4.40278	−0.155662
$801$	79.5212	2.80974
$802$	28.8556	1.01893
$803$	0.894318	0.0315598
$804$	4.65910	0.164314
$805$	−3.26508	−0.115079
$806$	−11.1227	−0.391779
$807$	4.46029	0.157010
$808$	13.7298	0.483014
$809$	−19.8200	−0.696836	−0.348418	−	0.937339i	$-0.613281\pi$
$809$	−19.8200	−0.696836	−0.348418	+	0.937339i	$0.613281\pi$
$810$	1.64817	0.0579109
$811$	30.1672	1.05931	0.529657	−	0.848212i	$-0.322321\pi$
$811$	30.1672	1.05931	0.529657	+	0.848212i	$0.322321\pi$
$812$	−7.95689	−0.279232
$813$	−36.0585	−1.26463
$814$	−1.05287	−0.0369031
$815$	−15.5971	−0.546342
$816$	−4.05712	−0.142028
$817$	0.362577	0.0126850
$818$	−2.16453	−0.0756809
$819$	51.6272	1.80400
$820$	3.71838	0.129851
$821$	29.3007	1.02260	0.511301	−	0.859402i	$-0.329164\pi$
$821$	29.3007	1.02260	0.511301	+	0.859402i	$0.329164\pi$
$822$	−10.2307	−0.356837
$823$	1.39176	0.0485138	0.0242569	−	0.999706i	$-0.492278\pi$
$823$	1.39176	0.0485138	0.0242569	+	0.999706i	$0.492278\pi$
$824$	−13.7670	−0.479596
$825$	−4.09896	−0.142708
$826$	12.3230	0.428773
$827$	−9.45946	−0.328937	−0.164469	−	0.986382i	$-0.552591\pi$
$827$	−9.45946	−0.328937	−0.164469	+	0.986382i	$0.552591\pi$
$828$	4.68838	0.162933
$829$	6.47558	0.224906	0.112453	−	0.993657i	$-0.464129\pi$
$829$	6.47558	0.224906	0.112453	+	0.993657i	$0.464129\pi$
$830$	−9.12465	−0.316721
$831$	42.2577	1.46590
$832$	−2.15313	−0.0746464
$833$	20.7496	0.718930
$834$	−14.2730	−0.494233
$835$	−10.1116	−0.349925
$836$	−0.0123889	−0.000428481 0
$837$	−31.8447	−1.10071
$838$	5.35225	0.184890
$839$	−14.6105	−0.504409	−0.252204	−	0.967674i	$-0.581156\pi$
$839$	−14.6105	−0.504409	−0.252204	+	0.967674i	$0.581156\pi$
$840$	10.2605	0.354022
$841$	−26.0700	−0.898964
$842$	3.13782	0.108137
$843$	5.60971	0.193209
$844$	−24.1823	−0.832389
$845$	−6.46370	−0.222358
$846$	−62.5505	−2.15053
$847$	50.6389	1.73997
$848$	8.34916	0.286711
$849$	66.0435	2.26661
$850$	−6.25385	−0.214505
$851$	2.93595	0.100643
$852$	7.94671	0.272250
$853$	−28.7336	−0.983818	−0.491909	−	0.870647i	$-0.663701\pi$
$853$	−28.7336	−0.983818	−0.491909	+	0.870647i	$0.663701\pi$
$854$	41.3445	1.41478
$855$	0.151514	0.00518168
$856$	−0.771043	−0.0263537
$857$	21.6888	0.740876	0.370438	−	0.928857i	$-0.379208\pi$
$857$	21.6888	0.740876	0.370438	+	0.928857i	$0.379208\pi$
$858$	−2.00455	−0.0684342
$859$	55.5267	1.89455	0.947273	−	0.320429i	$-0.103827\pi$
$859$	55.5267	1.89455	0.947273	+	0.320429i	$0.103827\pi$
$860$	7.37191	0.251380
$861$	63.8842	2.17717
$862$	4.95228	0.168675
$863$	18.4694	0.628707	0.314354	−	0.949306i	$-0.398212\pi$
$863$	18.4694	0.628707	0.314354	+	0.949306i	$0.398212\pi$
$864$	−6.16451	−0.209721
$865$	−0.192027	−0.00652911
$866$	−12.2375	−0.415849
$867$	42.7936	1.45335
$868$	−24.0129	−0.815052
$869$	−2.74161	−0.0930026
$870$	−3.77833	−0.128098
$871$	3.51216	0.119005
$872$	−4.04142	−0.136860
$873$	50.8339	1.72047
$874$	0.0345468	0.00116856
$875$	33.7776	1.14189
$876$	7.83687	0.264783
$877$	29.9793	1.01233	0.506165	−	0.862437i	$-0.331063\pi$
$877$	29.9793	1.01233	0.506165	+	0.862437i	$0.331063\pi$
$878$	−30.2706	−1.02158
$879$	83.5198	2.81705
$880$	−0.251891	−0.00849126
$881$	25.9113	0.872973	0.436486	−	0.899711i	$-0.356223\pi$
$881$	25.9113	0.872973	0.436486	+	0.899711i	$0.356223\pi$
$882$	75.3513	2.53721
$883$	6.37507	0.214538	0.107269	−	0.994230i	$-0.465789\pi$
$883$	6.37507	0.214538	0.107269	+	0.994230i	$0.465789\pi$
$884$	−3.05837	−0.102864
$885$	5.85160	0.196699
$886$	−14.3311	−0.481463
$887$	27.6680	0.929000	0.464500	−	0.885573i	$-0.346234\pi$
$887$	27.6680	0.929000	0.464500	+	0.885573i	$0.346234\pi$
$888$	−9.22626	−0.309613
$889$	−19.8477	−0.665671
$890$	11.9137	0.399348
$891$	−0.695161	−0.0232888
$892$	9.47379	0.317206
$893$	−0.460909	−0.0154237
$894$	9.73838	0.325700
$895$	−3.22925	−0.107942
$896$	−4.64843	−0.155293
$897$	5.58972	0.186635
$898$	−25.9061	−0.864496
$899$	8.84250	0.294914
$900$	−22.7106	−0.757021
$901$	11.8594	0.395094
$902$	−1.56833	−0.0522196
$903$	126.654	4.21479
$904$	−2.48631	−0.0826936
$905$	17.3523	0.576810
$906$	−19.0158	−0.631756
$907$	5.57349	0.185065	0.0925323	−	0.995710i	$-0.470504\pi$
$907$	5.57349	0.185065	0.0925323	+	0.995710i	$0.470504\pi$
$908$	−19.6010	−0.650481
$909$	70.8218	2.34901
$910$	7.73468	0.256402
$911$	−19.7877	−0.655594	−0.327797	−	0.944748i	$-0.606306\pi$
$911$	−19.7877	−0.655594	−0.327797	+	0.944748i	$0.606306\pi$
$912$	−0.108564	−0.00359490
$913$	3.84857	0.127369
$914$	−22.2930	−0.737386
$915$	19.6324	0.649029
$916$	−0.799017	−0.0264003
$917$	6.51396	0.215110
$918$	−8.75626	−0.289000
$919$	42.6447	1.40672	0.703359	−	0.710835i	$-0.251683\pi$
$919$	42.6447	1.40672	0.703359	+	0.710835i	$0.251683\pi$
$920$	0.702404	0.0231576
$921$	36.7323	1.21037
$922$	20.6565	0.680284
$923$	5.99045	0.197178
$924$	−4.32766	−0.142370
$925$	−14.2218	−0.467610
$926$	22.7977	0.749179
$927$	−71.0136	−2.33239
$928$	1.71174	0.0561905
$929$	−27.3039	−0.895812	−0.447906	−	0.894081i	$-0.647830\pi$
$929$	−27.3039	−0.895812	−0.447906	+	0.894081i	$0.647830\pi$
$930$	−11.4026	−0.373905
$931$	0.555233	0.0181970
$932$	2.37314	0.0777349
$933$	2.54412	0.0832906
$934$	−24.8504	−0.813129
$935$	−0.357794	−0.0117011
$936$	−11.1064	−0.363023
$937$	18.9366	0.618632	0.309316	−	0.950959i	$-0.399900\pi$
$937$	18.9366	0.618632	0.309316	+	0.950959i	$0.399900\pi$
$938$	7.58247	0.247576
$939$	63.4066	2.06919
$940$	−9.37119	−0.305654
$941$	48.4072	1.57803	0.789014	−	0.614375i	$-0.210592\pi$
$941$	48.4072	1.57803	0.789014	+	0.614375i	$0.210592\pi$
$942$	37.8489	1.23318
$943$	4.37331	0.142414
$944$	−2.65101	−0.0862829
$945$	22.1447	0.720368
$946$	−3.10930	−0.101092
$947$	−37.7633	−1.22714	−0.613572	−	0.789639i	$-0.710268\pi$
$947$	−37.7633	−1.22714	−0.613572	+	0.789639i	$0.710268\pi$
$948$	−24.0246	−0.780281
$949$	5.90765	0.191770
$950$	−0.167345	−0.00542940
$951$	−27.3331	−0.886337
$952$	−6.60278	−0.213997
$953$	−32.7368	−1.06045	−0.530225	−	0.847857i	$-0.677893\pi$
$953$	−32.7368	−1.06045	−0.530225	+	0.847857i	$0.677893\pi$
$954$	43.0670	1.39434
$955$	5.84096	0.189009
$956$	−11.5667	−0.374095
$957$	1.59361	0.0515142
$958$	−38.3607	−1.23938
$959$	−16.6500	−0.537657
$960$	−2.20731	−0.0712407
$961$	−4.31440	−0.139174
$962$	−6.95501	−0.224239
$963$	−3.97723	−0.128164
$964$	2.57079	0.0827997
$965$	7.04106	0.226660
$966$	12.0678	0.388274
$967$	−30.2956	−0.974241	−0.487121	−	0.873335i	$-0.661953\pi$
$967$	−30.2956	−0.974241	−0.487121	+	0.873335i	$0.661953\pi$
$968$	−10.8938	−0.350139
$969$	−0.154207	−0.00495385
$970$	7.61582	0.244529
$971$	8.66440	0.278054	0.139027	−	0.990289i	$-0.455603\pi$
$971$	8.66440	0.278054	0.139027	+	0.990289i	$0.455603\pi$
$972$	12.4019	0.397790
$973$	−23.2286	−0.744675
$974$	−1.88064	−0.0602596
$975$	−27.0767	−0.867149
$976$	−8.89428	−0.284699
$977$	0.452106	0.0144642	0.00723208	−	0.999974i	$-0.497698\pi$
$977$	0.452106	0.0144642	0.00723208	+	0.999974i	$0.497698\pi$
$978$	57.6470	1.84335
$979$	−5.02492	−0.160597
$980$	11.2890	0.360613
$981$	−20.8466	−0.665582
$982$	−13.1935	−0.421023
$983$	−24.2641	−0.773905	−0.386953	−	0.922100i	$-0.626472\pi$
$983$	−24.2641	−0.773905	−0.386953	+	0.922100i	$0.626472\pi$
$984$	−13.7432	−0.438116
$985$	2.94990	0.0939916
$986$	2.43140	0.0774316
$987$	−161.003	−5.12479
$988$	−0.0818383	−0.00260362
$989$	8.67033	0.275701
$990$	−1.29932	−0.0412950
$991$	−14.5127	−0.461011	−0.230505	−	0.973071i	$-0.574038\pi$
$991$	−14.5127	−0.461011	−0.230505	+	0.973071i	$0.574038\pi$
$992$	5.16581	0.164015
$993$	−4.83925	−0.153569
$994$	12.9329	0.410206
$995$	−8.07550	−0.256010
$996$	33.7248	1.06861
$997$	−9.27321	−0.293686	−0.146843	−	0.989160i	$-0.546911\pi$
$997$	−9.27321	−0.293686	−0.146843	+	0.989160i	$0.546911\pi$
$998$	19.2300	0.608715
$999$	−19.9125	−0.630004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.a.1.9	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.a.1.9	✓	75	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.85626 q^{3} +1.00000 q^{4} +0.772797 q^{5} -2.85626 q^{6} -4.64843 q^{7} +1.00000 q^{8} +5.15824 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.85626 q^{3} +1.00000 q^{4} +0.772797 q^{5} -2.85626 q^{6} -4.64843 q^{7} +1.00000 q^{8} +5.15824 q^{9} +0.772797 q^{10} -0.325948 q^{11} -2.85626 q^{12} -2.15313 q^{13} -4.64843 q^{14} -2.20731 q^{15} +1.00000 q^{16} +1.42043 q^{17} +5.15824 q^{18} +0.0380090 q^{19} +0.772797 q^{20} +13.2772 q^{21} -0.325948 q^{22} +0.908911 q^{23} -2.85626 q^{24} -4.40278 q^{25} -2.15313 q^{26} -6.16451 q^{27} -4.64843 q^{28} +1.71174 q^{29} -2.20731 q^{30} +5.16581 q^{31} +1.00000 q^{32} +0.930993 q^{33} +1.42043 q^{34} -3.59230 q^{35} +5.15824 q^{36} +3.23019 q^{37} +0.0380090 q^{38} +6.14991 q^{39} +0.772797 q^{40} +4.81159 q^{41} +13.2772 q^{42} +9.53926 q^{43} -0.325948 q^{44} +3.98627 q^{45} +0.908911 q^{46} -12.1263 q^{47} -2.85626 q^{48} +14.6079 q^{49} -4.40278 q^{50} -4.05712 q^{51} -2.15313 q^{52} +8.34916 q^{53} -6.16451 q^{54} -0.251891 q^{55} -4.64843 q^{56} -0.108564 q^{57} +1.71174 q^{58} -2.65101 q^{59} -2.20731 q^{60} -8.89428 q^{61} +5.16581 q^{62} -23.9777 q^{63} +1.00000 q^{64} -1.66393 q^{65} +0.930993 q^{66} -1.63119 q^{67} +1.42043 q^{68} -2.59609 q^{69} -3.59230 q^{70} -2.78220 q^{71} +5.15824 q^{72} -2.74375 q^{73} +3.23019 q^{74} +12.5755 q^{75} +0.0380090 q^{76} +1.51515 q^{77} +6.14991 q^{78} +8.41118 q^{79} +0.772797 q^{80} +2.13274 q^{81} +4.81159 q^{82} -11.8073 q^{83} +13.2772 q^{84} +1.09770 q^{85} +9.53926 q^{86} -4.88917 q^{87} -0.325948 q^{88} +15.4163 q^{89} +3.98627 q^{90} +10.0087 q^{91} +0.908911 q^{92} -14.7549 q^{93} -12.1263 q^{94} +0.0293732 q^{95} -2.85626 q^{96} +9.85488 q^{97} +14.6079 q^{98} -1.68132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * q^99

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