LMFDB - Embedded newform 8030.2.a.bb.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8030,2,Mod(1,8030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8030, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8030 = 2 \cdot 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8030.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:	$64.1198728231$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 23x^{4} - 32x^{3} - 16x^{2} + 17x - 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 23x^4 - 32x^3 - 16x^2 + 17*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.29392$ of defining polynomial
Character	$\chi$	$=$	8030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.29392 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.29392 q^{6} -0.0375670 q^{7} +1.00000 q^{8} -1.32576 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.29392 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.29392 q^{6} -0.0375670 q^{7} +1.00000 q^{8} -1.32576 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.29392 q^{12} -4.37567 q^{13} -0.0375670 q^{14} -1.29392 q^{15} +1.00000 q^{16} +2.78462 q^{17} -1.32576 q^{18} -2.93312 q^{19} -1.00000 q^{20} -0.0486088 q^{21} +1.00000 q^{22} +3.64937 q^{23} +1.29392 q^{24} +1.00000 q^{25} -4.37567 q^{26} -5.59721 q^{27} -0.0375670 q^{28} -4.56552 q^{29} -1.29392 q^{30} +8.26949 q^{31} +1.00000 q^{32} +1.29392 q^{33} +2.78462 q^{34} +0.0375670 q^{35} -1.32576 q^{36} -4.22468 q^{37} -2.93312 q^{38} -5.66179 q^{39} -1.00000 q^{40} -3.97221 q^{41} -0.0486088 q^{42} -11.2218 q^{43} +1.00000 q^{44} +1.32576 q^{45} +3.64937 q^{46} -0.238525 q^{47} +1.29392 q^{48} -6.99859 q^{49} +1.00000 q^{50} +3.60309 q^{51} -4.37567 q^{52} -10.4176 q^{53} -5.59721 q^{54} -1.00000 q^{55} -0.0375670 q^{56} -3.79524 q^{57} -4.56552 q^{58} -1.92118 q^{59} -1.29392 q^{60} +9.11616 q^{61} +8.26949 q^{62} +0.0498049 q^{63} +1.00000 q^{64} +4.37567 q^{65} +1.29392 q^{66} +9.42930 q^{67} +2.78462 q^{68} +4.72201 q^{69} +0.0375670 q^{70} -9.71798 q^{71} -1.32576 q^{72} -1.00000 q^{73} -4.22468 q^{74} +1.29392 q^{75} -2.93312 q^{76} -0.0375670 q^{77} -5.66179 q^{78} -5.80976 q^{79} -1.00000 q^{80} -3.26507 q^{81} -3.97221 q^{82} +8.90411 q^{83} -0.0486088 q^{84} -2.78462 q^{85} -11.2218 q^{86} -5.90744 q^{87} +1.00000 q^{88} -12.2019 q^{89} +1.32576 q^{90} +0.164381 q^{91} +3.64937 q^{92} +10.7001 q^{93} -0.238525 q^{94} +2.93312 q^{95} +1.29392 q^{96} -4.35126 q^{97} -6.99859 q^{98} -1.32576 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 3 q^{3} + 8 q^{4} - 8 q^{5} - 3 q^{6} - 4 q^{7} + 8 q^{8} + q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 3 * q^3 + 8 * q^4 - 8 * q^5 - 3 * q^6 - 4 * q^7 + 8 * q^8 + q^9	\( 8 q + 8 q^{2} - 3 q^{3} + 8 q^{4} - 8 q^{5} - 3 q^{6} - 4 q^{7} + 8 q^{8} + q^{9} - 8 q^{10} + 8 q^{11} - 3 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 8 q^{16} + 2 q^{17} + q^{18} - q^{19} - 8 q^{20} + 7 q^{21} + 8 q^{22} - 16 q^{23} - 3 q^{24} + 8 q^{25} - 7 q^{26} - 21 q^{27} - 4 q^{28} - 5 q^{29} + 3 q^{30} - 22 q^{31} + 8 q^{32} - 3 q^{33} + 2 q^{34} + 4 q^{35} + q^{36} - 9 q^{37} - q^{38} - 18 q^{39} - 8 q^{40} + 6 q^{41} + 7 q^{42} + 15 q^{43} + 8 q^{44} - q^{45} - 16 q^{46} - 7 q^{47} - 3 q^{48} - 6 q^{49} + 8 q^{50} + q^{51} - 7 q^{52} - 11 q^{53} - 21 q^{54} - 8 q^{55} - 4 q^{56} - 17 q^{57} - 5 q^{58} - 11 q^{59} + 3 q^{60} - 22 q^{61} - 22 q^{62} + q^{63} + 8 q^{64} + 7 q^{65} - 3 q^{66} + 21 q^{67} + 2 q^{68} + q^{69} + 4 q^{70} - 28 q^{71} + q^{72} - 8 q^{73} - 9 q^{74} - 3 q^{75} - q^{76} - 4 q^{77} - 18 q^{78} - 28 q^{79} - 8 q^{80} + 12 q^{81} + 6 q^{82} - 5 q^{83} + 7 q^{84} - 2 q^{85} + 15 q^{86} + 36 q^{87} + 8 q^{88} - 17 q^{89} - q^{90} - 29 q^{91} - 16 q^{92} + 42 q^{93} - 7 q^{94} + q^{95} - 3 q^{96} + q^{97} - 6 q^{98} + q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 3 * q^3 + 8 * q^4 - 8 * q^5 - 3 * q^6 - 4 * q^7 + 8 * q^8 + q^9 - 8 * q^10 + 8 * q^11 - 3 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 8 * q^16 + 2 * q^17 + q^18 - q^19 - 8 * q^20 + 7 * q^21 + 8 * q^22 - 16 * q^23 - 3 * q^24 + 8 * q^25 - 7 * q^26 - 21 * q^27 - 4 * q^28 - 5 * q^29 + 3 * q^30 - 22 * q^31 + 8 * q^32 - 3 * q^33 + 2 * q^34 + 4 * q^35 + q^36 - 9 * q^37 - q^38 - 18 * q^39 - 8 * q^40 + 6 * q^41 + 7 * q^42 + 15 * q^43 + 8 * q^44 - q^45 - 16 * q^46 - 7 * q^47 - 3 * q^48 - 6 * q^49 + 8 * q^50 + q^51 - 7 * q^52 - 11 * q^53 - 21 * q^54 - 8 * q^55 - 4 * q^56 - 17 * q^57 - 5 * q^58 - 11 * q^59 + 3 * q^60 - 22 * q^61 - 22 * q^62 + q^63 + 8 * q^64 + 7 * q^65 - 3 * q^66 + 21 * q^67 + 2 * q^68 + q^69 + 4 * q^70 - 28 * q^71 + q^72 - 8 * q^73 - 9 * q^74 - 3 * q^75 - q^76 - 4 * q^77 - 18 * q^78 - 28 * q^79 - 8 * q^80 + 12 * q^81 + 6 * q^82 - 5 * q^83 + 7 * q^84 - 2 * q^85 + 15 * q^86 + 36 * q^87 + 8 * q^88 - 17 * q^89 - q^90 - 29 * q^91 - 16 * q^92 + 42 * q^93 - 7 * q^94 + q^95 - 3 * q^96 + q^97 - 6 * q^98 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.29392	0.747047	0.373524	−	0.927621i	$-0.378150\pi$
$3$	1.29392	0.747047	0.373524	+	0.927621i	$0.378150\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.29392	0.528242
$7$	−0.0375670	−0.0141990	−0.00709950	−	0.999975i	$-0.502260\pi$
$7$	−0.0375670	−0.0141990	−0.00709950	+	0.999975i	$0.502260\pi$
$8$	1.00000	0.353553
$9$	−1.32576	−0.441921
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.29392	0.373524
$13$	−4.37567	−1.21359	−0.606797	−	0.794857i	$-0.707546\pi$
$13$	−4.37567	−1.21359	−0.606797	+	0.794857i	$0.707546\pi$
$14$	−0.0375670	−0.0100402
$15$	−1.29392	−0.334090
$16$	1.00000	0.250000
$17$	2.78462	0.675371	0.337685	−	0.941259i	$-0.390356\pi$
$17$	2.78462	0.675371	0.337685	+	0.941259i	$0.390356\pi$
$18$	−1.32576	−0.312485
$19$	−2.93312	−0.672905	−0.336452	−	0.941701i	$-0.609227\pi$
$19$	−2.93312	−0.672905	−0.336452	+	0.941701i	$0.609227\pi$
$20$	−1.00000	−0.223607
$21$	−0.0486088	−0.0106073
$22$	1.00000	0.213201
$23$	3.64937	0.760947	0.380473	−	0.924792i	$-0.375761\pi$
$23$	3.64937	0.760947	0.380473	+	0.924792i	$0.375761\pi$
$24$	1.29392	0.264121
$25$	1.00000	0.200000
$26$	−4.37567	−0.858140
$27$	−5.59721	−1.07718
$28$	−0.0375670	−0.00709950
$29$	−4.56552	−0.847797	−0.423898	−	0.905710i	$-0.639339\pi$
$29$	−4.56552	−0.847797	−0.423898	+	0.905710i	$0.639339\pi$
$30$	−1.29392	−0.236237
$31$	8.26949	1.48524	0.742622	−	0.669711i	$-0.233582\pi$
$31$	8.26949	1.48524	0.742622	+	0.669711i	$0.233582\pi$
$32$	1.00000	0.176777
$33$	1.29392	0.225243
$34$	2.78462	0.477559
$35$	0.0375670	0.00634998
$36$	−1.32576	−0.220960
$37$	−4.22468	−0.694532	−0.347266	−	0.937767i	$-0.612890\pi$
$37$	−4.22468	−0.694532	−0.347266	+	0.937767i	$0.612890\pi$
$38$	−2.93312	−0.475815
$39$	−5.66179	−0.906611
$40$	−1.00000	−0.158114
$41$	−3.97221	−0.620355	−0.310177	−	0.950679i	$-0.600388\pi$
$41$	−3.97221	−0.620355	−0.310177	+	0.950679i	$0.600388\pi$
$42$	−0.0486088	−0.00750051
$43$	−11.2218	−1.71130	−0.855651	−	0.517553i	$-0.826843\pi$
$43$	−11.2218	−1.71130	−0.855651	+	0.517553i	$0.826843\pi$
$44$	1.00000	0.150756
$45$	1.32576	0.197633
$46$	3.64937	0.538071
$47$	−0.238525	−0.0347925	−0.0173962	−	0.999849i	$-0.505538\pi$
$47$	−0.238525	−0.0347925	−0.0173962	+	0.999849i	$0.505538\pi$
$48$	1.29392	0.186762
$49$	−6.99859	−0.999798
$50$	1.00000	0.141421
$51$	3.60309	0.504534
$52$	−4.37567	−0.606797
$53$	−10.4176	−1.43096	−0.715481	−	0.698632i	$-0.753792\pi$
$53$	−10.4176	−1.43096	−0.715481	+	0.698632i	$0.753792\pi$
$54$	−5.59721	−0.761683
$55$	−1.00000	−0.134840
$56$	−0.0375670	−0.00502010
$57$	−3.79524	−0.502691
$58$	−4.56552	−0.599483
$59$	−1.92118	−0.250116	−0.125058	−	0.992149i	$-0.539912\pi$
$59$	−1.92118	−0.250116	−0.125058	+	0.992149i	$0.539912\pi$
$60$	−1.29392	−0.167045
$61$	9.11616	1.16720	0.583602	−	0.812040i	$-0.301643\pi$
$61$	9.11616	1.16720	0.583602	+	0.812040i	$0.301643\pi$
$62$	8.26949	1.05023
$63$	0.0498049	0.00627483
$64$	1.00000	0.125000
$65$	4.37567	0.542735
$66$	1.29392	0.159271
$67$	9.42930	1.15197	0.575986	−	0.817460i	$-0.304618\pi$
$67$	9.42930	1.15197	0.575986	+	0.817460i	$0.304618\pi$
$68$	2.78462	0.337685
$69$	4.72201	0.568463
$70$	0.0375670	0.00449012
$71$	−9.71798	−1.15331	−0.576656	−	0.816987i	$-0.695643\pi$
$71$	−9.71798	−1.15331	−0.576656	+	0.816987i	$0.695643\pi$
$72$	−1.32576	−0.156243
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	−4.22468	−0.491109
$75$	1.29392	0.149409
$76$	−2.93312	−0.336452
$77$	−0.0375670	−0.00428116
$78$	−5.66179	−0.641071
$79$	−5.80976	−0.653649	−0.326824	−	0.945085i	$-0.605979\pi$
$79$	−5.80976	−0.653649	−0.326824	+	0.945085i	$0.605979\pi$
$80$	−1.00000	−0.111803
$81$	−3.26507	−0.362786
$82$	−3.97221	−0.438657
$83$	8.90411	0.977353	0.488677	−	0.872465i	$-0.337480\pi$
$83$	8.90411	0.977353	0.488677	+	0.872465i	$0.337480\pi$
$84$	−0.0486088	−0.00530366
$85$	−2.78462	−0.302035
$86$	−11.2218	−1.21007
$87$	−5.90744	−0.633344
$88$	1.00000	0.106600
$89$	−12.2019	−1.29340	−0.646700	−	0.762745i	$-0.723851\pi$
$89$	−12.2019	−1.29340	−0.646700	+	0.762745i	$0.723851\pi$
$90$	1.32576	0.139748
$91$	0.164381	0.0172318
$92$	3.64937	0.380473
$93$	10.7001	1.10955
$94$	−0.238525	−0.0246020
$95$	2.93312	0.300932
$96$	1.29392	0.132061
$97$	−4.35126	−0.441803	−0.220902	−	0.975296i	$-0.570900\pi$
$97$	−4.35126	−0.441803	−0.220902	+	0.975296i	$0.570900\pi$
$98$	−6.99859	−0.706964
$99$	−1.32576	−0.133244
$100$	1.00000	0.100000
$101$	−8.29740	−0.825623	−0.412811	−	0.910817i	$-0.635453\pi$
$101$	−8.29740	−0.825623	−0.412811	+	0.910817i	$0.635453\pi$
$102$	3.60309	0.356759
$103$	−5.30554	−0.522770	−0.261385	−	0.965235i	$-0.584179\pi$
$103$	−5.30554	−0.522770	−0.261385	+	0.965235i	$0.584179\pi$
$104$	−4.37567	−0.429070
$105$	0.0486088	0.00474374
$106$	−10.4176	−1.01184
$107$	10.3803	1.00350	0.501752	−	0.865012i	$-0.332689\pi$
$107$	10.3803	1.00350	0.501752	+	0.865012i	$0.332689\pi$
$108$	−5.59721	−0.538591
$109$	−2.95033	−0.282590	−0.141295	−	0.989968i	$-0.545127\pi$
$109$	−2.95033	−0.282590	−0.141295	+	0.989968i	$0.545127\pi$
$110$	−1.00000	−0.0953463
$111$	−5.46641	−0.518848
$112$	−0.0375670	−0.00354975
$113$	4.53964	0.427053	0.213527	−	0.976937i	$-0.431505\pi$
$113$	4.53964	0.427053	0.213527	+	0.976937i	$0.431505\pi$
$114$	−3.79524	−0.355457
$115$	−3.64937	−0.340306
$116$	−4.56552	−0.423898
$117$	5.80110	0.536312
$118$	−1.92118	−0.176859
$119$	−0.104610	−0.00958959
$120$	−1.29392	−0.118119
$121$	1.00000	0.0909091
$122$	9.11616	0.825339
$123$	−5.13973	−0.463434
$124$	8.26949	0.742622
$125$	−1.00000	−0.0894427
$126$	0.0498049	0.00443697
$127$	5.20063	0.461481	0.230741	−	0.973015i	$-0.425885\pi$
$127$	5.20063	0.461481	0.230741	+	0.973015i	$0.425885\pi$
$128$	1.00000	0.0883883
$129$	−14.5201	−1.27842
$130$	4.37567	0.383772
$131$	0.501646	0.0438290	0.0219145	−	0.999760i	$-0.493024\pi$
$131$	0.501646	0.0438290	0.0219145	+	0.999760i	$0.493024\pi$
$132$	1.29392	0.112622
$133$	0.110189	0.00955457
$134$	9.42930	0.814567
$135$	5.59721	0.481731
$136$	2.78462	0.238780
$137$	−0.577059	−0.0493014	−0.0246507	−	0.999696i	$-0.507847\pi$
$137$	−0.577059	−0.0493014	−0.0246507	+	0.999696i	$0.507847\pi$
$138$	4.72201	0.401964
$139$	15.7189	1.33326	0.666628	−	0.745391i	$-0.267737\pi$
$139$	15.7189	1.33326	0.666628	+	0.745391i	$0.267737\pi$
$140$	0.0375670	0.00317499
$141$	−0.308633	−0.0259916
$142$	−9.71798	−0.815515
$143$	−4.37567	−0.365912
$144$	−1.32576	−0.110480
$145$	4.56552	0.379146
$146$	−1.00000	−0.0827606
$147$	−9.05564	−0.746897
$148$	−4.22468	−0.347266
$149$	−11.8340	−0.969477	−0.484739	−	0.874659i	$-0.661085\pi$
$149$	−11.8340	−0.969477	−0.484739	+	0.874659i	$0.661085\pi$
$150$	1.29392	0.105648
$151$	3.92710	0.319583	0.159791	−	0.987151i	$-0.448918\pi$
$151$	3.92710	0.319583	0.159791	+	0.987151i	$0.448918\pi$
$152$	−2.93312	−0.237908
$153$	−3.69175	−0.298460
$154$	−0.0375670	−0.00302724
$155$	−8.26949	−0.664221
$156$	−5.66179	−0.453306
$157$	−18.8763	−1.50649	−0.753245	−	0.657741i	$-0.771512\pi$
$157$	−18.8763	−1.50649	−0.753245	+	0.657741i	$0.771512\pi$
$158$	−5.80976	−0.462200
$159$	−13.4795	−1.06900
$160$	−1.00000	−0.0790569
$161$	−0.137096	−0.0108047
$162$	−3.26507	−0.256528
$163$	−19.3153	−1.51289	−0.756447	−	0.654055i	$-0.773066\pi$
$163$	−19.3153	−1.51289	−0.756447	+	0.654055i	$0.773066\pi$
$164$	−3.97221	−0.310177
$165$	−1.29392	−0.100732
$166$	8.90411	0.691093
$167$	−16.2899	−1.26055	−0.630275	−	0.776372i	$-0.717058\pi$
$167$	−16.2899	−1.26055	−0.630275	+	0.776372i	$0.717058\pi$
$168$	−0.0486088	−0.00375025
$169$	6.14650	0.472808
$170$	−2.78462	−0.213571
$171$	3.88862	0.297370
$172$	−11.2218	−0.855651
$173$	6.21651	0.472632	0.236316	−	0.971676i	$-0.424060\pi$
$173$	6.21651	0.472632	0.236316	+	0.971676i	$0.424060\pi$
$174$	−5.90744	−0.447842
$175$	−0.0375670	−0.00283980
$176$	1.00000	0.0753778
$177$	−2.48586	−0.186848
$178$	−12.2019	−0.914572
$179$	−18.8050	−1.40555	−0.702775	−	0.711412i	$-0.748056\pi$
$179$	−18.8050	−1.40555	−0.702775	+	0.711412i	$0.748056\pi$
$180$	1.32576	0.0988164
$181$	−8.40672	−0.624867	−0.312433	−	0.949940i	$-0.601144\pi$
$181$	−8.40672	−0.624867	−0.312433	+	0.949940i	$0.601144\pi$
$182$	0.164381	0.0121847
$183$	11.7956	0.871957
$184$	3.64937	0.269035
$185$	4.22468	0.310604
$186$	10.7001	0.784568
$187$	2.78462	0.203632
$188$	−0.238525	−0.0173962
$189$	0.210270	0.0152949
$190$	2.93312	0.212791
$191$	3.46476	0.250702	0.125351	−	0.992112i	$-0.459994\pi$
$191$	3.46476	0.250702	0.125351	+	0.992112i	$0.459994\pi$
$192$	1.29392	0.0933809
$193$	−23.0209	−1.65708	−0.828541	−	0.559928i	$-0.810829\pi$
$193$	−23.0209	−1.65708	−0.828541	+	0.559928i	$0.810829\pi$
$194$	−4.35126	−0.312402
$195$	5.66179	0.405449
$196$	−6.99859	−0.499899
$197$	−7.48275	−0.533124	−0.266562	−	0.963818i	$-0.585888\pi$
$197$	−7.48275	−0.533124	−0.266562	+	0.963818i	$0.585888\pi$
$198$	−1.32576	−0.0942178
$199$	−2.36522	−0.167666	−0.0838331	−	0.996480i	$-0.526716\pi$
$199$	−2.36522	−0.167666	−0.0838331	+	0.996480i	$0.526716\pi$
$200$	1.00000	0.0707107
$201$	12.2008	0.860577
$202$	−8.29740	−0.583803
$203$	0.171513	0.0120379
$204$	3.60309	0.252267
$205$	3.97221	0.277431
$206$	−5.30554	−0.369654
$207$	−4.83820	−0.336278
$208$	−4.37567	−0.303398
$209$	−2.93312	−0.202888
$210$	0.0486088	0.00335433
$211$	−1.58760	−0.109295	−0.0546475	−	0.998506i	$-0.517404\pi$
$211$	−1.58760	−0.109295	−0.0546475	+	0.998506i	$0.517404\pi$
$212$	−10.4176	−0.715481
$213$	−12.5743	−0.861578
$214$	10.3803	0.709584
$215$	11.2218	0.765318
$216$	−5.59721	−0.380842
$217$	−0.310660	−0.0210890
$218$	−2.95033	−0.199822
$219$	−1.29392	−0.0874353
$220$	−1.00000	−0.0674200
$221$	−12.1846	−0.819625
$222$	−5.46641	−0.366881
$223$	3.69357	0.247340	0.123670	−	0.992323i	$-0.460534\pi$
$223$	3.69357	0.247340	0.123670	+	0.992323i	$0.460534\pi$
$224$	−0.0375670	−0.00251005
$225$	−1.32576	−0.0883841
$226$	4.53964	0.301972
$227$	−9.84773	−0.653617	−0.326808	−	0.945091i	$-0.605973\pi$
$227$	−9.84773	−0.653617	−0.326808	+	0.945091i	$0.605973\pi$
$228$	−3.79524	−0.251346
$229$	7.01516	0.463575	0.231787	−	0.972766i	$-0.425543\pi$
$229$	7.01516	0.463575	0.231787	+	0.972766i	$0.425543\pi$
$230$	−3.64937	−0.240632
$231$	−0.0486088	−0.00319823
$232$	−4.56552	−0.299741
$233$	−18.9665	−1.24254	−0.621270	−	0.783597i	$-0.713383\pi$
$233$	−18.9665	−1.24254	−0.621270	+	0.783597i	$0.713383\pi$
$234$	5.80110	0.379230
$235$	0.238525	0.0155597
$236$	−1.92118	−0.125058
$237$	−7.51738	−0.488307
$238$	−0.104610	−0.00678086
$239$	22.3306	1.44445	0.722223	−	0.691660i	$-0.243120\pi$
$239$	22.3306	1.44445	0.722223	+	0.691660i	$0.243120\pi$
$240$	−1.29392	−0.0835224
$241$	1.13786	0.0732957	0.0366479	−	0.999328i	$-0.488332\pi$
$241$	1.13786	0.0732957	0.0366479	+	0.999328i	$0.488332\pi$
$242$	1.00000	0.0642824
$243$	12.5669	0.806165
$244$	9.11616	0.583602
$245$	6.99859	0.447123
$246$	−5.13973	−0.327697
$247$	12.8344	0.816632
$248$	8.26949	0.525113
$249$	11.5212	0.730129
$250$	−1.00000	−0.0632456
$251$	4.73228	0.298699	0.149349	−	0.988784i	$-0.452282\pi$
$251$	4.73228	0.298699	0.149349	+	0.988784i	$0.452282\pi$
$252$	0.0498049	0.00313741
$253$	3.64937	0.229434
$254$	5.20063	0.326316
$255$	−3.60309	−0.225634
$256$	1.00000	0.0625000
$257$	4.54212	0.283330	0.141665	−	0.989915i	$-0.454754\pi$
$257$	4.54212	0.283330	0.141665	+	0.989915i	$0.454754\pi$
$258$	−14.5201	−0.903982
$259$	0.158708	0.00986166
$260$	4.37567	0.271368
$261$	6.05280	0.374659
$262$	0.501646	0.0309918
$263$	18.2279	1.12398	0.561992	−	0.827143i	$-0.310035\pi$
$263$	18.2279	1.12398	0.561992	+	0.827143i	$0.310035\pi$
$264$	1.29392	0.0796355
$265$	10.4176	0.639945
$266$	0.110189	0.00675610
$267$	−15.7883	−0.966231
$268$	9.42930	0.575986
$269$	8.26288	0.503797	0.251898	−	0.967754i	$-0.418945\pi$
$269$	8.26288	0.503797	0.251898	+	0.967754i	$0.418945\pi$
$270$	5.59721	0.340635
$271$	−1.49092	−0.0905667	−0.0452834	−	0.998974i	$-0.514419\pi$
$271$	−1.49092	−0.0905667	−0.0452834	+	0.998974i	$0.514419\pi$
$272$	2.78462	0.168843
$273$	0.212696	0.0128730
$274$	−0.577059	−0.0348614
$275$	1.00000	0.0603023
$276$	4.72201	0.284232
$277$	14.7880	0.888527	0.444264	−	0.895896i	$-0.353465\pi$
$277$	14.7880	0.888527	0.444264	+	0.895896i	$0.353465\pi$
$278$	15.7189	0.942754
$279$	−10.9634	−0.656360
$280$	0.0375670	0.00224506
$281$	10.5072	0.626807	0.313403	−	0.949620i	$-0.398531\pi$
$281$	10.5072	0.626807	0.313403	+	0.949620i	$0.398531\pi$
$282$	−0.308633	−0.0183788
$283$	12.3491	0.734075	0.367038	−	0.930206i	$-0.380372\pi$
$283$	12.3491	0.734075	0.367038	+	0.930206i	$0.380372\pi$
$284$	−9.71798	−0.576656
$285$	3.79524	0.224810
$286$	−4.37567	−0.258739
$287$	0.149224	0.00880841
$288$	−1.32576	−0.0781213
$289$	−9.24587	−0.543875
$290$	4.56552	0.268097
$291$	−5.63019	−0.330048
$292$	−1.00000	−0.0585206
$293$	10.6852	0.624236	0.312118	−	0.950043i	$-0.398962\pi$
$293$	10.6852	0.624236	0.312118	+	0.950043i	$0.398962\pi$
$294$	−9.05564	−0.528136
$295$	1.92118	0.111855
$296$	−4.22468	−0.245554
$297$	−5.59721	−0.324783
$298$	−11.8340	−0.685524
$299$	−15.9685	−0.923480
$300$	1.29392	0.0747047
$301$	0.421568	0.0242988
$302$	3.92710	0.225979
$303$	−10.7362	−0.616779
$304$	−2.93312	−0.168226
$305$	−9.11616	−0.521990
$306$	−3.69175	−0.211043
$307$	14.2044	0.810687	0.405343	−	0.914164i	$-0.367152\pi$
$307$	14.2044	0.810687	0.405343	+	0.914164i	$0.367152\pi$
$308$	−0.0375670	−0.00214058
$309$	−6.86496	−0.390534
$310$	−8.26949	−0.469675
$311$	−18.1357	−1.02838	−0.514190	−	0.857676i	$-0.671907\pi$
$311$	−18.1357	−1.02838	−0.514190	+	0.857676i	$0.671907\pi$
$312$	−5.66179	−0.320535
$313$	−2.83328	−0.160146	−0.0800731	−	0.996789i	$-0.525515\pi$
$313$	−2.83328	−0.160146	−0.0800731	+	0.996789i	$0.525515\pi$
$314$	−18.8763	−1.06525
$315$	−0.0498049	−0.00280619
$316$	−5.80976	−0.326824
$317$	−19.9288	−1.11931	−0.559656	−	0.828725i	$-0.689067\pi$
$317$	−19.9288	−1.11931	−0.559656	+	0.828725i	$0.689067\pi$
$318$	−13.4795	−0.755894
$319$	−4.56552	−0.255620
$320$	−1.00000	−0.0559017
$321$	13.4313	0.749664
$322$	−0.137096	−0.00764006
$323$	−8.16765	−0.454460
$324$	−3.26507	−0.181393
$325$	−4.37567	−0.242719
$326$	−19.3153	−1.06978
$327$	−3.81750	−0.211108
$328$	−3.97221	−0.219328
$329$	0.00896068	0.000494018 0
$330$	−1.29392	−0.0712281
$331$	17.2233	0.946680	0.473340	−	0.880880i	$-0.343048\pi$
$331$	17.2233	0.946680	0.473340	+	0.880880i	$0.343048\pi$
$332$	8.90411	0.488677
$333$	5.60091	0.306928
$334$	−16.2899	−0.891344
$335$	−9.42930	−0.515178
$336$	−0.0486088	−0.00265183
$337$	28.7618	1.56675	0.783377	−	0.621546i	$-0.213495\pi$
$337$	28.7618	1.56675	0.783377	+	0.621546i	$0.213495\pi$
$338$	6.14650	0.334326
$339$	5.87394	0.319029
$340$	−2.78462	−0.151017
$341$	8.26949	0.447818
$342$	3.88862	0.210273
$343$	0.525885	0.0283951
$344$	−11.2218	−0.605037
$345$	−4.72201	−0.254224
$346$	6.21651	0.334202
$347$	19.4086	1.04191	0.520953	−	0.853585i	$-0.325577\pi$
$347$	19.4086	1.04191	0.520953	+	0.853585i	$0.325577\pi$
$348$	−5.90744	−0.316672
$349$	−19.9532	−1.06807	−0.534036	−	0.845462i	$-0.679325\pi$
$349$	−19.9532	−1.06807	−0.534036	+	0.845462i	$0.679325\pi$
$350$	−0.0375670	−0.00200804
$351$	24.4915	1.30726
$352$	1.00000	0.0533002
$353$	19.9143	1.05993	0.529966	−	0.848019i	$-0.322205\pi$
$353$	19.9143	1.05993	0.529966	+	0.848019i	$0.322205\pi$
$354$	−2.48586	−0.132122
$355$	9.71798	0.515777
$356$	−12.2019	−0.646700
$357$	−0.135357	−0.00716387
$358$	−18.8050	−0.993875
$359$	−6.95989	−0.367329	−0.183664	−	0.982989i	$-0.558796\pi$
$359$	−6.95989	−0.367329	−0.183664	+	0.982989i	$0.558796\pi$
$360$	1.32576	0.0698738
$361$	−10.3968	−0.547199
$362$	−8.40672	−0.441848
$363$	1.29392	0.0679134
$364$	0.164381	0.00861590
$365$	1.00000	0.0523424
$366$	11.7956	0.616567
$367$	25.3372	1.32259	0.661296	−	0.750125i	$-0.270007\pi$
$367$	25.3372	1.32259	0.661296	+	0.750125i	$0.270007\pi$
$368$	3.64937	0.190237
$369$	5.26620	0.274147
$370$	4.22468	0.219630
$371$	0.391357	0.0203182
$372$	10.7001	0.554773
$373$	−27.4327	−1.42041	−0.710206	−	0.703994i	$-0.751398\pi$
$373$	−27.4327	−1.42041	−0.710206	+	0.703994i	$0.751398\pi$
$374$	2.78462	0.143989
$375$	−1.29392	−0.0668179
$376$	−0.238525	−0.0123010
$377$	19.9772	1.02888
$378$	0.210270	0.0108151
$379$	23.1843	1.19090	0.595449	−	0.803393i	$-0.296974\pi$
$379$	23.1843	1.19090	0.595449	+	0.803393i	$0.296974\pi$
$380$	2.93312	0.150466
$381$	6.72921	0.344748
$382$	3.46476	0.177273
$383$	−1.47995	−0.0756217	−0.0378109	−	0.999285i	$-0.512038\pi$
$383$	−1.47995	−0.0756217	−0.0378109	+	0.999285i	$0.512038\pi$
$384$	1.29392	0.0660303
$385$	0.0375670	0.00191459
$386$	−23.0209	−1.17173
$387$	14.8774	0.756260
$388$	−4.35126	−0.220902
$389$	−7.54473	−0.382533	−0.191266	−	0.981538i	$-0.561259\pi$
$389$	−7.54473	−0.382533	−0.191266	+	0.981538i	$0.561259\pi$
$390$	5.66179	0.286696
$391$	10.1621	0.513921
$392$	−6.99859	−0.353482
$393$	0.649092	0.0327423
$394$	−7.48275	−0.376975
$395$	5.80976	0.292321
$396$	−1.32576	−0.0666220
$397$	−29.6376	−1.48747	−0.743733	−	0.668476i	$-0.766947\pi$
$397$	−29.6376	−1.48747	−0.743733	+	0.668476i	$0.766947\pi$
$398$	−2.36522	−0.118558
$399$	0.142576	0.00713772
$400$	1.00000	0.0500000
$401$	−26.2177	−1.30925	−0.654625	−	0.755954i	$-0.727173\pi$
$401$	−26.2177	−1.30925	−0.654625	+	0.755954i	$0.727173\pi$
$402$	12.2008	0.608520
$403$	−36.1846	−1.80248
$404$	−8.29740	−0.412811
$405$	3.26507	0.162243
$406$	0.171513	0.00851205
$407$	−4.22468	−0.209409
$408$	3.60309	0.178380
$409$	−3.80388	−0.188090	−0.0940448	−	0.995568i	$-0.529980\pi$
$409$	−3.80388	−0.188090	−0.0940448	+	0.995568i	$0.529980\pi$
$410$	3.97221	0.196173
$411$	−0.746670	−0.0368305
$412$	−5.30554	−0.261385
$413$	0.0721729	0.00355140
$414$	−4.83820	−0.237784
$415$	−8.90411	−0.437086
$416$	−4.37567	−0.214535
$417$	20.3390	0.996005
$418$	−2.93312	−0.143464
$419$	−10.6845	−0.521973	−0.260986	−	0.965343i	$-0.584048\pi$
$419$	−10.6845	−0.521973	−0.260986	+	0.965343i	$0.584048\pi$
$420$	0.0486088	0.00237187
$421$	15.0733	0.734626	0.367313	−	0.930097i	$-0.380278\pi$
$421$	15.0733	0.734626	0.367313	+	0.930097i	$0.380278\pi$
$422$	−1.58760	−0.0772833
$423$	0.316227	0.0153755
$424$	−10.4176	−0.505921
$425$	2.78462	0.135074
$426$	−12.5743	−0.609228
$427$	−0.342467	−0.0165731
$428$	10.3803	0.501752
$429$	−5.66179	−0.273354
$430$	11.2218	0.541161
$431$	−9.39539	−0.452560	−0.226280	−	0.974062i	$-0.572656\pi$
$431$	−9.39539	−0.452560	−0.226280	+	0.974062i	$0.572656\pi$
$432$	−5.59721	−0.269296
$433$	−11.1929	−0.537899	−0.268949	−	0.963154i	$-0.586676\pi$
$433$	−11.1929	−0.537899	−0.268949	+	0.963154i	$0.586676\pi$
$434$	−0.310660	−0.0149122
$435$	5.90744	0.283240
$436$	−2.95033	−0.141295
$437$	−10.7041	−0.512045
$438$	−1.29392	−0.0618261
$439$	21.3980	1.02127	0.510635	−	0.859797i	$-0.329410\pi$
$439$	21.3980	1.02127	0.510635	+	0.859797i	$0.329410\pi$
$440$	−1.00000	−0.0476731
$441$	9.27846	0.441831
$442$	−12.1846	−0.579562
$443$	−16.3701	−0.777766	−0.388883	−	0.921287i	$-0.627139\pi$
$443$	−16.3701	−0.777766	−0.388883	+	0.921287i	$0.627139\pi$
$444$	−5.46641	−0.259424
$445$	12.2019	0.578426
$446$	3.69357	0.174896
$447$	−15.3123	−0.724245
$448$	−0.0375670	−0.00177487
$449$	36.3135	1.71374	0.856870	−	0.515532i	$-0.172406\pi$
$449$	36.3135	1.71374	0.856870	+	0.515532i	$0.172406\pi$
$450$	−1.32576	−0.0624970
$451$	−3.97221	−0.187044
$452$	4.53964	0.213527
$453$	5.08137	0.238743
$454$	−9.84773	−0.462177
$455$	−0.164381	−0.00770630
$456$	−3.79524	−0.177728
$457$	17.2266	0.805827	0.402914	−	0.915238i	$-0.367998\pi$
$457$	17.2266	0.805827	0.402914	+	0.915238i	$0.367998\pi$
$458$	7.01516	0.327797
$459$	−15.5861	−0.727497
$460$	−3.64937	−0.170153
$461$	7.75109	0.361004	0.180502	−	0.983575i	$-0.442228\pi$
$461$	7.75109	0.361004	0.180502	+	0.983575i	$0.442228\pi$
$462$	−0.0486088	−0.00226149
$463$	35.8775	1.66737	0.833685	−	0.552240i	$-0.186227\pi$
$463$	35.8775	1.66737	0.833685	+	0.552240i	$0.186227\pi$
$464$	−4.56552	−0.211949
$465$	−10.7001	−0.496204
$466$	−18.9665	−0.878608
$467$	−4.39568	−0.203408	−0.101704	−	0.994815i	$-0.532429\pi$
$467$	−4.39568	−0.203408	−0.101704	+	0.994815i	$0.532429\pi$
$468$	5.80110	0.268156
$469$	−0.354231	−0.0163568
$470$	0.238525	0.0110023
$471$	−24.4244	−1.12542
$472$	−1.92118	−0.0884294
$473$	−11.2218	−0.515977
$474$	−7.51738	−0.345285
$475$	−2.93312	−0.134581
$476$	−0.104610	−0.00479479
$477$	13.8112	0.632371
$478$	22.3306	1.02138
$479$	6.22957	0.284636	0.142318	−	0.989821i	$-0.454544\pi$
$479$	6.22957	0.284636	0.142318	+	0.989821i	$0.454544\pi$
$480$	−1.29392	−0.0590593
$481$	18.4858	0.842880
$482$	1.13786	0.0518279
$483$	−0.177392	−0.00807161
$484$	1.00000	0.0454545
$485$	4.35126	0.197580
$486$	12.5669	0.570045
$487$	13.7353	0.622405	0.311203	−	0.950344i	$-0.399268\pi$
$487$	13.7353	0.622405	0.311203	+	0.950344i	$0.399268\pi$
$488$	9.11616	0.412669
$489$	−24.9926	−1.13020
$490$	6.99859	0.316164
$491$	7.73063	0.348878	0.174439	−	0.984668i	$-0.444189\pi$
$491$	7.73063	0.348878	0.174439	+	0.984668i	$0.444189\pi$
$492$	−5.13973	−0.231717
$493$	−12.7133	−0.572577
$494$	12.8344	0.577446
$495$	1.32576	0.0595886
$496$	8.26949	0.371311
$497$	0.365075	0.0163759
$498$	11.5212	0.516279
$499$	7.80995	0.349622	0.174811	−	0.984602i	$-0.444069\pi$
$499$	7.80995	0.349622	0.174811	+	0.984602i	$0.444069\pi$
$500$	−1.00000	−0.0447214
$501$	−21.0779	−0.941691
$502$	4.73228	0.211212
$503$	37.3917	1.66721	0.833606	−	0.552360i	$-0.186272\pi$
$503$	37.3917	1.66721	0.833606	+	0.552360i	$0.186272\pi$
$504$	0.0498049	0.00221849
$505$	8.29740	0.369230
$506$	3.64937	0.162234
$507$	7.95311	0.353210
$508$	5.20063	0.230741
$509$	−10.1088	−0.448064	−0.224032	−	0.974582i	$-0.571922\pi$
$509$	−10.1088	−0.448064	−0.224032	+	0.974582i	$0.571922\pi$
$510$	−3.60309	−0.159548
$511$	0.0375670	0.00166187
$512$	1.00000	0.0441942
$513$	16.4173	0.724841
$514$	4.54212	0.200345
$515$	5.30554	0.233790
$516$	−14.5201	−0.639212
$517$	−0.238525	−0.0104903
$518$	0.158708	0.00697325
$519$	8.04369	0.353079
$520$	4.37567	0.191886
$521$	−38.2059	−1.67383	−0.836916	−	0.547332i	$-0.815644\pi$
$521$	−38.2059	−1.67383	−0.836916	+	0.547332i	$0.815644\pi$
$522$	6.05280	0.264924
$523$	10.2621	0.448730	0.224365	−	0.974505i	$-0.427969\pi$
$523$	10.2621	0.448730	0.224365	+	0.974505i	$0.427969\pi$
$524$	0.501646	0.0219145
$525$	−0.0486088	−0.00212146
$526$	18.2279	0.794776
$527$	23.0274	1.00309
$528$	1.29392	0.0563108
$529$	−9.68208	−0.420960
$530$	10.4176	0.452510
$531$	2.54702	0.110531
$532$	0.110189	0.00477729
$533$	17.3811	0.752858
$534$	−15.7883	−0.683228
$535$	−10.3803	−0.448780
$536$	9.42930	0.407284
$537$	−24.3322	−1.05001
$538$	8.26288	0.356238
$539$	−6.99859	−0.301451
$540$	5.59721	0.240865
$541$	22.7768	0.979250	0.489625	−	0.871933i	$-0.337134\pi$
$541$	22.7768	0.979250	0.489625	+	0.871933i	$0.337134\pi$
$542$	−1.49092	−0.0640403
$543$	−10.8777	−0.466805
$544$	2.78462	0.119390
$545$	2.95033	0.126378
$546$	0.212696	0.00910256
$547$	−39.1362	−1.67334	−0.836672	−	0.547704i	$-0.815502\pi$
$547$	−39.1362	−1.67334	−0.836672	+	0.547704i	$0.815502\pi$
$548$	−0.577059	−0.0246507
$549$	−12.0859	−0.515812
$550$	1.00000	0.0426401
$551$	13.3912	0.570486
$552$	4.72201	0.200982
$553$	0.218255	0.00928116
$554$	14.7880	0.628284
$555$	5.46641	0.232036
$556$	15.7189	0.666628
$557$	−12.5489	−0.531714	−0.265857	−	0.964013i	$-0.585655\pi$
$557$	−12.5489	−0.531714	−0.265857	+	0.964013i	$0.585655\pi$
$558$	−10.9634	−0.464116
$559$	49.1028	2.07683
$560$	0.0375670	0.00158750
$561$	3.60309	0.152123
$562$	10.5072	0.443219
$563$	−24.7114	−1.04146	−0.520731	−	0.853721i	$-0.674341\pi$
$563$	−24.7114	−1.04146	−0.520731	+	0.853721i	$0.674341\pi$
$564$	−0.308633	−0.0129958
$565$	−4.53964	−0.190984
$566$	12.3491	0.519070
$567$	0.122659	0.00515119
$568$	−9.71798	−0.407757
$569$	16.7510	0.702238	0.351119	−	0.936331i	$-0.385801\pi$
$569$	16.7510	0.702238	0.351119	+	0.936331i	$0.385801\pi$
$570$	3.79524	0.158965
$571$	−31.0479	−1.29931	−0.649657	−	0.760227i	$-0.725088\pi$
$571$	−31.0479	−1.29931	−0.649657	+	0.760227i	$0.725088\pi$
$572$	−4.37567	−0.182956
$573$	4.48314	0.187286
$574$	0.149224	0.00622849
$575$	3.64937	0.152189
$576$	−1.32576	−0.0552401
$577$	36.6477	1.52567	0.762833	−	0.646596i	$-0.223808\pi$
$577$	36.6477	1.52567	0.762833	+	0.646596i	$0.223808\pi$
$578$	−9.24587	−0.384577
$579$	−29.7873	−1.23792
$580$	4.56552	0.189573
$581$	−0.334501	−0.0138774
$582$	−5.63019	−0.233379
$583$	−10.4176	−0.431451
$584$	−1.00000	−0.0413803
$585$	−5.80110	−0.239846
$586$	10.6852	0.441401
$587$	−22.3831	−0.923847	−0.461924	−	0.886920i	$-0.652841\pi$
$587$	−22.3831	−0.923847	−0.461924	+	0.886920i	$0.652841\pi$
$588$	−9.05564	−0.373448
$589$	−24.2554	−0.999427
$590$	1.92118	0.0790936
$591$	−9.68210	−0.398268
$592$	−4.22468	−0.173633
$593$	−2.86416	−0.117617	−0.0588084	−	0.998269i	$-0.518730\pi$
$593$	−2.86416	−0.117617	−0.0588084	+	0.998269i	$0.518730\pi$
$594$	−5.59721	−0.229656
$595$	0.104610	0.00428859
$596$	−11.8340	−0.484739
$597$	−3.06042	−0.125255
$598$	−15.9685	−0.652999
$599$	18.0255	0.736502	0.368251	−	0.929726i	$-0.379957\pi$
$599$	18.0255	0.736502	0.368251	+	0.929726i	$0.379957\pi$
$600$	1.29392	0.0528242
$601$	20.7672	0.847112	0.423556	−	0.905870i	$-0.360782\pi$
$601$	20.7672	0.847112	0.423556	+	0.905870i	$0.360782\pi$
$602$	0.421568	0.0171818
$603$	−12.5010	−0.509080
$604$	3.92710	0.159791
$605$	−1.00000	−0.0406558
$606$	−10.7362	−0.436129
$607$	−14.1932	−0.576086	−0.288043	−	0.957617i	$-0.593005\pi$
$607$	−14.1932	−0.576086	−0.288043	+	0.957617i	$0.593005\pi$
$608$	−2.93312	−0.118954
$609$	0.221925	0.00899285
$610$	−9.11616	−0.369103
$611$	1.04371	0.0422239
$612$	−3.69175	−0.149230
$613$	−42.9100	−1.73312	−0.866559	−	0.499075i	$-0.833673\pi$
$613$	−42.9100	−1.73312	−0.866559	+	0.499075i	$0.833673\pi$
$614$	14.2044	0.573242
$615$	5.13973	0.207254
$616$	−0.0375670	−0.00151362
$617$	18.2397	0.734303	0.367151	−	0.930161i	$-0.380333\pi$
$617$	18.2397	0.734303	0.367151	+	0.930161i	$0.380333\pi$
$618$	−6.86496	−0.276149
$619$	39.8205	1.60052	0.800261	−	0.599652i	$-0.204694\pi$
$619$	39.8205	1.60052	0.800261	+	0.599652i	$0.204694\pi$
$620$	−8.26949	−0.332111
$621$	−20.4263	−0.819679
$622$	−18.1357	−0.727174
$623$	0.458389	0.0183650
$624$	−5.66179	−0.226653
$625$	1.00000	0.0400000
$626$	−2.83328	−0.113240
$627$	−3.79524	−0.151567
$628$	−18.8763	−0.753245
$629$	−11.7641	−0.469067
$630$	−0.0498049	−0.00198428
$631$	−17.4591	−0.695034	−0.347517	−	0.937674i	$-0.612975\pi$
$631$	−17.4591	−0.695034	−0.347517	+	0.937674i	$0.612975\pi$
$632$	−5.80976	−0.231100
$633$	−2.05424	−0.0816486
$634$	−19.9288	−0.791474
$635$	−5.20063	−0.206381
$636$	−13.4795	−0.534498
$637$	30.6235	1.21335
$638$	−4.56552	−0.180751
$639$	12.8837	0.509672
$640$	−1.00000	−0.0395285
$641$	5.12799	0.202543	0.101272	−	0.994859i	$-0.467709\pi$
$641$	5.12799	0.202543	0.101272	+	0.994859i	$0.467709\pi$
$642$	13.4313	0.530093
$643$	17.5154	0.690740	0.345370	−	0.938467i	$-0.387753\pi$
$643$	17.5154	0.690740	0.345370	+	0.938467i	$0.387753\pi$
$644$	−0.137096	−0.00540234
$645$	14.5201	0.571729
$646$	−8.16765	−0.321352
$647$	−13.9774	−0.549509	−0.274755	−	0.961514i	$-0.588597\pi$
$647$	−13.9774	−0.549509	−0.274755	+	0.961514i	$0.588597\pi$
$648$	−3.26507	−0.128264
$649$	−1.92118	−0.0754128
$650$	−4.37567	−0.171628
$651$	−0.401970	−0.0157545
$652$	−19.3153	−0.756447
$653$	−42.5815	−1.66634	−0.833171	−	0.553015i	$-0.813477\pi$
$653$	−42.5815	−1.66634	−0.833171	+	0.553015i	$0.813477\pi$
$654$	−3.81750	−0.149276
$655$	−0.501646	−0.0196009
$656$	−3.97221	−0.155089
$657$	1.32576	0.0517229
$658$	0.00896068	0.000349324 0
$659$	26.4820	1.03159	0.515797	−	0.856711i	$-0.327496\pi$
$659$	26.4820	1.03159	0.515797	+	0.856711i	$0.327496\pi$
$660$	−1.29392	−0.0503659
$661$	−5.32648	−0.207176	−0.103588	−	0.994620i	$-0.533032\pi$
$661$	−5.32648	−0.207176	−0.103588	+	0.994620i	$0.533032\pi$
$662$	17.2233	0.669404
$663$	−15.7659	−0.612299
$664$	8.90411	0.345547
$665$	−0.110189	−0.00427293
$666$	5.60091	0.217031
$667$	−16.6613	−0.645128
$668$	−16.2899	−0.630275
$669$	4.77920	0.184774
$670$	−9.42930	−0.364286
$671$	9.11616	0.351926
$672$	−0.0486088	−0.00187513
$673$	46.2329	1.78215	0.891074	−	0.453859i	$-0.149953\pi$
$673$	46.2329	1.78215	0.891074	+	0.453859i	$0.149953\pi$
$674$	28.7618	1.10786
$675$	−5.59721	−0.215437
$676$	6.14650	0.236404
$677$	24.2837	0.933297	0.466648	−	0.884443i	$-0.345461\pi$
$677$	24.2837	0.933297	0.466648	+	0.884443i	$0.345461\pi$
$678$	5.87394	0.225587
$679$	0.163464	0.00627316
$680$	−2.78462	−0.106785
$681$	−12.7422	−0.488282
$682$	8.26949	0.316655
$683$	16.2831	0.623055	0.311527	−	0.950237i	$-0.399159\pi$
$683$	16.2831	0.623055	0.311527	+	0.950237i	$0.399159\pi$
$684$	3.88862	0.148685
$685$	0.577059	0.0220483
$686$	0.525885	0.0200784
$687$	9.07708	0.346312
$688$	−11.2218	−0.427826
$689$	45.5838	1.73660
$690$	−4.72201	−0.179764
$691$	−9.78749	−0.372334	−0.186167	−	0.982518i	$-0.559606\pi$
$691$	−9.78749	−0.372334	−0.186167	+	0.982518i	$0.559606\pi$
$692$	6.21651	0.236316
$693$	0.0498049	0.00189193
$694$	19.4086	0.736739
$695$	−15.7189	−0.596250
$696$	−5.90744	−0.223921
$697$	−11.0611	−0.418969
$698$	−19.9532	−0.755241
$699$	−24.5413	−0.928235
$700$	−0.0375670	−0.00141990
$701$	14.2978	0.540022	0.270011	−	0.962857i	$-0.412973\pi$
$701$	14.2978	0.540022	0.270011	+	0.962857i	$0.412973\pi$
$702$	24.4915	0.924373
$703$	12.3915	0.467354
$704$	1.00000	0.0376889
$705$	0.308633	0.0116238
$706$	19.9143	0.749485
$707$	0.311709	0.0117230
$708$	−2.48586	−0.0934242
$709$	−38.1953	−1.43445	−0.717227	−	0.696839i	$-0.754589\pi$
$709$	−38.1953	−1.43445	−0.717227	+	0.696839i	$0.754589\pi$
$710$	9.71798	0.364709
$711$	7.70236	0.288861
$712$	−12.2019	−0.457286
$713$	30.1784	1.13019
$714$	−0.135357	−0.00506562
$715$	4.37567	0.163641
$716$	−18.8050	−0.702775
$717$	28.8941	1.07907
$718$	−6.95989	−0.259741
$719$	19.8523	0.740364	0.370182	−	0.928959i	$-0.379295\pi$
$719$	19.8523	0.740364	0.370182	+	0.928959i	$0.379295\pi$
$720$	1.32576	0.0494082
$721$	0.199313	0.00742281
$722$	−10.3968	−0.386928
$723$	1.47230	0.0547554
$724$	−8.40672	−0.312433
$725$	−4.56552	−0.169559
$726$	1.29392	0.0480220
$727$	6.34027	0.235148	0.117574	−	0.993064i	$-0.462488\pi$
$727$	6.34027	0.235148	0.117574	+	0.993064i	$0.462488\pi$
$728$	0.164381	0.00609236
$729$	26.0558	0.965029
$730$	1.00000	0.0370117
$731$	−31.2484	−1.15576
$732$	11.7956	0.435979
$733$	−43.5176	−1.60736	−0.803679	−	0.595063i	$-0.797127\pi$
$733$	−43.5176	−1.60736	−0.803679	+	0.595063i	$0.797127\pi$
$734$	25.3372	0.935214
$735$	9.05564	0.334022
$736$	3.64937	0.134518
$737$	9.42930	0.347333
$738$	5.26620	0.193852
$739$	3.32137	0.122178	0.0610892	−	0.998132i	$-0.480543\pi$
$739$	3.32137	0.122178	0.0610892	+	0.998132i	$0.480543\pi$
$740$	4.22468	0.155302
$741$	16.6067	0.610063
$742$	0.391357	0.0143671
$743$	−4.65788	−0.170881	−0.0854405	−	0.996343i	$-0.527230\pi$
$743$	−4.65788	−0.170881	−0.0854405	+	0.996343i	$0.527230\pi$
$744$	10.7001	0.392284
$745$	11.8340	0.433563
$746$	−27.4327	−1.00438
$747$	−11.8047	−0.431913
$748$	2.78462	0.101816
$749$	−0.389958	−0.0142487
$750$	−1.29392	−0.0472474
$751$	4.80709	0.175413	0.0877066	−	0.996146i	$-0.472046\pi$
$751$	4.80709	0.175413	0.0877066	+	0.996146i	$0.472046\pi$
$752$	−0.238525	−0.00869811
$753$	6.12321	0.223142
$754$	19.9772	0.727528
$755$	−3.92710	−0.142922
$756$	0.210270	0.00764746
$757$	39.3645	1.43073	0.715364	−	0.698752i	$-0.246261\pi$
$757$	39.3645	1.43073	0.715364	+	0.698752i	$0.246261\pi$
$758$	23.1843	0.842093
$759$	4.72201	0.171398
$760$	2.93312	0.106396
$761$	−22.7263	−0.823826	−0.411913	−	0.911223i	$-0.635139\pi$
$761$	−22.7263	−0.823826	−0.411913	+	0.911223i	$0.635139\pi$
$762$	6.72921	0.243774
$763$	0.110835	0.00401250
$764$	3.46476	0.125351
$765$	3.69175	0.133475
$766$	−1.47995	−0.0534726
$767$	8.40644	0.303539
$768$	1.29392	0.0466904
$769$	−15.2746	−0.550816	−0.275408	−	0.961327i	$-0.588813\pi$
$769$	−15.2746	−0.550816	−0.275408	+	0.961327i	$0.588813\pi$
$770$	0.0375670	0.00135382
$771$	5.87716	0.211661
$772$	−23.0209	−0.828541
$773$	4.58011	0.164735	0.0823676	−	0.996602i	$-0.473752\pi$
$773$	4.58011	0.164735	0.0823676	+	0.996602i	$0.473752\pi$
$774$	14.8774	0.534757
$775$	8.26949	0.297049
$776$	−4.35126	−0.156201
$777$	0.205357	0.00736713
$778$	−7.54473	−0.270492
$779$	11.6510	0.417440
$780$	5.66179	0.202724
$781$	−9.71798	−0.347737
$782$	10.1621	0.363397
$783$	25.5542	0.913232
$784$	−6.99859	−0.249950
$785$	18.8763	0.673722
$786$	0.649092	0.0231523
$787$	6.57060	0.234217	0.117108	−	0.993119i	$-0.462638\pi$
$787$	6.57060	0.234217	0.117108	+	0.993119i	$0.462638\pi$
$788$	−7.48275	−0.266562
$789$	23.5856	0.839668
$790$	5.80976	0.206702
$791$	−0.170541	−0.00606372
$792$	−1.32576	−0.0471089
$793$	−39.8893	−1.41651
$794$	−29.6376	−1.05180
$795$	13.4795	0.478069
$796$	−2.36522	−0.0838331
$797$	−38.9548	−1.37985	−0.689925	−	0.723880i	$-0.742357\pi$
$797$	−38.9548	−1.37985	−0.689925	+	0.723880i	$0.742357\pi$
$798$	0.142576	0.00504713
$799$	−0.664203	−0.0234978
$800$	1.00000	0.0353553
$801$	16.1768	0.571580
$802$	−26.2177	−0.925779
$803$	−1.00000	−0.0352892
$804$	12.2008	0.430289
$805$	0.137096	0.00483200
$806$	−36.1846	−1.27455
$807$	10.6915	0.376360
$808$	−8.29740	−0.291902
$809$	−25.4159	−0.893577	−0.446788	−	0.894640i	$-0.647432\pi$
$809$	−25.4159	−0.893577	−0.446788	+	0.894640i	$0.647432\pi$
$810$	3.26507	0.114723
$811$	−25.8968	−0.909360	−0.454680	−	0.890655i	$-0.650246\pi$
$811$	−25.8968	−0.909360	−0.454680	+	0.890655i	$0.650246\pi$
$812$	0.171513	0.00601893
$813$	−1.92913	−0.0676576
$814$	−4.22468	−0.148075
$815$	19.3153	0.676587
$816$	3.60309	0.126133
$817$	32.9148	1.15154
$818$	−3.80388	−0.132999
$819$	−0.217930	−0.00761509
$820$	3.97221	0.138716
$821$	34.5454	1.20564	0.602821	−	0.797877i	$-0.294043\pi$
$821$	34.5454	1.20564	0.602821	+	0.797877i	$0.294043\pi$
$822$	−0.746670	−0.0260431
$823$	46.6843	1.62731	0.813656	−	0.581347i	$-0.197474\pi$
$823$	46.6843	1.62731	0.813656	+	0.581347i	$0.197474\pi$
$824$	−5.30554	−0.184827
$825$	1.29392	0.0450486
$826$	0.0721729	0.00251122
$827$	−7.69322	−0.267519	−0.133760	−	0.991014i	$-0.542705\pi$
$827$	−7.69322	−0.267519	−0.133760	+	0.991014i	$0.542705\pi$
$828$	−4.83820	−0.168139
$829$	−19.0644	−0.662133	−0.331067	−	0.943607i	$-0.607409\pi$
$829$	−19.0644	−0.662133	−0.331067	+	0.943607i	$0.607409\pi$
$830$	−8.90411	−0.309066
$831$	19.1346	0.663772
$832$	−4.37567	−0.151699
$833$	−19.4884	−0.675234
$834$	20.3390	0.704282
$835$	16.2899	0.563735
$836$	−2.93312	−0.101444
$837$	−46.2860	−1.59988
$838$	−10.6845	−0.369090
$839$	43.6663	1.50753	0.753764	−	0.657145i	$-0.228236\pi$
$839$	43.6663	1.50753	0.753764	+	0.657145i	$0.228236\pi$
$840$	0.0486088	0.00167716
$841$	−8.15599	−0.281241
$842$	15.0733	0.519459
$843$	13.5955	0.468254
$844$	−1.58760	−0.0546475
$845$	−6.14650	−0.211446
$846$	0.316227	0.0108721
$847$	−0.0375670	−0.00129082
$848$	−10.4176	−0.357740
$849$	15.9787	0.548389
$850$	2.78462	0.0955118
$851$	−15.4174	−0.528502
$852$	−12.5743	−0.430789
$853$	−26.4906	−0.907022	−0.453511	−	0.891251i	$-0.649829\pi$
$853$	−26.4906	−0.907022	−0.453511	+	0.891251i	$0.649829\pi$
$854$	−0.342467	−0.0117190
$855$	−3.88862	−0.132988
$856$	10.3803	0.354792
$857$	−11.1871	−0.382145	−0.191073	−	0.981576i	$-0.561197\pi$
$857$	−11.1871	−0.382145	−0.191073	+	0.981576i	$0.561197\pi$
$858$	−5.66179	−0.193290
$859$	29.4780	1.00578	0.502888	−	0.864351i	$-0.332271\pi$
$859$	29.4780	1.00578	0.502888	+	0.864351i	$0.332271\pi$
$860$	11.2218	0.382659
$861$	0.193084	0.00658030
$862$	−9.39539	−0.320008
$863$	1.71920	0.0585221	0.0292611	−	0.999572i	$-0.490685\pi$
$863$	1.71920	0.0585221	0.0292611	+	0.999572i	$0.490685\pi$
$864$	−5.59721	−0.190421
$865$	−6.21651	−0.211368
$866$	−11.1929	−0.380352
$867$	−11.9634	−0.406300
$868$	−0.310660	−0.0105445
$869$	−5.80976	−0.197083
$870$	5.90744	0.200281
$871$	−41.2595	−1.39803
$872$	−2.95033	−0.0999108
$873$	5.76873	0.195242
$874$	−10.7041	−0.362070
$875$	0.0375670	0.00127000
$876$	−1.29392	−0.0437176
$877$	16.2256	0.547899	0.273950	−	0.961744i	$-0.411670\pi$
$877$	16.2256	0.547899	0.273950	+	0.961744i	$0.411670\pi$
$878$	21.3980	0.722148
$879$	13.8258	0.466333
$880$	−1.00000	−0.0337100
$881$	4.46434	0.150407	0.0752037	−	0.997168i	$-0.476039\pi$
$881$	4.46434	0.150407	0.0752037	+	0.997168i	$0.476039\pi$
$882$	9.27846	0.312422
$883$	31.6359	1.06463	0.532316	−	0.846546i	$-0.321322\pi$
$883$	31.6359	1.06463	0.532316	+	0.846546i	$0.321322\pi$
$884$	−12.1846	−0.409813
$885$	2.48586	0.0835612
$886$	−16.3701	−0.549964
$887$	21.2753	0.714354	0.357177	−	0.934037i	$-0.383739\pi$
$887$	21.2753	0.714354	0.357177	+	0.934037i	$0.383739\pi$
$888$	−5.46641	−0.183441
$889$	−0.195372	−0.00655257
$890$	12.2019	0.409009
$891$	−3.26507	−0.109384
$892$	3.69357	0.123670
$893$	0.699623	0.0234120
$894$	−15.3123	−0.512119
$895$	18.8050	0.628582
$896$	−0.0375670	−0.00125503
$897$	−20.6620	−0.689883
$898$	36.3135	1.21180
$899$	−37.7545	−1.25918
$900$	−1.32576	−0.0441921
$901$	−29.0090	−0.966429
$902$	−3.97221	−0.132260
$903$	0.545477	0.0181523
$904$	4.53964	0.150986
$905$	8.40672	0.279449
$906$	5.08137	0.168817
$907$	−14.6743	−0.487253	−0.243626	−	0.969869i	$-0.578337\pi$
$907$	−14.6743	−0.487253	−0.243626	+	0.969869i	$0.578337\pi$
$908$	−9.84773	−0.326808
$909$	11.0004	0.364860
$910$	−0.164381	−0.00544917
$911$	27.3684	0.906754	0.453377	−	0.891319i	$-0.350219\pi$
$911$	27.3684	0.906754	0.453377	+	0.891319i	$0.350219\pi$
$912$	−3.79524	−0.125673
$913$	8.90411	0.294683
$914$	17.2266	0.569806
$915$	−11.7956	−0.389951
$916$	7.01516	0.231787
$917$	−0.0188453	−0.000622328 0
$918$	−15.5861	−0.514418
$919$	−37.6582	−1.24223	−0.621114	−	0.783720i	$-0.713320\pi$
$919$	−37.6582	−1.24223	−0.621114	+	0.783720i	$0.713320\pi$
$920$	−3.64937	−0.120316
$921$	18.3794	0.605621
$922$	7.75109	0.255268
$923$	42.5227	1.39965
$924$	−0.0486088	−0.00159911
$925$	−4.22468	−0.138906
$926$	35.8775	1.17901
$927$	7.03388	0.231023
$928$	−4.56552	−0.149871
$929$	−57.7730	−1.89547	−0.947735	−	0.319059i	$-0.896633\pi$
$929$	−57.7730	−1.89547	−0.947735	+	0.319059i	$0.896633\pi$
$930$	−10.7001	−0.350870
$931$	20.5277	0.672769
$932$	−18.9665	−0.621270
$933$	−23.4662	−0.768248
$934$	−4.39568	−0.143831
$935$	−2.78462	−0.0910670
$936$	5.80110	0.189615
$937$	−14.1297	−0.461598	−0.230799	−	0.973001i	$-0.574134\pi$
$937$	−14.1297	−0.461598	−0.230799	+	0.973001i	$0.574134\pi$
$938$	−0.354231	−0.0115660
$939$	−3.66604	−0.119637
$940$	0.238525	0.00777983
$941$	42.7754	1.39444	0.697220	−	0.716858i	$-0.254420\pi$
$941$	42.7754	1.39444	0.697220	+	0.716858i	$0.254420\pi$
$942$	−24.4244	−0.795791
$943$	−14.4961	−0.472057
$944$	−1.92118	−0.0625290
$945$	−0.210270	−0.00684009
$946$	−11.2218	−0.364851
$947$	38.3368	1.24578	0.622890	−	0.782310i	$-0.285958\pi$
$947$	38.3368	1.24578	0.622890	+	0.782310i	$0.285958\pi$
$948$	−7.51738	−0.244153
$949$	4.37567	0.142040
$950$	−2.93312	−0.0951631
$951$	−25.7863	−0.836179
$952$	−0.104610	−0.00339043
$953$	24.8550	0.805133	0.402566	−	0.915391i	$-0.368118\pi$
$953$	24.8550	0.805133	0.402566	+	0.915391i	$0.368118\pi$
$954$	13.8112	0.447154
$955$	−3.46476	−0.112117
$956$	22.3306	0.722223
$957$	−5.90744	−0.190960
$958$	6.22957	0.201268
$959$	0.0216784	0.000700031 0
$960$	−1.29392	−0.0417612
$961$	37.3844	1.20595
$962$	18.4858	0.596006
$963$	−13.7618	−0.443469
$964$	1.13786	0.0366479
$965$	23.0209	0.741070
$966$	−0.177392	−0.00570749
$967$	−48.8027	−1.56939	−0.784694	−	0.619883i	$-0.787180\pi$
$967$	−48.8027	−1.56939	−0.784694	+	0.619883i	$0.787180\pi$
$968$	1.00000	0.0321412
$969$	−10.5683	−0.339503
$970$	4.35126	0.139710
$971$	−19.0178	−0.610309	−0.305155	−	0.952303i	$-0.598708\pi$
$971$	−19.0178	−0.610309	−0.305155	+	0.952303i	$0.598708\pi$
$972$	12.5669	0.403082
$973$	−0.590510	−0.0189309
$974$	13.7353	0.440107
$975$	−5.66179	−0.181322
$976$	9.11616	0.291801
$977$	33.1224	1.05968	0.529840	−	0.848098i	$-0.322252\pi$
$977$	33.1224	1.05968	0.529840	+	0.848098i	$0.322252\pi$
$978$	−24.9926	−0.799174
$979$	−12.2019	−0.389975
$980$	6.99859	0.223562
$981$	3.91144	0.124883
$982$	7.73063	0.246694
$983$	−0.890369	−0.0283984	−0.0141992	−	0.999899i	$-0.504520\pi$
$983$	−0.890369	−0.0283984	−0.0141992	+	0.999899i	$0.504520\pi$
$984$	−5.13973	−0.163849
$985$	7.48275	0.238420
$986$	−12.7133	−0.404873
$987$	0.0115944	0.000369055 0
$988$	12.8344	0.408316
$989$	−40.9524	−1.30221
$990$	1.32576	0.0421355
$991$	11.3922	0.361885	0.180943	−	0.983494i	$-0.442085\pi$
$991$	11.3922	0.361885	0.180943	+	0.983494i	$0.442085\pi$
$992$	8.26949	0.262556
$993$	22.2857	0.707214
$994$	0.365075	0.0115795
$995$	2.36522	0.0749826
$996$	11.5212	0.365064
$997$	35.0185	1.10905	0.554524	−	0.832168i	$-0.312900\pi$
$997$	35.0185	1.10905	0.554524	+	0.832168i	$0.312900\pi$
$998$	7.80995	0.247220
$999$	23.6464	0.748138

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8030.2.a.bb.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8030.2.a.bb.1.7	✓	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.29392 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.29392 q^{6} -0.0375670 q^{7} +1.00000 q^{8} -1.32576 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.29392 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.29392 q^{6} -0.0375670 q^{7} +1.00000 q^{8} -1.32576 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.29392 q^{12} -4.37567 q^{13} -0.0375670 q^{14} -1.29392 q^{15} +1.00000 q^{16} +2.78462 q^{17} -1.32576 q^{18} -2.93312 q^{19} -1.00000 q^{20} -0.0486088 q^{21} +1.00000 q^{22} +3.64937 q^{23} +1.29392 q^{24} +1.00000 q^{25} -4.37567 q^{26} -5.59721 q^{27} -0.0375670 q^{28} -4.56552 q^{29} -1.29392 q^{30} +8.26949 q^{31} +1.00000 q^{32} +1.29392 q^{33} +2.78462 q^{34} +0.0375670 q^{35} -1.32576 q^{36} -4.22468 q^{37} -2.93312 q^{38} -5.66179 q^{39} -1.00000 q^{40} -3.97221 q^{41} -0.0486088 q^{42} -11.2218 q^{43} +1.00000 q^{44} +1.32576 q^{45} +3.64937 q^{46} -0.238525 q^{47} +1.29392 q^{48} -6.99859 q^{49} +1.00000 q^{50} +3.60309 q^{51} -4.37567 q^{52} -10.4176 q^{53} -5.59721 q^{54} -1.00000 q^{55} -0.0375670 q^{56} -3.79524 q^{57} -4.56552 q^{58} -1.92118 q^{59} -1.29392 q^{60} +9.11616 q^{61} +8.26949 q^{62} +0.0498049 q^{63} +1.00000 q^{64} +4.37567 q^{65} +1.29392 q^{66} +9.42930 q^{67} +2.78462 q^{68} +4.72201 q^{69} +0.0375670 q^{70} -9.71798 q^{71} -1.32576 q^{72} -1.00000 q^{73} -4.22468 q^{74} +1.29392 q^{75} -2.93312 q^{76} -0.0375670 q^{77} -5.66179 q^{78} -5.80976 q^{79} -1.00000 q^{80} -3.26507 q^{81} -3.97221 q^{82} +8.90411 q^{83} -0.0486088 q^{84} -2.78462 q^{85} -11.2218 q^{86} -5.90744 q^{87} +1.00000 q^{88} -12.2019 q^{89} +1.32576 q^{90} +0.164381 q^{91} +3.64937 q^{92} +10.7001 q^{93} -0.238525 q^{94} +2.93312 q^{95} +1.29392 q^{96} -4.35126 q^{97} -6.99859 q^{98} -1.32576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 3 q^{3} + 8 q^{4} - 8 q^{5} - 3 q^{6} - 4 q^{7} + 8 q^{8} + q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 3 * q^3 + 8 * q^4 - 8 * q^5 - 3 * q^6 - 4 * q^7 + 8 * q^8 + q^9	\( 8 q + 8 q^{2} - 3 q^{3} + 8 q^{4} - 8 q^{5} - 3 q^{6} - 4 q^{7} + 8 q^{8} + q^{9} - 8 q^{10} + 8 q^{11} - 3 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 8 q^{16} + 2 q^{17} + q^{18} - q^{19} - 8 q^{20} + 7 q^{21} + 8 q^{22} - 16 q^{23} - 3 q^{24} + 8 q^{25} - 7 q^{26} - 21 q^{27} - 4 q^{28} - 5 q^{29} + 3 q^{30} - 22 q^{31} + 8 q^{32} - 3 q^{33} + 2 q^{34} + 4 q^{35} + q^{36} - 9 q^{37} - q^{38} - 18 q^{39} - 8 q^{40} + 6 q^{41} + 7 q^{42} + 15 q^{43} + 8 q^{44} - q^{45} - 16 q^{46} - 7 q^{47} - 3 q^{48} - 6 q^{49} + 8 q^{50} + q^{51} - 7 q^{52} - 11 q^{53} - 21 q^{54} - 8 q^{55} - 4 q^{56} - 17 q^{57} - 5 q^{58} - 11 q^{59} + 3 q^{60} - 22 q^{61} - 22 q^{62} + q^{63} + 8 q^{64} + 7 q^{65} - 3 q^{66} + 21 q^{67} + 2 q^{68} + q^{69} + 4 q^{70} - 28 q^{71} + q^{72} - 8 q^{73} - 9 q^{74} - 3 q^{75} - q^{76} - 4 q^{77} - 18 q^{78} - 28 q^{79} - 8 q^{80} + 12 q^{81} + 6 q^{82} - 5 q^{83} + 7 q^{84} - 2 q^{85} + 15 q^{86} + 36 q^{87} + 8 q^{88} - 17 q^{89} - q^{90} - 29 q^{91} - 16 q^{92} + 42 q^{93} - 7 q^{94} + q^{95} - 3 q^{96} + q^{97} - 6 q^{98} + q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 3 * q^3 + 8 * q^4 - 8 * q^5 - 3 * q^6 - 4 * q^7 + 8 * q^8 + q^9 - 8 * q^10 + 8 * q^11 - 3 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 8 * q^16 + 2 * q^17 + q^18 - q^19 - 8 * q^20 + 7 * q^21 + 8 * q^22 - 16 * q^23 - 3 * q^24 + 8 * q^25 - 7 * q^26 - 21 * q^27 - 4 * q^28 - 5 * q^29 + 3 * q^30 - 22 * q^31 + 8 * q^32 - 3 * q^33 + 2 * q^34 + 4 * q^35 + q^36 - 9 * q^37 - q^38 - 18 * q^39 - 8 * q^40 + 6 * q^41 + 7 * q^42 + 15 * q^43 + 8 * q^44 - q^45 - 16 * q^46 - 7 * q^47 - 3 * q^48 - 6 * q^49 + 8 * q^50 + q^51 - 7 * q^52 - 11 * q^53 - 21 * q^54 - 8 * q^55 - 4 * q^56 - 17 * q^57 - 5 * q^58 - 11 * q^59 + 3 * q^60 - 22 * q^61 - 22 * q^62 + q^63 + 8 * q^64 + 7 * q^65 - 3 * q^66 + 21 * q^67 + 2 * q^68 + q^69 + 4 * q^70 - 28 * q^71 + q^72 - 8 * q^73 - 9 * q^74 - 3 * q^75 - q^76 - 4 * q^77 - 18 * q^78 - 28 * q^79 - 8 * q^80 + 12 * q^81 + 6 * q^82 - 5 * q^83 + 7 * q^84 - 2 * q^85 + 15 * q^86 + 36 * q^87 + 8 * q^88 - 17 * q^89 - q^90 - 29 * q^91 - 16 * q^92 + 42 * q^93 - 7 * q^94 + q^95 - 3 * q^96 + q^97 - 6 * q^98 + q^99

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