LMFDB - Embedded newform 8042.2.a.a.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.60859 q^{3} +1.00000 q^{4} +0.145596 q^{5} -2.60859 q^{6} +0.982128 q^{7} +1.00000 q^{8} +3.80473 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.60859 q^{3} +1.00000 q^{4} +0.145596 q^{5} -2.60859 q^{6} +0.982128 q^{7} +1.00000 q^{8} +3.80473 q^{9} +0.145596 q^{10} -4.17910 q^{11} -2.60859 q^{12} +2.84300 q^{13} +0.982128 q^{14} -0.379799 q^{15} +1.00000 q^{16} -1.12150 q^{17} +3.80473 q^{18} -4.48714 q^{19} +0.145596 q^{20} -2.56197 q^{21} -4.17910 q^{22} +6.95573 q^{23} -2.60859 q^{24} -4.97880 q^{25} +2.84300 q^{26} -2.09920 q^{27} +0.982128 q^{28} -7.09928 q^{29} -0.379799 q^{30} +5.43800 q^{31} +1.00000 q^{32} +10.9015 q^{33} -1.12150 q^{34} +0.142994 q^{35} +3.80473 q^{36} +11.1763 q^{37} -4.48714 q^{38} -7.41622 q^{39} +0.145596 q^{40} -0.660816 q^{41} -2.56197 q^{42} +7.34193 q^{43} -4.17910 q^{44} +0.553952 q^{45} +6.95573 q^{46} -9.13043 q^{47} -2.60859 q^{48} -6.03542 q^{49} -4.97880 q^{50} +2.92552 q^{51} +2.84300 q^{52} -2.95851 q^{53} -2.09920 q^{54} -0.608459 q^{55} +0.982128 q^{56} +11.7051 q^{57} -7.09928 q^{58} +14.3274 q^{59} -0.379799 q^{60} -9.35855 q^{61} +5.43800 q^{62} +3.73673 q^{63} +1.00000 q^{64} +0.413929 q^{65} +10.9015 q^{66} -3.06041 q^{67} -1.12150 q^{68} -18.1446 q^{69} +0.142994 q^{70} -6.29243 q^{71} +3.80473 q^{72} -13.1900 q^{73} +11.1763 q^{74} +12.9876 q^{75} -4.48714 q^{76} -4.10441 q^{77} -7.41622 q^{78} -7.99316 q^{79} +0.145596 q^{80} -5.93823 q^{81} -0.660816 q^{82} -5.16337 q^{83} -2.56197 q^{84} -0.163285 q^{85} +7.34193 q^{86} +18.5191 q^{87} -4.17910 q^{88} +11.6929 q^{89} +0.553952 q^{90} +2.79219 q^{91} +6.95573 q^{92} -14.1855 q^{93} -9.13043 q^{94} -0.653309 q^{95} -2.60859 q^{96} +8.46759 q^{97} -6.03542 q^{98} -15.9003 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	−2.60859	−1.50607	−0.753034	−	0.657981i	$-0.771411\pi$
$3$	−2.60859	−1.50607	−0.753034	+	0.657981i	$0.771411\pi$
$4$	1.00000	0.500000
$5$	0.145596	0.0651124	0.0325562	−	0.999470i	$-0.489635\pi$
$5$	0.145596	0.0651124	0.0325562	+	0.999470i	$0.489635\pi$
$6$	−2.60859	−1.06495
$7$	0.982128	0.371210	0.185605	−	0.982624i	$-0.440576\pi$
$7$	0.982128	0.371210	0.185605	+	0.982624i	$0.440576\pi$
$8$	1.00000	0.353553
$9$	3.80473	1.26824
$10$	0.145596	0.0460414
$11$	−4.17910	−1.26005	−0.630023	−	0.776577i	$-0.716954\pi$
$11$	−4.17910	−1.26005	−0.630023	+	0.776577i	$0.716954\pi$
$12$	−2.60859	−0.753034
$13$	2.84300	0.788507	0.394254	−	0.919002i	$-0.371003\pi$
$13$	2.84300	0.788507	0.394254	+	0.919002i	$0.371003\pi$
$14$	0.982128	0.262485
$15$	−0.379799	−0.0980637
$16$	1.00000	0.250000
$17$	−1.12150	−0.272003	−0.136001	−	0.990709i	$-0.543425\pi$
$17$	−1.12150	−0.272003	−0.136001	+	0.990709i	$0.543425\pi$
$18$	3.80473	0.896783
$19$	−4.48714	−1.02942	−0.514710	−	0.857364i	$-0.672101\pi$
$19$	−4.48714	−1.02942	−0.514710	+	0.857364i	$0.672101\pi$
$20$	0.145596	0.0325562
$21$	−2.56197	−0.559067
$22$	−4.17910	−0.890986
$23$	6.95573	1.45037	0.725185	−	0.688554i	$-0.241754\pi$
$23$	6.95573	1.45037	0.725185	+	0.688554i	$0.241754\pi$
$24$	−2.60859	−0.532476
$25$	−4.97880	−0.995760
$26$	2.84300	0.557559
$27$	−2.09920	−0.403991
$28$	0.982128	0.185605
$29$	−7.09928	−1.31830	−0.659152	−	0.752010i	$-0.729085\pi$
$29$	−7.09928	−1.31830	−0.659152	+	0.752010i	$0.729085\pi$
$30$	−0.379799	−0.0693415
$31$	5.43800	0.976693	0.488347	−	0.872650i	$-0.337600\pi$
$31$	5.43800	0.976693	0.488347	+	0.872650i	$0.337600\pi$
$32$	1.00000	0.176777
$33$	10.9015	1.89771
$34$	−1.12150	−0.192335
$35$	0.142994	0.0241704
$36$	3.80473	0.634121
$37$	11.1763	1.83738	0.918689	−	0.394982i	$-0.129249\pi$
$37$	11.1763	1.83738	0.918689	+	0.394982i	$0.129249\pi$
$38$	−4.48714	−0.727910
$39$	−7.41622	−1.18755
$40$	0.145596	0.0230207
$41$	−0.660816	−0.103202	−0.0516011	−	0.998668i	$-0.516432\pi$
$41$	−0.660816	−0.103202	−0.0516011	+	0.998668i	$0.516432\pi$
$42$	−2.56197	−0.395320
$43$	7.34193	1.11963	0.559817	−	0.828616i	$-0.310871\pi$
$43$	7.34193	1.11963	0.559817	+	0.828616i	$0.310871\pi$
$44$	−4.17910	−0.630023
$45$	0.553952	0.0825783
$46$	6.95573	1.02557
$47$	−9.13043	−1.33181	−0.665905	−	0.746036i	$-0.731955\pi$
$47$	−9.13043	−1.33181	−0.665905	+	0.746036i	$0.731955\pi$
$48$	−2.60859	−0.376517
$49$	−6.03542	−0.862203
$50$	−4.97880	−0.704109
$51$	2.92552	0.409655
$52$	2.84300	0.394254
$53$	−2.95851	−0.406382	−0.203191	−	0.979139i	$-0.565131\pi$
$53$	−2.95851	−0.406382	−0.203191	+	0.979139i	$0.565131\pi$
$54$	−2.09920	−0.285665
$55$	−0.608459	−0.0820446
$56$	0.982128	0.131242
$57$	11.7051	1.55038
$58$	−7.09928	−0.932182
$59$	14.3274	1.86527	0.932634	−	0.360823i	$-0.117504\pi$
$59$	14.3274	1.86527	0.932634	+	0.360823i	$0.117504\pi$
$60$	−0.379799	−0.0490319
$61$	−9.35855	−1.19824	−0.599120	−	0.800659i	$-0.704483\pi$
$61$	−9.35855	−1.19824	−0.599120	+	0.800659i	$0.704483\pi$
$62$	5.43800	0.690626
$63$	3.73673	0.470784
$64$	1.00000	0.125000
$65$	0.413929	0.0513416
$66$	10.9015	1.34189
$67$	−3.06041	−0.373889	−0.186944	−	0.982370i	$-0.559858\pi$
$67$	−3.06041	−0.373889	−0.186944	+	0.982370i	$0.559858\pi$
$68$	−1.12150	−0.136001
$69$	−18.1446	−2.18436
$70$	0.142994	0.0170910
$71$	−6.29243	−0.746774	−0.373387	−	0.927676i	$-0.621804\pi$
$71$	−6.29243	−0.746774	−0.373387	+	0.927676i	$0.621804\pi$
$72$	3.80473	0.448391
$73$	−13.1900	−1.54377	−0.771885	−	0.635762i	$-0.780686\pi$
$73$	−13.1900	−1.54377	−0.771885	+	0.635762i	$0.780686\pi$
$74$	11.1763	1.29922
$75$	12.9876	1.49968
$76$	−4.48714	−0.514710
$77$	−4.10441	−0.467741
$78$	−7.41622	−0.839722
$79$	−7.99316	−0.899301	−0.449651	−	0.893205i	$-0.648451\pi$
$79$	−7.99316	−0.899301	−0.449651	+	0.893205i	$0.648451\pi$
$80$	0.145596	0.0162781
$81$	−5.93823	−0.659804
$82$	−0.660816	−0.0729749
$83$	−5.16337	−0.566754	−0.283377	−	0.959009i	$-0.591455\pi$
$83$	−5.16337	−0.566754	−0.283377	+	0.959009i	$0.591455\pi$
$84$	−2.56197	−0.279534
$85$	−0.163285	−0.0177108
$86$	7.34193	0.791701
$87$	18.5191	1.98546
$88$	−4.17910	−0.445493
$89$	11.6929	1.23944	0.619722	−	0.784821i	$-0.287245\pi$
$89$	11.6929	1.23944	0.619722	+	0.784821i	$0.287245\pi$
$90$	0.553952	0.0583917
$91$	2.79219	0.292701
$92$	6.95573	0.725185
$93$	−14.1855	−1.47097
$94$	−9.13043	−0.941732
$95$	−0.653309	−0.0670281
$96$	−2.60859	−0.266238
$97$	8.46759	0.859753	0.429877	−	0.902888i	$-0.358557\pi$
$97$	8.46759	0.859753	0.429877	+	0.902888i	$0.358557\pi$
$98$	−6.03542	−0.609670
$99$	−15.9003	−1.59804
$100$	−4.97880	−0.497880
$101$	0.871496	0.0867171	0.0433585	−	0.999060i	$-0.486194\pi$
$101$	0.871496	0.0867171	0.0433585	+	0.999060i	$0.486194\pi$
$102$	2.92552	0.289670
$103$	9.82432	0.968019	0.484009	−	0.875063i	$-0.339180\pi$
$103$	9.82432	0.968019	0.484009	+	0.875063i	$0.339180\pi$
$104$	2.84300	0.278779
$105$	−0.373012	−0.0364022
$106$	−2.95851	−0.287356
$107$	7.36675	0.712170	0.356085	−	0.934453i	$-0.384111\pi$
$107$	7.36675	0.712170	0.356085	+	0.934453i	$0.384111\pi$
$108$	−2.09920	−0.201996
$109$	−10.7075	−1.02559	−0.512795	−	0.858511i	$-0.671390\pi$
$109$	−10.7075	−1.02559	−0.512795	+	0.858511i	$0.671390\pi$
$110$	−0.608459	−0.0580143
$111$	−29.1544	−2.76722
$112$	0.982128	0.0928024
$113$	−0.866005	−0.0814669	−0.0407335	−	0.999170i	$-0.512969\pi$
$113$	−0.866005	−0.0814669	−0.0407335	+	0.999170i	$0.512969\pi$
$114$	11.7051	1.09628
$115$	1.01273	0.0944371
$116$	−7.09928	−0.659152
$117$	10.8168	1.00002
$118$	14.3274	1.31894
$119$	−1.10145	−0.100970
$120$	−0.379799	−0.0346708
$121$	6.46485	0.587714
$122$	−9.35855	−0.847283
$123$	1.72380	0.155429
$124$	5.43800	0.488347
$125$	−1.45287	−0.129949
$126$	3.73673	0.332894
$127$	−11.5635	−1.02610	−0.513049	−	0.858359i	$-0.671484\pi$
$127$	−11.5635	−1.02610	−0.513049	+	0.858359i	$0.671484\pi$
$128$	1.00000	0.0883883
$129$	−19.1521	−1.68625
$130$	0.413929	0.0363040
$131$	−3.57634	−0.312466	−0.156233	−	0.987720i	$-0.549935\pi$
$131$	−3.57634	−0.312466	−0.156233	+	0.987720i	$0.549935\pi$
$132$	10.9015	0.948857
$133$	−4.40695	−0.382131
$134$	−3.06041	−0.264379
$135$	−0.305635	−0.0263048
$136$	−1.12150	−0.0961675
$137$	19.3456	1.65281	0.826404	−	0.563078i	$-0.190383\pi$
$137$	19.3456	1.65281	0.826404	+	0.563078i	$0.190383\pi$
$138$	−18.1446	−1.54457
$139$	8.96056	0.760025	0.380012	−	0.924981i	$-0.375920\pi$
$139$	8.96056	0.760025	0.380012	+	0.924981i	$0.375920\pi$
$140$	0.142994	0.0120852
$141$	23.8175	2.00580
$142$	−6.29243	−0.528049
$143$	−11.8812	−0.993555
$144$	3.80473	0.317061
$145$	−1.03363	−0.0858379
$146$	−13.1900	−1.09161
$147$	15.7439	1.29854
$148$	11.1763	0.918689
$149$	1.60782	0.131718	0.0658589	−	0.997829i	$-0.479021\pi$
$149$	1.60782	0.131718	0.0658589	+	0.997829i	$0.479021\pi$
$150$	12.9876	1.06044
$151$	−21.6866	−1.76483	−0.882415	−	0.470472i	$-0.844084\pi$
$151$	−21.6866	−1.76483	−0.882415	+	0.470472i	$0.844084\pi$
$152$	−4.48714	−0.363955
$153$	−4.26699	−0.344965
$154$	−4.10441	−0.330743
$155$	0.791749	0.0635948
$156$	−7.41622	−0.593773
$157$	−9.50045	−0.758218	−0.379109	−	0.925352i	$-0.623769\pi$
$157$	−9.50045	−0.758218	−0.379109	+	0.925352i	$0.623769\pi$
$158$	−7.99316	−0.635902
$159$	7.71752	0.612039
$160$	0.145596	0.0115104
$161$	6.83142	0.538392
$162$	−5.93823	−0.466552
$163$	22.8072	1.78640	0.893199	−	0.449661i	$-0.148455\pi$
$163$	22.8072	1.78640	0.893199	+	0.449661i	$0.148455\pi$
$164$	−0.660816	−0.0516011
$165$	1.58722	0.123565
$166$	−5.16337	−0.400756
$167$	19.8803	1.53839	0.769193	−	0.639017i	$-0.220659\pi$
$167$	19.8803	1.53839	0.769193	+	0.639017i	$0.220659\pi$
$168$	−2.56197	−0.197660
$169$	−4.91734	−0.378257
$170$	−0.163285	−0.0125234
$171$	−17.0723	−1.30555
$172$	7.34193	0.559817
$173$	10.6793	0.811929	0.405964	−	0.913889i	$-0.366936\pi$
$173$	10.6793	0.811929	0.405964	+	0.913889i	$0.366936\pi$
$174$	18.5191	1.40393
$175$	−4.88982	−0.369636
$176$	−4.17910	−0.315011
$177$	−37.3743	−2.80922
$178$	11.6929	0.876420
$179$	6.10198	0.456084	0.228042	−	0.973651i	$-0.426768\pi$
$179$	6.10198	0.456084	0.228042	+	0.973651i	$0.426768\pi$
$180$	0.553952	0.0412892
$181$	−21.8897	−1.62705	−0.813524	−	0.581531i	$-0.802454\pi$
$181$	−21.8897	−1.62705	−0.813524	+	0.581531i	$0.802454\pi$
$182$	2.79219	0.206971
$183$	24.4126	1.80463
$184$	6.95573	0.512783
$185$	1.62723	0.119636
$186$	−14.1855	−1.04013
$187$	4.68684	0.342736
$188$	−9.13043	−0.665905
$189$	−2.06168	−0.149965
$190$	−0.653309	−0.0473960
$191$	−24.0960	−1.74352	−0.871762	−	0.489929i	$-0.837023\pi$
$191$	−24.0960	−1.74352	−0.871762	+	0.489929i	$0.837023\pi$
$192$	−2.60859	−0.188259
$193$	−19.5552	−1.40761	−0.703807	−	0.710391i	$-0.748518\pi$
$193$	−19.5552	−1.40761	−0.703807	+	0.710391i	$0.748518\pi$
$194$	8.46759	0.607937
$195$	−1.07977	−0.0773240
$196$	−6.03542	−0.431102
$197$	4.14320	0.295191	0.147595	−	0.989048i	$-0.452847\pi$
$197$	4.14320	0.295191	0.147595	+	0.989048i	$0.452847\pi$
$198$	−15.9003	−1.12999
$199$	−25.5671	−1.81241	−0.906203	−	0.422843i	$-0.861032\pi$
$199$	−25.5671	−1.81241	−0.906203	+	0.422843i	$0.861032\pi$
$200$	−4.97880	−0.352054
$201$	7.98335	0.563102
$202$	0.871496	0.0613182
$203$	−6.97241	−0.489367
$204$	2.92552	0.204827
$205$	−0.0962120	−0.00671974
$206$	9.82432	0.684493
$207$	26.4647	1.83942
$208$	2.84300	0.197127
$209$	18.7522	1.29712
$210$	−0.373012	−0.0257402
$211$	21.2399	1.46222	0.731109	−	0.682260i	$-0.239003\pi$
$211$	21.2399	1.46222	0.731109	+	0.682260i	$0.239003\pi$
$212$	−2.95851	−0.203191
$213$	16.4143	1.12469
$214$	7.36675	0.503581
$215$	1.06895	0.0729021
$216$	−2.09920	−0.142832
$217$	5.34081	0.362558
$218$	−10.7075	−0.725202
$219$	34.4072	2.32502
$220$	−0.608459	−0.0410223
$221$	−3.18842	−0.214476
$222$	−29.1544	−1.95672
$223$	20.5761	1.37787	0.688937	−	0.724821i	$-0.258078\pi$
$223$	20.5761	1.37787	0.688937	+	0.724821i	$0.258078\pi$
$224$	0.982128	0.0656212
$225$	−18.9430	−1.26287
$226$	−0.866005	−0.0576058
$227$	13.3978	0.889242	0.444621	−	0.895719i	$-0.353338\pi$
$227$	13.3978	0.889242	0.444621	+	0.895719i	$0.353338\pi$
$228$	11.7051	0.775189
$229$	−19.0055	−1.25592	−0.627960	−	0.778246i	$-0.716110\pi$
$229$	−19.0055	−1.25592	−0.627960	+	0.778246i	$0.716110\pi$
$230$	1.01273	0.0667771
$231$	10.7067	0.704450
$232$	−7.09928	−0.466091
$233$	−8.74974	−0.573214	−0.286607	−	0.958048i	$-0.592527\pi$
$233$	−8.74974	−0.573214	−0.286607	+	0.958048i	$0.592527\pi$
$234$	10.8168	0.707120
$235$	−1.32935	−0.0867174
$236$	14.3274	0.932634
$237$	20.8509	1.35441
$238$	−1.10145	−0.0713966
$239$	−1.65843	−0.107275	−0.0536375	−	0.998560i	$-0.517082\pi$
$239$	−1.65843	−0.107275	−0.0536375	+	0.998560i	$0.517082\pi$
$240$	−0.379799	−0.0245159
$241$	−24.4901	−1.57754	−0.788772	−	0.614686i	$-0.789283\pi$
$241$	−24.4901	−1.57754	−0.788772	+	0.614686i	$0.789283\pi$
$242$	6.46485	0.415576
$243$	21.7880	1.39770
$244$	−9.35855	−0.599120
$245$	−0.878732	−0.0561401
$246$	1.72380	0.109905
$247$	−12.7570	−0.811705
$248$	5.43800	0.345313
$249$	13.4691	0.853570
$250$	−1.45287	−0.0918877
$251$	−5.75452	−0.363222	−0.181611	−	0.983370i	$-0.558131\pi$
$251$	−5.75452	−0.363222	−0.181611	+	0.983370i	$0.558131\pi$
$252$	3.73673	0.235392
$253$	−29.0687	−1.82753
$254$	−11.5635	−0.725561
$255$	0.425943	0.0266736
$256$	1.00000	0.0625000
$257$	−29.9548	−1.86853	−0.934266	−	0.356578i	$-0.883943\pi$
$257$	−29.9548	−1.86853	−0.934266	+	0.356578i	$0.883943\pi$
$258$	−19.1521	−1.19236
$259$	10.9766	0.682052
$260$	0.413929	0.0256708
$261$	−27.0108	−1.67193
$262$	−3.57634	−0.220947
$263$	2.00121	0.123400	0.0617001	−	0.998095i	$-0.480348\pi$
$263$	2.00121	0.123400	0.0617001	+	0.998095i	$0.480348\pi$
$264$	10.9015	0.670943
$265$	−0.430746	−0.0264605
$266$	−4.40695	−0.270207
$267$	−30.5019	−1.86669
$268$	−3.06041	−0.186944
$269$	−24.9755	−1.52278	−0.761391	−	0.648293i	$-0.775483\pi$
$269$	−24.9755	−1.52278	−0.761391	+	0.648293i	$0.775483\pi$
$270$	−0.305635	−0.0186003
$271$	20.3382	1.23545	0.617727	−	0.786392i	$-0.288053\pi$
$271$	20.3382	1.23545	0.617727	+	0.786392i	$0.288053\pi$
$272$	−1.12150	−0.0680007
$273$	−7.28368	−0.440828
$274$	19.3456	1.16871
$275$	20.8069	1.25470
$276$	−18.1446	−1.09218
$277$	−17.4865	−1.05066	−0.525331	−	0.850898i	$-0.676059\pi$
$277$	−17.4865	−1.05066	−0.525331	+	0.850898i	$0.676059\pi$
$278$	8.96056	0.537419
$279$	20.6901	1.23868
$280$	0.142994	0.00854551
$281$	9.51003	0.567321	0.283660	−	0.958925i	$-0.408451\pi$
$281$	9.51003	0.567321	0.283660	+	0.958925i	$0.408451\pi$
$282$	23.8175	1.41831
$283$	−14.0463	−0.834967	−0.417483	−	0.908685i	$-0.637088\pi$
$283$	−14.0463	−0.834967	−0.417483	+	0.908685i	$0.637088\pi$
$284$	−6.29243	−0.373387
$285$	1.70421	0.100949
$286$	−11.8812	−0.702549
$287$	−0.649006	−0.0383096
$288$	3.80473	0.224196
$289$	−15.7422	−0.926015
$290$	−1.03363	−0.0606966
$291$	−22.0884	−1.29485
$292$	−13.1900	−0.771885
$293$	17.7409	1.03643	0.518217	−	0.855249i	$-0.326596\pi$
$293$	17.7409	1.03643	0.518217	+	0.855249i	$0.326596\pi$
$294$	15.7439	0.918205
$295$	2.08601	0.121452
$296$	11.1763	0.649611
$297$	8.77276	0.509047
$298$	1.60782	0.0931386
$299$	19.7752	1.14363
$300$	12.9876	0.749842
$301$	7.21072	0.415619
$302$	−21.6866	−1.24792
$303$	−2.27337	−0.130602
$304$	−4.48714	−0.257355
$305$	−1.36257	−0.0780203
$306$	−4.26699	−0.243927
$307$	1.17197	0.0668877	0.0334438	−	0.999441i	$-0.489353\pi$
$307$	1.17197	0.0668877	0.0334438	+	0.999441i	$0.489353\pi$
$308$	−4.10441	−0.233870
$309$	−25.6276	−1.45790
$310$	0.791749	0.0449683
$311$	−32.6009	−1.84862	−0.924312	−	0.381637i	$-0.875361\pi$
$311$	−32.6009	−1.84862	−0.924312	+	0.381637i	$0.875361\pi$
$312$	−7.41622	−0.419861
$313$	−9.02642	−0.510204	−0.255102	−	0.966914i	$-0.582109\pi$
$313$	−9.02642	−0.510204	−0.255102	+	0.966914i	$0.582109\pi$
$314$	−9.50045	−0.536141
$315$	0.544052	0.0306539
$316$	−7.99316	−0.449651
$317$	5.36467	0.301310	0.150655	−	0.988586i	$-0.451862\pi$
$317$	5.36467	0.301310	0.150655	+	0.988586i	$0.451862\pi$
$318$	7.71752	0.432777
$319$	29.6686	1.66112
$320$	0.145596	0.00813905
$321$	−19.2168	−1.07258
$322$	6.83142	0.380700
$323$	5.03231	0.280005
$324$	−5.93823	−0.329902
$325$	−14.1547	−0.785164
$326$	22.8072	1.26317
$327$	27.9314	1.54461
$328$	−0.660816	−0.0364875
$329$	−8.96726	−0.494381
$330$	1.58722	0.0873735
$331$	−24.3500	−1.33840	−0.669198	−	0.743084i	$-0.733362\pi$
$331$	−24.3500	−1.33840	−0.669198	+	0.743084i	$0.733362\pi$
$332$	−5.16337	−0.283377
$333$	42.5229	2.33024
$334$	19.8803	1.08780
$335$	−0.445583	−0.0243448
$336$	−2.56197	−0.139767
$337$	10.3891	0.565929	0.282964	−	0.959130i	$-0.408682\pi$
$337$	10.3891	0.565929	0.282964	+	0.959130i	$0.408682\pi$
$338$	−4.91734	−0.267468
$339$	2.25905	0.122695
$340$	−0.163285	−0.00885538
$341$	−22.7259	−1.23068
$342$	−17.0723	−0.923167
$343$	−12.8025	−0.691268
$344$	7.34193	0.395851
$345$	−2.64178	−0.142229
$346$	10.6793	0.574120
$347$	−25.6316	−1.37598	−0.687988	−	0.725722i	$-0.741506\pi$
$347$	−25.6316	−1.37598	−0.687988	+	0.725722i	$0.741506\pi$
$348$	18.5191	0.992728
$349$	−0.863826	−0.0462395	−0.0231198	−	0.999733i	$-0.507360\pi$
$349$	−0.863826	−0.0462395	−0.0231198	+	0.999733i	$0.507360\pi$
$350$	−4.88982	−0.261372
$351$	−5.96803	−0.318550
$352$	−4.17910	−0.222747
$353$	−23.4397	−1.24757	−0.623783	−	0.781597i	$-0.714405\pi$
$353$	−23.4397	−1.24757	−0.623783	+	0.781597i	$0.714405\pi$
$354$	−37.3743	−1.98642
$355$	−0.916151	−0.0486242
$356$	11.6929	0.619722
$357$	2.87324	0.152068
$358$	6.10198	0.322500
$359$	−11.4780	−0.605788	−0.302894	−	0.953024i	$-0.597953\pi$
$359$	−11.4780	−0.605788	−0.302894	+	0.953024i	$0.597953\pi$
$360$	0.553952	0.0291958
$361$	1.13443	0.0597069
$362$	−21.8897	−1.15050
$363$	−16.8641	−0.885137
$364$	2.79219	0.146351
$365$	−1.92041	−0.100519
$366$	24.4126	1.27607
$367$	−34.6076	−1.80650	−0.903250	−	0.429114i	$-0.858826\pi$
$367$	−34.6076	−1.80650	−0.903250	+	0.429114i	$0.858826\pi$
$368$	6.95573	0.362593
$369$	−2.51422	−0.130885
$370$	1.62723	0.0845955
$371$	−2.90563	−0.150853
$372$	−14.1855	−0.735483
$373$	−22.3932	−1.15947	−0.579737	−	0.814804i	$-0.696845\pi$
$373$	−22.3932	−1.15947	−0.579737	+	0.814804i	$0.696845\pi$
$374$	4.68684	0.242351
$375$	3.78994	0.195712
$376$	−9.13043	−0.470866
$377$	−20.1833	−1.03949
$378$	−2.06168	−0.106042
$379$	−24.4406	−1.25543	−0.627715	−	0.778443i	$-0.716010\pi$
$379$	−24.4406	−1.25543	−0.627715	+	0.778443i	$0.716010\pi$
$380$	−0.653309	−0.0335140
$381$	30.1645	1.54537
$382$	−24.0960	−1.23286
$383$	2.87614	0.146964	0.0734820	−	0.997297i	$-0.476589\pi$
$383$	2.87614	0.146964	0.0734820	+	0.997297i	$0.476589\pi$
$384$	−2.60859	−0.133119
$385$	−0.597585	−0.0304557
$386$	−19.5552	−0.995333
$387$	27.9341	1.41997
$388$	8.46759	0.429877
$389$	−25.9085	−1.31361	−0.656807	−	0.754059i	$-0.728093\pi$
$389$	−25.9085	−1.31361	−0.656807	+	0.754059i	$0.728093\pi$
$390$	−1.07977	−0.0546763
$391$	−7.80083	−0.394505
$392$	−6.03542	−0.304835
$393$	9.32919	0.470595
$394$	4.14320	0.208731
$395$	−1.16377	−0.0585557
$396$	−15.9003	−0.799021
$397$	26.7674	1.34342	0.671710	−	0.740814i	$-0.265560\pi$
$397$	26.7674	1.34342	0.671710	+	0.740814i	$0.265560\pi$
$398$	−25.5671	−1.28156
$399$	11.4959	0.575515
$400$	−4.97880	−0.248940
$401$	−14.4337	−0.720787	−0.360393	−	0.932800i	$-0.617358\pi$
$401$	−14.4337	−0.720787	−0.360393	+	0.932800i	$0.617358\pi$
$402$	7.98335	0.398173
$403$	15.4602	0.770129
$404$	0.871496	0.0433585
$405$	−0.864582	−0.0429614
$406$	−6.97241	−0.346035
$407$	−46.7070	−2.31518
$408$	2.92552	0.144835
$409$	−7.13102	−0.352606	−0.176303	−	0.984336i	$-0.556414\pi$
$409$	−7.13102	−0.352606	−0.176303	+	0.984336i	$0.556414\pi$
$410$	−0.0962120	−0.00475157
$411$	−50.4647	−2.48924
$412$	9.82432	0.484009
$413$	14.0713	0.692406
$414$	26.4647	1.30067
$415$	−0.751766	−0.0369027
$416$	2.84300	0.139390
$417$	−23.3744	−1.14465
$418$	18.7522	0.917200
$419$	24.8397	1.21350	0.606749	−	0.794894i	$-0.292473\pi$
$419$	24.8397	1.21350	0.606749	+	0.794894i	$0.292473\pi$
$420$	−0.373012	−0.0182011
$421$	18.7183	0.912276	0.456138	−	0.889909i	$-0.349232\pi$
$421$	18.7183	0.912276	0.456138	+	0.889909i	$0.349232\pi$
$422$	21.2399	1.03394
$423$	−34.7388	−1.68906
$424$	−2.95851	−0.143678
$425$	5.58371	0.270850
$426$	16.4143	0.795278
$427$	−9.19130	−0.444798
$428$	7.36675	0.356085
$429$	30.9931	1.49636
$430$	1.06895	0.0515496
$431$	9.75181	0.469728	0.234864	−	0.972028i	$-0.424536\pi$
$431$	9.75181	0.469728	0.234864	+	0.972028i	$0.424536\pi$
$432$	−2.09920	−0.100998
$433$	−17.9369	−0.861992	−0.430996	−	0.902354i	$-0.641838\pi$
$433$	−17.9369	−0.861992	−0.430996	+	0.902354i	$0.641838\pi$
$434$	5.34081	0.256367
$435$	2.69630	0.129278
$436$	−10.7075	−0.512795
$437$	−31.2113	−1.49304
$438$	34.4072	1.64404
$439$	−33.8196	−1.61412	−0.807062	−	0.590467i	$-0.798944\pi$
$439$	−33.8196	−1.61412	−0.807062	+	0.590467i	$0.798944\pi$
$440$	−0.608459	−0.0290071
$441$	−22.9631	−1.09348
$442$	−3.18842	−0.151657
$443$	12.6160	0.599404	0.299702	−	0.954033i	$-0.403113\pi$
$443$	12.6160	0.599404	0.299702	+	0.954033i	$0.403113\pi$
$444$	−29.1544	−1.38361
$445$	1.70244	0.0807032
$446$	20.5761	0.974305
$447$	−4.19414	−0.198376
$448$	0.982128	0.0464012
$449$	−10.6555	−0.502865	−0.251432	−	0.967875i	$-0.580902\pi$
$449$	−10.6555	−0.502865	−0.251432	+	0.967875i	$0.580902\pi$
$450$	−18.9430	−0.892981
$451$	2.76161	0.130039
$452$	−0.866005	−0.0407335
$453$	56.5714	2.65795
$454$	13.3978	0.628789
$455$	0.406532	0.0190585
$456$	11.7051	0.548141
$457$	−3.43555	−0.160708	−0.0803540	−	0.996766i	$-0.525605\pi$
$457$	−3.43555	−0.160708	−0.0803540	+	0.996766i	$0.525605\pi$
$458$	−19.0055	−0.888069
$459$	2.35424	0.109887
$460$	1.01273	0.0472186
$461$	16.5076	0.768837	0.384418	−	0.923159i	$-0.374402\pi$
$461$	16.5076	0.768837	0.384418	+	0.923159i	$0.374402\pi$
$462$	10.7067	0.498121
$463$	−8.20063	−0.381116	−0.190558	−	0.981676i	$-0.561030\pi$
$463$	−8.20063	−0.381116	−0.190558	+	0.981676i	$0.561030\pi$
$464$	−7.09928	−0.329576
$465$	−2.06535	−0.0957782
$466$	−8.74974	−0.405324
$467$	−37.1932	−1.72110	−0.860548	−	0.509369i	$-0.829879\pi$
$467$	−37.1932	−1.72110	−0.860548	+	0.509369i	$0.829879\pi$
$468$	10.8168	0.500009
$469$	−3.00572	−0.138791
$470$	−1.32935	−0.0613185
$471$	24.7827	1.14193
$472$	14.3274	0.659472
$473$	−30.6827	−1.41079
$474$	20.8509	0.957712
$475$	22.3406	1.02506
$476$	−1.10145	−0.0504850
$477$	−11.2563	−0.515391
$478$	−1.65843	−0.0758549
$479$	−11.6550	−0.532528	−0.266264	−	0.963900i	$-0.585789\pi$
$479$	−11.6550	−0.532528	−0.266264	+	0.963900i	$0.585789\pi$
$480$	−0.379799	−0.0173354
$481$	31.7743	1.44879
$482$	−24.4901	−1.11549
$483$	−17.8204	−0.810854
$484$	6.46485	0.293857
$485$	1.23284	0.0559806
$486$	21.7880	0.988324
$487$	14.4079	0.652882	0.326441	−	0.945218i	$-0.394151\pi$
$487$	14.4079	0.652882	0.326441	+	0.945218i	$0.394151\pi$
$488$	−9.35855	−0.423642
$489$	−59.4946	−2.69044
$490$	−0.878732	−0.0396971
$491$	28.4748	1.28505	0.642525	−	0.766265i	$-0.277887\pi$
$491$	28.4748	1.28505	0.642525	+	0.766265i	$0.277887\pi$
$492$	1.72380	0.0777147
$493$	7.96182	0.358582
$494$	−12.7570	−0.573962
$495$	−2.31502	−0.104052
$496$	5.43800	0.244173
$497$	−6.17997	−0.277210
$498$	13.4691	0.603565
$499$	26.6036	1.19094	0.595469	−	0.803378i	$-0.296966\pi$
$499$	26.6036	1.19094	0.595469	+	0.803378i	$0.296966\pi$
$500$	−1.45287	−0.0649744
$501$	−51.8595	−2.31691
$502$	−5.75452	−0.256837
$503$	42.3254	1.88720	0.943598	−	0.331092i	$-0.107417\pi$
$503$	42.3254	1.88720	0.943598	+	0.331092i	$0.107417\pi$
$504$	3.73673	0.166447
$505$	0.126886	0.00564636
$506$	−29.0687	−1.29226
$507$	12.8273	0.569680
$508$	−11.5635	−0.513049
$509$	30.8707	1.36832	0.684160	−	0.729332i	$-0.260169\pi$
$509$	30.8707	1.36832	0.684160	+	0.729332i	$0.260169\pi$
$510$	0.425943	0.0188611
$511$	−12.9543	−0.573062
$512$	1.00000	0.0441942
$513$	9.41941	0.415877
$514$	−29.9548	−1.32125
$515$	1.43038	0.0630300
$516$	−19.1521	−0.843123
$517$	38.1570	1.67814
$518$	10.9766	0.482284
$519$	−27.8578	−1.22282
$520$	0.413929	0.0181520
$521$	−22.0811	−0.967389	−0.483695	−	0.875237i	$-0.660705\pi$
$521$	−22.0811	−0.967389	−0.483695	+	0.875237i	$0.660705\pi$
$522$	−27.0108	−1.18223
$523$	6.87619	0.300675	0.150338	−	0.988635i	$-0.451964\pi$
$523$	6.87619	0.300675	0.150338	+	0.988635i	$0.451964\pi$
$524$	−3.57634	−0.156233
$525$	12.7555	0.556697
$526$	2.00121	0.0872571
$527$	−6.09869	−0.265663
$528$	10.9015	0.474429
$529$	25.3822	1.10357
$530$	−0.430746	−0.0187104
$531$	54.5119	2.36561
$532$	−4.40695	−0.191065
$533$	−1.87870	−0.0813756
$534$	−30.5019	−1.31995
$535$	1.07257	0.0463711
$536$	−3.06041	−0.132190
$537$	−15.9176	−0.686893
$538$	−24.9755	−1.07677
$539$	25.2226	1.08642
$540$	−0.305635	−0.0131524
$541$	−9.52185	−0.409376	−0.204688	−	0.978827i	$-0.565618\pi$
$541$	−9.52185	−0.409376	−0.204688	+	0.978827i	$0.565618\pi$
$542$	20.3382	0.873599
$543$	57.1012	2.45045
$544$	−1.12150	−0.0480837
$545$	−1.55896	−0.0667787
$546$	−7.28368	−0.311713
$547$	−40.0218	−1.71121	−0.855605	−	0.517629i	$-0.826815\pi$
$547$	−40.0218	−1.71121	−0.855605	+	0.517629i	$0.826815\pi$
$548$	19.3456	0.826404
$549$	−35.6067	−1.51966
$550$	20.8069	0.887209
$551$	31.8555	1.35709
$552$	−18.1446	−0.772287
$553$	−7.85031	−0.333829
$554$	−17.4865	−0.742930
$555$	−4.24476	−0.180180
$556$	8.96056	0.380012
$557$	4.90683	0.207909	0.103955	−	0.994582i	$-0.466850\pi$
$557$	4.90683	0.207909	0.103955	+	0.994582i	$0.466850\pi$
$558$	20.6901	0.875882
$559$	20.8731	0.882840
$560$	0.142994	0.00604259
$561$	−12.2260	−0.516183
$562$	9.51003	0.401156
$563$	3.30350	0.139226	0.0696130	−	0.997574i	$-0.477824\pi$
$563$	3.30350	0.139226	0.0696130	+	0.997574i	$0.477824\pi$
$564$	23.8175	1.00290
$565$	−0.126087	−0.00530451
$566$	−14.0463	−0.590411
$567$	−5.83211	−0.244926
$568$	−6.29243	−0.264024
$569$	1.81288	0.0760001	0.0380000	−	0.999278i	$-0.487901\pi$
$569$	1.81288	0.0760001	0.0380000	+	0.999278i	$0.487901\pi$
$570$	1.70421	0.0713816
$571$	−9.18884	−0.384541	−0.192271	−	0.981342i	$-0.561585\pi$
$571$	−9.18884	−0.384541	−0.192271	+	0.981342i	$0.561585\pi$
$572$	−11.8812	−0.496777
$573$	62.8565	2.62587
$574$	−0.649006	−0.0270890
$575$	−34.6312	−1.44422
$576$	3.80473	0.158530
$577$	33.7871	1.40658	0.703289	−	0.710904i	$-0.251714\pi$
$577$	33.7871	1.40658	0.703289	+	0.710904i	$0.251714\pi$
$578$	−15.7422	−0.654791
$579$	51.0114	2.11996
$580$	−1.03363	−0.0429190
$581$	−5.07110	−0.210385
$582$	−22.0884	−0.915595
$583$	12.3639	0.512060
$584$	−13.1900	−0.545805
$585$	1.57489	0.0651136
$586$	17.7409	0.732869
$587$	−20.9975	−0.866660	−0.433330	−	0.901235i	$-0.642662\pi$
$587$	−20.9975	−0.866660	−0.433330	+	0.901235i	$0.642662\pi$
$588$	15.7439	0.649269
$589$	−24.4011	−1.00543
$590$	2.08601	0.0858796
$591$	−10.8079	−0.444577
$592$	11.1763	0.459344
$593$	19.0178	0.780967	0.390484	−	0.920610i	$-0.372308\pi$
$593$	19.0178	0.780967	0.390484	+	0.920610i	$0.372308\pi$
$594$	8.77276	0.359951
$595$	−0.160367	−0.00657440
$596$	1.60782	0.0658589
$597$	66.6941	2.72961
$598$	19.7752	0.808667
$599$	17.4150	0.711556	0.355778	−	0.934570i	$-0.384216\pi$
$599$	17.4150	0.711556	0.355778	+	0.934570i	$0.384216\pi$
$600$	12.9876	0.530218
$601$	−10.8429	−0.442290	−0.221145	−	0.975241i	$-0.570979\pi$
$601$	−10.8429	−0.442290	−0.221145	+	0.975241i	$0.570979\pi$
$602$	7.21072	0.293887
$603$	−11.6440	−0.474182
$604$	−21.6866	−0.882415
$605$	0.941255	0.0382675
$606$	−2.27337	−0.0923495
$607$	−18.5934	−0.754684	−0.377342	−	0.926074i	$-0.623162\pi$
$607$	−18.5934	−0.754684	−0.377342	+	0.926074i	$0.623162\pi$
$608$	−4.48714	−0.181978
$609$	18.1881	0.737020
$610$	−1.36257	−0.0551687
$611$	−25.9579	−1.05014
$612$	−4.26699	−0.172483
$613$	−18.2150	−0.735697	−0.367849	−	0.929886i	$-0.619906\pi$
$613$	−18.2150	−0.735697	−0.367849	+	0.929886i	$0.619906\pi$
$614$	1.17197	0.0472967
$615$	0.250977	0.0101204
$616$	−4.10441	−0.165371
$617$	−19.1359	−0.770381	−0.385191	−	0.922837i	$-0.625864\pi$
$617$	−19.1359	−0.770381	−0.385191	+	0.922837i	$0.625864\pi$
$618$	−25.6276	−1.03089
$619$	−16.7729	−0.674161	−0.337081	−	0.941476i	$-0.609439\pi$
$619$	−16.7729	−0.674161	−0.337081	+	0.941476i	$0.609439\pi$
$620$	0.791749	0.0317974
$621$	−14.6015	−0.585937
$622$	−32.6009	−1.30718
$623$	11.4839	0.460094
$624$	−7.41622	−0.296886
$625$	24.6825	0.987299
$626$	−9.02642	−0.360768
$627$	−48.9167	−1.95355
$628$	−9.50045	−0.379109
$629$	−12.5342	−0.499772
$630$	0.544052	0.0216756
$631$	−30.8133	−1.22666	−0.613330	−	0.789827i	$-0.710170\pi$
$631$	−30.8133	−1.22666	−0.613330	+	0.789827i	$0.710170\pi$
$632$	−7.99316	−0.317951
$633$	−55.4063	−2.20220
$634$	5.36467	0.213058
$635$	−1.68360	−0.0668117
$636$	7.71752	0.306020
$637$	−17.1587	−0.679853
$638$	29.6686	1.17459
$639$	−23.9410	−0.947090
$640$	0.145596	0.00575518
$641$	−6.21677	−0.245548	−0.122774	−	0.992435i	$-0.539179\pi$
$641$	−6.21677	−0.245548	−0.122774	+	0.992435i	$0.539179\pi$
$642$	−19.2168	−0.758427
$643$	36.3817	1.43475	0.717377	−	0.696685i	$-0.245342\pi$
$643$	36.3817	1.43475	0.717377	+	0.696685i	$0.245342\pi$
$644$	6.83142	0.269196
$645$	−2.78846	−0.109796
$646$	5.03231	0.197994
$647$	−6.22989	−0.244922	−0.122461	−	0.992473i	$-0.539079\pi$
$647$	−6.22989	−0.244922	−0.122461	+	0.992473i	$0.539079\pi$
$648$	−5.93823	−0.233276
$649$	−59.8756	−2.35032
$650$	−14.1547	−0.555195
$651$	−13.9320	−0.546037
$652$	22.8072	0.893199
$653$	−29.9552	−1.17224	−0.586119	−	0.810225i	$-0.699345\pi$
$653$	−29.9552	−1.17224	−0.586119	+	0.810225i	$0.699345\pi$
$654$	27.9314	1.09220
$655$	−0.520700	−0.0203454
$656$	−0.660816	−0.0258005
$657$	−50.1843	−1.95787
$658$	−8.96726	−0.349580
$659$	19.0976	0.743936	0.371968	−	0.928246i	$-0.378683\pi$
$659$	19.0976	0.743936	0.371968	+	0.928246i	$0.378683\pi$
$660$	1.58722	0.0617824
$661$	−1.69913	−0.0660884	−0.0330442	−	0.999454i	$-0.510520\pi$
$661$	−1.69913	−0.0660884	−0.0330442	+	0.999454i	$0.510520\pi$
$662$	−24.3500	−0.946389
$663$	8.31726	0.323016
$664$	−5.16337	−0.200378
$665$	−0.641633	−0.0248815
$666$	42.5229	1.64773
$667$	−49.3807	−1.91203
$668$	19.8803	0.769193
$669$	−53.6744	−2.07517
$670$	−0.445583	−0.0172144
$671$	39.1103	1.50984
$672$	−2.56197	−0.0988300
$673$	−5.19469	−0.200241	−0.100120	−	0.994975i	$-0.531923\pi$
$673$	−5.19469	−0.200241	−0.100120	+	0.994975i	$0.531923\pi$
$674$	10.3891	0.400172
$675$	10.4515	0.402279
$676$	−4.91734	−0.189128
$677$	22.3458	0.858819	0.429409	−	0.903110i	$-0.358722\pi$
$677$	22.3458	0.858819	0.429409	+	0.903110i	$0.358722\pi$
$678$	2.25905	0.0867583
$679$	8.31626	0.319149
$680$	−0.163285	−0.00626170
$681$	−34.9493	−1.33926
$682$	−22.7259	−0.870220
$683$	−15.1973	−0.581508	−0.290754	−	0.956798i	$-0.593906\pi$
$683$	−15.1973	−0.581508	−0.290754	+	0.956798i	$0.593906\pi$
$684$	−17.0723	−0.652777
$685$	2.81664	0.107618
$686$	−12.8025	−0.488800
$687$	49.5775	1.89150
$688$	7.34193	0.279909
$689$	−8.41104	−0.320435
$690$	−2.64178	−0.100571
$691$	−6.18654	−0.235347	−0.117674	−	0.993052i	$-0.537544\pi$
$691$	−6.18654	−0.235347	−0.117674	+	0.993052i	$0.537544\pi$
$692$	10.6793	0.405964
$693$	−15.6162	−0.593209
$694$	−25.6316	−0.972962
$695$	1.30462	0.0494870
$696$	18.5191	0.701965
$697$	0.741102	0.0280713
$698$	−0.863826	−0.0326963
$699$	22.8245	0.863300
$700$	−4.88982	−0.184818
$701$	−6.90189	−0.260681	−0.130340	−	0.991469i	$-0.541607\pi$
$701$	−6.90189	−0.260681	−0.130340	+	0.991469i	$0.541607\pi$
$702$	−5.96803	−0.225249
$703$	−50.1498	−1.89143
$704$	−4.17910	−0.157506
$705$	3.46773	0.130602
$706$	−23.4397	−0.882163
$707$	0.855921	0.0321902
$708$	−37.3743	−1.40461
$709$	−13.2084	−0.496052	−0.248026	−	0.968753i	$-0.579782\pi$
$709$	−13.2084	−0.496052	−0.248026	+	0.968753i	$0.579782\pi$
$710$	−0.916151	−0.0343825
$711$	−30.4118	−1.14053
$712$	11.6929	0.438210
$713$	37.8253	1.41657
$714$	2.87324	0.107528
$715$	−1.72985	−0.0646927
$716$	6.10198	0.228042
$717$	4.32616	0.161564
$718$	−11.4780	−0.428357
$719$	15.4681	0.576862	0.288431	−	0.957501i	$-0.406866\pi$
$719$	15.4681	0.576862	0.288431	+	0.957501i	$0.406866\pi$
$720$	0.553952	0.0206446
$721$	9.64874	0.359338
$722$	1.13443	0.0422191
$723$	63.8845	2.37589
$724$	−21.8897	−0.813524
$725$	35.3459	1.31271
$726$	−16.8641	−0.625887
$727$	45.9219	1.70315	0.851574	−	0.524234i	$-0.175648\pi$
$727$	45.9219	1.70315	0.851574	+	0.524234i	$0.175648\pi$
$728$	2.79219	0.103486
$729$	−39.0212	−1.44523
$730$	−1.92041	−0.0710774
$731$	−8.23395	−0.304544
$732$	24.4126	0.902315
$733$	30.4401	1.12433	0.562165	−	0.827025i	$-0.309968\pi$
$733$	30.4401	1.12433	0.562165	+	0.827025i	$0.309968\pi$
$734$	−34.6076	−1.27739
$735$	2.29225	0.0845509
$736$	6.95573	0.256392
$737$	12.7898	0.471117
$738$	−2.51422	−0.0925499
$739$	−50.0032	−1.83940	−0.919698	−	0.392626i	$-0.871567\pi$
$739$	−50.0032	−1.83940	−0.919698	+	0.392626i	$0.871567\pi$
$740$	1.62723	0.0598180
$741$	33.2776	1.22248
$742$	−2.90563	−0.106669
$743$	22.1897	0.814060	0.407030	−	0.913415i	$-0.366564\pi$
$743$	22.1897	0.814060	0.407030	+	0.913415i	$0.366564\pi$
$744$	−14.1855	−0.520065
$745$	0.234092	0.00857647
$746$	−22.3932	−0.819872
$747$	−19.6452	−0.718781
$748$	4.68684	0.171368
$749$	7.23509	0.264365
$750$	3.78994	0.138389
$751$	9.04184	0.329941	0.164971	−	0.986298i	$-0.447247\pi$
$751$	9.04184	0.329941	0.164971	+	0.986298i	$0.447247\pi$
$752$	−9.13043	−0.332953
$753$	15.0112	0.547037
$754$	−20.1833	−0.735032
$755$	−3.15748	−0.114912
$756$	−2.06168	−0.0749827
$757$	15.8333	0.575470	0.287735	−	0.957710i	$-0.407098\pi$
$757$	15.8333	0.575470	0.287735	+	0.957710i	$0.407098\pi$
$758$	−24.4406	−0.887723
$759$	75.8282	2.75239
$760$	−0.653309	−0.0236980
$761$	−39.2370	−1.42234	−0.711170	−	0.703020i	$-0.751835\pi$
$761$	−39.2370	−1.42234	−0.711170	+	0.703020i	$0.751835\pi$
$762$	30.1645	1.09274
$763$	−10.5161	−0.380709
$764$	−24.0960	−0.871762
$765$	−0.621255	−0.0224615
$766$	2.87614	0.103919
$767$	40.7328	1.47078
$768$	−2.60859	−0.0941293
$769$	5.25552	0.189519	0.0947595	−	0.995500i	$-0.469792\pi$
$769$	5.25552	0.189519	0.0947595	+	0.995500i	$0.469792\pi$
$770$	−0.597585	−0.0215355
$771$	78.1398	2.81414
$772$	−19.5552	−0.703807
$773$	−17.2388	−0.620038	−0.310019	−	0.950730i	$-0.600335\pi$
$773$	−17.2388	−0.620038	−0.310019	+	0.950730i	$0.600335\pi$
$774$	27.9341	1.00407
$775$	−27.0747	−0.972552
$776$	8.46759	0.303969
$777$	−28.6334	−1.02722
$778$	−25.9085	−0.928865
$779$	2.96517	0.106238
$780$	−1.07977	−0.0386620
$781$	26.2967	0.940969
$782$	−7.80083	−0.278957
$783$	14.9028	0.532583
$784$	−6.03542	−0.215551
$785$	−1.38322	−0.0493694
$786$	9.32919	0.332761
$787$	15.8568	0.565233	0.282616	−	0.959233i	$-0.408798\pi$
$787$	15.8568	0.565233	0.282616	+	0.959233i	$0.408798\pi$
$788$	4.14320	0.147595
$789$	−5.22034	−0.185849
$790$	−1.16377	−0.0414051
$791$	−0.850528	−0.0302413
$792$	−15.9003	−0.564993
$793$	−26.6064	−0.944820
$794$	26.7674	0.949941
$795$	1.12364	0.0398514
$796$	−25.5671	−0.906203
$797$	−30.1792	−1.06900	−0.534502	−	0.845167i	$-0.679501\pi$
$797$	−30.1792	−1.06900	−0.534502	+	0.845167i	$0.679501\pi$
$798$	11.4959	0.406951
$799$	10.2397	0.362256
$800$	−4.97880	−0.176027
$801$	44.4883	1.57192
$802$	−14.4337	−0.509673
$803$	55.1222	1.94522
$804$	7.98335	0.281551
$805$	0.994626	0.0350560
$806$	15.4602	0.544564
$807$	65.1507	2.29341
$808$	0.871496	0.0306591
$809$	34.5855	1.21596	0.607981	−	0.793952i	$-0.291980\pi$
$809$	34.5855	1.21596	0.607981	+	0.793952i	$0.291980\pi$
$810$	−0.864582	−0.0303783
$811$	−1.29947	−0.0456307	−0.0228154	−	0.999740i	$-0.507263\pi$
$811$	−1.29947	−0.0456307	−0.0228154	+	0.999740i	$0.507263\pi$
$812$	−6.97241	−0.244684
$813$	−53.0539	−1.86068
$814$	−46.7070	−1.63708
$815$	3.32063	0.116317
$816$	2.92552	0.102414
$817$	−32.9443	−1.15257
$818$	−7.13102	−0.249330
$819$	10.6235	0.371216
$820$	−0.0962120	−0.00335987
$821$	−36.5952	−1.27718	−0.638591	−	0.769546i	$-0.720482\pi$
$821$	−36.5952	−1.27718	−0.638591	+	0.769546i	$0.720482\pi$
$822$	−50.4647	−1.76016
$823$	−11.9860	−0.417805	−0.208903	−	0.977936i	$-0.566989\pi$
$823$	−11.9860	−0.417805	−0.208903	+	0.977936i	$0.566989\pi$
$824$	9.82432	0.342246
$825$	−54.2766	−1.88967
$826$	14.0713	0.489605
$827$	−24.9497	−0.867586	−0.433793	−	0.901013i	$-0.642825\pi$
$827$	−24.9497	−0.867586	−0.433793	+	0.901013i	$0.642825\pi$
$828$	26.4647	0.919711
$829$	27.9114	0.969402	0.484701	−	0.874680i	$-0.338928\pi$
$829$	27.9114	0.969402	0.484701	+	0.874680i	$0.338928\pi$
$830$	−0.751766	−0.0260942
$831$	45.6151	1.58237
$832$	2.84300	0.0985634
$833$	6.76870	0.234522
$834$	−23.3744	−0.809389
$835$	2.89449	0.100168
$836$	18.7522	0.648558
$837$	−11.4154	−0.394576
$838$	24.8397	0.858072
$839$	39.6354	1.36836	0.684182	−	0.729311i	$-0.260159\pi$
$839$	39.6354	1.36836	0.684182	+	0.729311i	$0.260159\pi$
$840$	−0.373012	−0.0128701
$841$	21.3998	0.737925
$842$	18.7183	0.645077
$843$	−24.8077	−0.854424
$844$	21.2399	0.731109
$845$	−0.715943	−0.0246292
$846$	−34.7388	−1.19434
$847$	6.34932	0.218165
$848$	−2.95851	−0.101596
$849$	36.6410	1.25752
$850$	5.58371	0.191520
$851$	77.7396	2.66488
$852$	16.4143	0.562346
$853$	32.9854	1.12940	0.564700	−	0.825296i	$-0.308992\pi$
$853$	32.9854	1.12940	0.564700	+	0.825296i	$0.308992\pi$
$854$	−9.19130	−0.314520
$855$	−2.48566	−0.0850078
$856$	7.36675	0.251790
$857$	42.3009	1.44497	0.722485	−	0.691386i	$-0.243000\pi$
$857$	42.3009	1.44497	0.722485	+	0.691386i	$0.243000\pi$
$858$	30.9931	1.05809
$859$	−0.398186	−0.0135859	−0.00679297	−	0.999977i	$-0.502162\pi$
$859$	−0.398186	−0.0135859	−0.00679297	+	0.999977i	$0.502162\pi$
$860$	1.06895	0.0364510
$861$	1.69299	0.0576969
$862$	9.75181	0.332148
$863$	14.5331	0.494713	0.247357	−	0.968925i	$-0.420438\pi$
$863$	14.5331	0.494713	0.247357	+	0.968925i	$0.420438\pi$
$864$	−2.09920	−0.0714162
$865$	1.55485	0.0528666
$866$	−17.9369	−0.609520
$867$	41.0650	1.39464
$868$	5.34081	0.181279
$869$	33.4042	1.13316
$870$	2.69630	0.0914132
$871$	−8.70076	−0.294814
$872$	−10.7075	−0.362601
$873$	32.2169	1.09038
$874$	−31.2113	−1.05574
$875$	−1.42691	−0.0482382
$876$	34.4072	1.16251
$877$	−10.6976	−0.361234	−0.180617	−	0.983554i	$-0.557809\pi$
$877$	−10.6976	−0.361234	−0.180617	+	0.983554i	$0.557809\pi$
$878$	−33.8196	−1.14136
$879$	−46.2786	−1.56094
$880$	−0.608459	−0.0205111
$881$	−5.07855	−0.171101	−0.0855504	−	0.996334i	$-0.527265\pi$
$881$	−5.07855	−0.171101	−0.0855504	+	0.996334i	$0.527265\pi$
$882$	−22.9631	−0.773209
$883$	−19.9889	−0.672679	−0.336339	−	0.941741i	$-0.609189\pi$
$883$	−19.9889	−0.672679	−0.336339	+	0.941741i	$0.609189\pi$
$884$	−3.18842	−0.107238
$885$	−5.44154	−0.182915
$886$	12.6160	0.423843
$887$	57.1958	1.92045	0.960223	−	0.279234i	$-0.0900806\pi$
$887$	57.1958	1.92045	0.960223	+	0.279234i	$0.0900806\pi$
$888$	−29.1544	−0.978359
$889$	−11.3569	−0.380898
$890$	1.70244	0.0570658
$891$	24.8165	0.831383
$892$	20.5761	0.688937
$893$	40.9695	1.37099
$894$	−4.19414	−0.140273
$895$	0.888423	0.0296967
$896$	0.982128	0.0328106
$897$	−51.5852	−1.72238
$898$	−10.6555	−0.355579
$899$	−38.6059	−1.28758
$900$	−18.9430	−0.631433
$901$	3.31795	0.110537
$902$	2.76161	0.0919517
$903$	−18.8098	−0.625951
$904$	−0.866005	−0.0288029
$905$	−3.18705	−0.105941
$906$	56.5714	1.87946
$907$	1.27533	0.0423466	0.0211733	−	0.999776i	$-0.493260\pi$
$907$	1.27533	0.0423466	0.0211733	+	0.999776i	$0.493260\pi$
$908$	13.3978	0.444621
$909$	3.31580	0.109978
$910$	0.406532	0.0134764
$911$	−2.03231	−0.0673333	−0.0336666	−	0.999433i	$-0.510718\pi$
$911$	−2.03231	−0.0673333	−0.0336666	+	0.999433i	$0.510718\pi$
$912$	11.7051	0.387595
$913$	21.5782	0.714136
$914$	−3.43555	−0.113638
$915$	3.55437	0.117504
$916$	−19.0055	−0.627960
$917$	−3.51242	−0.115990
$918$	2.35424	0.0777017
$919$	−45.1493	−1.48934	−0.744669	−	0.667433i	$-0.767393\pi$
$919$	−45.1493	−1.48934	−0.744669	+	0.667433i	$0.767393\pi$
$920$	1.01273	0.0333886
$921$	−3.05718	−0.100737
$922$	16.5076	0.543650
$923$	−17.8894	−0.588836
$924$	10.7067	0.352225
$925$	−55.6447	−1.82959
$926$	−8.20063	−0.269490
$927$	37.3788	1.22768
$928$	−7.09928	−0.233045
$929$	11.8446	0.388608	0.194304	−	0.980941i	$-0.437755\pi$
$929$	11.8446	0.388608	0.194304	+	0.980941i	$0.437755\pi$
$930$	−2.06535	−0.0677254
$931$	27.0818	0.887570
$932$	−8.74974	−0.286607
$933$	85.0422	2.78416
$934$	−37.1932	−1.21700
$935$	0.682384	0.0223163
$936$	10.8168	0.353560
$937$	−30.4979	−0.996322	−0.498161	−	0.867085i	$-0.665991\pi$
$937$	−30.4979	−0.996322	−0.498161	+	0.867085i	$0.665991\pi$
$938$	−3.00572	−0.0981402
$939$	23.5462	0.768402
$940$	−1.32935	−0.0433587
$941$	58.3892	1.90343	0.951717	−	0.306976i	$-0.0993173\pi$
$941$	58.3892	1.90343	0.951717	+	0.306976i	$0.0993173\pi$
$942$	24.7827	0.807465
$943$	−4.59646	−0.149681
$944$	14.3274	0.466317
$945$	−0.300173	−0.00976461
$946$	−30.6827	−0.997579
$947$	61.3679	1.99419	0.997095	−	0.0761692i	$-0.0242689\pi$
$947$	61.3679	1.99419	0.997095	+	0.0761692i	$0.0242689\pi$
$948$	20.8509	0.677205
$949$	−37.4991	−1.21727
$950$	22.3406	0.724824
$951$	−13.9942	−0.453793
$952$	−1.10145	−0.0356983
$953$	−0.00959351	−0.000310764 0	−0.000155382	−	1.00000i	$-0.500049\pi$
$953$	−0.00959351	−0.000310764 0	−0.000155382		1.00000i	$0.500049\pi$
$954$	−11.2563	−0.364436
$955$	−3.50827	−0.113525
$956$	−1.65843	−0.0536375
$957$	−77.3931	−2.50176
$958$	−11.6550	−0.376555
$959$	18.9999	0.613538
$960$	−0.379799	−0.0122580
$961$	−1.42819	−0.0460706
$962$	31.7743	1.02445
$963$	28.0285	0.903205
$964$	−24.4901	−0.788772
$965$	−2.84715	−0.0916531
$966$	−17.8204	−0.573361
$967$	12.8709	0.413900	0.206950	−	0.978352i	$-0.433646\pi$
$967$	12.8709	0.413900	0.206950	+	0.978352i	$0.433646\pi$
$968$	6.46485	0.207788
$969$	−13.1272	−0.421707
$970$	1.23284	0.0395843
$971$	−45.7703	−1.46884	−0.734419	−	0.678696i	$-0.762545\pi$
$971$	−45.7703	−1.46884	−0.734419	+	0.678696i	$0.762545\pi$
$972$	21.7880	0.698850
$973$	8.80042	0.282129
$974$	14.4079	0.461657
$975$	36.9239	1.18251
$976$	−9.35855	−0.299560
$977$	4.55087	0.145595	0.0727977	−	0.997347i	$-0.476807\pi$
$977$	4.55087	0.145595	0.0727977	+	0.997347i	$0.476807\pi$
$978$	−59.4946	−1.90243
$979$	−48.8658	−1.56176
$980$	−0.878732	−0.0280701
$981$	−40.7391	−1.30070
$982$	28.4748	0.908667
$983$	27.1554	0.866123	0.433062	−	0.901364i	$-0.357433\pi$
$983$	27.1554	0.866123	0.433062	+	0.901364i	$0.357433\pi$
$984$	1.72380	0.0549526
$985$	0.603232	0.0192206
$986$	7.96182	0.253556
$987$	23.3919	0.744572
$988$	−12.7570	−0.405853
$989$	51.0685	1.62388
$990$	−2.31502	−0.0735762
$991$	26.3209	0.836110	0.418055	−	0.908422i	$-0.362712\pi$
$991$	26.3209	0.836110	0.418055	+	0.908422i	$0.362712\pi$
$992$	5.43800	0.172657
$993$	63.5190	2.01572
$994$	−6.17997	−0.196017
$995$	−3.72247	−0.118010
$996$	13.4691	0.426785
$997$	50.0322	1.58454	0.792268	−	0.610173i	$-0.208900\pi$
$997$	50.0322	1.58454	0.792268	+	0.610173i	$0.208900\pi$
$998$	26.6036	0.842121
$999$	−23.4614	−0.742285

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.6	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.6	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.60859 q^{3} +1.00000 q^{4} +0.145596 q^{5} -2.60859 q^{6} +0.982128 q^{7} +1.00000 q^{8} +3.80473 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.60859 q^{3} +1.00000 q^{4} +0.145596 q^{5} -2.60859 q^{6} +0.982128 q^{7} +1.00000 q^{8} +3.80473 q^{9} +0.145596 q^{10} -4.17910 q^{11} -2.60859 q^{12} +2.84300 q^{13} +0.982128 q^{14} -0.379799 q^{15} +1.00000 q^{16} -1.12150 q^{17} +3.80473 q^{18} -4.48714 q^{19} +0.145596 q^{20} -2.56197 q^{21} -4.17910 q^{22} +6.95573 q^{23} -2.60859 q^{24} -4.97880 q^{25} +2.84300 q^{26} -2.09920 q^{27} +0.982128 q^{28} -7.09928 q^{29} -0.379799 q^{30} +5.43800 q^{31} +1.00000 q^{32} +10.9015 q^{33} -1.12150 q^{34} +0.142994 q^{35} +3.80473 q^{36} +11.1763 q^{37} -4.48714 q^{38} -7.41622 q^{39} +0.145596 q^{40} -0.660816 q^{41} -2.56197 q^{42} +7.34193 q^{43} -4.17910 q^{44} +0.553952 q^{45} +6.95573 q^{46} -9.13043 q^{47} -2.60859 q^{48} -6.03542 q^{49} -4.97880 q^{50} +2.92552 q^{51} +2.84300 q^{52} -2.95851 q^{53} -2.09920 q^{54} -0.608459 q^{55} +0.982128 q^{56} +11.7051 q^{57} -7.09928 q^{58} +14.3274 q^{59} -0.379799 q^{60} -9.35855 q^{61} +5.43800 q^{62} +3.73673 q^{63} +1.00000 q^{64} +0.413929 q^{65} +10.9015 q^{66} -3.06041 q^{67} -1.12150 q^{68} -18.1446 q^{69} +0.142994 q^{70} -6.29243 q^{71} +3.80473 q^{72} -13.1900 q^{73} +11.1763 q^{74} +12.9876 q^{75} -4.48714 q^{76} -4.10441 q^{77} -7.41622 q^{78} -7.99316 q^{79} +0.145596 q^{80} -5.93823 q^{81} -0.660816 q^{82} -5.16337 q^{83} -2.56197 q^{84} -0.163285 q^{85} +7.34193 q^{86} +18.5191 q^{87} -4.17910 q^{88} +11.6929 q^{89} +0.553952 q^{90} +2.79219 q^{91} +6.95573 q^{92} -14.1855 q^{93} -9.13043 q^{94} -0.653309 q^{95} -2.60859 q^{96} +8.46759 q^{97} -6.03542 q^{98} -15.9003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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