LMFDB - Embedded newform 8042.2.a.a.1.36

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			1.36
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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.126650 q^{3} +1.00000 q^{4} +2.50247 q^{5} -0.126650 q^{6} -3.12234 q^{7} +1.00000 q^{8} -2.98396 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.126650 q^{3} +1.00000 q^{4} +2.50247 q^{5} -0.126650 q^{6} -3.12234 q^{7} +1.00000 q^{8} -2.98396 q^{9} +2.50247 q^{10} +3.00996 q^{11} -0.126650 q^{12} +0.390555 q^{13} -3.12234 q^{14} -0.316937 q^{15} +1.00000 q^{16} -4.35775 q^{17} -2.98396 q^{18} -2.72254 q^{19} +2.50247 q^{20} +0.395443 q^{21} +3.00996 q^{22} +6.27644 q^{23} -0.126650 q^{24} +1.26236 q^{25} +0.390555 q^{26} +0.757866 q^{27} -3.12234 q^{28} -2.88480 q^{29} -0.316937 q^{30} -7.34113 q^{31} +1.00000 q^{32} -0.381211 q^{33} -4.35775 q^{34} -7.81357 q^{35} -2.98396 q^{36} +4.63215 q^{37} -2.72254 q^{38} -0.0494636 q^{39} +2.50247 q^{40} -0.369047 q^{41} +0.395443 q^{42} -7.60429 q^{43} +3.00996 q^{44} -7.46727 q^{45} +6.27644 q^{46} -6.02468 q^{47} -0.126650 q^{48} +2.74902 q^{49} +1.26236 q^{50} +0.551907 q^{51} +0.390555 q^{52} +2.88719 q^{53} +0.757866 q^{54} +7.53234 q^{55} -3.12234 q^{56} +0.344809 q^{57} -2.88480 q^{58} +13.7042 q^{59} -0.316937 q^{60} -12.8491 q^{61} -7.34113 q^{62} +9.31695 q^{63} +1.00000 q^{64} +0.977351 q^{65} -0.381211 q^{66} +1.22549 q^{67} -4.35775 q^{68} -0.794909 q^{69} -7.81357 q^{70} -8.76692 q^{71} -2.98396 q^{72} +3.21298 q^{73} +4.63215 q^{74} -0.159877 q^{75} -2.72254 q^{76} -9.39813 q^{77} -0.0494636 q^{78} -9.72775 q^{79} +2.50247 q^{80} +8.85590 q^{81} -0.369047 q^{82} -0.802753 q^{83} +0.395443 q^{84} -10.9051 q^{85} -7.60429 q^{86} +0.365359 q^{87} +3.00996 q^{88} -8.60368 q^{89} -7.46727 q^{90} -1.21945 q^{91} +6.27644 q^{92} +0.929752 q^{93} -6.02468 q^{94} -6.81307 q^{95} -0.126650 q^{96} -0.991788 q^{97} +2.74902 q^{98} -8.98161 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.126650	−0.0731212	−0.0365606	−	0.999331i	$-0.511640\pi$
$3$	−0.126650	−0.0731212	−0.0365606	+	0.999331i	$0.511640\pi$
$4$	1.00000	0.500000
$5$	2.50247	1.11914	0.559569	−	0.828784i	$-0.310967\pi$
$5$	2.50247	1.11914	0.559569	+	0.828784i	$0.310967\pi$
$6$	−0.126650	−0.0517045
$7$	−3.12234	−1.18013	−0.590067	−	0.807354i	$-0.700899\pi$
$7$	−3.12234	−1.18013	−0.590067	+	0.807354i	$0.700899\pi$
$8$	1.00000	0.353553
$9$	−2.98396	−0.994653
$10$	2.50247	0.791350
$11$	3.00996	0.907538	0.453769	−	0.891119i	$-0.350079\pi$
$11$	3.00996	0.907538	0.453769	+	0.891119i	$0.350079\pi$
$12$	−0.126650	−0.0365606
$13$	0.390555	0.108320	0.0541602	−	0.998532i	$-0.482752\pi$
$13$	0.390555	0.108320	0.0541602	+	0.998532i	$0.482752\pi$
$14$	−3.12234	−0.834481
$15$	−0.316937	−0.0818327
$16$	1.00000	0.250000
$17$	−4.35775	−1.05691	−0.528455	−	0.848962i	$-0.677228\pi$
$17$	−4.35775	−1.05691	−0.528455	+	0.848962i	$0.677228\pi$
$18$	−2.98396	−0.703326
$19$	−2.72254	−0.624593	−0.312297	−	0.949985i	$-0.601098\pi$
$19$	−2.72254	−0.624593	−0.312297	+	0.949985i	$0.601098\pi$
$20$	2.50247	0.559569
$21$	0.395443	0.0862928
$22$	3.00996	0.641726
$23$	6.27644	1.30873	0.654364	−	0.756179i	$-0.272936\pi$
$23$	6.27644	1.30873	0.654364	+	0.756179i	$0.272936\pi$
$24$	−0.126650	−0.0258522
$25$	1.26236	0.252471
$26$	0.390555	0.0765941
$27$	0.757866	0.145851
$28$	−3.12234	−0.590067
$29$	−2.88480	−0.535695	−0.267847	−	0.963461i	$-0.586312\pi$
$29$	−2.88480	−0.535695	−0.267847	+	0.963461i	$0.586312\pi$
$30$	−0.316937	−0.0578645
$31$	−7.34113	−1.31851	−0.659253	−	0.751921i	$-0.729127\pi$
$31$	−7.34113	−1.31851	−0.659253	+	0.751921i	$0.729127\pi$
$32$	1.00000	0.176777
$33$	−0.381211	−0.0663603
$34$	−4.35775	−0.747348
$35$	−7.81357	−1.32073
$36$	−2.98396	−0.497327
$37$	4.63215	0.761521	0.380761	−	0.924674i	$-0.375662\pi$
$37$	4.63215	0.761521	0.380761	+	0.924674i	$0.375662\pi$
$38$	−2.72254	−0.441654
$39$	−0.0494636	−0.00792051
$40$	2.50247	0.395675
$41$	−0.369047	−0.0576354	−0.0288177	−	0.999585i	$-0.509174\pi$
$41$	−0.369047	−0.0576354	−0.0288177	+	0.999585i	$0.509174\pi$
$42$	0.395443	0.0610183
$43$	−7.60429	−1.15964	−0.579822	−	0.814744i	$-0.696878\pi$
$43$	−7.60429	−1.15964	−0.579822	+	0.814744i	$0.696878\pi$
$44$	3.00996	0.453769
$45$	−7.46727	−1.11315
$46$	6.27644	0.925411
$47$	−6.02468	−0.878789	−0.439395	−	0.898294i	$-0.644807\pi$
$47$	−6.02468	−0.878789	−0.439395	+	0.898294i	$0.644807\pi$
$48$	−0.126650	−0.0182803
$49$	2.74902	0.392718
$50$	1.26236	0.178524
$51$	0.551907	0.0772824
$52$	0.390555	0.0541602
$53$	2.88719	0.396587	0.198293	−	0.980143i	$-0.436460\pi$
$53$	2.88719	0.396587	0.198293	+	0.980143i	$0.436460\pi$
$54$	0.757866	0.103133
$55$	7.53234	1.01566
$56$	−3.12234	−0.417241
$57$	0.344809	0.0456710
$58$	−2.88480	−0.378793
$59$	13.7042	1.78413	0.892064	−	0.451908i	$-0.149257\pi$
$59$	13.7042	1.78413	0.892064	+	0.451908i	$0.149257\pi$
$60$	−0.316937	−0.0409164
$61$	−12.8491	−1.64516	−0.822581	−	0.568648i	$-0.807467\pi$
$61$	−12.8491	−1.64516	−0.822581	+	0.568648i	$0.807467\pi$
$62$	−7.34113	−0.932325
$63$	9.31695	1.17382
$64$	1.00000	0.125000
$65$	0.977351	0.121225
$66$	−0.381211	−0.0469238
$67$	1.22549	0.149718	0.0748590	−	0.997194i	$-0.476149\pi$
$67$	1.22549	0.149718	0.0748590	+	0.997194i	$0.476149\pi$
$68$	−4.35775	−0.528455
$69$	−0.794909	−0.0956958
$70$	−7.81357	−0.933900
$71$	−8.76692	−1.04044	−0.520221	−	0.854032i	$-0.674150\pi$
$71$	−8.76692	−1.04044	−0.520221	+	0.854032i	$0.674150\pi$
$72$	−2.98396	−0.351663
$73$	3.21298	0.376051	0.188025	−	0.982164i	$-0.439791\pi$
$73$	3.21298	0.376051	0.188025	+	0.982164i	$0.439791\pi$
$74$	4.63215	0.538477
$75$	−0.159877	−0.0184610
$76$	−2.72254	−0.312297
$77$	−9.39813	−1.07102
$78$	−0.0494636	−0.00560065
$79$	−9.72775	−1.09446	−0.547229	−	0.836983i	$-0.684317\pi$
$79$	−9.72775	−1.09446	−0.547229	+	0.836983i	$0.684317\pi$
$80$	2.50247	0.279785
$81$	8.85590	0.983988
$82$	−0.369047	−0.0407544
$83$	−0.802753	−0.0881136	−0.0440568	−	0.999029i	$-0.514028\pi$
$83$	−0.802753	−0.0881136	−0.0440568	+	0.999029i	$0.514028\pi$
$84$	0.395443	0.0431464
$85$	−10.9051	−1.18283
$86$	−7.60429	−0.819991
$87$	0.365359	0.0391706
$88$	3.00996	0.320863
$89$	−8.60368	−0.911988	−0.455994	−	0.889983i	$-0.650716\pi$
$89$	−8.60368	−0.911988	−0.455994	+	0.889983i	$0.650716\pi$
$90$	−7.46727	−0.787119
$91$	−1.21945	−0.127833
$92$	6.27644	0.654364
$93$	0.929752	0.0964108
$94$	−6.02468	−0.621398
$95$	−6.81307	−0.699007
$96$	−0.126650	−0.0129261
$97$	−0.991788	−0.100701	−0.0503504	−	0.998732i	$-0.516034\pi$
$97$	−0.991788	−0.100701	−0.0503504	+	0.998732i	$0.516034\pi$
$98$	2.74902	0.277693
$99$	−8.98161	−0.902686
$100$	1.26236	0.126236
$101$	−12.0240	−1.19644	−0.598218	−	0.801334i	$-0.704124\pi$
$101$	−12.0240	−1.19644	−0.598218	+	0.801334i	$0.704124\pi$
$102$	0.551907	0.0546469
$103$	−12.8902	−1.27010	−0.635052	−	0.772469i	$-0.719021\pi$
$103$	−12.8902	−1.27010	−0.635052	+	0.772469i	$0.719021\pi$
$104$	0.390555	0.0382970
$105$	0.989585	0.0965736
$106$	2.88719	0.280429
$107$	−17.0471	−1.64800	−0.824002	−	0.566587i	$-0.808263\pi$
$107$	−17.0471	−1.64800	−0.824002	+	0.566587i	$0.808263\pi$
$108$	0.757866	0.0729257
$109$	14.0833	1.34893	0.674467	−	0.738305i	$-0.264374\pi$
$109$	14.0833	1.34893	0.674467	+	0.738305i	$0.264374\pi$
$110$	7.53234	0.718181
$111$	−0.586661	−0.0556834
$112$	−3.12234	−0.295034
$113$	3.86364	0.363461	0.181730	−	0.983348i	$-0.441830\pi$
$113$	3.86364	0.363461	0.181730	+	0.983348i	$0.441830\pi$
$114$	0.344809	0.0322943
$115$	15.7066	1.46465
$116$	−2.88480	−0.267847
$117$	−1.16540	−0.107741
$118$	13.7042	1.26157
$119$	13.6064	1.24729
$120$	−0.316937	−0.0289322
$121$	−1.94012	−0.176375
$122$	−12.8491	−1.16331
$123$	0.0467396	0.00421437
$124$	−7.34113	−0.659253
$125$	−9.35334	−0.836588
$126$	9.31695	0.830019
$127$	−0.943810	−0.0837496	−0.0418748	−	0.999123i	$-0.513333\pi$
$127$	−0.943810	−0.0837496	−0.0418748	+	0.999123i	$0.513333\pi$
$128$	1.00000	0.0883883
$129$	0.963080	0.0847945
$130$	0.977351	0.0857194
$131$	11.5999	1.01349	0.506744	−	0.862097i	$-0.330849\pi$
$131$	11.5999	1.01349	0.506744	+	0.862097i	$0.330849\pi$
$132$	−0.381211	−0.0331801
$133$	8.50070	0.737104
$134$	1.22549	0.105867
$135$	1.89654	0.163228
$136$	−4.35775	−0.373674
$137$	8.38292	0.716201	0.358100	−	0.933683i	$-0.383425\pi$
$137$	8.38292	0.716201	0.358100	+	0.933683i	$0.383425\pi$
$138$	−0.794909	−0.0676672
$139$	22.5822	1.91540	0.957699	−	0.287773i	$-0.0929147\pi$
$139$	22.5822	1.91540	0.957699	+	0.287773i	$0.0929147\pi$
$140$	−7.81357	−0.660367
$141$	0.763023	0.0642581
$142$	−8.76692	−0.735704
$143$	1.17555	0.0983048
$144$	−2.98396	−0.248663
$145$	−7.21914	−0.599517
$146$	3.21298	0.265908
$147$	−0.348163	−0.0287160
$148$	4.63215	0.380761
$149$	−15.9218	−1.30436	−0.652181	−	0.758063i	$-0.726146\pi$
$149$	−15.9218	−1.30436	−0.652181	+	0.758063i	$0.726146\pi$
$150$	−0.159877	−0.0130539
$151$	−10.8518	−0.883106	−0.441553	−	0.897235i	$-0.645572\pi$
$151$	−10.8518	−0.883106	−0.441553	+	0.897235i	$0.645572\pi$
$152$	−2.72254	−0.220827
$153$	13.0033	1.05126
$154$	−9.39813	−0.757323
$155$	−18.3710	−1.47559
$156$	−0.0494636	−0.00396026
$157$	21.2621	1.69690	0.848452	−	0.529272i	$-0.177535\pi$
$157$	21.2621	1.69690	0.848452	+	0.529272i	$0.177535\pi$
$158$	−9.72775	−0.773898
$159$	−0.365662	−0.0289989
$160$	2.50247	0.197838
$161$	−19.5972	−1.54448
$162$	8.85590	0.695785
$163$	6.92348	0.542289	0.271144	−	0.962539i	$-0.412598\pi$
$163$	6.92348	0.542289	0.271144	+	0.962539i	$0.412598\pi$
$164$	−0.369047	−0.0288177
$165$	−0.953968	−0.0742663
$166$	−0.802753	−0.0623057
$167$	10.5567	0.816900	0.408450	−	0.912781i	$-0.366069\pi$
$167$	10.5567	0.816900	0.408450	+	0.912781i	$0.366069\pi$
$168$	0.395443	0.0305091
$169$	−12.8475	−0.988267
$170$	−10.9051	−0.836385
$171$	8.12395	0.621254
$172$	−7.60429	−0.579822
$173$	−23.0655	−1.75364	−0.876819	−	0.480821i	$-0.840339\pi$
$173$	−23.0655	−1.75364	−0.876819	+	0.480821i	$0.840339\pi$
$174$	0.365359	0.0276978
$175$	−3.94151	−0.297950
$176$	3.00996	0.226884
$177$	−1.73563	−0.130458
$178$	−8.60368	−0.644873
$179$	8.05824	0.602301	0.301150	−	0.953577i	$-0.402629\pi$
$179$	8.05824	0.602301	0.301150	+	0.953577i	$0.402629\pi$
$180$	−7.46727	−0.556577
$181$	6.14897	0.457049	0.228525	−	0.973538i	$-0.426610\pi$
$181$	6.14897	0.457049	0.228525	+	0.973538i	$0.426610\pi$
$182$	−1.21945	−0.0903913
$183$	1.62734	0.120296
$184$	6.27644	0.462705
$185$	11.5918	0.852248
$186$	0.929752	0.0681727
$187$	−13.1167	−0.959185
$188$	−6.02468	−0.439395
$189$	−2.36632	−0.172124
$190$	−6.81307	−0.494272
$191$	−0.0544458	−0.00393956	−0.00196978	−	0.999998i	$-0.500627\pi$
$191$	−0.0544458	−0.00393956	−0.00196978	+	0.999998i	$0.500627\pi$
$192$	−0.126650	−0.00914015
$193$	−16.1070	−1.15941	−0.579705	−	0.814826i	$-0.696832\pi$
$193$	−16.1070	−1.15941	−0.579705	+	0.814826i	$0.696832\pi$
$194$	−0.991788	−0.0712062
$195$	−0.123781	−0.00886415
$196$	2.74902	0.196359
$197$	−8.52039	−0.607053	−0.303526	−	0.952823i	$-0.598164\pi$
$197$	−8.52039	−0.607053	−0.303526	+	0.952823i	$0.598164\pi$
$198$	−8.98161	−0.638295
$199$	−21.5290	−1.52615	−0.763076	−	0.646308i	$-0.776312\pi$
$199$	−21.5290	−1.52615	−0.763076	+	0.646308i	$0.776312\pi$
$200$	1.26236	0.0892620
$201$	−0.155208	−0.0109476
$202$	−12.0240	−0.846007
$203$	9.00735	0.632192
$204$	0.551907	0.0386412
$205$	−0.923529	−0.0645020
$206$	−12.8902	−0.898099
$207$	−18.7287	−1.30173
$208$	0.390555	0.0270801
$209$	−8.19474	−0.566842
$210$	0.989585	0.0682879
$211$	−1.17635	−0.0809831	−0.0404915	−	0.999180i	$-0.512892\pi$
$211$	−1.17635	−0.0809831	−0.0404915	+	0.999180i	$0.512892\pi$
$212$	2.88719	0.198293
$213$	1.11033	0.0760783
$214$	−17.0471	−1.16531
$215$	−19.0295	−1.29780
$216$	0.757866	0.0515663
$217$	22.9215	1.55602
$218$	14.0833	0.953840
$219$	−0.406922	−0.0274973
$220$	7.53234	0.507830
$221$	−1.70194	−0.114485
$222$	−0.586661	−0.0393741
$223$	4.45579	0.298382	0.149191	−	0.988808i	$-0.452333\pi$
$223$	4.45579	0.298382	0.149191	+	0.988808i	$0.452333\pi$
$224$	−3.12234	−0.208620
$225$	−3.76682	−0.251121
$226$	3.86364	0.257006
$227$	19.3013	1.28107	0.640536	−	0.767928i	$-0.278712\pi$
$227$	19.3013	1.28107	0.640536	+	0.767928i	$0.278712\pi$
$228$	0.344809	0.0228355
$229$	−26.1577	−1.72855	−0.864275	−	0.503020i	$-0.832222\pi$
$229$	−26.1577	−1.72855	−0.864275	+	0.503020i	$0.832222\pi$
$230$	15.7066	1.03566
$231$	1.19027	0.0783140
$232$	−2.88480	−0.189397
$233$	−20.2416	−1.32607	−0.663036	−	0.748588i	$-0.730732\pi$
$233$	−20.2416	−1.32607	−0.663036	+	0.748588i	$0.730732\pi$
$234$	−1.16540	−0.0761845
$235$	−15.0766	−0.983487
$236$	13.7042	0.892064
$237$	1.23202	0.0800280
$238$	13.6064	0.881971
$239$	−11.8533	−0.766728	−0.383364	−	0.923597i	$-0.625235\pi$
$239$	−11.8533	−0.766728	−0.383364	+	0.923597i	$0.625235\pi$
$240$	−0.316937	−0.0204582
$241$	5.11831	0.329699	0.164850	−	0.986319i	$-0.447286\pi$
$241$	5.11831	0.329699	0.164850	+	0.986319i	$0.447286\pi$
$242$	−1.94012	−0.124716
$243$	−3.39519	−0.217802
$244$	−12.8491	−0.822581
$245$	6.87935	0.439505
$246$	0.0467396	0.00298001
$247$	−1.06330	−0.0676562
$248$	−7.34113	−0.466162
$249$	0.101668	0.00644297
$250$	−9.35334	−0.591557
$251$	−19.6617	−1.24103	−0.620517	−	0.784193i	$-0.713077\pi$
$251$	−19.6617	−1.24103	−0.620517	+	0.784193i	$0.713077\pi$
$252$	9.31695	0.586912
$253$	18.8919	1.18772
$254$	−0.943810	−0.0592199
$255$	1.38113	0.0864898
$256$	1.00000	0.0625000
$257$	−8.77198	−0.547181	−0.273590	−	0.961846i	$-0.588211\pi$
$257$	−8.77198	−0.547181	−0.273590	+	0.961846i	$0.588211\pi$
$258$	0.963080	0.0599587
$259$	−14.4632	−0.898698
$260$	0.977351	0.0606127
$261$	8.60814	0.532831
$262$	11.5999	0.716644
$263$	−12.5090	−0.771335	−0.385667	−	0.922638i	$-0.626029\pi$
$263$	−12.5090	−0.771335	−0.385667	+	0.922638i	$0.626029\pi$
$264$	−0.381211	−0.0234619
$265$	7.22512	0.443835
$266$	8.50070	0.521211
$267$	1.08965	0.0666856
$268$	1.22549	0.0748590
$269$	27.5019	1.67682	0.838409	−	0.545042i	$-0.183486\pi$
$269$	27.5019	1.67682	0.838409	+	0.545042i	$0.183486\pi$
$270$	1.89654	0.115420
$271$	10.0589	0.611032	0.305516	−	0.952187i	$-0.401171\pi$
$271$	10.0589	0.611032	0.305516	+	0.952187i	$0.401171\pi$
$272$	−4.35775	−0.264227
$273$	0.154442	0.00934727
$274$	8.38292	0.506430
$275$	3.79964	0.229127
$276$	−0.794909	−0.0478479
$277$	−24.5466	−1.47486	−0.737430	−	0.675424i	$-0.763961\pi$
$277$	−24.5466	−1.47486	−0.737430	+	0.675424i	$0.763961\pi$
$278$	22.5822	1.35439
$279$	21.9056	1.31146
$280$	−7.81357	−0.466950
$281$	−4.34867	−0.259420	−0.129710	−	0.991552i	$-0.541405\pi$
$281$	−4.34867	−0.259420	−0.129710	+	0.991552i	$0.541405\pi$
$282$	0.763023	0.0454374
$283$	−7.70675	−0.458119	−0.229059	−	0.973412i	$-0.573565\pi$
$283$	−7.70675	−0.458119	−0.229059	+	0.973412i	$0.573565\pi$
$284$	−8.76692	−0.520221
$285$	0.862873	0.0511122
$286$	1.17555	0.0695120
$287$	1.15229	0.0680176
$288$	−2.98396	−0.175832
$289$	1.98997	0.117057
$290$	−7.21914	−0.423922
$291$	0.125610	0.00736337
$292$	3.21298	0.188025
$293$	−4.11838	−0.240598	−0.120299	−	0.992738i	$-0.538385\pi$
$293$	−4.11838	−0.240598	−0.120299	+	0.992738i	$0.538385\pi$
$294$	−0.348163	−0.0203053
$295$	34.2942	1.99669
$296$	4.63215	0.269238
$297$	2.28115	0.132366
$298$	−15.9218	−0.922324
$299$	2.45129	0.141762
$300$	−0.159877	−0.00923049
$301$	23.7432	1.36853
$302$	−10.8518	−0.624450
$303$	1.52284	0.0874848
$304$	−2.72254	−0.156148
$305$	−32.1546	−1.84116
$306$	13.0033	0.743352
$307$	−7.03580	−0.401554	−0.200777	−	0.979637i	$-0.564347\pi$
$307$	−7.03580	−0.401554	−0.200777	+	0.979637i	$0.564347\pi$
$308$	−9.39813	−0.535508
$309$	1.63253	0.0928715
$310$	−18.3710	−1.04340
$311$	3.45972	0.196183	0.0980913	−	0.995177i	$-0.468726\pi$
$311$	3.45972	0.196183	0.0980913	+	0.995177i	$0.468726\pi$
$312$	−0.0494636	−0.00280032
$313$	−20.3659	−1.15115	−0.575575	−	0.817749i	$-0.695222\pi$
$313$	−20.3659	−1.15115	−0.575575	+	0.817749i	$0.695222\pi$
$314$	21.2621	1.19989
$315$	23.3154	1.31367
$316$	−9.72775	−0.547229
$317$	3.95097	0.221908	0.110954	−	0.993826i	$-0.464609\pi$
$317$	3.95097	0.221908	0.110954	+	0.993826i	$0.464609\pi$
$318$	−0.365662	−0.0205053
$319$	−8.68316	−0.486163
$320$	2.50247	0.139892
$321$	2.15901	0.120504
$322$	−19.5972	−1.09211
$323$	11.8641	0.660139
$324$	8.85590	0.491994
$325$	0.493019	0.0273478
$326$	6.92348	0.383456
$327$	−1.78364	−0.0986356
$328$	−0.369047	−0.0203772
$329$	18.8111	1.03709
$330$	−0.953968	−0.0525142
$331$	14.1985	0.780418	0.390209	−	0.920726i	$-0.372403\pi$
$331$	14.1985	0.780418	0.390209	+	0.920726i	$0.372403\pi$
$332$	−0.802753	−0.0440568
$333$	−13.8222	−0.757450
$334$	10.5567	0.577635
$335$	3.06676	0.167555
$336$	0.395443	0.0215732
$337$	16.8297	0.916772	0.458386	−	0.888753i	$-0.348428\pi$
$337$	16.8297	0.916772	0.458386	+	0.888753i	$0.348428\pi$
$338$	−12.8475	−0.698810
$339$	−0.489329	−0.0265767
$340$	−10.9051	−0.591414
$341$	−22.0965	−1.19659
$342$	8.12395	0.439293
$343$	13.2730	0.716675
$344$	−7.60429	−0.409996
$345$	−1.98924	−0.107097
$346$	−23.0655	−1.24001
$347$	−6.72762	−0.361158	−0.180579	−	0.983561i	$-0.557797\pi$
$347$	−6.72762	−0.361158	−0.180579	+	0.983561i	$0.557797\pi$
$348$	0.365359	0.0195853
$349$	21.2764	1.13890	0.569449	−	0.822027i	$-0.307157\pi$
$349$	21.2764	1.13890	0.569449	+	0.822027i	$0.307157\pi$
$350$	−3.94151	−0.210682
$351$	0.295988	0.0157987
$352$	3.00996	0.160432
$353$	20.5331	1.09286	0.546432	−	0.837503i	$-0.315986\pi$
$353$	20.5331	1.09286	0.546432	+	0.837503i	$0.315986\pi$
$354$	−1.73563	−0.0922475
$355$	−21.9390	−1.16440
$356$	−8.60368	−0.455994
$357$	−1.72324	−0.0912037
$358$	8.05824	0.425891
$359$	27.2939	1.44052	0.720258	−	0.693706i	$-0.244023\pi$
$359$	27.2939	1.44052	0.720258	+	0.693706i	$0.244023\pi$
$360$	−7.46727	−0.393560
$361$	−11.5878	−0.609883
$362$	6.14897	0.323182
$363$	0.245716	0.0128967
$364$	−1.21945	−0.0639163
$365$	8.04038	0.420853
$366$	1.62734	0.0850623
$367$	10.6571	0.556294	0.278147	−	0.960539i	$-0.410280\pi$
$367$	10.6571	0.556294	0.278147	+	0.960539i	$0.410280\pi$
$368$	6.27644	0.327182
$369$	1.10122	0.0573273
$370$	11.5918	0.602630
$371$	−9.01481	−0.468026
$372$	0.929752	0.0482054
$373$	25.7862	1.33516	0.667579	−	0.744539i	$-0.267330\pi$
$373$	25.7862	1.33516	0.667579	+	0.744539i	$0.267330\pi$
$374$	−13.1167	−0.678246
$375$	1.18460	0.0611723
$376$	−6.02468	−0.310699
$377$	−1.12667	−0.0580267
$378$	−2.36632	−0.121710
$379$	9.98509	0.512900	0.256450	−	0.966558i	$-0.417447\pi$
$379$	9.98509	0.512900	0.256450	+	0.966558i	$0.417447\pi$
$380$	−6.81307	−0.349503
$381$	0.119533	0.00612387
$382$	−0.0544458	−0.00278569
$383$	29.2565	1.49494	0.747468	−	0.664298i	$-0.231269\pi$
$383$	29.2565	1.49494	0.747468	+	0.664298i	$0.231269\pi$
$384$	−0.126650	−0.00646306
$385$	−23.5185	−1.19862
$386$	−16.1070	−0.819827
$387$	22.6909	1.15344
$388$	−0.991788	−0.0503504
$389$	15.2945	0.775464	0.387732	−	0.921772i	$-0.373259\pi$
$389$	15.2945	0.775464	0.387732	+	0.921772i	$0.373259\pi$
$390$	−0.123781	−0.00626790
$391$	−27.3512	−1.38321
$392$	2.74902	0.138847
$393$	−1.46912	−0.0741075
$394$	−8.52039	−0.429251
$395$	−24.3434	−1.22485
$396$	−8.98161	−0.451343
$397$	6.29432	0.315903	0.157951	−	0.987447i	$-0.449511\pi$
$397$	6.29432	0.315903	0.157951	+	0.987447i	$0.449511\pi$
$398$	−21.5290	−1.07915
$399$	−1.07661	−0.0538979
$400$	1.26236	0.0631178
$401$	10.5056	0.524622	0.262311	−	0.964983i	$-0.415515\pi$
$401$	10.5056	0.524622	0.262311	+	0.964983i	$0.415515\pi$
$402$	−0.155208	−0.00774109
$403$	−2.86711	−0.142821
$404$	−12.0240	−0.598218
$405$	22.1616	1.10122
$406$	9.00735	0.447027
$407$	13.9426	0.691110
$408$	0.551907	0.0273235
$409$	24.4450	1.20873	0.604365	−	0.796708i	$-0.293427\pi$
$409$	24.4450	1.20873	0.604365	+	0.796708i	$0.293427\pi$
$410$	−0.923529	−0.0456098
$411$	−1.06169	−0.0523695
$412$	−12.8902	−0.635052
$413$	−42.7891	−2.10551
$414$	−18.7287	−0.920463
$415$	−2.00887	−0.0986114
$416$	0.390555	0.0191485
$417$	−2.86003	−0.140056
$418$	−8.19474	−0.400818
$419$	23.6498	1.15537	0.577684	−	0.816260i	$-0.303957\pi$
$419$	23.6498	1.15537	0.577684	+	0.816260i	$0.303957\pi$
$420$	0.989585	0.0482868
$421$	17.6998	0.862638	0.431319	−	0.902200i	$-0.358048\pi$
$421$	17.6998	0.862638	0.431319	+	0.902200i	$0.358048\pi$
$422$	−1.17635	−0.0572637
$423$	17.9774	0.874091
$424$	2.88719	0.140215
$425$	−5.50103	−0.266839
$426$	1.11033	0.0537955
$427$	40.1194	1.94151
$428$	−17.0471	−0.824002
$429$	−0.148884	−0.00718817
$430$	−19.0295	−0.917684
$431$	−35.5338	−1.71160	−0.855802	−	0.517304i	$-0.826936\pi$
$431$	−35.5338	−1.71160	−0.855802	+	0.517304i	$0.826936\pi$
$432$	0.757866	0.0364629
$433$	−3.27087	−0.157188	−0.0785939	−	0.996907i	$-0.525043\pi$
$433$	−3.27087	−0.157188	−0.0785939	+	0.996907i	$0.525043\pi$
$434$	22.9215	1.10027
$435$	0.914301	0.0438374
$436$	14.0833	0.674467
$437$	−17.0879	−0.817423
$438$	−0.406922	−0.0194435
$439$	22.7977	1.08807	0.544037	−	0.839061i	$-0.316895\pi$
$439$	22.7977	1.08807	0.544037	+	0.839061i	$0.316895\pi$
$440$	7.53234	0.359090
$441$	−8.20298	−0.390618
$442$	−1.70194	−0.0809530
$443$	−15.4272	−0.732967	−0.366484	−	0.930424i	$-0.619438\pi$
$443$	−15.4272	−0.732967	−0.366484	+	0.930424i	$0.619438\pi$
$444$	−0.586661	−0.0278417
$445$	−21.5304	−1.02064
$446$	4.45579	0.210988
$447$	2.01649	0.0953765
$448$	−3.12234	−0.147517
$449$	−13.6450	−0.643947	−0.321973	−	0.946749i	$-0.604346\pi$
$449$	−13.6450	−0.643947	−0.321973	+	0.946749i	$0.604346\pi$
$450$	−3.76682	−0.177569
$451$	−1.11082	−0.0523063
$452$	3.86364	0.181730
$453$	1.37437	0.0645737
$454$	19.3013	0.905855
$455$	−3.05163	−0.143062
$456$	0.344809	0.0161471
$457$	−12.0586	−0.564080	−0.282040	−	0.959403i	$-0.591011\pi$
$457$	−12.0586	−0.564080	−0.282040	+	0.959403i	$0.591011\pi$
$458$	−26.1577	−1.22227
$459$	−3.30259	−0.154152
$460$	15.7066	0.732324
$461$	−7.53956	−0.351153	−0.175576	−	0.984466i	$-0.556179\pi$
$461$	−7.53956	−0.351153	−0.175576	+	0.984466i	$0.556179\pi$
$462$	1.19027	0.0553764
$463$	−11.3090	−0.525572	−0.262786	−	0.964854i	$-0.584641\pi$
$463$	−11.3090	−0.525572	−0.262786	+	0.964854i	$0.584641\pi$
$464$	−2.88480	−0.133924
$465$	2.32668	0.107897
$466$	−20.2416	−0.937674
$467$	−8.17243	−0.378175	−0.189087	−	0.981960i	$-0.560553\pi$
$467$	−8.17243	−0.378175	−0.189087	+	0.981960i	$0.560553\pi$
$468$	−1.16540	−0.0538706
$469$	−3.82641	−0.176687
$470$	−15.0766	−0.695430
$471$	−2.69284	−0.124080
$472$	13.7042	0.630785
$473$	−22.8886	−1.05242
$474$	1.23202	0.0565883
$475$	−3.43681	−0.157692
$476$	13.6064	0.623647
$477$	−8.61527	−0.394466
$478$	−11.8533	−0.542159
$479$	31.2123	1.42612	0.713062	−	0.701101i	$-0.247308\pi$
$479$	31.2123	1.42612	0.713062	+	0.701101i	$0.247308\pi$
$480$	−0.316937	−0.0144661
$481$	1.80911	0.0824883
$482$	5.11831	0.233133
$483$	2.48198	0.112934
$484$	−1.94012	−0.0881875
$485$	−2.48192	−0.112698
$486$	−3.39519	−0.154009
$487$	26.7699	1.21306	0.606529	−	0.795061i	$-0.292561\pi$
$487$	26.7699	1.21306	0.606529	+	0.795061i	$0.292561\pi$
$488$	−12.8491	−0.581653
$489$	−0.876856	−0.0396528
$490$	6.87935	0.310777
$491$	−40.7714	−1.83999	−0.919995	−	0.391931i	$-0.871807\pi$
$491$	−40.7714	−1.83999	−0.919995	+	0.391931i	$0.871807\pi$
$492$	0.0467396	0.00210719
$493$	12.5713	0.566181
$494$	−1.06330	−0.0478402
$495$	−22.4762	−1.01023
$496$	−7.34113	−0.329627
$497$	27.3733	1.22786
$498$	0.101668	0.00455587
$499$	−22.6172	−1.01249	−0.506243	−	0.862391i	$-0.668966\pi$
$499$	−22.6172	−1.01249	−0.506243	+	0.862391i	$0.668966\pi$
$500$	−9.35334	−0.418294
$501$	−1.33700	−0.0597327
$502$	−19.6617	−0.877543
$503$	−21.1317	−0.942216	−0.471108	−	0.882076i	$-0.656146\pi$
$503$	−21.1317	−0.942216	−0.471108	+	0.882076i	$0.656146\pi$
$504$	9.31695	0.415010
$505$	−30.0898	−1.33898
$506$	18.8919	0.839846
$507$	1.62713	0.0722632
$508$	−0.943810	−0.0418748
$509$	−8.33660	−0.369513	−0.184757	−	0.982784i	$-0.559150\pi$
$509$	−8.33660	−0.369513	−0.184757	+	0.982784i	$0.559150\pi$
$510$	1.38113	0.0611575
$511$	−10.0320	−0.443790
$512$	1.00000	0.0441942
$513$	−2.06332	−0.0910978
$514$	−8.77198	−0.386915
$515$	−32.2572	−1.42142
$516$	0.963080	0.0423972
$517$	−18.1341	−0.797535
$518$	−14.4632	−0.635475
$519$	2.92124	0.128228
$520$	0.977351	0.0428597
$521$	3.41569	0.149644	0.0748222	−	0.997197i	$-0.476161\pi$
$521$	3.41569	0.149644	0.0748222	+	0.997197i	$0.476161\pi$
$522$	8.60814	0.376768
$523$	25.8922	1.13219	0.566093	−	0.824341i	$-0.308454\pi$
$523$	25.8922	1.13219	0.566093	+	0.824341i	$0.308454\pi$
$524$	11.5999	0.506744
$525$	0.499190	0.0217864
$526$	−12.5090	−0.545416
$527$	31.9908	1.39354
$528$	−0.381211	−0.0165901
$529$	16.3937	0.712771
$530$	7.22512	0.313839
$531$	−40.8926	−1.77459
$532$	8.50070	0.368552
$533$	−0.144133	−0.00624309
$534$	1.08965	0.0471539
$535$	−42.6598	−1.84434
$536$	1.22549	0.0529333
$537$	−1.02057	−0.0440409
$538$	27.5019	1.18569
$539$	8.27446	0.356406
$540$	1.89654	0.0816140
$541$	−34.1036	−1.46623	−0.733115	−	0.680105i	$-0.761934\pi$
$541$	−34.1036	−1.46623	−0.733115	+	0.680105i	$0.761934\pi$
$542$	10.0589	0.432065
$543$	−0.778764	−0.0334200
$544$	−4.35775	−0.186837
$545$	35.2430	1.50964
$546$	0.154442	0.00660952
$547$	15.5133	0.663300	0.331650	−	0.943403i	$-0.392395\pi$
$547$	15.5133	0.663300	0.331650	+	0.943403i	$0.392395\pi$
$548$	8.38292	0.358100
$549$	38.3413	1.63637
$550$	3.79964	0.162017
$551$	7.85400	0.334591
$552$	−0.794909	−0.0338336
$553$	30.3734	1.29161
$554$	−24.5466	−1.04288
$555$	−1.46810	−0.0623174
$556$	22.5822	0.957699
$557$	16.1369	0.683744	0.341872	−	0.939746i	$-0.388939\pi$
$557$	16.1369	0.683744	0.341872	+	0.939746i	$0.388939\pi$
$558$	21.9056	0.927340
$559$	−2.96989	−0.125613
$560$	−7.81357	−0.330184
$561$	1.66122	0.0701368
$562$	−4.34867	−0.183437
$563$	−19.6166	−0.826742	−0.413371	−	0.910563i	$-0.635649\pi$
$563$	−19.6166	−0.826742	−0.413371	+	0.910563i	$0.635649\pi$
$564$	0.763023	0.0321291
$565$	9.66865	0.406763
$566$	−7.70675	−0.323939
$567$	−27.6511	−1.16124
$568$	−8.76692	−0.367852
$569$	−29.3645	−1.23102	−0.615512	−	0.788128i	$-0.711051\pi$
$569$	−29.3645	−1.23102	−0.615512	+	0.788128i	$0.711051\pi$
$570$	0.862873	0.0361418
$571$	−15.5952	−0.652639	−0.326319	−	0.945260i	$-0.605808\pi$
$571$	−15.5952	−0.652639	−0.326319	+	0.945260i	$0.605808\pi$
$572$	1.17555	0.0491524
$573$	0.00689554	0.000288065 0
$574$	1.15229	0.0480957
$575$	7.92310	0.330416
$576$	−2.98396	−0.124332
$577$	5.18503	0.215856	0.107928	−	0.994159i	$-0.465578\pi$
$577$	5.18503	0.215856	0.107928	+	0.994159i	$0.465578\pi$
$578$	1.98997	0.0827716
$579$	2.03995	0.0847775
$580$	−7.21914	−0.299758
$581$	2.50647	0.103986
$582$	0.125610	0.00520669
$583$	8.69035	0.359917
$584$	3.21298	0.132954
$585$	−2.91638	−0.120577
$586$	−4.11838	−0.170129
$587$	13.9974	0.577733	0.288866	−	0.957369i	$-0.406722\pi$
$587$	13.9974	0.577733	0.288866	+	0.957369i	$0.406722\pi$
$588$	−0.348163	−0.0143580
$589$	19.9865	0.823531
$590$	34.2942	1.41187
$591$	1.07910	0.0443884
$592$	4.63215	0.190380
$593$	−13.0323	−0.535174	−0.267587	−	0.963534i	$-0.586226\pi$
$593$	−13.0323	−0.535174	−0.267587	+	0.963534i	$0.586226\pi$
$594$	2.28115	0.0935967
$595$	34.0496	1.39590
$596$	−15.9218	−0.652181
$597$	2.72664	0.111594
$598$	2.45129	0.100241
$599$	28.9347	1.18224	0.591120	−	0.806583i	$-0.298686\pi$
$599$	28.9347	1.18224	0.591120	+	0.806583i	$0.298686\pi$
$600$	−0.159877	−0.00652694
$601$	12.6610	0.516453	0.258226	−	0.966084i	$-0.416862\pi$
$601$	12.6610	0.516453	0.258226	+	0.966084i	$0.416862\pi$
$602$	23.7432	0.967700
$603$	−3.65683	−0.148917
$604$	−10.8518	−0.441553
$605$	−4.85510	−0.197388
$606$	1.52284	0.0618611
$607$	43.4709	1.76443	0.882214	−	0.470848i	$-0.156052\pi$
$607$	43.4709	1.76443	0.882214	+	0.470848i	$0.156052\pi$
$608$	−2.72254	−0.110414
$609$	−1.14078	−0.0462266
$610$	−32.1546	−1.30190
$611$	−2.35297	−0.0951908
$612$	13.0033	0.525629
$613$	−3.49037	−0.140975	−0.0704873	−	0.997513i	$-0.522455\pi$
$613$	−3.49037	−0.140975	−0.0704873	+	0.997513i	$0.522455\pi$
$614$	−7.03580	−0.283942
$615$	0.116965	0.00471647
$616$	−9.39813	−0.378662
$617$	−4.63108	−0.186440	−0.0932201	−	0.995646i	$-0.529716\pi$
$617$	−4.63108	−0.186440	−0.0932201	+	0.995646i	$0.529716\pi$
$618$	1.63253	0.0656701
$619$	−27.4710	−1.10415	−0.552076	−	0.833794i	$-0.686164\pi$
$619$	−27.4710	−1.10415	−0.552076	+	0.833794i	$0.686164\pi$
$620$	−18.3710	−0.737796
$621$	4.75670	0.190880
$622$	3.45972	0.138722
$623$	26.8636	1.07627
$624$	−0.0494636	−0.00198013
$625$	−29.7182	−1.18873
$626$	−20.3659	−0.813986
$627$	1.03786	0.0414482
$628$	21.2621	0.848452
$629$	−20.1858	−0.804859
$630$	23.3154	0.928907
$631$	6.00772	0.239164	0.119582	−	0.992824i	$-0.461845\pi$
$631$	6.00772	0.239164	0.119582	+	0.992824i	$0.461845\pi$
$632$	−9.72775	−0.386949
$633$	0.148984	0.00592158
$634$	3.95097	0.156913
$635$	−2.36186	−0.0937274
$636$	−0.365662	−0.0144994
$637$	1.07364	0.0425393
$638$	−8.68316	−0.343769
$639$	26.1601	1.03488
$640$	2.50247	0.0989188
$641$	−21.1296	−0.834570	−0.417285	−	0.908776i	$-0.637018\pi$
$641$	−21.1296	−0.834570	−0.417285	+	0.908776i	$0.637018\pi$
$642$	2.15901	0.0852092
$643$	−3.64782	−0.143856	−0.0719279	−	0.997410i	$-0.522915\pi$
$643$	−3.64782	−0.143856	−0.0719279	+	0.997410i	$0.522915\pi$
$644$	−19.5972	−0.772238
$645$	2.41008	0.0948968
$646$	11.8641	0.466788
$647$	32.0127	1.25855	0.629274	−	0.777183i	$-0.283352\pi$
$647$	32.0127	1.25855	0.629274	+	0.777183i	$0.283352\pi$
$648$	8.85590	0.347892
$649$	41.2490	1.61916
$650$	0.493019	0.0193378
$651$	−2.90300	−0.113778
$652$	6.92348	0.271144
$653$	−40.6625	−1.59125	−0.795624	−	0.605791i	$-0.792857\pi$
$653$	−40.6625	−1.59125	−0.795624	+	0.605791i	$0.792857\pi$
$654$	−1.78364	−0.0697459
$655$	29.0284	1.13423
$656$	−0.369047	−0.0144089
$657$	−9.58739	−0.374040
$658$	18.8111	0.733333
$659$	−28.9172	−1.12646	−0.563228	−	0.826302i	$-0.690441\pi$
$659$	−28.9172	−1.12646	−0.563228	+	0.826302i	$0.690441\pi$
$660$	−0.953968	−0.0371332
$661$	19.5479	0.760323	0.380162	−	0.924920i	$-0.375868\pi$
$661$	19.5479	0.760323	0.380162	+	0.924920i	$0.375868\pi$
$662$	14.1985	0.551839
$663$	0.215550	0.00837126
$664$	−0.802753	−0.0311529
$665$	21.2728	0.824922
$666$	−13.8222	−0.535598
$667$	−18.1063	−0.701079
$668$	10.5567	0.408450
$669$	−0.564324	−0.0218180
$670$	3.06676	0.118479
$671$	−38.6754	−1.49305
$672$	0.395443	0.0152546
$673$	−4.04802	−0.156040	−0.0780198	−	0.996952i	$-0.524860\pi$
$673$	−4.04802	−0.156040	−0.0780198	+	0.996952i	$0.524860\pi$
$674$	16.8297	0.648256
$675$	0.956696	0.0368233
$676$	−12.8475	−0.494133
$677$	−15.5233	−0.596609	−0.298305	−	0.954471i	$-0.596421\pi$
$677$	−15.5233	−0.596609	−0.298305	+	0.954471i	$0.596421\pi$
$678$	−0.489329	−0.0187926
$679$	3.09670	0.118841
$680$	−10.9051	−0.418193
$681$	−2.44450	−0.0936735
$682$	−22.0965	−0.846120
$683$	−19.5096	−0.746513	−0.373257	−	0.927728i	$-0.621759\pi$
$683$	−19.5096	−0.746513	−0.373257	+	0.927728i	$0.621759\pi$
$684$	8.12395	0.310627
$685$	20.9780	0.801528
$686$	13.2730	0.506766
$687$	3.31286	0.126394
$688$	−7.60429	−0.289911
$689$	1.12761	0.0429584
$690$	−1.98924	−0.0757289
$691$	16.2742	0.619099	0.309550	−	0.950883i	$-0.399822\pi$
$691$	16.2742	0.619099	0.309550	+	0.950883i	$0.399822\pi$
$692$	−23.0655	−0.876819
$693$	28.0437	1.06529
$694$	−6.72762	−0.255377
$695$	56.5113	2.14360
$696$	0.365359	0.0138489
$697$	1.60821	0.0609154
$698$	21.2764	0.805323
$699$	2.56359	0.0969639
$700$	−3.94151	−0.148975
$701$	−6.73472	−0.254367	−0.127183	−	0.991879i	$-0.540594\pi$
$701$	−6.73472	−0.254367	−0.127183	+	0.991879i	$0.540594\pi$
$702$	0.295988	0.0111714
$703$	−12.6112	−0.475641
$704$	3.00996	0.113442
$705$	1.90944	0.0719137
$706$	20.5331	0.772772
$707$	37.5431	1.41195
$708$	−1.73563	−0.0652288
$709$	4.15033	0.155869	0.0779344	−	0.996958i	$-0.475168\pi$
$709$	4.15033	0.155869	0.0779344	+	0.996958i	$0.475168\pi$
$710$	−21.9390	−0.823354
$711$	29.0272	1.08861
$712$	−8.60368	−0.322436
$713$	−46.0762	−1.72557
$714$	−1.72324	−0.0644907
$715$	2.94179	0.110017
$716$	8.05824	0.301150
$717$	1.50122	0.0560641
$718$	27.2939	1.01860
$719$	18.9536	0.706848	0.353424	−	0.935463i	$-0.385017\pi$
$719$	18.9536	0.706848	0.353424	+	0.935463i	$0.385017\pi$
$720$	−7.46727	−0.278289
$721$	40.2475	1.49889
$722$	−11.5878	−0.431252
$723$	−0.648232	−0.0241080
$724$	6.14897	0.228525
$725$	−3.64165	−0.135247
$726$	0.245716	0.00911938
$727$	−12.3450	−0.457849	−0.228925	−	0.973444i	$-0.573521\pi$
$727$	−12.3450	−0.457849	−0.228925	+	0.973444i	$0.573521\pi$
$728$	−1.21945	−0.0451957
$729$	−26.1377	−0.968063
$730$	8.04038	0.297588
$731$	33.1376	1.22564
$732$	1.62734	0.0601481
$733$	30.6547	1.13226	0.566128	−	0.824317i	$-0.308441\pi$
$733$	30.6547	1.13226	0.566128	+	0.824317i	$0.308441\pi$
$734$	10.6571	0.393359
$735$	−0.871267	−0.0321372
$736$	6.27644	0.231353
$737$	3.68869	0.135875
$738$	1.10122	0.0405365
$739$	−11.2582	−0.414140	−0.207070	−	0.978326i	$-0.566393\pi$
$739$	−11.2582	−0.414140	−0.207070	+	0.978326i	$0.566393\pi$
$740$	11.5918	0.426124
$741$	0.134667	0.00494710
$742$	−9.01481	−0.330944
$743$	15.1182	0.554634	0.277317	−	0.960778i	$-0.410555\pi$
$743$	15.1182	0.554634	0.277317	+	0.960778i	$0.410555\pi$
$744$	0.929752	0.0340864
$745$	−39.8438	−1.45976
$746$	25.7862	0.944100
$747$	2.39538	0.0876425
$748$	−13.1167	−0.479593
$749$	53.2268	1.94487
$750$	1.18460	0.0432554
$751$	−13.6534	−0.498219	−0.249110	−	0.968475i	$-0.580138\pi$
$751$	−13.6534	−0.498219	−0.249110	+	0.968475i	$0.580138\pi$
$752$	−6.02468	−0.219697
$753$	2.49014	0.0907458
$754$	−1.12667	−0.0410310
$755$	−27.1563	−0.988318
$756$	−2.36632	−0.0860622
$757$	−47.9357	−1.74225	−0.871127	−	0.491058i	$-0.836610\pi$
$757$	−47.9357	−1.74225	−0.871127	+	0.491058i	$0.836610\pi$
$758$	9.98509	0.362675
$759$	−2.39265	−0.0868476
$760$	−6.81307	−0.247136
$761$	46.4161	1.68258	0.841291	−	0.540582i	$-0.181796\pi$
$761$	46.4161	1.68258	0.841291	+	0.540582i	$0.181796\pi$
$762$	0.119533	0.00433023
$763$	−43.9728	−1.59192
$764$	−0.0544458	−0.00196978
$765$	32.5405	1.17650
$766$	29.2565	1.05708
$767$	5.35222	0.193257
$768$	−0.126650	−0.00457007
$769$	18.2813	0.659242	0.329621	−	0.944113i	$-0.393079\pi$
$769$	18.2813	0.659242	0.329621	+	0.944113i	$0.393079\pi$
$770$	−23.5185	−0.847550
$771$	1.11097	0.0400105
$772$	−16.1070	−0.579705
$773$	−42.5397	−1.53005	−0.765024	−	0.644002i	$-0.777273\pi$
$773$	−42.5397	−1.53005	−0.765024	+	0.644002i	$0.777273\pi$
$774$	22.6909	0.815607
$775$	−9.26712	−0.332885
$776$	−0.991788	−0.0356031
$777$	1.83176	0.0657139
$778$	15.2945	0.548336
$779$	1.00474	0.0359987
$780$	−0.123781	−0.00443208
$781$	−26.3881	−0.944240
$782$	−27.3512	−0.978075
$783$	−2.18630	−0.0781319
$784$	2.74902	0.0981794
$785$	53.2079	1.89907
$786$	−1.46912	−0.0524019
$787$	48.7724	1.73855	0.869274	−	0.494330i	$-0.164587\pi$
$787$	48.7724	1.73855	0.869274	+	0.494330i	$0.164587\pi$
$788$	−8.52039	−0.303526
$789$	1.58425	0.0564009
$790$	−24.3434	−0.866099
$791$	−12.0636	−0.428933
$792$	−8.98161	−0.319148
$793$	−5.01829	−0.178205
$794$	6.29432	0.223377
$795$	−0.915058	−0.0324538
$796$	−21.5290	−0.763076
$797$	24.1444	0.855240	0.427620	−	0.903959i	$-0.359352\pi$
$797$	24.1444	0.855240	0.427620	+	0.903959i	$0.359352\pi$
$798$	−1.07661	−0.0381116
$799$	26.2540	0.928800
$800$	1.26236	0.0446310
$801$	25.6730	0.907112
$802$	10.5056	0.370964
$803$	9.67094	0.341280
$804$	−0.155208	−0.00547378
$805$	−49.0414	−1.72848
$806$	−2.86711	−0.100990
$807$	−3.48310	−0.122611
$808$	−12.0240	−0.423004
$809$	−20.5369	−0.722040	−0.361020	−	0.932558i	$-0.617571\pi$
$809$	−20.5369	−0.722040	−0.361020	+	0.932558i	$0.617571\pi$
$810$	22.1616	0.778680
$811$	−8.92591	−0.313431	−0.156716	−	0.987644i	$-0.550091\pi$
$811$	−8.92591	−0.313431	−0.156716	+	0.987644i	$0.550091\pi$
$812$	9.00735	0.316096
$813$	−1.27395	−0.0446794
$814$	13.9426	0.488688
$815$	17.3258	0.606896
$816$	0.551907	0.0193206
$817$	20.7030	0.724305
$818$	24.4450	0.854701
$819$	3.63878	0.127149
$820$	−0.923529	−0.0322510
$821$	−28.7099	−1.00198	−0.500991	−	0.865453i	$-0.667031\pi$
$821$	−28.7099	−1.00198	−0.500991	+	0.865453i	$0.667031\pi$
$822$	−1.06169	−0.0370308
$823$	13.3817	0.466457	0.233228	−	0.972422i	$-0.425071\pi$
$823$	13.3817	0.466457	0.233228	+	0.972422i	$0.425071\pi$
$824$	−12.8902	−0.449050
$825$	−0.481223	−0.0167540
$826$	−42.7891	−1.48882
$827$	8.97382	0.312050	0.156025	−	0.987753i	$-0.450132\pi$
$827$	8.97382	0.312050	0.156025	+	0.987753i	$0.450132\pi$
$828$	−18.7287	−0.650866
$829$	−43.7005	−1.51778	−0.758890	−	0.651219i	$-0.774258\pi$
$829$	−43.7005	−1.51778	−0.758890	+	0.651219i	$0.774258\pi$
$830$	−2.00887	−0.0697288
$831$	3.10881	0.107844
$832$	0.390555	0.0135400
$833$	−11.9795	−0.415067
$834$	−2.86003	−0.0990347
$835$	26.4177	0.914224
$836$	−8.19474	−0.283421
$837$	−5.56360	−0.192306
$838$	23.6498	0.816969
$839$	44.3583	1.53142	0.765709	−	0.643187i	$-0.222388\pi$
$839$	44.3583	1.53142	0.765709	+	0.643187i	$0.222388\pi$
$840$	0.989585	0.0341439
$841$	−20.6779	−0.713031
$842$	17.6998	0.609977
$843$	0.550757	0.0189691
$844$	−1.17635	−0.0404915
$845$	−32.1504	−1.10601
$846$	17.9774	0.618075
$847$	6.05773	0.208146
$848$	2.88719	0.0991467
$849$	0.976057	0.0334982
$850$	−5.50103	−0.188684
$851$	29.0735	0.996625
$852$	1.11033	0.0380392
$853$	−16.0290	−0.548824	−0.274412	−	0.961612i	$-0.588483\pi$
$853$	−16.0290	−0.548824	−0.274412	+	0.961612i	$0.588483\pi$
$854$	40.1194	1.37286
$855$	20.3299	0.695269
$856$	−17.0471	−0.582657
$857$	−47.0530	−1.60730	−0.803649	−	0.595103i	$-0.797111\pi$
$857$	−47.0530	−1.60730	−0.803649	+	0.595103i	$0.797111\pi$
$858$	−0.148884	−0.00508280
$859$	57.1503	1.94994	0.974971	−	0.222332i	$-0.0713668\pi$
$859$	57.1503	1.94994	0.974971	+	0.222332i	$0.0713668\pi$
$860$	−19.0295	−0.648901
$861$	−0.145937	−0.00497353
$862$	−35.5338	−1.21029
$863$	44.6109	1.51857	0.759286	−	0.650757i	$-0.225548\pi$
$863$	44.6109	1.51857	0.759286	+	0.650757i	$0.225548\pi$
$864$	0.757866	0.0257831
$865$	−57.7207	−1.96256
$866$	−3.27087	−0.111149
$867$	−0.252028	−0.00855933
$868$	22.9215	0.778008
$869$	−29.2802	−0.993261
$870$	0.914301	0.0309977
$871$	0.478622	0.0162175
$872$	14.0833	0.476920
$873$	2.95946	0.100162
$874$	−17.0879	−0.578006
$875$	29.2043	0.987287
$876$	−0.406922	−0.0137486
$877$	−28.1902	−0.951917	−0.475959	−	0.879468i	$-0.657899\pi$
$877$	−28.1902	−0.951917	−0.475959	+	0.879468i	$0.657899\pi$
$878$	22.7977	0.769385
$879$	0.521591	0.0175928
$880$	7.53234	0.253915
$881$	20.8092	0.701079	0.350539	−	0.936548i	$-0.385998\pi$
$881$	20.8092	0.701079	0.350539	+	0.936548i	$0.385998\pi$
$882$	−8.20298	−0.276209
$883$	46.4087	1.56178	0.780888	−	0.624671i	$-0.214767\pi$
$883$	46.4087	1.56178	0.780888	+	0.624671i	$0.214767\pi$
$884$	−1.70194	−0.0572424
$885$	−4.34335	−0.146000
$886$	−15.4272	−0.518286
$887$	−37.3056	−1.25260	−0.626299	−	0.779583i	$-0.715431\pi$
$887$	−37.3056	−1.25260	−0.626299	+	0.779583i	$0.715431\pi$
$888$	−0.586661	−0.0196870
$889$	2.94690	0.0988358
$890$	−21.5304	−0.721702
$891$	26.6559	0.893007
$892$	4.45579	0.149191
$893$	16.4024	0.548886
$894$	2.01649	0.0674414
$895$	20.1655	0.674058
$896$	−3.12234	−0.104310
$897$	−0.310455	−0.0103658
$898$	−13.6450	−0.455339
$899$	21.1777	0.706317
$900$	−3.76682	−0.125561
$901$	−12.5817	−0.419156
$902$	−1.11082	−0.0369862
$903$	−3.00707	−0.100069
$904$	3.86364	0.128503
$905$	15.3876	0.511501
$906$	1.37437	0.0456605
$907$	−10.9397	−0.363246	−0.181623	−	0.983368i	$-0.558135\pi$
$907$	−10.9397	−0.363246	−0.181623	+	0.983368i	$0.558135\pi$
$908$	19.3013	0.640536
$909$	35.8792	1.19004
$910$	−3.05163	−0.101160
$911$	−7.46598	−0.247359	−0.123679	−	0.992322i	$-0.539469\pi$
$911$	−7.46598	−0.247359	−0.123679	+	0.992322i	$0.539469\pi$
$912$	0.344809	0.0114178
$913$	−2.41626	−0.0799665
$914$	−12.0586	−0.398865
$915$	4.07236	0.134628
$916$	−26.1577	−0.864275
$917$	−36.2189	−1.19605
$918$	−3.30259	−0.109002
$919$	45.6370	1.50543	0.752713	−	0.658349i	$-0.228745\pi$
$919$	45.6370	1.50543	0.752713	+	0.658349i	$0.228745\pi$
$920$	15.7066	0.517832
$921$	0.891081	0.0293621
$922$	−7.53956	−0.248302
$923$	−3.42396	−0.112701
$924$	1.19027	0.0391570
$925$	5.84742	0.192262
$926$	−11.3090	−0.371635
$927$	38.4637	1.26331
$928$	−2.88480	−0.0946984
$929$	−7.40149	−0.242835	−0.121417	−	0.992602i	$-0.538744\pi$
$929$	−7.40149	−0.242835	−0.121417	+	0.992602i	$0.538744\pi$
$930$	2.32668	0.0762947
$931$	−7.48433	−0.245289
$932$	−20.2416	−0.663036
$933$	−0.438172	−0.0143451
$934$	−8.17243	−0.267410
$935$	−32.8240	−1.07346
$936$	−1.16540	−0.0380923
$937$	−11.2189	−0.366505	−0.183253	−	0.983066i	$-0.558663\pi$
$937$	−11.2189	−0.366505	−0.183253	+	0.983066i	$0.558663\pi$
$938$	−3.82641	−0.124937
$939$	2.57934	0.0841735
$940$	−15.0766	−0.491744
$941$	−53.1502	−1.73265	−0.866323	−	0.499484i	$-0.833523\pi$
$941$	−53.1502	−1.73265	−0.866323	+	0.499484i	$0.833523\pi$
$942$	−2.69284	−0.0877376
$943$	−2.31630	−0.0754292
$944$	13.7042	0.446032
$945$	−5.92164	−0.192631
$946$	−22.8886	−0.744173
$947$	−9.21451	−0.299431	−0.149716	−	0.988729i	$-0.547836\pi$
$947$	−9.21451	−0.299431	−0.149716	+	0.988729i	$0.547836\pi$
$948$	1.23202	0.0400140
$949$	1.25484	0.0407339
$950$	−3.43681	−0.111505
$951$	−0.500389	−0.0162262
$952$	13.6064	0.440985
$953$	−7.72435	−0.250216	−0.125108	−	0.992143i	$-0.539928\pi$
$953$	−7.72435	−0.250216	−0.125108	+	0.992143i	$0.539928\pi$
$954$	−8.61527	−0.278930
$955$	−0.136249	−0.00440891
$956$	−11.8533	−0.383364
$957$	1.09972	0.0355488
$958$	31.2123	1.00842
$959$	−26.1743	−0.845213
$960$	−0.316937	−0.0102291
$961$	22.8922	0.738459
$962$	1.80911	0.0583280
$963$	50.8678	1.63919
$964$	5.11831	0.164850
$965$	−40.3074	−1.29754
$966$	2.48198	0.0798563
$967$	−1.21696	−0.0391348	−0.0195674	−	0.999809i	$-0.506229\pi$
$967$	−1.21696	−0.0391348	−0.0195674	+	0.999809i	$0.506229\pi$
$968$	−1.94012	−0.0623580
$969$	−1.50259	−0.0482701
$970$	−2.48192	−0.0796897
$971$	−41.6620	−1.33700	−0.668499	−	0.743713i	$-0.733063\pi$
$971$	−41.6620	−1.33700	−0.668499	+	0.743713i	$0.733063\pi$
$972$	−3.39519	−0.108901
$973$	−70.5094	−2.26043
$974$	26.7699	0.857762
$975$	−0.0624406	−0.00199970
$976$	−12.8491	−0.411291
$977$	−12.3489	−0.395077	−0.197539	−	0.980295i	$-0.563295\pi$
$977$	−12.3489	−0.395077	−0.197539	+	0.980295i	$0.563295\pi$
$978$	−0.876856	−0.0280388
$979$	−25.8967	−0.827664
$980$	6.87935	0.219753
$981$	−42.0240	−1.34172
$982$	−40.7714	−1.30107
$983$	−38.9411	−1.24203	−0.621013	−	0.783800i	$-0.713279\pi$
$983$	−38.9411	−1.24203	−0.621013	+	0.783800i	$0.713279\pi$
$984$	0.0467396	0.00149001
$985$	−21.3220	−0.679376
$986$	12.5713	0.400350
$987$	−2.38242	−0.0758332
$988$	−1.06330	−0.0338281
$989$	−47.7279	−1.51766
$990$	−22.4762	−0.714341
$991$	55.8994	1.77570	0.887852	−	0.460130i	$-0.152197\pi$
$991$	55.8994	1.77570	0.887852	+	0.460130i	$0.152197\pi$
$992$	−7.34113	−0.233081
$993$	−1.79823	−0.0570651
$994$	27.3733	0.868229
$995$	−53.8758	−1.70798
$996$	0.101668	0.00322149
$997$	39.7755	1.25970	0.629852	−	0.776715i	$-0.283116\pi$
$997$	39.7755	1.25970	0.629852	+	0.776715i	$0.283116\pi$
$998$	−22.6172	−0.715935
$999$	3.51055	0.111069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.36	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.36	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.126650 q^{3} +1.00000 q^{4} +2.50247 q^{5} -0.126650 q^{6} -3.12234 q^{7} +1.00000 q^{8} -2.98396 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.126650 q^{3} +1.00000 q^{4} +2.50247 q^{5} -0.126650 q^{6} -3.12234 q^{7} +1.00000 q^{8} -2.98396 q^{9} +2.50247 q^{10} +3.00996 q^{11} -0.126650 q^{12} +0.390555 q^{13} -3.12234 q^{14} -0.316937 q^{15} +1.00000 q^{16} -4.35775 q^{17} -2.98396 q^{18} -2.72254 q^{19} +2.50247 q^{20} +0.395443 q^{21} +3.00996 q^{22} +6.27644 q^{23} -0.126650 q^{24} +1.26236 q^{25} +0.390555 q^{26} +0.757866 q^{27} -3.12234 q^{28} -2.88480 q^{29} -0.316937 q^{30} -7.34113 q^{31} +1.00000 q^{32} -0.381211 q^{33} -4.35775 q^{34} -7.81357 q^{35} -2.98396 q^{36} +4.63215 q^{37} -2.72254 q^{38} -0.0494636 q^{39} +2.50247 q^{40} -0.369047 q^{41} +0.395443 q^{42} -7.60429 q^{43} +3.00996 q^{44} -7.46727 q^{45} +6.27644 q^{46} -6.02468 q^{47} -0.126650 q^{48} +2.74902 q^{49} +1.26236 q^{50} +0.551907 q^{51} +0.390555 q^{52} +2.88719 q^{53} +0.757866 q^{54} +7.53234 q^{55} -3.12234 q^{56} +0.344809 q^{57} -2.88480 q^{58} +13.7042 q^{59} -0.316937 q^{60} -12.8491 q^{61} -7.34113 q^{62} +9.31695 q^{63} +1.00000 q^{64} +0.977351 q^{65} -0.381211 q^{66} +1.22549 q^{67} -4.35775 q^{68} -0.794909 q^{69} -7.81357 q^{70} -8.76692 q^{71} -2.98396 q^{72} +3.21298 q^{73} +4.63215 q^{74} -0.159877 q^{75} -2.72254 q^{76} -9.39813 q^{77} -0.0494636 q^{78} -9.72775 q^{79} +2.50247 q^{80} +8.85590 q^{81} -0.369047 q^{82} -0.802753 q^{83} +0.395443 q^{84} -10.9051 q^{85} -7.60429 q^{86} +0.365359 q^{87} +3.00996 q^{88} -8.60368 q^{89} -7.46727 q^{90} -1.21945 q^{91} +6.27644 q^{92} +0.929752 q^{93} -6.02468 q^{94} -6.81307 q^{95} -0.126650 q^{96} -0.991788 q^{97} +2.74902 q^{98} -8.98161 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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