LMFDB - Embedded newform 8042.2.a.a.1.67

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			1.67
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)

$f(q)$	$=$	$q+1.00000 q^{2} +3.07757 q^{3} +1.00000 q^{4} -2.88236 q^{5} +3.07757 q^{6} -3.62182 q^{7} +1.00000 q^{8} +6.47145 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.07757 q^{3} +1.00000 q^{4} -2.88236 q^{5} +3.07757 q^{6} -3.62182 q^{7} +1.00000 q^{8} +6.47145 q^{9} -2.88236 q^{10} -2.46763 q^{11} +3.07757 q^{12} -2.59952 q^{13} -3.62182 q^{14} -8.87067 q^{15} +1.00000 q^{16} +5.74708 q^{17} +6.47145 q^{18} +2.97169 q^{19} -2.88236 q^{20} -11.1464 q^{21} -2.46763 q^{22} -2.72648 q^{23} +3.07757 q^{24} +3.30800 q^{25} -2.59952 q^{26} +10.6836 q^{27} -3.62182 q^{28} +4.95093 q^{29} -8.87067 q^{30} -4.81951 q^{31} +1.00000 q^{32} -7.59430 q^{33} +5.74708 q^{34} +10.4394 q^{35} +6.47145 q^{36} -4.74011 q^{37} +2.97169 q^{38} -8.00020 q^{39} -2.88236 q^{40} -6.06980 q^{41} -11.1464 q^{42} +0.943720 q^{43} -2.46763 q^{44} -18.6531 q^{45} -2.72648 q^{46} -7.10843 q^{47} +3.07757 q^{48} +6.11756 q^{49} +3.30800 q^{50} +17.6870 q^{51} -2.59952 q^{52} -11.0011 q^{53} +10.6836 q^{54} +7.11259 q^{55} -3.62182 q^{56} +9.14558 q^{57} +4.95093 q^{58} -12.6247 q^{59} -8.87067 q^{60} +2.56114 q^{61} -4.81951 q^{62} -23.4384 q^{63} +1.00000 q^{64} +7.49275 q^{65} -7.59430 q^{66} -5.84955 q^{67} +5.74708 q^{68} -8.39094 q^{69} +10.4394 q^{70} -6.02659 q^{71} +6.47145 q^{72} -4.97493 q^{73} -4.74011 q^{74} +10.1806 q^{75} +2.97169 q^{76} +8.93729 q^{77} -8.00020 q^{78} -8.56809 q^{79} -2.88236 q^{80} +13.4653 q^{81} -6.06980 q^{82} +2.94794 q^{83} -11.1464 q^{84} -16.5651 q^{85} +0.943720 q^{86} +15.2369 q^{87} -2.46763 q^{88} -7.53350 q^{89} -18.6531 q^{90} +9.41498 q^{91} -2.72648 q^{92} -14.8324 q^{93} -7.10843 q^{94} -8.56547 q^{95} +3.07757 q^{96} +5.01208 q^{97} +6.11756 q^{98} -15.9691 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.07757	1.77684	0.888419	−	0.459034i	$-0.151804\pi$
$3$	3.07757	1.77684	0.888419	+	0.459034i	$0.151804\pi$
$4$	1.00000	0.500000
$5$	−2.88236	−1.28903	−0.644516	−	0.764591i	$-0.722941\pi$
$5$	−2.88236	−1.28903	−0.644516	+	0.764591i	$0.722941\pi$
$6$	3.07757	1.25641
$7$	−3.62182	−1.36892	−0.684459	−	0.729051i	$-0.739962\pi$
$7$	−3.62182	−1.36892	−0.684459	+	0.729051i	$0.739962\pi$
$8$	1.00000	0.353553
$9$	6.47145	2.15715
$10$	−2.88236	−0.911483
$11$	−2.46763	−0.744017	−0.372009	−	0.928229i	$-0.621331\pi$
$11$	−2.46763	−0.744017	−0.372009	+	0.928229i	$0.621331\pi$
$12$	3.07757	0.888419
$13$	−2.59952	−0.720976	−0.360488	−	0.932764i	$-0.617390\pi$
$13$	−2.59952	−0.720976	−0.360488	+	0.932764i	$0.617390\pi$
$14$	−3.62182	−0.967971
$15$	−8.87067	−2.29040
$16$	1.00000	0.250000
$17$	5.74708	1.39387	0.696935	−	0.717134i	$-0.254546\pi$
$17$	5.74708	1.39387	0.696935	+	0.717134i	$0.254546\pi$
$18$	6.47145	1.52534
$19$	2.97169	0.681752	0.340876	−	0.940108i	$-0.389276\pi$
$19$	2.97169	0.681752	0.340876	+	0.940108i	$0.389276\pi$
$20$	−2.88236	−0.644516
$21$	−11.1464	−2.43234
$22$	−2.46763	−0.526100
$23$	−2.72648	−0.568511	−0.284255	−	0.958749i	$-0.591746\pi$
$23$	−2.72648	−0.568511	−0.284255	+	0.958749i	$0.591746\pi$
$24$	3.07757	0.628207
$25$	3.30800	0.661601
$26$	−2.59952	−0.509807
$27$	10.6836	2.05607
$28$	−3.62182	−0.684459
$29$	4.95093	0.919365	0.459683	−	0.888083i	$-0.347963\pi$
$29$	4.95093	0.919365	0.459683	+	0.888083i	$0.347963\pi$
$30$	−8.87067	−1.61956
$31$	−4.81951	−0.865609	−0.432805	−	0.901488i	$-0.642476\pi$
$31$	−4.81951	−0.865609	−0.432805	+	0.901488i	$0.642476\pi$
$32$	1.00000	0.176777
$33$	−7.59430	−1.32200
$34$	5.74708	0.985616
$35$	10.4394	1.76458
$36$	6.47145	1.07858
$37$	−4.74011	−0.779270	−0.389635	−	0.920969i	$-0.627399\pi$
$37$	−4.74011	−0.779270	−0.389635	+	0.920969i	$0.627399\pi$
$38$	2.97169	0.482071
$39$	−8.00020	−1.28106
$40$	−2.88236	−0.455741
$41$	−6.06980	−0.947944	−0.473972	−	0.880540i	$-0.657180\pi$
$41$	−6.06980	−0.947944	−0.473972	+	0.880540i	$0.657180\pi$
$42$	−11.1464	−1.71993
$43$	0.943720	0.143916	0.0719580	−	0.997408i	$-0.477075\pi$
$43$	0.943720	0.143916	0.0719580	+	0.997408i	$0.477075\pi$
$44$	−2.46763	−0.372009
$45$	−18.6531	−2.78063
$46$	−2.72648	−0.401998
$47$	−7.10843	−1.03687	−0.518435	−	0.855117i	$-0.673485\pi$
$47$	−7.10843	−1.03687	−0.518435	+	0.855117i	$0.673485\pi$
$48$	3.07757	0.444209
$49$	6.11756	0.873937
$50$	3.30800	0.467823
$51$	17.6870	2.47668
$52$	−2.59952	−0.360488
$53$	−11.0011	−1.51112	−0.755561	−	0.655078i	$-0.772636\pi$
$53$	−11.0011	−1.51112	−0.755561	+	0.655078i	$0.772636\pi$
$54$	10.6836	1.45386
$55$	7.11259	0.959062
$56$	−3.62182	−0.483986
$57$	9.14558	1.21136
$58$	4.95093	0.650089
$59$	−12.6247	−1.64359	−0.821796	−	0.569781i	$-0.807028\pi$
$59$	−12.6247	−1.64359	−0.821796	+	0.569781i	$0.807028\pi$
$60$	−8.87067	−1.14520
$61$	2.56114	0.327920	0.163960	−	0.986467i	$-0.447573\pi$
$61$	2.56114	0.327920	0.163960	+	0.986467i	$0.447573\pi$
$62$	−4.81951	−0.612078
$63$	−23.4384	−2.95296
$64$	1.00000	0.125000
$65$	7.49275	0.929361
$66$	−7.59430	−0.934794
$67$	−5.84955	−0.714636	−0.357318	−	0.933983i	$-0.616309\pi$
$67$	−5.84955	−0.714636	−0.357318	+	0.933983i	$0.616309\pi$
$68$	5.74708	0.696935
$69$	−8.39094	−1.01015
$70$	10.4394	1.24775
$71$	−6.02659	−0.715224	−0.357612	−	0.933870i	$-0.616409\pi$
$71$	−6.02659	−0.715224	−0.357612	+	0.933870i	$0.616409\pi$
$72$	6.47145	0.762668
$73$	−4.97493	−0.582272	−0.291136	−	0.956682i	$-0.594033\pi$
$73$	−4.97493	−0.582272	−0.291136	+	0.956682i	$0.594033\pi$
$74$	−4.74011	−0.551027
$75$	10.1806	1.17556
$76$	2.97169	0.340876
$77$	8.93729	1.01850
$78$	−8.00020	−0.905845
$79$	−8.56809	−0.963985	−0.481993	−	0.876175i	$-0.660087\pi$
$79$	−8.56809	−0.963985	−0.481993	+	0.876175i	$0.660087\pi$
$80$	−2.88236	−0.322258
$81$	13.4653	1.49615
$82$	−6.06980	−0.670298
$83$	2.94794	0.323578	0.161789	−	0.986825i	$-0.448274\pi$
$83$	2.94794	0.323578	0.161789	+	0.986825i	$0.448274\pi$
$84$	−11.1464	−1.21617
$85$	−16.5651	−1.79674
$86$	0.943720	0.101764
$87$	15.2369	1.63356
$88$	−2.46763	−0.263050
$89$	−7.53350	−0.798550	−0.399275	−	0.916831i	$-0.630738\pi$
$89$	−7.53350	−0.798550	−0.399275	+	0.916831i	$0.630738\pi$
$90$	−18.6531	−1.96620
$91$	9.41498	0.986958
$92$	−2.72648	−0.284255
$93$	−14.8324	−1.53805
$94$	−7.10843	−0.733178
$95$	−8.56547	−0.878799
$96$	3.07757	0.314103
$97$	5.01208	0.508899	0.254450	−	0.967086i	$-0.418106\pi$
$97$	5.01208	0.508899	0.254450	+	0.967086i	$0.418106\pi$
$98$	6.11756	0.617967
$99$	−15.9691	−1.60496
$100$	3.30800	0.330800
$101$	4.23423	0.421322	0.210661	−	0.977559i	$-0.432438\pi$
$101$	4.23423	0.421322	0.210661	+	0.977559i	$0.432438\pi$
$102$	17.6870	1.75128
$103$	−3.06039	−0.301550	−0.150775	−	0.988568i	$-0.548177\pi$
$103$	−3.06039	−0.301550	−0.150775	+	0.988568i	$0.548177\pi$
$104$	−2.59952	−0.254904
$105$	32.1280	3.13537
$106$	−11.0011	−1.06852
$107$	13.2356	1.27954	0.639768	−	0.768568i	$-0.279031\pi$
$107$	13.2356	1.27954	0.639768	+	0.768568i	$0.279031\pi$
$108$	10.6836	1.02803
$109$	5.91937	0.566972	0.283486	−	0.958976i	$-0.408509\pi$
$109$	5.91937	0.566972	0.283486	+	0.958976i	$0.408509\pi$
$110$	7.11259	0.678159
$111$	−14.5880	−1.38464
$112$	−3.62182	−0.342230
$113$	−3.66373	−0.344654	−0.172327	−	0.985040i	$-0.555129\pi$
$113$	−3.66373	−0.344654	−0.172327	+	0.985040i	$0.555129\pi$
$114$	9.14558	0.856562
$115$	7.85870	0.732828
$116$	4.95093	0.459683
$117$	−16.8226	−1.55525
$118$	−12.6247	−1.16220
$119$	−20.8149	−1.90810
$120$	−8.87067	−0.809778
$121$	−4.91082	−0.446438
$122$	2.56114	0.231875
$123$	−18.6803	−1.68434
$124$	−4.81951	−0.432805
$125$	4.87694	0.436207
$126$	−23.4384	−2.08806
$127$	−9.11417	−0.808752	−0.404376	−	0.914593i	$-0.632511\pi$
$127$	−9.11417	−0.808752	−0.404376	+	0.914593i	$0.632511\pi$
$128$	1.00000	0.0883883
$129$	2.90437	0.255715
$130$	7.49275	0.657157
$131$	−5.37444	−0.469567	−0.234784	−	0.972048i	$-0.575438\pi$
$131$	−5.37444	−0.469567	−0.234784	+	0.972048i	$0.575438\pi$
$132$	−7.59430	−0.660999
$133$	−10.7629	−0.933262
$134$	−5.84955	−0.505324
$135$	−30.7941	−2.65033
$136$	5.74708	0.492808
$137$	9.06751	0.774690	0.387345	−	0.921935i	$-0.373392\pi$
$137$	9.06751	0.774690	0.387345	+	0.921935i	$0.373392\pi$
$138$	−8.39094	−0.714284
$139$	−6.72284	−0.570224	−0.285112	−	0.958494i	$-0.592031\pi$
$139$	−6.72284	−0.570224	−0.285112	+	0.958494i	$0.592031\pi$
$140$	10.4394	0.882289
$141$	−21.8767	−1.84235
$142$	−6.02659	−0.505740
$143$	6.41464	0.536419
$144$	6.47145	0.539288
$145$	−14.2704	−1.18509
$146$	−4.97493	−0.411728
$147$	18.8272	1.55284
$148$	−4.74011	−0.389635
$149$	−11.6484	−0.954272	−0.477136	−	0.878829i	$-0.658325\pi$
$149$	−11.6484	−0.954272	−0.477136	+	0.878829i	$0.658325\pi$
$150$	10.1806	0.831244
$151$	3.65955	0.297810	0.148905	−	0.988851i	$-0.452425\pi$
$151$	3.65955	0.297810	0.148905	+	0.988851i	$0.452425\pi$
$152$	2.97169	0.241036
$153$	37.1919	3.00679
$154$	8.93729	0.720188
$155$	13.8916	1.11580
$156$	−8.00020	−0.640529
$157$	2.01280	0.160639	0.0803194	−	0.996769i	$-0.474406\pi$
$157$	2.01280	0.160639	0.0803194	+	0.996769i	$0.474406\pi$
$158$	−8.56809	−0.681640
$159$	−33.8568	−2.68502
$160$	−2.88236	−0.227871
$161$	9.87482	0.778245
$162$	13.4653	1.05794
$163$	12.9598	1.01509	0.507545	−	0.861625i	$-0.330553\pi$
$163$	12.9598	1.01509	0.507545	+	0.861625i	$0.330553\pi$
$164$	−6.06980	−0.473972
$165$	21.8895	1.70410
$166$	2.94794	0.228804
$167$	10.5632	0.817406	0.408703	−	0.912667i	$-0.365981\pi$
$167$	10.5632	0.817406	0.408703	+	0.912667i	$0.365981\pi$
$168$	−11.1464	−0.859964
$169$	−6.24251	−0.480193
$170$	−16.5651	−1.27049
$171$	19.2311	1.47064
$172$	0.943720	0.0719580
$173$	−2.86400	−0.217746	−0.108873	−	0.994056i	$-0.534724\pi$
$173$	−2.86400	−0.217746	−0.108873	+	0.994056i	$0.534724\pi$
$174$	15.2369	1.15510
$175$	−11.9810	−0.905678
$176$	−2.46763	−0.186004
$177$	−38.8534	−2.92040
$178$	−7.53350	−0.564660
$179$	−11.1504	−0.833423	−0.416712	−	0.909039i	$-0.636818\pi$
$179$	−11.1504	−0.833423	−0.416712	+	0.909039i	$0.636818\pi$
$180$	−18.6531	−1.39032
$181$	−26.6279	−1.97924	−0.989618	−	0.143724i	$-0.954092\pi$
$181$	−26.6279	−1.97924	−0.989618	+	0.143724i	$0.954092\pi$
$182$	9.41498	0.697884
$183$	7.88209	0.582661
$184$	−2.72648	−0.200999
$185$	13.6627	1.00450
$186$	−14.8324	−1.08756
$187$	−14.1816	−1.03706
$188$	−7.10843	−0.518435
$189$	−38.6942	−2.81459
$190$	−8.56547	−0.621405
$191$	23.6704	1.71273	0.856367	−	0.516368i	$-0.172716\pi$
$191$	23.6704	1.71273	0.856367	+	0.516368i	$0.172716\pi$
$192$	3.07757	0.222105
$193$	1.42708	0.102723	0.0513617	−	0.998680i	$-0.483644\pi$
$193$	1.42708	0.102723	0.0513617	+	0.998680i	$0.483644\pi$
$194$	5.01208	0.359846
$195$	23.0595	1.65132
$196$	6.11756	0.436968
$197$	−0.521513	−0.0371563	−0.0185781	−	0.999827i	$-0.505914\pi$
$197$	−0.521513	−0.0371563	−0.0185781	+	0.999827i	$0.505914\pi$
$198$	−15.9691	−1.13488
$199$	14.4011	1.02086	0.510432	−	0.859918i	$-0.329485\pi$
$199$	14.4011	1.02086	0.510432	+	0.859918i	$0.329485\pi$
$200$	3.30800	0.233911
$201$	−18.0024	−1.26979
$202$	4.23423	0.297919
$203$	−17.9314	−1.25854
$204$	17.6870	1.23834
$205$	17.4954	1.22193
$206$	−3.06039	−0.213228
$207$	−17.6443	−1.22636
$208$	−2.59952	−0.180244
$209$	−7.33301	−0.507235
$210$	32.1280	2.21704
$211$	17.5441	1.20778	0.603892	−	0.797066i	$-0.293616\pi$
$211$	17.5441	1.20778	0.603892	+	0.797066i	$0.293616\pi$
$212$	−11.0011	−0.755561
$213$	−18.5473	−1.27084
$214$	13.2356	0.904768
$215$	−2.72014	−0.185512
$216$	10.6836	0.726930
$217$	17.4554	1.18495
$218$	5.91937	0.400910
$219$	−15.3107	−1.03460
$220$	7.11259	0.479531
$221$	−14.9396	−1.00495
$222$	−14.5880	−0.979085
$223$	−20.3076	−1.35990	−0.679948	−	0.733261i	$-0.737998\pi$
$223$	−20.3076	−1.35990	−0.679948	+	0.733261i	$0.737998\pi$
$224$	−3.62182	−0.241993
$225$	21.4076	1.42717
$226$	−3.66373	−0.243707
$227$	−6.62127	−0.439469	−0.219735	−	0.975560i	$-0.570519\pi$
$227$	−6.62127	−0.439469	−0.219735	+	0.975560i	$0.570519\pi$
$228$	9.14558	0.605681
$229$	4.46536	0.295079	0.147540	−	0.989056i	$-0.452865\pi$
$229$	4.46536	0.295079	0.147540	+	0.989056i	$0.452865\pi$
$230$	7.85870	0.518188
$231$	27.5052	1.80971
$232$	4.95093	0.325045
$233$	−9.62117	−0.630304	−0.315152	−	0.949041i	$-0.602055\pi$
$233$	−9.62117	−0.630304	−0.315152	+	0.949041i	$0.602055\pi$
$234$	−16.8226	−1.09973
$235$	20.4891	1.33656
$236$	−12.6247	−0.821796
$237$	−26.3689	−1.71284
$238$	−20.8149	−1.34923
$239$	17.9755	1.16274	0.581369	−	0.813640i	$-0.302517\pi$
$239$	17.9755	1.16274	0.581369	+	0.813640i	$0.302517\pi$
$240$	−8.87067	−0.572600
$241$	−1.45939	−0.0940077	−0.0470039	−	0.998895i	$-0.514967\pi$
$241$	−1.45939	−0.0940077	−0.0470039	+	0.998895i	$0.514967\pi$
$242$	−4.91082	−0.315679
$243$	9.38958	0.602342
$244$	2.56114	0.163960
$245$	−17.6330	−1.12653
$246$	−18.6803	−1.19101
$247$	−7.72495	−0.491527
$248$	−4.81951	−0.306039
$249$	9.07249	0.574945
$250$	4.87694	0.308445
$251$	−4.58050	−0.289119	−0.144559	−	0.989496i	$-0.546176\pi$
$251$	−4.58050	−0.289119	−0.144559	+	0.989496i	$0.546176\pi$
$252$	−23.4384	−1.47648
$253$	6.72794	0.422982
$254$	−9.11417	−0.571874
$255$	−50.9804	−3.19252
$256$	1.00000	0.0625000
$257$	11.8043	0.736330	0.368165	−	0.929761i	$-0.379986\pi$
$257$	11.8043	0.736330	0.368165	+	0.929761i	$0.379986\pi$
$258$	2.90437	0.180818
$259$	17.1678	1.06676
$260$	7.49275	0.464680
$261$	32.0397	1.98321
$262$	−5.37444	−0.332034
$263$	8.87382	0.547183	0.273592	−	0.961846i	$-0.411788\pi$
$263$	8.87382	0.547183	0.273592	+	0.961846i	$0.411788\pi$
$264$	−7.59430	−0.467397
$265$	31.7093	1.94788
$266$	−10.7629	−0.659916
$267$	−23.1849	−1.41889
$268$	−5.84955	−0.357318
$269$	−5.72224	−0.348891	−0.174446	−	0.984667i	$-0.555813\pi$
$269$	−5.72224	−0.348891	−0.174446	+	0.984667i	$0.555813\pi$
$270$	−30.7941	−1.87407
$271$	24.7189	1.50157	0.750784	−	0.660548i	$-0.229676\pi$
$271$	24.7189	1.50157	0.750784	+	0.660548i	$0.229676\pi$
$272$	5.74708	0.348468
$273$	28.9753	1.75366
$274$	9.06751	0.547789
$275$	−8.16292	−0.492243
$276$	−8.39094	−0.505075
$277$	−1.48030	−0.0889424	−0.0444712	−	0.999011i	$-0.514160\pi$
$277$	−1.48030	−0.0889424	−0.0444712	+	0.999011i	$0.514160\pi$
$278$	−6.72284	−0.403209
$279$	−31.1892	−1.86725
$280$	10.4394	0.623873
$281$	−19.4816	−1.16217	−0.581087	−	0.813842i	$-0.697372\pi$
$281$	−19.4816	−1.16217	−0.581087	+	0.813842i	$0.697372\pi$
$282$	−21.8767	−1.30274
$283$	26.2373	1.55964	0.779822	−	0.626002i	$-0.215310\pi$
$283$	26.2373	1.55964	0.779822	+	0.626002i	$0.215310\pi$
$284$	−6.02659	−0.357612
$285$	−26.3609	−1.56148
$286$	6.41464	0.379306
$287$	21.9837	1.29766
$288$	6.47145	0.381334
$289$	16.0289	0.942876
$290$	−14.2704	−0.837985
$291$	15.4250	0.904231
$292$	−4.97493	−0.291136
$293$	31.2258	1.82423	0.912114	−	0.409936i	$-0.134449\pi$
$293$	31.2258	1.82423	0.912114	+	0.409936i	$0.134449\pi$
$294$	18.8272	1.09803
$295$	36.3889	2.11864
$296$	−4.74011	−0.275513
$297$	−26.3632	−1.52975
$298$	−11.6484	−0.674772
$299$	7.08754	0.409883
$300$	10.1806	0.587779
$301$	−3.41798	−0.197009
$302$	3.65955	0.210584
$303$	13.0312	0.748620
$304$	2.97169	0.170438
$305$	−7.38213	−0.422700
$306$	37.1919	2.12612
$307$	−0.342982	−0.0195750	−0.00978750	−	0.999952i	$-0.503116\pi$
$307$	−0.342982	−0.0195750	−0.00978750	+	0.999952i	$0.503116\pi$
$308$	8.93729	0.509250
$309$	−9.41858	−0.535805
$310$	13.8916	0.788988
$311$	−11.7911	−0.668610	−0.334305	−	0.942465i	$-0.608502\pi$
$311$	−11.7911	−0.668610	−0.334305	+	0.942465i	$0.608502\pi$
$312$	−8.00020	−0.452922
$313$	8.66594	0.489828	0.244914	−	0.969545i	$-0.421240\pi$
$313$	8.66594	0.489828	0.244914	+	0.969545i	$0.421240\pi$
$314$	2.01280	0.113589
$315$	67.5580	3.80646
$316$	−8.56809	−0.481993
$317$	−28.6565	−1.60951	−0.804756	−	0.593606i	$-0.797704\pi$
$317$	−28.6565	−1.60951	−0.804756	+	0.593606i	$0.797704\pi$
$318$	−33.8568	−1.89859
$319$	−12.2171	−0.684024
$320$	−2.88236	−0.161129
$321$	40.7336	2.27353
$322$	9.87482	0.550302
$323$	17.0785	0.950274
$324$	13.4653	0.748073
$325$	−8.59922	−0.476999
$326$	12.9598	0.717777
$327$	18.2173	1.00742
$328$	−6.06980	−0.335149
$329$	25.7454	1.41939
$330$	21.8895	1.20498
$331$	30.8971	1.69826	0.849130	−	0.528184i	$-0.177127\pi$
$331$	30.8971	1.69826	0.849130	+	0.528184i	$0.177127\pi$
$332$	2.94794	0.161789
$333$	−30.6754	−1.68100
$334$	10.5632	0.577994
$335$	16.8605	0.921189
$336$	−11.1464	−0.608086
$337$	−20.8945	−1.13820	−0.569099	−	0.822269i	$-0.692708\pi$
$337$	−20.8945	−1.13820	−0.569099	+	0.822269i	$0.692708\pi$
$338$	−6.24251	−0.339548
$339$	−11.2754	−0.612395
$340$	−16.5651	−0.898371
$341$	11.8928	0.644029
$342$	19.2311	1.03990
$343$	3.19604	0.172570
$344$	0.943720	0.0508820
$345$	24.1857	1.30212
$346$	−2.86400	−0.153969
$347$	−21.5113	−1.15479	−0.577394	−	0.816466i	$-0.695930\pi$
$347$	−21.5113	−1.15479	−0.577394	+	0.816466i	$0.695930\pi$
$348$	15.2369	0.816781
$349$	−8.80722	−0.471439	−0.235720	−	0.971821i	$-0.575745\pi$
$349$	−8.80722	−0.471439	−0.235720	+	0.971821i	$0.575745\pi$
$350$	−11.9810	−0.640411
$351$	−27.7723	−1.48238
$352$	−2.46763	−0.131525
$353$	−8.15052	−0.433809	−0.216904	−	0.976193i	$-0.569596\pi$
$353$	−8.15052	−0.433809	−0.216904	+	0.976193i	$0.569596\pi$
$354$	−38.8534	−2.06503
$355$	17.3708	0.921946
$356$	−7.53350	−0.399275
$357$	−64.0592	−3.39037
$358$	−11.1504	−0.589319
$359$	12.9908	0.685630	0.342815	−	0.939403i	$-0.388620\pi$
$359$	12.9908	0.685630	0.342815	+	0.939403i	$0.388620\pi$
$360$	−18.6531	−0.983102
$361$	−10.1691	−0.535215
$362$	−26.6279	−1.39953
$363$	−15.1134	−0.793248
$364$	9.41498	0.493479
$365$	14.3396	0.750567
$366$	7.88209	0.412004
$367$	15.0119	0.783615	0.391808	−	0.920047i	$-0.371850\pi$
$367$	15.0119	0.783615	0.391808	+	0.920047i	$0.371850\pi$
$368$	−2.72648	−0.142128
$369$	−39.2804	−2.04486
$370$	13.6627	0.710291
$371$	39.8441	2.06860
$372$	−14.8324	−0.769024
$373$	8.92506	0.462122	0.231061	−	0.972939i	$-0.425780\pi$
$373$	8.92506	0.462122	0.231061	+	0.972939i	$0.425780\pi$
$374$	−14.1816	−0.733315
$375$	15.0091	0.775068
$376$	−7.10843	−0.366589
$377$	−12.8700	−0.662841
$378$	−38.6942	−1.99021
$379$	−37.1127	−1.90635	−0.953176	−	0.302417i	$-0.902206\pi$
$379$	−37.1127	−1.90635	−0.953176	+	0.302417i	$0.902206\pi$
$380$	−8.56547	−0.439399
$381$	−28.0495	−1.43702
$382$	23.6704	1.21109
$383$	−18.6654	−0.953755	−0.476878	−	0.878970i	$-0.658232\pi$
$383$	−18.6654	−0.953755	−0.476878	+	0.878970i	$0.658232\pi$
$384$	3.07757	0.157052
$385$	−25.7605	−1.31288
$386$	1.42708	0.0726364
$387$	6.10724	0.310448
$388$	5.01208	0.254450
$389$	−28.7894	−1.45968	−0.729841	−	0.683617i	$-0.760406\pi$
$389$	−28.7894	−1.45968	−0.729841	+	0.683617i	$0.760406\pi$
$390$	23.0595	1.16766
$391$	−15.6693	−0.792430
$392$	6.11756	0.308983
$393$	−16.5402	−0.834345
$394$	−0.521513	−0.0262735
$395$	24.6963	1.24261
$396$	−15.9691	−0.802479
$397$	0.972479	0.0488073	0.0244037	−	0.999702i	$-0.492231\pi$
$397$	0.972479	0.0488073	0.0244037	+	0.999702i	$0.492231\pi$
$398$	14.4011	0.721860
$399$	−33.1236	−1.65825
$400$	3.30800	0.165400
$401$	30.5001	1.52310	0.761551	−	0.648105i	$-0.224438\pi$
$401$	30.5001	1.52310	0.761551	+	0.648105i	$0.224438\pi$
$402$	−18.0024	−0.897879
$403$	12.5284	0.624084
$404$	4.23423	0.210661
$405$	−38.8119	−1.92858
$406$	−17.9314	−0.889919
$407$	11.6968	0.579790
$408$	17.6870	0.875639
$409$	−17.1245	−0.846754	−0.423377	−	0.905954i	$-0.639155\pi$
$409$	−17.1245	−0.846754	−0.423377	+	0.905954i	$0.639155\pi$
$410$	17.4954	0.864035
$411$	27.9059	1.37650
$412$	−3.06039	−0.150775
$413$	45.7243	2.24994
$414$	−17.6443	−0.867169
$415$	−8.49702	−0.417102
$416$	−2.59952	−0.127452
$417$	−20.6900	−1.01320
$418$	−7.33301	−0.358669
$419$	13.2566	0.647630	0.323815	−	0.946120i	$-0.395035\pi$
$419$	13.2566	0.647630	0.323815	+	0.946120i	$0.395035\pi$
$420$	32.1280	1.56768
$421$	−19.0017	−0.926085	−0.463043	−	0.886336i	$-0.653242\pi$
$421$	−19.0017	−0.926085	−0.463043	+	0.886336i	$0.653242\pi$
$422$	17.5441	0.854033
$423$	−46.0018	−2.23669
$424$	−11.0011	−0.534262
$425$	19.0114	0.922186
$426$	−18.5473	−0.898618
$427$	−9.27598	−0.448896
$428$	13.2356	0.639768
$429$	19.7415	0.953129
$430$	−2.72014	−0.131177
$431$	−4.18314	−0.201495	−0.100747	−	0.994912i	$-0.532123\pi$
$431$	−4.18314	−0.201495	−0.100747	+	0.994912i	$0.532123\pi$
$432$	10.6836	0.514017
$433$	−28.7368	−1.38100	−0.690502	−	0.723330i	$-0.742610\pi$
$433$	−28.7368	−1.38100	−0.690502	+	0.723330i	$0.742610\pi$
$434$	17.4554	0.837885
$435$	−43.9181	−2.10571
$436$	5.91937	0.283486
$437$	−8.10225	−0.387583
$438$	−15.3107	−0.731574
$439$	37.8828	1.80805	0.904023	−	0.427484i	$-0.140600\pi$
$439$	37.8828	1.80805	0.904023	+	0.427484i	$0.140600\pi$
$440$	7.11259	0.339079
$441$	39.5895	1.88521
$442$	−14.9396	−0.710606
$443$	22.5756	1.07260	0.536300	−	0.844027i	$-0.319821\pi$
$443$	22.5756	1.07260	0.536300	+	0.844027i	$0.319821\pi$
$444$	−14.5880	−0.692318
$445$	21.7143	1.02936
$446$	−20.3076	−0.961591
$447$	−35.8487	−1.69559
$448$	−3.62182	−0.171115
$449$	10.2167	0.482158	0.241079	−	0.970506i	$-0.422499\pi$
$449$	10.2167	0.482158	0.241079	+	0.970506i	$0.422499\pi$
$450$	21.4076	1.00916
$451$	14.9780	0.705287
$452$	−3.66373	−0.172327
$453$	11.2625	0.529160
$454$	−6.62127	−0.310752
$455$	−27.1374	−1.27222
$456$	9.14558	0.428281
$457$	4.18290	0.195668	0.0978340	−	0.995203i	$-0.468809\pi$
$457$	4.18290	0.195668	0.0978340	+	0.995203i	$0.468809\pi$
$458$	4.46536	0.208652
$459$	61.3997	2.86589
$460$	7.85870	0.366414
$461$	−4.15466	−0.193502	−0.0967509	−	0.995309i	$-0.530845\pi$
$461$	−4.15466	−0.193502	−0.0967509	+	0.995309i	$0.530845\pi$
$462$	27.5052	1.27966
$463$	−0.821621	−0.0381839	−0.0190920	−	0.999818i	$-0.506078\pi$
$463$	−0.821621	−0.0381839	−0.0190920	+	0.999818i	$0.506078\pi$
$464$	4.95093	0.229841
$465$	42.7523	1.98259
$466$	−9.62117	−0.445692
$467$	3.80629	0.176134	0.0880670	−	0.996115i	$-0.471931\pi$
$467$	3.80629	0.176134	0.0880670	+	0.996115i	$0.471931\pi$
$468$	−16.8226	−0.777627
$469$	21.1860	0.978279
$470$	20.4891	0.945090
$471$	6.19453	0.285429
$472$	−12.6247	−0.581098
$473$	−2.32875	−0.107076
$474$	−26.3689	−1.21116
$475$	9.83035	0.451048
$476$	−20.8149	−0.954048
$477$	−71.1933	−3.25972
$478$	17.9755	0.822180
$479$	6.05347	0.276590	0.138295	−	0.990391i	$-0.455838\pi$
$479$	6.05347	0.276590	0.138295	+	0.990391i	$0.455838\pi$
$480$	−8.87067	−0.404889
$481$	12.3220	0.561835
$482$	−1.45939	−0.0664735
$483$	30.3905	1.38281
$484$	−4.91082	−0.223219
$485$	−14.4466	−0.655987
$486$	9.38958	0.425920
$487$	37.8035	1.71304	0.856519	−	0.516115i	$-0.172622\pi$
$487$	37.8035	1.71304	0.856519	+	0.516115i	$0.172622\pi$
$488$	2.56114	0.115937
$489$	39.8847	1.80365
$490$	−17.6330	−0.796578
$491$	−41.2286	−1.86062	−0.930311	−	0.366772i	$-0.880463\pi$
$491$	−41.2286	−1.86062	−0.930311	+	0.366772i	$0.880463\pi$
$492$	−18.6803	−0.842171
$493$	28.4534	1.28148
$494$	−7.72495	−0.347562
$495$	46.0288	2.06884
$496$	−4.81951	−0.216402
$497$	21.8272	0.979084
$498$	9.07249	0.406548
$499$	2.15362	0.0964092	0.0482046	−	0.998837i	$-0.484650\pi$
$499$	2.15362	0.0964092	0.0482046	+	0.998837i	$0.484650\pi$
$500$	4.87694	0.218103
$501$	32.5091	1.45240
$502$	−4.58050	−0.204438
$503$	25.1511	1.12143	0.560717	−	0.828007i	$-0.310526\pi$
$503$	25.1511	1.12143	0.560717	+	0.828007i	$0.310526\pi$
$504$	−23.4384	−1.04403
$505$	−12.2046	−0.543097
$506$	6.72794	0.299093
$507$	−19.2118	−0.853225
$508$	−9.11417	−0.404376
$509$	−29.0667	−1.28836	−0.644178	−	0.764875i	$-0.722801\pi$
$509$	−29.0667	−1.28836	−0.644178	+	0.764875i	$0.722801\pi$
$510$	−50.9804	−2.25745
$511$	18.0183	0.797083
$512$	1.00000	0.0441942
$513$	31.7484	1.40173
$514$	11.8043	0.520664
$515$	8.82116	0.388707
$516$	2.90437	0.127858
$517$	17.5409	0.771450
$518$	17.1678	0.754311
$519$	−8.81416	−0.386899
$520$	7.49275	0.328579
$521$	23.1566	1.01451	0.507254	−	0.861796i	$-0.330660\pi$
$521$	23.1566	1.01451	0.507254	+	0.861796i	$0.330660\pi$
$522$	32.0397	1.40234
$523$	−41.8465	−1.82982	−0.914911	−	0.403656i	$-0.867739\pi$
$523$	−41.8465	−1.82982	−0.914911	+	0.403656i	$0.867739\pi$
$524$	−5.37444	−0.234784
$525$	−36.8724	−1.60924
$526$	8.87382	0.386917
$527$	−27.6981	−1.20655
$528$	−7.59430	−0.330499
$529$	−15.5663	−0.676796
$530$	31.7093	1.37736
$531$	−81.7000	−3.54548
$532$	−10.7629	−0.466631
$533$	15.7786	0.683445
$534$	−23.1849	−1.00331
$535$	−38.1498	−1.64936
$536$	−5.84955	−0.252662
$537$	−34.3163	−1.48086
$538$	−5.72224	−0.246703
$539$	−15.0959	−0.650224
$540$	−30.7941	−1.32517
$541$	7.63999	0.328469	0.164234	−	0.986421i	$-0.447485\pi$
$541$	7.63999	0.328469	0.164234	+	0.986421i	$0.447485\pi$
$542$	24.7189	1.06177
$543$	−81.9493	−3.51678
$544$	5.74708	0.246404
$545$	−17.0617	−0.730845
$546$	28.9753	1.24003
$547$	−23.1971	−0.991838	−0.495919	−	0.868369i	$-0.665169\pi$
$547$	−23.1971	−0.991838	−0.495919	+	0.868369i	$0.665169\pi$
$548$	9.06751	0.387345
$549$	16.5743	0.707374
$550$	−8.16292	−0.348068
$551$	14.7126	0.626779
$552$	−8.39094	−0.357142
$553$	31.0320	1.31962
$554$	−1.48030	−0.0628917
$555$	42.0480	1.78484
$556$	−6.72284	−0.285112
$557$	−23.3714	−0.990279	−0.495140	−	0.868813i	$-0.664883\pi$
$557$	−23.3714	−0.990279	−0.495140	+	0.868813i	$0.664883\pi$
$558$	−31.1892	−1.32034
$559$	−2.45322	−0.103760
$560$	10.4394	0.441144
$561$	−43.6450	−1.84269
$562$	−19.4816	−0.821781
$563$	17.5954	0.741557	0.370779	−	0.928721i	$-0.379091\pi$
$563$	17.5954	0.741557	0.370779	+	0.928721i	$0.379091\pi$
$564$	−21.8767	−0.921175
$565$	10.5602	0.444270
$566$	26.2373	1.10283
$567$	−48.7689	−2.04810
$568$	−6.02659	−0.252870
$569$	5.84237	0.244925	0.122463	−	0.992473i	$-0.460921\pi$
$569$	5.84237	0.244925	0.122463	+	0.992473i	$0.460921\pi$
$570$	−26.3609	−1.10413
$571$	11.8478	0.495815	0.247907	−	0.968784i	$-0.420257\pi$
$571$	11.8478	0.495815	0.247907	+	0.968784i	$0.420257\pi$
$572$	6.41464	0.268210
$573$	72.8475	3.04325
$574$	21.9837	0.917583
$575$	−9.01921	−0.376127
$576$	6.47145	0.269644
$577$	28.4653	1.18502	0.592512	−	0.805561i	$-0.298136\pi$
$577$	28.4653	1.18502	0.592512	+	0.805561i	$0.298136\pi$
$578$	16.0289	0.666714
$579$	4.39194	0.182523
$580$	−14.2704	−0.592545
$581$	−10.6769	−0.442952
$582$	15.4250	0.639388
$583$	27.1467	1.12430
$584$	−4.97493	−0.205864
$585$	48.4889	2.00477
$586$	31.2258	1.28992
$587$	4.26585	0.176070	0.0880352	−	0.996117i	$-0.471941\pi$
$587$	4.26585	0.176070	0.0880352	+	0.996117i	$0.471941\pi$
$588$	18.8272	0.776422
$589$	−14.3221	−0.590131
$590$	36.3889	1.49811
$591$	−1.60499	−0.0660207
$592$	−4.74011	−0.194817
$593$	4.30699	0.176867	0.0884334	−	0.996082i	$-0.471814\pi$
$593$	4.30699	0.176867	0.0884334	+	0.996082i	$0.471814\pi$
$594$	−26.3632	−1.08170
$595$	59.9959	2.45959
$596$	−11.6484	−0.477136
$597$	44.3203	1.81391
$598$	7.08754	0.289831
$599$	−10.9608	−0.447846	−0.223923	−	0.974607i	$-0.571886\pi$
$599$	−10.9608	−0.447846	−0.223923	+	0.974607i	$0.571886\pi$
$600$	10.1806	0.415622
$601$	12.5472	0.511813	0.255907	−	0.966702i	$-0.417626\pi$
$601$	12.5472	0.511813	0.255907	+	0.966702i	$0.417626\pi$
$602$	−3.41798	−0.139307
$603$	−37.8551	−1.54158
$604$	3.65955	0.148905
$605$	14.1547	0.575472
$606$	13.0312	0.529354
$607$	−1.73668	−0.0704898	−0.0352449	−	0.999379i	$-0.511221\pi$
$607$	−1.73668	−0.0704898	−0.0352449	+	0.999379i	$0.511221\pi$
$608$	2.97169	0.120518
$609$	−55.1851	−2.23621
$610$	−7.38213	−0.298894
$611$	18.4785	0.747559
$612$	37.1919	1.50339
$613$	13.1450	0.530921	0.265461	−	0.964122i	$-0.414476\pi$
$613$	13.1450	0.530921	0.265461	+	0.964122i	$0.414476\pi$
$614$	−0.342982	−0.0138416
$615$	53.8433	2.17117
$616$	8.93729	0.360094
$617$	−26.5964	−1.07073	−0.535366	−	0.844620i	$-0.679826\pi$
$617$	−26.5964	−1.07073	−0.535366	+	0.844620i	$0.679826\pi$
$618$	−9.41858	−0.378871
$619$	28.9980	1.16553	0.582764	−	0.812642i	$-0.301971\pi$
$619$	28.9980	1.16553	0.582764	+	0.812642i	$0.301971\pi$
$620$	13.8916	0.557899
$621$	−29.1287	−1.16890
$622$	−11.7911	−0.472779
$623$	27.2850	1.09315
$624$	−8.00020	−0.320264
$625$	−30.5971	−1.22389
$626$	8.66594	0.346361
$627$	−22.5679	−0.901274
$628$	2.01280	0.0803194
$629$	−27.2418	−1.08620
$630$	67.5580	2.69157
$631$	34.1896	1.36107	0.680533	−	0.732717i	$-0.261748\pi$
$631$	34.1896	1.36107	0.680533	+	0.732717i	$0.261748\pi$
$632$	−8.56809	−0.340820
$633$	53.9932	2.14604
$634$	−28.6565	−1.13810
$635$	26.2703	1.04251
$636$	−33.8568	−1.34251
$637$	−15.9027	−0.630088
$638$	−12.2171	−0.483678
$639$	−39.0008	−1.54285
$640$	−2.88236	−0.113935
$641$	27.5946	1.08992	0.544960	−	0.838462i	$-0.316545\pi$
$641$	27.5946	1.08992	0.544960	+	0.838462i	$0.316545\pi$
$642$	40.7336	1.60763
$643$	−12.8490	−0.506717	−0.253358	−	0.967372i	$-0.581535\pi$
$643$	−12.8490	−0.506717	−0.253358	+	0.967372i	$0.581535\pi$
$644$	9.87482	0.389122
$645$	−8.37144	−0.329625
$646$	17.0785	0.671945
$647$	20.5506	0.807929	0.403964	−	0.914775i	$-0.367632\pi$
$647$	20.5506	0.807929	0.403964	+	0.914775i	$0.367632\pi$
$648$	13.4653	0.528968
$649$	31.1530	1.22286
$650$	−8.59922	−0.337289
$651$	53.7202	2.10546
$652$	12.9598	0.507545
$653$	27.8842	1.09119	0.545597	−	0.838047i	$-0.316303\pi$
$653$	27.8842	1.09119	0.545597	+	0.838047i	$0.316303\pi$
$654$	18.2173	0.712352
$655$	15.4911	0.605287
$656$	−6.06980	−0.236986
$657$	−32.1950	−1.25605
$658$	25.7454	1.00366
$659$	38.2925	1.49166	0.745832	−	0.666134i	$-0.232052\pi$
$659$	38.2925	1.49166	0.745832	+	0.666134i	$0.232052\pi$
$660$	21.8895	0.852048
$661$	−1.56320	−0.0608015	−0.0304008	−	0.999538i	$-0.509678\pi$
$661$	−1.56320	−0.0608015	−0.0304008	+	0.999538i	$0.509678\pi$
$662$	30.8971	1.20085
$663$	−45.9778	−1.78563
$664$	2.94794	0.114402
$665$	31.0226	1.20300
$666$	−30.6754	−1.18865
$667$	−13.4986	−0.522669
$668$	10.5632	0.408703
$669$	−62.4980	−2.41631
$670$	16.8605	0.651379
$671$	−6.31994	−0.243979
$672$	−11.1464	−0.429982
$673$	−21.9605	−0.846516	−0.423258	−	0.906009i	$-0.639114\pi$
$673$	−21.9605	−0.846516	−0.423258	+	0.906009i	$0.639114\pi$
$674$	−20.8945	−0.804827
$675$	35.3415	1.36030
$676$	−6.24251	−0.240097
$677$	−25.0217	−0.961664	−0.480832	−	0.876813i	$-0.659665\pi$
$677$	−25.0217	−0.961664	−0.480832	+	0.876813i	$0.659665\pi$
$678$	−11.2754	−0.433029
$679$	−18.1528	−0.696642
$680$	−16.5651	−0.635244
$681$	−20.3774	−0.780865
$682$	11.8928	0.455397
$683$	−11.6991	−0.447652	−0.223826	−	0.974629i	$-0.571855\pi$
$683$	−11.6991	−0.447652	−0.223826	+	0.974629i	$0.571855\pi$
$684$	19.2311	0.735320
$685$	−26.1358	−0.998599
$686$	3.19604	0.122025
$687$	13.7425	0.524308
$688$	0.943720	0.0359790
$689$	28.5976	1.08948
$690$	24.1857	0.920735
$691$	1.43087	0.0544328	0.0272164	−	0.999630i	$-0.491336\pi$
$691$	1.43087	0.0544328	0.0272164	+	0.999630i	$0.491336\pi$
$692$	−2.86400	−0.108873
$693$	57.8373	2.19706
$694$	−21.5113	−0.816559
$695$	19.3777	0.735036
$696$	15.2369	0.577551
$697$	−34.8836	−1.32131
$698$	−8.80722	−0.333358
$699$	−29.6098	−1.11995
$700$	−11.9810	−0.452839
$701$	27.2987	1.03106	0.515529	−	0.856872i	$-0.327596\pi$
$701$	27.2987	1.03106	0.515529	+	0.856872i	$0.327596\pi$
$702$	−27.7723	−1.04820
$703$	−14.0861	−0.531268
$704$	−2.46763	−0.0930022
$705$	63.0566	2.37485
$706$	−8.15052	−0.306749
$707$	−15.3356	−0.576755
$708$	−38.8534	−1.46020
$709$	−13.3618	−0.501815	−0.250907	−	0.968011i	$-0.580729\pi$
$709$	−13.3618	−0.501815	−0.250907	+	0.968011i	$0.580729\pi$
$710$	17.3708	0.651914
$711$	−55.4480	−2.07946
$712$	−7.53350	−0.282330
$713$	13.1403	0.492108
$714$	−64.0592	−2.39736
$715$	−18.4893	−0.691461
$716$	−11.1504	−0.416712
$717$	55.3209	2.06600
$718$	12.9908	0.484814
$719$	−31.2309	−1.16472	−0.582358	−	0.812933i	$-0.697870\pi$
$719$	−31.2309	−1.16472	−0.582358	+	0.812933i	$0.697870\pi$
$720$	−18.6531	−0.695158
$721$	11.0842	0.412797
$722$	−10.1691	−0.378454
$723$	−4.49139	−0.167036
$724$	−26.6279	−0.989618
$725$	16.3777	0.608253
$726$	−15.1134	−0.560911
$727$	15.6195	0.579294	0.289647	−	0.957134i	$-0.406462\pi$
$727$	15.6195	0.579294	0.289647	+	0.957134i	$0.406462\pi$
$728$	9.41498	0.348942
$729$	−11.4989	−0.425883
$730$	14.3396	0.530731
$731$	5.42363	0.200600
$732$	7.88209	0.291331
$733$	−22.6286	−0.835808	−0.417904	−	0.908491i	$-0.637235\pi$
$733$	−22.6286	−0.835808	−0.417904	+	0.908491i	$0.637235\pi$
$734$	15.0119	0.554100
$735$	−54.2669	−2.00166
$736$	−2.72648	−0.100499
$737$	14.4345	0.531702
$738$	−39.2804	−1.44593
$739$	−2.65210	−0.0975589	−0.0487795	−	0.998810i	$-0.515533\pi$
$739$	−2.65210	−0.0975589	−0.0487795	+	0.998810i	$0.515533\pi$
$740$	13.6627	0.502252
$741$	−23.7741	−0.873363
$742$	39.8441	1.46272
$743$	32.7352	1.20094	0.600470	−	0.799647i	$-0.294980\pi$
$743$	32.7352	1.20094	0.600470	+	0.799647i	$0.294980\pi$
$744$	−14.8324	−0.543782
$745$	33.5748	1.23009
$746$	8.92506	0.326770
$747$	19.0774	0.698006
$748$	−14.1816	−0.518532
$749$	−47.9370	−1.75158
$750$	15.0091	0.548056
$751$	42.6931	1.55789	0.778947	−	0.627089i	$-0.215754\pi$
$751$	42.6931	1.55789	0.778947	+	0.627089i	$0.215754\pi$
$752$	−7.10843	−0.259218
$753$	−14.0968	−0.513717
$754$	−12.8700	−0.468699
$755$	−10.5482	−0.383887
$756$	−38.6942	−1.40729
$757$	28.6405	1.04096	0.520479	−	0.853875i	$-0.325753\pi$
$757$	28.6405	1.04096	0.520479	+	0.853875i	$0.325753\pi$
$758$	−37.1127	−1.34799
$759$	20.7057	0.751570
$760$	−8.56547	−0.310702
$761$	−6.48505	−0.235083	−0.117542	−	0.993068i	$-0.537501\pi$
$761$	−6.48505	−0.235083	−0.117542	+	0.993068i	$0.537501\pi$
$762$	−28.0495	−1.01613
$763$	−21.4389	−0.776139
$764$	23.6704	0.856367
$765$	−107.201	−3.87584
$766$	−18.6654	−0.674407
$767$	32.8181	1.18499
$768$	3.07757	0.111052
$769$	−31.4703	−1.13485	−0.567424	−	0.823426i	$-0.692060\pi$
$769$	−31.4703	−1.13485	−0.567424	+	0.823426i	$0.692060\pi$
$770$	−25.7605	−0.928344
$771$	36.3285	1.30834
$772$	1.42708	0.0513617
$773$	23.6625	0.851082	0.425541	−	0.904939i	$-0.360084\pi$
$773$	23.6625	0.851082	0.425541	+	0.904939i	$0.360084\pi$
$774$	6.10724	0.219520
$775$	−15.9430	−0.572688
$776$	5.01208	0.179923
$777$	52.8352	1.89545
$778$	−28.7894	−1.03215
$779$	−18.0376	−0.646262
$780$	23.0595	0.825662
$781$	14.8714	0.532139
$782$	−15.6693	−0.560333
$783$	52.8940	1.89028
$784$	6.11756	0.218484
$785$	−5.80161	−0.207068
$786$	−16.5402	−0.589971
$787$	−49.6737	−1.77068	−0.885338	−	0.464948i	$-0.846073\pi$
$787$	−49.6737	−1.77068	−0.885338	+	0.464948i	$0.846073\pi$
$788$	−0.521513	−0.0185781
$789$	27.3098	0.972256
$790$	24.6963	0.878656
$791$	13.2693	0.471804
$792$	−15.9691	−0.567438
$793$	−6.65773	−0.236423
$794$	0.972479	0.0345120
$795$	97.5875	3.46107
$796$	14.4011	0.510432
$797$	23.3951	0.828698	0.414349	−	0.910118i	$-0.364009\pi$
$797$	23.3951	0.828698	0.414349	+	0.910118i	$0.364009\pi$
$798$	−33.1236	−1.17256
$799$	−40.8527	−1.44526
$800$	3.30800	0.116956
$801$	−48.7527	−1.72259
$802$	30.5001	1.07700
$803$	12.2763	0.433221
$804$	−18.0024	−0.634896
$805$	−28.4628	−1.00318
$806$	12.5284	0.441294
$807$	−17.6106	−0.619923
$808$	4.23423	0.148960
$809$	−16.6877	−0.586709	−0.293355	−	0.956004i	$-0.594772\pi$
$809$	−16.6877	−0.586709	−0.293355	+	0.956004i	$0.594772\pi$
$810$	−38.8119	−1.36371
$811$	−46.0952	−1.61862	−0.809310	−	0.587381i	$-0.800159\pi$
$811$	−46.0952	−1.61862	−0.809310	+	0.587381i	$0.800159\pi$
$812$	−17.9314	−0.629268
$813$	76.0743	2.66804
$814$	11.6968	0.409974
$815$	−37.3548	−1.30848
$816$	17.6870	0.619170
$817$	2.80444	0.0981150
$818$	−17.1245	−0.598746
$819$	60.9286	2.12902
$820$	17.4954	0.610965
$821$	−26.0637	−0.909630	−0.454815	−	0.890586i	$-0.650295\pi$
$821$	−26.0637	−0.909630	−0.454815	+	0.890586i	$0.650295\pi$
$822$	27.9059	0.973331
$823$	−48.8508	−1.70283	−0.851415	−	0.524492i	$-0.824255\pi$
$823$	−48.8508	−1.70283	−0.851415	+	0.524492i	$0.824255\pi$
$824$	−3.06039	−0.106614
$825$	−25.1220	−0.874635
$826$	45.7243	1.59095
$827$	44.9676	1.56368	0.781838	−	0.623481i	$-0.214282\pi$
$827$	44.9676	1.56368	0.781838	+	0.623481i	$0.214282\pi$
$828$	−17.6443	−0.613181
$829$	18.2414	0.633551	0.316775	−	0.948501i	$-0.397400\pi$
$829$	18.2414	0.633551	0.316775	+	0.948501i	$0.397400\pi$
$830$	−8.49702	−0.294936
$831$	−4.55572	−0.158036
$832$	−2.59952	−0.0901220
$833$	35.1581	1.21816
$834$	−20.6900	−0.716437
$835$	−30.4470	−1.05366
$836$	−7.33301	−0.253618
$837$	−51.4899	−1.77975
$838$	13.2566	0.457943
$839$	−45.9835	−1.58753	−0.793764	−	0.608226i	$-0.791881\pi$
$839$	−45.9835	−1.58753	−0.793764	+	0.608226i	$0.791881\pi$
$840$	32.1280	1.10852
$841$	−4.48826	−0.154768
$842$	−19.0017	−0.654841
$843$	−59.9560	−2.06499
$844$	17.5441	0.603892
$845$	17.9932	0.618984
$846$	−46.0018	−1.58158
$847$	17.7861	0.611137
$848$	−11.0011	−0.377781
$849$	80.7471	2.77123
$850$	19.0114	0.652084
$851$	12.9238	0.443023
$852$	−18.5473	−0.635419
$853$	36.7874	1.25958	0.629789	−	0.776766i	$-0.283141\pi$
$853$	36.7874	1.25958	0.629789	+	0.776766i	$0.283141\pi$
$854$	−9.27598	−0.317418
$855$	−55.4310	−1.89570
$856$	13.2356	0.452384
$857$	22.5650	0.770807	0.385403	−	0.922748i	$-0.374062\pi$
$857$	22.5650	0.770807	0.385403	+	0.922748i	$0.374062\pi$
$858$	19.7415	0.673964
$859$	44.6835	1.52458	0.762291	−	0.647234i	$-0.224075\pi$
$859$	44.6835	1.52458	0.762291	+	0.647234i	$0.224075\pi$
$860$	−2.72014	−0.0927561
$861$	67.6565	2.30573
$862$	−4.18314	−0.142478
$863$	57.0907	1.94339	0.971696	−	0.236235i	$-0.0759137\pi$
$863$	57.0907	1.94339	0.971696	+	0.236235i	$0.0759137\pi$
$864$	10.6836	0.363465
$865$	8.25507	0.280681
$866$	−28.7368	−0.976518
$867$	49.3301	1.67534
$868$	17.4554	0.592474
$869$	21.1428	0.717222
$870$	−43.9181	−1.48896
$871$	15.2060	0.515236
$872$	5.91937	0.200455
$873$	32.4354	1.09777
$874$	−8.10225	−0.274063
$875$	−17.6634	−0.597131
$876$	−15.3107	−0.517301
$877$	−14.6123	−0.493421	−0.246711	−	0.969089i	$-0.579350\pi$
$877$	−14.6123	−0.493421	−0.246711	+	0.969089i	$0.579350\pi$
$878$	37.8828	1.27848
$879$	96.0995	3.24136
$880$	7.11259	0.239765
$881$	14.7887	0.498244	0.249122	−	0.968472i	$-0.419858\pi$
$881$	14.7887	0.498244	0.249122	+	0.968472i	$0.419858\pi$
$882$	39.5895	1.33305
$883$	−42.4540	−1.42869	−0.714345	−	0.699794i	$-0.753275\pi$
$883$	−42.4540	−1.42869	−0.714345	+	0.699794i	$0.753275\pi$
$884$	−14.9396	−0.502474
$885$	111.989	3.76448
$886$	22.5756	0.758443
$887$	14.4591	0.485488	0.242744	−	0.970090i	$-0.421953\pi$
$887$	14.4591	0.485488	0.242744	+	0.970090i	$0.421953\pi$
$888$	−14.5880	−0.489543
$889$	33.0098	1.10711
$890$	21.7143	0.727864
$891$	−33.2274	−1.11316
$892$	−20.3076	−0.679948
$893$	−21.1240	−0.706888
$894$	−35.8487	−1.19896
$895$	32.1396	1.07431
$896$	−3.62182	−0.120996
$897$	21.8124	0.728295
$898$	10.2167	0.340937
$899$	−23.8611	−0.795811
$900$	21.4076	0.713586
$901$	−63.2244	−2.10631
$902$	14.9780	0.498713
$903$	−10.5191	−0.350053
$904$	−3.66373	−0.121854
$905$	76.7512	2.55130
$906$	11.2625	0.374173
$907$	55.2500	1.83455	0.917273	−	0.398258i	$-0.130385\pi$
$907$	55.2500	1.83455	0.917273	+	0.398258i	$0.130385\pi$
$908$	−6.62127	−0.219735
$909$	27.4016	0.908854
$910$	−27.1374	−0.899595
$911$	−34.6178	−1.14694	−0.573469	−	0.819227i	$-0.694403\pi$
$911$	−34.6178	−1.14694	−0.573469	+	0.819227i	$0.694403\pi$
$912$	9.14558	0.302840
$913$	−7.27441	−0.240748
$914$	4.18290	0.138358
$915$	−22.7190	−0.751068
$916$	4.46536	0.147540
$917$	19.4653	0.642799
$918$	61.3997	2.02649
$919$	32.0351	1.05674	0.528371	−	0.849014i	$-0.322803\pi$
$919$	32.0351	1.05674	0.528371	+	0.849014i	$0.322803\pi$
$920$	7.85870	0.259094
$921$	−1.05555	−0.0347816
$922$	−4.15466	−0.136826
$923$	15.6662	0.515660
$924$	27.5052	0.904853
$925$	−15.6803	−0.515566
$926$	−0.821621	−0.0270001
$927$	−19.8052	−0.650488
$928$	4.95093	0.162522
$929$	27.3486	0.897279	0.448640	−	0.893713i	$-0.351909\pi$
$929$	27.3486	0.897279	0.448640	+	0.893713i	$0.351909\pi$
$930$	42.7523	1.40190
$931$	18.1795	0.595808
$932$	−9.62117	−0.315152
$933$	−36.2879	−1.18801
$934$	3.80629	0.124546
$935$	40.8766	1.33681
$936$	−16.8226	−0.549865
$937$	26.1748	0.855092	0.427546	−	0.903994i	$-0.359378\pi$
$937$	26.1748	0.855092	0.427546	+	0.903994i	$0.359378\pi$
$938$	21.1860	0.691748
$939$	26.6701	0.870344
$940$	20.4891	0.668279
$941$	3.75279	0.122338	0.0611688	−	0.998127i	$-0.480517\pi$
$941$	3.75279	0.122338	0.0611688	+	0.998127i	$0.480517\pi$
$942$	6.19453	0.201829
$943$	16.5492	0.538916
$944$	−12.6247	−0.410898
$945$	111.531	3.62809
$946$	−2.32875	−0.0757142
$947$	−16.0759	−0.522398	−0.261199	−	0.965285i	$-0.584118\pi$
$947$	−16.0759	−0.522398	−0.261199	+	0.965285i	$0.584118\pi$
$948$	−26.3689	−0.856422
$949$	12.9324	0.419804
$950$	9.83035	0.318939
$951$	−88.1926	−2.85984
$952$	−20.8149	−0.674613
$953$	23.7534	0.769447	0.384723	−	0.923032i	$-0.374297\pi$
$953$	23.7534	0.769447	0.384723	+	0.923032i	$0.374297\pi$
$954$	−71.1933	−2.30497
$955$	−68.2268	−2.20777
$956$	17.9755	0.581369
$957$	−37.5989	−1.21540
$958$	6.05347	0.195579
$959$	−32.8409	−1.06049
$960$	−8.87067	−0.286300
$961$	−7.77233	−0.250720
$962$	12.3220	0.397277
$963$	85.6536	2.76015
$964$	−1.45939	−0.0470039
$965$	−4.11335	−0.132414
$966$	30.3905	0.977797
$967$	22.7047	0.730135	0.365068	−	0.930981i	$-0.381046\pi$
$967$	22.7047	0.730135	0.365068	+	0.930981i	$0.381046\pi$
$968$	−4.91082	−0.157840
$969$	52.5603	1.68848
$970$	−14.4466	−0.463853
$971$	30.6135	0.982435	0.491218	−	0.871037i	$-0.336552\pi$
$971$	30.6135	0.982435	0.491218	+	0.871037i	$0.336552\pi$
$972$	9.38958	0.301171
$973$	24.3489	0.780590
$974$	37.8035	1.21130
$975$	−26.4647	−0.847549
$976$	2.56114	0.0819801
$977$	14.6869	0.469877	0.234938	−	0.972010i	$-0.424511\pi$
$977$	14.6869	0.469877	0.234938	+	0.972010i	$0.424511\pi$
$978$	39.8847	1.27537
$979$	18.5899	0.594135
$980$	−17.6330	−0.563266
$981$	38.3069	1.22304
$982$	−41.2286	−1.31566
$983$	0.378924	0.0120858	0.00604291	−	0.999982i	$-0.498076\pi$
$983$	0.378924	0.0120858	0.00604291	+	0.999982i	$0.498076\pi$
$984$	−18.6803	−0.595505
$985$	1.50319	0.0478956
$986$	28.4534	0.906141
$987$	79.2334	2.52203
$988$	−7.72495	−0.245763
$989$	−2.57304	−0.0818178
$990$	46.0288	1.46289
$991$	−33.4890	−1.06381	−0.531907	−	0.846803i	$-0.678525\pi$
$991$	−33.4890	−1.06381	−0.531907	+	0.846803i	$0.678525\pi$
$992$	−4.81951	−0.153020
$993$	95.0882	3.01753
$994$	21.8272	0.692317
$995$	−41.5091	−1.31593
$996$	9.07249	0.287473
$997$	42.2771	1.33893	0.669464	−	0.742844i	$-0.266524\pi$
$997$	42.2771	1.33893	0.669464	+	0.742844i	$0.266524\pi$
$998$	2.15362	0.0681716
$999$	−50.6417	−1.60223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.67	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.67	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.07757 q^{3} +1.00000 q^{4} -2.88236 q^{5} +3.07757 q^{6} -3.62182 q^{7} +1.00000 q^{8} +6.47145 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.07757 q^{3} +1.00000 q^{4} -2.88236 q^{5} +3.07757 q^{6} -3.62182 q^{7} +1.00000 q^{8} +6.47145 q^{9} -2.88236 q^{10} -2.46763 q^{11} +3.07757 q^{12} -2.59952 q^{13} -3.62182 q^{14} -8.87067 q^{15} +1.00000 q^{16} +5.74708 q^{17} +6.47145 q^{18} +2.97169 q^{19} -2.88236 q^{20} -11.1464 q^{21} -2.46763 q^{22} -2.72648 q^{23} +3.07757 q^{24} +3.30800 q^{25} -2.59952 q^{26} +10.6836 q^{27} -3.62182 q^{28} +4.95093 q^{29} -8.87067 q^{30} -4.81951 q^{31} +1.00000 q^{32} -7.59430 q^{33} +5.74708 q^{34} +10.4394 q^{35} +6.47145 q^{36} -4.74011 q^{37} +2.97169 q^{38} -8.00020 q^{39} -2.88236 q^{40} -6.06980 q^{41} -11.1464 q^{42} +0.943720 q^{43} -2.46763 q^{44} -18.6531 q^{45} -2.72648 q^{46} -7.10843 q^{47} +3.07757 q^{48} +6.11756 q^{49} +3.30800 q^{50} +17.6870 q^{51} -2.59952 q^{52} -11.0011 q^{53} +10.6836 q^{54} +7.11259 q^{55} -3.62182 q^{56} +9.14558 q^{57} +4.95093 q^{58} -12.6247 q^{59} -8.87067 q^{60} +2.56114 q^{61} -4.81951 q^{62} -23.4384 q^{63} +1.00000 q^{64} +7.49275 q^{65} -7.59430 q^{66} -5.84955 q^{67} +5.74708 q^{68} -8.39094 q^{69} +10.4394 q^{70} -6.02659 q^{71} +6.47145 q^{72} -4.97493 q^{73} -4.74011 q^{74} +10.1806 q^{75} +2.97169 q^{76} +8.93729 q^{77} -8.00020 q^{78} -8.56809 q^{79} -2.88236 q^{80} +13.4653 q^{81} -6.06980 q^{82} +2.94794 q^{83} -11.1464 q^{84} -16.5651 q^{85} +0.943720 q^{86} +15.2369 q^{87} -2.46763 q^{88} -7.53350 q^{89} -18.6531 q^{90} +9.41498 q^{91} -2.72648 q^{92} -14.8324 q^{93} -7.10843 q^{94} -8.56547 q^{95} +3.07757 q^{96} +5.01208 q^{97} +6.11756 q^{98} -15.9691 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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