LMFDB - Embedded newform 8042.2.a.a.1.4

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			1.4
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)

$f(q)$	$=$	$q+1.00000 q^{2} -3.05107 q^{3} +1.00000 q^{4} -1.41145 q^{5} -3.05107 q^{6} -1.76739 q^{7} +1.00000 q^{8} +6.30900 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.05107 q^{3} +1.00000 q^{4} -1.41145 q^{5} -3.05107 q^{6} -1.76739 q^{7} +1.00000 q^{8} +6.30900 q^{9} -1.41145 q^{10} +0.990488 q^{11} -3.05107 q^{12} +2.81976 q^{13} -1.76739 q^{14} +4.30643 q^{15} +1.00000 q^{16} -1.33739 q^{17} +6.30900 q^{18} -3.59035 q^{19} -1.41145 q^{20} +5.39244 q^{21} +0.990488 q^{22} -1.70994 q^{23} -3.05107 q^{24} -3.00780 q^{25} +2.81976 q^{26} -10.0960 q^{27} -1.76739 q^{28} -7.08295 q^{29} +4.30643 q^{30} +8.88408 q^{31} +1.00000 q^{32} -3.02205 q^{33} -1.33739 q^{34} +2.49459 q^{35} +6.30900 q^{36} -3.20982 q^{37} -3.59035 q^{38} -8.60327 q^{39} -1.41145 q^{40} +8.34664 q^{41} +5.39244 q^{42} +4.65492 q^{43} +0.990488 q^{44} -8.90485 q^{45} -1.70994 q^{46} +6.28260 q^{47} -3.05107 q^{48} -3.87632 q^{49} -3.00780 q^{50} +4.08047 q^{51} +2.81976 q^{52} +3.23255 q^{53} -10.0960 q^{54} -1.39803 q^{55} -1.76739 q^{56} +10.9544 q^{57} -7.08295 q^{58} -6.97108 q^{59} +4.30643 q^{60} +3.30534 q^{61} +8.88408 q^{62} -11.1505 q^{63} +1.00000 q^{64} -3.97995 q^{65} -3.02205 q^{66} +7.59227 q^{67} -1.33739 q^{68} +5.21715 q^{69} +2.49459 q^{70} +2.97748 q^{71} +6.30900 q^{72} -7.64838 q^{73} -3.20982 q^{74} +9.17701 q^{75} -3.59035 q^{76} -1.75058 q^{77} -8.60327 q^{78} +1.60053 q^{79} -1.41145 q^{80} +11.8765 q^{81} +8.34664 q^{82} -3.29857 q^{83} +5.39244 q^{84} +1.88766 q^{85} +4.65492 q^{86} +21.6106 q^{87} +0.990488 q^{88} -9.31347 q^{89} -8.90485 q^{90} -4.98362 q^{91} -1.70994 q^{92} -27.1059 q^{93} +6.28260 q^{94} +5.06761 q^{95} -3.05107 q^{96} -10.1265 q^{97} -3.87632 q^{98} +6.24899 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.05107	−1.76153	−0.880767	−	0.473550i	$-0.842972\pi$
$3$	−3.05107	−1.76153	−0.880767	+	0.473550i	$0.842972\pi$
$4$	1.00000	0.500000
$5$	−1.41145	−0.631220	−0.315610	−	0.948889i	$-0.602209\pi$
$5$	−1.41145	−0.631220	−0.315610	+	0.948889i	$0.602209\pi$
$6$	−3.05107	−1.24559
$7$	−1.76739	−0.668012	−0.334006	−	0.942571i	$-0.608401\pi$
$7$	−1.76739	−0.668012	−0.334006	+	0.942571i	$0.608401\pi$
$8$	1.00000	0.353553
$9$	6.30900	2.10300
$10$	−1.41145	−0.446340
$11$	0.990488	0.298644	0.149322	−	0.988789i	$-0.452291\pi$
$11$	0.990488	0.298644	0.149322	+	0.988789i	$0.452291\pi$
$12$	−3.05107	−0.880767
$13$	2.81976	0.782060	0.391030	−	0.920378i	$-0.372119\pi$
$13$	2.81976	0.782060	0.391030	+	0.920378i	$0.372119\pi$
$14$	−1.76739	−0.472356
$15$	4.30643	1.11192
$16$	1.00000	0.250000
$17$	−1.33739	−0.324365	−0.162182	−	0.986761i	$-0.551853\pi$
$17$	−1.33739	−0.324365	−0.162182	+	0.986761i	$0.551853\pi$
$18$	6.30900	1.48705
$19$	−3.59035	−0.823683	−0.411842	−	0.911255i	$-0.635114\pi$
$19$	−3.59035	−0.823683	−0.411842	+	0.911255i	$0.635114\pi$
$20$	−1.41145	−0.315610
$21$	5.39244	1.17673
$22$	0.990488	0.211173
$23$	−1.70994	−0.356548	−0.178274	−	0.983981i	$-0.557051\pi$
$23$	−1.70994	−0.356548	−0.178274	+	0.983981i	$0.557051\pi$
$24$	−3.05107	−0.622796
$25$	−3.00780	−0.601561
$26$	2.81976	0.553000
$27$	−10.0960	−1.94297
$28$	−1.76739	−0.334006
$29$	−7.08295	−1.31527	−0.657636	−	0.753336i	$-0.728443\pi$
$29$	−7.08295	−1.31527	−0.657636	+	0.753336i	$0.728443\pi$
$30$	4.30643	0.786243
$31$	8.88408	1.59563	0.797814	−	0.602904i	$-0.205990\pi$
$31$	8.88408	1.59563	0.797814	+	0.602904i	$0.205990\pi$
$32$	1.00000	0.176777
$33$	−3.02205	−0.526071
$34$	−1.33739	−0.229361
$35$	2.49459	0.421663
$36$	6.30900	1.05150
$37$	−3.20982	−0.527691	−0.263845	−	0.964565i	$-0.584991\pi$
$37$	−3.20982	−0.527691	−0.263845	+	0.964565i	$0.584991\pi$
$38$	−3.59035	−0.582432
$39$	−8.60327	−1.37763
$40$	−1.41145	−0.223170
$41$	8.34664	1.30353	0.651763	−	0.758423i	$-0.274030\pi$
$41$	8.34664	1.30353	0.651763	+	0.758423i	$0.274030\pi$
$42$	5.39244	0.832071
$43$	4.65492	0.709869	0.354934	−	0.934891i	$-0.384503\pi$
$43$	4.65492	0.709869	0.354934	+	0.934891i	$0.384503\pi$
$44$	0.990488	0.149322
$45$	−8.90485	−1.32746
$46$	−1.70994	−0.252117
$47$	6.28260	0.916411	0.458206	−	0.888846i	$-0.348492\pi$
$47$	6.28260	0.916411	0.458206	+	0.888846i	$0.348492\pi$
$48$	−3.05107	−0.440383
$49$	−3.87632	−0.553760
$50$	−3.00780	−0.425368
$51$	4.08047	0.571379
$52$	2.81976	0.391030
$53$	3.23255	0.444024	0.222012	−	0.975044i	$-0.428738\pi$
$53$	3.23255	0.444024	0.222012	+	0.975044i	$0.428738\pi$
$54$	−10.0960	−1.37389
$55$	−1.39803	−0.188510
$56$	−1.76739	−0.236178
$57$	10.9544	1.45095
$58$	−7.08295	−0.930037
$59$	−6.97108	−0.907557	−0.453779	−	0.891114i	$-0.649924\pi$
$59$	−6.97108	−0.907557	−0.453779	+	0.891114i	$0.649924\pi$
$60$	4.30643	0.555958
$61$	3.30534	0.423206	0.211603	−	0.977356i	$-0.432132\pi$
$61$	3.30534	0.423206	0.211603	+	0.977356i	$0.432132\pi$
$62$	8.88408	1.12828
$63$	−11.1505	−1.40483
$64$	1.00000	0.125000
$65$	−3.97995	−0.493652
$66$	−3.02205	−0.371988
$67$	7.59227	0.927544	0.463772	−	0.885955i	$-0.346496\pi$
$67$	7.59227	0.927544	0.463772	+	0.885955i	$0.346496\pi$
$68$	−1.33739	−0.162182
$69$	5.21715	0.628071
$70$	2.49459	0.298161
$71$	2.97748	0.353362	0.176681	−	0.984268i	$-0.443464\pi$
$71$	2.97748	0.353362	0.176681	+	0.984268i	$0.443464\pi$
$72$	6.30900	0.743523
$73$	−7.64838	−0.895175	−0.447588	−	0.894240i	$-0.647717\pi$
$73$	−7.64838	−0.895175	−0.447588	+	0.894240i	$0.647717\pi$
$74$	−3.20982	−0.373134
$75$	9.17701	1.05967
$76$	−3.59035	−0.411842
$77$	−1.75058	−0.199497
$78$	−8.60327	−0.974128
$79$	1.60053	0.180074	0.0900370	−	0.995938i	$-0.471301\pi$
$79$	1.60053	0.180074	0.0900370	+	0.995938i	$0.471301\pi$
$80$	−1.41145	−0.157805
$81$	11.8765	1.31961
$82$	8.34664	0.921732
$83$	−3.29857	−0.362065	−0.181032	−	0.983477i	$-0.557944\pi$
$83$	−3.29857	−0.362065	−0.181032	+	0.983477i	$0.557944\pi$
$84$	5.39244	0.588363
$85$	1.88766	0.204746
$86$	4.65492	0.501953
$87$	21.6106	2.31689
$88$	0.990488	0.105586
$89$	−9.31347	−0.987226	−0.493613	−	0.869682i	$-0.664324\pi$
$89$	−9.31347	−0.987226	−0.493613	+	0.869682i	$0.664324\pi$
$90$	−8.90485	−0.938654
$91$	−4.98362	−0.522426
$92$	−1.70994	−0.178274
$93$	−27.1059	−2.81075
$94$	6.28260	0.648001
$95$	5.06761	0.519926
$96$	−3.05107	−0.311398
$97$	−10.1265	−1.02819	−0.514095	−	0.857733i	$-0.671872\pi$
$97$	−10.1265	−1.02819	−0.514095	+	0.857733i	$0.671872\pi$
$98$	−3.87632	−0.391567
$99$	6.24899	0.628048
$100$	−3.00780	−0.300780
$101$	18.6082	1.85159	0.925793	−	0.378032i	$-0.123399\pi$
$101$	18.6082	1.85159	0.925793	+	0.378032i	$0.123399\pi$
$102$	4.08047	0.404026
$103$	−1.10760	−0.109136	−0.0545678	−	0.998510i	$-0.517378\pi$
$103$	−1.10760	−0.109136	−0.0545678	+	0.998510i	$0.517378\pi$
$104$	2.81976	0.276500
$105$	−7.61116	−0.742773
$106$	3.23255	0.313973
$107$	17.0999	1.65311	0.826557	−	0.562853i	$-0.190296\pi$
$107$	17.0999	1.65311	0.826557	+	0.562853i	$0.190296\pi$
$108$	−10.0960	−0.971487
$109$	13.9731	1.33838	0.669190	−	0.743092i	$-0.266641\pi$
$109$	13.9731	1.33838	0.669190	+	0.743092i	$0.266641\pi$
$110$	−1.39803	−0.133297
$111$	9.79337	0.929545
$112$	−1.76739	−0.167003
$113$	−1.22548	−0.115284	−0.0576419	−	0.998337i	$-0.518358\pi$
$113$	−1.22548	−0.115284	−0.0576419	+	0.998337i	$0.518358\pi$
$114$	10.9544	1.02597
$115$	2.41350	0.225060
$116$	−7.08295	−0.657636
$117$	17.7899	1.64467
$118$	−6.97108	−0.641740
$119$	2.36370	0.216680
$120$	4.30643	0.393122
$121$	−10.0189	−0.910812
$122$	3.30534	0.299252
$123$	−25.4662	−2.29621
$124$	8.88408	0.797814
$125$	11.3026	1.01094
$126$	−11.1505	−0.993365
$127$	0.638189	0.0566301	0.0283151	−	0.999599i	$-0.490986\pi$
$127$	0.638189	0.0566301	0.0283151	+	0.999599i	$0.490986\pi$
$128$	1.00000	0.0883883
$129$	−14.2025	−1.25046
$130$	−3.97995	−0.349065
$131$	0.391940	0.0342440	0.0171220	−	0.999853i	$-0.494550\pi$
$131$	0.391940	0.0342440	0.0171220	+	0.999853i	$0.494550\pi$
$132$	−3.02205	−0.263035
$133$	6.34557	0.550231
$134$	7.59227	0.655872
$135$	14.2500	1.22644
$136$	−1.33739	−0.114680
$137$	−18.3245	−1.56557	−0.782786	−	0.622291i	$-0.786202\pi$
$137$	−18.3245	−1.56557	−0.782786	+	0.622291i	$0.786202\pi$
$138$	5.21715	0.444113
$139$	14.1675	1.20167	0.600836	−	0.799372i	$-0.294834\pi$
$139$	14.1675	1.20167	0.600836	+	0.799372i	$0.294834\pi$
$140$	2.49459	0.210831
$141$	−19.1686	−1.61429
$142$	2.97748	0.249865
$143$	2.79294	0.233557
$144$	6.30900	0.525750
$145$	9.99724	0.830226
$146$	−7.64838	−0.632985
$147$	11.8269	0.975466
$148$	−3.20982	−0.263845
$149$	−15.3214	−1.25518	−0.627588	−	0.778545i	$-0.715958\pi$
$149$	−15.3214	−1.25518	−0.627588	+	0.778545i	$0.715958\pi$
$150$	9.17701	0.749300
$151$	1.77251	0.144245	0.0721223	−	0.997396i	$-0.477023\pi$
$151$	1.77251	0.144245	0.0721223	+	0.997396i	$0.477023\pi$
$152$	−3.59035	−0.291216
$153$	−8.43760	−0.682139
$154$	−1.75058	−0.141066
$155$	−12.5394	−1.00719
$156$	−8.60327	−0.688813
$157$	4.77543	0.381121	0.190561	−	0.981675i	$-0.438969\pi$
$157$	4.77543	0.381121	0.190561	+	0.981675i	$0.438969\pi$
$158$	1.60053	0.127332
$159$	−9.86271	−0.782164
$160$	−1.41145	−0.111585
$161$	3.02214	0.238178
$162$	11.8765	0.933107
$163$	0.156336	0.0122452	0.00612258	−	0.999981i	$-0.498051\pi$
$163$	0.156336	0.0122452	0.00612258	+	0.999981i	$0.498051\pi$
$164$	8.34664	0.651763
$165$	4.26547	0.332066
$166$	−3.29857	−0.256019
$167$	17.1656	1.32831	0.664157	−	0.747593i	$-0.268791\pi$
$167$	17.1656	1.32831	0.664157	+	0.747593i	$0.268791\pi$
$168$	5.39244	0.416035
$169$	−5.04896	−0.388382
$170$	1.88766	0.144777
$171$	−22.6515	−1.73221
$172$	4.65492	0.354934
$173$	6.81045	0.517789	0.258894	−	0.965906i	$-0.416642\pi$
$173$	6.81045	0.517789	0.258894	+	0.965906i	$0.416642\pi$
$174$	21.6106	1.63829
$175$	5.31598	0.401850
$176$	0.990488	0.0746609
$177$	21.2692	1.59869
$178$	−9.31347	−0.698074
$179$	−1.76346	−0.131807	−0.0659034	−	0.997826i	$-0.520993\pi$
$179$	−1.76346	−0.131807	−0.0659034	+	0.997826i	$0.520993\pi$
$180$	−8.90485	−0.663728
$181$	−0.379324	−0.0281949	−0.0140975	−	0.999901i	$-0.504488\pi$
$181$	−0.379324	−0.0281949	−0.0140975	+	0.999901i	$0.504488\pi$
$182$	−4.98362	−0.369411
$183$	−10.0848	−0.745491
$184$	−1.70994	−0.126059
$185$	4.53050	0.333089
$186$	−27.1059	−1.98750
$187$	−1.32467	−0.0968694
$188$	6.28260	0.458206
$189$	17.8436	1.29793
$190$	5.06761	0.367643
$191$	13.9413	1.00876	0.504379	−	0.863482i	$-0.331721\pi$
$191$	13.9413	1.00876	0.504379	+	0.863482i	$0.331721\pi$
$192$	−3.05107	−0.220192
$193$	−14.6097	−1.05163	−0.525815	−	0.850599i	$-0.676240\pi$
$193$	−14.6097	−1.05163	−0.525815	+	0.850599i	$0.676240\pi$
$194$	−10.1265	−0.727041
$195$	12.1431	0.869585
$196$	−3.87632	−0.276880
$197$	−22.9184	−1.63287	−0.816433	−	0.577441i	$-0.804051\pi$
$197$	−22.9184	−1.63287	−0.816433	+	0.577441i	$0.804051\pi$
$198$	6.24899	0.444097
$199$	14.7608	1.04636	0.523182	−	0.852221i	$-0.324745\pi$
$199$	14.7608	1.04636	0.523182	+	0.852221i	$0.324745\pi$
$200$	−3.00780	−0.212684
$201$	−23.1645	−1.63390
$202$	18.6082	1.30927
$203$	12.5184	0.878617
$204$	4.08047	0.285690
$205$	−11.7809	−0.822812
$206$	−1.10760	−0.0771705
$207$	−10.7880	−0.749820
$208$	2.81976	0.195515
$209$	−3.55620	−0.245988
$210$	−7.61116	−0.525220
$211$	−23.7895	−1.63774	−0.818869	−	0.573981i	$-0.805399\pi$
$211$	−23.7895	−1.63774	−0.818869	+	0.573981i	$0.805399\pi$
$212$	3.23255	0.222012
$213$	−9.08449	−0.622459
$214$	17.0999	1.16893
$215$	−6.57019	−0.448084
$216$	−10.0960	−0.686945
$217$	−15.7017	−1.06590
$218$	13.9731	0.946377
$219$	23.3357	1.57688
$220$	−1.39803	−0.0942549
$221$	−3.77112	−0.253673
$222$	9.79337	0.657288
$223$	−28.4792	−1.90711	−0.953555	−	0.301219i	$-0.902606\pi$
$223$	−28.4792	−1.90711	−0.953555	+	0.301219i	$0.902606\pi$
$224$	−1.76739	−0.118089
$225$	−18.9762	−1.26508
$226$	−1.22548	−0.0815180
$227$	15.0721	1.00037	0.500185	−	0.865919i	$-0.333265\pi$
$227$	15.0721	1.00037	0.500185	+	0.865919i	$0.333265\pi$
$228$	10.9544	0.725473
$229$	19.3331	1.27757	0.638785	−	0.769385i	$-0.279437\pi$
$229$	19.3331	1.27757	0.638785	+	0.769385i	$0.279437\pi$
$230$	2.41350	0.159142
$231$	5.34115	0.351422
$232$	−7.08295	−0.465019
$233$	−27.3546	−1.79206	−0.896029	−	0.443995i	$-0.853561\pi$
$233$	−27.3546	−1.79206	−0.896029	+	0.443995i	$0.853561\pi$
$234$	17.7899	1.16296
$235$	−8.86758	−0.578457
$236$	−6.97108	−0.453779
$237$	−4.88333	−0.317206
$238$	2.36370	0.153216
$239$	−26.4726	−1.71237	−0.856185	−	0.516669i	$-0.827172\pi$
$239$	−26.4726	−1.71237	−0.856185	+	0.516669i	$0.827172\pi$
$240$	4.30643	0.277979
$241$	−11.3914	−0.733783	−0.366892	−	0.930264i	$-0.619578\pi$
$241$	−11.3914	−0.733783	−0.366892	+	0.930264i	$0.619578\pi$
$242$	−10.0189	−0.644041
$243$	−5.94806	−0.381568
$244$	3.30534	0.211603
$245$	5.47124	0.349544
$246$	−25.4662	−1.62366
$247$	−10.1239	−0.644170
$248$	8.88408	0.564139
$249$	10.0641	0.637789
$250$	11.3026	0.714841
$251$	−3.90272	−0.246338	−0.123169	−	0.992386i	$-0.539306\pi$
$251$	−3.90272	−0.246338	−0.123169	+	0.992386i	$0.539306\pi$
$252$	−11.1505	−0.702415
$253$	−1.69368	−0.106481
$254$	0.638189	0.0400435
$255$	−5.75938	−0.360666
$256$	1.00000	0.0625000
$257$	7.26231	0.453010	0.226505	−	0.974010i	$-0.427270\pi$
$257$	7.26231	0.453010	0.226505	+	0.974010i	$0.427270\pi$
$258$	−14.2025	−0.884207
$259$	5.67301	0.352504
$260$	−3.97995	−0.246826
$261$	−44.6864	−2.76602
$262$	0.391940	0.0242142
$263$	−8.73603	−0.538687	−0.269343	−	0.963044i	$-0.586807\pi$
$263$	−8.73603	−0.538687	−0.269343	+	0.963044i	$0.586807\pi$
$264$	−3.02205	−0.185994
$265$	−4.56258	−0.280277
$266$	6.34557	0.389072
$267$	28.4160	1.73903
$268$	7.59227	0.463772
$269$	24.1705	1.47370	0.736851	−	0.676056i	$-0.236312\pi$
$269$	24.1705	1.47370	0.736851	+	0.676056i	$0.236312\pi$
$270$	14.2500	0.867227
$271$	−20.6518	−1.25451	−0.627253	−	0.778815i	$-0.715821\pi$
$271$	−20.6518	−1.25451	−0.627253	+	0.778815i	$0.715821\pi$
$272$	−1.33739	−0.0810912
$273$	15.2054	0.920271
$274$	−18.3245	−1.10703
$275$	−2.97920	−0.179652
$276$	5.21715	0.314035
$277$	−13.1585	−0.790620	−0.395310	−	0.918548i	$-0.629363\pi$
$277$	−13.1585	−0.790620	−0.395310	+	0.918548i	$0.629363\pi$
$278$	14.1675	0.849710
$279$	56.0497	3.35561
$280$	2.49459	0.149080
$281$	−13.0492	−0.778451	−0.389226	−	0.921142i	$-0.627257\pi$
$281$	−13.0492	−0.778451	−0.389226	+	0.921142i	$0.627257\pi$
$282$	−19.1686	−1.14148
$283$	−13.1308	−0.780546	−0.390273	−	0.920699i	$-0.627619\pi$
$283$	−13.1308	−0.780546	−0.390273	+	0.920699i	$0.627619\pi$
$284$	2.97748	0.176681
$285$	−15.4616	−0.915867
$286$	2.79294	0.165150
$287$	−14.7518	−0.870771
$288$	6.30900	0.371762
$289$	−15.2114	−0.894788
$290$	9.99724	0.587058
$291$	30.8966	1.81119
$292$	−7.64838	−0.447588
$293$	−26.2268	−1.53218	−0.766092	−	0.642731i	$-0.777801\pi$
$293$	−26.2268	−1.53218	−0.766092	+	0.642731i	$0.777801\pi$
$294$	11.8269	0.689759
$295$	9.83934	0.572869
$296$	−3.20982	−0.186567
$297$	−9.99996	−0.580256
$298$	−15.3214	−0.887544
$299$	−4.82163	−0.278842
$300$	9.17701	0.529835
$301$	−8.22708	−0.474201
$302$	1.77251	0.101996
$303$	−56.7748	−3.26163
$304$	−3.59035	−0.205921
$305$	−4.66533	−0.267136
$306$	−8.43760	−0.482345
$307$	−34.2440	−1.95441	−0.977204	−	0.212301i	$-0.931904\pi$
$307$	−34.2440	−1.95441	−0.977204	+	0.212301i	$0.931904\pi$
$308$	−1.75058	−0.0997487
$309$	3.37937	0.192246
$310$	−12.5394	−0.712193
$311$	4.91937	0.278952	0.139476	−	0.990225i	$-0.455458\pi$
$311$	4.91937	0.278952	0.139476	+	0.990225i	$0.455458\pi$
$312$	−8.60327	−0.487064
$313$	−4.99586	−0.282383	−0.141191	−	0.989982i	$-0.545093\pi$
$313$	−4.99586	−0.282383	−0.141191	+	0.989982i	$0.545093\pi$
$314$	4.77543	0.269493
$315$	15.7384	0.886757
$316$	1.60053	0.0900370
$317$	−3.70493	−0.208090	−0.104045	−	0.994573i	$-0.533179\pi$
$317$	−3.70493	−0.208090	−0.104045	+	0.994573i	$0.533179\pi$
$318$	−9.86271	−0.553073
$319$	−7.01558	−0.392797
$320$	−1.41145	−0.0789025
$321$	−52.1731	−2.91202
$322$	3.02214	0.168417
$323$	4.80170	0.267174
$324$	11.8765	0.659806
$325$	−8.48128	−0.470457
$326$	0.156336	0.00865864
$327$	−42.6328	−2.35760
$328$	8.34664	0.460866
$329$	−11.1038	−0.612174
$330$	4.26547	0.234806
$331$	7.27379	0.399804	0.199902	−	0.979816i	$-0.435938\pi$
$331$	7.27379	0.399804	0.199902	+	0.979816i	$0.435938\pi$
$332$	−3.29857	−0.181032
$333$	−20.2508	−1.10973
$334$	17.1656	0.939260
$335$	−10.7161	−0.585484
$336$	5.39244	0.294181
$337$	20.5115	1.11733	0.558666	−	0.829393i	$-0.311313\pi$
$337$	20.5115	1.11733	0.558666	+	0.829393i	$0.311313\pi$
$338$	−5.04896	−0.274627
$339$	3.73903	0.203076
$340$	1.88766	0.102373
$341$	8.79958	0.476524
$342$	−22.6515	−1.22486
$343$	19.2227	1.03793
$344$	4.65492	0.250976
$345$	−7.36375	−0.396451
$346$	6.81045	0.366132
$347$	−19.7865	−1.06219	−0.531097	−	0.847311i	$-0.678220\pi$
$347$	−19.7865	−1.06219	−0.531097	+	0.847311i	$0.678220\pi$
$348$	21.6106	1.15845
$349$	17.9510	0.960897	0.480448	−	0.877023i	$-0.340474\pi$
$349$	17.9510	0.960897	0.480448	+	0.877023i	$0.340474\pi$
$350$	5.31598	0.284151
$351$	−28.4682	−1.51952
$352$	0.990488	0.0527932
$353$	−12.0914	−0.643562	−0.321781	−	0.946814i	$-0.604282\pi$
$353$	−12.0914	−0.643562	−0.321781	+	0.946814i	$0.604282\pi$
$354$	21.2692	1.13045
$355$	−4.20257	−0.223049
$356$	−9.31347	−0.493613
$357$	−7.21179	−0.381688
$358$	−1.76346	−0.0932015
$359$	−23.2548	−1.22734	−0.613671	−	0.789562i	$-0.710308\pi$
$359$	−23.2548	−1.22734	−0.613671	+	0.789562i	$0.710308\pi$
$360$	−8.90485	−0.469327
$361$	−6.10937	−0.321546
$362$	−0.379324	−0.0199368
$363$	30.5684	1.60443
$364$	−4.98362	−0.261213
$365$	10.7953	0.565053
$366$	−10.0848	−0.527142
$367$	−20.2384	−1.05643	−0.528217	−	0.849109i	$-0.677139\pi$
$367$	−20.2384	−1.05643	−0.528217	+	0.849109i	$0.677139\pi$
$368$	−1.70994	−0.0891369
$369$	52.6590	2.74132
$370$	4.53050	0.235530
$371$	−5.71318	−0.296614
$372$	−27.1059	−1.40538
$373$	−20.4964	−1.06126	−0.530631	−	0.847603i	$-0.678045\pi$
$373$	−20.4964	−1.06126	−0.530631	+	0.847603i	$0.678045\pi$
$374$	−1.32467	−0.0684970
$375$	−34.4851	−1.78080
$376$	6.28260	0.324000
$377$	−19.9722	−1.02862
$378$	17.8436	0.917775
$379$	−32.9203	−1.69100	−0.845500	−	0.533975i	$-0.820698\pi$
$379$	−32.9203	−1.69100	−0.845500	+	0.533975i	$0.820698\pi$
$380$	5.06761	0.259963
$381$	−1.94716	−0.0997559
$382$	13.9413	0.713300
$383$	14.0638	0.718628	0.359314	−	0.933217i	$-0.383011\pi$
$383$	14.0638	0.718628	0.359314	+	0.933217i	$0.383011\pi$
$384$	−3.05107	−0.155699
$385$	2.47086	0.125927
$386$	−14.6097	−0.743615
$387$	29.3679	1.49285
$388$	−10.1265	−0.514095
$389$	−25.6156	−1.29876	−0.649381	−	0.760464i	$-0.724972\pi$
$389$	−25.6156	−1.29876	−0.649381	+	0.760464i	$0.724972\pi$
$390$	12.1431	0.614890
$391$	2.28686	0.115652
$392$	−3.87632	−0.195784
$393$	−1.19584	−0.0603219
$394$	−22.9184	−1.15461
$395$	−2.25907	−0.113666
$396$	6.24899	0.314024
$397$	−22.3461	−1.12152	−0.560760	−	0.827978i	$-0.689491\pi$
$397$	−22.3461	−1.12152	−0.560760	+	0.827978i	$0.689491\pi$
$398$	14.7608	0.739892
$399$	−19.3607	−0.969250
$400$	−3.00780	−0.150390
$401$	39.6643	1.98074	0.990370	−	0.138449i	$-0.0442116\pi$
$401$	39.6643	1.98074	0.990370	+	0.138449i	$0.0442116\pi$
$402$	−23.1645	−1.15534
$403$	25.0510	1.24788
$404$	18.6082	0.925793
$405$	−16.7631	−0.832966
$406$	12.5184	0.621276
$407$	−3.17929	−0.157591
$408$	4.08047	0.202013
$409$	−7.75325	−0.383373	−0.191687	−	0.981456i	$-0.561396\pi$
$409$	−7.75325	−0.383373	−0.191687	+	0.981456i	$0.561396\pi$
$410$	−11.7809	−0.581816
$411$	55.9094	2.75781
$412$	−1.10760	−0.0545678
$413$	12.3206	0.606259
$414$	−10.7880	−0.530203
$415$	4.65577	0.228543
$416$	2.81976	0.138250
$417$	−43.2260	−2.11679
$418$	−3.55620	−0.173940
$419$	−15.5207	−0.758235	−0.379118	−	0.925348i	$-0.623772\pi$
$419$	−15.5207	−0.758235	−0.379118	+	0.925348i	$0.623772\pi$
$420$	−7.61116	−0.371387
$421$	7.02289	0.342275	0.171137	−	0.985247i	$-0.445256\pi$
$421$	7.02289	0.342275	0.171137	+	0.985247i	$0.445256\pi$
$422$	−23.7895	−1.15806
$423$	39.6369	1.92721
$424$	3.23255	0.156986
$425$	4.02261	0.195125
$426$	−9.08449	−0.440145
$427$	−5.84184	−0.282707
$428$	17.0999	0.826557
$429$	−8.52144	−0.411419
$430$	−6.57019	−0.316843
$431$	38.2424	1.84207	0.921035	−	0.389480i	$-0.127345\pi$
$431$	38.2424	1.84207	0.921035	+	0.389480i	$0.127345\pi$
$432$	−10.0960	−0.485743
$433$	−5.15882	−0.247917	−0.123958	−	0.992287i	$-0.539559\pi$
$433$	−5.15882	−0.247917	−0.123958	+	0.992287i	$0.539559\pi$
$434$	−15.7017	−0.753704
$435$	−30.5023	−1.46247
$436$	13.9731	0.669190
$437$	6.13930	0.293682
$438$	23.3357	1.11502
$439$	20.4598	0.976493	0.488246	−	0.872706i	$-0.337637\pi$
$439$	20.4598	0.976493	0.488246	+	0.872706i	$0.337637\pi$
$440$	−1.39803	−0.0666483
$441$	−24.4557	−1.16456
$442$	−3.77112	−0.179374
$443$	−26.2999	−1.24955	−0.624773	−	0.780806i	$-0.714809\pi$
$443$	−26.2999	−1.24955	−0.624773	+	0.780806i	$0.714809\pi$
$444$	9.79337	0.464773
$445$	13.1455	0.623157
$446$	−28.4792	−1.34853
$447$	46.7465	2.21104
$448$	−1.76739	−0.0835015
$449$	−7.78811	−0.367544	−0.183772	−	0.982969i	$-0.558831\pi$
$449$	−7.78811	−0.367544	−0.183772	+	0.982969i	$0.558831\pi$
$450$	−18.9762	−0.894549
$451$	8.26725	0.389290
$452$	−1.22548	−0.0576419
$453$	−5.40804	−0.254092
$454$	15.0721	0.707368
$455$	7.03414	0.329766
$456$	10.9544	0.512987
$457$	12.5807	0.588500	0.294250	−	0.955728i	$-0.404930\pi$
$457$	12.5807	0.588500	0.294250	+	0.955728i	$0.404930\pi$
$458$	19.3331	0.903378
$459$	13.5023	0.630232
$460$	2.41350	0.112530
$461$	−5.98896	−0.278934	−0.139467	−	0.990227i	$-0.544539\pi$
$461$	−5.98896	−0.278934	−0.139467	+	0.990227i	$0.544539\pi$
$462$	5.34115	0.248493
$463$	14.9361	0.694140	0.347070	−	0.937839i	$-0.387177\pi$
$463$	14.9361	0.694140	0.347070	+	0.937839i	$0.387177\pi$
$464$	−7.08295	−0.328818
$465$	38.2587	1.77420
$466$	−27.3546	−1.26718
$467$	22.9564	1.06230	0.531149	−	0.847279i	$-0.321761\pi$
$467$	22.9564	1.06230	0.531149	+	0.847279i	$0.321761\pi$
$468$	17.7899	0.822337
$469$	−13.4185	−0.619610
$470$	−8.86758	−0.409031
$471$	−14.5702	−0.671358
$472$	−6.97108	−0.320870
$473$	4.61064	0.211998
$474$	−4.88333	−0.224299
$475$	10.7991	0.495496
$476$	2.36370	0.108340
$477$	20.3941	0.933784
$478$	−26.4726	−1.21083
$479$	−4.91130	−0.224403	−0.112201	−	0.993685i	$-0.535790\pi$
$479$	−4.91130	−0.224403	−0.112201	+	0.993685i	$0.535790\pi$
$480$	4.30643	0.196561
$481$	−9.05091	−0.412686
$482$	−11.3914	−0.518863
$483$	−9.22076	−0.419559
$484$	−10.0189	−0.455406
$485$	14.2931	0.649015
$486$	−5.94806	−0.269809
$487$	−26.5750	−1.20423	−0.602115	−	0.798409i	$-0.705675\pi$
$487$	−26.5750	−1.20423	−0.602115	+	0.798409i	$0.705675\pi$
$488$	3.30534	0.149626
$489$	−0.476991	−0.0215703
$490$	5.47124	0.247165
$491$	24.0414	1.08497	0.542486	−	0.840065i	$-0.317483\pi$
$491$	24.0414	1.08497	0.542486	+	0.840065i	$0.317483\pi$
$492$	−25.4662	−1.14810
$493$	9.47267	0.426628
$494$	−10.1239	−0.455497
$495$	−8.82015	−0.396436
$496$	8.88408	0.398907
$497$	−5.26238	−0.236050
$498$	10.0641	0.450985
$499$	27.9188	1.24982	0.624908	−	0.780698i	$-0.285136\pi$
$499$	27.9188	1.24982	0.624908	+	0.780698i	$0.285136\pi$
$500$	11.3026	0.505469
$501$	−52.3734	−2.33987
$502$	−3.90272	−0.174187
$503$	−29.9901	−1.33719	−0.668596	−	0.743626i	$-0.733104\pi$
$503$	−29.9901	−1.33719	−0.668596	+	0.743626i	$0.733104\pi$
$504$	−11.1505	−0.496682
$505$	−26.2646	−1.16876
$506$	−1.69368	−0.0752932
$507$	15.4047	0.684147
$508$	0.638189	0.0283151
$509$	−9.34356	−0.414146	−0.207073	−	0.978325i	$-0.566394\pi$
$509$	−9.34356	−0.414146	−0.207073	+	0.978325i	$0.566394\pi$
$510$	−5.75938	−0.255030
$511$	13.5177	0.597988
$512$	1.00000	0.0441942
$513$	36.2482	1.60039
$514$	7.26231	0.320327
$515$	1.56333	0.0688886
$516$	−14.2025	−0.625229
$517$	6.22284	0.273680
$518$	5.67301	0.249258
$519$	−20.7791	−0.912103
$520$	−3.97995	−0.174532
$521$	−15.1620	−0.664259	−0.332130	−	0.943234i	$-0.607767\pi$
$521$	−15.1620	−0.664259	−0.332130	+	0.943234i	$0.607767\pi$
$522$	−44.6864	−1.95587
$523$	−5.22631	−0.228531	−0.114265	−	0.993450i	$-0.536451\pi$
$523$	−5.22631	−0.228531	−0.114265	+	0.993450i	$0.536451\pi$
$524$	0.391940	0.0171220
$525$	−16.2194	−0.707872
$526$	−8.73603	−0.380909
$527$	−11.8815	−0.517565
$528$	−3.02205	−0.131518
$529$	−20.0761	−0.872874
$530$	−4.56258	−0.198186
$531$	−43.9806	−1.90859
$532$	6.34557	0.275115
$533$	23.5355	1.01944
$534$	28.4160	1.22968
$535$	−24.1357	−1.04348
$536$	7.59227	0.327936
$537$	5.38042	0.232182
$538$	24.1705	1.04206
$539$	−3.83945	−0.165377
$540$	14.2500	0.613222
$541$	−5.57200	−0.239559	−0.119779	−	0.992801i	$-0.538219\pi$
$541$	−5.57200	−0.239559	−0.119779	+	0.992801i	$0.538219\pi$
$542$	−20.6518	−0.887070
$543$	1.15734	0.0496663
$544$	−1.33739	−0.0573401
$545$	−19.7223	−0.844812
$546$	15.2054	0.650730
$547$	−0.0877621	−0.00375244	−0.00187622	−	0.999998i	$-0.500597\pi$
$547$	−0.0877621	−0.00375244	−0.00187622	+	0.999998i	$0.500597\pi$
$548$	−18.3245	−0.782786
$549$	20.8534	0.890002
$550$	−2.97920	−0.127033
$551$	25.4303	1.08337
$552$	5.21715	0.222057
$553$	−2.82877	−0.120292
$554$	−13.1585	−0.559053
$555$	−13.8229	−0.586748
$556$	14.1675	0.600836
$557$	21.3248	0.903559	0.451780	−	0.892130i	$-0.350789\pi$
$557$	21.3248	0.903559	0.451780	+	0.892130i	$0.350789\pi$
$558$	56.0497	2.37277
$559$	13.1258	0.555160
$560$	2.49459	0.105416
$561$	4.04165	0.170639
$562$	−13.0492	−0.550448
$563$	−20.6875	−0.871876	−0.435938	−	0.899977i	$-0.643583\pi$
$563$	−20.6875	−0.871876	−0.435938	+	0.899977i	$0.643583\pi$
$564$	−19.1686	−0.807145
$565$	1.72971	0.0727695
$566$	−13.1308	−0.551930
$567$	−20.9905	−0.881517
$568$	2.97748	0.124932
$569$	−11.0933	−0.465056	−0.232528	−	0.972590i	$-0.574700\pi$
$569$	−11.0933	−0.465056	−0.232528	+	0.972590i	$0.574700\pi$
$570$	−15.4616	−0.647615
$571$	19.3218	0.808594	0.404297	−	0.914628i	$-0.367516\pi$
$571$	19.3218	0.808594	0.404297	+	0.914628i	$0.367516\pi$
$572$	2.79294	0.116779
$573$	−42.5359	−1.77696
$574$	−14.7518	−0.615728
$575$	5.14317	0.214485
$576$	6.30900	0.262875
$577$	1.00605	0.0418825	0.0209413	−	0.999781i	$-0.493334\pi$
$577$	1.00605	0.0418825	0.0209413	+	0.999781i	$0.493334\pi$
$578$	−15.2114	−0.632710
$579$	44.5752	1.85248
$580$	9.99724	0.415113
$581$	5.82987	0.241864
$582$	30.8966	1.28071
$583$	3.20180	0.132605
$584$	−7.64838	−0.316492
$585$	−25.1095	−1.03815
$586$	−26.2268	−1.08342
$587$	14.1937	0.585836	0.292918	−	0.956138i	$-0.405374\pi$
$587$	14.1937	0.585836	0.292918	+	0.956138i	$0.405374\pi$
$588$	11.8269	0.487733
$589$	−31.8970	−1.31429
$590$	9.83934	0.405079
$591$	69.9254	2.87635
$592$	−3.20982	−0.131923
$593$	16.2398	0.666889	0.333445	−	0.942770i	$-0.391789\pi$
$593$	16.2398	0.666889	0.333445	+	0.942770i	$0.391789\pi$
$594$	−9.99996	−0.410303
$595$	−3.33624	−0.136773
$596$	−15.3214	−0.627588
$597$	−45.0361	−1.84321
$598$	−4.82163	−0.197171
$599$	25.4165	1.03849	0.519245	−	0.854626i	$-0.326213\pi$
$599$	25.4165	1.03849	0.519245	+	0.854626i	$0.326213\pi$
$600$	9.17701	0.374650
$601$	29.4047	1.19944	0.599721	−	0.800209i	$-0.295278\pi$
$601$	29.4047	1.19944	0.599721	+	0.800209i	$0.295278\pi$
$602$	−8.22708	−0.335311
$603$	47.8997	1.95062
$604$	1.77251	0.0721223
$605$	14.1412	0.574923
$606$	−56.7748	−2.30632
$607$	11.3300	0.459872	0.229936	−	0.973206i	$-0.426148\pi$
$607$	11.3300	0.459872	0.229936	+	0.973206i	$0.426148\pi$
$608$	−3.59035	−0.145608
$609$	−38.1944	−1.54771
$610$	−4.66533	−0.188894
$611$	17.7154	0.716689
$612$	−8.43760	−0.341070
$613$	−45.0699	−1.82036	−0.910178	−	0.414218i	$-0.864055\pi$
$613$	−45.0699	−1.82036	−0.910178	+	0.414218i	$0.864055\pi$
$614$	−34.2440	−1.38198
$615$	35.9442	1.44941
$616$	−1.75058	−0.0705330
$617$	21.9085	0.882004	0.441002	−	0.897506i	$-0.354623\pi$
$617$	21.9085	0.882004	0.441002	+	0.897506i	$0.354623\pi$
$618$	3.37937	0.135938
$619$	6.95192	0.279421	0.139711	−	0.990192i	$-0.455383\pi$
$619$	6.95192	0.279421	0.139711	+	0.990192i	$0.455383\pi$
$620$	−12.5394	−0.503596
$621$	17.2636	0.692763
$622$	4.91937	0.197249
$623$	16.4606	0.659479
$624$	−8.60327	−0.344406
$625$	−0.914088	−0.0365635
$626$	−4.99586	−0.199675
$627$	10.8502	0.433316
$628$	4.77543	0.190561
$629$	4.29278	0.171164
$630$	15.7384	0.627032
$631$	−46.4204	−1.84796	−0.923982	−	0.382435i	$-0.875086\pi$
$631$	−46.4204	−1.84796	−0.923982	+	0.382435i	$0.875086\pi$
$632$	1.60053	0.0636658
$633$	72.5834	2.88493
$634$	−3.70493	−0.147142
$635$	−0.900773	−0.0357461
$636$	−9.86271	−0.391082
$637$	−10.9303	−0.433074
$638$	−7.01558	−0.277750
$639$	18.7849	0.743121
$640$	−1.41145	−0.0557925
$641$	−20.9759	−0.828498	−0.414249	−	0.910164i	$-0.635956\pi$
$641$	−20.9759	−0.828498	−0.414249	+	0.910164i	$0.635956\pi$
$642$	−52.1731	−2.05911
$643$	−3.99573	−0.157576	−0.0787880	−	0.996891i	$-0.525105\pi$
$643$	−3.99573	−0.157576	−0.0787880	+	0.996891i	$0.525105\pi$
$644$	3.02214	0.119089
$645$	20.0461	0.789314
$646$	4.80170	0.188920
$647$	−23.5850	−0.927221	−0.463610	−	0.886039i	$-0.653446\pi$
$647$	−23.5850	−0.927221	−0.463610	+	0.886039i	$0.653446\pi$
$648$	11.8765	0.466553
$649$	−6.90477	−0.271036
$650$	−8.48128	−0.332663
$651$	47.9068	1.87762
$652$	0.156336	0.00612258
$653$	−19.3043	−0.755436	−0.377718	−	0.925921i	$-0.623291\pi$
$653$	−19.3043	−0.755436	−0.377718	+	0.925921i	$0.623291\pi$
$654$	−42.6328	−1.66707
$655$	−0.553205	−0.0216155
$656$	8.34664	0.325882
$657$	−48.2537	−1.88255
$658$	−11.1038	−0.432872
$659$	−39.2456	−1.52879	−0.764396	−	0.644747i	$-0.776963\pi$
$659$	−39.2456	−1.52879	−0.764396	+	0.644747i	$0.776963\pi$
$660$	4.26547	0.166033
$661$	−35.8195	−1.39322	−0.696608	−	0.717452i	$-0.745308\pi$
$661$	−35.8195	−1.39322	−0.696608	+	0.717452i	$0.745308\pi$
$662$	7.27379	0.282704
$663$	11.5059	0.446853
$664$	−3.29857	−0.128009
$665$	−8.95646	−0.347317
$666$	−20.2508	−0.784701
$667$	12.1114	0.468957
$668$	17.1656	0.664157
$669$	86.8920	3.35944
$670$	−10.7161	−0.414000
$671$	3.27390	0.126388
$672$	5.39244	0.208018
$673$	−18.9883	−0.731945	−0.365973	−	0.930626i	$-0.619264\pi$
$673$	−18.9883	−0.731945	−0.365973	+	0.930626i	$0.619264\pi$
$674$	20.5115	0.790073
$675$	30.3668	1.16882
$676$	−5.04896	−0.194191
$677$	−9.51894	−0.365843	−0.182921	−	0.983128i	$-0.558555\pi$
$677$	−9.51894	−0.365843	−0.182921	+	0.983128i	$0.558555\pi$
$678$	3.73903	0.143597
$679$	17.8975	0.686844
$680$	1.88766	0.0723885
$681$	−45.9859	−1.76218
$682$	8.79958	0.336953
$683$	−9.13741	−0.349633	−0.174817	−	0.984601i	$-0.555933\pi$
$683$	−9.13741	−0.349633	−0.174817	+	0.984601i	$0.555933\pi$
$684$	−22.6515	−0.866103
$685$	25.8642	0.988221
$686$	19.2227	0.733928
$687$	−58.9867	−2.25048
$688$	4.65492	0.177467
$689$	9.11500	0.347254
$690$	−7.36375	−0.280333
$691$	−14.9149	−0.567390	−0.283695	−	0.958915i	$-0.591560\pi$
$691$	−14.9149	−0.567390	−0.283695	+	0.958915i	$0.591560\pi$
$692$	6.81045	0.258894
$693$	−11.0444	−0.419543
$694$	−19.7865	−0.751084
$695$	−19.9967	−0.758520
$696$	21.6106	0.819146
$697$	−11.1627	−0.422818
$698$	17.9510	0.679457
$699$	83.4607	3.15677
$700$	5.31598	0.200925
$701$	25.5660	0.965614	0.482807	−	0.875727i	$-0.339617\pi$
$701$	25.5660	0.965614	0.482807	+	0.875727i	$0.339617\pi$
$702$	−28.4682	−1.07446
$703$	11.5244	0.434650
$704$	0.990488	0.0373304
$705$	27.0556	1.01897
$706$	−12.0914	−0.455067
$707$	−32.8880	−1.23688
$708$	21.2692	0.799346
$709$	39.1418	1.47000	0.735000	−	0.678067i	$-0.237182\pi$
$709$	39.1418	1.47000	0.735000	+	0.678067i	$0.237182\pi$
$710$	−4.20257	−0.157720
$711$	10.0978	0.378696
$712$	−9.31347	−0.349037
$713$	−15.1913	−0.568917
$714$	−7.21179	−0.269894
$715$	−3.94210	−0.147426
$716$	−1.76346	−0.0659034
$717$	80.7696	3.01640
$718$	−23.2548	−0.867862
$719$	15.9034	0.593095	0.296548	−	0.955018i	$-0.404165\pi$
$719$	15.9034	0.593095	0.296548	+	0.955018i	$0.404165\pi$
$720$	−8.90485	−0.331864
$721$	1.95757	0.0729039
$722$	−6.10937	−0.227367
$723$	34.7558	1.29258
$724$	−0.379324	−0.0140975
$725$	21.3041	0.791216
$726$	30.5684	1.13450
$727$	3.70503	0.137412	0.0687059	−	0.997637i	$-0.478113\pi$
$727$	3.70503	0.137412	0.0687059	+	0.997637i	$0.478113\pi$
$728$	−4.98362	−0.184705
$729$	−17.4816	−0.647467
$730$	10.7953	0.399553
$731$	−6.22544	−0.230256
$732$	−10.0848	−0.372746
$733$	−11.0908	−0.409647	−0.204823	−	0.978799i	$-0.565662\pi$
$733$	−11.0908	−0.409647	−0.204823	+	0.978799i	$0.565662\pi$
$734$	−20.2384	−0.747011
$735$	−16.6931	−0.615734
$736$	−1.70994	−0.0630293
$737$	7.52006	0.277005
$738$	52.6590	1.93840
$739$	21.9608	0.807840	0.403920	−	0.914794i	$-0.367647\pi$
$739$	21.9608	0.807840	0.403920	+	0.914794i	$0.367647\pi$
$740$	4.53050	0.166545
$741$	30.8888	1.13473
$742$	−5.71318	−0.209738
$743$	9.16532	0.336243	0.168121	−	0.985766i	$-0.446230\pi$
$743$	9.16532	0.336243	0.168121	+	0.985766i	$0.446230\pi$
$744$	−27.1059	−0.993751
$745$	21.6254	0.792293
$746$	−20.4964	−0.750426
$747$	−20.8107	−0.761423
$748$	−1.32467	−0.0484347
$749$	−30.2223	−1.10430
$750$	−34.4851	−1.25922
$751$	−40.4684	−1.47671	−0.738357	−	0.674410i	$-0.764398\pi$
$751$	−40.4684	−1.47671	−0.738357	+	0.674410i	$0.764398\pi$
$752$	6.28260	0.229103
$753$	11.9075	0.433932
$754$	−19.9722	−0.727345
$755$	−2.50181	−0.0910501
$756$	17.8436	0.648965
$757$	6.71012	0.243883	0.121942	−	0.992537i	$-0.461088\pi$
$757$	6.71012	0.243883	0.121942	+	0.992537i	$0.461088\pi$
$758$	−32.9203	−1.19572
$759$	5.16752	0.187569
$760$	5.06761	0.183821
$761$	−21.8030	−0.790357	−0.395179	−	0.918604i	$-0.629317\pi$
$761$	−21.8030	−0.790357	−0.395179	+	0.918604i	$0.629317\pi$
$762$	−1.94716	−0.0705380
$763$	−24.6960	−0.894054
$764$	13.9413	0.504379
$765$	11.9093	0.430580
$766$	14.0638	0.508147
$767$	−19.6568	−0.709764
$768$	−3.05107	−0.110096
$769$	−12.0805	−0.435636	−0.217818	−	0.975989i	$-0.569894\pi$
$769$	−12.0805	−0.435636	−0.217818	+	0.975989i	$0.569894\pi$
$770$	2.47086	0.0890437
$771$	−22.1578	−0.797993
$772$	−14.6097	−0.525815
$773$	−14.0036	−0.503675	−0.251837	−	0.967770i	$-0.581035\pi$
$773$	−14.0036	−0.503675	−0.251837	+	0.967770i	$0.581035\pi$
$774$	29.3679	1.05561
$775$	−26.7216	−0.959867
$776$	−10.1265	−0.363520
$777$	−17.3087	−0.620948
$778$	−25.6156	−0.918363
$779$	−29.9674	−1.07369
$780$	12.1431	0.434793
$781$	2.94916	0.105529
$782$	2.28686	0.0817780
$783$	71.5094	2.55554
$784$	−3.87632	−0.138440
$785$	−6.74029	−0.240571
$786$	−1.19584	−0.0426541
$787$	6.43181	0.229269	0.114635	−	0.993408i	$-0.463430\pi$
$787$	6.43181	0.229269	0.114635	+	0.993408i	$0.463430\pi$
$788$	−22.9184	−0.816433
$789$	26.6542	0.948915
$790$	−2.25907	−0.0803743
$791$	2.16591	0.0770110
$792$	6.24899	0.222048
$793$	9.32027	0.330973
$794$	−22.3461	−0.793035
$795$	13.9207	0.493718
$796$	14.7608	0.523182
$797$	11.0103	0.390005	0.195003	−	0.980803i	$-0.437528\pi$
$797$	11.0103	0.390005	0.195003	+	0.980803i	$0.437528\pi$
$798$	−19.3607	−0.685363
$799$	−8.40229	−0.297252
$800$	−3.00780	−0.106342
$801$	−58.7587	−2.07614
$802$	39.6643	1.40059
$803$	−7.57563	−0.267338
$804$	−23.1645	−0.816950
$805$	−4.26561	−0.150343
$806$	25.0510	0.882382
$807$	−73.7458	−2.59597
$808$	18.6082	0.654634
$809$	40.3635	1.41911	0.709553	−	0.704652i	$-0.248897\pi$
$809$	40.3635	1.41911	0.709553	+	0.704652i	$0.248897\pi$
$810$	−16.7631	−0.588996
$811$	33.1330	1.16346	0.581728	−	0.813384i	$-0.302377\pi$
$811$	33.1330	1.16346	0.581728	+	0.813384i	$0.302377\pi$
$812$	12.5184	0.439309
$813$	63.0099	2.20986
$814$	−3.17929	−0.111434
$815$	−0.220660	−0.00772940
$816$	4.08047	0.142845
$817$	−16.7128	−0.584707
$818$	−7.75325	−0.271086
$819$	−31.4417	−1.09866
$820$	−11.7809	−0.411406
$821$	26.2748	0.916997	0.458499	−	0.888695i	$-0.348387\pi$
$821$	26.2748	0.916997	0.458499	+	0.888695i	$0.348387\pi$
$822$	55.9094	1.95006
$823$	15.7803	0.550066	0.275033	−	0.961435i	$-0.411311\pi$
$823$	15.7803	0.550066	0.275033	+	0.961435i	$0.411311\pi$
$824$	−1.10760	−0.0385852
$825$	9.08972	0.316464
$826$	12.3206	0.428690
$827$	27.4534	0.954647	0.477324	−	0.878728i	$-0.341607\pi$
$827$	27.4534	0.954647	0.477324	+	0.878728i	$0.341607\pi$
$828$	−10.7880	−0.374910
$829$	20.8442	0.723949	0.361974	−	0.932188i	$-0.382103\pi$
$829$	20.8442	0.723949	0.361974	+	0.932188i	$0.382103\pi$
$830$	4.65577	0.161604
$831$	40.1476	1.39270
$832$	2.81976	0.0977575
$833$	5.18415	0.179620
$834$	−43.2260	−1.49679
$835$	−24.2284	−0.838459
$836$	−3.55620	−0.122994
$837$	−89.6935	−3.10026
$838$	−15.5207	−0.536153
$839$	−3.90126	−0.134686	−0.0673432	−	0.997730i	$-0.521452\pi$
$839$	−3.90126	−0.134686	−0.0673432	+	0.997730i	$0.521452\pi$
$840$	−7.61116	−0.262610
$841$	21.1682	0.729939
$842$	7.02289	0.242025
$843$	39.8140	1.37127
$844$	−23.7895	−0.818869
$845$	7.12636	0.245154
$846$	39.6369	1.36275
$847$	17.7074	0.608434
$848$	3.23255	0.111006
$849$	40.0630	1.37496
$850$	4.02261	0.137974
$851$	5.48861	0.188147
$852$	−9.08449	−0.311229
$853$	23.7078	0.811738	0.405869	−	0.913931i	$-0.366969\pi$
$853$	23.7078	0.811738	0.405869	+	0.913931i	$0.366969\pi$
$854$	−5.84184	−0.199904
$855$	31.9716	1.09340
$856$	17.0999	0.584464
$857$	−16.0872	−0.549527	−0.274764	−	0.961512i	$-0.588600\pi$
$857$	−16.0872	−0.549527	−0.274764	+	0.961512i	$0.588600\pi$
$858$	−8.52144	−0.290917
$859$	48.9156	1.66898	0.834489	−	0.551024i	$-0.185763\pi$
$859$	48.9156	1.66898	0.834489	+	0.551024i	$0.185763\pi$
$860$	−6.57019	−0.224042
$861$	45.0087	1.53389
$862$	38.2424	1.30254
$863$	30.8015	1.04849	0.524247	−	0.851566i	$-0.324347\pi$
$863$	30.8015	1.04849	0.524247	+	0.851566i	$0.324347\pi$
$864$	−10.0960	−0.343472
$865$	−9.61262	−0.326839
$866$	−5.15882	−0.175304
$867$	46.4109	1.57620
$868$	−15.7017	−0.532949
$869$	1.58531	0.0537779
$870$	−30.5023	−1.03412
$871$	21.4084	0.725395
$872$	13.9731	0.473188
$873$	−63.8882	−2.16229
$874$	6.13930	0.207665
$875$	−19.9762	−0.675319
$876$	23.3357	0.788441
$877$	28.0998	0.948864	0.474432	−	0.880292i	$-0.342653\pi$
$877$	28.0998	0.948864	0.474432	+	0.880292i	$0.342653\pi$
$878$	20.4598	0.690485
$879$	80.0196	2.69899
$880$	−1.39803	−0.0471275
$881$	3.00848	0.101358	0.0506791	−	0.998715i	$-0.483861\pi$
$881$	3.00848	0.101358	0.0506791	+	0.998715i	$0.483861\pi$
$882$	−24.4557	−0.823466
$883$	−23.5000	−0.790837	−0.395418	−	0.918501i	$-0.629400\pi$
$883$	−23.5000	−0.790837	−0.395418	+	0.918501i	$0.629400\pi$
$884$	−3.77112	−0.126836
$885$	−30.0205	−1.00913
$886$	−26.2999	−0.883563
$887$	24.9535	0.837857	0.418929	−	0.908019i	$-0.362406\pi$
$887$	24.9535	0.837857	0.418929	+	0.908019i	$0.362406\pi$
$888$	9.79337	0.328644
$889$	−1.12793	−0.0378296
$890$	13.1455	0.440639
$891$	11.7635	0.394094
$892$	−28.4792	−0.953555
$893$	−22.5567	−0.754833
$894$	46.7465	1.56344
$895$	2.48903	0.0831992
$896$	−1.76739	−0.0590445
$897$	14.7111	0.491189
$898$	−7.78811	−0.259893
$899$	−62.9255	−2.09868
$900$	−18.9762	−0.632542
$901$	−4.32318	−0.144026
$902$	8.26725	0.275269
$903$	25.1014	0.835321
$904$	−1.22548	−0.0407590
$905$	0.535397	0.0177972
$906$	−5.40804	−0.179670
$907$	−19.1289	−0.635165	−0.317583	−	0.948231i	$-0.602871\pi$
$907$	−19.1289	−0.635165	−0.317583	+	0.948231i	$0.602871\pi$
$908$	15.0721	0.500185
$909$	117.399	3.89389
$910$	7.03414	0.233180
$911$	35.4600	1.17484	0.587421	−	0.809282i	$-0.300143\pi$
$911$	35.4600	1.17484	0.587421	+	0.809282i	$0.300143\pi$
$912$	10.9544	0.362737
$913$	−3.26719	−0.108128
$914$	12.5807	0.416133
$915$	14.2342	0.470569
$916$	19.3331	0.638785
$917$	−0.692713	−0.0228754
$918$	13.5023	0.445641
$919$	7.46159	0.246135	0.123067	−	0.992398i	$-0.460727\pi$
$919$	7.46159	0.246135	0.123067	+	0.992398i	$0.460727\pi$
$920$	2.41350	0.0795708
$921$	104.481	3.44276
$922$	−5.98896	−0.197236
$923$	8.39578	0.276350
$924$	5.34115	0.175711
$925$	9.65451	0.317438
$926$	14.9361	0.490831
$927$	−6.98788	−0.229512
$928$	−7.08295	−0.232509
$929$	−45.1928	−1.48273	−0.741364	−	0.671103i	$-0.765821\pi$
$929$	−45.1928	−1.48273	−0.741364	+	0.671103i	$0.765821\pi$
$930$	38.2587	1.25455
$931$	13.9173	0.456123
$932$	−27.3546	−0.896029
$933$	−15.0093	−0.491383
$934$	22.9564	0.751157
$935$	1.86971	0.0611459
$936$	17.7899	0.581480
$937$	−50.5444	−1.65122	−0.825608	−	0.564245i	$-0.809167\pi$
$937$	−50.5444	−1.65122	−0.825608	+	0.564245i	$0.809167\pi$
$938$	−13.4185	−0.438131
$939$	15.2427	0.497427
$940$	−8.86758	−0.289229
$941$	−34.3366	−1.11934	−0.559670	−	0.828716i	$-0.689072\pi$
$941$	−34.3366	−1.11934	−0.559670	+	0.828716i	$0.689072\pi$
$942$	−14.5702	−0.474721
$943$	−14.2723	−0.464769
$944$	−6.97108	−0.226889
$945$	−25.1854	−0.819280
$946$	4.61064	0.149905
$947$	−50.1975	−1.63120	−0.815600	−	0.578616i	$-0.803593\pi$
$947$	−50.1975	−1.63120	−0.815600	+	0.578616i	$0.803593\pi$
$948$	−4.88333	−0.158603
$949$	−21.5666	−0.700081
$950$	10.7991	0.350368
$951$	11.3040	0.366557
$952$	2.36370	0.0766078
$953$	−46.6564	−1.51135	−0.755675	−	0.654947i	$-0.772691\pi$
$953$	−46.6564	−1.51135	−0.755675	+	0.654947i	$0.772691\pi$
$954$	20.3941	0.660285
$955$	−19.6775	−0.636749
$956$	−26.4726	−0.856185
$957$	21.4050	0.691926
$958$	−4.91130	−0.158677
$959$	32.3867	1.04582
$960$	4.30643	0.138989
$961$	47.9268	1.54603
$962$	−9.05091	−0.291813
$963$	107.884	3.47650
$964$	−11.3914	−0.366892
$965$	20.6209	0.663811
$966$	−9.22076	−0.296673
$967$	−58.7466	−1.88916	−0.944581	−	0.328278i	$-0.893532\pi$
$967$	−58.7466	−1.88916	−0.944581	+	0.328278i	$0.893532\pi$
$968$	−10.0189	−0.322021
$969$	−14.6503	−0.470636
$970$	14.2931	0.458923
$971$	41.5009	1.33183	0.665914	−	0.746028i	$-0.268042\pi$
$971$	41.5009	1.33183	0.665914	+	0.746028i	$0.268042\pi$
$972$	−5.94806	−0.190784
$973$	−25.0396	−0.802731
$974$	−26.5750	−0.851519
$975$	25.8770	0.828726
$976$	3.30534	0.105801
$977$	26.0602	0.833739	0.416870	−	0.908966i	$-0.363127\pi$
$977$	26.0602	0.833739	0.416870	+	0.908966i	$0.363127\pi$
$978$	−0.476991	−0.0152525
$979$	−9.22489	−0.294829
$980$	5.47124	0.174772
$981$	88.1563	2.81461
$982$	24.0414	0.767191
$983$	−26.7644	−0.853653	−0.426826	−	0.904334i	$-0.640368\pi$
$983$	−26.7644	−0.853653	−0.426826	+	0.904334i	$0.640368\pi$
$984$	−25.4662	−0.811831
$985$	32.3481	1.03070
$986$	9.47267	0.301671
$987$	33.8785	1.07836
$988$	−10.1239	−0.322085
$989$	−7.95965	−0.253102
$990$	−8.82015	−0.280323
$991$	−15.1316	−0.480671	−0.240336	−	0.970690i	$-0.577257\pi$
$991$	−15.1316	−0.480671	−0.240336	+	0.970690i	$0.577257\pi$
$992$	8.88408	0.282070
$993$	−22.1928	−0.704267
$994$	−5.26238	−0.166913
$995$	−20.8341	−0.660487
$996$	10.0641	0.318895
$997$	6.59747	0.208944	0.104472	−	0.994528i	$-0.466685\pi$
$997$	6.59747	0.208944	0.104472	+	0.994528i	$0.466685\pi$
$998$	27.9188	0.883754
$999$	32.4063	1.02529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.4	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.4	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.05107 q^{3} +1.00000 q^{4} -1.41145 q^{5} -3.05107 q^{6} -1.76739 q^{7} +1.00000 q^{8} +6.30900 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.05107 q^{3} +1.00000 q^{4} -1.41145 q^{5} -3.05107 q^{6} -1.76739 q^{7} +1.00000 q^{8} +6.30900 q^{9} -1.41145 q^{10} +0.990488 q^{11} -3.05107 q^{12} +2.81976 q^{13} -1.76739 q^{14} +4.30643 q^{15} +1.00000 q^{16} -1.33739 q^{17} +6.30900 q^{18} -3.59035 q^{19} -1.41145 q^{20} +5.39244 q^{21} +0.990488 q^{22} -1.70994 q^{23} -3.05107 q^{24} -3.00780 q^{25} +2.81976 q^{26} -10.0960 q^{27} -1.76739 q^{28} -7.08295 q^{29} +4.30643 q^{30} +8.88408 q^{31} +1.00000 q^{32} -3.02205 q^{33} -1.33739 q^{34} +2.49459 q^{35} +6.30900 q^{36} -3.20982 q^{37} -3.59035 q^{38} -8.60327 q^{39} -1.41145 q^{40} +8.34664 q^{41} +5.39244 q^{42} +4.65492 q^{43} +0.990488 q^{44} -8.90485 q^{45} -1.70994 q^{46} +6.28260 q^{47} -3.05107 q^{48} -3.87632 q^{49} -3.00780 q^{50} +4.08047 q^{51} +2.81976 q^{52} +3.23255 q^{53} -10.0960 q^{54} -1.39803 q^{55} -1.76739 q^{56} +10.9544 q^{57} -7.08295 q^{58} -6.97108 q^{59} +4.30643 q^{60} +3.30534 q^{61} +8.88408 q^{62} -11.1505 q^{63} +1.00000 q^{64} -3.97995 q^{65} -3.02205 q^{66} +7.59227 q^{67} -1.33739 q^{68} +5.21715 q^{69} +2.49459 q^{70} +2.97748 q^{71} +6.30900 q^{72} -7.64838 q^{73} -3.20982 q^{74} +9.17701 q^{75} -3.59035 q^{76} -1.75058 q^{77} -8.60327 q^{78} +1.60053 q^{79} -1.41145 q^{80} +11.8765 q^{81} +8.34664 q^{82} -3.29857 q^{83} +5.39244 q^{84} +1.88766 q^{85} +4.65492 q^{86} +21.6106 q^{87} +0.990488 q^{88} -9.31347 q^{89} -8.90485 q^{90} -4.98362 q^{91} -1.70994 q^{92} -27.1059 q^{93} +6.28260 q^{94} +5.06761 q^{95} -3.05107 q^{96} -10.1265 q^{97} -3.87632 q^{98} +6.24899 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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