LMFDB - Embedded newform 8038.2.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.34219 q^{3} +1.00000 q^{4} -3.72898 q^{5} +2.34219 q^{6} -1.27056 q^{7} -1.00000 q^{8} +2.48587 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.34219 q^{3} +1.00000 q^{4} -3.72898 q^{5} +2.34219 q^{6} -1.27056 q^{7} -1.00000 q^{8} +2.48587 q^{9} +3.72898 q^{10} +0.244008 q^{11} -2.34219 q^{12} +0.143581 q^{13} +1.27056 q^{14} +8.73399 q^{15} +1.00000 q^{16} +3.78082 q^{17} -2.48587 q^{18} -3.24695 q^{19} -3.72898 q^{20} +2.97591 q^{21} -0.244008 q^{22} -3.61808 q^{23} +2.34219 q^{24} +8.90529 q^{25} -0.143581 q^{26} +1.20419 q^{27} -1.27056 q^{28} +5.02034 q^{29} -8.73399 q^{30} +1.50678 q^{31} -1.00000 q^{32} -0.571514 q^{33} -3.78082 q^{34} +4.73791 q^{35} +2.48587 q^{36} +2.81909 q^{37} +3.24695 q^{38} -0.336295 q^{39} +3.72898 q^{40} +3.59405 q^{41} -2.97591 q^{42} +2.44877 q^{43} +0.244008 q^{44} -9.26976 q^{45} +3.61808 q^{46} -13.1160 q^{47} -2.34219 q^{48} -5.38566 q^{49} -8.90529 q^{50} -8.85542 q^{51} +0.143581 q^{52} +12.4992 q^{53} -1.20419 q^{54} -0.909901 q^{55} +1.27056 q^{56} +7.60498 q^{57} -5.02034 q^{58} +2.59310 q^{59} +8.73399 q^{60} -4.56772 q^{61} -1.50678 q^{62} -3.15846 q^{63} +1.00000 q^{64} -0.535412 q^{65} +0.571514 q^{66} -7.06271 q^{67} +3.78082 q^{68} +8.47424 q^{69} -4.73791 q^{70} -9.62164 q^{71} -2.48587 q^{72} +6.05237 q^{73} -2.81909 q^{74} -20.8579 q^{75} -3.24695 q^{76} -0.310028 q^{77} +0.336295 q^{78} -2.60235 q^{79} -3.72898 q^{80} -10.2781 q^{81} -3.59405 q^{82} +1.67305 q^{83} +2.97591 q^{84} -14.0986 q^{85} -2.44877 q^{86} -11.7586 q^{87} -0.244008 q^{88} +10.4695 q^{89} +9.26976 q^{90} -0.182429 q^{91} -3.61808 q^{92} -3.52918 q^{93} +13.1160 q^{94} +12.1078 q^{95} +2.34219 q^{96} -9.13696 q^{97} +5.38566 q^{98} +0.606572 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) $ 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.34219	−1.35227	−0.676133	−	0.736780i	$-0.736346\pi$
$3$	−2.34219	−1.35227	−0.676133	+	0.736780i	$0.736346\pi$
$4$	1.00000	0.500000
$5$	−3.72898	−1.66765	−0.833825	−	0.552028i	$-0.813854\pi$
$5$	−3.72898	−1.66765	−0.833825	+	0.552028i	$0.813854\pi$
$6$	2.34219	0.956196
$7$	−1.27056	−0.480228	−0.240114	−	0.970745i	$-0.577185\pi$
$7$	−1.27056	−0.480228	−0.240114	+	0.970745i	$0.577185\pi$
$8$	−1.00000	−0.353553
$9$	2.48587	0.828623
$10$	3.72898	1.17921
$11$	0.244008	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$11$	0.244008	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$12$	−2.34219	−0.676133
$13$	0.143581	0.0398223	0.0199111	−	0.999802i	$-0.493662\pi$
$13$	0.143581	0.0398223	0.0199111	+	0.999802i	$0.493662\pi$
$14$	1.27056	0.339573
$15$	8.73399	2.25511
$16$	1.00000	0.250000
$17$	3.78082	0.916984	0.458492	−	0.888698i	$-0.348390\pi$
$17$	3.78082	0.916984	0.458492	+	0.888698i	$0.348390\pi$
$18$	−2.48587	−0.585925
$19$	−3.24695	−0.744901	−0.372450	−	0.928052i	$-0.621482\pi$
$19$	−3.24695	−0.744901	−0.372450	+	0.928052i	$0.621482\pi$
$20$	−3.72898	−0.833825
$21$	2.97591	0.649397
$22$	−0.244008	−0.0520227
$23$	−3.61808	−0.754421	−0.377211	−	0.926127i	$-0.623117\pi$
$23$	−3.61808	−0.754421	−0.377211	+	0.926127i	$0.623117\pi$
$24$	2.34219	0.478098
$25$	8.90529	1.78106
$26$	−0.143581	−0.0281586
$27$	1.20419	0.231747
$28$	−1.27056	−0.240114
$29$	5.02034	0.932253	0.466126	−	0.884718i	$-0.345649\pi$
$29$	5.02034	0.932253	0.466126	+	0.884718i	$0.345649\pi$
$30$	−8.73399	−1.59460
$31$	1.50678	0.270626	0.135313	−	0.990803i	$-0.456796\pi$
$31$	1.50678	0.270626	0.135313	+	0.990803i	$0.456796\pi$
$32$	−1.00000	−0.176777
$33$	−0.571514	−0.0994878
$34$	−3.78082	−0.648406
$35$	4.73791	0.800853
$36$	2.48587	0.414312
$37$	2.81909	0.463455	0.231728	−	0.972781i	$-0.425562\pi$
$37$	2.81909	0.463455	0.231728	+	0.972781i	$0.425562\pi$
$38$	3.24695	0.526725
$39$	−0.336295	−0.0538503
$40$	3.72898	0.589604
$41$	3.59405	0.561296	0.280648	−	0.959811i	$-0.409451\pi$
$41$	3.59405	0.561296	0.280648	+	0.959811i	$0.409451\pi$
$42$	−2.97591	−0.459193
$43$	2.44877	0.373434	0.186717	−	0.982414i	$-0.440215\pi$
$43$	2.44877	0.373434	0.186717	+	0.982414i	$0.440215\pi$
$44$	0.244008	0.0367856
$45$	−9.26976	−1.38185
$46$	3.61808	0.533457
$47$	−13.1160	−1.91316	−0.956579	−	0.291473i	$-0.905855\pi$
$47$	−13.1160	−1.91316	−0.956579	+	0.291473i	$0.905855\pi$
$48$	−2.34219	−0.338066
$49$	−5.38566	−0.769381
$50$	−8.90529	−1.25940
$51$	−8.85542	−1.24001
$52$	0.143581	0.0199111
$53$	12.4992	1.71690	0.858451	−	0.512896i	$-0.171427\pi$
$53$	12.4992	1.71690	0.858451	+	0.512896i	$0.171427\pi$
$54$	−1.20419	−0.163870
$55$	−0.909901	−0.122691
$56$	1.27056	0.169786
$57$	7.60498	1.00730
$58$	−5.02034	−0.659202
$59$	2.59310	0.337593	0.168797	−	0.985651i	$-0.446012\pi$
$59$	2.59310	0.337593	0.168797	+	0.985651i	$0.446012\pi$
$60$	8.73399	1.12755
$61$	−4.56772	−0.584837	−0.292418	−	0.956290i	$-0.594460\pi$
$61$	−4.56772	−0.584837	−0.292418	+	0.956290i	$0.594460\pi$
$62$	−1.50678	−0.191362
$63$	−3.15846	−0.397928
$64$	1.00000	0.125000
$65$	−0.535412	−0.0664097
$66$	0.571514	0.0703485
$67$	−7.06271	−0.862847	−0.431424	−	0.902149i	$-0.641989\pi$
$67$	−7.06271	−0.862847	−0.431424	+	0.902149i	$0.641989\pi$
$68$	3.78082	0.458492
$69$	8.47424	1.02018
$70$	−4.73791	−0.566289
$71$	−9.62164	−1.14188	−0.570940	−	0.820992i	$-0.693421\pi$
$71$	−9.62164	−1.14188	−0.570940	+	0.820992i	$0.693421\pi$
$72$	−2.48587	−0.292963
$73$	6.05237	0.708377	0.354188	−	0.935174i	$-0.384757\pi$
$73$	6.05237	0.708377	0.354188	+	0.935174i	$0.384757\pi$
$74$	−2.81909	−0.327712
$75$	−20.8579	−2.40846
$76$	−3.24695	−0.372450
$77$	−0.310028	−0.0353310
$78$	0.336295	0.0380779
$79$	−2.60235	−0.292787	−0.146394	−	0.989226i	$-0.546767\pi$
$79$	−2.60235	−0.292787	−0.146394	+	0.989226i	$0.546767\pi$
$80$	−3.72898	−0.416913
$81$	−10.2781	−1.14201
$82$	−3.59405	−0.396896
$83$	1.67305	0.183641	0.0918207	−	0.995776i	$-0.470731\pi$
$83$	1.67305	0.183641	0.0918207	+	0.995776i	$0.470731\pi$
$84$	2.97591	0.324698
$85$	−14.0986	−1.52921
$86$	−2.44877	−0.264058
$87$	−11.7586	−1.26065
$88$	−0.244008	−0.0260113
$89$	10.4695	1.10976	0.554882	−	0.831929i	$-0.312763\pi$
$89$	10.4695	1.10976	0.554882	+	0.831929i	$0.312763\pi$
$90$	9.26976	0.977118
$91$	−0.182429	−0.0191238
$92$	−3.61808	−0.377211
$93$	−3.52918	−0.365959
$94$	13.1160	1.35281
$95$	12.1078	1.24223
$96$	2.34219	0.239049
$97$	−9.13696	−0.927718	−0.463859	−	0.885909i	$-0.653535\pi$
$97$	−9.13696	−0.927718	−0.463859	+	0.885909i	$0.653535\pi$
$98$	5.38566	0.544034
$99$	0.606572	0.0609628
$100$	8.90529	0.890529
$101$	4.60701	0.458414	0.229207	−	0.973378i	$-0.426387\pi$
$101$	4.60701	0.458414	0.229207	+	0.973378i	$0.426387\pi$
$102$	8.85542	0.876817
$103$	1.40990	0.138921	0.0694607	−	0.997585i	$-0.477872\pi$
$103$	1.40990	0.138921	0.0694607	+	0.997585i	$0.477872\pi$
$104$	−0.143581	−0.0140793
$105$	−11.0971	−1.08297
$106$	−12.4992	−1.21403
$107$	3.65794	0.353626	0.176813	−	0.984244i	$-0.443421\pi$
$107$	3.65794	0.353626	0.176813	+	0.984244i	$0.443421\pi$
$108$	1.20419	0.115874
$109$	−17.5931	−1.68511	−0.842557	−	0.538607i	$-0.818951\pi$
$109$	−17.5931	−1.68511	−0.842557	+	0.538607i	$0.818951\pi$
$110$	0.909901	0.0867557
$111$	−6.60285	−0.626715
$112$	−1.27056	−0.120057
$113$	13.2005	1.24180	0.620900	−	0.783890i	$-0.286767\pi$
$113$	13.2005	1.24180	0.620900	+	0.783890i	$0.286767\pi$
$114$	−7.60498	−0.712272
$115$	13.4917	1.25811
$116$	5.02034	0.466126
$117$	0.356924	0.0329977
$118$	−2.59310	−0.238714
$119$	−4.80378	−0.440362
$120$	−8.73399	−0.797301
$121$	−10.9405	−0.994587
$122$	4.56772	0.413542
$123$	−8.41795	−0.759021
$124$	1.50678	0.135313
$125$	−14.5628	−1.30253
$126$	3.15846	0.281378
$127$	−1.25319	−0.111203	−0.0556013	−	0.998453i	$-0.517708\pi$
$127$	−1.25319	−0.111203	−0.0556013	+	0.998453i	$0.517708\pi$
$128$	−1.00000	−0.0883883
$129$	−5.73550	−0.504982
$130$	0.535412	0.0469587
$131$	−11.9265	−1.04203	−0.521013	−	0.853549i	$-0.674446\pi$
$131$	−11.9265	−1.04203	−0.521013	+	0.853549i	$0.674446\pi$
$132$	−0.571514	−0.0497439
$133$	4.12546	0.357723
$134$	7.06271	0.610125
$135$	−4.49041	−0.386473
$136$	−3.78082	−0.324203
$137$	−16.3003	−1.39263	−0.696316	−	0.717735i	$-0.745179\pi$
$137$	−16.3003	−1.39263	−0.696316	+	0.717735i	$0.745179\pi$
$138$	−8.47424	−0.721375
$139$	−8.10096	−0.687114	−0.343557	−	0.939132i	$-0.611632\pi$
$139$	−8.10096	−0.687114	−0.343557	+	0.939132i	$0.611632\pi$
$140$	4.73791	0.400427
$141$	30.7201	2.58710
$142$	9.62164	0.807430
$143$	0.0350350	0.00292977
$144$	2.48587	0.207156
$145$	−18.7207	−1.55467
$146$	−6.05237	−0.500898
$147$	12.6143	1.04041
$148$	2.81909	0.231728
$149$	−11.2750	−0.923684	−0.461842	−	0.886962i	$-0.652811\pi$
$149$	−11.2750	−0.923684	−0.461842	+	0.886962i	$0.652811\pi$
$150$	20.8579	1.70304
$151$	−1.75706	−0.142987	−0.0714937	−	0.997441i	$-0.522777\pi$
$151$	−1.75706	−0.142987	−0.0714937	+	0.997441i	$0.522777\pi$
$152$	3.24695	0.263362
$153$	9.39863	0.759834
$154$	0.310028	0.0249828
$155$	−5.61877	−0.451310
$156$	−0.336295	−0.0269252
$157$	14.6683	1.17066	0.585329	−	0.810796i	$-0.300966\pi$
$157$	14.6683	1.17066	0.585329	+	0.810796i	$0.300966\pi$
$158$	2.60235	0.207032
$159$	−29.2756	−2.32171
$160$	3.72898	0.294802
$161$	4.59700	0.362295
$162$	10.2781	0.807521
$163$	−15.0636	−1.17987	−0.589936	−	0.807450i	$-0.700847\pi$
$163$	−15.0636	−1.17987	−0.589936	+	0.807450i	$0.700847\pi$
$164$	3.59405	0.280648
$165$	2.13116	0.165911
$166$	−1.67305	−0.129854
$167$	17.0859	1.32215	0.661073	−	0.750322i	$-0.270102\pi$
$167$	17.0859	1.32215	0.661073	+	0.750322i	$0.270102\pi$
$168$	−2.97591	−0.229596
$169$	−12.9794	−0.998414
$170$	14.0986	1.08131
$171$	−8.07149	−0.617242
$172$	2.44877	0.186717
$173$	11.2167	0.852788	0.426394	−	0.904538i	$-0.359784\pi$
$173$	11.2167	0.852788	0.426394	+	0.904538i	$0.359784\pi$
$174$	11.7586	0.891417
$175$	−11.3148	−0.855315
$176$	0.244008	0.0183928
$177$	−6.07354	−0.456516
$178$	−10.4695	−0.784722
$179$	−7.12986	−0.532910	−0.266455	−	0.963847i	$-0.585852\pi$
$179$	−7.12986	−0.532910	−0.266455	+	0.963847i	$0.585852\pi$
$180$	−9.26976	−0.690927
$181$	−11.7899	−0.876333	−0.438167	−	0.898894i	$-0.644372\pi$
$181$	−11.7899	−0.876333	−0.438167	+	0.898894i	$0.644372\pi$
$182$	0.182429	0.0135226
$183$	10.6985	0.790855
$184$	3.61808	0.266728
$185$	−10.5123	−0.772881
$186$	3.52918	0.258772
$187$	0.922551	0.0674636
$188$	−13.1160	−0.956579
$189$	−1.53001	−0.111292
$190$	−12.1078	−0.878392
$191$	19.3582	1.40071	0.700356	−	0.713794i	$-0.253025\pi$
$191$	19.3582	1.40071	0.700356	+	0.713794i	$0.253025\pi$
$192$	−2.34219	−0.169033
$193$	3.85730	0.277655	0.138827	−	0.990317i	$-0.455667\pi$
$193$	3.85730	0.277655	0.138827	+	0.990317i	$0.455667\pi$
$194$	9.13696	0.655995
$195$	1.25404	0.0898035
$196$	−5.38566	−0.384690
$197$	−18.1391	−1.29235	−0.646177	−	0.763187i	$-0.723633\pi$
$197$	−18.1391	−1.29235	−0.646177	+	0.763187i	$0.723633\pi$
$198$	−0.606572	−0.0431072
$199$	−3.33831	−0.236646	−0.118323	−	0.992975i	$-0.537752\pi$
$199$	−3.33831	−0.236646	−0.118323	+	0.992975i	$0.537752\pi$
$200$	−8.90529	−0.629699
$201$	16.5422	1.16680
$202$	−4.60701	−0.324148
$203$	−6.37866	−0.447694
$204$	−8.85542	−0.620003
$205$	−13.4021	−0.936045
$206$	−1.40990	−0.0982322
$207$	−8.99407	−0.625131
$208$	0.143581	0.00995557
$209$	−0.792281	−0.0548033
$210$	11.0971	0.765773
$211$	−20.6390	−1.42085	−0.710425	−	0.703773i	$-0.751497\pi$
$211$	−20.6390	−1.42085	−0.710425	+	0.703773i	$0.751497\pi$
$212$	12.4992	0.858451
$213$	22.5358	1.54412
$214$	−3.65794	−0.250051
$215$	−9.13142	−0.622758
$216$	−1.20419	−0.0819350
$217$	−1.91447	−0.129962
$218$	17.5931	1.19156
$219$	−14.1758	−0.957914
$220$	−0.909901	−0.0613455
$221$	0.542855	0.0365164
$222$	6.60285	0.443154
$223$	−9.45437	−0.633111	−0.316556	−	0.948574i	$-0.602526\pi$
$223$	−9.45437	−0.633111	−0.316556	+	0.948574i	$0.602526\pi$
$224$	1.27056	0.0848932
$225$	22.1374	1.47583
$226$	−13.2005	−0.878085
$227$	9.01139	0.598107	0.299054	−	0.954236i	$-0.403329\pi$
$227$	9.01139	0.598107	0.299054	+	0.954236i	$0.403329\pi$
$228$	7.60498	0.503652
$229$	14.0082	0.925688	0.462844	−	0.886440i	$-0.346829\pi$
$229$	14.0082	0.925688	0.462844	+	0.886440i	$0.346829\pi$
$230$	−13.4917	−0.889619
$231$	0.726146	0.0477769
$232$	−5.02034	−0.329601
$233$	−16.9933	−1.11327	−0.556633	−	0.830758i	$-0.687907\pi$
$233$	−16.9933	−1.11327	−0.556633	+	0.830758i	$0.687907\pi$
$234$	−0.356924	−0.0233329
$235$	48.9091	3.19048
$236$	2.59310	0.168797
$237$	6.09520	0.395926
$238$	4.80378	0.311383
$239$	−27.6777	−1.79032	−0.895159	−	0.445746i	$-0.852938\pi$
$239$	−27.6777	−1.79032	−0.895159	+	0.445746i	$0.852938\pi$
$240$	8.73399	0.563777
$241$	1.46322	0.0942540	0.0471270	−	0.998889i	$-0.484993\pi$
$241$	1.46322	0.0942540	0.0471270	+	0.998889i	$0.484993\pi$
$242$	10.9405	0.703279
$243$	20.4606	1.31255
$244$	−4.56772	−0.292418
$245$	20.0830	1.28306
$246$	8.41795	0.536709
$247$	−0.466201	−0.0296637
$248$	−1.50678	−0.0956809
$249$	−3.91861	−0.248332
$250$	14.5628	0.921030
$251$	17.7429	1.11992	0.559960	−	0.828519i	$-0.310816\pi$
$251$	17.7429	1.11992	0.559960	+	0.828519i	$0.310816\pi$
$252$	−3.15846	−0.198964
$253$	−0.882840	−0.0555037
$254$	1.25319	0.0786321
$255$	33.0217	2.06790
$256$	1.00000	0.0625000
$257$	−8.13042	−0.507162	−0.253581	−	0.967314i	$-0.581608\pi$
$257$	−8.13042	−0.507162	−0.253581	+	0.967314i	$0.581608\pi$
$258$	5.73550	0.357076
$259$	−3.58183	−0.222564
$260$	−0.535412	−0.0332048
$261$	12.4799	0.772486
$262$	11.9265	0.736824
$263$	19.8007	1.22096	0.610481	−	0.792031i	$-0.290976\pi$
$263$	19.8007	1.22096	0.610481	+	0.792031i	$0.290976\pi$
$264$	0.571514	0.0351743
$265$	−46.6094	−2.86319
$266$	−4.12546	−0.252948
$267$	−24.5216	−1.50070
$268$	−7.06271	−0.431424
$269$	27.5524	1.67990	0.839948	−	0.542666i	$-0.182585\pi$
$269$	27.5524	1.67990	0.839948	+	0.542666i	$0.182585\pi$
$270$	4.49041	0.273278
$271$	4.08364	0.248064	0.124032	−	0.992278i	$-0.460418\pi$
$271$	4.08364	0.248064	0.124032	+	0.992278i	$0.460418\pi$
$272$	3.78082	0.229246
$273$	0.427285	0.0258605
$274$	16.3003	0.984740
$275$	2.17296	0.131035
$276$	8.47424	0.510089
$277$	−17.0523	−1.02457	−0.512286	−	0.858815i	$-0.671201\pi$
$277$	−17.0523	−1.02457	−0.512286	+	0.858815i	$0.671201\pi$
$278$	8.10096	0.485863
$279$	3.74567	0.224247
$280$	−4.73791	−0.283144
$281$	−23.7354	−1.41593	−0.707967	−	0.706246i	$-0.750387\pi$
$281$	−23.7354	−1.41593	−0.707967	+	0.706246i	$0.750387\pi$
$282$	−30.7201	−1.82936
$283$	18.6181	1.10673	0.553366	−	0.832938i	$-0.313343\pi$
$283$	18.6181	1.10673	0.553366	+	0.832938i	$0.313343\pi$
$284$	−9.62164	−0.570940
$285$	−28.3588	−1.67983
$286$	−0.0350350	−0.00207166
$287$	−4.56647	−0.269550
$288$	−2.48587	−0.146481
$289$	−2.70537	−0.159140
$290$	18.7207	1.09932
$291$	21.4005	1.25452
$292$	6.05237	0.354188
$293$	−20.9245	−1.22242	−0.611212	−	0.791467i	$-0.709318\pi$
$293$	−20.9245	−1.22242	−0.611212	+	0.791467i	$0.709318\pi$
$294$	−12.6143	−0.735679
$295$	−9.66962	−0.562987
$296$	−2.81909	−0.163856
$297$	0.293833	0.0170499
$298$	11.2750	0.653143
$299$	−0.519488	−0.0300428
$300$	−20.8579	−1.20423
$301$	−3.11132	−0.179334
$302$	1.75706	0.101107
$303$	−10.7905	−0.619898
$304$	−3.24695	−0.186225
$305$	17.0329	0.975303
$306$	−9.39863	−0.537284
$307$	−10.6690	−0.608914	−0.304457	−	0.952526i	$-0.598475\pi$
$307$	−10.6690	−0.608914	−0.304457	+	0.952526i	$0.598475\pi$
$308$	−0.310028	−0.0176655
$309$	−3.30225	−0.187859
$310$	5.61877	0.319124
$311$	5.36136	0.304015	0.152008	−	0.988379i	$-0.451426\pi$
$311$	5.36136	0.304015	0.152008	+	0.988379i	$0.451426\pi$
$312$	0.336295	0.0190390
$313$	2.11158	0.119354	0.0596769	−	0.998218i	$-0.480993\pi$
$313$	2.11158	0.119354	0.0596769	+	0.998218i	$0.480993\pi$
$314$	−14.6683	−0.827780
$315$	11.7778	0.663605
$316$	−2.60235	−0.146394
$317$	32.6446	1.83350	0.916751	−	0.399460i	$-0.130802\pi$
$317$	32.6446	1.83350	0.916751	+	0.399460i	$0.130802\pi$
$318$	29.2756	1.64170
$319$	1.22500	0.0685870
$320$	−3.72898	−0.208456
$321$	−8.56760	−0.478197
$322$	−4.59700	−0.256181
$323$	−12.2761	−0.683063
$324$	−10.2781	−0.571003
$325$	1.27863	0.0709258
$326$	15.0636	0.834296
$327$	41.2065	2.27872
$328$	−3.59405	−0.198448
$329$	16.6647	0.918753
$330$	−2.13116	−0.117317
$331$	−24.1153	−1.32549	−0.662747	−	0.748843i	$-0.730610\pi$
$331$	−24.1153	−1.32549	−0.662747	+	0.748843i	$0.730610\pi$
$332$	1.67305	0.0918207
$333$	7.00788	0.384030
$334$	−17.0859	−0.934898
$335$	26.3367	1.43893
$336$	2.97591	0.162349
$337$	12.9625	0.706112	0.353056	−	0.935602i	$-0.385142\pi$
$337$	12.9625	0.706112	0.353056	+	0.935602i	$0.385142\pi$
$338$	12.9794	0.705985
$339$	−30.9181	−1.67924
$340$	−14.0986	−0.764605
$341$	0.367667	0.0199103
$342$	8.07149	0.436456
$343$	15.7368	0.849707
$344$	−2.44877	−0.132029
$345$	−31.6003	−1.70130
$346$	−11.2167	−0.603012
$347$	15.9134	0.854276	0.427138	−	0.904186i	$-0.359522\pi$
$347$	15.9134	0.854276	0.427138	+	0.904186i	$0.359522\pi$
$348$	−11.7586	−0.630327
$349$	35.9816	1.92605	0.963026	−	0.269409i	$-0.0868283\pi$
$349$	35.9816	1.92605	0.963026	+	0.269409i	$0.0868283\pi$
$350$	11.3148	0.604799
$351$	0.172900	0.00922870
$352$	−0.244008	−0.0130057
$353$	1.33353	0.0709769	0.0354884	−	0.999370i	$-0.488701\pi$
$353$	1.33353	0.0709769	0.0354884	+	0.999370i	$0.488701\pi$
$354$	6.07354	0.322805
$355$	35.8789	1.90426
$356$	10.4695	0.554882
$357$	11.2514	0.595486
$358$	7.12986	0.376825
$359$	17.2023	0.907900	0.453950	−	0.891027i	$-0.350014\pi$
$359$	17.2023	0.907900	0.453950	+	0.891027i	$0.350014\pi$
$360$	9.26976	0.488559
$361$	−8.45733	−0.445123
$362$	11.7899	0.619661
$363$	25.6247	1.34495
$364$	−0.182429	−0.00956190
$365$	−22.5692	−1.18132
$366$	−10.6985	−0.559219
$367$	28.6016	1.49299	0.746496	−	0.665389i	$-0.231735\pi$
$367$	28.6016	1.49299	0.746496	+	0.665389i	$0.231735\pi$
$368$	−3.61808	−0.188605
$369$	8.93433	0.465102
$370$	10.5123	0.546509
$371$	−15.8811	−0.824505
$372$	−3.52918	−0.182979
$373$	5.18448	0.268442	0.134221	−	0.990951i	$-0.457147\pi$
$373$	5.18448	0.268442	0.134221	+	0.990951i	$0.457147\pi$
$374$	−0.922551	−0.0477040
$375$	34.1088	1.76137
$376$	13.1160	0.676404
$377$	0.720826	0.0371244
$378$	1.53001	0.0786950
$379$	−5.73617	−0.294647	−0.147324	−	0.989088i	$-0.547066\pi$
$379$	−5.73617	−0.294647	−0.147324	+	0.989088i	$0.547066\pi$
$380$	12.1078	0.621117
$381$	2.93521	0.150375
$382$	−19.3582	−0.990453
$383$	−19.2607	−0.984175	−0.492087	−	0.870546i	$-0.663766\pi$
$383$	−19.2607	−0.984175	−0.492087	+	0.870546i	$0.663766\pi$
$384$	2.34219	0.119525
$385$	1.15609	0.0589197
$386$	−3.85730	−0.196332
$387$	6.08733	0.309436
$388$	−9.13696	−0.463859
$389$	19.8857	1.00824	0.504122	−	0.863632i	$-0.331816\pi$
$389$	19.8857	1.00824	0.504122	+	0.863632i	$0.331816\pi$
$390$	−1.25404	−0.0635007
$391$	−13.6793	−0.691793
$392$	5.38566	0.272017
$393$	27.9343	1.40910
$394$	18.1391	0.913833
$395$	9.70411	0.488267
$396$	0.606572	0.0304814
$397$	27.7820	1.39434	0.697169	−	0.716907i	$-0.254443\pi$
$397$	27.7820	1.39434	0.697169	+	0.716907i	$0.254443\pi$
$398$	3.33831	0.167334
$399$	−9.66262	−0.483736
$400$	8.90529	0.445265
$401$	−34.5351	−1.72460	−0.862301	−	0.506396i	$-0.830977\pi$
$401$	−34.5351	−1.72460	−0.862301	+	0.506396i	$0.830977\pi$
$402$	−16.5422	−0.825052
$403$	0.216346	0.0107770
$404$	4.60701	0.229207
$405$	38.3267	1.90447
$406$	6.37866	0.316568
$407$	0.687880	0.0340969
$408$	8.85542	0.438409
$409$	1.44955	0.0716756	0.0358378	−	0.999358i	$-0.488590\pi$
$409$	1.44955	0.0716756	0.0358378	+	0.999358i	$0.488590\pi$
$410$	13.4021	0.661884
$411$	38.1786	1.88321
$412$	1.40990	0.0694607
$413$	−3.29470	−0.162122
$414$	8.99407	0.442034
$415$	−6.23878	−0.306250
$416$	−0.143581	−0.00703965
$417$	18.9740	0.929161
$418$	0.792281	0.0387518
$419$	32.6714	1.59610	0.798050	−	0.602591i	$-0.205865\pi$
$419$	32.6714	1.59610	0.798050	+	0.602591i	$0.205865\pi$
$420$	−11.0971	−0.541483
$421$	7.35191	0.358310	0.179155	−	0.983821i	$-0.442664\pi$
$421$	7.35191	0.358310	0.179155	+	0.983821i	$0.442664\pi$
$422$	20.6390	1.00469
$423$	−32.6045	−1.58529
$424$	−12.4992	−0.607016
$425$	33.6693	1.63320
$426$	−22.5358	−1.09186
$427$	5.80359	0.280855
$428$	3.65794	0.176813
$429$	−0.0820587	−0.00396183
$430$	9.13142	0.440356
$431$	−34.2268	−1.64864	−0.824322	−	0.566121i	$-0.808444\pi$
$431$	−34.2268	−1.64864	−0.824322	+	0.566121i	$0.808444\pi$
$432$	1.20419	0.0579368
$433$	−28.3299	−1.36145	−0.680725	−	0.732539i	$-0.738335\pi$
$433$	−28.3299	−1.36145	−0.680725	+	0.732539i	$0.738335\pi$
$434$	1.91447	0.0918973
$435$	43.8476	2.10233
$436$	−17.5931	−0.842557
$437$	11.7477	0.561969
$438$	14.1758	0.677347
$439$	−9.91361	−0.473151	−0.236575	−	0.971613i	$-0.576025\pi$
$439$	−9.91361	−0.473151	−0.236575	+	0.971613i	$0.576025\pi$
$440$	0.909901	0.0433778
$441$	−13.3881	−0.637527
$442$	−0.542855	−0.0258210
$443$	−9.64099	−0.458057	−0.229029	−	0.973420i	$-0.573555\pi$
$443$	−9.64099	−0.458057	−0.229029	+	0.973420i	$0.573555\pi$
$444$	−6.60285	−0.313357
$445$	−39.0405	−1.85070
$446$	9.45437	0.447677
$447$	26.4082	1.24907
$448$	−1.27056	−0.0600286
$449$	−3.99158	−0.188374	−0.0941872	−	0.995555i	$-0.530025\pi$
$449$	−3.99158	−0.188374	−0.0941872	+	0.995555i	$0.530025\pi$
$450$	−22.1374	−1.04357
$451$	0.876976	0.0412952
$452$	13.2005	0.620900
$453$	4.11537	0.193357
$454$	−9.01139	−0.422926
$455$	0.680275	0.0318918
$456$	−7.60498	−0.356136
$457$	33.1553	1.55094	0.775470	−	0.631384i	$-0.217513\pi$
$457$	33.1553	1.55094	0.775470	+	0.631384i	$0.217513\pi$
$458$	−14.0082	−0.654560
$459$	4.55284	0.212509
$460$	13.4917	0.629056
$461$	19.2205	0.895190	0.447595	−	0.894236i	$-0.352281\pi$
$461$	19.2205	0.895190	0.447595	+	0.894236i	$0.352281\pi$
$462$	−0.726146	−0.0337834
$463$	20.7312	0.963461	0.481730	−	0.876319i	$-0.340008\pi$
$463$	20.7312	0.963461	0.481730	+	0.876319i	$0.340008\pi$
$464$	5.02034	0.233063
$465$	13.1602	0.610291
$466$	16.9933	0.787199
$467$	−36.3213	−1.68075	−0.840375	−	0.542005i	$-0.817665\pi$
$467$	−36.3213	−1.68075	−0.840375	+	0.542005i	$0.817665\pi$
$468$	0.356924	0.0164988
$469$	8.97363	0.414364
$470$	−48.9091	−2.25601
$471$	−34.3560	−1.58304
$472$	−2.59310	−0.119357
$473$	0.597520	0.0274740
$474$	−6.09520	−0.279962
$475$	−28.9150	−1.32671
$476$	−4.80378	−0.220181
$477$	31.0715	1.42266
$478$	27.6777	1.26595
$479$	21.1110	0.964588	0.482294	−	0.876009i	$-0.339804\pi$
$479$	21.1110	0.964588	0.482294	+	0.876009i	$0.339804\pi$
$480$	−8.73399	−0.398650
$481$	0.404768	0.0184558
$482$	−1.46322	−0.0666476
$483$	−10.7671	−0.489919
$484$	−10.9405	−0.497294
$485$	34.0715	1.54711
$486$	−20.4606	−0.928113
$487$	−35.4120	−1.60467	−0.802334	−	0.596875i	$-0.796409\pi$
$487$	−35.4120	−1.60467	−0.802334	+	0.596875i	$0.796409\pi$
$488$	4.56772	0.206771
$489$	35.2819	1.59550
$490$	−20.0830	−0.907259
$491$	4.76378	0.214987	0.107493	−	0.994206i	$-0.465718\pi$
$491$	4.76378	0.214987	0.107493	+	0.994206i	$0.465718\pi$
$492$	−8.41795	−0.379510
$493$	18.9810	0.854861
$494$	0.466201	0.0209754
$495$	−2.26190	−0.101665
$496$	1.50678	0.0676566
$497$	12.2249	0.548363
$498$	3.91861	0.175597
$499$	−24.7578	−1.10831	−0.554156	−	0.832413i	$-0.686959\pi$
$499$	−24.7578	−1.10831	−0.554156	+	0.832413i	$0.686959\pi$
$500$	−14.5628	−0.651266
$501$	−40.0184	−1.78789
$502$	−17.7429	−0.791903
$503$	−22.3672	−0.997304	−0.498652	−	0.866802i	$-0.666171\pi$
$503$	−22.3672	−0.997304	−0.498652	+	0.866802i	$0.666171\pi$
$504$	3.15846	0.140689
$505$	−17.1794	−0.764475
$506$	0.882840	0.0392470
$507$	30.4002	1.35012
$508$	−1.25319	−0.0556013
$509$	−41.5912	−1.84350	−0.921749	−	0.387786i	$-0.873240\pi$
$509$	−41.5912	−1.84350	−0.921749	+	0.387786i	$0.873240\pi$
$510$	−33.0217	−1.46222
$511$	−7.68993	−0.340183
$512$	−1.00000	−0.0441942
$513$	−3.90995	−0.172629
$514$	8.13042	0.358617
$515$	−5.25748	−0.231672
$516$	−5.73550	−0.252491
$517$	−3.20040	−0.140753
$518$	3.58183	0.157377
$519$	−26.2716	−1.15320
$520$	0.535412	0.0234794
$521$	−11.2842	−0.494371	−0.247185	−	0.968968i	$-0.579506\pi$
$521$	−11.2842	−0.494371	−0.247185	+	0.968968i	$0.579506\pi$
$522$	−12.4799	−0.546230
$523$	−16.4953	−0.721288	−0.360644	−	0.932703i	$-0.617443\pi$
$523$	−16.4953	−0.721288	−0.360644	+	0.932703i	$0.617443\pi$
$524$	−11.9265	−0.521013
$525$	26.5013	1.15661
$526$	−19.8007	−0.863351
$527$	5.69688	0.248160
$528$	−0.571514	−0.0248720
$529$	−9.90951	−0.430848
$530$	46.6094	2.02458
$531$	6.44611	0.279737
$532$	4.12546	0.178861
$533$	0.516038	0.0223521
$534$	24.5216	1.06115
$535$	−13.6404	−0.589725
$536$	7.06271	0.305063
$537$	16.6995	0.720637
$538$	−27.5524	−1.18787
$539$	−1.31415	−0.0566043
$540$	−4.49041	−0.193237
$541$	15.5531	0.668679	0.334340	−	0.942453i	$-0.391487\pi$
$541$	15.5531	0.668679	0.334340	+	0.942453i	$0.391487\pi$
$542$	−4.08364	−0.175407
$543$	27.6141	1.18504
$544$	−3.78082	−0.162101
$545$	65.6043	2.81018
$546$	−0.427285	−0.0182861
$547$	28.4419	1.21609	0.608045	−	0.793903i	$-0.291954\pi$
$547$	28.4419	1.21609	0.608045	+	0.793903i	$0.291954\pi$
$548$	−16.3003	−0.696316
$549$	−11.3548	−0.484609
$550$	−2.17296	−0.0926555
$551$	−16.3008	−0.694436
$552$	−8.47424	−0.360688
$553$	3.30645	0.140605
$554$	17.0523	0.724481
$555$	24.6219	1.04514
$556$	−8.10096	−0.343557
$557$	−1.63889	−0.0694422	−0.0347211	−	0.999397i	$-0.511054\pi$
$557$	−1.63889	−0.0694422	−0.0347211	+	0.999397i	$0.511054\pi$
$558$	−3.74567	−0.158567
$559$	0.351598	0.0148710
$560$	4.73791	0.200213
$561$	−2.16079	−0.0912288
$562$	23.7354	1.00122
$563$	−25.9548	−1.09387	−0.546933	−	0.837176i	$-0.684205\pi$
$563$	−25.9548	−1.09387	−0.546933	+	0.837176i	$0.684205\pi$
$564$	30.7201	1.29355
$565$	−49.2244	−2.07089
$566$	−18.6181	−0.782578
$567$	13.0589	0.548424
$568$	9.62164	0.403715
$569$	29.8519	1.25146	0.625728	−	0.780041i	$-0.284802\pi$
$569$	29.8519	1.25146	0.625728	+	0.780041i	$0.284802\pi$
$570$	28.3588	1.18782
$571$	32.2856	1.35111	0.675554	−	0.737310i	$-0.263904\pi$
$571$	32.2856	1.35111	0.675554	+	0.737310i	$0.263904\pi$
$572$	0.0350350	0.00146489
$573$	−45.3407	−1.89413
$574$	4.56647	0.190601
$575$	−32.2200	−1.34367
$576$	2.48587	0.103578
$577$	26.4392	1.10068	0.550339	−	0.834941i	$-0.314499\pi$
$577$	26.4392	1.10068	0.550339	+	0.834941i	$0.314499\pi$
$578$	2.70537	0.112529
$579$	−9.03455	−0.375463
$580$	−18.7207	−0.777336
$581$	−2.12572	−0.0881898
$582$	−21.4005	−0.887080
$583$	3.04991	0.126315
$584$	−6.05237	−0.250449
$585$	−1.33096	−0.0550286
$586$	20.9245	0.864384
$587$	32.5467	1.34335	0.671674	−	0.740847i	$-0.265576\pi$
$587$	32.5467	1.34335	0.671674	+	0.740847i	$0.265576\pi$
$588$	12.6143	0.520204
$589$	−4.89245	−0.201590
$590$	9.66962	0.398092
$591$	42.4852	1.74761
$592$	2.81909	0.115864
$593$	−4.24703	−0.174405	−0.0872024	−	0.996191i	$-0.527793\pi$
$593$	−4.24703	−0.174405	−0.0872024	+	0.996191i	$0.527793\pi$
$594$	−0.293833	−0.0120561
$595$	17.9132	0.734370
$596$	−11.2750	−0.461842
$597$	7.81896	0.320009
$598$	0.519488	0.0212435
$599$	−21.1421	−0.863842	−0.431921	−	0.901912i	$-0.642164\pi$
$599$	−21.1421	−0.863842	−0.431921	+	0.901912i	$0.642164\pi$
$600$	20.8579	0.851521
$601$	−6.86448	−0.280008	−0.140004	−	0.990151i	$-0.544712\pi$
$601$	−6.86448	−0.280008	−0.140004	+	0.990151i	$0.544712\pi$
$602$	3.11132	0.126808
$603$	−17.5570	−0.714975
$604$	−1.75706	−0.0714937
$605$	40.7968	1.65862
$606$	10.7905	0.438334
$607$	12.3271	0.500341	0.250170	−	0.968202i	$-0.419513\pi$
$607$	12.3271	0.500341	0.250170	+	0.968202i	$0.419513\pi$
$608$	3.24695	0.131681
$609$	14.9401	0.605402
$610$	−17.0329	−0.689644
$611$	−1.88321	−0.0761863
$612$	9.39863	0.379917
$613$	−5.57584	−0.225206	−0.112603	−	0.993640i	$-0.535919\pi$
$613$	−5.57584	−0.225206	−0.112603	+	0.993640i	$0.535919\pi$
$614$	10.6690	0.430567
$615$	31.3904	1.26578
$616$	0.310028	0.0124914
$617$	−26.8502	−1.08095	−0.540474	−	0.841361i	$-0.681755\pi$
$617$	−26.8502	−1.08095	−0.540474	+	0.841361i	$0.681755\pi$
$618$	3.30225	0.132836
$619$	4.32963	0.174022	0.0870112	−	0.996207i	$-0.472268\pi$
$619$	4.32963	0.174022	0.0870112	+	0.996207i	$0.472268\pi$
$620$	−5.61877	−0.225655
$621$	−4.35687	−0.174835
$622$	−5.36136	−0.214971
$623$	−13.3022	−0.532940
$624$	−0.336295	−0.0134626
$625$	9.77777	0.391111
$626$	−2.11158	−0.0843959
$627$	1.85568	0.0741086
$628$	14.6683	0.585329
$629$	10.6585	0.424981
$630$	−11.7778	−0.469240
$631$	21.7582	0.866179	0.433090	−	0.901351i	$-0.357423\pi$
$631$	21.7582	0.866179	0.433090	+	0.901351i	$0.357423\pi$
$632$	2.60235	0.103516
$633$	48.3406	1.92137
$634$	−32.6446	−1.29648
$635$	4.67312	0.185447
$636$	−29.2756	−1.16085
$637$	−0.773281	−0.0306385
$638$	−1.22500	−0.0484983
$639$	−23.9182	−0.946187
$640$	3.72898	0.147401
$641$	17.4459	0.689071	0.344535	−	0.938773i	$-0.388037\pi$
$641$	17.4459	0.689071	0.344535	+	0.938773i	$0.388037\pi$
$642$	8.56760	0.338136
$643$	−30.5334	−1.20412	−0.602060	−	0.798451i	$-0.705653\pi$
$643$	−30.5334	−1.20412	−0.602060	+	0.798451i	$0.705653\pi$
$644$	4.59700	0.181147
$645$	21.3876	0.842134
$646$	12.2761	0.482998
$647$	22.5026	0.884669	0.442335	−	0.896850i	$-0.354150\pi$
$647$	22.5026	0.884669	0.442335	+	0.896850i	$0.354150\pi$
$648$	10.2781	0.403760
$649$	0.632738	0.0248371
$650$	−1.27863	−0.0501521
$651$	4.48405	0.175744
$652$	−15.0636	−0.589936
$653$	42.2708	1.65418	0.827092	−	0.562066i	$-0.189993\pi$
$653$	42.2708	1.65418	0.827092	+	0.562066i	$0.189993\pi$
$654$	−41.2065	−1.61130
$655$	44.4738	1.73774
$656$	3.59405	0.140324
$657$	15.0454	0.586977
$658$	−16.6647	−0.649656
$659$	−12.0661	−0.470029	−0.235014	−	0.971992i	$-0.575514\pi$
$659$	−12.0661	−0.470029	−0.235014	+	0.971992i	$0.575514\pi$
$660$	2.13116	0.0829555
$661$	22.6153	0.879632	0.439816	−	0.898088i	$-0.355044\pi$
$661$	22.6153	0.879632	0.439816	+	0.898088i	$0.355044\pi$
$662$	24.1153	0.937266
$663$	−1.27147	−0.0493799
$664$	−1.67305	−0.0649270
$665$	−15.3838	−0.596556
$666$	−7.00788	−0.271550
$667$	−18.1640	−0.703312
$668$	17.0859	0.661073
$669$	22.1440	0.856135
$670$	−26.3367	−1.01748
$671$	−1.11456	−0.0430271
$672$	−2.97591	−0.114798
$673$	−31.7303	−1.22311	−0.611556	−	0.791201i	$-0.709456\pi$
$673$	−31.7303	−1.22311	−0.611556	+	0.791201i	$0.709456\pi$
$674$	−12.9625	−0.499297
$675$	10.7237	0.412755
$676$	−12.9794	−0.499207
$677$	23.1555	0.889938	0.444969	−	0.895546i	$-0.353215\pi$
$677$	23.1555	0.889938	0.444969	+	0.895546i	$0.353215\pi$
$678$	30.9181	1.18740
$679$	11.6091	0.445516
$680$	14.0986	0.540657
$681$	−21.1064	−0.808800
$682$	−0.367667	−0.0140787
$683$	27.4412	1.05001	0.525004	−	0.851100i	$-0.324064\pi$
$683$	27.4412	1.05001	0.525004	+	0.851100i	$0.324064\pi$
$684$	−8.07149	−0.308621
$685$	60.7837	2.32242
$686$	−15.7368	−0.600834
$687$	−32.8099	−1.25178
$688$	2.44877	0.0933585
$689$	1.79466	0.0683709
$690$	31.6003	1.20300
$691$	−34.7674	−1.32262	−0.661308	−	0.750115i	$-0.729998\pi$
$691$	−34.7674	−1.32262	−0.661308	+	0.750115i	$0.729998\pi$
$692$	11.2167	0.426394
$693$	−0.770689	−0.0292761
$694$	−15.9134	−0.604064
$695$	30.2083	1.14587
$696$	11.7586	0.445708
$697$	13.5885	0.514699
$698$	−35.9816	−1.36192
$699$	39.8016	1.50543
$700$	−11.3148	−0.427657
$701$	47.7969	1.80527	0.902633	−	0.430412i	$-0.141632\pi$
$701$	47.7969	1.80527	0.902633	+	0.430412i	$0.141632\pi$
$702$	−0.172900	−0.00652568
$703$	−9.15343	−0.345228
$704$	0.244008	0.00919640
$705$	−114.555	−4.31438
$706$	−1.33353	−0.0501882
$707$	−5.85350	−0.220144
$708$	−6.07354	−0.228258
$709$	14.4273	0.541827	0.270914	−	0.962604i	$-0.412674\pi$
$709$	14.4273	0.541827	0.270914	+	0.962604i	$0.412674\pi$
$710$	−35.8789	−1.34651
$711$	−6.46910	−0.242610
$712$	−10.4695	−0.392361
$713$	−5.45166	−0.204166
$714$	−11.2514	−0.421073
$715$	−0.130645	−0.00488584
$716$	−7.12986	−0.266455
$717$	64.8264	2.42099
$718$	−17.2023	−0.641983
$719$	50.5941	1.88684	0.943421	−	0.331597i	$-0.107587\pi$
$719$	50.5941	1.88684	0.943421	+	0.331597i	$0.107587\pi$
$720$	−9.26976	−0.345463
$721$	−1.79137	−0.0667140
$722$	8.45733	0.314749
$723$	−3.42713	−0.127456
$724$	−11.7899	−0.438167
$725$	44.7076	1.66040
$726$	−25.6247	−0.951021
$727$	17.1904	0.637557	0.318779	−	0.947829i	$-0.396727\pi$
$727$	17.1904	0.637557	0.318779	+	0.947829i	$0.396727\pi$
$728$	0.182429	0.00676128
$729$	−17.0886	−0.632909
$730$	22.5692	0.835323
$731$	9.25837	0.342433
$732$	10.6985	0.395427
$733$	−27.7433	−1.02472	−0.512361	−	0.858770i	$-0.671229\pi$
$733$	−27.7433	−1.02472	−0.512361	+	0.858770i	$0.671229\pi$
$734$	−28.6016	−1.05571
$735$	−47.0384	−1.73504
$736$	3.61808	0.133364
$737$	−1.72336	−0.0634807
$738$	−8.93433	−0.328877
$739$	−5.24333	−0.192879	−0.0964396	−	0.995339i	$-0.530745\pi$
$739$	−5.24333	−0.192879	−0.0964396	+	0.995339i	$0.530745\pi$
$740$	−10.5123	−0.386441
$741$	1.09193	0.0401131
$742$	15.8811	0.583013
$743$	−1.62542	−0.0596309	−0.0298155	−	0.999555i	$-0.509492\pi$
$743$	−1.62542	−0.0596309	−0.0298155	+	0.999555i	$0.509492\pi$
$744$	3.52918	0.129386
$745$	42.0443	1.54038
$746$	−5.18448	−0.189817
$747$	4.15899	0.152169
$748$	0.922551	0.0337318
$749$	−4.64765	−0.169821
$750$	−34.1088	−1.24548
$751$	−33.1018	−1.20790	−0.603951	−	0.797021i	$-0.706408\pi$
$751$	−33.1018	−1.20790	−0.603951	+	0.797021i	$0.706408\pi$
$752$	−13.1160	−0.478290
$753$	−41.5572	−1.51443
$754$	−0.720826	−0.0262509
$755$	6.55204	0.238453
$756$	−1.53001	−0.0556458
$757$	40.1019	1.45753	0.728764	−	0.684764i	$-0.240095\pi$
$757$	40.1019	1.45753	0.728764	+	0.684764i	$0.240095\pi$
$758$	5.73617	0.208347
$759$	2.06778	0.0750558
$760$	−12.1078	−0.439196
$761$	28.5987	1.03670	0.518351	−	0.855168i	$-0.326546\pi$
$761$	28.5987	1.03670	0.518351	+	0.855168i	$0.326546\pi$
$762$	−2.93521	−0.106331
$763$	22.3532	0.809240
$764$	19.3582	0.700356
$765$	−35.0473	−1.26714
$766$	19.2607	0.695917
$767$	0.372321	0.0134437
$768$	−2.34219	−0.0845166
$769$	11.6340	0.419534	0.209767	−	0.977751i	$-0.432729\pi$
$769$	11.6340	0.419534	0.209767	+	0.977751i	$0.432729\pi$
$770$	−1.15609	−0.0416625
$771$	19.0430	0.685818
$772$	3.85730	0.138827
$773$	−20.2895	−0.729763	−0.364881	−	0.931054i	$-0.618890\pi$
$773$	−20.2895	−0.729763	−0.364881	+	0.931054i	$0.618890\pi$
$774$	−6.08733	−0.218804
$775$	13.4183	0.482001
$776$	9.13696	0.327998
$777$	8.38935	0.300966
$778$	−19.8857	−0.712936
$779$	−11.6697	−0.418110
$780$	1.25404	0.0449018
$781$	−2.34776	−0.0840094
$782$	13.6793	0.489171
$783$	6.04546	0.216047
$784$	−5.38566	−0.192345
$785$	−54.6978	−1.95225
$786$	−27.9343	−0.996382
$787$	33.8226	1.20564	0.602822	−	0.797875i	$-0.294043\pi$
$787$	33.8226	1.20564	0.602822	+	0.797875i	$0.294043\pi$
$788$	−18.1391	−0.646177
$789$	−46.3770	−1.65107
$790$	−9.70411	−0.345257
$791$	−16.7721	−0.596348
$792$	−0.606572	−0.0215536
$793$	−0.655839	−0.0232895
$794$	−27.7820	−0.985946
$795$	109.168	3.87180
$796$	−3.33831	−0.118323
$797$	18.9276	0.670450	0.335225	−	0.942138i	$-0.391188\pi$
$797$	18.9276	0.670450	0.335225	+	0.942138i	$0.391188\pi$
$798$	9.66262	0.342053
$799$	−49.5891	−1.75434
$800$	−8.90529	−0.314850
$801$	26.0258	0.919576
$802$	34.5351	1.21948
$803$	1.47683	0.0521161
$804$	16.5422	0.583400
$805$	−17.1421	−0.604181
$806$	−0.216346	−0.00762046
$807$	−64.5329	−2.27167
$808$	−4.60701	−0.162074
$809$	0.914466	0.0321509	0.0160755	−	0.999871i	$-0.494883\pi$
$809$	0.914466	0.0321509	0.0160755	+	0.999871i	$0.494883\pi$
$810$	−38.3267	−1.34666
$811$	−40.0607	−1.40672	−0.703360	−	0.710834i	$-0.748318\pi$
$811$	−40.0607	−1.40672	−0.703360	+	0.710834i	$0.748318\pi$
$812$	−6.37866	−0.223847
$813$	−9.56468	−0.335448
$814$	−0.687880	−0.0241102
$815$	56.1719	1.96762
$816$	−8.85542	−0.310002
$817$	−7.95103	−0.278171
$818$	−1.44955	−0.0506823
$819$	−0.453496	−0.0158464
$820$	−13.4021	−0.468022
$821$	14.8060	0.516732	0.258366	−	0.966047i	$-0.416816\pi$
$821$	14.8060	0.516732	0.258366	+	0.966047i	$0.416816\pi$
$822$	−38.1786	−1.33163
$823$	−45.6878	−1.59257	−0.796287	−	0.604918i	$-0.793206\pi$
$823$	−45.6878	−1.59257	−0.796287	+	0.604918i	$0.793206\pi$
$824$	−1.40990	−0.0491161
$825$	−5.08950	−0.177194
$826$	3.29470	0.114637
$827$	55.4752	1.92906	0.964531	−	0.263971i	$-0.0850324\pi$
$827$	55.4752	1.92906	0.964531	+	0.263971i	$0.0850324\pi$
$828$	−8.99407	−0.312566
$829$	−11.1614	−0.387651	−0.193826	−	0.981036i	$-0.562090\pi$
$829$	−11.1614	−0.387651	−0.193826	+	0.981036i	$0.562090\pi$
$830$	6.23878	0.216551
$831$	39.9397	1.38549
$832$	0.143581	0.00497779
$833$	−20.3622	−0.705510
$834$	−18.9740	−0.657016
$835$	−63.7129	−2.20488
$836$	−0.792281	−0.0274016
$837$	1.81446	0.0627169
$838$	−32.6714	−1.12861
$839$	39.1354	1.35111	0.675553	−	0.737312i	$-0.263905\pi$
$839$	39.1354	1.35111	0.675553	+	0.737312i	$0.263905\pi$
$840$	11.0971	0.382886
$841$	−3.79623	−0.130904
$842$	−7.35191	−0.253363
$843$	55.5928	1.91472
$844$	−20.6390	−0.710425
$845$	48.3999	1.66501
$846$	32.6045	1.12097
$847$	13.9006	0.477629
$848$	12.4992	0.429225
$849$	−43.6073	−1.49660
$850$	−33.6693	−1.15485
$851$	−10.1997	−0.349640
$852$	22.5358	0.772062
$853$	−13.9850	−0.478836	−0.239418	−	0.970917i	$-0.576957\pi$
$853$	−13.9850	−0.478836	−0.239418	+	0.970917i	$0.576957\pi$
$854$	−5.80359	−0.198595
$855$	30.0984	1.02934
$856$	−3.65794	−0.125026
$857$	−43.3844	−1.48198	−0.740992	−	0.671514i	$-0.765644\pi$
$857$	−43.3844	−1.48198	−0.740992	+	0.671514i	$0.765644\pi$
$858$	0.0820587	0.00280144
$859$	47.1702	1.60943	0.804713	−	0.593664i	$-0.202319\pi$
$859$	47.1702	1.60943	0.804713	+	0.593664i	$0.202319\pi$
$860$	−9.13142	−0.311379
$861$	10.6956	0.364503
$862$	34.2268	1.16577
$863$	20.9288	0.712425	0.356212	−	0.934405i	$-0.384068\pi$
$863$	20.9288	0.712425	0.356212	+	0.934405i	$0.384068\pi$
$864$	−1.20419	−0.0409675
$865$	−41.8267	−1.42215
$866$	28.3299	0.962690
$867$	6.33651	0.215199
$868$	−1.91447	−0.0649812
$869$	−0.634994	−0.0215407
$870$	−43.8476	−1.48657
$871$	−1.01407	−0.0343606
$872$	17.5931	0.595778
$873$	−22.7133	−0.768728
$874$	−11.7477	−0.397372
$875$	18.5029	0.625513
$876$	−14.1758	−0.478957
$877$	−29.1876	−0.985595	−0.492798	−	0.870144i	$-0.664026\pi$
$877$	−29.1876	−0.985595	−0.492798	+	0.870144i	$0.664026\pi$
$878$	9.91361	0.334568
$879$	49.0093	1.65304
$880$	−0.909901	−0.0306728
$881$	8.89471	0.299670	0.149835	−	0.988711i	$-0.452126\pi$
$881$	8.89471	0.299670	0.149835	+	0.988711i	$0.452126\pi$
$882$	13.3881	0.450799
$883$	10.4459	0.351533	0.175766	−	0.984432i	$-0.443760\pi$
$883$	10.4459	0.351533	0.175766	+	0.984432i	$0.443760\pi$
$884$	0.542855	0.0182582
$885$	22.6481	0.761308
$886$	9.64099	0.323895
$887$	−55.2988	−1.85675	−0.928376	−	0.371643i	$-0.878795\pi$
$887$	−55.2988	−1.85675	−0.928376	+	0.371643i	$0.878795\pi$
$888$	6.60285	0.221577
$889$	1.59226	0.0534026
$890$	39.0405	1.30864
$891$	−2.50793	−0.0840188
$892$	−9.45437	−0.316556
$893$	42.5868	1.42511
$894$	−26.4082	−0.883224
$895$	26.5871	0.888708
$896$	1.27056	0.0424466
$897$	1.21674	0.0406258
$898$	3.99158	0.133201
$899$	7.56456	0.252292
$900$	22.1374	0.737913
$901$	47.2574	1.57437
$902$	−0.876976	−0.0292001
$903$	7.28732	0.242507
$904$	−13.2005	−0.439042
$905$	43.9641	1.46142
$906$	−4.11537	−0.136724
$907$	−35.0587	−1.16410	−0.582052	−	0.813151i	$-0.697750\pi$
$907$	−35.0587	−1.16410	−0.582052	+	0.813151i	$0.697750\pi$
$908$	9.01139	0.299054
$909$	11.4524	0.379853
$910$	−0.680275	−0.0225509
$911$	6.87527	0.227788	0.113894	−	0.993493i	$-0.463668\pi$
$911$	6.87527	0.227788	0.113894	+	0.993493i	$0.463668\pi$
$912$	7.60498	0.251826
$913$	0.408238	0.0135107
$914$	−33.1553	−1.09668
$915$	−39.8944	−1.31887
$916$	14.0082	0.462844
$917$	15.1534	0.500411
$918$	−4.55284	−0.150266
$919$	−21.7069	−0.716043	−0.358022	−	0.933713i	$-0.616549\pi$
$919$	−21.7069	−0.716043	−0.358022	+	0.933713i	$0.616549\pi$
$920$	−13.4917	−0.444810
$921$	24.9889	0.823414
$922$	−19.2205	−0.632995
$923$	−1.38149	−0.0454722
$924$	0.726146	0.0238884
$925$	25.1048	0.825441
$926$	−20.7312	−0.681270
$927$	3.50482	0.115113
$928$	−5.02034	−0.164801
$929$	9.20780	0.302098	0.151049	−	0.988526i	$-0.451735\pi$
$929$	9.20780	0.302098	0.151049	+	0.988526i	$0.451735\pi$
$930$	−13.1602	−0.431541
$931$	17.4870	0.573112
$932$	−16.9933	−0.556633
$933$	−12.5574	−0.411109
$934$	36.3213	1.18847
$935$	−3.44018	−0.112506
$936$	−0.356924	−0.0116664
$937$	0.571543	0.0186715	0.00933574	−	0.999956i	$-0.497028\pi$
$937$	0.571543	0.0186715	0.00933574	+	0.999956i	$0.497028\pi$
$938$	−8.97363	−0.292999
$939$	−4.94574	−0.161398
$940$	48.9091	1.59524
$941$	−17.8951	−0.583365	−0.291682	−	0.956515i	$-0.594215\pi$
$941$	−17.8951	−0.583365	−0.291682	+	0.956515i	$0.594215\pi$
$942$	34.3560	1.11938
$943$	−13.0035	−0.423453
$944$	2.59310	0.0843983
$945$	5.70536	0.185595
$946$	−0.597520	−0.0194271
$947$	−25.2069	−0.819113	−0.409557	−	0.912285i	$-0.634317\pi$
$947$	−25.2069	−0.819113	−0.409557	+	0.912285i	$0.634317\pi$
$948$	6.09520	0.197963
$949$	0.869007	0.0282092
$950$	28.9150	0.938127
$951$	−76.4599	−2.47938
$952$	4.80378	0.155691
$953$	56.8256	1.84076	0.920381	−	0.391023i	$-0.127879\pi$
$953$	56.8256	1.84076	0.920381	+	0.391023i	$0.127879\pi$
$954$	−31.0715	−1.00598
$955$	−72.1864	−2.33590
$956$	−27.6777	−0.895159
$957$	−2.86919	−0.0927478
$958$	−21.1110	−0.682067
$959$	20.7106	0.668782
$960$	8.73399	0.281888
$961$	−28.7296	−0.926761
$962$	−0.404768	−0.0130502
$963$	9.09315	0.293023
$964$	1.46322	0.0471270
$965$	−14.3838	−0.463031
$966$	10.7671	0.346425
$967$	50.8509	1.63525	0.817627	−	0.575748i	$-0.195289\pi$
$967$	50.8509	1.63525	0.817627	+	0.575748i	$0.195289\pi$
$968$	10.9405	0.351640
$969$	28.7531	0.923682
$970$	−34.0715	−1.09397
$971$	7.22578	0.231886	0.115943	−	0.993256i	$-0.463011\pi$
$971$	7.22578	0.231886	0.115943	+	0.993256i	$0.463011\pi$
$972$	20.4606	0.656275
$973$	10.2928	0.329972
$974$	35.4120	1.13467
$975$	−2.99481	−0.0959106
$976$	−4.56772	−0.146209
$977$	3.00467	0.0961278	0.0480639	−	0.998844i	$-0.484695\pi$
$977$	3.00467	0.0961278	0.0480639	+	0.998844i	$0.484695\pi$
$978$	−35.2819	−1.12819
$979$	2.55464	0.0816467
$980$	20.0830	0.641529
$981$	−43.7342	−1.39632
$982$	−4.76378	−0.152018
$983$	25.5594	0.815219	0.407610	−	0.913156i	$-0.366362\pi$
$983$	25.5594	0.815219	0.407610	+	0.913156i	$0.366362\pi$
$984$	8.41795	0.268354
$985$	67.6402	2.15520
$986$	−18.9810	−0.604478
$987$	−39.0319	−1.24240
$988$	−0.466201	−0.0148318
$989$	−8.85985	−0.281727
$990$	2.26190	0.0718878
$991$	−1.24830	−0.0396535	−0.0198267	−	0.999803i	$-0.506311\pi$
$991$	−1.24830	−0.0396535	−0.0198267	+	0.999803i	$0.506311\pi$
$992$	−1.50678	−0.0478404
$993$	56.4826	1.79242
$994$	−12.2249	−0.387751
$995$	12.4485	0.394644
$996$	−3.91861	−0.124166
$997$	33.2212	1.05212	0.526062	−	0.850446i	$-0.323668\pi$
$997$	33.2212	1.05212	0.526062	+	0.850446i	$0.323668\pi$
$998$	24.7578	0.783695
$999$	3.39473	0.107404

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.9	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.9	✓	83	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.34219 q^{3} +1.00000 q^{4} -3.72898 q^{5} +2.34219 q^{6} -1.27056 q^{7} -1.00000 q^{8} +2.48587 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.34219 q^{3} +1.00000 q^{4} -3.72898 q^{5} +2.34219 q^{6} -1.27056 q^{7} -1.00000 q^{8} +2.48587 q^{9} +3.72898 q^{10} +0.244008 q^{11} -2.34219 q^{12} +0.143581 q^{13} +1.27056 q^{14} +8.73399 q^{15} +1.00000 q^{16} +3.78082 q^{17} -2.48587 q^{18} -3.24695 q^{19} -3.72898 q^{20} +2.97591 q^{21} -0.244008 q^{22} -3.61808 q^{23} +2.34219 q^{24} +8.90529 q^{25} -0.143581 q^{26} +1.20419 q^{27} -1.27056 q^{28} +5.02034 q^{29} -8.73399 q^{30} +1.50678 q^{31} -1.00000 q^{32} -0.571514 q^{33} -3.78082 q^{34} +4.73791 q^{35} +2.48587 q^{36} +2.81909 q^{37} +3.24695 q^{38} -0.336295 q^{39} +3.72898 q^{40} +3.59405 q^{41} -2.97591 q^{42} +2.44877 q^{43} +0.244008 q^{44} -9.26976 q^{45} +3.61808 q^{46} -13.1160 q^{47} -2.34219 q^{48} -5.38566 q^{49} -8.90529 q^{50} -8.85542 q^{51} +0.143581 q^{52} +12.4992 q^{53} -1.20419 q^{54} -0.909901 q^{55} +1.27056 q^{56} +7.60498 q^{57} -5.02034 q^{58} +2.59310 q^{59} +8.73399 q^{60} -4.56772 q^{61} -1.50678 q^{62} -3.15846 q^{63} +1.00000 q^{64} -0.535412 q^{65} +0.571514 q^{66} -7.06271 q^{67} +3.78082 q^{68} +8.47424 q^{69} -4.73791 q^{70} -9.62164 q^{71} -2.48587 q^{72} +6.05237 q^{73} -2.81909 q^{74} -20.8579 q^{75} -3.24695 q^{76} -0.310028 q^{77} +0.336295 q^{78} -2.60235 q^{79} -3.72898 q^{80} -10.2781 q^{81} -3.59405 q^{82} +1.67305 q^{83} +2.97591 q^{84} -14.0986 q^{85} -2.44877 q^{86} -11.7586 q^{87} -0.244008 q^{88} +10.4695 q^{89} +9.26976 q^{90} -0.182429 q^{91} -3.61808 q^{92} -3.52918 q^{93} +13.1160 q^{94} +12.1078 q^{95} +2.34219 q^{96} -9.13696 q^{97} +5.38566 q^{98} +0.606572 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

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