LMFDB - Embedded newform 4006.2.a.g.1.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4006,2,Mod(1,4006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, 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("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.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:	$31.9880710497$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.327102 q^{3} +1.00000 q^{4} -1.35704 q^{5} -0.327102 q^{6} +3.43826 q^{7} -1.00000 q^{8} -2.89300 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.327102 q^{3} +1.00000 q^{4} -1.35704 q^{5} -0.327102 q^{6} +3.43826 q^{7} -1.00000 q^{8} -2.89300 q^{9} +1.35704 q^{10} +5.09387 q^{11} +0.327102 q^{12} -3.01379 q^{13} -3.43826 q^{14} -0.443892 q^{15} +1.00000 q^{16} +7.56750 q^{17} +2.89300 q^{18} -6.52028 q^{19} -1.35704 q^{20} +1.12466 q^{21} -5.09387 q^{22} -5.33903 q^{23} -0.327102 q^{24} -3.15843 q^{25} +3.01379 q^{26} -1.92762 q^{27} +3.43826 q^{28} -4.11643 q^{29} +0.443892 q^{30} +3.27005 q^{31} -1.00000 q^{32} +1.66622 q^{33} -7.56750 q^{34} -4.66586 q^{35} -2.89300 q^{36} -9.67379 q^{37} +6.52028 q^{38} -0.985817 q^{39} +1.35704 q^{40} -8.83536 q^{41} -1.12466 q^{42} -1.26264 q^{43} +5.09387 q^{44} +3.92593 q^{45} +5.33903 q^{46} +6.16447 q^{47} +0.327102 q^{48} +4.82160 q^{49} +3.15843 q^{50} +2.47535 q^{51} -3.01379 q^{52} -6.20736 q^{53} +1.92762 q^{54} -6.91260 q^{55} -3.43826 q^{56} -2.13280 q^{57} +4.11643 q^{58} -0.844389 q^{59} -0.443892 q^{60} +11.2078 q^{61} -3.27005 q^{62} -9.94689 q^{63} +1.00000 q^{64} +4.08984 q^{65} -1.66622 q^{66} -3.77217 q^{67} +7.56750 q^{68} -1.74641 q^{69} +4.66586 q^{70} -0.624439 q^{71} +2.89300 q^{72} +7.05016 q^{73} +9.67379 q^{74} -1.03313 q^{75} -6.52028 q^{76} +17.5140 q^{77} +0.985817 q^{78} +5.34416 q^{79} -1.35704 q^{80} +8.04848 q^{81} +8.83536 q^{82} -7.15126 q^{83} +1.12466 q^{84} -10.2694 q^{85} +1.26264 q^{86} -1.34649 q^{87} -5.09387 q^{88} -12.9678 q^{89} -3.92593 q^{90} -10.3622 q^{91} -5.33903 q^{92} +1.06964 q^{93} -6.16447 q^{94} +8.84830 q^{95} -0.327102 q^{96} +13.2191 q^{97} -4.82160 q^{98} -14.7366 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9}+O(q^{10}) $ 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9} + 23 q^{10} - 28 q^{11} - q^{12} - 12 q^{13} - 12 q^{14} - 14 q^{15} + 40 q^{16} - 10 q^{17} - 29 q^{18} + 3 q^{19} - 23 q^{20} - 40 q^{21} + 28 q^{22} - 10 q^{23} + q^{24} + 43 q^{25} + 12 q^{26} - 7 q^{27} + 12 q^{28} - 37 q^{29} + 14 q^{30} - 16 q^{31} - 40 q^{32} - 11 q^{33} + 10 q^{34} - 24 q^{35} + 29 q^{36} - 17 q^{37} - 3 q^{38} - 10 q^{39} + 23 q^{40} - 58 q^{41} + 40 q^{42} + 37 q^{43} - 28 q^{44} - 66 q^{45} + 10 q^{46} - 34 q^{47} - q^{48} + 28 q^{49} - 43 q^{50} - 7 q^{51} - 12 q^{52} - 43 q^{53} + 7 q^{54} + 28 q^{55} - 12 q^{56} - 25 q^{57} + 37 q^{58} - 92 q^{59} - 14 q^{60} - 37 q^{61} + 16 q^{62} + 14 q^{63} + 40 q^{64} - 54 q^{65} + 11 q^{66} - 10 q^{67} - 10 q^{68} - 49 q^{69} + 24 q^{70} - 87 q^{71} - 29 q^{72} + 12 q^{73} + 17 q^{74} - 31 q^{75} + 3 q^{76} - 53 q^{77} + 10 q^{78} + 14 q^{79} - 23 q^{80} - 4 q^{81} + 58 q^{82} - 42 q^{83} - 40 q^{84} - 24 q^{85} - 37 q^{86} + 24 q^{87} + 28 q^{88} - 118 q^{89} + 66 q^{90} - 35 q^{91} - 10 q^{92} - 22 q^{93} + 34 q^{94} - 18 q^{95} + q^{96} + 2 q^{97} - 28 q^{98} - 48 q^{99}+O(q^{100}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9 + 23 * q^10 - 28 * q^11 - q^12 - 12 * q^13 - 12 * q^14 - 14 * q^15 + 40 * q^16 - 10 * q^17 - 29 * q^18 + 3 * q^19 - 23 * q^20 - 40 * q^21 + 28 * q^22 - 10 * q^23 + q^24 + 43 * q^25 + 12 * q^26 - 7 * q^27 + 12 * q^28 - 37 * q^29 + 14 * q^30 - 16 * q^31 - 40 * q^32 - 11 * q^33 + 10 * q^34 - 24 * q^35 + 29 * q^36 - 17 * q^37 - 3 * q^38 - 10 * q^39 + 23 * q^40 - 58 * q^41 + 40 * q^42 + 37 * q^43 - 28 * q^44 - 66 * q^45 + 10 * q^46 - 34 * q^47 - q^48 + 28 * q^49 - 43 * q^50 - 7 * q^51 - 12 * q^52 - 43 * q^53 + 7 * q^54 + 28 * q^55 - 12 * q^56 - 25 * q^57 + 37 * q^58 - 92 * q^59 - 14 * q^60 - 37 * q^61 + 16 * q^62 + 14 * q^63 + 40 * q^64 - 54 * q^65 + 11 * q^66 - 10 * q^67 - 10 * q^68 - 49 * q^69 + 24 * q^70 - 87 * q^71 - 29 * q^72 + 12 * q^73 + 17 * q^74 - 31 * q^75 + 3 * q^76 - 53 * q^77 + 10 * q^78 + 14 * q^79 - 23 * q^80 - 4 * q^81 + 58 * q^82 - 42 * q^83 - 40 * q^84 - 24 * q^85 - 37 * q^86 + 24 * q^87 + 28 * q^88 - 118 * q^89 + 66 * q^90 - 35 * q^91 - 10 * q^92 - 22 * q^93 + 34 * q^94 - 18 * q^95 + q^96 + 2 * q^97 - 28 * q^98 - 48 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.327102	0.188853	0.0944263	−	0.995532i	$-0.469898\pi$
$3$	0.327102	0.188853	0.0944263	+	0.995532i	$0.469898\pi$
$4$	1.00000	0.500000
$5$	−1.35704	−0.606888	−0.303444	−	0.952849i	$-0.598137\pi$
$5$	−1.35704	−0.606888	−0.303444	+	0.952849i	$0.598137\pi$
$6$	−0.327102	−0.133539
$7$	3.43826	1.29954	0.649769	−	0.760132i	$-0.274866\pi$
$7$	3.43826	1.29954	0.649769	+	0.760132i	$0.274866\pi$
$8$	−1.00000	−0.353553
$9$	−2.89300	−0.964335
$10$	1.35704	0.429135
$11$	5.09387	1.53586	0.767929	−	0.640535i	$-0.221287\pi$
$11$	5.09387	1.53586	0.767929	+	0.640535i	$0.221287\pi$
$12$	0.327102	0.0944263
$13$	−3.01379	−0.835874	−0.417937	−	0.908476i	$-0.637247\pi$
$13$	−3.01379	−0.835874	−0.417937	+	0.908476i	$0.637247\pi$
$14$	−3.43826	−0.918912
$15$	−0.443892	−0.114612
$16$	1.00000	0.250000
$17$	7.56750	1.83539	0.917695	−	0.397286i	$-0.130048\pi$
$17$	7.56750	1.83539	0.917695	+	0.397286i	$0.130048\pi$
$18$	2.89300	0.681888
$19$	−6.52028	−1.49585	−0.747927	−	0.663781i	$-0.768951\pi$
$19$	−6.52028	−1.49585	−0.747927	+	0.663781i	$0.768951\pi$
$20$	−1.35704	−0.303444
$21$	1.12466	0.245421
$22$	−5.09387	−1.08602
$23$	−5.33903	−1.11327	−0.556633	−	0.830759i	$-0.687907\pi$
$23$	−5.33903	−1.11327	−0.556633	+	0.830759i	$0.687907\pi$
$24$	−0.327102	−0.0667695
$25$	−3.15843	−0.631687
$26$	3.01379	0.591052
$27$	−1.92762	−0.370970
$28$	3.43826	0.649769
$29$	−4.11643	−0.764402	−0.382201	−	0.924079i	$-0.624834\pi$
$29$	−4.11643	−0.764402	−0.382201	+	0.924079i	$0.624834\pi$
$30$	0.443892	0.0810432
$31$	3.27005	0.587318	0.293659	−	0.955910i	$-0.405127\pi$
$31$	3.27005	0.587318	0.293659	+	0.955910i	$0.405127\pi$
$32$	−1.00000	−0.176777
$33$	1.66622	0.290051
$34$	−7.56750	−1.29782
$35$	−4.66586	−0.788675
$36$	−2.89300	−0.482167
$37$	−9.67379	−1.59036	−0.795180	−	0.606373i	$-0.792624\pi$
$37$	−9.67379	−1.59036	−0.795180	+	0.606373i	$0.792624\pi$
$38$	6.52028	1.05773
$39$	−0.985817	−0.157857
$40$	1.35704	0.214567
$41$	−8.83536	−1.37985	−0.689925	−	0.723881i	$-0.742357\pi$
$41$	−8.83536	−1.37985	−0.689925	+	0.723881i	$0.742357\pi$
$42$	−1.12466	−0.173539
$43$	−1.26264	−0.192551	−0.0962757	−	0.995355i	$-0.530693\pi$
$43$	−1.26264	−0.192551	−0.0962757	+	0.995355i	$0.530693\pi$
$44$	5.09387	0.767929
$45$	3.92593	0.585243
$46$	5.33903	0.787197
$47$	6.16447	0.899180	0.449590	−	0.893235i	$-0.351570\pi$
$47$	6.16447	0.899180	0.449590	+	0.893235i	$0.351570\pi$
$48$	0.327102	0.0472132
$49$	4.82160	0.688800
$50$	3.15843	0.446670
$51$	2.47535	0.346618
$52$	−3.01379	−0.417937
$53$	−6.20736	−0.852647	−0.426324	−	0.904571i	$-0.640191\pi$
$53$	−6.20736	−0.852647	−0.426324	+	0.904571i	$0.640191\pi$
$54$	1.92762	0.262315
$55$	−6.91260	−0.932095
$56$	−3.43826	−0.459456
$57$	−2.13280	−0.282496
$58$	4.11643	0.540514
$59$	−0.844389	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$59$	−0.844389	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$60$	−0.443892	−0.0573062
$61$	11.2078	1.43501	0.717505	−	0.696553i	$-0.245284\pi$
$61$	11.2078	1.43501	0.717505	+	0.696553i	$0.245284\pi$
$62$	−3.27005	−0.415297
$63$	−9.94689	−1.25319
$64$	1.00000	0.125000
$65$	4.08984	0.507282
$66$	−1.66622	−0.205097
$67$	−3.77217	−0.460844	−0.230422	−	0.973091i	$-0.574011\pi$
$67$	−3.77217	−0.460844	−0.230422	+	0.973091i	$0.574011\pi$
$68$	7.56750	0.917695
$69$	−1.74641	−0.210243
$70$	4.66586	0.557677
$71$	−0.624439	−0.0741073	−0.0370537	−	0.999313i	$-0.511797\pi$
$71$	−0.624439	−0.0741073	−0.0370537	+	0.999313i	$0.511797\pi$
$72$	2.89300	0.340944
$73$	7.05016	0.825159	0.412580	−	0.910922i	$-0.364628\pi$
$73$	7.05016	0.825159	0.412580	+	0.910922i	$0.364628\pi$
$74$	9.67379	1.12456
$75$	−1.03313	−0.119296
$76$	−6.52028	−0.747927
$77$	17.5140	1.99591
$78$	0.985817	0.111622
$79$	5.34416	0.601265	0.300632	−	0.953740i	$-0.402802\pi$
$79$	5.34416	0.601265	0.300632	+	0.953740i	$0.402802\pi$
$80$	−1.35704	−0.151722
$81$	8.04848	0.894276
$82$	8.83536	0.975702
$83$	−7.15126	−0.784953	−0.392476	−	0.919762i	$-0.628382\pi$
$83$	−7.15126	−0.784953	−0.392476	+	0.919762i	$0.628382\pi$
$84$	1.12466	0.122711
$85$	−10.2694	−1.11388
$86$	1.26264	0.136154
$87$	−1.34649	−0.144359
$88$	−5.09387	−0.543008
$89$	−12.9678	−1.37459	−0.687293	−	0.726380i	$-0.741201\pi$
$89$	−12.9678	−1.37459	−0.687293	+	0.726380i	$0.741201\pi$
$90$	−3.92593	−0.413830
$91$	−10.3622	−1.08625
$92$	−5.33903	−0.556633
$93$	1.06964	0.110917
$94$	−6.16447	−0.635817
$95$	8.84830	0.907817
$96$	−0.327102	−0.0333847
$97$	13.2191	1.34219	0.671097	−	0.741369i	$-0.265823\pi$
$97$	13.2191	1.34219	0.671097	+	0.741369i	$0.265823\pi$
$98$	−4.82160	−0.487055
$99$	−14.7366	−1.48108
$100$	−3.15843	−0.315843
$101$	−13.7305	−1.36624	−0.683119	−	0.730307i	$-0.739377\pi$
$101$	−13.7305	−1.36624	−0.683119	+	0.730307i	$0.739377\pi$
$102$	−2.47535	−0.245096
$103$	−4.16607	−0.410495	−0.205247	−	0.978710i	$-0.565800\pi$
$103$	−4.16607	−0.410495	−0.205247	+	0.978710i	$0.565800\pi$
$104$	3.01379	0.295526
$105$	−1.52621	−0.148943
$106$	6.20736	0.602913
$107$	−17.0673	−1.64996	−0.824979	−	0.565163i	$-0.808813\pi$
$107$	−17.0673	−1.64996	−0.824979	+	0.565163i	$0.808813\pi$
$108$	−1.92762	−0.185485
$109$	−0.0428325	−0.00410261	−0.00205131	−	0.999998i	$-0.500653\pi$
$109$	−0.0428325	−0.00410261	−0.00205131	+	0.999998i	$0.500653\pi$
$110$	6.91260	0.659090
$111$	−3.16432	−0.300344
$112$	3.43826	0.324885
$113$	−2.03801	−0.191719	−0.0958597	−	0.995395i	$-0.530560\pi$
$113$	−2.03801	−0.191719	−0.0958597	+	0.995395i	$0.530560\pi$
$114$	2.13280	0.199755
$115$	7.24530	0.675628
$116$	−4.11643	−0.382201
$117$	8.71890	0.806063
$118$	0.844389	0.0777323
$119$	26.0190	2.38516
$120$	0.443892	0.0405216
$121$	14.9475	1.35886
$122$	−11.2078	−1.01471
$123$	−2.89007	−0.260588
$124$	3.27005	0.293659
$125$	11.0713	0.990251
$126$	9.94689	0.886139
$127$	7.35375	0.652540	0.326270	−	0.945277i	$-0.394208\pi$
$127$	7.35375	0.652540	0.326270	+	0.945277i	$0.394208\pi$
$128$	−1.00000	−0.0883883
$129$	−0.413014	−0.0363638
$130$	−4.08984	−0.358703
$131$	−15.4731	−1.35189	−0.675947	−	0.736950i	$-0.736265\pi$
$131$	−15.4731	−1.35189	−0.675947	+	0.736950i	$0.736265\pi$
$132$	1.66622	0.145025
$133$	−22.4184	−1.94392
$134$	3.77217	0.325866
$135$	2.61586	0.225137
$136$	−7.56750	−0.648908
$137$	17.8197	1.52244	0.761220	−	0.648494i	$-0.224601\pi$
$137$	17.8197	1.52244	0.761220	+	0.648494i	$0.224601\pi$
$138$	1.74641	0.148664
$139$	14.5932	1.23778	0.618888	−	0.785479i	$-0.287583\pi$
$139$	14.5932	1.23778	0.618888	+	0.785479i	$0.287583\pi$
$140$	−4.66586	−0.394337
$141$	2.01641	0.169813
$142$	0.624439	0.0524018
$143$	−15.3518	−1.28379
$144$	−2.89300	−0.241084
$145$	5.58618	0.463907
$146$	−7.05016	−0.583476
$147$	1.57716	0.130082
$148$	−9.67379	−0.795180
$149$	9.30157	0.762014	0.381007	−	0.924572i	$-0.375577\pi$
$149$	9.30157	0.762014	0.381007	+	0.924572i	$0.375577\pi$
$150$	1.03313	0.0843548
$151$	−4.40710	−0.358645	−0.179323	−	0.983790i	$-0.557391\pi$
$151$	−4.40710	−0.358645	−0.179323	+	0.983790i	$0.557391\pi$
$152$	6.52028	0.528864
$153$	−21.8928	−1.76993
$154$	−17.5140	−1.41132
$155$	−4.43760	−0.356437
$156$	−0.985817	−0.0789285
$157$	−20.2617	−1.61706	−0.808530	−	0.588455i	$-0.799736\pi$
$157$	−20.2617	−1.61706	−0.808530	+	0.588455i	$0.799736\pi$
$158$	−5.34416	−0.425158
$159$	−2.03044	−0.161025
$160$	1.35704	0.107284
$161$	−18.3570	−1.44673
$162$	−8.04848	−0.632349
$163$	−19.8660	−1.55603	−0.778015	−	0.628246i	$-0.783773\pi$
$163$	−19.8660	−1.55603	−0.778015	+	0.628246i	$0.783773\pi$
$164$	−8.83536	−0.689925
$165$	−2.26113	−0.176029
$166$	7.15126	0.555045
$167$	−20.6337	−1.59668	−0.798341	−	0.602205i	$-0.794289\pi$
$167$	−20.6337	−1.59668	−0.798341	+	0.602205i	$0.794289\pi$
$168$	−1.12466	−0.0867695
$169$	−3.91708	−0.301314
$170$	10.2694	0.787629
$171$	18.8632	1.44250
$172$	−1.26264	−0.0962757
$173$	6.52758	0.496282	0.248141	−	0.968724i	$-0.420180\pi$
$173$	6.52758	0.496282	0.248141	+	0.968724i	$0.420180\pi$
$174$	1.34649	0.102078
$175$	−10.8595	−0.820901
$176$	5.09387	0.383965
$177$	−0.276201	−0.0207606
$178$	12.9678	0.971979
$179$	−5.61493	−0.419679	−0.209840	−	0.977736i	$-0.567294\pi$
$179$	−5.61493	−0.419679	−0.209840	+	0.977736i	$0.567294\pi$
$180$	3.92593	0.292622
$181$	5.81555	0.432267	0.216133	−	0.976364i	$-0.430655\pi$
$181$	5.81555	0.432267	0.216133	+	0.976364i	$0.430655\pi$
$182$	10.3622	0.768095
$183$	3.66609	0.271005
$184$	5.33903	0.393599
$185$	13.1278	0.965171
$186$	−1.06964	−0.0784299
$187$	38.5479	2.81890
$188$	6.16447	0.449590
$189$	−6.62764	−0.482090
$190$	−8.84830	−0.641923
$191$	−1.72592	−0.124883	−0.0624417	−	0.998049i	$-0.519889\pi$
$191$	−1.72592	−0.124883	−0.0624417	+	0.998049i	$0.519889\pi$
$192$	0.327102	0.0236066
$193$	13.2897	0.956610	0.478305	−	0.878194i	$-0.341251\pi$
$193$	13.2897	0.956610	0.478305	+	0.878194i	$0.341251\pi$
$194$	−13.2191	−0.949075
$195$	1.33780	0.0958016
$196$	4.82160	0.344400
$197$	5.96629	0.425081	0.212540	−	0.977152i	$-0.431826\pi$
$197$	5.96629	0.425081	0.212540	+	0.977152i	$0.431826\pi$
$198$	14.7366	1.04728
$199$	−16.8096	−1.19160	−0.595801	−	0.803132i	$-0.703165\pi$
$199$	−16.8096	−1.19160	−0.595801	+	0.803132i	$0.703165\pi$
$200$	3.15843	0.223335
$201$	−1.23389	−0.0870316
$202$	13.7305	0.966076
$203$	−14.1533	−0.993370
$204$	2.47535	0.173309
$205$	11.9900	0.837415
$206$	4.16607	0.290264
$207$	15.4458	1.07356
$208$	−3.01379	−0.208969
$209$	−33.2134	−2.29742
$210$	1.52621	0.105319
$211$	5.82566	0.401055	0.200528	−	0.979688i	$-0.435734\pi$
$211$	5.82566	0.401055	0.200528	+	0.979688i	$0.435734\pi$
$212$	−6.20736	−0.426324
$213$	−0.204256	−0.0139954
$214$	17.0673	1.16670
$215$	1.71346	0.116857
$216$	1.92762	0.131158
$217$	11.2433	0.763243
$218$	0.0428325	0.00290098
$219$	2.30613	0.155834
$220$	−6.91260	−0.466047
$221$	−22.8069	−1.53416
$222$	3.16432	0.212375
$223$	−25.5203	−1.70896	−0.854482	−	0.519480i	$-0.826126\pi$
$223$	−25.5203	−1.70896	−0.854482	+	0.519480i	$0.826126\pi$
$224$	−3.43826	−0.229728
$225$	9.13736	0.609157
$226$	2.03801	0.135566
$227$	−10.0721	−0.668508	−0.334254	−	0.942483i	$-0.608484\pi$
$227$	−10.0721	−0.668508	−0.334254	+	0.942483i	$0.608484\pi$
$228$	−2.13280	−0.141248
$229$	0.238643	0.0157700	0.00788499	−	0.999969i	$-0.497490\pi$
$229$	0.238643	0.0157700	0.00788499	+	0.999969i	$0.497490\pi$
$230$	−7.24530	−0.477741
$231$	5.72888	0.376932
$232$	4.11643	0.270257
$233$	−17.8074	−1.16660	−0.583301	−	0.812256i	$-0.698239\pi$
$233$	−17.8074	−1.16660	−0.583301	+	0.812256i	$0.698239\pi$
$234$	−8.71890	−0.569972
$235$	−8.36545	−0.545702
$236$	−0.844389	−0.0549650
$237$	1.74809	0.113550
$238$	−26.0190	−1.68656
$239$	−15.0051	−0.970598	−0.485299	−	0.874348i	$-0.661289\pi$
$239$	−15.0051	−0.970598	−0.485299	+	0.874348i	$0.661289\pi$
$240$	−0.443892	−0.0286531
$241$	13.3760	0.861626	0.430813	−	0.902441i	$-0.358227\pi$
$241$	13.3760	0.861626	0.430813	+	0.902441i	$0.358227\pi$
$242$	−14.9475	−0.960860
$243$	8.41552	0.539856
$244$	11.2078	0.717505
$245$	−6.54312	−0.418025
$246$	2.89007	0.184264
$247$	19.6507	1.25035
$248$	−3.27005	−0.207648
$249$	−2.33919	−0.148240
$250$	−11.0713	−0.700214
$251$	8.94357	0.564513	0.282256	−	0.959339i	$-0.408917\pi$
$251$	8.94357	0.564513	0.282256	+	0.959339i	$0.408917\pi$
$252$	−9.94689	−0.626595
$253$	−27.1963	−1.70982
$254$	−7.35375	−0.461415
$255$	−3.35916	−0.210358
$256$	1.00000	0.0625000
$257$	26.8059	1.67211	0.836055	−	0.548646i	$-0.184857\pi$
$257$	26.8059	1.67211	0.836055	+	0.548646i	$0.184857\pi$
$258$	0.413014	0.0257131
$259$	−33.2610	−2.06674
$260$	4.08984	0.253641
$261$	11.9089	0.737140
$262$	15.4731	0.955933
$263$	−9.16371	−0.565058	−0.282529	−	0.959259i	$-0.591173\pi$
$263$	−9.16371	−0.565058	−0.282529	+	0.959259i	$0.591173\pi$
$264$	−1.66622	−0.102548
$265$	8.42366	0.517462
$266$	22.4184	1.37456
$267$	−4.24180	−0.259594
$268$	−3.77217	−0.230422
$269$	−9.51626	−0.580217	−0.290108	−	0.956994i	$-0.593691\pi$
$269$	−9.51626	−0.580217	−0.290108	+	0.956994i	$0.593691\pi$
$270$	−2.61586	−0.159196
$271$	14.4233	0.876156	0.438078	−	0.898937i	$-0.355659\pi$
$271$	14.4233	0.876156	0.438078	+	0.898937i	$0.355659\pi$
$272$	7.56750	0.458847
$273$	−3.38949	−0.205141
$274$	−17.8197	−1.07653
$275$	−16.0886	−0.970181
$276$	−1.74641	−0.105122
$277$	−2.57423	−0.154671	−0.0773353	−	0.997005i	$-0.524641\pi$
$277$	−2.57423	−0.154671	−0.0773353	+	0.997005i	$0.524641\pi$
$278$	−14.5932	−0.875240
$279$	−9.46027	−0.566372
$280$	4.66586	0.278839
$281$	−25.1478	−1.50019	−0.750096	−	0.661328i	$-0.769993\pi$
$281$	−25.1478	−1.50019	−0.750096	+	0.661328i	$0.769993\pi$
$282$	−2.01641	−0.120076
$283$	10.7935	0.641604	0.320802	−	0.947146i	$-0.396048\pi$
$283$	10.7935	0.641604	0.320802	+	0.947146i	$0.396048\pi$
$284$	−0.624439	−0.0370537
$285$	2.89430	0.171444
$286$	15.3518	0.907773
$287$	−30.3782	−1.79317
$288$	2.89300	0.170472
$289$	40.2671	2.36865
$290$	−5.58618	−0.328032
$291$	4.32399	0.253477
$292$	7.05016	0.412580
$293$	1.06178	0.0620297	0.0310149	−	0.999519i	$-0.490126\pi$
$293$	1.06178	0.0620297	0.0310149	+	0.999519i	$0.490126\pi$
$294$	−1.57716	−0.0919817
$295$	1.14587	0.0667152
$296$	9.67379	0.562278
$297$	−9.81902	−0.569757
$298$	−9.30157	−0.538825
$299$	16.0907	0.930550
$300$	−1.03313	−0.0596478
$301$	−4.34129	−0.250228
$302$	4.40710	0.253600
$303$	−4.49129	−0.258018
$304$	−6.52028	−0.373964
$305$	−15.2095	−0.870891
$306$	21.8928	1.25153
$307$	−4.83023	−0.275676	−0.137838	−	0.990455i	$-0.544015\pi$
$307$	−4.83023	−0.275676	−0.137838	+	0.990455i	$0.544015\pi$
$308$	17.5140	0.997954
$309$	−1.36273	−0.0775230
$310$	4.43760	0.252039
$311$	−1.89337	−0.107363	−0.0536816	−	0.998558i	$-0.517096\pi$
$311$	−1.89337	−0.107363	−0.0536816	+	0.998558i	$0.517096\pi$
$312$	0.985817	0.0558109
$313$	10.4457	0.590426	0.295213	−	0.955432i	$-0.404609\pi$
$313$	10.4457	0.590426	0.295213	+	0.955432i	$0.404609\pi$
$314$	20.2617	1.14343
$315$	13.4984	0.760546
$316$	5.34416	0.300632
$317$	−14.7321	−0.827437	−0.413719	−	0.910405i	$-0.635770\pi$
$317$	−14.7321	−0.827437	−0.413719	+	0.910405i	$0.635770\pi$
$318$	2.03044	0.113862
$319$	−20.9686	−1.17401
$320$	−1.35704	−0.0758610
$321$	−5.58275	−0.311599
$322$	18.3570	1.02299
$323$	−49.3422	−2.74548
$324$	8.04848	0.447138
$325$	9.51885	0.528011
$326$	19.8660	1.10028
$327$	−0.0140106	−0.000774789 0
$328$	8.83536	0.487851
$329$	21.1950	1.16852
$330$	2.26113	0.124471
$331$	21.0618	1.15766	0.578831	−	0.815447i	$-0.303509\pi$
$331$	21.0618	1.15766	0.578831	+	0.815447i	$0.303509\pi$
$332$	−7.15126	−0.392476
$333$	27.9863	1.53364
$334$	20.6337	1.12903
$335$	5.11900	0.279681
$336$	1.12466	0.0613553
$337$	−18.3546	−0.999841	−0.499920	−	0.866071i	$-0.666637\pi$
$337$	−18.3546	−0.999841	−0.499920	+	0.866071i	$0.666637\pi$
$338$	3.91708	0.213061
$339$	−0.666636	−0.0362067
$340$	−10.2694	−0.556938
$341$	16.6572	0.902038
$342$	−18.8632	−1.02000
$343$	−7.48989	−0.404416
$344$	1.26264	0.0680772
$345$	2.36995	0.127594
$346$	−6.52758	−0.350925
$347$	−1.39981	−0.0751459	−0.0375729	−	0.999294i	$-0.511963\pi$
$347$	−1.39981	−0.0751459	−0.0375729	+	0.999294i	$0.511963\pi$
$348$	−1.34649	−0.0721797
$349$	−29.3737	−1.57234	−0.786169	−	0.618011i	$-0.787939\pi$
$349$	−29.3737	−1.57234	−0.786169	+	0.618011i	$0.787939\pi$
$350$	10.8595	0.580465
$351$	5.80942	0.310084
$352$	−5.09387	−0.271504
$353$	−8.59426	−0.457426	−0.228713	−	0.973494i	$-0.573452\pi$
$353$	−8.59426	−0.457426	−0.228713	+	0.973494i	$0.573452\pi$
$354$	0.276201	0.0146799
$355$	0.847391	0.0449749
$356$	−12.9678	−0.687293
$357$	8.51088	0.450444
$358$	5.61493	0.296758
$359$	−12.4447	−0.656805	−0.328403	−	0.944538i	$-0.606510\pi$
$359$	−12.4447	−0.656805	−0.328403	+	0.944538i	$0.606510\pi$
$360$	−3.92593	−0.206915
$361$	23.5140	1.23758
$362$	−5.81555	−0.305659
$363$	4.88936	0.256625
$364$	−10.3622	−0.543126
$365$	−9.56738	−0.500779
$366$	−3.66609	−0.191630
$367$	−1.40326	−0.0732496	−0.0366248	−	0.999329i	$-0.511661\pi$
$367$	−1.40326	−0.0732496	−0.0366248	+	0.999329i	$0.511661\pi$
$368$	−5.33903	−0.278316
$369$	25.5607	1.33064
$370$	−13.1278	−0.682479
$371$	−21.3425	−1.10805
$372$	1.06964	0.0554583
$373$	−0.681741	−0.0352992	−0.0176496	−	0.999844i	$-0.505618\pi$
$373$	−0.681741	−0.0352992	−0.0176496	+	0.999844i	$0.505618\pi$
$374$	−38.5479	−1.99326
$375$	3.62146	0.187012
$376$	−6.16447	−0.317908
$377$	12.4061	0.638944
$378$	6.62764	0.340889
$379$	4.69752	0.241295	0.120648	−	0.992695i	$-0.461503\pi$
$379$	4.69752	0.241295	0.120648	+	0.992695i	$0.461503\pi$
$380$	8.84830	0.453908
$381$	2.40543	0.123234
$382$	1.72592	0.0883060
$383$	22.8453	1.16734	0.583671	−	0.811990i	$-0.301616\pi$
$383$	22.8453	1.16734	0.583671	+	0.811990i	$0.301616\pi$
$384$	−0.327102	−0.0166924
$385$	−23.7673	−1.21129
$386$	−13.2897	−0.676425
$387$	3.65283	0.185684
$388$	13.2191	0.671097
$389$	−12.4159	−0.629511	−0.314755	−	0.949173i	$-0.601923\pi$
$389$	−12.4159	−0.629511	−0.314755	+	0.949173i	$0.601923\pi$
$390$	−1.33780	−0.0677420
$391$	−40.4031	−2.04327
$392$	−4.82160	−0.243528
$393$	−5.06130	−0.255309
$394$	−5.96629	−0.300577
$395$	−7.25226	−0.364901
$396$	−14.7366	−0.740541
$397$	−3.21342	−0.161277	−0.0806385	−	0.996743i	$-0.525696\pi$
$397$	−3.21342	−0.161277	−0.0806385	+	0.996743i	$0.525696\pi$
$398$	16.8096	0.842590
$399$	−7.33311	−0.367115
$400$	−3.15843	−0.157922
$401$	−15.3642	−0.767252	−0.383626	−	0.923488i	$-0.625325\pi$
$401$	−15.3642	−0.767252	−0.383626	+	0.923488i	$0.625325\pi$
$402$	1.23389	0.0615406
$403$	−9.85524	−0.490924
$404$	−13.7305	−0.683119
$405$	−10.9221	−0.542726
$406$	14.1533	0.702419
$407$	−49.2770	−2.44257
$408$	−2.47535	−0.122548
$409$	26.3954	1.30517	0.652585	−	0.757715i	$-0.273684\pi$
$409$	26.3954	1.30517	0.652585	+	0.757715i	$0.273684\pi$
$410$	−11.9900	−0.592142
$411$	5.82886	0.287517
$412$	−4.16607	−0.205247
$413$	−2.90322	−0.142858
$414$	−15.4458	−0.759122
$415$	9.70457	0.476379
$416$	3.01379	0.147763
$417$	4.77346	0.233757
$418$	33.2134	1.62452
$419$	−6.92104	−0.338115	−0.169057	−	0.985606i	$-0.554072\pi$
$419$	−6.92104	−0.338115	−0.169057	+	0.985606i	$0.554072\pi$
$420$	−1.52621	−0.0744716
$421$	−19.4347	−0.947187	−0.473594	−	0.880743i	$-0.657043\pi$
$421$	−19.4347	−0.947187	−0.473594	+	0.880743i	$0.657043\pi$
$422$	−5.82566	−0.283589
$423$	−17.8338	−0.867111
$424$	6.20736	0.301456
$425$	−23.9015	−1.15939
$426$	0.204256	0.00989622
$427$	38.5352	1.86485
$428$	−17.0673	−0.824979
$429$	−5.02162	−0.242446
$430$	−1.71346	−0.0826305
$431$	−14.6488	−0.705610	−0.352805	−	0.935697i	$-0.614772\pi$
$431$	−14.6488	−0.705610	−0.352805	+	0.935697i	$0.614772\pi$
$432$	−1.92762	−0.0927424
$433$	−5.26059	−0.252808	−0.126404	−	0.991979i	$-0.540344\pi$
$433$	−5.26059	−0.252808	−0.126404	+	0.991979i	$0.540344\pi$
$434$	−11.2433	−0.539694
$435$	1.82725	0.0876100
$436$	−0.0428325	−0.00205131
$437$	34.8120	1.66528
$438$	−2.30613	−0.110191
$439$	19.5468	0.932919	0.466460	−	0.884542i	$-0.345529\pi$
$439$	19.5468	0.932919	0.466460	+	0.884542i	$0.345529\pi$
$440$	6.91260	0.329545
$441$	−13.9489	−0.664234
$442$	22.8069	1.08481
$443$	7.29338	0.346519	0.173260	−	0.984876i	$-0.444570\pi$
$443$	7.29338	0.346519	0.173260	+	0.984876i	$0.444570\pi$
$444$	−3.16432	−0.150172
$445$	17.5979	0.834220
$446$	25.5203	1.20842
$447$	3.04256	0.143908
$448$	3.43826	0.162442
$449$	1.77964	0.0839862	0.0419931	−	0.999118i	$-0.486629\pi$
$449$	1.77964	0.0839862	0.0419931	+	0.999118i	$0.486629\pi$
$450$	−9.13736	−0.430739
$451$	−45.0061	−2.11926
$452$	−2.03801	−0.0958597
$453$	−1.44157	−0.0677311
$454$	10.0721	0.472707
$455$	14.0619	0.659233
$456$	2.13280	0.0998775
$457$	6.88855	0.322233	0.161116	−	0.986935i	$-0.448491\pi$
$457$	6.88855	0.322233	0.161116	+	0.986935i	$0.448491\pi$
$458$	−0.238643	−0.0111511
$459$	−14.5872	−0.680874
$460$	7.24530	0.337814
$461$	39.7757	1.85254	0.926270	−	0.376862i	$-0.122997\pi$
$461$	39.7757	1.85254	0.926270	+	0.376862i	$0.122997\pi$
$462$	−5.72888	−0.266531
$463$	−10.3390	−0.480496	−0.240248	−	0.970712i	$-0.577229\pi$
$463$	−10.3390	−0.480496	−0.240248	+	0.970712i	$0.577229\pi$
$464$	−4.11643	−0.191101
$465$	−1.45155	−0.0673140
$466$	17.8074	0.824913
$467$	−39.5476	−1.83005	−0.915023	−	0.403403i	$-0.867827\pi$
$467$	−39.5476	−1.83005	−0.915023	+	0.403403i	$0.867827\pi$
$468$	8.71890	0.403031
$469$	−12.9697	−0.598884
$470$	8.36545	0.385870
$471$	−6.62765	−0.305386
$472$	0.844389	0.0388661
$473$	−6.43174	−0.295732
$474$	−1.74809	−0.0802923
$475$	20.5939	0.944911
$476$	26.0190	1.19258
$477$	17.9579	0.822237
$478$	15.0051	0.686316
$479$	−30.1310	−1.37672	−0.688360	−	0.725369i	$-0.741669\pi$
$479$	−30.1310	−1.37672	−0.688360	+	0.725369i	$0.741669\pi$
$480$	0.443892	0.0202608
$481$	29.1547	1.32934
$482$	−13.3760	−0.609262
$483$	−6.00460	−0.273219
$484$	14.9475	0.679431
$485$	−17.9389	−0.814562
$486$	−8.41552	−0.381736
$487$	10.8284	0.490683	0.245341	−	0.969437i	$-0.421100\pi$
$487$	10.8284	0.490683	0.245341	+	0.969437i	$0.421100\pi$
$488$	−11.2078	−0.507353
$489$	−6.49823	−0.293860
$490$	6.54312	0.295588
$491$	−12.2359	−0.552197	−0.276099	−	0.961129i	$-0.589042\pi$
$491$	−12.2359	−0.552197	−0.276099	+	0.961129i	$0.589042\pi$
$492$	−2.89007	−0.130294
$493$	−31.1511	−1.40298
$494$	−19.6507	−0.884129
$495$	19.9982	0.898851
$496$	3.27005	0.146830
$497$	−2.14698	−0.0963053
$498$	2.33919	0.104822
$499$	25.1253	1.12476	0.562382	−	0.826878i	$-0.309885\pi$
$499$	25.1253	1.12476	0.562382	+	0.826878i	$0.309885\pi$
$500$	11.0713	0.495126
$501$	−6.74933	−0.301538
$502$	−8.94357	−0.399171
$503$	15.1427	0.675181	0.337590	−	0.941293i	$-0.390388\pi$
$503$	15.1427	0.675181	0.337590	+	0.941293i	$0.390388\pi$
$504$	9.94689	0.443070
$505$	18.6329	0.829154
$506$	27.1963	1.20902
$507$	−1.28129	−0.0569039
$508$	7.35375	0.326270
$509$	−19.8263	−0.878786	−0.439393	−	0.898295i	$-0.644806\pi$
$509$	−19.8263	−0.878786	−0.439393	+	0.898295i	$0.644806\pi$
$510$	3.35916	0.148746
$511$	24.2403	1.07233
$512$	−1.00000	−0.0441942
$513$	12.5686	0.554917
$514$	−26.8059	−1.18236
$515$	5.65353	0.249124
$516$	−0.413014	−0.0181819
$517$	31.4010	1.38101
$518$	33.2610	1.46140
$519$	2.13519	0.0937242
$520$	−4.08984	−0.179351
$521$	−38.8249	−1.70095	−0.850474	−	0.526017i	$-0.823685\pi$
$521$	−38.8249	−1.70095	−0.850474	+	0.526017i	$0.823685\pi$
$522$	−11.9089	−0.521237
$523$	−13.3213	−0.582501	−0.291251	−	0.956647i	$-0.594071\pi$
$523$	−13.3213	−0.582501	−0.291251	+	0.956647i	$0.594071\pi$
$524$	−15.4731	−0.675947
$525$	−3.55217	−0.155029
$526$	9.16371	0.399557
$527$	24.7461	1.07796
$528$	1.66622	0.0725127
$529$	5.50526	0.239359
$530$	−8.42366	−0.365901
$531$	2.44282	0.106009
$532$	−22.4184	−0.971960
$533$	26.6279	1.15338
$534$	4.24180	0.183561
$535$	23.1611	1.00134
$536$	3.77217	0.162933
$537$	−1.83666	−0.0792576
$538$	9.51626	0.410275
$539$	24.5606	1.05790
$540$	2.61586	0.112569
$541$	14.5379	0.625032	0.312516	−	0.949912i	$-0.398828\pi$
$541$	14.5379	0.625032	0.312516	+	0.949912i	$0.398828\pi$
$542$	−14.4233	−0.619536
$543$	1.90228	0.0816347
$544$	−7.56750	−0.324454
$545$	0.0581256	0.00248983
$546$	3.38949	0.145057
$547$	19.7359	0.843847	0.421923	−	0.906632i	$-0.361355\pi$
$547$	19.7359	0.843847	0.421923	+	0.906632i	$0.361355\pi$
$548$	17.8197	0.761220
$549$	−32.4242	−1.38383
$550$	16.0886	0.686022
$551$	26.8403	1.14343
$552$	1.74641	0.0743321
$553$	18.3746	0.781367
$554$	2.57423	0.109369
$555$	4.29412	0.182275
$556$	14.5932	0.618888
$557$	−39.4974	−1.67356	−0.836778	−	0.547542i	$-0.815564\pi$
$557$	−39.4974	−1.67356	−0.836778	+	0.547542i	$0.815564\pi$
$558$	9.46027	0.400485
$559$	3.80534	0.160949
$560$	−4.66586	−0.197169
$561$	12.6091	0.532356
$562$	25.1478	1.06080
$563$	−2.31242	−0.0974567	−0.0487284	−	0.998812i	$-0.515517\pi$
$563$	−2.31242	−0.0974567	−0.0487284	+	0.998812i	$0.515517\pi$
$564$	2.01641	0.0849063
$565$	2.76566	0.116352
$566$	−10.7935	−0.453683
$567$	27.6727	1.16215
$568$	0.624439	0.0262009
$569$	−37.9776	−1.59211	−0.796053	−	0.605227i	$-0.793082\pi$
$569$	−37.9776	−1.59211	−0.796053	+	0.605227i	$0.793082\pi$
$570$	−2.89430	−0.121229
$571$	17.3270	0.725111	0.362555	−	0.931962i	$-0.381904\pi$
$571$	17.3270	0.725111	0.362555	+	0.931962i	$0.381904\pi$
$572$	−15.3518	−0.641893
$573$	−0.564554	−0.0235846
$574$	30.3782	1.26796
$575$	16.8630	0.703235
$576$	−2.89300	−0.120542
$577$	17.7063	0.737122	0.368561	−	0.929604i	$-0.379851\pi$
$577$	17.7063	0.737122	0.368561	+	0.929604i	$0.379851\pi$
$578$	−40.2671	−1.67489
$579$	4.34708	0.180658
$580$	5.58618	0.231953
$581$	−24.5879	−1.02008
$582$	−4.32399	−0.179235
$583$	−31.6195	−1.30955
$584$	−7.05016	−0.291738
$585$	−11.8319	−0.489190
$586$	−1.06178	−0.0438616
$587$	44.8835	1.85254	0.926269	−	0.376863i	$-0.122997\pi$
$587$	44.8835	1.85254	0.926269	+	0.376863i	$0.122997\pi$
$588$	1.57716	0.0650409
$589$	−21.3216	−0.878543
$590$	−1.14587	−0.0471748
$591$	1.95159	0.0802776
$592$	−9.67379	−0.397590
$593$	25.8692	1.06232	0.531160	−	0.847272i	$-0.321756\pi$
$593$	25.8692	1.06232	0.531160	+	0.847272i	$0.321756\pi$
$594$	9.81902	0.402879
$595$	−35.3089	−1.44753
$596$	9.30157	0.381007
$597$	−5.49847	−0.225037
$598$	−16.0907	−0.657998
$599$	−8.77271	−0.358443	−0.179222	−	0.983809i	$-0.557358\pi$
$599$	−8.77271	−0.358443	−0.179222	+	0.983809i	$0.557358\pi$
$600$	1.03313	0.0421774
$601$	−2.23656	−0.0912312	−0.0456156	−	0.998959i	$-0.514525\pi$
$601$	−2.23656	−0.0912312	−0.0456156	+	0.998959i	$0.514525\pi$
$602$	4.34129	0.176938
$603$	10.9129	0.444408
$604$	−4.40710	−0.179323
$605$	−20.2844	−0.824677
$606$	4.49129	0.182446
$607$	−20.0272	−0.812878	−0.406439	−	0.913678i	$-0.633230\pi$
$607$	−20.0272	−0.812878	−0.406439	+	0.913678i	$0.633230\pi$
$608$	6.52028	0.264432
$609$	−4.62959	−0.187601
$610$	15.2095	0.615813
$611$	−18.5784	−0.751602
$612$	−21.8928	−0.884965
$613$	6.47300	0.261442	0.130721	−	0.991419i	$-0.458271\pi$
$613$	6.47300	0.261442	0.130721	+	0.991419i	$0.458271\pi$
$614$	4.83023	0.194932
$615$	3.92194	0.158148
$616$	−17.5140	−0.705660
$617$	21.9584	0.884011	0.442006	−	0.897012i	$-0.354267\pi$
$617$	21.9584	0.884011	0.442006	+	0.897012i	$0.354267\pi$
$618$	1.36273	0.0548170
$619$	−37.1174	−1.49188	−0.745938	−	0.666016i	$-0.767998\pi$
$619$	−37.1174	−1.49188	−0.745938	+	0.666016i	$0.767998\pi$
$620$	−4.43760	−0.178218
$621$	10.2916	0.412988
$622$	1.89337	0.0759173
$623$	−44.5867	−1.78633
$624$	−0.985817	−0.0394643
$625$	0.767865	0.0307146
$626$	−10.4457	−0.417494
$627$	−10.8642	−0.433874
$628$	−20.2617	−0.808530
$629$	−73.2064	−2.91893
$630$	−13.4984	−0.537787
$631$	33.5137	1.33416	0.667079	−	0.744987i	$-0.267544\pi$
$631$	33.5137	1.33416	0.667079	+	0.744987i	$0.267544\pi$
$632$	−5.34416	−0.212579
$633$	1.90559	0.0757403
$634$	14.7321	0.585086
$635$	−9.97936	−0.396019
$636$	−2.03044	−0.0805123
$637$	−14.5313	−0.575751
$638$	20.9686	0.830153
$639$	1.80651	0.0714643
$640$	1.35704	0.0536419
$641$	15.0210	0.593294	0.296647	−	0.954987i	$-0.404131\pi$
$641$	15.0210	0.593294	0.296647	+	0.954987i	$0.404131\pi$
$642$	5.58275	0.220334
$643$	−42.2721	−1.66705	−0.833524	−	0.552484i	$-0.813680\pi$
$643$	−42.2721	−1.66705	−0.833524	+	0.552484i	$0.813680\pi$
$644$	−18.3570	−0.723365
$645$	0.560478	0.0220688
$646$	49.3422	1.94134
$647$	32.2064	1.26616	0.633082	−	0.774085i	$-0.281790\pi$
$647$	32.2064	1.26616	0.633082	+	0.774085i	$0.281790\pi$
$648$	−8.04848	−0.316174
$649$	−4.30120	−0.168837
$650$	−9.51885	−0.373360
$651$	3.67770	0.144140
$652$	−19.8660	−0.778015
$653$	36.6516	1.43429	0.717144	−	0.696925i	$-0.245449\pi$
$653$	36.6516	1.43429	0.717144	+	0.696925i	$0.245449\pi$
$654$	0.0140106	0.000547858 0
$655$	20.9977	0.820449
$656$	−8.83536	−0.344963
$657$	−20.3962	−0.795730
$658$	−21.1950	−0.826268
$659$	45.4157	1.76914	0.884572	−	0.466403i	$-0.154450\pi$
$659$	45.4157	1.76914	0.884572	+	0.466403i	$0.154450\pi$
$660$	−2.26113	−0.0880143
$661$	36.1178	1.40482	0.702410	−	0.711773i	$-0.252107\pi$
$661$	36.1178	1.40482	0.702410	+	0.711773i	$0.252107\pi$
$662$	−21.0618	−0.818591
$663$	−7.46018	−0.289729
$664$	7.15126	0.277523
$665$	30.4227	1.17974
$666$	−27.9863	−1.08445
$667$	21.9778	0.850983
$668$	−20.6337	−0.798341
$669$	−8.34775	−0.322742
$670$	−5.11900	−0.197764
$671$	57.0910	2.20397
$672$	−1.12466	−0.0433848
$673$	−42.9928	−1.65725	−0.828625	−	0.559803i	$-0.810877\pi$
$673$	−42.9928	−1.65725	−0.828625	+	0.559803i	$0.810877\pi$
$674$	18.3546	0.706994
$675$	6.08824	0.234337
$676$	−3.91708	−0.150657
$677$	−15.0534	−0.578549	−0.289275	−	0.957246i	$-0.593414\pi$
$677$	−15.0534	−0.578549	−0.289275	+	0.957246i	$0.593414\pi$
$678$	0.666636	0.0256020
$679$	45.4506	1.74423
$680$	10.2694	0.393815
$681$	−3.29461	−0.126250
$682$	−16.6572	−0.637837
$683$	−11.1599	−0.427023	−0.213511	−	0.976941i	$-0.568490\pi$
$683$	−11.1599	−0.427023	−0.213511	+	0.976941i	$0.568490\pi$
$684$	18.8632	0.721252
$685$	−24.1821	−0.923951
$686$	7.48989	0.285965
$687$	0.0780607	0.00297820
$688$	−1.26264	−0.0481378
$689$	18.7077	0.712706
$690$	−2.36995	−0.0902226
$691$	37.0740	1.41036	0.705181	−	0.709027i	$-0.250866\pi$
$691$	37.0740	1.41036	0.705181	+	0.709027i	$0.250866\pi$
$692$	6.52758	0.248141
$693$	−50.6681	−1.92472
$694$	1.39981	0.0531361
$695$	−19.8036	−0.751192
$696$	1.34649	0.0510388
$697$	−66.8616	−2.53256
$698$	29.3737	1.11181
$699$	−5.82485	−0.220316
$700$	−10.8595	−0.410451
$701$	−34.9896	−1.32154	−0.660770	−	0.750588i	$-0.729770\pi$
$701$	−34.9896	−1.32154	−0.660770	+	0.750588i	$0.729770\pi$
$702$	−5.80942	−0.219263
$703$	63.0758	2.37895
$704$	5.09387	0.191982
$705$	−2.73636	−0.103057
$706$	8.59426	0.323449
$707$	−47.2091	−1.77548
$708$	−0.276201	−0.0103803
$709$	−34.2604	−1.28668	−0.643338	−	0.765582i	$-0.722451\pi$
$709$	−34.2604	−1.28668	−0.643338	+	0.765582i	$0.722451\pi$
$710$	−0.847391	−0.0318020
$711$	−15.4607	−0.579821
$712$	12.9678	0.485989
$713$	−17.4589	−0.653841
$714$	−8.51088	−0.318512
$715$	20.8331	0.779114
$716$	−5.61493	−0.209840
$717$	−4.90820	−0.183300
$718$	12.4447	0.464432
$719$	40.4326	1.50788	0.753941	−	0.656942i	$-0.228150\pi$
$719$	40.4326	1.50788	0.753941	+	0.656942i	$0.228150\pi$
$720$	3.92593	0.146311
$721$	−14.3240	−0.533454
$722$	−23.5140	−0.875102
$723$	4.37533	0.162720
$724$	5.81555	0.216133
$725$	13.0015	0.482863
$726$	−4.88936	−0.181461
$727$	1.39811	0.0518529	0.0259265	−	0.999664i	$-0.491746\pi$
$727$	1.39811	0.0518529	0.0259265	+	0.999664i	$0.491746\pi$
$728$	10.3622	0.384048
$729$	−21.3927	−0.792323
$730$	9.56738	0.354105
$731$	−9.55506	−0.353407
$732$	3.66609	0.135503
$733$	−46.7265	−1.72588	−0.862940	−	0.505306i	$-0.831380\pi$
$733$	−46.7265	−1.72588	−0.862940	+	0.505306i	$0.831380\pi$
$734$	1.40326	0.0517953
$735$	−2.14027	−0.0789451
$736$	5.33903	0.196799
$737$	−19.2149	−0.707791
$738$	−25.5607	−0.940903
$739$	−21.8405	−0.803416	−0.401708	−	0.915768i	$-0.631583\pi$
$739$	−21.8405	−0.803416	−0.401708	+	0.915768i	$0.631583\pi$
$740$	13.1278	0.482586
$741$	6.42780	0.236131
$742$	21.3425	0.783508
$743$	43.4503	1.59404	0.797019	−	0.603954i	$-0.206409\pi$
$743$	43.4503	1.59404	0.797019	+	0.603954i	$0.206409\pi$
$744$	−1.06964	−0.0392150
$745$	−12.6226	−0.462457
$746$	0.681741	0.0249603
$747$	20.6886	0.756957
$748$	38.5479	1.40945
$749$	−58.6817	−2.14418
$750$	−3.62146	−0.132237
$751$	−12.1447	−0.443168	−0.221584	−	0.975141i	$-0.571123\pi$
$751$	−12.1447	−0.443168	−0.221584	+	0.975141i	$0.571123\pi$
$752$	6.16447	0.224795
$753$	2.92546	0.106610
$754$	−12.4061	−0.451802
$755$	5.98063	0.217658
$756$	−6.62764	−0.241045
$757$	2.16307	0.0786182	0.0393091	−	0.999227i	$-0.487484\pi$
$757$	2.16307	0.0786182	0.0393091	+	0.999227i	$0.487484\pi$
$758$	−4.69752	−0.170622
$759$	−8.89598	−0.322904
$760$	−8.84830	−0.320962
$761$	35.0845	1.27181	0.635906	−	0.771766i	$-0.280627\pi$
$761$	35.0845	1.27181	0.635906	+	0.771766i	$0.280627\pi$
$762$	−2.40543	−0.0871395
$763$	−0.147269	−0.00533150
$764$	−1.72592	−0.0624417
$765$	29.7095	1.07415
$766$	−22.8453	−0.825435
$767$	2.54481	0.0918877
$768$	0.327102	0.0118033
$769$	−7.82990	−0.282353	−0.141177	−	0.989984i	$-0.545089\pi$
$769$	−7.82990	−0.282353	−0.141177	+	0.989984i	$0.545089\pi$
$770$	23.7673	0.856513
$771$	8.76829	0.315782
$772$	13.2897	0.478305
$773$	5.32656	0.191583	0.0957916	−	0.995401i	$-0.469462\pi$
$773$	5.32656	0.191583	0.0957916	+	0.995401i	$0.469462\pi$
$774$	−3.65283	−0.131298
$775$	−10.3282	−0.371001
$776$	−13.2191	−0.474537
$777$	−10.8797	−0.390308
$778$	12.4159	0.445131
$779$	57.6090	2.06406
$780$	1.33780	0.0479008
$781$	−3.18081	−0.113818
$782$	40.4031	1.44481
$783$	7.93490	0.283570
$784$	4.82160	0.172200
$785$	27.4960	0.981375
$786$	5.06130	0.180531
$787$	−8.13047	−0.289820	−0.144910	−	0.989445i	$-0.546289\pi$
$787$	−8.13047	−0.289820	−0.144910	+	0.989445i	$0.546289\pi$
$788$	5.96629	0.212540
$789$	−2.99747	−0.106713
$790$	7.25226	0.258024
$791$	−7.00718	−0.249147
$792$	14.7366	0.523641
$793$	−33.7779	−1.19949
$794$	3.21342	0.114040
$795$	2.75540	0.0977240
$796$	−16.8096	−0.595801
$797$	36.6430	1.29796	0.648981	−	0.760804i	$-0.275195\pi$
$797$	36.6430	1.29796	0.648981	+	0.760804i	$0.275195\pi$
$798$	7.33311	0.259589
$799$	46.6497	1.65035
$800$	3.15843	0.111667
$801$	37.5159	1.32556
$802$	15.3642	0.542529
$803$	35.9126	1.26733
$804$	−1.23389	−0.0435158
$805$	24.9112	0.878004
$806$	9.85524	0.347136
$807$	−3.11279	−0.109575
$808$	13.7305	0.483038
$809$	−46.0067	−1.61751	−0.808755	−	0.588145i	$-0.799858\pi$
$809$	−46.0067	−1.61751	−0.808755	+	0.588145i	$0.799858\pi$
$810$	10.9221	0.383765
$811$	−28.9640	−1.01706	−0.508532	−	0.861043i	$-0.669812\pi$
$811$	−28.9640	−1.01706	−0.508532	+	0.861043i	$0.669812\pi$
$812$	−14.1533	−0.496685
$813$	4.71791	0.165464
$814$	49.2770	1.72716
$815$	26.9591	0.944336
$816$	2.47535	0.0866545
$817$	8.23279	0.288029
$818$	−26.3954	−0.922895
$819$	29.9778	1.04751
$820$	11.9900	0.418708
$821$	−7.84762	−0.273884	−0.136942	−	0.990579i	$-0.543727\pi$
$821$	−7.84762	−0.273884	−0.136942	+	0.990579i	$0.543727\pi$
$822$	−5.82886	−0.203305
$823$	3.51670	0.122584	0.0612922	−	0.998120i	$-0.480478\pi$
$823$	3.51670	0.122584	0.0612922	+	0.998120i	$0.480478\pi$
$824$	4.16607	0.145132
$825$	−5.26263	−0.183221
$826$	2.90322	0.101016
$827$	−36.0081	−1.25212	−0.626062	−	0.779773i	$-0.715334\pi$
$827$	−36.0081	−1.25212	−0.626062	+	0.779773i	$0.715334\pi$
$828$	15.4458	0.536780
$829$	51.7804	1.79841	0.899204	−	0.437529i	$-0.144146\pi$
$829$	51.7804	1.79841	0.899204	+	0.437529i	$0.144146\pi$
$830$	−9.70457	−0.336851
$831$	−0.842037	−0.0292099
$832$	−3.01379	−0.104484
$833$	36.4875	1.26422
$834$	−4.77346	−0.165291
$835$	28.0008	0.969008
$836$	−33.2134	−1.14871
$837$	−6.30340	−0.217877
$838$	6.92104	0.239083
$839$	−9.40874	−0.324826	−0.162413	−	0.986723i	$-0.551928\pi$
$839$	−9.40874	−0.324826	−0.162413	+	0.986723i	$0.551928\pi$
$840$	1.52621	0.0526594
$841$	−12.0550	−0.415689
$842$	19.4347	0.669763
$843$	−8.22591	−0.283315
$844$	5.82566	0.200528
$845$	5.31565	0.182864
$846$	17.8338	0.613140
$847$	51.3933	1.76589
$848$	−6.20736	−0.213162
$849$	3.53056	0.121169
$850$	23.9015	0.819813
$851$	51.6487	1.77049
$852$	−0.204256	−0.00699768
$853$	4.01101	0.137335	0.0686673	−	0.997640i	$-0.478125\pi$
$853$	4.01101	0.137335	0.0686673	+	0.997640i	$0.478125\pi$
$854$	−38.5352	−1.31865
$855$	−25.5982	−0.875439
$856$	17.0673	0.583348
$857$	9.71744	0.331942	0.165971	−	0.986131i	$-0.446924\pi$
$857$	9.71744	0.331942	0.165971	+	0.986131i	$0.446924\pi$
$858$	5.02162	0.171435
$859$	42.2851	1.44275	0.721374	−	0.692545i	$-0.243511\pi$
$859$	42.2851	1.44275	0.721374	+	0.692545i	$0.243511\pi$
$860$	1.71346	0.0584286
$861$	−9.93678	−0.338645
$862$	14.6488	0.498941
$863$	27.5975	0.939430	0.469715	−	0.882818i	$-0.344357\pi$
$863$	27.5975	0.939430	0.469715	+	0.882818i	$0.344357\pi$
$864$	1.92762	0.0655788
$865$	−8.85820	−0.301188
$866$	5.26059	0.178762
$867$	13.1715	0.447327
$868$	11.2433	0.381621
$869$	27.2224	0.923458
$870$	−1.82725	−0.0619497
$871$	11.3685	0.385208
$872$	0.0428325	0.00145049
$873$	−38.2428	−1.29432
$874$	−34.8120	−1.17753
$875$	38.0661	1.28687
$876$	2.30613	0.0779168
$877$	−6.38198	−0.215504	−0.107752	−	0.994178i	$-0.534365\pi$
$877$	−6.38198	−0.215504	−0.107752	+	0.994178i	$0.534365\pi$
$878$	−19.5468	−0.659674
$879$	0.347310	0.0117145
$880$	−6.91260	−0.233024
$881$	−48.2077	−1.62416	−0.812079	−	0.583548i	$-0.801664\pi$
$881$	−48.2077	−1.62416	−0.812079	+	0.583548i	$0.801664\pi$
$882$	13.9489	0.469684
$883$	40.0318	1.34718	0.673589	−	0.739106i	$-0.264752\pi$
$883$	40.0318	1.34718	0.673589	+	0.739106i	$0.264752\pi$
$884$	−22.8069	−0.767078
$885$	0.374817	0.0125993
$886$	−7.29338	−0.245026
$887$	−5.09736	−0.171152	−0.0855762	−	0.996332i	$-0.527273\pi$
$887$	−5.09736	−0.171152	−0.0855762	+	0.996332i	$0.527273\pi$
$888$	3.16432	0.106188
$889$	25.2841	0.848001
$890$	−17.5979	−0.589883
$891$	40.9979	1.37348
$892$	−25.5203	−0.854482
$893$	−40.1941	−1.34504
$894$	−3.04256	−0.101759
$895$	7.61970	0.254698
$896$	−3.43826	−0.114864
$897$	5.26331	0.175737
$898$	−1.77964	−0.0593872
$899$	−13.4609	−0.448948
$900$	9.13736	0.304579
$901$	−46.9743	−1.56494
$902$	45.0061	1.49854
$903$	−1.42005	−0.0472562
$904$	2.03801	0.0677831
$905$	−7.89196	−0.262338
$906$	1.44157	0.0478931
$907$	−3.44384	−0.114351	−0.0571753	−	0.998364i	$-0.518209\pi$
$907$	−3.44384	−0.114351	−0.0571753	+	0.998364i	$0.518209\pi$
$908$	−10.0721	−0.334254
$909$	39.7225	1.31751
$910$	−14.0619	−0.466148
$911$	−12.3141	−0.407983	−0.203992	−	0.978973i	$-0.565392\pi$
$911$	−12.3141	−0.407983	−0.203992	+	0.978973i	$0.565392\pi$
$912$	−2.13280	−0.0706240
$913$	−36.4276	−1.20558
$914$	−6.88855	−0.227853
$915$	−4.97505	−0.164470
$916$	0.238643	0.00788499
$917$	−53.2006	−1.75684
$918$	14.5872	0.481451
$919$	−25.6432	−0.845893	−0.422946	−	0.906155i	$-0.639004\pi$
$919$	−25.6432	−0.845893	−0.422946	+	0.906155i	$0.639004\pi$
$920$	−7.24530	−0.238870
$921$	−1.57998	−0.0520621
$922$	−39.7757	−1.30994
$923$	1.88193	0.0619444
$924$	5.72888	0.188466
$925$	30.5540	1.00461
$926$	10.3390	0.339762
$927$	12.0524	0.395854
$928$	4.11643	0.135129
$929$	5.87944	0.192898	0.0964492	−	0.995338i	$-0.469251\pi$
$929$	5.87944	0.192898	0.0964492	+	0.995338i	$0.469251\pi$
$930$	1.45155	0.0475982
$931$	−31.4382	−1.03035
$932$	−17.8074	−0.583301
$933$	−0.619326	−0.0202758
$934$	39.5476	1.29404
$935$	−52.3111	−1.71076
$936$	−8.71890	−0.284986
$937$	58.4932	1.91089	0.955445	−	0.295170i	$-0.0953762\pi$
$937$	58.4932	1.91089	0.955445	+	0.295170i	$0.0953762\pi$
$938$	12.9697	0.423475
$939$	3.41681	0.111503
$940$	−8.36545	−0.272851
$941$	−29.6091	−0.965230	−0.482615	−	0.875833i	$-0.660313\pi$
$941$	−29.6091	−0.965230	−0.482615	+	0.875833i	$0.660313\pi$
$942$	6.62765	0.215941
$943$	47.1722	1.53614
$944$	−0.844389	−0.0274825
$945$	8.99399	0.292574
$946$	6.43174	0.209114
$947$	57.8622	1.88027	0.940135	−	0.340802i	$-0.110699\pi$
$947$	57.8622	1.88027	0.940135	+	0.340802i	$0.110699\pi$
$948$	1.74809	0.0567752
$949$	−21.2477	−0.689730
$950$	−20.5939	−0.668153
$951$	−4.81890	−0.156264
$952$	−26.0190	−0.843281
$953$	−6.38208	−0.206736	−0.103368	−	0.994643i	$-0.532962\pi$
$953$	−6.38208	−0.206736	−0.103368	+	0.994643i	$0.532962\pi$
$954$	−17.9579	−0.581409
$955$	2.34215	0.0757903
$956$	−15.0051	−0.485299
$957$	−6.85887	−0.221716
$958$	30.1310	0.973488
$959$	61.2687	1.97847
$960$	−0.443892	−0.0143266
$961$	−20.3068	−0.655057
$962$	−29.1547	−0.939987
$963$	49.3758	1.59111
$964$	13.3760	0.430813
$965$	−18.0346	−0.580555
$966$	6.00460	0.193195
$967$	16.6674	0.535987	0.267993	−	0.963421i	$-0.413639\pi$
$967$	16.6674	0.535987	0.267993	+	0.963421i	$0.413639\pi$
$968$	−14.9475	−0.480430
$969$	−16.1400	−0.518490
$970$	17.9389	0.575982
$971$	45.4640	1.45901	0.729505	−	0.683976i	$-0.239751\pi$
$971$	45.4640	1.45901	0.729505	+	0.683976i	$0.239751\pi$
$972$	8.41552	0.269928
$973$	50.1751	1.60854
$974$	−10.8284	−0.346965
$975$	3.11364	0.0997162
$976$	11.2078	0.358753
$977$	−2.91885	−0.0933823	−0.0466911	−	0.998909i	$-0.514868\pi$
$977$	−2.91885	−0.0933823	−0.0466911	+	0.998909i	$0.514868\pi$
$978$	6.49823	0.207791
$979$	−66.0563	−2.11117
$980$	−6.54312	−0.209012
$981$	0.123915	0.00395629
$982$	12.2359	0.390462
$983$	36.6515	1.16900	0.584501	−	0.811393i	$-0.301290\pi$
$983$	36.6515	1.16900	0.584501	+	0.811393i	$0.301290\pi$
$984$	2.89007	0.0921319
$985$	−8.09652	−0.257977
$986$	31.1511	0.992054
$987$	6.93294	0.220678
$988$	19.6507	0.625173
$989$	6.74129	0.214361
$990$	−19.9982	−0.635584
$991$	−14.4404	−0.458716	−0.229358	−	0.973342i	$-0.573663\pi$
$991$	−14.4404	−0.458716	−0.229358	+	0.973342i	$0.573663\pi$
$992$	−3.27005	−0.103824
$993$	6.88937	0.218628
$994$	2.14698	0.0680981
$995$	22.8114	0.723170
$996$	−2.33919	−0.0741202
$997$	37.6591	1.19268	0.596338	−	0.802733i	$-0.296622\pi$
$997$	37.6591	1.19268	0.596338	+	0.802733i	$0.296622\pi$
$998$	−25.1253	−0.795328
$999$	18.6473	0.589976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.g.1.23	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.g.1.23	✓	40	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:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.327102 q^{3} +1.00000 q^{4} -1.35704 q^{5} -0.327102 q^{6} +3.43826 q^{7} -1.00000 q^{8} -2.89300 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.327102 q^{3} +1.00000 q^{4} -1.35704 q^{5} -0.327102 q^{6} +3.43826 q^{7} -1.00000 q^{8} -2.89300 q^{9} +1.35704 q^{10} +5.09387 q^{11} +0.327102 q^{12} -3.01379 q^{13} -3.43826 q^{14} -0.443892 q^{15} +1.00000 q^{16} +7.56750 q^{17} +2.89300 q^{18} -6.52028 q^{19} -1.35704 q^{20} +1.12466 q^{21} -5.09387 q^{22} -5.33903 q^{23} -0.327102 q^{24} -3.15843 q^{25} +3.01379 q^{26} -1.92762 q^{27} +3.43826 q^{28} -4.11643 q^{29} +0.443892 q^{30} +3.27005 q^{31} -1.00000 q^{32} +1.66622 q^{33} -7.56750 q^{34} -4.66586 q^{35} -2.89300 q^{36} -9.67379 q^{37} +6.52028 q^{38} -0.985817 q^{39} +1.35704 q^{40} -8.83536 q^{41} -1.12466 q^{42} -1.26264 q^{43} +5.09387 q^{44} +3.92593 q^{45} +5.33903 q^{46} +6.16447 q^{47} +0.327102 q^{48} +4.82160 q^{49} +3.15843 q^{50} +2.47535 q^{51} -3.01379 q^{52} -6.20736 q^{53} +1.92762 q^{54} -6.91260 q^{55} -3.43826 q^{56} -2.13280 q^{57} +4.11643 q^{58} -0.844389 q^{59} -0.443892 q^{60} +11.2078 q^{61} -3.27005 q^{62} -9.94689 q^{63} +1.00000 q^{64} +4.08984 q^{65} -1.66622 q^{66} -3.77217 q^{67} +7.56750 q^{68} -1.74641 q^{69} +4.66586 q^{70} -0.624439 q^{71} +2.89300 q^{72} +7.05016 q^{73} +9.67379 q^{74} -1.03313 q^{75} -6.52028 q^{76} +17.5140 q^{77} +0.985817 q^{78} +5.34416 q^{79} -1.35704 q^{80} +8.04848 q^{81} +8.83536 q^{82} -7.15126 q^{83} +1.12466 q^{84} -10.2694 q^{85} +1.26264 q^{86} -1.34649 q^{87} -5.09387 q^{88} -12.9678 q^{89} -3.92593 q^{90} -10.3622 q^{91} -5.33903 q^{92} +1.06964 q^{93} -6.16447 q^{94} +8.84830 q^{95} -0.327102 q^{96} +13.2191 q^{97} -4.82160 q^{98} -14.7366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9}+O(q^{10}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9} + 23 q^{10} - 28 q^{11} - q^{12} - 12 q^{13} - 12 q^{14} - 14 q^{15} + 40 q^{16} - 10 q^{17} - 29 q^{18} + 3 q^{19} - 23 q^{20} - 40 q^{21} + 28 q^{22} - 10 q^{23} + q^{24} + 43 q^{25} + 12 q^{26} - 7 q^{27} + 12 q^{28} - 37 q^{29} + 14 q^{30} - 16 q^{31} - 40 q^{32} - 11 q^{33} + 10 q^{34} - 24 q^{35} + 29 q^{36} - 17 q^{37} - 3 q^{38} - 10 q^{39} + 23 q^{40} - 58 q^{41} + 40 q^{42} + 37 q^{43} - 28 q^{44} - 66 q^{45} + 10 q^{46} - 34 q^{47} - q^{48} + 28 q^{49} - 43 q^{50} - 7 q^{51} - 12 q^{52} - 43 q^{53} + 7 q^{54} + 28 q^{55} - 12 q^{56} - 25 q^{57} + 37 q^{58} - 92 q^{59} - 14 q^{60} - 37 q^{61} + 16 q^{62} + 14 q^{63} + 40 q^{64} - 54 q^{65} + 11 q^{66} - 10 q^{67} - 10 q^{68} - 49 q^{69} + 24 q^{70} - 87 q^{71} - 29 q^{72} + 12 q^{73} + 17 q^{74} - 31 q^{75} + 3 q^{76} - 53 q^{77} + 10 q^{78} + 14 q^{79} - 23 q^{80} - 4 q^{81} + 58 q^{82} - 42 q^{83} - 40 q^{84} - 24 q^{85} - 37 q^{86} + 24 q^{87} + 28 q^{88} - 118 q^{89} + 66 q^{90} - 35 q^{91} - 10 q^{92} - 22 q^{93} + 34 q^{94} - 18 q^{95} + q^{96} + 2 q^{97} - 28 q^{98} - 48 q^{99}+O(q^{100}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9 + 23 * q^10 - 28 * q^11 - q^12 - 12 * q^13 - 12 * q^14 - 14 * q^15 + 40 * q^16 - 10 * q^17 - 29 * q^18 + 3 * q^19 - 23 * q^20 - 40 * q^21 + 28 * q^22 - 10 * q^23 + q^24 + 43 * q^25 + 12 * q^26 - 7 * q^27 + 12 * q^28 - 37 * q^29 + 14 * q^30 - 16 * q^31 - 40 * q^32 - 11 * q^33 + 10 * q^34 - 24 * q^35 + 29 * q^36 - 17 * q^37 - 3 * q^38 - 10 * q^39 + 23 * q^40 - 58 * q^41 + 40 * q^42 + 37 * q^43 - 28 * q^44 - 66 * q^45 + 10 * q^46 - 34 * q^47 - q^48 + 28 * q^49 - 43 * q^50 - 7 * q^51 - 12 * q^52 - 43 * q^53 + 7 * q^54 + 28 * q^55 - 12 * q^56 - 25 * q^57 + 37 * q^58 - 92 * q^59 - 14 * q^60 - 37 * q^61 + 16 * q^62 + 14 * q^63 + 40 * q^64 - 54 * q^65 + 11 * q^66 - 10 * q^67 - 10 * q^68 - 49 * q^69 + 24 * q^70 - 87 * q^71 - 29 * q^72 + 12 * q^73 + 17 * q^74 - 31 * q^75 + 3 * q^76 - 53 * q^77 + 10 * q^78 + 14 * q^79 - 23 * q^80 - 4 * q^81 + 58 * q^82 - 42 * q^83 - 40 * q^84 - 24 * q^85 - 37 * q^86 + 24 * q^87 + 28 * q^88 - 118 * q^89 + 66 * q^90 - 35 * q^91 - 10 * q^92 - 22 * q^93 + 34 * q^94 - 18 * q^95 + q^96 + 2 * q^97 - 28 * q^98 - 48 * q^99

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