LMFDB - Embedded newform 4006.2.a.h.1.3

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:	$0$
Dimension:	$42$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.06826 q^{3} +1.00000 q^{4} -0.106963 q^{5} +3.06826 q^{6} +0.667572 q^{7} -1.00000 q^{8} +6.41420 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.06826 q^{3} +1.00000 q^{4} -0.106963 q^{5} +3.06826 q^{6} +0.667572 q^{7} -1.00000 q^{8} +6.41420 q^{9} +0.106963 q^{10} -3.81825 q^{11} -3.06826 q^{12} -2.35476 q^{13} -0.667572 q^{14} +0.328189 q^{15} +1.00000 q^{16} -7.95463 q^{17} -6.41420 q^{18} -0.927118 q^{19} -0.106963 q^{20} -2.04828 q^{21} +3.81825 q^{22} +2.46501 q^{23} +3.06826 q^{24} -4.98856 q^{25} +2.35476 q^{26} -10.4756 q^{27} +0.667572 q^{28} +2.73627 q^{29} -0.328189 q^{30} +0.338431 q^{31} -1.00000 q^{32} +11.7154 q^{33} +7.95463 q^{34} -0.0714054 q^{35} +6.41420 q^{36} -3.73972 q^{37} +0.927118 q^{38} +7.22500 q^{39} +0.106963 q^{40} +4.79754 q^{41} +2.04828 q^{42} -6.21046 q^{43} -3.81825 q^{44} -0.686081 q^{45} -2.46501 q^{46} -6.54563 q^{47} -3.06826 q^{48} -6.55435 q^{49} +4.98856 q^{50} +24.4069 q^{51} -2.35476 q^{52} +13.6546 q^{53} +10.4756 q^{54} +0.408411 q^{55} -0.667572 q^{56} +2.84464 q^{57} -2.73627 q^{58} -9.98584 q^{59} +0.328189 q^{60} +11.3728 q^{61} -0.338431 q^{62} +4.28194 q^{63} +1.00000 q^{64} +0.251871 q^{65} -11.7154 q^{66} -6.58187 q^{67} -7.95463 q^{68} -7.56328 q^{69} +0.0714054 q^{70} +3.82923 q^{71} -6.41420 q^{72} -12.5467 q^{73} +3.73972 q^{74} +15.3062 q^{75} -0.927118 q^{76} -2.54896 q^{77} -7.22500 q^{78} -17.3203 q^{79} -0.106963 q^{80} +12.8994 q^{81} -4.79754 q^{82} +14.6930 q^{83} -2.04828 q^{84} +0.850850 q^{85} +6.21046 q^{86} -8.39558 q^{87} +3.81825 q^{88} +8.59589 q^{89} +0.686081 q^{90} -1.57197 q^{91} +2.46501 q^{92} -1.03839 q^{93} +6.54563 q^{94} +0.0991671 q^{95} +3.06826 q^{96} -0.720909 q^{97} +6.55435 q^{98} -24.4910 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9}+O(q^{10}) $ 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9} - 27 q^{10} + 23 q^{11} + 15 q^{13} + 10 q^{14} + 6 q^{15} + 42 q^{16} + 14 q^{17} - 56 q^{18} - 4 q^{19} + 27 q^{20} + 26 q^{21} - 23 q^{22} + 12 q^{23} + 45 q^{25} - 15 q^{26} - 3 q^{27} - 10 q^{28} + 41 q^{29} - 6 q^{30} + 18 q^{31} - 42 q^{32} + 25 q^{33} - 14 q^{34} + 8 q^{35} + 56 q^{36} + 33 q^{37} + 4 q^{38} + 10 q^{39} - 27 q^{40} + 84 q^{41} - 26 q^{42} - 36 q^{43} + 23 q^{44} + 66 q^{45} - 12 q^{46} + 28 q^{47} + 58 q^{49} - 45 q^{50} + 17 q^{51} + 15 q^{52} + 68 q^{53} + 3 q^{54} - 28 q^{55} + 10 q^{56} - 9 q^{57} - 41 q^{58} + 59 q^{59} + 6 q^{60} + 41 q^{61} - 18 q^{62} - 28 q^{63} + 42 q^{64} + 44 q^{65} - 25 q^{66} + 14 q^{68} + 67 q^{69} - 8 q^{70} + 69 q^{71} - 56 q^{72} - 27 q^{73} - 33 q^{74} + 14 q^{75} - 4 q^{76} + 43 q^{77} - 10 q^{78} - 19 q^{79} + 27 q^{80} + 74 q^{81} - 84 q^{82} + 20 q^{83} + 26 q^{84} + 16 q^{85} + 36 q^{86} - 28 q^{87} - 23 q^{88} + 123 q^{89} - 66 q^{90} - 9 q^{91} + 12 q^{92} + 48 q^{93} - 28 q^{94} + 28 q^{95} + 10 q^{97} - 58 q^{98} + 75 q^{99}+O(q^{100}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9 - 27 * q^10 + 23 * q^11 + 15 * q^13 + 10 * q^14 + 6 * q^15 + 42 * q^16 + 14 * q^17 - 56 * q^18 - 4 * q^19 + 27 * q^20 + 26 * q^21 - 23 * q^22 + 12 * q^23 + 45 * q^25 - 15 * q^26 - 3 * q^27 - 10 * q^28 + 41 * q^29 - 6 * q^30 + 18 * q^31 - 42 * q^32 + 25 * q^33 - 14 * q^34 + 8 * q^35 + 56 * q^36 + 33 * q^37 + 4 * q^38 + 10 * q^39 - 27 * q^40 + 84 * q^41 - 26 * q^42 - 36 * q^43 + 23 * q^44 + 66 * q^45 - 12 * q^46 + 28 * q^47 + 58 * q^49 - 45 * q^50 + 17 * q^51 + 15 * q^52 + 68 * q^53 + 3 * q^54 - 28 * q^55 + 10 * q^56 - 9 * q^57 - 41 * q^58 + 59 * q^59 + 6 * q^60 + 41 * q^61 - 18 * q^62 - 28 * q^63 + 42 * q^64 + 44 * q^65 - 25 * q^66 + 14 * q^68 + 67 * q^69 - 8 * q^70 + 69 * q^71 - 56 * q^72 - 27 * q^73 - 33 * q^74 + 14 * q^75 - 4 * q^76 + 43 * q^77 - 10 * q^78 - 19 * q^79 + 27 * q^80 + 74 * q^81 - 84 * q^82 + 20 * q^83 + 26 * q^84 + 16 * q^85 + 36 * q^86 - 28 * q^87 - 23 * q^88 + 123 * q^89 - 66 * q^90 - 9 * q^91 + 12 * q^92 + 48 * q^93 - 28 * q^94 + 28 * q^95 + 10 * q^97 - 58 * q^98 + 75 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.06826	−1.77146	−0.885729	−	0.464202i	$-0.846341\pi$
$3$	−3.06826	−1.77146	−0.885729	+	0.464202i	$0.846341\pi$
$4$	1.00000	0.500000
$5$	−0.106963	−0.0478352	−0.0239176	−	0.999714i	$-0.507614\pi$
$5$	−0.106963	−0.0478352	−0.0239176	+	0.999714i	$0.507614\pi$
$6$	3.06826	1.25261
$7$	0.667572	0.252319	0.126159	−	0.992010i	$-0.459735\pi$
$7$	0.667572	0.252319	0.126159	+	0.992010i	$0.459735\pi$
$8$	−1.00000	−0.353553
$9$	6.41420	2.13807
$10$	0.106963	0.0338246
$11$	−3.81825	−1.15125	−0.575623	−	0.817715i	$-0.695240\pi$
$11$	−3.81825	−1.15125	−0.575623	+	0.817715i	$0.695240\pi$
$12$	−3.06826	−0.885729
$13$	−2.35476	−0.653092	−0.326546	−	0.945181i	$-0.605885\pi$
$13$	−2.35476	−0.653092	−0.326546	+	0.945181i	$0.605885\pi$
$14$	−0.667572	−0.178416
$15$	0.328189	0.0847381
$16$	1.00000	0.250000
$17$	−7.95463	−1.92928	−0.964641	−	0.263567i	$-0.915101\pi$
$17$	−7.95463	−1.92928	−0.964641	+	0.263567i	$0.915101\pi$
$18$	−6.41420	−1.51184
$19$	−0.927118	−0.212696	−0.106348	−	0.994329i	$-0.533916\pi$
$19$	−0.927118	−0.212696	−0.106348	+	0.994329i	$0.533916\pi$
$20$	−0.106963	−0.0239176
$21$	−2.04828	−0.446972
$22$	3.81825	0.814053
$23$	2.46501	0.513990	0.256995	−	0.966413i	$-0.417268\pi$
$23$	2.46501	0.513990	0.256995	+	0.966413i	$0.417268\pi$
$24$	3.06826	0.626305
$25$	−4.98856	−0.997712
$26$	2.35476	0.461806
$27$	−10.4756	−2.01604
$28$	0.667572	0.126159
$29$	2.73627	0.508112	0.254056	−	0.967189i	$-0.418235\pi$
$29$	2.73627	0.508112	0.254056	+	0.967189i	$0.418235\pi$
$30$	−0.328189	−0.0599189
$31$	0.338431	0.0607839	0.0303920	−	0.999538i	$-0.490324\pi$
$31$	0.338431	0.0607839	0.0303920	+	0.999538i	$0.490324\pi$
$32$	−1.00000	−0.176777
$33$	11.7154	2.03938
$34$	7.95463	1.36421
$35$	−0.0714054	−0.0120697
$36$	6.41420	1.06903
$37$	−3.73972	−0.614806	−0.307403	−	0.951579i	$-0.599460\pi$
$37$	−3.73972	−0.614806	−0.307403	+	0.951579i	$0.599460\pi$
$38$	0.927118	0.150398
$39$	7.22500	1.15693
$40$	0.106963	0.0169123
$41$	4.79754	0.749251	0.374625	−	0.927176i	$-0.377771\pi$
$41$	4.79754	0.749251	0.374625	+	0.927176i	$0.377771\pi$
$42$	2.04828	0.316057
$43$	−6.21046	−0.947086	−0.473543	−	0.880771i	$-0.657025\pi$
$43$	−6.21046	−0.947086	−0.473543	+	0.880771i	$0.657025\pi$
$44$	−3.81825	−0.575623
$45$	−0.686081	−0.102275
$46$	−2.46501	−0.363446
$47$	−6.54563	−0.954778	−0.477389	−	0.878692i	$-0.658417\pi$
$47$	−6.54563	−0.954778	−0.477389	+	0.878692i	$0.658417\pi$
$48$	−3.06826	−0.442865
$49$	−6.55435	−0.936335
$50$	4.98856	0.705489
$51$	24.4069	3.41764
$52$	−2.35476	−0.326546
$53$	13.6546	1.87561	0.937803	−	0.347169i	$-0.112857\pi$
$53$	13.6546	1.87561	0.937803	+	0.347169i	$0.112857\pi$
$54$	10.4756	1.42555
$55$	0.408411	0.0550701
$56$	−0.667572	−0.0892081
$57$	2.84464	0.376781
$58$	−2.73627	−0.359290
$59$	−9.98584	−1.30005	−0.650023	−	0.759915i	$-0.725241\pi$
$59$	−9.98584	−1.30005	−0.650023	+	0.759915i	$0.725241\pi$
$60$	0.328189	0.0423691
$61$	11.3728	1.45614	0.728072	−	0.685501i	$-0.240417\pi$
$61$	11.3728	1.45614	0.728072	+	0.685501i	$0.240417\pi$
$62$	−0.338431	−0.0429807
$63$	4.28194	0.539474
$64$	1.00000	0.125000
$65$	0.251871	0.0312408
$66$	−11.7154	−1.44206
$67$	−6.58187	−0.804104	−0.402052	−	0.915617i	$-0.631703\pi$
$67$	−6.58187	−0.804104	−0.402052	+	0.915617i	$0.631703\pi$
$68$	−7.95463	−0.964641
$69$	−7.56328	−0.910512
$70$	0.0714054	0.00853458
$71$	3.82923	0.454446	0.227223	−	0.973843i	$-0.427035\pi$
$71$	3.82923	0.454446	0.227223	+	0.973843i	$0.427035\pi$
$72$	−6.41420	−0.755921
$73$	−12.5467	−1.46848	−0.734238	−	0.678892i	$-0.762461\pi$
$73$	−12.5467	−1.46848	−0.734238	+	0.678892i	$0.762461\pi$
$74$	3.73972	0.434733
$75$	15.3062	1.76741
$76$	−0.927118	−0.106348
$77$	−2.54896	−0.290481
$78$	−7.22500	−0.818070
$79$	−17.3203	−1.94869	−0.974344	−	0.225063i	$-0.927741\pi$
$79$	−17.3203	−1.94869	−0.974344	+	0.225063i	$0.927741\pi$
$80$	−0.106963	−0.0119588
$81$	12.8994	1.43326
$82$	−4.79754	−0.529800
$83$	14.6930	1.61276	0.806381	−	0.591397i	$-0.201423\pi$
$83$	14.6930	1.61276	0.806381	+	0.591397i	$0.201423\pi$
$84$	−2.04828	−0.223486
$85$	0.850850	0.0922876
$86$	6.21046	0.669691
$87$	−8.39558	−0.900100
$88$	3.81825	0.407027
$89$	8.59589	0.911163	0.455581	−	0.890194i	$-0.349431\pi$
$89$	8.59589	0.911163	0.455581	+	0.890194i	$0.349431\pi$
$90$	0.686081	0.0723193
$91$	−1.57197	−0.164787
$92$	2.46501	0.256995
$93$	−1.03839	−0.107676
$94$	6.54563	0.675130
$95$	0.0991671	0.0101743
$96$	3.06826	0.313153
$97$	−0.720909	−0.0731972	−0.0365986	−	0.999330i	$-0.511652\pi$
$97$	−0.720909	−0.0731972	−0.0365986	+	0.999330i	$0.511652\pi$
$98$	6.55435	0.662089
$99$	−24.4910	−2.46144
$100$	−4.98856	−0.498856
$101$	1.63290	0.162480	0.0812400	−	0.996695i	$-0.474112\pi$
$101$	1.63290	0.162480	0.0812400	+	0.996695i	$0.474112\pi$
$102$	−24.4069	−2.41664
$103$	−5.51805	−0.543710	−0.271855	−	0.962338i	$-0.587637\pi$
$103$	−5.51805	−0.543710	−0.271855	+	0.962338i	$0.587637\pi$
$104$	2.35476	0.230903
$105$	0.219090	0.0213810
$106$	−13.6546	−1.32625
$107$	−13.4533	−1.30058	−0.650291	−	0.759686i	$-0.725353\pi$
$107$	−13.4533	−1.30058	−0.650291	+	0.759686i	$0.725353\pi$
$108$	−10.4756	−1.00802
$109$	−15.4912	−1.48379	−0.741895	−	0.670516i	$-0.766073\pi$
$109$	−15.4912	−1.48379	−0.741895	+	0.670516i	$0.766073\pi$
$110$	−0.408411	−0.0389404
$111$	11.4744	1.08910
$112$	0.667572	0.0630796
$113$	−5.87466	−0.552642	−0.276321	−	0.961065i	$-0.589115\pi$
$113$	−5.87466	−0.552642	−0.276321	+	0.961065i	$0.589115\pi$
$114$	−2.84464	−0.266425
$115$	−0.263664	−0.0245868
$116$	2.73627	0.254056
$117$	−15.1039	−1.39635
$118$	9.98584	0.919271
$119$	−5.31029	−0.486794
$120$	−0.328189	−0.0299594
$121$	3.57903	0.325366
$122$	−11.3728	−1.02965
$123$	−14.7201	−1.32727
$124$	0.338431	0.0303920
$125$	1.06840	0.0955610
$126$	−4.28194	−0.381466
$127$	−18.2439	−1.61888	−0.809442	−	0.587199i	$-0.800230\pi$
$127$	−18.2439	−1.61888	−0.809442	+	0.587199i	$0.800230\pi$
$128$	−1.00000	−0.0883883
$129$	19.0553	1.67772
$130$	−0.251871	−0.0220906
$131$	3.68039	0.321558	0.160779	−	0.986990i	$-0.448599\pi$
$131$	3.68039	0.321558	0.160779	+	0.986990i	$0.448599\pi$
$132$	11.7154	1.01969
$133$	−0.618918	−0.0536670
$134$	6.58187	0.568587
$135$	1.12050	0.0964376
$136$	7.95463	0.682104
$137$	−13.5500	−1.15766	−0.578829	−	0.815449i	$-0.696490\pi$
$137$	−13.5500	−1.15766	−0.578829	+	0.815449i	$0.696490\pi$
$138$	7.56328	0.643829
$139$	13.7248	1.16412	0.582060	−	0.813146i	$-0.302247\pi$
$139$	13.7248	1.16412	0.582060	+	0.813146i	$0.302247\pi$
$140$	−0.0714054	−0.00603486
$141$	20.0837	1.69135
$142$	−3.82923	−0.321342
$143$	8.99105	0.751869
$144$	6.41420	0.534517
$145$	−0.292679	−0.0243057
$146$	12.5467	1.03837
$147$	20.1104	1.65868
$148$	−3.73972	−0.307403
$149$	18.3532	1.50355	0.751776	−	0.659419i	$-0.229198\pi$
$149$	18.3532	1.50355	0.751776	+	0.659419i	$0.229198\pi$
$150$	−15.3062	−1.24974
$151$	−15.0601	−1.22557	−0.612787	−	0.790248i	$-0.709952\pi$
$151$	−15.0601	−1.22557	−0.612787	+	0.790248i	$0.709952\pi$
$152$	0.927118	0.0751992
$153$	−51.0226	−4.12493
$154$	2.54896	0.205401
$155$	−0.0361995	−0.00290761
$156$	7.22500	0.578463
$157$	23.3967	1.86726	0.933629	−	0.358241i	$-0.116624\pi$
$157$	23.3967	1.86726	0.933629	+	0.358241i	$0.116624\pi$
$158$	17.3203	1.37793
$159$	−41.8959	−3.32256
$160$	0.106963	0.00845615
$161$	1.64557	0.129689
$162$	−12.8994	−1.01347
$163$	−3.70687	−0.290345	−0.145172	−	0.989406i	$-0.546374\pi$
$163$	−3.70687	−0.290345	−0.145172	+	0.989406i	$0.546374\pi$
$164$	4.79754	0.374625
$165$	−1.25311	−0.0975544
$166$	−14.6930	−1.14039
$167$	−20.2818	−1.56945	−0.784725	−	0.619843i	$-0.787196\pi$
$167$	−20.2818	−1.56945	−0.784725	+	0.619843i	$0.787196\pi$
$168$	2.04828	0.158028
$169$	−7.45512	−0.573471
$170$	−0.850850	−0.0652572
$171$	−5.94672	−0.454757
$172$	−6.21046	−0.473543
$173$	17.5893	1.33729	0.668646	−	0.743581i	$-0.266874\pi$
$173$	17.5893	1.33729	0.668646	+	0.743581i	$0.266874\pi$
$174$	8.39558	0.636467
$175$	−3.33022	−0.251741
$176$	−3.81825	−0.287811
$177$	30.6391	2.30298
$178$	−8.59589	−0.644289
$179$	19.5079	1.45809	0.729046	−	0.684465i	$-0.239964\pi$
$179$	19.5079	1.45809	0.729046	+	0.684465i	$0.239964\pi$
$180$	−0.686081	−0.0511374
$181$	−13.1404	−0.976719	−0.488360	−	0.872643i	$-0.662405\pi$
$181$	−13.1404	−0.976719	−0.488360	+	0.872643i	$0.662405\pi$
$182$	1.57197	0.116522
$183$	−34.8948	−2.57950
$184$	−2.46501	−0.181723
$185$	0.400010	0.0294094
$186$	1.03839	0.0761386
$187$	30.3728	2.22108
$188$	−6.54563	−0.477389
$189$	−6.99325	−0.508684
$190$	−0.0991671	−0.00719434
$191$	−14.3974	−1.04176	−0.520880	−	0.853630i	$-0.674396\pi$
$191$	−14.3974	−1.04176	−0.520880	+	0.853630i	$0.674396\pi$
$192$	−3.06826	−0.221432
$193$	2.53319	0.182343	0.0911716	−	0.995835i	$-0.470939\pi$
$193$	2.53319	0.182343	0.0911716	+	0.995835i	$0.470939\pi$
$194$	0.720909	0.0517582
$195$	−0.772806	−0.0553418
$196$	−6.55435	−0.468168
$197$	10.1420	0.722590	0.361295	−	0.932452i	$-0.382335\pi$
$197$	10.1420	0.722590	0.361295	+	0.932452i	$0.382335\pi$
$198$	24.4910	1.74050
$199$	16.7840	1.18979	0.594894	−	0.803804i	$-0.297194\pi$
$199$	16.7840	1.18979	0.594894	+	0.803804i	$0.297194\pi$
$200$	4.98856	0.352744
$201$	20.1949	1.42444
$202$	−1.63290	−0.114891
$203$	1.82666	0.128206
$204$	24.4069	1.70882
$205$	−0.513159	−0.0358406
$206$	5.51805	0.384461
$207$	15.8111	1.09895
$208$	−2.35476	−0.163273
$209$	3.53997	0.244865
$210$	−0.219090	−0.0151187
$211$	9.77365	0.672846	0.336423	−	0.941711i	$-0.390783\pi$
$211$	9.77365	0.672846	0.336423	+	0.941711i	$0.390783\pi$
$212$	13.6546	0.937803
$213$	−11.7491	−0.805033
$214$	13.4533	0.919650
$215$	0.664288	0.0453041
$216$	10.4756	0.712777
$217$	0.225927	0.0153369
$218$	15.4912	1.04920
$219$	38.4964	2.60135
$220$	0.408411	0.0275350
$221$	18.7312	1.26000
$222$	−11.4744	−0.770112
$223$	−3.99870	−0.267772	−0.133886	−	0.990997i	$-0.542746\pi$
$223$	−3.99870	−0.267772	−0.133886	+	0.990997i	$0.542746\pi$
$224$	−0.667572	−0.0446040
$225$	−31.9976	−2.13317
$226$	5.87466	0.390777
$227$	16.4585	1.09239	0.546195	−	0.837658i	$-0.316076\pi$
$227$	16.4585	1.09239	0.546195	+	0.837658i	$0.316076\pi$
$228$	2.84464	0.188391
$229$	21.4549	1.41778	0.708889	−	0.705320i	$-0.249197\pi$
$229$	21.4549	1.41778	0.708889	+	0.705320i	$0.249197\pi$
$230$	0.263664	0.0173855
$231$	7.82086	0.514574
$232$	−2.73627	−0.179645
$233$	8.97475	0.587955	0.293978	−	0.955812i	$-0.405021\pi$
$233$	8.97475	0.587955	0.293978	+	0.955812i	$0.405021\pi$
$234$	15.1039	0.987371
$235$	0.700139	0.0456720
$236$	−9.98584	−0.650023
$237$	53.1432	3.45202
$238$	5.31029	0.344215
$239$	−14.7716	−0.955497	−0.477749	−	0.878497i	$-0.658547\pi$
$239$	−14.7716	−0.955497	−0.477749	+	0.878497i	$0.658547\pi$
$240$	0.328189	0.0211845
$241$	−10.8189	−0.696909	−0.348454	−	0.937326i	$-0.613293\pi$
$241$	−10.8189	−0.696909	−0.348454	+	0.937326i	$0.613293\pi$
$242$	−3.57903	−0.230069
$243$	−8.15163	−0.522927
$244$	11.3728	0.728072
$245$	0.701071	0.0447898
$246$	14.7201	0.938519
$247$	2.18314	0.138910
$248$	−0.338431	−0.0214904
$249$	−45.0818	−2.85694
$250$	−1.06840	−0.0675718
$251$	10.4545	0.659885	0.329942	−	0.944001i	$-0.392971\pi$
$251$	10.4545	0.659885	0.329942	+	0.944001i	$0.392971\pi$
$252$	4.28194	0.269737
$253$	−9.41202	−0.591729
$254$	18.2439	1.14472
$255$	−2.61063	−0.163484
$256$	1.00000	0.0625000
$257$	13.9499	0.870171	0.435086	−	0.900389i	$-0.356718\pi$
$257$	13.9499	0.870171	0.435086	+	0.900389i	$0.356718\pi$
$258$	−19.0553	−1.18633
$259$	−2.49653	−0.155127
$260$	0.251871	0.0156204
$261$	17.5510	1.08638
$262$	−3.68039	−0.227376
$263$	−9.40088	−0.579683	−0.289842	−	0.957075i	$-0.593603\pi$
$263$	−9.40088	−0.579683	−0.289842	+	0.957075i	$0.593603\pi$
$264$	−11.7154	−0.721031
$265$	−1.46054	−0.0897200
$266$	0.618918	0.0379483
$267$	−26.3744	−1.61409
$268$	−6.58187	−0.402052
$269$	−19.6147	−1.19593	−0.597964	−	0.801523i	$-0.704024\pi$
$269$	−19.6147	−1.19593	−0.597964	+	0.801523i	$0.704024\pi$
$270$	−1.12050	−0.0681917
$271$	15.9596	0.969479	0.484740	−	0.874658i	$-0.338914\pi$
$271$	15.9596	0.969479	0.484740	+	0.874658i	$0.338914\pi$
$272$	−7.95463	−0.482321
$273$	4.82321	0.291914
$274$	13.5500	0.818588
$275$	19.0476	1.14861
$276$	−7.56328	−0.455256
$277$	13.0159	0.782051	0.391025	−	0.920380i	$-0.372120\pi$
$277$	13.0159	0.782051	0.391025	+	0.920380i	$0.372120\pi$
$278$	−13.7248	−0.823157
$279$	2.17076	0.129960
$280$	0.0714054	0.00426729
$281$	33.2379	1.98280	0.991402	−	0.130849i	$-0.0417704\pi$
$281$	33.2379	1.98280	0.991402	+	0.130849i	$0.0417704\pi$
$282$	−20.0837	−1.19596
$283$	−27.7814	−1.65143	−0.825715	−	0.564087i	$-0.809228\pi$
$283$	−27.7814	−1.65143	−0.825715	+	0.564087i	$0.809228\pi$
$284$	3.82923	0.227223
$285$	−0.304270	−0.0180234
$286$	−8.99105	−0.531652
$287$	3.20271	0.189050
$288$	−6.41420	−0.377960
$289$	46.2762	2.72213
$290$	0.292679	0.0171867
$291$	2.21193	0.129666
$292$	−12.5467	−0.734238
$293$	−5.30253	−0.309777	−0.154888	−	0.987932i	$-0.549502\pi$
$293$	−5.30253	−0.309777	−0.154888	+	0.987932i	$0.549502\pi$
$294$	−20.1104	−1.17286
$295$	1.06811	0.0621879
$296$	3.73972	0.217367
$297$	39.9986	2.32095
$298$	−18.3532	−1.06317
$299$	−5.80450	−0.335683
$300$	15.3062	0.883703
$301$	−4.14593	−0.238968
$302$	15.0601	0.866612
$303$	−5.01017	−0.287827
$304$	−0.927118	−0.0531739
$305$	−1.21647	−0.0696549
$306$	51.0226	2.91677
$307$	16.3556	0.933462	0.466731	−	0.884399i	$-0.345432\pi$
$307$	16.3556	0.933462	0.466731	+	0.884399i	$0.345432\pi$
$308$	−2.54896	−0.145240
$309$	16.9308	0.963160
$310$	0.0361995	0.00205599
$311$	−32.9344	−1.86754	−0.933769	−	0.357877i	$-0.883501\pi$
$311$	−32.9344	−1.86754	−0.933769	+	0.357877i	$0.883501\pi$
$312$	−7.22500	−0.409035
$313$	11.6459	0.658265	0.329132	−	0.944284i	$-0.393244\pi$
$313$	11.6459	0.658265	0.329132	+	0.944284i	$0.393244\pi$
$314$	−23.3967	−1.32035
$315$	−0.458008	−0.0258059
$316$	−17.3203	−0.974344
$317$	5.83154	0.327532	0.163766	−	0.986499i	$-0.447636\pi$
$317$	5.83154	0.327532	0.163766	+	0.986499i	$0.447636\pi$
$318$	41.8959	2.34940
$319$	−10.4478	−0.584962
$320$	−0.106963	−0.00597940
$321$	41.2782	2.30393
$322$	−1.64557	−0.0917042
$323$	7.37489	0.410350
$324$	12.8994	0.716631
$325$	11.7468	0.651597
$326$	3.70687	0.205305
$327$	47.5311	2.62847
$328$	−4.79754	−0.264900
$329$	−4.36968	−0.240908
$330$	1.25311	0.0689814
$331$	−9.80851	−0.539124	−0.269562	−	0.962983i	$-0.586879\pi$
$331$	−9.80851	−0.539124	−0.269562	+	0.962983i	$0.586879\pi$
$332$	14.6930	0.806381
$333$	−23.9873	−1.31450
$334$	20.2818	1.10977
$335$	0.704016	0.0384645
$336$	−2.04828	−0.111743
$337$	21.9684	1.19670	0.598348	−	0.801236i	$-0.295824\pi$
$337$	21.9684	1.19670	0.598348	+	0.801236i	$0.295824\pi$
$338$	7.45512	0.405505
$339$	18.0250	0.978982
$340$	0.850850	0.0461438
$341$	−1.29221	−0.0699772
$342$	5.94672	0.321562
$343$	−9.04851	−0.488573
$344$	6.21046	0.334846
$345$	0.808990	0.0435546
$346$	−17.5893	−0.945608
$347$	−23.6806	−1.27124	−0.635621	−	0.772001i	$-0.719256\pi$
$347$	−23.6806	−1.27124	−0.635621	+	0.772001i	$0.719256\pi$
$348$	−8.39558	−0.450050
$349$	22.8609	1.22372	0.611858	−	0.790968i	$-0.290422\pi$
$349$	22.8609	1.22372	0.611858	+	0.790968i	$0.290422\pi$
$350$	3.33022	0.178008
$351$	24.6676	1.31666
$352$	3.81825	0.203513
$353$	22.4036	1.19242	0.596211	−	0.802828i	$-0.296672\pi$
$353$	22.4036	1.19242	0.596211	+	0.802828i	$0.296672\pi$
$354$	−30.6391	−1.62845
$355$	−0.409586	−0.0217385
$356$	8.59589	0.455581
$357$	16.2933	0.862335
$358$	−19.5079	−1.03103
$359$	30.1070	1.58899	0.794493	−	0.607274i	$-0.207737\pi$
$359$	30.1070	1.58899	0.794493	+	0.607274i	$0.207737\pi$
$360$	0.686081	0.0361596
$361$	−18.1405	−0.954761
$362$	13.1404	0.690645
$363$	−10.9814	−0.576373
$364$	−1.57197	−0.0823936
$365$	1.34203	0.0702449
$366$	34.8948	1.82398
$367$	−4.92878	−0.257280	−0.128640	−	0.991691i	$-0.541061\pi$
$367$	−4.92878	−0.257280	−0.128640	+	0.991691i	$0.541061\pi$
$368$	2.46501	0.128498
$369$	30.7724	1.60195
$370$	−0.400010	−0.0207956
$371$	9.11544	0.473250
$372$	−1.03839	−0.0538381
$373$	−12.2895	−0.636326	−0.318163	−	0.948036i	$-0.603066\pi$
$373$	−12.2895	−0.636326	−0.318163	+	0.948036i	$0.603066\pi$
$374$	−30.3728	−1.57054
$375$	−3.27814	−0.169282
$376$	6.54563	0.337565
$377$	−6.44325	−0.331844
$378$	6.99325	0.359694
$379$	−3.52810	−0.181226	−0.0906131	−	0.995886i	$-0.528883\pi$
$379$	−3.52810	−0.181226	−0.0906131	+	0.995886i	$0.528883\pi$
$380$	0.0991671	0.00508717
$381$	55.9770	2.86779
$382$	14.3974	0.736636
$383$	30.3382	1.55021	0.775106	−	0.631832i	$-0.217697\pi$
$383$	30.3382	1.55021	0.775106	+	0.631832i	$0.217697\pi$
$384$	3.06826	0.156576
$385$	0.272644	0.0138952
$386$	−2.53319	−0.128936
$387$	−39.8351	−2.02493
$388$	−0.720909	−0.0365986
$389$	−25.6361	−1.29980	−0.649900	−	0.760020i	$-0.725189\pi$
$389$	−25.6361	−1.29980	−0.649900	+	0.760020i	$0.725189\pi$
$390$	0.772806	0.0391325
$391$	−19.6083	−0.991632
$392$	6.55435	0.331045
$393$	−11.2924	−0.569626
$394$	−10.1420	−0.510948
$395$	1.85263	0.0932159
$396$	−24.4910	−1.23072
$397$	−17.1822	−0.862349	−0.431175	−	0.902269i	$-0.641901\pi$
$397$	−17.1822	−0.862349	−0.431175	+	0.902269i	$0.641901\pi$
$398$	−16.7840	−0.841308
$399$	1.89900	0.0950689
$400$	−4.98856	−0.249428
$401$	25.4170	1.26927	0.634633	−	0.772814i	$-0.281151\pi$
$401$	25.4170	1.26927	0.634633	+	0.772814i	$0.281151\pi$
$402$	−20.1949	−1.00723
$403$	−0.796922	−0.0396975
$404$	1.63290	0.0812400
$405$	−1.37975	−0.0685604
$406$	−1.82666	−0.0906555
$407$	14.2792	0.707792
$408$	−24.4069	−1.20832
$409$	27.6499	1.36720	0.683600	−	0.729857i	$-0.260413\pi$
$409$	27.6499	1.36720	0.683600	+	0.729857i	$0.260413\pi$
$410$	0.513159	0.0253431
$411$	41.5750	2.05074
$412$	−5.51805	−0.271855
$413$	−6.66627	−0.328026
$414$	−15.8111	−0.777072
$415$	−1.57160	−0.0771468
$416$	2.35476	0.115451
$417$	−42.1111	−2.06219
$418$	−3.53997	−0.173146
$419$	28.7510	1.40458	0.702289	−	0.711892i	$-0.252161\pi$
$419$	28.7510	1.40458	0.702289	+	0.711892i	$0.252161\pi$
$420$	0.219090	0.0106905
$421$	−17.6745	−0.861404	−0.430702	−	0.902494i	$-0.641734\pi$
$421$	−17.6745	−0.861404	−0.430702	+	0.902494i	$0.641734\pi$
$422$	−9.77365	−0.475774
$423$	−41.9850	−2.04138
$424$	−13.6546	−0.663127
$425$	39.6822	1.92487
$426$	11.7491	0.569244
$427$	7.59220	0.367412
$428$	−13.4533	−0.650291
$429$	−27.5868	−1.33191
$430$	−0.664288	−0.0320348
$431$	−7.99601	−0.385154	−0.192577	−	0.981282i	$-0.561685\pi$
$431$	−7.99601	−0.385154	−0.192577	+	0.981282i	$0.561685\pi$
$432$	−10.4756	−0.504010
$433$	24.1281	1.15952	0.579762	−	0.814786i	$-0.303146\pi$
$433$	24.1281	1.15952	0.579762	+	0.814786i	$0.303146\pi$
$434$	−0.225927	−0.0108448
$435$	0.898014	0.0430565
$436$	−15.4912	−0.741895
$437$	−2.28536	−0.109323
$438$	−38.4964	−1.83943
$439$	39.5356	1.88693	0.943465	−	0.331471i	$-0.107545\pi$
$439$	39.5356	1.88693	0.943465	+	0.331471i	$0.107545\pi$
$440$	−0.408411	−0.0194702
$441$	−42.0409	−2.00195
$442$	−18.7312	−0.890954
$443$	−38.5491	−1.83152	−0.915761	−	0.401724i	$-0.868411\pi$
$443$	−38.5491	−1.83152	−0.915761	+	0.401724i	$0.868411\pi$
$444$	11.4744	0.544551
$445$	−0.919441	−0.0435857
$446$	3.99870	0.189344
$447$	−56.3123	−2.66348
$448$	0.667572	0.0315398
$449$	−12.3690	−0.583729	−0.291864	−	0.956460i	$-0.594276\pi$
$449$	−12.3690	−0.583729	−0.291864	+	0.956460i	$0.594276\pi$
$450$	31.9976	1.50838
$451$	−18.3182	−0.862571
$452$	−5.87466	−0.276321
$453$	46.2083	2.17106
$454$	−16.4585	−0.772436
$455$	0.168142	0.00788263
$456$	−2.84464	−0.133212
$457$	−33.8872	−1.58518	−0.792588	−	0.609757i	$-0.791267\pi$
$457$	−33.8872	−1.58518	−0.792588	+	0.609757i	$0.791267\pi$
$458$	−21.4549	−1.00252
$459$	83.3299	3.88951
$460$	−0.263664	−0.0122934
$461$	−21.9892	−1.02414	−0.512069	−	0.858945i	$-0.671121\pi$
$461$	−21.9892	−1.02414	−0.512069	+	0.858945i	$0.671121\pi$
$462$	−7.82086	−0.363859
$463$	16.7251	0.777279	0.388640	−	0.921390i	$-0.372945\pi$
$463$	16.7251	0.777279	0.388640	+	0.921390i	$0.372945\pi$
$464$	2.73627	0.127028
$465$	0.111069	0.00515072
$466$	−8.97475	−0.415747
$467$	−30.9408	−1.43177	−0.715885	−	0.698218i	$-0.753976\pi$
$467$	−30.9408	−1.43177	−0.715885	+	0.698218i	$0.753976\pi$
$468$	−15.1039	−0.698177
$469$	−4.39388	−0.202890
$470$	−0.700139	−0.0322950
$471$	−71.7870	−3.30777
$472$	9.98584	0.459635
$473$	23.7131	1.09033
$474$	−53.1432	−2.44095
$475$	4.62498	0.212209
$476$	−5.31029	−0.243397
$477$	87.5834	4.01017
$478$	14.7716	0.675639
$479$	−6.09792	−0.278621	−0.139310	−	0.990249i	$-0.544489\pi$
$479$	−6.09792	−0.278621	−0.139310	+	0.990249i	$0.544489\pi$
$480$	−0.328189	−0.0149797
$481$	8.80612	0.401525
$482$	10.8189	0.492789
$483$	−5.04904	−0.229739
$484$	3.57903	0.162683
$485$	0.0771104	0.00350140
$486$	8.15163	0.369765
$487$	1.18138	0.0535336	0.0267668	−	0.999642i	$-0.491479\pi$
$487$	1.18138	0.0535336	0.0267668	+	0.999642i	$0.491479\pi$
$488$	−11.3728	−0.514825
$489$	11.3736	0.514334
$490$	−0.701071	−0.0316712
$491$	23.0462	1.04006	0.520030	−	0.854148i	$-0.325921\pi$
$491$	23.0462	1.04006	0.520030	+	0.854148i	$0.325921\pi$
$492$	−14.7201	−0.663633
$493$	−21.7660	−0.980292
$494$	−2.18314	−0.0982240
$495$	2.61963	0.117743
$496$	0.338431	0.0151960
$497$	2.55629	0.114665
$498$	45.0818	2.02016
$499$	−31.8401	−1.42536	−0.712679	−	0.701491i	$-0.752518\pi$
$499$	−31.8401	−1.42536	−0.712679	+	0.701491i	$0.752518\pi$
$500$	1.06840	0.0477805
$501$	62.2297	2.78022
$502$	−10.4545	−0.466609
$503$	−19.5249	−0.870570	−0.435285	−	0.900293i	$-0.643352\pi$
$503$	−19.5249	−0.870570	−0.435285	+	0.900293i	$0.643352\pi$
$504$	−4.28194	−0.190733
$505$	−0.174660	−0.00777226
$506$	9.41202	0.418415
$507$	22.8742	1.01588
$508$	−18.2439	−0.809442
$509$	31.8108	1.40999	0.704995	−	0.709212i	$-0.250949\pi$
$509$	31.8108	1.40999	0.704995	+	0.709212i	$0.250949\pi$
$510$	2.61063	0.115600
$511$	−8.37581	−0.370524
$512$	−1.00000	−0.0441942
$513$	9.71216	0.428802
$514$	−13.9499	−0.615304
$515$	0.590226	0.0260085
$516$	19.0553	0.838862
$517$	24.9928	1.09918
$518$	2.49653	0.109691
$519$	−53.9686	−2.36896
$520$	−0.251871	−0.0110453
$521$	21.4884	0.941425	0.470713	−	0.882287i	$-0.343997\pi$
$521$	21.4884	0.941425	0.470713	+	0.882287i	$0.343997\pi$
$522$	−17.5510	−0.768186
$523$	19.4615	0.850991	0.425495	−	0.904961i	$-0.360100\pi$
$523$	19.4615	0.850991	0.425495	+	0.904961i	$0.360100\pi$
$524$	3.68039	0.160779
$525$	10.2180	0.445949
$526$	9.40088	0.409898
$527$	−2.69209	−0.117269
$528$	11.7154	0.509846
$529$	−16.9237	−0.735814
$530$	1.46054	0.0634416
$531$	−64.0512	−2.77958
$532$	−0.618918	−0.0268335
$533$	−11.2970	−0.489330
$534$	26.3744	1.14133
$535$	1.43900	0.0622136
$536$	6.58187	0.284294
$537$	−59.8554	−2.58295
$538$	19.6147	0.845649
$539$	25.0261	1.07795
$540$	1.12050	0.0482188
$541$	26.3894	1.13457	0.567284	−	0.823523i	$-0.307994\pi$
$541$	26.3894	1.13457	0.567284	+	0.823523i	$0.307994\pi$
$542$	−15.9596	−0.685525
$543$	40.3182	1.73022
$544$	7.95463	0.341052
$545$	1.65698	0.0709774
$546$	−4.82321	−0.206414
$547$	33.1551	1.41761	0.708806	−	0.705404i	$-0.249234\pi$
$547$	33.1551	1.41761	0.708806	+	0.705404i	$0.249234\pi$
$548$	−13.5500	−0.578829
$549$	72.9477	3.11333
$550$	−19.0476	−0.812191
$551$	−2.53685	−0.108073
$552$	7.56328	0.321915
$553$	−11.5626	−0.491690
$554$	−13.0159	−0.552993
$555$	−1.22733	−0.0520975
$556$	13.7248	0.582060
$557$	−13.5830	−0.575531	−0.287765	−	0.957701i	$-0.592912\pi$
$557$	−13.5830	−0.575531	−0.287765	+	0.957701i	$0.592912\pi$
$558$	−2.17076	−0.0918957
$559$	14.6241	0.618534
$560$	−0.0714054	−0.00301743
$561$	−93.1915	−3.93455
$562$	−33.2379	−1.40205
$563$	−25.5788	−1.07802	−0.539010	−	0.842300i	$-0.681201\pi$
$563$	−25.5788	−1.07802	−0.539010	+	0.842300i	$0.681201\pi$
$564$	20.0837	0.845675
$565$	0.628370	0.0264357
$566$	27.7814	1.16774
$567$	8.61126	0.361639
$568$	−3.82923	−0.160671
$569$	−4.40611	−0.184714	−0.0923569	−	0.995726i	$-0.529440\pi$
$569$	−4.40611	−0.184714	−0.0923569	+	0.995726i	$0.529440\pi$
$570$	0.304270	0.0127445
$571$	13.2354	0.553883	0.276942	−	0.960887i	$-0.410679\pi$
$571$	13.2354	0.553883	0.276942	+	0.960887i	$0.410679\pi$
$572$	8.99105	0.375935
$573$	44.1750	1.84544
$574$	−3.20271	−0.133678
$575$	−12.2968	−0.512814
$576$	6.41420	0.267258
$577$	20.3852	0.848645	0.424322	−	0.905511i	$-0.360512\pi$
$577$	20.3852	0.848645	0.424322	+	0.905511i	$0.360512\pi$
$578$	−46.2762	−1.92484
$579$	−7.77248	−0.323013
$580$	−0.292679	−0.0121528
$581$	9.80861	0.406930
$582$	−2.21193	−0.0916876
$583$	−52.1367	−2.15928
$584$	12.5467	0.519185
$585$	1.61555	0.0667949
$586$	5.30253	0.219045
$587$	17.7744	0.733627	0.366814	−	0.930294i	$-0.380449\pi$
$587$	17.7744	0.733627	0.366814	+	0.930294i	$0.380449\pi$
$588$	20.1104	0.829340
$589$	−0.313765	−0.0129285
$590$	−1.06811	−0.0439735
$591$	−31.1184	−1.28004
$592$	−3.73972	−0.153701
$593$	21.2307	0.871841	0.435920	−	0.899985i	$-0.356423\pi$
$593$	21.2307	0.871841	0.435920	+	0.899985i	$0.356423\pi$
$594$	−39.9986	−1.64116
$595$	0.568004	0.0232859
$596$	18.3532	0.751776
$597$	−51.4977	−2.10766
$598$	5.80450	0.237364
$599$	23.5348	0.961607	0.480804	−	0.876828i	$-0.340345\pi$
$599$	23.5348	0.961607	0.480804	+	0.876828i	$0.340345\pi$
$600$	−15.3062	−0.624872
$601$	22.0860	0.900906	0.450453	−	0.892800i	$-0.351262\pi$
$601$	22.0860	0.900906	0.450453	+	0.892800i	$0.351262\pi$
$602$	4.14593	0.168976
$603$	−42.2175	−1.71923
$604$	−15.0601	−0.612787
$605$	−0.382823	−0.0155640
$606$	5.01017	0.203524
$607$	−1.41519	−0.0574410	−0.0287205	−	0.999587i	$-0.509143\pi$
$607$	−1.41519	−0.0574410	−0.0287205	+	0.999587i	$0.509143\pi$
$608$	0.927118	0.0375996
$609$	−5.60465	−0.227112
$610$	1.21647	0.0492535
$611$	15.4134	0.623558
$612$	−51.0226	−2.06247
$613$	0.349171	0.0141029	0.00705145	−	0.999975i	$-0.497755\pi$
$613$	0.349171	0.0141029	0.00705145	+	0.999975i	$0.497755\pi$
$614$	−16.3556	−0.660057
$615$	1.57450	0.0634901
$616$	2.54896	0.102700
$617$	18.8370	0.758347	0.379174	−	0.925326i	$-0.376208\pi$
$617$	18.8370	0.758347	0.379174	+	0.925326i	$0.376208\pi$
$618$	−16.9308	−0.681057
$619$	44.8422	1.80236	0.901180	−	0.433445i	$-0.142702\pi$
$619$	44.8422	1.80236	0.901180	+	0.433445i	$0.142702\pi$
$620$	−0.0361995	−0.00145381
$621$	−25.8226	−1.03622
$622$	32.9344	1.32055
$623$	5.73838	0.229903
$624$	7.22500	0.289231
$625$	24.8285	0.993141
$626$	−11.6459	−0.465464
$627$	−10.8615	−0.433768
$628$	23.3967	0.933629
$629$	29.7481	1.18613
$630$	0.458008	0.0182475
$631$	7.50462	0.298754	0.149377	−	0.988780i	$-0.452273\pi$
$631$	7.50462	0.298754	0.149377	+	0.988780i	$0.452273\pi$
$632$	17.3203	0.688965
$633$	−29.9881	−1.19192
$634$	−5.83154	−0.231600
$635$	1.95142	0.0774397
$636$	−41.8959	−1.66128
$637$	15.4339	0.611513
$638$	10.4478	0.413631
$639$	24.5615	0.971637
$640$	0.106963	0.00422808
$641$	−0.0634204	−0.00250495	−0.00125248	−	0.999999i	$-0.500399\pi$
$641$	−0.0634204	−0.00250495	−0.00125248	+	0.999999i	$0.500399\pi$
$642$	−41.2782	−1.62912
$643$	−9.78025	−0.385695	−0.192848	−	0.981229i	$-0.561772\pi$
$643$	−9.78025	−0.385695	−0.192848	+	0.981229i	$0.561772\pi$
$644$	1.64557	0.0648446
$645$	−2.03821	−0.0802543
$646$	−7.37489	−0.290161
$647$	9.65704	0.379657	0.189829	−	0.981817i	$-0.439207\pi$
$647$	9.65704	0.379657	0.189829	+	0.981817i	$0.439207\pi$
$648$	−12.8994	−0.506735
$649$	38.1284	1.49667
$650$	−11.7468	−0.460749
$651$	−0.693202	−0.0271687
$652$	−3.70687	−0.145172
$653$	10.9941	0.430231	0.215116	−	0.976589i	$-0.430987\pi$
$653$	10.9941	0.430231	0.215116	+	0.976589i	$0.430987\pi$
$654$	−47.5311	−1.85861
$655$	−0.393665	−0.0153818
$656$	4.79754	0.187313
$657$	−80.4769	−3.13970
$658$	4.36968	0.170348
$659$	3.58699	0.139729	0.0698646	−	0.997556i	$-0.477743\pi$
$659$	3.58699	0.139729	0.0698646	+	0.997556i	$0.477743\pi$
$660$	−1.25311	−0.0487772
$661$	−24.5823	−0.956140	−0.478070	−	0.878322i	$-0.658664\pi$
$661$	−24.5823	−0.956140	−0.478070	+	0.878322i	$0.658664\pi$
$662$	9.80851	0.381219
$663$	−57.4722	−2.23204
$664$	−14.6930	−0.570197
$665$	0.0662012	0.00256717
$666$	23.9873	0.929489
$667$	6.74493	0.261165
$668$	−20.2818	−0.784725
$669$	12.2690	0.474348
$670$	−0.704016	−0.0271985
$671$	−43.4244	−1.67638
$672$	2.04828	0.0790142
$673$	13.6654	0.526762	0.263381	−	0.964692i	$-0.415162\pi$
$673$	13.6654	0.526762	0.263381	+	0.964692i	$0.415162\pi$
$674$	−21.9684	−0.846192
$675$	52.2584	2.01143
$676$	−7.45512	−0.286735
$677$	−15.7106	−0.603809	−0.301904	−	0.953338i	$-0.597622\pi$
$677$	−15.7106	−0.603809	−0.301904	+	0.953338i	$0.597622\pi$
$678$	−18.0250	−0.692245
$679$	−0.481259	−0.0184690
$680$	−0.850850	−0.0326286
$681$	−50.4989	−1.93512
$682$	1.29221	0.0494814
$683$	−22.8506	−0.874356	−0.437178	−	0.899375i	$-0.644022\pi$
$683$	−22.8506	−0.874356	−0.437178	+	0.899375i	$0.644022\pi$
$684$	−5.94672	−0.227379
$685$	1.44935	0.0553768
$686$	9.04851	0.345474
$687$	−65.8291	−2.51154
$688$	−6.21046	−0.236772
$689$	−32.1533	−1.22494
$690$	−0.808990	−0.0307977
$691$	27.3049	1.03873	0.519363	−	0.854554i	$-0.326169\pi$
$691$	27.3049	1.03873	0.519363	+	0.854554i	$0.326169\pi$
$692$	17.5893	0.668646
$693$	−16.3495	−0.621067
$694$	23.6806	0.898903
$695$	−1.46804	−0.0556859
$696$	8.39558	0.318234
$697$	−38.1627	−1.44552
$698$	−22.8609	−0.865298
$699$	−27.5368	−1.04154
$700$	−3.33022	−0.125871
$701$	−41.8200	−1.57952	−0.789760	−	0.613415i	$-0.789795\pi$
$701$	−41.8200	−1.57952	−0.789760	+	0.613415i	$0.789795\pi$
$702$	−24.6676	−0.931018
$703$	3.46716	0.130766
$704$	−3.81825	−0.143906
$705$	−2.14820	−0.0809061
$706$	−22.4036	−0.843169
$707$	1.09008	0.0409967
$708$	30.6391	1.15149
$709$	−36.8584	−1.38425	−0.692123	−	0.721779i	$-0.743325\pi$
$709$	−36.8584	−1.38425	−0.692123	+	0.721779i	$0.743325\pi$
$710$	0.409586	0.0153715
$711$	−111.096	−4.16643
$712$	−8.59589	−0.322145
$713$	0.834235	0.0312423
$714$	−16.2933	−0.609763
$715$	−0.961707	−0.0359658
$716$	19.5079	0.729046
$717$	45.3232	1.69262
$718$	−30.1070	−1.12358
$719$	−48.1949	−1.79737	−0.898684	−	0.438597i	$-0.855475\pi$
$719$	−48.1949	−1.79737	−0.898684	+	0.438597i	$0.855475\pi$
$720$	−0.686081	−0.0255687
$721$	−3.68370	−0.137188
$722$	18.1405	0.675118
$723$	33.1953	1.23454
$724$	−13.1404	−0.488360
$725$	−13.6500	−0.506950
$726$	10.9814	0.407557
$727$	6.65977	0.246997	0.123499	−	0.992345i	$-0.460589\pi$
$727$	6.65977	0.246997	0.123499	+	0.992345i	$0.460589\pi$
$728$	1.57197	0.0582611
$729$	−13.6868	−0.506919
$730$	−1.34203	−0.0496706
$731$	49.4019	1.82720
$732$	−34.8948	−1.28975
$733$	21.7340	0.802762	0.401381	−	0.915911i	$-0.368530\pi$
$733$	21.7340	0.802762	0.401381	+	0.915911i	$0.368530\pi$
$734$	4.92878	0.181925
$735$	−2.15107	−0.0793433
$736$	−2.46501	−0.0908615
$737$	25.1312	0.925721
$738$	−30.7724	−1.13275
$739$	44.3674	1.63208	0.816041	−	0.577994i	$-0.196164\pi$
$739$	44.3674	1.63208	0.816041	+	0.577994i	$0.196164\pi$
$740$	0.400010	0.0147047
$741$	−6.69843	−0.246073
$742$	−9.11544	−0.334638
$743$	31.8967	1.17018	0.585089	−	0.810969i	$-0.301060\pi$
$743$	31.8967	1.17018	0.585089	+	0.810969i	$0.301060\pi$
$744$	1.03839	0.0380693
$745$	−1.96311	−0.0719227
$746$	12.2895	0.449950
$747$	94.2436	3.44819
$748$	30.3728	1.11054
$749$	−8.98106	−0.328161
$750$	3.27814	0.119701
$751$	34.4552	1.25729	0.628644	−	0.777694i	$-0.283610\pi$
$751$	34.4552	1.25729	0.628644	+	0.777694i	$0.283610\pi$
$752$	−6.54563	−0.238694
$753$	−32.0772	−1.16896
$754$	6.44325	0.234649
$755$	1.61087	0.0586256
$756$	−6.99325	−0.254342
$757$	5.67881	0.206400	0.103200	−	0.994661i	$-0.467092\pi$
$757$	5.67881	0.206400	0.103200	+	0.994661i	$0.467092\pi$
$758$	3.52810	0.128146
$759$	28.8785	1.04822
$760$	−0.0991671	−0.00359717
$761$	8.50861	0.308437	0.154218	−	0.988037i	$-0.450714\pi$
$761$	8.50861	0.308437	0.154218	+	0.988037i	$0.450714\pi$
$762$	−55.9770	−2.02783
$763$	−10.3415	−0.374388
$764$	−14.3974	−0.520880
$765$	5.45752	0.197317
$766$	−30.3382	−1.09616
$767$	23.5142	0.849049
$768$	−3.06826	−0.110716
$769$	−11.7503	−0.423726	−0.211863	−	0.977299i	$-0.567953\pi$
$769$	−11.7503	−0.423726	−0.211863	+	0.977299i	$0.567953\pi$
$770$	−0.272644	−0.00982539
$771$	−42.8019	−1.54147
$772$	2.53319	0.0911716
$773$	16.5013	0.593511	0.296755	−	0.954954i	$-0.404095\pi$
$773$	16.5013	0.593511	0.296755	+	0.954954i	$0.404095\pi$
$774$	39.8351	1.43184
$775$	−1.68828	−0.0606449
$776$	0.720909	0.0258791
$777$	7.66000	0.274801
$778$	25.6361	0.919097
$779$	−4.44789	−0.159362
$780$	−0.772806	−0.0276709
$781$	−14.6210	−0.523179
$782$	19.6083	0.701190
$783$	−28.6642	−1.02437
$784$	−6.55435	−0.234084
$785$	−2.50257	−0.0893207
$786$	11.2924	0.402786
$787$	−33.3218	−1.18779	−0.593897	−	0.804541i	$-0.702411\pi$
$787$	−33.3218	−1.18779	−0.593897	+	0.804541i	$0.702411\pi$
$788$	10.1420	0.361295
$789$	28.8443	1.02689
$790$	−1.85263	−0.0659136
$791$	−3.92176	−0.139442
$792$	24.4910	0.870250
$793$	−26.7803	−0.950996
$794$	17.1822	0.609773
$795$	4.48130	0.158935
$796$	16.7840	0.594894
$797$	−8.85273	−0.313580	−0.156790	−	0.987632i	$-0.550115\pi$
$797$	−8.85273	−0.313580	−0.156790	+	0.987632i	$0.550115\pi$
$798$	−1.89900	−0.0672239
$799$	52.0681	1.84204
$800$	4.98856	0.176372
$801$	55.1358	1.94813
$802$	−25.4170	−0.897506
$803$	47.9063	1.69058
$804$	20.1949	0.712218
$805$	−0.176015	−0.00620371
$806$	0.796922	0.0280704
$807$	60.1829	2.11854
$808$	−1.63290	−0.0574453
$809$	−38.2807	−1.34588	−0.672938	−	0.739699i	$-0.734968\pi$
$809$	−38.2807	−1.34588	−0.672938	+	0.739699i	$0.734968\pi$
$810$	1.37975	0.0484795
$811$	−43.2045	−1.51711	−0.758557	−	0.651607i	$-0.774095\pi$
$811$	−43.2045	−1.51711	−0.758557	+	0.651607i	$0.774095\pi$
$812$	1.82666	0.0641031
$813$	−48.9683	−1.71739
$814$	−14.2792	−0.500485
$815$	0.396497	0.0138887
$816$	24.4069	0.854411
$817$	5.75783	0.201441
$818$	−27.6499	−0.966757
$819$	−10.0829	−0.352326
$820$	−0.513159	−0.0179203
$821$	−10.0870	−0.352037	−0.176019	−	0.984387i	$-0.556322\pi$
$821$	−10.0870	−0.352037	−0.176019	+	0.984387i	$0.556322\pi$
$822$	−41.5750	−1.45009
$823$	18.8562	0.657284	0.328642	−	0.944455i	$-0.393409\pi$
$823$	18.8562	0.657284	0.328642	+	0.944455i	$0.393409\pi$
$824$	5.51805	0.192230
$825$	−58.4428	−2.03472
$826$	6.66627	0.231949
$827$	31.0403	1.07938	0.539689	−	0.841864i	$-0.318542\pi$
$827$	31.0403	1.07938	0.539689	+	0.841864i	$0.318542\pi$
$828$	15.8111	0.549473
$829$	9.78484	0.339841	0.169921	−	0.985458i	$-0.445649\pi$
$829$	9.78484	0.339841	0.169921	+	0.985458i	$0.445649\pi$
$830$	1.57160	0.0545510
$831$	−39.9362	−1.38537
$832$	−2.35476	−0.0816365
$833$	52.1374	1.80646
$834$	42.1111	1.45819
$835$	2.16940	0.0750750
$836$	3.53997	0.122432
$837$	−3.54528	−0.122543
$838$	−28.7510	−0.993187
$839$	−6.98006	−0.240978	−0.120489	−	0.992715i	$-0.538446\pi$
$839$	−6.98006	−0.240978	−0.120489	+	0.992715i	$0.538446\pi$
$840$	−0.219090	−0.00755933
$841$	−21.5128	−0.741822
$842$	17.6745	0.609104
$843$	−101.982	−3.51246
$844$	9.77365	0.336423
$845$	0.797421	0.0274321
$846$	41.9850	1.44347
$847$	2.38926	0.0820959
$848$	13.6546	0.468901
$849$	85.2404	2.92544
$850$	−39.6822	−1.36109
$851$	−9.21844	−0.316004
$852$	−11.7491	−0.402517
$853$	−56.6259	−1.93884	−0.969418	−	0.245417i	$-0.921075\pi$
$853$	−56.6259	−1.93884	−0.969418	+	0.245417i	$0.921075\pi$
$854$	−7.59220	−0.259800
$855$	0.636078	0.0217534
$856$	13.4533	0.459825
$857$	−17.2480	−0.589182	−0.294591	−	0.955623i	$-0.595183\pi$
$857$	−17.2480	−0.589182	−0.294591	+	0.955623i	$0.595183\pi$
$858$	27.5868	0.941799
$859$	−44.7454	−1.52669	−0.763347	−	0.645988i	$-0.776445\pi$
$859$	−44.7454	−1.52669	−0.763347	+	0.645988i	$0.776445\pi$
$860$	0.664288	0.0226520
$861$	−9.82673	−0.334894
$862$	7.99601	0.272345
$863$	−16.7031	−0.568579	−0.284289	−	0.958738i	$-0.591758\pi$
$863$	−16.7031	−0.568579	−0.284289	+	0.958738i	$0.591758\pi$
$864$	10.4756	0.356389
$865$	−1.88140	−0.0639697
$866$	−24.1281	−0.819907
$867$	−141.987	−4.82214
$868$	0.225927	0.00766846
$869$	66.1333	2.24342
$870$	−0.898014	−0.0304455
$871$	15.4987	0.525154
$872$	15.4912	0.524599
$873$	−4.62405	−0.156500
$874$	2.28536	0.0773033
$875$	0.713237	0.0241118
$876$	38.4964	1.30067
$877$	−32.7083	−1.10448	−0.552241	−	0.833685i	$-0.686227\pi$
$877$	−32.7083	−1.10448	−0.552241	+	0.833685i	$0.686227\pi$
$878$	−39.5356	−1.33426
$879$	16.2695	0.548757
$880$	0.408411	0.0137675
$881$	−16.2199	−0.546462	−0.273231	−	0.961948i	$-0.588092\pi$
$881$	−16.2199	−0.546462	−0.273231	+	0.961948i	$0.588092\pi$
$882$	42.0409	1.41559
$883$	28.0052	0.942451	0.471225	−	0.882013i	$-0.343812\pi$
$883$	28.0052	0.942451	0.471225	+	0.882013i	$0.343812\pi$
$884$	18.7312	0.629999
$885$	−3.27724	−0.110163
$886$	38.5491	1.29508
$887$	29.8987	1.00390	0.501950	−	0.864897i	$-0.332616\pi$
$887$	29.8987	1.00390	0.501950	+	0.864897i	$0.332616\pi$
$888$	−11.4744	−0.385056
$889$	−12.1791	−0.408475
$890$	0.919441	0.0308197
$891$	−49.2530	−1.65004
$892$	−3.99870	−0.133886
$893$	6.06857	0.203077
$894$	56.3123	1.88336
$895$	−2.08662	−0.0697481
$896$	−0.667572	−0.0223020
$897$	17.8097	0.594648
$898$	12.3690	0.412759
$899$	0.926037	0.0308851
$900$	−31.9976	−1.06659
$901$	−108.617	−3.61857
$902$	18.3182	0.609930
$903$	12.7208	0.423321
$904$	5.87466	0.195388
$905$	1.40553	0.0467216
$906$	−46.2083	−1.53517
$907$	13.5167	0.448815	0.224408	−	0.974495i	$-0.427955\pi$
$907$	13.5167	0.448815	0.224408	+	0.974495i	$0.427955\pi$
$908$	16.4585	0.546195
$909$	10.4738	0.347393
$910$	−0.168142	−0.00557386
$911$	37.4485	1.24073	0.620363	−	0.784315i	$-0.286985\pi$
$911$	37.4485	1.24073	0.620363	+	0.784315i	$0.286985\pi$
$912$	2.84464	0.0941953
$913$	−56.1014	−1.85668
$914$	33.8872	1.12089
$915$	3.73245	0.123391
$916$	21.4549	0.708889
$917$	2.45693	0.0811350
$918$	−83.3299	−2.75030
$919$	−35.4474	−1.16930	−0.584652	−	0.811284i	$-0.698769\pi$
$919$	−35.4474	−1.16930	−0.584652	+	0.811284i	$0.698769\pi$
$920$	0.263664	0.00869276
$921$	−50.1831	−1.65359
$922$	21.9892	0.724174
$923$	−9.01691	−0.296795
$924$	7.82086	0.257287
$925$	18.6558	0.613399
$926$	−16.7251	−0.549620
$927$	−35.3939	−1.16249
$928$	−2.73627	−0.0898224
$929$	9.96432	0.326919	0.163459	−	0.986550i	$-0.447735\pi$
$929$	9.96432	0.326919	0.163459	+	0.986550i	$0.447735\pi$
$930$	−0.111069	−0.00364211
$931$	6.07665	0.199154
$932$	8.97475	0.293978
$933$	101.051	3.30827
$934$	30.9408	1.01241
$935$	−3.24876	−0.106246
$936$	15.1039	0.493686
$937$	−16.6550	−0.544097	−0.272048	−	0.962284i	$-0.587701\pi$
$937$	−16.6550	−0.544097	−0.272048	+	0.962284i	$0.587701\pi$
$938$	4.39388	0.143465
$939$	−35.7326	−1.16609
$940$	0.700139	0.0228360
$941$	6.55358	0.213641	0.106820	−	0.994278i	$-0.465933\pi$
$941$	6.55358	0.213641	0.106820	+	0.994278i	$0.465933\pi$
$942$	71.7870	2.33895
$943$	11.8260	0.385107
$944$	−9.98584	−0.325011
$945$	0.748017	0.0243330
$946$	−23.7131	−0.770979
$947$	19.5084	0.633939	0.316970	−	0.948436i	$-0.397335\pi$
$947$	19.5084	0.633939	0.316970	+	0.948436i	$0.397335\pi$
$948$	53.1432	1.72601
$949$	29.5444	0.959050
$950$	−4.62498	−0.150054
$951$	−17.8926	−0.580209
$952$	5.31029	0.172108
$953$	−44.7063	−1.44818	−0.724090	−	0.689706i	$-0.757740\pi$
$953$	−44.7063	−1.44818	−0.724090	+	0.689706i	$0.757740\pi$
$954$	−87.5834	−2.83562
$955$	1.53999	0.0498328
$956$	−14.7716	−0.477749
$957$	32.0564	1.03624
$958$	6.09792	0.197015
$959$	−9.04563	−0.292099
$960$	0.328189	0.0105923
$961$	−30.8855	−0.996305
$962$	−8.80612	−0.283921
$963$	−86.2923	−2.78073
$964$	−10.8189	−0.348454
$965$	−0.270957	−0.00872242
$966$	5.04904	0.162450
$967$	−19.3530	−0.622351	−0.311176	−	0.950352i	$-0.600723\pi$
$967$	−19.3530	−0.622351	−0.311176	+	0.950352i	$0.600723\pi$
$968$	−3.57903	−0.115034
$969$	−22.6280	−0.726918
$970$	−0.0771104	−0.00247587
$971$	−1.69219	−0.0543049	−0.0271524	−	0.999631i	$-0.508644\pi$
$971$	−1.69219	−0.0543049	−0.0271524	+	0.999631i	$0.508644\pi$
$972$	−8.15163	−0.261464
$973$	9.16228	0.293729
$974$	−1.18138	−0.0378540
$975$	−36.0423	−1.15428
$976$	11.3728	0.364036
$977$	32.6591	1.04486	0.522428	−	0.852683i	$-0.325026\pi$
$977$	32.6591	1.04486	0.522428	+	0.852683i	$0.325026\pi$
$978$	−11.3736	−0.363689
$979$	−32.8213	−1.04897
$980$	0.701071	0.0223949
$981$	−99.3638	−3.17244
$982$	−23.0462	−0.735433
$983$	22.6335	0.721897	0.360949	−	0.932586i	$-0.382453\pi$
$983$	22.6335	0.721897	0.360949	+	0.932586i	$0.382453\pi$
$984$	14.7201	0.469260
$985$	−1.08482	−0.0345653
$986$	21.7660	0.693171
$987$	13.4073	0.426759
$988$	2.18314	0.0694549
$989$	−15.3088	−0.486793
$990$	−2.61963	−0.0832572
$991$	53.1251	1.68757	0.843787	−	0.536679i	$-0.180321\pi$
$991$	53.1251	1.68757	0.843787	+	0.536679i	$0.180321\pi$
$992$	−0.338431	−0.0107452
$993$	30.0950	0.955037
$994$	−2.55629	−0.0810806
$995$	−1.79527	−0.0569138
$996$	−45.0818	−1.42847
$997$	−20.2498	−0.641318	−0.320659	−	0.947195i	$-0.603904\pi$
$997$	−20.2498	−0.641318	−0.320659	+	0.947195i	$0.603904\pi$
$998$	31.8401	1.00788
$999$	39.1759	1.23947

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.h.1.3	✓	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.h.1.3	✓	42	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.06826 q^{3} +1.00000 q^{4} -0.106963 q^{5} +3.06826 q^{6} +0.667572 q^{7} -1.00000 q^{8} +6.41420 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.06826 q^{3} +1.00000 q^{4} -0.106963 q^{5} +3.06826 q^{6} +0.667572 q^{7} -1.00000 q^{8} +6.41420 q^{9} +0.106963 q^{10} -3.81825 q^{11} -3.06826 q^{12} -2.35476 q^{13} -0.667572 q^{14} +0.328189 q^{15} +1.00000 q^{16} -7.95463 q^{17} -6.41420 q^{18} -0.927118 q^{19} -0.106963 q^{20} -2.04828 q^{21} +3.81825 q^{22} +2.46501 q^{23} +3.06826 q^{24} -4.98856 q^{25} +2.35476 q^{26} -10.4756 q^{27} +0.667572 q^{28} +2.73627 q^{29} -0.328189 q^{30} +0.338431 q^{31} -1.00000 q^{32} +11.7154 q^{33} +7.95463 q^{34} -0.0714054 q^{35} +6.41420 q^{36} -3.73972 q^{37} +0.927118 q^{38} +7.22500 q^{39} +0.106963 q^{40} +4.79754 q^{41} +2.04828 q^{42} -6.21046 q^{43} -3.81825 q^{44} -0.686081 q^{45} -2.46501 q^{46} -6.54563 q^{47} -3.06826 q^{48} -6.55435 q^{49} +4.98856 q^{50} +24.4069 q^{51} -2.35476 q^{52} +13.6546 q^{53} +10.4756 q^{54} +0.408411 q^{55} -0.667572 q^{56} +2.84464 q^{57} -2.73627 q^{58} -9.98584 q^{59} +0.328189 q^{60} +11.3728 q^{61} -0.338431 q^{62} +4.28194 q^{63} +1.00000 q^{64} +0.251871 q^{65} -11.7154 q^{66} -6.58187 q^{67} -7.95463 q^{68} -7.56328 q^{69} +0.0714054 q^{70} +3.82923 q^{71} -6.41420 q^{72} -12.5467 q^{73} +3.73972 q^{74} +15.3062 q^{75} -0.927118 q^{76} -2.54896 q^{77} -7.22500 q^{78} -17.3203 q^{79} -0.106963 q^{80} +12.8994 q^{81} -4.79754 q^{82} +14.6930 q^{83} -2.04828 q^{84} +0.850850 q^{85} +6.21046 q^{86} -8.39558 q^{87} +3.81825 q^{88} +8.59589 q^{89} +0.686081 q^{90} -1.57197 q^{91} +2.46501 q^{92} -1.03839 q^{93} +6.54563 q^{94} +0.0991671 q^{95} +3.06826 q^{96} -0.720909 q^{97} +6.55435 q^{98} -24.4910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9}+O(q^{10}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9} - 27 q^{10} + 23 q^{11} + 15 q^{13} + 10 q^{14} + 6 q^{15} + 42 q^{16} + 14 q^{17} - 56 q^{18} - 4 q^{19} + 27 q^{20} + 26 q^{21} - 23 q^{22} + 12 q^{23} + 45 q^{25} - 15 q^{26} - 3 q^{27} - 10 q^{28} + 41 q^{29} - 6 q^{30} + 18 q^{31} - 42 q^{32} + 25 q^{33} - 14 q^{34} + 8 q^{35} + 56 q^{36} + 33 q^{37} + 4 q^{38} + 10 q^{39} - 27 q^{40} + 84 q^{41} - 26 q^{42} - 36 q^{43} + 23 q^{44} + 66 q^{45} - 12 q^{46} + 28 q^{47} + 58 q^{49} - 45 q^{50} + 17 q^{51} + 15 q^{52} + 68 q^{53} + 3 q^{54} - 28 q^{55} + 10 q^{56} - 9 q^{57} - 41 q^{58} + 59 q^{59} + 6 q^{60} + 41 q^{61} - 18 q^{62} - 28 q^{63} + 42 q^{64} + 44 q^{65} - 25 q^{66} + 14 q^{68} + 67 q^{69} - 8 q^{70} + 69 q^{71} - 56 q^{72} - 27 q^{73} - 33 q^{74} + 14 q^{75} - 4 q^{76} + 43 q^{77} - 10 q^{78} - 19 q^{79} + 27 q^{80} + 74 q^{81} - 84 q^{82} + 20 q^{83} + 26 q^{84} + 16 q^{85} + 36 q^{86} - 28 q^{87} - 23 q^{88} + 123 q^{89} - 66 q^{90} - 9 q^{91} + 12 q^{92} + 48 q^{93} - 28 q^{94} + 28 q^{95} + 10 q^{97} - 58 q^{98} + 75 q^{99}+O(q^{100}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9 - 27 * q^10 + 23 * q^11 + 15 * q^13 + 10 * q^14 + 6 * q^15 + 42 * q^16 + 14 * q^17 - 56 * q^18 - 4 * q^19 + 27 * q^20 + 26 * q^21 - 23 * q^22 + 12 * q^23 + 45 * q^25 - 15 * q^26 - 3 * q^27 - 10 * q^28 + 41 * q^29 - 6 * q^30 + 18 * q^31 - 42 * q^32 + 25 * q^33 - 14 * q^34 + 8 * q^35 + 56 * q^36 + 33 * q^37 + 4 * q^38 + 10 * q^39 - 27 * q^40 + 84 * q^41 - 26 * q^42 - 36 * q^43 + 23 * q^44 + 66 * q^45 - 12 * q^46 + 28 * q^47 + 58 * q^49 - 45 * q^50 + 17 * q^51 + 15 * q^52 + 68 * q^53 + 3 * q^54 - 28 * q^55 + 10 * q^56 - 9 * q^57 - 41 * q^58 + 59 * q^59 + 6 * q^60 + 41 * q^61 - 18 * q^62 - 28 * q^63 + 42 * q^64 + 44 * q^65 - 25 * q^66 + 14 * q^68 + 67 * q^69 - 8 * q^70 + 69 * q^71 - 56 * q^72 - 27 * q^73 - 33 * q^74 + 14 * q^75 - 4 * q^76 + 43 * q^77 - 10 * q^78 - 19 * q^79 + 27 * q^80 + 74 * q^81 - 84 * q^82 + 20 * q^83 + 26 * q^84 + 16 * q^85 + 36 * q^86 - 28 * q^87 - 23 * q^88 + 123 * q^89 - 66 * q^90 - 9 * q^91 + 12 * q^92 + 48 * q^93 - 28 * q^94 + 28 * q^95 + 10 * q^97 - 58 * q^98 + 75 * q^99

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