LMFDB - Embedded newform 2001.2.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,2,Mod(1,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2001.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:	$15.9780654445$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 $ x^20 - 2x^19 - 33x^18 + 64x^17 + 453x^16 - 846x^15 - 3353x^14 + 5985x^13 + 14484x^12 - 24566x^11 - 36791x^10 + 59410x^9 + 52109x^8 - 82362x^7 - 34967x^6 + 60661x^5 + 5201x^4 - 19624x^3 + 2768x^2 + 1408*x - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.97506$ of defining polynomial
Character	$\chi$	$=$	2001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.97506 q^{2} +1.00000 q^{3} +1.90086 q^{4} -3.88400 q^{5} -1.97506 q^{6} +2.64382 q^{7} +0.195801 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.97506 q^{2} +1.00000 q^{3} +1.90086 q^{4} -3.88400 q^{5} -1.97506 q^{6} +2.64382 q^{7} +0.195801 q^{8} +1.00000 q^{9} +7.67114 q^{10} +4.76488 q^{11} +1.90086 q^{12} +3.33092 q^{13} -5.22171 q^{14} -3.88400 q^{15} -4.18845 q^{16} -7.28945 q^{17} -1.97506 q^{18} +5.43957 q^{19} -7.38296 q^{20} +2.64382 q^{21} -9.41093 q^{22} +1.00000 q^{23} +0.195801 q^{24} +10.0855 q^{25} -6.57877 q^{26} +1.00000 q^{27} +5.02554 q^{28} +1.00000 q^{29} +7.67114 q^{30} +6.87216 q^{31} +7.88083 q^{32} +4.76488 q^{33} +14.3971 q^{34} -10.2686 q^{35} +1.90086 q^{36} -9.05053 q^{37} -10.7435 q^{38} +3.33092 q^{39} -0.760490 q^{40} +5.39284 q^{41} -5.22171 q^{42} -10.4237 q^{43} +9.05739 q^{44} -3.88400 q^{45} -1.97506 q^{46} +8.49635 q^{47} -4.18845 q^{48} -0.0102125 q^{49} -19.9194 q^{50} -7.28945 q^{51} +6.33163 q^{52} -7.62772 q^{53} -1.97506 q^{54} -18.5068 q^{55} +0.517662 q^{56} +5.43957 q^{57} -1.97506 q^{58} -10.2552 q^{59} -7.38296 q^{60} +0.452237 q^{61} -13.5729 q^{62} +2.64382 q^{63} -7.18823 q^{64} -12.9373 q^{65} -9.41093 q^{66} +12.0887 q^{67} -13.8562 q^{68} +1.00000 q^{69} +20.2811 q^{70} +12.0747 q^{71} +0.195801 q^{72} +6.12294 q^{73} +17.8753 q^{74} +10.0855 q^{75} +10.3399 q^{76} +12.5975 q^{77} -6.57877 q^{78} -9.83566 q^{79} +16.2679 q^{80} +1.00000 q^{81} -10.6512 q^{82} -10.0824 q^{83} +5.02554 q^{84} +28.3122 q^{85} +20.5874 q^{86} +1.00000 q^{87} +0.932967 q^{88} -0.490243 q^{89} +7.67114 q^{90} +8.80636 q^{91} +1.90086 q^{92} +6.87216 q^{93} -16.7808 q^{94} -21.1273 q^{95} +7.88083 q^{96} -0.789164 q^{97} +0.0201703 q^{98} +4.76488 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9}+O(q^{10}) $ 20 * q + 2 * q^2 + 20 * q^3 + 30 * q^4 - q^5 + 2 * q^6 + 9 * q^7 + 6 * q^8 + 20 * q^9	\( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9} + 7 q^{10} + 30 q^{12} + 21 q^{13} - q^{14} - q^{15} + 58 q^{16} - 4 q^{17} + 2 q^{18} + 7 q^{19} - 20 q^{20} + 9 q^{21} + 7 q^{22} + 20 q^{23} + 6 q^{24} + 47 q^{25} + 8 q^{26} + 20 q^{27} + 11 q^{28} + 20 q^{29} + 7 q^{30} + 28 q^{31} + 14 q^{32} + 16 q^{34} + 9 q^{35} + 30 q^{36} + 14 q^{37} - 20 q^{38} + 21 q^{39} + 34 q^{40} + 7 q^{41} - q^{42} + 3 q^{43} - q^{44} - q^{45} + 2 q^{46} + 3 q^{47} + 58 q^{48} + 35 q^{49} - 24 q^{50} - 4 q^{51} + 73 q^{52} - 19 q^{53} + 2 q^{54} + 29 q^{55} - 30 q^{56} + 7 q^{57} + 2 q^{58} + 20 q^{59} - 20 q^{60} + 15 q^{61} + 12 q^{62} + 9 q^{63} + 82 q^{64} - 28 q^{65} + 7 q^{66} + 20 q^{67} - 23 q^{68} + 20 q^{69} - 24 q^{70} + 63 q^{71} + 6 q^{72} + 19 q^{73} + 16 q^{74} + 47 q^{75} - 44 q^{76} - 7 q^{77} + 8 q^{78} + 32 q^{79} - 56 q^{80} + 20 q^{81} - 20 q^{82} - 21 q^{83} + 11 q^{84} + 4 q^{85} - 6 q^{86} + 20 q^{87} + 55 q^{88} - 13 q^{89} + 7 q^{90} + 70 q^{91} + 30 q^{92} + 28 q^{93} - 12 q^{94} + 9 q^{95} + 14 q^{96} - 9 q^{97} + 31 q^{98}+O(q^{100}) \) 20 * q + 2 * q^2 + 20 * q^3 + 30 * q^4 - q^5 + 2 * q^6 + 9 * q^7 + 6 * q^8 + 20 * q^9 + 7 * q^10 + 30 * q^12 + 21 * q^13 - q^14 - q^15 + 58 * q^16 - 4 * q^17 + 2 * q^18 + 7 * q^19 - 20 * q^20 + 9 * q^21 + 7 * q^22 + 20 * q^23 + 6 * q^24 + 47 * q^25 + 8 * q^26 + 20 * q^27 + 11 * q^28 + 20 * q^29 + 7 * q^30 + 28 * q^31 + 14 * q^32 + 16 * q^34 + 9 * q^35 + 30 * q^36 + 14 * q^37 - 20 * q^38 + 21 * q^39 + 34 * q^40 + 7 * q^41 - q^42 + 3 * q^43 - q^44 - q^45 + 2 * q^46 + 3 * q^47 + 58 * q^48 + 35 * q^49 - 24 * q^50 - 4 * q^51 + 73 * q^52 - 19 * q^53 + 2 * q^54 + 29 * q^55 - 30 * q^56 + 7 * q^57 + 2 * q^58 + 20 * q^59 - 20 * q^60 + 15 * q^61 + 12 * q^62 + 9 * q^63 + 82 * q^64 - 28 * q^65 + 7 * q^66 + 20 * q^67 - 23 * q^68 + 20 * q^69 - 24 * q^70 + 63 * q^71 + 6 * q^72 + 19 * q^73 + 16 * q^74 + 47 * q^75 - 44 * q^76 - 7 * q^77 + 8 * q^78 + 32 * q^79 - 56 * q^80 + 20 * q^81 - 20 * q^82 - 21 * q^83 + 11 * q^84 + 4 * q^85 - 6 * q^86 + 20 * q^87 + 55 * q^88 - 13 * q^89 + 7 * q^90 + 70 * q^91 + 30 * q^92 + 28 * q^93 - 12 * q^94 + 9 * q^95 + 14 * q^96 - 9 * q^97 + 31 * q^98

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 $

$2$	−1.97506	−1.39658	−0.698289	−	0.715816i	$-0.746055\pi$
$2$	−1.97506	−1.39658	−0.698289	+	0.715816i	$0.746055\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.90086	0.950432
$5$	−3.88400	−1.73698	−0.868489	−	0.495708i	$-0.834909\pi$
$5$	−3.88400	−1.73698	−0.868489	+	0.495708i	$0.834909\pi$
$6$	−1.97506	−0.806315
$7$	2.64382	0.999270	0.499635	−	0.866236i	$-0.333467\pi$
$7$	2.64382	0.999270	0.499635	+	0.866236i	$0.333467\pi$
$8$	0.195801	0.0692260
$9$	1.00000	0.333333
$10$	7.67114	2.42583
$11$	4.76488	1.43667	0.718333	−	0.695700i	$-0.244906\pi$
$11$	4.76488	1.43667	0.718333	+	0.695700i	$0.244906\pi$
$12$	1.90086	0.548732
$13$	3.33092	0.923832	0.461916	−	0.886924i	$-0.347162\pi$
$13$	3.33092	0.923832	0.461916	+	0.886924i	$0.347162\pi$
$14$	−5.22171	−1.39556
$15$	−3.88400	−1.00285
$16$	−4.18845	−1.04711
$17$	−7.28945	−1.76795	−0.883975	−	0.467533i	$-0.845143\pi$
$17$	−7.28945	−1.76795	−0.883975	+	0.467533i	$0.845143\pi$
$18$	−1.97506	−0.465526
$19$	5.43957	1.24792	0.623962	−	0.781455i	$-0.285522\pi$
$19$	5.43957	1.24792	0.623962	+	0.781455i	$0.285522\pi$
$20$	−7.38296	−1.65088
$21$	2.64382	0.576929
$22$	−9.41093	−2.00642
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0.195801	0.0399676
$25$	10.0855	2.01710
$26$	−6.57877	−1.29020
$27$	1.00000	0.192450
$28$	5.02554	0.949738
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	7.67114	1.40055
$31$	6.87216	1.23428	0.617138	−	0.786855i	$-0.288292\pi$
$31$	6.87216	1.23428	0.617138	+	0.786855i	$0.288292\pi$
$32$	7.88083	1.39315
$33$	4.76488	0.829459
$34$	14.3971	2.46908
$35$	−10.2686	−1.73571
$36$	1.90086	0.316811
$37$	−9.05053	−1.48790	−0.743949	−	0.668236i	$-0.767050\pi$
$37$	−9.05053	−1.48790	−0.743949	+	0.668236i	$0.767050\pi$
$38$	−10.7435	−1.74282
$39$	3.33092	0.533374
$40$	−0.760490	−0.120244
$41$	5.39284	0.842220	0.421110	−	0.907010i	$-0.361641\pi$
$41$	5.39284	0.842220	0.421110	+	0.907010i	$0.361641\pi$
$42$	−5.22171	−0.805727
$43$	−10.4237	−1.58960	−0.794799	−	0.606873i	$-0.792424\pi$
$43$	−10.4237	−1.58960	−0.794799	+	0.606873i	$0.792424\pi$
$44$	9.05739	1.36545
$45$	−3.88400	−0.578993
$46$	−1.97506	−0.291207
$47$	8.49635	1.23932	0.619660	−	0.784870i	$-0.287270\pi$
$47$	8.49635	1.23932	0.619660	+	0.784870i	$0.287270\pi$
$48$	−4.18845	−0.604550
$49$	−0.0102125	−0.00145893
$50$	−19.9194	−2.81703
$51$	−7.28945	−1.02073
$52$	6.33163	0.878039
$53$	−7.62772	−1.04775	−0.523874	−	0.851796i	$-0.675514\pi$
$53$	−7.62772	−1.04775	−0.523874	+	0.851796i	$0.675514\pi$
$54$	−1.97506	−0.268772
$55$	−18.5068	−2.49546
$56$	0.517662	0.0691755
$57$	5.43957	0.720489
$58$	−1.97506	−0.259338
$59$	−10.2552	−1.33511	−0.667554	−	0.744562i	$-0.732659\pi$
$59$	−10.2552	−1.33511	−0.667554	+	0.744562i	$0.732659\pi$
$60$	−7.38296	−0.953136
$61$	0.452237	0.0579030	0.0289515	−	0.999581i	$-0.490783\pi$
$61$	0.452237	0.0579030	0.0289515	+	0.999581i	$0.490783\pi$
$62$	−13.5729	−1.72376
$63$	2.64382	0.333090
$64$	−7.18823	−0.898528
$65$	−12.9373	−1.60468
$66$	−9.41093	−1.15841
$67$	12.0887	1.47687	0.738435	−	0.674325i	$-0.235565\pi$
$67$	12.0887	1.47687	0.738435	+	0.674325i	$0.235565\pi$
$68$	−13.8562	−1.68032
$69$	1.00000	0.120386
$70$	20.2811	2.42406
$71$	12.0747	1.43300	0.716501	−	0.697586i	$-0.245742\pi$
$71$	12.0747	1.43300	0.716501	+	0.697586i	$0.245742\pi$
$72$	0.195801	0.0230753
$73$	6.12294	0.716636	0.358318	−	0.933600i	$-0.383350\pi$
$73$	6.12294	0.716636	0.358318	+	0.933600i	$0.383350\pi$
$74$	17.8753	2.07797
$75$	10.0855	1.16457
$76$	10.3399	1.18607
$77$	12.5975	1.43562
$78$	−6.57877	−0.744899
$79$	−9.83566	−1.10660	−0.553299	−	0.832983i	$-0.686631\pi$
$79$	−9.83566	−1.10660	−0.553299	+	0.832983i	$0.686631\pi$
$80$	16.2679	1.81881
$81$	1.00000	0.111111
$82$	−10.6512	−1.17623
$83$	−10.0824	−1.10668	−0.553341	−	0.832955i	$-0.686648\pi$
$83$	−10.0824	−1.10668	−0.553341	+	0.832955i	$0.686648\pi$
$84$	5.02554	0.548332
$85$	28.3122	3.07089
$86$	20.5874	2.22000
$87$	1.00000	0.107211
$88$	0.932967	0.0994546
$89$	−0.490243	−0.0519657	−0.0259828	−	0.999662i	$-0.508272\pi$
$89$	−0.490243	−0.0519657	−0.0259828	+	0.999662i	$0.508272\pi$
$90$	7.67114	0.808609
$91$	8.80636	0.923157
$92$	1.90086	0.198179
$93$	6.87216	0.712609
$94$	−16.7808	−1.73081
$95$	−21.1273	−2.16762
$96$	7.88083	0.804334
$97$	−0.789164	−0.0801274	−0.0400637	−	0.999197i	$-0.512756\pi$
$97$	−0.789164	−0.0801274	−0.0400637	+	0.999197i	$0.512756\pi$
$98$	0.0201703	0.00203751
$99$	4.76488	0.478889
$100$	19.1711	1.91711
$101$	5.14905	0.512350	0.256175	−	0.966630i	$-0.417538\pi$
$101$	5.14905	0.512350	0.256175	+	0.966630i	$0.417538\pi$
$102$	14.3971	1.42553
$103$	−4.12758	−0.406702	−0.203351	−	0.979106i	$-0.565183\pi$
$103$	−4.12758	−0.406702	−0.203351	+	0.979106i	$0.565183\pi$
$104$	0.652197	0.0639532
$105$	−10.2686	−1.00211
$106$	15.0652	1.46326
$107$	11.9691	1.15710	0.578549	−	0.815648i	$-0.303619\pi$
$107$	11.9691	1.15710	0.578549	+	0.815648i	$0.303619\pi$
$108$	1.90086	0.182911
$109$	13.1260	1.25724	0.628622	−	0.777711i	$-0.283619\pi$
$109$	13.1260	1.25724	0.628622	+	0.777711i	$0.283619\pi$
$110$	36.5521	3.48510
$111$	−9.05053	−0.859038
$112$	−11.0735	−1.04635
$113$	−2.19956	−0.206917	−0.103459	−	0.994634i	$-0.532991\pi$
$113$	−2.19956	−0.206917	−0.103459	+	0.994634i	$0.532991\pi$
$114$	−10.7435	−1.00622
$115$	−3.88400	−0.362185
$116$	1.90086	0.176491
$117$	3.33092	0.307944
$118$	20.2545	1.86458
$119$	−19.2720	−1.76666
$120$	−0.760490	−0.0694230
$121$	11.7041	1.06401
$122$	−0.893196	−0.0808661
$123$	5.39284	0.486256
$124$	13.0630	1.17309
$125$	−19.7520	−1.76667
$126$	−5.22171	−0.465186
$127$	11.8159	1.04849	0.524245	−	0.851567i	$-0.324347\pi$
$127$	11.8159	1.04849	0.524245	+	0.851567i	$0.324347\pi$
$128$	−1.56448	−0.138282
$129$	−10.4237	−0.917754
$130$	25.5520	2.24106
$131$	13.0941	1.14403	0.572017	−	0.820242i	$-0.306161\pi$
$131$	13.0941	1.14403	0.572017	+	0.820242i	$0.306161\pi$
$132$	9.05739	0.788344
$133$	14.3813	1.24701
$134$	−23.8759	−2.06256
$135$	−3.88400	−0.334282
$136$	−1.42728	−0.122388
$137$	7.92555	0.677126	0.338563	−	0.940944i	$-0.390059\pi$
$137$	7.92555	0.677126	0.338563	+	0.940944i	$0.390059\pi$
$138$	−1.97506	−0.168128
$139$	3.89433	0.330313	0.165157	−	0.986267i	$-0.447187\pi$
$139$	3.89433	0.330313	0.165157	+	0.986267i	$0.447187\pi$
$140$	−19.5192	−1.64968
$141$	8.49635	0.715522
$142$	−23.8482	−2.00130
$143$	15.8714	1.32724
$144$	−4.18845	−0.349037
$145$	−3.88400	−0.322549
$146$	−12.0932	−1.00084
$147$	−0.0102125	−0.000842312 0
$148$	−17.2038	−1.41415
$149$	5.40852	0.443083	0.221542	−	0.975151i	$-0.428891\pi$
$149$	5.40852	0.443083	0.221542	+	0.975151i	$0.428891\pi$
$150$	−19.9194	−1.62641
$151$	9.81356	0.798616	0.399308	−	0.916817i	$-0.369250\pi$
$151$	9.81356	0.798616	0.399308	+	0.916817i	$0.369250\pi$
$152$	1.06507	0.0863887
$153$	−7.28945	−0.589317
$154$	−24.8808	−2.00495
$155$	−26.6915	−2.14391
$156$	6.33163	0.506936
$157$	17.1417	1.36806	0.684030	−	0.729454i	$-0.260226\pi$
$157$	17.1417	1.36806	0.684030	+	0.729454i	$0.260226\pi$
$158$	19.4260	1.54545
$159$	−7.62772	−0.604917
$160$	−30.6092	−2.41987
$161$	2.64382	0.208362
$162$	−1.97506	−0.155175
$163$	−9.30053	−0.728474	−0.364237	−	0.931306i	$-0.618670\pi$
$163$	−9.30053	−0.728474	−0.364237	+	0.931306i	$0.618670\pi$
$164$	10.2510	0.800472
$165$	−18.5068	−1.44075
$166$	19.9133	1.54557
$167$	−16.9187	−1.30921	−0.654603	−	0.755972i	$-0.727164\pi$
$167$	−16.9187	−1.30921	−0.654603	+	0.755972i	$0.727164\pi$
$168$	0.517662	0.0399385
$169$	−1.90496	−0.146535
$170$	−55.9184	−4.28874
$171$	5.43957	0.415974
$172$	−19.8140	−1.51080
$173$	3.35028	0.254717	0.127359	−	0.991857i	$-0.459350\pi$
$173$	3.35028	0.254717	0.127359	+	0.991857i	$0.459350\pi$
$174$	−1.97506	−0.149729
$175$	26.6642	2.01562
$176$	−19.9574	−1.50435
$177$	−10.2552	−0.770825
$178$	0.968260	0.0725742
$179$	9.84156	0.735593	0.367796	−	0.929906i	$-0.380112\pi$
$179$	9.84156	0.735593	0.367796	+	0.929906i	$0.380112\pi$
$180$	−7.38296	−0.550293
$181$	−3.83355	−0.284945	−0.142473	−	0.989799i	$-0.545505\pi$
$181$	−3.83355	−0.284945	−0.142473	+	0.989799i	$0.545505\pi$
$182$	−17.3931	−1.28926
$183$	0.452237	0.0334303
$184$	0.195801	0.0144346
$185$	35.1523	2.58445
$186$	−13.5729	−0.995215
$187$	−34.7333	−2.53995
$188$	16.1504	1.17789
$189$	2.64382	0.192310
$190$	41.7277	3.02725
$191$	25.4250	1.83969	0.919843	−	0.392287i	$-0.128316\pi$
$191$	25.4250	1.83969	0.919843	+	0.392287i	$0.128316\pi$
$192$	−7.18823	−0.518766
$193$	24.0004	1.72759	0.863795	−	0.503844i	$-0.168081\pi$
$193$	24.0004	1.72759	0.863795	+	0.503844i	$0.168081\pi$
$194$	1.55865	0.111904
$195$	−12.9373	−0.926460
$196$	−0.0194126	−0.00138661
$197$	23.4966	1.67406	0.837032	−	0.547154i	$-0.184289\pi$
$197$	23.4966	1.67406	0.837032	+	0.547154i	$0.184289\pi$
$198$	−9.41093	−0.668805
$199$	−6.12765	−0.434378	−0.217189	−	0.976130i	$-0.569689\pi$
$199$	−6.12765	−0.434378	−0.217189	+	0.976130i	$0.569689\pi$
$200$	1.97474	0.139635
$201$	12.0887	0.852671
$202$	−10.1697	−0.715536
$203$	2.64382	0.185560
$204$	−13.8562	−0.970131
$205$	−20.9458	−1.46292
$206$	8.15222	0.567992
$207$	1.00000	0.0695048
$208$	−13.9514	−0.967354
$209$	25.9189	1.79285
$210$	20.2811	1.39953
$211$	−7.19073	−0.495030	−0.247515	−	0.968884i	$-0.579614\pi$
$211$	−7.19073	−0.495030	−0.247515	+	0.968884i	$0.579614\pi$
$212$	−14.4992	−0.995812
$213$	12.0747	0.827344
$214$	−23.6397	−1.61598
$215$	40.4856	2.76110
$216$	0.195801	0.0133225
$217$	18.1688	1.23338
$218$	−25.9247	−1.75584
$219$	6.12294	0.413750
$220$	−35.1789	−2.37176
$221$	−24.2806	−1.63329
$222$	17.8753	1.19971
$223$	−8.86810	−0.593852	−0.296926	−	0.954900i	$-0.595961\pi$
$223$	−8.86810	−0.593852	−0.296926	+	0.954900i	$0.595961\pi$
$224$	20.8355	1.39213
$225$	10.0855	0.672365
$226$	4.34427	0.288976
$227$	10.6725	0.708359	0.354180	−	0.935177i	$-0.384760\pi$
$227$	10.6725	0.708359	0.354180	+	0.935177i	$0.384760\pi$
$228$	10.3399	0.684776
$229$	4.91406	0.324730	0.162365	−	0.986731i	$-0.448088\pi$
$229$	4.91406	0.324730	0.162365	+	0.986731i	$0.448088\pi$
$230$	7.67114	0.505820
$231$	12.5975	0.828854
$232$	0.195801	0.0128549
$233$	−22.8105	−1.49436	−0.747182	−	0.664619i	$-0.768594\pi$
$233$	−22.8105	−1.49436	−0.747182	+	0.664619i	$0.768594\pi$
$234$	−6.57877	−0.430068
$235$	−32.9998	−2.15267
$236$	−19.4936	−1.26893
$237$	−9.83566	−0.638895
$238$	38.0633	2.46728
$239$	−22.1209	−1.43088	−0.715441	−	0.698673i	$-0.753774\pi$
$239$	−22.1209	−1.43088	−0.715441	+	0.698673i	$0.753774\pi$
$240$	16.2679	1.05009
$241$	−5.77198	−0.371806	−0.185903	−	0.982568i	$-0.559521\pi$
$241$	−5.77198	−0.371806	−0.185903	+	0.982568i	$0.559521\pi$
$242$	−23.1163	−1.48597
$243$	1.00000	0.0641500
$244$	0.859641	0.0550329
$245$	0.0396654	0.00253413
$246$	−10.6512	−0.679094
$247$	18.1188	1.15287
$248$	1.34557	0.0854440
$249$	−10.0824	−0.638944
$250$	39.0114	2.46730
$251$	−18.2189	−1.14997	−0.574984	−	0.818165i	$-0.694992\pi$
$251$	−18.2189	−1.14997	−0.574984	+	0.818165i	$0.694992\pi$
$252$	5.02554	0.316579
$253$	4.76488	0.299565
$254$	−23.3371	−1.46430
$255$	28.3122	1.77298
$256$	17.4664	1.09165
$257$	6.94804	0.433407	0.216703	−	0.976237i	$-0.430470\pi$
$257$	6.94804	0.433407	0.216703	+	0.976237i	$0.430470\pi$
$258$	20.5874	1.28172
$259$	−23.9280	−1.48681
$260$	−24.5921	−1.52513
$261$	1.00000	0.0618984
$262$	−25.8616	−1.59773
$263$	7.20446	0.444246	0.222123	−	0.975019i	$-0.428701\pi$
$263$	7.20446	0.444246	0.222123	+	0.975019i	$0.428701\pi$
$264$	0.932967	0.0574201
$265$	29.6261	1.81992
$266$	−28.4038	−1.74155
$267$	−0.490243	−0.0300024
$268$	22.9790	1.40366
$269$	−4.24647	−0.258912	−0.129456	−	0.991585i	$-0.541323\pi$
$269$	−4.24647	−0.258912	−0.129456	+	0.991585i	$0.541323\pi$
$270$	7.67114	0.466851
$271$	19.9356	1.21100	0.605501	−	0.795845i	$-0.292973\pi$
$271$	19.9356	1.21100	0.605501	+	0.795845i	$0.292973\pi$
$272$	30.5315	1.85124
$273$	8.80636	0.532985
$274$	−15.6534	−0.945659
$275$	48.0561	2.89789
$276$	1.90086	0.114419
$277$	−2.38195	−0.143117	−0.0715586	−	0.997436i	$-0.522797\pi$
$277$	−2.38195	−0.143117	−0.0715586	+	0.997436i	$0.522797\pi$
$278$	−7.69154	−0.461308
$279$	6.87216	0.411425
$280$	−2.01060	−0.120156
$281$	13.4801	0.804155	0.402078	−	0.915606i	$-0.368288\pi$
$281$	13.4801	0.804155	0.402078	+	0.915606i	$0.368288\pi$
$282$	−16.7808	−0.999282
$283$	7.65647	0.455130	0.227565	−	0.973763i	$-0.426924\pi$
$283$	7.65647	0.455130	0.227565	+	0.973763i	$0.426924\pi$
$284$	22.9523	1.36197
$285$	−21.1273	−1.25147
$286$	−31.3471	−1.85359
$287$	14.2577	0.841605
$288$	7.88083	0.464382
$289$	36.1361	2.12565
$290$	7.67114	0.450465
$291$	−0.789164	−0.0462616
$292$	11.6389	0.681113
$293$	25.5856	1.49473	0.747364	−	0.664415i	$-0.231319\pi$
$293$	25.5856	1.49473	0.747364	+	0.664415i	$0.231319\pi$
$294$	0.0201703	0.00117636
$295$	39.8310	2.31905
$296$	−1.77210	−0.103001
$297$	4.76488	0.276486
$298$	−10.6821	−0.618800
$299$	3.33092	0.192632
$300$	19.1711	1.10684
$301$	−27.5584	−1.58844
$302$	−19.3824	−1.11533
$303$	5.14905	0.295805
$304$	−22.7833	−1.30671
$305$	−1.75649	−0.100576
$306$	14.3971	0.823027
$307$	−9.19667	−0.524882	−0.262441	−	0.964948i	$-0.584527\pi$
$307$	−9.19667	−0.524882	−0.262441	+	0.964948i	$0.584527\pi$
$308$	23.9461	1.36446
$309$	−4.12758	−0.234810
$310$	52.7173	2.99414
$311$	−10.3312	−0.585831	−0.292916	−	0.956138i	$-0.594626\pi$
$311$	−10.3312	−0.585831	−0.292916	+	0.956138i	$0.594626\pi$
$312$	0.652197	0.0369234
$313$	−17.7542	−1.00353	−0.501763	−	0.865005i	$-0.667315\pi$
$313$	−17.7542	−1.00353	−0.501763	+	0.865005i	$0.667315\pi$
$314$	−33.8560	−1.91060
$315$	−10.2686	−0.578570
$316$	−18.6962	−1.05175
$317$	2.01941	0.113421	0.0567106	−	0.998391i	$-0.481939\pi$
$317$	2.01941	0.113421	0.0567106	+	0.998391i	$0.481939\pi$
$318$	15.0652	0.844815
$319$	4.76488	0.266782
$320$	27.9191	1.56072
$321$	11.9691	0.668051
$322$	−5.22171	−0.290994
$323$	−39.6515	−2.20627
$324$	1.90086	0.105604
$325$	33.5939	1.86346
$326$	18.3691	1.01737
$327$	13.1260	0.725870
$328$	1.05592	0.0583035
$329$	22.4628	1.23842
$330$	36.5521	2.01212
$331$	−19.2584	−1.05854	−0.529268	−	0.848455i	$-0.677533\pi$
$331$	−19.2584	−1.05854	−0.529268	+	0.848455i	$0.677533\pi$
$332$	−19.1652	−1.05183
$333$	−9.05053	−0.495966
$334$	33.4154	1.82841
$335$	−46.9525	−2.56529
$336$	−11.0735	−0.604109
$337$	1.62764	0.0886634	0.0443317	−	0.999017i	$-0.485884\pi$
$337$	1.62764	0.0886634	0.0443317	+	0.999017i	$0.485884\pi$
$338$	3.76241	0.204648
$339$	−2.19956	−0.119464
$340$	53.8177	2.91867
$341$	32.7450	1.77324
$342$	−10.7435	−0.580941
$343$	−18.5337	−1.00073
$344$	−2.04096	−0.110041
$345$	−3.88400	−0.209108
$346$	−6.61700	−0.355732
$347$	24.2893	1.30392	0.651959	−	0.758254i	$-0.273947\pi$
$347$	24.2893	1.30392	0.651959	+	0.758254i	$0.273947\pi$
$348$	1.90086	0.101897
$349$	0.452922	0.0242443	0.0121222	−	0.999927i	$-0.496141\pi$
$349$	0.452922	0.0242443	0.0121222	+	0.999927i	$0.496141\pi$
$350$	−52.6634	−2.81498
$351$	3.33092	0.177791
$352$	37.5512	2.00149
$353$	−2.13217	−0.113484	−0.0567421	−	0.998389i	$-0.518071\pi$
$353$	−2.13217	−0.113484	−0.0567421	+	0.998389i	$0.518071\pi$
$354$	20.2545	1.07652
$355$	−46.8981	−2.48909
$356$	−0.931886	−0.0493898
$357$	−19.2720	−1.01998
$358$	−19.4377	−1.02731
$359$	4.24403	0.223991	0.111996	−	0.993709i	$-0.464276\pi$
$359$	4.24403	0.223991	0.111996	+	0.993709i	$0.464276\pi$
$360$	−0.760490	−0.0400814
$361$	10.5889	0.557313
$362$	7.57148	0.397948
$363$	11.7041	0.614305
$364$	16.7397	0.877398
$365$	−23.7815	−1.24478
$366$	−0.893196	−0.0466881
$367$	−18.1283	−0.946288	−0.473144	−	0.880985i	$-0.656881\pi$
$367$	−18.1283	−0.946288	−0.473144	+	0.880985i	$0.656881\pi$
$368$	−4.18845	−0.218338
$369$	5.39284	0.280740
$370$	−69.4279	−3.60938
$371$	−20.1663	−1.04698
$372$	13.0630	0.677287
$373$	7.46611	0.386581	0.193290	−	0.981142i	$-0.438084\pi$
$373$	7.46611	0.386581	0.193290	+	0.981142i	$0.438084\pi$
$374$	68.6005	3.54725
$375$	−19.7520	−1.01999
$376$	1.66359	0.0857932
$377$	3.33092	0.171551
$378$	−5.22171	−0.268576
$379$	−30.6908	−1.57648	−0.788239	−	0.615369i	$-0.789007\pi$
$379$	−30.6908	−1.57648	−0.788239	+	0.615369i	$0.789007\pi$
$380$	−40.1601	−2.06017
$381$	11.8159	0.605346
$382$	−50.2158	−2.56927
$383$	−16.3891	−0.837442	−0.418721	−	0.908115i	$-0.637521\pi$
$383$	−16.3891	−0.837442	−0.418721	+	0.908115i	$0.637521\pi$
$384$	−1.56448	−0.0798371
$385$	−48.9287	−2.49364
$386$	−47.4023	−2.41271
$387$	−10.4237	−0.529866
$388$	−1.50009	−0.0761556
$389$	0.876026	0.0444163	0.0222081	−	0.999753i	$-0.492930\pi$
$389$	0.876026	0.0444163	0.0222081	+	0.999753i	$0.492930\pi$
$390$	25.5520	1.29387
$391$	−7.28945	−0.368643
$392$	−0.00199961	−0.000100996 0
$393$	13.0941	0.660508
$394$	−46.4072	−2.33796
$395$	38.2017	1.92214
$396$	9.05739	0.455151
$397$	6.14689	0.308503	0.154252	−	0.988032i	$-0.450703\pi$
$397$	6.14689	0.308503	0.154252	+	0.988032i	$0.450703\pi$
$398$	12.1025	0.606643
$399$	14.3813	0.719963
$400$	−42.2425	−2.11212
$401$	8.06278	0.402636	0.201318	−	0.979526i	$-0.435477\pi$
$401$	8.06278	0.402636	0.201318	+	0.979526i	$0.435477\pi$
$402$	−23.8759	−1.19082
$403$	22.8906	1.14026
$404$	9.78764	0.486953
$405$	−3.88400	−0.192998
$406$	−5.22171	−0.259149
$407$	−43.1247	−2.13761
$408$	−1.42728	−0.0706608
$409$	3.21546	0.158994	0.0794970	−	0.996835i	$-0.474669\pi$
$409$	3.21546	0.158994	0.0794970	+	0.996835i	$0.474669\pi$
$410$	41.3692	2.04308
$411$	7.92555	0.390939
$412$	−7.84596	−0.386543
$413$	−27.1128	−1.33413
$414$	−1.97506	−0.0970689
$415$	39.1599	1.92228
$416$	26.2504	1.28703
$417$	3.89433	0.190706
$418$	−51.1914	−2.50385
$419$	29.0993	1.42160	0.710798	−	0.703396i	$-0.248334\pi$
$419$	29.0993	1.42160	0.710798	+	0.703396i	$0.248334\pi$
$420$	−19.5192	−0.952440
$421$	13.9597	0.680352	0.340176	−	0.940362i	$-0.389513\pi$
$421$	13.9597	0.680352	0.340176	+	0.940362i	$0.389513\pi$
$422$	14.2021	0.691348
$423$	8.49635	0.413107
$424$	−1.49351	−0.0725314
$425$	−73.5176	−3.56613
$426$	−23.8482	−1.15545
$427$	1.19563	0.0578608
$428$	22.7516	1.09974
$429$	15.8714	0.766281
$430$	−79.9616	−3.85609
$431$	13.7446	0.662055	0.331028	−	0.943621i	$-0.392605\pi$
$431$	13.7446	0.662055	0.331028	+	0.943621i	$0.392605\pi$
$432$	−4.18845	−0.201517
$433$	−24.8189	−1.19272	−0.596361	−	0.802716i	$-0.703387\pi$
$433$	−24.8189	−1.19272	−0.596361	+	0.802716i	$0.703387\pi$
$434$	−35.8844	−1.72251
$435$	−3.88400	−0.186224
$436$	24.9507	1.19492
$437$	5.43957	0.260210
$438$	−12.0932	−0.577834
$439$	−15.9294	−0.760268	−0.380134	−	0.924932i	$-0.624122\pi$
$439$	−15.9294	−0.760268	−0.380134	+	0.924932i	$0.624122\pi$
$440$	−3.62365	−0.172751
$441$	−0.0102125	−0.000486309 0
$442$	47.9556	2.28102
$443$	−37.1050	−1.76291	−0.881456	−	0.472265i	$-0.843436\pi$
$443$	−37.1050	−1.76291	−0.881456	+	0.472265i	$0.843436\pi$
$444$	−17.2038	−0.816457
$445$	1.90411	0.0902633
$446$	17.5150	0.829361
$447$	5.40852	0.255814
$448$	−19.0044	−0.897873
$449$	−13.0638	−0.616518	−0.308259	−	0.951303i	$-0.599746\pi$
$449$	−13.0638	−0.616518	−0.308259	+	0.951303i	$0.599746\pi$
$450$	−19.9194	−0.939011
$451$	25.6962	1.20999
$452$	−4.18107	−0.196661
$453$	9.81356	0.461081
$454$	−21.0788	−0.989279
$455$	−34.2039	−1.60350
$456$	1.06507	0.0498766
$457$	20.3373	0.951341	0.475670	−	0.879624i	$-0.342205\pi$
$457$	20.3373	0.951341	0.475670	+	0.879624i	$0.342205\pi$
$458$	−9.70557	−0.453512
$459$	−7.28945	−0.340242
$460$	−7.38296	−0.344232
$461$	−25.0320	−1.16585	−0.582927	−	0.812524i	$-0.698093\pi$
$461$	−25.0320	−1.16585	−0.582927	+	0.812524i	$0.698093\pi$
$462$	−24.8808	−1.15756
$463$	−22.0134	−1.02305	−0.511524	−	0.859269i	$-0.670919\pi$
$463$	−22.0134	−1.02305	−0.511524	+	0.859269i	$0.670919\pi$
$464$	−4.18845	−0.194444
$465$	−26.6915	−1.23779
$466$	45.0521	2.08700
$467$	11.1367	0.515345	0.257672	−	0.966232i	$-0.417044\pi$
$467$	11.1367	0.515345	0.257672	+	0.966232i	$0.417044\pi$
$468$	6.33163	0.292680
$469$	31.9603	1.47579
$470$	65.1767	3.00638
$471$	17.1417	0.789850
$472$	−2.00797	−0.0924241
$473$	−49.6676	−2.28372
$474$	19.4260	0.892267
$475$	54.8607	2.51718
$476$	−36.6334	−1.67909
$477$	−7.62772	−0.349249
$478$	43.6901	1.99834
$479$	26.1947	1.19687	0.598433	−	0.801173i	$-0.295790\pi$
$479$	26.1947	1.19687	0.598433	+	0.801173i	$0.295790\pi$
$480$	−30.6092	−1.39711
$481$	−30.1466	−1.37457
$482$	11.4000	0.519256
$483$	2.64382	0.120298
$484$	22.2479	1.01127
$485$	3.06511	0.139180
$486$	−1.97506	−0.0895906
$487$	41.1512	1.86474	0.932369	−	0.361508i	$-0.117738\pi$
$487$	41.1512	1.86474	0.932369	+	0.361508i	$0.117738\pi$
$488$	0.0885483	0.00400839
$489$	−9.30053	−0.420585
$490$	−0.0783415	−0.00353911
$491$	23.0875	1.04192	0.520962	−	0.853580i	$-0.325573\pi$
$491$	23.0875	1.04192	0.520962	+	0.853580i	$0.325573\pi$
$492$	10.2510	0.462153
$493$	−7.28945	−0.328300
$494$	−35.7857	−1.61008
$495$	−18.5068	−0.831819
$496$	−28.7837	−1.29242
$497$	31.9233	1.43196
$498$	19.9133	0.892335
$499$	−0.628061	−0.0281159	−0.0140579	−	0.999901i	$-0.504475\pi$
$499$	−0.628061	−0.0281159	−0.0140579	+	0.999901i	$0.504475\pi$
$500$	−37.5459	−1.67910
$501$	−16.9187	−0.755871
$502$	35.9835	1.60602
$503$	35.0299	1.56190	0.780952	−	0.624591i	$-0.214734\pi$
$503$	35.0299	1.56190	0.780952	+	0.624591i	$0.214734\pi$
$504$	0.517662	0.0230585
$505$	−19.9989	−0.889940
$506$	−9.41093	−0.418367
$507$	−1.90496	−0.0846022
$508$	22.4604	0.996519
$509$	2.65516	0.117688	0.0588439	−	0.998267i	$-0.481259\pi$
$509$	2.65516	0.117688	0.0588439	+	0.998267i	$0.481259\pi$
$510$	−55.9184	−2.47611
$511$	16.1879	0.716113
$512$	−31.3682	−1.38629
$513$	5.43957	0.240163
$514$	−13.7228	−0.605287
$515$	16.0315	0.706433
$516$	−19.8140	−0.872263
$517$	40.4841	1.78049
$518$	47.2592	2.07645
$519$	3.35028	0.147061
$520$	−2.53313	−0.111085
$521$	−8.23178	−0.360641	−0.180321	−	0.983608i	$-0.557713\pi$
$521$	−8.23178	−0.360641	−0.180321	+	0.983608i	$0.557713\pi$
$522$	−1.97506	−0.0864460
$523$	9.52821	0.416640	0.208320	−	0.978061i	$-0.433201\pi$
$523$	9.52821	0.416640	0.208320	+	0.978061i	$0.433201\pi$
$524$	24.8900	1.08733
$525$	26.6642	1.16372
$526$	−14.2293	−0.620425
$527$	−50.0942	−2.18214
$528$	−19.9574	−0.868536
$529$	1.00000	0.0434783
$530$	−58.5133	−2.54165
$531$	−10.2552	−0.445036
$532$	27.3368	1.18520
$533$	17.9631	0.778069
$534$	0.968260	0.0419007
$535$	−46.4881	−2.00985
$536$	2.36697	0.102238
$537$	9.84156	0.424695
$538$	8.38704	0.361591
$539$	−0.0486613	−0.00209599
$540$	−7.38296	−0.317712
$541$	44.6497	1.91964	0.959821	−	0.280612i	$-0.0905375\pi$
$541$	44.6497	1.91964	0.959821	+	0.280612i	$0.0905375\pi$
$542$	−39.3740	−1.69126
$543$	−3.83355	−0.164513
$544$	−57.4469	−2.46302
$545$	−50.9814	−2.18381
$546$	−17.3931	−0.744356
$547$	−35.0805	−1.49993	−0.749967	−	0.661476i	$-0.769931\pi$
$547$	−35.0805	−1.49993	−0.749967	+	0.661476i	$0.769931\pi$
$548$	15.0654	0.643562
$549$	0.452237	0.0193010
$550$	−94.9137	−4.04713
$551$	5.43957	0.231734
$552$	0.195801	0.00833383
$553$	−26.0037	−1.10579
$554$	4.70449	0.199874
$555$	35.1523	1.49213
$556$	7.40260	0.313940
$557$	−23.6013	−1.00002	−0.500009	−	0.866020i	$-0.666670\pi$
$557$	−23.6013	−1.00002	−0.500009	+	0.866020i	$0.666670\pi$
$558$	−13.5729	−0.574588
$559$	−34.7205	−1.46852
$560$	43.0095	1.81748
$561$	−34.7333	−1.46644
$562$	−26.6240	−1.12307
$563$	5.46440	0.230297	0.115149	−	0.993348i	$-0.463266\pi$
$563$	5.46440	0.230297	0.115149	+	0.993348i	$0.463266\pi$
$564$	16.1504	0.680055
$565$	8.54310	0.359411
$566$	−15.1220	−0.635625
$567$	2.64382	0.111030
$568$	2.36423	0.0992010
$569$	18.3374	0.768744	0.384372	−	0.923178i	$-0.374418\pi$
$569$	18.3374	0.768744	0.384372	+	0.923178i	$0.374418\pi$
$570$	41.7277	1.74778
$571$	−44.5394	−1.86391	−0.931957	−	0.362569i	$-0.881900\pi$
$571$	−44.5394	−1.86391	−0.931957	+	0.362569i	$0.881900\pi$
$572$	30.1695	1.26145
$573$	25.4250	1.06214
$574$	−28.1598	−1.17537
$575$	10.0855	0.420593
$576$	−7.18823	−0.299509
$577$	−9.61196	−0.400151	−0.200076	−	0.979780i	$-0.564119\pi$
$577$	−9.61196	−0.400151	−0.200076	+	0.979780i	$0.564119\pi$
$578$	−71.3709	−2.96864
$579$	24.0004	0.997424
$580$	−7.38296	−0.306561
$581$	−26.6560	−1.10588
$582$	1.55865	0.0646079
$583$	−36.3452	−1.50526
$584$	1.19888	0.0496098
$585$	−12.9373	−0.534892
$586$	−50.5332	−2.08751
$587$	−26.4163	−1.09032	−0.545158	−	0.838333i	$-0.683530\pi$
$587$	−26.4163	−1.09032	−0.545158	+	0.838333i	$0.683530\pi$
$588$	−0.0194126	−0.000800560 0
$589$	37.3816	1.54028
$590$	−78.6687	−3.23874
$591$	23.4966	0.966521
$592$	37.9077	1.55800
$593$	−14.0020	−0.574993	−0.287496	−	0.957782i	$-0.592823\pi$
$593$	−14.0020	−0.574993	−0.287496	+	0.957782i	$0.592823\pi$
$594$	−9.41093	−0.386135
$595$	74.8525	3.06865
$596$	10.2809	0.421120
$597$	−6.12765	−0.250788
$598$	−6.57877	−0.269026
$599$	27.8770	1.13902	0.569512	−	0.821983i	$-0.307132\pi$
$599$	27.8770	1.13902	0.569512	+	0.821983i	$0.307132\pi$
$600$	1.97474	0.0806186
$601$	−7.86933	−0.320997	−0.160498	−	0.987036i	$-0.551310\pi$
$601$	−7.86933	−0.320997	−0.160498	+	0.987036i	$0.551310\pi$
$602$	54.4294	2.21838
$603$	12.0887	0.492290
$604$	18.6542	0.759030
$605$	−45.4587	−1.84816
$606$	−10.1697	−0.413115
$607$	−42.6180	−1.72981	−0.864906	−	0.501933i	$-0.832622\pi$
$607$	−42.6180	−1.72981	−0.864906	+	0.501933i	$0.832622\pi$
$608$	42.8683	1.73854
$609$	2.64382	0.107133
$610$	3.46917	0.140463
$611$	28.3007	1.14492
$612$	−13.8562	−0.560106
$613$	−3.58163	−0.144660	−0.0723302	−	0.997381i	$-0.523044\pi$
$613$	−3.58163	−0.144660	−0.0723302	+	0.997381i	$0.523044\pi$
$614$	18.1640	0.733039
$615$	−20.9458	−0.844616
$616$	2.46660	0.0993820
$617$	25.1171	1.01118	0.505589	−	0.862775i	$-0.331275\pi$
$617$	25.1171	1.01118	0.505589	+	0.862775i	$0.331275\pi$
$618$	8.15222	0.327930
$619$	−1.91575	−0.0770004	−0.0385002	−	0.999259i	$-0.512258\pi$
$619$	−1.91575	−0.0770004	−0.0385002	+	0.999259i	$0.512258\pi$
$620$	−50.7369	−2.03764
$621$	1.00000	0.0401286
$622$	20.4048	0.818160
$623$	−1.29612	−0.0519278
$624$	−13.9514	−0.558502
$625$	26.2895	1.05158
$626$	35.0656	1.40150
$627$	25.9189	1.03510
$628$	32.5841	1.30025
$629$	65.9734	2.63053
$630$	20.2811	0.808019
$631$	41.3427	1.64583	0.822913	−	0.568168i	$-0.192348\pi$
$631$	41.3427	1.64583	0.822913	+	0.568168i	$0.192348\pi$
$632$	−1.92583	−0.0766054
$633$	−7.19073	−0.285806
$634$	−3.98845	−0.158402
$635$	−45.8930	−1.82121
$636$	−14.4992	−0.574933
$637$	−0.0340170	−0.00134780
$638$	−9.41093	−0.372582
$639$	12.0747	0.477667
$640$	6.07645	0.240193
$641$	−21.8011	−0.861090	−0.430545	−	0.902569i	$-0.641679\pi$
$641$	−21.8011	−0.861090	−0.430545	+	0.902569i	$0.641679\pi$
$642$	−23.6397	−0.932985
$643$	−2.00022	−0.0788810	−0.0394405	−	0.999222i	$-0.512558\pi$
$643$	−2.00022	−0.0788810	−0.0394405	+	0.999222i	$0.512558\pi$
$644$	5.02554	0.198034
$645$	40.4856	1.59412
$646$	78.3141	3.08123
$647$	47.7606	1.87766	0.938831	−	0.344379i	$-0.111910\pi$
$647$	47.7606	1.87766	0.938831	+	0.344379i	$0.111910\pi$
$648$	0.195801	0.00769178
$649$	−48.8646	−1.91810
$650$	−66.3501	−2.60246
$651$	18.1688	0.712089
$652$	−17.6790	−0.692365
$653$	−20.8681	−0.816631	−0.408316	−	0.912841i	$-0.633884\pi$
$653$	−20.8681	−0.816631	−0.408316	+	0.912841i	$0.633884\pi$
$654$	−25.9247	−1.01373
$655$	−50.8574	−1.98716
$656$	−22.5876	−0.881898
$657$	6.12294	0.238879
$658$	−44.3654	−1.72954
$659$	−38.0743	−1.48316	−0.741582	−	0.670863i	$-0.765924\pi$
$659$	−38.0743	−1.48316	−0.741582	+	0.670863i	$0.765924\pi$
$660$	−35.1789	−1.36934
$661$	24.0049	0.933684	0.466842	−	0.884341i	$-0.345392\pi$
$661$	24.0049	0.933684	0.466842	+	0.884341i	$0.345392\pi$
$662$	38.0364	1.47833
$663$	−24.2806	−0.942980
$664$	−1.97413	−0.0766112
$665$	−55.8568	−2.16603
$666$	17.8753	0.692656
$667$	1.00000	0.0387202
$668$	−32.1601	−1.24431
$669$	−8.86810	−0.342861
$670$	92.7341	3.58263
$671$	2.15486	0.0831873
$672$	20.8355	0.803747
$673$	−4.39075	−0.169251	−0.0846255	−	0.996413i	$-0.526969\pi$
$673$	−4.39075	−0.169251	−0.0846255	+	0.996413i	$0.526969\pi$
$674$	−3.21469	−0.123825
$675$	10.0855	0.388190
$676$	−3.62107	−0.139272
$677$	−11.7070	−0.449938	−0.224969	−	0.974366i	$-0.572228\pi$
$677$	−11.7070	−0.449938	−0.224969	+	0.974366i	$0.572228\pi$
$678$	4.34427	0.166841
$679$	−2.08641	−0.0800690
$680$	5.54356	0.212586
$681$	10.6725	0.408971
$682$	−64.6734	−2.47647
$683$	−29.3902	−1.12458	−0.562292	−	0.826939i	$-0.690080\pi$
$683$	−29.3902	−1.12458	−0.562292	+	0.826939i	$0.690080\pi$
$684$	10.3399	0.395355
$685$	−30.7829	−1.17615
$686$	36.6053	1.39760
$687$	4.91406	0.187483
$688$	43.6590	1.66449
$689$	−25.4073	−0.967942
$690$	7.67114	0.292035
$691$	−16.9740	−0.645723	−0.322861	−	0.946446i	$-0.604645\pi$
$691$	−16.9740	−0.645723	−0.322861	+	0.946446i	$0.604645\pi$
$692$	6.36842	0.242091
$693$	12.5975	0.478539
$694$	−47.9728	−1.82102
$695$	−15.1256	−0.573747
$696$	0.195801	0.00742181
$697$	−39.3108	−1.48900
$698$	−0.894547	−0.0338591
$699$	−22.8105	−0.862772
$700$	50.6850	1.91571
$701$	−43.4787	−1.64217	−0.821084	−	0.570808i	$-0.806630\pi$
$701$	−43.4787	−1.64217	−0.821084	+	0.570808i	$0.806630\pi$
$702$	−6.57877	−0.248300
$703$	−49.2310	−1.85678
$704$	−34.2510	−1.29088
$705$	−32.9998	−1.24285
$706$	4.21117	0.158490
$707$	13.6132	0.511976
$708$	−19.4936	−0.732616
$709$	9.31290	0.349753	0.174877	−	0.984590i	$-0.444047\pi$
$709$	9.31290	0.349753	0.174877	+	0.984590i	$0.444047\pi$
$710$	92.6267	3.47622
$711$	−9.83566	−0.368866
$712$	−0.0959900	−0.00359738
$713$	6.87216	0.257364
$714$	38.0633	1.42449
$715$	−61.6447	−2.30538
$716$	18.7075	0.699131
$717$	−22.1209	−0.826120
$718$	−8.38221	−0.312821
$719$	33.4838	1.24873	0.624367	−	0.781131i	$-0.285357\pi$
$719$	33.4838	1.24873	0.624367	+	0.781131i	$0.285357\pi$
$720$	16.2679	0.606270
$721$	−10.9126	−0.406406
$722$	−20.9138	−0.778331
$723$	−5.77198	−0.214662
$724$	−7.28705	−0.270821
$725$	10.0855	0.374565
$726$	−23.1163	−0.857926
$727$	−13.2045	−0.489726	−0.244863	−	0.969558i	$-0.578743\pi$
$727$	−13.2045	−0.489726	−0.244863	+	0.969558i	$0.578743\pi$
$728$	1.72429	0.0639065
$729$	1.00000	0.0370370
$730$	46.9699	1.73843
$731$	75.9829	2.81033
$732$	0.859641	0.0317732
$733$	6.77920	0.250395	0.125198	−	0.992132i	$-0.460043\pi$
$733$	6.77920	0.250395	0.125198	+	0.992132i	$0.460043\pi$
$734$	35.8044	1.32156
$735$	0.0396654	0.00146308
$736$	7.88083	0.290491
$737$	57.6012	2.12177
$738$	−10.6512	−0.392075
$739$	−32.2217	−1.18530	−0.592648	−	0.805462i	$-0.701917\pi$
$739$	−32.2217	−1.18530	−0.592648	+	0.805462i	$0.701917\pi$
$740$	66.8197	2.45634
$741$	18.1188	0.665610
$742$	39.8297	1.46219
$743$	−35.2349	−1.29264	−0.646321	−	0.763065i	$-0.723693\pi$
$743$	−35.2349	−1.29264	−0.646321	+	0.763065i	$0.723693\pi$
$744$	1.34557	0.0493311
$745$	−21.0067	−0.769626
$746$	−14.7460	−0.539890
$747$	−10.0824	−0.368894
$748$	−66.0234	−2.41405
$749$	31.6442	1.15625
$750$	39.0114	1.42450
$751$	−11.3638	−0.414670	−0.207335	−	0.978270i	$-0.566479\pi$
$751$	−11.3638	−0.414670	−0.207335	+	0.978270i	$0.566479\pi$
$752$	−35.5865	−1.29771
$753$	−18.2189	−0.663934
$754$	−6.57877	−0.239585
$755$	−38.1159	−1.38718
$756$	5.02554	0.182777
$757$	−18.9819	−0.689910	−0.344955	−	0.938619i	$-0.612106\pi$
$757$	−18.9819	−0.689910	−0.344955	+	0.938619i	$0.612106\pi$
$758$	60.6161	2.20168
$759$	4.76488	0.172954
$760$	−4.13674	−0.150055
$761$	3.79127	0.137434	0.0687168	−	0.997636i	$-0.478110\pi$
$761$	3.79127	0.137434	0.0687168	+	0.997636i	$0.478110\pi$
$762$	−23.3371	−0.845414
$763$	34.7028	1.25633
$764$	48.3294	1.74850
$765$	28.3122	1.02363
$766$	32.3694	1.16955
$767$	−34.1591	−1.23341
$768$	17.4664	0.630264
$769$	47.9385	1.72871	0.864354	−	0.502884i	$-0.167728\pi$
$769$	47.9385	1.72871	0.864354	+	0.502884i	$0.167728\pi$
$770$	96.6371	3.48256
$771$	6.94804	0.250228
$772$	45.6216	1.64196
$773$	−48.7564	−1.75365	−0.876823	−	0.480812i	$-0.840342\pi$
$773$	−48.7564	−1.75365	−0.876823	+	0.480812i	$0.840342\pi$
$774$	20.5874	0.739999
$775$	69.3090	2.48965
$776$	−0.154519	−0.00554690
$777$	−23.9280	−0.858412
$778$	−1.73020	−0.0620308
$779$	29.3347	1.05103
$780$	−24.5921	−0.880537
$781$	57.5345	2.05875
$782$	14.3971	0.514839
$783$	1.00000	0.0357371
$784$	0.0427745	0.00152766
$785$	−66.5786	−2.37629
$786$	−25.8616	−0.922452
$787$	7.87475	0.280705	0.140352	−	0.990102i	$-0.455176\pi$
$787$	7.87475	0.280705	0.140352	+	0.990102i	$0.455176\pi$
$788$	44.6638	1.59108
$789$	7.20446	0.256486
$790$	−75.4507	−2.68442
$791$	−5.81525	−0.206766
$792$	0.932967	0.0331515
$793$	1.50637	0.0534926
$794$	−12.1405	−0.430849
$795$	29.6261	1.05073
$796$	−11.6478	−0.412846
$797$	−5.88439	−0.208436	−0.104218	−	0.994554i	$-0.533234\pi$
$797$	−5.88439	−0.208436	−0.104218	+	0.994554i	$0.533234\pi$
$798$	−28.4038	−1.00549
$799$	−61.9337	−2.19106
$800$	79.4819	2.81011
$801$	−0.490243	−0.0173219
$802$	−15.9245	−0.562313
$803$	29.1751	1.02957
$804$	22.9790	0.810405
$805$	−10.2686	−0.361921
$806$	−45.2104	−1.59247
$807$	−4.24647	−0.149483
$808$	1.00819	0.0354679
$809$	−3.36443	−0.118287	−0.0591436	−	0.998249i	$-0.518837\pi$
$809$	−3.36443	−0.118287	−0.0591436	+	0.998249i	$0.518837\pi$
$810$	7.67114	0.269536
$811$	39.9912	1.40428	0.702141	−	0.712038i	$-0.252228\pi$
$811$	39.9912	1.40428	0.702141	+	0.712038i	$0.252228\pi$
$812$	5.02554	0.176362
$813$	19.9356	0.699172
$814$	85.1739	2.98534
$815$	36.1233	1.26534
$816$	30.5315	1.06881
$817$	−56.7004	−1.98370
$818$	−6.35072	−0.222048
$819$	8.80636	0.307719
$820$	−39.8151	−1.39040
$821$	41.2944	1.44119	0.720593	−	0.693359i	$-0.243870\pi$
$821$	41.2944	1.44119	0.720593	+	0.693359i	$0.243870\pi$
$822$	−15.6534	−0.545977
$823$	−12.7752	−0.445314	−0.222657	−	0.974897i	$-0.571473\pi$
$823$	−12.7752	−0.445314	−0.222657	+	0.974897i	$0.571473\pi$
$824$	−0.808183	−0.0281544
$825$	48.0561	1.67310
$826$	53.5494	1.86322
$827$	−0.0598083	−0.00207974	−0.00103987	−	0.999999i	$-0.500331\pi$
$827$	−0.0598083	−0.00207974	−0.00103987	+	0.999999i	$0.500331\pi$
$828$	1.90086	0.0660596
$829$	31.8700	1.10689	0.553446	−	0.832885i	$-0.313313\pi$
$829$	31.8700	1.10689	0.553446	+	0.832885i	$0.313313\pi$
$830$	−77.3432	−2.68462
$831$	−2.38195	−0.0826288
$832$	−23.9434	−0.830089
$833$	0.0744434	0.00257931
$834$	−7.69154	−0.266336
$835$	65.7122	2.27406
$836$	49.2683	1.70398
$837$	6.87216	0.237536
$838$	−57.4730	−1.98537
$839$	−10.1762	−0.351321	−0.175661	−	0.984451i	$-0.556206\pi$
$839$	−10.1762	−0.351321	−0.175661	+	0.984451i	$0.556206\pi$
$840$	−2.01060	−0.0693723
$841$	1.00000	0.0344828
$842$	−27.5712	−0.950165
$843$	13.4801	0.464279
$844$	−13.6686	−0.470492
$845$	7.39886	0.254529
$846$	−16.7808	−0.576936
$847$	30.9435	1.06323
$848$	31.9483	1.09711
$849$	7.65647	0.262769
$850$	145.202	4.98037
$851$	−9.05053	−0.310248
$852$	22.9523	0.786334
$853$	−24.6152	−0.842807	−0.421404	−	0.906873i	$-0.638462\pi$
$853$	−24.6152	−0.842807	−0.421404	+	0.906873i	$0.638462\pi$
$854$	−2.36145	−0.0808071
$855$	−21.1273	−0.722539
$856$	2.34356	0.0801012
$857$	21.6841	0.740715	0.370357	−	0.928889i	$-0.379235\pi$
$857$	21.6841	0.740715	0.370357	+	0.928889i	$0.379235\pi$
$858$	−31.3471	−1.07017
$859$	−24.9541	−0.851422	−0.425711	−	0.904859i	$-0.639976\pi$
$859$	−24.9541	−0.851422	−0.425711	+	0.904859i	$0.639976\pi$
$860$	76.9576	2.62423
$861$	14.2577	0.485901
$862$	−27.1465	−0.924612
$863$	24.6724	0.839859	0.419930	−	0.907557i	$-0.362055\pi$
$863$	24.6724	0.839859	0.419930	+	0.907557i	$0.362055\pi$
$864$	7.88083	0.268111
$865$	−13.0125	−0.442438
$866$	49.0189	1.66573
$867$	36.1361	1.22724
$868$	34.5363	1.17224
$869$	−46.8657	−1.58981
$870$	7.67114	0.260076
$871$	40.2665	1.36438
$872$	2.57008	0.0870339
$873$	−0.789164	−0.0267091
$874$	−10.7435	−0.363404
$875$	−52.2208	−1.76538
$876$	11.6389	0.393241
$877$	−38.5718	−1.30248	−0.651239	−	0.758873i	$-0.725750\pi$
$877$	−38.5718	−1.30248	−0.651239	+	0.758873i	$0.725750\pi$
$878$	31.4615	1.06177
$879$	25.5856	0.862982
$880$	77.5148	2.61302
$881$	−4.63783	−0.156253	−0.0781263	−	0.996943i	$-0.524894\pi$
$881$	−4.63783	−0.156253	−0.0781263	+	0.996943i	$0.524894\pi$
$882$	0.0201703	0.000679169 0
$883$	2.57549	0.0866721	0.0433360	−	0.999061i	$-0.486201\pi$
$883$	2.57549	0.0866721	0.0433360	+	0.999061i	$0.486201\pi$
$884$	−46.1541	−1.55233
$885$	39.8310	1.33891
$886$	73.2847	2.46205
$887$	−7.91018	−0.265598	−0.132799	−	0.991143i	$-0.542396\pi$
$887$	−7.91018	−0.265598	−0.132799	+	0.991143i	$0.542396\pi$
$888$	−1.77210	−0.0594678
$889$	31.2391	1.04773
$890$	−3.76073	−0.126060
$891$	4.76488	0.159630
$892$	−16.8571	−0.564416
$893$	46.2165	1.54658
$894$	−10.6821	−0.357264
$895$	−38.2247	−1.27771
$896$	−4.13621	−0.138181
$897$	3.33092	0.111216
$898$	25.8017	0.861015
$899$	6.87216	0.229199
$900$	19.1711	0.639037
$901$	55.6018	1.85237
$902$	−50.7516	−1.68984
$903$	−27.5584	−0.917085
$904$	−0.430676	−0.0143241
$905$	14.8895	0.494944
$906$	−19.3824	−0.643936
$907$	−20.5276	−0.681606	−0.340803	−	0.940135i	$-0.610699\pi$
$907$	−20.5276	−0.681606	−0.340803	+	0.940135i	$0.610699\pi$
$908$	20.2870	0.673247
$909$	5.14905	0.170783
$910$	67.5548	2.23942
$911$	49.7236	1.64742	0.823708	−	0.567014i	$-0.191901\pi$
$911$	49.7236	1.64742	0.823708	+	0.567014i	$0.191901\pi$
$912$	−22.7833	−0.754432
$913$	−48.0412	−1.58993
$914$	−40.1675	−1.32862
$915$	−1.75649	−0.0580678
$916$	9.34096	0.308634
$917$	34.6184	1.14320
$918$	14.3971	0.475175
$919$	45.2858	1.49384	0.746921	−	0.664913i	$-0.231531\pi$
$919$	45.2858	1.49384	0.746921	+	0.664913i	$0.231531\pi$
$920$	−0.760490	−0.0250726
$921$	−9.19667	−0.303041
$922$	49.4396	1.62821
$923$	40.2199	1.32385
$924$	23.9461	0.787769
$925$	−91.2789	−3.00123
$926$	43.4777	1.42877
$927$	−4.12758	−0.135567
$928$	7.88083	0.258701
$929$	19.7525	0.648057	0.324028	−	0.946047i	$-0.394963\pi$
$929$	19.7525	0.648057	0.324028	+	0.946047i	$0.394963\pi$
$930$	52.7173	1.72867
$931$	−0.0555516	−0.00182063
$932$	−43.3596	−1.42029
$933$	−10.3312	−0.338230
$934$	−21.9957	−0.719720
$935$	134.904	4.41185
$936$	0.652197	0.0213177
$937$	20.5188	0.670321	0.335160	−	0.942161i	$-0.391210\pi$
$937$	20.5188	0.670321	0.335160	+	0.942161i	$0.391210\pi$
$938$	−63.1236	−2.06106
$939$	−17.7542	−0.579386
$940$	−62.7282	−2.04597
$941$	−48.6773	−1.58683	−0.793417	−	0.608678i	$-0.791700\pi$
$941$	−48.6773	−1.58683	−0.793417	+	0.608678i	$0.791700\pi$
$942$	−33.8560	−1.10309
$943$	5.39284	0.175615
$944$	42.9532	1.39801
$945$	−10.2686	−0.334038
$946$	98.0966	3.18939
$947$	−52.1168	−1.69357	−0.846785	−	0.531936i	$-0.821465\pi$
$947$	−52.1168	−1.69357	−0.846785	+	0.531936i	$0.821465\pi$
$948$	−18.6962	−0.607226
$949$	20.3950	0.662051
$950$	−108.353	−3.51544
$951$	2.01941	0.0654838
$952$	−3.77347	−0.122299
$953$	49.9080	1.61668	0.808339	−	0.588717i	$-0.200367\pi$
$953$	49.9080	1.61668	0.808339	+	0.588717i	$0.200367\pi$
$954$	15.0652	0.487754
$955$	−98.7506	−3.19550
$956$	−42.0488	−1.35996
$957$	4.76488	0.154027
$958$	−51.7361	−1.67152
$959$	20.9537	0.676632
$960$	27.9191	0.901085
$961$	16.2265	0.523437
$962$	59.5414	1.91969
$963$	11.9691	0.385699
$964$	−10.9717	−0.353376
$965$	−93.2178	−3.00079
$966$	−5.22171	−0.168006
$967$	11.0712	0.356027	0.178014	−	0.984028i	$-0.443033\pi$
$967$	11.0712	0.356027	0.178014	+	0.984028i	$0.443033\pi$
$968$	2.29167	0.0736570
$969$	−39.6515	−1.27379
$970$	−6.05378	−0.194375
$971$	−0.191618	−0.00614932	−0.00307466	−	0.999995i	$-0.500979\pi$
$971$	−0.191618	−0.00614932	−0.00307466	+	0.999995i	$0.500979\pi$
$972$	1.90086	0.0609702
$973$	10.2959	0.330072
$974$	−81.2761	−2.60425
$975$	33.5939	1.07587
$976$	−1.89417	−0.0606309
$977$	9.08511	0.290659	0.145329	−	0.989383i	$-0.453576\pi$
$977$	9.08511	0.290659	0.145329	+	0.989383i	$0.453576\pi$
$978$	18.3691	0.587379
$979$	−2.33595	−0.0746573
$980$	0.0753984	0.00240851
$981$	13.1260	0.419081
$982$	−45.5992	−1.45513
$983$	−1.42571	−0.0454730	−0.0227365	−	0.999741i	$-0.507238\pi$
$983$	−1.42571	−0.0454730	−0.0227365	+	0.999741i	$0.507238\pi$
$984$	1.05592	0.0336615
$985$	−91.2609	−2.90781
$986$	14.3971	0.458497
$987$	22.4628	0.715000
$988$	34.4413	1.09573
$989$	−10.4237	−0.331454
$990$	36.5521	1.16170
$991$	44.4287	1.41132	0.705662	−	0.708549i	$-0.250650\pi$
$991$	44.4287	1.41132	0.705662	+	0.708549i	$0.250650\pi$
$992$	54.1583	1.71953
$993$	−19.2584	−0.611146
$994$	−63.0505	−1.99984
$995$	23.7998	0.754505
$996$	−19.1652	−0.607272
$997$	−5.76382	−0.182542	−0.0912709	−	0.995826i	$-0.529093\pi$
$997$	−5.76382	−0.182542	−0.0912709	+	0.995826i	$0.529093\pi$
$998$	1.24046	0.0392660
$999$	−9.05053	−0.286346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.2.a.o.1.4	✓	20
3.2	odd	2		6003.2.a.s.1.17		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.4	✓	20	1.1	even	1	trivial
6003.2.a.s.1.17		20	3.2	odd	2

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:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.97506 q^{2} +1.00000 q^{3} +1.90086 q^{4} -3.88400 q^{5} -1.97506 q^{6} +2.64382 q^{7} +0.195801 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.97506 q^{2} +1.00000 q^{3} +1.90086 q^{4} -3.88400 q^{5} -1.97506 q^{6} +2.64382 q^{7} +0.195801 q^{8} +1.00000 q^{9} +7.67114 q^{10} +4.76488 q^{11} +1.90086 q^{12} +3.33092 q^{13} -5.22171 q^{14} -3.88400 q^{15} -4.18845 q^{16} -7.28945 q^{17} -1.97506 q^{18} +5.43957 q^{19} -7.38296 q^{20} +2.64382 q^{21} -9.41093 q^{22} +1.00000 q^{23} +0.195801 q^{24} +10.0855 q^{25} -6.57877 q^{26} +1.00000 q^{27} +5.02554 q^{28} +1.00000 q^{29} +7.67114 q^{30} +6.87216 q^{31} +7.88083 q^{32} +4.76488 q^{33} +14.3971 q^{34} -10.2686 q^{35} +1.90086 q^{36} -9.05053 q^{37} -10.7435 q^{38} +3.33092 q^{39} -0.760490 q^{40} +5.39284 q^{41} -5.22171 q^{42} -10.4237 q^{43} +9.05739 q^{44} -3.88400 q^{45} -1.97506 q^{46} +8.49635 q^{47} -4.18845 q^{48} -0.0102125 q^{49} -19.9194 q^{50} -7.28945 q^{51} +6.33163 q^{52} -7.62772 q^{53} -1.97506 q^{54} -18.5068 q^{55} +0.517662 q^{56} +5.43957 q^{57} -1.97506 q^{58} -10.2552 q^{59} -7.38296 q^{60} +0.452237 q^{61} -13.5729 q^{62} +2.64382 q^{63} -7.18823 q^{64} -12.9373 q^{65} -9.41093 q^{66} +12.0887 q^{67} -13.8562 q^{68} +1.00000 q^{69} +20.2811 q^{70} +12.0747 q^{71} +0.195801 q^{72} +6.12294 q^{73} +17.8753 q^{74} +10.0855 q^{75} +10.3399 q^{76} +12.5975 q^{77} -6.57877 q^{78} -9.83566 q^{79} +16.2679 q^{80} +1.00000 q^{81} -10.6512 q^{82} -10.0824 q^{83} +5.02554 q^{84} +28.3122 q^{85} +20.5874 q^{86} +1.00000 q^{87} +0.932967 q^{88} -0.490243 q^{89} +7.67114 q^{90} +8.80636 q^{91} +1.90086 q^{92} +6.87216 q^{93} -16.7808 q^{94} -21.1273 q^{95} +7.88083 q^{96} -0.789164 q^{97} +0.0201703 q^{98} +4.76488 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9}+O(q^{10}) \) 20 * q + 2 * q^2 + 20 * q^3 + 30 * q^4 - q^5 + 2 * q^6 + 9 * q^7 + 6 * q^8 + 20 * q^9	\( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9} + 7 q^{10} + 30 q^{12} + 21 q^{13} - q^{14} - q^{15} + 58 q^{16} - 4 q^{17} + 2 q^{18} + 7 q^{19} - 20 q^{20} + 9 q^{21} + 7 q^{22} + 20 q^{23} + 6 q^{24} + 47 q^{25} + 8 q^{26} + 20 q^{27} + 11 q^{28} + 20 q^{29} + 7 q^{30} + 28 q^{31} + 14 q^{32} + 16 q^{34} + 9 q^{35} + 30 q^{36} + 14 q^{37} - 20 q^{38} + 21 q^{39} + 34 q^{40} + 7 q^{41} - q^{42} + 3 q^{43} - q^{44} - q^{45} + 2 q^{46} + 3 q^{47} + 58 q^{48} + 35 q^{49} - 24 q^{50} - 4 q^{51} + 73 q^{52} - 19 q^{53} + 2 q^{54} + 29 q^{55} - 30 q^{56} + 7 q^{57} + 2 q^{58} + 20 q^{59} - 20 q^{60} + 15 q^{61} + 12 q^{62} + 9 q^{63} + 82 q^{64} - 28 q^{65} + 7 q^{66} + 20 q^{67} - 23 q^{68} + 20 q^{69} - 24 q^{70} + 63 q^{71} + 6 q^{72} + 19 q^{73} + 16 q^{74} + 47 q^{75} - 44 q^{76} - 7 q^{77} + 8 q^{78} + 32 q^{79} - 56 q^{80} + 20 q^{81} - 20 q^{82} - 21 q^{83} + 11 q^{84} + 4 q^{85} - 6 q^{86} + 20 q^{87} + 55 q^{88} - 13 q^{89} + 7 q^{90} + 70 q^{91} + 30 q^{92} + 28 q^{93} - 12 q^{94} + 9 q^{95} + 14 q^{96} - 9 q^{97} + 31 q^{98}+O(q^{100}) \) 20 * q + 2 * q^2 + 20 * q^3 + 30 * q^4 - q^5 + 2 * q^6 + 9 * q^7 + 6 * q^8 + 20 * q^9 + 7 * q^10 + 30 * q^12 + 21 * q^13 - q^14 - q^15 + 58 * q^16 - 4 * q^17 + 2 * q^18 + 7 * q^19 - 20 * q^20 + 9 * q^21 + 7 * q^22 + 20 * q^23 + 6 * q^24 + 47 * q^25 + 8 * q^26 + 20 * q^27 + 11 * q^28 + 20 * q^29 + 7 * q^30 + 28 * q^31 + 14 * q^32 + 16 * q^34 + 9 * q^35 + 30 * q^36 + 14 * q^37 - 20 * q^38 + 21 * q^39 + 34 * q^40 + 7 * q^41 - q^42 + 3 * q^43 - q^44 - q^45 + 2 * q^46 + 3 * q^47 + 58 * q^48 + 35 * q^49 - 24 * q^50 - 4 * q^51 + 73 * q^52 - 19 * q^53 + 2 * q^54 + 29 * q^55 - 30 * q^56 + 7 * q^57 + 2 * q^58 + 20 * q^59 - 20 * q^60 + 15 * q^61 + 12 * q^62 + 9 * q^63 + 82 * q^64 - 28 * q^65 + 7 * q^66 + 20 * q^67 - 23 * q^68 + 20 * q^69 - 24 * q^70 + 63 * q^71 + 6 * q^72 + 19 * q^73 + 16 * q^74 + 47 * q^75 - 44 * q^76 - 7 * q^77 + 8 * q^78 + 32 * q^79 - 56 * q^80 + 20 * q^81 - 20 * q^82 - 21 * q^83 + 11 * q^84 + 4 * q^85 - 6 * q^86 + 20 * q^87 + 55 * q^88 - 13 * q^89 + 7 * q^90 + 70 * q^91 + 30 * q^92 + 28 * q^93 - 12 * q^94 + 9 * q^95 + 14 * q^96 - 9 * q^97 + 31 * q^98

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