LMFDB - Embedded newform 4002.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2389280.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 11x^{3} + 7x^{2} + 26x - 18 $ x^5 - x^4 - 11x^3 + 7x^2 + 26*x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61972$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.88781 q^{5} +1.00000 q^{6} -3.71597 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.88781 q^{5} +1.00000 q^{6} -3.71597 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.88781 q^{10} +5.50753 q^{11} -1.00000 q^{12} -3.61972 q^{13} +3.71597 q^{14} +2.88781 q^{15} +1.00000 q^{16} -4.95541 q^{17} -1.00000 q^{18} +4.95541 q^{19} -2.88781 q^{20} +3.71597 q^{21} -5.50753 q^{22} -1.00000 q^{23} +1.00000 q^{24} +3.33946 q^{25} +3.61972 q^{26} -1.00000 q^{27} -3.71597 q^{28} -1.00000 q^{29} -2.88781 q^{30} +6.02907 q^{31} -1.00000 q^{32} -5.50753 q^{33} +4.95541 q^{34} +10.7310 q^{35} +1.00000 q^{36} -1.34786 q^{37} -4.95541 q^{38} +3.61972 q^{39} +2.88781 q^{40} -1.77939 q^{41} -3.71597 q^{42} -7.39911 q^{43} +5.50753 q^{44} -2.88781 q^{45} +1.00000 q^{46} +13.6841 q^{47} -1.00000 q^{48} +6.80846 q^{49} -3.33946 q^{50} +4.95541 q^{51} -3.61972 q^{52} +2.70574 q^{53} +1.00000 q^{54} -15.9047 q^{55} +3.71597 q^{56} -4.95541 q^{57} +1.00000 q^{58} +8.87565 q^{59} +2.88781 q^{60} -5.99054 q^{61} -6.02907 q^{62} -3.71597 q^{63} +1.00000 q^{64} +10.4531 q^{65} +5.50753 q^{66} +9.67367 q^{67} -4.95541 q^{68} +1.00000 q^{69} -10.7310 q^{70} +3.39534 q^{71} -1.00000 q^{72} +13.9133 q^{73} +1.34786 q^{74} -3.33946 q^{75} +4.95541 q^{76} -20.4658 q^{77} -3.61972 q^{78} -4.99623 q^{79} -2.88781 q^{80} +1.00000 q^{81} +1.77939 q^{82} +9.98492 q^{83} +3.71597 q^{84} +14.3103 q^{85} +7.39911 q^{86} +1.00000 q^{87} -5.50753 q^{88} +11.1394 q^{89} +2.88781 q^{90} +13.4508 q^{91} -1.00000 q^{92} -6.02907 q^{93} -13.6841 q^{94} -14.3103 q^{95} +1.00000 q^{96} -6.70574 q^{97} -6.80846 q^{98} +5.50753 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9} - q^{10} + 5 q^{11} - 5 q^{12} - 11 q^{13} + 4 q^{14} - q^{15} + 5 q^{16} + 4 q^{17} - 5 q^{18} - 4 q^{19} + q^{20} + 4 q^{21} - 5 q^{22} - 5 q^{23} + 5 q^{24} + 12 q^{25} + 11 q^{26} - 5 q^{27} - 4 q^{28} - 5 q^{29} + q^{30} + 5 q^{31} - 5 q^{32} - 5 q^{33} - 4 q^{34} - 6 q^{35} + 5 q^{36} + 9 q^{37} + 4 q^{38} + 11 q^{39} - q^{40} + 5 q^{41} - 4 q^{42} - 16 q^{43} + 5 q^{44} + q^{45} + 5 q^{46} + 8 q^{47} - 5 q^{48} - 5 q^{49} - 12 q^{50} - 4 q^{51} - 11 q^{52} - 4 q^{53} + 5 q^{54} - 33 q^{55} + 4 q^{56} + 4 q^{57} + 5 q^{58} + 23 q^{59} - q^{60} + 7 q^{61} - 5 q^{62} - 4 q^{63} + 5 q^{64} - 5 q^{65} + 5 q^{66} + 5 q^{67} + 4 q^{68} + 5 q^{69} + 6 q^{70} - 21 q^{71} - 5 q^{72} - 8 q^{73} - 9 q^{74} - 12 q^{75} - 4 q^{76} - 11 q^{78} - 8 q^{79} + q^{80} + 5 q^{81} - 5 q^{82} - 18 q^{83} + 4 q^{84} - 10 q^{85} + 16 q^{86} + 5 q^{87} - 5 q^{88} + 32 q^{89} - q^{90} + 10 q^{91} - 5 q^{92} - 5 q^{93} - 8 q^{94} + 10 q^{95} + 5 q^{96} - 16 q^{97} + 5 q^{98} + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9 - q^10 + 5 * q^11 - 5 * q^12 - 11 * q^13 + 4 * q^14 - q^15 + 5 * q^16 + 4 * q^17 - 5 * q^18 - 4 * q^19 + q^20 + 4 * q^21 - 5 * q^22 - 5 * q^23 + 5 * q^24 + 12 * q^25 + 11 * q^26 - 5 * q^27 - 4 * q^28 - 5 * q^29 + q^30 + 5 * q^31 - 5 * q^32 - 5 * q^33 - 4 * q^34 - 6 * q^35 + 5 * q^36 + 9 * q^37 + 4 * q^38 + 11 * q^39 - q^40 + 5 * q^41 - 4 * q^42 - 16 * q^43 + 5 * q^44 + q^45 + 5 * q^46 + 8 * q^47 - 5 * q^48 - 5 * q^49 - 12 * q^50 - 4 * q^51 - 11 * q^52 - 4 * q^53 + 5 * q^54 - 33 * q^55 + 4 * q^56 + 4 * q^57 + 5 * q^58 + 23 * q^59 - q^60 + 7 * q^61 - 5 * q^62 - 4 * q^63 + 5 * q^64 - 5 * q^65 + 5 * q^66 + 5 * q^67 + 4 * q^68 + 5 * q^69 + 6 * q^70 - 21 * q^71 - 5 * q^72 - 8 * q^73 - 9 * q^74 - 12 * q^75 - 4 * q^76 - 11 * q^78 - 8 * q^79 + q^80 + 5 * q^81 - 5 * q^82 - 18 * q^83 + 4 * q^84 - 10 * q^85 + 16 * q^86 + 5 * q^87 - 5 * q^88 + 32 * q^89 - q^90 + 10 * q^91 - 5 * q^92 - 5 * q^93 - 8 * q^94 + 10 * q^95 + 5 * q^96 - 16 * q^97 + 5 * q^98 + 5 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.88781	−1.29147	−0.645734	−	0.763562i	$-0.723449\pi$
$5$	−2.88781	−1.29147	−0.645734	+	0.763562i	$0.723449\pi$
$6$	1.00000	0.408248
$7$	−3.71597	−1.40451	−0.702253	−	0.711927i	$-0.747822\pi$
$7$	−3.71597	−1.40451	−0.702253	+	0.711927i	$0.747822\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.88781	0.913206
$11$	5.50753	1.66058	0.830291	−	0.557329i	$-0.188174\pi$
$11$	5.50753	1.66058	0.830291	+	0.557329i	$0.188174\pi$
$12$	−1.00000	−0.288675
$13$	−3.61972	−1.00393	−0.501965	−	0.864888i	$-0.667389\pi$
$13$	−3.61972	−1.00393	−0.501965	+	0.864888i	$0.667389\pi$
$14$	3.71597	0.993136
$15$	2.88781	0.745630
$16$	1.00000	0.250000
$17$	−4.95541	−1.20186	−0.600932	−	0.799300i	$-0.705204\pi$
$17$	−4.95541	−1.20186	−0.600932	+	0.799300i	$0.705204\pi$
$18$	−1.00000	−0.235702
$19$	4.95541	1.13685	0.568425	−	0.822735i	$-0.307553\pi$
$19$	4.95541	1.13685	0.568425	+	0.822735i	$0.307553\pi$
$20$	−2.88781	−0.645734
$21$	3.71597	0.810892
$22$	−5.50753	−1.17421
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	3.33946	0.667892
$26$	3.61972	0.709885
$27$	−1.00000	−0.192450
$28$	−3.71597	−0.702253
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−2.88781	−0.527240
$31$	6.02907	1.08285	0.541426	−	0.840748i	$-0.317885\pi$
$31$	6.02907	1.08285	0.541426	+	0.840748i	$0.317885\pi$
$32$	−1.00000	−0.176777
$33$	−5.50753	−0.958738
$34$	4.95541	0.849846
$35$	10.7310	1.81388
$36$	1.00000	0.166667
$37$	−1.34786	−0.221586	−0.110793	−	0.993844i	$-0.535339\pi$
$37$	−1.34786	−0.221586	−0.110793	+	0.993844i	$0.535339\pi$
$38$	−4.95541	−0.803874
$39$	3.61972	0.579619
$40$	2.88781	0.456603
$41$	−1.77939	−0.277895	−0.138947	−	0.990300i	$-0.544372\pi$
$41$	−1.77939	−0.277895	−0.138947	+	0.990300i	$0.544372\pi$
$42$	−3.71597	−0.573387
$43$	−7.39911	−1.12835	−0.564177	−	0.825654i	$-0.690806\pi$
$43$	−7.39911	−1.12835	−0.564177	+	0.825654i	$0.690806\pi$
$44$	5.50753	0.830291
$45$	−2.88781	−0.430490
$46$	1.00000	0.147442
$47$	13.6841	1.99603	0.998016	−	0.0629568i	$-0.0200530\pi$
$47$	13.6841	1.99603	0.998016	+	0.0629568i	$0.0200530\pi$
$48$	−1.00000	−0.144338
$49$	6.80846	0.972637
$50$	−3.33946	−0.472271
$51$	4.95541	0.693896
$52$	−3.61972	−0.501965
$53$	2.70574	0.371662	0.185831	−	0.982582i	$-0.440502\pi$
$53$	2.70574	0.371662	0.185831	+	0.982582i	$0.440502\pi$
$54$	1.00000	0.136083
$55$	−15.9047	−2.14459
$56$	3.71597	0.496568
$57$	−4.95541	−0.656360
$58$	1.00000	0.131306
$59$	8.87565	1.15551	0.577756	−	0.816210i	$-0.303929\pi$
$59$	8.87565	1.15551	0.577756	+	0.816210i	$0.303929\pi$
$60$	2.88781	0.372815
$61$	−5.99054	−0.767009	−0.383505	−	0.923539i	$-0.625283\pi$
$61$	−5.99054	−0.767009	−0.383505	+	0.923539i	$0.625283\pi$
$62$	−6.02907	−0.765692
$63$	−3.71597	−0.468169
$64$	1.00000	0.125000
$65$	10.4531	1.29654
$66$	5.50753	0.677930
$67$	9.67367	1.18183	0.590914	−	0.806735i	$-0.298767\pi$
$67$	9.67367	1.18183	0.590914	+	0.806735i	$0.298767\pi$
$68$	−4.95541	−0.600932
$69$	1.00000	0.120386
$70$	−10.7310	−1.28260
$71$	3.39534	0.402953	0.201477	−	0.979493i	$-0.435426\pi$
$71$	3.39534	0.402953	0.201477	+	0.979493i	$0.435426\pi$
$72$	−1.00000	−0.117851
$73$	13.9133	1.62843	0.814215	−	0.580564i	$-0.197168\pi$
$73$	13.9133	1.62843	0.814215	+	0.580564i	$0.197168\pi$
$74$	1.34786	0.156685
$75$	−3.33946	−0.385608
$76$	4.95541	0.568425
$77$	−20.4658	−2.33230
$78$	−3.61972	−0.409852
$79$	−4.99623	−0.562120	−0.281060	−	0.959690i	$-0.590686\pi$
$79$	−4.99623	−0.562120	−0.281060	+	0.959690i	$0.590686\pi$
$80$	−2.88781	−0.322867
$81$	1.00000	0.111111
$82$	1.77939	0.196501
$83$	9.98492	1.09599	0.547994	−	0.836482i	$-0.315392\pi$
$83$	9.98492	1.09599	0.547994	+	0.836482i	$0.315392\pi$
$84$	3.71597	0.405446
$85$	14.3103	1.55217
$86$	7.39911	0.797867
$87$	1.00000	0.107211
$88$	−5.50753	−0.587105
$89$	11.1394	1.18078	0.590388	−	0.807120i	$-0.298975\pi$
$89$	11.1394	1.18078	0.590388	+	0.807120i	$0.298975\pi$
$90$	2.88781	0.304402
$91$	13.4508	1.41002
$92$	−1.00000	−0.104257
$93$	−6.02907	−0.625185
$94$	−13.6841	−1.41141
$95$	−14.3103	−1.46821
$96$	1.00000	0.102062
$97$	−6.70574	−0.680864	−0.340432	−	0.940269i	$-0.610573\pi$
$97$	−6.70574	−0.680864	−0.340432	+	0.940269i	$0.610573\pi$
$98$	−6.80846	−0.687758
$99$	5.50753	0.553528
$100$	3.33946	0.333946
$101$	−1.76290	−0.175415	−0.0877075	−	0.996146i	$-0.527954\pi$
$101$	−1.76290	−0.175415	−0.0877075	+	0.996146i	$0.527954\pi$
$102$	−4.95541	−0.490659
$103$	−7.96524	−0.784838	−0.392419	−	0.919787i	$-0.628362\pi$
$103$	−7.96524	−0.784838	−0.392419	+	0.919787i	$0.628362\pi$
$104$	3.61972	0.354943
$105$	−10.7310	−1.04724
$106$	−2.70574	−0.262804
$107$	−1.51548	−0.146507	−0.0732533	−	0.997313i	$-0.523338\pi$
$107$	−1.51548	−0.146507	−0.0732533	+	0.997313i	$0.523338\pi$
$108$	−1.00000	−0.0962250
$109$	−8.99827	−0.861878	−0.430939	−	0.902381i	$-0.641818\pi$
$109$	−8.99827	−0.861878	−0.430939	+	0.902381i	$0.641818\pi$
$110$	15.9047	1.51646
$111$	1.34786	0.127933
$112$	−3.71597	−0.351127
$113$	−9.12954	−0.858835	−0.429417	−	0.903106i	$-0.641281\pi$
$113$	−9.12954	−0.858835	−0.429417	+	0.903106i	$0.641281\pi$
$114$	4.95541	0.464117
$115$	2.88781	0.269290
$116$	−1.00000	−0.0928477
$117$	−3.61972	−0.334643
$118$	−8.87565	−0.817070
$119$	18.4142	1.68802
$120$	−2.88781	−0.263620
$121$	19.3329	1.75754
$122$	5.99054	0.542358
$123$	1.77939	0.160443
$124$	6.02907	0.541426
$125$	4.79533	0.428907
$126$	3.71597	0.331045
$127$	−5.91021	−0.524446	−0.262223	−	0.965007i	$-0.584456\pi$
$127$	−5.91021	−0.524446	−0.262223	+	0.965007i	$0.584456\pi$
$128$	−1.00000	−0.0883883
$129$	7.39911	0.651455
$130$	−10.4531	−0.916795
$131$	1.97363	0.172437	0.0862185	−	0.996276i	$-0.472522\pi$
$131$	1.97363	0.172437	0.0862185	+	0.996276i	$0.472522\pi$
$132$	−5.50753	−0.479369
$133$	−18.4142	−1.59671
$134$	−9.67367	−0.835678
$135$	2.88781	0.248543
$136$	4.95541	0.424923
$137$	−2.63621	−0.225227	−0.112613	−	0.993639i	$-0.535922\pi$
$137$	−2.63621	−0.225227	−0.112613	+	0.993639i	$0.535922\pi$
$138$	−1.00000	−0.0851257
$139$	−13.5488	−1.14919	−0.574595	−	0.818438i	$-0.694841\pi$
$139$	−13.5488	−1.14919	−0.574595	+	0.818438i	$0.694841\pi$
$140$	10.7310	0.906938
$141$	−13.6841	−1.15241
$142$	−3.39534	−0.284931
$143$	−19.9357	−1.66711
$144$	1.00000	0.0833333
$145$	2.88781	0.239820
$146$	−13.9133	−1.15147
$147$	−6.80846	−0.561552
$148$	−1.34786	−0.110793
$149$	−21.2105	−1.73763	−0.868815	−	0.495137i	$-0.835118\pi$
$149$	−21.2105	−1.73763	−0.868815	+	0.495137i	$0.835118\pi$
$150$	3.33946	0.272666
$151$	−9.47277	−0.770883	−0.385441	−	0.922732i	$-0.625951\pi$
$151$	−9.47277	−0.770883	−0.385441	+	0.922732i	$0.625951\pi$
$152$	−4.95541	−0.401937
$153$	−4.95541	−0.400621
$154$	20.4658	1.64918
$155$	−17.4108	−1.39847
$156$	3.61972	0.289809
$157$	10.1044	0.806419	0.403210	−	0.915108i	$-0.367895\pi$
$157$	10.1044	0.806419	0.403210	+	0.915108i	$0.367895\pi$
$158$	4.99623	0.397479
$159$	−2.70574	−0.214579
$160$	2.88781	0.228302
$161$	3.71597	0.292860
$162$	−1.00000	−0.0785674
$163$	−9.94758	−0.779154	−0.389577	−	0.920994i	$-0.627379\pi$
$163$	−9.94758	−0.779154	−0.389577	+	0.920994i	$0.627379\pi$
$164$	−1.77939	−0.138947
$165$	15.9047	1.23818
$166$	−9.98492	−0.774980
$167$	6.12496	0.473964	0.236982	−	0.971514i	$-0.423842\pi$
$167$	6.12496	0.473964	0.236982	+	0.971514i	$0.423842\pi$
$168$	−3.71597	−0.286694
$169$	0.102360	0.00787384
$170$	−14.3103	−1.09755
$171$	4.95541	0.378950
$172$	−7.39911	−0.564177
$173$	−5.20110	−0.395433	−0.197716	−	0.980259i	$-0.563352\pi$
$173$	−5.20110	−0.395433	−0.197716	+	0.980259i	$0.563352\pi$
$174$	−1.00000	−0.0758098
$175$	−12.4093	−0.938059
$176$	5.50753	0.415146
$177$	−8.87565	−0.667135
$178$	−11.1394	−0.834934
$179$	24.5654	1.83611	0.918053	−	0.396458i	$-0.129761\pi$
$179$	24.5654	1.83611	0.918053	+	0.396458i	$0.129761\pi$
$180$	−2.88781	−0.215245
$181$	−18.2013	−1.35289	−0.676447	−	0.736492i	$-0.736481\pi$
$181$	−18.2013	−1.35289	−0.676447	+	0.736492i	$0.736481\pi$
$182$	−13.4508	−0.997038
$183$	5.99054	0.442833
$184$	1.00000	0.0737210
$185$	3.89235	0.286171
$186$	6.02907	0.442073
$187$	−27.2921	−1.99579
$188$	13.6841	0.998016
$189$	3.71597	0.270297
$190$	14.3103	1.03818
$191$	−8.01593	−0.580012	−0.290006	−	0.957025i	$-0.593657\pi$
$191$	−8.01593	−0.580012	−0.290006	+	0.957025i	$0.593657\pi$
$192$	−1.00000	−0.0721688
$193$	−20.6982	−1.48989	−0.744945	−	0.667126i	$-0.767524\pi$
$193$	−20.6982	−1.48989	−0.744945	+	0.667126i	$0.767524\pi$
$194$	6.70574	0.481444
$195$	−10.4531	−0.748560
$196$	6.80846	0.486319
$197$	−8.35559	−0.595311	−0.297656	−	0.954673i	$-0.596205\pi$
$197$	−8.35559	−0.595311	−0.297656	+	0.954673i	$0.596205\pi$
$198$	−5.50753	−0.391403
$199$	1.72850	0.122530	0.0612650	−	0.998122i	$-0.480487\pi$
$199$	1.72850	0.122530	0.0612650	+	0.998122i	$0.480487\pi$
$200$	−3.33946	−0.236136
$201$	−9.67367	−0.682328
$202$	1.76290	0.124037
$203$	3.71597	0.260810
$204$	4.95541	0.346948
$205$	5.13856	0.358892
$206$	7.96524	0.554964
$207$	−1.00000	−0.0695048
$208$	−3.61972	−0.250982
$209$	27.2921	1.88783
$210$	10.7310	0.740512
$211$	−27.0978	−1.86549	−0.932746	−	0.360534i	$-0.882594\pi$
$211$	−27.0978	−1.86549	−0.932746	+	0.360534i	$0.882594\pi$
$212$	2.70574	0.185831
$213$	−3.39534	−0.232645
$214$	1.51548	0.103596
$215$	21.3672	1.45723
$216$	1.00000	0.0680414
$217$	−22.4039	−1.52087
$218$	8.99827	0.609440
$219$	−13.9133	−0.940174
$220$	−15.9047	−1.07230
$221$	17.9372	1.20659
$222$	−1.34786	−0.0904621
$223$	13.7789	0.922706	0.461353	−	0.887217i	$-0.347364\pi$
$223$	13.7789	0.922706	0.461353	+	0.887217i	$0.347364\pi$
$224$	3.71597	0.248284
$225$	3.33946	0.222631
$226$	9.12954	0.607288
$227$	−5.34593	−0.354822	−0.177411	−	0.984137i	$-0.556772\pi$
$227$	−5.34593	−0.354822	−0.177411	+	0.984137i	$0.556772\pi$
$228$	−4.95541	−0.328180
$229$	−13.1419	−0.868439	−0.434219	−	0.900807i	$-0.642976\pi$
$229$	−13.1419	−0.868439	−0.434219	+	0.900807i	$0.642976\pi$
$230$	−2.88781	−0.190417
$231$	20.4658	1.34655
$232$	1.00000	0.0656532
$233$	17.0564	1.11740	0.558701	−	0.829369i	$-0.311300\pi$
$233$	17.0564	1.11740	0.558701	+	0.829369i	$0.311300\pi$
$234$	3.61972	0.236628
$235$	−39.5171	−2.57781
$236$	8.87565	0.577756
$237$	4.99623	0.324540
$238$	−18.4142	−1.19361
$239$	−20.9889	−1.35766	−0.678828	−	0.734297i	$-0.737512\pi$
$239$	−20.9889	−1.35766	−0.678828	+	0.734297i	$0.737512\pi$
$240$	2.88781	0.186407
$241$	8.17097	0.526338	0.263169	−	0.964750i	$-0.415232\pi$
$241$	8.17097	0.526338	0.263169	+	0.964750i	$0.415232\pi$
$242$	−19.3329	−1.24277
$243$	−1.00000	−0.0641500
$244$	−5.99054	−0.383505
$245$	−19.6616	−1.25613
$246$	−1.77939	−0.113450
$247$	−17.9372	−1.14132
$248$	−6.02907	−0.382846
$249$	−9.98492	−0.632769
$250$	−4.79533	−0.303283
$251$	−6.36103	−0.401505	−0.200752	−	0.979642i	$-0.564339\pi$
$251$	−6.36103	−0.401505	−0.200752	+	0.979642i	$0.564339\pi$
$252$	−3.71597	−0.234084
$253$	−5.50753	−0.346255
$254$	5.91021	0.370840
$255$	−14.3103	−0.896145
$256$	1.00000	0.0625000
$257$	27.2143	1.69758	0.848790	−	0.528730i	$-0.177331\pi$
$257$	27.2143	1.69758	0.848790	+	0.528730i	$0.177331\pi$
$258$	−7.39911	−0.460649
$259$	5.00859	0.311219
$260$	10.4531	0.648272
$261$	−1.00000	−0.0618984
$262$	−1.97363	−0.121931
$263$	4.94742	0.305071	0.152536	−	0.988298i	$-0.451256\pi$
$263$	4.94742	0.305071	0.152536	+	0.988298i	$0.451256\pi$
$264$	5.50753	0.338965
$265$	−7.81366	−0.479989
$266$	18.4142	1.12905
$267$	−11.1394	−0.681721
$268$	9.67367	0.590914
$269$	13.7863	0.840566	0.420283	−	0.907393i	$-0.361931\pi$
$269$	13.7863	0.840566	0.420283	+	0.907393i	$0.361931\pi$
$270$	−2.88781	−0.175747
$271$	−15.9722	−0.970242	−0.485121	−	0.874447i	$-0.661224\pi$
$271$	−15.9722	−0.970242	−0.485121	+	0.874447i	$0.661224\pi$
$272$	−4.95541	−0.300466
$273$	−13.4508	−0.814078
$274$	2.63621	0.159259
$275$	18.3922	1.10909
$276$	1.00000	0.0601929
$277$	0.259238	0.0155761	0.00778804	−	0.999970i	$-0.497521\pi$
$277$	0.259238	0.0155761	0.00778804	+	0.999970i	$0.497521\pi$
$278$	13.5488	0.812601
$279$	6.02907	0.360951
$280$	−10.7310	−0.641302
$281$	−26.4700	−1.57907	−0.789533	−	0.613708i	$-0.789677\pi$
$281$	−26.4700	−1.57907	−0.789533	+	0.613708i	$0.789677\pi$
$282$	13.6841	0.814877
$283$	13.5133	0.803284	0.401642	−	0.915797i	$-0.368440\pi$
$283$	13.5133	0.803284	0.401642	+	0.915797i	$0.368440\pi$
$284$	3.39534	0.201477
$285$	14.3103	0.847669
$286$	19.9357	1.17882
$287$	6.61218	0.390305
$288$	−1.00000	−0.0589256
$289$	7.55609	0.444476
$290$	−2.88781	−0.169578
$291$	6.70574	0.393097
$292$	13.9133	0.814215
$293$	19.4076	1.13381	0.566903	−	0.823785i	$-0.308142\pi$
$293$	19.4076	1.13381	0.566903	+	0.823785i	$0.308142\pi$
$294$	6.80846	0.397077
$295$	−25.6312	−1.49231
$296$	1.34786	0.0783425
$297$	−5.50753	−0.319579
$298$	21.2105	1.22869
$299$	3.61972	0.209334
$300$	−3.33946	−0.192804
$301$	27.4949	1.58478
$302$	9.47277	0.545097
$303$	1.76290	0.101276
$304$	4.95541	0.284212
$305$	17.2995	0.990569
$306$	4.95541	0.283282
$307$	−3.55360	−0.202815	−0.101407	−	0.994845i	$-0.532335\pi$
$307$	−3.55360	−0.202815	−0.101407	+	0.994845i	$0.532335\pi$
$308$	−20.4658	−1.16615
$309$	7.96524	0.453127
$310$	17.4108	0.988868
$311$	−20.1790	−1.14424	−0.572122	−	0.820169i	$-0.693880\pi$
$311$	−20.1790	−1.14424	−0.572122	+	0.820169i	$0.693880\pi$
$312$	−3.61972	−0.204926
$313$	1.15591	0.0653357	0.0326679	−	0.999466i	$-0.489600\pi$
$313$	1.15591	0.0653357	0.0326679	+	0.999466i	$0.489600\pi$
$314$	−10.1044	−0.570224
$315$	10.7310	0.604625
$316$	−4.99623	−0.281060
$317$	−10.5653	−0.593404	−0.296702	−	0.954970i	$-0.595887\pi$
$317$	−10.5653	−0.593404	−0.296702	+	0.954970i	$0.595887\pi$
$318$	2.70574	0.151730
$319$	−5.50753	−0.308363
$320$	−2.88781	−0.161434
$321$	1.51548	0.0845857
$322$	−3.71597	−0.207083
$323$	−24.5561	−1.36634
$324$	1.00000	0.0555556
$325$	−12.0879	−0.670516
$326$	9.94758	0.550945
$327$	8.99827	0.497606
$328$	1.77939	0.0982506
$329$	−50.8498	−2.80344
$330$	−15.9047	−0.875526
$331$	−0.110346	−0.00606515	−0.00303257	−	0.999995i	$-0.500965\pi$
$331$	−0.110346	−0.00606515	−0.00303257	+	0.999995i	$0.500965\pi$
$332$	9.98492	0.547994
$333$	−1.34786	−0.0738620
$334$	−6.12496	−0.335143
$335$	−27.9358	−1.52629
$336$	3.71597	0.202723
$337$	−24.8678	−1.35463	−0.677317	−	0.735692i	$-0.736857\pi$
$337$	−24.8678	−1.35463	−0.677317	+	0.735692i	$0.736857\pi$
$338$	−0.102360	−0.00556765
$339$	9.12954	0.495848
$340$	14.3103	0.776085
$341$	33.2053	1.79817
$342$	−4.95541	−0.267958
$343$	0.711757	0.0384313
$344$	7.39911	0.398933
$345$	−2.88781	−0.155475
$346$	5.20110	0.279613
$347$	−0.159676	−0.00857183	−0.00428592	−	0.999991i	$-0.501364\pi$
$347$	−0.159676	−0.00857183	−0.00428592	+	0.999991i	$0.501364\pi$
$348$	1.00000	0.0536056
$349$	21.5953	1.15597	0.577985	−	0.816048i	$-0.303839\pi$
$349$	21.5953	1.15597	0.577985	+	0.816048i	$0.303839\pi$
$350$	12.4093	0.663308
$351$	3.61972	0.193206
$352$	−5.50753	−0.293552
$353$	−3.90456	−0.207819	−0.103909	−	0.994587i	$-0.533135\pi$
$353$	−3.90456	−0.207819	−0.103909	+	0.994587i	$0.533135\pi$
$354$	8.87565	0.471735
$355$	−9.80511	−0.520401
$356$	11.1394	0.590388
$357$	−18.4142	−0.974581
$358$	−24.5654	−1.29832
$359$	25.0273	1.32089	0.660446	−	0.750874i	$-0.270367\pi$
$359$	25.0273	1.32089	0.660446	+	0.750874i	$0.270367\pi$
$360$	2.88781	0.152201
$361$	5.55609	0.292426
$362$	18.2013	0.956640
$363$	−19.3329	−1.01471
$364$	13.4508	0.705012
$365$	−40.1790	−2.10307
$366$	−5.99054	−0.313130
$367$	−31.8680	−1.66350	−0.831748	−	0.555153i	$-0.812660\pi$
$367$	−31.8680	−1.66350	−0.831748	+	0.555153i	$0.812660\pi$
$368$	−1.00000	−0.0521286
$369$	−1.77939	−0.0926315
$370$	−3.89235	−0.202354
$371$	−10.0544	−0.522001
$372$	−6.02907	−0.312593
$373$	0.0879069	0.00455165	0.00227582	−	0.999997i	$-0.499276\pi$
$373$	0.0879069	0.00455165	0.00227582	+	0.999997i	$0.499276\pi$
$374$	27.2921	1.41124
$375$	−4.79533	−0.247630
$376$	−13.6841	−0.705704
$377$	3.61972	0.186425
$378$	−3.71597	−0.191129
$379$	20.7932	1.06807	0.534037	−	0.845461i	$-0.320674\pi$
$379$	20.7932	1.06807	0.534037	+	0.845461i	$0.320674\pi$
$380$	−14.3103	−0.734103
$381$	5.91021	0.302789
$382$	8.01593	0.410131
$383$	1.33991	0.0684661	0.0342331	−	0.999414i	$-0.489101\pi$
$383$	1.33991	0.0684661	0.0342331	+	0.999414i	$0.489101\pi$
$384$	1.00000	0.0510310
$385$	59.1015	3.01209
$386$	20.6982	1.05351
$387$	−7.39911	−0.376118
$388$	−6.70574	−0.340432
$389$	33.9141	1.71951	0.859757	−	0.510703i	$-0.170615\pi$
$389$	33.9141	1.71951	0.859757	+	0.510703i	$0.170615\pi$
$390$	10.4531	0.529312
$391$	4.95541	0.250606
$392$	−6.80846	−0.343879
$393$	−1.97363	−0.0995566
$394$	8.35559	0.420949
$395$	14.4282	0.725960
$396$	5.50753	0.276764
$397$	12.8962	0.647242	0.323621	−	0.946187i	$-0.395100\pi$
$397$	12.8962	0.647242	0.323621	+	0.946187i	$0.395100\pi$
$398$	−1.72850	−0.0866419
$399$	18.4142	0.921862
$400$	3.33946	0.166973
$401$	33.8000	1.68789	0.843945	−	0.536430i	$-0.180227\pi$
$401$	33.8000	1.68789	0.843945	+	0.536430i	$0.180227\pi$
$402$	9.67367	0.482479
$403$	−21.8235	−1.08711
$404$	−1.76290	−0.0877075
$405$	−2.88781	−0.143497
$406$	−3.71597	−0.184421
$407$	−7.42335	−0.367962
$408$	−4.95541	−0.245329
$409$	−21.4792	−1.06208	−0.531039	−	0.847348i	$-0.678198\pi$
$409$	−21.4792	−1.06208	−0.531039	+	0.847348i	$0.678198\pi$
$410$	−5.13856	−0.253775
$411$	2.63621	0.130035
$412$	−7.96524	−0.392419
$413$	−32.9817	−1.62292
$414$	1.00000	0.0491473
$415$	−28.8346	−1.41543
$416$	3.61972	0.177471
$417$	13.5488	0.663486
$418$	−27.2921	−1.33490
$419$	−13.3422	−0.651811	−0.325906	−	0.945402i	$-0.605669\pi$
$419$	−13.3422	−0.651811	−0.325906	+	0.945402i	$0.605669\pi$
$420$	−10.7310	−0.523621
$421$	−2.78087	−0.135531	−0.0677657	−	0.997701i	$-0.521587\pi$
$421$	−2.78087	−0.135531	−0.0677657	+	0.997701i	$0.521587\pi$
$422$	27.0978	1.31910
$423$	13.6841	0.665344
$424$	−2.70574	−0.131402
$425$	−16.5484	−0.802715
$426$	3.39534	0.164505
$427$	22.2607	1.07727
$428$	−1.51548	−0.0732533
$429$	19.9357	0.962505
$430$	−21.3672	−1.03042
$431$	7.88445	0.379781	0.189890	−	0.981805i	$-0.439187\pi$
$431$	7.88445	0.379781	0.189890	+	0.981805i	$0.439187\pi$
$432$	−1.00000	−0.0481125
$433$	0.0431567	0.00207398	0.00103699	−	0.999999i	$-0.499670\pi$
$433$	0.0431567	0.00207398	0.00103699	+	0.999999i	$0.499670\pi$
$434$	22.4039	1.07542
$435$	−2.88781	−0.138460
$436$	−8.99827	−0.430939
$437$	−4.95541	−0.237049
$438$	13.9133	0.664804
$439$	24.8532	1.18618	0.593089	−	0.805137i	$-0.297908\pi$
$439$	24.8532	1.18618	0.593089	+	0.805137i	$0.297908\pi$
$440$	15.9047	0.758228
$441$	6.80846	0.324212
$442$	−17.9372	−0.853185
$443$	−21.9452	−1.04265	−0.521323	−	0.853359i	$-0.674561\pi$
$443$	−21.9452	−1.04265	−0.521323	+	0.853359i	$0.674561\pi$
$444$	1.34786	0.0639664
$445$	−32.1685	−1.52493
$446$	−13.7789	−0.652451
$447$	21.2105	1.00322
$448$	−3.71597	−0.175563
$449$	−10.9510	−0.516811	−0.258405	−	0.966037i	$-0.583197\pi$
$449$	−10.9510	−0.516811	−0.258405	+	0.966037i	$0.583197\pi$
$450$	−3.33946	−0.157424
$451$	−9.80007	−0.461467
$452$	−9.12954	−0.429417
$453$	9.47277	0.445069
$454$	5.34593	0.250897
$455$	−38.8433	−1.82100
$456$	4.95541	0.232058
$457$	−22.7893	−1.06604	−0.533018	−	0.846104i	$-0.678942\pi$
$457$	−22.7893	−1.06604	−0.533018	+	0.846104i	$0.678942\pi$
$458$	13.1419	0.614079
$459$	4.95541	0.231299
$460$	2.88781	0.134645
$461$	−36.2538	−1.68851	−0.844254	−	0.535944i	$-0.819956\pi$
$461$	−36.2538	−1.68851	−0.844254	+	0.535944i	$0.819956\pi$
$462$	−20.4658	−0.952157
$463$	6.98071	0.324421	0.162210	−	0.986756i	$-0.448138\pi$
$463$	6.98071	0.324421	0.162210	+	0.986756i	$0.448138\pi$
$464$	−1.00000	−0.0464238
$465$	17.4108	0.807407
$466$	−17.0564	−0.790123
$467$	−30.8335	−1.42680	−0.713402	−	0.700755i	$-0.752847\pi$
$467$	−30.8335	−1.42680	−0.713402	+	0.700755i	$0.752847\pi$
$468$	−3.61972	−0.167322
$469$	−35.9471	−1.65988
$470$	39.5171	1.82279
$471$	−10.1044	−0.465586
$472$	−8.87565	−0.408535
$473$	−40.7508	−1.87373
$474$	−4.99623	−0.229484
$475$	16.5484	0.759293
$476$	18.4142	0.844012
$477$	2.70574	0.123887
$478$	20.9889	0.960008
$479$	−37.8452	−1.72919	−0.864595	−	0.502470i	$-0.832425\pi$
$479$	−37.8452	−1.72919	−0.864595	+	0.502470i	$0.832425\pi$
$480$	−2.88781	−0.131810
$481$	4.87886	0.222457
$482$	−8.17097	−0.372177
$483$	−3.71597	−0.169083
$484$	19.3329	0.878768
$485$	19.3649	0.879315
$486$	1.00000	0.0453609
$487$	0.692927	0.0313995	0.0156998	−	0.999877i	$-0.495002\pi$
$487$	0.692927	0.0313995	0.0156998	+	0.999877i	$0.495002\pi$
$488$	5.99054	0.271179
$489$	9.94758	0.449845
$490$	19.6616	0.888219
$491$	8.52194	0.384590	0.192295	−	0.981337i	$-0.438407\pi$
$491$	8.52194	0.384590	0.192295	+	0.981337i	$0.438407\pi$
$492$	1.77939	0.0802213
$493$	4.95541	0.223180
$494$	17.9372	0.807032
$495$	−15.9047	−0.714864
$496$	6.02907	0.270713
$497$	−12.6170	−0.565950
$498$	9.98492	0.447435
$499$	7.23643	0.323947	0.161974	−	0.986795i	$-0.448214\pi$
$499$	7.23643	0.323947	0.161974	+	0.986795i	$0.448214\pi$
$500$	4.79533	0.214453
$501$	−6.12496	−0.273643
$502$	6.36103	0.283907
$503$	0.408278	0.0182042	0.00910210	−	0.999959i	$-0.497103\pi$
$503$	0.408278	0.0182042	0.00910210	+	0.999959i	$0.497103\pi$
$504$	3.71597	0.165523
$505$	5.09092	0.226543
$506$	5.50753	0.244840
$507$	−0.102360	−0.00454597
$508$	−5.91021	−0.262223
$509$	−33.3664	−1.47894	−0.739470	−	0.673189i	$-0.764924\pi$
$509$	−33.3664	−1.47894	−0.739470	+	0.673189i	$0.764924\pi$
$510$	14.3103	0.633671
$511$	−51.7015	−2.28714
$512$	−1.00000	−0.0441942
$513$	−4.95541	−0.218787
$514$	−27.2143	−1.20037
$515$	23.0021	1.01359
$516$	7.39911	0.325728
$517$	75.3657	3.31458
$518$	−5.00859	−0.220065
$519$	5.20110	0.228303
$520$	−10.4531	−0.458397
$521$	−10.1756	−0.445802	−0.222901	−	0.974841i	$-0.571553\pi$
$521$	−10.1756	−0.445802	−0.222901	+	0.974841i	$0.571553\pi$
$522$	1.00000	0.0437688
$523$	9.53775	0.417057	0.208528	−	0.978016i	$-0.433133\pi$
$523$	9.53775	0.417057	0.208528	+	0.978016i	$0.433133\pi$
$524$	1.97363	0.0862185
$525$	12.4093	0.541588
$526$	−4.94742	−0.215718
$527$	−29.8765	−1.30144
$528$	−5.50753	−0.239685
$529$	1.00000	0.0434783
$530$	7.81366	0.339404
$531$	8.87565	0.385170
$532$	−18.4142	−0.798356
$533$	6.44090	0.278987
$534$	11.1394	0.482050
$535$	4.37641	0.189209
$536$	−9.67367	−0.417839
$537$	−24.5654	−1.06008
$538$	−13.7863	−0.594370
$539$	37.4978	1.61514
$540$	2.88781	0.124272
$541$	−19.3459	−0.831744	−0.415872	−	0.909423i	$-0.636524\pi$
$541$	−19.3459	−0.831744	−0.415872	+	0.909423i	$0.636524\pi$
$542$	15.9722	0.686065
$543$	18.2013	0.781093
$544$	4.95541	0.212461
$545$	25.9853	1.11309
$546$	13.4508	0.575640
$547$	−15.4328	−0.659861	−0.329930	−	0.944005i	$-0.607025\pi$
$547$	−15.4328	−0.659861	−0.329930	+	0.944005i	$0.607025\pi$
$548$	−2.63621	−0.112613
$549$	−5.99054	−0.255670
$550$	−18.3922	−0.784245
$551$	−4.95541	−0.211108
$552$	−1.00000	−0.0425628
$553$	18.5659	0.789501
$554$	−0.259238	−0.0110140
$555$	−3.89235	−0.165221
$556$	−13.5488	−0.574595
$557$	−39.6466	−1.67988	−0.839941	−	0.542678i	$-0.817410\pi$
$557$	−39.6466	−1.67988	−0.839941	+	0.542678i	$0.817410\pi$
$558$	−6.02907	−0.255231
$559$	26.7827	1.13279
$560$	10.7310	0.453469
$561$	27.2921	1.15227
$562$	26.4700	1.11657
$563$	−4.87087	−0.205283	−0.102641	−	0.994718i	$-0.532729\pi$
$563$	−4.87087	−0.205283	−0.102641	+	0.994718i	$0.532729\pi$
$564$	−13.6841	−0.576205
$565$	26.3644	1.10916
$566$	−13.5133	−0.568007
$567$	−3.71597	−0.156056
$568$	−3.39534	−0.142465
$569$	12.6800	0.531575	0.265787	−	0.964032i	$-0.414368\pi$
$569$	12.6800	0.531575	0.265787	+	0.964032i	$0.414368\pi$
$570$	−14.3103	−0.599392
$571$	−39.1654	−1.63902	−0.819511	−	0.573064i	$-0.805755\pi$
$571$	−39.1654	−1.63902	−0.819511	+	0.573064i	$0.805755\pi$
$572$	−19.9357	−0.833554
$573$	8.01593	0.334870
$574$	−6.61218	−0.275987
$575$	−3.33946	−0.139265
$576$	1.00000	0.0416667
$577$	3.95774	0.164763	0.0823814	−	0.996601i	$-0.473747\pi$
$577$	3.95774	0.164763	0.0823814	+	0.996601i	$0.473747\pi$
$578$	−7.55609	−0.314292
$579$	20.6982	0.860188
$580$	2.88781	0.119910
$581$	−37.1037	−1.53932
$582$	−6.70574	−0.277962
$583$	14.9019	0.617175
$584$	−13.9133	−0.575737
$585$	10.4531	0.432181
$586$	−19.4076	−0.801721
$587$	19.7477	0.815074	0.407537	−	0.913189i	$-0.366388\pi$
$587$	19.7477	0.815074	0.407537	+	0.913189i	$0.366388\pi$
$588$	−6.80846	−0.280776
$589$	29.8765	1.23104
$590$	25.6312	1.05522
$591$	8.35559	0.343703
$592$	−1.34786	−0.0553965
$593$	39.9052	1.63871	0.819356	−	0.573286i	$-0.194331\pi$
$593$	39.9052	1.63871	0.819356	+	0.573286i	$0.194331\pi$
$594$	5.50753	0.225977
$595$	−53.1767	−2.18003
$596$	−21.2105	−0.868815
$597$	−1.72850	−0.0707428
$598$	−3.61972	−0.148021
$599$	−34.3227	−1.40239	−0.701193	−	0.712972i	$-0.747349\pi$
$599$	−34.3227	−1.40239	−0.701193	+	0.712972i	$0.747349\pi$
$600$	3.33946	0.136333
$601$	−22.5528	−0.919949	−0.459975	−	0.887932i	$-0.652141\pi$
$601$	−22.5528	−0.919949	−0.459975	+	0.887932i	$0.652141\pi$
$602$	−27.4949	−1.12061
$603$	9.67367	0.393942
$604$	−9.47277	−0.385441
$605$	−55.8298	−2.26980
$606$	−1.76290	−0.0716129
$607$	−17.9582	−0.728901	−0.364451	−	0.931223i	$-0.618743\pi$
$607$	−17.9582	−0.728901	−0.364451	+	0.931223i	$0.618743\pi$
$608$	−4.95541	−0.200968
$609$	−3.71597	−0.150579
$610$	−17.2995	−0.700438
$611$	−49.5326	−2.00388
$612$	−4.95541	−0.200311
$613$	40.6855	1.64327	0.821637	−	0.570012i	$-0.193061\pi$
$613$	40.6855	1.64327	0.821637	+	0.570012i	$0.193061\pi$
$614$	3.55360	0.143412
$615$	−5.13856	−0.207207
$616$	20.4658	0.824592
$617$	23.4744	0.945045	0.472523	−	0.881319i	$-0.343344\pi$
$617$	23.4744	0.945045	0.472523	+	0.881319i	$0.343344\pi$
$618$	−7.96524	−0.320409
$619$	28.8906	1.16121	0.580605	−	0.814186i	$-0.302816\pi$
$619$	28.8906	1.16121	0.580605	+	0.814186i	$0.302816\pi$
$620$	−17.4108	−0.699235
$621$	1.00000	0.0401286
$622$	20.1790	0.809102
$623$	−41.3938	−1.65841
$624$	3.61972	0.144905
$625$	−30.5453	−1.22181
$626$	−1.15591	−0.0461993
$627$	−27.2921	−1.08994
$628$	10.1044	0.403210
$629$	6.67917	0.266316
$630$	−10.7310	−0.427535
$631$	−40.4437	−1.61004	−0.805020	−	0.593248i	$-0.797845\pi$
$631$	−40.4437	−1.61004	−0.805020	+	0.593248i	$0.797845\pi$
$632$	4.99623	0.198739
$633$	27.0978	1.07704
$634$	10.5653	0.419600
$635$	17.0676	0.677306
$636$	−2.70574	−0.107289
$637$	−24.6447	−0.976459
$638$	5.50753	0.218045
$639$	3.39534	0.134318
$640$	2.88781	0.114151
$641$	39.5362	1.56159	0.780794	−	0.624789i	$-0.214815\pi$
$641$	39.5362	1.56159	0.780794	+	0.624789i	$0.214815\pi$
$642$	−1.51548	−0.0598111
$643$	22.3465	0.881261	0.440631	−	0.897688i	$-0.354755\pi$
$643$	22.3465	0.881261	0.440631	+	0.897688i	$0.354755\pi$
$644$	3.71597	0.146430
$645$	−21.3672	−0.841335
$646$	24.5561	0.966146
$647$	−8.23332	−0.323685	−0.161843	−	0.986817i	$-0.551744\pi$
$647$	−8.23332	−0.323685	−0.161843	+	0.986817i	$0.551744\pi$
$648$	−1.00000	−0.0392837
$649$	48.8829	1.91882
$650$	12.0879	0.474127
$651$	22.4039	0.878076
$652$	−9.94758	−0.389577
$653$	−13.0292	−0.509870	−0.254935	−	0.966958i	$-0.582054\pi$
$653$	−13.0292	−0.509870	−0.254935	+	0.966958i	$0.582054\pi$
$654$	−8.99827	−0.351860
$655$	−5.69948	−0.222697
$656$	−1.77939	−0.0694737
$657$	13.9133	0.542810
$658$	50.8498	1.98233
$659$	−0.234991	−0.00915393	−0.00457697	−	0.999990i	$-0.501457\pi$
$659$	−0.234991	−0.00915393	−0.00457697	+	0.999990i	$0.501457\pi$
$660$	15.9047	0.619090
$661$	6.53996	0.254375	0.127188	−	0.991879i	$-0.459405\pi$
$661$	6.53996	0.254375	0.127188	+	0.991879i	$0.459405\pi$
$662$	0.110346	0.00428871
$663$	−17.9372	−0.696623
$664$	−9.98492	−0.387490
$665$	53.1767	2.06210
$666$	1.34786	0.0522283
$667$	1.00000	0.0387202
$668$	6.12496	0.236982
$669$	−13.7789	−0.532724
$670$	27.9358	1.07925
$671$	−32.9931	−1.27368
$672$	−3.71597	−0.143347
$673$	2.67081	0.102952	0.0514761	−	0.998674i	$-0.483607\pi$
$673$	2.67081	0.102952	0.0514761	+	0.998674i	$0.483607\pi$
$674$	24.8678	0.957870
$675$	−3.33946	−0.128536
$676$	0.102360	0.00393692
$677$	−24.0923	−0.925944	−0.462972	−	0.886373i	$-0.653217\pi$
$677$	−24.0923	−0.925944	−0.462972	+	0.886373i	$0.653217\pi$
$678$	−9.12954	−0.350618
$679$	24.9183	0.956278
$680$	−14.3103	−0.548775
$681$	5.34593	0.204856
$682$	−33.2053	−1.27150
$683$	−49.1611	−1.88110	−0.940548	−	0.339660i	$-0.889688\pi$
$683$	−49.1611	−1.88110	−0.940548	+	0.339660i	$0.889688\pi$
$684$	4.95541	0.189475
$685$	7.61289	0.290874
$686$	−0.711757	−0.0271750
$687$	13.1419	0.501393
$688$	−7.39911	−0.282089
$689$	−9.79401	−0.373122
$690$	2.88781	0.109937
$691$	−15.8128	−0.601549	−0.300774	−	0.953695i	$-0.597245\pi$
$691$	−15.8128	−0.601549	−0.300774	+	0.953695i	$0.597245\pi$
$692$	−5.20110	−0.197716
$693$	−20.4658	−0.777433
$694$	0.159676	0.00606120
$695$	39.1263	1.48414
$696$	−1.00000	−0.0379049
$697$	8.81763	0.333991
$698$	−21.5953	−0.817394
$699$	−17.0564	−0.645133
$700$	−12.4093	−0.469029
$701$	19.0238	0.718517	0.359259	−	0.933238i	$-0.383030\pi$
$701$	19.0238	0.718517	0.359259	+	0.933238i	$0.383030\pi$
$702$	−3.61972	−0.136617
$703$	−6.67917	−0.251910
$704$	5.50753	0.207573
$705$	39.5171	1.48830
$706$	3.90456	0.146950
$707$	6.55089	0.246371
$708$	−8.87565	−0.333567
$709$	−27.0229	−1.01487	−0.507433	−	0.861691i	$-0.669406\pi$
$709$	−27.0229	−1.01487	−0.507433	+	0.861691i	$0.669406\pi$
$710$	9.80511	0.367979
$711$	−4.99623	−0.187373
$712$	−11.1394	−0.417467
$713$	−6.02907	−0.225790
$714$	18.4142	0.689133
$715$	57.5706	2.15302
$716$	24.5654	0.918053
$717$	20.9889	0.783843
$718$	−25.0273	−0.934011
$719$	16.5250	0.616277	0.308138	−	0.951342i	$-0.400294\pi$
$719$	16.5250	0.616277	0.308138	+	0.951342i	$0.400294\pi$
$720$	−2.88781	−0.107622
$721$	29.5986	1.10231
$722$	−5.55609	−0.206776
$723$	−8.17097	−0.303882
$724$	−18.2013	−0.676447
$725$	−3.33946	−0.124024
$726$	19.3329	0.717511
$727$	35.7940	1.32752	0.663762	−	0.747944i	$-0.268959\pi$
$727$	35.7940	1.32752	0.663762	+	0.747944i	$0.268959\pi$
$728$	−13.4508	−0.498519
$729$	1.00000	0.0370370
$730$	40.1790	1.48709
$731$	36.6656	1.35613
$732$	5.99054	0.221417
$733$	30.3125	1.11962	0.559808	−	0.828622i	$-0.310875\pi$
$733$	30.3125	1.11962	0.559808	+	0.828622i	$0.310875\pi$
$734$	31.8680	1.17627
$735$	19.6616	0.725227
$736$	1.00000	0.0368605
$737$	53.2781	1.96252
$738$	1.77939	0.0655004
$739$	17.6301	0.648534	0.324267	−	0.945966i	$-0.394882\pi$
$739$	17.6301	0.648534	0.324267	+	0.945966i	$0.394882\pi$
$740$	3.89235	0.143086
$741$	17.9372	0.658939
$742$	10.0544	0.369110
$743$	25.1878	0.924050	0.462025	−	0.886867i	$-0.347123\pi$
$743$	25.1878	0.924050	0.462025	+	0.886867i	$0.347123\pi$
$744$	6.02907	0.221036
$745$	61.2519	2.24409
$746$	−0.0879069	−0.00321850
$747$	9.98492	0.365329
$748$	−27.2921	−0.997897
$749$	5.63147	0.205770
$750$	4.79533	0.175101
$751$	−10.4646	−0.381858	−0.190929	−	0.981604i	$-0.561150\pi$
$751$	−10.4646	−0.381858	−0.190929	+	0.981604i	$0.561150\pi$
$752$	13.6841	0.499008
$753$	6.36103	0.231809
$754$	−3.61972	−0.131822
$755$	27.3556	0.995571
$756$	3.71597	0.135149
$757$	22.8522	0.830577	0.415289	−	0.909690i	$-0.363681\pi$
$757$	22.8522	0.830577	0.415289	+	0.909690i	$0.363681\pi$
$758$	−20.7932	−0.755242
$759$	5.50753	0.199911
$760$	14.3103	0.519089
$761$	−43.1330	−1.56357	−0.781785	−	0.623548i	$-0.785690\pi$
$761$	−43.1330	−1.56357	−0.781785	+	0.623548i	$0.785690\pi$
$762$	−5.91021	−0.214104
$763$	33.4373	1.21051
$764$	−8.01593	−0.290006
$765$	14.3103	0.517390
$766$	−1.33991	−0.0484129
$767$	−32.1273	−1.16005
$768$	−1.00000	−0.0360844
$769$	42.7582	1.54190	0.770950	−	0.636895i	$-0.219782\pi$
$769$	42.7582	1.54190	0.770950	+	0.636895i	$0.219782\pi$
$770$	−59.1015	−2.12987
$771$	−27.2143	−0.980099
$772$	−20.6982	−0.744945
$773$	23.9548	0.861593	0.430796	−	0.902449i	$-0.358233\pi$
$773$	23.9548	0.861593	0.430796	+	0.902449i	$0.358233\pi$
$774$	7.39911	0.265956
$775$	20.1338	0.723229
$776$	6.70574	0.240722
$777$	−5.00859	−0.179682
$778$	−33.9141	−1.21588
$779$	−8.81763	−0.315924
$780$	−10.4531	−0.374280
$781$	18.7000	0.669137
$782$	−4.95541	−0.177205
$783$	1.00000	0.0357371
$784$	6.80846	0.243159
$785$	−29.1796	−1.04147
$786$	1.97363	0.0703971
$787$	−46.7137	−1.66516	−0.832582	−	0.553901i	$-0.813139\pi$
$787$	−46.7137	−1.66516	−0.832582	+	0.553901i	$0.813139\pi$
$788$	−8.35559	−0.297656
$789$	−4.94742	−0.176133
$790$	−14.4282	−0.513332
$791$	33.9251	1.20624
$792$	−5.50753	−0.195702
$793$	21.6840	0.770023
$794$	−12.8962	−0.457669
$795$	7.81366	0.277122
$796$	1.72850	0.0612650
$797$	−45.2735	−1.60367	−0.801836	−	0.597545i	$-0.796143\pi$
$797$	−45.2735	−1.60367	−0.801836	+	0.597545i	$0.796143\pi$
$798$	−18.4142	−0.651855
$799$	−67.8104	−2.39896
$800$	−3.33946	−0.118068
$801$	11.1394	0.393592
$802$	−33.8000	−1.19352
$803$	76.6280	2.70414
$804$	−9.67367	−0.341164
$805$	−10.7310	−0.378219
$806$	21.8235	0.768701
$807$	−13.7863	−0.485301
$808$	1.76290	0.0620186
$809$	20.9077	0.735076	0.367538	−	0.930009i	$-0.380201\pi$
$809$	20.9077	0.735076	0.367538	+	0.930009i	$0.380201\pi$
$810$	2.88781	0.101467
$811$	−15.5488	−0.545991	−0.272995	−	0.962015i	$-0.588014\pi$
$811$	−15.5488	−0.545991	−0.272995	+	0.962015i	$0.588014\pi$
$812$	3.71597	0.130405
$813$	15.9722	0.560169
$814$	7.42335	0.260188
$815$	28.7267	1.00625
$816$	4.95541	0.173474
$817$	−36.6656	−1.28277
$818$	21.4792	0.751002
$819$	13.4508	0.470008
$820$	5.13856	0.179446
$821$	29.8698	1.04246	0.521232	−	0.853415i	$-0.325473\pi$
$821$	29.8698	1.04246	0.521232	+	0.853415i	$0.325473\pi$
$822$	−2.63621	−0.0919485
$823$	8.60869	0.300080	0.150040	−	0.988680i	$-0.452060\pi$
$823$	8.60869	0.300080	0.150040	+	0.988680i	$0.452060\pi$
$824$	7.96524	0.277482
$825$	−18.3922	−0.640334
$826$	32.9817	1.14758
$827$	35.8295	1.24591	0.622957	−	0.782256i	$-0.285931\pi$
$827$	35.8295	1.24591	0.622957	+	0.782256i	$0.285931\pi$
$828$	−1.00000	−0.0347524
$829$	−43.0591	−1.49551	−0.747753	−	0.663977i	$-0.768867\pi$
$829$	−43.0591	−1.49551	−0.747753	+	0.663977i	$0.768867\pi$
$830$	28.8346	1.00086
$831$	−0.259238	−0.00899286
$832$	−3.61972	−0.125491
$833$	−33.7387	−1.16898
$834$	−13.5488	−0.469155
$835$	−17.6877	−0.612109
$836$	27.2921	0.943916
$837$	−6.02907	−0.208395
$838$	13.3422	0.460900
$839$	23.0922	0.797230	0.398615	−	0.917118i	$-0.369491\pi$
$839$	23.0922	0.797230	0.398615	+	0.917118i	$0.369491\pi$
$840$	10.7310	0.370256
$841$	1.00000	0.0344828
$842$	2.78087	0.0958352
$843$	26.4700	0.911674
$844$	−27.0978	−0.932746
$845$	−0.295596	−0.0101688
$846$	−13.6841	−0.470469
$847$	−71.8405	−2.46847
$848$	2.70574	0.0929154
$849$	−13.5133	−0.463776
$850$	16.5484	0.567605
$851$	1.34786	0.0462039
$852$	−3.39534	−0.116323
$853$	48.6314	1.66511	0.832553	−	0.553945i	$-0.186878\pi$
$853$	48.6314	1.66511	0.832553	+	0.553945i	$0.186878\pi$
$854$	−22.2607	−0.761744
$855$	−14.3103	−0.489402
$856$	1.51548	0.0517979
$857$	7.79582	0.266300	0.133150	−	0.991096i	$-0.457491\pi$
$857$	7.79582	0.266300	0.133150	+	0.991096i	$0.457491\pi$
$858$	−19.9357	−0.680594
$859$	14.8402	0.506340	0.253170	−	0.967422i	$-0.418527\pi$
$859$	14.8402	0.506340	0.253170	+	0.967422i	$0.418527\pi$
$860$	21.3672	0.728617
$861$	−6.61218	−0.225343
$862$	−7.88445	−0.268546
$863$	12.5125	0.425931	0.212966	−	0.977060i	$-0.431688\pi$
$863$	12.5125	0.425931	0.212966	+	0.977060i	$0.431688\pi$
$864$	1.00000	0.0340207
$865$	15.0198	0.510689
$866$	−0.0431567	−0.00146652
$867$	−7.55609	−0.256618
$868$	−22.4039	−0.760436
$869$	−27.5169	−0.933447
$870$	2.88781	0.0979060
$871$	−35.0160	−1.18647
$872$	8.99827	0.304720
$873$	−6.70574	−0.226955
$874$	4.95541	0.167619
$875$	−17.8193	−0.602402
$876$	−13.9133	−0.470087
$877$	16.0092	0.540592	0.270296	−	0.962777i	$-0.412878\pi$
$877$	16.0092	0.540592	0.270296	+	0.962777i	$0.412878\pi$
$878$	−24.8532	−0.838755
$879$	−19.4076	−0.654603
$880$	−15.9047	−0.536148
$881$	−56.3782	−1.89943	−0.949714	−	0.313118i	$-0.898626\pi$
$881$	−56.3782	−1.89943	−0.949714	+	0.313118i	$0.898626\pi$
$882$	−6.80846	−0.229253
$883$	37.3553	1.25711	0.628553	−	0.777767i	$-0.283648\pi$
$883$	37.3553	1.25711	0.628553	+	0.777767i	$0.283648\pi$
$884$	17.9372	0.603293
$885$	25.6312	0.861584
$886$	21.9452	0.737263
$887$	36.3244	1.21965	0.609827	−	0.792534i	$-0.291239\pi$
$887$	36.3244	1.21965	0.609827	+	0.792534i	$0.291239\pi$
$888$	−1.34786	−0.0452311
$889$	21.9622	0.736588
$890$	32.1685	1.07829
$891$	5.50753	0.184509
$892$	13.7789	0.461353
$893$	67.8104	2.26919
$894$	−21.2105	−0.709384
$895$	−70.9403	−2.37127
$896$	3.71597	0.124142
$897$	−3.61972	−0.120859
$898$	10.9510	0.365440
$899$	−6.02907	−0.201081
$900$	3.33946	0.111315
$901$	−13.4080	−0.446687
$902$	9.80007	0.326307
$903$	−27.4949	−0.914973
$904$	9.12954	0.303644
$905$	52.5620	1.74722
$906$	−9.47277	−0.314712
$907$	56.5181	1.87665	0.938326	−	0.345752i	$-0.112376\pi$
$907$	56.5181	1.87665	0.938326	+	0.345752i	$0.112376\pi$
$908$	−5.34593	−0.177411
$909$	−1.76290	−0.0584717
$910$	38.8433	1.28764
$911$	38.5719	1.27794	0.638972	−	0.769230i	$-0.279360\pi$
$911$	38.5719	1.27794	0.638972	+	0.769230i	$0.279360\pi$
$912$	−4.95541	−0.164090
$913$	54.9923	1.81998
$914$	22.7893	0.753802
$915$	−17.2995	−0.571905
$916$	−13.1419	−0.434219
$917$	−7.33396	−0.242189
$918$	−4.95541	−0.163553
$919$	−29.4673	−0.972036	−0.486018	−	0.873949i	$-0.661551\pi$
$919$	−29.4673	−0.972036	−0.486018	+	0.873949i	$0.661551\pi$
$920$	−2.88781	−0.0952084
$921$	3.55360	0.117095
$922$	36.2538	1.19395
$923$	−12.2902	−0.404536
$924$	20.4658	0.673277
$925$	−4.50111	−0.147996
$926$	−6.98071	−0.229400
$927$	−7.96524	−0.261613
$928$	1.00000	0.0328266
$929$	12.9544	0.425021	0.212511	−	0.977159i	$-0.431836\pi$
$929$	12.9544	0.425021	0.212511	+	0.977159i	$0.431836\pi$
$930$	−17.4108	−0.570923
$931$	33.7387	1.10574
$932$	17.0564	0.558701
$933$	20.1790	0.660629
$934$	30.8335	1.00890
$935$	78.8144	2.57751
$936$	3.61972	0.118314
$937$	23.5246	0.768516	0.384258	−	0.923226i	$-0.374457\pi$
$937$	23.5246	0.768516	0.384258	+	0.923226i	$0.374457\pi$
$938$	35.9471	1.17371
$939$	−1.15591	−0.0377216
$940$	−39.5171	−1.28891
$941$	−39.5272	−1.28855	−0.644275	−	0.764794i	$-0.722841\pi$
$941$	−39.5272	−1.28855	−0.644275	+	0.764794i	$0.722841\pi$
$942$	10.1044	0.329219
$943$	1.77939	0.0579450
$944$	8.87565	0.288878
$945$	−10.7310	−0.349081
$946$	40.7508	1.32492
$947$	42.4800	1.38041	0.690207	−	0.723612i	$-0.257519\pi$
$947$	42.4800	1.38041	0.690207	+	0.723612i	$0.257519\pi$
$948$	4.99623	0.162270
$949$	−50.3623	−1.63483
$950$	−16.5484	−0.536901
$951$	10.5653	0.342602
$952$	−18.4142	−0.596807
$953$	−47.5687	−1.54090	−0.770451	−	0.637499i	$-0.779969\pi$
$953$	−47.5687	−1.54090	−0.770451	+	0.637499i	$0.779969\pi$
$954$	−2.70574	−0.0876015
$955$	23.1485	0.749068
$956$	−20.9889	−0.678828
$957$	5.50753	0.178033
$958$	37.8452	1.22272
$959$	9.79610	0.316332
$960$	2.88781	0.0932037
$961$	5.34964	0.172569
$962$	−4.87886	−0.157301
$963$	−1.51548	−0.0488356
$964$	8.17097	0.263169
$965$	59.7725	1.92415
$966$	3.71597	0.119559
$967$	14.6186	0.470102	0.235051	−	0.971983i	$-0.424474\pi$
$967$	14.6186	0.470102	0.235051	+	0.971983i	$0.424474\pi$
$968$	−19.3329	−0.621383
$969$	24.5561	0.788855
$970$	−19.3649	−0.621770
$971$	1.48997	0.0478153	0.0239076	−	0.999714i	$-0.492389\pi$
$971$	1.48997	0.0478153	0.0239076	+	0.999714i	$0.492389\pi$
$972$	−1.00000	−0.0320750
$973$	50.3468	1.61405
$974$	−0.692927	−0.0222028
$975$	12.0879	0.387123
$976$	−5.99054	−0.191752
$977$	−37.4869	−1.19931	−0.599656	−	0.800258i	$-0.704696\pi$
$977$	−37.4869	−1.19931	−0.599656	+	0.800258i	$0.704696\pi$
$978$	−9.94758	−0.318088
$979$	61.3507	1.96078
$980$	−19.6616	−0.628065
$981$	−8.99827	−0.287293
$982$	−8.52194	−0.271946
$983$	−59.5677	−1.89991	−0.949957	−	0.312381i	$-0.898873\pi$
$983$	−59.5677	−1.89991	−0.949957	+	0.312381i	$0.898873\pi$
$984$	−1.77939	−0.0567250
$985$	24.1294	0.768826
$986$	−4.95541	−0.157812
$987$	50.8498	1.61857
$988$	−17.9372	−0.570658
$989$	7.39911	0.235278
$990$	15.9047	0.505485
$991$	−31.9628	−1.01533	−0.507666	−	0.861554i	$-0.669492\pi$
$991$	−31.9628	−1.01533	−0.507666	+	0.861554i	$0.669492\pi$
$992$	−6.02907	−0.191423
$993$	0.110346	0.00350171
$994$	12.6170	0.400187
$995$	−4.99158	−0.158244
$996$	−9.98492	−0.316384
$997$	32.0290	1.01437	0.507184	−	0.861838i	$-0.330686\pi$
$997$	32.0290	1.01437	0.507184	+	0.861838i	$0.330686\pi$
$998$	−7.23643	−0.229065
$999$	1.34786	0.0426442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.be.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.be.1.1	✓	5	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.88781 q^{5} +1.00000 q^{6} -3.71597 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.88781 q^{5} +1.00000 q^{6} -3.71597 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.88781 q^{10} +5.50753 q^{11} -1.00000 q^{12} -3.61972 q^{13} +3.71597 q^{14} +2.88781 q^{15} +1.00000 q^{16} -4.95541 q^{17} -1.00000 q^{18} +4.95541 q^{19} -2.88781 q^{20} +3.71597 q^{21} -5.50753 q^{22} -1.00000 q^{23} +1.00000 q^{24} +3.33946 q^{25} +3.61972 q^{26} -1.00000 q^{27} -3.71597 q^{28} -1.00000 q^{29} -2.88781 q^{30} +6.02907 q^{31} -1.00000 q^{32} -5.50753 q^{33} +4.95541 q^{34} +10.7310 q^{35} +1.00000 q^{36} -1.34786 q^{37} -4.95541 q^{38} +3.61972 q^{39} +2.88781 q^{40} -1.77939 q^{41} -3.71597 q^{42} -7.39911 q^{43} +5.50753 q^{44} -2.88781 q^{45} +1.00000 q^{46} +13.6841 q^{47} -1.00000 q^{48} +6.80846 q^{49} -3.33946 q^{50} +4.95541 q^{51} -3.61972 q^{52} +2.70574 q^{53} +1.00000 q^{54} -15.9047 q^{55} +3.71597 q^{56} -4.95541 q^{57} +1.00000 q^{58} +8.87565 q^{59} +2.88781 q^{60} -5.99054 q^{61} -6.02907 q^{62} -3.71597 q^{63} +1.00000 q^{64} +10.4531 q^{65} +5.50753 q^{66} +9.67367 q^{67} -4.95541 q^{68} +1.00000 q^{69} -10.7310 q^{70} +3.39534 q^{71} -1.00000 q^{72} +13.9133 q^{73} +1.34786 q^{74} -3.33946 q^{75} +4.95541 q^{76} -20.4658 q^{77} -3.61972 q^{78} -4.99623 q^{79} -2.88781 q^{80} +1.00000 q^{81} +1.77939 q^{82} +9.98492 q^{83} +3.71597 q^{84} +14.3103 q^{85} +7.39911 q^{86} +1.00000 q^{87} -5.50753 q^{88} +11.1394 q^{89} +2.88781 q^{90} +13.4508 q^{91} -1.00000 q^{92} -6.02907 q^{93} -13.6841 q^{94} -14.3103 q^{95} +1.00000 q^{96} -6.70574 q^{97} -6.80846 q^{98} +5.50753 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9} - q^{10} + 5 q^{11} - 5 q^{12} - 11 q^{13} + 4 q^{14} - q^{15} + 5 q^{16} + 4 q^{17} - 5 q^{18} - 4 q^{19} + q^{20} + 4 q^{21} - 5 q^{22} - 5 q^{23} + 5 q^{24} + 12 q^{25} + 11 q^{26} - 5 q^{27} - 4 q^{28} - 5 q^{29} + q^{30} + 5 q^{31} - 5 q^{32} - 5 q^{33} - 4 q^{34} - 6 q^{35} + 5 q^{36} + 9 q^{37} + 4 q^{38} + 11 q^{39} - q^{40} + 5 q^{41} - 4 q^{42} - 16 q^{43} + 5 q^{44} + q^{45} + 5 q^{46} + 8 q^{47} - 5 q^{48} - 5 q^{49} - 12 q^{50} - 4 q^{51} - 11 q^{52} - 4 q^{53} + 5 q^{54} - 33 q^{55} + 4 q^{56} + 4 q^{57} + 5 q^{58} + 23 q^{59} - q^{60} + 7 q^{61} - 5 q^{62} - 4 q^{63} + 5 q^{64} - 5 q^{65} + 5 q^{66} + 5 q^{67} + 4 q^{68} + 5 q^{69} + 6 q^{70} - 21 q^{71} - 5 q^{72} - 8 q^{73} - 9 q^{74} - 12 q^{75} - 4 q^{76} - 11 q^{78} - 8 q^{79} + q^{80} + 5 q^{81} - 5 q^{82} - 18 q^{83} + 4 q^{84} - 10 q^{85} + 16 q^{86} + 5 q^{87} - 5 q^{88} + 32 q^{89} - q^{90} + 10 q^{91} - 5 q^{92} - 5 q^{93} - 8 q^{94} + 10 q^{95} + 5 q^{96} - 16 q^{97} + 5 q^{98} + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9 - q^10 + 5 * q^11 - 5 * q^12 - 11 * q^13 + 4 * q^14 - q^15 + 5 * q^16 + 4 * q^17 - 5 * q^18 - 4 * q^19 + q^20 + 4 * q^21 - 5 * q^22 - 5 * q^23 + 5 * q^24 + 12 * q^25 + 11 * q^26 - 5 * q^27 - 4 * q^28 - 5 * q^29 + q^30 + 5 * q^31 - 5 * q^32 - 5 * q^33 - 4 * q^34 - 6 * q^35 + 5 * q^36 + 9 * q^37 + 4 * q^38 + 11 * q^39 - q^40 + 5 * q^41 - 4 * q^42 - 16 * q^43 + 5 * q^44 + q^45 + 5 * q^46 + 8 * q^47 - 5 * q^48 - 5 * q^49 - 12 * q^50 - 4 * q^51 - 11 * q^52 - 4 * q^53 + 5 * q^54 - 33 * q^55 + 4 * q^56 + 4 * q^57 + 5 * q^58 + 23 * q^59 - q^60 + 7 * q^61 - 5 * q^62 - 4 * q^63 + 5 * q^64 - 5 * q^65 + 5 * q^66 + 5 * q^67 + 4 * q^68 + 5 * q^69 + 6 * q^70 - 21 * q^71 - 5 * q^72 - 8 * q^73 - 9 * q^74 - 12 * q^75 - 4 * q^76 - 11 * q^78 - 8 * q^79 + q^80 + 5 * q^81 - 5 * q^82 - 18 * q^83 + 4 * q^84 - 10 * q^85 + 16 * q^86 + 5 * q^87 - 5 * q^88 + 32 * q^89 - q^90 + 10 * q^91 - 5 * q^92 - 5 * q^93 - 8 * q^94 + 10 * q^95 + 5 * q^96 - 16 * q^97 + 5 * q^98 + 5 * q^99

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