LMFDB - Embedded newform 8038.2.a.b.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.18666 q^{3} +1.00000 q^{4} -3.58998 q^{5} +2.18666 q^{6} -4.12584 q^{7} -1.00000 q^{8} +1.78150 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.18666 q^{3} +1.00000 q^{4} -3.58998 q^{5} +2.18666 q^{6} -4.12584 q^{7} -1.00000 q^{8} +1.78150 q^{9} +3.58998 q^{10} -1.26907 q^{11} -2.18666 q^{12} +1.37689 q^{13} +4.12584 q^{14} +7.85008 q^{15} +1.00000 q^{16} +4.40579 q^{17} -1.78150 q^{18} -2.03877 q^{19} -3.58998 q^{20} +9.02182 q^{21} +1.26907 q^{22} +3.03238 q^{23} +2.18666 q^{24} +7.88794 q^{25} -1.37689 q^{26} +2.66445 q^{27} -4.12584 q^{28} +1.75034 q^{29} -7.85008 q^{30} -3.88063 q^{31} -1.00000 q^{32} +2.77504 q^{33} -4.40579 q^{34} +14.8117 q^{35} +1.78150 q^{36} -7.30327 q^{37} +2.03877 q^{38} -3.01079 q^{39} +3.58998 q^{40} -6.07419 q^{41} -9.02182 q^{42} -0.744850 q^{43} -1.26907 q^{44} -6.39555 q^{45} -3.03238 q^{46} +4.24397 q^{47} -2.18666 q^{48} +10.0225 q^{49} -7.88794 q^{50} -9.63399 q^{51} +1.37689 q^{52} -9.55889 q^{53} -2.66445 q^{54} +4.55594 q^{55} +4.12584 q^{56} +4.45810 q^{57} -1.75034 q^{58} -10.8519 q^{59} +7.85008 q^{60} +11.5494 q^{61} +3.88063 q^{62} -7.35019 q^{63} +1.00000 q^{64} -4.94299 q^{65} -2.77504 q^{66} +6.48059 q^{67} +4.40579 q^{68} -6.63079 q^{69} -14.8117 q^{70} +0.697650 q^{71} -1.78150 q^{72} -8.72128 q^{73} +7.30327 q^{74} -17.2483 q^{75} -2.03877 q^{76} +5.23599 q^{77} +3.01079 q^{78} -13.1253 q^{79} -3.58998 q^{80} -11.1708 q^{81} +6.07419 q^{82} -14.3045 q^{83} +9.02182 q^{84} -15.8167 q^{85} +0.744850 q^{86} -3.82740 q^{87} +1.26907 q^{88} -2.06817 q^{89} +6.39555 q^{90} -5.68081 q^{91} +3.03238 q^{92} +8.48563 q^{93} -4.24397 q^{94} +7.31913 q^{95} +2.18666 q^{96} -8.82126 q^{97} -10.0225 q^{98} -2.26086 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) $ 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.18666	−1.26247	−0.631236	−	0.775591i	$-0.717452\pi$
$3$	−2.18666	−1.26247	−0.631236	+	0.775591i	$0.717452\pi$
$4$	1.00000	0.500000
$5$	−3.58998	−1.60549	−0.802743	−	0.596324i	$-0.796627\pi$
$5$	−3.58998	−1.60549	−0.802743	+	0.596324i	$0.796627\pi$
$6$	2.18666	0.892702
$7$	−4.12584	−1.55942	−0.779710	−	0.626141i	$-0.784633\pi$
$7$	−4.12584	−1.55942	−0.779710	+	0.626141i	$0.784633\pi$
$8$	−1.00000	−0.353553
$9$	1.78150	0.593834
$10$	3.58998	1.13525
$11$	−1.26907	−0.382640	−0.191320	−	0.981528i	$-0.561277\pi$
$11$	−1.26907	−0.382640	−0.191320	+	0.981528i	$0.561277\pi$
$12$	−2.18666	−0.631236
$13$	1.37689	0.381879	0.190940	−	0.981602i	$-0.438847\pi$
$13$	1.37689	0.381879	0.190940	+	0.981602i	$0.438847\pi$
$14$	4.12584	1.10268
$15$	7.85008	2.02688
$16$	1.00000	0.250000
$17$	4.40579	1.06856	0.534281	−	0.845307i	$-0.320582\pi$
$17$	4.40579	1.06856	0.534281	+	0.845307i	$0.320582\pi$
$18$	−1.78150	−0.419904
$19$	−2.03877	−0.467725	−0.233863	−	0.972270i	$-0.575137\pi$
$19$	−2.03877	−0.467725	−0.233863	+	0.972270i	$0.575137\pi$
$20$	−3.58998	−0.802743
$21$	9.02182	1.96872
$22$	1.26907	0.270567
$23$	3.03238	0.632294	0.316147	−	0.948710i	$-0.397611\pi$
$23$	3.03238	0.632294	0.316147	+	0.948710i	$0.397611\pi$
$24$	2.18666	0.446351
$25$	7.88794	1.57759
$26$	−1.37689	−0.270029
$27$	2.66445	0.512773
$28$	−4.12584	−0.779710
$29$	1.75034	0.325030	0.162515	−	0.986706i	$-0.448039\pi$
$29$	1.75034	0.325030	0.162515	+	0.986706i	$0.448039\pi$
$30$	−7.85008	−1.43322
$31$	−3.88063	−0.696981	−0.348491	−	0.937312i	$-0.613306\pi$
$31$	−3.88063	−0.696981	−0.348491	+	0.937312i	$0.613306\pi$
$32$	−1.00000	−0.176777
$33$	2.77504	0.483072
$34$	−4.40579	−0.755587
$35$	14.8117	2.50363
$36$	1.78150	0.296917
$37$	−7.30327	−1.20065	−0.600325	−	0.799756i	$-0.704962\pi$
$37$	−7.30327	−1.20065	−0.600325	+	0.799756i	$0.704962\pi$
$38$	2.03877	0.330732
$39$	−3.01079	−0.482112
$40$	3.58998	0.567625
$41$	−6.07419	−0.948630	−0.474315	−	0.880355i	$-0.657304\pi$
$41$	−6.07419	−0.948630	−0.474315	+	0.880355i	$0.657304\pi$
$42$	−9.02182	−1.39210
$43$	−0.744850	−0.113589	−0.0567943	−	0.998386i	$-0.518088\pi$
$43$	−0.744850	−0.113589	−0.0567943	+	0.998386i	$0.518088\pi$
$44$	−1.26907	−0.191320
$45$	−6.39555	−0.953393
$46$	−3.03238	−0.447100
$47$	4.24397	0.619046	0.309523	−	0.950892i	$-0.399831\pi$
$47$	4.24397	0.619046	0.309523	+	0.950892i	$0.399831\pi$
$48$	−2.18666	−0.315618
$49$	10.0225	1.43179
$50$	−7.88794	−1.11552
$51$	−9.63399	−1.34903
$52$	1.37689	0.190940
$53$	−9.55889	−1.31301	−0.656507	−	0.754320i	$-0.727967\pi$
$53$	−9.55889	−1.31301	−0.656507	+	0.754320i	$0.727967\pi$
$54$	−2.66445	−0.362585
$55$	4.55594	0.614323
$56$	4.12584	0.551338
$57$	4.45810	0.590490
$58$	−1.75034	−0.229831
$59$	−10.8519	−1.41280	−0.706400	−	0.707812i	$-0.749682\pi$
$59$	−10.8519	−1.41280	−0.706400	+	0.707812i	$0.749682\pi$
$60$	7.85008	1.01344
$61$	11.5494	1.47875	0.739373	−	0.673296i	$-0.235122\pi$
$61$	11.5494	1.47875	0.739373	+	0.673296i	$0.235122\pi$
$62$	3.88063	0.492840
$63$	−7.35019	−0.926037
$64$	1.00000	0.125000
$65$	−4.94299	−0.613102
$66$	−2.77504	−0.341583
$67$	6.48059	0.791730	0.395865	−	0.918309i	$-0.370445\pi$
$67$	6.48059	0.791730	0.395865	+	0.918309i	$0.370445\pi$
$68$	4.40579	0.534281
$69$	−6.63079	−0.798254
$70$	−14.8117	−1.77033
$71$	0.697650	0.0827958	0.0413979	−	0.999143i	$-0.486819\pi$
$71$	0.697650	0.0827958	0.0413979	+	0.999143i	$0.486819\pi$
$72$	−1.78150	−0.209952
$73$	−8.72128	−1.02075	−0.510374	−	0.859952i	$-0.670493\pi$
$73$	−8.72128	−1.02075	−0.510374	+	0.859952i	$0.670493\pi$
$74$	7.30327	0.848988
$75$	−17.2483	−1.99166
$76$	−2.03877	−0.233863
$77$	5.23599	0.596696
$78$	3.01079	0.340904
$79$	−13.1253	−1.47671	−0.738354	−	0.674414i	$-0.764396\pi$
$79$	−13.1253	−1.47671	−0.738354	+	0.674414i	$0.764396\pi$
$80$	−3.58998	−0.401372
$81$	−11.1708	−1.24120
$82$	6.07419	0.670782
$83$	−14.3045	−1.57013	−0.785063	−	0.619416i	$-0.787369\pi$
$83$	−14.3045	−1.57013	−0.785063	+	0.619416i	$0.787369\pi$
$84$	9.02182	0.984362
$85$	−15.8167	−1.71556
$86$	0.744850	0.0803192
$87$	−3.82740	−0.410341
$88$	1.26907	0.135284
$89$	−2.06817	−0.219226	−0.109613	−	0.993974i	$-0.534961\pi$
$89$	−2.06817	−0.219226	−0.109613	+	0.993974i	$0.534961\pi$
$90$	6.39555	0.674151
$91$	−5.68081	−0.595510
$92$	3.03238	0.316147
$93$	8.48563	0.879919
$94$	−4.24397	−0.437732
$95$	7.31913	0.750927
$96$	2.18666	0.223176
$97$	−8.82126	−0.895664	−0.447832	−	0.894118i	$-0.647804\pi$
$97$	−8.82126	−0.895664	−0.447832	+	0.894118i	$0.647804\pi$
$98$	−10.0225	−1.01243
$99$	−2.26086	−0.227225
$100$	7.88794	0.788794
$101$	−3.97446	−0.395473	−0.197737	−	0.980255i	$-0.563359\pi$
$101$	−3.97446	−0.395473	−0.197737	+	0.980255i	$0.563359\pi$
$102$	9.63399	0.953908
$103$	−5.46115	−0.538103	−0.269052	−	0.963126i	$-0.586710\pi$
$103$	−5.46115	−0.538103	−0.269052	+	0.963126i	$0.586710\pi$
$104$	−1.37689	−0.135015
$105$	−32.3881	−3.16076
$106$	9.55889	0.928441
$107$	−3.81579	−0.368886	−0.184443	−	0.982843i	$-0.559048\pi$
$107$	−3.81579	−0.368886	−0.184443	+	0.982843i	$0.559048\pi$
$108$	2.66445	0.256386
$109$	9.53022	0.912830	0.456415	−	0.889767i	$-0.349133\pi$
$109$	9.53022	0.912830	0.456415	+	0.889767i	$0.349133\pi$
$110$	−4.55594	−0.434392
$111$	15.9698	1.51579
$112$	−4.12584	−0.389855
$113$	−8.43027	−0.793053	−0.396526	−	0.918023i	$-0.629784\pi$
$113$	−8.43027	−0.793053	−0.396526	+	0.918023i	$0.629784\pi$
$114$	−4.45810	−0.417539
$115$	−10.8862	−1.01514
$116$	1.75034	0.162515
$117$	2.45292	0.226773
$118$	10.8519	0.999001
$119$	−18.1776	−1.66634
$120$	−7.85008	−0.716611
$121$	−9.38945	−0.853587
$122$	−11.5494	−1.04563
$123$	13.2822	1.19762
$124$	−3.88063	−0.348491
$125$	−10.3676	−0.927311
$126$	7.35019	0.654807
$127$	−14.3720	−1.27531	−0.637654	−	0.770323i	$-0.720095\pi$
$127$	−14.3720	−1.27531	−0.637654	+	0.770323i	$0.720095\pi$
$128$	−1.00000	−0.0883883
$129$	1.62874	0.143402
$130$	4.94299	0.433529
$131$	−6.49578	−0.567539	−0.283769	−	0.958893i	$-0.591585\pi$
$131$	−6.49578	−0.567539	−0.283769	+	0.958893i	$0.591585\pi$
$132$	2.77504	0.241536
$133$	8.41162	0.729380
$134$	−6.48059	−0.559838
$135$	−9.56530	−0.823250
$136$	−4.40579	−0.377794
$137$	13.9510	1.19192	0.595958	−	0.803015i	$-0.296772\pi$
$137$	13.9510	1.19192	0.595958	+	0.803015i	$0.296772\pi$
$138$	6.63079	0.564451
$139$	6.26782	0.531630	0.265815	−	0.964024i	$-0.414359\pi$
$139$	6.26782	0.531630	0.265815	+	0.964024i	$0.414359\pi$
$140$	14.8117	1.25181
$141$	−9.28013	−0.781528
$142$	−0.697650	−0.0585455
$143$	−1.74737	−0.146122
$144$	1.78150	0.148459
$145$	−6.28368	−0.521831
$146$	8.72128	0.721778
$147$	−21.9159	−1.80759
$148$	−7.30327	−0.600325
$149$	−2.81828	−0.230882	−0.115441	−	0.993314i	$-0.536828\pi$
$149$	−2.81828	−0.230882	−0.115441	+	0.993314i	$0.536828\pi$
$150$	17.2483	1.40832
$151$	−15.8995	−1.29388	−0.646942	−	0.762539i	$-0.723952\pi$
$151$	−15.8995	−1.29388	−0.646942	+	0.762539i	$0.723952\pi$
$152$	2.03877	0.165366
$153$	7.84893	0.634549
$154$	−5.23599	−0.421928
$155$	13.9314	1.11899
$156$	−3.01079	−0.241056
$157$	8.34616	0.666096	0.333048	−	0.942910i	$-0.391923\pi$
$157$	8.34616	0.666096	0.333048	+	0.942910i	$0.391923\pi$
$158$	13.1253	1.04419
$159$	20.9021	1.65764
$160$	3.58998	0.283813
$161$	−12.5111	−0.986013
$162$	11.1708	0.877658
$163$	4.46417	0.349661	0.174830	−	0.984599i	$-0.444062\pi$
$163$	4.46417	0.349661	0.174830	+	0.984599i	$0.444062\pi$
$164$	−6.07419	−0.474315
$165$	−9.96232	−0.775566
$166$	14.3045	1.11025
$167$	−7.22968	−0.559450	−0.279725	−	0.960080i	$-0.590243\pi$
$167$	−7.22968	−0.559450	−0.279725	+	0.960080i	$0.590243\pi$
$168$	−9.02182	−0.696049
$169$	−11.1042	−0.854168
$170$	15.8167	1.21309
$171$	−3.63207	−0.277751
$172$	−0.744850	−0.0567943
$173$	6.79447	0.516574	0.258287	−	0.966068i	$-0.416842\pi$
$173$	6.79447	0.516574	0.258287	+	0.966068i	$0.416842\pi$
$174$	3.82740	0.290155
$175$	−32.5444	−2.46012
$176$	−1.26907	−0.0956600
$177$	23.7295	1.78362
$178$	2.06817	0.155016
$179$	−1.06181	−0.0793634	−0.0396817	−	0.999212i	$-0.512634\pi$
$179$	−1.06181	−0.0793634	−0.0396817	+	0.999212i	$0.512634\pi$
$180$	−6.39555	−0.476696
$181$	19.0488	1.41589	0.707944	−	0.706269i	$-0.249623\pi$
$181$	19.0488	1.41589	0.707944	+	0.706269i	$0.249623\pi$
$182$	5.68081	0.421089
$183$	−25.2546	−1.86688
$184$	−3.03238	−0.223550
$185$	26.2186	1.92763
$186$	−8.48563	−0.622197
$187$	−5.59127	−0.408874
$188$	4.24397	0.309523
$189$	−10.9931	−0.799628
$190$	−7.31913	−0.530985
$191$	12.3749	0.895415	0.447707	−	0.894180i	$-0.352241\pi$
$191$	12.3749	0.895415	0.447707	+	0.894180i	$0.352241\pi$
$192$	−2.18666	−0.157809
$193$	8.89337	0.640159	0.320079	−	0.947391i	$-0.396290\pi$
$193$	8.89337	0.640159	0.320079	+	0.947391i	$0.396290\pi$
$194$	8.82126	0.633330
$195$	10.8087	0.774024
$196$	10.0225	0.715895
$197$	12.3051	0.876699	0.438350	−	0.898805i	$-0.355563\pi$
$197$	12.3051	0.876699	0.438350	+	0.898805i	$0.355563\pi$
$198$	2.26086	0.160672
$199$	−4.61160	−0.326908	−0.163454	−	0.986551i	$-0.552263\pi$
$199$	−4.61160	−0.326908	−0.163454	+	0.986551i	$0.552263\pi$
$200$	−7.88794	−0.557762
$201$	−14.1709	−0.999537
$202$	3.97446	0.279642
$203$	−7.22161	−0.506858
$204$	−9.63399	−0.674515
$205$	21.8062	1.52301
$206$	5.46115	0.380497
$207$	5.40219	0.375478
$208$	1.37689	0.0954698
$209$	2.58734	0.178970
$210$	32.3881	2.23499
$211$	−11.4463	−0.787998	−0.393999	−	0.919111i	$-0.628909\pi$
$211$	−11.4463	−0.787998	−0.393999	+	0.919111i	$0.628909\pi$
$212$	−9.55889	−0.656507
$213$	−1.52553	−0.104527
$214$	3.81579	0.260842
$215$	2.67400	0.182365
$216$	−2.66445	−0.181293
$217$	16.0108	1.08689
$218$	−9.53022	−0.645468
$219$	19.0705	1.28867
$220$	4.55594	0.307162
$221$	6.06627	0.408062
$222$	−15.9698	−1.07182
$223$	4.41687	0.295775	0.147888	−	0.989004i	$-0.452753\pi$
$223$	4.41687	0.295775	0.147888	+	0.989004i	$0.452753\pi$
$224$	4.12584	0.275669
$225$	14.0524	0.936826
$226$	8.43027	0.560773
$227$	24.0493	1.59621	0.798103	−	0.602522i	$-0.205837\pi$
$227$	24.0493	1.59621	0.798103	+	0.602522i	$0.205837\pi$
$228$	4.45810	0.295245
$229$	−14.0469	−0.928245	−0.464122	−	0.885771i	$-0.653630\pi$
$229$	−14.0469	−0.928245	−0.464122	+	0.885771i	$0.653630\pi$
$230$	10.8862	0.717813
$231$	−11.4493	−0.753312
$232$	−1.75034	−0.114915
$233$	0.819806	0.0537073	0.0268536	−	0.999639i	$-0.491451\pi$
$233$	0.819806	0.0537073	0.0268536	+	0.999639i	$0.491451\pi$
$234$	−2.45292	−0.160353
$235$	−15.2357	−0.993870
$236$	−10.8519	−0.706400
$237$	28.7005	1.86430
$238$	18.1776	1.17828
$239$	−25.7553	−1.66597	−0.832986	−	0.553295i	$-0.813370\pi$
$239$	−25.7553	−1.66597	−0.832986	+	0.553295i	$0.813370\pi$
$240$	7.85008	0.506720
$241$	−25.6221	−1.65046	−0.825232	−	0.564794i	$-0.808956\pi$
$241$	−25.6221	−1.65046	−0.825232	+	0.564794i	$0.808956\pi$
$242$	9.38945	0.603577
$243$	16.4334	1.05420
$244$	11.5494	0.739373
$245$	−35.9807	−2.29872
$246$	−13.2822	−0.846844
$247$	−2.80715	−0.178615
$248$	3.88063	0.246420
$249$	31.2792	1.98224
$250$	10.3676	0.655708
$251$	−7.11442	−0.449058	−0.224529	−	0.974467i	$-0.572084\pi$
$251$	−7.11442	−0.449058	−0.224529	+	0.974467i	$0.572084\pi$
$252$	−7.35019	−0.463018
$253$	−3.84831	−0.241941
$254$	14.3720	0.901779
$255$	34.5858	2.16585
$256$	1.00000	0.0625000
$257$	−5.95875	−0.371696	−0.185848	−	0.982578i	$-0.559503\pi$
$257$	−5.95875	−0.371696	−0.185848	+	0.982578i	$0.559503\pi$
$258$	−1.62874	−0.101401
$259$	30.1321	1.87232
$260$	−4.94299	−0.306551
$261$	3.11823	0.193014
$262$	6.49578	0.401310
$263$	−26.1639	−1.61334	−0.806669	−	0.591004i	$-0.798732\pi$
$263$	−26.1639	−1.61334	−0.806669	+	0.591004i	$0.798732\pi$
$264$	−2.77504	−0.170792
$265$	34.3162	2.10803
$266$	−8.41162	−0.515750
$267$	4.52240	0.276766
$268$	6.48059	0.395865
$269$	−6.08759	−0.371167	−0.185583	−	0.982628i	$-0.559418\pi$
$269$	−6.08759	−0.371167	−0.185583	+	0.982628i	$0.559418\pi$
$270$	9.56530	0.582126
$271$	−27.4243	−1.66591	−0.832953	−	0.553344i	$-0.813352\pi$
$271$	−27.4243	−1.66591	−0.832953	+	0.553344i	$0.813352\pi$
$272$	4.40579	0.267141
$273$	12.4220	0.751815
$274$	−13.9510	−0.842812
$275$	−10.0104	−0.603648
$276$	−6.63079	−0.399127
$277$	6.26218	0.376258	0.188129	−	0.982144i	$-0.439758\pi$
$277$	6.26218	0.376258	0.188129	+	0.982144i	$0.439758\pi$
$278$	−6.26782	−0.375919
$279$	−6.91335	−0.413891
$280$	−14.8117	−0.885166
$281$	−5.24276	−0.312757	−0.156379	−	0.987697i	$-0.549982\pi$
$281$	−5.24276	−0.312757	−0.156379	+	0.987697i	$0.549982\pi$
$282$	9.28013	0.552624
$283$	−1.84871	−0.109894	−0.0549472	−	0.998489i	$-0.517499\pi$
$283$	−1.84871	−0.109894	−0.0549472	+	0.998489i	$0.517499\pi$
$284$	0.697650	0.0413979
$285$	−16.0045	−0.948024
$286$	1.74737	0.103324
$287$	25.0611	1.47931
$288$	−1.78150	−0.104976
$289$	2.41102	0.141825
$290$	6.28368	0.368990
$291$	19.2891	1.13075
$292$	−8.72128	−0.510374
$293$	5.06081	0.295656	0.147828	−	0.989013i	$-0.452772\pi$
$293$	5.06081	0.295656	0.147828	+	0.989013i	$0.452772\pi$
$294$	21.9159	1.27816
$295$	38.9582	2.26823
$296$	7.30327	0.424494
$297$	−3.38138	−0.196207
$298$	2.81828	0.163258
$299$	4.17524	0.241460
$300$	−17.2483	−0.995830
$301$	3.07313	0.177132
$302$	15.8995	0.914914
$303$	8.69081	0.499274
$304$	−2.03877	−0.116931
$305$	−41.4620	−2.37411
$306$	−7.84893	−0.448694
$307$	−28.0673	−1.60189	−0.800943	−	0.598741i	$-0.795668\pi$
$307$	−28.0673	−1.60189	−0.800943	+	0.598741i	$0.795668\pi$
$308$	5.23599	0.298348
$309$	11.9417	0.679340
$310$	−13.9314	−0.791249
$311$	23.6224	1.33950	0.669751	−	0.742586i	$-0.266401\pi$
$311$	23.6224	1.33950	0.669751	+	0.742586i	$0.266401\pi$
$312$	3.01079	0.170452
$313$	2.30723	0.130412	0.0652061	−	0.997872i	$-0.479230\pi$
$313$	2.30723	0.130412	0.0652061	+	0.997872i	$0.479230\pi$
$314$	−8.34616	−0.471001
$315$	26.3870	1.48674
$316$	−13.1253	−0.738354
$317$	−7.69126	−0.431984	−0.215992	−	0.976395i	$-0.569299\pi$
$317$	−7.69126	−0.431984	−0.215992	+	0.976395i	$0.569299\pi$
$318$	−20.9021	−1.17213
$319$	−2.22131	−0.124369
$320$	−3.58998	−0.200686
$321$	8.34385	0.465708
$322$	12.5111	0.697216
$323$	−8.98239	−0.499793
$324$	−11.1708	−0.620598
$325$	10.8608	0.602448
$326$	−4.46417	−0.247248
$327$	−20.8394	−1.15242
$328$	6.07419	0.335391
$329$	−17.5099	−0.965353
$330$	9.96232	0.548408
$331$	−6.09623	−0.335079	−0.167540	−	0.985865i	$-0.553582\pi$
$331$	−6.09623	−0.335079	−0.167540	+	0.985865i	$0.553582\pi$
$332$	−14.3045	−0.785063
$333$	−13.0108	−0.712987
$334$	7.22968	0.395591
$335$	−23.2652	−1.27111
$336$	9.02182	0.492181
$337$	−10.5456	−0.574457	−0.287229	−	0.957862i	$-0.592734\pi$
$337$	−10.5456	−0.574457	−0.287229	+	0.957862i	$0.592734\pi$
$338$	11.1042	0.603988
$339$	18.4342	1.00121
$340$	−15.8167	−0.857781
$341$	4.92480	0.266693
$342$	3.63207	0.196400
$343$	−12.4705	−0.673343
$344$	0.744850	0.0401596
$345$	23.8044	1.28159
$346$	−6.79447	−0.365273
$347$	18.6432	1.00082	0.500410	−	0.865788i	$-0.333183\pi$
$347$	18.6432	1.00082	0.500410	+	0.865788i	$0.333183\pi$
$348$	−3.82740	−0.205170
$349$	−18.5265	−0.991703	−0.495851	−	0.868407i	$-0.665144\pi$
$349$	−18.5265	−0.991703	−0.495851	+	0.868407i	$0.665144\pi$
$350$	32.5444	1.73957
$351$	3.66864	0.195817
$352$	1.26907	0.0676418
$353$	30.1700	1.60579	0.802893	−	0.596123i	$-0.203293\pi$
$353$	30.1700	1.60579	0.802893	+	0.596123i	$0.203293\pi$
$354$	−23.7295	−1.26121
$355$	−2.50455	−0.132928
$356$	−2.06817	−0.109613
$357$	39.7483	2.10370
$358$	1.06181	0.0561184
$359$	−21.2182	−1.11985	−0.559927	−	0.828542i	$-0.689171\pi$
$359$	−21.2182	−1.11985	−0.559927	+	0.828542i	$0.689171\pi$
$360$	6.39555	0.337075
$361$	−14.8434	−0.781233
$362$	−19.0488	−1.00118
$363$	20.5316	1.07763
$364$	−5.68081	−0.297755
$365$	31.3092	1.63880
$366$	25.2546	1.32008
$367$	−22.0334	−1.15014	−0.575068	−	0.818106i	$-0.695024\pi$
$367$	−22.0334	−1.15014	−0.575068	+	0.818106i	$0.695024\pi$
$368$	3.03238	0.158074
$369$	−10.8212	−0.563329
$370$	−26.2186	−1.36304
$371$	39.4384	2.04754
$372$	8.48563	0.439960
$373$	17.2132	0.891268	0.445634	−	0.895215i	$-0.352978\pi$
$373$	17.2132	0.891268	0.445634	+	0.895215i	$0.352978\pi$
$374$	5.59127	0.289118
$375$	22.6706	1.17070
$376$	−4.24397	−0.218866
$377$	2.41002	0.124122
$378$	10.9931	0.565423
$379$	−20.0508	−1.02994	−0.514970	−	0.857208i	$-0.672197\pi$
$379$	−20.0508	−1.02994	−0.514970	+	0.857208i	$0.672197\pi$
$380$	7.31913	0.375463
$381$	31.4267	1.61004
$382$	−12.3749	−0.633154
$383$	25.0446	1.27972	0.639859	−	0.768493i	$-0.278993\pi$
$383$	25.0446	1.27972	0.639859	+	0.768493i	$0.278993\pi$
$384$	2.18666	0.111588
$385$	−18.7971	−0.957988
$386$	−8.89337	−0.452661
$387$	−1.32695	−0.0674528
$388$	−8.82126	−0.447832
$389$	3.98333	0.201963	0.100981	−	0.994888i	$-0.467802\pi$
$389$	3.98333	0.201963	0.100981	+	0.994888i	$0.467802\pi$
$390$	−10.8087	−0.547318
$391$	13.3600	0.675646
$392$	−10.0225	−0.506214
$393$	14.2041	0.716501
$394$	−12.3051	−0.619920
$395$	47.1194	2.37083
$396$	−2.26086	−0.113612
$397$	8.02163	0.402594	0.201297	−	0.979530i	$-0.435484\pi$
$397$	8.02163	0.402594	0.201297	+	0.979530i	$0.435484\pi$
$398$	4.61160	0.231159
$399$	−18.3934	−0.920821
$400$	7.88794	0.394397
$401$	−7.31139	−0.365113	−0.182557	−	0.983195i	$-0.558437\pi$
$401$	−7.31139	−0.365113	−0.182557	+	0.983195i	$0.558437\pi$
$402$	14.1709	0.706779
$403$	−5.34318	−0.266163
$404$	−3.97446	−0.197737
$405$	40.1028	1.99272
$406$	7.22161	0.358403
$407$	9.26838	0.459417
$408$	9.63399	0.476954
$409$	11.7205	0.579544	0.289772	−	0.957096i	$-0.406421\pi$
$409$	11.7205	0.579544	0.289772	+	0.957096i	$0.406421\pi$
$410$	−21.8062	−1.07693
$411$	−30.5062	−1.50476
$412$	−5.46115	−0.269052
$413$	44.7733	2.20315
$414$	−5.40219	−0.265503
$415$	51.3529	2.52082
$416$	−1.37689	−0.0675074
$417$	−13.7056	−0.671168
$418$	−2.58734	−0.126551
$419$	4.51569	0.220606	0.110303	−	0.993898i	$-0.464818\pi$
$419$	4.51569	0.220606	0.110303	+	0.993898i	$0.464818\pi$
$420$	−32.3881	−1.58038
$421$	−10.3457	−0.504216	−0.252108	−	0.967699i	$-0.581124\pi$
$421$	−10.3457	−0.504216	−0.252108	+	0.967699i	$0.581124\pi$
$422$	11.4463	0.557199
$423$	7.56063	0.367611
$424$	9.55889	0.464221
$425$	34.7526	1.68575
$426$	1.52553	0.0739120
$427$	−47.6509	−2.30599
$428$	−3.81579	−0.184443
$429$	3.82091	0.184475
$430$	−2.67400	−0.128951
$431$	17.0449	0.821023	0.410511	−	0.911855i	$-0.365350\pi$
$431$	17.0449	0.821023	0.410511	+	0.911855i	$0.365350\pi$
$432$	2.66445	0.128193
$433$	8.36063	0.401786	0.200893	−	0.979613i	$-0.435616\pi$
$433$	8.36063	0.401786	0.200893	+	0.979613i	$0.435616\pi$
$434$	−16.0108	−0.768545
$435$	13.7403	0.658797
$436$	9.53022	0.456415
$437$	−6.18231	−0.295740
$438$	−19.0705	−0.911225
$439$	−2.08393	−0.0994606	−0.0497303	−	0.998763i	$-0.515836\pi$
$439$	−2.08393	−0.0994606	−0.0497303	+	0.998763i	$0.515836\pi$
$440$	−4.55594	−0.217196
$441$	17.8552	0.850246
$442$	−6.06627	−0.288543
$443$	−20.6231	−0.979834	−0.489917	−	0.871769i	$-0.662973\pi$
$443$	−20.6231	−0.979834	−0.489917	+	0.871769i	$0.662973\pi$
$444$	15.9698	0.757894
$445$	7.42469	0.351964
$446$	−4.41687	−0.209145
$447$	6.16263	0.291482
$448$	−4.12584	−0.194927
$449$	12.7484	0.601633	0.300817	−	0.953682i	$-0.402741\pi$
$449$	12.7484	0.601633	0.300817	+	0.953682i	$0.402741\pi$
$450$	−14.0524	−0.662436
$451$	7.70859	0.362983
$452$	−8.43027	−0.396526
$453$	34.7669	1.63349
$454$	−24.0493	−1.12869
$455$	20.3940	0.956084
$456$	−4.45810	−0.208770
$457$	−25.9769	−1.21515	−0.607573	−	0.794264i	$-0.707857\pi$
$457$	−25.9769	−1.21515	−0.607573	+	0.794264i	$0.707857\pi$
$458$	14.0469	0.656368
$459$	11.7390	0.547930
$460$	−10.8862	−0.507570
$461$	17.3819	0.809556	0.404778	−	0.914415i	$-0.367349\pi$
$461$	17.3819	0.809556	0.404778	+	0.914415i	$0.367349\pi$
$462$	11.4493	0.532672
$463$	−42.7288	−1.98577	−0.992887	−	0.119063i	$-0.962011\pi$
$463$	−42.7288	−1.98577	−0.992887	+	0.119063i	$0.962011\pi$
$464$	1.75034	0.0812574
$465$	−30.4632	−1.41270
$466$	−0.819806	−0.0379768
$467$	6.76036	0.312832	0.156416	−	0.987691i	$-0.450006\pi$
$467$	6.76036	0.312832	0.156416	+	0.987691i	$0.450006\pi$
$468$	2.45292	0.113386
$469$	−26.7379	−1.23464
$470$	15.2357	0.702772
$471$	−18.2502	−0.840927
$472$	10.8519	0.499500
$473$	0.945269	0.0434635
$474$	−28.7005	−1.31826
$475$	−16.0817	−0.737878
$476$	−18.1776	−0.833168
$477$	−17.0292	−0.779712
$478$	25.7553	1.17802
$479$	−3.62391	−0.165581	−0.0827905	−	0.996567i	$-0.526383\pi$
$479$	−3.62391	−0.165581	−0.0827905	+	0.996567i	$0.526383\pi$
$480$	−7.85008	−0.358305
$481$	−10.0558	−0.458504
$482$	25.6221	1.16705
$483$	27.3576	1.24481
$484$	−9.38945	−0.426793
$485$	31.6681	1.43798
$486$	−16.4334	−0.745432
$487$	−10.4119	−0.471809	−0.235904	−	0.971776i	$-0.575805\pi$
$487$	−10.4119	−0.471809	−0.235904	+	0.971776i	$0.575805\pi$
$488$	−11.5494	−0.522816
$489$	−9.76164	−0.441437
$490$	35.9807	1.62544
$491$	−40.5488	−1.82994	−0.914970	−	0.403522i	$-0.867786\pi$
$491$	−40.5488	−1.82994	−0.914970	+	0.403522i	$0.867786\pi$
$492$	13.2822	0.598809
$493$	7.71163	0.347314
$494$	2.80715	0.126300
$495$	8.11642	0.364806
$496$	−3.88063	−0.174245
$497$	−2.87839	−0.129113
$498$	−31.2792	−1.40165
$499$	−3.17937	−0.142328	−0.0711640	−	0.997465i	$-0.522671\pi$
$499$	−3.17937	−0.142328	−0.0711640	+	0.997465i	$0.522671\pi$
$500$	−10.3676	−0.463655
$501$	15.8089	0.706289
$502$	7.11442	0.317532
$503$	22.6427	1.00959	0.504794	−	0.863240i	$-0.331568\pi$
$503$	22.6427	1.00959	0.504794	+	0.863240i	$0.331568\pi$
$504$	7.35019	0.327403
$505$	14.2682	0.634927
$506$	3.84831	0.171078
$507$	24.2811	1.07836
$508$	−14.3720	−0.637654
$509$	2.51129	0.111311	0.0556555	−	0.998450i	$-0.482275\pi$
$509$	2.51129	0.111311	0.0556555	+	0.998450i	$0.482275\pi$
$510$	−34.5858	−1.53149
$511$	35.9826	1.59178
$512$	−1.00000	−0.0441942
$513$	−5.43218	−0.239837
$514$	5.95875	0.262829
$515$	19.6054	0.863918
$516$	1.62874	0.0717012
$517$	−5.38590	−0.236872
$518$	−30.1321	−1.32393
$519$	−14.8572	−0.652160
$520$	4.94299	0.216764
$521$	−34.3961	−1.50692	−0.753459	−	0.657494i	$-0.771616\pi$
$521$	−34.3961	−1.50692	−0.753459	+	0.657494i	$0.771616\pi$
$522$	−3.11823	−0.136481
$523$	−29.7381	−1.30036	−0.650178	−	0.759782i	$-0.725306\pi$
$523$	−29.7381	−1.30036	−0.650178	+	0.759782i	$0.725306\pi$
$524$	−6.49578	−0.283769
$525$	71.1636	3.10583
$526$	26.1639	1.14080
$527$	−17.0972	−0.744768
$528$	2.77504	0.120768
$529$	−13.8047	−0.600204
$530$	−34.3162	−1.49060
$531$	−19.3327	−0.838969
$532$	8.41162	0.364690
$533$	−8.36347	−0.362262
$534$	−4.52240	−0.195703
$535$	13.6986	0.592242
$536$	−6.48059	−0.279919
$537$	2.32182	0.100194
$538$	6.08759	0.262455
$539$	−12.7193	−0.547860
$540$	−9.56530	−0.411625
$541$	−7.98110	−0.343134	−0.171567	−	0.985172i	$-0.554883\pi$
$541$	−7.98110	−0.343134	−0.171567	+	0.985172i	$0.554883\pi$
$542$	27.4243	1.17797
$543$	−41.6534	−1.78752
$544$	−4.40579	−0.188897
$545$	−34.2133	−1.46554
$546$	−12.4220	−0.531613
$547$	−7.00473	−0.299501	−0.149750	−	0.988724i	$-0.547847\pi$
$547$	−7.00473	−0.299501	−0.149750	+	0.988724i	$0.547847\pi$
$548$	13.9510	0.595958
$549$	20.5752	0.878130
$550$	10.0104	0.426844
$551$	−3.56853	−0.152025
$552$	6.63079	0.282225
$553$	54.1527	2.30281
$554$	−6.26218	−0.266054
$555$	−57.3313	−2.43358
$556$	6.26782	0.265815
$557$	33.5577	1.42188	0.710942	−	0.703251i	$-0.248269\pi$
$557$	33.5577	1.42188	0.710942	+	0.703251i	$0.248269\pi$
$558$	6.91335	0.292665
$559$	−1.02557	−0.0433771
$560$	14.8117	0.625907
$561$	12.2262	0.516192
$562$	5.24276	0.221153
$563$	−21.5367	−0.907665	−0.453833	−	0.891087i	$-0.649944\pi$
$563$	−21.5367	−0.907665	−0.453833	+	0.891087i	$0.649944\pi$
$564$	−9.28013	−0.390764
$565$	30.2645	1.27324
$566$	1.84871	0.0777071
$567$	46.0887	1.93554
$568$	−0.697650	−0.0292727
$569$	4.58749	0.192318	0.0961588	−	0.995366i	$-0.469344\pi$
$569$	4.58749	0.192318	0.0961588	+	0.995366i	$0.469344\pi$
$570$	16.0045	0.670354
$571$	−14.2182	−0.595012	−0.297506	−	0.954720i	$-0.596155\pi$
$571$	−14.2182	−0.595012	−0.297506	+	0.954720i	$0.596155\pi$
$572$	−1.74737	−0.0730611
$573$	−27.0597	−1.13044
$574$	−25.0611	−1.04603
$575$	23.9192	0.997500
$576$	1.78150	0.0742293
$577$	−10.4694	−0.435848	−0.217924	−	0.975966i	$-0.569929\pi$
$577$	−10.4694	−0.435848	−0.217924	+	0.975966i	$0.569929\pi$
$578$	−2.41102	−0.100285
$579$	−19.4468	−0.808182
$580$	−6.28368	−0.260915
$581$	59.0181	2.44849
$582$	−19.2891	−0.799561
$583$	12.1309	0.502411
$584$	8.72128	0.360889
$585$	−8.80594	−0.364081
$586$	−5.06081	−0.209060
$587$	0.386832	0.0159663	0.00798313	−	0.999968i	$-0.497459\pi$
$587$	0.386832	0.0159663	0.00798313	+	0.999968i	$0.497459\pi$
$588$	−21.9159	−0.903797
$589$	7.91170	0.325996
$590$	−38.9582	−1.60388
$591$	−26.9070	−1.10681
$592$	−7.30327	−0.300163
$593$	7.16130	0.294079	0.147040	−	0.989131i	$-0.453026\pi$
$593$	7.16130	0.294079	0.147040	+	0.989131i	$0.453026\pi$
$594$	3.38138	0.138740
$595$	65.2571	2.67528
$596$	−2.81828	−0.115441
$597$	10.0840	0.412711
$598$	−4.17524	−0.170738
$599$	32.8651	1.34283	0.671417	−	0.741080i	$-0.265686\pi$
$599$	32.8651	1.34283	0.671417	+	0.741080i	$0.265686\pi$
$600$	17.2483	0.704158
$601$	2.78026	0.113409	0.0567046	−	0.998391i	$-0.481941\pi$
$601$	2.78026	0.113409	0.0567046	+	0.998391i	$0.481941\pi$
$602$	−3.07313	−0.125251
$603$	11.5452	0.470156
$604$	−15.8995	−0.646942
$605$	33.7079	1.37042
$606$	−8.69081	−0.353040
$607$	7.83753	0.318115	0.159058	−	0.987269i	$-0.449154\pi$
$607$	7.83753	0.318115	0.159058	+	0.987269i	$0.449154\pi$
$608$	2.03877	0.0826829
$609$	15.7912	0.639893
$610$	41.4620	1.67875
$611$	5.84345	0.236401
$612$	7.84893	0.317274
$613$	−37.7465	−1.52457	−0.762284	−	0.647243i	$-0.775922\pi$
$613$	−37.7465	−1.52457	−0.762284	+	0.647243i	$0.775922\pi$
$614$	28.0673	1.13270
$615$	−47.6829	−1.92276
$616$	−5.23599	−0.210964
$617$	−35.0536	−1.41121	−0.705603	−	0.708608i	$-0.749324\pi$
$617$	−35.0536	−1.41121	−0.705603	+	0.708608i	$0.749324\pi$
$618$	−11.9417	−0.480366
$619$	16.3114	0.655612	0.327806	−	0.944745i	$-0.393691\pi$
$619$	16.3114	0.655612	0.327806	+	0.944745i	$0.393691\pi$
$620$	13.9314	0.559497
$621$	8.07961	0.324223
$622$	−23.6224	−0.947171
$623$	8.53294	0.341865
$624$	−3.01079	−0.120528
$625$	−2.22008	−0.0888033
$626$	−2.30723	−0.0922153
$627$	−5.65765	−0.225945
$628$	8.34616	0.333048
$629$	−32.1767	−1.28297
$630$	−26.3870	−1.05128
$631$	−44.1322	−1.75688	−0.878438	−	0.477856i	$-0.841414\pi$
$631$	−44.1322	−1.75688	−0.878438	+	0.477856i	$0.841414\pi$
$632$	13.1253	0.522095
$633$	25.0293	0.994825
$634$	7.69126	0.305459
$635$	51.5952	2.04749
$636$	20.9021	0.828821
$637$	13.7999	0.546771
$638$	2.22131	0.0879424
$639$	1.24287	0.0491670
$640$	3.58998	0.141906
$641$	38.6585	1.52692	0.763459	−	0.645856i	$-0.223499\pi$
$641$	38.6585	1.52692	0.763459	+	0.645856i	$0.223499\pi$
$642$	−8.34385	−0.329306
$643$	2.46077	0.0970432	0.0485216	−	0.998822i	$-0.484549\pi$
$643$	2.46077	0.0970432	0.0485216	+	0.998822i	$0.484549\pi$
$644$	−12.5111	−0.493006
$645$	−5.84713	−0.230231
$646$	8.98239	0.353407
$647$	42.7767	1.68173	0.840863	−	0.541248i	$-0.182048\pi$
$647$	42.7767	1.68173	0.840863	+	0.541248i	$0.182048\pi$
$648$	11.1708	0.438829
$649$	13.7719	0.540594
$650$	−10.8608	−0.425995
$651$	−35.0103	−1.37216
$652$	4.46417	0.174830
$653$	29.5255	1.15542	0.577712	−	0.816241i	$-0.303946\pi$
$653$	29.5255	1.15542	0.577712	+	0.816241i	$0.303946\pi$
$654$	20.8394	0.814885
$655$	23.3197	0.911176
$656$	−6.07419	−0.237157
$657$	−15.5370	−0.606155
$658$	17.5099	0.682607
$659$	3.47136	0.135225	0.0676126	−	0.997712i	$-0.478462\pi$
$659$	3.47136	0.135225	0.0676126	+	0.997712i	$0.478462\pi$
$660$	−9.96232	−0.387783
$661$	8.14261	0.316711	0.158355	−	0.987382i	$-0.449381\pi$
$661$	8.14261	0.316711	0.158355	+	0.987382i	$0.449381\pi$
$662$	6.09623	0.236937
$663$	−13.2649	−0.515166
$664$	14.3045	0.555123
$665$	−30.1975	−1.17101
$666$	13.0108	0.504158
$667$	5.30769	0.205514
$668$	−7.22968	−0.279725
$669$	−9.65821	−0.373408
$670$	23.2652	0.898812
$671$	−14.6570	−0.565827
$672$	−9.02182	−0.348024
$673$	−4.18019	−0.161135	−0.0805673	−	0.996749i	$-0.525673\pi$
$673$	−4.18019	−0.161135	−0.0805673	+	0.996749i	$0.525673\pi$
$674$	10.5456	0.406203
$675$	21.0170	0.808945
$676$	−11.1042	−0.427084
$677$	−2.22945	−0.0856848	−0.0428424	−	0.999082i	$-0.513641\pi$
$677$	−2.22945	−0.0856848	−0.0428424	+	0.999082i	$0.513641\pi$
$678$	−18.4342	−0.707960
$679$	36.3951	1.39672
$680$	15.8167	0.606543
$681$	−52.5877	−2.01516
$682$	−4.92480	−0.188580
$683$	26.1315	0.999895	0.499948	−	0.866056i	$-0.333353\pi$
$683$	26.1315	0.999895	0.499948	+	0.866056i	$0.333353\pi$
$684$	−3.63207	−0.138876
$685$	−50.0839	−1.91361
$686$	12.4705	0.476125
$687$	30.7158	1.17188
$688$	−0.744850	−0.0283971
$689$	−13.1615	−0.501413
$690$	−23.8044	−0.906218
$691$	−12.5927	−0.479049	−0.239525	−	0.970890i	$-0.576992\pi$
$691$	−12.5927	−0.479049	−0.239525	+	0.970890i	$0.576992\pi$
$692$	6.79447	0.258287
$693$	9.32792	0.354339
$694$	−18.6432	−0.707687
$695$	−22.5014	−0.853525
$696$	3.82740	0.145077
$697$	−26.7616	−1.01367
$698$	18.5265	0.701240
$699$	−1.79264	−0.0678039
$700$	−32.5444	−1.23006
$701$	−33.8967	−1.28026	−0.640130	−	0.768267i	$-0.721119\pi$
$701$	−33.8967	−1.28026	−0.640130	+	0.768267i	$0.721119\pi$
$702$	−3.66864	−0.138464
$703$	14.8897	0.561575
$704$	−1.26907	−0.0478300
$705$	33.3155	1.25473
$706$	−30.1700	−1.13546
$707$	16.3980	0.616709
$708$	23.7295	0.891810
$709$	−17.0681	−0.641005	−0.320502	−	0.947248i	$-0.603852\pi$
$709$	−17.0681	−0.641005	−0.320502	+	0.947248i	$0.603852\pi$
$710$	2.50455	0.0939940
$711$	−23.3827	−0.876919
$712$	2.06817	0.0775080
$713$	−11.7675	−0.440697
$714$	−39.7483	−1.48754
$715$	6.27301	0.234597
$716$	−1.06181	−0.0396817
$717$	56.3182	2.10324
$718$	21.2182	0.791856
$719$	29.2802	1.09197	0.545983	−	0.837796i	$-0.316156\pi$
$719$	29.2802	1.09197	0.545983	+	0.837796i	$0.316156\pi$
$720$	−6.39555	−0.238348
$721$	22.5318	0.839129
$722$	14.8434	0.552415
$723$	56.0269	2.08366
$724$	19.0488	0.707944
$725$	13.8066	0.512763
$726$	−20.5316	−0.761999
$727$	−2.45147	−0.0909199	−0.0454600	−	0.998966i	$-0.514475\pi$
$727$	−2.45147	−0.0909199	−0.0454600	+	0.998966i	$0.514475\pi$
$728$	5.68081	0.210545
$729$	−2.42198	−0.0897028
$730$	−31.3092	−1.15881
$731$	−3.28166	−0.121376
$732$	−25.2546	−0.933438
$733$	48.9593	1.80835	0.904176	−	0.427159i	$-0.140486\pi$
$733$	48.9593	1.80835	0.904176	+	0.427159i	$0.140486\pi$
$734$	22.0334	0.813268
$735$	78.6777	2.90207
$736$	−3.03238	−0.111775
$737$	−8.22434	−0.302947
$738$	10.8212	0.398333
$739$	5.94189	0.218576	0.109288	−	0.994010i	$-0.465143\pi$
$739$	5.94189	0.218576	0.109288	+	0.994010i	$0.465143\pi$
$740$	26.2186	0.963814
$741$	6.13829	0.225496
$742$	−39.4384	−1.44783
$743$	12.0667	0.442684	0.221342	−	0.975196i	$-0.428956\pi$
$743$	12.0667	0.442684	0.221342	+	0.975196i	$0.428956\pi$
$744$	−8.48563	−0.311098
$745$	10.1176	0.370678
$746$	−17.2132	−0.630222
$747$	−25.4835	−0.932394
$748$	−5.59127	−0.204437
$749$	15.7433	0.575249
$750$	−22.6706	−0.827812
$751$	−39.8317	−1.45348	−0.726740	−	0.686913i	$-0.758965\pi$
$751$	−39.8317	−1.45348	−0.726740	+	0.686913i	$0.758965\pi$
$752$	4.24397	0.154762
$753$	15.5568	0.566923
$754$	−2.41002	−0.0877676
$755$	57.0789	2.07731
$756$	−10.9931	−0.399814
$757$	−29.4729	−1.07121	−0.535605	−	0.844469i	$-0.679916\pi$
$757$	−29.4729	−1.07121	−0.535605	+	0.844469i	$0.679916\pi$
$758$	20.0508	0.728278
$759$	8.41496	0.305444
$760$	−7.31913	−0.265493
$761$	−8.94327	−0.324193	−0.162097	−	0.986775i	$-0.551826\pi$
$761$	−8.94327	−0.324193	−0.162097	+	0.986775i	$0.551826\pi$
$762$	−31.4267	−1.13847
$763$	−39.3202	−1.42349
$764$	12.3749	0.447707
$765$	−28.1775	−1.01876
$766$	−25.0446	−0.904897
$767$	−14.9419	−0.539519
$768$	−2.18666	−0.0789045
$769$	−10.8819	−0.392412	−0.196206	−	0.980563i	$-0.562862\pi$
$769$	−10.8819	−0.392412	−0.196206	+	0.980563i	$0.562862\pi$
$770$	18.7971	0.677400
$771$	13.0298	0.469256
$772$	8.89337	0.320079
$773$	−11.8217	−0.425198	−0.212599	−	0.977140i	$-0.568193\pi$
$773$	−11.8217	−0.425198	−0.212599	+	0.977140i	$0.568193\pi$
$774$	1.32695	0.0476963
$775$	−30.6102	−1.09955
$776$	8.82126	0.316665
$777$	−65.8888	−2.36375
$778$	−3.98333	−0.142809
$779$	12.3839	0.443698
$780$	10.8087	0.387012
$781$	−0.885369	−0.0316810
$782$	−13.3600	−0.477754
$783$	4.66368	0.166666
$784$	10.0225	0.357948
$785$	−29.9625	−1.06941
$786$	−14.2041	−0.506643
$787$	−41.5665	−1.48169	−0.740843	−	0.671678i	$-0.765574\pi$
$787$	−41.5665	−1.48169	−0.740843	+	0.671678i	$0.765574\pi$
$788$	12.3051	0.438350
$789$	57.2118	2.03679
$790$	−47.1194	−1.67643
$791$	34.7819	1.23670
$792$	2.26086	0.0803360
$793$	15.9022	0.564703
$794$	−8.02163	−0.284677
$795$	−75.0380	−2.66132
$796$	−4.61160	−0.163454
$797$	−53.1491	−1.88264	−0.941319	−	0.337519i	$-0.890412\pi$
$797$	−53.1491	−1.88264	−0.941319	+	0.337519i	$0.890412\pi$
$798$	18.3934	0.651119
$799$	18.6980	0.661489
$800$	−7.88794	−0.278881
$801$	−3.68445	−0.130184
$802$	7.31139	0.258174
$803$	11.0679	0.390579
$804$	−14.1709	−0.499768
$805$	44.9146	1.58303
$806$	5.34318	0.188206
$807$	13.3115	0.468588
$808$	3.97446	0.139821
$809$	−3.99245	−0.140367	−0.0701835	−	0.997534i	$-0.522358\pi$
$809$	−3.99245	−0.140367	−0.0701835	+	0.997534i	$0.522358\pi$
$810$	−40.1028	−1.40907
$811$	−20.9080	−0.734179	−0.367089	−	0.930186i	$-0.619646\pi$
$811$	−20.9080	−0.734179	−0.367089	+	0.930186i	$0.619646\pi$
$812$	−7.22161	−0.253429
$813$	59.9677	2.10316
$814$	−9.26838	−0.324857
$815$	−16.0263	−0.561376
$816$	−9.63399	−0.337257
$817$	1.51858	0.0531282
$818$	−11.7205	−0.409799
$819$	−10.1204	−0.353634
$820$	21.8062	0.761506
$821$	−7.99891	−0.279164	−0.139582	−	0.990211i	$-0.544576\pi$
$821$	−7.99891	−0.279164	−0.139582	+	0.990211i	$0.544576\pi$
$822$	30.5062	1.06403
$823$	2.03300	0.0708660	0.0354330	−	0.999372i	$-0.488719\pi$
$823$	2.03300	0.0708660	0.0354330	+	0.999372i	$0.488719\pi$
$824$	5.46115	0.190248
$825$	21.8893	0.762089
$826$	−44.7733	−1.55786
$827$	22.1559	0.770436	0.385218	−	0.922826i	$-0.374126\pi$
$827$	22.1559	0.770436	0.385218	+	0.922826i	$0.374126\pi$
$828$	5.40219	0.187739
$829$	−1.63086	−0.0566420	−0.0283210	−	0.999599i	$-0.509016\pi$
$829$	−1.63086	−0.0566420	−0.0283210	+	0.999599i	$0.509016\pi$
$830$	−51.3529	−1.78249
$831$	−13.6933	−0.475015
$832$	1.37689	0.0477349
$833$	44.1572	1.52996
$834$	13.7056	0.474587
$835$	25.9544	0.898189
$836$	2.58734	0.0894851
$837$	−10.3397	−0.357393
$838$	−4.51569	−0.155992
$839$	−19.7329	−0.681256	−0.340628	−	0.940198i	$-0.610640\pi$
$839$	−19.7329	−0.681256	−0.340628	+	0.940198i	$0.610640\pi$
$840$	32.3881	1.11750
$841$	−25.9363	−0.894356
$842$	10.3457	0.356535
$843$	11.4642	0.394847
$844$	−11.4463	−0.393999
$845$	39.8638	1.37136
$846$	−7.56063	−0.259940
$847$	38.7394	1.33110
$848$	−9.55889	−0.328254
$849$	4.04251	0.138739
$850$	−34.7526	−1.19201
$851$	−22.1463	−0.759165
$852$	−1.52553	−0.0522637
$853$	40.8974	1.40030	0.700150	−	0.713996i	$-0.253116\pi$
$853$	40.8974	1.40030	0.700150	+	0.713996i	$0.253116\pi$
$854$	47.6509	1.63058
$855$	13.0390	0.445926
$856$	3.81579	0.130421
$857$	57.0821	1.94989	0.974944	−	0.222450i	$-0.0714053\pi$
$857$	57.0821	1.94989	0.974944	+	0.222450i	$0.0714053\pi$
$858$	−3.82091	−0.130444
$859$	−28.4548	−0.970867	−0.485433	−	0.874274i	$-0.661338\pi$
$859$	−28.4548	−0.970867	−0.485433	+	0.874274i	$0.661338\pi$
$860$	2.67400	0.0911825
$861$	−54.8003	−1.86759
$862$	−17.0449	−0.580551
$863$	10.8655	0.369867	0.184933	−	0.982751i	$-0.440793\pi$
$863$	10.8655	0.369867	0.184933	+	0.982751i	$0.440793\pi$
$864$	−2.66445	−0.0906463
$865$	−24.3920	−0.829353
$866$	−8.36063	−0.284106
$867$	−5.27210	−0.179050
$868$	16.0108	0.543443
$869$	16.6569	0.565047
$870$	−13.7403	−0.465839
$871$	8.92303	0.302345
$872$	−9.53022	−0.322734
$873$	−15.7151	−0.531875
$874$	6.18231	0.209120
$875$	42.7752	1.44607
$876$	19.0705	0.644333
$877$	37.2513	1.25789	0.628944	−	0.777451i	$-0.283488\pi$
$877$	37.2513	1.25789	0.628944	+	0.777451i	$0.283488\pi$
$878$	2.08393	0.0703293
$879$	−11.0663	−0.373257
$880$	4.55594	0.153581
$881$	14.4937	0.488306	0.244153	−	0.969737i	$-0.421490\pi$
$881$	14.4937	0.488306	0.244153	+	0.969737i	$0.421490\pi$
$882$	−17.8552	−0.601215
$883$	23.6760	0.796759	0.398380	−	0.917221i	$-0.369573\pi$
$883$	23.6760	0.796759	0.398380	+	0.917221i	$0.369573\pi$
$884$	6.06627	0.204031
$885$	−85.1885	−2.86358
$886$	20.6231	0.692847
$887$	−38.4574	−1.29127	−0.645637	−	0.763644i	$-0.723408\pi$
$887$	−38.4574	−1.29127	−0.645637	+	0.763644i	$0.723408\pi$
$888$	−15.9698	−0.535912
$889$	59.2965	1.98874
$890$	−7.42469	−0.248876
$891$	14.1765	0.474931
$892$	4.41687	0.147888
$893$	−8.65246	−0.289543
$894$	−6.16263	−0.206109
$895$	3.81188	0.127417
$896$	4.12584	0.137835
$897$	−9.12984	−0.304837
$898$	−12.7484	−0.425419
$899$	−6.79241	−0.226540
$900$	14.0524	0.468413
$901$	−42.1145	−1.40304
$902$	−7.70859	−0.256668
$903$	−6.71990	−0.223624
$904$	8.43027	0.280386
$905$	−68.3848	−2.27319
$906$	−34.7669	−1.15505
$907$	46.6518	1.54905	0.774524	−	0.632544i	$-0.217989\pi$
$907$	46.6518	1.54905	0.774524	+	0.632544i	$0.217989\pi$
$908$	24.0493	0.798103
$909$	−7.08051	−0.234846
$910$	−20.3940	−0.676053
$911$	16.5537	0.548450	0.274225	−	0.961666i	$-0.411579\pi$
$911$	16.5537	0.548450	0.274225	+	0.961666i	$0.411579\pi$
$912$	4.45810	0.147622
$913$	18.1535	0.600793
$914$	25.9769	0.859238
$915$	90.6635	2.99724
$916$	−14.0469	−0.464122
$917$	26.8005	0.885031
$918$	−11.7390	−0.387445
$919$	−37.8797	−1.24953	−0.624767	−	0.780811i	$-0.714806\pi$
$919$	−37.8797	−1.24953	−0.624767	+	0.780811i	$0.714806\pi$
$920$	10.8862	0.358906
$921$	61.3738	2.02233
$922$	−17.3819	−0.572443
$923$	0.960584	0.0316180
$924$	−11.4493	−0.376656
$925$	−57.6078	−1.89413
$926$	42.7288	1.40415
$927$	−9.72905	−0.319544
$928$	−1.75034	−0.0574577
$929$	3.61362	0.118559	0.0592794	−	0.998241i	$-0.481120\pi$
$929$	3.61362	0.118559	0.0592794	+	0.998241i	$0.481120\pi$
$930$	30.4632	0.998929
$931$	−20.4336	−0.669685
$932$	0.819806	0.0268536
$933$	−51.6542	−1.69108
$934$	−6.76036	−0.221206
$935$	20.0725	0.656443
$936$	−2.45292	−0.0801763
$937$	−10.0794	−0.329279	−0.164640	−	0.986354i	$-0.552646\pi$
$937$	−10.0794	−0.329279	−0.164640	+	0.986354i	$0.552646\pi$
$938$	26.7379	0.873022
$939$	−5.04513	−0.164642
$940$	−15.2357	−0.496935
$941$	36.0114	1.17394	0.586970	−	0.809609i	$-0.300321\pi$
$941$	36.0114	1.17394	0.586970	+	0.809609i	$0.300321\pi$
$942$	18.2502	0.594625
$943$	−18.4192	−0.599813
$944$	−10.8519	−0.353200
$945$	39.4649	1.28379
$946$	−0.945269	−0.0307333
$947$	53.4845	1.73801	0.869007	−	0.494800i	$-0.164759\pi$
$947$	53.4845	1.73801	0.869007	+	0.494800i	$0.164759\pi$
$948$	28.7005	0.932150
$949$	−12.0082	−0.389803
$950$	16.0817	0.521758
$951$	16.8182	0.545368
$952$	18.1776	0.589139
$953$	13.4812	0.436699	0.218349	−	0.975871i	$-0.429933\pi$
$953$	13.4812	0.436699	0.218349	+	0.975871i	$0.429933\pi$
$954$	17.0292	0.551340
$955$	−44.4255	−1.43758
$956$	−25.7553	−0.832986
$957$	4.85725	0.157013
$958$	3.62391	0.117083
$959$	−57.5597	−1.85870
$960$	7.85008	0.253360
$961$	−15.9407	−0.514217
$962$	10.0558	0.324211
$963$	−6.79784	−0.219057
$964$	−25.6221	−0.825232
$965$	−31.9270	−1.02777
$966$	−27.3576	−0.880215
$967$	38.1851	1.22795	0.613975	−	0.789326i	$-0.289570\pi$
$967$	38.1851	1.22795	0.613975	+	0.789326i	$0.289570\pi$
$968$	9.38945	0.301788
$969$	19.6415	0.630975
$970$	−31.6681	−1.01680
$971$	−31.7178	−1.01787	−0.508937	−	0.860804i	$-0.669961\pi$
$971$	−31.7178	−1.01787	−0.508937	+	0.860804i	$0.669961\pi$
$972$	16.4334	0.527100
$973$	−25.8600	−0.829034
$974$	10.4119	0.333619
$975$	−23.7489	−0.760574
$976$	11.5494	0.369687
$977$	31.0749	0.994173	0.497087	−	0.867701i	$-0.334403\pi$
$977$	31.0749	0.994173	0.497087	+	0.867701i	$0.334403\pi$
$978$	9.76164	0.312143
$979$	2.62466	0.0838845
$980$	−35.9807	−1.14936
$981$	16.9781	0.542069
$982$	40.5488	1.29396
$983$	39.4633	1.25869	0.629343	−	0.777128i	$-0.283324\pi$
$983$	39.4633	1.25869	0.629343	+	0.777128i	$0.283324\pi$
$984$	−13.2822	−0.423422
$985$	−44.1749	−1.40753
$986$	−7.71163	−0.245588
$987$	38.2883	1.21873
$988$	−2.80715	−0.0893073
$989$	−2.25867	−0.0718214
$990$	−8.11642	−0.257957
$991$	19.2663	0.612014	0.306007	−	0.952029i	$-0.401007\pi$
$991$	19.2663	0.612014	0.306007	+	0.952029i	$0.401007\pi$
$992$	3.88063	0.123210
$993$	13.3304	0.423028
$994$	2.87839	0.0912970
$995$	16.5555	0.524846
$996$	31.2792	0.991119
$997$	−49.9181	−1.58092	−0.790462	−	0.612512i	$-0.790159\pi$
$997$	−49.9181	−1.58092	−0.790462	+	0.612512i	$0.790159\pi$
$998$	3.17937	0.100641
$999$	−19.4592	−0.615661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.11	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.11	✓	83	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.18666 q^{3} +1.00000 q^{4} -3.58998 q^{5} +2.18666 q^{6} -4.12584 q^{7} -1.00000 q^{8} +1.78150 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.18666 q^{3} +1.00000 q^{4} -3.58998 q^{5} +2.18666 q^{6} -4.12584 q^{7} -1.00000 q^{8} +1.78150 q^{9} +3.58998 q^{10} -1.26907 q^{11} -2.18666 q^{12} +1.37689 q^{13} +4.12584 q^{14} +7.85008 q^{15} +1.00000 q^{16} +4.40579 q^{17} -1.78150 q^{18} -2.03877 q^{19} -3.58998 q^{20} +9.02182 q^{21} +1.26907 q^{22} +3.03238 q^{23} +2.18666 q^{24} +7.88794 q^{25} -1.37689 q^{26} +2.66445 q^{27} -4.12584 q^{28} +1.75034 q^{29} -7.85008 q^{30} -3.88063 q^{31} -1.00000 q^{32} +2.77504 q^{33} -4.40579 q^{34} +14.8117 q^{35} +1.78150 q^{36} -7.30327 q^{37} +2.03877 q^{38} -3.01079 q^{39} +3.58998 q^{40} -6.07419 q^{41} -9.02182 q^{42} -0.744850 q^{43} -1.26907 q^{44} -6.39555 q^{45} -3.03238 q^{46} +4.24397 q^{47} -2.18666 q^{48} +10.0225 q^{49} -7.88794 q^{50} -9.63399 q^{51} +1.37689 q^{52} -9.55889 q^{53} -2.66445 q^{54} +4.55594 q^{55} +4.12584 q^{56} +4.45810 q^{57} -1.75034 q^{58} -10.8519 q^{59} +7.85008 q^{60} +11.5494 q^{61} +3.88063 q^{62} -7.35019 q^{63} +1.00000 q^{64} -4.94299 q^{65} -2.77504 q^{66} +6.48059 q^{67} +4.40579 q^{68} -6.63079 q^{69} -14.8117 q^{70} +0.697650 q^{71} -1.78150 q^{72} -8.72128 q^{73} +7.30327 q^{74} -17.2483 q^{75} -2.03877 q^{76} +5.23599 q^{77} +3.01079 q^{78} -13.1253 q^{79} -3.58998 q^{80} -11.1708 q^{81} +6.07419 q^{82} -14.3045 q^{83} +9.02182 q^{84} -15.8167 q^{85} +0.744850 q^{86} -3.82740 q^{87} +1.26907 q^{88} -2.06817 q^{89} +6.39555 q^{90} -5.68081 q^{91} +3.03238 q^{92} +8.48563 q^{93} -4.24397 q^{94} +7.31913 q^{95} +2.18666 q^{96} -8.82126 q^{97} -10.0225 q^{98} -2.26086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

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