LMFDB - Embedded newform 8043.2.a.p.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$41$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.551667 q^{2} +1.00000 q^{3} -1.69566 q^{4} -2.77566 q^{5} -0.551667 q^{6} -1.00000 q^{7} +2.03878 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.551667 q^{2} +1.00000 q^{3} -1.69566 q^{4} -2.77566 q^{5} -0.551667 q^{6} -1.00000 q^{7} +2.03878 q^{8} +1.00000 q^{9} +1.53124 q^{10} -4.32478 q^{11} -1.69566 q^{12} -1.98438 q^{13} +0.551667 q^{14} -2.77566 q^{15} +2.26660 q^{16} -3.75860 q^{17} -0.551667 q^{18} +1.82968 q^{19} +4.70659 q^{20} -1.00000 q^{21} +2.38584 q^{22} -0.779349 q^{23} +2.03878 q^{24} +2.70431 q^{25} +1.09472 q^{26} +1.00000 q^{27} +1.69566 q^{28} -1.51064 q^{29} +1.53124 q^{30} -7.00271 q^{31} -5.32796 q^{32} -4.32478 q^{33} +2.07350 q^{34} +2.77566 q^{35} -1.69566 q^{36} -8.78948 q^{37} -1.00938 q^{38} -1.98438 q^{39} -5.65896 q^{40} +3.26877 q^{41} +0.551667 q^{42} -11.7948 q^{43} +7.33337 q^{44} -2.77566 q^{45} +0.429942 q^{46} +1.87108 q^{47} +2.26660 q^{48} +1.00000 q^{49} -1.49188 q^{50} -3.75860 q^{51} +3.36484 q^{52} -3.28697 q^{53} -0.551667 q^{54} +12.0041 q^{55} -2.03878 q^{56} +1.82968 q^{57} +0.833369 q^{58} -3.22557 q^{59} +4.70659 q^{60} -1.89752 q^{61} +3.86316 q^{62} -1.00000 q^{63} -1.59393 q^{64} +5.50798 q^{65} +2.38584 q^{66} +5.36958 q^{67} +6.37331 q^{68} -0.779349 q^{69} -1.53124 q^{70} +1.14341 q^{71} +2.03878 q^{72} -9.96085 q^{73} +4.84887 q^{74} +2.70431 q^{75} -3.10252 q^{76} +4.32478 q^{77} +1.09472 q^{78} +1.11444 q^{79} -6.29132 q^{80} +1.00000 q^{81} -1.80328 q^{82} +7.56674 q^{83} +1.69566 q^{84} +10.4326 q^{85} +6.50682 q^{86} -1.51064 q^{87} -8.81726 q^{88} -13.1900 q^{89} +1.53124 q^{90} +1.98438 q^{91} +1.32151 q^{92} -7.00271 q^{93} -1.03221 q^{94} -5.07858 q^{95} -5.32796 q^{96} -10.4383 q^{97} -0.551667 q^{98} -4.32478 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9}+O(q^{10}) $ 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9	\( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9} + 18 q^{10} + 8 q^{11} + 45 q^{12} + 23 q^{13} - 7 q^{14} + 17 q^{15} + 37 q^{16} + 15 q^{17} + 7 q^{18} + 15 q^{19} + 53 q^{20} - 41 q^{21} + 13 q^{22} + 44 q^{23} + 12 q^{24} + 58 q^{25} + 9 q^{26} + 41 q^{27} - 45 q^{28} + 21 q^{29} + 18 q^{30} + 39 q^{31} + 61 q^{32} + 8 q^{33} + 9 q^{34} - 17 q^{35} + 45 q^{36} + 11 q^{37} + 44 q^{38} + 23 q^{39} + 24 q^{40} + 17 q^{41} - 7 q^{42} + 7 q^{43} + 30 q^{44} + 17 q^{45} - 12 q^{46} + 36 q^{47} + 37 q^{48} + 41 q^{49} + 28 q^{50} + 15 q^{51} + 58 q^{52} + 26 q^{53} + 7 q^{54} + 32 q^{55} - 12 q^{56} + 15 q^{57} - 4 q^{58} + 33 q^{59} + 53 q^{60} + 59 q^{61} - q^{62} - 41 q^{63} + 16 q^{64} + 72 q^{65} + 13 q^{66} + 12 q^{67} + 52 q^{68} + 44 q^{69} - 18 q^{70} + 33 q^{71} + 12 q^{72} + 18 q^{73} + 42 q^{74} + 58 q^{75} + 7 q^{76} - 8 q^{77} + 9 q^{78} + 22 q^{79} + 69 q^{80} + 41 q^{81} + 41 q^{82} + 32 q^{83} - 45 q^{84} - 44 q^{85} + 11 q^{86} + 21 q^{87} + 52 q^{88} + 63 q^{89} + 18 q^{90} - 23 q^{91} + 52 q^{92} + 39 q^{93} + 17 q^{94} + 37 q^{95} + 61 q^{96} + 8 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9 + 18 * q^10 + 8 * q^11 + 45 * q^12 + 23 * q^13 - 7 * q^14 + 17 * q^15 + 37 * q^16 + 15 * q^17 + 7 * q^18 + 15 * q^19 + 53 * q^20 - 41 * q^21 + 13 * q^22 + 44 * q^23 + 12 * q^24 + 58 * q^25 + 9 * q^26 + 41 * q^27 - 45 * q^28 + 21 * q^29 + 18 * q^30 + 39 * q^31 + 61 * q^32 + 8 * q^33 + 9 * q^34 - 17 * q^35 + 45 * q^36 + 11 * q^37 + 44 * q^38 + 23 * q^39 + 24 * q^40 + 17 * q^41 - 7 * q^42 + 7 * q^43 + 30 * q^44 + 17 * q^45 - 12 * q^46 + 36 * q^47 + 37 * q^48 + 41 * q^49 + 28 * q^50 + 15 * q^51 + 58 * q^52 + 26 * q^53 + 7 * q^54 + 32 * q^55 - 12 * q^56 + 15 * q^57 - 4 * q^58 + 33 * q^59 + 53 * q^60 + 59 * q^61 - q^62 - 41 * q^63 + 16 * q^64 + 72 * q^65 + 13 * q^66 + 12 * q^67 + 52 * q^68 + 44 * q^69 - 18 * q^70 + 33 * q^71 + 12 * q^72 + 18 * q^73 + 42 * q^74 + 58 * q^75 + 7 * q^76 - 8 * q^77 + 9 * q^78 + 22 * q^79 + 69 * q^80 + 41 * q^81 + 41 * q^82 + 32 * q^83 - 45 * q^84 - 44 * q^85 + 11 * q^86 + 21 * q^87 + 52 * q^88 + 63 * q^89 + 18 * q^90 - 23 * q^91 + 52 * q^92 + 39 * q^93 + 17 * q^94 + 37 * q^95 + 61 * q^96 + 8 * q^97 + 7 * q^98 + 8 * 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 $

$2$	−0.551667	−0.390088	−0.195044	−	0.980795i	$-0.562485\pi$
$2$	−0.551667	−0.390088	−0.195044	+	0.980795i	$0.562485\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.69566	−0.847832
$5$	−2.77566	−1.24131	−0.620657	−	0.784082i	$-0.713134\pi$
$5$	−2.77566	−1.24131	−0.620657	+	0.784082i	$0.713134\pi$
$6$	−0.551667	−0.225217
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.03878	0.720817
$9$	1.00000	0.333333
$10$	1.53124	0.484222
$11$	−4.32478	−1.30397	−0.651985	−	0.758232i	$-0.726064\pi$
$11$	−4.32478	−1.30397	−0.651985	+	0.758232i	$0.726064\pi$
$12$	−1.69566	−0.489496
$13$	−1.98438	−0.550368	−0.275184	−	0.961392i	$-0.588739\pi$
$13$	−1.98438	−0.550368	−0.275184	+	0.961392i	$0.588739\pi$
$14$	0.551667	0.147439
$15$	−2.77566	−0.716673
$16$	2.26660	0.566650
$17$	−3.75860	−0.911593	−0.455797	−	0.890084i	$-0.650646\pi$
$17$	−3.75860	−0.911593	−0.455797	+	0.890084i	$0.650646\pi$
$18$	−0.551667	−0.130029
$19$	1.82968	0.419758	0.209879	−	0.977727i	$-0.432693\pi$
$19$	1.82968	0.419758	0.209879	+	0.977727i	$0.432693\pi$
$20$	4.70659	1.05243
$21$	−1.00000	−0.218218
$22$	2.38584	0.508663
$23$	−0.779349	−0.162506	−0.0812528	−	0.996694i	$-0.525892\pi$
$23$	−0.779349	−0.162506	−0.0812528	+	0.996694i	$0.525892\pi$
$24$	2.03878	0.416164
$25$	2.70431	0.540862
$26$	1.09472	0.214692
$27$	1.00000	0.192450
$28$	1.69566	0.320450
$29$	−1.51064	−0.280518	−0.140259	−	0.990115i	$-0.544794\pi$
$29$	−1.51064	−0.280518	−0.140259	+	0.990115i	$0.544794\pi$
$30$	1.53124	0.279566
$31$	−7.00271	−1.25772	−0.628862	−	0.777517i	$-0.716479\pi$
$31$	−7.00271	−1.25772	−0.628862	+	0.777517i	$0.716479\pi$
$32$	−5.32796	−0.941860
$33$	−4.32478	−0.752848
$34$	2.07350	0.355601
$35$	2.77566	0.469173
$36$	−1.69566	−0.282611
$37$	−8.78948	−1.44498	−0.722491	−	0.691381i	$-0.757003\pi$
$37$	−8.78948	−1.44498	−0.722491	+	0.691381i	$0.757003\pi$
$38$	−1.00938	−0.163742
$39$	−1.98438	−0.317755
$40$	−5.65896	−0.894760
$41$	3.26877	0.510497	0.255248	−	0.966876i	$-0.417843\pi$
$41$	3.26877	0.510497	0.255248	+	0.966876i	$0.417843\pi$
$42$	0.551667	0.0851241
$43$	−11.7948	−1.79869	−0.899347	−	0.437235i	$-0.855958\pi$
$43$	−11.7948	−1.79869	−0.899347	+	0.437235i	$0.855958\pi$
$44$	7.33337	1.10555
$45$	−2.77566	−0.413772
$46$	0.429942	0.0633914
$47$	1.87108	0.272925	0.136462	−	0.990645i	$-0.456427\pi$
$47$	1.87108	0.272925	0.136462	+	0.990645i	$0.456427\pi$
$48$	2.26660	0.327155
$49$	1.00000	0.142857
$50$	−1.49188	−0.210984
$51$	−3.75860	−0.526309
$52$	3.36484	0.466620
$53$	−3.28697	−0.451500	−0.225750	−	0.974185i	$-0.572483\pi$
$53$	−3.28697	−0.451500	−0.225750	+	0.974185i	$0.572483\pi$
$54$	−0.551667	−0.0750724
$55$	12.0041	1.61864
$56$	−2.03878	−0.272443
$57$	1.82968	0.242347
$58$	0.833369	0.109427
$59$	−3.22557	−0.419934	−0.209967	−	0.977709i	$-0.567336\pi$
$59$	−3.22557	−0.419934	−0.209967	+	0.977709i	$0.567336\pi$
$60$	4.70659	0.607618
$61$	−1.89752	−0.242953	−0.121477	−	0.992594i	$-0.538763\pi$
$61$	−1.89752	−0.242953	−0.121477	+	0.992594i	$0.538763\pi$
$62$	3.86316	0.490622
$63$	−1.00000	−0.125988
$64$	−1.59393	−0.199242
$65$	5.50798	0.683180
$66$	2.38584	0.293677
$67$	5.36958	0.655998	0.327999	−	0.944678i	$-0.393626\pi$
$67$	5.36958	0.655998	0.327999	+	0.944678i	$0.393626\pi$
$68$	6.37331	0.772878
$69$	−0.779349	−0.0938226
$70$	−1.53124	−0.183019
$71$	1.14341	0.135698	0.0678489	−	0.997696i	$-0.478386\pi$
$71$	1.14341	0.135698	0.0678489	+	0.997696i	$0.478386\pi$
$72$	2.03878	0.240272
$73$	−9.96085	−1.16583	−0.582915	−	0.812533i	$-0.698088\pi$
$73$	−9.96085	−1.16583	−0.582915	+	0.812533i	$0.698088\pi$
$74$	4.84887	0.563670
$75$	2.70431	0.312267
$76$	−3.10252	−0.355884
$77$	4.32478	0.492855
$78$	1.09472	0.123952
$79$	1.11444	0.125384	0.0626922	−	0.998033i	$-0.480031\pi$
$79$	1.11444	0.125384	0.0626922	+	0.998033i	$0.480031\pi$
$80$	−6.29132	−0.703391
$81$	1.00000	0.111111
$82$	−1.80328	−0.199139
$83$	7.56674	0.830557	0.415279	−	0.909694i	$-0.363684\pi$
$83$	7.56674	0.830557	0.415279	+	0.909694i	$0.363684\pi$
$84$	1.69566	0.185012
$85$	10.4326	1.13157
$86$	6.50682	0.701649
$87$	−1.51064	−0.161957
$88$	−8.81726	−0.939924
$89$	−13.1900	−1.39813	−0.699067	−	0.715056i	$-0.746401\pi$
$89$	−13.1900	−1.39813	−0.699067	+	0.715056i	$0.746401\pi$
$90$	1.53124	0.161407
$91$	1.98438	0.208020
$92$	1.32151	0.137777
$93$	−7.00271	−0.726147
$94$	−1.03221	−0.106465
$95$	−5.07858	−0.521051
$96$	−5.32796	−0.543783
$97$	−10.4383	−1.05985	−0.529926	−	0.848044i	$-0.677780\pi$
$97$	−10.4383	−1.05985	−0.529926	+	0.848044i	$0.677780\pi$
$98$	−0.551667	−0.0557268
$99$	−4.32478	−0.434657
$100$	−4.58560	−0.458560
$101$	−13.6396	−1.35720	−0.678598	−	0.734510i	$-0.737412\pi$
$101$	−13.6396	−1.35720	−0.678598	+	0.734510i	$0.737412\pi$
$102$	2.07350	0.205307
$103$	−9.51399	−0.937441	−0.468721	−	0.883346i	$-0.655285\pi$
$103$	−9.51399	−0.937441	−0.468721	+	0.883346i	$0.655285\pi$
$104$	−4.04571	−0.396715
$105$	2.77566	0.270877
$106$	1.81332	0.176125
$107$	−2.77774	−0.268534	−0.134267	−	0.990945i	$-0.542868\pi$
$107$	−2.77774	−0.268534	−0.134267	+	0.990945i	$0.542868\pi$
$108$	−1.69566	−0.163165
$109$	−1.77487	−0.170002	−0.0850010	−	0.996381i	$-0.527089\pi$
$109$	−1.77487	−0.170002	−0.0850010	+	0.996381i	$0.527089\pi$
$110$	−6.62229	−0.631411
$111$	−8.78948	−0.834260
$112$	−2.26660	−0.214173
$113$	13.7697	1.29535	0.647674	−	0.761917i	$-0.275742\pi$
$113$	13.7697	1.29535	0.647674	+	0.761917i	$0.275742\pi$
$114$	−1.00938	−0.0945367
$115$	2.16321	0.201721
$116$	2.56153	0.237832
$117$	−1.98438	−0.183456
$118$	1.77944	0.163811
$119$	3.75860	0.344550
$120$	−5.65896	−0.516590
$121$	7.70373	0.700339
$122$	1.04680	0.0947730
$123$	3.26877	0.294735
$124$	11.8742	1.06634
$125$	6.37207	0.569935
$126$	0.551667	0.0491464
$127$	−19.9780	−1.77276	−0.886382	−	0.462956i	$-0.846789\pi$
$127$	−19.9780	−1.77276	−0.886382	+	0.462956i	$0.846789\pi$
$128$	11.5352	1.01958
$129$	−11.7948	−1.03848
$130$	−3.03857	−0.266500
$131$	14.8335	1.29601	0.648003	−	0.761638i	$-0.275604\pi$
$131$	14.8335	1.29601	0.648003	+	0.761638i	$0.275604\pi$
$132$	7.33337	0.638288
$133$	−1.82968	−0.158653
$134$	−2.96222	−0.255897
$135$	−2.77566	−0.238891
$136$	−7.66294	−0.657092
$137$	−6.48329	−0.553905	−0.276953	−	0.960884i	$-0.589325\pi$
$137$	−6.48329	−0.553905	−0.276953	+	0.960884i	$0.589325\pi$
$138$	0.429942	0.0365991
$139$	−22.0042	−1.86637	−0.933185	−	0.359397i	$-0.882983\pi$
$139$	−22.0042	−1.86637	−0.933185	+	0.359397i	$0.882983\pi$
$140$	−4.70659	−0.397779
$141$	1.87108	0.157573
$142$	−0.630782	−0.0529341
$143$	8.58201	0.717664
$144$	2.26660	0.188883
$145$	4.19302	0.348211
$146$	5.49508	0.454776
$147$	1.00000	0.0824786
$148$	14.9040	1.22510
$149$	−1.50172	−0.123026	−0.0615129	−	0.998106i	$-0.519593\pi$
$149$	−1.50172	−0.123026	−0.0615129	+	0.998106i	$0.519593\pi$
$150$	−1.49188	−0.121811
$151$	5.91034	0.480977	0.240488	−	0.970652i	$-0.422692\pi$
$151$	5.91034	0.480977	0.240488	+	0.970652i	$0.422692\pi$
$152$	3.73031	0.302568
$153$	−3.75860	−0.303864
$154$	−2.38584	−0.192257
$155$	19.4372	1.56123
$156$	3.36484	0.269403
$157$	9.59111	0.765454	0.382727	−	0.923862i	$-0.374985\pi$
$157$	9.59111	0.765454	0.382727	+	0.923862i	$0.374985\pi$
$158$	−0.614801	−0.0489109
$159$	−3.28697	−0.260674
$160$	14.7886	1.16914
$161$	0.779349	0.0614213
$162$	−0.551667	−0.0433431
$163$	−3.07725	−0.241029	−0.120514	−	0.992712i	$-0.538454\pi$
$163$	−3.07725	−0.241029	−0.120514	+	0.992712i	$0.538454\pi$
$164$	−5.54274	−0.432815
$165$	12.0041	0.934521
$166$	−4.17432	−0.323990
$167$	6.39751	0.495054	0.247527	−	0.968881i	$-0.420382\pi$
$167$	6.39751	0.495054	0.247527	+	0.968881i	$0.420382\pi$
$168$	−2.03878	−0.157295
$169$	−9.06223	−0.697095
$170$	−5.75533	−0.441413
$171$	1.82968	0.139919
$172$	20.0001	1.52499
$173$	6.78810	0.516090	0.258045	−	0.966133i	$-0.416922\pi$
$173$	6.78810	0.516090	0.258045	+	0.966133i	$0.416922\pi$
$174$	0.833369	0.0631776
$175$	−2.70431	−0.204427
$176$	−9.80254	−0.738895
$177$	−3.22557	−0.242449
$178$	7.27648	0.545395
$179$	−5.03353	−0.376223	−0.188112	−	0.982148i	$-0.560237\pi$
$179$	−5.03353	−0.376223	−0.188112	+	0.982148i	$0.560237\pi$
$180$	4.70659	0.350809
$181$	4.85907	0.361172	0.180586	−	0.983559i	$-0.442201\pi$
$181$	4.85907	0.361172	0.180586	+	0.983559i	$0.442201\pi$
$182$	−1.09472	−0.0811459
$183$	−1.89752	−0.140269
$184$	−1.58892	−0.117137
$185$	24.3966	1.79368
$186$	3.86316	0.283261
$187$	16.2551	1.18869
$188$	−3.17271	−0.231394
$189$	−1.00000	−0.0727393
$190$	2.80169	0.203256
$191$	9.61629	0.695811	0.347905	−	0.937530i	$-0.386893\pi$
$191$	9.61629	0.695811	0.347905	+	0.937530i	$0.386893\pi$
$192$	−1.59393	−0.115032
$193$	19.8888	1.43162	0.715812	−	0.698293i	$-0.246057\pi$
$193$	19.8888	1.43162	0.715812	+	0.698293i	$0.246057\pi$
$194$	5.75848	0.413435
$195$	5.50798	0.394434
$196$	−1.69566	−0.121119
$197$	10.9679	0.781427	0.390714	−	0.920512i	$-0.372228\pi$
$197$	10.9679	0.781427	0.390714	+	0.920512i	$0.372228\pi$
$198$	2.38584	0.169554
$199$	3.25742	0.230912	0.115456	−	0.993313i	$-0.463167\pi$
$199$	3.25742	0.230912	0.115456	+	0.993313i	$0.463167\pi$
$200$	5.51348	0.389862
$201$	5.36958	0.378741
$202$	7.52455	0.529425
$203$	1.51064	0.106026
$204$	6.37331	0.446221
$205$	−9.07302	−0.633687
$206$	5.24856	0.365684
$207$	−0.779349	−0.0541685
$208$	−4.49780	−0.311866
$209$	−7.91297	−0.547351
$210$	−1.53124	−0.105666
$211$	2.26840	0.156163	0.0780816	−	0.996947i	$-0.475121\pi$
$211$	2.26840	0.156163	0.0780816	+	0.996947i	$0.475121\pi$
$212$	5.57360	0.382796
$213$	1.14341	0.0783452
$214$	1.53239	0.104752
$215$	32.7385	2.23275
$216$	2.03878	0.138721
$217$	7.00271	0.475375
$218$	0.979140	0.0663157
$219$	−9.96085	−0.673092
$220$	−20.3550	−1.37233
$221$	7.45849	0.501712
$222$	4.84887	0.325435
$223$	6.48042	0.433961	0.216980	−	0.976176i	$-0.430379\pi$
$223$	6.48042	0.433961	0.216980	+	0.976176i	$0.430379\pi$
$224$	5.32796	0.355990
$225$	2.70431	0.180287
$226$	−7.59632	−0.505300
$227$	−27.5885	−1.83111	−0.915557	−	0.402189i	$-0.868250\pi$
$227$	−27.5885	−1.83111	−0.915557	+	0.402189i	$0.868250\pi$
$228$	−3.10252	−0.205470
$229$	14.4677	0.956054	0.478027	−	0.878345i	$-0.341352\pi$
$229$	14.4677	0.956054	0.478027	+	0.878345i	$0.341352\pi$
$230$	−1.19337	−0.0786887
$231$	4.32478	0.284550
$232$	−3.07985	−0.202202
$233$	−16.7898	−1.09994	−0.549970	−	0.835185i	$-0.685361\pi$
$233$	−16.7898	−1.09994	−0.549970	+	0.835185i	$0.685361\pi$
$234$	1.09472	0.0715640
$235$	−5.19348	−0.338785
$236$	5.46948	0.356033
$237$	1.11444	0.0723907
$238$	−2.07350	−0.134405
$239$	10.6101	0.686310	0.343155	−	0.939279i	$-0.388504\pi$
$239$	10.6101	0.686310	0.343155	+	0.939279i	$0.388504\pi$
$240$	−6.29132	−0.406103
$241$	−21.2276	−1.36739	−0.683697	−	0.729766i	$-0.739629\pi$
$241$	−21.2276	−1.36739	−0.683697	+	0.729766i	$0.739629\pi$
$242$	−4.24990	−0.273194
$243$	1.00000	0.0641500
$244$	3.21756	0.205983
$245$	−2.77566	−0.177331
$246$	−1.80328	−0.114973
$247$	−3.63078	−0.231021
$248$	−14.2770	−0.906588
$249$	7.56674	0.479523
$250$	−3.51526	−0.222325
$251$	12.0632	0.761420	0.380710	−	0.924695i	$-0.375680\pi$
$251$	12.0632	0.761420	0.380710	+	0.924695i	$0.375680\pi$
$252$	1.69566	0.106817
$253$	3.37051	0.211902
$254$	11.0212	0.691533
$255$	10.4326	0.653315
$256$	−3.17575	−0.198485
$257$	−20.3686	−1.27056	−0.635278	−	0.772284i	$-0.719114\pi$
$257$	−20.3686	−1.27056	−0.635278	+	0.772284i	$0.719114\pi$
$258$	6.50682	0.405097
$259$	8.78948	0.546152
$260$	−9.33967	−0.579222
$261$	−1.51064	−0.0935061
$262$	−8.18314	−0.505556
$263$	12.5700	0.775100	0.387550	−	0.921849i	$-0.373321\pi$
$263$	12.5700	0.775100	0.387550	+	0.921849i	$0.373321\pi$
$264$	−8.81726	−0.542665
$265$	9.12353	0.560454
$266$	1.00938	0.0618888
$267$	−13.1900	−0.807213
$268$	−9.10499	−0.556176
$269$	−15.0298	−0.916383	−0.458192	−	0.888853i	$-0.651503\pi$
$269$	−15.0298	−0.916383	−0.458192	+	0.888853i	$0.651503\pi$
$270$	1.53124	0.0931885
$271$	11.3180	0.687518	0.343759	−	0.939058i	$-0.388300\pi$
$271$	11.3180	0.687518	0.343759	+	0.939058i	$0.388300\pi$
$272$	−8.51923	−0.516554
$273$	1.98438	0.120100
$274$	3.57662	0.216072
$275$	−11.6955	−0.705268
$276$	1.32151	0.0795458
$277$	4.77073	0.286645	0.143323	−	0.989676i	$-0.454221\pi$
$277$	4.77073	0.286645	0.143323	+	0.989676i	$0.454221\pi$
$278$	12.1390	0.728048
$279$	−7.00271	−0.419241
$280$	5.65896	0.338188
$281$	1.64685	0.0982431	0.0491215	−	0.998793i	$-0.484358\pi$
$281$	1.64685	0.0982431	0.0491215	+	0.998793i	$0.484358\pi$
$282$	−1.03221	−0.0614673
$283$	11.8434	0.704019	0.352009	−	0.935997i	$-0.385499\pi$
$283$	11.8434	0.704019	0.352009	+	0.935997i	$0.385499\pi$
$284$	−1.93884	−0.115049
$285$	−5.07858	−0.300829
$286$	−4.73442	−0.279952
$287$	−3.26877	−0.192950
$288$	−5.32796	−0.313953
$289$	−2.87295	−0.168997
$290$	−2.31315	−0.135833
$291$	−10.4383	−0.611905
$292$	16.8902	0.988427
$293$	18.8811	1.10305	0.551524	−	0.834159i	$-0.314046\pi$
$293$	18.8811	1.10305	0.551524	+	0.834159i	$0.314046\pi$
$294$	−0.551667	−0.0321739
$295$	8.95310	0.521270
$296$	−17.9198	−1.04157
$297$	−4.32478	−0.250949
$298$	0.828451	0.0479909
$299$	1.54653	0.0894379
$300$	−4.58560	−0.264750
$301$	11.7948	0.679843
$302$	−3.26054	−0.187623
$303$	−13.6396	−0.783577
$304$	4.14715	0.237855
$305$	5.26689	0.301581
$306$	2.07350	0.118534
$307$	16.1044	0.919126	0.459563	−	0.888145i	$-0.348006\pi$
$307$	16.1044	0.919126	0.459563	+	0.888145i	$0.348006\pi$
$308$	−7.33337	−0.417858
$309$	−9.51399	−0.541232
$310$	−10.7228	−0.609017
$311$	28.5435	1.61855	0.809277	−	0.587427i	$-0.199859\pi$
$311$	28.5435	1.61855	0.809277	+	0.587427i	$0.199859\pi$
$312$	−4.04571	−0.229043
$313$	25.8004	1.45832	0.729162	−	0.684341i	$-0.239910\pi$
$313$	25.8004	1.45832	0.729162	+	0.684341i	$0.239910\pi$
$314$	−5.29110	−0.298594
$315$	2.77566	0.156391
$316$	−1.88972	−0.106305
$317$	8.04579	0.451897	0.225948	−	0.974139i	$-0.427452\pi$
$317$	8.04579	0.451897	0.225948	+	0.974139i	$0.427452\pi$
$318$	1.81332	0.101686
$319$	6.53317	0.365787
$320$	4.42423	0.247322
$321$	−2.77774	−0.155038
$322$	−0.429942	−0.0239597
$323$	−6.87703	−0.382648
$324$	−1.69566	−0.0942035
$325$	−5.36638	−0.297673
$326$	1.69762	0.0940223
$327$	−1.77487	−0.0981507
$328$	6.66430	0.367975
$329$	−1.87108	−0.103156
$330$	−6.62229	−0.364545
$331$	−1.31561	−0.0723125	−0.0361562	−	0.999346i	$-0.511511\pi$
$331$	−1.31561	−0.0723125	−0.0361562	+	0.999346i	$0.511511\pi$
$332$	−12.8306	−0.704173
$333$	−8.78948	−0.481661
$334$	−3.52930	−0.193115
$335$	−14.9041	−0.814300
$336$	−2.26660	−0.123653
$337$	9.71061	0.528971	0.264485	−	0.964390i	$-0.414798\pi$
$337$	9.71061	0.528971	0.264485	+	0.964390i	$0.414798\pi$
$338$	4.99934	0.271928
$339$	13.7697	0.747870
$340$	−17.6902	−0.959384
$341$	30.2852	1.64003
$342$	−1.00938	−0.0545808
$343$	−1.00000	−0.0539949
$344$	−24.0470	−1.29653
$345$	2.16321	0.116463
$346$	−3.74478	−0.201320
$347$	0.822246	0.0441405	0.0220702	−	0.999756i	$-0.492974\pi$
$347$	0.822246	0.0441405	0.0220702	+	0.999756i	$0.492974\pi$
$348$	2.56153	0.137312
$349$	−11.8570	−0.634693	−0.317347	−	0.948310i	$-0.602792\pi$
$349$	−11.8570	−0.634693	−0.317347	+	0.948310i	$0.602792\pi$
$350$	1.49188	0.0797443
$351$	−1.98438	−0.105918
$352$	23.0423	1.22816
$353$	−15.9333	−0.848046	−0.424023	−	0.905651i	$-0.639382\pi$
$353$	−15.9333	−0.848046	−0.424023	+	0.905651i	$0.639382\pi$
$354$	1.77944	0.0945763
$355$	−3.17372	−0.168444
$356$	22.3658	1.18538
$357$	3.75860	0.198926
$358$	2.77683	0.146760
$359$	−26.3698	−1.39175	−0.695873	−	0.718165i	$-0.744982\pi$
$359$	−26.3698	−1.39175	−0.695873	+	0.718165i	$0.744982\pi$
$360$	−5.65896	−0.298253
$361$	−15.6523	−0.823804
$362$	−2.68059	−0.140889
$363$	7.70373	0.404341
$364$	−3.36484	−0.176366
$365$	27.6480	1.44716
$366$	1.04680	0.0547172
$367$	2.27584	0.118798	0.0593990	−	0.998234i	$-0.481082\pi$
$367$	2.27584	0.118798	0.0593990	+	0.998234i	$0.481082\pi$
$368$	−1.76647	−0.0920837
$369$	3.26877	0.170166
$370$	−13.4588	−0.699691
$371$	3.28697	0.170651
$372$	11.8742	0.615650
$373$	9.36378	0.484838	0.242419	−	0.970172i	$-0.422059\pi$
$373$	9.36378	0.484838	0.242419	+	0.970172i	$0.422059\pi$
$374$	−8.96741	−0.463694
$375$	6.37207	0.329052
$376$	3.81471	0.196729
$377$	2.99768	0.154388
$378$	0.551667	0.0283747
$379$	−19.9959	−1.02712	−0.513559	−	0.858054i	$-0.671673\pi$
$379$	−19.9959	−1.02712	−0.513559	+	0.858054i	$0.671673\pi$
$380$	8.61156	0.441764
$381$	−19.9780	−1.02351
$382$	−5.30500	−0.271427
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	11.5352	0.588656
$385$	−12.0041	−0.611788
$386$	−10.9720	−0.558459
$387$	−11.7948	−0.599565
$388$	17.6999	0.898575
$389$	−1.93252	−0.0979826	−0.0489913	−	0.998799i	$-0.515601\pi$
$389$	−1.93252	−0.0979826	−0.0489913	+	0.998799i	$0.515601\pi$
$390$	−3.03857	−0.153864
$391$	2.92926	0.148139
$392$	2.03878	0.102974
$393$	14.8335	0.748249
$394$	−6.05061	−0.304825
$395$	−3.09331	−0.155641
$396$	7.33337	0.368516
$397$	28.9532	1.45312	0.726560	−	0.687103i	$-0.241118\pi$
$397$	28.9532	1.45312	0.726560	+	0.687103i	$0.241118\pi$
$398$	−1.79701	−0.0900761
$399$	−1.82968	−0.0915986
$400$	6.12958	0.306479
$401$	16.6952	0.833717	0.416859	−	0.908971i	$-0.363131\pi$
$401$	16.6952	0.833717	0.416859	+	0.908971i	$0.363131\pi$
$402$	−2.96222	−0.147742
$403$	13.8960	0.692211
$404$	23.1282	1.15067
$405$	−2.77566	−0.137924
$406$	−0.833369	−0.0413594
$407$	38.0126	1.88421
$408$	−7.66294	−0.379372
$409$	−13.9965	−0.692081	−0.346040	−	0.938220i	$-0.612474\pi$
$409$	−13.9965	−0.692081	−0.346040	+	0.938220i	$0.612474\pi$
$410$	5.00529	0.247194
$411$	−6.48329	−0.319797
$412$	16.1325	0.794792
$413$	3.22557	0.158720
$414$	0.429942	0.0211305
$415$	−21.0027	−1.03098
$416$	10.5727	0.518370
$417$	−22.0042	−1.07755
$418$	4.36533	0.213515
$419$	−6.97379	−0.340692	−0.170346	−	0.985384i	$-0.554488\pi$
$419$	−6.97379	−0.340692	−0.170346	+	0.985384i	$0.554488\pi$
$420$	−4.70659	−0.229658
$421$	−20.1459	−0.981853	−0.490927	−	0.871201i	$-0.663342\pi$
$421$	−20.1459	−0.981853	−0.490927	+	0.871201i	$0.663342\pi$
$422$	−1.25140	−0.0609173
$423$	1.87108	0.0909748
$424$	−6.70140	−0.325449
$425$	−10.1644	−0.493046
$426$	−0.630782	−0.0305615
$427$	1.89752	0.0918276
$428$	4.71011	0.227672
$429$	8.58201	0.414344
$430$	−18.0608	−0.870967
$431$	−24.7912	−1.19415	−0.597074	−	0.802186i	$-0.703670\pi$
$431$	−24.7912	−1.19415	−0.597074	+	0.802186i	$0.703670\pi$
$432$	2.26660	0.109052
$433$	−11.2800	−0.542084	−0.271042	−	0.962568i	$-0.587368\pi$
$433$	−11.2800	−0.542084	−0.271042	+	0.962568i	$0.587368\pi$
$434$	−3.86316	−0.185438
$435$	4.19302	0.201040
$436$	3.00959	0.144133
$437$	−1.42596	−0.0682129
$438$	5.49508	0.262565
$439$	5.68600	0.271378	0.135689	−	0.990751i	$-0.456675\pi$
$439$	5.68600	0.271378	0.135689	+	0.990751i	$0.456675\pi$
$440$	24.4738	1.16674
$441$	1.00000	0.0476190
$442$	−4.11461	−0.195712
$443$	27.5657	1.30968	0.654842	−	0.755766i	$-0.272735\pi$
$443$	27.5657	1.30968	0.654842	+	0.755766i	$0.272735\pi$
$444$	14.9040	0.707312
$445$	36.6109	1.73552
$446$	−3.57504	−0.169283
$447$	−1.50172	−0.0710290
$448$	1.59393	0.0753063
$449$	7.56731	0.357123	0.178562	−	0.983929i	$-0.442856\pi$
$449$	7.56731	0.357123	0.178562	+	0.983929i	$0.442856\pi$
$450$	−1.49188	−0.0703279
$451$	−14.1367	−0.665673
$452$	−23.3489	−1.09824
$453$	5.91034	0.277692
$454$	15.2197	0.714295
$455$	−5.50798	−0.258218
$456$	3.73031	0.174688
$457$	4.66527	0.218232	0.109116	−	0.994029i	$-0.465198\pi$
$457$	4.66527	0.218232	0.109116	+	0.994029i	$0.465198\pi$
$458$	−7.98137	−0.372945
$459$	−3.75860	−0.175436
$460$	−3.66808	−0.171025
$461$	32.3601	1.50716	0.753580	−	0.657356i	$-0.228325\pi$
$461$	32.3601	1.50716	0.753580	+	0.657356i	$0.228325\pi$
$462$	−2.38584	−0.110999
$463$	28.7803	1.33753	0.668767	−	0.743472i	$-0.266822\pi$
$463$	28.7803	1.33753	0.668767	+	0.743472i	$0.266822\pi$
$464$	−3.42401	−0.158956
$465$	19.4372	0.901377
$466$	9.26241	0.429073
$467$	22.9444	1.06174	0.530870	−	0.847453i	$-0.321865\pi$
$467$	22.9444	1.06174	0.530870	+	0.847453i	$0.321865\pi$
$468$	3.36484	0.155540
$469$	−5.36958	−0.247944
$470$	2.86507	0.132156
$471$	9.59111	0.441935
$472$	−6.57622	−0.302695
$473$	51.0101	2.34545
$474$	−0.614801	−0.0282387
$475$	4.94802	0.227031
$476$	−6.37331	−0.292120
$477$	−3.28697	−0.150500
$478$	−5.85325	−0.267721
$479$	6.80174	0.310780	0.155390	−	0.987853i	$-0.450337\pi$
$479$	6.80174	0.310780	0.155390	+	0.987853i	$0.450337\pi$
$480$	14.7886	0.675006
$481$	17.4417	0.795272
$482$	11.7106	0.533403
$483$	0.779349	0.0354616
$484$	−13.0629	−0.593770
$485$	28.9733	1.31561
$486$	−0.551667	−0.0250241
$487$	21.5229	0.975296	0.487648	−	0.873040i	$-0.337855\pi$
$487$	21.5229	0.975296	0.487648	+	0.873040i	$0.337855\pi$
$488$	−3.86863	−0.175125
$489$	−3.07725	−0.139158
$490$	1.53124	0.0691745
$491$	−26.6122	−1.20099	−0.600496	−	0.799628i	$-0.705030\pi$
$491$	−26.6122	−1.20099	−0.600496	+	0.799628i	$0.705030\pi$
$492$	−5.54274	−0.249886
$493$	5.67787	0.255719
$494$	2.00299	0.0901186
$495$	12.0041	0.539546
$496$	−15.8723	−0.712688
$497$	−1.14341	−0.0512890
$498$	−4.17432	−0.187056
$499$	−27.2781	−1.22113	−0.610567	−	0.791965i	$-0.709058\pi$
$499$	−27.2781	−1.22113	−0.610567	+	0.791965i	$0.709058\pi$
$500$	−10.8049	−0.483209
$501$	6.39751	0.285820
$502$	−6.65485	−0.297021
$503$	−34.8785	−1.55516	−0.777579	−	0.628786i	$-0.783552\pi$
$503$	−34.8785	−1.55516	−0.777579	+	0.628786i	$0.783552\pi$
$504$	−2.03878	−0.0908143
$505$	37.8591	1.68471
$506$	−1.85940	−0.0826606
$507$	−9.06223	−0.402468
$508$	33.8760	1.50300
$509$	44.0362	1.95187	0.975934	−	0.218065i	$-0.0699745\pi$
$509$	44.0362	1.95187	0.975934	+	0.218065i	$0.0699745\pi$
$510$	−5.75533	−0.254850
$511$	9.96085	0.440642
$512$	−21.3185	−0.942155
$513$	1.82968	0.0807824
$514$	11.2367	0.495628
$515$	26.4076	1.16366
$516$	20.0001	0.880454
$517$	−8.09200	−0.355886
$518$	−4.84887	−0.213047
$519$	6.78810	0.297965
$520$	11.2295	0.492448
$521$	11.4872	0.503263	0.251631	−	0.967823i	$-0.419033\pi$
$521$	11.4872	0.503263	0.251631	+	0.967823i	$0.419033\pi$
$522$	0.833369	0.0364756
$523$	8.40772	0.367644	0.183822	−	0.982960i	$-0.441153\pi$
$523$	8.40772	0.367644	0.183822	+	0.982960i	$0.441153\pi$
$524$	−25.1526	−1.09879
$525$	−2.70431	−0.118026
$526$	−6.93447	−0.302357
$527$	26.3203	1.14653
$528$	−9.80254	−0.426601
$529$	−22.3926	−0.973592
$530$	−5.03315	−0.218626
$531$	−3.22557	−0.139978
$532$	3.10252	0.134511
$533$	−6.48649	−0.280961
$534$	7.27648	0.314884
$535$	7.71006	0.333335
$536$	10.9474	0.472854
$537$	−5.03353	−0.217213
$538$	8.29145	0.357470
$539$	−4.32478	−0.186282
$540$	4.70659	0.202539
$541$	−2.91009	−0.125115	−0.0625573	−	0.998041i	$-0.519926\pi$
$541$	−2.91009	−0.125115	−0.0625573	+	0.998041i	$0.519926\pi$
$542$	−6.24376	−0.268192
$543$	4.85907	0.208523
$544$	20.0257	0.858593
$545$	4.92645	0.211026
$546$	−1.09472	−0.0468496
$547$	−22.2580	−0.951685	−0.475842	−	0.879531i	$-0.657857\pi$
$547$	−22.2580	−0.951685	−0.475842	+	0.879531i	$0.657857\pi$
$548$	10.9935	0.469618
$549$	−1.89752	−0.0809843
$550$	6.45205	0.275116
$551$	−2.76398	−0.117750
$552$	−1.58892	−0.0676289
$553$	−1.11444	−0.0473909
$554$	−2.63185	−0.111817
$555$	24.3966	1.03558
$556$	37.3117	1.58237
$557$	37.5853	1.59254	0.796270	−	0.604941i	$-0.206803\pi$
$557$	37.5853	1.59254	0.796270	+	0.604941i	$0.206803\pi$
$558$	3.86316	0.163541
$559$	23.4054	0.989945
$560$	6.29132	0.265857
$561$	16.2551	0.686291
$562$	−0.908516	−0.0383234
$563$	32.0638	1.35133	0.675664	−	0.737210i	$-0.263857\pi$
$563$	32.0638	1.35133	0.675664	+	0.737210i	$0.263857\pi$
$564$	−3.17271	−0.133595
$565$	−38.2202	−1.60794
$566$	−6.53363	−0.274629
$567$	−1.00000	−0.0419961
$568$	2.33116	0.0978132
$569$	42.9210	1.79934	0.899671	−	0.436568i	$-0.143806\pi$
$569$	42.9210	1.79934	0.899671	+	0.436568i	$0.143806\pi$
$570$	2.80169	0.117350
$571$	1.49190	0.0624340	0.0312170	−	0.999513i	$-0.490062\pi$
$571$	1.49190	0.0624340	0.0312170	+	0.999513i	$0.490062\pi$
$572$	−14.5522	−0.608458
$573$	9.61629	0.401726
$574$	1.80328	0.0752673
$575$	−2.10760	−0.0878931
$576$	−1.59393	−0.0664139
$577$	−43.3325	−1.80396	−0.901979	−	0.431780i	$-0.857886\pi$
$577$	−43.3325	−1.80396	−0.901979	+	0.431780i	$0.857886\pi$
$578$	1.58492	0.0659238
$579$	19.8888	0.826549
$580$	−7.10995	−0.295225
$581$	−7.56674	−0.313921
$582$	5.75848	0.238697
$583$	14.2154	0.588743
$584$	−20.3080	−0.840349
$585$	5.50798	0.227727
$586$	−10.4161	−0.430286
$587$	−19.7445	−0.814945	−0.407472	−	0.913218i	$-0.633590\pi$
$587$	−19.7445	−0.814945	−0.407472	+	0.913218i	$0.633590\pi$
$588$	−1.69566	−0.0699280
$589$	−12.8127	−0.527939
$590$	−4.93913	−0.203341
$591$	10.9679	0.451157
$592$	−19.9222	−0.818798
$593$	−19.6685	−0.807687	−0.403843	−	0.914828i	$-0.632326\pi$
$593$	−19.6685	−0.807687	−0.403843	+	0.914828i	$0.632326\pi$
$594$	2.38584	0.0978922
$595$	−10.4326	−0.427695
$596$	2.54642	0.104305
$597$	3.25742	0.133317
$598$	−0.853168	−0.0348886
$599$	13.8334	0.565216	0.282608	−	0.959236i	$-0.408800\pi$
$599$	13.8334	0.565216	0.282608	+	0.959236i	$0.408800\pi$
$600$	5.51348	0.225087
$601$	−20.6120	−0.840780	−0.420390	−	0.907344i	$-0.638107\pi$
$601$	−20.6120	−0.840780	−0.420390	+	0.907344i	$0.638107\pi$
$602$	−6.50682	−0.265198
$603$	5.36958	0.218666
$604$	−10.0220	−0.407787
$605$	−21.3830	−0.869341
$606$	7.52455	0.305664
$607$	−21.4059	−0.868839	−0.434419	−	0.900711i	$-0.643046\pi$
$607$	−21.4059	−0.868839	−0.434419	+	0.900711i	$0.643046\pi$
$608$	−9.74847	−0.395353
$609$	1.51064	0.0612141
$610$	−2.90557	−0.117643
$611$	−3.71293	−0.150209
$612$	6.37331	0.257626
$613$	−1.31022	−0.0529191	−0.0264596	−	0.999650i	$-0.508423\pi$
$613$	−1.31022	−0.0529191	−0.0264596	+	0.999650i	$0.508423\pi$
$614$	−8.88426	−0.358540
$615$	−9.07302	−0.365859
$616$	8.81726	0.355258
$617$	−0.791805	−0.0318769	−0.0159384	−	0.999873i	$-0.505074\pi$
$617$	−0.791805	−0.0318769	−0.0159384	+	0.999873i	$0.505074\pi$
$618$	5.24856	0.211128
$619$	−5.12902	−0.206153	−0.103076	−	0.994673i	$-0.532869\pi$
$619$	−5.12902	−0.206153	−0.103076	+	0.994673i	$0.532869\pi$
$620$	−32.9589	−1.32366
$621$	−0.779349	−0.0312742
$622$	−15.7465	−0.631378
$623$	13.1900	0.528445
$624$	−4.49780	−0.180056
$625$	−31.2083	−1.24833
$626$	−14.2332	−0.568874
$627$	−7.91297	−0.316013
$628$	−16.2633	−0.648976
$629$	33.0361	1.31724
$630$	−1.53124	−0.0610062
$631$	−49.3237	−1.96354	−0.981772	−	0.190061i	$-0.939131\pi$
$631$	−49.3237	−1.96354	−0.981772	+	0.190061i	$0.939131\pi$
$632$	2.27210	0.0903792
$633$	2.26840	0.0901608
$634$	−4.43860	−0.176279
$635$	55.4523	2.20056
$636$	5.57360	0.221007
$637$	−1.98438	−0.0786241
$638$	−3.60414	−0.142689
$639$	1.14341	0.0452326
$640$	−32.0180	−1.26562
$641$	−35.4724	−1.40107	−0.700537	−	0.713616i	$-0.747056\pi$
$641$	−35.4724	−1.40107	−0.700537	+	0.713616i	$0.747056\pi$
$642$	1.53239	0.0604785
$643$	43.3347	1.70896	0.854478	−	0.519488i	$-0.173877\pi$
$643$	43.3347	1.70896	0.854478	+	0.519488i	$0.173877\pi$
$644$	−1.32151	−0.0520749
$645$	32.7385	1.28908
$646$	3.79383	0.149266
$647$	−21.4001	−0.841324	−0.420662	−	0.907217i	$-0.638202\pi$
$647$	−21.4001	−0.841324	−0.420662	+	0.907217i	$0.638202\pi$
$648$	2.03878	0.0800907
$649$	13.9499	0.547581
$650$	2.96046	0.116119
$651$	7.00271	0.274458
$652$	5.21798	0.204352
$653$	2.75990	0.108003	0.0540016	−	0.998541i	$-0.482802\pi$
$653$	2.75990	0.108003	0.0540016	+	0.998541i	$0.482802\pi$
$654$	0.979140	0.0382874
$655$	−41.1727	−1.60875
$656$	7.40900	0.289273
$657$	−9.96085	−0.388610
$658$	1.03221	0.0402398
$659$	28.9373	1.12724	0.563619	−	0.826035i	$-0.309409\pi$
$659$	28.9373	1.12724	0.563619	+	0.826035i	$0.309409\pi$
$660$	−20.3550	−0.792316
$661$	−39.0214	−1.51775	−0.758877	−	0.651234i	$-0.774252\pi$
$661$	−39.0214	−1.51775	−0.758877	+	0.651234i	$0.774252\pi$
$662$	0.725779	0.0282082
$663$	7.45849	0.289664
$664$	15.4269	0.598679
$665$	5.07858	0.196939
$666$	4.84887	0.187890
$667$	1.17731	0.0455858
$668$	−10.8480	−0.419723
$669$	6.48042	0.250547
$670$	8.22213	0.317649
$671$	8.20638	0.316804
$672$	5.32796	0.205531
$673$	−5.54939	−0.213913	−0.106957	−	0.994264i	$-0.534111\pi$
$673$	−5.54939	−0.213913	−0.106957	+	0.994264i	$0.534111\pi$
$674$	−5.35703	−0.206345
$675$	2.70431	0.104089
$676$	15.3665	0.591019
$677$	28.8305	1.10804	0.554022	−	0.832502i	$-0.313092\pi$
$677$	28.8305	1.10804	0.554022	+	0.832502i	$0.313092\pi$
$678$	−7.59632	−0.291735
$679$	10.4383	0.400586
$680$	21.2697	0.815657
$681$	−27.5885	−1.05719
$682$	−16.7073	−0.639757
$683$	−13.6324	−0.521630	−0.260815	−	0.965389i	$-0.583991\pi$
$683$	−13.6324	−0.521630	−0.260815	+	0.965389i	$0.583991\pi$
$684$	−3.10252	−0.118628
$685$	17.9954	0.687571
$686$	0.551667	0.0210628
$687$	14.4677	0.551978
$688$	−26.7342	−1.01923
$689$	6.52260	0.248491
$690$	−1.19337	−0.0454310
$691$	−46.7827	−1.77970	−0.889849	−	0.456255i	$-0.849191\pi$
$691$	−46.7827	−1.77970	−0.889849	+	0.456255i	$0.849191\pi$
$692$	−11.5103	−0.437557
$693$	4.32478	0.164285
$694$	−0.453606	−0.0172187
$695$	61.0762	2.31675
$696$	−3.07985	−0.116741
$697$	−12.2860	−0.465366
$698$	6.54115	0.247586
$699$	−16.7898	−0.635050
$700$	4.58560	0.173319
$701$	17.3428	0.655028	0.327514	−	0.944846i	$-0.393789\pi$
$701$	17.3428	0.655028	0.327514	+	0.944846i	$0.393789\pi$
$702$	1.09472	0.0413175
$703$	−16.0819	−0.606542
$704$	6.89342	0.259805
$705$	−5.19348	−0.195598
$706$	8.78990	0.330812
$707$	13.6396	0.512972
$708$	5.46948	0.205556
$709$	10.8823	0.408695	0.204347	−	0.978898i	$-0.434493\pi$
$709$	10.8823	0.408695	0.204347	+	0.978898i	$0.434493\pi$
$710$	1.75084	0.0657078
$711$	1.11444	0.0417948
$712$	−26.8914	−1.00780
$713$	5.45755	0.204387
$714$	−2.07350	−0.0775986
$715$	−23.8208	−0.890847
$716$	8.53517	0.318974
$717$	10.6101	0.396241
$718$	14.5474	0.542903
$719$	44.1875	1.64792	0.823958	−	0.566650i	$-0.191761\pi$
$719$	44.1875	1.64792	0.823958	+	0.566650i	$0.191761\pi$
$720$	−6.29132	−0.234464
$721$	9.51399	0.354320
$722$	8.63485	0.321356
$723$	−21.2276	−0.789465
$724$	−8.23934	−0.306213
$725$	−4.08523	−0.151722
$726$	−4.24990	−0.157728
$727$	6.53355	0.242316	0.121158	−	0.992633i	$-0.461339\pi$
$727$	6.53355	0.242316	0.121158	+	0.992633i	$0.461339\pi$
$728$	4.04571	0.149944
$729$	1.00000	0.0370370
$730$	−15.2525	−0.564520
$731$	44.3320	1.63968
$732$	3.21756	0.118924
$733$	−15.1349	−0.559021	−0.279511	−	0.960143i	$-0.590172\pi$
$733$	−15.1349	−0.559021	−0.279511	+	0.960143i	$0.590172\pi$
$734$	−1.25551	−0.0463417
$735$	−2.77566	−0.102382
$736$	4.15234	0.153057
$737$	−23.2222	−0.855402
$738$	−1.80328	−0.0663795
$739$	3.71836	0.136782	0.0683910	−	0.997659i	$-0.478213\pi$
$739$	3.71836	0.136782	0.0683910	+	0.997659i	$0.478213\pi$
$740$	−41.3685	−1.52074
$741$	−3.63078	−0.133380
$742$	−1.81332	−0.0665689
$743$	−49.7007	−1.82334	−0.911670	−	0.410923i	$-0.865206\pi$
$743$	−49.7007	−1.82334	−0.911670	+	0.410923i	$0.865206\pi$
$744$	−14.2770	−0.523419
$745$	4.16828	0.152714
$746$	−5.16569	−0.189130
$747$	7.56674	0.276852
$748$	−27.5632	−1.00781
$749$	2.77774	0.101496
$750$	−3.51526	−0.128359
$751$	46.8358	1.70906	0.854531	−	0.519401i	$-0.173845\pi$
$751$	46.8358	1.70906	0.854531	+	0.519401i	$0.173845\pi$
$752$	4.24098	0.154653
$753$	12.0632	0.439606
$754$	−1.65372	−0.0602250
$755$	−16.4051	−0.597044
$756$	1.69566	0.0616707
$757$	15.1599	0.550997	0.275498	−	0.961302i	$-0.411157\pi$
$757$	15.1599	0.550997	0.275498	+	0.961302i	$0.411157\pi$
$758$	11.0311	0.400666
$759$	3.37051	0.122342
$760$	−10.3541	−0.375582
$761$	36.6653	1.32912	0.664558	−	0.747237i	$-0.268620\pi$
$761$	36.6653	1.32912	0.664558	+	0.747237i	$0.268620\pi$
$762$	11.0212	0.399257
$763$	1.77487	0.0642547
$764$	−16.3060	−0.589930
$765$	10.4326	0.377191
$766$	0.551667	0.0199326
$767$	6.40076	0.231118
$768$	−3.17575	−0.114595
$769$	44.1443	1.59189	0.795943	−	0.605372i	$-0.206976\pi$
$769$	44.1443	1.59189	0.795943	+	0.605372i	$0.206976\pi$
$770$	6.62229	0.238651
$771$	−20.3686	−0.733556
$772$	−33.7246	−1.21378
$773$	−43.8678	−1.57781	−0.788907	−	0.614512i	$-0.789353\pi$
$773$	−43.8678	−1.57781	−0.788907	+	0.614512i	$0.789353\pi$
$774$	6.50682	0.233883
$775$	−18.9375	−0.680254
$776$	−21.2814	−0.763958
$777$	8.78948	0.315321
$778$	1.06611	0.0382218
$779$	5.98081	0.214285
$780$	−9.33967	−0.334414
$781$	−4.94500	−0.176946
$782$	−1.61598	−0.0577872
$783$	−1.51064	−0.0539858
$784$	2.26660	0.0809500
$785$	−26.6217	−0.950169
$786$	−8.18314	−0.291883
$787$	13.8325	0.493077	0.246538	−	0.969133i	$-0.420707\pi$
$787$	13.8325	0.493077	0.246538	+	0.969133i	$0.420707\pi$
$788$	−18.5978	−0.662518
$789$	12.5700	0.447504
$790$	1.70648	0.0607138
$791$	−13.7697	−0.489596
$792$	−8.81726	−0.313308
$793$	3.76541	0.133714
$794$	−15.9725	−0.566844
$795$	9.12353	0.323578
$796$	−5.52348	−0.195775
$797$	−28.2553	−1.00086	−0.500428	−	0.865778i	$-0.666824\pi$
$797$	−28.2553	−1.00086	−0.500428	+	0.865778i	$0.666824\pi$
$798$	1.00938	0.0357315
$799$	−7.03262	−0.248796
$800$	−14.4085	−0.509416
$801$	−13.1900	−0.466045
$802$	−9.21019	−0.325223
$803$	43.0785	1.52021
$804$	−9.10499	−0.321108
$805$	−2.16321	−0.0762432
$806$	−7.66599	−0.270023
$807$	−15.0298	−0.529074
$808$	−27.8082	−0.978289
$809$	−4.28415	−0.150623	−0.0753114	−	0.997160i	$-0.523995\pi$
$809$	−4.28415	−0.150623	−0.0753114	+	0.997160i	$0.523995\pi$
$810$	1.53124	0.0538024
$811$	−25.5868	−0.898472	−0.449236	−	0.893413i	$-0.648304\pi$
$811$	−25.5868	−0.898472	−0.449236	+	0.893413i	$0.648304\pi$
$812$	−2.56153	−0.0898921
$813$	11.3180	0.396939
$814$	−20.9703	−0.735009
$815$	8.54141	0.299192
$816$	−8.51923	−0.298233
$817$	−21.5808	−0.755016
$818$	7.72139	0.269972
$819$	1.98438	0.0693399
$820$	15.3848	0.537260
$821$	−31.1945	−1.08870	−0.544349	−	0.838859i	$-0.683223\pi$
$821$	−31.1945	−1.08870	−0.544349	+	0.838859i	$0.683223\pi$
$822$	3.57662	0.124749
$823$	−52.5603	−1.83214	−0.916069	−	0.401021i	$-0.868656\pi$
$823$	−52.5603	−1.83214	−0.916069	+	0.401021i	$0.868656\pi$
$824$	−19.3969	−0.675723
$825$	−11.6955	−0.407187
$826$	−1.77944	−0.0619147
$827$	−31.3202	−1.08911	−0.544555	−	0.838725i	$-0.683302\pi$
$827$	−31.3202	−1.08911	−0.544555	+	0.838725i	$0.683302\pi$
$828$	1.32151	0.0459258
$829$	−13.0142	−0.452003	−0.226001	−	0.974127i	$-0.572565\pi$
$829$	−13.0142	−0.452003	−0.226001	+	0.974127i	$0.572565\pi$
$830$	11.5865	0.402174
$831$	4.77073	0.165495
$832$	3.16297	0.109656
$833$	−3.75860	−0.130228
$834$	12.1390	0.420339
$835$	−17.7573	−0.614518
$836$	13.4177	0.464062
$837$	−7.00271	−0.242049
$838$	3.84721	0.132900
$839$	−13.5924	−0.469262	−0.234631	−	0.972084i	$-0.575388\pi$
$839$	−13.5924	−0.469262	−0.234631	+	0.972084i	$0.575388\pi$
$840$	5.65896	0.195253
$841$	−26.7180	−0.921310
$842$	11.1139	0.383009
$843$	1.64685	0.0567207
$844$	−3.84644	−0.132400
$845$	25.1537	0.865314
$846$	−1.03221	−0.0354882
$847$	−7.70373	−0.264703
$848$	−7.45025	−0.255843
$849$	11.8434	0.406465
$850$	5.60737	0.192331
$851$	6.85007	0.234818
$852$	−1.93884	−0.0664235
$853$	23.7716	0.813925	0.406962	−	0.913445i	$-0.366588\pi$
$853$	23.7716	0.813925	0.406962	+	0.913445i	$0.366588\pi$
$854$	−1.04680	−0.0358208
$855$	−5.07858	−0.173684
$856$	−5.66319	−0.193564
$857$	−43.8154	−1.49670	−0.748352	−	0.663301i	$-0.769155\pi$
$857$	−43.8154	−1.49670	−0.748352	+	0.663301i	$0.769155\pi$
$858$	−4.73442	−0.161630
$859$	23.8338	0.813199	0.406599	−	0.913607i	$-0.366714\pi$
$859$	23.8338	0.813199	0.406599	+	0.913607i	$0.366714\pi$
$860$	−55.5134	−1.89299
$861$	−3.26877	−0.111400
$862$	13.6765	0.465823
$863$	0.253984	0.00864572	0.00432286	−	0.999991i	$-0.498624\pi$
$863$	0.253984	0.00864572	0.00432286	+	0.999991i	$0.498624\pi$
$864$	−5.32796	−0.181261
$865$	−18.8415	−0.640630
$866$	6.22283	0.211460
$867$	−2.87295	−0.0975707
$868$	−11.8742	−0.403038
$869$	−4.81971	−0.163498
$870$	−2.31315	−0.0784232
$871$	−10.6553	−0.361041
$872$	−3.61857	−0.122540
$873$	−10.4383	−0.353284
$874$	0.786656	0.0266090
$875$	−6.37207	−0.215415
$876$	16.8902	0.570669
$877$	33.1486	1.11935	0.559674	−	0.828713i	$-0.310927\pi$
$877$	33.1486	1.11935	0.559674	+	0.828713i	$0.310927\pi$
$878$	−3.13678	−0.105861
$879$	18.8811	0.636845
$880$	27.2086	0.917201
$881$	−21.1917	−0.713966	−0.356983	−	0.934111i	$-0.616195\pi$
$881$	−21.1917	−0.713966	−0.356983	+	0.934111i	$0.616195\pi$
$882$	−0.551667	−0.0185756
$883$	17.0538	0.573906	0.286953	−	0.957945i	$-0.407358\pi$
$883$	17.0538	0.573906	0.286953	+	0.957945i	$0.407358\pi$
$884$	−12.6471	−0.425367
$885$	8.95310	0.300955
$886$	−15.2071	−0.510892
$887$	−27.0899	−0.909590	−0.454795	−	0.890596i	$-0.650288\pi$
$887$	−27.0899	−0.909590	−0.454795	+	0.890596i	$0.650288\pi$
$888$	−17.9198	−0.601349
$889$	19.9780	0.670041
$890$	−20.1971	−0.677007
$891$	−4.32478	−0.144886
$892$	−10.9886	−0.367926
$893$	3.42347	0.114562
$894$	0.828451	0.0277076
$895$	13.9714	0.467012
$896$	−11.5352	−0.385366
$897$	1.54653	0.0516370
$898$	−4.17464	−0.139309
$899$	10.5785	0.352814
$900$	−4.58560	−0.152853
$901$	12.3544	0.411585
$902$	7.79878	0.259671
$903$	11.7948	0.392507
$904$	28.0734	0.933709
$905$	−13.4871	−0.448328
$906$	−3.26054	−0.108324
$907$	22.1800	0.736474	0.368237	−	0.929732i	$-0.379962\pi$
$907$	22.1800	0.736474	0.368237	+	0.929732i	$0.379962\pi$
$908$	46.7808	1.55248
$909$	−13.6396	−0.452399
$910$	3.03857	0.100728
$911$	36.1797	1.19869	0.599344	−	0.800491i	$-0.295428\pi$
$911$	36.1797	1.19869	0.599344	+	0.800491i	$0.295428\pi$
$912$	4.14715	0.137326
$913$	−32.7245	−1.08302
$914$	−2.57368	−0.0851297
$915$	5.26689	0.174118
$916$	−24.5324	−0.810573
$917$	−14.8335	−0.489844
$918$	2.07350	0.0684355
$919$	53.2945	1.75802	0.879011	−	0.476802i	$-0.158204\pi$
$919$	53.2945	1.75802	0.879011	+	0.476802i	$0.158204\pi$
$920$	4.41031	0.145403
$921$	16.1044	0.530657
$922$	−17.8520	−0.587925
$923$	−2.26896	−0.0746838
$924$	−7.33337	−0.241250
$925$	−23.7695	−0.781535
$926$	−15.8772	−0.521756
$927$	−9.51399	−0.312480
$928$	8.04862	0.264209
$929$	10.9119	0.358009	0.179005	−	0.983848i	$-0.442712\pi$
$929$	10.9119	0.358009	0.179005	+	0.983848i	$0.442712\pi$
$930$	−10.7228	−0.351616
$931$	1.82968	0.0599654
$932$	28.4699	0.932563
$933$	28.5435	0.934473
$934$	−12.6577	−0.414172
$935$	−45.1187	−1.47554
$936$	−4.04571	−0.132238
$937$	38.2647	1.25005	0.625026	−	0.780604i	$-0.285088\pi$
$937$	38.2647	1.25005	0.625026	+	0.780604i	$0.285088\pi$
$938$	2.96222	0.0967199
$939$	25.8004	0.841964
$940$	8.80639	0.287233
$941$	32.9825	1.07520	0.537599	−	0.843201i	$-0.319331\pi$
$941$	32.9825	1.07520	0.537599	+	0.843201i	$0.319331\pi$
$942$	−5.29110	−0.172393
$943$	−2.54752	−0.0829586
$944$	−7.31108	−0.237955
$945$	2.77566	0.0902923
$946$	−28.1406	−0.914930
$947$	−21.0631	−0.684458	−0.342229	−	0.939617i	$-0.611182\pi$
$947$	−21.0631	−0.684458	−0.342229	+	0.939617i	$0.611182\pi$
$948$	−1.88972	−0.0613751
$949$	19.7661	0.641636
$950$	−2.72966	−0.0885620
$951$	8.04579	0.260903
$952$	7.66294	0.248357
$953$	−5.23743	−0.169657	−0.0848284	−	0.996396i	$-0.527034\pi$
$953$	−5.23743	−0.169657	−0.0848284	+	0.996396i	$0.527034\pi$
$954$	1.81332	0.0587082
$955$	−26.6916	−0.863720
$956$	−17.9912	−0.581875
$957$	6.53317	0.211188
$958$	−3.75230	−0.121231
$959$	6.48329	0.209357
$960$	4.42423	0.142791
$961$	18.0379	0.581867
$962$	−9.62201	−0.310226
$963$	−2.77774	−0.0895113
$964$	35.9949	1.15932
$965$	−55.2045	−1.77710
$966$	−0.429942	−0.0138331
$967$	−47.8988	−1.54032	−0.770160	−	0.637850i	$-0.779824\pi$
$967$	−47.8988	−1.54032	−0.770160	+	0.637850i	$0.779824\pi$
$968$	15.7062	0.504816
$969$	−6.87703	−0.220922
$970$	−15.9836	−0.513203
$971$	16.4889	0.529155	0.264577	−	0.964364i	$-0.414768\pi$
$971$	16.4889	0.529155	0.264577	+	0.964364i	$0.414768\pi$
$972$	−1.69566	−0.0543884
$973$	22.0042	0.705421
$974$	−11.8735	−0.380451
$975$	−5.36638	−0.171862
$976$	−4.30093	−0.137669
$977$	−34.0030	−1.08785	−0.543926	−	0.839133i	$-0.683063\pi$
$977$	−34.0030	−1.08785	−0.543926	+	0.839133i	$0.683063\pi$
$978$	1.69762	0.0542838
$979$	57.0438	1.82313
$980$	4.70659	0.150347
$981$	−1.77487	−0.0566674
$982$	14.6811	0.468493
$983$	34.3652	1.09608	0.548039	−	0.836452i	$-0.315374\pi$
$983$	34.3652	1.09608	0.548039	+	0.836452i	$0.315374\pi$
$984$	6.66430	0.212450
$985$	−30.4431	−0.969997
$986$	−3.13230	−0.0997527
$987$	−1.87108	−0.0595570
$988$	6.15659	0.195867
$989$	9.19229	0.292298
$990$	−6.62229	−0.210470
$991$	−1.40953	−0.0447752	−0.0223876	−	0.999749i	$-0.507127\pi$
$991$	−1.40953	−0.0447752	−0.0223876	+	0.999749i	$0.507127\pi$
$992$	37.3102	1.18460
$993$	−1.31561	−0.0417496
$994$	0.630782	0.0200072
$995$	−9.04150	−0.286635
$996$	−12.8306	−0.406554
$997$	−12.4096	−0.393014	−0.196507	−	0.980502i	$-0.562960\pi$
$997$	−12.4096	−0.393014	−0.196507	+	0.980502i	$0.562960\pi$
$998$	15.0484	0.476349
$999$	−8.78948	−0.278087

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.p.1.15	✓	41

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.p.1.15	✓	41	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-0.551667 q^{2} +1.00000 q^{3} -1.69566 q^{4} -2.77566 q^{5} -0.551667 q^{6} -1.00000 q^{7} +2.03878 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.551667 q^{2} +1.00000 q^{3} -1.69566 q^{4} -2.77566 q^{5} -0.551667 q^{6} -1.00000 q^{7} +2.03878 q^{8} +1.00000 q^{9} +1.53124 q^{10} -4.32478 q^{11} -1.69566 q^{12} -1.98438 q^{13} +0.551667 q^{14} -2.77566 q^{15} +2.26660 q^{16} -3.75860 q^{17} -0.551667 q^{18} +1.82968 q^{19} +4.70659 q^{20} -1.00000 q^{21} +2.38584 q^{22} -0.779349 q^{23} +2.03878 q^{24} +2.70431 q^{25} +1.09472 q^{26} +1.00000 q^{27} +1.69566 q^{28} -1.51064 q^{29} +1.53124 q^{30} -7.00271 q^{31} -5.32796 q^{32} -4.32478 q^{33} +2.07350 q^{34} +2.77566 q^{35} -1.69566 q^{36} -8.78948 q^{37} -1.00938 q^{38} -1.98438 q^{39} -5.65896 q^{40} +3.26877 q^{41} +0.551667 q^{42} -11.7948 q^{43} +7.33337 q^{44} -2.77566 q^{45} +0.429942 q^{46} +1.87108 q^{47} +2.26660 q^{48} +1.00000 q^{49} -1.49188 q^{50} -3.75860 q^{51} +3.36484 q^{52} -3.28697 q^{53} -0.551667 q^{54} +12.0041 q^{55} -2.03878 q^{56} +1.82968 q^{57} +0.833369 q^{58} -3.22557 q^{59} +4.70659 q^{60} -1.89752 q^{61} +3.86316 q^{62} -1.00000 q^{63} -1.59393 q^{64} +5.50798 q^{65} +2.38584 q^{66} +5.36958 q^{67} +6.37331 q^{68} -0.779349 q^{69} -1.53124 q^{70} +1.14341 q^{71} +2.03878 q^{72} -9.96085 q^{73} +4.84887 q^{74} +2.70431 q^{75} -3.10252 q^{76} +4.32478 q^{77} +1.09472 q^{78} +1.11444 q^{79} -6.29132 q^{80} +1.00000 q^{81} -1.80328 q^{82} +7.56674 q^{83} +1.69566 q^{84} +10.4326 q^{85} +6.50682 q^{86} -1.51064 q^{87} -8.81726 q^{88} -13.1900 q^{89} +1.53124 q^{90} +1.98438 q^{91} +1.32151 q^{92} -7.00271 q^{93} -1.03221 q^{94} -5.07858 q^{95} -5.32796 q^{96} -10.4383 q^{97} -0.551667 q^{98} -4.32478 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9}+O(q^{10}) \) 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9	\( 41 q + 7 q^{2} + 41 q^{3} + 45 q^{4} + 17 q^{5} + 7 q^{6} - 41 q^{7} + 12 q^{8} + 41 q^{9} + 18 q^{10} + 8 q^{11} + 45 q^{12} + 23 q^{13} - 7 q^{14} + 17 q^{15} + 37 q^{16} + 15 q^{17} + 7 q^{18} + 15 q^{19} + 53 q^{20} - 41 q^{21} + 13 q^{22} + 44 q^{23} + 12 q^{24} + 58 q^{25} + 9 q^{26} + 41 q^{27} - 45 q^{28} + 21 q^{29} + 18 q^{30} + 39 q^{31} + 61 q^{32} + 8 q^{33} + 9 q^{34} - 17 q^{35} + 45 q^{36} + 11 q^{37} + 44 q^{38} + 23 q^{39} + 24 q^{40} + 17 q^{41} - 7 q^{42} + 7 q^{43} + 30 q^{44} + 17 q^{45} - 12 q^{46} + 36 q^{47} + 37 q^{48} + 41 q^{49} + 28 q^{50} + 15 q^{51} + 58 q^{52} + 26 q^{53} + 7 q^{54} + 32 q^{55} - 12 q^{56} + 15 q^{57} - 4 q^{58} + 33 q^{59} + 53 q^{60} + 59 q^{61} - q^{62} - 41 q^{63} + 16 q^{64} + 72 q^{65} + 13 q^{66} + 12 q^{67} + 52 q^{68} + 44 q^{69} - 18 q^{70} + 33 q^{71} + 12 q^{72} + 18 q^{73} + 42 q^{74} + 58 q^{75} + 7 q^{76} - 8 q^{77} + 9 q^{78} + 22 q^{79} + 69 q^{80} + 41 q^{81} + 41 q^{82} + 32 q^{83} - 45 q^{84} - 44 q^{85} + 11 q^{86} + 21 q^{87} + 52 q^{88} + 63 q^{89} + 18 q^{90} - 23 q^{91} + 52 q^{92} + 39 q^{93} + 17 q^{94} + 37 q^{95} + 61 q^{96} + 8 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) 41 * q + 7 * q^2 + 41 * q^3 + 45 * q^4 + 17 * q^5 + 7 * q^6 - 41 * q^7 + 12 * q^8 + 41 * q^9 + 18 * q^10 + 8 * q^11 + 45 * q^12 + 23 * q^13 - 7 * q^14 + 17 * q^15 + 37 * q^16 + 15 * q^17 + 7 * q^18 + 15 * q^19 + 53 * q^20 - 41 * q^21 + 13 * q^22 + 44 * q^23 + 12 * q^24 + 58 * q^25 + 9 * q^26 + 41 * q^27 - 45 * q^28 + 21 * q^29 + 18 * q^30 + 39 * q^31 + 61 * q^32 + 8 * q^33 + 9 * q^34 - 17 * q^35 + 45 * q^36 + 11 * q^37 + 44 * q^38 + 23 * q^39 + 24 * q^40 + 17 * q^41 - 7 * q^42 + 7 * q^43 + 30 * q^44 + 17 * q^45 - 12 * q^46 + 36 * q^47 + 37 * q^48 + 41 * q^49 + 28 * q^50 + 15 * q^51 + 58 * q^52 + 26 * q^53 + 7 * q^54 + 32 * q^55 - 12 * q^56 + 15 * q^57 - 4 * q^58 + 33 * q^59 + 53 * q^60 + 59 * q^61 - q^62 - 41 * q^63 + 16 * q^64 + 72 * q^65 + 13 * q^66 + 12 * q^67 + 52 * q^68 + 44 * q^69 - 18 * q^70 + 33 * q^71 + 12 * q^72 + 18 * q^73 + 42 * q^74 + 58 * q^75 + 7 * q^76 - 8 * q^77 + 9 * q^78 + 22 * q^79 + 69 * q^80 + 41 * q^81 + 41 * q^82 + 32 * q^83 - 45 * q^84 - 44 * q^85 + 11 * q^86 + 21 * q^87 + 52 * q^88 + 63 * q^89 + 18 * q^90 - 23 * q^91 + 52 * q^92 + 39 * q^93 + 17 * q^94 + 37 * q^95 + 61 * q^96 + 8 * q^97 + 7 * q^98 + 8 * q^99

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