LMFDB - Embedded newform 8031.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$1$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67330 q^{2} +1.00000 q^{3} +5.14651 q^{4} +3.81418 q^{5} -2.67330 q^{6} -0.936528 q^{7} -8.41154 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.67330 q^{2} +1.00000 q^{3} +5.14651 q^{4} +3.81418 q^{5} -2.67330 q^{6} -0.936528 q^{7} -8.41154 q^{8} +1.00000 q^{9} -10.1964 q^{10} -2.79168 q^{11} +5.14651 q^{12} -2.19689 q^{13} +2.50361 q^{14} +3.81418 q^{15} +12.1935 q^{16} -0.445928 q^{17} -2.67330 q^{18} +2.11846 q^{19} +19.6297 q^{20} -0.936528 q^{21} +7.46300 q^{22} +7.72992 q^{23} -8.41154 q^{24} +9.54800 q^{25} +5.87293 q^{26} +1.00000 q^{27} -4.81985 q^{28} -7.62389 q^{29} -10.1964 q^{30} -0.830764 q^{31} -15.7738 q^{32} -2.79168 q^{33} +1.19210 q^{34} -3.57209 q^{35} +5.14651 q^{36} -6.69881 q^{37} -5.66326 q^{38} -2.19689 q^{39} -32.0832 q^{40} -1.26425 q^{41} +2.50361 q^{42} -11.8036 q^{43} -14.3674 q^{44} +3.81418 q^{45} -20.6644 q^{46} -7.43177 q^{47} +12.1935 q^{48} -6.12292 q^{49} -25.5246 q^{50} -0.445928 q^{51} -11.3063 q^{52} +3.25814 q^{53} -2.67330 q^{54} -10.6480 q^{55} +7.87764 q^{56} +2.11846 q^{57} +20.3809 q^{58} +8.23363 q^{59} +19.6297 q^{60} -0.197122 q^{61} +2.22088 q^{62} -0.936528 q^{63} +17.7810 q^{64} -8.37934 q^{65} +7.46300 q^{66} -5.66572 q^{67} -2.29497 q^{68} +7.72992 q^{69} +9.54925 q^{70} -6.58177 q^{71} -8.41154 q^{72} +1.49320 q^{73} +17.9079 q^{74} +9.54800 q^{75} +10.9026 q^{76} +2.61449 q^{77} +5.87293 q^{78} -14.7878 q^{79} +46.5083 q^{80} +1.00000 q^{81} +3.37972 q^{82} -13.4719 q^{83} -4.81985 q^{84} -1.70085 q^{85} +31.5546 q^{86} -7.62389 q^{87} +23.4824 q^{88} -6.90565 q^{89} -10.1964 q^{90} +2.05745 q^{91} +39.7821 q^{92} -0.830764 q^{93} +19.8673 q^{94} +8.08018 q^{95} -15.7738 q^{96} -5.68913 q^{97} +16.3684 q^{98} -2.79168 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9}+O(q^{10}) $ 92 * q - 6 * q^2 + 92 * q^3 + 70 * q^4 - 18 * q^5 - 6 * q^6 - 42 * q^7 - 15 * q^8 + 92 * q^9	\( 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9} - 44 q^{10} - 24 q^{11} + 70 q^{12} - 48 q^{13} - 29 q^{14} - 18 q^{15} + 26 q^{16} - 69 q^{17} - 6 q^{18} - 74 q^{19} - 42 q^{20} - 42 q^{21} - 62 q^{22} - 19 q^{23} - 15 q^{24} + 16 q^{25} - 27 q^{26} + 92 q^{27} - 101 q^{28} - 54 q^{29} - 44 q^{30} - 67 q^{31} - 36 q^{32} - 24 q^{33} - 63 q^{34} - 31 q^{35} + 70 q^{36} - 70 q^{37} - 18 q^{38} - 48 q^{39} - 125 q^{40} - 98 q^{41} - 29 q^{42} - 159 q^{43} - 52 q^{44} - 18 q^{45} - 68 q^{46} - 15 q^{47} + 26 q^{48} - 28 q^{49} - 7 q^{50} - 69 q^{51} - 98 q^{52} - 23 q^{53} - 6 q^{54} - 93 q^{55} - 48 q^{56} - 74 q^{57} - 37 q^{58} - 36 q^{59} - 42 q^{60} - 172 q^{61} - 26 q^{62} - 42 q^{63} - 23 q^{64} - 66 q^{65} - 62 q^{66} - 143 q^{67} - 74 q^{68} - 19 q^{69} - 30 q^{70} - 9 q^{71} - 15 q^{72} - 134 q^{73} - 19 q^{74} + 16 q^{75} - 157 q^{76} - 25 q^{77} - 27 q^{78} - 138 q^{79} - 29 q^{80} + 92 q^{81} - 61 q^{82} - 24 q^{83} - 101 q^{84} - 84 q^{85} + 14 q^{86} - 54 q^{87} - 140 q^{88} - 148 q^{89} - 44 q^{90} - 115 q^{91} - 12 q^{92} - 67 q^{93} - 79 q^{94} - 10 q^{95} - 36 q^{96} - 165 q^{97} + 36 q^{98} - 24 q^{99}+O(q^{100}) \) 92 * q - 6 * q^2 + 92 * q^3 + 70 * q^4 - 18 * q^5 - 6 * q^6 - 42 * q^7 - 15 * q^8 + 92 * q^9 - 44 * q^10 - 24 * q^11 + 70 * q^12 - 48 * q^13 - 29 * q^14 - 18 * q^15 + 26 * q^16 - 69 * q^17 - 6 * q^18 - 74 * q^19 - 42 * q^20 - 42 * q^21 - 62 * q^22 - 19 * q^23 - 15 * q^24 + 16 * q^25 - 27 * q^26 + 92 * q^27 - 101 * q^28 - 54 * q^29 - 44 * q^30 - 67 * q^31 - 36 * q^32 - 24 * q^33 - 63 * q^34 - 31 * q^35 + 70 * q^36 - 70 * q^37 - 18 * q^38 - 48 * q^39 - 125 * q^40 - 98 * q^41 - 29 * q^42 - 159 * q^43 - 52 * q^44 - 18 * q^45 - 68 * q^46 - 15 * q^47 + 26 * q^48 - 28 * q^49 - 7 * q^50 - 69 * q^51 - 98 * q^52 - 23 * q^53 - 6 * q^54 - 93 * q^55 - 48 * q^56 - 74 * q^57 - 37 * q^58 - 36 * q^59 - 42 * q^60 - 172 * q^61 - 26 * q^62 - 42 * q^63 - 23 * q^64 - 66 * q^65 - 62 * q^66 - 143 * q^67 - 74 * q^68 - 19 * q^69 - 30 * q^70 - 9 * q^71 - 15 * q^72 - 134 * q^73 - 19 * q^74 + 16 * q^75 - 157 * q^76 - 25 * q^77 - 27 * q^78 - 138 * q^79 - 29 * q^80 + 92 * q^81 - 61 * q^82 - 24 * q^83 - 101 * q^84 - 84 * q^85 + 14 * q^86 - 54 * q^87 - 140 * q^88 - 148 * q^89 - 44 * q^90 - 115 * q^91 - 12 * q^92 - 67 * q^93 - 79 * q^94 - 10 * q^95 - 36 * q^96 - 165 * q^97 + 36 * q^98 - 24 * 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 $

$2$	−2.67330	−1.89031	−0.945153	−	0.326629i	$-0.894087\pi$
$2$	−2.67330	−1.89031	−0.945153	+	0.326629i	$0.894087\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.14651	2.57325
$5$	3.81418	1.70576	0.852878	−	0.522111i	$-0.174855\pi$
$5$	3.81418	1.70576	0.852878	+	0.522111i	$0.174855\pi$
$6$	−2.67330	−1.09137
$7$	−0.936528	−0.353974	−0.176987	−	0.984213i	$-0.556635\pi$
$7$	−0.936528	−0.353974	−0.176987	+	0.984213i	$0.556635\pi$
$8$	−8.41154	−2.97393
$9$	1.00000	0.333333
$10$	−10.1964	−3.22440
$11$	−2.79168	−0.841725	−0.420862	−	0.907125i	$-0.638272\pi$
$11$	−2.79168	−0.841725	−0.420862	+	0.907125i	$0.638272\pi$
$12$	5.14651	1.48567
$13$	−2.19689	−0.609307	−0.304654	−	0.952463i	$-0.598541\pi$
$13$	−2.19689	−0.609307	−0.304654	+	0.952463i	$0.598541\pi$
$14$	2.50361	0.669119
$15$	3.81418	0.984818
$16$	12.1935	3.04838
$17$	−0.445928	−0.108153	−0.0540767	−	0.998537i	$-0.517222\pi$
$17$	−0.445928	−0.108153	−0.0540767	+	0.998537i	$0.517222\pi$
$18$	−2.67330	−0.630102
$19$	2.11846	0.486007	0.243003	−	0.970025i	$-0.421867\pi$
$19$	2.11846	0.486007	0.243003	+	0.970025i	$0.421867\pi$
$20$	19.6297	4.38934
$21$	−0.936528	−0.204367
$22$	7.46300	1.59112
$23$	7.72992	1.61180	0.805900	−	0.592051i	$-0.201682\pi$
$23$	7.72992	1.61180	0.805900	+	0.592051i	$0.201682\pi$
$24$	−8.41154	−1.71700
$25$	9.54800	1.90960
$26$	5.87293	1.15178
$27$	1.00000	0.192450
$28$	−4.81985	−0.910865
$29$	−7.62389	−1.41572	−0.707861	−	0.706352i	$-0.750340\pi$
$29$	−7.62389	−1.41572	−0.707861	+	0.706352i	$0.750340\pi$
$30$	−10.1964	−1.86161
$31$	−0.830764	−0.149210	−0.0746048	−	0.997213i	$-0.523770\pi$
$31$	−0.830764	−0.149210	−0.0746048	+	0.997213i	$0.523770\pi$
$32$	−15.7738	−2.78844
$33$	−2.79168	−0.485970
$34$	1.19210	0.204443
$35$	−3.57209	−0.603793
$36$	5.14651	0.857751
$37$	−6.69881	−1.10128	−0.550639	−	0.834744i	$-0.685616\pi$
$37$	−6.69881	−1.10128	−0.550639	+	0.834744i	$0.685616\pi$
$38$	−5.66326	−0.918701
$39$	−2.19689	−0.351784
$40$	−32.0832	−5.07280
$41$	−1.26425	−0.197443	−0.0987216	−	0.995115i	$-0.531475\pi$
$41$	−1.26425	−0.197443	−0.0987216	+	0.995115i	$0.531475\pi$
$42$	2.50361	0.386316
$43$	−11.8036	−1.80004	−0.900019	−	0.435852i	$-0.856447\pi$
$43$	−11.8036	−1.80004	−0.900019	+	0.435852i	$0.856447\pi$
$44$	−14.3674	−2.16597
$45$	3.81418	0.568585
$46$	−20.6644	−3.04679
$47$	−7.43177	−1.08403	−0.542017	−	0.840367i	$-0.682339\pi$
$47$	−7.43177	−1.08403	−0.542017	+	0.840367i	$0.682339\pi$
$48$	12.1935	1.75998
$49$	−6.12292	−0.874702
$50$	−25.5246	−3.60973
$51$	−0.445928	−0.0624424
$52$	−11.3063	−1.56790
$53$	3.25814	0.447539	0.223770	−	0.974642i	$-0.428164\pi$
$53$	3.25814	0.447539	0.223770	+	0.974642i	$0.428164\pi$
$54$	−2.67330	−0.363789
$55$	−10.6480	−1.43578
$56$	7.87764	1.05269
$57$	2.11846	0.280596
$58$	20.3809	2.67615
$59$	8.23363	1.07193	0.535964	−	0.844241i	$-0.319948\pi$
$59$	8.23363	1.07193	0.535964	+	0.844241i	$0.319948\pi$
$60$	19.6297	2.53419
$61$	−0.197122	−0.0252389	−0.0126195	−	0.999920i	$-0.504017\pi$
$61$	−0.197122	−0.0252389	−0.0126195	+	0.999920i	$0.504017\pi$
$62$	2.22088	0.282052
$63$	−0.936528	−0.117991
$64$	17.7810	2.22262
$65$	−8.37934	−1.03933
$66$	7.46300	0.918631
$67$	−5.66572	−0.692177	−0.346089	−	0.938202i	$-0.612490\pi$
$67$	−5.66572	−0.692177	−0.346089	+	0.938202i	$0.612490\pi$
$68$	−2.29497	−0.278306
$69$	7.72992	0.930573
$70$	9.54925	1.14135
$71$	−6.58177	−0.781112	−0.390556	−	0.920579i	$-0.627717\pi$
$71$	−6.58177	−0.781112	−0.390556	+	0.920579i	$0.627717\pi$
$72$	−8.41154	−0.991310
$73$	1.49320	0.174766	0.0873830	−	0.996175i	$-0.472150\pi$
$73$	1.49320	0.174766	0.0873830	+	0.996175i	$0.472150\pi$
$74$	17.9079	2.08175
$75$	9.54800	1.10251
$76$	10.9026	1.25062
$77$	2.61449	0.297949
$78$	5.87293	0.664979
$79$	−14.7878	−1.66375	−0.831877	−	0.554959i	$-0.812734\pi$
$79$	−14.7878	−1.66375	−0.831877	+	0.554959i	$0.812734\pi$
$80$	46.5083	5.19979
$81$	1.00000	0.111111
$82$	3.37972	0.373228
$83$	−13.4719	−1.47873	−0.739367	−	0.673303i	$-0.764875\pi$
$83$	−13.4719	−1.47873	−0.739367	+	0.673303i	$0.764875\pi$
$84$	−4.81985	−0.525888
$85$	−1.70085	−0.184483
$86$	31.5546	3.40262
$87$	−7.62389	−0.817367
$88$	23.4824	2.50323
$89$	−6.90565	−0.731998	−0.365999	−	0.930615i	$-0.619273\pi$
$89$	−6.90565	−0.731998	−0.365999	+	0.930615i	$0.619273\pi$
$90$	−10.1964	−1.07480
$91$	2.05745	0.215679
$92$	39.7821	4.14757
$93$	−0.830764	−0.0861462
$94$	19.8673	2.04916
$95$	8.08018	0.829009
$96$	−15.7738	−1.60991
$97$	−5.68913	−0.577644	−0.288822	−	0.957383i	$-0.593264\pi$
$97$	−5.68913	−0.577644	−0.288822	+	0.957383i	$0.593264\pi$
$98$	16.3684	1.65345
$99$	−2.79168	−0.280575
$100$	49.1389	4.91389
$101$	−18.1502	−1.80602	−0.903008	−	0.429624i	$-0.858646\pi$
$101$	−18.1502	−1.80602	−0.903008	+	0.429624i	$0.858646\pi$
$102$	1.19210	0.118035
$103$	12.9918	1.28012	0.640061	−	0.768324i	$-0.278909\pi$
$103$	12.9918	1.28012	0.640061	+	0.768324i	$0.278909\pi$
$104$	18.4792	1.81204
$105$	−3.57209	−0.348600
$106$	−8.70996	−0.845986
$107$	16.7587	1.62012	0.810061	−	0.586346i	$-0.199434\pi$
$107$	16.7587	1.62012	0.810061	+	0.586346i	$0.199434\pi$
$108$	5.14651	0.495223
$109$	12.9996	1.24514	0.622569	−	0.782565i	$-0.286089\pi$
$109$	12.9996	1.24514	0.622569	+	0.782565i	$0.286089\pi$
$110$	28.4652	2.71405
$111$	−6.69881	−0.635823
$112$	−11.4196	−1.07905
$113$	4.64123	0.436610	0.218305	−	0.975881i	$-0.429947\pi$
$113$	4.64123	0.436610	0.218305	+	0.975881i	$0.429947\pi$
$114$	−5.66326	−0.530413
$115$	29.4834	2.74934
$116$	−39.2364	−3.64301
$117$	−2.19689	−0.203102
$118$	−22.0109	−2.02627
$119$	0.417624	0.0382835
$120$	−32.0832	−2.92878
$121$	−3.20650	−0.291500
$122$	0.526966	0.0477093
$123$	−1.26425	−0.113994
$124$	−4.27553	−0.383954
$125$	17.3469	1.55156
$126$	2.50361	0.223040
$127$	−2.69235	−0.238907	−0.119454	−	0.992840i	$-0.538114\pi$
$127$	−2.69235	−0.238907	−0.119454	+	0.992840i	$0.538114\pi$
$128$	−15.9862	−1.41299
$129$	−11.8036	−1.03925
$130$	22.4004	1.96465
$131$	−9.56029	−0.835286	−0.417643	−	0.908611i	$-0.637144\pi$
$131$	−9.56029	−0.835286	−0.417643	+	0.908611i	$0.637144\pi$
$132$	−14.3674	−1.25052
$133$	−1.98399	−0.172034
$134$	15.1461	1.30843
$135$	3.81418	0.328273
$136$	3.75094	0.321640
$137$	16.4727	1.40735	0.703677	−	0.710520i	$-0.251540\pi$
$137$	16.4727	1.40735	0.703677	+	0.710520i	$0.251540\pi$
$138$	−20.6644	−1.75907
$139$	−14.8238	−1.25734	−0.628670	−	0.777672i	$-0.716401\pi$
$139$	−14.8238	−1.25734	−0.628670	+	0.777672i	$0.716401\pi$
$140$	−18.3838	−1.55371
$141$	−7.43177	−0.625868
$142$	17.5950	1.47654
$143$	6.13302	0.512869
$144$	12.1935	1.01613
$145$	−29.0789	−2.41487
$146$	−3.99177	−0.330361
$147$	−6.12292	−0.505010
$148$	−34.4755	−2.83387
$149$	−2.47289	−0.202587	−0.101293	−	0.994857i	$-0.532298\pi$
$149$	−2.47289	−0.202587	−0.101293	+	0.994857i	$0.532298\pi$
$150$	−25.5246	−2.08408
$151$	−12.9083	−1.05046	−0.525230	−	0.850960i	$-0.676021\pi$
$151$	−12.9083	−1.05046	−0.525230	+	0.850960i	$0.676021\pi$
$152$	−17.8195	−1.44535
$153$	−0.445928	−0.0360511
$154$	−6.98930	−0.563214
$155$	−3.16869	−0.254515
$156$	−11.3063	−0.905229
$157$	−18.3221	−1.46226	−0.731130	−	0.682238i	$-0.761007\pi$
$157$	−18.3221	−1.46226	−0.731130	+	0.682238i	$0.761007\pi$
$158$	39.5321	3.14500
$159$	3.25814	0.258387
$160$	−60.1642	−4.75640
$161$	−7.23929	−0.570536
$162$	−2.67330	−0.210034
$163$	4.34913	0.340651	0.170325	−	0.985388i	$-0.445518\pi$
$163$	4.34913	0.340651	0.170325	+	0.985388i	$0.445518\pi$
$164$	−6.50649	−0.508071
$165$	−10.6480	−0.828946
$166$	36.0144	2.79526
$167$	6.99419	0.541226	0.270613	−	0.962688i	$-0.412774\pi$
$167$	6.99419	0.541226	0.270613	+	0.962688i	$0.412774\pi$
$168$	7.87764	0.607773
$169$	−8.17368	−0.628745
$170$	4.54688	0.348729
$171$	2.11846	0.162002
$172$	−60.7475	−4.63195
$173$	−3.45371	−0.262581	−0.131290	−	0.991344i	$-0.541912\pi$
$173$	−3.45371	−0.262581	−0.131290	+	0.991344i	$0.541912\pi$
$174$	20.3809	1.54507
$175$	−8.94197	−0.675949
$176$	−34.0405	−2.56590
$177$	8.23363	0.618877
$178$	18.4609	1.38370
$179$	4.14530	0.309834	0.154917	−	0.987927i	$-0.450489\pi$
$179$	4.14530	0.309834	0.154917	+	0.987927i	$0.450489\pi$
$180$	19.6297	1.46311
$181$	6.08645	0.452402	0.226201	−	0.974081i	$-0.427369\pi$
$181$	6.08645	0.452402	0.226201	+	0.974081i	$0.427369\pi$
$182$	−5.50016	−0.407699
$183$	−0.197122	−0.0145717
$184$	−65.0206	−4.79338
$185$	−25.5505	−1.87851
$186$	2.22088	0.162843
$187$	1.24489	0.0910353
$188$	−38.2476	−2.78950
$189$	−0.936528	−0.0681224
$190$	−21.6007	−1.56708
$191$	18.1500	1.31329	0.656645	−	0.754200i	$-0.271975\pi$
$191$	18.1500	1.31329	0.656645	+	0.754200i	$0.271975\pi$
$192$	17.7810	1.28323
$193$	5.01695	0.361128	0.180564	−	0.983563i	$-0.442208\pi$
$193$	5.01695	0.361128	0.180564	+	0.983563i	$0.442208\pi$
$194$	15.2087	1.09192
$195$	−8.37934	−0.600057
$196$	−31.5116	−2.25083
$197$	2.24907	0.160239	0.0801197	−	0.996785i	$-0.474470\pi$
$197$	2.24907	0.160239	0.0801197	+	0.996785i	$0.474470\pi$
$198$	7.46300	0.530372
$199$	−1.35599	−0.0961239	−0.0480619	−	0.998844i	$-0.515304\pi$
$199$	−1.35599	−0.0961239	−0.0480619	+	0.998844i	$0.515304\pi$
$200$	−80.3134	−5.67902
$201$	−5.66572	−0.399629
$202$	48.5209	3.41392
$203$	7.13999	0.501129
$204$	−2.29497	−0.160680
$205$	−4.82210	−0.336790
$206$	−34.7310	−2.41982
$207$	7.72992	0.537267
$208$	−26.7878	−1.85740
$209$	−5.91406	−0.409084
$210$	9.54925	0.658961
$211$	−5.91080	−0.406916	−0.203458	−	0.979084i	$-0.565218\pi$
$211$	−5.91080	−0.406916	−0.203458	+	0.979084i	$0.565218\pi$
$212$	16.7680	1.15163
$213$	−6.58177	−0.450975
$214$	−44.8009	−3.06252
$215$	−45.0212	−3.07042
$216$	−8.41154	−0.572333
$217$	0.778033	0.0528163
$218$	−34.7518	−2.35369
$219$	1.49320	0.100901
$220$	−54.8000	−3.69462
$221$	0.979654	0.0658986
$222$	17.9079	1.20190
$223$	12.8223	0.858642	0.429321	−	0.903152i	$-0.358753\pi$
$223$	12.8223	0.858642	0.429321	+	0.903152i	$0.358753\pi$
$224$	14.7726	0.987036
$225$	9.54800	0.636533
$226$	−12.4074	−0.825326
$227$	22.3742	1.48503	0.742514	−	0.669831i	$-0.233633\pi$
$227$	22.3742	1.48503	0.742514	+	0.669831i	$0.233633\pi$
$228$	10.9026	0.722045
$229$	−21.7280	−1.43583	−0.717914	−	0.696131i	$-0.754903\pi$
$229$	−21.7280	−1.43583	−0.717914	+	0.696131i	$0.754903\pi$
$230$	−78.8177	−5.19709
$231$	2.61449	0.172021
$232$	64.1287	4.21026
$233$	−2.32840	−0.152538	−0.0762692	−	0.997087i	$-0.524301\pi$
$233$	−2.32840	−0.152538	−0.0762692	+	0.997087i	$0.524301\pi$
$234$	5.87293	0.383926
$235$	−28.3461	−1.84910
$236$	42.3744	2.75834
$237$	−14.7878	−0.960569
$238$	−1.11643	−0.0723675
$239$	30.7927	1.99181	0.995907	−	0.0903806i	$-0.0288083\pi$
$239$	30.7927	1.99181	0.995907	+	0.0903806i	$0.0288083\pi$
$240$	46.5083	3.00210
$241$	20.3246	1.30922	0.654612	−	0.755965i	$-0.272832\pi$
$241$	20.3246	1.30922	0.654612	+	0.755965i	$0.272832\pi$
$242$	8.57191	0.551023
$243$	1.00000	0.0641500
$244$	−1.01449	−0.0649461
$245$	−23.3539	−1.49203
$246$	3.37972	0.215483
$247$	−4.65401	−0.296128
$248$	6.98801	0.443739
$249$	−13.4719	−0.853747
$250$	−46.3734	−2.93291
$251$	0.862449	0.0544373	0.0272187	−	0.999630i	$-0.491335\pi$
$251$	0.862449	0.0544373	0.0272187	+	0.999630i	$0.491335\pi$
$252$	−4.81985	−0.303622
$253$	−21.5795	−1.35669
$254$	7.19744	0.451608
$255$	−1.70085	−0.106511
$256$	7.17389	0.448368
$257$	−16.3170	−1.01783	−0.508914	−	0.860818i	$-0.669953\pi$
$257$	−16.3170	−1.01783	−0.508914	+	0.860818i	$0.669953\pi$
$258$	31.5546	1.96450
$259$	6.27362	0.389824
$260$	−43.1243	−2.67446
$261$	−7.62389	−0.471907
$262$	25.5575	1.57895
$263$	−5.63383	−0.347397	−0.173699	−	0.984799i	$-0.555572\pi$
$263$	−5.63383	−0.347397	−0.173699	+	0.984799i	$0.555572\pi$
$264$	23.4824	1.44524
$265$	12.4271	0.763392
$266$	5.30380	0.325197
$267$	−6.90565	−0.422619
$268$	−29.1586	−1.78115
$269$	−8.23547	−0.502125	−0.251063	−	0.967971i	$-0.580780\pi$
$269$	−8.23547	−0.502125	−0.251063	+	0.967971i	$0.580780\pi$
$270$	−10.1964	−0.620536
$271$	−17.0812	−1.03761	−0.518806	−	0.854892i	$-0.673623\pi$
$271$	−17.0812	−1.03761	−0.518806	+	0.854892i	$0.673623\pi$
$272$	−5.43743	−0.329693
$273$	2.05745	0.124522
$274$	−44.0363	−2.66033
$275$	−26.6550	−1.60736
$276$	39.7821	2.39460
$277$	25.1622	1.51185	0.755926	−	0.654657i	$-0.227187\pi$
$277$	25.1622	1.51185	0.755926	+	0.654657i	$0.227187\pi$
$278$	39.6285	2.37676
$279$	−0.830764	−0.0497365
$280$	30.0468	1.79564
$281$	26.7115	1.59348	0.796738	−	0.604325i	$-0.206557\pi$
$281$	26.7115	1.59348	0.796738	+	0.604325i	$0.206557\pi$
$282$	19.8673	1.18308
$283$	1.63793	0.0973646	0.0486823	−	0.998814i	$-0.484498\pi$
$283$	1.63793	0.0973646	0.0486823	+	0.998814i	$0.484498\pi$
$284$	−33.8731	−2.01000
$285$	8.08018	0.478628
$286$	−16.3954	−0.969479
$287$	1.18401	0.0698898
$288$	−15.7738	−0.929480
$289$	−16.8011	−0.988303
$290$	77.7366	4.56485
$291$	−5.68913	−0.333503
$292$	7.68477	0.449717
$293$	12.1836	0.711771	0.355885	−	0.934530i	$-0.384179\pi$
$293$	12.1836	0.711771	0.355885	+	0.934530i	$0.384179\pi$
$294$	16.3684	0.954622
$295$	31.4046	1.82844
$296$	56.3473	3.27512
$297$	−2.79168	−0.161990
$298$	6.61076	0.382951
$299$	−16.9818	−0.982082
$300$	49.1389	2.83703
$301$	11.0544	0.637167
$302$	34.5076	1.98569
$303$	−18.1502	−1.04270
$304$	25.8314	1.48153
$305$	−0.751860	−0.0430514
$306$	1.19210	0.0681476
$307$	−32.3835	−1.84822	−0.924112	−	0.382122i	$-0.875194\pi$
$307$	−32.3835	−1.84822	−0.924112	+	0.382122i	$0.875194\pi$
$308$	13.4555	0.766698
$309$	12.9918	0.739079
$310$	8.47084	0.481111
$311$	13.7651	0.780549	0.390275	−	0.920698i	$-0.372380\pi$
$311$	13.7651	0.780549	0.390275	+	0.920698i	$0.372380\pi$
$312$	18.4792	1.04618
$313$	−14.2273	−0.804177	−0.402088	−	0.915601i	$-0.631716\pi$
$313$	−14.2273	−0.804177	−0.402088	+	0.915601i	$0.631716\pi$
$314$	48.9803	2.76412
$315$	−3.57209	−0.201264
$316$	−76.1054	−4.28126
$317$	−10.1707	−0.571241	−0.285620	−	0.958343i	$-0.592200\pi$
$317$	−10.1707	−0.571241	−0.285620	+	0.958343i	$0.592200\pi$
$318$	−8.70996	−0.488430
$319$	21.2835	1.19165
$320$	67.8199	3.79125
$321$	16.7587	0.935377
$322$	19.3528	1.07849
$323$	−0.944678	−0.0525633
$324$	5.14651	0.285917
$325$	−20.9759	−1.16353
$326$	−11.6265	−0.643933
$327$	12.9996	0.718881
$328$	10.6343	0.587182
$329$	6.96005	0.383720
$330$	28.4652	1.56696
$331$	−15.9215	−0.875124	−0.437562	−	0.899188i	$-0.644158\pi$
$331$	−15.9215	−0.875124	−0.437562	+	0.899188i	$0.644158\pi$
$332$	−69.3332	−3.80516
$333$	−6.69881	−0.367093
$334$	−18.6975	−1.02308
$335$	−21.6101	−1.18068
$336$	−11.4196	−0.622989
$337$	26.9891	1.47019	0.735096	−	0.677964i	$-0.237137\pi$
$337$	26.9891	1.47019	0.735096	+	0.677964i	$0.237137\pi$
$338$	21.8507	1.18852
$339$	4.64123	0.252077
$340$	−8.75344	−0.474722
$341$	2.31923	0.125593
$342$	−5.66326	−0.306234
$343$	12.2900	0.663596
$344$	99.2868	5.35318
$345$	29.4834	1.58733
$346$	9.23279	0.496358
$347$	−5.41182	−0.290522	−0.145261	−	0.989393i	$-0.546402\pi$
$347$	−5.41182	−0.290522	−0.145261	+	0.989393i	$0.546402\pi$
$348$	−39.2364	−2.10329
$349$	14.1961	0.759902	0.379951	−	0.925007i	$-0.375941\pi$
$349$	14.1961	0.759902	0.379951	+	0.925007i	$0.375941\pi$
$350$	23.9045	1.27775
$351$	−2.19689	−0.117261
$352$	44.0355	2.34710
$353$	−14.1354	−0.752353	−0.376176	−	0.926548i	$-0.622761\pi$
$353$	−14.1354	−0.752353	−0.376176	+	0.926548i	$0.622761\pi$
$354$	−22.0109	−1.16987
$355$	−25.1041	−1.33239
$356$	−35.5400	−1.88362
$357$	0.417624	0.0221030
$358$	−11.0816	−0.585681
$359$	1.97547	0.104261	0.0521305	−	0.998640i	$-0.483399\pi$
$359$	1.97547	0.104261	0.0521305	+	0.998640i	$0.483399\pi$
$360$	−32.0832	−1.69093
$361$	−14.5121	−0.763797
$362$	−16.2709	−0.855179
$363$	−3.20650	−0.168297
$364$	10.5887	0.554997
$365$	5.69535	0.298108
$366$	0.526966	0.0275449
$367$	−33.2390	−1.73506	−0.867531	−	0.497383i	$-0.834294\pi$
$367$	−33.2390	−1.73506	−0.867531	+	0.497383i	$0.834294\pi$
$368$	94.2550	4.91338
$369$	−1.26425	−0.0658144
$370$	68.3040	3.55096
$371$	−3.05133	−0.158417
$372$	−4.27553	−0.221676
$373$	15.4289	0.798877	0.399438	−	0.916760i	$-0.369205\pi$
$373$	15.4289	0.798877	0.399438	+	0.916760i	$0.369205\pi$
$374$	−3.32796	−0.172085
$375$	17.3469	0.895791
$376$	62.5126	3.22384
$377$	16.7488	0.862609
$378$	2.50361	0.128772
$379$	−10.4035	−0.534394	−0.267197	−	0.963642i	$-0.586097\pi$
$379$	−10.4035	−0.534394	−0.267197	+	0.963642i	$0.586097\pi$
$380$	41.5847	2.13325
$381$	−2.69235	−0.137933
$382$	−48.5204	−2.48252
$383$	10.0230	0.512151	0.256075	−	0.966657i	$-0.417570\pi$
$383$	10.0230	0.512151	0.256075	+	0.966657i	$0.417570\pi$
$384$	−15.9862	−0.815793
$385$	9.97215	0.508228
$386$	−13.4118	−0.682642
$387$	−11.8036	−0.600012
$388$	−29.2792	−1.48642
$389$	−23.5901	−1.19607	−0.598033	−	0.801471i	$-0.704051\pi$
$389$	−23.5901	−1.19607	−0.598033	+	0.801471i	$0.704051\pi$
$390$	22.4004	1.13429
$391$	−3.44699	−0.174322
$392$	51.5032	2.60130
$393$	−9.56029	−0.482253
$394$	−6.01242	−0.302901
$395$	−56.4033	−2.83796
$396$	−14.3674	−0.721990
$397$	−24.9494	−1.25217	−0.626087	−	0.779754i	$-0.715344\pi$
$397$	−24.9494	−1.25217	−0.626087	+	0.779754i	$0.715344\pi$
$398$	3.62497	0.181703
$399$	−1.98399	−0.0993238
$400$	116.424	5.82119
$401$	28.2225	1.40936	0.704681	−	0.709524i	$-0.251090\pi$
$401$	28.2225	1.40936	0.704681	+	0.709524i	$0.251090\pi$
$402$	15.1461	0.755420
$403$	1.82510	0.0909145
$404$	−93.4103	−4.64734
$405$	3.81418	0.189528
$406$	−19.0873	−0.947286
$407$	18.7010	0.926972
$408$	3.75094	0.185699
$409$	−23.7215	−1.17295	−0.586477	−	0.809966i	$-0.699486\pi$
$409$	−23.7215	−1.17295	−0.586477	+	0.809966i	$0.699486\pi$
$410$	12.8909	0.636635
$411$	16.4727	0.812537
$412$	66.8625	3.29408
$413$	−7.71102	−0.379434
$414$	−20.6644	−1.01560
$415$	−51.3843	−2.52236
$416$	34.6533	1.69902
$417$	−14.8238	−0.725926
$418$	15.8100	0.773294
$419$	−13.2184	−0.645763	−0.322882	−	0.946439i	$-0.604652\pi$
$419$	−13.2184	−0.645763	−0.322882	+	0.946439i	$0.604652\pi$
$420$	−18.3838	−0.897037
$421$	29.3254	1.42923	0.714617	−	0.699516i	$-0.246601\pi$
$421$	29.3254	1.42923	0.714617	+	0.699516i	$0.246601\pi$
$422$	15.8013	0.769196
$423$	−7.43177	−0.361345
$424$	−27.4059	−1.33095
$425$	−4.25772	−0.206530
$426$	17.5950	0.852481
$427$	0.184610	0.00893392
$428$	86.2486	4.16898
$429$	6.13302	0.296105
$430$	120.355	5.80404
$431$	−5.82600	−0.280628	−0.140314	−	0.990107i	$-0.544811\pi$
$431$	−5.82600	−0.280628	−0.140314	+	0.990107i	$0.544811\pi$
$432$	12.1935	0.586661
$433$	−13.6528	−0.656111	−0.328055	−	0.944658i	$-0.606393\pi$
$433$	−13.6528	−0.656111	−0.328055	+	0.944658i	$0.606393\pi$
$434$	−2.07991	−0.0998390
$435$	−29.0789	−1.39423
$436$	66.9027	3.20406
$437$	16.3755	0.783346
$438$	−3.99177	−0.190734
$439$	−20.4617	−0.976582	−0.488291	−	0.872681i	$-0.662380\pi$
$439$	−20.4617	−0.976582	−0.488291	+	0.872681i	$0.662380\pi$
$440$	89.5661	4.26990
$441$	−6.12292	−0.291567
$442$	−2.61890	−0.124569
$443$	18.8409	0.895156	0.447578	−	0.894245i	$-0.352287\pi$
$443$	18.8409	0.895156	0.447578	+	0.894245i	$0.352287\pi$
$444$	−34.4755	−1.63613
$445$	−26.3394	−1.24861
$446$	−34.2777	−1.62309
$447$	−2.47289	−0.116964
$448$	−16.6524	−0.786751
$449$	−30.9417	−1.46023	−0.730114	−	0.683326i	$-0.760533\pi$
$449$	−30.9417	−1.46023	−0.730114	+	0.683326i	$0.760533\pi$
$450$	−25.5246	−1.20324
$451$	3.52940	0.166193
$452$	23.8861	1.12351
$453$	−12.9083	−0.606483
$454$	−59.8128	−2.80716
$455$	7.84748	0.367896
$456$	−17.8195	−0.834474
$457$	−4.48331	−0.209720	−0.104860	−	0.994487i	$-0.533439\pi$
$457$	−4.48331	−0.209720	−0.104860	+	0.994487i	$0.533439\pi$
$458$	58.0854	2.71415
$459$	−0.445928	−0.0208141
$460$	151.736	7.07474
$461$	1.47415	0.0686579	0.0343289	−	0.999411i	$-0.489071\pi$
$461$	1.47415	0.0686579	0.0343289	+	0.999411i	$0.489071\pi$
$462$	−6.98930	−0.325172
$463$	20.1688	0.937322	0.468661	−	0.883378i	$-0.344737\pi$
$463$	20.1688	0.937322	0.468661	+	0.883378i	$0.344737\pi$
$464$	−92.9621	−4.31566
$465$	−3.16869	−0.146944
$466$	6.22449	0.288344
$467$	16.7536	0.775265	0.387633	−	0.921814i	$-0.373293\pi$
$467$	16.7536	0.775265	0.387633	+	0.921814i	$0.373293\pi$
$468$	−11.3063	−0.522634
$469$	5.30610	0.245013
$470$	75.7776	3.49536
$471$	−18.3221	−0.844236
$472$	−69.2575	−3.18784
$473$	32.9520	1.51514
$474$	39.5321	1.81577
$475$	20.2270	0.928079
$476$	2.14930	0.0985132
$477$	3.25814	0.149180
$478$	−82.3180	−3.76514
$479$	25.5031	1.16526	0.582632	−	0.812736i	$-0.302023\pi$
$479$	25.5031	1.16526	0.582632	+	0.812736i	$0.302023\pi$
$480$	−60.1642	−2.74611
$481$	14.7165	0.671016
$482$	−54.3337	−2.47483
$483$	−7.23929	−0.329399
$484$	−16.5023	−0.750103
$485$	−21.6994	−0.985319
$486$	−2.67330	−0.121263
$487$	24.1662	1.09507	0.547537	−	0.836782i	$-0.315566\pi$
$487$	24.1662	1.09507	0.547537	+	0.836782i	$0.315566\pi$
$488$	1.65810	0.0750588
$489$	4.34913	0.196675
$490$	62.4319	2.82039
$491$	29.2935	1.32200	0.660999	−	0.750386i	$-0.270133\pi$
$491$	29.2935	1.32200	0.660999	+	0.750386i	$0.270133\pi$
$492$	−6.50649	−0.293335
$493$	3.39971	0.153115
$494$	12.4415	0.559772
$495$	−10.6480	−0.478592
$496$	−10.1299	−0.454848
$497$	6.16401	0.276493
$498$	36.0144	1.61384
$499$	−22.2400	−0.995598	−0.497799	−	0.867293i	$-0.665858\pi$
$499$	−22.2400	−0.995598	−0.497799	+	0.867293i	$0.665858\pi$
$500$	89.2760	3.99255
$501$	6.99419	0.312477
$502$	−2.30558	−0.102903
$503$	−18.4686	−0.823476	−0.411738	−	0.911302i	$-0.635078\pi$
$503$	−18.4686	−0.823476	−0.411738	+	0.911302i	$0.635078\pi$
$504$	7.87764	0.350898
$505$	−69.2283	−3.08062
$506$	57.6884	2.56456
$507$	−8.17368	−0.363006
$508$	−13.8562	−0.614769
$509$	−0.258410	−0.0114538	−0.00572690	−	0.999984i	$-0.501823\pi$
$509$	−0.258410	−0.0114538	−0.00572690	+	0.999984i	$0.501823\pi$
$510$	4.54688	0.201339
$511$	−1.39842	−0.0618627
$512$	12.7945	0.565442
$513$	2.11846	0.0935321
$514$	43.6202	1.92400
$515$	49.5532	2.18357
$516$	−60.7475	−2.67426
$517$	20.7471	0.912458
$518$	−16.7712	−0.736886
$519$	−3.45371	−0.151601
$520$	70.4832	3.09089
$521$	17.5813	0.770250	0.385125	−	0.922864i	$-0.374158\pi$
$521$	17.5813	0.770250	0.385125	+	0.922864i	$0.374158\pi$
$522$	20.3809	0.892048
$523$	−43.3693	−1.89641	−0.948205	−	0.317660i	$-0.897103\pi$
$523$	−43.3693	−1.89641	−0.948205	+	0.317660i	$0.897103\pi$
$524$	−49.2021	−2.14940
$525$	−8.94197	−0.390259
$526$	15.0609	0.656687
$527$	0.370461	0.0161375
$528$	−34.0405	−1.48142
$529$	36.7517	1.59790
$530$	−33.2214	−1.44304
$531$	8.23363	0.357309
$532$	−10.2106	−0.442687
$533$	2.77742	0.120304
$534$	18.4609	0.798879
$535$	63.9206	2.76353
$536$	47.6574	2.05849
$537$	4.14530	0.178883
$538$	22.0158	0.949170
$539$	17.0933	0.736258
$540$	19.6297	0.844729
$541$	3.02434	0.130027	0.0650133	−	0.997884i	$-0.479291\pi$
$541$	3.02434	0.130027	0.0650133	+	0.997884i	$0.479291\pi$
$542$	45.6632	1.96140
$543$	6.08645	0.261195
$544$	7.03398	0.301579
$545$	49.5830	2.12390
$546$	−5.50016	−0.235385
$547$	25.0509	1.07110	0.535550	−	0.844503i	$-0.320104\pi$
$547$	25.0509	1.07110	0.535550	+	0.844503i	$0.320104\pi$
$548$	84.7767	3.62148
$549$	−0.197122	−0.00841297
$550$	71.2567	3.03840
$551$	−16.1509	−0.688050
$552$	−65.0206	−2.76746
$553$	13.8492	0.588926
$554$	−67.2661	−2.85786
$555$	−25.5505	−1.08456
$556$	−76.2909	−3.23546
$557$	29.5273	1.25111	0.625556	−	0.780180i	$-0.284872\pi$
$557$	29.5273	1.25111	0.625556	+	0.780180i	$0.284872\pi$
$558$	2.22088	0.0940172
$559$	25.9313	1.09678
$560$	−43.5564	−1.84059
$561$	1.24489	0.0525593
$562$	−71.4078	−3.01216
$563$	44.1548	1.86090	0.930451	−	0.366416i	$-0.119415\pi$
$563$	44.1548	1.86090	0.930451	+	0.366416i	$0.119415\pi$
$564$	−38.2476	−1.61052
$565$	17.7025	0.744750
$566$	−4.37866	−0.184049
$567$	−0.936528	−0.0393305
$568$	55.3628	2.32297
$569$	−5.70238	−0.239056	−0.119528	−	0.992831i	$-0.538138\pi$
$569$	−5.70238	−0.239056	−0.119528	+	0.992831i	$0.538138\pi$
$570$	−21.6007	−0.904754
$571$	−5.35595	−0.224139	−0.112070	−	0.993700i	$-0.535748\pi$
$571$	−5.35595	−0.224139	−0.112070	+	0.993700i	$0.535748\pi$
$572$	31.5636	1.31974
$573$	18.1500	0.758228
$574$	−3.16520	−0.132113
$575$	73.8053	3.07789
$576$	17.7810	0.740874
$577$	−16.4560	−0.685071	−0.342535	−	0.939505i	$-0.611286\pi$
$577$	−16.4560	−0.685071	−0.342535	+	0.939505i	$0.611286\pi$
$578$	44.9144	1.86819
$579$	5.01695	0.208497
$580$	−149.655	−6.21408
$581$	12.6168	0.523433
$582$	15.2087	0.630422
$583$	−9.09569	−0.376705
$584$	−12.5601	−0.519742
$585$	−8.37934	−0.346443
$586$	−32.5702	−1.34546
$587$	−33.8746	−1.39815	−0.699077	−	0.715046i	$-0.746406\pi$
$587$	−33.8746	−1.39815	−0.699077	+	0.715046i	$0.746406\pi$
$588$	−31.5116	−1.29952
$589$	−1.75994	−0.0725169
$590$	−83.9537	−3.45632
$591$	2.24907	0.0925142
$592$	−81.6821	−3.35711
$593$	−27.5928	−1.13310	−0.566551	−	0.824027i	$-0.691723\pi$
$593$	−27.5928	−1.13310	−0.566551	+	0.824027i	$0.691723\pi$
$594$	7.46300	0.306210
$595$	1.59289	0.0653023
$596$	−12.7267	−0.521307
$597$	−1.35599	−0.0554972
$598$	45.3973	1.85643
$599$	−41.5732	−1.69864	−0.849318	−	0.527881i	$-0.822987\pi$
$599$	−41.5732	−1.69864	−0.849318	+	0.527881i	$0.822987\pi$
$600$	−80.3134	−3.27878
$601$	−13.1383	−0.535921	−0.267961	−	0.963430i	$-0.586350\pi$
$601$	−13.1383	−0.535921	−0.267961	+	0.963430i	$0.586350\pi$
$602$	−29.5517	−1.20444
$603$	−5.66572	−0.230726
$604$	−66.4325	−2.70310
$605$	−12.2302	−0.497227
$606$	48.5209	1.97103
$607$	5.35345	0.217290	0.108645	−	0.994081i	$-0.465349\pi$
$607$	5.35345	0.217290	0.108645	+	0.994081i	$0.465349\pi$
$608$	−33.4161	−1.35520
$609$	7.13999	0.289327
$610$	2.00995	0.0813803
$611$	16.3268	0.660510
$612$	−2.29497	−0.0927687
$613$	−43.3606	−1.75132	−0.875659	−	0.482930i	$-0.839572\pi$
$613$	−43.3606	−1.75132	−0.875659	+	0.482930i	$0.839572\pi$
$614$	86.5707	3.49371
$615$	−4.82210	−0.194446
$616$	−21.9919	−0.886079
$617$	−19.2017	−0.773031	−0.386515	−	0.922283i	$-0.626321\pi$
$617$	−19.2017	−0.773031	−0.386515	+	0.922283i	$0.626321\pi$
$618$	−34.7310	−1.39708
$619$	26.0960	1.04889	0.524444	−	0.851445i	$-0.324273\pi$
$619$	26.0960	1.04889	0.524444	+	0.851445i	$0.324273\pi$
$620$	−16.3077	−0.654932
$621$	7.72992	0.310191
$622$	−36.7983	−1.47548
$623$	6.46734	0.259108
$624$	−26.7878	−1.07237
$625$	18.4243	0.736973
$626$	38.0339	1.52014
$627$	−5.91406	−0.236185
$628$	−94.2947	−3.76277
$629$	2.98719	0.119107
$630$	9.54925	0.380451
$631$	−20.5182	−0.816817	−0.408408	−	0.912799i	$-0.633916\pi$
$631$	−20.5182	−0.816817	−0.408408	+	0.912799i	$0.633916\pi$
$632$	124.388	4.94789
$633$	−5.91080	−0.234933
$634$	27.1892	1.07982
$635$	−10.2691	−0.407517
$636$	16.7680	0.664895
$637$	13.4514	0.532962
$638$	−56.8971	−2.25258
$639$	−6.58177	−0.260371
$640$	−60.9743	−2.41022
$641$	−31.4239	−1.24117	−0.620585	−	0.784139i	$-0.713105\pi$
$641$	−31.4239	−1.24117	−0.620585	+	0.784139i	$0.713105\pi$
$642$	−44.8009	−1.76815
$643$	30.5023	1.20289	0.601447	−	0.798913i	$-0.294591\pi$
$643$	30.5023	1.20289	0.601447	+	0.798913i	$0.294591\pi$
$644$	−37.2570	−1.46813
$645$	−45.0212	−1.77271
$646$	2.52540	0.0993607
$647$	7.46937	0.293651	0.146826	−	0.989162i	$-0.453094\pi$
$647$	7.46937	0.293651	0.146826	+	0.989162i	$0.453094\pi$
$648$	−8.41154	−0.330437
$649$	−22.9857	−0.902267
$650$	56.0748	2.19943
$651$	0.778033	0.0304935
$652$	22.3829	0.876580
$653$	−6.28100	−0.245795	−0.122897	−	0.992419i	$-0.539219\pi$
$653$	−6.28100	−0.245795	−0.122897	+	0.992419i	$0.539219\pi$
$654$	−34.7518	−1.35890
$655$	−36.4647	−1.42479
$656$	−15.4157	−0.601882
$657$	1.49320	0.0582553
$658$	−18.6063	−0.725348
$659$	26.7073	1.04037	0.520184	−	0.854054i	$-0.325863\pi$
$659$	26.7073	1.04037	0.520184	+	0.854054i	$0.325863\pi$
$660$	−54.8000	−2.13309
$661$	2.05359	0.0798752	0.0399376	−	0.999202i	$-0.487284\pi$
$661$	2.05359	0.0798752	0.0399376	+	0.999202i	$0.487284\pi$
$662$	42.5628	1.65425
$663$	0.979654	0.0380466
$664$	113.319	4.39765
$665$	−7.56731	−0.293448
$666$	17.9079	0.693917
$667$	−58.9321	−2.28186
$668$	35.9956	1.39271
$669$	12.8223	0.495737
$670$	57.7701	2.23185
$671$	0.550303	0.0212442
$672$	14.7726	0.569865
$673$	10.6636	0.411053	0.205526	−	0.978652i	$-0.434109\pi$
$673$	10.6636	0.411053	0.205526	+	0.978652i	$0.434109\pi$
$674$	−72.1499	−2.77911
$675$	9.54800	0.367503
$676$	−42.0659	−1.61792
$677$	−7.76931	−0.298599	−0.149299	−	0.988792i	$-0.547702\pi$
$677$	−7.76931	−0.298599	−0.149299	+	0.988792i	$0.547702\pi$
$678$	−12.4074	−0.476502
$679$	5.32803	0.204471
$680$	14.3068	0.548640
$681$	22.3742	0.857381
$682$	−6.19999	−0.237410
$683$	50.0576	1.91540	0.957701	−	0.287764i	$-0.0929119\pi$
$683$	50.0576	1.91540	0.957701	+	0.287764i	$0.0929119\pi$
$684$	10.9026	0.416873
$685$	62.8298	2.40060
$686$	−32.8547	−1.25440
$687$	−21.7280	−0.828976
$688$	−143.928	−5.48720
$689$	−7.15776	−0.272689
$690$	−78.8177	−3.00054
$691$	43.9260	1.67102	0.835511	−	0.549473i	$-0.185172\pi$
$691$	43.9260	1.67102	0.835511	+	0.549473i	$0.185172\pi$
$692$	−17.7745	−0.675687
$693$	2.61449	0.0993163
$694$	14.4674	0.549175
$695$	−56.5408	−2.14472
$696$	64.1287	2.43079
$697$	0.563766	0.0213541
$698$	−37.9505	−1.43645
$699$	−2.32840	−0.0880680
$700$	−46.0199	−1.73939
$701$	−34.3939	−1.29904	−0.649520	−	0.760345i	$-0.725030\pi$
$701$	−34.3939	−1.29904	−0.649520	+	0.760345i	$0.725030\pi$
$702$	5.87293	0.221660
$703$	−14.1911	−0.535229
$704$	−49.6389	−1.87084
$705$	−28.3461	−1.06758
$706$	37.7882	1.42218
$707$	16.9982	0.639283
$708$	42.3744	1.59253
$709$	8.40417	0.315625	0.157813	−	0.987469i	$-0.449556\pi$
$709$	8.40417	0.315625	0.157813	+	0.987469i	$0.449556\pi$
$710$	67.1106	2.51862
$711$	−14.7878	−0.554585
$712$	58.0872	2.17691
$713$	−6.42174	−0.240496
$714$	−1.11643	−0.0417814
$715$	23.3925	0.874829
$716$	21.3338	0.797281
$717$	30.7927	1.14997
$718$	−5.28100	−0.197085
$719$	−49.6375	−1.85117	−0.925583	−	0.378545i	$-0.876425\pi$
$719$	−49.6375	−1.85117	−0.925583	+	0.378545i	$0.876425\pi$
$720$	46.5083	1.73326
$721$	−12.1672	−0.453130
$722$	38.7953	1.44381
$723$	20.3246	0.755880
$724$	31.3240	1.16415
$725$	−72.7929	−2.70346
$726$	8.57191	0.318134
$727$	8.38330	0.310920	0.155460	−	0.987842i	$-0.450314\pi$
$727$	8.38330	0.310920	0.155460	+	0.987842i	$0.450314\pi$
$728$	−17.3063	−0.641414
$729$	1.00000	0.0370370
$730$	−15.2253	−0.563515
$731$	5.26357	0.194680
$732$	−1.01449	−0.0374967
$733$	−10.2039	−0.376889	−0.188444	−	0.982084i	$-0.560344\pi$
$733$	−10.2039	−0.376889	−0.188444	+	0.982084i	$0.560344\pi$
$734$	88.8577	3.27980
$735$	−23.3539	−0.861423
$736$	−121.930	−4.49441
$737$	15.8169	0.582623
$738$	3.37972	0.124409
$739$	39.5971	1.45660	0.728301	−	0.685257i	$-0.240310\pi$
$739$	39.5971	1.45660	0.728301	+	0.685257i	$0.240310\pi$
$740$	−131.496	−4.83388
$741$	−4.65401	−0.170969
$742$	8.15712	0.299457
$743$	−38.4258	−1.40971	−0.704854	−	0.709353i	$-0.748987\pi$
$743$	−38.4258	−1.40971	−0.704854	+	0.709353i	$0.748987\pi$
$744$	6.98801	0.256193
$745$	−9.43205	−0.345564
$746$	−41.2459	−1.51012
$747$	−13.4719	−0.492911
$748$	6.40683	0.234257
$749$	−15.6950	−0.573481
$750$	−46.3734	−1.69332
$751$	8.97066	0.327344	0.163672	−	0.986515i	$-0.447666\pi$
$751$	8.97066	0.327344	0.163672	+	0.986515i	$0.447666\pi$
$752$	−90.6194	−3.30455
$753$	0.862449	0.0314294
$754$	−44.7746	−1.63059
$755$	−49.2345	−1.79183
$756$	−4.81985	−0.175296
$757$	24.4882	0.890039	0.445019	−	0.895521i	$-0.353197\pi$
$757$	24.4882	0.890039	0.445019	+	0.895521i	$0.353197\pi$
$758$	27.8117	1.01017
$759$	−21.5795	−0.783287
$760$	−67.9668	−2.46541
$761$	10.9805	0.398044	0.199022	−	0.979995i	$-0.436223\pi$
$761$	10.9805	0.398044	0.199022	+	0.979995i	$0.436223\pi$
$762$	7.19744	0.260736
$763$	−12.1745	−0.440747
$764$	93.4092	3.37943
$765$	−1.70085	−0.0614944
$766$	−26.7944	−0.968121
$767$	−18.0884	−0.653133
$768$	7.17389	0.258866
$769$	−5.53953	−0.199761	−0.0998803	−	0.994999i	$-0.531846\pi$
$769$	−5.53953	−0.199761	−0.0998803	+	0.994999i	$0.531846\pi$
$770$	−26.6585	−0.960705
$771$	−16.3170	−0.587643
$772$	25.8198	0.929274
$773$	3.50336	0.126007	0.0630036	−	0.998013i	$-0.479932\pi$
$773$	3.50336	0.126007	0.0630036	+	0.998013i	$0.479932\pi$
$774$	31.5546	1.13421
$775$	−7.93214	−0.284931
$776$	47.8544	1.71787
$777$	6.27362	0.225065
$778$	63.0633	2.26093
$779$	−2.67826	−0.0959588
$780$	−43.1243	−1.54410
$781$	18.3742	0.657481
$782$	9.21481	0.329521
$783$	−7.62389	−0.272456
$784$	−74.6599	−2.66643
$785$	−69.8837	−2.49426
$786$	25.5575	0.911605
$787$	14.8885	0.530718	0.265359	−	0.964150i	$-0.414510\pi$
$787$	14.8885	0.530718	0.265359	+	0.964150i	$0.414510\pi$
$788$	11.5748	0.412337
$789$	−5.63383	−0.200570
$790$	150.783	5.36461
$791$	−4.34664	−0.154549
$792$	23.4824	0.834410
$793$	0.433056	0.0153783
$794$	66.6970	2.36699
$795$	12.4271	0.440745
$796$	−6.97863	−0.247351
$797$	6.36284	0.225383	0.112692	−	0.993630i	$-0.464053\pi$
$797$	6.36284	0.225383	0.112692	+	0.993630i	$0.464053\pi$
$798$	5.30380	0.187752
$799$	3.31403	0.117242
$800$	−150.608	−5.32481
$801$	−6.90565	−0.243999
$802$	−75.4470	−2.66413
$803$	−4.16855	−0.147105
$804$	−29.1586	−1.02835
$805$	−27.6120	−0.973194
$806$	−4.87902	−0.171856
$807$	−8.23547	−0.289902
$808$	152.671	5.37096
$809$	−47.7007	−1.67707	−0.838534	−	0.544849i	$-0.816587\pi$
$809$	−47.7007	−1.67707	−0.838534	+	0.544849i	$0.816587\pi$
$810$	−10.1964	−0.358266
$811$	−0.953424	−0.0334792	−0.0167396	−	0.999860i	$-0.505329\pi$
$811$	−0.953424	−0.0334792	−0.0167396	+	0.999860i	$0.505329\pi$
$812$	36.7460	1.28953
$813$	−17.0812	−0.599065
$814$	−49.9932	−1.75226
$815$	16.5884	0.581066
$816$	−5.43743	−0.190348
$817$	−25.0055	−0.874831
$818$	63.4147	2.21724
$819$	2.05745	0.0718930
$820$	−24.8170	−0.866645
$821$	20.4971	0.715354	0.357677	−	0.933845i	$-0.383569\pi$
$821$	20.4971	0.715354	0.357677	+	0.933845i	$0.383569\pi$
$822$	−44.0363	−1.53594
$823$	−17.1805	−0.598873	−0.299437	−	0.954116i	$-0.596799\pi$
$823$	−17.1805	−0.598873	−0.299437	+	0.954116i	$0.596799\pi$
$824$	−109.281	−3.80699
$825$	−26.6550	−0.928008
$826$	20.6138	0.717247
$827$	17.0023	0.591229	0.295614	−	0.955307i	$-0.404476\pi$
$827$	17.0023	0.591229	0.295614	+	0.955307i	$0.404476\pi$
$828$	39.7821	1.38252
$829$	8.13061	0.282388	0.141194	−	0.989982i	$-0.454906\pi$
$829$	8.13061	0.282388	0.141194	+	0.989982i	$0.454906\pi$
$830$	137.365	4.76802
$831$	25.1622	0.872869
$832$	−39.0628	−1.35426
$833$	2.73038	0.0946020
$834$	39.6285	1.37222
$835$	26.6771	0.923200
$836$	−30.4367	−1.05268
$837$	−0.830764	−0.0287154
$838$	35.3368	1.22069
$839$	−14.1002	−0.486792	−0.243396	−	0.969927i	$-0.578261\pi$
$839$	−14.1002	−0.486792	−0.243396	+	0.969927i	$0.578261\pi$
$840$	30.0468	1.03671
$841$	29.1237	1.00427
$842$	−78.3955	−2.70169
$843$	26.7115	0.919994
$844$	−30.4200	−1.04710
$845$	−31.1759	−1.07248
$846$	19.8673	0.683052
$847$	3.00297	0.103183
$848$	39.7281	1.36427
$849$	1.63793	0.0562135
$850$	11.3821	0.390404
$851$	−51.7813	−1.77504
$852$	−33.8731	−1.16047
$853$	54.0075	1.84918	0.924591	−	0.380962i	$-0.124407\pi$
$853$	54.0075	1.84918	0.924591	+	0.380962i	$0.124407\pi$
$854$	−0.493518	−0.0168878
$855$	8.08018	0.276336
$856$	−140.966	−4.81813
$857$	−36.0649	−1.23195	−0.615977	−	0.787764i	$-0.711239\pi$
$857$	−36.0649	−1.23195	−0.615977	+	0.787764i	$0.711239\pi$
$858$	−16.3954	−0.559729
$859$	−4.04247	−0.137927	−0.0689637	−	0.997619i	$-0.521969\pi$
$859$	−4.04247	−0.137927	−0.0689637	+	0.997619i	$0.521969\pi$
$860$	−231.702	−7.90098
$861$	1.18401	0.0403509
$862$	15.5746	0.530473
$863$	−9.05691	−0.308301	−0.154150	−	0.988047i	$-0.549264\pi$
$863$	−9.05691	−0.308301	−0.154150	+	0.988047i	$0.549264\pi$
$864$	−15.7738	−0.536636
$865$	−13.1731	−0.447899
$866$	36.4979	1.24025
$867$	−16.8011	−0.570597
$868$	4.00415	0.135910
$869$	41.2828	1.40042
$870$	77.7366	2.63552
$871$	12.4469	0.421749
$872$	−109.347	−3.70295
$873$	−5.68913	−0.192548
$874$	−43.7765	−1.48076
$875$	−16.2459	−0.549211
$876$	7.68477	0.259644
$877$	−17.1130	−0.577864	−0.288932	−	0.957350i	$-0.593300\pi$
$877$	−17.1130	−0.577864	−0.288932	+	0.957350i	$0.593300\pi$
$878$	54.7001	1.84604
$879$	12.1836	0.410941
$880$	−129.837	−4.37679
$881$	−25.3936	−0.855532	−0.427766	−	0.903890i	$-0.640699\pi$
$881$	−25.3936	−0.855532	−0.427766	+	0.903890i	$0.640699\pi$
$882$	16.3684	0.551151
$883$	1.26342	0.0425174	0.0212587	−	0.999774i	$-0.493233\pi$
$883$	1.26342	0.0425174	0.0212587	+	0.999774i	$0.493233\pi$
$884$	5.04179	0.169574
$885$	31.4046	1.05565
$886$	−50.3672	−1.69212
$887$	−46.4029	−1.55806	−0.779029	−	0.626988i	$-0.784288\pi$
$887$	−46.4029	−1.55806	−0.779029	+	0.626988i	$0.784288\pi$
$888$	56.3473	1.89089
$889$	2.52146	0.0845670
$890$	70.4131	2.36025
$891$	−2.79168	−0.0935250
$892$	65.9898	2.20950
$893$	−15.7439	−0.526848
$894$	6.61076	0.221097
$895$	15.8109	0.528501
$896$	14.9715	0.500164
$897$	−16.9818	−0.567005
$898$	82.7162	2.76028
$899$	6.33366	0.211239
$900$	49.1389	1.63796
$901$	−1.45289	−0.0484029
$902$	−9.43512	−0.314155
$903$	11.0544	0.367868
$904$	−39.0399	−1.29845
$905$	23.2149	0.771688
$906$	34.5076	1.14644
$907$	−30.1347	−1.00061	−0.500304	−	0.865850i	$-0.666778\pi$
$907$	−30.1347	−1.00061	−0.500304	+	0.865850i	$0.666778\pi$
$908$	115.149	3.82135
$909$	−18.1502	−0.602005
$910$	−20.9786	−0.695435
$911$	59.7505	1.97962	0.989811	−	0.142386i	$-0.0454774\pi$
$911$	59.7505	1.97962	0.989811	+	0.142386i	$0.0454774\pi$
$912$	25.8314	0.855364
$913$	37.6093	1.24469
$914$	11.9852	0.396435
$915$	−0.751860	−0.0248557
$916$	−111.823	−3.69475
$917$	8.95347	0.295670
$918$	1.19210	0.0393450
$919$	31.0723	1.02498	0.512490	−	0.858693i	$-0.328723\pi$
$919$	31.0723	1.02498	0.512490	+	0.858693i	$0.328723\pi$
$920$	−248.000	−8.17633
$921$	−32.3835	−1.06707
$922$	−3.94083	−0.129784
$923$	14.4594	0.475937
$924$	13.4555	0.442653
$925$	−63.9603	−2.10300
$926$	−53.9170	−1.77182
$927$	12.9918	0.426707
$928$	120.258	3.94765
$929$	−52.3038	−1.71603	−0.858016	−	0.513623i	$-0.828303\pi$
$929$	−52.3038	−1.71603	−0.858016	+	0.513623i	$0.828303\pi$
$930$	8.47084	0.277770
$931$	−12.9711	−0.425111
$932$	−11.9831	−0.392520
$933$	13.7651	0.450650
$934$	−44.7874	−1.46549
$935$	4.74824	0.155284
$936$	18.4792	0.604012
$937$	−40.5626	−1.32512	−0.662561	−	0.749008i	$-0.730531\pi$
$937$	−40.5626	−1.32512	−0.662561	+	0.749008i	$0.730531\pi$
$938$	−14.1848	−0.463149
$939$	−14.2273	−0.464292
$940$	−145.884	−4.75820
$941$	−15.7766	−0.514301	−0.257150	−	0.966371i	$-0.582784\pi$
$941$	−15.7766	−0.514301	−0.257150	+	0.966371i	$0.582784\pi$
$942$	48.9803	1.59586
$943$	−9.77258	−0.318239
$944$	100.397	3.26764
$945$	−3.57209	−0.116200
$946$	−88.0905	−2.86407
$947$	30.8442	1.00230	0.501151	−	0.865360i	$-0.332910\pi$
$947$	30.8442	1.00230	0.501151	+	0.865360i	$0.332910\pi$
$948$	−76.1054	−2.47179
$949$	−3.28040	−0.106486
$950$	−54.0728	−1.75435
$951$	−10.1707	−0.329806
$952$	−3.51286	−0.113852
$953$	−1.08237	−0.0350614	−0.0175307	−	0.999846i	$-0.505580\pi$
$953$	−1.08237	−0.0350614	−0.0175307	+	0.999846i	$0.505580\pi$
$954$	−8.70996	−0.281995
$955$	69.2275	2.24015
$956$	158.475	5.12544
$957$	21.2835	0.687998
$958$	−68.1772	−2.20271
$959$	−15.4271	−0.498167
$960$	67.8199	2.18888
$961$	−30.3098	−0.977736
$962$	−39.3417	−1.26843
$963$	16.7587	0.540040
$964$	104.601	3.36896
$965$	19.1356	0.615996
$966$	19.3528	0.622665
$967$	−32.8111	−1.05514	−0.527568	−	0.849513i	$-0.676896\pi$
$967$	−32.8111	−1.05514	−0.527568	+	0.849513i	$0.676896\pi$
$968$	26.9716	0.866900
$969$	−0.944678	−0.0303474
$970$	58.0089	1.86255
$971$	11.5403	0.370344	0.185172	−	0.982706i	$-0.440716\pi$
$971$	11.5403	0.370344	0.185172	+	0.982706i	$0.440716\pi$
$972$	5.14651	0.165074
$973$	13.8829	0.445066
$974$	−64.6033	−2.07002
$975$	−20.9759	−0.671766
$976$	−2.40361	−0.0769378
$977$	−7.39302	−0.236524	−0.118262	−	0.992982i	$-0.537732\pi$
$977$	−7.39302	−0.236524	−0.118262	+	0.992982i	$0.537732\pi$
$978$	−11.6265	−0.371775
$979$	19.2784	0.616141
$980$	−120.191	−3.83937
$981$	12.9996	0.415046
$982$	−78.3103	−2.49898
$983$	−13.1953	−0.420865	−0.210432	−	0.977608i	$-0.567487\pi$
$983$	−13.1953	−0.420865	−0.210432	+	0.977608i	$0.567487\pi$
$984$	10.6343	0.339010
$985$	8.57836	0.273329
$986$	−9.08842	−0.289434
$987$	6.96005	0.221541
$988$	−23.9519	−0.762011
$989$	−91.2412	−2.90130
$990$	28.4652	0.904685
$991$	−26.8039	−0.851455	−0.425728	−	0.904851i	$-0.639982\pi$
$991$	−26.8039	−0.851455	−0.425728	+	0.904851i	$0.639982\pi$
$992$	13.1043	0.416062
$993$	−15.9215	−0.505253
$994$	−16.4782	−0.522657
$995$	−5.17201	−0.163964
$996$	−69.3332	−2.19691
$997$	29.6636	0.939455	0.469727	−	0.882812i	$-0.344352\pi$
$997$	29.6636	0.939455	0.469727	+	0.882812i	$0.344352\pi$
$998$	59.4540	1.88198
$999$	−6.69881	−0.211941

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.a.1.2	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.a.1.2	✓	92	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:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67330 q^{2} +1.00000 q^{3} +5.14651 q^{4} +3.81418 q^{5} -2.67330 q^{6} -0.936528 q^{7} -8.41154 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.67330 q^{2} +1.00000 q^{3} +5.14651 q^{4} +3.81418 q^{5} -2.67330 q^{6} -0.936528 q^{7} -8.41154 q^{8} +1.00000 q^{9} -10.1964 q^{10} -2.79168 q^{11} +5.14651 q^{12} -2.19689 q^{13} +2.50361 q^{14} +3.81418 q^{15} +12.1935 q^{16} -0.445928 q^{17} -2.67330 q^{18} +2.11846 q^{19} +19.6297 q^{20} -0.936528 q^{21} +7.46300 q^{22} +7.72992 q^{23} -8.41154 q^{24} +9.54800 q^{25} +5.87293 q^{26} +1.00000 q^{27} -4.81985 q^{28} -7.62389 q^{29} -10.1964 q^{30} -0.830764 q^{31} -15.7738 q^{32} -2.79168 q^{33} +1.19210 q^{34} -3.57209 q^{35} +5.14651 q^{36} -6.69881 q^{37} -5.66326 q^{38} -2.19689 q^{39} -32.0832 q^{40} -1.26425 q^{41} +2.50361 q^{42} -11.8036 q^{43} -14.3674 q^{44} +3.81418 q^{45} -20.6644 q^{46} -7.43177 q^{47} +12.1935 q^{48} -6.12292 q^{49} -25.5246 q^{50} -0.445928 q^{51} -11.3063 q^{52} +3.25814 q^{53} -2.67330 q^{54} -10.6480 q^{55} +7.87764 q^{56} +2.11846 q^{57} +20.3809 q^{58} +8.23363 q^{59} +19.6297 q^{60} -0.197122 q^{61} +2.22088 q^{62} -0.936528 q^{63} +17.7810 q^{64} -8.37934 q^{65} +7.46300 q^{66} -5.66572 q^{67} -2.29497 q^{68} +7.72992 q^{69} +9.54925 q^{70} -6.58177 q^{71} -8.41154 q^{72} +1.49320 q^{73} +17.9079 q^{74} +9.54800 q^{75} +10.9026 q^{76} +2.61449 q^{77} +5.87293 q^{78} -14.7878 q^{79} +46.5083 q^{80} +1.00000 q^{81} +3.37972 q^{82} -13.4719 q^{83} -4.81985 q^{84} -1.70085 q^{85} +31.5546 q^{86} -7.62389 q^{87} +23.4824 q^{88} -6.90565 q^{89} -10.1964 q^{90} +2.05745 q^{91} +39.7821 q^{92} -0.830764 q^{93} +19.8673 q^{94} +8.08018 q^{95} -15.7738 q^{96} -5.68913 q^{97} +16.3684 q^{98} -2.79168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9}+O(q^{10}) \) 92 * q - 6 * q^2 + 92 * q^3 + 70 * q^4 - 18 * q^5 - 6 * q^6 - 42 * q^7 - 15 * q^8 + 92 * q^9	\( 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9} - 44 q^{10} - 24 q^{11} + 70 q^{12} - 48 q^{13} - 29 q^{14} - 18 q^{15} + 26 q^{16} - 69 q^{17} - 6 q^{18} - 74 q^{19} - 42 q^{20} - 42 q^{21} - 62 q^{22} - 19 q^{23} - 15 q^{24} + 16 q^{25} - 27 q^{26} + 92 q^{27} - 101 q^{28} - 54 q^{29} - 44 q^{30} - 67 q^{31} - 36 q^{32} - 24 q^{33} - 63 q^{34} - 31 q^{35} + 70 q^{36} - 70 q^{37} - 18 q^{38} - 48 q^{39} - 125 q^{40} - 98 q^{41} - 29 q^{42} - 159 q^{43} - 52 q^{44} - 18 q^{45} - 68 q^{46} - 15 q^{47} + 26 q^{48} - 28 q^{49} - 7 q^{50} - 69 q^{51} - 98 q^{52} - 23 q^{53} - 6 q^{54} - 93 q^{55} - 48 q^{56} - 74 q^{57} - 37 q^{58} - 36 q^{59} - 42 q^{60} - 172 q^{61} - 26 q^{62} - 42 q^{63} - 23 q^{64} - 66 q^{65} - 62 q^{66} - 143 q^{67} - 74 q^{68} - 19 q^{69} - 30 q^{70} - 9 q^{71} - 15 q^{72} - 134 q^{73} - 19 q^{74} + 16 q^{75} - 157 q^{76} - 25 q^{77} - 27 q^{78} - 138 q^{79} - 29 q^{80} + 92 q^{81} - 61 q^{82} - 24 q^{83} - 101 q^{84} - 84 q^{85} + 14 q^{86} - 54 q^{87} - 140 q^{88} - 148 q^{89} - 44 q^{90} - 115 q^{91} - 12 q^{92} - 67 q^{93} - 79 q^{94} - 10 q^{95} - 36 q^{96} - 165 q^{97} + 36 q^{98} - 24 q^{99}+O(q^{100}) \) 92 * q - 6 * q^2 + 92 * q^3 + 70 * q^4 - 18 * q^5 - 6 * q^6 - 42 * q^7 - 15 * q^8 + 92 * q^9 - 44 * q^10 - 24 * q^11 + 70 * q^12 - 48 * q^13 - 29 * q^14 - 18 * q^15 + 26 * q^16 - 69 * q^17 - 6 * q^18 - 74 * q^19 - 42 * q^20 - 42 * q^21 - 62 * q^22 - 19 * q^23 - 15 * q^24 + 16 * q^25 - 27 * q^26 + 92 * q^27 - 101 * q^28 - 54 * q^29 - 44 * q^30 - 67 * q^31 - 36 * q^32 - 24 * q^33 - 63 * q^34 - 31 * q^35 + 70 * q^36 - 70 * q^37 - 18 * q^38 - 48 * q^39 - 125 * q^40 - 98 * q^41 - 29 * q^42 - 159 * q^43 - 52 * q^44 - 18 * q^45 - 68 * q^46 - 15 * q^47 + 26 * q^48 - 28 * q^49 - 7 * q^50 - 69 * q^51 - 98 * q^52 - 23 * q^53 - 6 * q^54 - 93 * q^55 - 48 * q^56 - 74 * q^57 - 37 * q^58 - 36 * q^59 - 42 * q^60 - 172 * q^61 - 26 * q^62 - 42 * q^63 - 23 * q^64 - 66 * q^65 - 62 * q^66 - 143 * q^67 - 74 * q^68 - 19 * q^69 - 30 * q^70 - 9 * q^71 - 15 * q^72 - 134 * q^73 - 19 * q^74 + 16 * q^75 - 157 * q^76 - 25 * q^77 - 27 * q^78 - 138 * q^79 - 29 * q^80 + 92 * q^81 - 61 * q^82 - 24 * q^83 - 101 * q^84 - 84 * q^85 + 14 * q^86 - 54 * q^87 - 140 * q^88 - 148 * q^89 - 44 * q^90 - 115 * q^91 - 12 * q^92 - 67 * q^93 - 79 * q^94 - 10 * q^95 - 36 * q^96 - 165 * q^97 + 36 * q^98 - 24 * q^99

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