LMFDB - Embedded newform 8038.2.a.a.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.12788 q^{3} +1.00000 q^{4} +3.19034 q^{5} -2.12788 q^{6} +2.73143 q^{7} +1.00000 q^{8} +1.52787 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.12788 q^{3} +1.00000 q^{4} +3.19034 q^{5} -2.12788 q^{6} +2.73143 q^{7} +1.00000 q^{8} +1.52787 q^{9} +3.19034 q^{10} -3.87931 q^{11} -2.12788 q^{12} +2.89555 q^{13} +2.73143 q^{14} -6.78865 q^{15} +1.00000 q^{16} -7.29312 q^{17} +1.52787 q^{18} +1.64593 q^{19} +3.19034 q^{20} -5.81216 q^{21} -3.87931 q^{22} -5.52793 q^{23} -2.12788 q^{24} +5.17826 q^{25} +2.89555 q^{26} +3.13251 q^{27} +2.73143 q^{28} -4.38749 q^{29} -6.78865 q^{30} -1.88667 q^{31} +1.00000 q^{32} +8.25471 q^{33} -7.29312 q^{34} +8.71420 q^{35} +1.52787 q^{36} -4.27144 q^{37} +1.64593 q^{38} -6.16139 q^{39} +3.19034 q^{40} +3.86135 q^{41} -5.81216 q^{42} -6.44048 q^{43} -3.87931 q^{44} +4.87442 q^{45} -5.52793 q^{46} -4.41164 q^{47} -2.12788 q^{48} +0.460728 q^{49} +5.17826 q^{50} +15.5189 q^{51} +2.89555 q^{52} -4.81937 q^{53} +3.13251 q^{54} -12.3763 q^{55} +2.73143 q^{56} -3.50234 q^{57} -4.38749 q^{58} -6.93691 q^{59} -6.78865 q^{60} -3.26404 q^{61} -1.88667 q^{62} +4.17328 q^{63} +1.00000 q^{64} +9.23780 q^{65} +8.25471 q^{66} -8.16346 q^{67} -7.29312 q^{68} +11.7628 q^{69} +8.71420 q^{70} -2.72005 q^{71} +1.52787 q^{72} +2.32453 q^{73} -4.27144 q^{74} -11.0187 q^{75} +1.64593 q^{76} -10.5961 q^{77} -6.16139 q^{78} -3.74700 q^{79} +3.19034 q^{80} -11.2492 q^{81} +3.86135 q^{82} -11.6500 q^{83} -5.81216 q^{84} -23.2675 q^{85} -6.44048 q^{86} +9.33604 q^{87} -3.87931 q^{88} +3.65537 q^{89} +4.87442 q^{90} +7.90901 q^{91} -5.52793 q^{92} +4.01460 q^{93} -4.41164 q^{94} +5.25108 q^{95} -2.12788 q^{96} -13.0759 q^{97} +0.460728 q^{98} -5.92709 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) $ 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.12788	−1.22853	−0.614266	−	0.789099i	$-0.710548\pi$
$3$	−2.12788	−1.22853	−0.614266	+	0.789099i	$0.710548\pi$
$4$	1.00000	0.500000
$5$	3.19034	1.42676	0.713381	−	0.700776i	$-0.247163\pi$
$5$	3.19034	1.42676	0.713381	+	0.700776i	$0.247163\pi$
$6$	−2.12788	−0.868703
$7$	2.73143	1.03238	0.516192	−	0.856473i	$-0.327349\pi$
$7$	2.73143	1.03238	0.516192	+	0.856473i	$0.327349\pi$
$8$	1.00000	0.353553
$9$	1.52787	0.509290
$10$	3.19034	1.00887
$11$	−3.87931	−1.16966	−0.584829	−	0.811157i	$-0.698838\pi$
$11$	−3.87931	−1.16966	−0.584829	+	0.811157i	$0.698838\pi$
$12$	−2.12788	−0.614266
$13$	2.89555	0.803082	0.401541	−	0.915841i	$-0.368475\pi$
$13$	2.89555	0.803082	0.401541	+	0.915841i	$0.368475\pi$
$14$	2.73143	0.730006
$15$	−6.78865	−1.75282
$16$	1.00000	0.250000
$17$	−7.29312	−1.76884	−0.884421	−	0.466690i	$-0.845446\pi$
$17$	−7.29312	−1.76884	−0.884421	+	0.466690i	$0.845446\pi$
$18$	1.52787	0.360122
$19$	1.64593	0.377602	0.188801	−	0.982015i	$-0.439540\pi$
$19$	1.64593	0.377602	0.188801	+	0.982015i	$0.439540\pi$
$20$	3.19034	0.713381
$21$	−5.81216	−1.26832
$22$	−3.87931	−0.827073
$23$	−5.52793	−1.15265	−0.576327	−	0.817219i	$-0.695515\pi$
$23$	−5.52793	−1.15265	−0.576327	+	0.817219i	$0.695515\pi$
$24$	−2.12788	−0.434352
$25$	5.17826	1.03565
$26$	2.89555	0.567865
$27$	3.13251	0.602853
$28$	2.73143	0.516192
$29$	−4.38749	−0.814736	−0.407368	−	0.913264i	$-0.633553\pi$
$29$	−4.38749	−0.814736	−0.407368	+	0.913264i	$0.633553\pi$
$30$	−6.78865	−1.23943
$31$	−1.88667	−0.338855	−0.169428	−	0.985543i	$-0.554192\pi$
$31$	−1.88667	−0.338855	−0.169428	+	0.985543i	$0.554192\pi$
$32$	1.00000	0.176777
$33$	8.25471	1.43696
$34$	−7.29312	−1.25076
$35$	8.71420	1.47297
$36$	1.52787	0.254645
$37$	−4.27144	−0.702220	−0.351110	−	0.936334i	$-0.614196\pi$
$37$	−4.27144	−0.702220	−0.351110	+	0.936334i	$0.614196\pi$
$38$	1.64593	0.267005
$39$	−6.16139	−0.986612
$40$	3.19034	0.504437
$41$	3.86135	0.603041	0.301520	−	0.953460i	$-0.402506\pi$
$41$	3.86135	0.603041	0.301520	+	0.953460i	$0.402506\pi$
$42$	−5.81216	−0.896836
$43$	−6.44048	−0.982163	−0.491082	−	0.871114i	$-0.663398\pi$
$43$	−6.44048	−0.982163	−0.491082	+	0.871114i	$0.663398\pi$
$44$	−3.87931	−0.584829
$45$	4.87442	0.726636
$46$	−5.52793	−0.815050
$47$	−4.41164	−0.643503	−0.321752	−	0.946824i	$-0.604272\pi$
$47$	−4.41164	−0.643503	−0.321752	+	0.946824i	$0.604272\pi$
$48$	−2.12788	−0.307133
$49$	0.460728	0.0658183
$50$	5.17826	0.732316
$51$	15.5189	2.17308
$52$	2.89555	0.401541
$53$	−4.81937	−0.661992	−0.330996	−	0.943632i	$-0.607385\pi$
$53$	−4.81937	−0.661992	−0.330996	+	0.943632i	$0.607385\pi$
$54$	3.13251	0.426281
$55$	−12.3763	−1.66882
$56$	2.73143	0.365003
$57$	−3.50234	−0.463897
$58$	−4.38749	−0.576105
$59$	−6.93691	−0.903109	−0.451555	−	0.892243i	$-0.649130\pi$
$59$	−6.93691	−0.903109	−0.451555	+	0.892243i	$0.649130\pi$
$60$	−6.78865	−0.876412
$61$	−3.26404	−0.417917	−0.208959	−	0.977924i	$-0.567007\pi$
$61$	−3.26404	−0.417917	−0.208959	+	0.977924i	$0.567007\pi$
$62$	−1.88667	−0.239607
$63$	4.17328	0.525783
$64$	1.00000	0.125000
$65$	9.23780	1.14581
$66$	8.25471	1.01608
$67$	−8.16346	−0.997325	−0.498663	−	0.866796i	$-0.666175\pi$
$67$	−8.16346	−0.997325	−0.498663	+	0.866796i	$0.666175\pi$
$68$	−7.29312	−0.884421
$69$	11.7628	1.41607
$70$	8.71420	1.04155
$71$	−2.72005	−0.322811	−0.161405	−	0.986888i	$-0.551603\pi$
$71$	−2.72005	−0.322811	−0.161405	+	0.986888i	$0.551603\pi$
$72$	1.52787	0.180061
$73$	2.32453	0.272065	0.136033	−	0.990704i	$-0.456565\pi$
$73$	2.32453	0.272065	0.136033	+	0.990704i	$0.456565\pi$
$74$	−4.27144	−0.496545
$75$	−11.0187	−1.27233
$76$	1.64593	0.188801
$77$	−10.5961	−1.20754
$78$	−6.16139	−0.697640
$79$	−3.74700	−0.421571	−0.210785	−	0.977532i	$-0.567602\pi$
$79$	−3.74700	−0.421571	−0.210785	+	0.977532i	$0.567602\pi$
$80$	3.19034	0.356691
$81$	−11.2492	−1.24991
$82$	3.86135	0.426414
$83$	−11.6500	−1.27876	−0.639378	−	0.768893i	$-0.720808\pi$
$83$	−11.6500	−1.27876	−0.639378	+	0.768893i	$0.720808\pi$
$84$	−5.81216	−0.634159
$85$	−23.2675	−2.52372
$86$	−6.44048	−0.694494
$87$	9.33604	1.00093
$88$	−3.87931	−0.413536
$89$	3.65537	0.387468	0.193734	−	0.981054i	$-0.437940\pi$
$89$	3.65537	0.387468	0.193734	+	0.981054i	$0.437940\pi$
$90$	4.87442	0.513809
$91$	7.90901	0.829090
$92$	−5.52793	−0.576327
$93$	4.01460	0.416295
$94$	−4.41164	−0.455026
$95$	5.25108	0.538749
$96$	−2.12788	−0.217176
$97$	−13.0759	−1.32765	−0.663827	−	0.747886i	$-0.731069\pi$
$97$	−13.0759	−1.32765	−0.663827	+	0.747886i	$0.731069\pi$
$98$	0.460728	0.0465406
$99$	−5.92709	−0.595695
$100$	5.17826	0.517826
$101$	13.0816	1.30166	0.650832	−	0.759222i	$-0.274420\pi$
$101$	13.0816	1.30166	0.650832	+	0.759222i	$0.274420\pi$
$102$	15.5189	1.53660
$103$	−0.866598	−0.0853884	−0.0426942	−	0.999088i	$-0.513594\pi$
$103$	−0.866598	−0.0853884	−0.0426942	+	0.999088i	$0.513594\pi$
$104$	2.89555	0.283932
$105$	−18.5428	−1.80959
$106$	−4.81937	−0.468099
$107$	−7.53538	−0.728472	−0.364236	−	0.931307i	$-0.618670\pi$
$107$	−7.53538	−0.728472	−0.364236	+	0.931307i	$0.618670\pi$
$108$	3.13251	0.301426
$109$	4.83129	0.462753	0.231377	−	0.972864i	$-0.425677\pi$
$109$	4.83129	0.462753	0.231377	+	0.972864i	$0.425677\pi$
$110$	−12.3763	−1.18004
$111$	9.08911	0.862700
$112$	2.73143	0.258096
$113$	10.3667	0.975222	0.487611	−	0.873061i	$-0.337869\pi$
$113$	10.3667	0.975222	0.487611	+	0.873061i	$0.337869\pi$
$114$	−3.50234	−0.328024
$115$	−17.6360	−1.64456
$116$	−4.38749	−0.407368
$117$	4.42403	0.409002
$118$	−6.93691	−0.638595
$119$	−19.9207	−1.82613
$120$	−6.78865	−0.619717
$121$	4.04908	0.368098
$122$	−3.26404	−0.295512
$123$	−8.21648	−0.740855
$124$	−1.88667	−0.169428
$125$	0.568704	0.0508664
$126$	4.17328	0.371785
$127$	−3.35020	−0.297282	−0.148641	−	0.988891i	$-0.547490\pi$
$127$	−3.35020	−0.297282	−0.148641	+	0.988891i	$0.547490\pi$
$128$	1.00000	0.0883883
$129$	13.7046	1.20662
$130$	9.23780	0.810209
$131$	7.42421	0.648656	0.324328	−	0.945945i	$-0.394862\pi$
$131$	7.42421	0.648656	0.324328	+	0.945945i	$0.394862\pi$
$132$	8.25471	0.718480
$133$	4.49575	0.389831
$134$	−8.16346	−0.705215
$135$	9.99378	0.860128
$136$	−7.29312	−0.625380
$137$	14.2495	1.21742	0.608709	−	0.793393i	$-0.291688\pi$
$137$	14.2495	1.21742	0.608709	+	0.793393i	$0.291688\pi$
$138$	11.7628	1.00131
$139$	9.33101	0.791446	0.395723	−	0.918370i	$-0.370494\pi$
$139$	9.33101	0.791446	0.395723	+	0.918370i	$0.370494\pi$
$140$	8.71420	0.736484
$141$	9.38743	0.790564
$142$	−2.72005	−0.228262
$143$	−11.2328	−0.939331
$144$	1.52787	0.127323
$145$	−13.9976	−1.16243
$146$	2.32453	0.192379
$147$	−0.980374	−0.0808599
$148$	−4.27144	−0.351110
$149$	−9.96166	−0.816091	−0.408046	−	0.912962i	$-0.633790\pi$
$149$	−9.96166	−0.816091	−0.408046	+	0.912962i	$0.633790\pi$
$150$	−11.0187	−0.899674
$151$	9.90963	0.806434	0.403217	−	0.915104i	$-0.367892\pi$
$151$	9.90963	0.806434	0.403217	+	0.915104i	$0.367892\pi$
$152$	1.64593	0.133503
$153$	−11.1429	−0.900853
$154$	−10.5961	−0.853857
$155$	−6.01911	−0.483466
$156$	−6.16139	−0.493306
$157$	9.77807	0.780375	0.390187	−	0.920735i	$-0.372410\pi$
$157$	9.77807	0.780375	0.390187	+	0.920735i	$0.372410\pi$
$158$	−3.74700	−0.298096
$159$	10.2550	0.813278
$160$	3.19034	0.252218
$161$	−15.0992	−1.18998
$162$	−11.2492	−0.883822
$163$	7.91081	0.619623	0.309811	−	0.950798i	$-0.399734\pi$
$163$	7.91081	0.619623	0.309811	+	0.950798i	$0.399734\pi$
$164$	3.86135	0.301520
$165$	26.3353	2.05020
$166$	−11.6500	−0.904217
$167$	−15.3744	−1.18971	−0.594854	−	0.803834i	$-0.702790\pi$
$167$	−15.3744	−1.18971	−0.594854	+	0.803834i	$0.702790\pi$
$168$	−5.81216	−0.448418
$169$	−4.61576	−0.355059
$170$	−23.2675	−1.78454
$171$	2.51477	0.192309
$172$	−6.44048	−0.491082
$173$	18.4526	1.40292	0.701462	−	0.712706i	$-0.252531\pi$
$173$	18.4526	1.40292	0.701462	+	0.712706i	$0.252531\pi$
$174$	9.33604	0.707764
$175$	14.1441	1.06919
$176$	−3.87931	−0.292414
$177$	14.7609	1.10950
$178$	3.65537	0.273981
$179$	−15.2616	−1.14071	−0.570353	−	0.821400i	$-0.693193\pi$
$179$	−15.2616	−1.14071	−0.570353	+	0.821400i	$0.693193\pi$
$180$	4.87442	0.363318
$181$	1.06850	0.0794213	0.0397106	−	0.999211i	$-0.487356\pi$
$181$	1.06850	0.0794213	0.0397106	+	0.999211i	$0.487356\pi$
$182$	7.90901	0.586255
$183$	6.94548	0.513425
$184$	−5.52793	−0.407525
$185$	−13.6273	−1.00190
$186$	4.01460	0.294365
$187$	28.2923	2.06894
$188$	−4.41164	−0.321752
$189$	8.55625	0.622376
$190$	5.25108	0.380953
$191$	12.2837	0.888814	0.444407	−	0.895825i	$-0.353414\pi$
$191$	12.2837	0.888814	0.444407	+	0.895825i	$0.353414\pi$
$192$	−2.12788	−0.153566
$193$	3.62087	0.260636	0.130318	−	0.991472i	$-0.458400\pi$
$193$	3.62087	0.260636	0.130318	+	0.991472i	$0.458400\pi$
$194$	−13.0759	−0.938794
$195$	−19.6569	−1.40766
$196$	0.460728	0.0329091
$197$	−7.20071	−0.513029	−0.256515	−	0.966540i	$-0.582574\pi$
$197$	−7.20071	−0.513029	−0.256515	+	0.966540i	$0.582574\pi$
$198$	−5.92709	−0.421220
$199$	15.5775	1.10426	0.552128	−	0.833759i	$-0.313816\pi$
$199$	15.5775	1.10426	0.552128	+	0.833759i	$0.313816\pi$
$200$	5.17826	0.366158
$201$	17.3709	1.22525
$202$	13.0816	0.920415
$203$	−11.9841	−0.841121
$204$	15.5189	1.08654
$205$	12.3190	0.860396
$206$	−0.866598	−0.0603787
$207$	−8.44597	−0.587035
$208$	2.89555	0.200771
$209$	−6.38508	−0.441665
$210$	−18.5428	−1.27957
$211$	−5.43094	−0.373882	−0.186941	−	0.982371i	$-0.559857\pi$
$211$	−5.43094	−0.373882	−0.186941	+	0.982371i	$0.559857\pi$
$212$	−4.81937	−0.330996
$213$	5.78795	0.396584
$214$	−7.53538	−0.515108
$215$	−20.5473	−1.40131
$216$	3.13251	0.213141
$217$	−5.15330	−0.349829
$218$	4.83129	0.327216
$219$	−4.94631	−0.334241
$220$	−12.3763	−0.834412
$221$	−21.1176	−1.42053
$222$	9.08911	0.610021
$223$	−2.23347	−0.149564	−0.0747822	−	0.997200i	$-0.523826\pi$
$223$	−2.23347	−0.149564	−0.0747822	+	0.997200i	$0.523826\pi$
$224$	2.73143	0.182502
$225$	7.91171	0.527447
$226$	10.3667	0.689586
$227$	9.86894	0.655024	0.327512	−	0.944847i	$-0.393790\pi$
$227$	9.86894	0.655024	0.327512	+	0.944847i	$0.393790\pi$
$228$	−3.50234	−0.231948
$229$	−1.20763	−0.0798027	−0.0399014	−	0.999204i	$-0.512704\pi$
$229$	−1.20763	−0.0798027	−0.0399014	+	0.999204i	$0.512704\pi$
$230$	−17.6360	−1.16288
$231$	22.5472	1.48350
$232$	−4.38749	−0.288053
$233$	1.62431	0.106412	0.0532062	−	0.998584i	$-0.483056\pi$
$233$	1.62431	0.106412	0.0532062	+	0.998584i	$0.483056\pi$
$234$	4.42403	0.289208
$235$	−14.0746	−0.918127
$236$	−6.93691	−0.451555
$237$	7.97317	0.517913
$238$	−19.9207	−1.29127
$239$	−11.0292	−0.713418	−0.356709	−	0.934215i	$-0.616101\pi$
$239$	−11.0292	−0.713418	−0.356709	+	0.934215i	$0.616101\pi$
$240$	−6.78865	−0.438206
$241$	21.4097	1.37912	0.689560	−	0.724229i	$-0.257804\pi$
$241$	21.4097	1.37912	0.689560	+	0.724229i	$0.257804\pi$
$242$	4.04908	0.260285
$243$	14.5394	0.932706
$244$	−3.26404	−0.208959
$245$	1.46988	0.0939071
$246$	−8.21648	−0.523864
$247$	4.76588	0.303246
$248$	−1.88667	−0.119803
$249$	24.7898	1.57099
$250$	0.568704	0.0359680
$251$	4.28151	0.270247	0.135123	−	0.990829i	$-0.456857\pi$
$251$	4.28151	0.270247	0.135123	+	0.990829i	$0.456857\pi$
$252$	4.17328	0.262892
$253$	21.4446	1.34821
$254$	−3.35020	−0.210210
$255$	49.5105	3.10047
$256$	1.00000	0.0625000
$257$	0.831506	0.0518679	0.0259339	−	0.999664i	$-0.491744\pi$
$257$	0.831506	0.0518679	0.0259339	+	0.999664i	$0.491744\pi$
$258$	13.7046	0.853208
$259$	−11.6671	−0.724961
$260$	9.23780	0.572904
$261$	−6.70351	−0.414937
$262$	7.42421	0.458669
$263$	−13.6650	−0.842619	−0.421309	−	0.906917i	$-0.638429\pi$
$263$	−13.6650	−0.842619	−0.421309	+	0.906917i	$0.638429\pi$
$264$	8.25471	0.508042
$265$	−15.3754	−0.944505
$266$	4.49575	0.275652
$267$	−7.77818	−0.476017
$268$	−8.16346	−0.498663
$269$	27.7599	1.69255	0.846276	−	0.532744i	$-0.178839\pi$
$269$	27.7599	1.69255	0.846276	+	0.532744i	$0.178839\pi$
$270$	9.99378	0.608202
$271$	−0.846461	−0.0514188	−0.0257094	−	0.999669i	$-0.508184\pi$
$271$	−0.846461	−0.0514188	−0.0257094	+	0.999669i	$0.508184\pi$
$272$	−7.29312	−0.442210
$273$	−16.8294	−1.01856
$274$	14.2495	0.860845
$275$	−20.0881	−1.21136
$276$	11.7628	0.708036
$277$	−6.24776	−0.375391	−0.187696	−	0.982227i	$-0.560102\pi$
$277$	−6.24776	−0.375391	−0.187696	+	0.982227i	$0.560102\pi$
$278$	9.33101	0.559637
$279$	−2.88258	−0.172576
$280$	8.71420	0.520773
$281$	−12.6174	−0.752693	−0.376347	−	0.926479i	$-0.622820\pi$
$281$	−12.6174	−0.752693	−0.376347	+	0.926479i	$0.622820\pi$
$282$	9.38743	0.559013
$283$	15.1361	0.899750	0.449875	−	0.893091i	$-0.351468\pi$
$283$	15.1361	0.899750	0.449875	+	0.893091i	$0.351468\pi$
$284$	−2.72005	−0.161405
$285$	−11.1737	−0.661870
$286$	−11.2328	−0.664207
$287$	10.5470	0.622570
$288$	1.52787	0.0900306
$289$	36.1896	2.12880
$290$	−13.9976	−0.821966
$291$	27.8239	1.63107
$292$	2.32453	0.136033
$293$	7.50075	0.438199	0.219099	−	0.975703i	$-0.429688\pi$
$293$	7.50075	0.438199	0.219099	+	0.975703i	$0.429688\pi$
$294$	−0.980374	−0.0571766
$295$	−22.1311	−1.28852
$296$	−4.27144	−0.248272
$297$	−12.1520	−0.705131
$298$	−9.96166	−0.577064
$299$	−16.0064	−0.925676
$300$	−11.0187	−0.636165
$301$	−17.5917	−1.01397
$302$	9.90963	0.570235
$303$	−27.8360	−1.59913
$304$	1.64593	0.0944006
$305$	−10.4134	−0.596269
$306$	−11.1429	−0.637000
$307$	2.07837	0.118619	0.0593094	−	0.998240i	$-0.481110\pi$
$307$	2.07837	0.118619	0.0593094	+	0.998240i	$0.481110\pi$
$308$	−10.5961	−0.603768
$309$	1.84402	0.104902
$310$	−6.01911	−0.341862
$311$	11.1900	0.634526	0.317263	−	0.948338i	$-0.397236\pi$
$311$	11.1900	0.634526	0.317263	+	0.948338i	$0.397236\pi$
$312$	−6.16139	−0.348820
$313$	5.95584	0.336644	0.168322	−	0.985732i	$-0.446165\pi$
$313$	5.95584	0.336644	0.168322	+	0.985732i	$0.446165\pi$
$314$	9.77807	0.551808
$315$	13.3142	0.750168
$316$	−3.74700	−0.210785
$317$	9.14961	0.513893	0.256947	−	0.966426i	$-0.417284\pi$
$317$	9.14961	0.513893	0.256947	+	0.966426i	$0.417284\pi$
$318$	10.2550	0.575074
$319$	17.0204	0.952962
$320$	3.19034	0.178345
$321$	16.0344	0.894951
$322$	−15.0992	−0.841445
$323$	−12.0040	−0.667919
$324$	−11.2492	−0.624957
$325$	14.9939	0.831714
$326$	7.91081	0.438139
$327$	−10.2804	−0.568507
$328$	3.86135	0.213207
$329$	−12.0501	−0.664343
$330$	26.3353	1.44971
$331$	−23.6280	−1.29871	−0.649356	−	0.760485i	$-0.724961\pi$
$331$	−23.6280	−1.29871	−0.649356	+	0.760485i	$0.724961\pi$
$332$	−11.6500	−0.639378
$333$	−6.52620	−0.357634
$334$	−15.3744	−0.841251
$335$	−26.0442	−1.42295
$336$	−5.81216	−0.317079
$337$	−0.232250	−0.0126515	−0.00632573	−	0.999980i	$-0.502014\pi$
$337$	−0.232250	−0.0126515	−0.00632573	+	0.999980i	$0.502014\pi$
$338$	−4.61576	−0.251064
$339$	−22.0592	−1.19809
$340$	−23.2675	−1.26186
$341$	7.31897	0.396345
$342$	2.51477	0.135983
$343$	−17.8616	−0.964435
$344$	−6.44048	−0.347247
$345$	37.5272	2.02040
$346$	18.4526	0.992018
$347$	−27.1639	−1.45823	−0.729117	−	0.684389i	$-0.760069\pi$
$347$	−27.1639	−1.45823	−0.729117	+	0.684389i	$0.760069\pi$
$348$	9.33604	0.500465
$349$	−10.7127	−0.573440	−0.286720	−	0.958014i	$-0.592565\pi$
$349$	−10.7127	−0.573440	−0.286720	+	0.958014i	$0.592565\pi$
$350$	14.1441	0.756032
$351$	9.07037	0.484140
$352$	−3.87931	−0.206768
$353$	−14.8824	−0.792110	−0.396055	−	0.918227i	$-0.629621\pi$
$353$	−14.8824	−0.792110	−0.396055	+	0.918227i	$0.629621\pi$
$354$	14.7609	0.784534
$355$	−8.67789	−0.460575
$356$	3.65537	0.193734
$357$	42.3888	2.24345
$358$	−15.2616	−0.806600
$359$	11.4625	0.604969	0.302485	−	0.953154i	$-0.402184\pi$
$359$	11.4625	0.604969	0.302485	+	0.953154i	$0.402184\pi$
$360$	4.87442	0.256905
$361$	−16.2909	−0.857416
$362$	1.06850	0.0561593
$363$	−8.61595	−0.452220
$364$	7.90901	0.414545
$365$	7.41602	0.388172
$366$	6.94548	0.363046
$367$	7.94715	0.414838	0.207419	−	0.978252i	$-0.433494\pi$
$367$	7.94715	0.414838	0.207419	+	0.978252i	$0.433494\pi$
$368$	−5.52793	−0.288164
$369$	5.89964	0.307123
$370$	−13.6273	−0.708451
$371$	−13.1638	−0.683430
$372$	4.01460	0.208147
$373$	9.57991	0.496029	0.248015	−	0.968756i	$-0.420222\pi$
$373$	9.57991	0.496029	0.248015	+	0.968756i	$0.420222\pi$
$374$	28.2923	1.46296
$375$	−1.21013	−0.0624910
$376$	−4.41164	−0.227513
$377$	−12.7042	−0.654300
$378$	8.55625	0.440086
$379$	−2.77082	−0.142327	−0.0711637	−	0.997465i	$-0.522671\pi$
$379$	−2.77082	−0.142327	−0.0711637	+	0.997465i	$0.522671\pi$
$380$	5.25108	0.269375
$381$	7.12882	0.365221
$382$	12.2837	0.628486
$383$	2.05656	0.105085	0.0525426	−	0.998619i	$-0.483267\pi$
$383$	2.05656	0.105085	0.0525426	+	0.998619i	$0.483267\pi$
$384$	−2.12788	−0.108588
$385$	−33.8051	−1.72287
$386$	3.62087	0.184297
$387$	−9.84021	−0.500206
$388$	−13.0759	−0.663827
$389$	−22.8002	−1.15601	−0.578007	−	0.816032i	$-0.696169\pi$
$389$	−22.8002	−1.15601	−0.578007	+	0.816032i	$0.696169\pi$
$390$	−19.6569	−0.995367
$391$	40.3159	2.03886
$392$	0.460728	0.0232703
$393$	−15.7978	−0.796895
$394$	−7.20071	−0.362766
$395$	−11.9542	−0.601481
$396$	−5.92709	−0.297847
$397$	12.2796	0.616298	0.308149	−	0.951338i	$-0.400290\pi$
$397$	12.2796	0.616298	0.308149	+	0.951338i	$0.400290\pi$
$398$	15.5775	0.780827
$399$	−9.56641	−0.478920
$400$	5.17826	0.258913
$401$	−30.3154	−1.51388	−0.756939	−	0.653485i	$-0.773306\pi$
$401$	−30.3154	−1.51388	−0.756939	+	0.653485i	$0.773306\pi$
$402$	17.3709	0.866380
$403$	−5.46295	−0.272129
$404$	13.0816	0.650832
$405$	−35.8888	−1.78333
$406$	−11.9841	−0.594762
$407$	16.5703	0.821357
$408$	15.5189	0.768299
$409$	15.4577	0.764334	0.382167	−	0.924093i	$-0.375178\pi$
$409$	15.4577	0.764334	0.382167	+	0.924093i	$0.375178\pi$
$410$	12.3190	0.608392
$411$	−30.3213	−1.49564
$412$	−0.866598	−0.0426942
$413$	−18.9477	−0.932356
$414$	−8.44597	−0.415097
$415$	−37.1675	−1.82448
$416$	2.89555	0.141966
$417$	−19.8553	−0.972317
$418$	−6.38508	−0.312305
$419$	−4.96947	−0.242775	−0.121387	−	0.992605i	$-0.538734\pi$
$419$	−4.96947	−0.242775	−0.121387	+	0.992605i	$0.538734\pi$
$420$	−18.5428	−0.904794
$421$	9.68711	0.472121	0.236060	−	0.971738i	$-0.424144\pi$
$421$	9.68711	0.472121	0.236060	+	0.971738i	$0.424144\pi$
$422$	−5.43094	−0.264374
$423$	−6.74041	−0.327730
$424$	−4.81937	−0.234049
$425$	−37.7657	−1.83190
$426$	5.78795	0.280427
$427$	−8.91550	−0.431451
$428$	−7.53538	−0.364236
$429$	23.9020	1.15400
$430$	−20.5473	−0.990879
$431$	−24.2169	−1.16649	−0.583244	−	0.812297i	$-0.698217\pi$
$431$	−24.2169	−1.16649	−0.583244	+	0.812297i	$0.698217\pi$
$432$	3.13251	0.150713
$433$	7.18634	0.345353	0.172677	−	0.984979i	$-0.444758\pi$
$433$	7.18634	0.345353	0.172677	+	0.984979i	$0.444758\pi$
$434$	−5.15330	−0.247367
$435$	29.7851	1.42809
$436$	4.83129	0.231377
$437$	−9.09860	−0.435245
$438$	−4.94631	−0.236344
$439$	20.3190	0.969775	0.484888	−	0.874576i	$-0.338860\pi$
$439$	20.3190	0.969775	0.484888	+	0.874576i	$0.338860\pi$
$440$	−12.3763	−0.590018
$441$	0.703933	0.0335206
$442$	−21.1176	−1.00446
$443$	−25.0281	−1.18912	−0.594560	−	0.804051i	$-0.702674\pi$
$443$	−25.0281	−1.18912	−0.594560	+	0.804051i	$0.702674\pi$
$444$	9.08911	0.431350
$445$	11.6619	0.552825
$446$	−2.23347	−0.105758
$447$	21.1972	1.00259
$448$	2.73143	0.129048
$449$	−25.7817	−1.21671	−0.608356	−	0.793664i	$-0.708171\pi$
$449$	−25.7817	−1.21671	−0.608356	+	0.793664i	$0.708171\pi$
$450$	7.91171	0.372961
$451$	−14.9794	−0.705351
$452$	10.3667	0.487611
$453$	−21.0865	−0.990730
$454$	9.86894	0.463172
$455$	25.2324	1.18291
$456$	−3.50234	−0.164012
$457$	10.5769	0.494766	0.247383	−	0.968918i	$-0.420429\pi$
$457$	10.5769	0.494766	0.247383	+	0.968918i	$0.420429\pi$
$458$	−1.20763	−0.0564290
$459$	−22.8458	−1.06635
$460$	−17.6360	−0.822282
$461$	17.0556	0.794359	0.397180	−	0.917741i	$-0.369989\pi$
$461$	17.0556	0.794359	0.397180	+	0.917741i	$0.369989\pi$
$462$	22.5472	1.04899
$463$	−11.3254	−0.526334	−0.263167	−	0.964750i	$-0.584767\pi$
$463$	−11.3254	−0.526334	−0.263167	+	0.964750i	$0.584767\pi$
$464$	−4.38749	−0.203684
$465$	12.8079	0.593954
$466$	1.62431	0.0752449
$467$	−16.5741	−0.766957	−0.383478	−	0.923550i	$-0.625274\pi$
$467$	−16.5741	−0.766957	−0.383478	+	0.923550i	$0.625274\pi$
$468$	4.42403	0.204501
$469$	−22.2979	−1.02962
$470$	−14.0746	−0.649214
$471$	−20.8066	−0.958715
$472$	−6.93691	−0.319297
$473$	24.9846	1.14879
$474$	7.97317	0.366220
$475$	8.52305	0.391065
$476$	−19.9207	−0.913063
$477$	−7.36338	−0.337146
$478$	−11.0292	−0.504463
$479$	−28.3479	−1.29525	−0.647623	−	0.761961i	$-0.724237\pi$
$479$	−28.3479	−1.29525	−0.647623	+	0.761961i	$0.724237\pi$
$480$	−6.78865	−0.309858
$481$	−12.3682	−0.563941
$482$	21.4097	0.975184
$483$	32.1292	1.46193
$484$	4.04908	0.184049
$485$	−41.7165	−1.89425
$486$	14.5394	0.659523
$487$	16.3153	0.739315	0.369658	−	0.929168i	$-0.379475\pi$
$487$	16.3153	0.739315	0.369658	+	0.929168i	$0.379475\pi$
$488$	−3.26404	−0.147756
$489$	−16.8333	−0.761226
$490$	1.46988	0.0664023
$491$	−12.1788	−0.549620	−0.274810	−	0.961499i	$-0.588615\pi$
$491$	−12.1788	−0.549620	−0.274810	+	0.961499i	$0.588615\pi$
$492$	−8.21648	−0.370427
$493$	31.9985	1.44114
$494$	4.76588	0.214427
$495$	−18.9094	−0.849915
$496$	−1.88667	−0.0847138
$497$	−7.42965	−0.333265
$498$	24.7898	1.11086
$499$	38.6642	1.73085	0.865423	−	0.501042i	$-0.167050\pi$
$499$	38.6642	1.73085	0.865423	+	0.501042i	$0.167050\pi$
$500$	0.568704	0.0254332
$501$	32.7149	1.46159
$502$	4.28151	0.191093
$503$	−20.0792	−0.895289	−0.447644	−	0.894212i	$-0.647737\pi$
$503$	−20.0792	−0.895289	−0.447644	+	0.894212i	$0.647737\pi$
$504$	4.17328	0.185892
$505$	41.7346	1.85716
$506$	21.4446	0.953329
$507$	9.82179	0.436201
$508$	−3.35020	−0.148641
$509$	14.1117	0.625492	0.312746	−	0.949837i	$-0.398751\pi$
$509$	14.1117	0.625492	0.312746	+	0.949837i	$0.398751\pi$
$510$	49.5105	2.19236
$511$	6.34929	0.280876
$512$	1.00000	0.0441942
$513$	5.15590	0.227639
$514$	0.831506	0.0366761
$515$	−2.76474	−0.121829
$516$	13.7046	0.603309
$517$	17.1141	0.752678
$518$	−11.6671	−0.512625
$519$	−39.2649	−1.72354
$520$	9.23780	0.405104
$521$	−6.09264	−0.266923	−0.133462	−	0.991054i	$-0.542609\pi$
$521$	−6.09264	−0.266923	−0.133462	+	0.991054i	$0.542609\pi$
$522$	−6.70351	−0.293405
$523$	−22.3511	−0.977345	−0.488673	−	0.872467i	$-0.662519\pi$
$523$	−22.3511	−0.977345	−0.488673	+	0.872467i	$0.662519\pi$
$524$	7.42421	0.324328
$525$	−30.0969	−1.31354
$526$	−13.6650	−0.595822
$527$	13.7597	0.599381
$528$	8.25471	0.359240
$529$	7.55806	0.328612
$530$	−15.3754	−0.667866
$531$	−10.5987	−0.459945
$532$	4.49575	0.194915
$533$	11.1807	0.484292
$534$	−7.77818	−0.336595
$535$	−24.0404	−1.03936
$536$	−8.16346	−0.352608
$537$	32.4748	1.40139
$538$	27.7599	1.19682
$539$	−1.78731	−0.0769848
$540$	9.99378	0.430064
$541$	−19.3282	−0.830985	−0.415493	−	0.909597i	$-0.636391\pi$
$541$	−19.3282	−0.830985	−0.415493	+	0.909597i	$0.636391\pi$
$542$	−0.846461	−0.0363586
$543$	−2.27365	−0.0975715
$544$	−7.29312	−0.312690
$545$	15.4134	0.660239
$546$	−16.8294	−0.720233
$547$	34.4510	1.47302	0.736509	−	0.676428i	$-0.236473\pi$
$547$	34.4510	1.47302	0.736509	+	0.676428i	$0.236473\pi$
$548$	14.2495	0.608709
$549$	−4.98703	−0.212841
$550$	−20.0881	−0.856559
$551$	−7.22150	−0.307646
$552$	11.7628	0.500657
$553$	−10.2347	−0.435223
$554$	−6.24776	−0.265442
$555$	28.9973	1.23087
$556$	9.33101	0.395723
$557$	34.3587	1.45583	0.727913	−	0.685669i	$-0.240490\pi$
$557$	34.3587	1.45583	0.727913	+	0.685669i	$0.240490\pi$
$558$	−2.88258	−0.122029
$559$	−18.6487	−0.788758
$560$	8.71420	0.368242
$561$	−60.2026	−2.54176
$562$	−12.6174	−0.532234
$563$	−40.9371	−1.72529	−0.862647	−	0.505807i	$-0.831195\pi$
$563$	−40.9371	−1.72529	−0.862647	+	0.505807i	$0.831195\pi$
$564$	9.38743	0.395282
$565$	33.0734	1.39141
$566$	15.1361	0.636220
$567$	−30.7265	−1.29039
$568$	−2.72005	−0.114131
$569$	16.6864	0.699529	0.349764	−	0.936838i	$-0.386262\pi$
$569$	16.6864	0.699529	0.349764	+	0.936838i	$0.386262\pi$
$570$	−11.1737	−0.468013
$571$	−21.3087	−0.891743	−0.445871	−	0.895097i	$-0.647106\pi$
$571$	−21.3087	−0.891743	−0.445871	+	0.895097i	$0.647106\pi$
$572$	−11.2328	−0.469666
$573$	−26.1381	−1.09194
$574$	10.5470	0.440224
$575$	−28.6251	−1.19375
$576$	1.52787	0.0636613
$577$	−19.8539	−0.826529	−0.413264	−	0.910611i	$-0.635611\pi$
$577$	−19.8539	−0.826529	−0.413264	+	0.910611i	$0.635611\pi$
$578$	36.1896	1.50529
$579$	−7.70478	−0.320200
$580$	−13.9976	−0.581217
$581$	−31.8212	−1.32017
$582$	27.8239	1.15334
$583$	18.6959	0.774303
$584$	2.32453	0.0961896
$585$	14.1142	0.583549
$586$	7.50075	0.309853
$587$	−43.1299	−1.78016	−0.890081	−	0.455802i	$-0.849352\pi$
$587$	−43.1299	−1.78016	−0.890081	+	0.455802i	$0.849352\pi$
$588$	−0.980374	−0.0404299
$589$	−3.10532	−0.127953
$590$	−22.1311	−0.911123
$591$	15.3222	0.630273
$592$	−4.27144	−0.175555
$593$	13.7834	0.566017	0.283008	−	0.959117i	$-0.408668\pi$
$593$	13.7834	0.566017	0.283008	+	0.959117i	$0.408668\pi$
$594$	−12.1520	−0.498603
$595$	−63.5537	−2.60545
$596$	−9.96166	−0.408046
$597$	−33.1470	−1.35661
$598$	−16.0064	−0.654552
$599$	6.41081	0.261938	0.130969	−	0.991386i	$-0.458191\pi$
$599$	6.41081	0.261938	0.130969	+	0.991386i	$0.458191\pi$
$600$	−11.0187	−0.449837
$601$	−33.2691	−1.35707	−0.678537	−	0.734566i	$-0.737386\pi$
$601$	−33.2691	−1.35707	−0.678537	+	0.734566i	$0.737386\pi$
$602$	−17.5917	−0.716985
$603$	−12.4727	−0.507928
$604$	9.90963	0.403217
$605$	12.9179	0.525189
$606$	−27.8360	−1.13076
$607$	17.6746	0.717388	0.358694	−	0.933455i	$-0.383222\pi$
$607$	17.6746	0.717388	0.358694	+	0.933455i	$0.383222\pi$
$608$	1.64593	0.0667513
$609$	25.5008	1.03334
$610$	−10.4134	−0.421626
$611$	−12.7741	−0.516786
$612$	−11.1429	−0.450427
$613$	−5.06813	−0.204700	−0.102350	−	0.994748i	$-0.532636\pi$
$613$	−5.06813	−0.204700	−0.102350	+	0.994748i	$0.532636\pi$
$614$	2.07837	0.0838761
$615$	−26.2133	−1.05702
$616$	−10.5961	−0.426929
$617$	16.1473	0.650066	0.325033	−	0.945703i	$-0.394625\pi$
$617$	16.1473	0.650066	0.325033	+	0.945703i	$0.394625\pi$
$618$	1.84402	0.0741772
$619$	−14.7408	−0.592483	−0.296242	−	0.955113i	$-0.595733\pi$
$619$	−14.7408	−0.592483	−0.296242	+	0.955113i	$0.595733\pi$
$620$	−6.01911	−0.241733
$621$	−17.3163	−0.694881
$622$	11.1900	0.448678
$623$	9.98439	0.400016
$624$	−6.16139	−0.246653
$625$	−24.0769	−0.963077
$626$	5.95584	0.238043
$627$	13.5867	0.542600
$628$	9.77807	0.390187
$629$	31.1521	1.24212
$630$	13.3142	0.530449
$631$	−14.7368	−0.586664	−0.293332	−	0.956011i	$-0.594764\pi$
$631$	−14.7368	−0.586664	−0.293332	+	0.956011i	$0.594764\pi$
$632$	−3.74700	−0.149048
$633$	11.5564	0.459325
$634$	9.14961	0.363378
$635$	−10.6883	−0.424151
$636$	10.2550	0.406639
$637$	1.33406	0.0528575
$638$	17.0204	0.673846
$639$	−4.15589	−0.164404
$640$	3.19034	0.126109
$641$	7.88543	0.311456	0.155728	−	0.987800i	$-0.450228\pi$
$641$	7.88543	0.311456	0.155728	+	0.987800i	$0.450228\pi$
$642$	16.0344	0.632826
$643$	−42.1441	−1.66200	−0.831000	−	0.556273i	$-0.812231\pi$
$643$	−42.1441	−1.66200	−0.831000	+	0.556273i	$0.812231\pi$
$644$	−15.0992	−0.594991
$645$	43.7222	1.72156
$646$	−12.0040	−0.472290
$647$	−6.24456	−0.245499	−0.122749	−	0.992438i	$-0.539171\pi$
$647$	−6.24456	−0.245499	−0.122749	+	0.992438i	$0.539171\pi$
$648$	−11.2492	−0.441911
$649$	26.9105	1.05633
$650$	14.9939	0.588110
$651$	10.9656	0.429776
$652$	7.91081	0.309811
$653$	−40.0000	−1.56532	−0.782661	−	0.622448i	$-0.786138\pi$
$653$	−40.0000	−1.56532	−0.782661	+	0.622448i	$0.786138\pi$
$654$	−10.2804	−0.401995
$655$	23.6857	0.925479
$656$	3.86135	0.150760
$657$	3.55157	0.138560
$658$	−12.0501	−0.469762
$659$	−25.8475	−1.00688	−0.503439	−	0.864031i	$-0.667932\pi$
$659$	−25.8475	−1.00688	−0.503439	+	0.864031i	$0.667932\pi$
$660$	26.3353	1.02510
$661$	39.1499	1.52275	0.761377	−	0.648310i	$-0.224524\pi$
$661$	39.1499	1.52275	0.761377	+	0.648310i	$0.224524\pi$
$662$	−23.6280	−0.918327
$663$	44.9358	1.74516
$664$	−11.6500	−0.452108
$665$	14.3430	0.556196
$666$	−6.52620	−0.252885
$667$	24.2537	0.939109
$668$	−15.3744	−0.594854
$669$	4.75256	0.183745
$670$	−26.0442	−1.00618
$671$	12.6622	0.488820
$672$	−5.81216	−0.224209
$673$	28.1144	1.08373	0.541865	−	0.840466i	$-0.317719\pi$
$673$	28.1144	1.08373	0.541865	+	0.840466i	$0.317719\pi$
$674$	−0.232250	−0.00894593
$675$	16.2210	0.624345
$676$	−4.61576	−0.177529
$677$	−9.76954	−0.375474	−0.187737	−	0.982219i	$-0.560115\pi$
$677$	−9.76954	−0.375474	−0.187737	+	0.982219i	$0.560115\pi$
$678$	−22.0592	−0.847178
$679$	−35.7159	−1.37065
$680$	−23.2675	−0.892269
$681$	−20.9999	−0.804718
$682$	7.31897	0.280258
$683$	24.1731	0.924957	0.462478	−	0.886631i	$-0.346960\pi$
$683$	24.1731	0.924957	0.462478	+	0.886631i	$0.346960\pi$
$684$	2.51477	0.0961546
$685$	45.4608	1.73697
$686$	−17.8616	−0.681958
$687$	2.56970	0.0980402
$688$	−6.44048	−0.245541
$689$	−13.9548	−0.531634
$690$	37.5272	1.42864
$691$	5.51561	0.209824	0.104912	−	0.994482i	$-0.466544\pi$
$691$	5.51561	0.209824	0.104912	+	0.994482i	$0.466544\pi$
$692$	18.4526	0.701462
$693$	−16.1894	−0.614986
$694$	−27.1639	−1.03113
$695$	29.7691	1.12921
$696$	9.33604	0.353882
$697$	−28.1613	−1.06668
$698$	−10.7127	−0.405483
$699$	−3.45634	−0.130731
$700$	14.1441	0.534595
$701$	11.4506	0.432482	0.216241	−	0.976340i	$-0.430620\pi$
$701$	11.4506	0.432482	0.216241	+	0.976340i	$0.430620\pi$
$702$	9.07037	0.342339
$703$	−7.03049	−0.265160
$704$	−3.87931	−0.146207
$705$	29.9491	1.12795
$706$	−14.8824	−0.560106
$707$	35.7314	1.34382
$708$	14.7609	0.554749
$709$	6.18216	0.232176	0.116088	−	0.993239i	$-0.462965\pi$
$709$	6.18216	0.232176	0.116088	+	0.993239i	$0.462965\pi$
$710$	−8.67789	−0.325675
$711$	−5.72494	−0.214702
$712$	3.65537	0.136991
$713$	10.4294	0.390583
$714$	42.3888	1.58636
$715$	−35.8363	−1.34020
$716$	−15.2616	−0.570353
$717$	23.4688	0.876457
$718$	11.4625	0.427778
$719$	32.4195	1.20904	0.604521	−	0.796589i	$-0.293364\pi$
$719$	32.4195	1.20904	0.604521	+	0.796589i	$0.293364\pi$
$720$	4.87442	0.181659
$721$	−2.36705	−0.0881537
$722$	−16.2909	−0.606285
$723$	−45.5572	−1.69429
$724$	1.06850	0.0397106
$725$	−22.7195	−0.843783
$726$	−8.61595	−0.319768
$727$	26.3235	0.976285	0.488142	−	0.872764i	$-0.337675\pi$
$727$	26.3235	0.976285	0.488142	+	0.872764i	$0.337675\pi$
$728$	7.90901	0.293128
$729$	2.80948	0.104055
$730$	7.41602	0.274479
$731$	46.9712	1.73729
$732$	6.94548	0.256712
$733$	50.3696	1.86044	0.930221	−	0.366999i	$-0.119615\pi$
$733$	50.3696	1.86044	0.930221	+	0.366999i	$0.119615\pi$
$734$	7.94715	0.293335
$735$	−3.12772	−0.115368
$736$	−5.52793	−0.203762
$737$	31.6686	1.16653
$738$	5.89964	0.217169
$739$	−3.06963	−0.112918	−0.0564590	−	0.998405i	$-0.517981\pi$
$739$	−3.06963	−0.112918	−0.0564590	+	0.998405i	$0.517981\pi$
$740$	−13.6273	−0.500951
$741$	−10.1412	−0.372547
$742$	−13.1638	−0.483258
$743$	16.0723	0.589635	0.294817	−	0.955554i	$-0.404741\pi$
$743$	16.0723	0.589635	0.294817	+	0.955554i	$0.404741\pi$
$744$	4.01460	0.147182
$745$	−31.7811	−1.16437
$746$	9.57991	0.350745
$747$	−17.7997	−0.651258
$748$	28.2923	1.03447
$749$	−20.5824	−0.752064
$750$	−1.21013	−0.0441878
$751$	−6.42663	−0.234511	−0.117256	−	0.993102i	$-0.537410\pi$
$751$	−6.42663	−0.234511	−0.117256	+	0.993102i	$0.537410\pi$
$752$	−4.41164	−0.160876
$753$	−9.11054	−0.332006
$754$	−12.7042	−0.462660
$755$	31.6151	1.15059
$756$	8.55625	0.311188
$757$	−22.8671	−0.831118	−0.415559	−	0.909566i	$-0.636414\pi$
$757$	−22.8671	−0.831118	−0.415559	+	0.909566i	$0.636414\pi$
$758$	−2.77082	−0.100641
$759$	−45.6315	−1.65632
$760$	5.25108	0.190477
$761$	−21.1587	−0.767002	−0.383501	−	0.923540i	$-0.625282\pi$
$761$	−21.1587	−0.767002	−0.383501	+	0.923540i	$0.625282\pi$
$762$	7.12882	0.258250
$763$	13.1963	0.477740
$764$	12.2837	0.444407
$765$	−35.5498	−1.28530
$766$	2.05656	0.0743065
$767$	−20.0862	−0.725271
$768$	−2.12788	−0.0767832
$769$	42.4240	1.52985	0.764925	−	0.644119i	$-0.222776\pi$
$769$	42.4240	1.52985	0.764925	+	0.644119i	$0.222776\pi$
$770$	−33.8051	−1.21825
$771$	−1.76934	−0.0637214
$772$	3.62087	0.130318
$773$	39.1814	1.40926	0.704629	−	0.709576i	$-0.251114\pi$
$773$	39.1814	1.40926	0.704629	+	0.709576i	$0.251114\pi$
$774$	−9.84021	−0.353699
$775$	−9.76965	−0.350936
$776$	−13.0759	−0.469397
$777$	24.8263	0.890638
$778$	−22.8002	−0.817425
$779$	6.35551	0.227710
$780$	−19.6569	−0.703831
$781$	10.5519	0.377578
$782$	40.3159	1.44169
$783$	−13.7439	−0.491166
$784$	0.460728	0.0164546
$785$	31.1953	1.11341
$786$	−15.7978	−0.563490
$787$	−11.1759	−0.398378	−0.199189	−	0.979961i	$-0.563831\pi$
$787$	−11.1759	−0.398378	−0.199189	+	0.979961i	$0.563831\pi$
$788$	−7.20071	−0.256515
$789$	29.0774	1.03518
$790$	−11.9542	−0.425312
$791$	28.3161	1.00680
$792$	−5.92709	−0.210610
$793$	−9.45120	−0.335622
$794$	12.2796	0.435788
$795$	32.7171	1.16035
$796$	15.5775	0.552128
$797$	24.9494	0.883754	0.441877	−	0.897076i	$-0.354313\pi$
$797$	24.9494	0.883754	0.441877	+	0.897076i	$0.354313\pi$
$798$	−9.56641	−0.338647
$799$	32.1746	1.13826
$800$	5.17826	0.183079
$801$	5.58493	0.197334
$802$	−30.3154	−1.07047
$803$	−9.01757	−0.318223
$804$	17.3709	0.612623
$805$	−48.1715	−1.69782
$806$	−5.46295	−0.192424
$807$	−59.0698	−2.07935
$808$	13.0816	0.460207
$809$	21.1005	0.741856	0.370928	−	0.928662i	$-0.379040\pi$
$809$	21.1005	0.741856	0.370928	+	0.928662i	$0.379040\pi$
$810$	−35.8888	−1.26100
$811$	−43.2506	−1.51873	−0.759367	−	0.650663i	$-0.774491\pi$
$811$	−43.2506	−1.51873	−0.759367	+	0.650663i	$0.774491\pi$
$812$	−11.9841	−0.420561
$813$	1.80117	0.0631697
$814$	16.5703	0.580787
$815$	25.2382	0.884055
$816$	15.5189	0.543269
$817$	−10.6006	−0.370867
$818$	15.4577	0.540465
$819$	12.0839	0.422247
$820$	12.3190	0.430198
$821$	−50.4631	−1.76118	−0.880588	−	0.473883i	$-0.842852\pi$
$821$	−50.4631	−1.76118	−0.880588	+	0.473883i	$0.842852\pi$
$822$	−30.3213	−1.05758
$823$	−52.7686	−1.83940	−0.919699	−	0.392623i	$-0.871568\pi$
$823$	−52.7686	−1.83940	−0.919699	+	0.392623i	$0.871568\pi$
$824$	−0.866598	−0.0301894
$825$	42.7450	1.48819
$826$	−18.9477	−0.659275
$827$	−31.5680	−1.09773	−0.548864	−	0.835912i	$-0.684939\pi$
$827$	−31.5680	−1.09773	−0.548864	+	0.835912i	$0.684939\pi$
$828$	−8.44597	−0.293518
$829$	3.01139	0.104590	0.0522949	−	0.998632i	$-0.483346\pi$
$829$	3.01139	0.104590	0.0522949	+	0.998632i	$0.483346\pi$
$830$	−37.1675	−1.29010
$831$	13.2945	0.461180
$832$	2.89555	0.100385
$833$	−3.36015	−0.116422
$834$	−19.8553	−0.687532
$835$	−49.0496	−1.69743
$836$	−6.38508	−0.220833
$837$	−5.91001	−0.204280
$838$	−4.96947	−0.171668
$839$	−32.0378	−1.10607	−0.553034	−	0.833159i	$-0.686530\pi$
$839$	−32.0378	−1.10607	−0.553034	+	0.833159i	$0.686530\pi$
$840$	−18.5428	−0.639786
$841$	−9.74995	−0.336205
$842$	9.68711	0.333840
$843$	26.8484	0.924707
$844$	−5.43094	−0.186941
$845$	−14.7258	−0.506584
$846$	−6.74041	−0.231740
$847$	11.0598	0.380019
$848$	−4.81937	−0.165498
$849$	−32.2079	−1.10537
$850$	−37.7657	−1.29535
$851$	23.6122	0.809417
$852$	5.78795	0.198292
$853$	−39.3138	−1.34608	−0.673040	−	0.739606i	$-0.735012\pi$
$853$	−39.3138	−1.34608	−0.673040	+	0.739606i	$0.735012\pi$
$854$	−8.91550	−0.305082
$855$	8.02296	0.274380
$856$	−7.53538	−0.257554
$857$	48.7562	1.66548	0.832740	−	0.553664i	$-0.186771\pi$
$857$	48.7562	1.66548	0.832740	+	0.553664i	$0.186771\pi$
$858$	23.9020	0.816000
$859$	20.9423	0.714543	0.357271	−	0.934001i	$-0.383707\pi$
$859$	20.9423	0.714543	0.357271	+	0.934001i	$0.383707\pi$
$860$	−20.5473	−0.700657
$861$	−22.4428	−0.764847
$862$	−24.2169	−0.824831
$863$	−15.8866	−0.540785	−0.270392	−	0.962750i	$-0.587153\pi$
$863$	−15.8866	−0.540785	−0.270392	+	0.962750i	$0.587153\pi$
$864$	3.13251	0.106570
$865$	58.8700	2.00164
$866$	7.18634	0.244202
$867$	−77.0071	−2.61530
$868$	−5.15330	−0.174915
$869$	14.5358	0.493093
$870$	29.7851	1.00981
$871$	−23.6377	−0.800934
$872$	4.83129	0.163608
$873$	−19.9783	−0.676161
$874$	−9.09860	−0.307765
$875$	1.55338	0.0525137
$876$	−4.94631	−0.167120
$877$	15.1081	0.510165	0.255082	−	0.966919i	$-0.417897\pi$
$877$	15.1081	0.510165	0.255082	+	0.966919i	$0.417897\pi$
$878$	20.3190	0.685735
$879$	−15.9607	−0.538341
$880$	−12.3763	−0.417206
$881$	7.19238	0.242317	0.121159	−	0.992633i	$-0.461339\pi$
$881$	7.19238	0.242317	0.121159	+	0.992633i	$0.461339\pi$
$882$	0.703933	0.0237026
$883$	8.60462	0.289569	0.144784	−	0.989463i	$-0.453751\pi$
$883$	8.60462	0.289569	0.144784	+	0.989463i	$0.453751\pi$
$884$	−21.1176	−0.710263
$885$	47.0923	1.58299
$886$	−25.0281	−0.840836
$887$	9.43426	0.316771	0.158386	−	0.987377i	$-0.449371\pi$
$887$	9.43426	0.316771	0.158386	+	0.987377i	$0.449371\pi$
$888$	9.08911	0.305010
$889$	−9.15085	−0.306910
$890$	11.6619	0.390906
$891$	43.6393	1.46197
$892$	−2.23347	−0.0747822
$893$	−7.26125	−0.242988
$894$	21.1972	0.708941
$895$	−48.6897	−1.62752
$896$	2.73143	0.0912508
$897$	34.0598	1.13722
$898$	−25.7817	−0.860346
$899$	8.27773	0.276078
$900$	7.91171	0.263724
$901$	35.1483	1.17096
$902$	−14.9794	−0.498759
$903$	37.4331	1.24569
$904$	10.3667	0.344793
$905$	3.40889	0.113315
$906$	−21.0865	−0.700552
$907$	27.4493	0.911440	0.455720	−	0.890123i	$-0.349382\pi$
$907$	27.4493	0.911440	0.455720	+	0.890123i	$0.349382\pi$
$908$	9.86894	0.327512
$909$	19.9869	0.662924
$910$	25.2324	0.836447
$911$	15.0707	0.499316	0.249658	−	0.968334i	$-0.419682\pi$
$911$	15.0707	0.499316	0.249658	+	0.968334i	$0.419682\pi$
$912$	−3.50234	−0.115974
$913$	45.1941	1.49571
$914$	10.5769	0.349852
$915$	22.1584	0.732535
$916$	−1.20763	−0.0399014
$917$	20.2787	0.669663
$918$	−22.8458	−0.754024
$919$	−27.7444	−0.915204	−0.457602	−	0.889157i	$-0.651292\pi$
$919$	−27.7444	−0.915204	−0.457602	+	0.889157i	$0.651292\pi$
$920$	−17.6360	−0.581441
$921$	−4.42252	−0.145727
$922$	17.0556	0.561697
$923$	−7.87606	−0.259244
$924$	22.5472	0.741748
$925$	−22.1186	−0.727255
$926$	−11.3254	−0.372174
$927$	−1.32405	−0.0434875
$928$	−4.38749	−0.144026
$929$	35.5423	1.16610	0.583052	−	0.812435i	$-0.301858\pi$
$929$	35.5423	1.16610	0.583052	+	0.812435i	$0.301858\pi$
$930$	12.8079	0.419989
$931$	0.758327	0.0248531
$932$	1.62431	0.0532062
$933$	−23.8110	−0.779536
$934$	−16.5741	−0.542320
$935$	90.2620	2.95188
$936$	4.42403	0.144604
$937$	11.7858	0.385025	0.192513	−	0.981294i	$-0.438336\pi$
$937$	11.7858	0.385025	0.192513	+	0.981294i	$0.438336\pi$
$938$	−22.2979	−0.728054
$939$	−12.6733	−0.413578
$940$	−14.0746	−0.459063
$941$	−35.4958	−1.15713	−0.578565	−	0.815636i	$-0.696387\pi$
$941$	−35.4958	−1.15713	−0.578565	+	0.815636i	$0.696387\pi$
$942$	−20.8066	−0.677914
$943$	−21.3453	−0.695098
$944$	−6.93691	−0.225777
$945$	27.2973	0.887983
$946$	24.9846	0.812320
$947$	34.3949	1.11768	0.558842	−	0.829274i	$-0.311246\pi$
$947$	34.3949	1.11768	0.558842	+	0.829274i	$0.311246\pi$
$948$	7.97317	0.258957
$949$	6.73079	0.218491
$950$	8.52305	0.276524
$951$	−19.4693	−0.631334
$952$	−19.9207	−0.645633
$953$	33.3193	1.07932	0.539659	−	0.841884i	$-0.318553\pi$
$953$	33.3193	1.07932	0.539659	+	0.841884i	$0.318553\pi$
$954$	−7.36338	−0.238398
$955$	39.1890	1.26813
$956$	−11.0292	−0.356709
$957$	−36.2175	−1.17074
$958$	−28.3479	−0.915878
$959$	38.9216	1.25684
$960$	−6.78865	−0.219103
$961$	−27.4405	−0.885177
$962$	−12.3682	−0.398766
$963$	−11.5131	−0.371004
$964$	21.4097	0.689560
$965$	11.5518	0.371866
$966$	32.1292	1.03374
$967$	31.3302	1.00751	0.503756	−	0.863846i	$-0.331951\pi$
$967$	31.3302	1.00751	0.503756	+	0.863846i	$0.331951\pi$
$968$	4.04908	0.130142
$969$	25.5430	0.820559
$970$	−41.7165	−1.33944
$971$	2.89051	0.0927609	0.0463804	−	0.998924i	$-0.485231\pi$
$971$	2.89051	0.0927609	0.0463804	+	0.998924i	$0.485231\pi$
$972$	14.5394	0.466353
$973$	25.4870	0.817077
$974$	16.3153	0.522775
$975$	−31.9053	−1.02179
$976$	−3.26404	−0.104479
$977$	−23.5135	−0.752265	−0.376132	−	0.926566i	$-0.622746\pi$
$977$	−23.5135	−0.752265	−0.376132	+	0.926566i	$0.622746\pi$
$978$	−16.8333	−0.538268
$979$	−14.1803	−0.453205
$980$	1.46988	0.0469535
$981$	7.38158	0.235676
$982$	−12.1788	−0.388640
$983$	48.3749	1.54292	0.771460	−	0.636277i	$-0.219527\pi$
$983$	48.3749	1.54292	0.771460	+	0.636277i	$0.219527\pi$
$984$	−8.21648	−0.261932
$985$	−22.9727	−0.731971
$986$	31.9985	1.01904
$987$	25.6411	0.816167
$988$	4.76588	0.151623
$989$	35.6025	1.13209
$990$	−18.9094	−0.600981
$991$	−18.8928	−0.600151	−0.300075	−	0.953916i	$-0.597012\pi$
$991$	−18.8928	−0.600151	−0.300075	+	0.953916i	$0.597012\pi$
$992$	−1.88667	−0.0599017
$993$	50.2775	1.59551
$994$	−7.42965	−0.235654
$995$	49.6974	1.57551
$996$	24.7898	0.785496
$997$	−36.7325	−1.16333	−0.581664	−	0.813429i	$-0.697598\pi$
$997$	−36.7325	−1.16333	−0.581664	+	0.813429i	$0.697598\pi$
$998$	38.6642	1.22389
$999$	−13.3803	−0.423335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.a.1.18	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.a.1.18	✓	75	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.12788 q^{3} +1.00000 q^{4} +3.19034 q^{5} -2.12788 q^{6} +2.73143 q^{7} +1.00000 q^{8} +1.52787 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.12788 q^{3} +1.00000 q^{4} +3.19034 q^{5} -2.12788 q^{6} +2.73143 q^{7} +1.00000 q^{8} +1.52787 q^{9} +3.19034 q^{10} -3.87931 q^{11} -2.12788 q^{12} +2.89555 q^{13} +2.73143 q^{14} -6.78865 q^{15} +1.00000 q^{16} -7.29312 q^{17} +1.52787 q^{18} +1.64593 q^{19} +3.19034 q^{20} -5.81216 q^{21} -3.87931 q^{22} -5.52793 q^{23} -2.12788 q^{24} +5.17826 q^{25} +2.89555 q^{26} +3.13251 q^{27} +2.73143 q^{28} -4.38749 q^{29} -6.78865 q^{30} -1.88667 q^{31} +1.00000 q^{32} +8.25471 q^{33} -7.29312 q^{34} +8.71420 q^{35} +1.52787 q^{36} -4.27144 q^{37} +1.64593 q^{38} -6.16139 q^{39} +3.19034 q^{40} +3.86135 q^{41} -5.81216 q^{42} -6.44048 q^{43} -3.87931 q^{44} +4.87442 q^{45} -5.52793 q^{46} -4.41164 q^{47} -2.12788 q^{48} +0.460728 q^{49} +5.17826 q^{50} +15.5189 q^{51} +2.89555 q^{52} -4.81937 q^{53} +3.13251 q^{54} -12.3763 q^{55} +2.73143 q^{56} -3.50234 q^{57} -4.38749 q^{58} -6.93691 q^{59} -6.78865 q^{60} -3.26404 q^{61} -1.88667 q^{62} +4.17328 q^{63} +1.00000 q^{64} +9.23780 q^{65} +8.25471 q^{66} -8.16346 q^{67} -7.29312 q^{68} +11.7628 q^{69} +8.71420 q^{70} -2.72005 q^{71} +1.52787 q^{72} +2.32453 q^{73} -4.27144 q^{74} -11.0187 q^{75} +1.64593 q^{76} -10.5961 q^{77} -6.16139 q^{78} -3.74700 q^{79} +3.19034 q^{80} -11.2492 q^{81} +3.86135 q^{82} -11.6500 q^{83} -5.81216 q^{84} -23.2675 q^{85} -6.44048 q^{86} +9.33604 q^{87} -3.87931 q^{88} +3.65537 q^{89} +4.87442 q^{90} +7.90901 q^{91} -5.52793 q^{92} +4.01460 q^{93} -4.41164 q^{94} +5.25108 q^{95} -2.12788 q^{96} -13.0759 q^{97} +0.460728 q^{98} -5.92709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * q^99

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