LMFDB - Embedded newform 8038.2.a.d.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.65852 q^{3} +1.00000 q^{4} +2.22462 q^{5} -2.65852 q^{6} +0.870094 q^{7} +1.00000 q^{8} +4.06774 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.65852 q^{3} +1.00000 q^{4} +2.22462 q^{5} -2.65852 q^{6} +0.870094 q^{7} +1.00000 q^{8} +4.06774 q^{9} +2.22462 q^{10} +1.52352 q^{11} -2.65852 q^{12} -1.27500 q^{13} +0.870094 q^{14} -5.91421 q^{15} +1.00000 q^{16} -4.63943 q^{17} +4.06774 q^{18} -7.12630 q^{19} +2.22462 q^{20} -2.31316 q^{21} +1.52352 q^{22} -3.72611 q^{23} -2.65852 q^{24} -0.0510459 q^{25} -1.27500 q^{26} -2.83860 q^{27} +0.870094 q^{28} -1.83349 q^{29} -5.91421 q^{30} +4.69212 q^{31} +1.00000 q^{32} -4.05031 q^{33} -4.63943 q^{34} +1.93563 q^{35} +4.06774 q^{36} +1.19120 q^{37} -7.12630 q^{38} +3.38962 q^{39} +2.22462 q^{40} +6.84391 q^{41} -2.31316 q^{42} +6.92422 q^{43} +1.52352 q^{44} +9.04919 q^{45} -3.72611 q^{46} +7.35612 q^{47} -2.65852 q^{48} -6.24294 q^{49} -0.0510459 q^{50} +12.3340 q^{51} -1.27500 q^{52} -3.33143 q^{53} -2.83860 q^{54} +3.38926 q^{55} +0.870094 q^{56} +18.9454 q^{57} -1.83349 q^{58} +4.38937 q^{59} -5.91421 q^{60} +5.59404 q^{61} +4.69212 q^{62} +3.53931 q^{63} +1.00000 q^{64} -2.83640 q^{65} -4.05031 q^{66} -7.96345 q^{67} -4.63943 q^{68} +9.90596 q^{69} +1.93563 q^{70} +7.81773 q^{71} +4.06774 q^{72} +13.0039 q^{73} +1.19120 q^{74} +0.135707 q^{75} -7.12630 q^{76} +1.32561 q^{77} +3.38962 q^{78} +4.85419 q^{79} +2.22462 q^{80} -4.65672 q^{81} +6.84391 q^{82} +11.6230 q^{83} -2.31316 q^{84} -10.3210 q^{85} +6.92422 q^{86} +4.87437 q^{87} +1.52352 q^{88} -11.1659 q^{89} +9.04919 q^{90} -1.10937 q^{91} -3.72611 q^{92} -12.4741 q^{93} +7.35612 q^{94} -15.8533 q^{95} -2.65852 q^{96} +11.0585 q^{97} -6.24294 q^{98} +6.19728 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) $ 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * 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
$3$	−2.65852	−1.53490	−0.767449	−	0.641110i	$-0.778474\pi$
$3$	−2.65852	−1.53490	−0.767449	+	0.641110i	$0.778474\pi$
$4$	1.00000	0.500000
$5$	2.22462	0.994882	0.497441	−	0.867498i	$-0.334273\pi$
$5$	2.22462	0.994882	0.497441	+	0.867498i	$0.334273\pi$
$6$	−2.65852	−1.08534
$7$	0.870094	0.328864	0.164432	−	0.986388i	$-0.447421\pi$
$7$	0.870094	0.328864	0.164432	+	0.986388i	$0.447421\pi$
$8$	1.00000	0.353553
$9$	4.06774	1.35591
$10$	2.22462	0.703488
$11$	1.52352	0.459359	0.229679	−	0.973266i	$-0.426232\pi$
$11$	1.52352	0.459359	0.229679	+	0.973266i	$0.426232\pi$
$12$	−2.65852	−0.767449
$13$	−1.27500	−0.353622	−0.176811	−	0.984245i	$-0.556578\pi$
$13$	−1.27500	−0.353622	−0.176811	+	0.984245i	$0.556578\pi$
$14$	0.870094	0.232542
$15$	−5.91421	−1.52704
$16$	1.00000	0.250000
$17$	−4.63943	−1.12523	−0.562614	−	0.826720i	$-0.690204\pi$
$17$	−4.63943	−1.12523	−0.562614	+	0.826720i	$0.690204\pi$
$18$	4.06774	0.958775
$19$	−7.12630	−1.63488	−0.817442	−	0.576010i	$-0.804609\pi$
$19$	−7.12630	−1.63488	−0.817442	+	0.576010i	$0.804609\pi$
$20$	2.22462	0.497441
$21$	−2.31316	−0.504773
$22$	1.52352	0.324816
$23$	−3.72611	−0.776949	−0.388474	−	0.921460i	$-0.626998\pi$
$23$	−3.72611	−0.776949	−0.388474	+	0.921460i	$0.626998\pi$
$24$	−2.65852	−0.542668
$25$	−0.0510459	−0.0102092
$26$	−1.27500	−0.250049
$27$	−2.83860	−0.546290
$28$	0.870094	0.164432
$29$	−1.83349	−0.340470	−0.170235	−	0.985403i	$-0.554453\pi$
$29$	−1.83349	−0.340470	−0.170235	+	0.985403i	$0.554453\pi$
$30$	−5.91421	−1.07978
$31$	4.69212	0.842730	0.421365	−	0.906891i	$-0.361551\pi$
$31$	4.69212	0.842730	0.421365	+	0.906891i	$0.361551\pi$
$32$	1.00000	0.176777
$33$	−4.05031	−0.705069
$34$	−4.63943	−0.795656
$35$	1.93563	0.327181
$36$	4.06774	0.677956
$37$	1.19120	0.195832	0.0979161	−	0.995195i	$-0.468782\pi$
$37$	1.19120	0.195832	0.0979161	+	0.995195i	$0.468782\pi$
$38$	−7.12630	−1.15604
$39$	3.38962	0.542774
$40$	2.22462	0.351744
$41$	6.84391	1.06884	0.534420	−	0.845219i	$-0.320530\pi$
$41$	6.84391	1.06884	0.534420	+	0.845219i	$0.320530\pi$
$42$	−2.31316	−0.356929
$43$	6.92422	1.05593	0.527967	−	0.849265i	$-0.322955\pi$
$43$	6.92422	1.05593	0.527967	+	0.849265i	$0.322955\pi$
$44$	1.52352	0.229679
$45$	9.04919	1.34897
$46$	−3.72611	−0.549386
$47$	7.35612	1.07300	0.536501	−	0.843900i	$-0.319746\pi$
$47$	7.35612	1.07300	0.536501	+	0.843900i	$0.319746\pi$
$48$	−2.65852	−0.383725
$49$	−6.24294	−0.891848
$50$	−0.0510459	−0.00721898
$51$	12.3340	1.72711
$52$	−1.27500	−0.176811
$53$	−3.33143	−0.457607	−0.228803	−	0.973473i	$-0.573481\pi$
$53$	−3.33143	−0.457607	−0.228803	+	0.973473i	$0.573481\pi$
$54$	−2.83860	−0.386285
$55$	3.38926	0.457008
$56$	0.870094	0.116271
$57$	18.9454	2.50938
$58$	−1.83349	−0.240749
$59$	4.38937	0.571447	0.285724	−	0.958312i	$-0.407766\pi$
$59$	4.38937	0.571447	0.285724	+	0.958312i	$0.407766\pi$
$60$	−5.91421	−0.763522
$61$	5.59404	0.716243	0.358121	−	0.933675i	$-0.383417\pi$
$61$	5.59404	0.716243	0.358121	+	0.933675i	$0.383417\pi$
$62$	4.69212	0.595900
$63$	3.53931	0.445911
$64$	1.00000	0.125000
$65$	−2.83640	−0.351812
$66$	−4.05031	−0.498559
$67$	−7.96345	−0.972890	−0.486445	−	0.873711i	$-0.661707\pi$
$67$	−7.96345	−0.972890	−0.486445	+	0.873711i	$0.661707\pi$
$68$	−4.63943	−0.562614
$69$	9.90596	1.19254
$70$	1.93563	0.231352
$71$	7.81773	0.927794	0.463897	−	0.885889i	$-0.346451\pi$
$71$	7.81773	0.927794	0.463897	+	0.885889i	$0.346451\pi$
$72$	4.06774	0.479387
$73$	13.0039	1.52199	0.760993	−	0.648760i	$-0.224712\pi$
$73$	13.0039	1.52199	0.760993	+	0.648760i	$0.224712\pi$
$74$	1.19120	0.138474
$75$	0.135707	0.0156700
$76$	−7.12630	−0.817442
$77$	1.32561	0.151067
$78$	3.38962	0.383799
$79$	4.85419	0.546139	0.273069	−	0.961994i	$-0.411961\pi$
$79$	4.85419	0.546139	0.273069	+	0.961994i	$0.411961\pi$
$80$	2.22462	0.248721
$81$	−4.65672	−0.517414
$82$	6.84391	0.755784
$83$	11.6230	1.27579	0.637897	−	0.770121i	$-0.279804\pi$
$83$	11.6230	1.27579	0.637897	+	0.770121i	$0.279804\pi$
$84$	−2.31316	−0.252387
$85$	−10.3210	−1.11947
$86$	6.92422	0.746658
$87$	4.87437	0.522587
$88$	1.52352	0.162408
$89$	−11.1659	−1.18359	−0.591794	−	0.806089i	$-0.701580\pi$
$89$	−11.1659	−1.18359	−0.591794	+	0.806089i	$0.701580\pi$
$90$	9.04919	0.953868
$91$	−1.10937	−0.116294
$92$	−3.72611	−0.388474
$93$	−12.4741	−1.29351
$94$	7.35612	0.758726
$95$	−15.8533	−1.62652
$96$	−2.65852	−0.271334
$97$	11.0585	1.12282	0.561410	−	0.827538i	$-0.310259\pi$
$97$	11.0585	1.12282	0.561410	+	0.827538i	$0.310259\pi$
$98$	−6.24294	−0.630632
$99$	6.19728	0.622850
$100$	−0.0510459	−0.00510459
$101$	13.2767	1.32108	0.660539	−	0.750791i	$-0.270328\pi$
$101$	13.2767	1.32108	0.660539	+	0.750791i	$0.270328\pi$
$102$	12.3340	1.22125
$103$	9.97564	0.982929	0.491464	−	0.870898i	$-0.336462\pi$
$103$	9.97564	0.982929	0.491464	+	0.870898i	$0.336462\pi$
$104$	−1.27500	−0.125024
$105$	−5.14592	−0.502190
$106$	−3.33143	−0.323577
$107$	19.4027	1.87573	0.937865	−	0.347001i	$-0.112800\pi$
$107$	19.4027	1.87573	0.937865	+	0.347001i	$0.112800\pi$
$108$	−2.83860	−0.273145
$109$	11.9618	1.14573	0.572866	−	0.819649i	$-0.305831\pi$
$109$	11.9618	1.14573	0.572866	+	0.819649i	$0.305831\pi$
$110$	3.38926	0.323153
$111$	−3.16683	−0.300582
$112$	0.870094	0.0822161
$113$	−6.56192	−0.617293	−0.308647	−	0.951177i	$-0.599876\pi$
$113$	−6.56192	−0.617293	−0.308647	+	0.951177i	$0.599876\pi$
$114$	18.9454	1.77440
$115$	−8.28920	−0.772972
$116$	−1.83349	−0.170235
$117$	−5.18638	−0.479481
$118$	4.38937	0.404074
$119$	−4.03674	−0.370047
$120$	−5.91421	−0.539891
$121$	−8.67889	−0.788990
$122$	5.59404	0.506460
$123$	−18.1947	−1.64056
$124$	4.69212	0.421365
$125$	−11.2367	−1.00504
$126$	3.53931	0.315307
$127$	8.76658	0.777908	0.388954	−	0.921257i	$-0.372836\pi$
$127$	8.76658	0.777908	0.388954	+	0.921257i	$0.372836\pi$
$128$	1.00000	0.0883883
$129$	−18.4082	−1.62075
$130$	−2.83640	−0.248769
$131$	−3.78337	−0.330555	−0.165277	−	0.986247i	$-0.552852\pi$
$131$	−3.78337	−0.330555	−0.165277	+	0.986247i	$0.552852\pi$
$132$	−4.05031	−0.352534
$133$	−6.20055	−0.537655
$134$	−7.96345	−0.687937
$135$	−6.31483	−0.543494
$136$	−4.63943	−0.397828
$137$	2.58059	0.220475	0.110237	−	0.993905i	$-0.464839\pi$
$137$	2.58059	0.220475	0.110237	+	0.993905i	$0.464839\pi$
$138$	9.90596	0.843251
$139$	−7.73441	−0.656024	−0.328012	−	0.944674i	$-0.606379\pi$
$139$	−7.73441	−0.656024	−0.328012	+	0.944674i	$0.606379\pi$
$140$	1.93563	0.163591
$141$	−19.5564	−1.64695
$142$	7.81773	0.656049
$143$	−1.94249	−0.162439
$144$	4.06774	0.338978
$145$	−4.07882	−0.338728
$146$	13.0039	1.07621
$147$	16.5970	1.36890
$148$	1.19120	0.0979161
$149$	10.2539	0.840028	0.420014	−	0.907518i	$-0.362025\pi$
$149$	10.2539	0.840028	0.420014	+	0.907518i	$0.362025\pi$
$150$	0.135707	0.0110804
$151$	20.7292	1.68692	0.843458	−	0.537196i	$-0.180516\pi$
$151$	20.7292	1.68692	0.843458	+	0.537196i	$0.180516\pi$
$152$	−7.12630	−0.578019
$153$	−18.8720	−1.52571
$154$	1.32561	0.106820
$155$	10.4382	0.838417
$156$	3.38962	0.271387
$157$	−1.90331	−0.151900	−0.0759502	−	0.997112i	$-0.524199\pi$
$157$	−1.90331	−0.151900	−0.0759502	+	0.997112i	$0.524199\pi$
$158$	4.85419	0.386178
$159$	8.85667	0.702380
$160$	2.22462	0.175872
$161$	−3.24207	−0.255511
$162$	−4.65672	−0.365867
$163$	3.32330	0.260301	0.130151	−	0.991494i	$-0.458454\pi$
$163$	3.32330	0.260301	0.130151	+	0.991494i	$0.458454\pi$
$164$	6.84391	0.534420
$165$	−9.01042	−0.701460
$166$	11.6230	0.902123
$167$	−1.65220	−0.127851	−0.0639255	−	0.997955i	$-0.520362\pi$
$167$	−1.65220	−0.127851	−0.0639255	+	0.997955i	$0.520362\pi$
$168$	−2.31316	−0.178464
$169$	−11.3744	−0.874951
$170$	−10.3210	−0.791584
$171$	−28.9879	−2.21676
$172$	6.92422	0.527967
$173$	−16.6726	−1.26759	−0.633796	−	0.773501i	$-0.718504\pi$
$173$	−16.6726	−1.26759	−0.633796	+	0.773501i	$0.718504\pi$
$174$	4.87437	0.369525
$175$	−0.0444147	−0.00335743
$176$	1.52352	0.114840
$177$	−11.6692	−0.877113
$178$	−11.1659	−0.836923
$179$	7.47895	0.559003	0.279501	−	0.960145i	$-0.409831\pi$
$179$	7.47895	0.559003	0.279501	+	0.960145i	$0.409831\pi$
$180$	9.04919	0.674487
$181$	−6.27028	−0.466067	−0.233033	−	0.972469i	$-0.574865\pi$
$181$	−6.27028	−0.466067	−0.233033	+	0.972469i	$0.574865\pi$
$182$	−1.10937	−0.0822321
$183$	−14.8719	−1.09936
$184$	−3.72611	−0.274693
$185$	2.64997	0.194830
$186$	−12.4741	−0.914646
$187$	−7.06827	−0.516883
$188$	7.35612	0.536501
$189$	−2.46985	−0.179655
$190$	−15.8533	−1.15012
$191$	−14.0567	−1.01710	−0.508552	−	0.861031i	$-0.669819\pi$
$191$	−14.0567	−1.01710	−0.508552	+	0.861031i	$0.669819\pi$
$192$	−2.65852	−0.191862
$193$	−12.3222	−0.886972	−0.443486	−	0.896281i	$-0.646258\pi$
$193$	−12.3222	−0.886972	−0.443486	+	0.896281i	$0.646258\pi$
$194$	11.0585	0.793953
$195$	7.54064	0.539996
$196$	−6.24294	−0.445924
$197$	6.98582	0.497719	0.248860	−	0.968540i	$-0.419944\pi$
$197$	6.98582	0.497719	0.248860	+	0.968540i	$0.419944\pi$
$198$	6.19728	0.440422
$199$	−15.2115	−1.07832	−0.539159	−	0.842204i	$-0.681258\pi$
$199$	−15.2115	−1.07832	−0.539159	+	0.842204i	$0.681258\pi$
$200$	−0.0510459	−0.00360949
$201$	21.1710	1.49329
$202$	13.2767	0.934144
$203$	−1.59531	−0.111969
$204$	12.3340	0.863555
$205$	15.2251	1.06337
$206$	9.97564	0.695036
$207$	−15.1569	−1.05347
$208$	−1.27500	−0.0884055
$209$	−10.8571	−0.750998
$210$	−5.14592	−0.355102
$211$	11.4467	0.788025	0.394012	−	0.919105i	$-0.371087\pi$
$211$	11.4467	0.788025	0.394012	+	0.919105i	$0.371087\pi$
$212$	−3.33143	−0.228803
$213$	−20.7836	−1.42407
$214$	19.4027	1.32634
$215$	15.4038	1.05053
$216$	−2.83860	−0.193143
$217$	4.08259	0.277144
$218$	11.9618	0.810155
$219$	−34.5710	−2.33609
$220$	3.38926	0.228504
$221$	5.91529	0.397905
$222$	−3.16683	−0.212544
$223$	16.3920	1.09769	0.548846	−	0.835923i	$-0.315067\pi$
$223$	16.3920	1.09769	0.548846	+	0.835923i	$0.315067\pi$
$224$	0.870094	0.0581356
$225$	−0.207641	−0.0138427
$226$	−6.56192	−0.436492
$227$	−27.6738	−1.83678	−0.918389	−	0.395679i	$-0.870509\pi$
$227$	−27.6738	−1.83678	−0.918389	+	0.395679i	$0.870509\pi$
$228$	18.9454	1.25469
$229$	24.1402	1.59523	0.797614	−	0.603168i	$-0.206095\pi$
$229$	24.1402	1.59523	0.797614	+	0.603168i	$0.206095\pi$
$230$	−8.28920	−0.546574
$231$	−3.52415	−0.231872
$232$	−1.83349	−0.120374
$233$	27.7496	1.81794	0.908968	−	0.416867i	$-0.136872\pi$
$233$	27.7496	1.81794	0.908968	+	0.416867i	$0.136872\pi$
$234$	−5.18638	−0.339044
$235$	16.3646	1.06751
$236$	4.38937	0.285724
$237$	−12.9050	−0.838267
$238$	−4.03674	−0.261663
$239$	−29.5337	−1.91038	−0.955188	−	0.296001i	$-0.904347\pi$
$239$	−29.5337	−1.91038	−0.955188	+	0.296001i	$0.904347\pi$
$240$	−5.91421	−0.381761
$241$	−8.07504	−0.520159	−0.260080	−	0.965587i	$-0.583749\pi$
$241$	−8.07504	−0.520159	−0.260080	+	0.965587i	$0.583749\pi$
$242$	−8.67889	−0.557900
$243$	20.8958	1.34047
$244$	5.59404	0.358121
$245$	−13.8882	−0.887284
$246$	−18.1947	−1.16005
$247$	9.08605	0.578131
$248$	4.69212	0.297950
$249$	−30.9001	−1.95821
$250$	−11.2367	−0.710670
$251$	13.8049	0.871356	0.435678	−	0.900103i	$-0.356509\pi$
$251$	13.8049	0.871356	0.435678	+	0.900103i	$0.356509\pi$
$252$	3.53931	0.222956
$253$	−5.67681	−0.356898
$254$	8.76658	0.550064
$255$	27.4386	1.71827
$256$	1.00000	0.0625000
$257$	28.4389	1.77397	0.886986	−	0.461796i	$-0.152795\pi$
$257$	28.4389	1.77397	0.886986	+	0.461796i	$0.152795\pi$
$258$	−18.4082	−1.14604
$259$	1.03646	0.0644022
$260$	−2.83640	−0.175906
$261$	−7.45815	−0.461648
$262$	−3.78337	−0.233738
$263$	26.3718	1.62615	0.813077	−	0.582157i	$-0.197791\pi$
$263$	26.3718	1.62615	0.813077	+	0.582157i	$0.197791\pi$
$264$	−4.05031	−0.249279
$265$	−7.41118	−0.455265
$266$	−6.20055	−0.380180
$267$	29.6849	1.81669
$268$	−7.96345	−0.486445
$269$	6.12815	0.373640	0.186820	−	0.982394i	$-0.440182\pi$
$269$	6.12815	0.373640	0.186820	+	0.982394i	$0.440182\pi$
$270$	−6.31483	−0.384308
$271$	29.4461	1.78872	0.894362	−	0.447345i	$-0.147630\pi$
$271$	29.4461	1.78872	0.894362	+	0.447345i	$0.147630\pi$
$272$	−4.63943	−0.281307
$273$	2.94929	0.178499
$274$	2.58059	0.155899
$275$	−0.0777694	−0.00468967
$276$	9.90596	0.596268
$277$	−10.7792	−0.647659	−0.323829	−	0.946115i	$-0.604970\pi$
$277$	−10.7792	−0.647659	−0.323829	+	0.946115i	$0.604970\pi$
$278$	−7.73441	−0.463879
$279$	19.0863	1.14267
$280$	1.93563	0.115676
$281$	−0.549010	−0.0327512	−0.0163756	−	0.999866i	$-0.505213\pi$
$281$	−0.549010	−0.0327512	−0.0163756	+	0.999866i	$0.505213\pi$
$282$	−19.5564	−1.16457
$283$	−5.91659	−0.351705	−0.175852	−	0.984417i	$-0.556268\pi$
$283$	−5.91659	−0.351705	−0.175852	+	0.984417i	$0.556268\pi$
$284$	7.81773	0.463897
$285$	42.1464	2.49654
$286$	−1.94249	−0.114862
$287$	5.95485	0.351503
$288$	4.06774	0.239694
$289$	4.52432	0.266137
$290$	−4.07882	−0.239517
$291$	−29.3992	−1.72341
$292$	13.0039	0.760993
$293$	−21.4970	−1.25587	−0.627935	−	0.778266i	$-0.716100\pi$
$293$	−21.4970	−1.25587	−0.627935	+	0.778266i	$0.716100\pi$
$294$	16.5970	0.967956
$295$	9.76470	0.568523
$296$	1.19120	0.0692371
$297$	−4.32467	−0.250943
$298$	10.2539	0.593990
$299$	4.75081	0.274746
$300$	0.135707	0.00783502
$301$	6.02472	0.347259
$302$	20.7292	1.19283
$303$	−35.2963	−2.02772
$304$	−7.12630	−0.408721
$305$	12.4446	0.712577
$306$	−18.8720	−1.07884
$307$	7.06554	0.403251	0.201626	−	0.979463i	$-0.435378\pi$
$307$	7.06554	0.403251	0.201626	+	0.979463i	$0.435378\pi$
$308$	1.32561	0.0755334
$309$	−26.5205	−1.50870
$310$	10.4382	0.592851
$311$	28.5179	1.61710	0.808551	−	0.588426i	$-0.200252\pi$
$311$	28.5179	1.61710	0.808551	+	0.588426i	$0.200252\pi$
$312$	3.38962	0.191900
$313$	−16.6228	−0.939576	−0.469788	−	0.882779i	$-0.655670\pi$
$313$	−16.6228	−0.939576	−0.469788	+	0.882779i	$0.655670\pi$
$314$	−1.90331	−0.107410
$315$	7.87364	0.443629
$316$	4.85419	0.273069
$317$	−15.6245	−0.877560	−0.438780	−	0.898595i	$-0.644589\pi$
$317$	−15.6245	−0.877560	−0.438780	+	0.898595i	$0.644589\pi$
$318$	8.85667	0.496658
$319$	−2.79336	−0.156398
$320$	2.22462	0.124360
$321$	−51.5825	−2.87905
$322$	−3.24207	−0.180673
$323$	33.0620	1.83962
$324$	−4.65672	−0.258707
$325$	0.0650836	0.00361019
$326$	3.32330	0.184061
$327$	−31.8007	−1.75858
$328$	6.84391	0.377892
$329$	6.40052	0.352872
$330$	−9.01042	−0.496007
$331$	−8.16959	−0.449041	−0.224521	−	0.974469i	$-0.572082\pi$
$331$	−8.16959	−0.449041	−0.224521	+	0.974469i	$0.572082\pi$
$332$	11.6230	0.637897
$333$	4.84549	0.265531
$334$	−1.65220	−0.0904043
$335$	−17.7157	−0.967911
$336$	−2.31316	−0.126193
$337$	−2.10469	−0.114650	−0.0573250	−	0.998356i	$-0.518257\pi$
$337$	−2.10469	−0.114650	−0.0573250	+	0.998356i	$0.518257\pi$
$338$	−11.3744	−0.618684
$339$	17.4450	0.947482
$340$	−10.3210	−0.559734
$341$	7.14855	0.387115
$342$	−28.9879	−1.56749
$343$	−11.5226	−0.622162
$344$	6.92422	0.373329
$345$	22.0370	1.18643
$346$	−16.6726	−0.896322
$347$	−2.83299	−0.152083	−0.0760413	−	0.997105i	$-0.524228\pi$
$347$	−2.83299	−0.152083	−0.0760413	+	0.997105i	$0.524228\pi$
$348$	4.87437	0.261293
$349$	2.01450	0.107834	0.0539168	−	0.998545i	$-0.482829\pi$
$349$	2.01450	0.107834	0.0539168	+	0.998545i	$0.482829\pi$
$350$	−0.0444147	−0.00237406
$351$	3.61923	0.193180
$352$	1.52352	0.0812039
$353$	33.3713	1.77618	0.888089	−	0.459672i	$-0.152033\pi$
$353$	33.3713	1.77618	0.888089	+	0.459672i	$0.152033\pi$
$354$	−11.6692	−0.620213
$355$	17.3915	0.923046
$356$	−11.1659	−0.591794
$357$	10.7318	0.567985
$358$	7.47895	0.395275
$359$	5.96357	0.314745	0.157372	−	0.987539i	$-0.449698\pi$
$359$	5.96357	0.314745	0.157372	+	0.987539i	$0.449698\pi$
$360$	9.04919	0.476934
$361$	31.7841	1.67285
$362$	−6.27028	−0.329559
$363$	23.0730	1.21102
$364$	−1.10937	−0.0581469
$365$	28.9287	1.51420
$366$	−14.8719	−0.777365
$367$	19.4642	1.01602	0.508011	−	0.861351i	$-0.330381\pi$
$367$	19.4642	1.01602	0.508011	+	0.861351i	$0.330381\pi$
$368$	−3.72611	−0.194237
$369$	27.8392	1.44925
$370$	2.64997	0.137766
$371$	−2.89865	−0.150491
$372$	−12.4741	−0.646753
$373$	4.31264	0.223300	0.111650	−	0.993748i	$-0.464386\pi$
$373$	4.31264	0.223300	0.111650	+	0.993748i	$0.464386\pi$
$374$	−7.06827	−0.365491
$375$	29.8730	1.54263
$376$	7.35612	0.379363
$377$	2.33770	0.120398
$378$	−2.46985	−0.127035
$379$	−5.60094	−0.287701	−0.143851	−	0.989599i	$-0.545948\pi$
$379$	−5.60094	−0.287701	−0.143851	+	0.989599i	$0.545948\pi$
$380$	−15.8533	−0.813259
$381$	−23.3062	−1.19401
$382$	−14.0567	−0.719201
$383$	0.178244	0.00910785	0.00455392	−	0.999990i	$-0.498550\pi$
$383$	0.178244	0.00910785	0.00455392	+	0.999990i	$0.498550\pi$
$384$	−2.65852	−0.135667
$385$	2.94897	0.150294
$386$	−12.3222	−0.627184
$387$	28.1659	1.43175
$388$	11.0585	0.561410
$389$	17.6836	0.896592	0.448296	−	0.893885i	$-0.352031\pi$
$389$	17.6836	0.896592	0.448296	+	0.893885i	$0.352031\pi$
$390$	7.54064	0.381835
$391$	17.2871	0.874244
$392$	−6.24294	−0.315316
$393$	10.0582	0.507368
$394$	6.98582	0.351941
$395$	10.7987	0.543344
$396$	6.19728	0.311425
$397$	2.90436	0.145766	0.0728829	−	0.997341i	$-0.476780\pi$
$397$	2.90436	0.145766	0.0728829	+	0.997341i	$0.476780\pi$
$398$	−15.2115	−0.762486
$399$	16.4843	0.825246
$400$	−0.0510459	−0.00255229
$401$	−25.7011	−1.28345	−0.641725	−	0.766935i	$-0.721781\pi$
$401$	−25.7011	−1.28345	−0.641725	+	0.766935i	$0.721781\pi$
$402$	21.1710	1.05591
$403$	−5.98247	−0.298008
$404$	13.2767	0.660539
$405$	−10.3595	−0.514766
$406$	−1.59531	−0.0791737
$407$	1.81482	0.0899572
$408$	12.3340	0.610625
$409$	15.8305	0.782768	0.391384	−	0.920227i	$-0.371996\pi$
$409$	15.8305	0.782768	0.391384	+	0.920227i	$0.371996\pi$
$410$	15.2251	0.751916
$411$	−6.86056	−0.338406
$412$	9.97564	0.491464
$413$	3.81916	0.187929
$414$	−15.1569	−0.744919
$415$	25.8569	1.26927
$416$	−1.27500	−0.0625122
$417$	20.5621	1.00693
$418$	−10.8571	−0.531036
$419$	19.2388	0.939875	0.469937	−	0.882700i	$-0.344277\pi$
$419$	19.2388	0.939875	0.469937	+	0.882700i	$0.344277\pi$
$420$	−5.14592	−0.251095
$421$	−23.0548	−1.12362	−0.561811	−	0.827265i	$-0.689895\pi$
$421$	−23.0548	−1.12362	−0.561811	+	0.827265i	$0.689895\pi$
$422$	11.4467	0.557218
$423$	29.9228	1.45490
$424$	−3.33143	−0.161788
$425$	0.236824	0.0114876
$426$	−20.7836	−1.00697
$427$	4.86733	0.235547
$428$	19.4027	0.937865
$429$	5.16416	0.249328
$430$	15.4038	0.742836
$431$	−24.2833	−1.16969	−0.584843	−	0.811147i	$-0.698844\pi$
$431$	−24.2833	−1.16969	−0.584843	+	0.811147i	$0.698844\pi$
$432$	−2.83860	−0.136572
$433$	33.1330	1.59227	0.796135	−	0.605119i	$-0.206875\pi$
$433$	33.1330	1.59227	0.796135	+	0.605119i	$0.206875\pi$
$434$	4.08259	0.195970
$435$	10.8436	0.519912
$436$	11.9618	0.572866
$437$	26.5534	1.27022
$438$	−34.5710	−1.65187
$439$	9.73331	0.464546	0.232273	−	0.972651i	$-0.425384\pi$
$439$	9.73331	0.464546	0.232273	+	0.972651i	$0.425384\pi$
$440$	3.38926	0.161577
$441$	−25.3946	−1.20927
$442$	5.91529	0.281362
$443$	−21.6988	−1.03094	−0.515470	−	0.856907i	$-0.672383\pi$
$443$	−21.6988	−1.03094	−0.515470	+	0.856907i	$0.672383\pi$
$444$	−3.16683	−0.150291
$445$	−24.8400	−1.17753
$446$	16.3920	0.776186
$447$	−27.2601	−1.28936
$448$	0.870094	0.0411081
$449$	−30.3765	−1.43355	−0.716777	−	0.697302i	$-0.754384\pi$
$449$	−30.3765	−1.43355	−0.716777	+	0.697302i	$0.754384\pi$
$450$	−0.207641	−0.00978830
$451$	10.4268	0.490981
$452$	−6.56192	−0.308647
$453$	−55.1089	−2.58924
$454$	−27.6738	−1.29880
$455$	−2.46794	−0.115699
$456$	18.9454	0.887200
$457$	−36.9498	−1.72844	−0.864220	−	0.503113i	$-0.832188\pi$
$457$	−36.9498	−1.72844	−0.864220	+	0.503113i	$0.832188\pi$
$458$	24.1402	1.12800
$459$	13.1695	0.614700
$460$	−8.28920	−0.386486
$461$	−36.5898	−1.70416	−0.852079	−	0.523413i	$-0.824659\pi$
$461$	−36.5898	−1.70416	−0.852079	+	0.523413i	$0.824659\pi$
$462$	−3.52415	−0.163958
$463$	14.6817	0.682317	0.341159	−	0.940006i	$-0.389181\pi$
$463$	14.6817	0.682317	0.341159	+	0.940006i	$0.389181\pi$
$464$	−1.83349	−0.0851175
$465$	−27.7502	−1.28689
$466$	27.7496	1.28547
$467$	−18.3108	−0.847321	−0.423660	−	0.905821i	$-0.639255\pi$
$467$	−18.3108	−0.847321	−0.423660	+	0.905821i	$0.639255\pi$
$468$	−5.18638	−0.239740
$469$	−6.92895	−0.319949
$470$	16.3646	0.754843
$471$	5.05998	0.233152
$472$	4.38937	0.202037
$473$	10.5492	0.485052
$474$	−12.9050	−0.592745
$475$	0.363768	0.0166908
$476$	−4.03674	−0.185024
$477$	−13.5514	−0.620475
$478$	−29.5337	−1.35084
$479$	24.4839	1.11870	0.559349	−	0.828932i	$-0.311051\pi$
$479$	24.4839	1.11870	0.559349	+	0.828932i	$0.311051\pi$
$480$	−5.91421	−0.269946
$481$	−1.51878	−0.0692506
$482$	−8.07504	−0.367808
$483$	8.61911	0.392183
$484$	−8.67889	−0.394495
$485$	24.6010	1.11707
$486$	20.8958	0.947853
$487$	8.64458	0.391723	0.195862	−	0.980632i	$-0.437250\pi$
$487$	8.64458	0.391723	0.195862	+	0.980632i	$0.437250\pi$
$488$	5.59404	0.253230
$489$	−8.83507	−0.399536
$490$	−13.8882	−0.627405
$491$	36.2805	1.63732	0.818658	−	0.574281i	$-0.194718\pi$
$491$	36.2805	1.63732	0.818658	+	0.574281i	$0.194718\pi$
$492$	−18.1947	−0.820280
$493$	8.50634	0.383106
$494$	9.08605	0.408801
$495$	13.7866	0.619663
$496$	4.69212	0.210683
$497$	6.80216	0.305118
$498$	−30.9001	−1.38467
$499$	−19.7389	−0.883636	−0.441818	−	0.897105i	$-0.645666\pi$
$499$	−19.7389	−0.883636	−0.441818	+	0.897105i	$0.645666\pi$
$500$	−11.2367	−0.502520
$501$	4.39241	0.196238
$502$	13.8049	0.616142
$503$	14.9999	0.668811	0.334405	−	0.942429i	$-0.391465\pi$
$503$	14.9999	0.668811	0.334405	+	0.942429i	$0.391465\pi$
$504$	3.53931	0.157654
$505$	29.5356	1.31432
$506$	−5.67681	−0.252365
$507$	30.2390	1.34296
$508$	8.76658	0.388954
$509$	15.8540	0.702714	0.351357	−	0.936242i	$-0.385720\pi$
$509$	15.8540	0.702714	0.351357	+	0.936242i	$0.385720\pi$
$510$	27.4386	1.21500
$511$	11.3146	0.500527
$512$	1.00000	0.0441942
$513$	20.2287	0.893120
$514$	28.4389	1.25439
$515$	22.1921	0.977899
$516$	−18.4082	−0.810375
$517$	11.2072	0.492892
$518$	1.03646	0.0455393
$519$	44.3244	1.94562
$520$	−2.83640	−0.124384
$521$	1.07631	0.0471540	0.0235770	−	0.999722i	$-0.492495\pi$
$521$	1.07631	0.0471540	0.0235770	+	0.999722i	$0.492495\pi$
$522$	−7.45815	−0.326434
$523$	9.18902	0.401808	0.200904	−	0.979611i	$-0.435612\pi$
$523$	9.18902	0.401808	0.200904	+	0.979611i	$0.435612\pi$
$524$	−3.78337	−0.165277
$525$	0.118077	0.00515332
$526$	26.3718	1.14986
$527$	−21.7688	−0.948263
$528$	−4.05031	−0.176267
$529$	−9.11607	−0.396351
$530$	−7.41118	−0.321921
$531$	17.8548	0.774832
$532$	−6.20055	−0.268828
$533$	−8.72601	−0.377965
$534$	29.6849	1.28459
$535$	43.1637	1.86613
$536$	−7.96345	−0.343969
$537$	−19.8829	−0.858012
$538$	6.12815	0.264203
$539$	−9.51124	−0.409678
$540$	−6.31483	−0.271747
$541$	−21.0256	−0.903961	−0.451981	−	0.892028i	$-0.649282\pi$
$541$	−21.0256	−0.903961	−0.451981	+	0.892028i	$0.649282\pi$
$542$	29.4461	1.26482
$543$	16.6697	0.715365
$544$	−4.63943	−0.198914
$545$	26.6105	1.13987
$546$	2.94929	0.126218
$547$	37.0720	1.58508	0.792542	−	0.609817i	$-0.208757\pi$
$547$	37.0720	1.58508	0.792542	+	0.609817i	$0.208757\pi$
$548$	2.58059	0.110237
$549$	22.7551	0.971163
$550$	−0.0777694	−0.00331610
$551$	13.0660	0.556629
$552$	9.90596	0.421625
$553$	4.22360	0.179606
$554$	−10.7792	−0.457964
$555$	−7.04501	−0.299044
$556$	−7.73441	−0.328012
$557$	6.14548	0.260393	0.130196	−	0.991488i	$-0.458439\pi$
$557$	6.14548	0.260393	0.130196	+	0.991488i	$0.458439\pi$
$558$	19.0863	0.807989
$559$	−8.82840	−0.373401
$560$	1.93563	0.0817954
$561$	18.7911	0.793363
$562$	−0.549010	−0.0231586
$563$	14.3206	0.603542	0.301771	−	0.953380i	$-0.402422\pi$
$563$	14.3206	0.603542	0.301771	+	0.953380i	$0.402422\pi$
$564$	−19.5564	−0.823474
$565$	−14.5978	−0.614134
$566$	−5.91659	−0.248693
$567$	−4.05179	−0.170159
$568$	7.81773	0.328025
$569$	−37.6939	−1.58021	−0.790105	−	0.612972i	$-0.789974\pi$
$569$	−37.6939	−1.58021	−0.790105	+	0.612972i	$0.789974\pi$
$570$	42.1464	1.76532
$571$	−45.2060	−1.89181	−0.945906	−	0.324441i	$-0.894824\pi$
$571$	−45.2060	−1.89181	−0.945906	+	0.324441i	$0.894824\pi$
$572$	−1.94249	−0.0812197
$573$	37.3699	1.56115
$574$	5.95485	0.248550
$575$	0.190203	0.00793200
$576$	4.06774	0.169489
$577$	30.5019	1.26981	0.634906	−	0.772590i	$-0.281039\pi$
$577$	30.5019	1.26981	0.634906	+	0.772590i	$0.281039\pi$
$578$	4.52432	0.188187
$579$	32.7589	1.36141
$580$	−4.07882	−0.169364
$581$	10.1131	0.419563
$582$	−29.3992	−1.21864
$583$	−5.07550	−0.210206
$584$	13.0039	0.538104
$585$	−11.5377	−0.477027
$586$	−21.4970	−0.888034
$587$	0.235846	0.00973442	0.00486721	−	0.999988i	$-0.498451\pi$
$587$	0.235846	0.00973442	0.00486721	+	0.999988i	$0.498451\pi$
$588$	16.5970	0.684448
$589$	−33.4375	−1.37777
$590$	9.76470	0.402006
$591$	−18.5720	−0.763949
$592$	1.19120	0.0489580
$593$	−3.06246	−0.125760	−0.0628801	−	0.998021i	$-0.520029\pi$
$593$	−3.06246	−0.125760	−0.0628801	+	0.998021i	$0.520029\pi$
$594$	−4.32467	−0.177443
$595$	−8.98023	−0.368154
$596$	10.2539	0.420014
$597$	40.4402	1.65511
$598$	4.75081	0.194275
$599$	26.6460	1.08872	0.544362	−	0.838850i	$-0.316772\pi$
$599$	26.6460	1.08872	0.544362	+	0.838850i	$0.316772\pi$
$600$	0.135707	0.00554020
$601$	−16.9795	−0.692610	−0.346305	−	0.938122i	$-0.612564\pi$
$601$	−16.9795	−0.692610	−0.346305	+	0.938122i	$0.612564\pi$
$602$	6.02472	0.245549
$603$	−32.3932	−1.31915
$604$	20.7292	0.843458
$605$	−19.3073	−0.784952
$606$	−35.2963	−1.43382
$607$	−3.29352	−0.133680	−0.0668398	−	0.997764i	$-0.521292\pi$
$607$	−3.29352	−0.133680	−0.0668398	+	0.997764i	$0.521292\pi$
$608$	−7.12630	−0.289010
$609$	4.24115	0.171860
$610$	12.4446	0.503868
$611$	−9.37908	−0.379437
$612$	−18.8720	−0.762855
$613$	48.5191	1.95967	0.979833	−	0.199817i	$-0.0640347\pi$
$613$	48.5191	1.95967	0.979833	+	0.199817i	$0.0640347\pi$
$614$	7.06554	0.285142
$615$	−40.4764	−1.63216
$616$	1.32561	0.0534102
$617$	−1.98916	−0.0800806	−0.0400403	−	0.999198i	$-0.512749\pi$
$617$	−1.98916	−0.0800806	−0.0400403	+	0.999198i	$0.512749\pi$
$618$	−26.5205	−1.06681
$619$	−42.8263	−1.72133	−0.860666	−	0.509170i	$-0.829953\pi$
$619$	−42.8263	−1.72133	−0.860666	+	0.509170i	$0.829953\pi$
$620$	10.4382	0.419209
$621$	10.5770	0.424439
$622$	28.5179	1.14346
$623$	−9.71542	−0.389240
$624$	3.38962	0.135693
$625$	−24.7422	−0.989687
$626$	−16.6228	−0.664381
$627$	28.8637	1.15271
$628$	−1.90331	−0.0759502
$629$	−5.52649	−0.220356
$630$	7.87364	0.313693
$631$	31.7116	1.26242	0.631210	−	0.775612i	$-0.282559\pi$
$631$	31.7116	1.26242	0.631210	+	0.775612i	$0.282559\pi$
$632$	4.85419	0.193089
$633$	−30.4313	−1.20954
$634$	−15.6245	−0.620529
$635$	19.5024	0.773927
$636$	8.85667	0.351190
$637$	7.95976	0.315377
$638$	−2.79336	−0.110590
$639$	31.8005	1.25801
$640$	2.22462	0.0879360
$641$	−30.7132	−1.21310	−0.606549	−	0.795046i	$-0.707447\pi$
$641$	−30.7132	−1.21310	−0.606549	+	0.795046i	$0.707447\pi$
$642$	−51.5825	−2.03580
$643$	38.5945	1.52202	0.761009	−	0.648741i	$-0.224704\pi$
$643$	38.5945	1.52202	0.761009	+	0.648741i	$0.224704\pi$
$644$	−3.24207	−0.127755
$645$	−40.9513	−1.61246
$646$	33.0620	1.30081
$647$	−43.0666	−1.69312	−0.846561	−	0.532291i	$-0.821331\pi$
$647$	−43.0666	−1.69312	−0.846561	+	0.532291i	$0.821331\pi$
$648$	−4.65672	−0.182933
$649$	6.68729	0.262499
$650$	0.0650836	0.00255279
$651$	−10.8536	−0.425388
$652$	3.32330	0.130151
$653$	−10.9416	−0.428178	−0.214089	−	0.976814i	$-0.568678\pi$
$653$	−10.9416	−0.428178	−0.214089	+	0.976814i	$0.568678\pi$
$654$	−31.8007	−1.24351
$655$	−8.41659	−0.328863
$656$	6.84391	0.267210
$657$	52.8963	2.06368
$658$	6.40052	0.249518
$659$	−41.7018	−1.62447	−0.812237	−	0.583328i	$-0.801750\pi$
$659$	−41.7018	−1.62447	−0.812237	+	0.583328i	$0.801750\pi$
$660$	−9.01042	−0.350730
$661$	−9.33133	−0.362947	−0.181473	−	0.983396i	$-0.558087\pi$
$661$	−9.33133	−0.362947	−0.181473	+	0.983396i	$0.558087\pi$
$662$	−8.16959	−0.317520
$663$	−15.7259	−0.610744
$664$	11.6230	0.451061
$665$	−13.7939	−0.534904
$666$	4.84549	0.187759
$667$	6.83178	0.264528
$668$	−1.65220	−0.0639255
$669$	−43.5786	−1.68485
$670$	−17.7157	−0.684416
$671$	8.52263	0.329012
$672$	−2.31316	−0.0892322
$673$	32.2477	1.24306	0.621529	−	0.783391i	$-0.286512\pi$
$673$	32.2477	1.24306	0.621529	+	0.783391i	$0.286512\pi$
$674$	−2.10469	−0.0810698
$675$	0.144899	0.00557717
$676$	−11.3744	−0.437476
$677$	27.8783	1.07145	0.535726	−	0.844392i	$-0.320038\pi$
$677$	27.8783	1.07145	0.535726	+	0.844392i	$0.320038\pi$
$678$	17.4450	0.669971
$679$	9.62192	0.369255
$680$	−10.3210	−0.395792
$681$	73.5715	2.81927
$682$	7.14855	0.273732
$683$	−51.1635	−1.95772	−0.978858	−	0.204542i	$-0.934430\pi$
$683$	−51.1635	−1.95772	−0.978858	+	0.204542i	$0.934430\pi$
$684$	−28.9879	−1.10838
$685$	5.74085	0.219347
$686$	−11.5226	−0.439935
$687$	−64.1772	−2.44851
$688$	6.92422	0.263983
$689$	4.24758	0.161820
$690$	22.0370	0.838935
$691$	−7.34794	−0.279529	−0.139764	−	0.990185i	$-0.544634\pi$
$691$	−7.34794	−0.279529	−0.139764	+	0.990185i	$0.544634\pi$
$692$	−16.6726	−0.633796
$693$	5.39221	0.204833
$694$	−2.83299	−0.107539
$695$	−17.2062	−0.652667
$696$	4.87437	0.184762
$697$	−31.7519	−1.20269
$698$	2.01450	0.0762499
$699$	−73.7729	−2.79035
$700$	−0.0444147	−0.00167872
$701$	−4.65777	−0.175921	−0.0879607	−	0.996124i	$-0.528035\pi$
$701$	−4.65777	−0.175921	−0.0879607	+	0.996124i	$0.528035\pi$
$702$	3.61923	0.136599
$703$	−8.48885	−0.320163
$704$	1.52352	0.0574198
$705$	−43.5057	−1.63852
$706$	33.3713	1.25595
$707$	11.5519	0.434456
$708$	−11.6692	−0.438557
$709$	15.0437	0.564980	0.282490	−	0.959270i	$-0.408840\pi$
$709$	15.0437	0.564980	0.282490	+	0.959270i	$0.408840\pi$
$710$	17.3915	0.652692
$711$	19.7456	0.740516
$712$	−11.1659	−0.418462
$713$	−17.4834	−0.654758
$714$	10.7318	0.401626
$715$	−4.32132	−0.161608
$716$	7.47895	0.279501
$717$	78.5159	2.93223
$718$	5.96357	0.222558
$719$	−35.9848	−1.34201	−0.671003	−	0.741455i	$-0.734136\pi$
$719$	−35.9848	−1.34201	−0.671003	+	0.741455i	$0.734136\pi$
$720$	9.04919	0.337243
$721$	8.67974	0.323250
$722$	31.7841	1.18288
$723$	21.4677	0.798391
$724$	−6.27028	−0.233033
$725$	0.0935920	0.00347592
$726$	23.0730	0.856320
$727$	28.9698	1.07443	0.537215	−	0.843445i	$-0.319476\pi$
$727$	28.9698	1.07443	0.537215	+	0.843445i	$0.319476\pi$
$728$	−1.10937	−0.0411160
$729$	−41.5818	−1.54007
$730$	28.9287	1.07070
$731$	−32.1244	−1.18817
$732$	−14.8719	−0.549680
$733$	39.4241	1.45616	0.728082	−	0.685490i	$-0.240412\pi$
$733$	39.4241	1.45616	0.728082	+	0.685490i	$0.240412\pi$
$734$	19.4642	0.718436
$735$	36.9221	1.36189
$736$	−3.72611	−0.137346
$737$	−12.1325	−0.446905
$738$	27.8392	1.02478
$739$	−13.8202	−0.508385	−0.254192	−	0.967154i	$-0.581810\pi$
$739$	−13.8202	−0.508385	−0.254192	+	0.967154i	$0.581810\pi$
$740$	2.64997	0.0974150
$741$	−24.1555	−0.887373
$742$	−2.89865	−0.106413
$743$	−4.60557	−0.168962	−0.0844810	−	0.996425i	$-0.526923\pi$
$743$	−4.60557	−0.168962	−0.0844810	+	0.996425i	$0.526923\pi$
$744$	−12.4741	−0.457323
$745$	22.8110	0.835729
$746$	4.31264	0.157897
$747$	47.2795	1.72987
$748$	−7.06827	−0.258441
$749$	16.8822	0.616861
$750$	29.8730	1.09081
$751$	21.0629	0.768596	0.384298	−	0.923209i	$-0.374443\pi$
$751$	21.0629	0.768596	0.384298	+	0.923209i	$0.374443\pi$
$752$	7.35612	0.268250
$753$	−36.7006	−1.33744
$754$	2.33770	0.0851341
$755$	46.1146	1.67828
$756$	−2.46985	−0.0898276
$757$	−16.1366	−0.586493	−0.293247	−	0.956037i	$-0.594736\pi$
$757$	−16.1366	−0.586493	−0.293247	+	0.956037i	$0.594736\pi$
$758$	−5.60094	−0.203435
$759$	15.0919	0.547802
$760$	−15.8533	−0.575061
$761$	27.3587	0.991751	0.495875	−	0.868394i	$-0.334847\pi$
$761$	27.3587	0.991751	0.495875	+	0.868394i	$0.334847\pi$
$762$	−23.3062	−0.844293
$763$	10.4079	0.376791
$764$	−14.0567	−0.508552
$765$	−41.9831	−1.51790
$766$	0.178244	0.00644022
$767$	−5.59646	−0.202076
$768$	−2.65852	−0.0959311
$769$	−19.7253	−0.711313	−0.355657	−	0.934617i	$-0.615743\pi$
$769$	−19.7253	−0.711313	−0.355657	+	0.934617i	$0.615743\pi$
$770$	2.94897	0.106274
$771$	−75.6055	−2.72287
$772$	−12.3222	−0.443486
$773$	−16.9467	−0.609530	−0.304765	−	0.952428i	$-0.598578\pi$
$773$	−16.9467	−0.609530	−0.304765	+	0.952428i	$0.598578\pi$
$774$	28.1659	1.01240
$775$	−0.239514	−0.00860358
$776$	11.0585	0.396977
$777$	−2.75544	−0.0988509
$778$	17.6836	0.633986
$779$	−48.7718	−1.74743
$780$	7.54064	0.269998
$781$	11.9105	0.426190
$782$	17.2871	0.618184
$783$	5.20454	0.185995
$784$	−6.24294	−0.222962
$785$	−4.23414	−0.151123
$786$	10.0582	0.358763
$787$	−49.5520	−1.76634	−0.883168	−	0.469056i	$-0.844594\pi$
$787$	−49.5520	−1.76634	−0.883168	+	0.469056i	$0.844594\pi$
$788$	6.98582	0.248860
$789$	−70.1099	−2.49598
$790$	10.7987	0.384202
$791$	−5.70948	−0.203006
$792$	6.19728	0.220211
$793$	−7.13241	−0.253279
$794$	2.90436	0.103072
$795$	19.7028	0.698785
$796$	−15.2115	−0.539159
$797$	2.26326	0.0801687	0.0400843	−	0.999196i	$-0.487237\pi$
$797$	2.26326	0.0801687	0.0400843	+	0.999196i	$0.487237\pi$
$798$	16.4843	0.583537
$799$	−34.1282	−1.20737
$800$	−0.0510459	−0.00180474
$801$	−45.4202	−1.60484
$802$	−25.7011	−0.907536
$803$	19.8116	0.699138
$804$	21.1710	0.746644
$805$	−7.21238	−0.254203
$806$	−5.98247	−0.210724
$807$	−16.2918	−0.573499
$808$	13.2767	0.467072
$809$	42.2135	1.48415	0.742073	−	0.670319i	$-0.233843\pi$
$809$	42.2135	1.48415	0.742073	+	0.670319i	$0.233843\pi$
$810$	−10.3595	−0.363994
$811$	2.90036	0.101845	0.0509227	−	0.998703i	$-0.483784\pi$
$811$	2.90036	0.101845	0.0509227	+	0.998703i	$0.483784\pi$
$812$	−1.59531	−0.0559843
$813$	−78.2831	−2.74551
$814$	1.81482	0.0636093
$815$	7.39310	0.258969
$816$	12.3340	0.431777
$817$	−49.3440	−1.72633
$818$	15.8305	0.553501
$819$	−4.51263	−0.157684
$820$	15.2251	0.531685
$821$	40.8683	1.42631	0.713157	−	0.701004i	$-0.247265\pi$
$821$	40.8683	1.42631	0.713157	+	0.701004i	$0.247265\pi$
$822$	−6.86056	−0.239290
$823$	28.0901	0.979159	0.489580	−	0.871959i	$-0.337150\pi$
$823$	28.0901	0.979159	0.489580	+	0.871959i	$0.337150\pi$
$824$	9.97564	0.347518
$825$	0.206752	0.00719817
$826$	3.81916	0.132886
$827$	46.8551	1.62931	0.814656	−	0.579944i	$-0.196926\pi$
$827$	46.8551	1.62931	0.814656	+	0.579944i	$0.196926\pi$
$828$	−15.1569	−0.526737
$829$	−40.5049	−1.40680	−0.703398	−	0.710797i	$-0.748335\pi$
$829$	−40.5049	−1.40680	−0.703398	+	0.710797i	$0.748335\pi$
$830$	25.8569	0.897506
$831$	28.6567	0.994091
$832$	−1.27500	−0.0442028
$833$	28.9637	1.00353
$834$	20.5621	0.712007
$835$	−3.67552	−0.127197
$836$	−10.8571	−0.375499
$837$	−13.3191	−0.460375
$838$	19.2388	0.664592
$839$	−26.4836	−0.914316	−0.457158	−	0.889385i	$-0.651133\pi$
$839$	−26.4836	−0.914316	−0.457158	+	0.889385i	$0.651133\pi$
$840$	−5.14592	−0.177551
$841$	−25.6383	−0.884080
$842$	−23.0548	−0.794521
$843$	1.45956	0.0502698
$844$	11.4467	0.394012
$845$	−25.3037	−0.870474
$846$	29.9228	1.02877
$847$	−7.55144	−0.259471
$848$	−3.33143	−0.114402
$849$	15.7294	0.539831
$850$	0.236824	0.00812299
$851$	−4.43855	−0.152152
$852$	−20.7836	−0.712035
$853$	−34.7334	−1.18925	−0.594624	−	0.804004i	$-0.702699\pi$
$853$	−34.7334	−1.18925	−0.594624	+	0.804004i	$0.702699\pi$
$854$	4.86733	0.166557
$855$	−64.4872	−2.20542
$856$	19.4027	0.663171
$857$	−46.1684	−1.57708	−0.788541	−	0.614982i	$-0.789163\pi$
$857$	−46.1684	−1.57708	−0.788541	+	0.614982i	$0.789163\pi$
$858$	5.16416	0.176301
$859$	12.5385	0.427808	0.213904	−	0.976855i	$-0.431382\pi$
$859$	12.5385	0.427808	0.213904	+	0.976855i	$0.431382\pi$
$860$	15.4038	0.525265
$861$	−15.8311	−0.539522
$862$	−24.2833	−0.827093
$863$	−52.0349	−1.77129	−0.885644	−	0.464366i	$-0.846282\pi$
$863$	−52.0349	−1.77129	−0.885644	+	0.464366i	$0.846282\pi$
$864$	−2.83860	−0.0965713
$865$	−37.0902	−1.26110
$866$	33.1330	1.12590
$867$	−12.0280	−0.408493
$868$	4.08259	0.138572
$869$	7.39545	0.250874
$870$	10.8436	0.367634
$871$	10.1534	0.344035
$872$	11.9618	0.405078
$873$	44.9830	1.52245
$874$	26.5534	0.898182
$875$	−9.77696	−0.330522
$876$	−34.5710	−1.16805
$877$	32.7127	1.10463	0.552314	−	0.833636i	$-0.313745\pi$
$877$	32.7127	1.10463	0.552314	+	0.833636i	$0.313745\pi$
$878$	9.73331	0.328483
$879$	57.1503	1.92763
$880$	3.38926	0.114252
$881$	48.9418	1.64889	0.824445	−	0.565942i	$-0.191488\pi$
$881$	48.9418	1.64889	0.824445	+	0.565942i	$0.191488\pi$
$882$	−25.3946	−0.855082
$883$	12.5377	0.421926	0.210963	−	0.977494i	$-0.432340\pi$
$883$	12.5377	0.421926	0.210963	+	0.977494i	$0.432340\pi$
$884$	5.91529	0.198953
$885$	−25.9597	−0.872624
$886$	−21.6988	−0.728985
$887$	−21.7493	−0.730270	−0.365135	−	0.930955i	$-0.618977\pi$
$887$	−21.7493	−0.730270	−0.365135	+	0.930955i	$0.618977\pi$
$888$	−3.16683	−0.106272
$889$	7.62775	0.255826
$890$	−24.8400	−0.832640
$891$	−7.09461	−0.237678
$892$	16.3920	0.548846
$893$	−52.4219	−1.75423
$894$	−27.2601	−0.911714
$895$	16.6378	0.556142
$896$	0.870094	0.0290678
$897$	−12.6301	−0.421707
$898$	−30.3765	−1.01368
$899$	−8.60295	−0.286924
$900$	−0.207641	−0.00692137
$901$	15.4559	0.514912
$902$	10.4268	0.347176
$903$	−16.0168	−0.533007
$904$	−6.56192	−0.218246
$905$	−13.9490	−0.463681
$906$	−55.1089	−1.83087
$907$	−27.9776	−0.928982	−0.464491	−	0.885578i	$-0.653763\pi$
$907$	−27.9776	−0.928982	−0.464491	+	0.885578i	$0.653763\pi$
$908$	−27.6738	−0.918389
$909$	54.0060	1.79127
$910$	−2.46794	−0.0818113
$911$	32.6714	1.08245	0.541226	−	0.840877i	$-0.317960\pi$
$911$	32.6714	1.08245	0.541226	+	0.840877i	$0.317960\pi$
$912$	18.9454	0.627345
$913$	17.7079	0.586047
$914$	−36.9498	−1.22219
$915$	−33.0843	−1.09373
$916$	24.1402	0.797614
$917$	−3.29189	−0.108708
$918$	13.1695	0.434658
$919$	−36.8236	−1.21470	−0.607349	−	0.794435i	$-0.707767\pi$
$919$	−36.8236	−1.21470	−0.607349	+	0.794435i	$0.707767\pi$
$920$	−8.28920	−0.273287
$921$	−18.7839	−0.618950
$922$	−36.5898	−1.20502
$923$	−9.96763	−0.328088
$924$	−3.52415	−0.115936
$925$	−0.0608059	−0.00199928
$926$	14.6817	0.482471
$927$	40.5783	1.33277
$928$	−1.83349	−0.0601872
$929$	16.6169	0.545183	0.272592	−	0.962130i	$-0.412119\pi$
$929$	16.6169	0.545183	0.272592	+	0.962130i	$0.412119\pi$
$930$	−27.7502	−0.909965
$931$	44.4890	1.45807
$932$	27.7496	0.908968
$933$	−75.8154	−2.48209
$934$	−18.3108	−0.599146
$935$	−15.7242	−0.514238
$936$	−5.18638	−0.169522
$937$	−21.1314	−0.690333	−0.345167	−	0.938541i	$-0.612178\pi$
$937$	−21.1314	−0.690333	−0.345167	+	0.938541i	$0.612178\pi$
$938$	−6.92895	−0.226238
$939$	44.1921	1.44215
$940$	16.3646	0.533755
$941$	−11.7639	−0.383492	−0.191746	−	0.981445i	$-0.561415\pi$
$941$	−11.7639	−0.383492	−0.191746	+	0.981445i	$0.561415\pi$
$942$	5.05998	0.164863
$943$	−25.5012	−0.830434
$944$	4.38937	0.142862
$945$	−5.49449	−0.178736
$946$	10.5492	0.342984
$947$	45.0423	1.46368	0.731838	−	0.681478i	$-0.238663\pi$
$947$	45.0423	1.46368	0.731838	+	0.681478i	$0.238663\pi$
$948$	−12.9050	−0.419134
$949$	−16.5800	−0.538208
$950$	0.363768	0.0118022
$951$	41.5381	1.34697
$952$	−4.03674	−0.130831
$953$	23.2489	0.753105	0.376552	−	0.926395i	$-0.377109\pi$
$953$	23.2489	0.753105	0.376552	+	0.926395i	$0.377109\pi$
$954$	−13.5514	−0.438742
$955$	−31.2708	−1.01190
$956$	−29.5337	−0.955188
$957$	7.42620	0.240055
$958$	24.4839	0.791039
$959$	2.24536	0.0725063
$960$	−5.91421	−0.190880
$961$	−8.98398	−0.289806
$962$	−1.51878	−0.0489676
$963$	78.9251	2.54333
$964$	−8.07504	−0.260080
$965$	−27.4123	−0.882433
$966$	8.61911	0.277315
$967$	22.5345	0.724662	0.362331	−	0.932050i	$-0.381981\pi$
$967$	22.5345	0.724662	0.362331	+	0.932050i	$0.381981\pi$
$968$	−8.67889	−0.278950
$969$	−87.8960	−2.82362
$970$	24.6010	0.789890
$971$	50.6263	1.62468	0.812338	−	0.583187i	$-0.198195\pi$
$971$	50.6263	1.62468	0.812338	+	0.583187i	$0.198195\pi$
$972$	20.8958	0.670233
$973$	−6.72966	−0.215743
$974$	8.64458	0.276990
$975$	−0.173026	−0.00554127
$976$	5.59404	0.179061
$977$	−10.8331	−0.346582	−0.173291	−	0.984871i	$-0.555440\pi$
$977$	−10.8331	−0.346582	−0.173291	+	0.984871i	$0.555440\pi$
$978$	−8.83507	−0.282514
$979$	−17.0116	−0.543692
$980$	−13.8882	−0.443642
$981$	48.6575	1.55351
$982$	36.2805	1.15776
$983$	20.0440	0.639304	0.319652	−	0.947535i	$-0.396434\pi$
$983$	20.0440	0.639304	0.319652	+	0.947535i	$0.396434\pi$
$984$	−18.1947	−0.580026
$985$	15.5408	0.495172
$986$	8.50634	0.270897
$987$	−17.0159	−0.541622
$988$	9.08605	0.289066
$989$	−25.8004	−0.820406
$990$	13.7866	0.438168
$991$	−26.6387	−0.846205	−0.423103	−	0.906082i	$-0.639059\pi$
$991$	−26.6387	−0.846205	−0.423103	+	0.906082i	$0.639059\pi$
$992$	4.69212	0.148975
$993$	21.7190	0.689232
$994$	6.80216	0.215751
$995$	−33.8400	−1.07280
$996$	−30.9001	−0.979107
$997$	17.1739	0.543904	0.271952	−	0.962311i	$-0.412331\pi$
$997$	17.1739	0.543904	0.271952	+	0.962311i	$0.412331\pi$
$998$	−19.7389	−0.624825
$999$	−3.38135	−0.106981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.d.1.8	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.d.1.8	✓	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.65852 q^{3} +1.00000 q^{4} +2.22462 q^{5} -2.65852 q^{6} +0.870094 q^{7} +1.00000 q^{8} +4.06774 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.65852 q^{3} +1.00000 q^{4} +2.22462 q^{5} -2.65852 q^{6} +0.870094 q^{7} +1.00000 q^{8} +4.06774 q^{9} +2.22462 q^{10} +1.52352 q^{11} -2.65852 q^{12} -1.27500 q^{13} +0.870094 q^{14} -5.91421 q^{15} +1.00000 q^{16} -4.63943 q^{17} +4.06774 q^{18} -7.12630 q^{19} +2.22462 q^{20} -2.31316 q^{21} +1.52352 q^{22} -3.72611 q^{23} -2.65852 q^{24} -0.0510459 q^{25} -1.27500 q^{26} -2.83860 q^{27} +0.870094 q^{28} -1.83349 q^{29} -5.91421 q^{30} +4.69212 q^{31} +1.00000 q^{32} -4.05031 q^{33} -4.63943 q^{34} +1.93563 q^{35} +4.06774 q^{36} +1.19120 q^{37} -7.12630 q^{38} +3.38962 q^{39} +2.22462 q^{40} +6.84391 q^{41} -2.31316 q^{42} +6.92422 q^{43} +1.52352 q^{44} +9.04919 q^{45} -3.72611 q^{46} +7.35612 q^{47} -2.65852 q^{48} -6.24294 q^{49} -0.0510459 q^{50} +12.3340 q^{51} -1.27500 q^{52} -3.33143 q^{53} -2.83860 q^{54} +3.38926 q^{55} +0.870094 q^{56} +18.9454 q^{57} -1.83349 q^{58} +4.38937 q^{59} -5.91421 q^{60} +5.59404 q^{61} +4.69212 q^{62} +3.53931 q^{63} +1.00000 q^{64} -2.83640 q^{65} -4.05031 q^{66} -7.96345 q^{67} -4.63943 q^{68} +9.90596 q^{69} +1.93563 q^{70} +7.81773 q^{71} +4.06774 q^{72} +13.0039 q^{73} +1.19120 q^{74} +0.135707 q^{75} -7.12630 q^{76} +1.32561 q^{77} +3.38962 q^{78} +4.85419 q^{79} +2.22462 q^{80} -4.65672 q^{81} +6.84391 q^{82} +11.6230 q^{83} -2.31316 q^{84} -10.3210 q^{85} +6.92422 q^{86} +4.87437 q^{87} +1.52352 q^{88} -11.1659 q^{89} +9.04919 q^{90} -1.10937 q^{91} -3.72611 q^{92} -12.4741 q^{93} +7.35612 q^{94} -15.8533 q^{95} -2.65852 q^{96} +11.0585 q^{97} -6.24294 q^{98} +6.19728 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

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