LMFDB - Embedded newform 8038.2.a.b.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.20646 q^{3} +1.00000 q^{4} +0.820064 q^{5} +2.20646 q^{6} +3.76837 q^{7} -1.00000 q^{8} +1.86848 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.20646 q^{3} +1.00000 q^{4} +0.820064 q^{5} +2.20646 q^{6} +3.76837 q^{7} -1.00000 q^{8} +1.86848 q^{9} -0.820064 q^{10} +4.54434 q^{11} -2.20646 q^{12} +0.840283 q^{13} -3.76837 q^{14} -1.80944 q^{15} +1.00000 q^{16} +0.875706 q^{17} -1.86848 q^{18} +4.66412 q^{19} +0.820064 q^{20} -8.31476 q^{21} -4.54434 q^{22} +2.02318 q^{23} +2.20646 q^{24} -4.32749 q^{25} -0.840283 q^{26} +2.49666 q^{27} +3.76837 q^{28} -2.79886 q^{29} +1.80944 q^{30} +5.76900 q^{31} -1.00000 q^{32} -10.0269 q^{33} -0.875706 q^{34} +3.09030 q^{35} +1.86848 q^{36} +9.52518 q^{37} -4.66412 q^{38} -1.85405 q^{39} -0.820064 q^{40} -3.04195 q^{41} +8.31476 q^{42} +0.646162 q^{43} +4.54434 q^{44} +1.53227 q^{45} -2.02318 q^{46} +8.52382 q^{47} -2.20646 q^{48} +7.20058 q^{49} +4.32749 q^{50} -1.93221 q^{51} +0.840283 q^{52} +1.01532 q^{53} -2.49666 q^{54} +3.72665 q^{55} -3.76837 q^{56} -10.2912 q^{57} +2.79886 q^{58} +0.211713 q^{59} -1.80944 q^{60} -4.11766 q^{61} -5.76900 q^{62} +7.04110 q^{63} +1.00000 q^{64} +0.689086 q^{65} +10.0269 q^{66} -9.83878 q^{67} +0.875706 q^{68} -4.46407 q^{69} -3.09030 q^{70} +14.5024 q^{71} -1.86848 q^{72} -3.81288 q^{73} -9.52518 q^{74} +9.54845 q^{75} +4.66412 q^{76} +17.1248 q^{77} +1.85405 q^{78} +8.40756 q^{79} +0.820064 q^{80} -11.1142 q^{81} +3.04195 q^{82} +8.21039 q^{83} -8.31476 q^{84} +0.718135 q^{85} -0.646162 q^{86} +6.17558 q^{87} -4.54434 q^{88} +3.57791 q^{89} -1.53227 q^{90} +3.16649 q^{91} +2.02318 q^{92} -12.7291 q^{93} -8.52382 q^{94} +3.82488 q^{95} +2.20646 q^{96} +2.20462 q^{97} -7.20058 q^{98} +8.49100 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) $ 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.20646	−1.27390	−0.636951	−	0.770904i	$-0.719805\pi$
$3$	−2.20646	−1.27390	−0.636951	+	0.770904i	$0.719805\pi$
$4$	1.00000	0.500000
$5$	0.820064	0.366744	0.183372	−	0.983044i	$-0.441299\pi$
$5$	0.820064	0.366744	0.183372	+	0.983044i	$0.441299\pi$
$6$	2.20646	0.900785
$7$	3.76837	1.42431	0.712154	−	0.702023i	$-0.247720\pi$
$7$	3.76837	1.42431	0.712154	+	0.702023i	$0.247720\pi$
$8$	−1.00000	−0.353553
$9$	1.86848	0.622825
$10$	−0.820064	−0.259327
$11$	4.54434	1.37017	0.685086	−	0.728462i	$-0.259765\pi$
$11$	4.54434	1.37017	0.685086	+	0.728462i	$0.259765\pi$
$12$	−2.20646	−0.636951
$13$	0.840283	0.233052	0.116526	−	0.993188i	$-0.462824\pi$
$13$	0.840283	0.233052	0.116526	+	0.993188i	$0.462824\pi$
$14$	−3.76837	−1.00714
$15$	−1.80944	−0.467196
$16$	1.00000	0.250000
$17$	0.875706	0.212390	0.106195	−	0.994345i	$-0.466133\pi$
$17$	0.875706	0.212390	0.106195	+	0.994345i	$0.466133\pi$
$18$	−1.86848	−0.440404
$19$	4.66412	1.07002	0.535012	−	0.844845i	$-0.320307\pi$
$19$	4.66412	1.07002	0.535012	+	0.844845i	$0.320307\pi$
$20$	0.820064	0.183372
$21$	−8.31476	−1.81443
$22$	−4.54434	−0.968857
$23$	2.02318	0.421862	0.210931	−	0.977501i	$-0.432350\pi$
$23$	2.02318	0.421862	0.210931	+	0.977501i	$0.432350\pi$
$24$	2.20646	0.450392
$25$	−4.32749	−0.865499
$26$	−0.840283	−0.164793
$27$	2.49666	0.480483
$28$	3.76837	0.712154
$29$	−2.79886	−0.519735	−0.259868	−	0.965644i	$-0.583679\pi$
$29$	−2.79886	−0.519735	−0.259868	+	0.965644i	$0.583679\pi$
$30$	1.80944	0.330357
$31$	5.76900	1.03614	0.518072	−	0.855337i	$-0.326650\pi$
$31$	5.76900	1.03614	0.518072	+	0.855337i	$0.326650\pi$
$32$	−1.00000	−0.176777
$33$	−10.0269	−1.74546
$34$	−0.875706	−0.150182
$35$	3.09030	0.522356
$36$	1.86848	0.311413
$37$	9.52518	1.56593	0.782965	−	0.622066i	$-0.213706\pi$
$37$	9.52518	1.56593	0.782965	+	0.622066i	$0.213706\pi$
$38$	−4.66412	−0.756621
$39$	−1.85405	−0.296886
$40$	−0.820064	−0.129664
$41$	−3.04195	−0.475073	−0.237537	−	0.971379i	$-0.576340\pi$
$41$	−3.04195	−0.475073	−0.237537	+	0.971379i	$0.576340\pi$
$42$	8.31476	1.28299
$43$	0.646162	0.0985388	0.0492694	−	0.998786i	$-0.484311\pi$
$43$	0.646162	0.0985388	0.0492694	+	0.998786i	$0.484311\pi$
$44$	4.54434	0.685086
$45$	1.53227	0.228417
$46$	−2.02318	−0.298302
$47$	8.52382	1.24333	0.621664	−	0.783284i	$-0.286457\pi$
$47$	8.52382	1.24333	0.621664	+	0.783284i	$0.286457\pi$
$48$	−2.20646	−0.318475
$49$	7.20058	1.02865
$50$	4.32749	0.612000
$51$	−1.93221	−0.270564
$52$	0.840283	0.116526
$53$	1.01532	0.139465	0.0697325	−	0.997566i	$-0.477785\pi$
$53$	1.01532	0.139465	0.0697325	+	0.997566i	$0.477785\pi$
$54$	−2.49666	−0.339753
$55$	3.72665	0.502502
$56$	−3.76837	−0.503569
$57$	−10.2912	−1.36310
$58$	2.79886	0.367508
$59$	0.211713	0.0275626	0.0137813	−	0.999905i	$-0.495613\pi$
$59$	0.211713	0.0275626	0.0137813	+	0.999905i	$0.495613\pi$
$60$	−1.80944	−0.233598
$61$	−4.11766	−0.527213	−0.263606	−	0.964630i	$-0.584912\pi$
$61$	−4.11766	−0.527213	−0.263606	+	0.964630i	$0.584912\pi$
$62$	−5.76900	−0.732664
$63$	7.04110	0.887096
$64$	1.00000	0.125000
$65$	0.689086	0.0854706
$66$	10.0269	1.23423
$67$	−9.83878	−1.20200	−0.600999	−	0.799249i	$-0.705231\pi$
$67$	−9.83878	−1.20200	−0.600999	+	0.799249i	$0.705231\pi$
$68$	0.875706	0.106195
$69$	−4.46407	−0.537411
$70$	−3.09030	−0.369362
$71$	14.5024	1.72112	0.860558	−	0.509353i	$-0.170115\pi$
$71$	14.5024	1.72112	0.860558	+	0.509353i	$0.170115\pi$
$72$	−1.86848	−0.220202
$73$	−3.81288	−0.446264	−0.223132	−	0.974788i	$-0.571628\pi$
$73$	−3.81288	−0.446264	−0.223132	+	0.974788i	$0.571628\pi$
$74$	−9.52518	−1.10728
$75$	9.54845	1.10256
$76$	4.66412	0.535012
$77$	17.1248	1.95155
$78$	1.85405	0.209930
$79$	8.40756	0.945924	0.472962	−	0.881083i	$-0.343185\pi$
$79$	8.40756	0.945924	0.472962	+	0.881083i	$0.343185\pi$
$80$	0.820064	0.0916860
$81$	−11.1142	−1.23491
$82$	3.04195	0.335928
$83$	8.21039	0.901207	0.450604	−	0.892724i	$-0.351209\pi$
$83$	8.21039	0.901207	0.450604	+	0.892724i	$0.351209\pi$
$84$	−8.31476	−0.907214
$85$	0.718135	0.0778927
$86$	−0.646162	−0.0696774
$87$	6.17558	0.662092
$88$	−4.54434	−0.484429
$89$	3.57791	0.379257	0.189629	−	0.981856i	$-0.439272\pi$
$89$	3.57791	0.379257	0.189629	+	0.981856i	$0.439272\pi$
$90$	−1.53227	−0.161516
$91$	3.16649	0.331939
$92$	2.02318	0.210931
$93$	−12.7291	−1.31994
$94$	−8.52382	−0.879165
$95$	3.82488	0.392425
$96$	2.20646	0.225196
$97$	2.20462	0.223846	0.111923	−	0.993717i	$-0.464299\pi$
$97$	2.20462	0.223846	0.111923	+	0.993717i	$0.464299\pi$
$98$	−7.20058	−0.727368
$99$	8.49100	0.853378
$100$	−4.32749	−0.432749
$101$	13.2595	1.31936	0.659682	−	0.751544i	$-0.270691\pi$
$101$	13.2595	1.31936	0.659682	+	0.751544i	$0.270691\pi$
$102$	1.93221	0.191318
$103$	−16.4731	−1.62315	−0.811574	−	0.584250i	$-0.801389\pi$
$103$	−16.4731	−1.62315	−0.811574	+	0.584250i	$0.801389\pi$
$104$	−0.840283	−0.0823965
$105$	−6.81864	−0.665431
$106$	−1.01532	−0.0986167
$107$	18.7669	1.81427	0.907133	−	0.420845i	$-0.138266\pi$
$107$	18.7669	1.81427	0.907133	+	0.420845i	$0.138266\pi$
$108$	2.49666	0.240242
$109$	2.44978	0.234646	0.117323	−	0.993094i	$-0.462569\pi$
$109$	2.44978	0.234646	0.117323	+	0.993094i	$0.462569\pi$
$110$	−3.72665	−0.355323
$111$	−21.0170	−1.99484
$112$	3.76837	0.356077
$113$	14.8404	1.39607	0.698034	−	0.716065i	$-0.254058\pi$
$113$	14.8404	1.39607	0.698034	+	0.716065i	$0.254058\pi$
$114$	10.2912	0.963861
$115$	1.65914	0.154715
$116$	−2.79886	−0.259868
$117$	1.57005	0.145151
$118$	−0.211713	−0.0194897
$119$	3.29998	0.302509
$120$	1.80944	0.165179
$121$	9.65107	0.877370
$122$	4.11766	0.372796
$123$	6.71196	0.605197
$124$	5.76900	0.518072
$125$	−7.64915	−0.684160
$126$	−7.04110	−0.627271
$127$	9.21212	0.817444	0.408722	−	0.912659i	$-0.365975\pi$
$127$	9.21212	0.817444	0.408722	+	0.912659i	$0.365975\pi$
$128$	−1.00000	−0.0883883
$129$	−1.42573	−0.125529
$130$	−0.689086	−0.0604368
$131$	2.00387	0.175079	0.0875394	−	0.996161i	$-0.472100\pi$
$131$	2.00387	0.175079	0.0875394	+	0.996161i	$0.472100\pi$
$132$	−10.0269	−0.872732
$133$	17.5761	1.52404
$134$	9.83878	0.849942
$135$	2.04743	0.176214
$136$	−0.875706	−0.0750912
$137$	−4.14970	−0.354533	−0.177266	−	0.984163i	$-0.556725\pi$
$137$	−4.14970	−0.354533	−0.177266	+	0.984163i	$0.556725\pi$
$138$	4.46407	0.380007
$139$	−9.30469	−0.789214	−0.394607	−	0.918850i	$-0.629119\pi$
$139$	−9.30469	−0.789214	−0.394607	+	0.918850i	$0.629119\pi$
$140$	3.09030	0.261178
$141$	−18.8075	−1.58388
$142$	−14.5024	−1.21701
$143$	3.81853	0.319322
$144$	1.86848	0.155706
$145$	−2.29525	−0.190610
$146$	3.81288	0.315557
$147$	−15.8878	−1.31040
$148$	9.52518	0.782965
$149$	−12.6280	−1.03453	−0.517265	−	0.855825i	$-0.673050\pi$
$149$	−12.6280	−1.03453	−0.517265	+	0.855825i	$0.673050\pi$
$150$	−9.54845	−0.779628
$151$	−10.3284	−0.840510	−0.420255	−	0.907406i	$-0.638059\pi$
$151$	−10.3284	−0.840510	−0.420255	+	0.907406i	$0.638059\pi$
$152$	−4.66412	−0.378310
$153$	1.63624	0.132282
$154$	−17.1248	−1.37995
$155$	4.73095	0.379999
$156$	−1.85405	−0.148443
$157$	−12.0497	−0.961669	−0.480835	−	0.876811i	$-0.659666\pi$
$157$	−12.0497	−0.961669	−0.480835	+	0.876811i	$0.659666\pi$
$158$	−8.40756	−0.668869
$159$	−2.24027	−0.177665
$160$	−0.820064	−0.0648318
$161$	7.62408	0.600862
$162$	11.1142	0.873216
$163$	−10.2221	−0.800657	−0.400329	−	0.916372i	$-0.631104\pi$
$163$	−10.2221	−0.800657	−0.400329	+	0.916372i	$0.631104\pi$
$164$	−3.04195	−0.237537
$165$	−8.22272	−0.640138
$166$	−8.21039	−0.637250
$167$	−2.80090	−0.216740	−0.108370	−	0.994111i	$-0.534563\pi$
$167$	−2.80090	−0.216740	−0.108370	+	0.994111i	$0.534563\pi$
$168$	8.31476	0.641497
$169$	−12.2939	−0.945687
$170$	−0.718135	−0.0550785
$171$	8.71481	0.666438
$172$	0.646162	0.0492694
$173$	−16.3112	−1.24011	−0.620057	−	0.784557i	$-0.712891\pi$
$173$	−16.3112	−1.24011	−0.620057	+	0.784557i	$0.712891\pi$
$174$	−6.17558	−0.468170
$175$	−16.3076	−1.23274
$176$	4.54434	0.342543
$177$	−0.467136	−0.0351121
$178$	−3.57791	−0.268175
$179$	−13.5499	−1.01277	−0.506383	−	0.862308i	$-0.669018\pi$
$179$	−13.5499	−1.01277	−0.506383	+	0.862308i	$0.669018\pi$
$180$	1.53227	0.114209
$181$	4.26215	0.316803	0.158402	−	0.987375i	$-0.449366\pi$
$181$	4.26215	0.316803	0.158402	+	0.987375i	$0.449366\pi$
$182$	−3.16649	−0.234716
$183$	9.08547	0.671617
$184$	−2.02318	−0.149151
$185$	7.81126	0.574295
$186$	12.7291	0.933342
$187$	3.97951	0.291011
$188$	8.52382	0.621664
$189$	9.40834	0.684356
$190$	−3.82488	−0.277486
$191$	15.7952	1.14290	0.571449	−	0.820638i	$-0.306382\pi$
$191$	15.7952	1.14290	0.571449	+	0.820638i	$0.306382\pi$
$192$	−2.20646	−0.159238
$193$	−2.68636	−0.193368	−0.0966842	−	0.995315i	$-0.530824\pi$
$193$	−2.68636	−0.193368	−0.0966842	+	0.995315i	$0.530824\pi$
$194$	−2.20462	−0.158283
$195$	−1.52044	−0.108881
$196$	7.20058	0.514327
$197$	−6.74241	−0.480377	−0.240189	−	0.970726i	$-0.577209\pi$
$197$	−6.74241	−0.480377	−0.240189	+	0.970726i	$0.577209\pi$
$198$	−8.49100	−0.603429
$199$	0.484017	0.0343111	0.0171555	−	0.999853i	$-0.494539\pi$
$199$	0.484017	0.0343111	0.0171555	+	0.999853i	$0.494539\pi$
$200$	4.32749	0.306000
$201$	21.7089	1.53123
$202$	−13.2595	−0.932932
$203$	−10.5471	−0.740264
$204$	−1.93221	−0.135282
$205$	−2.49460	−0.174230
$206$	16.4731	1.14774
$207$	3.78026	0.262746
$208$	0.840283	0.0582631
$209$	21.1954	1.46612
$210$	6.81864	0.470531
$211$	−11.5360	−0.794168	−0.397084	−	0.917782i	$-0.629978\pi$
$211$	−11.5360	−0.794168	−0.397084	+	0.917782i	$0.629978\pi$
$212$	1.01532	0.0697325
$213$	−31.9990	−2.19253
$214$	−18.7669	−1.28288
$215$	0.529894	0.0361385
$216$	−2.49666	−0.169876
$217$	21.7397	1.47579
$218$	−2.44978	−0.165920
$219$	8.41299	0.568497
$220$	3.72665	0.251251
$221$	0.735841	0.0494980
$222$	21.0170	1.41057
$223$	−21.5870	−1.44557	−0.722787	−	0.691071i	$-0.757139\pi$
$223$	−21.5870	−1.44557	−0.722787	+	0.691071i	$0.757139\pi$
$224$	−3.76837	−0.251785
$225$	−8.08582	−0.539055
$226$	−14.8404	−0.987169
$227$	−3.12335	−0.207304	−0.103652	−	0.994614i	$-0.533053\pi$
$227$	−3.12335	−0.207304	−0.103652	+	0.994614i	$0.533053\pi$
$228$	−10.2912	−0.681552
$229$	−17.5629	−1.16059	−0.580294	−	0.814407i	$-0.697062\pi$
$229$	−17.5629	−1.16059	−0.580294	+	0.814407i	$0.697062\pi$
$230$	−1.65914	−0.109400
$231$	−37.7851	−2.48608
$232$	2.79886	0.183754
$233$	−10.0961	−0.661415	−0.330708	−	0.943733i	$-0.607287\pi$
$233$	−10.0961	−0.661415	−0.330708	+	0.943733i	$0.607287\pi$
$234$	−1.57005	−0.102637
$235$	6.99008	0.455983
$236$	0.211713	0.0137813
$237$	−18.5510	−1.20501
$238$	−3.29998	−0.213906
$239$	−11.0713	−0.716145	−0.358072	−	0.933694i	$-0.616566\pi$
$239$	−11.0713	−0.716145	−0.358072	+	0.933694i	$0.616566\pi$
$240$	−1.80944	−0.116799
$241$	13.2019	0.850409	0.425204	−	0.905097i	$-0.360202\pi$
$241$	13.2019	0.850409	0.425204	+	0.905097i	$0.360202\pi$
$242$	−9.65107	−0.620394
$243$	17.0331	1.09268
$244$	−4.11766	−0.263606
$245$	5.90494	0.377253
$246$	−6.71196	−0.427939
$247$	3.91918	0.249372
$248$	−5.76900	−0.366332
$249$	−18.1159	−1.14805
$250$	7.64915	0.483774
$251$	−6.55068	−0.413475	−0.206738	−	0.978396i	$-0.566285\pi$
$251$	−6.55068	−0.413475	−0.206738	+	0.978396i	$0.566285\pi$
$252$	7.04110	0.443548
$253$	9.19403	0.578023
$254$	−9.21212	−0.578020
$255$	−1.58454	−0.0992277
$256$	1.00000	0.0625000
$257$	9.47534	0.591056	0.295528	−	0.955334i	$-0.404504\pi$
$257$	9.47534	0.591056	0.295528	+	0.955334i	$0.404504\pi$
$258$	1.42573	0.0887622
$259$	35.8944	2.23037
$260$	0.689086	0.0427353
$261$	−5.22961	−0.323704
$262$	−2.00387	−0.123799
$263$	−3.28494	−0.202558	−0.101279	−	0.994858i	$-0.532293\pi$
$263$	−3.28494	−0.202558	−0.101279	+	0.994858i	$0.532293\pi$
$264$	10.0269	0.617115
$265$	0.832628	0.0511480
$266$	−17.5761	−1.07766
$267$	−7.89452	−0.483137
$268$	−9.83878	−0.600999
$269$	−14.9104	−0.909101	−0.454550	−	0.890721i	$-0.650200\pi$
$269$	−14.9104	−0.909101	−0.454550	+	0.890721i	$0.650200\pi$
$270$	−2.04743	−0.124602
$271$	5.71899	0.347404	0.173702	−	0.984798i	$-0.444427\pi$
$271$	5.71899	0.347404	0.173702	+	0.984798i	$0.444427\pi$
$272$	0.875706	0.0530975
$273$	−6.98675	−0.422857
$274$	4.14970	0.250692
$275$	−19.6656	−1.18588
$276$	−4.46407	−0.268705
$277$	17.7367	1.06570	0.532848	−	0.846211i	$-0.321122\pi$
$277$	17.7367	1.06570	0.532848	+	0.846211i	$0.321122\pi$
$278$	9.30469	0.558058
$279$	10.7792	0.645336
$280$	−3.09030	−0.184681
$281$	−1.20985	−0.0721736	−0.0360868	−	0.999349i	$-0.511489\pi$
$281$	−1.20985	−0.0721736	−0.0360868	+	0.999349i	$0.511489\pi$
$282$	18.8075	1.11997
$283$	−15.1762	−0.902129	−0.451064	−	0.892491i	$-0.648956\pi$
$283$	−15.1762	−0.902129	−0.451064	+	0.892491i	$0.648956\pi$
$284$	14.5024	0.860558
$285$	−8.43946	−0.499910
$286$	−3.81853	−0.225795
$287$	−11.4632	−0.676651
$288$	−1.86848	−0.110101
$289$	−16.2331	−0.954891
$290$	2.29525	0.134781
$291$	−4.86442	−0.285157
$292$	−3.81288	−0.223132
$293$	−4.33939	−0.253510	−0.126755	−	0.991934i	$-0.540456\pi$
$293$	−4.33939	−0.253510	−0.126755	+	0.991934i	$0.540456\pi$
$294$	15.8878	0.926596
$295$	0.173618	0.0101084
$296$	−9.52518	−0.553640
$297$	11.3457	0.658344
$298$	12.6280	0.731523
$299$	1.70004	0.0983160
$300$	9.54845	0.551280
$301$	2.43497	0.140350
$302$	10.3284	0.594331
$303$	−29.2565	−1.68074
$304$	4.66412	0.267506
$305$	−3.37675	−0.193352
$306$	−1.63624	−0.0935374
$307$	18.0645	1.03099	0.515496	−	0.856892i	$-0.327608\pi$
$307$	18.0645	1.03099	0.515496	+	0.856892i	$0.327608\pi$
$308$	17.1248	0.975773
$309$	36.3474	2.06773
$310$	−4.73095	−0.268700
$311$	24.5086	1.38976	0.694879	−	0.719127i	$-0.255458\pi$
$311$	24.5086	1.38976	0.694879	+	0.719127i	$0.255458\pi$
$312$	1.85405	0.104965
$313$	18.2088	1.02922	0.514610	−	0.857424i	$-0.327937\pi$
$313$	18.2088	1.02922	0.514610	+	0.857424i	$0.327937\pi$
$314$	12.0497	0.680003
$315$	5.77416	0.325337
$316$	8.40756	0.472962
$317$	19.1210	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$317$	19.1210	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$318$	2.24027	0.125628
$319$	−12.7190	−0.712127
$320$	0.820064	0.0458430
$321$	−41.4085	−2.31120
$322$	−7.62408	−0.424873
$323$	4.08440	0.227262
$324$	−11.1142	−0.617457
$325$	−3.63632	−0.201707
$326$	10.2221	0.566150
$327$	−5.40534	−0.298916
$328$	3.04195	0.167964
$329$	32.1209	1.77088
$330$	8.22272	0.452646
$331$	8.37562	0.460366	0.230183	−	0.973147i	$-0.426068\pi$
$331$	8.37562	0.460366	0.230183	+	0.973147i	$0.426068\pi$
$332$	8.21039	0.450604
$333$	17.7976	0.975301
$334$	2.80090	0.153258
$335$	−8.06844	−0.440826
$336$	−8.31476	−0.453607
$337$	9.12869	0.497271	0.248636	−	0.968597i	$-0.420018\pi$
$337$	9.12869	0.497271	0.248636	+	0.968597i	$0.420018\pi$
$338$	12.2939	0.668701
$339$	−32.7448	−1.77845
$340$	0.718135	0.0389464
$341$	26.2163	1.41969
$342$	−8.71481	−0.471243
$343$	0.755861	0.0408127
$344$	−0.646162	−0.0348387
$345$	−3.66082	−0.197092
$346$	16.3112	0.876893
$347$	−17.4568	−0.937131	−0.468565	−	0.883429i	$-0.655229\pi$
$347$	−17.4568	−0.937131	−0.468565	+	0.883429i	$0.655229\pi$
$348$	6.17558	0.331046
$349$	−8.14021	−0.435735	−0.217868	−	0.975978i	$-0.569910\pi$
$349$	−8.14021	−0.435735	−0.217868	+	0.975978i	$0.569910\pi$
$350$	16.3076	0.871677
$351$	2.09790	0.111978
$352$	−4.54434	−0.242214
$353$	−1.55831	−0.0829405	−0.0414703	−	0.999140i	$-0.513204\pi$
$353$	−1.55831	−0.0829405	−0.0414703	+	0.999140i	$0.513204\pi$
$354$	0.467136	0.0248280
$355$	11.8929	0.631209
$356$	3.57791	0.189629
$357$	−7.28129	−0.385366
$358$	13.5499	0.716134
$359$	−30.0259	−1.58470	−0.792352	−	0.610064i	$-0.791144\pi$
$359$	−30.0259	−1.58470	−0.792352	+	0.610064i	$0.791144\pi$
$360$	−1.53227	−0.0807578
$361$	2.75405	0.144950
$362$	−4.26215	−0.224014
$363$	−21.2947	−1.11768
$364$	3.16649	0.165969
$365$	−3.12681	−0.163665
$366$	−9.08547	−0.474905
$367$	−4.49338	−0.234553	−0.117276	−	0.993099i	$-0.537416\pi$
$367$	−4.49338	−0.234553	−0.117276	+	0.993099i	$0.537416\pi$
$368$	2.02318	0.105466
$369$	−5.68382	−0.295888
$370$	−7.81126	−0.406088
$371$	3.82610	0.198641
$372$	−12.7291	−0.659972
$373$	−2.21177	−0.114521	−0.0572606	−	0.998359i	$-0.518237\pi$
$373$	−2.21177	−0.114521	−0.0572606	+	0.998359i	$0.518237\pi$
$374$	−3.97951	−0.205776
$375$	16.8776	0.871553
$376$	−8.52382	−0.439583
$377$	−2.35183	−0.121126
$378$	−9.40834	−0.483913
$379$	−20.1489	−1.03498	−0.517490	−	0.855689i	$-0.673134\pi$
$379$	−20.1489	−1.03498	−0.517490	+	0.855689i	$0.673134\pi$
$380$	3.82488	0.196212
$381$	−20.3262	−1.04134
$382$	−15.7952	−0.808150
$383$	8.13337	0.415596	0.207798	−	0.978172i	$-0.433370\pi$
$383$	8.13337	0.415596	0.207798	+	0.978172i	$0.433370\pi$
$384$	2.20646	0.112598
$385$	14.0434	0.715718
$386$	2.68636	0.136732
$387$	1.20734	0.0613725
$388$	2.20462	0.111923
$389$	−21.5042	−1.09031	−0.545153	−	0.838336i	$-0.683529\pi$
$389$	−21.5042	−1.09031	−0.545153	+	0.838336i	$0.683529\pi$
$390$	1.52044	0.0769906
$391$	1.77171	0.0895993
$392$	−7.20058	−0.363684
$393$	−4.42146	−0.223033
$394$	6.74241	0.339678
$395$	6.89474	0.346912
$396$	8.49100	0.426689
$397$	−12.4900	−0.626853	−0.313427	−	0.949612i	$-0.601477\pi$
$397$	−12.4900	−0.626853	−0.313427	+	0.949612i	$0.601477\pi$
$398$	−0.484017	−0.0242616
$399$	−38.7811	−1.94148
$400$	−4.32749	−0.216375
$401$	−32.0614	−1.60107	−0.800535	−	0.599286i	$-0.795451\pi$
$401$	−32.0614	−1.60107	−0.800535	+	0.599286i	$0.795451\pi$
$402$	−21.7089	−1.08274
$403$	4.84759	0.241476
$404$	13.2595	0.659682
$405$	−9.11438	−0.452897
$406$	10.5471	0.523445
$407$	43.2857	2.14559
$408$	1.93221	0.0956588
$409$	14.7645	0.730058	0.365029	−	0.930996i	$-0.381059\pi$
$409$	14.7645	0.730058	0.365029	+	0.930996i	$0.381059\pi$
$410$	2.49460	0.123199
$411$	9.15615	0.451640
$412$	−16.4731	−0.811574
$413$	0.797811	0.0392577
$414$	−3.78026	−0.185790
$415$	6.73305	0.330512
$416$	−0.840283	−0.0411982
$417$	20.5305	1.00538
$418$	−21.1954	−1.03670
$419$	27.7269	1.35455	0.677274	−	0.735731i	$-0.263161\pi$
$419$	27.7269	1.35455	0.677274	+	0.735731i	$0.263161\pi$
$420$	−6.81864	−0.332715
$421$	21.6861	1.05692	0.528458	−	0.848959i	$-0.322770\pi$
$421$	21.6861	1.05692	0.528458	+	0.848959i	$0.322770\pi$
$422$	11.5360	0.561562
$423$	15.9266	0.774376
$424$	−1.01532	−0.0493083
$425$	−3.78961	−0.183823
$426$	31.9990	1.55035
$427$	−15.5169	−0.750914
$428$	18.7669	0.907133
$429$	−8.42545	−0.406785
$430$	−0.529894	−0.0255538
$431$	−29.0119	−1.39746	−0.698728	−	0.715387i	$-0.746250\pi$
$431$	−29.0119	−1.39746	−0.698728	+	0.715387i	$0.746250\pi$
$432$	2.49666	0.120121
$433$	27.5784	1.32533	0.662667	−	0.748914i	$-0.269425\pi$
$433$	27.5784	1.32533	0.662667	+	0.748914i	$0.269425\pi$
$434$	−21.7397	−1.04354
$435$	5.06437	0.242818
$436$	2.44978	0.117323
$437$	9.43636	0.451402
$438$	−8.41299	−0.401988
$439$	−10.2648	−0.489910	−0.244955	−	0.969534i	$-0.578773\pi$
$439$	−10.2648	−0.489910	−0.244955	+	0.969534i	$0.578773\pi$
$440$	−3.72665	−0.177661
$441$	13.4541	0.640672
$442$	−0.735841	−0.0350004
$443$	26.0470	1.23753	0.618764	−	0.785577i	$-0.287634\pi$
$443$	26.0470	1.23753	0.618764	+	0.785577i	$0.287634\pi$
$444$	−21.0170	−0.997421
$445$	2.93411	0.139090
$446$	21.5870	1.02218
$447$	27.8633	1.31789
$448$	3.76837	0.178039
$449$	−14.7730	−0.697182	−0.348591	−	0.937275i	$-0.613340\pi$
$449$	−14.7730	−0.697182	−0.348591	+	0.937275i	$0.613340\pi$
$450$	8.08582	0.381169
$451$	−13.8237	−0.650932
$452$	14.8404	0.698034
$453$	22.7891	1.07073
$454$	3.12335	0.146586
$455$	2.59673	0.121736
$456$	10.2912	0.481930
$457$	0.364591	0.0170548	0.00852742	−	0.999964i	$-0.497286\pi$
$457$	0.364591	0.0170548	0.00852742	+	0.999964i	$0.497286\pi$
$458$	17.5629	0.820660
$459$	2.18634	0.102050
$460$	1.65914	0.0773577
$461$	−12.6478	−0.589067	−0.294534	−	0.955641i	$-0.595164\pi$
$461$	−12.6478	−0.589067	−0.294534	+	0.955641i	$0.595164\pi$
$462$	37.7851	1.75792
$463$	−8.33258	−0.387248	−0.193624	−	0.981076i	$-0.562024\pi$
$463$	−8.33258	−0.387248	−0.193624	+	0.981076i	$0.562024\pi$
$464$	−2.79886	−0.129934
$465$	−10.4387	−0.484082
$466$	10.0961	0.467691
$467$	25.4423	1.17733	0.588665	−	0.808377i	$-0.299654\pi$
$467$	25.4423	1.17733	0.588665	+	0.808377i	$0.299654\pi$
$468$	1.57005	0.0725755
$469$	−37.0761	−1.71202
$470$	−6.99008	−0.322429
$471$	26.5872	1.22507
$472$	−0.211713	−0.00974486
$473$	2.93638	0.135015
$474$	18.5510	0.852074
$475$	−20.1840	−0.926104
$476$	3.29998	0.151254
$477$	1.89710	0.0868624
$478$	11.0713	0.506391
$479$	−10.4189	−0.476053	−0.238027	−	0.971259i	$-0.576501\pi$
$479$	−10.4189	−0.476053	−0.238027	+	0.971259i	$0.576501\pi$
$480$	1.80944	0.0825893
$481$	8.00385	0.364944
$482$	−13.2019	−0.601330
$483$	−16.8222	−0.765439
$484$	9.65107	0.438685
$485$	1.80793	0.0820940
$486$	−17.0331	−0.772638
$487$	2.63907	0.119587	0.0597937	−	0.998211i	$-0.480956\pi$
$487$	2.63907	0.119587	0.0597937	+	0.998211i	$0.480956\pi$
$488$	4.11766	0.186398
$489$	22.5547	1.01996
$490$	−5.90494	−0.266758
$491$	−42.8951	−1.93583	−0.967913	−	0.251284i	$-0.919147\pi$
$491$	−42.8951	−1.93583	−0.967913	+	0.251284i	$0.919147\pi$
$492$	6.71196	0.302598
$493$	−2.45098	−0.110387
$494$	−3.91918	−0.176332
$495$	6.96317	0.312971
$496$	5.76900	0.259036
$497$	54.6503	2.45140
$498$	18.1159	0.811793
$499$	26.4140	1.18245	0.591226	−	0.806506i	$-0.298644\pi$
$499$	26.4140	1.18245	0.591226	+	0.806506i	$0.298644\pi$
$500$	−7.64915	−0.342080
$501$	6.18008	0.276105
$502$	6.55068	0.292371
$503$	−2.81698	−0.125603	−0.0628014	−	0.998026i	$-0.520003\pi$
$503$	−2.81698	−0.125603	−0.0628014	+	0.998026i	$0.520003\pi$
$504$	−7.04110	−0.313636
$505$	10.8736	0.483869
$506$	−9.19403	−0.408724
$507$	27.1261	1.20471
$508$	9.21212	0.408722
$509$	8.04449	0.356566	0.178283	−	0.983979i	$-0.442946\pi$
$509$	8.04449	0.356566	0.178283	+	0.983979i	$0.442946\pi$
$510$	1.58454	0.0701646
$511$	−14.3683	−0.635618
$512$	−1.00000	−0.0441942
$513$	11.6448	0.514128
$514$	−9.47534	−0.417939
$515$	−13.5090	−0.595279
$516$	−1.42573	−0.0627644
$517$	38.7352	1.70357
$518$	−35.8944	−1.57711
$519$	35.9899	1.57978
$520$	−0.689086	−0.0302184
$521$	−20.1007	−0.880628	−0.440314	−	0.897844i	$-0.645133\pi$
$521$	−20.1007	−0.880628	−0.440314	+	0.897844i	$0.645133\pi$
$522$	5.22961	0.228894
$523$	−1.90447	−0.0832766	−0.0416383	−	0.999133i	$-0.513258\pi$
$523$	−1.90447	−0.0832766	−0.0416383	+	0.999133i	$0.513258\pi$
$524$	2.00387	0.0875394
$525$	35.9821	1.57039
$526$	3.28494	0.143230
$527$	5.05195	0.220066
$528$	−10.0269	−0.436366
$529$	−18.9067	−0.822032
$530$	−0.832628	−0.0361671
$531$	0.395580	0.0171667
$532$	17.5761	0.762022
$533$	−2.55610	−0.110717
$534$	7.89452	0.341629
$535$	15.3901	0.665371
$536$	9.83878	0.424971
$537$	29.8973	1.29017
$538$	14.9104	0.642831
$539$	32.7219	1.40943
$540$	2.04743	0.0881072
$541$	2.16766	0.0931948	0.0465974	−	0.998914i	$-0.485162\pi$
$541$	2.16766	0.0931948	0.0465974	+	0.998914i	$0.485162\pi$
$542$	−5.71899	−0.245652
$543$	−9.40428	−0.403576
$544$	−0.875706	−0.0375456
$545$	2.00898	0.0860550
$546$	6.98675	0.299005
$547$	10.4037	0.444831	0.222415	−	0.974952i	$-0.428606\pi$
$547$	10.4037	0.444831	0.222415	+	0.974952i	$0.428606\pi$
$548$	−4.14970	−0.177266
$549$	−7.69376	−0.328362
$550$	19.6656	0.838545
$551$	−13.0542	−0.556129
$552$	4.46407	0.190003
$553$	31.6827	1.34729
$554$	−17.7367	−0.753561
$555$	−17.2353	−0.731596
$556$	−9.30469	−0.394607
$557$	35.4704	1.50293	0.751466	−	0.659772i	$-0.229347\pi$
$557$	35.4704	1.50293	0.751466	+	0.659772i	$0.229347\pi$
$558$	−10.7792	−0.456322
$559$	0.542959	0.0229647
$560$	3.09030	0.130589
$561$	−8.78064	−0.370719
$562$	1.20985	0.0510344
$563$	16.6191	0.700411	0.350206	−	0.936673i	$-0.386112\pi$
$563$	16.6191	0.700411	0.350206	+	0.936673i	$0.386112\pi$
$564$	−18.8075	−0.791939
$565$	12.1701	0.511999
$566$	15.1762	0.637901
$567$	−41.8825	−1.75890
$568$	−14.5024	−0.608506
$569$	24.4048	1.02310	0.511551	−	0.859253i	$-0.329071\pi$
$569$	24.4048	1.02310	0.511551	+	0.859253i	$0.329071\pi$
$570$	8.43946	0.353490
$571$	1.17540	0.0491891	0.0245946	−	0.999698i	$-0.492171\pi$
$571$	1.17540	0.0491891	0.0245946	+	0.999698i	$0.492171\pi$
$572$	3.81853	0.159661
$573$	−34.8514	−1.45594
$574$	11.4632	0.478465
$575$	−8.75530	−0.365121
$576$	1.86848	0.0778532
$577$	0.951735	0.0396212	0.0198106	−	0.999804i	$-0.493694\pi$
$577$	0.951735	0.0396212	0.0198106	+	0.999804i	$0.493694\pi$
$578$	16.2331	0.675210
$579$	5.92735	0.246332
$580$	−2.29525	−0.0953049
$581$	30.9397	1.28360
$582$	4.86442	0.201637
$583$	4.61397	0.191091
$584$	3.81288	0.157778
$585$	1.28754	0.0532333
$586$	4.33939	0.179259
$587$	27.8523	1.14959	0.574793	−	0.818299i	$-0.305082\pi$
$587$	27.8523	1.14959	0.574793	+	0.818299i	$0.305082\pi$
$588$	−15.8878	−0.655202
$589$	26.9073	1.10870
$590$	−0.173618	−0.00714774
$591$	14.8769	0.611953
$592$	9.52518	0.391483
$593$	−32.1841	−1.32164	−0.660821	−	0.750543i	$-0.729792\pi$
$593$	−32.1841	−1.32164	−0.660821	+	0.750543i	$0.729792\pi$
$594$	−11.3457	−0.465520
$595$	2.70620	0.110943
$596$	−12.6280	−0.517265
$597$	−1.06797	−0.0437089
$598$	−1.70004	−0.0695199
$599$	−2.85374	−0.116601	−0.0583003	−	0.998299i	$-0.518568\pi$
$599$	−2.85374	−0.116601	−0.0583003	+	0.998299i	$0.518568\pi$
$600$	−9.54845	−0.389814
$601$	34.5058	1.40752	0.703761	−	0.710436i	$-0.251502\pi$
$601$	34.5058	1.40752	0.703761	+	0.710436i	$0.251502\pi$
$602$	−2.43497	−0.0992422
$603$	−18.3835	−0.748636
$604$	−10.3284	−0.420255
$605$	7.91450	0.321770
$606$	29.2565	1.18846
$607$	−18.0771	−0.733727	−0.366864	−	0.930275i	$-0.619568\pi$
$607$	−18.0771	−0.733727	−0.366864	+	0.930275i	$0.619568\pi$
$608$	−4.66412	−0.189155
$609$	23.2719	0.943023
$610$	3.37675	0.136721
$611$	7.16242	0.289761
$612$	1.63624	0.0661409
$613$	35.3112	1.42620	0.713102	−	0.701060i	$-0.247289\pi$
$613$	35.3112	1.42620	0.713102	+	0.701060i	$0.247289\pi$
$614$	−18.0645	−0.729022
$615$	5.50424	0.221952
$616$	−17.1248	−0.689976
$617$	4.42331	0.178076	0.0890379	−	0.996028i	$-0.471621\pi$
$617$	4.42331	0.178076	0.0890379	+	0.996028i	$0.471621\pi$
$618$	−36.3474	−1.46211
$619$	−7.51064	−0.301878	−0.150939	−	0.988543i	$-0.548230\pi$
$619$	−7.51064	−0.301878	−0.150939	+	0.988543i	$0.548230\pi$
$620$	4.73095	0.190000
$621$	5.05120	0.202698
$622$	−24.5086	−0.982707
$623$	13.4829	0.540180
$624$	−1.85405	−0.0742215
$625$	15.3647	0.614587
$626$	−18.2088	−0.727769
$627$	−46.7668	−1.86769
$628$	−12.0497	−0.480835
$629$	8.34126	0.332588
$630$	−5.77416	−0.230048
$631$	6.26500	0.249406	0.124703	−	0.992194i	$-0.460202\pi$
$631$	6.26500	0.249406	0.124703	+	0.992194i	$0.460202\pi$
$632$	−8.40756	−0.334435
$633$	25.4537	1.01169
$634$	−19.1210	−0.759392
$635$	7.55453	0.299792
$636$	−2.24027	−0.0888324
$637$	6.05052	0.239730
$638$	12.7190	0.503550
$639$	27.0974	1.07195
$640$	−0.820064	−0.0324159
$641$	−15.2403	−0.601956	−0.300978	−	0.953631i	$-0.597313\pi$
$641$	−15.2403	−0.601956	−0.300978	+	0.953631i	$0.597313\pi$
$642$	41.4085	1.63426
$643$	−25.1206	−0.990659	−0.495330	−	0.868705i	$-0.664953\pi$
$643$	−25.1206	−0.990659	−0.495330	+	0.868705i	$0.664953\pi$
$644$	7.62408	0.300431
$645$	−1.16919	−0.0460369
$646$	−4.08440	−0.160699
$647$	−29.4767	−1.15885	−0.579425	−	0.815026i	$-0.696723\pi$
$647$	−29.4767	−1.15885	−0.579425	+	0.815026i	$0.696723\pi$
$648$	11.1142	0.436608
$649$	0.962095	0.0377655
$650$	3.63632	0.142628
$651$	−47.9678	−1.88001
$652$	−10.2221	−0.400329
$653$	30.8779	1.20835	0.604173	−	0.796853i	$-0.293504\pi$
$653$	30.8779	1.20835	0.604173	+	0.796853i	$0.293504\pi$
$654$	5.40534	0.211366
$655$	1.64330	0.0642091
$656$	−3.04195	−0.118768
$657$	−7.12429	−0.277945
$658$	−32.1209	−1.25220
$659$	34.2969	1.33602	0.668008	−	0.744154i	$-0.267147\pi$
$659$	34.2969	1.33602	0.668008	+	0.744154i	$0.267147\pi$
$660$	−8.22272	−0.320069
$661$	−5.29625	−0.206000	−0.103000	−	0.994681i	$-0.532844\pi$
$661$	−5.29625	−0.206000	−0.103000	+	0.994681i	$0.532844\pi$
$662$	−8.37562	−0.325528
$663$	−1.62361	−0.0630556
$664$	−8.21039	−0.318625
$665$	14.4136	0.558934
$666$	−17.7976	−0.689642
$667$	−5.66260	−0.219257
$668$	−2.80090	−0.108370
$669$	47.6310	1.84152
$670$	8.06844	0.311711
$671$	−18.7121	−0.722372
$672$	8.31476	0.320749
$673$	−34.7989	−1.34140	−0.670699	−	0.741729i	$-0.734006\pi$
$673$	−34.7989	−1.34140	−0.670699	+	0.741729i	$0.734006\pi$
$674$	−9.12869	−0.351624
$675$	−10.8043	−0.415858
$676$	−12.2939	−0.472843
$677$	−16.8818	−0.648820	−0.324410	−	0.945917i	$-0.605166\pi$
$677$	−16.8818	−0.648820	−0.324410	+	0.945917i	$0.605166\pi$
$678$	32.7448	1.25756
$679$	8.30782	0.318825
$680$	−0.718135	−0.0275392
$681$	6.89156	0.264085
$682$	−26.2163	−1.00388
$683$	44.3349	1.69643	0.848214	−	0.529654i	$-0.177678\pi$
$683$	44.3349	1.69643	0.848214	+	0.529654i	$0.177678\pi$
$684$	8.71481	0.333219
$685$	−3.40302	−0.130023
$686$	−0.755861	−0.0288589
$687$	38.7519	1.47848
$688$	0.646162	0.0246347
$689$	0.853156	0.0325027
$690$	3.66082	0.139365
$691$	19.3824	0.737340	0.368670	−	0.929560i	$-0.379813\pi$
$691$	19.3824	0.737340	0.368670	+	0.929560i	$0.379813\pi$
$692$	−16.3112	−0.620057
$693$	31.9972	1.21547
$694$	17.4568	0.662652
$695$	−7.63045	−0.289439
$696$	−6.17558	−0.234085
$697$	−2.66386	−0.100901
$698$	8.14021	0.308111
$699$	22.2766	0.842578
$700$	−16.3076	−0.616369
$701$	10.2683	0.387827	0.193914	−	0.981019i	$-0.437882\pi$
$701$	10.2683	0.387827	0.193914	+	0.981019i	$0.437882\pi$
$702$	−2.09790	−0.0791803
$703$	44.4266	1.67558
$704$	4.54434	0.171271
$705$	−15.4234	−0.580877
$706$	1.55831	0.0586478
$707$	49.9665	1.87918
$708$	−0.467136	−0.0175560
$709$	21.5733	0.810202	0.405101	−	0.914272i	$-0.367236\pi$
$709$	21.5733	0.810202	0.405101	+	0.914272i	$0.367236\pi$
$710$	−11.8929	−0.446332
$711$	15.7093	0.589146
$712$	−3.57791	−0.134088
$713$	11.6717	0.437110
$714$	7.28129	0.272495
$715$	3.13144	0.117109
$716$	−13.5499	−0.506383
$717$	24.4285	0.912298
$718$	30.0259	1.12056
$719$	−10.7998	−0.402764	−0.201382	−	0.979513i	$-0.564543\pi$
$719$	−10.7998	−0.402764	−0.201382	+	0.979513i	$0.564543\pi$
$720$	1.53227	0.0571044
$721$	−62.0768	−2.31186
$722$	−2.75405	−0.102495
$723$	−29.1295	−1.08334
$724$	4.26215	0.158402
$725$	12.1121	0.449830
$726$	21.2947	0.790321
$727$	19.2113	0.712509	0.356255	−	0.934389i	$-0.384054\pi$
$727$	19.2113	0.712509	0.356255	+	0.934389i	$0.384054\pi$
$728$	−3.16649	−0.117358
$729$	−4.24028	−0.157047
$730$	3.12681	0.115728
$731$	0.565848	0.0209286
$732$	9.08547	0.335809
$733$	−8.48466	−0.313388	−0.156694	−	0.987647i	$-0.550084\pi$
$733$	−8.48466	−0.313388	−0.156694	+	0.987647i	$0.550084\pi$
$734$	4.49338	0.165854
$735$	−13.0290	−0.480583
$736$	−2.02318	−0.0745754
$737$	−44.7108	−1.64694
$738$	5.68382	0.209224
$739$	−29.9532	−1.10185	−0.550923	−	0.834556i	$-0.685724\pi$
$739$	−29.9532	−1.10185	−0.550923	+	0.834556i	$0.685724\pi$
$740$	7.81126	0.287148
$741$	−8.64753	−0.317675
$742$	−3.82610	−0.140461
$743$	−22.0591	−0.809272	−0.404636	−	0.914478i	$-0.632602\pi$
$743$	−22.0591	−0.809272	−0.404636	+	0.914478i	$0.632602\pi$
$744$	12.7291	0.466671
$745$	−10.3558	−0.379407
$746$	2.21177	0.0809787
$747$	15.3409	0.561295
$748$	3.97951	0.145505
$749$	70.7206	2.58407
$750$	−16.8776	−0.616281
$751$	−3.79573	−0.138508	−0.0692541	−	0.997599i	$-0.522062\pi$
$751$	−3.79573	−0.138508	−0.0692541	+	0.997599i	$0.522062\pi$
$752$	8.52382	0.310832
$753$	14.4538	0.526727
$754$	2.35183	0.0856488
$755$	−8.46992	−0.308252
$756$	9.40834	0.342178
$757$	−3.44916	−0.125362	−0.0626810	−	0.998034i	$-0.519965\pi$
$757$	−3.44916	−0.125362	−0.0626810	+	0.998034i	$0.519965\pi$
$758$	20.1489	0.731841
$759$	−20.2863	−0.736345
$760$	−3.82488	−0.138743
$761$	23.6940	0.858908	0.429454	−	0.903089i	$-0.358706\pi$
$761$	23.6940	0.858908	0.429454	+	0.903089i	$0.358706\pi$
$762$	20.3262	0.736340
$763$	9.23166	0.334208
$764$	15.7952	0.571449
$765$	1.34182	0.0485136
$766$	−8.13337	−0.293870
$767$	0.177898	0.00642354
$768$	−2.20646	−0.0796189
$769$	−1.69748	−0.0612127	−0.0306064	−	0.999532i	$-0.509744\pi$
$769$	−1.69748	−0.0612127	−0.0306064	+	0.999532i	$0.509744\pi$
$770$	−14.0434	−0.506089
$771$	−20.9070	−0.752947
$772$	−2.68636	−0.0966842
$773$	40.2868	1.44902	0.724509	−	0.689266i	$-0.242067\pi$
$773$	40.2868	1.44902	0.724509	+	0.689266i	$0.242067\pi$
$774$	−1.20734	−0.0433969
$775$	−24.9653	−0.896781
$776$	−2.20462	−0.0791413
$777$	−79.1996	−2.84127
$778$	21.5042	0.770963
$779$	−14.1881	−0.508340
$780$	−1.52044	−0.0544406
$781$	65.9038	2.35822
$782$	−1.77171	−0.0633563
$783$	−6.98782	−0.249724
$784$	7.20058	0.257164
$785$	−9.88151	−0.352686
$786$	4.42146	0.157708
$787$	−55.5819	−1.98128	−0.990640	−	0.136501i	$-0.956414\pi$
$787$	−55.5819	−1.98128	−0.990640	+	0.136501i	$0.956414\pi$
$788$	−6.74241	−0.240189
$789$	7.24809	0.258039
$790$	−6.89474	−0.245304
$791$	55.9241	1.98843
$792$	−8.49100	−0.301715
$793$	−3.46000	−0.122868
$794$	12.4900	0.443252
$795$	−1.83716	−0.0651575
$796$	0.484017	0.0171555
$797$	41.0799	1.45512	0.727562	−	0.686042i	$-0.240653\pi$
$797$	41.0799	1.45512	0.727562	+	0.686042i	$0.240653\pi$
$798$	38.7811	1.37283
$799$	7.46437	0.264070
$800$	4.32749	0.153000
$801$	6.68524	0.236211
$802$	32.0614	1.13213
$803$	−17.3271	−0.611459
$804$	21.7089	0.765614
$805$	6.25224	0.220362
$806$	−4.84759	−0.170749
$807$	32.8992	1.15811
$808$	−13.2595	−0.466466
$809$	−13.8525	−0.487029	−0.243514	−	0.969897i	$-0.578300\pi$
$809$	−13.8525	−0.487029	−0.243514	+	0.969897i	$0.578300\pi$
$810$	9.11438	0.320247
$811$	−51.1215	−1.79512	−0.897559	−	0.440895i	$-0.854661\pi$
$811$	−51.1215	−1.79512	−0.897559	+	0.440895i	$0.854661\pi$
$812$	−10.5471	−0.370132
$813$	−12.6187	−0.442558
$814$	−43.2857	−1.51716
$815$	−8.38278	−0.293636
$816$	−1.93221	−0.0676410
$817$	3.01378	0.105439
$818$	−14.7645	−0.516229
$819$	5.91652	0.206740
$820$	−2.49460	−0.0871152
$821$	51.9537	1.81320	0.906598	−	0.421994i	$-0.138670\pi$
$821$	51.9537	1.81320	0.906598	+	0.421994i	$0.138670\pi$
$822$	−9.15615	−0.319357
$823$	−18.2886	−0.637499	−0.318750	−	0.947839i	$-0.603263\pi$
$823$	−18.2886	−0.637499	−0.318750	+	0.947839i	$0.603263\pi$
$824$	16.4731	0.573869
$825$	43.3915	1.51070
$826$	−0.797811	−0.0277594
$827$	43.1183	1.49937	0.749685	−	0.661795i	$-0.230205\pi$
$827$	43.1183	1.49937	0.749685	+	0.661795i	$0.230205\pi$
$828$	3.78026	0.131373
$829$	−23.2368	−0.807047	−0.403523	−	0.914969i	$-0.632215\pi$
$829$	−23.2368	−0.807047	−0.403523	+	0.914969i	$0.632215\pi$
$830$	−6.73305	−0.233707
$831$	−39.1354	−1.35759
$832$	0.840283	0.0291316
$833$	6.30559	0.218476
$834$	−20.5305	−0.710912
$835$	−2.29692	−0.0794881
$836$	21.1954	0.733058
$837$	14.4033	0.497849
$838$	−27.7269	−0.957810
$839$	3.81976	0.131873	0.0659364	−	0.997824i	$-0.478997\pi$
$839$	3.81976	0.131873	0.0659364	+	0.997824i	$0.478997\pi$
$840$	6.81864	0.235265
$841$	−21.1664	−0.729875
$842$	−21.6861	−0.747353
$843$	2.66949	0.0919421
$844$	−11.5360	−0.397084
$845$	−10.0818	−0.346825
$846$	−15.9266	−0.547567
$847$	36.3687	1.24965
$848$	1.01532	0.0348663
$849$	33.4856	1.14922
$850$	3.78961	0.129983
$851$	19.2712	0.660607
$852$	−31.9990	−1.09627
$853$	49.5706	1.69726	0.848632	−	0.528984i	$-0.177427\pi$
$853$	49.5706	1.69726	0.848632	+	0.528984i	$0.177427\pi$
$854$	15.5169	0.530976
$855$	7.14670	0.244412
$856$	−18.7669	−0.641440
$857$	20.6754	0.706257	0.353129	−	0.935575i	$-0.385118\pi$
$857$	20.6754	0.706257	0.353129	+	0.935575i	$0.385118\pi$
$858$	8.42545	0.287640
$859$	−19.3690	−0.660861	−0.330431	−	0.943830i	$-0.607194\pi$
$859$	−19.3690	−0.660861	−0.330431	+	0.943830i	$0.607194\pi$
$860$	0.529894	0.0180692
$861$	25.2931	0.861987
$862$	29.0119	0.988151
$863$	12.1093	0.412206	0.206103	−	0.978530i	$-0.433922\pi$
$863$	12.1093	0.412206	0.206103	+	0.978530i	$0.433922\pi$
$864$	−2.49666	−0.0849382
$865$	−13.3762	−0.454804
$866$	−27.5784	−0.937153
$867$	35.8178	1.21644
$868$	21.7397	0.737894
$869$	38.2068	1.29608
$870$	−5.06437	−0.171698
$871$	−8.26736	−0.280129
$872$	−2.44978	−0.0829599
$873$	4.11929	0.139417
$874$	−9.43636	−0.319190
$875$	−28.8248	−0.974455
$876$	8.41299	0.284248
$877$	10.4404	0.352549	0.176274	−	0.984341i	$-0.443595\pi$
$877$	10.4404	0.352549	0.176274	+	0.984341i	$0.443595\pi$
$878$	10.2648	0.346419
$879$	9.57471	0.322947
$880$	3.72665	0.125626
$881$	14.3535	0.483582	0.241791	−	0.970328i	$-0.422265\pi$
$881$	14.3535	0.483582	0.241791	+	0.970328i	$0.422265\pi$
$882$	−13.4541	−0.453024
$883$	35.3529	1.18972	0.594860	−	0.803829i	$-0.297207\pi$
$883$	35.3529	1.18972	0.594860	+	0.803829i	$0.297207\pi$
$884$	0.735841	0.0247490
$885$	−0.383082	−0.0128771
$886$	−26.0470	−0.875064
$887$	−10.1684	−0.341420	−0.170710	−	0.985321i	$-0.554606\pi$
$887$	−10.1684	−0.341420	−0.170710	+	0.985321i	$0.554606\pi$
$888$	21.0170	0.705283
$889$	34.7146	1.16429
$890$	−2.93411	−0.0983517
$891$	−50.5069	−1.69204
$892$	−21.5870	−0.722787
$893$	39.7562	1.33039
$894$	−27.8633	−0.931888
$895$	−11.1118	−0.371426
$896$	−3.76837	−0.125892
$897$	−3.75108	−0.125245
$898$	14.7730	0.492982
$899$	−16.1466	−0.538520
$900$	−8.08582	−0.269527
$901$	0.889123	0.0296210
$902$	13.8237	0.460279
$903$	−5.37268	−0.178792
$904$	−14.8404	−0.493585
$905$	3.49524	0.116186
$906$	−22.7891	−0.757119
$907$	28.2127	0.936786	0.468393	−	0.883520i	$-0.344833\pi$
$907$	28.2127	0.936786	0.468393	+	0.883520i	$0.344833\pi$
$908$	−3.12335	−0.103652
$909$	24.7750	0.821734
$910$	−2.59673	−0.0860807
$911$	34.5958	1.14621	0.573106	−	0.819481i	$-0.305738\pi$
$911$	34.5958	1.14621	0.573106	+	0.819481i	$0.305738\pi$
$912$	−10.2912	−0.340776
$913$	37.3108	1.23481
$914$	−0.364591	−0.0120596
$915$	7.45067	0.246312
$916$	−17.5629	−0.580294
$917$	7.55131	0.249366
$918$	−2.18634	−0.0721601
$919$	−24.2466	−0.799820	−0.399910	−	0.916554i	$-0.630959\pi$
$919$	−24.2466	−0.799820	−0.399910	+	0.916554i	$0.630959\pi$
$920$	−1.65914	−0.0547001
$921$	−39.8585	−1.31338
$922$	12.6478	0.416533
$923$	12.1861	0.401110
$924$	−37.7851	−1.24304
$925$	−41.2202	−1.35531
$926$	8.33258	0.273826
$927$	−30.7797	−1.01094
$928$	2.79886	0.0918771
$929$	13.0660	0.428683	0.214342	−	0.976759i	$-0.431239\pi$
$929$	13.0660	0.428683	0.214342	+	0.976759i	$0.431239\pi$
$930$	10.4387	0.342297
$931$	33.5844	1.10068
$932$	−10.0961	−0.330708
$933$	−54.0774	−1.77042
$934$	−25.4423	−0.832498
$935$	3.26345	0.106726
$936$	−1.57005	−0.0513186
$937$	−36.0525	−1.17778	−0.588892	−	0.808212i	$-0.700436\pi$
$937$	−36.0525	−1.17778	−0.588892	+	0.808212i	$0.700436\pi$
$938$	37.0761	1.21058
$939$	−40.1770	−1.31113
$940$	6.99008	0.227991
$941$	−42.3444	−1.38039	−0.690194	−	0.723625i	$-0.742475\pi$
$941$	−42.3444	−1.38039	−0.690194	+	0.723625i	$0.742475\pi$
$942$	−26.5872	−0.866257
$943$	−6.15442	−0.200416
$944$	0.211713	0.00689066
$945$	7.71545	0.250984
$946$	−2.93638	−0.0954700
$947$	−9.84702	−0.319985	−0.159993	−	0.987118i	$-0.551147\pi$
$947$	−9.84702	−0.319985	−0.159993	+	0.987118i	$0.551147\pi$
$948$	−18.5510	−0.602507
$949$	−3.20390	−0.104003
$950$	20.1840	0.654855
$951$	−42.1898	−1.36810
$952$	−3.29998	−0.106953
$953$	48.9246	1.58482	0.792411	−	0.609988i	$-0.208826\pi$
$953$	48.9246	1.58482	0.792411	+	0.609988i	$0.208826\pi$
$954$	−1.89710	−0.0614210
$955$	12.9530	0.419151
$956$	−11.0713	−0.358072
$957$	28.0640	0.907179
$958$	10.4189	0.336621
$959$	−15.6376	−0.504964
$960$	−1.80944	−0.0583995
$961$	2.28137	0.0735926
$962$	−8.00385	−0.258054
$963$	35.0655	1.12997
$964$	13.2019	0.425204
$965$	−2.20299	−0.0709167
$966$	16.8222	0.541247
$967$	−28.9255	−0.930182	−0.465091	−	0.885263i	$-0.653978\pi$
$967$	−28.9255	−0.930182	−0.465091	+	0.885263i	$0.653978\pi$
$968$	−9.65107	−0.310197
$969$	−9.01208	−0.289510
$970$	−1.80793	−0.0580492
$971$	41.1028	1.31905	0.659526	−	0.751682i	$-0.270757\pi$
$971$	41.1028	1.31905	0.659526	+	0.751682i	$0.270757\pi$
$972$	17.0331	0.546338
$973$	−35.0635	−1.12408
$974$	−2.63907	−0.0845611
$975$	8.02340	0.256954
$976$	−4.11766	−0.131803
$977$	15.5964	0.498972	0.249486	−	0.968378i	$-0.419738\pi$
$977$	15.5964	0.498972	0.249486	+	0.968378i	$0.419738\pi$
$978$	−22.5547	−0.721219
$979$	16.2592	0.519648
$980$	5.90494	0.188626
$981$	4.57735	0.146144
$982$	42.8951	1.36884
$983$	−45.3722	−1.44715	−0.723574	−	0.690247i	$-0.757502\pi$
$983$	−45.3722	−1.44715	−0.723574	+	0.690247i	$0.757502\pi$
$984$	−6.71196	−0.213969
$985$	−5.52921	−0.176175
$986$	2.45098	0.0780551
$987$	−70.8735	−2.25593
$988$	3.91918	0.124686
$989$	1.30730	0.0415698
$990$	−6.96317	−0.221304
$991$	5.10258	0.162089	0.0810444	−	0.996710i	$-0.474174\pi$
$991$	5.10258	0.162089	0.0810444	+	0.996710i	$0.474174\pi$
$992$	−5.76900	−0.183166
$993$	−18.4805	−0.586461
$994$	−54.6503	−1.73340
$995$	0.396925	0.0125834
$996$	−18.1159	−0.574025
$997$	−23.6959	−0.750458	−0.375229	−	0.926932i	$-0.622436\pi$
$997$	−23.6959	−0.750458	−0.375229	+	0.926932i	$0.622436\pi$
$998$	−26.4140	−0.836120
$999$	23.7812	0.752403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.10	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.10	✓	83	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.20646 q^{3} +1.00000 q^{4} +0.820064 q^{5} +2.20646 q^{6} +3.76837 q^{7} -1.00000 q^{8} +1.86848 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.20646 q^{3} +1.00000 q^{4} +0.820064 q^{5} +2.20646 q^{6} +3.76837 q^{7} -1.00000 q^{8} +1.86848 q^{9} -0.820064 q^{10} +4.54434 q^{11} -2.20646 q^{12} +0.840283 q^{13} -3.76837 q^{14} -1.80944 q^{15} +1.00000 q^{16} +0.875706 q^{17} -1.86848 q^{18} +4.66412 q^{19} +0.820064 q^{20} -8.31476 q^{21} -4.54434 q^{22} +2.02318 q^{23} +2.20646 q^{24} -4.32749 q^{25} -0.840283 q^{26} +2.49666 q^{27} +3.76837 q^{28} -2.79886 q^{29} +1.80944 q^{30} +5.76900 q^{31} -1.00000 q^{32} -10.0269 q^{33} -0.875706 q^{34} +3.09030 q^{35} +1.86848 q^{36} +9.52518 q^{37} -4.66412 q^{38} -1.85405 q^{39} -0.820064 q^{40} -3.04195 q^{41} +8.31476 q^{42} +0.646162 q^{43} +4.54434 q^{44} +1.53227 q^{45} -2.02318 q^{46} +8.52382 q^{47} -2.20646 q^{48} +7.20058 q^{49} +4.32749 q^{50} -1.93221 q^{51} +0.840283 q^{52} +1.01532 q^{53} -2.49666 q^{54} +3.72665 q^{55} -3.76837 q^{56} -10.2912 q^{57} +2.79886 q^{58} +0.211713 q^{59} -1.80944 q^{60} -4.11766 q^{61} -5.76900 q^{62} +7.04110 q^{63} +1.00000 q^{64} +0.689086 q^{65} +10.0269 q^{66} -9.83878 q^{67} +0.875706 q^{68} -4.46407 q^{69} -3.09030 q^{70} +14.5024 q^{71} -1.86848 q^{72} -3.81288 q^{73} -9.52518 q^{74} +9.54845 q^{75} +4.66412 q^{76} +17.1248 q^{77} +1.85405 q^{78} +8.40756 q^{79} +0.820064 q^{80} -11.1142 q^{81} +3.04195 q^{82} +8.21039 q^{83} -8.31476 q^{84} +0.718135 q^{85} -0.646162 q^{86} +6.17558 q^{87} -4.54434 q^{88} +3.57791 q^{89} -1.53227 q^{90} +3.16649 q^{91} +2.02318 q^{92} -12.7291 q^{93} -8.52382 q^{94} +3.82488 q^{95} +2.20646 q^{96} +2.20462 q^{97} -7.20058 q^{98} +8.49100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

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