LMFDB - Embedded newform 8038.2.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$84$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.22723 q^{3} +1.00000 q^{4} -4.01107 q^{5} +3.22723 q^{6} +2.38054 q^{7} -1.00000 q^{8} +7.41504 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.22723 q^{3} +1.00000 q^{4} -4.01107 q^{5} +3.22723 q^{6} +2.38054 q^{7} -1.00000 q^{8} +7.41504 q^{9} +4.01107 q^{10} +5.38222 q^{11} -3.22723 q^{12} -6.05302 q^{13} -2.38054 q^{14} +12.9447 q^{15} +1.00000 q^{16} -0.311510 q^{17} -7.41504 q^{18} +0.634072 q^{19} -4.01107 q^{20} -7.68256 q^{21} -5.38222 q^{22} -2.91431 q^{23} +3.22723 q^{24} +11.0887 q^{25} +6.05302 q^{26} -14.2483 q^{27} +2.38054 q^{28} +10.4573 q^{29} -12.9447 q^{30} +4.04978 q^{31} -1.00000 q^{32} -17.3697 q^{33} +0.311510 q^{34} -9.54851 q^{35} +7.41504 q^{36} +3.03943 q^{37} -0.634072 q^{38} +19.5345 q^{39} +4.01107 q^{40} -11.2167 q^{41} +7.68256 q^{42} -12.9395 q^{43} +5.38222 q^{44} -29.7422 q^{45} +2.91431 q^{46} +8.17795 q^{47} -3.22723 q^{48} -1.33303 q^{49} -11.0887 q^{50} +1.00532 q^{51} -6.05302 q^{52} -12.1765 q^{53} +14.2483 q^{54} -21.5885 q^{55} -2.38054 q^{56} -2.04630 q^{57} -10.4573 q^{58} -3.06613 q^{59} +12.9447 q^{60} +5.38097 q^{61} -4.04978 q^{62} +17.6518 q^{63} +1.00000 q^{64} +24.2791 q^{65} +17.3697 q^{66} -7.01888 q^{67} -0.311510 q^{68} +9.40516 q^{69} +9.54851 q^{70} -9.30452 q^{71} -7.41504 q^{72} +13.6826 q^{73} -3.03943 q^{74} -35.7858 q^{75} +0.634072 q^{76} +12.8126 q^{77} -19.5345 q^{78} +8.33989 q^{79} -4.01107 q^{80} +23.7376 q^{81} +11.2167 q^{82} -7.44115 q^{83} -7.68256 q^{84} +1.24949 q^{85} +12.9395 q^{86} -33.7481 q^{87} -5.38222 q^{88} -4.29789 q^{89} +29.7422 q^{90} -14.4095 q^{91} -2.91431 q^{92} -13.0696 q^{93} -8.17795 q^{94} -2.54331 q^{95} +3.22723 q^{96} -9.59444 q^{97} +1.33303 q^{98} +39.9093 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) $ 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * 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
$3$	−3.22723	−1.86324	−0.931622	−	0.363429i	$-0.881606\pi$
$3$	−3.22723	−1.86324	−0.931622	+	0.363429i	$0.881606\pi$
$4$	1.00000	0.500000
$5$	−4.01107	−1.79381	−0.896903	−	0.442228i	$-0.854188\pi$
$5$	−4.01107	−1.79381	−0.896903	+	0.442228i	$0.854188\pi$
$6$	3.22723	1.31751
$7$	2.38054	0.899759	0.449880	−	0.893089i	$-0.351467\pi$
$7$	2.38054	0.899759	0.449880	+	0.893089i	$0.351467\pi$
$8$	−1.00000	−0.353553
$9$	7.41504	2.47168
$10$	4.01107	1.26841
$11$	5.38222	1.62280	0.811400	−	0.584491i	$-0.198706\pi$
$11$	5.38222	1.62280	0.811400	+	0.584491i	$0.198706\pi$
$12$	−3.22723	−0.931622
$13$	−6.05302	−1.67881	−0.839403	−	0.543510i	$-0.817095\pi$
$13$	−6.05302	−1.67881	−0.839403	+	0.543510i	$0.817095\pi$
$14$	−2.38054	−0.636226
$15$	12.9447	3.34230
$16$	1.00000	0.250000
$17$	−0.311510	−0.0755523	−0.0377761	−	0.999286i	$-0.512027\pi$
$17$	−0.311510	−0.0755523	−0.0377761	+	0.999286i	$0.512027\pi$
$18$	−7.41504	−1.74774
$19$	0.634072	0.145466	0.0727331	−	0.997351i	$-0.476828\pi$
$19$	0.634072	0.145466	0.0727331	+	0.997351i	$0.476828\pi$
$20$	−4.01107	−0.896903
$21$	−7.68256	−1.67647
$22$	−5.38222	−1.14749
$23$	−2.91431	−0.607676	−0.303838	−	0.952724i	$-0.598268\pi$
$23$	−2.91431	−0.607676	−0.303838	+	0.952724i	$0.598268\pi$
$24$	3.22723	0.658756
$25$	11.0887	2.21774
$26$	6.05302	1.18709
$27$	−14.2483	−2.74210
$28$	2.38054	0.449880
$29$	10.4573	1.94187	0.970936	−	0.239340i	$-0.0769312\pi$
$29$	10.4573	1.94187	0.970936	+	0.239340i	$0.0769312\pi$
$30$	−12.9447	−2.36336
$31$	4.04978	0.727361	0.363681	−	0.931524i	$-0.381520\pi$
$31$	4.04978	0.727361	0.363681	+	0.931524i	$0.381520\pi$
$32$	−1.00000	−0.176777
$33$	−17.3697	−3.02367
$34$	0.311510	0.0534235
$35$	−9.54851	−1.61399
$36$	7.41504	1.23584
$37$	3.03943	0.499680	0.249840	−	0.968287i	$-0.419622\pi$
$37$	3.03943	0.499680	0.249840	+	0.968287i	$0.419622\pi$
$38$	−0.634072	−0.102860
$39$	19.5345	3.12802
$40$	4.01107	0.634206
$41$	−11.2167	−1.75175	−0.875877	−	0.482535i	$-0.839716\pi$
$41$	−11.2167	−1.75175	−0.875877	+	0.482535i	$0.839716\pi$
$42$	7.68256	1.18544
$43$	−12.9395	−1.97326	−0.986630	−	0.162975i	$-0.947891\pi$
$43$	−12.9395	−1.97326	−0.986630	+	0.162975i	$0.947891\pi$
$44$	5.38222	0.811400
$45$	−29.7422	−4.43371
$46$	2.91431	0.429692
$47$	8.17795	1.19288	0.596438	−	0.802659i	$-0.296582\pi$
$47$	8.17795	1.19288	0.596438	+	0.802659i	$0.296582\pi$
$48$	−3.22723	−0.465811
$49$	−1.33303	−0.190433
$50$	−11.0887	−1.56818
$51$	1.00532	0.140772
$52$	−6.05302	−0.839403
$53$	−12.1765	−1.67257	−0.836286	−	0.548294i	$-0.815277\pi$
$53$	−12.1765	−1.67257	−0.836286	+	0.548294i	$0.815277\pi$
$54$	14.2483	1.93895
$55$	−21.5885	−2.91099
$56$	−2.38054	−0.318113
$57$	−2.04630	−0.271039
$58$	−10.4573	−1.37311
$59$	−3.06613	−0.399176	−0.199588	−	0.979880i	$-0.563960\pi$
$59$	−3.06613	−0.399176	−0.199588	+	0.979880i	$0.563960\pi$
$60$	12.9447	1.67115
$61$	5.38097	0.688963	0.344481	−	0.938793i	$-0.388055\pi$
$61$	5.38097	0.688963	0.344481	+	0.938793i	$0.388055\pi$
$62$	−4.04978	−0.514322
$63$	17.6518	2.22392
$64$	1.00000	0.125000
$65$	24.2791	3.01145
$66$	17.3697	2.13806
$67$	−7.01888	−0.857493	−0.428747	−	0.903425i	$-0.641045\pi$
$67$	−7.01888	−0.857493	−0.428747	+	0.903425i	$0.641045\pi$
$68$	−0.311510	−0.0377761
$69$	9.40516	1.13225
$70$	9.54851	1.14127
$71$	−9.30452	−1.10424	−0.552122	−	0.833763i	$-0.686182\pi$
$71$	−9.30452	−1.10424	−0.552122	+	0.833763i	$0.686182\pi$
$72$	−7.41504	−0.873870
$73$	13.6826	1.60143	0.800716	−	0.599044i	$-0.204452\pi$
$73$	13.6826	1.60143	0.800716	+	0.599044i	$0.204452\pi$
$74$	−3.03943	−0.353327
$75$	−35.7858	−4.13219
$76$	0.634072	0.0727331
$77$	12.8126	1.46013
$78$	−19.5345	−2.21185
$79$	8.33989	0.938311	0.469156	−	0.883115i	$-0.344558\pi$
$79$	8.33989	0.938311	0.469156	+	0.883115i	$0.344558\pi$
$80$	−4.01107	−0.448451
$81$	23.7376	2.63752
$82$	11.2167	1.23868
$83$	−7.44115	−0.816772	−0.408386	−	0.912809i	$-0.633908\pi$
$83$	−7.44115	−0.816772	−0.408386	+	0.912809i	$0.633908\pi$
$84$	−7.68256	−0.838236
$85$	1.24949	0.135526
$86$	12.9395	1.39531
$87$	−33.7481	−3.61818
$88$	−5.38222	−0.573746
$89$	−4.29789	−0.455576	−0.227788	−	0.973711i	$-0.573149\pi$
$89$	−4.29789	−0.455576	−0.227788	+	0.973711i	$0.573149\pi$
$90$	29.7422	3.13511
$91$	−14.4095	−1.51052
$92$	−2.91431	−0.303838
$93$	−13.0696	−1.35525
$94$	−8.17795	−0.843491
$95$	−2.54331	−0.260938
$96$	3.22723	0.329378
$97$	−9.59444	−0.974168	−0.487084	−	0.873355i	$-0.661939\pi$
$97$	−9.59444	−0.974168	−0.487084	+	0.873355i	$0.661939\pi$
$98$	1.33303	0.134656
$99$	39.9093	4.01104
$100$	11.0887	1.10887
$101$	2.63434	0.262127	0.131063	−	0.991374i	$-0.458161\pi$
$101$	2.63434	0.262127	0.131063	+	0.991374i	$0.458161\pi$
$102$	−1.00532	−0.0995410
$103$	−10.2047	−1.00550	−0.502748	−	0.864433i	$-0.667678\pi$
$103$	−10.2047	−1.00550	−0.502748	+	0.864433i	$0.667678\pi$
$104$	6.05302	0.593547
$105$	30.8153	3.00726
$106$	12.1765	1.18269
$107$	−0.291305	−0.0281615	−0.0140808	−	0.999901i	$-0.504482\pi$
$107$	−0.291305	−0.0281615	−0.0140808	+	0.999901i	$0.504482\pi$
$108$	−14.2483	−1.37105
$109$	9.46430	0.906515	0.453258	−	0.891380i	$-0.350262\pi$
$109$	9.46430	0.906515	0.453258	+	0.891380i	$0.350262\pi$
$110$	21.5885	2.05838
$111$	−9.80896	−0.931026
$112$	2.38054	0.224940
$113$	13.0775	1.23022	0.615112	−	0.788440i	$-0.289111\pi$
$113$	13.0775	1.23022	0.615112	+	0.788440i	$0.289111\pi$
$114$	2.04630	0.191653
$115$	11.6895	1.09005
$116$	10.4573	0.970936
$117$	−44.8834	−4.14947
$118$	3.06613	0.282260
$119$	−0.741562	−0.0679788
$120$	−12.9447	−1.18168
$121$	17.9683	1.63348
$122$	−5.38097	−0.487170
$123$	36.1989	3.26394
$124$	4.04978	0.363681
$125$	−24.4222	−2.18439
$126$	−17.6518	−1.57255
$127$	10.8479	0.962595	0.481298	−	0.876557i	$-0.340166\pi$
$127$	10.8479	0.962595	0.481298	+	0.876557i	$0.340166\pi$
$128$	−1.00000	−0.0883883
$129$	41.7589	3.67667
$130$	−24.2791	−2.12942
$131$	−4.50012	−0.393178	−0.196589	−	0.980486i	$-0.562986\pi$
$131$	−4.50012	−0.393178	−0.196589	+	0.980486i	$0.562986\pi$
$132$	−17.3697	−1.51184
$133$	1.50943	0.130885
$134$	7.01888	0.606339
$135$	57.1511	4.91879
$136$	0.311510	0.0267118
$137$	7.17777	0.613238	0.306619	−	0.951832i	$-0.400802\pi$
$137$	7.17777	0.613238	0.306619	+	0.951832i	$0.400802\pi$
$138$	−9.40516	−0.800621
$139$	−1.08419	−0.0919598	−0.0459799	−	0.998942i	$-0.514641\pi$
$139$	−1.08419	−0.0919598	−0.0459799	+	0.998942i	$0.514641\pi$
$140$	−9.54851	−0.806997
$141$	−26.3921	−2.22262
$142$	9.30452	0.780818
$143$	−32.5787	−2.72437
$144$	7.41504	0.617920
$145$	−41.9450	−3.48334
$146$	−13.6826	−1.13238
$147$	4.30200	0.354823
$148$	3.03943	0.249840
$149$	−2.01539	−0.165108	−0.0825538	−	0.996587i	$-0.526308\pi$
$149$	−2.01539	−0.165108	−0.0825538	+	0.996587i	$0.526308\pi$
$150$	35.7858	2.92190
$151$	5.98783	0.487283	0.243641	−	0.969865i	$-0.421658\pi$
$151$	5.98783	0.487283	0.243641	+	0.969865i	$0.421658\pi$
$152$	−0.634072	−0.0514301
$153$	−2.30986	−0.186741
$154$	−12.8126	−1.03247
$155$	−16.2439	−1.30474
$156$	19.5345	1.56401
$157$	−13.0189	−1.03902	−0.519509	−	0.854465i	$-0.673885\pi$
$157$	−13.0189	−1.03902	−0.519509	+	0.854465i	$0.673885\pi$
$158$	−8.33989	−0.663486
$159$	39.2964	3.11641
$160$	4.01107	0.317103
$161$	−6.93763	−0.546762
$162$	−23.7376	−1.86501
$163$	18.3915	1.44053	0.720266	−	0.693698i	$-0.244020\pi$
$163$	18.3915	1.44053	0.720266	+	0.693698i	$0.244020\pi$
$164$	−11.2167	−0.875877
$165$	69.6710	5.42388
$166$	7.44115	0.577545
$167$	5.35896	0.414689	0.207344	−	0.978268i	$-0.433518\pi$
$167$	5.35896	0.414689	0.207344	+	0.978268i	$0.433518\pi$
$168$	7.68256	0.592722
$169$	23.6390	1.81839
$170$	−1.24949	−0.0958314
$171$	4.70167	0.359546
$172$	−12.9395	−0.986630
$173$	18.9522	1.44091	0.720453	−	0.693503i	$-0.243934\pi$
$173$	18.9522	1.44091	0.720453	+	0.693503i	$0.243934\pi$
$174$	33.7481	2.55844
$175$	26.3971	1.99543
$176$	5.38222	0.405700
$177$	9.89510	0.743762
$178$	4.29789	0.322141
$179$	−4.62908	−0.345993	−0.172997	−	0.984922i	$-0.555345\pi$
$179$	−4.62908	−0.345993	−0.172997	+	0.984922i	$0.555345\pi$
$180$	−29.7422	−2.21686
$181$	6.08062	0.451969	0.225984	−	0.974131i	$-0.427440\pi$
$181$	6.08062	0.451969	0.225984	+	0.974131i	$0.427440\pi$
$182$	14.4095	1.06810
$183$	−17.3656	−1.28371
$184$	2.91431	0.214846
$185$	−12.1914	−0.896329
$186$	13.0696	0.958307
$187$	−1.67661	−0.122606
$188$	8.17795	0.596438
$189$	−33.9188	−2.46723
$190$	2.54331	0.184511
$191$	−21.1353	−1.52930	−0.764649	−	0.644447i	$-0.777087\pi$
$191$	−21.1353	−1.52930	−0.764649	+	0.644447i	$0.777087\pi$
$192$	−3.22723	−0.232906
$193$	6.44714	0.464075	0.232038	−	0.972707i	$-0.425461\pi$
$193$	6.44714	0.464075	0.232038	+	0.972707i	$0.425461\pi$
$194$	9.59444	0.688841
$195$	−78.3543	−5.61107
$196$	−1.33303	−0.0952165
$197$	−0.232910	−0.0165941	−0.00829706	−	0.999966i	$-0.502641\pi$
$197$	−0.232910	−0.0165941	−0.00829706	+	0.999966i	$0.502641\pi$
$198$	−39.9093	−2.83623
$199$	22.8381	1.61895	0.809474	−	0.587156i	$-0.199753\pi$
$199$	22.8381	1.61895	0.809474	+	0.587156i	$0.199753\pi$
$200$	−11.0887	−0.784089
$201$	22.6516	1.59772
$202$	−2.63434	−0.185352
$203$	24.8940	1.74722
$204$	1.00532	0.0703861
$205$	44.9910	3.14230
$206$	10.2047	0.710993
$207$	−21.6097	−1.50198
$208$	−6.05302	−0.419701
$209$	3.41272	0.236062
$210$	−30.8153	−2.12646
$211$	14.9989	1.03257	0.516284	−	0.856418i	$-0.327315\pi$
$211$	14.9989	1.03257	0.516284	+	0.856418i	$0.327315\pi$
$212$	−12.1765	−0.836286
$213$	30.0279	2.05748
$214$	0.291305	0.0199132
$215$	51.9014	3.53965
$216$	14.2483	0.969477
$217$	9.64065	0.654450
$218$	−9.46430	−0.641003
$219$	−44.1571	−2.98386
$220$	−21.5885	−1.45549
$221$	1.88558	0.126838
$222$	9.80896	0.658334
$223$	25.1664	1.68527	0.842634	−	0.538487i	$-0.181004\pi$
$223$	25.1664	1.68527	0.842634	+	0.538487i	$0.181004\pi$
$224$	−2.38054	−0.159056
$225$	82.2231	5.48154
$226$	−13.0775	−0.869899
$227$	5.77255	0.383138	0.191569	−	0.981479i	$-0.438642\pi$
$227$	5.77255	0.383138	0.191569	+	0.981479i	$0.438642\pi$
$228$	−2.04630	−0.135519
$229$	9.22661	0.609712	0.304856	−	0.952399i	$-0.401392\pi$
$229$	9.22661	0.609712	0.304856	+	0.952399i	$0.401392\pi$
$230$	−11.6895	−0.770784
$231$	−41.3492	−2.72058
$232$	−10.4573	−0.686555
$233$	25.6640	1.68131	0.840654	−	0.541573i	$-0.182171\pi$
$233$	25.6640	1.68131	0.840654	+	0.541573i	$0.182171\pi$
$234$	44.8834	2.93412
$235$	−32.8023	−2.13979
$236$	−3.06613	−0.199588
$237$	−26.9148	−1.74830
$238$	0.741562	0.0480683
$239$	−19.8537	−1.28423	−0.642114	−	0.766610i	$-0.721942\pi$
$239$	−19.8537	−1.28423	−0.642114	+	0.766610i	$0.721942\pi$
$240$	12.9447	0.835574
$241$	9.54181	0.614642	0.307321	−	0.951606i	$-0.400567\pi$
$241$	9.54181	0.614642	0.307321	+	0.951606i	$0.400567\pi$
$242$	−17.9683	−1.15504
$243$	−33.8619	−2.17224
$244$	5.38097	0.344481
$245$	5.34688	0.341600
$246$	−36.1989	−2.30796
$247$	−3.83805	−0.244209
$248$	−4.04978	−0.257161
$249$	24.0143	1.52185
$250$	24.4222	1.54459
$251$	−2.67894	−0.169093	−0.0845467	−	0.996420i	$-0.526944\pi$
$251$	−2.67894	−0.169093	−0.0845467	+	0.996420i	$0.526944\pi$
$252$	17.6518	1.11196
$253$	−15.6855	−0.986136
$254$	−10.8479	−0.680658
$255$	−4.03239	−0.252518
$256$	1.00000	0.0625000
$257$	−21.5200	−1.34238	−0.671191	−	0.741285i	$-0.734217\pi$
$257$	−21.5200	−1.34238	−0.671191	+	0.741285i	$0.734217\pi$
$258$	−41.7589	−2.59980
$259$	7.23549	0.449592
$260$	24.2791	1.50573
$261$	77.5412	4.79968
$262$	4.50012	0.278019
$263$	−17.4380	−1.07527	−0.537637	−	0.843176i	$-0.680683\pi$
$263$	−17.4380	−1.07527	−0.537637	+	0.843176i	$0.680683\pi$
$264$	17.3697	1.06903
$265$	48.8408	3.00027
$266$	−1.50943	−0.0925494
$267$	13.8703	0.848849
$268$	−7.01888	−0.428747
$269$	−16.0073	−0.975982	−0.487991	−	0.872849i	$-0.662270\pi$
$269$	−16.0073	−0.975982	−0.487991	+	0.872849i	$0.662270\pi$
$270$	−57.1511	−3.47811
$271$	−3.12455	−0.189803	−0.0949014	−	0.995487i	$-0.530254\pi$
$271$	−3.12455	−0.189803	−0.0949014	+	0.995487i	$0.530254\pi$
$272$	−0.311510	−0.0188881
$273$	46.5027	2.81447
$274$	−7.17777	−0.433625
$275$	59.6818	3.59895
$276$	9.40516	0.566124
$277$	−1.79522	−0.107865	−0.0539323	−	0.998545i	$-0.517176\pi$
$277$	−1.79522	−0.107865	−0.0539323	+	0.998545i	$0.517176\pi$
$278$	1.08419	0.0650254
$279$	30.0292	1.79780
$280$	9.54851	0.570633
$281$	−17.2260	−1.02762	−0.513808	−	0.857905i	$-0.671766\pi$
$281$	−17.2260	−1.02762	−0.513808	+	0.857905i	$0.671766\pi$
$282$	26.3921	1.57163
$283$	10.7030	0.636226	0.318113	−	0.948053i	$-0.396951\pi$
$283$	10.7030	0.636226	0.318113	+	0.948053i	$0.396951\pi$
$284$	−9.30452	−0.552122
$285$	8.20785	0.486191
$286$	32.5787	1.92642
$287$	−26.7018	−1.57616
$288$	−7.41504	−0.436935
$289$	−16.9030	−0.994292
$290$	41.9450	2.46309
$291$	30.9635	1.81511
$292$	13.6826	0.800716
$293$	−9.31995	−0.544478	−0.272239	−	0.962230i	$-0.587764\pi$
$293$	−9.31995	−0.544478	−0.272239	+	0.962230i	$0.587764\pi$
$294$	−4.30200	−0.250898
$295$	12.2985	0.716043
$296$	−3.03943	−0.176664
$297$	−76.6877	−4.44987
$298$	2.01539	0.116749
$299$	17.6404	1.02017
$300$	−35.7858	−2.06609
$301$	−30.8031	−1.77546
$302$	−5.98783	−0.344561
$303$	−8.50163	−0.488406
$304$	0.634072	0.0363665
$305$	−21.5835	−1.23587
$306$	2.30986	0.132046
$307$	−24.6266	−1.40552	−0.702758	−	0.711429i	$-0.748048\pi$
$307$	−24.6266	−1.40552	−0.702758	+	0.711429i	$0.748048\pi$
$308$	12.8126	0.730065
$309$	32.9328	1.87348
$310$	16.2439	0.922594
$311$	−6.39346	−0.362540	−0.181270	−	0.983433i	$-0.558021\pi$
$311$	−6.39346	−0.362540	−0.181270	+	0.983433i	$0.558021\pi$
$312$	−19.5345	−1.10592
$313$	25.1396	1.42097	0.710487	−	0.703710i	$-0.248475\pi$
$313$	25.1396	1.42097	0.710487	+	0.703710i	$0.248475\pi$
$314$	13.0189	0.734697
$315$	−70.8026	−3.98927
$316$	8.33989	0.469156
$317$	−1.62509	−0.0912741	−0.0456371	−	0.998958i	$-0.514532\pi$
$317$	−1.62509	−0.0912741	−0.0456371	+	0.998958i	$0.514532\pi$
$318$	−39.2964	−2.20363
$319$	56.2835	3.15127
$320$	−4.01107	−0.224226
$321$	0.940110	0.0524718
$322$	6.93763	0.386619
$323$	−0.197520	−0.0109903
$324$	23.7376	1.31876
$325$	−67.1201	−3.72315
$326$	−18.3915	−1.01861
$327$	−30.5435	−1.68906
$328$	11.2167	0.619338
$329$	19.4679	1.07330
$330$	−69.6710	−3.83526
$331$	−17.9151	−0.984705	−0.492353	−	0.870396i	$-0.663863\pi$
$331$	−17.9151	−0.984705	−0.492353	+	0.870396i	$0.663863\pi$
$332$	−7.44115	−0.408386
$333$	22.5375	1.23505
$334$	−5.35896	−0.293229
$335$	28.1532	1.53818
$336$	−7.68256	−0.419118
$337$	12.2030	0.664742	0.332371	−	0.943149i	$-0.392151\pi$
$337$	12.2030	0.664742	0.332371	+	0.943149i	$0.392151\pi$
$338$	−23.6390	−1.28579
$339$	−42.2040	−2.29221
$340$	1.24949	0.0677630
$341$	21.7968	1.18036
$342$	−4.70167	−0.254237
$343$	−19.8371	−1.07110
$344$	12.9395	0.697653
$345$	−37.7248	−2.03103
$346$	−18.9522	−1.01887
$347$	−5.53908	−0.297353	−0.148677	−	0.988886i	$-0.547501\pi$
$347$	−5.53908	−0.297353	−0.148677	+	0.988886i	$0.547501\pi$
$348$	−33.7481	−1.80909
$349$	−29.9149	−1.60131	−0.800653	−	0.599129i	$-0.795514\pi$
$349$	−29.9149	−1.60131	−0.800653	+	0.599129i	$0.795514\pi$
$350$	−26.3971	−1.41098
$351$	86.2455	4.60345
$352$	−5.38222	−0.286873
$353$	−6.24503	−0.332389	−0.166195	−	0.986093i	$-0.553148\pi$
$353$	−6.24503	−0.332389	−0.166195	+	0.986093i	$0.553148\pi$
$354$	−9.89510	−0.525919
$355$	37.3211	1.98080
$356$	−4.29789	−0.227788
$357$	2.39319	0.126661
$358$	4.62908	0.244654
$359$	−22.4884	−1.18689	−0.593446	−	0.804874i	$-0.702233\pi$
$359$	−22.4884	−1.18689	−0.593446	+	0.804874i	$0.702233\pi$
$360$	29.7422	1.56755
$361$	−18.5980	−0.978840
$362$	−6.08062	−0.319590
$363$	−57.9878	−3.04357
$364$	−14.4095	−0.755261
$365$	−54.8821	−2.87266
$366$	17.3656	0.907717
$367$	7.42362	0.387510	0.193755	−	0.981050i	$-0.437933\pi$
$367$	7.42362	0.387510	0.193755	+	0.981050i	$0.437933\pi$
$368$	−2.91431	−0.151919
$369$	−83.1722	−4.32977
$370$	12.1914	0.633800
$371$	−28.9866	−1.50491
$372$	−13.0696	−0.677626
$373$	−15.7153	−0.813708	−0.406854	−	0.913493i	$-0.633374\pi$
$373$	−15.7153	−0.813708	−0.406854	+	0.913493i	$0.633374\pi$
$374$	1.67661	0.0866957
$375$	78.8161	4.07005
$376$	−8.17795	−0.421745
$377$	−63.2982	−3.26002
$378$	33.9188	1.74459
$379$	−18.0531	−0.927324	−0.463662	−	0.886012i	$-0.653465\pi$
$379$	−18.0531	−0.927324	−0.463662	+	0.886012i	$0.653465\pi$
$380$	−2.54331	−0.130469
$381$	−35.0087	−1.79355
$382$	21.1353	1.08138
$383$	3.69666	0.188891	0.0944453	−	0.995530i	$-0.469892\pi$
$383$	3.69666	0.188891	0.0944453	+	0.995530i	$0.469892\pi$
$384$	3.22723	0.164689
$385$	−51.3922	−2.61919
$386$	−6.44714	−0.328151
$387$	−95.9471	−4.87727
$388$	−9.59444	−0.487084
$389$	−9.48212	−0.480762	−0.240381	−	0.970679i	$-0.577272\pi$
$389$	−9.48212	−0.480762	−0.240381	+	0.970679i	$0.577272\pi$
$390$	78.3543	3.96762
$391$	0.907837	0.0459113
$392$	1.33303	0.0673282
$393$	14.5230	0.732586
$394$	0.232910	0.0117338
$395$	−33.4519	−1.68315
$396$	39.9093	2.00552
$397$	−10.0221	−0.502996	−0.251498	−	0.967858i	$-0.580923\pi$
$397$	−10.0221	−0.502996	−0.251498	+	0.967858i	$0.580923\pi$
$398$	−22.8381	−1.14477
$399$	−4.87130	−0.243870
$400$	11.0887	0.554435
$401$	11.8519	0.591855	0.295927	−	0.955210i	$-0.404371\pi$
$401$	11.8519	0.591855	0.295927	+	0.955210i	$0.404371\pi$
$402$	−22.6516	−1.12976
$403$	−24.5134	−1.22110
$404$	2.63434	0.131063
$405$	−95.2134	−4.73119
$406$	−24.8940	−1.23547
$407$	16.3589	0.810880
$408$	−1.00532	−0.0497705
$409$	4.01416	0.198487	0.0992436	−	0.995063i	$-0.468358\pi$
$409$	4.01416	0.198487	0.0992436	+	0.995063i	$0.468358\pi$
$410$	−44.9910	−2.22195
$411$	−23.1643	−1.14261
$412$	−10.2047	−0.502748
$413$	−7.29903	−0.359162
$414$	21.6097	1.06206
$415$	29.8470	1.46513
$416$	6.05302	0.296774
$417$	3.49893	0.171344
$418$	−3.41272	−0.166921
$419$	−0.160852	−0.00785813	−0.00392907	−	0.999992i	$-0.501251\pi$
$419$	−0.160852	−0.00785813	−0.00392907	+	0.999992i	$0.501251\pi$
$420$	30.8153	1.50363
$421$	−3.93140	−0.191605	−0.0958024	−	0.995400i	$-0.530542\pi$
$421$	−3.93140	−0.191605	−0.0958024	+	0.995400i	$0.530542\pi$
$422$	−14.9989	−0.730135
$423$	60.6398	2.94841
$424$	12.1765	0.591343
$425$	−3.45424	−0.167555
$426$	−30.0279	−1.45486
$427$	12.8096	0.619901
$428$	−0.291305	−0.0140808
$429$	105.139	5.07616
$430$	−51.9014	−2.50291
$431$	14.9520	0.720211	0.360105	−	0.932912i	$-0.382741\pi$
$431$	14.9520	0.720211	0.360105	+	0.932912i	$0.382741\pi$
$432$	−14.2483	−0.685524
$433$	−20.0552	−0.963789	−0.481895	−	0.876229i	$-0.660051\pi$
$433$	−20.0552	−0.963789	−0.481895	+	0.876229i	$0.660051\pi$
$434$	−9.64065	−0.462766
$435$	135.366	6.49031
$436$	9.46430	0.453258
$437$	−1.84788	−0.0883963
$438$	44.1571	2.10991
$439$	−1.11888	−0.0534010	−0.0267005	−	0.999643i	$-0.508500\pi$
$439$	−1.11888	−0.0534010	−0.0267005	+	0.999643i	$0.508500\pi$
$440$	21.5885	1.02919
$441$	−9.88447	−0.470689
$442$	−1.88558	−0.0896877
$443$	23.2106	1.10277	0.551384	−	0.834251i	$-0.314100\pi$
$443$	23.2106	1.10277	0.551384	+	0.834251i	$0.314100\pi$
$444$	−9.80896	−0.465513
$445$	17.2392	0.817214
$446$	−25.1664	−1.19166
$447$	6.50415	0.307636
$448$	2.38054	0.112470
$449$	9.38833	0.443063	0.221531	−	0.975153i	$-0.428895\pi$
$449$	9.38833	0.443063	0.221531	+	0.975153i	$0.428895\pi$
$450$	−82.2231	−3.87603
$451$	−60.3707	−2.84274
$452$	13.0775	0.615112
$453$	−19.3241	−0.907927
$454$	−5.77255	−0.270919
$455$	57.7973	2.70958
$456$	2.04630	0.0958267
$457$	−3.10394	−0.145196	−0.0725980	−	0.997361i	$-0.523129\pi$
$457$	−3.10394	−0.145196	−0.0725980	+	0.997361i	$0.523129\pi$
$458$	−9.22661	−0.431131
$459$	4.43850	0.207172
$460$	11.6895	0.545026
$461$	−13.6958	−0.637876	−0.318938	−	0.947776i	$-0.603326\pi$
$461$	−13.6958	−0.637876	−0.318938	+	0.947776i	$0.603326\pi$
$462$	41.3492	1.92374
$463$	34.2562	1.59202	0.796011	−	0.605282i	$-0.206940\pi$
$463$	34.2562	1.59202	0.796011	+	0.605282i	$0.206940\pi$
$464$	10.4573	0.485468
$465$	52.4230	2.43106
$466$	−25.6640	−1.18886
$467$	30.8409	1.42715	0.713574	−	0.700580i	$-0.247075\pi$
$467$	30.8409	1.42715	0.713574	+	0.700580i	$0.247075\pi$
$468$	−44.8834	−2.07473
$469$	−16.7087	−0.771538
$470$	32.8023	1.51306
$471$	42.0149	1.93595
$472$	3.06613	0.141130
$473$	−69.6434	−3.20221
$474$	26.9148	1.23624
$475$	7.03103	0.322606
$476$	−0.741562	−0.0339894
$477$	−90.2892	−4.13406
$478$	19.8537	0.908086
$479$	7.44367	0.340110	0.170055	−	0.985435i	$-0.445605\pi$
$479$	7.44367	0.340110	0.170055	+	0.985435i	$0.445605\pi$
$480$	−12.9447	−0.590840
$481$	−18.3978	−0.838865
$482$	−9.54181	−0.434618
$483$	22.3894	1.01875
$484$	17.9683	0.816739
$485$	38.4840	1.74747
$486$	33.8619	1.53601
$487$	39.9749	1.81144	0.905718	−	0.423881i	$-0.139332\pi$
$487$	39.9749	1.81144	0.905718	+	0.423881i	$0.139332\pi$
$488$	−5.38097	−0.243585
$489$	−59.3536	−2.68406
$490$	−5.34688	−0.241548
$491$	28.2410	1.27450	0.637248	−	0.770659i	$-0.280073\pi$
$491$	28.2410	1.27450	0.637248	+	0.770659i	$0.280073\pi$
$492$	36.1989	1.63197
$493$	−3.25755	−0.146713
$494$	3.83805	0.172682
$495$	−160.079	−7.19502
$496$	4.04978	0.181840
$497$	−22.1498	−0.993554
$498$	−24.0143	−1.07611
$499$	−39.0052	−1.74611	−0.873057	−	0.487619i	$-0.837866\pi$
$499$	−39.0052	−1.74611	−0.873057	+	0.487619i	$0.837866\pi$
$500$	−24.4222	−1.09219
$501$	−17.2946	−0.772666
$502$	2.67894	0.119567
$503$	−26.9801	−1.20299	−0.601493	−	0.798878i	$-0.705427\pi$
$503$	−26.9801	−1.20299	−0.601493	+	0.798878i	$0.705427\pi$
$504$	−17.6518	−0.786273
$505$	−10.5665	−0.470204
$506$	15.6855	0.697304
$507$	−76.2887	−3.38810
$508$	10.8479	0.481298
$509$	5.67140	0.251380	0.125690	−	0.992070i	$-0.459885\pi$
$509$	5.67140	0.251380	0.125690	+	0.992070i	$0.459885\pi$
$510$	4.03239	0.178557
$511$	32.5721	1.44090
$512$	−1.00000	−0.0441942
$513$	−9.03448	−0.398882
$514$	21.5200	0.949207
$515$	40.9316	1.80366
$516$	41.7589	1.83833
$517$	44.0155	1.93580
$518$	−7.23549	−0.317909
$519$	−61.1631	−2.68476
$520$	−24.2791	−1.06471
$521$	−30.4674	−1.33480	−0.667401	−	0.744699i	$-0.732593\pi$
$521$	−30.4674	−1.33480	−0.667401	+	0.744699i	$0.732593\pi$
$522$	−77.5412	−3.39389
$523$	36.6554	1.60283	0.801413	−	0.598111i	$-0.204082\pi$
$523$	36.6554	1.60283	0.801413	+	0.598111i	$0.204082\pi$
$524$	−4.50012	−0.196589
$525$	−85.1895	−3.71798
$526$	17.4380	0.760334
$527$	−1.26155	−0.0549538
$528$	−17.3697	−0.755918
$529$	−14.5068	−0.630730
$530$	−48.8408	−2.12151
$531$	−22.7354	−0.986634
$532$	1.50943	0.0654423
$533$	67.8949	2.94085
$534$	−13.8703	−0.600227
$535$	1.16845	0.0505163
$536$	7.01888	0.303170
$537$	14.9391	0.644670
$538$	16.0073	0.690123
$539$	−7.17466	−0.309035
$540$	57.1511	2.45939
$541$	−13.8853	−0.596977	−0.298489	−	0.954413i	$-0.596482\pi$
$541$	−13.8853	−0.596977	−0.298489	+	0.954413i	$0.596482\pi$
$542$	3.12455	0.134211
$543$	−19.6236	−0.842128
$544$	0.311510	0.0133559
$545$	−37.9620	−1.62611
$546$	−46.5027	−1.99013
$547$	−12.7209	−0.543906	−0.271953	−	0.962311i	$-0.587670\pi$
$547$	−12.7209	−0.543906	−0.271953	+	0.962311i	$0.587670\pi$
$548$	7.17777	0.306619
$549$	39.9001	1.70289
$550$	−59.6818	−2.54484
$551$	6.63068	0.282477
$552$	−9.40516	−0.400310
$553$	19.8534	0.844254
$554$	1.79522	0.0762717
$555$	39.3445	1.67008
$556$	−1.08419	−0.0459799
$557$	5.19959	0.220314	0.110157	−	0.993914i	$-0.464865\pi$
$557$	5.19959	0.220314	0.110157	+	0.993914i	$0.464865\pi$
$558$	−30.0292	−1.27124
$559$	78.3232	3.31272
$560$	−9.54851	−0.403498
$561$	5.41083	0.228445
$562$	17.2260	0.726634
$563$	−24.0821	−1.01494	−0.507469	−	0.861670i	$-0.669419\pi$
$563$	−24.0821	−1.01494	−0.507469	+	0.861670i	$0.669419\pi$
$564$	−26.3921	−1.11131
$565$	−52.4546	−2.20678
$566$	−10.7030	−0.449879
$567$	56.5084	2.37313
$568$	9.30452	0.390409
$569$	38.8496	1.62866	0.814329	−	0.580403i	$-0.197105\pi$
$569$	38.8496	1.62866	0.814329	+	0.580403i	$0.197105\pi$
$570$	−8.20785	−0.343789
$571$	−23.8687	−0.998874	−0.499437	−	0.866350i	$-0.666460\pi$
$571$	−23.8687	−0.998874	−0.499437	+	0.866350i	$0.666460\pi$
$572$	−32.5787	−1.36218
$573$	68.2086	2.84945
$574$	26.7018	1.11451
$575$	−32.3159	−1.34767
$576$	7.41504	0.308960
$577$	19.9409	0.830151	0.415075	−	0.909787i	$-0.363755\pi$
$577$	19.9409	0.830151	0.415075	+	0.909787i	$0.363755\pi$
$578$	16.9030	0.703071
$579$	−20.8064	−0.864685
$580$	−41.9450	−1.74167
$581$	−17.7139	−0.734898
$582$	−30.9635	−1.28348
$583$	−65.5366	−2.71425
$584$	−13.6826	−0.566192
$585$	180.030	7.44334
$586$	9.31995	0.385004
$587$	21.7626	0.898239	0.449119	−	0.893472i	$-0.351738\pi$
$587$	21.7626	0.898239	0.449119	+	0.893472i	$0.351738\pi$
$588$	4.30200	0.177412
$589$	2.56785	0.105806
$590$	−12.2985	−0.506319
$591$	0.751654	0.0309189
$592$	3.03943	0.124920
$593$	19.0100	0.780645	0.390323	−	0.920678i	$-0.372363\pi$
$593$	19.0100	0.780645	0.390323	+	0.920678i	$0.372363\pi$
$594$	76.6877	3.14654
$595$	2.97446	0.121941
$596$	−2.01539	−0.0825538
$597$	−73.7037	−3.01649
$598$	−17.6404	−0.721369
$599$	23.3934	0.955830	0.477915	−	0.878406i	$-0.341393\pi$
$599$	23.3934	0.955830	0.477915	+	0.878406i	$0.341393\pi$
$600$	35.7858	1.46095
$601$	−40.2440	−1.64159	−0.820794	−	0.571225i	$-0.806468\pi$
$601$	−40.2440	−1.64159	−0.820794	+	0.571225i	$0.806468\pi$
$602$	30.8031	1.25544
$603$	−52.0453	−2.11945
$604$	5.98783	0.243641
$605$	−72.0720	−2.93014
$606$	8.50163	0.345355
$607$	16.2980	0.661514	0.330757	−	0.943716i	$-0.392696\pi$
$607$	16.2980	0.661514	0.330757	+	0.943716i	$0.392696\pi$
$608$	−0.634072	−0.0257150
$609$	−80.3388	−3.25549
$610$	21.5835	0.873889
$611$	−49.5013	−2.00261
$612$	−2.30986	−0.0933704
$613$	35.2239	1.42268	0.711341	−	0.702847i	$-0.248088\pi$
$613$	35.2239	1.42268	0.711341	+	0.702847i	$0.248088\pi$
$614$	24.6266	0.993850
$615$	−145.196	−5.85488
$616$	−12.8126	−0.516234
$617$	6.33845	0.255176	0.127588	−	0.991827i	$-0.459276\pi$
$617$	6.33845	0.255176	0.127588	+	0.991827i	$0.459276\pi$
$618$	−32.9328	−1.32475
$619$	22.0137	0.884806	0.442403	−	0.896816i	$-0.354126\pi$
$619$	22.0137	0.884806	0.442403	+	0.896816i	$0.354126\pi$
$620$	−16.2439	−0.652372
$621$	41.5241	1.66631
$622$	6.39346	0.256354
$623$	−10.2313	−0.409908
$624$	19.5345	0.782006
$625$	42.5157	1.70063
$626$	−25.1396	−1.00478
$627$	−11.0136	−0.439842
$628$	−13.0189	−0.519509
$629$	−0.946814	−0.0377519
$630$	70.8026	2.82084
$631$	1.81445	0.0722322	0.0361161	−	0.999348i	$-0.488501\pi$
$631$	1.81445	0.0722322	0.0361161	+	0.999348i	$0.488501\pi$
$632$	−8.33989	−0.331743
$633$	−48.4050	−1.92393
$634$	1.62509	0.0645406
$635$	−43.5117	−1.72671
$636$	39.2964	1.55820
$637$	8.06886	0.319700
$638$	−56.2835	−2.22828
$639$	−68.9934	−2.72934
$640$	4.01107	0.158552
$641$	−2.92578	−0.115561	−0.0577807	−	0.998329i	$-0.518402\pi$
$641$	−2.92578	−0.115561	−0.0577807	+	0.998329i	$0.518402\pi$
$642$	−0.940110	−0.0371032
$643$	47.3207	1.86614	0.933072	−	0.359689i	$-0.117117\pi$
$643$	47.3207	1.86614	0.933072	+	0.359689i	$0.117117\pi$
$644$	−6.93763	−0.273381
$645$	−167.498	−6.59522
$646$	0.197520	0.00777131
$647$	−32.3853	−1.27320	−0.636600	−	0.771194i	$-0.719660\pi$
$647$	−32.3853	−1.27320	−0.636600	+	0.771194i	$0.719660\pi$
$648$	−23.7376	−0.932503
$649$	−16.5026	−0.647782
$650$	67.1201	2.63267
$651$	−31.1126	−1.21940
$652$	18.3915	0.720266
$653$	−22.7059	−0.888549	−0.444274	−	0.895891i	$-0.646538\pi$
$653$	−22.7059	−0.888549	−0.444274	+	0.895891i	$0.646538\pi$
$654$	30.5435	1.19435
$655$	18.0503	0.705284
$656$	−11.2167	−0.437938
$657$	101.457	3.95823
$658$	−19.4679	−0.758939
$659$	−13.3091	−0.518449	−0.259225	−	0.965817i	$-0.583467\pi$
$659$	−13.3091	−0.518449	−0.259225	+	0.965817i	$0.583467\pi$
$660$	69.6710	2.71194
$661$	19.5017	0.758528	0.379264	−	0.925288i	$-0.376177\pi$
$661$	19.5017	0.758528	0.379264	+	0.925288i	$0.376177\pi$
$662$	17.9151	0.696292
$663$	−6.08519	−0.236329
$664$	7.44115	0.288772
$665$	−6.05445	−0.234781
$666$	−22.5375	−0.873311
$667$	−30.4758	−1.18003
$668$	5.35896	0.207344
$669$	−81.2179	−3.14007
$670$	−28.1532	−1.08765
$671$	28.9616	1.11805
$672$	7.68256	0.296361
$673$	−26.6054	−1.02556	−0.512781	−	0.858520i	$-0.671385\pi$
$673$	−26.6054	−1.02556	−0.512781	+	0.858520i	$0.671385\pi$
$674$	−12.2030	−0.470044
$675$	−157.996	−6.08125
$676$	23.6390	0.909194
$677$	−30.8012	−1.18378	−0.591892	−	0.806017i	$-0.701619\pi$
$677$	−30.8012	−1.18378	−0.591892	+	0.806017i	$0.701619\pi$
$678$	42.2040	1.62083
$679$	−22.8400	−0.876517
$680$	−1.24949	−0.0479157
$681$	−18.6294	−0.713879
$682$	−21.7968	−0.834642
$683$	−8.22602	−0.314760	−0.157380	−	0.987538i	$-0.550305\pi$
$683$	−8.22602	−0.314760	−0.157380	+	0.987538i	$0.550305\pi$
$684$	4.70167	0.179773
$685$	−28.7905	−1.10003
$686$	19.8371	0.757384
$687$	−29.7764	−1.13604
$688$	−12.9395	−0.493315
$689$	73.7046	2.80792
$690$	37.7248	1.43616
$691$	17.9618	0.683297	0.341649	−	0.939828i	$-0.389015\pi$
$691$	17.9618	0.683297	0.341649	+	0.939828i	$0.389015\pi$
$692$	18.9522	0.720453
$693$	95.0058	3.60897
$694$	5.53908	0.210261
$695$	4.34876	0.164958
$696$	33.7481	1.27922
$697$	3.49411	0.132349
$698$	29.9149	1.13229
$699$	−82.8239	−3.13269
$700$	26.3971	0.997716
$701$	−32.0524	−1.21060	−0.605302	−	0.795996i	$-0.706947\pi$
$701$	−32.0524	−1.21060	−0.605302	+	0.795996i	$0.706947\pi$
$702$	−86.2455	−3.25513
$703$	1.92722	0.0726865
$704$	5.38222	0.202850
$705$	105.861	3.98695
$706$	6.24503	0.235035
$707$	6.27115	0.235851
$708$	9.89510	0.371881
$709$	−17.3523	−0.651680	−0.325840	−	0.945425i	$-0.605647\pi$
$709$	−17.3523	−0.651680	−0.325840	+	0.945425i	$0.605647\pi$
$710$	−37.3211	−1.40064
$711$	61.8406	2.31920
$712$	4.29789	0.161070
$713$	−11.8023	−0.442000
$714$	−2.39319	−0.0895630
$715$	130.675	4.88698
$716$	−4.62908	−0.172997
$717$	64.0724	2.39283
$718$	22.4884	0.839259
$719$	39.8504	1.48617	0.743084	−	0.669198i	$-0.233362\pi$
$719$	39.8504	1.48617	0.743084	+	0.669198i	$0.233362\pi$
$720$	−29.7422	−1.10843
$721$	−24.2926	−0.904704
$722$	18.5980	0.692144
$723$	−30.7936	−1.14523
$724$	6.08062	0.225984
$725$	115.958	4.30656
$726$	57.9878	2.15213
$727$	8.21209	0.304570	0.152285	−	0.988337i	$-0.451337\pi$
$727$	8.21209	0.304570	0.152285	+	0.988337i	$0.451337\pi$
$728$	14.4095	0.534050
$729$	38.0672	1.40990
$730$	54.8821	2.03128
$731$	4.03079	0.149084
$732$	−17.3656	−0.641853
$733$	−15.9100	−0.587647	−0.293824	−	0.955860i	$-0.594928\pi$
$733$	−15.9100	−0.587647	−0.293824	+	0.955860i	$0.594928\pi$
$734$	−7.42362	−0.274011
$735$	−17.2556	−0.636484
$736$	2.91431	0.107423
$737$	−37.7772	−1.39154
$738$	83.1722	3.06161
$739$	2.91200	0.107119	0.0535597	−	0.998565i	$-0.482943\pi$
$739$	2.91200	0.107119	0.0535597	+	0.998565i	$0.482943\pi$
$740$	−12.1914	−0.448164
$741$	12.3863	0.455022
$742$	28.9866	1.06413
$743$	−47.8108	−1.75401	−0.877004	−	0.480483i	$-0.840461\pi$
$743$	−47.8108	−1.75401	−0.877004	+	0.480483i	$0.840461\pi$
$744$	13.0696	0.479154
$745$	8.08389	0.296171
$746$	15.7153	0.575379
$747$	−55.1764	−2.01880
$748$	−1.67661	−0.0613031
$749$	−0.693464	−0.0253386
$750$	−78.8161	−2.87796
$751$	−39.7055	−1.44887	−0.724436	−	0.689342i	$-0.757900\pi$
$751$	−39.7055	−1.44887	−0.724436	+	0.689342i	$0.757900\pi$
$752$	8.17795	0.298219
$753$	8.64557	0.315062
$754$	63.2982	2.30519
$755$	−24.0176	−0.874091
$756$	−33.9188	−1.23361
$757$	47.3494	1.72094	0.860471	−	0.509500i	$-0.170170\pi$
$757$	47.3494	1.72094	0.860471	+	0.509500i	$0.170170\pi$
$758$	18.0531	0.655717
$759$	50.6206	1.83741
$760$	2.54331	0.0922555
$761$	22.2205	0.805494	0.402747	−	0.915311i	$-0.368056\pi$
$761$	22.2205	0.805494	0.402747	+	0.915311i	$0.368056\pi$
$762$	35.0087	1.26823
$763$	22.5301	0.815646
$764$	−21.1353	−0.764649
$765$	9.26500	0.334977
$766$	−3.69666	−0.133566
$767$	18.5593	0.670138
$768$	−3.22723	−0.116453
$769$	32.6336	1.17680	0.588398	−	0.808571i	$-0.299759\pi$
$769$	32.6336	1.17680	0.588398	+	0.808571i	$0.299759\pi$
$770$	51.3922	1.85205
$771$	69.4501	2.50118
$772$	6.44714	0.232038
$773$	−4.22249	−0.151872	−0.0759362	−	0.997113i	$-0.524195\pi$
$773$	−4.22249	−0.151872	−0.0759362	+	0.997113i	$0.524195\pi$
$774$	95.9471	3.44875
$775$	44.9067	1.61310
$776$	9.59444	0.344420
$777$	−23.3506	−0.837699
$778$	9.48212	0.339950
$779$	−7.11219	−0.254821
$780$	−78.3543	−2.80553
$781$	−50.0790	−1.79197
$782$	−0.907837	−0.0324642
$783$	−148.999	−5.32480
$784$	−1.33303	−0.0476083
$785$	52.2196	1.86380
$786$	−14.5230	−0.518017
$787$	7.31927	0.260904	0.130452	−	0.991455i	$-0.458357\pi$
$787$	7.31927	0.260904	0.130452	+	0.991455i	$0.458357\pi$
$788$	−0.232910	−0.00829706
$789$	56.2766	2.00350
$790$	33.4519	1.19017
$791$	31.1314	1.10691
$792$	−39.9093	−1.41812
$793$	−32.5711	−1.15663
$794$	10.0221	0.355672
$795$	−157.621	−5.59023
$796$	22.8381	0.809474
$797$	−13.7202	−0.485994	−0.242997	−	0.970027i	$-0.578130\pi$
$797$	−13.7202	−0.485994	−0.242997	+	0.970027i	$0.578130\pi$
$798$	4.87130	0.172442
$799$	−2.54751	−0.0901245
$800$	−11.0887	−0.392045
$801$	−31.8690	−1.12604
$802$	−11.8519	−0.418504
$803$	73.6430	2.59880
$804$	22.6516	0.798860
$805$	27.8273	0.980785
$806$	24.5134	0.863447
$807$	51.6593	1.81849
$808$	−2.63434	−0.0926758
$809$	48.2056	1.69482	0.847410	−	0.530939i	$-0.178161\pi$
$809$	48.2056	1.69482	0.847410	+	0.530939i	$0.178161\pi$
$810$	95.2134	3.34546
$811$	8.23162	0.289051	0.144526	−	0.989501i	$-0.453834\pi$
$811$	8.23162	0.289051	0.144526	+	0.989501i	$0.453834\pi$
$812$	24.8940	0.873609
$813$	10.0836	0.353649
$814$	−16.3589	−0.573379
$815$	−73.7695	−2.58403
$816$	1.00532	0.0351931
$817$	−8.20460	−0.287043
$818$	−4.01416	−0.140352
$819$	−106.847	−3.73352
$820$	44.9910	1.57115
$821$	−29.8241	−1.04087	−0.520434	−	0.853902i	$-0.674230\pi$
$821$	−29.8241	−1.04087	−0.520434	+	0.853902i	$0.674230\pi$
$822$	23.1643	0.807949
$823$	21.0697	0.734445	0.367223	−	0.930133i	$-0.380309\pi$
$823$	21.0697	0.734445	0.367223	+	0.930133i	$0.380309\pi$
$824$	10.2047	0.355496
$825$	−192.607	−6.70571
$826$	7.29903	0.253966
$827$	−26.2770	−0.913742	−0.456871	−	0.889533i	$-0.651030\pi$
$827$	−26.2770	−0.913742	−0.456871	+	0.889533i	$0.651030\pi$
$828$	−21.6097	−0.750990
$829$	−25.2319	−0.876341	−0.438170	−	0.898892i	$-0.644373\pi$
$829$	−25.2319	−0.876341	−0.438170	+	0.898892i	$0.644373\pi$
$830$	−29.8470	−1.03600
$831$	5.79361	0.200978
$832$	−6.05302	−0.209851
$833$	0.415252	0.0143876
$834$	−3.49893	−0.121158
$835$	−21.4952	−0.743871
$836$	3.41272	0.118031
$837$	−57.7026	−1.99449
$838$	0.160852	0.00555654
$839$	−31.9859	−1.10427	−0.552137	−	0.833753i	$-0.686188\pi$
$839$	−31.9859	−1.10427	−0.552137	+	0.833753i	$0.686188\pi$
$840$	−30.8153	−1.06323
$841$	80.3551	2.77086
$842$	3.93140	0.135485
$843$	55.5922	1.91470
$844$	14.9989	0.516284
$845$	−94.8179	−3.26183
$846$	−60.6398	−2.08484
$847$	42.7742	1.46974
$848$	−12.1765	−0.418143
$849$	−34.5410	−1.18544
$850$	3.45424	0.118479
$851$	−8.85786	−0.303643
$852$	30.0279	1.02874
$853$	−35.9442	−1.23070	−0.615352	−	0.788252i	$-0.710986\pi$
$853$	−35.9442	−1.23070	−0.615352	+	0.788252i	$0.710986\pi$
$854$	−12.8096	−0.438336
$855$	−18.8587	−0.644955
$856$	0.291305	0.00995661
$857$	45.6653	1.55990	0.779948	−	0.625845i	$-0.215246\pi$
$857$	45.6653	1.55990	0.779948	+	0.625845i	$0.215246\pi$
$858$	−105.139	−3.58939
$859$	17.5087	0.597390	0.298695	−	0.954349i	$-0.403449\pi$
$859$	17.5087	0.597390	0.298695	+	0.954349i	$0.403449\pi$
$860$	51.9014	1.76982
$861$	86.1729	2.93676
$862$	−14.9520	−0.509266
$863$	−19.6668	−0.669467	−0.334733	−	0.942313i	$-0.608646\pi$
$863$	−19.6668	−0.669467	−0.334733	+	0.942313i	$0.608646\pi$
$864$	14.2483	0.484739
$865$	−76.0185	−2.58471
$866$	20.0552	0.681502
$867$	54.5498	1.85261
$868$	9.64065	0.327225
$869$	44.8871	1.52269
$870$	−135.366	−4.58934
$871$	42.4854	1.43956
$872$	−9.46430	−0.320502
$873$	−71.1431	−2.40783
$874$	1.84788	0.0625056
$875$	−58.1380	−1.96542
$876$	−44.1571	−1.49193
$877$	39.6678	1.33949	0.669744	−	0.742592i	$-0.266404\pi$
$877$	39.6678	1.33949	0.669744	+	0.742592i	$0.266404\pi$
$878$	1.11888	0.0377602
$879$	30.0777	1.01449
$880$	−21.5885	−0.727747
$881$	−34.4695	−1.16131	−0.580654	−	0.814151i	$-0.697203\pi$
$881$	−34.4695	−1.16131	−0.580654	+	0.814151i	$0.697203\pi$
$882$	9.88447	0.332827
$883$	19.5788	0.658880	0.329440	−	0.944177i	$-0.393140\pi$
$883$	19.5788	0.658880	0.329440	+	0.944177i	$0.393140\pi$
$884$	1.88558	0.0634188
$885$	−39.6900	−1.33416
$886$	−23.2106	−0.779775
$887$	2.94746	0.0989662	0.0494831	−	0.998775i	$-0.484243\pi$
$887$	2.94746	0.0989662	0.0494831	+	0.998775i	$0.484243\pi$
$888$	9.80896	0.329167
$889$	25.8238	0.866104
$890$	−17.2392	−0.577858
$891$	127.761	4.28016
$892$	25.1664	0.842634
$893$	5.18541	0.173523
$894$	−6.50415	−0.217531
$895$	18.5676	0.620645
$896$	−2.38054	−0.0795282
$897$	−56.9296	−1.90083
$898$	−9.38833	−0.313293
$899$	42.3497	1.41244
$900$	82.2231	2.74077
$901$	3.79310	0.126367
$902$	60.3707	2.01012
$903$	99.4087	3.30811
$904$	−13.0775	−0.434950
$905$	−24.3898	−0.810744
$906$	19.3241	0.642001
$907$	4.84736	0.160954	0.0804770	−	0.996756i	$-0.474356\pi$
$907$	4.84736	0.160954	0.0804770	+	0.996756i	$0.474356\pi$
$908$	5.77255	0.191569
$909$	19.5337	0.647893
$910$	−57.7973	−1.91596
$911$	−31.5075	−1.04389	−0.521945	−	0.852979i	$-0.674793\pi$
$911$	−31.5075	−1.04389	−0.521945	+	0.852979i	$0.674793\pi$
$912$	−2.04630	−0.0677597
$913$	−40.0499	−1.32546
$914$	3.10394	0.102669
$915$	69.6548	2.30272
$916$	9.22661	0.304856
$917$	−10.7127	−0.353765
$918$	−4.43850	−0.146492
$919$	−6.30238	−0.207896	−0.103948	−	0.994583i	$-0.533148\pi$
$919$	−6.30238	−0.207896	−0.103948	+	0.994583i	$0.533148\pi$
$920$	−11.6895	−0.385392
$921$	79.4759	2.61882
$922$	13.6958	0.451047
$923$	56.3205	1.85381
$924$	−41.3492	−1.36029
$925$	33.7034	1.10816
$926$	−34.2562	−1.12573
$927$	−75.6679	−2.48526
$928$	−10.4573	−0.343278
$929$	32.6237	1.07035	0.535175	−	0.844741i	$-0.320246\pi$
$929$	32.6237	1.07035	0.535175	+	0.844741i	$0.320246\pi$
$930$	−52.4230	−1.71902
$931$	−0.845238	−0.0277016
$932$	25.6640	0.840654
$933$	20.6332	0.675500
$934$	−30.8409	−1.00915
$935$	6.72502	0.219932
$936$	44.8834	1.46706
$937$	−8.09231	−0.264364	−0.132182	−	0.991225i	$-0.542198\pi$
$937$	−8.09231	−0.264364	−0.132182	+	0.991225i	$0.542198\pi$
$938$	16.7087	0.545559
$939$	−81.1314	−2.64762
$940$	−32.8023	−1.06989
$941$	−19.6969	−0.642099	−0.321050	−	0.947062i	$-0.604036\pi$
$941$	−19.6969	−0.642099	−0.321050	+	0.947062i	$0.604036\pi$
$942$	−42.0149	−1.36892
$943$	32.6889	1.06450
$944$	−3.06613	−0.0997939
$945$	136.051	4.42573
$946$	69.6434	2.26430
$947$	−33.6474	−1.09339	−0.546696	−	0.837331i	$-0.684115\pi$
$947$	−33.6474	−1.09339	−0.546696	+	0.837331i	$0.684115\pi$
$948$	−26.9148	−0.874151
$949$	−82.8213	−2.68849
$950$	−7.03103	−0.228117
$951$	5.24454	0.170066
$952$	0.741562	0.0240342
$953$	−27.8562	−0.902351	−0.451175	−	0.892435i	$-0.648995\pi$
$953$	−27.8562	−0.902351	−0.451175	+	0.892435i	$0.648995\pi$
$954$	90.2892	2.92322
$955$	84.7752	2.74326
$956$	−19.8537	−0.642114
$957$	−181.640	−5.87158
$958$	−7.44367	−0.240494
$959$	17.0870	0.551767
$960$	12.9447	0.417787
$961$	−14.5993	−0.470946
$962$	18.3978	0.593167
$963$	−2.16004	−0.0696063
$964$	9.54181	0.307321
$965$	−25.8599	−0.832461
$966$	−22.3894	−0.720366
$967$	−25.4679	−0.818994	−0.409497	−	0.912312i	$-0.634296\pi$
$967$	−25.4679	−0.818994	−0.409497	+	0.912312i	$0.634296\pi$
$968$	−17.9683	−0.577522
$969$	0.637443	0.0204776
$970$	−38.4840	−1.23565
$971$	−4.88490	−0.156764	−0.0783820	−	0.996923i	$-0.524975\pi$
$971$	−4.88490	−0.156764	−0.0783820	+	0.996923i	$0.524975\pi$
$972$	−33.8619	−1.08612
$973$	−2.58096	−0.0827417
$974$	−39.9749	−1.28088
$975$	216.612	6.93714
$976$	5.38097	0.172241
$977$	−55.0118	−1.75998	−0.879992	−	0.474989i	$-0.842452\pi$
$977$	−55.0118	−1.75998	−0.879992	+	0.474989i	$0.842452\pi$
$978$	59.3536	1.89792
$979$	−23.1322	−0.739308
$980$	5.34688	0.170800
$981$	70.1781	2.24061
$982$	−28.2410	−0.901205
$983$	−57.4393	−1.83203	−0.916015	−	0.401145i	$-0.868612\pi$
$983$	−57.4393	−1.83203	−0.916015	+	0.401145i	$0.868612\pi$
$984$	−36.1989	−1.15398
$985$	0.934217	0.0297666
$986$	3.25755	0.103742
$987$	−62.8275	−1.99982
$988$	−3.83805	−0.122105
$989$	37.7098	1.19910
$990$	160.079	5.08765
$991$	−8.54958	−0.271586	−0.135793	−	0.990737i	$-0.543358\pi$
$991$	−8.54958	−0.271586	−0.135793	+	0.990737i	$0.543358\pi$
$992$	−4.04978	−0.128580
$993$	57.8163	1.83475
$994$	22.1498	0.702549
$995$	−91.6051	−2.90408
$996$	24.0143	0.760923
$997$	41.1596	1.30354	0.651769	−	0.758418i	$-0.274027\pi$
$997$	41.1596	1.30354	0.651769	+	0.758418i	$0.274027\pi$
$998$	39.0052	1.23469
$999$	−43.3069	−1.37017

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.c.1.5	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.c.1.5	✓	84	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.22723 q^{3} +1.00000 q^{4} -4.01107 q^{5} +3.22723 q^{6} +2.38054 q^{7} -1.00000 q^{8} +7.41504 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.22723 q^{3} +1.00000 q^{4} -4.01107 q^{5} +3.22723 q^{6} +2.38054 q^{7} -1.00000 q^{8} +7.41504 q^{9} +4.01107 q^{10} +5.38222 q^{11} -3.22723 q^{12} -6.05302 q^{13} -2.38054 q^{14} +12.9447 q^{15} +1.00000 q^{16} -0.311510 q^{17} -7.41504 q^{18} +0.634072 q^{19} -4.01107 q^{20} -7.68256 q^{21} -5.38222 q^{22} -2.91431 q^{23} +3.22723 q^{24} +11.0887 q^{25} +6.05302 q^{26} -14.2483 q^{27} +2.38054 q^{28} +10.4573 q^{29} -12.9447 q^{30} +4.04978 q^{31} -1.00000 q^{32} -17.3697 q^{33} +0.311510 q^{34} -9.54851 q^{35} +7.41504 q^{36} +3.03943 q^{37} -0.634072 q^{38} +19.5345 q^{39} +4.01107 q^{40} -11.2167 q^{41} +7.68256 q^{42} -12.9395 q^{43} +5.38222 q^{44} -29.7422 q^{45} +2.91431 q^{46} +8.17795 q^{47} -3.22723 q^{48} -1.33303 q^{49} -11.0887 q^{50} +1.00532 q^{51} -6.05302 q^{52} -12.1765 q^{53} +14.2483 q^{54} -21.5885 q^{55} -2.38054 q^{56} -2.04630 q^{57} -10.4573 q^{58} -3.06613 q^{59} +12.9447 q^{60} +5.38097 q^{61} -4.04978 q^{62} +17.6518 q^{63} +1.00000 q^{64} +24.2791 q^{65} +17.3697 q^{66} -7.01888 q^{67} -0.311510 q^{68} +9.40516 q^{69} +9.54851 q^{70} -9.30452 q^{71} -7.41504 q^{72} +13.6826 q^{73} -3.03943 q^{74} -35.7858 q^{75} +0.634072 q^{76} +12.8126 q^{77} -19.5345 q^{78} +8.33989 q^{79} -4.01107 q^{80} +23.7376 q^{81} +11.2167 q^{82} -7.44115 q^{83} -7.68256 q^{84} +1.24949 q^{85} +12.9395 q^{86} -33.7481 q^{87} -5.38222 q^{88} -4.29789 q^{89} +29.7422 q^{90} -14.4095 q^{91} -2.91431 q^{92} -13.0696 q^{93} -8.17795 q^{94} -2.54331 q^{95} +3.22723 q^{96} -9.59444 q^{97} +1.33303 q^{98} +39.9093 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * q^99

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