LMFDB - Embedded newform 8038.2.a.b.1.16

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			1.16
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.05749 q^{3} +1.00000 q^{4} +3.25924 q^{5} +2.05749 q^{6} -0.840215 q^{7} -1.00000 q^{8} +1.23326 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.05749 q^{3} +1.00000 q^{4} +3.25924 q^{5} +2.05749 q^{6} -0.840215 q^{7} -1.00000 q^{8} +1.23326 q^{9} -3.25924 q^{10} +1.62971 q^{11} -2.05749 q^{12} -4.10968 q^{13} +0.840215 q^{14} -6.70585 q^{15} +1.00000 q^{16} +7.91360 q^{17} -1.23326 q^{18} -2.91818 q^{19} +3.25924 q^{20} +1.72873 q^{21} -1.62971 q^{22} +4.53722 q^{23} +2.05749 q^{24} +5.62265 q^{25} +4.10968 q^{26} +3.63504 q^{27} -0.840215 q^{28} +0.0880813 q^{29} +6.70585 q^{30} +9.54850 q^{31} -1.00000 q^{32} -3.35310 q^{33} -7.91360 q^{34} -2.73846 q^{35} +1.23326 q^{36} +8.65462 q^{37} +2.91818 q^{38} +8.45561 q^{39} -3.25924 q^{40} +0.590601 q^{41} -1.72873 q^{42} -11.4938 q^{43} +1.62971 q^{44} +4.01950 q^{45} -4.53722 q^{46} -2.05727 q^{47} -2.05749 q^{48} -6.29404 q^{49} -5.62265 q^{50} -16.2821 q^{51} -4.10968 q^{52} +0.844203 q^{53} -3.63504 q^{54} +5.31160 q^{55} +0.840215 q^{56} +6.00413 q^{57} -0.0880813 q^{58} -6.65504 q^{59} -6.70585 q^{60} +6.70195 q^{61} -9.54850 q^{62} -1.03621 q^{63} +1.00000 q^{64} -13.3944 q^{65} +3.35310 q^{66} +6.68130 q^{67} +7.91360 q^{68} -9.33528 q^{69} +2.73846 q^{70} -1.33767 q^{71} -1.23326 q^{72} +2.38431 q^{73} -8.65462 q^{74} -11.5685 q^{75} -2.91818 q^{76} -1.36930 q^{77} -8.45561 q^{78} +7.65153 q^{79} +3.25924 q^{80} -11.1789 q^{81} -0.590601 q^{82} -7.64907 q^{83} +1.72873 q^{84} +25.7923 q^{85} +11.4938 q^{86} -0.181226 q^{87} -1.62971 q^{88} -9.78200 q^{89} -4.01950 q^{90} +3.45301 q^{91} +4.53722 q^{92} -19.6459 q^{93} +2.05727 q^{94} -9.51105 q^{95} +2.05749 q^{96} +14.1311 q^{97} +6.29404 q^{98} +2.00986 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.05749	−1.18789	−0.593946	−	0.804505i	$-0.702431\pi$
$3$	−2.05749	−1.18789	−0.593946	+	0.804505i	$0.702431\pi$
$4$	1.00000	0.500000
$5$	3.25924	1.45758	0.728788	−	0.684739i	$-0.240084\pi$
$5$	3.25924	1.45758	0.728788	+	0.684739i	$0.240084\pi$
$6$	2.05749	0.839967
$7$	−0.840215	−0.317571	−0.158786	−	0.987313i	$-0.550758\pi$
$7$	−0.840215	−0.317571	−0.158786	+	0.987313i	$0.550758\pi$
$8$	−1.00000	−0.353553
$9$	1.23326	0.411088
$10$	−3.25924	−1.03066
$11$	1.62971	0.491375	0.245687	−	0.969349i	$-0.420986\pi$
$11$	1.62971	0.491375	0.245687	+	0.969349i	$0.420986\pi$
$12$	−2.05749	−0.593946
$13$	−4.10968	−1.13982	−0.569909	−	0.821708i	$-0.693022\pi$
$13$	−4.10968	−1.13982	−0.569909	+	0.821708i	$0.693022\pi$
$14$	0.840215	0.224557
$15$	−6.70585	−1.73144
$16$	1.00000	0.250000
$17$	7.91360	1.91933	0.959665	−	0.281146i	$-0.0907146\pi$
$17$	7.91360	1.91933	0.959665	+	0.281146i	$0.0907146\pi$
$18$	−1.23326	−0.290683
$19$	−2.91818	−0.669476	−0.334738	−	0.942311i	$-0.608648\pi$
$19$	−2.91818	−0.669476	−0.334738	+	0.942311i	$0.608648\pi$
$20$	3.25924	0.728788
$21$	1.72873	0.377240
$22$	−1.62971	−0.347454
$23$	4.53722	0.946075	0.473038	−	0.881042i	$-0.343157\pi$
$23$	4.53722	0.946075	0.473038	+	0.881042i	$0.343157\pi$
$24$	2.05749	0.419983
$25$	5.62265	1.12453
$26$	4.10968	0.805974
$27$	3.63504	0.699564
$28$	−0.840215	−0.158786
$29$	0.0880813	0.0163563	0.00817814	−	0.999967i	$-0.497397\pi$
$29$	0.0880813	0.0163563	0.00817814	+	0.999967i	$0.497397\pi$
$30$	6.70585	1.22432
$31$	9.54850	1.71496	0.857481	−	0.514516i	$-0.172028\pi$
$31$	9.54850	1.71496	0.857481	+	0.514516i	$0.172028\pi$
$32$	−1.00000	−0.176777
$33$	−3.35310	−0.583700
$34$	−7.91360	−1.35717
$35$	−2.73846	−0.462885
$36$	1.23326	0.205544
$37$	8.65462	1.42281	0.711406	−	0.702782i	$-0.248059\pi$
$37$	8.65462	1.42281	0.711406	+	0.702782i	$0.248059\pi$
$38$	2.91818	0.473391
$39$	8.45561	1.35398
$40$	−3.25924	−0.515331
$41$	0.590601	0.0922364	0.0461182	−	0.998936i	$-0.485315\pi$
$41$	0.590601	0.0922364	0.0461182	+	0.998936i	$0.485315\pi$
$42$	−1.72873	−0.266749
$43$	−11.4938	−1.75279	−0.876396	−	0.481592i	$-0.840059\pi$
$43$	−11.4938	−1.75279	−0.876396	+	0.481592i	$0.840059\pi$
$44$	1.62971	0.245687
$45$	4.01950	0.599192
$46$	−4.53722	−0.668976
$47$	−2.05727	−0.300083	−0.150042	−	0.988680i	$-0.547941\pi$
$47$	−2.05727	−0.300083	−0.150042	+	0.988680i	$0.547941\pi$
$48$	−2.05749	−0.296973
$49$	−6.29404	−0.899148
$50$	−5.62265	−0.795163
$51$	−16.2821	−2.27996
$52$	−4.10968	−0.569909
$53$	0.844203	0.115960	0.0579801	−	0.998318i	$-0.481534\pi$
$53$	0.844203	0.115960	0.0579801	+	0.998318i	$0.481534\pi$
$54$	−3.63504	−0.494667
$55$	5.31160	0.716216
$56$	0.840215	0.112278
$57$	6.00413	0.795266
$58$	−0.0880813	−0.0115656
$59$	−6.65504	−0.866413	−0.433206	−	0.901295i	$-0.642618\pi$
$59$	−6.65504	−0.866413	−0.433206	+	0.901295i	$0.642618\pi$
$60$	−6.70585	−0.865722
$61$	6.70195	0.858096	0.429048	−	0.903282i	$-0.358849\pi$
$61$	6.70195	0.858096	0.429048	+	0.903282i	$0.358849\pi$
$62$	−9.54850	−1.21266
$63$	−1.03621	−0.130550
$64$	1.00000	0.125000
$65$	−13.3944	−1.66137
$66$	3.35310	0.412738
$67$	6.68130	0.816250	0.408125	−	0.912926i	$-0.366183\pi$
$67$	6.68130	0.816250	0.408125	+	0.912926i	$0.366183\pi$
$68$	7.91360	0.959665
$69$	−9.33528	−1.12384
$70$	2.73846	0.327309
$71$	−1.33767	−0.158752	−0.0793762	−	0.996845i	$-0.525293\pi$
$71$	−1.33767	−0.158752	−0.0793762	+	0.996845i	$0.525293\pi$
$72$	−1.23326	−0.145341
$73$	2.38431	0.279062	0.139531	−	0.990218i	$-0.455440\pi$
$73$	2.38431	0.279062	0.139531	+	0.990218i	$0.455440\pi$
$74$	−8.65462	−1.00608
$75$	−11.5685	−1.33582
$76$	−2.91818	−0.334738
$77$	−1.36930	−0.156046
$78$	−8.45561	−0.957410
$79$	7.65153	0.860864	0.430432	−	0.902623i	$-0.358361\pi$
$79$	7.65153	0.860864	0.430432	+	0.902623i	$0.358361\pi$
$80$	3.25924	0.364394
$81$	−11.1789	−1.24209
$82$	−0.590601	−0.0652210
$83$	−7.64907	−0.839595	−0.419797	−	0.907618i	$-0.637899\pi$
$83$	−7.64907	−0.839595	−0.419797	+	0.907618i	$0.637899\pi$
$84$	1.72873	0.188620
$85$	25.7923	2.79757
$86$	11.4938	1.23941
$87$	−0.181226	−0.0194295
$88$	−1.62971	−0.173727
$89$	−9.78200	−1.03689	−0.518445	−	0.855111i	$-0.673489\pi$
$89$	−9.78200	−1.03689	−0.518445	+	0.855111i	$0.673489\pi$
$90$	−4.01950	−0.423693
$91$	3.45301	0.361974
$92$	4.53722	0.473038
$93$	−19.6459	−2.03719
$94$	2.05727	0.212191
$95$	−9.51105	−0.975813
$96$	2.05749	0.209992
$97$	14.1311	1.43479	0.717396	−	0.696666i	$-0.245334\pi$
$97$	14.1311	1.43479	0.717396	+	0.696666i	$0.245334\pi$
$98$	6.29404	0.635794
$99$	2.00986	0.201998
$100$	5.62265	0.562265
$101$	16.6226	1.65401	0.827003	−	0.562198i	$-0.190044\pi$
$101$	16.6226	1.65401	0.827003	+	0.562198i	$0.190044\pi$
$102$	16.2821	1.61217
$103$	−3.57744	−0.352495	−0.176248	−	0.984346i	$-0.556396\pi$
$103$	−3.57744	−0.352495	−0.176248	+	0.984346i	$0.556396\pi$
$104$	4.10968	0.402987
$105$	5.63436	0.549857
$106$	−0.844203	−0.0819962
$107$	−10.7572	−1.03994	−0.519968	−	0.854186i	$-0.674056\pi$
$107$	−10.7572	−1.03994	−0.519968	+	0.854186i	$0.674056\pi$
$108$	3.63504	0.349782
$109$	1.58615	0.151925	0.0759627	−	0.997111i	$-0.475797\pi$
$109$	1.58615	0.151925	0.0759627	+	0.997111i	$0.475797\pi$
$110$	−5.31160	−0.506441
$111$	−17.8068	−1.69015
$112$	−0.840215	−0.0793928
$113$	11.2149	1.05501	0.527504	−	0.849553i	$-0.323128\pi$
$113$	11.2149	1.05501	0.527504	+	0.849553i	$0.323128\pi$
$114$	−6.00413	−0.562338
$115$	14.7879	1.37898
$116$	0.0880813	0.00817814
$117$	−5.06831	−0.468565
$118$	6.65504	0.612646
$119$	−6.64912	−0.609524
$120$	6.70585	0.612158
$121$	−8.34406	−0.758551
$122$	−6.70195	−0.606766
$123$	−1.21516	−0.109567
$124$	9.54850	0.857481
$125$	2.02936	0.181512
$126$	1.03621	0.0923126
$127$	2.06161	0.182939	0.0914694	−	0.995808i	$-0.470844\pi$
$127$	2.06161	0.182939	0.0914694	+	0.995808i	$0.470844\pi$
$128$	−1.00000	−0.0883883
$129$	23.6484	2.08213
$130$	13.3944	1.17477
$131$	−4.59796	−0.401726	−0.200863	−	0.979619i	$-0.564375\pi$
$131$	−4.59796	−0.401726	−0.200863	+	0.979619i	$0.564375\pi$
$132$	−3.35310	−0.291850
$133$	2.45190	0.212607
$134$	−6.68130	−0.577176
$135$	11.8475	1.01967
$136$	−7.91360	−0.678586
$137$	−7.64762	−0.653380	−0.326690	−	0.945132i	$-0.605933\pi$
$137$	−7.64762	−0.653380	−0.326690	+	0.945132i	$0.605933\pi$
$138$	9.33528	0.794671
$139$	13.7829	1.16905	0.584524	−	0.811377i	$-0.301281\pi$
$139$	13.7829	1.16905	0.584524	+	0.811377i	$0.301281\pi$
$140$	−2.73846	−0.231442
$141$	4.23280	0.356466
$142$	1.33767	0.112255
$143$	−6.69756	−0.560078
$144$	1.23326	0.102772
$145$	0.287078	0.0238405
$146$	−2.38431	−0.197327
$147$	12.9499	1.06809
$148$	8.65462	0.711406
$149$	16.8630	1.38147	0.690736	−	0.723107i	$-0.257287\pi$
$149$	16.8630	1.38147	0.690736	+	0.723107i	$0.257287\pi$
$150$	11.5685	0.944567
$151$	−1.88575	−0.153460	−0.0767302	−	0.997052i	$-0.524448\pi$
$151$	−1.88575	−0.153460	−0.0767302	+	0.997052i	$0.524448\pi$
$152$	2.91818	0.236696
$153$	9.75955	0.789013
$154$	1.36930	0.110342
$155$	31.1209	2.49969
$156$	8.45561	0.676991
$157$	14.6603	1.17002	0.585010	−	0.811026i	$-0.301091\pi$
$157$	14.6603	1.17002	0.585010	+	0.811026i	$0.301091\pi$
$158$	−7.65153	−0.608723
$159$	−1.73694	−0.137748
$160$	−3.25924	−0.257666
$161$	−3.81224	−0.300446
$162$	11.1789	0.878294
$163$	−6.62729	−0.519089	−0.259545	−	0.965731i	$-0.583573\pi$
$163$	−6.62729	−0.519089	−0.259545	+	0.965731i	$0.583573\pi$
$164$	0.590601	0.0461182
$165$	−10.9286	−0.850787
$166$	7.64907	0.593683
$167$	3.25610	0.251964	0.125982	−	0.992033i	$-0.459792\pi$
$167$	3.25610	0.251964	0.125982	+	0.992033i	$0.459792\pi$
$168$	−1.72873	−0.133375
$169$	3.88943	0.299187
$170$	−25.7923	−1.97818
$171$	−3.59888	−0.275214
$172$	−11.4938	−0.876396
$173$	−20.5804	−1.56470	−0.782348	−	0.622842i	$-0.785978\pi$
$173$	−20.5804	−1.56470	−0.782348	+	0.622842i	$0.785978\pi$
$174$	0.181226	0.0137387
$175$	−4.72423	−0.357118
$176$	1.62971	0.122844
$177$	13.6927	1.02920
$178$	9.78200	0.733192
$179$	9.36782	0.700184	0.350092	−	0.936715i	$-0.386150\pi$
$179$	9.36782	0.700184	0.350092	+	0.936715i	$0.386150\pi$
$180$	4.01950	0.299596
$181$	−11.2261	−0.834432	−0.417216	−	0.908807i	$-0.636994\pi$
$181$	−11.2261	−0.834432	−0.417216	+	0.908807i	$0.636994\pi$
$182$	−3.45301	−0.255954
$183$	−13.7892	−1.01933
$184$	−4.53722	−0.334488
$185$	28.2075	2.07386
$186$	19.6459	1.44051
$187$	12.8968	0.943110
$188$	−2.05727	−0.150042
$189$	−3.05422	−0.222162
$190$	9.51105	0.690004
$191$	−27.2442	−1.97132	−0.985660	−	0.168743i	$-0.946029\pi$
$191$	−27.2442	−1.97132	−0.985660	+	0.168743i	$0.946029\pi$
$192$	−2.05749	−0.148487
$193$	8.85412	0.637333	0.318667	−	0.947867i	$-0.396765\pi$
$193$	8.85412	0.637333	0.318667	+	0.947867i	$0.396765\pi$
$194$	−14.1311	−1.01455
$195$	27.5589	1.97353
$196$	−6.29404	−0.449574
$197$	10.8717	0.774579	0.387289	−	0.921958i	$-0.373411\pi$
$197$	10.8717	0.774579	0.387289	+	0.921958i	$0.373411\pi$
$198$	−2.00986	−0.142834
$199$	−15.2143	−1.07851	−0.539255	−	0.842142i	$-0.681294\pi$
$199$	−15.2143	−1.07851	−0.539255	+	0.842142i	$0.681294\pi$
$200$	−5.62265	−0.397581
$201$	−13.7467	−0.969617
$202$	−16.6226	−1.16956
$203$	−0.0740072	−0.00519429
$204$	−16.2821	−1.13998
$205$	1.92491	0.134442
$206$	3.57744	0.249252
$207$	5.59558	0.388920
$208$	−4.10968	−0.284955
$209$	−4.75577	−0.328964
$210$	−5.63436	−0.388808
$211$	23.7659	1.63611	0.818056	−	0.575138i	$-0.195052\pi$
$211$	23.7659	1.63611	0.818056	+	0.575138i	$0.195052\pi$
$212$	0.844203	0.0579801
$213$	2.75225	0.188581
$214$	10.7572	0.735346
$215$	−37.4611	−2.55483
$216$	−3.63504	−0.247333
$217$	−8.02279	−0.544623
$218$	−1.58615	−0.107427
$219$	−4.90569	−0.331496
$220$	5.31160	0.358108
$221$	−32.5223	−2.18769
$222$	17.8068	1.19511
$223$	−7.05002	−0.472104	−0.236052	−	0.971740i	$-0.575854\pi$
$223$	−7.05002	−0.472104	−0.236052	+	0.971740i	$0.575854\pi$
$224$	0.840215	0.0561392
$225$	6.93420	0.462280
$226$	−11.2149	−0.746003
$227$	16.3846	1.08749	0.543743	−	0.839252i	$-0.317007\pi$
$227$	16.3846	1.08749	0.543743	+	0.839252i	$0.317007\pi$
$228$	6.00413	0.397633
$229$	10.5321	0.695979	0.347990	−	0.937498i	$-0.386864\pi$
$229$	10.5321	0.695979	0.347990	+	0.937498i	$0.386864\pi$
$230$	−14.7879	−0.975084
$231$	2.81733	0.185366
$232$	−0.0880813	−0.00578282
$233$	6.40201	0.419410	0.209705	−	0.977765i	$-0.432750\pi$
$233$	6.40201	0.419410	0.209705	+	0.977765i	$0.432750\pi$
$234$	5.06831	0.331326
$235$	−6.70513	−0.437394
$236$	−6.65504	−0.433206
$237$	−15.7429	−1.02261
$238$	6.64912	0.430999
$239$	−15.1688	−0.981187	−0.490593	−	0.871389i	$-0.663220\pi$
$239$	−15.1688	−0.981187	−0.490593	+	0.871389i	$0.663220\pi$
$240$	−6.70585	−0.432861
$241$	1.63537	0.105343	0.0526717	−	0.998612i	$-0.483226\pi$
$241$	1.63537	0.105343	0.0526717	+	0.998612i	$0.483226\pi$
$242$	8.34406	0.536377
$243$	12.0952	0.775910
$244$	6.70195	0.429048
$245$	−20.5138	−1.31058
$246$	1.21516	0.0774755
$247$	11.9928	0.763082
$248$	−9.54850	−0.606331
$249$	15.7379	0.997348
$250$	−2.02936	−0.128348
$251$	−5.73299	−0.361863	−0.180931	−	0.983496i	$-0.557911\pi$
$251$	−5.73299	−0.361863	−0.180931	+	0.983496i	$0.557911\pi$
$252$	−1.03621	−0.0652748
$253$	7.39433	0.464877
$254$	−2.06161	−0.129357
$255$	−53.0674	−3.32321
$256$	1.00000	0.0625000
$257$	13.9925	0.872830	0.436415	−	0.899745i	$-0.356248\pi$
$257$	13.9925	0.872830	0.436415	+	0.899745i	$0.356248\pi$
$258$	−23.6484	−1.47229
$259$	−7.27174	−0.451844
$260$	−13.3944	−0.830687
$261$	0.108627	0.00672387
$262$	4.59796	0.284063
$263$	16.4843	1.01646	0.508232	−	0.861220i	$-0.330299\pi$
$263$	16.4843	1.01646	0.508232	+	0.861220i	$0.330299\pi$
$264$	3.35310	0.206369
$265$	2.75146	0.169021
$266$	−2.45190	−0.150336
$267$	20.1264	1.23171
$268$	6.68130	0.408125
$269$	1.85673	0.113207	0.0566034	−	0.998397i	$-0.481973\pi$
$269$	1.85673	0.113207	0.0566034	+	0.998397i	$0.481973\pi$
$270$	−11.8475	−0.721015
$271$	−22.4391	−1.36308	−0.681539	−	0.731781i	$-0.738689\pi$
$271$	−22.4391	−1.36308	−0.681539	+	0.731781i	$0.738689\pi$
$272$	7.91360	0.479832
$273$	−7.10453	−0.429986
$274$	7.64762	0.462009
$275$	9.16326	0.552565
$276$	−9.33528	−0.561918
$277$	−10.9916	−0.660421	−0.330210	−	0.943907i	$-0.607120\pi$
$277$	−10.9916	−0.660421	−0.330210	+	0.943907i	$0.607120\pi$
$278$	−13.7829	−0.826641
$279$	11.7758	0.705000
$280$	2.73846	0.163654
$281$	−25.2803	−1.50810	−0.754048	−	0.656820i	$-0.771901\pi$
$281$	−25.2803	−1.50810	−0.754048	+	0.656820i	$0.771901\pi$
$282$	−4.23280	−0.252060
$283$	18.3291	1.08955	0.544776	−	0.838581i	$-0.316615\pi$
$283$	18.3291	1.08955	0.544776	+	0.838581i	$0.316615\pi$
$284$	−1.33767	−0.0793762
$285$	19.5689	1.15916
$286$	6.69756	0.396035
$287$	−0.496232	−0.0292916
$288$	−1.23326	−0.0726707
$289$	45.6251	2.68383
$290$	−0.287078	−0.0168578
$291$	−29.0745	−1.70438
$292$	2.38431	0.139531
$293$	5.83471	0.340867	0.170434	−	0.985369i	$-0.445483\pi$
$293$	5.83471	0.340867	0.170434	+	0.985369i	$0.445483\pi$
$294$	−12.9499	−0.755255
$295$	−21.6904	−1.26286
$296$	−8.65462	−0.503040
$297$	5.92405	0.343748
$298$	−16.8630	−0.976848
$299$	−18.6465	−1.07835
$300$	−11.5685	−0.667910
$301$	9.65728	0.556636
$302$	1.88575	0.108513
$303$	−34.2007	−1.96478
$304$	−2.91818	−0.167369
$305$	21.8433	1.25074
$306$	−9.75955	−0.557916
$307$	5.46634	0.311981	0.155990	−	0.987759i	$-0.450143\pi$
$307$	5.46634	0.311981	0.155990	+	0.987759i	$0.450143\pi$
$308$	−1.36930	−0.0780232
$309$	7.36054	0.418726
$310$	−31.1209	−1.76755
$311$	−23.1972	−1.31539	−0.657696	−	0.753283i	$-0.728469\pi$
$311$	−23.1972	−1.31539	−0.657696	+	0.753283i	$0.728469\pi$
$312$	−8.45561	−0.478705
$313$	−19.4616	−1.10003	−0.550017	−	0.835153i	$-0.685379\pi$
$313$	−19.4616	−1.10003	−0.550017	+	0.835153i	$0.685379\pi$
$314$	−14.6603	−0.827329
$315$	−3.37724	−0.190286
$316$	7.65153	0.430432
$317$	13.2864	0.746237	0.373118	−	0.927784i	$-0.378289\pi$
$317$	13.2864	0.746237	0.373118	+	0.927784i	$0.378289\pi$
$318$	1.73694	0.0974026
$319$	0.143547	0.00803706
$320$	3.25924	0.182197
$321$	22.1328	1.23533
$322$	3.81224	0.212448
$323$	−23.0933	−1.28495
$324$	−11.1789	−0.621047
$325$	−23.1073	−1.28176
$326$	6.62729	0.367052
$327$	−3.26348	−0.180471
$328$	−0.590601	−0.0326105
$329$	1.72855	0.0952978
$330$	10.9286	0.601598
$331$	3.10725	0.170790	0.0853951	−	0.996347i	$-0.472785\pi$
$331$	3.10725	0.170790	0.0853951	+	0.996347i	$0.472785\pi$
$332$	−7.64907	−0.419797
$333$	10.6734	0.584900
$334$	−3.25610	−0.178166
$335$	21.7759	1.18975
$336$	1.72873	0.0943101
$337$	17.4340	0.949691	0.474846	−	0.880069i	$-0.342504\pi$
$337$	17.4340	0.949691	0.474846	+	0.880069i	$0.342504\pi$
$338$	−3.88943	−0.211557
$339$	−23.0745	−1.25324
$340$	25.7923	1.39879
$341$	15.5612	0.842689
$342$	3.59888	0.194605
$343$	11.1698	0.603115
$344$	11.4938	0.619705
$345$	−30.4259	−1.63808
$346$	20.5804	1.10641
$347$	12.2012	0.654997	0.327499	−	0.944852i	$-0.393794\pi$
$347$	12.2012	0.654997	0.327499	+	0.944852i	$0.393794\pi$
$348$	−0.181226	−0.00971475
$349$	−13.6843	−0.732505	−0.366252	−	0.930516i	$-0.619359\pi$
$349$	−13.6843	−0.732505	−0.366252	+	0.930516i	$0.619359\pi$
$350$	4.72423	0.252521
$351$	−14.9388	−0.797377
$352$	−1.62971	−0.0868636
$353$	0.839045	0.0446578	0.0223289	−	0.999751i	$-0.492892\pi$
$353$	0.839045	0.0446578	0.0223289	+	0.999751i	$0.492892\pi$
$354$	−13.6927	−0.727758
$355$	−4.35979	−0.231394
$356$	−9.78200	−0.518445
$357$	13.6805	0.724049
$358$	−9.36782	−0.495105
$359$	−28.5821	−1.50851	−0.754254	−	0.656583i	$-0.772001\pi$
$359$	−28.5821	−1.50851	−0.754254	+	0.656583i	$0.772001\pi$
$360$	−4.01950	−0.211846
$361$	−10.4842	−0.551801
$362$	11.2261	0.590032
$363$	17.1678	0.901077
$364$	3.45301	0.180987
$365$	7.77104	0.406755
$366$	13.7892	0.720772
$367$	−23.7761	−1.24110	−0.620552	−	0.784165i	$-0.713091\pi$
$367$	−23.7761	−1.24110	−0.620552	+	0.784165i	$0.713091\pi$
$368$	4.53722	0.236519
$369$	0.728367	0.0379173
$370$	−28.2075	−1.46644
$371$	−0.709312	−0.0368256
$372$	−19.6459	−1.01859
$373$	−13.1697	−0.681901	−0.340950	−	0.940081i	$-0.610749\pi$
$373$	−13.1697	−0.681901	−0.340950	+	0.940081i	$0.610749\pi$
$374$	−12.8968	−0.666879
$375$	−4.17539	−0.215616
$376$	2.05727	0.106095
$377$	−0.361985	−0.0186432
$378$	3.05422	0.157092
$379$	25.8227	1.32642	0.663211	−	0.748432i	$-0.269193\pi$
$379$	25.8227	1.32642	0.663211	+	0.748432i	$0.269193\pi$
$380$	−9.51105	−0.487907
$381$	−4.24175	−0.217311
$382$	27.2442	1.39393
$383$	7.32996	0.374543	0.187272	−	0.982308i	$-0.440036\pi$
$383$	7.32996	0.374543	0.187272	+	0.982308i	$0.440036\pi$
$384$	2.05749	0.104996
$385$	−4.46289	−0.227450
$386$	−8.85412	−0.450663
$387$	−14.1749	−0.720551
$388$	14.1311	0.717396
$389$	28.4100	1.44044	0.720221	−	0.693745i	$-0.244040\pi$
$389$	28.4100	1.44044	0.720221	+	0.693745i	$0.244040\pi$
$390$	−27.5589	−1.39550
$391$	35.9057	1.81583
$392$	6.29404	0.317897
$393$	9.46025	0.477207
$394$	−10.8717	−0.547710
$395$	24.9382	1.25478
$396$	2.00986	0.100999
$397$	23.6987	1.18940	0.594701	−	0.803947i	$-0.297270\pi$
$397$	23.6987	1.18940	0.594701	+	0.803947i	$0.297270\pi$
$398$	15.2143	0.762622
$399$	−5.04476	−0.252554
$400$	5.62265	0.281132
$401$	16.2519	0.811583	0.405792	−	0.913966i	$-0.366996\pi$
$401$	16.2519	0.811583	0.405792	+	0.913966i	$0.366996\pi$
$402$	13.7467	0.685623
$403$	−39.2412	−1.95475
$404$	16.6226	0.827003
$405$	−36.4346	−1.81045
$406$	0.0740072	0.00367292
$407$	14.1045	0.699133
$408$	16.2821	0.806086
$409$	−14.5480	−0.719353	−0.359677	−	0.933077i	$-0.617113\pi$
$409$	−14.5480	−0.719353	−0.359677	+	0.933077i	$0.617113\pi$
$410$	−1.92491	−0.0950646
$411$	15.7349	0.776145
$412$	−3.57744	−0.176248
$413$	5.59166	0.275148
$414$	−5.59558	−0.275008
$415$	−24.9302	−1.22377
$416$	4.10968	0.201493
$417$	−28.3581	−1.38870
$418$	4.75577	0.232612
$419$	32.8454	1.60461	0.802303	−	0.596918i	$-0.203608\pi$
$419$	32.8454	1.60461	0.802303	+	0.596918i	$0.203608\pi$
$420$	5.63436	0.274928
$421$	32.1834	1.56852	0.784262	−	0.620430i	$-0.213042\pi$
$421$	32.1834	1.56852	0.784262	+	0.620430i	$0.213042\pi$
$422$	−23.7659	−1.15691
$423$	−2.53715	−0.123360
$424$	−0.844203	−0.0409981
$425$	44.4954	2.15834
$426$	−2.75225	−0.133347
$427$	−5.63107	−0.272507
$428$	−10.7572	−0.519968
$429$	13.7802	0.665312
$430$	37.4611	1.80654
$431$	24.0853	1.16015	0.580074	−	0.814564i	$-0.303024\pi$
$431$	24.0853	1.16015	0.580074	+	0.814564i	$0.303024\pi$
$432$	3.63504	0.174891
$433$	24.8265	1.19308	0.596542	−	0.802582i	$-0.296541\pi$
$433$	24.8265	1.19308	0.596542	+	0.802582i	$0.296541\pi$
$434$	8.02279	0.385106
$435$	−0.590660	−0.0283200
$436$	1.58615	0.0759627
$437$	−13.2404	−0.633375
$438$	4.90569	0.234403
$439$	−13.9738	−0.666933	−0.333466	−	0.942762i	$-0.608218\pi$
$439$	−13.9738	−0.666933	−0.333466	+	0.942762i	$0.608218\pi$
$440$	−5.31160	−0.253221
$441$	−7.76221	−0.369629
$442$	32.5223	1.54693
$443$	30.3130	1.44021	0.720107	−	0.693863i	$-0.244093\pi$
$443$	30.3130	1.44021	0.720107	+	0.693863i	$0.244093\pi$
$444$	−17.8068	−0.845073
$445$	−31.8819	−1.51135
$446$	7.05002	0.333828
$447$	−34.6955	−1.64104
$448$	−0.840215	−0.0396964
$449$	18.9210	0.892935	0.446468	−	0.894800i	$-0.352682\pi$
$449$	18.9210	0.892935	0.446468	+	0.894800i	$0.352682\pi$
$450$	−6.93420	−0.326882
$451$	0.962506	0.0453226
$452$	11.2149	0.527504
$453$	3.87992	0.182294
$454$	−16.3846	−0.768969
$455$	11.2542	0.527605
$456$	−6.00413	−0.281169
$457$	−4.11466	−0.192475	−0.0962377	−	0.995358i	$-0.530681\pi$
$457$	−4.11466	−0.192475	−0.0962377	+	0.995358i	$0.530681\pi$
$458$	−10.5321	−0.492132
$459$	28.7663	1.34269
$460$	14.7879	0.689489
$461$	19.7002	0.917528	0.458764	−	0.888558i	$-0.348292\pi$
$461$	19.7002	0.917528	0.458764	+	0.888558i	$0.348292\pi$
$462$	−2.81733	−0.131074
$463$	−26.2834	−1.22149	−0.610747	−	0.791826i	$-0.709131\pi$
$463$	−26.2834	−1.22149	−0.610747	+	0.791826i	$0.709131\pi$
$464$	0.0880813	0.00408907
$465$	−64.0309	−2.96936
$466$	−6.40201	−0.296567
$467$	−19.5870	−0.906381	−0.453190	−	0.891414i	$-0.649714\pi$
$467$	−19.5870	−0.906381	−0.453190	+	0.891414i	$0.649714\pi$
$468$	−5.06831	−0.234283
$469$	−5.61372	−0.259218
$470$	6.70513	0.309284
$471$	−30.1634	−1.38986
$472$	6.65504	0.306323
$473$	−18.7315	−0.861277
$474$	15.7429	0.723097
$475$	−16.4079	−0.752846
$476$	−6.64912	−0.304762
$477$	1.04112	0.0476698
$478$	15.1688	0.693804
$479$	4.04884	0.184996	0.0924982	−	0.995713i	$-0.470515\pi$
$479$	4.04884	0.184996	0.0924982	+	0.995713i	$0.470515\pi$
$480$	6.70585	0.306079
$481$	−35.5677	−1.62175
$482$	−1.63537	−0.0744891
$483$	7.84364	0.356898
$484$	−8.34406	−0.379276
$485$	46.0565	2.09132
$486$	−12.0952	−0.548651
$487$	25.3280	1.14772	0.573861	−	0.818953i	$-0.305445\pi$
$487$	25.3280	1.14772	0.573861	+	0.818953i	$0.305445\pi$
$488$	−6.70195	−0.303383
$489$	13.6356	0.616622
$490$	20.5138	0.926718
$491$	−21.3526	−0.963628	−0.481814	−	0.876274i	$-0.660022\pi$
$491$	−21.3526	−0.963628	−0.481814	+	0.876274i	$0.660022\pi$
$492$	−1.21516	−0.0547834
$493$	0.697040	0.0313931
$494$	−11.9928	−0.539580
$495$	6.55060	0.294428
$496$	9.54850	0.428740
$497$	1.12393	0.0504152
$498$	−15.7379	−0.705232
$499$	−25.6223	−1.14701	−0.573506	−	0.819201i	$-0.694417\pi$
$499$	−25.6223	−1.14701	−0.573506	+	0.819201i	$0.694417\pi$
$500$	2.02936	0.0907558
$501$	−6.69938	−0.299306
$502$	5.73299	0.255876
$503$	−19.2605	−0.858785	−0.429393	−	0.903118i	$-0.641272\pi$
$503$	−19.2605	−0.858785	−0.429393	+	0.903118i	$0.641272\pi$
$504$	1.03621	0.0461563
$505$	54.1769	2.41084
$506$	−7.39433	−0.328718
$507$	−8.00246	−0.355402
$508$	2.06161	0.0914694
$509$	−12.8324	−0.568786	−0.284393	−	0.958708i	$-0.591792\pi$
$509$	−12.8324	−0.568786	−0.284393	+	0.958708i	$0.591792\pi$
$510$	53.0674	2.34987
$511$	−2.00333	−0.0886222
$512$	−1.00000	−0.0441942
$513$	−10.6077	−0.468342
$514$	−13.9925	−0.617184
$515$	−11.6597	−0.513789
$516$	23.6484	1.04106
$517$	−3.35274	−0.147453
$518$	7.27174	0.319502
$519$	42.3439	1.85869
$520$	13.3944	0.587384
$521$	−6.45763	−0.282914	−0.141457	−	0.989944i	$-0.545179\pi$
$521$	−6.45763	−0.282914	−0.141457	+	0.989944i	$0.545179\pi$
$522$	−0.108627	−0.00475449
$523$	−3.85944	−0.168762	−0.0843808	−	0.996434i	$-0.526891\pi$
$523$	−3.85944	−0.168762	−0.0843808	+	0.996434i	$0.526891\pi$
$524$	−4.59796	−0.200863
$525$	9.72006	0.424218
$526$	−16.4843	−0.718749
$527$	75.5630	3.29158
$528$	−3.35310	−0.145925
$529$	−2.41366	−0.104942
$530$	−2.75146	−0.119516
$531$	−8.20742	−0.356172
$532$	2.45190	0.106303
$533$	−2.42718	−0.105133
$534$	−20.1264	−0.870953
$535$	−35.0602	−1.51579
$536$	−6.68130	−0.288588
$537$	−19.2742	−0.831743
$538$	−1.85673	−0.0800492
$539$	−10.2574	−0.441819
$540$	11.8475	0.509834
$541$	12.8902	0.554194	0.277097	−	0.960842i	$-0.410628\pi$
$541$	12.8902	0.554194	0.277097	+	0.960842i	$0.410628\pi$
$542$	22.4391	0.963842
$543$	23.0976	0.991215
$544$	−7.91360	−0.339293
$545$	5.16964	0.221443
$546$	7.10453	0.304046
$547$	−3.30722	−0.141406	−0.0707032	−	0.997497i	$-0.522524\pi$
$547$	−3.30722	−0.141406	−0.0707032	+	0.997497i	$0.522524\pi$
$548$	−7.64762	−0.326690
$549$	8.26526	0.352753
$550$	−9.16326	−0.390723
$551$	−0.257037	−0.0109501
$552$	9.33528	0.397336
$553$	−6.42893	−0.273386
$554$	10.9916	0.466988
$555$	−58.0366	−2.46352
$556$	13.7829	0.584524
$557$	45.8642	1.94333	0.971663	−	0.236369i	$-0.0759575\pi$
$557$	45.8642	1.94333	0.971663	+	0.236369i	$0.0759575\pi$
$558$	−11.7758	−0.498510
$559$	47.2359	1.99786
$560$	−2.73846	−0.115721
$561$	−26.5351	−1.12031
$562$	25.2803	1.06638
$563$	32.2558	1.35942	0.679709	−	0.733482i	$-0.262106\pi$
$563$	32.2558	1.35942	0.679709	+	0.733482i	$0.262106\pi$
$564$	4.23280	0.178233
$565$	36.5520	1.53775
$566$	−18.3291	−0.770430
$567$	9.39264	0.394454
$568$	1.33767	0.0561275
$569$	4.19850	0.176010	0.0880052	−	0.996120i	$-0.471951\pi$
$569$	4.19850	0.176010	0.0880052	+	0.996120i	$0.471951\pi$
$570$	−19.5689	−0.819651
$571$	35.9455	1.50427	0.752136	−	0.659008i	$-0.229024\pi$
$571$	35.9455	1.50427	0.752136	+	0.659008i	$0.229024\pi$
$572$	−6.69756	−0.280039
$573$	56.0546	2.34172
$574$	0.496232	0.0207123
$575$	25.5112	1.06389
$576$	1.23326	0.0513860
$577$	6.95611	0.289587	0.144793	−	0.989462i	$-0.453748\pi$
$577$	6.95611	0.289587	0.144793	+	0.989462i	$0.453748\pi$
$578$	−45.6251	−1.89775
$579$	−18.2173	−0.757083
$580$	0.287078	0.0119203
$581$	6.42687	0.266631
$582$	29.0745	1.20518
$583$	1.37580	0.0569799
$584$	−2.38431	−0.0986634
$585$	−16.5188	−0.682970
$586$	−5.83471	−0.241029
$587$	27.5945	1.13895	0.569474	−	0.822009i	$-0.307147\pi$
$587$	27.5945	1.13895	0.569474	+	0.822009i	$0.307147\pi$
$588$	12.9499	0.534046
$589$	−27.8643	−1.14813
$590$	21.6904	0.892979
$591$	−22.3685	−0.920116
$592$	8.65462	0.355703
$593$	15.1726	0.623065	0.311532	−	0.950236i	$-0.399158\pi$
$593$	15.1726	0.623065	0.311532	+	0.950236i	$0.399158\pi$
$594$	−5.92405	−0.243067
$595$	−21.6711	−0.888428
$596$	16.8630	0.690736
$597$	31.3032	1.28115
$598$	18.6465	0.762512
$599$	−38.6432	−1.57892	−0.789460	−	0.613802i	$-0.789639\pi$
$599$	−38.6432	−1.57892	−0.789460	+	0.613802i	$0.789639\pi$
$600$	11.5685	0.472284
$601$	32.7905	1.33755	0.668776	−	0.743464i	$-0.266819\pi$
$601$	32.7905	1.33755	0.668776	+	0.743464i	$0.266819\pi$
$602$	−9.65728	−0.393601
$603$	8.23980	0.335550
$604$	−1.88575	−0.0767302
$605$	−27.1953	−1.10565
$606$	34.2007	1.38931
$607$	−27.9750	−1.13547	−0.567734	−	0.823212i	$-0.692180\pi$
$607$	−27.9750	−1.13547	−0.567734	+	0.823212i	$0.692180\pi$
$608$	2.91818	0.118348
$609$	0.152269	0.00617025
$610$	−21.8433	−0.884408
$611$	8.45470	0.342040
$612$	9.75955	0.394506
$613$	−34.7439	−1.40329	−0.701647	−	0.712525i	$-0.747551\pi$
$613$	−34.7439	−1.40329	−0.701647	+	0.712525i	$0.747551\pi$
$614$	−5.46634	−0.220604
$615$	−3.96048	−0.159702
$616$	1.36930	0.0551708
$617$	9.30770	0.374714	0.187357	−	0.982292i	$-0.440008\pi$
$617$	9.30770	0.374714	0.187357	+	0.982292i	$0.440008\pi$
$618$	−7.36054	−0.296084
$619$	26.1598	1.05145	0.525726	−	0.850654i	$-0.323794\pi$
$619$	26.1598	1.05145	0.525726	+	0.850654i	$0.323794\pi$
$620$	31.1209	1.24984
$621$	16.4930	0.661840
$622$	23.1972	0.930123
$623$	8.21898	0.329286
$624$	8.45561	0.338495
$625$	−21.4991	−0.859963
$626$	19.4616	0.777842
$627$	9.78495	0.390773
$628$	14.6603	0.585010
$629$	68.4892	2.73084
$630$	3.37724	0.134553
$631$	−0.529677	−0.0210861	−0.0105431	−	0.999944i	$-0.503356\pi$
$631$	−0.529677	−0.0210861	−0.0105431	+	0.999944i	$0.503356\pi$
$632$	−7.65153	−0.304361
$633$	−48.8981	−1.94352
$634$	−13.2864	−0.527669
$635$	6.71930	0.266647
$636$	−1.73694	−0.0688741
$637$	25.8665	1.02487
$638$	−0.143547	−0.00568306
$639$	−1.64970	−0.0652612
$640$	−3.25924	−0.128833
$641$	33.2730	1.31420	0.657102	−	0.753801i	$-0.271782\pi$
$641$	33.2730	1.31420	0.657102	+	0.753801i	$0.271782\pi$
$642$	−22.1328	−0.873511
$643$	18.0473	0.711717	0.355858	−	0.934540i	$-0.384189\pi$
$643$	18.0473	0.711717	0.355858	+	0.934540i	$0.384189\pi$
$644$	−3.81224	−0.150223
$645$	77.0759	3.03486
$646$	23.0933	0.908594
$647$	−12.7782	−0.502362	−0.251181	−	0.967940i	$-0.580819\pi$
$647$	−12.7782	−0.502362	−0.251181	+	0.967940i	$0.580819\pi$
$648$	11.1789	0.439147
$649$	−10.8458	−0.425733
$650$	23.1073	0.906341
$651$	16.5068	0.646953
$652$	−6.62729	−0.259545
$653$	−1.41907	−0.0555325	−0.0277663	−	0.999614i	$-0.508839\pi$
$653$	−1.41907	−0.0555325	−0.0277663	+	0.999614i	$0.508839\pi$
$654$	3.26348	0.127612
$655$	−14.9859	−0.585546
$656$	0.590601	0.0230591
$657$	2.94048	0.114719
$658$	−1.72855	−0.0673857
$659$	−37.5020	−1.46087	−0.730435	−	0.682982i	$-0.760683\pi$
$659$	−37.5020	−1.46087	−0.730435	+	0.682982i	$0.760683\pi$
$660$	−10.9286	−0.425394
$661$	16.9640	0.659821	0.329911	−	0.944012i	$-0.392981\pi$
$661$	16.9640	0.659821	0.329911	+	0.944012i	$0.392981\pi$
$662$	−3.10725	−0.120767
$663$	66.9143	2.59874
$664$	7.64907	0.296842
$665$	7.99133	0.309890
$666$	−10.6734	−0.413587
$667$	0.399644	0.0154743
$668$	3.25610	0.125982
$669$	14.5053	0.560809
$670$	−21.7759	−0.841278
$671$	10.9222	0.421647
$672$	−1.72873	−0.0666873
$673$	44.2191	1.70452	0.852261	−	0.523117i	$-0.175231\pi$
$673$	44.2191	1.70452	0.852261	+	0.523117i	$0.175231\pi$
$674$	−17.4340	−0.671533
$675$	20.4386	0.786681
$676$	3.88943	0.149593
$677$	−20.3479	−0.782035	−0.391017	−	0.920383i	$-0.627877\pi$
$677$	−20.3479	−0.782035	−0.391017	+	0.920383i	$0.627877\pi$
$678$	23.0745	0.886171
$679$	−11.8731	−0.455649
$680$	−25.7923	−0.989091
$681$	−33.7112	−1.29182
$682$	−15.5612	−0.595871
$683$	−17.5231	−0.670504	−0.335252	−	0.942129i	$-0.608821\pi$
$683$	−17.5231	−0.670504	−0.335252	+	0.942129i	$0.608821\pi$
$684$	−3.59888	−0.137607
$685$	−24.9254	−0.952351
$686$	−11.1698	−0.426467
$687$	−21.6696	−0.826748
$688$	−11.4938	−0.438198
$689$	−3.46940	−0.132174
$690$	30.4259	1.15829
$691$	−37.8147	−1.43854	−0.719270	−	0.694730i	$-0.755524\pi$
$691$	−37.8147	−1.43854	−0.719270	+	0.694730i	$0.755524\pi$
$692$	−20.5804	−0.782348
$693$	−1.68871	−0.0641488
$694$	−12.2012	−0.463153
$695$	44.9217	1.70398
$696$	0.181226	0.00686937
$697$	4.67378	0.177032
$698$	13.6843	0.517959
$699$	−13.1721	−0.498213
$700$	−4.72423	−0.178559
$701$	−31.1521	−1.17660	−0.588300	−	0.808643i	$-0.700203\pi$
$701$	−31.1521	−1.17660	−0.588300	+	0.808643i	$0.700203\pi$
$702$	14.9388	0.563830
$703$	−25.2557	−0.952539
$704$	1.62971	0.0614218
$705$	13.7957	0.519577
$706$	−0.839045	−0.0315779
$707$	−13.9665	−0.525265
$708$	13.6927	0.514602
$709$	−31.7640	−1.19292	−0.596462	−	0.802642i	$-0.703427\pi$
$709$	−31.7640	−1.19292	−0.596462	+	0.802642i	$0.703427\pi$
$710$	4.35979	0.163620
$711$	9.43634	0.353891
$712$	9.78200	0.366596
$713$	43.3236	1.62248
$714$	−13.6805	−0.511980
$715$	−21.8290	−0.816357
$716$	9.36782	0.350092
$717$	31.2096	1.16554
$718$	28.5821	1.06668
$719$	37.6530	1.40422	0.702110	−	0.712069i	$-0.252242\pi$
$719$	37.6530	1.40422	0.702110	+	0.712069i	$0.252242\pi$
$720$	4.01950	0.149798
$721$	3.00582	0.111942
$722$	10.4842	0.390182
$723$	−3.36476	−0.125137
$724$	−11.2261	−0.417216
$725$	0.495250	0.0183931
$726$	−17.1678	−0.637157
$727$	−6.27700	−0.232801	−0.116400	−	0.993202i	$-0.537136\pi$
$727$	−6.27700	−0.232801	−0.116400	+	0.993202i	$0.537136\pi$
$728$	−3.45301	−0.127977
$729$	8.65072	0.320397
$730$	−7.77104	−0.287619
$731$	−90.9575	−3.36419
$732$	−13.7892	−0.509663
$733$	17.9780	0.664033	0.332017	−	0.943274i	$-0.392271\pi$
$733$	17.9780	0.664033	0.332017	+	0.943274i	$0.392271\pi$
$734$	23.7761	0.877593
$735$	42.2069	1.55683
$736$	−4.53722	−0.167244
$737$	10.8885	0.401085
$738$	−0.728367	−0.0268115
$739$	−0.747465	−0.0274960	−0.0137480	−	0.999905i	$-0.504376\pi$
$739$	−0.747465	−0.0274960	−0.0137480	+	0.999905i	$0.504376\pi$
$740$	28.2075	1.03693
$741$	−24.6750	−0.906459
$742$	0.709312	0.0260396
$743$	33.4991	1.22896	0.614481	−	0.788932i	$-0.289366\pi$
$743$	33.4991	1.22896	0.614481	+	0.788932i	$0.289366\pi$
$744$	19.6459	0.720255
$745$	54.9606	2.01360
$746$	13.1697	0.482177
$747$	−9.43332	−0.345147
$748$	12.8968	0.471555
$749$	9.03834	0.330254
$750$	4.17539	0.152464
$751$	10.0368	0.366248	0.183124	−	0.983090i	$-0.441379\pi$
$751$	10.0368	0.366248	0.183124	+	0.983090i	$0.441379\pi$
$752$	−2.05727	−0.0750208
$753$	11.7956	0.429854
$754$	0.361985	0.0131827
$755$	−6.14612	−0.223680
$756$	−3.05422	−0.111081
$757$	16.9085	0.614551	0.307275	−	0.951621i	$-0.400583\pi$
$757$	16.9085	0.614551	0.307275	+	0.951621i	$0.400583\pi$
$758$	−25.8227	−0.937922
$759$	−15.2137	−0.552224
$760$	9.51105	0.345002
$761$	−34.5877	−1.25380	−0.626902	−	0.779098i	$-0.715677\pi$
$761$	−34.5877	−1.25380	−0.626902	+	0.779098i	$0.715677\pi$
$762$	4.24175	0.153662
$763$	−1.33270	−0.0482472
$764$	−27.2442	−0.985660
$765$	31.8087	1.15005
$766$	−7.32996	−0.264842
$767$	27.3501	0.987553
$768$	−2.05749	−0.0742433
$769$	−12.8439	−0.463164	−0.231582	−	0.972815i	$-0.574390\pi$
$769$	−12.8439	−0.463164	−0.231582	+	0.972815i	$0.574390\pi$
$770$	4.46289	0.160831
$771$	−28.7895	−1.03683
$772$	8.85412	0.318667
$773$	31.2124	1.12263	0.561316	−	0.827601i	$-0.310295\pi$
$773$	31.2124	1.12263	0.561316	+	0.827601i	$0.310295\pi$
$774$	14.1749	0.509506
$775$	53.6879	1.92853
$776$	−14.1311	−0.507275
$777$	14.9615	0.536742
$778$	−28.4100	−1.01855
$779$	−1.72348	−0.0617501
$780$	27.5589	0.986766
$781$	−2.18001	−0.0780069
$782$	−35.9057	−1.28399
$783$	0.320179	0.0114423
$784$	−6.29404	−0.224787
$785$	47.7815	1.70539
$786$	−9.46025	−0.337436
$787$	12.5654	0.447907	0.223954	−	0.974600i	$-0.428104\pi$
$787$	12.5654	0.447907	0.223954	+	0.974600i	$0.428104\pi$
$788$	10.8717	0.387289
$789$	−33.9162	−1.20745
$790$	−24.9382	−0.887260
$791$	−9.42292	−0.335040
$792$	−2.00986	−0.0714171
$793$	−27.5428	−0.978074
$794$	−23.6987	−0.841034
$795$	−5.66110	−0.200778
$796$	−15.2143	−0.539255
$797$	49.6701	1.75940	0.879702	−	0.475525i	$-0.157742\pi$
$797$	49.6701	1.75940	0.879702	+	0.475525i	$0.157742\pi$
$798$	5.04476	0.178582
$799$	−16.2804	−0.575959
$800$	−5.62265	−0.198791
$801$	−12.0638	−0.426253
$802$	−16.2519	−0.573876
$803$	3.88572	0.137124
$804$	−13.7467	−0.484809
$805$	−12.4250	−0.437924
$806$	39.2412	1.38221
$807$	−3.82020	−0.134477
$808$	−16.6226	−0.584779
$809$	−18.9142	−0.664989	−0.332494	−	0.943105i	$-0.607890\pi$
$809$	−18.9142	−0.664989	−0.332494	+	0.943105i	$0.607890\pi$
$810$	36.4346	1.28018
$811$	17.8409	0.626479	0.313240	−	0.949674i	$-0.398586\pi$
$811$	17.8409	0.626479	0.313240	+	0.949674i	$0.398586\pi$
$812$	−0.0740072	−0.00259714
$813$	46.1682	1.61919
$814$	−14.1045	−0.494362
$815$	−21.5999	−0.756613
$816$	−16.2821	−0.569989
$817$	33.5410	1.17345
$818$	14.5480	0.508660
$819$	4.25847	0.148803
$820$	1.92491	0.0672208
$821$	−45.7778	−1.59766	−0.798828	−	0.601559i	$-0.794546\pi$
$821$	−45.7778	−1.59766	−0.798828	+	0.601559i	$0.794546\pi$
$822$	−15.7349	−0.548817
$823$	36.2860	1.26485	0.632425	−	0.774621i	$-0.282060\pi$
$823$	36.2860	1.26485	0.632425	+	0.774621i	$0.282060\pi$
$824$	3.57744	0.124626
$825$	−18.8533	−0.656388
$826$	−5.59166	−0.194559
$827$	38.3069	1.33206	0.666030	−	0.745925i	$-0.267992\pi$
$827$	38.3069	1.33206	0.666030	+	0.745925i	$0.267992\pi$
$828$	5.59558	0.194460
$829$	−20.6153	−0.716000	−0.358000	−	0.933722i	$-0.616541\pi$
$829$	−20.6153	−0.716000	−0.358000	+	0.933722i	$0.616541\pi$
$830$	24.9302	0.865339
$831$	22.6151	0.784509
$832$	−4.10968	−0.142477
$833$	−49.8085	−1.72576
$834$	28.3581	0.981960
$835$	10.6124	0.367257
$836$	−4.75577	−0.164482
$837$	34.7092	1.19973
$838$	−32.8454	−1.13463
$839$	28.0663	0.968957	0.484478	−	0.874803i	$-0.339009\pi$
$839$	28.0663	0.968957	0.484478	+	0.874803i	$0.339009\pi$
$840$	−5.63436	−0.194404
$841$	−28.9922	−0.999732
$842$	−32.1834	−1.10911
$843$	52.0139	1.79145
$844$	23.7659	0.818056
$845$	12.6766	0.436088
$846$	2.53715	0.0872290
$847$	7.01080	0.240894
$848$	0.844203	0.0289900
$849$	−37.7120	−1.29427
$850$	−44.4954	−1.52618
$851$	39.2679	1.34609
$852$	2.75225	0.0942904
$853$	29.4892	1.00969	0.504846	−	0.863209i	$-0.331549\pi$
$853$	29.4892	1.00969	0.504846	+	0.863209i	$0.331549\pi$
$854$	5.63107	0.192691
$855$	−11.7296	−0.401145
$856$	10.7572	0.367673
$857$	−21.8781	−0.747342	−0.373671	−	0.927561i	$-0.621901\pi$
$857$	−21.8781	−0.747342	−0.373671	+	0.927561i	$0.621901\pi$
$858$	−13.7802	−0.470447
$859$	28.4044	0.969144	0.484572	−	0.874751i	$-0.338975\pi$
$859$	28.4044	0.969144	0.484572	+	0.874751i	$0.338975\pi$
$860$	−37.4611	−1.27741
$861$	1.02099	0.0347953
$862$	−24.0853	−0.820349
$863$	−40.5623	−1.38076	−0.690379	−	0.723448i	$-0.742556\pi$
$863$	−40.5623	−1.38076	−0.690379	+	0.723448i	$0.742556\pi$
$864$	−3.63504	−0.123667
$865$	−67.0763	−2.28066
$866$	−24.8265	−0.843638
$867$	−93.8731	−3.18810
$868$	−8.02279	−0.272311
$869$	12.4697	0.423007
$870$	0.590660	0.0200253
$871$	−27.4580	−0.930377
$872$	−1.58615	−0.0537137
$873$	17.4273	0.589825
$874$	13.2404	0.447864
$875$	−1.70510	−0.0576429
$876$	−4.90569	−0.165748
$877$	10.5295	0.355554	0.177777	−	0.984071i	$-0.443109\pi$
$877$	10.5295	0.355554	0.177777	+	0.984071i	$0.443109\pi$
$878$	13.9738	0.471593
$879$	−12.0048	−0.404913
$880$	5.31160	0.179054
$881$	41.9890	1.41465	0.707323	−	0.706891i	$-0.249903\pi$
$881$	41.9890	1.41465	0.707323	+	0.706891i	$0.249903\pi$
$882$	7.76221	0.261367
$883$	−11.6100	−0.390706	−0.195353	−	0.980733i	$-0.562585\pi$
$883$	−11.6100	−0.390706	−0.195353	+	0.980733i	$0.562585\pi$
$884$	−32.5223	−1.09384
$885$	44.6277	1.50014
$886$	−30.3130	−1.01838
$887$	−0.882503	−0.0296316	−0.0148158	−	0.999890i	$-0.504716\pi$
$887$	−0.882503	−0.0296316	−0.0148158	+	0.999890i	$0.504716\pi$
$888$	17.8068	0.597557
$889$	−1.73220	−0.0580961
$890$	31.8819	1.06868
$891$	−18.2182	−0.610334
$892$	−7.05002	−0.236052
$893$	6.00347	0.200899
$894$	34.6955	1.16039
$895$	30.5320	1.02057
$896$	0.840215	0.0280696
$897$	38.3650	1.28097
$898$	−18.9210	−0.631401
$899$	0.841044	0.0280504
$900$	6.93420	0.231140
$901$	6.68068	0.222566
$902$	−0.962506	−0.0320479
$903$	−19.8698	−0.661224
$904$	−11.2149	−0.373002
$905$	−36.5887	−1.21625
$906$	−3.87992	−0.128902
$907$	23.6704	0.785962	0.392981	−	0.919547i	$-0.371444\pi$
$907$	23.6704	0.785962	0.392981	+	0.919547i	$0.371444\pi$
$908$	16.3846	0.543743
$909$	20.5000	0.679941
$910$	−11.2542	−0.373073
$911$	7.27793	0.241128	0.120564	−	0.992706i	$-0.461530\pi$
$911$	7.27793	0.241128	0.120564	+	0.992706i	$0.461530\pi$
$912$	6.00413	0.198816
$913$	−12.4657	−0.412556
$914$	4.11466	0.136101
$915$	−44.9423	−1.48575
$916$	10.5321	0.347990
$917$	3.86327	0.127577
$918$	−28.7663	−0.949429
$919$	29.6435	0.977849	0.488924	−	0.872326i	$-0.337389\pi$
$919$	29.6435	0.977849	0.488924	+	0.872326i	$0.337389\pi$
$920$	−14.7879	−0.487542
$921$	−11.2469	−0.370599
$922$	−19.7002	−0.648790
$923$	5.49740	0.180949
$924$	2.81733	0.0926832
$925$	48.6619	1.59999
$926$	26.2834	0.863727
$927$	−4.41192	−0.144906
$928$	−0.0880813	−0.00289141
$929$	57.1017	1.87345	0.936723	−	0.350071i	$-0.113842\pi$
$929$	57.1017	1.87345	0.936723	+	0.350071i	$0.113842\pi$
$930$	64.0309	2.09965
$931$	18.3671	0.601959
$932$	6.40201	0.209705
$933$	47.7280	1.56254
$934$	19.5870	0.640908
$935$	42.0339	1.37466
$936$	5.06831	0.165663
$937$	56.0855	1.83223	0.916117	−	0.400912i	$-0.131307\pi$
$937$	56.0855	1.83223	0.916117	+	0.400912i	$0.131307\pi$
$938$	5.61372	0.183295
$939$	40.0420	1.30672
$940$	−6.70513	−0.218697
$941$	6.61023	0.215487	0.107744	−	0.994179i	$-0.465637\pi$
$941$	6.61023	0.215487	0.107744	+	0.994179i	$0.465637\pi$
$942$	30.1634	0.982777
$943$	2.67969	0.0872626
$944$	−6.65504	−0.216603
$945$	−9.95443	−0.323818
$946$	18.7315	0.609015
$947$	−15.8705	−0.515723	−0.257861	−	0.966182i	$-0.583018\pi$
$947$	−15.8705	−0.515723	−0.257861	+	0.966182i	$0.583018\pi$
$948$	−15.7429	−0.511307
$949$	−9.79874	−0.318081
$950$	16.4079	0.532343
$951$	−27.3366	−0.886449
$952$	6.64912	0.215499
$953$	−2.68068	−0.0868357	−0.0434178	−	0.999057i	$-0.513825\pi$
$953$	−2.68068	−0.0868357	−0.0434178	+	0.999057i	$0.513825\pi$
$954$	−1.04112	−0.0337076
$955$	−88.7954	−2.87335
$956$	−15.1688	−0.490593
$957$	−0.295345	−0.00954716
$958$	−4.04884	−0.130812
$959$	6.42564	0.207495
$960$	−6.70585	−0.216430
$961$	60.1739	1.94109
$962$	35.5677	1.14675
$963$	−13.2664	−0.427505
$964$	1.63537	0.0526717
$965$	28.8577	0.928962
$966$	−7.84364	−0.252365
$967$	−15.7123	−0.505275	−0.252637	−	0.967561i	$-0.581298\pi$
$967$	−15.7123	−0.505275	−0.252637	+	0.967561i	$0.581298\pi$
$968$	8.34406	0.268188
$969$	47.5142	1.52638
$970$	−46.0565	−1.47879
$971$	43.9625	1.41082	0.705411	−	0.708798i	$-0.250762\pi$
$971$	43.9625	1.41082	0.705411	+	0.708798i	$0.250762\pi$
$972$	12.0952	0.387955
$973$	−11.5806	−0.371256
$974$	−25.3280	−0.811562
$975$	47.5429	1.52259
$976$	6.70195	0.214524
$977$	−7.43338	−0.237815	−0.118907	−	0.992905i	$-0.537939\pi$
$977$	−7.43338	−0.237815	−0.118907	+	0.992905i	$0.537939\pi$
$978$	−13.6356	−0.436018
$979$	−15.9418	−0.509501
$980$	−20.5138	−0.655289
$981$	1.95614	0.0624547
$982$	21.3526	0.681388
$983$	22.5206	0.718296	0.359148	−	0.933281i	$-0.383067\pi$
$983$	22.5206	0.718296	0.359148	+	0.933281i	$0.383067\pi$
$984$	1.21516	0.0387377
$985$	35.4336	1.12901
$986$	−0.697040	−0.0221983
$987$	−3.55646	−0.113204
$988$	11.9928	0.381541
$989$	−52.1500	−1.65827
$990$	−6.55060	−0.208192
$991$	6.26173	0.198910	0.0994551	−	0.995042i	$-0.468290\pi$
$991$	6.26173	0.198910	0.0994551	+	0.995042i	$0.468290\pi$
$992$	−9.54850	−0.303165
$993$	−6.39314	−0.202880
$994$	−1.12393	−0.0356489
$995$	−49.5870	−1.57201
$996$	15.7379	0.498674
$997$	−9.35305	−0.296214	−0.148107	−	0.988971i	$-0.547318\pi$
$997$	−9.35305	−0.296214	−0.148107	+	0.988971i	$0.547318\pi$
$998$	25.6223	0.811061
$999$	31.4599	0.995348

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.16	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.16	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.05749 q^{3} +1.00000 q^{4} +3.25924 q^{5} +2.05749 q^{6} -0.840215 q^{7} -1.00000 q^{8} +1.23326 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.05749 q^{3} +1.00000 q^{4} +3.25924 q^{5} +2.05749 q^{6} -0.840215 q^{7} -1.00000 q^{8} +1.23326 q^{9} -3.25924 q^{10} +1.62971 q^{11} -2.05749 q^{12} -4.10968 q^{13} +0.840215 q^{14} -6.70585 q^{15} +1.00000 q^{16} +7.91360 q^{17} -1.23326 q^{18} -2.91818 q^{19} +3.25924 q^{20} +1.72873 q^{21} -1.62971 q^{22} +4.53722 q^{23} +2.05749 q^{24} +5.62265 q^{25} +4.10968 q^{26} +3.63504 q^{27} -0.840215 q^{28} +0.0880813 q^{29} +6.70585 q^{30} +9.54850 q^{31} -1.00000 q^{32} -3.35310 q^{33} -7.91360 q^{34} -2.73846 q^{35} +1.23326 q^{36} +8.65462 q^{37} +2.91818 q^{38} +8.45561 q^{39} -3.25924 q^{40} +0.590601 q^{41} -1.72873 q^{42} -11.4938 q^{43} +1.62971 q^{44} +4.01950 q^{45} -4.53722 q^{46} -2.05727 q^{47} -2.05749 q^{48} -6.29404 q^{49} -5.62265 q^{50} -16.2821 q^{51} -4.10968 q^{52} +0.844203 q^{53} -3.63504 q^{54} +5.31160 q^{55} +0.840215 q^{56} +6.00413 q^{57} -0.0880813 q^{58} -6.65504 q^{59} -6.70585 q^{60} +6.70195 q^{61} -9.54850 q^{62} -1.03621 q^{63} +1.00000 q^{64} -13.3944 q^{65} +3.35310 q^{66} +6.68130 q^{67} +7.91360 q^{68} -9.33528 q^{69} +2.73846 q^{70} -1.33767 q^{71} -1.23326 q^{72} +2.38431 q^{73} -8.65462 q^{74} -11.5685 q^{75} -2.91818 q^{76} -1.36930 q^{77} -8.45561 q^{78} +7.65153 q^{79} +3.25924 q^{80} -11.1789 q^{81} -0.590601 q^{82} -7.64907 q^{83} +1.72873 q^{84} +25.7923 q^{85} +11.4938 q^{86} -0.181226 q^{87} -1.62971 q^{88} -9.78200 q^{89} -4.01950 q^{90} +3.45301 q^{91} +4.53722 q^{92} -19.6459 q^{93} +2.05727 q^{94} -9.51105 q^{95} +2.05749 q^{96} +14.1311 q^{97} +6.29404 q^{98} +2.00986 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 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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