LMFDB - Embedded newform 8047.2.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$168$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59593 q^{2} -2.11181 q^{3} +4.73887 q^{4} -0.212258 q^{5} +5.48213 q^{6} +0.982044 q^{7} -7.10991 q^{8} +1.45976 q^{9} +O(q^{10})$	\(q-2.59593 q^{2} -2.11181 q^{3} +4.73887 q^{4} -0.212258 q^{5} +5.48213 q^{6} +0.982044 q^{7} -7.10991 q^{8} +1.45976 q^{9} +0.551008 q^{10} +5.95551 q^{11} -10.0076 q^{12} +1.00000 q^{13} -2.54932 q^{14} +0.448250 q^{15} +8.97913 q^{16} +3.91829 q^{17} -3.78943 q^{18} +0.758524 q^{19} -1.00586 q^{20} -2.07389 q^{21} -15.4601 q^{22} -0.567071 q^{23} +15.0148 q^{24} -4.95495 q^{25} -2.59593 q^{26} +3.25271 q^{27} +4.65378 q^{28} -7.16674 q^{29} -1.16363 q^{30} +1.08405 q^{31} -9.08938 q^{32} -12.5769 q^{33} -10.1716 q^{34} -0.208447 q^{35} +6.91759 q^{36} -1.14642 q^{37} -1.96908 q^{38} -2.11181 q^{39} +1.50914 q^{40} -5.10519 q^{41} +5.38369 q^{42} +11.4748 q^{43} +28.2223 q^{44} -0.309845 q^{45} +1.47208 q^{46} -1.47661 q^{47} -18.9622 q^{48} -6.03559 q^{49} +12.8627 q^{50} -8.27470 q^{51} +4.73887 q^{52} +11.0659 q^{53} -8.44381 q^{54} -1.26411 q^{55} -6.98225 q^{56} -1.60186 q^{57} +18.6044 q^{58} +6.24324 q^{59} +2.12420 q^{60} +6.94738 q^{61} -2.81411 q^{62} +1.43355 q^{63} +5.63716 q^{64} -0.212258 q^{65} +32.6488 q^{66} +7.94727 q^{67} +18.5683 q^{68} +1.19755 q^{69} +0.541115 q^{70} +5.45408 q^{71} -10.3787 q^{72} -5.30789 q^{73} +2.97604 q^{74} +10.4639 q^{75} +3.59455 q^{76} +5.84857 q^{77} +5.48213 q^{78} -13.4571 q^{79} -1.90589 q^{80} -11.2484 q^{81} +13.2527 q^{82} +10.1226 q^{83} -9.82791 q^{84} -0.831690 q^{85} -29.7879 q^{86} +15.1348 q^{87} -42.3431 q^{88} +0.0604180 q^{89} +0.804338 q^{90} +0.982044 q^{91} -2.68728 q^{92} -2.28930 q^{93} +3.83319 q^{94} -0.161003 q^{95} +19.1951 q^{96} +5.01786 q^{97} +15.6680 q^{98} +8.69358 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) $ 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * 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 $

$2$	−2.59593	−1.83560	−0.917801	−	0.397041i	$-0.870037\pi$
$2$	−2.59593	−1.83560	−0.917801	+	0.397041i	$0.870037\pi$
$3$	−2.11181	−1.21926	−0.609628	−	0.792688i	$-0.708681\pi$
$3$	−2.11181	−1.21926	−0.609628	+	0.792688i	$0.708681\pi$
$4$	4.73887	2.36943
$5$	−0.212258	−0.0949248	−0.0474624	−	0.998873i	$-0.515113\pi$
$5$	−0.212258	−0.0949248	−0.0474624	+	0.998873i	$0.515113\pi$
$6$	5.48213	2.23807
$7$	0.982044	0.371178	0.185589	−	0.982627i	$-0.440581\pi$
$7$	0.982044	0.371178	0.185589	+	0.982627i	$0.440581\pi$
$8$	−7.10991	−2.51373
$9$	1.45976	0.486585
$10$	0.551008	0.174244
$11$	5.95551	1.79565	0.897826	−	0.440350i	$-0.145146\pi$
$11$	5.95551	1.79565	0.897826	+	0.440350i	$0.145146\pi$
$12$	−10.0076	−2.88895
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.54932	−0.681335
$15$	0.448250	0.115738
$16$	8.97913	2.24478
$17$	3.91829	0.950325	0.475162	−	0.879898i	$-0.342389\pi$
$17$	3.91829	0.950325	0.475162	+	0.879898i	$0.342389\pi$
$18$	−3.78943	−0.893177
$19$	0.758524	0.174017	0.0870087	−	0.996208i	$-0.472269\pi$
$19$	0.758524	0.174017	0.0870087	+	0.996208i	$0.472269\pi$
$20$	−1.00586	−0.224918
$21$	−2.07389	−0.452561
$22$	−15.4601	−3.29610
$23$	−0.567071	−0.118243	−0.0591213	−	0.998251i	$-0.518830\pi$
$23$	−0.567071	−0.118243	−0.0591213	+	0.998251i	$0.518830\pi$
$24$	15.0148	3.06489
$25$	−4.95495	−0.990989
$26$	−2.59593	−0.509104
$27$	3.25271	0.625984
$28$	4.65378	0.879481
$29$	−7.16674	−1.33083	−0.665415	−	0.746474i	$-0.731745\pi$
$29$	−7.16674	−1.33083	−0.665415	+	0.746474i	$0.731745\pi$
$30$	−1.16363	−0.212448
$31$	1.08405	0.194700	0.0973502	−	0.995250i	$-0.468963\pi$
$31$	1.08405	0.194700	0.0973502	+	0.995250i	$0.468963\pi$
$32$	−9.08938	−1.60679
$33$	−12.5769	−2.18936
$34$	−10.1716	−1.74442
$35$	−0.208447	−0.0352340
$36$	6.91759	1.15293
$37$	−1.14642	−0.188471	−0.0942354	−	0.995550i	$-0.530041\pi$
$37$	−1.14642	−0.188471	−0.0942354	+	0.995550i	$0.530041\pi$
$38$	−1.96908	−0.319427
$39$	−2.11181	−0.338161
$40$	1.50914	0.238616
$41$	−5.10519	−0.797297	−0.398648	−	0.917104i	$-0.630521\pi$
$41$	−5.10519	−0.797297	−0.398648	+	0.917104i	$0.630521\pi$
$42$	5.38369	0.830722
$43$	11.4748	1.74989	0.874947	−	0.484219i	$-0.160896\pi$
$43$	11.4748	1.74989	0.874947	+	0.484219i	$0.160896\pi$
$44$	28.2223	4.25468
$45$	−0.309845	−0.0461890
$46$	1.47208	0.217046
$47$	−1.47661	−0.215386	−0.107693	−	0.994184i	$-0.534346\pi$
$47$	−1.47661	−0.215386	−0.107693	+	0.994184i	$0.534346\pi$
$48$	−18.9622	−2.73696
$49$	−6.03559	−0.862227
$50$	12.8627	1.81906
$51$	−8.27470	−1.15869
$52$	4.73887	0.657163
$53$	11.0659	1.52002	0.760011	−	0.649910i	$-0.225193\pi$
$53$	11.0659	1.52002	0.760011	+	0.649910i	$0.225193\pi$
$54$	−8.44381	−1.14906
$55$	−1.26411	−0.170452
$56$	−6.98225	−0.933043
$57$	−1.60186	−0.212172
$58$	18.6044	2.44287
$59$	6.24324	0.812800	0.406400	−	0.913695i	$-0.366784\pi$
$59$	6.24324	0.812800	0.406400	+	0.913695i	$0.366784\pi$
$60$	2.12420	0.274233
$61$	6.94738	0.889520	0.444760	−	0.895650i	$-0.353289\pi$
$61$	6.94738	0.889520	0.444760	+	0.895650i	$0.353289\pi$
$62$	−2.81411	−0.357392
$63$	1.43355	0.180610
$64$	5.63716	0.704646
$65$	−0.212258	−0.0263274
$66$	32.6488	4.01879
$67$	7.94727	0.970914	0.485457	−	0.874261i	$-0.338653\pi$
$67$	7.94727	0.970914	0.485457	+	0.874261i	$0.338653\pi$
$68$	18.5683	2.25173
$69$	1.19755	0.144168
$70$	0.541115	0.0646756
$71$	5.45408	0.647280	0.323640	−	0.946180i	$-0.395093\pi$
$71$	5.45408	0.647280	0.323640	+	0.946180i	$0.395093\pi$
$72$	−10.3787	−1.22315
$73$	−5.30789	−0.621242	−0.310621	−	0.950534i	$-0.600537\pi$
$73$	−5.30789	−0.621242	−0.310621	+	0.950534i	$0.600537\pi$
$74$	2.97604	0.345957
$75$	10.4639	1.20827
$76$	3.59455	0.412323
$77$	5.84857	0.666507
$78$	5.48213	0.620728
$79$	−13.4571	−1.51405	−0.757023	−	0.653389i	$-0.773347\pi$
$79$	−13.4571	−1.51405	−0.757023	+	0.653389i	$0.773347\pi$
$80$	−1.90589	−0.213085
$81$	−11.2484	−1.24982
$82$	13.2527	1.46352
$83$	10.1226	1.11110	0.555549	−	0.831484i	$-0.312508\pi$
$83$	10.1226	1.11110	0.555549	+	0.831484i	$0.312508\pi$
$84$	−9.82791	−1.07231
$85$	−0.831690	−0.0902094
$86$	−29.7879	−3.21211
$87$	15.1348	1.62262
$88$	−42.3431	−4.51379
$89$	0.0604180	0.00640429	0.00320215	−	0.999995i	$-0.498981\pi$
$89$	0.0604180	0.00640429	0.00320215	+	0.999995i	$0.498981\pi$
$90$	0.804338	0.0847846
$91$	0.982044	0.102946
$92$	−2.68728	−0.280168
$93$	−2.28930	−0.237390
$94$	3.83319	0.395363
$95$	−0.161003	−0.0165186
$96$	19.1951	1.95909
$97$	5.01786	0.509487	0.254743	−	0.967009i	$-0.418009\pi$
$97$	5.01786	0.509487	0.254743	+	0.967009i	$0.418009\pi$
$98$	15.6680	1.58271
$99$	8.69358	0.873738
$100$	−23.4808	−2.34808
$101$	−8.65114	−0.860821	−0.430410	−	0.902633i	$-0.641631\pi$
$101$	−8.65114	−0.860821	−0.430410	+	0.902633i	$0.641631\pi$
$102$	21.4806	2.12689
$103$	3.43622	0.338581	0.169291	−	0.985566i	$-0.445852\pi$
$103$	3.43622	0.338581	0.169291	+	0.985566i	$0.445852\pi$
$104$	−7.10991	−0.697184
$105$	0.440201	0.0429593
$106$	−28.7264	−2.79016
$107$	3.69234	0.356952	0.178476	−	0.983944i	$-0.442883\pi$
$107$	3.69234	0.356952	0.178476	+	0.983944i	$0.442883\pi$
$108$	15.4142	1.48323
$109$	−14.8541	−1.42277	−0.711384	−	0.702804i	$-0.751931\pi$
$109$	−14.8541	−1.42277	−0.711384	+	0.702804i	$0.751931\pi$
$110$	3.28153	0.312882
$111$	2.42103	0.229794
$112$	8.81790	0.833213
$113$	−8.18130	−0.769632	−0.384816	−	0.922993i	$-0.625735\pi$
$113$	−8.18130	−0.769632	−0.384816	+	0.922993i	$0.625735\pi$
$114$	4.15833	0.389463
$115$	0.120366	0.0112242
$116$	−33.9622	−3.15331
$117$	1.45976	0.134954
$118$	−16.2070	−1.49198
$119$	3.84793	0.352740
$120$	−3.18702	−0.290934
$121$	24.4680	2.22437
$122$	−18.0349	−1.63281
$123$	10.7812	0.972109
$124$	5.13715	0.461330
$125$	2.11302	0.188994
$126$	−3.72139	−0.331527
$127$	−3.49070	−0.309749	−0.154875	−	0.987934i	$-0.549497\pi$
$127$	−3.49070	−0.309749	−0.154875	+	0.987934i	$0.549497\pi$
$128$	3.54506	0.313342
$129$	−24.2327	−2.13357
$130$	0.551008	0.0483266
$131$	21.9797	1.92038	0.960188	−	0.279355i	$-0.0901206\pi$
$131$	21.9797	1.92038	0.960188	+	0.279355i	$0.0901206\pi$
$132$	−59.6003	−5.18754
$133$	0.744905	0.0645914
$134$	−20.6306	−1.78221
$135$	−0.690414	−0.0594214
$136$	−27.8587	−2.38886
$137$	14.3449	1.22557	0.612785	−	0.790249i	$-0.290049\pi$
$137$	14.3449	1.22557	0.612785	+	0.790249i	$0.290049\pi$
$138$	−3.10876	−0.264635
$139$	20.2106	1.71424	0.857122	−	0.515114i	$-0.172250\pi$
$139$	20.2106	1.71424	0.857122	+	0.515114i	$0.172250\pi$
$140$	−0.987803	−0.0834846
$141$	3.11833	0.262611
$142$	−14.1584	−1.18815
$143$	5.95551	0.498024
$144$	13.1073	1.09228
$145$	1.52120	0.126329
$146$	13.7789	1.14035
$147$	12.7460	1.05128
$148$	−5.43274	−0.446569
$149$	20.0144	1.63965	0.819823	−	0.572617i	$-0.194072\pi$
$149$	20.0144	1.63965	0.819823	+	0.572617i	$0.194072\pi$
$150$	−27.1636	−2.21790
$151$	−4.07723	−0.331800	−0.165900	−	0.986143i	$-0.553053\pi$
$151$	−4.07723	−0.331800	−0.165900	+	0.986143i	$0.553053\pi$
$152$	−5.39304	−0.437434
$153$	5.71975	0.462414
$154$	−15.1825	−1.22344
$155$	−0.230098	−0.0184819
$156$	−10.0076	−0.801249
$157$	−0.543734	−0.0433947	−0.0216973	−	0.999765i	$-0.506907\pi$
$157$	−0.543734	−0.0433947	−0.0216973	+	0.999765i	$0.506907\pi$
$158$	34.9338	2.77918
$159$	−23.3692	−1.85330
$160$	1.92930	0.152524
$161$	−0.556889	−0.0438890
$162$	29.2000	2.29417
$163$	11.6629	0.913508	0.456754	−	0.889593i	$-0.349012\pi$
$163$	11.6629	0.913508	0.456754	+	0.889593i	$0.349012\pi$
$164$	−24.1928	−1.88914
$165$	2.66956	0.207825
$166$	−26.2776	−2.03953
$167$	6.07425	0.470039	0.235020	−	0.971991i	$-0.424485\pi$
$167$	6.07425	0.470039	0.235020	+	0.971991i	$0.424485\pi$
$168$	14.7452	1.13762
$169$	1.00000	0.0769231
$170$	2.15901	0.165589
$171$	1.10726	0.0846743
$172$	54.3777	4.14626
$173$	−18.3402	−1.39438	−0.697191	−	0.716886i	$-0.745567\pi$
$173$	−18.3402	−1.39438	−0.697191	+	0.716886i	$0.745567\pi$
$174$	−39.2890	−2.97849
$175$	−4.86598	−0.367833
$176$	53.4752	4.03085
$177$	−13.1846	−0.991012
$178$	−0.156841	−0.0117557
$179$	4.73828	0.354155	0.177078	−	0.984197i	$-0.443336\pi$
$179$	4.73828	0.354155	0.177078	+	0.984197i	$0.443336\pi$
$180$	−1.46832	−0.109442
$181$	2.71067	0.201482	0.100741	−	0.994913i	$-0.467879\pi$
$181$	2.71067	0.201482	0.100741	+	0.994913i	$0.467879\pi$
$182$	−2.54932	−0.188968
$183$	−14.6716	−1.08455
$184$	4.03183	0.297230
$185$	0.243338	0.0178905
$186$	5.94288	0.435753
$187$	23.3354	1.70645
$188$	−6.99748	−0.510344
$189$	3.19430	0.232351
$190$	0.417953	0.0303215
$191$	3.66012	0.264837	0.132418	−	0.991194i	$-0.457726\pi$
$191$	3.66012	0.264837	0.132418	+	0.991194i	$0.457726\pi$
$192$	−11.9046	−0.859143
$193$	6.31340	0.454448	0.227224	−	0.973842i	$-0.427035\pi$
$193$	6.31340	0.454448	0.227224	+	0.973842i	$0.427035\pi$
$194$	−13.0260	−0.935215
$195$	0.448250	0.0320998
$196$	−28.6019	−2.04299
$197$	12.8232	0.913618	0.456809	−	0.889565i	$-0.348992\pi$
$197$	12.8232	0.913618	0.456809	+	0.889565i	$0.348992\pi$
$198$	−22.5680	−1.60383
$199$	−21.8014	−1.54546	−0.772731	−	0.634733i	$-0.781110\pi$
$199$	−21.8014	−1.54546	−0.772731	+	0.634733i	$0.781110\pi$
$200$	35.2292	2.49108
$201$	−16.7832	−1.18379
$202$	22.4578	1.58012
$203$	−7.03806	−0.493975
$204$	−39.2127	−2.74544
$205$	1.08362	0.0756832
$206$	−8.92021	−0.621500
$207$	−0.827786	−0.0575351
$208$	8.97913	0.622590
$209$	4.51740	0.312475
$210$	−1.14273	−0.0788561
$211$	−20.7232	−1.42664	−0.713322	−	0.700836i	$-0.752810\pi$
$211$	−20.7232	−1.42664	−0.713322	+	0.700836i	$0.752810\pi$
$212$	52.4400	3.60159
$213$	−11.5180	−0.789200
$214$	−9.58507	−0.655222
$215$	−2.43563	−0.166108
$216$	−23.1265	−1.57356
$217$	1.06458	0.0722685
$218$	38.5603	2.61164
$219$	11.2093	0.757452
$220$	−5.99043	−0.403875
$221$	3.91829	0.263573
$222$	−6.28483	−0.421810
$223$	2.05448	0.137578	0.0687890	−	0.997631i	$-0.478086\pi$
$223$	2.05448	0.137578	0.0687890	+	0.997631i	$0.478086\pi$
$224$	−8.92617	−0.596405
$225$	−7.23301	−0.482201
$226$	21.2381	1.41274
$227$	3.44064	0.228363	0.114182	−	0.993460i	$-0.463575\pi$
$227$	3.44064	0.228363	0.114182	+	0.993460i	$0.463575\pi$
$228$	−7.59101	−0.502727
$229$	1.41176	0.0932918	0.0466459	−	0.998911i	$-0.485147\pi$
$229$	1.41176	0.0932918	0.0466459	+	0.998911i	$0.485147\pi$
$230$	−0.312461	−0.0206031
$231$	−12.3511	−0.812642
$232$	50.9549	3.34535
$233$	25.4986	1.67047	0.835234	−	0.549894i	$-0.185332\pi$
$233$	25.4986	1.67047	0.835234	+	0.549894i	$0.185332\pi$
$234$	−3.78943	−0.247723
$235$	0.313424	0.0204455
$236$	29.5859	1.92588
$237$	28.4189	1.84601
$238$	−9.98898	−0.647489
$239$	−27.6219	−1.78671	−0.893357	−	0.449348i	$-0.851656\pi$
$239$	−27.6219	−1.78671	−0.893357	+	0.449348i	$0.851656\pi$
$240$	4.02489	0.259806
$241$	−8.85500	−0.570401	−0.285200	−	0.958468i	$-0.592060\pi$
$241$	−8.85500	−0.570401	−0.285200	+	0.958468i	$0.592060\pi$
$242$	−63.5174	−4.08305
$243$	13.9964	0.897867
$244$	32.9227	2.10766
$245$	1.28110	0.0818467
$246$	−27.9873	−1.78440
$247$	0.758524	0.0482638
$248$	−7.70748	−0.489425
$249$	−21.3770	−1.35471
$250$	−5.48526	−0.346918
$251$	28.6762	1.81003	0.905013	−	0.425384i	$-0.139861\pi$
$251$	28.6762	1.81003	0.905013	+	0.425384i	$0.139861\pi$
$252$	6.79338	0.427943
$253$	−3.37720	−0.212323
$254$	9.06162	0.568577
$255$	1.75637	0.109988
$256$	−20.4771	−1.27982
$257$	11.1245	0.693927	0.346963	−	0.937879i	$-0.387213\pi$
$257$	11.1245	0.693927	0.346963	+	0.937879i	$0.387213\pi$
$258$	62.9064	3.91638
$259$	−1.12584	−0.0699562
$260$	−1.00586	−0.0623810
$261$	−10.4617	−0.647562
$262$	−57.0578	−3.52505
$263$	−21.5525	−1.32899	−0.664493	−	0.747295i	$-0.731352\pi$
$263$	−21.5525	−1.32899	−0.664493	+	0.747295i	$0.731352\pi$
$264$	89.4208	5.50347
$265$	−2.34884	−0.144288
$266$	−1.93372	−0.118564
$267$	−0.127591	−0.00780847
$268$	37.6611	2.30052
$269$	−13.6804	−0.834108	−0.417054	−	0.908882i	$-0.636937\pi$
$269$	−13.6804	−0.834108	−0.417054	+	0.908882i	$0.636937\pi$
$270$	1.79227	0.109074
$271$	−15.9341	−0.967928	−0.483964	−	0.875088i	$-0.660803\pi$
$271$	−15.9341	−0.967928	−0.483964	+	0.875088i	$0.660803\pi$
$272$	35.1828	2.13327
$273$	−2.07389	−0.125518
$274$	−37.2385	−2.24966
$275$	−29.5092	−1.77947
$276$	5.67503	0.341596
$277$	−28.8414	−1.73291	−0.866455	−	0.499255i	$-0.833607\pi$
$277$	−28.8414	−1.73291	−0.866455	+	0.499255i	$0.833607\pi$
$278$	−52.4655	−3.14667
$279$	1.58244	0.0947384
$280$	1.48204	0.0885689
$281$	−12.1811	−0.726662	−0.363331	−	0.931660i	$-0.618361\pi$
$281$	−12.1811	−0.726662	−0.363331	+	0.931660i	$0.618361\pi$
$282$	−8.09498	−0.482049
$283$	−25.7489	−1.53061	−0.765306	−	0.643667i	$-0.777412\pi$
$283$	−25.7489	−1.53061	−0.765306	+	0.643667i	$0.777412\pi$
$284$	25.8461	1.53369
$285$	0.340009	0.0201404
$286$	−15.4601	−0.914174
$287$	−5.01352	−0.295939
$288$	−13.2683	−0.781840
$289$	−1.64701	−0.0968828
$290$	−3.94893	−0.231889
$291$	−10.5968	−0.621195
$292$	−25.1534	−1.47199
$293$	−11.2296	−0.656040	−0.328020	−	0.944671i	$-0.606381\pi$
$293$	−11.2296	−0.656040	−0.328020	+	0.944671i	$0.606381\pi$
$294$	−33.0879	−1.92972
$295$	−1.32518	−0.0771549
$296$	8.15097	0.473765
$297$	19.3715	1.12405
$298$	−51.9561	−3.00974
$299$	−0.567071	−0.0327946
$300$	49.5871	2.86291
$301$	11.2688	0.649522
$302$	10.5842	0.609053
$303$	18.2696	1.04956
$304$	6.81089	0.390631
$305$	−1.47464	−0.0844375
$306$	−14.8481	−0.848808
$307$	−32.0278	−1.82792	−0.913962	−	0.405800i	$-0.866993\pi$
$307$	−32.0278	−1.82792	−0.913962	+	0.405800i	$0.866993\pi$
$308$	27.7156	1.57924
$309$	−7.25667	−0.412817
$310$	0.597319	0.0339254
$311$	−18.2853	−1.03687	−0.518433	−	0.855118i	$-0.673484\pi$
$311$	−18.2853	−1.03687	−0.518433	+	0.855118i	$0.673484\pi$
$312$	15.0148	0.850046
$313$	11.3189	0.639785	0.319892	−	0.947454i	$-0.396353\pi$
$313$	11.3189	0.639785	0.319892	+	0.947454i	$0.396353\pi$
$314$	1.41150	0.0796554
$315$	−0.304282	−0.0171443
$316$	−63.7715	−3.58743
$317$	13.2211	0.742571	0.371286	−	0.928519i	$-0.378917\pi$
$317$	13.2211	0.742571	0.371286	+	0.928519i	$0.378917\pi$
$318$	60.6648	3.40191
$319$	−42.6815	−2.38971
$320$	−1.19654	−0.0668883
$321$	−7.79754	−0.435216
$322$	1.44565	0.0805628
$323$	2.97212	0.165373
$324$	−53.3046	−2.96137
$325$	−4.95495	−0.274851
$326$	−30.2761	−1.67684
$327$	31.3692	1.73472
$328$	36.2975	2.00419
$329$	−1.45010	−0.0799466
$330$	−6.92999	−0.381483
$331$	17.7265	0.974338	0.487169	−	0.873308i	$-0.338030\pi$
$331$	17.7265	0.974338	0.487169	+	0.873308i	$0.338030\pi$
$332$	47.9696	2.63267
$333$	−1.67350	−0.0917071
$334$	−15.7683	−0.862805
$335$	−1.68687	−0.0921638
$336$	−18.6218	−1.01590
$337$	26.7991	1.45984	0.729921	−	0.683531i	$-0.239557\pi$
$337$	26.7991	1.45984	0.729921	+	0.683531i	$0.239557\pi$
$338$	−2.59593	−0.141200
$339$	17.2774	0.938378
$340$	−3.94127	−0.213745
$341$	6.45604	0.349614
$342$	−2.87437	−0.155428
$343$	−12.8015	−0.691218
$344$	−81.5850	−4.39877
$345$	−0.254190	−0.0136851
$346$	47.6100	2.55953
$347$	18.8462	1.01172	0.505859	−	0.862616i	$-0.331176\pi$
$347$	18.8462	1.01172	0.505859	+	0.862616i	$0.331176\pi$
$348$	71.7219	3.84470
$349$	25.1581	1.34668	0.673341	−	0.739332i	$-0.264859\pi$
$349$	25.1581	1.34668	0.673341	+	0.739332i	$0.264859\pi$
$350$	12.6318	0.675196
$351$	3.25271	0.173617
$352$	−54.1318	−2.88524
$353$	6.14325	0.326972	0.163486	−	0.986546i	$-0.447726\pi$
$353$	6.14325	0.326972	0.163486	+	0.986546i	$0.447726\pi$
$354$	34.2262	1.81910
$355$	−1.15767	−0.0614429
$356$	0.286313	0.0151745
$357$	−8.12612	−0.430080
$358$	−12.3002	−0.650088
$359$	35.1218	1.85366	0.926829	−	0.375484i	$-0.122523\pi$
$359$	35.1218	1.85366	0.926829	+	0.375484i	$0.122523\pi$
$360$	2.20297	0.116107
$361$	−18.4246	−0.969718
$362$	−7.03671	−0.369841
$363$	−51.6719	−2.71207
$364$	4.65378	0.243924
$365$	1.12664	0.0589712
$366$	38.0864	1.99081
$367$	37.0472	1.93385	0.966923	−	0.255068i	$-0.0820977\pi$
$367$	37.0472	1.93385	0.966923	+	0.255068i	$0.0820977\pi$
$368$	−5.09181	−0.265429
$369$	−7.45233	−0.387953
$370$	−0.631688	−0.0328399
$371$	10.8672	0.564199
$372$	−10.8487	−0.562479
$373$	15.8011	0.818151	0.409076	−	0.912500i	$-0.365851\pi$
$373$	15.8011	0.818151	0.409076	+	0.912500i	$0.365851\pi$
$374$	−60.5771	−3.13237
$375$	−4.46230	−0.230432
$376$	10.4986	0.541424
$377$	−7.16674	−0.369106
$378$	−8.29220	−0.426505
$379$	−7.85370	−0.403418	−0.201709	−	0.979446i	$-0.564649\pi$
$379$	−7.85370	−0.403418	−0.201709	+	0.979446i	$0.564649\pi$
$380$	−0.762972	−0.0391397
$381$	7.37170	0.377664
$382$	−9.50142	−0.486135
$383$	−30.9635	−1.58216	−0.791081	−	0.611712i	$-0.790481\pi$
$383$	−30.9635	−1.58216	−0.791081	+	0.611712i	$0.790481\pi$
$384$	−7.48650	−0.382044
$385$	−1.24141	−0.0632680
$386$	−16.3892	−0.834186
$387$	16.7504	0.851473
$388$	23.7790	1.20719
$389$	18.9823	0.962439	0.481219	−	0.876600i	$-0.340194\pi$
$389$	18.9823	0.962439	0.481219	+	0.876600i	$0.340194\pi$
$390$	−1.16363	−0.0589225
$391$	−2.22195	−0.112369
$392$	42.9125	2.16741
$393$	−46.4170	−2.34143
$394$	−33.2883	−1.67704
$395$	2.85639	0.143720
$396$	41.1977	2.07026
$397$	−34.6265	−1.73785	−0.868926	−	0.494942i	$-0.835189\pi$
$397$	−34.6265	−1.73785	−0.868926	+	0.494942i	$0.835189\pi$
$398$	56.5951	2.83685
$399$	−1.57310	−0.0787535
$400$	−44.4911	−2.22455
$401$	−32.7088	−1.63340	−0.816701	−	0.577062i	$-0.804199\pi$
$401$	−32.7088	−1.63340	−0.816701	+	0.577062i	$0.804199\pi$
$402$	43.5679	2.17297
$403$	1.08405	0.0540002
$404$	−40.9966	−2.03966
$405$	2.38756	0.118639
$406$	18.2703	0.906741
$407$	−6.82753	−0.338428
$408$	58.8324	2.91264
$409$	−14.5354	−0.718730	−0.359365	−	0.933197i	$-0.617007\pi$
$409$	−14.5354	−0.718730	−0.359365	+	0.933197i	$0.617007\pi$
$410$	−2.81300	−0.138924
$411$	−30.2938	−1.49428
$412$	16.2838	0.802246
$413$	6.13114	0.301694
$414$	2.14888	0.105611
$415$	−2.14860	−0.105471
$416$	−9.08938	−0.445643
$417$	−42.6811	−2.09010
$418$	−11.7269	−0.573579
$419$	−22.1553	−1.08236	−0.541179	−	0.840907i	$-0.682022\pi$
$419$	−22.1553	−1.08236	−0.541179	+	0.840907i	$0.682022\pi$
$420$	2.08606	0.101789
$421$	−18.8034	−0.916420	−0.458210	−	0.888844i	$-0.651509\pi$
$421$	−18.8034	−0.916420	−0.458210	+	0.888844i	$0.651509\pi$
$422$	53.7960	2.61875
$423$	−2.15550	−0.104804
$424$	−78.6778	−3.82093
$425$	−19.4149	−0.941762
$426$	29.8999	1.44866
$427$	6.82263	0.330170
$428$	17.4975	0.845775
$429$	−12.5769	−0.607219
$430$	6.32272	0.304909
$431$	30.7966	1.48342	0.741710	−	0.670721i	$-0.234015\pi$
$431$	30.7966	1.48342	0.741710	+	0.670721i	$0.234015\pi$
$432$	29.2065	1.40520
$433$	−26.5124	−1.27411	−0.637053	−	0.770820i	$-0.719847\pi$
$433$	−26.5124	−1.27411	−0.637053	+	0.770820i	$0.719847\pi$
$434$	−2.76358	−0.132656
$435$	−3.21249	−0.154027
$436$	−70.3918	−3.37115
$437$	−0.430137	−0.0205763
$438$	−29.0985	−1.39038
$439$	18.9193	0.902971	0.451486	−	0.892278i	$-0.350894\pi$
$439$	18.9193	0.902971	0.451486	+	0.892278i	$0.350894\pi$
$440$	8.98768	0.428471
$441$	−8.81049	−0.419547
$442$	−10.1716	−0.483814
$443$	−15.5275	−0.737732	−0.368866	−	0.929483i	$-0.620254\pi$
$443$	−15.5275	−0.737732	−0.368866	+	0.929483i	$0.620254\pi$
$444$	11.4729	0.544482
$445$	−0.0128242	−0.000607926 0
$446$	−5.33329	−0.252538
$447$	−42.2667	−1.99915
$448$	5.53595	0.261549
$449$	29.0060	1.36888	0.684439	−	0.729070i	$-0.260047\pi$
$449$	29.0060	1.36888	0.684439	+	0.729070i	$0.260047\pi$
$450$	18.7764	0.885129
$451$	−30.4040	−1.43167
$452$	−38.7701	−1.82359
$453$	8.61034	0.404549
$454$	−8.93167	−0.419184
$455$	−0.208447	−0.00977215
$456$	11.3891	0.533344
$457$	38.3281	1.79291	0.896456	−	0.443134i	$-0.146133\pi$
$457$	38.3281	1.79291	0.896456	+	0.443134i	$0.146133\pi$
$458$	−3.66484	−0.171247
$459$	12.7451	0.594888
$460$	0.570397	0.0265949
$461$	1.79473	0.0835889	0.0417944	−	0.999126i	$-0.486693\pi$
$461$	1.79473	0.0835889	0.0417944	+	0.999126i	$0.486693\pi$
$462$	32.0626	1.49169
$463$	1.06332	0.0494166	0.0247083	−	0.999695i	$-0.492134\pi$
$463$	1.06332	0.0494166	0.0247083	+	0.999695i	$0.492134\pi$
$464$	−64.3510	−2.98742
$465$	0.485924	0.0225342
$466$	−66.1926	−3.06631
$467$	−23.5543	−1.08996	−0.544981	−	0.838449i	$-0.683463\pi$
$467$	−23.5543	−1.08996	−0.544981	+	0.838449i	$0.683463\pi$
$468$	6.91759	0.319766
$469$	7.80457	0.360382
$470$	−0.813627	−0.0375298
$471$	1.14826	0.0529092
$472$	−44.3889	−2.04316
$473$	68.3384	3.14220
$474$	−73.7737	−3.38854
$475$	−3.75845	−0.172449
$476$	18.2348	0.835793
$477$	16.1536	0.739621
$478$	71.7046	3.27969
$479$	−34.1377	−1.55979	−0.779895	−	0.625911i	$-0.784727\pi$
$479$	−34.1377	−1.55979	−0.779895	+	0.625911i	$0.784727\pi$
$480$	−4.07431	−0.185966
$481$	−1.14642	−0.0522724
$482$	22.9870	1.04703
$483$	1.17605	0.0535120
$484$	115.951	5.27049
$485$	−1.06508	−0.0483629
$486$	−36.3336	−1.64813
$487$	−16.4760	−0.746599	−0.373300	−	0.927711i	$-0.621774\pi$
$487$	−16.4760	−0.746599	−0.373300	+	0.927711i	$0.621774\pi$
$488$	−49.3952	−2.23602
$489$	−24.6298	−1.11380
$490$	−3.32566	−0.150238
$491$	25.6367	1.15697	0.578484	−	0.815694i	$-0.303645\pi$
$491$	25.6367	1.15697	0.578484	+	0.815694i	$0.303645\pi$
$492$	51.0907	2.30335
$493$	−28.0814	−1.26472
$494$	−1.96908	−0.0885930
$495$	−1.84529	−0.0829394
$496$	9.73379	0.437060
$497$	5.35614	0.240256
$498$	55.4933	2.48671
$499$	1.52661	0.0683405	0.0341702	−	0.999416i	$-0.489121\pi$
$499$	1.52661	0.0683405	0.0341702	+	0.999416i	$0.489121\pi$
$500$	10.0133	0.447809
$501$	−12.8277	−0.573098
$502$	−74.4415	−3.32249
$503$	−7.90478	−0.352457	−0.176228	−	0.984349i	$-0.556390\pi$
$503$	−7.90478	−0.352457	−0.176228	+	0.984349i	$0.556390\pi$
$504$	−10.1924	−0.454005
$505$	1.83628	0.0817133
$506$	8.76698	0.389740
$507$	−2.11181	−0.0937889
$508$	−16.5420	−0.733931
$509$	−0.349493	−0.0154910	−0.00774550	−	0.999970i	$-0.502465\pi$
$509$	−0.349493	−0.0154910	−0.00774550	+	0.999970i	$0.502465\pi$
$510$	−4.55943	−0.201895
$511$	−5.21258	−0.230591
$512$	46.0670	2.03589
$513$	2.46726	0.108932
$514$	−28.8784	−1.27377
$515$	−0.729367	−0.0321398
$516$	−114.835	−5.05535
$517$	−8.79398	−0.386759
$518$	2.92260	0.128412
$519$	38.7311	1.70011
$520$	1.50914	0.0661801
$521$	38.2101	1.67401	0.837007	−	0.547193i	$-0.184303\pi$
$521$	38.2101	1.67401	0.837007	+	0.547193i	$0.184303\pi$
$522$	27.1578	1.18867
$523$	30.1059	1.31644	0.658221	−	0.752825i	$-0.271309\pi$
$523$	30.1059	1.31644	0.658221	+	0.752825i	$0.271309\pi$
$524$	104.159	4.55020
$525$	10.2760	0.448483
$526$	55.9489	2.43949
$527$	4.24761	0.185029
$528$	−112.930	−4.91464
$529$	−22.6784	−0.986019
$530$	6.09742	0.264855
$531$	9.11360	0.395497
$532$	3.53000	0.153045
$533$	−5.10519	−0.221130
$534$	0.331219	0.0143332
$535$	−0.783731	−0.0338836
$536$	−56.5044	−2.44062
$537$	−10.0064	−0.431806
$538$	35.5134	1.53109
$539$	−35.9450	−1.54826
$540$	−3.27178	−0.140795
$541$	8.14455	0.350162	0.175081	−	0.984554i	$-0.443981\pi$
$541$	8.14455	0.350162	0.175081	+	0.984554i	$0.443981\pi$
$542$	41.3639	1.77673
$543$	−5.72443	−0.245659
$544$	−35.6148	−1.52697
$545$	3.15291	0.135056
$546$	5.38369	0.230401
$547$	25.4303	1.08732	0.543660	−	0.839306i	$-0.317038\pi$
$547$	25.4303	1.08732	0.543660	+	0.839306i	$0.317038\pi$
$548$	67.9787	2.90391
$549$	10.1415	0.432827
$550$	76.6039	3.26640
$551$	−5.43615	−0.231588
$552$	−8.51447	−0.362400
$553$	−13.2155	−0.561980
$554$	74.8703	3.18093
$555$	−0.513884	−0.0218132
$556$	95.7755	4.06179
$557$	−1.29288	−0.0547811	−0.0273906	−	0.999625i	$-0.508720\pi$
$557$	−1.29288	−0.0547811	−0.0273906	+	0.999625i	$0.508720\pi$
$558$	−4.10791	−0.173902
$559$	11.4748	0.485333
$560$	−1.87167	−0.0790926
$561$	−49.2800	−2.08060
$562$	31.6213	1.33386
$563$	19.6499	0.828146	0.414073	−	0.910244i	$-0.364106\pi$
$563$	19.6499	0.828146	0.414073	+	0.910244i	$0.364106\pi$
$564$	14.7774	0.622239
$565$	1.73655	0.0730571
$566$	66.8424	2.80959
$567$	−11.0464	−0.463906
$568$	−38.7780	−1.62709
$569$	−17.0337	−0.714091	−0.357046	−	0.934087i	$-0.616216\pi$
$569$	−17.0337	−0.714091	−0.357046	+	0.934087i	$0.616216\pi$
$570$	−0.882639	−0.0369697
$571$	−10.2598	−0.429361	−0.214680	−	0.976684i	$-0.568871\pi$
$571$	−10.2598	−0.429361	−0.214680	+	0.976684i	$0.568871\pi$
$572$	28.2223	1.18004
$573$	−7.72949	−0.322904
$574$	13.0148	0.543226
$575$	2.80981	0.117177
$576$	8.22888	0.342870
$577$	−39.8243	−1.65791	−0.828954	−	0.559316i	$-0.811064\pi$
$577$	−39.8243	−1.65791	−0.828954	+	0.559316i	$0.811064\pi$
$578$	4.27552	0.177838
$579$	−13.3327	−0.554089
$580$	7.20876	0.299328
$581$	9.94083	0.412415
$582$	27.5086	1.14027
$583$	65.9032	2.72943
$584$	37.7386	1.56164
$585$	−0.309845	−0.0128105
$586$	29.1513	1.20423
$587$	−4.34058	−0.179155	−0.0895774	−	0.995980i	$-0.528552\pi$
$587$	−4.34058	−0.179155	−0.0895774	+	0.995980i	$0.528552\pi$
$588$	60.4018	2.49093
$589$	0.822276	0.0338813
$590$	3.44008	0.141626
$591$	−27.0803	−1.11393
$592$	−10.2939	−0.423076
$593$	−17.3634	−0.713027	−0.356514	−	0.934290i	$-0.616035\pi$
$593$	−17.3634	−0.713027	−0.356514	+	0.934290i	$0.616035\pi$
$594$	−50.2872	−2.06331
$595$	−0.816756	−0.0334837
$596$	94.8457	3.88503
$597$	46.0406	1.88431
$598$	1.47208	0.0601978
$599$	−4.88829	−0.199730	−0.0998650	−	0.995001i	$-0.531841\pi$
$599$	−4.88829	−0.199730	−0.0998650	+	0.995001i	$0.531841\pi$
$600$	−74.3976	−3.03727
$601$	−7.64495	−0.311844	−0.155922	−	0.987769i	$-0.549835\pi$
$601$	−7.64495	−0.311844	−0.155922	+	0.987769i	$0.549835\pi$
$602$	−29.2530	−1.19226
$603$	11.6011	0.472432
$604$	−19.3214	−0.786178
$605$	−5.19355	−0.211148
$606$	−47.4267	−1.92658
$607$	−0.625947	−0.0254064	−0.0127032	−	0.999919i	$-0.504044\pi$
$607$	−0.625947	−0.0254064	−0.0127032	+	0.999919i	$0.504044\pi$
$608$	−6.89452	−0.279610
$609$	14.8631	0.602282
$610$	3.82806	0.154994
$611$	−1.47661	−0.0597374
$612$	27.1051	1.09566
$613$	2.43573	0.0983780	0.0491890	−	0.998789i	$-0.484336\pi$
$613$	2.43573	0.0983780	0.0491890	+	0.998789i	$0.484336\pi$
$614$	83.1420	3.35534
$615$	−2.28840	−0.0922772
$616$	−41.5828	−1.67542
$617$	−35.5416	−1.43085	−0.715426	−	0.698689i	$-0.753767\pi$
$617$	−35.5416	−1.43085	−0.715426	+	0.698689i	$0.753767\pi$
$618$	18.8378	0.757768
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−1.09040	−0.0437916
$621$	−1.84452	−0.0740180
$622$	47.4675	1.90327
$623$	0.0593331	0.00237713
$624$	−18.9622	−0.759097
$625$	24.3262	0.973049
$626$	−29.3832	−1.17439
$627$	−9.53990	−0.380987
$628$	−2.57668	−0.102821
$629$	−4.49202	−0.179108
$630$	0.789895	0.0314702
$631$	5.90919	0.235241	0.117621	−	0.993059i	$-0.462473\pi$
$631$	5.90919	0.235241	0.117621	+	0.993059i	$0.462473\pi$
$632$	95.6790	3.80591
$633$	43.7635	1.73944
$634$	−34.3211	−1.36307
$635$	0.740930	0.0294029
$636$	−110.743	−4.39126
$637$	−6.03559	−0.239139
$638$	110.798	4.38655
$639$	7.96162	0.314957
$640$	−0.752468	−0.0297439
$641$	27.3766	1.08131	0.540654	−	0.841245i	$-0.318177\pi$
$641$	27.3766	1.08131	0.540654	+	0.841245i	$0.318177\pi$
$642$	20.2419	0.798884
$643$	15.5152	0.611858	0.305929	−	0.952054i	$-0.401033\pi$
$643$	15.5152	0.611858	0.305929	+	0.952054i	$0.401033\pi$
$644$	−2.63902	−0.103992
$645$	5.14359	0.202529
$646$	−7.71542	−0.303559
$647$	30.9475	1.21667	0.608336	−	0.793679i	$-0.291837\pi$
$647$	30.9475	1.21667	0.608336	+	0.793679i	$0.291837\pi$
$648$	79.9750	3.14172
$649$	37.1816	1.45951
$650$	12.8627	0.504517
$651$	−2.24820	−0.0881138
$652$	55.2689	2.16450
$653$	29.3171	1.14726	0.573632	−	0.819113i	$-0.305534\pi$
$653$	29.3171	1.14726	0.573632	+	0.819113i	$0.305534\pi$
$654$	−81.4322	−3.18425
$655$	−4.66538	−0.182291
$656$	−45.8401	−1.78976
$657$	−7.74822	−0.302287
$658$	3.76436	0.146750
$659$	0.501103	0.0195202	0.00976011	−	0.999952i	$-0.496893\pi$
$659$	0.501103	0.0195202	0.00976011	+	0.999952i	$0.496893\pi$
$660$	12.6507	0.492427
$661$	41.4726	1.61310	0.806548	−	0.591169i	$-0.201333\pi$
$661$	41.4726	1.61310	0.806548	+	0.591169i	$0.201333\pi$
$662$	−46.0169	−1.78850
$663$	−8.27470	−0.321363
$664$	−71.9707	−2.79301
$665$	−0.158112	−0.00613133
$666$	4.34429	0.168338
$667$	4.06405	0.157361
$668$	28.7851	1.11373
$669$	−4.33867	−0.167743
$670$	4.37901	0.169176
$671$	41.3751	1.59727
$672$	18.8504	0.727171
$673$	23.3727	0.900951	0.450476	−	0.892789i	$-0.351254\pi$
$673$	23.3727	0.900951	0.450476	+	0.892789i	$0.351254\pi$
$674$	−69.5688	−2.67969
$675$	−16.1170	−0.620343
$676$	4.73887	0.182264
$677$	23.0344	0.885285	0.442642	−	0.896698i	$-0.354041\pi$
$677$	23.0344	0.885285	0.442642	+	0.896698i	$0.354041\pi$
$678$	−44.8509	−1.72249
$679$	4.92776	0.189110
$680$	5.91324	0.226762
$681$	−7.26599	−0.278433
$682$	−16.7595	−0.641753
$683$	23.2621	0.890101	0.445050	−	0.895506i	$-0.353186\pi$
$683$	23.2621	0.890101	0.445050	+	0.895506i	$0.353186\pi$
$684$	5.24716	0.200630
$685$	−3.04483	−0.116337
$686$	33.2319	1.26880
$687$	−2.98138	−0.113747
$688$	103.034	3.92813
$689$	11.0659	0.421578
$690$	0.659859	0.0251204
$691$	4.26416	0.162216	0.0811081	−	0.996705i	$-0.474154\pi$
$691$	4.26416	0.162216	0.0811081	+	0.996705i	$0.474154\pi$
$692$	−86.9119	−3.30389
$693$	8.53749	0.324312
$694$	−48.9235	−1.85711
$695$	−4.28988	−0.162724
$696$	−107.607	−4.07884
$697$	−20.0036	−0.757691
$698$	−65.3087	−2.47197
$699$	−53.8483	−2.03673
$700$	−23.0592	−0.871557
$701$	13.4000	0.506111	0.253055	−	0.967452i	$-0.418565\pi$
$701$	13.4000	0.506111	0.253055	+	0.967452i	$0.418565\pi$
$702$	−8.44381	−0.318691
$703$	−0.869589	−0.0327972
$704$	33.5722	1.26530
$705$	−0.661892	−0.0249283
$706$	−15.9475	−0.600191
$707$	−8.49581	−0.319518
$708$	−62.4798	−2.34814
$709$	−44.1210	−1.65700	−0.828500	−	0.559989i	$-0.810805\pi$
$709$	−44.1210	−1.65700	−0.828500	+	0.559989i	$0.810805\pi$
$710$	3.00524	0.112785
$711$	−19.6441	−0.736712
$712$	−0.429567	−0.0160987
$713$	−0.614732	−0.0230219
$714$	21.0949	0.789455
$715$	−1.26411	−0.0472749
$716$	22.4541	0.839147
$717$	58.3323	2.17846
$718$	−91.1738	−3.40258
$719$	20.8621	0.778026	0.389013	−	0.921232i	$-0.372816\pi$
$719$	20.8621	0.778026	0.389013	+	0.921232i	$0.372816\pi$
$720$	−2.78214	−0.103684
$721$	3.37453	0.125674
$722$	47.8291	1.78002
$723$	18.7001	0.695465
$724$	12.8455	0.477399
$725$	35.5108	1.31884
$726$	134.137	4.97829
$727$	−21.0850	−0.782000	−0.391000	−	0.920391i	$-0.627871\pi$
$727$	−21.0850	−0.782000	−0.391000	+	0.920391i	$0.627871\pi$
$728$	−6.98225	−0.258779
$729$	4.18745	0.155091
$730$	−2.92469	−0.108248
$731$	44.9617	1.66297
$732$	−69.5266	−2.56978
$733$	13.7852	0.509170	0.254585	−	0.967050i	$-0.418061\pi$
$733$	13.7852	0.509170	0.254585	+	0.967050i	$0.418061\pi$
$734$	−96.1720	−3.54977
$735$	−2.70545	−0.0997921
$736$	5.15433	0.189991
$737$	47.3300	1.74342
$738$	19.3457	0.712127
$739$	17.8872	0.657992	0.328996	−	0.944331i	$-0.393290\pi$
$739$	17.8872	0.657992	0.328996	+	0.944331i	$0.393290\pi$
$740$	1.15315	0.0423905
$741$	−1.60186	−0.0588459
$742$	−28.2106	−1.03564
$743$	−15.7558	−0.578023	−0.289012	−	0.957326i	$-0.593327\pi$
$743$	−15.7558	−0.578023	−0.289012	+	0.957326i	$0.593327\pi$
$744$	16.2768	0.596735
$745$	−4.24823	−0.155643
$746$	−41.0187	−1.50180
$747$	14.7765	0.540644
$748$	110.583	4.04333
$749$	3.62605	0.132493
$750$	11.5838	0.422982
$751$	−10.2173	−0.372834	−0.186417	−	0.982471i	$-0.559687\pi$
$751$	−10.2173	−0.372834	−0.186417	+	0.982471i	$0.559687\pi$
$752$	−13.2587	−0.483495
$753$	−60.5588	−2.20689
$754$	18.6044	0.677531
$755$	0.865426	0.0314961
$756$	15.1374	0.550541
$757$	−12.3155	−0.447616	−0.223808	−	0.974633i	$-0.571849\pi$
$757$	−12.3155	−0.447616	−0.223808	+	0.974633i	$0.571849\pi$
$758$	20.3877	0.740514
$759$	7.13201	0.258876
$760$	1.14472	0.0415233
$761$	31.9899	1.15963	0.579817	−	0.814747i	$-0.303124\pi$
$761$	31.9899	1.15963	0.579817	+	0.814747i	$0.303124\pi$
$762$	−19.1364	−0.693240
$763$	−14.5874	−0.528100
$764$	17.3448	0.627513
$765$	−1.21406	−0.0438946
$766$	80.3792	2.90422
$767$	6.24324	0.225430
$768$	43.2437	1.56042
$769$	−39.6147	−1.42854	−0.714272	−	0.699868i	$-0.753242\pi$
$769$	−39.6147	−1.42854	−0.714272	+	0.699868i	$0.753242\pi$
$770$	3.22261	0.116135
$771$	−23.4928	−0.846074
$772$	29.9183	1.07678
$773$	−50.8767	−1.82991	−0.914954	−	0.403559i	$-0.867773\pi$
$773$	−50.8767	−1.82991	−0.914954	+	0.403559i	$0.867773\pi$
$774$	−43.4830	−1.56296
$775$	−5.37139	−0.192946
$776$	−35.6766	−1.28071
$777$	2.37756	0.0852945
$778$	−49.2767	−1.76665
$779$	−3.87241	−0.138744
$780$	2.12420	0.0760585
$781$	32.4818	1.16229
$782$	5.76803	0.206264
$783$	−23.3113	−0.833078
$784$	−54.1943	−1.93551
$785$	0.115412	0.00411923
$786$	120.496	4.29793
$787$	−24.2460	−0.864277	−0.432138	−	0.901807i	$-0.642241\pi$
$787$	−24.2460	−0.864277	−0.432138	+	0.901807i	$0.642241\pi$
$788$	60.7676	2.16476
$789$	45.5149	1.62037
$790$	−7.41499	−0.263813
$791$	−8.03440	−0.285670
$792$	−61.8106	−2.19635
$793$	6.94738	0.246709
$794$	89.8880	3.19000
$795$	4.96030	0.175924
$796$	−103.314	−3.66187
$797$	−4.99338	−0.176875	−0.0884373	−	0.996082i	$-0.528187\pi$
$797$	−4.99338	−0.176875	−0.0884373	+	0.996082i	$0.528187\pi$
$798$	4.08366	0.144560
$799$	−5.78580	−0.204687
$800$	45.0374	1.59231
$801$	0.0881955	0.00311623
$802$	84.9099	2.99827
$803$	−31.6112	−1.11553
$804$	−79.5331	−2.80492
$805$	0.118204	0.00416616
$806$	−2.81411	−0.0991228
$807$	28.8904	1.01699
$808$	61.5089	2.16388
$809$	−2.27929	−0.0801356	−0.0400678	−	0.999197i	$-0.512757\pi$
$809$	−2.27929	−0.0801356	−0.0400678	+	0.999197i	$0.512757\pi$
$810$	−6.19795	−0.217774
$811$	6.13788	0.215530	0.107765	−	0.994176i	$-0.465631\pi$
$811$	6.13788	0.215530	0.107765	+	0.994176i	$0.465631\pi$
$812$	−33.3524	−1.17044
$813$	33.6499	1.18015
$814$	17.7238	0.621219
$815$	−2.47554	−0.0867145
$816$	−74.2995	−2.60100
$817$	8.70393	0.304512
$818$	37.7329	1.31930
$819$	1.43355	0.0500921
$820$	5.13513	0.179326
$821$	7.10551	0.247984	0.123992	−	0.992283i	$-0.460430\pi$
$821$	7.10551	0.247984	0.123992	+	0.992283i	$0.460430\pi$
$822$	78.6407	2.74291
$823$	26.4610	0.922372	0.461186	−	0.887304i	$-0.347424\pi$
$823$	26.4610	0.922372	0.461186	+	0.887304i	$0.347424\pi$
$824$	−24.4313	−0.851103
$825$	62.3179	2.16963
$826$	−15.9160	−0.553789
$827$	−18.5744	−0.645894	−0.322947	−	0.946417i	$-0.604673\pi$
$827$	−18.5744	−0.645894	−0.322947	+	0.946417i	$0.604673\pi$
$828$	−3.92277	−0.136326
$829$	−7.17700	−0.249268	−0.124634	−	0.992203i	$-0.539776\pi$
$829$	−7.17700	−0.249268	−0.124634	+	0.992203i	$0.539776\pi$
$830$	5.57763	0.193602
$831$	60.9076	2.11286
$832$	5.63716	0.195434
$833$	−23.6492	−0.819396
$834$	110.797	3.83659
$835$	−1.28931	−0.0446184
$836$	21.4073	0.740388
$837$	3.52609	0.121879
$838$	57.5138	1.98678
$839$	16.7380	0.577862	0.288931	−	0.957350i	$-0.406700\pi$
$839$	16.7380	0.577862	0.288931	+	0.957350i	$0.406700\pi$
$840$	−3.12979	−0.107988
$841$	22.3621	0.771108
$842$	48.8123	1.68218
$843$	25.7242	0.885987
$844$	−98.2045	−3.38034
$845$	−0.212258	−0.00730191
$846$	5.59552	0.192378
$847$	24.0287	0.825636
$848$	99.3624	3.41212
$849$	54.3768	1.86621
$850$	50.3998	1.72870
$851$	0.650103	0.0222853
$852$	−54.5822	−1.86996
$853$	−19.0758	−0.653143	−0.326572	−	0.945172i	$-0.605893\pi$
$853$	−19.0758	−0.653143	−0.326572	+	0.945172i	$0.605893\pi$
$854$	−17.7111	−0.606061
$855$	−0.235025	−0.00803769
$856$	−26.2522	−0.897283
$857$	35.0510	1.19732	0.598660	−	0.801003i	$-0.295700\pi$
$857$	35.0510	1.19732	0.598660	+	0.801003i	$0.295700\pi$
$858$	32.6488	1.11461
$859$	1.37456	0.0468995	0.0234498	−	0.999725i	$-0.492535\pi$
$859$	1.37456	0.0468995	0.0234498	+	0.999725i	$0.492535\pi$
$860$	−11.5421	−0.393583
$861$	10.5876	0.360825
$862$	−79.9459	−2.72297
$863$	−3.56863	−0.121478	−0.0607388	−	0.998154i	$-0.519346\pi$
$863$	−3.56863	−0.121478	−0.0607388	+	0.998154i	$0.519346\pi$
$864$	−29.5651	−1.00583
$865$	3.89287	0.132361
$866$	68.8245	2.33875
$867$	3.47817	0.118125
$868$	5.04491	0.171235
$869$	−80.1440	−2.71870
$870$	8.33941	0.282732
$871$	7.94727	0.269283
$872$	105.612	3.57646
$873$	7.32485	0.247909
$874$	1.11661	0.0377698
$875$	2.07508	0.0701505
$876$	53.1193	1.79473
$877$	−42.7850	−1.44475	−0.722374	−	0.691503i	$-0.756949\pi$
$877$	−42.7850	−1.44475	−0.722374	+	0.691503i	$0.756949\pi$
$878$	−49.1133	−1.65750
$879$	23.7148	0.799880
$880$	−11.3506	−0.382627
$881$	45.3084	1.52648	0.763240	−	0.646115i	$-0.223608\pi$
$881$	45.3084	1.52648	0.763240	+	0.646115i	$0.223608\pi$
$882$	22.8714	0.770121
$883$	−9.85097	−0.331512	−0.165756	−	0.986167i	$-0.553006\pi$
$883$	−9.85097	−0.331512	−0.165756	+	0.986167i	$0.553006\pi$
$884$	18.5683	0.624518
$885$	2.79853	0.0940716
$886$	40.3082	1.35418
$887$	50.4173	1.69285	0.846423	−	0.532511i	$-0.178752\pi$
$887$	50.4173	1.69285	0.846423	+	0.532511i	$0.178752\pi$
$888$	−17.2133	−0.577641
$889$	−3.42802	−0.114972
$890$	0.0332908	0.00111591
$891$	−66.9898	−2.24424
$892$	9.73590	0.325982
$893$	−1.12005	−0.0374810
$894$	109.722	3.66964
$895$	−1.00574	−0.0336181
$896$	3.48140	0.116306
$897$	1.19755	0.0399850
$898$	−75.2977	−2.51272
$899$	−7.76908	−0.259113
$900$	−34.2763	−1.14254
$901$	43.3595	1.44452
$902$	78.9267	2.62797
$903$	−23.7976	−0.791934
$904$	58.1683	1.93465
$905$	−0.575362	−0.0191257
$906$	−22.3519	−0.742591
$907$	−33.9009	−1.12566	−0.562831	−	0.826572i	$-0.690288\pi$
$907$	−33.9009	−1.12566	−0.562831	+	0.826572i	$0.690288\pi$
$908$	16.3047	0.541092
$909$	−12.6286	−0.418863
$910$	0.541115	0.0179378
$911$	−12.5356	−0.415324	−0.207662	−	0.978201i	$-0.566585\pi$
$911$	−12.5356	−0.415324	−0.207662	+	0.978201i	$0.566585\pi$
$912$	−14.3833	−0.476279
$913$	60.2851	1.99515
$914$	−99.4971	−3.29107
$915$	3.11416	0.102951
$916$	6.69015	0.221049
$917$	21.5850	0.712801
$918$	−33.0853	−1.09198
$919$	6.36020	0.209804	0.104902	−	0.994483i	$-0.466547\pi$
$919$	6.36020	0.209804	0.104902	+	0.994483i	$0.466547\pi$
$920$	−0.855789	−0.0282145
$921$	67.6367	2.22871
$922$	−4.65900	−0.153436
$923$	5.45408	0.179523
$924$	−58.5302	−1.92550
$925$	5.68046	0.186772
$926$	−2.76030	−0.0907092
$927$	5.01605	0.164749
$928$	65.1412	2.13836
$929$	10.7769	0.353578	0.176789	−	0.984249i	$-0.443429\pi$
$929$	10.7769	0.353578	0.176789	+	0.984249i	$0.443429\pi$
$930$	−1.26143	−0.0413638
$931$	−4.57814	−0.150043
$932$	120.834	3.95806
$933$	38.6152	1.26420
$934$	61.1453	2.00073
$935$	−4.95313	−0.161985
$936$	−10.3787	−0.339240
$937$	47.7347	1.55943	0.779713	−	0.626137i	$-0.215365\pi$
$937$	47.7347	1.55943	0.779713	+	0.626137i	$0.215365\pi$
$938$	−20.2601	−0.661517
$939$	−23.9035	−0.780061
$940$	1.48527	0.0484443
$941$	−30.1957	−0.984350	−0.492175	−	0.870496i	$-0.663798\pi$
$941$	−30.1957	−0.984350	−0.492175	+	0.870496i	$0.663798\pi$
$942$	−2.98082	−0.0971203
$943$	2.89501	0.0942744
$944$	56.0588	1.82456
$945$	−0.678018	−0.0220559
$946$	−177.402	−5.76783
$947$	44.3394	1.44084	0.720419	−	0.693539i	$-0.243950\pi$
$947$	44.3394	1.44084	0.720419	+	0.693539i	$0.243950\pi$
$948$	134.674	4.37399
$949$	−5.30789	−0.172301
$950$	9.75668	0.316548
$951$	−27.9205	−0.905385
$952$	−27.3585	−0.886694
$953$	22.2007	0.719151	0.359575	−	0.933116i	$-0.382922\pi$
$953$	22.2007	0.719151	0.359575	+	0.933116i	$0.382922\pi$
$954$	−41.9335	−1.35765
$955$	−0.776891	−0.0251396
$956$	−130.897	−4.23350
$957$	90.1355	2.91367
$958$	88.6190	2.86315
$959$	14.0874	0.454905
$960$	2.52686	0.0815540
$961$	−29.8248	−0.962092
$962$	2.97604	0.0959512
$963$	5.38992	0.173688
$964$	−41.9627	−1.35153
$965$	−1.34007	−0.0431384
$966$	−3.05294	−0.0982266
$967$	−1.21693	−0.0391338	−0.0195669	−	0.999809i	$-0.506229\pi$
$967$	−1.21693	−0.0391338	−0.0195669	+	0.999809i	$0.506229\pi$
$968$	−173.966	−5.59147
$969$	−6.27656	−0.201632
$970$	2.76488	0.0887751
$971$	32.2158	1.03385	0.516927	−	0.856029i	$-0.327076\pi$
$971$	32.2158	1.03385	0.516927	+	0.856029i	$0.327076\pi$
$972$	66.3269	2.12744
$973$	19.8477	0.636289
$974$	42.7706	1.37046
$975$	10.4639	0.335114
$976$	62.3814	1.99678
$977$	−10.6458	−0.340591	−0.170295	−	0.985393i	$-0.554472\pi$
$977$	−10.6458	−0.340591	−0.170295	+	0.985393i	$0.554472\pi$
$978$	63.9374	2.04449
$979$	0.359820	0.0114999
$980$	6.07098	0.193930
$981$	−21.6834	−0.692298
$982$	−66.5511	−2.12373
$983$	43.2404	1.37915	0.689577	−	0.724213i	$-0.257797\pi$
$983$	43.2404	1.37915	0.689577	+	0.724213i	$0.257797\pi$
$984$	−76.6535	−2.44362
$985$	−2.72184	−0.0867250
$986$	72.8973	2.32152
$987$	3.06234	0.0974754
$988$	3.59455	0.114358
$989$	−6.50704	−0.206912
$990$	4.79024	0.152244
$991$	5.17909	0.164519	0.0822597	−	0.996611i	$-0.473786\pi$
$991$	5.17909	0.164519	0.0822597	+	0.996611i	$0.473786\pi$
$992$	−9.85331	−0.312843
$993$	−37.4351	−1.18797
$994$	−13.9042	−0.441014
$995$	4.62754	0.146703
$996$	−101.303	−3.20990
$997$	37.7823	1.19658	0.598288	−	0.801281i	$-0.295848\pi$
$997$	37.7823	1.19658	0.598288	+	0.801281i	$0.295848\pi$
$998$	−3.96298	−0.125446
$999$	−3.72898	−0.117980

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.e.1.5	✓	168

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.e.1.5	✓	168	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.59593 q^{2} -2.11181 q^{3} +4.73887 q^{4} -0.212258 q^{5} +5.48213 q^{6} +0.982044 q^{7} -7.10991 q^{8} +1.45976 q^{9} +O(q^{10})\)	\(q-2.59593 q^{2} -2.11181 q^{3} +4.73887 q^{4} -0.212258 q^{5} +5.48213 q^{6} +0.982044 q^{7} -7.10991 q^{8} +1.45976 q^{9} +0.551008 q^{10} +5.95551 q^{11} -10.0076 q^{12} +1.00000 q^{13} -2.54932 q^{14} +0.448250 q^{15} +8.97913 q^{16} +3.91829 q^{17} -3.78943 q^{18} +0.758524 q^{19} -1.00586 q^{20} -2.07389 q^{21} -15.4601 q^{22} -0.567071 q^{23} +15.0148 q^{24} -4.95495 q^{25} -2.59593 q^{26} +3.25271 q^{27} +4.65378 q^{28} -7.16674 q^{29} -1.16363 q^{30} +1.08405 q^{31} -9.08938 q^{32} -12.5769 q^{33} -10.1716 q^{34} -0.208447 q^{35} +6.91759 q^{36} -1.14642 q^{37} -1.96908 q^{38} -2.11181 q^{39} +1.50914 q^{40} -5.10519 q^{41} +5.38369 q^{42} +11.4748 q^{43} +28.2223 q^{44} -0.309845 q^{45} +1.47208 q^{46} -1.47661 q^{47} -18.9622 q^{48} -6.03559 q^{49} +12.8627 q^{50} -8.27470 q^{51} +4.73887 q^{52} +11.0659 q^{53} -8.44381 q^{54} -1.26411 q^{55} -6.98225 q^{56} -1.60186 q^{57} +18.6044 q^{58} +6.24324 q^{59} +2.12420 q^{60} +6.94738 q^{61} -2.81411 q^{62} +1.43355 q^{63} +5.63716 q^{64} -0.212258 q^{65} +32.6488 q^{66} +7.94727 q^{67} +18.5683 q^{68} +1.19755 q^{69} +0.541115 q^{70} +5.45408 q^{71} -10.3787 q^{72} -5.30789 q^{73} +2.97604 q^{74} +10.4639 q^{75} +3.59455 q^{76} +5.84857 q^{77} +5.48213 q^{78} -13.4571 q^{79} -1.90589 q^{80} -11.2484 q^{81} +13.2527 q^{82} +10.1226 q^{83} -9.82791 q^{84} -0.831690 q^{85} -29.7879 q^{86} +15.1348 q^{87} -42.3431 q^{88} +0.0604180 q^{89} +0.804338 q^{90} +0.982044 q^{91} -2.68728 q^{92} -2.28930 q^{93} +3.83319 q^{94} -0.161003 q^{95} +19.1951 q^{96} +5.01786 q^{97} +15.6680 q^{98} +8.69358 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

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