LMFDB - Embedded newform 1849.4.a.j.1.45

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.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:	$109.094531601$
Analytic rank:	$1$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.45
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.80125 q^{2} +5.52870 q^{3} +15.0520 q^{4} -18.8098 q^{5} +26.5447 q^{6} -0.563586 q^{7} +33.8585 q^{8} +3.56656 q^{9} +O(q^{10})$	\(q+4.80125 q^{2} +5.52870 q^{3} +15.0520 q^{4} -18.8098 q^{5} +26.5447 q^{6} -0.563586 q^{7} +33.8585 q^{8} +3.56656 q^{9} -90.3106 q^{10} -21.9318 q^{11} +83.2181 q^{12} +90.1443 q^{13} -2.70592 q^{14} -103.994 q^{15} +42.1470 q^{16} -68.5734 q^{17} +17.1239 q^{18} -29.2074 q^{19} -283.125 q^{20} -3.11590 q^{21} -105.300 q^{22} -171.303 q^{23} +187.194 q^{24} +228.809 q^{25} +432.806 q^{26} -129.557 q^{27} -8.48310 q^{28} -9.36853 q^{29} -499.300 q^{30} +244.427 q^{31} -68.5095 q^{32} -121.255 q^{33} -329.238 q^{34} +10.6009 q^{35} +53.6839 q^{36} -213.593 q^{37} -140.232 q^{38} +498.381 q^{39} -636.872 q^{40} -191.608 q^{41} -14.9602 q^{42} -330.118 q^{44} -67.0863 q^{45} -822.469 q^{46} -341.454 q^{47} +233.018 q^{48} -342.682 q^{49} +1098.57 q^{50} -379.122 q^{51} +1356.85 q^{52} -488.826 q^{53} -622.034 q^{54} +412.533 q^{55} -19.0822 q^{56} -161.479 q^{57} -44.9807 q^{58} +319.938 q^{59} -1565.32 q^{60} +51.1903 q^{61} +1173.56 q^{62} -2.01006 q^{63} -666.107 q^{64} -1695.60 q^{65} -582.173 q^{66} -733.622 q^{67} -1032.17 q^{68} -947.083 q^{69} +50.8978 q^{70} +417.297 q^{71} +120.758 q^{72} -35.6291 q^{73} -1025.51 q^{74} +1265.01 q^{75} -439.630 q^{76} +12.3605 q^{77} +2392.85 q^{78} +32.5557 q^{79} -792.777 q^{80} -812.577 q^{81} -919.957 q^{82} -73.6881 q^{83} -46.9006 q^{84} +1289.85 q^{85} -51.7958 q^{87} -742.578 q^{88} +1128.32 q^{89} -322.098 q^{90} -50.8041 q^{91} -2578.45 q^{92} +1351.37 q^{93} -1639.41 q^{94} +549.385 q^{95} -378.768 q^{96} -309.477 q^{97} -1645.30 q^{98} -78.2211 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) $ 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} + 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} + 386 q^{18} - 12 q^{19} - 108 q^{20} - 408 q^{21} - 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} - 1493 q^{26} - 10 q^{27} + 242 q^{28} - 208 q^{29} + 48 q^{30} - 932 q^{31} + 1124 q^{32} - 254 q^{33} - 765 q^{34} - 1452 q^{35} + 747 q^{36} + 90 q^{37} - 1213 q^{38} + 1610 q^{39} - 1693 q^{40} - 1354 q^{41} + 16 q^{42} - 2704 q^{44} - 4508 q^{45} - 233 q^{46} - 3484 q^{47} + 376 q^{48} + 1324 q^{49} + 408 q^{50} - 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} + 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} - 1172 q^{61} + 1546 q^{62} + 3686 q^{63} + 606 q^{64} - 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} - 136 q^{69} + 1310 q^{70} - 162 q^{71} + 5814 q^{72} - 746 q^{73} - 4332 q^{74} - 236 q^{75} - 1338 q^{76} - 2024 q^{77} - 2782 q^{78} - 2656 q^{79} + 5713 q^{80} - 86 q^{81} + 4168 q^{82} - 3514 q^{83} - 4269 q^{84} + 7558 q^{85} - 10278 q^{87} - 11692 q^{88} - 2640 q^{89} - 8286 q^{90} + 5946 q^{91} - 4271 q^{92} + 2 q^{93} - 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} + 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 + 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 + 386 * q^18 - 12 * q^19 - 108 * q^20 - 408 * q^21 - 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 - 1493 * q^26 - 10 * q^27 + 242 * q^28 - 208 * q^29 + 48 * q^30 - 932 * q^31 + 1124 * q^32 - 254 * q^33 - 765 * q^34 - 1452 * q^35 + 747 * q^36 + 90 * q^37 - 1213 * q^38 + 1610 * q^39 - 1693 * q^40 - 1354 * q^41 + 16 * q^42 - 2704 * q^44 - 4508 * q^45 - 233 * q^46 - 3484 * q^47 + 376 * q^48 + 1324 * q^49 + 408 * q^50 - 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 + 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 - 1172 * q^61 + 1546 * q^62 + 3686 * q^63 + 606 * q^64 - 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 - 136 * q^69 + 1310 * q^70 - 162 * q^71 + 5814 * q^72 - 746 * q^73 - 4332 * q^74 - 236 * q^75 - 1338 * q^76 - 2024 * q^77 - 2782 * q^78 - 2656 * q^79 + 5713 * q^80 - 86 * q^81 + 4168 * q^82 - 3514 * q^83 - 4269 * q^84 + 7558 * q^85 - 10278 * q^87 - 11692 * q^88 - 2640 * q^89 - 8286 * q^90 + 5946 * q^91 - 4271 * q^92 + 2 * q^93 - 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 + 2826 * q^98 - 8174 * 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 $

$2$	4.80125	1.69750	0.848749	−	0.528795i	$-0.177356\pi$
$2$	4.80125	1.69750	0.848749	+	0.528795i	$0.177356\pi$
$3$	5.52870	1.06400	0.532000	−	0.846745i	$-0.321441\pi$
$3$	5.52870	1.06400	0.532000	+	0.846745i	$0.321441\pi$
$4$	15.0520	1.88150
$5$	−18.8098	−1.68240	−0.841200	−	0.540724i	$-0.818150\pi$
$5$	−18.8098	−1.68240	−0.841200	+	0.540724i	$0.818150\pi$
$6$	26.5447	1.80614
$7$	−0.563586	−0.0304308	−0.0152154	−	0.999884i	$-0.504843\pi$
$7$	−0.563586	−0.0304308	−0.0152154	+	0.999884i	$0.504843\pi$
$8$	33.8585	1.49635
$9$	3.56656	0.132095
$10$	−90.3106	−2.85587
$11$	−21.9318	−0.601154	−0.300577	−	0.953758i	$-0.597179\pi$
$11$	−21.9318	−0.601154	−0.300577	+	0.953758i	$0.597179\pi$
$12$	83.2181	2.00192
$13$	90.1443	1.92320	0.961598	−	0.274463i	$-0.0885001\pi$
$13$	90.1443	1.92320	0.961598	+	0.274463i	$0.0885001\pi$
$14$	−2.70592	−0.0516562
$15$	−103.994	−1.79007
$16$	42.1470	0.658547
$17$	−68.5734	−0.978323	−0.489161	−	0.872193i	$-0.662697\pi$
$17$	−68.5734	−0.978323	−0.489161	+	0.872193i	$0.662697\pi$
$18$	17.1239	0.224231
$19$	−29.2074	−0.352665	−0.176332	−	0.984331i	$-0.556423\pi$
$19$	−29.2074	−0.352665	−0.176332	+	0.984331i	$0.556423\pi$
$20$	−283.125	−3.16544
$21$	−3.11590	−0.0323783
$22$	−105.300	−1.02046
$23$	−171.303	−1.55301	−0.776503	−	0.630114i	$-0.783008\pi$
$23$	−171.303	−1.55301	−0.776503	+	0.630114i	$0.783008\pi$
$24$	187.194	1.59211
$25$	228.809	1.83047
$26$	432.806	3.26462
$27$	−129.557	−0.923451
$28$	−8.48310	−0.0572556
$29$	−9.36853	−0.0599894	−0.0299947	−	0.999550i	$-0.509549\pi$
$29$	−9.36853	−0.0599894	−0.0299947	+	0.999550i	$0.509549\pi$
$30$	−499.300	−3.03865
$31$	244.427	1.41614	0.708071	−	0.706141i	$-0.249565\pi$
$31$	244.427	1.41614	0.708071	+	0.706141i	$0.249565\pi$
$32$	−68.5095	−0.378465
$33$	−121.255	−0.639627
$34$	−329.238	−1.66070
$35$	10.6009	0.0511967
$36$	53.6839	0.248537
$37$	−213.593	−0.949041	−0.474520	−	0.880245i	$-0.657378\pi$
$37$	−213.593	−0.949041	−0.474520	+	0.880245i	$0.657378\pi$
$38$	−140.232	−0.598648
$39$	498.381	2.04628
$40$	−636.872	−2.51746
$41$	−191.608	−0.729856	−0.364928	−	0.931036i	$-0.618906\pi$
$41$	−191.608	−0.729856	−0.364928	+	0.931036i	$0.618906\pi$
$42$	−14.9602	−0.0549622
$43$	0	0
$43$	0	0
$44$	−330.118	−1.13107
$45$	−67.0863	−0.222236
$46$	−822.469	−2.63623
$47$	−341.454	−1.05971	−0.529853	−	0.848089i	$-0.677753\pi$
$47$	−341.454	−1.05971	−0.529853	+	0.848089i	$0.677753\pi$
$48$	233.018	0.700694
$49$	−342.682	−0.999074
$50$	1098.57	3.10722
$51$	−379.122	−1.04093
$52$	1356.85	3.61850
$53$	−488.826	−1.26690	−0.633448	−	0.773785i	$-0.718361\pi$
$53$	−488.826	−1.26690	−0.633448	+	0.773785i	$0.718361\pi$
$54$	−622.034	−1.56756
$55$	412.533	1.01138
$56$	−19.0822	−0.0455350
$57$	−161.479	−0.375235
$58$	−44.9807	−0.101832
$59$	319.938	0.705972	0.352986	−	0.935629i	$-0.385166\pi$
$59$	319.938	0.705972	0.352986	+	0.935629i	$0.385166\pi$
$60$	−1565.32	−3.36802
$61$	51.1903	0.107447	0.0537234	−	0.998556i	$-0.482891\pi$
$61$	51.1903	0.107447	0.0537234	+	0.998556i	$0.482891\pi$
$62$	1173.56	2.40390
$63$	−2.01006	−0.00401975
$64$	−666.107	−1.30099
$65$	−1695.60	−3.23558
$66$	−582.173	−1.08577
$67$	−733.622	−1.33771	−0.668853	−	0.743395i	$-0.733214\pi$
$67$	−733.622	−1.33771	−0.668853	+	0.743395i	$0.733214\pi$
$68$	−1032.17	−1.84072
$69$	−947.083	−1.65240
$70$	50.8978	0.0869064
$71$	417.297	0.697522	0.348761	−	0.937212i	$-0.386602\pi$
$71$	417.297	0.697522	0.348761	+	0.937212i	$0.386602\pi$
$72$	120.758	0.197660
$73$	−35.6291	−0.0571243	−0.0285621	−	0.999592i	$-0.509093\pi$
$73$	−35.6291	−0.0571243	−0.0285621	+	0.999592i	$0.509093\pi$
$74$	−1025.51	−1.61100
$75$	1265.01	1.94762
$76$	−439.630	−0.663539
$77$	12.3605	0.0182936
$78$	2392.85	3.47356
$79$	32.5557	0.0463645	0.0231823	−	0.999731i	$-0.492620\pi$
$79$	32.5557	0.0463645	0.0231823	+	0.999731i	$0.492620\pi$
$80$	−792.777	−1.10794
$81$	−812.577	−1.11465
$82$	−919.957	−1.23893
$83$	−73.6881	−0.0974497	−0.0487248	−	0.998812i	$-0.515516\pi$
$83$	−73.6881	−0.0974497	−0.0487248	+	0.998812i	$0.515516\pi$
$84$	−46.9006	−0.0609199
$85$	1289.85	1.64593
$86$	0	0
$87$	−51.7958	−0.0638287
$88$	−742.578	−0.899536
$89$	1128.32	1.34384	0.671918	−	0.740626i	$-0.265471\pi$
$89$	1128.32	1.34384	0.671918	+	0.740626i	$0.265471\pi$
$90$	−322.098	−0.377246
$91$	−50.8041	−0.0585243
$92$	−2578.45	−2.92198
$93$	1351.37	1.50678
$94$	−1639.41	−1.79885
$95$	549.385	0.593323
$96$	−378.768	−0.402686
$97$	−309.477	−0.323944	−0.161972	−	0.986795i	$-0.551785\pi$
$97$	−309.477	−0.323944	−0.161972	+	0.986795i	$0.551785\pi$
$98$	−1645.30	−1.69593
$99$	−78.2211	−0.0794093
$100$	3444.03	3.44403
$101$	331.449	0.326538	0.163269	−	0.986582i	$-0.447796\pi$
$101$	331.449	0.326538	0.163269	+	0.986582i	$0.447796\pi$
$102$	−1820.26	−1.76699
$103$	132.978	0.127211	0.0636053	−	0.997975i	$-0.479740\pi$
$103$	132.978	0.127211	0.0636053	+	0.997975i	$0.479740\pi$
$104$	3052.15	2.87777
$105$	58.6094	0.0544733
$106$	−2346.98	−2.15055
$107$	−273.667	−0.247256	−0.123628	−	0.992329i	$-0.539453\pi$
$107$	−273.667	−0.247256	−0.123628	+	0.992329i	$0.539453\pi$
$108$	−1950.09	−1.73747
$109$	1590.63	1.39775	0.698874	−	0.715245i	$-0.253685\pi$
$109$	1590.63	1.39775	0.698874	+	0.715245i	$0.253685\pi$
$110$	1980.68	1.71682
$111$	−1180.89	−1.00978
$112$	−23.7535	−0.0200401
$113$	−484.817	−0.403608	−0.201804	−	0.979426i	$-0.564680\pi$
$113$	−484.817	−0.403608	−0.201804	+	0.979426i	$0.564680\pi$
$114$	−775.300	−0.636961
$115$	3222.17	2.61278
$116$	−141.015	−0.112870
$117$	321.505	0.254044
$118$	1536.10	1.19839
$119$	38.6470	0.0297711
$120$	−3521.07	−2.67857
$121$	−849.995	−0.638614
$122$	245.778	0.182391
$123$	−1059.34	−0.776566
$124$	3679.12	2.66448
$125$	−1952.62	−1.39718
$126$	−9.65081	−0.00682351
$127$	401.864	0.280785	0.140392	−	0.990096i	$-0.455164\pi$
$127$	401.864	0.280785	0.140392	+	0.990096i	$0.455164\pi$
$128$	−2650.07	−1.82997
$129$	0	0
$130$	−8140.99	−5.49240
$131$	2186.29	1.45815	0.729073	−	0.684436i	$-0.239951\pi$
$131$	2186.29	1.45815	0.729073	+	0.684436i	$0.239951\pi$
$132$	−1825.12	−1.20346
$133$	16.4609	0.0107319
$134$	−3522.31	−2.27075
$135$	2436.93	1.55361
$136$	−2321.79	−1.46391
$137$	−830.062	−0.517642	−0.258821	−	0.965925i	$-0.583334\pi$
$137$	−830.062	−0.517642	−0.258821	+	0.965925i	$0.583334\pi$
$138$	−4547.18	−2.80494
$139$	−1919.05	−1.17102	−0.585511	−	0.810664i	$-0.699106\pi$
$139$	−1919.05	−1.17102	−0.585511	+	0.810664i	$0.699106\pi$
$140$	159.565	0.0963267
$141$	−1887.80	−1.12753
$142$	2003.55	1.18404
$143$	−1977.03	−1.15614
$144$	150.320	0.0869907
$145$	176.220	0.100926
$146$	−171.064	−0.0969684
$147$	−1894.59	−1.06301
$148$	−3215.01	−1.78562
$149$	−1690.89	−0.929686	−0.464843	−	0.885393i	$-0.653889\pi$
$149$	−1690.89	−0.929686	−0.464843	+	0.885393i	$0.653889\pi$
$150$	6073.65	3.30608
$151$	−1011.84	−0.545311	−0.272656	−	0.962112i	$-0.587902\pi$
$151$	−1011.84	−0.545311	−0.272656	+	0.962112i	$0.587902\pi$
$152$	−988.917	−0.527709
$153$	−244.571	−0.129231
$154$	59.3457	0.0310533
$155$	−4597.63	−2.38252
$156$	7501.64	3.85008
$157$	1236.95	0.628786	0.314393	−	0.949293i	$-0.398199\pi$
$157$	1236.95	0.628786	0.314393	+	0.949293i	$0.398199\pi$
$158$	156.308	0.0787037
$159$	−2702.58	−1.34798
$160$	1288.65	0.636729
$161$	96.5439	0.0472592
$162$	−3901.39	−1.89211
$163$	−1060.28	−0.509495	−0.254747	−	0.967008i	$-0.581992\pi$
$163$	−1060.28	−0.509495	−0.254747	+	0.967008i	$0.581992\pi$
$164$	−2884.08	−1.37323
$165$	2280.77	1.07611
$166$	−353.795	−0.165421
$167$	−1645.91	−0.762660	−0.381330	−	0.924439i	$-0.624534\pi$
$167$	−1645.91	−0.762660	−0.381330	+	0.924439i	$0.624534\pi$
$168$	−105.500	−0.0484493
$169$	5929.00	2.69868
$170$	6192.90	2.79396
$171$	−104.170	−0.0465852
$172$	0	0
$173$	2977.10	1.30835	0.654176	−	0.756342i	$-0.273015\pi$
$173$	2977.10	1.30835	0.654176	+	0.756342i	$0.273015\pi$
$174$	−248.685	−0.108349
$175$	−128.953	−0.0557026
$176$	−924.361	−0.395888
$177$	1768.84	0.751154
$178$	5417.33	2.28116
$179$	−4.36944	−0.00182451	−0.000912256	−	1.00000i	$-0.500290\pi$
$179$	−4.36944	−0.00182451	−0.000912256		1.00000i	$0.500290\pi$
$180$	−1009.78	−0.418138
$181$	1585.58	0.651134	0.325567	−	0.945519i	$-0.394445\pi$
$181$	1585.58	0.651134	0.325567	+	0.945519i	$0.394445\pi$
$182$	−243.923	−0.0993450
$183$	283.016	0.114323
$184$	−5800.06	−2.32384
$185$	4017.65	1.59667
$186$	6488.25	2.55775
$187$	1503.94	0.588122
$188$	−5139.57	−1.99384
$189$	73.0162	0.0281013
$190$	2637.73	1.00716
$191$	−632.532	−0.239625	−0.119813	−	0.992797i	$-0.538229\pi$
$191$	−632.532	−0.239625	−0.119813	+	0.992797i	$0.538229\pi$
$192$	−3682.71	−1.38425
$193$	2150.82	0.802174	0.401087	−	0.916040i	$-0.368633\pi$
$193$	2150.82	0.802174	0.401087	+	0.916040i	$0.368633\pi$
$194$	−1485.88	−0.549895
$195$	−9374.45	−3.44266
$196$	−5158.06	−1.87976
$197$	−865.753	−0.313109	−0.156554	−	0.987669i	$-0.550039\pi$
$197$	−865.753	−0.313109	−0.156554	+	0.987669i	$0.550039\pi$
$198$	−375.559	−0.134797
$199$	4687.11	1.66965	0.834825	−	0.550516i	$-0.185569\pi$
$199$	4687.11	1.66965	0.834825	+	0.550516i	$0.185569\pi$
$200$	7747.11	2.73902
$201$	−4055.98	−1.42332
$202$	1591.37	0.554298
$203$	5.27997	0.00182552
$204$	−5706.55	−1.95852
$205$	3604.10	1.22791
$206$	638.460	0.215940
$207$	−610.962	−0.205144
$208$	3799.32	1.26652
$209$	640.571	0.212006
$210$	281.399	0.0924683
$211$	1447.08	0.472138	0.236069	−	0.971736i	$-0.424141\pi$
$211$	1447.08	0.472138	0.236069	+	0.971736i	$0.424141\pi$
$212$	−7357.82	−2.38367
$213$	2307.11	0.742163
$214$	−1313.94	−0.419717
$215$	0	0
$216$	−4386.59	−1.38180
$217$	−137.756	−0.0430943
$218$	7637.00	2.37267
$219$	−196.983	−0.0607802
$220$	6209.45	1.90292
$221$	−6181.50	−1.88151
$222$	−5669.77	−1.71410
$223$	5483.79	1.64673	0.823367	−	0.567510i	$-0.192093\pi$
$223$	5483.79	1.64673	0.823367	+	0.567510i	$0.192093\pi$
$224$	38.6110	0.0115170
$225$	816.059	0.241795
$226$	−2327.73	−0.685125
$227$	−6340.18	−1.85380	−0.926900	−	0.375309i	$-0.877537\pi$
$227$	−6340.18	−1.85380	−0.926900	+	0.375309i	$0.877537\pi$
$228$	−2430.58	−0.706005
$229$	−5138.02	−1.48266	−0.741332	−	0.671139i	$-0.765805\pi$
$229$	−5138.02	−1.48266	−0.741332	+	0.671139i	$0.765805\pi$
$230$	15470.5	4.43518
$231$	68.3373	0.0194644
$232$	−317.204	−0.0897650
$233$	3941.80	1.10831	0.554154	−	0.832414i	$-0.313042\pi$
$233$	3941.80	1.10831	0.554154	+	0.832414i	$0.313042\pi$
$234$	1543.63	0.431239
$235$	6422.68	1.78285
$236$	4815.71	1.32829
$237$	179.991	0.0493318
$238$	185.554	0.0505364
$239$	−367.931	−0.0995794	−0.0497897	−	0.998760i	$-0.515855\pi$
$239$	−367.931	−0.0995794	−0.0497897	+	0.998760i	$0.515855\pi$
$240$	−4383.03	−1.17885
$241$	−812.124	−0.217068	−0.108534	−	0.994093i	$-0.534616\pi$
$241$	−812.124	−0.217068	−0.108534	+	0.994093i	$0.534616\pi$
$242$	−4081.04	−1.08405
$243$	−994.469	−0.262532
$244$	770.518	0.202161
$245$	6445.79	1.68084
$246$	−5086.17	−1.31822
$247$	−2632.88	−0.678243
$248$	8275.94	2.11904
$249$	−407.400	−0.103686
$250$	−9375.01	−2.37171
$251$	−3984.26	−1.00193	−0.500964	−	0.865468i	$-0.667021\pi$
$251$	−3984.26	−1.00193	−0.500964	+	0.865468i	$0.667021\pi$
$252$	−30.2555	−0.00756316
$253$	3756.99	0.933595
$254$	1929.45	0.476631
$255$	7131.20	1.75127
$256$	−7394.81	−1.80537
$257$	−2945.96	−0.715034	−0.357517	−	0.933907i	$-0.616377\pi$
$257$	−2945.96	−0.715034	−0.357517	+	0.933907i	$0.616377\pi$
$258$	0	0
$259$	120.378	0.0288800
$260$	−25522.1	−6.08776
$261$	−33.4134	−0.00792429
$262$	10496.9	2.47520
$263$	1825.81	0.428076	0.214038	−	0.976825i	$-0.431338\pi$
$263$	1825.81	0.428076	0.214038	+	0.976825i	$0.431338\pi$
$264$	−4105.50	−0.957105
$265$	9194.72	2.13142
$266$	79.0327	0.0182173
$267$	6238.13	1.42984
$268$	−11042.5	−2.51689
$269$	3509.27	0.795404	0.397702	−	0.917515i	$-0.369808\pi$
$269$	3509.27	0.795404	0.397702	+	0.917515i	$0.369808\pi$
$270$	11700.3	2.63726
$271$	−8225.07	−1.84368	−0.921840	−	0.387571i	$-0.873314\pi$
$271$	−8225.07	−1.84368	−0.921840	+	0.387571i	$0.873314\pi$
$272$	−2890.16	−0.644272
$273$	−280.881	−0.0622698
$274$	−3985.34	−0.878697
$275$	−5018.19	−1.10039
$276$	−14255.5	−3.10899
$277$	1286.60	0.279076	0.139538	−	0.990217i	$-0.455438\pi$
$277$	1286.60	0.279076	0.139538	+	0.990217i	$0.455438\pi$
$278$	−9213.87	−1.98781
$279$	871.764	0.187065
$280$	358.932	0.0766081
$281$	4799.08	1.01882	0.509411	−	0.860523i	$-0.329863\pi$
$281$	4799.08	1.01882	0.509411	+	0.860523i	$0.329863\pi$
$282$	−9063.80	−1.91398
$283$	792.682	0.166502	0.0832510	−	0.996529i	$-0.473470\pi$
$283$	792.682	0.166502	0.0832510	+	0.996529i	$0.473470\pi$
$284$	6281.17	1.31239
$285$	3037.38	0.631295
$286$	−9492.21	−1.96254
$287$	107.987	0.0222101
$288$	−244.343	−0.0499932
$289$	−210.694	−0.0428850
$290$	846.077	0.171322
$291$	−1711.01	−0.344677
$292$	−536.290	−0.107479
$293$	5226.15	1.04203	0.521015	−	0.853547i	$-0.325553\pi$
$293$	5226.15	1.04203	0.521015	+	0.853547i	$0.325553\pi$
$294$	−9096.40	−1.80447
$295$	−6017.97	−1.18773
$296$	−7231.95	−1.42010
$297$	2841.41	0.555136
$298$	−8118.39	−1.57814
$299$	−15442.0	−2.98673
$300$	19041.0	3.66445
$301$	0	0
$302$	−4858.08	−0.925665
$303$	1832.48	0.347437
$304$	−1231.00	−0.232246
$305$	−962.880	−0.180768
$306$	−1174.25	−0.219370
$307$	5434.88	1.01037	0.505187	−	0.863010i	$-0.331423\pi$
$307$	5434.88	1.01037	0.505187	+	0.863010i	$0.331423\pi$
$308$	186.050	0.0344194
$309$	735.195	0.135352
$310$	−22074.4	−4.04432
$311$	434.161	0.0791608	0.0395804	−	0.999216i	$-0.487398\pi$
$311$	434.161	0.0791608	0.0395804	+	0.999216i	$0.487398\pi$
$312$	16874.4	3.06195
$313$	−7130.42	−1.28765	−0.643826	−	0.765172i	$-0.722654\pi$
$313$	−7130.42	−1.28765	−0.643826	+	0.765172i	$0.722654\pi$
$314$	5938.91	1.06736
$315$	37.8089	0.00676282
$316$	490.028	0.0872350
$317$	−3967.42	−0.702942	−0.351471	−	0.936199i	$-0.614318\pi$
$317$	−3967.42	−0.702942	−0.351471	+	0.936199i	$0.614318\pi$
$318$	−12975.7	−2.28819
$319$	205.469	0.0360629
$320$	12529.3	2.18879
$321$	−1513.02	−0.263080
$322$	463.532	0.0802224
$323$	2002.85	0.345020
$324$	−12230.9	−2.09721
$325$	20625.8	3.52035
$326$	−5090.68	−0.864867
$327$	8794.11	1.48720
$328$	−6487.55	−1.09212
$329$	192.439	0.0322477
$330$	10950.6	1.82669
$331$	2977.86	0.494495	0.247248	−	0.968952i	$-0.420474\pi$
$331$	2977.86	0.494495	0.247248	+	0.968952i	$0.420474\pi$
$332$	−1109.15	−0.183352
$333$	−761.793	−0.125363
$334$	−7902.43	−1.29462
$335$	13799.3	2.25055
$336$	−131.326	−0.0213227
$337$	3415.37	0.552069	0.276035	−	0.961148i	$-0.410980\pi$
$337$	3415.37	0.552069	0.276035	+	0.961148i	$0.410980\pi$
$338$	28466.6	4.58101
$339$	−2680.41	−0.429439
$340$	19414.9	3.09682
$341$	−5360.73	−0.851320
$342$	−500.145	−0.0790782
$343$	386.441	0.0608334
$344$	0	0
$345$	17814.4	2.77999
$346$	14293.8	2.22093
$347$	4500.31	0.696222	0.348111	−	0.937453i	$-0.386823\pi$
$347$	4500.31	0.696222	0.348111	+	0.937453i	$0.386823\pi$
$348$	−779.632	−0.120094
$349$	746.598	0.114511	0.0572557	−	0.998360i	$-0.481765\pi$
$349$	746.598	0.114511	0.0572557	+	0.998360i	$0.481765\pi$
$350$	−619.137	−0.0945550
$351$	−11678.8	−1.77598
$352$	1502.54	0.227516
$353$	8674.41	1.30791	0.653955	−	0.756533i	$-0.273108\pi$
$353$	8674.41	1.30791	0.653955	+	0.756533i	$0.273108\pi$
$354$	8492.65	1.27508
$355$	−7849.28	−1.17351
$356$	16983.4	2.52843
$357$	213.668	0.0316764
$358$	−20.9788	−0.00309711
$359$	−6602.33	−0.970634	−0.485317	−	0.874338i	$-0.661296\pi$
$359$	−6602.33	−0.970634	−0.485317	+	0.874338i	$0.661296\pi$
$360$	−2271.44	−0.332543
$361$	−6005.93	−0.875628
$362$	7612.77	1.10530
$363$	−4699.37	−0.679485
$364$	−764.704	−0.110114
$365$	670.177	0.0961059
$366$	1358.83	0.194064
$367$	6165.86	0.876990	0.438495	−	0.898734i	$-0.355512\pi$
$367$	6165.86	0.876990	0.438495	+	0.898734i	$0.355512\pi$
$368$	−7219.91	−1.02273
$369$	−683.380	−0.0964102
$370$	19289.7	2.71034
$371$	275.496	0.0385526
$372$	20340.8	2.83500
$373$	13214.6	1.83438	0.917191	−	0.398447i	$-0.130451\pi$
$373$	13214.6	1.83438	0.917191	+	0.398447i	$0.130451\pi$
$374$	7220.79	0.998337
$375$	−10795.4	−1.48660
$376$	−11561.1	−1.58569
$377$	−844.520	−0.115371
$378$	350.569	0.0477020
$379$	−3031.81	−0.410907	−0.205453	−	0.978667i	$-0.565867\pi$
$379$	−3031.81	−0.410907	−0.205453	+	0.978667i	$0.565867\pi$
$380$	8269.35	1.11634
$381$	2221.78	0.298755
$382$	−3036.94	−0.406763
$383$	345.297	0.0460675	0.0230337	−	0.999735i	$-0.492667\pi$
$383$	345.297	0.0460675	0.0230337	+	0.999735i	$0.492667\pi$
$384$	−14651.5	−1.94708
$385$	−232.498	−0.0307771
$386$	10326.6	1.36169
$387$	0	0
$388$	−4658.25	−0.609502
$389$	−1815.47	−0.236628	−0.118314	−	0.992976i	$-0.537749\pi$
$389$	−1815.47	−0.236628	−0.118314	+	0.992976i	$0.537749\pi$
$390$	−45009.1	−5.84391
$391$	11746.8	1.51934
$392$	−11602.7	−1.49496
$393$	12087.4	1.55147
$394$	−4156.70	−0.531501
$395$	−612.365	−0.0780037
$396$	−1177.39	−0.149409
$397$	13316.0	1.68341	0.841704	−	0.539939i	$-0.181553\pi$
$397$	13316.0	1.68341	0.841704	+	0.539939i	$0.181553\pi$
$398$	22504.0	2.83423
$399$	91.0072	0.0114187
$400$	9643.60	1.20545
$401$	3895.31	0.485094	0.242547	−	0.970140i	$-0.422017\pi$
$401$	3895.31	0.485094	0.242547	+	0.970140i	$0.422017\pi$
$402$	−19473.8	−2.41608
$403$	22033.7	2.72352
$404$	4988.97	0.614383
$405$	15284.4	1.87528
$406$	25.3505	0.00309882
$407$	4684.49	0.570520
$408$	−12836.5	−1.55760
$409$	−7160.50	−0.865682	−0.432841	−	0.901470i	$-0.642489\pi$
$409$	−7160.50	−0.865682	−0.432841	+	0.901470i	$0.642489\pi$
$410$	17304.2	2.08437
$411$	−4589.17	−0.550771
$412$	2001.58	0.239347
$413$	−180.312	−0.0214833
$414$	−2933.38	−0.348232
$415$	1386.06	0.163949
$416$	−6175.74	−0.727862
$417$	−10609.9	−1.24597
$418$	3075.54	0.359879
$419$	8139.57	0.949032	0.474516	−	0.880247i	$-0.342623\pi$
$419$	8139.57	0.949032	0.474516	+	0.880247i	$0.342623\pi$
$420$	882.190	0.102492
$421$	−4012.83	−0.464545	−0.232272	−	0.972651i	$-0.574616\pi$
$421$	−4012.83	−0.464545	−0.232272	+	0.972651i	$0.574616\pi$
$422$	6947.80	0.801454
$423$	−1217.82	−0.139982
$424$	−16550.9	−1.89572
$425$	−15690.2	−1.79079
$426$	11077.0	1.25982
$427$	−28.8501	−0.00326969
$428$	−4119.24	−0.465212
$429$	−10930.4	−1.23013
$430$	0	0
$431$	4194.35	0.468758	0.234379	−	0.972145i	$-0.424694\pi$
$431$	4194.35	0.468758	0.234379	+	0.972145i	$0.424694\pi$
$432$	−5460.42	−0.608136
$433$	−7807.78	−0.866555	−0.433277	−	0.901261i	$-0.642643\pi$
$433$	−7807.78	−0.866555	−0.433277	+	0.901261i	$0.642643\pi$
$434$	−661.400	−0.0731526
$435$	974.269	0.107385
$436$	23942.2	2.62986
$437$	5003.31	0.547690
$438$	−945.764	−0.103174
$439$	−16558.6	−1.80022	−0.900112	−	0.435660i	$-0.856515\pi$
$439$	−16558.6	−1.80022	−0.900112	+	0.435660i	$0.856515\pi$
$440$	13967.8	1.51338
$441$	−1222.20	−0.131972
$442$	−29678.9	−3.19385
$443$	−13329.5	−1.42958	−0.714788	−	0.699341i	$-0.753477\pi$
$443$	−13329.5	−1.42958	−0.714788	+	0.699341i	$0.753477\pi$
$444$	−17774.8	−1.89990
$445$	−21223.4	−2.26087
$446$	26329.0	2.79533
$447$	−9348.44	−0.989185
$448$	375.409	0.0395902
$449$	2475.47	0.260189	0.130094	−	0.991502i	$-0.458472\pi$
$449$	2475.47	0.260189	0.130094	+	0.991502i	$0.458472\pi$
$450$	3918.10	0.410447
$451$	4202.31	0.438756
$452$	−7297.47	−0.759390
$453$	−5594.14	−0.580211
$454$	−30440.8	−3.14682
$455$	955.614	0.0984613
$456$	−5467.43	−0.561482
$457$	−11342.0	−1.16095	−0.580477	−	0.814276i	$-0.697134\pi$
$457$	−11342.0	−1.16095	−0.580477	+	0.814276i	$0.697134\pi$
$458$	−24668.9	−2.51682
$459$	8884.13	0.903433
$460$	48500.2	4.91594
$461$	−5072.43	−0.512465	−0.256233	−	0.966615i	$-0.582481\pi$
$461$	−5072.43	−0.512465	−0.256233	+	0.966615i	$0.582481\pi$
$462$	328.105	0.0330407
$463$	13775.2	1.38269	0.691346	−	0.722524i	$-0.257018\pi$
$463$	13775.2	1.38269	0.691346	+	0.722524i	$0.257018\pi$
$464$	−394.856	−0.0395059
$465$	−25418.9	−2.53500
$466$	18925.6	1.88135
$467$	−11957.0	−1.18481	−0.592403	−	0.805642i	$-0.701821\pi$
$467$	−11957.0	−1.18481	−0.592403	+	0.805642i	$0.701821\pi$
$468$	4839.30	0.477984
$469$	413.459	0.0407074
$470$	30836.9	3.02639
$471$	6838.73	0.669028
$472$	10832.6	1.05638
$473$	0	0
$474$	864.180	0.0837407
$475$	−6682.89	−0.645541
$476$	581.715	0.0560144
$477$	−1743.43	−0.167350
$478$	−1766.53	−0.169036
$479$	−15782.6	−1.50548	−0.752739	−	0.658319i	$-0.771268\pi$
$479$	−15782.6	−1.50548	−0.752739	+	0.658319i	$0.771268\pi$
$480$	7124.56	0.677479
$481$	−19254.2	−1.82519
$482$	−3899.21	−0.368473
$483$	533.763	0.0502837
$484$	−12794.1	−1.20155
$485$	5821.20	0.545004
$486$	−4774.70	−0.445647
$487$	2228.11	0.207321	0.103661	−	0.994613i	$-0.466944\pi$
$487$	2228.11	0.207321	0.103661	+	0.994613i	$0.466944\pi$
$488$	1733.23	0.160778
$489$	−5861.98	−0.542102
$490$	30947.8	2.85323
$491$	8782.43	0.807221	0.403611	−	0.914931i	$-0.367755\pi$
$491$	8782.43	0.807221	0.403611	+	0.914931i	$0.367755\pi$
$492$	−15945.2	−1.46111
$493$	642.432	0.0586890
$494$	−12641.1	−1.15132
$495$	1471.32	0.133598
$496$	10301.9	0.932597
$497$	−235.183	−0.0212261
$498$	−1956.03	−0.176008
$499$	−680.761	−0.0610722	−0.0305361	−	0.999534i	$-0.509721\pi$
$499$	−680.761	−0.0610722	−0.0305361	+	0.999534i	$0.509721\pi$
$500$	−29390.8	−2.62880
$501$	−9099.74	−0.811470
$502$	−19129.4	−1.70077
$503$	−5406.14	−0.479220	−0.239610	−	0.970869i	$-0.577020\pi$
$503$	−5406.14	−0.479220	−0.239610	+	0.970869i	$0.577020\pi$
$504$	−68.0577	−0.00601494
$505$	−6234.48	−0.549368
$506$	18038.2	1.58478
$507$	32779.7	2.87139
$508$	6048.86	0.528297
$509$	−21162.7	−1.84286	−0.921432	−	0.388539i	$-0.872980\pi$
$509$	−21162.7	−1.84286	−0.921432	+	0.388539i	$0.872980\pi$
$510$	34238.7	2.97278
$511$	20.0801	0.00173834
$512$	−14303.8	−1.23465
$513$	3784.00	0.325668
$514$	−14144.3	−1.21377
$515$	−2501.29	−0.214019
$516$	0	0
$517$	7488.71	0.637047
$518$	577.966	0.0490238
$519$	16459.5	1.39209
$520$	−57410.4	−4.84156
$521$	−172.007	−0.0144641	−0.00723203	−	0.999974i	$-0.502302\pi$
$521$	−172.007	−0.0144641	−0.00723203	+	0.999974i	$0.502302\pi$
$522$	−160.426	−0.0134515
$523$	−6460.04	−0.540111	−0.270055	−	0.962845i	$-0.587042\pi$
$523$	−6460.04	−0.540111	−0.270055	+	0.962845i	$0.587042\pi$
$524$	32908.1	2.74351
$525$	−712.944	−0.0592675
$526$	8766.15	0.726658
$527$	−16761.2	−1.38544
$528$	−5110.52	−0.421225
$529$	17177.7	1.41183
$530$	44146.2	3.61809
$531$	1141.08	0.0932552
$532$	247.769	0.0201920
$533$	−17272.3	−1.40366
$534$	29950.8	2.42715
$535$	5147.62	0.415983
$536$	−24839.4	−2.00167
$537$	−24.1574	−0.00194128
$538$	16848.9	1.35020
$539$	7515.65	0.600597
$540$	36680.7	2.92313
$541$	−8894.36	−0.706837	−0.353418	−	0.935465i	$-0.614981\pi$
$541$	−8894.36	−0.706837	−0.353418	+	0.935465i	$0.614981\pi$
$542$	−39490.6	−3.12964
$543$	8766.21	0.692806
$544$	4697.92	0.370261
$545$	−29919.4	−2.35157
$546$	−1348.58	−0.105703
$547$	7309.10	0.571325	0.285662	−	0.958330i	$-0.407786\pi$
$547$	7309.10	0.571325	0.285662	+	0.958330i	$0.407786\pi$
$548$	−12494.1	−0.973945
$549$	182.573	0.0141932
$550$	−24093.6	−1.86792
$551$	273.630	0.0211561
$552$	−32066.8	−2.47256
$553$	−18.3479	−0.00141091
$554$	6177.27	0.473731
$555$	22212.4	1.69885
$556$	−28885.6	−2.20328
$557$	−20239.8	−1.53965	−0.769826	−	0.638254i	$-0.779657\pi$
$557$	−20239.8	−1.53965	−0.769826	+	0.638254i	$0.779657\pi$
$558$	4185.56	0.317543
$559$	0	0
$560$	446.798	0.0337155
$561$	8314.83	0.625762
$562$	23041.6	1.72945
$563$	4935.99	0.369498	0.184749	−	0.982786i	$-0.440853\pi$
$563$	4935.99	0.369498	0.184749	+	0.982786i	$0.440853\pi$
$564$	−28415.2	−2.12144
$565$	9119.31	0.679031
$566$	3805.87	0.282637
$567$	457.957	0.0339195
$568$	14129.1	1.04374
$569$	10385.0	0.765132	0.382566	−	0.923928i	$-0.375040\pi$
$569$	10385.0	0.765132	0.382566	+	0.923928i	$0.375040\pi$
$570$	14583.2	1.07162
$571$	8599.14	0.630232	0.315116	−	0.949053i	$-0.397957\pi$
$571$	8599.14	0.630232	0.315116	+	0.949053i	$0.397957\pi$
$572$	−29758.3	−2.17527
$573$	−3497.08	−0.254961
$574$	518.475	0.0377016
$575$	−39195.6	−2.84273
$576$	−2375.71	−0.171854
$577$	−14110.2	−1.01805	−0.509026	−	0.860751i	$-0.669994\pi$
$577$	−14110.2	−1.01805	−0.509026	+	0.860751i	$0.669994\pi$
$578$	−1011.59	−0.0727972
$579$	11891.3	0.853513
$580$	2652.47	0.189893
$581$	41.5296	0.00296547
$582$	−8214.97	−0.585088
$583$	10720.9	0.761599
$584$	−1206.35	−0.0854778
$585$	−6047.45	−0.427404
$586$	25092.1	1.76885
$587$	−5528.80	−0.388753	−0.194376	−	0.980927i	$-0.562268\pi$
$587$	−5528.80	−0.388753	−0.194376	+	0.980927i	$0.562268\pi$
$588$	−28517.4	−2.00006
$589$	−7139.07	−0.499423
$590$	−28893.8	−2.01617
$591$	−4786.49	−0.333147
$592$	−9002.32	−0.624988
$593$	14338.2	0.992918	0.496459	−	0.868060i	$-0.334633\pi$
$593$	14338.2	0.992918	0.496459	+	0.868060i	$0.334633\pi$
$594$	13642.3	0.942343
$595$	−726.942	−0.0500869
$596$	−25451.3	−1.74921
$597$	25913.6	1.77651
$598$	−74140.9	−5.06998
$599$	17319.7	1.18141	0.590705	−	0.806888i	$-0.298850\pi$
$599$	17319.7	1.18141	0.590705	+	0.806888i	$0.298850\pi$
$600$	42831.5	2.91431
$601$	−28433.6	−1.92984	−0.964918	−	0.262550i	$-0.915437\pi$
$601$	−28433.6	−1.92984	−0.964918	+	0.262550i	$0.915437\pi$
$602$	0	0
$603$	−2616.51	−0.176704
$604$	−15230.2	−1.02600
$605$	15988.2	1.07440
$606$	8798.20	0.589773
$607$	−6746.30	−0.451110	−0.225555	−	0.974230i	$-0.572420\pi$
$607$	−6746.30	−0.451110	−0.225555	+	0.974230i	$0.572420\pi$
$608$	2000.98	0.133471
$609$	29.1914	0.00194236
$610$	−4623.03	−0.306854
$611$	−30780.2	−2.03802
$612$	−3681.29	−0.243149
$613$	−4756.54	−0.313401	−0.156701	−	0.987646i	$-0.550086\pi$
$613$	−4756.54	−0.313401	−0.156701	+	0.987646i	$0.550086\pi$
$614$	26094.2	1.71511
$615$	19926.0	1.30649
$616$	418.507	0.0273736
$617$	−13947.4	−0.910051	−0.455026	−	0.890478i	$-0.650370\pi$
$617$	−13947.4	−0.910051	−0.455026	+	0.890478i	$0.650370\pi$
$618$	3529.86	0.229760
$619$	10744.9	0.697699	0.348849	−	0.937179i	$-0.386572\pi$
$619$	10744.9	0.697699	0.348849	+	0.937179i	$0.386572\pi$
$620$	−69203.5	−4.48271
$621$	22193.4	1.43412
$622$	2084.52	0.134375
$623$	−635.903	−0.0408939
$624$	21005.3	1.34757
$625$	8127.28	0.520146
$626$	−34234.9	−2.18579
$627$	3541.52	0.225574
$628$	18618.6	1.18306
$629$	14646.8	0.928468
$630$	181.530	0.0114799
$631$	8390.63	0.529360	0.264680	−	0.964336i	$-0.414734\pi$
$631$	8390.63	0.529360	0.264680	+	0.964336i	$0.414734\pi$
$632$	1102.29	0.0693775
$633$	8000.48	0.502355
$634$	−19048.6	−1.19324
$635$	−7558.97	−0.472392
$636$	−40679.2	−2.53622
$637$	−30890.9	−1.92141
$638$	986.508	0.0612167
$639$	1488.32	0.0921390
$640$	49847.3	3.07873
$641$	6432.67	0.396373	0.198187	−	0.980164i	$-0.436495\pi$
$641$	6432.67	0.396373	0.198187	+	0.980164i	$0.436495\pi$
$642$	−7264.41	−0.446578
$643$	6600.52	0.404820	0.202410	−	0.979301i	$-0.435123\pi$
$643$	6600.52	0.404820	0.202410	+	0.979301i	$0.435123\pi$
$644$	1453.18	0.0889182
$645$	0	0
$646$	9616.17	0.585671
$647$	−16454.6	−0.999839	−0.499920	−	0.866072i	$-0.666637\pi$
$647$	−16454.6	−0.999839	−0.499920	+	0.866072i	$0.666637\pi$
$648$	−27512.6	−1.66790
$649$	−7016.82	−0.424398
$650$	99029.6	5.97579
$651$	−761.610	−0.0458523
$652$	−15959.4	−0.958615
$653$	−22219.0	−1.33154	−0.665771	−	0.746156i	$-0.731897\pi$
$653$	−22219.0	−1.33154	−0.665771	+	0.746156i	$0.731897\pi$
$654$	42222.7	2.52452
$655$	−41123.7	−2.45318
$656$	−8075.70	−0.480645
$657$	−127.073	−0.00754582
$658$	923.947	0.0547404
$659$	−29368.5	−1.73601	−0.868007	−	0.496551i	$-0.834599\pi$
$659$	−29368.5	−1.73601	−0.868007	+	0.496551i	$0.834599\pi$
$660$	34330.2	2.02470
$661$	−16178.2	−0.951981	−0.475990	−	0.879451i	$-0.657910\pi$
$661$	−16178.2	−0.951981	−0.475990	+	0.879451i	$0.657910\pi$
$662$	14297.4	0.839405
$663$	−34175.7	−2.00192
$664$	−2494.97	−0.145819
$665$	−309.625	−0.0180553
$666$	−3657.56	−0.212804
$667$	1604.86	0.0931639
$668$	−24774.3	−1.43495
$669$	30318.2	1.75212
$670$	66253.9	3.82031
$671$	−1122.70	−0.0645920
$672$	213.469	0.0122541
$673$	30064.0	1.72197	0.860983	−	0.508634i	$-0.169849\pi$
$673$	30064.0	1.72197	0.860983	+	0.508634i	$0.169849\pi$
$674$	16398.1	0.937136
$675$	−29643.6	−1.69035
$676$	89243.4	5.07757
$677$	27396.2	1.55528	0.777638	−	0.628713i	$-0.216418\pi$
$677$	27396.2	1.55528	0.777638	+	0.628713i	$0.216418\pi$
$678$	−12869.3	−0.728972
$679$	174.417	0.00985788
$680$	43672.4	2.46288
$681$	−35053.0	−1.97244
$682$	−25738.2	−1.44511
$683$	−27000.7	−1.51267	−0.756335	−	0.654185i	$-0.773012\pi$
$683$	−27000.7	−1.51267	−0.756335	+	0.654185i	$0.773012\pi$
$684$	−1567.97	−0.0876501
$685$	15613.3	0.870881
$686$	1855.40	0.103265
$687$	−28406.6	−1.57755
$688$	0	0
$689$	−44064.9	−2.43649
$690$	85531.6	4.71903
$691$	20353.7	1.12054	0.560270	−	0.828310i	$-0.310697\pi$
$691$	20353.7	1.12054	0.560270	+	0.828310i	$0.310697\pi$
$692$	44811.4	2.46167
$693$	44.0843	0.00241649
$694$	21607.1	1.18184
$695$	36097.0	1.97013
$696$	−1753.73	−0.0955100
$697$	13139.2	0.714035
$698$	3584.61	0.194383
$699$	21793.0	1.17924
$700$	−1941.01	−0.104804
$701$	−12725.1	−0.685622	−0.342811	−	0.939404i	$-0.611379\pi$
$701$	−12725.1	−0.685622	−0.342811	+	0.939404i	$0.611379\pi$
$702$	−56072.8	−3.01472
$703$	6238.49	0.334693
$704$	14608.9	0.782096
$705$	35509.1	1.89695
$706$	41648.0	2.22018
$707$	−186.800	−0.00993681
$708$	26624.6	1.41330
$709$	−29784.5	−1.57769	−0.788843	−	0.614595i	$-0.789320\pi$
$709$	−29784.5	−1.57769	−0.788843	+	0.614595i	$0.789320\pi$
$710$	−37686.4	−1.99203
$711$	116.112	0.00612451
$712$	38203.1	2.01085
$713$	−41871.1	−2.19928
$714$	1025.87	0.0537707
$715$	37187.5	1.94508
$716$	−65.7689	−0.00343282
$717$	−2034.18	−0.105952
$718$	−31699.4	−1.64765
$719$	−26156.8	−1.35673	−0.678363	−	0.734727i	$-0.737310\pi$
$719$	−26156.8	−1.35673	−0.678363	+	0.734727i	$0.737310\pi$
$720$	−2827.49	−0.146353
$721$	−74.9444	−0.00387112
$722$	−28836.0	−1.48638
$723$	−4489.99	−0.230961
$724$	23866.2	1.22511
$725$	−2143.60	−0.109809
$726$	−22562.9	−1.15342
$727$	13708.6	0.699346	0.349673	−	0.936872i	$-0.386293\pi$
$727$	13708.6	0.699346	0.349673	+	0.936872i	$0.386293\pi$
$728$	−1720.15	−0.0875728
$729$	16441.4	0.835312
$730$	3217.69	0.163140
$731$	0	0
$732$	4259.96	0.215099
$733$	19482.8	0.981736	0.490868	−	0.871234i	$-0.336680\pi$
$733$	19482.8	0.981736	0.490868	+	0.871234i	$0.336680\pi$
$734$	29603.8	1.48869
$735$	35636.8	1.78841
$736$	11735.9	0.587758
$737$	16089.7	0.804167
$738$	−3281.08	−0.163656
$739$	−9397.10	−0.467764	−0.233882	−	0.972265i	$-0.575143\pi$
$739$	−9397.10	−0.467764	−0.233882	+	0.972265i	$0.575143\pi$
$740$	60473.7	3.00413
$741$	−14556.4	−0.721650
$742$	1322.72	0.0654430
$743$	−21996.2	−1.08609	−0.543044	−	0.839705i	$-0.682728\pi$
$743$	−21996.2	−1.08609	−0.543044	+	0.839705i	$0.682728\pi$
$744$	45755.2	2.25466
$745$	31805.3	1.56410
$746$	63446.5	3.11386
$747$	−262.813	−0.0128726
$748$	22637.3	1.10655
$749$	154.235	0.00752419
$750$	−51831.6	−2.52350
$751$	−26869.9	−1.30559	−0.652795	−	0.757534i	$-0.726404\pi$
$751$	−26869.9	−1.30559	−0.652795	+	0.757534i	$0.726404\pi$
$752$	−14391.3	−0.697867
$753$	−22027.8	−1.06605
$754$	−4054.75	−0.195843
$755$	19032.4	0.917431
$756$	1099.04	0.0528727
$757$	1187.31	0.0570058	0.0285029	−	0.999594i	$-0.490926\pi$
$757$	1187.31	0.0570058	0.0285029	+	0.999594i	$0.490926\pi$
$758$	−14556.5	−0.697514
$759$	20771.3	0.993345
$760$	18601.3	0.887818
$761$	3960.08	0.188637	0.0943185	−	0.995542i	$-0.469933\pi$
$761$	3960.08	0.188637	0.0943185	+	0.995542i	$0.469933\pi$
$762$	10667.3	0.507135
$763$	−896.455	−0.0425345
$764$	−9520.88	−0.450855
$765$	4600.33	0.217419
$766$	1657.86	0.0781995
$767$	28840.6	1.35772
$768$	−40883.7	−1.92092
$769$	−18342.9	−0.860157	−0.430079	−	0.902791i	$-0.641514\pi$
$769$	−18342.9	−0.860157	−0.430079	+	0.902791i	$0.641514\pi$
$770$	−1116.28	−0.0522441
$771$	−16287.3	−0.760796
$772$	32374.2	1.50929
$773$	−10523.7	−0.489667	−0.244834	−	0.969565i	$-0.578733\pi$
$773$	−10523.7	−0.489667	−0.244834	+	0.969565i	$0.578733\pi$
$774$	0	0
$775$	55927.0	2.59220
$776$	−10478.4	−0.484734
$777$	665.535	0.0307284
$778$	−8716.55	−0.401675
$779$	5596.36	0.257394
$780$	−141104.	−6.47737
$781$	−9152.09	−0.419318
$782$	56399.4	2.57908
$783$	1213.75	0.0553973
$784$	−14443.0	−0.657938
$785$	−23266.8	−1.05787
$786$	58034.4	2.63361
$787$	1362.84	0.0617281	0.0308641	−	0.999524i	$-0.490174\pi$
$787$	1362.84	0.0617281	0.0308641	+	0.999524i	$0.490174\pi$
$788$	−13031.3	−0.589114
$789$	10094.3	0.455473
$790$	−2940.12	−0.132411
$791$	273.236	0.0122821
$792$	−2648.45	−0.118824
$793$	4614.52	0.206641
$794$	63933.7	2.85758
$795$	50834.9	2.26783
$796$	70550.4	3.14145
$797$	−30311.6	−1.34717	−0.673584	−	0.739111i	$-0.735246\pi$
$797$	−30311.6	−1.34717	−0.673584	+	0.739111i	$0.735246\pi$
$798$	436.948	0.0193832
$799$	23414.7	1.03673
$800$	−15675.5	−0.692768
$801$	4024.21	0.177514
$802$	18702.4	0.823446
$803$	781.411	0.0343405
$804$	−61050.7	−2.67797
$805$	−1815.97	−0.0795088
$806$	105789.	4.62317
$807$	19401.7	0.846309
$808$	11222.4	0.488615
$809$	39233.5	1.70504	0.852520	−	0.522694i	$-0.175073\pi$
$809$	39233.5	1.70504	0.852520	+	0.522694i	$0.175073\pi$
$810$	73384.3	3.18328
$811$	43510.5	1.88392	0.941961	−	0.335723i	$-0.108981\pi$
$811$	43510.5	1.88392	0.941961	+	0.335723i	$0.108981\pi$
$812$	79.4742	0.00343473
$813$	−45473.9	−1.96167
$814$	22491.4	0.968456
$815$	19943.7	0.857174
$816$	−15978.9	−0.685505
$817$	0	0
$818$	−34379.4	−1.46949
$819$	−181.196	−0.00773076
$820$	54249.0	2.31031
$821$	−20770.7	−0.882951	−0.441475	−	0.897273i	$-0.645545\pi$
$821$	−20770.7	−0.882951	−0.441475	+	0.897273i	$0.645545\pi$
$822$	−22033.7	−0.934933
$823$	−15244.0	−0.645652	−0.322826	−	0.946458i	$-0.604633\pi$
$823$	−15244.0	−0.645652	−0.322826	+	0.946458i	$0.604633\pi$
$824$	4502.43	0.190351
$825$	−27744.1	−1.17082
$826$	−865.725	−0.0364678
$827$	−32831.8	−1.38050	−0.690251	−	0.723570i	$-0.742500\pi$
$827$	−32831.8	−1.38050	−0.690251	+	0.723570i	$0.742500\pi$
$828$	−9196.21	−0.385979
$829$	−20720.4	−0.868092	−0.434046	−	0.900891i	$-0.642914\pi$
$829$	−20720.4	−0.868092	−0.434046	+	0.900891i	$0.642914\pi$
$830$	6654.82	0.278304
$831$	7113.21	0.296937
$832$	−60045.8	−2.50206
$833$	23498.9	0.977417
$834$	−50940.7	−2.11503
$835$	30959.2	1.28310
$836$	9641.88	0.398889
$837$	−31667.1	−1.30774
$838$	39080.1	1.61098
$839$	45367.8	1.86683	0.933415	−	0.358799i	$-0.116814\pi$
$839$	45367.8	1.86683	0.933415	+	0.358799i	$0.116814\pi$
$840$	1984.43	0.0815110
$841$	−24301.2	−0.996401
$842$	−19266.6	−0.788564
$843$	26532.7	1.08403
$844$	21781.5	0.888329
$845$	−111523.	−4.54026
$846$	−5847.04	−0.237619
$847$	479.045	0.0194335
$848$	−20602.6	−0.834311
$849$	4382.50	0.177158
$850$	−75332.5	−3.03986
$851$	36589.1	1.47387
$852$	34726.7	1.39638
$853$	9272.95	0.372216	0.186108	−	0.982529i	$-0.440413\pi$
$853$	9272.95	0.372216	0.186108	+	0.982529i	$0.440413\pi$
$854$	−138.517	−0.00555029
$855$	1959.41	0.0783748
$856$	−9265.95	−0.369981
$857$	8958.78	0.357090	0.178545	−	0.983932i	$-0.442861\pi$
$857$	8958.78	0.357090	0.178545	+	0.983932i	$0.442861\pi$
$858$	−52479.6	−2.08814
$859$	3992.34	0.158576	0.0792880	−	0.996852i	$-0.474735\pi$
$859$	3992.34	0.158576	0.0792880	+	0.996852i	$0.474735\pi$
$860$	0	0
$861$	597.030	0.0236315
$862$	20138.1	0.795717
$863$	−25382.8	−1.00121	−0.500603	−	0.865677i	$-0.666888\pi$
$863$	−25382.8	−1.00121	−0.500603	+	0.865677i	$0.666888\pi$
$864$	8875.85	0.349494
$865$	−55998.7	−2.20117
$866$	−37487.1	−1.47098
$867$	−1164.86	−0.0456296
$868$	−2073.50	−0.0810820
$869$	−714.005	−0.0278722
$870$	4677.71	0.182287
$871$	−66131.9	−2.57267
$872$	53856.3	2.09152
$873$	−1103.77	−0.0427914
$874$	24022.1	0.929704
$875$	1100.47	0.0425172
$876$	−2964.99	−0.114358
$877$	−34007.1	−1.30940	−0.654698	−	0.755891i	$-0.727204\pi$
$877$	−34007.1	−1.30940	−0.654698	+	0.755891i	$0.727204\pi$
$878$	−79501.9	−3.05588
$879$	28893.8	1.10872
$880$	17387.0	0.666042
$881$	−34430.4	−1.31667	−0.658337	−	0.752723i	$-0.728740\pi$
$881$	−34430.4	−1.31667	−0.658337	+	0.752723i	$0.728740\pi$
$882$	−5868.07	−0.224023
$883$	−4762.11	−0.181492	−0.0907461	−	0.995874i	$-0.528925\pi$
$883$	−4762.11	−0.181492	−0.0907461	+	0.995874i	$0.528925\pi$
$884$	−93044.0	−3.54006
$885$	−33271.6	−1.26374
$886$	−63998.2	−2.42670
$887$	−42087.0	−1.59317	−0.796587	−	0.604524i	$-0.793363\pi$
$887$	−42087.0	−1.59317	−0.796587	+	0.604524i	$0.793363\pi$
$888$	−39983.3	−1.51098
$889$	−226.485	−0.00854449
$890$	−101899.	−3.83782
$891$	17821.3	0.670074
$892$	82542.1	3.09833
$893$	9972.97	0.373721
$894$	−44884.2	−1.67914
$895$	82.1884	0.00306956
$896$	1493.54	0.0556873
$897$	−85374.2	−3.17788
$898$	11885.4	0.441670
$899$	−2289.92	−0.0849536
$900$	12283.3	0.454938
$901$	33520.5	1.23943
$902$	20176.3	0.744787
$903$	0	0
$904$	−16415.2	−0.603939
$905$	−29824.5	−1.09547
$906$	−26858.9	−0.984907
$907$	−25450.7	−0.931726	−0.465863	−	0.884857i	$-0.654256\pi$
$907$	−25450.7	−0.931726	−0.465863	+	0.884857i	$0.654256\pi$
$908$	−95432.5	−3.48793
$909$	1182.13	0.0431340
$910$	4588.14	0.167138
$911$	−19434.6	−0.706804	−0.353402	−	0.935472i	$-0.614975\pi$
$911$	−19434.6	−0.706804	−0.353402	+	0.935472i	$0.614975\pi$
$912$	−6805.85	−0.247110
$913$	1616.11	0.0585822
$914$	−54455.8	−1.97072
$915$	−5323.48	−0.192337
$916$	−77337.6	−2.78963
$917$	−1232.16	−0.0443725
$918$	42654.9	1.53358
$919$	479.135	0.0171983	0.00859913	−	0.999963i	$-0.497263\pi$
$919$	479.135	0.0171983	0.00859913	+	0.999963i	$0.497263\pi$
$920$	109098.	3.90962
$921$	30047.8	1.07504
$922$	−24354.0	−0.869909
$923$	37617.0	1.34147
$924$	1028.61	0.0366222
$925$	−48872.0	−1.73719
$926$	66138.0	2.34712
$927$	474.273	0.0168039
$928$	641.833	0.0227039
$929$	8375.21	0.295782	0.147891	−	0.989004i	$-0.452751\pi$
$929$	8375.21	0.295782	0.147891	+	0.989004i	$0.452751\pi$
$930$	−122043.	−4.30316
$931$	10008.8	0.352338
$932$	59332.1	2.08529
$933$	2400.35	0.0842271
$934$	−57408.6	−2.01121
$935$	−28288.8	−0.989457
$936$	10885.7	0.380138
$937$	15853.9	0.552746	0.276373	−	0.961050i	$-0.410867\pi$
$937$	15853.9	0.552746	0.276373	+	0.961050i	$0.410867\pi$
$938$	1985.12	0.0691008
$939$	−39422.0	−1.37006
$940$	96674.3	3.35444
$941$	−2647.73	−0.0917253	−0.0458626	−	0.998948i	$-0.514604\pi$
$941$	−2647.73	−0.0917253	−0.0458626	+	0.998948i	$0.514604\pi$
$942$	32834.5	1.13567
$943$	32823.0	1.13347
$944$	13484.4	0.464916
$945$	−1373.42	−0.0472776
$946$	0	0
$947$	−27521.2	−0.944369	−0.472185	−	0.881500i	$-0.656534\pi$
$947$	−27521.2	−0.944369	−0.472185	+	0.881500i	$0.656534\pi$
$948$	2709.22	0.0928179
$949$	−3211.76	−0.109861
$950$	−32086.3	−1.09581
$951$	−21934.7	−0.747930
$952$	1308.53	0.0445480
$953$	49753.8	1.69117	0.845585	−	0.533842i	$-0.179252\pi$
$953$	49753.8	1.69117	0.845585	+	0.533842i	$0.179252\pi$
$954$	−8370.64	−0.284077
$955$	11897.8	0.403145
$956$	−5538.11	−0.187359
$957$	1135.98	0.0383709
$958$	−75776.1	−2.55555
$959$	467.811	0.0157523
$960$	69271.0	2.32887
$961$	29953.7	1.00546
$962$	−92444.3	−3.09826
$963$	−976.049	−0.0326612
$964$	−12224.1	−0.408415
$965$	−40456.5	−1.34958
$966$	2562.73	0.0853566
$967$	52924.4	1.76002	0.880008	−	0.474960i	$-0.157537\pi$
$967$	52924.4	1.76002	0.880008	+	0.474960i	$0.157537\pi$
$968$	−28779.6	−0.955589
$969$	11073.1	0.367101
$970$	27949.0	0.925144
$971$	40927.9	1.35267	0.676333	−	0.736596i	$-0.263568\pi$
$971$	40927.9	1.35267	0.676333	+	0.736596i	$0.263568\pi$
$972$	−14968.8	−0.493954
$973$	1081.55	0.0356351
$974$	10697.7	0.351928
$975$	114034.	3.74565
$976$	2157.52	0.0707588
$977$	43543.0	1.42586	0.712930	−	0.701236i	$-0.247368\pi$
$977$	43543.0	1.42586	0.712930	+	0.701236i	$0.247368\pi$
$978$	−28144.8	−0.920218
$979$	−24746.0	−0.807852
$980$	97022.1	3.16251
$981$	5673.07	0.184635
$982$	42166.7	1.37026
$983$	46016.6	1.49309	0.746543	−	0.665337i	$-0.231712\pi$
$983$	46016.6	1.49309	0.746543	+	0.665337i	$0.231712\pi$
$984$	−35867.7	−1.16201
$985$	16284.6	0.526774
$986$	3084.48	0.0996245
$987$	1063.94	0.0343115
$988$	−39630.1	−1.27612
$989$	0	0
$990$	7064.19	0.226783
$991$	42967.2	1.37729	0.688646	−	0.725098i	$-0.258205\pi$
$991$	42967.2	1.37729	0.688646	+	0.725098i	$0.258205\pi$
$992$	−16745.6	−0.535960
$993$	16463.7	0.526143
$994$	−1129.17	−0.0360313
$995$	−88163.5	−2.80902
$996$	−6132.19	−0.195086
$997$	43345.3	1.37689	0.688445	−	0.725288i	$-0.258294\pi$
$997$	43345.3	1.37689	0.688445	+	0.725288i	$0.258294\pi$
$998$	−3268.50	−0.103670
$999$	27672.4	0.876392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.45	yes	50
43.42	odd	2		1849.4.a.i.1.6	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.6	✓	50	43.42	odd	2
1849.4.a.j.1.45	yes	50	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+4.80125 q^{2} +5.52870 q^{3} +15.0520 q^{4} -18.8098 q^{5} +26.5447 q^{6} -0.563586 q^{7} +33.8585 q^{8} +3.56656 q^{9} +O(q^{10})\)	\(q+4.80125 q^{2} +5.52870 q^{3} +15.0520 q^{4} -18.8098 q^{5} +26.5447 q^{6} -0.563586 q^{7} +33.8585 q^{8} +3.56656 q^{9} -90.3106 q^{10} -21.9318 q^{11} +83.2181 q^{12} +90.1443 q^{13} -2.70592 q^{14} -103.994 q^{15} +42.1470 q^{16} -68.5734 q^{17} +17.1239 q^{18} -29.2074 q^{19} -283.125 q^{20} -3.11590 q^{21} -105.300 q^{22} -171.303 q^{23} +187.194 q^{24} +228.809 q^{25} +432.806 q^{26} -129.557 q^{27} -8.48310 q^{28} -9.36853 q^{29} -499.300 q^{30} +244.427 q^{31} -68.5095 q^{32} -121.255 q^{33} -329.238 q^{34} +10.6009 q^{35} +53.6839 q^{36} -213.593 q^{37} -140.232 q^{38} +498.381 q^{39} -636.872 q^{40} -191.608 q^{41} -14.9602 q^{42} -330.118 q^{44} -67.0863 q^{45} -822.469 q^{46} -341.454 q^{47} +233.018 q^{48} -342.682 q^{49} +1098.57 q^{50} -379.122 q^{51} +1356.85 q^{52} -488.826 q^{53} -622.034 q^{54} +412.533 q^{55} -19.0822 q^{56} -161.479 q^{57} -44.9807 q^{58} +319.938 q^{59} -1565.32 q^{60} +51.1903 q^{61} +1173.56 q^{62} -2.01006 q^{63} -666.107 q^{64} -1695.60 q^{65} -582.173 q^{66} -733.622 q^{67} -1032.17 q^{68} -947.083 q^{69} +50.8978 q^{70} +417.297 q^{71} +120.758 q^{72} -35.6291 q^{73} -1025.51 q^{74} +1265.01 q^{75} -439.630 q^{76} +12.3605 q^{77} +2392.85 q^{78} +32.5557 q^{79} -792.777 q^{80} -812.577 q^{81} -919.957 q^{82} -73.6881 q^{83} -46.9006 q^{84} +1289.85 q^{85} -51.7958 q^{87} -742.578 q^{88} +1128.32 q^{89} -322.098 q^{90} -50.8041 q^{91} -2578.45 q^{92} +1351.37 q^{93} -1639.41 q^{94} +549.385 q^{95} -378.768 q^{96} -309.477 q^{97} -1645.30 q^{98} -78.2211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} + 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} + 386 q^{18} - 12 q^{19} - 108 q^{20} - 408 q^{21} - 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} - 1493 q^{26} - 10 q^{27} + 242 q^{28} - 208 q^{29} + 48 q^{30} - 932 q^{31} + 1124 q^{32} - 254 q^{33} - 765 q^{34} - 1452 q^{35} + 747 q^{36} + 90 q^{37} - 1213 q^{38} + 1610 q^{39} - 1693 q^{40} - 1354 q^{41} + 16 q^{42} - 2704 q^{44} - 4508 q^{45} - 233 q^{46} - 3484 q^{47} + 376 q^{48} + 1324 q^{49} + 408 q^{50} - 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} + 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} - 1172 q^{61} + 1546 q^{62} + 3686 q^{63} + 606 q^{64} - 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} - 136 q^{69} + 1310 q^{70} - 162 q^{71} + 5814 q^{72} - 746 q^{73} - 4332 q^{74} - 236 q^{75} - 1338 q^{76} - 2024 q^{77} - 2782 q^{78} - 2656 q^{79} + 5713 q^{80} - 86 q^{81} + 4168 q^{82} - 3514 q^{83} - 4269 q^{84} + 7558 q^{85} - 10278 q^{87} - 11692 q^{88} - 2640 q^{89} - 8286 q^{90} + 5946 q^{91} - 4271 q^{92} + 2 q^{93} - 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} + 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 + 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 + 386 * q^18 - 12 * q^19 - 108 * q^20 - 408 * q^21 - 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 - 1493 * q^26 - 10 * q^27 + 242 * q^28 - 208 * q^29 + 48 * q^30 - 932 * q^31 + 1124 * q^32 - 254 * q^33 - 765 * q^34 - 1452 * q^35 + 747 * q^36 + 90 * q^37 - 1213 * q^38 + 1610 * q^39 - 1693 * q^40 - 1354 * q^41 + 16 * q^42 - 2704 * q^44 - 4508 * q^45 - 233 * q^46 - 3484 * q^47 + 376 * q^48 + 1324 * q^49 + 408 * q^50 - 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 + 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 - 1172 * q^61 + 1546 * q^62 + 3686 * q^63 + 606 * q^64 - 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 - 136 * q^69 + 1310 * q^70 - 162 * q^71 + 5814 * q^72 - 746 * q^73 - 4332 * q^74 - 236 * q^75 - 1338 * q^76 - 2024 * q^77 - 2782 * q^78 - 2656 * q^79 + 5713 * q^80 - 86 * q^81 + 4168 * q^82 - 3514 * q^83 - 4269 * q^84 + 7558 * q^85 - 10278 * q^87 - 11692 * q^88 - 2640 * q^89 - 8286 * q^90 + 5946 * q^91 - 4271 * q^92 + 2 * q^93 - 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 + 2826 * q^98 - 8174 * q^99

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