LMFDB - Embedded newform 1849.4.a.i.1.1

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			1.1
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)

$f(q)$	$=$	$q-5.46223 q^{2} +9.09504 q^{3} +21.8360 q^{4} -6.15202 q^{5} -49.6792 q^{6} +2.59059 q^{7} -75.5754 q^{8} +55.7197 q^{9} +O(q^{10})$	\(q-5.46223 q^{2} +9.09504 q^{3} +21.8360 q^{4} -6.15202 q^{5} -49.6792 q^{6} +2.59059 q^{7} -75.5754 q^{8} +55.7197 q^{9} +33.6038 q^{10} -20.0476 q^{11} +198.599 q^{12} -34.1627 q^{13} -14.1504 q^{14} -55.9529 q^{15} +238.122 q^{16} +36.1588 q^{17} -304.354 q^{18} +142.784 q^{19} -134.335 q^{20} +23.5615 q^{21} +109.505 q^{22} -132.818 q^{23} -687.361 q^{24} -87.1526 q^{25} +186.604 q^{26} +261.207 q^{27} +56.5681 q^{28} -49.6805 q^{29} +305.628 q^{30} -51.3202 q^{31} -696.077 q^{32} -182.334 q^{33} -197.508 q^{34} -15.9374 q^{35} +1216.69 q^{36} -49.1882 q^{37} -779.922 q^{38} -310.711 q^{39} +464.941 q^{40} +71.4324 q^{41} -128.698 q^{42} -437.760 q^{44} -342.789 q^{45} +725.481 q^{46} -371.774 q^{47} +2165.73 q^{48} -336.289 q^{49} +476.048 q^{50} +328.865 q^{51} -745.976 q^{52} +260.514 q^{53} -1426.77 q^{54} +123.333 q^{55} -195.785 q^{56} +1298.63 q^{57} +271.366 q^{58} +29.4340 q^{59} -1221.79 q^{60} -461.029 q^{61} +280.323 q^{62} +144.347 q^{63} +1897.16 q^{64} +210.169 q^{65} +995.951 q^{66} +891.001 q^{67} +789.563 q^{68} -1207.98 q^{69} +87.0536 q^{70} +475.538 q^{71} -4211.04 q^{72} +1027.68 q^{73} +268.677 q^{74} -792.656 q^{75} +3117.84 q^{76} -51.9352 q^{77} +1697.17 q^{78} -307.975 q^{79} -1464.93 q^{80} +871.251 q^{81} -390.181 q^{82} -1145.51 q^{83} +514.489 q^{84} -222.450 q^{85} -451.846 q^{87} +1515.11 q^{88} -670.531 q^{89} +1872.39 q^{90} -88.5015 q^{91} -2900.20 q^{92} -466.759 q^{93} +2030.72 q^{94} -878.413 q^{95} -6330.85 q^{96} -125.357 q^{97} +1836.89 q^{98} -1117.05 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.46223	−1.93119	−0.965595	−	0.260049i	$-0.916261\pi$
$2$	−5.46223	−1.93119	−0.965595	+	0.260049i	$0.916261\pi$
$3$	9.09504	1.75034	0.875170	−	0.483815i	$-0.160749\pi$
$3$	9.09504	1.75034	0.875170	+	0.483815i	$0.160749\pi$
$4$	21.8360	2.72950
$5$	−6.15202	−0.550254	−0.275127	−	0.961408i	$-0.588720\pi$
$5$	−6.15202	−0.550254	−0.275127	+	0.961408i	$0.588720\pi$
$6$	−49.6792	−3.38024
$7$	2.59059	0.139879	0.0699394	−	0.997551i	$-0.477719\pi$
$7$	2.59059	0.139879	0.0699394	+	0.997551i	$0.477719\pi$
$8$	−75.5754	−3.33999
$9$	55.7197	2.06369
$10$	33.6038	1.06264
$11$	−20.0476	−0.549508	−0.274754	−	0.961515i	$-0.588596\pi$
$11$	−20.0476	−0.549508	−0.274754	+	0.961515i	$0.588596\pi$
$12$	198.599	4.77755
$13$	−34.1627	−0.728848	−0.364424	−	0.931233i	$-0.618734\pi$
$13$	−34.1627	−0.728848	−0.364424	+	0.931233i	$0.618734\pi$
$14$	−14.1504	−0.270132
$15$	−55.9529	−0.963131
$16$	238.122	3.72066
$17$	36.1588	0.515870	0.257935	−	0.966162i	$-0.416958\pi$
$17$	36.1588	0.515870	0.257935	+	0.966162i	$0.416958\pi$
$18$	−304.354	−3.98538
$19$	142.784	1.72405	0.862026	−	0.506864i	$-0.169195\pi$
$19$	142.784	1.72405	0.862026	+	0.506864i	$0.169195\pi$
$20$	−134.335	−1.50192
$21$	23.5615	0.244835
$22$	109.505	1.06121
$23$	−132.818	−1.20410	−0.602052	−	0.798457i	$-0.705650\pi$
$23$	−132.818	−1.20410	−0.602052	+	0.798457i	$0.705650\pi$
$24$	−687.361	−5.84612
$25$	−87.1526	−0.697221
$26$	186.604	1.40754
$27$	261.207	1.86182
$28$	56.5681	0.381799
$29$	−49.6805	−0.318118	−0.159059	−	0.987269i	$-0.550846\pi$
$29$	−49.6805	−0.318118	−0.159059	+	0.987269i	$0.550846\pi$
$30$	305.628	1.85999
$31$	−51.3202	−0.297335	−0.148667	−	0.988887i	$-0.547498\pi$
$31$	−51.3202	−0.297335	−0.148667	+	0.988887i	$0.547498\pi$
$32$	−696.077	−3.84532
$33$	−182.334	−0.961826
$34$	−197.508	−0.996244
$35$	−15.9374	−0.0769688
$36$	1216.69	5.63284
$37$	−49.1882	−0.218554	−0.109277	−	0.994011i	$-0.534853\pi$
$37$	−49.1882	−0.218554	−0.109277	+	0.994011i	$0.534853\pi$
$38$	−779.922	−3.32947
$39$	−310.711	−1.27573
$40$	464.941	1.83784
$41$	71.4324	0.272094	0.136047	−	0.990702i	$-0.456560\pi$
$41$	71.4324	0.272094	0.136047	+	0.990702i	$0.456560\pi$
$42$	−128.698	−0.472824
$43$	0	0
$43$	0	0
$44$	−437.760	−1.49988
$45$	−342.789	−1.13555
$46$	725.481	2.32536
$47$	−371.774	−1.15380	−0.576902	−	0.816813i	$-0.695739\pi$
$47$	−371.774	−1.15380	−0.576902	+	0.816813i	$0.695739\pi$
$48$	2165.73	6.51243
$49$	−336.289	−0.980434
$50$	476.048	1.34647
$51$	328.865	0.902948
$52$	−745.976	−1.98939
$53$	260.514	0.675177	0.337588	−	0.941294i	$-0.390389\pi$
$53$	260.514	0.675177	0.337588	+	0.941294i	$0.390389\pi$
$54$	−1426.77	−3.59554
$55$	123.333	0.302369
$56$	−195.785	−0.467194
$57$	1298.63	3.01768
$58$	271.366	0.614347
$59$	29.4340	0.0649489	0.0324745	−	0.999473i	$-0.489661\pi$
$59$	29.4340	0.0649489	0.0324745	+	0.999473i	$0.489661\pi$
$60$	−1221.79	−2.62886
$61$	−461.029	−0.967683	−0.483842	−	0.875156i	$-0.660759\pi$
$61$	−461.029	−0.967683	−0.483842	+	0.875156i	$0.660759\pi$
$62$	280.323	0.574210
$63$	144.347	0.288667
$64$	1897.16	3.70538
$65$	210.169	0.401051
$66$	995.951	1.85747
$67$	891.001	1.62467	0.812336	−	0.583189i	$-0.198195\pi$
$67$	891.001	1.62467	0.812336	+	0.583189i	$0.198195\pi$
$68$	789.563	1.40807
$69$	−1207.98	−2.10759
$70$	87.0536	0.148641
$71$	475.538	0.794873	0.397436	−	0.917630i	$-0.369900\pi$
$71$	475.538	0.794873	0.397436	+	0.917630i	$0.369900\pi$
$72$	−4211.04	−6.89271
$73$	1027.68	1.64768	0.823838	−	0.566826i	$-0.191829\pi$
$73$	1027.68	1.64768	0.823838	+	0.566826i	$0.191829\pi$
$74$	268.677	0.422069
$75$	−792.656	−1.22037
$76$	3117.84	4.70580
$77$	−51.9352	−0.0768645
$78$	1697.17	2.46368
$79$	−307.975	−0.438607	−0.219303	−	0.975657i	$-0.570378\pi$
$79$	−307.975	−0.438607	−0.219303	+	0.975657i	$0.570378\pi$
$80$	−1464.93	−2.04731
$81$	871.251	1.19513
$82$	−390.181	−0.525466
$83$	−1145.51	−1.51490	−0.757448	−	0.652895i	$-0.773554\pi$
$83$	−1145.51	−1.51490	−0.757448	+	0.652895i	$0.773554\pi$
$84$	514.489	0.668278
$85$	−222.450	−0.283859
$86$	0	0
$87$	−451.846	−0.556815
$88$	1515.11	1.83535
$89$	−670.531	−0.798608	−0.399304	−	0.916819i	$-0.630748\pi$
$89$	−670.531	−0.798608	−0.399304	+	0.916819i	$0.630748\pi$
$90$	1872.39	2.19297
$91$	−88.5015	−0.101950
$92$	−2900.20	−3.28660
$93$	−466.759	−0.520437
$94$	2030.72	2.22822
$95$	−878.413	−0.948666
$96$	−6330.85	−6.73062
$97$	−125.357	−0.131217	−0.0656085	−	0.997845i	$-0.520899\pi$
$97$	−125.357	−0.131217	−0.0656085	+	0.997845i	$0.520899\pi$
$98$	1836.89	1.89341
$99$	−1117.05	−1.13402
$100$	−1903.06	−1.90306
$101$	−1452.64	−1.43112	−0.715558	−	0.698554i	$-0.753827\pi$
$101$	−1452.64	−1.43112	−0.715558	+	0.698554i	$0.753827\pi$
$102$	−1796.34	−1.74377
$103$	840.149	0.803712	0.401856	−	0.915703i	$-0.368365\pi$
$103$	840.149	0.803712	0.401856	+	0.915703i	$0.368365\pi$
$104$	2581.86	2.43434
$105$	−144.951	−0.134722
$106$	−1422.99	−1.30390
$107$	−932.647	−0.842639	−0.421320	−	0.906912i	$-0.638433\pi$
$107$	−932.647	−0.842639	−0.421320	+	0.906912i	$0.638433\pi$
$108$	5703.70	5.08184
$109$	−74.7800	−0.0657122	−0.0328561	−	0.999460i	$-0.510460\pi$
$109$	−74.7800	−0.0657122	−0.0328561	+	0.999460i	$0.510460\pi$
$110$	−673.676	−0.583932
$111$	−447.368	−0.382543
$112$	616.878	0.520442
$113$	−2118.16	−1.76336	−0.881680	−	0.471849i	$-0.843587\pi$
$113$	−2118.16	−1.76336	−0.881680	+	0.471849i	$0.843587\pi$
$114$	−7093.42	−5.82771
$115$	817.097	0.662563
$116$	−1084.82	−0.868303
$117$	−1903.53	−1.50412
$118$	−160.776	−0.125429
$119$	93.6726	0.0721593
$120$	4228.66	3.21685
$121$	−929.092	−0.698041
$122$	2518.25	1.86878
$123$	649.680	0.476258
$124$	−1120.63	−0.811575
$125$	1305.17	0.933902
$126$	−788.456	−0.557470
$127$	−2054.14	−1.43524	−0.717619	−	0.696436i	$-0.754768\pi$
$127$	−2054.14	−1.43524	−0.717619	+	0.696436i	$0.754768\pi$
$128$	−4794.09	−3.31048
$129$	0	0
$130$	−1147.99	−0.774506
$131$	913.318	0.609137	0.304569	−	0.952490i	$-0.401488\pi$
$131$	913.318	0.609137	0.304569	+	0.952490i	$0.401488\pi$
$132$	−3981.44	−2.62530
$133$	369.896	0.241158
$134$	−4866.85	−3.13755
$135$	−1606.95	−1.02447
$136$	−2732.71	−1.72300
$137$	585.919	0.365390	0.182695	−	0.983170i	$-0.441518\pi$
$137$	585.919	0.365390	0.182695	+	0.983170i	$0.441518\pi$
$138$	6598.28	4.07016
$139$	−670.674	−0.409250	−0.204625	−	0.978840i	$-0.565598\pi$
$139$	−670.674	−0.409250	−0.204625	+	0.978840i	$0.565598\pi$
$140$	−348.008	−0.210086
$141$	−3381.30	−2.01955
$142$	−2597.50	−1.53505
$143$	684.881	0.400508
$144$	13268.1	7.67830
$145$	305.635	0.175046
$146$	−5613.40	−3.18198
$147$	−3058.56	−1.71609
$148$	−1074.07	−0.596542
$149$	−1919.37	−1.05531	−0.527653	−	0.849460i	$-0.676928\pi$
$149$	−1919.37	−1.05531	−0.527653	+	0.849460i	$0.676928\pi$
$150$	4329.67	2.35678
$151$	−2246.31	−1.21061	−0.605304	−	0.795994i	$-0.706948\pi$
$151$	−2246.31	−1.21061	−0.605304	+	0.795994i	$0.706948\pi$
$152$	−10791.0	−5.75832
$153$	2014.76	1.06460
$154$	283.682	0.148440
$155$	315.723	0.163610
$156$	−6784.67	−3.48211
$157$	1805.52	0.917808	0.458904	−	0.888486i	$-0.348242\pi$
$157$	1805.52	0.917808	0.458904	+	0.888486i	$0.348242\pi$
$158$	1682.23	0.847034
$159$	2369.39	1.18179
$160$	4282.28	2.11590
$161$	−344.076	−0.168429
$162$	−4758.98	−2.30803
$163$	3452.86	1.65920	0.829598	−	0.558361i	$-0.188570\pi$
$163$	3452.86	1.65920	0.829598	+	0.558361i	$0.188570\pi$
$164$	1559.80	0.742681
$165$	1121.72	0.529248
$166$	6257.06	2.92555
$167$	2197.72	1.01835	0.509175	−	0.860663i	$-0.329951\pi$
$167$	2197.72	1.01835	0.509175	+	0.860663i	$0.329951\pi$
$168$	−1780.67	−0.817748
$169$	−1029.91	−0.468781
$170$	1215.07	0.548187
$171$	7955.90	3.55791
$172$	0	0
$173$	−109.547	−0.0481427	−0.0240714	−	0.999710i	$-0.507663\pi$
$173$	−109.547	−0.0481427	−0.0240714	+	0.999710i	$0.507663\pi$
$174$	2468.09	1.07532
$175$	−225.777	−0.0975264
$176$	−4773.79	−2.04453
$177$	267.704	0.113683
$178$	3662.60	1.54227
$179$	2102.20	0.877799	0.438899	−	0.898536i	$-0.355368\pi$
$179$	2102.20	0.877799	0.438899	+	0.898536i	$0.355368\pi$
$180$	−7485.13	−3.09949
$181$	−2790.51	−1.14595	−0.572975	−	0.819573i	$-0.694211\pi$
$181$	−2790.51	−1.14595	−0.572975	+	0.819573i	$0.694211\pi$
$182$	483.416	0.196885
$183$	−4193.07	−1.69377
$184$	10037.7	4.02170
$185$	302.607	0.120260
$186$	2549.55	1.00506
$187$	−724.898	−0.283475
$188$	−8118.05	−3.14931
$189$	676.679	0.260429
$190$	4798.10	1.83206
$191$	−3833.56	−1.45229	−0.726144	−	0.687543i	$-0.758689\pi$
$191$	−3833.56	−1.45229	−0.726144	+	0.687543i	$0.758689\pi$
$192$	17254.7	6.48568
$193$	3873.73	1.44475	0.722375	−	0.691501i	$-0.243050\pi$
$193$	3873.73	1.44475	0.722375	+	0.691501i	$0.243050\pi$
$194$	684.728	0.253405
$195$	1911.50	0.701976
$196$	−7343.20	−2.67609
$197$	−2456.45	−0.888401	−0.444201	−	0.895927i	$-0.646512\pi$
$197$	−2456.45	−0.888401	−0.444201	+	0.895927i	$0.646512\pi$
$198$	6101.58	2.19000
$199$	162.945	0.0580446	0.0290223	−	0.999579i	$-0.490761\pi$
$199$	162.945	0.0580446	0.0290223	+	0.999579i	$0.490761\pi$
$200$	6586.59	2.32871
$201$	8103.68	2.84373
$202$	7934.64	2.76376
$203$	−128.702	−0.0444980
$204$	7181.10	2.46460
$205$	−439.454	−0.149721
$206$	−4589.09	−1.55212
$207$	−7400.56	−2.48490
$208$	−8134.90	−2.71180
$209$	−2862.49	−0.947381
$210$	791.756	0.260173
$211$	−975.959	−0.318426	−0.159213	−	0.987244i	$-0.550896\pi$
$211$	−975.959	−0.318426	−0.159213	+	0.987244i	$0.550896\pi$
$212$	5688.58	1.84289
$213$	4325.03	1.39130
$214$	5094.34	1.62730
$215$	0	0
$216$	−19740.8	−6.21847
$217$	−132.950	−0.0415908
$218$	408.466	0.126903
$219$	9346.74	2.88399
$220$	2693.11	0.825315
$221$	−1235.28	−0.375991
$222$	2443.63	0.738764
$223$	5259.83	1.57948	0.789740	−	0.613441i	$-0.210215\pi$
$223$	5259.83	1.57948	0.789740	+	0.613441i	$0.210215\pi$
$224$	−1803.25	−0.537878
$225$	−4856.12	−1.43885
$226$	11569.9	3.40538
$227$	−2907.49	−0.850119	−0.425060	−	0.905165i	$-0.639747\pi$
$227$	−2907.49	−0.850119	−0.425060	+	0.905165i	$0.639747\pi$
$228$	28356.9	8.23675
$229$	1315.09	0.379492	0.189746	−	0.981833i	$-0.439234\pi$
$229$	1315.09	0.379492	0.189746	+	0.981833i	$0.439234\pi$
$230$	−4463.17	−1.27953
$231$	−472.353	−0.134539
$232$	3754.62	1.06251
$233$	−1070.36	−0.300952	−0.150476	−	0.988614i	$-0.548081\pi$
$233$	−1070.36	−0.300952	−0.150476	+	0.988614i	$0.548081\pi$
$234$	10397.5	2.90474
$235$	2287.16	0.634885
$236$	642.721	0.177278
$237$	−2801.05	−0.767712
$238$	−511.661	−0.139353
$239$	−3598.65	−0.973964	−0.486982	−	0.873412i	$-0.661902\pi$
$239$	−3598.65	−0.973964	−0.486982	+	0.873412i	$0.661902\pi$
$240$	−13323.6	−3.58349
$241$	−4241.99	−1.13382	−0.566910	−	0.823780i	$-0.691861\pi$
$241$	−4241.99	−1.13382	−0.566910	+	0.823780i	$0.691861\pi$
$242$	5074.92	1.34805
$243$	871.487	0.230065
$244$	−10067.0	−2.64129
$245$	2068.86	0.539487
$246$	−3548.71	−0.919745
$247$	−4877.90	−1.25657
$248$	3878.54	0.993096
$249$	−10418.5	−2.65158
$250$	−7129.13	−1.80354
$251$	−2419.46	−0.608425	−0.304212	−	0.952604i	$-0.598393\pi$
$251$	−2419.46	−0.608425	−0.304212	+	0.952604i	$0.598393\pi$
$252$	3151.96	0.787915
$253$	2662.68	0.661665
$254$	11220.2	2.77172
$255$	−2023.19	−0.496851
$256$	11009.2	2.68779
$257$	2727.33	0.661969	0.330984	−	0.943636i	$-0.392619\pi$
$257$	2727.33	0.661969	0.330984	+	0.943636i	$0.392619\pi$
$258$	0	0
$259$	−127.426	−0.0305710
$260$	4589.26	1.09467
$261$	−2768.18	−0.656498
$262$	−4988.76	−1.17636
$263$	−1524.72	−0.357485	−0.178742	−	0.983896i	$-0.557203\pi$
$263$	−1524.72	−0.357485	−0.178742	+	0.983896i	$0.557203\pi$
$264$	13780.0	3.21249
$265$	−1602.69	−0.371518
$266$	−2020.46	−0.465723
$267$	−6098.50	−1.39784
$268$	19455.9	4.43454
$269$	−1365.12	−0.309415	−0.154707	−	0.987960i	$-0.549444\pi$
$269$	−1365.12	−0.309415	−0.154707	+	0.987960i	$0.549444\pi$
$270$	8777.53	1.97846
$271$	−8496.07	−1.90443	−0.952213	−	0.305434i	$-0.901198\pi$
$271$	−8496.07	−1.90443	−0.952213	+	0.305434i	$0.901198\pi$
$272$	8610.22	1.91938
$273$	−804.924	−0.178448
$274$	−3200.42	−0.705638
$275$	1747.20	0.383129
$276$	−26377.5	−5.75267
$277$	−85.6103	−0.0185698	−0.00928488	−	0.999957i	$-0.502956\pi$
$277$	−85.6103	−0.0185698	−0.00928488	+	0.999957i	$0.502956\pi$
$278$	3663.38	0.790340
$279$	−2859.55	−0.613607
$280$	1204.47	0.257075
$281$	−1787.07	−0.379387	−0.189693	−	0.981843i	$-0.560749\pi$
$281$	−1787.07	−0.379387	−0.189693	+	0.981843i	$0.560749\pi$
$282$	18469.4	3.90014
$283$	4265.75	0.896015	0.448008	−	0.894030i	$-0.352134\pi$
$283$	4265.75	0.896015	0.448008	+	0.894030i	$0.352134\pi$
$284$	10383.8	2.16960
$285$	−7989.20	−1.66049
$286$	−3740.98	−0.773457
$287$	185.052	0.0380602
$288$	−38785.2	−7.93555
$289$	−3605.54	−0.733878
$290$	−1669.45	−0.338047
$291$	−1140.12	−0.229674
$292$	22440.3	4.49733
$293$	7693.93	1.53408	0.767038	−	0.641601i	$-0.221730\pi$
$293$	7693.93	1.53408	0.767038	+	0.641601i	$0.221730\pi$
$294$	16706.6	3.31410
$295$	−181.079	−0.0357384
$296$	3717.41	0.729967
$297$	−5236.57	−1.02309
$298$	10484.0	2.03800
$299$	4537.41	0.877608
$300$	−17308.4	−3.33101
$301$	0	0
$302$	12269.8	2.33791
$303$	−13211.8	−2.50494
$304$	34000.2	6.41462
$305$	2836.26	0.532471
$306$	−11005.1	−2.05594
$307$	−2173.11	−0.403993	−0.201996	−	0.979386i	$-0.564743\pi$
$307$	−2173.11	−0.403993	−0.201996	+	0.979386i	$0.564743\pi$
$308$	−1134.06	−0.209801
$309$	7641.19	1.40677
$310$	−1724.55	−0.315961
$311$	9690.39	1.76685	0.883427	−	0.468568i	$-0.155230\pi$
$311$	9690.39	1.76685	0.883427	+	0.468568i	$0.155230\pi$
$312$	23482.1	4.26093
$313$	2920.33	0.527369	0.263685	−	0.964609i	$-0.415062\pi$
$313$	2920.33	0.527369	0.263685	+	0.964609i	$0.415062\pi$
$314$	−9862.15	−1.77246
$315$	−888.025	−0.158840
$316$	−6724.95	−1.19718
$317$	650.198	0.115201	0.0576006	−	0.998340i	$-0.481655\pi$
$317$	650.198	0.115201	0.0576006	+	0.998340i	$0.481655\pi$
$318$	−12942.1	−2.28226
$319$	995.976	0.174809
$320$	−11671.3	−2.03890
$321$	−8482.46	−1.47491
$322$	1879.42	0.325268
$323$	5162.91	0.889387
$324$	19024.6	3.26211
$325$	2977.37	0.508168
$326$	−18860.3	−3.20422
$327$	−680.127	−0.115019
$328$	−5398.53	−0.908793
$329$	−963.114	−0.161393
$330$	−6127.11	−1.02208
$331$	−2920.97	−0.485048	−0.242524	−	0.970145i	$-0.577975\pi$
$331$	−2920.97	−0.485048	−0.242524	+	0.970145i	$0.577975\pi$
$332$	−25013.4	−4.13491
$333$	−2740.75	−0.451027
$334$	−12004.4	−1.96663
$335$	−5481.46	−0.893982
$336$	5610.52	0.910950
$337$	−2962.81	−0.478916	−0.239458	−	0.970907i	$-0.576970\pi$
$337$	−2962.81	−0.478916	−0.239458	+	0.970907i	$0.576970\pi$
$338$	5625.62	0.905306
$339$	−19264.7	−3.08648
$340$	−4857.41	−0.774794
$341$	1028.85	0.163388
$342$	−43457.0	−6.87101
$343$	−1759.76	−0.277021
$344$	0	0
$345$	7431.53	1.15971
$346$	598.370	0.0929728
$347$	−7999.88	−1.23763	−0.618813	−	0.785538i	$-0.712386\pi$
$347$	−7999.88	−1.23763	−0.618813	+	0.785538i	$0.712386\pi$
$348$	−9866.49	−1.51983
$349$	−5314.28	−0.815091	−0.407546	−	0.913185i	$-0.633615\pi$
$349$	−5314.28	−0.815091	−0.407546	+	0.913185i	$0.633615\pi$
$350$	1233.25	0.188342
$351$	−8923.51	−1.35699
$352$	13954.7	2.11303
$353$	2313.24	0.348786	0.174393	−	0.984676i	$-0.444204\pi$
$353$	2313.24	0.348786	0.174393	+	0.984676i	$0.444204\pi$
$354$	−1462.26	−0.219543
$355$	−2925.52	−0.437382
$356$	−14641.7	−2.17980
$357$	851.955	0.126303
$358$	−11482.7	−1.69520
$359$	3108.14	0.456940	0.228470	−	0.973551i	$-0.426628\pi$
$359$	3108.14	0.456940	0.228470	+	0.973551i	$0.426628\pi$
$360$	25906.4	3.79274
$361$	13528.4	1.97236
$362$	15242.4	2.21305
$363$	−8450.13	−1.22181
$364$	−1932.52	−0.278273
$365$	−6322.28	−0.906639
$366$	22903.5	3.27100
$367$	645.178	0.0917657	0.0458828	−	0.998947i	$-0.485390\pi$
$367$	645.178	0.0917657	0.0458828	+	0.998947i	$0.485390\pi$
$368$	−31626.9	−4.48007
$369$	3980.19	0.561519
$370$	−1652.91	−0.232245
$371$	674.885	0.0944429
$372$	−10192.1	−1.42053
$373$	−11994.3	−1.66498	−0.832492	−	0.554037i	$-0.813087\pi$
$373$	−11994.3	−1.66498	−0.832492	+	0.554037i	$0.813087\pi$
$374$	3959.56	0.547444
$375$	11870.5	1.63465
$376$	28097.0	3.85370
$377$	1697.22	0.231860
$378$	−3696.18	−0.502939
$379$	−9712.13	−1.31630	−0.658151	−	0.752886i	$-0.728661\pi$
$379$	−9712.13	−1.31630	−0.658151	+	0.752886i	$0.728661\pi$
$380$	−19181.0	−2.58938
$381$	−18682.5	−2.51215
$382$	20939.8	2.80464
$383$	−11940.6	−1.59305	−0.796525	−	0.604605i	$-0.793331\pi$
$383$	−11940.6	−1.59305	−0.796525	+	0.604605i	$0.793331\pi$
$384$	−43602.4	−5.79447
$385$	319.506	0.0422950
$386$	−21159.2	−2.79009
$387$	0	0
$388$	−2737.29	−0.358157
$389$	2644.94	0.344740	0.172370	−	0.985032i	$-0.444857\pi$
$389$	2644.94	0.344740	0.172370	+	0.985032i	$0.444857\pi$
$390$	−10441.1	−1.35565
$391$	−4802.52	−0.621161
$392$	25415.2	3.27464
$393$	8306.66	1.06620
$394$	13417.7	1.71567
$395$	1894.67	0.241345
$396$	−24391.8	−3.09529
$397$	−8950.96	−1.13158	−0.565788	−	0.824551i	$-0.691428\pi$
$397$	−8950.96	−1.13158	−0.565788	+	0.824551i	$0.691428\pi$
$398$	−890.044	−0.112095
$399$	3364.22	0.422109
$400$	−20753.0	−2.59412
$401$	14096.5	1.75547	0.877736	−	0.479145i	$-0.159053\pi$
$401$	14096.5	1.75547	0.877736	+	0.479145i	$0.159053\pi$
$402$	−44264.2	−5.49179
$403$	1753.23	0.216712
$404$	−31719.7	−3.90623
$405$	−5359.96	−0.657626
$406$	702.999	0.0859341
$407$	986.106	0.120097
$408$	−24854.1	−3.01584
$409$	8473.30	1.02440	0.512198	−	0.858868i	$-0.328831\pi$
$409$	8473.30	1.02440	0.512198	+	0.858868i	$0.328831\pi$
$410$	2400.40	0.289140
$411$	5328.95	0.639557
$412$	18345.5	2.19373
$413$	76.2515	0.00908497
$414$	40423.6	4.79882
$415$	7047.22	0.833577
$416$	23779.8	2.80265
$417$	−6099.80	−0.716327
$418$	15635.6	1.82957
$419$	−500.922	−0.0584048	−0.0292024	−	0.999574i	$-0.509297\pi$
$419$	−500.922	−0.0584048	−0.0292024	+	0.999574i	$0.509297\pi$
$420$	−3165.15	−0.367722
$421$	−13129.3	−1.51991	−0.759955	−	0.649976i	$-0.774779\pi$
$421$	−13129.3	−1.51991	−0.759955	+	0.649976i	$0.774779\pi$
$422$	5330.92	0.614941
$423$	−20715.1	−2.38110
$424$	−19688.5	−2.25508
$425$	−3151.33	−0.359676
$426$	−23624.3	−2.68686
$427$	−1194.34	−0.135358
$428$	−20365.3	−2.29998
$429$	6229.01	0.701025
$430$	0	0
$431$	4610.52	0.515269	0.257635	−	0.966242i	$-0.417057\pi$
$431$	4610.52	0.515269	0.257635	+	0.966242i	$0.417057\pi$
$432$	62199.1	6.92722
$433$	11366.8	1.26155	0.630777	−	0.775964i	$-0.282736\pi$
$433$	11366.8	1.26155	0.630777	+	0.775964i	$0.282736\pi$
$434$	726.202	0.0803198
$435$	2779.76	0.306390
$436$	−1632.90	−0.179361
$437$	−18964.3	−2.07594
$438$	−51054.1	−5.56954
$439$	4689.83	0.509871	0.254935	−	0.966958i	$-0.417946\pi$
$439$	4689.83	0.509871	0.254935	+	0.966958i	$0.417946\pi$
$440$	−9320.98	−1.00991
$441$	−18737.9	−2.02331
$442$	6747.39	0.726110
$443$	−1822.08	−0.195417	−0.0977086	−	0.995215i	$-0.531151\pi$
$443$	−1822.08	−0.195417	−0.0977086	+	0.995215i	$0.531151\pi$
$444$	−9768.72	−1.04415
$445$	4125.12	0.439437
$446$	−28730.4	−3.05028
$447$	−17456.7	−1.84715
$448$	4914.75	0.518304
$449$	−12923.6	−1.35836	−0.679180	−	0.733972i	$-0.737664\pi$
$449$	−12923.6	−1.35836	−0.679180	+	0.733972i	$0.737664\pi$
$450$	26525.2	2.77869
$451$	−1432.05	−0.149518
$452$	−46252.1	−4.81309
$453$	−20430.2	−2.11898
$454$	15881.4	1.64174
$455$	544.463	0.0560985
$456$	−98144.4	−10.0790
$457$	−1087.83	−0.111349	−0.0556743	−	0.998449i	$-0.517731\pi$
$457$	−1087.83	−0.111349	−0.0556743	+	0.998449i	$0.517731\pi$
$458$	−7183.33	−0.732871
$459$	9444.91	0.960459
$460$	17842.1	1.80846
$461$	−1854.32	−0.187341	−0.0936707	−	0.995603i	$-0.529860\pi$
$461$	−1854.32	−0.187341	−0.0936707	+	0.995603i	$0.529860\pi$
$462$	2580.10	0.259821
$463$	19191.6	1.92637	0.963186	−	0.268834i	$-0.0866384\pi$
$463$	19191.6	1.92637	0.963186	+	0.268834i	$0.0866384\pi$
$464$	−11830.0	−1.18361
$465$	2871.51	0.286372
$466$	5846.58	0.581196
$467$	3701.95	0.366822	0.183411	−	0.983036i	$-0.441286\pi$
$467$	3701.95	0.366822	0.183411	+	0.983036i	$0.441286\pi$
$468$	−41565.5	−4.10548
$469$	2308.22	0.227257
$470$	−12493.0	−1.22608
$471$	16421.2	1.60648
$472$	−2224.49	−0.216929
$473$	0	0
$474$	15300.0	1.48260
$475$	−12444.0	−1.20205
$476$	2045.43	0.196959
$477$	14515.8	1.39336
$478$	19656.7	1.88091
$479$	8304.65	0.792169	0.396085	−	0.918214i	$-0.370369\pi$
$479$	8304.65	0.792169	0.396085	+	0.918214i	$0.370369\pi$
$480$	38947.5	3.70355
$481$	1680.40	0.159292
$482$	23170.7	2.18962
$483$	−3129.38	−0.294807
$484$	−20287.6	−1.90530
$485$	771.197	0.0722026
$486$	−4760.26	−0.444300
$487$	−15674.8	−1.45851	−0.729254	−	0.684243i	$-0.760133\pi$
$487$	−15674.8	−1.45851	−0.729254	+	0.684243i	$0.760133\pi$
$488$	34842.4	3.23205
$489$	31403.9	2.90416
$490$	−11300.6	−1.04185
$491$	−13069.6	−1.20127	−0.600636	−	0.799522i	$-0.705086\pi$
$491$	−13069.6	−1.20127	−0.600636	+	0.799522i	$0.705086\pi$
$492$	14186.4	1.29995
$493$	−1796.38	−0.164108
$494$	26644.2	2.42668
$495$	6872.10	0.623996
$496$	−12220.5	−1.10628
$497$	1231.92	0.111186
$498$	56908.2	5.12072
$499$	9756.69	0.875289	0.437645	−	0.899148i	$-0.355813\pi$
$499$	9756.69	0.875289	0.437645	+	0.899148i	$0.355813\pi$
$500$	28499.6	2.54908
$501$	19988.3	1.78246
$502$	13215.6	1.17498
$503$	−14365.4	−1.27340	−0.636700	−	0.771112i	$-0.719701\pi$
$503$	−14365.4	−1.27340	−0.636700	+	0.771112i	$0.719701\pi$
$504$	−10909.1	−0.964144
$505$	8936.65	0.787476
$506$	−14544.2	−1.27780
$507$	−9367.09	−0.820527
$508$	−44854.1	−3.91748
$509$	−13972.3	−1.21672	−0.608360	−	0.793662i	$-0.708172\pi$
$509$	−13972.3	−1.21672	−0.608360	+	0.793662i	$0.708172\pi$
$510$	11051.1	0.959513
$511$	2662.29	0.230475
$512$	−21782.0	−1.88015
$513$	37296.2	3.20988
$514$	−14897.3	−1.27839
$515$	−5168.62	−0.442246
$516$	0	0
$517$	7453.19	0.634025
$518$	696.032	0.0590384
$519$	−996.332	−0.0842662
$520$	−15883.6	−1.33951
$521$	13785.0	1.15918	0.579588	−	0.814910i	$-0.303213\pi$
$521$	13785.0	1.15918	0.579588	+	0.814910i	$0.303213\pi$
$522$	15120.4	1.26782
$523$	−8431.00	−0.704898	−0.352449	−	0.935831i	$-0.614651\pi$
$523$	−8431.00	−0.704898	−0.352449	+	0.935831i	$0.614651\pi$
$524$	19943.2	1.66264
$525$	−2053.45	−0.170704
$526$	8328.40	0.690372
$527$	−1855.68	−0.153386
$528$	−43417.8	−3.57863
$529$	5473.53	0.449867
$530$	8754.26	0.717473
$531$	1640.06	0.134035
$532$	8077.04	0.658241
$533$	−2440.32	−0.198315
$534$	33311.4	2.69949
$535$	5737.67	0.463665
$536$	−67337.7	−5.42639
$537$	19119.6	1.53645
$538$	7456.58	0.597539
$539$	6741.80	0.538756
$540$	−35089.3	−2.79630
$541$	375.733	0.0298596	0.0149298	−	0.999889i	$-0.495248\pi$
$541$	375.733	0.0298596	0.0149298	+	0.999889i	$0.495248\pi$
$542$	46407.5	3.67781
$543$	−25379.8	−2.00580
$544$	−25169.3	−1.98369
$545$	460.048	0.0361584
$546$	4396.68	0.344617
$547$	9627.22	0.752523	0.376261	−	0.926514i	$-0.377209\pi$
$547$	9627.22	0.752523	0.376261	+	0.926514i	$0.377209\pi$
$548$	12794.1	0.997331
$549$	−25688.4	−1.99700
$550$	−9543.64	−0.739894
$551$	−7093.60	−0.548453
$552$	91293.7	7.03934
$553$	−797.838	−0.0613518
$554$	467.623	0.0358618
$555$	2752.22	0.210496
$556$	−14644.8	−1.11705
$557$	−13673.1	−1.04012	−0.520061	−	0.854129i	$-0.674091\pi$
$557$	−13673.1	−1.04012	−0.520061	+	0.854129i	$0.674091\pi$
$558$	15619.5	1.18499
$559$	0	0
$560$	−3795.04	−0.286375
$561$	−6592.97	−0.496177
$562$	9761.39	0.732668
$563$	−5880.29	−0.440186	−0.220093	−	0.975479i	$-0.570636\pi$
$563$	−5880.29	−0.440186	−0.220093	+	0.975479i	$0.570636\pi$
$564$	−73834.0	−5.51236
$565$	13031.0	0.970295
$566$	−23300.5	−1.73038
$567$	2257.06	0.167174
$568$	−35939.0	−2.65487
$569$	−14311.3	−1.05441	−0.527207	−	0.849737i	$-0.676761\pi$
$569$	−14311.3	−1.05441	−0.527207	+	0.849737i	$0.676761\pi$
$570$	43638.9	3.20672
$571$	875.082	0.0641349	0.0320675	−	0.999486i	$-0.489791\pi$
$571$	875.082	0.0641349	0.0320675	+	0.999486i	$0.489791\pi$
$572$	14955.0	1.09318
$573$	−34866.4	−2.54200
$574$	−1010.80	−0.0735015
$575$	11575.4	0.839527
$576$	105709.	7.64677
$577$	1692.97	0.122147	0.0610737	−	0.998133i	$-0.480548\pi$
$577$	1692.97	0.122147	0.0610737	+	0.998133i	$0.480548\pi$
$578$	19694.3	1.41726
$579$	35231.7	2.52881
$580$	6673.85	0.477787
$581$	−2967.55	−0.211902
$582$	6227.62	0.443545
$583$	−5222.69	−0.371015
$584$	−77667.0	−5.50322
$585$	11710.6	0.827646
$586$	−42026.1	−2.96259
$587$	9558.71	0.672112	0.336056	−	0.941842i	$-0.390907\pi$
$587$	9558.71	0.672112	0.336056	+	0.941842i	$0.390907\pi$
$588$	−66786.7	−4.68407
$589$	−7327.73	−0.512621
$590$	989.095	0.0690176
$591$	−22341.5	−1.55501
$592$	−11712.8	−0.813164
$593$	380.686	0.0263624	0.0131812	−	0.999913i	$-0.495804\pi$
$593$	380.686	0.0263624	0.0131812	+	0.999913i	$0.495804\pi$
$594$	28603.4	1.97578
$595$	−576.276	−0.0397059
$596$	−41911.3	−2.88046
$597$	1481.99	0.101598
$598$	−24784.4	−1.69483
$599$	18267.5	1.24606	0.623031	−	0.782197i	$-0.285901\pi$
$599$	18267.5	1.24606	0.623031	+	0.782197i	$0.285901\pi$
$600$	59905.3	4.07604
$601$	3526.45	0.239346	0.119673	−	0.992813i	$-0.461815\pi$
$601$	3526.45	0.239346	0.119673	+	0.992813i	$0.461815\pi$
$602$	0	0
$603$	49646.3	3.35282
$604$	−49050.3	−3.30435
$605$	5715.80	0.384099
$606$	72165.8	4.83752
$607$	−1688.55	−0.112910	−0.0564548	−	0.998405i	$-0.517980\pi$
$607$	−1688.55	−0.112910	−0.0564548	+	0.998405i	$0.517980\pi$
$608$	−99389.0	−6.62953
$609$	−1170.55	−0.0778866
$610$	−15492.3	−1.02830
$611$	12700.8	0.840948
$612$	43994.2	2.90582
$613$	19844.4	1.30752	0.653759	−	0.756703i	$-0.273191\pi$
$613$	19844.4	1.30752	0.653759	+	0.756703i	$0.273191\pi$
$614$	11870.0	0.780187
$615$	−3996.85	−0.262063
$616$	3925.02	0.256727
$617$	12892.5	0.841218	0.420609	−	0.907242i	$-0.361816\pi$
$617$	12892.5	0.841218	0.420609	+	0.907242i	$0.361816\pi$
$618$	−41738.0	−2.71674
$619$	−17148.5	−1.11350	−0.556751	−	0.830679i	$-0.687952\pi$
$619$	−17148.5	−1.11350	−0.556751	+	0.830679i	$0.687952\pi$
$620$	6894.12	0.446572
$621$	−34692.8	−2.24183
$622$	−52931.2	−3.41213
$623$	−1737.07	−0.111708
$624$	−73987.2	−4.74657
$625$	2864.66	0.183338
$626$	−15951.5	−1.01845
$627$	−26034.5	−1.65824
$628$	39425.2	2.50515
$629$	−1778.58	−0.112745
$630$	4850.60	0.306750
$631$	−978.279	−0.0617190	−0.0308595	−	0.999524i	$-0.509824\pi$
$631$	−978.279	−0.0617190	−0.0308595	+	0.999524i	$0.509824\pi$
$632$	23275.4	1.46494
$633$	−8876.38	−0.557353
$634$	−3551.54	−0.222476
$635$	12637.1	0.789745
$636$	51737.9	3.22569
$637$	11488.5	0.714587
$638$	−5440.25	−0.337589
$639$	26496.8	1.64037
$640$	29493.3	1.82160
$641$	12867.5	0.792880	0.396440	−	0.918061i	$-0.370246\pi$
$641$	12867.5	0.792880	0.396440	+	0.918061i	$0.370246\pi$
$642$	46333.2	2.84833
$643$	13732.1	0.842211	0.421106	−	0.907012i	$-0.361642\pi$
$643$	13732.1	0.842211	0.421106	+	0.907012i	$0.361642\pi$
$644$	−7513.24	−0.459725
$645$	0	0
$646$	−28201.0	−1.71758
$647$	2460.71	0.149522	0.0747608	−	0.997201i	$-0.476181\pi$
$647$	2460.71	0.149522	0.0747608	+	0.997201i	$0.476181\pi$
$648$	−65845.2	−3.99173
$649$	−590.083	−0.0356899
$650$	−16263.1	−0.981369
$651$	−1209.18	−0.0727981
$652$	75396.6	4.52877
$653$	−26816.4	−1.60706	−0.803528	−	0.595267i	$-0.797046\pi$
$653$	−26816.4	−1.60706	−0.803528	+	0.595267i	$0.797046\pi$
$654$	3715.01	0.222123
$655$	−5618.75	−0.335180
$656$	17009.7	1.01237
$657$	57261.7	3.40029
$658$	5260.75	0.311680
$659$	−943.089	−0.0557474	−0.0278737	−	0.999611i	$-0.508874\pi$
$659$	−943.089	−0.0557474	−0.0278737	+	0.999611i	$0.508874\pi$
$660$	24493.9	1.44458
$661$	14083.2	0.828706	0.414353	−	0.910116i	$-0.364008\pi$
$661$	14083.2	0.828706	0.414353	+	0.910116i	$0.364008\pi$
$662$	15955.0	0.936721
$663$	−11234.9	−0.658112
$664$	86572.6	5.05974
$665$	−2275.61	−0.132698
$666$	14970.6	0.871020
$667$	6598.44	0.383048
$668$	47989.3	2.77958
$669$	47838.3	2.76463
$670$	29941.0	1.72645
$671$	9242.53	0.531750
$672$	−16400.6	−0.941470
$673$	−4292.74	−0.245873	−0.122937	−	0.992415i	$-0.539231\pi$
$673$	−4292.74	−0.245873	−0.122937	+	0.992415i	$0.539231\pi$
$674$	16183.6	0.924878
$675$	−22764.8	−1.29810
$676$	−22489.2	−1.27954
$677$	−19977.3	−1.13411	−0.567054	−	0.823681i	$-0.691917\pi$
$677$	−19977.3	−1.13411	−0.567054	+	0.823681i	$0.691917\pi$
$678$	105228.	5.96058
$679$	−324.748	−0.0183545
$680$	16811.7	0.948088
$681$	−26443.8	−1.48800
$682$	−5619.81	−0.315533
$683$	16248.8	0.910314	0.455157	−	0.890411i	$-0.349583\pi$
$683$	16248.8	0.910314	0.455157	+	0.890411i	$0.349583\pi$
$684$	173725.	9.71132
$685$	−3604.59	−0.201057
$686$	9612.21	0.534980
$687$	11960.8	0.664239
$688$	0	0
$689$	−8899.86	−0.492101
$690$	−40592.7	−2.23962
$691$	−27470.0	−1.51231	−0.756155	−	0.654392i	$-0.772925\pi$
$691$	−27470.0	−1.51231	−0.756155	+	0.654392i	$0.772925\pi$
$692$	−2392.06	−0.131405
$693$	−2893.81	−0.158625
$694$	43697.2	2.39009
$695$	4126.00	0.225191
$696$	34148.4	1.85976
$697$	2582.91	0.140365
$698$	29027.8	1.57410
$699$	−9735.00	−0.526769
$700$	−4930.06	−0.266198
$701$	−5373.47	−0.289519	−0.144760	−	0.989467i	$-0.546241\pi$
$701$	−5373.47	−0.289519	−0.144760	+	0.989467i	$0.546241\pi$
$702$	48742.3	2.62060
$703$	−7023.30	−0.376798
$704$	−38033.5	−2.03614
$705$	20801.8	1.11126
$706$	−12635.5	−0.673573
$707$	−3763.18	−0.200183
$708$	5845.57	0.310297
$709$	22366.6	1.18476	0.592380	−	0.805658i	$-0.298188\pi$
$709$	22366.6	1.18476	0.592380	+	0.805658i	$0.298188\pi$
$710$	15979.9	0.844667
$711$	−17160.3	−0.905150
$712$	50675.6	2.66735
$713$	6816.23	0.358022
$714$	−4653.58	−0.243916
$715$	−4213.40	−0.220381
$716$	45903.6	2.39595
$717$	−32729.8	−1.70477
$718$	−16977.4	−0.882439
$719$	−19417.5	−1.00716	−0.503582	−	0.863947i	$-0.667985\pi$
$719$	−19417.5	−1.00716	−0.503582	+	0.863947i	$0.667985\pi$
$720$	−81625.7	−4.22501
$721$	2176.48	0.112422
$722$	−73895.2	−3.80900
$723$	−38581.1	−1.98457
$724$	−60933.5	−3.12787
$725$	4329.78	0.221799
$726$	46156.6	2.35955
$727$	−5295.15	−0.270133	−0.135066	−	0.990837i	$-0.543125\pi$
$727$	−5295.15	−0.270133	−0.135066	+	0.990837i	$0.543125\pi$
$728$	6688.53	0.340513
$729$	−15597.6	−0.792439
$730$	34533.8	1.75089
$731$	0	0
$732$	−91559.9	−4.62316
$733$	−3812.57	−0.192115	−0.0960576	−	0.995376i	$-0.530623\pi$
$733$	−3812.57	−0.192115	−0.0960576	+	0.995376i	$0.530623\pi$
$734$	−3524.11	−0.177217
$735$	18816.3	0.944286
$736$	92451.3	4.63016
$737$	−17862.5	−0.892771
$738$	−21740.7	−1.08440
$739$	4166.76	0.207411	0.103706	−	0.994608i	$-0.466930\pi$
$739$	4166.76	0.207411	0.103706	+	0.994608i	$0.466930\pi$
$740$	6607.72	0.328249
$741$	−44364.6	−2.19943
$742$	−3686.38	−0.182387
$743$	−25570.4	−1.26257	−0.631284	−	0.775552i	$-0.717472\pi$
$743$	−25570.4	−1.26257	−0.631284	+	0.775552i	$0.717472\pi$
$744$	35275.5	1.73826
$745$	11808.0	0.580686
$746$	65515.4	3.21540
$747$	−63827.6	−3.12628
$748$	−15828.9	−0.773744
$749$	−2416.11	−0.117867
$750$	−64839.7	−3.15681
$751$	8697.56	0.422608	0.211304	−	0.977420i	$-0.432229\pi$
$751$	8697.56	0.422608	0.211304	+	0.977420i	$0.432229\pi$
$752$	−88527.7	−4.29292
$753$	−22005.0	−1.06495
$754$	−9270.59	−0.447765
$755$	13819.3	0.666141
$756$	14776.0	0.710842
$757$	−6943.97	−0.333399	−0.166699	−	0.986008i	$-0.553311\pi$
$757$	−6943.97	−0.333399	−0.166699	+	0.986008i	$0.553311\pi$
$758$	53049.9	2.54203
$759$	24217.2	1.15814
$760$	66386.4	3.16854
$761$	14571.8	0.694121	0.347060	−	0.937843i	$-0.387180\pi$
$761$	14571.8	0.694121	0.347060	+	0.937843i	$0.387180\pi$
$762$	102048.	4.85145
$763$	−193.724	−0.00919173
$764$	−83709.6	−3.96402
$765$	−12394.8	−0.585798
$766$	65222.6	3.07649
$767$	−1005.55	−0.0473379
$768$	100129.	4.70455
$769$	−32318.2	−1.51550	−0.757752	−	0.652543i	$-0.773702\pi$
$769$	−32318.2	−1.51550	−0.757752	+	0.652543i	$0.773702\pi$
$770$	−1745.22	−0.0816796
$771$	24805.1	1.15867
$772$	84586.6	3.94344
$773$	1785.68	0.0830874	0.0415437	−	0.999137i	$-0.486772\pi$
$773$	1785.68	0.0830874	0.0415437	+	0.999137i	$0.486772\pi$
$774$	0	0
$775$	4472.69	0.207308
$776$	9473.88	0.438264
$777$	−1158.95	−0.0535097
$778$	−14447.3	−0.665759
$779$	10199.4	0.469105
$780$	41739.5	1.91604
$781$	−9533.41	−0.436789
$782$	26232.5	1.19958
$783$	−12976.9	−0.592280
$784$	−80077.9	−3.64786
$785$	−11107.6	−0.505027
$786$	−45372.9	−2.05903
$787$	7591.08	0.343828	0.171914	−	0.985112i	$-0.445005\pi$
$787$	7591.08	0.343828	0.171914	+	0.985112i	$0.445005\pi$
$788$	−53639.1	−2.42489
$789$	−13867.4	−0.625720
$790$	−10349.1	−0.466083
$791$	−5487.28	−0.246656
$792$	84421.3	3.78760
$793$	15750.0	0.705293
$794$	48892.2	2.18529
$795$	−14576.5	−0.650284
$796$	3558.07	0.158433
$797$	−11200.4	−0.497790	−0.248895	−	0.968531i	$-0.580067\pi$
$797$	−11200.4	−0.497790	−0.248895	+	0.968531i	$0.580067\pi$
$798$	−18376.1	−0.815173
$799$	−13442.9	−0.595213
$800$	60664.9	2.68104
$801$	−37361.8	−1.64808
$802$	−76998.2	−3.39015
$803$	−20602.5	−0.905411
$804$	176952.	7.76196
$805$	2116.76	0.0926784
$806$	−9576.58	−0.418512
$807$	−12415.8	−0.541582
$808$	109784.	4.77991
$809$	−23832.0	−1.03571	−0.517854	−	0.855469i	$-0.673269\pi$
$809$	−23832.0	−1.03571	−0.517854	+	0.855469i	$0.673269\pi$
$810$	29277.3	1.27000
$811$	−23707.2	−1.02648	−0.513239	−	0.858246i	$-0.671555\pi$
$811$	−23707.2	−1.02648	−0.513239	+	0.858246i	$0.671555\pi$
$812$	−2810.33	−0.121457
$813$	−77272.1	−3.33339
$814$	−5386.34	−0.231930
$815$	−21242.1	−0.912978
$816$	78310.2	3.35957
$817$	0	0
$818$	−46283.1	−1.97830
$819$	−4931.27	−0.210394
$820$	−9595.91	−0.408663
$821$	−27578.1	−1.17233	−0.586165	−	0.810192i	$-0.699363\pi$
$821$	−27578.1	−1.17233	−0.586165	+	0.810192i	$0.699363\pi$
$822$	−29108.0	−1.23511
$823$	14566.9	0.616975	0.308488	−	0.951228i	$-0.400177\pi$
$823$	14566.9	0.616975	0.308488	+	0.951228i	$0.400177\pi$
$824$	−63494.6	−2.68439
$825$	15890.9	0.670606
$826$	−416.504	−0.0175448
$827$	46389.9	1.95059	0.975293	−	0.220917i	$-0.0709049\pi$
$827$	46389.9	1.95059	0.975293	+	0.220917i	$0.0709049\pi$
$828$	−161598.	−6.78253
$829$	17471.0	0.731957	0.365979	−	0.930623i	$-0.380734\pi$
$829$	17471.0	0.731957	0.365979	+	0.930623i	$0.380734\pi$
$830$	−38493.6	−1.60980
$831$	−778.629	−0.0325034
$832$	−64811.9	−2.70066
$833$	−12159.8	−0.505777
$834$	33318.5	1.38336
$835$	−13520.4	−0.560350
$836$	−62505.3	−2.58587
$837$	−13405.2	−0.553585
$838$	2736.15	0.112791
$839$	31457.8	1.29445	0.647225	−	0.762299i	$-0.275930\pi$
$839$	31457.8	1.29445	0.647225	+	0.762299i	$0.275930\pi$
$840$	10954.7	0.449969
$841$	−21920.9	−0.898801
$842$	71715.2	2.93524
$843$	−16253.5	−0.664056
$844$	−21311.0	−0.869142
$845$	6336.04	0.257949
$846$	113151.	4.59835
$847$	−2406.90	−0.0976411
$848$	62034.3	2.51211
$849$	38797.1	1.56833
$850$	17213.3	0.694602
$851$	6533.06	0.263161
$852$	94441.4	3.79755
$853$	12947.7	0.519720	0.259860	−	0.965646i	$-0.416324\pi$
$853$	12947.7	0.519720	0.259860	+	0.965646i	$0.416324\pi$
$854$	6523.74	0.261403
$855$	−48944.9	−1.95775
$856$	70485.2	2.81441
$857$	−22730.2	−0.906007	−0.453004	−	0.891509i	$-0.649648\pi$
$857$	−22730.2	−0.906007	−0.453004	+	0.891509i	$0.649648\pi$
$858$	−34024.3	−1.35381
$859$	−10019.6	−0.397980	−0.198990	−	0.980002i	$-0.563766\pi$
$859$	−10019.6	−0.397980	−0.198990	+	0.980002i	$0.563766\pi$
$860$	0	0
$861$	1683.06	0.0666183
$862$	−25183.8	−0.995084
$863$	−33745.3	−1.33106	−0.665530	−	0.746371i	$-0.731794\pi$
$863$	−33745.3	−1.33106	−0.665530	+	0.746371i	$0.731794\pi$
$864$	−181820.	−7.15930
$865$	673.934	0.0264907
$866$	−62088.0	−2.43630
$867$	−32792.5	−1.28454
$868$	−2903.09	−0.113522
$869$	6174.18	0.241018
$870$	−15183.7	−0.591697
$871$	−30439.0	−1.18414
$872$	5651.53	0.219478
$873$	−6984.84	−0.270791
$874$	103587.	4.00903
$875$	3381.15	0.130633
$876$	204095.	7.87185
$877$	−9357.18	−0.360285	−0.180142	−	0.983641i	$-0.557656\pi$
$877$	−9357.18	−0.360285	−0.180142	+	0.983641i	$0.557656\pi$
$878$	−25616.9	−0.984658
$879$	69976.6	2.68516
$880$	29368.5	1.12501
$881$	18561.7	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$881$	18561.7	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$882$	102351.	3.90740
$883$	−28177.1	−1.07388	−0.536940	−	0.843621i	$-0.680420\pi$
$883$	−28177.1	−1.07388	−0.536940	+	0.843621i	$0.680420\pi$
$884$	−26973.6	−1.02627
$885$	−1646.92	−0.0625543
$886$	9952.65	0.377388
$887$	13672.0	0.517543	0.258771	−	0.965939i	$-0.416682\pi$
$887$	13672.0	0.517543	0.258771	+	0.965939i	$0.416682\pi$
$888$	33810.0	1.27769
$889$	−5321.43	−0.200759
$890$	−22532.4	−0.848637
$891$	−17466.5	−0.656735
$892$	114854.	4.31119
$893$	−53083.5	−1.98922
$894$	95352.6	3.56719
$895$	−12932.8	−0.483012
$896$	−12419.5	−0.463066
$897$	41267.9	1.53611
$898$	70591.9	2.62325
$899$	2549.61	0.0945876
$900$	−106038.	−3.92734
$901$	9419.87	0.348304
$902$	7822.20	0.288748
$903$	0	0
$904$	160081.	5.88960
$905$	17167.3	0.630563
$906$	111595.	4.09215
$907$	−9828.52	−0.359813	−0.179907	−	0.983684i	$-0.557580\pi$
$907$	−9828.52	−0.359813	−0.179907	+	0.983684i	$0.557580\pi$
$908$	−63488.0	−2.32040
$909$	−80940.4	−2.95338
$910$	−2973.98	−0.108337
$911$	30471.1	1.10818	0.554091	−	0.832456i	$-0.313066\pi$
$911$	30471.1	1.10818	0.554091	+	0.832456i	$0.313066\pi$
$912$	309233.	11.2278
$913$	22964.8	0.832448
$914$	5941.95	0.215035
$915$	25795.9	0.932006
$916$	28716.3	1.03582
$917$	2366.03	0.0852053
$918$	−51590.3	−1.85483
$919$	42727.5	1.53368	0.766839	−	0.641839i	$-0.221828\pi$
$919$	42727.5	1.53368	0.766839	+	0.641839i	$0.221828\pi$
$920$	−61752.4	−2.21295
$921$	−19764.5	−0.707125
$922$	10128.7	0.361792
$923$	−16245.6	−0.579341
$924$	−10314.3	−0.367224
$925$	4286.88	0.152380
$926$	−104829.	−3.72019
$927$	46812.9	1.65861
$928$	34581.4	1.22327
$929$	−32717.0	−1.15545	−0.577723	−	0.816233i	$-0.696059\pi$
$929$	−32717.0	−1.15545	−0.577723	+	0.816233i	$0.696059\pi$
$930$	−15684.9	−0.553040
$931$	−48016.8	−1.69032
$932$	−23372.4	−0.821449
$933$	88134.5	3.09260
$934$	−20220.9	−0.708403
$935$	4459.59	0.155983
$936$	143860.	5.02374
$937$	−51528.4	−1.79654	−0.898271	−	0.439443i	$-0.855176\pi$
$937$	−51528.4	−1.79654	−0.898271	+	0.439443i	$0.855176\pi$
$938$	−12608.0	−0.438877
$939$	26560.5	0.923076
$940$	49942.4	1.73292
$941$	26909.6	0.932229	0.466114	−	0.884724i	$-0.345653\pi$
$941$	26909.6	0.932229	0.466114	+	0.884724i	$0.345653\pi$
$942$	−89696.6	−3.10241
$943$	−9487.49	−0.327630
$944$	7008.90	0.241653
$945$	−4162.94	−0.143302
$946$	0	0
$947$	18441.8	0.632818	0.316409	−	0.948623i	$-0.397523\pi$
$947$	18441.8	0.632818	0.316409	+	0.948623i	$0.397523\pi$
$948$	−61163.7	−2.09547
$949$	−35108.1	−1.20090
$950$	67972.2	2.32138
$951$	5913.58	0.201641
$952$	−7079.34	−0.241011
$953$	35819.6	1.21754	0.608768	−	0.793348i	$-0.291664\pi$
$953$	35819.6	1.21754	0.608768	+	0.793348i	$0.291664\pi$
$954$	−79288.5	−2.69084
$955$	23584.2	0.799126
$956$	−78580.1	−2.65843
$957$	9058.43	0.305975
$958$	−45361.9	−1.52983
$959$	1517.88	0.0511103
$960$	−106151.	−3.56877
$961$	−27157.2	−0.911592
$962$	−9178.73	−0.307624
$963$	−51966.8	−1.73895
$964$	−92628.1	−3.09476
$965$	−23831.2	−0.794979
$966$	17093.4	0.569329
$967$	−28557.3	−0.949680	−0.474840	−	0.880072i	$-0.657494\pi$
$967$	−28557.3	−0.949680	−0.474840	+	0.880072i	$0.657494\pi$
$968$	70216.5	2.33145
$969$	46956.9	1.55673
$970$	−4212.46	−0.139437
$971$	−32692.2	−1.08048	−0.540238	−	0.841513i	$-0.681666\pi$
$971$	−32692.2	−1.08048	−0.540238	+	0.841513i	$0.681666\pi$
$972$	19029.8	0.627963
$973$	−1737.44	−0.0572454
$974$	85619.5	2.81666
$975$	27079.3	0.889467
$976$	−109781.	−3.60042
$977$	−2318.74	−0.0759294	−0.0379647	−	0.999279i	$-0.512087\pi$
$977$	−2318.74	−0.0759294	−0.0379647	+	0.999279i	$0.512087\pi$
$978$	−171535.	−5.60848
$979$	13442.6	0.438842
$980$	45175.5	1.47253
$981$	−4166.72	−0.135610
$982$	71389.5	2.31989
$983$	26323.2	0.854099	0.427050	−	0.904228i	$-0.359553\pi$
$983$	26323.2	0.854099	0.427050	+	0.904228i	$0.359553\pi$
$984$	−49099.8	−1.59070
$985$	15112.2	0.488846
$986$	9812.27	0.316923
$987$	−8759.56	−0.282492
$988$	−106514.	−3.42981
$989$	0	0
$990$	−37537.0	−1.20506
$991$	57890.7	1.85566	0.927830	−	0.373004i	$-0.121672\pi$
$991$	57890.7	1.85566	0.927830	+	0.373004i	$0.121672\pi$
$992$	35722.8	1.14335
$993$	−26566.3	−0.849000
$994$	−6729.05	−0.214721
$995$	−1002.44	−0.0319392
$996$	−227498.	−7.23749
$997$	10032.0	0.318673	0.159337	−	0.987224i	$-0.449064\pi$
$997$	10032.0	0.318673	0.159337	+	0.987224i	$0.449064\pi$
$998$	−53293.3	−1.69035
$999$	−12848.3	−0.406908

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.1	✓	50
43.42	odd	2		1849.4.a.j.1.50	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.1	✓	50	1.1	even	1	trivial
1849.4.a.j.1.50	yes	50	43.42	odd	2

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)

\(f(q)\)	\(=\)	\(q-5.46223 q^{2} +9.09504 q^{3} +21.8360 q^{4} -6.15202 q^{5} -49.6792 q^{6} +2.59059 q^{7} -75.5754 q^{8} +55.7197 q^{9} +O(q^{10})\)	\(q-5.46223 q^{2} +9.09504 q^{3} +21.8360 q^{4} -6.15202 q^{5} -49.6792 q^{6} +2.59059 q^{7} -75.5754 q^{8} +55.7197 q^{9} +33.6038 q^{10} -20.0476 q^{11} +198.599 q^{12} -34.1627 q^{13} -14.1504 q^{14} -55.9529 q^{15} +238.122 q^{16} +36.1588 q^{17} -304.354 q^{18} +142.784 q^{19} -134.335 q^{20} +23.5615 q^{21} +109.505 q^{22} -132.818 q^{23} -687.361 q^{24} -87.1526 q^{25} +186.604 q^{26} +261.207 q^{27} +56.5681 q^{28} -49.6805 q^{29} +305.628 q^{30} -51.3202 q^{31} -696.077 q^{32} -182.334 q^{33} -197.508 q^{34} -15.9374 q^{35} +1216.69 q^{36} -49.1882 q^{37} -779.922 q^{38} -310.711 q^{39} +464.941 q^{40} +71.4324 q^{41} -128.698 q^{42} -437.760 q^{44} -342.789 q^{45} +725.481 q^{46} -371.774 q^{47} +2165.73 q^{48} -336.289 q^{49} +476.048 q^{50} +328.865 q^{51} -745.976 q^{52} +260.514 q^{53} -1426.77 q^{54} +123.333 q^{55} -195.785 q^{56} +1298.63 q^{57} +271.366 q^{58} +29.4340 q^{59} -1221.79 q^{60} -461.029 q^{61} +280.323 q^{62} +144.347 q^{63} +1897.16 q^{64} +210.169 q^{65} +995.951 q^{66} +891.001 q^{67} +789.563 q^{68} -1207.98 q^{69} +87.0536 q^{70} +475.538 q^{71} -4211.04 q^{72} +1027.68 q^{73} +268.677 q^{74} -792.656 q^{75} +3117.84 q^{76} -51.9352 q^{77} +1697.17 q^{78} -307.975 q^{79} -1464.93 q^{80} +871.251 q^{81} -390.181 q^{82} -1145.51 q^{83} +514.489 q^{84} -222.450 q^{85} -451.846 q^{87} +1515.11 q^{88} -670.531 q^{89} +1872.39 q^{90} -88.5015 q^{91} -2900.20 q^{92} -466.759 q^{93} +2030.72 q^{94} -878.413 q^{95} -6330.85 q^{96} -125.357 q^{97} +1836.89 q^{98} -1117.05 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) \) 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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