LMFDB - Embedded newform 1849.4.a.j.1.15

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			1.15
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)

$f(q)$	$=$	$q-2.34509 q^{2} +2.42890 q^{3} -2.50055 q^{4} -18.7081 q^{5} -5.69599 q^{6} -0.512587 q^{7} +24.6247 q^{8} -21.1004 q^{9} +O(q^{10})$	\(q-2.34509 q^{2} +2.42890 q^{3} -2.50055 q^{4} -18.7081 q^{5} -5.69599 q^{6} -0.512587 q^{7} +24.6247 q^{8} -21.1004 q^{9} +43.8722 q^{10} +5.04244 q^{11} -6.07358 q^{12} -48.6308 q^{13} +1.20206 q^{14} -45.4401 q^{15} -37.7429 q^{16} -6.35958 q^{17} +49.4825 q^{18} -123.762 q^{19} +46.7805 q^{20} -1.24502 q^{21} -11.8250 q^{22} +141.010 q^{23} +59.8111 q^{24} +224.993 q^{25} +114.044 q^{26} -116.831 q^{27} +1.28175 q^{28} +206.672 q^{29} +106.561 q^{30} +164.801 q^{31} -108.487 q^{32} +12.2476 q^{33} +14.9138 q^{34} +9.58954 q^{35} +52.7627 q^{36} +365.186 q^{37} +290.233 q^{38} -118.119 q^{39} -460.682 q^{40} -175.812 q^{41} +2.91969 q^{42} -12.6089 q^{44} +394.749 q^{45} -330.680 q^{46} +219.602 q^{47} -91.6737 q^{48} -342.737 q^{49} -527.630 q^{50} -15.4468 q^{51} +121.604 q^{52} +34.4084 q^{53} +273.980 q^{54} -94.3346 q^{55} -12.6223 q^{56} -300.605 q^{57} -484.665 q^{58} -771.285 q^{59} +113.625 q^{60} +432.289 q^{61} -386.473 q^{62} +10.8158 q^{63} +556.356 q^{64} +909.791 q^{65} -28.7217 q^{66} -644.829 q^{67} +15.9024 q^{68} +342.498 q^{69} -22.4883 q^{70} +919.149 q^{71} -519.593 q^{72} -98.7029 q^{73} -856.394 q^{74} +546.486 q^{75} +309.472 q^{76} -2.58469 q^{77} +277.001 q^{78} +411.930 q^{79} +706.098 q^{80} +285.940 q^{81} +412.295 q^{82} +271.229 q^{83} +3.11324 q^{84} +118.976 q^{85} +501.986 q^{87} +124.169 q^{88} -115.517 q^{89} -925.723 q^{90} +24.9276 q^{91} -352.601 q^{92} +400.285 q^{93} -514.986 q^{94} +2315.35 q^{95} -263.505 q^{96} +623.381 q^{97} +803.750 q^{98} -106.398 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.34509	−0.829115	−0.414557	−	0.910023i	$-0.636064\pi$
$2$	−2.34509	−0.829115	−0.414557	+	0.910023i	$0.636064\pi$
$3$	2.42890	0.467442	0.233721	−	0.972304i	$-0.424910\pi$
$3$	2.42890	0.467442	0.233721	+	0.972304i	$0.424910\pi$
$4$	−2.50055	−0.312569
$5$	−18.7081	−1.67330	−0.836652	−	0.547735i	$-0.815490\pi$
$5$	−18.7081	−1.67330	−0.836652	+	0.547735i	$0.815490\pi$
$6$	−5.69599	−0.387563
$7$	−0.512587	−0.0276771	−0.0138386	−	0.999904i	$-0.504405\pi$
$7$	−0.512587	−0.0276771	−0.0138386	+	0.999904i	$0.504405\pi$
$8$	24.6247	1.08827
$9$	−21.1004	−0.781498
$10$	43.8722	1.38736
$11$	5.04244	0.138214	0.0691070	−	0.997609i	$-0.477985\pi$
$11$	5.04244	0.138214	0.0691070	+	0.997609i	$0.477985\pi$
$12$	−6.07358	−0.146108
$13$	−48.6308	−1.03752	−0.518760	−	0.854920i	$-0.673606\pi$
$13$	−48.6308	−1.03752	−0.518760	+	0.854920i	$0.673606\pi$
$14$	1.20206	0.0229475
$15$	−45.4401	−0.782173
$16$	−37.7429	−0.589732
$17$	−6.35958	−0.0907308	−0.0453654	−	0.998970i	$-0.514445\pi$
$17$	−6.35958	−0.0907308	−0.0453654	+	0.998970i	$0.514445\pi$
$18$	49.4825	0.647951
$19$	−123.762	−1.49436	−0.747182	−	0.664620i	$-0.768594\pi$
$19$	−123.762	−1.49436	−0.747182	+	0.664620i	$0.768594\pi$
$20$	46.7805	0.523022
$21$	−1.24502	−0.0129375
$22$	−11.8250	−0.114595
$23$	141.010	1.27837	0.639185	−	0.769053i	$-0.279272\pi$
$23$	141.010	1.27837	0.639185	+	0.769053i	$0.279272\pi$
$24$	59.8111	0.508703
$25$	224.993	1.79995
$26$	114.044	0.860224
$27$	−116.831	−0.832747
$28$	1.28175	0.00865100
$29$	206.672	1.32338	0.661690	−	0.749777i	$-0.269839\pi$
$29$	206.672	1.32338	0.661690	+	0.749777i	$0.269839\pi$
$30$	106.561	0.648511
$31$	164.801	0.954809	0.477405	−	0.878684i	$-0.341578\pi$
$31$	164.801	0.954809	0.477405	+	0.878684i	$0.341578\pi$
$32$	−108.487	−0.599314
$33$	12.2476	0.0646071
$34$	14.9138	0.0752263
$35$	9.58954	0.0463122
$36$	52.7627	0.244272
$37$	365.186	1.62260	0.811300	−	0.584630i	$-0.198760\pi$
$37$	365.186	1.62260	0.811300	+	0.584630i	$0.198760\pi$
$38$	290.233	1.23900
$39$	−118.119	−0.484981
$40$	−460.682	−1.82101
$41$	−175.812	−0.669689	−0.334844	−	0.942273i	$-0.608684\pi$
$41$	−175.812	−0.669689	−0.334844	+	0.942273i	$0.608684\pi$
$42$	2.91969	0.0107266
$43$	0	0
$43$	0	0
$44$	−12.6089	−0.0432013
$45$	394.749	1.30768
$46$	−330.680	−1.05992
$47$	219.602	0.681537	0.340768	−	0.940147i	$-0.389313\pi$
$47$	219.602	0.681537	0.340768	+	0.940147i	$0.389313\pi$
$48$	−91.6737	−0.275666
$49$	−342.737	−0.999234
$50$	−527.630	−1.49236
$51$	−15.4468	−0.0424114
$52$	121.604	0.324296
$53$	34.4084	0.0891765	0.0445883	−	0.999005i	$-0.485802\pi$
$53$	34.4084	0.0891765	0.0445883	+	0.999005i	$0.485802\pi$
$54$	273.980	0.690443
$55$	−94.3346	−0.231274
$56$	−12.6223	−0.0301202
$57$	−300.605	−0.698529
$58$	−484.665	−1.09723
$59$	−771.285	−1.70191	−0.850956	−	0.525238i	$-0.823976\pi$
$59$	−771.285	−1.70191	−0.850956	+	0.525238i	$0.823976\pi$
$60$	113.625	0.244483
$61$	432.289	0.907360	0.453680	−	0.891165i	$-0.350111\pi$
$61$	432.289	0.907360	0.453680	+	0.891165i	$0.350111\pi$
$62$	−386.473	−0.791647
$63$	10.8158	0.0216296
$64$	556.356	1.08663
$65$	909.791	1.73609
$66$	−28.7217	−0.0535667
$67$	−644.829	−1.17580	−0.587899	−	0.808935i	$-0.700045\pi$
$67$	−644.829	−1.17580	−0.587899	+	0.808935i	$0.700045\pi$
$68$	15.9024	0.0283596
$69$	342.498	0.597564
$70$	−22.4883	−0.0383982
$71$	919.149	1.53638	0.768190	−	0.640222i	$-0.221158\pi$
$71$	919.149	1.53638	0.768190	+	0.640222i	$0.221158\pi$
$72$	−519.593	−0.850481
$73$	−98.7029	−0.158251	−0.0791253	−	0.996865i	$-0.525213\pi$
$73$	−98.7029	−0.158251	−0.0791253	+	0.996865i	$0.525213\pi$
$74$	−856.394	−1.34532
$75$	546.486	0.841371
$76$	309.472	0.467091
$77$	−2.58469	−0.00382536
$78$	277.001	0.402105
$79$	411.930	0.586655	0.293327	−	0.956012i	$-0.405237\pi$
$79$	411.930	0.586655	0.293327	+	0.956012i	$0.405237\pi$
$80$	706.098	0.986802
$81$	285.940	0.392237
$82$	412.295	0.555249
$83$	271.229	0.358689	0.179345	−	0.983786i	$-0.442602\pi$
$83$	271.229	0.358689	0.179345	+	0.983786i	$0.442602\pi$
$84$	3.11324	0.00404384
$85$	118.976	0.151820
$86$	0	0
$87$	501.986	0.618604
$88$	124.169	0.150414
$89$	−115.517	−0.137582	−0.0687908	−	0.997631i	$-0.521914\pi$
$89$	−115.517	−0.137582	−0.0687908	+	0.997631i	$0.521914\pi$
$90$	−925.723	−1.08422
$91$	24.9276	0.0287156
$92$	−352.601	−0.399578
$93$	400.285	0.446318
$94$	−514.986	−0.565072
$95$	2315.35	2.50053
$96$	−263.505	−0.280145
$97$	623.381	0.652523	0.326261	−	0.945280i	$-0.394211\pi$
$97$	623.381	0.652523	0.326261	+	0.945280i	$0.394211\pi$
$98$	803.750	0.828480
$99$	−106.398	−0.108014
$100$	−562.606	−0.562606
$101$	134.852	0.132854	0.0664270	−	0.997791i	$-0.478840\pi$
$101$	134.852	0.132854	0.0664270	+	0.997791i	$0.478840\pi$
$102$	36.2241	0.0351639
$103$	−246.042	−0.235371	−0.117686	−	0.993051i	$-0.537547\pi$
$103$	−246.042	−0.235371	−0.117686	+	0.993051i	$0.537547\pi$
$104$	−1197.52	−1.12910
$105$	23.2920	0.0216483
$106$	−80.6908	−0.0739376
$107$	−615.063	−0.555704	−0.277852	−	0.960624i	$-0.589623\pi$
$107$	−615.063	−0.555704	−0.277852	+	0.960624i	$0.589623\pi$
$108$	292.142	0.260291
$109$	1860.95	1.63529	0.817646	−	0.575721i	$-0.195279\pi$
$109$	1860.95	1.63529	0.817646	+	0.575721i	$0.195279\pi$
$110$	221.223	0.191753
$111$	887.001	0.758472
$112$	19.3465	0.0163221
$113$	2228.96	1.85560	0.927802	−	0.373072i	$-0.121696\pi$
$113$	2228.96	1.85560	0.927802	+	0.373072i	$0.121696\pi$
$114$	704.947	0.579161
$115$	−2638.02	−2.13910
$116$	−516.793	−0.413647
$117$	1026.13	0.810820
$118$	1808.73	1.41108
$119$	3.25984	0.00251117
$120$	−1118.95	−0.851215
$121$	−1305.57	−0.980897
$122$	−1013.76	−0.752306
$123$	−427.030	−0.313041
$124$	−412.092	−0.298443
$125$	−1870.68	−1.33855
$126$	−25.3641	−0.0179334
$127$	453.295	0.316720	0.158360	−	0.987381i	$-0.449379\pi$
$127$	453.295	0.316720	0.158360	+	0.987381i	$0.449379\pi$
$128$	−436.806	−0.301629
$129$	0	0
$130$	−2133.54	−1.43942
$131$	995.763	0.664124	0.332062	−	0.943258i	$-0.392256\pi$
$131$	995.763	0.664124	0.332062	+	0.943258i	$0.392256\pi$
$132$	−30.6257	−0.0201941
$133$	63.4388	0.0413597
$134$	1512.18	0.974871
$135$	2185.69	1.39344
$136$	−156.603	−0.0987397
$137$	2491.26	1.55359	0.776797	−	0.629751i	$-0.216843\pi$
$137$	2491.26	1.55359	0.776797	+	0.629751i	$0.216843\pi$
$138$	−803.189	−0.495449
$139$	−1951.81	−1.19101	−0.595506	−	0.803351i	$-0.703048\pi$
$139$	−1951.81	−1.19101	−0.595506	+	0.803351i	$0.703048\pi$
$140$	−23.9791	−0.0144757
$141$	533.391	0.318579
$142$	−2155.49	−1.27384
$143$	−245.218	−0.143400
$144$	796.391	0.460875
$145$	−3866.44	−2.21442
$146$	231.467	0.131208
$147$	−832.475	−0.467084
$148$	−913.165	−0.507174
$149$	2006.42	1.10317	0.551585	−	0.834119i	$-0.314023\pi$
$149$	2006.42	1.10317	0.551585	+	0.834119i	$0.314023\pi$
$150$	−1281.56	−0.697593
$151$	−3455.23	−1.86213	−0.931067	−	0.364847i	$-0.881121\pi$
$151$	−3455.23	−1.86213	−0.931067	+	0.364847i	$0.881121\pi$
$152$	−3047.60	−1.62627
$153$	134.190	0.0709059
$154$	6.06134	0.00317167
$155$	−3083.11	−1.59769
$156$	295.363	0.151590
$157$	380.367	0.193354	0.0966771	−	0.995316i	$-0.469179\pi$
$157$	380.367	0.193354	0.0966771	+	0.995316i	$0.469179\pi$
$158$	−966.013	−0.486404
$159$	83.5746	0.0416849
$160$	2029.59	1.00283
$161$	−72.2797	−0.0353816
$162$	−670.556	−0.325209
$163$	1067.78	0.513096	0.256548	−	0.966532i	$-0.417415\pi$
$163$	1067.78	0.513096	0.256548	+	0.966532i	$0.417415\pi$
$164$	439.627	0.209324
$165$	−229.129	−0.108107
$166$	−636.056	−0.297395
$167$	−4232.47	−1.96119	−0.980593	−	0.196055i	$-0.937187\pi$
$167$	−4232.47	−1.96119	−0.980593	+	0.196055i	$0.937187\pi$
$168$	−30.6584	−0.0140794
$169$	167.959	0.0764490
$170$	−279.009	−0.125876
$171$	2611.43	1.16784
$172$	0	0
$173$	−3617.39	−1.58974	−0.794869	−	0.606781i	$-0.792461\pi$
$173$	−3617.39	−1.58974	−0.794869	+	0.606781i	$0.792461\pi$
$174$	−1177.20	−0.512894
$175$	−115.329	−0.0498173
$176$	−190.316	−0.0815093
$177$	−1873.37	−0.795545
$178$	270.897	0.114071
$179$	−643.127	−0.268545	−0.134272	−	0.990944i	$-0.542870\pi$
$179$	−643.127	−0.268545	−0.134272	+	0.990944i	$0.542870\pi$
$180$	−987.090	−0.408741
$181$	3174.93	1.30382	0.651909	−	0.758297i	$-0.273968\pi$
$181$	3174.93	1.30382	0.651909	+	0.758297i	$0.273968\pi$
$182$	−58.4574	−0.0238085
$183$	1049.99	0.424138
$184$	3472.32	1.39121
$185$	−6831.94	−2.71510
$186$	−938.704	−0.370049
$187$	−32.0678	−0.0125403
$188$	−549.125	−0.213027
$189$	59.8862	0.0230480
$190$	−5429.71	−2.07322
$191$	464.296	0.175892	0.0879458	−	0.996125i	$-0.471970\pi$
$191$	464.296	0.175892	0.0879458	+	0.996125i	$0.471970\pi$
$192$	1351.33	0.507938
$193$	−3898.46	−1.45397	−0.726987	−	0.686651i	$-0.759080\pi$
$193$	−3898.46	−1.45397	−0.726987	+	0.686651i	$0.759080\pi$
$194$	−1461.88	−0.541017
$195$	2209.79	0.811521
$196$	857.031	0.312329
$197$	2153.97	0.779005	0.389503	−	0.921025i	$-0.372647\pi$
$197$	2153.97	0.779005	0.389503	+	0.921025i	$0.372647\pi$
$198$	249.512	0.0895560
$199$	−3038.98	−1.08255	−0.541275	−	0.840845i	$-0.682058\pi$
$199$	−3038.98	−1.08255	−0.541275	+	0.840845i	$0.682058\pi$
$200$	5540.40	1.95883
$201$	−1566.23	−0.549617
$202$	−316.240	−0.110151
$203$	−105.937	−0.0366274
$204$	38.6254	0.0132565
$205$	3289.11	1.12059
$206$	576.990	0.195150
$207$	−2975.36	−0.999043
$208$	1835.47	0.611860
$209$	−624.062	−0.206542
$210$	−54.6220	−0.0179489
$211$	969.498	0.316318	0.158159	−	0.987414i	$-0.449444\pi$
$211$	969.498	0.316318	0.158159	+	0.987414i	$0.449444\pi$
$212$	−86.0398	−0.0278738
$213$	2232.52	0.718169
$214$	1442.38	0.460743
$215$	0	0
$216$	−2876.94	−0.906254
$217$	−84.4748	−0.0264264
$218$	−4364.10	−1.35584
$219$	−239.740	−0.0739730
$220$	235.888	0.0722890
$221$	309.272	0.0941351
$222$	−2080.10	−0.628860
$223$	446.780	0.134164	0.0670821	−	0.997747i	$-0.478631\pi$
$223$	446.780	0.134164	0.0670821	+	0.997747i	$0.478631\pi$
$224$	55.6093	0.0165873
$225$	−4747.46	−1.40665
$226$	−5227.13	−1.53851
$227$	−552.010	−0.161402	−0.0807009	−	0.996738i	$-0.525716\pi$
$227$	−552.010	−0.161402	−0.0807009	+	0.996738i	$0.525716\pi$
$228$	751.678	0.218338
$229$	−393.795	−0.113636	−0.0568181	−	0.998385i	$-0.518096\pi$
$229$	−393.795	−0.113636	−0.0568181	+	0.998385i	$0.518096\pi$
$230$	6186.40	1.77356
$231$	−6.27796	−0.00178814
$232$	5089.25	1.44020
$233$	335.082	0.0942143	0.0471071	−	0.998890i	$-0.485000\pi$
$233$	335.082	0.0942143	0.0471071	+	0.998890i	$0.485000\pi$
$234$	−2406.37	−0.672263
$235$	−4108.34	−1.14042
$236$	1928.64	0.531964
$237$	1000.54	0.274227
$238$	−7.64462	−0.00208205
$239$	−5612.62	−1.51904	−0.759520	−	0.650485i	$-0.774566\pi$
$239$	−5612.62	−1.51904	−0.759520	+	0.650485i	$0.774566\pi$
$240$	1715.04	0.461273
$241$	2947.42	0.787800	0.393900	−	0.919153i	$-0.371126\pi$
$241$	2947.42	0.787800	0.393900	+	0.919153i	$0.371126\pi$
$242$	3061.69	0.813276
$243$	3848.96	1.01610
$244$	−1080.96	−0.283612
$245$	6411.97	1.67202
$246$	1001.42	0.259547
$247$	6018.64	1.55043
$248$	4058.18	1.03909
$249$	658.788	0.167667
$250$	4386.93	1.10981
$251$	1443.00	0.362874	0.181437	−	0.983403i	$-0.441925\pi$
$251$	1443.00	0.362874	0.181437	+	0.983403i	$0.441925\pi$
$252$	−27.0455	−0.00676073
$253$	711.032	0.176689
$254$	−1063.02	−0.262597
$255$	288.980	0.0709672
$256$	−3426.50	−0.836547
$257$	−7394.73	−1.79483	−0.897414	−	0.441189i	$-0.854557\pi$
$257$	−7394.73	−1.79483	−0.897414	+	0.441189i	$0.854557\pi$
$258$	0	0
$259$	−187.190	−0.0449089
$260$	−2274.98	−0.542646
$261$	−4360.87	−1.03422
$262$	−2335.16	−0.550635
$263$	−7763.59	−1.82024	−0.910121	−	0.414343i	$-0.864012\pi$
$263$	−7763.59	−1.82024	−0.910121	+	0.414343i	$0.864012\pi$
$264$	301.594	0.0703099
$265$	−643.716	−0.149219
$266$	−148.770	−0.0342919
$267$	−280.579	−0.0643114
$268$	1612.43	0.367517
$269$	6410.90	1.45308	0.726541	−	0.687123i	$-0.241127\pi$
$269$	6410.90	1.45308	0.726541	+	0.687123i	$0.241127\pi$
$270$	−5125.64	−1.15532
$271$	1239.58	0.277857	0.138929	−	0.990302i	$-0.455634\pi$
$271$	1239.58	0.277857	0.138929	+	0.990302i	$0.455634\pi$
$272$	240.029	0.0535069
$273$	60.5466	0.0134229
$274$	−5842.22	−1.28811
$275$	1134.52	0.248778
$276$	−856.433	−0.186780
$277$	1265.98	0.274605	0.137302	−	0.990529i	$-0.456157\pi$
$277$	1265.98	0.274605	0.137302	+	0.990529i	$0.456157\pi$
$278$	4577.18	0.987485
$279$	−3477.37	−0.746181
$280$	236.140	0.0504002
$281$	−2547.49	−0.540821	−0.270410	−	0.962745i	$-0.587159\pi$
$281$	−2547.49	−0.540821	−0.270410	+	0.962745i	$0.587159\pi$
$282$	−1250.85	−0.264139
$283$	−2198.22	−0.461733	−0.230866	−	0.972985i	$-0.574156\pi$
$283$	−2198.22	−0.461733	−0.230866	+	0.972985i	$0.574156\pi$
$284$	−2298.38	−0.480224
$285$	5623.76	1.16885
$286$	575.059	0.118895
$287$	90.1191	0.0185351
$288$	2289.13	0.468363
$289$	−4872.56	−0.991768
$290$	9067.16	1.83601
$291$	1514.13	0.305017
$292$	246.811	0.0494642
$293$	3476.05	0.693082	0.346541	−	0.938035i	$-0.387356\pi$
$293$	3476.05	0.693082	0.346541	+	0.938035i	$0.387356\pi$
$294$	1952.23	0.387266
$295$	14429.3	2.84781
$296$	8992.61	1.76583
$297$	−589.115	−0.115097
$298$	−4705.24	−0.914655
$299$	−6857.41	−1.32634
$300$	−1366.52	−0.262986
$301$	0	0
$302$	8102.82	1.54392
$303$	327.542	0.0621015
$304$	4671.13	0.881275
$305$	−8087.31	−1.51829
$306$	−314.688	−0.0587892
$307$	1861.37	0.346039	0.173019	−	0.984918i	$-0.444648\pi$
$307$	1861.37	0.346039	0.173019	+	0.984918i	$0.444648\pi$
$308$	6.46315	0.00119569
$309$	−597.611	−0.110022
$310$	7230.17	1.32467
$311$	−7750.28	−1.41311	−0.706556	−	0.707657i	$-0.749752\pi$
$311$	−7750.28	−1.41311	−0.706556	+	0.707657i	$0.749752\pi$
$312$	−2908.66	−0.527790
$313$	−1674.83	−0.302451	−0.151225	−	0.988499i	$-0.548322\pi$
$313$	−1674.83	−0.302451	−0.151225	+	0.988499i	$0.548322\pi$
$314$	−891.996	−0.160313
$315$	−202.344	−0.0361929
$316$	−1030.05	−0.183370
$317$	−4814.34	−0.852998	−0.426499	−	0.904488i	$-0.640253\pi$
$317$	−4814.34	−0.852998	−0.426499	+	0.904488i	$0.640253\pi$
$318$	−195.990	−0.0345616
$319$	1042.13	0.182910
$320$	−10408.4	−1.81827
$321$	−1493.93	−0.259760
$322$	169.502	0.0293354
$323$	787.073	0.135585
$324$	−715.008	−0.122601
$325$	−10941.6	−1.86748
$326$	−2504.03	−0.425415
$327$	4520.07	0.764404
$328$	−4329.33	−0.728802
$329$	−112.565	−0.0188630
$330$	537.329	0.0896333
$331$	10848.9	1.80154	0.900768	−	0.434301i	$-0.143005\pi$
$331$	10848.9	1.80154	0.900768	+	0.434301i	$0.143005\pi$
$332$	−678.220	−0.112115
$333$	−7705.58	−1.26806
$334$	9925.52	1.62605
$335$	12063.5	1.96747
$336$	46.9908	0.00762964
$337$	−8725.22	−1.41037	−0.705183	−	0.709026i	$-0.749135\pi$
$337$	−8725.22	−1.41037	−0.705183	+	0.709026i	$0.749135\pi$
$338$	−393.878	−0.0633850
$339$	5413.93	0.867388
$340$	−297.504	−0.0474542
$341$	830.999	0.131968
$342$	−6124.04	−0.968275
$343$	351.500	0.0553330
$344$	0	0
$345$	−6407.49	−0.999907
$346$	8483.10	1.31808
$347$	−5950.59	−0.920589	−0.460295	−	0.887766i	$-0.652256\pi$
$347$	−5950.59	−0.920589	−0.460295	+	0.887766i	$0.652256\pi$
$348$	−1255.24	−0.193356
$349$	−5162.99	−0.791887	−0.395944	−	0.918275i	$-0.629582\pi$
$349$	−5162.99	−0.791887	−0.395944	+	0.918275i	$0.629582\pi$
$350$	270.456	0.0413043
$351$	5681.60	0.863992
$352$	−547.042	−0.0828336
$353$	−6312.98	−0.951859	−0.475929	−	0.879484i	$-0.657888\pi$
$353$	−6312.98	−0.951859	−0.475929	+	0.879484i	$0.657888\pi$
$354$	4393.23	0.659598
$355$	−17195.5	−2.57083
$356$	288.855	0.0430037
$357$	7.91783	0.00117383
$358$	1508.19	0.222655
$359$	−2094.51	−0.307921	−0.153961	−	0.988077i	$-0.549203\pi$
$359$	−2094.51	−0.307921	−0.153961	+	0.988077i	$0.549203\pi$
$360$	9720.60	1.42311
$361$	8457.99	1.23312
$362$	−7445.51	−1.08101
$363$	−3171.11	−0.458513
$364$	−62.3325	−0.00897559
$365$	1846.54	0.264801
$366$	−2462.32	−0.351659
$367$	11071.4	1.57473	0.787363	−	0.616489i	$-0.211446\pi$
$367$	11071.4	1.57473	0.787363	+	0.616489i	$0.211446\pi$
$368$	−5322.10	−0.753896
$369$	3709.71	0.523360
$370$	16021.5	2.25113
$371$	−17.6373	−0.00246815
$372$	−1000.93	−0.139505
$373$	1517.88	0.210705	0.105353	−	0.994435i	$-0.466403\pi$
$373$	1517.88	0.210705	0.105353	+	0.994435i	$0.466403\pi$
$374$	75.2019	0.0103973
$375$	−4543.71	−0.625696
$376$	5407.64	0.741696
$377$	−10050.6	−1.37303
$378$	−140.439	−0.0191095
$379$	13541.0	1.83524	0.917618	−	0.397463i	$-0.130109\pi$
$379$	13541.0	1.83524	0.917618	+	0.397463i	$0.130109\pi$
$380$	−5789.64	−0.781585
$381$	1101.01	0.148048
$382$	−1088.82	−0.145834
$383$	1160.65	0.154847	0.0774235	−	0.996998i	$-0.475331\pi$
$383$	1160.65	0.154847	0.0774235	+	0.996998i	$0.475331\pi$
$384$	−1060.96	−0.140994
$385$	48.3547	0.00640100
$386$	9142.24	1.20551
$387$	0	0
$388$	−1558.79	−0.203958
$389$	−2125.41	−0.277024	−0.138512	−	0.990361i	$-0.544232\pi$
$389$	−2125.41	−0.277024	−0.138512	+	0.990361i	$0.544232\pi$
$390$	−5182.16	−0.672844
$391$	−896.761	−0.115988
$392$	−8439.82	−1.08744
$393$	2418.61	0.310440
$394$	−5051.26	−0.645885
$395$	−7706.43	−0.981652
$396$	266.053	0.0337618
$397$	1017.45	0.128626	0.0643128	−	0.997930i	$-0.479514\pi$
$397$	1017.45	0.128626	0.0643128	+	0.997930i	$0.479514\pi$
$398$	7126.69	0.897559
$399$	154.086	0.0193333
$400$	−8491.89	−1.06149
$401$	4536.33	0.564922	0.282461	−	0.959279i	$-0.408849\pi$
$401$	4536.33	0.564922	0.282461	+	0.959279i	$0.408849\pi$
$402$	3672.94	0.455696
$403$	−8014.40	−0.990634
$404$	−337.203	−0.0415260
$405$	−5349.40	−0.656331
$406$	248.433	0.0303683
$407$	1841.43	0.224266
$408$	−380.373	−0.0461551
$409$	−12621.0	−1.52584	−0.762918	−	0.646495i	$-0.776234\pi$
$409$	−12621.0	−1.52584	−0.762918	+	0.646495i	$0.776234\pi$
$410$	−7713.27	−0.929100
$411$	6051.01	0.726215
$412$	615.239	0.0735696
$413$	395.351	0.0471040
$414$	6977.50	0.828322
$415$	−5074.18	−0.600196
$416$	5275.84	0.621801
$417$	−4740.76	−0.556729
$418$	1463.48	0.171247
$419$	−7940.92	−0.925869	−0.462934	−	0.886392i	$-0.653203\pi$
$419$	−7940.92	−0.925869	−0.462934	+	0.886392i	$0.653203\pi$
$420$	−58.2429	−0.00676657
$421$	3246.56	0.375838	0.187919	−	0.982185i	$-0.439826\pi$
$421$	3246.56	0.375838	0.187919	+	0.982185i	$0.439826\pi$
$422$	−2273.56	−0.262264
$423$	−4633.70	−0.532619
$424$	847.298	0.0970482
$425$	−1430.86	−0.163311
$426$	−5235.47	−0.595444
$427$	−221.586	−0.0251131
$428$	1537.99	0.173696
$429$	−595.611	−0.0670312
$430$	0	0
$431$	−10024.0	−1.12028	−0.560141	−	0.828398i	$-0.689253\pi$
$431$	−10024.0	−1.12028	−0.560141	+	0.828398i	$0.689253\pi$
$432$	4409.55	0.491098
$433$	9624.64	1.06820	0.534100	−	0.845421i	$-0.320651\pi$
$433$	9624.64	1.06820	0.534100	+	0.845421i	$0.320651\pi$
$434$	198.101	0.0219105
$435$	−9391.21	−1.03511
$436$	−4653.40	−0.511141
$437$	−17451.6	−1.91035
$438$	562.211	0.0613321
$439$	−7330.44	−0.796954	−0.398477	−	0.917178i	$-0.630461\pi$
$439$	−7330.44	−0.796954	−0.398477	+	0.917178i	$0.630461\pi$
$440$	−2322.96	−0.251689
$441$	7231.91	0.780899
$442$	−725.270	−0.0780488
$443$	−1082.51	−0.116098	−0.0580492	−	0.998314i	$-0.518488\pi$
$443$	−1082.51	−0.116098	−0.0580492	+	0.998314i	$0.518488\pi$
$444$	−2217.99	−0.237074
$445$	2161.10	0.230216
$446$	−1047.74	−0.111238
$447$	4873.39	0.515668
$448$	−285.181	−0.0300749
$449$	5924.49	0.622703	0.311352	−	0.950295i	$-0.399218\pi$
$449$	5924.49	0.622703	0.311352	+	0.950295i	$0.399218\pi$
$450$	11133.2	1.16628
$451$	−886.523	−0.0925604
$452$	−5573.63	−0.580004
$453$	−8392.40	−0.870440
$454$	1294.51	0.133821
$455$	−466.347	−0.0480499
$456$	−7402.33	−0.760188
$457$	−9192.85	−0.940970	−0.470485	−	0.882408i	$-0.655921\pi$
$457$	−9192.85	−0.940970	−0.470485	+	0.882408i	$0.655921\pi$
$458$	923.485	0.0942175
$459$	742.997	0.0755559
$460$	6596.50	0.668616
$461$	−1014.04	−0.102448	−0.0512242	−	0.998687i	$-0.516312\pi$
$461$	−1014.04	−0.102448	−0.0512242	+	0.998687i	$0.516312\pi$
$462$	14.7224	0.00148257
$463$	11579.2	1.16227	0.581134	−	0.813808i	$-0.302609\pi$
$463$	11579.2	1.16227	0.581134	+	0.813808i	$0.302609\pi$
$464$	−7800.40	−0.780441
$465$	−7488.57	−0.746826
$466$	−785.797	−0.0781145
$467$	−11479.0	−1.13744	−0.568721	−	0.822530i	$-0.692562\pi$
$467$	−11479.0	−1.13744	−0.568721	+	0.822530i	$0.692562\pi$
$468$	−2565.89	−0.253437
$469$	330.531	0.0325427
$470$	9634.42	0.945538
$471$	923.875	0.0903820
$472$	−18992.7	−1.85214
$473$	0	0
$474$	−2346.35	−0.227366
$475$	−27845.6	−2.68977
$476$	−8.15139	−0.000784912 0
$477$	−726.032	−0.0696913
$478$	13162.1	1.25946
$479$	11310.1	1.07886	0.539429	−	0.842031i	$-0.318640\pi$
$479$	11310.1	1.07886	0.539429	+	0.842031i	$0.318640\pi$
$480$	4929.68	0.468767
$481$	−17759.3	−1.68348
$482$	−6911.96	−0.653177
$483$	−175.560	−0.0165389
$484$	3264.65	0.306597
$485$	−11662.3	−1.09187
$486$	−9026.17	−0.842460
$487$	13942.4	1.29731	0.648657	−	0.761081i	$-0.275331\pi$
$487$	13942.4	1.29731	0.648657	+	0.761081i	$0.275331\pi$
$488$	10645.0	0.987453
$489$	2593.52	0.239843
$490$	−15036.6	−1.38630
$491$	−6614.78	−0.607986	−0.303993	−	0.952674i	$-0.598320\pi$
$491$	−6614.78	−0.607986	−0.303993	+	0.952674i	$0.598320\pi$
$492$	1067.81	0.0978467
$493$	−1314.35	−0.120071
$494$	−14114.3	−1.28549
$495$	1990.50	0.180740
$496$	−6220.06	−0.563082
$497$	−471.144	−0.0425226
$498$	−1544.92	−0.139015
$499$	−1025.76	−0.0920229	−0.0460114	−	0.998941i	$-0.514651\pi$
$499$	−1025.76	−0.0920229	−0.0460114	+	0.998941i	$0.514651\pi$
$500$	4677.74	0.418389
$501$	−10280.2	−0.916741
$502$	−3383.96	−0.300864
$503$	−3411.78	−0.302433	−0.151217	−	0.988501i	$-0.548319\pi$
$503$	−3411.78	−0.302433	−0.151217	+	0.988501i	$0.548319\pi$
$504$	266.337	0.0235389
$505$	−2522.82	−0.222305
$506$	−1667.44	−0.146495
$507$	407.955	0.0357355
$508$	−1133.49	−0.0989967
$509$	−334.899	−0.0291634	−0.0145817	−	0.999894i	$-0.504642\pi$
$509$	−334.899	−0.0291634	−0.0145817	+	0.999894i	$0.504642\pi$
$510$	−677.685	−0.0588400
$511$	50.5938	0.00437992
$512$	11529.9	0.995223
$513$	14459.2	1.24443
$514$	17341.3	1.48812
$515$	4602.98	0.393847
$516$	0	0
$517$	1107.33	0.0941979
$518$	438.977	0.0372346
$519$	−8786.27	−0.743111
$520$	22403.4	1.88933
$521$	15973.0	1.34317	0.671585	−	0.740928i	$-0.265614\pi$
$521$	15973.0	1.34317	0.671585	+	0.740928i	$0.265614\pi$
$522$	10226.6	0.857486
$523$	4623.45	0.386557	0.193279	−	0.981144i	$-0.438088\pi$
$523$	4623.45	0.386557	0.193279	+	0.981144i	$0.438088\pi$
$524$	−2489.95	−0.207584
$525$	−280.122	−0.0232867
$526$	18206.3	1.50919
$527$	−1048.06	−0.0866307
$528$	−462.260	−0.0381009
$529$	7716.68	0.634230
$530$	1509.57	0.123720
$531$	16274.5	1.33004
$532$	−158.632	−0.0129277
$533$	8549.89	0.694816
$534$	657.983	0.0533216
$535$	11506.7	0.929862
$536$	−15878.8	−1.27959
$537$	−1562.09	−0.125529
$538$	−15034.1	−1.20477
$539$	−1728.23	−0.138108
$540$	−5465.42	−0.435545
$541$	5776.76	0.459080	0.229540	−	0.973299i	$-0.426278\pi$
$541$	5776.76	0.459080	0.229540	+	0.973299i	$0.426278\pi$
$542$	−2906.93	−0.230376
$543$	7711.60	0.609459
$544$	689.934	0.0543763
$545$	−34814.9	−2.73634
$546$	−141.987	−0.0111291
$547$	20861.6	1.63067	0.815334	−	0.578990i	$-0.196553\pi$
$547$	20861.6	1.63067	0.815334	+	0.578990i	$0.196553\pi$
$548$	−6229.51	−0.485605
$549$	−9121.49	−0.709100
$550$	−2660.54	−0.206265
$551$	−25578.1	−1.97761
$552$	8433.93	0.650311
$553$	−211.150	−0.0162369
$554$	−2968.85	−0.227679
$555$	−16594.1	−1.26915
$556$	4880.60	0.372273
$557$	−4932.51	−0.375219	−0.187610	−	0.982244i	$-0.560074\pi$
$557$	−4932.51	−0.375219	−0.187610	+	0.982244i	$0.560074\pi$
$558$	8154.75	0.618670
$559$	0	0
$560$	−361.937	−0.0273118
$561$	−77.8895	−0.00586185
$562$	5974.10	0.448403
$563$	−14078.4	−1.05388	−0.526939	−	0.849903i	$-0.676660\pi$
$563$	−14078.4	−1.05388	−0.526939	+	0.849903i	$0.676660\pi$
$564$	−1333.77	−0.0995778
$565$	−41699.7	−3.10499
$566$	5155.02	0.382830
$567$	−146.569	−0.0108560
$568$	22633.8	1.67200
$569$	−7165.14	−0.527905	−0.263953	−	0.964536i	$-0.585026\pi$
$569$	−7165.14	−0.527905	−0.263953	+	0.964536i	$0.585026\pi$
$570$	−13188.2	−0.969112
$571$	15291.1	1.12069	0.560344	−	0.828260i	$-0.310669\pi$
$571$	15291.1	1.12069	0.560344	+	0.828260i	$0.310669\pi$
$572$	613.180	0.0448223
$573$	1127.73	0.0822192
$574$	−211.337	−0.0153677
$575$	31726.2	2.30100
$576$	−11739.4	−0.849201
$577$	10271.5	0.741089	0.370545	−	0.928815i	$-0.379171\pi$
$577$	10271.5	0.741089	0.370545	+	0.928815i	$0.379171\pi$
$578$	11426.6	0.822290
$579$	−9468.97	−0.679649
$580$	9668.23	0.692157
$581$	−139.028	−0.00992749
$582$	−3550.77	−0.252894
$583$	173.502	0.0123254
$584$	−2430.53	−0.172219
$585$	−19197.0	−1.35675
$586$	−8151.66	−0.574645
$587$	5869.33	0.412697	0.206349	−	0.978479i	$-0.433842\pi$
$587$	5869.33	0.412697	0.206349	+	0.978479i	$0.433842\pi$
$588$	2081.64	0.145996
$589$	−20396.0	−1.42683
$590$	−33838.0	−2.36117
$591$	5231.78	0.364140
$592$	−13783.2	−0.956900
$593$	−13637.1	−0.944367	−0.472184	−	0.881500i	$-0.656534\pi$
$593$	−13637.1	−0.944367	−0.472184	+	0.881500i	$0.656534\pi$
$594$	1381.53	0.0954289
$595$	−60.9854	−0.00420195
$596$	−5017.15	−0.344816
$597$	−7381.38	−0.506030
$598$	16081.3	1.09968
$599$	2989.04	0.203888	0.101944	−	0.994790i	$-0.467494\pi$
$599$	2989.04	0.203888	0.101944	+	0.994790i	$0.467494\pi$
$600$	13457.1	0.915639
$601$	3752.57	0.254693	0.127347	−	0.991858i	$-0.459354\pi$
$601$	3752.57	0.254693	0.127347	+	0.991858i	$0.459354\pi$
$602$	0	0
$603$	13606.2	0.918883
$604$	8639.96	0.582045
$605$	24424.8	1.64134
$606$	−768.115	−0.0514893
$607$	−9165.95	−0.612906	−0.306453	−	0.951886i	$-0.599142\pi$
$607$	−9165.95	−0.612906	−0.306453	+	0.951886i	$0.599142\pi$
$608$	13426.6	0.895593
$609$	−257.312	−0.0171212
$610$	18965.5	1.25884
$611$	−10679.4	−0.707108
$612$	−335.548	−0.0221630
$613$	8403.46	0.553691	0.276846	−	0.960914i	$-0.410711\pi$
$613$	8403.46	0.553691	0.276846	+	0.960914i	$0.410711\pi$
$614$	−4365.08	−0.286906
$615$	7988.93	0.523812
$616$	−63.6474	−0.00416303
$617$	−14381.6	−0.938380	−0.469190	−	0.883097i	$-0.655454\pi$
$617$	−14381.6	−0.938380	−0.469190	+	0.883097i	$0.655454\pi$
$618$	1401.45	0.0912212
$619$	20695.9	1.34384	0.671921	−	0.740623i	$-0.265470\pi$
$619$	20695.9	1.34384	0.671921	+	0.740623i	$0.265470\pi$
$620$	7709.47	0.499386
$621$	−16474.3	−1.06456
$622$	18175.1	1.17163
$623$	59.2125	0.00380786
$624$	4458.17	0.286009
$625$	6872.81	0.439860
$626$	3927.64	0.250767
$627$	−1515.79	−0.0965465
$628$	−951.127	−0.0604365
$629$	−2322.43	−0.147220
$630$	474.514	0.0300081
$631$	18686.0	1.17889	0.589444	−	0.807809i	$-0.299347\pi$
$631$	18686.0	1.17889	0.589444	+	0.807809i	$0.299347\pi$
$632$	10143.7	0.638439
$633$	2354.81	0.147860
$634$	11290.1	0.707233
$635$	−8480.29	−0.529969
$636$	−208.982	−0.0130294
$637$	16667.6	1.03673
$638$	−2443.89	−0.151653
$639$	−19394.5	−1.20068
$640$	8171.81	0.504717
$641$	2741.94	0.168955	0.0844774	−	0.996425i	$-0.473078\pi$
$641$	2741.94	0.168955	0.0844774	+	0.996425i	$0.473078\pi$
$642$	3503.39	0.215371
$643$	−9202.37	−0.564395	−0.282198	−	0.959356i	$-0.591063\pi$
$643$	−9202.37	−0.564395	−0.282198	+	0.959356i	$0.591063\pi$
$644$	180.739	0.0110592
$645$	0	0
$646$	−1845.76	−0.112415
$647$	−8101.38	−0.492269	−0.246135	−	0.969236i	$-0.579161\pi$
$647$	−8101.38	−0.492269	−0.246135	+	0.969236i	$0.579161\pi$
$648$	7041.21	0.426859
$649$	−3889.16	−0.235228
$650$	25659.1	1.54836
$651$	−205.181	−0.0123528
$652$	−2670.02	−0.160378
$653$	−2590.36	−0.155235	−0.0776175	−	0.996983i	$-0.524731\pi$
$653$	−2590.36	−0.155235	−0.0776175	+	0.996983i	$0.524731\pi$
$654$	−10600.0	−0.633779
$655$	−18628.8	−1.11128
$656$	6635.66	0.394937
$657$	2082.67	0.123673
$658$	263.976	0.0156396
$659$	−5952.63	−0.351869	−0.175934	−	0.984402i	$-0.556295\pi$
$659$	−5952.63	−0.351869	−0.175934	+	0.984402i	$0.556295\pi$
$660$	572.949	0.0337909
$661$	−23326.7	−1.37262	−0.686311	−	0.727309i	$-0.740771\pi$
$661$	−23326.7	−1.37262	−0.686311	+	0.727309i	$0.740771\pi$
$662$	−25441.6	−1.49368
$663$	751.190	0.0440027
$664$	6678.94	0.390351
$665$	−1186.82	−0.0692073
$666$	18070.3	1.05137
$667$	29142.7	1.69177
$668$	10583.5	0.613005
$669$	1085.19	0.0627140
$670$	−28290.1	−1.63126
$671$	2179.79	0.125410
$672$	135.069	0.00775360
$673$	−20507.9	−1.17462	−0.587311	−	0.809362i	$-0.699813\pi$
$673$	−20507.9	−1.17462	−0.587311	+	0.809362i	$0.699813\pi$
$674$	20461.4	1.16935
$675$	−26286.2	−1.49890
$676$	−419.988	−0.0238956
$677$	−29831.9	−1.69355	−0.846774	−	0.531953i	$-0.821458\pi$
$677$	−29831.9	−1.69355	−0.846774	+	0.531953i	$0.821458\pi$
$678$	−12696.2	−0.719164
$679$	−319.537	−0.0180600
$680$	2929.74	0.165221
$681$	−1340.78	−0.0754460
$682$	−1948.77	−0.109417
$683$	−6248.79	−0.350078	−0.175039	−	0.984562i	$-0.556005\pi$
$683$	−6248.79	−0.350078	−0.175039	+	0.984562i	$0.556005\pi$
$684$	−6530.00	−0.365031
$685$	−46606.7	−2.59964
$686$	−824.300	−0.0458774
$687$	−956.489	−0.0531184
$688$	0	0
$689$	−1673.31	−0.0925225
$690$	15026.2	0.829037
$691$	17494.1	0.963107	0.481554	−	0.876417i	$-0.340073\pi$
$691$	17494.1	0.963107	0.481554	+	0.876417i	$0.340073\pi$
$692$	9045.45	0.496902
$693$	54.5382	0.00298951
$694$	13954.7	0.763274
$695$	36514.7	1.99292
$696$	12361.3	0.673208
$697$	1118.09	0.0607614
$698$	12107.7	0.656566
$699$	813.880	0.0440397
$700$	288.385	0.0155713
$701$	25115.2	1.35319	0.676596	−	0.736354i	$-0.263454\pi$
$701$	25115.2	1.35319	0.676596	+	0.736354i	$0.263454\pi$
$702$	−13323.9	−0.716349
$703$	−45196.1	−2.42476
$704$	2805.39	0.150188
$705$	−9978.74	−0.533080
$706$	14804.5	0.789200
$707$	−69.1233	−0.00367701
$708$	4684.46	0.248662
$709$	18394.0	0.974331	0.487165	−	0.873310i	$-0.338031\pi$
$709$	18394.0	0.974331	0.487165	+	0.873310i	$0.338031\pi$
$710$	40325.1	2.13151
$711$	−8691.90	−0.458469
$712$	−2844.57	−0.149726
$713$	23238.5	1.22060
$714$	−18.5680	−0.000973237 0
$715$	4587.57	0.239952
$716$	1608.17	0.0839387
$717$	−13632.5	−0.710063
$718$	4911.81	0.255302
$719$	30052.0	1.55876	0.779382	−	0.626549i	$-0.215533\pi$
$719$	30052.0	1.55876	0.779382	+	0.626549i	$0.215533\pi$
$720$	−14899.0	−0.771183
$721$	126.118	0.00651439
$722$	−19834.8	−1.02240
$723$	7158.99	0.368251
$724$	−7939.07	−0.407532
$725$	46499.8	2.38201
$726$	7436.54	0.380160
$727$	4166.51	0.212555	0.106277	−	0.994337i	$-0.466107\pi$
$727$	4166.51	0.212555	0.106277	+	0.994337i	$0.466107\pi$
$728$	613.835	0.0312503
$729$	1628.36	0.0827293
$730$	−4330.31	−0.219551
$731$	0	0
$732$	−2625.54	−0.132572
$733$	23201.4	1.16912	0.584559	−	0.811351i	$-0.301267\pi$
$733$	23201.4	1.16912	0.584559	+	0.811351i	$0.301267\pi$
$734$	−25963.5	−1.30563
$735$	15574.0	0.781574
$736$	−15297.8	−0.766145
$737$	−3251.51	−0.162512
$738$	−8699.62	−0.433926
$739$	27055.3	1.34675	0.673373	−	0.739303i	$-0.264845\pi$
$739$	27055.3	1.34675	0.673373	+	0.739303i	$0.264845\pi$
$740$	17083.6	0.848656
$741$	14618.7	0.724738
$742$	41.3611	0.00204638
$743$	16610.7	0.820172	0.410086	−	0.912047i	$-0.365499\pi$
$743$	16610.7	0.820172	0.410086	+	0.912047i	$0.365499\pi$
$744$	9856.91	0.485715
$745$	−37536.3	−1.84594
$746$	−3559.58	−0.174699
$747$	−5723.05	−0.280315
$748$	80.1871	0.00391969
$749$	315.273	0.0153803
$750$	10655.4	0.518774
$751$	−5199.15	−0.252623	−0.126311	−	0.991991i	$-0.540314\pi$
$751$	−5199.15	−0.252623	−0.126311	+	0.991991i	$0.540314\pi$
$752$	−8288.41	−0.401924
$753$	3504.90	0.169622
$754$	23569.7	1.13840
$755$	64640.7	3.11592
$756$	−149.748	−0.00720409
$757$	30028.9	1.44177	0.720884	−	0.693056i	$-0.243736\pi$
$757$	30028.9	1.44177	0.720884	+	0.693056i	$0.243736\pi$
$758$	−31754.9	−1.52162
$759$	1727.03	0.0825917
$760$	57014.9	2.72125
$761$	−13045.8	−0.621433	−0.310716	−	0.950503i	$-0.600569\pi$
$761$	−13045.8	−0.621433	−0.310716	+	0.950503i	$0.600569\pi$
$762$	−2581.97	−0.122749
$763$	−953.900	−0.0452602
$764$	−1161.00	−0.0549782
$765$	−2510.44	−0.118647
$766$	−2721.83	−0.128386
$767$	37508.2	1.76577
$768$	−8322.63	−0.391038
$769$	18935.3	0.887938	0.443969	−	0.896042i	$-0.353570\pi$
$769$	18935.3	0.887938	0.443969	+	0.896042i	$0.353570\pi$
$770$	−113.396	−0.00530716
$771$	−17961.1	−0.838978
$772$	9748.28	0.454467
$773$	7052.59	0.328155	0.164078	−	0.986447i	$-0.447535\pi$
$773$	7052.59	0.328155	0.164078	+	0.986447i	$0.447535\pi$
$774$	0	0
$775$	37079.1	1.71861
$776$	15350.6	0.710121
$777$	−454.665	−0.0209923
$778$	4984.28	0.229685
$779$	21758.8	1.00076
$780$	−5525.69	−0.253656
$781$	4634.76	0.212349
$782$	2102.99	0.0961670
$783$	−24145.7	−1.10204
$784$	12935.9	0.589281
$785$	−7115.95	−0.323540
$786$	−5671.86	−0.257390
$787$	31343.9	1.41968	0.709841	−	0.704362i	$-0.248767\pi$
$787$	31343.9	1.41968	0.709841	+	0.704362i	$0.248767\pi$
$788$	−5386.11	−0.243492
$789$	−18857.0	−0.850858
$790$	18072.3	0.813902
$791$	−1142.54	−0.0513578
$792$	−2620.02	−0.117548
$793$	−21022.6	−0.941405
$794$	−2386.01	−0.106645
$795$	−1563.52	−0.0697515
$796$	7599.12	0.338371
$797$	35516.9	1.57851	0.789255	−	0.614066i	$-0.210467\pi$
$797$	35516.9	1.57851	0.789255	+	0.614066i	$0.210467\pi$
$798$	−361.347	−0.0160295
$799$	−1396.58	−0.0618364
$800$	−24408.9	−1.07873
$801$	2437.46	0.107520
$802$	−10638.1	−0.468385
$803$	−497.704	−0.0218725
$804$	3916.42	0.171793
$805$	1352.22	0.0592042
$806$	18794.5	0.821350
$807$	15571.4	0.679232
$808$	3320.69	0.144581
$809$	−18244.6	−0.792886	−0.396443	−	0.918059i	$-0.629756\pi$
$809$	−18244.6	−0.792886	−0.396443	+	0.918059i	$0.629756\pi$
$810$	12544.8	0.544174
$811$	−26372.1	−1.14186	−0.570930	−	0.820999i	$-0.693417\pi$
$811$	−26372.1	−1.14186	−0.570930	+	0.820999i	$0.693417\pi$
$812$	264.902	0.0114486
$813$	3010.82	0.129882
$814$	−4318.32	−0.185942
$815$	−19976.1	−0.858565
$816$	583.006	0.0250114
$817$	0	0
$818$	29597.3	1.26509
$819$	−525.982	−0.0224412
$820$	−8224.58	−0.350262
$821$	−3948.12	−0.167832	−0.0839162	−	0.996473i	$-0.526743\pi$
$821$	−3948.12	−0.167832	−0.0839162	+	0.996473i	$0.526743\pi$
$822$	−14190.2	−0.602116
$823$	−36600.5	−1.55020	−0.775100	−	0.631839i	$-0.782301\pi$
$823$	−36600.5	−1.55020	−0.775100	+	0.631839i	$0.782301\pi$
$824$	−6058.71	−0.256147
$825$	2755.63	0.116289
$826$	−927.134	−0.0390546
$827$	8511.44	0.357886	0.178943	−	0.983859i	$-0.442732\pi$
$827$	8511.44	0.357886	0.178943	+	0.983859i	$0.442732\pi$
$828$	7440.04	0.312270
$829$	−20202.2	−0.846382	−0.423191	−	0.906041i	$-0.639090\pi$
$829$	−20202.2	−0.846382	−0.423191	+	0.906041i	$0.639090\pi$
$830$	11899.4	0.497632
$831$	3074.95	0.128362
$832$	−27056.1	−1.12740
$833$	2179.66	0.0906613
$834$	11117.5	0.461592
$835$	79181.4	3.28166
$836$	1560.50	0.0645585
$837$	−19253.9	−0.795115
$838$	18622.2	0.767652
$839$	5893.79	0.242522	0.121261	−	0.992621i	$-0.461306\pi$
$839$	5893.79	0.242522	0.121261	+	0.992621i	$0.461306\pi$
$840$	573.561	0.0235592
$841$	18324.3	0.751336
$842$	−7613.48	−0.311613
$843$	−6187.61	−0.252803
$844$	−2424.28	−0.0988709
$845$	−3142.19	−0.127922
$846$	10866.4	0.441603
$847$	669.221	0.0271484
$848$	−1298.67	−0.0525903
$849$	−5339.25	−0.215833
$850$	3355.50	0.135403
$851$	51494.7	2.07428
$852$	−5582.53	−0.224477
$853$	29634.1	1.18951	0.594756	−	0.803906i	$-0.297249\pi$
$853$	29634.1	1.18951	0.594756	+	0.803906i	$0.297249\pi$
$854$	519.639	0.0208217
$855$	−48854.9	−1.95415
$856$	−15145.8	−0.604756
$857$	18851.6	0.751411	0.375705	−	0.926739i	$-0.377401\pi$
$857$	18851.6	0.751411	0.375705	+	0.926739i	$0.377401\pi$
$858$	1396.76	0.0555765
$859$	8874.89	0.352511	0.176256	−	0.984344i	$-0.443601\pi$
$859$	8874.89	0.352511	0.176256	+	0.984344i	$0.443601\pi$
$860$	0	0
$861$	218.890	0.00866407
$862$	23507.3	0.928842
$863$	−34956.5	−1.37883	−0.689417	−	0.724365i	$-0.742133\pi$
$863$	−34956.5	−1.37883	−0.689417	+	0.724365i	$0.742133\pi$
$864$	12674.7	0.499077
$865$	67674.5	2.66012
$866$	−22570.7	−0.885660
$867$	−11835.0	−0.463594
$868$	211.233	0.00826005
$869$	2077.13	0.0810839
$870$	22023.2	0.858227
$871$	31358.6	1.21991
$872$	45825.4	1.77964
$873$	−13153.6	−0.509945
$874$	40925.6	1.58390
$875$	958.889	0.0370473
$876$	599.480	0.0231216
$877$	−21331.6	−0.821341	−0.410670	−	0.911784i	$-0.634705\pi$
$877$	−21331.6	−0.821341	−0.410670	+	0.911784i	$0.634705\pi$
$878$	17190.6	0.660767
$879$	8442.98	0.323976
$880$	3560.46	0.136390
$881$	28864.1	1.10381	0.551905	−	0.833907i	$-0.313901\pi$
$881$	28864.1	1.10381	0.551905	+	0.833907i	$0.313901\pi$
$882$	−16959.5	−0.647455
$883$	−14623.5	−0.557327	−0.278663	−	0.960389i	$-0.589891\pi$
$883$	−14623.5	−0.557327	−0.278663	+	0.960389i	$0.589891\pi$
$884$	−773.349	−0.0294237
$885$	35047.3	1.33119
$886$	2538.58	0.0962589
$887$	−31615.5	−1.19678	−0.598390	−	0.801205i	$-0.704193\pi$
$887$	−31615.5	−1.19678	−0.598390	+	0.801205i	$0.704193\pi$
$888$	21842.2	0.825422
$889$	−232.353	−0.00876589
$890$	−5067.98	−0.190875
$891$	1441.84	0.0542126
$892$	−1117.20	−0.0419355
$893$	−27178.3	−1.01846
$894$	−11428.6	−0.427548
$895$	12031.7	0.449357
$896$	223.901	0.00834823
$897$	−16656.0	−0.619985
$898$	−13893.5	−0.516293
$899$	34059.7	1.26358
$900$	11871.2	0.439676
$901$	−218.823	−0.00809106
$902$	2078.98	0.0767432
$903$	0	0
$904$	54887.7	2.01940
$905$	−59397.0	−2.18168
$906$	19681.0	0.721695
$907$	−32362.1	−1.18475	−0.592373	−	0.805664i	$-0.701809\pi$
$907$	−32362.1	−1.18475	−0.592373	+	0.805664i	$0.701809\pi$
$908$	1380.33	0.0504491
$909$	−2845.43	−0.103825
$910$	1093.63	0.0398389
$911$	35154.1	1.27849	0.639247	−	0.769002i	$-0.279246\pi$
$911$	35154.1	1.27849	0.639247	+	0.769002i	$0.279246\pi$
$912$	11345.7	0.411945
$913$	1367.66	0.0495759
$914$	21558.1	0.780172
$915$	−19643.3	−0.709712
$916$	984.703	0.0355191
$917$	−510.416	−0.0183810
$918$	−1742.40	−0.0626445
$919$	−7359.98	−0.264182	−0.132091	−	0.991238i	$-0.542169\pi$
$919$	−7359.98	−0.264182	−0.132091	+	0.991238i	$0.542169\pi$
$920$	−64960.6	−2.32792
$921$	4521.08	0.161753
$922$	2378.02	0.0849415
$923$	−44699.0	−1.59403
$924$	15.6983	0.000558915 0
$925$	82164.4	2.92059
$926$	−27154.2	−0.963654
$927$	5191.59	0.183942
$928$	−22421.3	−0.793121
$929$	−26410.1	−0.932708	−0.466354	−	0.884598i	$-0.654433\pi$
$929$	−26410.1	−0.932708	−0.466354	+	0.884598i	$0.654433\pi$
$930$	17561.4	0.619205
$931$	42417.8	1.49322
$932$	−837.888	−0.0294484
$933$	−18824.7	−0.660548
$934$	26919.3	0.943070
$935$	599.928	0.0209837
$936$	25268.2	0.882391
$937$	−26123.6	−0.910800	−0.455400	−	0.890287i	$-0.650504\pi$
$937$	−26123.6	−0.910800	−0.455400	+	0.890287i	$0.650504\pi$
$938$	−775.126	−0.0269816
$939$	−4068.00	−0.141378
$940$	10273.1	0.356459
$941$	6602.83	0.228742	0.114371	−	0.993438i	$-0.463515\pi$
$941$	6602.83	0.228742	0.114371	+	0.993438i	$0.463515\pi$
$942$	−2166.57	−0.0749370
$943$	−24791.2	−0.856110
$944$	29110.5	1.00367
$945$	−1120.36	−0.0385664
$946$	0	0
$947$	−9913.13	−0.340162	−0.170081	−	0.985430i	$-0.554403\pi$
$947$	−9913.13	−0.340162	−0.170081	+	0.985430i	$0.554403\pi$
$948$	−2501.89	−0.0857148
$949$	4800.00	0.164188
$950$	65300.4	2.23013
$951$	−11693.6	−0.398727
$952$	80.2727	0.00273283
$953$	12334.8	0.419268	0.209634	−	0.977780i	$-0.432773\pi$
$953$	12334.8	0.419268	0.209634	+	0.977780i	$0.432773\pi$
$954$	1702.61	0.0577821
$955$	−8686.10	−0.294320
$956$	14034.6	0.474804
$957$	2531.24	0.0854997
$958$	−26523.3	−0.894497
$959$	−1276.99	−0.0429990
$960$	−25280.9	−0.849935
$961$	−2631.71	−0.0883389
$962$	41647.2	1.39580
$963$	12978.1	0.434282
$964$	−7370.16	−0.246242
$965$	72932.7	2.43294
$966$	411.705	0.0137126
$967$	−43542.0	−1.44800	−0.724000	−	0.689800i	$-0.757699\pi$
$967$	−43542.0	−1.44800	−0.724000	+	0.689800i	$0.757699\pi$
$968$	−32149.4	−1.06748
$969$	1911.72	0.0633781
$970$	27349.1	0.905285
$971$	−47860.8	−1.58180	−0.790900	−	0.611946i	$-0.790387\pi$
$971$	−47860.8	−1.58180	−0.790900	+	0.611946i	$0.790387\pi$
$972$	−9624.52	−0.317599
$973$	1000.47	0.0329638
$974$	−32696.3	−1.07562
$975$	−26576.1	−0.872940
$976$	−16315.8	−0.535100
$977$	2411.64	0.0789716	0.0394858	−	0.999220i	$-0.487428\pi$
$977$	2411.64	0.0789716	0.0394858	+	0.999220i	$0.487428\pi$
$978$	−6082.04	−0.198857
$979$	−582.487	−0.0190157
$980$	−16033.4	−0.522621
$981$	−39266.9	−1.27798
$982$	15512.3	0.504090
$983$	−15548.6	−0.504500	−0.252250	−	0.967662i	$-0.581171\pi$
$983$	−15548.6	−0.504500	−0.252250	+	0.967662i	$0.581171\pi$
$984$	−10515.5	−0.340673
$985$	−40296.7	−1.30351
$986$	3082.26	0.0995530
$987$	−273.410	−0.00881735
$988$	−15049.9	−0.484617
$989$	0	0
$990$	−4667.91	−0.149854
$991$	−679.692	−0.0217872	−0.0108936	−	0.999941i	$-0.503468\pi$
$991$	−679.692	−0.0217872	−0.0108936	+	0.999941i	$0.503468\pi$
$992$	−17878.8	−0.572231
$993$	26350.8	0.842114
$994$	1104.88	0.0352561
$995$	56853.6	1.81144
$996$	−1647.33	−0.0524073
$997$	−42549.5	−1.35161	−0.675806	−	0.737080i	$-0.736204\pi$
$997$	−42549.5	−1.35161	−0.675806	+	0.737080i	$0.736204\pi$
$998$	2405.51	0.0762976
$999$	−42665.1	−1.35122

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.15	yes	50
43.42	odd	2		1849.4.a.i.1.36	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.36	✓	50	43.42	odd	2
1849.4.a.j.1.15	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)

\(f(q)\)	\(=\)	\(q-2.34509 q^{2} +2.42890 q^{3} -2.50055 q^{4} -18.7081 q^{5} -5.69599 q^{6} -0.512587 q^{7} +24.6247 q^{8} -21.1004 q^{9} +O(q^{10})\)	\(q-2.34509 q^{2} +2.42890 q^{3} -2.50055 q^{4} -18.7081 q^{5} -5.69599 q^{6} -0.512587 q^{7} +24.6247 q^{8} -21.1004 q^{9} +43.8722 q^{10} +5.04244 q^{11} -6.07358 q^{12} -48.6308 q^{13} +1.20206 q^{14} -45.4401 q^{15} -37.7429 q^{16} -6.35958 q^{17} +49.4825 q^{18} -123.762 q^{19} +46.7805 q^{20} -1.24502 q^{21} -11.8250 q^{22} +141.010 q^{23} +59.8111 q^{24} +224.993 q^{25} +114.044 q^{26} -116.831 q^{27} +1.28175 q^{28} +206.672 q^{29} +106.561 q^{30} +164.801 q^{31} -108.487 q^{32} +12.2476 q^{33} +14.9138 q^{34} +9.58954 q^{35} +52.7627 q^{36} +365.186 q^{37} +290.233 q^{38} -118.119 q^{39} -460.682 q^{40} -175.812 q^{41} +2.91969 q^{42} -12.6089 q^{44} +394.749 q^{45} -330.680 q^{46} +219.602 q^{47} -91.6737 q^{48} -342.737 q^{49} -527.630 q^{50} -15.4468 q^{51} +121.604 q^{52} +34.4084 q^{53} +273.980 q^{54} -94.3346 q^{55} -12.6223 q^{56} -300.605 q^{57} -484.665 q^{58} -771.285 q^{59} +113.625 q^{60} +432.289 q^{61} -386.473 q^{62} +10.8158 q^{63} +556.356 q^{64} +909.791 q^{65} -28.7217 q^{66} -644.829 q^{67} +15.9024 q^{68} +342.498 q^{69} -22.4883 q^{70} +919.149 q^{71} -519.593 q^{72} -98.7029 q^{73} -856.394 q^{74} +546.486 q^{75} +309.472 q^{76} -2.58469 q^{77} +277.001 q^{78} +411.930 q^{79} +706.098 q^{80} +285.940 q^{81} +412.295 q^{82} +271.229 q^{83} +3.11324 q^{84} +118.976 q^{85} +501.986 q^{87} +124.169 q^{88} -115.517 q^{89} -925.723 q^{90} +24.9276 q^{91} -352.601 q^{92} +400.285 q^{93} -514.986 q^{94} +2315.35 q^{95} -263.505 q^{96} +623.381 q^{97} +803.750 q^{98} -106.398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) \) 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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