LMFDB - Embedded newform 1849.4.a.i.1.35

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			1.35
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)

$f(q)$	$=$	$q+2.12255 q^{2} -9.87196 q^{3} -3.49478 q^{4} +15.5747 q^{5} -20.9537 q^{6} +16.3831 q^{7} -24.3983 q^{8} +70.4556 q^{9} +O(q^{10})$	\(q+2.12255 q^{2} -9.87196 q^{3} -3.49478 q^{4} +15.5747 q^{5} -20.9537 q^{6} +16.3831 q^{7} -24.3983 q^{8} +70.4556 q^{9} +33.0580 q^{10} +42.7049 q^{11} +34.5003 q^{12} -44.4137 q^{13} +34.7739 q^{14} -153.752 q^{15} -23.8283 q^{16} -80.6618 q^{17} +149.546 q^{18} -9.72641 q^{19} -54.4300 q^{20} -161.733 q^{21} +90.6434 q^{22} +81.9683 q^{23} +240.859 q^{24} +117.570 q^{25} -94.2703 q^{26} -428.992 q^{27} -57.2552 q^{28} -170.553 q^{29} -326.348 q^{30} -115.463 q^{31} +144.609 q^{32} -421.581 q^{33} -171.209 q^{34} +255.161 q^{35} -246.227 q^{36} -56.3932 q^{37} -20.6448 q^{38} +438.450 q^{39} -379.995 q^{40} +9.51295 q^{41} -343.287 q^{42} -149.244 q^{44} +1097.32 q^{45} +173.982 q^{46} +9.55328 q^{47} +235.232 q^{48} -74.5945 q^{49} +249.549 q^{50} +796.290 q^{51} +155.216 q^{52} +254.029 q^{53} -910.558 q^{54} +665.115 q^{55} -399.719 q^{56} +96.0187 q^{57} -362.007 q^{58} +288.741 q^{59} +537.331 q^{60} +67.9815 q^{61} -245.075 q^{62} +1154.28 q^{63} +497.567 q^{64} -691.728 q^{65} -894.828 q^{66} +273.745 q^{67} +281.895 q^{68} -809.187 q^{69} +541.592 q^{70} +113.470 q^{71} -1718.99 q^{72} +908.508 q^{73} -119.698 q^{74} -1160.65 q^{75} +33.9916 q^{76} +699.638 q^{77} +930.632 q^{78} -798.537 q^{79} -371.118 q^{80} +2332.69 q^{81} +20.1917 q^{82} -1273.00 q^{83} +565.221 q^{84} -1256.28 q^{85} +1683.69 q^{87} -1041.93 q^{88} -1394.34 q^{89} +2329.12 q^{90} -727.633 q^{91} -286.461 q^{92} +1139.84 q^{93} +20.2773 q^{94} -151.486 q^{95} -1427.58 q^{96} +253.183 q^{97} -158.331 q^{98} +3008.80 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.12255	0.750435	0.375218	−	0.926937i	$-0.377568\pi$
$2$	2.12255	0.750435	0.375218	+	0.926937i	$0.377568\pi$
$3$	−9.87196	−1.89986	−0.949930	−	0.312463i	$-0.898846\pi$
$3$	−9.87196	−1.89986	−0.949930	+	0.312463i	$0.898846\pi$
$4$	−3.49478	−0.436847
$5$	15.5747	1.39304	0.696520	−	0.717537i	$-0.254731\pi$
$5$	15.5747	1.39304	0.696520	+	0.717537i	$0.254731\pi$
$6$	−20.9537	−1.42572
$7$	16.3831	0.884603	0.442302	−	0.896866i	$-0.354162\pi$
$7$	16.3831	0.884603	0.442302	+	0.896866i	$0.354162\pi$
$8$	−24.3983	−1.07826
$9$	70.4556	2.60947
$10$	33.0580	1.04539
$11$	42.7049	1.17055	0.585274	−	0.810836i	$-0.300987\pi$
$11$	42.7049	1.17055	0.585274	+	0.810836i	$0.300987\pi$
$12$	34.5003	0.829948
$13$	−44.4137	−0.947549	−0.473774	−	0.880646i	$-0.657109\pi$
$13$	−44.4137	−0.947549	−0.473774	+	0.880646i	$0.657109\pi$
$14$	34.7739	0.663837
$15$	−153.752	−2.64658
$16$	−23.8283	−0.372318
$17$	−80.6618	−1.15079	−0.575393	−	0.817877i	$-0.695151\pi$
$17$	−80.6618	−1.15079	−0.575393	+	0.817877i	$0.695151\pi$
$18$	149.546	1.95824
$19$	−9.72641	−0.117442	−0.0587208	−	0.998274i	$-0.518702\pi$
$19$	−9.72641	−0.117442	−0.0587208	+	0.998274i	$0.518702\pi$
$20$	−54.4300	−0.608546
$21$	−161.733	−1.68062
$22$	90.6434	0.878420
$23$	81.9683	0.743111	0.371556	−	0.928411i	$-0.378824\pi$
$23$	81.9683	0.743111	0.371556	+	0.928411i	$0.378824\pi$
$24$	240.859	2.04854
$25$	117.570	0.940562
$26$	−94.2703	−0.711074
$27$	−428.992	−3.05776
$28$	−57.2552	−0.386436
$29$	−170.553	−1.09210	−0.546049	−	0.837753i	$-0.683869\pi$
$29$	−170.553	−1.09210	−0.546049	+	0.837753i	$0.683869\pi$
$30$	−326.348	−1.98609
$31$	−115.463	−0.668958	−0.334479	−	0.942403i	$-0.608560\pi$
$31$	−115.463	−0.668958	−0.334479	+	0.942403i	$0.608560\pi$
$32$	144.609	0.798860
$33$	−421.581	−2.22388
$34$	−171.209	−0.863590
$35$	255.161	1.23229
$36$	−246.227	−1.13994
$37$	−56.3932	−0.250567	−0.125284	−	0.992121i	$-0.539984\pi$
$37$	−56.3932	−0.250567	−0.125284	+	0.992121i	$0.539984\pi$
$38$	−20.6448	−0.0881323
$39$	438.450	1.80021
$40$	−379.995	−1.50206
$41$	9.51295	0.0362359	0.0181180	−	0.999836i	$-0.494233\pi$
$41$	9.51295	0.0362359	0.0181180	+	0.999836i	$0.494233\pi$
$42$	−343.287	−1.26120
$43$	0	0
$43$	0	0
$44$	−149.244	−0.511350
$45$	1097.32	3.63509
$46$	173.982	0.557657
$47$	9.55328	0.0296487	0.0148243	−	0.999890i	$-0.495281\pi$
$47$	9.55328	0.0296487	0.0148243	+	0.999890i	$0.495281\pi$
$48$	235.232	0.707351
$49$	−74.5945	−0.217477
$50$	249.549	0.705830
$51$	796.290	2.18633
$52$	155.216	0.413934
$53$	254.029	0.658370	0.329185	−	0.944265i	$-0.393226\pi$
$53$	254.029	0.658370	0.329185	+	0.944265i	$0.393226\pi$
$54$	−910.558	−2.29465
$55$	665.115	1.63062
$56$	−399.719	−0.953833
$57$	96.0187	0.223123
$58$	−362.007	−0.819549
$59$	288.741	0.637133	0.318566	−	0.947901i	$-0.396799\pi$
$59$	288.741	0.637133	0.318566	+	0.947901i	$0.396799\pi$
$60$	537.331	1.15615
$61$	67.9815	0.142691	0.0713454	−	0.997452i	$-0.477271\pi$
$61$	67.9815	0.142691	0.0713454	+	0.997452i	$0.477271\pi$
$62$	−245.075	−0.502009
$63$	1154.28	2.30834
$64$	497.567	0.971810
$65$	−691.728	−1.31997
$66$	−894.828	−1.66887
$67$	273.745	0.499153	0.249577	−	0.968355i	$-0.419709\pi$
$67$	273.745	0.499153	0.249577	+	0.968355i	$0.419709\pi$
$68$	281.895	0.502717
$69$	−809.187	−1.41181
$70$	541.592	0.924752
$71$	113.470	0.189668	0.0948339	−	0.995493i	$-0.469768\pi$
$71$	113.470	0.189668	0.0948339	+	0.995493i	$0.469768\pi$
$72$	−1718.99	−2.81369
$73$	908.508	1.45661	0.728307	−	0.685251i	$-0.240307\pi$
$73$	908.508	1.45661	0.728307	+	0.685251i	$0.240307\pi$
$74$	−119.698	−0.188035
$75$	−1160.65	−1.78694
$76$	33.9916	0.0513040
$77$	699.638	1.03547
$78$	930.632	1.35094
$79$	−798.537	−1.13725	−0.568623	−	0.822598i	$-0.692524\pi$
$79$	−798.537	−1.13725	−0.568623	+	0.822598i	$0.692524\pi$
$80$	−371.118	−0.518653
$81$	2332.69	3.19985
$82$	20.1917	0.0271927
$83$	−1273.00	−1.68349	−0.841744	−	0.539878i	$-0.818471\pi$
$83$	−1273.00	−1.68349	−0.841744	+	0.539878i	$0.818471\pi$
$84$	565.221	0.734175
$85$	−1256.28	−1.60309
$86$	0	0
$87$	1683.69	2.07483
$88$	−1041.93	−1.26215
$89$	−1394.34	−1.66067	−0.830335	−	0.557265i	$-0.811851\pi$
$89$	−1394.34	−1.66067	−0.830335	+	0.557265i	$0.811851\pi$
$90$	2329.12	2.72790
$91$	−727.633	−0.838205
$92$	−286.461	−0.324626
$93$	1139.84	1.27093
$94$	20.2773	0.0222494
$95$	−151.486	−0.163601
$96$	−1427.58	−1.51772
$97$	253.183	0.265018	0.132509	−	0.991182i	$-0.457697\pi$
$97$	253.183	0.265018	0.132509	+	0.991182i	$0.457697\pi$
$98$	−158.331	−0.163202
$99$	3008.80	3.05450
$100$	−410.882	−0.410882
$101$	623.621	0.614383	0.307191	−	0.951648i	$-0.400611\pi$
$101$	623.621	0.614383	0.307191	+	0.951648i	$0.400611\pi$
$102$	1690.17	1.64070
$103$	−1534.71	−1.46815	−0.734076	−	0.679067i	$-0.762385\pi$
$103$	−1534.71	−1.46815	−0.734076	+	0.679067i	$0.762385\pi$
$104$	1083.62	1.02170
$105$	−2518.94	−2.34117
$106$	539.190	0.494064
$107$	978.902	0.884430	0.442215	−	0.896909i	$-0.354193\pi$
$107$	978.902	0.884430	0.442215	+	0.896909i	$0.354193\pi$
$108$	1499.23	1.33577
$109$	−615.263	−0.540656	−0.270328	−	0.962768i	$-0.587132\pi$
$109$	−615.263	−0.540656	−0.270328	+	0.962768i	$0.587132\pi$
$110$	1411.74	1.22367
$111$	556.712	0.476043
$112$	−390.381	−0.329353
$113$	1889.47	1.57298	0.786488	−	0.617605i	$-0.211897\pi$
$113$	1889.47	1.57298	0.786488	+	0.617605i	$0.211897\pi$
$114$	203.805	0.167439
$115$	1276.63	1.03518
$116$	596.044	0.477080
$117$	−3129.19	−2.47260
$118$	612.867	0.478127
$119$	−1321.49	−1.01799
$120$	3751.29	2.85370
$121$	492.710	0.370181
$122$	144.294	0.107080
$123$	−93.9115	−0.0688432
$124$	403.516	0.292232
$125$	−115.717	−0.0828002
$126$	2450.02	1.73226
$127$	−333.272	−0.232859	−0.116430	−	0.993199i	$-0.537145\pi$
$127$	−333.272	−0.232859	−0.116430	+	0.993199i	$0.537145\pi$
$128$	−100.762	−0.0695797
$129$	0	0
$130$	−1468.23	−0.990554
$131$	1298.39	0.865959	0.432979	−	0.901404i	$-0.357462\pi$
$131$	1298.39	0.865959	0.432979	+	0.901404i	$0.357462\pi$
$132$	1473.33	0.971493
$133$	−159.349	−0.103889
$134$	581.038	0.374582
$135$	−6681.41	−4.25959
$136$	1968.01	1.24085
$137$	−1989.18	−1.24049	−0.620245	−	0.784408i	$-0.712967\pi$
$137$	−1989.18	−1.24049	−0.620245	+	0.784408i	$0.712967\pi$
$138$	−1717.54	−1.05947
$139$	−1554.18	−0.948374	−0.474187	−	0.880424i	$-0.657258\pi$
$139$	−1554.18	−0.948374	−0.474187	+	0.880424i	$0.657258\pi$
$140$	−891.731	−0.538322
$141$	−94.3096	−0.0563284
$142$	240.846	0.142333
$143$	−1896.68	−1.10915
$144$	−1678.84	−0.971550
$145$	−2656.30	−1.52134
$146$	1928.36	1.09309
$147$	736.394	0.413175
$148$	197.082	0.109460
$149$	2018.71	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$149$	2018.71	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$150$	−2463.54	−1.34098
$151$	−219.429	−0.118258	−0.0591289	−	0.998250i	$-0.518832\pi$
$151$	−219.429	−0.118258	−0.0591289	+	0.998250i	$0.518832\pi$
$152$	237.307	0.126633
$153$	−5683.08	−3.00294
$154$	1485.02	0.777053
$155$	−1798.29	−0.931885
$156$	−1532.28	−0.786416
$157$	2441.15	1.24092	0.620462	−	0.784236i	$-0.286945\pi$
$157$	2441.15	1.24092	0.620462	+	0.784236i	$0.286945\pi$
$158$	−1694.94	−0.853430
$159$	−2507.77	−1.25081
$160$	2252.24	1.11284
$161$	1342.89	0.657359
$162$	4951.26	2.40128
$163$	−2934.10	−1.40992	−0.704959	−	0.709248i	$-0.749035\pi$
$163$	−2934.10	−1.40992	−0.704959	+	0.709248i	$0.749035\pi$
$164$	−33.2456	−0.0158296
$165$	−6565.99	−3.09795
$166$	−2702.00	−1.26335
$167$	−4127.64	−1.91261	−0.956306	−	0.292368i	$-0.905557\pi$
$167$	−4127.64	−1.91261	−0.956306	+	0.292368i	$0.905557\pi$
$168$	3946.01	1.81215
$169$	−224.427	−0.102152
$170$	−2666.52	−1.20302
$171$	−685.280	−0.306460
$172$	0	0
$173$	−803.747	−0.353224	−0.176612	−	0.984281i	$-0.556514\pi$
$173$	−803.747	−0.353224	−0.176612	+	0.984281i	$0.556514\pi$
$174$	3573.72	1.55703
$175$	1926.16	0.832024
$176$	−1017.59	−0.435815
$177$	−2850.44	−1.21046
$178$	−2959.55	−1.24622
$179$	−3352.74	−1.39998	−0.699988	−	0.714155i	$-0.746811\pi$
$179$	−3352.74	−1.39998	−0.699988	+	0.714155i	$0.746811\pi$
$180$	−3834.90	−1.58798
$181$	−4226.02	−1.73546	−0.867728	−	0.497040i	$-0.834420\pi$
$181$	−4226.02	−1.73546	−0.867728	+	0.497040i	$0.834420\pi$
$182$	−1544.44	−0.629018
$183$	−671.111	−0.271093
$184$	−1999.88	−0.801268
$185$	−878.306	−0.349050
$186$	2419.37	0.953748
$187$	−3444.66	−1.34705
$188$	−33.3866	−0.0129519
$189$	−7028.21	−2.70491
$190$	−321.536	−0.122772
$191$	3645.85	1.38118	0.690588	−	0.723249i	$-0.257352\pi$
$191$	3645.85	1.38118	0.690588	+	0.723249i	$0.257352\pi$
$192$	−4911.96	−1.84630
$193$	3201.57	1.19406	0.597032	−	0.802218i	$-0.296347\pi$
$193$	3201.57	1.19406	0.597032	+	0.802218i	$0.296347\pi$
$194$	537.393	0.198879
$195$	6828.71	2.50776
$196$	260.691	0.0950041
$197$	92.9451	0.0336145	0.0168073	−	0.999859i	$-0.494650\pi$
$197$	92.9451	0.0336145	0.0168073	+	0.999859i	$0.494650\pi$
$198$	6386.33	2.29221
$199$	97.0214	0.0345611	0.0172806	−	0.999851i	$-0.494499\pi$
$199$	97.0214	0.0345611	0.0172806	+	0.999851i	$0.494499\pi$
$200$	−2868.51	−1.01417
$201$	−2702.40	−0.948321
$202$	1323.67	0.461054
$203$	−2794.18	−0.966074
$204$	−2782.86	−0.955093
$205$	148.161	0.0504781
$206$	−3257.51	−1.10175
$207$	5775.12	1.93912
$208$	1058.30	0.352789
$209$	−415.365	−0.137471
$210$	−5346.58	−1.75690
$211$	−106.820	−0.0348522	−0.0174261	−	0.999848i	$-0.505547\pi$
$211$	−106.820	−0.0348522	−0.0174261	+	0.999848i	$0.505547\pi$
$212$	−887.776	−0.287607
$213$	−1120.17	−0.360342
$214$	2077.77	0.663708
$215$	0	0
$216$	10466.7	3.29706
$217$	−1891.63	−0.591762
$218$	−1305.93	−0.405727
$219$	−8968.76	−2.76736
$220$	−2324.43	−0.712331
$221$	3582.48	1.09043
$222$	1181.65	0.357239
$223$	3792.84	1.13896	0.569478	−	0.822006i	$-0.307145\pi$
$223$	3792.84	1.13896	0.569478	+	0.822006i	$0.307145\pi$
$224$	2369.14	0.706675
$225$	8283.48	2.45436
$226$	4010.50	1.18042
$227$	−2142.14	−0.626339	−0.313170	−	0.949697i	$-0.601391\pi$
$227$	−2142.14	−0.626339	−0.313170	+	0.949697i	$0.601391\pi$
$228$	−335.564	−0.0974705
$229$	−866.711	−0.250104	−0.125052	−	0.992150i	$-0.539910\pi$
$229$	−866.711	−0.250104	−0.125052	+	0.992150i	$0.539910\pi$
$230$	2709.71	0.776839
$231$	−6906.80	−1.96725
$232$	4161.19	1.17757
$233$	−2698.47	−0.758724	−0.379362	−	0.925248i	$-0.623856\pi$
$233$	−2698.47	−0.758724	−0.379362	+	0.925248i	$0.623856\pi$
$234$	−6641.87	−1.85552
$235$	148.789	0.0413018
$236$	−1009.08	−0.278330
$237$	7883.13	2.16061
$238$	−2804.93	−0.763935
$239$	−1856.90	−0.502564	−0.251282	−	0.967914i	$-0.580852\pi$
$239$	−1856.90	−0.502564	−0.251282	+	0.967914i	$0.580852\pi$
$240$	3663.66	0.985369
$241$	4891.12	1.30732	0.653661	−	0.756788i	$-0.273232\pi$
$241$	4891.12	1.30732	0.653661	+	0.756788i	$0.273232\pi$
$242$	1045.80	0.277796
$243$	−11445.5	−3.02151
$244$	−237.580	−0.0623341
$245$	−1161.78	−0.302954
$246$	−199.332	−0.0516623
$247$	431.985	0.111282
$248$	2817.09	0.721311
$249$	12567.0	3.19839
$250$	−245.615	−0.0621362
$251$	−5679.81	−1.42831	−0.714156	−	0.699987i	$-0.753189\pi$
$251$	−5679.81	−1.42831	−0.714156	+	0.699987i	$0.753189\pi$
$252$	−4033.95	−1.00839
$253$	3500.45	0.869847
$254$	−707.387	−0.174746
$255$	12402.0	3.04565
$256$	−4194.41	−1.02403
$257$	3095.65	0.751366	0.375683	−	0.926748i	$-0.377408\pi$
$257$	3095.65	0.751366	0.375683	+	0.926748i	$0.377408\pi$
$258$	0	0
$259$	−923.895	−0.221653
$260$	2417.43	0.576627
$261$	−12016.4	−2.84980
$262$	2755.89	0.649846
$263$	−2790.13	−0.654170	−0.327085	−	0.944995i	$-0.606066\pi$
$263$	−2790.13	−0.654170	−0.327085	+	0.944995i	$0.606066\pi$
$264$	10285.8	2.39792
$265$	3956.42	0.917136
$266$	−338.225	−0.0779622
$267$	13764.9	3.15504
$268$	−956.678	−0.218054
$269$	−8193.14	−1.85704	−0.928521	−	0.371279i	$-0.878919\pi$
$269$	−8193.14	−1.85704	−0.928521	+	0.371279i	$0.878919\pi$
$270$	−14181.6	−3.19654
$271$	−4239.14	−0.950218	−0.475109	−	0.879927i	$-0.657591\pi$
$271$	−4239.14	−0.950218	−0.475109	+	0.879927i	$0.657591\pi$
$272$	1922.04	0.428458
$273$	7183.16	1.59247
$274$	−4222.14	−0.930907
$275$	5020.83	1.10097
$276$	2827.93	0.616744
$277$	8528.69	1.84996	0.924981	−	0.380014i	$-0.124081\pi$
$277$	8528.69	1.84996	0.924981	+	0.380014i	$0.124081\pi$
$278$	−3298.83	−0.711693
$279$	−8134.99	−1.74562
$280$	−6225.48	−1.32873
$281$	−1287.15	−0.273257	−0.136628	−	0.990622i	$-0.543627\pi$
$281$	−1287.15	−0.273257	−0.136628	+	0.990622i	$0.543627\pi$
$282$	−200.177	−0.0422708
$283$	−8974.14	−1.88501	−0.942504	−	0.334194i	$-0.891536\pi$
$283$	−8974.14	−1.88501	−0.942504	+	0.334194i	$0.891536\pi$
$284$	−396.552	−0.0828558
$285$	1495.46	0.310819
$286$	−4025.80	−0.832345
$287$	155.851	0.0320544
$288$	10188.5	2.08460
$289$	1593.32	0.324308
$290$	−5638.14	−1.14167
$291$	−2499.41	−0.503498
$292$	−3175.03	−0.636318
$293$	1173.60	0.234002	0.117001	−	0.993132i	$-0.462672\pi$
$293$	1173.60	0.234002	0.117001	+	0.993132i	$0.462672\pi$
$294$	1563.03	0.310061
$295$	4497.04	0.887552
$296$	1375.90	0.270177
$297$	−18320.1	−3.57925
$298$	4284.81	0.832928
$299$	−3640.51	−0.704134
$300$	4056.21	0.780617
$301$	0	0
$302$	−465.750	−0.0887447
$303$	−6156.37	−1.16724
$304$	231.764	0.0437256
$305$	1058.79	0.198774
$306$	−12062.6	−2.25351
$307$	−8027.77	−1.49241	−0.746204	−	0.665718i	$-0.768126\pi$
$307$	−8027.77	−1.49241	−0.746204	+	0.665718i	$0.768126\pi$
$308$	−2445.08	−0.452342
$309$	15150.6	2.78928
$310$	−3816.97	−0.699320
$311$	5900.91	1.07592	0.537958	−	0.842972i	$-0.319196\pi$
$311$	5900.91	1.07592	0.537958	+	0.842972i	$0.319196\pi$
$312$	−10697.4	−1.94109
$313$	−7645.72	−1.38071	−0.690354	−	0.723471i	$-0.742545\pi$
$313$	−7645.72	−1.38071	−0.690354	+	0.723471i	$0.742545\pi$
$314$	5181.47	0.931234
$315$	17977.5	3.21562
$316$	2790.71	0.496803
$317$	8873.30	1.57216	0.786079	−	0.618126i	$-0.212108\pi$
$317$	8873.30	1.57216	0.786079	+	0.618126i	$0.212108\pi$
$318$	−5322.87	−0.938653
$319$	−7283.45	−1.27835
$320$	7749.44	1.35377
$321$	−9663.68	−1.68029
$322$	2850.36	0.493305
$323$	784.549	0.135150
$324$	−8152.24	−1.39785
$325$	−5221.72	−0.891228
$326$	−6227.78	−1.05805
$327$	6073.85	1.02717
$328$	−232.099	−0.0390718
$329$	156.512	0.0262273
$330$	−13936.6	−2.32481
$331$	−3998.27	−0.663943	−0.331971	−	0.943289i	$-0.607714\pi$
$331$	−3998.27	−0.663943	−0.331971	+	0.943289i	$0.607714\pi$
$332$	4448.84	0.735426
$333$	−3973.22	−0.653847
$334$	−8761.12	−1.43529
$335$	4263.49	0.695341
$336$	3853.83	0.625725
$337$	3151.54	0.509423	0.254711	−	0.967017i	$-0.418020\pi$
$337$	3151.54	0.509423	0.254711	+	0.967017i	$0.418020\pi$
$338$	−476.359	−0.0766583
$339$	−18652.8	−2.98844
$340$	4390.42	0.700306
$341$	−4930.82	−0.783047
$342$	−1454.54	−0.229978
$343$	−6841.49	−1.07698
$344$	0	0
$345$	−12602.8	−1.96670
$346$	−1705.99	−0.265072
$347$	2639.21	0.408300	0.204150	−	0.978940i	$-0.434557\pi$
$347$	2639.21	0.408300	0.204150	+	0.978940i	$0.434557\pi$
$348$	−5884.12	−0.906386
$349$	−8673.47	−1.33032	−0.665158	−	0.746702i	$-0.731636\pi$
$349$	−8673.47	−1.33032	−0.665158	+	0.746702i	$0.731636\pi$
$350$	4088.38	0.624380
$351$	19053.1	2.89738
$352$	6175.52	0.935104
$353$	−799.760	−0.120586	−0.0602931	−	0.998181i	$-0.519204\pi$
$353$	−799.760	−0.120586	−0.0602931	+	0.998181i	$0.519204\pi$
$354$	−6050.20	−0.908374
$355$	1767.26	0.264215
$356$	4872.90	0.725459
$357$	13045.7	1.93404
$358$	−7116.36	−1.05059
$359$	−6892.85	−1.01334	−0.506672	−	0.862139i	$-0.669125\pi$
$359$	−6892.85	−1.01334	−0.506672	+	0.862139i	$0.669125\pi$
$360$	−26772.8	−3.91958
$361$	−6764.40	−0.986207
$362$	−8969.94	−1.30235
$363$	−4864.02	−0.703291
$364$	2542.91	0.366167
$365$	14149.7	2.02912
$366$	−1424.47	−0.203437
$367$	10785.9	1.53412	0.767058	−	0.641578i	$-0.221720\pi$
$367$	10785.9	1.53412	0.767058	+	0.641578i	$0.221720\pi$
$368$	−1953.17	−0.276673
$369$	670.241	0.0945565
$370$	−1864.25	−0.261940
$371$	4161.79	0.582397
$372$	−3983.49	−0.555200
$373$	2319.02	0.321916	0.160958	−	0.986961i	$-0.448542\pi$
$373$	2319.02	0.321916	0.160958	+	0.986961i	$0.448542\pi$
$374$	−7311.46	−1.01087
$375$	1142.35	0.157309
$376$	−233.083	−0.0319690
$377$	7574.88	1.03482
$378$	−14917.7	−2.02986
$379$	−9170.77	−1.24293	−0.621466	−	0.783441i	$-0.713462\pi$
$379$	−9170.77	−1.24293	−0.621466	+	0.783441i	$0.713462\pi$
$380$	529.408	0.0714686
$381$	3290.05	0.442400
$382$	7738.50	1.03648
$383$	6403.02	0.854253	0.427127	−	0.904192i	$-0.359526\pi$
$383$	6403.02	0.854253	0.427127	+	0.904192i	$0.359526\pi$
$384$	994.720	0.132192
$385$	10896.6	1.44245
$386$	6795.50	0.896067
$387$	0	0
$388$	−884.817	−0.115773
$389$	7196.53	0.937991	0.468995	−	0.883201i	$-0.344616\pi$
$389$	7196.53	0.937991	0.468995	+	0.883201i	$0.344616\pi$
$390$	14494.3	1.88191
$391$	−6611.71	−0.855162
$392$	1819.98	0.234497
$393$	−12817.6	−1.64520
$394$	197.281	0.0252255
$395$	−12436.9	−1.58423
$396$	−10515.1	−1.33435
$397$	10186.1	1.28773	0.643863	−	0.765141i	$-0.277331\pi$
$397$	10186.1	1.28773	0.643863	+	0.765141i	$0.277331\pi$
$398$	205.933	0.0259359
$399$	1573.08	0.197375
$400$	−2801.50	−0.350188
$401$	4639.83	0.577810	0.288905	−	0.957358i	$-0.406709\pi$
$401$	4639.83	0.577810	0.288905	+	0.957358i	$0.406709\pi$
$402$	−5735.98	−0.711654
$403$	5128.12	0.633870
$404$	−2179.42	−0.268391
$405$	36330.9	4.45752
$406$	−5930.79	−0.724976
$407$	−2408.27	−0.293301
$408$	−19428.1	−2.35743
$409$	−9793.28	−1.18398	−0.591988	−	0.805947i	$-0.701657\pi$
$409$	−9793.28	−1.18398	−0.591988	+	0.805947i	$0.701657\pi$
$410$	314.479	0.0378806
$411$	19637.1	2.35676
$412$	5363.48	0.641358
$413$	4730.46	0.563610
$414$	12258.0	1.45519
$415$	−19826.5	−2.34517
$416$	−6422.62	−0.756959
$417$	15342.8	1.80178
$418$	−881.635	−0.103163
$419$	10390.7	1.21150	0.605748	−	0.795657i	$-0.292874\pi$
$419$	10390.7	1.21150	0.605748	+	0.795657i	$0.292874\pi$
$420$	8803.13	1.02274
$421$	1278.81	0.148041	0.0740206	−	0.997257i	$-0.476417\pi$
$421$	1278.81	0.148041	0.0740206	+	0.997257i	$0.476417\pi$
$422$	−226.732	−0.0261543
$423$	673.082	0.0773673
$424$	−6197.87	−0.709895
$425$	−9483.42	−1.08238
$426$	−2377.62	−0.270414
$427$	1113.75	0.126225
$428$	−3421.04	−0.386361
$429$	18724.0	2.10723
$430$	0	0
$431$	−8345.12	−0.932646	−0.466323	−	0.884615i	$-0.654422\pi$
$431$	−8345.12	−0.932646	−0.466323	+	0.884615i	$0.654422\pi$
$432$	10222.2	1.13846
$433$	8533.68	0.947119	0.473560	−	0.880762i	$-0.342969\pi$
$433$	8533.68	0.947119	0.473560	+	0.880762i	$0.342969\pi$
$434$	−4015.09	−0.444079
$435$	26222.9	2.89033
$436$	2150.21	0.236184
$437$	−797.257	−0.0872722
$438$	−19036.6	−2.07673
$439$	−14123.5	−1.53549	−0.767744	−	0.640756i	$-0.778621\pi$
$439$	−14123.5	−1.53549	−0.767744	+	0.640756i	$0.778621\pi$
$440$	−16227.6	−1.75823
$441$	−5255.60	−0.567499
$442$	7604.01	0.818293
$443$	6033.41	0.647079	0.323539	−	0.946215i	$-0.395127\pi$
$443$	6033.41	0.647079	0.323539	+	0.946215i	$0.395127\pi$
$444$	−1945.58	−0.207958
$445$	−21716.4	−2.31338
$446$	8050.49	0.854713
$447$	−19928.6	−2.10871
$448$	8151.68	0.859667
$449$	−15345.6	−1.61293	−0.806464	−	0.591284i	$-0.798621\pi$
$449$	−15345.6	−1.61293	−0.806464	+	0.591284i	$0.798621\pi$
$450$	17582.1	1.84184
$451$	406.250	0.0424159
$452$	−6603.28	−0.687150
$453$	2166.20	0.224673
$454$	−4546.81	−0.470027
$455$	−11332.6	−1.16765
$456$	−2342.69	−0.240584
$457$	−16460.5	−1.68488	−0.842439	−	0.538792i	$-0.818881\pi$
$457$	−16460.5	−1.68488	−0.842439	+	0.538792i	$0.818881\pi$
$458$	−1839.64	−0.187687
$459$	34603.3	3.51883
$460$	−4461.53	−0.452217
$461$	−1574.00	−0.159021	−0.0795103	−	0.996834i	$-0.525336\pi$
$461$	−1574.00	−0.159021	−0.0795103	+	0.996834i	$0.525336\pi$
$462$	−14660.0	−1.47629
$463$	−10676.6	−1.07167	−0.535834	−	0.844323i	$-0.680003\pi$
$463$	−10676.6	−1.07167	−0.535834	+	0.844323i	$0.680003\pi$
$464$	4063.99	0.406608
$465$	17752.7	1.77045
$466$	−5727.64	−0.569373
$467$	−6834.42	−0.677215	−0.338607	−	0.940928i	$-0.609956\pi$
$467$	−6834.42	−0.677215	−0.338607	+	0.940928i	$0.609956\pi$
$468$	10935.8	1.08015
$469$	4484.79	0.441553
$470$	315.812	0.0309943
$471$	−24099.0	−2.35758
$472$	−7044.77	−0.686995
$473$	0	0
$474$	16732.3	1.62140
$475$	−1143.54	−0.110461
$476$	4618.31	0.444706
$477$	17897.8	1.71800
$478$	−3941.36	−0.377142
$479$	−3790.83	−0.361602	−0.180801	−	0.983520i	$-0.557869\pi$
$479$	−3790.83	−0.361602	−0.180801	+	0.983520i	$0.557869\pi$
$480$	−22234.0	−2.11425
$481$	2504.63	0.237425
$482$	10381.6	0.981060
$483$	−13257.0	−1.24889
$484$	−1721.91	−0.161712
$485$	3943.23	0.369181
$486$	−24293.6	−2.26745
$487$	6937.25	0.645497	0.322748	−	0.946485i	$-0.395393\pi$
$487$	6937.25	0.645497	0.322748	+	0.946485i	$0.395393\pi$
$488$	−1658.63	−0.153858
$489$	28965.4	2.67865
$490$	−2465.95	−0.227347
$491$	−3418.47	−0.314203	−0.157101	−	0.987582i	$-0.550215\pi$
$491$	−3418.47	−0.314203	−0.157101	+	0.987582i	$0.550215\pi$
$492$	328.200	0.0300739
$493$	13757.1	1.25677
$494$	916.911	0.0835097
$495$	46861.1	4.25505
$496$	2751.28	0.249065
$497$	1858.99	0.167781
$498$	26674.0	2.40018
$499$	8330.03	0.747301	0.373651	−	0.927570i	$-0.378106\pi$
$499$	8330.03	0.747301	0.373651	+	0.927570i	$0.378106\pi$
$500$	404.404	0.0361710
$501$	40747.9	3.63369
$502$	−12055.7	−1.07186
$503$	−1435.22	−0.127223	−0.0636116	−	0.997975i	$-0.520262\pi$
$503$	−1435.22	−0.127223	−0.0636116	+	0.997975i	$0.520262\pi$
$504$	−28162.4	−2.48900
$505$	9712.69	0.855860
$506$	7429.88	0.652764
$507$	2215.54	0.194074
$508$	1164.71	0.101724
$509$	−1358.45	−0.118295	−0.0591477	−	0.998249i	$-0.518838\pi$
$509$	−1358.45	−0.118295	−0.0591477	+	0.998249i	$0.518838\pi$
$510$	26323.8	2.28556
$511$	14884.2	1.28853
$512$	−8096.75	−0.698885
$513$	4172.55	0.359109
$514$	6570.67	0.563851
$515$	−23902.6	−2.04520
$516$	0	0
$517$	407.972	0.0347052
$518$	−1961.02	−0.166336
$519$	7934.56	0.671076
$520$	16876.9	1.42328
$521$	852.637	0.0716981	0.0358490	−	0.999357i	$-0.488586\pi$
$521$	852.637	0.0716981	0.0358490	+	0.999357i	$0.488586\pi$
$522$	−25505.4	−2.13859
$523$	−9749.01	−0.815094	−0.407547	−	0.913184i	$-0.633616\pi$
$523$	−9749.01	−0.815094	−0.407547	+	0.913184i	$0.633616\pi$
$524$	−4537.57	−0.378292
$525$	−19015.0	−1.58073
$526$	−5922.19	−0.490912
$527$	9313.42	0.769827
$528$	10045.6	0.827988
$529$	−5448.21	−0.447785
$530$	8397.71	0.688251
$531$	20343.4	1.66258
$532$	556.888	0.0453837
$533$	−422.505	−0.0343353
$534$	29216.6	2.36765
$535$	15246.1	1.23205
$536$	−6678.90	−0.538217
$537$	33098.1	2.65976
$538$	−17390.4	−1.39359
$539$	−3185.55	−0.254567
$540$	23350.0	1.86079
$541$	4019.47	0.319428	0.159714	−	0.987163i	$-0.448943\pi$
$541$	4019.47	0.319428	0.159714	+	0.987163i	$0.448943\pi$
$542$	−8997.78	−0.713077
$543$	41719.1	3.29712
$544$	−11664.4	−0.919317
$545$	−9582.52	−0.753156
$546$	15246.6	1.19505
$547$	4687.88	0.366434	0.183217	−	0.983073i	$-0.441349\pi$
$547$	4687.88	0.366434	0.183217	+	0.983073i	$0.441349\pi$
$548$	6951.74	0.541904
$549$	4789.68	0.372347
$550$	10657.0	0.826208
$551$	1658.87	0.128258
$552$	19742.8	1.52230
$553$	−13082.5	−1.00601
$554$	18102.6	1.38828
$555$	8670.60	0.663147
$556$	5431.52	0.414294
$557$	8917.98	0.678397	0.339198	−	0.940715i	$-0.389844\pi$
$557$	8917.98	0.678397	0.339198	+	0.940715i	$0.389844\pi$
$558$	−17266.9	−1.30998
$559$	0	0
$560$	−6080.06	−0.458803
$561$	34005.5	2.55920
$562$	−2732.05	−0.205061
$563$	4096.21	0.306634	0.153317	−	0.988177i	$-0.451004\pi$
$563$	4096.21	0.306634	0.153317	+	0.988177i	$0.451004\pi$
$564$	329.591	0.0246069
$565$	29427.9	2.19122
$566$	−19048.1	−1.41458
$567$	38216.7	2.83060
$568$	−2768.47	−0.204511
$569$	2683.32	0.197699	0.0988495	−	0.995102i	$-0.468484\pi$
$569$	2683.32	0.197699	0.0988495	+	0.995102i	$0.468484\pi$
$570$	3174.19	0.233249
$571$	10667.4	0.781815	0.390908	−	0.920430i	$-0.372161\pi$
$571$	10667.4	0.781815	0.390908	+	0.920430i	$0.372161\pi$
$572$	6628.48	0.484529
$573$	−35991.7	−2.62404
$574$	330.803	0.0240548
$575$	9637.02	0.698942
$576$	35056.4	2.53591
$577$	−21766.6	−1.57046	−0.785230	−	0.619205i	$-0.787455\pi$
$577$	−21766.6	−1.57046	−0.785230	+	0.619205i	$0.787455\pi$
$578$	3381.91	0.243372
$579$	−31605.8	−2.26855
$580$	9283.19	0.664592
$581$	−20855.6	−1.48922
$582$	−5305.12	−0.377843
$583$	10848.3	0.770653
$584$	−22166.0	−1.57061
$585$	−48736.1	−3.44443
$586$	2491.04	0.175604
$587$	24126.0	1.69640	0.848201	−	0.529675i	$-0.177686\pi$
$587$	24126.0	1.69640	0.848201	+	0.529675i	$0.177686\pi$
$588$	−2573.53	−0.180494
$589$	1123.04	0.0785635
$590$	9545.20	0.666050
$591$	−917.551	−0.0638629
$592$	1343.76	0.0932906
$593$	23547.7	1.63067	0.815337	−	0.578987i	$-0.196552\pi$
$593$	23547.7	1.63067	0.815337	+	0.578987i	$0.196552\pi$
$594$	−38885.3	−2.68600
$595$	−20581.7	−1.41810
$596$	−7054.94	−0.484868
$597$	−957.791	−0.0656613
$598$	−7727.17	−0.528407
$599$	−10000.3	−0.682139	−0.341070	−	0.940038i	$-0.610789\pi$
$599$	−10000.3	−0.682139	−0.341070	+	0.940038i	$0.610789\pi$
$600$	28317.8	1.92678
$601$	−4062.49	−0.275728	−0.137864	−	0.990451i	$-0.544024\pi$
$601$	−4062.49	−0.275728	−0.137864	+	0.990451i	$0.544024\pi$
$602$	0	0
$603$	19286.9	1.30252
$604$	766.857	0.0516605
$605$	7673.80	0.515676
$606$	−13067.2	−0.875939
$607$	−12695.2	−0.848902	−0.424451	−	0.905451i	$-0.639533\pi$
$607$	−12695.2	−0.848902	−0.424451	+	0.905451i	$0.639533\pi$
$608$	−1406.53	−0.0938195
$609$	27584.1	1.83541
$610$	2247.34	0.149167
$611$	−424.296	−0.0280936
$612$	19861.1	1.31182
$613$	−4025.43	−0.265230	−0.132615	−	0.991168i	$-0.542337\pi$
$613$	−4025.43	−0.265230	−0.132615	+	0.991168i	$0.542337\pi$
$614$	−17039.4	−1.11996
$615$	−1462.64	−0.0959013
$616$	−17070.0	−1.11651
$617$	−21979.7	−1.43415	−0.717074	−	0.696997i	$-0.754519\pi$
$617$	−21979.7	−1.43415	−0.717074	+	0.696997i	$0.754519\pi$
$618$	32158.0	2.09318
$619$	3508.41	0.227811	0.113906	−	0.993492i	$-0.463664\pi$
$619$	3508.41	0.227811	0.113906	+	0.993492i	$0.463664\pi$
$620$	6284.63	0.407091
$621$	−35163.7	−2.27226
$622$	12525.0	0.807405
$623$	−22843.6	−1.46903
$624$	−10447.5	−0.670250
$625$	−16498.5	−1.05591
$626$	−16228.4	−1.03613
$627$	4100.47	0.261176
$628$	−8531.28	−0.542094
$629$	4548.78	0.288349
$630$	38158.2	2.41311
$631$	−12163.9	−0.767412	−0.383706	−	0.923455i	$-0.625352\pi$
$631$	−12163.9	−0.767412	−0.383706	+	0.923455i	$0.625352\pi$
$632$	19482.9	1.22625
$633$	1054.53	0.0662143
$634$	18834.0	1.17980
$635$	−5190.60	−0.324382
$636$	8764.09	0.546413
$637$	3313.02	0.206070
$638$	−15459.5	−0.959321
$639$	7994.60	0.494932
$640$	−1569.34	−0.0969273
$641$	19508.5	1.20209	0.601045	−	0.799216i	$-0.294751\pi$
$641$	19508.5	1.20209	0.601045	+	0.799216i	$0.294751\pi$
$642$	−20511.7	−1.26095
$643$	8349.25	0.512072	0.256036	−	0.966667i	$-0.417583\pi$
$643$	8349.25	0.512072	0.256036	+	0.966667i	$0.417583\pi$
$644$	−4693.11	−0.287165
$645$	0	0
$646$	1665.25	0.101421
$647$	−24420.9	−1.48390	−0.741952	−	0.670453i	$-0.766100\pi$
$647$	−24420.9	−1.48390	−0.741952	+	0.670453i	$0.766100\pi$
$648$	−56913.6	−3.45027
$649$	12330.6	0.745794
$650$	−11083.4	−0.668809
$651$	18674.1	1.12427
$652$	10254.0	0.615919
$653$	17813.5	1.06753	0.533763	−	0.845634i	$-0.320777\pi$
$653$	17813.5	1.06753	0.533763	+	0.845634i	$0.320777\pi$
$654$	12892.1	0.770825
$655$	20221.9	1.20632
$656$	−226.678	−0.0134913
$657$	64009.5	3.80099
$658$	332.205	0.0196819
$659$	−10172.2	−0.601294	−0.300647	−	0.953735i	$-0.597203\pi$
$659$	−10172.2	−0.601294	−0.300647	+	0.953735i	$0.597203\pi$
$660$	22946.7	1.35333
$661$	−10378.7	−0.610716	−0.305358	−	0.952238i	$-0.598776\pi$
$661$	−10378.7	−0.610716	−0.305358	+	0.952238i	$0.598776\pi$
$662$	−8486.54	−0.498246
$663$	−35366.1	−2.07166
$664$	31058.9	1.81524
$665$	−2481.80	−0.144722
$666$	−8433.36	−0.490670
$667$	−13979.9	−0.811551
$668$	14425.2	0.835519
$669$	−37442.8	−2.16386
$670$	9049.47	0.521808
$671$	2903.15	0.167026
$672$	−23388.1	−1.34258
$673$	−20924.5	−1.19849	−0.599243	−	0.800567i	$-0.704532\pi$
$673$	−20924.5	−1.19849	−0.599243	+	0.800567i	$0.704532\pi$
$674$	6689.31	0.382289
$675$	−50436.7	−2.87601
$676$	784.324	0.0446247
$677$	−9393.92	−0.533290	−0.266645	−	0.963795i	$-0.585915\pi$
$677$	−9393.92	−0.533290	−0.266645	+	0.963795i	$0.585915\pi$
$678$	−39591.5	−2.24263
$679$	4147.91	0.234436
$680$	30651.0	1.72855
$681$	21147.1	1.18996
$682$	−10465.9	−0.587626
$683$	−4288.63	−0.240263	−0.120132	−	0.992758i	$-0.538332\pi$
$683$	−4288.63	−0.240263	−0.120132	+	0.992758i	$0.538332\pi$
$684$	2394.90	0.133876
$685$	−30980.8	−1.72805
$686$	−14521.4	−0.808207
$687$	8556.14	0.475163
$688$	0	0
$689$	−11282.4	−0.623838
$690$	−26750.1	−1.47588
$691$	−6084.89	−0.334993	−0.167496	−	0.985873i	$-0.553568\pi$
$691$	−6084.89	−0.334993	−0.167496	+	0.985873i	$0.553568\pi$
$692$	2808.92	0.154305
$693$	49293.4	2.70202
$694$	5601.86	0.306403
$695$	−24205.9	−1.32112
$696$	−41079.1	−2.23721
$697$	−767.332	−0.0416998
$698$	−18409.9	−0.998316
$699$	26639.2	1.44147
$700$	−6731.51	−0.363467
$701$	19240.3	1.03666	0.518329	−	0.855182i	$-0.326554\pi$
$701$	19240.3	1.03666	0.518329	+	0.855182i	$0.326554\pi$
$702$	40441.2	2.17429
$703$	548.504	0.0294270
$704$	21248.6	1.13755
$705$	−1468.84	−0.0784677
$706$	−1697.53	−0.0904922
$707$	10216.8	0.543485
$708$	9961.64	0.528787
$709$	−22023.7	−1.16660	−0.583299	−	0.812258i	$-0.698238\pi$
$709$	−22023.7	−1.16660	−0.583299	+	0.812258i	$0.698238\pi$
$710$	3751.10	0.198276
$711$	−56261.4	−2.96761
$712$	34019.4	1.79063
$713$	−9464.27	−0.497110
$714$	27690.1	1.45137
$715$	−29540.2	−1.54509
$716$	11717.1	0.611575
$717$	18331.2	0.954801
$718$	−14630.4	−0.760449
$719$	−9873.52	−0.512128	−0.256064	−	0.966660i	$-0.582426\pi$
$719$	−9873.52	−0.512128	−0.256064	+	0.966660i	$0.582426\pi$
$720$	−26147.4	−1.35341
$721$	−25143.3	−1.29873
$722$	−14357.8	−0.740085
$723$	−48284.9	−2.48373
$724$	14769.0	0.758129
$725$	−20051.9	−1.02719
$726$	−10324.1	−0.527774
$727$	−22548.2	−1.15030	−0.575150	−	0.818048i	$-0.695056\pi$
$727$	−22548.2	−1.15030	−0.575150	+	0.818048i	$0.695056\pi$
$728$	17753.0	0.903803
$729$	50006.4	2.54059
$730$	30033.5	1.52272
$731$	0	0
$732$	2345.38	0.118426
$733$	23342.2	1.17621	0.588106	−	0.808784i	$-0.299874\pi$
$733$	23342.2	1.17621	0.588106	+	0.808784i	$0.299874\pi$
$734$	22893.7	1.15125
$735$	11469.1	0.575570
$736$	11853.4	0.593642
$737$	11690.3	0.584282
$738$	1422.62	0.0709585
$739$	2588.54	0.128851	0.0644255	−	0.997923i	$-0.479479\pi$
$739$	2588.54	0.128851	0.0644255	+	0.997923i	$0.479479\pi$
$740$	3069.48	0.152482
$741$	−4264.54	−0.211420
$742$	8833.60	0.437051
$743$	−8452.43	−0.417348	−0.208674	−	0.977985i	$-0.566915\pi$
$743$	−8452.43	−0.417348	−0.208674	+	0.977985i	$0.566915\pi$
$744$	−27810.2	−1.37039
$745$	31440.7	1.54617
$746$	4922.25	0.241577
$747$	−89689.7	−4.39300
$748$	12038.3	0.588454
$749$	16037.4	0.782370
$750$	2424.70	0.118050
$751$	−3358.11	−0.163168	−0.0815841	−	0.996666i	$-0.525998\pi$
$751$	−3358.11	−0.163168	−0.0815841	+	0.996666i	$0.525998\pi$
$752$	−227.639	−0.0110387
$753$	56070.8	2.71359
$754$	16078.1	0.776563
$755$	−3417.54	−0.164738
$756$	24562.0	1.18163
$757$	−14867.1	−0.713810	−0.356905	−	0.934141i	$-0.616168\pi$
$757$	−14867.1	−0.713810	−0.356905	+	0.934141i	$0.616168\pi$
$758$	−19465.4	−0.932739
$759$	−34556.3	−1.65259
$760$	3695.98	0.176404
$761$	1721.42	0.0819991	0.0409995	−	0.999159i	$-0.486946\pi$
$761$	1721.42	0.0819991	0.0409995	+	0.999159i	$0.486946\pi$
$762$	6983.30	0.331992
$763$	−10079.9	−0.478266
$764$	−12741.4	−0.603362
$765$	−88512.0	−4.18321
$766$	13590.7	0.641062
$767$	−12824.0	−0.603714
$768$	41407.0	1.94550
$769$	13213.3	0.619616	0.309808	−	0.950799i	$-0.399735\pi$
$769$	13213.3	0.619616	0.309808	+	0.950799i	$0.399735\pi$
$770$	23128.7	1.08247
$771$	−30560.1	−1.42749
$772$	−11188.8	−0.521623
$773$	26452.9	1.23085	0.615424	−	0.788196i	$-0.288985\pi$
$773$	26452.9	1.23085	0.615424	+	0.788196i	$0.288985\pi$
$774$	0	0
$775$	−13575.0	−0.629196
$776$	−6177.21	−0.285759
$777$	9120.66	0.421109
$778$	15275.0	0.703901
$779$	−92.5268	−0.00425561
$780$	−23864.8	−1.09551
$781$	4845.73	0.222015
$782$	−14033.7	−0.641744
$783$	73165.8	3.33938
$784$	1777.46	0.0809704
$785$	38020.1	1.72866
$786$	−27206.1	−1.23462
$787$	22609.2	1.02405	0.512027	−	0.858970i	$-0.328895\pi$
$787$	22609.2	1.02405	0.512027	+	0.858970i	$0.328895\pi$
$788$	−324.822	−0.0146844
$789$	27544.0	1.24283
$790$	−26398.1	−1.18886
$791$	30955.3	1.39146
$792$	−73409.5	−3.29355
$793$	−3019.31	−0.135207
$794$	21620.6	0.966355
$795$	−39057.7	−1.74243
$796$	−339.068	−0.0150979
$797$	16151.3	0.717829	0.358915	−	0.933370i	$-0.383147\pi$
$797$	16151.3	0.717829	0.358915	+	0.933370i	$0.383147\pi$
$798$	3338.95	0.148117
$799$	−770.584	−0.0341193
$800$	17001.7	0.751377
$801$	−98239.0	−4.33346
$802$	9848.27	0.433609
$803$	38797.8	1.70504
$804$	9444.28	0.414271
$805$	20915.1	0.915727
$806$	10884.7	0.475678
$807$	80882.4	3.52812
$808$	−15215.3	−0.662465
$809$	8389.19	0.364584	0.182292	−	0.983244i	$-0.441648\pi$
$809$	8389.19	0.364584	0.182292	+	0.983244i	$0.441648\pi$
$810$	77114.2	3.34508
$811$	−687.716	−0.0297768	−0.0148884	−	0.999889i	$-0.504739\pi$
$811$	−687.716	−0.0297768	−0.0148884	+	0.999889i	$0.504739\pi$
$812$	9765.04	0.422027
$813$	41848.6	1.80528
$814$	−5111.67	−0.220103
$815$	−45697.7	−1.96407
$816$	−18974.3	−0.814010
$817$	0	0
$818$	−20786.7	−0.888498
$819$	−51265.8	−2.18727
$820$	−517.790	−0.0220512
$821$	−36524.3	−1.55263	−0.776313	−	0.630348i	$-0.782912\pi$
$821$	−36524.3	−1.55263	−0.776313	+	0.630348i	$0.782912\pi$
$822$	41680.8	1.76859
$823$	−2950.87	−0.124983	−0.0624915	−	0.998045i	$-0.519905\pi$
$823$	−2950.87	−0.124983	−0.0624915	+	0.998045i	$0.519905\pi$
$824$	37444.3	1.58305
$825$	−49565.4	−2.09169
$826$	10040.6	0.422953
$827$	−7907.73	−0.332501	−0.166251	−	0.986084i	$-0.553166\pi$
$827$	−7907.73	−0.332501	−0.166251	+	0.986084i	$0.553166\pi$
$828$	−20182.8	−0.847101
$829$	−36919.5	−1.54677	−0.773383	−	0.633939i	$-0.781437\pi$
$829$	−36919.5	−1.54677	−0.773383	+	0.633939i	$0.781437\pi$
$830$	−42082.7	−1.75989
$831$	−84194.9	−3.51467
$832$	−22098.8	−0.920838
$833$	6016.93	0.250269
$834$	32565.9	1.35212
$835$	−64286.6	−2.66435
$836$	1451.61	0.0600538
$837$	49532.5	2.04551
$838$	22054.7	0.909149
$839$	−21841.4	−0.898747	−0.449373	−	0.893344i	$-0.648353\pi$
$839$	−21841.4	−0.898747	−0.449373	+	0.893344i	$0.648353\pi$
$840$	61457.7	2.52440
$841$	4699.28	0.192680
$842$	2714.34	0.111095
$843$	12706.7	0.519149
$844$	373.313	0.0152251
$845$	−3495.38	−0.142302
$846$	1428.65	0.0580591
$847$	8072.11	0.327463
$848$	−6053.10	−0.245123
$849$	88592.4	3.58125
$850$	−20129.0	−0.812260
$851$	−4622.46	−0.186199
$852$	3914.75	0.157414
$853$	−43215.2	−1.73466	−0.867328	−	0.497737i	$-0.834165\pi$
$853$	−43215.2	−1.73466	−0.867328	+	0.497737i	$0.834165\pi$
$854$	2363.99	0.0947236
$855$	−10673.0	−0.426911
$856$	−23883.5	−0.953646
$857$	14793.5	0.589657	0.294828	−	0.955550i	$-0.404738\pi$
$857$	14793.5	0.589657	0.294828	+	0.955550i	$0.404738\pi$
$858$	39742.6	1.58134
$859$	16459.1	0.653758	0.326879	−	0.945066i	$-0.394003\pi$
$859$	16459.1	0.653758	0.326879	+	0.945066i	$0.394003\pi$
$860$	0	0
$861$	−1538.56	−0.0608989
$862$	−17712.9	−0.699890
$863$	−32877.1	−1.29681	−0.648406	−	0.761295i	$-0.724564\pi$
$863$	−32877.1	−1.29681	−0.648406	+	0.761295i	$0.724564\pi$
$864$	−62036.2	−2.44272
$865$	−12518.1	−0.492055
$866$	18113.2	0.710752
$867$	−15729.2	−0.616139
$868$	6610.84	0.258510
$869$	−34101.5	−1.33120
$870$	55659.5	2.16900
$871$	−12158.0	−0.472972
$872$	15011.3	0.582968
$873$	17838.1	0.691557
$874$	−1692.22	−0.0654921
$875$	−1895.80	−0.0732453
$876$	31343.8	1.20891
$877$	−13873.2	−0.534168	−0.267084	−	0.963673i	$-0.586060\pi$
$877$	−13873.2	−0.534168	−0.267084	+	0.963673i	$0.586060\pi$
$878$	−29977.9	−1.15228
$879$	−11585.8	−0.444572
$880$	−15848.6	−0.607108
$881$	36781.0	1.40657	0.703283	−	0.710910i	$-0.251717\pi$
$881$	36781.0	1.40657	0.703283	+	0.710910i	$0.251717\pi$
$882$	−11155.3	−0.425871
$883$	14471.7	0.551542	0.275771	−	0.961223i	$-0.411067\pi$
$883$	14471.7	0.551542	0.275771	+	0.961223i	$0.411067\pi$
$884$	−12520.0	−0.476349
$885$	−44394.6	−1.68622
$886$	12806.2	0.485591
$887$	29063.0	1.10016	0.550080	−	0.835112i	$-0.314597\pi$
$887$	29063.0	1.10016	0.550080	+	0.835112i	$0.314597\pi$
$888$	−13582.8	−0.513298
$889$	−5460.03	−0.205988
$890$	−46094.1	−1.73604
$891$	99617.4	3.74558
$892$	−13255.1	−0.497550
$893$	−92.9191	−0.00348199
$894$	−42299.5	−1.58245
$895$	−52217.8	−1.95022
$896$	−1650.79	−0.0615504
$897$	35939.0	1.33776
$898$	−32571.8	−1.21040
$899$	19692.5	0.730568
$900$	−28948.9	−1.07218
$901$	−20490.5	−0.757643
$902$	862.286	0.0318304
$903$	0	0
$904$	−46099.8	−1.69608
$905$	−65818.8	−2.41756
$906$	4597.87	0.168603
$907$	−5383.63	−0.197090	−0.0985450	−	0.995133i	$-0.531419\pi$
$907$	−5383.63	−0.197090	−0.0985450	+	0.995133i	$0.531419\pi$
$908$	7486.31	0.273614
$909$	43937.6	1.60321
$910$	−24054.1	−0.876248
$911$	−11716.3	−0.426100	−0.213050	−	0.977041i	$-0.568340\pi$
$911$	−11716.3	−0.426100	−0.213050	+	0.977041i	$0.568340\pi$
$912$	−2287.97	−0.0830725
$913$	−54363.2	−1.97060
$914$	−34938.2	−1.26439
$915$	−10452.3	−0.377643
$916$	3028.96	0.109257
$917$	21271.6	0.766030
$918$	73447.2	2.64065
$919$	18077.4	0.648879	0.324439	−	0.945906i	$-0.394824\pi$
$919$	18077.4	0.648879	0.324439	+	0.945906i	$0.394824\pi$
$920$	−31147.5	−1.11620
$921$	79249.8	2.83537
$922$	−3340.90	−0.119335
$923$	−5039.62	−0.179719
$924$	24137.7	0.859386
$925$	−6630.16	−0.235674
$926$	−22661.6	−0.804217
$927$	−108129.	−3.83110
$928$	−24663.5	−0.872435
$929$	2703.22	0.0954680	0.0477340	−	0.998860i	$-0.484800\pi$
$929$	2703.22	0.0954680	0.0477340	+	0.998860i	$0.484800\pi$
$930$	37680.9	1.32861
$931$	725.537	0.0255408
$932$	9430.55	0.331446
$933$	−58253.6	−2.04409
$934$	−14506.4	−0.508206
$935$	−53649.4	−1.87649
$936$	76346.8	2.66610
$937$	5610.28	0.195603	0.0978015	−	0.995206i	$-0.468819\pi$
$937$	5610.28	0.195603	0.0978015	+	0.995206i	$0.468819\pi$
$938$	9519.19	0.331357
$939$	75478.3	2.62315
$940$	−519.985	−0.0180426
$941$	−1782.55	−0.0617528	−0.0308764	−	0.999523i	$-0.509830\pi$
$941$	−1782.55	−0.0617528	−0.0308764	+	0.999523i	$0.509830\pi$
$942$	−51151.3	−1.76921
$943$	779.760	0.0269273
$944$	−6880.21	−0.237216
$945$	−109462.	−3.76804
$946$	0	0
$947$	−17721.3	−0.608094	−0.304047	−	0.952657i	$-0.598338\pi$
$947$	−17721.3	−0.608094	−0.304047	+	0.952657i	$0.598338\pi$
$948$	−27549.8	−0.943855
$949$	−40350.2	−1.38021
$950$	−2427.21	−0.0828939
$951$	−87596.9	−2.98688
$952$	32242.0	1.09766
$953$	−12699.3	−0.431658	−0.215829	−	0.976431i	$-0.569245\pi$
$953$	−12699.3	−0.431658	−0.215829	+	0.976431i	$0.569245\pi$
$954$	37989.0	1.28924
$955$	56782.9	1.92403
$956$	6489.45	0.219544
$957$	71901.9	2.42869
$958$	−8046.22	−0.271359
$959$	−32588.9	−1.09734
$960$	−76502.2	−2.57198
$961$	−16459.4	−0.552495
$962$	5316.21	0.178172
$963$	68969.1	2.30789
$964$	−17093.4	−0.571100
$965$	49863.4	1.66338
$966$	−28138.6	−0.937211
$967$	30667.6	1.01986	0.509929	−	0.860216i	$-0.329671\pi$
$967$	30667.6	1.01986	0.509929	+	0.860216i	$0.329671\pi$
$968$	−12021.3	−0.399151
$969$	−7745.04	−0.256766
$970$	8369.71	0.277047
$971$	38789.5	1.28199	0.640996	−	0.767544i	$-0.278521\pi$
$971$	38789.5	1.28199	0.640996	+	0.767544i	$0.278521\pi$
$972$	39999.3	1.31994
$973$	−25462.3	−0.838935
$974$	14724.7	0.484404
$975$	51548.6	1.69321
$976$	−1619.89	−0.0531263
$977$	34788.6	1.13919	0.569594	−	0.821926i	$-0.307101\pi$
$977$	34788.6	1.13919	0.569594	+	0.821926i	$0.307101\pi$
$978$	61480.4	2.01015
$979$	−59545.1	−1.94389
$980$	4060.18	0.132345
$981$	−43348.7	−1.41082
$982$	−7255.89	−0.235789
$983$	−34284.6	−1.11242	−0.556210	−	0.831042i	$-0.687745\pi$
$983$	−34284.6	−1.11242	−0.556210	+	0.831042i	$0.687745\pi$
$984$	2291.28	0.0742309
$985$	1447.59	0.0468264
$986$	29200.1	0.943126
$987$	−1545.08	−0.0498283
$988$	−1509.69	−0.0486131
$989$	0	0
$990$	99465.0	3.19314
$991$	−6213.41	−0.199168	−0.0995840	−	0.995029i	$-0.531751\pi$
$991$	−6213.41	−0.199168	−0.0995840	+	0.995029i	$0.531751\pi$
$992$	−16697.0	−0.534404
$993$	39470.8	1.26140
$994$	3945.80	0.125909
$995$	1511.08	0.0481450
$996$	−43918.7	−1.39721
$997$	−38148.6	−1.21181	−0.605907	−	0.795535i	$-0.707190\pi$
$997$	−38148.6	−1.21181	−0.605907	+	0.795535i	$0.707190\pi$
$998$	17680.9	0.560801
$999$	24192.3	0.766175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.35	✓	50
43.42	odd	2		1849.4.a.j.1.16	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.35	✓	50	1.1	even	1	trivial
1849.4.a.j.1.16	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)

\(f(q)\)	\(=\)	\(q+2.12255 q^{2} -9.87196 q^{3} -3.49478 q^{4} +15.5747 q^{5} -20.9537 q^{6} +16.3831 q^{7} -24.3983 q^{8} +70.4556 q^{9} +O(q^{10})\)	\(q+2.12255 q^{2} -9.87196 q^{3} -3.49478 q^{4} +15.5747 q^{5} -20.9537 q^{6} +16.3831 q^{7} -24.3983 q^{8} +70.4556 q^{9} +33.0580 q^{10} +42.7049 q^{11} +34.5003 q^{12} -44.4137 q^{13} +34.7739 q^{14} -153.752 q^{15} -23.8283 q^{16} -80.6618 q^{17} +149.546 q^{18} -9.72641 q^{19} -54.4300 q^{20} -161.733 q^{21} +90.6434 q^{22} +81.9683 q^{23} +240.859 q^{24} +117.570 q^{25} -94.2703 q^{26} -428.992 q^{27} -57.2552 q^{28} -170.553 q^{29} -326.348 q^{30} -115.463 q^{31} +144.609 q^{32} -421.581 q^{33} -171.209 q^{34} +255.161 q^{35} -246.227 q^{36} -56.3932 q^{37} -20.6448 q^{38} +438.450 q^{39} -379.995 q^{40} +9.51295 q^{41} -343.287 q^{42} -149.244 q^{44} +1097.32 q^{45} +173.982 q^{46} +9.55328 q^{47} +235.232 q^{48} -74.5945 q^{49} +249.549 q^{50} +796.290 q^{51} +155.216 q^{52} +254.029 q^{53} -910.558 q^{54} +665.115 q^{55} -399.719 q^{56} +96.0187 q^{57} -362.007 q^{58} +288.741 q^{59} +537.331 q^{60} +67.9815 q^{61} -245.075 q^{62} +1154.28 q^{63} +497.567 q^{64} -691.728 q^{65} -894.828 q^{66} +273.745 q^{67} +281.895 q^{68} -809.187 q^{69} +541.592 q^{70} +113.470 q^{71} -1718.99 q^{72} +908.508 q^{73} -119.698 q^{74} -1160.65 q^{75} +33.9916 q^{76} +699.638 q^{77} +930.632 q^{78} -798.537 q^{79} -371.118 q^{80} +2332.69 q^{81} +20.1917 q^{82} -1273.00 q^{83} +565.221 q^{84} -1256.28 q^{85} +1683.69 q^{87} -1041.93 q^{88} -1394.34 q^{89} +2329.12 q^{90} -727.633 q^{91} -286.461 q^{92} +1139.84 q^{93} +20.2773 q^{94} -151.486 q^{95} -1427.58 q^{96} +253.183 q^{97} -158.331 q^{98} +3008.80 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) \) 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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