LMFDB - Embedded newform 1849.4.a.k.1.19

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:	$60$
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q-2.74329 q^{2} +8.71284 q^{3} -0.474355 q^{4} -4.82785 q^{5} -23.9019 q^{6} +0.994355 q^{7} +23.2476 q^{8} +48.9136 q^{9} +O(q^{10})$	\(q-2.74329 q^{2} +8.71284 q^{3} -0.474355 q^{4} -4.82785 q^{5} -23.9019 q^{6} +0.994355 q^{7} +23.2476 q^{8} +48.9136 q^{9} +13.2442 q^{10} -12.5432 q^{11} -4.13298 q^{12} +63.4094 q^{13} -2.72780 q^{14} -42.0643 q^{15} -59.9801 q^{16} -12.4888 q^{17} -134.184 q^{18} -101.105 q^{19} +2.29011 q^{20} +8.66366 q^{21} +34.4097 q^{22} +30.6986 q^{23} +202.553 q^{24} -101.692 q^{25} -173.950 q^{26} +190.929 q^{27} -0.471677 q^{28} +261.499 q^{29} +115.395 q^{30} -327.096 q^{31} -21.4380 q^{32} -109.287 q^{33} +34.2604 q^{34} -4.80059 q^{35} -23.2024 q^{36} -391.553 q^{37} +277.359 q^{38} +552.476 q^{39} -112.236 q^{40} -237.136 q^{41} -23.7669 q^{42} +5.94995 q^{44} -236.147 q^{45} -84.2151 q^{46} +310.304 q^{47} -522.597 q^{48} -342.011 q^{49} +278.970 q^{50} -108.813 q^{51} -30.0786 q^{52} -356.920 q^{53} -523.775 q^{54} +60.5568 q^{55} +23.1164 q^{56} -880.908 q^{57} -717.368 q^{58} -219.204 q^{59} +19.9534 q^{60} -111.171 q^{61} +897.320 q^{62} +48.6375 q^{63} +538.652 q^{64} -306.131 q^{65} +299.807 q^{66} +720.344 q^{67} +5.92412 q^{68} +267.472 q^{69} +13.1694 q^{70} +491.134 q^{71} +1137.12 q^{72} +338.073 q^{73} +1074.15 q^{74} -886.025 q^{75} +47.9595 q^{76} -12.4724 q^{77} -1515.60 q^{78} +145.974 q^{79} +289.575 q^{80} +342.871 q^{81} +650.534 q^{82} +1053.81 q^{83} -4.10965 q^{84} +60.2939 q^{85} +2278.40 q^{87} -291.600 q^{88} -576.459 q^{89} +647.821 q^{90} +63.0514 q^{91} -14.5620 q^{92} -2849.94 q^{93} -851.253 q^{94} +488.118 q^{95} -186.786 q^{96} -646.327 q^{97} +938.236 q^{98} -613.534 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9}+O(q^{10}) $ 60 * q - 15 * q^2 - 37 * q^3 + 213 * q^4 - 51 * q^5 + 22 * q^6 - 54 * q^7 - 204 * q^8 + 387 * q^9	\( 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} - 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} - 625 q^{18} - 610 q^{19} - 345 q^{20} + 611 q^{21} - 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} - 1071 q^{26} - 1609 q^{27} - 46 q^{28} - 773 q^{29} - 375 q^{30} - 97 q^{31} - 1967 q^{32} - 500 q^{33} - 217 q^{34} + 247 q^{35} + 175 q^{36} - 228 q^{37} + 1253 q^{38} - 1493 q^{39} + 2220 q^{40} - 951 q^{41} - 2643 q^{42} - 1378 q^{44} - 1086 q^{45} + 565 q^{46} - 2 q^{47} - 2303 q^{48} + 1264 q^{49} - 3273 q^{50} - 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} - 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} - 2999 q^{61} - 5569 q^{62} - 2377 q^{63} + 2082 q^{64} - 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} - 1817 q^{69} - 2738 q^{70} - 8003 q^{71} - 1412 q^{72} + 1011 q^{73} - 1413 q^{74} - 7457 q^{75} - 5516 q^{76} - 4052 q^{77} + 1091 q^{78} - 4422 q^{79} - 1610 q^{80} + 2108 q^{81} - 4676 q^{82} - 297 q^{83} - 54 q^{84} - 4333 q^{85} + 1377 q^{87} - 3652 q^{88} - 2480 q^{89} - 1414 q^{90} - 4551 q^{91} - 3286 q^{92} - 4 q^{93} - 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} - 753 q^{98} + 5072 q^{99}+O(q^{100}) \) 60 * q - 15 * q^2 - 37 * q^3 + 213 * q^4 - 51 * q^5 + 22 * q^6 - 54 * q^7 - 204 * q^8 + 387 * q^9 + 27 * q^10 + 43 * q^11 - 432 * q^12 + 13 * q^13 + 96 * q^14 + 65 * q^15 + 717 * q^16 - 138 * q^17 - 625 * q^18 - 610 * q^19 - 345 * q^20 + 611 * q^21 - 118 * q^22 - 243 * q^23 + 258 * q^24 + 899 * q^25 - 1071 * q^26 - 1609 * q^27 - 46 * q^28 - 773 * q^29 - 375 * q^30 - 97 * q^31 - 1967 * q^32 - 500 * q^33 - 217 * q^34 + 247 * q^35 + 175 * q^36 - 228 * q^37 + 1253 * q^38 - 1493 * q^39 + 2220 * q^40 - 951 * q^41 - 2643 * q^42 - 1378 * q^44 - 1086 * q^45 + 565 * q^46 - 2 * q^47 - 2303 * q^48 + 1264 * q^49 - 3273 * q^50 - 3076 * q^51 - 2825 * q^52 - 39 * q^53 + 5201 * q^54 - 1306 * q^55 + 3683 * q^56 + 1342 * q^57 + 2588 * q^58 - 1065 * q^59 + 2803 * q^60 - 2999 * q^61 - 5569 * q^62 - 2377 * q^63 + 2082 * q^64 - 5578 * q^65 + 3338 * q^66 + 961 * q^67 - 3754 * q^68 - 1817 * q^69 - 2738 * q^70 - 8003 * q^71 - 1412 * q^72 + 1011 * q^73 - 1413 * q^74 - 7457 * q^75 - 5516 * q^76 - 4052 * q^77 + 1091 * q^78 - 4422 * q^79 - 1610 * q^80 + 2108 * q^81 - 4676 * q^82 - 297 * q^83 - 54 * q^84 - 4333 * q^85 + 1377 * q^87 - 3652 * q^88 - 2480 * q^89 - 1414 * q^90 - 4551 * q^91 - 3286 * q^92 - 4 * q^93 - 4609 * q^94 - 835 * q^95 + 5864 * q^96 + 3785 * q^97 - 753 * q^98 + 5072 * 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 $

$2$	−2.74329	−0.969900	−0.484950	−	0.874542i	$-0.661162\pi$
$2$	−2.74329	−0.969900	−0.484950	+	0.874542i	$0.661162\pi$
$3$	8.71284	1.67679	0.838393	−	0.545066i	$-0.183495\pi$
$3$	8.71284	1.67679	0.838393	+	0.545066i	$0.183495\pi$
$4$	−0.474355	−0.0592944
$5$	−4.82785	−0.431816	−0.215908	−	0.976414i	$-0.569271\pi$
$5$	−4.82785	−0.431816	−0.215908	+	0.976414i	$0.569271\pi$
$6$	−23.9019	−1.62632
$7$	0.994355	0.0536901	0.0268451	−	0.999640i	$-0.491454\pi$
$7$	0.994355	0.0536901	0.0268451	+	0.999640i	$0.491454\pi$
$8$	23.2476	1.02741
$9$	48.9136	1.81161
$10$	13.2442	0.418818
$11$	−12.5432	−0.343812	−0.171906	−	0.985113i	$-0.554992\pi$
$11$	−12.5432	−0.343812	−0.171906	+	0.985113i	$0.554992\pi$
$12$	−4.13298	−0.0994240
$13$	63.4094	1.35282	0.676408	−	0.736527i	$-0.263536\pi$
$13$	63.4094	1.35282	0.676408	+	0.736527i	$0.263536\pi$
$14$	−2.72780	−0.0520740
$15$	−42.0643	−0.724063
$16$	−59.9801	−0.937190
$17$	−12.4888	−0.178175	−0.0890875	−	0.996024i	$-0.528395\pi$
$17$	−12.4888	−0.178175	−0.0890875	+	0.996024i	$0.528395\pi$
$18$	−134.184	−1.75708
$19$	−101.105	−1.22079	−0.610394	−	0.792098i	$-0.708989\pi$
$19$	−101.105	−1.22079	−0.610394	+	0.792098i	$0.708989\pi$
$20$	2.29011	0.0256043
$21$	8.66366	0.0900269
$22$	34.4097	0.333463
$23$	30.6986	0.278308	0.139154	−	0.990271i	$-0.455562\pi$
$23$	30.6986	0.278308	0.139154	+	0.990271i	$0.455562\pi$
$24$	202.553	1.72275
$25$	−101.692	−0.813535
$26$	−173.950	−1.31210
$27$	190.929	1.36090
$28$	−0.471677	−0.00318352
$29$	261.499	1.67445	0.837227	−	0.546856i	$-0.184176\pi$
$29$	261.499	1.67445	0.837227	+	0.546856i	$0.184176\pi$
$30$	115.395	0.702269
$31$	−327.096	−1.89510	−0.947552	−	0.319602i	$-0.896451\pi$
$31$	−327.096	−1.89510	−0.947552	+	0.319602i	$0.896451\pi$
$32$	−21.4380	−0.118429
$33$	−109.287	−0.576499
$34$	34.2604	0.172812
$35$	−4.80059	−0.0231842
$36$	−23.2024	−0.107419
$37$	−391.553	−1.73976	−0.869878	−	0.493266i	$-0.835803\pi$
$37$	−391.553	−1.73976	−0.869878	+	0.493266i	$0.835803\pi$
$38$	277.359	1.18404
$39$	552.476	2.26838
$40$	−112.236	−0.443652
$41$	−237.136	−0.903280	−0.451640	−	0.892200i	$-0.649161\pi$
$41$	−237.136	−0.903280	−0.451640	+	0.892200i	$0.649161\pi$
$42$	−23.7669	−0.0873171
$43$	0	0
$43$	0	0
$44$	5.94995	0.0203861
$45$	−236.147	−0.782284
$46$	−84.2151	−0.269931
$47$	310.304	0.963031	0.481515	−	0.876438i	$-0.340087\pi$
$47$	310.304	0.963031	0.481515	+	0.876438i	$0.340087\pi$
$48$	−522.597	−1.57147
$49$	−342.011	−0.997117
$50$	278.970	0.789047
$51$	−108.813	−0.298761
$52$	−30.0786	−0.0802143
$53$	−356.920	−0.925033	−0.462517	−	0.886611i	$-0.653053\pi$
$53$	−356.920	−0.925033	−0.462517	+	0.886611i	$0.653053\pi$
$54$	−523.775	−1.31994
$55$	60.5568	0.148463
$56$	23.1164	0.0551617
$57$	−880.908	−2.04700
$58$	−717.368	−1.62405
$59$	−219.204	−0.483693	−0.241847	−	0.970315i	$-0.577753\pi$
$59$	−219.204	−0.483693	−0.241847	+	0.970315i	$0.577753\pi$
$60$	19.9534	0.0429329
$61$	−111.171	−0.233344	−0.116672	−	0.993171i	$-0.537223\pi$
$61$	−111.171	−0.233344	−0.116672	+	0.993171i	$0.537223\pi$
$62$	897.320	1.83806
$63$	48.6375	0.0972658
$64$	538.652	1.05205
$65$	−306.131	−0.584167
$66$	299.807	0.559146
$67$	720.344	1.31349	0.656746	−	0.754112i	$-0.271932\pi$
$67$	720.344	1.31349	0.656746	+	0.754112i	$0.271932\pi$
$68$	5.92412	0.0105648
$69$	267.472	0.466664
$70$	13.1694	0.0224864
$71$	491.134	0.820943	0.410471	−	0.911874i	$-0.365364\pi$
$71$	491.134	0.820943	0.410471	+	0.911874i	$0.365364\pi$
$72$	1137.12	1.86127
$73$	338.073	0.542034	0.271017	−	0.962575i	$-0.412640\pi$
$73$	338.073	0.542034	0.271017	+	0.962575i	$0.412640\pi$
$74$	1074.15	1.68739
$75$	−886.025	−1.36412
$76$	47.9595	0.0723859
$77$	−12.4724	−0.0184593
$78$	−1515.60	−2.20010
$79$	145.974	0.207890	0.103945	−	0.994583i	$-0.466853\pi$
$79$	145.974	0.207890	0.103945	+	0.994583i	$0.466853\pi$
$80$	289.575	0.404693
$81$	342.871	0.470331
$82$	650.534	0.876091
$83$	1053.81	1.39362	0.696810	−	0.717256i	$-0.254602\pi$
$83$	1053.81	1.39362	0.696810	+	0.717256i	$0.254602\pi$
$84$	−4.10965	−0.00533809
$85$	60.2939	0.0769388
$86$	0	0
$87$	2278.40	2.80770
$88$	−291.600	−0.353235
$89$	−576.459	−0.686567	−0.343284	−	0.939232i	$-0.611539\pi$
$89$	−576.459	−0.686567	−0.343284	+	0.939232i	$0.611539\pi$
$90$	647.821	0.758737
$91$	63.0514	0.0726328
$92$	−14.5620	−0.0165021
$93$	−2849.94	−3.17768
$94$	−851.253	−0.934043
$95$	488.118	0.527156
$96$	−186.786	−0.198581
$97$	−646.327	−0.676542	−0.338271	−	0.941049i	$-0.609842\pi$
$97$	−646.327	−0.676542	−0.338271	+	0.941049i	$0.609842\pi$
$98$	938.236	0.967104
$99$	−613.534	−0.622854
$100$	48.2381	0.0482381
$101$	−679.624	−0.669556	−0.334778	−	0.942297i	$-0.608661\pi$
$101$	−679.624	−0.669556	−0.334778	+	0.942297i	$0.608661\pi$
$102$	298.505	0.289769
$103$	−395.599	−0.378442	−0.189221	−	0.981935i	$-0.560596\pi$
$103$	−395.599	−0.378442	−0.189221	+	0.981935i	$0.560596\pi$
$104$	1474.12	1.38990
$105$	−41.8268	−0.0388750
$106$	979.136	0.897189
$107$	−30.1804	−0.0272677	−0.0136339	−	0.999907i	$-0.504340\pi$
$107$	−30.1804	−0.0272677	−0.0136339	+	0.999907i	$0.504340\pi$
$108$	−90.5683	−0.0806939
$109$	730.397	0.641829	0.320915	−	0.947108i	$-0.396010\pi$
$109$	730.397	0.641829	0.320915	+	0.947108i	$0.396010\pi$
$110$	−166.125	−0.143995
$111$	−3411.54	−2.91720
$112$	−59.6416	−0.0503178
$113$	−667.986	−0.556096	−0.278048	−	0.960567i	$-0.589687\pi$
$113$	−667.986	−0.556096	−0.278048	+	0.960567i	$0.589687\pi$
$114$	2416.59	1.98539
$115$	−148.208	−0.120178
$116$	−124.043	−0.0992857
$117$	3101.58	2.45078
$118$	601.340	0.469134
$119$	−12.4183	−0.00956623
$120$	−977.894	−0.743909
$121$	−1173.67	−0.881794
$122$	304.974	0.226320
$123$	−2066.13	−1.51461
$124$	155.160	0.112369
$125$	1094.43	0.783113
$126$	−133.427	−0.0943380
$127$	1845.47	1.28944	0.644722	−	0.764417i	$-0.276973\pi$
$127$	1845.47	1.28944	0.644722	+	0.764417i	$0.276973\pi$
$128$	−1306.17	−0.901958
$129$	0	0
$130$	839.806	0.566584
$131$	145.685	0.0971648	0.0485824	−	0.998819i	$-0.484530\pi$
$131$	145.685	0.0971648	0.0485824	+	0.998819i	$0.484530\pi$
$132$	51.8409	0.0341831
$133$	−100.534	−0.0655443
$134$	−1976.11	−1.27396
$135$	−921.778	−0.587660
$136$	−290.334	−0.183059
$137$	−2198.70	−1.37115	−0.685576	−	0.728002i	$-0.740449\pi$
$137$	−2198.70	−1.37115	−0.685576	+	0.728002i	$0.740449\pi$
$138$	−733.752	−0.452617
$139$	87.1247	0.0531642	0.0265821	−	0.999647i	$-0.491538\pi$
$139$	87.1247	0.0531642	0.0265821	+	0.999647i	$0.491538\pi$
$140$	2.27719	0.00137470
$141$	2703.63	1.61480
$142$	−1347.32	−0.796232
$143$	−795.359	−0.465114
$144$	−2933.84	−1.69783
$145$	−1262.48	−0.723056
$146$	−927.433	−0.525719
$147$	−2979.89	−1.67195
$148$	185.735	0.103158
$149$	−2665.55	−1.46557	−0.732786	−	0.680459i	$-0.761780\pi$
$149$	−2665.55	−1.46557	−0.732786	+	0.680459i	$0.761780\pi$
$150$	2430.62	1.32306
$151$	−563.495	−0.303686	−0.151843	−	0.988405i	$-0.548521\pi$
$151$	−563.495	−0.303686	−0.151843	+	0.988405i	$0.548521\pi$
$152$	−2350.44	−1.25425
$153$	−610.871	−0.322784
$154$	34.2155	0.0179037
$155$	1579.17	0.818336
$156$	−262.070	−0.134502
$157$	−1397.66	−0.710479	−0.355239	−	0.934775i	$-0.615601\pi$
$157$	−1397.66	−0.710479	−0.355239	+	0.934775i	$0.615601\pi$
$158$	−400.449	−0.201633
$159$	−3109.79	−1.55108
$160$	103.499	0.0511396
$161$	30.5253	0.0149424
$162$	−940.595	−0.456174
$163$	−1494.29	−0.718046	−0.359023	−	0.933329i	$-0.616890\pi$
$163$	−1494.29	−0.718046	−0.359023	+	0.933329i	$0.616890\pi$
$164$	112.487	0.0535594
$165$	527.622	0.248941
$166$	−2890.90	−1.35167
$167$	3668.76	1.69998	0.849992	−	0.526795i	$-0.176607\pi$
$167$	3668.76	1.69998	0.849992	+	0.526795i	$0.176607\pi$
$168$	201.409	0.0924945
$169$	1823.75	0.830109
$170$	−165.404	−0.0746229
$171$	−4945.39	−2.21160
$172$	0	0
$173$	−2656.47	−1.16744	−0.583721	−	0.811954i	$-0.698404\pi$
$173$	−2656.47	−1.16744	−0.583721	+	0.811954i	$0.698404\pi$
$174$	−6250.31	−2.72319
$175$	−101.118	−0.0436788
$176$	752.345	0.322217
$177$	−1909.89	−0.811050
$178$	1581.39	0.665902
$179$	−3270.38	−1.36559	−0.682793	−	0.730612i	$-0.739235\pi$
$179$	−3270.38	−1.36559	−0.682793	+	0.730612i	$0.739235\pi$
$180$	112.018	0.0463850
$181$	−1190.76	−0.488996	−0.244498	−	0.969650i	$-0.578623\pi$
$181$	−1190.76	−0.488996	−0.244498	+	0.969650i	$0.578623\pi$
$182$	−172.968	−0.0704466
$183$	−968.614	−0.391268
$184$	713.668	0.285937
$185$	1890.36	0.751255
$186$	7818.21	3.08204
$187$	156.650	0.0612586
$188$	−147.194	−0.0571023
$189$	189.852	0.0730671
$190$	−1339.05	−0.511289
$191$	−4355.24	−1.64992	−0.824959	−	0.565192i	$-0.808802\pi$
$191$	−4355.24	−1.64992	−0.824959	+	0.565192i	$0.808802\pi$
$192$	4693.19	1.76407
$193$	−2795.40	−1.04258	−0.521288	−	0.853381i	$-0.674548\pi$
$193$	−2795.40	−1.04258	−0.521288	+	0.853381i	$0.674548\pi$
$194$	1773.06	0.656178
$195$	−2667.27	−0.979524
$196$	162.235	0.0591235
$197$	1254.28	0.453622	0.226811	−	0.973939i	$-0.427170\pi$
$197$	1254.28	0.453622	0.226811	+	0.973939i	$0.427170\pi$
$198$	1683.10	0.604106
$199$	−276.720	−0.0985738	−0.0492869	−	0.998785i	$-0.515695\pi$
$199$	−276.720	−0.0985738	−0.0492869	+	0.998785i	$0.515695\pi$
$200$	−2364.09	−0.835834
$201$	6276.24	2.20245
$202$	1864.41	0.649402
$203$	260.023	0.0899016
$204$	51.6159	0.0177149
$205$	1144.86	0.390051
$206$	1085.24	0.367051
$207$	1501.58	0.504187
$208$	−3803.30	−1.26784
$209$	1268.18	0.419721
$210$	114.743	0.0377049
$211$	901.713	0.294201	0.147101	−	0.989122i	$-0.453006\pi$
$211$	901.713	0.294201	0.147101	+	0.989122i	$0.453006\pi$
$212$	169.307	0.0548493
$213$	4279.17	1.37655
$214$	82.7936	0.0264470
$215$	0	0
$216$	4438.66	1.39820
$217$	−325.250	−0.101748
$218$	−2003.69	−0.622510
$219$	2945.58	0.908876
$220$	−28.7254	−0.00880304
$221$	−791.906	−0.241038
$222$	9358.85	2.82939
$223$	4251.19	1.27660	0.638298	−	0.769789i	$-0.279639\pi$
$223$	4251.19	1.27660	0.638298	+	0.769789i	$0.279639\pi$
$224$	−21.3170	−0.00635848
$225$	−4974.11	−1.47381
$226$	1832.48	0.539357
$227$	−4524.40	−1.32288	−0.661442	−	0.749996i	$-0.730055\pi$
$227$	−4524.40	−1.32288	−0.661442	+	0.749996i	$0.730055\pi$
$228$	417.863	0.121376
$229$	1053.03	0.303869	0.151934	−	0.988391i	$-0.451450\pi$
$229$	1053.03	0.303869	0.151934	+	0.988391i	$0.451450\pi$
$230$	406.578	0.116561
$231$	−108.670	−0.0309523
$232$	6079.23	1.72035
$233$	−1770.11	−0.497698	−0.248849	−	0.968542i	$-0.580052\pi$
$233$	−1770.11	−0.497698	−0.248849	+	0.968542i	$0.580052\pi$
$234$	−8508.54	−2.37701
$235$	−1498.10	−0.415852
$236$	103.980	0.0286803
$237$	1271.85	0.348588
$238$	34.0670	0.00927829
$239$	−1654.10	−0.447677	−0.223839	−	0.974626i	$-0.571859\pi$
$239$	−1654.10	−0.447677	−0.223839	+	0.974626i	$0.571859\pi$
$240$	2523.02	0.678585
$241$	5811.47	1.55332	0.776659	−	0.629921i	$-0.216913\pi$
$241$	5811.47	1.55332	0.776659	+	0.629921i	$0.216913\pi$
$242$	3219.71	0.855251
$243$	−2167.71	−0.572259
$244$	52.7345	0.0138360
$245$	1651.18	0.430571
$246$	5668.00	1.46902
$247$	−6410.98	−1.65150
$248$	−7604.21	−1.94705
$249$	9181.66	2.33680
$250$	−3002.35	−0.759541
$251$	−527.609	−0.132679	−0.0663394	−	0.997797i	$-0.521132\pi$
$251$	−527.609	−0.132679	−0.0663394	+	0.997797i	$0.521132\pi$
$252$	−23.0714	−0.00576731
$253$	−385.059	−0.0956856
$254$	−5062.67	−1.25063
$255$	525.331	0.129010
$256$	−725.997	−0.177245
$257$	−7456.06	−1.80971	−0.904856	−	0.425717i	$-0.860022\pi$
$257$	−7456.06	−1.80971	−0.904856	+	0.425717i	$0.860022\pi$
$258$	0	0
$259$	−389.343	−0.0934077
$260$	145.215	0.0346378
$261$	12790.9	3.03346
$262$	−399.657	−0.0942401
$263$	1853.24	0.434509	0.217254	−	0.976115i	$-0.430290\pi$
$263$	1853.24	0.434509	0.217254	+	0.976115i	$0.430290\pi$
$264$	−2540.67	−0.592300
$265$	1723.16	0.399444
$266$	275.794	0.0635714
$267$	−5022.59	−1.15123
$268$	−341.699	−0.0778827
$269$	3490.83	0.791225	0.395613	−	0.918417i	$-0.370532\pi$
$269$	3490.83	0.791225	0.395613	+	0.918417i	$0.370532\pi$
$270$	2528.71	0.569971
$271$	−1052.34	−0.235886	−0.117943	−	0.993020i	$-0.537630\pi$
$271$	−1052.34	−0.235886	−0.117943	+	0.993020i	$0.537630\pi$
$272$	749.079	0.166984
$273$	549.357	0.121790
$274$	6031.68	1.32988
$275$	1275.54	0.279703
$276$	−126.876	−0.0276705
$277$	4652.94	1.00927	0.504636	−	0.863332i	$-0.331627\pi$
$277$	4652.94	1.00927	0.504636	+	0.863332i	$0.331627\pi$
$278$	−239.008	−0.0515639
$279$	−15999.4	−3.43320
$280$	−111.602	−0.0238197
$281$	5216.74	1.10749	0.553745	−	0.832687i	$-0.313198\pi$
$281$	5216.74	1.10749	0.553745	+	0.832687i	$0.313198\pi$
$282$	−7416.83	−1.56619
$283$	7572.67	1.59063	0.795316	−	0.606195i	$-0.207305\pi$
$283$	7572.67	1.59063	0.795316	+	0.606195i	$0.207305\pi$
$284$	−232.972	−0.0486773
$285$	4252.89	0.883928
$286$	2181.90	0.451114
$287$	−235.798	−0.0484972
$288$	−1048.61	−0.214548
$289$	−4757.03	−0.968254
$290$	3463.34	0.701291
$291$	−5631.34	−1.13442
$292$	−160.367	−0.0321396
$293$	−1902.15	−0.379266	−0.189633	−	0.981855i	$-0.560730\pi$
$293$	−1902.15	−0.379266	−0.189633	+	0.981855i	$0.560730\pi$
$294$	8174.70	1.62163
$295$	1058.28	0.208866
$296$	−9102.69	−1.78744
$297$	−2394.87	−0.467894
$298$	7312.38	1.42146
$299$	1946.58	0.376500
$300$	420.290	0.0808849
$301$	0	0
$302$	1545.83	0.294545
$303$	−5921.46	−1.12270
$304$	6064.27	1.14411
$305$	536.716	0.100762
$306$	1675.80	0.313068
$307$	6280.99	1.16767	0.583835	−	0.811872i	$-0.301551\pi$
$307$	6280.99	1.16767	0.583835	+	0.811872i	$0.301551\pi$
$308$	5.91636	0.00109453
$309$	−3446.79	−0.634567
$310$	−4332.12	−0.793704
$311$	−4512.31	−0.822733	−0.411366	−	0.911470i	$-0.634948\pi$
$311$	−4512.31	−0.822733	−0.411366	+	0.911470i	$0.634948\pi$
$312$	12843.7	2.33056
$313$	−7445.57	−1.34456	−0.672282	−	0.740295i	$-0.734686\pi$
$313$	−7445.57	−1.34456	−0.672282	+	0.740295i	$0.734686\pi$
$314$	3834.18	0.689093
$315$	−234.814	−0.0420009
$316$	−69.2435	−0.0123267
$317$	7984.88	1.41475	0.707374	−	0.706839i	$-0.249880\pi$
$317$	7984.88	1.41475	0.707374	+	0.706839i	$0.249880\pi$
$318$	8531.05	1.50440
$319$	−3280.04	−0.575696
$320$	−2600.53	−0.454294
$321$	−262.957	−0.0457222
$322$	−83.7397	−0.0144926
$323$	1262.67	0.217514
$324$	−162.643	−0.0278880
$325$	−6448.22	−1.10056
$326$	4099.26	0.696433
$327$	6363.84	1.07621
$328$	−5512.86	−0.928038
$329$	308.552	0.0517052
$330$	−1447.42	−0.241448
$331$	4174.58	0.693219	0.346610	−	0.938009i	$-0.387333\pi$
$331$	4174.58	0.693219	0.346610	+	0.938009i	$0.387333\pi$
$332$	−499.879	−0.0826338
$333$	−19152.3	−3.15177
$334$	−10064.5	−1.64881
$335$	−3477.71	−0.567187
$336$	−519.647	−0.0843723
$337$	−5608.08	−0.906503	−0.453252	−	0.891383i	$-0.649736\pi$
$337$	−5608.08	−0.906503	−0.453252	+	0.891383i	$0.649736\pi$
$338$	−5003.08	−0.805123
$339$	−5820.05	−0.932454
$340$	−28.6007	−0.00456204
$341$	4102.84	0.651558
$342$	13566.6	2.14503
$343$	−681.144	−0.107225
$344$	0	0
$345$	−1291.31	−0.201513
$346$	7287.46	1.13230
$347$	−10285.6	−1.59125	−0.795623	−	0.605792i	$-0.792856\pi$
$347$	−10285.6	−1.59125	−0.795623	+	0.605792i	$0.792856\pi$
$348$	−1080.77	−0.166481
$349$	−6951.00	−1.06613	−0.533064	−	0.846075i	$-0.678959\pi$
$349$	−6951.00	−1.06613	−0.533064	+	0.846075i	$0.678959\pi$
$350$	277.396	0.0423641
$351$	12106.7	1.84105
$352$	268.902	0.0407173
$353$	−5638.96	−0.850231	−0.425116	−	0.905139i	$-0.639767\pi$
$353$	−5638.96	−0.850231	−0.425116	+	0.905139i	$0.639767\pi$
$354$	5239.37	0.786637
$355$	−2371.12	−0.354496
$356$	273.446	0.0407096
$357$	−108.198	−0.0160405
$358$	8971.62	1.32448
$359$	8240.51	1.21147	0.605734	−	0.795667i	$-0.292879\pi$
$359$	8240.51	1.21147	0.605734	+	0.795667i	$0.292879\pi$
$360$	−5489.86	−0.803726
$361$	3363.14	0.490325
$362$	3266.59	0.474277
$363$	−10226.0	−1.47858
$364$	−29.9088	−0.00430672
$365$	−1632.17	−0.234059
$366$	2657.19	0.379491
$367$	−9830.95	−1.39829	−0.699143	−	0.714981i	$-0.746435\pi$
$367$	−9830.95	−1.39829	−0.699143	+	0.714981i	$0.746435\pi$
$368$	−1841.30	−0.260828
$369$	−11599.2	−1.63639
$370$	−5185.81	−0.728642
$371$	−354.905	−0.0496651
$372$	1351.88	0.188419
$373$	−4359.88	−0.605217	−0.302608	−	0.953115i	$-0.597857\pi$
$373$	−4359.88	−0.605217	−0.302608	+	0.953115i	$0.597857\pi$
$374$	−429.736	−0.0594147
$375$	9535.63	1.31311
$376$	7213.82	0.989427
$377$	16581.5	2.26523
$378$	−520.818	−0.0708677
$379$	−2357.07	−0.319458	−0.159729	−	0.987161i	$-0.551062\pi$
$379$	−2357.07	−0.319458	−0.159729	+	0.987161i	$0.551062\pi$
$380$	−231.541	−0.0312574
$381$	16079.3	2.16212
$382$	11947.7	1.60026
$383$	−1051.37	−0.140268	−0.0701340	−	0.997538i	$-0.522343\pi$
$383$	−1051.37	−0.140268	−0.0701340	+	0.997538i	$0.522343\pi$
$384$	−11380.5	−1.51239
$385$	60.2150	0.00797101
$386$	7668.60	1.01120
$387$	0	0
$388$	306.588	0.0401151
$389$	−6498.17	−0.846967	−0.423484	−	0.905904i	$-0.639193\pi$
$389$	−6498.17	−0.846967	−0.423484	+	0.905904i	$0.639193\pi$
$390$	7317.10	0.950040
$391$	−383.388	−0.0495876
$392$	−7950.95	−1.02445
$393$	1269.33	0.162925
$394$	−3440.84	−0.439968
$395$	−704.740	−0.0897704
$396$	291.033	0.0369317
$397$	5483.72	0.693249	0.346625	−	0.938004i	$-0.387328\pi$
$397$	5483.72	0.693249	0.346625	+	0.938004i	$0.387328\pi$
$398$	759.125	0.0956067
$399$	−875.935	−0.109904
$400$	6099.49	0.762437
$401$	13300.7	1.65637	0.828185	−	0.560454i	$-0.189373\pi$
$401$	13300.7	1.65637	0.828185	+	0.560454i	$0.189373\pi$
$402$	−17217.6	−2.13615
$403$	−20741.0	−2.56373
$404$	322.383	0.0397009
$405$	−1655.33	−0.203096
$406$	−713.318	−0.0871955
$407$	4911.35	0.598148
$408$	−2529.64	−0.306950
$409$	8978.83	1.08551	0.542756	−	0.839890i	$-0.317381\pi$
$409$	8978.83	1.08551	0.542756	+	0.839890i	$0.317381\pi$
$410$	−3140.68	−0.378310
$411$	−19156.9	−2.29913
$412$	187.654	0.0224395
$413$	−217.966	−0.0259695
$414$	−4119.26	−0.489011
$415$	−5087.62	−0.601787
$416$	−1359.37	−0.160213
$417$	759.103	0.0891450
$418$	−3478.98	−0.407088
$419$	−8801.11	−1.02616	−0.513082	−	0.858340i	$-0.671496\pi$
$419$	−8801.11	−1.02616	−0.513082	+	0.858340i	$0.671496\pi$
$420$	19.8408	0.00230507
$421$	−13439.0	−1.55576	−0.777881	−	0.628412i	$-0.783705\pi$
$421$	−13439.0	−1.55576	−0.777881	+	0.628412i	$0.783705\pi$
$422$	−2473.66	−0.285346
$423$	15178.1	1.74464
$424$	−8297.54	−0.950388
$425$	1270.01	0.144952
$426$	−11739.0	−1.33511
$427$	−110.543	−0.0125283
$428$	14.3162	0.00161682
$429$	−6929.83	−0.779896
$430$	0	0
$431$	−13200.3	−1.47526	−0.737629	−	0.675206i	$-0.764055\pi$
$431$	−13200.3	−1.47526	−0.737629	+	0.675206i	$0.764055\pi$
$432$	−11452.0	−1.27542
$433$	−5278.83	−0.585876	−0.292938	−	0.956131i	$-0.594633\pi$
$433$	−5278.83	−0.585876	−0.292938	+	0.956131i	$0.594633\pi$
$434$	892.255	0.0986857
$435$	−10999.8	−1.21241
$436$	−346.468	−0.0380569
$437$	−3103.77	−0.339756
$438$	−8080.58	−0.881518
$439$	10553.2	1.14732	0.573662	−	0.819092i	$-0.305522\pi$
$439$	10553.2	1.14732	0.573662	+	0.819092i	$0.305522\pi$
$440$	1407.80	0.152533
$441$	−16729.0	−1.80639
$442$	2172.43	0.233783
$443$	−3133.51	−0.336067	−0.168033	−	0.985781i	$-0.553742\pi$
$443$	−3133.51	−0.336067	−0.168033	+	0.985781i	$0.553742\pi$
$444$	1618.28	0.172974
$445$	2783.06	0.296471
$446$	−11662.3	−1.23817
$447$	−23224.5	−2.45745
$448$	535.611	0.0564849
$449$	15319.8	1.61021	0.805105	−	0.593132i	$-0.202109\pi$
$449$	15319.8	1.61021	0.805105	+	0.593132i	$0.202109\pi$
$450$	13645.4	1.42945
$451$	2974.46	0.310558
$452$	316.862	0.0329734
$453$	−4909.64	−0.509217
$454$	12411.7	1.28306
$455$	−304.403	−0.0313640
$456$	−20479.0	−2.10311
$457$	12631.2	1.29292	0.646458	−	0.762950i	$-0.276250\pi$
$457$	12631.2	1.29292	0.646458	+	0.762950i	$0.276250\pi$
$458$	−2888.76	−0.294722
$459$	−2384.48	−0.242479
$460$	70.3032	0.00712588
$461$	−919.083	−0.0928546	−0.0464273	−	0.998922i	$-0.514784\pi$
$461$	−919.083	−0.0928546	−0.0464273	+	0.998922i	$0.514784\pi$
$462$	298.114	0.0300206
$463$	−9655.13	−0.969140	−0.484570	−	0.874752i	$-0.661024\pi$
$463$	−9655.13	−0.969140	−0.484570	+	0.874752i	$0.661024\pi$
$464$	−15684.7	−1.56928
$465$	13759.1	1.37217
$466$	4855.93	0.482718
$467$	842.261	0.0834586	0.0417293	−	0.999129i	$-0.486713\pi$
$467$	842.261	0.0834586	0.0417293	+	0.999129i	$0.486713\pi$
$468$	−1471.25	−0.145317
$469$	716.277	0.0705216
$470$	4109.72	0.403335
$471$	−12177.6	−1.19132
$472$	−5095.96	−0.496951
$473$	0	0
$474$	−3489.05	−0.338095
$475$	10281.5	0.993155
$476$	5.89067	0.000567224 0
$477$	−17458.2	−1.67580
$478$	4537.68	0.434202
$479$	10036.2	0.957341	0.478671	−	0.877995i	$-0.341119\pi$
$479$	10036.2	0.957341	0.478671	+	0.877995i	$0.341119\pi$
$480$	901.773	0.0857502
$481$	−24828.2	−2.35357
$482$	−15942.6	−1.50656
$483$	265.962	0.0250552
$484$	556.735	0.0522854
$485$	3120.37	0.292141
$486$	5946.67	0.555034
$487$	−7004.26	−0.651732	−0.325866	−	0.945416i	$-0.605656\pi$
$487$	−7004.26	−0.651732	−0.325866	+	0.945416i	$0.605656\pi$
$488$	−2584.46	−0.239740
$489$	−13019.5	−1.20401
$490$	−4529.66	−0.417611
$491$	−8855.78	−0.813963	−0.406982	−	0.913436i	$-0.633419\pi$
$491$	−8855.78	−0.813963	−0.406982	+	0.913436i	$0.633419\pi$
$492$	980.080	0.0898077
$493$	−3265.80	−0.298346
$494$	17587.2	1.60179
$495$	2962.05	0.268958
$496$	19619.3	1.77607
$497$	488.362	0.0440765
$498$	−25188.0	−2.26646
$499$	6204.37	0.556604	0.278302	−	0.960494i	$-0.410228\pi$
$499$	6204.37	0.556604	0.278302	+	0.960494i	$0.410228\pi$
$500$	−519.150	−0.0464342
$501$	31965.3	2.85051
$502$	1447.38	0.128685
$503$	−7719.67	−0.684300	−0.342150	−	0.939645i	$-0.611155\pi$
$503$	−7719.67	−0.684300	−0.342150	+	0.939645i	$0.611155\pi$
$504$	1130.71	0.0999318
$505$	3281.12	0.289125
$506$	1056.33	0.0928055
$507$	15890.0	1.39192
$508$	−875.410	−0.0764568
$509$	−8025.63	−0.698880	−0.349440	−	0.936959i	$-0.613628\pi$
$509$	−8025.63	−0.698880	−0.349440	+	0.936959i	$0.613628\pi$
$510$	−1441.14	−0.125127
$511$	336.165	0.0291019
$512$	12441.0	1.07387
$513$	−19303.8	−1.66138
$514$	20454.1	1.75524
$515$	1909.89	0.163417
$516$	0	0
$517$	−3892.21	−0.331101
$518$	1068.08	0.0905962
$519$	−23145.4	−1.95755
$520$	−7116.82	−0.600179
$521$	−7825.91	−0.658079	−0.329040	−	0.944316i	$-0.606725\pi$
$521$	−7825.91	−0.658079	−0.329040	+	0.944316i	$0.606725\pi$
$522$	−35089.0	−2.94215
$523$	−6458.72	−0.540001	−0.270000	−	0.962860i	$-0.587024\pi$
$523$	−6458.72	−0.540001	−0.270000	+	0.962860i	$0.587024\pi$
$524$	−69.1066	−0.00576133
$525$	−881.023	−0.0732400
$526$	−5083.98	−0.421430
$527$	4085.03	0.337660
$528$	6555.06	0.540289
$529$	−11224.6	−0.922544
$530$	−4727.12	−0.387421
$531$	−10722.0	−0.876265
$532$	47.6887	0.00388641
$533$	−15036.7	−1.22197
$534$	13778.4	1.11658
$535$	145.706	0.0117746
$536$	16746.3	1.34949
$537$	−28494.3	−2.28980
$538$	−9576.36	−0.767409
$539$	4289.93	0.342820
$540$	437.250	0.0348449
$541$	−765.880	−0.0608646	−0.0304323	−	0.999537i	$-0.509688\pi$
$541$	−765.880	−0.0608646	−0.0304323	+	0.999537i	$0.509688\pi$
$542$	2886.88	0.228786
$543$	−10374.9	−0.819942
$544$	267.734	0.0211011
$545$	−3526.25	−0.277152
$546$	−1507.05	−0.118124
$547$	−6523.07	−0.509884	−0.254942	−	0.966956i	$-0.582056\pi$
$547$	−6523.07	−0.509884	−0.254942	+	0.966956i	$0.582056\pi$
$548$	1042.96	0.0813015
$549$	−5437.76	−0.422729
$550$	−3499.19	−0.271284
$551$	−26438.8	−2.04415
$552$	6218.08	0.479455
$553$	145.150	0.0111617
$554$	−12764.4	−0.978892
$555$	16470.4	1.25969
$556$	−41.3280	−0.00315234
$557$	−14942.4	−1.13668	−0.568340	−	0.822794i	$-0.692414\pi$
$557$	−14942.4	−1.13668	−0.568340	+	0.822794i	$0.692414\pi$
$558$	43891.1	3.32986
$559$	0	0
$560$	287.940	0.0217280
$561$	1364.86	0.102718
$562$	−14311.0	−1.07415
$563$	10019.2	0.750019	0.375009	−	0.927021i	$-0.377640\pi$
$563$	10019.2	0.750019	0.375009	+	0.927021i	$0.377640\pi$
$564$	−1282.48	−0.0957484
$565$	3224.93	0.240131
$566$	−20774.0	−1.54275
$567$	340.936	0.0252521
$568$	11417.7	0.843444
$569$	8279.60	0.610015	0.305008	−	0.952350i	$-0.401341\pi$
$569$	8279.60	0.610015	0.305008	+	0.952350i	$0.401341\pi$
$570$	−11666.9	−0.857322
$571$	−8993.21	−0.659114	−0.329557	−	0.944136i	$-0.606899\pi$
$571$	−8993.21	−0.659114	−0.329557	+	0.944136i	$0.606899\pi$
$572$	377.282	0.0275786
$573$	−37946.5	−2.76656
$574$	646.862	0.0470374
$575$	−3121.79	−0.226414
$576$	26347.4	1.90592
$577$	13036.3	0.940569	0.470284	−	0.882515i	$-0.344151\pi$
$577$	13036.3	0.940569	0.470284	+	0.882515i	$0.344151\pi$
$578$	13049.9	0.939109
$579$	−24355.9	−1.74818
$580$	598.863	0.0428731
$581$	1047.86	0.0748236
$582$	15448.4	1.10027
$583$	4476.93	0.318037
$584$	7859.40	0.556891
$585$	−14974.0	−1.05829
$586$	5218.15	0.367850
$587$	−5329.19	−0.374718	−0.187359	−	0.982292i	$-0.559993\pi$
$587$	−5329.19	−0.374718	−0.187359	+	0.982292i	$0.559993\pi$
$588$	1413.53	0.0991374
$589$	33070.9	2.31352
$590$	−2903.18	−0.202579
$591$	10928.3	0.760627
$592$	23485.4	1.63048
$593$	20729.4	1.43551	0.717754	−	0.696297i	$-0.245170\pi$
$593$	20729.4	1.43551	0.717754	+	0.696297i	$0.245170\pi$
$594$	6569.83	0.453811
$595$	59.9536	0.00413085
$596$	1264.42	0.0869002
$597$	−2411.02	−0.165287
$598$	−5340.03	−0.365167
$599$	22730.1	1.55046	0.775231	−	0.631678i	$-0.217633\pi$
$599$	22730.1	1.55046	0.775231	+	0.631678i	$0.217633\pi$
$600$	−20598.0	−1.40151
$601$	−6695.67	−0.454446	−0.227223	−	0.973843i	$-0.572965\pi$
$601$	−6695.67	−0.454446	−0.227223	+	0.973843i	$0.572965\pi$
$602$	0	0
$603$	35234.6	2.37954
$604$	267.297	0.0180069
$605$	5666.29	0.380772
$606$	16244.3	1.08891
$607$	5055.16	0.338027	0.169014	−	0.985614i	$-0.445942\pi$
$607$	5055.16	0.338027	0.169014	+	0.985614i	$0.445942\pi$
$608$	2167.48	0.144577
$609$	2265.54	0.150746
$610$	−1472.37	−0.0977286
$611$	19676.2	1.30280
$612$	289.770	0.0191393
$613$	−7732.10	−0.509456	−0.254728	−	0.967013i	$-0.581986\pi$
$613$	−7732.10	−0.509456	−0.254728	+	0.967013i	$0.581986\pi$
$614$	−17230.6	−1.13252
$615$	9974.97	0.654032
$616$	−289.954	−0.0189652
$617$	−19892.7	−1.29797	−0.648985	−	0.760801i	$-0.724806\pi$
$617$	−19892.7	−1.29797	−0.648985	+	0.760801i	$0.724806\pi$
$618$	9455.55	0.615466
$619$	7317.37	0.475137	0.237569	−	0.971371i	$-0.423650\pi$
$619$	7317.37	0.475137	0.237569	+	0.971371i	$0.423650\pi$
$620$	−749.088	−0.0485227
$621$	5861.26	0.378751
$622$	12378.6	0.797968
$623$	−573.205	−0.0368619
$624$	−33137.6	−2.12591
$625$	7427.72	0.475374
$626$	20425.4	1.30409
$627$	11049.4	0.703783
$628$	662.986	0.0421274
$629$	4890.03	0.309981
$630$	644.164	0.0407367
$631$	10233.3	0.645610	0.322805	−	0.946466i	$-0.395374\pi$
$631$	10233.3	0.645610	0.322805	+	0.946466i	$0.395374\pi$
$632$	3393.55	0.213589
$633$	7856.48	0.493313
$634$	−21904.8	−1.37216
$635$	−8909.67	−0.556802
$636$	1475.14	0.0919705
$637$	−21686.7	−1.34892
$638$	8998.11	0.558368
$639$	24023.1	1.48723
$640$	6306.01	0.389480
$641$	8872.37	0.546704	0.273352	−	0.961914i	$-0.411868\pi$
$641$	8872.37	0.546704	0.273352	+	0.961914i	$0.411868\pi$
$642$	721.367	0.0443459
$643$	−13442.3	−0.824438	−0.412219	−	0.911085i	$-0.635246\pi$
$643$	−13442.3	−0.824438	−0.412219	+	0.911085i	$0.635246\pi$
$644$	−14.4798	−0.000886001 0
$645$	0	0
$646$	−3463.88	−0.210967
$647$	15983.1	0.971192	0.485596	−	0.874183i	$-0.338603\pi$
$647$	15983.1	0.971192	0.485596	+	0.874183i	$0.338603\pi$
$648$	7970.94	0.483222
$649$	2749.52	0.166299
$650$	17689.3	1.06744
$651$	−2833.85	−0.170610
$652$	708.822	0.0425761
$653$	19700.1	1.18059	0.590294	−	0.807188i	$-0.299012\pi$
$653$	19700.1	1.18059	0.590294	+	0.807188i	$0.299012\pi$
$654$	−17457.9	−1.04382
$655$	−703.347	−0.0419573
$656$	14223.5	0.846545
$657$	16536.4	0.981956
$658$	−846.448	−0.0501489
$659$	−7122.89	−0.421044	−0.210522	−	0.977589i	$-0.567516\pi$
$659$	−7122.89	−0.421044	−0.210522	+	0.977589i	$0.567516\pi$
$660$	−250.280	−0.0147608
$661$	24964.5	1.46900	0.734498	−	0.678611i	$-0.237418\pi$
$661$	24964.5	1.46900	0.734498	+	0.678611i	$0.237418\pi$
$662$	−11452.1	−0.672353
$663$	−6899.75	−0.404169
$664$	24498.5	1.43182
$665$	485.362	0.0283031
$666$	52540.3	3.05690
$667$	8027.64	0.466014
$668$	−1740.30	−0.100800
$669$	37040.0	2.14058
$670$	9540.37	0.550115
$671$	1394.44	0.0802263
$672$	−185.731	−0.0106618
$673$	−501.559	−0.0287276	−0.0143638	−	0.999897i	$-0.504572\pi$
$673$	−501.559	−0.0287276	−0.0143638	+	0.999897i	$0.504572\pi$
$674$	15384.6	0.879217
$675$	−19416.0	−1.10714
$676$	−865.105	−0.0492208
$677$	6425.18	0.364756	0.182378	−	0.983228i	$-0.441621\pi$
$677$	6425.18	0.364756	0.182378	+	0.983228i	$0.441621\pi$
$678$	15966.1	0.904387
$679$	−642.678	−0.0363236
$680$	1401.69	0.0790476
$681$	−39420.3	−2.21819
$682$	−11255.3	−0.631946
$683$	−15668.9	−0.877825	−0.438913	−	0.898530i	$-0.644636\pi$
$683$	−15668.9	−0.877825	−0.438913	+	0.898530i	$0.644636\pi$
$684$	2345.87	0.131135
$685$	10615.0	0.592085
$686$	1868.58	0.103998
$687$	9174.85	0.509523
$688$	0	0
$689$	−22632.1	−1.25140
$690$	3542.45	0.195447
$691$	9872.73	0.543526	0.271763	−	0.962364i	$-0.412393\pi$
$691$	9872.73	0.543526	0.271763	+	0.962364i	$0.412393\pi$
$692$	1260.11	0.0692228
$693$	−610.071	−0.0334411
$694$	28216.5	1.54335
$695$	−420.625	−0.0229571
$696$	52967.4	2.88466
$697$	2961.54	0.160942
$698$	19068.6	1.03404
$699$	−15422.7	−0.834534
$700$	47.9657	0.00258991
$701$	18330.8	0.987652	0.493826	−	0.869561i	$-0.335598\pi$
$701$	18330.8	0.987652	0.493826	+	0.869561i	$0.335598\pi$
$702$	−33212.3	−1.78563
$703$	39587.9	2.12388
$704$	−6756.43	−0.361708
$705$	−13052.7	−0.697295
$706$	15469.3	0.824639
$707$	−675.788	−0.0359485
$708$	905.964	0.0480907
$709$	8213.04	0.435045	0.217523	−	0.976055i	$-0.430202\pi$
$709$	8213.04	0.435045	0.217523	+	0.976055i	$0.430202\pi$
$710$	6504.68	0.343826
$711$	7140.11	0.376617
$712$	−13401.3	−0.705386
$713$	−10041.4	−0.527423
$714$	296.820	0.0155577
$715$	3839.87	0.200843
$716$	1551.32	0.0809716
$717$	−14411.9	−0.750659
$718$	−22606.1	−1.17500
$719$	−10621.3	−0.550915	−0.275458	−	0.961313i	$-0.588829\pi$
$719$	−10621.3	−0.550915	−0.275458	+	0.961313i	$0.588829\pi$
$720$	14164.2	0.733148
$721$	−393.366	−0.0203186
$722$	−9226.08	−0.475567
$723$	50634.4	2.60458
$724$	564.842	0.0289947
$725$	−26592.3	−1.36223
$726$	28052.8	1.43407
$727$	11281.4	0.575521	0.287761	−	0.957702i	$-0.407089\pi$
$727$	11281.4	0.575521	0.287761	+	0.957702i	$0.407089\pi$
$728$	1465.80	0.0746236
$729$	−28144.5	−1.42989
$730$	4477.51	0.227014
$731$	0	0
$732$	459.467	0.0232000
$733$	16554.0	0.834154	0.417077	−	0.908871i	$-0.363054\pi$
$733$	16554.0	0.834154	0.417077	+	0.908871i	$0.363054\pi$
$734$	26969.2	1.35620
$735$	14386.5	0.721976
$736$	−658.115	−0.0329598
$737$	−9035.44	−0.451594
$738$	31819.9	1.58714
$739$	−6950.25	−0.345966	−0.172983	−	0.984925i	$-0.555341\pi$
$739$	−6950.25	−0.345966	−0.172983	+	0.984925i	$0.555341\pi$
$740$	−896.702	−0.0445452
$741$	−55857.9	−2.76922
$742$	973.609	0.0481702
$743$	7791.86	0.384731	0.192366	−	0.981323i	$-0.438384\pi$
$743$	7791.86	0.384731	0.192366	+	0.981323i	$0.438384\pi$
$744$	−66254.2	−3.26478
$745$	12868.9	0.632857
$746$	11960.4	0.586999
$747$	51545.5	2.52470
$748$	−74.3076	−0.00363229
$749$	−30.0100	−0.00146401
$750$	−26159.0	−1.27359
$751$	−37444.7	−1.81941	−0.909705	−	0.415254i	$-0.863693\pi$
$751$	−37444.7	−1.81941	−0.909705	+	0.415254i	$0.863693\pi$
$752$	−18612.1	−0.902542
$753$	−4596.97	−0.222474
$754$	−45487.9	−2.19704
$755$	2720.47	0.131136
$756$	−90.0571	−0.00433247
$757$	−20901.2	−1.00352	−0.501761	−	0.865006i	$-0.667314\pi$
$757$	−20901.2	−1.00352	−0.501761	+	0.865006i	$0.667314\pi$
$758$	6466.14	0.309843
$759$	−3354.96	−0.160444
$760$	11347.6	0.541605
$761$	6360.91	0.303000	0.151500	−	0.988457i	$-0.451590\pi$
$761$	6360.91	0.303000	0.151500	+	0.988457i	$0.451590\pi$
$762$	−44110.3	−2.09704
$763$	726.274	0.0344599
$764$	2065.93	0.0978309
$765$	2949.19	0.139383
$766$	2884.22	0.136046
$767$	−13899.6	−0.654347
$768$	−6325.50	−0.297203
$769$	21174.8	0.992954	0.496477	−	0.868050i	$-0.334627\pi$
$769$	21174.8	0.992954	0.496477	+	0.868050i	$0.334627\pi$
$770$	−165.187	−0.00773108
$771$	−64963.5	−3.03450
$772$	1326.01	0.0618189
$773$	−30679.2	−1.42750	−0.713749	−	0.700402i	$-0.753004\pi$
$773$	−30679.2	−1.42750	−0.713749	+	0.700402i	$0.753004\pi$
$774$	0	0
$775$	33263.0	1.54173
$776$	−15025.6	−0.695085
$777$	−3392.28	−0.156625
$778$	17826.4	0.821473
$779$	23975.6	1.10271
$780$	1265.23	0.0580803
$781$	−6160.41	−0.282250
$782$	1051.74	0.0480950
$783$	49927.9	2.27877
$784$	20513.9	0.934488
$785$	6747.68	0.306796
$786$	−3482.15	−0.158021
$787$	33191.5	1.50337	0.751683	−	0.659524i	$-0.229242\pi$
$787$	33191.5	1.50337	0.751683	+	0.659524i	$0.229242\pi$
$788$	−594.972	−0.0268972
$789$	16147.0	0.728578
$790$	1933.31	0.0870683
$791$	−664.215	−0.0298568
$792$	−14263.2	−0.639926
$793$	−7049.28	−0.315671
$794$	−15043.4	−0.672382
$795$	15013.6	0.669782
$796$	131.264	0.00584487
$797$	−5342.22	−0.237429	−0.118715	−	0.992928i	$-0.537877\pi$
$797$	−5342.22	−0.237429	−0.118715	+	0.992928i	$0.537877\pi$
$798$	2402.95	0.106596
$799$	−3875.31	−0.171588
$800$	2180.07	0.0963463
$801$	−28196.7	−1.24380
$802$	−36487.7	−1.60651
$803$	−4240.53	−0.186358
$804$	−2977.17	−0.130593
$805$	−147.371	−0.00645237
$806$	56898.5	2.48656
$807$	30415.0	1.32672
$808$	−15799.7	−0.687908
$809$	36955.4	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$809$	36955.4	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$810$	4541.05	0.196983
$811$	−2024.13	−0.0876409	−0.0438204	−	0.999039i	$-0.513953\pi$
$811$	−2024.13	−0.0876409	−0.0438204	+	0.999039i	$0.513953\pi$
$812$	−123.343	−0.00533066
$813$	−9168.88	−0.395531
$814$	−13473.3	−0.580144
$815$	7214.19	0.310064
$816$	6526.60	0.279996
$817$	0	0
$818$	−24631.5	−1.05284
$819$	3084.07	0.131583
$820$	−543.069	−0.0231278
$821$	−43872.0	−1.86497	−0.932487	−	0.361205i	$-0.882366\pi$
$821$	−43872.0	−1.86497	−0.932487	+	0.361205i	$0.882366\pi$
$822$	52553.0	2.22992
$823$	7900.39	0.334618	0.167309	−	0.985905i	$-0.446492\pi$
$823$	7900.39	0.334618	0.167309	+	0.985905i	$0.446492\pi$
$824$	−9196.74	−0.388815
$825$	11113.6	0.469002
$826$	597.945	0.0251879
$827$	20247.9	0.851378	0.425689	−	0.904870i	$-0.360032\pi$
$827$	20247.9	0.851378	0.425689	+	0.904870i	$0.360032\pi$
$828$	−712.280	−0.0298955
$829$	12684.9	0.531441	0.265721	−	0.964050i	$-0.414390\pi$
$829$	12684.9	0.531441	0.265721	+	0.964050i	$0.414390\pi$
$830$	13956.8	0.583673
$831$	40540.3	1.69233
$832$	34155.6	1.42324
$833$	4271.30	0.177661
$834$	−2082.44	−0.0864617
$835$	−17712.2	−0.734080
$836$	−601.567	−0.0248871
$837$	−62452.3	−2.57905
$838$	24144.0	0.995276
$839$	−35411.7	−1.45715	−0.728574	−	0.684967i	$-0.759817\pi$
$839$	−35411.7	−1.45715	−0.728574	+	0.684967i	$0.759817\pi$
$840$	−972.374	−0.0399406
$841$	43992.7	1.80379
$842$	36867.0	1.50893
$843$	45452.6	1.85702
$844$	−427.732	−0.0174445
$845$	−8804.79	−0.358454
$846$	−41637.8	−1.69213
$847$	−1167.04	−0.0473436
$848$	21408.1	0.866932
$849$	65979.5	2.66715
$850$	−3484.00	−0.140588
$851$	−12020.1	−0.484189
$852$	−2029.85	−0.0816214
$853$	7167.75	0.287713	0.143856	−	0.989599i	$-0.454050\pi$
$853$	7167.75	0.287713	0.143856	+	0.989599i	$0.454050\pi$
$854$	303.252	0.0121512
$855$	23875.6	0.955003
$856$	−701.622	−0.0280151
$857$	38853.9	1.54869	0.774344	−	0.632765i	$-0.218080\pi$
$857$	38853.9	1.54869	0.774344	+	0.632765i	$0.218080\pi$
$858$	19010.5	0.756421
$859$	−12846.7	−0.510273	−0.255137	−	0.966905i	$-0.582120\pi$
$859$	−12846.7	−0.510273	−0.255137	+	0.966905i	$0.582120\pi$
$860$	0	0
$861$	−2054.47	−0.0813195
$862$	36212.3	1.43085
$863$	−49634.2	−1.95778	−0.978892	−	0.204377i	$-0.934483\pi$
$863$	−49634.2	−1.95778	−0.978892	+	0.204377i	$0.934483\pi$
$864$	−4093.14	−0.161171
$865$	12825.0	0.504120
$866$	14481.4	0.568241
$867$	−41447.2	−1.62355
$868$	154.284	0.00603310
$869$	−1830.98	−0.0714752
$870$	30175.6	1.17592
$871$	45676.6	1.77691
$872$	16980.0	0.659421
$873$	−31614.2	−1.22563
$874$	8514.53	0.329529
$875$	1088.26	0.0420454
$876$	−1397.25	−0.0538912
$877$	17075.4	0.657463	0.328732	−	0.944423i	$-0.393379\pi$
$877$	17075.4	0.657463	0.328732	+	0.944423i	$0.393379\pi$
$878$	−28950.4	−1.11279
$879$	−16573.1	−0.635948
$880$	−3632.21	−0.139138
$881$	22263.7	0.851400	0.425700	−	0.904864i	$-0.360028\pi$
$881$	22263.7	0.851400	0.425700	+	0.904864i	$0.360028\pi$
$882$	45892.5	1.75202
$883$	−39190.7	−1.49363	−0.746813	−	0.665034i	$-0.768417\pi$
$883$	−39190.7	−1.49363	−0.746813	+	0.665034i	$0.768417\pi$
$884$	375.645	0.0142922
$885$	9220.64	0.350224
$886$	8596.13	0.325951
$887$	−16209.9	−0.613615	−0.306807	−	0.951772i	$-0.599261\pi$
$887$	−16209.9	−0.613615	−0.306807	+	0.951772i	$0.599261\pi$
$888$	−79310.3	−2.99716
$889$	1835.06	0.0692304
$890$	−7634.73	−0.287547
$891$	−4300.71	−0.161705
$892$	−2016.57	−0.0756950
$893$	−31373.1	−1.17566
$894$	63711.6	2.38348
$895$	15788.9	0.589682
$896$	−1298.80	−0.0484262
$897$	16960.2	0.631310
$898$	−42026.6	−1.56174
$899$	−85535.3	−3.17326
$900$	2359.50	0.0873887
$901$	4457.50	0.164818
$902$	−8159.80	−0.301210
$903$	0	0
$904$	−15529.1	−0.571338
$905$	5748.80	0.211156
$906$	13468.6	0.493889
$907$	−41480.8	−1.51857	−0.759287	−	0.650756i	$-0.774452\pi$
$907$	−41480.8	−1.51857	−0.759287	+	0.650756i	$0.774452\pi$
$908$	2146.17	0.0784396
$909$	−33242.9	−1.21298
$910$	835.065	0.0304199
$911$	3548.21	0.129042	0.0645211	−	0.997916i	$-0.479448\pi$
$911$	3548.21	0.129042	0.0645211	+	0.997916i	$0.479448\pi$
$912$	52837.0	1.91843
$913$	−13218.2	−0.479143
$914$	−34651.0	−1.25400
$915$	4676.32	0.168956
$916$	−499.509	−0.0180177
$917$	144.863	0.00521679
$918$	6541.31	0.235180
$919$	34616.9	1.24255	0.621276	−	0.783592i	$-0.286614\pi$
$919$	34616.9	1.24255	0.621276	+	0.783592i	$0.286614\pi$
$920$	−3445.48	−0.123472
$921$	54725.2	1.95793
$922$	2521.31	0.0900597
$923$	31142.5	1.11058
$924$	51.5483	0.00183530
$925$	39817.8	1.41535
$926$	26486.8	0.939969
$927$	−19350.2	−0.685591
$928$	−5606.01	−0.198304
$929$	17328.1	0.611966	0.305983	−	0.952037i	$-0.401015\pi$
$929$	17328.1	0.611966	0.305983	+	0.952037i	$0.401015\pi$
$930$	−37745.1	−1.33087
$931$	34578.9	1.21727
$932$	839.660	0.0295107
$933$	−39315.1	−1.37955
$934$	−2310.57	−0.0809465
$935$	−756.281	−0.0264524
$936$	72104.4	2.51795
$937$	−35808.9	−1.24848	−0.624240	−	0.781233i	$-0.714591\pi$
$937$	−35808.9	−1.24848	−0.624240	+	0.781233i	$0.714591\pi$
$938$	−1964.96	−0.0683989
$939$	−64872.0	−2.25455
$940$	710.631	0.0246577
$941$	19824.1	0.686766	0.343383	−	0.939195i	$-0.388427\pi$
$941$	19824.1	0.686766	0.343383	+	0.939195i	$0.388427\pi$
$942$	33406.6	1.15546
$943$	−7279.74	−0.251390
$944$	13147.9	0.453312
$945$	−916.575	−0.0315515
$946$	0	0
$947$	−2471.96	−0.0848234	−0.0424117	−	0.999100i	$-0.513504\pi$
$947$	−2471.96	−0.0848234	−0.0424117	+	0.999100i	$0.513504\pi$
$948$	−603.307	−0.0206693
$949$	21437.0	0.733272
$950$	−28205.2	−0.963260
$951$	69570.9	2.37223
$952$	−288.695	−0.00982844
$953$	−11993.6	−0.407673	−0.203836	−	0.979005i	$-0.565341\pi$
$953$	−11993.6	−0.407673	−0.203836	+	0.979005i	$0.565341\pi$
$954$	47893.0	1.62536
$955$	21026.5	0.712461
$956$	784.631	0.0265448
$957$	−28578.5	−0.965320
$958$	−27532.3	−0.928525
$959$	−2186.29	−0.0736173
$960$	−22658.0	−0.761754
$961$	77200.9	2.59142
$962$	68110.9	2.28273
$963$	−1476.23	−0.0493986
$964$	−2756.70	−0.0921031
$965$	13495.8	0.450201
$966$	−729.610	−0.0243011
$967$	4483.31	0.149094	0.0745468	−	0.997218i	$-0.476249\pi$
$967$	4483.31	0.149094	0.0745468	+	0.997218i	$0.476249\pi$
$968$	−27285.0	−0.905963
$969$	11001.5	0.364725
$970$	−8560.08	−0.283348
$971$	46934.8	1.55119	0.775597	−	0.631228i	$-0.217449\pi$
$971$	46934.8	1.55119	0.775597	+	0.631228i	$0.217449\pi$
$972$	1028.27	0.0339317
$973$	86.6329	0.00285439
$974$	19214.7	0.632115
$975$	−56182.3	−1.84541
$976$	6668.04	0.218687
$977$	6270.40	0.205331	0.102665	−	0.994716i	$-0.467263\pi$
$977$	6270.40	0.205331	0.102665	+	0.994716i	$0.467263\pi$
$978$	35716.2	1.16777
$979$	7230.66	0.236050
$980$	−783.245	−0.0255304
$981$	35726.3	1.16275
$982$	24294.0	0.789463
$983$	12981.2	0.421198	0.210599	−	0.977573i	$-0.432459\pi$
$983$	12981.2	0.421198	0.210599	+	0.977573i	$0.432459\pi$
$984$	−48032.6	−1.55612
$985$	−6055.45	−0.195881
$986$	8959.05	0.289365
$987$	2688.36	0.0866986
$988$	3041.08	0.0979248
$989$	0	0
$990$	−8125.77	−0.260862
$991$	−12043.5	−0.386050	−0.193025	−	0.981194i	$-0.561830\pi$
$991$	−12043.5	−0.386050	−0.193025	+	0.981194i	$0.561830\pi$
$992$	7012.28	0.224436
$993$	36372.4	1.16238
$994$	−1339.72	−0.0427498
$995$	1335.96	0.0425657
$996$	−4355.37	−0.138559
$997$	−32229.9	−1.02380	−0.511902	−	0.859044i	$-0.671059\pi$
$997$	−32229.9	−1.02380	−0.511902	+	0.859044i	$0.671059\pi$
$998$	−17020.4	−0.539850
$999$	−74759.1	−2.36764

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.k.1.19		60
43.12	odd	42		43.4.g.a.15.7	✓	120
43.18	odd	42		43.4.g.a.23.7	yes	120
43.42	odd	2		1849.4.a.l.1.42		60

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.15.7	✓	120	43.12	odd	42
43.4.g.a.23.7	yes	120	43.18	odd	42
1849.4.a.k.1.19		60	1.1	even	1	trivial
1849.4.a.l.1.42		60	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)

\(f(q)\)	\(=\)	\(q-2.74329 q^{2} +8.71284 q^{3} -0.474355 q^{4} -4.82785 q^{5} -23.9019 q^{6} +0.994355 q^{7} +23.2476 q^{8} +48.9136 q^{9} +O(q^{10})\)	\(q-2.74329 q^{2} +8.71284 q^{3} -0.474355 q^{4} -4.82785 q^{5} -23.9019 q^{6} +0.994355 q^{7} +23.2476 q^{8} +48.9136 q^{9} +13.2442 q^{10} -12.5432 q^{11} -4.13298 q^{12} +63.4094 q^{13} -2.72780 q^{14} -42.0643 q^{15} -59.9801 q^{16} -12.4888 q^{17} -134.184 q^{18} -101.105 q^{19} +2.29011 q^{20} +8.66366 q^{21} +34.4097 q^{22} +30.6986 q^{23} +202.553 q^{24} -101.692 q^{25} -173.950 q^{26} +190.929 q^{27} -0.471677 q^{28} +261.499 q^{29} +115.395 q^{30} -327.096 q^{31} -21.4380 q^{32} -109.287 q^{33} +34.2604 q^{34} -4.80059 q^{35} -23.2024 q^{36} -391.553 q^{37} +277.359 q^{38} +552.476 q^{39} -112.236 q^{40} -237.136 q^{41} -23.7669 q^{42} +5.94995 q^{44} -236.147 q^{45} -84.2151 q^{46} +310.304 q^{47} -522.597 q^{48} -342.011 q^{49} +278.970 q^{50} -108.813 q^{51} -30.0786 q^{52} -356.920 q^{53} -523.775 q^{54} +60.5568 q^{55} +23.1164 q^{56} -880.908 q^{57} -717.368 q^{58} -219.204 q^{59} +19.9534 q^{60} -111.171 q^{61} +897.320 q^{62} +48.6375 q^{63} +538.652 q^{64} -306.131 q^{65} +299.807 q^{66} +720.344 q^{67} +5.92412 q^{68} +267.472 q^{69} +13.1694 q^{70} +491.134 q^{71} +1137.12 q^{72} +338.073 q^{73} +1074.15 q^{74} -886.025 q^{75} +47.9595 q^{76} -12.4724 q^{77} -1515.60 q^{78} +145.974 q^{79} +289.575 q^{80} +342.871 q^{81} +650.534 q^{82} +1053.81 q^{83} -4.10965 q^{84} +60.2939 q^{85} +2278.40 q^{87} -291.600 q^{88} -576.459 q^{89} +647.821 q^{90} +63.0514 q^{91} -14.5620 q^{92} -2849.94 q^{93} -851.253 q^{94} +488.118 q^{95} -186.786 q^{96} -646.327 q^{97} +938.236 q^{98} -613.534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9}+O(q^{10}) \) 60 * q - 15 * q^2 - 37 * q^3 + 213 * q^4 - 51 * q^5 + 22 * q^6 - 54 * q^7 - 204 * q^8 + 387 * q^9	\( 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} - 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} - 625 q^{18} - 610 q^{19} - 345 q^{20} + 611 q^{21} - 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} - 1071 q^{26} - 1609 q^{27} - 46 q^{28} - 773 q^{29} - 375 q^{30} - 97 q^{31} - 1967 q^{32} - 500 q^{33} - 217 q^{34} + 247 q^{35} + 175 q^{36} - 228 q^{37} + 1253 q^{38} - 1493 q^{39} + 2220 q^{40} - 951 q^{41} - 2643 q^{42} - 1378 q^{44} - 1086 q^{45} + 565 q^{46} - 2 q^{47} - 2303 q^{48} + 1264 q^{49} - 3273 q^{50} - 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} - 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} - 2999 q^{61} - 5569 q^{62} - 2377 q^{63} + 2082 q^{64} - 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} - 1817 q^{69} - 2738 q^{70} - 8003 q^{71} - 1412 q^{72} + 1011 q^{73} - 1413 q^{74} - 7457 q^{75} - 5516 q^{76} - 4052 q^{77} + 1091 q^{78} - 4422 q^{79} - 1610 q^{80} + 2108 q^{81} - 4676 q^{82} - 297 q^{83} - 54 q^{84} - 4333 q^{85} + 1377 q^{87} - 3652 q^{88} - 2480 q^{89} - 1414 q^{90} - 4551 q^{91} - 3286 q^{92} - 4 q^{93} - 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} - 753 q^{98} + 5072 q^{99}+O(q^{100}) \) 60 * q - 15 * q^2 - 37 * q^3 + 213 * q^4 - 51 * q^5 + 22 * q^6 - 54 * q^7 - 204 * q^8 + 387 * q^9 + 27 * q^10 + 43 * q^11 - 432 * q^12 + 13 * q^13 + 96 * q^14 + 65 * q^15 + 717 * q^16 - 138 * q^17 - 625 * q^18 - 610 * q^19 - 345 * q^20 + 611 * q^21 - 118 * q^22 - 243 * q^23 + 258 * q^24 + 899 * q^25 - 1071 * q^26 - 1609 * q^27 - 46 * q^28 - 773 * q^29 - 375 * q^30 - 97 * q^31 - 1967 * q^32 - 500 * q^33 - 217 * q^34 + 247 * q^35 + 175 * q^36 - 228 * q^37 + 1253 * q^38 - 1493 * q^39 + 2220 * q^40 - 951 * q^41 - 2643 * q^42 - 1378 * q^44 - 1086 * q^45 + 565 * q^46 - 2 * q^47 - 2303 * q^48 + 1264 * q^49 - 3273 * q^50 - 3076 * q^51 - 2825 * q^52 - 39 * q^53 + 5201 * q^54 - 1306 * q^55 + 3683 * q^56 + 1342 * q^57 + 2588 * q^58 - 1065 * q^59 + 2803 * q^60 - 2999 * q^61 - 5569 * q^62 - 2377 * q^63 + 2082 * q^64 - 5578 * q^65 + 3338 * q^66 + 961 * q^67 - 3754 * q^68 - 1817 * q^69 - 2738 * q^70 - 8003 * q^71 - 1412 * q^72 + 1011 * q^73 - 1413 * q^74 - 7457 * q^75 - 5516 * q^76 - 4052 * q^77 + 1091 * q^78 - 4422 * q^79 - 1610 * q^80 + 2108 * q^81 - 4676 * q^82 - 297 * q^83 - 54 * q^84 - 4333 * q^85 + 1377 * q^87 - 3652 * q^88 - 2480 * q^89 - 1414 * q^90 - 4551 * q^91 - 3286 * q^92 - 4 * q^93 - 4609 * q^94 - 835 * q^95 + 5864 * q^96 + 3785 * q^97 - 753 * q^98 + 5072 * q^99

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