LMFDB - Embedded newform 1849.4.a.j.1.24

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			1.24
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)

$f(q)$	$=$	$q-0.0180033 q^{2} -4.14959 q^{3} -7.99968 q^{4} +13.3506 q^{5} +0.0747066 q^{6} +5.69523 q^{7} +0.288048 q^{8} -9.78088 q^{9} +O(q^{10})$	\(q-0.0180033 q^{2} -4.14959 q^{3} -7.99968 q^{4} +13.3506 q^{5} +0.0747066 q^{6} +5.69523 q^{7} +0.288048 q^{8} -9.78088 q^{9} -0.240356 q^{10} -47.5373 q^{11} +33.1954 q^{12} +76.9850 q^{13} -0.102533 q^{14} -55.3996 q^{15} +63.9922 q^{16} +7.02247 q^{17} +0.176089 q^{18} -132.370 q^{19} -106.800 q^{20} -23.6329 q^{21} +0.855831 q^{22} +93.7845 q^{23} -1.19528 q^{24} +53.2386 q^{25} -1.38599 q^{26} +152.626 q^{27} -45.5600 q^{28} +100.339 q^{29} +0.997377 q^{30} -169.100 q^{31} -3.45646 q^{32} +197.260 q^{33} -0.126428 q^{34} +76.0348 q^{35} +78.2439 q^{36} -268.545 q^{37} +2.38311 q^{38} -319.456 q^{39} +3.84561 q^{40} +157.369 q^{41} +0.425471 q^{42} +380.283 q^{44} -130.581 q^{45} -1.68843 q^{46} +394.707 q^{47} -265.542 q^{48} -310.564 q^{49} -0.958472 q^{50} -29.1404 q^{51} -615.855 q^{52} +447.422 q^{53} -2.74777 q^{54} -634.652 q^{55} +1.64050 q^{56} +549.282 q^{57} -1.80643 q^{58} -653.194 q^{59} +443.178 q^{60} -168.494 q^{61} +3.04437 q^{62} -55.7044 q^{63} -511.876 q^{64} +1027.80 q^{65} -3.55135 q^{66} +646.419 q^{67} -56.1775 q^{68} -389.167 q^{69} -1.36888 q^{70} -61.1604 q^{71} -2.81736 q^{72} +336.572 q^{73} +4.83471 q^{74} -220.918 q^{75} +1058.92 q^{76} -270.736 q^{77} +5.75129 q^{78} +664.672 q^{79} +854.335 q^{80} -369.250 q^{81} -2.83318 q^{82} -242.758 q^{83} +189.055 q^{84} +93.7542 q^{85} -416.364 q^{87} -13.6930 q^{88} +1437.09 q^{89} +2.35089 q^{90} +438.448 q^{91} -750.245 q^{92} +701.698 q^{93} -7.10606 q^{94} -1767.22 q^{95} +14.3429 q^{96} -1444.16 q^{97} +5.59120 q^{98} +464.957 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.0180033	−0.00636515	−0.00318257	−	0.999995i	$-0.501013\pi$
$2$	−0.0180033	−0.00636515	−0.00318257	+	0.999995i	$0.501013\pi$
$3$	−4.14959	−0.798589	−0.399295	−	0.916823i	$-0.630745\pi$
$3$	−4.14959	−0.798589	−0.399295	+	0.916823i	$0.630745\pi$
$4$	−7.99968	−0.999959
$5$	13.3506	1.19411	0.597057	−	0.802199i	$-0.296337\pi$
$5$	13.3506	1.19411	0.597057	+	0.802199i	$0.296337\pi$
$6$	0.0747066	0.00508314
$7$	5.69523	0.307514	0.153757	−	0.988109i	$-0.450863\pi$
$7$	5.69523	0.307514	0.153757	+	0.988109i	$0.450863\pi$
$8$	0.288048	0.0127300
$9$	−9.78088	−0.362255
$10$	−0.240356	−0.00760071
$11$	−47.5373	−1.30300	−0.651502	−	0.758647i	$-0.725861\pi$
$11$	−47.5373	−1.30300	−0.651502	+	0.758647i	$0.725861\pi$
$12$	33.1954	0.798557
$13$	76.9850	1.64245	0.821223	−	0.570607i	$-0.193292\pi$
$13$	76.9850	1.64245	0.821223	+	0.570607i	$0.193292\pi$
$14$	−0.102533	−0.00195737
$15$	−55.3996	−0.953607
$16$	63.9922	0.999878
$17$	7.02247	0.100188	0.0500941	−	0.998745i	$-0.484048\pi$
$17$	7.02247	0.100188	0.0500941	+	0.998745i	$0.484048\pi$
$18$	0.176089	0.00230581
$19$	−132.370	−1.59830	−0.799152	−	0.601128i	$-0.794718\pi$
$19$	−132.370	−1.59830	−0.799152	+	0.601128i	$0.794718\pi$
$20$	−106.800	−1.19407
$21$	−23.6329	−0.245577
$22$	0.855831	0.00829381
$23$	93.7845	0.850235	0.425118	−	0.905138i	$-0.360233\pi$
$23$	93.7845	0.850235	0.425118	+	0.905138i	$0.360233\pi$
$24$	−1.19528	−0.0101661
$25$	53.2386	0.425908
$26$	−1.38599	−0.0104544
$27$	152.626	1.08788
$28$	−45.5600	−0.307501
$29$	100.339	0.642497	0.321249	−	0.946995i	$-0.395898\pi$
$29$	100.339	0.642497	0.321249	+	0.946995i	$0.395898\pi$
$30$	0.997377	0.00606985
$31$	−169.100	−0.979720	−0.489860	−	0.871801i	$-0.662952\pi$
$31$	−169.100	−0.979720	−0.489860	+	0.871801i	$0.662952\pi$
$32$	−3.45646	−0.0190944
$33$	197.260	1.04056
$34$	−0.126428	−0.000637712 0
$35$	76.0348	0.367206
$36$	78.2439	0.362240
$37$	−268.545	−1.19320	−0.596601	−	0.802538i	$-0.703483\pi$
$37$	−268.545	−1.19320	−0.596601	+	0.802538i	$0.703483\pi$
$38$	2.38311	0.0101734
$39$	−319.456	−1.31164
$40$	3.84561	0.0152011
$41$	157.369	0.599438	0.299719	−	0.954027i	$-0.403107\pi$
$41$	157.369	0.599438	0.299719	+	0.954027i	$0.403107\pi$
$42$	0.425471	0.00156313
$43$	0	0
$43$	0	0
$44$	380.283	1.30295
$45$	−130.581	−0.432574
$46$	−1.68843	−0.00541187
$47$	394.707	1.22498	0.612489	−	0.790479i	$-0.290168\pi$
$47$	394.707	1.22498	0.612489	+	0.790479i	$0.290168\pi$
$48$	−265.542	−0.798492
$49$	−310.564	−0.905435
$50$	−0.958472	−0.00271097
$51$	−29.1404	−0.0800092
$52$	−615.855	−1.64238
$53$	447.422	1.15959	0.579794	−	0.814763i	$-0.303133\pi$
$53$	447.422	1.15959	0.579794	+	0.814763i	$0.303133\pi$
$54$	−2.74777	−0.00692453
$55$	−634.652	−1.55594
$56$	1.64050	0.00391466
$57$	549.282	1.27639
$58$	−1.80643	−0.00408959
$59$	−653.194	−1.44133	−0.720666	−	0.693283i	$-0.756164\pi$
$59$	−653.194	−1.44133	−0.720666	+	0.693283i	$0.756164\pi$
$60$	443.178	0.953568
$61$	−168.494	−0.353664	−0.176832	−	0.984241i	$-0.556585\pi$
$61$	−168.494	−0.353664	−0.176832	+	0.984241i	$0.556585\pi$
$62$	3.04437	0.00623606
$63$	−55.7044	−0.111398
$64$	−511.876	−0.999757
$65$	1027.80	1.96127
$66$	−3.55135	−0.00662335
$67$	646.419	1.17870	0.589348	−	0.807879i	$-0.299385\pi$
$67$	646.419	1.17870	0.589348	+	0.807879i	$0.299385\pi$
$68$	−56.1775	−0.100184
$69$	−389.167	−0.678989
$70$	−1.36888	−0.00233732
$71$	−61.1604	−0.102231	−0.0511155	−	0.998693i	$-0.516278\pi$
$71$	−61.1604	−0.102231	−0.0511155	+	0.998693i	$0.516278\pi$
$72$	−2.81736	−0.00461152
$73$	336.572	0.539627	0.269814	−	0.962913i	$-0.413038\pi$
$73$	336.572	0.539627	0.269814	+	0.962913i	$0.413038\pi$
$74$	4.83471	0.00759491
$75$	−220.918	−0.340126
$76$	1058.92	1.59824
$77$	−270.736	−0.400691
$78$	5.75129	0.00834878
$79$	664.672	0.946601	0.473301	−	0.880901i	$-0.343062\pi$
$79$	664.672	0.946601	0.473301	+	0.880901i	$0.343062\pi$
$80$	854.335	1.19397
$81$	−369.250	−0.506516
$82$	−2.83318	−0.00381551
$83$	−242.758	−0.321037	−0.160519	−	0.987033i	$-0.551317\pi$
$83$	−242.758	−0.321037	−0.160519	+	0.987033i	$0.551317\pi$
$84$	189.055	0.245567
$85$	93.7542	0.119636
$86$	0	0
$87$	−416.364	−0.513091
$88$	−13.6930	−0.0165873
$89$	1437.09	1.71158	0.855792	−	0.517320i	$-0.173070\pi$
$89$	1437.09	1.71158	0.855792	+	0.517320i	$0.173070\pi$
$90$	2.35089	0.00275339
$91$	438.448	0.505075
$92$	−750.245	−0.850201
$93$	701.698	0.782394
$94$	−7.10606	−0.00779717
$95$	−1767.22	−1.90856
$96$	14.3429	0.0152486
$97$	−1444.16	−1.51168	−0.755839	−	0.654758i	$-0.772771\pi$
$97$	−1444.16	−1.51168	−0.755839	+	0.654758i	$0.772771\pi$
$98$	5.59120	0.00576323
$99$	464.957	0.472020
$100$	−425.891	−0.425891
$101$	120.466	0.118681	0.0593404	−	0.998238i	$-0.481100\pi$
$101$	120.466	0.118681	0.0593404	+	0.998238i	$0.481100\pi$
$102$	0.524624	0.000509270 0
$103$	−224.367	−0.214636	−0.107318	−	0.994225i	$-0.534226\pi$
$103$	−224.367	−0.214636	−0.107318	+	0.994225i	$0.534226\pi$
$104$	22.1754	0.0209084
$105$	−315.513	−0.293247
$106$	−8.05510	−0.00738094
$107$	290.938	0.262860	0.131430	−	0.991325i	$-0.458043\pi$
$107$	290.938	0.262860	0.131430	+	0.991325i	$0.458043\pi$
$108$	−1220.96	−1.08784
$109$	873.020	0.767158	0.383579	−	0.923508i	$-0.374691\pi$
$109$	873.020	0.767158	0.383579	+	0.923508i	$0.374691\pi$
$110$	11.4259	0.00990375
$111$	1114.35	0.952879
$112$	364.451	0.307476
$113$	1726.96	1.43769	0.718843	−	0.695173i	$-0.244672\pi$
$113$	1726.96	1.43769	0.718843	+	0.695173i	$0.244672\pi$
$114$	−9.88892	−0.00812440
$115$	1252.08	1.01528
$116$	−802.676	−0.642471
$117$	−752.982	−0.594984
$118$	11.7597	0.00917428
$119$	39.9946	0.0308092
$120$	−15.9577	−0.0121394
$121$	928.796	0.697819
$122$	3.03346	0.00225112
$123$	−653.019	−0.478705
$124$	1352.75	0.979681
$125$	−958.058	−0.685531
$126$	1.00287	0.000709067 0
$127$	−496.351	−0.346804	−0.173402	−	0.984851i	$-0.555476\pi$
$127$	−496.351	−0.346804	−0.173402	+	0.984851i	$0.555476\pi$
$128$	36.8671	0.0254580
$129$	0	0
$130$	−18.5038	−0.0124838
$131$	−187.413	−0.124995	−0.0624975	−	0.998045i	$-0.519907\pi$
$131$	−187.413	−0.124995	−0.0624975	+	0.998045i	$0.519907\pi$
$132$	−1578.02	−1.04052
$133$	−753.879	−0.491500
$134$	−11.6377	−0.00750257
$135$	2037.64	1.29906
$136$	2.02281	0.00127540
$137$	−2099.47	−1.30927	−0.654633	−	0.755947i	$-0.727177\pi$
$137$	−2099.47	−1.30927	−0.654633	+	0.755947i	$0.727177\pi$
$138$	7.00631	0.00432186
$139$	−2945.77	−1.79753	−0.898767	−	0.438426i	$-0.855536\pi$
$139$	−2945.77	−1.79753	−0.898767	+	0.438426i	$0.855536\pi$
$140$	−608.253	−0.367191
$141$	−1637.87	−0.978255
$142$	1.10109	0.000650715 0
$143$	−3659.66	−2.14011
$144$	−625.901	−0.362211
$145$	1339.58	0.767215
$146$	−6.05943	−0.00343481
$147$	1288.72	0.723071
$148$	2148.27	1.19315
$149$	3106.95	1.70827	0.854133	−	0.520055i	$-0.174088\pi$
$149$	3106.95	1.70827	0.854133	+	0.520055i	$0.174088\pi$
$150$	3.97727	0.00216495
$151$	−1846.66	−0.995226	−0.497613	−	0.867399i	$-0.665790\pi$
$151$	−1846.66	−0.995226	−0.497613	+	0.867399i	$0.665790\pi$
$152$	−38.1289	−0.0203465
$153$	−68.6859	−0.0362937
$154$	4.87416	0.00255046
$155$	−2257.59	−1.16990
$156$	2555.55	1.31159
$157$	−1270.26	−0.645719	−0.322860	−	0.946447i	$-0.604644\pi$
$157$	−1270.26	−0.645719	−0.322860	+	0.946447i	$0.604644\pi$
$158$	−11.9663	−0.00602525
$159$	−1856.62	−0.926034
$160$	−46.1458	−0.0228009
$161$	534.124	0.261459
$162$	6.64774	0.00322405
$163$	1810.04	0.869773	0.434886	−	0.900485i	$-0.356789\pi$
$163$	1810.04	0.869773	0.434886	+	0.900485i	$0.356789\pi$
$164$	−1258.90	−0.599414
$165$	2633.55	1.24255
$166$	4.37045	0.00204345
$167$	−4200.89	−1.94656	−0.973278	−	0.229630i	$-0.926248\pi$
$167$	−4200.89	−1.94656	−0.973278	+	0.229630i	$0.926248\pi$
$168$	−6.80740	−0.00312620
$169$	3729.69	1.69763
$170$	−1.68789	−0.000761501 0
$171$	1294.70	0.578994
$172$	0	0
$173$	−1786.31	−0.785031	−0.392516	−	0.919745i	$-0.628395\pi$
$173$	−1786.31	−0.785031	−0.392516	+	0.919745i	$0.628395\pi$
$174$	7.49595	0.00326590
$175$	303.206	0.130973
$176$	−3042.02	−1.30285
$177$	2710.49	1.15103
$178$	−25.8724	−0.0108945
$179$	2188.39	0.913787	0.456894	−	0.889521i	$-0.348962\pi$
$179$	2188.39	0.913787	0.456894	+	0.889521i	$0.348962\pi$
$180$	1044.60	0.432556
$181$	1287.39	0.528680	0.264340	−	0.964430i	$-0.414846\pi$
$181$	1287.39	0.528680	0.264340	+	0.964430i	$0.414846\pi$
$182$	−7.89352	−0.00321487
$183$	699.183	0.282432
$184$	27.0144	0.0108235
$185$	−3585.23	−1.42482
$186$	−12.6329	−0.00498005
$187$	−333.829	−0.130546
$188$	−3157.53	−1.22493
$189$	869.239	0.334539
$190$	31.8159	0.0121483
$191$	2171.05	0.822468	0.411234	−	0.911530i	$-0.365098\pi$
$191$	2171.05	0.822468	0.411234	+	0.911530i	$0.365098\pi$
$192$	2124.07	0.798395
$193$	1864.44	0.695365	0.347683	−	0.937612i	$-0.386969\pi$
$193$	1864.44	0.695365	0.347683	+	0.937612i	$0.386969\pi$
$194$	25.9998	0.00962205
$195$	−4264.94	−1.56625
$196$	2484.41	0.905399
$197$	−1865.97	−0.674848	−0.337424	−	0.941353i	$-0.609556\pi$
$197$	−1865.97	−0.674848	−0.337424	+	0.941353i	$0.609556\pi$
$198$	−8.37078	−0.00300447
$199$	−2313.58	−0.824147	−0.412073	−	0.911151i	$-0.635195\pi$
$199$	−2313.58	−0.824147	−0.412073	+	0.911151i	$0.635195\pi$
$200$	15.3352	0.00542183
$201$	−2682.37	−0.941294
$202$	−2.16878	−0.000755421 0
$203$	571.452	0.197577
$204$	233.114	0.0800059
$205$	2100.98	0.715798
$206$	4.03936	0.00136619
$207$	−917.295	−0.308002
$208$	4926.44	1.64225
$209$	6292.52	2.08260
$210$	5.68030	0.00186656
$211$	−741.102	−0.241799	−0.120899	−	0.992665i	$-0.538578\pi$
$211$	−741.102	−0.241799	−0.120899	+	0.992665i	$0.538578\pi$
$212$	−3579.23	−1.15954
$213$	253.791	0.0816406
$214$	−5.23786	−0.00167314
$215$	0	0
$216$	43.9635	0.0138488
$217$	−963.066	−0.301277
$218$	−15.7173	−0.00488307
$219$	−1396.64	−0.430941
$220$	5077.01	1.55587
$221$	540.625	0.164554
$222$	−20.0621	−0.00606521
$223$	−2820.00	−0.846822	−0.423411	−	0.905938i	$-0.639167\pi$
$223$	−2820.00	−0.846822	−0.423411	+	0.905938i	$0.639167\pi$
$224$	−19.6853	−0.00587179
$225$	−520.720	−0.154287
$226$	−31.0910	−0.00915107
$227$	2015.04	0.589176	0.294588	−	0.955624i	$-0.404818\pi$
$227$	2015.04	0.589176	0.294588	+	0.955624i	$0.404818\pi$
$228$	−4394.08	−1.27634
$229$	−5723.24	−1.65154	−0.825769	−	0.564009i	$-0.809258\pi$
$229$	−5723.24	−1.65154	−0.825769	+	0.564009i	$0.809258\pi$
$230$	−22.5416	−0.00646239
$231$	1123.44	0.319988
$232$	28.9023	0.00817901
$233$	−6380.23	−1.79392	−0.896959	−	0.442114i	$-0.854229\pi$
$233$	−6380.23	−1.79392	−0.896959	+	0.442114i	$0.854229\pi$
$234$	13.5562	0.00378716
$235$	5269.58	1.46276
$236$	5225.34	1.44127
$237$	−2758.12	−0.755946
$238$	−0.720036	−0.000196105 0
$239$	1828.53	0.494887	0.247443	−	0.968902i	$-0.420410\pi$
$239$	1828.53	0.494887	0.247443	+	0.968902i	$0.420410\pi$
$240$	−3545.14	−0.953491
$241$	−5943.97	−1.58873	−0.794367	−	0.607439i	$-0.792197\pi$
$241$	−5943.97	−1.58873	−0.794367	+	0.607439i	$0.792197\pi$
$242$	−16.7214	−0.00444172
$243$	−2588.65	−0.683384
$244$	1347.90	0.353649
$245$	−4146.22	−1.08119
$246$	11.7565	0.00304703
$247$	−10190.5	−2.62513
$248$	−48.7090	−0.0124719
$249$	1007.34	0.256377
$250$	17.2483	0.00436350
$251$	−5413.50	−1.36134	−0.680671	−	0.732589i	$-0.738312\pi$
$251$	−5413.50	−1.36134	−0.680671	+	0.732589i	$0.738312\pi$
$252$	445.617	0.111394
$253$	−4458.26	−1.10786
$254$	8.93599	0.00220746
$255$	−389.041	−0.0955401
$256$	4094.34	0.999595
$257$	−886.594	−0.215191	−0.107596	−	0.994195i	$-0.534315\pi$
$257$	−886.594	−0.215191	−0.107596	+	0.994195i	$0.534315\pi$
$258$	0	0
$259$	−1529.42	−0.366926
$260$	−8222.04	−1.96119
$261$	−981.401	−0.232748
$262$	3.37406	0.000795611 0
$263$	−5738.13	−1.34535	−0.672677	−	0.739936i	$-0.734856\pi$
$263$	−5738.13	−1.34535	−0.672677	+	0.739936i	$0.734856\pi$
$264$	56.8204	0.0132464
$265$	5973.35	1.38468
$266$	13.5723	0.00312847
$267$	−5963.33	−1.36685
$268$	−5171.14	−1.17865
$269$	−6018.64	−1.36418	−0.682088	−	0.731270i	$-0.738928\pi$
$269$	−6018.64	−1.36418	−0.682088	+	0.731270i	$0.738928\pi$
$270$	−36.6844	−0.00826868
$271$	−5924.82	−1.32807	−0.664035	−	0.747701i	$-0.731158\pi$
$271$	−5924.82	−1.32807	−0.664035	+	0.747701i	$0.731158\pi$
$272$	449.383	0.100176
$273$	−1819.38	−0.403347
$274$	37.7974	0.00833367
$275$	−2530.82	−0.554960
$276$	3113.21	0.678961
$277$	−4026.86	−0.873467	−0.436733	−	0.899591i	$-0.643865\pi$
$277$	−4026.86	−0.873467	−0.436733	+	0.899591i	$0.643865\pi$
$278$	53.0338	0.0114416
$279$	1653.95	0.354909
$280$	21.9016	0.00467455
$281$	−2988.01	−0.634340	−0.317170	−	0.948369i	$-0.602733\pi$
$281$	−2988.01	−0.634340	−0.317170	+	0.948369i	$0.602733\pi$
$282$	29.4872	0.00622673
$283$	−715.323	−0.150253	−0.0751264	−	0.997174i	$-0.523936\pi$
$283$	−715.323	−0.150253	−0.0751264	+	0.997174i	$0.523936\pi$
$284$	489.263	0.102227
$285$	7333.25	1.52415
$286$	65.8862	0.0136221
$287$	896.255	0.184335
$288$	33.8072	0.00691704
$289$	−4863.68	−0.989962
$290$	−24.1169	−0.00488343
$291$	5992.70	1.20721
$292$	−2692.47	−0.539605
$293$	2561.88	0.510807	0.255403	−	0.966835i	$-0.417792\pi$
$293$	2561.88	0.510807	0.255403	+	0.966835i	$0.417792\pi$
$294$	−23.2012	−0.00460245
$295$	−8720.53	−1.72111
$296$	−77.3537	−0.0151895
$297$	−7255.41	−1.41751
$298$	−55.9356	−0.0108734
$299$	7220.00	1.39647
$300$	1767.27	0.340112
$301$	0	0
$302$	33.2461	0.00633476
$303$	−499.883	−0.0947773
$304$	−8470.66	−1.59811
$305$	−2249.50	−0.422315
$306$	1.23658	0.000231014 0
$307$	−8416.19	−1.56462	−0.782309	−	0.622891i	$-0.785958\pi$
$307$	−8416.19	−1.56462	−0.782309	+	0.622891i	$0.785958\pi$
$308$	2165.80	0.400675
$309$	931.032	0.171406
$310$	40.6442	0.00744657
$311$	−100.971	−0.0184102	−0.00920509	−	0.999958i	$-0.502930\pi$
$311$	−100.971	−0.0184102	−0.00920509	+	0.999958i	$0.502930\pi$
$312$	−92.0187	−0.0166972
$313$	4616.96	0.833758	0.416879	−	0.908962i	$-0.363124\pi$
$313$	4616.96	0.833758	0.416879	+	0.908962i	$0.363124\pi$
$314$	22.8690	0.00411010
$315$	−743.687	−0.133022
$316$	−5317.16	−0.946563
$317$	6990.10	1.23850	0.619248	−	0.785195i	$-0.287438\pi$
$317$	6990.10	1.23850	0.619248	+	0.785195i	$0.287438\pi$
$318$	33.4254	0.00589434
$319$	−4769.83	−0.837176
$320$	−6833.85	−1.19382
$321$	−1207.27	−0.209917
$322$	−9.61602	−0.00166422
$323$	−929.565	−0.160131
$324$	2953.88	0.506496
$325$	4098.57	0.699532
$326$	−32.5867	−0.00553623
$327$	−3622.68	−0.612644
$328$	45.3299	0.00763087
$329$	2247.95	0.376698
$330$	−47.4126	−0.00790903
$331$	−7239.49	−1.20217	−0.601085	−	0.799185i	$-0.705265\pi$
$331$	−7239.49	−1.20217	−0.601085	+	0.799185i	$0.705265\pi$
$332$	1941.98	0.321024
$333$	2626.61	0.432244
$334$	75.6301	0.0123901
$335$	8630.08	1.40750
$336$	−1512.32	−0.245547
$337$	6873.75	1.11109	0.555545	−	0.831487i	$-0.312510\pi$
$337$	6873.75	1.11109	0.555545	+	0.831487i	$0.312510\pi$
$338$	−67.1470	−0.0108057
$339$	−7166.17	−1.14812
$340$	−750.003	−0.119631
$341$	8038.58	1.27658
$342$	−23.3089	−0.00368538
$343$	−3722.20	−0.585947
$344$	0	0
$345$	−5195.62	−0.810790
$346$	32.1595	0.00499684
$347$	274.973	0.0425399	0.0212699	−	0.999774i	$-0.493229\pi$
$347$	274.973	0.0425399	0.0212699	+	0.999774i	$0.493229\pi$
$348$	3330.78	0.513071
$349$	3858.09	0.591744	0.295872	−	0.955228i	$-0.404390\pi$
$349$	3858.09	0.591744	0.295872	+	0.955228i	$0.404390\pi$
$350$	−5.45872	−0.000833660 0
$351$	11749.9	1.78679
$352$	164.311	0.0248801
$353$	−8034.04	−1.21136	−0.605678	−	0.795710i	$-0.707098\pi$
$353$	−8034.04	−1.21136	−0.605678	+	0.795710i	$0.707098\pi$
$354$	−48.7978	−0.00732649
$355$	−816.528	−0.122076
$356$	−11496.2	−1.71151
$357$	−165.961	−0.0246039
$358$	−39.3983	−0.00581639
$359$	−12330.0	−1.81268	−0.906341	−	0.422546i	$-0.861136\pi$
$359$	−12330.0	−1.81268	−0.906341	+	0.422546i	$0.861136\pi$
$360$	−37.6135	−0.00550668
$361$	10662.9	1.55458
$362$	−23.1774	−0.00336512
$363$	−3854.13	−0.557270
$364$	−3507.44	−0.505054
$365$	4493.44	0.644377
$366$	−12.5876	−0.00179772
$367$	−2953.15	−0.420036	−0.210018	−	0.977698i	$-0.567352\pi$
$367$	−2953.15	−0.420036	−0.210018	+	0.977698i	$0.567352\pi$
$368$	6001.48	0.850132
$369$	−1539.21	−0.217150
$370$	64.5462	0.00906918
$371$	2548.17	0.356589
$372$	−5613.35	−0.782363
$373$	5174.01	0.718230	0.359115	−	0.933293i	$-0.383079\pi$
$373$	5174.01	0.718230	0.359115	+	0.933293i	$0.383079\pi$
$374$	6.01004	0.000830941 0
$375$	3975.55	0.547458
$376$	113.695	0.0155940
$377$	7724.57	1.05527
$378$	−15.6492	−0.00212939
$379$	1962.08	0.265924	0.132962	−	0.991121i	$-0.457551\pi$
$379$	1962.08	0.265924	0.132962	+	0.991121i	$0.457551\pi$
$380$	14137.2	1.90848
$381$	2059.66	0.276954
$382$	−39.0861	−0.00523513
$383$	−11142.8	−1.48661	−0.743307	−	0.668951i	$-0.766744\pi$
$383$	−11142.8	−1.48661	−0.743307	+	0.668951i	$0.766744\pi$
$384$	−152.984	−0.0203305
$385$	−3614.49	−0.478471
$386$	−33.5662	−0.00442610
$387$	0	0
$388$	11552.9	1.51162
$389$	−1511.59	−0.197020	−0.0985099	−	0.995136i	$-0.531408\pi$
$389$	−1511.59	−0.197020	−0.0985099	+	0.995136i	$0.531408\pi$
$390$	76.7831	0.00996940
$391$	658.598	0.0851835
$392$	−89.4574	−0.0115262
$393$	777.687	0.0998196
$394$	33.5937	0.00429550
$395$	8873.78	1.13035
$396$	−3719.51	−0.472000
$397$	−4147.44	−0.524317	−0.262159	−	0.965025i	$-0.584434\pi$
$397$	−4147.44	−0.524317	−0.262159	+	0.965025i	$0.584434\pi$
$398$	41.6522	0.00524581
$399$	3128.29	0.392507
$400$	3406.85	0.425857
$401$	10249.0	1.27633	0.638166	−	0.769899i	$-0.279693\pi$
$401$	10249.0	1.27633	0.638166	+	0.769899i	$0.279693\pi$
$402$	48.2917	0.00599147
$403$	−13018.2	−1.60914
$404$	−963.685	−0.118676
$405$	−4929.71	−0.604838
$406$	−10.2880	−0.00125760
$407$	12765.9	1.55475
$408$	−8.39382	−0.00101852
$409$	11975.0	1.44774	0.723870	−	0.689937i	$-0.242362\pi$
$409$	11975.0	1.44774	0.723870	+	0.689937i	$0.242362\pi$
$410$	−37.8246	−0.00455616
$411$	8711.93	1.04557
$412$	1794.86	0.214628
$413$	−3720.09	−0.443229
$414$	16.5144	0.00196048
$415$	−3240.96	−0.383355
$416$	−266.095	−0.0313615
$417$	12223.8	1.43549
$418$	−113.286	−0.0132560
$419$	−8895.43	−1.03716	−0.518580	−	0.855029i	$-0.673539\pi$
$419$	−8895.43	−1.03716	−0.518580	+	0.855029i	$0.673539\pi$
$420$	2524.00	0.293235
$421$	−13596.3	−1.57398	−0.786988	−	0.616969i	$-0.788360\pi$
$421$	−13596.3	−1.57398	−0.786988	+	0.616969i	$0.788360\pi$
$422$	13.3423	0.00153909
$423$	−3860.59	−0.443755
$424$	128.879	0.0147616
$425$	373.866	0.0426710
$426$	−4.56908	−0.000519654 0
$427$	−959.614	−0.108756
$428$	−2327.41	−0.262850
$429$	15186.1	1.70907
$430$	0	0
$431$	2369.68	0.264834	0.132417	−	0.991194i	$-0.457726\pi$
$431$	2369.68	0.264834	0.132417	+	0.991194i	$0.457726\pi$
$432$	9766.86	1.08775
$433$	10808.9	1.19964	0.599820	−	0.800135i	$-0.295239\pi$
$433$	10808.9	1.19964	0.599820	+	0.800135i	$0.295239\pi$
$434$	17.3384	0.00191767
$435$	−5558.71	−0.612690
$436$	−6983.88	−0.767127
$437$	−12414.3	−1.35894
$438$	25.1441	0.00274300
$439$	−6203.61	−0.674447	−0.337223	−	0.941425i	$-0.609488\pi$
$439$	−6203.61	−0.674447	−0.337223	+	0.941425i	$0.609488\pi$
$440$	−182.810	−0.0198071
$441$	3037.59	0.327998
$442$	−9.73306	−0.00104741
$443$	492.806	0.0528531	0.0264265	−	0.999651i	$-0.491587\pi$
$443$	492.806	0.0528531	0.0264265	+	0.999651i	$0.491587\pi$
$444$	−8914.45	−0.952840
$445$	19186.0	2.04383
$446$	50.7695	0.00539014
$447$	−12892.6	−1.36420
$448$	−2915.25	−0.307439
$449$	10160.3	1.06791	0.533957	−	0.845511i	$-0.320704\pi$
$449$	10160.3	1.06791	0.533957	+	0.845511i	$0.320704\pi$
$450$	9.37471	0.000982062 0
$451$	−7480.92	−0.781070
$452$	−13815.1	−1.43763
$453$	7662.89	0.794777
$454$	−36.2775	−0.00375019
$455$	5853.54	0.603117
$456$	158.219	0.0162485
$457$	5086.43	0.520641	0.260320	−	0.965522i	$-0.416172\pi$
$457$	5086.43	0.520641	0.260320	+	0.965522i	$0.416172\pi$
$458$	103.037	0.0105123
$459$	1071.81	0.108993
$460$	−10016.2	−1.01524
$461$	−1930.12	−0.194999	−0.0974997	−	0.995236i	$-0.531085\pi$
$461$	−1930.12	−0.194999	−0.0974997	+	0.995236i	$0.531085\pi$
$462$	−20.2258	−0.00203677
$463$	−3106.92	−0.311859	−0.155929	−	0.987768i	$-0.549837\pi$
$463$	−3106.92	−0.311859	−0.155929	+	0.987768i	$0.549837\pi$
$464$	6420.89	0.642419
$465$	9368.09	0.934268
$466$	114.866	0.0114185
$467$	16618.6	1.64671	0.823357	−	0.567523i	$-0.192098\pi$
$467$	16618.6	1.64671	0.823357	+	0.567523i	$0.192098\pi$
$468$	6023.61	0.594960
$469$	3681.50	0.362465
$470$	−94.8701	−0.00931071
$471$	5271.07	0.515665
$472$	−188.151	−0.0183482
$473$	0	0
$474$	49.6554	0.00481170
$475$	−7047.20	−0.680732
$476$	−319.944	−0.0308080
$477$	−4376.18	−0.420066
$478$	−32.9197	−0.00315003
$479$	4267.07	0.407030	0.203515	−	0.979072i	$-0.434763\pi$
$479$	4267.07	0.407030	0.203515	+	0.979072i	$0.434763\pi$
$480$	191.486	0.0182086
$481$	−20673.9	−1.95977
$482$	107.011	0.0101125
$483$	−2216.40	−0.208798
$484$	−7430.07	−0.697790
$485$	−19280.5	−1.80512
$486$	46.6045	0.00434984
$487$	−7588.40	−0.706085	−0.353042	−	0.935607i	$-0.614853\pi$
$487$	−7588.40	−0.706085	−0.353042	+	0.935607i	$0.614853\pi$
$488$	−48.5344	−0.00450215
$489$	−7510.91	−0.694591
$490$	74.6459	0.00688195
$491$	15704.9	1.44349	0.721743	−	0.692161i	$-0.243341\pi$
$491$	15704.9	1.44349	0.721743	+	0.692161i	$0.243341\pi$
$492$	5223.94	0.478686
$493$	704.625	0.0643706
$494$	183.463	0.0167093
$495$	6207.46	0.563645
$496$	−10821.1	−0.979601
$497$	−348.323	−0.0314374
$498$	−18.1356	−0.00163188
$499$	8093.69	0.726098	0.363049	−	0.931770i	$-0.381736\pi$
$499$	8093.69	0.726098	0.363049	+	0.931770i	$0.381736\pi$
$500$	7664.16	0.685503
$501$	17432.0	1.55450
$502$	97.4611	0.00866514
$503$	−9867.83	−0.874722	−0.437361	−	0.899286i	$-0.644087\pi$
$503$	−9867.83	−0.874722	−0.437361	+	0.899286i	$0.644087\pi$
$504$	−16.0455	−0.00141810
$505$	1608.29	0.141719
$506$	80.2636	0.00705169
$507$	−15476.7	−1.35571
$508$	3970.65	0.346790
$509$	−5383.79	−0.468825	−0.234413	−	0.972137i	$-0.575317\pi$
$509$	−5383.79	−0.468825	−0.234413	+	0.972137i	$0.575317\pi$
$510$	7.00405	0.000608127 0
$511$	1916.86	0.165943
$512$	−368.649	−0.0318206
$513$	−20203.1	−1.73877
$514$	15.9617	0.00136972
$515$	−2995.44	−0.256300
$516$	0	0
$517$	−18763.3	−1.59615
$518$	27.5348	0.00233554
$519$	7412.45	0.626918
$520$	296.054	0.0249670
$521$	−298.734	−0.0251205	−0.0125602	−	0.999921i	$-0.503998\pi$
$521$	−298.734	−0.0251205	−0.0125602	+	0.999921i	$0.503998\pi$
$522$	17.6685	0.00148147
$523$	4948.95	0.413771	0.206886	−	0.978365i	$-0.433667\pi$
$523$	4948.95	0.413771	0.206886	+	0.978365i	$0.433667\pi$
$524$	1499.24	0.124990
$525$	−1258.18	−0.104593
$526$	103.306	0.00856338
$527$	−1187.50	−0.0981563
$528$	12623.1	1.04044
$529$	−3371.47	−0.277100
$530$	−107.540	−0.00881369
$531$	6388.81	0.522130
$532$	6030.78	0.491481
$533$	12115.1	0.984545
$534$	107.360	0.00870021
$535$	3884.20	0.313885
$536$	186.199	0.0150048
$537$	−9080.92	−0.729741
$538$	108.356	0.00868317
$539$	14763.4	1.17979
$540$	−16300.5	−1.29900
$541$	23749.8	1.88740	0.943701	−	0.330799i	$-0.107318\pi$
$541$	23749.8	1.88740	0.943701	+	0.330799i	$0.107318\pi$
$542$	106.667	0.00845336
$543$	−5342.15	−0.422198
$544$	−24.2728	−0.00191303
$545$	11655.3	0.916074
$546$	32.7549	0.00256736
$547$	−487.096	−0.0380744	−0.0190372	−	0.999819i	$-0.506060\pi$
$547$	−487.096	−0.0380744	−0.0190372	+	0.999819i	$0.506060\pi$
$548$	16795.0	1.30921
$549$	1648.02	0.128116
$550$	45.5632	0.00353240
$551$	−13281.8	−1.02691
$552$	−112.099	−0.00864355
$553$	3785.46	0.291093
$554$	72.4969	0.00555974
$555$	14877.3	1.13785
$556$	23565.2	1.79746
$557$	−18770.9	−1.42791	−0.713956	−	0.700190i	$-0.753098\pi$
$557$	−18770.9	−1.42791	−0.713956	+	0.700190i	$0.753098\pi$
$558$	−29.7767	−0.00225904
$559$	0	0
$560$	4865.63	0.367162
$561$	1385.26	0.104252
$562$	53.7942	0.00403767
$563$	−19245.5	−1.44068	−0.720339	−	0.693622i	$-0.756014\pi$
$563$	−19245.5	−1.44068	−0.720339	+	0.693622i	$0.756014\pi$
$564$	13102.5	0.978215
$565$	23055.9	1.71676
$566$	12.8782	0.000956381 0
$567$	−2102.97	−0.155761
$568$	−17.6171	−0.00130140
$569$	13782.8	1.01548	0.507738	−	0.861512i	$-0.330482\pi$
$569$	13782.8	1.01548	0.507738	+	0.861512i	$0.330482\pi$
$570$	−132.023	−0.00970146
$571$	1113.80	0.0816307	0.0408153	−	0.999167i	$-0.487004\pi$
$571$	1113.80	0.0816307	0.0408153	+	0.999167i	$0.487004\pi$
$572$	29276.1	2.14003
$573$	−9008.96	−0.656815
$574$	−16.1356	−0.00117332
$575$	4992.95	0.362122
$576$	5006.60	0.362167
$577$	355.943	0.0256813	0.0128406	−	0.999918i	$-0.495913\pi$
$577$	355.943	0.0256813	0.0128406	+	0.999918i	$0.495913\pi$
$578$	87.5626	0.00630125
$579$	−7736.68	−0.555311
$580$	−10716.2	−0.767184
$581$	−1382.56	−0.0987234
$582$	−107.889	−0.00768406
$583$	−21269.2	−1.51095
$584$	96.9489	0.00686947
$585$	−10052.8	−0.710479
$586$	−46.1223	−0.00325136
$587$	2791.25	0.196265	0.0981324	−	0.995173i	$-0.468713\pi$
$587$	2791.25	0.196265	0.0981324	+	0.995173i	$0.468713\pi$
$588$	−10309.3	−0.723042
$589$	22383.8	1.56589
$590$	156.999	0.0109551
$591$	7743.02	0.538926
$592$	−17184.8	−1.19306
$593$	20640.7	1.42937	0.714683	−	0.699449i	$-0.246571\pi$
$593$	20640.7	1.42937	0.714683	+	0.699449i	$0.246571\pi$
$594$	130.622	0.00902269
$595$	533.952	0.0367897
$596$	−24854.6	−1.70820
$597$	9600.40	0.658155
$598$	−129.984	−0.00888871
$599$	3539.11	0.241409	0.120705	−	0.992688i	$-0.461485\pi$
$599$	3539.11	0.241409	0.120705	+	0.992688i	$0.461485\pi$
$600$	−63.6350	−0.00432981
$601$	−19466.5	−1.32123	−0.660613	−	0.750727i	$-0.729704\pi$
$601$	−19466.5	−1.32123	−0.660613	+	0.750727i	$0.729704\pi$
$602$	0	0
$603$	−6322.55	−0.426988
$604$	14772.7	0.995185
$605$	12400.0	0.833275
$606$	8.99957	0.000603271 0
$607$	156.745	0.0104812	0.00524060	−	0.999986i	$-0.498332\pi$
$607$	156.745	0.0104812	0.00524060	+	0.999986i	$0.498332\pi$
$608$	457.532	0.0305187
$609$	−2371.29	−0.157783
$610$	40.4985	0.00268810
$611$	30386.6	2.01196
$612$	549.465	0.0362922
$613$	−8239.90	−0.542914	−0.271457	−	0.962451i	$-0.587506\pi$
$613$	−8239.90	−0.542914	−0.271457	+	0.962451i	$0.587506\pi$
$614$	151.520	0.00995902
$615$	−8718.19	−0.571628
$616$	−77.9849	−0.00510081
$617$	−675.335	−0.0440647	−0.0220324	−	0.999757i	$-0.507014\pi$
$617$	−675.335	−0.0440647	−0.0220324	+	0.999757i	$0.507014\pi$
$618$	−16.7617	−0.00109103
$619$	−3826.89	−0.248491	−0.124245	−	0.992252i	$-0.539651\pi$
$619$	−3826.89	−0.248491	−0.124245	+	0.992252i	$0.539651\pi$
$620$	18060.0	1.16985
$621$	14313.9	0.924956
$622$	1.81783	0.000117184 0
$623$	8184.54	0.526335
$624$	−20442.7	−1.31148
$625$	−19445.5	−1.24451
$626$	−83.1208	−0.00530699
$627$	−26111.4	−1.66314
$628$	10161.7	0.645693
$629$	−1885.85	−0.119545
$630$	13.3889	0.000846706 0
$631$	−13119.3	−0.827687	−0.413843	−	0.910348i	$-0.635814\pi$
$631$	−13119.3	−0.827687	−0.413843	+	0.910348i	$0.635814\pi$
$632$	191.457	0.0120503
$633$	3075.27	0.193098
$634$	−125.845	−0.00788321
$635$	−6626.59	−0.414123
$636$	14852.4	0.925997
$637$	−23908.8	−1.48713
$638$	85.8729	0.00532875
$639$	598.203	0.0370337
$640$	492.198	0.0303998
$641$	19467.8	1.19958	0.599791	−	0.800156i	$-0.295250\pi$
$641$	19467.8	1.19958	0.599791	+	0.800156i	$0.295250\pi$
$642$	21.7350	0.00133616
$643$	−9713.64	−0.595752	−0.297876	−	0.954605i	$-0.596278\pi$
$643$	−9713.64	−0.595752	−0.297876	+	0.954605i	$0.596278\pi$
$644$	−4272.82	−0.261448
$645$	0	0
$646$	16.7353	0.00101926
$647$	−536.886	−0.0326232	−0.0163116	−	0.999867i	$-0.505192\pi$
$647$	−536.886	−0.0326232	−0.0163116	+	0.999867i	$0.505192\pi$
$648$	−106.362	−0.00644797
$649$	31051.1	1.87806
$650$	−73.7880	−0.00445262
$651$	3996.33	0.240597
$652$	−14479.7	−0.869737
$653$	−1067.84	−0.0639938	−0.0319969	−	0.999488i	$-0.510187\pi$
$653$	−1067.84	−0.0639938	−0.0319969	+	0.999488i	$0.510187\pi$
$654$	65.2203	0.00389957
$655$	−2502.07	−0.149258
$656$	10070.4	0.599365
$657$	−3291.97	−0.195483
$658$	−40.4706	−0.00239773
$659$	9950.77	0.588205	0.294102	−	0.955774i	$-0.404979\pi$
$659$	9950.77	0.588205	0.294102	+	0.955774i	$0.404979\pi$
$660$	−21067.5	−1.24250
$661$	−10362.8	−0.609785	−0.304892	−	0.952387i	$-0.598620\pi$
$661$	−10362.8	−0.609785	−0.304892	+	0.952387i	$0.598620\pi$
$662$	130.335	0.00765199
$663$	−2243.37	−0.131411
$664$	−69.9258	−0.00408682
$665$	−10064.7	−0.586908
$666$	−47.2877	−0.00275129
$667$	9410.20	0.546274
$668$	33605.8	1.94648
$669$	11701.9	0.676263
$670$	−155.370	−0.00895892
$671$	8009.77	0.460825
$672$	81.6860	0.00468915
$673$	−32597.3	−1.86707	−0.933533	−	0.358493i	$-0.883291\pi$
$673$	−32597.3	−1.86707	−0.933533	+	0.358493i	$0.883291\pi$
$674$	−123.751	−0.00707225
$675$	8125.57	0.463338
$676$	−29836.3	−1.69756
$677$	5463.36	0.310154	0.155077	−	0.987902i	$-0.450437\pi$
$677$	5463.36	0.310154	0.155077	+	0.987902i	$0.450437\pi$
$678$	129.015	0.00730795
$679$	−8224.85	−0.464861
$680$	27.0057	0.00152297
$681$	−8361.60	−0.470510
$682$	−144.721	−0.00812561
$683$	−2908.18	−0.162926	−0.0814629	−	0.996676i	$-0.525959\pi$
$683$	−2908.18	−0.162926	−0.0814629	+	0.996676i	$0.525959\pi$
$684$	−10357.2	−0.578970
$685$	−28029.1	−1.56341
$686$	67.0121	0.00372964
$687$	23749.1	1.31890
$688$	0	0
$689$	34444.8	1.90456
$690$	93.5385	0.00516080
$691$	−26270.3	−1.44626	−0.723132	−	0.690709i	$-0.757298\pi$
$691$	−26270.3	−1.44626	−0.723132	+	0.690709i	$0.757298\pi$
$692$	14289.9	0.785000
$693$	2648.04	0.145152
$694$	−4.95044	−0.000270773 0
$695$	−39327.9	−2.14646
$696$	−119.933	−0.00653167
$697$	1105.12	0.0600566
$698$	−69.4585	−0.00376654
$699$	26475.4	1.43260
$700$	−2425.55	−0.130967
$701$	−2836.62	−0.152836	−0.0764178	−	0.997076i	$-0.524348\pi$
$701$	−2836.62	−0.152836	−0.0764178	+	0.997076i	$0.524348\pi$
$702$	−211.537	−0.0113732
$703$	35547.3	1.90710
$704$	24333.2	1.30269
$705$	−21866.6	−1.16815
$706$	144.640	0.00771046
$707$	686.079	0.0364960
$708$	−21683.0	−1.15099
$709$	−5284.45	−0.279918	−0.139959	−	0.990157i	$-0.544697\pi$
$709$	−5284.45	−0.279918	−0.139959	+	0.990157i	$0.544697\pi$
$710$	14.7002	0.000777028 0
$711$	−6501.08	−0.342911
$712$	413.950	0.0217885
$713$	−15859.0	−0.832993
$714$	2.98786	0.000156607 0
$715$	−48858.7	−2.55554
$716$	−17506.4	−0.913750
$717$	−7587.66	−0.395211
$718$	221.982	0.0115380
$719$	8442.51	0.437904	0.218952	−	0.975736i	$-0.429736\pi$
$719$	8442.51	0.437904	0.218952	+	0.975736i	$0.429736\pi$
$720$	−8356.15	−0.432521
$721$	−1277.82	−0.0660036
$722$	−191.967	−0.00989512
$723$	24665.0	1.26875
$724$	−10298.7	−0.528658
$725$	5341.88	0.273645
$726$	69.3872	0.00354711
$727$	19342.0	0.986731	0.493366	−	0.869822i	$-0.335766\pi$
$727$	19342.0	0.986731	0.493366	+	0.869822i	$0.335766\pi$
$728$	126.294	0.00642962
$729$	20711.6	1.05226
$730$	−80.8970	−0.00410155
$731$	0	0
$732$	−5593.24	−0.282421
$733$	−907.180	−0.0457127	−0.0228564	−	0.999739i	$-0.507276\pi$
$733$	−907.180	−0.0457127	−0.0228564	+	0.999739i	$0.507276\pi$
$734$	53.1666	0.00267359
$735$	17205.1	0.863429
$736$	−324.162	−0.0162347
$737$	−30729.0	−1.53584
$738$	27.7110	0.00138219
$739$	26048.6	1.29663	0.648317	−	0.761370i	$-0.275473\pi$
$739$	26048.6	1.29663	0.648317	+	0.761370i	$0.275473\pi$
$740$	28680.7	1.42476
$741$	42286.5	2.09640
$742$	−45.8756	−0.00226974
$743$	30883.3	1.52489	0.762447	−	0.647050i	$-0.223998\pi$
$743$	30883.3	1.52489	0.762447	+	0.647050i	$0.223998\pi$
$744$	202.122	0.00995990
$745$	41479.7	2.03986
$746$	−93.1494	−0.00457164
$747$	2374.38	0.116297
$748$	2670.53	0.130540
$749$	1656.96	0.0808331
$750$	−71.5732	−0.00348465
$751$	4168.40	0.202540	0.101270	−	0.994859i	$-0.467709\pi$
$751$	4168.40	0.202540	0.101270	+	0.994859i	$0.467709\pi$
$752$	25258.2	1.22483
$753$	22463.8	1.08715
$754$	−139.068	−0.00671693
$755$	−24654.0	−1.18841
$756$	−6953.63	−0.334525
$757$	21816.2	1.04746	0.523728	−	0.851886i	$-0.324541\pi$
$757$	21816.2	1.04746	0.523728	+	0.851886i	$0.324541\pi$
$758$	−35.3239	−0.00169264
$759$	18500.0	0.884725
$760$	−509.044	−0.0242960
$761$	11079.6	0.527774	0.263887	−	0.964554i	$-0.414995\pi$
$761$	11079.6	0.527774	0.263887	+	0.964554i	$0.414995\pi$
$762$	−37.0807	−0.00176285
$763$	4972.05	0.235911
$764$	−17367.7	−0.822435
$765$	−916.999	−0.0433388
$766$	200.609	0.00946251
$767$	−50286.1	−2.36731
$768$	−16989.8	−0.798266
$769$	23438.6	1.09911	0.549555	−	0.835457i	$-0.314797\pi$
$769$	23438.6	1.09911	0.549555	+	0.835457i	$0.314797\pi$
$770$	65.0729	0.00304554
$771$	3679.00	0.171850
$772$	−14914.9	−0.695337
$773$	−37721.7	−1.75518	−0.877591	−	0.479411i	$-0.840850\pi$
$773$	−37721.7	−1.75518	−0.877591	+	0.479411i	$0.840850\pi$
$774$	0	0
$775$	−9002.66	−0.417271
$776$	−415.988	−0.0192437
$777$	6346.49	0.293023
$778$	27.2137	0.00125406
$779$	−20831.0	−0.958085
$780$	34118.1	1.56618
$781$	2907.40	0.133207
$782$	−11.8570	−0.000542205 0
$783$	15314.3	0.698961
$784$	−19873.7	−0.905325
$785$	−16958.8	−0.771062
$786$	−14.0010	−0.000635367 0
$787$	−11707.4	−0.530270	−0.265135	−	0.964211i	$-0.585417\pi$
$787$	−11707.4	−0.530270	−0.265135	+	0.964211i	$0.585417\pi$
$788$	14927.2	0.674820
$789$	23810.9	1.07439
$790$	−159.758	−0.00719484
$791$	9835.42	0.442108
$792$	133.930	0.00600882
$793$	−12971.5	−0.580874
$794$	74.6678	0.00333735
$795$	−24787.0	−1.10579
$796$	18507.9	0.824113
$797$	1717.85	0.0763482	0.0381741	−	0.999271i	$-0.487846\pi$
$797$	1717.85	0.0763482	0.0381741	+	0.999271i	$0.487846\pi$
$798$	−56.3197	−0.00249836
$799$	2771.82	0.122728
$800$	−184.017	−0.00813247
$801$	−14056.0	−0.620030
$802$	−184.516	−0.00812404
$803$	−15999.7	−0.703136
$804$	21458.1	0.941256
$805$	7130.88	0.312212
$806$	234.371	0.0102424
$807$	24974.9	1.08942
$808$	34.6998	0.00151081
$809$	−20559.3	−0.893482	−0.446741	−	0.894663i	$-0.647415\pi$
$809$	−20559.3	−0.893482	−0.446741	+	0.894663i	$0.647415\pi$
$810$	88.7514	0.00384988
$811$	12479.9	0.540354	0.270177	−	0.962811i	$-0.412918\pi$
$811$	12479.9	0.540354	0.270177	+	0.962811i	$0.412918\pi$
$812$	−4571.43	−0.197569
$813$	24585.6	1.06058
$814$	−229.829	−0.00989619
$815$	24165.1	1.03861
$816$	−1864.76	−0.0799995
$817$	0	0
$818$	−215.590	−0.00921507
$819$	−4288.40	−0.182966
$820$	−16807.1	−0.715769
$821$	−29309.3	−1.24592	−0.622962	−	0.782252i	$-0.714071\pi$
$821$	−29309.3	−1.24592	−0.622962	+	0.782252i	$0.714071\pi$
$822$	−156.844	−0.00665518
$823$	−21999.6	−0.931782	−0.465891	−	0.884842i	$-0.654266\pi$
$823$	−21999.6	−0.931782	−0.465891	+	0.884842i	$0.654266\pi$
$824$	−64.6284	−0.00273233
$825$	10501.9	0.443185
$826$	66.9740	0.00282122
$827$	30534.9	1.28392	0.641960	−	0.766738i	$-0.278121\pi$
$827$	30534.9	1.28392	0.641960	+	0.766738i	$0.278121\pi$
$828$	7338.06	0.307990
$829$	26025.4	1.09035	0.545174	−	0.838323i	$-0.316464\pi$
$829$	26025.4	1.09035	0.545174	+	0.838323i	$0.316464\pi$
$830$	58.3481	0.00244011
$831$	16709.8	0.697541
$832$	−39406.8	−1.64205
$833$	−2180.93	−0.0907139
$834$	−220.069	−0.00913712
$835$	−56084.4	−2.32441
$836$	−50338.1	−2.08251
$837$	−25809.1	−1.06582
$838$	160.148	0.00660168
$839$	18134.8	0.746224	0.373112	−	0.927786i	$-0.378291\pi$
$839$	18134.8	0.746224	0.373112	+	0.927786i	$0.378291\pi$
$840$	−90.8829	−0.00373304
$841$	−14321.2	−0.587198
$842$	244.779	0.0100186
$843$	12399.0	0.506578
$844$	5928.58	0.241789
$845$	49793.7	2.02716
$846$	69.5035	0.00282456
$847$	5289.71	0.214589
$848$	28631.5	1.15945
$849$	2968.30	0.119990
$850$	−6.73084	−0.000271607 0
$851$	−25185.3	−1.01450
$852$	−2030.24	−0.0816373
$853$	−14999.0	−0.602059	−0.301029	−	0.953615i	$-0.597330\pi$
$853$	−14999.0	−0.602059	−0.301029	+	0.953615i	$0.597330\pi$
$854$	17.2763	0.000692250 0
$855$	17285.0	0.691385
$856$	83.8041	0.00334622
$857$	−19754.8	−0.787409	−0.393705	−	0.919237i	$-0.628807\pi$
$857$	−19754.8	−0.787409	−0.393705	+	0.919237i	$0.628807\pi$
$858$	−273.401	−0.0108785
$859$	−31691.4	−1.25879	−0.629393	−	0.777087i	$-0.716696\pi$
$859$	−31691.4	−1.25879	−0.629393	+	0.777087i	$0.716696\pi$
$860$	0	0
$861$	−3719.09	−0.147208
$862$	−42.6622	−0.00168571
$863$	−11419.1	−0.450420	−0.225210	−	0.974310i	$-0.572307\pi$
$863$	−11419.1	−0.450420	−0.225210	+	0.974310i	$0.572307\pi$
$864$	−527.544	−0.0207725
$865$	−23848.3	−0.937417
$866$	−194.597	−0.00763588
$867$	20182.3	0.790573
$868$	7704.22	0.301265
$869$	−31596.7	−1.23342
$870$	100.075	0.00389986
$871$	49764.6	1.93594
$872$	251.472	0.00976594
$873$	14125.2	0.547613
$874$	223.498	0.00864982
$875$	−5456.36	−0.210810
$876$	11172.6	0.430923
$877$	−21841.3	−0.840966	−0.420483	−	0.907301i	$-0.638139\pi$
$877$	−21841.3	−0.840966	−0.420483	+	0.907301i	$0.638139\pi$
$878$	111.686	0.00429295
$879$	−10630.7	−0.407925
$880$	−40612.8	−1.55575
$881$	−18076.0	−0.691257	−0.345628	−	0.938372i	$-0.612334\pi$
$881$	−18076.0	−0.691257	−0.345628	+	0.938372i	$0.612334\pi$
$882$	−54.6869	−0.00208776
$883$	−26624.2	−1.01469	−0.507347	−	0.861742i	$-0.669374\pi$
$883$	−26624.2	−1.01469	−0.507347	+	0.861742i	$0.669374\pi$
$884$	−4324.82	−0.164547
$885$	36186.6	1.37446
$886$	−8.87215	−0.000336417 0
$887$	−31435.2	−1.18996	−0.594978	−	0.803742i	$-0.702839\pi$
$887$	−31435.2	−1.18996	−0.594978	+	0.803742i	$0.702839\pi$
$888$	320.986	0.0121302
$889$	−2826.84	−0.106647
$890$	−345.412	−0.0130092
$891$	17553.2	0.659993
$892$	22559.1	0.846787
$893$	−52247.5	−1.95789
$894$	232.110	0.00868335
$895$	29216.3	1.09117
$896$	209.967	0.00782868
$897$	−29960.1	−1.11520
$898$	−182.919	−0.00679743
$899$	−16967.3	−0.629467
$900$	4165.59	0.154281
$901$	3142.01	0.116177
$902$	134.682	0.00497163
$903$	0	0
$904$	497.446	0.0183018
$905$	17187.5	0.631304
$906$	−137.958	−0.00505887
$907$	17535.3	0.641950	0.320975	−	0.947088i	$-0.395989\pi$
$907$	17535.3	0.641950	0.320975	+	0.947088i	$0.395989\pi$
$908$	−16119.7	−0.589152
$909$	−1178.26	−0.0429927
$910$	−105.383	−0.00383893
$911$	10181.4	0.370278	0.185139	−	0.982712i	$-0.440726\pi$
$911$	10181.4	0.370278	0.185139	+	0.982712i	$0.440726\pi$
$912$	35149.8	1.27623
$913$	11540.0	0.418313
$914$	−91.5727	−0.00331396
$915$	9334.51	0.337256
$916$	45784.0	1.65147
$917$	−1067.36	−0.0384376
$918$	−19.2961	−0.000693756 0
$919$	24370.5	0.874764	0.437382	−	0.899276i	$-0.355906\pi$
$919$	24370.5	0.874764	0.437382	+	0.899276i	$0.355906\pi$
$920$	360.659	0.0129245
$921$	34923.8	1.24949
$922$	34.7487	0.00124120
$923$	−4708.44	−0.167909
$924$	−8987.19	−0.319975
$925$	−14296.9	−0.508195
$926$	55.9349	0.00198503
$927$	2194.51	0.0777531
$928$	−346.816	−0.0122681
$929$	10550.6	0.372608	0.186304	−	0.982492i	$-0.440349\pi$
$929$	10550.6	0.372608	0.186304	+	0.982492i	$0.440349\pi$
$930$	−168.657	−0.00594675
$931$	41109.4	1.44716
$932$	51039.8	1.79385
$933$	418.991	0.0147022
$934$	−299.190	−0.0104816
$935$	−4456.82	−0.155886
$936$	−216.895	−0.00757417
$937$	−26213.4	−0.913931	−0.456966	−	0.889484i	$-0.651064\pi$
$937$	−26213.4	−0.913931	−0.456966	+	0.889484i	$0.651064\pi$
$938$	−66.2794	−0.00230714
$939$	−19158.5	−0.665830
$940$	−42154.9	−1.46270
$941$	−12985.5	−0.449856	−0.224928	−	0.974375i	$-0.572215\pi$
$941$	−12985.5	−0.449856	−0.224928	+	0.974375i	$0.572215\pi$
$942$	−94.8969	−0.00328228
$943$	14758.8	0.509664
$944$	−41799.3	−1.44116
$945$	11604.9	0.399477
$946$	0	0
$947$	19335.2	0.663474	0.331737	−	0.943372i	$-0.392365\pi$
$947$	19335.2	0.663474	0.331737	+	0.943372i	$0.392365\pi$
$948$	22064.1	0.755915
$949$	25911.0	0.886309
$950$	126.873	0.00433296
$951$	−29006.1	−0.989050
$952$	11.5203	0.000392202 0
$953$	−29785.2	−1.01242	−0.506210	−	0.862410i	$-0.668954\pi$
$953$	−29785.2	−1.01242	−0.506210	+	0.862410i	$0.668954\pi$
$954$	78.7860	0.00267378
$955$	28984.8	0.982121
$956$	−14627.7	−0.494867
$957$	19792.8	0.668560
$958$	−76.8216	−0.00259081
$959$	−11956.9	−0.402617
$960$	28357.7	0.953375
$961$	−1196.05	−0.0401482
$962$	372.200	0.0124742
$963$	−2845.63	−0.0952225
$964$	47549.8	1.58867
$965$	24891.4	0.830346
$966$	39.9026	0.00132903
$967$	31419.2	1.04485	0.522426	−	0.852684i	$-0.325027\pi$
$967$	31419.2	1.04485	0.522426	+	0.852684i	$0.325027\pi$
$968$	267.538	0.00888325
$969$	3857.31	0.127879
$970$	347.113	0.0114898
$971$	−4021.58	−0.132913	−0.0664565	−	0.997789i	$-0.521169\pi$
$971$	−4021.58	−0.132913	−0.0664565	+	0.997789i	$0.521169\pi$
$972$	20708.4	0.683356
$973$	−16776.9	−0.552766
$974$	136.617	0.00449433
$975$	−17007.4	−0.558639
$976$	−10782.3	−0.353621
$977$	3043.64	0.0996671	0.0498335	−	0.998758i	$-0.484131\pi$
$977$	3043.64	0.0996671	0.0498335	+	0.998758i	$0.484131\pi$
$978$	135.222	0.00442117
$979$	−68315.3	−2.23020
$980$	33168.4	1.08115
$981$	−8538.91	−0.277907
$982$	−282.741	−0.00918800
$983$	20451.1	0.663569	0.331785	−	0.943355i	$-0.392349\pi$
$983$	20451.1	0.663569	0.331785	+	0.943355i	$0.392349\pi$
$984$	−188.101	−0.00609393
$985$	−24911.9	−0.805845
$986$	−12.6856	−0.000409728 0
$987$	−9328.08	−0.300827
$988$	81520.8	2.62502
$989$	0	0
$990$	−111.755	−0.00358768
$991$	−945.176	−0.0302972	−0.0151486	−	0.999885i	$-0.504822\pi$
$991$	−945.176	−0.0302972	−0.0151486	+	0.999885i	$0.504822\pi$
$992$	584.488	0.0187072
$993$	30040.9	0.960040
$994$	6.27097	0.000200104 0
$995$	−30887.7	−0.984125
$996$	−8058.43	−0.256367
$997$	2417.17	0.0767828	0.0383914	−	0.999263i	$-0.487777\pi$
$997$	2417.17	0.0767828	0.0383914	+	0.999263i	$0.487777\pi$
$998$	−145.713	−0.00462172
$999$	−40986.8	−1.29806

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.24	yes	50
43.42	odd	2		1849.4.a.i.1.27	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.27	✓	50	43.42	odd	2
1849.4.a.j.1.24	yes	50	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0180033 q^{2} -4.14959 q^{3} -7.99968 q^{4} +13.3506 q^{5} +0.0747066 q^{6} +5.69523 q^{7} +0.288048 q^{8} -9.78088 q^{9} +O(q^{10})\)	\(q-0.0180033 q^{2} -4.14959 q^{3} -7.99968 q^{4} +13.3506 q^{5} +0.0747066 q^{6} +5.69523 q^{7} +0.288048 q^{8} -9.78088 q^{9} -0.240356 q^{10} -47.5373 q^{11} +33.1954 q^{12} +76.9850 q^{13} -0.102533 q^{14} -55.3996 q^{15} +63.9922 q^{16} +7.02247 q^{17} +0.176089 q^{18} -132.370 q^{19} -106.800 q^{20} -23.6329 q^{21} +0.855831 q^{22} +93.7845 q^{23} -1.19528 q^{24} +53.2386 q^{25} -1.38599 q^{26} +152.626 q^{27} -45.5600 q^{28} +100.339 q^{29} +0.997377 q^{30} -169.100 q^{31} -3.45646 q^{32} +197.260 q^{33} -0.126428 q^{34} +76.0348 q^{35} +78.2439 q^{36} -268.545 q^{37} +2.38311 q^{38} -319.456 q^{39} +3.84561 q^{40} +157.369 q^{41} +0.425471 q^{42} +380.283 q^{44} -130.581 q^{45} -1.68843 q^{46} +394.707 q^{47} -265.542 q^{48} -310.564 q^{49} -0.958472 q^{50} -29.1404 q^{51} -615.855 q^{52} +447.422 q^{53} -2.74777 q^{54} -634.652 q^{55} +1.64050 q^{56} +549.282 q^{57} -1.80643 q^{58} -653.194 q^{59} +443.178 q^{60} -168.494 q^{61} +3.04437 q^{62} -55.7044 q^{63} -511.876 q^{64} +1027.80 q^{65} -3.55135 q^{66} +646.419 q^{67} -56.1775 q^{68} -389.167 q^{69} -1.36888 q^{70} -61.1604 q^{71} -2.81736 q^{72} +336.572 q^{73} +4.83471 q^{74} -220.918 q^{75} +1058.92 q^{76} -270.736 q^{77} +5.75129 q^{78} +664.672 q^{79} +854.335 q^{80} -369.250 q^{81} -2.83318 q^{82} -242.758 q^{83} +189.055 q^{84} +93.7542 q^{85} -416.364 q^{87} -13.6930 q^{88} +1437.09 q^{89} +2.35089 q^{90} +438.448 q^{91} -750.245 q^{92} +701.698 q^{93} -7.10606 q^{94} -1767.22 q^{95} +14.3429 q^{96} -1444.16 q^{97} +5.59120 q^{98} +464.957 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) \) 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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