LMFDB - Embedded newform 1849.4.a.j.1.46

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			1.46
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)

$f(q)$	$=$	$q+4.85844 q^{2} +5.73018 q^{3} +15.6044 q^{4} -14.4569 q^{5} +27.8397 q^{6} +13.1859 q^{7} +36.9457 q^{8} +5.83494 q^{9} +O(q^{10})$	\(q+4.85844 q^{2} +5.73018 q^{3} +15.6044 q^{4} -14.4569 q^{5} +27.8397 q^{6} +13.1859 q^{7} +36.9457 q^{8} +5.83494 q^{9} -70.2380 q^{10} -56.6297 q^{11} +89.4162 q^{12} -68.5859 q^{13} +64.0631 q^{14} -82.8407 q^{15} +54.6629 q^{16} +23.3842 q^{17} +28.3487 q^{18} +119.755 q^{19} -225.592 q^{20} +75.5578 q^{21} -275.132 q^{22} -49.8515 q^{23} +211.705 q^{24} +84.0024 q^{25} -333.220 q^{26} -121.280 q^{27} +205.759 q^{28} -290.660 q^{29} -402.477 q^{30} +58.9050 q^{31} -29.9890 q^{32} -324.498 q^{33} +113.611 q^{34} -190.628 q^{35} +91.0510 q^{36} -147.330 q^{37} +581.820 q^{38} -393.009 q^{39} -534.121 q^{40} +216.626 q^{41} +367.093 q^{42} -883.674 q^{44} -84.3552 q^{45} -242.201 q^{46} -233.374 q^{47} +313.228 q^{48} -169.131 q^{49} +408.121 q^{50} +133.996 q^{51} -1070.24 q^{52} +712.940 q^{53} -589.229 q^{54} +818.690 q^{55} +487.164 q^{56} +686.215 q^{57} -1412.15 q^{58} +70.0677 q^{59} -1292.68 q^{60} -328.266 q^{61} +286.186 q^{62} +76.9392 q^{63} -583.003 q^{64} +991.541 q^{65} -1576.55 q^{66} +307.039 q^{67} +364.898 q^{68} -285.658 q^{69} -926.155 q^{70} -1135.63 q^{71} +215.576 q^{72} -203.220 q^{73} -715.792 q^{74} +481.349 q^{75} +1868.70 q^{76} -746.715 q^{77} -1909.41 q^{78} +255.296 q^{79} -790.257 q^{80} -852.497 q^{81} +1052.47 q^{82} +665.658 q^{83} +1179.04 q^{84} -338.064 q^{85} -1665.53 q^{87} -2092.22 q^{88} +834.098 q^{89} -409.835 q^{90} -904.369 q^{91} -777.905 q^{92} +337.536 q^{93} -1133.83 q^{94} -1731.28 q^{95} -171.842 q^{96} +817.923 q^{97} -821.713 q^{98} -330.431 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.85844	1.71772	0.858859	−	0.512212i	$-0.171174\pi$
$2$	4.85844	1.71772	0.858859	+	0.512212i	$0.171174\pi$
$3$	5.73018	1.10277	0.551387	−	0.834250i	$-0.314099\pi$
$3$	5.73018	1.10277	0.551387	+	0.834250i	$0.314099\pi$
$4$	15.6044	1.95055
$5$	−14.4569	−1.29307	−0.646533	−	0.762886i	$-0.723782\pi$
$5$	−14.4569	−1.29307	−0.646533	+	0.762886i	$0.723782\pi$
$6$	27.8397	1.89425
$7$	13.1859	0.711974	0.355987	−	0.934491i	$-0.384145\pi$
$7$	13.1859	0.711974	0.355987	+	0.934491i	$0.384145\pi$
$8$	36.9457	1.63278
$9$	5.83494	0.216109
$10$	−70.2380	−2.22112
$11$	−56.6297	−1.55223	−0.776113	−	0.630594i	$-0.782811\pi$
$11$	−56.6297	−1.55223	−0.776113	+	0.630594i	$0.782811\pi$
$12$	89.4162	2.15102
$13$	−68.5859	−1.46325	−0.731627	−	0.681705i	$-0.761239\pi$
$13$	−68.5859	−1.46325	−0.731627	+	0.681705i	$0.761239\pi$
$14$	64.0631	1.22297
$15$	−82.8407	−1.42596
$16$	54.6629	0.854108
$17$	23.3842	0.333618	0.166809	−	0.985989i	$-0.446654\pi$
$17$	23.3842	0.333618	0.166809	+	0.985989i	$0.446654\pi$
$18$	28.3487	0.371214
$19$	119.755	1.44598	0.722989	−	0.690859i	$-0.242768\pi$
$19$	119.755	1.44598	0.722989	+	0.690859i	$0.242768\pi$
$20$	−225.592	−2.52220
$21$	75.5578	0.785146
$22$	−275.132	−2.66629
$23$	−49.8515	−0.451946	−0.225973	−	0.974134i	$-0.572556\pi$
$23$	−49.8515	−0.451946	−0.225973	+	0.974134i	$0.572556\pi$
$24$	211.705	1.80059
$25$	84.0024	0.672019
$26$	−333.220	−2.51346
$27$	−121.280	−0.864454
$28$	205.759	1.38874
$29$	−290.660	−1.86118	−0.930590	−	0.366064i	$-0.880705\pi$
$29$	−290.660	−1.86118	−0.930590	+	0.366064i	$0.880705\pi$
$30$	−402.477	−2.44939
$31$	58.9050	0.341279	0.170639	−	0.985334i	$-0.445417\pi$
$31$	58.9050	0.341279	0.170639	+	0.985334i	$0.445417\pi$
$32$	−29.9890	−0.165667
$33$	−324.498	−1.71175
$34$	113.611	0.573062
$35$	−190.628	−0.920629
$36$	91.0510	0.421532
$37$	−147.330	−0.654617	−0.327309	−	0.944918i	$-0.606142\pi$
$37$	−147.330	−0.654617	−0.327309	+	0.944918i	$0.606142\pi$
$38$	581.820	2.48378
$39$	−393.009	−1.61364
$40$	−534.121	−2.11130
$41$	216.626	0.825154	0.412577	−	0.910923i	$-0.364629\pi$
$41$	216.626	0.825154	0.412577	+	0.910923i	$0.364629\pi$
$42$	367.093	1.34866
$43$	0	0
$43$	0	0
$44$	−883.674	−3.02770
$45$	−84.3552	−0.279443
$46$	−242.201	−0.776316
$47$	−233.374	−0.724277	−0.362139	−	0.932124i	$-0.617953\pi$
$47$	−233.374	−0.724277	−0.362139	+	0.932124i	$0.617953\pi$
$48$	313.228	0.941888
$49$	−169.131	−0.493093
$50$	408.121	1.15434
$51$	133.996	0.367905
$52$	−1070.24	−2.85416
$53$	712.940	1.84773	0.923866	−	0.382716i	$-0.125011\pi$
$53$	712.940	1.84773	0.923866	+	0.382716i	$0.125011\pi$
$54$	−589.229	−1.48489
$55$	818.690	2.00713
$56$	487.164	1.16250
$57$	686.215	1.59459
$58$	−1412.15	−3.19698
$59$	70.0677	0.154611	0.0773054	−	0.997007i	$-0.475368\pi$
$59$	70.0677	0.154611	0.0773054	+	0.997007i	$0.475368\pi$
$60$	−1292.68	−2.78141
$61$	−328.266	−0.689019	−0.344509	−	0.938783i	$-0.611955\pi$
$61$	−328.266	−0.689019	−0.344509	+	0.938783i	$0.611955\pi$
$62$	286.186	0.586221
$63$	76.9392	0.153864
$64$	−583.003	−1.13868
$65$	991.541	1.89208
$66$	−1576.55	−2.94031
$67$	307.039	0.559863	0.279932	−	0.960020i	$-0.409688\pi$
$67$	307.039	0.559863	0.279932	+	0.960020i	$0.409688\pi$
$68$	364.898	0.650740
$69$	−285.658	−0.498394
$70$	−926.155	−1.58138
$71$	−1135.63	−1.89823	−0.949114	−	0.314933i	$-0.898018\pi$
$71$	−1135.63	−1.89823	−0.949114	+	0.314933i	$0.898018\pi$
$72$	215.576	0.352859
$73$	−203.220	−0.325824	−0.162912	−	0.986641i	$-0.552089\pi$
$73$	−203.220	−0.325824	−0.162912	+	0.986641i	$0.552089\pi$
$74$	−715.792	−1.12445
$75$	481.349	0.741085
$76$	1868.70	2.82046
$77$	−746.715	−1.10514
$78$	−1909.41	−2.77177
$79$	255.296	0.363583	0.181791	−	0.983337i	$-0.441810\pi$
$79$	255.296	0.363583	0.181791	+	0.983337i	$0.441810\pi$
$80$	−790.257	−1.10442
$81$	−852.497	−1.16941
$82$	1052.47	1.41738
$83$	665.658	0.880306	0.440153	−	0.897923i	$-0.354924\pi$
$83$	665.658	0.880306	0.440153	+	0.897923i	$0.354924\pi$
$84$	1179.04	1.53147
$85$	−338.064	−0.431390
$86$	0	0
$87$	−1665.53	−2.05246
$88$	−2092.22	−2.53445
$89$	834.098	0.993419	0.496709	−	0.867917i	$-0.334542\pi$
$89$	834.098	0.993419	0.496709	+	0.867917i	$0.334542\pi$
$90$	−409.835	−0.480004
$91$	−904.369	−1.04180
$92$	−777.905	−0.881545
$93$	337.536	0.376353
$94$	−1133.83	−1.24410
$95$	−1731.28	−1.86974
$96$	−171.842	−0.182693
$97$	817.923	0.856160	0.428080	−	0.903741i	$-0.359190\pi$
$97$	817.923	0.856160	0.428080	+	0.903741i	$0.359190\pi$
$98$	−821.713	−0.846995
$99$	−330.431	−0.335450
$100$	1310.81	1.31081
$101$	−642.462	−0.632944	−0.316472	−	0.948602i	$-0.602498\pi$
$101$	−642.462	−0.632944	−0.316472	+	0.948602i	$0.602498\pi$
$102$	651.010	0.631957
$103$	76.9496	0.0736123	0.0368062	−	0.999322i	$-0.488282\pi$
$103$	76.9496	0.0736123	0.0368062	+	0.999322i	$0.488282\pi$
$104$	−2533.95	−2.38918
$105$	−1092.33	−1.01524
$106$	3463.77	3.17388
$107$	−843.742	−0.762315	−0.381157	−	0.924510i	$-0.624474\pi$
$107$	−843.742	−0.762315	−0.381157	+	0.924510i	$0.624474\pi$
$108$	−1892.50	−1.68616
$109$	−1162.50	−1.02154	−0.510769	−	0.859718i	$-0.670639\pi$
$109$	−1162.50	−1.02154	−0.510769	+	0.859718i	$0.670639\pi$
$110$	3977.56	3.44768
$111$	−844.225	−0.721894
$112$	720.782	0.608103
$113$	−884.391	−0.736252	−0.368126	−	0.929776i	$-0.620001\pi$
$113$	−884.391	−0.736252	−0.368126	+	0.929776i	$0.620001\pi$
$114$	3333.93	2.73905
$115$	720.699	0.584396
$116$	−4535.58	−3.63033
$117$	−400.195	−0.316222
$118$	340.420	0.265578
$119$	308.343	0.237527
$120$	−3060.61	−2.32828
$121$	1875.92	1.40941
$122$	−1594.86	−1.18354
$123$	1241.31	0.909958
$124$	919.179	0.665683
$125$	592.699	0.424101
$126$	373.804	0.264295
$127$	−582.571	−0.407046	−0.203523	−	0.979070i	$-0.565239\pi$
$127$	−582.571	−0.407046	−0.203523	+	0.979070i	$0.565239\pi$
$128$	−2592.57	−1.79026
$129$	0	0
$130$	4817.34	3.25007
$131$	903.044	0.602285	0.301142	−	0.953579i	$-0.402632\pi$
$131$	903.044	0.602285	0.301142	+	0.953579i	$0.402632\pi$
$132$	−5063.61	−3.33887
$133$	1579.08	1.02950
$134$	1491.73	0.961687
$135$	1753.33	1.11780
$136$	863.947	0.544727
$137$	−2121.22	−1.32283	−0.661415	−	0.750020i	$-0.730044\pi$
$137$	−2121.22	−1.32283	−0.661415	+	0.750020i	$0.730044\pi$
$138$	−1387.85	−0.856100
$139$	1042.17	0.635939	0.317970	−	0.948101i	$-0.396999\pi$
$139$	1042.17	0.635939	0.317970	+	0.948101i	$0.396999\pi$
$140$	−2974.64	−1.79574
$141$	−1337.27	−0.798713
$142$	−5517.38	−3.26062
$143$	3884.00	2.27130
$144$	318.955	0.184580
$145$	4202.05	2.40663
$146$	−987.334	−0.559673
$147$	−969.151	−0.543770
$148$	−2298.99	−1.27687
$149$	323.453	0.177841	0.0889205	−	0.996039i	$-0.471658\pi$
$149$	323.453	0.177841	0.0889205	+	0.996039i	$0.471658\pi$
$150$	2338.60	1.27297
$151$	864.578	0.465949	0.232975	−	0.972483i	$-0.425154\pi$
$151$	864.578	0.465949	0.232975	+	0.972483i	$0.425154\pi$
$152$	4424.42	2.36097
$153$	136.446	0.0720979
$154$	−3627.87	−1.89833
$155$	−851.584	−0.441296
$156$	−6132.69	−3.14749
$157$	577.261	0.293442	0.146721	−	0.989178i	$-0.453128\pi$
$157$	577.261	0.293442	0.146721	+	0.989178i	$0.453128\pi$
$158$	1240.34	0.624533
$159$	4085.27	2.03763
$160$	433.548	0.214219
$161$	−657.339	−0.321774
$162$	−4141.80	−2.00871
$163$	1803.66	0.866707	0.433353	−	0.901224i	$-0.357330\pi$
$163$	1803.66	0.866707	0.433353	+	0.901224i	$0.357330\pi$
$164$	3380.33	1.60951
$165$	4691.24	2.21341
$166$	3234.06	1.51212
$167$	−2552.51	−1.18275	−0.591374	−	0.806397i	$-0.701414\pi$
$167$	−2552.51	−1.18275	−0.591374	+	0.806397i	$0.701414\pi$
$168$	2791.53	1.28197
$169$	2507.03	1.14111
$170$	−1642.46	−0.741007
$171$	698.761	0.312489
$172$	0	0
$173$	−676.365	−0.297243	−0.148622	−	0.988894i	$-0.547484\pi$
$173$	−676.365	−0.297243	−0.148622	+	0.988894i	$0.547484\pi$
$174$	−8091.89	−3.52554
$175$	1107.65	0.478460
$176$	−3095.54	−1.32577
$177$	401.500	0.170501
$178$	4052.42	1.70641
$179$	1530.14	0.638927	0.319464	−	0.947598i	$-0.396497\pi$
$179$	1530.14	0.638927	0.319464	+	0.947598i	$0.396497\pi$
$180$	−1316.32	−0.545069
$181$	2614.23	1.07356	0.536780	−	0.843722i	$-0.319641\pi$
$181$	2614.23	1.07356	0.536780	+	0.843722i	$0.319641\pi$
$182$	−4393.82	−1.78952
$183$	−1881.02	−0.759832
$184$	−1841.80	−0.737930
$185$	2129.93	0.846463
$186$	1639.90	0.646469
$187$	−1324.24	−0.517851
$188$	−3641.66	−1.41274
$189$	−1599.18	−0.615469
$190$	−8411.33	−3.21169
$191$	3942.47	1.49355	0.746773	−	0.665079i	$-0.231602\pi$
$191$	3942.47	1.49355	0.746773	+	0.665079i	$0.231602\pi$
$192$	−3340.71	−1.25570
$193$	−80.0135	−0.0298420	−0.0149210	−	0.999889i	$-0.504750\pi$
$193$	−80.0135	−0.0298420	−0.0149210	+	0.999889i	$0.504750\pi$
$194$	3973.83	1.47064
$195$	5681.70	2.08654
$196$	−2639.19	−0.961806
$197$	2777.66	1.00457	0.502284	−	0.864703i	$-0.332493\pi$
$197$	2777.66	1.00457	0.502284	+	0.864703i	$0.332493\pi$
$198$	−1605.38	−0.576208
$199$	−2219.08	−0.790484	−0.395242	−	0.918577i	$-0.629339\pi$
$199$	−2219.08	−0.790484	−0.395242	+	0.918577i	$0.629339\pi$
$200$	3103.53	1.09726
$201$	1759.39	0.617402
$202$	−3121.36	−1.08722
$203$	−3832.62	−1.32511
$204$	2090.93	0.717619
$205$	−3131.75	−1.06698
$206$	373.855	0.126445
$207$	−290.881	−0.0976696
$208$	−3749.11	−1.24978
$209$	−6781.66	−2.24449
$210$	−5307.03	−1.74390
$211$	3231.50	1.05434	0.527169	−	0.849760i	$-0.323253\pi$
$211$	3231.50	1.05434	0.527169	+	0.849760i	$0.323253\pi$
$212$	11125.0	3.60410
$213$	−6507.35	−2.09332
$214$	−4099.27	−1.30944
$215$	0	0
$216$	−4480.76	−1.41147
$217$	776.717	0.242982
$218$	−5647.96	−1.75471
$219$	−1164.49	−0.359310
$220$	12775.2	3.91502
$221$	−1603.83	−0.488168
$222$	−4101.61	−1.24001
$223$	−2987.09	−0.896996	−0.448498	−	0.893784i	$-0.648041\pi$
$223$	−2987.09	−0.896996	−0.448498	+	0.893784i	$0.648041\pi$
$224$	−395.433	−0.117951
$225$	490.149	0.145229
$226$	−4296.76	−1.26467
$227$	2015.14	0.589206	0.294603	−	0.955620i	$-0.404813\pi$
$227$	2015.14	0.589206	0.294603	+	0.955620i	$0.404813\pi$
$228$	10708.0	3.11033
$229$	−4594.18	−1.32573	−0.662865	−	0.748739i	$-0.730659\pi$
$229$	−4594.18	−1.32573	−0.662865	+	0.748739i	$0.730659\pi$
$230$	3501.47	1.00383
$231$	−4278.81	−1.21872
$232$	−10738.6	−3.03890
$233$	−919.717	−0.258595	−0.129298	−	0.991606i	$-0.541272\pi$
$233$	−919.717	−0.258595	−0.129298	+	0.991606i	$0.541272\pi$
$234$	−1944.32	−0.543181
$235$	3373.86	0.936538
$236$	1093.37	0.301577
$237$	1462.89	0.400950
$238$	1498.07	0.408005
$239$	1476.81	0.399695	0.199847	−	0.979827i	$-0.435955\pi$
$239$	1476.81	0.399695	0.199847	+	0.979827i	$0.435955\pi$
$240$	−4528.32	−1.21792
$241$	5049.60	1.34968	0.674841	−	0.737963i	$-0.264212\pi$
$241$	5049.60	1.34968	0.674841	+	0.737963i	$0.264212\pi$
$242$	9114.05	2.42096
$243$	−1610.41	−0.425135
$244$	−5122.41	−1.34397
$245$	2445.11	0.637602
$246$	6030.81	1.56305
$247$	−8213.48	−2.11583
$248$	2176.28	0.557235
$249$	3814.34	0.970778
$250$	2879.59	0.728486
$251$	−292.300	−0.0735052	−0.0367526	−	0.999324i	$-0.511701\pi$
$251$	−292.300	−0.0735052	−0.0367526	+	0.999324i	$0.511701\pi$
$252$	1200.59	0.300120
$253$	2823.08	0.701523
$254$	−2830.39	−0.699190
$255$	−1937.17	−0.475726
$256$	−7931.84	−1.93648
$257$	4.22507	0.00102550	0.000512749	−	1.00000i	$-0.499837\pi$
$257$	4.22507	0.00102550	0.000512749		1.00000i	$0.499837\pi$
$258$	0	0
$259$	−1942.68	−0.466070
$260$	15472.4	3.69061
$261$	−1695.98	−0.402217
$262$	4387.38	1.03456
$263$	−8246.98	−1.93358	−0.966788	−	0.255580i	$-0.917733\pi$
$263$	−8246.98	−1.93358	−0.966788	+	0.255580i	$0.917733\pi$
$264$	−11988.8	−2.79492
$265$	−10306.9	−2.38924
$266$	7671.85	1.76839
$267$	4779.53	1.09552
$268$	4791.18	1.09204
$269$	3136.78	0.710976	0.355488	−	0.934681i	$-0.384315\pi$
$269$	3136.78	0.710976	0.355488	+	0.934681i	$0.384315\pi$
$270$	8518.44	1.92006
$271$	−1246.03	−0.279303	−0.139652	−	0.990201i	$-0.544598\pi$
$271$	−1246.03	−0.279303	−0.139652	+	0.990201i	$0.544598\pi$
$272$	1278.25	0.284946
$273$	−5182.20	−1.14887
$274$	−10305.8	−2.27225
$275$	−4757.03	−1.04313
$276$	−4457.53	−0.972145
$277$	−1353.84	−0.293662	−0.146831	−	0.989162i	$-0.546907\pi$
$277$	−1353.84	−0.293662	−0.146831	+	0.989162i	$0.546907\pi$
$278$	5063.31	1.09236
$279$	343.707	0.0737534
$280$	−7042.88	−1.50319
$281$	−5032.93	−1.06847	−0.534234	−	0.845337i	$-0.679400\pi$
$281$	−5032.93	−1.06847	−0.534234	+	0.845337i	$0.679400\pi$
$282$	−6497.05	−1.37196
$283$	−1540.76	−0.323636	−0.161818	−	0.986821i	$-0.551736\pi$
$283$	−1540.76	−0.323636	−0.161818	+	0.986821i	$0.551736\pi$
$284$	−17720.8	−3.70260
$285$	−9920.55	−2.06190
$286$	18870.2	3.90146
$287$	2856.42	0.587488
$288$	−174.984	−0.0358022
$289$	−4366.18	−0.888699
$290$	20415.4	4.13391
$291$	4686.85	0.944150
$292$	−3171.14	−0.635537
$293$	−2032.88	−0.405331	−0.202665	−	0.979248i	$-0.564960\pi$
$293$	−2032.88	−0.405331	−0.202665	+	0.979248i	$0.564960\pi$
$294$	−4708.56	−0.934044
$295$	−1012.96	−0.199922
$296$	−5443.19	−1.06885
$297$	6868.02	1.34183
$298$	1571.48	0.305481
$299$	3419.11	0.661312
$300$	7511.17	1.44553
$301$	0	0
$302$	4200.50	0.800369
$303$	−3681.42	−0.697994
$304$	6546.14	1.23502
$305$	4745.71	0.890947
$306$	662.913	0.123844
$307$	−9698.57	−1.80302	−0.901509	−	0.432761i	$-0.857539\pi$
$307$	−9698.57	−1.80302	−0.901509	+	0.432761i	$0.857539\pi$
$308$	−11652.1	−2.15564
$309$	440.935	0.0811777
$310$	−4137.37	−0.758022
$311$	−8337.39	−1.52016	−0.760081	−	0.649829i	$-0.774841\pi$
$311$	−8337.39	−1.52016	−0.760081	+	0.649829i	$0.774841\pi$
$312$	−14520.0	−2.63472
$313$	9108.79	1.64492	0.822459	−	0.568825i	$-0.192602\pi$
$313$	9108.79	1.64492	0.822459	+	0.568825i	$0.192602\pi$
$314$	2804.59	0.504051
$315$	−1112.30	−0.198956
$316$	3983.75	0.709188
$317$	3842.07	0.680733	0.340366	−	0.940293i	$-0.389449\pi$
$317$	3842.07	0.680733	0.340366	+	0.940293i	$0.389449\pi$
$318$	19848.0	3.50007
$319$	16460.0	2.88897
$320$	8428.43	1.47239
$321$	−4834.79	−0.840660
$322$	−3193.64	−0.552716
$323$	2800.37	0.482405
$324$	−13302.7	−2.28099
$325$	−5761.38	−0.983335
$326$	8762.95	1.48876
$327$	−6661.36	−1.12653
$328$	8003.40	1.34730
$329$	−3077.25	−0.515666
$330$	22792.1	3.80201
$331$	−2360.71	−0.392014	−0.196007	−	0.980603i	$-0.562798\pi$
$331$	−2360.71	−0.392014	−0.196007	+	0.980603i	$0.562798\pi$
$332$	10387.2	1.71709
$333$	−859.659	−0.141469
$334$	−12401.2	−2.03163
$335$	−4438.84	−0.723940
$336$	4130.21	0.670599
$337$	−66.5688	−0.0107603	−0.00538017	−	0.999986i	$-0.501713\pi$
$337$	−66.5688	−0.0107603	−0.00538017	+	0.999986i	$0.501713\pi$
$338$	12180.2	1.96011
$339$	−5067.72	−0.811919
$340$	−5275.29	−0.841450
$341$	−3335.77	−0.529742
$342$	3394.89	0.536768
$343$	−6752.93	−1.06304
$344$	0	0
$345$	4129.73	0.644456
$346$	−3286.08	−0.510580
$347$	2886.72	0.446592	0.223296	−	0.974751i	$-0.428318\pi$
$347$	2886.72	0.446592	0.223296	+	0.974751i	$0.428318\pi$
$348$	−25989.7	−4.00343
$349$	7743.50	1.18768	0.593840	−	0.804583i	$-0.297611\pi$
$349$	7743.50	1.18768	0.593840	+	0.804583i	$0.297611\pi$
$350$	5381.45	0.821859
$351$	8318.07	1.26492
$352$	1698.27	0.257153
$353$	−2857.45	−0.430840	−0.215420	−	0.976521i	$-0.569112\pi$
$353$	−2857.45	−0.430840	−0.215420	+	0.976521i	$0.569112\pi$
$354$	1950.67	0.292872
$355$	16417.7	2.45453
$356$	13015.6	1.93772
$357$	1766.86	0.261939
$358$	7434.09	1.09750
$359$	−2242.44	−0.329670	−0.164835	−	0.986321i	$-0.552709\pi$
$359$	−2242.44	−0.329670	−0.164835	+	0.986321i	$0.552709\pi$
$360$	−3116.56	−0.456270
$361$	7482.16	1.09085
$362$	12701.1	1.84407
$363$	10749.4	1.55426
$364$	−14112.2	−2.03208
$365$	2937.94	0.421312
$366$	−9138.84	−1.30518
$367$	11678.8	1.66111	0.830557	−	0.556934i	$-0.188022\pi$
$367$	11678.8	1.66111	0.830557	+	0.556934i	$0.188022\pi$
$368$	−2725.03	−0.386011
$369$	1264.00	0.178323
$370$	10348.1	1.45398
$371$	9400.78	1.31554
$372$	5267.06	0.734097
$373$	−4698.81	−0.652266	−0.326133	−	0.945324i	$-0.605746\pi$
$373$	−4698.81	−0.652266	−0.326133	+	0.945324i	$0.605746\pi$
$374$	−6433.75	−0.889522
$375$	3396.27	0.467687
$376$	−8622.15	−1.18259
$377$	19935.2	2.72338
$378$	−7769.54	−1.05720
$379$	9794.57	1.32748	0.663738	−	0.747965i	$-0.268969\pi$
$379$	9794.57	1.32748	0.663738	+	0.747965i	$0.268969\pi$
$380$	−27015.7	−3.64704
$381$	−3338.24	−0.448879
$382$	19154.3	2.56549
$383$	−4659.25	−0.621610	−0.310805	−	0.950474i	$-0.600599\pi$
$383$	−4659.25	−0.621610	−0.310805	+	0.950474i	$0.600599\pi$
$384$	−14855.9	−1.97425
$385$	10795.2	1.42902
$386$	−388.741	−0.0512601
$387$	0	0
$388$	12763.2	1.66999
$389$	−15017.9	−1.95743	−0.978713	−	0.205236i	$-0.934204\pi$
$389$	−15017.9	−1.95743	−0.978713	+	0.205236i	$0.934204\pi$
$390$	27604.2	3.58409
$391$	−1165.74	−0.150777
$392$	−6248.66	−0.805115
$393$	5174.60	0.664184
$394$	13495.1	1.72557
$395$	−3690.79	−0.470137
$396$	−5156.19	−0.654313
$397$	5330.02	0.673819	0.336909	−	0.941537i	$-0.390618\pi$
$397$	5330.02	0.673819	0.336909	+	0.941537i	$0.390618\pi$
$398$	−10781.3	−1.35783
$399$	9048.39	1.13530
$400$	4591.82	0.573977
$401$	1881.10	0.234259	0.117130	−	0.993117i	$-0.462631\pi$
$401$	1881.10	0.234259	0.117130	+	0.993117i	$0.462631\pi$
$402$	8547.89	1.06052
$403$	−4040.05	−0.499378
$404$	−10025.3	−1.23459
$405$	12324.5	1.51212
$406$	−18620.6	−2.27617
$407$	8343.23	1.01611
$408$	4950.57	0.600710
$409$	−8907.66	−1.07691	−0.538454	−	0.842655i	$-0.680991\pi$
$409$	−8907.66	−1.07691	−0.538454	+	0.842655i	$0.680991\pi$
$410$	−15215.4	−1.83277
$411$	−12154.9	−1.45878
$412$	1200.76	0.143585
$413$	923.908	0.110079
$414$	−1413.23	−0.167769
$415$	−9623.36	−1.13829
$416$	2056.82	0.242413
$417$	5971.81	0.701297
$418$	−32948.3	−3.85539
$419$	14129.0	1.64736	0.823682	−	0.567052i	$-0.191916\pi$
$419$	14129.0	1.64736	0.823682	+	0.567052i	$0.191916\pi$
$420$	−17045.2	−1.98029
$421$	−631.853	−0.0731464	−0.0365732	−	0.999331i	$-0.511644\pi$
$421$	−631.853	−0.0731464	−0.0365732	+	0.999331i	$0.511644\pi$
$422$	15700.0	1.81106
$423$	−1361.72	−0.156523
$424$	26340.0	3.01695
$425$	1964.33	0.224198
$426$	−31615.6	−3.59572
$427$	−4328.50	−0.490563
$428$	−13166.1	−1.48694
$429$	22256.0	2.50473
$430$	0	0
$431$	−1391.25	−0.155485	−0.0777426	−	0.996973i	$-0.524771\pi$
$431$	−1391.25	−0.155485	−0.0777426	+	0.996973i	$0.524771\pi$
$432$	−6629.50	−0.738337
$433$	11215.1	1.24472	0.622359	−	0.782732i	$-0.286175\pi$
$433$	11215.1	1.24472	0.622359	+	0.782732i	$0.286175\pi$
$434$	3773.63	0.417374
$435$	24078.5	2.65396
$436$	−18140.2	−1.99257
$437$	−5969.95	−0.653504
$438$	−5657.60	−0.617193
$439$	4321.01	0.469774	0.234887	−	0.972023i	$-0.424528\pi$
$439$	4321.01	0.469774	0.234887	+	0.972023i	$0.424528\pi$
$440$	30247.1	3.27721
$441$	−986.870	−0.106562
$442$	−7792.10	−0.838535
$443$	−10333.2	−1.10823	−0.554113	−	0.832441i	$-0.686943\pi$
$443$	−10333.2	−1.10823	−0.554113	+	0.832441i	$0.686943\pi$
$444$	−13173.6	−1.40809
$445$	−12058.5	−1.28456
$446$	−14512.6	−1.54079
$447$	1853.44	0.196118
$448$	−7687.44	−0.810709
$449$	−8737.55	−0.918375	−0.459188	−	0.888339i	$-0.651859\pi$
$449$	−8737.55	−0.918375	−0.459188	+	0.888339i	$0.651859\pi$
$450$	2381.36	0.249463
$451$	−12267.5	−1.28083
$452$	−13800.4	−1.43610
$453$	4954.18	0.513836
$454$	9790.45	1.01209
$455$	13074.4	1.34711
$456$	25352.7	2.60362
$457$	−11206.6	−1.14709	−0.573545	−	0.819174i	$-0.694432\pi$
$457$	−11206.6	−1.14709	−0.573545	+	0.819174i	$0.694432\pi$
$458$	−22320.5	−2.27723
$459$	−2836.03	−0.288398
$460$	11246.1	1.13990
$461$	−14525.7	−1.46752	−0.733762	−	0.679407i	$-0.762237\pi$
$461$	−14525.7	−1.46752	−0.733762	+	0.679407i	$0.762237\pi$
$462$	−20788.4	−2.09342
$463$	−4930.50	−0.494903	−0.247451	−	0.968900i	$-0.579593\pi$
$463$	−4930.50	−0.494903	−0.247451	+	0.968900i	$0.579593\pi$
$464$	−15888.3	−1.58965
$465$	−4879.73	−0.486649
$466$	−4468.39	−0.444193
$467$	−6108.04	−0.605238	−0.302619	−	0.953112i	$-0.597861\pi$
$467$	−6108.04	−0.605238	−0.302619	+	0.953112i	$0.597861\pi$
$468$	−6244.81	−0.616809
$469$	4048.60	0.398608
$470$	16391.7	1.60871
$471$	3307.81	0.323600
$472$	2588.70	0.252446
$473$	0	0
$474$	7107.37	0.688718
$475$	10059.7	0.971725
$476$	4811.52	0.463310
$477$	4159.96	0.399311
$478$	7175.00	0.686563
$479$	−10162.4	−0.969382	−0.484691	−	0.874685i	$-0.661068\pi$
$479$	−10162.4	−0.969382	−0.484691	+	0.874685i	$0.661068\pi$
$480$	2484.31	0.236235
$481$	10104.7	0.957871
$482$	24533.2	2.31837
$483$	−3766.67	−0.354843
$484$	29272.7	2.74912
$485$	−11824.6	−1.10707
$486$	−7824.08	−0.730263
$487$	13884.5	1.29192	0.645961	−	0.763371i	$-0.276457\pi$
$487$	13884.5	1.29192	0.645961	+	0.763371i	$0.276457\pi$
$488$	−12128.0	−1.12502
$489$	10335.3	0.955781
$490$	11879.4	1.09522
$491$	−14658.8	−1.34733	−0.673667	−	0.739035i	$-0.735282\pi$
$491$	−14658.8	−1.34733	−0.673667	+	0.739035i	$0.735282\pi$
$492$	19369.9	1.77492
$493$	−6796.86	−0.620923
$494$	−39904.7	−3.63441
$495$	4777.01	0.433759
$496$	3219.92	0.291489
$497$	−14974.3	−1.35149
$498$	18531.7	1.66752
$499$	−15383.3	−1.38006	−0.690029	−	0.723782i	$-0.742402\pi$
$499$	−15383.3	−1.38006	−0.690029	+	0.723782i	$0.742402\pi$
$500$	9248.73	0.827232
$501$	−14626.3	−1.30430
$502$	−1420.12	−0.126261
$503$	12847.1	1.13882	0.569409	−	0.822054i	$-0.307172\pi$
$503$	12847.1	1.13882	0.569409	+	0.822054i	$0.307172\pi$
$504$	2842.57	0.251227
$505$	9288.02	0.818438
$506$	13715.7	1.20502
$507$	14365.7	1.25839
$508$	−9090.69	−0.793965
$509$	−14988.6	−1.30522	−0.652611	−	0.757693i	$-0.726326\pi$
$509$	−14988.6	−1.30522	−0.652611	+	0.757693i	$0.726326\pi$
$510$	−9411.60	−0.817162
$511$	−2679.65	−0.231978
$512$	−17795.8	−1.53607
$513$	−14523.8	−1.24998
$514$	20.5273	0.00176151
$515$	−1112.45	−0.0951856
$516$	0	0
$517$	13215.9	1.12424
$518$	−9438.39	−0.800577
$519$	−3875.69	−0.327792
$520$	36633.2	3.08937
$521$	−10203.3	−0.857997	−0.428998	−	0.903305i	$-0.641133\pi$
$521$	−10203.3	−0.857997	−0.428998	+	0.903305i	$0.641133\pi$
$522$	−8239.83	−0.690896
$523$	−5487.11	−0.458766	−0.229383	−	0.973336i	$-0.573671\pi$
$523$	−5487.11	−0.458766	−0.229383	+	0.973336i	$0.573671\pi$
$524$	14091.5	1.17479
$525$	6347.03	0.527633
$526$	−40067.4	−3.32134
$527$	1377.45	0.113857
$528$	−17738.0	−1.46202
$529$	−9681.83	−0.795745
$530$	−50075.5	−4.10404
$531$	408.841	0.0334128
$532$	24640.6	2.00809
$533$	−14857.5	−1.20741
$534$	23221.1	1.88179
$535$	12197.9	0.985723
$536$	11343.8	0.914136
$537$	8767.97	0.704592
$538$	15239.8	1.22126
$539$	9577.84	0.765393
$540$	27359.7	2.18032
$541$	13269.0	1.05449	0.527245	−	0.849713i	$-0.323225\pi$
$541$	13269.0	1.05449	0.527245	+	0.849713i	$0.323225\pi$
$542$	−6053.78	−0.479764
$543$	14980.0	1.18389
$544$	−701.269	−0.0552696
$545$	16806.2	1.32092
$546$	−25177.4	−1.97343
$547$	9377.89	0.733034	0.366517	−	0.930411i	$-0.380550\pi$
$547$	9377.89	0.733034	0.366517	+	0.930411i	$0.380550\pi$
$548$	−33100.4	−2.58025
$549$	−1915.41	−0.148903
$550$	−23111.7	−1.79180
$551$	−34807.9	−2.69122
$552$	−10553.8	−0.813770
$553$	3366.32	0.258861
$554$	−6577.55	−0.504429
$555$	12204.9	0.933457
$556$	16262.4	1.24043
$557$	10526.2	0.800735	0.400368	−	0.916355i	$-0.368882\pi$
$557$	10526.2	0.800735	0.400368	+	0.916355i	$0.368882\pi$
$558$	1669.88	0.126688
$559$	0	0
$560$	−10420.3	−0.786317
$561$	−7588.14	−0.571072
$562$	−24452.2	−1.83533
$563$	20928.1	1.56663	0.783315	−	0.621625i	$-0.213527\pi$
$563$	20928.1	1.56663	0.783315	+	0.621625i	$0.213527\pi$
$564$	−20867.4	−1.55793
$565$	12785.6	0.952022
$566$	−7485.71	−0.555915
$567$	−11241.0	−0.832586
$568$	−41956.5	−3.09940
$569$	−16353.3	−1.20486	−0.602431	−	0.798171i	$-0.705801\pi$
$569$	−16353.3	−1.20486	−0.602431	+	0.798171i	$0.705801\pi$
$570$	−48198.4	−3.54177
$571$	4027.42	0.295171	0.147585	−	0.989049i	$-0.452850\pi$
$571$	4027.42	0.295171	0.147585	+	0.989049i	$0.452850\pi$
$572$	60607.6	4.43030
$573$	22591.1	1.64704
$574$	13877.7	1.00914
$575$	−4187.65	−0.303716
$576$	−3401.79	−0.246078
$577$	−18455.6	−1.33157	−0.665786	−	0.746143i	$-0.731904\pi$
$577$	−18455.6	−1.33157	−0.665786	+	0.746143i	$0.731904\pi$
$578$	−21212.8	−1.52653
$579$	−458.492	−0.0329089
$580$	65570.6	4.69426
$581$	8777.32	0.626755
$582$	22770.8	1.62178
$583$	−40373.5	−2.86810
$584$	−7508.12	−0.532000
$585$	5785.58	0.408896
$586$	−9876.61	−0.696244
$587$	12216.2	0.858974	0.429487	−	0.903073i	$-0.358694\pi$
$587$	12216.2	0.858974	0.429487	+	0.903073i	$0.358694\pi$
$588$	−15123.1	−1.06065
$589$	7054.14	0.493482
$590$	−4921.42	−0.343409
$591$	15916.5	1.10781
$592$	−8053.47	−0.559114
$593$	−15656.9	−1.08424	−0.542118	−	0.840302i	$-0.682378\pi$
$593$	−15656.9	−1.08424	−0.542118	+	0.840302i	$0.682378\pi$
$594$	33367.9	2.30488
$595$	−4457.69	−0.307138
$596$	5047.31	0.346889
$597$	−12715.7	−0.871724
$598$	16611.5	1.13595
$599$	−7086.75	−0.483400	−0.241700	−	0.970351i	$-0.577705\pi$
$599$	−7086.75	−0.483400	−0.241700	+	0.970351i	$0.577705\pi$
$600$	17783.8	1.21003
$601$	4349.45	0.295204	0.147602	−	0.989047i	$-0.452845\pi$
$601$	4349.45	0.295204	0.147602	+	0.989047i	$0.452845\pi$
$602$	0	0
$603$	1791.56	0.120991
$604$	13491.2	0.908859
$605$	−27120.0	−1.82246
$606$	−17886.0	−1.19896
$607$	−11528.1	−0.770860	−0.385430	−	0.922737i	$-0.625947\pi$
$607$	−11528.1	−0.770860	−0.385430	+	0.922737i	$0.625947\pi$
$608$	−3591.32	−0.239551
$609$	−21961.6	−1.46130
$610$	23056.8	1.53040
$611$	16006.1	1.05980
$612$	2129.16	0.140631
$613$	19355.5	1.27530	0.637652	−	0.770324i	$-0.279906\pi$
$613$	19355.5	1.27530	0.637652	+	0.770324i	$0.279906\pi$
$614$	−47119.9	−3.09708
$615$	−17945.5	−1.17664
$616$	−27587.9	−1.80446
$617$	−13935.0	−0.909238	−0.454619	−	0.890686i	$-0.650225\pi$
$617$	−13935.0	−0.909238	−0.454619	+	0.890686i	$0.650225\pi$
$618$	2142.26	0.139440
$619$	15495.3	1.00615	0.503075	−	0.864243i	$-0.332202\pi$
$619$	15495.3	1.00615	0.503075	+	0.864243i	$0.332202\pi$
$620$	−13288.5	−0.860772
$621$	6045.97	0.390687
$622$	−40506.7	−2.61121
$623$	10998.4	0.707288
$624$	−21483.0	−1.37822
$625$	−19068.9	−1.22041
$626$	44254.5	2.82550
$627$	−38860.1	−2.47516
$628$	9007.83	0.572375
$629$	−3445.19	−0.218392
$630$	−5404.06	−0.341750
$631$	−13393.6	−0.844993	−0.422496	−	0.906365i	$-0.638846\pi$
$631$	−13393.6	−0.844993	−0.422496	+	0.906365i	$0.638846\pi$
$632$	9432.09	0.593652
$633$	18517.0	1.16270
$634$	18666.5	1.16931
$635$	8422.18	0.526337
$636$	63748.3	3.97451
$637$	11600.0	0.721521
$638$	79969.8	4.96244
$639$	−6626.32	−0.410224
$640$	37480.6	2.31492
$641$	2195.86	0.135306	0.0676530	−	0.997709i	$-0.478449\pi$
$641$	2195.86	0.135306	0.0676530	+	0.997709i	$0.478449\pi$
$642$	−23489.6	−1.44402
$643$	−19987.0	−1.22583	−0.612915	−	0.790149i	$-0.710003\pi$
$643$	−19987.0	−1.22583	−0.612915	+	0.790149i	$0.710003\pi$
$644$	−10257.4	−0.627637
$645$	0	0
$646$	13605.4	0.828635
$647$	27112.0	1.64743	0.823713	−	0.567008i	$-0.191899\pi$
$647$	27112.0	1.64743	0.823713	+	0.567008i	$0.191899\pi$
$648$	−31496.1	−1.90939
$649$	−3967.91	−0.239991
$650$	−27991.3	−1.68909
$651$	4450.73	0.267954
$652$	28145.0	1.69056
$653$	15361.6	0.920591	0.460296	−	0.887766i	$-0.347743\pi$
$653$	15361.6	0.920591	0.460296	+	0.887766i	$0.347743\pi$
$654$	−32363.8	−1.93505
$655$	−13055.2	−0.778794
$656$	11841.4	0.704771
$657$	−1185.78	−0.0704134
$658$	−14950.6	−0.885769
$659$	−276.285	−0.0163316	−0.00816580	−	0.999967i	$-0.502599\pi$
$659$	−276.285	−0.0163316	−0.00816580	+	0.999967i	$0.502599\pi$
$660$	73204.2	4.31738
$661$	1668.69	0.0981916	0.0490958	−	0.998794i	$-0.484366\pi$
$661$	1668.69	0.0981916	0.0490958	+	0.998794i	$0.484366\pi$
$662$	−11469.4	−0.673369
$663$	−9190.22	−0.538339
$664$	24593.2	1.43735
$665$	−22828.6	−1.33121
$666$	−4176.60	−0.243003
$667$	14489.8	0.841153
$668$	−39830.4	−2.30701
$669$	−17116.5	−0.989184
$670$	−21565.9	−1.24352
$671$	18589.6	1.06951
$672$	−2265.90	−0.130073
$673$	−188.158	−0.0107770	−0.00538851	−	0.999985i	$-0.501715\pi$
$673$	−188.158	−0.0107770	−0.00538851	+	0.999985i	$0.501715\pi$
$674$	−323.421	−0.0184832
$675$	−10187.8	−0.580930
$676$	39120.7	2.22580
$677$	15768.2	0.895154	0.447577	−	0.894245i	$-0.352287\pi$
$677$	15768.2	0.895154	0.447577	+	0.894245i	$0.352287\pi$
$678$	−24621.2	−1.39465
$679$	10785.1	0.609563
$680$	−12490.0	−0.704367
$681$	11547.1	0.649761
$682$	−16206.6	−0.909947
$683$	7660.10	0.429144	0.214572	−	0.976708i	$-0.431164\pi$
$683$	7660.10	0.429144	0.214572	+	0.976708i	$0.431164\pi$
$684$	10903.8	0.609526
$685$	30666.2	1.71051
$686$	−32808.7	−1.82601
$687$	−26325.5	−1.46198
$688$	0	0
$689$	−48897.6	−2.70370
$690$	20064.1	1.10699
$691$	−5645.69	−0.310814	−0.155407	−	0.987851i	$-0.549669\pi$
$691$	−5645.69	−0.310814	−0.155407	+	0.987851i	$0.549669\pi$
$692$	−10554.3	−0.579789
$693$	−4357.04	−0.238832
$694$	14025.0	0.767118
$695$	−15066.5	−0.822311
$696$	−61534.3	−3.35122
$697$	5065.64	0.275286
$698$	37621.3	2.04010
$699$	−5270.14	−0.285172
$700$	17284.3	0.933262
$701$	−16807.3	−0.905568	−0.452784	−	0.891620i	$-0.649569\pi$
$701$	−16807.3	−0.905568	−0.452784	+	0.891620i	$0.649569\pi$
$702$	40412.8	2.17277
$703$	−17643.4	−0.946562
$704$	33015.3	1.76749
$705$	19332.8	1.03279
$706$	−13882.7	−0.740062
$707$	−8471.46	−0.450640
$708$	6265.19	0.332571
$709$	2473.18	0.131005	0.0655023	−	0.997852i	$-0.479135\pi$
$709$	2473.18	0.131005	0.0655023	+	0.997852i	$0.479135\pi$
$710$	79764.3	4.21620
$711$	1489.64	0.0785735
$712$	30816.3	1.62204
$713$	−2936.50	−0.154240
$714$	8584.18	0.449937
$715$	−56150.6	−2.93694
$716$	23877.0	1.24626
$717$	8462.39	0.440773
$718$	−10894.8	−0.566280
$719$	1318.06	0.0683664	0.0341832	−	0.999416i	$-0.489117\pi$
$719$	1318.06	0.0683664	0.0341832	+	0.999416i	$0.489117\pi$
$720$	−4611.10	−0.238675
$721$	1014.65	0.0524100
$722$	36351.6	1.87378
$723$	28935.1	1.48839
$724$	40793.6	2.09404
$725$	−24416.1	−1.25075
$726$	52225.1	2.66977
$727$	29579.7	1.50901	0.754505	−	0.656294i	$-0.227877\pi$
$727$	29579.7	1.50901	0.754505	+	0.656294i	$0.227877\pi$
$728$	−33412.6	−1.70103
$729$	13789.5	0.700578
$730$	14273.8	0.723695
$731$	0	0
$732$	−29352.3	−1.48209
$733$	−20648.4	−1.04047	−0.520237	−	0.854022i	$-0.674156\pi$
$733$	−20648.4	−1.04047	−0.520237	+	0.854022i	$0.674156\pi$
$734$	56740.8	2.85332
$735$	14010.9	0.703131
$736$	1495.00	0.0748727
$737$	−17387.5	−0.869035
$738$	6141.07	0.306309
$739$	−412.419	−0.0205292	−0.0102646	−	0.999947i	$-0.503267\pi$
$739$	−412.419	−0.0205292	−0.0102646	+	0.999947i	$0.503267\pi$
$740$	33236.4	1.65107
$741$	−47064.7	−2.33328
$742$	45673.1	2.25972
$743$	5178.96	0.255717	0.127858	−	0.991792i	$-0.459190\pi$
$743$	5178.96	0.255717	0.127858	+	0.991792i	$0.459190\pi$
$744$	12470.5	0.614504
$745$	−4676.14	−0.229960
$746$	−22828.9	−1.12041
$747$	3884.07	0.190242
$748$	−20664.0	−1.01010
$749$	−11125.5	−0.542748
$750$	16500.6	0.803355
$751$	34622.1	1.68226	0.841132	−	0.540831i	$-0.181890\pi$
$751$	34622.1	1.68226	0.841132	+	0.540831i	$0.181890\pi$
$752$	−12756.9	−0.618611
$753$	−1674.93	−0.0810596
$754$	96853.8	4.67800
$755$	−12499.1	−0.602503
$756$	−24954.4	−1.20051
$757$	18507.3	0.888585	0.444292	−	0.895882i	$-0.353455\pi$
$757$	18507.3	0.888585	0.444292	+	0.895882i	$0.353455\pi$
$758$	47586.4	2.28023
$759$	16176.7	0.773620
$760$	−63963.4	−3.05289
$761$	20023.6	0.953817	0.476909	−	0.878953i	$-0.341757\pi$
$761$	20023.6	0.953817	0.476909	+	0.878953i	$0.341757\pi$
$762$	−16218.6	−0.771048
$763$	−15328.7	−0.727309
$764$	61520.0	2.91324
$765$	−1972.58	−0.0932273
$766$	−22636.7	−1.06775
$767$	−4805.66	−0.226235
$768$	−45450.8	−2.13550
$769$	−16391.2	−0.768636	−0.384318	−	0.923201i	$-0.625563\pi$
$769$	−16391.2	−0.768636	−0.384318	+	0.923201i	$0.625563\pi$
$770$	52447.8	2.45466
$771$	24.2104	0.00113089
$772$	−1248.57	−0.0582084
$773$	−21244.9	−0.988519	−0.494260	−	0.869314i	$-0.664561\pi$
$773$	−21244.9	−0.988519	−0.494260	+	0.869314i	$0.664561\pi$
$774$	0	0
$775$	4948.16	0.229346
$776$	30218.7	1.39792
$777$	−11131.9	−0.513970
$778$	−72963.6	−3.36230
$779$	25942.0	1.19315
$780$	88659.8	4.06991
$781$	64310.2	2.94648
$782$	−5663.67	−0.258993
$783$	35251.1	1.60890
$784$	−9245.20	−0.421155
$785$	−8345.41	−0.379440
$786$	25140.5	1.14088
$787$	2206.07	0.0999212	0.0499606	−	0.998751i	$-0.484090\pi$
$787$	2206.07	0.0999212	0.0499606	+	0.998751i	$0.484090\pi$
$788$	43343.8	1.95947
$789$	−47256.6	−2.13230
$790$	−17931.5	−0.807562
$791$	−11661.5	−0.524192
$792$	−12208.0	−0.547717
$793$	22514.4	1.00821
$794$	25895.6	1.15743
$795$	−59060.4	−2.63479
$796$	−34627.5	−1.54188
$797$	−39524.6	−1.75663	−0.878314	−	0.478085i	$-0.841331\pi$
$797$	−39524.6	−1.75663	−0.878314	+	0.478085i	$0.841331\pi$
$798$	43961.1	1.95013
$799$	−5457.26	−0.241632
$800$	−2519.15	−0.111332
$801$	4866.92	0.214687
$802$	9139.23	0.402391
$803$	11508.3	0.505752
$804$	27454.3	1.20428
$805$	9503.09	0.416075
$806$	−19628.3	−0.857790
$807$	17974.3	0.784046
$808$	−23736.2	−1.03346
$809$	−39225.2	−1.70468	−0.852339	−	0.522990i	$-0.824816\pi$
$809$	−39225.2	−1.70468	−0.852339	+	0.522990i	$0.824816\pi$
$810$	59877.7	2.59739
$811$	10801.0	0.467664	0.233832	−	0.972277i	$-0.424873\pi$
$811$	10801.0	0.467664	0.233832	+	0.972277i	$0.424873\pi$
$812$	−59805.9	−2.58470
$813$	−7139.99	−0.308008
$814$	40535.1	1.74540
$815$	−26075.3	−1.12071
$816$	7324.60	0.314231
$817$	0	0
$818$	−43277.3	−1.84982
$819$	−5276.94	−0.225142
$820$	−48869.1	−2.08120
$821$	25868.4	1.09965	0.549826	−	0.835279i	$-0.314694\pi$
$821$	25868.4	1.09965	0.549826	+	0.835279i	$0.314694\pi$
$822$	−59054.0	−2.50577
$823$	45014.4	1.90657	0.953283	−	0.302079i	$-0.0976806\pi$
$823$	45014.4	1.90657	0.953283	+	0.302079i	$0.0976806\pi$
$824$	2842.96	0.120193
$825$	−27258.6	−1.15033
$826$	4488.75	0.189084
$827$	46882.2	1.97129	0.985644	−	0.168839i	$-0.0540017\pi$
$827$	46882.2	1.97129	0.985644	+	0.168839i	$0.0540017\pi$
$828$	−4539.03	−0.190510
$829$	−26628.6	−1.11562	−0.557810	−	0.829969i	$-0.688358\pi$
$829$	−26628.6	−1.11562	−0.557810	+	0.829969i	$0.688358\pi$
$830$	−46754.5	−1.95527
$831$	−7757.75	−0.323843
$832$	39985.8	1.66618
$833$	−3955.00	−0.164505
$834$	29013.7	1.20463
$835$	36901.4	1.52937
$836$	−105824.	−4.37799
$837$	−7143.97	−0.295020
$838$	68644.8	2.82971
$839$	−23832.3	−0.980670	−0.490335	−	0.871534i	$-0.663126\pi$
$839$	−23832.3	−0.980670	−0.490335	+	0.871534i	$0.663126\pi$
$840$	−40357.0	−1.65768
$841$	60094.2	2.46399
$842$	−3069.82	−0.125645
$843$	−28839.6	−1.17828
$844$	50425.7	2.05654
$845$	−36243.9	−1.47553
$846$	−6615.84	−0.268862
$847$	24735.8	1.00346
$848$	38971.4	1.57816
$849$	−8828.85	−0.356897
$850$	9543.58	0.385108
$851$	7344.60	0.295852
$852$	−101543.	−4.08313
$853$	−8283.40	−0.332495	−0.166248	−	0.986084i	$-0.553165\pi$
$853$	−8283.40	−0.332495	−0.166248	+	0.986084i	$0.553165\pi$
$854$	−21029.7	−0.842649
$855$	−10101.9	−0.404069
$856$	−31172.6	−1.24470
$857$	−16912.0	−0.674098	−0.337049	−	0.941487i	$-0.609429\pi$
$857$	−16912.0	−0.674098	−0.337049	+	0.941487i	$0.609429\pi$
$858$	108129.	4.30242
$859$	35911.2	1.42640	0.713199	−	0.700962i	$-0.247246\pi$
$859$	35911.2	1.42640	0.713199	+	0.700962i	$0.247246\pi$
$860$	0	0
$861$	16367.8	0.647866
$862$	−6759.30	−0.267080
$863$	−12877.6	−0.507949	−0.253975	−	0.967211i	$-0.581738\pi$
$863$	−12877.6	−0.507949	−0.253975	+	0.967211i	$0.581738\pi$
$864$	3637.05	0.143212
$865$	9778.15	0.384355
$866$	54487.8	2.13807
$867$	−25019.0	−0.980033
$868$	12120.2	0.473949
$869$	−14457.3	−0.564363
$870$	116984.	4.55876
$871$	−21058.6	−0.819222
$872$	−42949.5	−1.66795
$873$	4772.53	0.185024
$874$	−29004.6	−1.12254
$875$	7815.29	0.301949
$876$	−18171.2	−0.700853
$877$	−40122.1	−1.54484	−0.772421	−	0.635110i	$-0.780955\pi$
$877$	−40122.1	−1.54484	−0.772421	+	0.635110i	$0.780955\pi$
$878$	20993.4	0.806939
$879$	−11648.7	−0.446988
$880$	44752.0	1.71431
$881$	−32354.6	−1.23729	−0.618646	−	0.785670i	$-0.712318\pi$
$881$	−32354.6	−1.23729	−0.618646	+	0.785670i	$0.712318\pi$
$882$	−4794.65	−0.183043
$883$	2575.95	0.0981741	0.0490870	−	0.998795i	$-0.484369\pi$
$883$	2575.95	0.0981741	0.0490870	+	0.998795i	$0.484369\pi$
$884$	−25026.8	−0.952199
$885$	−5804.46	−0.220469
$886$	−50203.2	−1.90362
$887$	−171.714	−0.00650009	−0.00325004	−	0.999995i	$-0.501035\pi$
$887$	−171.714	−0.00650009	−0.00325004	+	0.999995i	$0.501035\pi$
$888$	−31190.5	−1.17870
$889$	−7681.75	−0.289806
$890$	−58585.4	−2.20650
$891$	48276.6	1.81518
$892$	−46611.8	−1.74964
$893$	−27947.5	−1.04729
$894$	9004.85	0.336876
$895$	−22121.1	−0.826175
$896$	−34185.5	−1.27462
$897$	19592.1	0.729277
$898$	−42450.9	−1.57751
$899$	−17121.3	−0.635181
$900$	7648.50	0.283278
$901$	16671.5	0.616437
$902$	−59600.8	−2.20010
$903$	0	0
$904$	−32674.4	−1.20214
$905$	−37793.7	−1.38818
$906$	24069.6	0.882626
$907$	6826.29	0.249904	0.124952	−	0.992163i	$-0.460122\pi$
$907$	6826.29	0.249904	0.124952	+	0.992163i	$0.460122\pi$
$908$	31445.2	1.14928
$909$	−3748.73	−0.136785
$910$	63521.1	2.31396
$911$	−2888.99	−0.105067	−0.0525337	−	0.998619i	$-0.516730\pi$
$911$	−2888.99	−0.105067	−0.0525337	+	0.998619i	$0.516730\pi$
$912$	37510.5	1.36195
$913$	−37696.0	−1.36643
$914$	−54446.4	−1.97038
$915$	27193.8	0.982512
$916$	−71689.6	−2.58591
$917$	11907.5	0.428811
$918$	−13778.7	−0.495386
$919$	16257.1	0.583539	0.291770	−	0.956489i	$-0.405756\pi$
$919$	16257.1	0.583539	0.291770	+	0.956489i	$0.405756\pi$
$920$	26626.7	0.954193
$921$	−55574.5	−1.98832
$922$	−70572.1	−2.52079
$923$	77888.0	2.77759
$924$	−66768.5	−2.37719
$925$	−12376.0	−0.439915
$926$	−23954.6	−0.850103
$927$	448.996	0.0159083
$928$	8716.60	0.308337
$929$	−46132.6	−1.62924	−0.814618	−	0.579998i	$-0.803053\pi$
$929$	−46132.6	−1.62924	−0.814618	+	0.579998i	$0.803053\pi$
$930$	−23707.9	−0.835926
$931$	−20254.2	−0.713002
$932$	−14351.7	−0.504404
$933$	−47774.7	−1.67639
$934$	−29675.5	−1.03963
$935$	19144.4	0.669615
$936$	−14785.5	−0.516323
$937$	14113.4	0.492066	0.246033	−	0.969262i	$-0.420873\pi$
$937$	14113.4	0.492066	0.246033	+	0.969262i	$0.420873\pi$
$938$	19669.9	0.684696
$939$	52195.0	1.81397
$940$	52647.2	1.82677
$941$	−27642.6	−0.957622	−0.478811	−	0.877918i	$-0.658932\pi$
$941$	−27642.6	−0.957622	−0.478811	+	0.877918i	$0.658932\pi$
$942$	16070.8	0.555854
$943$	−10799.1	−0.372925
$944$	3830.11	0.132054
$945$	23119.3	0.795841
$946$	0	0
$947$	15987.3	0.548592	0.274296	−	0.961645i	$-0.411555\pi$
$947$	15987.3	0.548592	0.274296	+	0.961645i	$0.411555\pi$
$948$	22827.6	0.782074
$949$	13938.0	0.476763
$950$	48874.3	1.66915
$951$	22015.8	0.750694
$952$	11391.9	0.387831
$953$	−44654.8	−1.51785	−0.758925	−	0.651178i	$-0.774275\pi$
$953$	−44654.8	−1.51785	−0.758925	+	0.651178i	$0.774275\pi$
$954$	20210.9	0.685904
$955$	−56996.0	−1.93125
$956$	23044.8	0.779626
$957$	94318.6	3.18588
$958$	−49373.6	−1.66512
$959$	−27970.2	−0.941820
$960$	48296.4	1.62371
$961$	−26321.2	−0.883529
$962$	49093.2	1.64535
$963$	−4923.19	−0.164743
$964$	78796.2	2.63263
$965$	1156.75	0.0385876
$966$	−18300.1	−0.609521
$967$	−27849.4	−0.926140	−0.463070	−	0.886322i	$-0.653252\pi$
$967$	−27849.4	−0.926140	−0.463070	+	0.886322i	$0.653252\pi$
$968$	69307.2	2.30126
$969$	16046.6	0.531983
$970$	−57449.3	−1.90164
$971$	25393.9	0.839268	0.419634	−	0.907694i	$-0.362159\pi$
$971$	25393.9	0.839268	0.419634	+	0.907694i	$0.362159\pi$
$972$	−25129.6	−0.829250
$973$	13742.0	0.452772
$974$	67456.9	2.21916
$975$	−33013.7	−1.08440
$976$	−17944.0	−0.588497
$977$	−45241.2	−1.48147	−0.740734	−	0.671798i	$-0.765522\pi$
$977$	−45241.2	−1.48147	−0.740734	+	0.671798i	$0.765522\pi$
$978$	50213.3	1.64176
$979$	−47234.7	−1.54201
$980$	38154.6	1.24368
$981$	−6783.14	−0.220764
$982$	−71218.7	−2.31434
$983$	−28920.1	−0.938361	−0.469180	−	0.883102i	$-0.655451\pi$
$983$	−28920.1	−0.938361	−0.469180	+	0.883102i	$0.655451\pi$
$984$	45860.9	1.48577
$985$	−40156.4	−1.29897
$986$	−33022.1	−1.06657
$987$	−17633.2	−0.568663
$988$	−128167.	−4.12705
$989$	0	0
$990$	23208.8	0.745075
$991$	−483.939	−0.0155124	−0.00775622	−	0.999970i	$-0.502469\pi$
$991$	−483.939	−0.0155124	−0.00775622	+	0.999970i	$0.502469\pi$
$992$	−1766.50	−0.0565388
$993$	−13527.3	−0.432302
$994$	−72751.8	−2.32148
$995$	32081.0	1.02215
$996$	59520.6	1.89356
$997$	−35131.0	−1.11596	−0.557978	−	0.829856i	$-0.688423\pi$
$997$	−35131.0	−1.11596	−0.557978	+	0.829856i	$0.688423\pi$
$998$	−74738.6	−2.37055
$999$	17868.1	0.565886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.46	yes	50
43.42	odd	2		1849.4.a.i.1.5	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.5	✓	50	43.42	odd	2
1849.4.a.j.1.46	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)

\(f(q)\)	\(=\)	\(q+4.85844 q^{2} +5.73018 q^{3} +15.6044 q^{4} -14.4569 q^{5} +27.8397 q^{6} +13.1859 q^{7} +36.9457 q^{8} +5.83494 q^{9} +O(q^{10})\)	\(q+4.85844 q^{2} +5.73018 q^{3} +15.6044 q^{4} -14.4569 q^{5} +27.8397 q^{6} +13.1859 q^{7} +36.9457 q^{8} +5.83494 q^{9} -70.2380 q^{10} -56.6297 q^{11} +89.4162 q^{12} -68.5859 q^{13} +64.0631 q^{14} -82.8407 q^{15} +54.6629 q^{16} +23.3842 q^{17} +28.3487 q^{18} +119.755 q^{19} -225.592 q^{20} +75.5578 q^{21} -275.132 q^{22} -49.8515 q^{23} +211.705 q^{24} +84.0024 q^{25} -333.220 q^{26} -121.280 q^{27} +205.759 q^{28} -290.660 q^{29} -402.477 q^{30} +58.9050 q^{31} -29.9890 q^{32} -324.498 q^{33} +113.611 q^{34} -190.628 q^{35} +91.0510 q^{36} -147.330 q^{37} +581.820 q^{38} -393.009 q^{39} -534.121 q^{40} +216.626 q^{41} +367.093 q^{42} -883.674 q^{44} -84.3552 q^{45} -242.201 q^{46} -233.374 q^{47} +313.228 q^{48} -169.131 q^{49} +408.121 q^{50} +133.996 q^{51} -1070.24 q^{52} +712.940 q^{53} -589.229 q^{54} +818.690 q^{55} +487.164 q^{56} +686.215 q^{57} -1412.15 q^{58} +70.0677 q^{59} -1292.68 q^{60} -328.266 q^{61} +286.186 q^{62} +76.9392 q^{63} -583.003 q^{64} +991.541 q^{65} -1576.55 q^{66} +307.039 q^{67} +364.898 q^{68} -285.658 q^{69} -926.155 q^{70} -1135.63 q^{71} +215.576 q^{72} -203.220 q^{73} -715.792 q^{74} +481.349 q^{75} +1868.70 q^{76} -746.715 q^{77} -1909.41 q^{78} +255.296 q^{79} -790.257 q^{80} -852.497 q^{81} +1052.47 q^{82} +665.658 q^{83} +1179.04 q^{84} -338.064 q^{85} -1665.53 q^{87} -2092.22 q^{88} +834.098 q^{89} -409.835 q^{90} -904.369 q^{91} -777.905 q^{92} +337.536 q^{93} -1133.83 q^{94} -1731.28 q^{95} -171.842 q^{96} +817.923 q^{97} -821.713 q^{98} -330.431 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) \) 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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