LMFDB - Embedded newform 1849.4.a.h.1.11

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:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-1.56619 q^{2} -5.42418 q^{3} -5.54704 q^{4} +8.12470 q^{5} +8.49531 q^{6} -30.7370 q^{7} +21.2173 q^{8} +2.42171 q^{9} +O(q^{10})$	\(q-1.56619 q^{2} -5.42418 q^{3} -5.54704 q^{4} +8.12470 q^{5} +8.49531 q^{6} -30.7370 q^{7} +21.2173 q^{8} +2.42171 q^{9} -12.7249 q^{10} +9.06614 q^{11} +30.0881 q^{12} -16.1006 q^{13} +48.1400 q^{14} -44.0698 q^{15} +11.1459 q^{16} -13.4280 q^{17} -3.79286 q^{18} -76.7126 q^{19} -45.0680 q^{20} +166.723 q^{21} -14.1993 q^{22} -38.4314 q^{23} -115.086 q^{24} -58.9893 q^{25} +25.2166 q^{26} +133.317 q^{27} +170.499 q^{28} +251.400 q^{29} +69.0219 q^{30} -120.743 q^{31} -187.195 q^{32} -49.1764 q^{33} +21.0308 q^{34} -249.729 q^{35} -13.4333 q^{36} -391.816 q^{37} +120.147 q^{38} +87.3323 q^{39} +172.384 q^{40} -338.720 q^{41} -261.120 q^{42} -50.2902 q^{44} +19.6756 q^{45} +60.1911 q^{46} +120.370 q^{47} -60.4574 q^{48} +601.761 q^{49} +92.3886 q^{50} +72.8358 q^{51} +89.3103 q^{52} -480.889 q^{53} -208.800 q^{54} +73.6597 q^{55} -652.155 q^{56} +416.103 q^{57} -393.741 q^{58} +357.983 q^{59} +244.457 q^{60} -592.360 q^{61} +189.106 q^{62} -74.4359 q^{63} +204.016 q^{64} -130.812 q^{65} +77.0197 q^{66} -672.610 q^{67} +74.4856 q^{68} +208.459 q^{69} +391.123 q^{70} +417.963 q^{71} +51.3820 q^{72} -922.640 q^{73} +613.660 q^{74} +319.968 q^{75} +425.528 q^{76} -278.666 q^{77} -136.779 q^{78} -107.902 q^{79} +90.5571 q^{80} -788.521 q^{81} +530.501 q^{82} -1003.60 q^{83} -924.817 q^{84} -109.098 q^{85} -1363.64 q^{87} +192.359 q^{88} -188.544 q^{89} -30.8158 q^{90} +494.882 q^{91} +213.181 q^{92} +654.930 q^{93} -188.522 q^{94} -623.267 q^{95} +1015.38 q^{96} +481.540 q^{97} -942.474 q^{98} +21.9555 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 6 q^{2} + 2 q^{3} + 114 q^{4} + 27 q^{5} + 8 q^{6} + 48 q^{7} + 90 q^{8} + 216 q^{9}+O(q^{10}) $ 30 * q + 6 * q^2 + 2 * q^3 + 114 * q^4 + 27 * q^5 + 8 * q^6 + 48 * q^7 + 90 * q^8 + 216 * q^9	\( 30 q + 6 q^{2} + 2 q^{3} + 114 q^{4} + 27 q^{5} + 8 q^{6} + 48 q^{7} + 90 q^{8} + 216 q^{9} - 27 q^{10} + 80 q^{11} - 36 q^{12} - 13 q^{13} + 36 q^{14} + 16 q^{15} + 318 q^{16} + 66 q^{17} + 80 q^{18} + 254 q^{19} + 312 q^{20} - 548 q^{21} + 305 q^{22} - 105 q^{23} + 123 q^{24} + 523 q^{25} + 549 q^{26} - 10 q^{27} + 578 q^{28} + 793 q^{29} + 1560 q^{30} - 359 q^{31} + 676 q^{32} + 208 q^{33} + 1007 q^{34} - 514 q^{35} + 776 q^{36} + 510 q^{37} - 2066 q^{38} + 898 q^{39} - 1248 q^{40} - 270 q^{41} - 915 q^{42} + 3256 q^{44} + 807 q^{45} + 1960 q^{46} + 1421 q^{47} - 632 q^{48} + 386 q^{49} - 141 q^{50} + 209 q^{51} + 2825 q^{52} - 21 q^{53} + 2368 q^{54} + 2258 q^{55} + 2521 q^{56} - 1723 q^{57} - 347 q^{58} + 1752 q^{59} + 2711 q^{60} + 1759 q^{61} + 395 q^{62} + 2204 q^{63} + 222 q^{64} + 1151 q^{65} + 160 q^{66} - 3001 q^{67} + 1921 q^{68} + 1660 q^{69} + 1597 q^{70} + 727 q^{71} + 9100 q^{72} + 4623 q^{73} - 2649 q^{74} + 1027 q^{75} + 874 q^{76} + 3556 q^{77} - 4979 q^{78} + 546 q^{79} + 5809 q^{80} - 410 q^{81} - 4397 q^{82} - 492 q^{83} - 10611 q^{84} - 1723 q^{85} + 5937 q^{87} + 3974 q^{88} + 5218 q^{89} + 10492 q^{90} + 1104 q^{91} + 1060 q^{92} + 1997 q^{93} - 2134 q^{94} + 6346 q^{95} - 11984 q^{96} + 2590 q^{97} + 6270 q^{98} - 2693 q^{99}+O(q^{100}) \) 30 * q + 6 * q^2 + 2 * q^3 + 114 * q^4 + 27 * q^5 + 8 * q^6 + 48 * q^7 + 90 * q^8 + 216 * q^9 - 27 * q^10 + 80 * q^11 - 36 * q^12 - 13 * q^13 + 36 * q^14 + 16 * q^15 + 318 * q^16 + 66 * q^17 + 80 * q^18 + 254 * q^19 + 312 * q^20 - 548 * q^21 + 305 * q^22 - 105 * q^23 + 123 * q^24 + 523 * q^25 + 549 * q^26 - 10 * q^27 + 578 * q^28 + 793 * q^29 + 1560 * q^30 - 359 * q^31 + 676 * q^32 + 208 * q^33 + 1007 * q^34 - 514 * q^35 + 776 * q^36 + 510 * q^37 - 2066 * q^38 + 898 * q^39 - 1248 * q^40 - 270 * q^41 - 915 * q^42 + 3256 * q^44 + 807 * q^45 + 1960 * q^46 + 1421 * q^47 - 632 * q^48 + 386 * q^49 - 141 * q^50 + 209 * q^51 + 2825 * q^52 - 21 * q^53 + 2368 * q^54 + 2258 * q^55 + 2521 * q^56 - 1723 * q^57 - 347 * q^58 + 1752 * q^59 + 2711 * q^60 + 1759 * q^61 + 395 * q^62 + 2204 * q^63 + 222 * q^64 + 1151 * q^65 + 160 * q^66 - 3001 * q^67 + 1921 * q^68 + 1660 * q^69 + 1597 * q^70 + 727 * q^71 + 9100 * q^72 + 4623 * q^73 - 2649 * q^74 + 1027 * q^75 + 874 * q^76 + 3556 * q^77 - 4979 * q^78 + 546 * q^79 + 5809 * q^80 - 410 * q^81 - 4397 * q^82 - 492 * q^83 - 10611 * q^84 - 1723 * q^85 + 5937 * q^87 + 3974 * q^88 + 5218 * q^89 + 10492 * q^90 + 1104 * q^91 + 1060 * q^92 + 1997 * q^93 - 2134 * q^94 + 6346 * q^95 - 11984 * q^96 + 2590 * q^97 + 6270 * q^98 - 2693 * 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 $

$2$	−1.56619	−0.553733	−0.276867	−	0.960908i	$-0.589296\pi$
$2$	−1.56619	−0.553733	−0.276867	+	0.960908i	$0.589296\pi$
$3$	−5.42418	−1.04388	−0.521942	−	0.852981i	$-0.674792\pi$
$3$	−5.42418	−1.04388	−0.521942	+	0.852981i	$0.674792\pi$
$4$	−5.54704	−0.693380
$5$	8.12470	0.726695	0.363348	−	0.931654i	$-0.381634\pi$
$5$	8.12470	0.726695	0.363348	+	0.931654i	$0.381634\pi$
$6$	8.49531	0.578033
$7$	−30.7370	−1.65964	−0.829820	−	0.558031i	$-0.811557\pi$
$7$	−30.7370	−1.65964	−0.829820	+	0.558031i	$0.811557\pi$
$8$	21.2173	0.937680
$9$	2.42171	0.0896928
$10$	−12.7249	−0.402395
$11$	9.06614	0.248504	0.124252	−	0.992251i	$-0.460347\pi$
$11$	9.06614	0.248504	0.124252	+	0.992251i	$0.460347\pi$
$12$	30.0881	0.723808
$13$	−16.1006	−0.343499	−0.171750	−	0.985141i	$-0.554942\pi$
$13$	−16.1006	−0.343499	−0.171750	+	0.985141i	$0.554942\pi$
$14$	48.1400	0.918998
$15$	−44.0698	−0.758585
$16$	11.1459	0.174155
$17$	−13.4280	−0.191574	−0.0957872	−	0.995402i	$-0.530537\pi$
$17$	−13.4280	−0.191574	−0.0957872	+	0.995402i	$0.530537\pi$
$18$	−3.79286	−0.0496659
$19$	−76.7126	−0.926267	−0.463134	−	0.886288i	$-0.653275\pi$
$19$	−76.7126	−0.926267	−0.463134	+	0.886288i	$0.653275\pi$
$20$	−45.0680	−0.503876
$21$	166.723	1.73247
$22$	−14.1993	−0.137605
$23$	−38.4314	−0.348413	−0.174207	−	0.984709i	$-0.555736\pi$
$23$	−38.4314	−0.348413	−0.174207	+	0.984709i	$0.555736\pi$
$24$	−115.086	−0.978829
$25$	−58.9893	−0.471914
$26$	25.2166	0.190207
$27$	133.317	0.950255
$28$	170.499	1.15076
$29$	251.400	1.60979	0.804893	−	0.593420i	$-0.202223\pi$
$29$	251.400	1.60979	0.804893	+	0.593420i	$0.202223\pi$
$30$	69.0219	0.420054
$31$	−120.743	−0.699549	−0.349775	−	0.936834i	$-0.613742\pi$
$31$	−120.743	−0.699549	−0.349775	+	0.936834i	$0.613742\pi$
$32$	−187.195	−1.03412
$33$	−49.1764	−0.259409
$34$	21.0308	0.106081
$35$	−249.729	−1.20605
$36$	−13.4333	−0.0621911
$37$	−391.816	−1.74092	−0.870462	−	0.492235i	$-0.836180\pi$
$37$	−391.816	−1.74092	−0.870462	+	0.492235i	$0.836180\pi$
$38$	120.147	0.512905
$39$	87.3323	0.358573
$40$	172.384	0.681408
$41$	−338.720	−1.29022	−0.645112	−	0.764088i	$-0.723189\pi$
$41$	−338.720	−1.29022	−0.645112	+	0.764088i	$0.723189\pi$
$42$	−261.120	−0.959327
$43$	0	0
$43$	0	0
$44$	−50.2902	−0.172308
$45$	19.6756	0.0651793
$46$	60.1911	0.192928
$47$	120.370	0.373568	0.186784	−	0.982401i	$-0.440194\pi$
$47$	120.370	0.373568	0.186784	+	0.982401i	$0.440194\pi$
$48$	−60.4574	−0.181797
$49$	601.761	1.75440
$50$	92.3886	0.261315
$51$	72.8358	0.199981
$52$	89.3103	0.238175
$53$	−480.889	−1.24633	−0.623163	−	0.782092i	$-0.714153\pi$
$53$	−480.889	−1.24633	−0.623163	+	0.782092i	$0.714153\pi$
$54$	−208.800	−0.526188
$55$	73.6597	0.180587
$56$	−652.155	−1.55621
$57$	416.103	0.966915
$58$	−393.741	−0.891392
$59$	357.983	0.789923	0.394961	−	0.918698i	$-0.370758\pi$
$59$	357.983	0.789923	0.394961	+	0.918698i	$0.370758\pi$
$60$	244.457	0.525987
$61$	−592.360	−1.24334	−0.621672	−	0.783278i	$-0.713546\pi$
$61$	−592.360	−1.24334	−0.621672	+	0.783278i	$0.713546\pi$
$62$	189.106	0.387364
$63$	−74.4359	−0.148858
$64$	204.016	0.398469
$65$	−130.812	−0.249619
$66$	77.0197	0.143644
$67$	−672.610	−1.22645	−0.613227	−	0.789907i	$-0.710129\pi$
$67$	−672.610	−1.22645	−0.613227	+	0.789907i	$0.710129\pi$
$68$	74.4856	0.132834
$69$	208.459	0.363703
$70$	391.123	0.667831
$71$	417.963	0.698635	0.349318	−	0.937004i	$-0.386413\pi$
$71$	417.963	0.698635	0.349318	+	0.937004i	$0.386413\pi$
$72$	51.3820	0.0841032
$73$	−922.640	−1.47927	−0.739636	−	0.673007i	$-0.765002\pi$
$73$	−922.640	−1.47927	−0.739636	+	0.673007i	$0.765002\pi$
$74$	613.660	0.964008
$75$	319.968	0.492623
$76$	425.528	0.642255
$77$	−278.666	−0.412427
$78$	−136.779	−0.198554
$79$	−107.902	−0.153670	−0.0768350	−	0.997044i	$-0.524481\pi$
$79$	−107.902	−0.153670	−0.0768350	+	0.997044i	$0.524481\pi$
$80$	90.5571	0.126557
$81$	−788.521	−1.08165
$82$	530.501	0.714440
$83$	−1003.60	−1.32722	−0.663609	−	0.748080i	$-0.730976\pi$
$83$	−1003.60	−1.32722	−0.663609	+	0.748080i	$0.730976\pi$
$84$	−924.817	−1.20126
$85$	−109.098	−0.139216
$86$	0	0
$87$	−1363.64	−1.68043
$88$	192.359	0.233017
$89$	−188.544	−0.224558	−0.112279	−	0.993677i	$-0.535815\pi$
$89$	−188.544	−0.224558	−0.112279	+	0.993677i	$0.535815\pi$
$90$	−30.8158	−0.0360919
$91$	494.882	0.570085
$92$	213.181	0.241583
$93$	654.930	0.730248
$94$	−188.522	−0.206857
$95$	−623.267	−0.673114
$96$	1015.38	1.07950
$97$	481.540	0.504051	0.252026	−	0.967721i	$-0.418903\pi$
$97$	481.540	0.504051	0.252026	+	0.967721i	$0.418903\pi$
$98$	−942.474	−0.971472
$99$	21.9555	0.0222890
$100$	327.216	0.327216
$101$	−1370.30	−1.35000	−0.674999	−	0.737818i	$-0.735856\pi$
$101$	−1370.30	−1.35000	−0.674999	+	0.737818i	$0.735856\pi$
$102$	−114.075	−0.110736
$103$	−823.195	−0.787494	−0.393747	−	0.919219i	$-0.628821\pi$
$103$	−823.195	−0.787494	−0.393747	+	0.919219i	$0.628821\pi$
$104$	−341.610	−0.322092
$105$	1354.57	1.25898
$106$	753.166	0.690132
$107$	−1936.29	−1.74943	−0.874713	−	0.484640i	$-0.838950\pi$
$107$	−1936.29	−1.74943	−0.874713	+	0.484640i	$0.838950\pi$
$108$	−739.515	−0.658887
$109$	−1765.66	−1.55155	−0.775777	−	0.631007i	$-0.782642\pi$
$109$	−1765.66	−1.55155	−0.775777	+	0.631007i	$0.782642\pi$
$110$	−115.365	−0.0999969
$111$	2125.28	1.81732
$112$	−342.591	−0.289034
$113$	1935.29	1.61112	0.805559	−	0.592515i	$-0.201865\pi$
$113$	1935.29	1.61112	0.805559	+	0.592515i	$0.201865\pi$
$114$	−651.698	−0.535413
$115$	−312.244	−0.253190
$116$	−1394.52	−1.11619
$117$	−38.9908	−0.0308094
$118$	−560.671	−0.437406
$119$	412.736	0.317945
$120$	−935.042	−0.711310
$121$	−1248.81	−0.938246
$122$	927.751	0.688481
$123$	1837.28	1.34684
$124$	669.764	0.485053
$125$	−1494.86	−1.06963
$126$	116.581	0.0824275
$127$	216.846	0.151511	0.0757556	−	0.997126i	$-0.475863\pi$
$127$	216.846	0.151511	0.0757556	+	0.997126i	$0.475863\pi$
$128$	1178.03	0.813470
$129$	0	0
$130$	204.877	0.138222
$131$	−629.545	−0.419875	−0.209937	−	0.977715i	$-0.567326\pi$
$131$	−629.545	−0.419875	−0.209937	+	0.977715i	$0.567326\pi$
$132$	272.783	0.179869
$133$	2357.91	1.53727
$134$	1053.44	0.679128
$135$	1083.16	0.690545
$136$	−284.906	−0.179636
$137$	−1799.38	−1.12212	−0.561062	−	0.827774i	$-0.689607\pi$
$137$	−1799.38	−1.12212	−0.561062	+	0.827774i	$0.689607\pi$
$138$	−326.487	−0.201394
$139$	1364.33	0.832524	0.416262	−	0.909245i	$-0.363340\pi$
$139$	1364.33	0.832524	0.416262	+	0.909245i	$0.363340\pi$
$140$	1385.25	0.836252
$141$	−652.906	−0.389962
$142$	−654.611	−0.386857
$143$	−145.970	−0.0853610
$144$	26.9921	0.0156204
$145$	2042.55	1.16982
$146$	1445.03	0.819122
$147$	−3264.06	−1.83139
$148$	2173.42	1.20712
$149$	−61.1081	−0.0335984	−0.0167992	−	0.999859i	$-0.505348\pi$
$149$	−61.1081	−0.0335984	−0.0167992	+	0.999859i	$0.505348\pi$
$150$	−501.132	−0.272782
$151$	1992.02	1.07357	0.536784	−	0.843720i	$-0.319639\pi$
$151$	1992.02	1.07357	0.536784	+	0.843720i	$0.319639\pi$
$152$	−1627.63	−0.868543
$153$	−32.5186	−0.0171828
$154$	436.445	0.228375
$155$	−980.998	−0.508359
$156$	−484.435	−0.248627
$157$	517.016	0.262818	0.131409	−	0.991328i	$-0.458050\pi$
$157$	517.016	0.262818	0.131409	+	0.991328i	$0.458050\pi$
$158$	168.996	0.0850922
$159$	2608.43	1.30102
$160$	−1520.90	−0.751487
$161$	1181.27	0.578241
$162$	1234.98	0.598944
$163$	1550.46	0.745041	0.372521	−	0.928024i	$-0.378494\pi$
$163$	1550.46	0.745041	0.372521	+	0.928024i	$0.378494\pi$
$164$	1878.89	0.894615
$165$	−399.543	−0.188512
$166$	1571.83	0.734924
$167$	−2645.09	−1.22565	−0.612825	−	0.790219i	$-0.709967\pi$
$167$	−2645.09	−1.22565	−0.612825	+	0.790219i	$0.709967\pi$
$168$	3537.40	1.62450
$169$	−1937.77	−0.882008
$170$	170.869	0.0770887
$171$	−185.775	−0.0830795
$172$	0	0
$173$	−2035.37	−0.894488	−0.447244	−	0.894412i	$-0.647594\pi$
$173$	−2035.37	−0.894488	−0.447244	+	0.894412i	$0.647594\pi$
$174$	2135.72	0.930509
$175$	1813.15	0.783208
$176$	101.050	0.0432782
$177$	−1941.76	−0.824587
$178$	295.297	0.124345
$179$	−3581.88	−1.49566	−0.747828	−	0.663893i	$-0.768903\pi$
$179$	−3581.88	−1.49566	−0.747828	+	0.663893i	$0.768903\pi$
$180$	−109.141	−0.0451940
$181$	142.547	0.0585384	0.0292692	−	0.999572i	$-0.490682\pi$
$181$	142.547	0.0585384	0.0292692	+	0.999572i	$0.490682\pi$
$182$	−775.081	−0.315675
$183$	3213.07	1.29791
$184$	−815.411	−0.326700
$185$	−3183.39	−1.26512
$186$	−1025.75	−0.404362
$187$	−121.740	−0.0476070
$188$	−667.694	−0.259025
$189$	−4097.76	−1.57708
$190$	976.157	0.372726
$191$	613.948	0.232585	0.116292	−	0.993215i	$-0.462899\pi$
$191$	613.948	0.232585	0.116292	+	0.993215i	$0.462899\pi$
$192$	−1106.62	−0.415956
$193$	−251.603	−0.0938381	−0.0469190	−	0.998899i	$-0.514940\pi$
$193$	−251.603	−0.0938381	−0.0469190	+	0.998899i	$0.514940\pi$
$194$	−754.185	−0.279110
$195$	709.548	0.260573
$196$	−3337.99	−1.21647
$197$	−2046.10	−0.739992	−0.369996	−	0.929033i	$-0.620641\pi$
$197$	−2046.10	−0.739992	−0.369996	+	0.929033i	$0.620641\pi$
$198$	−34.3866	−0.0123422
$199$	602.944	0.214782	0.107391	−	0.994217i	$-0.465750\pi$
$199$	602.944	0.214782	0.107391	+	0.994217i	$0.465750\pi$
$200$	−1251.59	−0.442505
$201$	3648.36	1.28027
$202$	2146.15	0.747539
$203$	−7727.27	−2.67167
$204$	−404.023	−0.138663
$205$	−2752.00	−0.937599
$206$	1289.28	0.436061
$207$	−93.0696	−0.0312502
$208$	−179.455	−0.0598220
$209$	−695.488	−0.230181
$210$	−2121.52	−0.697138
$211$	4349.61	1.41914	0.709572	−	0.704633i	$-0.248888\pi$
$211$	4349.61	1.41914	0.709572	+	0.704633i	$0.248888\pi$
$212$	2667.51	0.864176
$213$	−2267.11	−0.729294
$214$	3032.61	0.968716
$215$	0	0
$216$	2828.63	0.891035
$217$	3711.26	1.16100
$218$	2765.36	0.859147
$219$	5004.56	1.54419
$220$	−408.593	−0.125215
$221$	216.198	0.0658057
$222$	−3328.60	−1.00631
$223$	−2364.75	−0.710113	−0.355057	−	0.934845i	$-0.615538\pi$
$223$	−2364.75	−0.710113	−0.355057	+	0.934845i	$0.615538\pi$
$224$	5753.80	1.71626
$225$	−142.855	−0.0423273
$226$	−3031.03	−0.892130
$227$	6612.47	1.93341	0.966707	−	0.255886i	$-0.0823671\pi$
$227$	6612.47	1.93341	0.966707	+	0.255886i	$0.0823671\pi$
$228$	−2308.14	−0.670439
$229$	2165.70	0.624950	0.312475	−	0.949926i	$-0.398842\pi$
$229$	2165.70	0.624950	0.312475	+	0.949926i	$0.398842\pi$
$230$	489.034	0.140200
$231$	1511.53	0.430526
$232$	5334.02	1.50946
$233$	−2568.88	−0.722286	−0.361143	−	0.932510i	$-0.617613\pi$
$233$	−2568.88	−0.722286	−0.361143	+	0.932510i	$0.617613\pi$
$234$	61.0671	0.0170602
$235$	977.966	0.271470
$236$	−1985.75	−0.547716
$237$	585.280	0.160414
$238$	−646.424	−0.176057
$239$	952.472	0.257784	0.128892	−	0.991659i	$-0.458858\pi$
$239$	952.472	0.257784	0.128892	+	0.991659i	$0.458858\pi$
$240$	−491.198	−0.132111
$241$	2756.95	0.736892	0.368446	−	0.929649i	$-0.379890\pi$
$241$	2756.95	0.736892	0.368446	+	0.929649i	$0.379890\pi$
$242$	1955.87	0.519538
$243$	677.520	0.178860
$244$	3285.85	0.862109
$245$	4889.13	1.27492
$246$	−2877.53	−0.745792
$247$	1235.12	0.318172
$248$	−2561.83	−0.655954
$249$	5443.69	1.38546
$250$	2341.24	0.592291
$251$	−725.383	−0.182413	−0.0912067	−	0.995832i	$-0.529072\pi$
$251$	−725.383	−0.182413	−0.0912067	+	0.995832i	$0.529072\pi$
$252$	412.898	0.103215
$253$	−348.425	−0.0865822
$254$	−339.622	−0.0838968
$255$	591.769	0.145326
$256$	−3477.15	−0.848915
$257$	−3866.44	−0.938452	−0.469226	−	0.883078i	$-0.655467\pi$
$257$	−3866.44	−0.938452	−0.469226	+	0.883078i	$0.655467\pi$
$258$	0	0
$259$	12043.2	2.88931
$260$	725.620	0.173081
$261$	608.816	0.144386
$262$	985.989	0.232499
$263$	1275.72	0.299103	0.149552	−	0.988754i	$-0.452217\pi$
$263$	1275.72	0.299103	0.149552	+	0.988754i	$0.452217\pi$
$264$	−1043.39	−0.243243
$265$	−3907.08	−0.905699
$266$	−3692.95	−0.851238
$267$	1022.70	0.234412
$268$	3730.99	0.850398
$269$	−702.374	−0.159199	−0.0795994	−	0.996827i	$-0.525364\pi$
$269$	−702.374	−0.159199	−0.0795994	+	0.996827i	$0.525364\pi$
$270$	−1696.44	−0.382378
$271$	6411.84	1.43724	0.718619	−	0.695404i	$-0.244775\pi$
$271$	6411.84	1.43724	0.718619	+	0.695404i	$0.244775\pi$
$272$	−149.667	−0.0333636
$273$	−2684.33	−0.595102
$274$	2818.17	0.621358
$275$	−534.805	−0.117273
$276$	−1156.33	−0.252184
$277$	2089.16	0.453161	0.226580	−	0.973992i	$-0.427245\pi$
$277$	2089.16	0.453161	0.226580	+	0.973992i	$0.427245\pi$
$278$	−2136.80	−0.460996
$279$	−292.403	−0.0627445
$280$	−5298.56	−1.13089
$281$	−3390.64	−0.719817	−0.359908	−	0.932988i	$-0.617192\pi$
$281$	−3390.64	−0.719817	−0.359908	+	0.932988i	$0.617192\pi$
$282$	1022.58	0.215935
$283$	6836.92	1.43609	0.718044	−	0.695998i	$-0.245038\pi$
$283$	6836.92	1.43609	0.718044	+	0.695998i	$0.245038\pi$
$284$	−2318.46	−0.484419
$285$	3380.71	0.702653
$286$	228.617	0.0472672
$287$	10411.2	2.14131
$288$	−453.331	−0.0927527
$289$	−4732.69	−0.963299
$290$	−3199.03	−0.647770
$291$	−2611.96	−0.526171
$292$	5117.92	1.02570
$293$	6814.59	1.35875	0.679373	−	0.733793i	$-0.262252\pi$
$293$	6814.59	1.35875	0.679373	+	0.733793i	$0.262252\pi$
$294$	5112.15	1.01410
$295$	2908.51	0.574033
$296$	−8313.28	−1.63243
$297$	1208.67	0.236142
$298$	95.7071	0.0186046
$299$	618.767	0.119680
$300$	−1774.88	−0.341575
$301$	0	0
$302$	−3119.90	−0.594470
$303$	7432.75	1.40924
$304$	−855.032	−0.161314
$305$	−4812.75	−0.903532
$306$	50.9305	0.00951471
$307$	1604.28	0.298244	0.149122	−	0.988819i	$-0.452355\pi$
$307$	1604.28	0.298244	0.149122	+	0.988819i	$0.452355\pi$
$308$	1545.77	0.285969
$309$	4465.16	0.822052
$310$	1536.43	0.281495
$311$	−2318.26	−0.422690	−0.211345	−	0.977412i	$-0.567784\pi$
$311$	−2318.26	−0.422690	−0.211345	+	0.977412i	$0.567784\pi$
$312$	1852.95	0.336227
$313$	−4380.98	−0.791142	−0.395571	−	0.918435i	$-0.629453\pi$
$313$	−4380.98	−0.791142	−0.395571	+	0.918435i	$0.629453\pi$
$314$	−809.748	−0.145531
$315$	−604.769	−0.108174
$316$	598.537	0.106552
$317$	9306.43	1.64890	0.824450	−	0.565934i	$-0.191484\pi$
$317$	9306.43	1.64890	0.824450	+	0.565934i	$0.191484\pi$
$318$	−4085.31	−0.720417
$319$	2279.23	0.400038
$320$	1657.57	0.289566
$321$	10502.8	1.82620
$322$	−1850.09	−0.320191
$323$	1030.10	0.177449
$324$	4373.96	0.749993
$325$	949.760	0.162102
$326$	−2428.33	−0.412554
$327$	9577.24	1.61964
$328$	−7186.72	−1.20982
$329$	−3699.79	−0.619989
$330$	625.762	0.104385
$331$	−5183.44	−0.860749	−0.430374	−	0.902651i	$-0.641618\pi$
$331$	−5183.44	−0.860749	−0.430374	+	0.902651i	$0.641618\pi$
$332$	5566.99	0.920266
$333$	−948.864	−0.156148
$334$	4142.73	0.678683
$335$	−5464.75	−0.891258
$336$	1858.28	0.301718
$337$	3000.12	0.484947	0.242474	−	0.970158i	$-0.422041\pi$
$337$	3000.12	0.484947	0.242474	+	0.970158i	$0.422041\pi$
$338$	3034.93	0.488397
$339$	−10497.3	−1.68182
$340$	605.173	0.0965297
$341$	−1094.67	−0.173841
$342$	290.960	0.0460039
$343$	−7953.52	−1.25204
$344$	0	0
$345$	1693.67	0.264301
$346$	3187.78	0.495307
$347$	9148.76	1.41536	0.707682	−	0.706531i	$-0.249741\pi$
$347$	9148.76	1.41536	0.707682	+	0.706531i	$0.249741\pi$
$348$	7564.15	1.16518
$349$	2096.34	0.321531	0.160766	−	0.986993i	$-0.448604\pi$
$349$	2096.34	0.321531	0.160766	+	0.986993i	$0.448604\pi$
$350$	−2839.75	−0.433688
$351$	−2146.48	−0.326412
$352$	−1697.14	−0.256982
$353$	319.106	0.0481142	0.0240571	−	0.999711i	$-0.492342\pi$
$353$	319.106	0.0481142	0.0240571	+	0.999711i	$0.492342\pi$
$354$	3041.18	0.456601
$355$	3395.82	0.507695
$356$	1045.86	0.155704
$357$	−2238.75	−0.331897
$358$	5609.92	0.828194
$359$	9112.72	1.33970	0.669848	−	0.742498i	$-0.266359\pi$
$359$	9112.72	1.33970	0.669848	+	0.742498i	$0.266359\pi$
$360$	417.463	0.0611174
$361$	−974.175	−0.142029
$362$	−223.257	−0.0324147
$363$	6773.74	0.979419
$364$	−2745.13	−0.395285
$365$	−7496.17	−1.07498
$366$	−5032.29	−0.718694
$367$	−2726.93	−0.387860	−0.193930	−	0.981015i	$-0.562123\pi$
$367$	−2726.93	−0.387860	−0.193930	+	0.981015i	$0.562123\pi$
$368$	−428.353	−0.0606779
$369$	−820.280	−0.115724
$370$	4985.81	0.700540
$371$	14781.1	2.06845
$372$	−3632.92	−0.506339
$373$	−13249.6	−1.83924	−0.919622	−	0.392804i	$-0.871505\pi$
$373$	−13249.6	−1.83924	−0.919622	+	0.392804i	$0.871505\pi$
$374$	190.669	0.0263616
$375$	8108.37	1.11657
$376$	2553.92	0.350288
$377$	−4047.68	−0.552960
$378$	6417.89	0.873282
$379$	8778.93	1.18982	0.594912	−	0.803791i	$-0.297187\pi$
$379$	8778.93	1.18982	0.594912	+	0.803791i	$0.297187\pi$
$380$	3457.28	0.466724
$381$	−1176.21	−0.158160
$382$	−961.562	−0.128790
$383$	−11141.2	−1.48639	−0.743194	−	0.669076i	$-0.766690\pi$
$383$	−11141.2	−1.48639	−0.743194	+	0.669076i	$0.766690\pi$
$384$	−6389.85	−0.849168
$385$	−2264.07	−0.299709
$386$	394.058	0.0519613
$387$	0	0
$388$	−2671.12	−0.349499
$389$	11673.9	1.52157	0.760786	−	0.649003i	$-0.224814\pi$
$389$	11673.9	1.52157	0.760786	+	0.649003i	$0.224814\pi$
$390$	−1111.29	−0.144288
$391$	516.057	0.0667471
$392$	12767.7	1.64507
$393$	3414.76	0.438300
$394$	3204.58	0.409758
$395$	−876.672	−0.111671
$396$	−121.788	−0.0154548
$397$	9942.51	1.25693	0.628464	−	0.777839i	$-0.283684\pi$
$397$	9942.51	1.25693	0.628464	+	0.777839i	$0.283684\pi$
$398$	−944.327	−0.118932
$399$	−12789.7	−1.60473
$400$	−657.489	−0.0821861
$401$	4884.85	0.608323	0.304162	−	0.952620i	$-0.401624\pi$
$401$	4884.85	0.608323	0.304162	+	0.952620i	$0.401624\pi$
$402$	−5714.03	−0.708930
$403$	1944.02	0.240295
$404$	7601.10	0.936061
$405$	−6406.50	−0.786028
$406$	12102.4	1.47939
$407$	−3552.26	−0.432627
$408$	1545.38	0.187519
$409$	10707.9	1.29455	0.647276	−	0.762256i	$-0.275908\pi$
$409$	10707.9	1.29455	0.647276	+	0.762256i	$0.275908\pi$
$410$	4310.16	0.519180
$411$	9760.14	1.17137
$412$	4566.29	0.546032
$413$	−11003.3	−1.31099
$414$	145.765	0.0173043
$415$	−8153.92	−0.964483
$416$	3013.94	0.355218
$417$	−7400.36	−0.869058
$418$	1089.27	0.127459
$419$	−5076.87	−0.591936	−0.295968	−	0.955198i	$-0.595642\pi$
$419$	−5076.87	−0.591936	−0.295968	+	0.955198i	$0.595642\pi$
$420$	−7513.86	−0.872950
$421$	−3748.60	−0.433956	−0.216978	−	0.976176i	$-0.569620\pi$
$421$	−3748.60	−0.433956	−0.216978	+	0.976176i	$0.569620\pi$
$422$	−6812.33	−0.785827
$423$	291.500	0.0335064
$424$	−10203.2	−1.16865
$425$	792.107	0.0904067
$426$	3550.73	0.403834
$427$	18207.4	2.06350
$428$	10740.7	1.21302
$429$	791.767	0.0891069
$430$	0	0
$431$	−472.408	−0.0527960	−0.0263980	−	0.999652i	$-0.508404\pi$
$431$	−472.408	−0.0527960	−0.0263980	+	0.999652i	$0.508404\pi$
$432$	1485.94	0.165491
$433$	500.116	0.0555058	0.0277529	−	0.999615i	$-0.491165\pi$
$433$	500.116	0.0555058	0.0277529	+	0.999615i	$0.491165\pi$
$434$	−5812.56	−0.642884
$435$	−11079.1	−1.22116
$436$	9794.17	1.07582
$437$	2948.18	0.322724
$438$	−7838.12	−0.855068
$439$	−15113.3	−1.64309	−0.821544	−	0.570144i	$-0.806887\pi$
$439$	−15113.3	−1.64309	−0.821544	+	0.570144i	$0.806887\pi$
$440$	1562.86	0.169333
$441$	1457.29	0.157357
$442$	−338.608	−0.0364388
$443$	−8694.15	−0.932441	−0.466221	−	0.884668i	$-0.654385\pi$
$443$	−8694.15	−0.932441	−0.466221	+	0.884668i	$0.654385\pi$
$444$	−11789.0	−1.26009
$445$	−1531.86	−0.163185
$446$	3703.65	0.393213
$447$	331.461	0.0350729
$448$	−6270.84	−0.661316
$449$	−15475.4	−1.62657	−0.813287	−	0.581863i	$-0.802324\pi$
$449$	−15475.4	−1.62657	−0.813287	+	0.581863i	$0.802324\pi$
$450$	223.738	0.0234380
$451$	−3070.88	−0.320626
$452$	−10735.1	−1.11712
$453$	−10805.1	−1.12068
$454$	−10356.4	−1.07060
$455$	4020.77	0.414278
$456$	8828.57	0.906658
$457$	7336.32	0.750938	0.375469	−	0.926835i	$-0.377482\pi$
$457$	7336.32	0.750938	0.375469	+	0.926835i	$0.377482\pi$
$458$	−3391.91	−0.346056
$459$	−1790.18	−0.182045
$460$	1732.03	0.175557
$461$	−1976.62	−0.199698	−0.0998488	−	0.995003i	$-0.531836\pi$
$461$	−1976.62	−0.199698	−0.0998488	+	0.995003i	$0.531836\pi$
$462$	−2367.35	−0.238397
$463$	7644.41	0.767313	0.383656	−	0.923476i	$-0.374665\pi$
$463$	7644.41	0.767313	0.383656	+	0.923476i	$0.374665\pi$
$464$	2802.08	0.280352
$465$	5321.11	0.530668
$466$	4023.36	0.399954
$467$	−4008.28	−0.397176	−0.198588	−	0.980083i	$-0.563635\pi$
$467$	−4008.28	−0.397176	−0.198588	+	0.980083i	$0.563635\pi$
$468$	216.283	0.0213626
$469$	20674.0	2.03547
$470$	−1531.69	−0.150322
$471$	−2804.39	−0.274351
$472$	7595.43	0.740695
$473$	0	0
$474$	−916.662	−0.0888263
$475$	4525.22	0.437119
$476$	−2289.46	−0.220456
$477$	−1164.57	−0.111786
$478$	−1491.76	−0.142743
$479$	−10236.8	−0.976473	−0.488236	−	0.872711i	$-0.662360\pi$
$479$	−10236.8	−0.976473	−0.488236	+	0.872711i	$0.662360\pi$
$480$	8249.65	0.784465
$481$	6308.46	0.598006
$482$	−4317.92	−0.408042
$483$	−6407.39	−0.603616
$484$	6927.17	0.650560
$485$	3912.37	0.366292
$486$	−1061.13	−0.0990406
$487$	4057.87	0.377576	0.188788	−	0.982018i	$-0.439544\pi$
$487$	4057.87	0.377576	0.188788	+	0.982018i	$0.439544\pi$
$488$	−12568.3	−1.16586
$489$	−8410.00	−0.777737
$490$	−7657.32	−0.705964
$491$	−3884.88	−0.357072	−0.178536	−	0.983933i	$-0.557136\pi$
$491$	−3884.88	−0.357072	−0.178536	+	0.983933i	$0.557136\pi$
$492$	−10191.4	−0.933874
$493$	−3375.80	−0.308394
$494$	−1934.43	−0.176182
$495$	178.382	0.0161973
$496$	−1345.79	−0.121830
$497$	−12846.9	−1.15948
$498$	−8525.87	−0.767176
$499$	−8005.88	−0.718222	−0.359111	−	0.933295i	$-0.616920\pi$
$499$	−8005.88	−0.718222	−0.359111	+	0.933295i	$0.616920\pi$
$500$	8292.03	0.741662
$501$	14347.5	1.27944
$502$	1136.09	0.101008
$503$	3167.37	0.280767	0.140384	−	0.990097i	$-0.455166\pi$
$503$	3167.37	0.280767	0.140384	+	0.990097i	$0.455166\pi$
$504$	−1579.33	−0.139581
$505$	−11133.3	−0.981037
$506$	545.701	0.0479434
$507$	10510.8	0.920714
$508$	−1202.85	−0.105055
$509$	−2437.53	−0.212262	−0.106131	−	0.994352i	$-0.533846\pi$
$509$	−2437.53	−0.212262	−0.106131	+	0.994352i	$0.533846\pi$
$510$	−926.825	−0.0804716
$511$	28359.2	2.45506
$512$	−3978.34	−0.343398
$513$	−10227.1	−0.880190
$514$	6055.60	0.519652
$515$	−6688.21	−0.572268
$516$	0	0
$517$	1091.29	0.0928332
$518$	−18862.1	−1.59991
$519$	11040.2	0.933741
$520$	−2775.48	−0.234063
$521$	12207.6	1.02654	0.513268	−	0.858228i	$-0.328435\pi$
$521$	12207.6	1.02654	0.513268	+	0.858228i	$0.328435\pi$
$522$	−953.525	−0.0799514
$523$	−18236.1	−1.52468	−0.762340	−	0.647177i	$-0.775950\pi$
$523$	−18236.1	−1.52468	−0.762340	+	0.647177i	$0.775950\pi$
$524$	3492.11	0.291133
$525$	−9834.85	−0.817577
$526$	−1998.02	−0.165623
$527$	1621.33	0.134016
$528$	−548.115	−0.0451774
$529$	−10690.0	−0.878608
$530$	6119.25	0.501515
$531$	866.930	0.0708504
$532$	−13079.4	−1.06591
$533$	5453.58	0.443191
$534$	−1601.74	−0.129802
$535$	−15731.8	−1.27130
$536$	−14271.0	−1.15002
$537$	19428.8	1.56129
$538$	1100.05	0.0881537
$539$	5455.65	0.435977
$540$	−6008.33	−0.478810
$541$	−4277.33	−0.339920	−0.169960	−	0.985451i	$-0.554364\pi$
$541$	−4277.33	−0.339920	−0.169960	+	0.985451i	$0.554364\pi$
$542$	−10042.2	−0.795846
$543$	−773.202	−0.0611073
$544$	2513.65	0.198110
$545$	−14345.4	−1.12751
$546$	4204.18	0.329528
$547$	10280.6	0.803596	0.401798	−	0.915728i	$-0.368385\pi$
$547$	10280.6	0.803596	0.401798	+	0.915728i	$0.368385\pi$
$548$	9981.21	0.778058
$549$	−1434.52	−0.111519
$550$	837.609	0.0649377
$551$	−19285.5	−1.49109
$552$	4422.93	0.341037
$553$	3316.58	0.255037
$554$	−3272.03	−0.250930
$555$	17267.3	1.32064
$556$	−7567.98	−0.577255
$557$	−19282.4	−1.46683	−0.733413	−	0.679784i	$-0.762074\pi$
$557$	−19282.4	−1.46683	−0.733413	+	0.679784i	$0.762074\pi$
$558$	457.960	0.0347437
$559$	0	0
$560$	−2783.45	−0.210040
$561$	660.340	0.0496962
$562$	5310.40	0.398586
$563$	14877.6	1.11371	0.556854	−	0.830610i	$-0.312008\pi$
$563$	14877.6	1.11371	0.556854	+	0.830610i	$0.312008\pi$
$564$	3621.69	0.270391
$565$	15723.6	1.17079
$566$	−10707.9	−0.795209
$567$	24236.8	1.79515
$568$	8868.04	0.655096
$569$	−10794.5	−0.795307	−0.397654	−	0.917536i	$-0.630175\pi$
$569$	−10794.5	−0.795307	−0.397654	+	0.917536i	$0.630175\pi$
$570$	−5294.85	−0.389082
$571$	7995.91	0.586022	0.293011	−	0.956109i	$-0.405343\pi$
$571$	7995.91	0.586022	0.293011	+	0.956109i	$0.405343\pi$
$572$	809.700	0.0591875
$573$	−3330.16	−0.242792
$574$	−16306.0	−1.18571
$575$	2267.04	0.164421
$576$	494.067	0.0357398
$577$	11565.4	0.834444	0.417222	−	0.908805i	$-0.363004\pi$
$577$	11565.4	0.834444	0.417222	+	0.908805i	$0.363004\pi$
$578$	7412.31	0.533411
$579$	1364.74	0.0979560
$580$	−11330.1	−0.811132
$581$	30847.5	2.20270
$582$	4090.83	0.291358
$583$	−4359.81	−0.309717
$584$	−19575.9	−1.38708
$585$	−316.788	−0.0223890
$586$	−10673.0	−0.752382
$587$	6354.81	0.446833	0.223417	−	0.974723i	$-0.428279\pi$
$587$	6354.81	0.446833	0.223417	+	0.974723i	$0.428279\pi$
$588$	18105.8	1.26985
$589$	9262.49	0.647970
$590$	−4555.28	−0.317861
$591$	11098.4	0.772465
$592$	−4367.15	−0.303190
$593$	−21747.7	−1.50602	−0.753010	−	0.658009i	$-0.771399\pi$
$593$	−21747.7	−1.50602	−0.753010	+	0.658009i	$0.771399\pi$
$594$	−1893.01	−0.130760
$595$	3353.35	0.231049
$596$	338.969	0.0232965
$597$	−3270.47	−0.224207
$598$	−969.110	−0.0662706
$599$	10874.2	0.741750	0.370875	−	0.928683i	$-0.379058\pi$
$599$	10874.2	0.741750	0.370875	+	0.928683i	$0.379058\pi$
$600$	6788.86	0.461923
$601$	16092.5	1.09223	0.546113	−	0.837712i	$-0.316107\pi$
$601$	16092.5	1.09223	0.546113	+	0.837712i	$0.316107\pi$
$602$	0	0
$603$	−1628.86	−0.110004
$604$	−11049.8	−0.744390
$605$	−10146.2	−0.681819
$606$	−11641.1	−0.780344
$607$	−23235.8	−1.55372	−0.776862	−	0.629671i	$-0.783190\pi$
$607$	−23235.8	−1.55372	−0.776862	+	0.629671i	$0.783190\pi$
$608$	14360.2	0.957868
$609$	41914.1	2.78891
$610$	7537.70	0.500316
$611$	−1938.02	−0.128320
$612$	180.382	0.0119142
$613$	−14997.4	−0.988153	−0.494076	−	0.869418i	$-0.664494\pi$
$613$	−14997.4	−0.988153	−0.494076	+	0.869418i	$0.664494\pi$
$614$	−2512.61	−0.165148
$615$	14927.3	0.978745
$616$	−5912.53	−0.386725
$617$	2016.56	0.131578	0.0657890	−	0.997834i	$-0.479044\pi$
$617$	2016.56	0.131578	0.0657890	+	0.997834i	$0.479044\pi$
$618$	−6993.30	−0.455197
$619$	17984.1	1.16776	0.583879	−	0.811841i	$-0.301534\pi$
$619$	17984.1	1.16776	0.583879	+	0.811841i	$0.301534\pi$
$620$	5441.63	0.352486
$621$	−5123.57	−0.331081
$622$	3630.85	0.234058
$623$	5795.27	0.372685
$624$	973.397	0.0624472
$625$	−4771.61	−0.305383
$626$	6861.46	0.438082
$627$	3772.45	0.240282
$628$	−2867.91	−0.182233
$629$	5261.31	0.333517
$630$	947.185	0.0598996
$631$	−7502.61	−0.473335	−0.236667	−	0.971591i	$-0.576055\pi$
$631$	−7502.61	−0.473335	−0.236667	+	0.971591i	$0.576055\pi$
$632$	−2289.39	−0.144093
$633$	−23593.1	−1.48142
$634$	−14575.7	−0.913051
$635$	1761.80	0.110103
$636$	−14469.1	−0.902100
$637$	−9688.68	−0.602637
$638$	−3569.71	−0.221515
$639$	1012.18	0.0626625
$640$	9571.14	0.591145
$641$	−17658.5	−1.08809	−0.544047	−	0.839055i	$-0.683109\pi$
$641$	−17658.5	−1.08809	−0.544047	+	0.839055i	$0.683109\pi$
$642$	−16449.4	−1.01123
$643$	−12908.4	−0.791694	−0.395847	−	0.918316i	$-0.629549\pi$
$643$	−12908.4	−0.791694	−0.395847	+	0.918316i	$0.629549\pi$
$644$	−6552.52	−0.400940
$645$	0	0
$646$	−1613.33	−0.0982595
$647$	−20030.3	−1.21712	−0.608558	−	0.793510i	$-0.708252\pi$
$647$	−20030.3	−1.21712	−0.608558	+	0.793510i	$0.708252\pi$
$648$	−16730.3	−1.01424
$649$	3245.53	0.196299
$650$	−1487.51	−0.0897613
$651$	−20130.6	−1.21195
$652$	−8600.48	−0.516597
$653$	−3145.07	−0.188478	−0.0942391	−	0.995550i	$-0.530042\pi$
$653$	−3145.07	−0.188478	−0.0942391	+	0.995550i	$0.530042\pi$
$654$	−14999.8	−0.896849
$655$	−5114.86	−0.305121
$656$	−3775.34	−0.224699
$657$	−2234.36	−0.132680
$658$	5794.60	0.343308
$659$	−16132.7	−0.953627	−0.476813	−	0.879005i	$-0.658208\pi$
$659$	−16132.7	−0.953627	−0.476813	+	0.879005i	$0.658208\pi$
$660$	2216.28	0.130710
$661$	−9134.09	−0.537481	−0.268740	−	0.963213i	$-0.586607\pi$
$661$	−9134.09	−0.537481	−0.268740	+	0.963213i	$0.586607\pi$
$662$	8118.28	0.476625
$663$	−1172.70	−0.0686935
$664$	−21293.6	−1.24451
$665$	19157.3	1.11713
$666$	1486.10	0.0864645
$667$	−9661.66	−0.560871
$668$	14672.4	0.849841
$669$	12826.8	0.741275
$670$	8558.86	0.493519
$671$	−5370.42	−0.308976
$672$	−31209.7	−1.79158
$673$	−5116.23	−0.293041	−0.146520	−	0.989208i	$-0.546807\pi$
$673$	−5116.23	−0.293041	−0.146520	+	0.989208i	$0.546807\pi$
$674$	−4698.78	−0.268531
$675$	−7864.27	−0.448439
$676$	10748.9	0.611567
$677$	−4972.81	−0.282305	−0.141153	−	0.989988i	$-0.545081\pi$
$677$	−4972.81	−0.282305	−0.141153	+	0.989988i	$0.545081\pi$
$678$	16440.9	0.931280
$679$	−14801.1	−0.836543
$680$	−2314.77	−0.130540
$681$	−35867.2	−2.01826
$682$	1714.47	0.0962614
$683$	−7581.73	−0.424754	−0.212377	−	0.977188i	$-0.568120\pi$
$683$	−7581.73	−0.424754	−0.212377	+	0.977188i	$0.568120\pi$
$684$	1030.50	0.0576056
$685$	−14619.4	−0.815443
$686$	12456.8	0.693296
$687$	−11747.2	−0.652375
$688$	0	0
$689$	7742.58	0.428112
$690$	−2652.61	−0.146352
$691$	−7653.70	−0.421361	−0.210681	−	0.977555i	$-0.567568\pi$
$691$	−7653.70	−0.421361	−0.210681	+	0.977555i	$0.567568\pi$
$692$	11290.3	0.620219
$693$	−674.846	−0.0369918
$694$	−14328.7	−0.783734
$695$	11084.8	0.604991
$696$	−28932.7	−1.57571
$697$	4548.33	0.247174
$698$	−3283.27	−0.178043
$699$	13934.0	0.753982
$700$	−10057.6	−0.543060
$701$	5942.69	0.320189	0.160094	−	0.987102i	$-0.448820\pi$
$701$	5942.69	0.320189	0.160094	+	0.987102i	$0.448820\pi$
$702$	3361.80	0.180745
$703$	30057.3	1.61256
$704$	1849.64	0.0990213
$705$	−5304.66	−0.283383
$706$	−499.782	−0.0266424
$707$	42118.8	2.24051
$708$	10771.0	0.571752
$709$	−5113.78	−0.270877	−0.135439	−	0.990786i	$-0.543244\pi$
$709$	−5113.78	−0.270877	−0.135439	+	0.990786i	$0.543244\pi$
$710$	−5318.52	−0.281127
$711$	−261.307	−0.0137831
$712$	−4000.39	−0.210563
$713$	4640.31	0.243732
$714$	3506.32	0.183783
$715$	−1185.96	−0.0620314
$716$	19868.8	1.03706
$717$	−5166.38	−0.269096
$718$	−14272.3	−0.741835
$719$	36844.4	1.91108	0.955538	−	0.294869i	$-0.0952761\pi$
$719$	36844.4	1.91108	0.955538	+	0.294869i	$0.0952761\pi$
$720$	219.303	0.0113513
$721$	25302.5	1.30696
$722$	1525.75	0.0786460
$723$	−14954.2	−0.769230
$724$	−790.715	−0.0405893
$725$	−14829.9	−0.759681
$726$	−10609.0	−0.542337
$727$	−21420.6	−1.09277	−0.546387	−	0.837533i	$-0.683997\pi$
$727$	−21420.6	−1.09277	−0.546387	+	0.837533i	$0.683997\pi$
$728$	10500.1	0.534558
$729$	17615.1	0.894939
$730$	11740.5	0.595252
$731$	0	0
$732$	−17823.0	−0.899942
$733$	−35541.1	−1.79091	−0.895457	−	0.445148i	$-0.853151\pi$
$733$	−35541.1	−1.79091	−0.895457	+	0.445148i	$0.853151\pi$
$734$	4270.90	0.214771
$735$	−26519.5	−1.33087
$736$	7194.17	0.360300
$737$	−6097.98	−0.304779
$738$	1284.72	0.0640801
$739$	5111.15	0.254420	0.127210	−	0.991876i	$-0.459398\pi$
$739$	5111.15	0.254420	0.127210	+	0.991876i	$0.459398\pi$
$740$	17658.4	0.877209
$741$	−6699.49	−0.332135
$742$	−23150.0	−1.14537
$743$	37864.8	1.86962	0.934808	−	0.355153i	$-0.115571\pi$
$743$	37864.8	1.86962	0.934808	+	0.355153i	$0.115571\pi$
$744$	13895.8	0.684739
$745$	−496.485	−0.0244158
$746$	20751.4	1.01845
$747$	−2430.42	−0.119042
$748$	675.297	0.0330098
$749$	59515.8	2.90342
$750$	−12699.3	−0.618283
$751$	4814.57	0.233937	0.116968	−	0.993136i	$-0.462682\pi$
$751$	4814.57	0.233937	0.116968	+	0.993136i	$0.462682\pi$
$752$	1341.63	0.0650587
$753$	3934.60	0.190418
$754$	6339.45	0.306192
$755$	16184.6	0.780156
$756$	22730.4	1.09352
$757$	−13342.1	−0.640589	−0.320295	−	0.947318i	$-0.603782\pi$
$757$	−13342.1	−0.640589	−0.320295	+	0.947318i	$0.603782\pi$
$758$	−13749.5	−0.658845
$759$	1889.92	0.0903817
$760$	−13224.0	−0.631166
$761$	−17924.5	−0.853826	−0.426913	−	0.904293i	$-0.640399\pi$
$761$	−17924.5	−0.853826	−0.426913	+	0.904293i	$0.640399\pi$
$762$	1842.17	0.0875785
$763$	54271.0	2.57502
$764$	−3405.59	−0.161270
$765$	−264.204	−0.0124867
$766$	17449.2	0.823062
$767$	−5763.73	−0.271338
$768$	18860.7	0.886168
$769$	4472.87	0.209748	0.104874	−	0.994486i	$-0.466556\pi$
$769$	4472.87	0.209748	0.104874	+	0.994486i	$0.466556\pi$
$770$	3545.98	0.165959
$771$	20972.3	0.979635
$772$	1395.65	0.0650654
$773$	33168.7	1.54333	0.771665	−	0.636029i	$-0.219424\pi$
$773$	33168.7	1.54333	0.771665	+	0.636029i	$0.219424\pi$
$774$	0	0
$775$	7122.52	0.330127
$776$	10217.0	0.472639
$777$	−65324.7	−3.01610
$778$	−18283.6	−0.842544
$779$	25984.1	1.19509
$780$	−3935.89	−0.180676
$781$	3789.31	0.173614
$782$	−808.245	−0.0369601
$783$	33515.9	1.52971
$784$	6707.17	0.305538
$785$	4200.60	0.190988
$786$	−5348.18	−0.242701
$787$	6452.90	0.292276	0.146138	−	0.989264i	$-0.453316\pi$
$787$	6452.90	0.292276	0.146138	+	0.989264i	$0.453316\pi$
$788$	11349.8	0.513095
$789$	−6919.72	−0.312229
$790$	1373.04	0.0618361
$791$	−59484.8	−2.67388
$792$	465.837	0.0209000
$793$	9537.33	0.427088
$794$	−15571.9	−0.696002
$795$	21192.7	0.945444
$796$	−3344.55	−0.148925
$797$	8676.60	0.385622	0.192811	−	0.981236i	$-0.438240\pi$
$797$	8676.60	0.385622	0.192811	+	0.981236i	$0.438240\pi$
$798$	20031.2	0.888593
$799$	−1616.32	−0.0715661
$800$	11042.5	0.488014
$801$	−456.598	−0.0201412
$802$	−7650.62	−0.336849
$803$	−8364.79	−0.367605
$804$	−20237.6	−0.887716
$805$	9597.43	0.420205
$806$	−3044.72	−0.133059
$807$	3809.80	0.166185
$808$	−29074.0	−1.26587
$809$	10277.3	0.446637	0.223319	−	0.974745i	$-0.428311\pi$
$809$	10277.3	0.446637	0.223319	+	0.974745i	$0.428311\pi$
$810$	10033.8	0.435250
$811$	31979.8	1.38466	0.692332	−	0.721579i	$-0.256584\pi$
$811$	31979.8	1.38466	0.692332	+	0.721579i	$0.256584\pi$
$812$	42863.4	1.85248
$813$	−34779.0	−1.50031
$814$	5563.53	0.239560
$815$	12597.1	0.541418
$816$	811.821	0.0348277
$817$	0	0
$818$	−16770.7	−0.716837
$819$	1198.46	0.0511325
$820$	15265.4	0.650112
$821$	4889.45	0.207848	0.103924	−	0.994585i	$-0.466860\pi$
$821$	4889.45	0.207848	0.103924	+	0.994585i	$0.466860\pi$
$822$	−15286.3	−0.648625
$823$	−27193.5	−1.15177	−0.575884	−	0.817531i	$-0.695342\pi$
$823$	−27193.5	−1.15177	−0.575884	+	0.817531i	$0.695342\pi$
$824$	−17466.0	−0.738417
$825$	2900.88	0.122419
$826$	17233.3	0.725937
$827$	−5190.02	−0.218228	−0.109114	−	0.994029i	$-0.534801\pi$
$827$	−5190.02	−0.218228	−0.109114	+	0.994029i	$0.534801\pi$
$828$	516.260	0.0216682
$829$	22048.5	0.923733	0.461867	−	0.886949i	$-0.347180\pi$
$829$	22048.5	0.923733	0.461867	+	0.886949i	$0.347180\pi$
$830$	12770.6	0.534066
$831$	−11332.0	−0.473047
$832$	−3284.78	−0.136874
$833$	−8080.44	−0.336099
$834$	11590.4	0.481226
$835$	−21490.6	−0.890674
$836$	3857.89	0.159603
$837$	−16097.1	−0.664750
$838$	7951.36	0.327775
$839$	−24727.9	−1.01752	−0.508761	−	0.860908i	$-0.669896\pi$
$839$	−24727.9	−1.01752	−0.508761	+	0.860908i	$0.669896\pi$
$840$	28740.3	1.18052
$841$	38812.9	1.59141
$842$	5871.03	0.240296
$843$	18391.4	0.751405
$844$	−24127.4	−0.984006
$845$	−15743.8	−0.640951
$846$	−456.545	−0.0185536
$847$	38384.5	1.55715
$848$	−5359.95	−0.217054
$849$	−37084.7	−1.49911
$850$	−1240.59	−0.0500612
$851$	15058.1	0.606562
$852$	12575.7	0.505677
$853$	−23989.1	−0.962922	−0.481461	−	0.876468i	$-0.659894\pi$
$853$	−23989.1	−0.962922	−0.481461	+	0.876468i	$0.659894\pi$
$854$	−28516.3	−1.14263
$855$	−1509.37	−0.0603735
$856$	−41082.9	−1.64040
$857$	−10224.3	−0.407534	−0.203767	−	0.979019i	$-0.565319\pi$
$857$	−10224.3	−0.407534	−0.203767	+	0.979019i	$0.565319\pi$
$858$	−1240.06	−0.0493414
$859$	−20256.5	−0.804590	−0.402295	−	0.915510i	$-0.631787\pi$
$859$	−20256.5	−0.804590	−0.402295	+	0.915510i	$0.631787\pi$
$860$	0	0
$861$	−56472.3	−2.23527
$862$	739.882	0.0292349
$863$	−21978.0	−0.866905	−0.433453	−	0.901176i	$-0.642705\pi$
$863$	−21978.0	−0.866905	−0.433453	+	0.901176i	$0.642705\pi$
$864$	−24956.3	−0.982673
$865$	−16536.8	−0.650020
$866$	−783.278	−0.0307354
$867$	25670.9	1.00557
$868$	−20586.5	−0.805014
$869$	−978.256	−0.0381876
$870$	17352.1	0.676197
$871$	10829.4	0.421286
$872$	−37462.5	−1.45486
$873$	1166.15	0.0452097
$874$	−4617.42	−0.178703
$875$	45947.4	1.77521
$876$	−27760.5	−1.07071
$877$	−11390.5	−0.438575	−0.219287	−	0.975660i	$-0.570373\pi$
$877$	−11390.5	−0.438575	−0.219287	+	0.975660i	$0.570373\pi$
$878$	23670.3	0.909833
$879$	−36963.5	−1.41837
$880$	821.004	0.0314500
$881$	36825.6	1.40827	0.704135	−	0.710066i	$-0.251335\pi$
$881$	36825.6	1.40827	0.704135	+	0.710066i	$0.251335\pi$
$882$	−2282.39	−0.0871340
$883$	−27495.1	−1.04789	−0.523943	−	0.851754i	$-0.675539\pi$
$883$	−27495.1	−1.04789	−0.523943	+	0.851754i	$0.675539\pi$
$884$	−1199.26	−0.0456283
$885$	−15776.2	−0.599224
$886$	13616.7	0.516324
$887$	−39402.6	−1.49155	−0.745777	−	0.666195i	$-0.767922\pi$
$887$	−39402.6	−1.49155	−0.745777	+	0.666195i	$0.767922\pi$
$888$	45092.7	1.70407
$889$	−6665.17	−0.251454
$890$	2399.20	0.0903610
$891$	−7148.85	−0.268794
$892$	13117.3	0.492378
$893$	−9233.86	−0.346024
$894$	−519.132	−0.0194210
$895$	−29101.7	−1.08689
$896$	−36209.1	−1.35007
$897$	−3356.30	−0.124932
$898$	24237.5	0.900687
$899$	−30354.7	−1.12612
$900$	792.420	0.0293489
$901$	6457.38	0.238764
$902$	4809.60	0.177541
$903$	0	0
$904$	41061.5	1.51071
$905$	1158.15	0.0425396
$906$	16922.9	0.620557
$907$	−842.302	−0.0308359	−0.0154180	−	0.999881i	$-0.504908\pi$
$907$	−842.302	−0.0308359	−0.0154180	+	0.999881i	$0.504908\pi$
$908$	−36679.6	−1.34059
$909$	−3318.46	−0.121085
$910$	−6297.30	−0.229399
$911$	−7588.34	−0.275975	−0.137987	−	0.990434i	$-0.544063\pi$
$911$	−7588.34	−0.275975	−0.137987	+	0.990434i	$0.544063\pi$
$912$	4637.84	0.168393
$913$	−9098.75	−0.329819
$914$	−11490.1	−0.415819
$915$	26105.2	0.943182
$916$	−12013.2	−0.433328
$917$	19350.3	0.696841
$918$	2803.77	0.100804
$919$	1126.30	0.0404278	0.0202139	−	0.999796i	$-0.493565\pi$
$919$	1126.30	0.0404278	0.0202139	+	0.999796i	$0.493565\pi$
$920$	−6624.97	−0.237412
$921$	−8701.89	−0.311332
$922$	3095.78	0.110579
$923$	−6729.44	−0.239981
$924$	−8384.53	−0.298518
$925$	23113.0	0.821567
$926$	−11972.6	−0.424887
$927$	−1993.54	−0.0706325
$928$	−47060.8	−1.66470
$929$	12483.2	0.440861	0.220431	−	0.975403i	$-0.429254\pi$
$929$	12483.2	0.440861	0.220431	+	0.975403i	$0.429254\pi$
$930$	−8333.89	−0.293848
$931$	−46162.6	−1.62505
$932$	14249.6	0.500818
$933$	12574.7	0.441239
$934$	6277.74	0.219929
$935$	−989.102	−0.0345958
$936$	−827.279	−0.0288894
$937$	50746.3	1.76927	0.884636	−	0.466282i	$-0.154407\pi$
$937$	50746.3	1.76927	0.884636	+	0.466282i	$0.154407\pi$
$938$	−32379.5	−1.12711
$939$	23763.2	0.825860
$940$	−5424.82	−0.188232
$941$	21706.6	0.751983	0.375991	−	0.926623i	$-0.377302\pi$
$941$	21706.6	0.751983	0.375991	+	0.926623i	$0.377302\pi$
$942$	4392.22	0.151917
$943$	13017.5	0.449531
$944$	3990.05	0.137569
$945$	−33293.1	−1.14606
$946$	0	0
$947$	16594.0	0.569412	0.284706	−	0.958615i	$-0.408104\pi$
$947$	16594.0	0.569412	0.284706	+	0.958615i	$0.408104\pi$
$948$	−3246.57	−0.111228
$949$	14855.0	0.508129
$950$	−7087.37	−0.242047
$951$	−50479.8	−1.72126
$952$	8757.13	0.298131
$953$	5575.02	0.189499	0.0947495	−	0.995501i	$-0.469795\pi$
$953$	5575.02	0.189499	0.0947495	+	0.995501i	$0.469795\pi$
$954$	1823.95	0.0618998
$955$	4988.14	0.169018
$956$	−5283.40	−0.178742
$957$	−12362.9	−0.417593
$958$	16032.8	0.540705
$959$	55307.4	1.86232
$960$	−8990.96	−0.302273
$961$	−15212.2	−0.510631
$962$	−9880.27	−0.331136
$963$	−4689.13	−0.156911
$964$	−15292.9	−0.510946
$965$	−2044.20	−0.0681917
$966$	10035.2	0.334242
$967$	25230.5	0.839046	0.419523	−	0.907745i	$-0.362197\pi$
$967$	25230.5	0.839046	0.419523	+	0.907745i	$0.362197\pi$
$968$	−26496.3	−0.879775
$969$	−5587.43	−0.185236
$970$	−6127.52	−0.202828
$971$	37852.9	1.25104	0.625519	−	0.780209i	$-0.284887\pi$
$971$	37852.9	1.25104	0.625519	+	0.780209i	$0.284887\pi$
$972$	−3758.23	−0.124018
$973$	−41935.3	−1.38169
$974$	−6355.41	−0.209076
$975$	−5151.67	−0.169216
$976$	−6602.39	−0.216534
$977$	−24861.4	−0.814111	−0.407055	−	0.913404i	$-0.633444\pi$
$977$	−24861.4	−0.814111	−0.407055	+	0.913404i	$0.633444\pi$
$978$	13171.7	0.430659
$979$	−1709.37	−0.0558035
$980$	−27120.2	−0.884002
$981$	−4275.90	−0.139163
$982$	6084.47	0.197722
$983$	−38582.6	−1.25187	−0.625937	−	0.779873i	$-0.715283\pi$
$983$	−38582.6	−1.25187	−0.625937	+	0.779873i	$0.715283\pi$
$984$	38982.0	1.26291
$985$	−16623.9	−0.537748
$986$	5287.15	0.170768
$987$	20068.3	0.647196
$988$	−6851.23	−0.220614
$989$	0	0
$990$	−279.381	−0.00896900
$991$	−45385.9	−1.45483	−0.727413	−	0.686200i	$-0.759277\pi$
$991$	−45385.9	−1.45483	−0.727413	+	0.686200i	$0.759277\pi$
$992$	22602.4	0.723415
$993$	28115.9	0.898521
$994$	20120.8	0.642044
$995$	4898.74	0.156081
$996$	−30196.3	−0.960650
$997$	55850.4	1.77412	0.887061	−	0.461653i	$-0.152743\pi$
$997$	55850.4	1.77412	0.887061	+	0.461653i	$0.152743\pi$
$998$	12538.8	0.397703
$999$	−52235.8	−1.65432

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.h.1.11		30
43.21	even	7		43.4.e.a.11.4	yes	60
43.41	even	7		43.4.e.a.4.4	✓	60
43.42	odd	2		1849.4.a.g.1.20		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.4.4	✓	60	43.41	even	7
43.4.e.a.11.4	yes	60	43.21	even	7
1849.4.a.g.1.20		30	43.42	odd	2
1849.4.a.h.1.11		30	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)

\(f(q)\)	\(=\)	\(q-1.56619 q^{2} -5.42418 q^{3} -5.54704 q^{4} +8.12470 q^{5} +8.49531 q^{6} -30.7370 q^{7} +21.2173 q^{8} +2.42171 q^{9} +O(q^{10})\)	\(q-1.56619 q^{2} -5.42418 q^{3} -5.54704 q^{4} +8.12470 q^{5} +8.49531 q^{6} -30.7370 q^{7} +21.2173 q^{8} +2.42171 q^{9} -12.7249 q^{10} +9.06614 q^{11} +30.0881 q^{12} -16.1006 q^{13} +48.1400 q^{14} -44.0698 q^{15} +11.1459 q^{16} -13.4280 q^{17} -3.79286 q^{18} -76.7126 q^{19} -45.0680 q^{20} +166.723 q^{21} -14.1993 q^{22} -38.4314 q^{23} -115.086 q^{24} -58.9893 q^{25} +25.2166 q^{26} +133.317 q^{27} +170.499 q^{28} +251.400 q^{29} +69.0219 q^{30} -120.743 q^{31} -187.195 q^{32} -49.1764 q^{33} +21.0308 q^{34} -249.729 q^{35} -13.4333 q^{36} -391.816 q^{37} +120.147 q^{38} +87.3323 q^{39} +172.384 q^{40} -338.720 q^{41} -261.120 q^{42} -50.2902 q^{44} +19.6756 q^{45} +60.1911 q^{46} +120.370 q^{47} -60.4574 q^{48} +601.761 q^{49} +92.3886 q^{50} +72.8358 q^{51} +89.3103 q^{52} -480.889 q^{53} -208.800 q^{54} +73.6597 q^{55} -652.155 q^{56} +416.103 q^{57} -393.741 q^{58} +357.983 q^{59} +244.457 q^{60} -592.360 q^{61} +189.106 q^{62} -74.4359 q^{63} +204.016 q^{64} -130.812 q^{65} +77.0197 q^{66} -672.610 q^{67} +74.4856 q^{68} +208.459 q^{69} +391.123 q^{70} +417.963 q^{71} +51.3820 q^{72} -922.640 q^{73} +613.660 q^{74} +319.968 q^{75} +425.528 q^{76} -278.666 q^{77} -136.779 q^{78} -107.902 q^{79} +90.5571 q^{80} -788.521 q^{81} +530.501 q^{82} -1003.60 q^{83} -924.817 q^{84} -109.098 q^{85} -1363.64 q^{87} +192.359 q^{88} -188.544 q^{89} -30.8158 q^{90} +494.882 q^{91} +213.181 q^{92} +654.930 q^{93} -188.522 q^{94} -623.267 q^{95} +1015.38 q^{96} +481.540 q^{97} -942.474 q^{98} +21.9555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} + 2 q^{3} + 114 q^{4} + 27 q^{5} + 8 q^{6} + 48 q^{7} + 90 q^{8} + 216 q^{9}+O(q^{10}) \) 30 * q + 6 * q^2 + 2 * q^3 + 114 * q^4 + 27 * q^5 + 8 * q^6 + 48 * q^7 + 90 * q^8 + 216 * q^9	\( 30 q + 6 q^{2} + 2 q^{3} + 114 q^{4} + 27 q^{5} + 8 q^{6} + 48 q^{7} + 90 q^{8} + 216 q^{9} - 27 q^{10} + 80 q^{11} - 36 q^{12} - 13 q^{13} + 36 q^{14} + 16 q^{15} + 318 q^{16} + 66 q^{17} + 80 q^{18} + 254 q^{19} + 312 q^{20} - 548 q^{21} + 305 q^{22} - 105 q^{23} + 123 q^{24} + 523 q^{25} + 549 q^{26} - 10 q^{27} + 578 q^{28} + 793 q^{29} + 1560 q^{30} - 359 q^{31} + 676 q^{32} + 208 q^{33} + 1007 q^{34} - 514 q^{35} + 776 q^{36} + 510 q^{37} - 2066 q^{38} + 898 q^{39} - 1248 q^{40} - 270 q^{41} - 915 q^{42} + 3256 q^{44} + 807 q^{45} + 1960 q^{46} + 1421 q^{47} - 632 q^{48} + 386 q^{49} - 141 q^{50} + 209 q^{51} + 2825 q^{52} - 21 q^{53} + 2368 q^{54} + 2258 q^{55} + 2521 q^{56} - 1723 q^{57} - 347 q^{58} + 1752 q^{59} + 2711 q^{60} + 1759 q^{61} + 395 q^{62} + 2204 q^{63} + 222 q^{64} + 1151 q^{65} + 160 q^{66} - 3001 q^{67} + 1921 q^{68} + 1660 q^{69} + 1597 q^{70} + 727 q^{71} + 9100 q^{72} + 4623 q^{73} - 2649 q^{74} + 1027 q^{75} + 874 q^{76} + 3556 q^{77} - 4979 q^{78} + 546 q^{79} + 5809 q^{80} - 410 q^{81} - 4397 q^{82} - 492 q^{83} - 10611 q^{84} - 1723 q^{85} + 5937 q^{87} + 3974 q^{88} + 5218 q^{89} + 10492 q^{90} + 1104 q^{91} + 1060 q^{92} + 1997 q^{93} - 2134 q^{94} + 6346 q^{95} - 11984 q^{96} + 2590 q^{97} + 6270 q^{98} - 2693 q^{99}+O(q^{100}) \) 30 * q + 6 * q^2 + 2 * q^3 + 114 * q^4 + 27 * q^5 + 8 * q^6 + 48 * q^7 + 90 * q^8 + 216 * q^9 - 27 * q^10 + 80 * q^11 - 36 * q^12 - 13 * q^13 + 36 * q^14 + 16 * q^15 + 318 * q^16 + 66 * q^17 + 80 * q^18 + 254 * q^19 + 312 * q^20 - 548 * q^21 + 305 * q^22 - 105 * q^23 + 123 * q^24 + 523 * q^25 + 549 * q^26 - 10 * q^27 + 578 * q^28 + 793 * q^29 + 1560 * q^30 - 359 * q^31 + 676 * q^32 + 208 * q^33 + 1007 * q^34 - 514 * q^35 + 776 * q^36 + 510 * q^37 - 2066 * q^38 + 898 * q^39 - 1248 * q^40 - 270 * q^41 - 915 * q^42 + 3256 * q^44 + 807 * q^45 + 1960 * q^46 + 1421 * q^47 - 632 * q^48 + 386 * q^49 - 141 * q^50 + 209 * q^51 + 2825 * q^52 - 21 * q^53 + 2368 * q^54 + 2258 * q^55 + 2521 * q^56 - 1723 * q^57 - 347 * q^58 + 1752 * q^59 + 2711 * q^60 + 1759 * q^61 + 395 * q^62 + 2204 * q^63 + 222 * q^64 + 1151 * q^65 + 160 * q^66 - 3001 * q^67 + 1921 * q^68 + 1660 * q^69 + 1597 * q^70 + 727 * q^71 + 9100 * q^72 + 4623 * q^73 - 2649 * q^74 + 1027 * q^75 + 874 * q^76 + 3556 * q^77 - 4979 * q^78 + 546 * q^79 + 5809 * q^80 - 410 * q^81 - 4397 * q^82 - 492 * q^83 - 10611 * q^84 - 1723 * q^85 + 5937 * q^87 + 3974 * q^88 + 5218 * q^89 + 10492 * q^90 + 1104 * q^91 + 1060 * q^92 + 1997 * q^93 - 2134 * q^94 + 6346 * q^95 - 11984 * q^96 + 2590 * q^97 + 6270 * q^98 - 2693 * q^99

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