LMFDB - Embedded newform 1849.4.a.b.1.2

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:	$4$
Coefficient field:	4.4.45868.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 11x + 1 $ x^4 - x^3 - 10x^2 + 11x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.18808$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.132290 q^{2} +3.71054 q^{3} -7.98250 q^{4} -2.43196 q^{5} -0.490867 q^{6} +30.9271 q^{7} +2.11433 q^{8} -13.2319 q^{9} +O(q^{10})$	\(q-0.132290 q^{2} +3.71054 q^{3} -7.98250 q^{4} -2.43196 q^{5} -0.490867 q^{6} +30.9271 q^{7} +2.11433 q^{8} -13.2319 q^{9} +0.321724 q^{10} +57.4761 q^{11} -29.6194 q^{12} -27.0029 q^{13} -4.09135 q^{14} -9.02388 q^{15} +63.5803 q^{16} -91.1332 q^{17} +1.75045 q^{18} -77.7008 q^{19} +19.4131 q^{20} +114.756 q^{21} -7.60352 q^{22} +46.4541 q^{23} +7.84529 q^{24} -119.086 q^{25} +3.57222 q^{26} -149.282 q^{27} -246.876 q^{28} +251.724 q^{29} +1.19377 q^{30} -241.365 q^{31} -25.3256 q^{32} +213.267 q^{33} +12.0560 q^{34} -75.2135 q^{35} +105.624 q^{36} +23.0183 q^{37} +10.2790 q^{38} -100.195 q^{39} -5.14196 q^{40} -4.94980 q^{41} -15.1811 q^{42} -458.803 q^{44} +32.1795 q^{45} -6.14542 q^{46} -485.378 q^{47} +235.917 q^{48} +613.485 q^{49} +15.7538 q^{50} -338.153 q^{51} +215.551 q^{52} +111.940 q^{53} +19.7485 q^{54} -139.780 q^{55} +65.3900 q^{56} -288.312 q^{57} -33.3006 q^{58} +544.546 q^{59} +72.0331 q^{60} -367.434 q^{61} +31.9302 q^{62} -409.224 q^{63} -505.292 q^{64} +65.6700 q^{65} -28.2131 q^{66} -272.873 q^{67} +727.471 q^{68} +172.370 q^{69} +9.95000 q^{70} +37.7297 q^{71} -27.9766 q^{72} -667.567 q^{73} -3.04510 q^{74} -441.872 q^{75} +620.247 q^{76} +1777.57 q^{77} +13.2549 q^{78} +499.520 q^{79} -154.625 q^{80} -196.655 q^{81} +0.654810 q^{82} -432.180 q^{83} -916.041 q^{84} +221.632 q^{85} +934.031 q^{87} +121.523 q^{88} +615.323 q^{89} -4.25703 q^{90} -835.122 q^{91} -370.820 q^{92} -895.594 q^{93} +64.2108 q^{94} +188.965 q^{95} -93.9718 q^{96} -1309.43 q^{97} -81.1580 q^{98} -760.518 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 11 q^{3} + 2 q^{4} + 27 q^{5} - 27 q^{6} + 20 q^{7} + 66 q^{8} - 9 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 11 * q^3 + 2 * q^4 + 27 * q^5 - 27 * q^6 + 20 * q^7 + 66 * q^8 - 9 * q^9	\( 4 q + 4 q^{2} + 11 q^{3} + 2 q^{4} + 27 q^{5} - 27 q^{6} + 20 q^{7} + 66 q^{8} - 9 q^{9} - 3 q^{10} - 62 q^{11} - 61 q^{12} - 2 q^{13} + 112 q^{14} + 92 q^{15} + 202 q^{16} - 207 q^{17} - 299 q^{18} - 99 q^{19} - 81 q^{20} - 90 q^{21} - 202 q^{22} - 103 q^{23} + 209 q^{24} - 101 q^{25} + 50 q^{26} + 218 q^{27} + 80 q^{28} + 99 q^{29} - 300 q^{30} + 131 q^{31} + 342 q^{32} + 32 q^{33} - 53 q^{34} - 374 q^{35} - 379 q^{36} + 449 q^{37} - 609 q^{38} - 98 q^{39} + 133 q^{40} - 491 q^{41} + 394 q^{42} - 764 q^{44} + 338 q^{45} - 1061 q^{46} + 19 q^{47} - 237 q^{48} + 236 q^{49} - 599 q^{50} - 1649 q^{51} + 224 q^{52} - 1220 q^{53} - 322 q^{54} - 1360 q^{55} + 344 q^{56} + 232 q^{57} - 771 q^{58} + 816 q^{59} - 156 q^{60} - 372 q^{61} + 97 q^{62} - 1914 q^{63} + 434 q^{64} + 350 q^{65} + 812 q^{66} + 110 q^{67} - 1697 q^{68} + 1238 q^{69} + 718 q^{70} - 468 q^{71} + 315 q^{72} - 628 q^{73} + 395 q^{74} - 62 q^{75} - 1671 q^{76} + 2044 q^{77} - 90 q^{78} + 1095 q^{79} + 31 q^{80} + 2056 q^{81} + 2287 q^{82} - 980 q^{83} - 610 q^{84} + 152 q^{85} - 507 q^{87} - 1816 q^{88} + 738 q^{89} - 2398 q^{90} - 852 q^{91} - 2517 q^{92} + 35 q^{93} + 2233 q^{94} + 1149 q^{95} - 1551 q^{96} - 1765 q^{97} - 1652 q^{98} - 58 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 11 * q^3 + 2 * q^4 + 27 * q^5 - 27 * q^6 + 20 * q^7 + 66 * q^8 - 9 * q^9 - 3 * q^10 - 62 * q^11 - 61 * q^12 - 2 * q^13 + 112 * q^14 + 92 * q^15 + 202 * q^16 - 207 * q^17 - 299 * q^18 - 99 * q^19 - 81 * q^20 - 90 * q^21 - 202 * q^22 - 103 * q^23 + 209 * q^24 - 101 * q^25 + 50 * q^26 + 218 * q^27 + 80 * q^28 + 99 * q^29 - 300 * q^30 + 131 * q^31 + 342 * q^32 + 32 * q^33 - 53 * q^34 - 374 * q^35 - 379 * q^36 + 449 * q^37 - 609 * q^38 - 98 * q^39 + 133 * q^40 - 491 * q^41 + 394 * q^42 - 764 * q^44 + 338 * q^45 - 1061 * q^46 + 19 * q^47 - 237 * q^48 + 236 * q^49 - 599 * q^50 - 1649 * q^51 + 224 * q^52 - 1220 * q^53 - 322 * q^54 - 1360 * q^55 + 344 * q^56 + 232 * q^57 - 771 * q^58 + 816 * q^59 - 156 * q^60 - 372 * q^61 + 97 * q^62 - 1914 * q^63 + 434 * q^64 + 350 * q^65 + 812 * q^66 + 110 * q^67 - 1697 * q^68 + 1238 * q^69 + 718 * q^70 - 468 * q^71 + 315 * q^72 - 628 * q^73 + 395 * q^74 - 62 * q^75 - 1671 * q^76 + 2044 * q^77 - 90 * q^78 + 1095 * q^79 + 31 * q^80 + 2056 * q^81 + 2287 * q^82 - 980 * q^83 - 610 * q^84 + 152 * q^85 - 507 * q^87 - 1816 * q^88 + 738 * q^89 - 2398 * q^90 - 852 * q^91 - 2517 * q^92 + 35 * q^93 + 2233 * q^94 + 1149 * q^95 - 1551 * q^96 - 1765 * q^97 - 1652 * q^98 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.132290	−0.0467716	−0.0233858	−	0.999727i	$-0.507445\pi$
$2$	−0.132290	−0.0467716	−0.0233858	+	0.999727i	$0.507445\pi$
$3$	3.71054	0.714093	0.357047	−	0.934087i	$-0.383784\pi$
$3$	3.71054	0.714093	0.357047	+	0.934087i	$0.383784\pi$
$4$	−7.98250	−0.997812
$5$	−2.43196	−0.217521	−0.108761	−	0.994068i	$-0.534688\pi$
$5$	−2.43196	−0.217521	−0.108761	+	0.994068i	$0.534688\pi$
$6$	−0.490867	−0.0333993
$7$	30.9271	1.66991	0.834953	−	0.550321i	$-0.185495\pi$
$7$	30.9271	1.66991	0.834953	+	0.550321i	$0.185495\pi$
$8$	2.11433	0.0934409
$9$	−13.2319	−0.490071
$10$	0.321724	0.0101738
$11$	57.4761	1.57543	0.787713	−	0.616042i	$-0.211265\pi$
$11$	57.4761	1.57543	0.787713	+	0.616042i	$0.211265\pi$
$12$	−29.6194	−0.712531
$13$	−27.0029	−0.576097	−0.288049	−	0.957616i	$-0.593006\pi$
$13$	−27.0029	−0.576097	−0.288049	+	0.957616i	$0.593006\pi$
$14$	−4.09135	−0.0781042
$15$	−9.02388	−0.155330
$16$	63.5803	0.993442
$17$	−91.1332	−1.30018	−0.650090	−	0.759857i	$-0.725269\pi$
$17$	−91.1332	−1.30018	−0.650090	+	0.759857i	$0.725269\pi$
$18$	1.75045	0.0229214
$19$	−77.7008	−0.938200	−0.469100	−	0.883145i	$-0.655422\pi$
$19$	−77.7008	−0.938200	−0.469100	+	0.883145i	$0.655422\pi$
$20$	19.4131	0.217045
$21$	114.756	1.19247
$22$	−7.60352	−0.0736852
$23$	46.4541	0.421146	0.210573	−	0.977578i	$-0.432467\pi$
$23$	46.4541	0.421146	0.210573	+	0.977578i	$0.432467\pi$
$24$	7.84529	0.0667255
$25$	−119.086	−0.952685
$26$	3.57222	0.0269450
$27$	−149.282	−1.06405
$28$	−246.876	−1.66625
$29$	251.724	1.61186	0.805931	−	0.592010i	$-0.201665\pi$
$29$	251.724	1.61186	0.805931	+	0.592010i	$0.201665\pi$
$30$	1.19377	0.00726505
$31$	−241.365	−1.39840	−0.699200	−	0.714926i	$-0.746460\pi$
$31$	−241.365	−1.39840	−0.699200	+	0.714926i	$0.746460\pi$
$32$	−25.3256	−0.139906
$33$	213.267	1.12500
$34$	12.0560	0.0608115
$35$	−75.2135	−0.363240
$36$	105.624	0.488999
$37$	23.0183	0.102275	0.0511377	−	0.998692i	$-0.483715\pi$
$37$	23.0183	0.102275	0.0511377	+	0.998692i	$0.483715\pi$
$38$	10.2790	0.0438811
$39$	−100.195	−0.411387
$40$	−5.14196	−0.0203254
$41$	−4.94980	−0.0188544	−0.00942719	−	0.999956i	$-0.503001\pi$
$41$	−4.94980	−0.0188544	−0.00942719	+	0.999956i	$0.503001\pi$
$42$	−15.1811	−0.0557737
$43$	0	0
$43$	0	0
$44$	−458.803	−1.57198
$45$	32.1795	0.106601
$46$	−6.14542	−0.0196977
$47$	−485.378	−1.50638	−0.753189	−	0.657805i	$-0.771485\pi$
$47$	−485.378	−1.50638	−0.753189	+	0.657805i	$0.771485\pi$
$48$	235.917	0.709410
$49$	613.485	1.78859
$50$	15.7538	0.0445586
$51$	−338.153	−0.928450
$52$	215.551	0.574837
$53$	111.940	0.290115	0.145058	−	0.989423i	$-0.453663\pi$
$53$	111.940	0.290115	0.145058	+	0.989423i	$0.453663\pi$
$54$	19.7485	0.0497673
$55$	−139.780	−0.342689
$56$	65.3900	0.156038
$57$	−288.312	−0.669962
$58$	−33.3006	−0.0753893
$59$	544.546	1.20159	0.600795	−	0.799403i	$-0.294851\pi$
$59$	544.546	1.20159	0.600795	+	0.799403i	$0.294851\pi$
$60$	72.0331	0.154991
$61$	−367.434	−0.771231	−0.385615	−	0.922660i	$-0.626011\pi$
$61$	−367.434	−0.771231	−0.385615	+	0.922660i	$0.626011\pi$
$62$	31.9302	0.0654054
$63$	−409.224	−0.818372
$64$	−505.292	−0.986898
$65$	65.6700	0.125313
$66$	−28.2131	−0.0526181
$67$	−272.873	−0.497562	−0.248781	−	0.968560i	$-0.580030\pi$
$67$	−272.873	−0.497562	−0.248781	+	0.968560i	$0.580030\pi$
$68$	727.471	1.29734
$69$	172.370	0.300738
$70$	9.95000	0.0169893
$71$	37.7297	0.0630661	0.0315331	−	0.999503i	$-0.489961\pi$
$71$	37.7297	0.0630661	0.0315331	+	0.999503i	$0.489961\pi$
$72$	−27.9766	−0.0457926
$73$	−667.567	−1.07031	−0.535157	−	0.844753i	$-0.679747\pi$
$73$	−667.567	−1.07031	−0.535157	+	0.844753i	$0.679747\pi$
$74$	−3.04510	−0.00478358
$75$	−441.872	−0.680306
$76$	620.247	0.936147
$77$	1777.57	2.63081
$78$	13.2549	0.0192412
$79$	499.520	0.711398	0.355699	−	0.934601i	$-0.384243\pi$
$79$	499.520	0.711398	0.355699	+	0.934601i	$0.384243\pi$
$80$	−154.625	−0.216095
$81$	−196.655	−0.269760
$82$	0.654810	0.000881849 0
$83$	−432.180	−0.571541	−0.285771	−	0.958298i	$-0.592250\pi$
$83$	−432.180	−0.571541	−0.285771	+	0.958298i	$0.592250\pi$
$84$	−916.041	−1.18986
$85$	221.632	0.282817
$86$	0	0
$87$	934.031	1.15102
$88$	121.523	0.147209
$89$	615.323	0.732855	0.366427	−	0.930447i	$-0.380581\pi$
$89$	615.323	0.732855	0.366427	+	0.930447i	$0.380581\pi$
$90$	−4.25703	−0.00498589
$91$	−835.122	−0.962028
$92$	−370.820	−0.420225
$93$	−895.594	−0.998589
$94$	64.2108	0.0704557
$95$	188.965	0.204078
$96$	−93.9718	−0.0999058
$97$	−1309.43	−1.37064	−0.685320	−	0.728242i	$-0.740338\pi$
$97$	−1309.43	−1.37064	−0.685320	+	0.728242i	$0.740338\pi$
$98$	−81.1580	−0.0836551
$99$	−760.518	−0.772070
$100$	950.600	0.950600
$101$	309.272	0.304691	0.152345	−	0.988327i	$-0.451317\pi$
$101$	309.272	0.304691	0.152345	+	0.988327i	$0.451317\pi$
$102$	44.7343	0.0434251
$103$	−1004.70	−0.961131	−0.480566	−	0.876959i	$-0.659569\pi$
$103$	−1004.70	−0.961131	−0.480566	+	0.876959i	$0.659569\pi$
$104$	−57.0930	−0.0538310
$105$	−279.083	−0.259387
$106$	−14.8085	−0.0135692
$107$	441.417	0.398817	0.199408	−	0.979916i	$-0.436098\pi$
$107$	441.417	0.398817	0.199408	+	0.979916i	$0.436098\pi$
$108$	1191.64	1.06172
$109$	254.812	0.223913	0.111957	−	0.993713i	$-0.464288\pi$
$109$	254.812	0.223913	0.111957	+	0.993713i	$0.464288\pi$
$110$	18.4915	0.0160281
$111$	85.4104	0.0730342
$112$	1966.35	1.65896
$113$	1569.73	1.30679	0.653396	−	0.757017i	$-0.273344\pi$
$113$	1569.73	1.30679	0.653396	+	0.757017i	$0.273344\pi$
$114$	38.1408	0.0313352
$115$	−112.975	−0.0916082
$116$	−2009.39	−1.60833
$117$	357.300	0.282328
$118$	−72.0380	−0.0562003
$119$	−2818.49	−2.17118
$120$	−19.0794	−0.0145142
$121$	1972.50	1.48197
$122$	48.6078	0.0360717
$123$	−18.3664	−0.0134638
$124$	1926.70	1.39534
$125$	593.607	0.424750
$126$	54.1363	0.0382766
$127$	1261.51	0.881427	0.440713	−	0.897648i	$-0.354726\pi$
$127$	1261.51	0.881427	0.440713	+	0.897648i	$0.354726\pi$
$128$	269.450	0.186065
$129$	0	0
$130$	−8.68749	−0.00586110
$131$	778.971	0.519534	0.259767	−	0.965671i	$-0.416354\pi$
$131$	778.971	0.519534	0.259767	+	0.965671i	$0.416354\pi$
$132$	−1702.41	−1.12254
$133$	−2403.06	−1.56671
$134$	36.0983	0.0232718
$135$	363.048	0.231453
$136$	−192.685	−0.121490
$137$	−1898.45	−1.18391	−0.591954	−	0.805971i	$-0.701643\pi$
$137$	−1898.45	−1.18391	−0.591954	+	0.805971i	$0.701643\pi$
$138$	−22.8028	−0.0140660
$139$	−3168.52	−1.93346	−0.966729	−	0.255805i	$-0.917660\pi$
$139$	−3168.52	−1.93346	−0.966729	+	0.255805i	$0.917660\pi$
$140$	600.392	0.362445
$141$	−1801.02	−1.07569
$142$	−4.99127	−0.00294970
$143$	−1552.02	−0.907599
$144$	−841.288	−0.486857
$145$	−612.183	−0.350614
$146$	88.3125	0.0500603
$147$	2276.36	1.27722
$148$	−183.744	−0.102052
$149$	925.613	0.508921	0.254460	−	0.967083i	$-0.418102\pi$
$149$	925.613	0.508921	0.254460	+	0.967083i	$0.418102\pi$
$150$	58.4552	0.0318190
$151$	−823.904	−0.444029	−0.222014	−	0.975043i	$-0.571263\pi$
$151$	−823.904	−0.444029	−0.222014	+	0.975043i	$0.571263\pi$
$152$	−164.285	−0.0876662
$153$	1205.87	0.637180
$154$	−235.155	−0.123047
$155$	586.990	0.304182
$156$	799.809	0.410487
$157$	−642.320	−0.326514	−0.163257	−	0.986584i	$-0.552200\pi$
$157$	−642.320	−0.326514	−0.163257	+	0.986584i	$0.552200\pi$
$158$	−66.0816	−0.0332732
$159$	415.357	0.207169
$160$	61.5910	0.0304325
$161$	1436.69	0.703274
$162$	26.0155	0.0126171
$163$	−78.7938	−0.0378626	−0.0189313	−	0.999821i	$-0.506026\pi$
$163$	−78.7938	−0.0378626	−0.0189313	+	0.999821i	$0.506026\pi$
$164$	39.5118	0.0188131
$165$	−518.658	−0.244712
$166$	57.1731	0.0267319
$167$	−319.779	−0.148175	−0.0740877	−	0.997252i	$-0.523604\pi$
$167$	−319.779	−0.148175	−0.0740877	+	0.997252i	$0.523604\pi$
$168$	242.632	0.111425
$169$	−1467.84	−0.668112
$170$	−29.3198	−0.0132278
$171$	1028.13	0.459784
$172$	0	0
$173$	−2391.17	−1.05085	−0.525426	−	0.850839i	$-0.676094\pi$
$173$	−2391.17	−1.05085	−0.525426	+	0.850839i	$0.676094\pi$
$174$	−123.563	−0.0538350
$175$	−3682.97	−1.59089
$176$	3654.35	1.56510
$177$	2020.56	0.858048
$178$	−81.4011	−0.0342768
$179$	−3584.04	−1.49656	−0.748278	−	0.663385i	$-0.769119\pi$
$179$	−3584.04	−1.49656	−0.748278	+	0.663385i	$0.769119\pi$
$180$	−256.873	−0.106368
$181$	330.039	0.135534	0.0677670	−	0.997701i	$-0.478413\pi$
$181$	330.039	0.135534	0.0677670	+	0.997701i	$0.478413\pi$
$182$	110.478	0.0449956
$183$	−1363.38	−0.550731
$184$	98.2192	0.0393523
$185$	−55.9797	−0.0222471
$186$	118.478	0.0467056
$187$	−5237.98	−2.04834
$188$	3874.53	1.50308
$189$	−4616.86	−1.77686
$190$	−24.9982	−0.00954507
$191$	−3076.65	−1.16554	−0.582771	−	0.812637i	$-0.698031\pi$
$191$	−3076.65	−1.16554	−0.582771	+	0.812637i	$0.698031\pi$
$192$	−1874.91	−0.704738
$193$	1036.86	0.386707	0.193354	−	0.981129i	$-0.438064\pi$
$193$	1036.86	0.386707	0.193354	+	0.981129i	$0.438064\pi$
$194$	173.224	0.0641070
$195$	243.671	0.0894854
$196$	−4897.15	−1.78467
$197$	−4086.58	−1.47795	−0.738977	−	0.673731i	$-0.764691\pi$
$197$	−4086.58	−1.47795	−0.738977	+	0.673731i	$0.764691\pi$
$198$	100.609	0.0361110
$199$	−1092.91	−0.389319	−0.194659	−	0.980871i	$-0.562360\pi$
$199$	−1092.91	−0.389319	−0.194659	+	0.980871i	$0.562360\pi$
$200$	−251.786	−0.0890197
$201$	−1012.50	−0.355306
$202$	−40.9137	−0.0142509
$203$	7785.09	2.69166
$204$	2699.31	0.926419
$205$	12.0377	0.00410123
$206$	132.912	0.0449536
$207$	−614.677	−0.206391
$208$	−1716.85	−0.572319
$209$	−4465.94	−1.47806
$210$	36.9198	0.0121320
$211$	−5236.96	−1.70866	−0.854330	−	0.519731i	$-0.826032\pi$
$211$	−5236.96	−1.70866	−0.854330	+	0.519731i	$0.826032\pi$
$212$	−893.559	−0.289481
$213$	139.998	0.0450351
$214$	−58.3951	−0.0186533
$215$	0	0
$216$	−315.631	−0.0994257
$217$	−7464.72	−2.33520
$218$	−33.7091	−0.0104728
$219$	−2477.03	−0.764304
$220$	1115.79	0.341939
$221$	2460.86	0.749030
$222$	−11.2989	−0.00341592
$223$	3098.28	0.930388	0.465194	−	0.885209i	$-0.345985\pi$
$223$	3098.28	0.930388	0.465194	+	0.885209i	$0.345985\pi$
$224$	−783.249	−0.233630
$225$	1575.73	0.466883
$226$	−207.659	−0.0611207
$227$	−252.410	−0.0738020	−0.0369010	−	0.999319i	$-0.511749\pi$
$227$	−252.410	−0.0738020	−0.0369010	+	0.999319i	$0.511749\pi$
$228$	2301.45	0.668497
$229$	−3780.06	−1.09080	−0.545400	−	0.838176i	$-0.683622\pi$
$229$	−3780.06	−1.09080	−0.545400	+	0.838176i	$0.683622\pi$
$230$	14.9454	0.00428466
$231$	6595.74	1.87865
$232$	532.227	0.150614
$233$	−5493.20	−1.54451	−0.772257	−	0.635310i	$-0.780872\pi$
$233$	−5493.20	−1.54451	−0.772257	+	0.635310i	$0.780872\pi$
$234$	−47.2672	−0.0132049
$235$	1180.42	0.327669
$236$	−4346.84	−1.19896
$237$	1853.49	0.508004
$238$	372.858	0.101549
$239$	−6634.15	−1.79551	−0.897756	−	0.440493i	$-0.854804\pi$
$239$	−6634.15	−1.79551	−0.897756	+	0.440493i	$0.854804\pi$
$240$	−573.741	−0.154312
$241$	4967.50	1.32774	0.663869	−	0.747849i	$-0.268913\pi$
$241$	4967.50	1.32774	0.663869	+	0.747849i	$0.268913\pi$
$242$	−260.942	−0.0693141
$243$	3300.92	0.871416
$244$	2933.04	0.769543
$245$	−1491.97	−0.389056
$246$	2.42970	0.000629723 0
$247$	2098.15	0.540494
$248$	−510.324	−0.130668
$249$	−1603.62	−0.408134
$250$	−78.5283	−0.0198663
$251$	−4714.86	−1.18565	−0.592827	−	0.805330i	$-0.701988\pi$
$251$	−4714.86	−1.18565	−0.592827	+	0.805330i	$0.701988\pi$
$252$	3266.63	0.816582
$253$	2670.00	0.663485
$254$	−166.886	−0.0412257
$255$	822.376	0.201958
$256$	4006.69	0.978196
$257$	1572.21	0.381602	0.190801	−	0.981629i	$-0.438892\pi$
$257$	1572.21	0.381602	0.190801	+	0.981629i	$0.438892\pi$
$258$	0	0
$259$	711.890	0.170790
$260$	−524.211	−0.125039
$261$	−3330.79	−0.789926
$262$	−103.050	−0.0242995
$263$	8379.06	1.96454	0.982272	−	0.187462i	$-0.0600261\pi$
$263$	8379.06	1.96454	0.982272	+	0.187462i	$0.0600261\pi$
$264$	450.916	0.105121
$265$	−272.233	−0.0631062
$266$	317.901	0.0732773
$267$	2283.18	0.523327
$268$	2178.20	0.496474
$269$	912.352	0.206792	0.103396	−	0.994640i	$-0.467029\pi$
$269$	912.352	0.206792	0.103396	+	0.994640i	$0.467029\pi$
$270$	−48.0277	−0.0108254
$271$	5724.87	1.28325	0.641625	−	0.767018i	$-0.278260\pi$
$271$	5724.87	1.28325	0.641625	+	0.767018i	$0.278260\pi$
$272$	−5794.28	−1.29165
$273$	−3098.75	−0.686978
$274$	251.146	0.0553733
$275$	−6844.57	−1.50088
$276$	−1375.94	−0.300080
$277$	−2764.17	−0.599578	−0.299789	−	0.954006i	$-0.596916\pi$
$277$	−2764.17	−0.599578	−0.299789	+	0.954006i	$0.596916\pi$
$278$	419.164	0.0904309
$279$	3193.72	0.685315
$280$	−159.026	−0.0339415
$281$	−6620.94	−1.40559	−0.702797	−	0.711390i	$-0.748066\pi$
$281$	−6620.94	−1.40559	−0.702797	+	0.711390i	$0.748066\pi$
$282$	238.256	0.0503119
$283$	−1927.22	−0.404811	−0.202405	−	0.979302i	$-0.564876\pi$
$283$	−1927.22	−0.404811	−0.202405	+	0.979302i	$0.564876\pi$
$284$	−301.178	−0.0629282
$285$	701.163	0.145731
$286$	205.317	0.0424498
$287$	−153.083	−0.0314850
$288$	335.107	0.0685637
$289$	3392.27	0.690467
$290$	80.9857	0.0163988
$291$	−4858.68	−0.978765
$292$	5328.86	1.06797
$293$	4848.29	0.966689	0.483344	−	0.875430i	$-0.339422\pi$
$293$	4848.29	0.966689	0.483344	+	0.875430i	$0.339422\pi$
$294$	−301.140	−0.0597375
$295$	−1324.31	−0.261371
$296$	48.6682	0.00955670
$297$	−8580.15	−1.67633
$298$	−122.449	−0.0238030
$299$	−1254.40	−0.242621
$300$	3527.24	0.678818
$301$	0	0
$302$	108.994	0.0207679
$303$	1147.57	0.217578
$304$	−4940.24	−0.932047
$305$	893.584	0.167759
$306$	−159.524	−0.0298019
$307$	2408.94	0.447835	0.223918	−	0.974608i	$-0.428115\pi$
$307$	2408.94	0.447835	0.223918	+	0.974608i	$0.428115\pi$
$308$	−14189.4	−2.62506
$309$	−3728.00	−0.686337
$310$	−77.6530	−0.0142271
$311$	−5325.15	−0.970937	−0.485468	−	0.874254i	$-0.661351\pi$
$311$	−5325.15	−0.970937	−0.485468	+	0.874254i	$0.661351\pi$
$312$	−211.846	−0.0384404
$313$	−1232.32	−0.222540	−0.111270	−	0.993790i	$-0.535492\pi$
$313$	−1232.32	−0.222540	−0.111270	+	0.993790i	$0.535492\pi$
$314$	84.9726	0.0152716
$315$	995.218	0.178013
$316$	−3987.42	−0.709842
$317$	−3291.92	−0.583258	−0.291629	−	0.956532i	$-0.594197\pi$
$317$	−3291.92	−0.583258	−0.291629	+	0.956532i	$0.594197\pi$
$318$	−54.9476	−0.00968965
$319$	14468.1	2.53937
$320$	1228.85	0.214671
$321$	1637.90	0.284793
$322$	−190.060	−0.0328933
$323$	7081.13	1.21983
$324$	1569.80	0.269170
$325$	3215.66	0.548839
$326$	10.4236	0.00177089
$327$	945.489	0.159895
$328$	−10.4655	−0.00176177
$329$	−15011.3	−2.51551
$330$	68.6132	0.0114456
$331$	−11506.5	−1.91074	−0.955369	−	0.295414i	$-0.904543\pi$
$331$	−11506.5	−1.91074	−0.955369	+	0.295414i	$0.904543\pi$
$332$	3449.88	0.570291
$333$	−304.576	−0.0501221
$334$	42.3036	0.00693040
$335$	663.615	0.108230
$336$	7296.23	1.18465
$337$	7227.26	1.16823	0.584115	−	0.811671i	$-0.301442\pi$
$337$	7227.26	1.16823	0.584115	+	0.811671i	$0.301442\pi$
$338$	194.181	0.0312487
$339$	5824.53	0.933171
$340$	−1769.18	−0.282198
$341$	−13872.7	−2.20308
$342$	−136.011	−0.0215048
$343$	8365.32	1.31687
$344$	0	0
$345$	−419.197	−0.0654168
$346$	316.328	0.0491500
$347$	2385.97	0.369123	0.184562	−	0.982821i	$-0.440913\pi$
$347$	2385.97	0.369123	0.184562	+	0.982821i	$0.440913\pi$
$348$	−7455.91	−1.14850
$349$	−4263.54	−0.653931	−0.326965	−	0.945036i	$-0.606026\pi$
$349$	−4263.54	−0.653931	−0.326965	+	0.945036i	$0.606026\pi$
$350$	487.220	0.0744087
$351$	4031.05	0.612996
$352$	−1455.62	−0.220411
$353$	−4737.79	−0.714354	−0.357177	−	0.934037i	$-0.616261\pi$
$353$	−4737.79	−0.714354	−0.357177	+	0.934037i	$0.616261\pi$
$354$	−267.300	−0.0401323
$355$	−91.7572	−0.0137182
$356$	−4911.81	−0.731251
$357$	−10458.1	−1.55042
$358$	474.133	0.0699964
$359$	7306.54	1.07416	0.537081	−	0.843530i	$-0.319527\pi$
$359$	7306.54	1.07416	0.537081	+	0.843530i	$0.319527\pi$
$360$	68.0379	0.00996087
$361$	−821.582	−0.119782
$362$	−43.6609	−0.00633914
$363$	7319.04	1.05826
$364$	6666.36	0.959924
$365$	1623.50	0.232816
$366$	180.361	0.0257586
$367$	−2659.87	−0.378321	−0.189161	−	0.981946i	$-0.560577\pi$
$367$	−2659.87	−0.378321	−0.189161	+	0.981946i	$0.560577\pi$
$368$	2953.57	0.418384
$369$	65.4953	0.00923997
$370$	7.40555	0.00104053
$371$	3461.97	0.484465
$372$	7149.08	0.996404
$373$	−4081.31	−0.566547	−0.283274	−	0.959039i	$-0.591420\pi$
$373$	−4081.31	−0.566547	−0.283274	+	0.959039i	$0.591420\pi$
$374$	692.933	0.0958040
$375$	2202.60	0.303311
$376$	−1026.25	−0.140757
$377$	−6797.28	−0.928588
$378$	610.765	0.0831067
$379$	6532.37	0.885344	0.442672	−	0.896684i	$-0.354031\pi$
$379$	6532.37	0.885344	0.442672	+	0.896684i	$0.354031\pi$
$380$	−1508.42	−0.203632
$381$	4680.89	0.629421
$382$	407.010	0.0545143
$383$	13420.7	1.79052	0.895258	−	0.445548i	$-0.146991\pi$
$383$	13420.7	1.79052	0.895258	+	0.445548i	$0.146991\pi$
$384$	999.806	0.132867
$385$	−4322.98	−0.572258
$386$	−137.166	−0.0180869
$387$	0	0
$388$	10452.5	1.36764
$389$	6680.03	0.870671	0.435335	−	0.900268i	$-0.356630\pi$
$389$	6680.03	0.870671	0.435335	+	0.900268i	$0.356630\pi$
$390$	−32.2353	−0.00418538
$391$	−4233.52	−0.547566
$392$	1297.11	0.167127
$393$	2890.40	0.370996
$394$	540.614	0.0691262
$395$	−1214.81	−0.154744
$396$	6070.84	0.770381
$397$	1343.73	0.169873	0.0849367	−	0.996386i	$-0.472931\pi$
$397$	1343.73	0.169873	0.0849367	+	0.996386i	$0.472931\pi$
$398$	144.581	0.0182091
$399$	−8916.65	−1.11877
$400$	−7571.49	−0.946437
$401$	−3601.71	−0.448531	−0.224266	−	0.974528i	$-0.571998\pi$
$401$	−3601.71	−0.448531	−0.224266	+	0.974528i	$0.571998\pi$
$402$	133.944	0.0166182
$403$	6517.56	0.805614
$404$	−2468.77	−0.304024
$405$	478.258	0.0586786
$406$	−1029.89	−0.125893
$407$	1323.00	0.161127
$408$	−714.966	−0.0867552
$409$	9576.98	1.15783	0.578913	−	0.815389i	$-0.303477\pi$
$409$	9576.98	1.15783	0.578913	+	0.815389i	$0.303477\pi$
$410$	−1.59247	−0.000191821 0
$411$	−7044.27	−0.845422
$412$	8020.05	0.959028
$413$	16841.2	2.00654
$414$	81.3156	0.00965325
$415$	1051.04	0.124322
$416$	683.866	0.0805993
$417$	−11756.9	−1.38067
$418$	590.799	0.0691314
$419$	−2324.86	−0.271066	−0.135533	−	0.990773i	$-0.543275\pi$
$419$	−2324.86	−0.271066	−0.135533	+	0.990773i	$0.543275\pi$
$420$	2227.78	0.258820
$421$	5759.27	0.666721	0.333361	−	0.942799i	$-0.391817\pi$
$421$	5759.27	0.666721	0.333361	+	0.942799i	$0.391817\pi$
$422$	692.798	0.0799168
$423$	6422.48	0.738231
$424$	236.677	0.0271086
$425$	10852.7	1.23866
$426$	−18.5203	−0.00210636
$427$	−11363.7	−1.28788
$428$	−3523.61	−0.397945
$429$	−5758.84	−0.648110
$430$	0	0
$431$	3848.55	0.430112	0.215056	−	0.976602i	$-0.431007\pi$
$431$	3848.55	0.430112	0.215056	+	0.976602i	$0.431007\pi$
$432$	−9491.39	−1.05707
$433$	−11834.5	−1.31347	−0.656734	−	0.754122i	$-0.728063\pi$
$433$	−11834.5	−1.31347	−0.656734	+	0.754122i	$0.728063\pi$
$434$	987.508	0.109221
$435$	−2271.53	−0.250371
$436$	−2034.04	−0.223423
$437$	−3609.53	−0.395119
$438$	327.687	0.0357477
$439$	6385.94	0.694269	0.347135	−	0.937815i	$-0.387155\pi$
$439$	6385.94	0.694269	0.347135	+	0.937815i	$0.387155\pi$
$440$	−295.540	−0.0320211
$441$	−8117.58	−0.876534
$442$	−325.548	−0.0350333
$443$	16117.1	1.72855	0.864273	−	0.503023i	$-0.167779\pi$
$443$	16117.1	1.72855	0.864273	+	0.503023i	$0.167779\pi$
$444$	−681.788	−0.0728744
$445$	−1496.44	−0.159411
$446$	−409.872	−0.0435157
$447$	3434.52	0.363417
$448$	−15627.2	−1.64803
$449$	−8306.13	−0.873030	−0.436515	−	0.899697i	$-0.643787\pi$
$449$	−8306.13	−0.873030	−0.436515	+	0.899697i	$0.643787\pi$
$450$	−208.453	−0.0218369
$451$	−284.495	−0.0297037
$452$	−12530.3	−1.30393
$453$	−3057.13	−0.317078
$454$	33.3914	0.00345184
$455$	2030.98	0.209261
$456$	−609.585	−0.0626019
$457$	2707.96	0.277184	0.138592	−	0.990350i	$-0.455742\pi$
$457$	2707.96	0.277184	0.138592	+	0.990350i	$0.455742\pi$
$458$	500.064	0.0510185
$459$	13604.6	1.38346
$460$	901.820	0.0914078
$461$	−6196.52	−0.626031	−0.313016	−	0.949748i	$-0.601339\pi$
$461$	−6196.52	−0.626031	−0.313016	+	0.949748i	$0.601339\pi$
$462$	−872.550	−0.0878674
$463$	11030.1	1.10715	0.553575	−	0.832800i	$-0.313263\pi$
$463$	11030.1	1.10715	0.553575	+	0.832800i	$0.313263\pi$
$464$	16004.7	1.60129
$465$	2178.05	0.217214
$466$	726.696	0.0722394
$467$	−11555.3	−1.14500	−0.572500	−	0.819905i	$-0.694026\pi$
$467$	−11555.3	−1.14500	−0.572500	+	0.819905i	$0.694026\pi$
$468$	−2852.15	−0.281711
$469$	−8439.15	−0.830883
$470$	−156.158	−0.0153256
$471$	−2383.35	−0.233162
$472$	1151.35	0.112278
$473$	0	0
$474$	−245.198	−0.0237602
$475$	9253.05	0.893808
$476$	22498.6	2.16643
$477$	−1481.18	−0.142177
$478$	877.632	0.0839790
$479$	8325.68	0.794176	0.397088	−	0.917781i	$-0.370021\pi$
$479$	8325.68	0.794176	0.397088	+	0.917781i	$0.370021\pi$
$480$	228.536	0.0217316
$481$	−621.562	−0.0589205
$482$	−657.151	−0.0621004
$483$	5330.90	0.502204
$484$	−15745.5	−1.47873
$485$	3184.47	0.298143
$486$	−436.679	−0.0407575
$487$	10138.3	0.943351	0.471676	−	0.881772i	$-0.343649\pi$
$487$	10138.3	0.943351	0.471676	+	0.881772i	$0.343649\pi$
$488$	−776.875	−0.0720645
$489$	−292.367	−0.0270374
$490$	197.373	0.0181968
$491$	2881.74	0.264870	0.132435	−	0.991192i	$-0.457720\pi$
$491$	2881.74	0.264870	0.132435	+	0.991192i	$0.457720\pi$
$492$	146.610	0.0134343
$493$	−22940.4	−2.09571
$494$	−277.564	−0.0252798
$495$	1849.55	0.167942
$496$	−15346.1	−1.38923
$497$	1166.87	0.105315
$498$	212.143	0.0190891
$499$	−11286.4	−1.01252	−0.506260	−	0.862381i	$-0.668972\pi$
$499$	−11286.4	−1.01252	−0.506260	+	0.862381i	$0.668972\pi$
$500$	−4738.46	−0.423821
$501$	−1186.55	−0.105811
$502$	623.729	0.0554549
$503$	2558.17	0.226765	0.113383	−	0.993551i	$-0.463831\pi$
$503$	2558.17	0.226765	0.113383	+	0.993551i	$0.463831\pi$
$504$	−865.234	−0.0764694
$505$	−752.138	−0.0662767
$506$	−353.215	−0.0310322
$507$	−5446.49	−0.477095
$508$	−10070.0	−0.879498
$509$	−9563.36	−0.832787	−0.416393	−	0.909185i	$-0.636706\pi$
$509$	−9563.36	−0.832787	−0.416393	+	0.909185i	$0.636706\pi$
$510$	−108.792	−0.00944588
$511$	−20645.9	−1.78732
$512$	−2685.65	−0.231816
$513$	11599.3	0.998291
$514$	−207.988	−0.0178481
$515$	2443.40	0.209066
$516$	0	0
$517$	−27897.7	−2.37319
$518$	−94.1759	−0.00798813
$519$	−8872.53	−0.750406
$520$	138.848	0.0117094
$521$	−2877.00	−0.241926	−0.120963	−	0.992657i	$-0.538598\pi$
$521$	−2877.00	−0.241926	−0.120963	+	0.992657i	$0.538598\pi$
$522$	440.630	0.0369461
$523$	−3325.92	−0.278073	−0.139036	−	0.990287i	$-0.544401\pi$
$523$	−3325.92	−0.278073	−0.139036	+	0.990287i	$0.544401\pi$
$524$	−6218.13	−0.518398
$525$	−13665.8	−1.13605
$526$	−1108.47	−0.0918849
$527$	21996.4	1.81817
$528$	13559.6	1.11762
$529$	−10009.0	−0.822636
$530$	36.0138	0.00295158
$531$	−7205.38	−0.588864
$532$	19182.4	1.56328
$533$	133.659	0.0108619
$534$	−302.042	−0.0244768
$535$	−1073.51	−0.0867511
$536$	−576.941	−0.0464927
$537$	−13298.7	−1.06868
$538$	−120.695	−0.00967200
$539$	35260.7	2.81779
$540$	−2898.03	−0.230947
$541$	−5538.22	−0.440123	−0.220062	−	0.975486i	$-0.570626\pi$
$541$	−5538.22	−0.440123	−0.220062	+	0.975486i	$0.570626\pi$
$542$	−757.343	−0.0600197
$543$	1224.62	0.0967839
$544$	2308.01	0.181903
$545$	−619.692	−0.0487059
$546$	409.934	0.0321311
$547$	−8315.46	−0.649988	−0.324994	−	0.945716i	$-0.605362\pi$
$547$	−8315.46	−0.649988	−0.324994	+	0.945716i	$0.605362\pi$
$548$	15154.4	1.18132
$549$	4861.85	0.377957
$550$	905.469	0.0701988
$551$	−19559.2	−1.51225
$552$	364.446	0.0281012
$553$	15448.7	1.18797
$554$	365.672	0.0280432
$555$	−207.715	−0.0158865
$556$	25292.7	1.92923
$557$	14618.8	1.11206	0.556030	−	0.831162i	$-0.312324\pi$
$557$	14618.8	1.11206	0.556030	+	0.831162i	$0.312324\pi$
$558$	−422.497	−0.0320533
$559$	0	0
$560$	−4782.10	−0.360858
$561$	−19435.7	−1.46270
$562$	875.884	0.0657419
$563$	−11435.0	−0.856002	−0.428001	−	0.903778i	$-0.640782\pi$
$563$	−11435.0	−0.856002	−0.428001	+	0.903778i	$0.640782\pi$
$564$	14376.6	1.07334
$565$	−3817.51	−0.284255
$566$	254.952	0.0189336
$567$	−6081.97	−0.450474
$568$	79.7729	0.00589296
$569$	−12715.1	−0.936809	−0.468405	−	0.883514i	$-0.655171\pi$
$569$	−12715.1	−0.936809	−0.468405	+	0.883514i	$0.655171\pi$
$570$	−92.7569	−0.00681607
$571$	−10477.2	−0.767880	−0.383940	−	0.923358i	$-0.625433\pi$
$571$	−10477.2	−0.767880	−0.383940	+	0.923358i	$0.625433\pi$
$572$	12389.0	0.905613
$573$	−11416.0	−0.832306
$574$	20.2514	0.00147261
$575$	−5532.02	−0.401219
$576$	6685.98	0.483650
$577$	−4578.85	−0.330364	−0.165182	−	0.986263i	$-0.552821\pi$
$577$	−4578.85	−0.330364	−0.165182	+	0.986263i	$0.552821\pi$
$578$	−448.763	−0.0322943
$579$	3847.29	0.276145
$580$	4886.75	0.349847
$581$	−13366.1	−0.954420
$582$	642.755	0.0457784
$583$	6433.86	0.457055
$584$	−1411.46	−0.100011
$585$	−868.940	−0.0614124
$586$	−641.380	−0.0452136
$587$	−16380.1	−1.15175	−0.575877	−	0.817536i	$-0.695339\pi$
$587$	−16380.1	−1.15175	−0.575877	+	0.817536i	$0.695339\pi$
$588$	−18171.0	−1.27442
$589$	18754.3	1.31198
$590$	175.194	0.0122248
$591$	−15163.4	−1.05540
$592$	1463.51	0.101605
$593$	−8930.17	−0.618412	−0.309206	−	0.950995i	$-0.600063\pi$
$593$	−8930.17	−0.618412	−0.309206	+	0.950995i	$0.600063\pi$
$594$	1135.07	0.0784047
$595$	6854.45	0.472277
$596$	−7388.71	−0.507807
$597$	−4055.29	−0.278010
$598$	165.944	0.0113478
$599$	4858.09	0.331380	0.165690	−	0.986178i	$-0.447015\pi$
$599$	4858.09	0.331380	0.165690	+	0.986178i	$0.447015\pi$
$600$	−934.260	−0.0635684
$601$	−13436.8	−0.911981	−0.455990	−	0.889985i	$-0.650715\pi$
$601$	−13436.8	−0.911981	−0.455990	+	0.889985i	$0.650715\pi$
$602$	0	0
$603$	3610.62	0.243841
$604$	6576.81	0.443058
$605$	−4797.04	−0.322360
$606$	−151.812	−0.0101765
$607$	−16795.4	−1.12307	−0.561536	−	0.827452i	$-0.689790\pi$
$607$	−16795.4	−1.12307	−0.561536	+	0.827452i	$0.689790\pi$
$608$	1967.82	0.131260
$609$	28886.9	1.92209
$610$	−118.212	−0.00784636
$611$	13106.6	0.867819
$612$	−9625.83	−0.635786
$613$	16383.5	1.07948	0.539742	−	0.841830i	$-0.318522\pi$
$613$	16383.5	1.07948	0.539742	+	0.841830i	$0.318522\pi$
$614$	−318.679	−0.0209460
$615$	44.6664	0.00292866
$616$	3758.36	0.245826
$617$	1621.64	0.105810	0.0529050	−	0.998600i	$-0.483152\pi$
$617$	1621.64	0.105810	0.0529050	+	0.998600i	$0.483152\pi$
$618$	493.177	0.0321011
$619$	−22067.9	−1.43293	−0.716465	−	0.697623i	$-0.754241\pi$
$619$	−22067.9	−1.43293	−0.716465	+	0.697623i	$0.754241\pi$
$620$	−4685.65	−0.303516
$621$	−6934.77	−0.448120
$622$	704.464	0.0454123
$623$	19030.1	1.22380
$624$	−6370.45	−0.408689
$625$	13442.1	0.860292
$626$	163.024	0.0104085
$627$	−16571.0	−1.05548
$628$	5127.32	0.325800
$629$	−2097.73	−0.132976
$630$	−131.657	−0.00832596
$631$	13513.9	0.852582	0.426291	−	0.904586i	$-0.359820\pi$
$631$	13513.9	0.852582	0.426291	+	0.904586i	$0.359820\pi$
$632$	1056.15	0.0664736
$633$	−19431.9	−1.22014
$634$	435.489	0.0272799
$635$	−3067.95	−0.191729
$636$	−3315.59	−0.206716
$637$	−16565.9	−1.03040
$638$	−1913.99	−0.118770
$639$	−499.236	−0.0309069
$640$	−655.293	−0.0404730
$641$	26991.5	1.66319	0.831593	−	0.555386i	$-0.187430\pi$
$641$	26991.5	1.66319	0.831593	+	0.555386i	$0.187430\pi$
$642$	−216.677	−0.0133202
$643$	−9851.96	−0.604236	−0.302118	−	0.953271i	$-0.597694\pi$
$643$	−9851.96	−0.604236	−0.302118	+	0.953271i	$0.597694\pi$
$644$	−11468.4	−0.701736
$645$	0	0
$646$	−936.763	−0.0570533
$647$	19739.3	1.19943	0.599716	−	0.800213i	$-0.295280\pi$
$647$	19739.3	1.19943	0.599716	+	0.800213i	$0.295280\pi$
$648$	−415.793	−0.0252066
$649$	31298.4	1.89302
$650$	−425.400	−0.0256701
$651$	−27698.1	−1.66755
$652$	628.971	0.0377798
$653$	29887.8	1.79112	0.895560	−	0.444941i	$-0.146775\pi$
$653$	29887.8	1.79112	0.895560	+	0.444941i	$0.146775\pi$
$654$	−125.079	−0.00747855
$655$	−1894.43	−0.113010
$656$	−314.710	−0.0187307
$657$	8833.19	0.524529
$658$	1985.85	0.117654
$659$	21046.1	1.24407	0.622034	−	0.782990i	$-0.286307\pi$
$659$	21046.1	1.24407	0.622034	+	0.782990i	$0.286307\pi$
$660$	4140.18	0.244176
$661$	−26741.1	−1.57354	−0.786768	−	0.617249i	$-0.788247\pi$
$661$	−26741.1	−1.57354	−0.786768	+	0.617249i	$0.788247\pi$
$662$	1522.20	0.0893683
$663$	9131.13	0.534877
$664$	−913.769	−0.0534053
$665$	5844.15	0.340792
$666$	40.2924	0.00234429
$667$	11693.6	0.678829
$668$	2552.64	0.147851
$669$	11496.3	0.664384
$670$	−87.7897	−0.00506211
$671$	−21118.7	−1.21502
$672$	−2906.27	−0.166833
$673$	18729.3	1.07275	0.536376	−	0.843979i	$-0.319793\pi$
$673$	18729.3	1.07275	0.536376	+	0.843979i	$0.319793\pi$
$674$	−956.094	−0.0546400
$675$	17777.3	1.01370
$676$	11717.1	0.666651
$677$	−9741.75	−0.553036	−0.276518	−	0.961009i	$-0.589181\pi$
$677$	−9741.75	−0.553036	−0.276518	+	0.961009i	$0.589181\pi$
$678$	−770.527	−0.0436459
$679$	−40496.8	−2.28884
$680$	468.603	0.0264266
$681$	−936.578	−0.0527015
$682$	1835.22	0.103041
$683$	6993.85	0.391819	0.195909	−	0.980622i	$-0.437234\pi$
$683$	6993.85	0.391819	0.195909	+	0.980622i	$0.437234\pi$
$684$	−8207.05	−0.458778
$685$	4616.96	0.257525
$686$	−1106.65	−0.0615919
$687$	−14026.1	−0.778934
$688$	0	0
$689$	−3022.70	−0.167135
$690$	55.4556	0.00305965
$691$	23393.5	1.28789	0.643944	−	0.765072i	$-0.277297\pi$
$691$	23393.5	1.28789	0.643944	+	0.765072i	$0.277297\pi$
$692$	19087.5	1.04855
$693$	−23520.6	−1.28928
$694$	−315.641	−0.0172645
$695$	7705.72	0.420568
$696$	1974.85	0.107552
$697$	451.091	0.0245141
$698$	564.024	0.0305854
$699$	−20382.7	−1.10293
$700$	29399.3	1.58741
$701$	−28391.0	−1.52969	−0.764845	−	0.644214i	$-0.777184\pi$
$701$	−28391.0	−1.52969	−0.764845	+	0.644214i	$0.777184\pi$
$702$	−533.268	−0.0286708
$703$	−1788.54	−0.0959547
$704$	−29042.2	−1.55479
$705$	4380.00	0.233986
$706$	626.762	0.0334115
$707$	9564.90	0.508805
$708$	−16129.1	−0.856171
$709$	9229.67	0.488896	0.244448	−	0.969662i	$-0.421393\pi$
$709$	9229.67	0.488896	0.244448	+	0.969662i	$0.421393\pi$
$710$	12.1386	0.000641623 0
$711$	−6609.61	−0.348635
$712$	1300.99	0.0684786
$713$	−11212.4	−0.588931
$714$	1383.50	0.0725158
$715$	3774.46	0.197422
$716$	28609.6	1.49328
$717$	−24616.3	−1.28216
$718$	−966.583	−0.0502403
$719$	−9470.91	−0.491245	−0.245623	−	0.969366i	$-0.578992\pi$
$719$	−9470.91	−0.491245	−0.245623	+	0.969366i	$0.578992\pi$
$720$	2045.98	0.105902
$721$	−31072.6	−1.60500
$722$	108.687	0.00560238
$723$	18432.1	0.948129
$724$	−2634.54	−0.135237
$725$	−29976.7	−1.53560
$726$	−968.236	−0.0494967
$727$	23901.5	1.21934	0.609669	−	0.792656i	$-0.291302\pi$
$727$	23901.5	1.21934	0.609669	+	0.792656i	$0.291302\pi$
$728$	−1765.72	−0.0898928
$729$	17557.9	0.892032
$730$	−214.773	−0.0108892
$731$	0	0
$732$	10883.2	0.549526
$733$	5181.65	0.261103	0.130552	−	0.991442i	$-0.458325\pi$
$733$	5181.65	0.261103	0.130552	+	0.991442i	$0.458325\pi$
$734$	351.874	0.0176947
$735$	−5536.02	−0.277822
$736$	−1176.48	−0.0589208
$737$	−15683.6	−0.783873
$738$	−8.66438	−0.000432168 0
$739$	−7776.68	−0.387104	−0.193552	−	0.981090i	$-0.562001\pi$
$739$	−7776.68	−0.387104	−0.193552	+	0.981090i	$0.562001\pi$
$740$	446.858	0.0221984
$741$	7785.26	0.385963
$742$	−457.985	−0.0226592
$743$	−19598.9	−0.967716	−0.483858	−	0.875147i	$-0.660765\pi$
$743$	−19598.9	−0.967716	−0.483858	+	0.875147i	$0.660765\pi$
$744$	−1893.58	−0.0933090
$745$	−2251.06	−0.110701
$746$	539.917	0.0264983
$747$	5718.57	0.280096
$748$	41812.2	2.04386
$749$	13651.8	0.665987
$750$	−291.382	−0.0141864
$751$	11015.1	0.535215	0.267607	−	0.963528i	$-0.413767\pi$
$751$	11015.1	0.535215	0.267607	+	0.963528i	$0.413767\pi$
$752$	−30860.5	−1.49650
$753$	−17494.7	−0.846667
$754$	899.213	0.0434316
$755$	2003.70	0.0965857
$756$	36854.1	1.77298
$757$	−20914.1	−1.00414	−0.502071	−	0.864826i	$-0.667428\pi$
$757$	−20914.1	−1.00414	−0.502071	+	0.864826i	$0.667428\pi$
$758$	−864.168	−0.0414090
$759$	9907.15	0.473790
$760$	399.534	0.0190693
$761$	−1286.70	−0.0612917	−0.0306459	−	0.999530i	$-0.509756\pi$
$761$	−1286.70	−0.0612917	−0.0306459	+	0.999530i	$0.509756\pi$
$762$	−619.235	−0.0294390
$763$	7880.59	0.373914
$764$	24559.3	1.16299
$765$	−2932.62	−0.138600
$766$	−1775.43	−0.0837453
$767$	−14704.3	−0.692233
$768$	14867.0	0.698523
$769$	18574.3	0.871009	0.435504	−	0.900187i	$-0.356570\pi$
$769$	18574.3	0.871009	0.435504	+	0.900187i	$0.356570\pi$
$770$	571.887	0.0267654
$771$	5833.74	0.272499
$772$	−8276.70	−0.385861
$773$	22741.4	1.05815	0.529076	−	0.848575i	$-0.322539\pi$
$773$	22741.4	1.05815	0.529076	+	0.848575i	$0.322539\pi$
$774$	0	0
$775$	28743.1	1.33223
$776$	−2768.55	−0.128074
$777$	2641.49	0.121960
$778$	−883.701	−0.0407227
$779$	384.604	0.0176892
$780$	−1945.11	−0.0892897
$781$	2168.56	0.0993561
$782$	560.052	0.0256105
$783$	−37577.9	−1.71510
$784$	39005.6	1.77686
$785$	1562.10	0.0710237
$786$	−382.371	−0.0173521
$787$	10618.1	0.480935	0.240467	−	0.970657i	$-0.422699\pi$
$787$	10618.1	0.480935	0.240467	+	0.970657i	$0.422699\pi$
$788$	32621.1	1.47472
$789$	31090.8	1.40287
$790$	160.708	0.00723763
$791$	48547.1	2.18222
$792$	−1607.98	−0.0721429
$793$	9921.78	0.444304
$794$	−177.762	−0.00794525
$795$	−1010.13	−0.0450638
$796$	8724.16	0.388467
$797$	−35165.0	−1.56287	−0.781436	−	0.623986i	$-0.785512\pi$
$797$	−35165.0	−1.56287	−0.781436	+	0.623986i	$0.785512\pi$
$798$	1179.58	0.0523269
$799$	44234.1	1.95856
$800$	3015.92	0.133286
$801$	−8141.89	−0.359150
$802$	476.471	0.0209785
$803$	−38369.2	−1.68620
$804$	8082.31	0.354529
$805$	−3493.98	−0.152977
$806$	−862.208	−0.0376799
$807$	3385.32	0.147669
$808$	653.903	0.0284706
$809$	−16359.8	−0.710975	−0.355488	−	0.934681i	$-0.615685\pi$
$809$	−16359.8	−0.710975	−0.355488	+	0.934681i	$0.615685\pi$
$810$	−63.2688	−0.00274449
$811$	35937.9	1.55604	0.778021	−	0.628238i	$-0.216224\pi$
$811$	35937.9	1.55604	0.778021	+	0.628238i	$0.216224\pi$
$812$	−62144.5	−2.68577
$813$	21242.3	0.916361
$814$	−175.020	−0.00753618
$815$	191.623	0.00823592
$816$	−21499.9	−0.922361
$817$	0	0
$818$	−1266.94	−0.0541534
$819$	11050.3	0.471462
$820$	−96.0911	−0.00409225
$821$	24331.7	1.03433	0.517164	−	0.855886i	$-0.326988\pi$
$821$	24331.7	1.03433	0.517164	+	0.855886i	$0.326988\pi$
$822$	931.887	0.0395417
$823$	9100.87	0.385463	0.192732	−	0.981251i	$-0.438265\pi$
$823$	9100.87	0.385463	0.192732	+	0.981251i	$0.438265\pi$
$824$	−2124.27	−0.0898089
$825$	−25397.0	−1.07177
$826$	−2227.93	−0.0938492
$827$	14558.4	0.612146	0.306073	−	0.952008i	$-0.400985\pi$
$827$	14558.4	0.612146	0.306073	+	0.952008i	$0.400985\pi$
$828$	4906.66	0.205940
$829$	43404.7	1.81847	0.909233	−	0.416287i	$-0.136669\pi$
$829$	43404.7	1.81847	0.909233	+	0.416287i	$0.136669\pi$
$830$	−139.043	−0.00581475
$831$	−10256.6	−0.428154
$832$	13644.4	0.568549
$833$	−55908.9	−2.32548
$834$	1555.32	0.0645761
$835$	777.691	0.0322313
$836$	35649.4	1.47483
$837$	36031.4	1.48797
$838$	307.556	0.0126782
$839$	−23744.3	−0.977048	−0.488524	−	0.872550i	$-0.662465\pi$
$839$	−23744.3	−0.977048	−0.488524	+	0.872550i	$0.662465\pi$
$840$	−590.071	−0.0242374
$841$	38976.0	1.59810
$842$	−761.894	−0.0311836
$843$	−24567.2	−1.00373
$844$	41804.0	1.70492
$845$	3569.74	0.145329
$846$	−849.631	−0.0345283
$847$	61003.7	2.47475
$848$	7117.17	0.288213
$849$	−7151.03	−0.289073
$850$	−1435.70	−0.0579342
$851$	1069.30	0.0430729
$852$	−1117.53	−0.0449366
$853$	24503.2	0.983558	0.491779	−	0.870720i	$-0.336347\pi$
$853$	24503.2	0.983558	0.491779	+	0.870720i	$0.336347\pi$
$854$	1503.30	0.0602363
$855$	−2500.37	−0.100013
$856$	933.300	0.0372658
$857$	5417.22	0.215926	0.107963	−	0.994155i	$-0.465567\pi$
$857$	5417.22	0.215926	0.107963	+	0.994155i	$0.465567\pi$
$858$	761.837	0.0303132
$859$	−2383.54	−0.0946743	−0.0473371	−	0.998879i	$-0.515074\pi$
$859$	−2383.54	−0.0946743	−0.0473371	+	0.998879i	$0.515074\pi$
$860$	0	0
$861$	−568.020	−0.0224833
$862$	−509.125	−0.0201170
$863$	41532.4	1.63821	0.819107	−	0.573641i	$-0.194469\pi$
$863$	41532.4	1.63821	0.819107	+	0.573641i	$0.194469\pi$
$864$	3780.66	0.148867
$865$	5815.23	0.228583
$866$	1565.59	0.0614330
$867$	12587.1	0.493058
$868$	59587.1	2.33009
$869$	28710.5	1.12076
$870$	300.501	0.0117103
$871$	7368.35	0.286644
$872$	538.755	0.0209227
$873$	17326.2	0.671711
$874$	477.504	0.0184804
$875$	18358.5	0.709293
$876$	19772.9	0.762632
$877$	−3909.83	−0.150542	−0.0752711	−	0.997163i	$-0.523982\pi$
$877$	−3909.83	−0.150542	−0.0752711	+	0.997163i	$0.523982\pi$
$878$	−844.796	−0.0324721
$879$	17989.7	0.690306
$880$	−8887.23	−0.340441
$881$	41556.2	1.58917	0.794587	−	0.607150i	$-0.207687\pi$
$881$	41556.2	1.58917	0.794587	+	0.607150i	$0.207687\pi$
$882$	1073.88	0.0409969
$883$	3449.18	0.131454	0.0657271	−	0.997838i	$-0.479063\pi$
$883$	3449.18	0.131454	0.0657271	+	0.997838i	$0.479063\pi$
$884$	−19643.8	−0.747391
$885$	−4913.92	−0.186644
$886$	−2132.13	−0.0808469
$887$	−27108.0	−1.02615	−0.513077	−	0.858343i	$-0.671494\pi$
$887$	−27108.0	−1.02615	−0.513077	+	0.858343i	$0.671494\pi$
$888$	180.585	0.00682438
$889$	39014.9	1.47190
$890$	197.964	0.00745593
$891$	−11303.0	−0.424987
$892$	−24732.1	−0.928352
$893$	37714.3	1.41328
$894$	−454.353	−0.0169976
$895$	8716.24	0.325533
$896$	8333.32	0.310710
$897$	−4654.49	−0.173254
$898$	1098.82	0.0408330
$899$	−60757.3	−2.25403
$900$	−12578.3	−0.465861
$901$	−10201.4	−0.377202
$902$	37.6359	0.00138929
$903$	0	0
$904$	3318.91	0.122108
$905$	−802.643	−0.0294815
$906$	404.428	0.0148303
$907$	5950.44	0.217840	0.108920	−	0.994051i	$-0.465261\pi$
$907$	5950.44	0.217840	0.108920	+	0.994051i	$0.465261\pi$
$908$	2014.86	0.0736406
$909$	−4092.26	−0.149320
$910$	−268.679	−0.00978750
$911$	18881.6	0.686693	0.343346	−	0.939209i	$-0.388440\pi$
$911$	18881.6	0.686693	0.343346	+	0.939209i	$0.388440\pi$
$912$	−18331.0	−0.665569
$913$	−24840.0	−0.900421
$914$	−358.236	−0.0129643
$915$	3315.68	0.119796
$916$	30174.3	1.08841
$917$	24091.3	0.867574
$918$	−1799.75	−0.0647064
$919$	−17663.8	−0.634031	−0.317015	−	0.948420i	$-0.602681\pi$
$919$	−17663.8	−0.634031	−0.317015	+	0.948420i	$0.602681\pi$
$920$	−238.865	−0.00855995
$921$	8938.46	0.319796
$922$	819.738	0.0292805
$923$	−1018.81	−0.0363322
$924$	−52650.5	−1.87454
$925$	−2741.15	−0.0974362
$926$	−1459.17	−0.0517831
$927$	13294.2	0.471022
$928$	−6375.07	−0.225509
$929$	−19271.8	−0.680609	−0.340305	−	0.940315i	$-0.610530\pi$
$929$	−19271.8	−0.680609	−0.340305	+	0.940315i	$0.610530\pi$
$930$	−288.134	−0.0101595
$931$	−47668.3	−1.67805
$932$	43849.5	1.54114
$933$	−19759.2	−0.693339
$934$	1528.65	0.0535535
$935$	12738.6	0.445557
$936$	755.449	0.0263810
$937$	−27235.9	−0.949582	−0.474791	−	0.880099i	$-0.657476\pi$
$937$	−27235.9	−0.949582	−0.474791	+	0.880099i	$0.657476\pi$
$938$	1116.42	0.0388617
$939$	−4572.58	−0.158914
$940$	−9422.71	−0.326952
$941$	23033.3	0.797943	0.398972	−	0.916963i	$-0.369367\pi$
$941$	23033.3	0.797943	0.398972	+	0.916963i	$0.369367\pi$
$942$	315.294	0.0109053
$943$	−229.939	−0.00794044
$944$	34622.4	1.19371
$945$	11228.0	0.386505
$946$	0	0
$947$	28775.6	0.987415	0.493707	−	0.869628i	$-0.335641\pi$
$947$	28775.6	0.987415	0.493707	+	0.869628i	$0.335641\pi$
$948$	−14795.5	−0.506893
$949$	18026.3	0.616604
$950$	−1224.09	−0.0418048
$951$	−12214.8	−0.416501
$952$	−5959.20	−0.202877
$953$	52529.2	1.78551	0.892753	−	0.450546i	$-0.148771\pi$
$953$	52529.2	1.78551	0.892753	+	0.450546i	$0.148771\pi$
$954$	195.945	0.00664985
$955$	7482.29	0.253530
$956$	52957.1	1.79158
$957$	53684.5	1.81335
$958$	−1101.41	−0.0371449
$959$	−58713.5	−1.97702
$960$	4559.70	0.153295
$961$	28466.0	0.955524
$962$	82.2265	0.00275581
$963$	−5840.79	−0.195448
$964$	−39653.1	−1.32483
$965$	−2521.59	−0.0841170
$966$	−705.225	−0.0234889
$967$	24200.7	0.804802	0.402401	−	0.915464i	$-0.368176\pi$
$967$	24200.7	0.804802	0.402401	+	0.915464i	$0.368176\pi$
$968$	4170.51	0.138477
$969$	26274.8	0.871071
$970$	−421.274	−0.0139446
$971$	−48798.8	−1.61280	−0.806399	−	0.591371i	$-0.798587\pi$
$971$	−48798.8	−1.61280	−0.806399	+	0.591371i	$0.798587\pi$
$972$	−26349.6	−0.869509
$973$	−97993.2	−3.22869
$974$	−1341.20	−0.0441221
$975$	11931.8	0.391922
$976$	−23361.5	−0.766173
$977$	−53524.3	−1.75271	−0.876354	−	0.481668i	$-0.840031\pi$
$977$	−53524.3	−1.75271	−0.876354	+	0.481668i	$0.840031\pi$
$978$	38.6773	0.00126458
$979$	35366.3	1.15456
$980$	11909.7	0.388204
$981$	−3371.65	−0.109733
$982$	−381.226	−0.0123884
$983$	36466.4	1.18321	0.591607	−	0.806227i	$-0.298494\pi$
$983$	36466.4	1.18321	0.591607	+	0.806227i	$0.298494\pi$
$984$	−38.8326	−0.00125807
$985$	9938.40	0.321486
$986$	3034.79	0.0980197
$987$	−55700.2	−1.79631
$988$	−16748.5	−0.539312
$989$	0	0
$990$	−244.677	−0.00785490
$991$	10340.0	0.331444	0.165722	−	0.986173i	$-0.447005\pi$
$991$	10340.0	0.331444	0.165722	+	0.986173i	$0.447005\pi$
$992$	6112.72	0.195644
$993$	−42695.3	−1.36445
$994$	−154.365	−0.00492573
$995$	2657.92	0.0846851
$996$	12800.9	0.407241
$997$	−39788.0	−1.26389	−0.631945	−	0.775014i	$-0.717743\pi$
$997$	−39788.0	−1.26389	−0.631945	+	0.775014i	$0.717743\pi$
$998$	1493.07	0.0473572
$999$	−3436.22	−0.108826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.b.1.2		4
43.42	odd	2		43.4.a.a.1.3	✓	4
129.128	even	2		387.4.a.e.1.2		4
172.171	even	2		688.4.a.f.1.3		4
215.214	odd	2		1075.4.a.a.1.2		4
301.300	even	2		2107.4.a.b.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.a.1.3	✓	4	43.42	odd	2
387.4.a.e.1.2		4	129.128	even	2
688.4.a.f.1.3		4	172.171	even	2
1075.4.a.a.1.2		4	215.214	odd	2
1849.4.a.b.1.2		4	1.1	even	1	trivial
2107.4.a.b.1.3		4	301.300	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.132290 q^{2} +3.71054 q^{3} -7.98250 q^{4} -2.43196 q^{5} -0.490867 q^{6} +30.9271 q^{7} +2.11433 q^{8} -13.2319 q^{9} +O(q^{10})\)	\(q-0.132290 q^{2} +3.71054 q^{3} -7.98250 q^{4} -2.43196 q^{5} -0.490867 q^{6} +30.9271 q^{7} +2.11433 q^{8} -13.2319 q^{9} +0.321724 q^{10} +57.4761 q^{11} -29.6194 q^{12} -27.0029 q^{13} -4.09135 q^{14} -9.02388 q^{15} +63.5803 q^{16} -91.1332 q^{17} +1.75045 q^{18} -77.7008 q^{19} +19.4131 q^{20} +114.756 q^{21} -7.60352 q^{22} +46.4541 q^{23} +7.84529 q^{24} -119.086 q^{25} +3.57222 q^{26} -149.282 q^{27} -246.876 q^{28} +251.724 q^{29} +1.19377 q^{30} -241.365 q^{31} -25.3256 q^{32} +213.267 q^{33} +12.0560 q^{34} -75.2135 q^{35} +105.624 q^{36} +23.0183 q^{37} +10.2790 q^{38} -100.195 q^{39} -5.14196 q^{40} -4.94980 q^{41} -15.1811 q^{42} -458.803 q^{44} +32.1795 q^{45} -6.14542 q^{46} -485.378 q^{47} +235.917 q^{48} +613.485 q^{49} +15.7538 q^{50} -338.153 q^{51} +215.551 q^{52} +111.940 q^{53} +19.7485 q^{54} -139.780 q^{55} +65.3900 q^{56} -288.312 q^{57} -33.3006 q^{58} +544.546 q^{59} +72.0331 q^{60} -367.434 q^{61} +31.9302 q^{62} -409.224 q^{63} -505.292 q^{64} +65.6700 q^{65} -28.2131 q^{66} -272.873 q^{67} +727.471 q^{68} +172.370 q^{69} +9.95000 q^{70} +37.7297 q^{71} -27.9766 q^{72} -667.567 q^{73} -3.04510 q^{74} -441.872 q^{75} +620.247 q^{76} +1777.57 q^{77} +13.2549 q^{78} +499.520 q^{79} -154.625 q^{80} -196.655 q^{81} +0.654810 q^{82} -432.180 q^{83} -916.041 q^{84} +221.632 q^{85} +934.031 q^{87} +121.523 q^{88} +615.323 q^{89} -4.25703 q^{90} -835.122 q^{91} -370.820 q^{92} -895.594 q^{93} +64.2108 q^{94} +188.965 q^{95} -93.9718 q^{96} -1309.43 q^{97} -81.1580 q^{98} -760.518 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 11 q^{3} + 2 q^{4} + 27 q^{5} - 27 q^{6} + 20 q^{7} + 66 q^{8} - 9 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 11 * q^3 + 2 * q^4 + 27 * q^5 - 27 * q^6 + 20 * q^7 + 66 * q^8 - 9 * q^9	\( 4 q + 4 q^{2} + 11 q^{3} + 2 q^{4} + 27 q^{5} - 27 q^{6} + 20 q^{7} + 66 q^{8} - 9 q^{9} - 3 q^{10} - 62 q^{11} - 61 q^{12} - 2 q^{13} + 112 q^{14} + 92 q^{15} + 202 q^{16} - 207 q^{17} - 299 q^{18} - 99 q^{19} - 81 q^{20} - 90 q^{21} - 202 q^{22} - 103 q^{23} + 209 q^{24} - 101 q^{25} + 50 q^{26} + 218 q^{27} + 80 q^{28} + 99 q^{29} - 300 q^{30} + 131 q^{31} + 342 q^{32} + 32 q^{33} - 53 q^{34} - 374 q^{35} - 379 q^{36} + 449 q^{37} - 609 q^{38} - 98 q^{39} + 133 q^{40} - 491 q^{41} + 394 q^{42} - 764 q^{44} + 338 q^{45} - 1061 q^{46} + 19 q^{47} - 237 q^{48} + 236 q^{49} - 599 q^{50} - 1649 q^{51} + 224 q^{52} - 1220 q^{53} - 322 q^{54} - 1360 q^{55} + 344 q^{56} + 232 q^{57} - 771 q^{58} + 816 q^{59} - 156 q^{60} - 372 q^{61} + 97 q^{62} - 1914 q^{63} + 434 q^{64} + 350 q^{65} + 812 q^{66} + 110 q^{67} - 1697 q^{68} + 1238 q^{69} + 718 q^{70} - 468 q^{71} + 315 q^{72} - 628 q^{73} + 395 q^{74} - 62 q^{75} - 1671 q^{76} + 2044 q^{77} - 90 q^{78} + 1095 q^{79} + 31 q^{80} + 2056 q^{81} + 2287 q^{82} - 980 q^{83} - 610 q^{84} + 152 q^{85} - 507 q^{87} - 1816 q^{88} + 738 q^{89} - 2398 q^{90} - 852 q^{91} - 2517 q^{92} + 35 q^{93} + 2233 q^{94} + 1149 q^{95} - 1551 q^{96} - 1765 q^{97} - 1652 q^{98} - 58 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 11 * q^3 + 2 * q^4 + 27 * q^5 - 27 * q^6 + 20 * q^7 + 66 * q^8 - 9 * q^9 - 3 * q^10 - 62 * q^11 - 61 * q^12 - 2 * q^13 + 112 * q^14 + 92 * q^15 + 202 * q^16 - 207 * q^17 - 299 * q^18 - 99 * q^19 - 81 * q^20 - 90 * q^21 - 202 * q^22 - 103 * q^23 + 209 * q^24 - 101 * q^25 + 50 * q^26 + 218 * q^27 + 80 * q^28 + 99 * q^29 - 300 * q^30 + 131 * q^31 + 342 * q^32 + 32 * q^33 - 53 * q^34 - 374 * q^35 - 379 * q^36 + 449 * q^37 - 609 * q^38 - 98 * q^39 + 133 * q^40 - 491 * q^41 + 394 * q^42 - 764 * q^44 + 338 * q^45 - 1061 * q^46 + 19 * q^47 - 237 * q^48 + 236 * q^49 - 599 * q^50 - 1649 * q^51 + 224 * q^52 - 1220 * q^53 - 322 * q^54 - 1360 * q^55 + 344 * q^56 + 232 * q^57 - 771 * q^58 + 816 * q^59 - 156 * q^60 - 372 * q^61 + 97 * q^62 - 1914 * q^63 + 434 * q^64 + 350 * q^65 + 812 * q^66 + 110 * q^67 - 1697 * q^68 + 1238 * q^69 + 718 * q^70 - 468 * q^71 + 315 * q^72 - 628 * q^73 + 395 * q^74 - 62 * q^75 - 1671 * q^76 + 2044 * q^77 - 90 * q^78 + 1095 * q^79 + 31 * q^80 + 2056 * q^81 + 2287 * q^82 - 980 * q^83 - 610 * q^84 + 152 * q^85 - 507 * q^87 - 1816 * q^88 + 738 * q^89 - 2398 * q^90 - 852 * q^91 - 2517 * q^92 + 35 * q^93 + 2233 * q^94 + 1149 * q^95 - 1551 * q^96 - 1765 * q^97 - 1652 * q^98 - 58 * q^99

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