LMFDB - Embedded newform 1849.4.a.g.1.8

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

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-3.54418 q^{2} -3.02897 q^{3} +4.56122 q^{4} +20.5936 q^{5} +10.7352 q^{6} +15.8438 q^{7} +12.1877 q^{8} -17.8253 q^{9} +O(q^{10})$	\(q-3.54418 q^{2} -3.02897 q^{3} +4.56122 q^{4} +20.5936 q^{5} +10.7352 q^{6} +15.8438 q^{7} +12.1877 q^{8} -17.8253 q^{9} -72.9874 q^{10} -34.9365 q^{11} -13.8158 q^{12} -56.8312 q^{13} -56.1534 q^{14} -62.3774 q^{15} -79.6850 q^{16} +107.838 q^{17} +63.1762 q^{18} +82.7064 q^{19} +93.9319 q^{20} -47.9905 q^{21} +123.821 q^{22} -34.1859 q^{23} -36.9161 q^{24} +299.096 q^{25} +201.420 q^{26} +135.775 q^{27} +72.2672 q^{28} -201.807 q^{29} +221.077 q^{30} -43.9890 q^{31} +184.917 q^{32} +105.822 q^{33} -382.196 q^{34} +326.282 q^{35} -81.3052 q^{36} -170.895 q^{37} -293.126 q^{38} +172.140 q^{39} +250.988 q^{40} +28.1282 q^{41} +170.087 q^{42} -159.353 q^{44} -367.088 q^{45} +121.161 q^{46} +37.3245 q^{47} +241.364 q^{48} -91.9730 q^{49} -1060.05 q^{50} -326.637 q^{51} -259.219 q^{52} -249.787 q^{53} -481.210 q^{54} -719.469 q^{55} +193.099 q^{56} -250.515 q^{57} +715.240 q^{58} +8.65004 q^{59} -284.517 q^{60} -133.760 q^{61} +155.905 q^{62} -282.422 q^{63} -17.8983 q^{64} -1170.36 q^{65} -375.051 q^{66} -660.668 q^{67} +491.870 q^{68} +103.548 q^{69} -1156.40 q^{70} +573.469 q^{71} -217.249 q^{72} -403.354 q^{73} +605.683 q^{74} -905.954 q^{75} +377.242 q^{76} -553.529 q^{77} -610.096 q^{78} +104.296 q^{79} -1641.00 q^{80} +70.0267 q^{81} -99.6914 q^{82} -1157.13 q^{83} -218.895 q^{84} +2220.76 q^{85} +611.267 q^{87} -425.795 q^{88} +194.033 q^{89} +1301.03 q^{90} -900.424 q^{91} -155.929 q^{92} +133.241 q^{93} -132.285 q^{94} +1703.22 q^{95} -560.108 q^{96} -1349.88 q^{97} +325.969 q^{98} +622.755 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.54418	−1.25306	−0.626529	−	0.779398i	$-0.715525\pi$
$2$	−3.54418	−1.25306	−0.626529	+	0.779398i	$0.715525\pi$
$3$	−3.02897	−0.582926	−0.291463	−	0.956582i	$-0.594142\pi$
$3$	−3.02897	−0.582926	−0.291463	+	0.956582i	$0.594142\pi$
$4$	4.56122	0.570152
$5$	20.5936	1.84195	0.920974	−	0.389624i	$-0.127395\pi$
$5$	20.5936	1.84195	0.920974	+	0.389624i	$0.127395\pi$
$6$	10.7352	0.730439
$7$	15.8438	0.855487	0.427743	−	0.903900i	$-0.359309\pi$
$7$	15.8438	0.855487	0.427743	+	0.903900i	$0.359309\pi$
$8$	12.1877	0.538624
$9$	−17.8253	−0.660198
$10$	−72.9874	−2.30807
$11$	−34.9365	−0.957614	−0.478807	−	0.877920i	$-0.658931\pi$
$11$	−34.9365	−0.957614	−0.478807	+	0.877920i	$0.658931\pi$
$12$	−13.8158	−0.332356
$13$	−56.8312	−1.21247	−0.606236	−	0.795285i	$-0.707321\pi$
$13$	−56.8312	−1.21247	−0.606236	+	0.795285i	$0.707321\pi$
$14$	−56.1534	−1.07197
$15$	−62.3774	−1.07372
$16$	−79.6850	−1.24508
$17$	107.838	1.53850	0.769248	−	0.638950i	$-0.220631\pi$
$17$	107.838	1.53850	0.769248	+	0.638950i	$0.220631\pi$
$18$	63.1762	0.827265
$19$	82.7064	0.998639	0.499319	−	0.866418i	$-0.333583\pi$
$19$	82.7064	0.998639	0.499319	+	0.866418i	$0.333583\pi$
$20$	93.9319	1.05019
$21$	−47.9905	−0.498685
$22$	123.821	1.19995
$23$	−34.1859	−0.309924	−0.154962	−	0.987920i	$-0.549526\pi$
$23$	−34.1859	−0.309924	−0.154962	+	0.987920i	$0.549526\pi$
$24$	−36.9161	−0.313978
$25$	299.096	2.39277
$26$	201.420	1.51930
$27$	135.775	0.967772
$28$	72.2672	0.487757
$29$	−201.807	−1.29223	−0.646113	−	0.763241i	$-0.723607\pi$
$29$	−201.807	−1.29223	−0.646113	+	0.763241i	$0.723607\pi$
$30$	221.077	1.34543
$31$	−43.9890	−0.254860	−0.127430	−	0.991848i	$-0.540673\pi$
$31$	−43.9890	−0.254860	−0.127430	+	0.991848i	$0.540673\pi$
$32$	184.917	1.02153
$33$	105.822	0.558218
$34$	−382.196	−1.92782
$35$	326.282	1.57576
$36$	−81.3052	−0.376413
$37$	−170.895	−0.759324	−0.379662	−	0.925125i	$-0.623960\pi$
$37$	−170.895	−0.759324	−0.379662	+	0.925125i	$0.623960\pi$
$38$	−293.126	−1.25135
$39$	172.140	0.706781
$40$	250.988	0.992117
$41$	28.1282	0.107144	0.0535718	−	0.998564i	$-0.482939\pi$
$41$	28.1282	0.107144	0.0535718	+	0.998564i	$0.482939\pi$
$42$	170.087	0.624881
$43$	0	0
$43$	0	0
$44$	−159.353	−0.545986
$45$	−367.088	−1.21605
$46$	121.161	0.388352
$47$	37.3245	0.115837	0.0579185	−	0.998321i	$-0.481554\pi$
$47$	37.3245	0.115837	0.0579185	+	0.998321i	$0.481554\pi$
$48$	241.364	0.725788
$49$	−91.9730	−0.268143
$50$	−1060.05	−2.99828
$51$	−326.637	−0.896829
$52$	−259.219	−0.691294
$53$	−249.787	−0.647375	−0.323687	−	0.946164i	$-0.604923\pi$
$53$	−249.787	−0.647375	−0.323687	+	0.946164i	$0.604923\pi$
$54$	−481.210	−1.21267
$55$	−719.469	−1.76388
$56$	193.099	0.460786
$57$	−250.515	−0.582132
$58$	715.240	1.61923
$59$	8.65004	0.0190871	0.00954355	−	0.999954i	$-0.496962\pi$
$59$	8.65004	0.0190871	0.00954355	+	0.999954i	$0.496962\pi$
$60$	−284.517	−0.612183
$61$	−133.760	−0.280758	−0.140379	−	0.990098i	$-0.544832\pi$
$61$	−133.760	−0.280758	−0.140379	+	0.990098i	$0.544832\pi$
$62$	155.905	0.319354
$63$	−282.422	−0.564790
$64$	−17.8983	−0.0349576
$65$	−1170.36	−2.23331
$66$	−375.051	−0.699479
$67$	−660.668	−1.20468	−0.602339	−	0.798240i	$-0.705764\pi$
$67$	−660.668	−1.20468	−0.602339	+	0.798240i	$0.705764\pi$
$68$	491.870	0.877177
$69$	103.548	0.180663
$70$	−1156.40	−1.97452
$71$	573.469	0.958566	0.479283	−	0.877660i	$-0.340897\pi$
$71$	573.469	0.958566	0.479283	+	0.877660i	$0.340897\pi$
$72$	−217.249	−0.355598
$73$	−403.354	−0.646699	−0.323349	−	0.946280i	$-0.604809\pi$
$73$	−403.354	−0.646699	−0.323349	+	0.946280i	$0.604809\pi$
$74$	605.683	0.951476
$75$	−905.954	−1.39481
$76$	377.242	0.569376
$77$	−553.529	−0.819226
$78$	−610.096	−0.885637
$79$	104.296	0.148535	0.0742675	−	0.997238i	$-0.476338\pi$
$79$	104.296	0.148535	0.0742675	+	0.997238i	$0.476338\pi$
$80$	−1641.00	−2.29337
$81$	70.0267	0.0960585
$82$	−99.6914	−0.134257
$83$	−1157.13	−1.53026	−0.765128	−	0.643878i	$-0.777324\pi$
$83$	−1157.13	−1.53026	−0.765128	+	0.643878i	$0.777324\pi$
$84$	−218.895	−0.284326
$85$	2220.76	2.83383
$86$	0	0
$87$	611.267	0.753272
$88$	−425.795	−0.515794
$89$	194.033	0.231095	0.115547	−	0.993302i	$-0.463138\pi$
$89$	194.033	0.231095	0.115547	+	0.993302i	$0.463138\pi$
$90$	1301.03	1.52378
$91$	−900.424	−1.03725
$92$	−155.929	−0.176704
$93$	133.241	0.148564
$94$	−132.285	−0.145150
$95$	1703.22	1.83944
$96$	−560.108	−0.595477
$97$	−1349.88	−1.41298	−0.706490	−	0.707723i	$-0.749723\pi$
$97$	−1349.88	−1.41298	−0.706490	+	0.707723i	$0.749723\pi$
$98$	325.969	0.335998
$99$	622.755	0.632215
$100$	1364.24	1.36424
$101$	−272.354	−0.268319	−0.134160	−	0.990960i	$-0.542833\pi$
$101$	−272.354	−0.268319	−0.134160	+	0.990960i	$0.542833\pi$
$102$	1157.66	1.12378
$103$	1116.00	1.06760	0.533798	−	0.845612i	$-0.320764\pi$
$103$	1116.00	1.06760	0.533798	+	0.845612i	$0.320764\pi$
$104$	−692.640	−0.653067
$105$	−988.297	−0.918552
$106$	885.290	0.811197
$107$	897.726	0.811088	0.405544	−	0.914075i	$-0.367082\pi$
$107$	897.726	0.811088	0.405544	+	0.914075i	$0.367082\pi$
$108$	619.298	0.551777
$109$	−402.268	−0.353489	−0.176744	−	0.984257i	$-0.556557\pi$
$109$	−402.268	−0.353489	−0.176744	+	0.984257i	$0.556557\pi$
$110$	2549.93	2.21024
$111$	517.636	0.442629
$112$	−1262.52	−1.06515
$113$	701.523	0.584016	0.292008	−	0.956416i	$-0.405677\pi$
$113$	701.523	0.584016	0.292008	+	0.956416i	$0.405677\pi$
$114$	887.871	0.729445
$115$	−704.011	−0.570864
$116$	−920.484	−0.736766
$117$	1013.04	0.800471
$118$	−30.6573	−0.0239172
$119$	1708.56	1.31616
$120$	−760.235	−0.578331
$121$	−110.439	−0.0829745
$122$	474.070	0.351806
$123$	−85.1995	−0.0624567
$124$	−200.643	−0.145309
$125$	3585.27	2.56541
$126$	1000.95	0.707714
$127$	−2646.00	−1.84878	−0.924389	−	0.381450i	$-0.875425\pi$
$127$	−2646.00	−1.84878	−0.924389	+	0.381450i	$0.875425\pi$
$128$	−1415.90	−0.977727
$129$	0	0
$130$	4147.96	2.79847
$131$	415.081	0.276838	0.138419	−	0.990374i	$-0.455798\pi$
$131$	415.081	0.276838	0.138419	+	0.990374i	$0.455798\pi$
$132$	482.676	0.318269
$133$	1310.39	0.854322
$134$	2341.53	1.50953
$135$	2796.09	1.78259
$136$	1314.29	0.828671
$137$	2290.38	1.42832	0.714162	−	0.699981i	$-0.246808\pi$
$137$	2290.38	1.42832	0.714162	+	0.699981i	$0.246808\pi$
$138$	−366.993	−0.226381
$139$	2202.98	1.34428	0.672138	−	0.740426i	$-0.265376\pi$
$139$	2202.98	1.34428	0.672138	+	0.740426i	$0.265376\pi$
$140$	1488.24	0.898424
$141$	−113.055	−0.0675244
$142$	−2032.48	−1.20114
$143$	1985.49	1.16108
$144$	1420.41	0.821998
$145$	−4155.93	−2.38021
$146$	1429.56	0.810350
$147$	278.583	0.156307
$148$	−779.489	−0.432930
$149$	2075.02	1.14089	0.570444	−	0.821337i	$-0.306771\pi$
$149$	2075.02	1.14089	0.570444	+	0.821337i	$0.306771\pi$
$150$	3210.87	1.74777
$151$	−3452.62	−1.86073	−0.930366	−	0.366631i	$-0.880511\pi$
$151$	−3452.62	−1.86073	−0.930366	+	0.366631i	$0.880511\pi$
$152$	1008.00	0.537891
$153$	−1922.24	−1.01571
$154$	1961.81	1.02654
$155$	−905.891	−0.469438
$156$	785.168	0.402973
$157$	−2437.56	−1.23910	−0.619550	−	0.784957i	$-0.712685\pi$
$157$	−2437.56	−1.23910	−0.619550	+	0.784957i	$0.712685\pi$
$158$	−369.645	−0.186123
$159$	756.597	0.377371
$160$	3808.10	1.88161
$161$	−541.636	−0.265136
$162$	−248.187	−0.120367
$163$	221.185	0.106285	0.0531427	−	0.998587i	$-0.483076\pi$
$163$	221.185	0.106285	0.0531427	+	0.998587i	$0.483076\pi$
$164$	128.299	0.0610881
$165$	2179.25	1.02821
$166$	4101.07	1.91750
$167$	−3743.77	−1.73474	−0.867369	−	0.497665i	$-0.834191\pi$
$167$	−3743.77	−1.73474	−0.867369	+	0.497665i	$0.834191\pi$
$168$	−584.892	−0.268604
$169$	1032.79	0.470090
$170$	−7870.79	−3.55095
$171$	−1474.27	−0.659299
$172$	0	0
$173$	−260.450	−0.114460	−0.0572302	−	0.998361i	$-0.518227\pi$
$173$	−260.450	−0.114460	−0.0572302	+	0.998361i	$0.518227\pi$
$174$	−2166.44	−0.943893
$175$	4738.83	2.04698
$176$	2783.92	1.19231
$177$	−26.2007	−0.0111264
$178$	−687.688	−0.289575
$179$	−1459.14	−0.609282	−0.304641	−	0.952467i	$-0.598537\pi$
$179$	−1459.14	−0.609282	−0.304641	+	0.952467i	$0.598537\pi$
$180$	−1674.37	−0.693333
$181$	−195.999	−0.0804889	−0.0402445	−	0.999190i	$-0.512814\pi$
$181$	−195.999	−0.0804889	−0.0402445	+	0.999190i	$0.512814\pi$
$182$	3191.27	1.29974
$183$	405.156	0.163661
$184$	−416.646	−0.166932
$185$	−3519.34	−1.39863
$186$	−472.231	−0.186160
$187$	−3767.47	−1.47329
$188$	170.245	0.0660447
$189$	2151.19	0.827916
$190$	−6036.53	−2.30492
$191$	−1960.00	−0.742518	−0.371259	−	0.928529i	$-0.621074\pi$
$191$	−1960.00	−0.742518	−0.371259	+	0.928529i	$0.621074\pi$
$192$	54.2135	0.0203777
$193$	3858.41	1.43904	0.719519	−	0.694472i	$-0.244362\pi$
$193$	3858.41	1.43904	0.719519	+	0.694472i	$0.244362\pi$
$194$	4784.20	1.77055
$195$	3544.98	1.30185
$196$	−419.509	−0.152882
$197$	−2045.33	−0.739714	−0.369857	−	0.929089i	$-0.620593\pi$
$197$	−2045.33	−0.739714	−0.369857	+	0.929089i	$0.620593\pi$
$198$	−2207.16	−0.792201
$199$	−490.264	−0.174643	−0.0873213	−	0.996180i	$-0.527831\pi$
$199$	−490.264	−0.174643	−0.0873213	+	0.996180i	$0.527831\pi$
$200$	3645.29	1.28880
$201$	2001.14	0.702238
$202$	965.271	0.336219
$203$	−3197.39	−1.10548
$204$	−1489.86	−0.511329
$205$	579.261	0.197353
$206$	−3955.29	−1.33776
$207$	609.375	0.204611
$208$	4528.60	1.50962
$209$	−2889.47	−0.956311
$210$	3502.70	1.15100
$211$	1752.89	0.571914	0.285957	−	0.958243i	$-0.407689\pi$
$211$	1752.89	0.571914	0.285957	+	0.958243i	$0.407689\pi$
$212$	−1139.33	−0.369102
$213$	−1737.02	−0.558773
$214$	−3181.70	−1.01634
$215$	0	0
$216$	1654.78	0.521265
$217$	−696.954	−0.218029
$218$	1425.71	0.442941
$219$	1221.75	0.376977
$220$	−3281.65	−1.00568
$221$	−6128.54	−1.86539
$222$	−1834.60	−0.554640
$223$	−1883.52	−0.565604	−0.282802	−	0.959178i	$-0.591264\pi$
$223$	−1883.52	−0.565604	−0.282802	+	0.959178i	$0.591264\pi$
$224$	2929.79	0.873906
$225$	−5331.49	−1.57970
$226$	−2486.33	−0.731805
$227$	−1692.30	−0.494811	−0.247406	−	0.968912i	$-0.579578\pi$
$227$	−1692.30	−0.494811	−0.247406	+	0.968912i	$0.579578\pi$
$228$	−1142.65	−0.331904
$229$	2575.82	0.743298	0.371649	−	0.928373i	$-0.378793\pi$
$229$	2575.82	0.743298	0.371649	+	0.928373i	$0.378793\pi$
$230$	2495.14	0.715325
$231$	1676.62	0.477548
$232$	−2459.55	−0.696024
$233$	3630.28	1.02072	0.510360	−	0.859961i	$-0.329512\pi$
$233$	3630.28	1.02072	0.510360	+	0.859961i	$0.329512\pi$
$234$	−3590.38	−1.00304
$235$	768.646	0.213366
$236$	39.4547	0.0108826
$237$	−315.911	−0.0865848
$238$	−6055.45	−1.64923
$239$	5539.96	1.49937	0.749686	−	0.661794i	$-0.230204\pi$
$239$	5539.96	1.49937	0.749686	+	0.661794i	$0.230204\pi$
$240$	4970.55	1.33686
$241$	−5377.78	−1.43740	−0.718700	−	0.695320i	$-0.755263\pi$
$241$	−5377.78	−1.43740	−0.718700	+	0.695320i	$0.755263\pi$
$242$	391.416	0.103972
$243$	−3878.02	−1.02377
$244$	−610.109	−0.160075
$245$	−1894.05	−0.493905
$246$	301.962	0.0782618
$247$	−4700.30	−1.21082
$248$	−536.123	−0.137274
$249$	3504.90	0.892025
$250$	−12706.9	−3.21461
$251$	−300.948	−0.0756800	−0.0378400	−	0.999284i	$-0.512048\pi$
$251$	−300.948	−0.0756800	−0.0378400	+	0.999284i	$0.512048\pi$
$252$	−1288.19	−0.322016
$253$	1194.34	0.296788
$254$	9377.92	2.31663
$255$	−6726.63	−1.65191
$256$	5161.39	1.26011
$257$	5975.65	1.45039	0.725196	−	0.688542i	$-0.241749\pi$
$257$	5975.65	1.45039	0.725196	+	0.688542i	$0.241749\pi$
$258$	0	0
$259$	−2707.63	−0.649591
$260$	−5338.26	−1.27333
$261$	3597.27	0.853125
$262$	−1471.12	−0.346894
$263$	−7486.94	−1.75538	−0.877690	−	0.479229i	$-0.840916\pi$
$263$	−7486.94	−1.75538	−0.877690	+	0.479229i	$0.840916\pi$
$264$	1289.72	0.300670
$265$	−5144.01	−1.19243
$266$	−4644.24	−1.07051
$267$	−587.720	−0.134711
$268$	−3013.45	−0.686850
$269$	−2932.20	−0.664608	−0.332304	−	0.943172i	$-0.607826\pi$
$269$	−2932.20	−0.664608	−0.332304	+	0.943172i	$0.607826\pi$
$270$	−9909.84	−2.23368
$271$	1045.83	0.234428	0.117214	−	0.993107i	$-0.462604\pi$
$271$	1045.83	0.234428	0.117214	+	0.993107i	$0.462604\pi$
$272$	−8593.04	−1.91555
$273$	2727.36	0.604642
$274$	−8117.52	−1.78977
$275$	−10449.4	−2.29135
$276$	472.305	0.103005
$277$	2637.01	0.571996	0.285998	−	0.958230i	$-0.407675\pi$
$277$	2637.01	0.571996	0.285998	+	0.958230i	$0.407675\pi$
$278$	−7807.76	−1.68445
$279$	784.118	0.168258
$280$	3976.61	0.848743
$281$	5808.37	1.23309	0.616545	−	0.787319i	$-0.288532\pi$
$281$	5808.37	1.23309	0.616545	+	0.787319i	$0.288532\pi$
$282$	400.687	0.0846119
$283$	4759.15	0.999654	0.499827	−	0.866125i	$-0.333397\pi$
$283$	4759.15	0.999654	0.499827	+	0.866125i	$0.333397\pi$
$284$	2615.71	0.546529
$285$	−5159.01	−1.07226
$286$	−7036.92	−1.45490
$287$	445.658	0.0916599
$288$	−3296.20	−0.674412
$289$	6715.94	1.36697
$290$	14729.4	2.98254
$291$	4088.73	0.823663
$292$	−1839.78	−0.368717
$293$	3049.33	0.607999	0.303999	−	0.952672i	$-0.401678\pi$
$293$	3049.33	0.607999	0.303999	+	0.952672i	$0.401678\pi$
$294$	−987.350	−0.195862
$295$	178.135	0.0351574
$296$	−2082.81	−0.408990
$297$	−4743.49	−0.926752
$298$	−7354.25	−1.42960
$299$	1942.83	0.375774
$300$	−4132.25	−0.795253
$301$	0	0
$302$	12236.7	2.33160
$303$	824.952	0.156410
$304$	−6590.46	−1.24338
$305$	−2754.60	−0.517142
$306$	6812.77	1.27275
$307$	−1717.19	−0.319236	−0.159618	−	0.987179i	$-0.551026\pi$
$307$	−1717.19	−0.319236	−0.159618	+	0.987179i	$0.551026\pi$
$308$	−2524.76	−0.467084
$309$	−3380.32	−0.622329
$310$	3210.64	0.588233
$311$	−4501.75	−0.820807	−0.410403	−	0.911904i	$-0.634612\pi$
$311$	−4501.75	−0.820807	−0.410403	+	0.911904i	$0.634612\pi$
$312$	2097.99	0.380689
$313$	637.908	0.115197	0.0575986	−	0.998340i	$-0.481656\pi$
$313$	637.908	0.115197	0.0575986	+	0.998340i	$0.481656\pi$
$314$	8639.16	1.55266
$315$	−5816.08	−1.04031
$316$	475.718	0.0846875
$317$	7175.13	1.27128	0.635640	−	0.771986i	$-0.280736\pi$
$317$	7175.13	1.27128	0.635640	+	0.771986i	$0.280736\pi$
$318$	−2681.52	−0.472868
$319$	7050.43	1.23746
$320$	−368.591	−0.0643902
$321$	−2719.19	−0.472804
$322$	1919.65	0.332230
$323$	8918.85	1.53640
$324$	319.407	0.0547680
$325$	−16998.0	−2.90117
$326$	−783.918	−0.133182
$327$	1218.46	0.206058
$328$	342.817	0.0577101
$329$	591.363	0.0990970
$330$	−7723.66	−1.28840
$331$	−4586.76	−0.761666	−0.380833	−	0.924644i	$-0.624363\pi$
$331$	−4586.76	−0.761666	−0.380833	+	0.924644i	$0.624363\pi$
$332$	−5277.91	−0.872478
$333$	3046.26	0.501304
$334$	13268.6	2.17373
$335$	−13605.5	−2.21895
$336$	3824.13	0.620902
$337$	−5787.80	−0.935553	−0.467776	−	0.883847i	$-0.654945\pi$
$337$	−5787.80	−0.935553	−0.467776	+	0.883847i	$0.654945\pi$
$338$	−3660.38	−0.589049
$339$	−2124.89	−0.340438
$340$	10129.4	1.61571
$341$	1536.82	0.244057
$342$	5225.07	0.826139
$343$	−6891.64	−1.08488
$344$	0	0
$345$	2132.43	0.332771
$346$	923.082	0.143425
$347$	−734.552	−0.113639	−0.0568196	−	0.998384i	$-0.518096\pi$
$347$	−734.552	−0.113639	−0.0568196	+	0.998384i	$0.518096\pi$
$348$	2788.12	0.429480
$349$	5789.06	0.887912	0.443956	−	0.896049i	$-0.353575\pi$
$349$	5789.06	0.887912	0.443956	+	0.896049i	$0.353575\pi$
$350$	−16795.3	−2.56499
$351$	−7716.24	−1.17340
$352$	−6460.35	−0.978233
$353$	−8702.40	−1.31213	−0.656065	−	0.754705i	$-0.727780\pi$
$353$	−8702.40	−1.31213	−0.656065	+	0.754705i	$0.727780\pi$
$354$	92.8601	0.0139420
$355$	11809.8	1.76563
$356$	885.026	0.131759
$357$	−5175.18	−0.767226
$358$	5171.47	0.763466
$359$	−9717.63	−1.42863	−0.714313	−	0.699826i	$-0.753261\pi$
$359$	−9717.63	−1.42863	−0.714313	+	0.699826i	$0.753261\pi$
$360$	−4473.95	−0.654993
$361$	−18.6587	−0.00272032
$362$	694.656	0.100857
$363$	334.517	0.0483680
$364$	−4107.03	−0.591393
$365$	−8306.51	−1.19118
$366$	−1435.95	−0.205077
$367$	−3428.85	−0.487696	−0.243848	−	0.969813i	$-0.578410\pi$
$367$	−3428.85	−0.487696	−0.243848	+	0.969813i	$0.578410\pi$
$368$	2724.10	0.385880
$369$	−501.394	−0.0707359
$370$	12473.2	1.75257
$371$	−3957.58	−0.553820
$372$	607.742	0.0847042
$373$	−1643.28	−0.228113	−0.114056	−	0.993474i	$-0.536384\pi$
$373$	−1643.28	−0.228113	−0.114056	+	0.993474i	$0.536384\pi$
$374$	13352.6	1.84611
$375$	−10859.7	−1.49544
$376$	454.899	0.0623926
$377$	11468.9	1.56679
$378$	−7624.21	−1.03743
$379$	25.5605	0.00346426	0.00173213	−	0.999998i	$-0.499449\pi$
$379$	25.5605	0.00346426	0.00173213	+	0.999998i	$0.499449\pi$
$380$	7768.76	1.04876
$381$	8014.67	1.07770
$382$	6946.61	0.930418
$383$	−13048.1	−1.74080	−0.870401	−	0.492343i	$-0.836140\pi$
$383$	−13048.1	−1.74080	−0.870401	+	0.492343i	$0.836140\pi$
$384$	4288.72	0.569942
$385$	−11399.1	−1.50897
$386$	−13674.9	−1.80320
$387$	0	0
$388$	−6157.08	−0.805614
$389$	293.352	0.0382353	0.0191176	−	0.999817i	$-0.493914\pi$
$389$	293.352	0.0382353	0.0191176	+	0.999817i	$0.493914\pi$
$390$	−12564.1	−1.63130
$391$	−3686.52	−0.476817
$392$	−1120.94	−0.144428
$393$	−1257.27	−0.161376
$394$	7249.02	0.926904
$395$	2147.84	0.273594
$396$	2840.52	0.360459
$397$	1885.00	0.238300	0.119150	−	0.992876i	$-0.461983\pi$
$397$	1885.00	0.238300	0.119150	+	0.992876i	$0.461983\pi$
$398$	1737.58	0.218837
$399$	−3969.12	−0.498006
$400$	−23833.5	−2.97919
$401$	−4235.37	−0.527442	−0.263721	−	0.964599i	$-0.584950\pi$
$401$	−4235.37	−0.527442	−0.263721	+	0.964599i	$0.584950\pi$
$402$	−7092.41	−0.879944
$403$	2499.95	0.309010
$404$	−1242.26	−0.152983
$405$	1442.10	0.176935
$406$	11332.1	1.38523
$407$	5970.48	0.727139
$408$	−3980.94	−0.483054
$409$	2188.61	0.264596	0.132298	−	0.991210i	$-0.457764\pi$
$409$	2188.61	0.264596	0.132298	+	0.991210i	$0.457764\pi$
$410$	−2053.00	−0.247294
$411$	−6937.49	−0.832607
$412$	5090.30	0.608692
$413$	137.050	0.0163288
$414$	−2159.74	−0.256389
$415$	−23829.4	−2.81865
$416$	−10509.0	−1.23858
$417$	−6672.76	−0.783613
$418$	10240.8	1.19831
$419$	5269.22	0.614364	0.307182	−	0.951651i	$-0.400614\pi$
$419$	5269.22	0.614364	0.307182	+	0.951651i	$0.400614\pi$
$420$	−4507.84	−0.523714
$421$	2504.31	0.289911	0.144955	−	0.989438i	$-0.453696\pi$
$421$	2504.31	0.289911	0.144955	+	0.989438i	$0.453696\pi$
$422$	−6212.55	−0.716640
$423$	−665.322	−0.0764753
$424$	−3044.32	−0.348691
$425$	32253.8	3.68127
$426$	6156.31	0.700174
$427$	−2119.27	−0.240185
$428$	4094.72	0.462444
$429$	−6013.98	−0.676824
$430$	0	0
$431$	6649.90	0.743189	0.371594	−	0.928395i	$-0.378811\pi$
$431$	6649.90	0.743189	0.371594	+	0.928395i	$0.378811\pi$
$432$	−10819.2	−1.20495
$433$	3329.19	0.369493	0.184747	−	0.982786i	$-0.440854\pi$
$433$	3329.19	0.369493	0.184747	+	0.982786i	$0.440854\pi$
$434$	2470.13	0.273203
$435$	12588.2	1.38749
$436$	−1834.83	−0.201542
$437$	−2827.39	−0.309502
$438$	−4330.09	−0.472374
$439$	−843.207	−0.0916722	−0.0458361	−	0.998949i	$-0.514595\pi$
$439$	−843.207	−0.0916722	−0.0458361	+	0.998949i	$0.514595\pi$
$440$	−8768.65	−0.950066
$441$	1639.45	0.177027
$442$	21720.6	2.33743
$443$	−1625.78	−0.174364	−0.0871818	−	0.996192i	$-0.527786\pi$
$443$	−1625.78	−0.174364	−0.0871818	+	0.996192i	$0.527786\pi$
$444$	2361.05	0.252366
$445$	3995.84	0.425665
$446$	6675.53	0.708734
$447$	−6285.18	−0.665053
$448$	−283.578	−0.0299058
$449$	−4164.91	−0.437760	−0.218880	−	0.975752i	$-0.570240\pi$
$449$	−4164.91	−0.437760	−0.218880	+	0.975752i	$0.570240\pi$
$450$	18895.8	1.97946
$451$	−982.701	−0.102602
$452$	3199.80	0.332978
$453$	10457.9	1.08467
$454$	5997.83	0.620027
$455$	−18543.0	−1.91057
$456$	−3053.20	−0.313550
$457$	6127.25	0.627179	0.313589	−	0.949559i	$-0.398468\pi$
$457$	6127.25	0.627179	0.313589	+	0.949559i	$0.398468\pi$
$458$	−9129.18	−0.931394
$459$	14641.6	1.48891
$460$	−3211.15	−0.325479
$461$	9381.48	0.947807	0.473903	−	0.880577i	$-0.342845\pi$
$461$	9381.48	0.947807	0.473903	+	0.880577i	$0.342845\pi$
$462$	−5942.25	−0.598395
$463$	−5046.82	−0.506578	−0.253289	−	0.967391i	$-0.581512\pi$
$463$	−5046.82	−0.506578	−0.253289	+	0.967391i	$0.581512\pi$
$464$	16081.0	1.60892
$465$	2743.92	0.273648
$466$	−12866.4	−1.27902
$467$	−3049.24	−0.302146	−0.151073	−	0.988523i	$-0.548273\pi$
$467$	−3049.24	−0.302146	−0.151073	+	0.988523i	$0.548273\pi$
$468$	4620.67	0.456390
$469$	−10467.5	−1.03059
$470$	−2724.22	−0.267359
$471$	7383.30	0.722303
$472$	105.424	0.0102808
$473$	0	0
$474$	1119.64	0.108496
$475$	24737.2	2.38951
$476$	7793.11	0.750413
$477$	4452.53	0.427395
$478$	−19634.6	−1.87880
$479$	3061.63	0.292045	0.146022	−	0.989281i	$-0.453353\pi$
$479$	3061.63	0.292045	0.146022	+	0.989281i	$0.453353\pi$
$480$	−11534.6	−1.09684
$481$	9712.17	0.920659
$482$	19059.8	1.80114
$483$	1640.60	0.154554
$484$	−503.737	−0.0473081
$485$	−27798.8	−2.60264
$486$	13744.4	1.28284
$487$	−14099.8	−1.31196	−0.655980	−	0.754778i	$-0.727744\pi$
$487$	−14099.8	−1.31196	−0.655980	+	0.754778i	$0.727744\pi$
$488$	−1630.23	−0.151223
$489$	−669.962	−0.0619565
$490$	6712.87	0.618891
$491$	−11124.3	−1.02247	−0.511236	−	0.859440i	$-0.670812\pi$
$491$	−11124.3	−1.02247	−0.511236	+	0.859440i	$0.670812\pi$
$492$	−388.613	−0.0356098
$493$	−21762.4	−1.98809
$494$	16658.7	1.51723
$495$	12824.8	1.16451
$496$	3505.26	0.317320
$497$	9085.94	0.820041
$498$	−12422.0	−1.11776
$499$	−19272.6	−1.72898	−0.864489	−	0.502653i	$-0.832357\pi$
$499$	−19272.6	−1.72898	−0.864489	+	0.502653i	$0.832357\pi$
$500$	16353.2	1.46267
$501$	11339.8	1.01122
$502$	1066.61	0.0948314
$503$	−19144.3	−1.69702	−0.848512	−	0.529177i	$-0.822501\pi$
$503$	−19144.3	−1.69702	−0.848512	+	0.529177i	$0.822501\pi$
$504$	−3442.06	−0.304210
$505$	−5608.75	−0.494230
$506$	−4232.94	−0.371892
$507$	−3128.28	−0.274027
$508$	−12069.0	−1.05409
$509$	−7714.02	−0.671745	−0.335872	−	0.941907i	$-0.609031\pi$
$509$	−7714.02	−0.671745	−0.335872	+	0.941907i	$0.609031\pi$
$510$	23840.4	2.06994
$511$	−6390.67	−0.553242
$512$	−6965.70	−0.601257
$513$	11229.4	0.966455
$514$	−21178.8	−1.81742
$515$	22982.4	1.96646
$516$	0	0
$517$	−1303.99	−0.110927
$518$	9596.34	0.813975
$519$	788.896	0.0667219
$520$	−14264.0	−1.20291
$521$	−21995.0	−1.84956	−0.924778	−	0.380508i	$-0.875749\pi$
$521$	−21995.0	−1.84956	−0.924778	+	0.380508i	$0.875749\pi$
$522$	−12749.4	−1.06901
$523$	18985.1	1.58730	0.793652	−	0.608372i	$-0.208177\pi$
$523$	18985.1	1.58730	0.793652	+	0.608372i	$0.208177\pi$
$524$	1893.27	0.157840
$525$	−14353.8	−1.19324
$526$	26535.1	2.19959
$527$	−4743.66	−0.392101
$528$	−8432.41	−0.695025
$529$	−10998.3	−0.903947
$530$	18231.3	1.49418
$531$	−154.190	−0.0126013
$532$	5976.95	0.487094
$533$	−1598.56	−0.129909
$534$	2082.99	0.168801
$535$	18487.4	1.49398
$536$	−8052.00	−0.648868
$537$	4419.70	0.355166
$538$	10392.3	0.832792
$539$	3213.22	0.256777
$540$	12753.6	1.01634
$541$	−18655.6	−1.48256	−0.741282	−	0.671194i	$-0.765782\pi$
$541$	−18655.6	−1.48256	−0.741282	+	0.671194i	$0.765782\pi$
$542$	−3706.62	−0.293751
$543$	593.676	0.0469191
$544$	19941.0	1.57162
$545$	−8284.14	−0.651107
$546$	−9666.25	−0.757651
$547$	7000.58	0.547209	0.273604	−	0.961842i	$-0.411784\pi$
$547$	7000.58	0.547209	0.273604	+	0.961842i	$0.411784\pi$
$548$	10446.9	0.814362
$549$	2384.32	0.185356
$550$	37034.5	2.87120
$551$	−16690.7	−1.29047
$552$	1262.01	0.0973092
$553$	1652.45	0.127070
$554$	−9346.05	−0.716743
$555$	10660.0	0.815300
$556$	10048.3	0.766441
$557$	18391.7	1.39907	0.699533	−	0.714600i	$-0.253391\pi$
$557$	18391.7	1.39907	0.699533	+	0.714600i	$0.253391\pi$
$558$	−2779.06	−0.210837
$559$	0	0
$560$	−25999.8	−1.96195
$561$	11411.6	0.858817
$562$	−20585.9	−1.54513
$563$	6904.46	0.516853	0.258426	−	0.966031i	$-0.416796\pi$
$563$	6904.46	0.516853	0.258426	+	0.966031i	$0.416796\pi$
$564$	−515.668	−0.0384992
$565$	14446.9	1.07573
$566$	−16867.3	−1.25262
$567$	1109.49	0.0821768
$568$	6989.25	0.516307
$569$	4400.65	0.324227	0.162113	−	0.986772i	$-0.448169\pi$
$569$	4400.65	0.324227	0.162113	+	0.986772i	$0.448169\pi$
$570$	18284.5	1.34360
$571$	12310.8	0.902264	0.451132	−	0.892457i	$-0.351020\pi$
$571$	12310.8	0.902264	0.451132	+	0.892457i	$0.351020\pi$
$572$	9056.23	0.661993
$573$	5936.80	0.432833
$574$	−1579.49	−0.114855
$575$	−10224.9	−0.741577
$576$	319.043	0.0230790
$577$	−13225.7	−0.954232	−0.477116	−	0.878840i	$-0.658318\pi$
$577$	−13225.7	−0.954232	−0.477116	+	0.878840i	$0.658318\pi$
$578$	−23802.5	−1.71290
$579$	−11687.0	−0.838853
$580$	−18956.1	−1.35708
$581$	−18333.3	−1.30911
$582$	−14491.2	−1.03210
$583$	8726.68	0.619935
$584$	−4915.94	−0.348327
$585$	20862.0	1.47443
$586$	−10807.4	−0.761857
$587$	−1471.43	−0.103462	−0.0517310	−	0.998661i	$-0.516474\pi$
$587$	−1471.43	−0.103462	−0.0517310	+	0.998661i	$0.516474\pi$
$588$	1270.68	0.0891189
$589$	−3638.17	−0.254513
$590$	−631.344	−0.0440543
$591$	6195.24	0.431198
$592$	13617.8	0.945418
$593$	4243.81	0.293883	0.146941	−	0.989145i	$-0.453057\pi$
$593$	4243.81	0.293883	0.146941	+	0.989145i	$0.453057\pi$
$594$	16811.8	1.16127
$595$	35185.4	2.42430
$596$	9464.62	0.650480
$597$	1484.99	0.101804
$598$	−6885.73	−0.470867
$599$	−20221.6	−1.37935	−0.689675	−	0.724119i	$-0.742246\pi$
$599$	−20221.6	−1.37935	−0.689675	+	0.724119i	$0.742246\pi$
$600$	−11041.5	−0.751277
$601$	−9067.23	−0.615408	−0.307704	−	0.951482i	$-0.599561\pi$
$601$	−9067.23	−0.615408	−0.307704	+	0.951482i	$0.599561\pi$
$602$	0	0
$603$	11776.6	0.795325
$604$	−15748.2	−1.06090
$605$	−2274.34	−0.152835
$606$	−2923.78	−0.195991
$607$	−19133.3	−1.27940	−0.639699	−	0.768625i	$-0.720941\pi$
$607$	−19133.3	−1.27940	−0.639699	+	0.768625i	$0.720941\pi$
$608$	15293.8	1.02014
$609$	9684.81	0.644414
$610$	9762.82	0.648008
$611$	−2121.20	−0.140449
$612$	−8767.76	−0.579110
$613$	−25944.6	−1.70945	−0.854723	−	0.519084i	$-0.826273\pi$
$613$	−25944.6	−1.70945	−0.854723	+	0.519084i	$0.826273\pi$
$614$	6086.05	0.400021
$615$	−1754.56	−0.115042
$616$	−6746.22	−0.441255
$617$	12175.4	0.794428	0.397214	−	0.917726i	$-0.369977\pi$
$617$	12175.4	0.794428	0.397214	+	0.917726i	$0.369977\pi$
$618$	11980.5	0.779814
$619$	16300.3	1.05842	0.529211	−	0.848490i	$-0.322488\pi$
$619$	16300.3	1.05842	0.529211	+	0.848490i	$0.322488\pi$
$620$	−4131.97	−0.267651
$621$	−4641.58	−0.299936
$622$	15955.0	1.02852
$623$	3074.22	0.197699
$624$	−13717.0	−0.879998
$625$	36446.6	2.33258
$626$	−2260.86	−0.144349
$627$	8752.13	0.557458
$628$	−11118.2	−0.706475
$629$	−18428.9	−1.16822
$630$	20613.2	1.30357
$631$	−407.086	−0.0256828	−0.0128414	−	0.999918i	$-0.504088\pi$
$631$	−407.086	−0.0256828	−0.0128414	+	0.999918i	$0.504088\pi$
$632$	1271.13	0.0800045
$633$	−5309.45	−0.333383
$634$	−25430.0	−1.59299
$635$	−54490.8	−3.40535
$636$	3451.00	0.215159
$637$	5226.93	0.325116
$638$	−24988.0	−1.55060
$639$	−10222.3	−0.632843
$640$	−29158.5	−1.80092
$641$	−20991.1	−1.29345	−0.646723	−	0.762725i	$-0.723861\pi$
$641$	−20991.1	−1.29345	−0.646723	+	0.762725i	$0.723861\pi$
$642$	9637.28	0.592451
$643$	1636.46	0.100367	0.0501834	−	0.998740i	$-0.484019\pi$
$643$	1636.46	0.100367	0.0501834	+	0.998740i	$0.484019\pi$
$644$	−2470.52	−0.151168
$645$	0	0
$646$	−31610.0	−1.92520
$647$	−18731.5	−1.13820	−0.569098	−	0.822270i	$-0.692707\pi$
$647$	−18731.5	−1.13820	−0.569098	+	0.822270i	$0.692707\pi$
$648$	853.462	0.0517394
$649$	−302.202	−0.0182781
$650$	60244.0	3.63533
$651$	2111.05	0.127095
$652$	1008.87	0.0605988
$653$	−27858.6	−1.66951	−0.834754	−	0.550622i	$-0.814390\pi$
$653$	−27858.6	−1.66951	−0.834754	+	0.550622i	$0.814390\pi$
$654$	−4318.43	−0.258202
$655$	8548.01	0.509921
$656$	−2241.40	−0.133402
$657$	7189.92	0.426949
$658$	−2095.90	−0.124174
$659$	−29068.8	−1.71830	−0.859151	−	0.511723i	$-0.829007\pi$
$659$	−29068.8	−1.71830	−0.859151	+	0.511723i	$0.829007\pi$
$660$	9940.03	0.586235
$661$	−22256.5	−1.30965	−0.654823	−	0.755782i	$-0.727257\pi$
$661$	−22256.5	−1.30965	−0.654823	+	0.755782i	$0.727257\pi$
$662$	16256.3	0.954411
$663$	18563.2	1.08738
$664$	−14102.7	−0.824232
$665$	26985.6	1.57362
$666$	−10796.5	−0.628162
$667$	6898.94	0.400492
$668$	−17076.1	−0.989065
$669$	5705.12	0.329705
$670$	48220.4	2.78048
$671$	4673.12	0.268858
$672$	−8874.25	−0.509422
$673$	14047.5	0.804593	0.402296	−	0.915510i	$-0.368212\pi$
$673$	14047.5	0.804593	0.402296	+	0.915510i	$0.368212\pi$
$674$	20513.0	1.17230
$675$	40609.7	2.31566
$676$	4710.76	0.268023
$677$	−10748.8	−0.610207	−0.305103	−	0.952319i	$-0.598691\pi$
$677$	−10748.8	−0.610207	−0.305103	+	0.952319i	$0.598691\pi$
$678$	7531.01	0.426588
$679$	−21387.2	−1.20879
$680$	27065.9	1.52637
$681$	5125.94	0.288438
$682$	−5446.77	−0.305818
$683$	3265.39	0.182938	0.0914690	−	0.995808i	$-0.470844\pi$
$683$	3265.39	0.182938	0.0914690	+	0.995808i	$0.470844\pi$
$684$	−6724.46	−0.375901
$685$	47167.2	2.63090
$686$	24425.2	1.35942
$687$	−7802.09	−0.433287
$688$	0	0
$689$	14195.7	0.784924
$690$	−7557.71	−0.416981
$691$	23042.9	1.26858	0.634292	−	0.773094i	$-0.281292\pi$
$691$	23042.9	1.26858	0.634292	+	0.773094i	$0.281292\pi$
$692$	−1187.97	−0.0652598
$693$	9866.83	0.540851
$694$	2603.38	0.142396
$695$	45367.3	2.47608
$696$	7449.92	0.405730
$697$	3033.28	0.164840
$698$	−20517.5	−1.11260
$699$	−10996.0	−0.595003
$700$	21614.8	1.16709
$701$	−3749.94	−0.202045	−0.101022	−	0.994884i	$-0.532211\pi$
$701$	−3749.94	−0.202045	−0.101022	+	0.994884i	$0.532211\pi$
$702$	27347.7	1.47033
$703$	−14134.1	−0.758290
$704$	625.305	0.0334760
$705$	−2328.21	−0.124376
$706$	30842.9	1.64417
$707$	−4315.13	−0.229543
$708$	−119.507	−0.00634372
$709$	−3372.61	−0.178648	−0.0893238	−	0.996003i	$-0.528471\pi$
$709$	−3372.61	−0.178648	−0.0893238	+	0.996003i	$0.528471\pi$
$710$	−41856.0	−2.21243
$711$	−1859.12	−0.0980624
$712$	2364.81	0.124473
$713$	1503.80	0.0789871
$714$	18341.8	0.961377
$715$	40888.3	2.13865
$716$	−6655.47	−0.347384
$717$	−16780.4	−0.874023
$718$	34441.0	1.79015
$719$	−24951.3	−1.29420	−0.647099	−	0.762406i	$-0.724018\pi$
$719$	−24951.3	−1.29420	−0.647099	+	0.762406i	$0.724018\pi$
$720$	29251.4	1.51408
$721$	17681.7	0.913314
$722$	66.1297	0.00340872
$723$	16289.1	0.837897
$724$	−893.994	−0.0458909
$725$	−60359.7	−3.09200
$726$	−1185.59	−0.0606078
$727$	−6345.25	−0.323703	−0.161852	−	0.986815i	$-0.551747\pi$
$727$	−6345.25	−0.323703	−0.161852	+	0.986815i	$0.551747\pi$
$728$	−10974.1	−0.558690
$729$	9855.70	0.500722
$730$	29439.8	1.49262
$731$	0	0
$732$	1848.00	0.0933117
$733$	33572.9	1.69174	0.845868	−	0.533393i	$-0.179083\pi$
$733$	33572.9	1.69174	0.845868	+	0.533393i	$0.179083\pi$
$734$	12152.5	0.611111
$735$	5737.04	0.287910
$736$	−6321.55	−0.316597
$737$	23081.4	1.15362
$738$	1777.03	0.0886361
$739$	31399.5	1.56299	0.781494	−	0.623912i	$-0.214458\pi$
$739$	31399.5	1.56299	0.781494	+	0.623912i	$0.214458\pi$
$740$	−16052.5	−0.797434
$741$	14237.1	0.705819
$742$	14026.4	0.693968
$743$	7125.78	0.351843	0.175922	−	0.984404i	$-0.443709\pi$
$743$	7125.78	0.351843	0.175922	+	0.984404i	$0.443709\pi$
$744$	1623.90	0.0800203
$745$	42732.1	2.10146
$746$	5824.10	0.285838
$747$	20626.2	1.01027
$748$	−17184.2	−0.839998
$749$	14223.4	0.693875
$750$	38488.7	1.87388
$751$	−5062.80	−0.245998	−0.122999	−	0.992407i	$-0.539251\pi$
$751$	−5062.80	−0.245998	−0.122999	+	0.992407i	$0.539251\pi$
$752$	−2974.20	−0.144226
$753$	911.563	0.0441158
$754$	−40647.9	−1.96328
$755$	−71102.0	−3.42737
$756$	9812.05	0.472038
$757$	−11723.0	−0.562855	−0.281428	−	0.959582i	$-0.590808\pi$
$757$	−11723.0	−0.562855	−0.281428	+	0.959582i	$0.590808\pi$
$758$	−90.5910	−0.00434092
$759$	−3617.61	−0.173005
$760$	20758.3	0.990767
$761$	3477.10	0.165630	0.0828151	−	0.996565i	$-0.473609\pi$
$761$	3477.10	0.165630	0.0828151	+	0.996565i	$0.473609\pi$
$762$	−28405.4	−1.35042
$763$	−6373.46	−0.302405
$764$	−8940.01	−0.423348
$765$	−39585.9	−1.87089
$766$	46244.9	2.18132
$767$	−491.592	−0.0231426
$768$	−15633.7	−0.734548
$769$	−40055.7	−1.87834	−0.939172	−	0.343447i	$-0.888405\pi$
$769$	−40055.7	−1.87834	−0.939172	+	0.343447i	$0.888405\pi$
$770$	40400.6	1.89083
$771$	−18100.1	−0.845471
$772$	17599.0	0.820471
$773$	25647.3	1.19336	0.596682	−	0.802478i	$-0.296485\pi$
$773$	25647.3	1.19336	0.596682	+	0.802478i	$0.296485\pi$
$774$	0	0
$775$	−13156.9	−0.609821
$776$	−16451.8	−0.761065
$777$	8201.34	0.378663
$778$	−1039.69	−0.0479110
$779$	2326.38	0.106998
$780$	16169.4	0.742255
$781$	−20035.0	−0.917937
$782$	13065.7	0.597479
$783$	−27400.2	−1.25058
$784$	7328.87	0.333859
$785$	−50198.2	−2.28236
$786$	4455.98	0.202213
$787$	26018.7	1.17848	0.589242	−	0.807957i	$-0.299427\pi$
$787$	26018.7	1.17848	0.589242	+	0.807957i	$0.299427\pi$
$788$	−9329.19	−0.421750
$789$	22677.7	1.02326
$790$	−7612.33	−0.342828
$791$	11114.8	0.499617
$792$	7589.94	0.340526
$793$	7601.76	0.340411
$794$	−6680.77	−0.298604
$795$	15581.1	0.695098
$796$	−2236.20	−0.0995728
$797$	−5578.67	−0.247938	−0.123969	−	0.992286i	$-0.539562\pi$
$797$	−5578.67	−0.247938	−0.123969	+	0.992286i	$0.539562\pi$
$798$	14067.3	0.624030
$799$	4024.98	0.178215
$800$	55308.0	2.44429
$801$	−3458.70	−0.152568
$802$	15010.9	0.660915
$803$	14091.8	0.619288
$804$	9127.65	0.400382
$805$	−11154.2	−0.488366
$806$	−8860.26	−0.387208
$807$	8881.56	0.387417
$808$	−3319.36	−0.144523
$809$	−25428.5	−1.10509	−0.552546	−	0.833483i	$-0.686343\pi$
$809$	−25428.5	−1.10509	−0.552546	+	0.833483i	$0.686343\pi$
$810$	−5111.07	−0.221709
$811$	31125.3	1.34766	0.673832	−	0.738884i	$-0.264647\pi$
$811$	31125.3	1.34766	0.673832	+	0.738884i	$0.264647\pi$
$812$	−14584.0	−0.630293
$813$	−3167.80	−0.136654
$814$	−21160.5	−0.911147
$815$	4554.99	0.195772
$816$	26028.1	1.11662
$817$	0	0
$818$	−7756.82	−0.331554
$819$	16050.4	0.684793
$820$	2642.13	0.112521
$821$	6826.92	0.290209	0.145104	−	0.989416i	$-0.453648\pi$
$821$	6826.92	0.290209	0.145104	+	0.989416i	$0.453648\pi$
$822$	24587.7	1.04330
$823$	32634.5	1.38222	0.691111	−	0.722749i	$-0.257122\pi$
$823$	32634.5	1.38222	0.691111	+	0.722749i	$0.257122\pi$
$824$	13601.4	0.575033
$825$	31650.9	1.33569
$826$	−485.729	−0.0204609
$827$	−28135.7	−1.18304	−0.591521	−	0.806290i	$-0.701472\pi$
$827$	−28135.7	−1.18304	−0.591521	+	0.806290i	$0.701472\pi$
$828$	2779.49	0.116659
$829$	11861.8	0.496955	0.248478	−	0.968638i	$-0.420070\pi$
$829$	11861.8	0.496955	0.248478	+	0.968638i	$0.420070\pi$
$830$	84455.8	3.53193
$831$	−7987.44	−0.333431
$832$	1017.18	0.0423852
$833$	−9918.14	−0.412537
$834$	23649.5	0.981911
$835$	−77097.6	−3.19530
$836$	−13179.5	−0.545243
$837$	−5972.59	−0.246646
$838$	−18675.1	−0.769833
$839$	−30920.7	−1.27235	−0.636175	−	0.771545i	$-0.719485\pi$
$839$	−30920.7	−1.27235	−0.636175	+	0.771545i	$0.719485\pi$
$840$	−12045.0	−0.494754
$841$	16337.0	0.669850
$842$	−8875.71	−0.363275
$843$	−17593.4	−0.718800
$844$	7995.30	0.326078
$845$	21268.8	0.865880
$846$	2358.02	0.0958279
$847$	−1749.78	−0.0709836
$848$	19904.3	0.806032
$849$	−14415.3	−0.582724
$850$	−114313.	−4.61284
$851$	5842.20	0.235333
$852$	−7922.92	−0.318586
$853$	−21418.7	−0.859743	−0.429872	−	0.902890i	$-0.641441\pi$
$853$	−21418.7	−0.859743	−0.429872	+	0.902890i	$0.641441\pi$
$854$	7511.09	0.300965
$855$	−30360.5	−1.21439
$856$	10941.2	0.436871
$857$	−7917.10	−0.315569	−0.157785	−	0.987474i	$-0.550435\pi$
$857$	−7917.10	−0.315569	−0.157785	+	0.987474i	$0.550435\pi$
$858$	21314.6	0.848099
$859$	27239.7	1.08196	0.540982	−	0.841034i	$-0.318053\pi$
$859$	27239.7	1.08196	0.540982	+	0.841034i	$0.318053\pi$
$860$	0	0
$861$	−1349.89	−0.0534309
$862$	−23568.4	−0.931258
$863$	−12643.6	−0.498718	−0.249359	−	0.968411i	$-0.580220\pi$
$863$	−12643.6	−0.498718	−0.249359	+	0.968411i	$0.580220\pi$
$864$	25107.0	0.988609
$865$	−5363.60	−0.210830
$866$	−11799.3	−0.462996
$867$	−20342.4	−0.796844
$868$	−3178.96	−0.124310
$869$	−3643.75	−0.142239
$870$	−44614.8	−1.73860
$871$	37546.5	1.46064
$872$	−4902.70	−0.190397
$873$	24062.0	0.932846
$874$	10020.8	0.387824
$875$	56804.4	2.19468
$876$	5572.65	0.214934
$877$	−19888.4	−0.765773	−0.382886	−	0.923795i	$-0.625070\pi$
$877$	−19888.4	−0.765773	−0.382886	+	0.923795i	$0.625070\pi$
$878$	2988.48	0.114870
$879$	−9236.33	−0.354418
$880$	57330.9	2.19616
$881$	−7276.04	−0.278247	−0.139124	−	0.990275i	$-0.544429\pi$
$881$	−7276.04	−0.278247	−0.139124	+	0.990275i	$0.544429\pi$
$882$	−5810.50	−0.221825
$883$	−20575.3	−0.784160	−0.392080	−	0.919931i	$-0.628244\pi$
$883$	−20575.3	−0.784160	−0.392080	+	0.919931i	$0.628244\pi$
$884$	−27953.6	−1.06355
$885$	−539.567	−0.0204942
$886$	5762.05	0.218488
$887$	−35258.8	−1.33469	−0.667347	−	0.744747i	$-0.732570\pi$
$887$	−35258.8	−1.33469	−0.667347	+	0.744747i	$0.732570\pi$
$888$	6308.78	0.238411
$889$	−41922.8	−1.58161
$890$	−14162.0	−0.533382
$891$	−2446.49	−0.0919870
$892$	−8591.13	−0.322480
$893$	3086.97	0.115679
$894$	22275.8	0.833349
$895$	−30049.0	−1.12227
$896$	−22433.3	−0.836432
$897$	−5884.76	−0.219048
$898$	14761.2	0.548539
$899$	8877.27	0.329337
$900$	−24318.1	−0.900670
$901$	−26936.4	−0.995984
$902$	3482.87	0.128566
$903$	0	0
$904$	8549.93	0.314565
$905$	−4036.33	−0.148256
$906$	−37064.7	−1.35915
$907$	−29990.1	−1.09791	−0.548955	−	0.835852i	$-0.684974\pi$
$907$	−29990.1	−1.09791	−0.548955	+	0.835852i	$0.684974\pi$
$908$	−7718.97	−0.282118
$909$	4854.80	0.177144
$910$	65719.7	2.39405
$911$	−31712.6	−1.15333	−0.576665	−	0.816980i	$-0.695646\pi$
$911$	−31712.6	−1.15333	−0.576665	+	0.816980i	$0.695646\pi$
$912$	19962.3	0.724801
$913$	40426.0	1.46539
$914$	−21716.1	−0.785891
$915$	8343.62	0.301455
$916$	11748.9	0.423793
$917$	6576.47	0.236831
$918$	−51892.5	−1.86569
$919$	40847.8	1.46621	0.733104	−	0.680117i	$-0.238071\pi$
$919$	40847.8	1.46621	0.733104	+	0.680117i	$0.238071\pi$
$920$	−8580.25	−0.307481
$921$	5201.33	0.186091
$922$	−33249.6	−1.18766
$923$	−32590.9	−1.16224
$924$	7647.44	0.272275
$925$	−51114.1	−1.81689
$926$	17886.8	0.634771
$927$	−19893.0	−0.704824
$928$	−37317.5	−1.32005
$929$	−28741.6	−1.01505	−0.507526	−	0.861637i	$-0.669440\pi$
$929$	−28741.6	−1.01505	−0.507526	+	0.861637i	$0.669440\pi$
$930$	−9724.94	−0.342896
$931$	−7606.75	−0.267778
$932$	16558.5	0.581965
$933$	13635.7	0.478469
$934$	10807.1	0.378606
$935$	−77585.8	−2.71372
$936$	12346.5	0.431153
$937$	44340.2	1.54592	0.772962	−	0.634452i	$-0.218774\pi$
$937$	44340.2	1.54592	0.772962	+	0.634452i	$0.218774\pi$
$938$	37098.7	1.29138
$939$	−1932.21	−0.0671514
$940$	3505.96	0.121651
$941$	20231.3	0.700872	0.350436	−	0.936587i	$-0.386033\pi$
$941$	20231.3	0.700872	0.350436	+	0.936587i	$0.386033\pi$
$942$	−26167.8	−0.905087
$943$	−961.587	−0.0332064
$944$	−689.279	−0.0237649
$945$	44300.8	1.52498
$946$	0	0
$947$	15976.9	0.548235	0.274117	−	0.961696i	$-0.411614\pi$
$947$	15976.9	0.548235	0.274117	+	0.961696i	$0.411614\pi$
$948$	−1440.94	−0.0493665
$949$	22923.1	0.784104
$950$	−87673.0	−2.99420
$951$	−21733.3	−0.741061
$952$	20823.4	0.708917
$953$	−12716.9	−0.432255	−0.216128	−	0.976365i	$-0.569343\pi$
$953$	−12716.9	−0.432255	−0.216128	+	0.976365i	$0.569343\pi$
$954$	−15780.6	−0.535551
$955$	−40363.6	−1.36768
$956$	25268.9	0.854870
$957$	−21355.5	−0.721344
$958$	−10851.0	−0.365949
$959$	36288.4	1.22191
$960$	1116.45	0.0375347
$961$	−27856.0	−0.935047
$962$	−34421.7	−1.15364
$963$	−16002.3	−0.535478
$964$	−24529.2	−0.819537
$965$	79458.6	2.65063
$966$	−5814.58	−0.193666
$967$	−28688.5	−0.954044	−0.477022	−	0.878891i	$-0.658284\pi$
$967$	−28688.5	−0.954044	−0.477022	+	0.878891i	$0.658284\pi$
$968$	−1346.00	−0.0446921
$969$	−27014.9	−0.895609
$970$	98524.0	3.26125
$971$	23625.4	0.780819	0.390410	−	0.920641i	$-0.372333\pi$
$971$	23625.4	0.780819	0.390410	+	0.920641i	$0.372333\pi$
$972$	−17688.5	−0.583703
$973$	34903.6	1.15001
$974$	49972.3	1.64396
$975$	51486.5	1.69117
$976$	10658.7	0.349566
$977$	26135.9	0.855845	0.427922	−	0.903815i	$-0.359246\pi$
$977$	26135.9	0.855845	0.427922	+	0.903815i	$0.359246\pi$
$978$	2374.47	0.0776350
$979$	−6778.83	−0.221300
$980$	−8639.19	−0.281601
$981$	7170.56	0.233372
$982$	39426.6	1.28122
$983$	43801.4	1.42121	0.710603	−	0.703593i	$-0.248422\pi$
$983$	43801.4	1.42121	0.710603	+	0.703593i	$0.248422\pi$
$984$	−1038.38	−0.0336407
$985$	−42120.7	−1.36251
$986$	77129.7	2.49119
$987$	−1791.22	−0.0577662
$988$	−21439.1	−0.690353
$989$	0	0
$990$	−45453.3	−1.45919
$991$	−33981.6	−1.08926	−0.544632	−	0.838675i	$-0.683331\pi$
$991$	−33981.6	−1.08926	−0.544632	+	0.838675i	$0.683331\pi$
$992$	−8134.30	−0.260347
$993$	13893.2	0.443994
$994$	−32202.2	−1.02756
$995$	−10096.3	−0.321682
$996$	15986.6	0.508590
$997$	19933.8	0.633209	0.316605	−	0.948558i	$-0.397457\pi$
$997$	19933.8	0.633209	0.316605	+	0.948558i	$0.397457\pi$
$998$	68305.5	2.16651
$999$	−23203.2	−0.734852

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.g.1.8		30
43.32	odd	14		43.4.e.a.35.3	yes	60
43.39	odd	14		43.4.e.a.16.3	✓	60
43.42	odd	2		1849.4.a.h.1.23		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.16.3	✓	60	43.39	odd	14
43.4.e.a.35.3	yes	60	43.32	odd	14
1849.4.a.g.1.8		30	1.1	even	1	trivial
1849.4.a.h.1.23		30	43.42	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.54418 q^{2} -3.02897 q^{3} +4.56122 q^{4} +20.5936 q^{5} +10.7352 q^{6} +15.8438 q^{7} +12.1877 q^{8} -17.8253 q^{9} +O(q^{10})\)	\(q-3.54418 q^{2} -3.02897 q^{3} +4.56122 q^{4} +20.5936 q^{5} +10.7352 q^{6} +15.8438 q^{7} +12.1877 q^{8} -17.8253 q^{9} -72.9874 q^{10} -34.9365 q^{11} -13.8158 q^{12} -56.8312 q^{13} -56.1534 q^{14} -62.3774 q^{15} -79.6850 q^{16} +107.838 q^{17} +63.1762 q^{18} +82.7064 q^{19} +93.9319 q^{20} -47.9905 q^{21} +123.821 q^{22} -34.1859 q^{23} -36.9161 q^{24} +299.096 q^{25} +201.420 q^{26} +135.775 q^{27} +72.2672 q^{28} -201.807 q^{29} +221.077 q^{30} -43.9890 q^{31} +184.917 q^{32} +105.822 q^{33} -382.196 q^{34} +326.282 q^{35} -81.3052 q^{36} -170.895 q^{37} -293.126 q^{38} +172.140 q^{39} +250.988 q^{40} +28.1282 q^{41} +170.087 q^{42} -159.353 q^{44} -367.088 q^{45} +121.161 q^{46} +37.3245 q^{47} +241.364 q^{48} -91.9730 q^{49} -1060.05 q^{50} -326.637 q^{51} -259.219 q^{52} -249.787 q^{53} -481.210 q^{54} -719.469 q^{55} +193.099 q^{56} -250.515 q^{57} +715.240 q^{58} +8.65004 q^{59} -284.517 q^{60} -133.760 q^{61} +155.905 q^{62} -282.422 q^{63} -17.8983 q^{64} -1170.36 q^{65} -375.051 q^{66} -660.668 q^{67} +491.870 q^{68} +103.548 q^{69} -1156.40 q^{70} +573.469 q^{71} -217.249 q^{72} -403.354 q^{73} +605.683 q^{74} -905.954 q^{75} +377.242 q^{76} -553.529 q^{77} -610.096 q^{78} +104.296 q^{79} -1641.00 q^{80} +70.0267 q^{81} -99.6914 q^{82} -1157.13 q^{83} -218.895 q^{84} +2220.76 q^{85} +611.267 q^{87} -425.795 q^{88} +194.033 q^{89} +1301.03 q^{90} -900.424 q^{91} -155.929 q^{92} +133.241 q^{93} -132.285 q^{94} +1703.22 q^{95} -560.108 q^{96} -1349.88 q^{97} +325.969 q^{98} +622.755 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9}+O(q^{10}) \) 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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