LMFDB - Embedded newform 1045.4.a.g.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.28558 q^{2} +0.0269626 q^{3} -2.77614 q^{4} -5.00000 q^{5} +0.0616250 q^{6} -20.0645 q^{7} -24.6297 q^{8} -26.9993 q^{9} +O(q^{10})$	\(q+2.28558 q^{2} +0.0269626 q^{3} -2.77614 q^{4} -5.00000 q^{5} +0.0616250 q^{6} -20.0645 q^{7} -24.6297 q^{8} -26.9993 q^{9} -11.4279 q^{10} -11.0000 q^{11} -0.0748520 q^{12} +58.5804 q^{13} -45.8589 q^{14} -0.134813 q^{15} -34.0839 q^{16} -33.3772 q^{17} -61.7089 q^{18} -19.0000 q^{19} +13.8807 q^{20} -0.540990 q^{21} -25.1413 q^{22} -167.385 q^{23} -0.664080 q^{24} +25.0000 q^{25} +133.890 q^{26} -1.45596 q^{27} +55.7018 q^{28} -206.641 q^{29} -0.308125 q^{30} +243.701 q^{31} +119.136 q^{32} -0.296588 q^{33} -76.2860 q^{34} +100.322 q^{35} +74.9538 q^{36} +288.156 q^{37} -43.4259 q^{38} +1.57948 q^{39} +123.148 q^{40} -229.770 q^{41} -1.23647 q^{42} -545.592 q^{43} +30.5376 q^{44} +134.996 q^{45} -382.571 q^{46} +190.321 q^{47} -0.918990 q^{48} +59.5835 q^{49} +57.1394 q^{50} -0.899934 q^{51} -162.627 q^{52} +738.828 q^{53} -3.32771 q^{54} +55.0000 q^{55} +494.182 q^{56} -0.512289 q^{57} -472.295 q^{58} +661.271 q^{59} +0.374260 q^{60} -70.6345 q^{61} +556.997 q^{62} +541.726 q^{63} +544.966 q^{64} -292.902 q^{65} -0.677876 q^{66} +149.356 q^{67} +92.6597 q^{68} -4.51313 q^{69} +229.295 q^{70} +533.744 q^{71} +664.984 q^{72} +157.573 q^{73} +658.601 q^{74} +0.674065 q^{75} +52.7467 q^{76} +220.709 q^{77} +3.61002 q^{78} -84.9020 q^{79} +170.420 q^{80} +728.941 q^{81} -525.156 q^{82} +971.702 q^{83} +1.50187 q^{84} +166.886 q^{85} -1246.99 q^{86} -5.57159 q^{87} +270.927 q^{88} +89.9418 q^{89} +308.544 q^{90} -1175.39 q^{91} +464.684 q^{92} +6.57081 q^{93} +434.994 q^{94} +95.0000 q^{95} +3.21222 q^{96} -932.367 q^{97} +136.183 q^{98} +296.992 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * 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 $

$2$	2.28558	0.808073	0.404037	−	0.914743i	$-0.367607\pi$
$2$	2.28558	0.808073	0.404037	+	0.914743i	$0.367607\pi$
$3$	0.0269626	0.00518895	0.00259448	−	0.999997i	$-0.499174\pi$
$3$	0.0269626	0.00518895	0.00259448	+	0.999997i	$0.499174\pi$
$4$	−2.77614	−0.347018
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0.0616250	0.00419305
$7$	−20.0645	−1.08338	−0.541690	−	0.840578i	$-0.682215\pi$
$7$	−20.0645	−1.08338	−0.541690	+	0.840578i	$0.682215\pi$
$8$	−24.6297	−1.08849
$9$	−26.9993	−0.999973
$10$	−11.4279	−0.361381
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−0.0748520	−0.00180066
$13$	58.5804	1.24979	0.624895	−	0.780708i	$-0.285142\pi$
$13$	58.5804	1.24979	0.624895	+	0.780708i	$0.285142\pi$
$14$	−45.8589	−0.875451
$15$	−0.134813	−0.00232057
$16$	−34.0839	−0.532561
$17$	−33.3772	−0.476185	−0.238093	−	0.971242i	$-0.576522\pi$
$17$	−33.3772	−0.476185	−0.238093	+	0.971242i	$0.576522\pi$
$18$	−61.7089	−0.808051
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	13.8807	0.155191
$21$	−0.540990	−0.00562161
$22$	−25.1413	−0.243643
$23$	−167.385	−1.51749	−0.758743	−	0.651390i	$-0.774186\pi$
$23$	−167.385	−1.51749	−0.758743	+	0.651390i	$0.774186\pi$
$24$	−0.664080	−0.00564812
$25$	25.0000	0.200000
$26$	133.890	1.00992
$27$	−1.45596	−0.0103778
$28$	55.7018	0.375952
$29$	−206.641	−1.32318	−0.661592	−	0.749864i	$-0.730119\pi$
$29$	−206.641	−1.32318	−0.661592	+	0.749864i	$0.730119\pi$
$30$	−0.308125	−0.00187519
$31$	243.701	1.41194	0.705968	−	0.708244i	$-0.250512\pi$
$31$	243.701	1.41194	0.705968	+	0.708244i	$0.250512\pi$
$32$	119.136	0.658141
$33$	−0.296588	−0.00156453
$34$	−76.2860	−0.384793
$35$	100.322	0.484502
$36$	74.9538	0.347008
$37$	288.156	1.28034	0.640169	−	0.768234i	$-0.278864\pi$
$37$	288.156	1.28034	0.640169	+	0.768234i	$0.278864\pi$
$38$	−43.4259	−0.185385
$39$	1.57948	0.00648511
$40$	123.148	0.486787
$41$	−229.770	−0.875219	−0.437610	−	0.899165i	$-0.644175\pi$
$41$	−229.770	−0.875219	−0.437610	+	0.899165i	$0.644175\pi$
$42$	−1.23647	−0.00454267
$43$	−545.592	−1.93493	−0.967465	−	0.253006i	$-0.918581\pi$
$43$	−545.592	−1.93493	−0.967465	+	0.253006i	$0.918581\pi$
$44$	30.5376	0.104630
$45$	134.996	0.447202
$46$	−382.571	−1.22624
$47$	190.321	0.590664	0.295332	−	0.955395i	$-0.404570\pi$
$47$	190.321	0.590664	0.295332	+	0.955395i	$0.404570\pi$
$48$	−0.918990	−0.00276343
$49$	59.5835	0.173713
$50$	57.1394	0.161615
$51$	−0.899934	−0.00247090
$52$	−162.627	−0.433700
$53$	738.828	1.91483	0.957413	−	0.288721i	$-0.0932300\pi$
$53$	738.828	1.91483	0.957413	+	0.288721i	$0.0932300\pi$
$54$	−3.32771	−0.00838599
$55$	55.0000	0.134840
$56$	494.182	1.17925
$57$	−0.512289	−0.00119043
$58$	−472.295	−1.06923
$59$	661.271	1.45916	0.729578	−	0.683898i	$-0.239716\pi$
$59$	661.271	1.45916	0.729578	+	0.683898i	$0.239716\pi$
$60$	0.374260	0.000805279 0
$61$	−70.6345	−0.148259	−0.0741296	−	0.997249i	$-0.523618\pi$
$61$	−70.6345	−0.148259	−0.0741296	+	0.997249i	$0.523618\pi$
$62$	556.997	1.14095
$63$	541.726	1.08335
$64$	544.966	1.06439
$65$	−292.902	−0.558923
$66$	−0.677876	−0.00126425
$67$	149.356	0.272340	0.136170	−	0.990686i	$-0.456521\pi$
$67$	149.356	0.272340	0.136170	+	0.990686i	$0.456521\pi$
$68$	92.6597	0.165245
$69$	−4.51313	−0.00787416
$70$	229.295	0.391513
$71$	533.744	0.892165	0.446083	−	0.894992i	$-0.352819\pi$
$71$	533.744	0.892165	0.446083	+	0.894992i	$0.352819\pi$
$72$	664.984	1.08846
$73$	157.573	0.252637	0.126319	−	0.991990i	$-0.459684\pi$
$73$	157.573	0.252637	0.126319	+	0.991990i	$0.459684\pi$
$74$	658.601	1.03461
$75$	0.674065	0.00103779
$76$	52.7467	0.0796113
$77$	220.709	0.326651
$78$	3.61002	0.00524044
$79$	−84.9020	−0.120914	−0.0604571	−	0.998171i	$-0.519256\pi$
$79$	−84.9020	−0.120914	−0.0604571	+	0.998171i	$0.519256\pi$
$80$	170.420	0.238169
$81$	728.941	0.999919
$82$	−525.156	−0.707241
$83$	971.702	1.28504	0.642519	−	0.766270i	$-0.277890\pi$
$83$	971.702	1.28504	0.642519	+	0.766270i	$0.277890\pi$
$84$	1.50187	0.00195080
$85$	166.886	0.212957
$86$	−1246.99	−1.56356
$87$	−5.57159	−0.00686594
$88$	270.927	0.328192
$89$	89.9418	0.107121	0.0535607	−	0.998565i	$-0.482943\pi$
$89$	89.9418	0.107121	0.0535607	+	0.998565i	$0.482943\pi$
$90$	308.544	0.361372
$91$	−1175.39	−1.35400
$92$	464.684	0.526594
$93$	6.57081	0.00732647
$94$	434.994	0.477300
$95$	95.0000	0.102598
$96$	3.21222	0.00341506
$97$	−932.367	−0.975954	−0.487977	−	0.872856i	$-0.662265\pi$
$97$	−932.367	−0.975954	−0.487977	+	0.872856i	$0.662265\pi$
$98$	136.183	0.140373
$99$	296.992	0.301503
$100$	−69.4035	−0.0694035
$101$	−10.8120	−0.0106519	−0.00532593	−	0.999986i	$-0.501695\pi$
$101$	−10.8120	−0.0106519	−0.00532593	+	0.999986i	$0.501695\pi$
$102$	−2.05687	−0.00199667
$103$	−985.016	−0.942296	−0.471148	−	0.882054i	$-0.656160\pi$
$103$	−985.016	−0.942296	−0.471148	+	0.882054i	$0.656160\pi$
$104$	−1442.82	−1.36038
$105$	2.70495	0.00251406
$106$	1688.65	1.54732
$107$	1764.21	1.59395	0.796977	−	0.604010i	$-0.206431\pi$
$107$	1764.21	1.59395	0.796977	+	0.604010i	$0.206431\pi$
$108$	4.04195	0.00360127
$109$	−687.627	−0.604245	−0.302123	−	0.953269i	$-0.597695\pi$
$109$	−687.627	−0.604245	−0.302123	+	0.953269i	$0.597695\pi$
$110$	125.707	0.108961
$111$	7.76942	0.00664361
$112$	683.876	0.576966
$113$	−1070.35	−0.891062	−0.445531	−	0.895266i	$-0.646985\pi$
$113$	−1070.35	−0.891062	−0.445531	+	0.895266i	$0.646985\pi$
$114$	−1.17088	−0.000961952 0
$115$	836.924	0.678640
$116$	573.666	0.459168
$117$	−1581.63	−1.24976
$118$	1511.39	1.17911
$119$	669.695	0.515890
$120$	3.32040	0.00252591
$121$	121.000	0.0909091
$122$	−161.440	−0.119804
$123$	−6.19518	−0.00454147
$124$	−676.549	−0.489967
$125$	−125.000	−0.0894427
$126$	1238.16	0.875427
$127$	704.544	0.492269	0.246134	−	0.969236i	$-0.420840\pi$
$127$	704.544	0.492269	0.246134	+	0.969236i	$0.420840\pi$
$128$	292.472	0.201962
$129$	−14.7106	−0.0100403
$130$	−669.450	−0.451651
$131$	−2183.01	−1.45596	−0.727979	−	0.685600i	$-0.759540\pi$
$131$	−2183.01	−1.45596	−0.727979	+	0.685600i	$0.759540\pi$
$132$	0.823372	0.000542919 0
$133$	381.225	0.248544
$134$	341.365	0.220070
$135$	7.27980	0.00464108
$136$	822.069	0.518322
$137$	−337.725	−0.210612	−0.105306	−	0.994440i	$-0.533582\pi$
$137$	−337.725	−0.210612	−0.105306	+	0.994440i	$0.533582\pi$
$138$	−10.3151	−0.00636290
$139$	−1514.55	−0.924187	−0.462094	−	0.886831i	$-0.652902\pi$
$139$	−1514.55	−0.924187	−0.462094	+	0.886831i	$0.652902\pi$
$140$	−278.509	−0.168131
$141$	5.13155	0.00306493
$142$	1219.91	0.720935
$143$	−644.384	−0.376826
$144$	920.241	0.532547
$145$	1033.21	0.591746
$146$	360.145	0.204149
$147$	1.60652	0.000901387 0
$148$	−799.961	−0.444300
$149$	−1641.30	−0.902421	−0.451210	−	0.892418i	$-0.649008\pi$
$149$	−1641.30	−0.902421	−0.451210	+	0.892418i	$0.649008\pi$
$150$	1.54063	0.000838611 0
$151$	750.577	0.404511	0.202255	−	0.979333i	$-0.435173\pi$
$151$	750.577	0.404511	0.202255	+	0.979333i	$0.435173\pi$
$152$	467.964	0.249716
$153$	901.159	0.476172
$154$	504.448	0.263958
$155$	−1218.51	−0.631437
$156$	−4.38486	−0.00225045
$157$	1283.67	0.652534	0.326267	−	0.945278i	$-0.394209\pi$
$157$	1283.67	0.652534	0.326267	+	0.945278i	$0.394209\pi$
$158$	−194.050	−0.0977076
$159$	19.9207	0.00993594
$160$	−595.681	−0.294329
$161$	3358.49	1.64401
$162$	1666.05	0.808008
$163$	−3057.91	−1.46941	−0.734706	−	0.678386i	$-0.762680\pi$
$163$	−3057.91	−1.46941	−0.734706	+	0.678386i	$0.762680\pi$
$164$	637.873	0.303717
$165$	1.48294	0.000699678 0
$166$	2220.90	1.03840
$167$	968.309	0.448683	0.224341	−	0.974511i	$-0.427977\pi$
$167$	968.309	0.448683	0.224341	+	0.974511i	$0.427977\pi$
$168$	13.3244	0.00611906
$169$	1234.66	0.561977
$170$	381.430	0.172084
$171$	512.986	0.229410
$172$	1514.64	0.671455
$173$	−142.248	−0.0625141	−0.0312571	−	0.999511i	$-0.509951\pi$
$173$	−142.248	−0.0625141	−0.0312571	+	0.999511i	$0.509951\pi$
$174$	−12.7343	−0.00554818
$175$	−501.612	−0.216676
$176$	374.923	0.160573
$177$	17.8296	0.00757149
$178$	205.569	0.0865620
$179$	2573.56	1.07462	0.537309	−	0.843385i	$-0.319441\pi$
$179$	2573.56	1.07462	0.537309	+	0.843385i	$0.319441\pi$
$180$	−374.769	−0.155187
$181$	4021.04	1.65128	0.825639	−	0.564199i	$-0.190815\pi$
$181$	4021.04	1.65128	0.825639	+	0.564199i	$0.190815\pi$
$182$	−2686.43	−1.09413
$183$	−1.90449	−0.000769310 0
$184$	4122.64	1.65177
$185$	−1440.78	−0.572584
$186$	15.0181	0.00592032
$187$	367.149	0.143575
$188$	−528.359	−0.204971
$189$	29.2131	0.0112431
$190$	217.130	0.0829066
$191$	5129.28	1.94315	0.971576	−	0.236728i	$-0.0760750\pi$
$191$	5129.28	1.94315	0.971576	+	0.236728i	$0.0760750\pi$
$192$	14.6937	0.00552305
$193$	−2195.46	−0.818821	−0.409411	−	0.912350i	$-0.634266\pi$
$193$	−2195.46	−0.818821	−0.409411	+	0.912350i	$0.634266\pi$
$194$	−2131.00	−0.788643
$195$	−7.89740	−0.00290023
$196$	−165.412	−0.0602814
$197$	−2938.32	−1.06267	−0.531337	−	0.847161i	$-0.678310\pi$
$197$	−2938.32	−1.06267	−0.531337	+	0.847161i	$0.678310\pi$
$198$	678.798	0.243637
$199$	−5526.58	−1.96869	−0.984344	−	0.176258i	$-0.943601\pi$
$199$	−5526.58	−1.96869	−0.984344	+	0.176258i	$0.943601\pi$
$200$	−615.742	−0.217698
$201$	4.02703	0.00141316
$202$	−24.7117	−0.00860748
$203$	4146.15	1.43351
$204$	2.49835	0.000857447 0
$205$	1148.85	0.391410
$206$	−2251.33	−0.761444
$207$	4519.27	1.51744
$208$	−1996.65	−0.665590
$209$	209.000	0.0691714
$210$	6.18237	0.00203154
$211$	−1991.13	−0.649646	−0.324823	−	0.945775i	$-0.605305\pi$
$211$	−1991.13	−0.649646	−0.324823	+	0.945775i	$0.605305\pi$
$212$	−2051.09	−0.664479
$213$	14.3911	0.00462940
$214$	4032.25	1.28803
$215$	2727.96	0.865327
$216$	35.8599	0.0112961
$217$	−4889.74	−1.52966
$218$	−1571.62	−0.488275
$219$	4.24858	0.00131092
$220$	−152.688	−0.0467919
$221$	−1955.25	−0.595132
$222$	17.7576	0.00536852
$223$	3578.87	1.07470	0.537351	−	0.843358i	$-0.319425\pi$
$223$	3578.87	1.07470	0.537351	+	0.843358i	$0.319425\pi$
$224$	−2390.41	−0.713017
$225$	−674.982	−0.199995
$226$	−2446.37	−0.720044
$227$	−5467.29	−1.59858	−0.799288	−	0.600948i	$-0.794790\pi$
$227$	−5467.29	−1.59858	−0.799288	+	0.600948i	$0.794790\pi$
$228$	1.42219	0.000413099 0
$229$	−797.553	−0.230148	−0.115074	−	0.993357i	$-0.536710\pi$
$229$	−797.553	−0.230148	−0.115074	+	0.993357i	$0.536710\pi$
$230$	1912.85	0.548391
$231$	5.95089	0.00169498
$232$	5089.51	1.44027
$233$	475.737	0.133762	0.0668810	−	0.997761i	$-0.478695\pi$
$233$	475.737	0.133762	0.0668810	+	0.997761i	$0.478695\pi$
$234$	−3614.93	−1.00990
$235$	−951.606	−0.264153
$236$	−1835.78	−0.506353
$237$	−2.28918	−0.000627418 0
$238$	1530.64	0.416877
$239$	6384.37	1.72791	0.863955	−	0.503569i	$-0.167980\pi$
$239$	6384.37	1.72791	0.863955	+	0.503569i	$0.167980\pi$
$240$	4.59495	0.00123585
$241$	−1497.86	−0.400356	−0.200178	−	0.979760i	$-0.564152\pi$
$241$	−1497.86	−0.400356	−0.200178	+	0.979760i	$0.564152\pi$
$242$	276.555	0.0734612
$243$	58.9651	0.0155663
$244$	196.091	0.0514486
$245$	−297.917	−0.0776867
$246$	−14.1596	−0.00366984
$247$	−1113.03	−0.286722
$248$	−6002.28	−1.53688
$249$	26.1996	0.00666800
$250$	−285.697	−0.0722763
$251$	626.913	0.157651	0.0788255	−	0.996888i	$-0.474883\pi$
$251$	626.913	0.157651	0.0788255	+	0.996888i	$0.474883\pi$
$252$	−1503.91	−0.375942
$253$	1841.23	0.457539
$254$	1610.29	0.397789
$255$	4.49967	0.00110502
$256$	−3691.26	−0.901187
$257$	−2838.21	−0.688882	−0.344441	−	0.938808i	$-0.611932\pi$
$257$	−2838.21	−0.688882	−0.344441	+	0.938808i	$0.611932\pi$
$258$	−33.6221	−0.00811326
$259$	−5781.69	−1.38709
$260$	813.137	0.193956
$261$	5579.17	1.32315
$262$	−4989.43	−1.17652
$263$	8215.76	1.92626	0.963129	−	0.269041i	$-0.0867068\pi$
$263$	8215.76	1.92626	0.963129	+	0.269041i	$0.0867068\pi$
$264$	7.30488	0.00170297
$265$	−3694.14	−0.856337
$266$	871.319	0.200842
$267$	2.42506	0.000555848 0
$268$	−414.634	−0.0945067
$269$	−346.177	−0.0784640	−0.0392320	−	0.999230i	$-0.512491\pi$
$269$	−346.177	−0.0784640	−0.0392320	+	0.999230i	$0.512491\pi$
$270$	16.6385	0.00375033
$271$	−2164.25	−0.485124	−0.242562	−	0.970136i	$-0.577988\pi$
$271$	−2164.25	−0.485124	−0.242562	+	0.970136i	$0.577988\pi$
$272$	1137.62	0.253598
$273$	−31.6914	−0.00702583
$274$	−771.896	−0.170190
$275$	−275.000	−0.0603023
$276$	12.5291	0.00273247
$277$	−970.377	−0.210485	−0.105242	−	0.994447i	$-0.533562\pi$
$277$	−970.377	−0.210485	−0.105242	+	0.994447i	$0.533562\pi$
$278$	−3461.61	−0.746811
$279$	−6579.75	−1.41190
$280$	−2470.91	−0.527375
$281$	5667.77	1.20324	0.601620	−	0.798782i	$-0.294522\pi$
$281$	5667.77	1.20324	0.601620	+	0.798782i	$0.294522\pi$
$282$	11.7286	0.00247668
$283$	8057.08	1.69238	0.846190	−	0.532881i	$-0.178891\pi$
$283$	8057.08	1.69238	0.846190	+	0.532881i	$0.178891\pi$
$284$	−1481.75	−0.309597
$285$	2.56145	0.000532375 0
$286$	−1472.79	−0.304503
$287$	4610.21	0.948195
$288$	−3216.59	−0.658123
$289$	−3798.97	−0.773248
$290$	2361.47	0.478174
$291$	−25.1390	−0.00506418
$292$	−437.445	−0.0876696
$293$	272.493	0.0543318	0.0271659	−	0.999631i	$-0.491352\pi$
$293$	272.493	0.0543318	0.0271659	+	0.999631i	$0.491352\pi$
$294$	3.67183	0.000728387 0
$295$	−3306.36	−0.652554
$296$	−7097.18	−1.39363
$297$	16.0156	0.00312901
$298$	−3751.32	−0.729222
$299$	−9805.47	−1.89654
$300$	−1.87130	−0.000360132 0
$301$	10947.0	2.09626
$302$	1715.50	0.326874
$303$	−0.291520	−5.52719e−5 0
$304$	647.594	0.122178
$305$	353.172	0.0663036
$306$	2059.67	0.384782
$307$	1492.84	0.277528	0.138764	−	0.990325i	$-0.455687\pi$
$307$	1492.84	0.277528	0.138764	+	0.990325i	$0.455687\pi$
$308$	−612.720	−0.113354
$309$	−26.5586	−0.00488953
$310$	−2784.99	−0.510247
$311$	−4574.03	−0.833986	−0.416993	−	0.908910i	$-0.636916\pi$
$311$	−4574.03	−0.833986	−0.416993	+	0.908910i	$0.636916\pi$
$312$	−38.9021	−0.00705896
$313$	6519.30	1.17729	0.588646	−	0.808391i	$-0.299661\pi$
$313$	6519.30	1.17729	0.588646	+	0.808391i	$0.299661\pi$
$314$	2933.92	0.527295
$315$	−2708.63	−0.484489
$316$	235.700	0.0419594
$317$	9069.02	1.60684	0.803418	−	0.595416i	$-0.203013\pi$
$317$	9069.02	1.60684	0.803418	+	0.595416i	$0.203013\pi$
$318$	45.5303	0.00802897
$319$	2273.06	0.398955
$320$	−2724.83	−0.476008
$321$	47.5678	0.00827095
$322$	7676.09	1.32848
$323$	634.166	0.109244
$324$	−2023.64	−0.346990
$325$	1464.51	0.249958
$326$	−6989.09	−1.18739
$327$	−18.5402	−0.00313540
$328$	5659.16	0.952666
$329$	−3818.70	−0.639914
$330$	3.38938	0.000565391 0
$331$	7094.74	1.17813	0.589067	−	0.808084i	$-0.299496\pi$
$331$	7094.74	1.17813	0.589067	+	0.808084i	$0.299496\pi$
$332$	−2697.58	−0.445931
$333$	−7779.99	−1.28030
$334$	2213.14	0.362568
$335$	−746.781	−0.121794
$336$	18.4391	0.00299385
$337$	1374.88	0.222239	0.111119	−	0.993807i	$-0.464556\pi$
$337$	1374.88	0.222239	0.111119	+	0.993807i	$0.464556\pi$
$338$	2821.92	0.454119
$339$	−28.8594	−0.00462368
$340$	−463.299	−0.0738997
$341$	−2680.71	−0.425715
$342$	1172.47	0.185380
$343$	5686.61	0.895183
$344$	13437.8	2.10615
$345$	22.5656	0.00352143
$346$	−325.119	−0.0505160
$347$	8582.30	1.32773	0.663864	−	0.747853i	$-0.268915\pi$
$347$	8582.30	1.32773	0.663864	+	0.747853i	$0.268915\pi$
$348$	15.4675	0.00238260
$349$	−10999.2	−1.68703	−0.843515	−	0.537106i	$-0.819518\pi$
$349$	−10999.2	−1.68703	−0.843515	+	0.537106i	$0.819518\pi$
$350$	−1146.47	−0.175090
$351$	−85.2907	−0.0129700
$352$	−1310.50	−0.198437
$353$	−4161.82	−0.627511	−0.313755	−	0.949504i	$-0.601587\pi$
$353$	−4161.82	−0.627511	−0.313755	+	0.949504i	$0.601587\pi$
$354$	40.7509	0.00611832
$355$	−2668.72	−0.398988
$356$	−249.691	−0.0371731
$357$	18.0567	0.00267693
$358$	5882.06	0.868370
$359$	3256.43	0.478740	0.239370	−	0.970928i	$-0.423059\pi$
$359$	3256.43	0.478740	0.239370	+	0.970928i	$0.423059\pi$
$360$	−3324.92	−0.486774
$361$	361.000	0.0526316
$362$	9190.39	1.33435
$363$	3.26247	0.000471723 0
$364$	3263.04	0.469862
$365$	−787.865	−0.112983
$366$	−4.35285	−0.000621659 0
$367$	3969.49	0.564594	0.282297	−	0.959327i	$-0.408904\pi$
$367$	3969.49	0.564594	0.282297	+	0.959327i	$0.408904\pi$
$368$	5705.13	0.808154
$369$	6203.61	0.875196
$370$	−3293.01	−0.462690
$371$	−14824.2	−2.07449
$372$	−18.2415	−0.00254241
$373$	−11348.1	−1.57529	−0.787644	−	0.616130i	$-0.788699\pi$
$373$	−11348.1	−1.57529	−0.787644	+	0.616130i	$0.788699\pi$
$374$	839.146	0.116019
$375$	−3.37032	−0.000464114 0
$376$	−4687.55	−0.642931
$377$	−12105.1	−1.65370
$378$	66.7687	0.00908522
$379$	−11728.5	−1.58959	−0.794795	−	0.606878i	$-0.792422\pi$
$379$	−11728.5	−1.58959	−0.794795	+	0.606878i	$0.792422\pi$
$380$	−263.733	−0.0356033
$381$	18.9963	0.00255436
$382$	11723.4	1.57021
$383$	−13765.4	−1.83650	−0.918250	−	0.396001i	$-0.870398\pi$
$383$	−13765.4	−1.83650	−0.918250	+	0.396001i	$0.870398\pi$
$384$	7.88580	0.00104797
$385$	−1103.55	−0.146083
$386$	−5017.89	−0.661667
$387$	14730.6	1.93488
$388$	2588.38	0.338673
$389$	612.548	0.0798391	0.0399195	−	0.999203i	$-0.487290\pi$
$389$	612.548	0.0798391	0.0399195	+	0.999203i	$0.487290\pi$
$390$	−18.0501	−0.00234360
$391$	5586.83	0.722604
$392$	−1467.52	−0.189084
$393$	−58.8596	−0.00755489
$394$	−6715.76	−0.858718
$395$	424.510	0.0540745
$396$	−824.492	−0.104627
$397$	5227.65	0.660877	0.330438	−	0.943828i	$-0.392803\pi$
$397$	5227.65	0.660877	0.330438	+	0.943828i	$0.392803\pi$
$398$	−12631.4	−1.59084
$399$	10.2788	0.00128969
$400$	−852.098	−0.106512
$401$	−1490.42	−0.185606	−0.0928028	−	0.995685i	$-0.529583\pi$
$401$	−1490.42	−0.185606	−0.0928028	+	0.995685i	$0.529583\pi$
$402$	9.20408	0.00114193
$403$	14276.1	1.76462
$404$	30.0157	0.00369638
$405$	−3644.71	−0.447177
$406$	9476.35	1.15838
$407$	−3169.71	−0.386036
$408$	22.1651	0.00268955
$409$	−3613.03	−0.436805	−0.218402	−	0.975859i	$-0.570084\pi$
$409$	−3613.03	−0.436805	−0.218402	+	0.975859i	$0.570084\pi$
$410$	2625.78	0.316288
$411$	−9.10594	−0.00109285
$412$	2734.54	0.326994
$413$	−13268.1	−1.58082
$414$	10329.1	1.22621
$415$	−4858.51	−0.574686
$416$	6979.05	0.822538
$417$	−40.8361	−0.00479556
$418$	477.685	0.0558956
$419$	−9764.01	−1.13843	−0.569216	−	0.822188i	$-0.692753\pi$
$419$	−9764.01	−1.13843	−0.569216	+	0.822188i	$0.692753\pi$
$420$	−7.50933	−0.000872423 0
$421$	12642.3	1.46353	0.731765	−	0.681557i	$-0.238697\pi$
$421$	12642.3	1.46353	0.731765	+	0.681557i	$0.238697\pi$
$422$	−4550.89	−0.524961
$423$	−5138.53	−0.590648
$424$	−18197.1	−2.08427
$425$	−834.429	−0.0952370
$426$	32.8920	0.00374090
$427$	1417.24	0.160621
$428$	−4897.71	−0.553130
$429$	−17.3743	−0.00195533
$430$	6234.96	0.699247
$431$	−7103.88	−0.793926	−0.396963	−	0.917835i	$-0.629936\pi$
$431$	−7103.88	−0.793926	−0.396963	+	0.917835i	$0.629936\pi$
$432$	49.6248	0.00552679
$433$	11097.5	1.23167	0.615836	−	0.787874i	$-0.288818\pi$
$433$	11097.5	1.23167	0.615836	+	0.787874i	$0.288818\pi$
$434$	−11175.9	−1.23608
$435$	27.8579	0.00307054
$436$	1908.95	0.209684
$437$	3180.31	0.348135
$438$	9.71044	0.00105932
$439$	9940.72	1.08074	0.540370	−	0.841428i	$-0.318284\pi$
$439$	9940.72	1.08074	0.540370	+	0.841428i	$0.318284\pi$
$440$	−1354.63	−0.146772
$441$	−1608.71	−0.173708
$442$	−4468.87	−0.480910
$443$	11075.5	1.18784	0.593922	−	0.804523i	$-0.297579\pi$
$443$	11075.5	1.18784	0.593922	+	0.804523i	$0.297579\pi$
$444$	−21.5690	−0.00230545
$445$	−449.709	−0.0479062
$446$	8179.78	0.868439
$447$	−44.2538	−0.00468262
$448$	−10934.5	−1.15314
$449$	2338.16	0.245756	0.122878	−	0.992422i	$-0.460788\pi$
$449$	2338.16	0.245756	0.122878	+	0.992422i	$0.460788\pi$
$450$	−1542.72	−0.161610
$451$	2527.47	0.263889
$452$	2971.44	0.309214
$453$	20.2375	0.00209899
$454$	−12495.9	−1.29177
$455$	5876.93	0.605527
$456$	12.6175	0.00129577
$457$	−5978.62	−0.611965	−0.305983	−	0.952037i	$-0.598985\pi$
$457$	−5978.62	−0.611965	−0.305983	+	0.952037i	$0.598985\pi$
$458$	−1822.87	−0.185976
$459$	48.5958	0.00494174
$460$	−2323.42	−0.235500
$461$	17184.4	1.73613	0.868065	−	0.496450i	$-0.165364\pi$
$461$	17184.4	1.73613	0.868065	+	0.496450i	$0.165364\pi$
$462$	13.6012	0.00136967
$463$	−14553.8	−1.46085	−0.730425	−	0.682993i	$-0.760678\pi$
$463$	−14553.8	−1.46085	−0.730425	+	0.682993i	$0.760678\pi$
$464$	7043.15	0.704676
$465$	−32.8541	−0.00327650
$466$	1087.33	0.108090
$467$	1703.75	0.168822	0.0844112	−	0.996431i	$-0.473099\pi$
$467$	1703.75	0.168822	0.0844112	+	0.996431i	$0.473099\pi$
$468$	4390.82	0.433688
$469$	−2996.75	−0.295047
$470$	−2174.97	−0.213455
$471$	34.6110	0.00338597
$472$	−16286.9	−1.58828
$473$	6001.51	0.583403
$474$	−5.23209	−0.000507000 0
$475$	−475.000	−0.0458831
$476$	−1859.17	−0.179023
$477$	−19947.8	−1.91478
$478$	14592.0	1.39628
$479$	10564.9	1.00777	0.503885	−	0.863771i	$-0.331904\pi$
$479$	10564.9	1.00777	0.503885	+	0.863771i	$0.331904\pi$
$480$	−16.0611	−0.00152726
$481$	16880.3	1.60015
$482$	−3423.47	−0.323517
$483$	90.5536	0.00853071
$484$	−335.913	−0.0315471
$485$	4661.84	0.436460
$486$	134.769	0.0125787
$487$	−1566.32	−0.145743	−0.0728713	−	0.997341i	$-0.523216\pi$
$487$	−1566.32	−0.145743	−0.0728713	+	0.997341i	$0.523216\pi$
$488$	1739.71	0.161379
$489$	−82.4492	−0.00762471
$490$	−680.913	−0.0627765
$491$	−9746.65	−0.895846	−0.447923	−	0.894072i	$-0.647836\pi$
$491$	−9746.65	−0.895846	−0.447923	+	0.894072i	$0.647836\pi$
$492$	17.1987	0.00157597
$493$	6897.10	0.630081
$494$	−2543.91	−0.231692
$495$	−1484.96	−0.134836
$496$	−8306.28	−0.751942
$497$	−10709.3	−0.966554
$498$	59.8812	0.00538823
$499$	−4523.95	−0.405851	−0.202926	−	0.979194i	$-0.565045\pi$
$499$	−4523.95	−0.405851	−0.202926	+	0.979194i	$0.565045\pi$
$500$	347.018	0.0310382
$501$	26.1081	0.00232819
$502$	1432.86	0.127394
$503$	17165.9	1.52165	0.760824	−	0.648959i	$-0.224795\pi$
$503$	17165.9	1.52165	0.760824	+	0.648959i	$0.224795\pi$
$504$	−13342.6	−1.17922
$505$	54.0601	0.00476365
$506$	4208.28	0.369725
$507$	33.2897	0.00291607
$508$	−1955.91	−0.170826
$509$	425.922	0.0370897	0.0185449	−	0.999828i	$-0.494097\pi$
$509$	425.922	0.0370897	0.0185449	+	0.999828i	$0.494097\pi$
$510$	10.2843	0.000892938 0
$511$	−3161.62	−0.273702
$512$	−10776.4	−0.930187
$513$	27.6632	0.00238082
$514$	−6486.95	−0.556667
$515$	4925.08	0.421408
$516$	40.8386	0.00348415
$517$	−2093.53	−0.178092
$518$	−13214.5	−1.12087
$519$	−3.83538	−0.000324383 0
$520$	7214.09	0.608382
$521$	20868.2	1.75481	0.877403	−	0.479754i	$-0.159274\pi$
$521$	20868.2	1.75481	0.877403	+	0.479754i	$0.159274\pi$
$522$	12751.6	1.06920
$523$	−13737.6	−1.14857	−0.574286	−	0.818655i	$-0.694720\pi$
$523$	−13737.6	−1.14857	−0.574286	+	0.818655i	$0.694720\pi$
$524$	6060.34	0.505243
$525$	−13.5248	−0.00112432
$526$	18777.8	1.55656
$527$	−8134.05	−0.672343
$528$	10.1089	0.000833207 0
$529$	15850.7	1.30276
$530$	−8443.24	−0.691983
$531$	−17853.8	−1.45912
$532$	−1058.34	−0.0862493
$533$	−13460.0	−1.09384
$534$	5.54267	0.000449166 0
$535$	−8821.07	−0.712838
$536$	−3678.60	−0.296439
$537$	69.3897	0.00557614
$538$	−791.215	−0.0634046
$539$	−655.418	−0.0523764
$540$	−20.2098	−0.00161054
$541$	9462.34	0.751973	0.375987	−	0.926625i	$-0.377304\pi$
$541$	9462.34	0.751973	0.375987	+	0.926625i	$0.377304\pi$
$542$	−4946.55	−0.392016
$543$	108.418	0.00856840
$544$	−3976.43	−0.313397
$545$	3438.14	0.270227
$546$	−72.4332	−0.00567739
$547$	−23541.5	−1.84015	−0.920077	−	0.391739i	$-0.871874\pi$
$547$	−23541.5	−1.84015	−0.920077	+	0.391739i	$0.871874\pi$
$548$	937.573	0.0730860
$549$	1907.08	0.148255
$550$	−628.533	−0.0487286
$551$	3926.19	0.303559
$552$	111.157	0.00857094
$553$	1703.52	0.130996
$554$	−2217.87	−0.170087
$555$	−38.8471	−0.00297111
$556$	4204.59	0.320709
$557$	16934.9	1.28825	0.644124	−	0.764921i	$-0.277222\pi$
$557$	16934.9	1.28825	0.644124	+	0.764921i	$0.277222\pi$
$558$	−15038.5	−1.14092
$559$	−31961.0	−2.41826
$560$	−3419.38	−0.258027
$561$	9.89928	0.000745005 0
$562$	12954.1	0.972306
$563$	−6516.02	−0.487776	−0.243888	−	0.969803i	$-0.578423\pi$
$563$	−6516.02	−0.487776	−0.243888	+	0.969803i	$0.578423\pi$
$564$	−14.2459	−0.00106358
$565$	5351.75	0.398495
$566$	18415.1	1.36757
$567$	−14625.8	−1.08329
$568$	−13145.9	−0.971112
$569$	3536.36	0.260548	0.130274	−	0.991478i	$-0.458414\pi$
$569$	3536.36	0.260548	0.130274	+	0.991478i	$0.458414\pi$
$570$	5.85438	0.000430198 0
$571$	14235.5	1.04332	0.521662	−	0.853152i	$-0.325312\pi$
$571$	14235.5	1.04332	0.521662	+	0.853152i	$0.325312\pi$
$572$	1788.90	0.130765
$573$	138.299	0.0100829
$574$	10537.0	0.766211
$575$	−4184.62	−0.303497
$576$	−14713.7	−1.06436
$577$	−9671.84	−0.697823	−0.348912	−	0.937156i	$-0.613449\pi$
$577$	−9671.84	−0.697823	−0.348912	+	0.937156i	$0.613449\pi$
$578$	−8682.83	−0.624841
$579$	−59.1952	−0.00424882
$580$	−2868.33	−0.205346
$581$	−19496.7	−1.39218
$582$	−57.4572	−0.00409223
$583$	−8127.11	−0.577342
$584$	−3880.97	−0.274993
$585$	7908.14	0.558908
$586$	622.804	0.0439041
$587$	7071.60	0.497233	0.248617	−	0.968602i	$-0.420024\pi$
$587$	7071.60	0.497233	0.248617	+	0.968602i	$0.420024\pi$
$588$	−4.45994	−0.000312797 0
$589$	−4630.32	−0.323920
$590$	−7556.93	−0.527312
$591$	−79.2248	−0.00551417
$592$	−9821.47	−0.681858
$593$	3578.65	0.247820	0.123910	−	0.992293i	$-0.460457\pi$
$593$	3578.65	0.247820	0.123910	+	0.992293i	$0.460457\pi$
$594$	36.6048	0.00252847
$595$	−3348.48	−0.230713
$596$	4556.49	0.313156
$597$	−149.011	−0.0102154
$598$	−22411.2	−1.53254
$599$	21997.0	1.50046	0.750229	−	0.661178i	$-0.229943\pi$
$599$	21997.0	1.50046	0.750229	+	0.661178i	$0.229943\pi$
$600$	−16.6020	−0.00112962
$601$	15549.0	1.05534	0.527668	−	0.849451i	$-0.323067\pi$
$601$	15549.0	1.05534	0.527668	+	0.849451i	$0.323067\pi$
$602$	25020.2	1.69394
$603$	−4032.51	−0.272332
$604$	−2083.71	−0.140372
$605$	−605.000	−0.0406558
$606$	−0.666292	−4.46638e−5 0
$607$	23573.3	1.57629	0.788146	−	0.615488i	$-0.211041\pi$
$607$	23573.3	1.57629	0.788146	+	0.615488i	$0.211041\pi$
$608$	−2263.59	−0.150988
$609$	111.791	0.00743842
$610$	807.202	0.0535781
$611$	11149.1	0.738206
$612$	−2501.74	−0.165240
$613$	273.266	0.0180050	0.00900252	−	0.999959i	$-0.497134\pi$
$613$	273.266	0.0180050	0.00900252	+	0.999959i	$0.497134\pi$
$614$	3412.01	0.224263
$615$	30.9759	0.00203101
$616$	−5436.00	−0.355556
$617$	−17456.8	−1.13903	−0.569517	−	0.821979i	$-0.692870\pi$
$617$	−17456.8	−1.13903	−0.569517	+	0.821979i	$0.692870\pi$
$618$	−60.7017	−0.00395110
$619$	1111.15	0.0721500	0.0360750	−	0.999349i	$-0.488514\pi$
$619$	1111.15	0.0721500	0.0360750	+	0.999349i	$0.488514\pi$
$620$	3382.74	0.219120
$621$	243.706	0.0157481
$622$	−10454.3	−0.673922
$623$	−1804.64	−0.116053
$624$	−53.8348	−0.00345371
$625$	625.000	0.0400000
$626$	14900.4	0.951339
$627$	5.63518	0.000358927 0
$628$	−3563.64	−0.226441
$629$	−9617.81	−0.609678
$630$	−6190.79	−0.391503
$631$	22433.9	1.41534	0.707669	−	0.706544i	$-0.249747\pi$
$631$	22433.9	1.41534	0.707669	+	0.706544i	$0.249747\pi$
$632$	2091.11	0.131614
$633$	−53.6861	−0.00337098
$634$	20727.9	1.29844
$635$	−3522.72	−0.220149
$636$	−55.3027	−0.00344795
$637$	3490.42	0.217105
$638$	5195.24	0.322385
$639$	−14410.7	−0.892141
$640$	−1462.36	−0.0903201
$641$	−2123.97	−0.130876	−0.0654382	−	0.997857i	$-0.520845\pi$
$641$	−2123.97	−0.130876	−0.0654382	+	0.997857i	$0.520845\pi$
$642$	108.720	0.00668353
$643$	−15043.7	−0.922656	−0.461328	−	0.887230i	$-0.652627\pi$
$643$	−15043.7	−0.922656	−0.461328	+	0.887230i	$0.652627\pi$
$644$	−9323.65	−0.570502
$645$	73.5528	0.00449014
$646$	1449.43	0.0882775
$647$	7262.41	0.441290	0.220645	−	0.975354i	$-0.429184\pi$
$647$	7262.41	0.441290	0.220645	+	0.975354i	$0.429184\pi$
$648$	−17953.6	−1.08840
$649$	−7273.99	−0.439952
$650$	3347.25	0.201984
$651$	−131.840	−0.00793735
$652$	8489.20	0.509912
$653$	13376.7	0.801641	0.400821	−	0.916157i	$-0.368725\pi$
$653$	13376.7	0.801641	0.400821	+	0.916157i	$0.368725\pi$
$654$	−42.3751	−0.00253363
$655$	10915.0	0.651124
$656$	7831.45	0.466108
$657$	−4254.36	−0.252631
$658$	−8727.92	−0.517097
$659$	11015.8	0.651159	0.325579	−	0.945515i	$-0.394441\pi$
$659$	11015.8	0.651159	0.325579	+	0.945515i	$0.394441\pi$
$660$	−4.11686	−0.000242801 0
$661$	−13576.0	−0.798858	−0.399429	−	0.916764i	$-0.630792\pi$
$661$	−13576.0	−0.798858	−0.399429	+	0.916764i	$0.630792\pi$
$662$	16215.6	0.952018
$663$	−52.7185	−0.00308811
$664$	−23932.7	−1.39875
$665$	−1906.13	−0.111152
$666$	−17781.8	−1.03458
$667$	34588.6	2.00791
$668$	−2688.16	−0.155701
$669$	96.4956	0.00557658
$670$	−1706.82	−0.0984185
$671$	776.979	0.0447019
$672$	−64.4515	−0.00369981
$673$	17746.4	1.01645	0.508227	−	0.861223i	$-0.330301\pi$
$673$	17746.4	1.01645	0.508227	+	0.861223i	$0.330301\pi$
$674$	3142.39	0.179585
$675$	−36.3990	−0.00207555
$676$	−3427.60	−0.195016
$677$	−31790.1	−1.80471	−0.902357	−	0.430989i	$-0.858165\pi$
$677$	−31790.1	−1.80471	−0.902357	+	0.430989i	$0.858165\pi$
$678$	−65.9604	−0.00373627
$679$	18707.5	1.05733
$680$	−4110.35	−0.231801
$681$	−147.412	−0.00829494
$682$	−6126.97	−0.344009
$683$	8620.15	0.482930	0.241465	−	0.970410i	$-0.422372\pi$
$683$	8620.15	0.482930	0.241465	+	0.970410i	$0.422372\pi$
$684$	−1424.12	−0.0796092
$685$	1688.63	0.0941884
$686$	12997.2	0.723374
$687$	−21.5041	−0.00119423
$688$	18595.9	1.03047
$689$	43280.8	2.39313
$690$	51.5755	0.00284557
$691$	−586.780	−0.0323041	−0.0161521	−	0.999870i	$-0.505142\pi$
$691$	−586.780	−0.0323041	−0.0161521	+	0.999870i	$0.505142\pi$
$692$	394.901	0.0216935
$693$	−5958.99	−0.326643
$694$	19615.5	1.07290
$695$	7572.73	0.413309
$696$	137.226	0.00747350
$697$	7669.06	0.416766
$698$	−25139.5	−1.36324
$699$	12.8271	0.000694085 0
$700$	1392.55	0.0751904
$701$	−21501.2	−1.15847	−0.579237	−	0.815159i	$-0.696649\pi$
$701$	−21501.2	−1.15847	−0.579237	+	0.815159i	$0.696649\pi$
$702$	−194.938	−0.0104807
$703$	−5474.95	−0.293729
$704$	−5994.63	−0.320925
$705$	−25.6578	−0.00137068
$706$	−9512.16	−0.507075
$707$	216.938	0.0115400
$708$	−49.4975	−0.00262744
$709$	−742.991	−0.0393563	−0.0196781	−	0.999806i	$-0.506264\pi$
$709$	−742.991	−0.0393563	−0.0196781	+	0.999806i	$0.506264\pi$
$710$	−6099.56	−0.322412
$711$	2292.29	0.120911
$712$	−2215.24	−0.116601
$713$	−40791.9	−2.14259
$714$	41.2700	0.00216315
$715$	3221.92	0.168522
$716$	−7144.56	−0.372912
$717$	172.139	0.00896604
$718$	7442.82	0.386857
$719$	36831.7	1.91042	0.955210	−	0.295929i	$-0.0956291\pi$
$719$	36831.7	1.91042	0.955210	+	0.295929i	$0.0956291\pi$
$720$	−4601.20	−0.238162
$721$	19763.8	1.02087
$722$	825.093	0.0425302
$723$	−40.3862	−0.00207743
$724$	−11163.0	−0.573023
$725$	−5166.03	−0.264637
$726$	7.45663	0.000381187 0
$727$	−13374.3	−0.682290	−0.341145	−	0.940011i	$-0.610815\pi$
$727$	−13374.3	−0.682290	−0.341145	+	0.940011i	$0.610815\pi$
$728$	28949.4	1.47381
$729$	−19679.8	−0.999838
$730$	−1800.73	−0.0912984
$731$	18210.3	0.921385
$732$	5.28713	0.000266964 0
$733$	−5875.85	−0.296084	−0.148042	−	0.988981i	$-0.547297\pi$
$733$	−5875.85	−0.296084	−0.148042	+	0.988981i	$0.547297\pi$
$734$	9072.58	0.456233
$735$	−8.03262	−0.000403113 0
$736$	−19941.6	−0.998719
$737$	−1642.92	−0.0821135
$738$	14178.8	0.707222
$739$	5502.20	0.273886	0.136943	−	0.990579i	$-0.456272\pi$
$739$	5502.20	0.273886	0.136943	+	0.990579i	$0.456272\pi$
$740$	3999.80	0.198697
$741$	−30.0101	−0.00148779
$742$	−33881.8	−1.67634
$743$	−32509.7	−1.60520	−0.802601	−	0.596516i	$-0.796551\pi$
$743$	−32509.7	−1.60520	−0.802601	+	0.596516i	$0.796551\pi$
$744$	−161.837	−0.00797478
$745$	8206.51	0.403575
$746$	−25937.0	−1.27295
$747$	−26235.2	−1.28500
$748$	−1019.26	−0.0498232
$749$	−35398.0	−1.72686
$750$	−7.70313	−0.000375038 0
$751$	23653.6	1.14931	0.574656	−	0.818395i	$-0.305136\pi$
$751$	23653.6	1.14931	0.574656	+	0.818395i	$0.305136\pi$
$752$	−6486.89	−0.314565
$753$	16.9032	0.000818044 0
$754$	−27667.2	−1.33631
$755$	−3752.89	−0.180903
$756$	−81.0997	−0.00390154
$757$	13429.6	0.644791	0.322396	−	0.946605i	$-0.395512\pi$
$757$	13429.6	0.644791	0.322396	+	0.946605i	$0.395512\pi$
$758$	−26806.5	−1.28451
$759$	49.6444	0.00237415
$760$	−2339.82	−0.111677
$761$	13353.2	0.636076	0.318038	−	0.948078i	$-0.396976\pi$
$761$	13353.2	0.636076	0.318038	+	0.948078i	$0.396976\pi$
$762$	43.4175	0.00206411
$763$	13796.9	0.654628
$764$	−14239.6	−0.674308
$765$	−4505.79	−0.212951
$766$	−31461.9	−1.48403
$767$	38737.6	1.82364
$768$	−99.5260	−0.00467622
$769$	5114.45	0.239833	0.119917	−	0.992784i	$-0.461737\pi$
$769$	5114.45	0.239833	0.119917	+	0.992784i	$0.461737\pi$
$770$	−2522.24	−0.118046
$771$	−76.5255	−0.00357458
$772$	6094.90	0.284145
$773$	2357.32	0.109686	0.0548428	−	0.998495i	$-0.482534\pi$
$773$	2357.32	0.109686	0.0548428	+	0.998495i	$0.482534\pi$
$774$	33667.9	1.56352
$775$	6092.53	0.282387
$776$	22963.9	1.06232
$777$	−155.889	−0.00719755
$778$	1400.02	0.0645158
$779$	4365.62	0.200789
$780$	21.9243	0.00100643
$781$	−5871.18	−0.268998
$782$	12769.1	0.583917
$783$	300.862	0.0137317
$784$	−2030.84	−0.0925126
$785$	−6418.34	−0.291822
$786$	−134.528	−0.00610491
$787$	9055.79	0.410170	0.205085	−	0.978744i	$-0.434253\pi$
$787$	9055.79	0.410170	0.205085	+	0.978744i	$0.434253\pi$
$788$	8157.20	0.368767
$789$	221.518	0.00999526
$790$	970.250	0.0436961
$791$	21476.0	0.965359
$792$	−7314.82	−0.328183
$793$	−4137.80	−0.185293
$794$	11948.2	0.534037
$795$	−99.6036	−0.00444349
$796$	15342.6	0.683170
$797$	30715.5	1.36512	0.682559	−	0.730831i	$-0.260867\pi$
$797$	30715.5	1.36512	0.682559	+	0.730831i	$0.260867\pi$
$798$	23.4930	0.00104216
$799$	−6352.38	−0.281265
$800$	2978.40	0.131628
$801$	−2428.36	−0.107119
$802$	−3406.46	−0.149983
$803$	−1733.30	−0.0761730
$804$	−11.1796	−0.000490391 0
$805$	−16792.5	−0.735225
$806$	32629.1	1.42595
$807$	−9.33384	−0.000407146 0
$808$	266.297	0.0115944
$809$	17658.3	0.767406	0.383703	−	0.923456i	$-0.374649\pi$
$809$	17658.3	0.767406	0.383703	+	0.923456i	$0.374649\pi$
$810$	−8330.25	−0.361352
$811$	39003.2	1.68876	0.844382	−	0.535741i	$-0.179968\pi$
$811$	39003.2	1.68876	0.844382	+	0.535741i	$0.179968\pi$
$812$	−11510.3	−0.497454
$813$	−58.3537	−0.00251729
$814$	−7244.62	−0.311945
$815$	15289.6	0.657141
$816$	30.6733	0.00131591
$817$	10366.2	0.443903
$818$	−8257.86	−0.352970
$819$	31734.6	1.35396
$820$	−3189.37	−0.135826
$821$	19334.9	0.821917	0.410959	−	0.911654i	$-0.365194\pi$
$821$	19334.9	0.821917	0.410959	+	0.911654i	$0.365194\pi$
$822$	−20.8123	−0.000883106 0
$823$	−6125.62	−0.259448	−0.129724	−	0.991550i	$-0.541409\pi$
$823$	−6125.62	−0.259448	−0.129724	+	0.991550i	$0.541409\pi$
$824$	24260.6	1.02568
$825$	−7.41471	−0.000312906 0
$826$	−30325.2	−1.27742
$827$	14751.2	0.620251	0.310126	−	0.950696i	$-0.399629\pi$
$827$	14751.2	0.620251	0.310126	+	0.950696i	$0.399629\pi$
$828$	−12546.1	−0.526580
$829$	20969.9	0.878548	0.439274	−	0.898353i	$-0.355236\pi$
$829$	20969.9	0.878548	0.439274	+	0.898353i	$0.355236\pi$
$830$	−11104.5	−0.464389
$831$	−26.1639	−0.00109220
$832$	31924.3	1.33026
$833$	−1988.73	−0.0827194
$834$	−93.3339	−0.00387517
$835$	−4841.55	−0.200657
$836$	−580.214	−0.0240037
$837$	−354.819	−0.0146527
$838$	−22316.4	−0.919937
$839$	−12357.6	−0.508500	−0.254250	−	0.967139i	$-0.581829\pi$
$839$	−12357.6	−0.508500	−0.254250	+	0.967139i	$0.581829\pi$
$840$	−66.6221	−0.00273653
$841$	18311.7	0.750817
$842$	28894.9	1.18264
$843$	152.818	0.00624356
$844$	5527.67	0.225439
$845$	−6173.32	−0.251324
$846$	−11744.5	−0.477287
$847$	−2427.80	−0.0984891
$848$	−25182.1	−1.01976
$849$	217.240	0.00878168
$850$	−1907.15	−0.0769585
$851$	−48232.9	−1.94289
$852$	−39.9518	−0.00160648
$853$	−42430.6	−1.70316	−0.851580	−	0.524224i	$-0.824355\pi$
$853$	−42430.6	−1.70316	−0.851580	+	0.524224i	$0.824355\pi$
$854$	3239.22	0.129794
$855$	−2564.93	−0.102595
$856$	−43452.1	−1.73500
$857$	−35194.4	−1.40282	−0.701411	−	0.712757i	$-0.747446\pi$
$857$	−35194.4	−1.40282	−0.701411	+	0.712757i	$0.747446\pi$
$858$	−39.7102	−0.00158005
$859$	45292.6	1.79903	0.899513	−	0.436895i	$-0.143922\pi$
$859$	45292.6	1.79903	0.899513	+	0.436895i	$0.143922\pi$
$860$	−7573.20	−0.300284
$861$	124.303	0.00492014
$862$	−16236.5	−0.641550
$863$	9326.97	0.367896	0.183948	−	0.982936i	$-0.441112\pi$
$863$	9326.97	0.367896	0.183948	+	0.982936i	$0.441112\pi$
$864$	−173.458	−0.00683003
$865$	711.242	0.0279572
$866$	25364.3	0.995281
$867$	−102.430	−0.00401235
$868$	13574.6	0.530820
$869$	933.922	0.0364570
$870$	63.6714	0.00248122
$871$	8749.34	0.340368
$872$	16936.0	0.657714
$873$	25173.2	0.975928
$874$	7268.85	0.281319
$875$	2508.06	0.0969005
$876$	−11.7946	−0.000454914 0
$877$	−24062.3	−0.926483	−0.463241	−	0.886232i	$-0.653314\pi$
$877$	−24062.3	−0.926483	−0.463241	+	0.886232i	$0.653314\pi$
$878$	22720.3	0.873317
$879$	7.34712	0.000281925 0
$880$	−1874.61	−0.0718105
$881$	−41178.8	−1.57474	−0.787372	−	0.616479i	$-0.788559\pi$
$881$	−41178.8	−1.57474	−0.787372	+	0.616479i	$0.788559\pi$
$882$	−3676.83	−0.140369
$883$	−30834.7	−1.17516	−0.587582	−	0.809165i	$-0.699920\pi$
$883$	−30834.7	−1.17516	−0.587582	+	0.809165i	$0.699920\pi$
$884$	5428.04	0.206521
$885$	−89.1480	−0.00338607
$886$	25314.0	0.959865
$887$	−28880.3	−1.09324	−0.546622	−	0.837380i	$-0.684086\pi$
$887$	−28880.3	−1.09324	−0.546622	+	0.837380i	$0.684086\pi$
$888$	−191.358	−0.00723149
$889$	−14136.3	−0.533314
$890$	−1027.84	−0.0387117
$891$	−8018.35	−0.301487
$892$	−9935.45	−0.372941
$893$	−3616.10	−0.135508
$894$	−101.145	−0.00378390
$895$	−12867.8	−0.480584
$896$	−5868.30	−0.218801
$897$	−264.381	−0.00984105
$898$	5344.04	0.198589
$899$	−50358.7	−1.86825
$900$	1873.85	0.0694017
$901$	−24660.0	−0.911812
$902$	5776.72	0.213241
$903$	295.160	0.0108774
$904$	26362.4	0.969912
$905$	−20105.2	−0.738474
$906$	46.2544	0.00169614
$907$	−25207.0	−0.922807	−0.461403	−	0.887190i	$-0.652654\pi$
$907$	−25207.0	−0.922807	−0.461403	+	0.887190i	$0.652654\pi$
$908$	15178.0	0.554734
$909$	291.917	0.0106516
$910$	13432.2	0.489310
$911$	14228.1	0.517452	0.258726	−	0.965951i	$-0.416697\pi$
$911$	14228.1	0.517452	0.258726	+	0.965951i	$0.416697\pi$
$912$	17.4608	0.000633975 0
$913$	−10688.7	−0.387453
$914$	−13664.6	−0.494513
$915$	9.52244	0.000344046 0
$916$	2214.12	0.0798653
$917$	43801.0	1.57736
$918$	111.069	0.00399329
$919$	6879.68	0.246942	0.123471	−	0.992348i	$-0.460597\pi$
$919$	6879.68	0.246942	0.123471	+	0.992348i	$0.460597\pi$
$920$	−20613.2	−0.738692
$921$	40.2510	0.00144008
$922$	39276.2	1.40292
$923$	31266.9	1.11502
$924$	−16.5205	−0.000588188 0
$925$	7203.89	0.256067
$926$	−33263.9	−1.18047
$927$	26594.7	0.942271
$928$	−24618.5	−0.870841
$929$	13176.7	0.465355	0.232677	−	0.972554i	$-0.425251\pi$
$929$	13176.7	0.465355	0.232677	+	0.972554i	$0.425251\pi$
$930$	−75.0904	−0.00264765
$931$	−1132.09	−0.0398524
$932$	−1320.71	−0.0464178
$933$	−123.328	−0.00432751
$934$	3894.05	0.136421
$935$	−1835.74	−0.0642088
$936$	38955.0	1.36035
$937$	28381.2	0.989512	0.494756	−	0.869032i	$-0.335257\pi$
$937$	28381.2	0.989512	0.494756	+	0.869032i	$0.335257\pi$
$938$	−6849.31	−0.238420
$939$	175.777	0.00610892
$940$	2641.79	0.0916657
$941$	−25544.1	−0.884923	−0.442462	−	0.896787i	$-0.645895\pi$
$941$	−25544.1	−0.884923	−0.442462	+	0.896787i	$0.645895\pi$
$942$	79.1060	0.00273611
$943$	38460.0	1.32813
$944$	−22538.7	−0.777090
$945$	−146.065	−0.00502805
$946$	13716.9	0.471433
$947$	31599.7	1.08432	0.542160	−	0.840275i	$-0.317606\pi$
$947$	31599.7	1.08432	0.542160	+	0.840275i	$0.317606\pi$
$948$	6.35508	0.000217725 0
$949$	9230.69	0.315744
$950$	−1085.65	−0.0370769
$951$	244.524	0.00833779
$952$	−16494.4	−0.561540
$953$	41121.1	1.39774	0.698868	−	0.715250i	$-0.253687\pi$
$953$	41121.1	1.39774	0.698868	+	0.715250i	$0.253687\pi$
$954$	−45592.3	−1.54728
$955$	−25646.4	−0.869004
$956$	−17723.9	−0.599615
$957$	61.2875	0.00207016
$958$	24146.8	0.814352
$959$	6776.28	0.228173
$960$	−73.4685	−0.00246998
$961$	29599.2	0.993562
$962$	38581.1	1.29304
$963$	−47632.5	−1.59391
$964$	4158.27	0.138930
$965$	10977.3	0.366188
$966$	206.967	0.00689344
$967$	14974.3	0.497975	0.248988	−	0.968507i	$-0.419902\pi$
$967$	14974.3	0.497975	0.248988	+	0.968507i	$0.419902\pi$
$968$	−2980.19	−0.0989535
$969$	17.0988	0.000566864 0
$970$	10655.0	0.352692
$971$	−14571.1	−0.481574	−0.240787	−	0.970578i	$-0.577406\pi$
$971$	−14571.1	−0.481574	−0.240787	+	0.970578i	$0.577406\pi$
$972$	−163.695	−0.00540178
$973$	30388.6	1.00125
$974$	−3579.94	−0.117771
$975$	39.4870	0.00129702
$976$	2407.50	0.0789571
$977$	36310.8	1.18903	0.594516	−	0.804083i	$-0.297344\pi$
$977$	36310.8	1.18903	0.594516	+	0.804083i	$0.297344\pi$
$978$	−188.444	−0.00616132
$979$	−989.360	−0.0322983
$980$	827.061	0.0269587
$981$	18565.4	0.604229
$982$	−22276.7	−0.723909
$983$	−19993.3	−0.648714	−0.324357	−	0.945935i	$-0.605148\pi$
$983$	−19993.3	−0.648714	−0.324357	+	0.945935i	$0.605148\pi$
$984$	152.585	0.00494334
$985$	14691.6	0.475242
$986$	15763.9	0.509151
$987$	−102.962	−0.00332048
$988$	3089.92	0.0994975
$989$	91323.8	2.93623
$990$	−3393.99	−0.108958
$991$	53419.2	1.71233	0.856164	−	0.516705i	$-0.172841\pi$
$991$	53419.2	1.71233	0.856164	+	0.516705i	$0.172841\pi$
$992$	29033.6	0.929252
$993$	191.293	0.00611328
$994$	−24476.9	−0.781046
$995$	27632.9	0.880424
$996$	−72.7338	−0.00231391
$997$	−35502.2	−1.12775	−0.563874	−	0.825861i	$-0.690690\pi$
$997$	−35502.2	−1.12775	−0.563874	+	0.825861i	$0.690690\pi$
$998$	−10339.8	−0.327957
$999$	−419.543	−0.0132870

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.g.1.15	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.g.1.15	✓	23	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+2.28558 q^{2} +0.0269626 q^{3} -2.77614 q^{4} -5.00000 q^{5} +0.0616250 q^{6} -20.0645 q^{7} -24.6297 q^{8} -26.9993 q^{9} +O(q^{10})\)	\(q+2.28558 q^{2} +0.0269626 q^{3} -2.77614 q^{4} -5.00000 q^{5} +0.0616250 q^{6} -20.0645 q^{7} -24.6297 q^{8} -26.9993 q^{9} -11.4279 q^{10} -11.0000 q^{11} -0.0748520 q^{12} +58.5804 q^{13} -45.8589 q^{14} -0.134813 q^{15} -34.0839 q^{16} -33.3772 q^{17} -61.7089 q^{18} -19.0000 q^{19} +13.8807 q^{20} -0.540990 q^{21} -25.1413 q^{22} -167.385 q^{23} -0.664080 q^{24} +25.0000 q^{25} +133.890 q^{26} -1.45596 q^{27} +55.7018 q^{28} -206.641 q^{29} -0.308125 q^{30} +243.701 q^{31} +119.136 q^{32} -0.296588 q^{33} -76.2860 q^{34} +100.322 q^{35} +74.9538 q^{36} +288.156 q^{37} -43.4259 q^{38} +1.57948 q^{39} +123.148 q^{40} -229.770 q^{41} -1.23647 q^{42} -545.592 q^{43} +30.5376 q^{44} +134.996 q^{45} -382.571 q^{46} +190.321 q^{47} -0.918990 q^{48} +59.5835 q^{49} +57.1394 q^{50} -0.899934 q^{51} -162.627 q^{52} +738.828 q^{53} -3.32771 q^{54} +55.0000 q^{55} +494.182 q^{56} -0.512289 q^{57} -472.295 q^{58} +661.271 q^{59} +0.374260 q^{60} -70.6345 q^{61} +556.997 q^{62} +541.726 q^{63} +544.966 q^{64} -292.902 q^{65} -0.677876 q^{66} +149.356 q^{67} +92.6597 q^{68} -4.51313 q^{69} +229.295 q^{70} +533.744 q^{71} +664.984 q^{72} +157.573 q^{73} +658.601 q^{74} +0.674065 q^{75} +52.7467 q^{76} +220.709 q^{77} +3.61002 q^{78} -84.9020 q^{79} +170.420 q^{80} +728.941 q^{81} -525.156 q^{82} +971.702 q^{83} +1.50187 q^{84} +166.886 q^{85} -1246.99 q^{86} -5.57159 q^{87} +270.927 q^{88} +89.9418 q^{89} +308.544 q^{90} -1175.39 q^{91} +464.684 q^{92} +6.57081 q^{93} +434.994 q^{94} +95.0000 q^{95} +3.21222 q^{96} -932.367 q^{97} +136.183 q^{98} +296.992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * q^99

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