LMFDB - Embedded newform 8007.2.a.f.1.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.514170 q^{2} -1.00000 q^{3} -1.73563 q^{4} -2.23060 q^{5} +0.514170 q^{6} +1.18904 q^{7} +1.92075 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.514170 q^{2} -1.00000 q^{3} -1.73563 q^{4} -2.23060 q^{5} +0.514170 q^{6} +1.18904 q^{7} +1.92075 q^{8} +1.00000 q^{9} +1.14691 q^{10} -3.76470 q^{11} +1.73563 q^{12} -1.73173 q^{13} -0.611367 q^{14} +2.23060 q^{15} +2.48367 q^{16} -1.00000 q^{17} -0.514170 q^{18} +1.01761 q^{19} +3.87149 q^{20} -1.18904 q^{21} +1.93569 q^{22} +2.15719 q^{23} -1.92075 q^{24} -0.0244263 q^{25} +0.890405 q^{26} -1.00000 q^{27} -2.06373 q^{28} -1.45443 q^{29} -1.14691 q^{30} -4.31473 q^{31} -5.11852 q^{32} +3.76470 q^{33} +0.514170 q^{34} -2.65226 q^{35} -1.73563 q^{36} -1.11433 q^{37} -0.523226 q^{38} +1.73173 q^{39} -4.28442 q^{40} +2.21959 q^{41} +0.611367 q^{42} +9.50603 q^{43} +6.53412 q^{44} -2.23060 q^{45} -1.10916 q^{46} +5.01983 q^{47} -2.48367 q^{48} -5.58619 q^{49} +0.0125592 q^{50} +1.00000 q^{51} +3.00565 q^{52} +13.2050 q^{53} +0.514170 q^{54} +8.39753 q^{55} +2.28384 q^{56} -1.01761 q^{57} +0.747824 q^{58} +3.08150 q^{59} -3.87149 q^{60} -9.73285 q^{61} +2.21851 q^{62} +1.18904 q^{63} -2.33555 q^{64} +3.86280 q^{65} -1.93569 q^{66} -1.07604 q^{67} +1.73563 q^{68} -2.15719 q^{69} +1.36371 q^{70} -13.7666 q^{71} +1.92075 q^{72} +5.31205 q^{73} +0.572955 q^{74} +0.0244263 q^{75} -1.76620 q^{76} -4.47636 q^{77} -0.890405 q^{78} -9.97851 q^{79} -5.54007 q^{80} +1.00000 q^{81} -1.14124 q^{82} +14.0231 q^{83} +2.06373 q^{84} +2.23060 q^{85} -4.88771 q^{86} +1.45443 q^{87} -7.23103 q^{88} +7.10670 q^{89} +1.14691 q^{90} -2.05909 q^{91} -3.74409 q^{92} +4.31473 q^{93} -2.58104 q^{94} -2.26989 q^{95} +5.11852 q^{96} +1.44167 q^{97} +2.87225 q^{98} -3.76470 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.514170	−0.363573	−0.181786	−	0.983338i	$-0.558188\pi$
$2$	−0.514170	−0.363573	−0.181786	+	0.983338i	$0.558188\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.73563	−0.867815
$5$	−2.23060	−0.997554	−0.498777	−	0.866730i	$-0.666217\pi$
$5$	−2.23060	−0.997554	−0.498777	+	0.866730i	$0.666217\pi$
$6$	0.514170	0.209909
$7$	1.18904	0.449414	0.224707	−	0.974426i	$-0.427858\pi$
$7$	1.18904	0.449414	0.224707	+	0.974426i	$0.427858\pi$
$8$	1.92075	0.679087
$9$	1.00000	0.333333
$10$	1.14691	0.362684
$11$	−3.76470	−1.13510	−0.567549	−	0.823339i	$-0.692108\pi$
$11$	−3.76470	−1.13510	−0.567549	+	0.823339i	$0.692108\pi$
$12$	1.73563	0.501033
$13$	−1.73173	−0.480296	−0.240148	−	0.970736i	$-0.577196\pi$
$13$	−1.73173	−0.480296	−0.240148	+	0.970736i	$0.577196\pi$
$14$	−0.611367	−0.163395
$15$	2.23060	0.575938
$16$	2.48367	0.620917
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.514170	−0.121191
$19$	1.01761	0.233456	0.116728	−	0.993164i	$-0.462759\pi$
$19$	1.01761	0.233456	0.116728	+	0.993164i	$0.462759\pi$
$20$	3.87149	0.865692
$21$	−1.18904	−0.259469
$22$	1.93569	0.412691
$23$	2.15719	0.449806	0.224903	−	0.974381i	$-0.427793\pi$
$23$	2.15719	0.449806	0.224903	+	0.974381i	$0.427793\pi$
$24$	−1.92075	−0.392071
$25$	−0.0244263	−0.00488525
$26$	0.890405	0.174623
$27$	−1.00000	−0.192450
$28$	−2.06373	−0.390008
$29$	−1.45443	−0.270081	−0.135040	−	0.990840i	$-0.543116\pi$
$29$	−1.45443	−0.270081	−0.135040	+	0.990840i	$0.543116\pi$
$30$	−1.14691	−0.209396
$31$	−4.31473	−0.774949	−0.387475	−	0.921880i	$-0.626652\pi$
$31$	−4.31473	−0.774949	−0.387475	+	0.921880i	$0.626652\pi$
$32$	−5.11852	−0.904836
$33$	3.76470	0.655349
$34$	0.514170	0.0881794
$35$	−2.65226	−0.448314
$36$	−1.73563	−0.289272
$37$	−1.11433	−0.183195	−0.0915974	−	0.995796i	$-0.529197\pi$
$37$	−1.11433	−0.183195	−0.0915974	+	0.995796i	$0.529197\pi$
$38$	−0.523226	−0.0848784
$39$	1.73173	0.277299
$40$	−4.28442	−0.677426
$41$	2.21959	0.346641	0.173321	−	0.984865i	$-0.444550\pi$
$41$	2.21959	0.346641	0.173321	+	0.984865i	$0.444550\pi$
$42$	0.611367	0.0943359
$43$	9.50603	1.44966	0.724828	−	0.688930i	$-0.241919\pi$
$43$	9.50603	1.44966	0.724828	+	0.688930i	$0.241919\pi$
$44$	6.53412	0.985055
$45$	−2.23060	−0.332518
$46$	−1.10916	−0.163537
$47$	5.01983	0.732217	0.366109	−	0.930572i	$-0.380690\pi$
$47$	5.01983	0.732217	0.366109	+	0.930572i	$0.380690\pi$
$48$	−2.48367	−0.358487
$49$	−5.58619	−0.798028
$50$	0.0125592	0.00177615
$51$	1.00000	0.140028
$52$	3.00565	0.416808
$53$	13.2050	1.81384	0.906921	−	0.421301i	$-0.138426\pi$
$53$	13.2050	1.81384	0.906921	+	0.421301i	$0.138426\pi$
$54$	0.514170	0.0699696
$55$	8.39753	1.13232
$56$	2.28384	0.305191
$57$	−1.01761	−0.134786
$58$	0.747824	0.0981941
$59$	3.08150	0.401177	0.200589	−	0.979676i	$-0.435715\pi$
$59$	3.08150	0.401177	0.200589	+	0.979676i	$0.435715\pi$
$60$	−3.87149	−0.499808
$61$	−9.73285	−1.24616	−0.623082	−	0.782156i	$-0.714120\pi$
$61$	−9.73285	−1.24616	−0.623082	+	0.782156i	$0.714120\pi$
$62$	2.21851	0.281751
$63$	1.18904	0.149805
$64$	−2.33555	−0.291943
$65$	3.86280	0.479122
$66$	−1.93569	−0.238267
$67$	−1.07604	−0.131459	−0.0657294	−	0.997837i	$-0.520937\pi$
$67$	−1.07604	−0.131459	−0.0657294	+	0.997837i	$0.520937\pi$
$68$	1.73563	0.210476
$69$	−2.15719	−0.259696
$70$	1.36371	0.162995
$71$	−13.7666	−1.63380	−0.816898	−	0.576782i	$-0.804308\pi$
$71$	−13.7666	−1.63380	−0.816898	+	0.576782i	$0.804308\pi$
$72$	1.92075	0.226362
$73$	5.31205	0.621729	0.310864	−	0.950454i	$-0.399381\pi$
$73$	5.31205	0.621729	0.310864	+	0.950454i	$0.399381\pi$
$74$	0.572955	0.0666047
$75$	0.0244263	0.00282050
$76$	−1.76620	−0.202597
$77$	−4.47636	−0.510129
$78$	−0.890405	−0.100819
$79$	−9.97851	−1.12267	−0.561335	−	0.827589i	$-0.689712\pi$
$79$	−9.97851	−1.12267	−0.561335	+	0.827589i	$0.689712\pi$
$80$	−5.54007	−0.619399
$81$	1.00000	0.111111
$82$	−1.14124	−0.126029
$83$	14.0231	1.53923	0.769617	−	0.638506i	$-0.220447\pi$
$83$	14.0231	1.53923	0.769617	+	0.638506i	$0.220447\pi$
$84$	2.06373	0.225171
$85$	2.23060	0.241942
$86$	−4.88771	−0.527056
$87$	1.45443	0.155931
$88$	−7.23103	−0.770831
$89$	7.10670	0.753308	0.376654	−	0.926354i	$-0.377075\pi$
$89$	7.10670	0.753308	0.376654	+	0.926354i	$0.377075\pi$
$90$	1.14691	0.120895
$91$	−2.05909	−0.215852
$92$	−3.74409	−0.390348
$93$	4.31473	0.447417
$94$	−2.58104	−0.266214
$95$	−2.26989	−0.232885
$96$	5.11852	0.522407
$97$	1.44167	0.146380	0.0731899	−	0.997318i	$-0.476682\pi$
$97$	1.44167	0.146380	0.0731899	+	0.997318i	$0.476682\pi$
$98$	2.87225	0.290141
$99$	−3.76470	−0.378366
$100$	0.0423949	0.00423949
$101$	10.1000	1.00499	0.502495	−	0.864580i	$-0.332416\pi$
$101$	10.1000	1.00499	0.502495	+	0.864580i	$0.332416\pi$
$102$	−0.514170	−0.0509104
$103$	10.6269	1.04710	0.523549	−	0.851996i	$-0.324608\pi$
$103$	10.6269	1.04710	0.523549	+	0.851996i	$0.324608\pi$
$104$	−3.32622	−0.326163
$105$	2.65226	0.258834
$106$	−6.78960	−0.659464
$107$	−10.1754	−0.983689	−0.491845	−	0.870683i	$-0.663677\pi$
$107$	−10.1754	−0.983689	−0.491845	+	0.870683i	$0.663677\pi$
$108$	1.73563	0.167011
$109$	−1.10382	−0.105727	−0.0528636	−	0.998602i	$-0.516835\pi$
$109$	−1.10382	−0.105727	−0.0528636	+	0.998602i	$0.516835\pi$
$110$	−4.31776	−0.411682
$111$	1.11433	0.105768
$112$	2.95317	0.279049
$113$	−0.726543	−0.0683475	−0.0341737	−	0.999416i	$-0.510880\pi$
$113$	−0.726543	−0.0683475	−0.0341737	+	0.999416i	$0.510880\pi$
$114$	0.523226	0.0490046
$115$	−4.81184	−0.448706
$116$	2.52435	0.234380
$117$	−1.73173	−0.160099
$118$	−1.58442	−0.145857
$119$	−1.18904	−0.108999
$120$	4.28442	0.391112
$121$	3.17293	0.288449
$122$	5.00434	0.453072
$123$	−2.21959	−0.200133
$124$	7.48878	0.672512
$125$	11.2075	1.00243
$126$	−0.611367	−0.0544649
$127$	16.5326	1.46703	0.733517	−	0.679671i	$-0.237877\pi$
$127$	16.5326	1.46703	0.733517	+	0.679671i	$0.237877\pi$
$128$	11.4379	1.01098
$129$	−9.50603	−0.836959
$130$	−1.98614	−0.174196
$131$	5.81654	0.508194	0.254097	−	0.967179i	$-0.418222\pi$
$131$	5.81654	0.508194	0.254097	+	0.967179i	$0.418222\pi$
$132$	−6.53412	−0.568722
$133$	1.20998	0.104918
$134$	0.553265	0.0477948
$135$	2.23060	0.191979
$136$	−1.92075	−0.164703
$137$	21.1478	1.80678	0.903389	−	0.428821i	$-0.141071\pi$
$137$	21.1478	1.80678	0.903389	+	0.428821i	$0.141071\pi$
$138$	1.10916	0.0944183
$139$	−15.9119	−1.34963	−0.674814	−	0.737988i	$-0.735776\pi$
$139$	−15.9119	−1.34963	−0.674814	+	0.737988i	$0.735776\pi$
$140$	4.60335	0.389054
$141$	−5.01983	−0.422746
$142$	7.07838	0.594004
$143$	6.51945	0.545184
$144$	2.48367	0.206972
$145$	3.24425	0.269420
$146$	−2.73130	−0.226044
$147$	5.58619	0.460741
$148$	1.93407	0.158979
$149$	20.2298	1.65729	0.828644	−	0.559776i	$-0.189113\pi$
$149$	20.2298	1.65729	0.828644	+	0.559776i	$0.189113\pi$
$150$	−0.0125592	−0.00102546
$151$	10.1850	0.828841	0.414421	−	0.910085i	$-0.363984\pi$
$151$	10.1850	0.828841	0.414421	+	0.910085i	$0.363984\pi$
$152$	1.95458	0.158537
$153$	−1.00000	−0.0808452
$154$	2.30161	0.185469
$155$	9.62444	0.773054
$156$	−3.00565	−0.240644
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	5.13065	0.408172
$159$	−13.2050	−1.04722
$160$	11.4174	0.902623
$161$	2.56498	0.202149
$162$	−0.514170	−0.0403970
$163$	−7.74482	−0.606621	−0.303311	−	0.952892i	$-0.598092\pi$
$163$	−7.74482	−0.606621	−0.303311	+	0.952892i	$0.598092\pi$
$164$	−3.85238	−0.300820
$165$	−8.39753	−0.653747
$166$	−7.21024	−0.559624
$167$	2.88188	0.223007	0.111503	−	0.993764i	$-0.464433\pi$
$167$	2.88188	0.223007	0.111503	+	0.993764i	$0.464433\pi$
$168$	−2.28384	−0.176202
$169$	−10.0011	−0.769315
$170$	−1.14691	−0.0879637
$171$	1.01761	0.0778188
$172$	−16.4989	−1.25803
$173$	−11.1660	−0.848938	−0.424469	−	0.905443i	$-0.639539\pi$
$173$	−11.1660	−0.848938	−0.424469	+	0.905443i	$0.639539\pi$
$174$	−0.747824	−0.0566924
$175$	−0.0290437	−0.00219550
$176$	−9.35026	−0.704802
$177$	−3.08150	−0.231620
$178$	−3.65405	−0.273883
$179$	8.15506	0.609538	0.304769	−	0.952426i	$-0.401421\pi$
$179$	8.15506	0.609538	0.304769	+	0.952426i	$0.401421\pi$
$180$	3.87149	0.288564
$181$	6.14306	0.456610	0.228305	−	0.973590i	$-0.426682\pi$
$181$	6.14306	0.456610	0.228305	+	0.973590i	$0.426682\pi$
$182$	1.05872	0.0784778
$183$	9.73285	0.719473
$184$	4.14343	0.305457
$185$	2.48563	0.182747
$186$	−2.21851	−0.162669
$187$	3.76470	0.275302
$188$	−8.71256	−0.635429
$189$	−1.18904	−0.0864897
$190$	1.16711	0.0846708
$191$	−6.14219	−0.444433	−0.222217	−	0.974997i	$-0.571329\pi$
$191$	−6.14219	−0.444433	−0.222217	+	0.974997i	$0.571329\pi$
$192$	2.33555	0.168554
$193$	−1.60323	−0.115403	−0.0577015	−	0.998334i	$-0.518377\pi$
$193$	−1.60323	−0.115403	−0.0577015	+	0.998334i	$0.518377\pi$
$194$	−0.741265	−0.0532197
$195$	−3.86280	−0.276621
$196$	9.69556	0.692540
$197$	5.34589	0.380879	0.190440	−	0.981699i	$-0.439009\pi$
$197$	5.34589	0.380879	0.190440	+	0.981699i	$0.439009\pi$
$198$	1.93569	0.137564
$199$	−4.94252	−0.350366	−0.175183	−	0.984536i	$-0.556052\pi$
$199$	−4.94252	−0.350366	−0.175183	+	0.984536i	$0.556052\pi$
$200$	−0.0469167	−0.00331751
$201$	1.07604	0.0758977
$202$	−5.19313	−0.365387
$203$	−1.72937	−0.121378
$204$	−1.73563	−0.121518
$205$	−4.95101	−0.345794
$206$	−5.46402	−0.380697
$207$	2.15719	0.149935
$208$	−4.30105	−0.298224
$209$	−3.83100	−0.264996
$210$	−1.36371	−0.0941052
$211$	5.63736	0.388092	0.194046	−	0.980992i	$-0.437839\pi$
$211$	5.63736	0.388092	0.194046	+	0.980992i	$0.437839\pi$
$212$	−22.9189	−1.57408
$213$	13.7666	0.943273
$214$	5.23186	0.357643
$215$	−21.2041	−1.44611
$216$	−1.92075	−0.130690
$217$	−5.13038	−0.348273
$218$	0.567553	0.0384395
$219$	−5.31205	−0.358955
$220$	−14.5750	−0.982646
$221$	1.73173	0.116489
$222$	−0.572955	−0.0384542
$223$	−2.41484	−0.161710	−0.0808550	−	0.996726i	$-0.525765\pi$
$223$	−2.41484	−0.161710	−0.0808550	+	0.996726i	$0.525765\pi$
$224$	−6.08611	−0.406645
$225$	−0.0244263	−0.00162842
$226$	0.373567	0.0248493
$227$	22.6950	1.50632	0.753162	−	0.657836i	$-0.228528\pi$
$227$	22.6950	1.50632	0.753162	+	0.657836i	$0.228528\pi$
$228$	1.76620	0.116969
$229$	2.88354	0.190550	0.0952750	−	0.995451i	$-0.469627\pi$
$229$	2.88354	0.190550	0.0952750	+	0.995451i	$0.469627\pi$
$230$	2.47410	0.163137
$231$	4.47636	0.294523
$232$	−2.79359	−0.183408
$233$	17.2482	1.12997	0.564983	−	0.825102i	$-0.308883\pi$
$233$	17.2482	1.12997	0.564983	+	0.825102i	$0.308883\pi$
$234$	0.890405	0.0582076
$235$	−11.1972	−0.730426
$236$	−5.34835	−0.348148
$237$	9.97851	0.648174
$238$	0.611367	0.0396290
$239$	−27.3485	−1.76903	−0.884515	−	0.466512i	$-0.845511\pi$
$239$	−27.3485	−1.76903	−0.884515	+	0.466512i	$0.845511\pi$
$240$	5.54007	0.357610
$241$	2.84765	0.183433	0.0917165	−	0.995785i	$-0.470765\pi$
$241$	2.84765	0.183433	0.0917165	+	0.995785i	$0.470765\pi$
$242$	−1.63143	−0.104872
$243$	−1.00000	−0.0641500
$244$	16.8926	1.08144
$245$	12.4606	0.796076
$246$	1.14124	0.0727631
$247$	−1.76223	−0.112128
$248$	−8.28752	−0.526258
$249$	−14.0231	−0.888677
$250$	−5.76255	−0.364456
$251$	−24.5200	−1.54769	−0.773845	−	0.633375i	$-0.781669\pi$
$251$	−24.5200	−1.54769	−0.773845	+	0.633375i	$0.781669\pi$
$252$	−2.06373	−0.130003
$253$	−8.12118	−0.510574
$254$	−8.50058	−0.533374
$255$	−2.23060	−0.139686
$256$	−1.20994	−0.0756210
$257$	−28.6769	−1.78881	−0.894406	−	0.447255i	$-0.852402\pi$
$257$	−28.6769	−1.78881	−0.894406	+	0.447255i	$0.852402\pi$
$258$	4.88771	0.304296
$259$	−1.32498	−0.0823302
$260$	−6.70440	−0.415789
$261$	−1.45443	−0.0900269
$262$	−2.99069	−0.184765
$263$	−2.63094	−0.162231	−0.0811154	−	0.996705i	$-0.525848\pi$
$263$	−2.63094	−0.162231	−0.0811154	+	0.996705i	$0.525848\pi$
$264$	7.23103	0.445039
$265$	−29.4550	−1.80941
$266$	−0.622134	−0.0381455
$267$	−7.10670	−0.434923
$268$	1.86760	0.114082
$269$	8.89967	0.542622	0.271311	−	0.962492i	$-0.412543\pi$
$269$	8.89967	0.542622	0.271311	+	0.962492i	$0.412543\pi$
$270$	−1.14691	−0.0697985
$271$	−5.43976	−0.330442	−0.165221	−	0.986257i	$-0.552834\pi$
$271$	−5.43976	−0.330442	−0.165221	+	0.986257i	$0.552834\pi$
$272$	−2.48367	−0.150595
$273$	2.05909	0.124622
$274$	−10.8736	−0.656896
$275$	0.0919574	0.00554524
$276$	3.74409	0.225368
$277$	6.91104	0.415244	0.207622	−	0.978209i	$-0.433428\pi$
$277$	6.91104	0.415244	0.207622	+	0.978209i	$0.433428\pi$
$278$	8.18141	0.490688
$279$	−4.31473	−0.258316
$280$	−5.09433	−0.304444
$281$	−27.8390	−1.66073	−0.830367	−	0.557216i	$-0.811869\pi$
$281$	−27.8390	−1.66073	−0.830367	+	0.557216i	$0.811869\pi$
$282$	2.58104	0.153699
$283$	12.7768	0.759504	0.379752	−	0.925088i	$-0.376009\pi$
$283$	12.7768	0.759504	0.379752	+	0.925088i	$0.376009\pi$
$284$	23.8937	1.41783
$285$	2.26989	0.134456
$286$	−3.35210	−0.198214
$287$	2.63917	0.155785
$288$	−5.11852	−0.301612
$289$	1.00000	0.0588235
$290$	−1.66810	−0.0979539
$291$	−1.44167	−0.0845124
$292$	−9.21976	−0.539545
$293$	−14.6412	−0.855346	−0.427673	−	0.903934i	$-0.640666\pi$
$293$	−14.6412	−0.855346	−0.427673	+	0.903934i	$0.640666\pi$
$294$	−2.87225	−0.167513
$295$	−6.87360	−0.400196
$296$	−2.14035	−0.124405
$297$	3.76470	0.218450
$298$	−10.4015	−0.602545
$299$	−3.73569	−0.216040
$300$	−0.0423949	−0.00244767
$301$	11.3030	0.651495
$302$	−5.23680	−0.301344
$303$	−10.1000	−0.580231
$304$	2.52741	0.144957
$305$	21.7101	1.24312
$306$	0.514170	0.0293931
$307$	−31.3082	−1.78685	−0.893426	−	0.449209i	$-0.851706\pi$
$307$	−31.3082	−1.78685	−0.893426	+	0.449209i	$0.851706\pi$
$308$	7.76930	0.442697
$309$	−10.6269	−0.604542
$310$	−4.94860	−0.281061
$311$	9.90880	0.561876	0.280938	−	0.959726i	$-0.409354\pi$
$311$	9.90880	0.561876	0.280938	+	0.959726i	$0.409354\pi$
$312$	3.32622	0.188310
$313$	24.9421	1.40981	0.704907	−	0.709300i	$-0.250989\pi$
$313$	24.9421	1.40981	0.704907	+	0.709300i	$0.250989\pi$
$314$	0.514170	0.0290163
$315$	−2.65226	−0.149438
$316$	17.3190	0.974269
$317$	6.05204	0.339917	0.169958	−	0.985451i	$-0.445637\pi$
$317$	6.05204	0.339917	0.169958	+	0.985451i	$0.445637\pi$
$318$	6.78960	0.380742
$319$	5.47549	0.306568
$320$	5.20967	0.291229
$321$	10.1754	0.567933
$322$	−1.31884	−0.0734959
$323$	−1.01761	−0.0566215
$324$	−1.73563	−0.0964239
$325$	0.0422998	0.00234637
$326$	3.98215	0.220551
$327$	1.10382	0.0610416
$328$	4.26327	0.235400
$329$	5.96876	0.329068
$330$	4.31776	0.237685
$331$	20.4687	1.12506	0.562532	−	0.826776i	$-0.309827\pi$
$331$	20.4687	1.12506	0.562532	+	0.826776i	$0.309827\pi$
$332$	−24.3389	−1.33577
$333$	−1.11433	−0.0610650
$334$	−1.48178	−0.0810791
$335$	2.40021	0.131137
$336$	−2.95317	−0.161109
$337$	−26.6402	−1.45119	−0.725593	−	0.688125i	$-0.758434\pi$
$337$	−26.6402	−1.45119	−0.725593	+	0.688125i	$0.758434\pi$
$338$	5.14226	0.279702
$339$	0.726543	0.0394604
$340$	−3.87149	−0.209961
$341$	16.2437	0.879643
$342$	−0.523226	−0.0282928
$343$	−14.9654	−0.808058
$344$	18.2587	0.984442
$345$	4.81184	0.259061
$346$	5.74124	0.308651
$347$	15.8160	0.849048	0.424524	−	0.905417i	$-0.360441\pi$
$347$	15.8160	0.849048	0.424524	+	0.905417i	$0.360441\pi$
$348$	−2.52435	−0.135319
$349$	−29.5438	−1.58144	−0.790721	−	0.612177i	$-0.790294\pi$
$349$	−29.5438	−1.58144	−0.790721	+	0.612177i	$0.790294\pi$
$350$	0.0149334	0.000798224 0
$351$	1.73173	0.0924331
$352$	19.2697	1.02708
$353$	−11.7051	−0.623002	−0.311501	−	0.950246i	$-0.600832\pi$
$353$	−11.7051	−0.623002	−0.311501	+	0.950246i	$0.600832\pi$
$354$	1.58442	0.0842107
$355$	30.7078	1.62980
$356$	−12.3346	−0.653732
$357$	1.18904	0.0629305
$358$	−4.19309	−0.221611
$359$	20.6109	1.08780	0.543902	−	0.839149i	$-0.316946\pi$
$359$	20.6109	1.08780	0.543902	+	0.839149i	$0.316946\pi$
$360$	−4.28442	−0.225809
$361$	−17.9645	−0.945498
$362$	−3.15857	−0.166011
$363$	−3.17293	−0.166536
$364$	3.57382	0.187319
$365$	−11.8491	−0.620208
$366$	−5.00434	−0.261581
$367$	−17.4966	−0.913313	−0.456656	−	0.889643i	$-0.650953\pi$
$367$	−17.4966	−0.913313	−0.456656	+	0.889643i	$0.650953\pi$
$368$	5.35775	0.279292
$369$	2.21959	0.115547
$370$	−1.27803	−0.0664418
$371$	15.7012	0.815165
$372$	−7.48878	−0.388275
$373$	6.81511	0.352873	0.176436	−	0.984312i	$-0.443543\pi$
$373$	6.81511	0.352873	0.176436	+	0.984312i	$0.443543\pi$
$374$	−1.93569	−0.100092
$375$	−11.2075	−0.578752
$376$	9.64182	0.497239
$377$	2.51869	0.129719
$378$	0.611367	0.0314453
$379$	16.5901	0.852175	0.426088	−	0.904682i	$-0.359891\pi$
$379$	16.5901	0.852175	0.426088	+	0.904682i	$0.359891\pi$
$380$	3.93968	0.202101
$381$	−16.5326	−0.846993
$382$	3.15813	0.161584
$383$	−14.5530	−0.743623	−0.371812	−	0.928308i	$-0.621263\pi$
$383$	−14.5530	−0.743623	−0.371812	+	0.928308i	$0.621263\pi$
$384$	−11.4379	−0.583689
$385$	9.98497	0.508881
$386$	0.824332	0.0419574
$387$	9.50603	0.483219
$388$	−2.50221	−0.127030
$389$	28.2560	1.43264	0.716318	−	0.697774i	$-0.245826\pi$
$389$	28.2560	1.43264	0.716318	+	0.697774i	$0.245826\pi$
$390$	1.98614	0.100572
$391$	−2.15719	−0.109094
$392$	−10.7297	−0.541930
$393$	−5.81654	−0.293406
$394$	−2.74870	−0.138477
$395$	22.2581	1.11992
$396$	6.53412	0.328352
$397$	−17.0174	−0.854080	−0.427040	−	0.904233i	$-0.640444\pi$
$397$	−17.0174	−0.854080	−0.427040	+	0.904233i	$0.640444\pi$
$398$	2.54129	0.127383
$399$	−1.20998	−0.0605747
$400$	−0.0606667	−0.00303334
$401$	−20.1447	−1.00598	−0.502990	−	0.864292i	$-0.667767\pi$
$401$	−20.1447	−1.00598	−0.502990	+	0.864292i	$0.667767\pi$
$402$	−0.553265	−0.0275944
$403$	7.47197	0.372205
$404$	−17.5299	−0.872145
$405$	−2.23060	−0.110839
$406$	0.889190	0.0441298
$407$	4.19512	0.207944
$408$	1.92075	0.0950912
$409$	−17.6560	−0.873032	−0.436516	−	0.899696i	$-0.643788\pi$
$409$	−17.6560	−0.873032	−0.436516	+	0.899696i	$0.643788\pi$
$410$	2.54566	0.125721
$411$	−21.1478	−1.04314
$412$	−18.4443	−0.908687
$413$	3.66402	0.180295
$414$	−1.10916	−0.0545124
$415$	−31.2799	−1.53547
$416$	8.86392	0.434589
$417$	15.9119	0.779208
$418$	1.96979	0.0963454
$419$	−26.2708	−1.28341	−0.641706	−	0.766951i	$-0.721773\pi$
$419$	−26.2708	−1.28341	−0.641706	+	0.766951i	$0.721773\pi$
$420$	−4.60335	−0.224620
$421$	0.854664	0.0416538	0.0208269	−	0.999783i	$-0.493370\pi$
$421$	0.854664	0.0416538	0.0208269	+	0.999783i	$0.493370\pi$
$422$	−2.89856	−0.141100
$423$	5.01983	0.244072
$424$	25.3634	1.23176
$425$	0.0244263	0.00118485
$426$	−7.07838	−0.342949
$427$	−11.5727	−0.560043
$428$	17.6607	0.853660
$429$	−6.51945	−0.314762
$430$	10.9025	0.525767
$431$	−38.6065	−1.85961	−0.929805	−	0.368053i	$-0.880024\pi$
$431$	−38.6065	−1.85961	−0.929805	+	0.368053i	$0.880024\pi$
$432$	−2.48367	−0.119496
$433$	−2.39340	−0.115019	−0.0575096	−	0.998345i	$-0.518316\pi$
$433$	−2.39340	−0.115019	−0.0575096	+	0.998345i	$0.518316\pi$
$434$	2.63788	0.126622
$435$	−3.24425	−0.155550
$436$	1.91583	0.0917516
$437$	2.19519	0.105010
$438$	2.73130	0.130506
$439$	−20.9687	−1.00078	−0.500391	−	0.865800i	$-0.666810\pi$
$439$	−20.9687	−1.00078	−0.500391	+	0.865800i	$0.666810\pi$
$440$	16.1295	0.768945
$441$	−5.58619	−0.266009
$442$	−0.890405	−0.0423523
$443$	−13.0580	−0.620403	−0.310201	−	0.950671i	$-0.600397\pi$
$443$	−13.0580	−0.620403	−0.310201	+	0.950671i	$0.600397\pi$
$444$	−1.93407	−0.0917867
$445$	−15.8522	−0.751466
$446$	1.24164	0.0587934
$447$	−20.2298	−0.956836
$448$	−2.77705	−0.131203
$449$	−19.2732	−0.909559	−0.454780	−	0.890604i	$-0.650282\pi$
$449$	−19.2732	−0.909559	−0.454780	+	0.890604i	$0.650282\pi$
$450$	0.0125592	0.000592048 0
$451$	−8.35607	−0.393472
$452$	1.26101	0.0593129
$453$	−10.1850	−0.478532
$454$	−11.6691	−0.547658
$455$	4.59301	0.215324
$456$	−1.95458	−0.0915315
$457$	12.6683	0.592598	0.296299	−	0.955095i	$-0.404248\pi$
$457$	12.6683	0.592598	0.296299	+	0.955095i	$0.404248\pi$
$458$	−1.48263	−0.0692788
$459$	1.00000	0.0466760
$460$	8.35156	0.389394
$461$	−3.79484	−0.176743	−0.0883716	−	0.996088i	$-0.528166\pi$
$461$	−3.79484	−0.176743	−0.0883716	+	0.996088i	$0.528166\pi$
$462$	−2.30161	−0.107081
$463$	−29.9544	−1.39210	−0.696048	−	0.717995i	$-0.745060\pi$
$463$	−29.9544	−1.39210	−0.696048	+	0.717995i	$0.745060\pi$
$464$	−3.61232	−0.167698
$465$	−9.62444	−0.446323
$466$	−8.86850	−0.410825
$467$	20.5814	0.952391	0.476196	−	0.879339i	$-0.342015\pi$
$467$	20.5814	0.952391	0.476196	+	0.879339i	$0.342015\pi$
$468$	3.00565	0.138936
$469$	−1.27945	−0.0590793
$470$	5.75727	0.265563
$471$	1.00000	0.0460776
$472$	5.91879	0.272434
$473$	−35.7873	−1.64550
$474$	−5.13065	−0.235658
$475$	−0.0248565	−0.00114049
$476$	2.06373	0.0945907
$477$	13.2050	0.604614
$478$	14.0618	0.643172
$479$	−30.3123	−1.38500	−0.692502	−	0.721416i	$-0.743491\pi$
$479$	−30.3123	−1.38500	−0.692502	+	0.721416i	$0.743491\pi$
$480$	−11.4174	−0.521129
$481$	1.92972	0.0879878
$482$	−1.46417	−0.0666913
$483$	−2.56498	−0.116711
$484$	−5.50704	−0.250320
$485$	−3.21580	−0.146022
$486$	0.514170	0.0233232
$487$	9.18808	0.416352	0.208176	−	0.978091i	$-0.433247\pi$
$487$	9.18808	0.416352	0.208176	+	0.978091i	$0.433247\pi$
$488$	−18.6944	−0.846254
$489$	7.74482	0.350233
$490$	−6.40684	−0.289432
$491$	−21.0280	−0.948981	−0.474491	−	0.880260i	$-0.657368\pi$
$491$	−21.0280	−0.948981	−0.474491	+	0.880260i	$0.657368\pi$
$492$	3.85238	0.173679
$493$	1.45443	0.0655042
$494$	0.906088	0.0407668
$495$	8.39753	0.377441
$496$	−10.7164	−0.481179
$497$	−16.3690	−0.734250
$498$	7.21024	0.323099
$499$	18.9397	0.847859	0.423930	−	0.905695i	$-0.360650\pi$
$499$	18.9397	0.847859	0.423930	+	0.905695i	$0.360650\pi$
$500$	−19.4520	−0.869921
$501$	−2.88188	−0.128753
$502$	12.6075	0.562698
$503$	17.8046	0.793870	0.396935	−	0.917847i	$-0.370074\pi$
$503$	17.8046	0.793870	0.396935	+	0.917847i	$0.370074\pi$
$504$	2.28384	0.101730
$505$	−22.5291	−1.00253
$506$	4.17566	0.185631
$507$	10.0011	0.444164
$508$	−28.6945	−1.27311
$509$	24.6273	1.09158	0.545792	−	0.837921i	$-0.316229\pi$
$509$	24.6273	1.09158	0.545792	+	0.837921i	$0.316229\pi$
$510$	1.14691	0.0507859
$511$	6.31623	0.279413
$512$	−22.2537	−0.983485
$513$	−1.01761	−0.0449287
$514$	14.7448	0.650364
$515$	−23.7043	−1.04454
$516$	16.4989	0.726325
$517$	−18.8981	−0.831138
$518$	0.681265	0.0299330
$519$	11.1660	0.490134
$520$	7.41947	0.325365
$521$	22.4184	0.982169	0.491084	−	0.871112i	$-0.336601\pi$
$521$	22.4184	0.982169	0.491084	+	0.871112i	$0.336601\pi$
$522$	0.747824	0.0327314
$523$	−42.0016	−1.83660	−0.918300	−	0.395884i	$-0.870438\pi$
$523$	−42.0016	−1.83660	−0.918300	+	0.395884i	$0.870438\pi$
$524$	−10.0954	−0.441018
$525$	0.0290437	0.00126757
$526$	1.35275	0.0589827
$527$	4.31473	0.187953
$528$	9.35026	0.406918
$529$	−18.3465	−0.797675
$530$	15.1449	0.657851
$531$	3.08150	0.133726
$532$	−2.10007	−0.0910498
$533$	−3.84373	−0.166491
$534$	3.65405	0.158126
$535$	22.6971	0.981283
$536$	−2.06679	−0.0892719
$537$	−8.15506	−0.351917
$538$	−4.57594	−0.197283
$539$	21.0303	0.905840
$540$	−3.87149	−0.166603
$541$	−39.4376	−1.69556	−0.847778	−	0.530352i	$-0.822060\pi$
$541$	−39.4376	−1.69556	−0.847778	+	0.530352i	$0.822060\pi$
$542$	2.79696	0.120140
$543$	−6.14306	−0.263624
$544$	5.11852	0.219455
$545$	2.46219	0.105469
$546$	−1.05872	−0.0453092
$547$	10.3572	0.442842	0.221421	−	0.975178i	$-0.428931\pi$
$547$	10.3572	0.442842	0.221421	+	0.975178i	$0.428931\pi$
$548$	−36.7047	−1.56795
$549$	−9.73285	−0.415388
$550$	−0.0472817	−0.00201610
$551$	−1.48005	−0.0630521
$552$	−4.14343	−0.176356
$553$	−11.8648	−0.504543
$554$	−3.55345	−0.150972
$555$	−2.48563	−0.105509
$556$	27.6171	1.17123
$557$	34.5040	1.46198	0.730992	−	0.682386i	$-0.239058\pi$
$557$	34.5040	1.46198	0.730992	+	0.682386i	$0.239058\pi$
$558$	2.21851	0.0939168
$559$	−16.4619	−0.696265
$560$	−6.58734	−0.278366
$561$	−3.76470	−0.158946
$562$	14.3140	0.603798
$563$	−34.1976	−1.44126	−0.720629	−	0.693321i	$-0.756147\pi$
$563$	−34.1976	−1.44126	−0.720629	+	0.693321i	$0.756147\pi$
$564$	8.71256	0.366865
$565$	1.62063	0.0681803
$566$	−6.56946	−0.276135
$567$	1.18904	0.0499348
$568$	−26.4422	−1.10949
$569$	−22.0129	−0.922828	−0.461414	−	0.887185i	$-0.652658\pi$
$569$	−22.0129	−0.922828	−0.461414	+	0.887185i	$0.652658\pi$
$570$	−1.16711	−0.0488847
$571$	−15.6963	−0.656870	−0.328435	−	0.944527i	$-0.606521\pi$
$571$	−15.6963	−0.656870	−0.328435	+	0.944527i	$0.606521\pi$
$572$	−11.3153	−0.473119
$573$	6.14219	0.256594
$574$	−1.35698	−0.0566393
$575$	−0.0526922	−0.00219742
$576$	−2.33555	−0.0973144
$577$	−32.1369	−1.33788	−0.668938	−	0.743319i	$-0.733251\pi$
$577$	−32.1369	−1.33788	−0.668938	+	0.743319i	$0.733251\pi$
$578$	−0.514170	−0.0213866
$579$	1.60323	0.0666279
$580$	−5.63082	−0.233807
$581$	16.6740	0.691752
$582$	0.741265	0.0307264
$583$	−49.7127	−2.05889
$584$	10.2031	0.422208
$585$	3.86280	0.159707
$586$	7.52804	0.310981
$587$	30.0234	1.23920	0.619600	−	0.784918i	$-0.287295\pi$
$587$	30.0234	1.23920	0.619600	+	0.784918i	$0.287295\pi$
$588$	−9.69556	−0.399838
$589$	−4.39073	−0.180917
$590$	3.53420	0.145501
$591$	−5.34589	−0.219901
$592$	−2.76763	−0.113749
$593$	8.34511	0.342692	0.171346	−	0.985211i	$-0.445188\pi$
$593$	8.34511	0.342692	0.171346	+	0.985211i	$0.445188\pi$
$594$	−1.93569	−0.0794224
$595$	2.65226	0.108732
$596$	−35.1114	−1.43822
$597$	4.94252	0.202284
$598$	1.92078	0.0785464
$599$	−23.3385	−0.953585	−0.476793	−	0.879016i	$-0.658201\pi$
$599$	−23.3385	−0.953585	−0.476793	+	0.879016i	$0.658201\pi$
$600$	0.0469167	0.00191537
$601$	43.3537	1.76843	0.884216	−	0.467078i	$-0.154693\pi$
$601$	43.3537	1.76843	0.884216	+	0.467078i	$0.154693\pi$
$602$	−5.81167	−0.236866
$603$	−1.07604	−0.0438196
$604$	−17.6773	−0.719281
$605$	−7.07754	−0.287743
$606$	5.19313	0.210956
$607$	33.8996	1.37594	0.687970	−	0.725739i	$-0.258502\pi$
$607$	33.8996	1.37594	0.687970	+	0.725739i	$0.258502\pi$
$608$	−5.20867	−0.211240
$609$	1.72937	0.0700776
$610$	−11.1627	−0.451964
$611$	−8.69300	−0.351681
$612$	1.73563	0.0701587
$613$	−2.02264	−0.0816936	−0.0408468	−	0.999165i	$-0.513006\pi$
$613$	−2.02264	−0.0816936	−0.0408468	+	0.999165i	$0.513006\pi$
$614$	16.0977	0.649651
$615$	4.95101	0.199644
$616$	−8.59796	−0.346422
$617$	−44.5900	−1.79513	−0.897563	−	0.440886i	$-0.854664\pi$
$617$	−44.5900	−1.79513	−0.897563	+	0.440886i	$0.854664\pi$
$618$	5.46402	0.219795
$619$	−29.7208	−1.19458	−0.597290	−	0.802025i	$-0.703756\pi$
$619$	−29.7208	−1.19458	−0.597290	+	0.802025i	$0.703756\pi$
$620$	−16.7045	−0.670867
$621$	−2.15719	−0.0865652
$622$	−5.09481	−0.204283
$623$	8.45012	0.338547
$624$	4.30105	0.172180
$625$	−24.8773	−0.995091
$626$	−12.8245	−0.512570
$627$	3.83100	0.152996
$628$	1.73563	0.0692592
$629$	1.11433	0.0444313
$630$	1.36371	0.0543317
$631$	44.1891	1.75914	0.879570	−	0.475770i	$-0.157830\pi$
$631$	44.1891	1.75914	0.879570	+	0.475770i	$0.157830\pi$
$632$	−19.1662	−0.762390
$633$	−5.63736	−0.224065
$634$	−3.11178	−0.123584
$635$	−36.8777	−1.46345
$636$	22.9189	0.908795
$637$	9.67380	0.383290
$638$	−2.81533	−0.111460
$639$	−13.7666	−0.544599
$640$	−25.5134	−1.00851
$641$	19.6845	0.777489	0.388745	−	0.921346i	$-0.372909\pi$
$641$	19.6845	0.777489	0.388745	+	0.921346i	$0.372909\pi$
$642$	−5.23186	−0.206485
$643$	3.57002	0.140788	0.0703939	−	0.997519i	$-0.477574\pi$
$643$	3.57002	0.140788	0.0703939	+	0.997519i	$0.477574\pi$
$644$	−4.45186	−0.175428
$645$	21.2041	0.834912
$646$	0.523226	0.0205860
$647$	34.1186	1.34134	0.670670	−	0.741755i	$-0.266007\pi$
$647$	34.1186	1.34134	0.670670	+	0.741755i	$0.266007\pi$
$648$	1.92075	0.0754541
$649$	−11.6009	−0.455376
$650$	−0.0217493	−0.000853076 0
$651$	5.13038	0.201075
$652$	13.4421	0.526435
$653$	−23.5821	−0.922838	−0.461419	−	0.887182i	$-0.652659\pi$
$653$	−23.5821	−0.922838	−0.461419	+	0.887182i	$0.652659\pi$
$654$	−0.567553	−0.0221931
$655$	−12.9744	−0.506951
$656$	5.51272	0.215235
$657$	5.31205	0.207243
$658$	−3.06895	−0.119640
$659$	17.8392	0.694916	0.347458	−	0.937696i	$-0.387045\pi$
$659$	17.8392	0.694916	0.347458	+	0.937696i	$0.387045\pi$
$660$	14.5750	0.567331
$661$	−33.9635	−1.32103	−0.660513	−	0.750815i	$-0.729661\pi$
$661$	−33.9635	−1.32103	−0.660513	+	0.750815i	$0.729661\pi$
$662$	−10.5244	−0.409043
$663$	−1.73173	−0.0672550
$664$	26.9348	1.04527
$665$	−2.69898	−0.104662
$666$	0.572955	0.0222016
$667$	−3.13749	−0.121484
$668$	−5.00187	−0.193528
$669$	2.41484	0.0933633
$670$	−1.23411	−0.0476780
$671$	36.6412	1.41452
$672$	6.08611	0.234777
$673$	−7.16175	−0.276065	−0.138033	−	0.990428i	$-0.544078\pi$
$673$	−7.16175	−0.276065	−0.138033	+	0.990428i	$0.544078\pi$
$674$	13.6976	0.527612
$675$	0.0244263	0.000940167 0
$676$	17.3582	0.667623
$677$	38.5677	1.48228	0.741139	−	0.671352i	$-0.234286\pi$
$677$	38.5677	1.48228	0.741139	+	0.671352i	$0.234286\pi$
$678$	−0.373567	−0.0143467
$679$	1.71420	0.0657850
$680$	4.28442	0.164300
$681$	−22.6950	−0.869676
$682$	−8.35200	−0.319815
$683$	4.09135	0.156551	0.0782756	−	0.996932i	$-0.475059\pi$
$683$	4.09135	0.156551	0.0782756	+	0.996932i	$0.475059\pi$
$684$	−1.76620	−0.0675323
$685$	−47.1723	−1.80236
$686$	7.69478	0.293788
$687$	−2.88354	−0.110014
$688$	23.6098	0.900116
$689$	−22.8675	−0.871182
$690$	−2.47410	−0.0941874
$691$	−40.2715	−1.53200	−0.766001	−	0.642840i	$-0.777756\pi$
$691$	−40.2715	−1.53200	−0.766001	+	0.642840i	$0.777756\pi$
$692$	19.3801	0.736721
$693$	−4.47636	−0.170043
$694$	−8.13211	−0.308691
$695$	35.4930	1.34633
$696$	2.79359	0.105891
$697$	−2.21959	−0.0840729
$698$	15.1905	0.574970
$699$	−17.2482	−0.652386
$700$	0.0504091	0.00190529
$701$	−51.0135	−1.92675	−0.963377	−	0.268152i	$-0.913587\pi$
$701$	−51.0135	−1.92675	−0.963377	+	0.268152i	$0.913587\pi$
$702$	−0.890405	−0.0336062
$703$	−1.13396	−0.0427680
$704$	8.79262	0.331384
$705$	11.1972	0.421712
$706$	6.01843	0.226507
$707$	12.0093	0.451656
$708$	5.34835	0.201003
$709$	29.9318	1.12411	0.562056	−	0.827099i	$-0.310010\pi$
$709$	29.9318	1.12411	0.562056	+	0.827099i	$0.310010\pi$
$710$	−15.7890	−0.592552
$711$	−9.97851	−0.374223
$712$	13.6502	0.511562
$713$	−9.30772	−0.348577
$714$	−0.611367	−0.0228798
$715$	−14.5423	−0.543850
$716$	−14.1542	−0.528966
$717$	27.3485	1.02135
$718$	−10.5975	−0.395496
$719$	16.1469	0.602177	0.301088	−	0.953596i	$-0.402650\pi$
$719$	16.1469	0.602177	0.301088	+	0.953596i	$0.402650\pi$
$720$	−5.54007	−0.206466
$721$	12.6358	0.470580
$722$	9.23679	0.343758
$723$	−2.84765	−0.105905
$724$	−10.6621	−0.396253
$725$	0.0355263	0.00131941
$726$	1.63143	0.0605479
$727$	−1.25104	−0.0463986	−0.0231993	−	0.999731i	$-0.507385\pi$
$727$	−1.25104	−0.0463986	−0.0231993	+	0.999731i	$0.507385\pi$
$728$	−3.95500	−0.146582
$729$	1.00000	0.0370370
$730$	6.09243	0.225491
$731$	−9.50603	−0.351593
$732$	−16.8926	−0.624370
$733$	48.0206	1.77368	0.886841	−	0.462075i	$-0.152895\pi$
$733$	48.0206	1.77368	0.886841	+	0.462075i	$0.152895\pi$
$734$	8.99620	0.332056
$735$	−12.4606	−0.459615
$736$	−11.0416	−0.407000
$737$	4.05095	0.149219
$738$	−1.14124	−0.0420098
$739$	3.29332	0.121147	0.0605733	−	0.998164i	$-0.480707\pi$
$739$	3.29332	0.121147	0.0605733	+	0.998164i	$0.480707\pi$
$740$	−4.31412	−0.158590
$741$	1.76223	0.0647373
$742$	−8.07308	−0.296372
$743$	32.2405	1.18279	0.591395	−	0.806382i	$-0.298577\pi$
$743$	32.2405	1.18279	0.591395	+	0.806382i	$0.298577\pi$
$744$	8.28752	0.303835
$745$	−45.1245	−1.65323
$746$	−3.50412	−0.128295
$747$	14.0231	0.513078
$748$	−6.53412	−0.238911
$749$	−12.0989	−0.442083
$750$	5.76255	0.210419
$751$	37.0044	1.35031	0.675155	−	0.737675i	$-0.264077\pi$
$751$	37.0044	1.35031	0.675155	+	0.737675i	$0.264077\pi$
$752$	12.4676	0.454646
$753$	24.5200	0.893559
$754$	−1.29503	−0.0471623
$755$	−22.7186	−0.826814
$756$	2.06373	0.0750570
$757$	−48.5832	−1.76579	−0.882893	−	0.469574i	$-0.844407\pi$
$757$	−48.5832	−1.76579	−0.882893	+	0.469574i	$0.844407\pi$
$758$	−8.53012	−0.309828
$759$	8.12118	0.294780
$760$	−4.35988	−0.158149
$761$	−39.8327	−1.44393	−0.721967	−	0.691928i	$-0.756762\pi$
$761$	−39.8327	−1.44393	−0.721967	+	0.691928i	$0.756762\pi$
$762$	8.50058	0.307944
$763$	−1.31249	−0.0475152
$764$	10.6606	0.385686
$765$	2.23060	0.0806475
$766$	7.48271	0.270361
$767$	−5.33634	−0.192684
$768$	1.20994	0.0436598
$769$	17.3906	0.627123	0.313561	−	0.949568i	$-0.398478\pi$
$769$	17.3906	0.627123	0.313561	+	0.949568i	$0.398478\pi$
$770$	−5.13397	−0.185015
$771$	28.6769	1.03277
$772$	2.78261	0.100148
$773$	−39.0106	−1.40311	−0.701556	−	0.712614i	$-0.747511\pi$
$773$	−39.0106	−1.40311	−0.701556	+	0.712614i	$0.747511\pi$
$774$	−4.88771	−0.175685
$775$	0.105393	0.00378582
$776$	2.76909	0.0994046
$777$	1.32498	0.0475334
$778$	−14.5284	−0.520867
$779$	2.25868	0.0809256
$780$	6.70440	0.240056
$781$	51.8271	1.85452
$782$	1.10916	0.0396636
$783$	1.45443	0.0519771
$784$	−13.8742	−0.495509
$785$	2.23060	0.0796135
$786$	2.99069	0.106674
$787$	24.0086	0.855815	0.427907	−	0.903823i	$-0.359251\pi$
$787$	24.0086	0.855815	0.427907	+	0.903823i	$0.359251\pi$
$788$	−9.27849	−0.330533
$789$	2.63094	0.0936640
$790$	−11.4444	−0.407174
$791$	−0.863887	−0.0307163
$792$	−7.23103	−0.256944
$793$	16.8547	0.598528
$794$	8.74985	0.310521
$795$	29.4550	1.04466
$796$	8.57838	0.304052
$797$	−39.0448	−1.38304	−0.691518	−	0.722359i	$-0.743058\pi$
$797$	−39.0448	−1.38304	−0.691518	+	0.722359i	$0.743058\pi$
$798$	0.622134	0.0220233
$799$	−5.01983	−0.177589
$800$	0.125026	0.00442035
$801$	7.10670	0.251103
$802$	10.3578	0.365747
$803$	−19.9983	−0.705724
$804$	−1.86760	−0.0658652
$805$	−5.72145	−0.201655
$806$	−3.84186	−0.135324
$807$	−8.89967	−0.313283
$808$	19.3996	0.682476
$809$	−13.5099	−0.474982	−0.237491	−	0.971390i	$-0.576325\pi$
$809$	−13.5099	−0.474982	−0.237491	+	0.971390i	$0.576325\pi$
$810$	1.14691	0.0402982
$811$	−24.7830	−0.870249	−0.435125	−	0.900370i	$-0.643296\pi$
$811$	−24.7830	−0.870249	−0.435125	+	0.900370i	$0.643296\pi$
$812$	3.00155	0.105334
$813$	5.43976	0.190781
$814$	−2.15700	−0.0756029
$815$	17.2756	0.605138
$816$	2.48367	0.0869458
$817$	9.67346	0.338431
$818$	9.07818	0.317411
$819$	−2.05909	−0.0719506
$820$	8.59312	0.300085
$821$	2.27618	0.0794391	0.0397196	−	0.999211i	$-0.487354\pi$
$821$	2.27618	0.0794391	0.0397196	+	0.999211i	$0.487354\pi$
$822$	10.8736	0.379259
$823$	6.39602	0.222951	0.111476	−	0.993767i	$-0.464442\pi$
$823$	6.39602	0.222951	0.111476	+	0.993767i	$0.464442\pi$
$824$	20.4116	0.711071
$825$	−0.0919574	−0.00320155
$826$	−1.88393	−0.0655502
$827$	14.9349	0.519336	0.259668	−	0.965698i	$-0.416387\pi$
$827$	14.9349	0.519336	0.259668	+	0.965698i	$0.416387\pi$
$828$	−3.74409	−0.130116
$829$	48.6569	1.68993	0.844963	−	0.534825i	$-0.179623\pi$
$829$	48.6569	1.68993	0.844963	+	0.534825i	$0.179623\pi$
$830$	16.0832	0.558255
$831$	−6.91104	−0.239741
$832$	4.04454	0.140219
$833$	5.58619	0.193550
$834$	−8.18141	−0.283299
$835$	−6.42832	−0.222461
$836$	6.64920	0.229967
$837$	4.31473	0.149139
$838$	13.5076	0.466614
$839$	4.42332	0.152710	0.0763549	−	0.997081i	$-0.475672\pi$
$839$	4.42332	0.152710	0.0763549	+	0.997081i	$0.475672\pi$
$840$	5.09433	0.175771
$841$	−26.8846	−0.927056
$842$	−0.439442	−0.0151442
$843$	27.8390	0.958826
$844$	−9.78436	−0.336792
$845$	22.3084	0.767434
$846$	−2.58104	−0.0887381
$847$	3.77273	0.129633
$848$	32.7968	1.12625
$849$	−12.7768	−0.438500
$850$	−0.0125592	−0.000430778 0
$851$	−2.40383	−0.0824021
$852$	−23.8937	−0.818586
$853$	−51.7029	−1.77027	−0.885136	−	0.465332i	$-0.845935\pi$
$853$	−51.7029	−1.77027	−0.885136	+	0.465332i	$0.845935\pi$
$854$	5.95034	0.203617
$855$	−2.26989	−0.0776285
$856$	−19.5443	−0.668010
$857$	48.2602	1.64854	0.824268	−	0.566199i	$-0.191587\pi$
$857$	48.2602	1.64854	0.824268	+	0.566199i	$0.191587\pi$
$858$	3.35210	0.114439
$859$	27.6188	0.942342	0.471171	−	0.882042i	$-0.343831\pi$
$859$	27.6188	0.942342	0.471171	+	0.882042i	$0.343831\pi$
$860$	36.8025	1.25496
$861$	−2.63917	−0.0899427
$862$	19.8503	0.676104
$863$	35.1708	1.19723	0.598613	−	0.801038i	$-0.295719\pi$
$863$	35.1708	1.19723	0.598613	+	0.801038i	$0.295719\pi$
$864$	5.11852	0.174136
$865$	24.9069	0.846861
$866$	1.23061	0.0418179
$867$	−1.00000	−0.0339618
$868$	8.90443	0.302236
$869$	37.5660	1.27434
$870$	1.66810	0.0565537
$871$	1.86341	0.0631392
$872$	−2.12017	−0.0717980
$873$	1.44167	0.0487932
$874$	−1.12870	−0.0381788
$875$	13.3261	0.450505
$876$	9.21976	0.311507
$877$	−54.2884	−1.83319	−0.916594	−	0.399820i	$-0.869073\pi$
$877$	−54.2884	−1.83319	−0.916594	+	0.399820i	$0.869073\pi$
$878$	10.7815	0.363857
$879$	14.6412	0.493834
$880$	20.8567	0.703078
$881$	−7.11623	−0.239752	−0.119876	−	0.992789i	$-0.538250\pi$
$881$	−7.11623	−0.239752	−0.119876	+	0.992789i	$0.538250\pi$
$882$	2.87225	0.0967137
$883$	−16.2851	−0.548037	−0.274019	−	0.961724i	$-0.588353\pi$
$883$	−16.2851	−0.548037	−0.274019	+	0.961724i	$0.588353\pi$
$884$	−3.00565	−0.101091
$885$	6.87360	0.231053
$886$	6.71401	0.225562
$887$	1.96917	0.0661184	0.0330592	−	0.999453i	$-0.489475\pi$
$887$	1.96917	0.0661184	0.0330592	+	0.999453i	$0.489475\pi$
$888$	2.14035	0.0718254
$889$	19.6579	0.659305
$890$	8.15072	0.273213
$891$	−3.76470	−0.126122
$892$	4.19128	0.140334
$893$	5.10824	0.170941
$894$	10.4015	0.347880
$895$	−18.1907	−0.608047
$896$	13.6001	0.454347
$897$	3.73569	0.124731
$898$	9.90971	0.330691
$899$	6.27548	0.209299
$900$	0.0423949	0.00141316
$901$	−13.2050	−0.439921
$902$	4.29644	0.143056
$903$	−11.3030	−0.376141
$904$	−1.39551	−0.0464139
$905$	−13.7027	−0.455493
$906$	5.23680	0.173981
$907$	−6.69902	−0.222437	−0.111219	−	0.993796i	$-0.535475\pi$
$907$	−6.69902	−0.222437	−0.111219	+	0.993796i	$0.535475\pi$
$908$	−39.3902	−1.30721
$909$	10.1000	0.334997
$910$	−2.36159	−0.0782859
$911$	−46.8026	−1.55064	−0.775319	−	0.631570i	$-0.782411\pi$
$911$	−46.8026	−1.55064	−0.775319	+	0.631570i	$0.782411\pi$
$912$	−2.52741	−0.0836910
$913$	−52.7926	−1.74718
$914$	−6.51366	−0.215453
$915$	−21.7101	−0.717714
$916$	−5.00476	−0.165362
$917$	6.91608	0.228389
$918$	−0.514170	−0.0169701
$919$	19.5255	0.644088	0.322044	−	0.946725i	$-0.395630\pi$
$919$	19.5255	0.644088	0.322044	+	0.946725i	$0.395630\pi$
$920$	−9.24232	−0.304710
$921$	31.3082	1.03164
$922$	1.95119	0.0642590
$923$	23.8401	0.784707
$924$	−7.76930	−0.255591
$925$	0.0272189	0.000894953 0
$926$	15.4016	0.506129
$927$	10.6269	0.349033
$928$	7.44453	0.244379
$929$	−13.3463	−0.437877	−0.218939	−	0.975739i	$-0.570259\pi$
$929$	−13.3463	−0.437877	−0.218939	+	0.975739i	$0.570259\pi$
$930$	4.94860	0.162271
$931$	−5.68458	−0.186305
$932$	−29.9365	−0.980601
$933$	−9.90880	−0.324400
$934$	−10.5823	−0.346264
$935$	−8.39753	−0.274629
$936$	−3.32622	−0.108721
$937$	18.4543	0.602876	0.301438	−	0.953486i	$-0.402533\pi$
$937$	18.4543	0.602876	0.301438	+	0.953486i	$0.402533\pi$
$938$	0.657853	0.0214796
$939$	−24.9421	−0.813956
$940$	19.4342	0.633875
$941$	17.5919	0.573480	0.286740	−	0.958008i	$-0.407428\pi$
$941$	17.5919	0.573480	0.286740	+	0.958008i	$0.407428\pi$
$942$	−0.514170	−0.0167526
$943$	4.78808	0.155921
$944$	7.65343	0.249098
$945$	2.65226	0.0862781
$946$	18.4007	0.598260
$947$	−21.4608	−0.697384	−0.348692	−	0.937237i	$-0.613374\pi$
$947$	−21.4608	−0.697384	−0.348692	+	0.937237i	$0.613374\pi$
$948$	−17.3190	−0.562495
$949$	−9.19906	−0.298614
$950$	0.0127804	0.000414652 0
$951$	−6.05204	−0.196251
$952$	−2.28384	−0.0740196
$953$	−45.1751	−1.46336	−0.731682	−	0.681646i	$-0.761265\pi$
$953$	−45.1751	−1.46336	−0.731682	+	0.681646i	$0.761265\pi$
$954$	−6.78960	−0.219821
$955$	13.7008	0.443346
$956$	47.4669	1.53519
$957$	−5.47549	−0.176997
$958$	15.5857	0.503550
$959$	25.1455	0.811991
$960$	−5.20967	−0.168141
$961$	−12.3831	−0.399454
$962$	−0.992206	−0.0319900
$963$	−10.1754	−0.327896
$964$	−4.94246	−0.159186
$965$	3.57616	0.115121
$966$	1.31884	0.0424329
$967$	−6.97023	−0.224148	−0.112074	−	0.993700i	$-0.535749\pi$
$967$	−6.97023	−0.224148	−0.112074	+	0.993700i	$0.535749\pi$
$968$	6.09441	0.195882
$969$	1.01761	0.0326904
$970$	1.65346	0.0530896
$971$	54.9614	1.76380	0.881898	−	0.471441i	$-0.156266\pi$
$971$	54.9614	1.76380	0.881898	+	0.471441i	$0.156266\pi$
$972$	1.73563	0.0556703
$973$	−18.9198	−0.606541
$974$	−4.72423	−0.151374
$975$	−0.0422998	−0.00135468
$976$	−24.1732	−0.773765
$977$	54.5244	1.74439	0.872196	−	0.489157i	$-0.162695\pi$
$977$	54.5244	1.74439	0.872196	+	0.489157i	$0.162695\pi$
$978$	−3.98215	−0.127335
$979$	−26.7546	−0.855079
$980$	−21.6269	−0.690846
$981$	−1.10382	−0.0352424
$982$	10.8120	0.345024
$983$	−32.0460	−1.02211	−0.511055	−	0.859548i	$-0.670745\pi$
$983$	−32.0460	−1.02211	−0.511055	+	0.859548i	$0.670745\pi$
$984$	−4.26327	−0.135908
$985$	−11.9245	−0.379948
$986$	−0.747824	−0.0238156
$987$	−5.96876	−0.189988
$988$	3.05859	0.0973066
$989$	20.5063	0.652064
$990$	−4.31776	−0.137227
$991$	−41.8283	−1.32872	−0.664360	−	0.747413i	$-0.731296\pi$
$991$	−41.8283	−1.32872	−0.664360	+	0.747413i	$0.731296\pi$
$992$	22.0851	0.701201
$993$	−20.4687	−0.649556
$994$	8.41645	0.266954
$995$	11.0248	0.349509
$996$	24.3389	0.771207
$997$	46.0383	1.45805	0.729023	−	0.684489i	$-0.239975\pi$
$997$	46.0383	1.45805	0.729023	+	0.684489i	$0.239975\pi$
$998$	−9.73824	−0.308259
$999$	1.11433	0.0352559

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.21	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.21	✓	48	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-0.514170 q^{2} -1.00000 q^{3} -1.73563 q^{4} -2.23060 q^{5} +0.514170 q^{6} +1.18904 q^{7} +1.92075 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.514170 q^{2} -1.00000 q^{3} -1.73563 q^{4} -2.23060 q^{5} +0.514170 q^{6} +1.18904 q^{7} +1.92075 q^{8} +1.00000 q^{9} +1.14691 q^{10} -3.76470 q^{11} +1.73563 q^{12} -1.73173 q^{13} -0.611367 q^{14} +2.23060 q^{15} +2.48367 q^{16} -1.00000 q^{17} -0.514170 q^{18} +1.01761 q^{19} +3.87149 q^{20} -1.18904 q^{21} +1.93569 q^{22} +2.15719 q^{23} -1.92075 q^{24} -0.0244263 q^{25} +0.890405 q^{26} -1.00000 q^{27} -2.06373 q^{28} -1.45443 q^{29} -1.14691 q^{30} -4.31473 q^{31} -5.11852 q^{32} +3.76470 q^{33} +0.514170 q^{34} -2.65226 q^{35} -1.73563 q^{36} -1.11433 q^{37} -0.523226 q^{38} +1.73173 q^{39} -4.28442 q^{40} +2.21959 q^{41} +0.611367 q^{42} +9.50603 q^{43} +6.53412 q^{44} -2.23060 q^{45} -1.10916 q^{46} +5.01983 q^{47} -2.48367 q^{48} -5.58619 q^{49} +0.0125592 q^{50} +1.00000 q^{51} +3.00565 q^{52} +13.2050 q^{53} +0.514170 q^{54} +8.39753 q^{55} +2.28384 q^{56} -1.01761 q^{57} +0.747824 q^{58} +3.08150 q^{59} -3.87149 q^{60} -9.73285 q^{61} +2.21851 q^{62} +1.18904 q^{63} -2.33555 q^{64} +3.86280 q^{65} -1.93569 q^{66} -1.07604 q^{67} +1.73563 q^{68} -2.15719 q^{69} +1.36371 q^{70} -13.7666 q^{71} +1.92075 q^{72} +5.31205 q^{73} +0.572955 q^{74} +0.0244263 q^{75} -1.76620 q^{76} -4.47636 q^{77} -0.890405 q^{78} -9.97851 q^{79} -5.54007 q^{80} +1.00000 q^{81} -1.14124 q^{82} +14.0231 q^{83} +2.06373 q^{84} +2.23060 q^{85} -4.88771 q^{86} +1.45443 q^{87} -7.23103 q^{88} +7.10670 q^{89} +1.14691 q^{90} -2.05909 q^{91} -3.74409 q^{92} +4.31473 q^{93} -2.58104 q^{94} -2.26989 q^{95} +5.11852 q^{96} +1.44167 q^{97} +2.87225 q^{98} -3.76470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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