LMFDB - Embedded newform 8041.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, 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("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$1$
Dimension:	$62$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46781 q^{2} +0.627778 q^{3} +4.09007 q^{4} +2.55941 q^{5} -1.54924 q^{6} -1.57206 q^{7} -5.15790 q^{8} -2.60589 q^{9} +O(q^{10})$	\(q-2.46781 q^{2} +0.627778 q^{3} +4.09007 q^{4} +2.55941 q^{5} -1.54924 q^{6} -1.57206 q^{7} -5.15790 q^{8} -2.60589 q^{9} -6.31614 q^{10} +1.00000 q^{11} +2.56766 q^{12} +1.60114 q^{13} +3.87954 q^{14} +1.60674 q^{15} +4.54856 q^{16} -1.00000 q^{17} +6.43085 q^{18} +1.64531 q^{19} +10.4682 q^{20} -0.986904 q^{21} -2.46781 q^{22} +6.42975 q^{23} -3.23802 q^{24} +1.55059 q^{25} -3.95130 q^{26} -3.51926 q^{27} -6.42983 q^{28} +5.94207 q^{29} -3.96513 q^{30} -7.44518 q^{31} -0.909163 q^{32} +0.627778 q^{33} +2.46781 q^{34} -4.02355 q^{35} -10.6583 q^{36} -7.67397 q^{37} -4.06030 q^{38} +1.00516 q^{39} -13.2012 q^{40} -0.846412 q^{41} +2.43549 q^{42} +1.00000 q^{43} +4.09007 q^{44} -6.66956 q^{45} -15.8674 q^{46} +6.73853 q^{47} +2.85548 q^{48} -4.52863 q^{49} -3.82656 q^{50} -0.627778 q^{51} +6.54878 q^{52} -9.57017 q^{53} +8.68485 q^{54} +2.55941 q^{55} +8.10852 q^{56} +1.03289 q^{57} -14.6639 q^{58} -9.76541 q^{59} +6.57170 q^{60} +9.28774 q^{61} +18.3733 q^{62} +4.09662 q^{63} -6.85348 q^{64} +4.09797 q^{65} -1.54924 q^{66} -7.59000 q^{67} -4.09007 q^{68} +4.03646 q^{69} +9.92934 q^{70} -7.64523 q^{71} +13.4409 q^{72} -16.2964 q^{73} +18.9379 q^{74} +0.973428 q^{75} +6.72943 q^{76} -1.57206 q^{77} -2.48054 q^{78} -2.30454 q^{79} +11.6416 q^{80} +5.60837 q^{81} +2.08878 q^{82} -9.83490 q^{83} -4.03651 q^{84} -2.55941 q^{85} -2.46781 q^{86} +3.73030 q^{87} -5.15790 q^{88} -8.29280 q^{89} +16.4592 q^{90} -2.51708 q^{91} +26.2982 q^{92} -4.67392 q^{93} -16.6294 q^{94} +4.21102 q^{95} -0.570752 q^{96} +13.0446 q^{97} +11.1758 q^{98} -2.60589 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 62 q - 7 q^{2} - 8 q^{3} + 49 q^{4} - 13 q^{5} - 2 q^{6} - 11 q^{7} - 9 q^{8} + 40 q^{9}+O(q^{10}) $ 62 * q - 7 * q^2 - 8 * q^3 + 49 * q^4 - 13 * q^5 - 2 * q^6 - 11 * q^7 - 9 * q^8 + 40 * q^9	\( 62 q - 7 q^{2} - 8 q^{3} + 49 q^{4} - 13 q^{5} - 2 q^{6} - 11 q^{7} - 9 q^{8} + 40 q^{9} - 7 q^{10} + 62 q^{11} - 17 q^{12} - 31 q^{14} - 20 q^{15} + 27 q^{16} - 62 q^{17} + 3 q^{18} - 29 q^{20} - 18 q^{21} - 7 q^{22} - 50 q^{23} - 31 q^{24} + 35 q^{25} - 32 q^{26} - 14 q^{27} - 13 q^{28} - 26 q^{29} - 10 q^{30} - 58 q^{31} - 5 q^{32} - 8 q^{33} + 7 q^{34} - 32 q^{35} - 29 q^{36} - 41 q^{37} - 10 q^{38} - 53 q^{39} - 31 q^{40} - 55 q^{41} - 7 q^{42} + 62 q^{43} + 49 q^{44} - 34 q^{45} - 39 q^{46} - 31 q^{47} - 30 q^{48} + 35 q^{49} - 40 q^{50} + 8 q^{51} + 13 q^{52} - 74 q^{53} + 48 q^{54} - 13 q^{55} - 75 q^{56} - 43 q^{57} - 46 q^{58} - 65 q^{59} - 8 q^{60} - 14 q^{61} - 29 q^{62} - 23 q^{63} - 15 q^{64} - 9 q^{65} - 2 q^{66} - q^{67} - 49 q^{68} - 59 q^{69} - 31 q^{70} - 141 q^{71} + 9 q^{72} - 4 q^{73} - 94 q^{74} - 43 q^{75} + 34 q^{76} - 11 q^{77} - 11 q^{78} - 63 q^{79} - 41 q^{80} - 30 q^{81} + 38 q^{82} - 44 q^{83} - 16 q^{84} + 13 q^{85} - 7 q^{86} - 8 q^{87} - 9 q^{88} - 58 q^{89} - 55 q^{90} - 78 q^{91} - 104 q^{92} - 5 q^{94} - 99 q^{95} - 148 q^{96} - 26 q^{97} + 16 q^{98} + 40 q^{99}+O(q^{100}) \) 62 * q - 7 * q^2 - 8 * q^3 + 49 * q^4 - 13 * q^5 - 2 * q^6 - 11 * q^7 - 9 * q^8 + 40 * q^9 - 7 * q^10 + 62 * q^11 - 17 * q^12 - 31 * q^14 - 20 * q^15 + 27 * q^16 - 62 * q^17 + 3 * q^18 - 29 * q^20 - 18 * q^21 - 7 * q^22 - 50 * q^23 - 31 * q^24 + 35 * q^25 - 32 * q^26 - 14 * q^27 - 13 * q^28 - 26 * q^29 - 10 * q^30 - 58 * q^31 - 5 * q^32 - 8 * q^33 + 7 * q^34 - 32 * q^35 - 29 * q^36 - 41 * q^37 - 10 * q^38 - 53 * q^39 - 31 * q^40 - 55 * q^41 - 7 * q^42 + 62 * q^43 + 49 * q^44 - 34 * q^45 - 39 * q^46 - 31 * q^47 - 30 * q^48 + 35 * q^49 - 40 * q^50 + 8 * q^51 + 13 * q^52 - 74 * q^53 + 48 * q^54 - 13 * q^55 - 75 * q^56 - 43 * q^57 - 46 * q^58 - 65 * q^59 - 8 * q^60 - 14 * q^61 - 29 * q^62 - 23 * q^63 - 15 * q^64 - 9 * q^65 - 2 * q^66 - q^67 - 49 * q^68 - 59 * q^69 - 31 * q^70 - 141 * q^71 + 9 * q^72 - 4 * q^73 - 94 * q^74 - 43 * q^75 + 34 * q^76 - 11 * q^77 - 11 * q^78 - 63 * q^79 - 41 * q^80 - 30 * q^81 + 38 * q^82 - 44 * q^83 - 16 * q^84 + 13 * q^85 - 7 * q^86 - 8 * q^87 - 9 * q^88 - 58 * q^89 - 55 * q^90 - 78 * q^91 - 104 * q^92 - 5 * q^94 - 99 * q^95 - 148 * q^96 - 26 * q^97 + 16 * q^98 + 40 * 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 $

$2$	−2.46781	−1.74500	−0.872502	−	0.488611i	$-0.837504\pi$
$2$	−2.46781	−1.74500	−0.872502	+	0.488611i	$0.837504\pi$
$3$	0.627778	0.362448	0.181224	−	0.983442i	$-0.441994\pi$
$3$	0.627778	0.362448	0.181224	+	0.983442i	$0.441994\pi$
$4$	4.09007	2.04504
$5$	2.55941	1.14460	0.572302	−	0.820043i	$-0.306050\pi$
$5$	2.55941	1.14460	0.572302	+	0.820043i	$0.306050\pi$
$6$	−1.54924	−0.632473
$7$	−1.57206	−0.594182	−0.297091	−	0.954849i	$-0.596016\pi$
$7$	−1.57206	−0.594182	−0.297091	+	0.954849i	$0.596016\pi$
$8$	−5.15790	−1.82359
$9$	−2.60589	−0.868632
$10$	−6.31614	−1.99734
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.56766	0.741219
$13$	1.60114	0.444076	0.222038	−	0.975038i	$-0.428729\pi$
$13$	1.60114	0.444076	0.222038	+	0.975038i	$0.428729\pi$
$14$	3.87954	1.03685
$15$	1.60674	0.414859
$16$	4.54856	1.13714
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	6.43085	1.51577
$19$	1.64531	0.377460	0.188730	−	0.982029i	$-0.439563\pi$
$19$	1.64531	0.377460	0.188730	+	0.982029i	$0.439563\pi$
$20$	10.4682	2.34076
$21$	−0.986904	−0.215360
$22$	−2.46781	−0.526138
$23$	6.42975	1.34070	0.670348	−	0.742047i	$-0.266145\pi$
$23$	6.42975	1.34070	0.670348	+	0.742047i	$0.266145\pi$
$24$	−3.23802	−0.660957
$25$	1.55059	0.310119
$26$	−3.95130	−0.774914
$27$	−3.51926	−0.677281
$28$	−6.42983	−1.21512
$29$	5.94207	1.10341	0.551707	−	0.834038i	$-0.313977\pi$
$29$	5.94207	1.10341	0.551707	+	0.834038i	$0.313977\pi$
$30$	−3.96513	−0.723931
$31$	−7.44518	−1.33719	−0.668597	−	0.743625i	$-0.733105\pi$
$31$	−7.44518	−1.33719	−0.668597	+	0.743625i	$0.733105\pi$
$32$	−0.909163	−0.160719
$33$	0.627778	0.109282
$34$	2.46781	0.423225
$35$	−4.02355	−0.680103
$36$	−10.6583	−1.77638
$37$	−7.67397	−1.26159	−0.630797	−	0.775948i	$-0.717272\pi$
$37$	−7.67397	−1.26159	−0.630797	+	0.775948i	$0.717272\pi$
$38$	−4.06030	−0.658668
$39$	1.00516	0.160954
$40$	−13.2012	−2.08729
$41$	−0.846412	−0.132187	−0.0660937	−	0.997813i	$-0.521054\pi$
$41$	−0.846412	−0.132187	−0.0660937	+	0.997813i	$0.521054\pi$
$42$	2.43549	0.375804
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	4.09007	0.616602
$45$	−6.66956	−0.994239
$46$	−15.8674	−2.33952
$47$	6.73853	0.982915	0.491458	−	0.870902i	$-0.336464\pi$
$47$	6.73853	0.982915	0.491458	+	0.870902i	$0.336464\pi$
$48$	2.85548	0.412154
$49$	−4.52863	−0.646948
$50$	−3.82656	−0.541158
$51$	−0.627778	−0.0879065
$52$	6.54878	0.908152
$53$	−9.57017	−1.31456	−0.657282	−	0.753645i	$-0.728294\pi$
$53$	−9.57017	−1.31456	−0.657282	+	0.753645i	$0.728294\pi$
$54$	8.68485	1.18186
$55$	2.55941	0.345111
$56$	8.10852	1.08355
$57$	1.03289	0.136809
$58$	−14.6639	−1.92546
$59$	−9.76541	−1.27135	−0.635674	−	0.771957i	$-0.719278\pi$
$59$	−9.76541	−1.27135	−0.635674	+	0.771957i	$0.719278\pi$
$60$	6.57170	0.848402
$61$	9.28774	1.18917	0.594586	−	0.804032i	$-0.297316\pi$
$61$	9.28774	1.18917	0.594586	+	0.804032i	$0.297316\pi$
$62$	18.3733	2.33341
$63$	4.09662	0.516125
$64$	−6.85348	−0.856684
$65$	4.09797	0.508291
$66$	−1.54924	−0.190698
$67$	−7.59000	−0.927266	−0.463633	−	0.886027i	$-0.653454\pi$
$67$	−7.59000	−0.927266	−0.463633	+	0.886027i	$0.653454\pi$
$68$	−4.09007	−0.495994
$69$	4.03646	0.485932
$70$	9.92934	1.18678
$71$	−7.64523	−0.907322	−0.453661	−	0.891174i	$-0.649882\pi$
$71$	−7.64523	−0.907322	−0.453661	+	0.891174i	$0.649882\pi$
$72$	13.4409	1.58403
$73$	−16.2964	−1.90735	−0.953674	−	0.300843i	$-0.902732\pi$
$73$	−16.2964	−1.90735	−0.953674	+	0.300843i	$0.902732\pi$
$74$	18.9379	2.20148
$75$	0.973428	0.112402
$76$	6.72943	0.771919
$77$	−1.57206	−0.179153
$78$	−2.48054	−0.280866
$79$	−2.30454	−0.259281	−0.129640	−	0.991561i	$-0.541382\pi$
$79$	−2.30454	−0.259281	−0.129640	+	0.991561i	$0.541382\pi$
$80$	11.6416	1.30157
$81$	5.60837	0.623152
$82$	2.08878	0.230667
$83$	−9.83490	−1.07952	−0.539760	−	0.841819i	$-0.681485\pi$
$83$	−9.83490	−1.07952	−0.539760	+	0.841819i	$0.681485\pi$
$84$	−4.03651	−0.440419
$85$	−2.55941	−0.277607
$86$	−2.46781	−0.266111
$87$	3.73030	0.399930
$88$	−5.15790	−0.549834
$89$	−8.29280	−0.879035	−0.439518	−	0.898234i	$-0.644851\pi$
$89$	−8.29280	−0.879035	−0.439518	+	0.898234i	$0.644851\pi$
$90$	16.4592	1.73495
$91$	−2.51708	−0.263862
$92$	26.2982	2.74177
$93$	−4.67392	−0.484663
$94$	−16.6294	−1.71519
$95$	4.21102	0.432042
$96$	−0.570752	−0.0582522
$97$	13.0446	1.32448	0.662241	−	0.749291i	$-0.269605\pi$
$97$	13.0446	1.32448	0.662241	+	0.749291i	$0.269605\pi$
$98$	11.1758	1.12893
$99$	−2.60589	−0.261902
$100$	6.34204	0.634204
$101$	18.0186	1.79292	0.896458	−	0.443130i	$-0.146132\pi$
$101$	18.0186	1.79292	0.896458	+	0.443130i	$0.146132\pi$
$102$	1.54924	0.153397
$103$	7.75532	0.764154	0.382077	−	0.924131i	$-0.375209\pi$
$103$	7.75532	0.764154	0.382077	+	0.924131i	$0.375209\pi$
$104$	−8.25851	−0.809814
$105$	−2.52589	−0.246502
$106$	23.6173	2.29392
$107$	−7.03849	−0.680436	−0.340218	−	0.940347i	$-0.610501\pi$
$107$	−7.03849	−0.680436	−0.340218	+	0.940347i	$0.610501\pi$
$108$	−14.3940	−1.38507
$109$	13.9240	1.33368	0.666839	−	0.745202i	$-0.267647\pi$
$109$	13.9240	1.33368	0.666839	+	0.745202i	$0.267647\pi$
$110$	−6.31614	−0.602220
$111$	−4.81755	−0.457262
$112$	−7.15060	−0.675668
$113$	16.1658	1.52075	0.760376	−	0.649483i	$-0.225015\pi$
$113$	16.1658	1.52075	0.760376	+	0.649483i	$0.225015\pi$
$114$	−2.54897	−0.238733
$115$	16.4564	1.53457
$116$	24.3035	2.25652
$117$	−4.17240	−0.385738
$118$	24.0992	2.21851
$119$	1.57206	0.144110
$120$	−8.28742	−0.756534
$121$	1.00000	0.0909091
$122$	−22.9203	−2.07511
$123$	−0.531359	−0.0479110
$124$	−30.4514	−2.73461
$125$	−8.82846	−0.789641
$126$	−10.1097	−0.900641
$127$	8.31541	0.737873	0.368937	−	0.929455i	$-0.379722\pi$
$127$	8.31541	0.737873	0.368937	+	0.929455i	$0.379722\pi$
$128$	18.7314	1.65564
$129$	0.627778	0.0552728
$130$	−10.1130	−0.886970
$131$	10.0164	0.875135	0.437568	−	0.899186i	$-0.355840\pi$
$131$	10.0164	0.875135	0.437568	+	0.899186i	$0.355840\pi$
$132$	2.56766	0.223486
$133$	−2.58652	−0.224280
$134$	18.7307	1.61808
$135$	−9.00723	−0.775219
$136$	5.15790	0.442286
$137$	−12.6180	−1.07803	−0.539015	−	0.842296i	$-0.681204\pi$
$137$	−12.6180	−1.07803	−0.539015	+	0.842296i	$0.681204\pi$
$138$	−9.96120	−0.847954
$139$	−7.84191	−0.665143	−0.332571	−	0.943078i	$-0.607916\pi$
$139$	−7.84191	−0.665143	−0.332571	+	0.943078i	$0.607916\pi$
$140$	−16.4566	−1.39084
$141$	4.23030	0.356255
$142$	18.8670	1.58328
$143$	1.60114	0.133894
$144$	−11.8531	−0.987755
$145$	15.2082	1.26297
$146$	40.2163	3.32833
$147$	−2.84298	−0.234485
$148$	−31.3871	−2.58000
$149$	−22.9115	−1.87698	−0.938491	−	0.345304i	$-0.887776\pi$
$149$	−22.9115	−1.87698	−0.938491	+	0.345304i	$0.887776\pi$
$150$	−2.40223	−0.196142
$151$	−11.5221	−0.937656	−0.468828	−	0.883289i	$-0.655324\pi$
$151$	−11.5221	−0.937656	−0.468828	+	0.883289i	$0.655324\pi$
$152$	−8.48633	−0.688333
$153$	2.60589	0.210674
$154$	3.87954	0.312622
$155$	−19.0553	−1.53056
$156$	4.11118	0.329158
$157$	−6.86458	−0.547853	−0.273926	−	0.961751i	$-0.588322\pi$
$157$	−6.86458	−0.547853	−0.273926	+	0.961751i	$0.588322\pi$
$158$	5.68716	0.452446
$159$	−6.00794	−0.476461
$160$	−2.32692	−0.183959
$161$	−10.1079	−0.796618
$162$	−13.8404	−1.08740
$163$	10.1237	0.792951	0.396475	−	0.918045i	$-0.370233\pi$
$163$	10.1237	0.792951	0.396475	+	0.918045i	$0.370233\pi$
$164$	−3.46189	−0.270328
$165$	1.60674	0.125085
$166$	24.2706	1.88377
$167$	−13.9911	−1.08266	−0.541332	−	0.840809i	$-0.682080\pi$
$167$	−13.9911	−1.08266	−0.541332	+	0.840809i	$0.682080\pi$
$168$	5.09035	0.392729
$169$	−10.4364	−0.802797
$170$	6.31614	0.484426
$171$	−4.28750	−0.327873
$172$	4.09007	0.311865
$173$	16.9549	1.28905	0.644527	−	0.764581i	$-0.277054\pi$
$173$	16.9549	1.28905	0.644527	+	0.764581i	$0.277054\pi$
$174$	−9.20566	−0.697880
$175$	−2.43762	−0.184267
$176$	4.54856	0.342860
$177$	−6.13051	−0.460797
$178$	20.4650	1.53392
$179$	−2.69915	−0.201744	−0.100872	−	0.994899i	$-0.532163\pi$
$179$	−2.69915	−0.201744	−0.100872	+	0.994899i	$0.532163\pi$
$180$	−27.2790	−2.03326
$181$	1.13995	0.0847320	0.0423660	−	0.999102i	$-0.486510\pi$
$181$	1.13995	0.0847320	0.0423660	+	0.999102i	$0.486510\pi$
$182$	6.21168	0.460440
$183$	5.83064	0.431013
$184$	−33.1640	−2.44488
$185$	−19.6409	−1.44402
$186$	11.5343	0.845739
$187$	−1.00000	−0.0731272
$188$	27.5611	2.01010
$189$	5.53248	0.402429
$190$	−10.3920	−0.753914
$191$	−0.653572	−0.0472908	−0.0236454	−	0.999720i	$-0.507527\pi$
$191$	−0.653572	−0.0472908	−0.0236454	+	0.999720i	$0.507527\pi$
$192$	−4.30246	−0.310503
$193$	−0.326366	−0.0234923	−0.0117462	−	0.999931i	$-0.503739\pi$
$193$	−0.326366	−0.0234923	−0.0117462	+	0.999931i	$0.503739\pi$
$194$	−32.1917	−2.31123
$195$	2.57262	0.184229
$196$	−18.5224	−1.32303
$197$	−0.870570	−0.0620255	−0.0310128	−	0.999519i	$-0.509873\pi$
$197$	−0.870570	−0.0620255	−0.0310128	+	0.999519i	$0.509873\pi$
$198$	6.43085	0.457020
$199$	−17.8243	−1.26353	−0.631766	−	0.775159i	$-0.717670\pi$
$199$	−17.8243	−1.26353	−0.631766	+	0.775159i	$0.717670\pi$
$200$	−7.99780	−0.565530
$201$	−4.76483	−0.336085
$202$	−44.4664	−3.12864
$203$	−9.34128	−0.655629
$204$	−2.56766	−0.179772
$205$	−2.16632	−0.151302
$206$	−19.1386	−1.33345
$207$	−16.7553	−1.16457
$208$	7.28287	0.504976
$209$	1.64531	0.113808
$210$	6.23342	0.430147
$211$	13.0280	0.896887	0.448444	−	0.893811i	$-0.351979\pi$
$211$	13.0280	0.896887	0.448444	+	0.893811i	$0.351979\pi$
$212$	−39.1427	−2.68833
$213$	−4.79951	−0.328857
$214$	17.3696	1.18736
$215$	2.55941	0.174550
$216$	18.1520	1.23509
$217$	11.7043	0.794537
$218$	−34.3618	−2.32727
$219$	−10.2305	−0.691314
$220$	10.4682	0.705765
$221$	−1.60114	−0.107704
$222$	11.8888	0.797923
$223$	8.52328	0.570761	0.285380	−	0.958414i	$-0.407880\pi$
$223$	8.52328	0.570761	0.285380	+	0.958414i	$0.407880\pi$
$224$	1.42926	0.0954962
$225$	−4.04068	−0.269379
$226$	−39.8941	−2.65372
$227$	−18.2877	−1.21380	−0.606898	−	0.794780i	$-0.707586\pi$
$227$	−18.2877	−1.21380	−0.606898	+	0.794780i	$0.707586\pi$
$228$	4.22459	0.279780
$229$	−8.79117	−0.580937	−0.290468	−	0.956885i	$-0.593811\pi$
$229$	−8.79117	−0.580937	−0.290468	+	0.956885i	$0.593811\pi$
$230$	−40.6112	−2.67782
$231$	−0.986904	−0.0649335
$232$	−30.6486	−2.01218
$233$	−23.8941	−1.56535	−0.782677	−	0.622429i	$-0.786146\pi$
$233$	−23.8941	−1.56535	−0.782677	+	0.622429i	$0.786146\pi$
$234$	10.2967	0.673115
$235$	17.2467	1.12505
$236$	−39.9413	−2.59995
$237$	−1.44674	−0.0939758
$238$	−3.87954	−0.251473
$239$	9.36238	0.605602	0.302801	−	0.953054i	$-0.402078\pi$
$239$	9.36238	0.605602	0.302801	+	0.953054i	$0.402078\pi$
$240$	7.30836	0.471753
$241$	−19.9654	−1.28608	−0.643041	−	0.765832i	$-0.722327\pi$
$241$	−19.9654	−1.28608	−0.643041	+	0.765832i	$0.722327\pi$
$242$	−2.46781	−0.158637
$243$	14.0786	0.903142
$244$	37.9875	2.43190
$245$	−11.5906	−0.740499
$246$	1.31129	0.0836049
$247$	2.63437	0.167621
$248$	38.4015	2.43850
$249$	−6.17413	−0.391270
$250$	21.7869	1.37793
$251$	−0.0172756	−0.00109043	−0.000545213	−	1.00000i	$-0.500174\pi$
$251$	−0.0172756	−0.00109043	−0.000545213		1.00000i	$0.500174\pi$
$252$	16.7555	1.05550
$253$	6.42975	0.404235
$254$	−20.5208	−1.28759
$255$	−1.60674	−0.100618
$256$	−32.5185	−2.03241
$257$	−13.7413	−0.857156	−0.428578	−	0.903505i	$-0.640985\pi$
$257$	−13.7413	−0.857156	−0.428578	+	0.903505i	$0.640985\pi$
$258$	−1.54924	−0.0964512
$259$	12.0639	0.749616
$260$	16.7610	1.03947
$261$	−15.4844	−0.958461
$262$	−24.7185	−1.52711
$263$	7.02176	0.432980	0.216490	−	0.976285i	$-0.430539\pi$
$263$	7.02176	0.432980	0.216490	+	0.976285i	$0.430539\pi$
$264$	−3.23802	−0.199286
$265$	−24.4940	−1.50466
$266$	6.38303	0.391369
$267$	−5.20604	−0.318604
$268$	−31.0437	−1.89629
$269$	−4.92920	−0.300539	−0.150269	−	0.988645i	$-0.548014\pi$
$269$	−4.92920	−0.300539	−0.150269	+	0.988645i	$0.548014\pi$
$270$	22.2281	1.35276
$271$	3.85173	0.233976	0.116988	−	0.993133i	$-0.462676\pi$
$271$	3.85173	0.233976	0.116988	+	0.993133i	$0.462676\pi$
$272$	−4.54856	−0.275797
$273$	−1.58017	−0.0956362
$274$	31.1389	1.88117
$275$	1.55059	0.0935043
$276$	16.5094	0.993750
$277$	18.4126	1.10630	0.553152	−	0.833080i	$-0.313425\pi$
$277$	18.4126	1.10630	0.553152	+	0.833080i	$0.313425\pi$
$278$	19.3523	1.16068
$279$	19.4014	1.16153
$280$	20.7530	1.24023
$281$	6.50215	0.387886	0.193943	−	0.981013i	$-0.437872\pi$
$281$	6.50215	0.387886	0.193943	+	0.981013i	$0.437872\pi$
$282$	−10.4396	−0.621667
$283$	22.6413	1.34588	0.672942	−	0.739695i	$-0.265030\pi$
$283$	22.6413	1.34588	0.672942	+	0.739695i	$0.265030\pi$
$284$	−31.2696	−1.85551
$285$	2.64359	0.156593
$286$	−3.95130	−0.233645
$287$	1.33061	0.0785434
$288$	2.36918	0.139605
$289$	1.00000	0.0588235
$290$	−37.5309	−2.20389
$291$	8.18914	0.480056
$292$	−66.6534	−3.90060
$293$	−11.7989	−0.689297	−0.344649	−	0.938732i	$-0.612002\pi$
$293$	−11.7989	−0.689297	−0.344649	+	0.938732i	$0.612002\pi$
$294$	7.01592	0.409177
$295$	−24.9937	−1.45519
$296$	39.5816	2.30063
$297$	−3.51926	−0.204208
$298$	56.5411	3.27534
$299$	10.2949	0.595371
$300$	3.98139	0.229866
$301$	−1.57206	−0.0906119
$302$	28.4344	1.63621
$303$	11.3117	0.649838
$304$	7.48378	0.429224
$305$	23.7711	1.36113
$306$	−6.43085	−0.367627
$307$	−27.6929	−1.58052	−0.790259	−	0.612773i	$-0.790054\pi$
$307$	−27.6929	−1.58052	−0.790259	+	0.612773i	$0.790054\pi$
$308$	−6.42983	−0.366374
$309$	4.86862	0.276966
$310$	47.0248	2.67083
$311$	−6.42269	−0.364197	−0.182099	−	0.983280i	$-0.558289\pi$
$311$	−6.42269	−0.364197	−0.182099	+	0.983280i	$0.558289\pi$
$312$	−5.18451	−0.293515
$313$	−2.32983	−0.131690	−0.0658449	−	0.997830i	$-0.520974\pi$
$313$	−2.32983	−0.131690	−0.0658449	+	0.997830i	$0.520974\pi$
$314$	16.9405	0.956005
$315$	10.4849	0.590759
$316$	−9.42573	−0.530239
$317$	17.1999	0.966040	0.483020	−	0.875609i	$-0.339540\pi$
$317$	17.1999	0.966040	0.483020	+	0.875609i	$0.339540\pi$
$318$	14.8264	0.831426
$319$	5.94207	0.332692
$320$	−17.5409	−0.980565
$321$	−4.41861	−0.246623
$322$	24.9445	1.39010
$323$	−1.64531	−0.0915474
$324$	22.9387	1.27437
$325$	2.48271	0.137716
$326$	−24.9834	−1.38370
$327$	8.74118	0.483389
$328$	4.36571	0.241056
$329$	−10.5934	−0.584031
$330$	−3.96513	−0.218273
$331$	−27.1320	−1.49131	−0.745655	−	0.666333i	$-0.767863\pi$
$331$	−27.1320	−1.49131	−0.745655	+	0.666333i	$0.767863\pi$
$332$	−40.2255	−2.20766
$333$	19.9976	1.09586
$334$	34.5274	1.88925
$335$	−19.4259	−1.06135
$336$	−4.48899	−0.244894
$337$	−15.1809	−0.826958	−0.413479	−	0.910514i	$-0.635686\pi$
$337$	−15.1809	−0.826958	−0.413479	+	0.910514i	$0.635686\pi$
$338$	25.7549	1.40088
$339$	10.1485	0.551193
$340$	−10.4682	−0.567717
$341$	−7.44518	−0.403179
$342$	10.5807	0.572140
$343$	18.1237	0.978587
$344$	−5.15790	−0.278095
$345$	10.3310	0.556200
$346$	−41.8413	−2.24940
$347$	20.4104	1.09569	0.547844	−	0.836580i	$-0.315449\pi$
$347$	20.4104	1.09569	0.547844	+	0.836580i	$0.315449\pi$
$348$	15.2572	0.817872
$349$	36.3331	1.94486	0.972432	−	0.233186i	$-0.0749151\pi$
$349$	36.3331	1.94486	0.972432	+	0.233186i	$0.0749151\pi$
$350$	6.01558	0.321546
$351$	−5.63482	−0.300764
$352$	−0.909163	−0.0484585
$353$	−4.62425	−0.246124	−0.123062	−	0.992399i	$-0.539271\pi$
$353$	−4.62425	−0.246124	−0.123062	+	0.992399i	$0.539271\pi$
$354$	15.1289	0.804093
$355$	−19.5673	−1.03852
$356$	−33.9182	−1.79766
$357$	0.986904	0.0522325
$358$	6.66098	0.352044
$359$	14.7164	0.776700	0.388350	−	0.921512i	$-0.373045\pi$
$359$	14.7164	0.776700	0.388350	+	0.921512i	$0.373045\pi$
$360$	34.4009	1.81309
$361$	−16.2930	−0.857524
$362$	−2.81318	−0.147858
$363$	0.627778	0.0329498
$364$	−10.2951	−0.539608
$365$	−41.7092	−2.18316
$366$	−14.3889	−0.752119
$367$	29.3364	1.53135	0.765673	−	0.643230i	$-0.222406\pi$
$367$	29.3364	1.53135	0.765673	+	0.643230i	$0.222406\pi$
$368$	29.2461	1.52456
$369$	2.20566	0.114822
$370$	48.4699	2.51983
$371$	15.0449	0.781090
$372$	−19.1167	−0.991154
$373$	−8.27992	−0.428718	−0.214359	−	0.976755i	$-0.568766\pi$
$373$	−8.27992	−0.428718	−0.214359	+	0.976755i	$0.568766\pi$
$374$	2.46781	0.127607
$375$	−5.54231	−0.286204
$376$	−34.7566	−1.79244
$377$	9.51408	0.490000
$378$	−13.6531	−0.702239
$379$	1.20495	0.0618939	0.0309469	−	0.999521i	$-0.490148\pi$
$379$	1.20495	0.0618939	0.0309469	+	0.999521i	$0.490148\pi$
$380$	17.2234	0.883541
$381$	5.22023	0.267441
$382$	1.61289	0.0825227
$383$	−17.4490	−0.891601	−0.445801	−	0.895132i	$-0.647081\pi$
$383$	−17.4490	−0.891601	−0.445801	+	0.895132i	$0.647081\pi$
$384$	11.7592	0.600082
$385$	−4.02355	−0.205059
$386$	0.805408	0.0409942
$387$	−2.60589	−0.132465
$388$	53.3535	2.70862
$389$	−1.81708	−0.0921298	−0.0460649	−	0.998938i	$-0.514668\pi$
$389$	−1.81708	−0.0921298	−0.0460649	+	0.998938i	$0.514668\pi$
$390$	−6.34873	−0.321480
$391$	−6.42975	−0.325167
$392$	23.3582	1.17977
$393$	6.28806	0.317191
$394$	2.14840	0.108235
$395$	−5.89827	−0.296774
$396$	−10.6583	−0.535600
$397$	−15.1356	−0.759632	−0.379816	−	0.925062i	$-0.624013\pi$
$397$	−15.1356	−0.759632	−0.379816	+	0.925062i	$0.624013\pi$
$398$	43.9870	2.20487
$399$	−1.62376	−0.0812897
$400$	7.05296	0.352648
$401$	−0.306093	−0.0152856	−0.00764278	−	0.999971i	$-0.502433\pi$
$401$	−0.306093	−0.0152856	−0.00764278	+	0.999971i	$0.502433\pi$
$402$	11.7587	0.586470
$403$	−11.9208	−0.593816
$404$	73.6973	3.66658
$405$	14.3541	0.713263
$406$	23.0525	1.14408
$407$	−7.67397	−0.380385
$408$	3.23802	0.160306
$409$	17.4192	0.861322	0.430661	−	0.902514i	$-0.358281\pi$
$409$	17.4192	0.861322	0.430661	+	0.902514i	$0.358281\pi$
$410$	5.34605	0.264023
$411$	−7.92132	−0.390730
$412$	31.7198	1.56272
$413$	15.3518	0.755413
$414$	41.3488	2.03218
$415$	−25.1716	−1.23562
$416$	−1.45570	−0.0713713
$417$	−4.92298	−0.241079
$418$	−4.06030	−0.198596
$419$	−23.2451	−1.13560	−0.567799	−	0.823167i	$-0.692205\pi$
$419$	−23.2451	−1.13560	−0.567799	+	0.823167i	$0.692205\pi$
$420$	−10.3311	−0.504106
$421$	36.3732	1.77272	0.886361	−	0.462995i	$-0.153225\pi$
$421$	36.3732	1.77272	0.886361	+	0.462995i	$0.153225\pi$
$422$	−32.1507	−1.56507
$423$	−17.5599	−0.853791
$424$	49.3620	2.39723
$425$	−1.55059	−0.0752148
$426$	11.8443	0.573856
$427$	−14.6009	−0.706585
$428$	−28.7879	−1.39152
$429$	1.00516	0.0485296
$430$	−6.31614	−0.304591
$431$	−1.03255	−0.0497362	−0.0248681	−	0.999691i	$-0.507917\pi$
$431$	−1.03255	−0.0497362	−0.0248681	+	0.999691i	$0.507917\pi$
$432$	−16.0075	−0.770163
$433$	6.33714	0.304544	0.152272	−	0.988339i	$-0.451341\pi$
$433$	6.33714	0.304544	0.152272	+	0.988339i	$0.451341\pi$
$434$	−28.8839	−1.38647
$435$	9.54738	0.457762
$436$	56.9502	2.72742
$437$	10.5789	0.506059
$438$	25.2469	1.20634
$439$	−18.7318	−0.894020	−0.447010	−	0.894529i	$-0.647511\pi$
$439$	−18.7318	−0.894020	−0.447010	+	0.894529i	$0.647511\pi$
$440$	−13.2012	−0.629342
$441$	11.8011	0.561959
$442$	3.95130	0.187944
$443$	−11.3233	−0.537985	−0.268992	−	0.963142i	$-0.586691\pi$
$443$	−11.3233	−0.537985	−0.268992	+	0.963142i	$0.586691\pi$
$444$	−19.7041	−0.935117
$445$	−21.2247	−1.00615
$446$	−21.0338	−0.995980
$447$	−14.3833	−0.680308
$448$	10.7741	0.509027
$449$	−10.3206	−0.487059	−0.243530	−	0.969893i	$-0.578305\pi$
$449$	−10.3206	−0.487059	−0.243530	+	0.969893i	$0.578305\pi$
$450$	9.97163	0.470067
$451$	−0.846412	−0.0398560
$452$	66.1194	3.11000
$453$	−7.23333	−0.339852
$454$	45.1305	2.11808
$455$	−6.44225	−0.302018
$456$	−5.32753	−0.249485
$457$	−32.5305	−1.52171	−0.760856	−	0.648921i	$-0.775221\pi$
$457$	−32.5305	−1.52171	−0.760856	+	0.648921i	$0.775221\pi$
$458$	21.6949	1.01374
$459$	3.51926	0.164265
$460$	67.3079	3.13825
$461$	−26.0006	−1.21097	−0.605484	−	0.795858i	$-0.707020\pi$
$461$	−26.0006	−1.21097	−0.605484	+	0.795858i	$0.707020\pi$
$462$	2.43549	0.113309
$463$	27.3769	1.27231	0.636157	−	0.771559i	$-0.280523\pi$
$463$	27.3769	1.27231	0.636157	+	0.771559i	$0.280523\pi$
$464$	27.0278	1.25474
$465$	−11.9625	−0.554747
$466$	58.9660	2.73155
$467$	−4.98292	−0.230582	−0.115291	−	0.993332i	$-0.536780\pi$
$467$	−4.98292	−0.230582	−0.115291	+	0.993332i	$0.536780\pi$
$468$	−17.0654	−0.788849
$469$	11.9319	0.550965
$470$	−42.5615	−1.96321
$471$	−4.30943	−0.198568
$472$	50.3690	2.31842
$473$	1.00000	0.0459800
$474$	3.57027	0.163988
$475$	2.55120	0.117057
$476$	6.42983	0.294711
$477$	24.9389	1.14187
$478$	−23.1045	−1.05678
$479$	−21.7557	−0.994044	−0.497022	−	0.867738i	$-0.665573\pi$
$479$	−21.7557	−0.994044	−0.497022	+	0.867738i	$0.665573\pi$
$480$	−1.46079	−0.0666757
$481$	−12.2871	−0.560243
$482$	49.2707	2.24422
$483$	−6.34555	−0.288732
$484$	4.09007	0.185912
$485$	33.3866	1.51601
$486$	−34.7432	−1.57599
$487$	−30.5732	−1.38540	−0.692702	−	0.721224i	$-0.743580\pi$
$487$	−30.5732	−1.38540	−0.692702	+	0.721224i	$0.743580\pi$
$488$	−47.9052	−2.16857
$489$	6.35545	0.287403
$490$	28.6035	1.29217
$491$	−4.06467	−0.183436	−0.0917180	−	0.995785i	$-0.529236\pi$
$491$	−4.06467	−0.183436	−0.0917180	+	0.995785i	$0.529236\pi$
$492$	−2.17330	−0.0979798
$493$	−5.94207	−0.267617
$494$	−6.50111	−0.292499
$495$	−6.66956	−0.299774
$496$	−33.8648	−1.52058
$497$	12.0187	0.539114
$498$	15.2366	0.682767
$499$	−37.4795	−1.67781	−0.838907	−	0.544274i	$-0.816805\pi$
$499$	−37.4795	−1.67781	−0.838907	+	0.544274i	$0.816805\pi$
$500$	−36.1090	−1.61485
$501$	−8.78331	−0.392409
$502$	0.0426328	0.00190280
$503$	23.0058	1.02578	0.512888	−	0.858455i	$-0.328576\pi$
$503$	23.0058	1.02578	0.512888	+	0.858455i	$0.328576\pi$
$504$	−21.1299	−0.941203
$505$	46.1170	2.05218
$506$	−15.8674	−0.705392
$507$	−6.55171	−0.290972
$508$	34.0106	1.50898
$509$	24.5258	1.08709	0.543545	−	0.839380i	$-0.317082\pi$
$509$	24.5258	1.08709	0.543545	+	0.839380i	$0.317082\pi$
$510$	3.96513	0.175579
$511$	25.6189	1.13331
$512$	42.7866	1.89092
$513$	−5.79026	−0.255646
$514$	33.9108	1.49574
$515$	19.8491	0.874654
$516$	2.56766	0.113035
$517$	6.73853	0.296360
$518$	−29.7715	−1.30808
$519$	10.6439	0.467215
$520$	−21.1369	−0.926916
$521$	−36.5953	−1.60327	−0.801634	−	0.597815i	$-0.796036\pi$
$521$	−36.5953	−1.60327	−0.801634	+	0.597815i	$0.796036\pi$
$522$	38.2125	1.67252
$523$	17.2519	0.754373	0.377187	−	0.926137i	$-0.376892\pi$
$523$	17.2519	0.754373	0.377187	+	0.926137i	$0.376892\pi$
$524$	40.9677	1.78968
$525$	−1.53029	−0.0667871
$526$	−17.3284	−0.755552
$527$	7.44518	0.324317
$528$	2.85548	0.124269
$529$	18.3417	0.797467
$530$	60.4465	2.62563
$531$	25.4476	1.10433
$532$	−10.5791	−0.458660
$533$	−1.35522	−0.0587012
$534$	12.8475	0.555966
$535$	−18.0144	−0.778830
$536$	39.1485	1.69096
$537$	−1.69447	−0.0731217
$538$	12.1643	0.524441
$539$	−4.52863	−0.195062
$540$	−36.8402	−1.58535
$541$	1.62369	0.0698077	0.0349039	−	0.999391i	$-0.488888\pi$
$541$	1.62369	0.0698077	0.0349039	+	0.999391i	$0.488888\pi$
$542$	−9.50533	−0.408289
$543$	0.715637	0.0307109
$544$	0.909163	0.0389800
$545$	35.6373	1.52653
$546$	3.89955	0.166886
$547$	−9.89290	−0.422990	−0.211495	−	0.977379i	$-0.567833\pi$
$547$	−9.89290	−0.422990	−0.211495	+	0.977379i	$0.567833\pi$
$548$	−51.6087	−2.20461
$549$	−24.2029	−1.03295
$550$	−3.82656	−0.163165
$551$	9.77654	0.416494
$552$	−20.8196	−0.886143
$553$	3.62287	0.154060
$554$	−45.4387	−1.93051
$555$	−12.3301	−0.523384
$556$	−32.0740	−1.36024
$557$	−30.4287	−1.28930	−0.644652	−	0.764476i	$-0.722998\pi$
$557$	−30.4287	−1.28930	−0.644652	+	0.764476i	$0.722998\pi$
$558$	−47.8788	−2.02687
$559$	1.60114	0.0677210
$560$	−18.3013	−0.773372
$561$	−0.627778	−0.0265048
$562$	−16.0461	−0.676862
$563$	−35.4378	−1.49352	−0.746762	−	0.665091i	$-0.768393\pi$
$563$	−35.4378	−1.49352	−0.746762	+	0.665091i	$0.768393\pi$
$564$	17.3022	0.728555
$565$	41.3750	1.74066
$566$	−55.8743	−2.34857
$567$	−8.81669	−0.370266
$568$	39.4333	1.65459
$569$	−13.6847	−0.573692	−0.286846	−	0.957977i	$-0.592607\pi$
$569$	−13.6847	−0.573692	−0.286846	+	0.957977i	$0.592607\pi$
$570$	−6.52386	−0.273255
$571$	19.9283	0.833974	0.416987	−	0.908912i	$-0.363086\pi$
$571$	19.9283	0.833974	0.416987	+	0.908912i	$0.363086\pi$
$572$	6.54878	0.273818
$573$	−0.410298	−0.0171405
$574$	−3.28369	−0.137058
$575$	9.96993	0.415775
$576$	17.8594	0.744143
$577$	39.5275	1.64555	0.822776	−	0.568366i	$-0.192424\pi$
$577$	39.5275	1.64555	0.822776	+	0.568366i	$0.192424\pi$
$578$	−2.46781	−0.102647
$579$	−0.204885	−0.00851474
$580$	62.2027	2.58283
$581$	15.4610	0.641432
$582$	−20.2092	−0.837699
$583$	−9.57017	−0.396356
$584$	84.0551	3.47823
$585$	−10.6789	−0.441518
$586$	29.1173	1.20283
$587$	18.5946	0.767482	0.383741	−	0.923441i	$-0.374636\pi$
$587$	18.5946	0.767482	0.383741	+	0.923441i	$0.374636\pi$
$588$	−11.6280	−0.479530
$589$	−12.2496	−0.504737
$590$	61.6797	2.53931
$591$	−0.546525	−0.0224810
$592$	−34.9055	−1.43461
$593$	22.4883	0.923485	0.461742	−	0.887014i	$-0.347224\pi$
$593$	22.4883	0.923485	0.461742	+	0.887014i	$0.347224\pi$
$594$	8.68485	0.356344
$595$	4.02355	0.164949
$596$	−93.7097	−3.83850
$597$	−11.1897	−0.457964
$598$	−25.4059	−1.03892
$599$	−34.4269	−1.40664	−0.703322	−	0.710872i	$-0.748301\pi$
$599$	−34.4269	−1.40664	−0.703322	+	0.710872i	$0.748301\pi$
$600$	−5.02085	−0.204975
$601$	−33.4772	−1.36556	−0.682782	−	0.730622i	$-0.739230\pi$
$601$	−33.4772	−1.36556	−0.682782	+	0.730622i	$0.739230\pi$
$602$	3.87954	0.158118
$603$	19.7787	0.805452
$604$	−47.1263	−1.91754
$605$	2.55941	0.104055
$606$	−27.9150	−1.13397
$607$	35.3298	1.43399	0.716996	−	0.697078i	$-0.245517\pi$
$607$	35.3298	1.43399	0.716996	+	0.697078i	$0.245517\pi$
$608$	−1.49585	−0.0606648
$609$	−5.86425	−0.237631
$610$	−58.6626	−2.37518
$611$	10.7893	0.436489
$612$	10.6583	0.430836
$613$	15.7118	0.634594	0.317297	−	0.948326i	$-0.397225\pi$
$613$	15.7118	0.634594	0.317297	+	0.948326i	$0.397225\pi$
$614$	68.3408	2.75801
$615$	−1.35997	−0.0548391
$616$	8.10852	0.326702
$617$	6.06723	0.244257	0.122129	−	0.992514i	$-0.461028\pi$
$617$	6.06723	0.244257	0.122129	+	0.992514i	$0.461028\pi$
$618$	−12.0148	−0.483306
$619$	−2.26454	−0.0910197	−0.0455099	−	0.998964i	$-0.514491\pi$
$619$	−2.26454	−0.0910197	−0.0455099	+	0.998964i	$0.514491\pi$
$620$	−77.9376	−3.13005
$621$	−22.6280	−0.908029
$622$	15.8500	0.635526
$623$	13.0368	0.522307
$624$	4.57203	0.183028
$625$	−30.3486	−1.21395
$626$	5.74957	0.229799
$627$	1.03289	0.0412496
$628$	−28.0766	−1.12038
$629$	7.67397	0.305981
$630$	−25.8748	−1.03088
$631$	−7.15709	−0.284919	−0.142460	−	0.989801i	$-0.545501\pi$
$631$	−7.15709	−0.284919	−0.142460	+	0.989801i	$0.545501\pi$
$632$	11.8866	0.472823
$633$	8.17871	0.325075
$634$	−42.4459	−1.68574
$635$	21.2826	0.844573
$636$	−24.5729	−0.974380
$637$	−7.25097	−0.287294
$638$	−14.6639	−0.580549
$639$	19.9227	0.788128
$640$	47.9413	1.89505
$641$	12.4541	0.491906	0.245953	−	0.969282i	$-0.420899\pi$
$641$	12.4541	0.491906	0.245953	+	0.969282i	$0.420899\pi$
$642$	10.9043	0.430357
$643$	−26.3261	−1.03820	−0.519100	−	0.854714i	$-0.673733\pi$
$643$	−26.3261	−1.03820	−0.519100	+	0.854714i	$0.673733\pi$
$644$	−41.3423	−1.62911
$645$	1.60674	0.0632654
$646$	4.06030	0.159751
$647$	−48.7360	−1.91601	−0.958005	−	0.286751i	$-0.907425\pi$
$647$	−48.7360	−1.91601	−0.958005	+	0.286751i	$0.907425\pi$
$648$	−28.9274	−1.13638
$649$	−9.76541	−0.383326
$650$	−6.12686	−0.240315
$651$	7.34768	0.287978
$652$	41.4067	1.62161
$653$	33.8196	1.32346	0.661732	−	0.749740i	$-0.269822\pi$
$653$	33.8196	1.32346	0.661732	+	0.749740i	$0.269822\pi$
$654$	−21.5716	−0.843515
$655$	25.6360	1.00168
$656$	−3.84995	−0.150315
$657$	42.4667	1.65678
$658$	26.1424	1.01914
$659$	−29.9702	−1.16747	−0.583736	−	0.811944i	$-0.698410\pi$
$659$	−29.9702	−1.16747	−0.583736	+	0.811944i	$0.698410\pi$
$660$	6.57170	0.255803
$661$	−34.3365	−1.33553	−0.667767	−	0.744370i	$-0.732750\pi$
$661$	−34.3365	−1.33553	−0.667767	+	0.744370i	$0.732750\pi$
$662$	66.9565	2.60234
$663$	−1.00516	−0.0390372
$664$	50.7274	1.96861
$665$	−6.61997	−0.256711
$666$	−49.3501	−1.91228
$667$	38.2061	1.47934
$668$	−57.2247	−2.21409
$669$	5.35073	0.206871
$670$	47.9395	1.85206
$671$	9.28774	0.358549
$672$	0.897256	0.0346124
$673$	−40.8202	−1.57350	−0.786752	−	0.617269i	$-0.788239\pi$
$673$	−40.8202	−1.57350	−0.786752	+	0.617269i	$0.788239\pi$
$674$	37.4636	1.44304
$675$	−5.45694	−0.210038
$676$	−42.6855	−1.64175
$677$	−40.8683	−1.57070	−0.785349	−	0.619053i	$-0.787517\pi$
$677$	−40.8683	−1.57070	−0.785349	+	0.619053i	$0.787517\pi$
$678$	−25.0447	−0.961834
$679$	−20.5069	−0.786984
$680$	13.2012	0.506243
$681$	−11.4806	−0.439938
$682$	18.3733	0.703549
$683$	30.8291	1.17964	0.589821	−	0.807534i	$-0.299198\pi$
$683$	30.8291	1.17964	0.589821	+	0.807534i	$0.299198\pi$
$684$	−17.5362	−0.670513
$685$	−32.2947	−1.23392
$686$	−44.7258	−1.70764
$687$	−5.51890	−0.210559
$688$	4.54856	0.173412
$689$	−15.3232	−0.583766
$690$	−25.4948	−0.970571
$691$	21.6710	0.824405	0.412203	−	0.911092i	$-0.364760\pi$
$691$	21.6710	0.824405	0.412203	+	0.911092i	$0.364760\pi$
$692$	69.3466	2.63616
$693$	4.09662	0.155618
$694$	−50.3690	−1.91198
$695$	−20.0707	−0.761325
$696$	−19.2405	−0.729310
$697$	0.846412	0.0320601
$698$	−89.6630	−3.39379
$699$	−15.0002	−0.567359
$700$	−9.97006	−0.376833
$701$	1.27043	0.0479834	0.0239917	−	0.999712i	$-0.492362\pi$
$701$	1.27043	0.0479834	0.0239917	+	0.999712i	$0.492362\pi$
$702$	13.9056	0.524835
$703$	−12.6260	−0.476200
$704$	−6.85348	−0.258300
$705$	10.8271	0.407771
$706$	11.4117	0.429487
$707$	−28.3262	−1.06532
$708$	−25.0742	−0.942348
$709$	46.2593	1.73730	0.868651	−	0.495424i	$-0.164987\pi$
$709$	46.2593	1.73730	0.868651	+	0.495424i	$0.164987\pi$
$710$	48.2883	1.81223
$711$	6.00539	0.225220
$712$	42.7735	1.60300
$713$	−47.8707	−1.79277
$714$	−2.43549	−0.0911458
$715$	4.09797	0.153256
$716$	−11.0397	−0.412574
$717$	5.87749	0.219499
$718$	−36.3171	−1.35534
$719$	−3.69725	−0.137884	−0.0689422	−	0.997621i	$-0.521962\pi$
$719$	−3.69725	−0.137884	−0.0689422	+	0.997621i	$0.521962\pi$
$720$	−30.3369	−1.13059
$721$	−12.1918	−0.454047
$722$	40.2079	1.49638
$723$	−12.5338	−0.466138
$724$	4.66249	0.173280
$725$	9.21373	0.342189
$726$	−1.54924	−0.0574975
$727$	−1.69767	−0.0629632	−0.0314816	−	0.999504i	$-0.510023\pi$
$727$	−1.69767	−0.0629632	−0.0314816	+	0.999504i	$0.510023\pi$
$728$	12.9829	0.481177
$729$	−7.98689	−0.295811
$730$	102.930	3.80962
$731$	−1.00000	−0.0369863
$732$	23.8477	0.881438
$733$	49.0014	1.80991	0.904953	−	0.425511i	$-0.139906\pi$
$733$	49.0014	1.80991	0.904953	+	0.425511i	$0.139906\pi$
$734$	−72.3965	−2.67220
$735$	−7.27635	−0.268392
$736$	−5.84569	−0.215475
$737$	−7.59000	−0.279581
$738$	−5.44315	−0.200365
$739$	−35.5066	−1.30613	−0.653066	−	0.757301i	$-0.726518\pi$
$739$	−35.5066	−1.30613	−0.653066	+	0.757301i	$0.726518\pi$
$740$	−80.3326	−2.95308
$741$	1.65380	0.0607538
$742$	−37.1278	−1.36301
$743$	32.0486	1.17575	0.587875	−	0.808952i	$-0.299965\pi$
$743$	32.0486	1.17575	0.587875	+	0.808952i	$0.299965\pi$
$744$	24.1076	0.883828
$745$	−58.6399	−2.14840
$746$	20.4333	0.748115
$747$	25.6287	0.937706
$748$	−4.09007	−0.149548
$749$	11.0649	0.404303
$750$	13.6774	0.499426
$751$	−5.95496	−0.217300	−0.108650	−	0.994080i	$-0.534653\pi$
$751$	−5.95496	−0.217300	−0.108650	+	0.994080i	$0.534653\pi$
$752$	30.6506	1.11771
$753$	−0.0108452	−0.000395222 0
$754$	−23.4789	−0.855052
$755$	−29.4898	−1.07325
$756$	22.6282	0.822981
$757$	−13.7240	−0.498806	−0.249403	−	0.968400i	$-0.580234\pi$
$757$	−13.7240	−0.498806	−0.249403	+	0.968400i	$0.580234\pi$
$758$	−2.97357	−0.108005
$759$	4.03646	0.146514
$760$	−21.7200	−0.787868
$761$	−25.3207	−0.917876	−0.458938	−	0.888468i	$-0.651770\pi$
$761$	−25.3207	−0.917876	−0.458938	+	0.888468i	$0.651770\pi$
$762$	−12.8825	−0.466685
$763$	−21.8893	−0.792448
$764$	−2.67316	−0.0967115
$765$	6.66956	0.241138
$766$	43.0607	1.55585
$767$	−15.6358	−0.564575
$768$	−20.4144	−0.736641
$769$	−7.39528	−0.266680	−0.133340	−	0.991070i	$-0.542570\pi$
$769$	−7.39528	−0.266680	−0.133340	+	0.991070i	$0.542570\pi$
$770$	9.92934	0.357828
$771$	−8.62646	−0.310674
$772$	−1.33486	−0.0480427
$773$	20.5549	0.739308	0.369654	−	0.929169i	$-0.379476\pi$
$773$	20.5549	0.739308	0.369654	+	0.929169i	$0.379476\pi$
$774$	6.43085	0.231152
$775$	−11.5444	−0.414689
$776$	−67.2829	−2.41532
$777$	7.57347	0.271697
$778$	4.48421	0.160767
$779$	−1.39261	−0.0498954
$780$	10.5222	0.376755
$781$	−7.64523	−0.273568
$782$	15.8674	0.567417
$783$	−20.9117	−0.747322
$784$	−20.5987	−0.735669
$785$	−17.5693	−0.627075
$786$	−15.5177	−0.553499
$787$	−16.0451	−0.571945	−0.285972	−	0.958238i	$-0.592317\pi$
$787$	−16.0451	−0.571945	−0.285972	+	0.958238i	$0.592317\pi$
$788$	−3.56070	−0.126845
$789$	4.40811	0.156933
$790$	14.5558	0.517872
$791$	−25.4136	−0.903604
$792$	13.4409	0.477603
$793$	14.8710	0.528083
$794$	37.3516	1.32556
$795$	−15.3768	−0.545359
$796$	−72.9028	−2.58397
$797$	−43.1414	−1.52815	−0.764074	−	0.645129i	$-0.776804\pi$
$797$	−43.1414	−1.52815	−0.764074	+	0.645129i	$0.776804\pi$
$798$	4.00713	0.141851
$799$	−6.73853	−0.238392
$800$	−1.40974	−0.0498419
$801$	21.6102	0.763558
$802$	0.755379	0.0266733
$803$	−16.2964	−0.575087
$804$	−19.4885	−0.687307
$805$	−25.8704	−0.911812
$806$	29.4182	1.03621
$807$	−3.09444	−0.108930
$808$	−92.9380	−3.26955
$809$	−26.9704	−0.948230	−0.474115	−	0.880463i	$-0.657232\pi$
$809$	−26.9704	−0.948230	−0.474115	+	0.880463i	$0.657232\pi$
$810$	−35.4232	−1.24465
$811$	−2.87523	−0.100963	−0.0504814	−	0.998725i	$-0.516076\pi$
$811$	−2.87523	−0.100963	−0.0504814	+	0.998725i	$0.516076\pi$
$812$	−38.2065	−1.34079
$813$	2.41803	0.0848041
$814$	18.9379	0.663772
$815$	25.9108	0.907615
$816$	−2.85548	−0.0999619
$817$	1.64531	0.0575620
$818$	−42.9871	−1.50301
$819$	6.55925	0.229199
$820$	−8.86040	−0.309419
$821$	−22.7685	−0.794626	−0.397313	−	0.917683i	$-0.630057\pi$
$821$	−22.7685	−0.794626	−0.397313	+	0.917683i	$0.630057\pi$
$822$	19.5483	0.681825
$823$	−5.21487	−0.181779	−0.0908895	−	0.995861i	$-0.528971\pi$
$823$	−5.21487	−0.181779	−0.0908895	+	0.995861i	$0.528971\pi$
$824$	−40.0011	−1.39351
$825$	0.973428	0.0338904
$826$	−37.8853	−1.31820
$827$	−30.4563	−1.05907	−0.529535	−	0.848288i	$-0.677634\pi$
$827$	−30.4563	−1.05907	−0.529535	+	0.848288i	$0.677634\pi$
$828$	−68.5303	−2.38159
$829$	18.7541	0.651357	0.325678	−	0.945481i	$-0.394407\pi$
$829$	18.7541	0.651357	0.325678	+	0.945481i	$0.394407\pi$
$830$	62.1186	2.15617
$831$	11.5590	0.400978
$832$	−10.9734	−0.380433
$833$	4.52863	0.156908
$834$	12.1490	0.420684
$835$	−35.8090	−1.23922
$836$	6.72943	0.232742
$837$	26.2015	0.905657
$838$	57.3645	1.98162
$839$	−10.9681	−0.378661	−0.189331	−	0.981913i	$-0.560632\pi$
$839$	−10.9681	−0.378661	−0.189331	+	0.981913i	$0.560632\pi$
$840$	13.0283	0.449519
$841$	6.30820	0.217524
$842$	−89.7621	−3.09340
$843$	4.08191	0.140588
$844$	53.2856	1.83417
$845$	−26.7109	−0.918884
$846$	43.3344	1.48987
$847$	−1.57206	−0.0540166
$848$	−43.5305	−1.49484
$849$	14.2137	0.487813
$850$	3.82656	0.131250
$851$	−49.3417	−1.69141
$852$	−19.6303	−0.672524
$853$	1.39484	0.0477585	0.0238793	−	0.999715i	$-0.492398\pi$
$853$	1.39484	0.0477585	0.0238793	+	0.999715i	$0.492398\pi$
$854$	36.0321	1.23299
$855$	−10.9735	−0.375285
$856$	36.3038	1.24084
$857$	−53.2054	−1.81746	−0.908731	−	0.417382i	$-0.862948\pi$
$857$	−53.2054	−1.81746	−0.908731	+	0.417382i	$0.862948\pi$
$858$	−2.48054	−0.0846843
$859$	−43.5309	−1.48526	−0.742628	−	0.669704i	$-0.766421\pi$
$859$	−43.5309	−1.48526	−0.742628	+	0.669704i	$0.766421\pi$
$860$	10.4682	0.356962
$861$	0.835327	0.0284679
$862$	2.54814	0.0867898
$863$	−20.2841	−0.690478	−0.345239	−	0.938515i	$-0.612202\pi$
$863$	−20.2841	−0.690478	−0.345239	+	0.938515i	$0.612202\pi$
$864$	3.19958	0.108852
$865$	43.3945	1.47546
$866$	−15.6388	−0.531430
$867$	0.627778	0.0213205
$868$	47.8713	1.62486
$869$	−2.30454	−0.0781761
$870$	−23.5611	−0.798796
$871$	−12.1526	−0.411776
$872$	−71.8186	−2.43209
$873$	−33.9930	−1.15049
$874$	−26.1068	−0.883074
$875$	13.8788	0.469191
$876$	−41.8435	−1.41376
$877$	−2.88291	−0.0973490	−0.0486745	−	0.998815i	$-0.515500\pi$
$877$	−2.88291	−0.0973490	−0.0486745	+	0.998815i	$0.515500\pi$
$878$	46.2265	1.56007
$879$	−7.40707	−0.249834
$880$	11.6416	0.392439
$881$	43.2319	1.45652	0.728259	−	0.685302i	$-0.240330\pi$
$881$	43.2319	1.45652	0.728259	+	0.685302i	$0.240330\pi$
$882$	−29.1229	−0.980621
$883$	19.1411	0.644150	0.322075	−	0.946714i	$-0.395620\pi$
$883$	19.1411	0.644150	0.322075	+	0.946714i	$0.395620\pi$
$884$	−6.54878	−0.220259
$885$	−15.6905	−0.527431
$886$	27.9437	0.938785
$887$	34.8388	1.16977	0.584886	−	0.811116i	$-0.301139\pi$
$887$	34.8388	1.16977	0.584886	+	0.811116i	$0.301139\pi$
$888$	24.8484	0.833859
$889$	−13.0723	−0.438431
$890$	52.3785	1.75573
$891$	5.60837	0.187888
$892$	34.8608	1.16723
$893$	11.0870	0.371011
$894$	35.4953	1.18714
$895$	−6.90824	−0.230917
$896$	−29.4468	−0.983749
$897$	6.46293	0.215791
$898$	25.4693	0.849920
$899$	−44.2398	−1.47548
$900$	−16.5267	−0.550890
$901$	9.57017	0.318829
$902$	2.08878	0.0695488
$903$	−0.986904	−0.0328421
$904$	−83.3817	−2.77323
$905$	2.91761	0.0969846
$906$	17.8505	0.593042
$907$	−12.0233	−0.399228	−0.199614	−	0.979875i	$-0.563969\pi$
$907$	−12.0233	−0.399228	−0.199614	+	0.979875i	$0.563969\pi$
$908$	−74.7980	−2.48226
$909$	−46.9545	−1.55738
$910$	15.8982	0.527022
$911$	−32.5946	−1.07991	−0.539954	−	0.841695i	$-0.681558\pi$
$911$	−32.5946	−1.07991	−0.539954	+	0.841695i	$0.681558\pi$
$912$	4.69815	0.155571
$913$	−9.83490	−0.325488
$914$	80.2790	2.65539
$915$	14.9230	0.493339
$916$	−35.9565	−1.18804
$917$	−15.7463	−0.519990
$918$	−8.68485	−0.286643
$919$	25.6771	0.847008	0.423504	−	0.905894i	$-0.360800\pi$
$919$	25.6771	0.847008	0.423504	+	0.905894i	$0.360800\pi$
$920$	−84.8804	−2.79843
$921$	−17.3850	−0.572855
$922$	64.1644	2.11314
$923$	−12.2411	−0.402920
$924$	−4.03651	−0.132791
$925$	−11.8992	−0.391243
$926$	−67.5610	−2.22019
$927$	−20.2095	−0.663768
$928$	−5.40231	−0.177339
$929$	18.1024	0.593919	0.296960	−	0.954890i	$-0.404027\pi$
$929$	18.1024	0.593919	0.296960	+	0.954890i	$0.404027\pi$
$930$	29.5211	0.968036
$931$	−7.45100	−0.244197
$932$	−97.7285	−3.20120
$933$	−4.03202	−0.132003
$934$	12.2969	0.402366
$935$	−2.55941	−0.0837017
$936$	21.5208	0.703430
$937$	−0.598406	−0.0195491	−0.00977453	−	0.999952i	$-0.503111\pi$
$937$	−0.598406	−0.0195491	−0.00977453	+	0.999952i	$0.503111\pi$
$938$	−29.4457	−0.961435
$939$	−1.46262	−0.0477307
$940$	70.5401	2.30077
$941$	9.15848	0.298558	0.149279	−	0.988795i	$-0.452305\pi$
$941$	9.15848	0.298558	0.149279	+	0.988795i	$0.452305\pi$
$942$	10.6348	0.346502
$943$	−5.44222	−0.177223
$944$	−44.4185	−1.44570
$945$	14.1599	0.460621
$946$	−2.46781	−0.0802353
$947$	53.0361	1.72344	0.861721	−	0.507383i	$-0.169387\pi$
$947$	53.0361	1.72344	0.861721	+	0.507383i	$0.169387\pi$
$948$	−5.91727	−0.192184
$949$	−26.0928	−0.847007
$950$	−6.29588	−0.204265
$951$	10.7977	0.350139
$952$	−8.10852	−0.262799
$953$	29.8478	0.966866	0.483433	−	0.875381i	$-0.339390\pi$
$953$	29.8478	0.966866	0.483433	+	0.875381i	$0.339390\pi$
$954$	−61.5443	−1.99257
$955$	−1.67276	−0.0541293
$956$	38.2928	1.23848
$957$	3.73030	0.120584
$958$	53.6889	1.73461
$959$	19.8363	0.640547
$960$	−11.0118	−0.355403
$961$	24.4308	0.788089
$962$	30.3222	0.977626
$963$	18.3416	0.591048
$964$	−81.6598	−2.63008
$965$	−0.835305	−0.0268894
$966$	15.6596	0.503839
$967$	15.5423	0.499806	0.249903	−	0.968271i	$-0.419601\pi$
$967$	15.5423	0.499806	0.249903	+	0.968271i	$0.419601\pi$
$968$	−5.15790	−0.165781
$969$	−1.03289	−0.0331811
$970$	−82.3917	−2.64544
$971$	11.8525	0.380364	0.190182	−	0.981749i	$-0.439092\pi$
$971$	11.8525	0.380364	0.190182	+	0.981749i	$0.439092\pi$
$972$	57.5825	1.84696
$973$	12.3279	0.395216
$974$	75.4487	2.41753
$975$	1.55859	0.0499149
$976$	42.2458	1.35225
$977$	28.7149	0.918672	0.459336	−	0.888263i	$-0.348087\pi$
$977$	28.7149	0.918672	0.459336	+	0.888263i	$0.348087\pi$
$978$	−15.6840	−0.501520
$979$	−8.29280	−0.265039
$980$	−47.4066	−1.51435
$981$	−36.2845	−1.15847
$982$	10.0308	0.320097
$983$	56.3425	1.79705	0.898523	−	0.438926i	$-0.144641\pi$
$983$	56.3425	1.79705	0.898523	+	0.438926i	$0.144641\pi$
$984$	2.74070	0.0873702
$985$	−2.22815	−0.0709947
$986$	14.6639	0.466993
$987$	−6.65027	−0.211681
$988$	10.7748	0.342791
$989$	6.42975	0.204454
$990$	16.4592	0.523107
$991$	−4.94251	−0.157004	−0.0785019	−	0.996914i	$-0.525014\pi$
$991$	−4.94251	−0.157004	−0.0785019	+	0.996914i	$0.525014\pi$
$992$	6.76888	0.214912
$993$	−17.0329	−0.540522
$994$	−29.6600	−0.940757
$995$	−45.6198	−1.44624
$996$	−25.2527	−0.800161
$997$	25.5386	0.808817	0.404409	−	0.914578i	$-0.367477\pi$
$997$	25.5386	0.808817	0.404409	+	0.914578i	$0.367477\pi$
$998$	92.4923	2.92779
$999$	27.0067	0.854453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.d.1.4	✓	62

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.d.1.4	✓	62	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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46781 q^{2} +0.627778 q^{3} +4.09007 q^{4} +2.55941 q^{5} -1.54924 q^{6} -1.57206 q^{7} -5.15790 q^{8} -2.60589 q^{9} +O(q^{10})\)	\(q-2.46781 q^{2} +0.627778 q^{3} +4.09007 q^{4} +2.55941 q^{5} -1.54924 q^{6} -1.57206 q^{7} -5.15790 q^{8} -2.60589 q^{9} -6.31614 q^{10} +1.00000 q^{11} +2.56766 q^{12} +1.60114 q^{13} +3.87954 q^{14} +1.60674 q^{15} +4.54856 q^{16} -1.00000 q^{17} +6.43085 q^{18} +1.64531 q^{19} +10.4682 q^{20} -0.986904 q^{21} -2.46781 q^{22} +6.42975 q^{23} -3.23802 q^{24} +1.55059 q^{25} -3.95130 q^{26} -3.51926 q^{27} -6.42983 q^{28} +5.94207 q^{29} -3.96513 q^{30} -7.44518 q^{31} -0.909163 q^{32} +0.627778 q^{33} +2.46781 q^{34} -4.02355 q^{35} -10.6583 q^{36} -7.67397 q^{37} -4.06030 q^{38} +1.00516 q^{39} -13.2012 q^{40} -0.846412 q^{41} +2.43549 q^{42} +1.00000 q^{43} +4.09007 q^{44} -6.66956 q^{45} -15.8674 q^{46} +6.73853 q^{47} +2.85548 q^{48} -4.52863 q^{49} -3.82656 q^{50} -0.627778 q^{51} +6.54878 q^{52} -9.57017 q^{53} +8.68485 q^{54} +2.55941 q^{55} +8.10852 q^{56} +1.03289 q^{57} -14.6639 q^{58} -9.76541 q^{59} +6.57170 q^{60} +9.28774 q^{61} +18.3733 q^{62} +4.09662 q^{63} -6.85348 q^{64} +4.09797 q^{65} -1.54924 q^{66} -7.59000 q^{67} -4.09007 q^{68} +4.03646 q^{69} +9.92934 q^{70} -7.64523 q^{71} +13.4409 q^{72} -16.2964 q^{73} +18.9379 q^{74} +0.973428 q^{75} +6.72943 q^{76} -1.57206 q^{77} -2.48054 q^{78} -2.30454 q^{79} +11.6416 q^{80} +5.60837 q^{81} +2.08878 q^{82} -9.83490 q^{83} -4.03651 q^{84} -2.55941 q^{85} -2.46781 q^{86} +3.73030 q^{87} -5.15790 q^{88} -8.29280 q^{89} +16.4592 q^{90} -2.51708 q^{91} +26.2982 q^{92} -4.67392 q^{93} -16.6294 q^{94} +4.21102 q^{95} -0.570752 q^{96} +13.0446 q^{97} +11.1758 q^{98} -2.60589 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 62 q - 7 q^{2} - 8 q^{3} + 49 q^{4} - 13 q^{5} - 2 q^{6} - 11 q^{7} - 9 q^{8} + 40 q^{9}+O(q^{10}) \) 62 * q - 7 * q^2 - 8 * q^3 + 49 * q^4 - 13 * q^5 - 2 * q^6 - 11 * q^7 - 9 * q^8 + 40 * q^9	\( 62 q - 7 q^{2} - 8 q^{3} + 49 q^{4} - 13 q^{5} - 2 q^{6} - 11 q^{7} - 9 q^{8} + 40 q^{9} - 7 q^{10} + 62 q^{11} - 17 q^{12} - 31 q^{14} - 20 q^{15} + 27 q^{16} - 62 q^{17} + 3 q^{18} - 29 q^{20} - 18 q^{21} - 7 q^{22} - 50 q^{23} - 31 q^{24} + 35 q^{25} - 32 q^{26} - 14 q^{27} - 13 q^{28} - 26 q^{29} - 10 q^{30} - 58 q^{31} - 5 q^{32} - 8 q^{33} + 7 q^{34} - 32 q^{35} - 29 q^{36} - 41 q^{37} - 10 q^{38} - 53 q^{39} - 31 q^{40} - 55 q^{41} - 7 q^{42} + 62 q^{43} + 49 q^{44} - 34 q^{45} - 39 q^{46} - 31 q^{47} - 30 q^{48} + 35 q^{49} - 40 q^{50} + 8 q^{51} + 13 q^{52} - 74 q^{53} + 48 q^{54} - 13 q^{55} - 75 q^{56} - 43 q^{57} - 46 q^{58} - 65 q^{59} - 8 q^{60} - 14 q^{61} - 29 q^{62} - 23 q^{63} - 15 q^{64} - 9 q^{65} - 2 q^{66} - q^{67} - 49 q^{68} - 59 q^{69} - 31 q^{70} - 141 q^{71} + 9 q^{72} - 4 q^{73} - 94 q^{74} - 43 q^{75} + 34 q^{76} - 11 q^{77} - 11 q^{78} - 63 q^{79} - 41 q^{80} - 30 q^{81} + 38 q^{82} - 44 q^{83} - 16 q^{84} + 13 q^{85} - 7 q^{86} - 8 q^{87} - 9 q^{88} - 58 q^{89} - 55 q^{90} - 78 q^{91} - 104 q^{92} - 5 q^{94} - 99 q^{95} - 148 q^{96} - 26 q^{97} + 16 q^{98} + 40 q^{99}+O(q^{100}) \) 62 * q - 7 * q^2 - 8 * q^3 + 49 * q^4 - 13 * q^5 - 2 * q^6 - 11 * q^7 - 9 * q^8 + 40 * q^9 - 7 * q^10 + 62 * q^11 - 17 * q^12 - 31 * q^14 - 20 * q^15 + 27 * q^16 - 62 * q^17 + 3 * q^18 - 29 * q^20 - 18 * q^21 - 7 * q^22 - 50 * q^23 - 31 * q^24 + 35 * q^25 - 32 * q^26 - 14 * q^27 - 13 * q^28 - 26 * q^29 - 10 * q^30 - 58 * q^31 - 5 * q^32 - 8 * q^33 + 7 * q^34 - 32 * q^35 - 29 * q^36 - 41 * q^37 - 10 * q^38 - 53 * q^39 - 31 * q^40 - 55 * q^41 - 7 * q^42 + 62 * q^43 + 49 * q^44 - 34 * q^45 - 39 * q^46 - 31 * q^47 - 30 * q^48 + 35 * q^49 - 40 * q^50 + 8 * q^51 + 13 * q^52 - 74 * q^53 + 48 * q^54 - 13 * q^55 - 75 * q^56 - 43 * q^57 - 46 * q^58 - 65 * q^59 - 8 * q^60 - 14 * q^61 - 29 * q^62 - 23 * q^63 - 15 * q^64 - 9 * q^65 - 2 * q^66 - q^67 - 49 * q^68 - 59 * q^69 - 31 * q^70 - 141 * q^71 + 9 * q^72 - 4 * q^73 - 94 * q^74 - 43 * q^75 + 34 * q^76 - 11 * q^77 - 11 * q^78 - 63 * q^79 - 41 * q^80 - 30 * q^81 + 38 * q^82 - 44 * q^83 - 16 * q^84 + 13 * q^85 - 7 * q^86 - 8 * q^87 - 9 * q^88 - 58 * q^89 - 55 * q^90 - 78 * q^91 - 104 * q^92 - 5 * q^94 - 99 * q^95 - 148 * q^96 - 26 * q^97 + 16 * q^98 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more