LMFDB - Embedded newform 6015.2.a.b.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6015,2,Mod(1,6015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6015, 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("6015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6015 = 3 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6015.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:	$48.0300168158$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	6015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.13202 q^{2} +1.00000 q^{3} -0.718528 q^{4} +1.00000 q^{5} -1.13202 q^{6} -1.31252 q^{7} +3.07743 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.13202 q^{2} +1.00000 q^{3} -0.718528 q^{4} +1.00000 q^{5} -1.13202 q^{6} -1.31252 q^{7} +3.07743 q^{8} +1.00000 q^{9} -1.13202 q^{10} -1.41913 q^{11} -0.718528 q^{12} -5.18667 q^{13} +1.48580 q^{14} +1.00000 q^{15} -2.04666 q^{16} -4.33294 q^{17} -1.13202 q^{18} +3.03198 q^{19} -0.718528 q^{20} -1.31252 q^{21} +1.60649 q^{22} +8.40076 q^{23} +3.07743 q^{24} +1.00000 q^{25} +5.87142 q^{26} +1.00000 q^{27} +0.943081 q^{28} -5.89019 q^{29} -1.13202 q^{30} +1.81212 q^{31} -3.83800 q^{32} -1.41913 q^{33} +4.90498 q^{34} -1.31252 q^{35} -0.718528 q^{36} +10.8240 q^{37} -3.43227 q^{38} -5.18667 q^{39} +3.07743 q^{40} +9.22953 q^{41} +1.48580 q^{42} -3.23445 q^{43} +1.01969 q^{44} +1.00000 q^{45} -9.50984 q^{46} -10.2544 q^{47} -2.04666 q^{48} -5.27730 q^{49} -1.13202 q^{50} -4.33294 q^{51} +3.72677 q^{52} +4.93614 q^{53} -1.13202 q^{54} -1.41913 q^{55} -4.03918 q^{56} +3.03198 q^{57} +6.66782 q^{58} +0.00794662 q^{59} -0.718528 q^{60} +3.00642 q^{61} -2.05136 q^{62} -1.31252 q^{63} +8.43802 q^{64} -5.18667 q^{65} +1.60649 q^{66} +11.6866 q^{67} +3.11334 q^{68} +8.40076 q^{69} +1.48580 q^{70} -0.963271 q^{71} +3.07743 q^{72} +2.55437 q^{73} -12.2529 q^{74} +1.00000 q^{75} -2.17857 q^{76} +1.86263 q^{77} +5.87142 q^{78} -11.2283 q^{79} -2.04666 q^{80} +1.00000 q^{81} -10.4480 q^{82} -5.33874 q^{83} +0.943081 q^{84} -4.33294 q^{85} +3.66146 q^{86} -5.89019 q^{87} -4.36728 q^{88} -8.80509 q^{89} -1.13202 q^{90} +6.80760 q^{91} -6.03618 q^{92} +1.81212 q^{93} +11.6082 q^{94} +3.03198 q^{95} -3.83800 q^{96} -14.3021 q^{97} +5.97401 q^{98} -1.41913 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 5 q^{2} + 23 q^{3} + 9 q^{4} + 23 q^{5} - 5 q^{6} - 16 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) $ 23 * q - 5 * q^2 + 23 * q^3 + 9 * q^4 + 23 * q^5 - 5 * q^6 - 16 * q^7 - 12 * q^8 + 23 * q^9	\( 23 q - 5 q^{2} + 23 q^{3} + 9 q^{4} + 23 q^{5} - 5 q^{6} - 16 q^{7} - 12 q^{8} + 23 q^{9} - 5 q^{10} - 13 q^{11} + 9 q^{12} - 18 q^{13} - 6 q^{14} + 23 q^{15} - 11 q^{16} - 34 q^{17} - 5 q^{18} - 35 q^{19} + 9 q^{20} - 16 q^{21} - 11 q^{22} - 14 q^{23} - 12 q^{24} + 23 q^{25} - 6 q^{26} + 23 q^{27} - 26 q^{28} - 43 q^{29} - 5 q^{30} - 21 q^{31} - 14 q^{32} - 13 q^{33} - 12 q^{34} - 16 q^{35} + 9 q^{36} - 18 q^{37} + 6 q^{38} - 18 q^{39} - 12 q^{40} - 45 q^{41} - 6 q^{42} - 43 q^{43} - 11 q^{44} + 23 q^{45} - 29 q^{46} - 14 q^{47} - 11 q^{48} - 25 q^{49} - 5 q^{50} - 34 q^{51} - 20 q^{52} - 3 q^{53} - 5 q^{54} - 13 q^{55} + 3 q^{56} - 35 q^{57} + 10 q^{58} - 9 q^{59} + 9 q^{60} - 67 q^{61} - 7 q^{62} - 16 q^{63} - 8 q^{64} - 18 q^{65} - 11 q^{66} - 32 q^{67} - 24 q^{68} - 14 q^{69} - 6 q^{70} - 8 q^{71} - 12 q^{72} - 39 q^{73} - 16 q^{74} + 23 q^{75} - 48 q^{76} - 26 q^{77} - 6 q^{78} - 59 q^{79} - 11 q^{80} + 23 q^{81} - q^{82} - 23 q^{83} - 26 q^{84} - 34 q^{85} - 7 q^{86} - 43 q^{87} + 17 q^{88} - 51 q^{89} - 5 q^{90} - 37 q^{91} + 11 q^{92} - 21 q^{93} + 8 q^{94} - 35 q^{95} - 14 q^{96} - 29 q^{97} + 32 q^{98} - 13 q^{99}+O(q^{100}) \) 23 * q - 5 * q^2 + 23 * q^3 + 9 * q^4 + 23 * q^5 - 5 * q^6 - 16 * q^7 - 12 * q^8 + 23 * q^9 - 5 * q^10 - 13 * q^11 + 9 * q^12 - 18 * q^13 - 6 * q^14 + 23 * q^15 - 11 * q^16 - 34 * q^17 - 5 * q^18 - 35 * q^19 + 9 * q^20 - 16 * q^21 - 11 * q^22 - 14 * q^23 - 12 * q^24 + 23 * q^25 - 6 * q^26 + 23 * q^27 - 26 * q^28 - 43 * q^29 - 5 * q^30 - 21 * q^31 - 14 * q^32 - 13 * q^33 - 12 * q^34 - 16 * q^35 + 9 * q^36 - 18 * q^37 + 6 * q^38 - 18 * q^39 - 12 * q^40 - 45 * q^41 - 6 * q^42 - 43 * q^43 - 11 * q^44 + 23 * q^45 - 29 * q^46 - 14 * q^47 - 11 * q^48 - 25 * q^49 - 5 * q^50 - 34 * q^51 - 20 * q^52 - 3 * q^53 - 5 * q^54 - 13 * q^55 + 3 * q^56 - 35 * q^57 + 10 * q^58 - 9 * q^59 + 9 * q^60 - 67 * q^61 - 7 * q^62 - 16 * q^63 - 8 * q^64 - 18 * q^65 - 11 * q^66 - 32 * q^67 - 24 * q^68 - 14 * q^69 - 6 * q^70 - 8 * q^71 - 12 * q^72 - 39 * q^73 - 16 * q^74 + 23 * q^75 - 48 * q^76 - 26 * q^77 - 6 * q^78 - 59 * q^79 - 11 * q^80 + 23 * q^81 - q^82 - 23 * q^83 - 26 * q^84 - 34 * q^85 - 7 * q^86 - 43 * q^87 + 17 * q^88 - 51 * q^89 - 5 * q^90 - 37 * q^91 + 11 * q^92 - 21 * q^93 + 8 * q^94 - 35 * q^95 - 14 * q^96 - 29 * q^97 + 32 * q^98 - 13 * 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 $

$2$	−1.13202	−0.800460	−0.400230	−	0.916415i	$-0.631070\pi$
$2$	−1.13202	−0.800460	−0.400230	+	0.916415i	$0.631070\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.718528	−0.359264
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.13202	−0.462146
$7$	−1.31252	−0.496085	−0.248042	−	0.968749i	$-0.579787\pi$
$7$	−1.31252	−0.496085	−0.248042	+	0.968749i	$0.579787\pi$
$8$	3.07743	1.08804
$9$	1.00000	0.333333
$10$	−1.13202	−0.357977
$11$	−1.41913	−0.427884	−0.213942	−	0.976846i	$-0.568630\pi$
$11$	−1.41913	−0.427884	−0.213942	+	0.976846i	$0.568630\pi$
$12$	−0.718528	−0.207421
$13$	−5.18667	−1.43852	−0.719262	−	0.694739i	$-0.755520\pi$
$13$	−5.18667	−1.43852	−0.719262	+	0.694739i	$0.755520\pi$
$14$	1.48580	0.397096
$15$	1.00000	0.258199
$16$	−2.04666	−0.511665
$17$	−4.33294	−1.05089	−0.525446	−	0.850827i	$-0.676101\pi$
$17$	−4.33294	−1.05089	−0.525446	+	0.850827i	$0.676101\pi$
$18$	−1.13202	−0.266820
$19$	3.03198	0.695585	0.347792	−	0.937572i	$-0.386931\pi$
$19$	3.03198	0.695585	0.347792	+	0.937572i	$0.386931\pi$
$20$	−0.718528	−0.160668
$21$	−1.31252	−0.286415
$22$	1.60649	0.342504
$23$	8.40076	1.75168	0.875840	−	0.482602i	$-0.160308\pi$
$23$	8.40076	1.75168	0.875840	+	0.482602i	$0.160308\pi$
$24$	3.07743	0.628178
$25$	1.00000	0.200000
$26$	5.87142	1.15148
$27$	1.00000	0.192450
$28$	0.943081	0.178226
$29$	−5.89019	−1.09378	−0.546891	−	0.837204i	$-0.684189\pi$
$29$	−5.89019	−1.09378	−0.546891	+	0.837204i	$0.684189\pi$
$30$	−1.13202	−0.206678
$31$	1.81212	0.325467	0.162733	−	0.986670i	$-0.447969\pi$
$31$	1.81212	0.325467	0.162733	+	0.986670i	$0.447969\pi$
$32$	−3.83800	−0.678469
$33$	−1.41913	−0.247039
$34$	4.90498	0.841196
$35$	−1.31252	−0.221856
$36$	−0.718528	−0.119755
$37$	10.8240	1.77945	0.889723	−	0.456500i	$-0.150897\pi$
$37$	10.8240	1.77945	0.889723	+	0.456500i	$0.150897\pi$
$38$	−3.43227	−0.556788
$39$	−5.18667	−0.830532
$40$	3.07743	0.486585
$41$	9.22953	1.44141	0.720705	−	0.693242i	$-0.243818\pi$
$41$	9.22953	1.44141	0.720705	+	0.693242i	$0.243818\pi$
$42$	1.48580	0.229264
$43$	−3.23445	−0.493248	−0.246624	−	0.969111i	$-0.579321\pi$
$43$	−3.23445	−0.493248	−0.246624	+	0.969111i	$0.579321\pi$
$44$	1.01969	0.153723
$45$	1.00000	0.149071
$46$	−9.50984	−1.40215
$47$	−10.2544	−1.49576	−0.747882	−	0.663832i	$-0.768929\pi$
$47$	−10.2544	−1.49576	−0.747882	+	0.663832i	$0.768929\pi$
$48$	−2.04666	−0.295410
$49$	−5.27730	−0.753900
$50$	−1.13202	−0.160092
$51$	−4.33294	−0.606733
$52$	3.72677	0.516810
$53$	4.93614	0.678031	0.339015	−	0.940781i	$-0.389906\pi$
$53$	4.93614	0.678031	0.339015	+	0.940781i	$0.389906\pi$
$54$	−1.13202	−0.154049
$55$	−1.41913	−0.191355
$56$	−4.03918	−0.539758
$57$	3.03198	0.401596
$58$	6.66782	0.875528
$59$	0.00794662	0.00103456	0.000517281	−	1.00000i	$-0.499835\pi$
$59$	0.00794662	0.00103456	0.000517281		1.00000i	$0.499835\pi$
$60$	−0.718528	−0.0927616
$61$	3.00642	0.384933	0.192467	−	0.981304i	$-0.438351\pi$
$61$	3.00642	0.384933	0.192467	+	0.981304i	$0.438351\pi$
$62$	−2.05136	−0.260523
$63$	−1.31252	−0.165362
$64$	8.43802	1.05475
$65$	−5.18667	−0.643327
$66$	1.60649	0.197745
$67$	11.6866	1.42775	0.713875	−	0.700273i	$-0.246938\pi$
$67$	11.6866	1.42775	0.713875	+	0.700273i	$0.246938\pi$
$68$	3.11334	0.377548
$69$	8.40076	1.01133
$70$	1.48580	0.177587
$71$	−0.963271	−0.114319	−0.0571596	−	0.998365i	$-0.518204\pi$
$71$	−0.963271	−0.114319	−0.0571596	+	0.998365i	$0.518204\pi$
$72$	3.07743	0.362679
$73$	2.55437	0.298967	0.149483	−	0.988764i	$-0.452239\pi$
$73$	2.55437	0.298967	0.149483	+	0.988764i	$0.452239\pi$
$74$	−12.2529	−1.42438
$75$	1.00000	0.115470
$76$	−2.17857	−0.249899
$77$	1.86263	0.212267
$78$	5.87142	0.664808
$79$	−11.2283	−1.26329	−0.631644	−	0.775259i	$-0.717620\pi$
$79$	−11.2283	−1.26329	−0.631644	+	0.775259i	$0.717620\pi$
$80$	−2.04666	−0.228824
$81$	1.00000	0.111111
$82$	−10.4480	−1.15379
$83$	−5.33874	−0.586003	−0.293001	−	0.956112i	$-0.594654\pi$
$83$	−5.33874	−0.586003	−0.293001	+	0.956112i	$0.594654\pi$
$84$	0.943081	0.102899
$85$	−4.33294	−0.469973
$86$	3.66146	0.394826
$87$	−5.89019	−0.631495
$88$	−4.36728	−0.465553
$89$	−8.80509	−0.933338	−0.466669	−	0.884432i	$-0.654546\pi$
$89$	−8.80509	−0.933338	−0.466669	+	0.884432i	$0.654546\pi$
$90$	−1.13202	−0.119326
$91$	6.80760	0.713630
$92$	−6.03618	−0.629315
$93$	1.81212	0.187908
$94$	11.6082	1.19730
$95$	3.03198	0.311075
$96$	−3.83800	−0.391714
$97$	−14.3021	−1.45216	−0.726079	−	0.687612i	$-0.758659\pi$
$97$	−14.3021	−1.45216	−0.726079	+	0.687612i	$0.758659\pi$
$98$	5.97401	0.603466
$99$	−1.41913	−0.142628
$100$	−0.718528	−0.0718528
$101$	−12.0390	−1.19793	−0.598965	−	0.800775i	$-0.704421\pi$
$101$	−12.0390	−1.19793	−0.598965	+	0.800775i	$0.704421\pi$
$102$	4.90498	0.485665
$103$	0.181013	0.0178357	0.00891787	−	0.999960i	$-0.497161\pi$
$103$	0.181013	0.0178357	0.00891787	+	0.999960i	$0.497161\pi$
$104$	−15.9616	−1.56517
$105$	−1.31252	−0.128089
$106$	−5.58781	−0.542736
$107$	−15.6355	−1.51154	−0.755772	−	0.654835i	$-0.772738\pi$
$107$	−15.6355	−1.51154	−0.755772	+	0.654835i	$0.772738\pi$
$108$	−0.718528	−0.0691404
$109$	1.83395	0.175661	0.0878303	−	0.996135i	$-0.472007\pi$
$109$	1.83395	0.175661	0.0878303	+	0.996135i	$0.472007\pi$
$110$	1.60649	0.153172
$111$	10.8240	1.02736
$112$	2.68628	0.253829
$113$	15.9593	1.50133	0.750664	−	0.660685i	$-0.229734\pi$
$113$	15.9593	1.50133	0.750664	+	0.660685i	$0.229734\pi$
$114$	−3.43227	−0.321461
$115$	8.40076	0.783375
$116$	4.23227	0.392956
$117$	−5.18667	−0.479508
$118$	−0.00899574	−0.000828125 0
$119$	5.68706	0.521332
$120$	3.07743	0.280930
$121$	−8.98607	−0.816915
$122$	−3.40334	−0.308124
$123$	9.22953	0.832199
$124$	−1.30206	−0.116928
$125$	1.00000	0.0894427
$126$	1.48580	0.132365
$127$	−2.39967	−0.212936	−0.106468	−	0.994316i	$-0.533954\pi$
$127$	−2.39967	−0.212936	−0.106468	+	0.994316i	$0.533954\pi$
$128$	−1.87602	−0.165818
$129$	−3.23445	−0.284777
$130$	5.87142	0.514958
$131$	−7.24950	−0.633392	−0.316696	−	0.948527i	$-0.602573\pi$
$131$	−7.24950	−0.633392	−0.316696	+	0.948527i	$0.602573\pi$
$132$	1.01969	0.0887522
$133$	−3.97953	−0.345069
$134$	−13.2295	−1.14286
$135$	1.00000	0.0860663
$136$	−13.3343	−1.14341
$137$	−5.94955	−0.508304	−0.254152	−	0.967164i	$-0.581796\pi$
$137$	−5.94955	−0.508304	−0.254152	+	0.967164i	$0.581796\pi$
$138$	−9.50984	−0.809531
$139$	1.70772	0.144847	0.0724234	−	0.997374i	$-0.476927\pi$
$139$	1.70772	0.144847	0.0724234	+	0.997374i	$0.476927\pi$
$140$	0.943081	0.0797049
$141$	−10.2544	−0.863580
$142$	1.09044	0.0915079
$143$	7.36056	0.615521
$144$	−2.04666	−0.170555
$145$	−5.89019	−0.489154
$146$	−2.89160	−0.239311
$147$	−5.27730	−0.435264
$148$	−7.77731	−0.639291
$149$	−2.31878	−0.189962	−0.0949810	−	0.995479i	$-0.530279\pi$
$149$	−2.31878	−0.189962	−0.0949810	+	0.995479i	$0.530279\pi$
$150$	−1.13202	−0.0924291
$151$	−15.7524	−1.28191	−0.640955	−	0.767578i	$-0.721462\pi$
$151$	−15.7524	−1.28191	−0.640955	+	0.767578i	$0.721462\pi$
$152$	9.33072	0.756821
$153$	−4.33294	−0.350297
$154$	−2.10854	−0.169911
$155$	1.81212	0.145553
$156$	3.72677	0.298380
$157$	−13.6892	−1.09251	−0.546257	−	0.837618i	$-0.683948\pi$
$157$	−13.6892	−1.09251	−0.546257	+	0.837618i	$0.683948\pi$
$158$	12.7107	1.01121
$159$	4.93614	0.391461
$160$	−3.83800	−0.303420
$161$	−11.0261	−0.868982
$162$	−1.13202	−0.0889400
$163$	−14.7624	−1.15628	−0.578141	−	0.815937i	$-0.696222\pi$
$163$	−14.7624	−1.15628	−0.578141	+	0.815937i	$0.696222\pi$
$164$	−6.63168	−0.517847
$165$	−1.41913	−0.110479
$166$	6.04357	0.469072
$167$	−2.06932	−0.160129	−0.0800643	−	0.996790i	$-0.525513\pi$
$167$	−2.06932	−0.160129	−0.0800643	+	0.996790i	$0.525513\pi$
$168$	−4.03918	−0.311630
$169$	13.9016	1.06935
$170$	4.90498	0.376195
$171$	3.03198	0.231862
$172$	2.32404	0.177206
$173$	5.34517	0.406386	0.203193	−	0.979139i	$-0.434868\pi$
$173$	5.34517	0.406386	0.203193	+	0.979139i	$0.434868\pi$
$174$	6.66782	0.505486
$175$	−1.31252	−0.0992170
$176$	2.90448	0.218933
$177$	0.00794662	0.000597305 0
$178$	9.96755	0.747100
$179$	3.98827	0.298097	0.149049	−	0.988830i	$-0.452379\pi$
$179$	3.98827	0.298097	0.149049	+	0.988830i	$0.452379\pi$
$180$	−0.718528	−0.0535559
$181$	8.87340	0.659555	0.329777	−	0.944059i	$-0.393026\pi$
$181$	8.87340	0.659555	0.329777	+	0.944059i	$0.393026\pi$
$182$	−7.70634	−0.571232
$183$	3.00642	0.222241
$184$	25.8528	1.90589
$185$	10.8240	0.795793
$186$	−2.05136	−0.150413
$187$	6.14900	0.449660
$188$	7.36810	0.537374
$189$	−1.31252	−0.0954716
$190$	−3.43227	−0.249003
$191$	1.03488	0.0748811	0.0374406	−	0.999299i	$-0.488080\pi$
$191$	1.03488	0.0748811	0.0374406	+	0.999299i	$0.488080\pi$
$192$	8.43802	0.608961
$193$	10.8843	0.783467	0.391733	−	0.920079i	$-0.371876\pi$
$193$	10.8843	0.783467	0.391733	+	0.920079i	$0.371876\pi$
$194$	16.1903	1.16239
$195$	−5.18667	−0.371425
$196$	3.79189	0.270849
$197$	−13.6277	−0.970936	−0.485468	−	0.874254i	$-0.661351\pi$
$197$	−13.6277	−0.970936	−0.485468	+	0.874254i	$0.661351\pi$
$198$	1.60649	0.114168
$199$	−19.1675	−1.35875	−0.679374	−	0.733793i	$-0.737748\pi$
$199$	−19.1675	−1.35875	−0.679374	+	0.733793i	$0.737748\pi$
$200$	3.07743	0.217607
$201$	11.6866	0.824312
$202$	13.6285	0.958895
$203$	7.73098	0.542608
$204$	3.11334	0.217977
$205$	9.22953	0.644618
$206$	−0.204910	−0.0142768
$207$	8.40076	0.583893
$208$	10.6154	0.736043
$209$	−4.30278	−0.297629
$210$	1.48580	0.102530
$211$	6.64052	0.457152	0.228576	−	0.973526i	$-0.426593\pi$
$211$	6.64052	0.457152	0.228576	+	0.973526i	$0.426593\pi$
$212$	−3.54675	−0.243592
$213$	−0.963271	−0.0660022
$214$	17.6998	1.20993
$215$	−3.23445	−0.220587
$216$	3.07743	0.209393
$217$	−2.37844	−0.161459
$218$	−2.07607	−0.140609
$219$	2.55437	0.172609
$220$	1.01969	0.0687472
$221$	22.4735	1.51173
$222$	−12.2529	−0.822364
$223$	26.3520	1.76466	0.882330	−	0.470631i	$-0.155974\pi$
$223$	26.3520	1.76466	0.882330	+	0.470631i	$0.155974\pi$
$224$	5.03744	0.336578
$225$	1.00000	0.0666667
$226$	−18.0663	−1.20175
$227$	−1.94414	−0.129037	−0.0645186	−	0.997917i	$-0.520551\pi$
$227$	−1.94414	−0.129037	−0.0645186	+	0.997917i	$0.520551\pi$
$228$	−2.17857	−0.144279
$229$	−17.2987	−1.14313	−0.571567	−	0.820556i	$-0.693664\pi$
$229$	−17.2987	−1.14313	−0.571567	+	0.820556i	$0.693664\pi$
$230$	−9.50984	−0.627060
$231$	1.86263	0.122552
$232$	−18.1267	−1.19007
$233$	3.34893	0.219396	0.109698	−	0.993965i	$-0.465012\pi$
$233$	3.34893	0.219396	0.109698	+	0.993965i	$0.465012\pi$
$234$	5.87142	0.383827
$235$	−10.2544	−0.668926
$236$	−0.00570987	−0.000371681 0
$237$	−11.2283	−0.729360
$238$	−6.43787	−0.417305
$239$	3.82090	0.247153	0.123577	−	0.992335i	$-0.460564\pi$
$239$	3.82090	0.247153	0.123577	+	0.992335i	$0.460564\pi$
$240$	−2.04666	−0.132111
$241$	−7.31028	−0.470896	−0.235448	−	0.971887i	$-0.575656\pi$
$241$	−7.31028	−0.470896	−0.235448	+	0.971887i	$0.575656\pi$
$242$	10.1724	0.653908
$243$	1.00000	0.0641500
$244$	−2.16020	−0.138293
$245$	−5.27730	−0.337154
$246$	−10.4480	−0.666142
$247$	−15.7259	−1.00062
$248$	5.57668	0.354120
$249$	−5.33874	−0.338329
$250$	−1.13202	−0.0715953
$251$	−4.45056	−0.280917	−0.140458	−	0.990087i	$-0.544858\pi$
$251$	−4.45056	−0.280917	−0.140458	+	0.990087i	$0.544858\pi$
$252$	0.943081	0.0594085
$253$	−11.9218	−0.749515
$254$	2.71647	0.170447
$255$	−4.33294	−0.271339
$256$	−14.7523	−0.922022
$257$	20.8451	1.30028	0.650142	−	0.759813i	$-0.274710\pi$
$257$	20.8451	1.30028	0.650142	+	0.759813i	$0.274710\pi$
$258$	3.66146	0.227953
$259$	−14.2066	−0.882757
$260$	3.72677	0.231124
$261$	−5.89019	−0.364594
$262$	8.20659	0.507005
$263$	−4.96179	−0.305957	−0.152979	−	0.988229i	$-0.548887\pi$
$263$	−4.96179	−0.305957	−0.152979	+	0.988229i	$0.548887\pi$
$264$	−4.36728	−0.268787
$265$	4.93614	0.303225
$266$	4.50491	0.276214
$267$	−8.80509	−0.538863
$268$	−8.39718	−0.512939
$269$	2.75209	0.167798	0.0838991	−	0.996474i	$-0.473263\pi$
$269$	2.75209	0.167798	0.0838991	+	0.996474i	$0.473263\pi$
$270$	−1.13202	−0.0688926
$271$	−14.1301	−0.858341	−0.429170	−	0.903224i	$-0.641194\pi$
$271$	−14.1301	−0.858341	−0.429170	+	0.903224i	$0.641194\pi$
$272$	8.86805	0.537705
$273$	6.80760	0.412014
$274$	6.73501	0.406877
$275$	−1.41913	−0.0855768
$276$	−6.03618	−0.363335
$277$	24.3032	1.46024	0.730118	−	0.683321i	$-0.239465\pi$
$277$	24.3032	1.46024	0.730118	+	0.683321i	$0.239465\pi$
$278$	−1.93317	−0.115944
$279$	1.81212	0.108489
$280$	−4.03918	−0.241387
$281$	16.0169	0.955488	0.477744	−	0.878499i	$-0.341455\pi$
$281$	16.0169	0.955488	0.477744	+	0.878499i	$0.341455\pi$
$282$	11.6082	0.691261
$283$	−21.9405	−1.30423	−0.652114	−	0.758121i	$-0.726118\pi$
$283$	−21.9405	−1.30423	−0.652114	+	0.758121i	$0.726118\pi$
$284$	0.692137	0.0410708
$285$	3.03198	0.179599
$286$	−8.33231	−0.492700
$287$	−12.1139	−0.715062
$288$	−3.83800	−0.226156
$289$	1.77434	0.104373
$290$	6.66782	0.391548
$291$	−14.3021	−0.838404
$292$	−1.83539	−0.107408
$293$	−9.85009	−0.575448	−0.287724	−	0.957713i	$-0.592899\pi$
$293$	−9.85009	−0.575448	−0.287724	+	0.957713i	$0.592899\pi$
$294$	5.97401	0.348411
$295$	0.00794662	0.000462670 0
$296$	33.3100	1.93610
$297$	−1.41913	−0.0823463
$298$	2.62491	0.152057
$299$	−43.5720	−2.51983
$300$	−0.718528	−0.0414842
$301$	4.24527	0.244693
$302$	17.8320	1.02612
$303$	−12.0390	−0.691625
$304$	−6.20544	−0.355907
$305$	3.00642	0.172147
$306$	4.90498	0.280399
$307$	−22.9685	−1.31088	−0.655441	−	0.755247i	$-0.727517\pi$
$307$	−22.9685	−1.31088	−0.655441	+	0.755247i	$0.727517\pi$
$308$	−1.33835	−0.0762598
$309$	0.181013	0.0102975
$310$	−2.05136	−0.116509
$311$	26.9286	1.52698	0.763491	−	0.645818i	$-0.223484\pi$
$311$	26.9286	1.52698	0.763491	+	0.645818i	$0.223484\pi$
$312$	−15.9616	−0.903649
$313$	−31.8804	−1.80199	−0.900993	−	0.433834i	$-0.857160\pi$
$313$	−31.8804	−1.80199	−0.900993	+	0.433834i	$0.857160\pi$
$314$	15.4964	0.874513
$315$	−1.31252	−0.0739520
$316$	8.06788	0.453854
$317$	−31.0403	−1.74340	−0.871699	−	0.490042i	$-0.836981\pi$
$317$	−31.0403	−1.74340	−0.871699	+	0.490042i	$0.836981\pi$
$318$	−5.58781	−0.313349
$319$	8.35895	0.468011
$320$	8.43802	0.471700
$321$	−15.6355	−0.872690
$322$	12.4818	0.695585
$323$	−13.1374	−0.730984
$324$	−0.718528	−0.0399182
$325$	−5.18667	−0.287705
$326$	16.7114	0.925557
$327$	1.83395	0.101418
$328$	28.4033	1.56831
$329$	13.4591	0.742026
$330$	1.60649	0.0884341
$331$	−19.4092	−1.06683	−0.533414	−	0.845854i	$-0.679091\pi$
$331$	−19.4092	−1.06683	−0.533414	+	0.845854i	$0.679091\pi$
$332$	3.83603	0.210530
$333$	10.8240	0.593149
$334$	2.34251	0.128177
$335$	11.6866	0.638509
$336$	2.68628	0.146548
$337$	−8.15176	−0.444055	−0.222027	−	0.975040i	$-0.571267\pi$
$337$	−8.15176	−0.444055	−0.222027	+	0.975040i	$0.571267\pi$
$338$	−15.7369	−0.855972
$339$	15.9593	0.866792
$340$	3.11334	0.168844
$341$	−2.57164	−0.139262
$342$	−3.43227	−0.185596
$343$	16.1142	0.870083
$344$	−9.95379	−0.536672
$345$	8.40076	0.452282
$346$	−6.05085	−0.325296
$347$	−15.6284	−0.838977	−0.419489	−	0.907761i	$-0.637791\pi$
$347$	−15.6284	−0.838977	−0.419489	+	0.907761i	$0.637791\pi$
$348$	4.23227	0.226873
$349$	10.5394	0.564163	0.282081	−	0.959390i	$-0.408975\pi$
$349$	10.5394	0.564163	0.282081	+	0.959390i	$0.408975\pi$
$350$	1.48580	0.0794192
$351$	−5.18667	−0.276844
$352$	5.44662	0.290306
$353$	−3.08782	−0.164348	−0.0821742	−	0.996618i	$-0.526186\pi$
$353$	−3.08782	−0.164348	−0.0821742	+	0.996618i	$0.526186\pi$
$354$	−0.00899574	−0.000478118 0
$355$	−0.963271	−0.0511251
$356$	6.32671	0.335315
$357$	5.68706	0.300991
$358$	−4.51480	−0.238615
$359$	9.93406	0.524299	0.262150	−	0.965027i	$-0.415569\pi$
$359$	9.93406	0.524299	0.262150	+	0.965027i	$0.415569\pi$
$360$	3.07743	0.162195
$361$	−9.80708	−0.516162
$362$	−10.0449	−0.527947
$363$	−8.98607	−0.471646
$364$	−4.89145	−0.256382
$365$	2.55437	0.133702
$366$	−3.40334	−0.177895
$367$	3.53906	0.184738	0.0923688	−	0.995725i	$-0.470556\pi$
$367$	3.53906	0.184738	0.0923688	+	0.995725i	$0.470556\pi$
$368$	−17.1935	−0.896273
$369$	9.22953	0.480470
$370$	−12.2529	−0.637000
$371$	−6.47877	−0.336361
$372$	−1.30206	−0.0675087
$373$	3.25658	0.168619	0.0843096	−	0.996440i	$-0.473132\pi$
$373$	3.25658	0.168619	0.0843096	+	0.996440i	$0.473132\pi$
$374$	−6.96080	−0.359934
$375$	1.00000	0.0516398
$376$	−31.5573	−1.62745
$377$	30.5505	1.57343
$378$	1.48580	0.0764212
$379$	−24.2404	−1.24515	−0.622573	−	0.782562i	$-0.713913\pi$
$379$	−24.2404	−1.24515	−0.622573	+	0.782562i	$0.713913\pi$
$380$	−2.17857	−0.111758
$381$	−2.39967	−0.122939
$382$	−1.17150	−0.0599393
$383$	−5.55069	−0.283627	−0.141814	−	0.989893i	$-0.545293\pi$
$383$	−5.55069	−0.283627	−0.141814	+	0.989893i	$0.545293\pi$
$384$	−1.87602	−0.0957351
$385$	1.86263	0.0949286
$386$	−12.3212	−0.627133
$387$	−3.23445	−0.164416
$388$	10.2765	0.521708
$389$	20.2372	1.02607	0.513034	−	0.858369i	$-0.328522\pi$
$389$	20.2372	1.02607	0.513034	+	0.858369i	$0.328522\pi$
$390$	5.87142	0.297311
$391$	−36.4000	−1.84083
$392$	−16.2405	−0.820270
$393$	−7.24950	−0.365689
$394$	15.4269	0.777196
$395$	−11.2283	−0.564960
$396$	1.01969	0.0512411
$397$	−12.5721	−0.630975	−0.315488	−	0.948930i	$-0.602168\pi$
$397$	−12.5721	−0.630975	−0.315488	+	0.948930i	$0.602168\pi$
$398$	21.6980	1.08762
$399$	−3.97953	−0.199226
$400$	−2.04666	−0.102333
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	−13.2295	−0.659828
$403$	−9.39888	−0.468191
$404$	8.65039	0.430373
$405$	1.00000	0.0496904
$406$	−8.75163	−0.434336
$407$	−15.3606	−0.761396
$408$	−13.3343	−0.660147
$409$	−36.1167	−1.78586	−0.892928	−	0.450200i	$-0.851353\pi$
$409$	−36.1167	−1.78586	−0.892928	+	0.450200i	$0.851353\pi$
$410$	−10.4480	−0.515991
$411$	−5.94955	−0.293469
$412$	−0.130063	−0.00640774
$413$	−0.0104301	−0.000513231 0
$414$	−9.50984	−0.467383
$415$	−5.33874	−0.262068
$416$	19.9064	0.975993
$417$	1.70772	0.0836273
$418$	4.87084	0.238240
$419$	38.7135	1.89128	0.945638	−	0.325220i	$-0.105438\pi$
$419$	38.7135	1.89128	0.945638	+	0.325220i	$0.105438\pi$
$420$	0.943081	0.0460176
$421$	8.63276	0.420735	0.210367	−	0.977622i	$-0.432534\pi$
$421$	8.63276	0.420735	0.210367	+	0.977622i	$0.432534\pi$
$422$	−7.51721	−0.365932
$423$	−10.2544	−0.498588
$424$	15.1906	0.737722
$425$	−4.33294	−0.210178
$426$	1.09044	0.0528321
$427$	−3.94599	−0.190960
$428$	11.2346	0.543043
$429$	7.36056	0.355371
$430$	3.66146	0.176571
$431$	−26.2260	−1.26326	−0.631631	−	0.775269i	$-0.717614\pi$
$431$	−26.2260	−1.26326	−0.631631	+	0.775269i	$0.717614\pi$
$432$	−2.04666	−0.0984700
$433$	6.83877	0.328650	0.164325	−	0.986406i	$-0.447455\pi$
$433$	6.83877	0.328650	0.164325	+	0.986406i	$0.447455\pi$
$434$	2.69245	0.129242
$435$	−5.89019	−0.282413
$436$	−1.31775	−0.0631086
$437$	25.4710	1.21844
$438$	−2.89160	−0.138166
$439$	0.689773	0.0329211	0.0164605	−	0.999865i	$-0.494760\pi$
$439$	0.689773	0.0329211	0.0164605	+	0.999865i	$0.494760\pi$
$440$	−4.36728	−0.208202
$441$	−5.27730	−0.251300
$442$	−25.4405	−1.21008
$443$	16.8073	0.798539	0.399270	−	0.916834i	$-0.369264\pi$
$443$	16.8073	0.798539	0.399270	+	0.916834i	$0.369264\pi$
$444$	−7.77731	−0.369095
$445$	−8.80509	−0.417401
$446$	−29.8310	−1.41254
$447$	−2.31878	−0.109675
$448$	−11.0750	−0.523247
$449$	27.7244	1.30839	0.654197	−	0.756324i	$-0.273007\pi$
$449$	27.7244	1.30839	0.654197	+	0.756324i	$0.273007\pi$
$450$	−1.13202	−0.0533640
$451$	−13.0979	−0.616756
$452$	−11.4672	−0.539373
$453$	−15.7524	−0.740112
$454$	2.20081	0.103289
$455$	6.80760	0.319145
$456$	9.33072	0.436951
$457$	9.30411	0.435228	0.217614	−	0.976035i	$-0.430173\pi$
$457$	9.30411	0.435228	0.217614	+	0.976035i	$0.430173\pi$
$458$	19.5825	0.915032
$459$	−4.33294	−0.202244
$460$	−6.03618	−0.281438
$461$	−20.5875	−0.958856	−0.479428	−	0.877581i	$-0.659156\pi$
$461$	−20.5875	−0.958856	−0.479428	+	0.877581i	$0.659156\pi$
$462$	−2.10854	−0.0980982
$463$	−26.5199	−1.23248	−0.616242	−	0.787557i	$-0.711346\pi$
$463$	−26.5199	−1.23248	−0.616242	+	0.787557i	$0.711346\pi$
$464$	12.0552	0.559650
$465$	1.81212	0.0840351
$466$	−3.79106	−0.175617
$467$	1.75506	0.0812144	0.0406072	−	0.999175i	$-0.487071\pi$
$467$	1.75506	0.0812144	0.0406072	+	0.999175i	$0.487071\pi$
$468$	3.72677	0.172270
$469$	−15.3389	−0.708285
$470$	11.6082	0.535448
$471$	−13.6892	−0.630763
$472$	0.0244552	0.00112564
$473$	4.59010	0.211053
$474$	12.7107	0.583823
$475$	3.03198	0.139117
$476$	−4.08631	−0.187296
$477$	4.93614	0.226010
$478$	−4.32534	−0.197836
$479$	5.25313	0.240022	0.120011	−	0.992773i	$-0.461707\pi$
$479$	5.25313	0.240022	0.120011	+	0.992773i	$0.461707\pi$
$480$	−3.83800	−0.175180
$481$	−56.1403	−2.55978
$482$	8.27539	0.376934
$483$	−11.0261	−0.501707
$484$	6.45674	0.293488
$485$	−14.3021	−0.649425
$486$	−1.13202	−0.0513495
$487$	10.5098	0.476243	0.238121	−	0.971235i	$-0.423468\pi$
$487$	10.5098	0.476243	0.238121	+	0.971235i	$0.423468\pi$
$488$	9.25207	0.418821
$489$	−14.7624	−0.667580
$490$	5.97401	0.269878
$491$	−44.1318	−1.99164	−0.995820	−	0.0913324i	$-0.970887\pi$
$491$	−44.1318	−1.99164	−0.995820	+	0.0913324i	$0.970887\pi$
$492$	−6.63168	−0.298979
$493$	25.5218	1.14945
$494$	17.8021	0.800952
$495$	−1.41913	−0.0637852
$496$	−3.70880	−0.166530
$497$	1.26431	0.0567120
$498$	6.04357	0.270819
$499$	−38.8733	−1.74021	−0.870103	−	0.492870i	$-0.835948\pi$
$499$	−38.8733	−1.74021	−0.870103	+	0.492870i	$0.835948\pi$
$500$	−0.718528	−0.0321336
$501$	−2.06932	−0.0924503
$502$	5.03813	0.224863
$503$	−18.0719	−0.805785	−0.402893	−	0.915247i	$-0.631995\pi$
$503$	−18.0719	−0.805785	−0.402893	+	0.915247i	$0.631995\pi$
$504$	−4.03918	−0.179919
$505$	−12.0390	−0.535730
$506$	13.4957	0.599957
$507$	13.9016	0.617390
$508$	1.72423	0.0765002
$509$	−9.99911	−0.443203	−0.221601	−	0.975137i	$-0.571128\pi$
$509$	−9.99911	−0.443203	−0.221601	+	0.975137i	$0.571128\pi$
$510$	4.90498	0.217196
$511$	−3.35266	−0.148313
$512$	20.4520	0.903859
$513$	3.03198	0.133865
$514$	−23.5971	−1.04082
$515$	0.181013	0.00797638
$516$	2.32404	0.102310
$517$	14.5524	0.640013
$518$	16.0822	0.706611
$519$	5.34517	0.234627
$520$	−15.9616	−0.699964
$521$	−8.09742	−0.354754	−0.177377	−	0.984143i	$-0.556761\pi$
$521$	−8.09742	−0.354754	−0.177377	+	0.984143i	$0.556761\pi$
$522$	6.66782	0.291843
$523$	36.0706	1.57726	0.788628	−	0.614870i	$-0.210792\pi$
$523$	36.0706	1.57726	0.788628	+	0.614870i	$0.210792\pi$
$524$	5.20897	0.227555
$525$	−1.31252	−0.0572830
$526$	5.61686	0.244907
$527$	−7.85181	−0.342030
$528$	2.90448	0.126401
$529$	47.5727	2.06838
$530$	−5.58781	−0.242719
$531$	0.00794662	0.000344854 0
$532$	2.85941	0.123971
$533$	−47.8705	−2.07350
$534$	9.96755	0.431338
$535$	−15.6355	−0.675983
$536$	35.9648	1.55344
$537$	3.98827	0.172107
$538$	−3.11543	−0.134316
$539$	7.48917	0.322582
$540$	−0.718528	−0.0309205
$541$	−34.6255	−1.48867	−0.744333	−	0.667808i	$-0.767233\pi$
$541$	−34.6255	−1.48867	−0.744333	+	0.667808i	$0.767233\pi$
$542$	15.9955	0.687067
$543$	8.87340	0.380794
$544$	16.6298	0.712997
$545$	1.83395	0.0785578
$546$	−7.70634	−0.329801
$547$	24.1981	1.03463	0.517317	−	0.855794i	$-0.326931\pi$
$547$	24.1981	1.03463	0.517317	+	0.855794i	$0.326931\pi$
$548$	4.27492	0.182615
$549$	3.00642	0.128311
$550$	1.60649	0.0685008
$551$	−17.8590	−0.760818
$552$	25.8528	1.10037
$553$	14.7374	0.626698
$554$	−27.5117	−1.16886
$555$	10.8240	0.459451
$556$	−1.22704	−0.0520382
$557$	19.3817	0.821228	0.410614	−	0.911809i	$-0.365314\pi$
$557$	19.3817	0.821228	0.410614	+	0.911809i	$0.365314\pi$
$558$	−2.05136	−0.0868410
$559$	16.7760	0.709550
$560$	2.68628	0.113516
$561$	6.14900	0.259611
$562$	−18.1315	−0.764830
$563$	−17.4606	−0.735878	−0.367939	−	0.929850i	$-0.619936\pi$
$563$	−17.4606	−0.735878	−0.367939	+	0.929850i	$0.619936\pi$
$564$	7.36810	0.310253
$565$	15.9593	0.671414
$566$	24.8371	1.04398
$567$	−1.31252	−0.0551206
$568$	−2.96440	−0.124383
$569$	19.4319	0.814627	0.407314	−	0.913288i	$-0.366466\pi$
$569$	19.4319	0.814627	0.407314	+	0.913288i	$0.366466\pi$
$570$	−3.43227	−0.143762
$571$	29.1286	1.21899	0.609496	−	0.792789i	$-0.291372\pi$
$571$	29.1286	1.21899	0.609496	+	0.792789i	$0.291372\pi$
$572$	−5.28877	−0.221135
$573$	1.03488	0.0432326
$574$	13.7132	0.572379
$575$	8.40076	0.350336
$576$	8.43802	0.351584
$577$	−5.16182	−0.214889	−0.107445	−	0.994211i	$-0.534267\pi$
$577$	−5.16182	−0.214889	−0.107445	+	0.994211i	$0.534267\pi$
$578$	−2.00859	−0.0835465
$579$	10.8843	0.452335
$580$	4.23227	0.175735
$581$	7.00719	0.290707
$582$	16.1903	0.671108
$583$	−7.00502	−0.290118
$584$	7.86091	0.325287
$585$	−5.18667	−0.214442
$586$	11.1505	0.460623
$587$	18.7235	0.772803	0.386401	−	0.922331i	$-0.373718\pi$
$587$	18.7235	0.772803	0.386401	+	0.922331i	$0.373718\pi$
$588$	3.79189	0.156375
$589$	5.49432	0.226390
$590$	−0.00899574	−0.000370349 0
$591$	−13.6277	−0.560570
$592$	−22.1530	−0.910481
$593$	−35.5475	−1.45976	−0.729881	−	0.683574i	$-0.760425\pi$
$593$	−35.5475	−1.45976	−0.729881	+	0.683574i	$0.760425\pi$
$594$	1.60649	0.0659149
$595$	5.68706	0.233147
$596$	1.66611	0.0682465
$597$	−19.1675	−0.784473
$598$	49.3244	2.01702
$599$	−7.07008	−0.288876	−0.144438	−	0.989514i	$-0.546137\pi$
$599$	−7.07008	−0.288876	−0.144438	+	0.989514i	$0.546137\pi$
$600$	3.07743	0.125636
$601$	−15.4562	−0.630473	−0.315236	−	0.949013i	$-0.602084\pi$
$601$	−15.4562	−0.630473	−0.315236	+	0.949013i	$0.602084\pi$
$602$	−4.80573	−0.195867
$603$	11.6866	0.475917
$604$	11.3185	0.460545
$605$	−8.98607	−0.365336
$606$	13.6285	0.553618
$607$	−21.2895	−0.864115	−0.432057	−	0.901846i	$-0.642212\pi$
$607$	−21.2895	−0.864115	−0.432057	+	0.901846i	$0.642212\pi$
$608$	−11.6368	−0.471933
$609$	7.73098	0.313275
$610$	−3.40334	−0.137797
$611$	53.1864	2.15169
$612$	3.11334	0.125849
$613$	11.8020	0.476677	0.238339	−	0.971182i	$-0.423397\pi$
$613$	11.8020	0.476677	0.238339	+	0.971182i	$0.423397\pi$
$614$	26.0008	1.04931
$615$	9.22953	0.372171
$616$	5.73213	0.230954
$617$	27.3762	1.10213	0.551063	−	0.834464i	$-0.314222\pi$
$617$	27.3762	1.10213	0.551063	+	0.834464i	$0.314222\pi$
$618$	−0.204910	−0.00824271
$619$	−34.3742	−1.38162	−0.690808	−	0.723038i	$-0.742745\pi$
$619$	−34.3742	−1.38162	−0.690808	+	0.723038i	$0.742745\pi$
$620$	−1.30206	−0.0522920
$621$	8.40076	0.337111
$622$	−30.4838	−1.22229
$623$	11.5568	0.463015
$624$	10.6154	0.424954
$625$	1.00000	0.0400000
$626$	36.0893	1.44242
$627$	−4.30278	−0.171836
$628$	9.83604	0.392501
$629$	−46.8995	−1.87001
$630$	1.48580	0.0591956
$631$	−40.1489	−1.59830	−0.799151	−	0.601130i	$-0.794717\pi$
$631$	−40.1489	−1.59830	−0.799151	+	0.601130i	$0.794717\pi$
$632$	−34.5545	−1.37450
$633$	6.64052	0.263937
$634$	35.1383	1.39552
$635$	−2.39967	−0.0952278
$636$	−3.54675	−0.140638
$637$	27.3716	1.08450
$638$	−9.46251	−0.374624
$639$	−0.963271	−0.0381064
$640$	−1.87602	−0.0741561
$641$	2.43086	0.0960133	0.0480066	−	0.998847i	$-0.484713\pi$
$641$	2.43086	0.0960133	0.0480066	+	0.998847i	$0.484713\pi$
$642$	17.6998	0.698554
$643$	22.1850	0.874892	0.437446	−	0.899245i	$-0.355883\pi$
$643$	22.1850	0.874892	0.437446	+	0.899245i	$0.355883\pi$
$644$	7.92259	0.312194
$645$	−3.23445	−0.127356
$646$	14.8718	0.585123
$647$	7.99065	0.314145	0.157072	−	0.987587i	$-0.449794\pi$
$647$	7.99065	0.314145	0.157072	+	0.987587i	$0.449794\pi$
$648$	3.07743	0.120893
$649$	−0.0112773	−0.000442672 0
$650$	5.87142	0.230296
$651$	−2.37844	−0.0932185
$652$	10.6072	0.415411
$653$	−45.0620	−1.76341	−0.881706	−	0.471798i	$-0.843605\pi$
$653$	−45.0620	−1.76341	−0.881706	+	0.471798i	$0.843605\pi$
$654$	−2.07607	−0.0811808
$655$	−7.24950	−0.283261
$656$	−18.8897	−0.737520
$657$	2.55437	0.0996556
$658$	−15.2360	−0.593962
$659$	26.7750	1.04300	0.521502	−	0.853250i	$-0.325372\pi$
$659$	26.7750	1.04300	0.521502	+	0.853250i	$0.325372\pi$
$660$	1.01969	0.0396912
$661$	0.529639	0.0206006	0.0103003	−	0.999947i	$-0.496721\pi$
$661$	0.529639	0.0206006	0.0103003	+	0.999947i	$0.496721\pi$
$662$	21.9717	0.853953
$663$	22.4735	0.872799
$664$	−16.4296	−0.637592
$665$	−3.97953	−0.154320
$666$	−12.2529	−0.474792
$667$	−49.4821	−1.91595
$668$	1.48686	0.0575285
$669$	26.3520	1.01883
$670$	−13.2295	−0.511101
$671$	−4.26651	−0.164707
$672$	5.03744	0.194324
$673$	−3.19610	−0.123201	−0.0616003	−	0.998101i	$-0.519620\pi$
$673$	−3.19610	−0.123201	−0.0616003	+	0.998101i	$0.519620\pi$
$674$	9.22796	0.355448
$675$	1.00000	0.0384900
$676$	−9.98866	−0.384179
$677$	33.1775	1.27512	0.637558	−	0.770403i	$-0.279945\pi$
$677$	33.1775	1.27512	0.637558	+	0.770403i	$0.279945\pi$
$678$	−18.0663	−0.693832
$679$	18.7717	0.720394
$680$	−13.3343	−0.511348
$681$	−1.94414	−0.0744997
$682$	2.91115	0.111474
$683$	39.1472	1.49793	0.748964	−	0.662611i	$-0.230552\pi$
$683$	39.1472	1.49793	0.748964	+	0.662611i	$0.230552\pi$
$684$	−2.17857	−0.0832995
$685$	−5.94955	−0.227320
$686$	−18.2416	−0.696467
$687$	−17.2987	−0.659988
$688$	6.61982	0.252378
$689$	−25.6021	−0.975363
$690$	−9.50984	−0.362033
$691$	−16.9362	−0.644284	−0.322142	−	0.946691i	$-0.604403\pi$
$691$	−16.9362	−0.644284	−0.322142	+	0.946691i	$0.604403\pi$
$692$	−3.84066	−0.146000
$693$	1.86263	0.0707556
$694$	17.6917	0.671567
$695$	1.70772	0.0647774
$696$	−18.1267	−0.687089
$697$	−39.9910	−1.51477
$698$	−11.9309	−0.451590
$699$	3.34893	0.126668
$700$	0.943081	0.0356451
$701$	−47.6673	−1.80037	−0.900185	−	0.435508i	$-0.856569\pi$
$701$	−47.6673	−1.80037	−0.900185	+	0.435508i	$0.856569\pi$
$702$	5.87142	0.221603
$703$	32.8180	1.23776
$704$	−11.9746	−0.451311
$705$	−10.2544	−0.386205
$706$	3.49548	0.131554
$707$	15.8015	0.594275
$708$	−0.00570987	−0.000214590 0
$709$	−34.5494	−1.29753	−0.648765	−	0.760989i	$-0.724714\pi$
$709$	−34.5494	−1.29753	−0.648765	+	0.760989i	$0.724714\pi$
$710$	1.09044	0.0409236
$711$	−11.2283	−0.421096
$712$	−27.0971	−1.01551
$713$	15.2232	0.570113
$714$	−6.43787	−0.240931
$715$	7.36056	0.275269
$716$	−2.86568	−0.107096
$717$	3.82090	0.142694
$718$	−11.2456	−0.419681
$719$	35.7464	1.33312	0.666558	−	0.745453i	$-0.267767\pi$
$719$	35.7464	1.33312	0.666558	+	0.745453i	$0.267767\pi$
$720$	−2.04666	−0.0762745
$721$	−0.237583	−0.00884804
$722$	11.1018	0.413167
$723$	−7.31028	−0.271872
$724$	−6.37579	−0.236954
$725$	−5.89019	−0.218756
$726$	10.1724	0.377534
$727$	−42.3457	−1.57051	−0.785257	−	0.619169i	$-0.787469\pi$
$727$	−42.3457	−1.57051	−0.785257	+	0.619169i	$0.787469\pi$
$728$	20.9499	0.776455
$729$	1.00000	0.0370370
$730$	−2.89160	−0.107023
$731$	14.0147	0.518351
$732$	−2.16020	−0.0798433
$733$	−7.76897	−0.286953	−0.143477	−	0.989654i	$-0.545828\pi$
$733$	−7.76897	−0.286953	−0.143477	+	0.989654i	$0.545828\pi$
$734$	−4.00629	−0.147875
$735$	−5.27730	−0.194656
$736$	−32.2421	−1.18846
$737$	−16.5849	−0.610911
$738$	−10.4480	−0.384597
$739$	−36.4612	−1.34125	−0.670623	−	0.741799i	$-0.733973\pi$
$739$	−36.4612	−1.34125	−0.670623	+	0.741799i	$0.733973\pi$
$740$	−7.77731	−0.285900
$741$	−15.7259	−0.577705
$742$	7.33410	0.269243
$743$	4.96968	0.182320	0.0911599	−	0.995836i	$-0.470943\pi$
$743$	4.96968	0.182320	0.0911599	+	0.995836i	$0.470943\pi$
$744$	5.57668	0.204451
$745$	−2.31878	−0.0849536
$746$	−3.68651	−0.134973
$747$	−5.33874	−0.195334
$748$	−4.41823	−0.161547
$749$	20.5219	0.749854
$750$	−1.13202	−0.0413356
$751$	−4.97211	−0.181435	−0.0907174	−	0.995877i	$-0.528916\pi$
$751$	−4.97211	−0.181435	−0.0907174	+	0.995877i	$0.528916\pi$
$752$	20.9874	0.765330
$753$	−4.45056	−0.162187
$754$	−34.5838	−1.25947
$755$	−15.7524	−0.573288
$756$	0.943081	0.0342995
$757$	12.1797	0.442681	0.221340	−	0.975197i	$-0.428957\pi$
$757$	12.1797	0.442681	0.221340	+	0.975197i	$0.428957\pi$
$758$	27.4406	0.996689
$759$	−11.9218	−0.432733
$760$	9.33072	0.338461
$761$	−12.5028	−0.453227	−0.226614	−	0.973985i	$-0.572765\pi$
$761$	−12.5028	−0.453227	−0.226614	+	0.973985i	$0.572765\pi$
$762$	2.71647	0.0984074
$763$	−2.40709	−0.0871426
$764$	−0.743589	−0.0269021
$765$	−4.33294	−0.156658
$766$	6.28350	0.227032
$767$	−0.0412165	−0.00148824
$768$	−14.7523	−0.532329
$769$	−22.1090	−0.797271	−0.398636	−	0.917109i	$-0.630516\pi$
$769$	−22.1090	−0.797271	−0.398636	+	0.917109i	$0.630516\pi$
$770$	−2.10854	−0.0759865
$771$	20.8451	0.750719
$772$	−7.82065	−0.281471
$773$	33.7642	1.21441	0.607207	−	0.794544i	$-0.292290\pi$
$773$	33.7642	1.21441	0.607207	+	0.794544i	$0.292290\pi$
$774$	3.66146	0.131609
$775$	1.81212	0.0650933
$776$	−44.0137	−1.58000
$777$	−14.2066	−0.509660
$778$	−22.9089	−0.821326
$779$	27.9838	1.00262
$780$	3.72677	0.133440
$781$	1.36701	0.0489153
$782$	41.2055	1.47351
$783$	−5.89019	−0.210498
$784$	10.8008	0.385744
$785$	−13.6892	−0.488587
$786$	8.20659	0.292719
$787$	25.6503	0.914333	0.457166	−	0.889381i	$-0.348864\pi$
$787$	25.6503	0.914333	0.457166	+	0.889381i	$0.348864\pi$
$788$	9.79192	0.348823
$789$	−4.96179	−0.176645
$790$	12.7107	0.452227
$791$	−20.9469	−0.744786
$792$	−4.36728	−0.155184
$793$	−15.5933	−0.553736
$794$	14.2319	0.505070
$795$	4.93614	0.175067
$796$	13.7724	0.488149
$797$	−6.70821	−0.237617	−0.118809	−	0.992917i	$-0.537907\pi$
$797$	−6.70821	−0.237617	−0.118809	+	0.992917i	$0.537907\pi$
$798$	4.50491	0.159472
$799$	44.4318	1.57189
$800$	−3.83800	−0.135694
$801$	−8.80509	−0.311113
$802$	1.13202	0.0399731
$803$	−3.62499	−0.127923
$804$	−8.39718	−0.296146
$805$	−11.0261	−0.388620
$806$	10.6397	0.374768
$807$	2.75209	0.0968783
$808$	−37.0493	−1.30339
$809$	−53.2076	−1.87068	−0.935339	−	0.353753i	$-0.884905\pi$
$809$	−53.2076	−1.87068	−0.935339	+	0.353753i	$0.884905\pi$
$810$	−1.13202	−0.0397752
$811$	26.5097	0.930882	0.465441	−	0.885079i	$-0.345896\pi$
$811$	26.5097	0.930882	0.465441	+	0.885079i	$0.345896\pi$
$812$	−5.55493	−0.194940
$813$	−14.1301	−0.495563
$814$	17.3885	0.609467
$815$	−14.7624	−0.517105
$816$	8.86805	0.310444
$817$	−9.80679	−0.343096
$818$	40.8849	1.42951
$819$	6.80760	0.237877
$820$	−6.63168	−0.231588
$821$	19.0963	0.666464	0.333232	−	0.942845i	$-0.391861\pi$
$821$	19.0963	0.666464	0.333232	+	0.942845i	$0.391861\pi$
$822$	6.73501	0.234911
$823$	38.8395	1.35386	0.676930	−	0.736048i	$-0.263310\pi$
$823$	38.8395	1.35386	0.676930	+	0.736048i	$0.263310\pi$
$824$	0.557055	0.0194059
$825$	−1.41913	−0.0494078
$826$	0.0118071	0.000410820 0
$827$	−28.2915	−0.983793	−0.491897	−	0.870654i	$-0.663696\pi$
$827$	−28.2915	−0.983793	−0.491897	+	0.870654i	$0.663696\pi$
$828$	−6.03618	−0.209772
$829$	−15.3807	−0.534194	−0.267097	−	0.963670i	$-0.586064\pi$
$829$	−15.3807	−0.534194	−0.267097	+	0.963670i	$0.586064\pi$
$830$	6.04357	0.209775
$831$	24.3032	0.843068
$832$	−43.7652	−1.51729
$833$	22.8662	0.792267
$834$	−1.93317	−0.0669403
$835$	−2.06932	−0.0716117
$836$	3.09167	0.106928
$837$	1.81212	0.0626361
$838$	−43.8245	−1.51389
$839$	−13.5540	−0.467935	−0.233967	−	0.972244i	$-0.575171\pi$
$839$	−13.5540	−0.467935	−0.233967	+	0.972244i	$0.575171\pi$
$840$	−4.03918	−0.139365
$841$	5.69437	0.196357
$842$	−9.77246	−0.336781
$843$	16.0169	0.551651
$844$	−4.77140	−0.164238
$845$	13.9016	0.478228
$846$	11.6082	0.399100
$847$	11.7944	0.405259
$848$	−10.1026	−0.346925
$849$	−21.9405	−0.752997
$850$	4.90498	0.168239
$851$	90.9294	3.11702
$852$	0.692137	0.0237122
$853$	33.9775	1.16337	0.581684	−	0.813415i	$-0.302394\pi$
$853$	33.9775	1.16337	0.581684	+	0.813415i	$0.302394\pi$
$854$	4.46694	0.152855
$855$	3.03198	0.103692
$856$	−48.1173	−1.64461
$857$	18.2764	0.624310	0.312155	−	0.950031i	$-0.398949\pi$
$857$	18.2764	0.624310	0.312155	+	0.950031i	$0.398949\pi$
$858$	−8.33231	−0.284460
$859$	21.8146	0.744305	0.372152	−	0.928172i	$-0.378620\pi$
$859$	21.8146	0.744305	0.372152	+	0.928172i	$0.378620\pi$
$860$	2.32404	0.0792491
$861$	−12.1139	−0.412841
$862$	29.6884	1.01119
$863$	5.22907	0.178000	0.0889998	−	0.996032i	$-0.471633\pi$
$863$	5.22907	0.178000	0.0889998	+	0.996032i	$0.471633\pi$
$864$	−3.83800	−0.130571
$865$	5.34517	0.181741
$866$	−7.74163	−0.263071
$867$	1.77434	0.0602599
$868$	1.70898	0.0580065
$869$	15.9345	0.540541
$870$	6.66782	0.226060
$871$	−60.6147	−2.05385
$872$	5.64386	0.191125
$873$	−14.3021	−0.484053
$874$	−28.8337	−0.975313
$875$	−1.31252	−0.0443712
$876$	−1.83539	−0.0620121
$877$	2.14066	0.0722851	0.0361426	−	0.999347i	$-0.488493\pi$
$877$	2.14066	0.0722851	0.0361426	+	0.999347i	$0.488493\pi$
$878$	−0.780838	−0.0263520
$879$	−9.85009	−0.332235
$880$	2.90448	0.0979099
$881$	−31.2235	−1.05195	−0.525974	−	0.850501i	$-0.676299\pi$
$881$	−31.2235	−1.05195	−0.525974	+	0.850501i	$0.676299\pi$
$882$	5.97401	0.201155
$883$	26.5535	0.893596	0.446798	−	0.894635i	$-0.352564\pi$
$883$	26.5535	0.893596	0.446798	+	0.894635i	$0.352564\pi$
$884$	−16.1479	−0.543111
$885$	0.00794662	0.000267123 0
$886$	−19.0262	−0.639199
$887$	10.7176	0.359861	0.179931	−	0.983679i	$-0.442413\pi$
$887$	10.7176	0.359861	0.179931	+	0.983679i	$0.442413\pi$
$888$	33.3100	1.11781
$889$	3.14960	0.105634
$890$	9.96755	0.334113
$891$	−1.41913	−0.0475427
$892$	−18.9346	−0.633979
$893$	−31.0913	−1.04043
$894$	2.62491	0.0877901
$895$	3.98827	0.133313
$896$	2.46230	0.0822598
$897$	−43.5720	−1.45483
$898$	−31.3846	−1.04732
$899$	−10.6737	−0.355989
$900$	−0.718528	−0.0239509
$901$	−21.3880	−0.712537
$902$	14.8271	0.493689
$903$	4.24527	0.141274
$904$	49.1137	1.63350
$905$	8.87340	0.294962
$906$	17.8320	0.592430
$907$	−39.9300	−1.32585	−0.662927	−	0.748684i	$-0.730686\pi$
$907$	−39.9300	−1.32585	−0.662927	+	0.748684i	$0.730686\pi$
$908$	1.39692	0.0463585
$909$	−12.0390	−0.399310
$910$	−7.70634	−0.255463
$911$	50.1995	1.66318	0.831592	−	0.555388i	$-0.187430\pi$
$911$	50.1995	1.66318	0.831592	+	0.555388i	$0.187430\pi$
$912$	−6.20544	−0.205483
$913$	7.57637	0.250741
$914$	−10.5324	−0.348382
$915$	3.00642	0.0993893
$916$	12.4296	0.410687
$917$	9.51510	0.314216
$918$	4.90498	0.161888
$919$	−44.3679	−1.46356	−0.731780	−	0.681541i	$-0.761310\pi$
$919$	−44.3679	−1.46356	−0.731780	+	0.681541i	$0.761310\pi$
$920$	25.8528	0.852340
$921$	−22.9685	−0.756838
$922$	23.3055	0.767526
$923$	4.99617	0.164451
$924$	−1.33835	−0.0440286
$925$	10.8240	0.355889
$926$	30.0211	0.986553
$927$	0.181013	0.00594524
$928$	22.6066	0.742096
$929$	−24.4283	−0.801467	−0.400734	−	0.916195i	$-0.631245\pi$
$929$	−24.4283	−0.801467	−0.400734	+	0.916195i	$0.631245\pi$
$930$	−2.05136	−0.0672667
$931$	−16.0007	−0.524401
$932$	−2.40630	−0.0788209
$933$	26.9286	0.881604
$934$	−1.98676	−0.0650089
$935$	6.14900	0.201094
$936$	−15.9616	−0.521722
$937$	−23.4905	−0.767402	−0.383701	−	0.923457i	$-0.625351\pi$
$937$	−23.4905	−0.767402	−0.383701	+	0.923457i	$0.625351\pi$
$938$	17.3640	0.566954
$939$	−31.8804	−1.04038
$940$	7.36810	0.240321
$941$	−12.3651	−0.403090	−0.201545	−	0.979479i	$-0.564596\pi$
$941$	−12.3651	−0.403090	−0.201545	+	0.979479i	$0.564596\pi$
$942$	15.4964	0.504900
$943$	77.5351	2.52489
$944$	−0.0162640	−0.000529349 0
$945$	−1.31252	−0.0426962
$946$	−5.19609	−0.168940
$947$	−29.4470	−0.956898	−0.478449	−	0.878115i	$-0.658801\pi$
$947$	−29.4470	−0.956898	−0.478449	+	0.878115i	$0.658801\pi$
$948$	8.06788	0.262033
$949$	−13.2487	−0.430071
$950$	−3.43227	−0.111358
$951$	−31.0403	−1.00655
$952$	17.5015	0.567228
$953$	−11.3945	−0.369105	−0.184553	−	0.982823i	$-0.559084\pi$
$953$	−11.3945	−0.369105	−0.184553	+	0.982823i	$0.559084\pi$
$954$	−5.58781	−0.180912
$955$	1.03488	0.0334878
$956$	−2.74542	−0.0887933
$957$	8.35895	0.270206
$958$	−5.94666	−0.192128
$959$	7.80888	0.252162
$960$	8.43802	0.272336
$961$	−27.7162	−0.894071
$962$	63.5520	2.04900
$963$	−15.6355	−0.503848
$964$	5.25264	0.169176
$965$	10.8843	0.350377
$966$	12.4818	0.401596
$967$	−8.68544	−0.279305	−0.139652	−	0.990201i	$-0.544599\pi$
$967$	−8.68544	−0.279305	−0.139652	+	0.990201i	$0.544599\pi$
$968$	−27.6540	−0.888834
$969$	−13.1374	−0.422034
$970$	16.1903	0.519838
$971$	−26.5332	−0.851490	−0.425745	−	0.904843i	$-0.639988\pi$
$971$	−26.5332	−0.851490	−0.425745	+	0.904843i	$0.639988\pi$
$972$	−0.718528	−0.0230468
$973$	−2.24141	−0.0718563
$974$	−11.8973	−0.381213
$975$	−5.18667	−0.166106
$976$	−6.15313	−0.196957
$977$	38.7083	1.23839	0.619194	−	0.785238i	$-0.287460\pi$
$977$	38.7083	1.23839	0.619194	+	0.785238i	$0.287460\pi$
$978$	16.7114	0.534371
$979$	12.4956	0.399360
$980$	3.79189	0.121127
$981$	1.83395	0.0585536
$982$	49.9581	1.59423
$983$	56.8867	1.81440	0.907202	−	0.420696i	$-0.138214\pi$
$983$	56.8867	1.81440	0.907202	+	0.420696i	$0.138214\pi$
$984$	28.4033	0.905463
$985$	−13.6277	−0.434216
$986$	−28.8913	−0.920085
$987$	13.4591	0.428409
$988$	11.2995	0.359485
$989$	−27.1718	−0.864013
$990$	1.60649	0.0510575
$991$	43.7113	1.38854	0.694268	−	0.719716i	$-0.255728\pi$
$991$	43.7113	1.38854	0.694268	+	0.719716i	$0.255728\pi$
$992$	−6.95492	−0.220819
$993$	−19.4092	−0.615934
$994$	−1.43123	−0.0453957
$995$	−19.1675	−0.607650
$996$	3.83603	0.121549
$997$	20.9654	0.663981	0.331990	−	0.943283i	$-0.392280\pi$
$997$	20.9654	0.663981	0.331990	+	0.943283i	$0.392280\pi$
$998$	44.0053	1.39296
$999$	10.8240	0.342455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6015.2.a.b.1.8	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6015.2.a.b.1.8	✓	23	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6015 = 3 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6015.a (trivial)

\(f(q)\)	\(=\)	\(q-1.13202 q^{2} +1.00000 q^{3} -0.718528 q^{4} +1.00000 q^{5} -1.13202 q^{6} -1.31252 q^{7} +3.07743 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.13202 q^{2} +1.00000 q^{3} -0.718528 q^{4} +1.00000 q^{5} -1.13202 q^{6} -1.31252 q^{7} +3.07743 q^{8} +1.00000 q^{9} -1.13202 q^{10} -1.41913 q^{11} -0.718528 q^{12} -5.18667 q^{13} +1.48580 q^{14} +1.00000 q^{15} -2.04666 q^{16} -4.33294 q^{17} -1.13202 q^{18} +3.03198 q^{19} -0.718528 q^{20} -1.31252 q^{21} +1.60649 q^{22} +8.40076 q^{23} +3.07743 q^{24} +1.00000 q^{25} +5.87142 q^{26} +1.00000 q^{27} +0.943081 q^{28} -5.89019 q^{29} -1.13202 q^{30} +1.81212 q^{31} -3.83800 q^{32} -1.41913 q^{33} +4.90498 q^{34} -1.31252 q^{35} -0.718528 q^{36} +10.8240 q^{37} -3.43227 q^{38} -5.18667 q^{39} +3.07743 q^{40} +9.22953 q^{41} +1.48580 q^{42} -3.23445 q^{43} +1.01969 q^{44} +1.00000 q^{45} -9.50984 q^{46} -10.2544 q^{47} -2.04666 q^{48} -5.27730 q^{49} -1.13202 q^{50} -4.33294 q^{51} +3.72677 q^{52} +4.93614 q^{53} -1.13202 q^{54} -1.41913 q^{55} -4.03918 q^{56} +3.03198 q^{57} +6.66782 q^{58} +0.00794662 q^{59} -0.718528 q^{60} +3.00642 q^{61} -2.05136 q^{62} -1.31252 q^{63} +8.43802 q^{64} -5.18667 q^{65} +1.60649 q^{66} +11.6866 q^{67} +3.11334 q^{68} +8.40076 q^{69} +1.48580 q^{70} -0.963271 q^{71} +3.07743 q^{72} +2.55437 q^{73} -12.2529 q^{74} +1.00000 q^{75} -2.17857 q^{76} +1.86263 q^{77} +5.87142 q^{78} -11.2283 q^{79} -2.04666 q^{80} +1.00000 q^{81} -10.4480 q^{82} -5.33874 q^{83} +0.943081 q^{84} -4.33294 q^{85} +3.66146 q^{86} -5.89019 q^{87} -4.36728 q^{88} -8.80509 q^{89} -1.13202 q^{90} +6.80760 q^{91} -6.03618 q^{92} +1.81212 q^{93} +11.6082 q^{94} +3.03198 q^{95} -3.83800 q^{96} -14.3021 q^{97} +5.97401 q^{98} -1.41913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 5 q^{2} + 23 q^{3} + 9 q^{4} + 23 q^{5} - 5 q^{6} - 16 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) 23 * q - 5 * q^2 + 23 * q^3 + 9 * q^4 + 23 * q^5 - 5 * q^6 - 16 * q^7 - 12 * q^8 + 23 * q^9	\( 23 q - 5 q^{2} + 23 q^{3} + 9 q^{4} + 23 q^{5} - 5 q^{6} - 16 q^{7} - 12 q^{8} + 23 q^{9} - 5 q^{10} - 13 q^{11} + 9 q^{12} - 18 q^{13} - 6 q^{14} + 23 q^{15} - 11 q^{16} - 34 q^{17} - 5 q^{18} - 35 q^{19} + 9 q^{20} - 16 q^{21} - 11 q^{22} - 14 q^{23} - 12 q^{24} + 23 q^{25} - 6 q^{26} + 23 q^{27} - 26 q^{28} - 43 q^{29} - 5 q^{30} - 21 q^{31} - 14 q^{32} - 13 q^{33} - 12 q^{34} - 16 q^{35} + 9 q^{36} - 18 q^{37} + 6 q^{38} - 18 q^{39} - 12 q^{40} - 45 q^{41} - 6 q^{42} - 43 q^{43} - 11 q^{44} + 23 q^{45} - 29 q^{46} - 14 q^{47} - 11 q^{48} - 25 q^{49} - 5 q^{50} - 34 q^{51} - 20 q^{52} - 3 q^{53} - 5 q^{54} - 13 q^{55} + 3 q^{56} - 35 q^{57} + 10 q^{58} - 9 q^{59} + 9 q^{60} - 67 q^{61} - 7 q^{62} - 16 q^{63} - 8 q^{64} - 18 q^{65} - 11 q^{66} - 32 q^{67} - 24 q^{68} - 14 q^{69} - 6 q^{70} - 8 q^{71} - 12 q^{72} - 39 q^{73} - 16 q^{74} + 23 q^{75} - 48 q^{76} - 26 q^{77} - 6 q^{78} - 59 q^{79} - 11 q^{80} + 23 q^{81} - q^{82} - 23 q^{83} - 26 q^{84} - 34 q^{85} - 7 q^{86} - 43 q^{87} + 17 q^{88} - 51 q^{89} - 5 q^{90} - 37 q^{91} + 11 q^{92} - 21 q^{93} + 8 q^{94} - 35 q^{95} - 14 q^{96} - 29 q^{97} + 32 q^{98} - 13 q^{99}+O(q^{100}) \) 23 * q - 5 * q^2 + 23 * q^3 + 9 * q^4 + 23 * q^5 - 5 * q^6 - 16 * q^7 - 12 * q^8 + 23 * q^9 - 5 * q^10 - 13 * q^11 + 9 * q^12 - 18 * q^13 - 6 * q^14 + 23 * q^15 - 11 * q^16 - 34 * q^17 - 5 * q^18 - 35 * q^19 + 9 * q^20 - 16 * q^21 - 11 * q^22 - 14 * q^23 - 12 * q^24 + 23 * q^25 - 6 * q^26 + 23 * q^27 - 26 * q^28 - 43 * q^29 - 5 * q^30 - 21 * q^31 - 14 * q^32 - 13 * q^33 - 12 * q^34 - 16 * q^35 + 9 * q^36 - 18 * q^37 + 6 * q^38 - 18 * q^39 - 12 * q^40 - 45 * q^41 - 6 * q^42 - 43 * q^43 - 11 * q^44 + 23 * q^45 - 29 * q^46 - 14 * q^47 - 11 * q^48 - 25 * q^49 - 5 * q^50 - 34 * q^51 - 20 * q^52 - 3 * q^53 - 5 * q^54 - 13 * q^55 + 3 * q^56 - 35 * q^57 + 10 * q^58 - 9 * q^59 + 9 * q^60 - 67 * q^61 - 7 * q^62 - 16 * q^63 - 8 * q^64 - 18 * q^65 - 11 * q^66 - 32 * q^67 - 24 * q^68 - 14 * q^69 - 6 * q^70 - 8 * q^71 - 12 * q^72 - 39 * q^73 - 16 * q^74 + 23 * q^75 - 48 * q^76 - 26 * q^77 - 6 * q^78 - 59 * q^79 - 11 * q^80 + 23 * q^81 - q^82 - 23 * q^83 - 26 * q^84 - 34 * q^85 - 7 * q^86 - 43 * q^87 + 17 * q^88 - 51 * q^89 - 5 * q^90 - 37 * q^91 + 11 * q^92 - 21 * q^93 + 8 * q^94 - 35 * q^95 - 14 * q^96 - 29 * q^97 + 32 * q^98 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more