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Properties

Label 50.12.d.b.21.4
Level $50$
Weight $12$
Character 50.21
Analytic conductor $38.417$
Analytic rank $0$
Dimension $56$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,12,Mod(11,50)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("50.11"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.4171590280\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 21.4
Character \(\chi\) \(=\) 50.21
Dual form 50.12.d.b.31.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(9.88854 - 30.4338i) q^{2} +(-342.324 - 248.713i) q^{3} +(-828.433 - 601.892i) q^{4} +(-4234.41 + 5558.59i) q^{5} +(-10954.4 + 7958.81i) q^{6} +40593.8 q^{7} +(-26509.9 + 19260.5i) q^{8} +(586.044 + 1803.66i) q^{9} +(127297. + 183836. i) q^{10} +(1391.14 - 4281.48i) q^{11} +(133894. + 412084. i) q^{12} +(538247. + 1.65655e6i) q^{13} +(401414. - 1.23542e6i) q^{14} +(2.83203e6 - 849685. i) q^{15} +(324028. + 997255. i) q^{16} +(2.37525e6 - 1.72572e6i) q^{17} +60687.3 q^{18} +(1.23089e7 - 8.94294e6i) q^{19} +(6.85360e6 - 2.05626e6i) q^{20} +(-1.38962e7 - 1.00962e7i) q^{21} +(-116546. - 84675.3i) q^{22} +(-25630.5 + 78882.7i) q^{23} +1.38653e7 q^{24} +(-1.29677e7 - 4.70747e7i) q^{25} +5.57377e7 q^{26} +(-2.29151e7 + 7.05253e7i) q^{27} +(-3.36293e7 - 2.44331e7i) q^{28} +(-4.76181e7 - 3.45966e7i) q^{29} +(2.14549e6 - 9.45916e7i) q^{30} +(-5.13973e7 + 3.73423e7i) q^{31} +3.35544e7 q^{32} +(-1.54108e6 + 1.11966e6i) q^{33} +(-2.90325e7 - 8.93529e7i) q^{34} +(-1.71891e8 + 2.25644e8i) q^{35} +(600109. - 1.84694e6i) q^{36} +(-2.32315e8 - 7.14992e8i) q^{37} +(-1.50451e8 - 4.63040e8i) q^{38} +(2.27751e8 - 7.00946e8i) q^{39} +(5.19215e6 - 2.28914e8i) q^{40} +(2.49872e8 + 7.69026e8i) q^{41} +(-4.44679e8 + 3.23078e8i) q^{42} -1.04186e9 q^{43} +(-3.72946e6 + 2.70961e6i) q^{44} +(-1.25073e7 - 4.37984e6i) q^{45} +(2.14725e6 + 1.56007e6i) q^{46} +(1.39738e9 + 1.01526e9i) q^{47} +(1.37108e8 - 4.21974e8i) q^{48} -3.29469e8 q^{49} +(-1.56089e9 - 7.08437e7i) q^{50} -1.24231e9 q^{51} +(5.51165e8 - 1.69631e9i) q^{52} +(-1.98170e9 - 1.43979e9i) q^{53} +(1.91976e9 + 1.39478e9i) q^{54} +(1.79084e7 + 2.58623e7i) q^{55} +(-1.07614e9 + 7.81859e8i) q^{56} -6.43785e9 q^{57} +(-1.52378e9 + 1.10709e9i) q^{58} +(-1.94546e9 - 5.98752e9i) q^{59} +(-2.85757e9 - 1.00067e9i) q^{60} +(2.30043e9 - 7.07999e9i) q^{61} +(6.28225e8 + 1.93348e9i) q^{62} +(2.37897e7 + 7.32173e7i) q^{63} +(3.31804e8 - 1.02119e9i) q^{64} +(-1.14873e10 - 4.02263e9i) q^{65} +(1.88365e7 + 5.79727e7i) q^{66} +(1.36817e10 - 9.94034e9i) q^{67} -3.00644e9 q^{68} +(2.83931e7 - 2.06288e7i) q^{69} +(5.16747e9 + 7.46259e9i) q^{70} +(3.67595e8 + 2.67073e8i) q^{71} +(-5.02754e7 - 3.65272e7i) q^{72} +(8.19810e9 - 2.52311e10i) q^{73} -2.40572e10 q^{74} +(-7.26892e9 + 1.93400e10i) q^{75} -1.55798e10 q^{76} +(5.64716e7 - 1.73802e8i) q^{77} +(-1.90803e10 - 1.38627e10i) q^{78} +(6.24701e8 + 4.53872e8i) q^{79} +(-6.91540e9 - 2.42165e9i) q^{80} +(2.56567e10 - 1.86407e10i) q^{81} +2.58753e10 q^{82} +(4.97986e10 - 3.61808e10i) q^{83} +(5.43527e9 + 1.67281e10i) q^{84} +(-4.65211e8 + 2.05105e10i) q^{85} +(-1.03025e10 + 3.17077e10i) q^{86} +(7.69620e9 + 2.36865e10i) q^{87} +(4.55848e7 + 1.40296e8i) q^{88} +(3.26706e9 - 1.00550e10i) q^{89} +(-2.56975e8 + 3.37336e8i) q^{90} +(2.18495e10 + 6.72458e10i) q^{91} +(6.87121e7 - 4.99222e7i) q^{92} +2.68820e10 q^{93} +(4.47162e10 - 3.24882e10i) q^{94} +(-2.41079e9 + 1.06288e11i) q^{95} +(-1.14865e10 - 8.34541e9i) q^{96} +(5.63245e10 + 4.09222e10i) q^{97} +(-3.25797e9 + 1.00270e10i) q^{98} +8.53759e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 448 q^{2} - 263 q^{3} - 14336 q^{4} + 1770 q^{5} - 8416 q^{6} - 111844 q^{7} - 458752 q^{8} - 1174523 q^{9} + 304960 q^{10} + 207277 q^{11} + 1026048 q^{12} + 893677 q^{13} - 1270048 q^{14} + 4696640 q^{15}+ \cdots - 505737997606 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(e\left(\frac{3}{5}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.88854 − 30.4338i 0.218508 − 0.672499i
\(3\) −342.324 − 248.713i −0.813336 − 0.590923i 0.101460 − 0.994840i \(-0.467649\pi\)
−0.914796 + 0.403916i \(0.867649\pi\)
\(4\) −828.433 − 601.892i −0.404508 − 0.293893i
\(5\) −4234.41 + 5558.59i −0.605979 + 0.795480i
\(6\) −10954.4 + 7958.81i −0.575115 + 0.417846i
\(7\) 40593.8 0.912895 0.456447 − 0.889750i \(-0.349122\pi\)
0.456447 + 0.889750i \(0.349122\pi\)
\(8\) −26509.9 + 19260.5i −0.286031 + 0.207813i
\(9\) 586.044 + 1803.66i 0.00330823 + 0.0101817i
\(10\) 127297. + 183836.i 0.402548 + 0.581339i
\(11\) 1391.14 − 4281.48i 0.00260442 − 0.00801557i −0.949746 − 0.313022i \(-0.898659\pi\)
0.952350 + 0.305006i \(0.0986586\pi\)
\(12\) 133894. + 412084.i 0.155333 + 0.478067i
\(13\) 538247. + 1.65655e6i 0.402062 + 1.23742i 0.923324 + 0.384023i \(0.125462\pi\)
−0.521262 + 0.853397i \(0.674538\pi\)
\(14\) 401414. − 1.23542e6i 0.199475 − 0.613920i
\(15\) 2.83203e6 − 849685.i 0.962933 − 0.288906i
\(16\) 324028. + 997255.i 0.0772542 + 0.237764i
\(17\) 2.37525e6 − 1.72572e6i 0.405734 − 0.294783i −0.366139 − 0.930560i \(-0.619320\pi\)
0.771872 + 0.635778i \(0.219320\pi\)
\(18\) 60687.3 0.00757005
\(19\) 1.23089e7 − 8.94294e6i 1.14045 − 0.828582i 0.153264 − 0.988185i \(-0.451021\pi\)
0.987181 + 0.159603i \(0.0510214\pi\)
\(20\) 6.85360e6 − 2.05626e6i 0.478910 − 0.143686i
\(21\) −1.38962e7 − 1.00962e7i −0.742490 − 0.539451i
\(22\) −116546. − 84675.3i −0.00482137 − 0.00350293i
\(23\) −25630.5 + 78882.7i −0.000830337 + 0.00255552i −0.951471 − 0.307739i \(-0.900428\pi\)
0.950640 + 0.310294i \(0.100428\pi\)
\(24\) 1.38653e7 0.355441
\(25\) −1.29677e7 − 4.70747e7i −0.265578 − 0.964089i
\(26\) 5.57377e7 0.920017
\(27\) −2.29151e7 + 7.05253e7i −0.307341 + 0.945898i
\(28\) −3.36293e7 − 2.44331e7i −0.369274 − 0.268293i
\(29\) −4.76181e7 − 3.45966e7i −0.431105 − 0.313216i 0.350985 − 0.936381i \(-0.385847\pi\)
−0.782091 + 0.623165i \(0.785847\pi\)
\(30\) 2.14549e6 − 9.45916e7i 0.0161198 − 0.710699i
\(31\) −5.13973e7 + 3.73423e7i −0.322442 + 0.234268i −0.737217 − 0.675656i \(-0.763860\pi\)
0.414775 + 0.909924i \(0.363860\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −1.54108e6 + 1.11966e6i −0.00685485 + 0.00498034i
\(34\) −2.90325e7 − 8.93529e7i −0.109585 − 0.337268i
\(35\) −1.71891e8 + 2.25644e8i −0.553195 + 0.726190i
\(36\) 600109. − 1.84694e6i 0.00165412 − 0.00509085i
\(37\) −2.32315e8 − 7.14992e8i −0.550767 − 1.69509i −0.706869 − 0.707345i \(-0.749893\pi\)
0.156102 − 0.987741i \(-0.450107\pi\)
\(38\) −1.50451e8 − 4.63040e8i −0.308024 − 0.948000i
\(39\) 2.27751e8 − 7.00946e8i 0.404208 − 1.24403i
\(40\) 5.19215e6 − 2.28914e8i 0.00801711 − 0.353462i
\(41\) 2.49872e8 + 7.69026e8i 0.336826 + 1.03664i 0.965815 + 0.259231i \(0.0834690\pi\)
−0.628989 + 0.777414i \(0.716531\pi\)
\(42\) −4.44679e8 + 3.23078e8i −0.525020 + 0.381449i
\(43\) −1.04186e9 −1.08077 −0.540383 − 0.841419i \(-0.681721\pi\)
−0.540383 + 0.841419i \(0.681721\pi\)
\(44\) −3.72946e6 + 2.70961e6i −0.00340923 + 0.00247695i
\(45\) −1.25073e7 − 4.37984e6i −0.0101041 − 0.00353826i
\(46\) 2.14725e6 + 1.56007e6i 0.00153715 + 0.00111680i
\(47\) 1.39738e9 + 1.01526e9i 0.888745 + 0.645711i 0.935550 − 0.353193i \(-0.114904\pi\)
−0.0468057 + 0.998904i \(0.514904\pi\)
\(48\) 1.37108e8 − 4.21974e8i 0.0776667 − 0.239033i
\(49\) −3.29469e8 −0.166623
\(50\) −1.56089e9 − 7.08437e7i −0.706380 − 0.0320602i
\(51\) −1.24231e9 −0.504192
\(52\) 5.51165e8 − 1.69631e9i 0.201031 − 0.618710i
\(53\) −1.98170e9 − 1.43979e9i −0.650910 − 0.472914i 0.212671 − 0.977124i \(-0.431784\pi\)
−0.863581 + 0.504210i \(0.831784\pi\)
\(54\) 1.91976e9 + 1.39478e9i 0.568958 + 0.413372i
\(55\) 1.79084e7 + 2.58623e7i 0.00479801 + 0.00692903i
\(56\) −1.07614e9 + 7.81859e8i −0.261116 + 0.189712i
\(57\) −6.43785e9 −1.41719
\(58\) −1.52378e9 + 1.10709e9i −0.304837 + 0.221477i
\(59\) −1.94546e9 − 5.98752e9i −0.354272 − 1.09034i −0.956430 − 0.291961i \(-0.905692\pi\)
0.602158 − 0.798377i \(-0.294308\pi\)
\(60\) −2.85757e9 − 1.00067e9i −0.474422 − 0.166134i
\(61\) 2.30043e9 − 7.07999e9i 0.348734 − 1.07329i −0.610820 − 0.791770i \(-0.709160\pi\)
0.959554 − 0.281524i \(-0.0908400\pi\)
\(62\) 6.28225e8 + 1.93348e9i 0.0870885 + 0.268031i
\(63\) 2.37897e7 + 7.32173e7i 0.00302007 + 0.00929482i
\(64\) 3.31804e8 − 1.02119e9i 0.0386271 − 0.118882i
\(65\) −1.14873e10 − 4.02263e9i −1.22798 − 0.430018i
\(66\) 1.88365e7 + 5.79727e7i 0.00185143 + 0.00569812i
\(67\) 1.36817e10 − 9.94034e9i 1.23802 − 0.899477i 0.240558 − 0.970635i \(-0.422669\pi\)
0.997465 + 0.0711581i \(0.0226695\pi\)
\(68\) −3.00644e9 −0.250757
\(69\) 2.83931e7 − 2.06288e7i 0.00218546 − 0.00158783i
\(70\) 5.16747e9 + 7.46259e9i 0.367484 + 0.530701i
\(71\) 3.67595e8 + 2.67073e8i 0.0241795 + 0.0175675i 0.599809 − 0.800143i \(-0.295243\pi\)
−0.575630 + 0.817710i \(0.695243\pi\)
\(72\) −5.02754e7 − 3.65272e7i −0.00306215 − 0.00222478i
\(73\) 8.19810e9 − 2.52311e10i 0.462847 − 1.42450i −0.398824 − 0.917028i \(-0.630581\pi\)
0.861670 − 0.507468i \(-0.169419\pi\)
\(74\) −2.40572e10 −1.26029
\(75\) −7.26892e9 + 1.93400e10i −0.353698 + 0.941065i
\(76\) −1.55798e10 −0.704834
\(77\) 5.64716e7 − 1.73802e8i 0.00237756 − 0.00731737i
\(78\) −1.90803e10 − 1.38627e10i −0.748283 − 0.543659i
\(79\) 6.24701e8 + 4.53872e8i 0.0228414 + 0.0165953i 0.599147 − 0.800639i \(-0.295506\pi\)
−0.576306 + 0.817234i \(0.695506\pi\)
\(80\) −6.91540e9 − 2.42165e9i −0.235951 − 0.0826259i
\(81\) 2.56567e10 − 1.86407e10i 0.817585 − 0.594011i
\(82\) 2.58753e10 0.770741
\(83\) 4.97986e10 − 3.61808e10i 1.38767 − 1.00820i 0.391557 − 0.920154i \(-0.371937\pi\)
0.996116 − 0.0880498i \(-0.0280635\pi\)
\(84\) 5.43527e9 + 1.67281e10i 0.141803 + 0.436425i
\(85\) −4.65211e8 + 2.05105e10i −0.0113722 + 0.501385i
\(86\) −1.03025e10 + 3.17077e10i −0.236156 + 0.726814i
\(87\) 7.69620e9 + 2.36865e10i 0.165547 + 0.509500i
\(88\) 4.55848e7 + 1.40296e8i 0.000920800 + 0.00283393i
\(89\) 3.26706e9 − 1.00550e10i 0.0620171 − 0.190869i −0.915248 − 0.402892i \(-0.868005\pi\)
0.977265 + 0.212023i \(0.0680051\pi\)
\(90\) −2.56975e8 + 3.37336e8i −0.00458729 + 0.00602183i
\(91\) 2.18495e10 + 6.72458e10i 0.367040 + 1.12963i
\(92\) 6.87121e7 − 4.99222e7i 0.00108693 − 0.000789698i
\(93\) 2.68820e10 0.400688
\(94\) 4.47162e10 − 3.24882e10i 0.628437 − 0.456586i
\(95\) −2.41079e9 + 1.06288e11i −0.0319654 + 1.40931i
\(96\) −1.14865e10 − 8.34541e9i −0.143779 − 0.104461i
\(97\) 5.63245e10 + 4.09222e10i 0.665968 + 0.483854i 0.868673 − 0.495386i \(-0.164973\pi\)
−0.202705 + 0.979240i \(0.564973\pi\)
\(98\) −3.25797e9 + 1.00270e10i −0.0364085 + 0.112054i
\(99\) 8.53759e6 9.02281e−5
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 50.12.d.b.21.4 ✓ 56
25.6 even 5 inner 50.12.d.b.31.4 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.21.4 ✓ 56 1.1 even 1 trivial
50.12.d.b.31.4 yes 56 25.6 even 5 inner