Properties

Label 50.22.a.b.1.1
Level $50$
Weight $22$
Character 50.1
Self dual yes
Analytic conductor $139.739$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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] = [50,22,Mod(1,50)] 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("50.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1024,21924] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(139.738672144\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 50.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} +21924.0 q^{3} +1.04858e6 q^{4} -2.24502e7 q^{6} +7.22753e8 q^{7} -1.07374e9 q^{8} -9.97969e9 q^{9} +4.99764e10 q^{11} +2.29890e10 q^{12} +2.43514e10 q^{13} -7.40099e11 q^{14} +1.09951e12 q^{16} +5.76887e12 q^{17} +1.02192e13 q^{18} -3.02502e13 q^{19} +1.58456e13 q^{21} -5.11758e13 q^{22} +1.45464e14 q^{23} -2.35407e13 q^{24} -2.49358e13 q^{26} -4.48128e14 q^{27} +7.57862e14 q^{28} -1.16711e15 q^{29} -7.43191e15 q^{31} -1.12590e15 q^{32} +1.09568e15 q^{33} -5.90733e15 q^{34} -1.04645e16 q^{36} +5.46217e16 q^{37} +3.09762e16 q^{38} +5.33880e14 q^{39} -2.08896e16 q^{41} -1.62259e16 q^{42} -7.61939e16 q^{43} +5.24041e16 q^{44} -1.48955e17 q^{46} -5.08990e17 q^{47} +2.41057e16 q^{48} -3.61736e16 q^{49} +1.26477e17 q^{51} +2.55343e16 q^{52} -1.34149e18 q^{53} +4.58883e17 q^{54} -7.76050e17 q^{56} -6.63206e17 q^{57} +1.19512e18 q^{58} +2.30640e18 q^{59} +5.26834e18 q^{61} +7.61027e18 q^{62} -7.21285e18 q^{63} +1.15292e18 q^{64} -1.12198e18 q^{66} +1.09050e19 q^{67} +6.04910e18 q^{68} +3.18915e18 q^{69} -1.33678e19 q^{71} +1.07156e19 q^{72} +1.79529e19 q^{73} -5.59326e19 q^{74} -3.17197e19 q^{76} +3.61206e19 q^{77} -5.46693e17 q^{78} +1.18985e20 q^{79} +9.45663e19 q^{81} +2.13909e19 q^{82} -1.34353e20 q^{83} +1.66154e19 q^{84} +7.80226e19 q^{86} -2.55877e19 q^{87} -5.36617e19 q^{88} +3.47081e20 q^{89} +1.76001e19 q^{91} +1.52530e20 q^{92} -1.62937e20 q^{93} +5.21205e20 q^{94} -2.46842e19 q^{96} +3.76522e20 q^{97} +3.70418e19 q^{98} -4.98749e20 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1024.00 −0.707107
\(3\) 21924.0 0.214361 0.107181 − 0.994240i \(-0.465818\pi\)
0.107181 + 0.994240i \(0.465818\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 0 0
\(6\) −2.24502e7 −0.151576
\(7\) 7.22753e8 0.967076 0.483538 − 0.875323i \(-0.339351\pi\)
0.483538 + 0.875323i \(0.339351\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) −9.97969e9 −0.954049
\(10\) 0 0
\(11\) 4.99764e10 0.580954 0.290477 − 0.956882i \(-0.406186\pi\)
0.290477 + 0.956882i \(0.406186\pi\)
\(12\) 2.29890e10 0.107181
\(13\) 2.43514e10 0.0489913 0.0244956 − 0.999700i \(-0.492202\pi\)
0.0244956 + 0.999700i \(0.492202\pi\)
\(14\) −7.40099e11 −0.683826
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 5.76887e12 0.694029 0.347014 − 0.937860i \(-0.387196\pi\)
0.347014 + 0.937860i \(0.387196\pi\)
\(18\) 1.02192e13 0.674615
\(19\) −3.02502e13 −1.13192 −0.565960 − 0.824433i \(-0.691494\pi\)
−0.565960 + 0.824433i \(0.691494\pi\)
\(20\) 0 0
\(21\) 1.58456e13 0.207304
\(22\) −5.11758e13 −0.410797
\(23\) 1.45464e14 0.732173 0.366086 − 0.930581i \(-0.380697\pi\)
0.366086 + 0.930581i \(0.380697\pi\)
\(24\) −2.35407e13 −0.0757882
\(25\) 0 0
\(26\) −2.49358e13 −0.0346421
\(27\) −4.48128e14 −0.418873
\(28\) 7.57862e14 0.483538
\(29\) −1.16711e15 −0.515148 −0.257574 − 0.966259i \(-0.582923\pi\)
−0.257574 + 0.966259i \(0.582923\pi\)
\(30\) 0 0
\(31\) −7.43191e15 −1.62856 −0.814278 − 0.580476i \(-0.802867\pi\)
−0.814278 + 0.580476i \(0.802867\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) 1.09568e15 0.124534
\(34\) −5.90733e15 −0.490752
\(35\) 0 0
\(36\) −1.04645e16 −0.477025
\(37\) 5.46217e16 1.86744 0.933721 − 0.358003i \(-0.116542\pi\)
0.933721 + 0.358003i \(0.116542\pi\)
\(38\) 3.09762e16 0.800388
\(39\) 5.33880e14 0.0105018
\(40\) 0 0
\(41\) −2.08896e16 −0.243052 −0.121526 − 0.992588i \(-0.538779\pi\)
−0.121526 + 0.992588i \(0.538779\pi\)
\(42\) −1.62259e16 −0.146586
\(43\) −7.61939e16 −0.537652 −0.268826 − 0.963189i \(-0.586636\pi\)
−0.268826 + 0.963189i \(0.586636\pi\)
\(44\) 5.24041e16 0.290477
\(45\) 0 0
\(46\) −1.48955e17 −0.517724
\(47\) −5.08990e17 −1.41150 −0.705751 − 0.708460i \(-0.749390\pi\)
−0.705751 + 0.708460i \(0.749390\pi\)
\(48\) 2.41057e16 0.0535904
\(49\) −3.61736e16 −0.0647639
\(50\) 0 0
\(51\) 1.26477e17 0.148773
\(52\) 2.55343e16 0.0244956
\(53\) −1.34149e18 −1.05364 −0.526819 − 0.849978i \(-0.676615\pi\)
−0.526819 + 0.849978i \(0.676615\pi\)
\(54\) 4.58883e17 0.296188
\(55\) 0 0
\(56\) −7.76050e17 −0.341913
\(57\) −6.63206e17 −0.242640
\(58\) 1.19512e18 0.364265
\(59\) 2.30640e18 0.587475 0.293737 − 0.955886i \(-0.405101\pi\)
0.293737 + 0.955886i \(0.405101\pi\)
\(60\) 0 0
\(61\) 5.26834e18 0.945607 0.472803 − 0.881168i \(-0.343242\pi\)
0.472803 + 0.881168i \(0.343242\pi\)
\(62\) 7.61027e18 1.15156
\(63\) −7.21285e18 −0.922638
\(64\) 1.15292e18 0.125000
\(65\) 0 0
\(66\) −1.12198e18 −0.0880589
\(67\) 1.09050e19 0.730869 0.365434 − 0.930837i \(-0.380920\pi\)
0.365434 + 0.930837i \(0.380920\pi\)
\(68\) 6.04910e18 0.347014
\(69\) 3.18915e18 0.156950
\(70\) 0 0
\(71\) −1.33678e19 −0.487357 −0.243679 − 0.969856i \(-0.578354\pi\)
−0.243679 + 0.969856i \(0.578354\pi\)
\(72\) 1.07156e19 0.337307
\(73\) 1.79529e19 0.488926 0.244463 − 0.969659i \(-0.421388\pi\)
0.244463 + 0.969659i \(0.421388\pi\)
\(74\) −5.59326e19 −1.32048
\(75\) 0 0
\(76\) −3.17197e19 −0.565960
\(77\) 3.61206e19 0.561827
\(78\) −5.46693e17 −0.00742592
\(79\) 1.18985e20 1.41387 0.706933 − 0.707281i \(-0.250078\pi\)
0.706933 + 0.707281i \(0.250078\pi\)
\(80\) 0 0
\(81\) 9.45663e19 0.864259
\(82\) 2.13909e19 0.171864
\(83\) −1.34353e20 −0.950447 −0.475224 − 0.879865i \(-0.657633\pi\)
−0.475224 + 0.879865i \(0.657633\pi\)
\(84\) 1.66154e19 0.103652
\(85\) 0 0
\(86\) 7.80226e19 0.380177
\(87\) −2.55877e19 −0.110428
\(88\) −5.36617e19 −0.205398
\(89\) 3.47081e20 1.17988 0.589938 − 0.807449i \(-0.299152\pi\)
0.589938 + 0.807449i \(0.299152\pi\)
\(90\) 0 0
\(91\) 1.76001e19 0.0473783
\(92\) 1.52530e20 0.366086
\(93\) −1.62937e20 −0.349099
\(94\) 5.21205e20 0.998083
\(95\) 0 0
\(96\) −2.46842e19 −0.0378941
\(97\) 3.76522e20 0.518426 0.259213 − 0.965820i \(-0.416537\pi\)
0.259213 + 0.965820i \(0.416537\pi\)
\(98\) 3.70418e19 0.0457950
\(99\) −4.98749e20 −0.554259
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 50.22.a.b.1.1 1
5.2 odd 4 50.22.b.b.49.1 2
5.3 odd 4 50.22.b.b.49.2 2
5.4 even 2 10.22.a.a.1.1 ✓ 1
20.19 odd 2 80.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.a.1.1 ✓ 1 5.4 even 2
50.22.a.b.1.1 1 1.1 even 1 trivial
50.22.b.b.49.1 2 5.2 odd 4
50.22.b.b.49.2 2 5.3 odd 4
80.22.a.a.1.1 1 20.19 odd 2