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Properties

Label 49.16.a.a.1.1
Level $49$
Weight $16$
Character 49.1
Self dual yes
Analytic conductor $69.920$
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] = [49,16,Mod(1,49)] 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("49.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,216,3348] 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: \(69.9198174990\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+216.000 q^{2} +3348.00 q^{3} +13888.0 q^{4} -52110.0 q^{5} +723168. q^{6} -4.07808e6 q^{8} -3.13980e6 q^{9} -1.12558e7 q^{10} +2.05869e7 q^{11} +4.64970e7 q^{12} +1.90073e8 q^{13} -1.74464e8 q^{15} -1.33595e9 q^{16} -1.64653e9 q^{17} -6.78197e8 q^{18} -1.56326e9 q^{19} -7.23704e8 q^{20} +4.44676e9 q^{22} +9.45112e9 q^{23} -1.36534e10 q^{24} -2.78021e10 q^{25} +4.10558e10 q^{26} -5.85522e10 q^{27} -3.69026e10 q^{29} -3.76843e10 q^{30} -7.15885e10 q^{31} -1.54934e11 q^{32} +6.89248e10 q^{33} -3.55650e11 q^{34} -4.36056e10 q^{36} -1.03365e12 q^{37} -3.37664e11 q^{38} +6.36366e11 q^{39} +2.12509e11 q^{40} -1.64197e12 q^{41} -4.92403e11 q^{43} +2.85910e11 q^{44} +1.63615e11 q^{45} +2.04144e12 q^{46} +3.41068e12 q^{47} -4.47275e12 q^{48} -6.00526e12 q^{50} -5.51258e12 q^{51} +2.63974e12 q^{52} +6.79715e12 q^{53} -1.26473e13 q^{54} -1.07278e12 q^{55} -5.23379e12 q^{57} -7.97095e12 q^{58} -9.85886e12 q^{59} -2.42296e12 q^{60} -4.93184e12 q^{61} -1.54631e13 q^{62} +1.03106e13 q^{64} -9.90472e12 q^{65} +1.48878e13 q^{66} -2.88378e13 q^{67} -2.28670e13 q^{68} +3.16423e13 q^{69} +1.25050e14 q^{71} +1.28044e13 q^{72} +8.21715e13 q^{73} -2.23269e14 q^{74} -9.30815e13 q^{75} -2.17105e13 q^{76} +1.37455e14 q^{78} -2.54131e13 q^{79} +6.96162e13 q^{80} -1.50980e14 q^{81} -3.54666e14 q^{82} +2.81737e14 q^{83} +8.58006e13 q^{85} -1.06359e14 q^{86} -1.23550e14 q^{87} -8.39548e13 q^{88} -7.15619e14 q^{89} +3.53409e13 q^{90} +1.31257e14 q^{92} -2.39678e14 q^{93} +7.36708e14 q^{94} +8.14613e13 q^{95} -5.18719e14 q^{96} -6.12786e14 q^{97} -6.46387e13 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 216.000 1.19324 0.596621 − 0.802523i \(-0.296509\pi\)
0.596621 + 0.802523i \(0.296509\pi\)
\(3\) 3348.00 0.883845 0.441922 − 0.897053i \(-0.354297\pi\)
0.441922 + 0.897053i \(0.354297\pi\)
\(4\) 13888.0 0.423828
\(5\) −52110.0 −0.298295 −0.149148 − 0.988815i \(-0.547653\pi\)
−0.149148 + 0.988815i \(0.547653\pi\)
\(6\) 723168. 1.05464
\(7\) 0 0
\(8\) −4.07808e6 −0.687513
\(9\) −3.13980e6 −0.218818
\(10\) −1.12558e7 −0.355938
\(11\) 2.05869e7 0.318526 0.159263 − 0.987236i \(-0.449088\pi\)
0.159263 + 0.987236i \(0.449088\pi\)
\(12\) 4.64970e7 0.374598
\(13\) 1.90073e8 0.840129 0.420065 − 0.907494i \(-0.362007\pi\)
0.420065 + 0.907494i \(0.362007\pi\)
\(14\) 0 0
\(15\) −1.74464e8 −0.263647
\(16\) −1.33595e9 −1.24420
\(17\) −1.64653e9 −0.973200 −0.486600 − 0.873625i \(-0.661763\pi\)
−0.486600 + 0.873625i \(0.661763\pi\)
\(18\) −6.78197e8 −0.261103
\(19\) −1.56326e9 −0.401216 −0.200608 − 0.979672i \(-0.564292\pi\)
−0.200608 + 0.979672i \(0.564292\pi\)
\(20\) −7.23704e8 −0.126426
\(21\) 0 0
\(22\) 4.44676e9 0.380079
\(23\) 9.45112e9 0.578794 0.289397 − 0.957209i \(-0.406545\pi\)
0.289397 + 0.957209i \(0.406545\pi\)
\(24\) −1.36534e10 −0.607655
\(25\) −2.78021e10 −0.911020
\(26\) 4.10558e10 1.00248
\(27\) −5.85522e10 −1.07725
\(28\) 0 0
\(29\) −3.69026e10 −0.397257 −0.198629 − 0.980075i \(-0.563649\pi\)
−0.198629 + 0.980075i \(0.563649\pi\)
\(30\) −3.76843e10 −0.314594
\(31\) −7.15885e10 −0.467337 −0.233669 − 0.972316i \(-0.575073\pi\)
−0.233669 + 0.972316i \(0.575073\pi\)
\(32\) −1.54934e11 −0.797117
\(33\) 6.89248e10 0.281528
\(34\) −3.55650e11 −1.16126
\(35\) 0 0
\(36\) −4.36056e10 −0.0927413
\(37\) −1.03365e12 −1.79003 −0.895017 − 0.446031i \(-0.852837\pi\)
−0.895017 + 0.446031i \(0.852837\pi\)
\(38\) −3.37664e11 −0.478748
\(39\) 6.36366e11 0.742544
\(40\) 2.12509e11 0.205082
\(41\) −1.64197e12 −1.31670 −0.658351 − 0.752711i \(-0.728746\pi\)
−0.658351 + 0.752711i \(0.728746\pi\)
\(42\) 0 0
\(43\) −4.92403e11 −0.276253 −0.138127 − 0.990415i \(-0.544108\pi\)
−0.138127 + 0.990415i \(0.544108\pi\)
\(44\) 2.85910e11 0.135000
\(45\) 1.63615e11 0.0652724
\(46\) 2.04144e12 0.690642
\(47\) 3.41068e12 0.981991 0.490996 − 0.871162i \(-0.336633\pi\)
0.490996 + 0.871162i \(0.336633\pi\)
\(48\) −4.47275e12 −1.09968
\(49\) 0 0
\(50\) −6.00526e12 −1.08707
\(51\) −5.51258e12 −0.860158
\(52\) 2.63974e12 0.356070
\(53\) 6.79715e12 0.794800 0.397400 − 0.917645i \(-0.369913\pi\)
0.397400 + 0.917645i \(0.369913\pi\)
\(54\) −1.26473e13 −1.28542
\(55\) −1.07278e12 −0.0950147
\(56\) 0 0
\(57\) −5.23379e12 −0.354613
\(58\) −7.97095e12 −0.474024
\(59\) −9.85886e12 −0.515747 −0.257873 − 0.966179i \(-0.583022\pi\)
−0.257873 + 0.966179i \(0.583022\pi\)
\(60\) −2.42296e12 −0.111741
\(61\) −4.93184e12 −0.200926 −0.100463 − 0.994941i \(-0.532032\pi\)
−0.100463 + 0.994941i \(0.532032\pi\)
\(62\) −1.54631e13 −0.557647
\(63\) 0 0
\(64\) 1.03106e13 0.293044
\(65\) −9.90472e12 −0.250606
\(66\) 1.48878e13 0.335931
\(67\) −2.88378e13 −0.581302 −0.290651 − 0.956829i \(-0.593872\pi\)
−0.290651 + 0.956829i \(0.593872\pi\)
\(68\) −2.28670e13 −0.412470
\(69\) 3.16423e13 0.511564
\(70\) 0 0
\(71\) 1.25050e14 1.63172 0.815862 − 0.578247i \(-0.196263\pi\)
0.815862 + 0.578247i \(0.196263\pi\)
\(72\) 1.28044e13 0.150440
\(73\) 8.21715e13 0.870562 0.435281 − 0.900295i \(-0.356649\pi\)
0.435281 + 0.900295i \(0.356649\pi\)
\(74\) −2.23269e14 −2.13595
\(75\) −9.30815e13 −0.805200
\(76\) −2.17105e13 −0.170047
\(77\) 0 0
\(78\) 1.37455e14 0.886035
\(79\) −2.54131e13 −0.148886 −0.0744430 − 0.997225i \(-0.523718\pi\)
−0.0744430 + 0.997225i \(0.523718\pi\)
\(80\) 6.96162e13 0.371138
\(81\) −1.50980e14 −0.733300
\(82\) −3.54666e14 −1.57114
\(83\) 2.81737e14 1.13961 0.569807 − 0.821779i \(-0.307018\pi\)
0.569807 + 0.821779i \(0.307018\pi\)
\(84\) 0 0
\(85\) 8.58006e13 0.290301
\(86\) −1.06359e14 −0.329637
\(87\) −1.23550e14 −0.351114
\(88\) −8.39548e13 −0.218991
\(89\) −7.15619e14 −1.71497 −0.857485 − 0.514509i \(-0.827974\pi\)
−0.857485 + 0.514509i \(0.827974\pi\)
\(90\) 3.53409e13 0.0778858
\(91\) 0 0
\(92\) 1.31257e14 0.245309
\(93\) −2.39678e14 −0.413054
\(94\) 7.36708e14 1.17175
\(95\) 8.14613e13 0.119681
\(96\) −5.18719e14 −0.704528
\(97\) −6.12786e14 −0.770054 −0.385027 − 0.922905i \(-0.625808\pi\)
−0.385027 + 0.922905i \(0.625808\pi\)
\(98\) 0 0
\(99\) −6.46387e13 −0.0696993
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 49.16.a.a.1.1 1
7.2 even 3 49.16.c.b.18.1 2
7.3 odd 6 49.16.c.c.30.1 2
7.4 even 3 49.16.c.b.30.1 2
7.5 odd 6 49.16.c.c.18.1 2
7.6 odd 2 1.16.a.a.1.1 ✓ 1
21.20 even 2 9.16.a.a.1.1 1
28.27 even 2 16.16.a.d.1.1 1
35.13 even 4 25.16.b.a.24.1 2
35.27 even 4 25.16.b.a.24.2 2
35.34 odd 2 25.16.a.a.1.1 1
56.13 odd 2 64.16.a.i.1.1 1
56.27 even 2 64.16.a.c.1.1 1
77.76 even 2 121.16.a.a.1.1 1
84.83 odd 2 144.16.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.16.a.a.1.1 ✓ 1 7.6 odd 2
9.16.a.a.1.1 1 21.20 even 2
16.16.a.d.1.1 1 28.27 even 2
25.16.a.a.1.1 1 35.34 odd 2
25.16.b.a.24.1 2 35.13 even 4
25.16.b.a.24.2 2 35.27 even 4
49.16.a.a.1.1 1 1.1 even 1 trivial
49.16.c.b.18.1 2 7.2 even 3
49.16.c.b.30.1 2 7.4 even 3
49.16.c.c.18.1 2 7.5 odd 6
49.16.c.c.30.1 2 7.3 odd 6
64.16.a.c.1.1 1 56.27 even 2
64.16.a.i.1.1 1 56.13 odd 2
121.16.a.a.1.1 1 77.76 even 2
144.16.a.f.1.1 1 84.83 odd 2