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Properties

Label 25.16.a.a.1.1
Level $25$
Weight $16$
Character 25.1
Self dual yes
Analytic conductor $35.673$
Analytic rank $0$
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] = [25,16,Mod(1,25)] 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("25.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-216] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.6733762750\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 25.1

$q$-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)
 
\(f(q)\) \(=\) \(q-216.000 q^{2} +3348.00 q^{3} +13888.0 q^{4} -723168. q^{6} -2.82246e6 q^{7} +4.07808e6 q^{8} -3.13980e6 q^{9} +2.05869e7 q^{11} +4.64970e7 q^{12} +1.90073e8 q^{13} +6.09650e8 q^{14} -1.33595e9 q^{16} -1.64653e9 q^{17} +6.78197e8 q^{18} +1.56326e9 q^{19} -9.44958e9 q^{21} -4.44676e9 q^{22} -9.45112e9 q^{23} +1.36534e10 q^{24} -4.10558e10 q^{26} -5.85522e10 q^{27} -3.91983e10 q^{28} -3.69026e10 q^{29} +7.15885e10 q^{31} +1.54934e11 q^{32} +6.89248e10 q^{33} +3.55650e11 q^{34} -4.36056e10 q^{36} +1.03365e12 q^{37} -3.37664e11 q^{38} +6.36366e11 q^{39} +1.64197e12 q^{41} +2.04111e12 q^{42} +4.92403e11 q^{43} +2.85910e11 q^{44} +2.04144e12 q^{46} +3.41068e12 q^{47} -4.47275e12 q^{48} +3.21870e12 q^{49} -5.51258e12 q^{51} +2.63974e12 q^{52} -6.79715e12 q^{53} +1.26473e13 q^{54} -1.15102e13 q^{56} +5.23379e12 q^{57} +7.97095e12 q^{58} +9.85886e12 q^{59} +4.93184e12 q^{61} -1.54631e13 q^{62} +8.86196e12 q^{63} +1.03106e13 q^{64} -1.48878e13 q^{66} +2.88378e13 q^{67} -2.28670e13 q^{68} -3.16423e13 q^{69} +1.25050e14 q^{71} -1.28044e13 q^{72} +8.21715e13 q^{73} -2.23269e14 q^{74} +2.17105e13 q^{76} -5.81055e13 q^{77} -1.37455e14 q^{78} -2.54131e13 q^{79} -1.50980e14 q^{81} -3.54666e14 q^{82} +2.81737e14 q^{83} -1.31236e14 q^{84} -1.06359e14 q^{86} -1.23550e14 q^{87} +8.39548e13 q^{88} +7.15619e14 q^{89} -5.36474e14 q^{91} -1.31257e14 q^{92} +2.39678e14 q^{93} -7.36708e14 q^{94} +5.18719e14 q^{96} -6.12786e14 q^{97} -6.95238e14 q^{98} -6.46387e13 q^{99} +O(q^{100})\)

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 \(-0.703491\pi\)
−0.596621 + 0.802523i \(0.703491\pi\)
\(3\) 3348.00 0.883845 0.441922 − 0.897053i \(-0.354297\pi\)
0.441922 + 0.897053i \(0.354297\pi\)
\(4\) 13888.0 0.423828
\(5\) 0 0
\(6\) −723168. −1.05464
\(7\) −2.82246e6 −1.29536 −0.647682 − 0.761911i \(-0.724261\pi\)
−0.647682 + 0.761911i \(0.724261\pi\)
\(8\) 4.07808e6 0.687513
\(9\) −3.13980e6 −0.218818
\(10\) 0 0
\(11\) 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 \(0.362007\pi\)
\(14\) 6.09650e8 1.54568
\(15\) 0 0
\(16\) −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 \(-0.435708\pi\)
0.200608 + 0.979672i \(0.435708\pi\)
\(20\) 0 0
\(21\) −9.44958e9 −1.14490
\(22\) −4.44676e9 −0.380079
\(23\) −9.45112e9 −0.578794 −0.289397 − 0.957209i \(-0.593455\pi\)
−0.289397 + 0.957209i \(0.593455\pi\)
\(24\) 1.36534e10 0.607655
\(25\) 0 0
\(26\) −4.10558e10 −1.00248
\(27\) −5.85522e10 −1.07725
\(28\) −3.91983e10 −0.549012
\(29\) −3.69026e10 −0.397257 −0.198629 − 0.980075i \(-0.563649\pi\)
−0.198629 + 0.980075i \(0.563649\pi\)
\(30\) 0 0
\(31\) 7.15885e10 0.467337 0.233669 − 0.972316i \(-0.424927\pi\)
0.233669 + 0.972316i \(0.424927\pi\)
\(32\) 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 \(-0.147163\pi\)
0.895017 + 0.446031i \(0.147163\pi\)
\(38\) −3.37664e11 −0.478748
\(39\) 6.36366e11 0.742544
\(40\) 0 0
\(41\) 1.64197e12 1.31670 0.658351 − 0.752711i \(-0.271254\pi\)
0.658351 + 0.752711i \(0.271254\pi\)
\(42\) 2.04111e12 1.36614
\(43\) 4.92403e11 0.276253 0.138127 − 0.990415i \(-0.455892\pi\)
0.138127 + 0.990415i \(0.455892\pi\)
\(44\) 2.85910e11 0.135000
\(45\) 0 0
\(46\) 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
\(49\) 3.21870e12 0.677968
\(50\) 0 0
\(51\) −5.51258e12 −0.860158
\(52\) 2.63974e12 0.356070
\(53\) −6.79715e12 −0.794800 −0.397400 − 0.917645i \(-0.630087\pi\)
−0.397400 + 0.917645i \(0.630087\pi\)
\(54\) 1.26473e13 1.28542
\(55\) 0 0
\(56\) −1.15102e13 −0.890580
\(57\) 5.23379e12 0.354613
\(58\) 7.97095e12 0.474024
\(59\) 9.85886e12 0.515747 0.257873 − 0.966179i \(-0.416978\pi\)
0.257873 + 0.966179i \(0.416978\pi\)
\(60\) 0 0
\(61\) 4.93184e12 0.200926 0.100463 − 0.994941i \(-0.467968\pi\)
0.100463 + 0.994941i \(0.467968\pi\)
\(62\) −1.54631e13 −0.557647
\(63\) 8.86196e12 0.283449
\(64\) 1.03106e13 0.293044
\(65\) 0 0
\(66\) −1.48878e13 −0.335931
\(67\) 2.88378e13 0.581302 0.290651 − 0.956829i \(-0.406128\pi\)
0.290651 + 0.956829i \(0.406128\pi\)
\(68\) −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
\(75\) 0 0
\(76\) 2.17105e13 0.170047
\(77\) −5.81055e13 −0.412607
\(78\) −1.37455e14 −0.886035
\(79\) −2.54131e13 −0.148886 −0.0744430 − 0.997225i \(-0.523718\pi\)
−0.0744430 + 0.997225i \(0.523718\pi\)
\(80\) 0 0
\(81\) −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 \(0.307018\pi\)
\(84\) −1.31236e14 −0.485241
\(85\) 0 0
\(86\) −1.06359e14 −0.329637
\(87\) −1.23550e14 −0.351114
\(88\) 8.39548e13 0.218991
\(89\) 7.15619e14 1.71497 0.857485 − 0.514509i \(-0.172026\pi\)
0.857485 + 0.514509i \(0.172026\pi\)
\(90\) 0 0
\(91\) −5.36474e14 −1.08827
\(92\) −1.31257e14 −0.245309
\(93\) 2.39678e14 0.413054
\(94\) −7.36708e14 −1.17175
\(95\) 0 0
\(96\) 5.18719e14 0.704528
\(97\) −6.12786e14 −0.770054 −0.385027 − 0.922905i \(-0.625808\pi\)
−0.385027 + 0.922905i \(0.625808\pi\)
\(98\) −6.95238e14 −0.808981
\(99\) −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 25.16.a.a.1.1 1
5.2 odd 4 25.16.b.a.24.1 2
5.3 odd 4 25.16.b.a.24.2 2
5.4 even 2 1.16.a.a.1.1 ✓ 1
15.14 odd 2 9.16.a.a.1.1 1
20.19 odd 2 16.16.a.d.1.1 1
35.4 even 6 49.16.c.c.30.1 2
35.9 even 6 49.16.c.c.18.1 2
35.19 odd 6 49.16.c.b.18.1 2
35.24 odd 6 49.16.c.b.30.1 2
35.34 odd 2 49.16.a.a.1.1 1
40.19 odd 2 64.16.a.c.1.1 1
40.29 even 2 64.16.a.i.1.1 1
55.54 odd 2 121.16.a.a.1.1 1
60.59 even 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 5.4 even 2
9.16.a.a.1.1 1 15.14 odd 2
16.16.a.d.1.1 1 20.19 odd 2
25.16.a.a.1.1 1 1.1 even 1 trivial
25.16.b.a.24.1 2 5.2 odd 4
25.16.b.a.24.2 2 5.3 odd 4
49.16.a.a.1.1 1 35.34 odd 2
49.16.c.b.18.1 2 35.19 odd 6
49.16.c.b.30.1 2 35.24 odd 6
49.16.c.c.18.1 2 35.9 even 6
49.16.c.c.30.1 2 35.4 even 6
64.16.a.c.1.1 1 40.19 odd 2
64.16.a.i.1.1 1 40.29 even 2
121.16.a.a.1.1 1 55.54 odd 2
144.16.a.f.1.1 1 60.59 even 2