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Properties

Label 80.22.a.c.1.1
Level $80$
Weight $22$
Character 80.1
Self dual yes
Analytic conductor $223.582$
Analytic rank $0$
Dimension $2$
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] = [80,22,Mod(1,80)] 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("80.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 80.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-30972] 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: \(223.581875430\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1179649}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 294912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(543.558\) of defining polynomial
Character \(\chi\) \(=\) 80.1

$q$-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)
 
\(f(q)\) \(=\) \(q-145820. q^{3} -9.76562e6 q^{5} +2.36011e8 q^{7} +1.08031e10 q^{9} -1.49202e11 q^{11} +4.89350e11 q^{13} +1.42402e12 q^{15} +5.46288e12 q^{17} +1.12049e13 q^{19} -3.44151e13 q^{21} -1.78791e13 q^{23} +9.53674e13 q^{25} -4.99812e13 q^{27} -2.82120e15 q^{29} -4.24797e15 q^{31} +2.17567e16 q^{33} -2.30479e15 q^{35} +5.74457e16 q^{37} -7.13570e16 q^{39} -3.51283e16 q^{41} +2.62144e17 q^{43} -1.05499e17 q^{45} -1.08406e17 q^{47} -5.02845e17 q^{49} -7.96597e17 q^{51} -2.47953e18 q^{53} +1.45705e18 q^{55} -1.63389e18 q^{57} -2.14259e17 q^{59} +3.64272e18 q^{61} +2.54965e18 q^{63} -4.77881e18 q^{65} +1.13372e19 q^{67} +2.60712e18 q^{69} -2.44087e19 q^{71} -4.12244e19 q^{73} -1.39065e19 q^{75} -3.52133e19 q^{77} -6.83832e19 q^{79} -1.05716e20 q^{81} -1.39311e20 q^{83} -5.33484e19 q^{85} +4.11387e20 q^{87} -4.04817e20 q^{89} +1.15492e20 q^{91} +6.19438e20 q^{93} -1.09423e20 q^{95} -9.18963e20 q^{97} -1.61185e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 30972 q^{3} - 19531250 q^{5} + 439959356 q^{7} + 13532817186 q^{9} - 105191777184 q^{11} + 308456648932 q^{13} + 302460937500 q^{15} + 18393996100404 q^{17} - 23821216715800 q^{19} - 10991999204616 q^{21}+ \cdots - 14\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −145820. −1.42575 −0.712876 − 0.701290i \(-0.752608\pi\)
−0.712876 + 0.701290i \(0.752608\pi\)
\(4\) 0 0
\(5\) −9.76562e6 −0.447214
\(6\) 0 0
\(7\) 2.36011e8 0.315793 0.157896 − 0.987456i \(-0.449529\pi\)
0.157896 + 0.987456i \(0.449529\pi\)
\(8\) 0 0
\(9\) 1.08031e10 1.03277
\(10\) 0 0
\(11\) −1.49202e11 −1.73441 −0.867206 − 0.497949i \(-0.834087\pi\)
−0.867206 + 0.497949i \(0.834087\pi\)
\(12\) 0 0
\(13\) 4.89350e11 0.984497 0.492248 − 0.870455i \(-0.336175\pi\)
0.492248 + 0.870455i \(0.336175\pi\)
\(14\) 0 0
\(15\) 1.42402e12 0.637615
\(16\) 0 0
\(17\) 5.46288e12 0.657216 0.328608 − 0.944466i \(-0.393421\pi\)
0.328608 + 0.944466i \(0.393421\pi\)
\(18\) 0 0
\(19\) 1.12049e13 0.419270 0.209635 − 0.977780i \(-0.432772\pi\)
0.209635 + 0.977780i \(0.432772\pi\)
\(20\) 0 0
\(21\) −3.44151e13 −0.450242
\(22\) 0 0
\(23\) −1.78791e13 −0.0899916 −0.0449958 − 0.998987i \(-0.514327\pi\)
−0.0449958 + 0.998987i \(0.514327\pi\)
\(24\) 0 0
\(25\) 9.53674e13 0.200000
\(26\) 0 0
\(27\) −4.99812e13 −0.0467183
\(28\) 0 0
\(29\) −2.82120e15 −1.24525 −0.622623 − 0.782522i \(-0.713933\pi\)
−0.622623 + 0.782522i \(0.713933\pi\)
\(30\) 0 0
\(31\) −4.24797e15 −0.930858 −0.465429 − 0.885085i \(-0.654100\pi\)
−0.465429 + 0.885085i \(0.654100\pi\)
\(32\) 0 0
\(33\) 2.17567e16 2.47284
\(34\) 0 0
\(35\) −2.30479e15 −0.141227
\(36\) 0 0
\(37\) 5.74457e16 1.96399 0.981995 − 0.188909i \(-0.0604951\pi\)
0.981995 + 0.188909i \(0.0604951\pi\)
\(38\) 0 0
\(39\) −7.13570e16 −1.40365
\(40\) 0 0
\(41\) −3.51283e16 −0.408721 −0.204361 − 0.978896i \(-0.565512\pi\)
−0.204361 + 0.978896i \(0.565512\pi\)
\(42\) 0 0
\(43\) 2.62144e17 1.84978 0.924891 − 0.380233i \(-0.124156\pi\)
0.924891 + 0.380233i \(0.124156\pi\)
\(44\) 0 0
\(45\) −1.05499e17 −0.461868
\(46\) 0 0
\(47\) −1.08406e17 −0.300626 −0.150313 − 0.988638i \(-0.548028\pi\)
−0.150313 + 0.988638i \(0.548028\pi\)
\(48\) 0 0
\(49\) −5.02845e17 −0.900275
\(50\) 0 0
\(51\) −7.96597e17 −0.937027
\(52\) 0 0
\(53\) −2.47953e18 −1.94748 −0.973740 − 0.227662i \(-0.926892\pi\)
−0.973740 + 0.227662i \(0.926892\pi\)
\(54\) 0 0
\(55\) 1.45705e18 0.775653
\(56\) 0 0
\(57\) −1.63389e18 −0.597775
\(58\) 0 0
\(59\) −2.14259e17 −0.0545748 −0.0272874 − 0.999628i \(-0.508687\pi\)
−0.0272874 + 0.999628i \(0.508687\pi\)
\(60\) 0 0
\(61\) 3.64272e18 0.653826 0.326913 − 0.945054i \(-0.393992\pi\)
0.326913 + 0.945054i \(0.393992\pi\)
\(62\) 0 0
\(63\) 2.54965e18 0.326141
\(64\) 0 0
\(65\) −4.77881e18 −0.440280
\(66\) 0 0
\(67\) 1.13372e19 0.759835 0.379918 − 0.925020i \(-0.375952\pi\)
0.379918 + 0.925020i \(0.375952\pi\)
\(68\) 0 0
\(69\) 2.60712e18 0.128306
\(70\) 0 0
\(71\) −2.44087e19 −0.889882 −0.444941 − 0.895560i \(-0.646775\pi\)
−0.444941 + 0.895560i \(0.646775\pi\)
\(72\) 0 0
\(73\) −4.12244e19 −1.12270 −0.561350 − 0.827578i \(-0.689718\pi\)
−0.561350 + 0.827578i \(0.689718\pi\)
\(74\) 0 0
\(75\) −1.39065e19 −0.285150
\(76\) 0 0
\(77\) −3.52133e19 −0.547715
\(78\) 0 0
\(79\) −6.83832e19 −0.812578 −0.406289 − 0.913745i \(-0.633177\pi\)
−0.406289 + 0.913745i \(0.633177\pi\)
\(80\) 0 0
\(81\) −1.05716e20 −0.966159
\(82\) 0 0
\(83\) −1.39311e20 −0.985522 −0.492761 − 0.870165i \(-0.664012\pi\)
−0.492761 + 0.870165i \(0.664012\pi\)
\(84\) 0 0
\(85\) −5.33484e19 −0.293916
\(86\) 0 0
\(87\) 4.11387e20 1.77541
\(88\) 0 0
\(89\) −4.04817e20 −1.37614 −0.688072 − 0.725643i \(-0.741542\pi\)
−0.688072 + 0.725643i \(0.741542\pi\)
\(90\) 0 0
\(91\) 1.15492e20 0.310897
\(92\) 0 0
\(93\) 6.19438e20 1.32717
\(94\) 0 0
\(95\) −1.09423e20 −0.187503
\(96\) 0 0
\(97\) −9.18963e20 −1.26530 −0.632652 − 0.774436i \(-0.718034\pi\)
−0.632652 + 0.774436i \(0.718034\pi\)
\(98\) 0 0
\(99\) −1.61185e21 −1.79124
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 80.22.a.c.1.1 2
4.3 odd 2 10.22.a.d.1.2 ✓ 2
20.3 even 4 50.22.b.e.49.2 4
20.7 even 4 50.22.b.e.49.3 4
20.19 odd 2 50.22.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.d.1.2 ✓ 2 4.3 odd 2
50.22.a.d.1.1 2 20.19 odd 2
50.22.b.e.49.2 4 20.3 even 4
50.22.b.e.49.3 4 20.7 even 4
80.22.a.c.1.1 2 1.1 even 1 trivial