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Properties

Label 288.2.w.b
Level $288$
Weight $2$
Character orbit 288.w
Analytic conductor $2.300$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,2,Mod(35,288)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([4, 3, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("288.35"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.w (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,4] 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: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{2} + 4 q^{8} + 8 q^{10} + 8 q^{11} - 12 q^{14} + 8 q^{16} - 32 q^{20} + 16 q^{22} + 36 q^{26} + 16 q^{29} + 24 q^{32} - 24 q^{35} - 32 q^{38} - 32 q^{40} - 8 q^{44} - 32 q^{46} - 8 q^{50} - 56 q^{52}+ \cdots + 44 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −1.32473 + 0.495070i 0 1.50981 − 1.31167i 0.0505677 − 0.0209458i 0 1.44150 + 1.44150i −1.35072 + 2.48506i 0 −0.0566189 + 0.0527821i
35.2 −1.16775 − 0.797726i 0 0.727265 + 1.86308i −1.65625 + 0.686041i 0 −0.456585 − 0.456585i 0.636970 − 2.75577i 0 2.48135 + 0.520112i
35.3 −0.430512 + 1.34709i 0 −1.62932 − 1.15988i −1.78959 + 0.741273i 0 −2.33709 − 2.33709i 2.26391 − 1.69550i 0 −0.228123 − 2.72987i
35.4 −0.349793 − 1.37027i 0 −1.75529 + 0.958624i 2.97412 − 1.23192i 0 0.237717 + 0.237717i 1.92756 + 2.06990i 0 −2.72839 − 3.64443i
35.5 0.345448 + 1.37137i 0 −1.76133 + 0.947475i 1.64859 − 0.682869i 0 2.51270 + 2.51270i −1.90779 − 2.08814i 0 1.50597 + 2.02494i
35.6 1.19734 − 0.752583i 0 0.867238 − 1.80219i 0.0913223 − 0.0378270i 0 −3.05457 − 3.05457i −0.317923 − 2.81050i 0 0.0808758 − 0.114019i
35.7 1.33776 + 0.458699i 0 1.57919 + 1.22726i 2.70076 − 1.11869i 0 −1.06647 − 1.06647i 1.54963 + 2.36614i 0 4.12611 − 0.257702i
35.8 1.39224 + 0.248345i 0 1.87665 + 0.691511i −4.01952 + 1.66494i 0 2.72280 + 2.72280i 2.44101 + 1.42881i 0 −6.00960 + 1.31976i
107.1 −1.32473 − 0.495070i 0 1.50981 + 1.31167i 0.0505677 + 0.0209458i 0 1.44150 − 1.44150i −1.35072 − 2.48506i 0 −0.0566189 − 0.0527821i
107.2 −1.16775 + 0.797726i 0 0.727265 − 1.86308i −1.65625 − 0.686041i 0 −0.456585 + 0.456585i 0.636970 + 2.75577i 0 2.48135 − 0.520112i
107.3 −0.430512 − 1.34709i 0 −1.62932 + 1.15988i −1.78959 − 0.741273i 0 −2.33709 + 2.33709i 2.26391 + 1.69550i 0 −0.228123 + 2.72987i
107.4 −0.349793 + 1.37027i 0 −1.75529 − 0.958624i 2.97412 + 1.23192i 0 0.237717 − 0.237717i 1.92756 − 2.06990i 0 −2.72839 + 3.64443i
107.5 0.345448 − 1.37137i 0 −1.76133 − 0.947475i 1.64859 + 0.682869i 0 2.51270 − 2.51270i −1.90779 + 2.08814i 0 1.50597 − 2.02494i
107.6 1.19734 + 0.752583i 0 0.867238 + 1.80219i 0.0913223 + 0.0378270i 0 −3.05457 + 3.05457i −0.317923 + 2.81050i 0 0.0808758 + 0.114019i
107.7 1.33776 − 0.458699i 0 1.57919 − 1.22726i 2.70076 + 1.11869i 0 −1.06647 + 1.06647i 1.54963 − 2.36614i 0 4.12611 + 0.257702i
107.8 1.39224 − 0.248345i 0 1.87665 − 0.691511i −4.01952 − 1.66494i 0 2.72280 − 2.72280i 2.44101 − 1.42881i 0 −6.00960 − 1.31976i
179.1 −1.24371 + 0.673185i 0 1.09364 − 1.67450i −1.11994 − 2.70378i 0 1.57144 + 1.57144i −0.232933 + 2.81882i 0 3.21304 + 2.60880i
179.2 −1.24152 − 0.677214i 0 1.08276 + 1.68155i −0.0963530 − 0.232617i 0 0.617536 + 0.617536i −0.205502 − 2.82095i 0 −0.0379068 + 0.354051i
179.3 −0.450008 + 1.34071i 0 −1.59499 − 1.20666i 0.739921 + 1.78633i 0 0.385417 + 0.385417i 2.33553 − 1.59540i 0 −2.72791 + 0.188155i
179.4 0.290763 − 1.38400i 0 −1.83091 − 0.804831i 1.12344 + 2.71222i 0 3.03150 + 3.03150i −1.64625 + 2.29997i 0 4.08037 − 0.766228i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
96.o even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.w.b yes 32
3.b odd 2 1 288.2.w.a ✓ 32
4.b odd 2 1 1152.2.w.a 32
12.b even 2 1 1152.2.w.b 32
32.g even 8 1 1152.2.w.b 32
32.h odd 8 1 288.2.w.a ✓ 32
96.o even 8 1 inner 288.2.w.b yes 32
96.p odd 8 1 1152.2.w.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.w.a ✓ 32 3.b odd 2 1
288.2.w.a ✓ 32 32.h odd 8 1
288.2.w.b yes 32 1.a even 1 1 trivial
288.2.w.b yes 32 96.o even 8 1 inner
1152.2.w.a 32 4.b odd 2 1
1152.2.w.a 32 96.p odd 8 1
1152.2.w.b 32 12.b even 2 1
1152.2.w.b 32 32.g even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 16 T_{5}^{29} - 224 T_{5}^{27} + 896 T_{5}^{26} - 6080 T_{5}^{25} + 48512 T_{5}^{24} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display