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Properties

Label 738.2.ba.d
Level $738$
Weight $2$
Character orbit 738.ba
Analytic conductor $5.893$
Analytic rank $0$
Dimension $64$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [738,2,Mod(17,738)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(738, base_ring=CyclotomicField(40)) chi = DirichletCharacter(H, H._module([20, 33])) 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("738.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 738 = 2 \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 738.ba (of order \(40\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64,0,0,0,0,0,-4,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.89295966917\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(4\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 4 q^{7} + 8 q^{11} + 16 q^{13} - 4 q^{14} + 16 q^{16} + 16 q^{17} + 4 q^{19} - 4 q^{22} - 48 q^{23} + 40 q^{25} + 20 q^{26} + 4 q^{28} - 4 q^{29} + 40 q^{31} + 4 q^{34} + 52 q^{35} + 8 q^{37} - 16 q^{38}+ \cdots + 64 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0.987688 − 0.156434i 0 0.951057 − 0.309017i −1.22555 + 2.40528i 0 −3.01188 − 1.84568i 0.891007 − 0.453990i 0 −0.834195 + 2.56739i
17.2 0.987688 − 0.156434i 0 0.951057 − 0.309017i −0.664363 + 1.30389i 0 −1.15147 − 0.705622i 0.891007 − 0.453990i 0 −0.452211 + 1.39176i
17.3 0.987688 − 0.156434i 0 0.951057 − 0.309017i 0.825861 − 1.62084i 0 3.92357 + 2.40437i 0.891007 − 0.453990i 0 0.562138 − 1.73008i
17.4 0.987688 − 0.156434i 0 0.951057 − 0.309017i 1.06405 − 2.08832i 0 −1.59404 − 0.976830i 0.891007 − 0.453990i 0 0.724268 − 2.22907i
35.1 0.891007 + 0.453990i 0 0.587785 + 0.809017i −2.57126 − 0.407247i 0 0.193635 + 2.46037i 0.156434 + 0.987688i 0 −2.10612 − 1.53019i
35.2 0.891007 + 0.453990i 0 0.587785 + 0.809017i −2.42465 − 0.384027i 0 −0.225585 − 2.86633i 0.156434 + 0.987688i 0 −1.98603 − 1.44294i
35.3 0.891007 + 0.453990i 0 0.587785 + 0.809017i 1.33031 + 0.210701i 0 −0.125683 − 1.59696i 0.156434 + 0.987688i 0 1.08966 + 0.791685i
35.4 0.891007 + 0.453990i 0 0.587785 + 0.809017i 3.66559 + 0.580573i 0 0.273584 + 3.47621i 0.156434 + 0.987688i 0 3.00249 + 2.18144i
53.1 −0.987688 − 0.156434i 0 0.951057 + 0.309017i −1.33996 − 2.62982i 0 0.701084 + 1.14406i −0.891007 − 0.453990i 0 0.912069 + 2.80706i
53.2 −0.987688 − 0.156434i 0 0.951057 + 0.309017i −0.119754 − 0.235030i 0 1.02092 + 1.66599i −0.891007 − 0.453990i 0 0.0815126 + 0.250870i
53.3 −0.987688 − 0.156434i 0 0.951057 + 0.309017i 0.128614 + 0.252420i 0 −2.05782 − 3.35806i −0.891007 − 0.453990i 0 −0.0875438 − 0.269432i
53.4 −0.987688 − 0.156434i 0 0.951057 + 0.309017i 1.33110 + 2.61243i 0 −0.623970 − 1.01823i −0.891007 − 0.453990i 0 −0.906038 − 2.78850i
71.1 0.156434 + 0.987688i 0 −0.951057 + 0.309017i −3.89209 − 1.98312i 0 2.58756e−5 0 0.000107780i −0.453990 − 0.891007i 0 1.34984 − 4.15440i
71.2 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 0.270262 + 0.137705i 0 −0.981321 − 4.08750i −0.453990 − 0.891007i 0 −0.0937316 + 0.288476i
71.3 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 0.379411 + 0.193319i 0 −0.349510 − 1.45581i −0.453990 − 0.891007i 0 −0.131586 + 0.404981i
71.4 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 3.24241 + 1.65209i 0 0.672558 + 2.80140i −0.453990 − 0.891007i 0 −1.12453 + 3.46094i
89.1 −0.891007 + 0.453990i 0 0.587785 − 0.809017i −2.93619 + 0.465047i 0 4.14775 + 0.326435i −0.156434 + 0.987688i 0 2.40504 − 1.74736i
89.2 −0.891007 + 0.453990i 0 0.587785 − 0.809017i −1.54573 + 0.244820i 0 −1.04101 − 0.0819295i −0.156434 + 0.987688i 0 1.26611 − 0.919884i
89.3 −0.891007 + 0.453990i 0 0.587785 − 0.809017i 1.56662 − 0.248128i 0 −3.71285 − 0.292207i −0.156434 + 0.987688i 0 −1.28322 + 0.932313i
89.4 −0.891007 + 0.453990i 0 0.587785 − 0.809017i 2.91531 − 0.461739i 0 3.01030 + 0.236916i −0.156434 + 0.987688i 0 −2.38793 + 1.73493i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
123.o even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 738.2.ba.d yes 64
3.b odd 2 1 738.2.ba.c ✓ 64
41.h odd 40 1 738.2.ba.c ✓ 64
123.o even 40 1 inner 738.2.ba.d yes 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
738.2.ba.c ✓ 64 3.b odd 2 1
738.2.ba.c ✓ 64 41.h odd 40 1
738.2.ba.d yes 64 1.a even 1 1 trivial
738.2.ba.d yes 64 123.o even 40 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{64} - 20 T_{5}^{62} + 22 T_{5}^{60} - 568 T_{5}^{59} + 5920 T_{5}^{58} + 8104 T_{5}^{57} + \cdots + 1943350309681 \) acting on \(S_{2}^{\mathrm{new}}(738, [\chi])\). Copy content Toggle raw display