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Properties

Label 738.2.ba.a
Level $738$
Weight $2$
Character orbit 738.ba
Analytic conductor $5.893$
Analytic rank $0$
Dimension $48$
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 = [48,0,0,0,-4,0,4,0,0,0,-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: \(48\)
Relative dimension: \(3\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{5} + 4 q^{7} - 8 q^{11} - 4 q^{13} + 4 q^{14} + 12 q^{16} - 16 q^{17} - 4 q^{19} + 16 q^{20} + 20 q^{22} - 40 q^{25} - 20 q^{26} - 4 q^{28} + 32 q^{29} - 40 q^{31} - 4 q^{34} - 52 q^{35} - 24 q^{37}+ \cdots + 16 q^{97}+O(q^{100}) \) 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 −1.48149 + 2.90758i 0 2.16613 + 1.32741i 0.891007 − 0.453990i 0 −1.00840 + 3.10354i
17.2 0.987688 − 0.156434i 0 0.951057 − 0.309017i −0.172522 + 0.338593i 0 −0.262313 − 0.160746i 0.891007 − 0.453990i 0 −0.117430 + 0.361413i
17.3 0.987688 − 0.156434i 0 0.951057 − 0.309017i 1.79605 − 3.52494i 0 −0.0699988 − 0.0428953i 0.891007 − 0.453990i 0 1.22251 − 3.76251i
35.1 0.891007 + 0.453990i 0 0.587785 + 0.809017i −3.14624 − 0.498316i 0 0.0181761 + 0.230950i 0.156434 + 0.987688i 0 −2.57709 − 1.87237i
35.2 0.891007 + 0.453990i 0 0.587785 + 0.809017i 1.00092 + 0.158531i 0 0.204016 + 2.59227i 0.156434 + 0.987688i 0 0.819859 + 0.595662i
35.3 0.891007 + 0.453990i 0 0.587785 + 0.809017i 3.04212 + 0.481824i 0 −0.338143 − 4.29651i 0.156434 + 0.987688i 0 2.49180 + 1.81040i
53.1 −0.987688 − 0.156434i 0 0.951057 + 0.309017i −1.70058 − 3.33758i 0 −0.496661 − 0.810478i −0.891007 − 0.453990i 0 1.15753 + 3.56252i
53.2 −0.987688 − 0.156434i 0 0.951057 + 0.309017i 0.329149 + 0.645992i 0 −0.847255 − 1.38259i −0.891007 − 0.453990i 0 −0.224042 − 0.689529i
53.3 −0.987688 − 0.156434i 0 0.951057 + 0.309017i 1.51347 + 2.97036i 0 2.30370 + 3.75930i −0.891007 − 0.453990i 0 −1.03017 − 3.17055i
71.1 0.156434 + 0.987688i 0 −0.951057 + 0.309017i −2.46374 − 1.25534i 0 0.486233 + 2.02531i −0.453990 − 0.891007i 0 0.854470 − 2.62979i
71.2 0.156434 + 0.987688i 0 −0.951057 + 0.309017i −0.973712 − 0.496131i 0 −0.502389 − 2.09260i −0.453990 − 0.891007i 0 0.337701 − 1.03934i
71.3 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 1.67738 + 0.854668i 0 0.674403 + 2.80909i −0.453990 − 0.891007i 0 −0.581745 + 1.79043i
89.1 −0.891007 + 0.453990i 0 0.587785 − 0.809017i −2.53900 + 0.402138i 0 1.91317 + 0.150570i −0.156434 + 0.987688i 0 2.07970 − 1.51099i
89.2 −0.891007 + 0.453990i 0 0.587785 − 0.809017i 0.699688 − 0.110820i 0 −2.26358 − 0.178148i −0.156434 + 0.987688i 0 −0.573115 + 0.416393i
89.3 −0.891007 + 0.453990i 0 0.587785 − 0.809017i 2.73611 − 0.433358i 0 −2.05379 − 0.161637i −0.156434 + 0.987688i 0 −2.24116 + 1.62829i
179.1 −0.156434 + 0.987688i 0 −0.951057 − 0.309017i −2.39103 + 1.21829i 0 2.33211 + 0.559890i 0.453990 − 0.891007i 0 −0.829251 − 2.55217i
179.2 −0.156434 + 0.987688i 0 −0.951057 − 0.309017i 0.176229 − 0.0897932i 0 −0.0776566 − 0.0186437i 0.453990 − 0.891007i 0 0.0611194 + 0.188106i
179.3 −0.156434 + 0.987688i 0 −0.951057 − 0.309017i 0.454725 − 0.231694i 0 −2.47024 − 0.593051i 0.453990 − 0.891007i 0 0.157707 + 0.485372i
233.1 −0.453990 + 0.891007i 0 −0.587785 − 0.809017i −0.524441 + 3.31119i 0 2.69206 − 3.15200i 0.987688 − 0.156434i 0 −2.71220 − 1.97053i
233.2 −0.453990 + 0.891007i 0 −0.587785 − 0.809017i 0.101091 − 0.638263i 0 −0.804009 + 0.941374i 0.987688 − 0.156434i 0 0.522802 + 0.379838i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.3
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.a ✓ 48
3.b odd 2 1 738.2.ba.b yes 48
41.h odd 40 1 738.2.ba.b yes 48
123.o even 40 1 inner 738.2.ba.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
738.2.ba.a ✓ 48 1.a even 1 1 trivial
738.2.ba.a ✓ 48 123.o even 40 1 inner
738.2.ba.b yes 48 3.b odd 2 1
738.2.ba.b yes 48 41.h odd 40 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} + 4 T_{5}^{47} + 28 T_{5}^{46} + 72 T_{5}^{45} + 106 T_{5}^{44} + 44 T_{5}^{43} + \cdots + 57138481 \) acting on \(S_{2}^{\mathrm{new}}(738, [\chi])\). Copy content Toggle raw display