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Properties

Label 738.2.u.f
Level $738$
Weight $2$
Character orbit 738.u
Analytic conductor $5.893$
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] = [738,2,Mod(289,738)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(738, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([0, 13])) 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("738.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 738 = 2 \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 738.u (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,8,0,0,4,0,0,-4,4] 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: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 246)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{4} + 4 q^{7} - 4 q^{10} + 4 q^{11} - 12 q^{13} - 4 q^{14} - 8 q^{16} + 4 q^{17} - 4 q^{19} - 4 q^{22} - 40 q^{23} + 12 q^{25} - 8 q^{26} - 4 q^{28} - 16 q^{29} - 4 q^{31} - 16 q^{34} + 8 q^{35}+ \cdots + 56 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0.951057 − 0.309017i 0 0.809017 − 0.587785i −1.84265 − 2.53619i 0 0.0346438 + 0.0679922i 0.587785 − 0.809017i 0 −2.53619 − 1.84265i
289.2 0.951057 − 0.309017i 0 0.809017 − 0.587785i −0.761749 − 1.04846i 0 1.24897 + 2.45124i 0.587785 − 0.809017i 0 −1.04846 − 0.761749i
289.3 0.951057 − 0.309017i 0 0.809017 − 0.587785i 0.0603073 + 0.0830058i 0 −2.01841 − 3.96135i 0.587785 − 0.809017i 0 0.0830058 + 0.0603073i
289.4 0.951057 − 0.309017i 0 0.809017 − 0.587785i 1.36852 + 1.88361i 0 0.450716 + 0.884580i 0.587785 − 0.809017i 0 1.88361 + 1.36852i
307.1 −0.951057 − 0.309017i 0 0.809017 + 0.587785i −1.96776 + 2.70839i 0 0.898760 + 0.457941i −0.587785 − 0.809017i 0 2.70839 − 1.96776i
307.2 −0.951057 − 0.309017i 0 0.809017 + 0.587785i −0.763524 + 1.05090i 0 0.161080 + 0.0820744i −0.587785 − 0.809017i 0 1.05090 − 0.763524i
307.3 −0.951057 − 0.309017i 0 0.809017 + 0.587785i 1.83795 − 2.52972i 0 −0.370027 − 0.188538i −0.587785 − 0.809017i 0 −2.52972 + 1.83795i
307.4 −0.951057 − 0.309017i 0 0.809017 + 0.587785i 2.06891 − 2.84761i 0 2.83033 + 1.44213i −0.587785 − 0.809017i 0 −2.84761 + 2.06891i
361.1 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i −3.52711 + 1.14603i 0 −0.702798 + 4.43729i 0.951057 + 0.309017i 0 1.14603 − 3.52711i
361.2 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i −2.20810 + 0.717455i 0 0.160226 − 1.01163i 0.951057 + 0.309017i 0 0.717455 − 2.20810i
361.3 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i −0.166180 + 0.0539952i 0 0.415581 − 2.62387i 0.951057 + 0.309017i 0 0.0539952 − 0.166180i
361.4 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i 3.99928 − 1.29944i 0 0.684527 − 4.32193i 0.951057 + 0.309017i 0 −1.29944 + 3.99928i
415.1 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i −3.52711 − 1.14603i 0 −0.702798 − 4.43729i 0.951057 − 0.309017i 0 1.14603 + 3.52711i
415.2 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i −2.20810 − 0.717455i 0 0.160226 + 1.01163i 0.951057 − 0.309017i 0 0.717455 + 2.20810i
415.3 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i −0.166180 − 0.0539952i 0 0.415581 + 2.62387i 0.951057 − 0.309017i 0 0.0539952 + 0.166180i
415.4 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i 3.99928 + 1.29944i 0 0.684527 + 4.32193i 0.951057 − 0.309017i 0 −1.29944 − 3.99928i
487.1 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −2.14845 − 0.698074i 0 −0.0615361 + 0.00974637i −0.951057 + 0.309017i 0 −0.698074 − 2.14845i
487.2 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −0.134590 − 0.0437310i 0 −1.21079 + 0.191771i −0.951057 + 0.309017i 0 −0.0437310 − 0.134590i
487.3 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 1.70227 + 0.553100i 0 −4.40876 + 0.698278i −0.951057 + 0.309017i 0 0.553100 + 1.70227i
487.4 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 2.48289 + 0.806739i 0 3.88748 − 0.615717i −0.951057 + 0.309017i 0 0.806739 + 2.48289i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.g even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 738.2.u.f 32
3.b odd 2 1 246.2.n.b ✓ 32
41.g even 20 1 inner 738.2.u.f 32
123.m odd 20 1 246.2.n.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
246.2.n.b ✓ 32 3.b odd 2 1
246.2.n.b ✓ 32 123.m odd 20 1
738.2.u.f 32 1.a even 1 1 trivial
738.2.u.f 32 41.g even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} - 26 T_{5}^{30} + 448 T_{5}^{28} - 6332 T_{5}^{26} + 1700 T_{5}^{25} + 99624 T_{5}^{24} + \cdots + 192721 \) acting on \(S_{2}^{\mathrm{new}}(738, [\chi])\). Copy content Toggle raw display