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Properties

Label 54.7.b.a
Level $54$
Weight $7$
Character orbit 54.b
Analytic conductor $12.423$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [54,7,Mod(53,54)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("54.53"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 54.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-64,0,0,-410] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.4229205155\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 32 q^{4} + 6 \beta q^{5} - 205 q^{7} - 32 \beta q^{8} - 192 q^{10} - 186 \beta q^{11} - 2041 q^{13} - 205 \beta q^{14} + 1024 q^{16} - 1458 \beta q^{17} - 1501 q^{19} - 192 \beta q^{20} + \cdots - 75624 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{4} - 410 q^{7} - 384 q^{10} - 4082 q^{13} + 2048 q^{16} - 3002 q^{19} + 11904 q^{22} + 28946 q^{25} + 13120 q^{28} - 69980 q^{31} + 93312 q^{34} - 115250 q^{37} + 12288 q^{40} - 125132 q^{43}+ \cdots + 1052302 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
− 1.41421i
1.41421i
− 5.65685i 0 −32.0000 − 33.9411i 0 −205.000 181.019i 0 −192.000
53.2 5.65685i 0 −32.0000 33.9411i 0 −205.000 − 181.019i 0 −192.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.7.b.a ✓ 2
3.b odd 2 1 inner 54.7.b.a ✓ 2
4.b odd 2 1 432.7.e.f 2
9.c even 3 2 162.7.d.c 4
9.d odd 6 2 162.7.d.c 4
12.b even 2 1 432.7.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.7.b.a ✓ 2 1.a even 1 1 trivial
54.7.b.a ✓ 2 3.b odd 2 1 inner
162.7.d.c 4 9.c even 3 2
162.7.d.c 4 9.d odd 6 2
432.7.e.f 2 4.b odd 2 1
432.7.e.f 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 1152 \) acting on \(S_{7}^{\mathrm{new}}(54, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 32 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1152 \) Copy content Toggle raw display
$7$ \( (T + 205)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1107072 \) Copy content Toggle raw display
$13$ \( (T + 2041)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 68024448 \) Copy content Toggle raw display
$19$ \( (T + 1501)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 50320512 \) Copy content Toggle raw display
$29$ \( T^{2} + 808985088 \) Copy content Toggle raw display
$31$ \( (T + 34990)^{2} \) Copy content Toggle raw display
$37$ \( (T + 57625)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 18046960128 \) Copy content Toggle raw display
$43$ \( (T + 62566)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2693192832 \) Copy content Toggle raw display
$53$ \( T^{2} + 5957079552 \) Copy content Toggle raw display
$59$ \( T^{2} + 137850302592 \) Copy content Toggle raw display
$61$ \( (T + 61297)^{2} \) Copy content Toggle raw display
$67$ \( (T - 67691)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 259614885888 \) Copy content Toggle raw display
$73$ \( (T - 423983)^{2} \) Copy content Toggle raw display
$79$ \( (T + 707533)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 741235917312 \) Copy content Toggle raw display
$89$ \( T^{2} + 445764373632 \) Copy content Toggle raw display
$97$ \( (T - 526151)^{2} \) Copy content Toggle raw display
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