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Properties

Label 576.7.g.e
Level $576$
Weight $7$
Character orbit 576.g
Analytic conductor $132.511$
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] = [576,7,Mod(127,576)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) 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("576.127"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 576.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-180,0,0,0,0,0,0,0,-3524] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(132.511152165\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 48)
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 \(\beta = 108\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 90 q^{5} + \beta q^{7} + 9 \beta q^{11} - 1762 q^{13} + 1638 q^{17} - 67 \beta q^{19} - 72 \beta q^{23} - 7525 q^{25} - 16002 q^{29} - 85 \beta q^{31} - 90 \beta q^{35} - 61130 q^{37} + 98550 q^{41} + \cdots + 557026 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 180 q^{5} - 3524 q^{13} + 3276 q^{17} - 15050 q^{25} - 32004 q^{29} - 122260 q^{37} + 197100 q^{41} + 165314 q^{49} - 550692 q^{53} - 213268 q^{61} + 317160 q^{65} + 201956 q^{73} - 629856 q^{77}+ \cdots + 1114052 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0 −90.0000 0 − 187.061i 0 0 0
127.2 0 0 0 −90.0000 0 187.061i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.7.g.e 2
3.b odd 2 1 192.7.g.c 2
4.b odd 2 1 inner 576.7.g.e 2
8.b even 2 1 144.7.g.e 2
8.d odd 2 1 144.7.g.e 2
12.b even 2 1 192.7.g.c 2
24.f even 2 1 48.7.g.a ✓ 2
24.h odd 2 1 48.7.g.a ✓ 2
48.i odd 4 2 768.7.b.c 4
48.k even 4 2 768.7.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.g.a ✓ 2 24.f even 2 1
48.7.g.a ✓ 2 24.h odd 2 1
144.7.g.e 2 8.b even 2 1
144.7.g.e 2 8.d odd 2 1
192.7.g.c 2 3.b odd 2 1
192.7.g.c 2 12.b even 2 1
576.7.g.e 2 1.a even 1 1 trivial
576.7.g.e 2 4.b odd 2 1 inner
768.7.b.c 4 48.i odd 4 2
768.7.b.c 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 90 \) acting on \(S_{7}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 90)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 34992 \) Copy content Toggle raw display
$11$ \( T^{2} + 2834352 \) Copy content Toggle raw display
$13$ \( (T + 1762)^{2} \) Copy content Toggle raw display
$17$ \( (T - 1638)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 157079088 \) Copy content Toggle raw display
$23$ \( T^{2} + 181398528 \) Copy content Toggle raw display
$29$ \( (T + 16002)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 252817200 \) Copy content Toggle raw display
$37$ \( (T + 61130)^{2} \) Copy content Toggle raw display
$41$ \( (T - 98550)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2420361648 \) Copy content Toggle raw display
$47$ \( T^{2} + 31846779072 \) Copy content Toggle raw display
$53$ \( (T + 275346)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 64626059952 \) Copy content Toggle raw display
$61$ \( (T + 106634)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 304311462192 \) Copy content Toggle raw display
$71$ \( T^{2} + 5487305472 \) Copy content Toggle raw display
$73$ \( (T - 100978)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 5968550448 \) Copy content Toggle raw display
$83$ \( T^{2} + 415768260528 \) Copy content Toggle raw display
$89$ \( (T - 819054)^{2} \) Copy content Toggle raw display
$97$ \( (T - 557026)^{2} \) Copy content Toggle raw display
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