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Properties

Label 392.6.a.d
Level $392$
Weight $6$
Character orbit 392.a
Self dual yes
Analytic conductor $62.870$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [392,6,Mod(1,392)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("392.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,6,0,-82,0,0,0,222] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(62.8704573667\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{345}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 86 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{345}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 3) q^{3} + (3 \beta - 41) q^{5} + (6 \beta + 111) q^{9} + ( - 6 \beta + 170) q^{11} + ( - 39 \beta - 455) q^{13} + ( - 32 \beta + 912) q^{15} + (6 \beta - 1608) q^{17} + ( - 45 \beta + 337) q^{19}+ \cdots + (354 \beta + 6450) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 82 q^{5} + 222 q^{9} + 340 q^{11} - 910 q^{13} + 1824 q^{15} - 3216 q^{17} + 674 q^{19} - 1104 q^{23} + 3322 q^{25} + 3348 q^{27} + 8064 q^{29} + 6212 q^{31} - 3120 q^{33} - 8512 q^{37}+ \cdots + 12900 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.78709
9.78709
0 −15.5742 0 −96.7225 0 0 0 −0.445054 0
1.2 0 21.5742 0 14.7225 0 0 0 222.445 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.6.a.d 2
4.b odd 2 1 784.6.a.u 2
7.b odd 2 1 56.6.a.e ✓ 2
7.c even 3 2 392.6.i.i 4
7.d odd 6 2 392.6.i.j 4
21.c even 2 1 504.6.a.i 2
28.d even 2 1 112.6.a.i 2
56.e even 2 1 448.6.a.t 2
56.h odd 2 1 448.6.a.v 2
84.h odd 2 1 1008.6.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.e ✓ 2 7.b odd 2 1
112.6.a.i 2 28.d even 2 1
392.6.a.d 2 1.a even 1 1 trivial
392.6.i.i 4 7.c even 3 2
392.6.i.j 4 7.d odd 6 2
448.6.a.t 2 56.e even 2 1
448.6.a.v 2 56.h odd 2 1
504.6.a.i 2 21.c even 2 1
784.6.a.u 2 4.b odd 2 1
1008.6.a.bd 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 6T_{3} - 336 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(392))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 6T - 336 \) Copy content Toggle raw display
$5$ \( T^{2} + 82T - 1424 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 340T + 16480 \) Copy content Toggle raw display
$13$ \( T^{2} + 910T - 317720 \) Copy content Toggle raw display
$17$ \( T^{2} + 3216 T + 2573244 \) Copy content Toggle raw display
$19$ \( T^{2} - 674T - 585056 \) Copy content Toggle raw display
$23$ \( T^{2} + 1104 T - 1483776 \) Copy content Toggle raw display
$29$ \( T^{2} - 8064 T + 11773404 \) Copy content Toggle raw display
$31$ \( T^{2} - 6212 T - 5567264 \) Copy content Toggle raw display
$37$ \( T^{2} + 8512 T - 34360964 \) Copy content Toggle raw display
$41$ \( T^{2} - 1304 T - 158165876 \) Copy content Toggle raw display
$43$ \( T^{2} + 10004 T - 194677376 \) Copy content Toggle raw display
$47$ \( T^{2} - 12748 T - 135251744 \) Copy content Toggle raw display
$53$ \( T^{2} + 11220 T - 172017180 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 1715115024 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 2636833536 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 1158165296 \) Copy content Toggle raw display
$71$ \( T^{2} - 89720 T + 967897600 \) Copy content Toggle raw display
$73$ \( T^{2} - 55588 T - 243399884 \) Copy content Toggle raw display
$79$ \( T^{2} - 48824 T - 724002176 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 3779136224 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 7628401980 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 1150801884 \) Copy content Toggle raw display
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