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Properties

Label 6.28.a.a
Level $6$
Weight $28$
Character orbit 6.a
Self dual yes
Analytic conductor $27.711$
Analytic rank $1$
Dimension $1$
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] = [6,28,Mod(1,6)] 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("6.1"); S:= CuspForms(chi, 28); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 28, names="a")
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-8192,1594323] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(27.7113344903\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 8192 q^{2} + 1594323 q^{3} + 67108864 q^{4} + 1992850350 q^{5} - 13060694016 q^{6} - 321751224088 q^{7} - 549755813888 q^{8} + 2541865828329 q^{9} - 16325430067200 q^{10} - 25382713790940 q^{11} + 106993205379072 q^{12}+ \cdots - 64\!\cdots\!60 q^{99}+O(q^{100}) \) 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
0
−8192.00 1.59432e6 6.71089e7 1.99285e9 −1.30607e10 −3.21751e11 −5.49756e11 2.54187e12 −1.63254e13
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)

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 6.28.a.a ✓ 1
3.b odd 2 1 18.28.a.d 1
4.b odd 2 1 48.28.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.28.a.a ✓ 1 1.a even 1 1 trivial
18.28.a.d 1 3.b odd 2 1
48.28.a.b 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 1992850350 \) acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(6))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8192 \) Copy content Toggle raw display
$3$ \( T - 1594323 \) Copy content Toggle raw display
$5$ \( T - 1992850350 \) Copy content Toggle raw display
$7$ \( T + 321751224088 \) Copy content Toggle raw display
$11$ \( T + 25382713790940 \) Copy content Toggle raw display
$13$ \( T + 618784763833834 \) Copy content Toggle raw display
$17$ \( T - 42\!\cdots\!82 \) Copy content Toggle raw display
$19$ \( T - 42\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T + 30\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T + 10\!\cdots\!14 \) Copy content Toggle raw display
$31$ \( T - 55\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T + 28\!\cdots\!66 \) Copy content Toggle raw display
$41$ \( T + 34\!\cdots\!06 \) Copy content Toggle raw display
$43$ \( T - 20\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T + 81\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T + 11\!\cdots\!78 \) Copy content Toggle raw display
$59$ \( T + 11\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T + 19\!\cdots\!30 \) Copy content Toggle raw display
$67$ \( T + 19\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T - 22\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T - 14\!\cdots\!66 \) Copy content Toggle raw display
$79$ \( T + 42\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T + 97\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T + 92\!\cdots\!78 \) Copy content Toggle raw display
$97$ \( T - 12\!\cdots\!54 \) Copy content Toggle raw display
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