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Properties

Label 8464.2.a.bz
Level $8464$
Weight $2$
Character orbit 8464.a
Self dual yes
Analytic conductor $67.585$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,18,0,0,0,12] 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: yes
Analytic conductor: \(67.5853802708\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.197448192.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 18x^{4} - 6x^{3} + 48x^{2} - 23 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2116)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + \beta_{3} q^{5} + ( - \beta_{3} + \beta_1) q^{7} + (\beta_{5} - \beta_{2} + 3) q^{9} + (\beta_{4} - \beta_{3} - \beta_1) q^{11} + (\beta_{5} + 2) q^{13} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{15}+ \cdots + (4 \beta_{4} + 2 \beta_{3} + 8 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{9} + 12 q^{13} + 18 q^{25} + 24 q^{27} + 24 q^{29} + 12 q^{31} - 48 q^{35} + 36 q^{39} + 12 q^{41} + 12 q^{47} + 18 q^{49} - 36 q^{55} + 12 q^{59} - 24 q^{71} - 24 q^{73} - 60 q^{75} + 24 q^{77}+ \cdots + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 18x^{4} - 6x^{3} + 48x^{2} - 23 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -12\nu^{5} - 27\nu^{4} + 200\nu^{3} + 522\nu^{2} - 207\nu - 600 ) / 179 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -24\nu^{5} - 54\nu^{4} + 400\nu^{3} + 1044\nu^{2} - 56\nu - 1200 ) / 179 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -27\nu^{5} - 16\nu^{4} + 450\nu^{3} + 369\nu^{2} - 600\nu + 82 ) / 179 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -35\nu^{5} - 34\nu^{4} + 643\nu^{3} + 717\nu^{2} - 1454\nu - 497 ) / 179 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 42\nu^{5} + 5\nu^{4} - 700\nu^{3} - 395\nu^{2} + 1172\nu + 310 ) / 179 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - 4\beta_{3} + \beta_{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{4} - 3\beta_{3} + 6\beta_{2} - 14\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -36\beta_{5} - 64\beta_{3} + 21\beta_{2} - 24\beta _1 + 152 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -6\beta_{5} + 100\beta_{4} - 130\beta_{3} + 179\beta_{2} - 408\beta _1 + 180 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−3.55932
−0.730889
4.08329
1.25487
−1.93819
0.890237
0 −2.74657 0 −3.03363 0 4.44785 0 4.54364 0
1.2 0 −2.74657 0 3.03363 0 −4.44785 0 4.54364 0
1.3 0 −0.454904 0 −3.77465 0 2.36043 0 −2.79306 0
1.4 0 −0.454904 0 3.77465 0 −2.36043 0 −2.79306 0
1.5 0 3.20147 0 −0.741015 0 2.15523 0 7.24943 0
1.6 0 3.20147 0 0.741015 0 −2.15523 0 7.24943 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(23\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8464.2.a.bz 6
4.b odd 2 1 2116.2.a.g ✓ 6
23.b odd 2 1 inner 8464.2.a.bz 6
92.b even 2 1 2116.2.a.g ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2116.2.a.g ✓ 6 4.b odd 2 1
2116.2.a.g ✓ 6 92.b even 2 1
8464.2.a.bz 6 1.a even 1 1 trivial
8464.2.a.bz 6 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8464))\):

\( T_{3}^{3} - 9T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{6} - 24T_{5}^{4} + 144T_{5}^{2} - 72 \) Copy content Toggle raw display
\( T_{7}^{6} - 30T_{7}^{4} + 228T_{7}^{2} - 512 \) Copy content Toggle raw display
\( T_{13}^{3} - 6T_{13}^{2} + 3T_{13} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 9 T - 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} - 24 T^{4} + \cdots - 72 \) Copy content Toggle raw display
$7$ \( T^{6} - 30 T^{4} + \cdots - 512 \) Copy content Toggle raw display
$11$ \( T^{6} - 54 T^{4} + \cdots - 2592 \) Copy content Toggle raw display
$13$ \( (T^{3} - 6 T^{2} + 3 T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 96 T^{4} + \cdots - 25992 \) Copy content Toggle raw display
$19$ \( T^{6} - 60 T^{4} + \cdots - 1568 \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( (T^{3} - 12 T^{2} + \cdots - 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} + \cdots + 144)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 144 T^{4} + \cdots - 55112 \) Copy content Toggle raw display
$41$ \( (T^{3} - 6 T^{2} + 3 T + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 8)^{3} \) Copy content Toggle raw display
$47$ \( (T^{3} - 6 T^{2} - 15 T + 84)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} - 198 T^{4} + \cdots - 648 \) Copy content Toggle raw display
$59$ \( (T^{3} - 6 T^{2} - 60 T - 96)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - 102 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$67$ \( T^{6} - 342 T^{4} + \cdots - 729632 \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} + \cdots - 276)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 12 T^{2} + \cdots + 24)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 324 T^{4} + \cdots - 67712 \) Copy content Toggle raw display
$83$ \( T^{6} - 294 T^{4} + \cdots - 152352 \) Copy content Toggle raw display
$89$ \( T^{6} - 42 T^{4} + \cdots - 1152 \) Copy content Toggle raw display
$97$ \( T^{6} - 408 T^{4} + \cdots - 1976072 \) Copy content Toggle raw display
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