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Properties

Label 8424.2.a.ba
Level $8424$
Weight $2$
Character orbit 8424.a
Self dual yes
Analytic conductor $67.266$
Analytic rank $1$
Dimension $8$
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] = [8424,2,Mod(1,8424)] 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("8424.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8424, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8424 = 2^{3} \cdot 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8424.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,1,0,-2] 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: yes
Analytic conductor: \(67.2659786627\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 17x^{6} + 48x^{5} + 76x^{4} - 168x^{3} - 153x^{2} + 144x + 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 936)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{5} - \beta_{3} q^{7} + ( - \beta_1 + 1) q^{11} - q^{13} + (\beta_{6} + \beta_{5} + \beta_{4}) q^{17} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{19} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots - 1) q^{23}+ \cdots + (\beta_{7} + 3 \beta_{5} - \beta_{4} + \cdots - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{5} - 2 q^{7} + 5 q^{11} - 8 q^{13} + 4 q^{17} - 9 q^{19} - 15 q^{23} + 9 q^{25} + 4 q^{29} - 4 q^{31} - 9 q^{35} - 5 q^{37} - 11 q^{43} - 33 q^{47} + 6 q^{49} + 10 q^{53} + q^{55} + 11 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 17x^{6} + 48x^{5} + 76x^{4} - 168x^{3} - 153x^{2} + 144x + 108 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 15\nu^{5} - 16\nu^{4} - 44\nu^{3} + 48\nu^{2} + 21\nu - 36 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} - 31\nu^{5} - 3\nu^{4} + 104\nu^{3} + 54\nu^{2} - 72\nu - 54 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 31\nu^{5} - 3\nu^{4} + 104\nu^{3} + 63\nu^{2} - 81\nu - 99 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 3\nu^{6} + 79\nu^{5} - 42\nu^{4} - 284\nu^{3} + 42\nu^{2} + 297\nu + 90 ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 3\nu^{6} - 85\nu^{5} - 48\nu^{4} + 362\nu^{3} + 222\nu^{2} - 369\nu - 234 ) / 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + \nu^{6} - 16\nu^{5} - 16\nu^{4} + 60\nu^{3} + 69\nu^{2} - 54\nu - 63 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} - 2\beta_{5} - 2\beta_{4} + \beta_{2} + 9\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 11\beta_{4} - 14\beta_{3} - 2\beta_{2} + 12\beta _1 + 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15\beta_{7} - 16\beta_{6} - 26\beta_{5} - 27\beta_{4} + \beta_{3} + 12\beta_{2} + 94\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{7} + 3\beta_{5} + 120\beta_{4} - 165\beta_{3} - 31\beta_{2} + 125\beta _1 + 486 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 182\beta_{7} - 196\beta_{6} - 299\beta_{5} - 325\beta_{4} + 26\beta_{3} + 131\beta_{2} + 1016\beta _1 + 136 \) 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
2.10793
−3.38756
3.27655
−1.10115
−0.679943
1.15435
3.26539
−1.63558
0 0 0 −3.76619 0 −3.19190 0 0 0
1.2 0 0 0 −1.44569 0 4.19661 0 0 0
1.3 0 0 0 −1.42619 0 −0.447398 0 0 0
1.4 0 0 0 −1.15244 0 0.694280 0 0 0
1.5 0 0 0 −0.432935 0 1.00463 0 0 0
1.6 0 0 0 2.55570 0 −3.49023 0 0 0
1.7 0 0 0 2.87832 0 2.78194 0 0 0
1.8 0 0 0 3.78944 0 −3.54794 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(13\) \( +1 \)

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 8424.2.a.ba 8
3.b odd 2 1 8424.2.a.y 8
9.c even 3 2 2808.2.q.e 16
9.d odd 6 2 936.2.q.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.q.f ✓ 16 9.d odd 6 2
2808.2.q.e 16 9.c even 3 2
8424.2.a.y 8 3.b odd 2 1
8424.2.a.ba 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8424))\):

\( T_{5}^{8} - T_{5}^{7} - 24T_{5}^{6} + 13T_{5}^{5} + 167T_{5}^{4} + 45T_{5}^{3} - 390T_{5}^{2} - 414T_{5} - 108 \) Copy content Toggle raw display
\( T_{7}^{8} + 2T_{7}^{7} - 29T_{7}^{6} - 53T_{7}^{5} + 239T_{7}^{4} + 295T_{7}^{3} - 612T_{7}^{2} + 12T_{7} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} + \cdots - 108 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{8} - 5 T^{7} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( (T + 1)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + \cdots + 13689 \) Copy content Toggle raw display
$19$ \( T^{8} + 9 T^{7} + \cdots - 10092 \) Copy content Toggle raw display
$23$ \( T^{8} + 15 T^{7} + \cdots + 15346 \) Copy content Toggle raw display
$29$ \( T^{8} - 4 T^{7} + \cdots + 2916 \) Copy content Toggle raw display
$31$ \( T^{8} + 4 T^{7} + \cdots - 193244 \) Copy content Toggle raw display
$37$ \( T^{8} + 5 T^{7} + \cdots + 96 \) Copy content Toggle raw display
$41$ \( T^{8} - 84 T^{6} + \cdots - 54 \) Copy content Toggle raw display
$43$ \( T^{8} + 11 T^{7} + \cdots - 10528669 \) Copy content Toggle raw display
$47$ \( T^{8} + 33 T^{7} + \cdots + 1834524 \) Copy content Toggle raw display
$53$ \( T^{8} - 10 T^{7} + \cdots - 183546 \) Copy content Toggle raw display
$59$ \( T^{8} - 11 T^{7} + \cdots + 1945134 \) Copy content Toggle raw display
$61$ \( T^{8} + 24 T^{7} + \cdots - 3767958 \) Copy content Toggle raw display
$67$ \( T^{8} + 18 T^{7} + \cdots + 160236 \) Copy content Toggle raw display
$71$ \( T^{8} + 15 T^{7} + \cdots - 3942256 \) Copy content Toggle raw display
$73$ \( T^{8} + 13 T^{7} + \cdots - 1056606 \) Copy content Toggle raw display
$79$ \( T^{8} + 12 T^{7} + \cdots + 729042 \) Copy content Toggle raw display
$83$ \( T^{8} - 8 T^{7} + \cdots - 12168 \) Copy content Toggle raw display
$89$ \( T^{8} + 17 T^{7} + \cdots + 55296 \) Copy content Toggle raw display
$97$ \( T^{8} + 16 T^{7} + \cdots - 28497344 \) Copy content Toggle raw display
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