↑

Properties

Label 8464.2.a.by
Level $8464$
Weight $2$
Character orbit 8464.a
Self dual yes
Analytic conductor $67.585$
Analytic rank $1$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

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 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")
 
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);
 
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,-4,0,0,0,0,0,10,0,0,0,-4] 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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.26849792.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 14x^{4} - 2x^{3} + 28x^{2} - 4x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4232)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + (\beta_{4} - \beta_{3}) q^{5} - \beta_{2} q^{7} + ( - \beta_{5} + 2 \beta_1 + 2) q^{9} + (\beta_{4} + \beta_{2}) q^{11} + ( - \beta_1 - 1) q^{13} + ( - 3 \beta_{4} + 3 \beta_{3} + 2 \beta_{2}) q^{15}+ \cdots + (2 \beta_{4} - \beta_{3} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} + 10 q^{9} - 4 q^{13} + 14 q^{25} - 40 q^{27} - 12 q^{29} - 4 q^{31} - 16 q^{35} + 28 q^{39} + 16 q^{41} - 28 q^{47} + 2 q^{49} + 28 q^{55} - 36 q^{59} - 32 q^{71} + 20 q^{73} - 80 q^{75}+ \cdots - 20 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 14x^{4} - 2x^{3} + 28x^{2} - 4x - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -6\nu^{5} + 23\nu^{4} + 80\nu^{3} - 261\nu^{2} - 228\nu + 191 ) / 101 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{5} + 3\nu^{4} - 73\nu^{3} - 56\nu^{2} - 201\nu - 19 ) / 101 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{5} + 16\nu^{4} - 120\nu^{3} - 265\nu^{2} + 140\nu + 370 ) / 101 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24\nu^{5} + 9\nu^{4} - 320\nu^{3} - 168\nu^{2} + 407\nu + 44 ) / 101 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 42\nu^{5} + 41\nu^{4} - 560\nu^{3} - 597\nu^{2} + 788\nu + 279 ) / 101 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 4\beta_{3} + \beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{5} - 11\beta_{4} + 3\beta_{2} - 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17\beta_{5} - 8\beta_{4} - 48\beta_{3} + 15\beta _1 + 104 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 117\beta_{5} - 248\beta_{4} - 10\beta_{3} + 80\beta_{2} - 115\beta _1 + 54 ) / 2 \) 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
−0.446592
−3.27502
0.700694
3.52912
1.16011
−1.66832
0 −3.32340 0 −4.04579 0 −0.654213 0 8.04502 0
1.2 0 −3.32340 0 4.04579 0 0.654213 0 8.04502 0
1.3 0 −0.357926 0 −1.57672 0 2.08290 0 −2.87189 0
1.4 0 −0.357926 0 1.57672 0 −2.08290 0 −2.87189 0
1.5 0 1.68133 0 −1.77357 0 4.15133 0 −0.173127 0
1.6 0 1.68133 0 1.77357 0 −4.15133 0 −0.173127 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.by 6
4.b odd 2 1 4232.2.a.v ✓ 6
23.b odd 2 1 inner 8464.2.a.by 6
92.b even 2 1 4232.2.a.v ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4232.2.a.v ✓ 6 4.b odd 2 1
4232.2.a.v ✓ 6 92.b even 2 1
8464.2.a.by 6 1.a even 1 1 trivial
8464.2.a.by 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8464))\):

\( T_{3}^{3} + 2T_{3}^{2} - 5T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{6} - 22T_{5}^{4} + 100T_{5}^{2} - 128 \) Copy content Toggle raw display
\( T_{7}^{6} - 22T_{7}^{4} + 84T_{7}^{2} - 32 \) Copy content Toggle raw display
\( T_{13}^{3} + 2T_{13}^{2} - 5T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} + 2 T^{2} - 5 T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} - 22 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$7$ \( T^{6} - 22 T^{4} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( T^{6} - 32 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$13$ \( (T^{3} + 2 T^{2} - 5 T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 16 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$19$ \( T^{6} - 86 T^{4} + \cdots - 10952 \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( (T^{3} + 6 T^{2} - 49 T - 82)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 2 T^{2} - 31 T + 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 78 T^{4} + \cdots - 8192 \) Copy content Toggle raw display
$41$ \( (T^{3} - 8 T^{2} - 29 T - 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} - 162 T^{4} + \cdots - 107648 \) Copy content Toggle raw display
$47$ \( (T^{3} + 14 T^{2} + \cdots + 56)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} - 68 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$59$ \( (T^{3} + 18 T^{2} + \cdots + 112)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - 252 T^{4} + \cdots - 199712 \) Copy content Toggle raw display
$67$ \( T^{6} - 192 T^{4} + \cdots - 81608 \) Copy content Toggle raw display
$71$ \( (T^{3} + 16 T^{2} + \cdots + 116)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 10 T^{2} + \cdots + 886)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 220 T^{4} + \cdots - 43808 \) Copy content Toggle raw display
$83$ \( T^{6} - 368 T^{4} + \cdots - 1031048 \) Copy content Toggle raw display
$89$ \( T^{6} - 234 T^{4} + \cdots - 93312 \) Copy content Toggle raw display
$97$ \( T^{6} - 344 T^{4} + \cdots - 1404488 \) Copy content Toggle raw display
show more
show less