↑

Properties

Label 9464.2.a.be
Level $9464$
Weight $2$
Character orbit 9464.a
Self dual yes
Analytic conductor $75.570$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9464,2,Mod(1,9464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9464, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9464.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9464.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(75.5704204729\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.415174304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 10x^{4} + 4x^{3} + 22x^{2} + 4x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 728)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{5} + \beta_{4} + \beta_1) q^{5} + q^{7} + (\beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{5} + \beta_{4} + \beta_1) q^{5} + q^{7} + (\beta_{2} + \beta_1) q^{9} - \beta_{4} q^{11} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{15} + (\beta_{4} + \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{19}+ \cdots + (\beta_{5} + 3 \beta_{4} + \beta_{3} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{3} + q^{5} + 6 q^{7} + 3 q^{9} - q^{11} + 2 q^{15} - 2 q^{17} - 9 q^{19} + q^{21} + 10 q^{23} + 13 q^{25} + 13 q^{27} + 5 q^{29} + 10 q^{31} + 9 q^{33} + q^{35} + 13 q^{37} + 7 q^{41} + 3 q^{43} - 3 q^{45} - 2 q^{47} + 6 q^{49} + 6 q^{51} + 3 q^{53} - 20 q^{55} - 10 q^{57} - 4 q^{59} + 3 q^{61} + 3 q^{63} - 13 q^{67} - 15 q^{69} - 6 q^{71} - 24 q^{73} + 29 q^{75} - q^{77} + 20 q^{79} + 2 q^{81} - 15 q^{83} + 16 q^{85} + 4 q^{87} + q^{89} + 37 q^{93} - 27 q^{95} + 6 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 10x^{4} + 4x^{3} + 22x^{2} + 4x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 12\nu^{2} + 4\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - \nu^{4} - 10\nu^{3} + 4\nu^{2} + 20\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} + 9\nu^{3} - 12\nu^{2} - 18\nu + 4 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{3} + 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{5} + 2\beta_{4} + \beta_{3} + 8\beta_{2} + 13\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{5} + 4\beta_{4} + 11\beta_{3} + 4\beta_{2} + 59\beta _1 + 22 \) 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.

comment: embeddings in the coefficient field
 
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
−2.42473
−1.25097
−0.649715
0.344051
1.93919
3.04217
0 −2.42473 0 0.0443237 0 1.00000 0 2.87931 0
1.2 0 −1.25097 0 −2.55830 0 1.00000 0 −1.43508 0
1.3 0 −0.649715 0 1.23828 0 1.00000 0 −2.57787 0
1.4 0 0.344051 0 4.20126 0 1.00000 0 −2.88163 0
1.5 0 1.93919 0 −3.73905 0 1.00000 0 0.760458 0
1.6 0 3.04217 0 1.81349 0 1.00000 0 6.25481 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
\(2\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 9464.2.a.be 6
13.b even 2 1 9464.2.a.bd 6
13.d odd 4 2 728.2.k.b ✓ 12
52.f even 4 2 1456.2.k.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.k.b ✓ 12 13.d odd 4 2
1456.2.k.f 12 52.f even 4 2
9464.2.a.bd 6 13.b even 2 1
9464.2.a.be 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9464))\):

\( T_{3}^{6} - T_{3}^{5} - 10T_{3}^{4} + 4T_{3}^{3} + 22T_{3}^{2} + 4T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{6} - T_{5}^{5} - 21T_{5}^{4} + 17T_{5}^{3} + 84T_{5}^{2} - 94T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{6} + T_{11}^{5} - 22T_{11}^{4} + 24T_{11}^{3} + 46T_{11}^{2} - 64T_{11} + 16 \) Copy content Toggle raw display
\( T_{17}^{6} + 2T_{17}^{5} - 32T_{17}^{4} - 10T_{17}^{3} + 228T_{17}^{2} - 40T_{17} - 304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} - 10 T^{4} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} - 21 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + T^{5} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} + \cdots - 304 \) Copy content Toggle raw display
$19$ \( T^{6} + 9 T^{5} + \cdots + 2336 \) Copy content Toggle raw display
$23$ \( T^{6} - 10 T^{5} + \cdots + 3232 \) Copy content Toggle raw display
$29$ \( T^{6} - 5 T^{5} + \cdots + 5872 \) Copy content Toggle raw display
$31$ \( T^{6} - 10 T^{5} + \cdots - 113320 \) Copy content Toggle raw display
$37$ \( T^{6} - 13 T^{5} + \cdots + 187736 \) Copy content Toggle raw display
$41$ \( T^{6} - 7 T^{5} + \cdots - 127648 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + \cdots - 100480 \) Copy content Toggle raw display
$47$ \( T^{6} + 2 T^{5} + \cdots + 40 \) Copy content Toggle raw display
$53$ \( T^{6} - 3 T^{5} + \cdots - 24712 \) Copy content Toggle raw display
$59$ \( T^{6} + 4 T^{5} + \cdots - 1520 \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} + \cdots - 84752 \) Copy content Toggle raw display
$67$ \( T^{6} + 13 T^{5} + \cdots - 4864 \) Copy content Toggle raw display
$71$ \( T^{6} + 6 T^{5} + \cdots - 79616 \) Copy content Toggle raw display
$73$ \( T^{6} + 24 T^{5} + \cdots + 132556 \) Copy content Toggle raw display
$79$ \( T^{6} - 20 T^{5} + \cdots + 97772 \) Copy content Toggle raw display
$83$ \( T^{6} + 15 T^{5} + \cdots - 50012 \) Copy content Toggle raw display
$89$ \( T^{6} - T^{5} + \cdots + 57656 \) Copy content Toggle raw display
$97$ \( T^{6} - 6 T^{5} + \cdots - 2228 \) Copy content Toggle raw display
show more
show less