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Properties

Label 3584.2.a.l
Level $3584$
Weight $2$
Character orbit 3584.a
Self dual yes
Analytic conductor $28.618$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3584,2,Mod(1,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3584.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.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: \(28.6183840844\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.18857984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 9x^{4} + 16x^{3} + 6x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + \beta_1 q^{5} + q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} + \beta_1 q^{5} + q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{9} + (\beta_{5} + \beta_{3} + \beta_1 + 1) q^{11} + \beta_{4} q^{13} + (\beta_{4} + \beta_{3} + \beta_{2} + \cdots - 1) q^{15}+ \cdots + ( - \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + 6 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} + 6 q^{7} + 10 q^{9} + 8 q^{11} - 8 q^{15} + 20 q^{19} + 4 q^{21} + 2 q^{25} + 16 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{33} + 20 q^{37} - 4 q^{41} + 24 q^{43} + 8 q^{47} + 6 q^{49} + 24 q^{51} + 4 q^{53} + 16 q^{55} - 4 q^{57} + 12 q^{59} - 8 q^{61} + 10 q^{63} - 8 q^{65} + 16 q^{67} - 40 q^{69} - 8 q^{71} + 16 q^{73} + 28 q^{75} + 8 q^{77} - 24 q^{79} + 10 q^{81} + 12 q^{83} - 8 q^{87} + 16 q^{89} + 24 q^{93} - 16 q^{95} + 16 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 9x^{4} + 16x^{3} + 6x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 6\nu^{4} - 18\nu^{3} - 52\nu^{2} + 65\nu + 14 ) / 19 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{5} + \nu^{4} + 35\nu^{3} + 4\nu^{2} - 81\nu - 23 ) / 19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{5} - 8\nu^{4} - 52\nu^{3} + 63\nu^{2} + 78\nu - 44 ) / 19 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -16\nu^{5} + 18\nu^{4} + 155\nu^{3} - 118\nu^{2} - 147\nu + 42 ) / 19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\nu^{5} - 31\nu^{4} - 154\nu^{3} + 237\nu^{2} + 117\nu - 123 ) / 19 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + 3\beta_{2} + 3\beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{5} + 7\beta_{4} - 13\beta_{3} - 5\beta_{2} + 9\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{5} + 9\beta_{4} + \beta_{3} + 27\beta_{2} + 33\beta _1 + 62 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 83\beta_{5} + 59\beta_{4} - 123\beta_{3} - 31\beta_{2} + 93\beta _1 + 16 ) / 2 \) 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.

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
0.752719
0.194171
1.62476
3.17133
−0.951290
−2.79169
0 −2.53584 0 1.47134 0 1.00000 0 3.43049 0
1.2 0 −1.01685 0 1.29145 0 1.00000 0 −1.96601 0
1.3 0 −0.101362 0 −2.19640 0 1.00000 0 −2.98973 0
1.4 0 1.81529 0 2.66965 0 1.00000 0 0.295267 0
1.5 0 2.61578 0 −3.96111 0 1.00000 0 3.84231 0
1.6 0 3.22299 0 0.725063 0 1.00000 0 7.38766 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
\(2\) \(1\)
\(7\) \(-1\)

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 3584.2.a.l yes 6
4.b odd 2 1 3584.2.a.e ✓ 6
8.b even 2 1 3584.2.a.f yes 6
8.d odd 2 1 3584.2.a.k yes 6
16.e even 4 2 3584.2.b.i 12
16.f odd 4 2 3584.2.b.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.e ✓ 6 4.b odd 2 1
3584.2.a.f yes 6 8.b even 2 1
3584.2.a.k yes 6 8.d odd 2 1
3584.2.a.l yes 6 1.a even 1 1 trivial
3584.2.b.i 12 16.e even 4 2
3584.2.b.k 12 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3584))\):

\( T_{3}^{6} - 4T_{3}^{5} - 6T_{3}^{4} + 32T_{3}^{3} - 2T_{3}^{2} - 40T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{6} - 16T_{5}^{4} + 16T_{5}^{3} + 46T_{5}^{2} - 80T_{5} + 32 \) Copy content Toggle raw display
\( T_{23}^{6} - 76T_{23}^{4} + 96T_{23}^{3} + 1060T_{23}^{2} - 832T_{23} - 2848 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{6} - 16 T^{4} + \cdots + 32 \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 8 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{6} - 24 T^{4} + \cdots - 32 \) Copy content Toggle raw display
$17$ \( T^{6} - 48 T^{4} + \cdots - 896 \) Copy content Toggle raw display
$19$ \( T^{6} - 20 T^{5} + \cdots - 548 \) Copy content Toggle raw display
$23$ \( T^{6} - 76 T^{4} + \cdots - 2848 \) Copy content Toggle raw display
$29$ \( T^{6} - 4 T^{5} + \cdots + 128 \) Copy content Toggle raw display
$31$ \( T^{6} + 8 T^{5} + \cdots + 11776 \) Copy content Toggle raw display
$37$ \( T^{6} - 20 T^{5} + \cdots + 128 \) Copy content Toggle raw display
$41$ \( T^{6} + 4 T^{5} + \cdots + 5696 \) Copy content Toggle raw display
$43$ \( T^{6} - 24 T^{5} + \cdots - 46016 \) Copy content Toggle raw display
$47$ \( T^{6} - 8 T^{5} + \cdots - 179200 \) Copy content Toggle raw display
$53$ \( T^{6} - 4 T^{5} + \cdots - 48896 \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + \cdots + 15548 \) Copy content Toggle raw display
$61$ \( T^{6} + 8 T^{5} + \cdots + 49664 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} + \cdots + 4352 \) Copy content Toggle raw display
$71$ \( T^{6} + 8 T^{5} + \cdots - 51200 \) Copy content Toggle raw display
$73$ \( T^{6} - 16 T^{5} + \cdots + 899200 \) Copy content Toggle raw display
$79$ \( T^{6} + 24 T^{5} + \cdots - 21632 \) Copy content Toggle raw display
$83$ \( T^{6} - 12 T^{5} + \cdots - 269732 \) Copy content Toggle raw display
$89$ \( T^{6} - 16 T^{5} + \cdots + 6272 \) Copy content Toggle raw display
$97$ \( T^{6} - 16 T^{5} + \cdots + 232576 \) Copy content Toggle raw display
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