↑

Properties

Label 2816.2.a.p
Level $2816$
Weight $2$
Character orbit 2816.a
Self dual yes
Analytic conductor $22.486$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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] = [2816,2,Mod(1,2816)] 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("2816.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2816, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2816 = 2^{8} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2816.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,-4,0,0,0,5,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.4858732092\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.380224.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 8x^{3} - 2x^{2} + 5x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 88)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} + \beta_1) q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{9} + q^{11} + ( - \beta_{3} - \beta_1 - 2) q^{13} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{15}+ \cdots + ( - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{5} + 5 q^{9} + 5 q^{11} - 10 q^{13} - 4 q^{15} - 2 q^{17} - 12 q^{21} - 6 q^{23} + 3 q^{25} - 12 q^{27} - 10 q^{29} + 2 q^{31} - 16 q^{37} + 12 q^{39} - 2 q^{41} + 4 q^{43} - 20 q^{45} + 2 q^{47}+ \cdots + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 8x^{3} - 2x^{2} + 5x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{4} + 8\nu^{2} + 2\nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{4} - \nu^{3} - 15\nu^{2} + 3\nu + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 4\nu^{4} - 2\nu^{3} - 31\nu^{2} + 8\nu + 15 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{4} - 7\beta_{3} - 9\beta_{2} + 5\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{3} - 9\beta_{2} + 8\beta _1 + 20 \) 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
−2.56491
−0.479072
2.82726
−0.658084
0.874805
0 −3.05779 0 −0.699283 0 3.27803 0 6.35006 0
1.2 0 −1.33544 0 −1.93119 0 −1.83930 0 −1.21660 0
1.3 0 0.229967 0 2.51595 0 1.47743 0 −2.94712 0
1.4 0 1.81026 0 0.282461 0 −3.84939 0 0.277041 0
1.5 0 2.35300 0 −4.16794 0 0.933222 0 2.53661 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(11\) \( -1 \)

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 2816.2.a.p 5
4.b odd 2 1 2816.2.a.o 5
8.b even 2 1 2816.2.a.q 5
8.d odd 2 1 2816.2.a.r 5
16.e even 4 2 352.2.c.a 10
16.f odd 4 2 88.2.c.a ✓ 10
48.i odd 4 2 3168.2.f.g 10
48.k even 4 2 792.2.f.g 10
176.i even 4 2 968.2.c.d 10
176.l odd 4 2 3872.2.c.f 10
176.v odd 20 8 968.2.o.g 40
176.x even 20 8 968.2.o.h 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.c.a ✓ 10 16.f odd 4 2
352.2.c.a 10 16.e even 4 2
792.2.f.g 10 48.k even 4 2
968.2.c.d 10 176.i even 4 2
968.2.o.g 40 176.v odd 20 8
968.2.o.h 40 176.x even 20 8
2816.2.a.o 5 4.b odd 2 1
2816.2.a.p 5 1.a even 1 1 trivial
2816.2.a.q 5 8.b even 2 1
2816.2.a.r 5 8.d odd 2 1
3168.2.f.g 10 48.i odd 4 2
3872.2.c.f 10 176.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2816))\):

\( T_{3}^{5} - 10T_{3}^{3} + 4T_{3}^{2} + 17T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{5} + 4T_{5}^{4} - 6T_{5}^{3} - 24T_{5}^{2} - 7T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{5} - 16T_{7}^{3} + 8T_{7}^{2} + 40T_{7} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 10 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{5} + 4 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{5} - 16 T^{3} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( (T - 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 10 T^{4} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( T^{5} + 2 T^{4} + \cdots + 464 \) Copy content Toggle raw display
$19$ \( T^{5} - 48 T^{3} + \cdots - 512 \) Copy content Toggle raw display
$23$ \( T^{5} + 6 T^{4} + \cdots + 314 \) Copy content Toggle raw display
$29$ \( T^{5} + 10 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$31$ \( T^{5} - 2 T^{4} + \cdots + 226 \) Copy content Toggle raw display
$37$ \( T^{5} + 16 T^{4} + \cdots + 424 \) Copy content Toggle raw display
$41$ \( T^{5} + 2 T^{4} + \cdots + 464 \) Copy content Toggle raw display
$43$ \( T^{5} - 4 T^{4} + \cdots - 10048 \) Copy content Toggle raw display
$47$ \( T^{5} - 2 T^{4} + \cdots + 224 \) Copy content Toggle raw display
$53$ \( T^{5} + 8 T^{4} + \cdots - 6656 \) Copy content Toggle raw display
$59$ \( T^{5} + 8 T^{4} + \cdots + 428 \) Copy content Toggle raw display
$61$ \( T^{5} + 30 T^{4} + \cdots - 256 \) Copy content Toggle raw display
$67$ \( T^{5} - 90 T^{3} + \cdots + 668 \) Copy content Toggle raw display
$71$ \( T^{5} + 6 T^{4} + \cdots + 83746 \) Copy content Toggle raw display
$73$ \( T^{5} - 2 T^{4} + \cdots - 12752 \) Copy content Toggle raw display
$79$ \( T^{5} + 8 T^{4} + \cdots - 11008 \) Copy content Toggle raw display
$83$ \( T^{5} - 200 T^{3} + \cdots - 896 \) Copy content Toggle raw display
$89$ \( T^{5} - 2 T^{4} + \cdots - 3566 \) Copy content Toggle raw display
$97$ \( T^{5} + 10 T^{4} + \cdots + 20462 \) Copy content Toggle raw display
show more
show less