Properties

Label 1792.2.a.x
Level $1792$
Weight $2$
Character orbit 1792.a
Self dual yes
Analytic conductor $14.309$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,2,Mod(1,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, 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("1792.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.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: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.9248.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_1 q^{5} + q^{7} + ( - \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_1 q^{5} + q^{7} + ( - \beta_{3} + 2) q^{9} + (\beta_{2} - \beta_1) q^{11} + \beta_1 q^{13} + ( - \beta_{3} + 1) q^{15} + 2 q^{17} + \beta_{2} q^{19} + \beta_{2} q^{21} + (\beta_{3} - 1) q^{23} + (\beta_{3} + 2) q^{25} + (2 \beta_{2} + 2 \beta_1) q^{27} - 2 \beta_{2} q^{29} + (2 \beta_{3} - 2) q^{31} + 4 q^{33} + \beta_1 q^{35} - 2 \beta_{2} q^{37} + ( - \beta_{3} + 1) q^{39} + (2 \beta_{3} + 4) q^{41} + (\beta_{2} - \beta_1) q^{43} + (4 \beta_{2} - \beta_1) q^{45} + q^{49} + 2 \beta_{2} q^{51} + (2 \beta_{2} - 2 \beta_1) q^{53} + ( - 2 \beta_{3} - 6) q^{55} + ( - \beta_{3} + 5) q^{57} + ( - \beta_{2} + 2 \beta_1) q^{59} + \beta_1 q^{61} + ( - \beta_{3} + 2) q^{63} + (\beta_{3} + 7) q^{65} + ( - \beta_{2} - 5 \beta_1) q^{67} + ( - 4 \beta_{2} - 2 \beta_1) q^{69} + 8 q^{71} + 6 q^{73} + ( - \beta_{2} - 2 \beta_1) q^{75} + (\beta_{2} - \beta_1) q^{77} + ( - \beta_{3} + 6) q^{81} + ( - 3 \beta_{2} + 2 \beta_1) q^{83} + 2 \beta_1 q^{85} + (2 \beta_{3} - 10) q^{87} + ( - 2 \beta_{3} + 8) q^{89} + \beta_1 q^{91} + ( - 8 \beta_{2} - 4 \beta_1) q^{93} + ( - \beta_{3} + 1) q^{95} + ( - 2 \beta_{3} + 4) q^{97} + (\beta_{2} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} + 8 q^{9} + 4 q^{15} + 8 q^{17} - 4 q^{23} + 8 q^{25} - 8 q^{31} + 16 q^{33} + 4 q^{39} + 16 q^{41} + 4 q^{49} - 24 q^{55} + 20 q^{57} + 8 q^{63} + 28 q^{65} + 32 q^{71} + 24 q^{73} + 24 q^{81} - 40 q^{87} + 32 q^{89} + 4 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display

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
0.662153
2.13578
−2.13578
−0.662153
0 −3.02045 0 −1.69614 0 1.00000 0 6.12311 0
1.2 0 −0.936426 0 3.33513 0 1.00000 0 −2.12311 0
1.3 0 0.936426 0 −3.33513 0 1.00000 0 −2.12311 0
1.4 0 3.02045 0 1.69614 0 1.00000 0 6.12311 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.a.x 4
4.b odd 2 1 1792.2.a.v 4
8.b even 2 1 inner 1792.2.a.x 4
8.d odd 2 1 1792.2.a.v 4
16.e even 4 2 56.2.b.b 4
16.f odd 4 2 224.2.b.b 4
48.i odd 4 2 504.2.c.d 4
48.k even 4 2 2016.2.c.c 4
112.j even 4 2 1568.2.b.d 4
112.l odd 4 2 392.2.b.c 4
112.u odd 12 4 1568.2.t.d 8
112.v even 12 4 1568.2.t.e 8
112.w even 12 4 392.2.p.f 8
112.x odd 12 4 392.2.p.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.b 4 16.e even 4 2
224.2.b.b 4 16.f odd 4 2
392.2.b.c 4 112.l odd 4 2
392.2.p.e 8 112.x odd 12 4
392.2.p.f 8 112.w even 12 4
504.2.c.d 4 48.i odd 4 2
1568.2.b.d 4 112.j even 4 2
1568.2.t.d 8 112.u odd 12 4
1568.2.t.e 8 112.v even 12 4
1792.2.a.v 4 4.b odd 2 1
1792.2.a.v 4 8.d odd 2 1
1792.2.a.x 4 1.a even 1 1 trivial
1792.2.a.x 4 8.b even 2 1 inner
2016.2.c.c 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1792))\):

\( T_{3}^{4} - 10T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{4} - 14T_{5}^{2} + 32 \) Copy content Toggle raw display
\( T_{23}^{2} + 2T_{23} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 10T^{2} + 8 \) Copy content Toggle raw display
$5$ \( T^{4} - 14T^{2} + 32 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 20T^{2} + 32 \) Copy content Toggle raw display
$13$ \( T^{4} - 14T^{2} + 32 \) Copy content Toggle raw display
$17$ \( (T - 2)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 10T^{2} + 8 \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 40T^{2} + 128 \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 40T^{2} + 128 \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T - 52)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 20T^{2} + 32 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 80T^{2} + 512 \) Copy content Toggle raw display
$59$ \( T^{4} - 58T^{2} + 8 \) Copy content Toggle raw display
$61$ \( T^{4} - 14T^{2} + 32 \) Copy content Toggle raw display
$67$ \( T^{4} - 380 T^{2} + 32768 \) Copy content Toggle raw display
$71$ \( (T - 8)^{4} \) Copy content Toggle raw display
$73$ \( (T - 6)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 122T^{2} + 2888 \) Copy content Toggle raw display
$89$ \( (T^{2} - 16 T - 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T - 52)^{2} \) Copy content Toggle raw display
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