Properties

Label 1248.2.a.p
Level $1248$
Weight $2$
Character orbit 1248.a
Self dual yes
Analytic conductor $9.965$
Analytic rank $0$
Dimension $3$
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] = [1248,2,Mod(1,1248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1248, 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("1248.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.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: \(9.96533017226\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + (\beta_1 + 1) q^{5} - \beta_{2} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta_1 + 1) q^{5} - \beta_{2} q^{7} + q^{9} + ( - \beta_{2} - \beta_1 + 1) q^{11} + q^{13} + (\beta_1 + 1) q^{15} + 2 q^{17} + \beta_{2} q^{19} - \beta_{2} q^{21} + 2 \beta_{2} q^{23} + (2 \beta_{2} + 2 \beta_1 + 5) q^{25} + q^{27} + 2 q^{29} + (\beta_{2} + 2 \beta_1 - 2) q^{31} + ( - \beta_{2} - \beta_1 + 1) q^{33} + ( - 2 \beta_1 + 2) q^{35} - 2 \beta_1 q^{37} + q^{39} + ( - \beta_1 + 3) q^{41} + (2 \beta_{2} + 2 \beta_1 + 2) q^{43} + (\beta_1 + 1) q^{45} + ( - \beta_{2} - \beta_1 - 3) q^{47} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{49} + 2 q^{51} + ( - 2 \beta_1 + 4) q^{53} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{55} + \beta_{2} q^{57} + ( - \beta_{2} + \beta_1 + 7) q^{59} + (2 \beta_{2} - 2 \beta_1) q^{61} - \beta_{2} q^{63} + (\beta_1 + 1) q^{65} + ( - 3 \beta_{2} - 2 \beta_1 - 6) q^{67} + 2 \beta_{2} q^{69} + (\beta_{2} - \beta_1 - 3) q^{71} + (2 \beta_1 + 8) q^{73} + (2 \beta_{2} + 2 \beta_1 + 5) q^{75} + ( - 4 \beta_{2} + 8) q^{77} - 12 q^{79} + q^{81} + (\beta_{2} - \beta_1 + 1) q^{83} + (2 \beta_1 + 2) q^{85} + 2 q^{87} + ( - 3 \beta_1 - 3) q^{89} - \beta_{2} q^{91} + (\beta_{2} + 2 \beta_1 - 2) q^{93} + (2 \beta_1 - 2) q^{95} + 2 \beta_1 q^{97} + ( - \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 2 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 2 q^{5} + 3 q^{9} + 4 q^{11} + 3 q^{13} + 2 q^{15} + 6 q^{17} + 13 q^{25} + 3 q^{27} + 6 q^{29} - 8 q^{31} + 4 q^{33} + 8 q^{35} + 2 q^{37} + 3 q^{39} + 10 q^{41} + 4 q^{43} + 2 q^{45} - 8 q^{47} + 11 q^{49} + 6 q^{51} + 14 q^{53} - 16 q^{55} + 20 q^{59} + 2 q^{61} + 2 q^{65} - 16 q^{67} - 8 q^{71} + 22 q^{73} + 13 q^{75} + 24 q^{77} - 36 q^{79} + 3 q^{81} + 4 q^{83} + 4 q^{85} + 6 q^{87} - 6 q^{89} - 8 q^{93} - 8 q^{95} - 2 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 5 ) / 2 \) 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
−1.48119
0.311108
2.17009
0 1.00000 0 −2.96239 0 −3.35026 0 1.00000 0
1.2 0 1.00000 0 0.622216 0 4.42864 0 1.00000 0
1.3 0 1.00000 0 4.34017 0 −1.07838 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(13\) \( -1 \)

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 1248.2.a.p yes 3
3.b odd 2 1 3744.2.a.z 3
4.b odd 2 1 1248.2.a.o 3
8.b even 2 1 2496.2.a.bk 3
8.d odd 2 1 2496.2.a.bl 3
12.b even 2 1 3744.2.a.ba 3
24.f even 2 1 7488.2.a.cx 3
24.h odd 2 1 7488.2.a.cy 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.a.o 3 4.b odd 2 1
1248.2.a.p yes 3 1.a even 1 1 trivial
2496.2.a.bk 3 8.b even 2 1
2496.2.a.bl 3 8.d odd 2 1
3744.2.a.z 3 3.b odd 2 1
3744.2.a.ba 3 12.b even 2 1
7488.2.a.cx 3 24.f even 2 1
7488.2.a.cy 3 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1248))\):

\( T_{5}^{3} - 2T_{5}^{2} - 12T_{5} + 8 \) Copy content Toggle raw display
\( T_{7}^{3} - 16T_{7} - 16 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} - 16T_{11} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( T^{3} - 16T - 16 \) Copy content Toggle raw display
$11$ \( T^{3} - 4 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( (T - 2)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 16T + 16 \) Copy content Toggle raw display
$23$ \( T^{3} - 64T + 128 \) Copy content Toggle raw display
$29$ \( (T - 2)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} + 8 T^{2} + \cdots - 272 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} + \cdots + 40 \) Copy content Toggle raw display
$41$ \( T^{3} - 10 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$43$ \( T^{3} - 4 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{3} + 8T^{2} - 32 \) Copy content Toggle raw display
$53$ \( T^{3} - 14 T^{2} + \cdots + 152 \) Copy content Toggle raw display
$59$ \( T^{3} - 20 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$61$ \( T^{3} - 2 T^{2} + \cdots - 536 \) Copy content Toggle raw display
$67$ \( T^{3} + 16 T^{2} + \cdots - 1040 \) Copy content Toggle raw display
$71$ \( T^{3} + 8 T^{2} + \cdots - 160 \) Copy content Toggle raw display
$73$ \( T^{3} - 22 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$79$ \( (T + 12)^{3} \) Copy content Toggle raw display
$83$ \( T^{3} - 4 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$89$ \( T^{3} + 6 T^{2} + \cdots - 216 \) Copy content Toggle raw display
$97$ \( T^{3} + 2 T^{2} + \cdots - 40 \) Copy content Toggle raw display
show more
show less