Properties

Label 7744.2.a.dc
Level $7744$
Weight $2$
Character orbit 7744.a
Self dual yes
Analytic conductor $61.836$
Analytic rank $0$
Dimension $2$
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] = [7744,2,Mod(1,7744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7744, 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("7744.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7744 = 2^{6} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7744.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: \(61.8361513253\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 484)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{3} - 3 q^{5} - 2 \beta q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} - 3 q^{5} - 2 \beta q^{7} + q^{9} + 3 \beta q^{13} - 6 q^{15} - 3 \beta q^{17} - 2 \beta q^{19} - 4 \beta q^{21} + 6 q^{23} + 4 q^{25} - 4 q^{27} - 3 \beta q^{29} + 2 q^{31} + 6 \beta q^{35} - q^{37} + 6 \beta q^{39} + 3 \beta q^{41} - 3 q^{45} - 6 q^{47} + 5 q^{49} - 6 \beta q^{51} + 3 q^{53} - 4 \beta q^{57} - 8 \beta q^{61} - 2 \beta q^{63} - 9 \beta q^{65} - 2 q^{67} + 12 q^{69} + 4 \beta q^{73} + 8 q^{75} + 2 \beta q^{79} - 11 q^{81} + 6 \beta q^{83} + 9 \beta q^{85} - 6 \beta q^{87} + 15 q^{89} - 18 q^{91} + 4 q^{93} + 6 \beta q^{95} + 5 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} - 6 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} - 6 q^{5} + 2 q^{9} - 12 q^{15} + 12 q^{23} + 8 q^{25} - 8 q^{27} + 4 q^{31} - 2 q^{37} - 6 q^{45} - 12 q^{47} + 10 q^{49} + 6 q^{53} - 4 q^{67} + 24 q^{69} + 16 q^{75} - 22 q^{81} + 30 q^{89} - 36 q^{91} + 8 q^{93} + 10 q^{97}+O(q^{100}) \) 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
1.73205
−1.73205
0 2.00000 0 −3.00000 0 −3.46410 0 1.00000 0
1.2 0 2.00000 0 −3.00000 0 3.46410 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7744.2.a.dc 2
4.b odd 2 1 7744.2.a.bl 2
8.b even 2 1 1936.2.a.m 2
8.d odd 2 1 484.2.a.e 2
11.b odd 2 1 inner 7744.2.a.dc 2
24.f even 2 1 4356.2.a.m 2
44.c even 2 1 7744.2.a.bl 2
88.b odd 2 1 1936.2.a.m 2
88.g even 2 1 484.2.a.e 2
88.k even 10 4 484.2.e.f 8
88.l odd 10 4 484.2.e.f 8
264.p odd 2 1 4356.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
484.2.a.e 2 8.d odd 2 1
484.2.a.e 2 88.g even 2 1
484.2.e.f 8 88.k even 10 4
484.2.e.f 8 88.l odd 10 4
1936.2.a.m 2 8.b even 2 1
1936.2.a.m 2 88.b odd 2 1
4356.2.a.m 2 24.f even 2 1
4356.2.a.m 2 264.p odd 2 1
7744.2.a.bl 2 4.b odd 2 1
7744.2.a.bl 2 44.c even 2 1
7744.2.a.dc 2 1.a even 1 1 trivial
7744.2.a.dc 2 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7744))\):

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display
\( T_{13}^{2} - 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 12 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 27 \) Copy content Toggle raw display
$17$ \( T^{2} - 27 \) Copy content Toggle raw display
$19$ \( T^{2} - 12 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 27 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 27 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 192 \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 48 \) Copy content Toggle raw display
$79$ \( T^{2} - 12 \) Copy content Toggle raw display
$83$ \( T^{2} - 108 \) Copy content Toggle raw display
$89$ \( (T - 15)^{2} \) Copy content Toggle raw display
$97$ \( (T - 5)^{2} \) Copy content Toggle raw display
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