Properties

Label 7744.2.a.dm
Level $7744$
Weight $2$
Character orbit 7744.a
Self dual yes
Analytic conductor $61.836$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: 4.4.22000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 15x^{2} + 55 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 352)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - 3 \beta_{2} - 2) q^{5} + \beta_1 q^{7} + (\beta_{2} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - 3 \beta_{2} - 2) q^{5} + \beta_1 q^{7} + (\beta_{2} + 5) q^{9} - 3 \beta_{2} q^{13} + ( - 3 \beta_{3} - 2 \beta_1) q^{15} + ( - \beta_{2} + 4) q^{17} - \beta_1 q^{19} + (\beta_{2} + 8) q^{21} + 2 \beta_{3} q^{23} + (3 \beta_{2} + 8) q^{25} + (\beta_{3} + 2 \beta_1) q^{27} + ( - \beta_{2} - 3) q^{29} + (\beta_{3} - \beta_1) q^{31} + ( - 3 \beta_{3} - 2 \beta_1) q^{35} + ( - \beta_{2} - 1) q^{37} - 3 \beta_{3} q^{39} + ( - \beta_{2} + 7) q^{41} + (2 \beta_{3} + 2 \beta_1) q^{43} + ( - 14 \beta_{2} - 13) q^{45} + (2 \beta_{3} + 3 \beta_1) q^{47} + (\beta_{2} + 1) q^{49} + ( - \beta_{3} + 4 \beta_1) q^{51} + ( - \beta_{2} - 2) q^{53} + ( - \beta_{2} - 8) q^{57} + ( - 2 \beta_{3} + \beta_1) q^{59} + ( - \beta_{2} + 2) q^{61} + (\beta_{3} + 5 \beta_1) q^{63} + ( - 3 \beta_{2} + 9) q^{65} + ( - 2 \beta_{3} + 2 \beta_1) q^{67} + (14 \beta_{2} + 2) q^{69} + (\beta_{3} - 3 \beta_1) q^{71} + ( - 7 \beta_{2} - 7) q^{73} + (3 \beta_{3} + 8 \beta_1) q^{75} + (\beta_{3} + 3 \beta_1) q^{79} + (6 \beta_{2} + 2) q^{81} + ( - \beta_{3} + 3 \beta_1) q^{83} + ( - 13 \beta_{2} - 5) q^{85} + ( - \beta_{3} - 3 \beta_1) q^{87} + (2 \beta_{2} - 4) q^{89} - 3 \beta_{3} q^{91} + (6 \beta_{2} - 7) q^{93} + (3 \beta_{3} + 2 \beta_1) q^{95} - 11 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} + 18 q^{9} + 6 q^{13} + 18 q^{17} + 30 q^{21} + 26 q^{25} - 10 q^{29} - 2 q^{37} + 30 q^{41} - 24 q^{45} + 2 q^{49} - 6 q^{53} - 30 q^{57} + 10 q^{61} + 42 q^{65} - 20 q^{69} - 14 q^{73} - 4 q^{81} + 6 q^{85} - 20 q^{89} - 40 q^{93} + 22 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 15x^{2} + 55 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 8\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 8\beta_1 \) 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
−2.93565
−2.52626
2.52626
2.93565
0 −2.93565 0 −3.85410 0 −2.93565 0 5.61803 0
1.2 0 −2.52626 0 2.85410 0 −2.52626 0 3.38197 0
1.3 0 2.52626 0 2.85410 0 2.52626 0 3.38197 0
1.4 0 2.93565 0 −3.85410 0 2.93565 0 5.61803 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
4.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.dm 4
4.b odd 2 1 inner 7744.2.a.dm 4
8.b even 2 1 3872.2.a.bj 4
8.d odd 2 1 3872.2.a.bj 4
11.b odd 2 1 7744.2.a.dl 4
11.c even 5 2 704.2.m.j 8
44.c even 2 1 7744.2.a.dl 4
44.h odd 10 2 704.2.m.j 8
88.b odd 2 1 3872.2.a.bk 4
88.g even 2 1 3872.2.a.bk 4
88.l odd 10 2 352.2.m.d 8
88.o even 10 2 352.2.m.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.m.d 8 88.l odd 10 2
352.2.m.d 8 88.o even 10 2
704.2.m.j 8 11.c even 5 2
704.2.m.j 8 44.h odd 10 2
3872.2.a.bj 4 8.b even 2 1
3872.2.a.bj 4 8.d odd 2 1
3872.2.a.bk 4 88.b odd 2 1
3872.2.a.bk 4 88.g even 2 1
7744.2.a.dl 4 11.b odd 2 1
7744.2.a.dl 4 44.c even 2 1
7744.2.a.dm 4 1.a even 1 1 trivial
7744.2.a.dm 4 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7744))\):

\( T_{3}^{4} - 15T_{3}^{2} + 55 \) Copy content Toggle raw display
\( T_{5}^{2} + T_{5} - 11 \) Copy content Toggle raw display
\( T_{7}^{4} - 15T_{7}^{2} + 55 \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 15T^{2} + 55 \) Copy content Toggle raw display
$5$ \( (T^{2} + T - 11)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 15T^{2} + 55 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 3 T - 9)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 9 T + 19)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 15T^{2} + 55 \) Copy content Toggle raw display
$23$ \( T^{4} - 80T^{2} + 880 \) Copy content Toggle raw display
$29$ \( (T^{2} + 5 T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 45T^{2} + 55 \) Copy content Toggle raw display
$37$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 15 T + 55)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 100T^{2} + 880 \) Copy content Toggle raw display
$47$ \( T^{4} - 155T^{2} + 55 \) Copy content Toggle raw display
$53$ \( (T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 115T^{2} + 55 \) Copy content Toggle raw display
$61$ \( (T^{2} - 5 T + 5)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 180T^{2} + 880 \) Copy content Toggle raw display
$71$ \( T^{4} - 185T^{2} + 6655 \) Copy content Toggle raw display
$73$ \( (T^{2} + 7 T - 49)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 125T^{2} + 1375 \) Copy content Toggle raw display
$83$ \( T^{4} - 185T^{2} + 6655 \) Copy content Toggle raw display
$89$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 11 T - 121)^{2} \) Copy content Toggle raw display
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