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Properties

Label 7744.2.a.dh
Level $7744$
Weight $2$
Character orbit 7744.a
Self dual yes
Analytic conductor $61.836$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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: \(4\)
Coefficient field: 4.4.5225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} + (2 \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{3} - 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} + (2 \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{3} - 2 \beta_{2}) q^{9} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{13} + (\beta_{3} + \beta_{2} + 4) q^{15} + ( - \beta_{3} + 2 \beta_1 + 3) q^{17} + ( - \beta_{2} + 3) q^{19} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{21} + (2 \beta_{2} + 2 \beta_1) q^{23} + (6 \beta_{2} - \beta_1 + 6) q^{25} + (2 \beta_{2} + 2 \beta_1 - 3) q^{27} + ( - 2 \beta_{2} - \beta_1 - 3) q^{29} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{31} + ( - \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 8) q^{35} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{37} + (\beta_{3} - 5 \beta_{2} - 2 \beta_1 + 2) q^{39} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{41} + ( - \beta_{3} + \beta_1 + 1) q^{43} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{45} + ( - 2 \beta_{3} - \beta_1 + 3) q^{47} + ( - 3 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{49} + ( - 2 \beta_{3} - 8 \beta_{2} + \cdots + 1) q^{51}+ \cdots + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9} + q^{13} + 16 q^{15} + 12 q^{17} + 14 q^{19} - q^{21} - 2 q^{23} + 11 q^{25} - 14 q^{27} - 9 q^{29} + 11 q^{31} + 18 q^{35} - 13 q^{37} + 18 q^{39} + 8 q^{41} + 3 q^{43} - 22 q^{45} + 7 q^{47} - q^{49} + 14 q^{51} - 11 q^{53} - 7 q^{57} - 10 q^{59} - 17 q^{61} - 15 q^{63} + 5 q^{65} + 5 q^{67} + 4 q^{69} - 5 q^{71} + 6 q^{73} - 7 q^{75} - 7 q^{79} - 8 q^{81} + 20 q^{83} - 13 q^{85} + 3 q^{87} + 11 q^{89} - 6 q^{91} + 10 q^{93} - 6 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} + x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 4\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} + 2\nu - 11 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 4\beta_{2} + 6\beta _1 + 5 \) 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.

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
−1.88301
3.10522
−1.48718
1.26498
0 −3.04678 0 −1.78180 0 −0.353057 0 6.28284 0
1.2 0 −1.91913 0 −3.40632 0 −0.869151 0 0.683063 0
1.3 0 0.919131 0 4.02435 0 3.72325 0 −2.15520 0
1.4 0 2.04678 0 0.163765 0 −3.50105 0 1.18929 0
\(n\): e.g. 2-40 or 990-1000
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 7744.2.a.dh 4
4.b odd 2 1 7744.2.a.ds 4
8.b even 2 1 968.2.a.m 4
8.d odd 2 1 1936.2.a.bc 4
11.b odd 2 1 7744.2.a.di 4
11.d odd 10 2 704.2.m.l 8
24.h odd 2 1 8712.2.a.cd 4
44.c even 2 1 7744.2.a.dr 4
44.g even 10 2 704.2.m.i 8
88.b odd 2 1 968.2.a.n 4
88.g even 2 1 1936.2.a.bb 4
88.k even 10 2 176.2.m.d 8
88.o even 10 2 968.2.i.p 8
88.o even 10 2 968.2.i.t 8
88.p odd 10 2 88.2.i.b ✓ 8
88.p odd 10 2 968.2.i.s 8
264.m even 2 1 8712.2.a.ce 4
264.u even 10 2 792.2.r.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.i.b ✓ 8 88.p odd 10 2
176.2.m.d 8 88.k even 10 2
704.2.m.i 8 44.g even 10 2
704.2.m.l 8 11.d odd 10 2
792.2.r.g 8 264.u even 10 2
968.2.a.m 4 8.b even 2 1
968.2.a.n 4 88.b odd 2 1
968.2.i.p 8 88.o even 10 2
968.2.i.s 8 88.p odd 10 2
968.2.i.t 8 88.o even 10 2
1936.2.a.bb 4 88.g even 2 1
1936.2.a.bc 4 8.d odd 2 1
7744.2.a.dh 4 1.a even 1 1 trivial
7744.2.a.di 4 11.b odd 2 1
7744.2.a.dr 4 44.c even 2 1
7744.2.a.ds 4 4.b odd 2 1
8712.2.a.cd 4 24.h odd 2 1
8712.2.a.ce 4 264.m even 2 1

Hecke kernels

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

\( T_{3}^{4} + 2T_{3}^{3} - 7T_{3}^{2} - 8T_{3} + 11 \) Copy content Toggle raw display
\( T_{5}^{4} + T_{5}^{3} - 15T_{5}^{2} - 22T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} + T_{7}^{3} - 13T_{7}^{2} - 16T_{7} - 4 \) Copy content Toggle raw display
\( T_{13}^{4} - T_{13}^{3} - 23T_{13}^{2} + 66T_{13} - 44 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 11 \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} - 15 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} - 13 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{3} + \cdots - 44 \) Copy content Toggle raw display
$17$ \( T^{4} - 12 T^{3} + \cdots - 199 \) Copy content Toggle raw display
$19$ \( (T^{2} - 7 T + 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$29$ \( T^{4} + 9 T^{3} + \cdots - 44 \) Copy content Toggle raw display
$31$ \( T^{4} - 11 T^{3} + \cdots - 236 \) Copy content Toggle raw display
$37$ \( T^{4} + 13 T^{3} + \cdots - 44 \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} + \cdots - 11 \) Copy content Toggle raw display
$43$ \( T^{4} - 3 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$47$ \( T^{4} - 7 T^{3} + \cdots + 176 \) Copy content Toggle raw display
$53$ \( T^{4} + 11 T^{3} + \cdots - 176 \) Copy content Toggle raw display
$59$ \( (T^{2} + 5 T - 25)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 17 T^{3} + \cdots - 1936 \) Copy content Toggle raw display
$67$ \( T^{4} - 5 T^{3} + \cdots - 176 \) Copy content Toggle raw display
$71$ \( T^{4} + 5 T^{3} + \cdots - 716 \) Copy content Toggle raw display
$73$ \( T^{4} - 6 T^{3} + \cdots + 781 \) Copy content Toggle raw display
$79$ \( T^{4} + 7 T^{3} + \cdots + 844 \) Copy content Toggle raw display
$83$ \( T^{4} - 20 T^{3} + \cdots - 11 \) Copy content Toggle raw display
$89$ \( T^{4} - 11 T^{3} + \cdots - 124 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + \cdots - 491 \) Copy content Toggle raw display
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