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Properties

Label 5600.2.a.bx
Level $5600$
Weight $2$
Character orbit 5600.a
Self dual yes
Analytic conductor $44.716$
Analytic rank $0$
Dimension $5$
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] = [5600,2,Mod(1,5600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5600, 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("5600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5600 = 2^{5} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5600.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: \(44.7162251319\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.504568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1120)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - q^{7} + (\beta_{3} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - q^{7} + (\beta_{3} + \beta_{2} + 1) q^{9} + (\beta_{3} + \beta_1 + 1) q^{11} + (2 \beta_{2} - \beta_1 - 1) q^{13} + ( - \beta_{4} + \beta_{2} - 1) q^{17} + ( - \beta_{4} + \beta_{2} - \beta_1 + 2) q^{19} - \beta_{2} q^{21} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \cdots - 2) q^{23}+ \cdots + (4 \beta_{2} - 2 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{3} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{3} - 5 q^{7} + 7 q^{9} + 4 q^{11} - 4 q^{17} + 12 q^{19} - 2 q^{21} - 8 q^{23} + 14 q^{27} - 12 q^{29} + 12 q^{31} + 8 q^{33} - 12 q^{37} + 32 q^{39} - 2 q^{41} + 8 q^{43} - 14 q^{47} + 5 q^{49} + 12 q^{51} - 8 q^{53} + 8 q^{57} + 16 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{67} + 4 q^{69} + 4 q^{71} + 12 q^{73} - 4 q^{77} + 32 q^{79} + q^{81} + 24 q^{83} - 2 q^{87} + 2 q^{89} + 20 q^{93} + 12 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 6\nu^{2} - 3\nu + 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} - \beta_{3} - 3\beta_{2} + 5\beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} + 3\beta_{2} + 9\beta _1 + 23 ) / 2 \) 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
0.271831
1.28447
−1.23118
2.52064
−1.84576
0 −2.19794 0 0 0 −1.00000 0 1.83094 0
1.2 0 −1.63460 0 0 0 −1.00000 0 −0.328072 0
1.3 0 0.746976 0 0 0 −1.00000 0 −2.44203 0
1.4 0 1.83297 0 0 0 −1.00000 0 0.359777 0
1.5 0 3.25260 0 0 0 −1.00000 0 7.57939 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

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 5600.2.a.bx 5
4.b odd 2 1 5600.2.a.bu 5
5.b even 2 1 5600.2.a.bv 5
5.c odd 4 2 1120.2.g.c yes 10
20.d odd 2 1 5600.2.a.bw 5
20.e even 4 2 1120.2.g.b ✓ 10
40.i odd 4 2 2240.2.g.n 10
40.k even 4 2 2240.2.g.o 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.g.b ✓ 10 20.e even 4 2
1120.2.g.c yes 10 5.c odd 4 2
2240.2.g.n 10 40.i odd 4 2
2240.2.g.o 10 40.k even 4 2
5600.2.a.bu 5 4.b odd 2 1
5600.2.a.bv 5 5.b even 2 1
5600.2.a.bw 5 20.d odd 2 1
5600.2.a.bx 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5600))\):

\( T_{3}^{5} - 2T_{3}^{4} - 9T_{3}^{3} + 12T_{3}^{2} + 18T_{3} - 16 \) Copy content Toggle raw display
\( T_{11}^{5} - 4T_{11}^{4} - 25T_{11}^{3} + 152T_{11}^{2} - 248T_{11} + 128 \) Copy content Toggle raw display
\( T_{13}^{5} - 47T_{13}^{3} - 80T_{13}^{2} + 378T_{13} + 824 \) Copy content Toggle raw display
\( T_{19}^{5} - 12T_{19}^{4} + 30T_{19}^{3} + 72T_{19}^{2} - 184T_{19} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 2 T^{4} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( (T + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 4 T^{4} + \cdots + 128 \) Copy content Toggle raw display
$13$ \( T^{5} - 47 T^{3} + \cdots + 824 \) Copy content Toggle raw display
$17$ \( T^{5} + 4 T^{4} + \cdots - 32 \) Copy content Toggle raw display
$19$ \( T^{5} - 12 T^{4} + \cdots - 256 \) Copy content Toggle raw display
$23$ \( T^{5} + 8 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$29$ \( T^{5} + 12 T^{4} + \cdots + 4744 \) Copy content Toggle raw display
$31$ \( T^{5} - 12 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{5} + 12 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{5} + 2 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$43$ \( T^{5} - 8 T^{4} + \cdots - 15104 \) Copy content Toggle raw display
$47$ \( T^{5} + 14 T^{4} + \cdots - 64 \) Copy content Toggle raw display
$53$ \( T^{5} + 8 T^{4} + \cdots + 1856 \) Copy content Toggle raw display
$59$ \( T^{5} - 16 T^{4} + \cdots - 58112 \) Copy content Toggle raw display
$61$ \( T^{5} + 10 T^{4} + \cdots + 1648 \) Copy content Toggle raw display
$67$ \( T^{5} - 4 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$71$ \( T^{5} - 4 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$73$ \( T^{5} - 12 T^{4} + \cdots - 2048 \) Copy content Toggle raw display
$79$ \( T^{5} - 32 T^{4} + \cdots - 3904 \) Copy content Toggle raw display
$83$ \( T^{5} - 24 T^{4} + \cdots + 27392 \) Copy content Toggle raw display
$89$ \( T^{5} - 2 T^{4} + \cdots - 3104 \) Copy content Toggle raw display
$97$ \( T^{5} - 12 T^{4} + \cdots - 3424 \) Copy content Toggle raw display
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