Properties

Label 5520.2.a.cc
Level $5520$
Weight $2$
Character orbit 5520.a
Self dual yes
Analytic conductor $44.077$
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] = [5520,2,Mod(1,5520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5520.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.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: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.20087896.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4 \) 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 2760)
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.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + ( - \beta_1 + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + ( - \beta_1 + 1) q^{7} + q^{9} + ( - \beta_{2} + 1) q^{11} + ( - \beta_{4} + 1) q^{13} + q^{15} + (\beta_{3} + \beta_1 + 2) q^{17} + (\beta_{3} + 1) q^{19} + ( - \beta_1 + 1) q^{21} + q^{23} + q^{25} + q^{27} + (\beta_{2} - \beta_1 + 2) q^{29} + ( - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{31} + ( - \beta_{2} + 1) q^{33} + ( - \beta_1 + 1) q^{35} + (\beta_1 + 1) q^{37} + ( - \beta_{4} + 1) q^{39} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots + 2) q^{41}+ \cdots + ( - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + 5 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{3} + 5 q^{5} + 4 q^{7} + 5 q^{9} + 4 q^{11} + 4 q^{13} + 5 q^{15} + 10 q^{17} + 4 q^{19} + 4 q^{21} + 5 q^{23} + 5 q^{25} + 5 q^{27} + 10 q^{29} - 6 q^{31} + 4 q^{33} + 4 q^{35} + 6 q^{37} + 4 q^{39} + 12 q^{41} + 2 q^{43} + 5 q^{45} - 4 q^{47} + 19 q^{49} + 10 q^{51} + 4 q^{55} + 4 q^{57} - 6 q^{59} + 16 q^{61} + 4 q^{63} + 4 q^{65} + 4 q^{67} + 5 q^{69} - 8 q^{71} + 14 q^{73} + 5 q^{75} + 16 q^{77} - 18 q^{79} + 5 q^{81} + 20 q^{83} + 10 q^{85} + 10 q^{87} + 18 q^{89} - 20 q^{91} - 6 q^{93} + 4 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} - \nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + 4\nu^{3} + 13\nu^{2} - 36\nu - 20 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 21\nu^{2} - 16\nu + 60 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 21\nu^{2} + 60 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{3} + 4\beta _1 + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{4} - 11\beta_{3} + 4\beta_{2} + 8\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 37\beta_{4} - 21\beta_{3} + 84\beta _1 + 216 ) / 2 \) 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
4.50582
−3.43588
−2.39144
2.27399
0.0475116
0 1.00000 0 1.00000 0 −2.89831 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 −2.62057 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 0.944775 0 1.00000 0
1.4 0 1.00000 0 1.00000 0 3.55148 0 1.00000 0
1.5 0 1.00000 0 1.00000 0 5.02263 0 1.00000 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
\(2\) \( +1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)
\(23\) \( -1 \)

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 5520.2.a.cc 5
4.b odd 2 1 2760.2.a.w 5
12.b even 2 1 8280.2.a.br 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.w 5 4.b odd 2 1
5520.2.a.cc 5 1.a even 1 1 trivial
8280.2.a.br 5 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7}^{5} - 4T_{7}^{4} - 19T_{7}^{3} + 54T_{7}^{2} + 104T_{7} - 128 \) Copy content Toggle raw display
\( T_{11}^{5} - 4T_{11}^{4} - 46T_{11}^{3} + 160T_{11}^{2} + 408T_{11} - 1312 \) Copy content Toggle raw display
\( T_{13}^{5} - 4T_{13}^{4} - 66T_{13}^{3} + 248T_{13}^{2} + 968T_{13} - 3584 \) Copy content Toggle raw display
\( T_{17}^{5} - 10T_{17}^{4} - 25T_{17}^{3} + 584T_{17}^{2} - 2148T_{17} + 2416 \) Copy content Toggle raw display
\( T_{19}^{5} - 4T_{19}^{4} - 54T_{19}^{3} + 120T_{19}^{2} + 256T_{19} - 512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 4 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$11$ \( T^{5} - 4 T^{4} + \cdots - 1312 \) Copy content Toggle raw display
$13$ \( T^{5} - 4 T^{4} + \cdots - 3584 \) Copy content Toggle raw display
$17$ \( T^{5} - 10 T^{4} + \cdots + 2416 \) Copy content Toggle raw display
$19$ \( T^{5} - 4 T^{4} + \cdots - 512 \) Copy content Toggle raw display
$23$ \( (T - 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 10 T^{4} + \cdots + 1424 \) Copy content Toggle raw display
$31$ \( T^{5} + 6 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{5} - 6 T^{4} + \cdots - 112 \) Copy content Toggle raw display
$41$ \( T^{5} - 12 T^{4} + \cdots + 1912 \) Copy content Toggle raw display
$43$ \( T^{5} - 2 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$47$ \( T^{5} + 4 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$53$ \( T^{5} - 141 T^{3} + \cdots - 2344 \) Copy content Toggle raw display
$59$ \( T^{5} + 6 T^{4} + \cdots + 53248 \) Copy content Toggle raw display
$61$ \( T^{5} - 16 T^{4} + \cdots + 25856 \) Copy content Toggle raw display
$67$ \( T^{5} - 4 T^{4} + \cdots + 6176 \) Copy content Toggle raw display
$71$ \( T^{5} + 8 T^{4} + \cdots + 6464 \) Copy content Toggle raw display
$73$ \( T^{5} - 14 T^{4} + \cdots - 656 \) Copy content Toggle raw display
$79$ \( T^{5} + 18 T^{4} + \cdots + 32768 \) Copy content Toggle raw display
$83$ \( T^{5} - 20 T^{4} + \cdots + 9056 \) Copy content Toggle raw display
$89$ \( T^{5} - 18 T^{4} + \cdots - 3328 \) Copy content Toggle raw display
$97$ \( T^{5} - 4 T^{4} + \cdots - 13952 \) Copy content Toggle raw display
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