Properties

Label 5520.2.a.cb
Level $5520$
Weight $2$
Character orbit 5520.a
Self dual yes
Analytic conductor $44.077$
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] = [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: \(4\)
Coefficient field: 4.4.54764.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + \beta_1 q^{7} + q^{9} + \beta_{3} q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{13} - q^{15} + (2 \beta_{3} + \beta_{2} + 1) q^{17} + ( - \beta_{2} - \beta_1 + 1) q^{19} + \beta_1 q^{21} - q^{23} + q^{25} + q^{27} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} - \beta_{2} + 1) q^{31} + \beta_{3} q^{33} - \beta_1 q^{35} + (2 \beta_{2} + \beta_1) q^{37} + ( - \beta_{2} + \beta_1 - 1) q^{39} + (\beta_{3} - \beta_1 + 2) q^{41} + (2 \beta_{2} + 6) q^{43} - q^{45} + (\beta_{2} + \beta_1 - 1) q^{47} + ( - 3 \beta_{3} - \beta_{2} + 2) q^{49} + (2 \beta_{3} + \beta_{2} + 1) q^{51} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{53} - \beta_{3} q^{55} + ( - \beta_{2} - \beta_1 + 1) q^{57} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{59} + (\beta_{3} + 2 \beta_{2}) q^{61} + \beta_1 q^{63} + (\beta_{2} - \beta_1 + 1) q^{65} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 7) q^{67} - q^{69} + ( - \beta_{3} + \beta_1) q^{71} + ( - \beta_{2} + \beta_1 - 5) q^{73} + q^{75} + ( - \beta_{2} - 3 \beta_1 + 1) q^{77} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 + 3) q^{79} + q^{81} + ( - 2 \beta_{3} + \beta_{2} + 7) q^{83} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{85} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{87} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{89} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 9) q^{91} + ( - \beta_{3} - \beta_{2} + 1) q^{93} + (\beta_{2} + \beta_1 - 1) q^{95} + ( - 2 \beta_{3} - 2 \beta_{2} - 4) q^{97} + \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9} - 2 q^{13} - 4 q^{15} + 2 q^{17} + 6 q^{19} - 4 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} + 6 q^{31} - 4 q^{37} - 2 q^{39} + 8 q^{41} + 20 q^{43} - 4 q^{45} - 6 q^{47} + 10 q^{49} + 2 q^{51} - 4 q^{53} + 6 q^{57} + 4 q^{59} - 4 q^{61} + 2 q^{65} + 26 q^{67} - 4 q^{69} - 18 q^{73} + 4 q^{75} + 6 q^{77} + 18 q^{79} + 4 q^{81} + 26 q^{83} - 2 q^{85} + 4 q^{87} + 14 q^{89} + 38 q^{91} + 6 q^{93} - 6 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 11\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} + 9\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 8\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_{2} + \beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{3} + 11\beta_{2} - 7\beta _1 + 15 ) / 2 \) 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
0.655762
3.36007
−0.339102
−2.67673
0 1.00000 0 −1.00000 0 −4.46569 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −0.512641 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 0.845563 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 4.13277 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
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.cb 4
4.b odd 2 1 2760.2.a.v 4
12.b even 2 1 8280.2.a.bq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.v 4 4.b odd 2 1
5520.2.a.cb 4 1.a even 1 1 trivial
8280.2.a.bq 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7}^{4} - 19T_{7}^{2} + 6T_{7} + 8 \) Copy content Toggle raw display
\( T_{11}^{4} - 22T_{11}^{2} - 12T_{11} + 80 \) Copy content Toggle raw display
\( T_{13}^{4} + 2T_{13}^{3} - 38T_{13}^{2} - 56T_{13} + 296 \) Copy content Toggle raw display
\( T_{17}^{4} - 2T_{17}^{3} - 71T_{17}^{2} + 58T_{17} + 1024 \) Copy content Toggle raw display
\( T_{19}^{4} - 6T_{19}^{3} - 26T_{19}^{2} + 112T_{19} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 19 T^{2} + 6 T + 8 \) Copy content Toggle raw display
$11$ \( T^{4} - 22 T^{2} - 12 T + 80 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} - 38 T^{2} - 56 T + 296 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} - 71 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} - 26 T^{2} + 112 T + 256 \) Copy content Toggle raw display
$23$ \( (T + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} - 75 T^{2} + 400 T - 164 \) Copy content Toggle raw display
$31$ \( T^{4} - 6 T^{3} - 11 T^{2} + 104 T - 128 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} - 95 T^{2} + \cdots + 2104 \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} - 11 T^{2} + 204 T - 380 \) Copy content Toggle raw display
$43$ \( T^{4} - 20 T^{3} + 68 T^{2} + \cdots - 2048 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} - 26 T^{2} - 112 T + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} - 147 T^{2} + \cdots + 1468 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} - 75 T^{2} - 84 T + 320 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} - 62 T^{2} - 364 T - 400 \) Copy content Toggle raw display
$67$ \( T^{4} - 26 T^{3} + 141 T^{2} + \cdots - 6976 \) Copy content Toggle raw display
$71$ \( T^{4} - 35 T^{2} - 96 T - 64 \) Copy content Toggle raw display
$73$ \( T^{4} + 18 T^{3} + 82 T^{2} + \cdots - 152 \) Copy content Toggle raw display
$79$ \( T^{4} - 18 T^{3} - 56 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$83$ \( T^{4} - 26 T^{3} + 109 T^{2} + \cdots - 9176 \) Copy content Toggle raw display
$89$ \( T^{4} - 14 T^{3} + 24 T^{2} + 48 T - 64 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} - 44 T^{2} - 128 T + 64 \) Copy content Toggle raw display
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