Properties

Label 5520.2.a.bi
Level $5520$
Weight $2$
Character orbit 5520.a
Self dual yes
Analytic conductor $44.077$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} + q^{7} + q^{9} - \beta q^{11} + (\beta + 2) q^{13} + q^{15} + ( - \beta - 3) q^{17} + ( - \beta - 2) q^{19} - q^{21} + q^{23} + q^{25} - q^{27} + (3 \beta + 3) q^{29} + (2 \beta - 5) q^{31} + \beta q^{33} - q^{35} + (2 \beta - 1) q^{37} + ( - \beta - 2) q^{39} + (\beta + 3) q^{41} - 2 q^{43} - q^{45} + ( - \beta - 6) q^{47} - 6 q^{49} + (\beta + 3) q^{51} + ( - \beta + 3) q^{53} + \beta q^{55} + (\beta + 2) q^{57} + ( - 3 \beta - 3) q^{59} + ( - 3 \beta + 8) q^{61} + q^{63} + ( - \beta - 2) q^{65} + 7 q^{67} - q^{69} + (3 \beta - 3) q^{71} + ( - 3 \beta + 2) q^{73} - q^{75} - \beta q^{77} + 4 q^{79} + q^{81} + ( - 5 \beta - 3) q^{83} + (\beta + 3) q^{85} + ( - 3 \beta - 3) q^{87} + ( - 2 \beta - 12) q^{89} + (\beta + 2) q^{91} + ( - 2 \beta + 5) q^{93} + (\beta + 2) q^{95} + ( - 2 \beta + 8) q^{97} - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 10 q^{31} - 2 q^{35} - 2 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} - 12 q^{47} - 12 q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{57} - 6 q^{59} + 16 q^{61} + 2 q^{63} - 4 q^{65} + 14 q^{67} - 2 q^{69} - 6 q^{71} + 4 q^{73} - 2 q^{75} + 8 q^{79} + 2 q^{81} - 6 q^{83} + 6 q^{85} - 6 q^{87} - 24 q^{89} + 4 q^{91} + 10 q^{93} + 4 q^{95} + 16 q^{97}+O(q^{100}) \) 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
2.44949
−2.44949
0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.00000 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.bi 2
4.b odd 2 1 345.2.a.i 2
12.b even 2 1 1035.2.a.k 2
20.d odd 2 1 1725.2.a.y 2
20.e even 4 2 1725.2.b.m 4
60.h even 2 1 5175.2.a.bl 2
92.b even 2 1 7935.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.i 2 4.b odd 2 1
1035.2.a.k 2 12.b even 2 1
1725.2.a.y 2 20.d odd 2 1
1725.2.b.m 4 20.e even 4 2
5175.2.a.bl 2 60.h even 2 1
5520.2.a.bi 2 1.a even 1 1 trivial
7935.2.a.t 2 92.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 6 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} + 3 \) Copy content Toggle raw display
\( T_{19}^{2} + 4T_{19} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 3 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 45 \) Copy content Toggle raw display
$31$ \( T^{2} + 10T + 1 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 23 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 3 \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 30 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 3 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 45 \) Copy content Toggle raw display
$61$ \( T^{2} - 16T + 10 \) Copy content Toggle raw display
$67$ \( (T - 7)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 6T - 45 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 50 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 141 \) Copy content Toggle raw display
$89$ \( T^{2} + 24T + 120 \) Copy content Toggle raw display
$97$ \( T^{2} - 16T + 40 \) Copy content Toggle raw display
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