Properties

Label 4560.2.a.bs
Level $4560$
Weight $2$
Character orbit 4560.a
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
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] = [4560,2,Mod(1,4560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4560, 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("4560.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.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: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1772.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 12x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - \beta_{2} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} - \beta_{2} q^{7} + q^{9} + ( - \beta_{2} - 2) q^{11} + ( - \beta_1 + 1) q^{13} - q^{15} + 4 q^{17} - q^{19} + \beta_{2} q^{21} + (\beta_{2} - \beta_1 + 1) q^{23} + q^{25} - q^{27} - \beta_{2} q^{29} + 2 q^{31} + (\beta_{2} + 2) q^{33} - \beta_{2} q^{35} + ( - 2 \beta_{2} + \beta_1 - 1) q^{37} + (\beta_1 - 1) q^{39} + ( - 2 \beta_{2} + \beta_1 + 1) q^{41} + ( - \beta_1 - 1) q^{43} + q^{45} + ( - \beta_{2} + \beta_1 - 1) q^{47} + ( - \beta_{2} - \beta_1 + 2) q^{49} - 4 q^{51} + ( - \beta_{2} + \beta_1 - 3) q^{53} + ( - \beta_{2} - 2) q^{55} + q^{57} + (3 \beta_{2} - \beta_1 - 1) q^{59} + (\beta_{2} + \beta_1 + 5) q^{61} - \beta_{2} q^{63} + ( - \beta_1 + 1) q^{65} - 4 q^{67} + ( - \beta_{2} + \beta_1 - 1) q^{69} + (2 \beta_1 + 2) q^{71} + 6 q^{73} - q^{75} + (\beta_{2} - \beta_1 + 9) q^{77} + (3 \beta_{2} + \beta_1 + 1) q^{79} + q^{81} + (2 \beta_{2} + 2) q^{83} + 4 q^{85} + \beta_{2} q^{87} + (2 \beta_{2} - \beta_1 + 3) q^{89} + ( - 5 \beta_{2} + \beta_1 + 3) q^{91} - 2 q^{93} - q^{95} + ( - 2 \beta_{2} + \beta_1 + 11) q^{97} + ( - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 6 q^{11} + 4 q^{13} - 3 q^{15} + 12 q^{17} - 3 q^{19} + 4 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{31} + 6 q^{33} - 4 q^{37} - 4 q^{39} + 2 q^{41} - 2 q^{43} + 3 q^{45} - 4 q^{47} + 7 q^{49} - 12 q^{51} - 10 q^{53} - 6 q^{55} + 3 q^{57} - 2 q^{59} + 14 q^{61} + 4 q^{65} - 12 q^{67} - 4 q^{69} + 4 q^{71} + 18 q^{73} - 3 q^{75} + 28 q^{77} + 2 q^{79} + 3 q^{81} + 6 q^{83} + 12 q^{85} + 10 q^{89} + 8 q^{91} - 6 q^{93} - 3 q^{95} + 32 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 12x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 3\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 8 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + \beta _1 + 17 ) / 2 \) 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
−3.32803
3.67370
0.654334
0 −1.00000 0 1.00000 0 −3.20191 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −0.911179 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 4.11309 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(19\) \(1\)

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 4560.2.a.bs 3
4.b odd 2 1 2280.2.a.u 3
12.b even 2 1 6840.2.a.bh 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.u 3 4.b odd 2 1
4560.2.a.bs 3 1.a even 1 1 trivial
6840.2.a.bh 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4560))\):

\( T_{7}^{3} - 14T_{7} - 12 \) Copy content Toggle raw display
\( T_{11}^{3} + 6T_{11}^{2} - 2T_{11} - 32 \) Copy content Toggle raw display
\( T_{13}^{3} - 4T_{13}^{2} - 38T_{13} + 164 \) Copy content Toggle raw display
\( T_{17} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 14T - 12 \) Copy content Toggle raw display
$11$ \( T^{3} + 6 T^{2} - 2 T - 32 \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} - 38 T + 164 \) Copy content Toggle raw display
$17$ \( (T - 4)^{3} \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 4 T^{2} - 44 T + 32 \) Copy content Toggle raw display
$29$ \( T^{3} - 14T - 12 \) Copy content Toggle raw display
$31$ \( (T - 2)^{3} \) Copy content Toggle raw display
$37$ \( T^{3} + 4 T^{2} - 78 T + 180 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} - 82 T + 344 \) Copy content Toggle raw display
$43$ \( T^{3} + 2 T^{2} - 42 T + 80 \) Copy content Toggle raw display
$47$ \( T^{3} + 4 T^{2} - 44 T - 32 \) Copy content Toggle raw display
$53$ \( T^{3} + 10 T^{2} - 16 T - 96 \) Copy content Toggle raw display
$59$ \( T^{3} + 2 T^{2} - 144 T - 688 \) Copy content Toggle raw display
$61$ \( T^{3} - 14T^{2} + 144 \) Copy content Toggle raw display
$67$ \( (T + 4)^{3} \) Copy content Toggle raw display
$71$ \( T^{3} - 4 T^{2} - 168 T - 640 \) Copy content Toggle raw display
$73$ \( (T - 6)^{3} \) Copy content Toggle raw display
$79$ \( T^{3} - 2 T^{2} - 192 T + 1024 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} - 44 T + 200 \) Copy content Toggle raw display
$89$ \( T^{3} - 10 T^{2} - 50 T - 48 \) Copy content Toggle raw display
$97$ \( T^{3} - 32 T^{2} + 258 T - 36 \) Copy content Toggle raw display
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