Properties

Label 2760.2.a.s
Level $2760$
Weight $2$
Character orbit 2760.a
Self dual yes
Analytic conductor $22.039$
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] = [2760,2,Mod(1,2760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, 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("2760.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.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: \(22.0387109579\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1436.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 11x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + (\beta_{2} - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + (\beta_{2} - 1) q^{7} + q^{9} + (\beta_{2} + \beta_1 + 1) q^{11} + (\beta_{2} + \beta_1 - 1) q^{13} - q^{15} + ( - \beta_1 + 2) q^{17} + ( - \beta_{2} - \beta_1 + 1) q^{19} + (\beta_{2} - 1) q^{21} + q^{23} + q^{25} + q^{27} - \beta_1 q^{29} + ( - \beta_{2} + 1) q^{31} + (\beta_{2} + \beta_1 + 1) q^{33} + ( - \beta_{2} + 1) q^{35} + ( - \beta_{2} + 2 \beta_1 + 3) q^{37} + (\beta_{2} + \beta_1 - 1) q^{39} + (2 \beta_{2} + 3 \beta_1) q^{41} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{43} - q^{45} + ( - 3 \beta_{2} + \beta_1 + 3) q^{47} + ( - \beta_{2} - 2 \beta_1 + 2) q^{49} + ( - \beta_1 + 2) q^{51} + ( - 3 \beta_1 + 2) q^{53} + ( - \beta_{2} - \beta_1 - 1) q^{55} + ( - \beta_{2} - \beta_1 + 1) q^{57} + (2 \beta_{2} - \beta_1 + 2) q^{59} + (\beta_{2} + \beta_1 + 1) q^{61} + (\beta_{2} - 1) q^{63} + ( - \beta_{2} - \beta_1 + 1) q^{65} + ( - \beta_{2} + 2 \beta_1 + 3) q^{67} + q^{69} + ( - \beta_1 + 6) q^{71} + ( - \beta_{2} - 3 \beta_1 + 3) q^{73} + q^{75} + ( - \beta_{2} - 3 \beta_1 + 5) q^{77} - 8 q^{79} + q^{81} + (\beta_1 + 8) q^{83} + (\beta_1 - 2) q^{85} - \beta_1 q^{87} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{89} + ( - 3 \beta_{2} - 3 \beta_1 + 7) q^{91} + ( - \beta_{2} + 1) q^{93} + (\beta_{2} + \beta_1 - 1) q^{95} + (4 \beta_1 + 4) q^{97} + (\beta_{2} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} + 4 q^{11} - 2 q^{13} - 3 q^{15} + 6 q^{17} + 2 q^{19} - 2 q^{21} + 3 q^{23} + 3 q^{25} + 3 q^{27} + 2 q^{31} + 4 q^{33} + 2 q^{35} + 8 q^{37} - 2 q^{39} + 2 q^{41} + 10 q^{43} - 3 q^{45} + 6 q^{47} + 5 q^{49} + 6 q^{51} + 6 q^{53} - 4 q^{55} + 2 q^{57} + 8 q^{59} + 4 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} + 3 q^{69} + 18 q^{71} + 8 q^{73} + 3 q^{75} + 14 q^{77} - 24 q^{79} + 3 q^{81} + 24 q^{83} - 6 q^{85} + 4 q^{89} + 18 q^{91} + 2 q^{93} - 2 q^{95} + 12 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 11x - 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 7 \) 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
−1.28282
3.76644
−2.48361
0 1.00000 0 −1.00000 0 −3.78872 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −1.34683 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 3.13555 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 2760.2.a.s 3
3.b odd 2 1 8280.2.a.bn 3
4.b odd 2 1 5520.2.a.bw 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.s 3 1.a even 1 1 trivial
5520.2.a.bw 3 4.b odd 2 1
8280.2.a.bn 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2760))\):

\( T_{7}^{3} + 2T_{7}^{2} - 11T_{7} - 16 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} - 10T_{11} + 36 \) 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
$7$ \( T^{3} + 2 T^{2} - 11 T - 16 \) Copy content Toggle raw display
$11$ \( T^{3} - 4 T^{2} - 10 T + 36 \) Copy content Toggle raw display
$13$ \( T^{3} + 2 T^{2} - 14 T + 8 \) Copy content Toggle raw display
$17$ \( T^{3} - 6T^{2} + T + 26 \) Copy content Toggle raw display
$19$ \( T^{3} - 2 T^{2} - 14 T - 8 \) Copy content Toggle raw display
$23$ \( (T - 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 11T + 12 \) Copy content Toggle raw display
$31$ \( T^{3} - 2 T^{2} - 11 T + 16 \) Copy content Toggle raw display
$37$ \( T^{3} - 8 T^{2} - 51 T + 214 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} - 99 T + 82 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} - 28 T + 24 \) Copy content Toggle raw display
$47$ \( T^{3} - 6 T^{2} - 134 T + 936 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 87 T + 514 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 55 T - 72 \) Copy content Toggle raw display
$61$ \( T^{3} - 4 T^{2} - 10 T + 36 \) Copy content Toggle raw display
$67$ \( T^{3} - 8 T^{2} - 51 T + 214 \) Copy content Toggle raw display
$71$ \( T^{3} - 18 T^{2} + 97 T - 138 \) Copy content Toggle raw display
$73$ \( T^{3} - 8 T^{2} - 66 T + 484 \) Copy content Toggle raw display
$79$ \( (T + 8)^{3} \) Copy content Toggle raw display
$83$ \( T^{3} - 24 T^{2} + 181 T - 436 \) Copy content Toggle raw display
$89$ \( T^{3} - 4 T^{2} - 120 T + 576 \) Copy content Toggle raw display
$97$ \( T^{3} - 12 T^{2} - 128 T - 128 \) Copy content Toggle raw display
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