Properties

Label 3120.2.a.bi
Level $3120$
Weight $2$
Character orbit 3120.a
Self dual yes
Analytic conductor $24.913$
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] = [3120,2,Mod(1,3120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3120, 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("3120.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.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: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.940.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 7x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
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 \( q - q^{3} + q^{5} - \beta_{2} q^{7} + q^{9} + ( - \beta_1 - 2) q^{11} - q^{13} - q^{15} + ( - \beta_{2} - 2 \beta_1 + 2) q^{17} + ( - \beta_{2} - \beta_1 - 2) q^{19} + \beta_{2} q^{21} + \beta_{2} q^{23} + q^{25} - q^{27} + ( - \beta_{2} + \beta_1 + 4) q^{29} + 2 \beta_1 q^{31} + (\beta_1 + 2) q^{33} - \beta_{2} q^{35} + (\beta_1 + 4) q^{37} + q^{39} + \beta_1 q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + q^{45} + (2 \beta_1 + 4) q^{47} + ( - 2 \beta_{2} - \beta_1 + 7) q^{49} + (\beta_{2} + 2 \beta_1 - 2) q^{51} + (2 \beta_{2} - \beta_1) q^{53} + ( - \beta_1 - 2) q^{55} + (\beta_{2} + \beta_1 + 2) q^{57} + (2 \beta_{2} - 2 \beta_1 - 4) q^{59} + ( - 2 \beta_{2} + \beta_1) q^{61} - \beta_{2} q^{63} - q^{65} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{67} - \beta_{2} q^{69} + ( - \beta_1 + 6) q^{71} + (\beta_{2} - \beta_1 + 8) q^{73} - q^{75} + (2 \beta_{2} + 3 \beta_1 - 2) q^{77} + (\beta_1 - 6) q^{79} + q^{81} + (2 \beta_{2} - 2 \beta_1) q^{83} + ( - \beta_{2} - 2 \beta_1 + 2) q^{85} + (\beta_{2} - \beta_1 - 4) q^{87} + ( - \beta_1 + 4) q^{89} + \beta_{2} q^{91} - 2 \beta_1 q^{93} + ( - \beta_{2} - \beta_1 - 2) q^{95} + (3 \beta_{2} + 2) q^{97} + ( - \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} - 5 q^{11} - 3 q^{13} - 3 q^{15} + 7 q^{17} - 6 q^{19} + q^{21} + q^{23} + 3 q^{25} - 3 q^{27} + 10 q^{29} - 2 q^{31} + 5 q^{33} - q^{35} + 11 q^{37} + 3 q^{39} - q^{41} + 3 q^{45} + 10 q^{47} + 20 q^{49} - 7 q^{51} + 3 q^{53} - 5 q^{55} + 6 q^{57} - 8 q^{59} - 3 q^{61} - q^{63} - 3 q^{65} + 12 q^{67} - q^{69} + 19 q^{71} + 26 q^{73} - 3 q^{75} - 7 q^{77} - 19 q^{79} + 3 q^{81} + 4 q^{83} + 7 q^{85} - 10 q^{87} + 13 q^{89} + q^{91} + 2 q^{93} - 6 q^{95} + 9 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 7x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 5 \) 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
−0.602705
2.89511
−2.29240
0 −1.00000 0 1.00000 0 −3.43134 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −2.40857 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 4.83991 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(1\)

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 3120.2.a.bi 3
3.b odd 2 1 9360.2.a.cy 3
4.b odd 2 1 1560.2.a.q 3
12.b even 2 1 4680.2.a.bh 3
20.d odd 2 1 7800.2.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.q 3 4.b odd 2 1
3120.2.a.bi 3 1.a even 1 1 trivial
4680.2.a.bh 3 12.b even 2 1
7800.2.a.bi 3 20.d odd 2 1
9360.2.a.cy 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3120))\):

\( T_{7}^{3} + T_{7}^{2} - 20T_{7} - 40 \) Copy content Toggle raw display
\( T_{11}^{3} + 5T_{11}^{2} - 8T_{11} - 32 \) Copy content Toggle raw display
\( T_{17}^{3} - 7T_{17}^{2} - 52T_{17} + 356 \) Copy content Toggle raw display
\( T_{19}^{3} + 6T_{19}^{2} - 16T_{19} - 16 \) Copy content Toggle raw display
\( T_{31}^{3} + 2T_{31}^{2} - 64T_{31} + 32 \) 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
$7$ \( T^{3} + T^{2} - 20 T - 40 \) Copy content Toggle raw display
$11$ \( T^{3} + 5 T^{2} - 8 T - 32 \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 7 T^{2} - 52 T + 356 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} - 16 T - 16 \) Copy content Toggle raw display
$23$ \( T^{3} - T^{2} - 20 T + 40 \) Copy content Toggle raw display
$29$ \( T^{3} - 10 T^{2} - 12 T + 184 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 64 T + 32 \) Copy content Toggle raw display
$37$ \( T^{3} - 11 T^{2} + 24 T + 20 \) Copy content Toggle raw display
$41$ \( T^{3} + T^{2} - 16T + 4 \) Copy content Toggle raw display
$43$ \( T^{3} - 112T - 256 \) Copy content Toggle raw display
$47$ \( T^{3} - 10 T^{2} - 32 T + 256 \) Copy content Toggle raw display
$53$ \( T^{3} - 3 T^{2} - 112 T + 164 \) Copy content Toggle raw display
$59$ \( T^{3} + 8 T^{2} - 160 T - 1024 \) Copy content Toggle raw display
$61$ \( T^{3} + 3 T^{2} - 112 T - 164 \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} - 64 T + 640 \) Copy content Toggle raw display
$71$ \( T^{3} - 19 T^{2} + 104 T - 160 \) Copy content Toggle raw display
$73$ \( T^{3} - 26 T^{2} + 180 T - 328 \) Copy content Toggle raw display
$79$ \( T^{3} + 19 T^{2} + 104 T + 160 \) Copy content Toggle raw display
$83$ \( T^{3} - 4 T^{2} - 176 T - 320 \) Copy content Toggle raw display
$89$ \( T^{3} - 13 T^{2} + 40 T - 20 \) Copy content Toggle raw display
$97$ \( T^{3} - 9 T^{2} - 156 T + 1420 \) Copy content Toggle raw display
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