Properties

Label 1080.6.a.g
Level $1080$
Weight $6$
Character orbit 1080.a
Self dual yes
Analytic conductor $173.215$
Analytic rank $1$
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] = [1080,6,Mod(1,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1080.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: \(173.214525398\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.15881.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 29x - 55 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 11 \)
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.

\(f(q)\) \(=\) \( q + 25 q^{5} + (\beta_{2} - 10) q^{7} + (3 \beta_{2} - \beta_1 - 220) q^{11} + ( - 3 \beta_{2} + \beta_1 - 426) q^{13} + ( - 7 \beta_{2} - 2 \beta_1 + 577) q^{17} + (3 \beta_{2} + 9 \beta_1 + 337) q^{19}+ \cdots + (284 \beta_{2} - 92 \beta_1 + 4920) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 75 q^{5} - 30 q^{7} - 660 q^{11} - 1278 q^{13} + 1731 q^{17} + 1011 q^{19} + 2433 q^{23} + 1875 q^{25} + 3786 q^{29} + 4131 q^{31} - 750 q^{35} - 7122 q^{37} + 17352 q^{41} + 11556 q^{43} - 19548 q^{47}+ \cdots + 14760 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 29x - 55 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -12\nu^{2} + 232 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -18\nu^{2} + 66\nu + 348 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{2} - 3\beta_1 ) / 132 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta _1 + 232 ) / 12 \) 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
−3.82250
6.15876
−2.33626
0 0 0 25.0000 0 −177.292 0 0 0
1.2 0 0 0 25.0000 0 61.7318 0 0 0
1.3 0 0 0 25.0000 0 85.5605 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1080.6.a.g yes 3
3.b odd 2 1 1080.6.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.6.a.e 3 3.b odd 2 1
1080.6.a.g yes 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1080))\):

\( T_{7}^{3} + 30T_{7}^{2} - 20832T_{7} + 936424 \) Copy content Toggle raw display
\( T_{11}^{3} + 660T_{11}^{2} - 114084T_{11} - 16969720 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 30 T^{2} + \cdots + 936424 \) Copy content Toggle raw display
$11$ \( T^{3} + 660 T^{2} + \cdots - 16969720 \) Copy content Toggle raw display
$13$ \( T^{3} + 1278 T^{2} + \cdots - 62570968 \) Copy content Toggle raw display
$17$ \( T^{3} - 1731 T^{2} + \cdots + 362051059 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots + 1066478795 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 16297632125 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 100231188440 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 36775432739 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 11280290472 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 19821902472 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 74089212824 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 48232251200 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 6200694992835 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 9865368652616 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 15693428262193 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 20150295569000 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 46987871536152 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 119813275420344 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 306616858223481 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 623768894560265 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 152901460706184 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 36671348002304 \) Copy content Toggle raw display
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