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Properties

Label 1080.2.d.d
Level $1080$
Weight $2$
Character orbit 1080.d
Analytic conductor $8.624$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1080,2,Mod(109,1080)]
 
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, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + 2 q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{5} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{7} - 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + 2 q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{5} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{7} - 2 \beta_{3} q^{8} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{10} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{11}+ \cdots + (2 \beta_{3} + 12) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 6 q^{5} + 4 q^{10} + 16 q^{16} - 12 q^{20} + 2 q^{25} - 20 q^{31} - 6 q^{35} + 8 q^{40} - 8 q^{49} - 24 q^{50} + 12 q^{53} - 18 q^{55} + 24 q^{62} + 32 q^{64} - 60 q^{77} - 40 q^{79} - 24 q^{80} + 60 q^{83} + 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.707107 + 1.22474i
−0.707107 − 1.22474i
0.707107 + 1.22474i
0.707107 − 1.22474i
−1.41421 0 2.00000 −2.20711 − 0.358719i 0 − 4.18154i −2.82843 0 3.12132 + 0.507306i
109.2 −1.41421 0 2.00000 −2.20711 + 0.358719i 0 4.18154i −2.82843 0 3.12132 − 0.507306i
109.3 1.41421 0 2.00000 −0.792893 − 2.09077i 0 − 0.717439i 2.82843 0 −1.12132 − 2.95680i
109.4 1.41421 0 2.00000 −0.792893 + 2.09077i 0 0.717439i 2.82843 0 −1.12132 + 2.95680i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
15.d odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.d.d ✓ 4
3.b odd 2 1 1080.2.d.e yes 4
4.b odd 2 1 4320.2.d.a 4
5.b even 2 1 1080.2.d.e yes 4
8.b even 2 1 1080.2.d.e yes 4
8.d odd 2 1 4320.2.d.f 4
12.b even 2 1 4320.2.d.f 4
15.d odd 2 1 inner 1080.2.d.d ✓ 4
20.d odd 2 1 4320.2.d.f 4
24.f even 2 1 4320.2.d.a 4
24.h odd 2 1 CM 1080.2.d.d ✓ 4
40.e odd 2 1 4320.2.d.a 4
40.f even 2 1 inner 1080.2.d.d ✓ 4
60.h even 2 1 4320.2.d.a 4
120.i odd 2 1 1080.2.d.e yes 4
120.m even 2 1 4320.2.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.d.d ✓ 4 1.a even 1 1 trivial
1080.2.d.d ✓ 4 15.d odd 2 1 inner
1080.2.d.d ✓ 4 24.h odd 2 1 CM
1080.2.d.d ✓ 4 40.f even 2 1 inner
1080.2.d.e yes 4 3.b odd 2 1
1080.2.d.e yes 4 5.b even 2 1
1080.2.d.e yes 4 8.b even 2 1
1080.2.d.e yes 4 120.i odd 2 1
4320.2.d.a 4 4.b odd 2 1
4320.2.d.a 4 24.f even 2 1
4320.2.d.a 4 40.e odd 2 1
4320.2.d.a 4 60.h even 2 1
4320.2.d.f 4 8.d odd 2 1
4320.2.d.f 4 12.b even 2 1
4320.2.d.f 4 20.d odd 2 1
4320.2.d.f 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1080, [\chi])\):

\( T_{7}^{4} + 18T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{4} + 54T_{11}^{2} + 441 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{53}^{2} - 6T_{53} - 41 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 18T^{2} + 9 \) Copy content Toggle raw display
$11$ \( T^{4} + 54T^{2} + 441 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 6 T - 41)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 342 T^{2} + 15129 \) Copy content Toggle raw display
$79$ \( (T + 10)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 30 T + 217)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 198T^{2} + 8649 \) Copy content Toggle raw display
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