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Properties

Label 1080.2.k.a
Level $1080$
Weight $2$
Character orbit 1080.k
Analytic conductor $8.624$
Analytic rank $0$
Dimension $12$
CM no
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(541,1080)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.541");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.k (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: \(12\)
Coefficient field: 12.0.4097540240769024.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6x^{9} + 18x^{6} - 48x^{3} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} - \beta_{8} q^{4} - \beta_{7} q^{5} + (\beta_{5} - \beta_{3}) q^{7} + (\beta_{7} - \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} - \beta_{8} q^{4} - \beta_{7} q^{5} + (\beta_{5} - \beta_{3}) q^{7} + (\beta_{7} - \beta_{4}) q^{8} + \beta_{3} q^{10} + ( - \beta_{2} + \beta_1) q^{11} + (\beta_{10} - \beta_{8} + \beta_{5}) q^{13} + ( - 2 \beta_{7} + \beta_{6} + \beta_{2}) q^{14} - 2 \beta_{10} q^{16} + (\beta_{9} - \beta_{6} - \beta_{4}) q^{17} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_{5}) q^{19}+ \cdots + ( - 3 \beta_{9} - \beta_{7} + \cdots + \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{22} - 12 q^{25} - 12 q^{28} - 12 q^{31} + 24 q^{34} + 12 q^{40} - 12 q^{46} - 36 q^{49} - 36 q^{52} + 24 q^{58} + 72 q^{64} - 24 q^{70} + 72 q^{73} - 36 q^{76} + 60 q^{79} + 24 q^{82} - 60 q^{94} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6x^{9} + 18x^{6} - 48x^{3} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} + 6\nu^{7} - 18\nu^{4} + 48\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} + 2\nu^{8} - 10\nu^{5} + 24\nu^{2} ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} + 6\nu^{6} - 10\nu^{3} + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{10} + 10\nu^{7} - 22\nu^{4} + 64\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{10} + 14\nu^{7} - 42\nu^{4} + 112\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{9} - 10\nu^{6} + 30\nu^{3} - 72 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{10} - 10\nu^{7} + 30\nu^{4} - 64\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} + 3\nu^{8} - 8\nu^{5} + 22\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3\nu^{11} + 10\nu^{8} - 30\nu^{5} + 72\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( \nu^{9} - 3\nu^{6} + 8\nu^{3} - 21 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{6} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + 3\beta_{7} + \beta_{4} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + 2\beta_{5} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} + 2\beta_{9} - 2\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} + 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{8} - 5\beta_{6} + 6\beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{10} - 12\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5\beta_{11} - 9\beta_{7} + 5\beta_{4} + 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -6\beta_{6} + 14\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 14\beta_{10} - 20\beta_{9} - 20\beta_{3} + 14\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
−0.732533 − 1.20971i
−0.732533 + 1.20971i
1.23925 − 0.681372i
1.23925 + 0.681372i
−0.0295379 − 1.41391i
−0.0295379 + 1.41391i
1.41391 + 0.0295379i
1.41391 − 0.0295379i
−0.681372 + 1.23925i
−0.681372 − 1.23925i
−1.20971 − 0.732533i
−1.20971 + 0.732533i
−1.40397 − 0.169938i 0 1.94224 + 0.477176i 1.00000i 0 −0.339877 −2.64575 − 1.00000i 0 0.169938 − 1.40397i
541.2 −1.40397 + 0.169938i 0 1.94224 − 0.477176i − 1.00000i 0 −0.339877 −2.64575 + 1.00000i 0 0.169938 + 1.40397i
541.3 −0.849154 − 1.13090i 0 −0.557875 + 1.92062i 1.00000i 0 −2.26180 2.64575 − 1.00000i 0 1.13090 − 0.849154i
541.4 −0.849154 + 1.13090i 0 −0.557875 − 1.92062i − 1.00000i 0 −2.26180 2.64575 + 1.00000i 0 1.13090 + 0.849154i
541.5 −0.554812 − 1.30084i 0 −1.38437 + 1.44344i − 1.00000i 0 2.60168 2.64575 + 1.00000i 0 −1.30084 + 0.554812i
541.6 −0.554812 + 1.30084i 0 −1.38437 − 1.44344i 1.00000i 0 2.60168 2.64575 − 1.00000i 0 −1.30084 − 0.554812i
541.7 0.554812 − 1.30084i 0 −1.38437 − 1.44344i − 1.00000i 0 2.60168 −2.64575 + 1.00000i 0 −1.30084 − 0.554812i
541.8 0.554812 + 1.30084i 0 −1.38437 + 1.44344i 1.00000i 0 2.60168 −2.64575 − 1.00000i 0 −1.30084 + 0.554812i
541.9 0.849154 − 1.13090i 0 −0.557875 − 1.92062i 1.00000i 0 −2.26180 −2.64575 − 1.00000i 0 1.13090 + 0.849154i
541.10 0.849154 + 1.13090i 0 −0.557875 + 1.92062i − 1.00000i 0 −2.26180 −2.64575 + 1.00000i 0 1.13090 − 0.849154i
541.11 1.40397 − 0.169938i 0 1.94224 − 0.477176i 1.00000i 0 −0.339877 2.64575 − 1.00000i 0 0.169938 + 1.40397i
541.12 1.40397 + 0.169938i 0 1.94224 + 0.477176i − 1.00000i 0 −0.339877 2.64575 + 1.00000i 0 0.169938 − 1.40397i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.k.a ✓ 12
3.b odd 2 1 inner 1080.2.k.a ✓ 12
4.b odd 2 1 4320.2.k.a 12
8.b even 2 1 inner 1080.2.k.a ✓ 12
8.d odd 2 1 4320.2.k.a 12
12.b even 2 1 4320.2.k.a 12
24.f even 2 1 4320.2.k.a 12
24.h odd 2 1 inner 1080.2.k.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.k.a ✓ 12 1.a even 1 1 trivial
1080.2.k.a ✓ 12 3.b odd 2 1 inner
1080.2.k.a ✓ 12 8.b even 2 1 inner
1080.2.k.a ✓ 12 24.h odd 2 1 inner
4320.2.k.a 12 4.b odd 2 1
4320.2.k.a 12 8.d odd 2 1
4320.2.k.a 12 12.b even 2 1
4320.2.k.a 12 24.f even 2 1

Hecke kernels

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

\( T_{7}^{3} - 6T_{7} - 2 \) Copy content Toggle raw display
\( T_{17}^{6} - 33T_{17}^{4} + 99T_{17}^{2} - 63 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 12T^{6} + 64 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} - 6 T - 2)^{4} \) Copy content Toggle raw display
$11$ \( (T^{6} + 24 T^{4} + 144 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 24 T^{4} + \cdots + 252)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 33 T^{4} + \cdots - 63)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 45 T^{4} + 39 T^{2} + 7)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 45 T^{4} + 123 T^{2} - 7)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 24 T^{4} + \cdots + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 3 T^{2} - 3 T - 7)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 96 T^{4} + \cdots + 7168)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 96 T^{4} + \cdots - 14812)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 84 T^{4} + \cdots + 1372)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 180 T^{4} + \cdots - 5488)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 99 T^{4} + \cdots + 4489)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 192 T^{4} + \cdots + 131044)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 213 T^{4} + \cdots + 329623)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 384 T^{4} + \cdots + 790272)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 336 T^{4} + \cdots - 231868)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 18 T^{2} + \cdots + 722)^{4} \) Copy content Toggle raw display
$79$ \( (T^{3} - 15 T^{2} + \cdots + 677)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 267 T^{4} + \cdots + 159201)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 516 T^{4} + \cdots - 3388)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 18 T^{2} + \cdots - 56)^{4} \) Copy content Toggle raw display
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