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Properties

Label 1083.2.d.a
Level $1083$
Weight $2$
Character orbit 1083.d
Analytic conductor $8.648$
Analytic rank $0$
Dimension $6$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1083,2,Mod(1082,1083)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1083, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1083.1082");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1083 = 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1083.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: \(8.64779853890\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 19^{2} \)
Twist minimal: no (minimal twist has level 57)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 2 q^{4} - \beta_{3} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - 2 q^{4} - \beta_{3} q^{7} - 3 q^{9} - 2 \beta_1 q^{12} + \beta_{2} q^{13} + 4 q^{16} + \beta_{5} q^{21} + 5 q^{25} - 3 \beta_1 q^{27} + 2 \beta_{3} q^{28} + (\beta_{5} + \beta_{2}) q^{31} + 6 q^{36} + (\beta_{5} - \beta_{2}) q^{37} + \beta_{4} q^{39} + (\beta_{4} + \beta_{3}) q^{43} + 4 \beta_1 q^{48} + ( - \beta_{4} + \beta_{3} + 7) q^{49} - 2 \beta_{2} q^{52} + (\beta_{4} + 2 \beta_{3}) q^{61} + 3 \beta_{3} q^{63} - 8 q^{64} + (\beta_{5} + 2 \beta_{2}) q^{67} + (\beta_{4} - 2 \beta_{3}) q^{73} + 5 \beta_1 q^{75} + ( - \beta_{5} + 2 \beta_{2}) q^{79} + 9 q^{81} - 2 \beta_{5} q^{84} + (2 \beta_{5} + \beta_{2} - \beta_1) q^{91} + (\beta_{4} + 3 \beta_{3}) q^{93} + 3 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{4} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 12 q^{4} - 18 q^{9} + 24 q^{16} + 30 q^{25} + 36 q^{36} + 42 q^{49} - 48 q^{64} + 54 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{18}^{3} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{18}^{5} + 3\zeta_{18}^{4} + 4\zeta_{18}^{2} - 4\zeta_{18} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{18}^{5} - 2\zeta_{18}^{4} + 3\zeta_{18}^{2} + 3\zeta_{18} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 7\zeta_{18}^{5} - 5\zeta_{18}^{4} - 2\zeta_{18}^{2} - 2\zeta_{18} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -5\zeta_{18}^{5} - 4\zeta_{18}^{4} + \zeta_{18}^{2} - \zeta_{18} \) Copy content Toggle raw display

Display \(\zeta_{18}^j\) in terms of \(\beta_i\)

\(\zeta_{18}\)\(=\) \( ( -3\beta_{5} - 2\beta_{4} + 5\beta_{3} - 4\beta_{2} ) / 38 \) Copy content Toggle raw display
\(\zeta_{18}^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} + 7\beta_{3} + 5\beta_{2} ) / 38 \) Copy content Toggle raw display
\(\zeta_{18}^{3}\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{18}^{4}\)\(=\) \( ( -4\beta_{5} - 3\beta_{4} - 2\beta_{3} + \beta_{2} ) / 38 \) Copy content Toggle raw display
\(\zeta_{18}^{5}\)\(=\) \( ( -4\beta_{5} + 3\beta_{4} + 2\beta_{3} + \beta_{2} ) / 38 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{18}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1083\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(724\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1082.1
0.939693 − 0.342020i
−0.766044 − 0.642788i
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.766044 + 0.642788i
−0.173648 − 0.984808i
0 − 1.73205i −2.00000 0 0 −4.94356 0 −3.00000 0
1082.2 0 − 1.73205i −2.00000 0 0 0.837496 0 −3.00000 0
1082.3 0 − 1.73205i −2.00000 0 0 4.10607 0 −3.00000 0
1082.4 0 1.73205i −2.00000 0 0 −4.94356 0 −3.00000 0
1082.5 0 1.73205i −2.00000 0 0 0.837496 0 −3.00000 0
1082.6 0 1.73205i −2.00000 0 0 4.10607 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1082.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1083.2.d.a 6
3.b odd 2 1 CM 1083.2.d.a 6
19.b odd 2 1 inner 1083.2.d.a 6
19.e even 9 1 57.2.j.a ✓ 6
19.f odd 18 1 57.2.j.a ✓ 6
57.d even 2 1 inner 1083.2.d.a 6
57.j even 18 1 57.2.j.a ✓ 6
57.l odd 18 1 57.2.j.a ✓ 6
76.k even 18 1 912.2.cc.a 6
76.l odd 18 1 912.2.cc.a 6
228.u odd 18 1 912.2.cc.a 6
228.v even 18 1 912.2.cc.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.j.a ✓ 6 19.e even 9 1
57.2.j.a ✓ 6 19.f odd 18 1
57.2.j.a ✓ 6 57.j even 18 1
57.2.j.a ✓ 6 57.l odd 18 1
912.2.cc.a 6 76.k even 18 1
912.2.cc.a 6 76.l odd 18 1
912.2.cc.a 6 228.u odd 18 1
912.2.cc.a 6 228.v even 18 1
1083.2.d.a 6 1.a even 1 1 trivial
1083.2.d.a 6 3.b odd 2 1 CM
1083.2.d.a 6 19.b odd 2 1 inner
1083.2.d.a 6 57.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(1083, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} - 21 T + 17)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 78 T^{4} + \cdots + 8427 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + 186 T^{4} + \cdots + 118803 \) Copy content Toggle raw display
$37$ \( T^{6} + 222 T^{4} + \cdots + 98283 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( (T^{3} - 129 T - 71)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( (T^{3} - 183 T + 719)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 402 T^{4} + \cdots + 189003 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( (T^{3} - 219 T + 271)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 474 T^{4} + \cdots + 48387 \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( (T^{2} + 27)^{3} \) Copy content Toggle raw display
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