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Properties

Label 1386.2.k.t
Level $1386$
Weight $2$
Character orbit 1386.k
Analytic conductor $11.067$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,2,Mod(793,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.793");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.k (of order \(3\), degree \(2\), 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: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{7} - q^{8} + (2 \beta_{2} - \beta_1 + 2) q^{10} + (\beta_{2} + 1) q^{11}+ \cdots + (2 \beta_{3} + 4 \beta_1 + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} - 2 q^{7} - 4 q^{8} + 4 q^{10} + 2 q^{11} - 4 q^{13} + 2 q^{14} - 2 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} - 4 q^{23} - 2 q^{25} - 2 q^{26} + 4 q^{28} + 12 q^{29}+ \cdots + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
793.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.70711 + 2.95680i 0 −2.62132 − 0.358719i −1.00000 0 1.70711 + 2.95680i
793.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.292893 + 0.507306i 0 1.62132 + 2.09077i −1.00000 0 0.292893 + 0.507306i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.70711 − 2.95680i 0 −2.62132 + 0.358719i −1.00000 0 1.70711 − 2.95680i
991.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.292893 − 0.507306i 0 1.62132 − 2.09077i −1.00000 0 0.292893 − 0.507306i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.k.t 4
3.b odd 2 1 154.2.e.e ✓ 4
7.c even 3 1 inner 1386.2.k.t 4
7.c even 3 1 9702.2.a.cx 2
7.d odd 6 1 9702.2.a.ch 2
12.b even 2 1 1232.2.q.f 4
21.c even 2 1 1078.2.e.m 4
21.g even 6 1 1078.2.a.x 2
21.g even 6 1 1078.2.e.m 4
21.h odd 6 1 154.2.e.e ✓ 4
21.h odd 6 1 1078.2.a.t 2
84.j odd 6 1 8624.2.a.bh 2
84.n even 6 1 1232.2.q.f 4
84.n even 6 1 8624.2.a.cc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e ✓ 4 3.b odd 2 1
154.2.e.e ✓ 4 21.h odd 6 1
1078.2.a.t 2 21.h odd 6 1
1078.2.a.x 2 21.g even 6 1
1078.2.e.m 4 21.c even 2 1
1078.2.e.m 4 21.g even 6 1
1232.2.q.f 4 12.b even 2 1
1232.2.q.f 4 84.n even 6 1
1386.2.k.t 4 1.a even 1 1 trivial
1386.2.k.t 4 7.c even 3 1 inner
8624.2.a.bh 2 84.j odd 6 1
8624.2.a.cc 2 84.n even 6 1
9702.2.a.ch 2 7.d odd 6 1
9702.2.a.cx 2 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{4} + 4T_{5}^{3} + 14T_{5}^{2} + 8T_{5} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 7 \) Copy content Toggle raw display
\( T_{17}^{4} + 4T_{17}^{3} + 44T_{17}^{2} - 112T_{17} + 784 \) Copy content Toggle raw display
\( T_{23}^{4} + 4T_{23}^{3} + 30T_{23}^{2} - 56T_{23} + 196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots + 784 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T - 23)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 16 T^{3} + \cdots + 3844 \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T + 14)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4 T^{3} + \cdots + 4624 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + \cdots + 8836 \) Copy content Toggle raw display
$59$ \( T^{4} + 14 T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$61$ \( T^{4} + 18 T^{3} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{4} + 14 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$71$ \( (T^{2} - 8 T - 34)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 16 T^{3} + \cdots + 3844 \) Copy content Toggle raw display
$79$ \( T^{4} - 18 T^{3} + \cdots + 3969 \) Copy content Toggle raw display
$83$ \( (T^{2} + 4 T - 196)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} + \cdots + 3136 \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 7)^{2} \) Copy content Toggle raw display
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