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Properties

Label 1568.2.t.a
Level $1568$
Weight $2$
Character orbit 1568.t
Analytic conductor $12.521$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1568,2,Mod(177,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.t (of order \(6\), degree \(2\), not 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: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 392)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 3 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta_1 - 3) q^{9} + \beta_{2} q^{11} + (8 \beta_1 + 8) q^{23} + 5 \beta_1 q^{25} + (2 \beta_{3} - 2 \beta_{2}) q^{29} + 2 \beta_{3} q^{37} + (\beta_{3} - \beta_{2}) q^{43} + 2 \beta_{2} q^{53} + 3 \beta_{2} q^{67} + 16 q^{71} + ( - 8 \beta_1 - 8) q^{79} + 9 \beta_1 q^{81} + (3 \beta_{3} - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{9} + 16 q^{23} - 10 q^{25} + 64 q^{71} - 16 q^{79} - 18 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - x^{2} - 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} - \nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + \nu^{2} + 3\nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{3} + \nu^{2} + 3\nu + 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} + 10 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(-1\) \(1\) \(-1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
−0.895644 + 1.09445i
1.39564 − 0.228425i
−0.895644 − 1.09445i
1.39564 + 0.228425i
0 0 0 0 0 0 0 −1.50000 + 2.59808i 0
177.2 0 0 0 0 0 0 0 −1.50000 + 2.59808i 0
753.1 0 0 0 0 0 0 0 −1.50000 − 2.59808i 0
753.2 0 0 0 0 0 0 0 −1.50000 − 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
56.h odd 2 1 inner
56.j odd 6 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.t.a 4
4.b odd 2 1 392.2.p.c 4
7.b odd 2 1 CM 1568.2.t.a 4
7.c even 3 1 1568.2.b.b 2
7.c even 3 1 inner 1568.2.t.a 4
7.d odd 6 1 1568.2.b.b 2
7.d odd 6 1 inner 1568.2.t.a 4
8.b even 2 1 inner 1568.2.t.a 4
8.d odd 2 1 392.2.p.c 4
28.d even 2 1 392.2.p.c 4
28.f even 6 1 392.2.b.a ✓ 2
28.f even 6 1 392.2.p.c 4
28.g odd 6 1 392.2.b.a ✓ 2
28.g odd 6 1 392.2.p.c 4
56.e even 2 1 392.2.p.c 4
56.h odd 2 1 inner 1568.2.t.a 4
56.j odd 6 1 1568.2.b.b 2
56.j odd 6 1 inner 1568.2.t.a 4
56.k odd 6 1 392.2.b.a ✓ 2
56.k odd 6 1 392.2.p.c 4
56.m even 6 1 392.2.b.a ✓ 2
56.m even 6 1 392.2.p.c 4
56.p even 6 1 1568.2.b.b 2
56.p even 6 1 inner 1568.2.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.b.a ✓ 2 28.f even 6 1
392.2.b.a ✓ 2 28.g odd 6 1
392.2.b.a ✓ 2 56.k odd 6 1
392.2.b.a ✓ 2 56.m even 6 1
392.2.p.c 4 4.b odd 2 1
392.2.p.c 4 8.d odd 2 1
392.2.p.c 4 28.d even 2 1
392.2.p.c 4 28.f even 6 1
392.2.p.c 4 28.g odd 6 1
392.2.p.c 4 56.e even 2 1
392.2.p.c 4 56.k odd 6 1
392.2.p.c 4 56.m even 6 1
1568.2.b.b 2 7.c even 3 1
1568.2.b.b 2 7.d odd 6 1
1568.2.b.b 2 56.j odd 6 1
1568.2.b.b 2 56.p even 6 1
1568.2.t.a 4 1.a even 1 1 trivial
1568.2.t.a 4 7.b odd 2 1 CM
1568.2.t.a 4 7.c even 3 1 inner
1568.2.t.a 4 7.d odd 6 1 inner
1568.2.t.a 4 8.b even 2 1 inner
1568.2.t.a 4 56.h odd 2 1 inner
1568.2.t.a 4 56.j odd 6 1 inner
1568.2.t.a 4 56.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 28T^{2} + 784 \) 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
$23$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 112)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 112 T^{2} + 12544 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 112 T^{2} + 12544 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 252 T^{2} + 63504 \) Copy content Toggle raw display
$71$ \( (T - 16)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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