Properties

Label 1568.1.c.a
Level $1568$
Weight $1$
Character orbit 1568.c
Analytic conductor $0.783$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1568,1,Mod(97,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.97");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1568.c (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: \(0.782533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.3373232128.1

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + q^{9} + \beta_{3} q^{13} - \beta_{3} q^{17} + (\beta_{2} - 1) q^{25} + \beta_{2} q^{29} - \beta_{2} q^{37} - \beta_1 q^{41} - \beta_1 q^{45} - \beta_{3} q^{61} + \beta_{2} q^{65} + \beta_1 q^{73} + q^{81} - \beta_{2} q^{85} + \beta_{3} q^{89} + \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} - 4 q^{25} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
1.84776i
0.765367i
0.765367i
1.84776i
0 0 0 1.84776i 0 0 0 1.00000 0
97.2 0 0 0 0.765367i 0 0 0 1.00000 0
97.3 0 0 0 0.765367i 0 0 0 1.00000 0
97.4 0 0 0 1.84776i 0 0 0 1.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.1.c.a 4
4.b odd 2 1 CM 1568.1.c.a 4
7.b odd 2 1 inner 1568.1.c.a 4
7.c even 3 2 1568.1.s.a 8
7.d odd 6 2 1568.1.s.a 8
8.b even 2 1 3136.1.c.a 4
8.d odd 2 1 3136.1.c.a 4
28.d even 2 1 inner 1568.1.c.a 4
28.f even 6 2 1568.1.s.a 8
28.g odd 6 2 1568.1.s.a 8
56.e even 2 1 3136.1.c.a 4
56.h odd 2 1 3136.1.c.a 4
56.j odd 6 2 3136.1.s.a 8
56.k odd 6 2 3136.1.s.a 8
56.m even 6 2 3136.1.s.a 8
56.p even 6 2 3136.1.s.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.1.c.a 4 1.a even 1 1 trivial
1568.1.c.a 4 4.b odd 2 1 CM
1568.1.c.a 4 7.b odd 2 1 inner
1568.1.c.a 4 28.d even 2 1 inner
1568.1.s.a 8 7.c even 3 2
1568.1.s.a 8 7.d odd 6 2
1568.1.s.a 8 28.f even 6 2
1568.1.s.a 8 28.g odd 6 2
3136.1.c.a 4 8.b even 2 1
3136.1.c.a 4 8.d odd 2 1
3136.1.c.a 4 56.e even 2 1
3136.1.c.a 4 56.h odd 2 1
3136.1.s.a 8 56.j odd 6 2
3136.1.s.a 8 56.k odd 6 2
3136.1.s.a 8 56.m even 6 2
3136.1.s.a 8 56.p even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1568, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4T^{2} + 2 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 4T^{2} + 2 \) Copy content Toggle raw display
$17$ \( T^{4} + 4T^{2} + 2 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 4T^{2} + 2 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 4T^{2} + 2 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 4T^{2} + 2 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 4T^{2} + 2 \) Copy content Toggle raw display
$97$ \( T^{4} + 4T^{2} + 2 \) Copy content Toggle raw display
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