Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1568,1,Mod(79,1568)]
 
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([3, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.79");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1568.o (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: \(0.782533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 392)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.2744.1
Artin image: $C_6\times D_8$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{3} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{3} + ( - \beta_{2} - 1) q^{9} + ( - \beta_{3} - \beta_1) q^{17} - \beta_1 q^{19} + \beta_{2} q^{25} - \beta_{3} q^{41} + ( - 2 \beta_{2} - 2) q^{51} - 2 q^{57} + (\beta_{3} + \beta_1) q^{59} + 2 \beta_{2} q^{67} + (\beta_{3} + \beta_1) q^{73} + \beta_1 q^{75} - \beta_{2} q^{81} - \beta_{3} q^{83} - \beta_1 q^{89} + \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{9} - 2 q^{25} - 4 q^{51} - 8 q^{57} - 4 q^{67} + 2 q^{81}+O(q^{100}) \) 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
\(\nu\)\(=\) \( \beta_1 \) 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

Character values

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

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(-1\) \(-1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 −0.707107 1.22474i 0 0 0 0 0 −0.500000 + 0.866025i 0
79.2 0 0.707107 + 1.22474i 0 0 0 0 0 −0.500000 + 0.866025i 0
655.1 0 −0.707107 + 1.22474i 0 0 0 0 0 −0.500000 0.866025i 0
655.2 0 0.707107 1.22474i 0 0 0 0 0 −0.500000 0.866025i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
56.e even 2 1 inner
56.k odd 6 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.1.o.b 4
4.b odd 2 1 392.1.k.b 4
7.b odd 2 1 inner 1568.1.o.b 4
7.c even 3 1 1568.1.g.b 2
7.c even 3 1 inner 1568.1.o.b 4
7.d odd 6 1 1568.1.g.b 2
7.d odd 6 1 inner 1568.1.o.b 4
8.b even 2 1 392.1.k.b 4
8.d odd 2 1 CM 1568.1.o.b 4
12.b even 2 1 3528.1.bx.b 4
24.h odd 2 1 3528.1.bx.b 4
28.d even 2 1 392.1.k.b 4
28.f even 6 1 392.1.g.b 2
28.f even 6 1 392.1.k.b 4
28.g odd 6 1 392.1.g.b 2
28.g odd 6 1 392.1.k.b 4
56.e even 2 1 inner 1568.1.o.b 4
56.h odd 2 1 392.1.k.b 4
56.j odd 6 1 392.1.g.b 2
56.j odd 6 1 392.1.k.b 4
56.k odd 6 1 1568.1.g.b 2
56.k odd 6 1 inner 1568.1.o.b 4
56.m even 6 1 1568.1.g.b 2
56.m even 6 1 inner 1568.1.o.b 4
56.p even 6 1 392.1.g.b 2
56.p even 6 1 392.1.k.b 4
84.h odd 2 1 3528.1.bx.b 4
84.j odd 6 1 3528.1.g.c 2
84.j odd 6 1 3528.1.bx.b 4
84.n even 6 1 3528.1.g.c 2
84.n even 6 1 3528.1.bx.b 4
168.i even 2 1 3528.1.bx.b 4
168.s odd 6 1 3528.1.g.c 2
168.s odd 6 1 3528.1.bx.b 4
168.ba even 6 1 3528.1.g.c 2
168.ba even 6 1 3528.1.bx.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.1.g.b 2 28.f even 6 1
392.1.g.b 2 28.g odd 6 1
392.1.g.b 2 56.j odd 6 1
392.1.g.b 2 56.p even 6 1
392.1.k.b 4 4.b odd 2 1
392.1.k.b 4 8.b even 2 1
392.1.k.b 4 28.d even 2 1
392.1.k.b 4 28.f even 6 1
392.1.k.b 4 28.g odd 6 1
392.1.k.b 4 56.h odd 2 1
392.1.k.b 4 56.j odd 6 1
392.1.k.b 4 56.p even 6 1
1568.1.g.b 2 7.c even 3 1
1568.1.g.b 2 7.d odd 6 1
1568.1.g.b 2 56.k odd 6 1
1568.1.g.b 2 56.m even 6 1
1568.1.o.b 4 1.a even 1 1 trivial
1568.1.o.b 4 7.b odd 2 1 inner
1568.1.o.b 4 7.c even 3 1 inner
1568.1.o.b 4 7.d odd 6 1 inner
1568.1.o.b 4 8.d odd 2 1 CM
1568.1.o.b 4 56.e even 2 1 inner
1568.1.o.b 4 56.k odd 6 1 inner
1568.1.o.b 4 56.m even 6 1 inner
3528.1.g.c 2 84.j odd 6 1
3528.1.g.c 2 84.n even 6 1
3528.1.g.c 2 168.s odd 6 1
3528.1.g.c 2 168.ba even 6 1
3528.1.bx.b 4 12.b even 2 1
3528.1.bx.b 4 24.h odd 2 1
3528.1.bx.b 4 84.h odd 2 1
3528.1.bx.b 4 84.j odd 6 1
3528.1.bx.b 4 84.n even 6 1
3528.1.bx.b 4 168.i even 2 1
3528.1.bx.b 4 168.s odd 6 1
3528.1.bx.b 4 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2T_{3}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1568, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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