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Properties

Label 1568.1.bl.a
Level $1568$
Weight $1$
Character orbit 1568.bl
Analytic conductor $0.783$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
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] = [1568,1,Mod(117,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([0, 15, 20]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.117");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1568.bl (of order \(24\), degree \(8\), 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: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.5156108238848.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} + \zeta_{24}^{3} q^{8} + \zeta_{24}^{11} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} + \zeta_{24}^{3} q^{8} + \zeta_{24}^{11} q^{9} + ( - \zeta_{24}^{10} + \zeta_{24}^{7}) q^{11} - \zeta_{24}^{8} q^{16} + \zeta_{24}^{4} q^{18} + ( - \zeta_{24}^{3} + 1) q^{22} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{23} + \zeta_{24} q^{25} + ( - \zeta_{24}^{9} + \zeta_{24}^{6}) q^{29} - \zeta_{24} q^{32} - \zeta_{24}^{9} q^{36} + ( - \zeta_{24}^{11} + \zeta_{24}^{2}) q^{37} + (\zeta_{24}^{9} - 1) q^{43} + (\zeta_{24}^{8} - \zeta_{24}^{5}) q^{44} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{46} - \zeta_{24}^{6} q^{50} + (\zeta_{24}^{4} + \zeta_{24}) q^{53} + ( - \zeta_{24}^{11} - \zeta_{24}^{2}) q^{58} + \zeta_{24}^{6} q^{64} + ( - \zeta_{24}^{7} - \zeta_{24}^{4}) q^{67} - \zeta_{24}^{2} q^{72} + ( - \zeta_{24}^{7} - \zeta_{24}^{4}) q^{74} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{79} - \zeta_{24}^{10} q^{81} + (\zeta_{24}^{5} + \zeta_{24}^{2}) q^{86} + (\zeta_{24}^{10} + \zeta_{24}) q^{88} + (\zeta_{24}^{6} - 1) q^{92} + (\zeta_{24}^{9} - \zeta_{24}^{6}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{16} + 4 q^{18} + 8 q^{22} + 4 q^{23} - 8 q^{43} - 4 q^{44} + 4 q^{53} - 4 q^{67} - 4 q^{74} - 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(\zeta_{24}^{9}\) \(1\) \(-\zeta_{24}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
0.258819 + 0.965926i
0.258819 − 0.965926i
0.965926 − 0.258819i
−0.965926 − 0.258819i
−0.258819 − 0.965926i
0.965926 − 0.258819i 0 0.866025 − 0.500000i 0 0 0 0.707107 − 0.707107i 0.258819 + 0.965926i 0
325.1 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 0 0 0 −0.707107 + 0.707107i 0.965926 + 0.258819i 0
509.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 0 0 0 0.707107 + 0.707107i −0.965926 + 0.258819i 0
717.1 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 0 0 0 −0.707107 − 0.707107i −0.258819 + 0.965926i 0
901.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 0 0 0 −0.707107 + 0.707107i −0.258819 − 0.965926i 0
1109.1 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i 0 0 0 0.707107 − 0.707107i −0.965926 − 0.258819i 0
1293.1 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 0 −0.707107 − 0.707107i 0.965926 − 0.258819i 0
1501.1 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 0 0.707107 + 0.707107i 0.258819 − 0.965926i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.1
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
32.g even 8 1 inner
224.v odd 8 1 inner
224.bc odd 24 1 inner
224.bd even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.1.bl.a 8
7.b odd 2 1 CM 1568.1.bl.a 8
7.c even 3 1 224.1.v.a ✓ 4
7.c even 3 1 inner 1568.1.bl.a 8
7.d odd 6 1 224.1.v.a ✓ 4
7.d odd 6 1 inner 1568.1.bl.a 8
21.g even 6 1 2016.1.dp.b 4
21.h odd 6 1 2016.1.dp.b 4
28.f even 6 1 896.1.v.a 4
28.g odd 6 1 896.1.v.a 4
32.g even 8 1 inner 1568.1.bl.a 8
56.j odd 6 1 1792.1.v.a 4
56.k odd 6 1 1792.1.v.b 4
56.m even 6 1 1792.1.v.b 4
56.p even 6 1 1792.1.v.a 4
112.u odd 12 1 3584.1.v.a 4
112.u odd 12 1 3584.1.v.c 4
112.v even 12 1 3584.1.v.a 4
112.v even 12 1 3584.1.v.c 4
112.w even 12 1 3584.1.v.b 4
112.w even 12 1 3584.1.v.d 4
112.x odd 12 1 3584.1.v.b 4
112.x odd 12 1 3584.1.v.d 4
224.v odd 8 1 inner 1568.1.bl.a 8
224.bc odd 24 1 224.1.v.a ✓ 4
224.bc odd 24 1 inner 1568.1.bl.a 8
224.bc odd 24 1 1792.1.v.a 4
224.bc odd 24 1 3584.1.v.b 4
224.bc odd 24 1 3584.1.v.d 4
224.bd even 24 1 224.1.v.a ✓ 4
224.bd even 24 1 inner 1568.1.bl.a 8
224.bd even 24 1 1792.1.v.a 4
224.bd even 24 1 3584.1.v.b 4
224.bd even 24 1 3584.1.v.d 4
224.be even 24 1 896.1.v.a 4
224.be even 24 1 1792.1.v.b 4
224.be even 24 1 3584.1.v.a 4
224.be even 24 1 3584.1.v.c 4
224.bf odd 24 1 896.1.v.a 4
224.bf odd 24 1 1792.1.v.b 4
224.bf odd 24 1 3584.1.v.a 4
224.bf odd 24 1 3584.1.v.c 4
672.ce odd 24 1 2016.1.dp.b 4
672.cl even 24 1 2016.1.dp.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.v.a ✓ 4 7.c even 3 1
224.1.v.a ✓ 4 7.d odd 6 1
224.1.v.a ✓ 4 224.bc odd 24 1
224.1.v.a ✓ 4 224.bd even 24 1
896.1.v.a 4 28.f even 6 1
896.1.v.a 4 28.g odd 6 1
896.1.v.a 4 224.be even 24 1
896.1.v.a 4 224.bf odd 24 1
1568.1.bl.a 8 1.a even 1 1 trivial
1568.1.bl.a 8 7.b odd 2 1 CM
1568.1.bl.a 8 7.c even 3 1 inner
1568.1.bl.a 8 7.d odd 6 1 inner
1568.1.bl.a 8 32.g even 8 1 inner
1568.1.bl.a 8 224.v odd 8 1 inner
1568.1.bl.a 8 224.bc odd 24 1 inner
1568.1.bl.a 8 224.bd even 24 1 inner
1792.1.v.a 4 56.j odd 6 1
1792.1.v.a 4 56.p even 6 1
1792.1.v.a 4 224.bc odd 24 1
1792.1.v.a 4 224.bd even 24 1
1792.1.v.b 4 56.k odd 6 1
1792.1.v.b 4 56.m even 6 1
1792.1.v.b 4 224.be even 24 1
1792.1.v.b 4 224.bf odd 24 1
2016.1.dp.b 4 21.g even 6 1
2016.1.dp.b 4 21.h odd 6 1
2016.1.dp.b 4 672.ce odd 24 1
2016.1.dp.b 4 672.cl even 24 1
3584.1.v.a 4 112.u odd 12 1
3584.1.v.a 4 112.v even 12 1
3584.1.v.a 4 224.be even 24 1
3584.1.v.a 4 224.bf odd 24 1
3584.1.v.b 4 112.w even 12 1
3584.1.v.b 4 112.x odd 12 1
3584.1.v.b 4 224.bc odd 24 1
3584.1.v.b 4 224.bd even 24 1
3584.1.v.c 4 112.u odd 12 1
3584.1.v.c 4 112.v even 12 1
3584.1.v.c 4 224.be even 24 1
3584.1.v.c 4 224.bf odd 24 1
3584.1.v.d 4 112.w even 12 1
3584.1.v.d 4 112.x odd 12 1
3584.1.v.d 4 224.bc odd 24 1
3584.1.v.d 4 224.bd even 24 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 2 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 2 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} + 6 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 4 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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