Properties

Label 1568.4.a.o
Level $1568$
Weight $4$
Character orbit 1568.a
Self dual yes
Analytic conductor $92.515$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1568,4,Mod(1,1568)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1568.a (trivial)

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: yes
Analytic conductor: \(92.5149948890\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 8 q^{3} + 10 q^{5} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{3} + 10 q^{5} + 37 q^{9} + 40 q^{11} + 50 q^{13} + 80 q^{15} + 30 q^{17} + 40 q^{19} - 48 q^{23} - 25 q^{25} + 80 q^{27} - 34 q^{29} + 320 q^{31} + 320 q^{33} + 310 q^{37} + 400 q^{39} - 410 q^{41} - 152 q^{43} + 370 q^{45} - 416 q^{47} + 240 q^{51} - 410 q^{53} + 400 q^{55} + 320 q^{57} - 200 q^{59} - 30 q^{61} + 500 q^{65} - 776 q^{67} - 384 q^{69} - 400 q^{71} + 630 q^{73} - 200 q^{75} + 1120 q^{79} - 359 q^{81} + 552 q^{83} + 300 q^{85} - 272 q^{87} + 326 q^{89} + 2560 q^{93} + 400 q^{95} + 110 q^{97} + 1480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 8.00000 0 10.0000 0 0 0 37.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.4.a.o 1
4.b odd 2 1 1568.4.a.c 1
7.b odd 2 1 32.4.a.a 1
21.c even 2 1 288.4.a.h 1
28.d even 2 1 32.4.a.c yes 1
35.c odd 2 1 800.4.a.k 1
35.f even 4 2 800.4.c.b 2
56.e even 2 1 64.4.a.a 1
56.h odd 2 1 64.4.a.e 1
84.h odd 2 1 288.4.a.i 1
112.j even 4 2 256.4.b.c 2
112.l odd 4 2 256.4.b.e 2
140.c even 2 1 800.4.a.a 1
140.j odd 4 2 800.4.c.a 2
168.e odd 2 1 576.4.a.h 1
168.i even 2 1 576.4.a.g 1
280.c odd 2 1 1600.4.a.e 1
280.n even 2 1 1600.4.a.bw 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.a.a 1 7.b odd 2 1
32.4.a.c yes 1 28.d even 2 1
64.4.a.a 1 56.e even 2 1
64.4.a.e 1 56.h odd 2 1
256.4.b.c 2 112.j even 4 2
256.4.b.e 2 112.l odd 4 2
288.4.a.h 1 21.c even 2 1
288.4.a.i 1 84.h odd 2 1
576.4.a.g 1 168.i even 2 1
576.4.a.h 1 168.e odd 2 1
800.4.a.a 1 140.c even 2 1
800.4.a.k 1 35.c odd 2 1
800.4.c.a 2 140.j odd 4 2
800.4.c.b 2 35.f even 4 2
1568.4.a.c 1 4.b odd 2 1
1568.4.a.o 1 1.a even 1 1 trivial
1600.4.a.e 1 280.c odd 2 1
1600.4.a.bw 1 280.n even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1568))\):

\( T_{3} - 8 \) Copy content Toggle raw display
\( T_{5} - 10 \) Copy content Toggle raw display
\( T_{11} - 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 8 \) Copy content Toggle raw display
$5$ \( T - 10 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 40 \) Copy content Toggle raw display
$13$ \( T - 50 \) Copy content Toggle raw display
$17$ \( T - 30 \) Copy content Toggle raw display
$19$ \( T - 40 \) Copy content Toggle raw display
$23$ \( T + 48 \) Copy content Toggle raw display
$29$ \( T + 34 \) Copy content Toggle raw display
$31$ \( T - 320 \) Copy content Toggle raw display
$37$ \( T - 310 \) Copy content Toggle raw display
$41$ \( T + 410 \) Copy content Toggle raw display
$43$ \( T + 152 \) Copy content Toggle raw display
$47$ \( T + 416 \) Copy content Toggle raw display
$53$ \( T + 410 \) Copy content Toggle raw display
$59$ \( T + 200 \) Copy content Toggle raw display
$61$ \( T + 30 \) Copy content Toggle raw display
$67$ \( T + 776 \) Copy content Toggle raw display
$71$ \( T + 400 \) Copy content Toggle raw display
$73$ \( T - 630 \) Copy content Toggle raw display
$79$ \( T - 1120 \) Copy content Toggle raw display
$83$ \( T - 552 \) Copy content Toggle raw display
$89$ \( T - 326 \) Copy content Toggle raw display
$97$ \( T - 110 \) Copy content Toggle raw display
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