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Properties

Label 1568.4.a.t
Level $1568$
Weight $4$
Character orbit 1568.a
Self dual yes
Analytic conductor $92.515$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1568,4,Mod(1,1568)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1568.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,2,0,0,0,-40,0,0,0,-120,0,0,0,50,0,0,0,0,0,0,0,-248] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(92.5149948890\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} - 20 q^{9} - \beta q^{11} - 60 q^{13} + \beta q^{15} + 25 q^{17} + 37 \beta q^{19} + 59 \beta q^{23} - 124 q^{25} - 47 \beta q^{27} + 52 q^{29} - 85 \beta q^{31} - 7 q^{33} + 95 q^{37} + \cdots + 20 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 40 q^{9} - 120 q^{13} + 50 q^{17} - 248 q^{25} + 104 q^{29} - 14 q^{33} + 190 q^{37} + 128 q^{41} - 40 q^{45} + 406 q^{53} + 518 q^{57} + 302 q^{61} - 120 q^{65} + 826 q^{69} + 670 q^{73}+ \cdots - 128 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−2.64575
2.64575
0 −2.64575 0 1.00000 0 0 0 −20.0000 0
1.2 0 2.64575 0 1.00000 0 0 0 −20.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.4.a.t 2
4.b odd 2 1 inner 1568.4.a.t 2
7.b odd 2 1 1568.4.a.q 2
7.c even 3 2 224.4.i.a ✓ 4
28.d even 2 1 1568.4.a.q 2
28.g odd 6 2 224.4.i.a ✓ 4
56.k odd 6 2 448.4.i.i 4
56.p even 6 2 448.4.i.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.4.i.a ✓ 4 7.c even 3 2
224.4.i.a ✓ 4 28.g odd 6 2
448.4.i.i 4 56.k odd 6 2
448.4.i.i 4 56.p even 6 2
1568.4.a.q 2 7.b odd 2 1
1568.4.a.q 2 28.d even 2 1
1568.4.a.t 2 1.a even 1 1 trivial
1568.4.a.t 2 4.b odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} - 7 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 7 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 7 \) Copy content Toggle raw display
$13$ \( (T + 60)^{2} \) Copy content Toggle raw display
$17$ \( (T - 25)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 9583 \) Copy content Toggle raw display
$23$ \( T^{2} - 24367 \) Copy content Toggle raw display
$29$ \( (T - 52)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 50575 \) Copy content Toggle raw display
$37$ \( (T - 95)^{2} \) Copy content Toggle raw display
$41$ \( (T - 64)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 40432 \) Copy content Toggle raw display
$47$ \( T^{2} - 190575 \) Copy content Toggle raw display
$53$ \( (T - 203)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 22743 \) Copy content Toggle raw display
$61$ \( (T - 151)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 794983 \) Copy content Toggle raw display
$71$ \( T^{2} - 790272 \) Copy content Toggle raw display
$73$ \( (T - 335)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 63 \) Copy content Toggle raw display
$83$ \( T^{2} - 9072 \) Copy content Toggle raw display
$89$ \( (T - 1263)^{2} \) Copy content Toggle raw display
$97$ \( (T + 64)^{2} \) Copy content Toggle raw display
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