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Properties

Label 1568.4.a.u
Level $1568$
Weight $4$
Character orbit 1568.a
Self dual yes
Analytic conductor $92.515$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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 = [2,0,6,0,6,0,0,0,38,0,-44,0,-14,0,-56,0,-96,0,170,0,0,0,-152, 0,-158] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 3) q^{3} + ( - \beta + 3) q^{5} + (6 \beta + 19) q^{9} + (2 \beta - 22) q^{11} + (\beta - 7) q^{13} - 28 q^{15} + ( - 10 \beta - 48) q^{17} + ( - 9 \beta + 85) q^{19} + ( - 16 \beta - 76) q^{23}+ \cdots + ( - 94 \beta + 26) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 6 q^{5} + 38 q^{9} - 44 q^{11} - 14 q^{13} - 56 q^{15} - 96 q^{17} + 170 q^{19} - 152 q^{23} - 158 q^{25} + 396 q^{27} - 128 q^{29} + 68 q^{31} + 16 q^{33} - 256 q^{37} + 32 q^{39} - 88 q^{41}+ \cdots + 52 q^{99}+O(q^{100}) \) 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.

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
−2.54138
3.54138
0 −3.08276 0 9.08276 0 0 0 −17.4966 0
1.2 0 9.08276 0 −3.08276 0 0 0 55.4966 0
\(n\): e.g. 2-40 or 80-90
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.u 2
4.b odd 2 1 1568.4.a.p 2
7.b odd 2 1 224.4.a.c ✓ 2
21.c even 2 1 2016.4.a.p 2
28.d even 2 1 224.4.a.d yes 2
56.e even 2 1 448.4.a.q 2
56.h odd 2 1 448.4.a.t 2
84.h odd 2 1 2016.4.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.4.a.c ✓ 2 7.b odd 2 1
224.4.a.d yes 2 28.d even 2 1
448.4.a.q 2 56.e even 2 1
448.4.a.t 2 56.h odd 2 1
1568.4.a.p 2 4.b odd 2 1
1568.4.a.u 2 1.a even 1 1 trivial
2016.4.a.o 2 84.h odd 2 1
2016.4.a.p 2 21.c even 2 1

Hecke kernels

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

\( T_{3}^{2} - 6T_{3} - 28 \) Copy content Toggle raw display
\( T_{5}^{2} - 6T_{5} - 28 \) Copy content Toggle raw display
\( T_{11}^{2} + 44T_{11} + 336 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 6T - 28 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T - 28 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 44T + 336 \) Copy content Toggle raw display
$13$ \( T^{2} + 14T + 12 \) Copy content Toggle raw display
$17$ \( T^{2} + 96T - 1396 \) Copy content Toggle raw display
$19$ \( T^{2} - 170T + 4228 \) Copy content Toggle raw display
$23$ \( T^{2} + 152T - 3696 \) Copy content Toggle raw display
$29$ \( T^{2} + 128T - 38676 \) Copy content Toggle raw display
$31$ \( T^{2} - 68T - 23856 \) Copy content Toggle raw display
$37$ \( T^{2} + 256T + 16236 \) Copy content Toggle raw display
$41$ \( T^{2} + 88T - 122532 \) Copy content Toggle raw display
$43$ \( T^{2} + 724T + 119056 \) Copy content Toggle raw display
$47$ \( T^{2} + 244T - 3024 \) Copy content Toggle raw display
$53$ \( T^{2} - 188T - 142716 \) Copy content Toggle raw display
$59$ \( T^{2} - 138T - 35532 \) Copy content Toggle raw display
$61$ \( T^{2} - 358T + 29044 \) Copy content Toggle raw display
$67$ \( T^{2} + 200T - 757232 \) Copy content Toggle raw display
$71$ \( T^{2} - 1400 T + 460992 \) Copy content Toggle raw display
$73$ \( T^{2} - 628T - 399276 \) Copy content Toggle raw display
$79$ \( T^{2} + 200T - 985152 \) Copy content Toggle raw display
$83$ \( T^{2} - 618T - 839916 \) Copy content Toggle raw display
$89$ \( T^{2} - 908T - 30684 \) Copy content Toggle raw display
$97$ \( T^{2} + 1520 T + 524172 \) Copy content Toggle raw display
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