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Properties

Label 1680.4.a.bk
Level $1680$
Weight $4$
Character orbit 1680.a
Self dual yes
Analytic conductor $99.123$
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] = [1680,4,Mod(1,1680)] 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("1680.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1680.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,6,0,-10,0,14,0,18,0,26,0,14,0,-30,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(99.1232088096\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
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 \(\beta = \sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} - 5 q^{5} + 7 q^{7} + 9 q^{9} + (5 \beta + 13) q^{11} + ( - 11 \beta + 7) q^{13} - 15 q^{15} + (14 \beta + 8) q^{17} + (13 \beta - 87) q^{19} + 21 q^{21} + ( - 28 \beta - 92) q^{23} + 25 q^{25}+ \cdots + (45 \beta + 117) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9} + 26 q^{11} + 14 q^{13} - 30 q^{15} + 16 q^{17} - 174 q^{19} + 42 q^{21} - 184 q^{23} + 50 q^{25} + 54 q^{27} - 32 q^{29} - 330 q^{31} + 78 q^{33} - 70 q^{35}+ \cdots + 234 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
0 3.00000 0 −5.00000 0 7.00000 0 9.00000 0
1.2 0 3.00000 0 −5.00000 0 7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( -1 \)

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 1680.4.a.bk 2
4.b odd 2 1 105.4.a.c ✓ 2
12.b even 2 1 315.4.a.m 2
20.d odd 2 1 525.4.a.p 2
20.e even 4 2 525.4.d.i 4
28.d even 2 1 735.4.a.k 2
60.h even 2 1 1575.4.a.m 2
84.h odd 2 1 2205.4.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.c ✓ 2 4.b odd 2 1
315.4.a.m 2 12.b even 2 1
525.4.a.p 2 20.d odd 2 1
525.4.d.i 4 20.e even 4 2
735.4.a.k 2 28.d even 2 1
1575.4.a.m 2 60.h even 2 1
1680.4.a.bk 2 1.a even 1 1 trivial
2205.4.a.bh 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1680))\):

\( T_{11}^{2} - 26T_{11} - 256 \) Copy content Toggle raw display
\( T_{13}^{2} - 14T_{13} - 2008 \) Copy content Toggle raw display
\( T_{17}^{2} - 16T_{17} - 3268 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 26T - 256 \) Copy content Toggle raw display
$13$ \( T^{2} - 14T - 2008 \) Copy content Toggle raw display
$17$ \( T^{2} - 16T - 3268 \) Copy content Toggle raw display
$19$ \( T^{2} + 174T + 4696 \) Copy content Toggle raw display
$23$ \( T^{2} + 184T - 4864 \) Copy content Toggle raw display
$29$ \( T^{2} + 32T - 29732 \) Copy content Toggle raw display
$31$ \( T^{2} + 330T + 25848 \) Copy content Toggle raw display
$37$ \( T^{2} + 132T + 1908 \) Copy content Toggle raw display
$41$ \( T^{2} - 200T - 73300 \) Copy content Toggle raw display
$43$ \( T^{2} + 364T + 13472 \) Copy content Toggle raw display
$47$ \( T^{2} + 292T - 28256 \) Copy content Toggle raw display
$53$ \( T^{2} - 34T - 194344 \) Copy content Toggle raw display
$59$ \( T^{2} + 364T + 27616 \) Copy content Toggle raw display
$61$ \( T^{2} - 792T - 113076 \) Copy content Toggle raw display
$67$ \( T^{2} - 788T + 151904 \) Copy content Toggle raw display
$71$ \( T^{2} + 454T - 411296 \) Copy content Toggle raw display
$73$ \( T^{2} - 778T + 16664 \) Copy content Toggle raw display
$79$ \( T^{2} + 408T + 8704 \) Copy content Toggle raw display
$83$ \( T^{2} + 1136T + 26416 \) Copy content Toggle raw display
$89$ \( T^{2} - 36T - 2252108 \) Copy content Toggle raw display
$97$ \( T^{2} + 498T - 28592 \) Copy content Toggle raw display
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