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Properties

Label 1680.4.a.bg
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,-6,0,10,0,14,0,18,0,22,0,-22,0,-30,0,116] 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: \(\Q(\sqrt{65}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \sqrt{65}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + 5 q^{5} + 7 q^{7} + 9 q^{9} + ( - \beta + 11) q^{11} + ( - \beta - 11) q^{13} - 15 q^{15} + (8 \beta + 58) q^{17} + ( - 7 \beta - 51) q^{19} - 21 q^{21} + ( - 10 \beta - 130) q^{23} + 25 q^{25}+ \cdots + ( - 9 \beta + 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} - 22 q^{13} - 30 q^{15} + 116 q^{17} - 102 q^{19} - 42 q^{21} - 260 q^{23} + 50 q^{25} - 54 q^{27} - 196 q^{29} - 150 q^{31} - 66 q^{33} + 70 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) \) 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
4.53113
−3.53113
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.bg 2
4.b odd 2 1 105.4.a.f ✓ 2
12.b even 2 1 315.4.a.i 2
20.d odd 2 1 525.4.a.k 2
20.e even 4 2 525.4.d.h 4
28.d even 2 1 735.4.a.p 2
60.h even 2 1 1575.4.a.w 2
84.h odd 2 1 2205.4.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.f ✓ 2 4.b odd 2 1
315.4.a.i 2 12.b even 2 1
525.4.a.k 2 20.d odd 2 1
525.4.d.h 4 20.e even 4 2
735.4.a.p 2 28.d even 2 1
1575.4.a.w 2 60.h even 2 1
1680.4.a.bg 2 1.a even 1 1 trivial
2205.4.a.z 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1680))\):

\( T_{11}^{2} - 22T_{11} + 56 \) Copy content Toggle raw display
\( T_{13}^{2} + 22T_{13} + 56 \) Copy content Toggle raw display
\( T_{17}^{2} - 116T_{17} - 796 \) 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
$11$ \( T^{2} - 22T + 56 \) Copy content Toggle raw display
$13$ \( T^{2} + 22T + 56 \) Copy content Toggle raw display
$17$ \( T^{2} - 116T - 796 \) Copy content Toggle raw display
$19$ \( T^{2} + 102T - 584 \) Copy content Toggle raw display
$23$ \( T^{2} + 260T + 10400 \) Copy content Toggle raw display
$29$ \( T^{2} + 196T - 27836 \) Copy content Toggle raw display
$31$ \( T^{2} + 150T - 23040 \) Copy content Toggle raw display
$37$ \( T^{2} + 96T - 18756 \) Copy content Toggle raw display
$41$ \( T^{2} + 176T - 154756 \) Copy content Toggle raw display
$43$ \( T^{2} - 344T + 12944 \) Copy content Toggle raw display
$47$ \( T^{2} + 560T + 40960 \) Copy content Toggle raw display
$53$ \( T^{2} - 326T - 93616 \) Copy content Toggle raw display
$59$ \( T^{2} - 844T + 63424 \) Copy content Toggle raw display
$61$ \( T^{2} + 204T + 1044 \) Copy content Toggle raw display
$67$ \( T^{2} - 104T - 63856 \) Copy content Toggle raw display
$71$ \( T^{2} + 1670 T + 668560 \) Copy content Toggle raw display
$73$ \( T^{2} + 386T - 625816 \) Copy content Toggle raw display
$79$ \( T^{2} - 888T + 21376 \) Copy content Toggle raw display
$83$ \( T^{2} + 928T - 542864 \) Copy content Toggle raw display
$89$ \( T^{2} - 588T + 85396 \) Copy content Toggle raw display
$97$ \( T^{2} - 522 T - 1534064 \) Copy content Toggle raw display
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