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Properties

Label 1280.4.a.h
Level $1280$
Weight $4$
Character orbit 1280.a
Self dual yes
Analytic conductor $75.522$
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] = [1280,4,Mod(1,1280)] 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("1280.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-10,0,0,0,142,0,0,0,172] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(75.5224448073\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 640)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 7 \beta q^{3} - 5 q^{5} + 13 \beta q^{7} + 71 q^{9} - 38 \beta q^{11} + 86 q^{13} - 35 \beta q^{15} - 46 q^{17} + 4 \beta q^{19} + 182 q^{21} - 83 \beta q^{23} + 25 q^{25} + 308 \beta q^{27} + 152 q^{29} + \cdots - 2698 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{5} + 142 q^{9} + 172 q^{13} - 92 q^{17} + 364 q^{21} + 50 q^{25} + 304 q^{29} - 1064 q^{33} - 100 q^{37} + 480 q^{41} - 710 q^{45} - 10 q^{49} + 652 q^{53} + 112 q^{57} + 380 q^{61} - 860 q^{65}+ \cdots + 3196 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 −9.89949 0 −5.00000 0 −18.3848 0 71.0000 0
1.2 0 9.89949 0 −5.00000 0 18.3848 0 71.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)

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 1280.4.a.h 2
4.b odd 2 1 inner 1280.4.a.h 2
8.b even 2 1 1280.4.a.n 2
8.d odd 2 1 1280.4.a.n 2
16.e even 4 2 640.4.d.a ✓ 4
16.f odd 4 2 640.4.d.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.d.a ✓ 4 16.e even 4 2
640.4.d.a ✓ 4 16.f odd 4 2
1280.4.a.h 2 1.a even 1 1 trivial
1280.4.a.h 2 4.b odd 2 1 inner
1280.4.a.n 2 8.b even 2 1
1280.4.a.n 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1280))\):

\( T_{3}^{2} - 98 \) Copy content Toggle raw display
\( T_{7}^{2} - 338 \) Copy content Toggle raw display
\( T_{13} - 86 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 98 \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 338 \) Copy content Toggle raw display
$11$ \( T^{2} - 2888 \) Copy content Toggle raw display
$13$ \( (T - 86)^{2} \) Copy content Toggle raw display
$17$ \( (T + 46)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 32 \) Copy content Toggle raw display
$23$ \( T^{2} - 13778 \) Copy content Toggle raw display
$29$ \( (T - 152)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 60552 \) Copy content Toggle raw display
$37$ \( (T + 50)^{2} \) Copy content Toggle raw display
$41$ \( (T - 240)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 14450 \) Copy content Toggle raw display
$47$ \( T^{2} - 144722 \) Copy content Toggle raw display
$53$ \( (T - 326)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 111392 \) Copy content Toggle raw display
$61$ \( (T - 190)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 21218 \) Copy content Toggle raw display
$71$ \( T^{2} - 273800 \) Copy content Toggle raw display
$73$ \( (T + 678)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 123008 \) Copy content Toggle raw display
$83$ \( T^{2} - 589698 \) Copy content Toggle raw display
$89$ \( (T + 730)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1598)^{2} \) Copy content Toggle raw display
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