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Properties

Label 1280.2.c.c
Level $1280$
Weight $2$
Character orbit 1280.c
Analytic conductor $10.221$
Analytic rank $0$
Dimension $2$
CM discriminant -40
Inner twists $4$

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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] = [1280,2,Mod(769,1280)] 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("1280.769"); S:= CuspForms(chi, 2); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,6,0,4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} + 2 \beta q^{7} + 3 q^{9} + 2 q^{11} - 2 \beta q^{13} + 6 q^{19} + 2 \beta q^{23} - 5 q^{25} - 10 q^{35} + 2 \beta q^{37} - 2 q^{41} + 3 \beta q^{45} + 6 \beta q^{47} - 13 q^{49} - 6 \beta q^{53} + \cdots + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{9} + 4 q^{11} + 12 q^{19} - 10 q^{25} - 20 q^{35} - 4 q^{41} - 26 q^{49} - 28 q^{59} + 20 q^{65} + 18 q^{81} + 28 q^{89} + 40 q^{91} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
− 2.23607i
2.23607i
0 0 0 − 2.23607i 0 − 4.47214i 0 3.00000 0
769.2 0 0 0 2.23607i 0 4.47214i 0 3.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
5.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.c.c 2
4.b odd 2 1 1280.2.c.b 2
5.b even 2 1 inner 1280.2.c.c 2
5.c odd 4 2 6400.2.a.bj 2
8.b even 2 1 1280.2.c.b 2
8.d odd 2 1 inner 1280.2.c.c 2
16.e even 4 2 320.2.f.a ✓ 4
16.f odd 4 2 320.2.f.a ✓ 4
20.d odd 2 1 1280.2.c.b 2
20.e even 4 2 6400.2.a.bi 2
40.e odd 2 1 CM 1280.2.c.c 2
40.f even 2 1 1280.2.c.b 2
40.i odd 4 2 6400.2.a.bi 2
40.k even 4 2 6400.2.a.bj 2
48.i odd 4 2 2880.2.d.e 4
48.k even 4 2 2880.2.d.e 4
80.i odd 4 2 1600.2.d.g 4
80.j even 4 2 1600.2.d.g 4
80.k odd 4 2 320.2.f.a ✓ 4
80.q even 4 2 320.2.f.a ✓ 4
80.s even 4 2 1600.2.d.g 4
80.t odd 4 2 1600.2.d.g 4
240.t even 4 2 2880.2.d.e 4
240.bm odd 4 2 2880.2.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.f.a ✓ 4 16.e even 4 2
320.2.f.a ✓ 4 16.f odd 4 2
320.2.f.a ✓ 4 80.k odd 4 2
320.2.f.a ✓ 4 80.q even 4 2
1280.2.c.b 2 4.b odd 2 1
1280.2.c.b 2 8.b even 2 1
1280.2.c.b 2 20.d odd 2 1
1280.2.c.b 2 40.f even 2 1
1280.2.c.c 2 1.a even 1 1 trivial
1280.2.c.c 2 5.b even 2 1 inner
1280.2.c.c 2 8.d odd 2 1 inner
1280.2.c.c 2 40.e odd 2 1 CM
1600.2.d.g 4 80.i odd 4 2
1600.2.d.g 4 80.j even 4 2
1600.2.d.g 4 80.s even 4 2
1600.2.d.g 4 80.t odd 4 2
2880.2.d.e 4 48.i odd 4 2
2880.2.d.e 4 48.k even 4 2
2880.2.d.e 4 240.t even 4 2
2880.2.d.e 4 240.bm odd 4 2
6400.2.a.bi 2 20.e even 4 2
6400.2.a.bi 2 40.i odd 4 2
6400.2.a.bj 2 5.c odd 4 2
6400.2.a.bj 2 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} + 20 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 20 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 20 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 20 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 20 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 180 \) Copy content Toggle raw display
$53$ \( T^{2} + 180 \) Copy content Toggle raw display
$59$ \( (T + 14)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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