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Properties

Label 1280.2.f.e
Level $1280$
Weight $2$
Character orbit 1280.f
Analytic conductor $10.221$
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] = [1280,2,Mod(129,1280)] 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("1280.129"); 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, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,4,0,-4,0,0,0,2,0,0,0,-8] 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: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{3} + ( - i - 2) q^{5} + 2 i q^{7} + q^{9} + 4 i q^{11} - 4 q^{13} + ( - 2 i - 4) q^{15} + 4 i q^{19} + 4 i q^{21} + 2 i q^{23} + (4 i + 3) q^{25} - 4 q^{27} + 2 i q^{29} + 8 i q^{33} + ( - 4 i + 2) q^{35}+ \cdots + 4 i q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} - 4 q^{5} + 2 q^{9} - 8 q^{13} - 8 q^{15} + 6 q^{25} - 8 q^{27} + 4 q^{35} - 8 q^{37} - 16 q^{39} - 4 q^{41} + 12 q^{43} - 4 q^{45} + 6 q^{49} - 8 q^{53} + 8 q^{55} + 16 q^{65} - 28 q^{67}+ \cdots + 8 q^{95}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
− 1.00000i
0 2.00000 0 −2.00000 − 1.00000i 0 2.00000i 0 1.00000 0
129.2 0 2.00000 0 −2.00000 + 1.00000i 0 − 2.00000i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.f.e 2
4.b odd 2 1 1280.2.f.a 2
5.b even 2 1 1280.2.f.b 2
8.b even 2 1 1280.2.f.b 2
8.d odd 2 1 1280.2.f.f 2
16.e even 4 1 80.2.c.a 2
16.e even 4 1 320.2.c.b 2
16.f odd 4 1 40.2.c.a ✓ 2
16.f odd 4 1 320.2.c.c 2
20.d odd 2 1 1280.2.f.f 2
40.e odd 2 1 1280.2.f.a 2
40.f even 2 1 inner 1280.2.f.e 2
48.i odd 4 1 720.2.f.e 2
48.i odd 4 1 2880.2.f.i 2
48.k even 4 1 360.2.f.c 2
48.k even 4 1 2880.2.f.h 2
80.i odd 4 1 400.2.a.b 1
80.i odd 4 1 1600.2.a.d 1
80.j even 4 1 200.2.a.b 1
80.j even 4 1 1600.2.a.f 1
80.k odd 4 1 40.2.c.a ✓ 2
80.k odd 4 1 320.2.c.c 2
80.q even 4 1 80.2.c.a 2
80.q even 4 1 320.2.c.b 2
80.s even 4 1 200.2.a.d 1
80.s even 4 1 1600.2.a.v 1
80.t odd 4 1 400.2.a.g 1
80.t odd 4 1 1600.2.a.u 1
112.j even 4 1 1960.2.g.b 2
240.t even 4 1 360.2.f.c 2
240.t even 4 1 2880.2.f.h 2
240.z odd 4 1 1800.2.a.s 1
240.bb even 4 1 3600.2.a.k 1
240.bd odd 4 1 1800.2.a.j 1
240.bf even 4 1 3600.2.a.bb 1
240.bm odd 4 1 720.2.f.e 2
240.bm odd 4 1 2880.2.f.i 2
560.u odd 4 1 9800.2.a.d 1
560.be even 4 1 1960.2.g.b 2
560.bm odd 4 1 9800.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a ✓ 2 16.f odd 4 1
40.2.c.a ✓ 2 80.k odd 4 1
80.2.c.a 2 16.e even 4 1
80.2.c.a 2 80.q even 4 1
200.2.a.b 1 80.j even 4 1
200.2.a.d 1 80.s even 4 1
320.2.c.b 2 16.e even 4 1
320.2.c.b 2 80.q even 4 1
320.2.c.c 2 16.f odd 4 1
320.2.c.c 2 80.k odd 4 1
360.2.f.c 2 48.k even 4 1
360.2.f.c 2 240.t even 4 1
400.2.a.b 1 80.i odd 4 1
400.2.a.g 1 80.t odd 4 1
720.2.f.e 2 48.i odd 4 1
720.2.f.e 2 240.bm odd 4 1
1280.2.f.a 2 4.b odd 2 1
1280.2.f.a 2 40.e odd 2 1
1280.2.f.b 2 5.b even 2 1
1280.2.f.b 2 8.b even 2 1
1280.2.f.e 2 1.a even 1 1 trivial
1280.2.f.e 2 40.f even 2 1 inner
1280.2.f.f 2 8.d odd 2 1
1280.2.f.f 2 20.d odd 2 1
1600.2.a.d 1 80.i odd 4 1
1600.2.a.f 1 80.j even 4 1
1600.2.a.u 1 80.t odd 4 1
1600.2.a.v 1 80.s even 4 1
1800.2.a.j 1 240.bd odd 4 1
1800.2.a.s 1 240.z odd 4 1
1960.2.g.b 2 112.j even 4 1
1960.2.g.b 2 560.be even 4 1
2880.2.f.h 2 48.k even 4 1
2880.2.f.h 2 240.t even 4 1
2880.2.f.i 2 48.i odd 4 1
2880.2.f.i 2 240.bm odd 4 1
3600.2.a.k 1 240.bb even 4 1
3600.2.a.bb 1 240.bf even 4 1
9800.2.a.d 1 560.u odd 4 1
9800.2.a.bf 1 560.bm odd 4 1

Hecke kernels

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

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 4 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T - 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( (T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 100 \) Copy content Toggle raw display
$67$ \( (T + 14)^{2} \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T - 16)^{2} \) Copy content Toggle raw display
$83$ \( (T - 2)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 256 \) Copy content Toggle raw display
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