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Properties

Label 1280.3.g.e
Level $1280$
Weight $3$
Character orbit 1280.g
Analytic conductor $34.877$
Analytic rank $0$
Dimension $8$
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] = [1280,3,Mod(1151,1280)] 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("1280.1151"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(34.8774738381\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 20)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + \beta_1 q^{5} + ( - \beta_{5} - \beta_{3}) q^{7} + (\beta_{6} + 1) q^{9} + (\beta_{7} - 2 \beta_{4}) q^{11} + ( - 2 \beta_{2} - 2 \beta_1) q^{13} + ( - \beta_{5} - \beta_{3}) q^{15}+ \cdots + ( - 3 \beta_{7} + 10 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} - 48 q^{17} - 40 q^{25} + 160 q^{33} + 224 q^{41} - 8 q^{49} + 80 q^{65} - 528 q^{73} - 552 q^{81} + 176 q^{89} - 528 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{20}^{7} - \zeta_{20}^{5} + 2\zeta_{20}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\zeta_{20}^{5} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\zeta_{20}^{6} + 4\zeta_{20}^{4} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{20}^{7} + 2\zeta_{20}^{5} - 2\zeta_{20}^{3} + 4\zeta_{20} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{20}^{6} - 6\zeta_{20}^{4} + 8\zeta_{20}^{2} - 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -4\zeta_{20}^{6} + 4\zeta_{20}^{4} + 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -8\zeta_{20}^{7} + 8\zeta_{20}^{3} \) Copy content Toggle raw display

Display \(\zeta_{20}^j\) in terms of \(\beta_i\)

\(\zeta_{20}\)\(=\) \( ( 2\beta_{4} + \beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( ( 2\beta_{6} + 2\beta_{5} + \beta_{3} + 4 ) / 16 \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( \beta_{7} - 2\beta_{2} + 4\beta_1 ) / 16 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( ( \beta_{6} + \beta_{3} - 2 ) / 8 \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( ( -\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( ( -\beta_{6} + \beta_{3} + 2 ) / 8 \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( -\beta_{7} - 2\beta_{2} + 4\beta_1 ) / 16 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{20}^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
0.951057 − 0.309017i
0.951057 + 0.309017i
0.587785 − 0.809017i
0.587785 + 0.809017i
−0.587785 − 0.809017i
−0.587785 + 0.809017i
−0.951057 − 0.309017i
−0.951057 + 0.309017i
0 −3.80423 0 − 2.23607i 0 8.50651i 0 5.47214 0
1151.2 0 −3.80423 0 2.23607i 0 − 8.50651i 0 5.47214 0
1151.3 0 −2.35114 0 − 2.23607i 0 5.25731i 0 −3.47214 0
1151.4 0 −2.35114 0 2.23607i 0 − 5.25731i 0 −3.47214 0
1151.5 0 2.35114 0 − 2.23607i 0 − 5.25731i 0 −3.47214 0
1151.6 0 2.35114 0 2.23607i 0 5.25731i 0 −3.47214 0
1151.7 0 3.80423 0 − 2.23607i 0 − 8.50651i 0 5.47214 0
1151.8 0 3.80423 0 2.23607i 0 8.50651i 0 5.47214 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.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 1280.3.g.e 8
4.b odd 2 1 inner 1280.3.g.e 8
8.b even 2 1 inner 1280.3.g.e 8
8.d odd 2 1 inner 1280.3.g.e 8
16.e even 4 1 20.3.b.a ✓ 4
16.e even 4 1 320.3.b.c 4
16.f odd 4 1 20.3.b.a ✓ 4
16.f odd 4 1 320.3.b.c 4
48.i odd 4 1 180.3.c.a 4
48.i odd 4 1 2880.3.e.e 4
48.k even 4 1 180.3.c.a 4
48.k even 4 1 2880.3.e.e 4
80.i odd 4 1 100.3.d.b 8
80.i odd 4 1 1600.3.h.n 8
80.j even 4 1 100.3.d.b 8
80.j even 4 1 1600.3.h.n 8
80.k odd 4 1 100.3.b.f 4
80.k odd 4 1 1600.3.b.s 4
80.q even 4 1 100.3.b.f 4
80.q even 4 1 1600.3.b.s 4
80.s even 4 1 100.3.d.b 8
80.s even 4 1 1600.3.h.n 8
80.t odd 4 1 100.3.d.b 8
80.t odd 4 1 1600.3.h.n 8
240.t even 4 1 900.3.c.k 4
240.z odd 4 1 900.3.f.e 8
240.bb even 4 1 900.3.f.e 8
240.bd odd 4 1 900.3.f.e 8
240.bf even 4 1 900.3.f.e 8
240.bm odd 4 1 900.3.c.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a ✓ 4 16.e even 4 1
20.3.b.a ✓ 4 16.f odd 4 1
100.3.b.f 4 80.k odd 4 1
100.3.b.f 4 80.q even 4 1
100.3.d.b 8 80.i odd 4 1
100.3.d.b 8 80.j even 4 1
100.3.d.b 8 80.s even 4 1
100.3.d.b 8 80.t odd 4 1
180.3.c.a 4 48.i odd 4 1
180.3.c.a 4 48.k even 4 1
320.3.b.c 4 16.e even 4 1
320.3.b.c 4 16.f odd 4 1
900.3.c.k 4 240.t even 4 1
900.3.c.k 4 240.bm odd 4 1
900.3.f.e 8 240.z odd 4 1
900.3.f.e 8 240.bb even 4 1
900.3.f.e 8 240.bd odd 4 1
900.3.f.e 8 240.bf even 4 1
1280.3.g.e 8 1.a even 1 1 trivial
1280.3.g.e 8 4.b odd 2 1 inner
1280.3.g.e 8 8.b even 2 1 inner
1280.3.g.e 8 8.d odd 2 1 inner
1600.3.b.s 4 80.k odd 4 1
1600.3.b.s 4 80.q even 4 1
1600.3.h.n 8 80.i odd 4 1
1600.3.h.n 8 80.j even 4 1
1600.3.h.n 8 80.s even 4 1
1600.3.h.n 8 80.t odd 4 1
2880.3.e.e 4 48.i odd 4 1
2880.3.e.e 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 20T_{3}^{2} + 80 \) acting on \(S_{3}^{\mathrm{new}}(1280, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 20 T^{2} + 80)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 100 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 400 T^{2} + 1280)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 72 T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12 T - 284)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 320 T^{2} + 20480)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 260 T^{2} + 80)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 168 T^{2} + 5776)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2320 T^{2} + 154880)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1032 T^{2} + 234256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 56 T + 604)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 500 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 4100 T^{2} + 3561680)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 4872 T^{2} + 2062096)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 5760 T^{2} + 1658880)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8808 T^{2} + 5550736)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 10420 T^{2} + 19920080)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8080 T^{2} + 10138880)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 132 T - 764)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 13120 T^{2} + 2478080)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 6260 T^{2} + 2620880)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 44 T - 7516)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 132 T + 3636)^{4} \) Copy content Toggle raw display
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