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Properties

Label 1280.4.a.ba
Level $1280$
Weight $4$
Character orbit 1280.a
Self dual yes
Analytic conductor $75.522$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,4,Mod(1,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(75.5224448073\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 21x^{4} + 38x^{3} + 93x^{2} - 108x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} - 5 q^{5} + (\beta_{4} + 2) q^{7} + ( - \beta_{5} + 2 \beta_1 + 9) q^{9} + (\beta_{5} - \beta_{3} - 7) q^{11} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{13} + (5 \beta_1 + 5) q^{15}+ \cdots + (21 \beta_{5} - 30 \beta_{4} + \cdots - 917) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} - 30 q^{5} + 14 q^{7} + 54 q^{9} - 44 q^{11} + 30 q^{15} - 152 q^{19} - 4 q^{21} + 302 q^{23} + 150 q^{25} - 216 q^{27} + 132 q^{31} - 116 q^{33} - 70 q^{35} + 68 q^{37} + 300 q^{39} - 20 q^{41}+ \cdots - 5516 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 21x^{4} + 38x^{3} + 93x^{2} - 108x - 60 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 8\nu^{4} - 21\nu^{3} - 140\nu^{2} + 101\nu + 342 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{5} + 8\nu^{4} + 31\nu^{3} - 124\nu^{2} + 273\nu + 78 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 13\nu^{3} + 4\nu^{2} - 21\nu - 42 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -13\nu^{5} + 24\nu^{4} + 273\nu^{3} - 356\nu^{2} - 1057\nu + 258 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} - 55\nu^{3} + 20\nu^{2} + 159\nu - 94 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} - 2\beta _1 + 6 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} - 8\beta _1 + 122 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} - 4\beta_{3} + 2\beta_{2} - 7\beta _1 + 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 34\beta_{5} + 38\beta_{4} + 19\beta_{3} + 15\beta_{2} - 80\beta _1 + 1474 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8\beta_{5} + 60\beta_{4} - 247\beta_{3} + 87\beta_{2} - 354\beta _1 + 470 ) / 16 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
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
−0.430422
1.46129
−3.90720
3.78252
−2.05190
3.14570
0 −9.57890 0 −5.00000 0 21.5703 0 64.7554 0
1.2 0 −6.25785 0 −5.00000 0 −34.6280 0 12.1606 0
1.3 0 −1.51777 0 −5.00000 0 5.13620 0 −24.6964 0
1.4 0 −0.888401 0 −5.00000 0 26.6173 0 −26.2107 0
1.5 0 4.24443 0 −5.00000 0 −14.6308 0 −8.98481 0
1.6 0 7.99849 0 −5.00000 0 9.93501 0 36.9759 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

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 1280.4.a.ba 6
4.b odd 2 1 1280.4.a.bc 6
8.b even 2 1 1280.4.a.bd 6
8.d odd 2 1 1280.4.a.bb 6
16.e even 4 2 160.4.d.a 12
16.f odd 4 2 40.4.d.a ✓ 12
48.i odd 4 2 1440.4.k.c 12
48.k even 4 2 360.4.k.c 12
80.i odd 4 2 800.4.f.c 12
80.j even 4 2 200.4.f.c 12
80.k odd 4 2 200.4.d.b 12
80.q even 4 2 800.4.d.d 12
80.s even 4 2 200.4.f.b 12
80.t odd 4 2 800.4.f.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.d.a ✓ 12 16.f odd 4 2
160.4.d.a 12 16.e even 4 2
200.4.d.b 12 80.k odd 4 2
200.4.f.b 12 80.s even 4 2
200.4.f.c 12 80.j even 4 2
360.4.k.c 12 48.k even 4 2
800.4.d.d 12 80.q even 4 2
800.4.f.b 12 80.t odd 4 2
800.4.f.c 12 80.i odd 4 2
1280.4.a.ba 6 1.a even 1 1 trivial
1280.4.a.bb 6 8.d odd 2 1
1280.4.a.bc 6 4.b odd 2 1
1280.4.a.bd 6 8.b even 2 1
1440.4.k.c 12 48.i odd 4 2

Hecke kernels

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

\( T_{3}^{6} + 6T_{3}^{5} - 90T_{3}^{4} - 432T_{3}^{3} + 1428T_{3}^{2} + 4632T_{3} + 2744 \) Copy content Toggle raw display
\( T_{7}^{6} - 14T_{7}^{5} - 1258T_{7}^{4} + 23408T_{7}^{3} + 166612T_{7}^{2} - 4186552T_{7} + 14843128 \) Copy content Toggle raw display
\( T_{13}^{6} - 5572T_{13}^{4} - 163840T_{13}^{3} + 649392T_{13}^{2} + 34406400T_{13} - 157790400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 6 T^{5} + \cdots + 2744 \) Copy content Toggle raw display
$5$ \( (T + 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 14 T^{5} + \cdots + 14843128 \) Copy content Toggle raw display
$11$ \( T^{6} + 44 T^{5} + \cdots - 513918400 \) Copy content Toggle raw display
$13$ \( T^{6} - 5572 T^{4} + \cdots - 157790400 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 7473839808 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 42102357568 \) Copy content Toggle raw display
$23$ \( T^{6} - 302 T^{5} + \cdots + 881168216 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 33576038400 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 1437816300032 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 45951464886848 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 71667547865600 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 508806074248 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 72048375466472 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 929403110278976 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 248373422305600 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 93747278347656 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 36\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 378730163491776 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 28\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 62\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 33\!\cdots\!28 \) Copy content Toggle raw display
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