Properties

Label 560.6.a.s
Level $560$
Weight $6$
Character orbit 560.a
Self dual yes
Analytic conductor $89.815$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,6,Mod(1,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 560.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: \(89.8149390953\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 463x - 1890 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 280)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{3} - 25 q^{5} - 49 q^{7} + (\beta_{2} + 2 \beta_1 + 70) q^{9} + (\beta_{2} + 4 \beta_1 + 193) q^{11} + ( - 2 \beta_{2} + 23 \beta_1 + 26) q^{13} + ( - 25 \beta_1 + 50) q^{15} + ( - 4 \beta_{2} - 13 \beta_1 - 416) q^{17}+ \cdots + (116 \beta_{2} + 750 \beta_1 + 52228) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{3} - 75 q^{5} - 147 q^{7} + 209 q^{9} + 578 q^{11} + 80 q^{13} + 150 q^{15} - 1244 q^{17} - 944 q^{19} + 294 q^{21} - 1096 q^{23} + 1875 q^{25} + 3006 q^{27} + 1868 q^{29} + 6620 q^{31} + 2662 q^{33}+ \cdots + 156568 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 463x - 1890 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 6\nu - 309 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 6\beta _1 + 309 \) 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.

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
−19.0769
−4.24759
23.3245
0 −21.0769 0 −25.0000 0 −49.0000 0 201.235 0
1.2 0 −6.24759 0 −25.0000 0 −49.0000 0 −203.968 0
1.3 0 21.3245 0 −25.0000 0 −49.0000 0 211.733 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)

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 560.6.a.s 3
4.b odd 2 1 280.6.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.6.a.e 3 4.b odd 2 1
560.6.a.s 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 6T_{3}^{2} - 451T_{3} - 2808 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(560))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 6 T^{2} + \cdots - 2808 \) Copy content Toggle raw display
$5$ \( (T + 25)^{3} \) Copy content Toggle raw display
$7$ \( (T + 49)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 578 T^{2} + \cdots + 9760932 \) Copy content Toggle raw display
$13$ \( T^{3} - 80 T^{2} + \cdots + 128496494 \) Copy content Toggle raw display
$17$ \( T^{3} + 1244 T^{2} + \cdots - 651843290 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots + 1721529888 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 31363349376 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 3100258778 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 16721033600 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 15436371144 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 330620399712 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 1675236784 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 1323866448632 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 13966928161632 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 363808567296 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 4786876164960 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 705690174528 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 336586973184 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 3081736975464 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 116381435931432 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 41224716201984 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 175059765854496 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 426037755101066 \) Copy content Toggle raw display
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