Properties

Label 560.6.a.v
Level $560$
Weight $6$
Character orbit 560.a
Self dual yes
Analytic conductor $89.815$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 82x^{2} + 58x + 1168 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 35)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 3) q^{3} + 25 q^{5} + 49 q^{7} + ( - 3 \beta_{3} + 2 \beta_{2} + \cdots + 196) q^{9} + ( - 8 \beta_{3} + \beta_{2} + \cdots - 196) q^{11} + (5 \beta_{3} + 5 \beta_{2} - 34 \beta_1) q^{13}+ \cdots + ( - 1533 \beta_{3} - 755 \beta_{2} + \cdots + 102587) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} + 100 q^{5} + 196 q^{7} + 774 q^{9} - 770 q^{11} + 58 q^{13} - 350 q^{15} + 2006 q^{17} - 564 q^{19} - 686 q^{21} + 6340 q^{23} + 2500 q^{25} + 7438 q^{27} + 8066 q^{29} + 5856 q^{31} - 8130 q^{33}+ \cdots + 420668 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 82x^{2} + 58x + 1168 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} - \nu^{2} + 58\nu + 42 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 15\nu^{2} + 22\nu + 592 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{2} + 8\nu - 332 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 + 5 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{3} - \beta_{2} - \beta _1 + 659 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 53\beta_{3} + 59\beta_{2} - 37\beta _1 + 303 ) / 16 \) 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
−3.86448
8.05190
−7.92431
4.73688
0 −26.2268 0 25.0000 0 49.0000 0 444.847 0
1.2 0 −15.9755 0 25.0000 0 49.0000 0 12.2177 0
1.3 0 −0.133419 0 25.0000 0 49.0000 0 −242.982 0
1.4 0 28.3358 0 25.0000 0 49.0000 0 559.917 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.v 4
4.b odd 2 1 35.6.a.d 4
12.b even 2 1 315.6.a.l 4
20.d odd 2 1 175.6.a.f 4
20.e even 4 2 175.6.b.f 8
28.d even 2 1 245.6.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.d 4 4.b odd 2 1
175.6.a.f 4 20.d odd 2 1
175.6.b.f 8 20.e even 4 2
245.6.a.e 4 28.d even 2 1
315.6.a.l 4 12.b even 2 1
560.6.a.v 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 14T_{3}^{3} - 775T_{3}^{2} - 11976T_{3} - 1584 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(560))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 14 T^{3} + \cdots - 1584 \) Copy content Toggle raw display
$5$ \( (T - 25)^{4} \) Copy content Toggle raw display
$7$ \( (T - 49)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 20510223536 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 430417376452 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 617989800868 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 5221683964480 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 19307649395456 \) Copy content Toggle raw display
$29$ \( T^{4} - 8066 T^{3} + \cdots + 831691300 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 738407035699200 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 57\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 225655412958976 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 16\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 25\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 74\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 17\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 32\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 29\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 21\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 75\!\cdots\!24 \) Copy content Toggle raw display
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