Properties

Label 560.2.e.d
Level $560$
Weight $2$
Character orbit 560.e
Analytic conductor $4.472$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,2,Mod(559,560)]
 
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([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.e (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12745506816.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + \beta_{4} q^{5} + \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + \beta_{4} q^{5} + \beta_1 q^{7} + q^{9} + 2 \beta_{3} q^{11} + ( - \beta_{6} + \beta_{4}) q^{13} + ( - \beta_{3} - \beta_1) q^{15} + ( - 2 \beta_{6} + 2 \beta_{4}) q^{17} + (\beta_{5} + 2 \beta_{2}) q^{19} + (\beta_{6} + \beta_{4}) q^{21} + ( - \beta_{7} - 2) q^{25} + 4 \beta_{5} q^{27} - 6 q^{29} + ( - 2 \beta_{6} + 2 \beta_{4}) q^{33} + (3 \beta_{5} - \beta_{2}) q^{35} + 2 \beta_{7} q^{37} - 2 \beta_{3} q^{39} + (2 \beta_{6} + 2 \beta_{4}) q^{41} - 2 \beta_1 q^{43} + \beta_{4} q^{45} - 2 \beta_{5} q^{47} + 7 q^{49} - 4 \beta_{3} q^{51} - 2 \beta_{7} q^{53} + ( - 4 \beta_{5} - 2 \beta_{2}) q^{55} + 2 \beta_{7} q^{57} + (\beta_{5} + 2 \beta_{2}) q^{59} + ( - 3 \beta_{6} - 3 \beta_{4}) q^{61} + \beta_1 q^{63} + ( - \beta_{7} + 3) q^{65} + 2 \beta_1 q^{67} + (4 \beta_{6} - 4 \beta_{4}) q^{73} + ( - \beta_{5} + 2 \beta_{2}) q^{75} + 2 \beta_{7} q^{77} + 4 \beta_{3} q^{79} - 5 q^{81} + 7 \beta_{5} q^{83} + ( - 2 \beta_{7} + 6) q^{85} - 6 \beta_{5} q^{87} + ( - 2 \beta_{6} - 2 \beta_{4}) q^{89} + ( - \beta_{5} - 2 \beta_{2}) q^{91} + (7 \beta_{3} - 3 \beta_1) q^{95} + (6 \beta_{6} - 6 \beta_{4}) q^{97} + 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 16 q^{25} - 48 q^{29} + 56 q^{49} + 24 q^{65} - 40 q^{81} + 48 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 148 ) / 55 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 11\nu^{5} - 88\nu^{3} + 336\nu ) / 99 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\nu^{6} - 110\nu^{4} + 880\nu^{2} - 657 ) / 495 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 5\nu^{5} + 40\nu^{3} + 48\nu ) / 45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{7} + 55\nu^{5} - 341\nu^{3} + 81\nu ) / 297 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 8\nu^{5} + 55\nu^{3} - 45\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 8\nu^{4} - 46\nu^{2} + 36 ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 4\beta_{3} + \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{6} + 11\beta_{5} + 5\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{7} + 23\beta_{3} - 8\beta _1 - 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -24\beta_{6} + 24\beta_{5} + 55\beta_{4} - 31\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -55\beta _1 - 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -368\beta_{6} - 368\beta_{5} + 165\beta_{4} - 203\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
2.23256 1.28897i
−1.00781 + 0.581861i
−2.23256 1.28897i
1.00781 + 0.581861i
−1.00781 0.581861i
2.23256 + 1.28897i
1.00781 0.581861i
−2.23256 + 1.28897i
0 1.41421i 0 −1.22474 1.87083i 0 2.64575 0 1.00000 0
559.2 0 1.41421i 0 −1.22474 + 1.87083i 0 −2.64575 0 1.00000 0
559.3 0 1.41421i 0 1.22474 1.87083i 0 2.64575 0 1.00000 0
559.4 0 1.41421i 0 1.22474 + 1.87083i 0 −2.64575 0 1.00000 0
559.5 0 1.41421i 0 −1.22474 1.87083i 0 −2.64575 0 1.00000 0
559.6 0 1.41421i 0 −1.22474 + 1.87083i 0 2.64575 0 1.00000 0
559.7 0 1.41421i 0 1.22474 1.87083i 0 −2.64575 0 1.00000 0
559.8 0 1.41421i 0 1.22474 + 1.87083i 0 2.64575 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.e.d 8
4.b odd 2 1 inner 560.2.e.d 8
5.b even 2 1 inner 560.2.e.d 8
5.c odd 4 2 2800.2.k.k 8
7.b odd 2 1 inner 560.2.e.d 8
8.b even 2 1 2240.2.e.e 8
8.d odd 2 1 2240.2.e.e 8
20.d odd 2 1 inner 560.2.e.d 8
20.e even 4 2 2800.2.k.k 8
28.d even 2 1 inner 560.2.e.d 8
35.c odd 2 1 inner 560.2.e.d 8
35.f even 4 2 2800.2.k.k 8
40.e odd 2 1 2240.2.e.e 8
40.f even 2 1 2240.2.e.e 8
56.e even 2 1 2240.2.e.e 8
56.h odd 2 1 2240.2.e.e 8
140.c even 2 1 inner 560.2.e.d 8
140.j odd 4 2 2800.2.k.k 8
280.c odd 2 1 2240.2.e.e 8
280.n even 2 1 2240.2.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.d 8 1.a even 1 1 trivial
560.2.e.d 8 4.b odd 2 1 inner
560.2.e.d 8 5.b even 2 1 inner
560.2.e.d 8 7.b odd 2 1 inner
560.2.e.d 8 20.d odd 2 1 inner
560.2.e.d 8 28.d even 2 1 inner
560.2.e.d 8 35.c odd 2 1 inner
560.2.e.d 8 140.c even 2 1 inner
2240.2.e.e 8 8.b even 2 1
2240.2.e.e 8 8.d odd 2 1
2240.2.e.e 8 40.e odd 2 1
2240.2.e.e 8 40.f even 2 1
2240.2.e.e 8 56.e even 2 1
2240.2.e.e 8 56.h odd 2 1
2240.2.e.e 8 280.c odd 2 1
2240.2.e.e 8 280.n even 2 1
2800.2.k.k 8 5.c odd 4 2
2800.2.k.k 8 20.e even 4 2
2800.2.k.k 8 35.f even 4 2
2800.2.k.k 8 140.j odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 4 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 42)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T + 6)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} + 84)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 56)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 84)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 42)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 126)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} - 96)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 98)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 56)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 216)^{4} \) Copy content Toggle raw display
show more
show less