↑

Properties

Label 560.3.p.e
Level $560$
Weight $3$
Character orbit 560.p
Analytic conductor $15.259$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,3,Mod(209,560)]
 
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, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.209");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 560.p (of order \(2\), degree \(1\), not 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: \(15.2588948042\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-41})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 40x^{2} + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} + 2 \beta_{2}) q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} + 2 \beta_{2}) q^{7} - 7 q^{9} + 2 q^{11} - 11 \beta_{2} q^{13} + ( - \beta_{3} - 3) q^{15} - 10 \beta_{2} q^{17} + ( - \beta_{2} - 2 \beta_1) q^{19} + (\beta_{2} + 2 \beta_1 + 4) q^{21} + 4 \beta_{3} q^{23} + (3 \beta_{3} - 16) q^{25} - 16 \beta_{2} q^{27} + 14 q^{29} + (4 \beta_{2} + 8 \beta_1) q^{31} + 2 \beta_{2} q^{33} + ( - 2 \beta_{3} - 22 \beta_{2} + \cdots - 6) q^{35}+ \cdots - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 28 q^{9} + 8 q^{11} - 12 q^{15} + 16 q^{21} - 64 q^{25} + 56 q^{29} - 24 q^{35} - 88 q^{39} - 132 q^{49} - 80 q^{51} + 132 q^{65} + 320 q^{71} + 48 q^{79} + 124 q^{81} + 120 q^{85} - 176 q^{91} + 164 q^{95} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 40x^{2} + 441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 19\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 21\beta_{2} - 19\beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.707107 − 4.52769i
0.707107 + 4.52769i
−0.707107 − 4.52769i
−0.707107 + 4.52769i
0 −1.41421 0 2.12132 − 4.52769i 0 −2.82843 + 6.40312i 0 −7.00000 0
209.2 0 −1.41421 0 2.12132 + 4.52769i 0 −2.82843 − 6.40312i 0 −7.00000 0
209.3 0 1.41421 0 −2.12132 − 4.52769i 0 2.82843 − 6.40312i 0 −7.00000 0
209.4 0 1.41421 0 −2.12132 + 4.52769i 0 2.82843 + 6.40312i 0 −7.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.3.p.e 4
4.b odd 2 1 140.3.h.c ✓ 4
5.b even 2 1 inner 560.3.p.e 4
7.b odd 2 1 inner 560.3.p.e 4
12.b even 2 1 1260.3.p.c 4
20.d odd 2 1 140.3.h.c ✓ 4
20.e even 4 2 700.3.d.c 4
28.d even 2 1 140.3.h.c ✓ 4
28.f even 6 2 980.3.n.c 8
28.g odd 6 2 980.3.n.c 8
35.c odd 2 1 inner 560.3.p.e 4
60.h even 2 1 1260.3.p.c 4
84.h odd 2 1 1260.3.p.c 4
140.c even 2 1 140.3.h.c ✓ 4
140.j odd 4 2 700.3.d.c 4
140.p odd 6 2 980.3.n.c 8
140.s even 6 2 980.3.n.c 8
420.o odd 2 1 1260.3.p.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.3.h.c ✓ 4 4.b odd 2 1
140.3.h.c ✓ 4 20.d odd 2 1
140.3.h.c ✓ 4 28.d even 2 1
140.3.h.c ✓ 4 140.c even 2 1
560.3.p.e 4 1.a even 1 1 trivial
560.3.p.e 4 5.b even 2 1 inner
560.3.p.e 4 7.b odd 2 1 inner
560.3.p.e 4 35.c odd 2 1 inner
700.3.d.c 4 20.e even 4 2
700.3.d.c 4 140.j odd 4 2
980.3.n.c 8 28.f even 6 2
980.3.n.c 8 28.g odd 6 2
980.3.n.c 8 140.p odd 6 2
980.3.n.c 8 140.s even 6 2
1260.3.p.c 4 12.b even 2 1
1260.3.p.c 4 60.h even 2 1
1260.3.p.c 4 84.h odd 2 1
1260.3.p.c 4 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 \) acting on \(S_{3}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 32T^{2} + 625 \) Copy content Toggle raw display
$7$ \( T^{4} + 66T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T - 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 242)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 200)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 82)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 656)^{2} \) Copy content Toggle raw display
$29$ \( (T - 14)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1312)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 164)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 328)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4100)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 3528)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4100)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 9922)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6642)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4100)^{2} \) Copy content Toggle raw display
$71$ \( (T - 80)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 14112)^{2} \) Copy content Toggle raw display
$79$ \( (T - 12)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 16072)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 25992)^{2} \) Copy content Toggle raw display
show more
show less