↑

Properties

Label 560.2.cc.c
Level $560$
Weight $2$
Character orbit 560.cc
Analytic conductor $4.472$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,2,Mod(159,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.159");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.cc (of order \(6\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{3} + \beta_1 - 2) q^{5} + (2 \beta_{2} + 1) q^{7} + (\beta_{3} - 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{3} + \beta_1 - 2) q^{5} + (2 \beta_{2} + 1) q^{7} + (\beta_{3} - 2 \beta_1 + 1) q^{9} + (\beta_{2} + 1) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{13} + ( - \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{15} + ( - 2 \beta_{3} + 4 \beta_{2} + \cdots - 5) q^{17}+ \cdots + (3 \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} - 6 q^{5} + 8 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} - 6 q^{5} + 8 q^{7} + q^{9} + 6 q^{11} - 6 q^{13} + q^{15} - 9 q^{17} + 3 q^{21} - 2 q^{25} + 12 q^{29} - 3 q^{31} + 3 q^{33} - 12 q^{35} - 30 q^{37} + 12 q^{39} - 16 q^{43} + 15 q^{45} + 18 q^{47} + 4 q^{49} + 3 q^{51} - 12 q^{53} - 9 q^{55} - 15 q^{59} - 21 q^{61} - q^{63} + 9 q^{65} + q^{67} + 15 q^{73} - 18 q^{75} + 6 q^{77} + 15 q^{79} + 4 q^{81} - 3 q^{85} + 42 q^{87} - 9 q^{89} - 12 q^{91} + 45 q^{93} + 33 q^{95} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(\beta_{2}\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1
1.68614 − 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 − 1.26217i
0 −0.686141 + 0.396143i 0 −1.50000 − 1.65831i 0 2.00000 − 1.73205i 0 −1.18614 + 2.05446i 0
159.2 0 2.18614 − 1.26217i 0 −1.50000 + 1.65831i 0 2.00000 − 1.73205i 0 1.68614 − 2.92048i 0
479.1 0 −0.686141 − 0.396143i 0 −1.50000 + 1.65831i 0 2.00000 + 1.73205i 0 −1.18614 − 2.05446i 0
479.2 0 2.18614 + 1.26217i 0 −1.50000 − 1.65831i 0 2.00000 + 1.73205i 0 1.68614 + 2.92048i 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
140.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.cc.c yes 4
4.b odd 2 1 560.2.cc.a ✓ 4
5.b even 2 1 560.2.cc.b yes 4
7.d odd 6 1 560.2.cc.d yes 4
20.d odd 2 1 560.2.cc.d yes 4
28.f even 6 1 560.2.cc.b yes 4
35.i odd 6 1 560.2.cc.a ✓ 4
140.s even 6 1 inner 560.2.cc.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.cc.a ✓ 4 4.b odd 2 1
560.2.cc.a ✓ 4 35.i odd 6 1
560.2.cc.b yes 4 5.b even 2 1
560.2.cc.b yes 4 28.f even 6 1
560.2.cc.c yes 4 1.a even 1 1 trivial
560.2.cc.c yes 4 140.s even 6 1 inner
560.2.cc.d yes 4 7.d odd 6 1
560.2.cc.d yes 4 20.d odd 2 1

Hecke kernels

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

\( T_{3}^{4} - 3T_{3}^{3} + T_{3}^{2} + 6T_{3} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 3 \) Copy content Toggle raw display
\( T_{19}^{4} + 33T_{19}^{2} + 1089 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3 T - 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 9 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( T^{4} + 33T^{2} + 1089 \) Copy content Toggle raw display
$23$ \( T^{4} + 33T^{2} + 1089 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T - 24)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 3 T^{3} + \cdots + 5184 \) Copy content Toggle raw display
$37$ \( (T^{2} + 15 T + 75)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 19T^{2} + 16 \) Copy content Toggle raw display
$43$ \( (T + 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 18 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + \cdots + 7569 \) Copy content Toggle raw display
$59$ \( T^{4} + 15 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$61$ \( T^{4} + 21 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$67$ \( T^{4} - T^{3} + \cdots + 5476 \) Copy content Toggle raw display
$71$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 15 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$79$ \( T^{4} - 15 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$83$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$89$ \( T^{4} + 9 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$97$ \( (T + 6)^{4} \) Copy content Toggle raw display
show more
show less