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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,2,Mod(81,560)]
 
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([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.q (of order \(3\), degree \(2\), 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: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 3) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 3) q^{7} - \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} - q^{13} + 2 q^{15} + ( - 6 \zeta_{6} + 6) q^{17} - \zeta_{6} q^{19} + (6 \zeta_{6} - 2) q^{21} + 9 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 4 q^{27} + 6 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} + 6 \zeta_{6} q^{33} + ( - \zeta_{6} - 2) q^{35} + 7 \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 2) q^{39} + 3 q^{41} - 2 q^{43} + (\zeta_{6} - 1) q^{45} + 9 \zeta_{6} q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + 12 \zeta_{6} q^{51} + (9 \zeta_{6} - 9) q^{53} - 3 q^{55} + 2 q^{57} - 8 \zeta_{6} q^{61} + ( - \zeta_{6} - 2) q^{63} + \zeta_{6} q^{65} + ( - 8 \zeta_{6} + 8) q^{67} - 18 q^{69} + ( - 4 \zeta_{6} + 4) q^{73} - 2 \zeta_{6} q^{75} + ( - 9 \zeta_{6} + 3) q^{77} - 10 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} - 6 q^{85} + (12 \zeta_{6} - 12) q^{87} - 6 \zeta_{6} q^{89} + (2 \zeta_{6} - 3) q^{91} + 16 \zeta_{6} q^{93} + (\zeta_{6} - 1) q^{95} - 10 q^{97} - 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - q^{5} + 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - q^{5} + 4 q^{7} - q^{9} + 3 q^{11} - 2 q^{13} + 4 q^{15} + 6 q^{17} - q^{19} + 2 q^{21} + 9 q^{23} - q^{25} - 8 q^{27} + 12 q^{29} + 8 q^{31} + 6 q^{33} - 5 q^{35} + 7 q^{37} + 2 q^{39} + 6 q^{41} - 4 q^{43} - q^{45} + 9 q^{47} + 2 q^{49} + 12 q^{51} - 9 q^{53} - 6 q^{55} + 4 q^{57} - 8 q^{61} - 5 q^{63} + q^{65} + 8 q^{67} - 36 q^{69} + 4 q^{73} - 2 q^{75} - 3 q^{77} - 10 q^{79} + 11 q^{81} - 12 q^{85} - 12 q^{87} - 6 q^{89} - 4 q^{91} + 16 q^{93} - q^{95} - 20 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 −1.00000 + 1.73205i 0 −0.500000 − 0.866025i 0 2.00000 − 1.73205i 0 −0.500000 − 0.866025i 0
401.1 0 −1.00000 − 1.73205i 0 −0.500000 + 0.866025i 0 2.00000 + 1.73205i 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.q.b 2
4.b odd 2 1 70.2.e.d ✓ 2
7.c even 3 1 inner 560.2.q.b 2
7.c even 3 1 3920.2.a.bh 1
7.d odd 6 1 3920.2.a.e 1
12.b even 2 1 630.2.k.d 2
20.d odd 2 1 350.2.e.b 2
20.e even 4 2 350.2.j.d 4
28.d even 2 1 490.2.e.g 2
28.f even 6 1 490.2.a.d 1
28.f even 6 1 490.2.e.g 2
28.g odd 6 1 70.2.e.d ✓ 2
28.g odd 6 1 490.2.a.a 1
84.j odd 6 1 4410.2.a.bg 1
84.n even 6 1 630.2.k.d 2
84.n even 6 1 4410.2.a.x 1
140.p odd 6 1 350.2.e.b 2
140.p odd 6 1 2450.2.a.bf 1
140.s even 6 1 2450.2.a.v 1
140.w even 12 2 350.2.j.d 4
140.w even 12 2 2450.2.c.q 2
140.x odd 12 2 2450.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.d ✓ 2 4.b odd 2 1
70.2.e.d ✓ 2 28.g odd 6 1
350.2.e.b 2 20.d odd 2 1
350.2.e.b 2 140.p odd 6 1
350.2.j.d 4 20.e even 4 2
350.2.j.d 4 140.w even 12 2
490.2.a.a 1 28.g odd 6 1
490.2.a.d 1 28.f even 6 1
490.2.e.g 2 28.d even 2 1
490.2.e.g 2 28.f even 6 1
560.2.q.b 2 1.a even 1 1 trivial
560.2.q.b 2 7.c even 3 1 inner
630.2.k.d 2 12.b even 2 1
630.2.k.d 2 84.n even 6 1
2450.2.a.v 1 140.s even 6 1
2450.2.a.bf 1 140.p odd 6 1
2450.2.c.e 2 140.x odd 12 2
2450.2.c.q 2 140.w even 12 2
3920.2.a.e 1 7.d odd 6 1
3920.2.a.bh 1 7.c even 3 1
4410.2.a.x 1 84.n even 6 1
4410.2.a.bg 1 84.j odd 6 1

Hecke kernels

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

\( T_{3}^{2} + 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( (T - 3)^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$53$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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