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Properties

Label 560.2.q.g
Level $560$
Weight $2$
Character orbit 560.q
Analytic conductor $4.472$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [560,2,Mod(81,560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,1,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{5} + (3 \zeta_{6} - 1) q^{7} + 2 \zeta_{6} q^{9} + (6 \zeta_{6} - 6) q^{11} - 4 q^{13} + q^{15} + 2 \zeta_{6} q^{19} + (\zeta_{6} + 2) q^{21} - 3 \zeta_{6} q^{23} + \cdots - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{5} + q^{7} + 2 q^{9} - 6 q^{11} - 8 q^{13} + 2 q^{15} + 2 q^{19} + 5 q^{21} - 3 q^{23} - q^{25} + 10 q^{27} - 6 q^{29} + 8 q^{31} + 6 q^{33} - 4 q^{35} + 4 q^{37} - 4 q^{39} + 18 q^{41}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 0.500000 − 0.866025i 0 0.500000 + 0.866025i 0 0.500000 + 2.59808i 0 1.00000 + 1.73205i 0
401.1 0 0.500000 + 0.866025i 0 0.500000 − 0.866025i 0 0.500000 − 2.59808i 0 1.00000 − 1.73205i 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.g 2
4.b odd 2 1 70.2.e.c ✓ 2
7.c even 3 1 inner 560.2.q.g 2
7.c even 3 1 3920.2.a.p 1
7.d odd 6 1 3920.2.a.bc 1
12.b even 2 1 630.2.k.b 2
20.d odd 2 1 350.2.e.e 2
20.e even 4 2 350.2.j.b 4
28.d even 2 1 490.2.e.h 2
28.f even 6 1 490.2.a.b 1
28.f even 6 1 490.2.e.h 2
28.g odd 6 1 70.2.e.c ✓ 2
28.g odd 6 1 490.2.a.c 1
84.j odd 6 1 4410.2.a.bd 1
84.n even 6 1 630.2.k.b 2
84.n even 6 1 4410.2.a.bm 1
140.p odd 6 1 350.2.e.e 2
140.p odd 6 1 2450.2.a.w 1
140.s even 6 1 2450.2.a.bc 1
140.w even 12 2 350.2.j.b 4
140.w even 12 2 2450.2.c.g 2
140.x odd 12 2 2450.2.c.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.c ✓ 2 4.b odd 2 1
70.2.e.c ✓ 2 28.g odd 6 1
350.2.e.e 2 20.d odd 2 1
350.2.e.e 2 140.p odd 6 1
350.2.j.b 4 20.e even 4 2
350.2.j.b 4 140.w even 12 2
490.2.a.b 1 28.f even 6 1
490.2.a.c 1 28.g odd 6 1
490.2.e.h 2 28.d even 2 1
490.2.e.h 2 28.f even 6 1
560.2.q.g 2 1.a even 1 1 trivial
560.2.q.g 2 7.c even 3 1 inner
630.2.k.b 2 12.b even 2 1
630.2.k.b 2 84.n even 6 1
2450.2.a.w 1 140.p odd 6 1
2450.2.a.bc 1 140.s even 6 1
2450.2.c.g 2 140.w even 12 2
2450.2.c.l 2 140.x odd 12 2
3920.2.a.p 1 7.c even 3 1
3920.2.a.bc 1 7.d odd 6 1
4410.2.a.bd 1 84.j odd 6 1
4410.2.a.bm 1 84.n even 6 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 36 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$41$ \( (T - 9)^{2} \) Copy content Toggle raw display
$43$ \( (T - 7)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$83$ \( (T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$97$ \( (T - 14)^{2} \) Copy content Toggle raw display
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