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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(26,567)] 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("567.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,0,0,4] 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: \(4.52751779461\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 3) q^{7} + ( - 4 \zeta_{6} - 4) q^{13} + (4 \zeta_{6} - 4) q^{16} + ( - 3 \zeta_{6} + 6) q^{19} - 5 q^{25} + ( - 2 \zeta_{6} - 4) q^{28} + ( - \zeta_{6} + 2) q^{31} + \cdots + ( - 11 \zeta_{6} + 22) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 4 q^{7} - 12 q^{13} - 4 q^{16} + 9 q^{19} - 10 q^{25} - 10 q^{28} + 3 q^{31} - 10 q^{37} - 13 q^{43} + 2 q^{49} + 15 q^{61} + 16 q^{64} + 16 q^{67} + 27 q^{73} - 18 q^{76} + 4 q^{79} - 36 q^{91}+ \cdots + 33 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(\zeta_{6}\) \(1 - \zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 −1.00000 + 1.73205i 0 0 2.00000 + 1.73205i 0 0 0
458.1 0 0 −1.00000 − 1.73205i 0 0 2.00000 − 1.73205i 0 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
63.k odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.s.b 2
3.b odd 2 1 CM 567.2.s.b 2
7.d odd 6 1 567.2.i.a 2
9.c even 3 1 189.2.p.a ✓ 2
9.c even 3 1 567.2.i.a 2
9.d odd 6 1 189.2.p.a ✓ 2
9.d odd 6 1 567.2.i.a 2
21.g even 6 1 567.2.i.a 2
63.g even 3 1 1323.2.c.a 2
63.i even 6 1 189.2.p.a ✓ 2
63.k odd 6 1 inner 567.2.s.b 2
63.k odd 6 1 1323.2.c.a 2
63.n odd 6 1 1323.2.c.a 2
63.s even 6 1 inner 567.2.s.b 2
63.s even 6 1 1323.2.c.a 2
63.t odd 6 1 189.2.p.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.a ✓ 2 9.c even 3 1
189.2.p.a ✓ 2 9.d odd 6 1
189.2.p.a ✓ 2 63.i even 6 1
189.2.p.a ✓ 2 63.t odd 6 1
567.2.i.a 2 7.d odd 6 1
567.2.i.a 2 9.c even 3 1
567.2.i.a 2 9.d odd 6 1
567.2.i.a 2 21.g even 6 1
567.2.s.b 2 1.a even 1 1 trivial
567.2.s.b 2 3.b odd 2 1 CM
567.2.s.b 2 63.k odd 6 1 inner
567.2.s.b 2 63.s even 6 1 inner
1323.2.c.a 2 63.g even 3 1
1323.2.c.a 2 63.k odd 6 1
1323.2.c.a 2 63.n odd 6 1
1323.2.c.a 2 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} + 12T_{13} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$67$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 27T + 243 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 33T + 363 \) Copy content Toggle raw display
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