Properties

Label 567.2.g.g
Level $567$
Weight $2$
Character orbit 567.g
Analytic conductor $4.528$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,2,Mod(109,567)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(567, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.g (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: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 189)
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 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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (4 \beta_1 - 4) q^{4} + \beta_{3} q^{5} + ( - 2 \beta_1 - 1) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (4 \beta_1 - 4) q^{4} + \beta_{3} q^{5} + ( - 2 \beta_1 - 1) q^{7} + 2 \beta_{3} q^{8} - 6 \beta_1 q^{10} + 2 \beta_{3} q^{11} + 4 \beta_1 q^{13} + ( - 2 \beta_{3} + 3 \beta_{2}) q^{14} - 4 \beta_1 q^{16} + \beta_{2} q^{17} + ( - \beta_1 + 1) q^{19} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{20} - 12 \beta_1 q^{22} + \beta_{3} q^{23} + q^{25} + (4 \beta_{3} - 4 \beta_{2}) q^{26} + ( - 4 \beta_1 + 12) q^{28} + (3 \beta_{3} - 3 \beta_{2}) q^{29} + ( - 7 \beta_1 + 7) q^{31} + ( - 6 \beta_1 + 6) q^{34} + ( - \beta_{3} - 2 \beta_{2}) q^{35} + (8 \beta_1 - 8) q^{37} - \beta_{3} q^{38} + 12 q^{40} - 3 \beta_{2} q^{41} + ( - \beta_1 + 1) q^{43} + ( - 8 \beta_{3} + 8 \beta_{2}) q^{44} - 6 \beta_1 q^{46} - \beta_{2} q^{47} + (8 \beta_1 - 3) q^{49} - \beta_{2} q^{50} - 16 q^{52} + \beta_{2} q^{53} + 12 q^{55} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{56} - 18 q^{58} + (4 \beta_{3} - 4 \beta_{2}) q^{59} - 5 \beta_1 q^{61} - 7 \beta_{3} q^{62} - 8 q^{64} + 4 \beta_{2} q^{65} + (2 \beta_1 - 2) q^{67} - 4 \beta_{3} q^{68} + (18 \beta_1 - 12) q^{70} + \beta_1 q^{73} + 8 \beta_{3} q^{74} + 4 \beta_1 q^{76} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{77} + 4 \beta_1 q^{79} - 4 \beta_{2} q^{80} + (18 \beta_1 - 18) q^{82} + ( - 6 \beta_{3} + 6 \beta_{2}) q^{83} + 6 \beta_1 q^{85} - \beta_{3} q^{86} + 24 q^{88} + ( - \beta_{3} + \beta_{2}) q^{89} + ( - 12 \beta_1 + 8) q^{91} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{92} + (6 \beta_1 - 6) q^{94} + (\beta_{3} - \beta_{2}) q^{95} + ( - \beta_1 + 1) q^{97} + (8 \beta_{3} - 5 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 8 q^{7} - 12 q^{10} + 8 q^{13} - 8 q^{16} + 2 q^{19} - 24 q^{22} + 4 q^{25} + 40 q^{28} + 14 q^{31} + 12 q^{34} - 16 q^{37} + 48 q^{40} + 2 q^{43} - 12 q^{46} + 4 q^{49} - 64 q^{52} + 48 q^{55} - 72 q^{58} - 10 q^{61} - 32 q^{64} - 4 q^{67} - 12 q^{70} + 2 q^{73} + 8 q^{76} + 8 q^{79} - 36 q^{82} + 12 q^{85} + 96 q^{88} + 8 q^{91} - 12 q^{94} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
−1.22474 + 2.12132i 0 −2.00000 3.46410i 2.44949 0 −2.00000 + 1.73205i 4.89898 0 −3.00000 + 5.19615i
109.2 1.22474 2.12132i 0 −2.00000 3.46410i −2.44949 0 −2.00000 + 1.73205i −4.89898 0 −3.00000 + 5.19615i
541.1 −1.22474 2.12132i 0 −2.00000 + 3.46410i 2.44949 0 −2.00000 1.73205i 4.89898 0 −3.00000 5.19615i
541.2 1.22474 + 2.12132i 0 −2.00000 + 3.46410i −2.44949 0 −2.00000 1.73205i −4.89898 0 −3.00000 5.19615i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.g even 3 1 inner
63.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.g.g 4
3.b odd 2 1 inner 567.2.g.g 4
7.c even 3 1 567.2.h.g 4
9.c even 3 1 189.2.e.d 4
9.c even 3 1 567.2.h.g 4
9.d odd 6 1 189.2.e.d 4
9.d odd 6 1 567.2.h.g 4
21.h odd 6 1 567.2.h.g 4
63.g even 3 1 inner 567.2.g.g 4
63.g even 3 1 1323.2.a.v 2
63.h even 3 1 189.2.e.d 4
63.j odd 6 1 189.2.e.d 4
63.k odd 6 1 1323.2.a.u 2
63.n odd 6 1 inner 567.2.g.g 4
63.n odd 6 1 1323.2.a.v 2
63.s even 6 1 1323.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.d 4 9.c even 3 1
189.2.e.d 4 9.d odd 6 1
189.2.e.d 4 63.h even 3 1
189.2.e.d 4 63.j odd 6 1
567.2.g.g 4 1.a even 1 1 trivial
567.2.g.g 4 3.b odd 2 1 inner
567.2.g.g 4 63.g even 3 1 inner
567.2.g.g 4 63.n odd 6 1 inner
567.2.h.g 4 7.c even 3 1
567.2.h.g 4 9.c even 3 1
567.2.h.g 4 9.d odd 6 1
567.2.h.g 4 21.h odd 6 1
1323.2.a.u 2 63.k odd 6 1
1323.2.a.u 2 63.s even 6 1
1323.2.a.v 2 63.g even 3 1
1323.2.a.v 2 63.n odd 6 1

Hecke kernels

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

\( T_{2}^{4} + 6T_{2}^{2} + 36 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 54T^{2} + 2916 \) Copy content Toggle raw display
$31$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 54T^{2} + 2916 \) Copy content Toggle raw display
$43$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 216 T^{2} + 46656 \) Copy content Toggle raw display
$89$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$97$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
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