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Properties

Label 567.2.a.g
Level $567$
Weight $2$
Character orbit 567.a
Self dual yes
Analytic conductor $4.528$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,2,Mod(1,567)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(567, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.a (trivial)

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: yes
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} + 2) q^{5} - q^{7} + (\beta_{2} + \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} + 2) q^{5} - q^{7} + (\beta_{2} + \beta_1 + 2) q^{8} + (3 \beta_1 - 1) q^{10} + ( - \beta_{2} - 2 \beta_1 + 1) q^{11} + q^{13} - \beta_1 q^{14} + ( - \beta_{2} + 2 \beta_1) q^{16} + ( - \beta_{2} - \beta_1 + 4) q^{17} + ( - \beta_{2} - \beta_1 - 1) q^{19} + (\beta_{2} + 2 \beta_1 + 5) q^{20} + ( - 2 \beta_{2} - 2 \beta_1 - 5) q^{22} + ( - 2 \beta_{2} + \beta_1 - 1) q^{23} + (2 \beta_{2} - \beta_1 + 3) q^{25} + \beta_1 q^{26} + ( - \beta_{2} - \beta_1 - 1) q^{28} - \beta_1 q^{29} + ( - \beta_{2} + 2 \beta_1 - 2) q^{31} + ( - \beta_1 + 3) q^{32} + ( - \beta_{2} + 2 \beta_1 - 2) q^{34} + ( - \beta_{2} - 2) q^{35} + 3 \beta_{2} q^{37} + ( - \beta_{2} - 3 \beta_1 - 2) q^{38} + (2 \beta_{2} + 2 \beta_1 + 7) q^{40} + ( - \beta_{2} + 7) q^{41} + ( - \beta_{2} - 4 \beta_1) q^{43} + ( - 5 \beta_1 - 6) q^{44} + (\beta_{2} - 2 \beta_1 + 5) q^{46} + (3 \beta_{2} + 3 \beta_1 + 3) q^{47} + q^{49} + ( - \beta_{2} + 4 \beta_1 - 5) q^{50} + (\beta_{2} + \beta_1 + 1) q^{52} + ( - 2 \beta_{2} + \beta_1 + 5) q^{53} + (\beta_{2} - 5 \beta_1) q^{55} + ( - \beta_{2} - \beta_1 - 2) q^{56} + ( - \beta_{2} - \beta_1 - 3) q^{58} + (2 \beta_{2} - \beta_1 + 4) q^{59} + (2 \beta_{2} - \beta_1 - 1) q^{61} + (2 \beta_{2} - \beta_1 + 7) q^{62} + (\beta_{2} - 2 \beta_1 - 3) q^{64} + (\beta_{2} + 2) q^{65} + ( - \beta_{2} - 7 \beta_1 + 2) q^{67} + (4 \beta_{2} + \beta_1 - 1) q^{68} + ( - 3 \beta_1 + 1) q^{70} + (2 \beta_{2} - \beta_1 - 2) q^{71} + ( - 5 \beta_{2} + \beta_1 - 1) q^{73} + (3 \beta_1 - 3) q^{74} + ( - \beta_{2} - 4 \beta_1 - 6) q^{76} + (\beta_{2} + 2 \beta_1 - 1) q^{77} + ( - \beta_{2} + 5 \beta_1 + 3) q^{79} + (7 \beta_1 - 6) q^{80} + (6 \beta_1 + 1) q^{82} + ( - \beta_{2} - \beta_1 + 4) q^{83} + (4 \beta_{2} - 2 \beta_1 + 5) q^{85} + ( - 4 \beta_{2} - 5 \beta_1 - 11) q^{86} + ( - \beta_{2} - 7 \beta_1 - 5) q^{88} + (\beta_{2} + 6 \beta_1 - 1) q^{89} - q^{91} + (2 \beta_{2} + 2 \beta_1 - 5) q^{92} + (3 \beta_{2} + 9 \beta_1 + 6) q^{94} + ( - \beta_{2} - 2 \beta_1 - 5) q^{95} + ( - 2 \beta_{2} - 5 \beta_1 + 2) q^{97} + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{4} + 5 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 3 q^{4} + 5 q^{5} - 3 q^{7} + 6 q^{8} + 2 q^{11} + 3 q^{13} - q^{14} + 3 q^{16} + 12 q^{17} - 3 q^{19} + 16 q^{20} - 15 q^{22} + 6 q^{25} + q^{26} - 3 q^{28} - q^{29} - 3 q^{31} + 8 q^{32} - 3 q^{34} - 5 q^{35} - 3 q^{37} - 8 q^{38} + 21 q^{40} + 22 q^{41} - 3 q^{43} - 23 q^{44} + 12 q^{46} + 9 q^{47} + 3 q^{49} - 10 q^{50} + 3 q^{52} + 18 q^{53} - 6 q^{55} - 6 q^{56} - 9 q^{58} + 9 q^{59} - 6 q^{61} + 18 q^{62} - 12 q^{64} + 5 q^{65} - 6 q^{68} - 9 q^{71} + 3 q^{73} - 6 q^{74} - 21 q^{76} - 2 q^{77} + 15 q^{79} - 11 q^{80} + 9 q^{82} + 12 q^{83} + 9 q^{85} - 34 q^{86} - 21 q^{88} + 2 q^{89} - 3 q^{91} - 15 q^{92} + 24 q^{94} - 16 q^{95} + 3 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 1 \) : Copy content Toggle raw display

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

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

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.69963
0.239123
2.46050
−1.69963 0 0.888736 3.58836 0 −1.00000 1.88874 0 −6.09888
1.2 0.239123 0 −1.94282 −1.18194 0 −1.00000 −0.942820 0 −0.282630
1.3 2.46050 0 4.05408 2.59358 0 −1.00000 5.05408 0 6.38151
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.a.g 3
3.b odd 2 1 567.2.a.d 3
4.b odd 2 1 9072.2.a.cd 3
7.b odd 2 1 3969.2.a.p 3
9.c even 3 2 189.2.f.a 6
9.d odd 6 2 63.2.f.b ✓ 6
12.b even 2 1 9072.2.a.bq 3
21.c even 2 1 3969.2.a.m 3
36.f odd 6 2 3024.2.r.g 6
36.h even 6 2 1008.2.r.k 6
63.g even 3 2 1323.2.g.c 6
63.h even 3 2 1323.2.h.d 6
63.i even 6 2 441.2.h.b 6
63.j odd 6 2 441.2.h.c 6
63.k odd 6 2 1323.2.g.b 6
63.l odd 6 2 1323.2.f.c 6
63.n odd 6 2 441.2.g.e 6
63.o even 6 2 441.2.f.d 6
63.s even 6 2 441.2.g.d 6
63.t odd 6 2 1323.2.h.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b ✓ 6 9.d odd 6 2
189.2.f.a 6 9.c even 3 2
441.2.f.d 6 63.o even 6 2
441.2.g.d 6 63.s even 6 2
441.2.g.e 6 63.n odd 6 2
441.2.h.b 6 63.i even 6 2
441.2.h.c 6 63.j odd 6 2
567.2.a.d 3 3.b odd 2 1
567.2.a.g 3 1.a even 1 1 trivial
1008.2.r.k 6 36.h even 6 2
1323.2.f.c 6 63.l odd 6 2
1323.2.g.b 6 63.k odd 6 2
1323.2.g.c 6 63.g even 3 2
1323.2.h.d 6 63.h even 3 2
1323.2.h.e 6 63.t odd 6 2
3024.2.r.g 6 36.f odd 6 2
3969.2.a.m 3 21.c even 2 1
3969.2.a.p 3 7.b odd 2 1
9072.2.a.bq 3 12.b even 2 1
9072.2.a.cd 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(567))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 4T + 1 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 5 T^{2} + 2 T + 11 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 19 T + 47 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 12 T^{2} + 39 T - 27 \) Copy content Toggle raw display
$19$ \( T^{3} + 3 T^{2} - 6 T - 7 \) Copy content Toggle raw display
$23$ \( T^{3} - 33T + 9 \) Copy content Toggle raw display
$29$ \( T^{3} + T^{2} - 4T - 1 \) Copy content Toggle raw display
$31$ \( T^{3} + 3 T^{2} - 24 T + 27 \) Copy content Toggle raw display
$37$ \( T^{3} + 3 T^{2} - 54 T + 81 \) Copy content Toggle raw display
$41$ \( T^{3} - 22 T^{2} + 155 T - 353 \) Copy content Toggle raw display
$43$ \( T^{3} + 3 T^{2} - 66 T + 121 \) Copy content Toggle raw display
$47$ \( T^{3} - 9 T^{2} - 54 T + 189 \) Copy content Toggle raw display
$53$ \( T^{3} - 18 T^{2} + 75 T - 9 \) Copy content Toggle raw display
$59$ \( T^{3} - 9 T^{2} - 6 T + 63 \) Copy content Toggle raw display
$61$ \( T^{3} + 6 T^{2} - 21 T - 67 \) Copy content Toggle raw display
$67$ \( T^{3} - 207T + 683 \) Copy content Toggle raw display
$71$ \( T^{3} + 9 T^{2} - 6 T - 81 \) Copy content Toggle raw display
$73$ \( T^{3} - 3 T^{2} - 168 T - 243 \) Copy content Toggle raw display
$79$ \( T^{3} - 15 T^{2} - 48 T + 769 \) Copy content Toggle raw display
$83$ \( T^{3} - 12 T^{2} + 39 T - 27 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 151 T - 379 \) Copy content Toggle raw display
$97$ \( T^{3} - 3 T^{2} - 114 T + 603 \) Copy content Toggle raw display
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