Properties

Label 567.2.c.a
Level $567$
Weight $2$
Character orbit 567.c
Analytic conductor $4.528$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,2,Mod(566,567)]
 
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([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.566");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.c (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.12745506816.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 16x^{6} + 79x^{4} + 120x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{4} q^{7} + (\beta_{3} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{4} q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{11} + ( - \beta_{5} - \beta_1) q^{14} + (\beta_{6} - 2 \beta_{2} + 5) q^{16} + (\beta_{6} - \beta_{4} - 2) q^{22} + (\beta_{7} - \beta_{5} + \beta_1) q^{23} - 5 q^{25} + (\beta_{6} - 2 \beta_{4} + 4) q^{28} + ( - \beta_{7} + \beta_{5} + 2 \beta_{3} - \beta_1) q^{29} + (\beta_{7} + \beta_{5} - 3 \beta_{3} + 4 \beta_1) q^{32} + (3 \beta_{4} - 2 \beta_{2}) q^{37} + ( - 3 \beta_{4} + 4 \beta_{2}) q^{43} + ( - \beta_{7} + 2 \beta_{5} - \beta_{3}) q^{44} + (\beta_{6} - 3 \beta_{4} + \beta_{2} - 5) q^{46} + 7 q^{49} - 5 \beta_1 q^{50} + (\beta_{7} - 2 \beta_{5} - \beta_{3} - 3 \beta_1) q^{53} + (\beta_{7} + \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{56} + (\beta_{6} + 3 \beta_{4} - 5 \beta_{2} + 7) q^{58} + ( - 2 \beta_{6} + 5 \beta_{4} + 4 \beta_{2} - 10) q^{64} + ( - 2 \beta_{6} + 1) q^{67} + (2 \beta_{7} + \beta_{5} + \beta_{3}) q^{71} + ( - 3 \beta_{5} - 2 \beta_{3} + \beta_1) q^{74} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3} + 2 \beta_1) q^{77} + ( - 2 \beta_{6} - 5) q^{79} + (3 \beta_{5} + 4 \beta_{3} - 5 \beta_1) q^{86} + ( - \beta_{6} + 5 \beta_{4} + \beta_{2} - 4) q^{88} + (3 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{92} + 7 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 36 q^{16} - 20 q^{22} - 40 q^{25} + 28 q^{28} - 44 q^{46} + 56 q^{49} + 52 q^{58} - 72 q^{64} + 16 q^{67} - 32 q^{79} - 28 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 16x^{6} + 79x^{4} + 120x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 12\nu^{4} + 36\nu^{2} + 16 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 12\nu^{5} - 36\nu^{3} - 21\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{4} + 8\nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 17\nu^{5} + 91\nu^{3} + 151\nu ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 8\beta_{2} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + \beta_{5} - 11\beta_{3} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{6} + 5\beta_{4} + 60\beta_{2} - 172 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -12\beta_{7} - 17\beta_{5} + 96\beta_{3} - 285\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
566.1
2.81442i
2.29678i
1.65070i
0.281155i
0.281155i
1.65070i
2.29678i
2.81442i
2.81442i 0 −5.92095 0 0 −2.64575 11.0352i 0 0
566.2 2.29678i 0 −3.27520 0 0 2.64575 2.92886i 0 0
566.3 1.65070i 0 −0.724799 0 0 −2.64575 2.10497i 0 0
566.4 0.281155i 0 1.92095 0 0 2.64575 1.10240i 0 0
566.5 0.281155i 0 1.92095 0 0 2.64575 1.10240i 0 0
566.6 1.65070i 0 −0.724799 0 0 −2.64575 2.10497i 0 0
566.7 2.29678i 0 −3.27520 0 0 2.64575 2.92886i 0 0
566.8 2.81442i 0 −5.92095 0 0 −2.64575 11.0352i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 566.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.c.a 8
3.b odd 2 1 inner 567.2.c.a 8
7.b odd 2 1 CM 567.2.c.a 8
9.c even 3 2 567.2.o.h 16
9.d odd 6 2 567.2.o.h 16
21.c even 2 1 inner 567.2.c.a 8
63.l odd 6 2 567.2.o.h 16
63.o even 6 2 567.2.o.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.c.a 8 1.a even 1 1 trivial
567.2.c.a 8 3.b odd 2 1 inner
567.2.c.a 8 7.b odd 2 1 CM
567.2.c.a 8 21.c even 2 1 inner
567.2.o.h 16 9.c even 3 2
567.2.o.h 16 9.d odd 6 2
567.2.o.h 16 63.l odd 6 2
567.2.o.h 16 63.o even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 16T_{2}^{6} + 79T_{2}^{4} + 120T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 16 T^{6} + 79 T^{4} + 120 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 88 T^{6} + 2626 T^{4} + \cdots + 106929 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 92 T^{2} + 324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 116 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 110 T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 230 T^{2} + 10201)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 424 T^{6} + \cdots + 70408881 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T - 185)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + 568 T^{6} + 97906 T^{4} + \cdots + 4524129 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 173)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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