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Properties

Label 567.2.o.e
Level $567$
Weight $2$
Character orbit 567.o
Analytic conductor $4.528$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,2,Mod(188,567)]
 
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([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.188");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.o (of order \(6\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{4} + \beta_{2}) q^{5} + (\beta_{7} - \beta_{6} - \beta_1) q^{7} + (2 \beta_{5} - 2 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{4} + \beta_{2}) q^{5} + (\beta_{7} - \beta_{6} - \beta_1) q^{7} + (2 \beta_{5} - 2 \beta_{3}) q^{8} + \beta_{7} q^{10} + \beta_{3} q^{11} + \beta_{6} q^{13} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{14} + 4 \beta_1 q^{16} - 3 \beta_{4} q^{17} + 3 \beta_{7} q^{19} + ( - 2 \beta_1 + 2) q^{22} + 2 \beta_{5} q^{23} + 2 \beta_1 q^{25} + 2 \beta_{4} q^{26} - 5 \beta_{3} q^{29} + \beta_{6} q^{31} + (3 \beta_{7} - 3 \beta_{6}) q^{34} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3}) q^{35} + 5 q^{37} + 6 \beta_{2} q^{38} - 2 \beta_{6} q^{40} + (5 \beta_{4} - 5 \beta_{2}) q^{41} - 5 \beta_1 q^{43} + 4 q^{46} - 5 \beta_{2} q^{47} + (2 \beta_{6} - 5 \beta_1 + 5) q^{49} + 2 \beta_{5} q^{50} + ( - 8 \beta_{5} + 8 \beta_{3}) q^{53} + \beta_{7} q^{55} + (2 \beta_{3} - 4 \beta_{2}) q^{56} + (10 \beta_1 - 10) q^{58} + (5 \beta_{4} - 5 \beta_{2}) q^{59} + ( - \beta_{7} + \beta_{6}) q^{61} + 2 \beta_{4} q^{62} - 8 q^{64} - 3 \beta_{3} q^{65} + (2 \beta_1 - 2) q^{67} + ( - \beta_{7} + \beta_{6} - 6 \beta_1) q^{70} + (10 \beta_{5} - 10 \beta_{3}) q^{71} + 5 \beta_{3} q^{74} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{77} + 13 \beta_1 q^{79} - 4 \beta_{4} q^{80} - 5 \beta_{7} q^{82} + \beta_{2} q^{83} + ( - 9 \beta_1 + 9) q^{85} - 5 \beta_{5} q^{86} + 4 \beta_1 q^{88} + 6 \beta_{4} q^{89} + ( - \beta_{7} - 6) q^{91} - 5 \beta_{6} q^{94} - 9 \beta_{5} q^{95} + ( - 7 \beta_{7} + 7 \beta_{6}) q^{97} + ( - 5 \beta_{5} + 4 \beta_{4} + 5 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} + 16 q^{16} + 8 q^{22} + 8 q^{25} + 40 q^{37} - 20 q^{43} + 32 q^{46} + 20 q^{49} - 40 q^{58} - 64 q^{64} - 8 q^{67} - 24 q^{70} + 52 q^{79} + 36 q^{85} + 16 q^{88} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{6} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + 3\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + 3\beta_{5} - 3\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} - \beta_{6} + 3\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} - 3\beta_{3} ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1
−0.965926 + 0.258819i
−0.258819 − 0.965926i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.965926 − 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 − 0.965926i
−1.22474 − 0.707107i 0 0 −0.866025 − 1.50000i 0 1.62132 + 2.09077i 2.82843i 0 2.44949i
188.2 −1.22474 − 0.707107i 0 0 0.866025 + 1.50000i 0 −2.62132 − 0.358719i 2.82843i 0 − 2.44949i
188.3 1.22474 + 0.707107i 0 0 −0.866025 − 1.50000i 0 −2.62132 − 0.358719i − 2.82843i 0 − 2.44949i
188.4 1.22474 + 0.707107i 0 0 0.866025 + 1.50000i 0 1.62132 + 2.09077i − 2.82843i 0 2.44949i
377.1 −1.22474 + 0.707107i 0 0 −0.866025 + 1.50000i 0 1.62132 − 2.09077i − 2.82843i 0 − 2.44949i
377.2 −1.22474 + 0.707107i 0 0 0.866025 − 1.50000i 0 −2.62132 + 0.358719i − 2.82843i 0 2.44949i
377.3 1.22474 − 0.707107i 0 0 −0.866025 + 1.50000i 0 −2.62132 + 0.358719i 2.82843i 0 2.44949i
377.4 1.22474 − 0.707107i 0 0 0.866025 − 1.50000i 0 1.62132 − 2.09077i 2.82843i 0 − 2.44949i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
21.c even 2 1 inner
63.l odd 6 1 inner
63.o 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.o.e 8
3.b odd 2 1 inner 567.2.o.e 8
7.b odd 2 1 inner 567.2.o.e 8
9.c even 3 1 189.2.c.c ✓ 4
9.c even 3 1 inner 567.2.o.e 8
9.d odd 6 1 189.2.c.c ✓ 4
9.d odd 6 1 inner 567.2.o.e 8
21.c even 2 1 inner 567.2.o.e 8
36.f odd 6 1 3024.2.k.g 4
36.h even 6 1 3024.2.k.g 4
63.l odd 6 1 189.2.c.c ✓ 4
63.l odd 6 1 inner 567.2.o.e 8
63.o even 6 1 189.2.c.c ✓ 4
63.o even 6 1 inner 567.2.o.e 8
252.s odd 6 1 3024.2.k.g 4
252.bi even 6 1 3024.2.k.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.c.c ✓ 4 9.c even 3 1
189.2.c.c ✓ 4 9.d odd 6 1
189.2.c.c ✓ 4 63.l odd 6 1
189.2.c.c ✓ 4 63.o even 6 1
567.2.o.e 8 1.a even 1 1 trivial
567.2.o.e 8 3.b odd 2 1 inner
567.2.o.e 8 7.b odd 2 1 inner
567.2.o.e 8 9.c even 3 1 inner
567.2.o.e 8 9.d odd 6 1 inner
567.2.o.e 8 21.c even 2 1 inner
567.2.o.e 8 63.l odd 6 1 inner
567.2.o.e 8 63.o even 6 1 inner
3024.2.k.g 4 36.f odd 6 1
3024.2.k.g 4 36.h even 6 1
3024.2.k.g 4 252.s odd 6 1
3024.2.k.g 4 252.bi even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} - 3 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 27)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 54)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 50 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$37$ \( (T - 5)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 75 T^{2} + 5625)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 5 T + 25)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 75 T^{2} + 5625)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 128)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 75 T^{2} + 5625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 200)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 13 T + 169)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 294 T^{2} + 86436)^{2} \) Copy content Toggle raw display
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