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Properties

Label 567.2.s.e
Level $567$
Weight $2$
Character orbit 567.s
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(26,567)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//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);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.s (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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 \beta_{2} q^{4} + (3 \beta_{2} - 2) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 \beta_{2} q^{4} + (3 \beta_{2} - 2) q^{7} + \beta_{3} q^{8} + 2 \beta_{3} q^{11} + (\beta_{2} + 1) q^{13} + (3 \beta_{3} - 2 \beta_1) q^{14} + ( - \beta_{2} + 1) q^{16} + ( - 4 \beta_{3} + 2 \beta_1) q^{17} + (2 \beta_{2} - 4) q^{19} + (10 \beta_{2} - 10) q^{22} + 2 \beta_{3} q^{23} - 5 q^{25} + (\beta_{3} + \beta_1) q^{26} + (3 \beta_{2} - 9) q^{28} + ( - 2 \beta_{3} + 2 \beta_1) q^{29} + ( - \beta_{2} + 2) q^{31} + ( - 3 \beta_{3} + 3 \beta_1) q^{32} + ( - 10 \beta_{2} + 20) q^{34} - 5 \beta_{2} q^{37} + (2 \beta_{3} - 4 \beta_1) q^{38} + ( - 4 \beta_{3} + 2 \beta_1) q^{41} + 7 \beta_{2} q^{43} + (6 \beta_{3} - 6 \beta_1) q^{44} + (10 \beta_{2} - 10) q^{46} + ( - 4 \beta_{3} + 2 \beta_1) q^{47} + ( - 3 \beta_{2} - 5) q^{49} - 5 \beta_1 q^{50} + (6 \beta_{2} - 3) q^{52} - 2 \beta_1 q^{53} + (\beta_{3} - 3 \beta_1) q^{56} + 10 q^{58} + ( - 2 \beta_{3} - 2 \beta_1) q^{59} + ( - 5 \beta_{2} - 5) q^{61} + ( - \beta_{3} + 2 \beta_1) q^{62} + 13 q^{64} + \beta_{2} q^{67} + ( - 6 \beta_{3} + 12 \beta_1) q^{68} - 4 \beta_{3} q^{71} + (4 \beta_{2} + 4) q^{73} - 5 \beta_{3} q^{74} + ( - 6 \beta_{2} - 6) q^{76} + (2 \beta_{3} - 6 \beta_1) q^{77} + (11 \beta_{2} - 11) q^{79} + ( - 10 \beta_{2} + 20) q^{82} + ( - 2 \beta_{3} - 2 \beta_1) q^{83} + 7 \beta_{3} q^{86} - 10 q^{88} + (4 \beta_{3} + 4 \beta_1) q^{89} + (4 \beta_{2} - 5) q^{91} + (6 \beta_{3} - 6 \beta_1) q^{92} + ( - 10 \beta_{2} + 20) q^{94} + ( - \beta_{2} + 2) q^{97} + ( - 3 \beta_{3} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{4} - 2 q^{7} + 6 q^{13} + 2 q^{16} - 12 q^{19} - 20 q^{22} - 20 q^{25} - 30 q^{28} + 6 q^{31} + 60 q^{34} - 10 q^{37} + 14 q^{43} - 20 q^{46} - 26 q^{49} + 40 q^{58} - 30 q^{61} + 52 q^{64} + 2 q^{67} + 24 q^{73} - 36 q^{76} - 22 q^{79} + 60 q^{82} - 40 q^{88} - 12 q^{91} + 60 q^{94} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(\beta_{2}\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.93649 + 1.11803i
1.93649 − 1.11803i
−1.93649 − 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i 0 1.50000 − 2.59808i 0 0 −0.500000 − 2.59808i 2.23607i 0 0
26.2 1.93649 − 1.11803i 0 1.50000 − 2.59808i 0 0 −0.500000 − 2.59808i − 2.23607i 0 0
458.1 −1.93649 − 1.11803i 0 1.50000 + 2.59808i 0 0 −0.500000 + 2.59808i − 2.23607i 0 0
458.2 1.93649 + 1.11803i 0 1.50000 + 2.59808i 0 0 −0.500000 + 2.59808i 2.23607i 0 0
\(n\): 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.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.e 4
3.b odd 2 1 inner 567.2.s.e 4
7.d odd 6 1 567.2.i.c 4
9.c even 3 1 189.2.p.c ✓ 4
9.c even 3 1 567.2.i.c 4
9.d odd 6 1 189.2.p.c ✓ 4
9.d odd 6 1 567.2.i.c 4
21.g even 6 1 567.2.i.c 4
63.g even 3 1 1323.2.c.b 4
63.i even 6 1 189.2.p.c ✓ 4
63.k odd 6 1 inner 567.2.s.e 4
63.k odd 6 1 1323.2.c.b 4
63.n odd 6 1 1323.2.c.b 4
63.s even 6 1 inner 567.2.s.e 4
63.s even 6 1 1323.2.c.b 4
63.t odd 6 1 189.2.p.c ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.c ✓ 4 9.c even 3 1
189.2.p.c ✓ 4 9.d odd 6 1
189.2.p.c ✓ 4 63.i even 6 1
189.2.p.c ✓ 4 63.t odd 6 1
567.2.i.c 4 7.d odd 6 1
567.2.i.c 4 9.c even 3 1
567.2.i.c 4 9.d odd 6 1
567.2.i.c 4 21.g even 6 1
567.2.s.e 4 1.a even 1 1 trivial
567.2.s.e 4 3.b odd 2 1 inner
567.2.s.e 4 63.k odd 6 1 inner
567.2.s.e 4 63.s even 6 1 inner
1323.2.c.b 4 63.g even 3 1
1323.2.c.b 4 63.k odd 6 1
1323.2.c.b 4 63.n odd 6 1
1323.2.c.b 4 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 \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{4} - 5T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{11}^{2} + 20 \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 60T^{2} + 3600 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 20T^{2} + 400 \) Copy content Toggle raw display
$31$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 60T^{2} + 3600 \) Copy content Toggle raw display
$43$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 60T^{2} + 3600 \) Copy content Toggle raw display
$53$ \( T^{4} - 20T^{2} + 400 \) Copy content Toggle raw display
$59$ \( T^{4} + 60T^{2} + 3600 \) Copy content Toggle raw display
$61$ \( (T^{2} + 15 T + 75)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 12 T + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 60T^{2} + 3600 \) Copy content Toggle raw display
$89$ \( T^{4} + 240 T^{2} + 57600 \) Copy content Toggle raw display
$97$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
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