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Properties

Label 576.2.s.e
Level $576$
Weight $2$
Character orbit 576.s
Analytic conductor $4.599$
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] = [576,2,Mod(191,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.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: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 144)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_1 - 2) q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_1 - 2) q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} + 3 q^{9} + (3 \beta_{3} - 3 \beta_{2}) q^{11} + \beta_1 q^{13} + (2 \beta_{3} - \beta_{2}) q^{15} + (4 \beta_1 - 2) q^{17} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{19} + (3 \beta_1 + 3) q^{21} + 3 \beta_{2} q^{23} + (2 \beta_1 - 2) q^{25} - 3 \beta_{3} q^{27} + (5 \beta_1 + 5) q^{29} + ( - 2 \beta_{3} + \beta_{2}) q^{31} + (9 \beta_1 - 9) q^{33} + 3 \beta_{3} q^{35} + 4 q^{37} - \beta_{2} q^{39} + ( - 3 \beta_1 + 6) q^{41} + (\beta_{3} + \beta_{2}) q^{43} + (3 \beta_1 - 6) q^{45} + (3 \beta_{3} - 3 \beta_{2}) q^{47} + 2 \beta_1 q^{49} + (2 \beta_{3} - 4 \beta_{2}) q^{51} + ( - 12 \beta_1 + 6) q^{53} + ( - 3 \beta_{3} + 6 \beta_{2}) q^{55} + ( - 12 \beta_1 + 6) q^{57} + 3 \beta_{2} q^{59} + (7 \beta_1 - 7) q^{61} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{63} + ( - \beta_1 - 1) q^{65} + ( - 6 \beta_{3} + 3 \beta_{2}) q^{67} - 9 \beta_1 q^{69} + 6 \beta_{3} q^{71} + 4 q^{73} + (2 \beta_{3} - 2 \beta_{2}) q^{75} + (9 \beta_1 - 18) q^{77} + (5 \beta_{3} + 5 \beta_{2}) q^{79} + 9 q^{81} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{83} - 6 \beta_1 q^{85} + ( - 5 \beta_{3} - 5 \beta_{2}) q^{87} + ( - 4 \beta_1 + 2) q^{89} + (\beta_{3} - 2 \beta_{2}) q^{91} + ( - 3 \beta_1 + 6) q^{93} - 6 \beta_{2} q^{95} + (\beta_1 - 1) q^{97} + (9 \beta_{3} - 9 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{5} + 12 q^{9} + 2 q^{13} + 18 q^{21} - 4 q^{25} + 30 q^{29} - 18 q^{33} + 16 q^{37} + 18 q^{41} - 18 q^{45} + 4 q^{49} - 14 q^{61} - 6 q^{65} - 18 q^{69} + 16 q^{73} - 54 q^{77} + 36 q^{81} - 12 q^{85} + 18 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(\beta_{1}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 −1.73205 0 −1.50000 + 0.866025i 0 −2.59808 − 1.50000i 0 3.00000 0
191.2 0 1.73205 0 −1.50000 + 0.866025i 0 2.59808 + 1.50000i 0 3.00000 0
383.1 0 −1.73205 0 −1.50000 − 0.866025i 0 −2.59808 + 1.50000i 0 3.00000 0
383.2 0 1.73205 0 −1.50000 − 0.866025i 0 2.59808 − 1.50000i 0 3.00000 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
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.s.e 4
3.b odd 2 1 1728.2.s.e 4
4.b odd 2 1 inner 576.2.s.e 4
8.b even 2 1 144.2.s.e ✓ 4
8.d odd 2 1 144.2.s.e ✓ 4
9.c even 3 1 1728.2.s.e 4
9.c even 3 1 5184.2.c.f 4
9.d odd 6 1 inner 576.2.s.e 4
9.d odd 6 1 5184.2.c.f 4
12.b even 2 1 1728.2.s.e 4
24.f even 2 1 432.2.s.e 4
24.h odd 2 1 432.2.s.e 4
36.f odd 6 1 1728.2.s.e 4
36.f odd 6 1 5184.2.c.f 4
36.h even 6 1 inner 576.2.s.e 4
36.h even 6 1 5184.2.c.f 4
72.j odd 6 1 144.2.s.e ✓ 4
72.j odd 6 1 1296.2.c.f 4
72.l even 6 1 144.2.s.e ✓ 4
72.l even 6 1 1296.2.c.f 4
72.n even 6 1 432.2.s.e 4
72.n even 6 1 1296.2.c.f 4
72.p odd 6 1 432.2.s.e 4
72.p odd 6 1 1296.2.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.e ✓ 4 8.b even 2 1
144.2.s.e ✓ 4 8.d odd 2 1
144.2.s.e ✓ 4 72.j odd 6 1
144.2.s.e ✓ 4 72.l even 6 1
432.2.s.e 4 24.f even 2 1
432.2.s.e 4 24.h odd 2 1
432.2.s.e 4 72.n even 6 1
432.2.s.e 4 72.p odd 6 1
576.2.s.e 4 1.a even 1 1 trivial
576.2.s.e 4 4.b odd 2 1 inner
576.2.s.e 4 9.d odd 6 1 inner
576.2.s.e 4 36.h even 6 1 inner
1296.2.c.f 4 72.j odd 6 1
1296.2.c.f 4 72.l even 6 1
1296.2.c.f 4 72.n even 6 1
1296.2.c.f 4 72.p odd 6 1
1728.2.s.e 4 3.b odd 2 1
1728.2.s.e 4 9.c even 3 1
1728.2.s.e 4 12.b even 2 1
1728.2.s.e 4 36.f odd 6 1
5184.2.c.f 4 9.c even 3 1
5184.2.c.f 4 9.d odd 6 1
5184.2.c.f 4 36.f odd 6 1
5184.2.c.f 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 3 \) Copy content Toggle raw display
\( T_{7}^{4} - 9T_{7}^{2} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$11$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$29$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$37$ \( (T - 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$47$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$53$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$61$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$71$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$73$ \( (T - 4)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - 225 T^{2} + 50625 \) Copy content Toggle raw display
$83$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
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