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Properties

Label 576.2.r.a
Level $576$
Weight $2$
Character orbit 576.r
Analytic conductor $4.599$
Analytic rank $1$
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(97,576)]
 
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([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.r (of order \(6\), degree \(2\), 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: \(1\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{2} - 2) q^{5} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{2} - 2) q^{5} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{11} + ( - 2 \zeta_{12}^{2} - 2) q^{13} + 6 \zeta_{12}^{3} q^{15} - 3 q^{17} + 7 \zeta_{12}^{3} q^{19} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{23} + 7 \zeta_{12}^{2} q^{25} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + (4 \zeta_{12}^{2} - 8) q^{29} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{31} + ( - 3 \zeta_{12}^{2} - 3) q^{33} + (12 \zeta_{12}^{2} - 6) q^{37} + 6 \zeta_{12}^{3} q^{39} + (3 \zeta_{12}^{2} - 3) q^{41} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{43} + ( - 6 \zeta_{12}^{2} + 12) q^{45} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{47} + ( - 7 \zeta_{12}^{2} + 7) q^{49} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{51} + ( - 16 \zeta_{12}^{2} + 8) q^{53} + (6 \zeta_{12}^{3} - 12 \zeta_{12}) q^{55} + ( - 7 \zeta_{12}^{2} + 14) q^{57} - 9 \zeta_{12} q^{59} + (4 \zeta_{12}^{2} - 8) q^{61} + 12 \zeta_{12}^{2} q^{65} - 5 \zeta_{12} q^{67} - 6 q^{69} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{71} - 7 q^{73} + ( - 14 \zeta_{12}^{3} + 7 \zeta_{12}) q^{75} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}) q^{79} - 9 \zeta_{12}^{2} q^{81} + (12 \zeta_{12}^{3} - 12 \zeta_{12}) q^{83} + (6 \zeta_{12}^{2} + 6) q^{85} + 12 \zeta_{12} q^{87} - 6 q^{89} + 12 q^{93} + ( - 28 \zeta_{12}^{3} + 14 \zeta_{12}) q^{95} - \zeta_{12}^{2} q^{97} + 9 \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{5} - 6 q^{9} - 12 q^{13} - 12 q^{17} + 14 q^{25} - 24 q^{29} - 18 q^{33} - 6 q^{41} + 36 q^{45} + 14 q^{49} + 42 q^{57} - 24 q^{61} + 24 q^{65} - 24 q^{69} - 28 q^{73} - 18 q^{81} + 36 q^{85} - 24 q^{89} + 48 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0 −0.866025 + 1.50000i 0 −3.00000 + 1.73205i 0 0 0 −1.50000 − 2.59808i 0
97.2 0 0.866025 − 1.50000i 0 −3.00000 + 1.73205i 0 0 0 −1.50000 − 2.59808i 0
481.1 0 −0.866025 − 1.50000i 0 −3.00000 − 1.73205i 0 0 0 −1.50000 + 2.59808i 0
481.2 0 0.866025 + 1.50000i 0 −3.00000 − 1.73205i 0 0 0 −1.50000 + 2.59808i 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
72.n even 6 1 inner
72.p odd 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 6T_{5} + 12 \) acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T + 3)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$37$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$47$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$53$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$71$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 7)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 300 T^{2} + 90000 \) Copy content Toggle raw display
$83$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
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