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Properties

Label 576.2.bb.d
Level $576$
Weight $2$
Character orbit 576.bb
Analytic conductor $4.599$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,2,Mod(49,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(12))
 
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("576.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.bb (of order \(12\), degree \(4\), 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: \(4\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} + 2) q^{3} + (2 \zeta_{12}^{3} - 2 \zeta_{12} - 2) q^{5} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{7} + ( - 3 \zeta_{12}^{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} + 2) q^{3} + (2 \zeta_{12}^{3} - 2 \zeta_{12} - 2) q^{5} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{7} + ( - 3 \zeta_{12}^{2} + 3) q^{9} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} + 3) q^{11} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} - 1) q^{13} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{15} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 4) q^{17} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} + 1) q^{19} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - \zeta_{12} + 3) q^{21} + ( - \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{23} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12} + 8) q^{25} + ( - 6 \zeta_{12}^{2} + 3) q^{27} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 2) q^{29} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{31} + (\zeta_{12}^{3} - 4 \zeta_{12}^{2} - 5 \zeta_{12} + 5) q^{33} + (2 \zeta_{12}^{3} - 2) q^{35} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{37} + ( - 5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 5) q^{39} - 3 \zeta_{12} q^{41} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 5 \zeta_{12} + 3) q^{43} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6) q^{45} + (5 \zeta_{12}^{3} + \zeta_{12}^{2} + 5 \zeta_{12}) q^{47} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{49} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 3 \zeta_{12} - 8) q^{51} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} - 6) q^{53} + (8 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{55} + ( - 3 \zeta_{12} + 3) q^{57} + ( - \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{59} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6) q^{61} + (3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{63} + (6 \zeta_{12}^{3} + 14 \zeta_{12}^{2} + 6 \zeta_{12}) q^{65} + ( - 5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{67} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} - 3) q^{69} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{71} + (8 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{73} + ( - 6 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 3 \zeta_{12} + 12) q^{75} + (4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 7 \zeta_{12} + 7) q^{77} + 12 \zeta_{12}^{2} q^{79} - 9 \zeta_{12}^{2} q^{81} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{83} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{85} + (3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{87} + 2 \zeta_{12}^{3} q^{89} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 6) q^{91} + ( - 6 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{93} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{95} + (\zeta_{12}^{3} - 10 \zeta_{12}^{2} + \zeta_{12}) q^{97} + (6 \zeta_{12}^{3} - 9 \zeta_{12}^{2} - 9 \zeta_{12} + 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 8 q^{5} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} - 8 q^{5} + 6 q^{7} + 6 q^{9} + 10 q^{11} - 10 q^{13} - 12 q^{15} - 16 q^{17} + 6 q^{19} + 6 q^{21} - 6 q^{23} + 24 q^{25} - 6 q^{29} + 8 q^{31} + 12 q^{33} - 8 q^{35} + 12 q^{37} - 24 q^{39} + 16 q^{43} - 12 q^{45} + 2 q^{47} - 6 q^{49} - 24 q^{51} - 16 q^{53} + 12 q^{57} - 6 q^{59} - 12 q^{61} + 28 q^{65} - 16 q^{67} - 12 q^{69} + 24 q^{75} + 22 q^{77} + 24 q^{79} - 18 q^{81} + 2 q^{83} + 20 q^{85} - 6 q^{87} - 20 q^{91} - 20 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(1\) \(\zeta_{12}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
0 1.50000 − 0.866025i 0 −3.73205 + 1.00000i 0 0.633975 − 0.366025i 0 1.50000 − 2.59808i 0
241.1 0 1.50000 + 0.866025i 0 −0.267949 + 1.00000i 0 2.36603 + 1.36603i 0 1.50000 + 2.59808i 0
337.1 0 1.50000 − 0.866025i 0 −0.267949 − 1.00000i 0 2.36603 − 1.36603i 0 1.50000 − 2.59808i 0
529.1 0 1.50000 + 0.866025i 0 −3.73205 − 1.00000i 0 0.633975 + 0.366025i 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
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.bb.d 4
3.b odd 2 1 1728.2.bc.d 4
4.b odd 2 1 144.2.x.b ✓ 4
9.c even 3 1 576.2.bb.c 4
9.d odd 6 1 1728.2.bc.a 4
12.b even 2 1 432.2.y.c 4
16.e even 4 1 576.2.bb.c 4
16.f odd 4 1 144.2.x.c yes 4
36.f odd 6 1 144.2.x.c yes 4
36.h even 6 1 432.2.y.b 4
48.i odd 4 1 1728.2.bc.a 4
48.k even 4 1 432.2.y.b 4
144.u even 12 1 432.2.y.c 4
144.v odd 12 1 144.2.x.b ✓ 4
144.w odd 12 1 1728.2.bc.d 4
144.x even 12 1 inner 576.2.bb.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.b ✓ 4 4.b odd 2 1
144.2.x.b ✓ 4 144.v odd 12 1
144.2.x.c yes 4 16.f odd 4 1
144.2.x.c yes 4 36.f odd 6 1
432.2.y.b 4 36.h even 6 1
432.2.y.b 4 48.k even 4 1
432.2.y.c 4 12.b even 2 1
432.2.y.c 4 144.u even 12 1
576.2.bb.c 4 9.c even 3 1
576.2.bb.c 4 16.e even 4 1
576.2.bb.d 4 1.a even 1 1 trivial
576.2.bb.d 4 144.x even 12 1 inner
1728.2.bc.a 4 9.d odd 6 1
1728.2.bc.a 4 48.i odd 4 1
1728.2.bc.d 4 3.b odd 2 1
1728.2.bc.d 4 144.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 8T_{5}^{3} + 20T_{5}^{2} + 16T_{5} + 16 \) 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
$3$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 8 T^{3} + 20 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + 14 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$11$ \( T^{4} - 10 T^{3} + 41 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{4} + 10 T^{3} + 74 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$17$ \( (T^{2} + 8 T + 13)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + 18 T^{2} - 18 T + 9 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 18 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$41$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$43$ \( T^{4} - 16 T^{3} + 65 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} + 78 T^{2} + \cdots + 5476 \) Copy content Toggle raw display
$53$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + 45 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + 180 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$67$ \( T^{4} + 16 T^{3} + 113 T^{2} + \cdots + 1369 \) Copy content Toggle raw display
$71$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$73$ \( T^{4} + 134T^{2} + 3721 \) Copy content Toggle raw display
$79$ \( (T^{2} - 12 T + 144)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$89$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 20 T^{3} + 303 T^{2} + \cdots + 9409 \) Copy content Toggle raw display
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