Properties

Label 576.3.o.c
Level $576$
Weight $3$
Character orbit 576.o
Analytic conductor $15.695$
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] = [576,3,Mod(319,576)]
 
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, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.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: \(15.6948632272\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
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.

\(f(q)\) \(=\) \( q - 3 \zeta_{12}^{3} q^{3} + ( - 7 \zeta_{12}^{2} + 7) q^{5} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{7} - 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \zeta_{12}^{3} q^{3} + ( - 7 \zeta_{12}^{2} + 7) q^{5} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{7} - 9 q^{9} + (13 \zeta_{12}^{3} - 13 \zeta_{12}) q^{11} + ( - \zeta_{12}^{2} + 1) q^{13} - 21 \zeta_{12} q^{15} - 24 q^{17} + 24 \zeta_{12}^{3} q^{19} + 15 \zeta_{12}^{2} q^{21} + 13 \zeta_{12} q^{23} - 24 \zeta_{12}^{2} q^{25} + 27 \zeta_{12}^{3} q^{27} - 55 \zeta_{12}^{2} q^{29} - 19 \zeta_{12} q^{31} + 39 \zeta_{12}^{2} q^{33} + 35 \zeta_{12}^{3} q^{35} + 48 q^{37} - 3 \zeta_{12} q^{39} + ( - 7 \zeta_{12}^{2} + 7) q^{41} + ( - 53 \zeta_{12}^{3} + 53 \zeta_{12}) q^{43} + (63 \zeta_{12}^{2} - 63) q^{45} + (37 \zeta_{12}^{3} - 37 \zeta_{12}) q^{47} + (24 \zeta_{12}^{2} - 24) q^{49} + 72 \zeta_{12}^{3} q^{51} + 48 q^{53} + 91 \zeta_{12}^{3} q^{55} + 72 q^{57} - 83 \zeta_{12} q^{59} + 25 \zeta_{12}^{2} q^{61} + ( - 45 \zeta_{12}^{3} + 45 \zeta_{12}) q^{63} - 7 \zeta_{12}^{2} q^{65} - 101 \zeta_{12} q^{67} + ( - 39 \zeta_{12}^{2} + 39) q^{69} - 2 \zeta_{12}^{3} q^{71} - 120 q^{73} + (72 \zeta_{12}^{3} - 72 \zeta_{12}) q^{75} + ( - 65 \zeta_{12}^{2} + 65) q^{77} + (115 \zeta_{12}^{3} - 115 \zeta_{12}) q^{79} + 81 q^{81} + (59 \zeta_{12}^{3} - 59 \zeta_{12}) q^{83} + (168 \zeta_{12}^{2} - 168) q^{85} + (165 \zeta_{12}^{3} - 165 \zeta_{12}) q^{87} + 120 q^{89} + 5 \zeta_{12}^{3} q^{91} + (57 \zeta_{12}^{2} - 57) q^{93} + 168 \zeta_{12} q^{95} - 25 \zeta_{12}^{2} q^{97} + ( - 117 \zeta_{12}^{3} + 117 \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{5} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 14 q^{5} - 36 q^{9} + 2 q^{13} - 96 q^{17} + 30 q^{21} - 48 q^{25} - 110 q^{29} + 78 q^{33} + 192 q^{37} + 14 q^{41} - 126 q^{45} - 48 q^{49} + 192 q^{53} + 288 q^{57} + 50 q^{61} - 14 q^{65} + 78 q^{69} - 480 q^{73} + 130 q^{77} + 324 q^{81} - 336 q^{85} + 480 q^{89} - 114 q^{93} - 50 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 3.00000i 0 3.50000 6.06218i 0 −4.33013 + 2.50000i 0 −9.00000 0
319.2 0 3.00000i 0 3.50000 6.06218i 0 4.33013 2.50000i 0 −9.00000 0
511.1 0 3.00000i 0 3.50000 + 6.06218i 0 4.33013 + 2.50000i 0 −9.00000 0
511.2 0 3.00000i 0 3.50000 + 6.06218i 0 −4.33013 2.50000i 0 −9.00000 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.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.3.o.c 4
3.b odd 2 1 1728.3.o.c 4
4.b odd 2 1 inner 576.3.o.c 4
8.b even 2 1 288.3.o.a 4
8.d odd 2 1 288.3.o.a 4
9.c even 3 1 inner 576.3.o.c 4
9.d odd 6 1 1728.3.o.c 4
12.b even 2 1 1728.3.o.c 4
24.f even 2 1 864.3.o.a 4
24.h odd 2 1 864.3.o.a 4
36.f odd 6 1 inner 576.3.o.c 4
36.h even 6 1 1728.3.o.c 4
72.j odd 6 1 864.3.o.a 4
72.j odd 6 1 2592.3.g.a 2
72.l even 6 1 864.3.o.a 4
72.l even 6 1 2592.3.g.a 2
72.n even 6 1 288.3.o.a 4
72.n even 6 1 2592.3.g.b 2
72.p odd 6 1 288.3.o.a 4
72.p odd 6 1 2592.3.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.o.a 4 8.b even 2 1
288.3.o.a 4 8.d odd 2 1
288.3.o.a 4 72.n even 6 1
288.3.o.a 4 72.p odd 6 1
576.3.o.c 4 1.a even 1 1 trivial
576.3.o.c 4 4.b odd 2 1 inner
576.3.o.c 4 9.c even 3 1 inner
576.3.o.c 4 36.f odd 6 1 inner
864.3.o.a 4 24.f even 2 1
864.3.o.a 4 24.h odd 2 1
864.3.o.a 4 72.j odd 6 1
864.3.o.a 4 72.l even 6 1
1728.3.o.c 4 3.b odd 2 1
1728.3.o.c 4 9.d odd 6 1
1728.3.o.c 4 12.b even 2 1
1728.3.o.c 4 36.h even 6 1
2592.3.g.a 2 72.j odd 6 1
2592.3.g.a 2 72.l even 6 1
2592.3.g.b 2 72.n even 6 1
2592.3.g.b 2 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} - 7T_{5} + 49 \) Copy content Toggle raw display
\( T_{7}^{4} - 25T_{7}^{2} + 625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$11$ \( T^{4} - 169 T^{2} + 28561 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T + 24)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 576)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 169 T^{2} + 28561 \) Copy content Toggle raw display
$29$ \( (T^{2} + 55 T + 3025)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 361 T^{2} + 130321 \) Copy content Toggle raw display
$37$ \( (T - 48)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 2809 T^{2} + \cdots + 7890481 \) Copy content Toggle raw display
$47$ \( T^{4} - 1369 T^{2} + \cdots + 1874161 \) Copy content Toggle raw display
$53$ \( (T - 48)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 6889 T^{2} + \cdots + 47458321 \) Copy content Toggle raw display
$61$ \( (T^{2} - 25 T + 625)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 10201 T^{2} + \cdots + 104060401 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T + 120)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - 13225 T^{2} + \cdots + 174900625 \) Copy content Toggle raw display
$83$ \( T^{4} - 3481 T^{2} + \cdots + 12117361 \) Copy content Toggle raw display
$89$ \( (T - 120)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 25 T + 625)^{2} \) Copy content Toggle raw display
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