Properties

Label 576.3.o.b
Level $576$
Weight $3$
Character orbit 576.o
Analytic conductor $15.695$
Analytic rank $0$
Dimension $2$
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] = [576,3,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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{6} q^{3} + ( - 4 \zeta_{6} + 4) q^{5} + (2 \zeta_{6} - 4) q^{7} + (9 \zeta_{6} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \zeta_{6} q^{3} + ( - 4 \zeta_{6} + 4) q^{5} + (2 \zeta_{6} - 4) q^{7} + (9 \zeta_{6} - 9) q^{9} + ( - 7 \zeta_{6} + 14) q^{11} + (22 \zeta_{6} - 22) q^{13} + 12 q^{15} - 11 q^{17} + (18 \zeta_{6} - 9) q^{19} + ( - 6 \zeta_{6} - 6) q^{21} + (14 \zeta_{6} + 14) q^{23} + 9 \zeta_{6} q^{25} - 27 q^{27} + 34 \zeta_{6} q^{29} + (4 \zeta_{6} + 4) q^{31} + (21 \zeta_{6} + 21) q^{33} + (16 \zeta_{6} - 8) q^{35} + 16 q^{37} - 66 q^{39} + (13 \zeta_{6} - 13) q^{41} + ( - 29 \zeta_{6} + 58) q^{43} + 36 \zeta_{6} q^{45} + ( - 2 \zeta_{6} + 4) q^{47} + (37 \zeta_{6} - 37) q^{49} - 33 \zeta_{6} q^{51} - 52 q^{53} + ( - 56 \zeta_{6} + 28) q^{55} + (27 \zeta_{6} - 54) q^{57} + (31 \zeta_{6} + 31) q^{59} - 16 \zeta_{6} q^{61} + ( - 36 \zeta_{6} + 18) q^{63} + 88 \zeta_{6} q^{65} + ( - 67 \zeta_{6} - 67) q^{67} + (84 \zeta_{6} - 42) q^{69} - 25 q^{73} + (27 \zeta_{6} - 27) q^{75} + (42 \zeta_{6} - 42) q^{77} + ( - 16 \zeta_{6} + 32) q^{79} - 81 \zeta_{6} q^{81} + (20 \zeta_{6} - 40) q^{83} + (44 \zeta_{6} - 44) q^{85} + (102 \zeta_{6} - 102) q^{87} - 2 q^{89} + ( - 88 \zeta_{6} + 44) q^{91} + (24 \zeta_{6} - 12) q^{93} + (36 \zeta_{6} + 36) q^{95} + 43 \zeta_{6} q^{97} + (126 \zeta_{6} - 63) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 4 q^{5} - 6 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + 4 q^{5} - 6 q^{7} - 9 q^{9} + 21 q^{11} - 22 q^{13} + 24 q^{15} - 22 q^{17} - 18 q^{21} + 42 q^{23} + 9 q^{25} - 54 q^{27} + 34 q^{29} + 12 q^{31} + 63 q^{33} + 32 q^{37} - 132 q^{39} - 13 q^{41} + 87 q^{43} + 36 q^{45} + 6 q^{47} - 37 q^{49} - 33 q^{51} - 104 q^{53} - 81 q^{57} + 93 q^{59} - 16 q^{61} + 88 q^{65} - 201 q^{67} - 50 q^{73} - 27 q^{75} - 42 q^{77} + 48 q^{79} - 81 q^{81} - 60 q^{83} - 44 q^{85} - 102 q^{87} - 4 q^{89} + 108 q^{95} + 43 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(-1\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 + 2.59808i 0 2.00000 3.46410i 0 −3.00000 + 1.73205i 0 −4.50000 + 7.79423i 0
511.1 0 1.50000 2.59808i 0 2.00000 + 3.46410i 0 −3.00000 1.73205i 0 −4.50000 7.79423i 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
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.b 2
3.b odd 2 1 1728.3.o.a 2
4.b odd 2 1 576.3.o.a 2
8.b even 2 1 36.3.f.b yes 2
8.d odd 2 1 36.3.f.a 2
9.c even 3 1 576.3.o.a 2
9.d odd 6 1 1728.3.o.b 2
12.b even 2 1 1728.3.o.b 2
24.f even 2 1 108.3.f.b 2
24.h odd 2 1 108.3.f.a 2
36.f odd 6 1 inner 576.3.o.b 2
36.h even 6 1 1728.3.o.a 2
72.j odd 6 1 108.3.f.b 2
72.j odd 6 1 324.3.d.c 2
72.l even 6 1 108.3.f.a 2
72.l even 6 1 324.3.d.c 2
72.n even 6 1 36.3.f.a 2
72.n even 6 1 324.3.d.b 2
72.p odd 6 1 36.3.f.b yes 2
72.p odd 6 1 324.3.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.a 2 8.d odd 2 1
36.3.f.a 2 72.n even 6 1
36.3.f.b yes 2 8.b even 2 1
36.3.f.b yes 2 72.p odd 6 1
108.3.f.a 2 24.h odd 2 1
108.3.f.a 2 72.l even 6 1
108.3.f.b 2 24.f even 2 1
108.3.f.b 2 72.j odd 6 1
324.3.d.b 2 72.n even 6 1
324.3.d.b 2 72.p odd 6 1
324.3.d.c 2 72.j odd 6 1
324.3.d.c 2 72.l even 6 1
576.3.o.a 2 4.b odd 2 1
576.3.o.a 2 9.c even 3 1
576.3.o.b 2 1.a even 1 1 trivial
576.3.o.b 2 36.f odd 6 1 inner
1728.3.o.a 2 3.b odd 2 1
1728.3.o.a 2 36.h even 6 1
1728.3.o.b 2 9.d odd 6 1
1728.3.o.b 2 12.b even 2 1

Hecke kernels

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

\( T_{5}^{2} - 4T_{5} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$11$ \( T^{2} - 21T + 147 \) Copy content Toggle raw display
$13$ \( T^{2} + 22T + 484 \) Copy content Toggle raw display
$17$ \( (T + 11)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 243 \) Copy content Toggle raw display
$23$ \( T^{2} - 42T + 588 \) Copy content Toggle raw display
$29$ \( T^{2} - 34T + 1156 \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$37$ \( (T - 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$43$ \( T^{2} - 87T + 2523 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$53$ \( (T + 52)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 93T + 2883 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$67$ \( T^{2} + 201T + 13467 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 25)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 48T + 768 \) Copy content Toggle raw display
$83$ \( T^{2} + 60T + 1200 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 43T + 1849 \) Copy content Toggle raw display
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