Properties

Label 384.3.b.a
Level $384$
Weight $3$
Character orbit 384.b
Analytic conductor $10.463$
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] = [384,3,Mod(319,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), 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: \(10.4632421514\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} + 3 q^{9} + 4 \beta_1 q^{11} + \beta_{2} q^{15} - 18 q^{17} - 12 \beta_1 q^{19} + 3 \beta_{3} q^{21} + 6 \beta_{2} q^{23} + 9 q^{25} + 3 \beta_1 q^{27} + \beta_{3} q^{29} + 7 \beta_{2} q^{31} + 12 q^{33} - 16 \beta_1 q^{35} - 18 \beta_{3} q^{37} + 18 q^{41} + 36 \beta_1 q^{43} + 3 \beta_{3} q^{45} + 6 \beta_{2} q^{47} + q^{49} - 18 \beta_1 q^{51} + 11 \beta_{3} q^{53} + 4 \beta_{2} q^{55} - 36 q^{57} + 36 \beta_1 q^{59} - 18 \beta_{3} q^{61} + 3 \beta_{2} q^{63} - 12 \beta_1 q^{67} + 18 \beta_{3} q^{69} - 6 \beta_{2} q^{71} - 82 q^{73} + 9 \beta_1 q^{75} + 12 \beta_{3} q^{77} - 9 \beta_{2} q^{79} + 9 q^{81} - 76 \beta_1 q^{83} - 18 \beta_{3} q^{85} + \beta_{2} q^{87} + 126 q^{89} + 21 \beta_{3} q^{93} - 12 \beta_{2} q^{95} + 110 q^{97} + 12 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} - 72 q^{17} + 36 q^{25} + 48 q^{33} + 72 q^{41} + 4 q^{49} - 144 q^{57} - 328 q^{73} + 36 q^{81} + 504 q^{89} + 440 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 8\zeta_{12}^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 4\beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 4 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0 −1.73205 0 4.00000i 0 6.92820i 0 3.00000 0
319.2 0 −1.73205 0 4.00000i 0 6.92820i 0 3.00000 0
319.3 0 1.73205 0 4.00000i 0 6.92820i 0 3.00000 0
319.4 0 1.73205 0 4.00000i 0 6.92820i 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
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.b.a 4
3.b odd 2 1 1152.3.b.h 4
4.b odd 2 1 inner 384.3.b.a 4
8.b even 2 1 inner 384.3.b.a 4
8.d odd 2 1 inner 384.3.b.a 4
12.b even 2 1 1152.3.b.h 4
16.e even 4 1 768.3.g.a 2
16.e even 4 1 768.3.g.b 2
16.f odd 4 1 768.3.g.a 2
16.f odd 4 1 768.3.g.b 2
24.f even 2 1 1152.3.b.h 4
24.h odd 2 1 1152.3.b.h 4
48.i odd 4 1 2304.3.g.g 2
48.i odd 4 1 2304.3.g.n 2
48.k even 4 1 2304.3.g.g 2
48.k even 4 1 2304.3.g.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.b.a 4 1.a even 1 1 trivial
384.3.b.a 4 4.b odd 2 1 inner
384.3.b.a 4 8.b even 2 1 inner
384.3.b.a 4 8.d odd 2 1 inner
768.3.g.a 2 16.e even 4 1
768.3.g.a 2 16.f odd 4 1
768.3.g.b 2 16.e even 4 1
768.3.g.b 2 16.f odd 4 1
1152.3.b.h 4 3.b odd 2 1
1152.3.b.h 4 12.b even 2 1
1152.3.b.h 4 24.f even 2 1
1152.3.b.h 4 24.h odd 2 1
2304.3.g.g 2 48.i odd 4 1
2304.3.g.g 2 48.k even 4 1
2304.3.g.n 2 48.i odd 4 1
2304.3.g.n 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 16 \) acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T + 18)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 432)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1728)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2352)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 5184)^{2} \) Copy content Toggle raw display
$41$ \( (T - 18)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 3888)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 1728)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 1936)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 3888)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 5184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 432)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1728)^{2} \) Copy content Toggle raw display
$73$ \( (T + 82)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 3888)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 17328)^{2} \) Copy content Toggle raw display
$89$ \( (T - 126)^{4} \) Copy content Toggle raw display
$97$ \( (T - 110)^{4} \) Copy content Toggle raw display
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