Properties

Label 384.3.h.d
Level $384$
Weight $3$
Character orbit 384.h
Self dual yes
Analytic conductor $10.463$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,3,Mod(65,384)]
 
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([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.h (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: yes
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 4\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + \beta q^{5} - \beta q^{7} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + \beta q^{5} - \beta q^{7} + 9 q^{9} + 10 q^{11} + 3 \beta q^{15} - 3 \beta q^{21} + 71 q^{25} + 27 q^{27} - 3 \beta q^{29} - 5 \beta q^{31} + 30 q^{33} - 96 q^{35} + 9 \beta q^{45} + 47 q^{49} + 5 \beta q^{53} + 10 \beta q^{55} - 10 q^{59} - 9 \beta q^{63} - 50 q^{73} + 213 q^{75} - 10 \beta q^{77} + 15 \beta q^{79} + 81 q^{81} - 134 q^{83} - 9 \beta q^{87} - 15 \beta q^{93} - 190 q^{97} + 90 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 18 q^{9} + 20 q^{11} + 142 q^{25} + 54 q^{27} + 60 q^{33} - 192 q^{35} + 94 q^{49} - 20 q^{59} - 100 q^{73} + 426 q^{75} + 162 q^{81} - 268 q^{83} - 380 q^{97} + 180 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−2.44949
2.44949
0 3.00000 0 −9.79796 0 9.79796 0 9.00000 0
65.2 0 3.00000 0 9.79796 0 −9.79796 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
8.d odd 2 1 inner
12.b even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} - 96 \) Copy content Toggle raw display
\( T_{11} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 96 \) Copy content Toggle raw display
$7$ \( T^{2} - 96 \) Copy content Toggle raw display
$11$ \( (T - 10)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 864 \) Copy content Toggle raw display
$31$ \( T^{2} - 2400 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2400 \) Copy content Toggle raw display
$59$ \( (T + 10)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 50)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 21600 \) Copy content Toggle raw display
$83$ \( (T + 134)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 190)^{2} \) Copy content Toggle raw display
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