Properties

Label 384.8.a.p
Level $384$
Weight $8$
Character orbit 384.a
Self dual yes
Analytic conductor $119.956$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,8,Mod(1,384)]
 
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, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.a (trivial)

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: \(119.955849786\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 620x^{2} - 700x + 83625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - 27 q^{3} + ( - \beta_{2} + 84) q^{5} + (\beta_{3} + 170) q^{7} + 729 q^{9} + ( - \beta_{3} + 6 \beta_{2} + \cdots - 964) q^{11} + ( - \beta_{3} - 9 \beta_{2} + \cdots + 2670) q^{13} + (27 \beta_{2} - 2268) q^{15}+ \cdots + ( - 729 \beta_{3} + 4374 \beta_{2} + \cdots - 702756) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 108 q^{3} + 336 q^{5} + 680 q^{7} + 2916 q^{9} - 3856 q^{11} + 10680 q^{13} - 9072 q^{15} + 26232 q^{17} + 15456 q^{19} - 18360 q^{21} - 11312 q^{23} + 159052 q^{25} - 78732 q^{27} - 1856 q^{29} + 71752 q^{31}+ \cdots - 2811024 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 620x^{2} - 700x + 83625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 16\nu^{3} - 100\nu^{2} + 680\nu + 22600 ) / 25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\nu^{3} - 220\nu^{2} - 5320\nu + 59800 ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 48\nu^{3} - 1020\nu^{2} - 14760\nu + 291000 ) / 25 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 6\beta_{2} + 3\beta_1 ) / 768 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -25\beta_{3} + 70\beta_{2} + 5\beta _1 + 119040 ) / 384 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -355\beta_{3} + 1130\beta_{2} + 1135\beta _1 + 403200 ) / 768 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−16.8500
21.8313
−17.7724
12.7912
0 −27.0000 0 −333.320 0 988.659 0 729.000 0
1.2 0 −27.0000 0 −127.307 0 −547.276 0 729.000 0
1.3 0 −27.0000 0 282.264 0 −1362.23 0 729.000 0
1.4 0 −27.0000 0 514.363 0 1600.84 0 729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.8.a.p yes 4
4.b odd 2 1 384.8.a.t yes 4
8.b even 2 1 384.8.a.q yes 4
8.d odd 2 1 384.8.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.8.a.m 4 8.d odd 2 1
384.8.a.p yes 4 1.a even 1 1 trivial
384.8.a.q yes 4 8.b even 2 1
384.8.a.t yes 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(384))\):

\( T_{5}^{4} - 336T_{5}^{3} - 179328T_{5}^{2} + 33072640T_{5} + 6160819200 \) Copy content Toggle raw display
\( T_{7}^{4} - 680T_{7}^{3} - 2616456T_{7}^{2} + 1091635168T_{7} + 1179914752144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T + 27)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 6160819200 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 1179914752144 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 616855133477120 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots - 85261472264688 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 66\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 78\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 99\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 14\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 31\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 93\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 14\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 27\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 21\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 53\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 84\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 39\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
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