Properties

Label 384.7.h.a
Level $384$
Weight $7$
Character orbit 384.h
Self dual yes
Analytic conductor $88.341$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.65");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(88.3407681100\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(\beta = 12\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 27 q^{3} + 7 \beta q^{5} - 17 \beta q^{7} + 729 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 27 q^{3} + 7 \beta q^{5} - 17 \beta q^{7} + 729 q^{9} + 2630 q^{11} - 189 \beta q^{15} + 459 \beta q^{21} + 26711 q^{25} - 19683 q^{27} + 1659 \beta q^{29} - 805 \beta q^{31} - 71010 q^{33} - 102816 q^{35} + 5103 \beta q^{45} + 132047 q^{49} - 10045 \beta q^{53} + 18410 \beta q^{55} - 103430 q^{59} - 12393 \beta q^{63} + 674350 q^{73} - 721197 q^{75} - 44710 \beta q^{77} - 14385 \beta q^{79} + 531441 q^{81} - 363274 q^{83} - 44793 \beta q^{87} + 21735 \beta q^{93} - 1495870 q^{97} + 1917270 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 54 q^{3} + 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 54 q^{3} + 1458 q^{9} + 5260 q^{11} + 53422 q^{25} - 39366 q^{27} - 142020 q^{33} - 205632 q^{35} + 264094 q^{49} - 206860 q^{59} + 1348700 q^{73} - 1442394 q^{75} + 1062882 q^{81} - 726548 q^{83} - 2991740 q^{97} + 3834540 q^{99}+O(q^{100}) \) 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 −27.0000 0 −205.757 0 499.696 0 729.000 0
65.2 0 −27.0000 0 205.757 0 −499.696 0 729.000 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 384.7.h.a 2
3.b odd 2 1 384.7.h.d yes 2
4.b odd 2 1 384.7.h.d yes 2
8.b even 2 1 384.7.h.d yes 2
8.d odd 2 1 inner 384.7.h.a 2
12.b even 2 1 inner 384.7.h.a 2
24.f even 2 1 384.7.h.d yes 2
24.h odd 2 1 CM 384.7.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.7.h.a 2 1.a even 1 1 trivial
384.7.h.a 2 8.d odd 2 1 inner
384.7.h.a 2 12.b even 2 1 inner
384.7.h.a 2 24.h odd 2 1 CM
384.7.h.d yes 2 3.b odd 2 1
384.7.h.d yes 2 4.b odd 2 1
384.7.h.d yes 2 8.b even 2 1
384.7.h.d yes 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} - 42336 \) Copy content Toggle raw display
\( T_{11} - 2630 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 42336 \) Copy content Toggle raw display
$7$ \( T^{2} - 249696 \) Copy content Toggle raw display
$11$ \( (T - 2630)^{2} \) 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
$29$ \( T^{2} - 2377970784 \) Copy content Toggle raw display
$31$ \( T^{2} - 559893600 \) 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
$53$ \( T^{2} - 87179349600 \) Copy content Toggle raw display
$59$ \( (T + 103430)^{2} \) 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
$73$ \( (T - 674350)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 178785986400 \) Copy content Toggle raw display
$83$ \( (T + 363274)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 1495870)^{2} \) Copy content Toggle raw display
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