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Properties

Label 384.3.g.b
Level $384$
Weight $3$
Character orbit 384.g
Analytic conductor $10.463$
Analytic rank $0$
Dimension $8$
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] = [384,3,Mod(127,384)]
 
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, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.g (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: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} + 2) q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} + 2) q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} - 3 q^{9} + ( - \beta_{7} - \beta_{4}) q^{11} + (\beta_{6} - \beta_{5} - 6) q^{13} + ( - \beta_{7} + 2 \beta_1) q^{15} + (\beta_{6} + 2 \beta_{5} + 2) q^{17} + (\beta_{7} - 3 \beta_{4} + \cdots - 4 \beta_1) q^{19}+ \cdots + (3 \beta_{7} + 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{5} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{5} - 24 q^{9} - 48 q^{13} + 16 q^{17} - 8 q^{25} + 80 q^{29} + 16 q^{37} + 80 q^{41} - 48 q^{45} - 88 q^{49} - 176 q^{53} + 96 q^{57} + 272 q^{61} - 160 q^{65} - 16 q^{73} - 320 q^{77} + 72 q^{81} - 32 q^{85} - 240 q^{89} + 192 q^{93} + 400 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -4\zeta_{24}^{5} - 4\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{24}^{6} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} - 4\zeta_{24}^{2} - 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -4\zeta_{24}^{7} + 2\zeta_{24}^{6} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -8\zeta_{24}^{7} + 4\zeta_{24}^{5} + 4\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 4\zeta_{24}^{6} - 4\zeta_{24}^{5} + 4\zeta_{24}^{3} - 8\zeta_{24}^{2} + 4\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -4\zeta_{24}^{7} - 6\zeta_{24}^{6} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - 2\zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 2\beta_{5} - 3\beta_{4} - 2\beta_{3} + 2\beta_{2} ) / 32 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{4} - 2\beta_{3} ) / 16 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} - 2\beta_{3} - 2\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 2\beta_{5} - 3\beta_{4} + 2\beta_{3} - 2\beta_{2} ) / 32 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{7} + \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 2\beta_{5} - 3\beta_{4} - 2\beta_{3} - 2\beta_{2} ) / 32 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
0 − 1.73205i 0 −4.29253 0 − 2.75787i 0 −3.00000 0
127.2 0 − 1.73205i 0 1.36433 0 − 1.24213i 0 −3.00000 0
127.3 0 − 1.73205i 0 2.63567 0 12.5558i 0 −3.00000 0
127.4 0 − 1.73205i 0 8.29253 0 − 8.55583i 0 −3.00000 0
127.5 0 1.73205i 0 −4.29253 0 2.75787i 0 −3.00000 0
127.6 0 1.73205i 0 1.36433 0 1.24213i 0 −3.00000 0
127.7 0 1.73205i 0 2.63567 0 − 12.5558i 0 −3.00000 0
127.8 0 1.73205i 0 8.29253 0 8.55583i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 8T_{5}^{3} - 16T_{5}^{2} + 128T_{5} - 128 \) acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 8 T^{3} + \cdots - 128)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 240 T^{6} + \cdots + 135424 \) Copy content Toggle raw display
$11$ \( (T^{4} + 224 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 24 T^{3} + \cdots - 35696)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 8 T^{3} + \cdots + 70288)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 22351446016 \) Copy content Toggle raw display
$23$ \( T^{8} + 960 T^{6} + \cdots + 34668544 \) Copy content Toggle raw display
$29$ \( (T^{4} - 40 T^{3} + \cdots - 777344)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 1115025664 \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} + \cdots - 553712)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 40 T^{3} + \cdots - 3231344)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 10858553933824 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 102236225536 \) Copy content Toggle raw display
$53$ \( (T^{4} + 88 T^{3} + \cdots - 3148928)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 81909160935424 \) Copy content Toggle raw display
$61$ \( (T^{4} - 136 T^{3} + \cdots - 13242608)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9312 T^{2} + 20793600)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 11090924732416 \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} + \cdots - 2276336)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 3915755776 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 86180178755584 \) Copy content Toggle raw display
$89$ \( (T^{4} + 120 T^{3} + \cdots - 1654256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 200 T^{3} + \cdots + 161817616)^{2} \) Copy content Toggle raw display
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