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Properties

Label 384.3.b.c
Level $384$
Weight $3$
Character orbit 384.b
Analytic conductor $10.463$
Analytic rank $0$
Dimension $8$
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: \(8\)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{5} - \beta_{3}) q^{5} + \beta_1 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{5} - \beta_{3}) q^{5} + \beta_1 q^{7} + 3 q^{9} + (\beta_{6} + 4 \beta_{2}) q^{11} + ( - \beta_{5} - 2 \beta_{3}) q^{13} + ( - \beta_{4} + 3 \beta_1) q^{15} + ( - \beta_{7} - 2) q^{17} + ( - \beta_{6} - 4 \beta_{2}) q^{19} - \beta_{3} q^{21} + (4 \beta_{4} - 2 \beta_1) q^{23} + ( - 2 \beta_{7} - 15) q^{25} - 3 \beta_{2} q^{27} + ( - 5 \beta_{5} - 5 \beta_{3}) q^{29} + ( - 4 \beta_{4} - 9 \beta_1) q^{31} + ( - \beta_{7} - 12) q^{33} + (\beta_{6} + 8 \beta_{2}) q^{35} + (\beta_{5} - 8 \beta_{3}) q^{37} + (\beta_{4} + 6 \beta_1) q^{39} + ( - \beta_{7} + 18) q^{41} + ( - 5 \beta_{6} - 4 \beta_{2}) q^{43} + (3 \beta_{5} - 3 \beta_{3}) q^{45} + ( - 8 \beta_{4} + 6 \beta_1) q^{47} + 41 q^{49} + (3 \beta_{6} + 2 \beta_{2}) q^{51} + (9 \beta_{5} - 7 \beta_{3}) q^{53} + (12 \beta_{4} - 28 \beta_1) q^{55} + (\beta_{7} + 12) q^{57} - 20 \beta_{2} q^{59} + (11 \beta_{5} - 4 \beta_{3}) q^{61} + 3 \beta_1 q^{63} + ( - \beta_{7} - 32) q^{65} + ( - 4 \beta_{6} + 28 \beta_{2}) q^{67} + ( - 12 \beta_{5} + 2 \beta_{3}) q^{69} + (4 \beta_{4} + 34 \beta_1) q^{71} - 10 q^{73} + (6 \beta_{6} + 15 \beta_{2}) q^{75} + ( - 8 \beta_{5} + 4 \beta_{3}) q^{77} + ( - 12 \beta_{4} - 17 \beta_1) q^{79} + 9 q^{81} + (3 \beta_{6} - 44 \beta_{2}) q^{83} + ( - 26 \beta_{5} + 18 \beta_{3}) q^{85} + (5 \beta_{4} + 15 \beta_1) q^{87} + (2 \beta_{7} - 34) q^{89} + ( - \beta_{6} + 16 \beta_{2}) q^{91} + (12 \beta_{5} + 9 \beta_{3}) q^{93} + ( - 12 \beta_{4} + 28 \beta_1) q^{95} + (2 \beta_{7} - 66) q^{97} + (3 \beta_{6} + 12 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} - 16 q^{17} - 120 q^{25} - 96 q^{33} + 144 q^{41} + 328 q^{49} + 96 q^{57} - 256 q^{65} - 80 q^{73} + 72 q^{81} - 272 q^{89} - 528 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

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

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} - 4\beta_{3} + 4\beta_1 ) / 32 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{5} + 4\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} - 4\beta_1 ) / 16 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + 4 ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{6} - 4\beta_{3} - 4\beta_1 ) / 32 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 4\beta_{3} - 4\beta_1 ) / 32 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
0.965926 − 0.258819i
−0.965926 − 0.258819i
−0.965926 + 0.258819i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 − 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
0 −1.73205 0 − 8.89898i 0 2.82843i 0 3.00000 0
319.2 0 −1.73205 0 − 0.898979i 0 2.82843i 0 3.00000 0
319.3 0 −1.73205 0 0.898979i 0 − 2.82843i 0 3.00000 0
319.4 0 −1.73205 0 8.89898i 0 − 2.82843i 0 3.00000 0
319.5 0 1.73205 0 − 8.89898i 0 − 2.82843i 0 3.00000 0
319.6 0 1.73205 0 − 0.898979i 0 − 2.82843i 0 3.00000 0
319.7 0 1.73205 0 0.898979i 0 2.82843i 0 3.00000 0
319.8 0 1.73205 0 8.89898i 0 2.82843i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.8
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.c ✓ 8
3.b odd 2 1 1152.3.b.j 8
4.b odd 2 1 inner 384.3.b.c ✓ 8
8.b even 2 1 inner 384.3.b.c ✓ 8
8.d odd 2 1 inner 384.3.b.c ✓ 8
12.b even 2 1 1152.3.b.j 8
16.e even 4 1 768.3.g.c 4
16.e even 4 1 768.3.g.g 4
16.f odd 4 1 768.3.g.c 4
16.f odd 4 1 768.3.g.g 4
24.f even 2 1 1152.3.b.j 8
24.h odd 2 1 1152.3.b.j 8
48.i odd 4 1 2304.3.g.o 4
48.i odd 4 1 2304.3.g.x 4
48.k even 4 1 2304.3.g.o 4
48.k even 4 1 2304.3.g.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.b.c ✓ 8 1.a even 1 1 trivial
384.3.b.c ✓ 8 4.b odd 2 1 inner
384.3.b.c ✓ 8 8.b even 2 1 inner
384.3.b.c ✓ 8 8.d odd 2 1 inner
768.3.g.c 4 16.e even 4 1
768.3.g.c 4 16.f odd 4 1
768.3.g.g 4 16.e even 4 1
768.3.g.g 4 16.f odd 4 1
1152.3.b.j 8 3.b odd 2 1
1152.3.b.j 8 12.b even 2 1
1152.3.b.j 8 24.f even 2 1
1152.3.b.j 8 24.h odd 2 1
2304.3.g.o 4 48.i odd 4 1
2304.3.g.o 4 48.k even 4 1
2304.3.g.x 4 48.i odd 4 1
2304.3.g.x 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 80T_{5}^{2} + 64 \) 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
$5$ \( (T^{4} + 80 T^{2} + 64)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 352 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 224 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 380)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 352 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1600 T^{2} + 541696)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2000 T^{2} + 40000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2832 T^{2} + 14400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3104 T^{2} + 2310400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 36 T - 60)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 6496 T^{2} + 9935104)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 6720 T^{2} + 7750656)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 4944 T^{2} + 14400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 1200)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4640 T^{2} + 2408704)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 8800 T^{2} + 92416)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 20032 T^{2} + 71910400)^{2} \) Copy content Toggle raw display
$73$ \( (T + 10)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 18448 T^{2} + 21160000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 13920 T^{2} + 21678336)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 68 T - 380)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 132 T + 2820)^{4} \) Copy content Toggle raw display
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