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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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
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

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{3} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{5} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{3} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{5} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{7} + 3 q^{9} + ( 8 \zeta_{24} + 8 \zeta_{24}^{2} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{11} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{13} + ( 4 + 6 \zeta_{24} - 6 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - 6 \zeta_{24}^{5} ) q^{15} + ( -2 - 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 16 \zeta_{24}^{7} ) q^{17} + ( -8 \zeta_{24} - 8 \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{19} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{21} + ( -16 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 32 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{23} + ( -15 - 16 \zeta_{24} - 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} + 32 \zeta_{24}^{7} ) q^{25} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{27} + ( 10 \zeta_{24} - 10 \zeta_{24}^{3} + 10 \zeta_{24}^{5} - 20 \zeta_{24}^{6} + 20 \zeta_{24}^{7} ) q^{29} + ( 16 - 18 \zeta_{24} + 18 \zeta_{24}^{3} - 32 \zeta_{24}^{4} + 18 \zeta_{24}^{5} ) q^{31} + ( -12 - 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 16 \zeta_{24}^{7} ) q^{33} + ( 8 \zeta_{24} + 16 \zeta_{24}^{2} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 8 \zeta_{24}^{6} ) q^{35} + ( 16 \zeta_{24} - 16 \zeta_{24}^{3} + 16 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 32 \zeta_{24}^{7} ) q^{37} + ( -4 + 12 \zeta_{24} - 12 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 12 \zeta_{24}^{5} ) q^{39} + ( 18 - 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 16 \zeta_{24}^{7} ) q^{41} + ( -40 \zeta_{24} - 8 \zeta_{24}^{2} - 40 \zeta_{24}^{3} + 40 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{43} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 12 \zeta_{24}^{6} + 12 \zeta_{24}^{7} ) q^{45} + ( 32 + 12 \zeta_{24} - 12 \zeta_{24}^{3} - 64 \zeta_{24}^{4} - 12 \zeta_{24}^{5} ) q^{47} + 41 q^{49} + ( 24 \zeta_{24} + 4 \zeta_{24}^{2} + 24 \zeta_{24}^{3} - 24 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{51} + ( 14 \zeta_{24} - 14 \zeta_{24}^{3} + 14 \zeta_{24}^{5} + 36 \zeta_{24}^{6} + 28 \zeta_{24}^{7} ) q^{53} + ( -48 - 56 \zeta_{24} + 56 \zeta_{24}^{3} + 96 \zeta_{24}^{4} + 56 \zeta_{24}^{5} ) q^{55} + ( 12 + 8 \zeta_{24} + 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - 16 \zeta_{24}^{7} ) q^{57} + ( -40 \zeta_{24}^{2} + 20 \zeta_{24}^{6} ) q^{59} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + 44 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{61} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{63} + ( -32 - 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 16 \zeta_{24}^{7} ) q^{65} + ( -32 \zeta_{24} + 56 \zeta_{24}^{2} - 32 \zeta_{24}^{3} + 32 \zeta_{24}^{5} - 28 \zeta_{24}^{6} ) q^{67} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 48 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{69} + ( -16 + 68 \zeta_{24} - 68 \zeta_{24}^{3} + 32 \zeta_{24}^{4} - 68 \zeta_{24}^{5} ) q^{71} -10 q^{73} + ( 48 \zeta_{24} + 30 \zeta_{24}^{2} + 48 \zeta_{24}^{3} - 48 \zeta_{24}^{5} - 15 \zeta_{24}^{6} ) q^{75} + ( -8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 32 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{77} + ( 48 - 34 \zeta_{24} + 34 \zeta_{24}^{3} - 96 \zeta_{24}^{4} + 34 \zeta_{24}^{5} ) q^{79} + 9 q^{81} + ( 24 \zeta_{24} - 88 \zeta_{24}^{2} + 24 \zeta_{24}^{3} - 24 \zeta_{24}^{5} + 44 \zeta_{24}^{6} ) q^{83} + ( -36 \zeta_{24} + 36 \zeta_{24}^{3} - 36 \zeta_{24}^{5} - 104 \zeta_{24}^{6} - 72 \zeta_{24}^{7} ) q^{85} + ( -20 + 30 \zeta_{24} - 30 \zeta_{24}^{3} + 40 \zeta_{24}^{4} - 30 \zeta_{24}^{5} ) q^{87} + ( -34 + 16 \zeta_{24} + 16 \zeta_{24}^{3} + 16 \zeta_{24}^{5} - 32 \zeta_{24}^{7} ) q^{89} + ( -8 \zeta_{24} + 32 \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - 16 \zeta_{24}^{6} ) q^{91} + ( -18 \zeta_{24} + 18 \zeta_{24}^{3} - 18 \zeta_{24}^{5} + 48 \zeta_{24}^{6} - 36 \zeta_{24}^{7} ) q^{93} + ( 48 + 56 \zeta_{24} - 56 \zeta_{24}^{3} - 96 \zeta_{24}^{4} - 56 \zeta_{24}^{5} ) q^{95} + ( -66 + 16 \zeta_{24} + 16 \zeta_{24}^{3} + 16 \zeta_{24}^{5} - 32 \zeta_{24}^{7} ) q^{97} + ( 24 \zeta_{24} + 24 \zeta_{24}^{2} + 24 \zeta_{24}^{3} - 24 \zeta_{24}^{5} - 12 \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 16q^{17} - 120q^{25} - 96q^{33} + 144q^{41} + 328q^{49} + 96q^{57} - 256q^{65} - 80q^{73} + 72q^{81} - 272q^{89} - 528q^{97} + O(q^{100}) \)

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.

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 \( T_{5}^{4} + 80 T_{5}^{2} + 64 \) acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -3 + T^{2} )^{4} \)
$5$ \( ( 64 + 80 T^{2} + T^{4} )^{2} \)
$7$ \( ( 8 + T^{2} )^{4} \)
$11$ \( ( 6400 - 352 T^{2} + T^{4} )^{2} \)
$13$ \( ( 6400 + 224 T^{2} + T^{4} )^{2} \)
$17$ \( ( -380 + 4 T + T^{2} )^{4} \)
$19$ \( ( 6400 - 352 T^{2} + T^{4} )^{2} \)
$23$ \( ( 541696 + 1600 T^{2} + T^{4} )^{2} \)
$29$ \( ( 40000 + 2000 T^{2} + T^{4} )^{2} \)
$31$ \( ( 14400 + 2832 T^{2} + T^{4} )^{2} \)
$37$ \( ( 2310400 + 3104 T^{2} + T^{4} )^{2} \)
$41$ \( ( -60 - 36 T + T^{2} )^{4} \)
$43$ \( ( 9935104 - 6496 T^{2} + T^{4} )^{2} \)
$47$ \( ( 7750656 + 6720 T^{2} + T^{4} )^{2} \)
$53$ \( ( 14400 + 4944 T^{2} + T^{4} )^{2} \)
$59$ \( ( -1200 + T^{2} )^{4} \)
$61$ \( ( 2408704 + 4640 T^{2} + T^{4} )^{2} \)
$67$ \( ( 92416 - 8800 T^{2} + T^{4} )^{2} \)
$71$ \( ( 71910400 + 20032 T^{2} + T^{4} )^{2} \)
$73$ \( ( 10 + T )^{8} \)
$79$ \( ( 21160000 + 18448 T^{2} + T^{4} )^{2} \)
$83$ \( ( 21678336 - 13920 T^{2} + T^{4} )^{2} \)
$89$ \( ( -380 + 68 T + T^{2} )^{4} \)
$97$ \( ( 2820 + 132 T + T^{2} )^{4} \)
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