Properties

Label 384.3.b.a
Level $384$
Weight $3$
Character orbit 384.b
Analytic conductor $10.463$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 4 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 4 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} + 3 q^{9} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{11} + ( -4 + 8 \zeta_{12}^{2} ) q^{15} -18 q^{17} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{19} + 12 \zeta_{12}^{3} q^{21} + ( -24 + 48 \zeta_{12}^{2} ) q^{23} + 9 q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + 4 \zeta_{12}^{3} q^{29} + ( -28 + 56 \zeta_{12}^{2} ) q^{31} + 12 q^{33} + ( -32 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{35} -72 \zeta_{12}^{3} q^{37} + 18 q^{41} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{43} + 12 \zeta_{12}^{3} q^{45} + ( -24 + 48 \zeta_{12}^{2} ) q^{47} + q^{49} + ( -36 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{51} + 44 \zeta_{12}^{3} q^{53} + ( -16 + 32 \zeta_{12}^{2} ) q^{55} -36 q^{57} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{59} -72 \zeta_{12}^{3} q^{61} + ( -12 + 24 \zeta_{12}^{2} ) q^{63} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{67} + 72 \zeta_{12}^{3} q^{69} + ( 24 - 48 \zeta_{12}^{2} ) q^{71} -82 q^{73} + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{75} + 48 \zeta_{12}^{3} q^{77} + ( 36 - 72 \zeta_{12}^{2} ) q^{79} + 9 q^{81} + ( -152 \zeta_{12} + 76 \zeta_{12}^{3} ) q^{83} -72 \zeta_{12}^{3} q^{85} + ( -4 + 8 \zeta_{12}^{2} ) q^{87} + 126 q^{89} + 84 \zeta_{12}^{3} q^{93} + ( 48 - 96 \zeta_{12}^{2} ) q^{95} + 110 q^{97} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} - 72q^{17} + 36q^{25} + 48q^{33} + 72q^{41} + 4q^{49} - 144q^{57} - 328q^{73} + 36q^{81} + 504q^{89} + 440q^{97} + O(q^{100}) \)

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
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0 −1.73205 0 4.00000i 0 6.92820i 0 3.00000 0
319.2 0 −1.73205 0 4.00000i 0 6.92820i 0 3.00000 0
319.3 0 1.73205 0 4.00000i 0 6.92820i 0 3.00000 0
319.4 0 1.73205 0 4.00000i 0 6.92820i 0 3.00000 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
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.a 4
3.b odd 2 1 1152.3.b.h 4
4.b odd 2 1 inner 384.3.b.a 4
8.b even 2 1 inner 384.3.b.a 4
8.d odd 2 1 inner 384.3.b.a 4
12.b even 2 1 1152.3.b.h 4
16.e even 4 1 768.3.g.a 2
16.e even 4 1 768.3.g.b 2
16.f odd 4 1 768.3.g.a 2
16.f odd 4 1 768.3.g.b 2
24.f even 2 1 1152.3.b.h 4
24.h odd 2 1 1152.3.b.h 4
48.i odd 4 1 2304.3.g.g 2
48.i odd 4 1 2304.3.g.n 2
48.k even 4 1 2304.3.g.g 2
48.k even 4 1 2304.3.g.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.b.a 4 1.a even 1 1 trivial
384.3.b.a 4 4.b odd 2 1 inner
384.3.b.a 4 8.b even 2 1 inner
384.3.b.a 4 8.d odd 2 1 inner
768.3.g.a 2 16.e even 4 1
768.3.g.a 2 16.f odd 4 1
768.3.g.b 2 16.e even 4 1
768.3.g.b 2 16.f odd 4 1
1152.3.b.h 4 3.b odd 2 1
1152.3.b.h 4 12.b even 2 1
1152.3.b.h 4 24.f even 2 1
1152.3.b.h 4 24.h odd 2 1
2304.3.g.g 2 48.i odd 4 1
2304.3.g.g 2 48.k even 4 1
2304.3.g.n 2 48.i odd 4 1
2304.3.g.n 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 16 \) acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 16 + T^{2} )^{2} \)
$7$ \( ( 48 + T^{2} )^{2} \)
$11$ \( ( -48 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( 18 + T )^{4} \)
$19$ \( ( -432 + T^{2} )^{2} \)
$23$ \( ( 1728 + T^{2} )^{2} \)
$29$ \( ( 16 + T^{2} )^{2} \)
$31$ \( ( 2352 + T^{2} )^{2} \)
$37$ \( ( 5184 + T^{2} )^{2} \)
$41$ \( ( -18 + T )^{4} \)
$43$ \( ( -3888 + T^{2} )^{2} \)
$47$ \( ( 1728 + T^{2} )^{2} \)
$53$ \( ( 1936 + T^{2} )^{2} \)
$59$ \( ( -3888 + T^{2} )^{2} \)
$61$ \( ( 5184 + T^{2} )^{2} \)
$67$ \( ( -432 + T^{2} )^{2} \)
$71$ \( ( 1728 + T^{2} )^{2} \)
$73$ \( ( 82 + T )^{4} \)
$79$ \( ( 3888 + T^{2} )^{2} \)
$83$ \( ( -17328 + T^{2} )^{2} \)
$89$ \( ( -126 + T )^{4} \)
$97$ \( ( -110 + T )^{4} \)
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