Properties

Label 384.2.d.c
Level $384$
Weight $2$
Character orbit 384.d
Analytic conductor $3.066$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{7} - q^{9} +O(q^{10})\) \( q + \zeta_{8}^{2} q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{7} - q^{9} -4 \zeta_{8}^{2} q^{11} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{13} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{15} -2 q^{17} + 4 \zeta_{8}^{2} q^{19} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{21} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{23} -3 q^{25} -\zeta_{8}^{2} q^{27} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{31} + 4 q^{33} + 8 \zeta_{8}^{2} q^{35} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{39} + 10 q^{41} -12 \zeta_{8}^{2} q^{43} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{45} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{47} + q^{49} -2 \zeta_{8}^{2} q^{51} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{53} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{55} -4 q^{57} + 4 \zeta_{8}^{2} q^{59} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{61} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{63} -16 q^{65} + 4 \zeta_{8}^{2} q^{67} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{69} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} -2 q^{73} -3 \zeta_{8}^{2} q^{75} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{77} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{79} + q^{81} -4 \zeta_{8}^{2} q^{83} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{85} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{87} + 6 q^{89} + 16 \zeta_{8}^{2} q^{91} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{93} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{95} + 14 q^{97} + 4 \zeta_{8}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} - 8q^{17} - 12q^{25} + 16q^{33} + 40q^{41} + 4q^{49} - 16q^{57} - 64q^{65} - 8q^{73} + 4q^{81} + 24q^{89} + 56q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 1.00000i 0 2.82843i 0 2.82843 0 −1.00000 0
193.2 0 1.00000i 0 2.82843i 0 −2.82843 0 −1.00000 0
193.3 0 1.00000i 0 2.82843i 0 −2.82843 0 −1.00000 0
193.4 0 1.00000i 0 2.82843i 0 2.82843 0 −1.00000 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 384.2.d.c 4
3.b odd 2 1 1152.2.d.h 4
4.b odd 2 1 inner 384.2.d.c 4
8.b even 2 1 inner 384.2.d.c 4
8.d odd 2 1 inner 384.2.d.c 4
12.b even 2 1 1152.2.d.h 4
16.e even 4 1 768.2.a.i 2
16.e even 4 1 768.2.a.l 2
16.f odd 4 1 768.2.a.i 2
16.f odd 4 1 768.2.a.l 2
24.f even 2 1 1152.2.d.h 4
24.h odd 2 1 1152.2.d.h 4
48.i odd 4 1 2304.2.a.r 2
48.i odd 4 1 2304.2.a.x 2
48.k even 4 1 2304.2.a.r 2
48.k even 4 1 2304.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.d.c 4 1.a even 1 1 trivial
384.2.d.c 4 4.b odd 2 1 inner
384.2.d.c 4 8.b even 2 1 inner
384.2.d.c 4 8.d odd 2 1 inner
768.2.a.i 2 16.e even 4 1
768.2.a.i 2 16.f odd 4 1
768.2.a.l 2 16.e even 4 1
768.2.a.l 2 16.f odd 4 1
1152.2.d.h 4 3.b odd 2 1
1152.2.d.h 4 12.b even 2 1
1152.2.d.h 4 24.f even 2 1
1152.2.d.h 4 24.h odd 2 1
2304.2.a.r 2 48.i odd 4 1
2304.2.a.r 2 48.k even 4 1
2304.2.a.x 2 48.i odd 4 1
2304.2.a.x 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} + 8 \)
\( T_{7}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( 8 + T^{2} )^{2} \)
$7$ \( ( -8 + T^{2} )^{2} \)
$11$ \( ( 16 + T^{2} )^{2} \)
$13$ \( ( 32 + T^{2} )^{2} \)
$17$ \( ( 2 + T )^{4} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( ( -32 + T^{2} )^{2} \)
$29$ \( ( 8 + T^{2} )^{2} \)
$31$ \( ( -72 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( ( -10 + T )^{4} \)
$43$ \( ( 144 + T^{2} )^{2} \)
$47$ \( ( -32 + T^{2} )^{2} \)
$53$ \( ( 8 + T^{2} )^{2} \)
$59$ \( ( 16 + T^{2} )^{2} \)
$61$ \( ( 128 + T^{2} )^{2} \)
$67$ \( ( 16 + T^{2} )^{2} \)
$71$ \( ( -32 + T^{2} )^{2} \)
$73$ \( ( 2 + T )^{4} \)
$79$ \( ( -72 + T^{2} )^{2} \)
$83$ \( ( 16 + T^{2} )^{2} \)
$89$ \( ( -6 + T )^{4} \)
$97$ \( ( -14 + T )^{4} \)
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