Properties

Label 384.4.f.d
Level $384$
Weight $4$
Character orbit 384.f
Analytic conductor $22.657$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.6567334422\)
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: \( 2^{3}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( -3 \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{3} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{5} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{7} + ( 9 + 18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -3 \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{3} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{5} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{7} + ( 9 + 18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{9} + 30 \zeta_{8}^{2} q^{11} + 72 \zeta_{8}^{2} q^{13} + ( 24 + 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{15} + ( -36 \zeta_{8} - 36 \zeta_{8}^{3} ) q^{17} + ( -18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{19} + ( -36 \zeta_{8} + 72 \zeta_{8}^{2} + 36 \zeta_{8}^{3} ) q^{21} + 144 q^{23} -93 q^{25} + ( 27 \zeta_{8} - 135 \zeta_{8}^{2} - 27 \zeta_{8}^{3} ) q^{27} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29} + ( -156 \zeta_{8} - 156 \zeta_{8}^{3} ) q^{31} + ( 90 - 90 \zeta_{8} - 90 \zeta_{8}^{3} ) q^{33} + 96 \zeta_{8}^{2} q^{35} -72 \zeta_{8}^{2} q^{37} + ( 216 - 216 \zeta_{8} - 216 \zeta_{8}^{3} ) q^{39} + ( 216 \zeta_{8} + 216 \zeta_{8}^{3} ) q^{41} + ( 162 \zeta_{8} - 162 \zeta_{8}^{3} ) q^{43} + ( -36 \zeta_{8} - 144 \zeta_{8}^{2} + 36 \zeta_{8}^{3} ) q^{45} + 576 q^{47} + 55 q^{49} + ( -108 \zeta_{8} + 216 \zeta_{8}^{2} + 108 \zeta_{8}^{3} ) q^{51} + ( 364 \zeta_{8} - 364 \zeta_{8}^{3} ) q^{53} + ( -120 \zeta_{8} - 120 \zeta_{8}^{3} ) q^{55} + ( 108 + 54 \zeta_{8} + 54 \zeta_{8}^{3} ) q^{57} -414 \zeta_{8}^{2} q^{59} -504 \zeta_{8}^{2} q^{61} + ( 432 - 108 \zeta_{8} - 108 \zeta_{8}^{3} ) q^{63} + ( -288 \zeta_{8} - 288 \zeta_{8}^{3} ) q^{65} + ( -558 \zeta_{8} + 558 \zeta_{8}^{3} ) q^{67} + ( -432 \zeta_{8} - 432 \zeta_{8}^{2} + 432 \zeta_{8}^{3} ) q^{69} + 720 q^{71} -178 q^{73} + ( 279 \zeta_{8} + 279 \zeta_{8}^{2} - 279 \zeta_{8}^{3} ) q^{75} + ( 360 \zeta_{8} - 360 \zeta_{8}^{3} ) q^{77} + ( -684 \zeta_{8} - 684 \zeta_{8}^{3} ) q^{79} + ( -567 + 324 \zeta_{8} + 324 \zeta_{8}^{3} ) q^{81} + 438 \zeta_{8}^{2} q^{83} + 288 \zeta_{8}^{2} q^{85} + ( 24 + 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{87} + ( -612 \zeta_{8} - 612 \zeta_{8}^{3} ) q^{89} + ( 864 \zeta_{8} - 864 \zeta_{8}^{3} ) q^{91} + ( -468 \zeta_{8} + 936 \zeta_{8}^{2} + 468 \zeta_{8}^{3} ) q^{93} + 144 q^{95} + 650 q^{97} + ( -540 \zeta_{8} + 270 \zeta_{8}^{2} + 540 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 36q^{9} + O(q^{10}) \) \( 4q + 36q^{9} + 96q^{15} + 576q^{23} - 372q^{25} + 360q^{33} + 864q^{39} + 2304q^{47} + 220q^{49} + 432q^{57} + 1728q^{63} + 2880q^{71} - 712q^{73} - 2268q^{81} + 96q^{87} + 576q^{95} + 2600q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0 −4.24264 3.00000i 0 −5.65685 0 16.9706i 0 9.00000 + 25.4558i 0
191.2 0 −4.24264 + 3.00000i 0 −5.65685 0 16.9706i 0 9.00000 25.4558i 0
191.3 0 4.24264 3.00000i 0 5.65685 0 16.9706i 0 9.00000 25.4558i 0
191.4 0 4.24264 + 3.00000i 0 5.65685 0 16.9706i 0 9.00000 + 25.4558i 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
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.4.f.d yes 4
3.b odd 2 1 384.4.f.c 4
4.b odd 2 1 384.4.f.c 4
8.b even 2 1 inner 384.4.f.d yes 4
8.d odd 2 1 384.4.f.c 4
12.b even 2 1 inner 384.4.f.d yes 4
16.e even 4 1 768.4.c.c 2
16.e even 4 1 768.4.c.h 2
16.f odd 4 1 768.4.c.b 2
16.f odd 4 1 768.4.c.i 2
24.f even 2 1 inner 384.4.f.d yes 4
24.h odd 2 1 384.4.f.c 4
48.i odd 4 1 768.4.c.b 2
48.i odd 4 1 768.4.c.i 2
48.k even 4 1 768.4.c.c 2
48.k even 4 1 768.4.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.f.c 4 3.b odd 2 1
384.4.f.c 4 4.b odd 2 1
384.4.f.c 4 8.d odd 2 1
384.4.f.c 4 24.h odd 2 1
384.4.f.d yes 4 1.a even 1 1 trivial
384.4.f.d yes 4 8.b even 2 1 inner
384.4.f.d yes 4 12.b even 2 1 inner
384.4.f.d yes 4 24.f even 2 1 inner
768.4.c.b 2 16.f odd 4 1
768.4.c.b 2 48.i odd 4 1
768.4.c.c 2 16.e even 4 1
768.4.c.c 2 48.k even 4 1
768.4.c.h 2 16.e even 4 1
768.4.c.h 2 48.k even 4 1
768.4.c.i 2 16.f odd 4 1
768.4.c.i 2 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} - 32 \)
\( T_{23} - 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 729 - 18 T^{2} + T^{4} \)
$5$ \( ( -32 + T^{2} )^{2} \)
$7$ \( ( 288 + T^{2} )^{2} \)
$11$ \( ( 900 + T^{2} )^{2} \)
$13$ \( ( 5184 + T^{2} )^{2} \)
$17$ \( ( 2592 + T^{2} )^{2} \)
$19$ \( ( -648 + T^{2} )^{2} \)
$23$ \( ( -144 + T )^{4} \)
$29$ \( ( -32 + T^{2} )^{2} \)
$31$ \( ( 48672 + T^{2} )^{2} \)
$37$ \( ( 5184 + T^{2} )^{2} \)
$41$ \( ( 93312 + T^{2} )^{2} \)
$43$ \( ( -52488 + T^{2} )^{2} \)
$47$ \( ( -576 + T )^{4} \)
$53$ \( ( -264992 + T^{2} )^{2} \)
$59$ \( ( 171396 + T^{2} )^{2} \)
$61$ \( ( 254016 + T^{2} )^{2} \)
$67$ \( ( -622728 + T^{2} )^{2} \)
$71$ \( ( -720 + T )^{4} \)
$73$ \( ( 178 + T )^{4} \)
$79$ \( ( 935712 + T^{2} )^{2} \)
$83$ \( ( 191844 + T^{2} )^{2} \)
$89$ \( ( 749088 + T^{2} )^{2} \)
$97$ \( ( -650 + T )^{4} \)
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