Properties

Label 384.4.d.a
Level $384$
Weight $4$
Character orbit 384.d
Analytic conductor $22.657$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 i q^{3} + 8 i q^{5} - 12 q^{7} - 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 i q^{3} + 8 i q^{5} - 12 q^{7} - 9 q^{9} - 12 i q^{11} + 20 i q^{13} + 24 q^{15} + 62 q^{17} - 108 i q^{19} + 36 i q^{21} + 72 q^{23} + 61 q^{25} + 27 i q^{27} - 128 i q^{29} + 204 q^{31} - 36 q^{33} - 96 i q^{35} + 228 i q^{37} + 60 q^{39} - 22 q^{41} - 204 i q^{43} - 72 i q^{45} + 600 q^{47} - 199 q^{49} - 186 i q^{51} - 256 i q^{53} + 96 q^{55} - 324 q^{57} - 828 i q^{59} - 84 i q^{61} + 108 q^{63} - 160 q^{65} - 348 i q^{67} - 216 i q^{69} - 456 q^{71} + 822 q^{73} - 183 i q^{75} + 144 i q^{77} + 1356 q^{79} + 81 q^{81} - 108 i q^{83} + 496 i q^{85} - 384 q^{87} - 938 q^{89} - 240 i q^{91} - 612 i q^{93} + 864 q^{95} + 1278 q^{97} + 108 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 24 q^{7} - 18 q^{9} + 48 q^{15} + 124 q^{17} + 144 q^{23} + 122 q^{25} + 408 q^{31} - 72 q^{33} + 120 q^{39} - 44 q^{41} + 1200 q^{47} - 398 q^{49} + 192 q^{55} - 648 q^{57} + 216 q^{63} - 320 q^{65} - 912 q^{71} + 1644 q^{73} + 2712 q^{79} + 162 q^{81} - 768 q^{87} - 1876 q^{89} + 1728 q^{95} + 2556 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
1.00000i
1.00000i
0 3.00000i 0 8.00000i 0 −12.0000 0 −9.00000 0
193.2 0 3.00000i 0 8.00000i 0 −12.0000 0 −9.00000 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
8.b 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.d.a 2
3.b odd 2 1 1152.4.d.b 2
4.b odd 2 1 384.4.d.b yes 2
8.b even 2 1 inner 384.4.d.a 2
8.d odd 2 1 384.4.d.b yes 2
12.b even 2 1 1152.4.d.g 2
16.e even 4 1 768.4.a.a 1
16.e even 4 1 768.4.a.d 1
16.f odd 4 1 768.4.a.b 1
16.f odd 4 1 768.4.a.c 1
24.f even 2 1 1152.4.d.g 2
24.h odd 2 1 1152.4.d.b 2
48.i odd 4 1 2304.4.a.f 1
48.i odd 4 1 2304.4.a.l 1
48.k even 4 1 2304.4.a.e 1
48.k even 4 1 2304.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.a 2 1.a even 1 1 trivial
384.4.d.a 2 8.b even 2 1 inner
384.4.d.b yes 2 4.b odd 2 1
384.4.d.b yes 2 8.d odd 2 1
768.4.a.a 1 16.e even 4 1
768.4.a.b 1 16.f odd 4 1
768.4.a.c 1 16.f odd 4 1
768.4.a.d 1 16.e even 4 1
1152.4.d.b 2 3.b odd 2 1
1152.4.d.b 2 24.h odd 2 1
1152.4.d.g 2 12.b even 2 1
1152.4.d.g 2 24.f even 2 1
2304.4.a.e 1 48.k even 4 1
2304.4.a.f 1 48.i odd 4 1
2304.4.a.k 1 48.k even 4 1
2304.4.a.l 1 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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{7} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 64 \) Copy content Toggle raw display
$7$ \( (T + 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 144 \) Copy content Toggle raw display
$13$ \( T^{2} + 400 \) Copy content Toggle raw display
$17$ \( (T - 62)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 11664 \) Copy content Toggle raw display
$23$ \( (T - 72)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16384 \) Copy content Toggle raw display
$31$ \( (T - 204)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 51984 \) Copy content Toggle raw display
$41$ \( (T + 22)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 41616 \) Copy content Toggle raw display
$47$ \( (T - 600)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 65536 \) Copy content Toggle raw display
$59$ \( T^{2} + 685584 \) Copy content Toggle raw display
$61$ \( T^{2} + 7056 \) Copy content Toggle raw display
$67$ \( T^{2} + 121104 \) Copy content Toggle raw display
$71$ \( (T + 456)^{2} \) Copy content Toggle raw display
$73$ \( (T - 822)^{2} \) Copy content Toggle raw display
$79$ \( (T - 1356)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 11664 \) Copy content Toggle raw display
$89$ \( (T + 938)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1278)^{2} \) Copy content Toggle raw display
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