Properties

Label 384.4.d.e
Level $384$
Weight $4$
Character orbit 384.d
Analytic conductor $22.657$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + (\beta_{2} - 4 \beta_1) q^{5} + ( - \beta_{3} + 8) q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} + (\beta_{2} - 4 \beta_1) q^{5} + ( - \beta_{3} + 8) q^{7} - 9 q^{9} + ( - 4 \beta_{2} - 4 \beta_1) q^{11} + ( - 2 \beta_{2} - 36 \beta_1) q^{13} + ( - 3 \beta_{3} + 12) q^{15} + ( - 4 \beta_{3} - 18) q^{17} + (4 \beta_{2} - 68 \beta_1) q^{19} + ( - 3 \beta_{2} + 24 \beta_1) q^{21} + (2 \beta_{3} + 128) q^{23} + (8 \beta_{3} - 99) q^{25} - 27 \beta_1 q^{27} + (9 \beta_{2} - 76 \beta_1) q^{29} + (13 \beta_{3} + 40) q^{31} + (12 \beta_{3} + 12) q^{33} + (12 \beta_{2} - 240 \beta_1) q^{35} + ( - 4 \beta_{2} + 68 \beta_1) q^{37} + (6 \beta_{3} + 108) q^{39} + ( - 20 \beta_{3} + 218) q^{41} + (4 \beta_{2} - 356 \beta_1) q^{43} + ( - 9 \beta_{2} + 36 \beta_1) q^{45} + ( - 14 \beta_{3} + 112) q^{47} + ( - 16 \beta_{3} - 71) q^{49} + ( - 12 \beta_{2} - 54 \beta_1) q^{51} + (15 \beta_{2} + 172 \beta_1) q^{53} + ( - 12 \beta_{3} + 816) q^{55} + ( - 12 \beta_{3} + 204) q^{57} - 324 \beta_1 q^{59} - 324 \beta_1 q^{61} + (9 \beta_{3} - 72) q^{63} + (28 \beta_{3} + 272) q^{65} + ( - 48 \beta_{2} - 228 \beta_1) q^{67} + (6 \beta_{2} + 384 \beta_1) q^{69} + ( - 2 \beta_{3} + 1024) q^{71} + (48 \beta_{3} - 330) q^{73} + (24 \beta_{2} - 297 \beta_1) q^{75} + ( - 28 \beta_{2} + 800 \beta_1) q^{77} + ( - 59 \beta_{3} - 248) q^{79} + 81 q^{81} + ( - 28 \beta_{2} - 388 \beta_1) q^{83} + ( - 2 \beta_{2} - 760 \beta_1) q^{85} + ( - 27 \beta_{3} + 228) q^{87} + (8 \beta_{3} - 266) q^{89} + (20 \beta_{2} + 128 \beta_1) q^{91} + (39 \beta_{2} + 120 \beta_1) q^{93} + (84 \beta_{3} - 1104) q^{95} + ( - 88 \beta_{3} - 610) q^{97} + (36 \beta_{2} + 36 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{7} - 36 q^{9} + 48 q^{15} - 72 q^{17} + 512 q^{23} - 396 q^{25} + 160 q^{31} + 48 q^{33} + 432 q^{39} + 872 q^{41} + 448 q^{47} - 284 q^{49} + 3264 q^{55} + 816 q^{57} - 288 q^{63} + 1088 q^{65} + 4096 q^{71} - 1320 q^{73} - 992 q^{79} + 324 q^{81} + 912 q^{87} - 1064 q^{89} - 4416 q^{95} - 2440 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{3} + 40\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} + 28 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 4\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 28 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 10\beta_1 ) / 2 \) 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
1.30278i
2.30278i
2.30278i
1.30278i
0 3.00000i 0 10.4222i 0 −6.42221 0 −9.00000 0
193.2 0 3.00000i 0 18.4222i 0 22.4222 0 −9.00000 0
193.3 0 3.00000i 0 18.4222i 0 22.4222 0 −9.00000 0
193.4 0 3.00000i 0 10.4222i 0 −6.42221 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.e yes 4
3.b odd 2 1 1152.4.d.o 4
4.b odd 2 1 384.4.d.c 4
8.b even 2 1 inner 384.4.d.e yes 4
8.d odd 2 1 384.4.d.c 4
12.b even 2 1 1152.4.d.i 4
16.e even 4 1 768.4.a.e 2
16.e even 4 1 768.4.a.p 2
16.f odd 4 1 768.4.a.j 2
16.f odd 4 1 768.4.a.k 2
24.f even 2 1 1152.4.d.i 4
24.h odd 2 1 1152.4.d.o 4
48.i odd 4 1 2304.4.a.s 2
48.i odd 4 1 2304.4.a.bp 2
48.k even 4 1 2304.4.a.t 2
48.k even 4 1 2304.4.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.c 4 4.b odd 2 1
384.4.d.c 4 8.d odd 2 1
384.4.d.e yes 4 1.a even 1 1 trivial
384.4.d.e yes 4 8.b even 2 1 inner
768.4.a.e 2 16.e even 4 1
768.4.a.j 2 16.f odd 4 1
768.4.a.k 2 16.f odd 4 1
768.4.a.p 2 16.e even 4 1
1152.4.d.i 4 12.b even 2 1
1152.4.d.i 4 24.f even 2 1
1152.4.d.o 4 3.b odd 2 1
1152.4.d.o 4 24.h odd 2 1
2304.4.a.s 2 48.i odd 4 1
2304.4.a.t 2 48.k even 4 1
2304.4.a.bp 2 48.i odd 4 1
2304.4.a.bq 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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{4} + 448T_{5}^{2} + 36864 \) Copy content Toggle raw display
\( T_{7}^{2} - 16T_{7} - 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 448 T^{2} + 36864 \) Copy content Toggle raw display
$7$ \( (T^{2} - 16 T - 144)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 6688 T^{2} + \cdots + 10969344 \) Copy content Toggle raw display
$13$ \( T^{4} + 4256 T^{2} + 215296 \) Copy content Toggle raw display
$17$ \( (T^{2} + 36 T - 3004)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 15904 T^{2} + \cdots + 1679616 \) Copy content Toggle raw display
$23$ \( (T^{2} - 256 T + 15552)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 45248 T^{2} + \cdots + 122589184 \) Copy content Toggle raw display
$31$ \( (T^{2} - 80 T - 33552)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 15904 T^{2} + \cdots + 1679616 \) Copy content Toggle raw display
$41$ \( (T^{2} - 436 T - 35676)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 260128 T^{2} + \cdots + 15229534464 \) Copy content Toggle raw display
$47$ \( (T^{2} - 224 T - 28224)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 152768 T^{2} + \cdots + 296390656 \) Copy content Toggle raw display
$59$ \( (T^{2} + 104976)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 104976)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 1062432 T^{2} + \cdots + 182540853504 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2048 T + 1047744)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 660 T - 370332)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 496 T - 662544)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 627232 T^{2} + \cdots + 156950784 \) Copy content Toggle raw display
$89$ \( (T^{2} + 532 T + 57444)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1220 T - 1238652)^{2} \) Copy content Toggle raw display
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