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 $2$

Related objects

Downloads

Learn more

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: \(x^{4} + 7 x^{2} + 9\)
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.

\(f(q)\) \(=\) \( q + 3 \beta_{1} q^{3} + ( -4 \beta_{1} + \beta_{2} ) q^{5} + ( 8 - \beta_{3} ) q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 \beta_{1} q^{3} + ( -4 \beta_{1} + \beta_{2} ) q^{5} + ( 8 - \beta_{3} ) q^{7} -9 q^{9} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{11} + ( -36 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 12 - 3 \beta_{3} ) q^{15} + ( -18 - 4 \beta_{3} ) q^{17} + ( -68 \beta_{1} + 4 \beta_{2} ) q^{19} + ( 24 \beta_{1} - 3 \beta_{2} ) q^{21} + ( 128 + 2 \beta_{3} ) q^{23} + ( -99 + 8 \beta_{3} ) q^{25} -27 \beta_{1} q^{27} + ( -76 \beta_{1} + 9 \beta_{2} ) q^{29} + ( 40 + 13 \beta_{3} ) q^{31} + ( 12 + 12 \beta_{3} ) q^{33} + ( -240 \beta_{1} + 12 \beta_{2} ) q^{35} + ( 68 \beta_{1} - 4 \beta_{2} ) q^{37} + ( 108 + 6 \beta_{3} ) q^{39} + ( 218 - 20 \beta_{3} ) q^{41} + ( -356 \beta_{1} + 4 \beta_{2} ) q^{43} + ( 36 \beta_{1} - 9 \beta_{2} ) q^{45} + ( 112 - 14 \beta_{3} ) q^{47} + ( -71 - 16 \beta_{3} ) q^{49} + ( -54 \beta_{1} - 12 \beta_{2} ) q^{51} + ( 172 \beta_{1} + 15 \beta_{2} ) q^{53} + ( 816 - 12 \beta_{3} ) q^{55} + ( 204 - 12 \beta_{3} ) q^{57} -324 \beta_{1} q^{59} -324 \beta_{1} q^{61} + ( -72 + 9 \beta_{3} ) q^{63} + ( 272 + 28 \beta_{3} ) q^{65} + ( -228 \beta_{1} - 48 \beta_{2} ) q^{67} + ( 384 \beta_{1} + 6 \beta_{2} ) q^{69} + ( 1024 - 2 \beta_{3} ) q^{71} + ( -330 + 48 \beta_{3} ) q^{73} + ( -297 \beta_{1} + 24 \beta_{2} ) q^{75} + ( 800 \beta_{1} - 28 \beta_{2} ) q^{77} + ( -248 - 59 \beta_{3} ) q^{79} + 81 q^{81} + ( -388 \beta_{1} - 28 \beta_{2} ) q^{83} + ( -760 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 228 - 27 \beta_{3} ) q^{87} + ( -266 + 8 \beta_{3} ) q^{89} + ( 128 \beta_{1} + 20 \beta_{2} ) q^{91} + ( 120 \beta_{1} + 39 \beta_{2} ) q^{93} + ( -1104 + 84 \beta_{3} ) q^{95} + ( -610 - 88 \beta_{3} ) q^{97} + ( 36 \beta_{1} + 36 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{7} - 36q^{9} + O(q^{10}) \) \( 4q + 32q^{7} - 36q^{9} + 48q^{15} - 72q^{17} + 512q^{23} - 396q^{25} + 160q^{31} + 48q^{33} + 432q^{39} + 872q^{41} + 448q^{47} - 284q^{49} + 3264q^{55} + 816q^{57} - 288q^{63} + 1088q^{65} + 4096q^{71} - 1320q^{73} - 992q^{79} + 324q^{81} + 912q^{87} - 1064q^{89} - 4416q^{95} - 2440q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{3} + 40 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 8 \nu^{2} + 28 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 4 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 28\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} + 10 \beta_{1}\)\()/2\)

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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{4} + 448 T_{5}^{2} + 36864 \)
\( T_{7}^{2} - 16 T_{7} - 144 \)

Hecke characteristic polynomials

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