# Properties

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

# Related objects

## 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
 − 2.30278i 1.30278i − 1.30278i 2.30278i
0 3.00000i 0 18.4222i 0 −22.4222 0 −9.00000 0
193.2 0 3.00000i 0 10.4222i 0 6.42221 0 −9.00000 0
193.3 0 3.00000i 0 10.4222i 0 6.42221 0 −9.00000 0
193.4 0 3.00000i 0 18.4222i 0 −22.4222 0 −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.c 4
3.b odd 2 1 1152.4.d.i 4
4.b odd 2 1 384.4.d.e yes 4
8.b even 2 1 inner 384.4.d.c 4
8.d odd 2 1 384.4.d.e yes 4
12.b even 2 1 1152.4.d.o 4
16.e even 4 1 768.4.a.j 2
16.e even 4 1 768.4.a.k 2
16.f odd 4 1 768.4.a.e 2
16.f odd 4 1 768.4.a.p 2
24.f even 2 1 1152.4.d.o 4
24.h odd 2 1 1152.4.d.i 4
48.i odd 4 1 2304.4.a.t 2
48.i odd 4 1 2304.4.a.bq 2
48.k even 4 1 2304.4.a.s 2
48.k even 4 1 2304.4.a.bp 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.c 4 1.a even 1 1 trivial
384.4.d.c 4 8.b even 2 1 inner
384.4.d.e yes 4 4.b odd 2 1
384.4.d.e yes 4 8.d odd 2 1
768.4.a.e 2 16.f odd 4 1
768.4.a.j 2 16.e even 4 1
768.4.a.k 2 16.e even 4 1
768.4.a.p 2 16.f odd 4 1
1152.4.d.i 4 3.b odd 2 1
1152.4.d.i 4 24.h odd 2 1
1152.4.d.o 4 12.b even 2 1
1152.4.d.o 4 24.f even 2 1
2304.4.a.s 2 48.k even 4 1
2304.4.a.t 2 48.i odd 4 1
2304.4.a.bp 2 48.k even 4 1
2304.4.a.bq 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 $$S_{4}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{4} + 448 T_{5}^{2} + 36864$$ $$T_{7}^{2} + 16 T_{7} - 144$$

## Hecke characteristic polynomials

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