# Properties

 Label 384.2.j.a Level $384$ Weight $2$ Character orbit 384.j Analytic conductor $3.066$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 384.j (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( \beta_{3} - \beta_{5} ) q^{5} + ( \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + \beta_{4} q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( \beta_{3} - \beta_{5} ) q^{5} + ( \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + \beta_{4} q^{9} + ( -1 + \beta_{1} + \beta_{4} + \beta_{7} ) q^{11} + ( -\beta_{1} - 2 \beta_{2} + \beta_{7} ) q^{13} + ( 1 + \beta_{1} ) q^{15} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{17} + ( -1 - \beta_{1} - \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{19} + ( -\beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{21} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{23} + ( 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{25} -\beta_{3} q^{27} + ( 2 + 3 \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{29} + ( -3 - \beta_{1} - \beta_{5} + \beta_{6} ) q^{31} + ( \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{33} + ( 3 + \beta_{1} + 3 \beta_{4} - \beta_{7} ) q^{35} + ( 2 + \beta_{1} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{37} + ( 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{39} + ( -3 \beta_{2} - 3 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{41} + ( -1 - \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{43} + ( -\beta_{2} + \beta_{6} ) q^{45} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( -1 + 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{49} + ( 1 + \beta_{1} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{51} + ( -2 - 5 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{53} + ( 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} ) q^{55} + ( \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{57} + ( 4 - 4 \beta_{4} ) q^{59} + ( -2 + \beta_{1} + 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{61} + ( -1 + \beta_{1} + \beta_{5} - \beta_{6} ) q^{63} + ( -2 + 2 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + \beta_{5} - \beta_{6} ) q^{65} + ( -2 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{7} ) q^{67} + ( -2 + 2 \beta_{4} ) q^{69} + ( -2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( -4 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{73} + ( 2 + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{75} + ( -2 + 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{77} + ( 3 - 3 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + \beta_{5} - \beta_{6} ) q^{79} - q^{81} + ( -5 + \beta_{1} - 5 \beta_{4} - 4 \beta_{6} - \beta_{7} ) q^{83} + ( 2 + 2 \beta_{1} - 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{85} + ( -2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{7} ) q^{87} + ( 6 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{89} + ( -1 - \beta_{1} + 4 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{91} + ( \beta_{1} + 3 \beta_{2} - \beta_{6} - \beta_{7} ) q^{93} + ( 6 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{95} + ( 6 \beta_{2} - 6 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{97} + ( -1 + \beta_{1} - \beta_{4} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{11} + 8q^{15} - 8q^{19} + 16q^{29} - 24q^{31} + 24q^{35} + 16q^{37} - 8q^{43} - 8q^{49} + 8q^{51} - 16q^{53} + 32q^{59} - 16q^{61} - 8q^{63} - 16q^{65} - 16q^{67} - 16q^{69} + 16q^{75} - 16q^{77} + 24q^{79} - 8q^{81} - 40q^{83} + 16q^{85} - 8q^{91} + 48q^{95} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{6} + 3 \nu^{5} - 11 \nu^{4} + 17 \nu^{3} - 24 \nu^{2} + 16 \nu - 5$$ $$\beta_{2}$$ $$=$$ $$5 \nu^{7} - 17 \nu^{6} + 60 \nu^{5} - 105 \nu^{4} + 155 \nu^{3} - 133 \nu^{2} + 77 \nu - 19$$ $$\beta_{3}$$ $$=$$ $$5 \nu^{7} - 18 \nu^{6} + 63 \nu^{5} - 115 \nu^{4} + 170 \nu^{3} - 152 \nu^{2} + 89 \nu - 23$$ $$\beta_{4}$$ $$=$$ $$8 \nu^{7} - 28 \nu^{6} + 98 \nu^{5} - 175 \nu^{4} + 256 \nu^{3} - 223 \nu^{2} + 126 \nu - 31$$ $$\beta_{5}$$ $$=$$ $$9 \nu^{7} - 31 \nu^{6} + 108 \nu^{5} - 190 \nu^{4} + 275 \nu^{3} - 236 \nu^{2} + 131 \nu - 33$$ $$\beta_{6}$$ $$=$$ $$9 \nu^{7} - 32 \nu^{6} + 111 \nu^{5} - 200 \nu^{4} + 290 \nu^{3} - 253 \nu^{2} + 141 \nu - 33$$ $$\beta_{7}$$ $$=$$ $$10 \nu^{7} - 35 \nu^{6} + 123 \nu^{5} - 220 \nu^{4} + 325 \nu^{3} - 285 \nu^{2} + 168 \nu - 43$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{3} - \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} + 2 \beta_{6} - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} - 5$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{7} - \beta_{6} + 3 \beta_{5} - 6 \beta_{4} + 12 \beta_{3} + 4 \beta_{2} - 2 \beta_{1} + 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$3 \beta_{7} - 10 \beta_{6} + 5 \beta_{5} + 10 \beta_{4} + 6 \beta_{3} - 19 \beta_{2} - 5 \beta_{1} + 26$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$22 \beta_{7} - 9 \beta_{6} - 11 \beta_{5} + 45 \beta_{4} - 48 \beta_{3} - 32 \beta_{2} + 5 \beta_{1} - 6$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$7 \beta_{7} + 33 \beta_{6} - 30 \beta_{5} - 83 \beta_{3} + 64 \beta_{2} + 35 \beta_{1} - 118$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.5 + 1.44392i 0.5 − 0.0297061i 0.5 − 2.10607i 0.5 + 0.691860i 0.5 − 1.44392i 0.5 + 0.0297061i 0.5 + 2.10607i 0.5 − 0.691860i
0 −0.707107 + 0.707107i 0 −1.74912 1.74912i 0 2.55765i 0 1.00000i 0
97.2 0 −0.707107 + 0.707107i 0 0.334904 + 0.334904i 0 4.55765i 0 1.00000i 0
97.3 0 0.707107 0.707107i 0 −1.27133 1.27133i 0 0.158942i 0 1.00000i 0
97.4 0 0.707107 0.707107i 0 2.68554 + 2.68554i 0 2.15894i 0 1.00000i 0
289.1 0 −0.707107 0.707107i 0 −1.74912 + 1.74912i 0 2.55765i 0 1.00000i 0
289.2 0 −0.707107 0.707107i 0 0.334904 0.334904i 0 4.55765i 0 1.00000i 0
289.3 0 0.707107 + 0.707107i 0 −1.27133 + 1.27133i 0 0.158942i 0 1.00000i 0
289.4 0 0.707107 + 0.707107i 0 2.68554 2.68554i 0 2.15894i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.2.j.a 8
3.b odd 2 1 1152.2.k.f 8
4.b odd 2 1 384.2.j.b 8
8.b even 2 1 192.2.j.a 8
8.d odd 2 1 48.2.j.a 8
12.b even 2 1 1152.2.k.c 8
16.e even 4 1 192.2.j.a 8
16.e even 4 1 inner 384.2.j.a 8
16.f odd 4 1 48.2.j.a 8
16.f odd 4 1 384.2.j.b 8
24.f even 2 1 144.2.k.b 8
24.h odd 2 1 576.2.k.b 8
32.g even 8 1 3072.2.a.n 4
32.g even 8 1 3072.2.a.o 4
32.g even 8 2 3072.2.d.i 8
32.h odd 8 1 3072.2.a.i 4
32.h odd 8 1 3072.2.a.t 4
32.h odd 8 2 3072.2.d.f 8
48.i odd 4 1 576.2.k.b 8
48.i odd 4 1 1152.2.k.f 8
48.k even 4 1 144.2.k.b 8
48.k even 4 1 1152.2.k.c 8
96.o even 8 1 9216.2.a.y 4
96.o even 8 1 9216.2.a.bo 4
96.p odd 8 1 9216.2.a.x 4
96.p odd 8 1 9216.2.a.bn 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 8.d odd 2 1
48.2.j.a 8 16.f odd 4 1
144.2.k.b 8 24.f even 2 1
144.2.k.b 8 48.k even 4 1
192.2.j.a 8 8.b even 2 1
192.2.j.a 8 16.e even 4 1
384.2.j.a 8 1.a even 1 1 trivial
384.2.j.a 8 16.e even 4 1 inner
384.2.j.b 8 4.b odd 2 1
384.2.j.b 8 16.f odd 4 1
576.2.k.b 8 24.h odd 2 1
576.2.k.b 8 48.i odd 4 1
1152.2.k.c 8 12.b even 2 1
1152.2.k.c 8 48.k even 4 1
1152.2.k.f 8 3.b odd 2 1
1152.2.k.f 8 48.i odd 4 1
3072.2.a.i 4 32.h odd 8 1
3072.2.a.n 4 32.g even 8 1
3072.2.a.o 4 32.g even 8 1
3072.2.a.t 4 32.h odd 8 1
3072.2.d.f 8 32.h odd 8 2
3072.2.d.i 8 32.g even 8 2
9216.2.a.x 4 96.p odd 8 1
9216.2.a.y 4 96.o even 8 1
9216.2.a.bn 4 96.p odd 8 1
9216.2.a.bo 4 96.o even 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{8} + 8 T_{11}^{7} + 32 T_{11}^{6} + 256 T_{11}^{3} + 2048 T_{11}^{2} - 2048 T_{11} + 1024$$ acting on $$S_{2}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 + T^{4} )^{2}$$
$5$ $$64 - 128 T + 128 T^{2} + 192 T^{3} + 128 T^{4} + 16 T^{5} + T^{8}$$
$7$ $$16 + 640 T^{2} + 264 T^{4} + 32 T^{6} + T^{8}$$
$11$ $$1024 - 2048 T + 2048 T^{2} + 256 T^{3} + 32 T^{6} + 8 T^{7} + T^{8}$$
$13$ $$16 - 256 T + 2048 T^{2} + 1792 T^{3} + 776 T^{4} + 64 T^{5} + T^{8}$$
$17$ $$( 16 + 64 T - 32 T^{2} + T^{4} )^{2}$$
$19$ $$256 - 1536 T + 4608 T^{2} + 1408 T^{3} + 224 T^{4} - 32 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$23$ $$( 8 + T^{2} )^{4}$$
$29$ $$61504 - 35712 T + 10368 T^{2} - 1088 T^{3} + 896 T^{4} - 464 T^{5} + 128 T^{6} - 16 T^{7} + T^{8}$$
$31$ $$( -28 + 24 T + 40 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$37$ $$1106704 - 639616 T + 184832 T^{2} - 24128 T^{3} + 2248 T^{4} - 416 T^{5} + 128 T^{6} - 16 T^{7} + T^{8}$$
$41$ $$12544 + 22528 T^{2} + 3872 T^{4} + 128 T^{6} + T^{8}$$
$43$ $$12544 + 39424 T + 61952 T^{2} + 29056 T^{3} + 6624 T^{4} - 288 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$47$ $$( -8 + T^{2} )^{4}$$
$53$ $$18496 - 36992 T + 36992 T^{2} + 1088 T^{3} - 128 T^{4} + 80 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$59$ $$( 32 - 8 T + T^{2} )^{4}$$
$61$ $$1106704 + 639616 T + 184832 T^{2} + 24128 T^{3} + 2248 T^{4} + 416 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$67$ $$65536 - 65536 T + 32768 T^{2} + 12288 T^{3} + 3584 T^{4} - 768 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$71$ $$4096 + 40960 T^{2} + 4224 T^{4} + 128 T^{6} + T^{8}$$
$73$ $$4096 + 16384 T^{2} + 8320 T^{4} + 256 T^{6} + T^{8}$$
$79$ $$( -10108 + 2888 T - 168 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$83$ $$1024 - 10240 T + 51200 T^{2} - 60160 T^{3} + 33792 T^{4} + 7680 T^{5} + 800 T^{6} + 40 T^{7} + T^{8}$$
$89$ $$3625216 + 1901824 T^{2} + 62304 T^{4} + 464 T^{6} + T^{8}$$
$97$ $$( 512 + 768 T - 224 T^{2} + T^{4} )^{2}$$