# Properties

 Label 384.3.e.a Level $384$ Weight $3$ Character orbit 384.e Analytic conductor $10.463$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.4632421514$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 18 x^{6} + 99 x^{4} + 170 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{10}\cdot 3$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + \beta_{3} q^{5} + ( -1 - \beta_{2} ) q^{7} + ( \beta_{2} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + \beta_{3} q^{5} + ( -1 - \beta_{2} ) q^{7} + ( \beta_{2} + \beta_{7} ) q^{9} + ( -\beta_{4} - \beta_{5} + \beta_{7} ) q^{11} + ( 1 - \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{13} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{15} + ( 1 - 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( 3 - \beta_{1} + 2 \beta_{2} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{19} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{21} + ( -4 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{23} + ( -2 - 5 \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{25} + ( 6 - \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{27} + ( -2 + 6 \beta_{1} + 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{29} + ( 9 - 4 \beta_{1} + \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{31} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{33} + ( 1 - 3 \beta_{1} - 8 \beta_{3} - 5 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{35} + ( -1 + 9 \beta_{1} - 4 \beta_{2} - 3 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{37} + ( 13 - \beta_{1} + 2 \beta_{3} - 5 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{39} + ( -2 + 6 \beta_{1} - 6 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{41} + ( -17 - \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{43} + ( -10 + 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{45} + ( 2 - 6 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{47} + ( 2 + 9 \beta_{1} + 2 \beta_{2} + 3 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} ) q^{49} + ( -23 + \beta_{1} + 4 \beta_{3} + 5 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{51} + ( 4 - 12 \beta_{1} + \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{53} + ( -22 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{55} + ( -1 - 4 \beta_{1} - \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{57} + ( 1 - 3 \beta_{1} - 8 \beta_{4} - \beta_{6} ) q^{59} + ( -13 - 11 \beta_{1} + 4 \beta_{2} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{61} + ( -31 - 3 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} ) q^{63} + ( 4 - 12 \beta_{1} + 2 \beta_{3} - 6 \beta_{4} - 4 \beta_{6} ) q^{65} + ( 37 - \beta_{1} - 8 \beta_{2} - 4 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{67} + ( 8 + 2 \beta_{1} - 6 \beta_{2} + 6 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} ) q^{69} + ( 2 - 6 \beta_{1} - 4 \beta_{3} + 10 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{71} + ( -2 - 12 \beta_{1} + 4 \beta_{2} + 4 \beta_{6} ) q^{73} + ( 43 + 10 \beta_{2} + 9 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{75} + ( -4 + 12 \beta_{1} - 2 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{77} + ( 41 + 16 \beta_{1} - 3 \beta_{2} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{79} + ( 10 - 3 \beta_{1} + 4 \beta_{3} + 8 \beta_{4} - 5 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{81} + ( -2 + 6 \beta_{1} + 8 \beta_{3} - 5 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{83} + ( 16 + 8 \beta_{1} + 16 \beta_{2} + 8 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} ) q^{85} + ( 48 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} + 3 \beta_{6} - 7 \beta_{7} ) q^{87} + ( -5 + 15 \beta_{1} + 16 \beta_{3} - 8 \beta_{4} + \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{89} + ( -40 - 2 \beta_{1} + 10 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{91} + ( 27 - 11 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} + 8 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{93} + ( -8 + 24 \beta_{1} + 4 \beta_{3} + 16 \beta_{4} + 2 \beta_{5} + 8 \beta_{6} - 2 \beta_{7} ) q^{95} + ( 5 + 13 \beta_{1} - 2 \beta_{2} - 5 \beta_{5} - \beta_{6} - 5 \beta_{7} ) q^{97} + ( -55 + 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 10 \beta_{4} - 4 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} - 8q^{7} + O(q^{10})$$ $$8q - 4q^{3} - 8q^{7} - 16q^{15} + 24q^{19} + 16q^{21} - 40q^{25} + 44q^{27} + 56q^{31} + 8q^{33} + 32q^{37} + 104q^{39} - 136q^{43} - 80q^{45} + 72q^{49} - 176q^{51} - 192q^{55} - 40q^{57} - 160q^{61} - 264q^{63} + 280q^{67} + 80q^{69} - 80q^{73} + 348q^{75} + 408q^{79} + 72q^{81} + 192q^{85} + 368q^{87} - 336q^{91} + 160q^{93} + 96q^{97} - 432q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 18 x^{6} + 99 x^{4} + 170 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 16 \nu^{4} + 70 \nu^{2} + 6 \nu + 63$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} + 11 \nu^{2} + 18$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{7} + 33 \nu^{5} + 159 \nu^{3} + 193 \nu$$$$)/9$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} + 36 \nu^{5} + 180 \nu^{3} + 178 \nu$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 16 \nu^{5} - 3 \nu^{4} - 70 \nu^{3} - 27 \nu^{2} - 57 \nu - 27$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 16 \nu^{4} - 70 \nu^{2} + 6 \nu - 61$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 16 \nu^{5} - 3 \nu^{4} + 70 \nu^{3} - 27 \nu^{2} + 63 \nu - 27$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + 3 \beta_{1} - 1$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{7} - \beta_{6} + 3 \beta_{5} + 6 \beta_{2} - 3 \beta_{1} - 53$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{7} - 13 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 12 \beta_{3} - 39 \beta_{1} + 13$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$($$$$-33 \beta_{7} + 11 \beta_{6} - 33 \beta_{5} - 54 \beta_{2} + 33 \beta_{1} + 367$$$$)/12$$ $$\nu^{5}$$ $$=$$ $$($$$$21 \beta_{7} + 101 \beta_{6} - 21 \beta_{5} + 57 \beta_{4} - 120 \beta_{3} + 303 \beta_{1} - 101$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$159 \beta_{7} - 59 \beta_{6} + 159 \beta_{5} + 222 \beta_{2} - 141 \beta_{1} - 1453$$$$)/6$$ $$\nu^{7}$$ $$=$$ $$($$$$-54 \beta_{7} - 413 \beta_{6} + 54 \beta_{5} - 351 \beta_{4} + 540 \beta_{3} - 1239 \beta_{1} + 413$$$$)/6$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.888828i − 0.888828i 2.55118i − 2.55118i 2.98985i − 2.98985i 1.32750i − 1.32750i
0 −2.86531 0.888828i 0 8.59176i 0 −10.9340 0 7.41997 + 5.09353i 0
257.2 0 −2.86531 + 0.888828i 0 8.59176i 0 −10.9340 0 7.41997 5.09353i 0
257.3 0 −1.57844 2.55118i 0 1.31534i 0 10.2329 0 −4.01705 + 8.05378i 0
257.4 0 −1.57844 + 2.55118i 0 1.31534i 0 10.2329 0 −4.01705 8.05378i 0
257.5 0 −0.246559 2.98985i 0 6.63641i 0 −0.578158 0 −8.87842 + 1.47435i 0
257.6 0 −0.246559 + 2.98985i 0 6.63641i 0 −0.578158 0 −8.87842 1.47435i 0
257.7 0 2.69031 1.32750i 0 0.640013i 0 −2.72077 0 5.47550 7.14275i 0
257.8 0 2.69031 + 1.32750i 0 0.640013i 0 −2.72077 0 5.47550 + 7.14275i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.e.a 8
3.b odd 2 1 inner 384.3.e.a 8
4.b odd 2 1 384.3.e.d yes 8
8.b even 2 1 384.3.e.c yes 8
8.d odd 2 1 384.3.e.b yes 8
12.b even 2 1 384.3.e.d yes 8
16.e even 4 2 768.3.h.h 16
16.f odd 4 2 768.3.h.g 16
24.f even 2 1 384.3.e.b yes 8
24.h odd 2 1 384.3.e.c yes 8
48.i odd 4 2 768.3.h.h 16
48.k even 4 2 768.3.h.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.e.a 8 1.a even 1 1 trivial
384.3.e.a 8 3.b odd 2 1 inner
384.3.e.b yes 8 8.d odd 2 1
384.3.e.b yes 8 24.f even 2 1
384.3.e.c yes 8 8.b even 2 1
384.3.e.c yes 8 24.h odd 2 1
384.3.e.d yes 8 4.b odd 2 1
384.3.e.d yes 8 12.b even 2 1
768.3.h.g 16 16.f odd 4 2
768.3.h.g 16 48.k even 4 2
768.3.h.h 16 16.e even 4 2
768.3.h.h 16 48.i odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{7}^{4} + 4 T_{7}^{3} - 108 T_{7}^{2} - 368 T_{7} - 176$$ $$T_{13}^{4} - 360 T_{13}^{2} + 256 T_{13} + 7824$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$6561 + 2916 T + 648 T^{2} - 36 T^{3} - 66 T^{4} - 4 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$5$ $$2304 + 7040 T^{2} + 3504 T^{4} + 120 T^{6} + T^{8}$$
$7$ $$( -176 - 368 T - 108 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$11$ $$20214016 + 3167360 T^{2} + 69552 T^{4} + 488 T^{6} + T^{8}$$
$13$ $$( 7824 + 256 T - 360 T^{2} + T^{4} )^{2}$$
$17$ $$991494144 + 60416000 T^{2} + 480000 T^{4} + 1248 T^{6} + T^{8}$$
$19$ $$( -98352 + 17584 T - 780 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$23$ $$48132849664 + 478846976 T^{2} + 1671936 T^{4} + 2336 T^{6} + T^{8}$$
$29$ $$78767790336 + 896411520 T^{2} + 3074736 T^{4} + 3704 T^{6} + T^{8}$$
$31$ $$( 50256 + 17520 T - 828 T^{2} - 28 T^{3} + T^{4} )^{2}$$
$37$ $$( -488432 + 90944 T - 3528 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$41$ $$3916979306496 + 31405105152 T^{2} + 34597632 T^{4} + 10976 T^{6} + T^{8}$$
$43$ $$( -917424 - 92496 T - 588 T^{2} + 68 T^{3} + T^{4} )^{2}$$
$47$ $$424144797696 + 3025141760 T^{2} + 6905856 T^{4} + 5376 T^{6} + T^{8}$$
$53$ $$870911869798656 + 673606710144 T^{2} + 188831664 T^{4} + 22776 T^{6} + T^{8}$$
$59$ $$12745356964096 + 31548679808 T^{2} + 26455344 T^{4} + 8840 T^{6} + T^{8}$$
$61$ $$( -4584688 - 262208 T - 2184 T^{2} + 80 T^{3} + T^{4} )^{2}$$
$67$ $$( -6784816 + 335312 T + 900 T^{2} - 140 T^{3} + T^{4} )^{2}$$
$71$ $$1706597351424 + 36124434432 T^{2} + 77642496 T^{4} + 22688 T^{6} + T^{8}$$
$73$ $$( -899312 - 192608 T - 5544 T^{2} + 40 T^{3} + T^{4} )^{2}$$
$79$ $$( -38762928 + 676400 T + 7524 T^{2} - 204 T^{3} + T^{4} )^{2}$$
$83$ $$56638749360384 + 102766133376 T^{2} + 59473584 T^{4} + 13416 T^{6} + T^{8}$$
$89$ $$9213001971859456 + 4688908992512 T^{2} + 730308096 T^{4} + 45632 T^{6} + T^{8}$$
$97$ $$( 8416272 + 274112 T - 8136 T^{2} - 48 T^{3} + T^{4} )^{2}$$