# Properties

 Label 384.5.b.a Level $384$ Weight $5$ Character orbit 384.b Analytic conductor $39.694$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 384.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.6940658242$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - 2x^{3} + 5x^{2} - 4x + 61$$ x^4 - 2*x^3 + 5*x^2 - 4*x + 61 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}\cdot 3$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \beta_1 q^{3} + \beta_{2} q^{5} + \beta_{3} q^{7} + 27 q^{9}+O(q^{10})$$ q + 3*b1 * q^3 + b2 * q^5 + b3 * q^7 + 27 * q^9 $$q + 3 \beta_1 q^{3} + \beta_{2} q^{5} + \beta_{3} q^{7} + 27 q^{9} - 52 \beta_1 q^{11} + 2 \beta_{2} q^{13} + 3 \beta_{3} q^{15} - 338 q^{17} - 4 \beta_1 q^{19} + 9 \beta_{2} q^{21} + 14 \beta_{3} q^{23} - 287 q^{25} + 81 \beta_1 q^{27} - 43 \beta_{2} q^{29} - 25 \beta_{3} q^{31} - 468 q^{33} - 912 \beta_1 q^{35} + 8 \beta_{2} q^{37} + 6 \beta_{3} q^{39} + 578 q^{41} - 1172 \beta_1 q^{43} + 27 \beta_{2} q^{45} - 42 \beta_{3} q^{47} - 335 q^{49} - 1014 \beta_1 q^{51} - 81 \beta_{2} q^{53} - 52 \beta_{3} q^{55} - 36 q^{57} - 692 \beta_1 q^{59} + 212 \beta_{2} q^{61} + 27 \beta_{3} q^{63} - 1824 q^{65} - 4772 \beta_1 q^{67} + 126 \beta_{2} q^{69} + 82 \beta_{3} q^{71} + 8734 q^{73} - 861 \beta_1 q^{75} - 156 \beta_{2} q^{77} + 215 \beta_{3} q^{79} + 729 q^{81} - 7620 \beta_1 q^{83} - 338 \beta_{2} q^{85} - 129 \beta_{3} q^{87} + 910 q^{89} - 1824 \beta_1 q^{91} - 225 \beta_{2} q^{93} - 4 \beta_{3} q^{95} + 5422 q^{97} - 1404 \beta_1 q^{99}+O(q^{100})$$ q + 3*b1 * q^3 + b2 * q^5 + b3 * q^7 + 27 * q^9 - 52*b1 * q^11 + 2*b2 * q^13 + 3*b3 * q^15 - 338 * q^17 - 4*b1 * q^19 + 9*b2 * q^21 + 14*b3 * q^23 - 287 * q^25 + 81*b1 * q^27 - 43*b2 * q^29 - 25*b3 * q^31 - 468 * q^33 - 912*b1 * q^35 + 8*b2 * q^37 + 6*b3 * q^39 + 578 * q^41 - 1172*b1 * q^43 + 27*b2 * q^45 - 42*b3 * q^47 - 335 * q^49 - 1014*b1 * q^51 - 81*b2 * q^53 - 52*b3 * q^55 - 36 * q^57 - 692*b1 * q^59 + 212*b2 * q^61 + 27*b3 * q^63 - 1824 * q^65 - 4772*b1 * q^67 + 126*b2 * q^69 + 82*b3 * q^71 + 8734 * q^73 - 861*b1 * q^75 - 156*b2 * q^77 + 215*b3 * q^79 + 729 * q^81 - 7620*b1 * q^83 - 338*b2 * q^85 - 129*b3 * q^87 + 910 * q^89 - 1824*b1 * q^91 - 225*b2 * q^93 - 4*b3 * q^95 + 5422 * q^97 - 1404*b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 108 q^{9}+O(q^{10})$$ 4 * q + 108 * q^9 $$4 q + 108 q^{9} - 1352 q^{17} - 1148 q^{25} - 1872 q^{33} + 2312 q^{41} - 1340 q^{49} - 144 q^{57} - 7296 q^{65} + 34936 q^{73} + 2916 q^{81} + 3640 q^{89} + 21688 q^{97}+O(q^{100})$$ 4 * q + 108 * q^9 - 1352 * q^17 - 1148 * q^25 - 1872 * q^33 + 2312 * q^41 - 1340 * q^49 - 144 * q^57 - 7296 * q^65 + 34936 * q^73 + 2916 * q^81 + 3640 * q^89 + 21688 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} + 5x^{2} - 4x + 61$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{3} + 3\nu^{2} + 7\nu - 4 ) / 31$$ (-2*v^3 + 3*v^2 + 7*v - 4) / 31 $$\beta_{2}$$ $$=$$ $$4\nu^{2} - 4\nu + 8$$ 4*v^2 - 4*v + 8 $$\beta_{3}$$ $$=$$ $$( 48\nu^{3} - 72\nu^{2} + 576\nu - 276 ) / 31$$ (48*v^3 - 72*v^2 + 576*v - 276) / 31
 $$\nu$$ $$=$$ $$( \beta_{3} + 24\beta _1 + 12 ) / 24$$ (b3 + 24*b1 + 12) / 24 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 6\beta_{2} + 24\beta _1 - 36 ) / 24$$ (b3 + 6*b2 + 24*b1 - 36) / 24 $$\nu^{3}$$ $$=$$ $$( 5\beta_{3} + 9\beta_{2} - 252\beta _1 - 60 ) / 24$$ (5*b3 + 9*b2 - 252*b1 - 60) / 24

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 −1.23205 + 2.17945i −1.23205 − 2.17945i 2.23205 − 2.17945i 2.23205 + 2.17945i
0 −5.19615 0 30.1993i 0 52.3068i 0 27.0000 0
319.2 0 −5.19615 0 30.1993i 0 52.3068i 0 27.0000 0
319.3 0 5.19615 0 30.1993i 0 52.3068i 0 27.0000 0
319.4 0 5.19615 0 30.1993i 0 52.3068i 0 27.0000 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.5.b.a 4
3.b odd 2 1 1152.5.b.j 4
4.b odd 2 1 inner 384.5.b.a 4
8.b even 2 1 inner 384.5.b.a 4
8.d odd 2 1 inner 384.5.b.a 4
12.b even 2 1 1152.5.b.j 4
16.e even 4 2 768.5.g.d 4
16.f odd 4 2 768.5.g.d 4
24.f even 2 1 1152.5.b.j 4
24.h odd 2 1 1152.5.b.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.b.a 4 1.a even 1 1 trivial
384.5.b.a 4 4.b odd 2 1 inner
384.5.b.a 4 8.b even 2 1 inner
384.5.b.a 4 8.d odd 2 1 inner
768.5.g.d 4 16.e even 4 2
768.5.g.d 4 16.f odd 4 2
1152.5.b.j 4 3.b odd 2 1
1152.5.b.j 4 12.b even 2 1
1152.5.b.j 4 24.f even 2 1
1152.5.b.j 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 912$$ acting on $$S_{5}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 27)^{2}$$
$5$ $$(T^{2} + 912)^{2}$$
$7$ $$(T^{2} + 2736)^{2}$$
$11$ $$(T^{2} - 8112)^{2}$$
$13$ $$(T^{2} + 3648)^{2}$$
$17$ $$(T + 338)^{4}$$
$19$ $$(T^{2} - 48)^{2}$$
$23$ $$(T^{2} + 536256)^{2}$$
$29$ $$(T^{2} + 1686288)^{2}$$
$31$ $$(T^{2} + 1710000)^{2}$$
$37$ $$(T^{2} + 58368)^{2}$$
$41$ $$(T - 578)^{4}$$
$43$ $$(T^{2} - 4120752)^{2}$$
$47$ $$(T^{2} + 4826304)^{2}$$
$53$ $$(T^{2} + 5983632)^{2}$$
$59$ $$(T^{2} - 1436592)^{2}$$
$61$ $$(T^{2} + 40988928)^{2}$$
$67$ $$(T^{2} - 68315952)^{2}$$
$71$ $$(T^{2} + 18396864)^{2}$$
$73$ $$(T - 8734)^{4}$$
$79$ $$(T^{2} + 126471600)^{2}$$
$83$ $$(T^{2} - 174193200)^{2}$$
$89$ $$(T - 910)^{4}$$
$97$ $$(T - 5422)^{4}$$