# Properties

 Label 384.5.h.a Level $384$ Weight $5$ Character orbit 384.h Self dual yes Analytic conductor $39.694$ Analytic rank $0$ Dimension $2$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$39.6940658242$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 8\sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 9 q^{3} + \beta q^{5} - 5 \beta q^{7} + 81 q^{9} +O(q^{10})$$ q - 9 * q^3 + b * q^5 - 5*b * q^7 + 81 * q^9 $$q - 9 q^{3} + \beta q^{5} - 5 \beta q^{7} + 81 q^{9} - 142 q^{11} - 9 \beta q^{15} + 45 \beta q^{21} - 241 q^{25} - 729 q^{27} - 75 \beta q^{29} + 95 \beta q^{31} + 1278 q^{33} - 1920 q^{35} + 81 \beta q^{45} + 7199 q^{49} - 235 \beta q^{53} - 142 \beta q^{55} + 6862 q^{59} - 405 \beta q^{63} + 8158 q^{73} + 2169 q^{75} + 710 \beta q^{77} + 435 \beta q^{79} + 6561 q^{81} + 4178 q^{83} + 675 \beta q^{87} - 855 \beta q^{93} + 17282 q^{97} - 11502 q^{99} +O(q^{100})$$ q - 9 * q^3 + b * q^5 - 5*b * q^7 + 81 * q^9 - 142 * q^11 - 9*b * q^15 + 45*b * q^21 - 241 * q^25 - 729 * q^27 - 75*b * q^29 + 95*b * q^31 + 1278 * q^33 - 1920 * q^35 + 81*b * q^45 + 7199 * q^49 - 235*b * q^53 - 142*b * q^55 + 6862 * q^59 - 405*b * q^63 + 8158 * q^73 + 2169 * q^75 + 710*b * q^77 + 435*b * q^79 + 6561 * q^81 + 4178 * q^83 + 675*b * q^87 - 855*b * q^93 + 17282 * q^97 - 11502 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{3} + 162 q^{9}+O(q^{10})$$ 2 * q - 18 * q^3 + 162 * q^9 $$2 q - 18 q^{3} + 162 q^{9} - 284 q^{11} - 482 q^{25} - 1458 q^{27} + 2556 q^{33} - 3840 q^{35} + 14398 q^{49} + 13724 q^{59} + 16316 q^{73} + 4338 q^{75} + 13122 q^{81} + 8356 q^{83} + 34564 q^{97} - 23004 q^{99}+O(q^{100})$$ 2 * q - 18 * q^3 + 162 * q^9 - 284 * q^11 - 482 * q^25 - 1458 * q^27 + 2556 * q^33 - 3840 * q^35 + 14398 * q^49 + 13724 * q^59 + 16316 * q^73 + 4338 * q^75 + 13122 * q^81 + 8356 * q^83 + 34564 * q^97 - 23004 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −2.44949 2.44949
0 −9.00000 0 −19.5959 0 97.9796 0 81.0000 0
65.2 0 −9.00000 0 19.5959 0 −97.9796 0 81.0000 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
8.d odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.5.h.a 2
3.b odd 2 1 384.5.h.d yes 2
4.b odd 2 1 384.5.h.d yes 2
8.b even 2 1 384.5.h.d yes 2
8.d odd 2 1 inner 384.5.h.a 2
12.b even 2 1 inner 384.5.h.a 2
24.f even 2 1 384.5.h.d yes 2
24.h odd 2 1 CM 384.5.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.h.a 2 1.a even 1 1 trivial
384.5.h.a 2 8.d odd 2 1 inner
384.5.h.a 2 12.b even 2 1 inner
384.5.h.a 2 24.h odd 2 1 CM
384.5.h.d yes 2 3.b odd 2 1
384.5.h.d yes 2 4.b odd 2 1
384.5.h.d yes 2 8.b even 2 1
384.5.h.d yes 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{2} - 384$$ T5^2 - 384 $$T_{11} + 142$$ T11 + 142

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 9)^{2}$$
$5$ $$T^{2} - 384$$
$7$ $$T^{2} - 9600$$
$11$ $$(T + 142)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 2160000$$
$31$ $$T^{2} - 3465600$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 21206400$$
$59$ $$(T - 6862)^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$(T - 8158)^{2}$$
$79$ $$T^{2} - 72662400$$
$83$ $$(T - 4178)^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 17282)^{2}$$