# Properties

 Label 384.6.f.a Level $384$ Weight $6$ Character orbit 384.f Analytic conductor $61.587$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 384.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$61.5873868082$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 9 \beta_{2} q^{3} + 11 \beta_{1} q^{5} -19 \beta_{3} q^{7} -243 q^{9} +O(q^{10})$$ $$q + 9 \beta_{2} q^{3} + 11 \beta_{1} q^{5} -19 \beta_{3} q^{7} -243 q^{9} -178 \beta_{2} q^{11} -99 \beta_{3} q^{15} -513 \beta_{1} q^{21} -2157 q^{25} -2187 \beta_{2} q^{27} -3109 \beta_{1} q^{29} + 1661 \beta_{3} q^{31} + 4806 q^{33} + 1672 \beta_{2} q^{35} -2673 \beta_{1} q^{45} + 8143 q^{49} -5225 \beta_{1} q^{53} + 1958 \beta_{3} q^{55} + 16746 \beta_{2} q^{59} + 4617 \beta_{3} q^{63} -90706 q^{73} -19413 \beta_{2} q^{75} + 10146 \beta_{1} q^{77} + 22377 \beta_{3} q^{79} + 59049 q^{81} + 550 \beta_{2} q^{83} + 27981 \beta_{3} q^{87} + 44847 \beta_{1} q^{93} + 90242 q^{97} + 43254 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 972q^{9} + O(q^{10})$$ $$4q - 972q^{9} - 8628q^{25} + 19224q^{33} + 32572q^{49} - 362824q^{73} + 236196q^{81} + 360968q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 + 1.22474i −0.707107 − 1.22474i
0 15.5885i 0 −31.1127 0 93.0806i 0 −243.000 0
191.2 0 15.5885i 0 31.1127 0 93.0806i 0 −243.000 0
191.3 0 15.5885i 0 −31.1127 0 93.0806i 0 −243.000 0
191.4 0 15.5885i 0 31.1127 0 93.0806i 0 −243.000 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 $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.6.f.a 4
3.b odd 2 1 inner 384.6.f.a 4
4.b odd 2 1 inner 384.6.f.a 4
8.b even 2 1 inner 384.6.f.a 4
8.d odd 2 1 inner 384.6.f.a 4
12.b even 2 1 inner 384.6.f.a 4
24.f even 2 1 inner 384.6.f.a 4
24.h odd 2 1 CM 384.6.f.a 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{2} - 968$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 243 + T^{2} )^{2}$$
$5$ $$( -968 + T^{2} )^{2}$$
$7$ $$( 8664 + T^{2} )^{2}$$
$11$ $$( 95052 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( -77327048 + T^{2} )^{2}$$
$31$ $$( 66214104 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( -218405000 + T^{2} )^{2}$$
$59$ $$( 841285548 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( 90706 + T )^{4}$$
$79$ $$( 12017523096 + T^{2} )^{2}$$
$83$ $$( 907500 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( -90242 + T )^{4}$$