# Properties

 Label 384.2.c.a Level $384$ Weight $2$ Character orbit 384.c Analytic conductor $3.066$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{3} ) q^{3} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{3} ) q^{3} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( 2 - \beta_{1} + \beta_{3} ) q^{11} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{13} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{15} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{17} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{19} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{21} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{23} + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{25} + ( -3 + \beta_{1} + 2 \beta_{2} ) q^{27} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{29} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{33} + 4 q^{35} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{37} + ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{41} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{43} + ( \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{45} + 8 q^{47} + ( -1 - 2 \beta_{1} + 2 \beta_{3} ) q^{49} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{51} + ( \beta_{1} + \beta_{3} ) q^{53} + ( 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{55} + ( -2 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{57} + ( 2 + \beta_{1} - \beta_{3} ) q^{59} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{61} + ( -4 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{63} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{67} + ( -8 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{69} + ( -4 - 6 \beta_{1} + 6 \beta_{3} ) q^{71} -2 q^{73} + ( -5 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{75} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{77} + ( 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{79} + ( 1 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{81} + ( 6 + \beta_{1} - \beta_{3} ) q^{83} -8 q^{85} + ( 2 + 7 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{87} + 2 \beta_{2} q^{89} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{91} + ( 2 - 3 \beta_{1} + \beta_{3} ) q^{93} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{95} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{97} + ( -2 + 5 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + O(q^{10})$$ $$4q - 2q^{3} + 12q^{11} - 12q^{15} + 8q^{21} + 8q^{23} - 4q^{25} - 14q^{27} + 4q^{33} + 16q^{35} + 16q^{37} - 20q^{39} - 8q^{45} + 32q^{47} + 4q^{49} - 16q^{51} - 4q^{57} + 4q^{59} - 16q^{61} - 8q^{63} - 24q^{69} + 8q^{71} - 8q^{73} - 18q^{75} + 4q^{81} + 20q^{83} - 32q^{85} - 4q^{87} + 16q^{93} - 8q^{95} - 16q^{97} - 20q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## Character values

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

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
383.1
 − 0.618034i 0.618034i − 1.61803i 1.61803i
0 −1.61803 0.618034i 0 1.23607i 0 3.23607i 0 2.23607 + 2.00000i 0
383.2 0 −1.61803 + 0.618034i 0 1.23607i 0 3.23607i 0 2.23607 2.00000i 0
383.3 0 0.618034 1.61803i 0 3.23607i 0 1.23607i 0 −2.23607 2.00000i 0
383.4 0 0.618034 + 1.61803i 0 3.23607i 0 1.23607i 0 −2.23607 + 2.00000i 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
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 384.2.c.a 4
3.b odd 2 1 384.2.c.d yes 4
4.b odd 2 1 384.2.c.d yes 4
8.b even 2 1 384.2.c.c yes 4
8.d odd 2 1 384.2.c.b yes 4
12.b even 2 1 inner 384.2.c.a 4
16.e even 4 1 768.2.f.c 4
16.e even 4 1 768.2.f.e 4
16.f odd 4 1 768.2.f.b 4
16.f odd 4 1 768.2.f.f 4
24.f even 2 1 384.2.c.c yes 4
24.h odd 2 1 384.2.c.b yes 4
48.i odd 4 1 768.2.f.b 4
48.i odd 4 1 768.2.f.f 4
48.k even 4 1 768.2.f.c 4
48.k even 4 1 768.2.f.e 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{11}^{2} - 6 T_{11} + 4$$ $$T_{23}^{2} - 4 T_{23} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 6 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$16 + 12 T^{2} + T^{4}$$
$7$ $$16 + 12 T^{2} + T^{4}$$
$11$ $$( 4 - 6 T + T^{2} )^{2}$$
$13$ $$( -20 + T^{2} )^{2}$$
$17$ $$256 + 48 T^{2} + T^{4}$$
$19$ $$16 + 28 T^{2} + T^{4}$$
$23$ $$( -16 - 4 T + T^{2} )^{2}$$
$29$ $$1936 + 108 T^{2} + T^{4}$$
$31$ $$16 + 28 T^{2} + T^{4}$$
$37$ $$( -4 - 8 T + T^{2} )^{2}$$
$41$ $$256 + 48 T^{2} + T^{4}$$
$43$ $$400 + 60 T^{2} + T^{4}$$
$47$ $$( -8 + T )^{4}$$
$53$ $$16 + 12 T^{2} + T^{4}$$
$59$ $$( -4 - 2 T + T^{2} )^{2}$$
$61$ $$( -4 + 8 T + T^{2} )^{2}$$
$67$ $$1296 + 108 T^{2} + T^{4}$$
$71$ $$( -176 - 4 T + T^{2} )^{2}$$
$73$ $$( 2 + T )^{4}$$
$79$ $$16 + 188 T^{2} + T^{4}$$
$83$ $$( 20 - 10 T + T^{2} )^{2}$$
$89$ $$( 16 + T^{2} )^{2}$$
$97$ $$( -4 + 8 T + T^{2} )^{2}$$