# Properties

 Label 384.3.h.e Level $384$ Weight $3$ Character orbit 384.h Analytic conductor $10.463$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.4632421514$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} -4 q^{5} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{7} + ( 1 - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} -4 q^{5} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{7} + ( 1 - \beta_{2} ) q^{9} + ( -\beta_{1} - \beta_{3} ) q^{11} + 2 \beta_{2} q^{13} -4 \beta_{3} q^{15} + 2 \beta_{2} q^{17} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{19} + ( -20 + 2 \beta_{2} ) q^{21} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{23} -9 q^{25} + ( -9 \beta_{1} + 2 \beta_{3} ) q^{27} -52 q^{29} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{31} + ( -10 + \beta_{2} ) q^{33} + ( 8 \beta_{1} + 8 \beta_{3} ) q^{35} + 6 \beta_{2} q^{37} + ( 18 \beta_{1} - 2 \beta_{3} ) q^{39} -4 \beta_{2} q^{41} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{43} + ( -4 + 4 \beta_{2} ) q^{45} + ( -16 \beta_{1} + 16 \beta_{3} ) q^{47} + 31 q^{49} + ( 18 \beta_{1} - 2 \beta_{3} ) q^{51} -20 q^{53} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{55} + ( 40 + 5 \beta_{2} ) q^{57} + ( -23 \beta_{1} - 23 \beta_{3} ) q^{59} + 2 \beta_{2} q^{61} + ( 18 \beta_{1} - 22 \beta_{3} ) q^{63} -8 \beta_{2} q^{65} + ( 11 \beta_{1} - 11 \beta_{3} ) q^{67} + ( -32 - 4 \beta_{2} ) q^{69} + ( -20 \beta_{1} + 20 \beta_{3} ) q^{71} -50 q^{73} -9 \beta_{3} q^{75} + 40 q^{77} + ( -18 \beta_{1} - 18 \beta_{3} ) q^{79} + ( -79 - 2 \beta_{2} ) q^{81} + ( 23 \beta_{1} + 23 \beta_{3} ) q^{83} -8 \beta_{2} q^{85} -52 \beta_{3} q^{87} + 18 \beta_{2} q^{89} + ( -40 \beta_{1} + 40 \beta_{3} ) q^{91} + ( 60 - 6 \beta_{2} ) q^{93} + ( -20 \beta_{1} + 20 \beta_{3} ) q^{95} + 50 q^{97} + ( 9 \beta_{1} - 11 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{5} + 4q^{9} + O(q^{10})$$ $$4q - 16q^{5} + 4q^{9} - 80q^{21} - 36q^{25} - 208q^{29} - 40q^{33} - 16q^{45} + 124q^{49} - 80q^{53} + 160q^{57} - 128q^{69} - 200q^{73} + 160q^{77} - 316q^{81} + 240q^{93} + 200q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} + 2 \nu^{2} + 4 \nu + 3$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{3} + 16 \nu$$ $$\beta_{3}$$ $$=$$ $$-2 \nu^{3} + 2 \nu^{2} - 4 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{1} - 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3} - \beta_{2} + 2 \beta_{1}$$$$)/4$$

## 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
 − 1.61803i 1.61803i 0.618034i − 0.618034i
0 −2.23607 2.00000i 0 −4.00000 0 8.94427 0 1.00000 + 8.94427i 0
65.2 0 −2.23607 + 2.00000i 0 −4.00000 0 8.94427 0 1.00000 8.94427i 0
65.3 0 2.23607 2.00000i 0 −4.00000 0 −8.94427 0 1.00000 8.94427i 0
65.4 0 2.23607 + 2.00000i 0 −4.00000 0 −8.94427 0 1.00000 + 8.94427i 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
24.f even 2 1 inner
24.h 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.h.e 4
3.b odd 2 1 384.3.h.f yes 4
4.b odd 2 1 inner 384.3.h.e 4
8.b even 2 1 384.3.h.f yes 4
8.d odd 2 1 384.3.h.f yes 4
12.b even 2 1 384.3.h.f yes 4
16.e even 4 1 768.3.e.h 4
16.e even 4 1 768.3.e.m 4
16.f odd 4 1 768.3.e.h 4
16.f odd 4 1 768.3.e.m 4
24.f even 2 1 inner 384.3.h.e 4
24.h odd 2 1 inner 384.3.h.e 4
48.i odd 4 1 768.3.e.h 4
48.i odd 4 1 768.3.e.m 4
48.k even 4 1 768.3.e.h 4
48.k even 4 1 768.3.e.m 4

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

## Hecke kernels

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

 $$T_{5} + 4$$ $$T_{11}^{2} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 2 T^{2} + T^{4}$$
$5$ $$( 4 + T )^{4}$$
$7$ $$( -80 + T^{2} )^{2}$$
$11$ $$( -20 + T^{2} )^{2}$$
$13$ $$( 320 + T^{2} )^{2}$$
$17$ $$( 320 + T^{2} )^{2}$$
$19$ $$( 400 + T^{2} )^{2}$$
$23$ $$( 256 + T^{2} )^{2}$$
$29$ $$( 52 + T )^{4}$$
$31$ $$( -720 + T^{2} )^{2}$$
$37$ $$( 2880 + T^{2} )^{2}$$
$41$ $$( 1280 + T^{2} )^{2}$$
$43$ $$( 1296 + T^{2} )^{2}$$
$47$ $$( 4096 + T^{2} )^{2}$$
$53$ $$( 20 + T )^{4}$$
$59$ $$( -10580 + T^{2} )^{2}$$
$61$ $$( 320 + T^{2} )^{2}$$
$67$ $$( 1936 + T^{2} )^{2}$$
$71$ $$( 6400 + T^{2} )^{2}$$
$73$ $$( 50 + T )^{4}$$
$79$ $$( -6480 + T^{2} )^{2}$$
$83$ $$( -10580 + T^{2} )^{2}$$
$89$ $$( 25920 + T^{2} )^{2}$$
$97$ $$( -50 + T )^{4}$$