# Properties

 Label 384.3.h.b Level $384$ Weight $3$ Character orbit 384.h Analytic conductor $10.463$ Analytic rank $0$ Dimension $2$ CM discriminant -8 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: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{3} + ( -7 - 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{3} + ( -7 - 2 \beta ) q^{9} -14 q^{11} -12 \beta q^{17} + 6 \beta q^{19} -25 q^{25} + ( 23 - 5 \beta ) q^{27} + ( 14 - 14 \beta ) q^{33} -24 \beta q^{41} -30 \beta q^{43} -49 q^{49} + ( 96 + 12 \beta ) q^{51} + ( -48 - 6 \beta ) q^{57} -82 q^{59} + 42 \beta q^{67} + 142 q^{73} + ( 25 - 25 \beta ) q^{75} + ( 17 + 28 \beta ) q^{81} -158 q^{83} + 36 \beta q^{89} -94 q^{97} + ( 98 + 28 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 14q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 14q^{9} - 28q^{11} - 50q^{25} + 46q^{27} + 28q^{33} - 98q^{49} + 192q^{51} - 96q^{57} - 164q^{59} + 284q^{73} + 50q^{75} + 34q^{81} - 316q^{83} - 188q^{97} + 196q^{99} + O(q^{100})$$

## 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
 − 1.41421i 1.41421i
0 −1.00000 2.82843i 0 0 0 0 0 −7.00000 + 5.65685i 0
65.2 0 −1.00000 + 2.82843i 0 0 0 0 0 −7.00000 5.65685i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
12.b 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.b 2
3.b odd 2 1 384.3.h.c yes 2
4.b odd 2 1 384.3.h.c yes 2
8.b even 2 1 384.3.h.c yes 2
8.d odd 2 1 CM 384.3.h.b 2
12.b even 2 1 inner 384.3.h.b 2
16.e even 4 2 768.3.e.k 4
16.f odd 4 2 768.3.e.k 4
24.f even 2 1 384.3.h.c yes 2
24.h odd 2 1 inner 384.3.h.b 2
48.i odd 4 2 768.3.e.k 4
48.k even 4 2 768.3.e.k 4

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

## Hecke kernels

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

 $$T_{5}$$ $$T_{11} + 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 14 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$1152 + T^{2}$$
$19$ $$288 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$4608 + T^{2}$$
$43$ $$7200 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$( 82 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$14112 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -142 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$( 158 + T )^{2}$$
$89$ $$10368 + T^{2}$$
$97$ $$( 94 + T )^{2}$$