# Properties

 Label 384.5.h.f Level $384$ Weight $5$ Character orbit 384.h Analytic conductor $39.694$ Analytic rank $0$ Dimension $4$ CM no 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: no Analytic conductor: $$39.6940658242$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}\cdot 3^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 3 - \beta_{1} ) q^{3} -\beta_{2} q^{5} + \beta_{2} q^{7} + ( -63 - 6 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( 3 - \beta_{1} ) q^{3} -\beta_{2} q^{5} + \beta_{2} q^{7} + ( -63 - 6 \beta_{1} ) q^{9} -134 q^{11} -\beta_{3} q^{13} + ( -3 \beta_{2} + \beta_{3} ) q^{15} + 28 \beta_{1} q^{17} + 58 \beta_{1} q^{19} + ( 3 \beta_{2} - \beta_{3} ) q^{21} -2 \beta_{3} q^{23} + 815 q^{25} + ( -621 + 45 \beta_{1} ) q^{27} -21 \beta_{2} q^{29} + 29 \beta_{2} q^{31} + ( -402 + 134 \beta_{1} ) q^{33} -1440 q^{35} + 5 \beta_{3} q^{37} + ( -72 \beta_{2} - 3 \beta_{3} ) q^{39} + 152 \beta_{1} q^{41} -50 \beta_{1} q^{43} + ( 63 \beta_{2} + 6 \beta_{3} ) q^{45} + 8 \beta_{3} q^{47} -961 q^{49} + ( 2016 + 84 \beta_{1} ) q^{51} + 43 \beta_{2} q^{53} + 134 \beta_{2} q^{55} + ( 4176 + 174 \beta_{1} ) q^{57} + 1382 q^{59} -\beta_{3} q^{61} + ( -63 \beta_{2} - 6 \beta_{3} ) q^{63} + 1440 \beta_{1} q^{65} + 22 \beta_{1} q^{67} + ( -144 \beta_{2} - 6 \beta_{3} ) q^{69} + 6 \beta_{3} q^{71} + 4894 q^{73} + ( 2445 - 815 \beta_{1} ) q^{75} -134 \beta_{2} q^{77} -39 \beta_{2} q^{79} + ( 1377 + 756 \beta_{1} ) q^{81} + 5914 q^{83} -28 \beta_{3} q^{85} + ( -63 \beta_{2} + 21 \beta_{3} ) q^{87} + 1068 \beta_{1} q^{89} -1440 \beta_{1} q^{91} + ( 87 \beta_{2} - 29 \beta_{3} ) q^{93} -58 \beta_{3} q^{95} + 5858 q^{97} + ( 8442 + 804 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{3} - 252q^{9} + O(q^{10})$$ $$4q + 12q^{3} - 252q^{9} - 536q^{11} + 3260q^{25} - 2484q^{27} - 1608q^{33} - 5760q^{35} - 3844q^{49} + 8064q^{51} + 16704q^{57} + 5528q^{59} + 19576q^{73} + 9780q^{75} + 5508q^{81} + 23656q^{83} + 23432q^{97} + 33768q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} - 2 \nu$$ $$\beta_{2}$$ $$=$$ $$-4 \nu^{3} + 28 \nu$$ $$\beta_{3}$$ $$=$$ $$144 \nu^{2} - 288$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 288$$$$)/144$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} + 14 \beta_{1}$$$$)/24$$

## 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
 1.58114 + 0.707107i −1.58114 + 0.707107i 1.58114 − 0.707107i −1.58114 − 0.707107i
0 3.00000 8.48528i 0 −37.9473 0 37.9473 0 −63.0000 50.9117i 0
65.2 0 3.00000 8.48528i 0 37.9473 0 −37.9473 0 −63.0000 50.9117i 0
65.3 0 3.00000 + 8.48528i 0 −37.9473 0 37.9473 0 −63.0000 + 50.9117i 0
65.4 0 3.00000 + 8.48528i 0 37.9473 0 −37.9473 0 −63.0000 + 50.9117i 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 inner
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 384.5.h.f yes 4
3.b odd 2 1 384.5.h.e 4
4.b odd 2 1 384.5.h.e 4
8.b even 2 1 384.5.h.e 4
8.d odd 2 1 inner 384.5.h.f yes 4
12.b even 2 1 inner 384.5.h.f yes 4
24.f even 2 1 384.5.h.e 4
24.h odd 2 1 inner 384.5.h.f yes 4

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

## Hecke kernels

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

 $$T_{5}^{2} - 1440$$ $$T_{11} + 134$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 81 - 6 T + T^{2} )^{2}$$
$5$ $$( -1440 + T^{2} )^{2}$$
$7$ $$( -1440 + T^{2} )^{2}$$
$11$ $$( 134 + T )^{4}$$
$13$ $$( 103680 + T^{2} )^{2}$$
$17$ $$( 56448 + T^{2} )^{2}$$
$19$ $$( 242208 + T^{2} )^{2}$$
$23$ $$( 414720 + T^{2} )^{2}$$
$29$ $$( -635040 + T^{2} )^{2}$$
$31$ $$( -1211040 + T^{2} )^{2}$$
$37$ $$( 2592000 + T^{2} )^{2}$$
$41$ $$( 1663488 + T^{2} )^{2}$$
$43$ $$( 180000 + T^{2} )^{2}$$
$47$ $$( 6635520 + T^{2} )^{2}$$
$53$ $$( -2662560 + T^{2} )^{2}$$
$59$ $$( -1382 + T )^{4}$$
$61$ $$( 103680 + T^{2} )^{2}$$
$67$ $$( 34848 + T^{2} )^{2}$$
$71$ $$( 3732480 + T^{2} )^{2}$$
$73$ $$( -4894 + T )^{4}$$
$79$ $$( -2190240 + T^{2} )^{2}$$
$83$ $$( -5914 + T )^{4}$$
$89$ $$( 82124928 + T^{2} )^{2}$$
$97$ $$( -5858 + T )^{4}$$