# Properties

 Label 384.5.h.e 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: $$x^{4} - 4x^{2} + 9$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 3) q^{3} + \beta_{2} q^{5} + \beta_{2} q^{7} + (6 \beta_1 - 63) q^{9}+O(q^{10})$$ q + (-b1 - 3) * q^3 + b2 * q^5 + b2 * q^7 + (6*b1 - 63) * q^9 $$q + ( - \beta_1 - 3) q^{3} + \beta_{2} q^{5} + \beta_{2} q^{7} + (6 \beta_1 - 63) q^{9} + 134 q^{11} - \beta_{3} q^{13} + ( - \beta_{3} - 3 \beta_{2}) q^{15} - 28 \beta_1 q^{17} + 58 \beta_1 q^{19} + ( - \beta_{3} - 3 \beta_{2}) q^{21} + 2 \beta_{3} q^{23} + 815 q^{25} + (45 \beta_1 + 621) q^{27} + 21 \beta_{2} q^{29} + 29 \beta_{2} q^{31} + ( - 134 \beta_1 - 402) q^{33} + 1440 q^{35} + 5 \beta_{3} q^{37} + (3 \beta_{3} - 72 \beta_{2}) q^{39} - 152 \beta_1 q^{41} - 50 \beta_1 q^{43} + (6 \beta_{3} - 63 \beta_{2}) q^{45} - 8 \beta_{3} q^{47} - 961 q^{49} + (84 \beta_1 - 2016) q^{51} - 43 \beta_{2} q^{53} + 134 \beta_{2} q^{55} + ( - 174 \beta_1 + 4176) q^{57} - 1382 q^{59} - \beta_{3} q^{61} + (6 \beta_{3} - 63 \beta_{2}) q^{63} - 1440 \beta_1 q^{65} + 22 \beta_1 q^{67} + ( - 6 \beta_{3} + 144 \beta_{2}) q^{69} - 6 \beta_{3} q^{71} + 4894 q^{73} + ( - 815 \beta_1 - 2445) q^{75} + 134 \beta_{2} q^{77} - 39 \beta_{2} q^{79} + ( - 756 \beta_1 + 1377) q^{81} - 5914 q^{83} - 28 \beta_{3} q^{85} + ( - 21 \beta_{3} - 63 \beta_{2}) q^{87} - 1068 \beta_1 q^{89} - 1440 \beta_1 q^{91} + ( - 29 \beta_{3} - 87 \beta_{2}) q^{93} + 58 \beta_{3} q^{95} + 5858 q^{97} + (804 \beta_1 - 8442) q^{99}+O(q^{100})$$ q + (-b1 - 3) * q^3 + b2 * q^5 + b2 * q^7 + (6*b1 - 63) * q^9 + 134 * q^11 - b3 * q^13 + (-b3 - 3*b2) * q^15 - 28*b1 * q^17 + 58*b1 * q^19 + (-b3 - 3*b2) * q^21 + 2*b3 * q^23 + 815 * q^25 + (45*b1 + 621) * q^27 + 21*b2 * q^29 + 29*b2 * q^31 + (-134*b1 - 402) * q^33 + 1440 * q^35 + 5*b3 * q^37 + (3*b3 - 72*b2) * q^39 - 152*b1 * q^41 - 50*b1 * q^43 + (6*b3 - 63*b2) * q^45 - 8*b3 * q^47 - 961 * q^49 + (84*b1 - 2016) * q^51 - 43*b2 * q^53 + 134*b2 * q^55 + (-174*b1 + 4176) * q^57 - 1382 * q^59 - b3 * q^61 + (6*b3 - 63*b2) * q^63 - 1440*b1 * q^65 + 22*b1 * q^67 + (-6*b3 + 144*b2) * q^69 - 6*b3 * q^71 + 4894 * q^73 + (-815*b1 - 2445) * q^75 + 134*b2 * q^77 - 39*b2 * q^79 + (-756*b1 + 1377) * q^81 - 5914 * q^83 - 28*b3 * q^85 + (-21*b3 - 63*b2) * q^87 - 1068*b1 * q^89 - 1440*b1 * q^91 + (-29*b3 - 87*b2) * q^93 + 58*b3 * q^95 + 5858 * q^97 + (804*b1 - 8442) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} - 252 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 - 252 * q^9 $$4 q - 12 q^{3} - 252 q^{9} + 536 q^{11} + 3260 q^{25} + 2484 q^{27} - 1608 q^{33} + 5760 q^{35} - 3844 q^{49} - 8064 q^{51} + 16704 q^{57} - 5528 q^{59} + 19576 q^{73} - 9780 q^{75} + 5508 q^{81} - 23656 q^{83} + 23432 q^{97} - 33768 q^{99}+O(q^{100})$$ 4 * q - 12 * q^3 - 252 * q^9 + 536 * q^11 + 3260 * q^25 + 2484 * q^27 - 1608 * q^33 + 5760 * q^35 - 3844 * q^49 - 8064 * q^51 + 16704 * q^57 - 5528 * q^59 + 19576 * q^73 - 9780 * q^75 + 5508 * q^81 - 23656 * q^83 + 23432 * q^97 - 33768 * q^99

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

 $$\beta_{1}$$ $$=$$ $$2\nu^{3} - 2\nu$$ 2*v^3 - 2*v $$\beta_{2}$$ $$=$$ $$-4\nu^{3} + 28\nu$$ -4*v^3 + 28*v $$\beta_{3}$$ $$=$$ $$144\nu^{2} - 288$$ 144*v^2 - 288
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta_1 ) / 24$$ (b2 + 2*b1) / 24 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 288 ) / 144$$ (b3 + 288) / 144 $$\nu^{3}$$ $$=$$ $$( \beta_{2} + 14\beta_1 ) / 24$$ (b2 + 14*b1) / 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.e 4
3.b odd 2 1 384.5.h.f yes 4
4.b odd 2 1 384.5.h.f yes 4
8.b even 2 1 384.5.h.f yes 4
8.d odd 2 1 inner 384.5.h.e 4
12.b even 2 1 inner 384.5.h.e 4
24.f even 2 1 384.5.h.f yes 4
24.h odd 2 1 inner 384.5.h.e 4

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

## Hecke kernels

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

 $$T_{5}^{2} - 1440$$ T5^2 - 1440 $$T_{11} - 134$$ T11 - 134

## Hecke characteristic polynomials

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