# Properties

 Label 3584.2.a.a Level $3584$ Weight $2$ Character orbit 3584.a Self dual yes Analytic conductor $28.618$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3584 = 2^{9} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3584.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.6183840844$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - q^{7} - q^{9}+O(q^{10})$$ q + b * q^3 - q^7 - q^9 $$q + \beta q^{3} - q^{7} - q^{9} - \beta q^{11} - \beta q^{19} - \beta q^{21} + 4 q^{23} - 5 q^{25} - 4 \beta q^{27} + 2 \beta q^{29} + 4 q^{31} - 2 q^{33} + 2 \beta q^{37} - 6 q^{41} - 3 \beta q^{43} - 4 q^{47} + q^{49} - 6 \beta q^{53} - 2 q^{57} - 5 \beta q^{59} - 4 \beta q^{61} + q^{63} + \beta q^{67} + 4 \beta q^{69} - 8 q^{71} - 8 q^{73} - 5 \beta q^{75} + \beta q^{77} - 4 q^{79} - 5 q^{81} + 5 \beta q^{83} + 4 q^{87} - 8 q^{89} + 4 \beta q^{93} - 8 q^{97} + \beta q^{99} +O(q^{100})$$ q + b * q^3 - q^7 - q^9 - b * q^11 - b * q^19 - b * q^21 + 4 * q^23 - 5 * q^25 - 4*b * q^27 + 2*b * q^29 + 4 * q^31 - 2 * q^33 + 2*b * q^37 - 6 * q^41 - 3*b * q^43 - 4 * q^47 + q^49 - 6*b * q^53 - 2 * q^57 - 5*b * q^59 - 4*b * q^61 + q^63 + b * q^67 + 4*b * q^69 - 8 * q^71 - 8 * q^73 - 5*b * q^75 + b * q^77 - 4 * q^79 - 5 * q^81 + 5*b * q^83 + 4 * q^87 - 8 * q^89 + 4*b * q^93 - 8 * q^97 + b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^7 - 2 * q^9 $$2 q - 2 q^{7} - 2 q^{9} + 8 q^{23} - 10 q^{25} + 8 q^{31} - 4 q^{33} - 12 q^{41} - 8 q^{47} + 2 q^{49} - 4 q^{57} + 2 q^{63} - 16 q^{71} - 16 q^{73} - 8 q^{79} - 10 q^{81} + 8 q^{87} - 16 q^{89} - 16 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 - 2 * q^9 + 8 * q^23 - 10 * q^25 + 8 * q^31 - 4 * q^33 - 12 * q^41 - 8 * q^47 + 2 * q^49 - 4 * q^57 + 2 * q^63 - 16 * q^71 - 16 * q^73 - 8 * q^79 - 10 * q^81 + 8 * q^87 - 16 * q^89 - 16 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 −1.41421 0 0 0 −1.00000 0 −1.00000 0
1.2 0 1.41421 0 0 0 −1.00000 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.a.a 2
4.b odd 2 1 3584.2.a.b yes 2
8.b even 2 1 inner 3584.2.a.a 2
8.d odd 2 1 3584.2.a.b yes 2
16.e even 4 2 3584.2.b.b 2
16.f odd 4 2 3584.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.a 2 1.a even 1 1 trivial
3584.2.a.a 2 8.b even 2 1 inner
3584.2.a.b yes 2 4.b odd 2 1
3584.2.a.b yes 2 8.d odd 2 1
3584.2.b.a 2 16.f odd 4 2
3584.2.b.b 2 16.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3584))$$:

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{5}$$ T5 $$T_{23} - 4$$ T23 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 2$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 2$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} - 8$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} - 8$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} - 18$$
$47$ $$(T + 4)^{2}$$
$53$ $$T^{2} - 72$$
$59$ $$T^{2} - 50$$
$61$ $$T^{2} - 32$$
$67$ $$T^{2} - 2$$
$71$ $$(T + 8)^{2}$$
$73$ $$(T + 8)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} - 50$$
$89$ $$(T + 8)^{2}$$
$97$ $$(T + 8)^{2}$$