# Properties

 Label 3584.2.a.c Level $3584$ Weight $2$ Character orbit 3584.a Self dual yes Analytic conductor $28.618$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ Defining polynomial: $$x^{4} - 4x^{2} + 2$$ x^4 - 4*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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{3} q^{5} - q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10})$$ q + b1 * q^3 - b3 * q^5 - q^7 + (b2 - 1) * q^9 $$q + \beta_1 q^{3} - \beta_{3} q^{5} - q^{7} + (\beta_{2} - 1) q^{9} - 2 \beta_{3} q^{11} + 3 \beta_1 q^{13} - \beta_{2} q^{15} + ( - 2 \beta_{2} - 2) q^{17} + ( - \beta_{3} + 2 \beta_1) q^{19} - \beta_1 q^{21} + (\beta_{2} + 2) q^{23} + ( - \beta_{2} - 3) q^{25} + (\beta_{3} - 3 \beta_1) q^{27} + ( - 4 \beta_{3} + 2 \beta_1) q^{29} + (2 \beta_{2} + 4) q^{31} - 2 \beta_{2} q^{33} + \beta_{3} q^{35} + (2 \beta_{3} + 4 \beta_1) q^{37} + (3 \beta_{2} + 6) q^{39} + (6 \beta_{2} + 2) q^{41} + 2 \beta_1 q^{43} + (2 \beta_{3} - \beta_1) q^{45} + 4 q^{47} + q^{49} + ( - 2 \beta_{3} - 4 \beta_1) q^{51} + 4 \beta_{3} q^{53} + ( - 2 \beta_{2} + 4) q^{55} + (\beta_{2} + 4) q^{57} + ( - 9 \beta_{3} - 2 \beta_1) q^{59} + \beta_{3} q^{61} + ( - \beta_{2} + 1) q^{63} - 3 \beta_{2} q^{65} + (6 \beta_{3} + 6 \beta_1) q^{67} + (\beta_{3} + 3 \beta_1) q^{69} + ( - 2 \beta_{2} + 2) q^{71} + (6 \beta_{2} - 2) q^{73} + ( - \beta_{3} - 4 \beta_1) q^{75} + 2 \beta_{3} q^{77} + ( - 2 \beta_{2} + 6) q^{79} + ( - 5 \beta_{2} - 3) q^{81} - 3 \beta_1 q^{83} + 2 \beta_1 q^{85} + ( - 2 \beta_{2} + 4) q^{87} + 6 q^{89} - 3 \beta_1 q^{91} + (2 \beta_{3} + 6 \beta_1) q^{93} + ( - 3 \beta_{2} + 2) q^{95} - 2 q^{97} + (4 \beta_{3} - 2 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 - b3 * q^5 - q^7 + (b2 - 1) * q^9 - 2*b3 * q^11 + 3*b1 * q^13 - b2 * q^15 + (-2*b2 - 2) * q^17 + (-b3 + 2*b1) * q^19 - b1 * q^21 + (b2 + 2) * q^23 + (-b2 - 3) * q^25 + (b3 - 3*b1) * q^27 + (-4*b3 + 2*b1) * q^29 + (2*b2 + 4) * q^31 - 2*b2 * q^33 + b3 * q^35 + (2*b3 + 4*b1) * q^37 + (3*b2 + 6) * q^39 + (6*b2 + 2) * q^41 + 2*b1 * q^43 + (2*b3 - b1) * q^45 + 4 * q^47 + q^49 + (-2*b3 - 4*b1) * q^51 + 4*b3 * q^53 + (-2*b2 + 4) * q^55 + (b2 + 4) * q^57 + (-9*b3 - 2*b1) * q^59 + b3 * q^61 + (-b2 + 1) * q^63 - 3*b2 * q^65 + (6*b3 + 6*b1) * q^67 + (b3 + 3*b1) * q^69 + (-2*b2 + 2) * q^71 + (6*b2 - 2) * q^73 + (-b3 - 4*b1) * q^75 + 2*b3 * q^77 + (-2*b2 + 6) * q^79 + (-5*b2 - 3) * q^81 - 3*b1 * q^83 + 2*b1 * q^85 + (-2*b2 + 4) * q^87 + 6 * q^89 - 3*b1 * q^91 + (2*b3 + 6*b1) * q^93 + (-3*b2 + 2) * q^95 - 2 * q^97 + (4*b3 - 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^7 - 4 * q^9 $$4 q - 4 q^{7} - 4 q^{9} - 8 q^{17} + 8 q^{23} - 12 q^{25} + 16 q^{31} + 24 q^{39} + 8 q^{41} + 16 q^{47} + 4 q^{49} + 16 q^{55} + 16 q^{57} + 4 q^{63} + 8 q^{71} - 8 q^{73} + 24 q^{79} - 12 q^{81} + 16 q^{87} + 24 q^{89} + 8 q^{95} - 8 q^{97}+O(q^{100})$$ 4 * q - 4 * q^7 - 4 * q^9 - 8 * q^17 + 8 * q^23 - 12 * q^25 + 16 * q^31 + 24 * q^39 + 8 * q^41 + 16 * q^47 + 4 * q^49 + 16 * q^55 + 16 * q^57 + 4 * q^63 + 8 * q^71 - 8 * q^73 + 24 * q^79 - 12 * q^81 + 16 * q^87 + 24 * q^89 + 8 * q^95 - 8 * q^97

Basis of coefficient ring in terms of $$\nu = \zeta_{16} + \zeta_{16}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1

## 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
 −1.84776 −0.765367 0.765367 1.84776
0 −1.84776 0 0.765367 0 −1.00000 0 0.414214 0
1.2 0 −0.765367 0 −1.84776 0 −1.00000 0 −2.41421 0
1.3 0 0.765367 0 1.84776 0 −1.00000 0 −2.41421 0
1.4 0 1.84776 0 −0.765367 0 −1.00000 0 0.414214 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.c 4
4.b odd 2 1 3584.2.a.d yes 4
8.b even 2 1 inner 3584.2.a.c 4
8.d odd 2 1 3584.2.a.d yes 4
16.e even 4 2 3584.2.b.d 4
16.f odd 4 2 3584.2.b.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.c 4 1.a even 1 1 trivial
3584.2.a.c 4 8.b even 2 1 inner
3584.2.a.d yes 4 4.b odd 2 1
3584.2.a.d yes 4 8.d odd 2 1
3584.2.b.c 4 16.f odd 4 2
3584.2.b.d 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3584))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 4T^{2} + 2$$
$5$ $$T^{4} - 4T^{2} + 2$$
$7$ $$(T + 1)^{4}$$
$11$ $$T^{4} - 16T^{2} + 32$$
$13$ $$T^{4} - 36T^{2} + 162$$
$17$ $$(T^{2} + 4 T - 4)^{2}$$
$19$ $$T^{4} - 20T^{2} + 98$$
$23$ $$(T^{2} - 4 T + 2)^{2}$$
$29$ $$T^{4} - 80T^{2} + 32$$
$31$ $$(T^{2} - 8 T + 8)^{2}$$
$37$ $$T^{4} - 80T^{2} + 32$$
$41$ $$(T^{2} - 4 T - 68)^{2}$$
$43$ $$T^{4} - 16T^{2} + 32$$
$47$ $$(T - 4)^{4}$$
$53$ $$T^{4} - 64T^{2} + 512$$
$59$ $$T^{4} - 340 T^{2} + 25538$$
$61$ $$T^{4} - 4T^{2} + 2$$
$67$ $$T^{4} - 288 T^{2} + 10368$$
$71$ $$(T^{2} - 4 T - 4)^{2}$$
$73$ $$(T^{2} + 4 T - 68)^{2}$$
$79$ $$(T^{2} - 12 T + 28)^{2}$$
$83$ $$T^{4} - 36T^{2} + 162$$
$89$ $$(T - 6)^{4}$$
$97$ $$(T + 2)^{4}$$
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