# Properties

 Label 1536.2.a.d Level $1536$ Weight $2$ Character orbit 1536.a Self dual yes Analytic conductor $12.265$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1536 = 2^{9} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1536.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.2650217505$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{5} + 3 \beta q^{7} + q^{9}+O(q^{10})$$ q - q^3 + b * q^5 + 3*b * q^7 + q^9 $$q - q^{3} + \beta q^{5} + 3 \beta q^{7} + q^{9} + 6 q^{11} + 4 \beta q^{13} - \beta q^{15} - 6 q^{17} + 4 q^{19} - 3 \beta q^{21} - 2 \beta q^{23} - 3 q^{25} - q^{27} + \beta q^{29} + \beta q^{31} - 6 q^{33} + 6 q^{35} - 6 \beta q^{37} - 4 \beta q^{39} - 2 q^{41} + \beta q^{45} - 2 \beta q^{47} + 11 q^{49} + 6 q^{51} - 7 \beta q^{53} + 6 \beta q^{55} - 4 q^{57} + 4 q^{59} - 6 \beta q^{61} + 3 \beta q^{63} + 8 q^{65} + 8 q^{67} + 2 \beta q^{69} + 2 \beta q^{71} + 8 q^{73} + 3 q^{75} + 18 \beta q^{77} - 9 \beta q^{79} + q^{81} + 2 q^{83} - 6 \beta q^{85} - \beta q^{87} + 2 q^{89} + 24 q^{91} - \beta q^{93} + 4 \beta q^{95} + 2 q^{97} + 6 q^{99} +O(q^{100})$$ q - q^3 + b * q^5 + 3*b * q^7 + q^9 + 6 * q^11 + 4*b * q^13 - b * q^15 - 6 * q^17 + 4 * q^19 - 3*b * q^21 - 2*b * q^23 - 3 * q^25 - q^27 + b * q^29 + b * q^31 - 6 * q^33 + 6 * q^35 - 6*b * q^37 - 4*b * q^39 - 2 * q^41 + b * q^45 - 2*b * q^47 + 11 * q^49 + 6 * q^51 - 7*b * q^53 + 6*b * q^55 - 4 * q^57 + 4 * q^59 - 6*b * q^61 + 3*b * q^63 + 8 * q^65 + 8 * q^67 + 2*b * q^69 + 2*b * q^71 + 8 * q^73 + 3 * q^75 + 18*b * q^77 - 9*b * q^79 + q^81 + 2 * q^83 - 6*b * q^85 - b * q^87 + 2 * q^89 + 24 * q^91 - b * q^93 + 4*b * q^95 + 2 * q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} + 12 q^{11} - 12 q^{17} + 8 q^{19} - 6 q^{25} - 2 q^{27} - 12 q^{33} + 12 q^{35} - 4 q^{41} + 22 q^{49} + 12 q^{51} - 8 q^{57} + 8 q^{59} + 16 q^{65} + 16 q^{67} + 16 q^{73} + 6 q^{75} + 2 q^{81} + 4 q^{83} + 4 q^{89} + 48 q^{91} + 4 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 + 12 * q^11 - 12 * q^17 + 8 * q^19 - 6 * q^25 - 2 * q^27 - 12 * q^33 + 12 * q^35 - 4 * q^41 + 22 * q^49 + 12 * q^51 - 8 * q^57 + 8 * q^59 + 16 * q^65 + 16 * q^67 + 16 * q^73 + 6 * q^75 + 2 * q^81 + 4 * q^83 + 4 * q^89 + 48 * q^91 + 4 * q^97 + 12 * q^99

## 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 −1.00000 0 −1.41421 0 −4.24264 0 1.00000 0
1.2 0 −1.00000 0 1.41421 0 4.24264 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
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1536.2.a.d 2
3.b odd 2 1 4608.2.a.g 2
4.b odd 2 1 1536.2.a.i yes 2
8.b even 2 1 1536.2.a.i yes 2
8.d odd 2 1 inner 1536.2.a.d 2
12.b even 2 1 4608.2.a.l 2
16.e even 4 2 1536.2.d.c 4
16.f odd 4 2 1536.2.d.c 4
24.f even 2 1 4608.2.a.g 2
24.h odd 2 1 4608.2.a.l 2
48.i odd 4 2 4608.2.d.n 4
48.k even 4 2 4608.2.d.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.d 2 1.a even 1 1 trivial
1536.2.a.d 2 8.d odd 2 1 inner
1536.2.a.i yes 2 4.b odd 2 1
1536.2.a.i yes 2 8.b even 2 1
1536.2.d.c 4 16.e even 4 2
1536.2.d.c 4 16.f odd 4 2
4608.2.a.g 2 3.b odd 2 1
4608.2.a.g 2 24.f even 2 1
4608.2.a.l 2 12.b even 2 1
4608.2.a.l 2 24.h odd 2 1
4608.2.d.n 4 48.i odd 4 2
4608.2.d.n 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1536))$$:

 $$T_{5}^{2} - 2$$ T5^2 - 2 $$T_{7}^{2} - 18$$ T7^2 - 18 $$T_{11} - 6$$ T11 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - 2$$
$7$ $$T^{2} - 18$$
$11$ $$(T - 6)^{2}$$
$13$ $$T^{2} - 32$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} - 8$$
$29$ $$T^{2} - 2$$
$31$ $$T^{2} - 2$$
$37$ $$T^{2} - 72$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 8$$
$53$ $$T^{2} - 98$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} - 72$$
$67$ $$(T - 8)^{2}$$
$71$ $$T^{2} - 8$$
$73$ $$(T - 8)^{2}$$
$79$ $$T^{2} - 162$$
$83$ $$(T - 2)^{2}$$
$89$ $$(T - 2)^{2}$$
$97$ $$(T - 2)^{2}$$