# Properties

 Label 1536.2.a.l Level $1536$ Weight $2$ Character orbit 1536.a Self dual yes Analytic conductor $12.265$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + (\beta + 2) q^{5} + ( - \beta + 2) q^{7} + q^{9}+O(q^{10})$$ q + q^3 + (b + 2) * q^5 + (-b + 2) * q^7 + q^9 $$q + q^{3} + (\beta + 2) q^{5} + ( - \beta + 2) q^{7} + q^{9} + 2 q^{11} - 2 \beta q^{13} + (\beta + 2) q^{15} + ( - 4 \beta - 2) q^{17} + 4 \beta q^{19} + ( - \beta + 2) q^{21} + (2 \beta + 4) q^{23} + (4 \beta + 1) q^{25} + q^{27} + (\beta + 2) q^{29} + (\beta + 6) q^{31} + 2 q^{33} + 2 q^{35} + (4 \beta - 4) q^{37} - 2 \beta q^{39} + (4 \beta - 6) q^{41} + ( - 4 \beta - 4) q^{43} + (\beta + 2) q^{45} + ( - 6 \beta + 4) q^{47} + ( - 4 \beta - 1) q^{49} + ( - 4 \beta - 2) q^{51} + ( - 7 \beta + 2) q^{53} + (2 \beta + 4) q^{55} + 4 \beta q^{57} - 4 q^{59} + (4 \beta - 4) q^{61} + ( - \beta + 2) q^{63} + ( - 4 \beta - 4) q^{65} - 8 q^{67} + (2 \beta + 4) q^{69} + (2 \beta + 12) q^{71} + (4 \beta + 4) q^{73} + (4 \beta + 1) q^{75} + ( - 2 \beta + 4) q^{77} + (3 \beta + 10) q^{79} + q^{81} + ( - 8 \beta - 2) q^{83} + ( - 10 \beta - 12) q^{85} + (\beta + 2) q^{87} + 2 q^{89} + ( - 4 \beta + 4) q^{91} + (\beta + 6) q^{93} + (8 \beta + 8) q^{95} + ( - 8 \beta + 2) q^{97} + 2 q^{99} +O(q^{100})$$ q + q^3 + (b + 2) * q^5 + (-b + 2) * q^7 + q^9 + 2 * q^11 - 2*b * q^13 + (b + 2) * q^15 + (-4*b - 2) * q^17 + 4*b * q^19 + (-b + 2) * q^21 + (2*b + 4) * q^23 + (4*b + 1) * q^25 + q^27 + (b + 2) * q^29 + (b + 6) * q^31 + 2 * q^33 + 2 * q^35 + (4*b - 4) * q^37 - 2*b * q^39 + (4*b - 6) * q^41 + (-4*b - 4) * q^43 + (b + 2) * q^45 + (-6*b + 4) * q^47 + (-4*b - 1) * q^49 + (-4*b - 2) * q^51 + (-7*b + 2) * q^53 + (2*b + 4) * q^55 + 4*b * q^57 - 4 * q^59 + (4*b - 4) * q^61 + (-b + 2) * q^63 + (-4*b - 4) * q^65 - 8 * q^67 + (2*b + 4) * q^69 + (2*b + 12) * q^71 + (4*b + 4) * q^73 + (4*b + 1) * q^75 + (-2*b + 4) * q^77 + (3*b + 10) * q^79 + q^81 + (-8*b - 2) * q^83 + (-10*b - 12) * q^85 + (b + 2) * q^87 + 2 * q^89 + (-4*b + 4) * q^91 + (b + 6) * q^93 + (8*b + 8) * q^95 + (-8*b + 2) * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{9} + 4 q^{11} + 4 q^{15} - 4 q^{17} + 4 q^{21} + 8 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 12 q^{31} + 4 q^{33} + 4 q^{35} - 8 q^{37} - 12 q^{41} - 8 q^{43} + 4 q^{45} + 8 q^{47} - 2 q^{49} - 4 q^{51} + 4 q^{53} + 8 q^{55} - 8 q^{59} - 8 q^{61} + 4 q^{63} - 8 q^{65} - 16 q^{67} + 8 q^{69} + 24 q^{71} + 8 q^{73} + 2 q^{75} + 8 q^{77} + 20 q^{79} + 2 q^{81} - 4 q^{83} - 24 q^{85} + 4 q^{87} + 4 q^{89} + 8 q^{91} + 12 q^{93} + 16 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^9 + 4 * q^11 + 4 * q^15 - 4 * q^17 + 4 * q^21 + 8 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 12 * q^31 + 4 * q^33 + 4 * q^35 - 8 * q^37 - 12 * q^41 - 8 * q^43 + 4 * q^45 + 8 * q^47 - 2 * q^49 - 4 * q^51 + 4 * q^53 + 8 * q^55 - 8 * q^59 - 8 * q^61 + 4 * q^63 - 8 * q^65 - 16 * q^67 + 8 * q^69 + 24 * q^71 + 8 * q^73 + 2 * q^75 + 8 * q^77 + 20 * q^79 + 2 * q^81 - 4 * q^83 - 24 * q^85 + 4 * q^87 + 4 * q^89 + 8 * q^91 + 12 * q^93 + 16 * q^95 + 4 * q^97 + 4 * 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 0.585786 0 3.41421 0 1.00000 0
1.2 0 1.00000 0 3.41421 0 0.585786 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

This newform does not admit any (nontrivial) inner twists.

## Twists

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

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

 $$T_{5}^{2} - 4T_{5} + 2$$ T5^2 - 4*T5 + 2 $$T_{7}^{2} - 4T_{7} + 2$$ T7^2 - 4*T7 + 2 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 4T + 2$$
$7$ $$T^{2} - 4T + 2$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} - 8$$
$17$ $$T^{2} + 4T - 28$$
$19$ $$T^{2} - 32$$
$23$ $$T^{2} - 8T + 8$$
$29$ $$T^{2} - 4T + 2$$
$31$ $$T^{2} - 12T + 34$$
$37$ $$T^{2} + 8T - 16$$
$41$ $$T^{2} + 12T + 4$$
$43$ $$T^{2} + 8T - 16$$
$47$ $$T^{2} - 8T - 56$$
$53$ $$T^{2} - 4T - 94$$
$59$ $$(T + 4)^{2}$$
$61$ $$T^{2} + 8T - 16$$
$67$ $$(T + 8)^{2}$$
$71$ $$T^{2} - 24T + 136$$
$73$ $$T^{2} - 8T - 16$$
$79$ $$T^{2} - 20T + 82$$
$83$ $$T^{2} + 4T - 124$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} - 4T - 124$$