# Properties

 Label 512.2.a.f Level $512$ Weight $2$ Character orbit 512.a Self dual yes Analytic conductor $4.088$ Analytic rank $0$ Dimension $2$ CM discriminant -8 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$512 = 2^{9}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 512.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.08834058349$$ Analytic rank: $$0$$ 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: $N(\mathrm{U}(1))$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta + 2) q^{3} + (4 \beta + 3) q^{9}+O(q^{10})$$ q + (b + 2) * q^3 + (4*b + 3) * q^9 $$q + (\beta + 2) q^{3} + (4 \beta + 3) q^{9} + ( - 3 \beta + 2) q^{11} - 4 \beta q^{17} + ( - \beta + 6) q^{19} - 5 q^{25} + (8 \beta + 8) q^{27} + ( - 4 \beta - 2) q^{33} - 6 q^{41} + ( - 5 \beta + 6) q^{43} - 7 q^{49} + ( - 8 \beta - 8) q^{51} + (4 \beta + 10) q^{57} + (3 \beta - 10) q^{59} + (7 \beta + 6) q^{67} - 12 \beta q^{73} + ( - 5 \beta - 10) q^{75} + (12 \beta + 23) q^{81} + (9 \beta + 2) q^{83} + 4 \beta q^{89} + 12 \beta q^{97} + ( - \beta - 18) q^{99} +O(q^{100})$$ q + (b + 2) * q^3 + (4*b + 3) * q^9 + (-3*b + 2) * q^11 - 4*b * q^17 + (-b + 6) * q^19 - 5 * q^25 + (8*b + 8) * q^27 + (-4*b - 2) * q^33 - 6 * q^41 + (-5*b + 6) * q^43 - 7 * q^49 + (-8*b - 8) * q^51 + (4*b + 10) * q^57 + (3*b - 10) * q^59 + (7*b + 6) * q^67 - 12*b * q^73 + (-5*b - 10) * q^75 + (12*b + 23) * q^81 + (9*b + 2) * q^83 + 4*b * q^89 + 12*b * q^97 + (-b - 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} + 6 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 + 6 * q^9 $$2 q + 4 q^{3} + 6 q^{9} + 4 q^{11} + 12 q^{19} - 10 q^{25} + 16 q^{27} - 4 q^{33} - 12 q^{41} + 12 q^{43} - 14 q^{49} - 16 q^{51} + 20 q^{57} - 20 q^{59} + 12 q^{67} - 20 q^{75} + 46 q^{81} + 4 q^{83} - 36 q^{99}+O(q^{100})$$ 2 * q + 4 * q^3 + 6 * q^9 + 4 * q^11 + 12 * q^19 - 10 * q^25 + 16 * q^27 - 4 * q^33 - 12 * q^41 + 12 * q^43 - 14 * q^49 - 16 * q^51 + 20 * q^57 - 20 * q^59 + 12 * q^67 - 20 * q^75 + 46 * q^81 + 4 * q^83 - 36 * 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 0.585786 0 0 0 0 0 −2.65685 0
1.2 0 3.41421 0 0 0 0 0 8.65685 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
$$2$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.a.f yes 2
3.b odd 2 1 4608.2.a.i 2
4.b odd 2 1 512.2.a.a 2
8.b even 2 1 512.2.a.a 2
8.d odd 2 1 CM 512.2.a.f yes 2
12.b even 2 1 4608.2.a.k 2
16.e even 4 2 512.2.b.c 4
16.f odd 4 2 512.2.b.c 4
24.f even 2 1 4608.2.a.i 2
24.h odd 2 1 4608.2.a.k 2
32.g even 8 2 1024.2.e.g 4
32.g even 8 2 1024.2.e.o 4
32.h odd 8 2 1024.2.e.g 4
32.h odd 8 2 1024.2.e.o 4
48.i odd 4 2 4608.2.d.k 4
48.k even 4 2 4608.2.d.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.a 2 4.b odd 2 1
512.2.a.a 2 8.b even 2 1
512.2.a.f yes 2 1.a even 1 1 trivial
512.2.a.f yes 2 8.d odd 2 1 CM
512.2.b.c 4 16.e even 4 2
512.2.b.c 4 16.f odd 4 2
1024.2.e.g 4 32.g even 8 2
1024.2.e.g 4 32.h odd 8 2
1024.2.e.o 4 32.g even 8 2
1024.2.e.o 4 32.h odd 8 2
4608.2.a.i 2 3.b odd 2 1
4608.2.a.i 2 24.f even 2 1
4608.2.a.k 2 12.b even 2 1
4608.2.a.k 2 24.h odd 2 1
4608.2.d.k 4 48.i odd 4 2
4608.2.d.k 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(512))$$:

 $$T_{3}^{2} - 4T_{3} + 2$$ T3^2 - 4*T3 + 2 $$T_{5}$$ T5 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 4T + 2$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 4T - 14$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 32$$
$19$ $$T^{2} - 12T + 34$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} - 12T - 14$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 20T + 82$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 12T - 62$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 288$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 4T - 158$$
$89$ $$T^{2} - 32$$
$97$ $$T^{2} - 288$$