# Properties

 Label 510.2.a.e Level $510$ Weight $2$ Character orbit 510.a Self dual yes Analytic conductor $4.072$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$510 = 2 \cdot 3 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 510.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.07237050309$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} + 4q^{11} - q^{12} - 2q^{13} - q^{15} + q^{16} + q^{17} + q^{18} + 4q^{19} + q^{20} + 4q^{22} - q^{24} + q^{25} - 2q^{26} - q^{27} - 2q^{29} - q^{30} + 8q^{31} + q^{32} - 4q^{33} + q^{34} + q^{36} + 6q^{37} + 4q^{38} + 2q^{39} + q^{40} - 6q^{41} - 4q^{43} + 4q^{44} + q^{45} - q^{48} - 7q^{49} + q^{50} - q^{51} - 2q^{52} - 10q^{53} - q^{54} + 4q^{55} - 4q^{57} - 2q^{58} - 4q^{59} - q^{60} - 2q^{61} + 8q^{62} + q^{64} - 2q^{65} - 4q^{66} + 4q^{67} + q^{68} + q^{72} - 6q^{73} + 6q^{74} - q^{75} + 4q^{76} + 2q^{78} + 8q^{79} + q^{80} + q^{81} - 6q^{82} - 12q^{83} + q^{85} - 4q^{86} + 2q^{87} + 4q^{88} - 6q^{89} + q^{90} - 8q^{93} + 4q^{95} - q^{96} - 14q^{97} - 7q^{98} + 4q^{99} + O(q^{100})$$

## 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
 0
1.00000 −1.00000 1.00000 1.00000 −1.00000 0 1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$17$$ $$-1$$

## 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 510.2.a.e 1
3.b odd 2 1 1530.2.a.b 1
4.b odd 2 1 4080.2.a.ba 1
5.b even 2 1 2550.2.a.l 1
5.c odd 4 2 2550.2.d.k 2
15.d odd 2 1 7650.2.a.bx 1
17.b even 2 1 8670.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.e 1 1.a even 1 1 trivial
1530.2.a.b 1 3.b odd 2 1
2550.2.a.l 1 5.b even 2 1
2550.2.d.k 2 5.c odd 4 2
4080.2.a.ba 1 4.b odd 2 1
7650.2.a.bx 1 15.d odd 2 1
8670.2.a.v 1 17.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(510))$$:

 $$T_{7}$$ $$T_{11} - 4$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$-4 + T$$
$13$ $$2 + T$$
$17$ $$-1 + T$$
$19$ $$-4 + T$$
$23$ $$T$$
$29$ $$2 + T$$
$31$ $$-8 + T$$
$37$ $$-6 + T$$
$41$ $$6 + T$$
$43$ $$4 + T$$
$47$ $$T$$
$53$ $$10 + T$$
$59$ $$4 + T$$
$61$ $$2 + T$$
$67$ $$-4 + T$$
$71$ $$T$$
$73$ $$6 + T$$
$79$ $$-8 + T$$
$83$ $$12 + T$$
$89$ $$6 + T$$
$97$ $$14 + T$$