# Properties

 Label 510.2.a.g 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 2q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 2q^{7} + q^{8} + q^{9} + q^{10} + q^{12} - 4q^{13} + 2q^{14} + q^{15} + q^{16} - q^{17} + q^{18} - 4q^{19} + q^{20} + 2q^{21} + q^{24} + q^{25} - 4q^{26} + q^{27} + 2q^{28} + 6q^{29} + q^{30} - 4q^{31} + q^{32} - q^{34} + 2q^{35} + q^{36} + 2q^{37} - 4q^{38} - 4q^{39} + q^{40} + 2q^{42} + 2q^{43} + q^{45} + q^{48} - 3q^{49} + q^{50} - q^{51} - 4q^{52} - 6q^{53} + q^{54} + 2q^{56} - 4q^{57} + 6q^{58} + 6q^{59} + q^{60} - 10q^{61} - 4q^{62} + 2q^{63} + q^{64} - 4q^{65} + 2q^{67} - q^{68} + 2q^{70} - 6q^{71} + q^{72} - 16q^{73} + 2q^{74} + q^{75} - 4q^{76} - 4q^{78} + 8q^{79} + q^{80} + q^{81} + 2q^{84} - q^{85} + 2q^{86} + 6q^{87} + 6q^{89} + q^{90} - 8q^{91} - 4q^{93} - 4q^{95} + q^{96} - 4q^{97} - 3q^{98} + O(q^{100})$$

## 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 2.00000 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.g 1
3.b odd 2 1 1530.2.a.c 1
4.b odd 2 1 4080.2.a.l 1
5.b even 2 1 2550.2.a.b 1
5.c odd 4 2 2550.2.d.r 2
15.d odd 2 1 7650.2.a.bp 1
17.b even 2 1 8670.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.g 1 1.a even 1 1 trivial
1530.2.a.c 1 3.b odd 2 1
2550.2.a.b 1 5.b even 2 1
2550.2.d.r 2 5.c odd 4 2
4080.2.a.l 1 4.b odd 2 1
7650.2.a.bp 1 15.d odd 2 1
8670.2.a.n 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(510))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$-1 + T$$
$7$ $$-2 + T$$
$11$ $$T$$
$13$ $$4 + T$$
$17$ $$1 + T$$
$19$ $$4 + T$$
$23$ $$T$$
$29$ $$-6 + T$$
$31$ $$4 + T$$
$37$ $$-2 + T$$
$41$ $$T$$
$43$ $$-2 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$-6 + T$$
$61$ $$10 + T$$
$67$ $$-2 + T$$
$71$ $$6 + T$$
$73$ $$16 + T$$
$79$ $$-8 + T$$
$83$ $$T$$
$89$ $$-6 + T$$
$97$ $$4 + T$$