# Properties

 Label 1710.2.a.s Level $1710$ Weight $2$ Character orbit 1710.a Self dual yes Analytic conductor $13.654$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1710.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.6544187456$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} + 2q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + q^{5} + 2q^{7} + q^{8} + q^{10} + 6q^{11} + 2q^{14} + q^{16} - 2q^{17} - q^{19} + q^{20} + 6q^{22} - 4q^{23} + q^{25} + 2q^{28} + 8q^{29} - 8q^{31} + q^{32} - 2q^{34} + 2q^{35} - 4q^{37} - q^{38} + q^{40} + 4q^{41} - 6q^{43} + 6q^{44} - 4q^{46} + 12q^{47} - 3q^{49} + q^{50} - 6q^{53} + 6q^{55} + 2q^{56} + 8q^{58} + 4q^{59} + 2q^{61} - 8q^{62} + q^{64} - 8q^{67} - 2q^{68} + 2q^{70} + 6q^{73} - 4q^{74} - q^{76} + 12q^{77} + 8q^{79} + q^{80} + 4q^{82} - 4q^{83} - 2q^{85} - 6q^{86} + 6q^{88} + 4q^{89} - 4q^{92} + 12q^{94} - q^{95} + 12q^{97} - 3q^{98} + O(q^{100})$$

## 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 0 1.00000 1.00000 0 2.00000 1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$1$$

## 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 1710.2.a.s 1
3.b odd 2 1 570.2.a.b 1
5.b even 2 1 8550.2.a.g 1
12.b even 2 1 4560.2.a.r 1
15.d odd 2 1 2850.2.a.y 1
15.e even 4 2 2850.2.d.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.b 1 3.b odd 2 1
1710.2.a.s 1 1.a even 1 1 trivial
2850.2.a.y 1 15.d odd 2 1
2850.2.d.j 2 15.e even 4 2
4560.2.a.r 1 12.b even 2 1
8550.2.a.g 1 5.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1710))$$:

 $$T_{7} - 2$$ $$T_{11} - 6$$ $$T_{13}$$ $$T_{53} + 6$$

## Hecke characteristic polynomials

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