# Properties

 Label 1710.2.a.f 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} + 4 q^{7} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{5} + 4 q^{7} - q^{8} + q^{10} + 4 q^{11} - 2 q^{13} - 4 q^{14} + q^{16} + 2 q^{17} - q^{19} - q^{20} - 4 q^{22} + 8 q^{23} + q^{25} + 2 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} - 2 q^{34} - 4 q^{35} - 10 q^{37} + q^{38} + q^{40} + 2 q^{41} + 12 q^{43} + 4 q^{44} - 8 q^{46} + 9 q^{49} - q^{50} - 2 q^{52} - 6 q^{53} - 4 q^{55} - 4 q^{56} + 6 q^{58} - 10 q^{61} - 4 q^{62} + q^{64} + 2 q^{65} - 4 q^{67} + 2 q^{68} + 4 q^{70} + 8 q^{71} + 2 q^{73} + 10 q^{74} - q^{76} + 16 q^{77} - 12 q^{79} - q^{80} - 2 q^{82} + 8 q^{83} - 2 q^{85} - 12 q^{86} - 4 q^{88} - 6 q^{89} - 8 q^{91} + 8 q^{92} + q^{95} + 18 q^{97} - 9 q^{98} + O(q^{100})$$

## 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 4.00000 −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.f 1
3.b odd 2 1 570.2.a.m 1
5.b even 2 1 8550.2.a.t 1
12.b even 2 1 4560.2.a.k 1
15.d odd 2 1 2850.2.a.a 1
15.e even 4 2 2850.2.d.k 2

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

 $$T_{7} - 4$$ $$T_{11} - 4$$ $$T_{13} + 2$$ $$T_{53} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$-4 + T$$
$11$ $$-4 + T$$
$13$ $$2 + T$$
$17$ $$-2 + T$$
$19$ $$1 + T$$
$23$ $$-8 + T$$
$29$ $$6 + T$$
$31$ $$-4 + T$$
$37$ $$10 + T$$
$41$ $$-2 + T$$
$43$ $$-12 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$4 + T$$
$71$ $$-8 + T$$
$73$ $$-2 + T$$
$79$ $$12 + T$$
$83$ $$-8 + T$$
$89$ $$6 + T$$
$97$ $$-18 + T$$