# Properties

 Label 1710.2.a.h Level $1710$ Weight $2$ Character orbit 1710.a Self dual yes Analytic conductor $13.654$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} - 2 q^{7} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} + q^{5} - 2 q^{7} - q^{8} - q^{10} - 2 q^{11} - 4 q^{13} + 2 q^{14} + q^{16} + 6 q^{17} + q^{19} + q^{20} + 2 q^{22} + 8 q^{23} + q^{25} + 4 q^{26} - 2 q^{28} - 6 q^{29} - 8 q^{31} - q^{32} - 6 q^{34} - 2 q^{35} - 8 q^{37} - q^{38} - q^{40} - 12 q^{41} - 2 q^{44} - 8 q^{46} - 3 q^{49} - q^{50} - 4 q^{52} + 10 q^{53} - 2 q^{55} + 2 q^{56} + 6 q^{58} - 6 q^{59} - 6 q^{61} + 8 q^{62} + q^{64} - 4 q^{65} + 12 q^{67} + 6 q^{68} + 2 q^{70} + 12 q^{71} - 10 q^{73} + 8 q^{74} + q^{76} + 4 q^{77} - 8 q^{79} + q^{80} + 12 q^{82} - 8 q^{83} + 6 q^{85} + 2 q^{88} - 8 q^{89} + 8 q^{91} + 8 q^{92} + q^{95} - 14 q^{97} + 3 q^{98} + O(q^{100})$$

## 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 −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.h 1
3.b odd 2 1 1710.2.a.m yes 1
5.b even 2 1 8550.2.a.bg 1
15.d odd 2 1 8550.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.a.h 1 1.a even 1 1 trivial
1710.2.a.m yes 1 3.b odd 2 1
8550.2.a.p 1 15.d odd 2 1
8550.2.a.bg 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1710))$$:

 $$T_{7} + 2$$ $$T_{11} + 2$$ $$T_{13} + 4$$ $$T_{53} - 10$$

## Hecke characteristic polynomials

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