# Properties

 Label 1710.2.a.o 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} + 2 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - q^5 + 2 * q^7 + q^8 $$q + q^{2} + q^{4} - q^{5} + 2 q^{7} + q^{8} - q^{10} + 2 q^{13} + 2 q^{14} + q^{16} + q^{19} - q^{20} + q^{25} + 2 q^{26} + 2 q^{28} + 6 q^{29} + 2 q^{31} + q^{32} - 2 q^{35} + 2 q^{37} + q^{38} - q^{40} + 8 q^{43} - 3 q^{49} + q^{50} + 2 q^{52} - 6 q^{53} + 2 q^{56} + 6 q^{58} + 6 q^{59} + 2 q^{61} + 2 q^{62} + q^{64} - 2 q^{65} - 4 q^{67} - 2 q^{70} + 14 q^{73} + 2 q^{74} + q^{76} + 2 q^{79} - q^{80} - 6 q^{83} + 8 q^{86} + 12 q^{89} + 4 q^{91} - q^{95} - 10 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 + q^4 - q^5 + 2 * q^7 + q^8 - q^10 + 2 * q^13 + 2 * q^14 + q^16 + q^19 - q^20 + q^25 + 2 * q^26 + 2 * q^28 + 6 * q^29 + 2 * q^31 + q^32 - 2 * q^35 + 2 * q^37 + q^38 - q^40 + 8 * q^43 - 3 * q^49 + q^50 + 2 * q^52 - 6 * q^53 + 2 * q^56 + 6 * q^58 + 6 * q^59 + 2 * q^61 + 2 * q^62 + q^64 - 2 * q^65 - 4 * q^67 - 2 * q^70 + 14 * q^73 + 2 * q^74 + q^76 + 2 * q^79 - q^80 - 6 * q^83 + 8 * q^86 + 12 * q^89 + 4 * q^91 - q^95 - 10 * q^97 - 3 * q^98

## 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.o 1
3.b odd 2 1 570.2.a.f 1
5.b even 2 1 8550.2.a.e 1
12.b even 2 1 4560.2.a.m 1
15.d odd 2 1 2850.2.a.q 1
15.e even 4 2 2850.2.d.d 2

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

 $$T_{7} - 2$$ T7 - 2 $$T_{11}$$ T11 $$T_{13} - 2$$ T13 - 2 $$T_{53} + 6$$ T53 + 6

## Hecke characteristic polynomials

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