# Properties

 Label 1710.2.a.b 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 $$q - q^{2} + q^{4} - q^{5} - 2 q^{7} - q^{8} + q^{10} - 2 q^{11} + 4 q^{13} + 2 q^{14} + q^{16} + 2 q^{17} - q^{19} - q^{20} + 2 q^{22} - 4 q^{23} + q^{25} - 4 q^{26} - 2 q^{28} - 8 q^{31} - q^{32} - 2 q^{34} + 2 q^{35} + 8 q^{37} + q^{38} + q^{40} + 8 q^{41} - 6 q^{43} - 2 q^{44} + 4 q^{46} + 12 q^{47} - 3 q^{49} - q^{50} + 4 q^{52} + 6 q^{53} + 2 q^{55} + 2 q^{56} + 2 q^{61} + 8 q^{62} + q^{64} - 4 q^{65} + 8 q^{67} + 2 q^{68} - 2 q^{70} + 8 q^{71} + 14 q^{73} - 8 q^{74} - q^{76} + 4 q^{77} - q^{80} - 8 q^{82} - 4 q^{83} - 2 q^{85} + 6 q^{86} + 2 q^{88} - 8 q^{91} - 4 q^{92} - 12 q^{94} + q^{95} - 12 q^{97} + 3 q^{98}+O(q^{100})$$ 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 + 2 * q^17 - q^19 - q^20 + 2 * q^22 - 4 * q^23 + q^25 - 4 * q^26 - 2 * q^28 - 8 * q^31 - q^32 - 2 * q^34 + 2 * q^35 + 8 * q^37 + q^38 + q^40 + 8 * q^41 - 6 * q^43 - 2 * q^44 + 4 * q^46 + 12 * q^47 - 3 * q^49 - q^50 + 4 * q^52 + 6 * q^53 + 2 * q^55 + 2 * q^56 + 2 * q^61 + 8 * q^62 + q^64 - 4 * q^65 + 8 * q^67 + 2 * q^68 - 2 * q^70 + 8 * q^71 + 14 * q^73 - 8 * q^74 - q^76 + 4 * q^77 - q^80 - 8 * q^82 - 4 * q^83 - 2 * q^85 + 6 * q^86 + 2 * q^88 - 8 * q^91 - 4 * q^92 - 12 * q^94 + q^95 - 12 * 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.b 1
3.b odd 2 1 570.2.a.l 1
5.b even 2 1 8550.2.a.bf 1
12.b even 2 1 4560.2.a.o 1
15.d odd 2 1 2850.2.a.e 1
15.e even 4 2 2850.2.d.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.l 1 3.b odd 2 1
1710.2.a.b 1 1.a even 1 1 trivial
2850.2.a.e 1 15.d odd 2 1
2850.2.d.q 2 15.e even 4 2
4560.2.a.o 1 12.b even 2 1
8550.2.a.bf 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 $$T_{11} + 2$$ T11 + 2 $$T_{13} - 4$$ T13 - 4 $$T_{53} - 6$$ T53 - 6

## Hecke characteristic polynomials

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