# Properties

 Label 1710.2.a.x Level $1710$ Weight $2$ Character orbit 1710.a Self dual yes Analytic conductor $13.654$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{17}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} + (\beta + 1) q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - q^5 + (b + 1) * q^7 + q^8 $$q + q^{2} + q^{4} - q^{5} + (\beta + 1) q^{7} + q^{8} - q^{10} + 2 q^{11} + 4 q^{13} + (\beta + 1) q^{14} + q^{16} + ( - \beta - 1) q^{17} - q^{19} - q^{20} + 2 q^{22} + q^{25} + 4 q^{26} + (\beta + 1) q^{28} - 2 q^{29} + ( - \beta + 5) q^{31} + q^{32} + ( - \beta - 1) q^{34} + ( - \beta - 1) q^{35} - q^{38} - q^{40} + (\beta - 1) q^{41} + ( - 2 \beta + 2) q^{43} + 2 q^{44} + ( - 2 \beta + 2) q^{47} + (2 \beta + 11) q^{49} + q^{50} + 4 q^{52} + (2 \beta + 4) q^{53} - 2 q^{55} + (\beta + 1) q^{56} - 2 q^{58} + (\beta + 1) q^{59} + ( - 2 \beta + 4) q^{61} + ( - \beta + 5) q^{62} + q^{64} - 4 q^{65} + ( - 2 \beta - 2) q^{67} + ( - \beta - 1) q^{68} + ( - \beta - 1) q^{70} + ( - 2 \beta - 2) q^{71} + 6 q^{73} - q^{76} + (2 \beta + 2) q^{77} + ( - \beta + 5) q^{79} - q^{80} + (\beta - 1) q^{82} + (\beta + 11) q^{83} + (\beta + 1) q^{85} + ( - 2 \beta + 2) q^{86} + 2 q^{88} + (3 \beta + 1) q^{89} + (4 \beta + 4) q^{91} + ( - 2 \beta + 2) q^{94} + q^{95} - 6 q^{97} + (2 \beta + 11) q^{98}+O(q^{100})$$ q + q^2 + q^4 - q^5 + (b + 1) * q^7 + q^8 - q^10 + 2 * q^11 + 4 * q^13 + (b + 1) * q^14 + q^16 + (-b - 1) * q^17 - q^19 - q^20 + 2 * q^22 + q^25 + 4 * q^26 + (b + 1) * q^28 - 2 * q^29 + (-b + 5) * q^31 + q^32 + (-b - 1) * q^34 + (-b - 1) * q^35 - q^38 - q^40 + (b - 1) * q^41 + (-2*b + 2) * q^43 + 2 * q^44 + (-2*b + 2) * q^47 + (2*b + 11) * q^49 + q^50 + 4 * q^52 + (2*b + 4) * q^53 - 2 * q^55 + (b + 1) * q^56 - 2 * q^58 + (b + 1) * q^59 + (-2*b + 4) * q^61 + (-b + 5) * q^62 + q^64 - 4 * q^65 + (-2*b - 2) * q^67 + (-b - 1) * q^68 + (-b - 1) * q^70 + (-2*b - 2) * q^71 + 6 * q^73 - q^76 + (2*b + 2) * q^77 + (-b + 5) * q^79 - q^80 + (b - 1) * q^82 + (b + 11) * q^83 + (b + 1) * q^85 + (-2*b + 2) * q^86 + 2 * q^88 + (3*b + 1) * q^89 + (4*b + 4) * q^91 + (-2*b + 2) * q^94 + q^95 - 6 * q^97 + (2*b + 11) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} + 8 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{19} - 2 q^{20} + 4 q^{22} + 2 q^{25} + 8 q^{26} + 2 q^{28} - 4 q^{29} + 10 q^{31} + 2 q^{32} - 2 q^{34} - 2 q^{35} - 2 q^{38} - 2 q^{40} - 2 q^{41} + 4 q^{43} + 4 q^{44} + 4 q^{47} + 22 q^{49} + 2 q^{50} + 8 q^{52} + 8 q^{53} - 4 q^{55} + 2 q^{56} - 4 q^{58} + 2 q^{59} + 8 q^{61} + 10 q^{62} + 2 q^{64} - 8 q^{65} - 4 q^{67} - 2 q^{68} - 2 q^{70} - 4 q^{71} + 12 q^{73} - 2 q^{76} + 4 q^{77} + 10 q^{79} - 2 q^{80} - 2 q^{82} + 22 q^{83} + 2 q^{85} + 4 q^{86} + 4 q^{88} + 2 q^{89} + 8 q^{91} + 4 q^{94} + 2 q^{95} - 12 q^{97} + 22 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8 - 2 * q^10 + 4 * q^11 + 8 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 - 2 * q^19 - 2 * q^20 + 4 * q^22 + 2 * q^25 + 8 * q^26 + 2 * q^28 - 4 * q^29 + 10 * q^31 + 2 * q^32 - 2 * q^34 - 2 * q^35 - 2 * q^38 - 2 * q^40 - 2 * q^41 + 4 * q^43 + 4 * q^44 + 4 * q^47 + 22 * q^49 + 2 * q^50 + 8 * q^52 + 8 * q^53 - 4 * q^55 + 2 * q^56 - 4 * q^58 + 2 * q^59 + 8 * q^61 + 10 * q^62 + 2 * q^64 - 8 * q^65 - 4 * q^67 - 2 * q^68 - 2 * q^70 - 4 * q^71 + 12 * q^73 - 2 * q^76 + 4 * q^77 + 10 * q^79 - 2 * q^80 - 2 * q^82 + 22 * q^83 + 2 * q^85 + 4 * q^86 + 4 * q^88 + 2 * q^89 + 8 * q^91 + 4 * q^94 + 2 * q^95 - 12 * q^97 + 22 * 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
 −1.56155 2.56155
1.00000 0 1.00000 −1.00000 0 −3.12311 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 5.12311 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.x yes 2
3.b odd 2 1 1710.2.a.v 2
5.b even 2 1 8550.2.a.bp 2
15.d odd 2 1 8550.2.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.a.v 2 3.b odd 2 1
1710.2.a.x yes 2 1.a even 1 1 trivial
8550.2.a.bp 2 5.b even 2 1
8550.2.a.bx 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1710))$$:

 $$T_{7}^{2} - 2T_{7} - 16$$ T7^2 - 2*T7 - 16 $$T_{11} - 2$$ T11 - 2 $$T_{13} - 4$$ T13 - 4 $$T_{53}^{2} - 8T_{53} - 52$$ T53^2 - 8*T53 - 52

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 2T - 16$$
$11$ $$(T - 2)^{2}$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2} + 2T - 16$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2} - 10T + 8$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 2T - 16$$
$43$ $$T^{2} - 4T - 64$$
$47$ $$T^{2} - 4T - 64$$
$53$ $$T^{2} - 8T - 52$$
$59$ $$T^{2} - 2T - 16$$
$61$ $$T^{2} - 8T - 52$$
$67$ $$T^{2} + 4T - 64$$
$71$ $$T^{2} + 4T - 64$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2} - 10T + 8$$
$83$ $$T^{2} - 22T + 104$$
$89$ $$T^{2} - 2T - 152$$
$97$ $$(T + 6)^{2}$$