# Properties

 Label 1710.2.a.a Level $1710$ Weight $2$ Character orbit 1710.a Self dual yes Analytic conductor $13.654$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} - 4 q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - q^5 - 4 * q^7 - q^8 $$q - q^{2} + q^{4} - q^{5} - 4 q^{7} - q^{8} + q^{10} + 6 q^{11} + 4 q^{14} + q^{16} - 4 q^{17} - q^{19} - q^{20} - 6 q^{22} + 4 q^{23} + q^{25} - 4 q^{28} + 10 q^{29} - 2 q^{31} - q^{32} + 4 q^{34} + 4 q^{35} - 4 q^{37} + q^{38} + q^{40} - 10 q^{41} - 12 q^{43} + 6 q^{44} - 4 q^{46} + 9 q^{49} - q^{50} + 6 q^{53} - 6 q^{55} + 4 q^{56} - 10 q^{58} - 4 q^{59} - 10 q^{61} + 2 q^{62} + q^{64} - 8 q^{67} - 4 q^{68} - 4 q^{70} + 6 q^{73} + 4 q^{74} - q^{76} - 24 q^{77} - 10 q^{79} - q^{80} + 10 q^{82} - 14 q^{83} + 4 q^{85} + 12 q^{86} - 6 q^{88} - 10 q^{89} + 4 q^{92} + q^{95} - 6 q^{97} - 9 q^{98}+O(q^{100})$$ q - q^2 + q^4 - q^5 - 4 * q^7 - q^8 + q^10 + 6 * q^11 + 4 * q^14 + q^16 - 4 * q^17 - q^19 - q^20 - 6 * q^22 + 4 * q^23 + q^25 - 4 * q^28 + 10 * q^29 - 2 * q^31 - q^32 + 4 * q^34 + 4 * q^35 - 4 * q^37 + q^38 + q^40 - 10 * q^41 - 12 * q^43 + 6 * q^44 - 4 * q^46 + 9 * q^49 - q^50 + 6 * q^53 - 6 * q^55 + 4 * q^56 - 10 * q^58 - 4 * q^59 - 10 * q^61 + 2 * q^62 + q^64 - 8 * q^67 - 4 * q^68 - 4 * q^70 + 6 * q^73 + 4 * q^74 - q^76 - 24 * q^77 - 10 * q^79 - q^80 + 10 * q^82 - 14 * q^83 + 4 * q^85 + 12 * q^86 - 6 * q^88 - 10 * q^89 + 4 * q^92 + q^95 - 6 * q^97 - 9 * 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 −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.a 1
3.b odd 2 1 1710.2.a.p yes 1
5.b even 2 1 8550.2.a.bl 1
15.d odd 2 1 8550.2.a.q 1

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

 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 6$$ T11 - 6 $$T_{13}$$ T13 $$T_{53} - 6$$ T53 - 6

## Hecke characteristic polynomials

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