Properties

 Label 1710.2.a.c 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: 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 $$q - q^{2} + q^{4} - q^{5} - 2 q^{7} - q^{8} + q^{10} + 6 q^{13} + 2 q^{14} + q^{16} - 8 q^{17} + q^{19} - q^{20} + 4 q^{23} + q^{25} - 6 q^{26} - 2 q^{28} - 2 q^{29} - 2 q^{31} - q^{32} + 8 q^{34} + 2 q^{35} - 2 q^{37} - q^{38} + q^{40} + 12 q^{41} + 4 q^{43} - 4 q^{46} - 12 q^{47} - 3 q^{49} - q^{50} + 6 q^{52} - 10 q^{53} + 2 q^{56} + 2 q^{58} - 6 q^{59} - 14 q^{61} + 2 q^{62} + q^{64} - 6 q^{65} - 12 q^{67} - 8 q^{68} - 2 q^{70} + 8 q^{71} - 10 q^{73} + 2 q^{74} + q^{76} + 14 q^{79} - q^{80} - 12 q^{82} - 2 q^{83} + 8 q^{85} - 4 q^{86} - 12 q^{91} + 4 q^{92} + 12 q^{94} - q^{95} + 2 q^{97} + 3 q^{98}+O(q^{100})$$ q - q^2 + q^4 - q^5 - 2 * q^7 - q^8 + q^10 + 6 * q^13 + 2 * q^14 + q^16 - 8 * q^17 + q^19 - q^20 + 4 * q^23 + q^25 - 6 * q^26 - 2 * q^28 - 2 * q^29 - 2 * q^31 - q^32 + 8 * q^34 + 2 * q^35 - 2 * q^37 - q^38 + q^40 + 12 * q^41 + 4 * q^43 - 4 * q^46 - 12 * q^47 - 3 * q^49 - q^50 + 6 * q^52 - 10 * q^53 + 2 * q^56 + 2 * q^58 - 6 * q^59 - 14 * q^61 + 2 * q^62 + q^64 - 6 * q^65 - 12 * q^67 - 8 * q^68 - 2 * q^70 + 8 * q^71 - 10 * q^73 + 2 * q^74 + q^76 + 14 * q^79 - q^80 - 12 * q^82 - 2 * q^83 + 8 * q^85 - 4 * q^86 - 12 * q^91 + 4 * q^92 + 12 * q^94 - q^95 + 2 * 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.c 1
3.b odd 2 1 570.2.a.h 1
5.b even 2 1 8550.2.a.bh 1
12.b even 2 1 4560.2.a.bc 1
15.d odd 2 1 2850.2.a.n 1
15.e even 4 2 2850.2.d.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.h 1 3.b odd 2 1
1710.2.a.c 1 1.a even 1 1 trivial
2850.2.a.n 1 15.d odd 2 1
2850.2.d.e 2 15.e even 4 2
4560.2.a.bc 1 12.b even 2 1
8550.2.a.bh 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 $$T_{13} - 6$$ T13 - 6 $$T_{53} + 10$$ T53 + 10

Hecke characteristic polynomials

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