Properties

 Label 1170.2.a.a Level $1170$ Weight $2$ Character orbit 1170.a Self dual yes Analytic conductor $9.342$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1170.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$9.34249703649$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} - 4q^{11} + q^{13} + q^{16} + 6q^{17} + 4q^{19} - q^{20} + 4q^{22} - 8q^{23} + q^{25} - q^{26} - 6q^{29} - 8q^{31} - q^{32} - 6q^{34} - 10q^{37} - 4q^{38} + q^{40} + 6q^{41} + 4q^{43} - 4q^{44} + 8q^{46} - 7q^{49} - q^{50} + q^{52} + 10q^{53} + 4q^{55} + 6q^{58} - 4q^{59} - 2q^{61} + 8q^{62} + q^{64} - q^{65} - 12q^{67} + 6q^{68} - 16q^{71} + 2q^{73} + 10q^{74} + 4q^{76} - 16q^{79} - q^{80} - 6q^{82} + 12q^{83} - 6q^{85} - 4q^{86} + 4q^{88} - 10q^{89} - 8q^{92} - 4q^{95} - 6q^{97} + 7q^{98} + O(q^{100})$$

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 0 −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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$-1$$

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 1170.2.a.a 1
3.b odd 2 1 390.2.a.f 1
4.b odd 2 1 9360.2.a.p 1
5.b even 2 1 5850.2.a.bo 1
5.c odd 4 2 5850.2.e.e 2
12.b even 2 1 3120.2.a.w 1
15.d odd 2 1 1950.2.a.k 1
15.e even 4 2 1950.2.e.g 2
39.d odd 2 1 5070.2.a.a 1
39.f even 4 2 5070.2.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.f 1 3.b odd 2 1
1170.2.a.a 1 1.a even 1 1 trivial
1950.2.a.k 1 15.d odd 2 1
1950.2.e.g 2 15.e even 4 2
3120.2.a.w 1 12.b even 2 1
5070.2.a.a 1 39.d odd 2 1
5070.2.b.d 2 39.f even 4 2
5850.2.a.bo 1 5.b even 2 1
5850.2.e.e 2 5.c odd 4 2
9360.2.a.p 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1170))$$:

 $$T_{7}$$ $$T_{11} + 4$$ $$T_{17} - 6$$ $$T_{31} + 8$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$4 + T$$
$13$ $$-1 + T$$
$17$ $$-6 + T$$
$19$ $$-4 + T$$
$23$ $$8 + T$$
$29$ $$6 + T$$
$31$ $$8 + T$$
$37$ $$10 + T$$
$41$ $$-6 + T$$
$43$ $$-4 + T$$
$47$ $$T$$
$53$ $$-10 + T$$
$59$ $$4 + T$$
$61$ $$2 + T$$
$67$ $$12 + T$$
$71$ $$16 + T$$
$73$ $$-2 + T$$
$79$ $$16 + T$$
$83$ $$-12 + T$$
$89$ $$10 + T$$
$97$ $$6 + T$$