# Properties

 Label 1170.2.a.e 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} + 2 q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + q^5 + 2 * q^7 - q^8 $$q - q^{2} + q^{4} + q^{5} + 2 q^{7} - q^{8} - q^{10} - 4 q^{11} - q^{13} - 2 q^{14} + q^{16} - 8 q^{17} - 6 q^{19} + q^{20} + 4 q^{22} - 6 q^{23} + q^{25} + q^{26} + 2 q^{28} + 4 q^{29} - q^{32} + 8 q^{34} + 2 q^{35} - 2 q^{37} + 6 q^{38} - q^{40} + 2 q^{41} - 4 q^{43} - 4 q^{44} + 6 q^{46} - 3 q^{49} - q^{50} - q^{52} + 10 q^{53} - 4 q^{55} - 2 q^{56} - 4 q^{58} - 4 q^{59} - 10 q^{61} + q^{64} - q^{65} + 12 q^{67} - 8 q^{68} - 2 q^{70} + 8 q^{71} - 8 q^{73} + 2 q^{74} - 6 q^{76} - 8 q^{77} + 8 q^{79} + q^{80} - 2 q^{82} - 12 q^{83} - 8 q^{85} + 4 q^{86} + 4 q^{88} + 14 q^{89} - 2 q^{91} - 6 q^{92} - 6 q^{95} - 16 q^{97} + 3 q^{98}+O(q^{100})$$ q - q^2 + q^4 + q^5 + 2 * q^7 - q^8 - q^10 - 4 * q^11 - q^13 - 2 * q^14 + q^16 - 8 * q^17 - 6 * q^19 + q^20 + 4 * q^22 - 6 * q^23 + q^25 + q^26 + 2 * q^28 + 4 * q^29 - q^32 + 8 * q^34 + 2 * q^35 - 2 * q^37 + 6 * q^38 - q^40 + 2 * q^41 - 4 * q^43 - 4 * q^44 + 6 * q^46 - 3 * q^49 - q^50 - q^52 + 10 * q^53 - 4 * q^55 - 2 * q^56 - 4 * q^58 - 4 * q^59 - 10 * q^61 + q^64 - q^65 + 12 * q^67 - 8 * q^68 - 2 * q^70 + 8 * q^71 - 8 * q^73 + 2 * q^74 - 6 * q^76 - 8 * q^77 + 8 * q^79 + q^80 - 2 * q^82 - 12 * q^83 - 8 * q^85 + 4 * q^86 + 4 * q^88 + 14 * q^89 - 2 * q^91 - 6 * q^92 - 6 * q^95 - 16 * 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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.e 1
3.b odd 2 1 390.2.a.e 1
4.b odd 2 1 9360.2.a.bh 1
5.b even 2 1 5850.2.a.bi 1
5.c odd 4 2 5850.2.e.i 2
12.b even 2 1 3120.2.a.o 1
15.d odd 2 1 1950.2.a.h 1
15.e even 4 2 1950.2.e.f 2
39.d odd 2 1 5070.2.a.e 1
39.f even 4 2 5070.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.e 1 3.b odd 2 1
1170.2.a.e 1 1.a even 1 1 trivial
1950.2.a.h 1 15.d odd 2 1
1950.2.e.f 2 15.e even 4 2
3120.2.a.o 1 12.b even 2 1
5070.2.a.e 1 39.d odd 2 1
5070.2.b.e 2 39.f even 4 2
5850.2.a.bi 1 5.b even 2 1
5850.2.e.i 2 5.c odd 4 2
9360.2.a.bh 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1170))$$:

 $$T_{7} - 2$$ T7 - 2 $$T_{11} + 4$$ T11 + 4 $$T_{17} + 8$$ T17 + 8 $$T_{31}$$ T31

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T - 2$$
$11$ $$T + 4$$
$13$ $$T + 1$$
$17$ $$T + 8$$
$19$ $$T + 6$$
$23$ $$T + 6$$
$29$ $$T - 4$$
$31$ $$T$$
$37$ $$T + 2$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T - 10$$
$59$ $$T + 4$$
$61$ $$T + 10$$
$67$ $$T - 12$$
$71$ $$T - 8$$
$73$ $$T + 8$$
$79$ $$T - 8$$
$83$ $$T + 12$$
$89$ $$T - 14$$
$97$ $$T + 16$$