# Properties

 Label 1470.2.a.n Level 1470 Weight 2 Character orbit 1470.a Self dual yes Analytic conductor 11.738 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} + 2q^{11} - q^{12} + 2q^{13} - q^{15} + q^{16} - 4q^{17} + q^{18} + q^{20} + 2q^{22} + 8q^{23} - q^{24} + q^{25} + 2q^{26} - q^{27} - q^{30} - 2q^{31} + q^{32} - 2q^{33} - 4q^{34} + q^{36} + 8q^{37} - 2q^{39} + q^{40} - 2q^{41} - 2q^{43} + 2q^{44} + q^{45} + 8q^{46} + 10q^{47} - q^{48} + q^{50} + 4q^{51} + 2q^{52} - 2q^{53} - q^{54} + 2q^{55} + 4q^{59} - q^{60} - 10q^{61} - 2q^{62} + q^{64} + 2q^{65} - 2q^{66} + 2q^{67} - 4q^{68} - 8q^{69} - 12q^{71} + q^{72} + 10q^{73} + 8q^{74} - q^{75} - 2q^{78} + 16q^{79} + q^{80} + q^{81} - 2q^{82} + 16q^{83} - 4q^{85} - 2q^{86} + 2q^{88} + 14q^{89} + q^{90} + 8q^{92} + 2q^{93} + 10q^{94} - q^{96} + 6q^{97} + 2q^{99} + O(q^{100})$$

## 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 −1.00000 1.00000 1.00000 −1.00000 0 1.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 1470.2.a.n 1
3.b odd 2 1 4410.2.a.e 1
5.b even 2 1 7350.2.a.bh 1
7.b odd 2 1 1470.2.a.p yes 1
7.c even 3 2 1470.2.i.g 2
7.d odd 6 2 1470.2.i.c 2
21.c even 2 1 4410.2.a.n 1
35.c odd 2 1 7350.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.n 1 1.a even 1 1 trivial
1470.2.a.p yes 1 7.b odd 2 1
1470.2.i.c 2 7.d odd 6 2
1470.2.i.g 2 7.c even 3 2
4410.2.a.e 1 3.b odd 2 1
4410.2.a.n 1 21.c even 2 1
7350.2.a.o 1 35.c odd 2 1
7350.2.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1470))$$:

 $$T_{11} - 2$$ $$T_{13} - 2$$ $$T_{17} + 4$$ $$T_{19}$$ $$T_{31} + 2$$

## Hecke characteristic polynomials

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