# Properties

 Label 1470.2.a.o 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: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} - 5q^{11} + q^{12} + 5q^{13} - q^{15} + q^{16} + 4q^{17} + q^{18} + 7q^{19} - q^{20} - 5q^{22} + q^{23} + q^{24} + q^{25} + 5q^{26} + q^{27} - q^{30} + 2q^{31} + q^{32} - 5q^{33} + 4q^{34} + q^{36} + q^{37} + 7q^{38} + 5q^{39} - q^{40} - 5q^{41} + 12q^{43} - 5q^{44} - q^{45} + q^{46} + 11q^{47} + q^{48} + q^{50} + 4q^{51} + 5q^{52} - 9q^{53} + q^{54} + 5q^{55} + 7q^{57} - 4q^{59} - q^{60} - 4q^{61} + 2q^{62} + q^{64} - 5q^{65} - 5q^{66} - 12q^{67} + 4q^{68} + q^{69} + 2q^{71} + q^{72} - 10q^{73} + q^{74} + q^{75} + 7q^{76} + 5q^{78} - 12q^{79} - q^{80} + q^{81} - 5q^{82} + 12q^{83} - 4q^{85} + 12q^{86} - 5q^{88} - 14q^{89} - q^{90} + q^{92} + 2q^{93} + 11q^{94} - 7q^{95} + q^{96} + 8q^{97} - 5q^{99} + O(q^{100})$$

## 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.o 1
3.b odd 2 1 4410.2.a.u 1
5.b even 2 1 7350.2.a.a 1
7.b odd 2 1 1470.2.a.l 1
7.c even 3 2 1470.2.i.e 2
7.d odd 6 2 210.2.i.b 2
21.c even 2 1 4410.2.a.j 1
21.g even 6 2 630.2.k.g 2
28.f even 6 2 1680.2.bg.d 2
35.c odd 2 1 7350.2.a.u 1
35.i odd 6 2 1050.2.i.p 2
35.k even 12 4 1050.2.o.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.b 2 7.d odd 6 2
630.2.k.g 2 21.g even 6 2
1050.2.i.p 2 35.i odd 6 2
1050.2.o.g 4 35.k even 12 4
1470.2.a.l 1 7.b odd 2 1
1470.2.a.o 1 1.a even 1 1 trivial
1470.2.i.e 2 7.c even 3 2
1680.2.bg.d 2 28.f even 6 2
4410.2.a.j 1 21.c even 2 1
4410.2.a.u 1 3.b odd 2 1
7350.2.a.a 1 5.b even 2 1
7350.2.a.u 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1470))$$:

 $$T_{11} + 5$$ $$T_{13} - 5$$ $$T_{17} - 4$$ $$T_{19} - 7$$ $$T_{31} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$5 + T$$
$13$ $$-5 + T$$
$17$ $$-4 + T$$
$19$ $$-7 + T$$
$23$ $$-1 + T$$
$29$ $$T$$
$31$ $$-2 + T$$
$37$ $$-1 + T$$
$41$ $$5 + T$$
$43$ $$-12 + T$$
$47$ $$-11 + T$$
$53$ $$9 + T$$
$59$ $$4 + T$$
$61$ $$4 + T$$
$67$ $$12 + T$$
$71$ $$-2 + T$$
$73$ $$10 + T$$
$79$ $$12 + T$$
$83$ $$-12 + T$$
$89$ $$14 + T$$
$97$ $$-8 + T$$