# Properties

 Label 1470.2.a.k Level $1470$ Weight $2$ Character orbit 1470.a Self dual yes Analytic conductor $11.738$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - q^{11} - q^{12} - 7q^{13} + q^{15} + q^{16} + 4q^{17} + q^{18} - q^{19} - q^{20} - q^{22} + q^{23} - q^{24} + q^{25} - 7q^{26} - q^{27} - 8q^{29} + q^{30} - 6q^{31} + q^{32} + q^{33} + 4q^{34} + q^{36} - 3q^{37} - q^{38} + 7q^{39} - q^{40} - 9q^{41} - 4q^{43} - q^{44} - q^{45} + q^{46} + 3q^{47} - q^{48} + q^{50} - 4q^{51} - 7q^{52} - q^{53} - q^{54} + q^{55} + q^{57} - 8q^{58} - 12q^{59} + q^{60} + 4q^{61} - 6q^{62} + q^{64} + 7q^{65} + q^{66} + 12q^{67} + 4q^{68} - q^{69} - 14q^{71} + q^{72} + 14q^{73} - 3q^{74} - q^{75} - q^{76} + 7q^{78} + 4q^{79} - q^{80} + q^{81} - 9q^{82} - 12q^{83} - 4q^{85} - 4q^{86} + 8q^{87} - q^{88} + 2q^{89} - q^{90} + q^{92} + 6q^{93} + 3q^{94} + q^{95} - q^{96} + 16q^{97} - q^{99} + O(q^{100})$$

## 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.k 1
3.b odd 2 1 4410.2.a.q 1
5.b even 2 1 7350.2.a.ba 1
7.b odd 2 1 1470.2.a.r 1
7.c even 3 2 1470.2.i.i 2
7.d odd 6 2 210.2.i.a 2
21.c even 2 1 4410.2.a.g 1
21.g even 6 2 630.2.k.h 2
28.f even 6 2 1680.2.bg.k 2
35.c odd 2 1 7350.2.a.j 1
35.i odd 6 2 1050.2.i.s 2
35.k even 12 4 1050.2.o.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.a 2 7.d odd 6 2
630.2.k.h 2 21.g even 6 2
1050.2.i.s 2 35.i odd 6 2
1050.2.o.j 4 35.k even 12 4
1470.2.a.k 1 1.a even 1 1 trivial
1470.2.a.r 1 7.b odd 2 1
1470.2.i.i 2 7.c even 3 2
1680.2.bg.k 2 28.f even 6 2
4410.2.a.g 1 21.c even 2 1
4410.2.a.q 1 3.b odd 2 1
7350.2.a.j 1 35.c odd 2 1
7350.2.a.ba 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1470))$$:

 $$T_{11} + 1$$ $$T_{13} + 7$$ $$T_{17} - 4$$ $$T_{19} + 1$$ $$T_{31} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$1 + T$$
$13$ $$7 + T$$
$17$ $$-4 + T$$
$19$ $$1 + T$$
$23$ $$-1 + T$$
$29$ $$8 + T$$
$31$ $$6 + T$$
$37$ $$3 + T$$
$41$ $$9 + T$$
$43$ $$4 + T$$
$47$ $$-3 + T$$
$53$ $$1 + T$$
$59$ $$12 + T$$
$61$ $$-4 + T$$
$67$ $$-12 + T$$
$71$ $$14 + T$$
$73$ $$-14 + T$$
$79$ $$-4 + T$$
$83$ $$12 + T$$
$89$ $$-2 + T$$
$97$ $$-16 + T$$