# Properties

 Label 1470.2.a.f 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

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

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.c 2 7.c even 3 2
630.2.k.a 2 21.h odd 6 2
1050.2.i.i 2 35.j even 6 2
1050.2.o.c 4 35.l odd 12 4
1470.2.a.e 1 7.b odd 2 1
1470.2.a.f 1 1.a even 1 1 trivial
1470.2.i.p 2 7.d odd 6 2
1680.2.bg.n 2 28.g odd 6 2
4410.2.a.w 1 21.c even 2 1
4410.2.a.bh 1 3.b odd 2 1
7350.2.a.cd 1 5.b even 2 1
7350.2.a.cx 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1470))$$:

 $$T_{11} - 3$$ $$T_{13} - 5$$ $$T_{17}$$ $$T_{19} - 5$$ $$T_{31} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$-3 + T$$
$13$ $$-5 + T$$
$17$ $$T$$
$19$ $$-5 + T$$
$23$ $$9 + T$$
$29$ $$T$$
$31$ $$10 + T$$
$37$ $$1 + T$$
$41$ $$-9 + T$$
$43$ $$-8 + T$$
$47$ $$-3 + T$$
$53$ $$3 + T$$
$59$ $$-12 + T$$
$61$ $$-8 + T$$
$67$ $$-8 + T$$
$71$ $$6 + T$$
$73$ $$-2 + T$$
$79$ $$-8 + T$$
$83$ $$T$$
$89$ $$-6 + T$$
$97$ $$-8 + T$$