# Properties

 Label 1470.2.a.g 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

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

## 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.g 1
3.b odd 2 1 4410.2.a.bc 1
5.b even 2 1 7350.2.a.bo 1
7.b odd 2 1 210.2.a.a 1
7.c even 3 2 1470.2.i.n 2
7.d odd 6 2 1470.2.i.t 2
21.c even 2 1 630.2.a.i 1
28.d even 2 1 1680.2.a.o 1
35.c odd 2 1 1050.2.a.q 1
35.f even 4 2 1050.2.g.f 2
56.e even 2 1 6720.2.a.z 1
56.h odd 2 1 6720.2.a.cg 1
84.h odd 2 1 5040.2.a.bg 1
105.g even 2 1 3150.2.a.t 1
105.k odd 4 2 3150.2.g.t 2
140.c even 2 1 8400.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.a 1 7.b odd 2 1
630.2.a.i 1 21.c even 2 1
1050.2.a.q 1 35.c odd 2 1
1050.2.g.f 2 35.f even 4 2
1470.2.a.g 1 1.a even 1 1 trivial
1470.2.i.n 2 7.c even 3 2
1470.2.i.t 2 7.d odd 6 2
1680.2.a.o 1 28.d even 2 1
3150.2.a.t 1 105.g even 2 1
3150.2.g.t 2 105.k odd 4 2
4410.2.a.bc 1 3.b odd 2 1
5040.2.a.bg 1 84.h odd 2 1
6720.2.a.z 1 56.e even 2 1
6720.2.a.cg 1 56.h odd 2 1
7350.2.a.bo 1 5.b even 2 1
8400.2.a.m 1 140.c even 2 1

## Hecke kernels

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

 $$T_{11} + 4$$ $$T_{13} - 2$$ $$T_{17} - 6$$ $$T_{19}$$ $$T_{31} - 8$$

## Hecke characteristic polynomials

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