# Properties

 Label 1470.2.a.r 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

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

## 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

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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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

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