# Properties

 Label 1470.2.a.t Level $1470$ Weight $2$ Character orbit 1470.a Self dual yes Analytic conductor $11.738$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−1.00000 1.00000 1.00000 −1.00000 −1.00000 0 −1.00000 1.00000 1.00000
1.2 −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.t yes 2
3.b odd 2 1 4410.2.a.bz 2
5.b even 2 1 7350.2.a.dh 2
7.b odd 2 1 1470.2.a.s 2
7.c even 3 2 1470.2.i.w 4
7.d odd 6 2 1470.2.i.x 4
21.c even 2 1 4410.2.a.bw 2
35.c odd 2 1 7350.2.a.dl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.s 2 7.b odd 2 1
1470.2.a.t yes 2 1.a even 1 1 trivial
1470.2.i.w 4 7.c even 3 2
1470.2.i.x 4 7.d odd 6 2
4410.2.a.bw 2 21.c even 2 1
4410.2.a.bz 2 3.b odd 2 1
7350.2.a.dh 2 5.b even 2 1
7350.2.a.dl 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1470))$$:

 $$T_{11}^{2} + 4 T_{11} - 14$$ $$T_{13}^{2} - 32$$ $$T_{17}^{2} + 8 T_{17} + 14$$ $$T_{19}^{2} - 8 T_{19} + 8$$ $$T_{31}^{2} - 12 T_{31} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-14 + 4 T + T^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$14 + 8 T + T^{2}$$
$19$ $$8 - 8 T + T^{2}$$
$23$ $$28 - 12 T + T^{2}$$
$29$ $$14 - 8 T + T^{2}$$
$31$ $$18 - 12 T + T^{2}$$
$37$ $$-2 - 8 T + T^{2}$$
$41$ $$-68 + 4 T + T^{2}$$
$43$ $$-46 - 4 T + T^{2}$$
$47$ $$-14 + 4 T + T^{2}$$
$53$ $$4 - 12 T + T^{2}$$
$59$ $$28 - 12 T + T^{2}$$
$61$ $$4 - 12 T + T^{2}$$
$67$ $$-46 - 4 T + T^{2}$$
$71$ $$-56 - 8 T + T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$56 + 16 T + T^{2}$$
$89$ $$-4 + 4 T + T^{2}$$
$97$ $$-68 + 4 T + T^{2}$$