# Properties

 Label 1470.2.a.v 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: $$\Z[a_1, \ldots, a_{11}]$$ 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.

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

## 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.v yes 2
3.b odd 2 1 4410.2.a.bn 2
5.b even 2 1 7350.2.a.dd 2
7.b odd 2 1 1470.2.a.u 2
7.c even 3 2 1470.2.i.u 4
7.d odd 6 2 1470.2.i.v 4
21.c even 2 1 4410.2.a.br 2
35.c odd 2 1 7350.2.a.df 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.u 2 7.b odd 2 1
1470.2.a.v yes 2 1.a even 1 1 trivial
1470.2.i.u 4 7.c even 3 2
1470.2.i.v 4 7.d odd 6 2
4410.2.a.bn 2 3.b odd 2 1
4410.2.a.br 2 21.c even 2 1
7350.2.a.dd 2 5.b even 2 1
7350.2.a.df 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1470))$$:

 $$T_{11}^{2} - 4 T_{11} + 2$$ $$T_{13}$$ $$T_{17}^{2} - 2$$ $$T_{19}^{2} - 8$$ $$T_{31}^{2} + 4 T_{31} - 46$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$2 - 4 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$-4 - 4 T + T^{2}$$
$29$ $$-2 - 8 T + T^{2}$$
$31$ $$-46 + 4 T + T^{2}$$
$37$ $$-2 + T^{2}$$
$41$ $$28 + 12 T + T^{2}$$
$43$ $$34 - 12 T + T^{2}$$
$47$ $$-46 + 4 T + T^{2}$$
$53$ $$-124 - 4 T + T^{2}$$
$59$ $$-36 + 12 T + T^{2}$$
$61$ $$4 + 12 T + T^{2}$$
$67$ $$-94 + 4 T + T^{2}$$
$71$ $$56 - 16 T + T^{2}$$
$73$ $$-28 - 4 T + T^{2}$$
$79$ $$-128 + T^{2}$$
$83$ $$56 + 16 T + T^{2}$$
$89$ $$-68 - 4 T + T^{2}$$
$97$ $$28 - 12 T + T^{2}$$