# Properties

 Label 1470.2.i.w Level $1470$ Weight $2$ Character orbit 1470.i Analytic conductor $11.738$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{3} + ( -1 - \beta_{2} ) q^{4} -\beta_{2} q^{5} - q^{6} - q^{8} + \beta_{2} q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{3} + ( -1 - \beta_{2} ) q^{4} -\beta_{2} q^{5} - q^{6} - q^{8} + \beta_{2} q^{9} + ( -1 - \beta_{2} ) q^{10} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{11} + \beta_{2} q^{12} -4 \beta_{3} q^{13} - q^{15} + \beta_{2} q^{16} + ( 4 + \beta_{1} + 4 \beta_{2} ) q^{17} + ( 1 + \beta_{2} ) q^{18} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{19} - q^{20} + ( 2 - 3 \beta_{3} ) q^{22} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{23} + ( 1 + \beta_{2} ) q^{24} + ( -1 - \beta_{2} ) q^{25} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{26} + q^{27} + ( 4 - \beta_{3} ) q^{29} + \beta_{2} q^{30} + ( -6 + 3 \beta_{1} - 6 \beta_{2} ) q^{31} + ( 1 + \beta_{2} ) q^{32} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{33} + ( 4 - \beta_{3} ) q^{34} + q^{36} + ( 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{37} + ( 4 - 2 \beta_{1} + 4 \beta_{2} ) q^{38} -4 \beta_{1} q^{39} + \beta_{2} q^{40} + ( -2 - 6 \beta_{3} ) q^{41} + ( 2 + 5 \beta_{3} ) q^{43} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{44} + ( 1 + \beta_{2} ) q^{45} + ( 6 + 2 \beta_{1} + 6 \beta_{2} ) q^{46} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{47} + q^{48} - q^{50} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{51} -4 \beta_{1} q^{52} + ( -6 - 4 \beta_{1} - 6 \beta_{2} ) q^{53} -\beta_{2} q^{54} + ( 2 - 3 \beta_{3} ) q^{55} + ( 4 + 2 \beta_{3} ) q^{57} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{58} + ( -6 - 2 \beta_{1} - 6 \beta_{2} ) q^{59} + ( 1 + \beta_{2} ) q^{60} + ( -4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{61} + ( -6 - 3 \beta_{3} ) q^{62} + q^{64} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{65} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{66} + ( -2 + 5 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{68} + ( 6 - 2 \beta_{3} ) q^{69} + ( 4 - 6 \beta_{3} ) q^{71} -\beta_{2} q^{72} + ( -2 - 2 \beta_{2} ) q^{73} + ( 4 + 3 \beta_{1} + 4 \beta_{2} ) q^{74} + \beta_{2} q^{75} + ( 4 + 2 \beta_{3} ) q^{76} + 4 \beta_{3} q^{78} + 8 \beta_{2} q^{79} + ( 1 + \beta_{2} ) q^{80} + ( -1 - \beta_{2} ) q^{81} + ( -6 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{82} + ( -8 + 2 \beta_{3} ) q^{83} + ( 4 - \beta_{3} ) q^{85} + ( 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{86} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{87} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{88} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{89} + q^{90} + ( 6 - 2 \beta_{3} ) q^{92} + ( -3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{93} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 4 - 2 \beta_{1} + 4 \beta_{2} ) q^{95} -\beta_{2} q^{96} + ( -2 + 6 \beta_{3} ) q^{97} + ( -2 + 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{3} - 2q^{4} + 2q^{5} - 4q^{6} - 4q^{8} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{3} - 2q^{4} + 2q^{5} - 4q^{6} - 4q^{8} - 2q^{9} - 2q^{10} + 4q^{11} - 2q^{12} - 4q^{15} - 2q^{16} + 8q^{17} + 2q^{18} - 8q^{19} - 4q^{20} + 8q^{22} - 12q^{23} + 2q^{24} - 2q^{25} + 4q^{27} + 16q^{29} - 2q^{30} - 12q^{31} + 2q^{32} + 4q^{33} + 16q^{34} + 4q^{36} - 8q^{37} + 8q^{38} - 2q^{40} - 8q^{41} + 8q^{43} + 4q^{44} + 2q^{45} + 12q^{46} + 4q^{47} + 4q^{48} - 4q^{50} + 8q^{51} - 12q^{53} + 2q^{54} + 8q^{55} + 16q^{57} + 8q^{58} - 12q^{59} + 2q^{60} - 12q^{61} - 24q^{62} + 4q^{64} - 4q^{66} - 4q^{67} + 8q^{68} + 24q^{69} + 16q^{71} + 2q^{72} - 4q^{73} + 8q^{74} - 2q^{75} + 16q^{76} - 16q^{79} + 2q^{80} - 2q^{81} - 4q^{82} - 32q^{83} + 16q^{85} + 4q^{86} - 8q^{87} - 4q^{88} + 4q^{89} + 4q^{90} + 24q^{92} - 12q^{93} - 4q^{94} + 8q^{95} + 2q^{96} - 8q^{97} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$$.

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.00000 0 −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
361.2 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.00000 0 −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
961.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i −1.00000 0 −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
961.2 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i −1.00000 0 −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.i.w 4
7.b odd 2 1 1470.2.i.x 4
7.c even 3 1 1470.2.a.t yes 2
7.c even 3 1 inner 1470.2.i.w 4
7.d odd 6 1 1470.2.a.s 2
7.d odd 6 1 1470.2.i.x 4
21.g even 6 1 4410.2.a.bw 2
21.h odd 6 1 4410.2.a.bz 2
35.i odd 6 1 7350.2.a.dl 2
35.j even 6 1 7350.2.a.dh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.s 2 7.d odd 6 1
1470.2.a.t yes 2 7.c even 3 1
1470.2.i.w 4 1.a even 1 1 trivial
1470.2.i.w 4 7.c even 3 1 inner
1470.2.i.x 4 7.b odd 2 1
1470.2.i.x 4 7.d odd 6 1
4410.2.a.bw 2 21.g even 6 1
4410.2.a.bz 2 21.h odd 6 1
7350.2.a.dh 2 35.j even 6 1
7350.2.a.dl 2 35.i odd 6 1

## Hecke kernels

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

 $$T_{11}^{4} - 4 T_{11}^{3} + 30 T_{11}^{2} + 56 T_{11} + 196$$ $$T_{13}^{2} - 32$$ $$T_{17}^{4} - 8 T_{17}^{3} + 50 T_{17}^{2} - 112 T_{17} + 196$$ $$T_{19}^{4} + 8 T_{19}^{3} + 56 T_{19}^{2} + 64 T_{19} + 64$$ $$T_{31}^{4} + 12 T_{31}^{3} + 126 T_{31}^{2} + 216 T_{31} + 324$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$196 + 56 T + 30 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$( -32 + T^{2} )^{2}$$
$17$ $$196 - 112 T + 50 T^{2} - 8 T^{3} + T^{4}$$
$19$ $$64 + 64 T + 56 T^{2} + 8 T^{3} + T^{4}$$
$23$ $$784 + 336 T + 116 T^{2} + 12 T^{3} + T^{4}$$
$29$ $$( 14 - 8 T + T^{2} )^{2}$$
$31$ $$324 + 216 T + 126 T^{2} + 12 T^{3} + T^{4}$$
$37$ $$4 - 16 T + 66 T^{2} + 8 T^{3} + T^{4}$$
$41$ $$( -68 + 4 T + T^{2} )^{2}$$
$43$ $$( -46 - 4 T + T^{2} )^{2}$$
$47$ $$196 + 56 T + 30 T^{2} - 4 T^{3} + T^{4}$$
$53$ $$16 + 48 T + 140 T^{2} + 12 T^{3} + T^{4}$$
$59$ $$784 + 336 T + 116 T^{2} + 12 T^{3} + T^{4}$$
$61$ $$16 + 48 T + 140 T^{2} + 12 T^{3} + T^{4}$$
$67$ $$2116 - 184 T + 62 T^{2} + 4 T^{3} + T^{4}$$
$71$ $$( -56 - 8 T + T^{2} )^{2}$$
$73$ $$( 4 + 2 T + T^{2} )^{2}$$
$79$ $$( 64 + 8 T + T^{2} )^{2}$$
$83$ $$( 56 + 16 T + T^{2} )^{2}$$
$89$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$97$ $$( -68 + 4 T + T^{2} )^{2}$$