# Properties

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

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{2} + \beta_{2} q^{3} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + q^{6} + q^{8} + ( -1 - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{2} + \beta_{2} q^{3} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + q^{6} + q^{8} + ( -1 - \beta_{2} ) q^{9} + \beta_{2} q^{10} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{11} + ( -1 - \beta_{2} ) q^{12} + q^{15} + ( -1 - \beta_{2} ) q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} + \beta_{2} q^{18} + 2 \beta_{1} q^{19} + q^{20} + ( 2 + \beta_{3} ) q^{22} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{23} + \beta_{2} q^{24} + \beta_{2} q^{25} + q^{27} + ( 4 - 3 \beta_{3} ) q^{29} + ( -1 - \beta_{2} ) q^{30} + ( 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{31} + \beta_{2} q^{32} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{33} -\beta_{3} q^{34} + q^{36} + \beta_{1} q^{37} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{38} + ( -1 - \beta_{2} ) q^{40} + ( -6 + 2 \beta_{3} ) q^{41} + ( 6 + \beta_{3} ) q^{43} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{44} + \beta_{2} q^{45} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{46} + ( 2 + 5 \beta_{1} + 2 \beta_{2} ) q^{47} + q^{48} + q^{50} -\beta_{1} q^{51} + ( -8 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{53} + ( -1 - \beta_{2} ) q^{54} + ( 2 + \beta_{3} ) q^{55} + 2 \beta_{3} q^{57} + ( -4 - 3 \beta_{1} - 4 \beta_{2} ) q^{58} + ( 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{59} + \beta_{2} q^{60} + ( 6 + 4 \beta_{1} + 6 \beta_{2} ) q^{61} + ( -2 - 5 \beta_{3} ) q^{62} + q^{64} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{66} + ( 7 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} ) q^{67} -\beta_{1} q^{68} + ( 2 - 2 \beta_{3} ) q^{69} + ( 8 - 2 \beta_{3} ) q^{71} + ( -1 - \beta_{2} ) q^{72} + ( 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{73} + ( -\beta_{1} - \beta_{3} ) q^{74} + ( -1 - \beta_{2} ) q^{75} + 2 \beta_{3} q^{76} + 8 \beta_{1} q^{79} + \beta_{2} q^{80} + \beta_{2} q^{81} + ( 6 + 2 \beta_{1} + 6 \beta_{2} ) q^{82} + ( -8 - 2 \beta_{3} ) q^{83} -\beta_{3} q^{85} + ( -6 + \beta_{1} - 6 \beta_{2} ) q^{86} + ( 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{87} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{88} + ( -2 + 6 \beta_{1} - 2 \beta_{2} ) q^{89} + q^{90} + ( 2 - 2 \beta_{3} ) q^{92} + ( 2 - 5 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -5 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{94} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{95} + ( -1 - \beta_{2} ) q^{96} + ( 6 - 2 \beta_{3} ) q^{97} + ( 2 + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{3} - 2q^{4} - 2q^{5} + 4q^{6} + 4q^{8} - 2q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{3} - 2q^{4} - 2q^{5} + 4q^{6} + 4q^{8} - 2q^{9} - 2q^{10} - 4q^{11} - 2q^{12} + 4q^{15} - 2q^{16} - 2q^{18} + 4q^{20} + 8q^{22} - 4q^{23} - 2q^{24} - 2q^{25} + 4q^{27} + 16q^{29} - 2q^{30} + 4q^{31} - 2q^{32} - 4q^{33} + 4q^{36} - 2q^{40} - 24q^{41} + 24q^{43} - 4q^{44} - 2q^{45} - 4q^{46} + 4q^{47} + 4q^{48} + 4q^{50} - 4q^{53} - 2q^{54} + 8q^{55} - 8q^{58} + 12q^{59} - 2q^{60} + 12q^{61} - 8q^{62} + 4q^{64} - 4q^{66} + 4q^{67} + 8q^{69} + 32q^{71} - 2q^{72} - 4q^{73} - 2q^{75} - 2q^{80} - 2q^{81} + 12q^{82} - 32q^{83} - 12q^{86} - 8q^{87} - 4q^{88} - 4q^{89} + 4q^{90} + 8q^{92} + 4q^{93} + 4q^{94} - 2q^{96} + 24q^{97} + 8q^{99} + O(q^{100})$$

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 $$\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$$.

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

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.u 2 7.d odd 6 1
1470.2.a.v yes 2 7.c even 3 1
1470.2.i.u 4 1.a even 1 1 trivial
1470.2.i.u 4 7.c even 3 1 inner
1470.2.i.v 4 7.b odd 2 1
1470.2.i.v 4 7.d odd 6 1
4410.2.a.bn 2 21.h odd 6 1
4410.2.a.br 2 21.g even 6 1
7350.2.a.dd 2 35.j even 6 1
7350.2.a.df 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 $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

 $$T_{11}^{4} + 4 T_{11}^{3} + 14 T_{11}^{2} + 8 T_{11} + 4$$ $$T_{13}$$ $$T_{17}^{4} + 2 T_{17}^{2} + 4$$ $$T_{19}^{4} + 8 T_{19}^{2} + 64$$ $$T_{31}^{4} - 4 T_{31}^{3} + 62 T_{31}^{2} + 184 T_{31} + 2116$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$4 + 2 T^{2} + T^{4}$$
$19$ $$64 + 8 T^{2} + T^{4}$$
$23$ $$16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4}$$
$29$ $$( -2 - 8 T + T^{2} )^{2}$$
$31$ $$2116 + 184 T + 62 T^{2} - 4 T^{3} + T^{4}$$
$37$ $$4 + 2 T^{2} + T^{4}$$
$41$ $$( 28 + 12 T + T^{2} )^{2}$$
$43$ $$( 34 - 12 T + T^{2} )^{2}$$
$47$ $$2116 + 184 T + 62 T^{2} - 4 T^{3} + T^{4}$$
$53$ $$15376 - 496 T + 140 T^{2} + 4 T^{3} + T^{4}$$
$59$ $$1296 + 432 T + 180 T^{2} - 12 T^{3} + T^{4}$$
$61$ $$16 - 48 T + 140 T^{2} - 12 T^{3} + T^{4}$$
$67$ $$8836 + 376 T + 110 T^{2} - 4 T^{3} + T^{4}$$
$71$ $$( 56 - 16 T + T^{2} )^{2}$$
$73$ $$784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4}$$
$79$ $$16384 + 128 T^{2} + T^{4}$$
$83$ $$( 56 + 16 T + T^{2} )^{2}$$
$89$ $$4624 - 272 T + 84 T^{2} + 4 T^{3} + T^{4}$$
$97$ $$( 28 - 12 T + T^{2} )^{2}$$