Properties

 Label 1470.2.i.r Level $1470$ Weight $2$ Character orbit 1470.i Analytic conductor $11.738$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + q^{6} - q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + q^{6} - q^{8} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{10} + ( -6 + 6 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} -6 q^{13} + q^{15} -\zeta_{6} q^{16} + ( 1 - \zeta_{6} ) q^{18} -4 \zeta_{6} q^{19} - q^{20} -6 q^{22} + ( -1 + \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} -6 \zeta_{6} q^{26} - q^{27} -8 q^{29} + \zeta_{6} q^{30} + ( 2 - 2 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} + 6 \zeta_{6} q^{33} + q^{36} -4 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} + ( -6 + 6 \zeta_{6} ) q^{39} -\zeta_{6} q^{40} + 10 q^{41} -6 q^{43} -6 \zeta_{6} q^{44} + ( 1 - \zeta_{6} ) q^{45} -2 \zeta_{6} q^{47} - q^{48} - q^{50} + ( 6 - 6 \zeta_{6} ) q^{52} + ( -10 + 10 \zeta_{6} ) q^{53} -\zeta_{6} q^{54} -6 q^{55} -4 q^{57} -8 \zeta_{6} q^{58} + ( -4 + 4 \zeta_{6} ) q^{59} + ( -1 + \zeta_{6} ) q^{60} + 14 \zeta_{6} q^{61} + 2 q^{62} + q^{64} -6 \zeta_{6} q^{65} + ( -6 + 6 \zeta_{6} ) q^{66} + ( -14 + 14 \zeta_{6} ) q^{67} + 8 q^{71} + \zeta_{6} q^{72} + ( 6 - 6 \zeta_{6} ) q^{73} + ( 4 - 4 \zeta_{6} ) q^{74} + \zeta_{6} q^{75} + 4 q^{76} -6 q^{78} + 8 \zeta_{6} q^{79} + ( 1 - \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 10 \zeta_{6} q^{82} + 8 q^{83} -6 \zeta_{6} q^{86} + ( -8 + 8 \zeta_{6} ) q^{87} + ( 6 - 6 \zeta_{6} ) q^{88} -18 \zeta_{6} q^{89} + q^{90} -2 \zeta_{6} q^{93} + ( 2 - 2 \zeta_{6} ) q^{94} + ( 4 - 4 \zeta_{6} ) q^{95} -\zeta_{6} q^{96} -2 q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + q^{3} - q^{4} + q^{5} + 2q^{6} - 2q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} + q^{3} - q^{4} + q^{5} + 2q^{6} - 2q^{8} - q^{9} - q^{10} - 6q^{11} + q^{12} - 12q^{13} + 2q^{15} - q^{16} + q^{18} - 4q^{19} - 2q^{20} - 12q^{22} - q^{24} - q^{25} - 6q^{26} - 2q^{27} - 16q^{29} + q^{30} + 2q^{31} + q^{32} + 6q^{33} + 2q^{36} - 4q^{37} + 4q^{38} - 6q^{39} - q^{40} + 20q^{41} - 12q^{43} - 6q^{44} + q^{45} - 2q^{47} - 2q^{48} - 2q^{50} + 6q^{52} - 10q^{53} - q^{54} - 12q^{55} - 8q^{57} - 8q^{58} - 4q^{59} - q^{60} + 14q^{61} + 4q^{62} + 2q^{64} - 6q^{65} - 6q^{66} - 14q^{67} + 16q^{71} + q^{72} + 6q^{73} + 4q^{74} + q^{75} + 8q^{76} - 12q^{78} + 8q^{79} + q^{80} - q^{81} + 10q^{82} + 16q^{83} - 6q^{86} - 8q^{87} + 6q^{88} - 18q^{89} + 2q^{90} - 2q^{93} + 2q^{94} + 4q^{95} - q^{96} - 4q^{97} + 12q^{99} + O(q^{100})$$

Character values

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

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

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
 0.5 + 0.866025i 0.5 − 0.866025i
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
 $$n$$: 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.r 2
7.b odd 2 1 1470.2.i.k 2
7.c even 3 1 1470.2.a.c 1
7.c even 3 1 inner 1470.2.i.r 2
7.d odd 6 1 1470.2.a.i yes 1
7.d odd 6 1 1470.2.i.k 2
21.g even 6 1 4410.2.a.v 1
21.h odd 6 1 4410.2.a.be 1
35.i odd 6 1 7350.2.a.cf 1
35.j even 6 1 7350.2.a.da 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.c 1 7.c even 3 1
1470.2.a.i yes 1 7.d odd 6 1
1470.2.i.k 2 7.b odd 2 1
1470.2.i.k 2 7.d odd 6 1
1470.2.i.r 2 1.a even 1 1 trivial
1470.2.i.r 2 7.c even 3 1 inner
4410.2.a.v 1 21.g even 6 1
4410.2.a.be 1 21.h odd 6 1
7350.2.a.cf 1 35.i odd 6 1
7350.2.a.da 1 35.j even 6 1

Hecke kernels

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

 $$T_{11}^{2} + 6 T_{11} + 36$$ $$T_{13} + 6$$ $$T_{17}$$ $$T_{19}^{2} + 4 T_{19} + 16$$ $$T_{31}^{2} - 2 T_{31} + 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$36 + 6 T + T^{2}$$
$13$ $$( 6 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$16 + 4 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 8 + T )^{2}$$
$31$ $$4 - 2 T + T^{2}$$
$37$ $$16 + 4 T + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$( 6 + T )^{2}$$
$47$ $$4 + 2 T + T^{2}$$
$53$ $$100 + 10 T + T^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$196 - 14 T + T^{2}$$
$67$ $$196 + 14 T + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$36 - 6 T + T^{2}$$
$79$ $$64 - 8 T + T^{2}$$
$83$ $$( -8 + T )^{2}$$
$89$ $$324 + 18 T + T^{2}$$
$97$ $$( 2 + T )^{2}$$