# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.n (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + \zeta_{12} q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + q^{6} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + \zeta_{12} q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + q^{6} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{10} + ( 5 - 5 \zeta_{12}^{2} ) q^{11} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{12} -\zeta_{12}^{3} q^{13} + ( 1 + 2 \zeta_{12}^{3} ) q^{15} -\zeta_{12}^{2} q^{16} + 2 \zeta_{12} q^{17} + \zeta_{12} q^{18} -7 \zeta_{12}^{2} q^{19} + ( 2 - \zeta_{12}^{3} ) q^{20} -5 \zeta_{12}^{3} q^{22} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{23} + ( 1 - \zeta_{12}^{2} ) q^{24} + ( -3 + 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} -\zeta_{12}^{2} q^{26} + \zeta_{12}^{3} q^{27} + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{30} + ( -6 + 6 \zeta_{12}^{2} ) q^{31} -\zeta_{12} q^{32} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{33} + 2 q^{34} + q^{36} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{37} -7 \zeta_{12} q^{38} + ( 1 - \zeta_{12}^{2} ) q^{39} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{40} + 9 q^{41} + 10 \zeta_{12}^{3} q^{43} -5 \zeta_{12}^{2} q^{44} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{45} + ( 3 - 3 \zeta_{12}^{2} ) q^{46} + ( -13 \zeta_{12} + 13 \zeta_{12}^{3} ) q^{47} -\zeta_{12}^{3} q^{48} + ( 4 + 3 \zeta_{12}^{3} ) q^{50} + 2 \zeta_{12}^{2} q^{51} -\zeta_{12} q^{52} + \zeta_{12} q^{53} + \zeta_{12}^{2} q^{54} + ( 10 - 5 \zeta_{12}^{3} ) q^{55} -7 \zeta_{12}^{3} q^{57} + ( -4 + 4 \zeta_{12}^{2} ) q^{59} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{60} -2 \zeta_{12}^{2} q^{61} + 6 \zeta_{12}^{3} q^{62} - q^{64} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} + ( 5 - 5 \zeta_{12}^{2} ) q^{66} + 6 \zeta_{12} q^{67} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{68} + 3 q^{69} -2 q^{71} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{72} + 4 \zeta_{12} q^{73} + ( 5 - 5 \zeta_{12}^{2} ) q^{74} + ( -3 \zeta_{12} + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{75} -7 q^{76} -\zeta_{12}^{3} q^{78} -14 \zeta_{12}^{2} q^{79} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} ) q^{80} + ( -1 + \zeta_{12}^{2} ) q^{81} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{82} + 10 \zeta_{12}^{3} q^{83} + ( 2 + 4 \zeta_{12}^{3} ) q^{85} + 10 \zeta_{12}^{2} q^{86} -5 \zeta_{12} q^{88} -10 \zeta_{12}^{2} q^{89} + ( 1 + 2 \zeta_{12}^{3} ) q^{90} -3 \zeta_{12}^{3} q^{92} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{93} + ( -13 + 13 \zeta_{12}^{2} ) q^{94} + ( 14 - 7 \zeta_{12} - 14 \zeta_{12}^{2} ) q^{95} -\zeta_{12}^{2} q^{96} + 8 \zeta_{12}^{3} q^{97} + 5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 4q^{5} + 4q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{4} + 4q^{5} + 4q^{6} + 2q^{9} + 2q^{10} + 10q^{11} + 4q^{15} - 2q^{16} - 14q^{19} + 8q^{20} + 2q^{24} - 6q^{25} - 2q^{26} + 4q^{30} - 12q^{31} + 8q^{34} + 4q^{36} + 2q^{39} - 2q^{40} + 36q^{41} - 10q^{44} - 4q^{45} + 6q^{46} + 16q^{50} + 4q^{51} + 2q^{54} + 40q^{55} - 8q^{59} + 2q^{60} - 4q^{61} - 4q^{64} - 2q^{65} + 10q^{66} + 12q^{69} - 8q^{71} + 10q^{74} + 8q^{75} - 28q^{76} - 28q^{79} + 4q^{80} - 2q^{81} + 8q^{85} + 20q^{86} - 20q^{89} + 4q^{90} - 26q^{94} + 28q^{95} - 2q^{96} + 20q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.133975 2.23205i 1.00000 0 1.00000i 0.500000 0.866025i −1.23205 + 1.86603i
79.2 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 1.86603 1.23205i 1.00000 0 1.00000i 0.500000 0.866025i 2.23205 0.133975i
949.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0.133975 + 2.23205i 1.00000 0 1.00000i 0.500000 + 0.866025i −1.23205 1.86603i
949.2 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 1.86603 + 1.23205i 1.00000 0 1.00000i 0.500000 + 0.866025i 2.23205 + 0.133975i
 $$n$$: 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.n.i 4
5.b even 2 1 inner 1470.2.n.i 4
7.b odd 2 1 210.2.n.a 4
7.c even 3 1 1470.2.g.a 2
7.c even 3 1 inner 1470.2.n.i 4
7.d odd 6 1 210.2.n.a 4
7.d odd 6 1 1470.2.g.f 2
21.c even 2 1 630.2.u.c 4
21.g even 6 1 630.2.u.c 4
28.d even 2 1 1680.2.di.a 4
28.f even 6 1 1680.2.di.a 4
35.c odd 2 1 210.2.n.a 4
35.f even 4 1 1050.2.i.f 2
35.f even 4 1 1050.2.i.o 2
35.i odd 6 1 210.2.n.a 4
35.i odd 6 1 1470.2.g.f 2
35.j even 6 1 1470.2.g.a 2
35.j even 6 1 inner 1470.2.n.i 4
35.k even 12 1 1050.2.i.f 2
35.k even 12 1 1050.2.i.o 2
35.k even 12 1 7350.2.a.t 1
35.k even 12 1 7350.2.a.bn 1
35.l odd 12 1 7350.2.a.b 1
35.l odd 12 1 7350.2.a.ch 1
105.g even 2 1 630.2.u.c 4
105.p even 6 1 630.2.u.c 4
140.c even 2 1 1680.2.di.a 4
140.s even 6 1 1680.2.di.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a 4 7.b odd 2 1
210.2.n.a 4 7.d odd 6 1
210.2.n.a 4 35.c odd 2 1
210.2.n.a 4 35.i odd 6 1
630.2.u.c 4 21.c even 2 1
630.2.u.c 4 21.g even 6 1
630.2.u.c 4 105.g even 2 1
630.2.u.c 4 105.p even 6 1
1050.2.i.f 2 35.f even 4 1
1050.2.i.f 2 35.k even 12 1
1050.2.i.o 2 35.f even 4 1
1050.2.i.o 2 35.k even 12 1
1470.2.g.a 2 7.c even 3 1
1470.2.g.a 2 35.j even 6 1
1470.2.g.f 2 7.d odd 6 1
1470.2.g.f 2 35.i odd 6 1
1470.2.n.i 4 1.a even 1 1 trivial
1470.2.n.i 4 5.b even 2 1 inner
1470.2.n.i 4 7.c even 3 1 inner
1470.2.n.i 4 35.j even 6 1 inner
1680.2.di.a 4 28.d even 2 1
1680.2.di.a 4 28.f even 6 1
1680.2.di.a 4 140.c even 2 1
1680.2.di.a 4 140.s even 6 1
7350.2.a.b 1 35.l odd 12 1
7350.2.a.t 1 35.k even 12 1
7350.2.a.bn 1 35.k even 12 1
7350.2.a.ch 1 35.l odd 12 1

## Hecke kernels

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

 $$T_{11}^{2} - 5 T_{11} + 25$$ $$T_{17}^{4} - 4 T_{17}^{2} + 16$$ $$T_{19}^{2} + 7 T_{19} + 49$$ $$T_{31}^{2} + 6 T_{31} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$25 - 20 T + 11 T^{2} - 4 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 25 - 5 T + T^{2} )^{2}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$16 - 4 T^{2} + T^{4}$$
$19$ $$( 49 + 7 T + T^{2} )^{2}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$T^{4}$$
$31$ $$( 36 + 6 T + T^{2} )^{2}$$
$37$ $$625 - 25 T^{2} + T^{4}$$
$41$ $$( -9 + T )^{4}$$
$43$ $$( 100 + T^{2} )^{2}$$
$47$ $$28561 - 169 T^{2} + T^{4}$$
$53$ $$1 - T^{2} + T^{4}$$
$59$ $$( 16 + 4 T + T^{2} )^{2}$$
$61$ $$( 4 + 2 T + T^{2} )^{2}$$
$67$ $$1296 - 36 T^{2} + T^{4}$$
$71$ $$( 2 + T )^{4}$$
$73$ $$256 - 16 T^{2} + T^{4}$$
$79$ $$( 196 + 14 T + T^{2} )^{2}$$
$83$ $$( 100 + T^{2} )^{2}$$
$89$ $$( 100 + 10 T + T^{2} )^{2}$$
$97$ $$( 64 + T^{2} )^{2}$$