# Properties

 Label 1470.2.n.b 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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} -\zeta_{12} q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + q^{6} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} -\zeta_{12} q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + q^{6} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{10} + ( -2 + 2 \zeta_{12}^{2} ) q^{11} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{12} + 6 \zeta_{12}^{3} q^{13} + ( -2 + \zeta_{12}^{3} ) q^{15} -\zeta_{12}^{2} q^{16} + 4 \zeta_{12} q^{17} -\zeta_{12} q^{18} -6 \zeta_{12}^{2} q^{19} + ( -1 - 2 \zeta_{12}^{3} ) q^{20} -2 \zeta_{12}^{3} q^{22} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{23} + ( 1 - \zeta_{12}^{2} ) q^{24} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{25} -6 \zeta_{12}^{2} q^{26} -\zeta_{12}^{3} q^{27} -6 q^{29} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{30} + ( 2 - 2 \zeta_{12}^{2} ) q^{31} + \zeta_{12} q^{32} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{33} -4 q^{34} + q^{36} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{37} + 6 \zeta_{12} q^{38} + ( 6 - 6 \zeta_{12}^{2} ) q^{39} + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{40} + 2 q^{41} + 4 \zeta_{12}^{3} q^{43} + 2 \zeta_{12}^{2} q^{44} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{45} + ( 8 - 8 \zeta_{12}^{2} ) q^{46} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{47} + \zeta_{12}^{3} q^{48} + ( 4 + 3 \zeta_{12}^{3} ) q^{50} -4 \zeta_{12}^{2} q^{51} + 6 \zeta_{12} q^{52} -6 \zeta_{12} q^{53} + \zeta_{12}^{2} q^{54} + ( 2 + 4 \zeta_{12}^{3} ) q^{55} + 6 \zeta_{12}^{3} q^{57} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{58} + ( -8 + 8 \zeta_{12}^{2} ) q^{59} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{60} + 10 \zeta_{12}^{2} q^{61} + 2 \zeta_{12}^{3} q^{62} - q^{64} + ( 6 \zeta_{12} + 12 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{65} + ( -2 + 2 \zeta_{12}^{2} ) q^{66} + 8 \zeta_{12} q^{67} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{68} + 8 q^{69} -6 q^{71} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{72} + 14 \zeta_{12} q^{73} + ( 4 - 4 \zeta_{12}^{2} ) q^{74} + ( -3 \zeta_{12} + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{75} -6 q^{76} + 6 \zeta_{12}^{3} q^{78} -12 \zeta_{12}^{2} q^{79} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{80} + ( -1 + \zeta_{12}^{2} ) q^{81} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{82} -8 \zeta_{12}^{3} q^{83} + ( 8 - 4 \zeta_{12}^{3} ) q^{85} -4 \zeta_{12}^{2} q^{86} + 6 \zeta_{12} q^{87} -2 \zeta_{12} q^{88} -10 \zeta_{12}^{2} q^{89} + ( -2 + \zeta_{12}^{3} ) q^{90} + 8 \zeta_{12}^{3} q^{92} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{93} + ( 8 - 8 \zeta_{12}^{2} ) q^{94} + ( -6 - 12 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{95} -\zeta_{12}^{2} q^{96} + 10 \zeta_{12}^{3} q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 2q^{5} + 4q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 2q^{5} + 4q^{6} + 2q^{9} - 4q^{10} - 4q^{11} - 8q^{15} - 2q^{16} - 12q^{19} - 4q^{20} + 2q^{24} + 6q^{25} - 12q^{26} - 24q^{29} - 2q^{30} + 4q^{31} - 16q^{34} + 4q^{36} + 12q^{39} + 4q^{40} + 8q^{41} + 4q^{44} + 2q^{45} + 16q^{46} + 16q^{50} - 8q^{51} + 2q^{54} + 8q^{55} - 16q^{59} - 4q^{60} + 20q^{61} - 4q^{64} + 24q^{65} - 4q^{66} + 32q^{69} - 24q^{71} + 8q^{74} + 8q^{75} - 24q^{76} - 24q^{79} - 2q^{80} - 2q^{81} + 32q^{85} - 8q^{86} - 20q^{89} - 8q^{90} + 16q^{94} - 12q^{95} - 2q^{96} - 8q^{99} + O(q^{100})$$

## 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 1.23205 + 1.86603i 1.00000 0 1.00000i 0.500000 0.866025i −0.133975 2.23205i
79.2 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −2.23205 0.133975i 1.00000 0 1.00000i 0.500000 0.866025i −1.86603 1.23205i
949.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 1.23205 1.86603i 1.00000 0 1.00000i 0.500000 + 0.866025i −0.133975 + 2.23205i
949.2 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −2.23205 + 0.133975i 1.00000 0 1.00000i 0.500000 + 0.866025i −1.86603 + 1.23205i
 $$n$$: 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.b 4
5.b even 2 1 inner 1470.2.n.b 4
7.b odd 2 1 1470.2.n.f 4
7.c even 3 1 210.2.g.b 2
7.c even 3 1 inner 1470.2.n.b 4
7.d odd 6 1 1470.2.g.b 2
7.d odd 6 1 1470.2.n.f 4
21.h odd 6 1 630.2.g.c 2
28.g odd 6 1 1680.2.t.e 2
35.c odd 2 1 1470.2.n.f 4
35.i odd 6 1 1470.2.g.b 2
35.i odd 6 1 1470.2.n.f 4
35.j even 6 1 210.2.g.b 2
35.j even 6 1 inner 1470.2.n.b 4
35.k even 12 1 7350.2.a.bk 1
35.k even 12 1 7350.2.a.bz 1
35.l odd 12 1 1050.2.a.d 1
35.l odd 12 1 1050.2.a.p 1
84.n even 6 1 5040.2.t.h 2
105.o odd 6 1 630.2.g.c 2
105.x even 12 1 3150.2.a.d 1
105.x even 12 1 3150.2.a.bk 1
140.p odd 6 1 1680.2.t.e 2
140.w even 12 1 8400.2.a.w 1
140.w even 12 1 8400.2.a.bp 1
420.ba even 6 1 5040.2.t.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.g.b 2 7.c even 3 1
210.2.g.b 2 35.j even 6 1
630.2.g.c 2 21.h odd 6 1
630.2.g.c 2 105.o odd 6 1
1050.2.a.d 1 35.l odd 12 1
1050.2.a.p 1 35.l odd 12 1
1470.2.g.b 2 7.d odd 6 1
1470.2.g.b 2 35.i odd 6 1
1470.2.n.b 4 1.a even 1 1 trivial
1470.2.n.b 4 5.b even 2 1 inner
1470.2.n.b 4 7.c even 3 1 inner
1470.2.n.b 4 35.j even 6 1 inner
1470.2.n.f 4 7.b odd 2 1
1470.2.n.f 4 7.d odd 6 1
1470.2.n.f 4 35.c odd 2 1
1470.2.n.f 4 35.i odd 6 1
1680.2.t.e 2 28.g odd 6 1
1680.2.t.e 2 140.p odd 6 1
3150.2.a.d 1 105.x even 12 1
3150.2.a.bk 1 105.x even 12 1
5040.2.t.h 2 84.n even 6 1
5040.2.t.h 2 420.ba even 6 1
7350.2.a.bk 1 35.k even 12 1
7350.2.a.bz 1 35.k even 12 1
8400.2.a.w 1 140.w even 12 1
8400.2.a.bp 1 140.w even 12 1

## Hecke kernels

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

 $$T_{11}^{2} + 2 T_{11} + 4$$ $$T_{17}^{4} - 16 T_{17}^{2} + 256$$ $$T_{19}^{2} + 6 T_{19} + 36$$ $$T_{31}^{2} - 2 T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$25 + 10 T - T^{2} + 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 4 + 2 T + T^{2} )^{2}$$
$13$ $$( 36 + T^{2} )^{2}$$
$17$ $$256 - 16 T^{2} + T^{4}$$
$19$ $$( 36 + 6 T + T^{2} )^{2}$$
$23$ $$4096 - 64 T^{2} + T^{4}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$( 4 - 2 T + T^{2} )^{2}$$
$37$ $$256 - 16 T^{2} + T^{4}$$
$41$ $$( -2 + T )^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$4096 - 64 T^{2} + T^{4}$$
$53$ $$1296 - 36 T^{2} + T^{4}$$
$59$ $$( 64 + 8 T + T^{2} )^{2}$$
$61$ $$( 100 - 10 T + T^{2} )^{2}$$
$67$ $$4096 - 64 T^{2} + T^{4}$$
$71$ $$( 6 + T )^{4}$$
$73$ $$38416 - 196 T^{2} + T^{4}$$
$79$ $$( 144 + 12 T + T^{2} )^{2}$$
$83$ $$( 64 + T^{2} )^{2}$$
$89$ $$( 100 + 10 T + T^{2} )^{2}$$
$97$ $$( 100 + T^{2} )^{2}$$