# Properties

 Label 1470.2.g.a Level 1470 Weight 2 Character orbit 1470.g 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
589.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 −2.00000 1.00000i 1.00000 0 1.00000i −1.00000 −1.00000 + 2.00000i
589.2 1.00000i 1.00000i −1.00000 −2.00000 + 1.00000i 1.00000 0 1.00000i −1.00000 −1.00000 2.00000i
 $$n$$: 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

## Twists

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

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

## Hecke kernels

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

 $$T_{11} + 5$$ $$T_{17}^{2} + 4$$ $$T_{19} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$1 + 4 T + 5 T^{2}$$
$7$ 1
$11$ $$( 1 + 5 T + 11 T^{2} )^{2}$$
$13$ $$1 - 25 T^{2} + 169 T^{4}$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 - 7 T + 19 T^{2} )^{2}$$
$23$ $$1 - 37 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 - 6 T + 31 T^{2} )^{2}$$
$37$ $$1 - 49 T^{2} + 1369 T^{4}$$
$41$ $$( 1 - 9 T + 41 T^{2} )^{2}$$
$43$ $$1 + 14 T^{2} + 1849 T^{4}$$
$47$ $$1 + 75 T^{2} + 2209 T^{4}$$
$53$ $$1 - 105 T^{2} + 2809 T^{4}$$
$59$ $$( 1 - 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 2 T + 61 T^{2} )^{2}$$
$67$ $$1 - 98 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 2 T + 71 T^{2} )^{2}$$
$73$ $$1 - 130 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 14 T + 79 T^{2} )^{2}$$
$83$ $$1 - 66 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 10 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$