# Properties

 Label 1470.2.g.f 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.f 2
5.b even 2 1 inner 1470.2.g.f 2
5.c odd 4 1 7350.2.a.t 1
5.c odd 4 1 7350.2.a.bn 1
7.b odd 2 1 1470.2.g.a 2
7.c even 3 2 210.2.n.a 4
7.d odd 6 2 1470.2.n.i 4
21.h odd 6 2 630.2.u.c 4
28.g odd 6 2 1680.2.di.a 4
35.c odd 2 1 1470.2.g.a 2
35.f even 4 1 7350.2.a.b 1
35.f even 4 1 7350.2.a.ch 1
35.i odd 6 2 1470.2.n.i 4
35.j even 6 2 210.2.n.a 4
35.l odd 12 2 1050.2.i.f 2
35.l odd 12 2 1050.2.i.o 2
105.o odd 6 2 630.2.u.c 4
140.p odd 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.c even 3 2
210.2.n.a 4 35.j even 6 2
630.2.u.c 4 21.h odd 6 2
630.2.u.c 4 105.o odd 6 2
1050.2.i.f 2 35.l odd 12 2
1050.2.i.o 2 35.l odd 12 2
1470.2.g.a 2 7.b odd 2 1
1470.2.g.a 2 35.c odd 2 1
1470.2.g.f 2 1.a even 1 1 trivial
1470.2.g.f 2 5.b even 2 1 inner
1470.2.n.i 4 7.d odd 6 2
1470.2.n.i 4 35.i odd 6 2
1680.2.di.a 4 28.g odd 6 2
1680.2.di.a 4 140.p odd 6 2
7350.2.a.b 1 35.f even 4 1
7350.2.a.t 1 5.c odd 4 1
7350.2.a.bn 1 5.c odd 4 1
7350.2.a.ch 1 35.f even 4 1

## Hecke kernels

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

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