# Properties

 Label 1470.2.g.d 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: yes 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.

 $$f(q)$$ $$=$$ $$q -i q^{2} + i q^{3} - q^{4} + ( 1 - 2 i ) q^{5} + q^{6} + i q^{8} - q^{9} +O(q^{10})$$ $$q -i q^{2} + i q^{3} - q^{4} + ( 1 - 2 i ) q^{5} + q^{6} + i q^{8} - q^{9} + ( -2 - i ) q^{10} -i q^{12} -2 i q^{13} + ( 2 + i ) q^{15} + q^{16} + 2 i q^{17} + i q^{18} + 2 q^{19} + ( -1 + 2 i ) q^{20} -8 i q^{23} - q^{24} + ( -3 - 4 i ) q^{25} -2 q^{26} -i q^{27} + 8 q^{29} + ( 1 - 2 i ) q^{30} -4 q^{31} -i q^{32} + 2 q^{34} + q^{36} -6 i q^{37} -2 i q^{38} + 2 q^{39} + ( 2 + i ) q^{40} -10 q^{41} -2 i q^{43} + ( -1 + 2 i ) q^{45} -8 q^{46} + 6 i q^{47} + i q^{48} + ( -4 + 3 i ) q^{50} -2 q^{51} + 2 i q^{52} -6 i q^{53} - q^{54} + 2 i q^{57} -8 i q^{58} + 12 q^{59} + ( -2 - i ) q^{60} -2 q^{61} + 4 i q^{62} - q^{64} + ( -4 - 2 i ) q^{65} -14 i q^{67} -2 i q^{68} + 8 q^{69} + 6 q^{71} -i q^{72} -10 i q^{73} -6 q^{74} + ( 4 - 3 i ) q^{75} -2 q^{76} -2 i q^{78} -4 q^{79} + ( 1 - 2 i ) q^{80} + q^{81} + 10 i q^{82} + 12 i q^{83} + ( 4 + 2 i ) q^{85} -2 q^{86} + 8 i q^{87} -14 q^{89} + ( 2 + i ) q^{90} + 8 i q^{92} -4 i q^{93} + 6 q^{94} + ( 2 - 4 i ) q^{95} + q^{96} -14 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{5} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{5} + 2q^{6} - 2q^{9} - 4q^{10} + 4q^{15} + 2q^{16} + 4q^{19} - 2q^{20} - 2q^{24} - 6q^{25} - 4q^{26} + 16q^{29} + 2q^{30} - 8q^{31} + 4q^{34} + 2q^{36} + 4q^{39} + 4q^{40} - 20q^{41} - 2q^{45} - 16q^{46} - 8q^{50} - 4q^{51} - 2q^{54} + 24q^{59} - 4q^{60} - 4q^{61} - 2q^{64} - 8q^{65} + 16q^{69} + 12q^{71} - 12q^{74} + 8q^{75} - 4q^{76} - 8q^{79} + 2q^{80} + 2q^{81} + 8q^{85} - 4q^{86} - 28q^{89} + 4q^{90} + 12q^{94} + 4q^{95} + 2q^{96} + O(q^{100})$$

## 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 1.00000 2.00000i 1.00000 0 1.00000i −1.00000 −2.00000 1.00000i
589.2 1.00000i 1.00000i −1.00000 1.00000 + 2.00000i 1.00000 0 1.00000i −1.00000 −2.00000 + 1.00000i
 $$n$$: 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.d yes 2
5.b even 2 1 inner 1470.2.g.d yes 2
5.c odd 4 1 7350.2.a.m 1
5.c odd 4 1 7350.2.a.cr 1
7.b odd 2 1 1470.2.g.c 2
7.c even 3 2 1470.2.n.d 4
7.d odd 6 2 1470.2.n.e 4
35.c odd 2 1 1470.2.g.c 2
35.f even 4 1 7350.2.a.bc 1
35.f even 4 1 7350.2.a.bx 1
35.i odd 6 2 1470.2.n.e 4
35.j even 6 2 1470.2.n.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.c 2 7.b odd 2 1
1470.2.g.c 2 35.c odd 2 1
1470.2.g.d yes 2 1.a even 1 1 trivial
1470.2.g.d yes 2 5.b even 2 1 inner
1470.2.n.d 4 7.c even 3 2
1470.2.n.d 4 35.j even 6 2
1470.2.n.e 4 7.d odd 6 2
1470.2.n.e 4 35.i odd 6 2
7350.2.a.m 1 5.c odd 4 1
7350.2.a.bc 1 35.f even 4 1
7350.2.a.bx 1 35.f even 4 1
7350.2.a.cr 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 $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( -8 + T )^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$196 + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 14 + T )^{2}$$
$97$ $$196 + T^{2}$$