# Properties

 Label 1470.2.g.i Level $1470$ Weight $2$ Character orbit 1470.g Analytic conductor $11.738$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.29160000.2 Defining polynomial: $$x^{6} - 20 x^{3} + 125$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 20 x^{3} + 125$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 20 \nu^{2}$$$$)/25$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 10$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} - 10 \nu$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - 15 \nu^{2}$$$$)/25$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} + 10$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{4} + 10 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$20 \beta_{5} + 15 \beta_{2}$$

## 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
 −1.40280 + 1.74131i −0.806615 − 2.08551i 2.20942 + 0.344208i −1.40280 − 1.74131i −0.806615 + 2.08551i 2.20942 − 0.344208i
1.00000i 1.00000i −1.00000 −1.40280 1.74131i 1.00000 0 1.00000i −1.00000 −1.74131 + 1.40280i
589.2 1.00000i 1.00000i −1.00000 −0.806615 + 2.08551i 1.00000 0 1.00000i −1.00000 2.08551 + 0.806615i
589.3 1.00000i 1.00000i −1.00000 2.20942 0.344208i 1.00000 0 1.00000i −1.00000 −0.344208 2.20942i
589.4 1.00000i 1.00000i −1.00000 −1.40280 + 1.74131i 1.00000 0 1.00000i −1.00000 −1.74131 1.40280i
589.5 1.00000i 1.00000i −1.00000 −0.806615 2.08551i 1.00000 0 1.00000i −1.00000 2.08551 0.806615i
589.6 1.00000i 1.00000i −1.00000 2.20942 + 0.344208i 1.00000 0 1.00000i −1.00000 −0.344208 + 2.20942i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 589.6 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.i 6
5.b even 2 1 inner 1470.2.g.i 6
5.c odd 4 1 7350.2.a.dn 3
5.c odd 4 1 7350.2.a.dq 3
7.b odd 2 1 1470.2.g.h 6
7.c even 3 2 210.2.n.b 12
7.d odd 6 2 1470.2.n.j 12
21.h odd 6 2 630.2.u.f 12
28.g odd 6 2 1680.2.di.c 12
35.c odd 2 1 1470.2.g.h 6
35.f even 4 1 7350.2.a.do 3
35.f even 4 1 7350.2.a.dp 3
35.i odd 6 2 1470.2.n.j 12
35.j even 6 2 210.2.n.b 12
35.l odd 12 2 1050.2.i.u 6
35.l odd 12 2 1050.2.i.v 6
105.o odd 6 2 630.2.u.f 12
140.p odd 6 2 1680.2.di.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.b 12 7.c even 3 2
210.2.n.b 12 35.j even 6 2
630.2.u.f 12 21.h odd 6 2
630.2.u.f 12 105.o odd 6 2
1050.2.i.u 6 35.l odd 12 2
1050.2.i.v 6 35.l odd 12 2
1470.2.g.h 6 7.b odd 2 1
1470.2.g.h 6 35.c odd 2 1
1470.2.g.i 6 1.a even 1 1 trivial
1470.2.g.i 6 5.b even 2 1 inner
1470.2.n.j 12 7.d odd 6 2
1470.2.n.j 12 35.i odd 6 2
1680.2.di.c 12 28.g odd 6 2
1680.2.di.c 12 140.p odd 6 2
7350.2.a.dn 3 5.c odd 4 1
7350.2.a.do 3 35.f even 4 1
7350.2.a.dp 3 35.f even 4 1
7350.2.a.dq 3 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 $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

 $$T_{11}^{3} - 3 T_{11}^{2} - 27 T_{11} + 49$$ $$T_{17}^{6} + 132 T_{17}^{4} + 5568 T_{17}^{2} + 73984$$ $$T_{19}^{3} + 3 T_{19}^{2} - 12 T_{19} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$( 1 + T^{2} )^{3}$$
$5$ $$125 - 20 T^{3} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$( 49 - 27 T - 3 T^{2} + T^{3} )^{2}$$
$13$ $$576 + 288 T^{2} + 33 T^{4} + T^{6}$$
$17$ $$73984 + 5568 T^{2} + 132 T^{4} + T^{6}$$
$19$ $$( -24 - 12 T + 3 T^{2} + T^{3} )^{2}$$
$23$ $$64 + 288 T^{2} + 57 T^{4} + T^{6}$$
$29$ $$( 24 + 33 T + 12 T^{2} + T^{3} )^{2}$$
$31$ $$( -10 - 15 T + T^{3} )^{2}$$
$37$ $$64 + 288 T^{2} + 57 T^{4} + T^{6}$$
$41$ $$( -128 - 48 T + 9 T^{2} + T^{3} )^{2}$$
$43$ $$( 4 + T^{2} )^{3}$$
$47$ $$163216 + 19848 T^{2} + 273 T^{4} + T^{6}$$
$53$ $$660969 + 23283 T^{2} + 267 T^{4} + T^{6}$$
$59$ $$( 436 - 27 T - 12 T^{2} + T^{3} )^{2}$$
$61$ $$( 712 - 108 T - 6 T^{2} + T^{3} )^{2}$$
$67$ $$153664 + 16368 T^{2} + 252 T^{4} + T^{6}$$
$71$ $$( -6 + T )^{6}$$
$73$ $$( 16 + T^{2} )^{3}$$
$79$ $$( 402 + 177 T + 24 T^{2} + T^{3} )^{2}$$
$83$ $$7396 + 2793 T^{2} + 198 T^{4} + T^{6}$$
$89$ $$( 392 - 108 T - 6 T^{2} + T^{3} )^{2}$$
$97$ $$12544 + 8313 T^{2} + 342 T^{4} + T^{6}$$