# Properties

 Label 1470.2.d.e Level $1470$ Weight $2$ Character orbit 1470.d Analytic conductor $11.738$ Analytic rank $0$ Dimension $8$ 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3317760000.3 Defining polynomial: $$x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( \beta_{2} + \beta_{5} ) q^{3} + q^{4} + ( \beta_{2} - \beta_{6} ) q^{5} + ( -\beta_{2} - \beta_{5} ) q^{6} - q^{8} + ( -2 - \beta_{7} ) q^{9} +O(q^{10})$$ $$q - q^{2} + ( \beta_{2} + \beta_{5} ) q^{3} + q^{4} + ( \beta_{2} - \beta_{6} ) q^{5} + ( -\beta_{2} - \beta_{5} ) q^{6} - q^{8} + ( -2 - \beta_{7} ) q^{9} + ( -\beta_{2} + \beta_{6} ) q^{10} + ( \beta_{4} - \beta_{7} ) q^{11} + ( \beta_{2} + \beta_{5} ) q^{12} + ( 2 \beta_{2} + \beta_{3} ) q^{13} + ( 1 - \beta_{1} - \beta_{4} - \beta_{7} ) q^{15} + q^{16} + ( \beta_{2} + 2 \beta_{5} ) q^{17} + ( 2 + \beta_{7} ) q^{18} + ( -\beta_{2} - 2 \beta_{5} + \beta_{6} ) q^{19} + ( \beta_{2} - \beta_{6} ) q^{20} + ( -\beta_{4} + \beta_{7} ) q^{22} + ( 1 + 2 \beta_{1} + \beta_{4} ) q^{23} + ( -\beta_{2} - \beta_{5} ) q^{24} + ( -1 - 2 \beta_{4} ) q^{25} + ( -2 \beta_{2} - \beta_{3} ) q^{26} + ( -4 \beta_{2} - \beta_{5} ) q^{27} + ( \beta_{4} - 2 \beta_{7} ) q^{29} + ( -1 + \beta_{1} + \beta_{4} + \beta_{7} ) q^{30} + ( \beta_{2} + 2 \beta_{5} + 2 \beta_{6} ) q^{31} - q^{32} + ( -2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{33} + ( -\beta_{2} - 2 \beta_{5} ) q^{34} + ( -2 - \beta_{7} ) q^{36} + ( -2 \beta_{4} - \beta_{7} ) q^{37} + ( \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{38} + ( 2 + \beta_{1} - 2 \beta_{4} - 2 \beta_{7} ) q^{39} + ( -\beta_{2} + \beta_{6} ) q^{40} + ( 2 \beta_{2} + \beta_{3} ) q^{41} + ( -\beta_{4} + 2 \beta_{7} ) q^{43} + ( \beta_{4} - \beta_{7} ) q^{44} + ( -\beta_{2} - \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{45} + ( -1 - 2 \beta_{1} - \beta_{4} ) q^{46} + ( 3 \beta_{2} + 6 \beta_{5} + \beta_{6} ) q^{47} + ( \beta_{2} + \beta_{5} ) q^{48} + ( 1 + 2 \beta_{4} ) q^{50} + ( -5 - \beta_{7} ) q^{51} + ( 2 \beta_{2} + \beta_{3} ) q^{52} -5 q^{53} + ( 4 \beta_{2} + \beta_{5} ) q^{54} + ( 4 \beta_{2} - \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{55} + ( 5 + \beta_{1} + \beta_{4} + \beta_{7} ) q^{57} + ( -\beta_{4} + 2 \beta_{7} ) q^{58} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 1 - \beta_{1} - \beta_{4} - \beta_{7} ) q^{60} + 2 \beta_{6} q^{61} + ( -\beta_{2} - 2 \beta_{5} - 2 \beta_{6} ) q^{62} + q^{64} + ( 4 + 2 \beta_{1} - \beta_{4} - 3 \beta_{7} ) q^{65} + ( 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{66} + ( 4 \beta_{4} - 2 \beta_{7} ) q^{67} + ( \beta_{2} + 2 \beta_{5} ) q^{68} + ( \beta_{2} + \beta_{3} + \beta_{5} - 5 \beta_{6} ) q^{69} + ( -\beta_{4} + 2 \beta_{7} ) q^{71} + ( 2 + \beta_{7} ) q^{72} + ( -3 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 2 \beta_{4} + \beta_{7} ) q^{74} + ( -\beta_{2} - 2 \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{75} + ( -\beta_{2} - 2 \beta_{5} + \beta_{6} ) q^{76} + ( -2 - \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{78} + ( 6 - 2 \beta_{1} - \beta_{4} ) q^{79} + ( \beta_{2} - \beta_{6} ) q^{80} + ( -1 + 4 \beta_{7} ) q^{81} + ( -2 \beta_{2} - \beta_{3} ) q^{82} + ( -2 \beta_{2} - 4 \beta_{5} + 6 \beta_{6} ) q^{83} + ( -2 \beta_{1} - \beta_{4} - 2 \beta_{7} ) q^{85} + ( \beta_{4} - 2 \beta_{7} ) q^{86} + ( -4 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{87} + ( -\beta_{4} + \beta_{7} ) q^{88} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{89} + ( \beta_{2} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{90} + ( 1 + 2 \beta_{1} + \beta_{4} ) q^{92} + ( -5 + 2 \beta_{1} + 2 \beta_{4} - \beta_{7} ) q^{93} + ( -3 \beta_{2} - 6 \beta_{5} - \beta_{6} ) q^{94} + ( 3 + 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{95} + ( -\beta_{2} - \beta_{5} ) q^{96} + ( 3 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -5 + 2 \beta_{1} - \beta_{4} + 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{2} + 8q^{4} - 8q^{8} - 16q^{9} + O(q^{10})$$ $$8q - 8q^{2} + 8q^{4} - 8q^{8} - 16q^{9} + 8q^{15} + 8q^{16} + 16q^{18} + 8q^{23} - 8q^{25} - 8q^{30} - 8q^{32} - 16q^{36} + 16q^{39} - 8q^{46} + 8q^{50} - 40q^{51} - 40q^{53} + 40q^{57} + 8q^{60} + 8q^{64} + 32q^{65} + 16q^{72} - 16q^{78} + 48q^{79} - 8q^{81} + 8q^{92} - 40q^{93} + 24q^{95} - 40q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{7} + 28 \nu^{5} - 49 \nu^{3} + 180 \nu$$$$)/189$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu$$$$)/189$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} + 2 \nu^{2} + 18$$$$)/9$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{7} - \nu^{5} - 5 \nu^{3} + 63 \nu$$$$)/27$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{7} - 28 \nu^{5} + 49 \nu^{3} - 324 \nu$$$$)/189$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{6} + 14 \nu^{4} - 56 \nu^{2} + 225$$$$)/63$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} + 22$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - 2 \beta_{6} + \beta_{3} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + 4 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{7} + \beta_{6} + 4 \beta_{3} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{5} - 7 \beta_{4} + 7 \beta_{2} + 5 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{7} + 22$$ $$\nu^{7}$$ $$=$$ $$($$$$29 \beta_{5} + 8 \beta_{4} + 8 \beta_{2} + 29 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1469.1
 1.01575 − 1.40294i −1.72286 − 0.178197i −1.72286 + 0.178197i 1.01575 + 1.40294i 1.72286 − 0.178197i −1.01575 − 1.40294i −1.01575 + 1.40294i 1.72286 + 0.178197i
−1.00000 −0.707107 1.58114i 1.00000 −1.41421 1.73205i 0.707107 + 1.58114i 0 −1.00000 −2.00000 + 2.23607i 1.41421 + 1.73205i
1469.2 −1.00000 −0.707107 1.58114i 1.00000 −1.41421 + 1.73205i 0.707107 + 1.58114i 0 −1.00000 −2.00000 + 2.23607i 1.41421 1.73205i
1469.3 −1.00000 −0.707107 + 1.58114i 1.00000 −1.41421 1.73205i 0.707107 1.58114i 0 −1.00000 −2.00000 2.23607i 1.41421 + 1.73205i
1469.4 −1.00000 −0.707107 + 1.58114i 1.00000 −1.41421 + 1.73205i 0.707107 1.58114i 0 −1.00000 −2.00000 2.23607i 1.41421 1.73205i
1469.5 −1.00000 0.707107 1.58114i 1.00000 1.41421 1.73205i −0.707107 + 1.58114i 0 −1.00000 −2.00000 2.23607i −1.41421 + 1.73205i
1469.6 −1.00000 0.707107 1.58114i 1.00000 1.41421 + 1.73205i −0.707107 + 1.58114i 0 −1.00000 −2.00000 2.23607i −1.41421 1.73205i
1469.7 −1.00000 0.707107 + 1.58114i 1.00000 1.41421 1.73205i −0.707107 1.58114i 0 −1.00000 −2.00000 + 2.23607i −1.41421 + 1.73205i
1469.8 −1.00000 0.707107 + 1.58114i 1.00000 1.41421 + 1.73205i −0.707107 1.58114i 0 −1.00000 −2.00000 + 2.23607i −1.41421 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1469.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g 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.d.e 8
3.b odd 2 1 1470.2.d.f 8
5.b even 2 1 1470.2.d.f 8
7.b odd 2 1 inner 1470.2.d.e 8
7.c even 3 1 210.2.t.f yes 8
7.d odd 6 1 210.2.t.f yes 8
15.d odd 2 1 inner 1470.2.d.e 8
21.c even 2 1 1470.2.d.f 8
21.g even 6 1 210.2.t.e 8
21.h odd 6 1 210.2.t.e 8
35.c odd 2 1 1470.2.d.f 8
35.i odd 6 1 210.2.t.e 8
35.j even 6 1 210.2.t.e 8
35.k even 12 2 1050.2.s.i 16
35.l odd 12 2 1050.2.s.i 16
105.g even 2 1 inner 1470.2.d.e 8
105.o odd 6 1 210.2.t.f yes 8
105.p even 6 1 210.2.t.f yes 8
105.w odd 12 2 1050.2.s.i 16
105.x even 12 2 1050.2.s.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.e 8 21.g even 6 1
210.2.t.e 8 21.h odd 6 1
210.2.t.e 8 35.i odd 6 1
210.2.t.e 8 35.j even 6 1
210.2.t.f yes 8 7.c even 3 1
210.2.t.f yes 8 7.d odd 6 1
210.2.t.f yes 8 105.o odd 6 1
210.2.t.f yes 8 105.p even 6 1
1050.2.s.i 16 35.k even 12 2
1050.2.s.i 16 35.l odd 12 2
1050.2.s.i 16 105.w odd 12 2
1050.2.s.i 16 105.x even 12 2
1470.2.d.e 8 1.a even 1 1 trivial
1470.2.d.e 8 7.b odd 2 1 inner
1470.2.d.e 8 15.d odd 2 1 inner
1470.2.d.e 8 105.g even 2 1 inner
1470.2.d.f 8 3.b odd 2 1
1470.2.d.f 8 5.b even 2 1
1470.2.d.f 8 21.c even 2 1
1470.2.d.f 8 35.c odd 2 1

## Hecke kernels

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

 $$T_{11}^{4} + 22 T_{11}^{2} + 1$$ $$T_{13}^{4} - 46 T_{13}^{2} + 49$$ $$T_{23}^{2} - 2 T_{23} - 29$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{8}$$
$3$ $$( 9 + 4 T^{2} + T^{4} )^{2}$$
$5$ $$( 25 + 2 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 1 + 22 T^{2} + T^{4} )^{2}$$
$13$ $$( 49 - 46 T^{2} + T^{4} )^{2}$$
$17$ $$( 10 + T^{2} )^{4}$$
$19$ $$( 49 + 26 T^{2} + T^{4} )^{2}$$
$23$ $$( -29 - 2 T + T^{2} )^{4}$$
$29$ $$( 196 + 52 T^{2} + T^{4} )^{2}$$
$31$ $$( 4 + 44 T^{2} + T^{4} )^{2}$$
$37$ $$( 361 + 58 T^{2} + T^{4} )^{2}$$
$41$ $$( 49 - 46 T^{2} + T^{4} )^{2}$$
$43$ $$( 196 + 52 T^{2} + T^{4} )^{2}$$
$47$ $$( 7569 + 186 T^{2} + T^{4} )^{2}$$
$53$ $$( 5 + T )^{8}$$
$59$ $$( 2704 - 136 T^{2} + T^{4} )^{2}$$
$61$ $$( 12 + T^{2} )^{4}$$
$67$ $$( 5776 + 232 T^{2} + T^{4} )^{2}$$
$71$ $$( 196 + 52 T^{2} + T^{4} )^{2}$$
$73$ $$( 1764 - 156 T^{2} + T^{4} )^{2}$$
$79$ $$( 6 - 12 T + T^{2} )^{4}$$
$83$ $$( 4624 + 296 T^{2} + T^{4} )^{2}$$
$89$ $$( 2704 - 136 T^{2} + T^{4} )^{2}$$
$97$ $$( 1764 - 156 T^{2} + T^{4} )^{2}$$