# Properties

 Label 1470.2.i.f Level $1470$ Weight $2$ Character orbit 1470.i Analytic conductor $11.738$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} - q^{6} + q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} - q^{6} + q^{8} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{10} + ( -4 + 4 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} -2 q^{13} - q^{15} -\zeta_{6} q^{16} + ( -2 + 2 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} + 4 \zeta_{6} q^{19} + q^{20} + 4 q^{22} + 8 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} + 2 \zeta_{6} q^{26} - q^{27} + 6 q^{29} + \zeta_{6} q^{30} + ( 8 - 8 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + 4 \zeta_{6} q^{33} + 2 q^{34} + q^{36} + 2 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} + ( -2 + 2 \zeta_{6} ) q^{39} -\zeta_{6} q^{40} + 2 q^{41} -12 q^{43} -4 \zeta_{6} q^{44} + ( -1 + \zeta_{6} ) q^{45} + ( 8 - 8 \zeta_{6} ) q^{46} + 8 \zeta_{6} q^{47} - q^{48} + q^{50} + 2 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + \zeta_{6} q^{54} + 4 q^{55} + 4 q^{57} -6 \zeta_{6} q^{58} + ( -4 + 4 \zeta_{6} ) q^{59} + ( 1 - \zeta_{6} ) q^{60} + 2 \zeta_{6} q^{61} -8 q^{62} + q^{64} + 2 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{66} + ( -12 + 12 \zeta_{6} ) q^{67} -2 \zeta_{6} q^{68} + 8 q^{69} + 8 q^{71} -\zeta_{6} q^{72} + ( 14 - 14 \zeta_{6} ) q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} + \zeta_{6} q^{75} -4 q^{76} + 2 q^{78} + ( -1 + \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -2 \zeta_{6} q^{82} + 12 q^{83} + 2 q^{85} + 12 \zeta_{6} q^{86} + ( 6 - 6 \zeta_{6} ) q^{87} + ( -4 + 4 \zeta_{6} ) q^{88} -2 \zeta_{6} q^{89} + q^{90} -8 q^{92} -8 \zeta_{6} q^{93} + ( 8 - 8 \zeta_{6} ) q^{94} + ( 4 - 4 \zeta_{6} ) q^{95} + \zeta_{6} q^{96} + 10 q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{3} - q^{4} - q^{5} - 2q^{6} + 2q^{8} - q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{3} - q^{4} - q^{5} - 2q^{6} + 2q^{8} - q^{9} - q^{10} - 4q^{11} + q^{12} - 4q^{13} - 2q^{15} - q^{16} - 2q^{17} - q^{18} + 4q^{19} + 2q^{20} + 8q^{22} + 8q^{23} + q^{24} - q^{25} + 2q^{26} - 2q^{27} + 12q^{29} + q^{30} + 8q^{31} - q^{32} + 4q^{33} + 4q^{34} + 2q^{36} + 2q^{37} + 4q^{38} - 2q^{39} - q^{40} + 4q^{41} - 24q^{43} - 4q^{44} - q^{45} + 8q^{46} + 8q^{47} - 2q^{48} + 2q^{50} + 2q^{51} + 2q^{52} - 6q^{53} + q^{54} + 8q^{55} + 8q^{57} - 6q^{58} - 4q^{59} + q^{60} + 2q^{61} - 16q^{62} + 2q^{64} + 2q^{65} + 4q^{66} - 12q^{67} - 2q^{68} + 16q^{69} + 16q^{71} - q^{72} + 14q^{73} + 2q^{74} + q^{75} - 8q^{76} + 4q^{78} - q^{80} - q^{81} - 2q^{82} + 24q^{83} + 4q^{85} + 12q^{86} + 6q^{87} - 4q^{88} - 2q^{89} + 2q^{90} - 16q^{92} - 8q^{93} + 8q^{94} + 4q^{95} + q^{96} + 20q^{97} + 8q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 0 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
961.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 0 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.i.f 2
7.b odd 2 1 1470.2.i.b 2
7.c even 3 1 210.2.a.c 1
7.c even 3 1 inner 1470.2.i.f 2
7.d odd 6 1 1470.2.a.q 1
7.d odd 6 1 1470.2.i.b 2
21.g even 6 1 4410.2.a.l 1
21.h odd 6 1 630.2.a.b 1
28.g odd 6 1 1680.2.a.q 1
35.i odd 6 1 7350.2.a.p 1
35.j even 6 1 1050.2.a.h 1
35.l odd 12 2 1050.2.g.d 2
56.k odd 6 1 6720.2.a.k 1
56.p even 6 1 6720.2.a.bp 1
84.n even 6 1 5040.2.a.i 1
105.o odd 6 1 3150.2.a.w 1
105.x even 12 2 3150.2.g.e 2
140.p odd 6 1 8400.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.c 1 7.c even 3 1
630.2.a.b 1 21.h odd 6 1
1050.2.a.h 1 35.j even 6 1
1050.2.g.d 2 35.l odd 12 2
1470.2.a.q 1 7.d odd 6 1
1470.2.i.b 2 7.b odd 2 1
1470.2.i.b 2 7.d odd 6 1
1470.2.i.f 2 1.a even 1 1 trivial
1470.2.i.f 2 7.c even 3 1 inner
1680.2.a.q 1 28.g odd 6 1
3150.2.a.w 1 105.o odd 6 1
3150.2.g.e 2 105.x even 12 2
4410.2.a.l 1 21.g even 6 1
5040.2.a.i 1 84.n even 6 1
6720.2.a.k 1 56.k odd 6 1
6720.2.a.bp 1 56.p even 6 1
7350.2.a.p 1 35.i odd 6 1
8400.2.a.p 1 140.p odd 6 1

## Hecke kernels

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

 $$T_{11}^{2} + 4 T_{11} + 16$$ $$T_{13} + 2$$ $$T_{17}^{2} + 2 T_{17} + 4$$ $$T_{19}^{2} - 4 T_{19} + 16$$ $$T_{31}^{2} - 8 T_{31} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 + 4 T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$4 + 2 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$64 - 8 T + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$64 - 8 T + T^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$( 12 + T )^{2}$$
$47$ $$64 - 8 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$144 + 12 T + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$196 - 14 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$4 + 2 T + T^{2}$$
$97$ $$( -10 + T )^{2}$$
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