# Properties

 Label 1470.2.i.h Level 1470 Weight 2 Character orbit 1470.i Analytic conductor 11.738 Analytic rank 1 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.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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.

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

## 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.h 2
7.b odd 2 1 1470.2.i.d 2
7.c even 3 1 1470.2.a.m 1
7.c even 3 1 inner 1470.2.i.h 2
7.d odd 6 1 210.2.a.d 1
7.d odd 6 1 1470.2.i.d 2
21.g even 6 1 630.2.a.f 1
21.h odd 6 1 4410.2.a.f 1
28.f even 6 1 1680.2.a.b 1
35.i odd 6 1 1050.2.a.a 1
35.j even 6 1 7350.2.a.bd 1
35.k even 12 2 1050.2.g.h 2
56.j odd 6 1 6720.2.a.bb 1
56.m even 6 1 6720.2.a.cc 1
84.j odd 6 1 5040.2.a.ba 1
105.p even 6 1 3150.2.a.ba 1
105.w odd 12 2 3150.2.g.o 2
140.s even 6 1 8400.2.a.cn 1

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

## Hecke kernels

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

 $$T_{11}$$ $$T_{13} + 2$$ $$T_{17}^{2} + 6 T_{17} + 36$$ $$T_{19}^{2} + 4 T_{19} + 16$$ $$T_{31}^{2} + 4 T_{31} + 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ 1
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 2 T + 13 T^{2} )^{2}$$
$17$ $$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$
$37$ $$1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$
$47$ $$1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4}$$
$53$ $$1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4}$$
$59$ $$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$
$61$ $$1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4}$$
$67$ $$1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4}$$
$79$ $$1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 12 T + 83 T^{2} )^{2}$$
$89$ $$1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 14 T + 97 T^{2} )^{2}$$