# Properties

 Label 1470.2.i.q Level $1470$ Weight $2$ Character orbit 1470.i 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.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 30) 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.

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

## 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.q 2
7.b odd 2 1 1470.2.i.o 2
7.c even 3 1 1470.2.a.d 1
7.c even 3 1 inner 1470.2.i.q 2
7.d odd 6 1 30.2.a.a 1
7.d odd 6 1 1470.2.i.o 2
21.g even 6 1 90.2.a.c 1
21.h odd 6 1 4410.2.a.z 1
28.f even 6 1 240.2.a.b 1
35.i odd 6 1 150.2.a.b 1
35.j even 6 1 7350.2.a.ct 1
35.k even 12 2 150.2.c.a 2
56.j odd 6 1 960.2.a.e 1
56.m even 6 1 960.2.a.p 1
63.i even 6 1 810.2.e.b 2
63.k odd 6 1 810.2.e.l 2
63.s even 6 1 810.2.e.b 2
63.t odd 6 1 810.2.e.l 2
77.i even 6 1 3630.2.a.w 1
84.j odd 6 1 720.2.a.j 1
91.s odd 6 1 5070.2.a.w 1
91.bb even 12 2 5070.2.b.k 2
105.p even 6 1 450.2.a.d 1
105.w odd 12 2 450.2.c.b 2
112.v even 12 2 3840.2.k.f 2
112.x odd 12 2 3840.2.k.y 2
119.h odd 6 1 8670.2.a.g 1
140.s even 6 1 1200.2.a.k 1
140.x odd 12 2 1200.2.f.e 2
168.ba even 6 1 2880.2.a.a 1
168.be odd 6 1 2880.2.a.q 1
280.ba even 6 1 4800.2.a.d 1
280.bk odd 6 1 4800.2.a.cq 1
280.bp odd 12 2 4800.2.f.w 2
280.bv even 12 2 4800.2.f.p 2
420.be odd 6 1 3600.2.a.f 1
420.br even 12 2 3600.2.f.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 7.d odd 6 1
90.2.a.c 1 21.g even 6 1
150.2.a.b 1 35.i odd 6 1
150.2.c.a 2 35.k even 12 2
240.2.a.b 1 28.f even 6 1
450.2.a.d 1 105.p even 6 1
450.2.c.b 2 105.w odd 12 2
720.2.a.j 1 84.j odd 6 1
810.2.e.b 2 63.i even 6 1
810.2.e.b 2 63.s even 6 1
810.2.e.l 2 63.k odd 6 1
810.2.e.l 2 63.t odd 6 1
960.2.a.e 1 56.j odd 6 1
960.2.a.p 1 56.m even 6 1
1200.2.a.k 1 140.s even 6 1
1200.2.f.e 2 140.x odd 12 2
1470.2.a.d 1 7.c even 3 1
1470.2.i.o 2 7.b odd 2 1
1470.2.i.o 2 7.d odd 6 1
1470.2.i.q 2 1.a even 1 1 trivial
1470.2.i.q 2 7.c even 3 1 inner
2880.2.a.a 1 168.ba even 6 1
2880.2.a.q 1 168.be odd 6 1
3600.2.a.f 1 420.be odd 6 1
3600.2.f.i 2 420.br even 12 2
3630.2.a.w 1 77.i even 6 1
3840.2.k.f 2 112.v even 12 2
3840.2.k.y 2 112.x odd 12 2
4410.2.a.z 1 21.h odd 6 1
4800.2.a.d 1 280.ba even 6 1
4800.2.a.cq 1 280.bk odd 6 1
4800.2.f.p 2 280.bv even 12 2
4800.2.f.w 2 280.bp odd 12 2
5070.2.a.w 1 91.s odd 6 1
5070.2.b.k 2 91.bb even 12 2
7350.2.a.ct 1 35.j even 6 1
8670.2.a.g 1 119.h odd 6 1

## Hecke kernels

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

 $$T_{11}$$ $$T_{13} + 2$$ $$T_{17}^{2} - 6 T_{17} + 36$$ $$T_{19}^{2} + 4 T_{19} + 16$$ $$T_{31}^{2} - 8 T_{31} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$16 + 4 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$64 - 8 T + T^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$100 + 10 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 - 2 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$324 - 18 T + T^{2}$$
$97$ $$( 2 + T )^{2}$$