# Properties

 Label 490.2.e.h Level $490$ Weight $2$ Character orbit 490.e Analytic conductor $3.913$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$490 = 2 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 490.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.91266969904$$ 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 70) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + q^{6} - q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + q^{6} - q^{8} + 2 \zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{10} + ( 6 - 6 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} + 4 q^{13} - q^{15} -\zeta_{6} q^{16} + ( -2 + 2 \zeta_{6} ) q^{18} + 2 \zeta_{6} q^{19} + q^{20} + 6 q^{22} + 3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{26} + 5 q^{27} -3 q^{29} -\zeta_{6} q^{30} + ( 8 - 8 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} -6 \zeta_{6} q^{33} -2 q^{36} + 4 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{38} + ( 4 - 4 \zeta_{6} ) q^{39} + \zeta_{6} q^{40} -9 q^{41} -7 q^{43} + 6 \zeta_{6} q^{44} + ( 2 - 2 \zeta_{6} ) q^{45} + ( -3 + 3 \zeta_{6} ) q^{46} - q^{48} - q^{50} + ( -4 + 4 \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} + 5 \zeta_{6} q^{54} -6 q^{55} + 2 q^{57} -3 \zeta_{6} q^{58} + ( -6 + 6 \zeta_{6} ) q^{59} + ( 1 - \zeta_{6} ) q^{60} + 5 \zeta_{6} q^{61} + 8 q^{62} + q^{64} -4 \zeta_{6} q^{65} + ( 6 - 6 \zeta_{6} ) q^{66} + ( -5 + 5 \zeta_{6} ) q^{67} + 3 q^{69} -6 q^{71} -2 \zeta_{6} q^{72} + ( -16 + 16 \zeta_{6} ) q^{73} + ( -4 + 4 \zeta_{6} ) q^{74} + \zeta_{6} q^{75} -2 q^{76} + 4 q^{78} -2 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -9 \zeta_{6} q^{82} -3 q^{83} -7 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{87} + ( -6 + 6 \zeta_{6} ) q^{88} -15 \zeta_{6} q^{89} + 2 q^{90} -3 q^{92} -8 \zeta_{6} q^{93} + ( 2 - 2 \zeta_{6} ) q^{95} -\zeta_{6} q^{96} -14 q^{97} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + q^{3} - q^{4} - q^{5} + 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + q^{2} + q^{3} - q^{4} - q^{5} + 2q^{6} - 2q^{8} + 2q^{9} + q^{10} + 6q^{11} + q^{12} + 8q^{13} - 2q^{15} - q^{16} - 2q^{18} + 2q^{19} + 2q^{20} + 12q^{22} + 3q^{23} - q^{24} - q^{25} + 4q^{26} + 10q^{27} - 6q^{29} - q^{30} + 8q^{31} + q^{32} - 6q^{33} - 4q^{36} + 4q^{37} - 2q^{38} + 4q^{39} + q^{40} - 18q^{41} - 14q^{43} + 6q^{44} + 2q^{45} - 3q^{46} - 2q^{48} - 2q^{50} - 4q^{52} + 6q^{53} + 5q^{54} - 12q^{55} + 4q^{57} - 3q^{58} - 6q^{59} + q^{60} + 5q^{61} + 16q^{62} + 2q^{64} - 4q^{65} + 6q^{66} - 5q^{67} + 6q^{69} - 12q^{71} - 2q^{72} - 16q^{73} - 4q^{74} + q^{75} - 4q^{76} + 8q^{78} - 2q^{79} - q^{80} - q^{81} - 9q^{82} - 6q^{83} - 7q^{86} - 3q^{87} - 6q^{88} - 15q^{89} + 4q^{90} - 6q^{92} - 8q^{93} + 2q^{95} - q^{96} - 28q^{97} + 24q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/490\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 1.00000 + 1.73205i 0.500000 0.866025i
471.1 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 0 −1.00000 1.00000 1.73205i 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 490.2.e.h 2
7.b odd 2 1 70.2.e.c 2
7.c even 3 1 490.2.a.b 1
7.c even 3 1 inner 490.2.e.h 2
7.d odd 6 1 70.2.e.c 2
7.d odd 6 1 490.2.a.c 1
21.c even 2 1 630.2.k.b 2
21.g even 6 1 630.2.k.b 2
21.g even 6 1 4410.2.a.bm 1
21.h odd 6 1 4410.2.a.bd 1
28.d even 2 1 560.2.q.g 2
28.f even 6 1 560.2.q.g 2
28.f even 6 1 3920.2.a.p 1
28.g odd 6 1 3920.2.a.bc 1
35.c odd 2 1 350.2.e.e 2
35.f even 4 2 350.2.j.b 4
35.i odd 6 1 350.2.e.e 2
35.i odd 6 1 2450.2.a.w 1
35.j even 6 1 2450.2.a.bc 1
35.k even 12 2 350.2.j.b 4
35.k even 12 2 2450.2.c.g 2
35.l odd 12 2 2450.2.c.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.c 2 7.b odd 2 1
70.2.e.c 2 7.d odd 6 1
350.2.e.e 2 35.c odd 2 1
350.2.e.e 2 35.i odd 6 1
350.2.j.b 4 35.f even 4 2
350.2.j.b 4 35.k even 12 2
490.2.a.b 1 7.c even 3 1
490.2.a.c 1 7.d odd 6 1
490.2.e.h 2 1.a even 1 1 trivial
490.2.e.h 2 7.c even 3 1 inner
560.2.q.g 2 28.d even 2 1
560.2.q.g 2 28.f even 6 1
630.2.k.b 2 21.c even 2 1
630.2.k.b 2 21.g even 6 1
2450.2.a.w 1 35.i odd 6 1
2450.2.a.bc 1 35.j even 6 1
2450.2.c.g 2 35.k even 12 2
2450.2.c.l 2 35.l odd 12 2
3920.2.a.p 1 28.f even 6 1
3920.2.a.bc 1 28.g odd 6 1
4410.2.a.bd 1 21.h odd 6 1
4410.2.a.bm 1 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(490, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{11}^{2} - 6 T_{11} + 36$$ $$T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$36 - 6 T + T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$4 - 2 T + T^{2}$$
$23$ $$9 - 3 T + T^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$64 - 8 T + T^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$( 9 + T )^{2}$$
$43$ $$( 7 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$36 + 6 T + T^{2}$$
$61$ $$25 - 5 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$256 + 16 T + T^{2}$$
$79$ $$4 + 2 T + T^{2}$$
$83$ $$( 3 + T )^{2}$$
$89$ $$225 + 15 T + T^{2}$$
$97$ $$( 14 + T )^{2}$$