# Properties

 Label 490.2.e.f 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} -3 q^{6} + q^{8} -6 \zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} -3 q^{6} + q^{8} -6 \zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{10} + ( 2 - 2 \zeta_{6} ) q^{11} + 3 \zeta_{6} q^{12} -3 q^{15} -\zeta_{6} q^{16} + ( -4 + 4 \zeta_{6} ) q^{17} + ( -6 + 6 \zeta_{6} ) q^{18} -6 \zeta_{6} q^{19} + q^{20} -2 q^{22} -3 \zeta_{6} q^{23} + ( 3 - 3 \zeta_{6} ) q^{24} + ( -1 + \zeta_{6} ) q^{25} -9 q^{27} + 9 q^{29} + 3 \zeta_{6} q^{30} + ( -4 + 4 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} -6 \zeta_{6} q^{33} + 4 q^{34} + 6 q^{36} + 4 \zeta_{6} q^{37} + ( -6 + 6 \zeta_{6} ) q^{38} -\zeta_{6} q^{40} + 7 q^{41} -5 q^{43} + 2 \zeta_{6} q^{44} + ( -6 + 6 \zeta_{6} ) q^{45} + ( -3 + 3 \zeta_{6} ) q^{46} + 8 \zeta_{6} q^{47} -3 q^{48} + q^{50} + 12 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{53} + 9 \zeta_{6} q^{54} -2 q^{55} -18 q^{57} -9 \zeta_{6} q^{58} + ( 10 - 10 \zeta_{6} ) q^{59} + ( 3 - 3 \zeta_{6} ) q^{60} + \zeta_{6} q^{61} + 4 q^{62} + q^{64} + ( -6 + 6 \zeta_{6} ) q^{66} + ( 9 - 9 \zeta_{6} ) q^{67} -4 \zeta_{6} q^{68} -9 q^{69} + 2 q^{71} -6 \zeta_{6} q^{72} + ( -4 + 4 \zeta_{6} ) q^{73} + ( 4 - 4 \zeta_{6} ) q^{74} + 3 \zeta_{6} q^{75} + 6 q^{76} -10 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} -7 \zeta_{6} q^{82} + 7 q^{83} + 4 q^{85} + 5 \zeta_{6} q^{86} + ( 27 - 27 \zeta_{6} ) q^{87} + ( 2 - 2 \zeta_{6} ) q^{88} + \zeta_{6} q^{89} + 6 q^{90} + 3 q^{92} + 12 \zeta_{6} q^{93} + ( 8 - 8 \zeta_{6} ) q^{94} + ( -6 + 6 \zeta_{6} ) q^{95} + 3 \zeta_{6} q^{96} -14 q^{97} -12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 3q^{3} - q^{4} - q^{5} - 6q^{6} + 2q^{8} - 6q^{9} + O(q^{10})$$ $$2q - q^{2} + 3q^{3} - q^{4} - q^{5} - 6q^{6} + 2q^{8} - 6q^{9} - q^{10} + 2q^{11} + 3q^{12} - 6q^{15} - q^{16} - 4q^{17} - 6q^{18} - 6q^{19} + 2q^{20} - 4q^{22} - 3q^{23} + 3q^{24} - q^{25} - 18q^{27} + 18q^{29} + 3q^{30} - 4q^{31} - q^{32} - 6q^{33} + 8q^{34} + 12q^{36} + 4q^{37} - 6q^{38} - q^{40} + 14q^{41} - 10q^{43} + 2q^{44} - 6q^{45} - 3q^{46} + 8q^{47} - 6q^{48} + 2q^{50} + 12q^{51} + 2q^{53} + 9q^{54} - 4q^{55} - 36q^{57} - 9q^{58} + 10q^{59} + 3q^{60} + q^{61} + 8q^{62} + 2q^{64} - 6q^{66} + 9q^{67} - 4q^{68} - 18q^{69} + 4q^{71} - 6q^{72} - 4q^{73} + 4q^{74} + 3q^{75} + 12q^{76} - 10q^{79} - q^{80} - 9q^{81} - 7q^{82} + 14q^{83} + 8q^{85} + 5q^{86} + 27q^{87} + 2q^{88} + q^{89} + 12q^{90} + 6q^{92} + 12q^{93} + 8q^{94} - 6q^{95} + 3q^{96} - 28q^{97} - 24q^{99} + O(q^{100})$$

## 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 1.50000 2.59808i −0.500000 + 0.866025i −0.500000 0.866025i −3.00000 0 1.00000 −3.00000 5.19615i −0.500000 + 0.866025i
471.1 −0.500000 + 0.866025i 1.50000 + 2.59808i −0.500000 0.866025i −0.500000 + 0.866025i −3.00000 0 1.00000 −3.00000 + 5.19615i −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.f 2
7.b odd 2 1 70.2.e.a 2
7.c even 3 1 490.2.a.e 1
7.c even 3 1 inner 490.2.e.f 2
7.d odd 6 1 70.2.e.a 2
7.d odd 6 1 490.2.a.k 1
21.c even 2 1 630.2.k.f 2
21.g even 6 1 630.2.k.f 2
21.g even 6 1 4410.2.a.r 1
21.h odd 6 1 4410.2.a.h 1
28.d even 2 1 560.2.q.i 2
28.f even 6 1 560.2.q.i 2
28.f even 6 1 3920.2.a.b 1
28.g odd 6 1 3920.2.a.bk 1
35.c odd 2 1 350.2.e.l 2
35.f even 4 2 350.2.j.f 4
35.i odd 6 1 350.2.e.l 2
35.i odd 6 1 2450.2.a.b 1
35.j even 6 1 2450.2.a.q 1
35.k even 12 2 350.2.j.f 4
35.k even 12 2 2450.2.c.s 2
35.l odd 12 2 2450.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a 2 7.b odd 2 1
70.2.e.a 2 7.d odd 6 1
350.2.e.l 2 35.c odd 2 1
350.2.e.l 2 35.i odd 6 1
350.2.j.f 4 35.f even 4 2
350.2.j.f 4 35.k even 12 2
490.2.a.e 1 7.c even 3 1
490.2.a.k 1 7.d odd 6 1
490.2.e.f 2 1.a even 1 1 trivial
490.2.e.f 2 7.c even 3 1 inner
560.2.q.i 2 28.d even 2 1
560.2.q.i 2 28.f even 6 1
630.2.k.f 2 21.c even 2 1
630.2.k.f 2 21.g even 6 1
2450.2.a.b 1 35.i odd 6 1
2450.2.a.q 1 35.j even 6 1
2450.2.c.a 2 35.l odd 12 2
2450.2.c.s 2 35.k even 12 2
3920.2.a.b 1 28.f even 6 1
3920.2.a.bk 1 28.g odd 6 1
4410.2.a.h 1 21.h odd 6 1
4410.2.a.r 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 $$S_{2}^{\mathrm{new}}(490, [\chi])$$:

 $$T_{3}^{2} - 3 T_{3} + 9$$ $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$16 + 4 T + T^{2}$$
$19$ $$36 + 6 T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$( -9 + T )^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$( -7 + T )^{2}$$
$43$ $$( 5 + T )^{2}$$
$47$ $$64 - 8 T + T^{2}$$
$53$ $$4 - 2 T + T^{2}$$
$59$ $$100 - 10 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$81 - 9 T + T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$16 + 4 T + T^{2}$$
$79$ $$100 + 10 T + T^{2}$$
$83$ $$( -7 + T )^{2}$$
$89$ $$1 - T + T^{2}$$
$97$ $$( 14 + T )^{2}$$