# Properties

 Label 490.2.e.d 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + q^{8} + 3 \zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{10} + ( -4 + 4 \zeta_{6} ) q^{11} -6 q^{13} -\zeta_{6} q^{16} + ( -2 + 2 \zeta_{6} ) q^{17} + ( 3 - 3 \zeta_{6} ) q^{18} - q^{20} + 4 q^{22} + ( -1 + \zeta_{6} ) q^{25} + 6 \zeta_{6} q^{26} + 6 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + 2 q^{34} -3 q^{36} + 10 \zeta_{6} q^{37} + \zeta_{6} q^{40} + 2 q^{41} + 4 q^{43} -4 \zeta_{6} q^{44} + ( -3 + 3 \zeta_{6} ) q^{45} -8 \zeta_{6} q^{47} + q^{50} + ( 6 - 6 \zeta_{6} ) q^{52} + ( 2 - 2 \zeta_{6} ) q^{53} -4 q^{55} -6 \zeta_{6} q^{58} + ( 8 - 8 \zeta_{6} ) q^{59} + 14 \zeta_{6} q^{61} + 8 q^{62} + q^{64} -6 \zeta_{6} q^{65} + ( 12 - 12 \zeta_{6} ) q^{67} -2 \zeta_{6} q^{68} -16 q^{71} + 3 \zeta_{6} q^{72} + ( -2 + 2 \zeta_{6} ) q^{73} + ( 10 - 10 \zeta_{6} ) q^{74} + 8 \zeta_{6} q^{79} + ( 1 - \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} -2 \zeta_{6} q^{82} + 8 q^{83} -2 q^{85} -4 \zeta_{6} q^{86} + ( -4 + 4 \zeta_{6} ) q^{88} -10 \zeta_{6} q^{89} + 3 q^{90} + ( -8 + 8 \zeta_{6} ) q^{94} + 2 q^{97} -12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + q^{5} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{4} + q^{5} + 2q^{8} + 3q^{9} + q^{10} - 4q^{11} - 12q^{13} - q^{16} - 2q^{17} + 3q^{18} - 2q^{20} + 8q^{22} - q^{25} + 6q^{26} + 12q^{29} - 8q^{31} - q^{32} + 4q^{34} - 6q^{36} + 10q^{37} + q^{40} + 4q^{41} + 8q^{43} - 4q^{44} - 3q^{45} - 8q^{47} + 2q^{50} + 6q^{52} + 2q^{53} - 8q^{55} - 6q^{58} + 8q^{59} + 14q^{61} + 16q^{62} + 2q^{64} - 6q^{65} + 12q^{67} - 2q^{68} - 32q^{71} + 3q^{72} - 2q^{73} + 10q^{74} + 8q^{79} + q^{80} - 9q^{81} - 2q^{82} + 16q^{83} - 4q^{85} - 4q^{86} - 4q^{88} - 10q^{89} + 6q^{90} - 8q^{94} + 4q^{97} - 24q^{99} + O(q^{100})$$

## 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 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0 1.00000 1.50000 + 2.59808i 0.500000 0.866025i
471.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 0 1.00000 1.50000 2.59808i 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.d 2
7.b odd 2 1 490.2.e.c 2
7.c even 3 1 70.2.a.a 1
7.c even 3 1 inner 490.2.e.d 2
7.d odd 6 1 490.2.a.h 1
7.d odd 6 1 490.2.e.c 2
21.g even 6 1 4410.2.a.b 1
21.h odd 6 1 630.2.a.d 1
28.f even 6 1 3920.2.a.t 1
28.g odd 6 1 560.2.a.d 1
35.i odd 6 1 2450.2.a.l 1
35.j even 6 1 350.2.a.b 1
35.k even 12 2 2450.2.c.k 2
35.l odd 12 2 350.2.c.b 2
56.k odd 6 1 2240.2.a.q 1
56.p even 6 1 2240.2.a.n 1
77.h odd 6 1 8470.2.a.j 1
84.n even 6 1 5040.2.a.bm 1
105.o odd 6 1 3150.2.a.bj 1
105.x even 12 2 3150.2.g.c 2
140.p odd 6 1 2800.2.a.m 1
140.w even 12 2 2800.2.g.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 7.c even 3 1
350.2.a.b 1 35.j even 6 1
350.2.c.b 2 35.l odd 12 2
490.2.a.h 1 7.d odd 6 1
490.2.e.c 2 7.b odd 2 1
490.2.e.c 2 7.d odd 6 1
490.2.e.d 2 1.a even 1 1 trivial
490.2.e.d 2 7.c even 3 1 inner
560.2.a.d 1 28.g odd 6 1
630.2.a.d 1 21.h odd 6 1
2240.2.a.n 1 56.p even 6 1
2240.2.a.q 1 56.k odd 6 1
2450.2.a.l 1 35.i odd 6 1
2450.2.c.k 2 35.k even 12 2
2800.2.a.m 1 140.p odd 6 1
2800.2.g.n 2 140.w even 12 2
3150.2.a.bj 1 105.o odd 6 1
3150.2.g.c 2 105.x even 12 2
3920.2.a.t 1 28.f even 6 1
4410.2.a.b 1 21.g even 6 1
5040.2.a.bm 1 84.n even 6 1
8470.2.a.j 1 77.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 $$S_{2}^{\mathrm{new}}(490, [\chi])$$:

 $$T_{3}$$ $$T_{11}^{2} + 4 T_{11} + 16$$ $$T_{13} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 + 4 T + T^{2}$$
$13$ $$( 6 + T )^{2}$$
$17$ $$4 + 2 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$100 - 10 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$64 + 8 T + T^{2}$$
$53$ $$4 - 2 T + T^{2}$$
$59$ $$64 - 8 T + T^{2}$$
$61$ $$196 - 14 T + T^{2}$$
$67$ $$144 - 12 T + T^{2}$$
$71$ $$( 16 + T )^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$64 - 8 T + T^{2}$$
$83$ $$( -8 + T )^{2}$$
$89$ $$100 + 10 T + T^{2}$$
$97$ $$( -2 + T )^{2}$$