# Properties

 Label 490.2.e.c 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: $$\Q(\sqrt{-3})$$ 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.c 2
7.b odd 2 1 490.2.e.d 2
7.c even 3 1 490.2.a.h 1
7.c even 3 1 inner 490.2.e.c 2
7.d odd 6 1 70.2.a.a 1
7.d odd 6 1 490.2.e.d 2
21.g even 6 1 630.2.a.d 1
21.h odd 6 1 4410.2.a.b 1
28.f even 6 1 560.2.a.d 1
28.g odd 6 1 3920.2.a.t 1
35.i odd 6 1 350.2.a.b 1
35.j even 6 1 2450.2.a.l 1
35.k even 12 2 350.2.c.b 2
35.l odd 12 2 2450.2.c.k 2
56.j odd 6 1 2240.2.a.n 1
56.m even 6 1 2240.2.a.q 1
77.i even 6 1 8470.2.a.j 1
84.j odd 6 1 5040.2.a.bm 1
105.p even 6 1 3150.2.a.bj 1
105.w odd 12 2 3150.2.g.c 2
140.s even 6 1 2800.2.a.m 1
140.x odd 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.d odd 6 1
350.2.a.b 1 35.i odd 6 1
350.2.c.b 2 35.k even 12 2
490.2.a.h 1 7.c even 3 1
490.2.e.c 2 1.a even 1 1 trivial
490.2.e.c 2 7.c even 3 1 inner
490.2.e.d 2 7.b odd 2 1
490.2.e.d 2 7.d odd 6 1
560.2.a.d 1 28.f even 6 1
630.2.a.d 1 21.g even 6 1
2240.2.a.n 1 56.j odd 6 1
2240.2.a.q 1 56.m even 6 1
2450.2.a.l 1 35.j even 6 1
2450.2.c.k 2 35.l odd 12 2
2800.2.a.m 1 140.s even 6 1
2800.2.g.n 2 140.x odd 12 2
3150.2.a.bj 1 105.p even 6 1
3150.2.g.c 2 105.w odd 12 2
3920.2.a.t 1 28.g odd 6 1
4410.2.a.b 1 21.h odd 6 1
5040.2.a.bm 1 84.j odd 6 1
8470.2.a.j 1 77.i even 6 1

## Hecke kernels

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$5$ $$1 + T + T^{2}$$
$7$ 1
$11$ $$1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 6 T + 13 T^{2} )^{2}$$
$17$ $$1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4}$$
$19$ $$1 - 19 T^{2} + 361 T^{4}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} )$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 4 T + 43 T^{2} )^{2}$$
$47$ $$1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4}$$
$53$ $$1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4}$$
$59$ $$1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4}$$
$61$ $$( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 16 T + 71 T^{2} )^{2}$$
$73$ $$1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 8 T + 83 T^{2} )^{2}$$
$89$ $$1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 2 T + 97 T^{2} )^{2}$$