# Properties

 Label 490.2.c.a Level $490$ Weight $2$ Character orbit 490.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.91266969904$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + ( -2 - i ) q^{5} -i q^{8} + 3 q^{9} +O(q^{10})$$ $$q + i q^{2} - q^{4} + ( -2 - i ) q^{5} -i q^{8} + 3 q^{9} + ( 1 - 2 i ) q^{10} + 3 q^{11} + 5 i q^{13} + q^{16} -2 i q^{17} + 3 i q^{18} + 5 q^{19} + ( 2 + i ) q^{20} + 3 i q^{22} + 7 i q^{23} + ( 3 + 4 i ) q^{25} -5 q^{26} + 4 q^{29} -2 q^{31} + i q^{32} + 2 q^{34} -3 q^{36} + i q^{37} + 5 i q^{38} + ( -1 + 2 i ) q^{40} + 3 q^{41} -2 i q^{43} -3 q^{44} + ( -6 - 3 i ) q^{45} -7 q^{46} -7 i q^{47} + ( -4 + 3 i ) q^{50} -5 i q^{52} -9 i q^{53} + ( -6 - 3 i ) q^{55} + 4 i q^{58} + 4 q^{59} + 6 q^{61} -2 i q^{62} - q^{64} + ( 5 - 10 i ) q^{65} + 2 i q^{67} + 2 i q^{68} -6 q^{71} -3 i q^{72} + 16 i q^{73} - q^{74} -5 q^{76} -14 q^{79} + ( -2 - i ) q^{80} + 9 q^{81} + 3 i q^{82} + 6 i q^{83} + ( -2 + 4 i ) q^{85} + 2 q^{86} -3 i q^{88} -2 q^{89} + ( 3 - 6 i ) q^{90} -7 i q^{92} + 7 q^{94} + ( -10 - 5 i ) q^{95} -12 i q^{97} + 9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 4q^{5} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 4q^{5} + 6q^{9} + 2q^{10} + 6q^{11} + 2q^{16} + 10q^{19} + 4q^{20} + 6q^{25} - 10q^{26} + 8q^{29} - 4q^{31} + 4q^{34} - 6q^{36} - 2q^{40} + 6q^{41} - 6q^{44} - 12q^{45} - 14q^{46} - 8q^{50} - 12q^{55} + 8q^{59} + 12q^{61} - 2q^{64} + 10q^{65} - 12q^{71} - 2q^{74} - 10q^{76} - 28q^{79} - 4q^{80} + 18q^{81} - 4q^{85} + 4q^{86} - 4q^{89} + 6q^{90} + 14q^{94} - 20q^{95} + 18q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 −2.00000 + 1.00000i 0 0 1.00000i 3.00000 1.00000 + 2.00000i
99.2 1.00000i 0 −1.00000 −2.00000 1.00000i 0 0 1.00000i 3.00000 1.00000 2.00000i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.c.a 2
5.b even 2 1 inner 490.2.c.a 2
5.c odd 4 1 2450.2.a.k 1
5.c odd 4 1 2450.2.a.ba 1
7.b odd 2 1 490.2.c.d 2
7.c even 3 2 70.2.i.b 4
7.d odd 6 2 490.2.i.a 4
21.h odd 6 2 630.2.u.a 4
28.g odd 6 2 560.2.bw.d 4
35.c odd 2 1 490.2.c.d 2
35.f even 4 1 2450.2.a.j 1
35.f even 4 1 2450.2.a.bb 1
35.i odd 6 2 490.2.i.a 4
35.j even 6 2 70.2.i.b 4
35.l odd 12 2 350.2.e.c 2
35.l odd 12 2 350.2.e.j 2
105.o odd 6 2 630.2.u.a 4
140.p odd 6 2 560.2.bw.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.b 4 7.c even 3 2
70.2.i.b 4 35.j even 6 2
350.2.e.c 2 35.l odd 12 2
350.2.e.j 2 35.l odd 12 2
490.2.c.a 2 1.a even 1 1 trivial
490.2.c.a 2 5.b even 2 1 inner
490.2.c.d 2 7.b odd 2 1
490.2.c.d 2 35.c odd 2 1
490.2.i.a 4 7.d odd 6 2
490.2.i.a 4 35.i odd 6 2
560.2.bw.d 4 28.g odd 6 2
560.2.bw.d 4 140.p odd 6 2
630.2.u.a 4 21.h odd 6 2
630.2.u.a 4 105.o odd 6 2
2450.2.a.j 1 35.f even 4 1
2450.2.a.k 1 5.c odd 4 1
2450.2.a.ba 1 5.c odd 4 1
2450.2.a.bb 1 35.f even 4 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{11} - 3$$ $$T_{19} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 + 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$25 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -5 + T )^{2}$$
$23$ $$49 + T^{2}$$
$29$ $$( -4 + T )^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$1 + T^{2}$$
$41$ $$( -3 + T )^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$49 + T^{2}$$
$53$ $$81 + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$256 + T^{2}$$
$79$ $$( 14 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$( 2 + T )^{2}$$
$97$ $$144 + T^{2}$$