# Properties

 Label 490.2.c.g Level $490$ Weight $2$ Character orbit 490.c Analytic conductor $3.913$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{2} q^{2} - q^{4} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{2} q^{8} + 3 q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{2} q^{2} - q^{4} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{2} q^{8} + 3 q^{9} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{10} -4 q^{11} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{13} + q^{16} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} -3 \zeta_{8}^{2} q^{18} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{19} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{20} + 4 \zeta_{8}^{2} q^{22} + ( -4 + 3 \zeta_{8}^{2} ) q^{25} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{26} + 4 q^{29} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{31} -\zeta_{8}^{2} q^{32} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{34} -3 q^{36} + 6 \zeta_{8}^{2} q^{37} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{38} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{40} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{41} -12 \zeta_{8}^{2} q^{43} + 4 q^{44} + ( 6 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{45} + ( 3 + 4 \zeta_{8}^{2} ) q^{50} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{52} -12 \zeta_{8}^{2} q^{53} + ( -8 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{55} -4 \zeta_{8}^{2} q^{58} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{59} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{61} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{62} - q^{64} + ( -9 + 3 \zeta_{8}^{2} ) q^{65} + 12 \zeta_{8}^{2} q^{67} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{68} + 8 q^{71} + 3 \zeta_{8}^{2} q^{72} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{73} + 6 q^{74} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{76} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{80} + 9 q^{81} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{82} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{83} + ( -9 + 3 \zeta_{8}^{2} ) q^{85} -12 q^{86} -4 \zeta_{8}^{2} q^{88} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{89} + ( 3 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{90} + ( 4 + 12 \zeta_{8}^{2} ) q^{95} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{97} -12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 12q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 12q^{9} - 16q^{11} + 4q^{16} - 16q^{25} + 16q^{29} - 12q^{36} + 16q^{44} + 12q^{50} - 4q^{64} - 36q^{65} + 32q^{71} + 24q^{74} + 36q^{81} - 36q^{85} - 48q^{86} + 16q^{95} - 48q^{99} + O(q^{100})$$

## 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
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
1.00000i 0 −1.00000 −0.707107 2.12132i 0 0 1.00000i 3.00000 −2.12132 + 0.707107i
99.2 1.00000i 0 −1.00000 0.707107 + 2.12132i 0 0 1.00000i 3.00000 2.12132 0.707107i
99.3 1.00000i 0 −1.00000 −0.707107 + 2.12132i 0 0 1.00000i 3.00000 −2.12132 0.707107i
99.4 1.00000i 0 −1.00000 0.707107 2.12132i 0 0 1.00000i 3.00000 2.12132 + 0.707107i
 $$n$$: 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
7.b odd 2 1 inner
35.c odd 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.g 4
5.b even 2 1 inner 490.2.c.g 4
5.c odd 4 1 2450.2.a.bi 2
5.c odd 4 1 2450.2.a.bo 2
7.b odd 2 1 inner 490.2.c.g 4
7.c even 3 2 490.2.i.d 8
7.d odd 6 2 490.2.i.d 8
35.c odd 2 1 inner 490.2.c.g 4
35.f even 4 1 2450.2.a.bi 2
35.f even 4 1 2450.2.a.bo 2
35.i odd 6 2 490.2.i.d 8
35.j even 6 2 490.2.i.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.c.g 4 1.a even 1 1 trivial
490.2.c.g 4 5.b even 2 1 inner
490.2.c.g 4 7.b odd 2 1 inner
490.2.c.g 4 35.c odd 2 1 inner
490.2.i.d 8 7.c even 3 2
490.2.i.d 8 7.d odd 6 2
490.2.i.d 8 35.i odd 6 2
490.2.i.d 8 35.j even 6 2
2450.2.a.bi 2 5.c odd 4 1
2450.2.a.bi 2 35.f even 4 1
2450.2.a.bo 2 5.c odd 4 1
2450.2.a.bo 2 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 $$S_{2}^{\mathrm{new}}(490, [\chi])$$:

 $$T_{3}$$ $$T_{11} + 4$$ $$T_{19}^{2} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$25 + 8 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 4 + T )^{4}$$
$13$ $$( 18 + T^{2} )^{2}$$
$17$ $$( 18 + T^{2} )^{2}$$
$19$ $$( -32 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$( -4 + T )^{4}$$
$31$ $$( -32 + T^{2} )^{2}$$
$37$ $$( 36 + T^{2} )^{2}$$
$41$ $$( -2 + T^{2} )^{2}$$
$43$ $$( 144 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( 144 + T^{2} )^{2}$$
$59$ $$( -128 + T^{2} )^{2}$$
$61$ $$( -50 + T^{2} )^{2}$$
$67$ $$( 144 + T^{2} )^{2}$$
$71$ $$( -8 + T )^{4}$$
$73$ $$( 18 + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$( 288 + T^{2} )^{2}$$
$89$ $$( -18 + T^{2} )^{2}$$
$97$ $$( 18 + T^{2} )^{2}$$