# Properties

 Label 490.2.e.i Level $490$ Weight $2$ Character orbit 490.e Analytic conductor $3.913$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{2} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{3} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + ( -2 + \beta_{3} ) q^{6} - q^{8} + ( -3 - 4 \beta_{1} - 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{2} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{3} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + ( -2 + \beta_{3} ) q^{6} - q^{8} + ( -3 - 4 \beta_{1} - 3 \beta_{2} ) q^{9} -\beta_{2} q^{10} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{11} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{12} + ( -2 + 2 \beta_{3} ) q^{13} + ( 2 - \beta_{3} ) q^{15} + ( -1 - \beta_{2} ) q^{16} + ( -\beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{17} + ( -4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{18} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{19} + q^{20} + ( -2 - 2 \beta_{3} ) q^{22} + ( 4 - 2 \beta_{1} + 4 \beta_{2} ) q^{23} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{24} + \beta_{2} q^{25} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{26} + ( 8 - 8 \beta_{3} ) q^{27} + ( -2 + 2 \beta_{3} ) q^{29} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{30} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{31} -\beta_{2} q^{32} + 2 \beta_{1} q^{33} + ( -4 - \beta_{3} ) q^{34} + ( 3 - 4 \beta_{3} ) q^{36} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{37} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{38} + ( -6 \beta_{1} - 8 \beta_{2} - 6 \beta_{3} ) q^{39} + ( 1 + \beta_{2} ) q^{40} + ( 4 + 5 \beta_{3} ) q^{41} + ( -6 + 2 \beta_{3} ) q^{43} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{44} + ( 4 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{45} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{46} + ( -8 + 2 \beta_{1} - 8 \beta_{2} ) q^{47} + ( 2 - \beta_{3} ) q^{48} - q^{50} + ( -6 - 2 \beta_{1} - 6 \beta_{2} ) q^{51} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{52} + ( 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{53} + ( 8 + 8 \beta_{1} + 8 \beta_{2} ) q^{54} + ( 2 + 2 \beta_{3} ) q^{55} + 2 q^{57} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -\beta_{1} + 10 \beta_{2} - \beta_{3} ) q^{59} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{60} + ( 2 - 8 \beta_{1} + 2 \beta_{2} ) q^{61} -2 \beta_{3} q^{62} + q^{64} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{65} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{66} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{67} + ( -4 + \beta_{1} - 4 \beta_{2} ) q^{68} -4 q^{69} + ( 4 + 6 \beta_{3} ) q^{71} + ( 3 + 4 \beta_{1} + 3 \beta_{2} ) q^{72} + ( -\beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{73} + ( 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{74} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{75} + ( 2 + \beta_{3} ) q^{76} + ( 8 - 6 \beta_{3} ) q^{78} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{79} + \beta_{2} q^{80} + ( 12 \beta_{1} + 23 \beta_{2} + 12 \beta_{3} ) q^{81} + ( 4 - 5 \beta_{1} + 4 \beta_{2} ) q^{82} + ( 2 + 3 \beta_{3} ) q^{83} + ( 4 + \beta_{3} ) q^{85} + ( -6 - 2 \beta_{1} - 6 \beta_{2} ) q^{86} + ( -6 \beta_{1} - 8 \beta_{2} - 6 \beta_{3} ) q^{87} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{88} -9 \beta_{1} q^{89} + ( -3 + 4 \beta_{3} ) q^{90} + ( -4 - 2 \beta_{3} ) q^{92} + ( 4 + 4 \beta_{1} + 4 \beta_{2} ) q^{93} + ( 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{94} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{95} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{96} + ( -12 - 3 \beta_{3} ) q^{97} + ( -10 - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 4q^{3} - 2q^{4} - 2q^{5} - 8q^{6} - 4q^{8} - 6q^{9} + O(q^{10})$$ $$4q + 2q^{2} - 4q^{3} - 2q^{4} - 2q^{5} - 8q^{6} - 4q^{8} - 6q^{9} + 2q^{10} - 4q^{11} - 4q^{12} - 8q^{13} + 8q^{15} - 2q^{16} - 8q^{17} + 6q^{18} - 4q^{19} + 4q^{20} - 8q^{22} + 8q^{23} + 4q^{24} - 2q^{25} - 4q^{26} + 32q^{27} - 8q^{29} + 4q^{30} + 2q^{32} - 16q^{34} + 12q^{36} + 4q^{37} + 4q^{38} + 16q^{39} + 2q^{40} + 16q^{41} - 24q^{43} - 4q^{44} - 6q^{45} - 8q^{46} - 16q^{47} + 8q^{48} - 4q^{50} - 12q^{51} + 4q^{52} + 4q^{53} + 16q^{54} + 8q^{55} + 8q^{57} - 4q^{58} - 20q^{59} - 4q^{60} + 4q^{61} + 4q^{64} + 4q^{65} + 8q^{67} - 8q^{68} - 16q^{69} + 16q^{71} + 6q^{72} + 16q^{73} - 4q^{74} - 4q^{75} + 8q^{76} + 32q^{78} + 8q^{79} - 2q^{80} - 46q^{81} + 8q^{82} + 8q^{83} + 16q^{85} - 12q^{86} + 16q^{87} + 4q^{88} - 12q^{90} - 16q^{92} + 8q^{93} + 16q^{94} - 4q^{95} + 4q^{96} - 48q^{97} - 40q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
0.500000 + 0.866025i −1.70711 + 2.95680i −0.500000 + 0.866025i −0.500000 0.866025i −3.41421 0 −1.00000 −4.32843 7.49706i 0.500000 0.866025i
361.2 0.500000 + 0.866025i −0.292893 + 0.507306i −0.500000 + 0.866025i −0.500000 0.866025i −0.585786 0 −1.00000 1.32843 + 2.30090i 0.500000 0.866025i
471.1 0.500000 0.866025i −1.70711 2.95680i −0.500000 0.866025i −0.500000 + 0.866025i −3.41421 0 −1.00000 −4.32843 + 7.49706i 0.500000 + 0.866025i
471.2 0.500000 0.866025i −0.292893 0.507306i −0.500000 0.866025i −0.500000 + 0.866025i −0.585786 0 −1.00000 1.32843 2.30090i 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.i 4
7.b odd 2 1 490.2.e.j 4
7.c even 3 1 490.2.a.m yes 2
7.c even 3 1 inner 490.2.e.i 4
7.d odd 6 1 490.2.a.l 2
7.d odd 6 1 490.2.e.j 4
21.g even 6 1 4410.2.a.by 2
21.h odd 6 1 4410.2.a.bt 2
28.f even 6 1 3920.2.a.ca 2
28.g odd 6 1 3920.2.a.bm 2
35.i odd 6 1 2450.2.a.bs 2
35.j even 6 1 2450.2.a.bn 2
35.k even 12 2 2450.2.c.w 4
35.l odd 12 2 2450.2.c.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.l 2 7.d odd 6 1
490.2.a.m yes 2 7.c even 3 1
490.2.e.i 4 1.a even 1 1 trivial
490.2.e.i 4 7.c even 3 1 inner
490.2.e.j 4 7.b odd 2 1
490.2.e.j 4 7.d odd 6 1
2450.2.a.bn 2 35.j even 6 1
2450.2.a.bs 2 35.i odd 6 1
2450.2.c.t 4 35.l odd 12 2
2450.2.c.w 4 35.k even 12 2
3920.2.a.bm 2 28.g odd 6 1
3920.2.a.ca 2 28.f even 6 1
4410.2.a.bt 2 21.h odd 6 1
4410.2.a.by 2 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 $$S_{2}^{\mathrm{new}}(490, [\chi])$$:

 $$T_{3}^{4} + 4 T_{3}^{3} + 14 T_{3}^{2} + 8 T_{3} + 4$$ $$T_{11}^{4} + 4 T_{11}^{3} + 20 T_{11}^{2} - 16 T_{11} + 16$$ $$T_{13}^{2} + 4 T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$( -4 + 4 T + T^{2} )^{2}$$
$17$ $$196 + 112 T + 50 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$29$ $$( -4 + 4 T + T^{2} )^{2}$$
$31$ $$64 + 8 T^{2} + T^{4}$$
$37$ $$784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$( -34 - 8 T + T^{2} )^{2}$$
$43$ $$( 28 + 12 T + T^{2} )^{2}$$
$47$ $$3136 + 896 T + 200 T^{2} + 16 T^{3} + T^{4}$$
$53$ $$4624 + 272 T + 84 T^{2} - 4 T^{3} + T^{4}$$
$59$ $$9604 + 1960 T + 302 T^{2} + 20 T^{3} + T^{4}$$
$61$ $$15376 + 496 T + 140 T^{2} - 4 T^{3} + T^{4}$$
$67$ $$256 + 128 T + 80 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$( -56 - 8 T + T^{2} )^{2}$$
$73$ $$3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4}$$
$79$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$83$ $$( -14 - 4 T + T^{2} )^{2}$$
$89$ $$26244 + 162 T^{2} + T^{4}$$
$97$ $$( 126 + 24 T + T^{2} )^{2}$$