# Properties

 Label 490.2.i.d Level $490$ Weight $2$ Character orbit 490.i Analytic conductor $3.913$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$490 = 2 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 490.i (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.91266969904$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{5} -\zeta_{24}^{6} q^{8} + ( -3 + 3 \zeta_{24}^{4} ) q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{5} -\zeta_{24}^{6} q^{8} + ( -3 + 3 \zeta_{24}^{4} ) q^{9} + ( -\zeta_{24} - 2 \zeta_{24}^{7} ) q^{10} + 4 \zeta_{24}^{4} q^{11} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{13} + ( -1 + \zeta_{24}^{4} ) q^{16} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{17} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{18} + ( -4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} + ( -2 \zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{20} -4 \zeta_{24}^{6} q^{22} + ( -3 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{25} + ( -3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{26} + 4 q^{29} + ( -4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{31} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{32} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{34} -3 q^{36} + 6 \zeta_{24}^{2} q^{37} + ( 4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{38} + ( 2 \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{40} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{41} + 12 \zeta_{24}^{6} q^{43} + ( -4 + 4 \zeta_{24}^{4} ) q^{44} + ( -6 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{45} + ( 3 - 4 \zeta_{24}^{6} ) q^{50} + ( 3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{52} + ( 12 \zeta_{24}^{2} - 12 \zeta_{24}^{6} ) q^{53} + ( -8 \zeta_{24} + 4 \zeta_{24}^{3} + 8 \zeta_{24}^{5} ) q^{55} -4 \zeta_{24}^{2} q^{58} + ( 8 \zeta_{24} + 8 \zeta_{24}^{7} ) q^{59} + ( 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{61} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{62} - q^{64} + ( 9 + 3 \zeta_{24}^{2} - 9 \zeta_{24}^{4} ) q^{65} + ( -12 \zeta_{24}^{2} + 12 \zeta_{24}^{6} ) q^{67} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{68} + 8 q^{71} + 3 \zeta_{24}^{2} q^{72} + ( 3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{73} -6 \zeta_{24}^{4} q^{74} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{76} + ( -2 \zeta_{24} + \zeta_{24}^{7} ) q^{80} -9 \zeta_{24}^{4} q^{81} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{82} + ( 12 \zeta_{24} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} ) q^{83} + ( -9 - 3 \zeta_{24}^{6} ) q^{85} + ( 12 - 12 \zeta_{24}^{4} ) q^{86} + ( 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{88} + ( 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{89} + ( 3 \zeta_{24} + 6 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{90} + ( -12 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + 12 \zeta_{24}^{6} ) q^{95} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{97} -12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} - 12q^{9} + O(q^{10})$$ $$8q + 4q^{4} - 12q^{9} + 16q^{11} - 4q^{16} + 16q^{25} + 32q^{29} - 24q^{36} - 16q^{44} + 24q^{50} - 8q^{64} + 36q^{65} + 64q^{71} - 24q^{74} - 36q^{81} - 72q^{85} + 48q^{86} - 16q^{95} - 96q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-1 + \zeta_{24}^{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i
−0.866025 0.500000i 0 0.500000 + 0.866025i −1.48356 1.67303i 0 0 1.00000i −1.50000 + 2.59808i 0.448288 + 2.19067i
79.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.48356 + 1.67303i 0 0 1.00000i −1.50000 + 2.59808i −0.448288 2.19067i
79.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.19067 0.448288i 0 0 1.00000i −1.50000 + 2.59808i −1.67303 1.48356i
79.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.19067 + 0.448288i 0 0 1.00000i −1.50000 + 2.59808i 1.67303 + 1.48356i
459.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −1.48356 + 1.67303i 0 0 1.00000i −1.50000 2.59808i 0.448288 2.19067i
459.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.48356 1.67303i 0 0 1.00000i −1.50000 2.59808i −0.448288 + 2.19067i
459.3 0.866025 0.500000i 0 0.500000 0.866025i −2.19067 + 0.448288i 0 0 1.00000i −1.50000 2.59808i −1.67303 + 1.48356i
459.4 0.866025 0.500000i 0 0.500000 0.866025i 2.19067 0.448288i 0 0 1.00000i −1.50000 2.59808i 1.67303 1.48356i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 459.4 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
7.c even 3 1 inner
7.d odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.i.d 8
5.b even 2 1 inner 490.2.i.d 8
7.b odd 2 1 inner 490.2.i.d 8
7.c even 3 1 490.2.c.g 4
7.c even 3 1 inner 490.2.i.d 8
7.d odd 6 1 490.2.c.g 4
7.d odd 6 1 inner 490.2.i.d 8
35.c odd 2 1 inner 490.2.i.d 8
35.i odd 6 1 490.2.c.g 4
35.i odd 6 1 inner 490.2.i.d 8
35.j even 6 1 490.2.c.g 4
35.j even 6 1 inner 490.2.i.d 8
35.k even 12 1 2450.2.a.bi 2
35.k even 12 1 2450.2.a.bo 2
35.l odd 12 1 2450.2.a.bi 2
35.l odd 12 1 2450.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.c.g 4 7.c even 3 1
490.2.c.g 4 7.d odd 6 1
490.2.c.g 4 35.i odd 6 1
490.2.c.g 4 35.j even 6 1
490.2.i.d 8 1.a even 1 1 trivial
490.2.i.d 8 5.b even 2 1 inner
490.2.i.d 8 7.b odd 2 1 inner
490.2.i.d 8 7.c even 3 1 inner
490.2.i.d 8 7.d odd 6 1 inner
490.2.i.d 8 35.c odd 2 1 inner
490.2.i.d 8 35.i odd 6 1 inner
490.2.i.d 8 35.j even 6 1 inner
2450.2.a.bi 2 35.k even 12 1
2450.2.a.bi 2 35.l odd 12 1
2450.2.a.bo 2 35.k even 12 1
2450.2.a.bo 2 35.l odd 12 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{11}^{2} - 4 T_{11} + 16$$ $$T_{19}^{4} + 32 T_{19}^{2} + 1024$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$625 - 200 T^{2} + 39 T^{4} - 8 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 16 - 4 T + T^{2} )^{4}$$
$13$ $$( 18 + T^{2} )^{4}$$
$17$ $$( 324 - 18 T^{2} + T^{4} )^{2}$$
$19$ $$( 1024 + 32 T^{2} + T^{4} )^{2}$$
$23$ $$T^{8}$$
$29$ $$( -4 + T )^{8}$$
$31$ $$( 1024 + 32 T^{2} + T^{4} )^{2}$$
$37$ $$( 1296 - 36 T^{2} + T^{4} )^{2}$$
$41$ $$( -2 + T^{2} )^{4}$$
$43$ $$( 144 + T^{2} )^{4}$$
$47$ $$T^{8}$$
$53$ $$( 20736 - 144 T^{2} + T^{4} )^{2}$$
$59$ $$( 16384 + 128 T^{2} + T^{4} )^{2}$$
$61$ $$( 2500 + 50 T^{2} + T^{4} )^{2}$$
$67$ $$( 20736 - 144 T^{2} + T^{4} )^{2}$$
$71$ $$( -8 + T )^{8}$$
$73$ $$( 324 - 18 T^{2} + T^{4} )^{2}$$
$79$ $$T^{8}$$
$83$ $$( 288 + T^{2} )^{4}$$
$89$ $$( 324 + 18 T^{2} + T^{4} )^{2}$$
$97$ $$( 18 + T^{2} )^{4}$$