# Properties

 Label 2240.2.g.k Level $2240$ Weight $2$ Character orbit 2240.g Analytic conductor $17.886$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{2} q^{5} + \beta_{1} q^{7} + 2 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{2} q^{5} + \beta_{1} q^{7} + 2 q^{9} + \beta_{3} q^{11} -\beta_{2} q^{13} + \beta_{3} q^{15} + \beta_{2} q^{17} + 2 \beta_{3} q^{19} - q^{21} + 4 \beta_{1} q^{23} -5 q^{25} + 5 \beta_{1} q^{27} - q^{29} + \beta_{2} q^{33} + \beta_{3} q^{35} -4 \beta_{2} q^{37} + \beta_{3} q^{39} -6 \beta_{1} q^{43} -2 \beta_{2} q^{45} -3 \beta_{1} q^{47} - q^{49} -\beta_{3} q^{51} -2 \beta_{2} q^{53} -5 \beta_{1} q^{55} + 2 \beta_{2} q^{57} + 2 \beta_{3} q^{59} + 10 q^{61} + 2 \beta_{1} q^{63} -5 q^{65} + 2 \beta_{1} q^{67} -4 q^{69} + 4 \beta_{3} q^{71} + 6 \beta_{2} q^{73} -5 \beta_{1} q^{75} + \beta_{2} q^{77} + 7 \beta_{3} q^{79} + q^{81} + 4 \beta_{1} q^{83} + 5 q^{85} -\beta_{1} q^{87} + 14 q^{89} + \beta_{3} q^{91} -10 \beta_{1} q^{95} + \beta_{2} q^{97} + 2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{9} + O(q^{10})$$ $$4q + 8q^{9} - 4q^{21} - 20q^{25} - 4q^{29} - 4q^{49} + 40q^{61} - 20q^{65} - 16q^{69} + 4q^{81} + 20q^{85} + 56q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$$.

 $$n$$ $$897$$ $$1471$$ $$1541$$ $$1921$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 1.61803i − 0.618034i 0.618034i − 1.61803i
0 1.00000i 0 2.23607i 0 1.00000i 0 2.00000 0
449.2 0 1.00000i 0 2.23607i 0 1.00000i 0 2.00000 0
449.3 0 1.00000i 0 2.23607i 0 1.00000i 0 2.00000 0
449.4 0 1.00000i 0 2.23607i 0 1.00000i 0 2.00000 0
 $$n$$: 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.g.k 4
4.b odd 2 1 inner 2240.2.g.k 4
5.b even 2 1 inner 2240.2.g.k 4
8.b even 2 1 1120.2.g.a 4
8.d odd 2 1 1120.2.g.a 4
20.d odd 2 1 inner 2240.2.g.k 4
40.e odd 2 1 1120.2.g.a 4
40.f even 2 1 1120.2.g.a 4
40.i odd 4 1 5600.2.a.w 2
40.i odd 4 1 5600.2.a.bl 2
40.k even 4 1 5600.2.a.w 2
40.k even 4 1 5600.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.g.a 4 8.b even 2 1
1120.2.g.a 4 8.d odd 2 1
1120.2.g.a 4 40.e odd 2 1
1120.2.g.a 4 40.f even 2 1
2240.2.g.k 4 1.a even 1 1 trivial
2240.2.g.k 4 4.b odd 2 1 inner
2240.2.g.k 4 5.b even 2 1 inner
2240.2.g.k 4 20.d odd 2 1 inner
5600.2.a.w 2 40.i odd 4 1
5600.2.a.w 2 40.k even 4 1
5600.2.a.bl 2 40.i odd 4 1
5600.2.a.bl 2 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3}^{2} + 1$$ $$T_{11}^{2} - 5$$ $$T_{19}^{2} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -5 + T^{2} )^{2}$$
$13$ $$( 5 + T^{2} )^{2}$$
$17$ $$( 5 + T^{2} )^{2}$$
$19$ $$( -20 + T^{2} )^{2}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$( 1 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$( 80 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( 36 + T^{2} )^{2}$$
$47$ $$( 9 + T^{2} )^{2}$$
$53$ $$( 20 + T^{2} )^{2}$$
$59$ $$( -20 + T^{2} )^{2}$$
$61$ $$( -10 + T )^{4}$$
$67$ $$( 4 + T^{2} )^{2}$$
$71$ $$( -80 + T^{2} )^{2}$$
$73$ $$( 180 + T^{2} )^{2}$$
$79$ $$( -245 + T^{2} )^{2}$$
$83$ $$( 16 + T^{2} )^{2}$$
$89$ $$( -14 + T )^{4}$$
$97$ $$( 5 + T^{2} )^{2}$$