# Properties

 Label 2240.2.e.c Level $2240$ Weight $2$ Character orbit 2240.e Analytic conductor $17.886$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 560) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{3} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + \beta_{3} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{9} + ( -2 \beta_{1} + \beta_{2} ) q^{15} + ( 3 - \beta_{3} ) q^{21} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{23} -5 q^{25} + 4 \beta_{2} q^{27} -6 q^{29} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{35} + 2 \beta_{3} q^{41} + ( -2 \beta_{1} + \beta_{2} ) q^{43} + \beta_{3} q^{45} -7 \beta_{2} q^{47} + ( -2 + 3 \beta_{3} ) q^{49} + 6 \beta_{3} q^{61} + ( -\beta_{1} - \beta_{2} ) q^{63} + ( -10 \beta_{1} + 5 \beta_{2} ) q^{67} -6 \beta_{3} q^{69} -5 \beta_{2} q^{75} -5 q^{81} -11 \beta_{2} q^{83} -6 \beta_{2} q^{87} -8 \beta_{3} q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{9} + O(q^{10})$$ $$4q + 4q^{9} + 12q^{21} - 20q^{25} - 24q^{29} - 8q^{49} - 20q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2239.1
 1.58114 − 0.707107i −1.58114 − 0.707107i −1.58114 + 0.707107i 1.58114 + 0.707107i
0 1.41421i 0 2.23607i 0 −1.58114 + 2.12132i 0 1.00000 0
2239.2 0 1.41421i 0 2.23607i 0 1.58114 + 2.12132i 0 1.00000 0
2239.3 0 1.41421i 0 2.23607i 0 1.58114 2.12132i 0 1.00000 0
2239.4 0 1.41421i 0 2.23607i 0 −1.58114 2.12132i 0 1.00000 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.e.c 4
4.b odd 2 1 inner 2240.2.e.c 4
5.b even 2 1 inner 2240.2.e.c 4
7.b odd 2 1 inner 2240.2.e.c 4
8.b even 2 1 560.2.e.b 4
8.d odd 2 1 560.2.e.b 4
20.d odd 2 1 CM 2240.2.e.c 4
28.d even 2 1 inner 2240.2.e.c 4
35.c odd 2 1 inner 2240.2.e.c 4
40.e odd 2 1 560.2.e.b 4
40.f even 2 1 560.2.e.b 4
40.i odd 4 2 2800.2.k.g 4
40.k even 4 2 2800.2.k.g 4
56.e even 2 1 560.2.e.b 4
56.h odd 2 1 560.2.e.b 4
140.c even 2 1 inner 2240.2.e.c 4
280.c odd 2 1 560.2.e.b 4
280.n even 2 1 560.2.e.b 4
280.s even 4 2 2800.2.k.g 4
280.y odd 4 2 2800.2.k.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.b 4 8.b even 2 1
560.2.e.b 4 8.d odd 2 1
560.2.e.b 4 40.e odd 2 1
560.2.e.b 4 40.f even 2 1
560.2.e.b 4 56.e even 2 1
560.2.e.b 4 56.h odd 2 1
560.2.e.b 4 280.c odd 2 1
560.2.e.b 4 280.n even 2 1
2240.2.e.c 4 1.a even 1 1 trivial
2240.2.e.c 4 4.b odd 2 1 inner
2240.2.e.c 4 5.b even 2 1 inner
2240.2.e.c 4 7.b odd 2 1 inner
2240.2.e.c 4 20.d odd 2 1 CM
2240.2.e.c 4 28.d even 2 1 inner
2240.2.e.c 4 35.c odd 2 1 inner
2240.2.e.c 4 140.c even 2 1 inner
2800.2.k.g 4 40.i odd 4 2
2800.2.k.g 4 40.k even 4 2
2800.2.k.g 4 280.s even 4 2
2800.2.k.g 4 280.y odd 4 2

## Hecke kernels

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

 $$T_{3}^{2} + 2$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 2 + T^{2} )^{2}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$49 + 4 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( -90 + T^{2} )^{2}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 20 + T^{2} )^{2}$$
$43$ $$( -10 + T^{2} )^{2}$$
$47$ $$( 98 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 180 + T^{2} )^{2}$$
$67$ $$( -250 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 242 + T^{2} )^{2}$$
$89$ $$( 320 + T^{2} )^{2}$$
$97$ $$T^{4}$$