# Properties

 Label 2240.2.e.b Level $2240$ Weight $2$ Character orbit 2240.e Analytic conductor $17.886$ Analytic rank $0$ Dimension $4$ CM discriminant -35 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: $$\Q(\sqrt{5}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - x^{3} + 5 x^{2} + 2 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{5} + \beta_{2} q^{7} -4 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{5} + \beta_{2} q^{7} -4 q^{9} + \beta_{3} q^{11} + 3 \beta_{1} q^{13} -\beta_{3} q^{15} + \beta_{1} q^{17} -7 q^{21} + 5 q^{25} -\beta_{2} q^{27} + 9 q^{29} + 7 \beta_{1} q^{33} -\beta_{3} q^{35} -3 \beta_{3} q^{39} -4 \beta_{1} q^{45} + 3 \beta_{2} q^{47} -7 q^{49} -\beta_{3} q^{51} -5 \beta_{2} q^{55} -4 \beta_{2} q^{63} + 15 q^{65} -2 \beta_{3} q^{71} -6 \beta_{1} q^{73} + 5 \beta_{2} q^{75} + 7 \beta_{1} q^{77} + 3 \beta_{3} q^{79} -5 q^{81} -6 \beta_{2} q^{83} + 5 q^{85} + 9 \beta_{2} q^{87} -3 \beta_{3} q^{91} -3 \beta_{1} q^{97} -4 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{9} + O(q^{10})$$ $$4q - 16q^{9} - 28q^{21} + 20q^{25} + 36q^{29} - 28q^{49} + 60q^{65} - 20q^{81} + 20q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 2 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} + \nu + 7$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{3} + 4 \nu^{2} - 8 \nu + 1$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} + 7 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{2} + 3 \beta_{1} - 9$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} - 2 \beta_{2} + 5 \beta_{1} - 10$$$$)/2$$

## 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
 0.809017 − 2.14046i −0.309017 + 0.817582i 0.809017 + 2.14046i −0.309017 − 0.817582i
0 2.64575i 0 −2.23607 0 2.64575i 0 −4.00000 0
2239.2 0 2.64575i 0 2.23607 0 2.64575i 0 −4.00000 0
2239.3 0 2.64575i 0 −2.23607 0 2.64575i 0 −4.00000 0
2239.4 0 2.64575i 0 2.23607 0 2.64575i 0 −4.00000 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 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.b 4
4.b odd 2 1 inner 2240.2.e.b 4
5.b even 2 1 inner 2240.2.e.b 4
7.b odd 2 1 inner 2240.2.e.b 4
8.b even 2 1 560.2.e.a 4
8.d odd 2 1 560.2.e.a 4
20.d odd 2 1 inner 2240.2.e.b 4
28.d even 2 1 inner 2240.2.e.b 4
35.c odd 2 1 CM 2240.2.e.b 4
40.e odd 2 1 560.2.e.a 4
40.f even 2 1 560.2.e.a 4
40.i odd 4 2 2800.2.k.j 4
40.k even 4 2 2800.2.k.j 4
56.e even 2 1 560.2.e.a 4
56.h odd 2 1 560.2.e.a 4
140.c even 2 1 inner 2240.2.e.b 4
280.c odd 2 1 560.2.e.a 4
280.n even 2 1 560.2.e.a 4
280.s even 4 2 2800.2.k.j 4
280.y odd 4 2 2800.2.k.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.a 4 8.b even 2 1
560.2.e.a 4 8.d odd 2 1
560.2.e.a 4 40.e odd 2 1
560.2.e.a 4 40.f even 2 1
560.2.e.a 4 56.e even 2 1
560.2.e.a 4 56.h odd 2 1
560.2.e.a 4 280.c odd 2 1
560.2.e.a 4 280.n even 2 1
2240.2.e.b 4 1.a even 1 1 trivial
2240.2.e.b 4 4.b odd 2 1 inner
2240.2.e.b 4 5.b even 2 1 inner
2240.2.e.b 4 7.b odd 2 1 inner
2240.2.e.b 4 20.d odd 2 1 inner
2240.2.e.b 4 28.d even 2 1 inner
2240.2.e.b 4 35.c odd 2 1 CM
2240.2.e.b 4 140.c even 2 1 inner
2800.2.k.j 4 40.i odd 4 2
2800.2.k.j 4 40.k even 4 2
2800.2.k.j 4 280.s even 4 2
2800.2.k.j 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 $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3}^{2} + 7$$ $$T_{11}^{2} + 35$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 7 + T^{2} )^{2}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$( 7 + T^{2} )^{2}$$
$11$ $$( 35 + T^{2} )^{2}$$
$13$ $$( -45 + T^{2} )^{2}$$
$17$ $$( -5 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( -9 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$( 63 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$( 140 + T^{2} )^{2}$$
$73$ $$( -180 + T^{2} )^{2}$$
$79$ $$( 315 + T^{2} )^{2}$$
$83$ $$( 252 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( -45 + T^{2} )^{2}$$