Properties

 Label 2240.2.e.a 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 140) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{1} + \beta_{2} ) q^{3} -\beta_{3} q^{5} + ( \beta_{1} + 2 \beta_{2} ) q^{7} -7 q^{9} +O(q^{10})$$ $$q + ( 2 \beta_{1} + \beta_{2} ) q^{3} -\beta_{3} q^{5} + ( \beta_{1} + 2 \beta_{2} ) q^{7} -7 q^{9} -5 \beta_{2} q^{15} + ( -5 - 3 \beta_{3} ) q^{21} + \beta_{2} q^{23} -5 q^{25} + ( -8 \beta_{1} - 4 \beta_{2} ) q^{27} -6 q^{29} + ( 3 \beta_{1} - \beta_{2} ) q^{35} -2 \beta_{3} q^{41} -9 \beta_{2} q^{43} + 7 \beta_{3} q^{45} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 2 - 3 \beta_{3} ) q^{49} -6 \beta_{3} q^{61} + ( -7 \beta_{1} - 14 \beta_{2} ) q^{63} + 3 \beta_{2} q^{67} -2 \beta_{3} q^{69} + ( -10 \beta_{1} - 5 \beta_{2} ) q^{75} + 19 q^{81} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{83} + ( -12 \beta_{1} - 6 \beta_{2} ) q^{87} -8 \beta_{3} q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 28q^{9} + O(q^{10})$$ $$4q - 28q^{9} - 20q^{21} - 20q^{25} - 24q^{29} + 8q^{49} + 76q^{81} + O(q^{100})$$

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
 −0.707107 − 1.58114i 0.707107 − 1.58114i 0.707107 + 1.58114i −0.707107 + 1.58114i
0 3.16228i 0 2.23607i 0 2.12132 1.58114i 0 −7.00000 0
2239.2 0 3.16228i 0 2.23607i 0 −2.12132 1.58114i 0 −7.00000 0
2239.3 0 3.16228i 0 2.23607i 0 −2.12132 + 1.58114i 0 −7.00000 0
2239.4 0 3.16228i 0 2.23607i 0 2.12132 + 1.58114i 0 −7.00000 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.a 4
4.b odd 2 1 inner 2240.2.e.a 4
5.b even 2 1 inner 2240.2.e.a 4
7.b odd 2 1 inner 2240.2.e.a 4
8.b even 2 1 140.2.c.a 4
8.d odd 2 1 140.2.c.a 4
20.d odd 2 1 CM 2240.2.e.a 4
28.d even 2 1 inner 2240.2.e.a 4
35.c odd 2 1 inner 2240.2.e.a 4
40.e odd 2 1 140.2.c.a 4
40.f even 2 1 140.2.c.a 4
40.i odd 4 2 700.2.g.d 4
40.k even 4 2 700.2.g.d 4
56.e even 2 1 140.2.c.a 4
56.h odd 2 1 140.2.c.a 4
56.j odd 6 2 980.2.s.b 8
56.k odd 6 2 980.2.s.b 8
56.m even 6 2 980.2.s.b 8
56.p even 6 2 980.2.s.b 8
140.c even 2 1 inner 2240.2.e.a 4
280.c odd 2 1 140.2.c.a 4
280.n even 2 1 140.2.c.a 4
280.s even 4 2 700.2.g.d 4
280.y odd 4 2 700.2.g.d 4
280.ba even 6 2 980.2.s.b 8
280.bf even 6 2 980.2.s.b 8
280.bi odd 6 2 980.2.s.b 8
280.bk odd 6 2 980.2.s.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.c.a 4 8.b even 2 1
140.2.c.a 4 8.d odd 2 1
140.2.c.a 4 40.e odd 2 1
140.2.c.a 4 40.f even 2 1
140.2.c.a 4 56.e even 2 1
140.2.c.a 4 56.h odd 2 1
140.2.c.a 4 280.c odd 2 1
140.2.c.a 4 280.n even 2 1
700.2.g.d 4 40.i odd 4 2
700.2.g.d 4 40.k even 4 2
700.2.g.d 4 280.s even 4 2
700.2.g.d 4 280.y odd 4 2
980.2.s.b 8 56.j odd 6 2
980.2.s.b 8 56.k odd 6 2
980.2.s.b 8 56.m even 6 2
980.2.s.b 8 56.p even 6 2
980.2.s.b 8 280.ba even 6 2
980.2.s.b 8 280.bf even 6 2
980.2.s.b 8 280.bi odd 6 2
980.2.s.b 8 280.bk odd 6 2
2240.2.e.a 4 1.a even 1 1 trivial
2240.2.e.a 4 4.b odd 2 1 inner
2240.2.e.a 4 5.b even 2 1 inner
2240.2.e.a 4 7.b odd 2 1 inner
2240.2.e.a 4 20.d odd 2 1 CM
2240.2.e.a 4 28.d even 2 1 inner
2240.2.e.a 4 35.c odd 2 1 inner
2240.2.e.a 4 140.c even 2 1 inner

Hecke kernels

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

 $$T_{3}^{2} + 10$$ $$T_{11}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 10 + T^{2} )^{2}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$49 - 4 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( -2 + T^{2} )^{2}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 20 + T^{2} )^{2}$$
$43$ $$( -162 + T^{2} )^{2}$$
$47$ $$( 90 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 180 + T^{2} )^{2}$$
$67$ $$( -18 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 90 + T^{2} )^{2}$$
$89$ $$( 320 + T^{2} )^{2}$$
$97$ $$T^{4}$$