# Properties

 Label 2240.2.a.be Level $2240$ Weight $2$ Character orbit 2240.a Self dual yes Analytic conductor $17.886$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} - q^{5} + q^{7} + ( 1 + \beta ) q^{9} +O(q^{10})$$ $$q -\beta q^{3} - q^{5} + q^{7} + ( 1 + \beta ) q^{9} + \beta q^{11} + ( -2 + 3 \beta ) q^{13} + \beta q^{15} + ( 6 - \beta ) q^{17} + ( 4 - 2 \beta ) q^{19} -\beta q^{21} -2 \beta q^{23} + q^{25} + ( -4 + \beta ) q^{27} + ( -2 - \beta ) q^{29} -4 \beta q^{31} + ( -4 - \beta ) q^{33} - q^{35} + ( 2 - 4 \beta ) q^{37} + ( -12 - \beta ) q^{39} + ( 2 + 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} + ( -1 - \beta ) q^{45} + ( 4 + \beta ) q^{47} + q^{49} + ( 4 - 5 \beta ) q^{51} + ( 10 - 2 \beta ) q^{53} -\beta q^{55} + ( 8 - 2 \beta ) q^{57} + 4 q^{59} + ( 10 + 2 \beta ) q^{61} + ( 1 + \beta ) q^{63} + ( 2 - 3 \beta ) q^{65} + ( 4 + 4 \beta ) q^{67} + ( 8 + 2 \beta ) q^{69} + ( 2 + 4 \beta ) q^{73} -\beta q^{75} + \beta q^{77} + ( -4 - 3 \beta ) q^{79} -7 q^{81} -12 q^{83} + ( -6 + \beta ) q^{85} + ( 4 + 3 \beta ) q^{87} + ( 2 - 2 \beta ) q^{89} + ( -2 + 3 \beta ) q^{91} + ( 16 + 4 \beta ) q^{93} + ( -4 + 2 \beta ) q^{95} + ( 6 + 3 \beta ) q^{97} + ( 4 + 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + O(q^{10})$$ $$2q - q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + q^{11} - q^{13} + q^{15} + 11q^{17} + 6q^{19} - q^{21} - 2q^{23} + 2q^{25} - 7q^{27} - 5q^{29} - 4q^{31} - 9q^{33} - 2q^{35} - 25q^{39} + 6q^{41} + 6q^{43} - 3q^{45} + 9q^{47} + 2q^{49} + 3q^{51} + 18q^{53} - q^{55} + 14q^{57} + 8q^{59} + 22q^{61} + 3q^{63} + q^{65} + 12q^{67} + 18q^{69} + 8q^{73} - q^{75} + q^{77} - 11q^{79} - 14q^{81} - 24q^{83} - 11q^{85} + 11q^{87} + 2q^{89} - q^{91} + 36q^{93} - 6q^{95} + 15q^{97} + 10q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.56155 −1.56155
0 −2.56155 0 −1.00000 0 1.00000 0 3.56155 0
1.2 0 1.56155 0 −1.00000 0 1.00000 0 −0.561553 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.a.be 2
4.b odd 2 1 2240.2.a.bi 2
8.b even 2 1 280.2.a.d 2
8.d odd 2 1 560.2.a.g 2
24.f even 2 1 5040.2.a.bq 2
24.h odd 2 1 2520.2.a.w 2
40.e odd 2 1 2800.2.a.bn 2
40.f even 2 1 1400.2.a.p 2
40.i odd 4 2 1400.2.g.k 4
40.k even 4 2 2800.2.g.u 4
56.e even 2 1 3920.2.a.bu 2
56.h odd 2 1 1960.2.a.r 2
56.j odd 6 2 1960.2.q.u 4
56.p even 6 2 1960.2.q.s 4
280.c odd 2 1 9800.2.a.by 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.d 2 8.b even 2 1
560.2.a.g 2 8.d odd 2 1
1400.2.a.p 2 40.f even 2 1
1400.2.g.k 4 40.i odd 4 2
1960.2.a.r 2 56.h odd 2 1
1960.2.q.s 4 56.p even 6 2
1960.2.q.u 4 56.j odd 6 2
2240.2.a.be 2 1.a even 1 1 trivial
2240.2.a.bi 2 4.b odd 2 1
2520.2.a.w 2 24.h odd 2 1
2800.2.a.bn 2 40.e odd 2 1
2800.2.g.u 4 40.k even 4 2
3920.2.a.bu 2 56.e even 2 1
5040.2.a.bq 2 24.f even 2 1
9800.2.a.by 2 280.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2240))$$:

 $$T_{3}^{2} + T_{3} - 4$$ $$T_{11}^{2} - T_{11} - 4$$ $$T_{13}^{2} + T_{13} - 38$$ $$T_{19}^{2} - 6 T_{19} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 + T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-4 - T + T^{2}$$
$13$ $$-38 + T + T^{2}$$
$17$ $$26 - 11 T + T^{2}$$
$19$ $$-8 - 6 T + T^{2}$$
$23$ $$-16 + 2 T + T^{2}$$
$29$ $$2 + 5 T + T^{2}$$
$31$ $$-64 + 4 T + T^{2}$$
$37$ $$-68 + T^{2}$$
$41$ $$-8 - 6 T + T^{2}$$
$43$ $$-8 - 6 T + T^{2}$$
$47$ $$16 - 9 T + T^{2}$$
$53$ $$64 - 18 T + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$104 - 22 T + T^{2}$$
$67$ $$-32 - 12 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$-52 - 8 T + T^{2}$$
$79$ $$-8 + 11 T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$-16 - 2 T + T^{2}$$
$97$ $$18 - 15 T + T^{2}$$