# Properties

 Label 2240.2.a.bb Level $2240$ Weight $2$ Character orbit 2240.a Self dual yes Analytic conductor $17.886$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + q^{5} + q^{7} + 6q^{9} + O(q^{10})$$ $$q + 3q^{3} + q^{5} + q^{7} + 6q^{9} - 5q^{11} + 3q^{13} + 3q^{15} - q^{17} + 6q^{19} + 3q^{21} - 6q^{23} + q^{25} + 9q^{27} + 9q^{29} + 4q^{31} - 15q^{33} + q^{35} - 2q^{37} + 9q^{39} - 4q^{41} + 10q^{43} + 6q^{45} + q^{47} + q^{49} - 3q^{51} - 4q^{53} - 5q^{55} + 18q^{57} - 8q^{59} + 8q^{61} + 6q^{63} + 3q^{65} + 12q^{67} - 18q^{69} - 8q^{71} + 2q^{73} + 3q^{75} - 5q^{77} - 13q^{79} + 9q^{81} - 4q^{83} - q^{85} + 27q^{87} + 4q^{89} + 3q^{91} + 12q^{93} + 6q^{95} - 13q^{97} - 30q^{99} + O(q^{100})$$

## 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
 0
0 3.00000 0 1.00000 0 1.00000 0 6.00000 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.bb 1
4.b odd 2 1 2240.2.a.c 1
8.b even 2 1 560.2.a.a 1
8.d odd 2 1 140.2.a.b 1
24.f even 2 1 1260.2.a.h 1
24.h odd 2 1 5040.2.a.bd 1
40.e odd 2 1 700.2.a.b 1
40.f even 2 1 2800.2.a.be 1
40.i odd 4 2 2800.2.g.c 2
40.k even 4 2 700.2.e.a 2
56.e even 2 1 980.2.a.b 1
56.h odd 2 1 3920.2.a.bl 1
56.k odd 6 2 980.2.i.b 2
56.m even 6 2 980.2.i.j 2
120.m even 2 1 6300.2.a.bf 1
120.q odd 4 2 6300.2.k.p 2
168.e odd 2 1 8820.2.a.n 1
280.n even 2 1 4900.2.a.u 1
280.y odd 4 2 4900.2.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.b 1 8.d odd 2 1
560.2.a.a 1 8.b even 2 1
700.2.a.b 1 40.e odd 2 1
700.2.e.a 2 40.k even 4 2
980.2.a.b 1 56.e even 2 1
980.2.i.b 2 56.k odd 6 2
980.2.i.j 2 56.m even 6 2
1260.2.a.h 1 24.f even 2 1
2240.2.a.c 1 4.b odd 2 1
2240.2.a.bb 1 1.a even 1 1 trivial
2800.2.a.be 1 40.f even 2 1
2800.2.g.c 2 40.i odd 4 2
3920.2.a.bl 1 56.h odd 2 1
4900.2.a.u 1 280.n even 2 1
4900.2.e.a 2 280.y odd 4 2
5040.2.a.bd 1 24.h odd 2 1
6300.2.a.bf 1 120.m even 2 1
6300.2.k.p 2 120.q odd 4 2
8820.2.a.n 1 168.e odd 2 1

## Hecke kernels

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

 $$T_{3} - 3$$ $$T_{11} + 5$$ $$T_{13} - 3$$ $$T_{19} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$-1 + T$$
$7$ $$-1 + T$$
$11$ $$5 + T$$
$13$ $$-3 + T$$
$17$ $$1 + T$$
$19$ $$-6 + T$$
$23$ $$6 + T$$
$29$ $$-9 + T$$
$31$ $$-4 + T$$
$37$ $$2 + T$$
$41$ $$4 + T$$
$43$ $$-10 + T$$
$47$ $$-1 + T$$
$53$ $$4 + T$$
$59$ $$8 + T$$
$61$ $$-8 + T$$
$67$ $$-12 + T$$
$71$ $$8 + T$$
$73$ $$-2 + T$$
$79$ $$13 + T$$
$83$ $$4 + T$$
$89$ $$-4 + T$$
$97$ $$13 + T$$