# Properties

 Label 2240.2.b.a Level $2240$ Weight $2$ Character orbit 2240.b Analytic conductor $17.886$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{5} - q^{7} + 3 q^{9} +O(q^{10})$$ $$q + i q^{5} - q^{7} + 3 q^{9} -2 i q^{11} -2 i q^{13} + 2 q^{17} -4 i q^{19} - q^{25} + 4 i q^{29} -4 q^{31} -i q^{35} -8 i q^{37} + 2 q^{41} -6 i q^{43} + 3 i q^{45} + q^{49} -4 i q^{53} + 2 q^{55} + 2 i q^{61} -3 q^{63} + 2 q^{65} -2 i q^{67} + 8 q^{71} + 10 q^{73} + 2 i q^{77} + 8 q^{79} + 9 q^{81} -12 i q^{83} + 2 i q^{85} + 6 q^{89} + 2 i q^{91} + 4 q^{95} + 2 q^{97} -6 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{7} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{7} + 6q^{9} + 4q^{17} - 2q^{25} - 8q^{31} + 4q^{41} + 2q^{49} + 4q^{55} - 6q^{63} + 4q^{65} + 16q^{71} + 20q^{73} + 16q^{79} + 18q^{81} + 12q^{89} + 8q^{95} + 4q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1121.1
 − 1.00000i 1.00000i
0 0 0 1.00000i 0 −1.00000 0 3.00000 0
1121.2 0 0 0 1.00000i 0 −1.00000 0 3.00000 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
8.b 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.b.a 2
4.b odd 2 1 2240.2.b.b yes 2
8.b even 2 1 inner 2240.2.b.a 2
8.d odd 2 1 2240.2.b.b yes 2
16.e even 4 1 8960.2.a.j 1
16.e even 4 1 8960.2.a.l 1
16.f odd 4 1 8960.2.a.i 1
16.f odd 4 1 8960.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.b.a 2 1.a even 1 1 trivial
2240.2.b.a 2 8.b even 2 1 inner
2240.2.b.b yes 2 4.b odd 2 1
2240.2.b.b yes 2 8.d odd 2 1
8960.2.a.i 1 16.f odd 4 1
8960.2.a.j 1 16.e even 4 1
8960.2.a.k 1 16.f odd 4 1
8960.2.a.l 1 16.e even 4 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{23}$$ $$T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$16 + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$36 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$16 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$4 + T^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$( -2 + T )^{2}$$