# Properties

 Label 2240.2.g.d Level $2240$ Weight $2$ Character orbit 2240.g 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.g (of order $$2$$, degree $$1$$, not 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 i ) q^{5} -i q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 i ) q^{5} -i q^{7} + 3 q^{9} -4 i q^{13} -4 i q^{17} + 4 q^{19} + 8 i q^{23} + ( -3 - 4 i ) q^{25} + 2 q^{29} -8 q^{31} + ( 2 + i ) q^{35} -8 i q^{37} + 6 q^{41} -8 i q^{43} + ( -3 + 6 i ) q^{45} -8 i q^{47} - q^{49} -4 q^{59} + 6 q^{61} -3 i q^{63} + ( 8 + 4 i ) q^{65} + 8 i q^{67} + 12 q^{71} -4 i q^{73} + 4 q^{79} + 9 q^{81} + ( 8 + 4 i ) q^{85} + 10 q^{89} -4 q^{91} + ( -4 + 8 i ) q^{95} + 12 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{5} + 6q^{9} + 8q^{19} - 6q^{25} + 4q^{29} - 16q^{31} + 4q^{35} + 12q^{41} - 6q^{45} - 2q^{49} - 8q^{59} + 12q^{61} + 16q^{65} + 24q^{71} + 8q^{79} + 18q^{81} + 16q^{85} + 20q^{89} - 8q^{91} - 8q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 0 0 −1.00000 2.00000i 0 1.00000i 0 3.00000 0
449.2 0 0 0 −1.00000 + 2.00000i 0 1.00000i 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
5.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.g.d 2
4.b odd 2 1 2240.2.g.c 2
5.b even 2 1 inner 2240.2.g.d 2
8.b even 2 1 140.2.e.b 2
8.d odd 2 1 560.2.g.c 2
20.d odd 2 1 2240.2.g.c 2
24.f even 2 1 5040.2.t.g 2
24.h odd 2 1 1260.2.k.b 2
40.e odd 2 1 560.2.g.c 2
40.f even 2 1 140.2.e.b 2
40.i odd 4 1 700.2.a.f 1
40.i odd 4 1 700.2.a.h 1
40.k even 4 1 2800.2.a.o 1
40.k even 4 1 2800.2.a.s 1
56.h odd 2 1 980.2.e.a 2
56.j odd 6 2 980.2.q.e 4
56.p even 6 2 980.2.q.d 4
120.i odd 2 1 1260.2.k.b 2
120.m even 2 1 5040.2.t.g 2
120.w even 4 1 6300.2.a.g 1
120.w even 4 1 6300.2.a.y 1
280.c odd 2 1 980.2.e.a 2
280.s even 4 1 4900.2.a.l 1
280.s even 4 1 4900.2.a.m 1
280.bf even 6 2 980.2.q.d 4
280.bk odd 6 2 980.2.q.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.b 2 8.b even 2 1
140.2.e.b 2 40.f even 2 1
560.2.g.c 2 8.d odd 2 1
560.2.g.c 2 40.e odd 2 1
700.2.a.f 1 40.i odd 4 1
700.2.a.h 1 40.i odd 4 1
980.2.e.a 2 56.h odd 2 1
980.2.e.a 2 280.c odd 2 1
980.2.q.d 4 56.p even 6 2
980.2.q.d 4 280.bf even 6 2
980.2.q.e 4 56.j odd 6 2
980.2.q.e 4 280.bk odd 6 2
1260.2.k.b 2 24.h odd 2 1
1260.2.k.b 2 120.i odd 2 1
2240.2.g.c 2 4.b odd 2 1
2240.2.g.c 2 20.d odd 2 1
2240.2.g.d 2 1.a even 1 1 trivial
2240.2.g.d 2 5.b even 2 1 inner
2800.2.a.o 1 40.k even 4 1
2800.2.a.s 1 40.k even 4 1
4900.2.a.l 1 280.s even 4 1
4900.2.a.m 1 280.s even 4 1
5040.2.t.g 2 24.f even 2 1
5040.2.t.g 2 120.m even 2 1
6300.2.a.g 1 120.w even 4 1
6300.2.a.y 1 120.w even 4 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{11}$$ $$T_{19} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$144 + T^{2}$$