# Properties

 Label 2240.2.g.e 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + ( 2 + i ) q^{5} + i q^{7} -6 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} + ( 2 + i ) q^{5} + i q^{7} -6 q^{9} -3 q^{11} + i q^{13} + ( -3 + 6 i ) q^{15} -5 i q^{17} -8 q^{19} -3 q^{21} -2 i q^{23} + ( 3 + 4 i ) q^{25} -9 i q^{27} - q^{29} -2 q^{31} -9 i q^{33} + ( -1 + 2 i ) q^{35} -10 i q^{37} -3 q^{39} -6 q^{41} -4 i q^{43} + ( -12 - 6 i ) q^{45} + 11 i q^{47} - q^{49} + 15 q^{51} + 6 i q^{53} + ( -6 - 3 i ) q^{55} -24 i q^{57} -10 q^{59} -6 i q^{63} + ( -1 + 2 i ) q^{65} + 10 i q^{67} + 6 q^{69} + 10 i q^{73} + ( -12 + 9 i ) q^{75} -3 i q^{77} + 7 q^{79} + 9 q^{81} + 12 i q^{83} + ( 5 - 10 i ) q^{85} -3 i q^{87} -8 q^{89} - q^{91} -6 i q^{93} + ( -16 - 8 i ) q^{95} + 3 i q^{97} + 18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} - 12 q^{9} + O(q^{10})$$ $$2 q + 4 q^{5} - 12 q^{9} - 6 q^{11} - 6 q^{15} - 16 q^{19} - 6 q^{21} + 6 q^{25} - 2 q^{29} - 4 q^{31} - 2 q^{35} - 6 q^{39} - 12 q^{41} - 24 q^{45} - 2 q^{49} + 30 q^{51} - 12 q^{55} - 20 q^{59} - 2 q^{65} + 12 q^{69} - 24 q^{75} + 14 q^{79} + 18 q^{81} + 10 q^{85} - 16 q^{89} - 2 q^{91} - 32 q^{95} + 36 q^{99} + O(q^{100})$$

## 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 3.00000i 0 2.00000 1.00000i 0 1.00000i 0 −6.00000 0
449.2 0 3.00000i 0 2.00000 + 1.00000i 0 1.00000i 0 −6.00000 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.e 2
4.b odd 2 1 2240.2.g.f 2
5.b even 2 1 inner 2240.2.g.e 2
8.b even 2 1 140.2.e.a 2
8.d odd 2 1 560.2.g.a 2
20.d odd 2 1 2240.2.g.f 2
24.f even 2 1 5040.2.t.s 2
24.h odd 2 1 1260.2.k.c 2
40.e odd 2 1 560.2.g.a 2
40.f even 2 1 140.2.e.a 2
40.i odd 4 1 700.2.a.a 1
40.i odd 4 1 700.2.a.j 1
40.k even 4 1 2800.2.a.a 1
40.k even 4 1 2800.2.a.bf 1
56.h odd 2 1 980.2.e.b 2
56.j odd 6 2 980.2.q.c 4
56.p even 6 2 980.2.q.f 4
120.i odd 2 1 1260.2.k.c 2
120.m even 2 1 5040.2.t.s 2
120.w even 4 1 6300.2.a.c 1
120.w even 4 1 6300.2.a.t 1
280.c odd 2 1 980.2.e.b 2
280.s even 4 1 4900.2.a.b 1
280.s even 4 1 4900.2.a.w 1
280.bf even 6 2 980.2.q.f 4
280.bk odd 6 2 980.2.q.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 8.b even 2 1
140.2.e.a 2 40.f even 2 1
560.2.g.a 2 8.d odd 2 1
560.2.g.a 2 40.e odd 2 1
700.2.a.a 1 40.i odd 4 1
700.2.a.j 1 40.i odd 4 1
980.2.e.b 2 56.h odd 2 1
980.2.e.b 2 280.c odd 2 1
980.2.q.c 4 56.j odd 6 2
980.2.q.c 4 280.bk odd 6 2
980.2.q.f 4 56.p even 6 2
980.2.q.f 4 280.bf even 6 2
1260.2.k.c 2 24.h odd 2 1
1260.2.k.c 2 120.i odd 2 1
2240.2.g.e 2 1.a even 1 1 trivial
2240.2.g.e 2 5.b even 2 1 inner
2240.2.g.f 2 4.b odd 2 1
2240.2.g.f 2 20.d odd 2 1
2800.2.a.a 1 40.k even 4 1
2800.2.a.bf 1 40.k even 4 1
4900.2.a.b 1 280.s even 4 1
4900.2.a.w 1 280.s even 4 1
5040.2.t.s 2 24.f even 2 1
5040.2.t.s 2 120.m even 2 1
6300.2.a.c 1 120.w even 4 1
6300.2.a.t 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 $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3}^{2} + 9$$ $$T_{11} + 3$$ $$T_{19} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$5 - 4 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$25 + T^{2}$$
$19$ $$( 8 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$121 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 10 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$100 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( -7 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 8 + T )^{2}$$
$97$ $$9 + T^{2}$$