# Properties

 Label 280.2.g.a Level $280$ Weight $2$ Character orbit 280.g Analytic conductor $2.236$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + ( 2 - i ) q^{5} + i q^{7} + 2 q^{9} +O(q^{10})$$ $$q + i q^{3} + ( 2 - i ) q^{5} + i q^{7} + 2 q^{9} - q^{11} -i q^{13} + ( 1 + 2 i ) q^{15} + 3 i q^{17} + 4 q^{19} - q^{21} + 2 i q^{23} + ( 3 - 4 i ) q^{25} + 5 i q^{27} + q^{29} -6 q^{31} -i q^{33} + ( 1 + 2 i ) q^{35} -2 i q^{37} + q^{39} -10 q^{41} + ( 4 - 2 i ) q^{45} -9 i q^{47} - q^{49} -3 q^{51} -14 i q^{53} + ( -2 + i ) q^{55} + 4 i q^{57} -6 q^{59} -4 q^{61} + 2 i q^{63} + ( -1 - 2 i ) q^{65} -10 i q^{67} -2 q^{69} -16 q^{71} + 10 i q^{73} + ( 4 + 3 i ) q^{75} -i q^{77} + 11 q^{79} + q^{81} + 4 i q^{83} + ( 3 + 6 i ) q^{85} + i q^{87} -12 q^{89} + q^{91} -6 i q^{93} + ( 8 - 4 i ) q^{95} + 19 i q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} + 4q^{9} + O(q^{10})$$ $$2q + 4q^{5} + 4q^{9} - 2q^{11} + 2q^{15} + 8q^{19} - 2q^{21} + 6q^{25} + 2q^{29} - 12q^{31} + 2q^{35} + 2q^{39} - 20q^{41} + 8q^{45} - 2q^{49} - 6q^{51} - 4q^{55} - 12q^{59} - 8q^{61} - 2q^{65} - 4q^{69} - 32q^{71} + 8q^{75} + 22q^{79} + 2q^{81} + 6q^{85} - 24q^{89} + 2q^{91} + 16q^{95} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$$.

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 − 1.00000i 1.00000i
0 1.00000i 0 2.00000 + 1.00000i 0 1.00000i 0 2.00000 0
169.2 0 1.00000i 0 2.00000 1.00000i 0 1.00000i 0 2.00000 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 280.2.g.a 2
3.b odd 2 1 2520.2.t.a 2
4.b odd 2 1 560.2.g.d 2
5.b even 2 1 inner 280.2.g.a 2
5.c odd 4 1 1400.2.a.d 1
5.c odd 4 1 1400.2.a.j 1
7.b odd 2 1 1960.2.g.a 2
8.b even 2 1 2240.2.g.b 2
8.d odd 2 1 2240.2.g.a 2
12.b even 2 1 5040.2.t.a 2
15.d odd 2 1 2520.2.t.a 2
20.d odd 2 1 560.2.g.d 2
20.e even 4 1 2800.2.a.k 1
20.e even 4 1 2800.2.a.u 1
35.c odd 2 1 1960.2.g.a 2
35.f even 4 1 9800.2.a.p 1
35.f even 4 1 9800.2.a.bb 1
40.e odd 2 1 2240.2.g.a 2
40.f even 2 1 2240.2.g.b 2
60.h even 2 1 5040.2.t.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.a 2 1.a even 1 1 trivial
280.2.g.a 2 5.b even 2 1 inner
560.2.g.d 2 4.b odd 2 1
560.2.g.d 2 20.d odd 2 1
1400.2.a.d 1 5.c odd 4 1
1400.2.a.j 1 5.c odd 4 1
1960.2.g.a 2 7.b odd 2 1
1960.2.g.a 2 35.c odd 2 1
2240.2.g.a 2 8.d odd 2 1
2240.2.g.a 2 40.e odd 2 1
2240.2.g.b 2 8.b even 2 1
2240.2.g.b 2 40.f even 2 1
2520.2.t.a 2 3.b odd 2 1
2520.2.t.a 2 15.d odd 2 1
2800.2.a.k 1 20.e even 4 1
2800.2.a.u 1 20.e even 4 1
5040.2.t.a 2 12.b even 2 1
5040.2.t.a 2 60.h even 2 1
9800.2.a.p 1 35.f even 4 1
9800.2.a.bb 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 - 4 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$( 6 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$81 + T^{2}$$
$53$ $$196 + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$( 4 + T )^{2}$$
$67$ $$100 + T^{2}$$
$71$ $$( 16 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( -11 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( 12 + T )^{2}$$
$97$ $$361 + T^{2}$$