# Properties

 Label 280.2.b.a Level $280$ Weight $2$ Character orbit 280.b 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.b (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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + i ) q^{2} + 2 i q^{3} + 2 i q^{4} -i q^{5} + ( -2 + 2 i ) q^{6} - q^{7} + ( -2 + 2 i ) q^{8} - q^{9} +O(q^{10})$$ $$q + ( 1 + i ) q^{2} + 2 i q^{3} + 2 i q^{4} -i q^{5} + ( -2 + 2 i ) q^{6} - q^{7} + ( -2 + 2 i ) q^{8} - q^{9} + ( 1 - i ) q^{10} + 4 i q^{11} -4 q^{12} -2 i q^{13} + ( -1 - i ) q^{14} + 2 q^{15} -4 q^{16} + 4 q^{17} + ( -1 - i ) q^{18} -4 i q^{19} + 2 q^{20} -2 i q^{21} + ( -4 + 4 i ) q^{22} + ( -4 - 4 i ) q^{24} - q^{25} + ( 2 - 2 i ) q^{26} + 4 i q^{27} -2 i q^{28} -8 i q^{29} + ( 2 + 2 i ) q^{30} + 8 q^{31} + ( -4 - 4 i ) q^{32} -8 q^{33} + ( 4 + 4 i ) q^{34} + i q^{35} -2 i q^{36} + 6 i q^{37} + ( 4 - 4 i ) q^{38} + 4 q^{39} + ( 2 + 2 i ) q^{40} + 6 q^{41} + ( 2 - 2 i ) q^{42} + 4 i q^{43} -8 q^{44} + i q^{45} -8 i q^{48} + q^{49} + ( -1 - i ) q^{50} + 8 i q^{51} + 4 q^{52} + 6 i q^{53} + ( -4 + 4 i ) q^{54} + 4 q^{55} + ( 2 - 2 i ) q^{56} + 8 q^{57} + ( 8 - 8 i ) q^{58} -8 i q^{59} + 4 i q^{60} -10 i q^{61} + ( 8 + 8 i ) q^{62} + q^{63} -8 i q^{64} -2 q^{65} + ( -8 - 8 i ) q^{66} + 4 i q^{67} + 8 i q^{68} + ( -1 + i ) q^{70} + 10 q^{71} + ( 2 - 2 i ) q^{72} -16 q^{73} + ( -6 + 6 i ) q^{74} -2 i q^{75} + 8 q^{76} -4 i q^{77} + ( 4 + 4 i ) q^{78} -10 q^{79} + 4 i q^{80} -11 q^{81} + ( 6 + 6 i ) q^{82} -14 i q^{83} + 4 q^{84} -4 i q^{85} + ( -4 + 4 i ) q^{86} + 16 q^{87} + ( -8 - 8 i ) q^{88} -6 q^{89} + ( -1 + i ) q^{90} + 2 i q^{91} + 16 i q^{93} -4 q^{95} + ( 8 - 8 i ) q^{96} -12 q^{97} + ( 1 + i ) q^{98} -4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 4q^{6} - 2q^{7} - 4q^{8} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 4q^{6} - 2q^{7} - 4q^{8} - 2q^{9} + 2q^{10} - 8q^{12} - 2q^{14} + 4q^{15} - 8q^{16} + 8q^{17} - 2q^{18} + 4q^{20} - 8q^{22} - 8q^{24} - 2q^{25} + 4q^{26} + 4q^{30} + 16q^{31} - 8q^{32} - 16q^{33} + 8q^{34} + 8q^{38} + 8q^{39} + 4q^{40} + 12q^{41} + 4q^{42} - 16q^{44} + 2q^{49} - 2q^{50} + 8q^{52} - 8q^{54} + 8q^{55} + 4q^{56} + 16q^{57} + 16q^{58} + 16q^{62} + 2q^{63} - 4q^{65} - 16q^{66} - 2q^{70} + 20q^{71} + 4q^{72} - 32q^{73} - 12q^{74} + 16q^{76} + 8q^{78} - 20q^{79} - 22q^{81} + 12q^{82} + 8q^{84} - 8q^{86} + 32q^{87} - 16q^{88} - 12q^{89} - 2q^{90} - 8q^{95} + 16q^{96} - 24q^{97} + 2q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
141.1
 − 1.00000i 1.00000i
1.00000 1.00000i 2.00000i 2.00000i 1.00000i −2.00000 2.00000i −1.00000 −2.00000 2.00000i −1.00000 1.00000 + 1.00000i
141.2 1.00000 + 1.00000i 2.00000i 2.00000i 1.00000i −2.00000 + 2.00000i −1.00000 −2.00000 + 2.00000i −1.00000 1.00000 1.00000i
 $$n$$: 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 280.2.b.a 2
4.b odd 2 1 1120.2.b.b 2
8.b even 2 1 inner 280.2.b.a 2
8.d odd 2 1 1120.2.b.b 2
16.e even 4 1 8960.2.a.d 1
16.e even 4 1 8960.2.a.r 1
16.f odd 4 1 8960.2.a.g 1
16.f odd 4 1 8960.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.b.a 2 1.a even 1 1 trivial
280.2.b.a 2 8.b even 2 1 inner
1120.2.b.b 2 4.b odd 2 1
1120.2.b.b 2 8.d odd 2 1
8960.2.a.d 1 16.e even 4 1
8960.2.a.g 1 16.f odd 4 1
8960.2.a.m 1 16.f odd 4 1
8960.2.a.r 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 $$S_{2}^{\mathrm{new}}(280, [\chi])$$:

 $$T_{3}^{2} + 4$$ $$T_{13}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 - 2 T + T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$( -4 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$64 + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$64 + T^{2}$$
$61$ $$100 + T^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -10 + T )^{2}$$
$73$ $$( 16 + T )^{2}$$
$79$ $$( 10 + T )^{2}$$
$83$ $$196 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$( 12 + T )^{2}$$