# Properties

 Label 280.2.b.b 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + i ) q^{2} + 2 i q^{3} + 2 i q^{4} + i q^{5} + ( -2 + 2 i ) q^{6} + q^{7} + ( -2 + 2 i ) q^{8} - q^{9} +O(q^{10})$$ $$q + ( 1 + i ) q^{2} + 2 i q^{3} + 2 i q^{4} + i q^{5} + ( -2 + 2 i ) q^{6} + q^{7} + ( -2 + 2 i ) q^{8} - q^{9} + ( -1 + i ) q^{10} -4 i q^{11} -4 q^{12} -6 i q^{13} + ( 1 + i ) q^{14} -2 q^{15} -4 q^{16} + ( -1 - i ) q^{18} + 8 i q^{19} -2 q^{20} + 2 i q^{21} + ( 4 - 4 i ) q^{22} + 8 q^{23} + ( -4 - 4 i ) q^{24} - q^{25} + ( 6 - 6 i ) q^{26} + 4 i q^{27} + 2 i q^{28} + ( -2 - 2 i ) q^{30} -8 q^{31} + ( -4 - 4 i ) q^{32} + 8 q^{33} + i q^{35} -2 i q^{36} -10 i q^{37} + ( -8 + 8 i ) q^{38} + 12 q^{39} + ( -2 - 2 i ) q^{40} -2 q^{41} + ( -2 + 2 i ) q^{42} + 4 i q^{43} + 8 q^{44} -i q^{45} + ( 8 + 8 i ) q^{46} + 8 q^{47} -8 i q^{48} + q^{49} + ( -1 - i ) q^{50} + 12 q^{52} -2 i q^{53} + ( -4 + 4 i ) q^{54} + 4 q^{55} + ( -2 + 2 i ) q^{56} -16 q^{57} + 4 i q^{59} -4 i q^{60} + 2 i q^{61} + ( -8 - 8 i ) q^{62} - q^{63} -8 i q^{64} + 6 q^{65} + ( 8 + 8 i ) q^{66} -12 i q^{67} + 16 i q^{69} + ( -1 + i ) q^{70} + 2 q^{71} + ( 2 - 2 i ) q^{72} + 4 q^{73} + ( 10 - 10 i ) q^{74} -2 i q^{75} -16 q^{76} -4 i q^{77} + ( 12 + 12 i ) q^{78} -2 q^{79} -4 i q^{80} -11 q^{81} + ( -2 - 2 i ) q^{82} -6 i q^{83} -4 q^{84} + ( -4 + 4 i ) q^{86} + ( 8 + 8 i ) q^{88} + 2 q^{89} + ( 1 - i ) q^{90} -6 i q^{91} + 16 i q^{92} -16 i q^{93} + ( 8 + 8 i ) q^{94} -8 q^{95} + ( 8 - 8 i ) q^{96} -16 q^{97} + ( 1 + i ) q^{98} + 4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 4q^{6} + 2q^{7} - 4q^{8} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 4q^{6} + 2q^{7} - 4q^{8} - 2q^{9} - 2q^{10} - 8q^{12} + 2q^{14} - 4q^{15} - 8q^{16} - 2q^{18} - 4q^{20} + 8q^{22} + 16q^{23} - 8q^{24} - 2q^{25} + 12q^{26} - 4q^{30} - 16q^{31} - 8q^{32} + 16q^{33} - 16q^{38} + 24q^{39} - 4q^{40} - 4q^{41} - 4q^{42} + 16q^{44} + 16q^{46} + 16q^{47} + 2q^{49} - 2q^{50} + 24q^{52} - 8q^{54} + 8q^{55} - 4q^{56} - 32q^{57} - 16q^{62} - 2q^{63} + 12q^{65} + 16q^{66} - 2q^{70} + 4q^{71} + 4q^{72} + 8q^{73} + 20q^{74} - 32q^{76} + 24q^{78} - 4q^{79} - 22q^{81} - 4q^{82} - 8q^{84} - 8q^{86} + 16q^{88} + 4q^{89} + 2q^{90} + 16q^{94} - 16q^{95} + 16q^{96} - 32q^{97} + 2q^{98} + O(q^{100})$$

## 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.b 2
4.b odd 2 1 1120.2.b.a 2
8.b even 2 1 inner 280.2.b.b 2
8.d odd 2 1 1120.2.b.a 2
16.e even 4 1 8960.2.a.f 1
16.e even 4 1 8960.2.a.n 1
16.f odd 4 1 8960.2.a.b 1
16.f odd 4 1 8960.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.b.b 2 1.a even 1 1 trivial
280.2.b.b 2 8.b even 2 1 inner
1120.2.b.a 2 4.b odd 2 1
1120.2.b.a 2 8.d odd 2 1
8960.2.a.b 1 16.f odd 4 1
8960.2.a.f 1 16.e even 4 1
8960.2.a.n 1 16.e even 4 1
8960.2.a.t 1 16.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$:

 $$T_{3}^{2} + 4$$ $$T_{13}^{2} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 - 2 T + T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$64 + T^{2}$$
$23$ $$( -8 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$4 + T^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$( -4 + T )^{2}$$
$79$ $$( 2 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$( -2 + T )^{2}$$
$97$ $$( 16 + T )^{2}$$