# Properties

 Label 280.2.bl.a Level $280$ Weight $2$ Character orbit 280.bl Analytic conductor $2.236$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.bl (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + 2 \zeta_{12} q^{4} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( -1 - 2 \zeta_{12}^{2} ) q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} -3 \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + 2 \zeta_{12} q^{4} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( -1 - 2 \zeta_{12}^{2} ) q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} -3 \zeta_{12}^{2} q^{9} + ( -1 - \zeta_{12} + \zeta_{12}^{2} ) q^{10} + 3 \zeta_{12} q^{11} -\zeta_{12}^{3} q^{13} + ( -2 + 3 \zeta_{12} + 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{14} + 4 \zeta_{12}^{2} q^{16} + ( 6 - 6 \zeta_{12}^{2} ) q^{17} + ( -3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{18} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{19} + 2 q^{20} + ( -3 - 3 \zeta_{12}^{3} ) q^{22} -3 \zeta_{12}^{2} q^{23} + ( 1 - \zeta_{12}^{2} ) q^{25} + ( -\zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{26} + ( -2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{28} -10 \zeta_{12}^{3} q^{29} + ( 4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{32} + ( -6 + 6 \zeta_{12}^{3} ) q^{34} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{35} -6 \zeta_{12}^{3} q^{36} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{37} + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{38} + ( -2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{40} + 9 q^{41} + 6 \zeta_{12}^{3} q^{43} + 6 \zeta_{12}^{2} q^{44} -3 \zeta_{12} q^{45} + ( -3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{46} -5 \zeta_{12}^{2} q^{47} + ( -3 + 8 \zeta_{12}^{2} ) q^{49} + ( -1 + \zeta_{12}^{3} ) q^{50} + ( 2 - 2 \zeta_{12}^{2} ) q^{52} + 9 \zeta_{12} q^{53} + 3 q^{55} + ( 2 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{56} + ( -10 \zeta_{12} + 10 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{58} + 8 \zeta_{12} q^{59} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{61} + ( -6 + 9 \zeta_{12}^{2} ) q^{63} + 8 \zeta_{12}^{3} q^{64} -\zeta_{12}^{2} q^{65} -2 \zeta_{12} q^{67} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{68} + ( 3 + \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{70} -10 q^{71} + ( -6 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{72} + ( 8 - 8 \zeta_{12}^{2} ) q^{73} + ( 5 + 5 \zeta_{12} - 5 \zeta_{12}^{2} ) q^{74} -2 q^{76} + ( -3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} + 4 \zeta_{12}^{2} q^{79} + 4 \zeta_{12} q^{80} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( -9 \zeta_{12} - 9 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{82} + 16 \zeta_{12}^{3} q^{83} -6 \zeta_{12}^{3} q^{85} + ( 6 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{86} + ( 6 - 6 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{88} + 2 \zeta_{12}^{2} q^{89} + ( 3 + 3 \zeta_{12}^{3} ) q^{90} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{91} -6 \zeta_{12}^{3} q^{92} + ( -5 + 5 \zeta_{12} + 5 \zeta_{12}^{2} ) q^{94} + ( -1 + \zeta_{12}^{2} ) q^{95} + 8 q^{97} + ( 8 - 5 \zeta_{12} - 5 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{98} -9 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 8q^{7} - 8q^{8} - 6q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 8q^{7} - 8q^{8} - 6q^{9} - 2q^{10} - 2q^{14} + 8q^{16} + 12q^{17} - 6q^{18} + 8q^{20} - 12q^{22} - 6q^{23} + 2q^{25} + 2q^{26} + 8q^{32} - 24q^{34} + 2q^{38} - 4q^{40} + 36q^{41} + 12q^{44} - 6q^{46} - 10q^{47} + 4q^{49} - 4q^{50} + 4q^{52} + 12q^{55} + 16q^{56} + 20q^{58} - 6q^{63} - 2q^{65} + 10q^{70} - 40q^{71} + 12q^{72} + 16q^{73} + 10q^{74} - 8q^{76} + 8q^{79} - 18q^{81} - 18q^{82} - 12q^{86} + 12q^{88} + 4q^{89} + 12q^{90} - 10q^{94} - 2q^{95} + 32q^{97} + 22q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
221.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−1.36603 0.366025i 0 1.73205 + 1.00000i 0.866025 0.500000i 0 −2.00000 1.73205i −2.00000 2.00000i −1.50000 2.59808i −1.36603 + 0.366025i
221.2 0.366025 1.36603i 0 −1.73205 1.00000i −0.866025 + 0.500000i 0 −2.00000 1.73205i −2.00000 + 2.00000i −1.50000 2.59808i 0.366025 + 1.36603i
261.1 −1.36603 + 0.366025i 0 1.73205 1.00000i 0.866025 + 0.500000i 0 −2.00000 + 1.73205i −2.00000 + 2.00000i −1.50000 + 2.59808i −1.36603 0.366025i
261.2 0.366025 + 1.36603i 0 −1.73205 + 1.00000i −0.866025 0.500000i 0 −2.00000 + 1.73205i −2.00000 2.00000i −1.50000 + 2.59808i 0.366025 1.36603i
 $$n$$: 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
7.c even 3 1 inner
8.b even 2 1 inner
56.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.bl.a 4
4.b odd 2 1 1120.2.cb.a 4
7.c even 3 1 inner 280.2.bl.a 4
8.b even 2 1 inner 280.2.bl.a 4
8.d odd 2 1 1120.2.cb.a 4
28.g odd 6 1 1120.2.cb.a 4
56.k odd 6 1 1120.2.cb.a 4
56.p even 6 1 inner 280.2.bl.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bl.a 4 1.a even 1 1 trivial
280.2.bl.a 4 7.c even 3 1 inner
280.2.bl.a 4 8.b even 2 1 inner
280.2.bl.a 4 56.p even 6 1 inner
1120.2.cb.a 4 4.b odd 2 1
1120.2.cb.a 4 8.d odd 2 1
1120.2.cb.a 4 28.g odd 6 1
1120.2.cb.a 4 56.k odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$1 - T^{2} + T^{4}$$
$7$ $$( 7 + 4 T + T^{2} )^{2}$$
$11$ $$81 - 9 T^{2} + T^{4}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$( 36 - 6 T + T^{2} )^{2}$$
$19$ $$1 - T^{2} + T^{4}$$
$23$ $$( 9 + 3 T + T^{2} )^{2}$$
$29$ $$( 100 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$625 - 25 T^{2} + T^{4}$$
$41$ $$( -9 + T )^{4}$$
$43$ $$( 36 + T^{2} )^{2}$$
$47$ $$( 25 + 5 T + T^{2} )^{2}$$
$53$ $$6561 - 81 T^{2} + T^{4}$$
$59$ $$4096 - 64 T^{2} + T^{4}$$
$61$ $$4096 - 64 T^{2} + T^{4}$$
$67$ $$16 - 4 T^{2} + T^{4}$$
$71$ $$( 10 + T )^{4}$$
$73$ $$( 64 - 8 T + T^{2} )^{2}$$
$79$ $$( 16 - 4 T + T^{2} )^{2}$$
$83$ $$( 256 + T^{2} )^{2}$$
$89$ $$( 4 - 2 T + T^{2} )^{2}$$
$97$ $$( -8 + T )^{4}$$