# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.bj (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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -2 + \zeta_{12}^{2} ) q^{3} + 2 \zeta_{12} q^{4} + ( 1 - \zeta_{12}^{2} ) q^{5} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -2 + \zeta_{12}^{2} ) q^{3} + 2 \zeta_{12} q^{4} + ( 1 - \zeta_{12}^{2} ) q^{5} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( 1 - \zeta_{12}^{3} ) q^{10} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{11} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{12} + ( 3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{13} + ( -1 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{14} + ( -1 + 2 \zeta_{12}^{2} ) q^{15} + 4 \zeta_{12}^{2} q^{16} + ( 2 - 3 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{17} + ( -3 - 3 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{19} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{20} + ( -\zeta_{12} - 4 \zeta_{12}^{3} ) q^{21} + ( 4 - 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{22} + ( 4 - \zeta_{12} + 4 \zeta_{12}^{2} ) q^{23} + ( -4 - 2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{24} -\zeta_{12}^{2} q^{25} + ( 2 + 4 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{26} + ( 3 - 6 \zeta_{12}^{2} ) q^{27} + ( -6 + 4 \zeta_{12}^{2} ) q^{28} + ( 2 - 4 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{29} + ( -2 + \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{30} + ( 3 \zeta_{12} - 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{31} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{32} + ( 2 - 6 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{33} + ( -2 - 2 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{34} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{35} + 2 \zeta_{12} q^{37} + ( -6 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{38} + ( -6 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{39} + ( 2 + 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{40} + ( 3 - 6 \zeta_{12}^{2} ) q^{41} + ( -1 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{42} + ( -3 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{43} + ( -4 + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{44} + ( -5 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{46} + ( 6 - 2 \zeta_{12} - 6 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{47} + ( -4 - 4 \zeta_{12}^{2} ) q^{48} + ( -3 - 5 \zeta_{12}^{2} ) q^{49} + ( 1 - \zeta_{12} - \zeta_{12}^{2} ) q^{50} + ( -3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{51} + ( 2 + 6 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{52} + ( 6 - 5 \zeta_{12} - 3 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{53} + ( 6 - 3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{54} + ( -2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{55} + ( -4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{56} + ( 9 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{57} + ( 4 + 5 \zeta_{12} - 9 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{58} + ( 2 + 3 \zeta_{12} - \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{59} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{60} + ( 6 - \zeta_{12} - 6 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{61} + ( 6 - 6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{62} + 8 \zeta_{12}^{3} q^{64} + ( 3 + \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} + ( -8 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{66} + ( -6 \zeta_{12} - \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{67} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{68} + ( -12 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{69} + ( 2 - \zeta_{12} + \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{70} + ( -1 + 2 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{71} + ( -10 - 3 \zeta_{12} + 5 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{73} + ( 2 + 2 \zeta_{12}^{3} ) q^{74} + ( 1 + \zeta_{12}^{2} ) q^{75} + ( -6 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{76} + ( -10 + 6 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{77} + ( -6 - 6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{78} + ( -1 - 3 \zeta_{12} - \zeta_{12}^{2} ) q^{79} + 4 q^{80} + 9 \zeta_{12}^{2} q^{81} + ( 6 - 3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{82} + ( -5 + 10 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{83} + ( 8 - 10 \zeta_{12}^{2} ) q^{84} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{85} + ( 8 + \zeta_{12} - 7 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{86} + ( 7 \zeta_{12} + 6 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{87} + ( -8 + 8 \zeta_{12} ) q^{88} + ( 5 + 6 \zeta_{12} + 5 \zeta_{12}^{2} ) q^{89} + ( -4 - 3 \zeta_{12} + 5 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{91} + ( 8 \zeta_{12} - 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{92} + ( 3 - 9 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{93} + ( 4 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{94} + ( -6 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{95} + ( 4 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{96} + ( -2 + 4 \zeta_{12}^{2} ) q^{97} + ( 5 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 6q^{3} + 2q^{5} - 6q^{6} + 8q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 6q^{3} + 2q^{5} - 6q^{6} + 8q^{8} + 4q^{10} - 4q^{11} + 12q^{13} + 2q^{14} + 8q^{16} + 6q^{17} - 18q^{19} + 16q^{22} + 24q^{23} - 12q^{24} - 2q^{25} + 12q^{26} - 16q^{28} - 6q^{30} - 6q^{31} - 8q^{32} + 12q^{33} - 12q^{38} - 18q^{39} + 4q^{40} - 12q^{42} - 12q^{43} - 4q^{46} + 12q^{47} - 24q^{48} - 22q^{49} + 2q^{50} - 6q^{51} + 12q^{52} + 18q^{53} + 18q^{54} - 8q^{55} - 20q^{56} + 36q^{57} - 2q^{58} + 6q^{59} + 12q^{61} + 24q^{62} + 6q^{65} - 24q^{66} - 2q^{67} - 24q^{68} - 48q^{69} + 10q^{70} - 30q^{73} + 8q^{74} + 6q^{75} - 12q^{76} - 36q^{77} - 24q^{78} - 6q^{79} + 16q^{80} + 18q^{81} + 18q^{82} + 12q^{84} + 18q^{86} + 12q^{87} - 32q^{88} + 30q^{89} - 6q^{91} - 4q^{92} + 18q^{93} + 24q^{94} - 18q^{95} + 4q^{98} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
131.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.366025 + 1.36603i −1.50000 + 0.866025i −1.73205 1.00000i 0.500000 0.866025i −0.633975 2.36603i 0.866025 2.50000i 2.00000 2.00000i 0 1.00000 + 1.00000i
131.2 1.36603 + 0.366025i −1.50000 + 0.866025i 1.73205 + 1.00000i 0.500000 0.866025i −2.36603 + 0.633975i −0.866025 + 2.50000i 2.00000 + 2.00000i 0 1.00000 1.00000i
171.1 −0.366025 1.36603i −1.50000 0.866025i −1.73205 + 1.00000i 0.500000 + 0.866025i −0.633975 + 2.36603i 0.866025 + 2.50000i 2.00000 + 2.00000i 0 1.00000 1.00000i
171.2 1.36603 0.366025i −1.50000 0.866025i 1.73205 1.00000i 0.500000 + 0.866025i −2.36603 0.633975i −0.866025 2.50000i 2.00000 2.00000i 0 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
56.m 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.bj.d yes 4
4.b odd 2 1 1120.2.bz.d 4
7.d odd 6 1 280.2.bj.a 4
8.b even 2 1 1120.2.bz.a 4
8.d odd 2 1 280.2.bj.a 4
28.f even 6 1 1120.2.bz.a 4
56.j odd 6 1 1120.2.bz.d 4
56.m even 6 1 inner 280.2.bj.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bj.a 4 7.d odd 6 1
280.2.bj.a 4 8.d odd 2 1
280.2.bj.d yes 4 1.a even 1 1 trivial
280.2.bj.d yes 4 56.m even 6 1 inner
1120.2.bz.a 4 8.b even 2 1
1120.2.bz.a 4 28.f even 6 1
1120.2.bz.d 4 4.b odd 2 1
1120.2.bz.d 4 56.j odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} + 3 T_{3} + 3$$ $$T_{13}^{2} - 6 T_{13} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$( 3 + 3 T + T^{2} )^{2}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$49 + 11 T^{2} + T^{4}$$
$11$ $$64 - 32 T + 24 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$( 6 - 6 T + T^{2} )^{2}$$
$17$ $$36 + 36 T + 6 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$324 + 324 T + 126 T^{2} + 18 T^{3} + T^{4}$$
$23$ $$2209 - 1128 T + 239 T^{2} - 24 T^{3} + T^{4}$$
$29$ $$1369 + 122 T^{2} + T^{4}$$
$31$ $$324 - 108 T + 54 T^{2} + 6 T^{3} + T^{4}$$
$37$ $$16 - 4 T^{2} + T^{4}$$
$41$ $$( 27 + T^{2} )^{2}$$
$43$ $$( -39 + 6 T + T^{2} )^{2}$$
$47$ $$576 - 288 T + 120 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$4 - 36 T + 110 T^{2} - 18 T^{3} + T^{4}$$
$59$ $$36 + 36 T + 6 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$1089 - 396 T + 111 T^{2} - 12 T^{3} + T^{4}$$
$67$ $$11449 - 214 T + 111 T^{2} + 2 T^{3} + T^{4}$$
$71$ $$2116 + 104 T^{2} + T^{4}$$
$73$ $$4356 + 1980 T + 366 T^{2} + 30 T^{3} + T^{4}$$
$79$ $$36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4}$$
$83$ $$1521 + 222 T^{2} + T^{4}$$
$89$ $$1521 - 1170 T + 339 T^{2} - 30 T^{3} + T^{4}$$
$97$ $$( 12 + T^{2} )^{2}$$
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