# Properties

 Label 280.2.bj.b Level $280$ Weight $2$ Character orbit 280.bj Analytic conductor $2.236$ Analytic rank $1$ Dimension $4$ 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.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$1$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{3} -2 q^{4} + ( -1 + \beta_{2} ) q^{5} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{6} + ( -2 - \beta_{2} ) q^{7} -2 \beta_{3} q^{8} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{3} -2 q^{4} + ( -1 + \beta_{2} ) q^{5} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{6} + ( -2 - \beta_{2} ) q^{7} -2 \beta_{3} q^{8} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{9} -\beta_{1} q^{10} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{11} + ( 4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{12} + ( -2 + 2 \beta_{1} - \beta_{3} ) q^{13} + ( \beta_{1} - 3 \beta_{3} ) q^{14} + ( 1 - 2 \beta_{2} - \beta_{3} ) q^{15} + 4 q^{16} + ( -4 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{17} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{18} -3 \beta_{1} q^{19} + ( 2 - 2 \beta_{2} ) q^{20} + ( 5 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{21} + ( -8 + 4 \beta_{2} ) q^{22} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{23} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{24} -\beta_{2} q^{25} + ( -2 + 4 \beta_{2} - 2 \beta_{3} ) q^{26} + ( -3 + 6 \beta_{2} + 5 \beta_{3} ) q^{27} + ( 4 + 2 \beta_{2} ) q^{28} + ( 5 - 10 \beta_{2} - \beta_{3} ) q^{29} + ( 2 + 2 \beta_{1} - \beta_{3} ) q^{30} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{31} + 4 \beta_{3} q^{32} + ( -4 - 6 \beta_{1} - 4 \beta_{2} ) q^{33} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{34} + ( 3 - 2 \beta_{2} ) q^{35} + ( -4 - 4 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} ) q^{36} + ( -2 - 2 \beta_{2} ) q^{37} + ( 6 - 6 \beta_{2} ) q^{38} + ( -\beta_{1} + \beta_{3} ) q^{39} + 2 \beta_{1} q^{40} + ( -3 + 6 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -2 + \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{42} + ( -7 + 2 \beta_{1} - \beta_{3} ) q^{43} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{44} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{45} + ( 4 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{46} + ( 6 - 6 \beta_{2} ) q^{47} + ( -8 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{48} + ( 3 + 5 \beta_{2} ) q^{49} + ( \beta_{1} - \beta_{3} ) q^{50} + ( 4 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{51} + ( 4 - 4 \beta_{1} + 2 \beta_{3} ) q^{52} + ( -4 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{53} + ( -10 - 6 \beta_{1} + 3 \beta_{3} ) q^{54} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{55} + ( -2 \beta_{1} + 6 \beta_{3} ) q^{56} + ( 6 + 6 \beta_{1} - 3 \beta_{3} ) q^{57} + ( 2 + 10 \beta_{1} - 5 \beta_{3} ) q^{58} + ( -4 - 5 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{59} + ( -2 + 4 \beta_{2} + 2 \beta_{3} ) q^{60} + ( -7 - 3 \beta_{1} + 7 \beta_{2} + 6 \beta_{3} ) q^{61} + ( 4 + 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{62} + ( -6 - 8 \beta_{1} + 4 \beta_{2} + 10 \beta_{3} ) q^{63} -8 q^{64} + ( 2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 12 + 4 \beta_{1} - 12 \beta_{2} - 8 \beta_{3} ) q^{66} + ( 3 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{67} + ( 8 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{68} + ( 1 + 2 \beta_{1} - \beta_{3} ) q^{69} + ( 2 \beta_{1} + \beta_{3} ) q^{70} -\beta_{3} q^{71} + ( -8 - 4 \beta_{1} - 8 \beta_{2} ) q^{72} + ( -8 + 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{73} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{74} + ( 1 + \beta_{1} + \beta_{2} ) q^{75} + 6 \beta_{1} q^{76} + ( -2 \beta_{1} - 8 \beta_{3} ) q^{77} -2 \beta_{2} q^{78} + ( -2 + 3 \beta_{1} - 2 \beta_{2} ) q^{79} + ( -4 + 4 \beta_{2} ) q^{80} + ( -2 \beta_{1} - 13 \beta_{2} - 2 \beta_{3} ) q^{81} + ( 4 - 6 \beta_{1} + 3 \beta_{3} ) q^{82} + ( 3 - 6 \beta_{2} - 5 \beta_{3} ) q^{83} + ( -10 - 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{84} + ( 2 - 4 \beta_{2} + \beta_{3} ) q^{85} + ( -2 + 4 \beta_{2} - 7 \beta_{3} ) q^{86} + ( 6 \beta_{1} + 17 \beta_{2} + 6 \beta_{3} ) q^{87} + ( 16 - 8 \beta_{2} ) q^{88} + ( 3 - 4 \beta_{1} + 3 \beta_{2} ) q^{89} + ( -8 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{90} + ( 4 - 5 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{91} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{92} + ( 6 + 7 \beta_{1} + 6 \beta_{2} ) q^{93} + 6 \beta_{1} q^{94} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{95} + ( -4 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{96} + ( 2 - 4 \beta_{2} + 6 \beta_{3} ) q^{97} + ( -5 \beta_{1} + 8 \beta_{3} ) q^{98} + ( 24 + 8 \beta_{1} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{3} - 8q^{4} - 2q^{5} - 4q^{6} - 10q^{7} + 4q^{9} + O(q^{10})$$ $$4q - 6q^{3} - 8q^{4} - 2q^{5} - 4q^{6} - 10q^{7} + 4q^{9} + 12q^{12} - 8q^{13} + 16q^{16} - 12q^{17} + 24q^{18} + 4q^{20} + 18q^{21} - 24q^{22} + 6q^{23} + 8q^{24} - 2q^{25} + 20q^{28} + 8q^{30} - 8q^{31} - 24q^{33} + 4q^{34} + 8q^{35} - 8q^{36} - 12q^{37} + 12q^{38} + 4q^{42} - 28q^{43} + 4q^{45} + 8q^{46} + 12q^{47} - 24q^{48} + 22q^{49} + 8q^{51} + 16q^{52} - 12q^{53} - 40q^{54} + 24q^{57} + 8q^{58} - 12q^{59} - 14q^{61} + 12q^{62} - 16q^{63} - 32q^{64} + 4q^{65} + 24q^{66} - 10q^{67} + 24q^{68} + 4q^{69} - 48q^{72} - 24q^{73} + 6q^{75} - 4q^{78} - 12q^{79} - 8q^{80} - 26q^{81} + 16q^{82} - 36q^{84} + 34q^{87} + 48q^{88} + 18q^{89} - 24q^{90} + 20q^{91} - 12q^{92} + 36q^{93} - 16q^{96} + 96q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1 - \beta_{2}$$

## 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
 −1.22474 − 0.707107i 1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
1.41421i −0.275255 + 0.158919i −2.00000 −0.500000 + 0.866025i 0.224745 + 0.389270i −2.50000 0.866025i 2.82843i −1.44949 + 2.51059i 1.22474 + 0.707107i
131.2 1.41421i −2.72474 + 1.57313i −2.00000 −0.500000 + 0.866025i −2.22474 3.85337i −2.50000 0.866025i 2.82843i 3.44949 5.97469i −1.22474 0.707107i
171.1 1.41421i −2.72474 1.57313i −2.00000 −0.500000 0.866025i −2.22474 + 3.85337i −2.50000 + 0.866025i 2.82843i 3.44949 + 5.97469i −1.22474 + 0.707107i
171.2 1.41421i −0.275255 0.158919i −2.00000 −0.500000 0.866025i 0.224745 0.389270i −2.50000 + 0.866025i 2.82843i −1.44949 2.51059i 1.22474 0.707107i
 $$n$$: 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.b 4
4.b odd 2 1 1120.2.bz.b 4
7.d odd 6 1 280.2.bj.c yes 4
8.b even 2 1 1120.2.bz.c 4
8.d odd 2 1 280.2.bj.c yes 4
28.f even 6 1 1120.2.bz.c 4
56.j odd 6 1 1120.2.bz.b 4
56.m even 6 1 inner 280.2.bj.b 4

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

## Hecke kernels

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

 $$T_{3}^{4} + 6 T_{3}^{3} + 13 T_{3}^{2} + 6 T_{3} + 1$$ $$T_{13}^{2} + 4 T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$1 + 6 T + 13 T^{2} + 6 T^{3} + T^{4}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$( 7 + 5 T + T^{2} )^{2}$$
$11$ $$576 + 24 T^{2} + T^{4}$$
$13$ $$( -2 + 4 T + T^{2} )^{2}$$
$17$ $$100 + 120 T + 58 T^{2} + 12 T^{3} + T^{4}$$
$19$ $$324 - 18 T^{2} + T^{4}$$
$23$ $$25 + 30 T + 7 T^{2} - 6 T^{3} + T^{4}$$
$29$ $$5329 + 154 T^{2} + T^{4}$$
$31$ $$100 + 80 T + 54 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$( 12 + 6 T + T^{2} )^{2}$$
$41$ $$361 + 70 T^{2} + T^{4}$$
$43$ $$( 43 + 14 T + T^{2} )^{2}$$
$47$ $$( 36 - 6 T + T^{2} )^{2}$$
$53$ $$100 + 120 T + 58 T^{2} + 12 T^{3} + T^{4}$$
$59$ $$1444 - 456 T + 10 T^{2} + 12 T^{3} + T^{4}$$
$61$ $$25 - 70 T + 201 T^{2} + 14 T^{3} + T^{4}$$
$67$ $$841 - 290 T + 129 T^{2} + 10 T^{3} + T^{4}$$
$71$ $$( 2 + T^{2} )^{2}$$
$73$ $$900 + 720 T + 222 T^{2} + 24 T^{3} + T^{4}$$
$79$ $$36 - 72 T + 42 T^{2} + 12 T^{3} + T^{4}$$
$83$ $$529 + 154 T^{2} + T^{4}$$
$89$ $$25 + 90 T + 103 T^{2} - 18 T^{3} + T^{4}$$
$97$ $$3600 + 168 T^{2} + T^{4}$$