# Properties

 Label 280.2.bv.d Level $280$ Weight $2$ Character orbit 280.bv 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.bv (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$4$$ 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: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

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

## Character values

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

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$1$$ $$-1$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
117.1
 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i
1.00000 1.00000i 1.86603 + 0.500000i 2.00000i −0.133975 + 2.23205i 2.36603 1.36603i 0.866025 2.50000i −2.00000 2.00000i 0.633975 + 0.366025i 2.09808 + 2.36603i
157.1 1.00000 1.00000i 0.133975 + 0.500000i 2.00000i −1.86603 1.23205i 0.633975 + 0.366025i −0.866025 2.50000i −2.00000 2.00000i 2.36603 1.36603i −3.09808 + 0.633975i
173.1 1.00000 + 1.00000i 0.133975 0.500000i 2.00000i −1.86603 + 1.23205i 0.633975 0.366025i −0.866025 + 2.50000i −2.00000 + 2.00000i 2.36603 + 1.36603i −3.09808 0.633975i
213.1 1.00000 + 1.00000i 1.86603 0.500000i 2.00000i −0.133975 2.23205i 2.36603 + 1.36603i 0.866025 + 2.50000i −2.00000 + 2.00000i 0.633975 0.366025i 2.09808 2.36603i
 $$n$$: 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
280.bv even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.bv.d yes 4
5.c odd 4 1 280.2.bv.a 4
7.d odd 6 1 280.2.bv.b yes 4
8.b even 2 1 280.2.bv.c yes 4
35.k even 12 1 280.2.bv.c yes 4
40.i odd 4 1 280.2.bv.b yes 4
56.j odd 6 1 280.2.bv.a 4
280.bv even 12 1 inner 280.2.bv.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bv.a 4 5.c odd 4 1
280.2.bv.a 4 56.j odd 6 1
280.2.bv.b yes 4 7.d odd 6 1
280.2.bv.b yes 4 40.i odd 4 1
280.2.bv.c yes 4 8.b even 2 1
280.2.bv.c yes 4 35.k even 12 1
280.2.bv.d yes 4 1.a even 1 1 trivial
280.2.bv.d yes 4 280.bv even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 4 T_{3}^{3} + 5 T_{3}^{2} - 2 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 - 2 T + T^{2} )^{2}$$
$3$ $$1 - 2 T + 5 T^{2} - 4 T^{3} + T^{4}$$
$5$ $$25 + 20 T + 11 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$49 + 11 T^{2} + T^{4}$$
$11$ $$4 - 12 T + 14 T^{2} - 6 T^{3} + T^{4}$$
$13$ $$( 18 + 6 T + T^{2} )^{2}$$
$17$ $$324 - 108 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$( 12 - 6 T + T^{2} )^{2}$$
$23$ $$81 - 54 T + 45 T^{2} - 12 T^{3} + T^{4}$$
$29$ $$( -11 + 2 T + T^{2} )^{2}$$
$31$ $$4 + 36 T + 110 T^{2} + 18 T^{3} + T^{4}$$
$37$ $$144 - 144 T + 36 T^{2} + T^{4}$$
$41$ $$9 + 42 T^{2} + T^{4}$$
$43$ $$9 - 18 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$47$ $$36 - 36 T + 90 T^{2} - 18 T^{3} + T^{4}$$
$53$ $$1296 + 432 T + 180 T^{2} + 24 T^{3} + T^{4}$$
$59$ $$1936 - 1056 T + 236 T^{2} - 24 T^{3} + T^{4}$$
$61$ $$1521 + 234 T + 75 T^{2} - 6 T^{3} + T^{4}$$
$67$ $$1089 + 594 T + 117 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$( 132 - 24 T + T^{2} )^{2}$$
$73$ $$8836 + 2068 T + 290 T^{2} + 26 T^{3} + T^{4}$$
$79$ $$144 - 288 T + 204 T^{2} - 24 T^{3} + T^{4}$$
$83$ $$4761 + 414 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$89$ $$13689 - 2808 T + 459 T^{2} - 24 T^{3} + T^{4}$$
$97$ $$14884 + 3904 T + 512 T^{2} + 32 T^{3} + T^{4}$$
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