# Properties

 Label 360.2.bf.a Level $360$ Weight $2$ Character orbit 360.bf Analytic conductor $2.875$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 360.bf (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.87461447277$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$217$$ $$271$$ $$281$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−1.36603 + 0.366025i −0.866025 + 1.50000i 1.73205 1.00000i 0.866025 0.500000i 0.633975 2.36603i 2.00000 3.46410i −2.00000 + 2.00000i −1.50000 2.59808i −1.00000 + 1.00000i
61.2 0.366025 + 1.36603i 0.866025 1.50000i −1.73205 + 1.00000i −0.866025 + 0.500000i 2.36603 + 0.633975i 2.00000 3.46410i −2.00000 2.00000i −1.50000 2.59808i −1.00000 1.00000i
301.1 −1.36603 0.366025i −0.866025 1.50000i 1.73205 + 1.00000i 0.866025 + 0.500000i 0.633975 + 2.36603i 2.00000 + 3.46410i −2.00000 2.00000i −1.50000 + 2.59808i −1.00000 1.00000i
301.2 0.366025 1.36603i 0.866025 + 1.50000i −1.73205 1.00000i −0.866025 0.500000i 2.36603 0.633975i 2.00000 + 3.46410i −2.00000 + 2.00000i −1.50000 + 2.59808i −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
9.c even 3 1 inner
72.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.bf.a 4
3.b odd 2 1 1080.2.bf.a 4
4.b odd 2 1 1440.2.bv.a 4
8.b even 2 1 inner 360.2.bf.a 4
8.d odd 2 1 1440.2.bv.a 4
9.c even 3 1 inner 360.2.bf.a 4
9.d odd 6 1 1080.2.bf.a 4
12.b even 2 1 4320.2.bv.a 4
24.f even 2 1 4320.2.bv.a 4
24.h odd 2 1 1080.2.bf.a 4
36.f odd 6 1 1440.2.bv.a 4
36.h even 6 1 4320.2.bv.a 4
72.j odd 6 1 1080.2.bf.a 4
72.l even 6 1 4320.2.bv.a 4
72.n even 6 1 inner 360.2.bf.a 4
72.p odd 6 1 1440.2.bv.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bf.a 4 1.a even 1 1 trivial
360.2.bf.a 4 8.b even 2 1 inner
360.2.bf.a 4 9.c even 3 1 inner
360.2.bf.a 4 72.n even 6 1 inner
1080.2.bf.a 4 3.b odd 2 1
1080.2.bf.a 4 9.d odd 6 1
1080.2.bf.a 4 24.h odd 2 1
1080.2.bf.a 4 72.j odd 6 1
1440.2.bv.a 4 4.b odd 2 1
1440.2.bv.a 4 8.d odd 2 1
1440.2.bv.a 4 36.f odd 6 1
1440.2.bv.a 4 72.p odd 6 1
4320.2.bv.a 4 12.b even 2 1
4320.2.bv.a 4 24.f even 2 1
4320.2.bv.a 4 36.h even 6 1
4320.2.bv.a 4 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 4 T_{7} + 16$$ acting on $$S_{2}^{\mathrm{new}}(360, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$1 - T^{2} + T^{4}$$
$7$ $$( 16 - 4 T + T^{2} )^{2}$$
$11$ $$625 - 25 T^{2} + T^{4}$$
$13$ $$1296 - 36 T^{2} + T^{4}$$
$17$ $$( 3 + T )^{4}$$
$19$ $$( 49 + T^{2} )^{2}$$
$23$ $$( 4 + 2 T + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 16 + 4 T + T^{2} )^{2}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( 25 - 5 T + T^{2} )^{2}$$
$43$ $$625 - 25 T^{2} + T^{4}$$
$47$ $$( 4 + 2 T + T^{2} )^{2}$$
$53$ $$( 64 + T^{2} )^{2}$$
$59$ $$2401 - 49 T^{2} + T^{4}$$
$61$ $$256 - 16 T^{2} + T^{4}$$
$67$ $$81 - 9 T^{2} + T^{4}$$
$71$ $$( -2 + T )^{4}$$
$73$ $$( -1 + T )^{4}$$
$79$ $$( 100 + 10 T + T^{2} )^{2}$$
$83$ $$20736 - 144 T^{2} + T^{4}$$
$89$ $$( -2 + T )^{4}$$
$97$ $$( 49 - 7 T + T^{2} )^{2}$$
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