# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 360.br (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

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

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

## Character values

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

 $$n$$ $$181$$ $$217$$ $$271$$ $$281$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{3}$$ $$1$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
77.1
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
−1.00000 1.00000i 1.50000 + 0.866025i 2.00000i −2.23205 0.133975i −0.633975 2.36603i 1.13397 4.23205i 2.00000 2.00000i 1.50000 + 2.59808i 2.09808 + 2.36603i
173.1 −1.00000 + 1.00000i 1.50000 0.866025i 2.00000i −2.23205 + 0.133975i −0.633975 + 2.36603i 1.13397 + 4.23205i 2.00000 + 2.00000i 1.50000 2.59808i 2.09808 2.36603i
293.1 −1.00000 + 1.00000i 1.50000 + 0.866025i 2.00000i 1.23205 + 1.86603i −2.36603 + 0.633975i 2.86603 + 0.767949i 2.00000 + 2.00000i 1.50000 + 2.59808i −3.09808 0.633975i
317.1 −1.00000 1.00000i 1.50000 0.866025i 2.00000i 1.23205 1.86603i −2.36603 0.633975i 2.86603 0.767949i 2.00000 2.00000i 1.50000 2.59808i −3.09808 + 0.633975i
 $$n$$: 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
360.br even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.br.b yes 4
5.c odd 4 1 360.2.br.a 4
8.b even 2 1 360.2.br.c yes 4
9.d odd 6 1 360.2.br.d yes 4
40.i odd 4 1 360.2.br.d yes 4
45.l even 12 1 360.2.br.c yes 4
72.j odd 6 1 360.2.br.a 4
360.br even 12 1 inner 360.2.br.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.br.a 4 5.c odd 4 1
360.2.br.a 4 72.j odd 6 1
360.2.br.b yes 4 1.a even 1 1 trivial
360.2.br.b yes 4 360.br even 12 1 inner
360.2.br.c yes 4 8.b even 2 1
360.2.br.c yes 4 45.l even 12 1
360.2.br.d yes 4 9.d odd 6 1
360.2.br.d yes 4 40.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(360, [\chi])$$:

 $$T_{7}^{4} - 8 T_{7}^{3} + 41 T_{7}^{2} - 130 T_{7} + 169$$ $$T_{11}^{2} + 2 T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + 2 T + T^{2} )^{2}$$
$3$ $$( 3 - 3 T + T^{2} )^{2}$$
$5$ $$25 + 10 T - T^{2} + 2 T^{3} + T^{4}$$
$7$ $$169 - 130 T + 41 T^{2} - 8 T^{3} + T^{4}$$
$11$ $$( 4 + 2 T + T^{2} )^{2}$$
$13$ $$144 - 144 T + 36 T^{2} + T^{4}$$
$17$ $$( 18 - 6 T + T^{2} )^{2}$$
$19$ $$( 6 - 6 T + T^{2} )^{2}$$
$23$ $$9 - 18 T + 9 T^{2} + T^{4}$$
$29$ $$529 + 414 T + 131 T^{2} + 18 T^{3} + T^{4}$$
$31$ $$324 + 108 T + 54 T^{2} - 6 T^{3} + T^{4}$$
$37$ $$36 - 72 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$41$ $$9 + 36 T + 51 T^{2} + 12 T^{3} + T^{4}$$
$43$ $$36 + 108 T + 90 T^{2} + 6 T^{3} + T^{4}$$
$47$ $$81 + 108 T + 45 T^{2} + 6 T^{3} + T^{4}$$
$53$ $$2916 + T^{4}$$
$59$ $$484 + 396 T + 86 T^{2} - 18 T^{3} + T^{4}$$
$61$ $$1521 + 1170 T + 339 T^{2} + 30 T^{3} + T^{4}$$
$67$ $$1521 - 468 T + 45 T^{2} - 6 T^{3} + T^{4}$$
$71$ $$( 144 + T^{2} )^{2}$$
$73$ $$16 - 80 T + 200 T^{2} + 20 T^{3} + T^{4}$$
$79$ $$324 - 324 T + 126 T^{2} - 18 T^{3} + T^{4}$$
$83$ $$14641 - 3146 T + 365 T^{2} - 28 T^{3} + T^{4}$$
$89$ $$( -39 + 12 T + T^{2} )^{2}$$
$97$ $$2500 + 500 T + 50 T^{2} + 10 T^{3} + T^{4}$$