# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 360.bi (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: $$\Z[a_1, \ldots, a_{5}]$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} -\zeta_{12} q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} -\zeta_{12} q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -2 + 2 \zeta_{12}^{2} ) q^{11} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + ( 2 + \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{15} + 6 \zeta_{12}^{3} q^{17} -2 q^{19} + ( -1 + 2 \zeta_{12}^{2} ) q^{21} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{23} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -7 + 7 \zeta_{12}^{2} ) q^{29} + 6 \zeta_{12}^{2} q^{31} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{33} + ( 2 - \zeta_{12}^{3} ) q^{35} + 2 \zeta_{12}^{3} q^{37} + ( 2 + 2 \zeta_{12}^{2} ) q^{39} -5 \zeta_{12}^{2} q^{41} -12 \zeta_{12} q^{43} + ( -3 - 6 \zeta_{12}^{3} ) q^{45} + 9 \zeta_{12} q^{47} -6 \zeta_{12}^{2} q^{49} + ( 12 - 6 \zeta_{12}^{2} ) q^{51} + 8 \zeta_{12}^{3} q^{53} + ( -2 - 4 \zeta_{12}^{3} ) q^{55} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{57} -12 \zeta_{12}^{2} q^{59} + ( 7 - 7 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} + ( 4 - 2 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{65} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{67} + ( -1 - \zeta_{12}^{2} ) q^{69} -10 q^{71} -4 \zeta_{12}^{3} q^{73} + ( -4 - 6 \zeta_{12} + 8 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{75} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{77} + ( 4 - 4 \zeta_{12}^{2} ) q^{79} -9 \zeta_{12}^{2} q^{81} -5 \zeta_{12} q^{83} + ( -6 \zeta_{12} - 12 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{85} + ( 14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{87} + 15 q^{89} + 2 q^{91} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{93} + ( 4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{95} + 16 \zeta_{12} q^{97} -6 \zeta_{12}^{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} - 6q^{9} + O(q^{10})$$ $$4q + 2q^{5} - 6q^{9} - 4q^{11} + 12q^{15} - 8q^{19} + 6q^{25} - 14q^{29} + 12q^{31} + 8q^{35} + 12q^{39} - 10q^{41} - 12q^{45} - 12q^{49} + 36q^{51} - 8q^{55} - 24q^{59} + 14q^{61} + 8q^{65} - 6q^{69} - 40q^{71} + 8q^{79} - 18q^{81} - 24q^{85} + 60q^{89} + 8q^{91} - 4q^{95} - 12q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 + 1.50000i 0 −1.23205 1.86603i 0 −0.866025 + 0.500000i 0 −1.50000 2.59808i 0
49.2 0 0.866025 1.50000i 0 2.23205 + 0.133975i 0 0.866025 0.500000i 0 −1.50000 2.59808i 0
169.1 0 −0.866025 1.50000i 0 −1.23205 + 1.86603i 0 −0.866025 0.500000i 0 −1.50000 + 2.59808i 0
169.2 0 0.866025 + 1.50000i 0 2.23205 0.133975i 0 0.866025 + 0.500000i 0 −1.50000 + 2.59808i 0
 $$n$$: 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
5.b even 2 1 inner
9.c even 3 1 inner
45.j 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.bi.a 4
3.b odd 2 1 1080.2.bi.a 4
4.b odd 2 1 720.2.by.b 4
5.b even 2 1 inner 360.2.bi.a 4
9.c even 3 1 inner 360.2.bi.a 4
9.c even 3 1 3240.2.f.b 2
9.d odd 6 1 1080.2.bi.a 4
9.d odd 6 1 3240.2.f.e 2
12.b even 2 1 2160.2.by.a 4
15.d odd 2 1 1080.2.bi.a 4
20.d odd 2 1 720.2.by.b 4
36.f odd 6 1 720.2.by.b 4
36.h even 6 1 2160.2.by.a 4
45.h odd 6 1 1080.2.bi.a 4
45.h odd 6 1 3240.2.f.e 2
45.j even 6 1 inner 360.2.bi.a 4
45.j even 6 1 3240.2.f.b 2
60.h even 2 1 2160.2.by.a 4
180.n even 6 1 2160.2.by.a 4
180.p odd 6 1 720.2.by.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bi.a 4 1.a even 1 1 trivial
360.2.bi.a 4 5.b even 2 1 inner
360.2.bi.a 4 9.c even 3 1 inner
360.2.bi.a 4 45.j even 6 1 inner
720.2.by.b 4 4.b odd 2 1
720.2.by.b 4 20.d odd 2 1
720.2.by.b 4 36.f odd 6 1
720.2.by.b 4 180.p odd 6 1
1080.2.bi.a 4 3.b odd 2 1
1080.2.bi.a 4 9.d odd 6 1
1080.2.bi.a 4 15.d odd 2 1
1080.2.bi.a 4 45.h odd 6 1
2160.2.by.a 4 12.b even 2 1
2160.2.by.a 4 36.h even 6 1
2160.2.by.a 4 60.h even 2 1
2160.2.by.a 4 180.n even 6 1
3240.2.f.b 2 9.c even 3 1
3240.2.f.b 2 45.j even 6 1
3240.2.f.e 2 9.d odd 6 1
3240.2.f.e 2 45.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - T_{7}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(360, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$25 - 10 T - T^{2} - 2 T^{3} + T^{4}$$
$7$ $$1 - T^{2} + T^{4}$$
$11$ $$( 4 + 2 T + T^{2} )^{2}$$
$13$ $$16 - 4 T^{2} + T^{4}$$
$17$ $$( 36 + T^{2} )^{2}$$
$19$ $$( 2 + T )^{4}$$
$23$ $$1 - T^{2} + T^{4}$$
$29$ $$( 49 + 7 T + T^{2} )^{2}$$
$31$ $$( 36 - 6 T + T^{2} )^{2}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( 25 + 5 T + T^{2} )^{2}$$
$43$ $$20736 - 144 T^{2} + T^{4}$$
$47$ $$6561 - 81 T^{2} + T^{4}$$
$53$ $$( 64 + T^{2} )^{2}$$
$59$ $$( 144 + 12 T + T^{2} )^{2}$$
$61$ $$( 49 - 7 T + T^{2} )^{2}$$
$67$ $$625 - 25 T^{2} + T^{4}$$
$71$ $$( 10 + T )^{4}$$
$73$ $$( 16 + T^{2} )^{2}$$
$79$ $$( 16 - 4 T + T^{2} )^{2}$$
$83$ $$625 - 25 T^{2} + T^{4}$$
$89$ $$( -15 + T )^{4}$$
$97$ $$65536 - 256 T^{2} + T^{4}$$