# Properties

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

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.87461447277$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{5} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{5} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{11} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{13} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{15} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} -2 q^{19} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{21} + ( \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{23} + ( -1 + \zeta_{12}^{2} ) q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -5 - 2 \zeta_{12} + 5 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{29} + 2 \zeta_{12}^{2} q^{31} + ( 6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{33} + ( -2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{35} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{37} + ( -6 + 2 \zeta_{12} + 6 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{39} + ( 4 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{41} + ( 8 + 2 \zeta_{12} - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{43} -3 q^{45} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{47} + ( -4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{49} + ( -4 \zeta_{12} - 6 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{51} + 6 q^{53} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{55} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{57} + ( 6 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{59} + ( 5 - 2 \zeta_{12} - 5 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{61} + ( -3 \zeta_{12} - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{63} + ( -2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{65} + ( -\zeta_{12} - 8 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{67} + ( 3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{69} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{73} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{75} + ( -6 \zeta_{12} - 10 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{77} + ( -12 + 2 \zeta_{12} + 12 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( 8 - 3 \zeta_{12} - 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{83} + ( 2 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{85} + ( 6 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{87} + ( -3 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{89} + ( 2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{91} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{93} -2 \zeta_{12}^{2} q^{95} + ( 2 - 4 \zeta_{12} - 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{97} + ( -6 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} - 4q^{7} - 6q^{9} + O(q^{10})$$ $$4q + 2q^{5} - 4q^{7} - 6q^{9} - 4q^{11} + 4q^{13} + 16q^{17} - 8q^{19} + 12q^{21} - 4q^{23} - 2q^{25} - 10q^{29} + 4q^{31} + 24q^{33} - 8q^{35} - 12q^{39} - 6q^{41} + 16q^{43} - 12q^{45} - 4q^{47} - 12q^{51} + 24q^{53} - 8q^{55} - 8q^{59} + 10q^{61} - 12q^{63} - 4q^{65} - 16q^{67} + 6q^{69} + 24q^{71} - 20q^{77} - 24q^{79} - 18q^{81} + 16q^{83} + 8q^{85} + 24q^{87} - 12q^{89} + 8q^{91} - 4q^{95} + 4q^{97} - 12q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 + 1.50000i 0 0.500000 0.866025i 0 −1.86603 3.23205i 0 −1.50000 2.59808i 0
121.2 0 0.866025 1.50000i 0 0.500000 0.866025i 0 −0.133975 0.232051i 0 −1.50000 2.59808i 0
241.1 0 −0.866025 1.50000i 0 0.500000 + 0.866025i 0 −1.86603 + 3.23205i 0 −1.50000 + 2.59808i 0
241.2 0 0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 −0.133975 + 0.232051i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.q.b 4
3.b odd 2 1 1080.2.q.b 4
4.b odd 2 1 720.2.q.h 4
9.c even 3 1 inner 360.2.q.b 4
9.c even 3 1 3240.2.a.k 2
9.d odd 6 1 1080.2.q.b 4
9.d odd 6 1 3240.2.a.p 2
12.b even 2 1 2160.2.q.h 4
36.f odd 6 1 720.2.q.h 4
36.f odd 6 1 6480.2.a.ba 2
36.h even 6 1 2160.2.q.h 4
36.h even 6 1 6480.2.a.bk 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.b 4 1.a even 1 1 trivial
360.2.q.b 4 9.c even 3 1 inner
720.2.q.h 4 4.b odd 2 1
720.2.q.h 4 36.f odd 6 1
1080.2.q.b 4 3.b odd 2 1
1080.2.q.b 4 9.d odd 6 1
2160.2.q.h 4 12.b even 2 1
2160.2.q.h 4 36.h even 6 1
3240.2.a.k 2 9.c even 3 1
3240.2.a.p 2 9.d odd 6 1
6480.2.a.ba 2 36.f odd 6 1
6480.2.a.bk 2 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 4 T_{7}^{3} + 15 T_{7}^{2} + 4 T_{7} + 1$$ acting on $$S_{2}^{\mathrm{new}}(360, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$1 + 4 T + 15 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$64 - 32 T + 24 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$64 + 32 T + 24 T^{2} - 4 T^{3} + T^{4}$$
$17$ $$( 4 - 8 T + T^{2} )^{2}$$
$19$ $$( 2 + T )^{4}$$
$23$ $$1 + 4 T + 15 T^{2} + 4 T^{3} + T^{4}$$
$29$ $$169 + 130 T + 87 T^{2} + 10 T^{3} + T^{4}$$
$31$ $$( 4 - 2 T + T^{2} )^{2}$$
$37$ $$( -108 + T^{2} )^{2}$$
$41$ $$1521 - 234 T + 75 T^{2} + 6 T^{3} + T^{4}$$
$43$ $$2704 - 832 T + 204 T^{2} - 16 T^{3} + T^{4}$$
$47$ $$1 + 4 T + 15 T^{2} + 4 T^{3} + T^{4}$$
$53$ $$( -6 + T )^{4}$$
$59$ $$8464 - 736 T + 156 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$169 - 130 T + 87 T^{2} - 10 T^{3} + T^{4}$$
$67$ $$3721 + 976 T + 195 T^{2} + 16 T^{3} + T^{4}$$
$71$ $$( 24 - 12 T + T^{2} )^{2}$$
$73$ $$( -48 + T^{2} )^{2}$$
$79$ $$17424 + 3168 T + 444 T^{2} + 24 T^{3} + T^{4}$$
$83$ $$1369 - 592 T + 219 T^{2} - 16 T^{3} + T^{4}$$
$89$ $$( -39 + 6 T + T^{2} )^{2}$$
$97$ $$1936 + 176 T + 60 T^{2} - 4 T^{3} + T^{4}$$
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