# Properties

 Label 360.2.w.b Level $360$ Weight $2$ Character orbit 360.w 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.w (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.87461447277$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{2} + 2 q^{4} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( -1 + \zeta_{8}^{2} ) q^{7} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{2} + 2 q^{4} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( -1 + \zeta_{8}^{2} ) q^{7} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( 3 + \zeta_{8}^{2} ) q^{10} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{11} + ( 2 + 2 \zeta_{8}^{2} ) q^{13} -2 \zeta_{8}^{3} q^{14} + 4 q^{16} + 4 \zeta_{8} q^{17} -6 \zeta_{8}^{2} q^{19} + ( -4 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{20} -6 q^{22} + 2 \zeta_{8} q^{23} + ( 4 + 3 \zeta_{8}^{2} ) q^{25} -4 \zeta_{8} q^{26} + ( -2 + 2 \zeta_{8}^{2} ) q^{28} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{29} + 6 \zeta_{8}^{2} q^{31} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{32} + ( -4 - 4 \zeta_{8}^{2} ) q^{34} + ( \zeta_{8} - 3 \zeta_{8}^{3} ) q^{35} + ( 8 - 8 \zeta_{8}^{2} ) q^{37} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{38} + ( 6 + 2 \zeta_{8}^{2} ) q^{40} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{41} + ( -2 + 2 \zeta_{8}^{2} ) q^{43} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{44} + ( -2 - 2 \zeta_{8}^{2} ) q^{46} + 2 \zeta_{8}^{3} q^{47} + 5 \zeta_{8}^{2} q^{49} + ( -7 \zeta_{8} + \zeta_{8}^{3} ) q^{50} + ( 4 + 4 \zeta_{8}^{2} ) q^{52} + 2 \zeta_{8} q^{53} + ( -9 - 3 \zeta_{8}^{2} ) q^{55} -4 \zeta_{8}^{3} q^{56} -6 q^{58} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{59} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{62} + 8 q^{64} + ( -6 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{65} + ( -8 - 8 \zeta_{8}^{2} ) q^{67} + 8 \zeta_{8} q^{68} + ( -4 + 2 \zeta_{8}^{2} ) q^{70} + ( -10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{71} + ( -5 + 5 \zeta_{8}^{2} ) q^{73} + 16 \zeta_{8}^{3} q^{74} -12 \zeta_{8}^{2} q^{76} + 6 \zeta_{8}^{3} q^{77} -14 q^{79} + ( -8 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{80} -12 q^{82} -2 \zeta_{8}^{3} q^{83} + ( -4 - 8 \zeta_{8}^{2} ) q^{85} -4 \zeta_{8}^{3} q^{86} -12 q^{88} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{89} -4 q^{91} + 4 \zeta_{8} q^{92} + ( 2 - 2 \zeta_{8}^{2} ) q^{94} + ( 6 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{95} + ( 7 + 7 \zeta_{8}^{2} ) q^{97} + ( -5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} - 4q^{7} + O(q^{10})$$ $$4q + 8q^{4} - 4q^{7} + 12q^{10} + 8q^{13} + 16q^{16} - 24q^{22} + 16q^{25} - 8q^{28} - 16q^{34} + 32q^{37} + 24q^{40} - 8q^{43} - 8q^{46} + 16q^{52} - 36q^{55} - 24q^{58} + 32q^{64} - 32q^{67} - 16q^{70} - 20q^{73} - 56q^{79} - 48q^{82} - 16q^{85} - 48q^{88} - 16q^{91} + 8q^{94} + 28q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$181$$ $$217$$ $$271$$ $$281$$ $$\chi(n)$$ $$-1$$ $$\zeta_{8}^{2}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
−1.41421 0 2.00000 −2.12132 + 0.707107i 0 −1.00000 1.00000i −2.82843 0 3.00000 1.00000i
163.2 1.41421 0 2.00000 2.12132 0.707107i 0 −1.00000 1.00000i 2.82843 0 3.00000 1.00000i
307.1 −1.41421 0 2.00000 −2.12132 0.707107i 0 −1.00000 + 1.00000i −2.82843 0 3.00000 + 1.00000i
307.2 1.41421 0 2.00000 2.12132 + 0.707107i 0 −1.00000 + 1.00000i 2.82843 0 3.00000 + 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
3.b odd 2 1 inner
40.k even 4 1 inner
120.q odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.w.b yes 4
3.b odd 2 1 inner 360.2.w.b yes 4
4.b odd 2 1 1440.2.bi.b 4
5.c odd 4 1 360.2.w.a 4
8.b even 2 1 1440.2.bi.a 4
8.d odd 2 1 360.2.w.a 4
12.b even 2 1 1440.2.bi.b 4
15.e even 4 1 360.2.w.a 4
20.e even 4 1 1440.2.bi.a 4
24.f even 2 1 360.2.w.a 4
24.h odd 2 1 1440.2.bi.a 4
40.i odd 4 1 1440.2.bi.b 4
40.k even 4 1 inner 360.2.w.b yes 4
60.l odd 4 1 1440.2.bi.a 4
120.q odd 4 1 inner 360.2.w.b yes 4
120.w even 4 1 1440.2.bi.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.w.a 4 5.c odd 4 1
360.2.w.a 4 8.d odd 2 1
360.2.w.a 4 15.e even 4 1
360.2.w.a 4 24.f even 2 1
360.2.w.b yes 4 1.a even 1 1 trivial
360.2.w.b yes 4 3.b odd 2 1 inner
360.2.w.b yes 4 40.k even 4 1 inner
360.2.w.b yes 4 120.q odd 4 1 inner
1440.2.bi.a 4 8.b even 2 1
1440.2.bi.a 4 20.e even 4 1
1440.2.bi.a 4 24.h odd 2 1
1440.2.bi.a 4 60.l odd 4 1
1440.2.bi.b 4 4.b odd 2 1
1440.2.bi.b 4 12.b even 2 1
1440.2.bi.b 4 40.i odd 4 1
1440.2.bi.b 4 120.w even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$25 - 8 T^{2} + T^{4}$$
$7$ $$( 2 + 2 T + T^{2} )^{2}$$
$11$ $$( -18 + T^{2} )^{2}$$
$13$ $$( 8 - 4 T + T^{2} )^{2}$$
$17$ $$256 + T^{4}$$
$19$ $$( 36 + T^{2} )^{2}$$
$23$ $$16 + T^{4}$$
$29$ $$( -18 + T^{2} )^{2}$$
$31$ $$( 36 + T^{2} )^{2}$$
$37$ $$( 128 - 16 T + T^{2} )^{2}$$
$41$ $$( -72 + T^{2} )^{2}$$
$43$ $$( 8 + 4 T + T^{2} )^{2}$$
$47$ $$16 + T^{4}$$
$53$ $$16 + T^{4}$$
$59$ $$( 98 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( 128 + 16 T + T^{2} )^{2}$$
$71$ $$( 200 + T^{2} )^{2}$$
$73$ $$( 50 + 10 T + T^{2} )^{2}$$
$79$ $$( 14 + T )^{4}$$
$83$ $$16 + T^{4}$$
$89$ $$( 8 + T^{2} )^{2}$$
$97$ $$( 98 - 14 T + T^{2} )^{2}$$