# Properties

 Label 360.2.d.b Level $360$ Weight $2$ Character orbit 360.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.87461447277$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{3} ) q^{4} + ( -\beta_{2} + \beta_{3} ) q^{5} + ( 2 \beta_{1} + \beta_{2} ) q^{7} + 2 \beta_{2} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{3} ) q^{4} + ( -\beta_{2} + \beta_{3} ) q^{5} + ( 2 \beta_{1} + \beta_{2} ) q^{7} + 2 \beta_{2} q^{8} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{10} -2 \beta_{3} q^{11} + ( -3 + \beta_{3} ) q^{14} + ( -2 - 2 \beta_{3} ) q^{16} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{17} -2 \beta_{3} q^{19} + ( -3 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{20} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{22} + ( 2 \beta_{1} + \beta_{2} ) q^{23} + ( -1 + 4 \beta_{1} + 2 \beta_{2} ) q^{25} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{28} + 4 q^{31} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{32} + ( -6 + 2 \beta_{3} ) q^{34} + ( 3 \beta_{2} + 2 \beta_{3} ) q^{35} -6 \beta_{2} q^{37} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{38} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{40} + 3 \beta_{2} q^{43} + ( 6 + 2 \beta_{3} ) q^{44} + ( -3 + \beta_{3} ) q^{46} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{47} + q^{49} + ( -6 - \beta_{1} + 2 \beta_{3} ) q^{50} -4 \beta_{2} q^{53} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{55} -4 \beta_{3} q^{56} -6 \beta_{3} q^{59} -2 \beta_{3} q^{61} + 4 \beta_{1} q^{62} + 8 q^{64} -3 \beta_{2} q^{67} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{68} + ( -3 + 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{70} + 12 q^{71} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{73} + ( 6 + 6 \beta_{3} ) q^{74} + ( 6 + 2 \beta_{3} ) q^{76} -6 \beta_{2} q^{77} -4 q^{79} + ( 6 - 4 \beta_{1} - 2 \beta_{3} ) q^{80} + 7 \beta_{2} q^{83} + ( 6 \beta_{2} + 4 \beta_{3} ) q^{85} + ( -3 - 3 \beta_{3} ) q^{86} + ( 8 \beta_{1} + 4 \beta_{2} ) q^{88} -6 q^{89} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{92} + ( 9 - 3 \beta_{3} ) q^{94} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{95} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{10} - 12q^{14} - 8q^{16} - 12q^{20} - 4q^{25} + 16q^{31} - 24q^{34} - 16q^{40} + 24q^{44} - 12q^{46} + 4q^{49} - 24q^{50} + 24q^{55} + 32q^{64} - 12q^{70} + 48q^{71} + 24q^{74} + 24q^{76} - 16q^{79} + 24q^{80} - 12q^{86} - 24q^{89} + 36q^{94} + 24q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2}$$

## Character values

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

 $$n$$ $$181$$ $$217$$ $$271$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i 0.707107 + 1.22474i
−0.707107 1.22474i 0 −1.00000 + 1.73205i −1.41421 + 1.73205i 0 2.44949i 2.82843 0 3.12132 + 0.507306i
109.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i −1.41421 1.73205i 0 2.44949i 2.82843 0 3.12132 0.507306i
109.3 0.707107 1.22474i 0 −1.00000 1.73205i 1.41421 1.73205i 0 2.44949i −2.82843 0 −1.12132 2.95680i
109.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 1.41421 + 1.73205i 0 2.44949i −2.82843 0 −1.12132 + 2.95680i
 $$n$$: 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
8.b even 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.d.b 4
3.b odd 2 1 40.2.f.a 4
4.b odd 2 1 1440.2.d.c 4
5.b even 2 1 inner 360.2.d.b 4
5.c odd 4 2 1800.2.k.m 4
8.b even 2 1 inner 360.2.d.b 4
8.d odd 2 1 1440.2.d.c 4
12.b even 2 1 160.2.f.a 4
15.d odd 2 1 40.2.f.a 4
15.e even 4 2 200.2.d.e 4
20.d odd 2 1 1440.2.d.c 4
20.e even 4 2 7200.2.k.l 4
24.f even 2 1 160.2.f.a 4
24.h odd 2 1 40.2.f.a 4
40.e odd 2 1 1440.2.d.c 4
40.f even 2 1 inner 360.2.d.b 4
40.i odd 4 2 1800.2.k.m 4
40.k even 4 2 7200.2.k.l 4
48.i odd 4 2 1280.2.c.i 4
48.k even 4 2 1280.2.c.k 4
60.h even 2 1 160.2.f.a 4
60.l odd 4 2 800.2.d.f 4
120.i odd 2 1 40.2.f.a 4
120.m even 2 1 160.2.f.a 4
120.q odd 4 2 800.2.d.f 4
120.w even 4 2 200.2.d.e 4
240.t even 4 2 1280.2.c.k 4
240.z odd 4 2 6400.2.a.cm 4
240.bb even 4 2 6400.2.a.co 4
240.bd odd 4 2 6400.2.a.cm 4
240.bf even 4 2 6400.2.a.co 4
240.bm odd 4 2 1280.2.c.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.f.a 4 3.b odd 2 1
40.2.f.a 4 15.d odd 2 1
40.2.f.a 4 24.h odd 2 1
40.2.f.a 4 120.i odd 2 1
160.2.f.a 4 12.b even 2 1
160.2.f.a 4 24.f even 2 1
160.2.f.a 4 60.h even 2 1
160.2.f.a 4 120.m even 2 1
200.2.d.e 4 15.e even 4 2
200.2.d.e 4 120.w even 4 2
360.2.d.b 4 1.a even 1 1 trivial
360.2.d.b 4 5.b even 2 1 inner
360.2.d.b 4 8.b even 2 1 inner
360.2.d.b 4 40.f even 2 1 inner
800.2.d.f 4 60.l odd 4 2
800.2.d.f 4 120.q odd 4 2
1280.2.c.i 4 48.i odd 4 2
1280.2.c.i 4 240.bm odd 4 2
1280.2.c.k 4 48.k even 4 2
1280.2.c.k 4 240.t even 4 2
1440.2.d.c 4 4.b odd 2 1
1440.2.d.c 4 8.d odd 2 1
1440.2.d.c 4 20.d odd 2 1
1440.2.d.c 4 40.e odd 2 1
1800.2.k.m 4 5.c odd 4 2
1800.2.k.m 4 40.i odd 4 2
6400.2.a.cm 4 240.z odd 4 2
6400.2.a.cm 4 240.bd odd 4 2
6400.2.a.co 4 240.bb even 4 2
6400.2.a.co 4 240.bf even 4 2
7200.2.k.l 4 20.e even 4 2
7200.2.k.l 4 40.k even 4 2

## Hecke kernels

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

 $$T_{7}^{2} + 6$$ $$T_{11}^{2} + 12$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 + 2 T^{2} + T^{4}$$
$7$ $$( 6 + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 24 + T^{2} )^{2}$$
$19$ $$( 12 + T^{2} )^{2}$$
$23$ $$( 6 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$( -72 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -18 + T^{2} )^{2}$$
$47$ $$( 54 + T^{2} )^{2}$$
$53$ $$( -32 + T^{2} )^{2}$$
$59$ $$( 108 + T^{2} )^{2}$$
$61$ $$( 12 + T^{2} )^{2}$$
$67$ $$( -18 + T^{2} )^{2}$$
$71$ $$( -12 + T )^{4}$$
$73$ $$( 24 + T^{2} )^{2}$$
$79$ $$( 4 + T )^{4}$$
$83$ $$( -98 + T^{2} )^{2}$$
$89$ $$( 6 + T )^{4}$$
$97$ $$( 24 + T^{2} )^{2}$$