# Properties

 Label 360.2.d.c Level $360$ Weight $2$ Character orbit 360.d Analytic conductor $2.875$ Analytic rank $0$ Dimension $4$ CM discriminant -15 Inner twists $8$

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## 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: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + \beta_{3} q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + \beta_{3} q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{8} + ( 2 + \beta_{3} ) q^{10} + ( -4 + \beta_{3} ) q^{16} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{17} + ( -2 + 4 \beta_{3} ) q^{19} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{20} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{23} + 5 q^{25} -8 q^{31} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{32} + ( 8 - 4 \beta_{3} ) q^{34} + ( 8 \beta_{1} - 2 \beta_{2} ) q^{38} + ( -4 + 3 \beta_{3} ) q^{40} + ( 4 - 2 \beta_{3} ) q^{46} + ( -6 \beta_{1} + 6 \beta_{2} ) q^{47} + 7 q^{49} -5 \beta_{2} q^{50} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 4 - 8 \beta_{3} ) q^{61} + 8 \beta_{2} q^{62} + ( -4 - 3 \beta_{3} ) q^{64} + ( -8 \beta_{1} - 4 \beta_{2} ) q^{68} + ( -16 + 2 \beta_{3} ) q^{76} -16 q^{79} + ( 6 \beta_{1} + \beta_{2} ) q^{80} + ( -8 \beta_{1} - 8 \beta_{2} ) q^{83} + ( 4 - 8 \beta_{3} ) q^{85} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{92} + ( 12 - 6 \beta_{3} ) q^{94} + ( 10 \beta_{1} - 10 \beta_{2} ) q^{95} -7 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + O(q^{10})$$ $$4q + 2q^{4} + 10q^{10} - 14q^{16} + 20q^{25} - 32q^{31} + 24q^{34} - 10q^{40} + 12q^{46} + 28q^{49} - 22q^{64} - 60q^{76} - 64q^{79} + 36q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

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

## 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
 −0.309017 + 0.535233i −0.309017 − 0.535233i 0.809017 − 1.40126i 0.809017 + 1.40126i
−1.11803 0.866025i 0 0.500000 + 1.93649i −2.23607 0 0 1.11803 2.59808i 0 2.50000 + 1.93649i
109.2 −1.11803 + 0.866025i 0 0.500000 1.93649i −2.23607 0 0 1.11803 + 2.59808i 0 2.50000 1.93649i
109.3 1.11803 0.866025i 0 0.500000 1.93649i 2.23607 0 0 −1.11803 2.59808i 0 2.50000 1.93649i
109.4 1.11803 + 0.866025i 0 0.500000 + 1.93649i 2.23607 0 0 −1.11803 + 2.59808i 0 2.50000 + 1.93649i
 $$n$$: 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner
40.f even 2 1 inner
120.i odd 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.c 4
3.b odd 2 1 inner 360.2.d.c 4
4.b odd 2 1 1440.2.d.d 4
5.b even 2 1 inner 360.2.d.c 4
5.c odd 4 2 1800.2.k.l 4
8.b even 2 1 inner 360.2.d.c 4
8.d odd 2 1 1440.2.d.d 4
12.b even 2 1 1440.2.d.d 4
15.d odd 2 1 CM 360.2.d.c 4
15.e even 4 2 1800.2.k.l 4
20.d odd 2 1 1440.2.d.d 4
20.e even 4 2 7200.2.k.n 4
24.f even 2 1 1440.2.d.d 4
24.h odd 2 1 inner 360.2.d.c 4
40.e odd 2 1 1440.2.d.d 4
40.f even 2 1 inner 360.2.d.c 4
40.i odd 4 2 1800.2.k.l 4
40.k even 4 2 7200.2.k.n 4
60.h even 2 1 1440.2.d.d 4
60.l odd 4 2 7200.2.k.n 4
120.i odd 2 1 inner 360.2.d.c 4
120.m even 2 1 1440.2.d.d 4
120.q odd 4 2 7200.2.k.n 4
120.w even 4 2 1800.2.k.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.d.c 4 1.a even 1 1 trivial
360.2.d.c 4 3.b odd 2 1 inner
360.2.d.c 4 5.b even 2 1 inner
360.2.d.c 4 8.b even 2 1 inner
360.2.d.c 4 15.d odd 2 1 CM
360.2.d.c 4 24.h odd 2 1 inner
360.2.d.c 4 40.f even 2 1 inner
360.2.d.c 4 120.i odd 2 1 inner
1440.2.d.d 4 4.b odd 2 1
1440.2.d.d 4 8.d odd 2 1
1440.2.d.d 4 12.b even 2 1
1440.2.d.d 4 20.d odd 2 1
1440.2.d.d 4 24.f even 2 1
1440.2.d.d 4 40.e odd 2 1
1440.2.d.d 4 60.h even 2 1
1440.2.d.d 4 120.m even 2 1
1800.2.k.l 4 5.c odd 4 2
1800.2.k.l 4 15.e even 4 2
1800.2.k.l 4 40.i odd 4 2
1800.2.k.l 4 120.w even 4 2
7200.2.k.n 4 20.e even 4 2
7200.2.k.n 4 40.k even 4 2
7200.2.k.n 4 60.l odd 4 2
7200.2.k.n 4 120.q odd 4 2

## Hecke kernels

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

 $$T_{7}$$ $$T_{11}$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$( 48 + T^{2} )^{2}$$
$19$ $$( 60 + T^{2} )^{2}$$
$23$ $$( 12 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 8 + T )^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$( 108 + T^{2} )^{2}$$
$53$ $$( -20 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 240 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 16 + T )^{4}$$
$83$ $$( -320 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$
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