# Properties

 Label 360.2.d.a Level $360$ Weight $2$ Character orbit 360.d Analytic conductor $2.875$ Analytic rank $0$ Dimension $4$ CM discriminant -120 Inner twists $8$

# 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: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -2 q^{4} + \beta_{2} q^{5} + 2 \beta_{1} q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} -2 q^{4} + \beta_{2} q^{5} + 2 \beta_{1} q^{8} + \beta_{3} q^{10} + 2 \beta_{2} q^{11} + 2 \beta_{3} q^{13} + 4 q^{16} -2 \beta_{1} q^{17} -2 \beta_{2} q^{20} + 2 \beta_{3} q^{22} + 4 \beta_{1} q^{23} -5 q^{25} -4 \beta_{2} q^{26} + 2 \beta_{2} q^{29} + 2 q^{31} -4 \beta_{1} q^{32} -4 q^{34} + 2 \beta_{3} q^{37} -2 \beta_{3} q^{40} -4 \beta_{3} q^{43} -4 \beta_{2} q^{44} + 8 q^{46} -8 \beta_{1} q^{47} + 7 q^{49} + 5 \beta_{1} q^{50} -4 \beta_{3} q^{52} -10 q^{55} + 2 \beta_{3} q^{58} + 2 \beta_{2} q^{59} -2 \beta_{1} q^{62} -8 q^{64} + 10 \beta_{1} q^{65} -4 \beta_{3} q^{67} + 4 \beta_{1} q^{68} -4 \beta_{2} q^{74} + 14 q^{79} + 4 \beta_{2} q^{80} + 2 \beta_{3} q^{85} + 8 \beta_{2} q^{86} -4 \beta_{3} q^{88} -8 \beta_{1} q^{92} -16 q^{94} -7 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + O(q^{10})$$ $$4q - 8q^{4} + 16q^{16} - 20q^{25} + 8q^{31} - 16q^{34} + 32q^{46} + 28q^{49} - 40q^{55} - 32q^{64} + 56q^{79} - 64q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 7 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} + 7 \beta_{1}$$$$)/2$$

## 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
 −1.58114 + 0.707107i 1.58114 + 0.707107i 1.58114 − 0.707107i −1.58114 − 0.707107i
1.41421i 0 −2.00000 2.23607i 0 0 2.82843i 0 −3.16228
109.2 1.41421i 0 −2.00000 2.23607i 0 0 2.82843i 0 3.16228
109.3 1.41421i 0 −2.00000 2.23607i 0 0 2.82843i 0 3.16228
109.4 1.41421i 0 −2.00000 2.23607i 0 0 2.82843i 0 −3.16228
 $$n$$: 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
120.i odd 2 1 CM by $$\Q(\sqrt{-30})$$
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
24.h odd 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.a 4
3.b odd 2 1 inner 360.2.d.a 4
4.b odd 2 1 1440.2.d.a 4
5.b even 2 1 inner 360.2.d.a 4
5.c odd 4 2 1800.2.k.o 4
8.b even 2 1 inner 360.2.d.a 4
8.d odd 2 1 1440.2.d.a 4
12.b even 2 1 1440.2.d.a 4
15.d odd 2 1 inner 360.2.d.a 4
15.e even 4 2 1800.2.k.o 4
20.d odd 2 1 1440.2.d.a 4
20.e even 4 2 7200.2.k.m 4
24.f even 2 1 1440.2.d.a 4
24.h odd 2 1 inner 360.2.d.a 4
40.e odd 2 1 1440.2.d.a 4
40.f even 2 1 inner 360.2.d.a 4
40.i odd 4 2 1800.2.k.o 4
40.k even 4 2 7200.2.k.m 4
60.h even 2 1 1440.2.d.a 4
60.l odd 4 2 7200.2.k.m 4
120.i odd 2 1 CM 360.2.d.a 4
120.m even 2 1 1440.2.d.a 4
120.q odd 4 2 7200.2.k.m 4
120.w even 4 2 1800.2.k.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.d.a 4 1.a even 1 1 trivial
360.2.d.a 4 3.b odd 2 1 inner
360.2.d.a 4 5.b even 2 1 inner
360.2.d.a 4 8.b even 2 1 inner
360.2.d.a 4 15.d odd 2 1 inner
360.2.d.a 4 24.h odd 2 1 inner
360.2.d.a 4 40.f even 2 1 inner
360.2.d.a 4 120.i odd 2 1 CM
1440.2.d.a 4 4.b odd 2 1
1440.2.d.a 4 8.d odd 2 1
1440.2.d.a 4 12.b even 2 1
1440.2.d.a 4 20.d odd 2 1
1440.2.d.a 4 24.f even 2 1
1440.2.d.a 4 40.e odd 2 1
1440.2.d.a 4 60.h even 2 1
1440.2.d.a 4 120.m even 2 1
1800.2.k.o 4 5.c odd 4 2
1800.2.k.o 4 15.e even 4 2
1800.2.k.o 4 40.i odd 4 2
1800.2.k.o 4 120.w even 4 2
7200.2.k.m 4 20.e even 4 2
7200.2.k.m 4 40.k even 4 2
7200.2.k.m 4 60.l odd 4 2
7200.2.k.m 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 $$S_{2}^{\mathrm{new}}(360, [\chi])$$:

 $$T_{7}$$ $$T_{11}^{2} + 20$$ $$T_{13}^{2} - 40$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( 20 + T^{2} )^{2}$$
$13$ $$( -40 + T^{2} )^{2}$$
$17$ $$( 8 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( 32 + T^{2} )^{2}$$
$29$ $$( 20 + T^{2} )^{2}$$
$31$ $$( -2 + T )^{4}$$
$37$ $$( -40 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -160 + T^{2} )^{2}$$
$47$ $$( 128 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$( 20 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( -160 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$( -14 + T )^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$