# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.87461447277$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Defining polynomial: $$x^{4} - 3 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{3} q^{5} + 2 q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{3} q^{5} + 2 q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} + ( 1 - \beta_{2} ) q^{10} -2 \beta_{3} q^{11} + 2 \beta_{1} q^{14} + ( -1 + 3 \beta_{2} ) q^{16} + ( -2 + 4 \beta_{2} ) q^{19} + ( \beta_{1} - 2 \beta_{3} ) q^{20} + ( 2 - 2 \beta_{2} ) q^{22} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{23} - q^{25} + ( 2 + 2 \beta_{2} ) q^{28} -6 \beta_{3} q^{29} + 4 q^{31} + ( -\beta_{1} + 6 \beta_{3} ) q^{32} -2 \beta_{3} q^{35} + ( 4 - 8 \beta_{2} ) q^{37} + ( -2 \beta_{1} + 8 \beta_{3} ) q^{38} + ( 3 - \beta_{2} ) q^{40} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{41} + ( -4 + 8 \beta_{2} ) q^{43} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{44} + ( -6 - 2 \beta_{2} ) q^{46} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{47} -3 q^{49} -\beta_{1} q^{50} + 2 \beta_{3} q^{53} -2 q^{55} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{56} + ( 6 - 6 \beta_{2} ) q^{58} -10 \beta_{3} q^{59} + ( 4 - 8 \beta_{2} ) q^{61} + 4 \beta_{1} q^{62} + ( -7 + 5 \beta_{2} ) q^{64} + ( 2 - 2 \beta_{2} ) q^{70} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{71} -14 q^{73} + ( 4 \beta_{1} - 16 \beta_{3} ) q^{74} + ( -10 + 6 \beta_{2} ) q^{76} -4 \beta_{3} q^{77} + 4 q^{79} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{80} + ( -12 - 4 \beta_{2} ) q^{82} + 12 \beta_{3} q^{83} + ( -4 \beta_{1} + 16 \beta_{3} ) q^{86} + ( 6 - 2 \beta_{2} ) q^{88} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{89} + ( -6 \beta_{1} - 4 \beta_{3} ) q^{92} + ( -6 - 2 \beta_{2} ) q^{94} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{95} + 2 q^{97} -3 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{4} + 8q^{7} + O(q^{10})$$ $$4q + 6q^{4} + 8q^{7} + 2q^{10} + 2q^{16} + 4q^{22} - 4q^{25} + 12q^{28} + 16q^{31} + 10q^{40} - 28q^{46} - 12q^{49} - 8q^{55} + 12q^{58} - 18q^{64} + 4q^{70} - 56q^{73} - 28q^{76} + 16q^{79} - 56q^{82} + 20q^{88} - 28q^{94} + 8q^{97} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 −1.32288 − 0.500000i −1.32288 + 0.500000i 1.32288 − 0.500000i 1.32288 + 0.500000i
−1.32288 0.500000i 0 1.50000 + 1.32288i 1.00000i 0 2.00000 −1.32288 2.50000i 0 0.500000 1.32288i
181.2 −1.32288 + 0.500000i 0 1.50000 1.32288i 1.00000i 0 2.00000 −1.32288 + 2.50000i 0 0.500000 + 1.32288i
181.3 1.32288 0.500000i 0 1.50000 1.32288i 1.00000i 0 2.00000 1.32288 2.50000i 0 0.500000 + 1.32288i
181.4 1.32288 + 0.500000i 0 1.50000 + 1.32288i 1.00000i 0 2.00000 1.32288 + 2.50000i 0 0.500000 1.32288i
 $$n$$: 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
8.b even 2 1 inner
24.h 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.k.d 4
3.b odd 2 1 inner 360.2.k.d 4
4.b odd 2 1 1440.2.k.d 4
5.b even 2 1 1800.2.k.n 4
5.c odd 4 1 1800.2.d.k 4
5.c odd 4 1 1800.2.d.o 4
8.b even 2 1 inner 360.2.k.d 4
8.d odd 2 1 1440.2.k.d 4
12.b even 2 1 1440.2.k.d 4
15.d odd 2 1 1800.2.k.n 4
15.e even 4 1 1800.2.d.k 4
15.e even 4 1 1800.2.d.o 4
20.d odd 2 1 7200.2.k.o 4
20.e even 4 1 7200.2.d.k 4
20.e even 4 1 7200.2.d.l 4
24.f even 2 1 1440.2.k.d 4
24.h odd 2 1 inner 360.2.k.d 4
40.e odd 2 1 7200.2.k.o 4
40.f even 2 1 1800.2.k.n 4
40.i odd 4 1 1800.2.d.k 4
40.i odd 4 1 1800.2.d.o 4
40.k even 4 1 7200.2.d.k 4
40.k even 4 1 7200.2.d.l 4
60.h even 2 1 7200.2.k.o 4
60.l odd 4 1 7200.2.d.k 4
60.l odd 4 1 7200.2.d.l 4
120.i odd 2 1 1800.2.k.n 4
120.m even 2 1 7200.2.k.o 4
120.q odd 4 1 7200.2.d.k 4
120.q odd 4 1 7200.2.d.l 4
120.w even 4 1 1800.2.d.k 4
120.w even 4 1 1800.2.d.o 4

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

## Hecke kernels

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

 $$T_{7} - 2$$ $$T_{11}^{2} + 4$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 3 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( -2 + T )^{4}$$
$11$ $$( 4 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$( 28 + T^{2} )^{2}$$
$23$ $$( -28 + T^{2} )^{2}$$
$29$ $$( 36 + T^{2} )^{2}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$( 112 + T^{2} )^{2}$$
$41$ $$( -112 + T^{2} )^{2}$$
$43$ $$( 112 + T^{2} )^{2}$$
$47$ $$( -28 + T^{2} )^{2}$$
$53$ $$( 4 + T^{2} )^{2}$$
$59$ $$( 100 + T^{2} )^{2}$$
$61$ $$( 112 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$( -112 + T^{2} )^{2}$$
$73$ $$( 14 + T )^{4}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$( 144 + T^{2} )^{2}$$
$89$ $$( -112 + T^{2} )^{2}$$
$97$ $$( -2 + T )^{4}$$