# Properties

 Label 360.2.b.a Level $360$ Weight $2$ Character orbit 360.b Analytic conductor $2.875$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 360.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.87461447277$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -2 q^{4} - q^{5} -3 \beta q^{7} -2 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} -2 q^{4} - q^{5} -3 \beta q^{7} -2 \beta q^{8} -\beta q^{10} -\beta q^{11} -3 \beta q^{13} + 6 q^{14} + 4 q^{16} + 2 \beta q^{17} -4 q^{19} + 2 q^{20} + 2 q^{22} + 6 q^{23} + q^{25} + 6 q^{26} + 6 \beta q^{28} -6 q^{29} -6 \beta q^{31} + 4 \beta q^{32} -4 q^{34} + 3 \beta q^{35} + 3 \beta q^{37} -4 \beta q^{38} + 2 \beta q^{40} -7 \beta q^{41} + 8 q^{43} + 2 \beta q^{44} + 6 \beta q^{46} -11 q^{49} + \beta q^{50} + 6 \beta q^{52} -12 q^{53} + \beta q^{55} -12 q^{56} -6 \beta q^{58} -\beta q^{59} + 6 \beta q^{61} + 12 q^{62} -8 q^{64} + 3 \beta q^{65} + 8 q^{67} -4 \beta q^{68} -6 q^{70} + 14 q^{73} -6 q^{74} + 8 q^{76} -6 q^{77} -6 \beta q^{79} -4 q^{80} + 14 q^{82} + 2 \beta q^{83} -2 \beta q^{85} + 8 \beta q^{86} -4 q^{88} + 5 \beta q^{89} -18 q^{91} -12 q^{92} + 4 q^{95} -10 q^{97} -11 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} - 2q^{5} + O(q^{10})$$ $$2q - 4q^{4} - 2q^{5} + 12q^{14} + 8q^{16} - 8q^{19} + 4q^{20} + 4q^{22} + 12q^{23} + 2q^{25} + 12q^{26} - 12q^{29} - 8q^{34} + 16q^{43} - 22q^{49} - 24q^{53} - 24q^{56} + 24q^{62} - 16q^{64} + 16q^{67} - 12q^{70} + 28q^{73} - 12q^{74} + 16q^{76} - 12q^{77} - 8q^{80} + 28q^{82} - 8q^{88} - 36q^{91} - 24q^{92} + 8q^{95} - 20q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 − 1.41421i 1.41421i
1.41421i 0 −2.00000 −1.00000 0 4.24264i 2.82843i 0 1.41421i
251.2 1.41421i 0 −2.00000 −1.00000 0 4.24264i 2.82843i 0 1.41421i
 $$n$$: 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
24.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.b.a 2
3.b odd 2 1 360.2.b.b yes 2
4.b odd 2 1 1440.2.b.a 2
5.b even 2 1 1800.2.b.b 2
5.c odd 4 2 1800.2.m.a 4
8.b even 2 1 1440.2.b.b 2
8.d odd 2 1 360.2.b.b yes 2
12.b even 2 1 1440.2.b.b 2
15.d odd 2 1 1800.2.b.a 2
15.e even 4 2 1800.2.m.b 4
20.d odd 2 1 7200.2.b.b 2
20.e even 4 2 7200.2.m.b 4
24.f even 2 1 inner 360.2.b.a 2
24.h odd 2 1 1440.2.b.a 2
40.e odd 2 1 1800.2.b.a 2
40.f even 2 1 7200.2.b.a 2
40.i odd 4 2 7200.2.m.a 4
40.k even 4 2 1800.2.m.b 4
60.h even 2 1 7200.2.b.a 2
60.l odd 4 2 7200.2.m.a 4
120.i odd 2 1 7200.2.b.b 2
120.m even 2 1 1800.2.b.b 2
120.q odd 4 2 1800.2.m.a 4
120.w even 4 2 7200.2.m.b 4

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

## Hecke kernels

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

 $$T_{7}^{2} + 18$$ $$T_{23} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$18 + T^{2}$$
$11$ $$2 + T^{2}$$
$13$ $$18 + T^{2}$$
$17$ $$8 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$72 + T^{2}$$
$37$ $$18 + T^{2}$$
$41$ $$98 + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$( 12 + T )^{2}$$
$59$ $$2 + T^{2}$$
$61$ $$72 + T^{2}$$
$67$ $$( -8 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$( -14 + T )^{2}$$
$79$ $$72 + T^{2}$$
$83$ $$8 + T^{2}$$
$89$ $$50 + T^{2}$$
$97$ $$( 10 + T )^{2}$$