# Properties

 Label 360.1.p.b Level 360 Weight 1 Character orbit 360.p Self dual yes Analytic conductor 0.180 Analytic rank 0 Dimension 1 Projective image $$D_{2}$$ CM/RM discs -15, -40, 24 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$0.179663404548$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{6}, \sqrt{-10})$$ Artin image $D_4$ Artin field Galois closure of 4.0.5400.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} + q^{16} - 2q^{19} - q^{20} - 2q^{23} + q^{25} + q^{32} - 2q^{38} - q^{40} - 2q^{46} + 2q^{47} - q^{49} + q^{50} + 2q^{53} + q^{64} - 2q^{76} - q^{80} - 2q^{92} + 2q^{94} + 2q^{95} - q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0
1.00000 0 1.00000 −1.00000 0 0 1.00000 0 −1.00000
 $$n$$: 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 $$\Q(\sqrt{-15})$$
24.f even 2 1 RM by $$\Q(\sqrt{6})$$
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.1.p.b yes 1
3.b odd 2 1 360.1.p.a 1
4.b odd 2 1 1440.1.p.a 1
5.b even 2 1 360.1.p.a 1
5.c odd 4 2 1800.1.g.c 2
8.b even 2 1 1440.1.p.b 1
8.d odd 2 1 360.1.p.a 1
9.c even 3 2 3240.1.z.d 2
9.d odd 6 2 3240.1.z.f 2
12.b even 2 1 1440.1.p.b 1
15.d odd 2 1 CM 360.1.p.b yes 1
15.e even 4 2 1800.1.g.c 2
20.d odd 2 1 1440.1.p.b 1
24.f even 2 1 RM 360.1.p.b yes 1
24.h odd 2 1 1440.1.p.a 1
40.e odd 2 1 CM 360.1.p.b yes 1
40.f even 2 1 1440.1.p.a 1
40.k even 4 2 1800.1.g.c 2
45.h odd 6 2 3240.1.z.d 2
45.j even 6 2 3240.1.z.f 2
60.h even 2 1 1440.1.p.a 1
72.l even 6 2 3240.1.z.d 2
72.p odd 6 2 3240.1.z.f 2
120.i odd 2 1 1440.1.p.b 1
120.m even 2 1 360.1.p.a 1
120.q odd 4 2 1800.1.g.c 2
360.z odd 6 2 3240.1.z.d 2
360.bd even 6 2 3240.1.z.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.1.p.a 1 3.b odd 2 1
360.1.p.a 1 5.b even 2 1
360.1.p.a 1 8.d odd 2 1
360.1.p.a 1 120.m even 2 1
360.1.p.b yes 1 1.a even 1 1 trivial
360.1.p.b yes 1 15.d odd 2 1 CM
360.1.p.b yes 1 24.f even 2 1 RM
360.1.p.b yes 1 40.e odd 2 1 CM
1440.1.p.a 1 4.b odd 2 1
1440.1.p.a 1 24.h odd 2 1
1440.1.p.a 1 40.f even 2 1
1440.1.p.a 1 60.h even 2 1
1440.1.p.b 1 8.b even 2 1
1440.1.p.b 1 12.b even 2 1
1440.1.p.b 1 20.d odd 2 1
1440.1.p.b 1 120.i odd 2 1
1800.1.g.c 2 5.c odd 4 2
1800.1.g.c 2 15.e even 4 2
1800.1.g.c 2 40.k even 4 2
1800.1.g.c 2 120.q odd 4 2
3240.1.z.d 2 9.c even 3 2
3240.1.z.d 2 45.h odd 6 2
3240.1.z.d 2 72.l even 6 2
3240.1.z.d 2 360.z odd 6 2
3240.1.z.f 2 9.d odd 6 2
3240.1.z.f 2 45.j even 6 2
3240.1.z.f 2 72.p odd 6 2
3240.1.z.f 2 360.bd even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23} + 2$$ acting on $$S_{1}^{\mathrm{new}}(360, [\chi])$$.

## Hecke characteristic polynomials

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