# Properties

 Label 360.1.u.a Level $360$ Weight $1$ Character orbit 360.u Analytic conductor $0.180$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 360.u (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.179663404548$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.9000.2

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{5} + ( - \zeta_{8}^{2} - 1) q^{7} - \zeta_{8} q^{8} +O(q^{10})$$ q - z^3 * q^2 - z^2 * q^4 + z * q^5 + (-z^2 - 1) * q^7 - z * q^8 $$q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{5} + ( - \zeta_{8}^{2} - 1) q^{7} - \zeta_{8} q^{8} + q^{10} + (\zeta_{8}^{3} + \zeta_{8}) q^{11} + (\zeta_{8}^{3} - \zeta_{8}) q^{14} - q^{16} - \zeta_{8}^{3} q^{20} + (\zeta_{8}^{2} + 1) q^{22} + \zeta_{8}^{2} q^{25} + (\zeta_{8}^{2} - 1) q^{28} + (\zeta_{8}^{3} - \zeta_{8}) q^{29} + \zeta_{8}^{3} q^{32} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{35} - \zeta_{8}^{2} q^{40} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{44} + \zeta_{8}^{2} q^{49} + \zeta_{8} q^{50} + (\zeta_{8}^{2} - 1) q^{55} + (\zeta_{8}^{3} + \zeta_{8}) q^{56} + (\zeta_{8}^{2} - 1) q^{58} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{59} + \zeta_{8}^{2} q^{64} + ( - \zeta_{8}^{2} - 1) q^{70} + ( - \zeta_{8}^{2} + 1) q^{73} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{77} - \zeta_{8} q^{80} - \zeta_{8} q^{83} + ( - \zeta_{8}^{2} + 1) q^{88} + ( - \zeta_{8}^{2} - 1) q^{97} + \zeta_{8} q^{98} +O(q^{100})$$ q - z^3 * q^2 - z^2 * q^4 + z * q^5 + (-z^2 - 1) * q^7 - z * q^8 + q^10 + (z^3 + z) * q^11 + (z^3 - z) * q^14 - q^16 - z^3 * q^20 + (z^2 + 1) * q^22 + z^2 * q^25 + (z^2 - 1) * q^28 + (z^3 - z) * q^29 + z^3 * q^32 + (-z^3 - z) * q^35 - z^2 * q^40 + (-z^3 + z) * q^44 + z^2 * q^49 + z * q^50 + (z^2 - 1) * q^55 + (z^3 + z) * q^56 + (z^2 - 1) * q^58 + (-z^3 + z) * q^59 + z^2 * q^64 + (-z^2 - 1) * q^70 + (-z^2 + 1) * q^73 + (-2*z^3 + z) * q^77 - z * q^80 - z * q^83 + (-z^2 + 1) * q^88 + (-z^2 - 1) * q^97 + z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{7}+O(q^{10})$$ 4 * q - 4 * q^7 $$4 q - 4 q^{7} + 4 q^{10} - 4 q^{16} + 4 q^{22} - 4 q^{28} - 4 q^{55} - 4 q^{58} - 4 q^{70} + 4 q^{73} + 4 q^{88} - 4 q^{97}+O(q^{100})$$ 4 * q - 4 * q^7 + 4 * q^10 - 4 * q^16 + 4 * q^22 - 4 * q^28 - 4 * q^55 - 4 * q^58 - 4 * q^70 + 4 * q^73 + 4 * q^88 - 4 * q^97

## Character values

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

 $$n$$ $$181$$ $$217$$ $$271$$ $$281$$ $$\chi(n)$$ $$-1$$ $$\zeta_{8}^{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
−0.707107 + 0.707107i 0 1.00000i −0.707107 0.707107i 0 −1.00000 1.00000i 0.707107 + 0.707107i 0 1.00000
37.2 0.707107 0.707107i 0 1.00000i 0.707107 + 0.707107i 0 −1.00000 1.00000i −0.707107 0.707107i 0 1.00000
253.1 −0.707107 0.707107i 0 1.00000i −0.707107 + 0.707107i 0 −1.00000 + 1.00000i 0.707107 0.707107i 0 1.00000
253.2 0.707107 + 0.707107i 0 1.00000i 0.707107 0.707107i 0 −1.00000 + 1.00000i −0.707107 + 0.707107i 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
5.c odd 4 1 inner
8.b even 2 1 inner
15.e even 4 1 inner
40.i odd 4 1 inner
120.w even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.1.u.a 4
3.b odd 2 1 inner 360.1.u.a 4
4.b odd 2 1 1440.1.y.a 4
5.b even 2 1 1800.1.u.b 4
5.c odd 4 1 inner 360.1.u.a 4
5.c odd 4 1 1800.1.u.b 4
8.b even 2 1 inner 360.1.u.a 4
8.d odd 2 1 1440.1.y.a 4
9.c even 3 2 3240.1.bv.c 8
9.d odd 6 2 3240.1.bv.c 8
12.b even 2 1 1440.1.y.a 4
15.d odd 2 1 1800.1.u.b 4
15.e even 4 1 inner 360.1.u.a 4
15.e even 4 1 1800.1.u.b 4
20.e even 4 1 1440.1.y.a 4
24.f even 2 1 1440.1.y.a 4
24.h odd 2 1 CM 360.1.u.a 4
40.f even 2 1 1800.1.u.b 4
40.i odd 4 1 inner 360.1.u.a 4
40.i odd 4 1 1800.1.u.b 4
40.k even 4 1 1440.1.y.a 4
45.k odd 12 2 3240.1.bv.c 8
45.l even 12 2 3240.1.bv.c 8
60.l odd 4 1 1440.1.y.a 4
72.j odd 6 2 3240.1.bv.c 8
72.n even 6 2 3240.1.bv.c 8
120.i odd 2 1 1800.1.u.b 4
120.q odd 4 1 1440.1.y.a 4
120.w even 4 1 inner 360.1.u.a 4
120.w even 4 1 1800.1.u.b 4
360.br even 12 2 3240.1.bv.c 8
360.bu odd 12 2 3240.1.bv.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.1.u.a 4 1.a even 1 1 trivial
360.1.u.a 4 3.b odd 2 1 inner
360.1.u.a 4 5.c odd 4 1 inner
360.1.u.a 4 8.b even 2 1 inner
360.1.u.a 4 15.e even 4 1 inner
360.1.u.a 4 24.h odd 2 1 CM
360.1.u.a 4 40.i odd 4 1 inner
360.1.u.a 4 120.w even 4 1 inner
1440.1.y.a 4 4.b odd 2 1
1440.1.y.a 4 8.d odd 2 1
1440.1.y.a 4 12.b even 2 1
1440.1.y.a 4 20.e even 4 1
1440.1.y.a 4 24.f even 2 1
1440.1.y.a 4 40.k even 4 1
1440.1.y.a 4 60.l odd 4 1
1440.1.y.a 4 120.q odd 4 1
1800.1.u.b 4 5.b even 2 1
1800.1.u.b 4 5.c odd 4 1
1800.1.u.b 4 15.d odd 2 1
1800.1.u.b 4 15.e even 4 1
1800.1.u.b 4 40.f even 2 1
1800.1.u.b 4 40.i odd 4 1
1800.1.u.b 4 120.i odd 2 1
1800.1.u.b 4 120.w even 4 1
3240.1.bv.c 8 9.c even 3 2
3240.1.bv.c 8 9.d odd 6 2
3240.1.bv.c 8 45.k odd 12 2
3240.1.bv.c 8 45.l even 12 2
3240.1.bv.c 8 72.j odd 6 2
3240.1.bv.c 8 72.n even 6 2
3240.1.bv.c 8 360.br even 12 2
3240.1.bv.c 8 360.bu odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(360, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 1$$
$7$ $$(T^{2} + 2 T + 2)^{2}$$
$11$ $$(T^{2} + 2)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$(T^{2} - 2)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} - 2)^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - 2 T + 2)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 16$$
$89$ $$T^{4}$$
$97$ $$(T^{2} + 2 T + 2)^{2}$$