# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.87461447277$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 40) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 - 1) * q^4 + (b3 - b2) * q^5 + (b2 + 2*b1) * q^7 + 2*b2 * q^8 $$q + \beta_1 q^{2} + (\beta_{3} - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{10} - 2 \beta_{3} q^{11} + (\beta_{3} - 3) q^{14} + ( - 2 \beta_{3} - 2) q^{16} + (2 \beta_{2} + 4 \beta_1) q^{17} - 2 \beta_{3} q^{19} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 3) q^{20} + ( - 4 \beta_{2} - 2 \beta_1) q^{22} + (\beta_{2} + 2 \beta_1) q^{23} + (2 \beta_{2} + 4 \beta_1 - 1) q^{25} + (2 \beta_{2} - 2 \beta_1) q^{28} + 4 q^{31} + ( - 4 \beta_{2} - 4 \beta_1) q^{32} + (2 \beta_{3} - 6) q^{34} + (2 \beta_{3} + 3 \beta_{2}) q^{35} - 6 \beta_{2} q^{37} + ( - 4 \beta_{2} - 2 \beta_1) q^{38} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{40} + 3 \beta_{2} q^{43} + (2 \beta_{3} + 6) q^{44} + (\beta_{3} - 3) q^{46} + ( - 3 \beta_{2} - 6 \beta_1) q^{47} + q^{49} + (2 \beta_{3} - \beta_1 - 6) q^{50} - 4 \beta_{2} q^{53} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{55} - 4 \beta_{3} q^{56} - 6 \beta_{3} q^{59} - 2 \beta_{3} q^{61} + 4 \beta_1 q^{62} + 8 q^{64} - 3 \beta_{2} q^{67} + (4 \beta_{2} - 4 \beta_1) q^{68} + ( - 3 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 3) q^{70} + 12 q^{71} + ( - 2 \beta_{2} - 4 \beta_1) q^{73} + (6 \beta_{3} + 6) q^{74} + (2 \beta_{3} + 6) q^{76} - 6 \beta_{2} q^{77} - 4 q^{79} + ( - 2 \beta_{3} - 4 \beta_1 + 6) q^{80} + 7 \beta_{2} q^{83} + (4 \beta_{3} + 6 \beta_{2}) q^{85} + ( - 3 \beta_{3} - 3) q^{86} + (4 \beta_{2} + 8 \beta_1) q^{88} - 6 q^{89} + (2 \beta_{2} - 2 \beta_1) q^{92} + ( - 3 \beta_{3} + 9) q^{94} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{95} + (2 \beta_{2} + 4 \beta_1) q^{97} + \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 - 1) * q^4 + (b3 - b2) * q^5 + (b2 + 2*b1) * q^7 + 2*b2 * q^8 + (b3 + 2*b2 + b1 + 1) * q^10 - 2*b3 * q^11 + (b3 - 3) * q^14 + (-2*b3 - 2) * q^16 + (2*b2 + 4*b1) * q^17 - 2*b3 * q^19 + (-b3 + 2*b2 + 2*b1 - 3) * q^20 + (-4*b2 - 2*b1) * q^22 + (b2 + 2*b1) * q^23 + (2*b2 + 4*b1 - 1) * q^25 + (2*b2 - 2*b1) * q^28 + 4 * q^31 + (-4*b2 - 4*b1) * q^32 + (2*b3 - 6) * q^34 + (2*b3 + 3*b2) * q^35 - 6*b2 * q^37 + (-4*b2 - 2*b1) * q^38 + (-2*b2 - 4*b1 - 4) * q^40 + 3*b2 * q^43 + (2*b3 + 6) * q^44 + (b3 - 3) * q^46 + (-3*b2 - 6*b1) * q^47 + q^49 + (2*b3 - b1 - 6) * q^50 - 4*b2 * q^53 + (-2*b2 - 4*b1 + 6) * q^55 - 4*b3 * q^56 - 6*b3 * q^59 - 2*b3 * q^61 + 4*b1 * q^62 + 8 * q^64 - 3*b2 * q^67 + (4*b2 - 4*b1) * q^68 + (-3*b3 + 4*b2 + 2*b1 - 3) * q^70 + 12 * q^71 + (-2*b2 - 4*b1) * q^73 + (6*b3 + 6) * q^74 + (2*b3 + 6) * q^76 - 6*b2 * q^77 - 4 * q^79 + (-2*b3 - 4*b1 + 6) * q^80 + 7*b2 * q^83 + (4*b3 + 6*b2) * q^85 + (-3*b3 - 3) * q^86 + (4*b2 + 8*b1) * q^88 - 6 * q^89 + (2*b2 - 2*b1) * q^92 + (-3*b3 + 9) * q^94 + (-2*b2 - 4*b1 + 6) * q^95 + (2*b2 + 4*b1) * q^97 + b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} + 4 q^{10} - 12 q^{14} - 8 q^{16} - 12 q^{20} - 4 q^{25} + 16 q^{31} - 24 q^{34} - 16 q^{40} + 24 q^{44} - 12 q^{46} + 4 q^{49} - 24 q^{50} + 24 q^{55} + 32 q^{64} - 12 q^{70} + 48 q^{71} + 24 q^{74} + 24 q^{76} - 16 q^{79} + 24 q^{80} - 12 q^{86} - 24 q^{89} + 36 q^{94} + 24 q^{95}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^10 - 12 * q^14 - 8 * q^16 - 12 * q^20 - 4 * q^25 + 16 * q^31 - 24 * q^34 - 16 * q^40 + 24 * q^44 - 12 * q^46 + 4 * q^49 - 24 * q^50 + 24 * q^55 + 32 * q^64 - 12 * q^70 + 48 * q^71 + 24 * q^74 + 24 * q^76 - 16 * q^79 + 24 * q^80 - 12 * q^86 - 24 * q^89 + 36 * q^94 + 24 * q^95

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 1$$ b3 - 1 $$\nu^{3}$$ $$=$$ $$2\beta_{2}$$ 2*b2

## 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
 −0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i 0.707107 + 1.22474i
−0.707107 1.22474i 0 −1.00000 + 1.73205i −1.41421 + 1.73205i 0 2.44949i 2.82843 0 3.12132 + 0.507306i
109.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i −1.41421 1.73205i 0 2.44949i 2.82843 0 3.12132 0.507306i
109.3 0.707107 1.22474i 0 −1.00000 1.73205i 1.41421 1.73205i 0 2.44949i −2.82843 0 −1.12132 2.95680i
109.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 1.41421 + 1.73205i 0 2.44949i −2.82843 0 −1.12132 + 2.95680i
 $$n$$: 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
5.b even 2 1 inner
8.b even 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.b 4
3.b odd 2 1 40.2.f.a 4
4.b odd 2 1 1440.2.d.c 4
5.b even 2 1 inner 360.2.d.b 4
5.c odd 4 2 1800.2.k.m 4
8.b even 2 1 inner 360.2.d.b 4
8.d odd 2 1 1440.2.d.c 4
12.b even 2 1 160.2.f.a 4
15.d odd 2 1 40.2.f.a 4
15.e even 4 2 200.2.d.e 4
20.d odd 2 1 1440.2.d.c 4
20.e even 4 2 7200.2.k.l 4
24.f even 2 1 160.2.f.a 4
24.h odd 2 1 40.2.f.a 4
40.e odd 2 1 1440.2.d.c 4
40.f even 2 1 inner 360.2.d.b 4
40.i odd 4 2 1800.2.k.m 4
40.k even 4 2 7200.2.k.l 4
48.i odd 4 2 1280.2.c.i 4
48.k even 4 2 1280.2.c.k 4
60.h even 2 1 160.2.f.a 4
60.l odd 4 2 800.2.d.f 4
120.i odd 2 1 40.2.f.a 4
120.m even 2 1 160.2.f.a 4
120.q odd 4 2 800.2.d.f 4
120.w even 4 2 200.2.d.e 4
240.t even 4 2 1280.2.c.k 4
240.z odd 4 2 6400.2.a.cm 4
240.bb even 4 2 6400.2.a.co 4
240.bd odd 4 2 6400.2.a.cm 4
240.bf even 4 2 6400.2.a.co 4
240.bm odd 4 2 1280.2.c.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.f.a 4 3.b odd 2 1
40.2.f.a 4 15.d odd 2 1
40.2.f.a 4 24.h odd 2 1
40.2.f.a 4 120.i odd 2 1
160.2.f.a 4 12.b even 2 1
160.2.f.a 4 24.f even 2 1
160.2.f.a 4 60.h even 2 1
160.2.f.a 4 120.m even 2 1
200.2.d.e 4 15.e even 4 2
200.2.d.e 4 120.w even 4 2
360.2.d.b 4 1.a even 1 1 trivial
360.2.d.b 4 5.b even 2 1 inner
360.2.d.b 4 8.b even 2 1 inner
360.2.d.b 4 40.f even 2 1 inner
800.2.d.f 4 60.l odd 4 2
800.2.d.f 4 120.q odd 4 2
1280.2.c.i 4 48.i odd 4 2
1280.2.c.i 4 240.bm odd 4 2
1280.2.c.k 4 48.k even 4 2
1280.2.c.k 4 240.t even 4 2
1440.2.d.c 4 4.b odd 2 1
1440.2.d.c 4 8.d odd 2 1
1440.2.d.c 4 20.d odd 2 1
1440.2.d.c 4 40.e odd 2 1
1800.2.k.m 4 5.c odd 4 2
1800.2.k.m 4 40.i odd 4 2
6400.2.a.cm 4 240.z odd 4 2
6400.2.a.cm 4 240.bd odd 4 2
6400.2.a.co 4 240.bb even 4 2
6400.2.a.co 4 240.bf even 4 2
7200.2.k.l 4 20.e even 4 2
7200.2.k.l 4 40.k even 4 2

## Hecke kernels

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

 $$T_{7}^{2} + 6$$ T7^2 + 6 $$T_{11}^{2} + 12$$ T11^2 + 12 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 2T^{2} + 25$$
$7$ $$(T^{2} + 6)^{2}$$
$11$ $$(T^{2} + 12)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 24)^{2}$$
$19$ $$(T^{2} + 12)^{2}$$
$23$ $$(T^{2} + 6)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T - 4)^{4}$$
$37$ $$(T^{2} - 72)^{2}$$
$41$ $$T^{4}$$
$43$ $$(T^{2} - 18)^{2}$$
$47$ $$(T^{2} + 54)^{2}$$
$53$ $$(T^{2} - 32)^{2}$$
$59$ $$(T^{2} + 108)^{2}$$
$61$ $$(T^{2} + 12)^{2}$$
$67$ $$(T^{2} - 18)^{2}$$
$71$ $$(T - 12)^{4}$$
$73$ $$(T^{2} + 24)^{2}$$
$79$ $$(T + 4)^{4}$$
$83$ $$(T^{2} - 98)^{2}$$
$89$ $$(T + 6)^{4}$$
$97$ $$(T^{2} + 24)^{2}$$