# Properties

 Label 5400.2.a.bu Level $5400$ Weight $2$ Character orbit 5400.a Self dual yes Analytic conductor $43.119$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5400 = 2^{3} \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.1192170915$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4 q^{7}+O(q^{10})$$ q + 4 * q^7 $$q + 4 q^{7} + 2 q^{11} - 4 q^{13} - q^{17} - 5 q^{19} - 5 q^{23} + 8 q^{29} + 7 q^{31} + 6 q^{37} + 6 q^{41} + 2 q^{43} - 8 q^{47} + 9 q^{49} - 9 q^{53} + 4 q^{59} + 13 q^{61} + 10 q^{67} - 6 q^{71} + 6 q^{73} + 8 q^{77} + 9 q^{79} + 17 q^{83} - 6 q^{89} - 16 q^{91} + 8 q^{97}+O(q^{100})$$ q + 4 * q^7 + 2 * q^11 - 4 * q^13 - q^17 - 5 * q^19 - 5 * q^23 + 8 * q^29 + 7 * q^31 + 6 * q^37 + 6 * q^41 + 2 * q^43 - 8 * q^47 + 9 * q^49 - 9 * q^53 + 4 * q^59 + 13 * q^61 + 10 * q^67 - 6 * q^71 + 6 * q^73 + 8 * q^77 + 9 * q^79 + 17 * q^83 - 6 * q^89 - 16 * q^91 + 8 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 4.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5400.2.a.bu 1
3.b odd 2 1 5400.2.a.br 1
5.b even 2 1 1080.2.a.a 1
5.c odd 4 2 5400.2.f.s 2
15.d odd 2 1 1080.2.a.g yes 1
15.e even 4 2 5400.2.f.l 2
20.d odd 2 1 2160.2.a.l 1
40.e odd 2 1 8640.2.a.cg 1
40.f even 2 1 8640.2.a.bf 1
45.h odd 6 2 3240.2.q.l 2
45.j even 6 2 3240.2.q.w 2
60.h even 2 1 2160.2.a.w 1
120.i odd 2 1 8640.2.a.a 1
120.m even 2 1 8640.2.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.a 1 5.b even 2 1
1080.2.a.g yes 1 15.d odd 2 1
2160.2.a.l 1 20.d odd 2 1
2160.2.a.w 1 60.h even 2 1
3240.2.q.l 2 45.h odd 6 2
3240.2.q.w 2 45.j even 6 2
5400.2.a.br 1 3.b odd 2 1
5400.2.a.bu 1 1.a even 1 1 trivial
5400.2.f.l 2 15.e even 4 2
5400.2.f.s 2 5.c odd 4 2
8640.2.a.a 1 120.i odd 2 1
8640.2.a.bd 1 120.m even 2 1
8640.2.a.bf 1 40.f even 2 1
8640.2.a.cg 1 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5400))$$:

 $$T_{7} - 4$$ T7 - 4 $$T_{11} - 2$$ T11 - 2 $$T_{13} + 4$$ T13 + 4 $$T_{17} + 1$$ T17 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 4$$
$11$ $$T - 2$$
$13$ $$T + 4$$
$17$ $$T + 1$$
$19$ $$T + 5$$
$23$ $$T + 5$$
$29$ $$T - 8$$
$31$ $$T - 7$$
$37$ $$T - 6$$
$41$ $$T - 6$$
$43$ $$T - 2$$
$47$ $$T + 8$$
$53$ $$T + 9$$
$59$ $$T - 4$$
$61$ $$T - 13$$
$67$ $$T - 10$$
$71$ $$T + 6$$
$73$ $$T - 6$$
$79$ $$T - 9$$
$83$ $$T - 17$$
$89$ $$T + 6$$
$97$ $$T - 8$$