# Properties

 Label 216.2.a.d Level $216$ Weight $2$ Character orbit 216.a Self dual yes Analytic conductor $1.725$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 216.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.72476868366$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{5} - 3q^{7} + O(q^{10})$$ $$q + 4q^{5} - 3q^{7} + 4q^{11} + q^{13} - 4q^{17} - q^{19} + 4q^{23} + 11q^{25} - 4q^{31} - 12q^{35} - 9q^{37} - 8q^{43} - 12q^{47} + 2q^{49} - 8q^{53} + 16q^{55} + 4q^{59} - 5q^{61} + 4q^{65} + 11q^{67} + 8q^{71} + q^{73} - 12q^{77} - 5q^{79} + 8q^{83} - 16q^{85} + 12q^{89} - 3q^{91} - 4q^{95} + 5q^{97} + O(q^{100})$$

## 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 4.00000 0 −3.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$

## 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 216.2.a.d yes 1
3.b odd 2 1 216.2.a.a 1
4.b odd 2 1 432.2.a.h 1
5.b even 2 1 5400.2.a.bp 1
5.c odd 4 2 5400.2.f.v 2
8.b even 2 1 1728.2.a.a 1
8.d odd 2 1 1728.2.a.b 1
9.c even 3 2 648.2.i.a 2
9.d odd 6 2 648.2.i.h 2
12.b even 2 1 432.2.a.a 1
15.d odd 2 1 5400.2.a.bn 1
15.e even 4 2 5400.2.f.e 2
24.f even 2 1 1728.2.a.bb 1
24.h odd 2 1 1728.2.a.ba 1
36.f odd 6 2 1296.2.i.a 2
36.h even 6 2 1296.2.i.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.a 1 3.b odd 2 1
216.2.a.d yes 1 1.a even 1 1 trivial
432.2.a.a 1 12.b even 2 1
432.2.a.h 1 4.b odd 2 1
648.2.i.a 2 9.c even 3 2
648.2.i.h 2 9.d odd 6 2
1296.2.i.a 2 36.f odd 6 2
1296.2.i.q 2 36.h even 6 2
1728.2.a.a 1 8.b even 2 1
1728.2.a.b 1 8.d odd 2 1
1728.2.a.ba 1 24.h odd 2 1
1728.2.a.bb 1 24.f even 2 1
5400.2.a.bn 1 15.d odd 2 1
5400.2.a.bp 1 5.b even 2 1
5400.2.f.e 2 15.e even 4 2
5400.2.f.v 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(216))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$3 + T$$
$11$ $$-4 + T$$
$13$ $$-1 + T$$
$17$ $$4 + T$$
$19$ $$1 + T$$
$23$ $$-4 + T$$
$29$ $$T$$
$31$ $$4 + T$$
$37$ $$9 + T$$
$41$ $$T$$
$43$ $$8 + T$$
$47$ $$12 + T$$
$53$ $$8 + T$$
$59$ $$-4 + T$$
$61$ $$5 + T$$
$67$ $$-11 + T$$
$71$ $$-8 + T$$
$73$ $$-1 + T$$
$79$ $$5 + T$$
$83$ $$-8 + T$$
$89$ $$-12 + T$$
$97$ $$-5 + T$$