Properties

 Label 216.4.a.a Level $216$ Weight $4$ Character orbit 216.a Self dual yes Analytic conductor $12.744$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 216.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$12.7444125612$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 4q^{5} + 3q^{7} + O(q^{10})$$ $$q - 4q^{5} + 3q^{7} - 28q^{11} - 11q^{13} - 44q^{17} + 29q^{19} - 172q^{23} - 109q^{25} - 192q^{29} + 116q^{31} - 12q^{35} - 69q^{37} - 384q^{41} + 328q^{43} - 156q^{47} - 334q^{49} + 392q^{53} + 112q^{55} - 412q^{59} - 425q^{61} + 44q^{65} + 257q^{67} + 1000q^{71} - 359q^{73} - 84q^{77} + 877q^{79} + 328q^{83} + 176q^{85} + 1572q^{89} - 33q^{91} - 116q^{95} - 1483q^{97} + O(q^{100})$$

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 216.4.a.a 1
3.b odd 2 1 216.4.a.d yes 1
4.b odd 2 1 432.4.a.d 1
8.b even 2 1 1728.4.a.x 1
8.d odd 2 1 1728.4.a.w 1
9.c even 3 2 648.4.i.i 2
9.d odd 6 2 648.4.i.d 2
12.b even 2 1 432.4.a.k 1
24.f even 2 1 1728.4.a.i 1
24.h odd 2 1 1728.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.a 1 1.a even 1 1 trivial
216.4.a.d yes 1 3.b odd 2 1
432.4.a.d 1 4.b odd 2 1
432.4.a.k 1 12.b even 2 1
648.4.i.d 2 9.d odd 6 2
648.4.i.i 2 9.c even 3 2
1728.4.a.i 1 24.f even 2 1
1728.4.a.j 1 24.h odd 2 1
1728.4.a.w 1 8.d odd 2 1
1728.4.a.x 1 8.b even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$4 + T$$
$7$ $$-3 + T$$
$11$ $$28 + T$$
$13$ $$11 + T$$
$17$ $$44 + T$$
$19$ $$-29 + T$$
$23$ $$172 + T$$
$29$ $$192 + T$$
$31$ $$-116 + T$$
$37$ $$69 + T$$
$41$ $$384 + T$$
$43$ $$-328 + T$$
$47$ $$156 + T$$
$53$ $$-392 + T$$
$59$ $$412 + T$$
$61$ $$425 + T$$
$67$ $$-257 + T$$
$71$ $$-1000 + T$$
$73$ $$359 + T$$
$79$ $$-877 + T$$
$83$ $$-328 + T$$
$89$ $$-1572 + T$$
$97$ $$1483 + T$$