# Properties

 Label 216.2.d.a Level $216$ Weight $2$ Character orbit 216.d Analytic conductor $1.725$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 216.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.72476868366$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} + ( -\zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( -1 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} + ( -\zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( -1 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( 2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{10} + ( 2 \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{11} + ( -\zeta_{8} - 6 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{14} + 4 q^{16} + ( 2 \zeta_{8} - 6 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{20} + ( -4 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{22} + ( -6 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{25} + ( 2 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{28} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( 5 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{31} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{32} + ( -8 \zeta_{8} + 3 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{35} + ( -4 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{40} + ( -4 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{44} + ( 12 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{49} + ( -6 \zeta_{8} + 12 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{50} + ( 5 \zeta_{8} + 3 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{53} + ( -5 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{55} + ( 2 \zeta_{8} + 12 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{56} -4 q^{58} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{59} + ( 5 \zeta_{8} - 6 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{62} -8 q^{64} + ( 16 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{70} + ( -7 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{73} + ( -11 \zeta_{8} - 15 \zeta_{8}^{2} - 11 \zeta_{8}^{3} ) q^{77} -10 q^{79} + ( -4 \zeta_{8} + 12 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{80} + ( 2 \zeta_{8} - 15 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{83} + ( 8 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{88} + ( -1 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{97} + ( 12 \zeta_{8} + 12 \zeta_{8}^{2} + 12 \zeta_{8}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} - 4q^{7} + O(q^{10})$$ $$4q - 8q^{4} - 4q^{7} + 8q^{10} + 16q^{16} - 16q^{22} - 24q^{25} + 8q^{28} + 20q^{31} - 16q^{40} + 48q^{49} - 20q^{55} - 16q^{58} - 32q^{64} + 64q^{70} - 28q^{73} - 40q^{79} + 32q^{88} - 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$$.

 $$n$$ $$55$$ $$109$$ $$137$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
1.41421i 0 −2.00000 1.58579i 0 −5.24264 2.82843i 0 −2.24264
109.2 1.41421i 0 −2.00000 4.41421i 0 3.24264 2.82843i 0 6.24264
109.3 1.41421i 0 −2.00000 4.41421i 0 3.24264 2.82843i 0 6.24264
109.4 1.41421i 0 −2.00000 1.58579i 0 −5.24264 2.82843i 0 −2.24264
 $$n$$: 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.d.a 4
3.b odd 2 1 inner 216.2.d.a 4
4.b odd 2 1 864.2.d.b 4
8.b even 2 1 inner 216.2.d.a 4
8.d odd 2 1 864.2.d.b 4
9.c even 3 2 648.2.n.p 8
9.d odd 6 2 648.2.n.p 8
12.b even 2 1 864.2.d.b 4
16.e even 4 1 6912.2.a.z 2
16.e even 4 1 6912.2.a.bz 2
16.f odd 4 1 6912.2.a.y 2
16.f odd 4 1 6912.2.a.by 2
24.f even 2 1 864.2.d.b 4
24.h odd 2 1 CM 216.2.d.a 4
36.f odd 6 2 2592.2.r.o 8
36.h even 6 2 2592.2.r.o 8
48.i odd 4 1 6912.2.a.z 2
48.i odd 4 1 6912.2.a.bz 2
48.k even 4 1 6912.2.a.y 2
48.k even 4 1 6912.2.a.by 2
72.j odd 6 2 648.2.n.p 8
72.l even 6 2 2592.2.r.o 8
72.n even 6 2 648.2.n.p 8
72.p odd 6 2 2592.2.r.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.a 4 1.a even 1 1 trivial
216.2.d.a 4 3.b odd 2 1 inner
216.2.d.a 4 8.b even 2 1 inner
216.2.d.a 4 24.h odd 2 1 CM
648.2.n.p 8 9.c even 3 2
648.2.n.p 8 9.d odd 6 2
648.2.n.p 8 72.j odd 6 2
648.2.n.p 8 72.n even 6 2
864.2.d.b 4 4.b odd 2 1
864.2.d.b 4 8.d odd 2 1
864.2.d.b 4 12.b even 2 1
864.2.d.b 4 24.f even 2 1
2592.2.r.o 8 36.f odd 6 2
2592.2.r.o 8 36.h even 6 2
2592.2.r.o 8 72.l even 6 2
2592.2.r.o 8 72.p odd 6 2
6912.2.a.y 2 16.f odd 4 1
6912.2.a.y 2 48.k even 4 1
6912.2.a.z 2 16.e even 4 1
6912.2.a.z 2 48.i odd 4 1
6912.2.a.by 2 16.f odd 4 1
6912.2.a.by 2 48.k even 4 1
6912.2.a.bz 2 16.e even 4 1
6912.2.a.bz 2 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 22 T_{5}^{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$49 + 22 T^{2} + T^{4}$$
$7$ $$( -17 + 2 T + T^{2} )^{2}$$
$11$ $$1 + 34 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 8 + T^{2} )^{2}$$
$31$ $$( 7 - 10 T + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$1681 + 118 T^{2} + T^{4}$$
$59$ $$( 128 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -23 + 14 T + T^{2} )^{2}$$
$79$ $$( 10 + T )^{4}$$
$83$ $$47089 + 466 T^{2} + T^{4}$$
$89$ $$T^{4}$$
$97$ $$( -287 + 2 T + T^{2} )^{2}$$