# Properties

 Label 216.2.l.a Level $216$ Weight $2$ Character orbit 216.l Analytic conductor $1.725$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 216.l (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.72476868366$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{3} q^{8} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{11} + ( -4 + 4 \beta_{2} ) q^{16} + ( 3 - 6 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -1 - 6 \beta_{1} + 3 \beta_{3} ) q^{19} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{22} + ( 5 - 5 \beta_{2} ) q^{25} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( 4 + 3 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{34} + ( -6 - \beta_{1} - 6 \beta_{2} ) q^{38} + ( -6 + 4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{41} + ( -5 + 3 \beta_{1} + 5 \beta_{2} - 6 \beta_{3} ) q^{43} + ( -6 + 12 \beta_{2} - 2 \beta_{3} ) q^{44} -7 \beta_{2} q^{49} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{50} + ( -6 - 5 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} ) q^{59} -8 q^{64} + ( 3 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} ) q^{67} + ( 12 + 4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -1 + 12 \beta_{1} - 6 \beta_{3} ) q^{73} + ( -6 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{76} + ( 8 - 6 \beta_{1} + 3 \beta_{3} ) q^{82} + 2 \beta_{1} q^{83} + ( 12 - 5 \beta_{1} - 6 \beta_{2} + 5 \beta_{3} ) q^{86} + ( 4 - 6 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} ) q^{88} + 4 \beta_{3} q^{89} + ( -5 - 6 \beta_{1} + 5 \beta_{2} + 12 \beta_{3} ) q^{97} -7 \beta_{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + O(q^{10})$$ $$4 q + 4 q^{4} + 18 q^{11} - 8 q^{16} - 4 q^{19} - 4 q^{22} + 10 q^{25} + 8 q^{34} - 36 q^{38} - 18 q^{41} - 10 q^{43} - 14 q^{49} - 18 q^{59} - 32 q^{64} + 14 q^{67} + 36 q^{68} - 4 q^{73} - 4 q^{76} + 32 q^{82} + 36 q^{86} + 8 q^{88} - 10 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$55$$ $$109$$ $$137$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 0 0 0 2.82843i 0 0
35.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 0 0 0 2.82843i 0 0
179.1 −1.22474 + 0.707107i 0 1.00000 1.73205i 0 0 0 2.82843i 0 0
179.2 1.22474 0.707107i 0 1.00000 1.73205i 0 0 0 2.82843i 0 0
 $$n$$: 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
9.d odd 6 1 inner
72.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.l.a 4
3.b odd 2 1 72.2.l.a 4
4.b odd 2 1 864.2.p.a 4
8.b even 2 1 864.2.p.a 4
8.d odd 2 1 CM 216.2.l.a 4
9.c even 3 1 72.2.l.a 4
9.c even 3 1 648.2.f.a 4
9.d odd 6 1 inner 216.2.l.a 4
9.d odd 6 1 648.2.f.a 4
12.b even 2 1 288.2.p.a 4
24.f even 2 1 72.2.l.a 4
24.h odd 2 1 288.2.p.a 4
36.f odd 6 1 288.2.p.a 4
36.f odd 6 1 2592.2.f.a 4
36.h even 6 1 864.2.p.a 4
36.h even 6 1 2592.2.f.a 4
72.j odd 6 1 864.2.p.a 4
72.j odd 6 1 2592.2.f.a 4
72.l even 6 1 inner 216.2.l.a 4
72.l even 6 1 648.2.f.a 4
72.n even 6 1 288.2.p.a 4
72.n even 6 1 2592.2.f.a 4
72.p odd 6 1 72.2.l.a 4
72.p odd 6 1 648.2.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.a 4 3.b odd 2 1
72.2.l.a 4 9.c even 3 1
72.2.l.a 4 24.f even 2 1
72.2.l.a 4 72.p odd 6 1
216.2.l.a 4 1.a even 1 1 trivial
216.2.l.a 4 8.d odd 2 1 CM
216.2.l.a 4 9.d odd 6 1 inner
216.2.l.a 4 72.l even 6 1 inner
288.2.p.a 4 12.b even 2 1
288.2.p.a 4 24.h odd 2 1
288.2.p.a 4 36.f odd 6 1
288.2.p.a 4 72.n even 6 1
648.2.f.a 4 9.c even 3 1
648.2.f.a 4 9.d odd 6 1
648.2.f.a 4 72.l even 6 1
648.2.f.a 4 72.p odd 6 1
864.2.p.a 4 4.b odd 2 1
864.2.p.a 4 8.b even 2 1
864.2.p.a 4 36.h even 6 1
864.2.p.a 4 72.j odd 6 1
2592.2.f.a 4 36.f odd 6 1
2592.2.f.a 4 36.h even 6 1
2592.2.f.a 4 72.j odd 6 1
2592.2.f.a 4 72.n even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$625 - 450 T + 133 T^{2} - 18 T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$361 + 70 T^{2} + T^{4}$$
$19$ $$( -53 + 2 T + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$25 - 90 T + 103 T^{2} + 18 T^{3} + T^{4}$$
$43$ $$841 - 290 T + 129 T^{2} + 10 T^{3} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$529 - 414 T + 85 T^{2} + 18 T^{3} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$25 + 70 T + 201 T^{2} - 14 T^{3} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -215 + 2 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$64 - 8 T^{2} + T^{4}$$
$89$ $$( 32 + T^{2} )^{2}$$
$97$ $$36481 - 1910 T + 291 T^{2} + 10 T^{3} + T^{4}$$