# Properties

 Label 216.3.h.a Level $216$ Weight $3$ Character orbit 216.h Self dual yes Analytic conductor $5.886$ Analytic rank $0$ Dimension $2$ CM discriminant -24 Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$5.88557371018$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + ( -1 + \beta ) q^{5} + ( 5 + \beta ) q^{7} -8 q^{8} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + ( -1 + \beta ) q^{5} + ( 5 + \beta ) q^{7} -8 q^{8} + ( 2 - 2 \beta ) q^{10} + ( 5 - 2 \beta ) q^{11} + ( -10 - 2 \beta ) q^{14} + 16 q^{16} + ( -4 + 4 \beta ) q^{20} + ( -10 + 4 \beta ) q^{22} + ( 48 - 2 \beta ) q^{25} + ( 20 + 4 \beta ) q^{28} + 50 q^{29} + ( -19 - 5 \beta ) q^{31} -32 q^{32} + ( 67 + 4 \beta ) q^{35} + ( 8 - 8 \beta ) q^{40} + ( 20 - 8 \beta ) q^{44} + ( 48 + 10 \beta ) q^{49} + ( -96 + 4 \beta ) q^{50} + ( 47 - 5 \beta ) q^{53} + ( -149 + 7 \beta ) q^{55} + ( -40 - 8 \beta ) q^{56} -100 q^{58} -10 q^{59} + ( 38 + 10 \beta ) q^{62} + 64 q^{64} + ( -134 - 8 \beta ) q^{70} + ( -25 - 14 \beta ) q^{73} + ( -119 - 5 \beta ) q^{77} -58 q^{79} + ( -16 + 16 \beta ) q^{80} + ( -67 + 10 \beta ) q^{83} + ( -40 + 16 \beta ) q^{88} + ( 95 + 4 \beta ) q^{97} + ( -96 - 20 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} - 2q^{5} + 10q^{7} - 16q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} - 2q^{5} + 10q^{7} - 16q^{8} + 4q^{10} + 10q^{11} - 20q^{14} + 32q^{16} - 8q^{20} - 20q^{22} + 96q^{25} + 40q^{28} + 100q^{29} - 38q^{31} - 64q^{32} + 134q^{35} + 16q^{40} + 40q^{44} + 96q^{49} - 192q^{50} + 94q^{53} - 298q^{55} - 80q^{56} - 200q^{58} - 20q^{59} + 76q^{62} + 128q^{64} - 268q^{70} - 50q^{73} - 238q^{77} - 116q^{79} - 32q^{80} - 134q^{83} - 80q^{88} + 190q^{97} - 192q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −1.41421 1.41421
−2.00000 0 4.00000 −9.48528 0 −3.48528 −8.00000 0 18.9706
53.2 −2.00000 0 4.00000 7.48528 0 13.4853 −8.00000 0 −14.9706
 $$n$$: 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 $$\Q(\sqrt{-6})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.h.a 2
3.b odd 2 1 216.3.h.b yes 2
4.b odd 2 1 864.3.h.a 2
8.b even 2 1 216.3.h.b yes 2
8.d odd 2 1 864.3.h.b 2
12.b even 2 1 864.3.h.b 2
24.f even 2 1 864.3.h.a 2
24.h odd 2 1 CM 216.3.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.h.a 2 1.a even 1 1 trivial
216.3.h.a 2 24.h odd 2 1 CM
216.3.h.b yes 2 3.b odd 2 1
216.3.h.b yes 2 8.b even 2 1
864.3.h.a 2 4.b odd 2 1
864.3.h.a 2 24.f even 2 1
864.3.h.b 2 8.d odd 2 1
864.3.h.b 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2 T_{5} - 71$$ acting on $$S_{3}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-71 + 2 T + T^{2}$$
$7$ $$-47 - 10 T + T^{2}$$
$11$ $$-263 - 10 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -50 + T )^{2}$$
$31$ $$-1439 + 38 T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$409 - 94 T + T^{2}$$
$59$ $$( 10 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$-13487 + 50 T + T^{2}$$
$79$ $$( 58 + T )^{2}$$
$83$ $$-2711 + 134 T + T^{2}$$
$89$ $$T^{2}$$
$97$ $$7873 - 190 T + T^{2}$$