# Properties

 Label 216.3.e.a Level $216$ Weight $3$ Character orbit 216.e Analytic conductor $5.886$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no 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: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{5} -3 q^{7} +O(q^{10})$$ $$q + \beta q^{5} -3 q^{7} + 7 \beta q^{11} + 7 q^{13} + 5 \beta q^{17} -19 q^{19} + \beta q^{23} + 17 q^{25} + 18 \beta q^{29} -10 q^{31} -3 \beta q^{35} + 63 q^{37} -18 \beta q^{41} -50 q^{43} + 15 \beta q^{47} -40 q^{49} -26 \beta q^{53} -56 q^{55} -35 \beta q^{59} + 79 q^{61} + 7 \beta q^{65} + 77 q^{67} -28 \beta q^{71} -17 q^{73} -21 \beta q^{77} -11 q^{79} + 14 \beta q^{83} -40 q^{85} -15 \beta q^{89} -21 q^{91} -19 \beta q^{95} -97 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{7} + O(q^{10})$$ $$2q - 6q^{7} + 14q^{13} - 38q^{19} + 34q^{25} - 20q^{31} + 126q^{37} - 100q^{43} - 80q^{49} - 112q^{55} + 158q^{61} + 154q^{67} - 34q^{73} - 22q^{79} - 80q^{85} - 42q^{91} - 194q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 − 1.41421i 1.41421i
0 0 0 2.82843i 0 −3.00000 0 0 0
161.2 0 0 0 2.82843i 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.e.a 2
3.b odd 2 1 inner 216.3.e.a 2
4.b odd 2 1 432.3.e.f 2
8.b even 2 1 1728.3.e.h 2
8.d odd 2 1 1728.3.e.j 2
9.c even 3 2 648.3.m.c 4
9.d odd 6 2 648.3.m.c 4
12.b even 2 1 432.3.e.f 2
24.f even 2 1 1728.3.e.j 2
24.h odd 2 1 1728.3.e.h 2
36.f odd 6 2 1296.3.q.g 4
36.h even 6 2 1296.3.q.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.e.a 2 1.a even 1 1 trivial
216.3.e.a 2 3.b odd 2 1 inner
432.3.e.f 2 4.b odd 2 1
432.3.e.f 2 12.b even 2 1
648.3.m.c 4 9.c even 3 2
648.3.m.c 4 9.d odd 6 2
1296.3.q.g 4 36.f odd 6 2
1296.3.q.g 4 36.h even 6 2
1728.3.e.h 2 8.b even 2 1
1728.3.e.h 2 24.h odd 2 1
1728.3.e.j 2 8.d odd 2 1
1728.3.e.j 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 8$$ acting on $$S_{3}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$8 + T^{2}$$
$7$ $$( 3 + T )^{2}$$
$11$ $$392 + T^{2}$$
$13$ $$( -7 + T )^{2}$$
$17$ $$200 + T^{2}$$
$19$ $$( 19 + T )^{2}$$
$23$ $$8 + T^{2}$$
$29$ $$2592 + T^{2}$$
$31$ $$( 10 + T )^{2}$$
$37$ $$( -63 + T )^{2}$$
$41$ $$2592 + T^{2}$$
$43$ $$( 50 + T )^{2}$$
$47$ $$1800 + T^{2}$$
$53$ $$5408 + T^{2}$$
$59$ $$9800 + T^{2}$$
$61$ $$( -79 + T )^{2}$$
$67$ $$( -77 + T )^{2}$$
$71$ $$6272 + T^{2}$$
$73$ $$( 17 + T )^{2}$$
$79$ $$( 11 + T )^{2}$$
$83$ $$1568 + T^{2}$$
$89$ $$1800 + T^{2}$$
$97$ $$( 97 + T )^{2}$$