# Properties

 Label 216.3.e.c Level $216$ Weight $3$ Character orbit 216.e Analytic conductor $5.886$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{8} - 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{5} + ( -3 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{7} +O(q^{10})$$ $$q + ( 2 \zeta_{8} - 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{5} + ( -3 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{7} + ( -4 \zeta_{8} + 9 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{11} + ( -2 + 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{13} + ( -8 \zeta_{8} - 6 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{17} + ( 8 + 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{19} + ( 20 \zeta_{8} + 12 \zeta_{8}^{2} + 20 \zeta_{8}^{3} ) q^{23} + ( 8 + 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{25} -42 \zeta_{8}^{2} q^{29} + ( -1 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{31} + ( -24 \zeta_{8} + 33 \zeta_{8}^{2} - 24 \zeta_{8}^{3} ) q^{35} + ( -36 + 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{37} + ( 36 \zeta_{8} - 12 \zeta_{8}^{2} + 36 \zeta_{8}^{3} ) q^{41} + ( -14 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{43} + ( -24 \zeta_{8} + 42 \zeta_{8}^{2} - 24 \zeta_{8}^{3} ) q^{47} + ( 32 - 36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{49} + ( -34 \zeta_{8} - 51 \zeta_{8}^{2} - 34 \zeta_{8}^{3} ) q^{53} + ( 43 - 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{55} + ( 56 \zeta_{8} + 18 \zeta_{8}^{2} + 56 \zeta_{8}^{3} ) q^{59} + ( 52 - 48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{61} + ( -40 \zeta_{8} + 54 \zeta_{8}^{2} - 40 \zeta_{8}^{3} ) q^{65} + ( -58 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{67} + ( -20 \zeta_{8} - 18 \zeta_{8}^{2} - 20 \zeta_{8}^{3} ) q^{71} + ( -71 - 36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{73} + ( 66 \zeta_{8} - 75 \zeta_{8}^{2} + 66 \zeta_{8}^{3} ) q^{77} + ( -74 - 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{79} + ( 28 \zeta_{8} - 69 \zeta_{8}^{2} + 28 \zeta_{8}^{3} ) q^{83} + ( 14 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{85} + ( -12 \zeta_{8} + 90 \zeta_{8}^{2} - 12 \zeta_{8}^{3} ) q^{89} + ( 150 - 48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{91} + ( -20 \zeta_{8} + 24 \zeta_{8}^{2} - 20 \zeta_{8}^{3} ) q^{95} + ( 11 + 48 \zeta_{8} - 48 \zeta_{8}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{7} + O(q^{10})$$ $$4q - 12q^{7} - 8q^{13} + 32q^{19} + 32q^{25} - 4q^{31} - 144q^{37} - 56q^{43} + 128q^{49} + 172q^{55} + 208q^{61} - 232q^{67} - 284q^{73} - 296q^{79} + 56q^{85} + 600q^{91} + 44q^{97} + O(q^{100})$$

## 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
 −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
0 0 0 5.82843i 0 −11.4853 0 0 0
161.2 0 0 0 0.171573i 0 5.48528 0 0 0
161.3 0 0 0 0.171573i 0 5.48528 0 0 0
161.4 0 0 0 5.82843i 0 −11.4853 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.c 4
3.b odd 2 1 inner 216.3.e.c 4
4.b odd 2 1 432.3.e.h 4
8.b even 2 1 1728.3.e.p 4
8.d odd 2 1 1728.3.e.s 4
9.c even 3 2 648.3.m.f 8
9.d odd 6 2 648.3.m.f 8
12.b even 2 1 432.3.e.h 4
24.f even 2 1 1728.3.e.s 4
24.h odd 2 1 1728.3.e.p 4
36.f odd 6 2 1296.3.q.l 8
36.h even 6 2 1296.3.q.l 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 34 T_{5}^{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$1 + 34 T^{2} + T^{4}$$
$7$ $$( -63 + 6 T + T^{2} )^{2}$$
$11$ $$2401 + 226 T^{2} + T^{4}$$
$13$ $$( -284 + 4 T + T^{2} )^{2}$$
$17$ $$8464 + 328 T^{2} + T^{4}$$
$19$ $$( -224 - 16 T + T^{2} )^{2}$$
$23$ $$430336 + 1888 T^{2} + T^{4}$$
$29$ $$( 1764 + T^{2} )^{2}$$
$31$ $$( -71 + 2 T + T^{2} )^{2}$$
$37$ $$( 1008 + 72 T + T^{2} )^{2}$$
$41$ $$5992704 + 5472 T^{2} + T^{4}$$
$43$ $$( -92 + 28 T + T^{2} )^{2}$$
$47$ $$374544 + 5832 T^{2} + T^{4}$$
$53$ $$83521 + 9826 T^{2} + T^{4}$$
$59$ $$35378704 + 13192 T^{2} + T^{4}$$
$61$ $$( -1904 - 104 T + T^{2} )^{2}$$
$67$ $$( 3076 + 116 T + T^{2} )^{2}$$
$71$ $$226576 + 2248 T^{2} + T^{4}$$
$73$ $$( 2449 + 142 T + T^{2} )^{2}$$
$79$ $$( 4324 + 148 T + T^{2} )^{2}$$
$83$ $$10195249 + 12658 T^{2} + T^{4}$$
$89$ $$61027344 + 16776 T^{2} + T^{4}$$
$97$ $$( -4487 - 22 T + T^{2} )^{2}$$