# Properties

 Label 216.2.n.a Level $216$ Weight $2$ Character orbit 216.n Analytic conductor $1.725$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 216.n (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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + 2 \zeta_{12} q^{4} + 2 \zeta_{12} q^{5} -4 \zeta_{12}^{2} q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + 2 \zeta_{12} q^{4} + 2 \zeta_{12} q^{5} -4 \zeta_{12}^{2} q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( 2 + 2 \zeta_{12}^{3} ) q^{10} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{11} + 2 \zeta_{12} q^{13} + ( 4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{14} + 4 \zeta_{12}^{2} q^{16} -5 q^{17} -\zeta_{12}^{3} q^{19} + 4 \zeta_{12}^{2} q^{20} + ( -3 - 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{22} + ( 2 - 2 \zeta_{12}^{2} ) q^{23} -\zeta_{12}^{2} q^{25} + ( 2 + 2 \zeta_{12}^{3} ) q^{26} -8 \zeta_{12}^{3} q^{28} + ( 4 - 4 \zeta_{12}^{2} ) q^{31} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{32} + ( -5 \zeta_{12} - 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{34} -8 \zeta_{12}^{3} q^{35} -2 \zeta_{12}^{3} q^{37} + ( \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{38} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{40} + ( -5 + 5 \zeta_{12}^{2} ) q^{41} + ( -11 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{43} -6 q^{44} + ( 2 - 2 \zeta_{12}^{3} ) q^{46} -6 \zeta_{12}^{2} q^{47} + ( -9 + 9 \zeta_{12}^{2} ) q^{49} + ( 1 - \zeta_{12} - \zeta_{12}^{2} ) q^{50} + 4 \zeta_{12}^{2} q^{52} -6 q^{55} + ( 8 \zeta_{12} - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{56} + \zeta_{12} q^{59} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{61} + ( 4 - 4 \zeta_{12}^{3} ) q^{62} + 8 \zeta_{12}^{3} q^{64} + 4 \zeta_{12}^{2} q^{65} + 3 \zeta_{12} q^{67} -10 \zeta_{12} q^{68} + ( 8 \zeta_{12} - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{70} + 6 q^{71} + 9 q^{73} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{74} + ( 2 - 2 \zeta_{12}^{2} ) q^{76} + 12 \zeta_{12} q^{77} + 14 \zeta_{12}^{2} q^{79} + 8 \zeta_{12}^{3} q^{80} + ( -5 + 5 \zeta_{12}^{3} ) q^{82} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{83} -10 \zeta_{12} q^{85} + ( -11 - 11 \zeta_{12} + 11 \zeta_{12}^{2} ) q^{86} + ( -6 \zeta_{12} - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{88} + 14 q^{89} -8 \zeta_{12}^{3} q^{91} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{92} + ( 6 - 6 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{94} + ( 2 - 2 \zeta_{12}^{2} ) q^{95} -\zeta_{12}^{2} q^{97} + ( -9 + 9 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 8q^{7} + 8q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 8q^{7} + 8q^{8} + 8q^{10} + 8q^{14} + 8q^{16} - 20q^{17} + 8q^{20} - 6q^{22} + 4q^{23} - 2q^{25} + 8q^{26} + 8q^{31} - 8q^{32} - 10q^{34} - 2q^{38} - 8q^{40} - 10q^{41} - 24q^{44} + 8q^{46} - 12q^{47} - 18q^{49} + 2q^{50} + 8q^{52} - 24q^{55} - 16q^{56} + 16q^{62} + 8q^{65} - 16q^{70} + 24q^{71} + 36q^{73} - 4q^{74} + 4q^{76} + 28q^{79} - 20q^{82} - 22q^{86} - 12q^{88} + 56q^{89} + 12q^{94} + 4q^{95} - 2q^{97} - 36q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$55$$ $$109$$ $$137$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.366025 + 1.36603i 0 −1.73205 1.00000i −1.73205 1.00000i 0 −2.00000 3.46410i 2.00000 2.00000i 0 2.00000 2.00000i
37.2 1.36603 + 0.366025i 0 1.73205 + 1.00000i 1.73205 + 1.00000i 0 −2.00000 3.46410i 2.00000 + 2.00000i 0 2.00000 + 2.00000i
181.1 −0.366025 1.36603i 0 −1.73205 + 1.00000i −1.73205 + 1.00000i 0 −2.00000 + 3.46410i 2.00000 + 2.00000i 0 2.00000 + 2.00000i
181.2 1.36603 0.366025i 0 1.73205 1.00000i 1.73205 1.00000i 0 −2.00000 + 3.46410i 2.00000 2.00000i 0 2.00000 2.00000i
 $$n$$: 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.b even 2 1 inner
9.c even 3 1 inner
72.n 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.n.a 4
3.b odd 2 1 72.2.n.a 4
4.b odd 2 1 864.2.r.a 4
8.b even 2 1 inner 216.2.n.a 4
8.d odd 2 1 864.2.r.a 4
9.c even 3 1 inner 216.2.n.a 4
9.c even 3 1 648.2.d.a 2
9.d odd 6 1 72.2.n.a 4
9.d odd 6 1 648.2.d.d 2
12.b even 2 1 288.2.r.a 4
24.f even 2 1 288.2.r.a 4
24.h odd 2 1 72.2.n.a 4
36.f odd 6 1 864.2.r.a 4
36.f odd 6 1 2592.2.d.a 2
36.h even 6 1 288.2.r.a 4
36.h even 6 1 2592.2.d.b 2
72.j odd 6 1 72.2.n.a 4
72.j odd 6 1 648.2.d.d 2
72.l even 6 1 288.2.r.a 4
72.l even 6 1 2592.2.d.b 2
72.n even 6 1 inner 216.2.n.a 4
72.n even 6 1 648.2.d.a 2
72.p odd 6 1 864.2.r.a 4
72.p odd 6 1 2592.2.d.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 4 T_{5}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4}$$
$3$ 1
$5$ $$( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} )( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} )$$
$7$ $$( 1 - T + 7 T^{2} )^{2}( 1 + 5 T + 7 T^{2} )^{2}$$
$11$ $$1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - T^{2} + 169 T^{4} )( 1 + 23 T^{2} + 169 T^{4} )$$
$17$ $$( 1 + 5 T + 17 T^{2} )^{4}$$
$19$ $$( 1 - 37 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 29 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 5 T - 16 T^{2} + 205 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 35 T^{2} - 624 T^{4} - 64715 T^{6} + 3418801 T^{8}$$
$47$ $$( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{4}$$
$59$ $$1 + 117 T^{2} + 10208 T^{4} + 407277 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} )( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )$$
$67$ $$1 + 125 T^{2} + 11136 T^{4} + 561125 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{4}$$
$73$ $$( 1 - 9 T + 73 T^{2} )^{4}$$
$79$ $$( 1 - 14 T + 117 T^{2} - 1106 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 14 T + 89 T^{2} )^{4}$$
$97$ $$( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2}$$