# Properties

 Label 216.2.f.a Level $216$ Weight $2$ Character orbit 216.f Analytic conductor $1.725$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.72476868366$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.23123460096.3 Defining polynomial: $$x^{8} + 2 x^{6} + 6 x^{4} + 8 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( -1 - \beta_{1} - \beta_{4} ) q^{4} + \beta_{2} q^{5} + \beta_{6} q^{7} + ( \beta_{2} - \beta_{3} + \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( -1 - \beta_{1} - \beta_{4} ) q^{4} + \beta_{2} q^{5} + \beta_{6} q^{7} + ( \beta_{2} - \beta_{3} + \beta_{7} ) q^{8} + ( -\beta_{4} - \beta_{6} ) q^{10} + ( \beta_{2} + \beta_{5} + \beta_{7} ) q^{11} + ( 1 + 2 \beta_{1} + \beta_{4} + \beta_{6} ) q^{13} + ( \beta_{2} - \beta_{5} ) q^{14} + ( -2 + \beta_{4} - \beta_{6} ) q^{16} + ( -\beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{17} + ( 2 + \beta_{4} ) q^{19} + 2 \beta_{5} q^{20} + ( -2 - 2 \beta_{1} + \beta_{4} - \beta_{6} ) q^{22} + ( -\beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{7} ) q^{23} + ( 1 + 2 \beta_{4} ) q^{25} + ( \beta_{3} - 2 \beta_{7} ) q^{26} + ( -1 + \beta_{1} - 2 \beta_{4} - \beta_{6} ) q^{28} + ( 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{29} + ( 2 + 4 \beta_{1} + 2 \beta_{4} ) q^{31} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{32} + ( -4 - 3 \beta_{4} + \beta_{6} ) q^{34} + ( \beta_{2} - 4 \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{35} + ( -1 - 2 \beta_{1} - \beta_{4} + \beta_{6} ) q^{37} + ( -\beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{38} + ( 2 - 2 \beta_{1} + 2 \beta_{4} ) q^{40} + ( -2 \beta_{2} - 2 \beta_{5} - 2 \beta_{7} ) q^{41} + 2 q^{43} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{44} + ( -2 + 2 \beta_{1} + 3 \beta_{4} + \beta_{6} ) q^{46} + ( -3 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{7} ) q^{47} + ( -2 - 4 \beta_{4} ) q^{49} + ( -2 \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{50} + ( 5 + \beta_{1} - 3 \beta_{4} ) q^{52} + ( 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{53} + ( -2 - 4 \beta_{1} - 2 \beta_{4} ) q^{55} + ( \beta_{2} - \beta_{3} + 4 \beta_{5} - \beta_{7} ) q^{56} + ( 4 - 4 \beta_{1} - 4 \beta_{4} ) q^{58} + ( \beta_{2} + \beta_{5} + \beta_{7} ) q^{59} + ( -1 - 2 \beta_{1} - \beta_{4} - \beta_{6} ) q^{61} + ( -2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 4 \beta_{7} ) q^{62} + ( 2 + 2 \beta_{1} + 4 \beta_{4} + 2 \beta_{6} ) q^{64} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{65} + ( -4 - 3 \beta_{4} ) q^{67} + ( 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} ) q^{68} + ( 10 + 2 \beta_{1} + 5 \beta_{4} - \beta_{6} ) q^{70} + ( -2 \beta_{2} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{71} + ( -1 + 4 \beta_{4} ) q^{73} + ( 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{74} + ( -3 - \beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{76} + ( -\beta_{2} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{77} + ( -2 - 4 \beta_{1} - 2 \beta_{4} - 3 \beta_{6} ) q^{79} + ( -2 \beta_{2} + 2 \beta_{3} - 4 \beta_{5} + 2 \beta_{7} ) q^{80} + ( 4 + 4 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} ) q^{82} + ( 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{83} + ( 2 + 4 \beta_{1} + 2 \beta_{4} - 4 \beta_{6} ) q^{85} + 2 \beta_{3} q^{86} + ( -6 + 2 \beta_{1} + 4 \beta_{4} + 2 \beta_{6} ) q^{88} + ( \beta_{2} + 6 \beta_{3} - 2 \beta_{5} + 4 \beta_{7} ) q^{89} + ( -6 + \beta_{4} ) q^{91} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{92} + ( -2 + 2 \beta_{1} + 5 \beta_{4} + 3 \beta_{6} ) q^{94} + ( 3 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{7} ) q^{95} + ( -1 - 6 \beta_{4} ) q^{97} + ( 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{4} + O(q^{10})$$ $$8q - 4q^{4} - 16q^{16} + 16q^{19} - 8q^{22} + 8q^{25} - 12q^{28} - 32q^{34} + 24q^{40} + 16q^{43} - 24q^{46} - 16q^{49} + 36q^{52} + 48q^{58} + 8q^{64} - 32q^{67} + 72q^{70} - 8q^{73} - 20q^{76} + 16q^{82} - 56q^{88} - 48q^{91} - 24q^{94} - 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 2 x^{6} + 6 x^{4} + 8 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 2 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{5} + 6 \nu^{3} + 8 \nu$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} + 2 \nu^{2} + 4$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{5} + 2 \nu^{3} - 8 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 2 \nu^{4} - 2 \nu^{2} + 4$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{5} + 2 \nu^{3} + 8 \nu$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{5}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + \beta_{4} - 2$$ $$\nu^{5}$$ $$=$$ $$-\beta_{7} + \beta_{5} + 2 \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{6} + 2 \beta_{4} - 2 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-2 \beta_{7} - 4 \beta_{5} + 2 \beta_{3} - 4 \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 1.08766 − 0.903873i 1.08766 + 0.903873i 0.563016 − 1.29731i 0.563016 + 1.29731i −0.563016 − 1.29731i −0.563016 + 1.29731i −1.08766 − 0.903873i −1.08766 + 0.903873i
−1.08766 0.903873i 0 0.366025 + 1.96622i −1.59245 0 1.43937i 1.37910 2.46943i 0 1.73205 + 1.43937i
107.2 −1.08766 + 0.903873i 0 0.366025 1.96622i −1.59245 0 1.43937i 1.37910 + 2.46943i 0 1.73205 1.43937i
107.3 −0.563016 1.29731i 0 −1.36603 + 1.46081i 3.07638 0 3.99102i 2.66422 + 0.949697i 0 −1.73205 3.99102i
107.4 −0.563016 + 1.29731i 0 −1.36603 1.46081i 3.07638 0 3.99102i 2.66422 0.949697i 0 −1.73205 + 3.99102i
107.5 0.563016 1.29731i 0 −1.36603 1.46081i −3.07638 0 3.99102i −2.66422 + 0.949697i 0 −1.73205 + 3.99102i
107.6 0.563016 + 1.29731i 0 −1.36603 + 1.46081i −3.07638 0 3.99102i −2.66422 0.949697i 0 −1.73205 3.99102i
107.7 1.08766 0.903873i 0 0.366025 1.96622i 1.59245 0 1.43937i −1.37910 2.46943i 0 1.73205 1.43937i
107.8 1.08766 + 0.903873i 0 0.366025 + 1.96622i 1.59245 0 1.43937i −1.37910 + 2.46943i 0 1.73205 + 1.43937i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 107.8 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
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.f.a 8
3.b odd 2 1 inner 216.2.f.a 8
4.b odd 2 1 864.2.f.a 8
8.b even 2 1 864.2.f.a 8
8.d odd 2 1 inner 216.2.f.a 8
9.c even 3 2 648.2.l.f 16
9.d odd 6 2 648.2.l.f 16
12.b even 2 1 864.2.f.a 8
24.f even 2 1 inner 216.2.f.a 8
24.h odd 2 1 864.2.f.a 8
36.f odd 6 2 2592.2.p.f 16
36.h even 6 2 2592.2.p.f 16
72.j odd 6 2 2592.2.p.f 16
72.l even 6 2 648.2.l.f 16
72.n even 6 2 2592.2.p.f 16
72.p odd 6 2 648.2.l.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.f.a 8 1.a even 1 1 trivial
216.2.f.a 8 3.b odd 2 1 inner
216.2.f.a 8 8.d odd 2 1 inner
216.2.f.a 8 24.f even 2 1 inner
648.2.l.f 16 9.c even 3 2
648.2.l.f 16 9.d odd 6 2
648.2.l.f 16 72.l even 6 2
648.2.l.f 16 72.p odd 6 2
864.2.f.a 8 4.b odd 2 1
864.2.f.a 8 8.b even 2 1
864.2.f.a 8 12.b even 2 1
864.2.f.a 8 24.h odd 2 1
2592.2.p.f 16 36.f odd 6 2
2592.2.p.f 16 36.h even 6 2
2592.2.p.f 16 72.j odd 6 2
2592.2.p.f 16 72.n even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 12 T_{5}^{2} + 24$$ acting on $$S_{2}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 8 T^{2} + 6 T^{4} + 2 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 24 - 12 T^{2} + T^{4} )^{2}$$
$7$ $$( 33 + 18 T^{2} + T^{4} )^{2}$$
$11$ $$( 88 + 28 T^{2} + T^{4} )^{2}$$
$13$ $$( 33 + 30 T^{2} + T^{4} )^{2}$$
$17$ $$( 88 + 52 T^{2} + T^{4} )^{2}$$
$19$ $$( 1 - 4 T + T^{2} )^{4}$$
$23$ $$( 24 - 36 T^{2} + T^{4} )^{2}$$
$29$ $$( 1536 - 96 T^{2} + T^{4} )^{2}$$
$31$ $$( 2112 + 96 T^{2} + T^{4} )^{2}$$
$37$ $$( 297 + 54 T^{2} + T^{4} )^{2}$$
$41$ $$( 1408 + 112 T^{2} + T^{4} )^{2}$$
$43$ $$( -2 + T )^{8}$$
$47$ $$( 4056 - 132 T^{2} + T^{4} )^{2}$$
$53$ $$( 3456 - 144 T^{2} + T^{4} )^{2}$$
$59$ $$( 88 + 28 T^{2} + T^{4} )^{2}$$
$61$ $$( 33 + 30 T^{2} + T^{4} )^{2}$$
$67$ $$( -11 + 8 T + T^{2} )^{4}$$
$71$ $$( 3456 - 144 T^{2} + T^{4} )^{2}$$
$73$ $$( -47 + 2 T + T^{2} )^{4}$$
$79$ $$( 5577 + 186 T^{2} + T^{4} )^{2}$$
$83$ $$( 5632 + 160 T^{2} + T^{4} )^{2}$$
$89$ $$( 46552 + 436 T^{2} + T^{4} )^{2}$$
$97$ $$( -107 + 2 T + T^{2} )^{4}$$