# Properties

 Label 216.3.h.e Level $216$ Weight $3$ Character orbit 216.h Analytic conductor $5.886$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: no Analytic conductor: $$5.88557371018$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.629407744.1 Defining polynomial: $$x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{5} + \beta_{7} ) q^{4} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{5} + ( -6 + \beta_{5} ) q^{7} -2 \beta_{6} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{5} + \beta_{7} ) q^{4} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{5} + ( -6 + \beta_{5} ) q^{7} -2 \beta_{6} q^{8} + ( -7 + 2 \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{10} + ( -5 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} ) q^{11} + ( \beta_{4} + 7 \beta_{7} ) q^{13} + ( -5 \beta_{1} - 2 \beta_{3} - \beta_{6} ) q^{14} + ( -4 \beta_{4} - 4 \beta_{5} ) q^{16} + ( -4 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{17} + ( 7 \beta_{4} + 2 \beta_{7} ) q^{19} + ( -12 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 6 \beta_{6} ) q^{20} + ( -11 - 8 \beta_{4} - \beta_{5} - 9 \beta_{7} ) q^{22} + ( 3 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} - 8 \beta_{6} ) q^{23} + ( 17 + 14 \beta_{5} ) q^{25} + ( -\beta_{1} + 2 \beta_{2} + 14 \beta_{3} - 6 \beta_{6} ) q^{26} + ( 13 - 2 \beta_{4} - 7 \beta_{5} - 5 \beta_{7} ) q^{28} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{29} + ( 26 + 6 \beta_{5} ) q^{31} + ( -8 \beta_{2} + 8 \beta_{3} ) q^{32} + ( 13 + 2 \beta_{4} + \beta_{5} - 5 \beta_{7} ) q^{34} + ( 5 \beta_{1} + 6 \beta_{2} - 5 \beta_{3} ) q^{35} + ( 5 \beta_{4} + \beta_{7} ) q^{37} + ( -7 \beta_{1} + 14 \beta_{2} + 4 \beta_{3} + 5 \beta_{6} ) q^{38} + ( 4 \beta_{4} + 4 \beta_{5} - 16 \beta_{7} ) q^{40} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{41} + ( -6 \beta_{4} + 10 \beta_{7} ) q^{43} + ( -4 \beta_{1} - 16 \beta_{2} - 16 \beta_{3} + 2 \beta_{6} ) q^{44} + ( 13 - 8 \beta_{4} - 17 \beta_{5} + 7 \beta_{7} ) q^{46} + ( 11 \beta_{1} + 2 \beta_{2} + 15 \beta_{3} + 4 \beta_{6} ) q^{47} + ( -6 - 12 \beta_{5} ) q^{49} + ( 31 \beta_{1} - 28 \beta_{3} - 14 \beta_{6} ) q^{50} + ( -53 - 16 \beta_{4} - 11 \beta_{5} - 3 \beta_{7} ) q^{52} + ( -2 \beta_{1} - 20 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 14 + 22 \beta_{5} ) q^{55} + ( 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 10 \beta_{6} ) q^{56} + ( -8 + 4 \beta_{4} - 4 \beta_{5} ) q^{58} + ( 7 \beta_{1} + 12 \beta_{2} - 7 \beta_{3} ) q^{59} + ( 19 \beta_{4} - 3 \beta_{7} ) q^{61} + ( 32 \beta_{1} - 12 \beta_{3} - 6 \beta_{6} ) q^{62} + ( -40 + 16 \beta_{4} - 8 \beta_{5} + 8 \beta_{7} ) q^{64} + ( -42 \beta_{1} + 19 \beta_{2} - 4 \beta_{3} + 38 \beta_{6} ) q^{65} + ( 17 \beta_{4} + 36 \beta_{7} ) q^{67} + ( 12 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} + 6 \beta_{6} ) q^{68} + ( 21 - 12 \beta_{4} + 11 \beta_{5} - \beta_{7} ) q^{70} + ( 6 \beta_{1} - 12 \beta_{2} - 18 \beta_{3} - 24 \beta_{6} ) q^{71} + ( -41 + 20 \beta_{5} ) q^{73} + ( -5 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} + 4 \beta_{6} ) q^{74} + ( 5 - 18 \beta_{4} + 17 \beta_{5} - 21 \beta_{7} ) q^{76} + ( 32 \beta_{1} - 37 \beta_{2} - 32 \beta_{3} ) q^{77} + ( -24 - 5 \beta_{5} ) q^{79} + ( 8 \beta_{2} - 40 \beta_{3} + 16 \beta_{6} ) q^{80} + ( -10 + 2 \beta_{5} + 2 \beta_{7} ) q^{82} + ( -2 \beta_{1} + 30 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -16 \beta_{4} + 22 \beta_{7} ) q^{85} + ( 6 \beta_{1} - 12 \beta_{2} + 20 \beta_{3} - 16 \beta_{6} ) q^{86} + ( 52 + 36 \beta_{4} - 16 \beta_{5} + 12 \beta_{7} ) q^{88} + ( 40 \beta_{1} - \beta_{2} + 38 \beta_{3} - 2 \beta_{6} ) q^{89} + ( -21 \beta_{4} - 38 \beta_{7} ) q^{91} + ( 4 \beta_{1} - 16 \beta_{2} + 48 \beta_{3} + 2 \beta_{6} ) q^{92} + ( -69 + 4 \beta_{4} + 21 \beta_{5} + 9 \beta_{7} ) q^{94} + ( 35 \beta_{1} - 8 \beta_{2} + 19 \beta_{3} - 16 \beta_{6} ) q^{95} + ( -9 - 2 \beta_{5} ) q^{97} + ( -18 \beta_{1} + 24 \beta_{3} + 12 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} - 48q^{7} + O(q^{10})$$ $$8q - 8q^{4} - 48q^{7} - 56q^{10} - 88q^{22} + 136q^{25} + 104q^{28} + 208q^{31} + 104q^{34} + 104q^{46} - 48q^{49} - 424q^{52} + 112q^{55} - 64q^{58} - 320q^{64} + 168q^{70} - 328q^{73} + 40q^{76} - 192q^{79} - 80q^{82} + 416q^{88} - 552q^{94} - 72q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{5} + 2 \nu^{3} + 16 \nu$$$$)/12$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} - 10 \nu^{3} + 28 \nu$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{5} - 2 \nu^{3} - 4 \nu$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + 2 \nu^{4} + 2 \nu^{2} + 4$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{5} + 4 \nu^{3} - 4 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{6} + 2 \nu^{4} - 14 \nu^{2} + 32$$$$)/12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + 2 \beta_{2} + 2 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{7} + 3 \beta_{5} - 2 \beta_{4} + 3$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{6} - 3 \beta_{3} - \beta_{2} + 2 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{6} - 6 \beta_{3} + 2 \beta_{2} - 4 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-3 \beta_{7} - 3 \beta_{5} + 4 \beta_{4} + 15$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$8 \beta_{6} + 6 \beta_{3} + 10 \beta_{2} - 8 \beta_{1}$$$$)/3$$

## 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
 −1.38255 + 0.297594i −1.38255 − 0.297594i 0.767178 − 1.18804i 0.767178 + 1.18804i −0.767178 − 1.18804i −0.767178 + 1.18804i 1.38255 + 0.297594i 1.38255 − 0.297594i
−1.68014 1.08495i 0 1.64575 + 3.64575i 8.89047 0 −3.35425 1.19038 7.91094i 0 −14.9373 9.64575i
53.2 −1.68014 + 1.08495i 0 1.64575 3.64575i 8.89047 0 −3.35425 1.19038 + 7.91094i 0 −14.9373 + 9.64575i
53.3 −0.420861 1.95522i 0 −3.64575 + 1.64575i −2.22699 0 −8.64575 4.75216 + 6.43560i 0 0.937254 + 4.35425i
53.4 −0.420861 + 1.95522i 0 −3.64575 1.64575i −2.22699 0 −8.64575 4.75216 6.43560i 0 0.937254 4.35425i
53.5 0.420861 1.95522i 0 −3.64575 1.64575i 2.22699 0 −8.64575 −4.75216 + 6.43560i 0 0.937254 4.35425i
53.6 0.420861 + 1.95522i 0 −3.64575 + 1.64575i 2.22699 0 −8.64575 −4.75216 6.43560i 0 0.937254 + 4.35425i
53.7 1.68014 1.08495i 0 1.64575 3.64575i −8.89047 0 −3.35425 −1.19038 7.91094i 0 −14.9373 + 9.64575i
53.8 1.68014 + 1.08495i 0 1.64575 + 3.64575i −8.89047 0 −3.35425 −1.19038 + 7.91094i 0 −14.9373 9.64575i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.8 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.b even 2 1 inner
24.h 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.h.e 8
3.b odd 2 1 inner 216.3.h.e 8
4.b odd 2 1 864.3.h.f 8
8.b even 2 1 inner 216.3.h.e 8
8.d odd 2 1 864.3.h.f 8
12.b even 2 1 864.3.h.f 8
24.f even 2 1 864.3.h.f 8
24.h odd 2 1 inner 216.3.h.e 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 84 T_{5}^{2} + 392$$ acting on $$S_{3}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + 64 T^{2} + 8 T^{4} + 4 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 392 - 84 T^{2} + T^{4} )^{2}$$
$7$ $$( 29 + 12 T + T^{2} )^{4}$$
$11$ $$( 25992 - 460 T^{2} + T^{4} )^{2}$$
$13$ $$( 106929 + 718 T^{2} + T^{4} )^{2}$$
$17$ $$( 5832 + 236 T^{2} + T^{4} )^{2}$$
$19$ $$( 110889 + 778 T^{2} + T^{4} )^{2}$$
$23$ $$( 763848 + 1748 T^{2} + T^{4} )^{2}$$
$29$ $$( 4608 - 160 T^{2} + T^{4} )^{2}$$
$31$ $$( 424 - 52 T + T^{2} )^{4}$$
$37$ $$( 35721 + 406 T^{2} + T^{4} )^{2}$$
$41$ $$( 1152 + 80 T^{2} + T^{4} )^{2}$$
$43$ $$( 7056 + 2968 T^{2} + T^{4} )^{2}$$
$47$ $$( 622728 + 4212 T^{2} + T^{4} )^{2}$$
$53$ $$( 14709888 - 8368 T^{2} + T^{4} )^{2}$$
$59$ $$( 4135688 - 4140 T^{2} + T^{4} )^{2}$$
$61$ $$( 12510369 + 7326 T^{2} + T^{4} )^{2}$$
$67$ $$( 78269409 + 18594 T^{2} + T^{4} )^{2}$$
$71$ $$( 68024448 + 16848 T^{2} + T^{4} )^{2}$$
$73$ $$( -1119 + 82 T + T^{2} )^{4}$$
$79$ $$( 401 + 48 T + T^{2} )^{4}$$
$83$ $$( 48255488 - 17568 T^{2} + T^{4} )^{2}$$
$89$ $$( 182710728 + 29900 T^{2} + T^{4} )^{2}$$
$97$ $$( 53 + 18 T + T^{2} )^{4}$$