# Properties

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

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.72476868366$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + \beta_{4} q^{5} + ( -\beta_{2} - \beta_{6} ) q^{7} + \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + \beta_{4} q^{5} + ( -\beta_{2} - \beta_{6} ) q^{7} + \beta_{3} q^{8} + ( -2 + \beta_{2} + \beta_{5} - \beta_{6} ) q^{10} + ( \beta_{1} - \beta_{4} - \beta_{7} ) q^{11} + ( 1 - \beta_{2} - \beta_{5} + \beta_{6} ) q^{13} + ( -\beta_{3} - \beta_{7} ) q^{14} + ( \beta_{2} + \beta_{5} + \beta_{6} ) q^{16} + ( 2 \beta_{3} - \beta_{4} ) q^{17} + ( 2 - 2 \beta_{2} - \beta_{5} + 2 \beta_{6} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{4} ) q^{20} + ( -2 + \beta_{2} - \beta_{5} + \beta_{6} ) q^{22} + ( 3 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{23} + ( -1 - 2 \beta_{2} - 2 \beta_{6} ) q^{25} + ( \beta_{1} - 2 \beta_{4} ) q^{26} + ( -4 - \beta_{5} - \beta_{6} ) q^{28} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{29} + 2 q^{31} + ( 2 \beta_{4} + 2 \beta_{7} ) q^{32} + ( 2 + \beta_{2} + \beta_{5} + 3 \beta_{6} ) q^{34} + ( \beta_{1} - 3 \beta_{4} - \beta_{7} ) q^{35} + ( 1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{37} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{38} + ( -4 + 2 \beta_{5} - 2 \beta_{6} ) q^{40} + ( -6 \beta_{1} - 2 \beta_{7} ) q^{41} + ( -2 + 2 \beta_{2} - 2 \beta_{6} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{44} + ( 2 + \beta_{2} - \beta_{5} - 3 \beta_{6} ) q^{46} + ( -3 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{47} + ( -\beta_{1} - 2 \beta_{3} - 2 \beta_{7} ) q^{50} + ( 4 - \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{52} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{53} + ( 2 + 2 \beta_{2} + 2 \beta_{6} ) q^{55} + ( -4 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{56} + ( 4 - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} ) q^{58} + ( -\beta_{1} - 3 \beta_{4} + \beta_{7} ) q^{59} + ( -3 + 3 \beta_{2} - \beta_{5} - 3 \beta_{6} ) q^{61} + 2 \beta_{1} q^{62} + ( 4 + 2 \beta_{5} - 2 \beta_{6} ) q^{64} + ( 6 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{65} + \beta_{5} q^{67} + ( 2 \beta_{1} + 2 \beta_{4} + 4 \beta_{7} ) q^{68} + ( 2 - \beta_{2} - 3 \beta_{5} + 3 \beta_{6} ) q^{70} + ( 1 + 4 \beta_{2} + 4 \beta_{6} ) q^{73} + ( \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{74} + ( 8 - 2 \beta_{2} - 3 \beta_{5} + \beta_{6} ) q^{76} + ( 2 \beta_{1} + \beta_{4} - 2 \beta_{7} ) q^{77} + ( -6 - \beta_{2} - \beta_{6} ) q^{79} + ( -4 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} ) q^{80} + ( -8 - 4 \beta_{2} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{7} ) q^{83} + ( -2 + 2 \beta_{2} + 4 \beta_{5} - 2 \beta_{6} ) q^{85} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{7} ) q^{86} + ( 4 - 2 \beta_{2} + 4 \beta_{6} ) q^{88} + ( -2 \beta_{3} + \beta_{4} ) q^{89} + ( -2 + 2 \beta_{2} + 3 \beta_{5} - 2 \beta_{6} ) q^{91} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{7} ) q^{92} + ( -6 - 3 \beta_{2} - \beta_{5} - 3 \beta_{6} ) q^{94} + ( 9 \beta_{1} + 2 \beta_{3} - \beta_{4} + 3 \beta_{7} ) q^{95} + ( -3 + 2 \beta_{2} + 2 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + O(q^{10})$$ $$8q + 4q^{4} - 8q^{10} - 16q^{22} - 8q^{25} - 28q^{28} + 16q^{31} + 8q^{34} - 24q^{40} + 32q^{46} + 20q^{52} + 16q^{55} + 32q^{58} + 40q^{64} + 8q^{73} + 52q^{76} - 48q^{79} - 80q^{82} + 8q^{88} - 48q^{94} - 24q^{97} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −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
−1.38255 0.297594i 0 1.82288 + 0.822876i 3.36028i 0 −2.64575 −2.27533 1.68014i 0 −1.00000 + 4.64575i
109.2 −1.38255 + 0.297594i 0 1.82288 0.822876i 3.36028i 0 −2.64575 −2.27533 + 1.68014i 0 −1.00000 4.64575i
109.3 −0.767178 1.18804i 0 −0.822876 + 1.82288i 0.841723i 0 2.64575 2.79694 0.420861i 0 −1.00000 + 0.645751i
109.4 −0.767178 + 1.18804i 0 −0.822876 1.82288i 0.841723i 0 2.64575 2.79694 + 0.420861i 0 −1.00000 0.645751i
109.5 0.767178 1.18804i 0 −0.822876 1.82288i 0.841723i 0 2.64575 −2.79694 0.420861i 0 −1.00000 0.645751i
109.6 0.767178 + 1.18804i 0 −0.822876 + 1.82288i 0.841723i 0 2.64575 −2.79694 + 0.420861i 0 −1.00000 + 0.645751i
109.7 1.38255 0.297594i 0 1.82288 0.822876i 3.36028i 0 −2.64575 2.27533 1.68014i 0 −1.00000 4.64575i
109.8 1.38255 + 0.297594i 0 1.82288 + 0.822876i 3.36028i 0 −2.64575 2.27533 + 1.68014i 0 −1.00000 + 4.64575i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.8 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 216.2.d.c 8
3.b odd 2 1 inner 216.2.d.c 8
4.b odd 2 1 864.2.d.c 8
8.b even 2 1 inner 216.2.d.c 8
8.d odd 2 1 864.2.d.c 8
9.c even 3 2 648.2.n.q 16
9.d odd 6 2 648.2.n.q 16
12.b even 2 1 864.2.d.c 8
16.e even 4 1 6912.2.a.cc 4
16.e even 4 1 6912.2.a.cj 4
16.f odd 4 1 6912.2.a.cd 4
16.f odd 4 1 6912.2.a.ci 4
24.f even 2 1 864.2.d.c 8
24.h odd 2 1 inner 216.2.d.c 8
36.f odd 6 2 2592.2.r.q 16
36.h even 6 2 2592.2.r.q 16
48.i odd 4 1 6912.2.a.cc 4
48.i odd 4 1 6912.2.a.cj 4
48.k even 4 1 6912.2.a.cd 4
48.k even 4 1 6912.2.a.ci 4
72.j odd 6 2 648.2.n.q 16
72.l even 6 2 2592.2.r.q 16
72.n even 6 2 648.2.n.q 16
72.p odd 6 2 2592.2.r.q 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.c 8 1.a even 1 1 trivial
216.2.d.c 8 3.b odd 2 1 inner
216.2.d.c 8 8.b even 2 1 inner
216.2.d.c 8 24.h odd 2 1 inner
648.2.n.q 16 9.c even 3 2
648.2.n.q 16 9.d odd 6 2
648.2.n.q 16 72.j odd 6 2
648.2.n.q 16 72.n even 6 2
864.2.d.c 8 4.b odd 2 1
864.2.d.c 8 8.d odd 2 1
864.2.d.c 8 12.b even 2 1
864.2.d.c 8 24.f even 2 1
2592.2.r.q 16 36.f odd 6 2
2592.2.r.q 16 36.h even 6 2
2592.2.r.q 16 72.l even 6 2
2592.2.r.q 16 72.p odd 6 2
6912.2.a.cc 4 16.e even 4 1
6912.2.a.cc 4 48.i odd 4 1
6912.2.a.cd 4 16.f odd 4 1
6912.2.a.cd 4 48.k even 4 1
6912.2.a.ci 4 16.f odd 4 1
6912.2.a.ci 4 48.k even 4 1
6912.2.a.cj 4 16.e even 4 1
6912.2.a.cj 4 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 12 T_{5}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 8 T^{2} + 2 T^{4} - 2 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 8 + 12 T^{2} + T^{4} )^{2}$$
$7$ $$( -7 + T^{2} )^{4}$$
$11$ $$( 72 + 20 T^{2} + T^{4} )^{2}$$
$13$ $$( 9 + 22 T^{2} + T^{4} )^{2}$$
$17$ $$( 648 - 52 T^{2} + T^{4} )^{2}$$
$19$ $$( 729 + 58 T^{2} + T^{4} )^{2}$$
$23$ $$( 72 - 76 T^{2} + T^{4} )^{2}$$
$29$ $$( 1152 + 80 T^{2} + T^{4} )^{2}$$
$31$ $$( -2 + T )^{8}$$
$37$ $$( 81 + 46 T^{2} + T^{4} )^{2}$$
$41$ $$( 4608 - 160 T^{2} + T^{4} )^{2}$$
$43$ $$( 576 + 64 T^{2} + T^{4} )^{2}$$
$47$ $$( 648 - 108 T^{2} + T^{4} )^{2}$$
$53$ $$( 1152 + 80 T^{2} + T^{4} )^{2}$$
$59$ $$( 6728 + 180 T^{2} + T^{4} )^{2}$$
$61$ $$( 729 + 198 T^{2} + T^{4} )^{2}$$
$67$ $$( 9 + T^{2} )^{4}$$
$71$ $$T^{8}$$
$73$ $$( -111 - 2 T + T^{2} )^{4}$$
$79$ $$( 29 + 12 T + T^{2} )^{4}$$
$83$ $$( 512 + 96 T^{2} + T^{4} )^{2}$$
$89$ $$( 648 - 52 T^{2} + T^{4} )^{2}$$
$97$ $$( -19 + 6 T + T^{2} )^{4}$$
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