# Properties

 Label 216.2.d.b Level $216$ Weight $2$ Character orbit 216.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.72476868366$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Defining polynomial: $$x^{4} - 3 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{3} q^{5} + q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{3} q^{5} + q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} + ( 1 - \beta_{2} ) q^{10} -3 \beta_{3} q^{11} + ( -2 + 4 \beta_{2} ) q^{13} + \beta_{1} q^{14} + ( -1 + 3 \beta_{2} ) q^{16} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{17} + ( 2 - 4 \beta_{2} ) q^{19} + ( \beta_{1} - 2 \beta_{3} ) q^{20} + ( 3 - 3 \beta_{2} ) q^{22} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{23} + 4 q^{25} + ( -2 \beta_{1} + 8 \beta_{3} ) q^{26} + ( 1 + \beta_{2} ) q^{28} -6 \beta_{3} q^{29} -7 q^{31} + ( -\beta_{1} + 6 \beta_{3} ) q^{32} + ( -6 - 2 \beta_{2} ) q^{34} -\beta_{3} q^{35} + ( -2 + 4 \beta_{2} ) q^{37} + ( 2 \beta_{1} - 8 \beta_{3} ) q^{38} + ( 3 - \beta_{2} ) q^{40} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{41} + ( 4 - 8 \beta_{2} ) q^{43} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{44} + ( -6 - 2 \beta_{2} ) q^{46} -6 q^{49} + 4 \beta_{1} q^{50} + ( -10 + 6 \beta_{2} ) q^{52} + 9 \beta_{3} q^{53} -3 q^{55} + ( \beta_{1} + 2 \beta_{3} ) q^{56} + ( 6 - 6 \beta_{2} ) q^{58} + 4 \beta_{3} q^{59} -7 \beta_{1} q^{62} + ( -7 + 5 \beta_{2} ) q^{64} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -6 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 1 - \beta_{2} ) q^{70} + ( 12 \beta_{1} - 6 \beta_{3} ) q^{71} + 3 q^{73} + ( -2 \beta_{1} + 8 \beta_{3} ) q^{74} + ( 10 - 6 \beta_{2} ) q^{76} -3 \beta_{3} q^{77} + 4 q^{79} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{80} + ( 6 + 2 \beta_{2} ) q^{82} + 7 \beta_{3} q^{83} + ( -2 + 4 \beta_{2} ) q^{85} + ( 4 \beta_{1} - 16 \beta_{3} ) q^{86} + ( 9 - 3 \beta_{2} ) q^{88} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{89} + ( -2 + 4 \beta_{2} ) q^{91} + ( -6 \beta_{1} - 4 \beta_{3} ) q^{92} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{95} + 7 q^{97} -6 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{4} + 4q^{7} + O(q^{10})$$ $$4q + 6q^{4} + 4q^{7} + 2q^{10} + 2q^{16} + 6q^{22} + 16q^{25} + 6q^{28} - 28q^{31} - 28q^{34} + 10q^{40} - 28q^{46} - 24q^{49} - 28q^{52} - 12q^{55} + 12q^{58} - 18q^{64} + 2q^{70} + 12q^{73} + 28q^{76} + 16q^{79} + 28q^{82} + 30q^{88} + 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 3 x^{2} + 4$$:

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

## 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
 −1.32288 − 0.500000i −1.32288 + 0.500000i 1.32288 − 0.500000i 1.32288 + 0.500000i
−1.32288 0.500000i 0 1.50000 + 1.32288i 1.00000i 0 1.00000 −1.32288 2.50000i 0 0.500000 1.32288i
109.2 −1.32288 + 0.500000i 0 1.50000 1.32288i 1.00000i 0 1.00000 −1.32288 + 2.50000i 0 0.500000 + 1.32288i
109.3 1.32288 0.500000i 0 1.50000 1.32288i 1.00000i 0 1.00000 1.32288 2.50000i 0 0.500000 + 1.32288i
109.4 1.32288 + 0.500000i 0 1.50000 + 1.32288i 1.00000i 0 1.00000 1.32288 + 2.50000i 0 0.500000 1.32288i
 $$n$$: 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
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.b 4
3.b odd 2 1 inner 216.2.d.b 4
4.b odd 2 1 864.2.d.a 4
8.b even 2 1 inner 216.2.d.b 4
8.d odd 2 1 864.2.d.a 4
9.c even 3 2 648.2.n.n 8
9.d odd 6 2 648.2.n.n 8
12.b even 2 1 864.2.d.a 4
16.e even 4 1 6912.2.a.bc 2
16.e even 4 1 6912.2.a.bu 2
16.f odd 4 1 6912.2.a.bd 2
16.f odd 4 1 6912.2.a.bv 2
24.f even 2 1 864.2.d.a 4
24.h odd 2 1 inner 216.2.d.b 4
36.f odd 6 2 2592.2.r.p 8
36.h even 6 2 2592.2.r.p 8
48.i odd 4 1 6912.2.a.bc 2
48.i odd 4 1 6912.2.a.bu 2
48.k even 4 1 6912.2.a.bd 2
48.k even 4 1 6912.2.a.bv 2
72.j odd 6 2 648.2.n.n 8
72.l even 6 2 2592.2.r.p 8
72.n even 6 2 648.2.n.n 8
72.p odd 6 2 2592.2.r.p 8

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 3 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( -1 + T )^{4}$$
$11$ $$( 9 + T^{2} )^{2}$$
$13$ $$( 28 + T^{2} )^{2}$$
$17$ $$( -28 + T^{2} )^{2}$$
$19$ $$( 28 + T^{2} )^{2}$$
$23$ $$( -28 + T^{2} )^{2}$$
$29$ $$( 36 + T^{2} )^{2}$$
$31$ $$( 7 + T )^{4}$$
$37$ $$( 28 + T^{2} )^{2}$$
$41$ $$( -28 + T^{2} )^{2}$$
$43$ $$( 112 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( 81 + T^{2} )^{2}$$
$59$ $$( 16 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$( -252 + T^{2} )^{2}$$
$73$ $$( -3 + T )^{4}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$( 49 + T^{2} )^{2}$$
$89$ $$( -112 + T^{2} )^{2}$$
$97$ $$( -7 + T )^{4}$$