Properties

 Label 216.3.p.a Level $216$ Weight $3$ Character orbit 216.p Analytic conductor $5.886$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 216.p (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$5.88557371018$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q -2 \beta_{1} q^{2} + ( -4 + 4 \beta_{1} ) q^{4} + 8 q^{8} +O(q^{10})$$ $$q -2 \beta_{1} q^{2} + ( -4 + 4 \beta_{1} ) q^{4} + 8 q^{8} + ( -7 \beta_{1} + \beta_{2} ) q^{11} -16 \beta_{1} q^{16} + ( 1 + 2 \beta_{3} ) q^{17} + ( 17 + \beta_{3} ) q^{19} + ( -14 + 14 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{22} -25 \beta_{1} q^{25} + ( -32 + 32 \beta_{1} ) q^{32} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{34} + ( -34 \beta_{1} - 2 \beta_{2} ) q^{38} + ( 23 - 23 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{41} + ( 7 \beta_{1} + 5 \beta_{2} ) q^{43} + ( 28 - 4 \beta_{3} ) q^{44} + ( -49 + 49 \beta_{1} ) q^{49} + ( -50 + 50 \beta_{1} ) q^{50} + ( 41 - 41 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{59} + 64 q^{64} + ( 31 - 31 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{67} + ( -4 + 4 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{68} + ( 71 - 2 \beta_{3} ) q^{73} + ( -68 + 68 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{76} + ( -46 - 8 \beta_{3} ) q^{82} + 158 \beta_{1} q^{83} + ( 14 - 14 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{86} + ( -56 \beta_{1} + 8 \beta_{2} ) q^{88} -146 q^{89} + ( -47 \beta_{1} - 10 \beta_{2} ) q^{97} + 98 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 8q^{4} + 32q^{8} + O(q^{10})$$ $$4q - 4q^{2} - 8q^{4} + 32q^{8} - 14q^{11} - 32q^{16} + 4q^{17} + 68q^{19} - 28q^{22} - 50q^{25} - 64q^{32} - 4q^{34} - 68q^{38} + 46q^{41} + 14q^{43} + 112q^{44} - 98q^{49} - 100q^{50} + 82q^{59} + 256q^{64} + 62q^{67} - 8q^{68} + 284q^{73} - 136q^{76} - 184q^{82} + 316q^{83} + 28q^{86} - 112q^{88} - 584q^{89} - 94q^{97} + 392q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$3 \nu^{3} + 6 \nu$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{3} + 12 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/18$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2}$$$$)/9$$

Character values

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

 $$n$$ $$55$$ $$109$$ $$137$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1 + \beta_{1}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.00000 + 1.73205i 0 −2.00000 3.46410i 0 0 0 8.00000 0 0
19.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 0 0 0 8.00000 0 0
91.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i 0 0 0 8.00000 0 0
91.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 0 0 0 8.00000 0 0
 $$n$$: 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.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
9.c even 3 1 inner
72.p odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.p.a 4
3.b odd 2 1 72.3.p.a 4
4.b odd 2 1 864.3.t.a 4
8.b even 2 1 864.3.t.a 4
8.d odd 2 1 CM 216.3.p.a 4
9.c even 3 1 inner 216.3.p.a 4
9.c even 3 1 648.3.b.b 2
9.d odd 6 1 72.3.p.a 4
9.d odd 6 1 648.3.b.a 2
12.b even 2 1 288.3.t.a 4
24.f even 2 1 72.3.p.a 4
24.h odd 2 1 288.3.t.a 4
36.f odd 6 1 864.3.t.a 4
36.f odd 6 1 2592.3.b.a 2
36.h even 6 1 288.3.t.a 4
36.h even 6 1 2592.3.b.b 2
72.j odd 6 1 288.3.t.a 4
72.j odd 6 1 2592.3.b.b 2
72.l even 6 1 72.3.p.a 4
72.l even 6 1 648.3.b.a 2
72.n even 6 1 864.3.t.a 4
72.n even 6 1 2592.3.b.a 2
72.p odd 6 1 inner 216.3.p.a 4
72.p odd 6 1 648.3.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.p.a 4 3.b odd 2 1
72.3.p.a 4 9.d odd 6 1
72.3.p.a 4 24.f even 2 1
72.3.p.a 4 72.l even 6 1
216.3.p.a 4 1.a even 1 1 trivial
216.3.p.a 4 8.d odd 2 1 CM
216.3.p.a 4 9.c even 3 1 inner
216.3.p.a 4 72.p odd 6 1 inner
288.3.t.a 4 12.b even 2 1
288.3.t.a 4 24.h odd 2 1
288.3.t.a 4 36.h even 6 1
288.3.t.a 4 72.j odd 6 1
648.3.b.a 2 9.d odd 6 1
648.3.b.a 2 72.l even 6 1
648.3.b.b 2 9.c even 3 1
648.3.b.b 2 72.p odd 6 1
864.3.t.a 4 4.b odd 2 1
864.3.t.a 4 8.b even 2 1
864.3.t.a 4 36.f odd 6 1
864.3.t.a 4 72.n even 6 1
2592.3.b.a 2 36.f odd 6 1
2592.3.b.a 2 72.n even 6 1
2592.3.b.b 2 36.h even 6 1
2592.3.b.b 2 72.j odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$27889 - 2338 T + 363 T^{2} + 14 T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -863 - 2 T + T^{2} )^{2}$$
$19$ $$( 73 - 34 T + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$8567329 + 134642 T + 5043 T^{2} - 46 T^{3} + T^{4}$$
$43$ $$28633201 + 74914 T + 5547 T^{2} - 14 T^{3} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$13830961 + 304958 T + 10443 T^{2} - 82 T^{3} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$92602129 + 596626 T + 13467 T^{2} - 62 T^{3} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$( 4177 - 142 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$( 24964 - 158 T + T^{2} )^{2}$$
$89$ $$( 146 + T )^{4}$$
$97$ $$376010881 - 1822754 T + 28227 T^{2} + 94 T^{3} + T^{4}$$