Properties

Label 216.1.r.a
Level $216$
Weight $1$
Character orbit 216.r
Analytic conductor $0.108$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 216.r (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.107798042729\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.128536820158464.7

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{18}^{2} q^{2} + \zeta_{18}^{8} q^{3} + \zeta_{18}^{4} q^{4} -\zeta_{18} q^{6} + \zeta_{18}^{6} q^{8} -\zeta_{18}^{7} q^{9} +O(q^{10})\) \( q + \zeta_{18}^{2} q^{2} + \zeta_{18}^{8} q^{3} + \zeta_{18}^{4} q^{4} -\zeta_{18} q^{6} + \zeta_{18}^{6} q^{8} -\zeta_{18}^{7} q^{9} + ( -\zeta_{18}^{3} + \zeta_{18}^{4} ) q^{11} -\zeta_{18}^{3} q^{12} + \zeta_{18}^{8} q^{16} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{17} + q^{18} + ( -\zeta_{18}^{5} - \zeta_{18}^{7} ) q^{19} + ( -\zeta_{18}^{5} + \zeta_{18}^{6} ) q^{22} -\zeta_{18}^{5} q^{24} + \zeta_{18}^{2} q^{25} + \zeta_{18}^{6} q^{27} -\zeta_{18} q^{32} + ( \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{33} + ( -\zeta_{18}^{3} - \zeta_{18}^{7} ) q^{34} + \zeta_{18}^{2} q^{36} + ( 1 - \zeta_{18}^{7} ) q^{38} + ( \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{41} + ( -\zeta_{18} + \zeta_{18}^{6} ) q^{43} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{44} -\zeta_{18}^{7} q^{48} -\zeta_{18} q^{49} + \zeta_{18}^{4} q^{50} + ( 1 + \zeta_{18}^{4} ) q^{51} + \zeta_{18}^{8} q^{54} + ( \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{57} + ( 1 + \zeta_{18}^{2} ) q^{59} -\zeta_{18}^{3} q^{64} + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{66} + ( \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{67} + ( 1 - \zeta_{18}^{5} ) q^{68} + \zeta_{18}^{4} q^{72} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{73} -\zeta_{18} q^{75} + ( 1 + \zeta_{18}^{2} ) q^{76} -\zeta_{18}^{5} q^{81} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{82} -\zeta_{18}^{2} q^{83} + ( -\zeta_{18}^{3} + \zeta_{18}^{8} ) q^{86} + ( 1 - \zeta_{18} ) q^{88} -\zeta_{18}^{6} q^{89} + q^{96} + ( -\zeta_{18}^{3} + \zeta_{18}^{4} ) q^{97} -\zeta_{18}^{3} q^{98} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{8} + O(q^{10}) \) \( 6q - 3q^{8} - 3q^{11} - 3q^{12} + 6q^{18} - 3q^{22} - 3q^{27} - 3q^{33} - 3q^{34} + 6q^{38} - 3q^{41} - 3q^{43} + 6q^{51} - 3q^{57} + 6q^{59} - 3q^{64} - 3q^{67} + 6q^{68} + 6q^{76} - 3q^{86} + 6q^{88} + 3q^{89} + 6q^{96} - 3q^{97} - 3q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{18}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.939693 + 0.342020i
−0.766044 0.642788i
−0.173648 0.984808i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0.939693 0.342020i
0.766044 + 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i 0 −0.939693 0.342020i 0 −0.500000 + 0.866025i 0.766044 0.642788i 0
67.1 0.173648 + 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i 0 0.766044 + 0.642788i 0 −0.500000 0.866025i 0.173648 0.984808i 0
115.1 −0.939693 + 0.342020i 0.173648 0.984808i 0.766044 0.642788i 0 0.173648 + 0.984808i 0 −0.500000 + 0.866025i −0.939693 0.342020i 0
139.1 −0.939693 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i 0 0.173648 0.984808i 0 −0.500000 0.866025i −0.939693 + 0.342020i 0
187.1 0.173648 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i 0 0.766044 0.642788i 0 −0.500000 + 0.866025i 0.173648 + 0.984808i 0
211.1 0.766044 0.642788i −0.939693 0.342020i 0.173648 0.984808i 0 −0.939693 + 0.342020i 0 −0.500000 0.866025i 0.766044 + 0.642788i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
27.e even 9 1 inner
216.r odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.1.r.a 6
3.b odd 2 1 648.1.r.a 6
4.b odd 2 1 864.1.bd.a 6
8.b even 2 1 864.1.bd.a 6
8.d odd 2 1 CM 216.1.r.a 6
9.c even 3 1 1944.1.r.a 6
9.c even 3 1 1944.1.r.b 6
9.d odd 6 1 1944.1.r.c 6
9.d odd 6 1 1944.1.r.d 6
12.b even 2 1 2592.1.bd.a 6
24.f even 2 1 648.1.r.a 6
24.h odd 2 1 2592.1.bd.a 6
27.e even 9 1 inner 216.1.r.a 6
27.e even 9 1 1944.1.r.a 6
27.e even 9 1 1944.1.r.b 6
27.f odd 18 1 648.1.r.a 6
27.f odd 18 1 1944.1.r.c 6
27.f odd 18 1 1944.1.r.d 6
72.l even 6 1 1944.1.r.c 6
72.l even 6 1 1944.1.r.d 6
72.p odd 6 1 1944.1.r.a 6
72.p odd 6 1 1944.1.r.b 6
108.j odd 18 1 864.1.bd.a 6
108.l even 18 1 2592.1.bd.a 6
216.r odd 18 1 inner 216.1.r.a 6
216.r odd 18 1 1944.1.r.a 6
216.r odd 18 1 1944.1.r.b 6
216.t even 18 1 864.1.bd.a 6
216.v even 18 1 648.1.r.a 6
216.v even 18 1 1944.1.r.c 6
216.v even 18 1 1944.1.r.d 6
216.x odd 18 1 2592.1.bd.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.1.r.a 6 1.a even 1 1 trivial
216.1.r.a 6 8.d odd 2 1 CM
216.1.r.a 6 27.e even 9 1 inner
216.1.r.a 6 216.r odd 18 1 inner
648.1.r.a 6 3.b odd 2 1
648.1.r.a 6 24.f even 2 1
648.1.r.a 6 27.f odd 18 1
648.1.r.a 6 216.v even 18 1
864.1.bd.a 6 4.b odd 2 1
864.1.bd.a 6 8.b even 2 1
864.1.bd.a 6 108.j odd 18 1
864.1.bd.a 6 216.t even 18 1
1944.1.r.a 6 9.c even 3 1
1944.1.r.a 6 27.e even 9 1
1944.1.r.a 6 72.p odd 6 1
1944.1.r.a 6 216.r odd 18 1
1944.1.r.b 6 9.c even 3 1
1944.1.r.b 6 27.e even 9 1
1944.1.r.b 6 72.p odd 6 1
1944.1.r.b 6 216.r odd 18 1
1944.1.r.c 6 9.d odd 6 1
1944.1.r.c 6 27.f odd 18 1
1944.1.r.c 6 72.l even 6 1
1944.1.r.c 6 216.v even 18 1
1944.1.r.d 6 9.d odd 6 1
1944.1.r.d 6 27.f odd 18 1
1944.1.r.d 6 72.l even 6 1
1944.1.r.d 6 216.v even 18 1
2592.1.bd.a 6 12.b even 2 1
2592.1.bd.a 6 24.h odd 2 1
2592.1.bd.a 6 108.l even 18 1
2592.1.bd.a 6 216.x odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{3} + T^{6} \)
$3$ \( 1 + T^{3} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( T^{6} \)
$11$ \( 1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
$19$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
$23$ \( T^{6} \)
$29$ \( T^{6} \)
$31$ \( T^{6} \)
$37$ \( T^{6} \)
$41$ \( 1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
$43$ \( 1 - 3 T + 3 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
$47$ \( T^{6} \)
$53$ \( T^{6} \)
$59$ \( 1 - 3 T + 12 T^{2} - 19 T^{3} + 15 T^{4} - 6 T^{5} + T^{6} \)
$61$ \( T^{6} \)
$67$ \( 1 - 3 T + 3 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
$71$ \( T^{6} \)
$73$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
$79$ \( T^{6} \)
$83$ \( 1 - T^{3} + T^{6} \)
$89$ \( ( 1 - T + T^{2} )^{3} \)
$97$ \( 1 + 6 T + 12 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
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