Properties

Label 216.2.n.a
Level $216$
Weight $2$
Character orbit 216.n
Analytic conductor $1.725$
Analytic rank $0$
Dimension $4$
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.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + 2 \zeta_{12} q^{4} + 2 \zeta_{12} q^{5} -4 \zeta_{12}^{2} q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + 2 \zeta_{12} q^{4} + 2 \zeta_{12} q^{5} -4 \zeta_{12}^{2} q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( 2 + 2 \zeta_{12}^{3} ) q^{10} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{11} + 2 \zeta_{12} q^{13} + ( 4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{14} + 4 \zeta_{12}^{2} q^{16} -5 q^{17} -\zeta_{12}^{3} q^{19} + 4 \zeta_{12}^{2} q^{20} + ( -3 - 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{22} + ( 2 - 2 \zeta_{12}^{2} ) q^{23} -\zeta_{12}^{2} q^{25} + ( 2 + 2 \zeta_{12}^{3} ) q^{26} -8 \zeta_{12}^{3} q^{28} + ( 4 - 4 \zeta_{12}^{2} ) q^{31} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{32} + ( -5 \zeta_{12} - 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{34} -8 \zeta_{12}^{3} q^{35} -2 \zeta_{12}^{3} q^{37} + ( \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{38} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{40} + ( -5 + 5 \zeta_{12}^{2} ) q^{41} + ( -11 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{43} -6 q^{44} + ( 2 - 2 \zeta_{12}^{3} ) q^{46} -6 \zeta_{12}^{2} q^{47} + ( -9 + 9 \zeta_{12}^{2} ) q^{49} + ( 1 - \zeta_{12} - \zeta_{12}^{2} ) q^{50} + 4 \zeta_{12}^{2} q^{52} -6 q^{55} + ( 8 \zeta_{12} - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{56} + \zeta_{12} q^{59} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{61} + ( 4 - 4 \zeta_{12}^{3} ) q^{62} + 8 \zeta_{12}^{3} q^{64} + 4 \zeta_{12}^{2} q^{65} + 3 \zeta_{12} q^{67} -10 \zeta_{12} q^{68} + ( 8 \zeta_{12} - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{70} + 6 q^{71} + 9 q^{73} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{74} + ( 2 - 2 \zeta_{12}^{2} ) q^{76} + 12 \zeta_{12} q^{77} + 14 \zeta_{12}^{2} q^{79} + 8 \zeta_{12}^{3} q^{80} + ( -5 + 5 \zeta_{12}^{3} ) q^{82} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{83} -10 \zeta_{12} q^{85} + ( -11 - 11 \zeta_{12} + 11 \zeta_{12}^{2} ) q^{86} + ( -6 \zeta_{12} - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{88} + 14 q^{89} -8 \zeta_{12}^{3} q^{91} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{92} + ( 6 - 6 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{94} + ( 2 - 2 \zeta_{12}^{2} ) q^{95} -\zeta_{12}^{2} q^{97} + ( -9 + 9 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 8q^{7} + 8q^{8} + O(q^{10}) \) \( 4q + 2q^{2} - 8q^{7} + 8q^{8} + 8q^{10} + 8q^{14} + 8q^{16} - 20q^{17} + 8q^{20} - 6q^{22} + 4q^{23} - 2q^{25} + 8q^{26} + 8q^{31} - 8q^{32} - 10q^{34} - 2q^{38} - 8q^{40} - 10q^{41} - 24q^{44} + 8q^{46} - 12q^{47} - 18q^{49} + 2q^{50} + 8q^{52} - 24q^{55} - 16q^{56} + 16q^{62} + 8q^{65} - 16q^{70} + 24q^{71} + 36q^{73} - 4q^{74} + 4q^{76} + 28q^{79} - 20q^{82} - 22q^{86} - 12q^{88} + 56q^{89} + 12q^{94} + 4q^{95} - 2q^{97} - 36q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.366025 + 1.36603i 0 −1.73205 1.00000i −1.73205 1.00000i 0 −2.00000 3.46410i 2.00000 2.00000i 0 2.00000 2.00000i
37.2 1.36603 + 0.366025i 0 1.73205 + 1.00000i 1.73205 + 1.00000i 0 −2.00000 3.46410i 2.00000 + 2.00000i 0 2.00000 + 2.00000i
181.1 −0.366025 1.36603i 0 −1.73205 + 1.00000i −1.73205 + 1.00000i 0 −2.00000 + 3.46410i 2.00000 + 2.00000i 0 2.00000 + 2.00000i
181.2 1.36603 0.366025i 0 1.73205 1.00000i 1.73205 1.00000i 0 −2.00000 + 3.46410i 2.00000 2.00000i 0 2.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.n.a 4
3.b odd 2 1 72.2.n.a 4
4.b odd 2 1 864.2.r.a 4
8.b even 2 1 inner 216.2.n.a 4
8.d odd 2 1 864.2.r.a 4
9.c even 3 1 inner 216.2.n.a 4
9.c even 3 1 648.2.d.a 2
9.d odd 6 1 72.2.n.a 4
9.d odd 6 1 648.2.d.d 2
12.b even 2 1 288.2.r.a 4
24.f even 2 1 288.2.r.a 4
24.h odd 2 1 72.2.n.a 4
36.f odd 6 1 864.2.r.a 4
36.f odd 6 1 2592.2.d.a 2
36.h even 6 1 288.2.r.a 4
36.h even 6 1 2592.2.d.b 2
72.j odd 6 1 72.2.n.a 4
72.j odd 6 1 648.2.d.d 2
72.l even 6 1 288.2.r.a 4
72.l even 6 1 2592.2.d.b 2
72.n even 6 1 inner 216.2.n.a 4
72.n even 6 1 648.2.d.a 2
72.p odd 6 1 864.2.r.a 4
72.p odd 6 1 2592.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.a 4 3.b odd 2 1
72.2.n.a 4 9.d odd 6 1
72.2.n.a 4 24.h odd 2 1
72.2.n.a 4 72.j odd 6 1
216.2.n.a 4 1.a even 1 1 trivial
216.2.n.a 4 8.b even 2 1 inner
216.2.n.a 4 9.c even 3 1 inner
216.2.n.a 4 72.n even 6 1 inner
288.2.r.a 4 12.b even 2 1
288.2.r.a 4 24.f even 2 1
288.2.r.a 4 36.h even 6 1
288.2.r.a 4 72.l even 6 1
648.2.d.a 2 9.c even 3 1
648.2.d.a 2 72.n even 6 1
648.2.d.d 2 9.d odd 6 1
648.2.d.d 2 72.j odd 6 1
864.2.r.a 4 4.b odd 2 1
864.2.r.a 4 8.d odd 2 1
864.2.r.a 4 36.f odd 6 1
864.2.r.a 4 72.p odd 6 1
2592.2.d.a 2 36.f odd 6 1
2592.2.d.a 2 72.p odd 6 1
2592.2.d.b 2 36.h even 6 1
2592.2.d.b 2 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 4 T_{5}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 16 - 4 T^{2} + T^{4} \)
$7$ \( ( 16 + 4 T + T^{2} )^{2} \)
$11$ \( 81 - 9 T^{2} + T^{4} \)
$13$ \( 16 - 4 T^{2} + T^{4} \)
$17$ \( ( 5 + T )^{4} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( ( 4 - 2 T + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 16 - 4 T + T^{2} )^{2} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( 25 + 5 T + T^{2} )^{2} \)
$43$ \( 14641 - 121 T^{2} + T^{4} \)
$47$ \( ( 36 + 6 T + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( 1 - T^{2} + T^{4} \)
$61$ \( 20736 - 144 T^{2} + T^{4} \)
$67$ \( 81 - 9 T^{2} + T^{4} \)
$71$ \( ( -6 + T )^{4} \)
$73$ \( ( -9 + T )^{4} \)
$79$ \( ( 196 - 14 T + T^{2} )^{2} \)
$83$ \( 256 - 16 T^{2} + T^{4} \)
$89$ \( ( -14 + T )^{4} \)
$97$ \( ( 1 + T + T^{2} )^{2} \)
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