Properties

Label 216.2.d.a
Level $216$
Weight $2$
Character orbit 216.d
Analytic conductor $1.725$
Analytic rank $0$
Dimension $4$
CM discriminant -24
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} + ( -\zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( -1 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} + ( -\zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( -1 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( 2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{10} + ( 2 \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{11} + ( -\zeta_{8} - 6 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{14} + 4 q^{16} + ( 2 \zeta_{8} - 6 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{20} + ( -4 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{22} + ( -6 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{25} + ( 2 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{28} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( 5 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{31} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{32} + ( -8 \zeta_{8} + 3 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{35} + ( -4 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{40} + ( -4 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{44} + ( 12 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{49} + ( -6 \zeta_{8} + 12 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{50} + ( 5 \zeta_{8} + 3 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{53} + ( -5 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{55} + ( 2 \zeta_{8} + 12 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{56} -4 q^{58} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{59} + ( 5 \zeta_{8} - 6 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{62} -8 q^{64} + ( 16 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{70} + ( -7 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{73} + ( -11 \zeta_{8} - 15 \zeta_{8}^{2} - 11 \zeta_{8}^{3} ) q^{77} -10 q^{79} + ( -4 \zeta_{8} + 12 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{80} + ( 2 \zeta_{8} - 15 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{83} + ( 8 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{88} + ( -1 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{97} + ( 12 \zeta_{8} + 12 \zeta_{8}^{2} + 12 \zeta_{8}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} - 4q^{7} + O(q^{10}) \) \( 4q - 8q^{4} - 4q^{7} + 8q^{10} + 16q^{16} - 16q^{22} - 24q^{25} + 8q^{28} + 20q^{31} - 16q^{40} + 48q^{49} - 20q^{55} - 16q^{58} - 32q^{64} + 64q^{70} - 28q^{73} - 40q^{79} + 32q^{88} - 4q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
1.41421i 0 −2.00000 1.58579i 0 −5.24264 2.82843i 0 −2.24264
109.2 1.41421i 0 −2.00000 4.41421i 0 3.24264 2.82843i 0 6.24264
109.3 1.41421i 0 −2.00000 4.41421i 0 3.24264 2.82843i 0 6.24264
109.4 1.41421i 0 −2.00000 1.58579i 0 −5.24264 2.82843i 0 −2.24264
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 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.a 4
3.b odd 2 1 inner 216.2.d.a 4
4.b odd 2 1 864.2.d.b 4
8.b even 2 1 inner 216.2.d.a 4
8.d odd 2 1 864.2.d.b 4
9.c even 3 2 648.2.n.p 8
9.d odd 6 2 648.2.n.p 8
12.b even 2 1 864.2.d.b 4
16.e even 4 1 6912.2.a.z 2
16.e even 4 1 6912.2.a.bz 2
16.f odd 4 1 6912.2.a.y 2
16.f odd 4 1 6912.2.a.by 2
24.f even 2 1 864.2.d.b 4
24.h odd 2 1 CM 216.2.d.a 4
36.f odd 6 2 2592.2.r.o 8
36.h even 6 2 2592.2.r.o 8
48.i odd 4 1 6912.2.a.z 2
48.i odd 4 1 6912.2.a.bz 2
48.k even 4 1 6912.2.a.y 2
48.k even 4 1 6912.2.a.by 2
72.j odd 6 2 648.2.n.p 8
72.l even 6 2 2592.2.r.o 8
72.n even 6 2 648.2.n.p 8
72.p odd 6 2 2592.2.r.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.a 4 1.a even 1 1 trivial
216.2.d.a 4 3.b odd 2 1 inner
216.2.d.a 4 8.b even 2 1 inner
216.2.d.a 4 24.h odd 2 1 CM
648.2.n.p 8 9.c even 3 2
648.2.n.p 8 9.d odd 6 2
648.2.n.p 8 72.j odd 6 2
648.2.n.p 8 72.n even 6 2
864.2.d.b 4 4.b odd 2 1
864.2.d.b 4 8.d odd 2 1
864.2.d.b 4 12.b even 2 1
864.2.d.b 4 24.f even 2 1
2592.2.r.o 8 36.f odd 6 2
2592.2.r.o 8 36.h even 6 2
2592.2.r.o 8 72.l even 6 2
2592.2.r.o 8 72.p odd 6 2
6912.2.a.y 2 16.f odd 4 1
6912.2.a.y 2 48.k even 4 1
6912.2.a.z 2 16.e even 4 1
6912.2.a.z 2 48.i odd 4 1
6912.2.a.by 2 16.f odd 4 1
6912.2.a.by 2 48.k even 4 1
6912.2.a.bz 2 16.e even 4 1
6912.2.a.bz 2 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 22 T_{5}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 49 + 22 T^{2} + T^{4} \)
$7$ \( ( -17 + 2 T + T^{2} )^{2} \)
$11$ \( 1 + 34 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 8 + T^{2} )^{2} \)
$31$ \( ( 7 - 10 T + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 1681 + 118 T^{2} + T^{4} \)
$59$ \( ( 128 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -23 + 14 T + T^{2} )^{2} \)
$79$ \( ( 10 + T )^{4} \)
$83$ \( 47089 + 466 T^{2} + T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( -287 + 2 T + T^{2} )^{2} \)
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