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 $4$

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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:

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

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