Properties

Label 216.2.i.b
Level $216$
Weight $2$
Character orbit 216.i
Analytic conductor $1.725$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{3} ) q^{5} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{3} ) q^{5} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + ( 1 - \beta_{1} ) q^{11} + ( -3 \beta_{1} + \beta_{3} ) q^{13} + ( 3 + \beta_{2} ) q^{17} + ( 3 - \beta_{2} ) q^{19} + ( 3 \beta_{1} - \beta_{3} ) q^{23} + ( -3 + 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{25} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{29} + ( -3 \beta_{1} - \beta_{3} ) q^{31} + ( -8 - \beta_{2} ) q^{35} + ( 4 + 2 \beta_{2} ) q^{37} + ( -5 \beta_{1} - 2 \beta_{3} ) q^{41} + ( -3 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{47} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{49} + ( 4 - 2 \beta_{2} ) q^{53} + \beta_{2} q^{55} + 7 \beta_{1} q^{59} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{61} + ( -8 + 11 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{65} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{67} + 4 q^{71} + ( -5 - 3 \beta_{2} ) q^{73} + ( \beta_{1} + \beta_{3} ) q^{77} + ( 4 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{79} + ( 12 - 13 \beta_{1} - \beta_{2} + \beta_{3} ) q^{83} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{85} -6 q^{89} + ( -4 + \beta_{2} ) q^{91} + ( -12 \beta_{1} + 4 \beta_{3} ) q^{95} + ( 3 - 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{5} - 3q^{7} + O(q^{10}) \) \( 4q - q^{5} - 3q^{7} + 2q^{11} - 5q^{13} + 10q^{17} + 14q^{19} + 5q^{23} - 7q^{25} - 3q^{29} - 7q^{31} - 30q^{35} + 12q^{37} - 12q^{41} - 8q^{43} + 3q^{47} - 7q^{49} + 20q^{53} - 2q^{55} + 14q^{59} + q^{61} - 19q^{65} - 4q^{67} + 16q^{71} - 14q^{73} + 3q^{77} + 7q^{79} + 25q^{83} + 14q^{85} - 24q^{89} - 18q^{91} - 20q^{95} + 8q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + \nu^{2} + 2 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11\)\()/3\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
0 0 0 −1.68614 2.92048i 0 0.686141 1.18843i 0 0 0
73.2 0 0 0 1.18614 + 2.05446i 0 −2.18614 + 3.78651i 0 0 0
145.1 0 0 0 −1.68614 + 2.92048i 0 0.686141 + 1.18843i 0 0 0
145.2 0 0 0 1.18614 2.05446i 0 −2.18614 3.78651i 0 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.i.b 4
3.b odd 2 1 72.2.i.b 4
4.b odd 2 1 432.2.i.d 4
8.b even 2 1 1728.2.i.i 4
8.d odd 2 1 1728.2.i.j 4
9.c even 3 1 inner 216.2.i.b 4
9.c even 3 1 648.2.a.g 2
9.d odd 6 1 72.2.i.b 4
9.d odd 6 1 648.2.a.f 2
12.b even 2 1 144.2.i.d 4
24.f even 2 1 576.2.i.l 4
24.h odd 2 1 576.2.i.j 4
36.f odd 6 1 432.2.i.d 4
36.f odd 6 1 1296.2.a.p 2
36.h even 6 1 144.2.i.d 4
36.h even 6 1 1296.2.a.n 2
72.j odd 6 1 576.2.i.j 4
72.j odd 6 1 5184.2.a.bt 2
72.l even 6 1 576.2.i.l 4
72.l even 6 1 5184.2.a.bs 2
72.n even 6 1 1728.2.i.i 4
72.n even 6 1 5184.2.a.bp 2
72.p odd 6 1 1728.2.i.j 4
72.p odd 6 1 5184.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.b 4 3.b odd 2 1
72.2.i.b 4 9.d odd 6 1
144.2.i.d 4 12.b even 2 1
144.2.i.d 4 36.h even 6 1
216.2.i.b 4 1.a even 1 1 trivial
216.2.i.b 4 9.c even 3 1 inner
432.2.i.d 4 4.b odd 2 1
432.2.i.d 4 36.f odd 6 1
576.2.i.j 4 24.h odd 2 1
576.2.i.j 4 72.j odd 6 1
576.2.i.l 4 24.f even 2 1
576.2.i.l 4 72.l even 6 1
648.2.a.f 2 9.d odd 6 1
648.2.a.g 2 9.c even 3 1
1296.2.a.n 2 36.h even 6 1
1296.2.a.p 2 36.f odd 6 1
1728.2.i.i 4 8.b even 2 1
1728.2.i.i 4 72.n even 6 1
1728.2.i.j 4 8.d odd 2 1
1728.2.i.j 4 72.p odd 6 1
5184.2.a.bo 2 72.p odd 6 1
5184.2.a.bp 2 72.n even 6 1
5184.2.a.bs 2 72.l even 6 1
5184.2.a.bt 2 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + T_{5}^{3} + 9 T_{5}^{2} - 8 T_{5} + 64 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 64 - 8 T + 9 T^{2} + T^{3} + T^{4} \)
$7$ \( 36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( 4 - 10 T + 27 T^{2} + 5 T^{3} + T^{4} \)
$17$ \( ( -2 - 5 T + T^{2} )^{2} \)
$19$ \( ( 4 - 7 T + T^{2} )^{2} \)
$23$ \( 4 + 10 T + 27 T^{2} - 5 T^{3} + T^{4} \)
$29$ \( 36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4} \)
$31$ \( 16 + 28 T + 45 T^{2} + 7 T^{3} + T^{4} \)
$37$ \( ( -24 - 6 T + T^{2} )^{2} \)
$41$ \( 9 + 36 T + 141 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( 289 - 136 T + 81 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} \)
$53$ \( ( -8 - 10 T + T^{2} )^{2} \)
$59$ \( ( 49 - 7 T + T^{2} )^{2} \)
$61$ \( 64 + 8 T + 9 T^{2} - T^{3} + T^{4} \)
$67$ \( 841 - 116 T + 45 T^{2} + 4 T^{3} + T^{4} \)
$71$ \( ( -4 + T )^{4} \)
$73$ \( ( -62 + 7 T + T^{2} )^{2} \)
$79$ \( 16 - 28 T + 45 T^{2} - 7 T^{3} + T^{4} \)
$83$ \( 21904 - 3700 T + 477 T^{2} - 25 T^{3} + T^{4} \)
$89$ \( ( 6 + T )^{4} \)
$97$ \( 289 + 136 T + 81 T^{2} - 8 T^{3} + T^{4} \)
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