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: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{7} + ( - \beta_1 + 1) q^{11} + (\beta_{3} - 3 \beta_1) q^{13} + (\beta_{2} + 3) q^{17} + ( - \beta_{2} + 3) q^{19} + ( - \beta_{3} + 3 \beta_1) q^{23} + ( - \beta_{3} + \beta_{2} + 4 \beta_1 - 3) q^{25} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{29} + ( - \beta_{3} - 3 \beta_1) q^{31} + ( - \beta_{2} - 8) q^{35} + (2 \beta_{2} + 4) q^{37} + ( - 2 \beta_{3} - 5 \beta_1) q^{41} + ( - 2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 - 3) q^{43} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{47} + ( - 3 \beta_{3} - 2 \beta_1) q^{49} + ( - 2 \beta_{2} + 4) q^{53} + \beta_{2} q^{55} + 7 \beta_1 q^{59} + (\beta_{3} - \beta_{2} - \beta_1) q^{61} + ( - 3 \beta_{3} + 3 \beta_{2} + 11 \beta_1 - 8) q^{65} + (2 \beta_{3} - 3 \beta_1) q^{67} + 4 q^{71} + ( - 3 \beta_{2} - 5) q^{73} + (\beta_{3} + \beta_1) q^{77} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{79} + (\beta_{3} - \beta_{2} - 13 \beta_1 + 12) q^{83} + (2 \beta_{3} + 6 \beta_1) q^{85} - 6 q^{89} + (\beta_{2} - 4) q^{91} + (4 \beta_{3} - 12 \beta_1) q^{95} + (2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{5} - 3 q^{7} + 2 q^{11} - 5 q^{13} + 10 q^{17} + 14 q^{19} + 5 q^{23} - 7 q^{25} - 3 q^{29} - 7 q^{31} - 30 q^{35} + 12 q^{37} - 12 q^{41} - 8 q^{43} + 3 q^{47} - 7 q^{49} + 20 q^{53} - 2 q^{55} + 14 q^{59} + q^{61} - 19 q^{65} - 4 q^{67} + 16 q^{71} - 14 q^{73} + 3 q^{77} + 7 q^{79} + 25 q^{83} + 14 q^{85} - 24 q^{89} - 18 q^{91} - 20 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 \) Copy content Toggle raw display

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 \( T_{5}^{4} + T_{5}^{3} + 9T_{5}^{2} - 8T_{5} + 64 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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