Properties

Label 216.2.i.a
Level $216$
Weight $2$
Character orbit 216.i
Analytic conductor $1.725$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 3) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 3) q^{7} + ( - 5 \zeta_{6} + 5) q^{11} + 5 \zeta_{6} q^{13} + 2 q^{17} - 4 q^{19} - \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + (9 \zeta_{6} - 9) q^{29} + \zeta_{6} q^{31} - 3 q^{35} - 6 q^{37} + 3 \zeta_{6} q^{41} + (\zeta_{6} - 1) q^{43} + (3 \zeta_{6} - 3) q^{47} - 2 \zeta_{6} q^{49} - 2 q^{53} - 5 q^{55} + 11 \zeta_{6} q^{59} + (7 \zeta_{6} - 7) q^{61} + ( - 5 \zeta_{6} + 5) q^{65} + \zeta_{6} q^{67} - 4 q^{71} - 2 q^{73} - 15 \zeta_{6} q^{77} + (\zeta_{6} - 1) q^{79} + ( - \zeta_{6} + 1) q^{83} - 2 \zeta_{6} q^{85} + 18 q^{89} + 15 q^{91} + 4 \zeta_{6} q^{95} + ( - 13 \zeta_{6} + 13) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 3 q^{7} + 5 q^{11} + 5 q^{13} + 4 q^{17} - 8 q^{19} - q^{23} + 4 q^{25} - 9 q^{29} + q^{31} - 6 q^{35} - 12 q^{37} + 3 q^{41} - q^{43} - 3 q^{47} - 2 q^{49} - 4 q^{53} - 10 q^{55} + 11 q^{59} - 7 q^{61} + 5 q^{65} + q^{67} - 8 q^{71} - 4 q^{73} - 15 q^{77} - q^{79} + q^{83} - 2 q^{85} + 36 q^{89} + 30 q^{91} + 4 q^{95} + 13 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −0.500000 0.866025i 0 1.50000 2.59808i 0 0 0
145.1 0 0 0 −0.500000 + 0.866025i 0 1.50000 + 2.59808i 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.a 2
3.b odd 2 1 72.2.i.a 2
4.b odd 2 1 432.2.i.a 2
8.b even 2 1 1728.2.i.h 2
8.d odd 2 1 1728.2.i.g 2
9.c even 3 1 inner 216.2.i.a 2
9.c even 3 1 648.2.a.c 1
9.d odd 6 1 72.2.i.a 2
9.d odd 6 1 648.2.a.a 1
12.b even 2 1 144.2.i.b 2
24.f even 2 1 576.2.i.c 2
24.h odd 2 1 576.2.i.d 2
36.f odd 6 1 432.2.i.a 2
36.f odd 6 1 1296.2.a.i 1
36.h even 6 1 144.2.i.b 2
36.h even 6 1 1296.2.a.e 1
72.j odd 6 1 576.2.i.d 2
72.j odd 6 1 5184.2.a.s 1
72.l even 6 1 576.2.i.c 2
72.l even 6 1 5184.2.a.x 1
72.n even 6 1 1728.2.i.h 2
72.n even 6 1 5184.2.a.i 1
72.p odd 6 1 1728.2.i.g 2
72.p odd 6 1 5184.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.a 2 3.b odd 2 1
72.2.i.a 2 9.d odd 6 1
144.2.i.b 2 12.b even 2 1
144.2.i.b 2 36.h even 6 1
216.2.i.a 2 1.a even 1 1 trivial
216.2.i.a 2 9.c even 3 1 inner
432.2.i.a 2 4.b odd 2 1
432.2.i.a 2 36.f odd 6 1
576.2.i.c 2 24.f even 2 1
576.2.i.c 2 72.l even 6 1
576.2.i.d 2 24.h odd 2 1
576.2.i.d 2 72.j odd 6 1
648.2.a.a 1 9.d odd 6 1
648.2.a.c 1 9.c even 3 1
1296.2.a.e 1 36.h even 6 1
1296.2.a.i 1 36.f odd 6 1
1728.2.i.g 2 8.d odd 2 1
1728.2.i.g 2 72.p odd 6 1
1728.2.i.h 2 8.b even 2 1
1728.2.i.h 2 72.n even 6 1
5184.2.a.i 1 72.n even 6 1
5184.2.a.n 1 72.p odd 6 1
5184.2.a.s 1 72.j odd 6 1
5184.2.a.x 1 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$11$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$31$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$89$ \( (T - 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
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