Properties

Label 216.2.l.a
Level 216
Weight 2
Character orbit 216.l
Analytic conductor 1.725
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: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 216.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{3} q^{8} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{11} + ( -4 + 4 \beta_{2} ) q^{16} + ( 3 - 6 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -1 - 6 \beta_{1} + 3 \beta_{3} ) q^{19} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{22} + ( 5 - 5 \beta_{2} ) q^{25} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( 4 + 3 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{34} + ( -6 - \beta_{1} - 6 \beta_{2} ) q^{38} + ( -6 + 4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{41} + ( -5 + 3 \beta_{1} + 5 \beta_{2} - 6 \beta_{3} ) q^{43} + ( -6 + 12 \beta_{2} - 2 \beta_{3} ) q^{44} -7 \beta_{2} q^{49} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{50} + ( -6 - 5 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} ) q^{59} -8 q^{64} + ( 3 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} ) q^{67} + ( 12 + 4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -1 + 12 \beta_{1} - 6 \beta_{3} ) q^{73} + ( -6 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{76} + ( 8 - 6 \beta_{1} + 3 \beta_{3} ) q^{82} + 2 \beta_{1} q^{83} + ( 12 - 5 \beta_{1} - 6 \beta_{2} + 5 \beta_{3} ) q^{86} + ( 4 - 6 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} ) q^{88} + 4 \beta_{3} q^{89} + ( -5 - 6 \beta_{1} + 5 \beta_{2} + 12 \beta_{3} ) q^{97} -7 \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + O(q^{10}) \) \( 4q + 4q^{4} + 18q^{11} - 8q^{16} - 4q^{19} - 4q^{22} + 10q^{25} + 8q^{34} - 36q^{38} - 18q^{41} - 10q^{43} - 14q^{49} - 18q^{59} - 32q^{64} + 14q^{67} + 36q^{68} - 4q^{73} - 4q^{76} + 32q^{82} + 36q^{86} + 8q^{88} - 10q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 0 0 0 2.82843i 0 0
35.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 0 0 0 2.82843i 0 0
179.1 −1.22474 + 0.707107i 0 1.00000 1.73205i 0 0 0 2.82843i 0 0
179.2 1.22474 0.707107i 0 1.00000 1.73205i 0 0 0 2.82843i 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.d odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.l.a 4
3.b odd 2 1 72.2.l.a 4
4.b odd 2 1 864.2.p.a 4
8.b even 2 1 864.2.p.a 4
8.d odd 2 1 CM 216.2.l.a 4
9.c even 3 1 72.2.l.a 4
9.c even 3 1 648.2.f.a 4
9.d odd 6 1 inner 216.2.l.a 4
9.d odd 6 1 648.2.f.a 4
12.b even 2 1 288.2.p.a 4
24.f even 2 1 72.2.l.a 4
24.h odd 2 1 288.2.p.a 4
36.f odd 6 1 288.2.p.a 4
36.f odd 6 1 2592.2.f.a 4
36.h even 6 1 864.2.p.a 4
36.h even 6 1 2592.2.f.a 4
72.j odd 6 1 864.2.p.a 4
72.j odd 6 1 2592.2.f.a 4
72.l even 6 1 inner 216.2.l.a 4
72.l even 6 1 648.2.f.a 4
72.n even 6 1 288.2.p.a 4
72.n even 6 1 2592.2.f.a 4
72.p odd 6 1 72.2.l.a 4
72.p odd 6 1 648.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.a 4 3.b odd 2 1
72.2.l.a 4 9.c even 3 1
72.2.l.a 4 24.f even 2 1
72.2.l.a 4 72.p odd 6 1
216.2.l.a 4 1.a even 1 1 trivial
216.2.l.a 4 8.d odd 2 1 CM
216.2.l.a 4 9.d odd 6 1 inner
216.2.l.a 4 72.l even 6 1 inner
288.2.p.a 4 12.b even 2 1
288.2.p.a 4 24.h odd 2 1
288.2.p.a 4 36.f odd 6 1
288.2.p.a 4 72.n even 6 1
648.2.f.a 4 9.c even 3 1
648.2.f.a 4 9.d odd 6 1
648.2.f.a 4 72.l even 6 1
648.2.f.a 4 72.p odd 6 1
864.2.p.a 4 4.b odd 2 1
864.2.p.a 4 8.b even 2 1
864.2.p.a 4 36.h even 6 1
864.2.p.a 4 72.j odd 6 1
2592.2.f.a 4 36.f odd 6 1
2592.2.f.a 4 36.h even 6 1
2592.2.f.a 4 72.j odd 6 1
2592.2.f.a 4 72.n even 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + 7 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )^{2}( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} ) \)
$13$ \( ( 1 + 13 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} )( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} ) \)
$19$ \( ( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 29 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 31 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 37 T^{2} )^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2}( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} ) \)
$43$ \( ( 1 + 10 T + 43 T^{2} )^{2}( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} ) \)
$47$ \( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 6 T + 59 T^{2} )^{2}( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} ) \)
$61$ \( ( 1 + 61 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 14 T + 67 T^{2} )^{2}( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} ) \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( ( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 79 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 18 T + 241 T^{2} - 1494 T^{3} + 6889 T^{4} )( 1 + 18 T + 241 T^{2} + 1494 T^{3} + 6889 T^{4} ) \)
$89$ \( ( 1 - 18 T + 89 T^{2} )^{2}( 1 + 18 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{2}( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} ) \)
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