Properties

Label 216.3.h.b
Level $216$
Weight $3$
Character orbit 216.h
Self dual yes
Analytic conductor $5.886$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + ( 1 + \beta ) q^{5} + ( 5 - \beta ) q^{7} + 8 q^{8} +O(q^{10})\) \( q + 2 q^{2} + 4 q^{4} + ( 1 + \beta ) q^{5} + ( 5 - \beta ) q^{7} + 8 q^{8} + ( 2 + 2 \beta ) q^{10} + ( -5 - 2 \beta ) q^{11} + ( 10 - 2 \beta ) q^{14} + 16 q^{16} + ( 4 + 4 \beta ) q^{20} + ( -10 - 4 \beta ) q^{22} + ( 48 + 2 \beta ) q^{25} + ( 20 - 4 \beta ) q^{28} -50 q^{29} + ( -19 + 5 \beta ) q^{31} + 32 q^{32} + ( -67 + 4 \beta ) q^{35} + ( 8 + 8 \beta ) q^{40} + ( -20 - 8 \beta ) q^{44} + ( 48 - 10 \beta ) q^{49} + ( 96 + 4 \beta ) q^{50} + ( -47 - 5 \beta ) q^{53} + ( -149 - 7 \beta ) q^{55} + ( 40 - 8 \beta ) q^{56} -100 q^{58} + 10 q^{59} + ( -38 + 10 \beta ) q^{62} + 64 q^{64} + ( -134 + 8 \beta ) q^{70} + ( -25 + 14 \beta ) q^{73} + ( 119 - 5 \beta ) q^{77} -58 q^{79} + ( 16 + 16 \beta ) q^{80} + ( 67 + 10 \beta ) q^{83} + ( -40 - 16 \beta ) q^{88} + ( 95 - 4 \beta ) q^{97} + ( 96 - 20 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} + 2q^{5} + 10q^{7} + 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 2q^{5} + 10q^{7} + 16q^{8} + 4q^{10} - 10q^{11} + 20q^{14} + 32q^{16} + 8q^{20} - 20q^{22} + 96q^{25} + 40q^{28} - 100q^{29} - 38q^{31} + 64q^{32} - 134q^{35} + 16q^{40} - 40q^{44} + 96q^{49} + 192q^{50} - 94q^{53} - 298q^{55} + 80q^{56} - 200q^{58} + 20q^{59} - 76q^{62} + 128q^{64} - 268q^{70} - 50q^{73} + 238q^{77} - 116q^{79} + 32q^{80} + 134q^{83} - 80q^{88} + 190q^{97} + 192q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.41421
1.41421
2.00000 0 4.00000 −7.48528 0 13.4853 8.00000 0 −14.9706
53.2 2.00000 0 4.00000 9.48528 0 −3.48528 8.00000 0 18.9706
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.h.b yes 2
3.b odd 2 1 216.3.h.a 2
4.b odd 2 1 864.3.h.b 2
8.b even 2 1 216.3.h.a 2
8.d odd 2 1 864.3.h.a 2
12.b even 2 1 864.3.h.a 2
24.f even 2 1 864.3.h.b 2
24.h odd 2 1 CM 216.3.h.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.h.a 2 3.b odd 2 1
216.3.h.a 2 8.b even 2 1
216.3.h.b yes 2 1.a even 1 1 trivial
216.3.h.b yes 2 24.h odd 2 1 CM
864.3.h.a 2 8.d odd 2 1
864.3.h.a 2 12.b even 2 1
864.3.h.b 2 4.b odd 2 1
864.3.h.b 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 2 T_{5} - 71 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -71 - 2 T + T^{2} \)
$7$ \( -47 - 10 T + T^{2} \)
$11$ \( -263 + 10 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 50 + T )^{2} \)
$31$ \( -1439 + 38 T + T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 409 + 94 T + T^{2} \)
$59$ \( ( -10 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( -13487 + 50 T + T^{2} \)
$79$ \( ( 58 + T )^{2} \)
$83$ \( -2711 - 134 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 7873 - 190 T + T^{2} \)
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