Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{3} q^{5} + q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{3} q^{5} + q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} + ( 1 - \beta_{2} ) q^{10} -3 \beta_{3} q^{11} + ( -2 + 4 \beta_{2} ) q^{13} + \beta_{1} q^{14} + ( -1 + 3 \beta_{2} ) q^{16} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{17} + ( 2 - 4 \beta_{2} ) q^{19} + ( \beta_{1} - 2 \beta_{3} ) q^{20} + ( 3 - 3 \beta_{2} ) q^{22} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{23} + 4 q^{25} + ( -2 \beta_{1} + 8 \beta_{3} ) q^{26} + ( 1 + \beta_{2} ) q^{28} -6 \beta_{3} q^{29} -7 q^{31} + ( -\beta_{1} + 6 \beta_{3} ) q^{32} + ( -6 - 2 \beta_{2} ) q^{34} -\beta_{3} q^{35} + ( -2 + 4 \beta_{2} ) q^{37} + ( 2 \beta_{1} - 8 \beta_{3} ) q^{38} + ( 3 - \beta_{2} ) q^{40} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{41} + ( 4 - 8 \beta_{2} ) q^{43} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{44} + ( -6 - 2 \beta_{2} ) q^{46} -6 q^{49} + 4 \beta_{1} q^{50} + ( -10 + 6 \beta_{2} ) q^{52} + 9 \beta_{3} q^{53} -3 q^{55} + ( \beta_{1} + 2 \beta_{3} ) q^{56} + ( 6 - 6 \beta_{2} ) q^{58} + 4 \beta_{3} q^{59} -7 \beta_{1} q^{62} + ( -7 + 5 \beta_{2} ) q^{64} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -6 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 1 - \beta_{2} ) q^{70} + ( 12 \beta_{1} - 6 \beta_{3} ) q^{71} + 3 q^{73} + ( -2 \beta_{1} + 8 \beta_{3} ) q^{74} + ( 10 - 6 \beta_{2} ) q^{76} -3 \beta_{3} q^{77} + 4 q^{79} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{80} + ( 6 + 2 \beta_{2} ) q^{82} + 7 \beta_{3} q^{83} + ( -2 + 4 \beta_{2} ) q^{85} + ( 4 \beta_{1} - 16 \beta_{3} ) q^{86} + ( 9 - 3 \beta_{2} ) q^{88} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{89} + ( -2 + 4 \beta_{2} ) q^{91} + ( -6 \beta_{1} - 4 \beta_{3} ) q^{92} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{95} + 7 q^{97} -6 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} + 4q^{7} + O(q^{10}) \) \( 4q + 6q^{4} + 4q^{7} + 2q^{10} + 2q^{16} + 6q^{22} + 16q^{25} + 6q^{28} - 28q^{31} - 28q^{34} + 10q^{40} - 28q^{46} - 24q^{49} - 28q^{52} - 12q^{55} + 12q^{58} - 18q^{64} + 2q^{70} + 12q^{73} + 28q^{76} + 16q^{79} + 28q^{82} + 30q^{88} + 28q^{97} + O(q^{100}) \)

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.32288 0.500000i
−1.32288 + 0.500000i
1.32288 0.500000i
1.32288 + 0.500000i
−1.32288 0.500000i 0 1.50000 + 1.32288i 1.00000i 0 1.00000 −1.32288 2.50000i 0 0.500000 1.32288i
109.2 −1.32288 + 0.500000i 0 1.50000 1.32288i 1.00000i 0 1.00000 −1.32288 + 2.50000i 0 0.500000 + 1.32288i
109.3 1.32288 0.500000i 0 1.50000 1.32288i 1.00000i 0 1.00000 1.32288 2.50000i 0 0.500000 + 1.32288i
109.4 1.32288 + 0.500000i 0 1.50000 + 1.32288i 1.00000i 0 1.00000 1.32288 + 2.50000i 0 0.500000 1.32288i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.d.b 4
3.b odd 2 1 inner 216.2.d.b 4
4.b odd 2 1 864.2.d.a 4
8.b even 2 1 inner 216.2.d.b 4
8.d odd 2 1 864.2.d.a 4
9.c even 3 2 648.2.n.n 8
9.d odd 6 2 648.2.n.n 8
12.b even 2 1 864.2.d.a 4
16.e even 4 1 6912.2.a.bc 2
16.e even 4 1 6912.2.a.bu 2
16.f odd 4 1 6912.2.a.bd 2
16.f odd 4 1 6912.2.a.bv 2
24.f even 2 1 864.2.d.a 4
24.h odd 2 1 inner 216.2.d.b 4
36.f odd 6 2 2592.2.r.p 8
36.h even 6 2 2592.2.r.p 8
48.i odd 4 1 6912.2.a.bc 2
48.i odd 4 1 6912.2.a.bu 2
48.k even 4 1 6912.2.a.bd 2
48.k even 4 1 6912.2.a.bv 2
72.j odd 6 2 648.2.n.n 8
72.l even 6 2 2592.2.r.p 8
72.n even 6 2 648.2.n.n 8
72.p odd 6 2 2592.2.r.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.b 4 1.a even 1 1 trivial
216.2.d.b 4 3.b odd 2 1 inner
216.2.d.b 4 8.b even 2 1 inner
216.2.d.b 4 24.h odd 2 1 inner
648.2.n.n 8 9.c even 3 2
648.2.n.n 8 9.d odd 6 2
648.2.n.n 8 72.j odd 6 2
648.2.n.n 8 72.n even 6 2
864.2.d.a 4 4.b odd 2 1
864.2.d.a 4 8.d odd 2 1
864.2.d.a 4 12.b even 2 1
864.2.d.a 4 24.f even 2 1
2592.2.r.p 8 36.f odd 6 2
2592.2.r.p 8 36.h even 6 2
2592.2.r.p 8 72.l even 6 2
2592.2.r.p 8 72.p odd 6 2
6912.2.a.bc 2 16.e even 4 1
6912.2.a.bc 2 48.i odd 4 1
6912.2.a.bd 2 16.f odd 4 1
6912.2.a.bd 2 48.k even 4 1
6912.2.a.bu 2 16.e even 4 1
6912.2.a.bu 2 48.i odd 4 1
6912.2.a.bv 2 16.f odd 4 1
6912.2.a.bv 2 48.k even 4 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( ( 1 - 9 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - T + 7 T^{2} )^{4} \)
$11$ \( ( 1 - 13 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 2 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 6 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 10 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 18 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 7 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 46 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 54 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 26 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 - 25 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 102 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 3 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 117 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 66 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 7 T + 97 T^{2} )^{4} \)
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