Properties

Label 216.2.d.c
Level $216$
Weight $2$
Character orbit 216.d
Analytic conductor $1.725$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.629407744.1
Defining polynomial: \(x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + \beta_{4} q^{5} + ( -\beta_{2} - \beta_{6} ) q^{7} + \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + \beta_{4} q^{5} + ( -\beta_{2} - \beta_{6} ) q^{7} + \beta_{3} q^{8} + ( -2 + \beta_{2} + \beta_{5} - \beta_{6} ) q^{10} + ( \beta_{1} - \beta_{4} - \beta_{7} ) q^{11} + ( 1 - \beta_{2} - \beta_{5} + \beta_{6} ) q^{13} + ( -\beta_{3} - \beta_{7} ) q^{14} + ( \beta_{2} + \beta_{5} + \beta_{6} ) q^{16} + ( 2 \beta_{3} - \beta_{4} ) q^{17} + ( 2 - 2 \beta_{2} - \beta_{5} + 2 \beta_{6} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{4} ) q^{20} + ( -2 + \beta_{2} - \beta_{5} + \beta_{6} ) q^{22} + ( 3 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{23} + ( -1 - 2 \beta_{2} - 2 \beta_{6} ) q^{25} + ( \beta_{1} - 2 \beta_{4} ) q^{26} + ( -4 - \beta_{5} - \beta_{6} ) q^{28} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{29} + 2 q^{31} + ( 2 \beta_{4} + 2 \beta_{7} ) q^{32} + ( 2 + \beta_{2} + \beta_{5} + 3 \beta_{6} ) q^{34} + ( \beta_{1} - 3 \beta_{4} - \beta_{7} ) q^{35} + ( 1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{37} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{38} + ( -4 + 2 \beta_{5} - 2 \beta_{6} ) q^{40} + ( -6 \beta_{1} - 2 \beta_{7} ) q^{41} + ( -2 + 2 \beta_{2} - 2 \beta_{6} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{44} + ( 2 + \beta_{2} - \beta_{5} - 3 \beta_{6} ) q^{46} + ( -3 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{47} + ( -\beta_{1} - 2 \beta_{3} - 2 \beta_{7} ) q^{50} + ( 4 - \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{52} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{53} + ( 2 + 2 \beta_{2} + 2 \beta_{6} ) q^{55} + ( -4 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{56} + ( 4 - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} ) q^{58} + ( -\beta_{1} - 3 \beta_{4} + \beta_{7} ) q^{59} + ( -3 + 3 \beta_{2} - \beta_{5} - 3 \beta_{6} ) q^{61} + 2 \beta_{1} q^{62} + ( 4 + 2 \beta_{5} - 2 \beta_{6} ) q^{64} + ( 6 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{65} + \beta_{5} q^{67} + ( 2 \beta_{1} + 2 \beta_{4} + 4 \beta_{7} ) q^{68} + ( 2 - \beta_{2} - 3 \beta_{5} + 3 \beta_{6} ) q^{70} + ( 1 + 4 \beta_{2} + 4 \beta_{6} ) q^{73} + ( \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{74} + ( 8 - 2 \beta_{2} - 3 \beta_{5} + \beta_{6} ) q^{76} + ( 2 \beta_{1} + \beta_{4} - 2 \beta_{7} ) q^{77} + ( -6 - \beta_{2} - \beta_{6} ) q^{79} + ( -4 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} ) q^{80} + ( -8 - 4 \beta_{2} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{7} ) q^{83} + ( -2 + 2 \beta_{2} + 4 \beta_{5} - 2 \beta_{6} ) q^{85} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{7} ) q^{86} + ( 4 - 2 \beta_{2} + 4 \beta_{6} ) q^{88} + ( -2 \beta_{3} + \beta_{4} ) q^{89} + ( -2 + 2 \beta_{2} + 3 \beta_{5} - 2 \beta_{6} ) q^{91} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{7} ) q^{92} + ( -6 - 3 \beta_{2} - \beta_{5} - 3 \beta_{6} ) q^{94} + ( 9 \beta_{1} + 2 \beta_{3} - \beta_{4} + 3 \beta_{7} ) q^{95} + ( -3 + 2 \beta_{2} + 2 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} + O(q^{10}) \) \( 8q + 4q^{4} - 8q^{10} - 16q^{22} - 8q^{25} - 28q^{28} + 16q^{31} + 8q^{34} - 24q^{40} + 32q^{46} + 20q^{52} + 16q^{55} + 32q^{58} + 40q^{64} + 8q^{73} + 52q^{76} - 48q^{79} - 80q^{82} + 8q^{88} - 48q^{94} - 24q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{3} - 4 \nu \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{4} - 2 \nu^{2} + 4 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{5} - 2 \nu^{3} + 4 \nu \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{2}\)
\(\nu^{5}\)\(=\)\(2 \beta_{7} + 2 \beta_{4}\)
\(\nu^{6}\)\(=\)\(-2 \beta_{6} + 2 \beta_{5} + 4\)
\(\nu^{7}\)\(=\)\(4 \beta_{4} - 2 \beta_{3} + 4 \beta_{1}\)

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
−1.38255 0.297594i
−1.38255 + 0.297594i
−0.767178 1.18804i
−0.767178 + 1.18804i
0.767178 1.18804i
0.767178 + 1.18804i
1.38255 0.297594i
1.38255 + 0.297594i
−1.38255 0.297594i 0 1.82288 + 0.822876i 3.36028i 0 −2.64575 −2.27533 1.68014i 0 −1.00000 + 4.64575i
109.2 −1.38255 + 0.297594i 0 1.82288 0.822876i 3.36028i 0 −2.64575 −2.27533 + 1.68014i 0 −1.00000 4.64575i
109.3 −0.767178 1.18804i 0 −0.822876 + 1.82288i 0.841723i 0 2.64575 2.79694 0.420861i 0 −1.00000 + 0.645751i
109.4 −0.767178 + 1.18804i 0 −0.822876 1.82288i 0.841723i 0 2.64575 2.79694 + 0.420861i 0 −1.00000 0.645751i
109.5 0.767178 1.18804i 0 −0.822876 1.82288i 0.841723i 0 2.64575 −2.79694 0.420861i 0 −1.00000 0.645751i
109.6 0.767178 + 1.18804i 0 −0.822876 + 1.82288i 0.841723i 0 2.64575 −2.79694 + 0.420861i 0 −1.00000 + 0.645751i
109.7 1.38255 0.297594i 0 1.82288 0.822876i 3.36028i 0 −2.64575 2.27533 1.68014i 0 −1.00000 4.64575i
109.8 1.38255 + 0.297594i 0 1.82288 + 0.822876i 3.36028i 0 −2.64575 2.27533 + 1.68014i 0 −1.00000 + 4.64575i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
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.c 8
3.b odd 2 1 inner 216.2.d.c 8
4.b odd 2 1 864.2.d.c 8
8.b even 2 1 inner 216.2.d.c 8
8.d odd 2 1 864.2.d.c 8
9.c even 3 2 648.2.n.q 16
9.d odd 6 2 648.2.n.q 16
12.b even 2 1 864.2.d.c 8
16.e even 4 1 6912.2.a.cc 4
16.e even 4 1 6912.2.a.cj 4
16.f odd 4 1 6912.2.a.cd 4
16.f odd 4 1 6912.2.a.ci 4
24.f even 2 1 864.2.d.c 8
24.h odd 2 1 inner 216.2.d.c 8
36.f odd 6 2 2592.2.r.q 16
36.h even 6 2 2592.2.r.q 16
48.i odd 4 1 6912.2.a.cc 4
48.i odd 4 1 6912.2.a.cj 4
48.k even 4 1 6912.2.a.cd 4
48.k even 4 1 6912.2.a.ci 4
72.j odd 6 2 648.2.n.q 16
72.l even 6 2 2592.2.r.q 16
72.n even 6 2 648.2.n.q 16
72.p odd 6 2 2592.2.r.q 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.c 8 1.a even 1 1 trivial
216.2.d.c 8 3.b odd 2 1 inner
216.2.d.c 8 8.b even 2 1 inner
216.2.d.c 8 24.h odd 2 1 inner
648.2.n.q 16 9.c even 3 2
648.2.n.q 16 9.d odd 6 2
648.2.n.q 16 72.j odd 6 2
648.2.n.q 16 72.n even 6 2
864.2.d.c 8 4.b odd 2 1
864.2.d.c 8 8.d odd 2 1
864.2.d.c 8 12.b even 2 1
864.2.d.c 8 24.f even 2 1
2592.2.r.q 16 36.f odd 6 2
2592.2.r.q 16 36.h even 6 2
2592.2.r.q 16 72.l even 6 2
2592.2.r.q 16 72.p odd 6 2
6912.2.a.cc 4 16.e even 4 1
6912.2.a.cc 4 48.i odd 4 1
6912.2.a.cd 4 16.f odd 4 1
6912.2.a.cd 4 48.k even 4 1
6912.2.a.ci 4 16.f odd 4 1
6912.2.a.ci 4 48.k even 4 1
6912.2.a.cj 4 16.e even 4 1
6912.2.a.cj 4 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 12 T_{5}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 2 T^{4} - 8 T^{6} + 16 T^{8} \)
$3$ 1
$5$ \( ( 1 - 8 T^{2} + 38 T^{4} - 200 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 + 7 T^{2} + 49 T^{4} )^{4} \)
$11$ \( ( 1 - 24 T^{2} + 358 T^{4} - 2904 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 30 T^{2} + 451 T^{4} - 5070 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 16 T^{2} + 614 T^{4} + 4624 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 18 T^{2} + 691 T^{4} - 6498 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 16 T^{2} - 250 T^{4} + 8464 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 36 T^{2} + 1558 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 2 T + 31 T^{2} )^{8} \)
$37$ \( ( 1 - 102 T^{2} + 4891 T^{4} - 139638 T^{6} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 + 4 T^{2} + 1574 T^{4} + 6724 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 108 T^{2} + 6166 T^{4} - 199692 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 80 T^{2} + 3750 T^{4} + 176720 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 132 T^{2} + 9526 T^{4} - 370788 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 56 T^{2} + 6374 T^{4} - 194936 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 46 T^{2} - 1101 T^{4} - 171166 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 125 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{8} \)
$73$ \( ( 1 - 2 T + 35 T^{2} - 146 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 + 12 T + 187 T^{2} + 948 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 236 T^{2} + 25910 T^{4} - 1625804 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 304 T^{2} + 38918 T^{4} + 2407984 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 6 T + 175 T^{2} + 582 T^{3} + 9409 T^{4} )^{4} \)
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