Properties

Label 216.3.h.f
Level $216$
Weight $3$
Character orbit 216.h
Analytic conductor $5.886$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.242095489024.11
Defining polynomial: \(x^{8} - 2 x^{6} + 2 x^{4} - 32 x^{2} + 256\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} - \beta_{5} ) q^{5} + ( 6 - \beta_{2} - \beta_{7} ) q^{7} + ( \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} - \beta_{5} ) q^{5} + ( 6 - \beta_{2} - \beta_{7} ) q^{7} + ( \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{8} + ( 4 + \beta_{2} - \beta_{4} - \beta_{7} ) q^{10} + ( \beta_{1} + \beta_{6} ) q^{11} + ( 1 - \beta_{2} + \beta_{4} + \beta_{7} ) q^{13} + ( 6 \beta_{1} - \beta_{3} + \beta_{5} - 3 \beta_{6} ) q^{14} + ( \beta_{2} + 5 \beta_{4} + \beta_{7} ) q^{16} + ( 7 \beta_{1} + 2 \beta_{3} + 3 \beta_{5} - 4 \beta_{6} ) q^{17} + ( 2 - 2 \beta_{2} - 5 \beta_{4} + 2 \beta_{7} ) q^{19} + ( 6 \beta_{1} + 4 \beta_{5} ) q^{20} + ( 4 + \beta_{2} + \beta_{4} + \beta_{7} ) q^{22} + ( 7 \beta_{1} - 2 \beta_{3} + 4 \beta_{5} - 3 \beta_{6} ) q^{23} + ( -7 - 2 \beta_{2} - 2 \beta_{7} ) q^{25} + ( -\beta_{1} - 4 \beta_{5} ) q^{26} + ( -16 + 6 \beta_{2} - 5 \beta_{4} - \beta_{7} ) q^{28} + ( 4 \beta_{1} + 6 \beta_{5} + 10 \beta_{6} ) q^{29} + ( -10 - 6 \beta_{2} - 6 \beta_{7} ) q^{31} + ( 6 \beta_{3} - 6 \beta_{5} - 2 \beta_{6} ) q^{32} + ( -28 + 7 \beta_{2} + 5 \beta_{4} - 3 \beta_{7} ) q^{34} + ( 5 \beta_{1} - 10 \beta_{5} - 5 \beta_{6} ) q^{35} + ( 7 - 7 \beta_{2} - 7 \beta_{4} + 7 \beta_{7} ) q^{37} + ( -2 \beta_{1} - 7 \beta_{3} - \beta_{5} + 7 \beta_{6} ) q^{38} + ( -16 + 6 \beta_{2} + 4 \beta_{4} + 4 \beta_{7} ) q^{40} + ( -10 \beta_{1} + 12 \beta_{3} - 8 \beta_{5} + 2 \beta_{6} ) q^{41} + ( 10 - 10 \beta_{2} + 6 \beta_{4} + 10 \beta_{7} ) q^{43} + ( 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{44} + ( -28 + 7 \beta_{2} - 5 \beta_{4} + 3 \beta_{7} ) q^{46} + ( -3 \beta_{1} - 14 \beta_{3} + 2 \beta_{5} + 5 \beta_{6} ) q^{47} + ( 18 - 12 \beta_{2} - 12 \beta_{7} ) q^{49} + ( -7 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 6 \beta_{6} ) q^{50} + ( 16 - \beta_{2} - 4 \beta_{4} - 4 \beta_{7} ) q^{52} + ( -14 \beta_{1} + 12 \beta_{5} - 2 \beta_{6} ) q^{53} + ( 2 + 2 \beta_{2} + 2 \beta_{7} ) q^{55} + ( -9 \beta_{1} + \beta_{3} + 13 \beta_{5} + 9 \beta_{6} ) q^{56} + ( 16 + 4 \beta_{2} + 16 \beta_{4} + 16 \beta_{7} ) q^{58} + ( -11 \beta_{1} - 8 \beta_{5} - 19 \beta_{6} ) q^{59} + ( 3 - 3 \beta_{2} - 5 \beta_{4} + 3 \beta_{7} ) q^{61} + ( -10 \beta_{1} - 6 \beta_{3} + 6 \beta_{5} - 18 \beta_{6} ) q^{62} + ( 16 + 10 \beta_{4} - 14 \beta_{7} ) q^{64} + ( -13 \beta_{1} + 2 \beta_{3} - 7 \beta_{5} + 6 \beta_{6} ) q^{65} + ( -12 + 12 \beta_{2} - 7 \beta_{4} - 12 \beta_{7} ) q^{67} + ( -18 \beta_{1} + 12 \beta_{3} + 8 \beta_{5} - 4 \beta_{6} ) q^{68} + ( 20 + 5 \beta_{2} - 15 \beta_{4} - 15 \beta_{7} ) q^{70} + ( -30 \beta_{1} - 12 \beta_{3} - 12 \beta_{5} + 18 \beta_{6} ) q^{71} + ( 31 + 4 \beta_{2} + 4 \beta_{7} ) q^{73} + ( -7 \beta_{1} - 14 \beta_{3} - 14 \beta_{5} + 14 \beta_{6} ) q^{74} + ( 32 - 2 \beta_{2} - 15 \beta_{4} + 13 \beta_{7} ) q^{76} + ( -\beta_{1} + 3 \beta_{5} + 2 \beta_{6} ) q^{77} + ( 24 + 5 \beta_{2} + 5 \beta_{7} ) q^{79} + ( -14 \beta_{1} + 10 \beta_{3} - 6 \beta_{5} + 10 \beta_{6} ) q^{80} + ( 40 - 10 \beta_{2} + 30 \beta_{4} - 18 \beta_{7} ) q^{82} + ( -20 \beta_{1} + 10 \beta_{5} - 10 \beta_{6} ) q^{83} + ( -14 + 14 \beta_{2} - 4 \beta_{4} - 14 \beta_{7} ) q^{85} + ( -10 \beta_{1} - 4 \beta_{3} - 36 \beta_{5} + 4 \beta_{6} ) q^{86} + ( 16 + 4 \beta_{2} + 6 \beta_{4} - 2 \beta_{7} ) q^{88} + ( 5 \beta_{1} + 22 \beta_{3} - 3 \beta_{5} - 8 \beta_{6} ) q^{89} + ( 10 - 10 \beta_{2} + 15 \beta_{4} + 10 \beta_{7} ) q^{91} + ( -24 \beta_{1} + 2 \beta_{3} + 6 \beta_{5} + 18 \beta_{6} ) q^{92} + ( 12 - 3 \beta_{2} - 35 \beta_{4} + 21 \beta_{7} ) q^{94} + ( -5 \beta_{1} - 10 \beta_{3} + 5 \beta_{6} ) q^{95} + ( -9 + 14 \beta_{2} + 14 \beta_{7} ) q^{97} + ( 18 \beta_{1} - 12 \beta_{3} + 12 \beta_{5} - 36 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} + 48q^{7} + O(q^{10}) \) \( 8q + 4q^{4} + 48q^{7} + 40q^{10} + 32q^{22} - 56q^{25} - 100q^{28} - 80q^{31} - 184q^{34} - 120q^{40} - 208q^{46} + 144q^{49} + 140q^{52} + 16q^{55} + 80q^{58} + 184q^{64} + 240q^{70} + 248q^{73} + 196q^{76} + 192q^{79} + 352q^{82} + 152q^{88} - 72q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{6} + 2 x^{4} - 32 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 14 \nu^{5} + 66 \nu^{3} - 96 \nu \)\()/160\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 14 \nu^{4} - 14 \nu^{2} - 16 \)\()/80\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 6 \nu^{5} + 26 \nu^{3} - 56 \nu \)\()/80\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} - 2 \nu^{5} + 42 \nu^{3} + 48 \nu \)\()/160\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{4} - 2 \nu^{2} + 16 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{3} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{7} + 5 \beta_{4} + \beta_{2}\)
\(\nu^{5}\)\(=\)\(-2 \beta_{6} - 6 \beta_{5} + 6 \beta_{3}\)
\(\nu^{6}\)\(=\)\(-14 \beta_{7} + 10 \beta_{4} + 16\)
\(\nu^{7}\)\(=\)\(-38 \beta_{6} + 18 \beta_{5} + 10 \beta_{3} + 30 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.90839 0.598380i
−1.90839 + 0.598380i
−0.926315 1.77255i
−0.926315 + 1.77255i
0.926315 1.77255i
0.926315 + 1.77255i
1.90839 0.598380i
1.90839 + 0.598380i
−1.90839 0.598380i 0 3.28388 + 2.28388i −2.62001 0 0.432236 −4.90029 6.32354i 0 5.00000 + 1.56776i
53.2 −1.90839 + 0.598380i 0 3.28388 2.28388i −2.62001 0 0.432236 −4.90029 + 6.32354i 0 5.00000 1.56776i
53.3 −0.926315 1.77255i 0 −2.28388 + 3.28388i −5.39773 0 11.5678 7.93645 + 1.00639i 0 5.00000 + 9.56776i
53.4 −0.926315 + 1.77255i 0 −2.28388 3.28388i −5.39773 0 11.5678 7.93645 1.00639i 0 5.00000 9.56776i
53.5 0.926315 1.77255i 0 −2.28388 3.28388i 5.39773 0 11.5678 −7.93645 + 1.00639i 0 5.00000 9.56776i
53.6 0.926315 + 1.77255i 0 −2.28388 + 3.28388i 5.39773 0 11.5678 −7.93645 1.00639i 0 5.00000 + 9.56776i
53.7 1.90839 0.598380i 0 3.28388 2.28388i 2.62001 0 0.432236 4.90029 6.32354i 0 5.00000 1.56776i
53.8 1.90839 + 0.598380i 0 3.28388 + 2.28388i 2.62001 0 0.432236 4.90029 + 6.32354i 0 5.00000 + 1.56776i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.8
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 216.3.h.f 8
3.b odd 2 1 inner 216.3.h.f 8
4.b odd 2 1 864.3.h.e 8
8.b even 2 1 inner 216.3.h.f 8
8.d odd 2 1 864.3.h.e 8
12.b even 2 1 864.3.h.e 8
24.f even 2 1 864.3.h.e 8
24.h odd 2 1 inner 216.3.h.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.h.f 8 1.a even 1 1 trivial
216.3.h.f 8 3.b odd 2 1 inner
216.3.h.f 8 8.b even 2 1 inner
216.3.h.f 8 24.h odd 2 1 inner
864.3.h.e 8 4.b odd 2 1
864.3.h.e 8 8.d odd 2 1
864.3.h.e 8 12.b even 2 1
864.3.h.e 8 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 36 T_{5}^{2} + 200 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 - 32 T^{2} + 2 T^{4} - 2 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 200 - 36 T^{2} + T^{4} )^{2} \)
$7$ \( ( 5 - 12 T + T^{2} )^{4} \)
$11$ \( ( 72 - 28 T^{2} + T^{4} )^{2} \)
$13$ \( ( 225 + 94 T^{2} + T^{4} )^{2} \)
$17$ \( ( 145800 + 764 T^{2} + T^{4} )^{2} \)
$19$ \( ( 2025 + 586 T^{2} + T^{4} )^{2} \)
$23$ \( ( 8712 + 932 T^{2} + T^{4} )^{2} \)
$29$ \( ( 2880000 - 3616 T^{2} + T^{4} )^{2} \)
$31$ \( ( -1016 + 20 T + T^{2} )^{4} \)
$37$ \( ( 1750329 + 3430 T^{2} + T^{4} )^{2} \)
$41$ \( ( 15235200 + 7952 T^{2} + T^{4} )^{2} \)
$43$ \( ( 5363856 + 7768 T^{2} + T^{4} )^{2} \)
$47$ \( ( 5604552 + 7812 T^{2} + T^{4} )^{2} \)
$53$ \( ( 6480000 - 5488 T^{2} + T^{4} )^{2} \)
$59$ \( ( 31331528 - 11196 T^{2} + T^{4} )^{2} \)
$61$ \( ( 18225 + 846 T^{2} + T^{4} )^{2} \)
$67$ \( ( 11390625 + 11106 T^{2} + T^{4} )^{2} \)
$71$ \( ( 58320000 + 16272 T^{2} + T^{4} )^{2} \)
$73$ \( ( 465 - 62 T + T^{2} )^{4} \)
$79$ \( ( -199 - 48 T + T^{2} )^{4} \)
$83$ \( ( 8000000 - 7200 T^{2} + T^{4} )^{2} \)
$89$ \( ( 32805000 + 19292 T^{2} + T^{4} )^{2} \)
$97$ \( ( -5995 + 18 T + T^{2} )^{4} \)
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