Properties

Label 216.2.f.b
Level $216$
Weight $2$
Character orbit 216.f
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.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.170772624.1
Defining polynomial: \(x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 6 x^{4} - 12 x^{3} + 20 x^{2} - 24 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} -\beta_{3} q^{4} + \beta_{2} q^{5} + ( \beta_{1} - \beta_{3} ) q^{7} + ( -\beta_{2} + \beta_{4} + \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{4} q^{2} -\beta_{3} q^{4} + \beta_{2} q^{5} + ( \beta_{1} - \beta_{3} ) q^{7} + ( -\beta_{2} + \beta_{4} + \beta_{5} ) q^{8} + ( 1 + \beta_{7} ) q^{10} + ( -\beta_{4} - \beta_{6} ) q^{11} + ( -\beta_{1} + \beta_{7} ) q^{13} + ( \beta_{5} + \beta_{6} ) q^{14} + ( \beta_{1} - \beta_{3} - \beta_{7} ) q^{16} + ( -\beta_{4} - \beta_{5} - \beta_{6} ) q^{17} + ( -2 - \beta_{1} - 2 \beta_{3} - \beta_{7} ) q^{19} + ( -\beta_{2} - 2 \beta_{5} + \beta_{6} ) q^{20} + ( 3 - \beta_{1} + 2 \beta_{3} ) q^{22} + ( 2 \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{23} + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{7} ) q^{25} + ( -2 \beta_{2} - 2 \beta_{5} ) q^{26} + ( -2 + 2 \beta_{1} - \beta_{3} ) q^{28} + ( -3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{29} + ( 2 \beta_{1} - \beta_{3} - \beta_{7} ) q^{31} + ( \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{32} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{34} -\beta_{5} q^{35} + ( -3 \beta_{1} + 4 \beta_{3} - \beta_{7} ) q^{37} + ( -2 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{38} + ( -6 - \beta_{1} - \beta_{3} - \beta_{7} ) q^{40} + ( -\beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{41} -4 q^{43} + ( \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{44} + ( -2 + 2 \beta_{3} + 2 \beta_{7} ) q^{46} + ( 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{47} + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{7} ) q^{49} + ( 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} ) q^{50} + ( -4 - 2 \beta_{1} - 2 \beta_{7} ) q^{52} + ( -\beta_{2} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{53} + ( \beta_{1} + \beta_{3} - 2 \beta_{7} ) q^{55} + ( \beta_{2} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{56} + ( -4 + 2 \beta_{3} ) q^{58} + ( 2 \beta_{4} + 2 \beta_{6} ) q^{59} + ( -2 \beta_{1} + 2 \beta_{7} ) q^{61} + ( 2 \beta_{2} + 3 \beta_{5} + \beta_{6} ) q^{62} + ( 4 + 3 \beta_{1} - \beta_{3} + \beta_{7} ) q^{64} + ( 3 \beta_{4} - 5 \beta_{5} + 3 \beta_{6} ) q^{65} -4 q^{67} + ( 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{68} + ( -1 - \beta_{1} ) q^{70} + ( -2 \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{71} + ( 1 - \beta_{1} - 2 \beta_{3} - \beta_{7} ) q^{73} + ( 2 \beta_{2} - 2 \beta_{5} - 4 \beta_{6} ) q^{74} + ( 8 + 2 \beta_{1} - 2 \beta_{7} ) q^{76} + ( -\beta_{2} + 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{77} + ( \beta_{1} - 2 \beta_{3} + \beta_{7} ) q^{79} + ( -\beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{80} + ( 6 + 2 \beta_{1} + 2 \beta_{3} ) q^{82} + ( -\beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{83} + ( \beta_{1} - \beta_{7} ) q^{85} -4 \beta_{4} q^{86} + ( 2 - 3 \beta_{1} - \beta_{3} + \beta_{7} ) q^{88} + ( 2 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} ) q^{89} + ( 2 - \beta_{1} - 2 \beta_{3} - \beta_{7} ) q^{91} + ( -6 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} ) q^{92} + ( 8 - 4 \beta_{3} ) q^{94} + ( -6 \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{95} - q^{97} + ( 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{4} + O(q^{10}) \) \( 8q + 2q^{4} + 6q^{10} + 2q^{16} - 8q^{19} + 22q^{22} + 8q^{25} - 18q^{28} + 16q^{34} - 42q^{40} - 32q^{43} - 24q^{46} + 8q^{49} - 24q^{52} - 36q^{58} + 26q^{64} - 32q^{67} - 6q^{70} + 16q^{73} + 64q^{76} + 40q^{82} + 22q^{88} + 24q^{91} + 72q^{94} - 8q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 6 x^{4} - 12 x^{3} + 20 x^{2} - 24 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + \nu^{4} - 2 \nu^{2} - 4 \nu \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} - \nu^{5} + 2 \nu^{4} + 8 \nu^{2} - 4 \nu + 4 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{5} - \nu^{4} + 2 \nu^{3} - 6 \nu^{2} + 4 \nu - 12 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{7} + 3 \nu^{6} - 5 \nu^{5} + 5 \nu^{4} - 6 \nu^{3} + 14 \nu^{2} - 16 \nu + 20 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{7} - 5 \nu^{6} + 7 \nu^{5} - 6 \nu^{4} + 10 \nu^{3} - 24 \nu^{2} + 28 \nu - 28 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} + 6 \nu^{6} - 10 \nu^{5} + 7 \nu^{4} - 8 \nu^{3} + 22 \nu^{2} - 36 \nu + 40 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 8 \nu^{6} - 16 \nu^{5} + 11 \nu^{4} - 16 \nu^{3} + 42 \nu^{2} - 44 \nu + 64 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{1} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{4} + 2 \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{7} - \beta_{6} + 3 \beta_{5} + 9 \beta_{4} + 2 \beta_{2} + 3 \beta_{1}\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-3 \beta_{7} - \beta_{6} - 5 \beta_{5} + \beta_{4} + 2 \beta_{2} + \beta_{1} + 16\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{7} + 8 \beta_{3} - 3 \beta_{1} + 8\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(7 \beta_{7} - 5 \beta_{6} - \beta_{5} - 11 \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 11 \beta_{1} + 8\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.02187 0.977642i
−1.02187 + 0.977642i
0.774115 + 1.18353i
0.774115 1.18353i
1.41203 + 0.0786378i
1.41203 0.0786378i
0.335728 + 1.37379i
0.335728 1.37379i
−1.35760 0.396143i 0 1.68614 + 1.07561i 0.505408 0 3.42703i −1.86301 2.12819i 0 −0.686141 0.200214i
107.2 −1.35760 + 0.396143i 0 1.68614 1.07561i 0.505408 0 3.42703i −1.86301 + 2.12819i 0 −0.686141 + 0.200214i
107.3 −0.637910 1.26217i 0 −1.18614 + 1.61030i −3.42703 0 0.505408i 2.78912 + 0.469882i 0 2.18614 + 4.32550i
107.4 −0.637910 + 1.26217i 0 −1.18614 1.61030i −3.42703 0 0.505408i 2.78912 0.469882i 0 2.18614 4.32550i
107.5 0.637910 1.26217i 0 −1.18614 1.61030i 3.42703 0 0.505408i −2.78912 + 0.469882i 0 2.18614 4.32550i
107.6 0.637910 + 1.26217i 0 −1.18614 + 1.61030i 3.42703 0 0.505408i −2.78912 0.469882i 0 2.18614 + 4.32550i
107.7 1.35760 0.396143i 0 1.68614 1.07561i −0.505408 0 3.42703i 1.86301 2.12819i 0 −0.686141 + 0.200214i
107.8 1.35760 + 0.396143i 0 1.68614 + 1.07561i −0.505408 0 3.42703i 1.86301 + 2.12819i 0 −0.686141 0.200214i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.8
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 12 T_{5}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} - 4 T^{6} + 16 T^{8} \)
$3$ 1
$5$ \( ( 1 + 8 T^{2} + 33 T^{4} + 200 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 - 16 T^{2} + 129 T^{4} - 784 T^{6} + 2401 T^{8} )^{2} \)
$11$ \( ( 1 - 11 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 - 2 T - 6 T^{2} - 26 T^{3} + 169 T^{4} )^{2}( 1 + 2 T - 6 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 40 T^{2} + 846 T^{4} - 11560 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 2 T + 6 T^{2} + 38 T^{3} + 361 T^{4} )^{4} \)
$23$ \( ( 1 + 32 T^{2} + 1182 T^{4} + 16928 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 80 T^{2} + 3150 T^{4} + 67280 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 64 T^{2} + 2913 T^{4} - 61504 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 32 T^{2} + 1806 T^{4} + 43808 T^{6} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 88 T^{2} + 4110 T^{4} - 147928 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{8} \)
$47$ \( ( 1 + 44 T^{2} + 2790 T^{4} + 97196 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 176 T^{2} + 13065 T^{4} + 494384 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 74 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 - 100 T^{2} + 7830 T^{4} - 372100 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{8} \)
$71$ \( ( 1 + 176 T^{2} + 16638 T^{4} + 887216 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 4 T + 117 T^{2} - 292 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 256 T^{2} + 28734 T^{4} - 1597696 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 - 214 T^{2} + 23115 T^{4} - 1474246 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 52 T^{2} - 2490 T^{4} - 411892 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + T + 97 T^{2} )^{8} \)
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