Properties

Label 216.2.f.a
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: 8.0.23123460096.3
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 2 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{5} + 6 \nu^{3} + 8 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{4} + 2 \nu^{2} + 4 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{5} + 2 \nu^{3} - 8 \nu \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{4} - 2 \nu^{2} + 4 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{5} + 2 \nu^{3} + 8 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{5}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{6} + \beta_{4} - 2\)
\(\nu^{5}\)\(=\)\(-\beta_{7} + \beta_{5} + 2 \beta_{2}\)
\(\nu^{6}\)\(=\)\(-2 \beta_{6} + 2 \beta_{4} - 2 \beta_{1}\)
\(\nu^{7}\)\(=\)\(-2 \beta_{7} - 4 \beta_{5} + 2 \beta_{3} - 4 \beta_{2}\)

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
1.08766 0.903873i
1.08766 + 0.903873i
0.563016 1.29731i
0.563016 + 1.29731i
−0.563016 1.29731i
−0.563016 + 1.29731i
−1.08766 0.903873i
−1.08766 + 0.903873i
−1.08766 0.903873i 0 0.366025 + 1.96622i −1.59245 0 1.43937i 1.37910 2.46943i 0 1.73205 + 1.43937i
107.2 −1.08766 + 0.903873i 0 0.366025 1.96622i −1.59245 0 1.43937i 1.37910 + 2.46943i 0 1.73205 1.43937i
107.3 −0.563016 1.29731i 0 −1.36603 + 1.46081i 3.07638 0 3.99102i 2.66422 + 0.949697i 0 −1.73205 3.99102i
107.4 −0.563016 + 1.29731i 0 −1.36603 1.46081i 3.07638 0 3.99102i 2.66422 0.949697i 0 −1.73205 + 3.99102i
107.5 0.563016 1.29731i 0 −1.36603 1.46081i −3.07638 0 3.99102i −2.66422 + 0.949697i 0 −1.73205 + 3.99102i
107.6 0.563016 + 1.29731i 0 −1.36603 + 1.46081i −3.07638 0 3.99102i −2.66422 0.949697i 0 −1.73205 3.99102i
107.7 1.08766 0.903873i 0 0.366025 1.96622i 1.59245 0 1.43937i −1.37910 2.46943i 0 1.73205 1.43937i
107.8 1.08766 + 0.903873i 0 0.366025 + 1.96622i 1.59245 0 1.43937i −1.37910 + 2.46943i 0 1.73205 + 1.43937i
\(n\): 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.a 8
3.b odd 2 1 inner 216.2.f.a 8
4.b odd 2 1 864.2.f.a 8
8.b even 2 1 864.2.f.a 8
8.d odd 2 1 inner 216.2.f.a 8
9.c even 3 2 648.2.l.f 16
9.d odd 6 2 648.2.l.f 16
12.b even 2 1 864.2.f.a 8
24.f even 2 1 inner 216.2.f.a 8
24.h odd 2 1 864.2.f.a 8
36.f odd 6 2 2592.2.p.f 16
36.h even 6 2 2592.2.p.f 16
72.j odd 6 2 2592.2.p.f 16
72.l even 6 2 648.2.l.f 16
72.n even 6 2 2592.2.p.f 16
72.p odd 6 2 648.2.l.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.f.a 8 1.a even 1 1 trivial
216.2.f.a 8 3.b odd 2 1 inner
216.2.f.a 8 8.d odd 2 1 inner
216.2.f.a 8 24.f even 2 1 inner
648.2.l.f 16 9.c even 3 2
648.2.l.f 16 9.d odd 6 2
648.2.l.f 16 72.l even 6 2
648.2.l.f 16 72.p odd 6 2
864.2.f.a 8 4.b odd 2 1
864.2.f.a 8 8.b even 2 1
864.2.f.a 8 12.b even 2 1
864.2.f.a 8 24.h odd 2 1
2592.2.p.f 16 36.f odd 6 2
2592.2.p.f 16 36.h even 6 2
2592.2.p.f 16 72.j odd 6 2
2592.2.p.f 16 72.n even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 12 T_{5}^{2} + 24 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 6 T^{4} + 8 T^{6} + 16 T^{8} \)
$3$ \( \)
$5$ \( ( 1 + 8 T^{2} + 54 T^{4} + 200 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 - 10 T^{2} + 75 T^{4} - 490 T^{6} + 2401 T^{8} )^{2} \)
$11$ \( ( 1 - 16 T^{2} + 198 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 22 T^{2} + 267 T^{4} - 3718 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 16 T^{2} + 54 T^{4} - 4624 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 4 T + 39 T^{2} - 76 T^{3} + 361 T^{4} )^{4} \)
$23$ \( ( 1 + 56 T^{2} + 1542 T^{4} + 29624 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 20 T^{2} + 1014 T^{4} + 16820 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 28 T^{2} + 1926 T^{4} - 26908 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 94 T^{2} + 4515 T^{4} - 128686 T^{6} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 52 T^{2} + 2310 T^{4} - 87412 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 2 T + 43 T^{2} )^{8} \)
$47$ \( ( 1 + 56 T^{2} + 4902 T^{4} + 123704 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 68 T^{2} + 5046 T^{4} + 191012 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 208 T^{2} + 17670 T^{4} - 724048 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 214 T^{2} + 18699 T^{4} - 796294 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 8 T + 123 T^{2} + 536 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 + 140 T^{2} + 13254 T^{4} + 705740 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 2 T + 99 T^{2} + 146 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 130 T^{2} + 13635 T^{4} - 811330 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 - 172 T^{2} + 20406 T^{4} - 1184908 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 80 T^{2} + 16470 T^{4} + 633680 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 2 T + 87 T^{2} + 194 T^{3} + 9409 T^{4} )^{4} \)
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