Properties

Label 216.3.r.a
Level $216$
Weight $3$
Character orbit 216.r
Analytic conductor $5.886$
Analytic rank $0$
Dimension $12$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.r (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.101559956668416.1
Defining polynomial: \(x^{12} - 8 x^{6} + 64\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + ( \beta_{5} - \beta_{8} - \beta_{11} ) q^{3} + 4 \beta_{4} q^{4} + ( 2 \beta_{1} - 2 \beta_{10} ) q^{6} + 8 \beta_{6} q^{8} + ( 7 \beta_{4} - 2 \beta_{7} - 7 \beta_{10} ) q^{9} +O(q^{10})\) \( q + 2 \beta_{2} q^{2} + ( \beta_{5} - \beta_{8} - \beta_{11} ) q^{3} + 4 \beta_{4} q^{4} + ( 2 \beta_{1} - 2 \beta_{10} ) q^{6} + 8 \beta_{6} q^{8} + ( 7 \beta_{4} - 2 \beta_{7} - 7 \beta_{10} ) q^{9} + ( -7 + 3 \beta_{1} - 3 \beta_{3} + 7 \beta_{4} + 7 \beta_{6} - 3 \beta_{7} ) q^{11} + ( 4 + 4 \beta_{3} - 4 \beta_{6} ) q^{12} + 16 \beta_{8} q^{16} + ( -6 \beta_{1} + \beta_{2} + 6 \beta_{5} - \beta_{8} - \beta_{10} ) q^{17} + ( 14 - 4 \beta_{9} ) q^{18} + ( -17 \beta_{2} + 17 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} + 17 \beta_{8} - 17 \beta_{10} ) q^{19} + ( -14 \beta_{2} + 6 \beta_{3} - 6 \beta_{5} + 14 \beta_{6} + 14 \beta_{8} - 6 \beta_{9} ) q^{22} + ( 8 \beta_{2} + 8 \beta_{5} - 8 \beta_{8} ) q^{24} -25 \beta_{2} q^{25} + ( 5 \beta_{3} - 23 \beta_{6} - 5 \beta_{9} ) q^{27} + 32 \beta_{10} q^{32} + ( 31 - 17 \beta_{2} + 4 \beta_{3} - 31 \beta_{6} + 10 \beta_{11} ) q^{33} + ( 2 - 12 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 12 \beta_{7} - 2 \beta_{10} ) q^{34} + ( 28 \beta_{2} - 8 \beta_{11} ) q^{36} + ( 34 - 34 \beta_{4} - 6 \beta_{7} - 6 \beta_{9} + 34 \beta_{10} ) q^{38} + ( -12 \beta_{3} - 12 \beta_{5} + 23 \beta_{6} - 23 \beta_{8} + 12 \beta_{9} + 12 \beta_{11} ) q^{41} + ( 15 \beta_{1} - 15 \beta_{3} - 7 \beta_{6} + 15 \beta_{9} - 7 \beta_{10} ) q^{43} + ( -28 \beta_{4} + 12 \beta_{5} - 12 \beta_{7} + 28 \beta_{8} + 28 \beta_{10} - 12 \beta_{11} ) q^{44} + ( 16 \beta_{4} + 16 \beta_{7} - 16 \beta_{10} ) q^{48} -49 \beta_{10} q^{49} -50 \beta_{4} q^{50} + ( -49 + 7 \beta_{1} + 47 \beta_{4} - 7 \beta_{7} + 5 \beta_{9} ) q^{51} + ( 10 \beta_{5} - 46 \beta_{8} - 10 \beta_{11} ) q^{54} + ( -20 \beta_{1} + 14 \beta_{3} - 7 \beta_{4} - 41 \beta_{6} + 20 \beta_{7} - 14 \beta_{9} ) q^{57} + ( -41 + 41 \beta_{2} + 15 \beta_{9} + 15 \beta_{11} ) q^{59} + ( -64 + 64 \beta_{6} ) q^{64} + ( -20 \beta_{1} + 62 \beta_{2} - 34 \beta_{4} + 8 \beta_{5} + 20 \beta_{7} - 62 \beta_{8} ) q^{66} + ( -31 - 31 \beta_{2} + 21 \beta_{3} + 31 \beta_{6} - 21 \beta_{11} ) q^{67} + ( 4 + 4 \beta_{2} - 24 \beta_{5} - 4 \beta_{8} + 24 \beta_{9} ) q^{68} + ( 16 \beta_{1} + 56 \beta_{4} - 16 \beta_{7} ) q^{72} + ( 6 \beta_{1} - 71 \beta_{4} - 6 \beta_{5} - 6 \beta_{7} - 71 \beta_{8} + 6 \beta_{11} ) q^{73} + ( -25 \beta_{1} + 25 \beta_{10} ) q^{75} + ( -68 + 68 \beta_{2} - 12 \beta_{9} - 12 \beta_{11} ) q^{76} + ( 17 \beta_{2} - 28 \beta_{5} - 17 \beta_{8} ) q^{81} + ( -24 \beta_{1} - 24 \beta_{5} + 46 \beta_{8} - 46 \beta_{10} + 24 \beta_{11} ) q^{82} + ( 79 \beta_{2} + 18 \beta_{5} - 9 \beta_{11} ) q^{83} + ( 14 + 30 \beta_{3} - 30 \beta_{5} - 14 \beta_{6} - 14 \beta_{8} + 30 \beta_{11} ) q^{86} + ( -56 + 24 \beta_{1} - 24 \beta_{9} + 56 \beta_{10} ) q^{88} + ( -18 \beta_{3} + 73 \beta_{6} - 18 \beta_{9} ) q^{89} + ( 32 + 32 \beta_{9} ) q^{96} + ( 47 + 30 \beta_{1} - 30 \beta_{3} - 47 \beta_{4} - 47 \beta_{6} - 30 \beta_{7} ) q^{97} + ( 98 - 98 \beta_{6} ) q^{98} + ( -7 \beta_{1} + \beta_{2} + 97 \beta_{10} - 35 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 48q^{8} + O(q^{10}) \) \( 12q + 48q^{8} - 42q^{11} + 24q^{12} + 168q^{18} + 84q^{22} - 138q^{27} + 186q^{33} + 12q^{34} + 408q^{38} + 138q^{41} - 42q^{43} - 588q^{51} - 246q^{57} - 492q^{59} - 384q^{64} - 186q^{67} + 48q^{68} - 816q^{76} + 84q^{86} - 672q^{88} + 438q^{89} + 384q^{96} + 282q^{97} + 588q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 8 x^{6} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/4\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/2\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/8\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/4\)
\(\beta_{8}\)\(=\)\( \nu^{8} \)\(/16\)
\(\beta_{9}\)\(=\)\( \nu^{9} \)\(/8\)
\(\beta_{10}\)\(=\)\( \nu^{10} \)\(/32\)
\(\beta_{11}\)\(=\)\( \nu^{11} \)\(/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)
\(\nu^{4}\)\(=\)\(4 \beta_{4}\)
\(\nu^{5}\)\(=\)\(2 \beta_{5}\)
\(\nu^{6}\)\(=\)\(8 \beta_{6}\)
\(\nu^{7}\)\(=\)\(4 \beta_{7}\)
\(\nu^{8}\)\(=\)\(16 \beta_{8}\)
\(\nu^{9}\)\(=\)\(8 \beta_{9}\)
\(\nu^{10}\)\(=\)\(32 \beta_{10}\)
\(\nu^{11}\)\(=\)\(16 \beta_{11}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.483690 + 1.32893i
0.483690 1.32893i
−0.909039 + 1.08335i
0.909039 1.08335i
−1.39273 + 0.245576i
1.39273 0.245576i
−1.39273 0.245576i
1.39273 + 0.245576i
−0.909039 1.08335i
0.909039 + 1.08335i
−0.483690 1.32893i
0.483690 + 1.32893i
−1.53209 1.28558i −0.0276864 2.99987i 0.694593 + 3.93923i 0 −3.81414 + 4.63166i 0 4.00000 6.92820i −8.99847 + 0.166112i 0
43.2 −1.53209 1.28558i 1.90707 + 2.31583i 0.694593 + 3.93923i 0 0.0553729 5.99974i 0 4.00000 6.92820i −1.72616 + 8.83292i 0
67.1 −0.347296 1.96962i −2.58412 1.52391i −3.75877 + 1.36808i 0 −2.10407 + 5.61898i 0 4.00000 + 6.92820i 4.35538 + 7.87596i 0
67.2 −0.347296 1.96962i 1.05203 + 2.80949i −3.75877 + 1.36808i 0 5.16824 3.04783i 0 4.00000 + 6.92820i −6.78645 + 5.91135i 0
115.1 1.87939 0.684040i −2.95911 + 0.493657i 3.06418 2.57115i 0 −5.22362 + 2.95192i 0 4.00000 6.92820i 8.51261 2.92156i 0
115.2 1.87939 0.684040i 2.61181 + 1.47596i 3.06418 2.57115i 0 5.91821 + 0.987313i 0 4.00000 6.92820i 4.64309 + 7.70985i 0
139.1 1.87939 + 0.684040i −2.95911 0.493657i 3.06418 + 2.57115i 0 −5.22362 2.95192i 0 4.00000 + 6.92820i 8.51261 + 2.92156i 0
139.2 1.87939 + 0.684040i 2.61181 1.47596i 3.06418 + 2.57115i 0 5.91821 0.987313i 0 4.00000 + 6.92820i 4.64309 7.70985i 0
187.1 −0.347296 + 1.96962i −2.58412 + 1.52391i −3.75877 1.36808i 0 −2.10407 5.61898i 0 4.00000 6.92820i 4.35538 7.87596i 0
187.2 −0.347296 + 1.96962i 1.05203 2.80949i −3.75877 1.36808i 0 5.16824 + 3.04783i 0 4.00000 6.92820i −6.78645 5.91135i 0
211.1 −1.53209 + 1.28558i −0.0276864 + 2.99987i 0.694593 3.93923i 0 −3.81414 4.63166i 0 4.00000 + 6.92820i −8.99847 0.166112i 0
211.2 −1.53209 + 1.28558i 1.90707 2.31583i 0.694593 3.93923i 0 0.0553729 + 5.99974i 0 4.00000 + 6.92820i −1.72616 8.83292i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
27.e even 9 1 inner
216.r odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.r.a 12
8.d odd 2 1 CM 216.3.r.a 12
27.e even 9 1 inner 216.3.r.a 12
216.r odd 18 1 inner 216.3.r.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.r.a 12 1.a even 1 1 trivial
216.3.r.a 12 8.d odd 2 1 CM
216.3.r.a 12 27.e even 9 1 inner
216.3.r.a 12 216.r odd 18 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 64 - 8 T^{3} + T^{6} )^{2} \)
$3$ \( 531441 + 33534 T^{3} + 1387 T^{6} + 46 T^{9} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( T^{12} \)
$11$ \( 22970736721 + 48900247284 T + 690516137724 T^{2} + 163841915104 T^{3} + 14388660540 T^{4} - 298294542 T^{5} - 75882714 T^{6} - 1837332 T^{7} + 132300 T^{8} + 11788 T^{9} + 813 T^{10} + 42 T^{11} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( 4821029113844161 + 103903401640902 T + 54431835312123 T^{2} - 885174096326 T^{3} + 447221295459 T^{4} - 3892245642 T^{5} + 1167540540 T^{6} - 4489326 T^{7} + 2255067 T^{8} - 3452 T^{9} + 1734 T^{10} + T^{12} \)
$19$ \( 18218876644427761 + 362819843335914 T + 165538788960627 T^{2} - 2482705257818 T^{3} + 1089979364259 T^{4} - 8733331494 T^{5} + 2276683260 T^{6} - 8064018 T^{7} + 3518667 T^{8} - 4964 T^{9} + 2166 T^{10} + T^{12} \)
$23$ \( T^{12} \)
$29$ \( T^{12} \)
$31$ \( T^{12} \)
$37$ \( T^{12} \)
$41$ \( 15040089624557904481 - 1951843817068062036 T + 316923585999309564 T^{2} - 21055133887393376 T^{3} + 685279134137340 T^{4} - 1994264720802 T^{5} - 162850387434 T^{6} + 1958659668 T^{7} + 8284140 T^{8} - 314732 T^{9} + 7653 T^{10} - 138 T^{11} + T^{12} \)
$43$ \( \)\(17\!\cdots\!01\)\( - 4757011452706844058 T + 711080357124210135 T^{2} - 4267850865432014 T^{3} - 52514102476827 T^{4} - 3797567241426 T^{5} - 65658437484 T^{6} + 2406802356 T^{7} + 55360890 T^{8} - 133364 T^{9} - 4371 T^{10} + 42 T^{11} + T^{12} \)
$47$ \( T^{12} \)
$53$ \( T^{12} \)
$59$ \( \)\(11\!\cdots\!81\)\( + \)\(28\!\cdots\!46\)\( T + 33197558443207983033 T^{2} + 1643273285974746910 T^{3} + 47723084889306048 T^{4} + 998878636701198 T^{5} + 16835044360719 T^{6} + 226963368720 T^{7} + 2411871597 T^{8} + 19895578 T^{9} + 121746 T^{10} + 492 T^{11} + T^{12} \)
$61$ \( T^{12} \)
$67$ \( \)\(71\!\cdots\!61\)\( - \)\(17\!\cdots\!54\)\( T + 17306321595561576519 T^{2} - 922662021313604990 T^{3} + 4505790785672997 T^{4} + 296160068199294 T^{5} + 4618572424788 T^{6} + 37196390484 T^{7} + 185104314 T^{8} + 236716 T^{9} + 9597 T^{10} + 186 T^{11} + T^{12} \)
$71$ \( T^{12} \)
$73$ \( \)\(10\!\cdots\!21\)\( + \)\(96\!\cdots\!38\)\( T + \)\(11\!\cdots\!23\)\( T^{2} - 2302329478177954454 T^{3} + 67680046493478819 T^{4} - 454786981486938 T^{5} + 8319466687740 T^{6} - 28447299774 T^{7} + 766752507 T^{8} - 1186268 T^{9} + 31974 T^{10} + T^{12} \)
$79$ \( T^{12} \)
$83$ \( \)\(62\!\cdots\!29\)\( - 151345094418253982 T^{3} + 3559345255083 T^{6} - 1907534 T^{9} + T^{12} \)
$89$ \( ( 5987809 + 357262 T + 23763 T^{2} - 146 T^{3} + T^{4} )^{3} \)
$97$ \( \)\(67\!\cdots\!41\)\( - \)\(44\!\cdots\!36\)\( T + \)\(18\!\cdots\!40\)\( T^{2} - 40884674212570337120 T^{3} + 703538789684325372 T^{4} - 2459859509429010 T^{5} - 17654687344266 T^{6} + 154074865140 T^{7} - 45567252 T^{8} - 1337996 T^{9} + 24789 T^{10} - 282 T^{11} + T^{12} \)
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