# 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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