# Properties

 Label 216.2.v.a Level $216$ Weight $2$ Character orbit 216.v Analytic conductor $1.725$ Analytic rank $0$ Dimension $12$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.72476868366$$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{11} q^{2} + ( \beta_{5} - \beta_{8} - \beta_{11} ) q^{3} -2 \beta_{4} q^{4} + ( -\beta_{1} - 2 \beta_{10} ) q^{6} + ( -2 \beta_{3} + 2 \beta_{9} ) q^{8} + ( \beta_{4} - 2 \beta_{7} - \beta_{10} ) q^{9} +O(q^{10})$$ $$q -\beta_{11} q^{2} + ( \beta_{5} - \beta_{8} - \beta_{11} ) q^{3} -2 \beta_{4} q^{4} + ( -\beta_{1} - 2 \beta_{10} ) q^{6} + ( -2 \beta_{3} + 2 \beta_{9} ) q^{8} + ( \beta_{4} - 2 \beta_{7} - \beta_{10} ) q^{9} + ( 3 - \beta_{1} - \beta_{3} + 3 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{11} + ( -2 - 2 \beta_{3} + 2 \beta_{6} ) q^{12} + 4 \beta_{8} q^{16} + ( -2 \beta_{1} + 3 \beta_{2} - 2 \beta_{5} - 3 \beta_{8} + 3 \beta_{10} ) q^{17} + ( -4 - \beta_{9} ) q^{18} + ( \beta_{2} - \beta_{4} + 3 \beta_{5} + 3 \beta_{7} - \beta_{8} + \beta_{10} ) q^{19} + ( -2 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} ) q^{22} + ( -4 \beta_{2} + 2 \beta_{5} + 4 \beta_{8} ) q^{24} + 5 \beta_{2} q^{25} + ( -\beta_{3} - 5 \beta_{6} + \beta_{9} ) q^{27} + 4 \beta_{1} q^{32} + ( 1 - 5 \beta_{2} + 4 \beta_{3} - \beta_{6} - 2 \beta_{11} ) q^{33} + ( -4 + 3 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} - 3 \beta_{7} + 4 \beta_{10} ) q^{34} + ( -2 \beta_{2} + 4 \beta_{11} ) q^{36} + ( 6 + 6 \beta_{4} - \beta_{7} + \beta_{9} - 6 \beta_{10} ) q^{38} + ( 4 \beta_{3} - 4 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} - 4 \beta_{9} + 4 \beta_{11} ) q^{41} + ( 3 \beta_{1} - 3 \beta_{3} + 5 \beta_{6} + 3 \beta_{9} + 5 \beta_{10} ) q^{43} + ( -6 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 6 \beta_{8} + 6 \beta_{10} - 2 \beta_{11} ) q^{44} + ( 4 \beta_{4} + 4 \beta_{7} - 4 \beta_{10} ) q^{48} -7 \beta_{10} q^{49} + ( 5 \beta_{1} - 5 \beta_{7} ) q^{50} + ( -1 + \beta_{1} - 7 \beta_{4} - \beta_{7} + 5 \beta_{9} ) q^{51} + ( -5 \beta_{5} + 2 \beta_{8} + 5 \beta_{11} ) q^{54} + ( 4 \beta_{1} + 2 \beta_{3} + 5 \beta_{4} + 7 \beta_{6} - 4 \beta_{7} - 2 \beta_{9} ) q^{57} + ( 3 + 3 \beta_{2} - 5 \beta_{9} + 5 \beta_{11} ) q^{59} + ( 8 - 8 \beta_{6} ) q^{64} + ( -5 \beta_{1} + 8 \beta_{2} - 4 \beta_{4} - \beta_{5} + 5 \beta_{7} - 8 \beta_{8} ) q^{66} + ( -7 - 7 \beta_{2} - 3 \beta_{3} + 7 \beta_{6} + 3 \beta_{11} ) q^{67} + ( -6 + 6 \beta_{2} + 4 \beta_{5} - 6 \beta_{8} + 4 \beta_{9} ) q^{68} + ( -2 \beta_{1} + 8 \beta_{4} + 2 \beta_{7} ) q^{72} + ( -6 \beta_{1} + \beta_{4} + 6 \beta_{5} + 6 \beta_{7} + \beta_{8} - 6 \beta_{11} ) q^{73} + ( 5 \beta_{1} - 5 \beta_{10} ) q^{75} + ( -2 + 2 \beta_{2} - 6 \beta_{9} - 6 \beta_{11} ) q^{76} + ( -7 \beta_{2} - 4 \beta_{5} + 7 \beta_{8} ) q^{81} + ( -3 \beta_{1} - 3 \beta_{5} - 8 \beta_{8} + 8 \beta_{10} + 3 \beta_{11} ) q^{82} + ( -9 \beta_{2} + 18 \beta_{8} + \beta_{11} ) q^{83} + ( 6 + 5 \beta_{3} + 5 \beta_{5} - 6 \beta_{6} + 6 \beta_{8} - 5 \beta_{11} ) q^{86} + ( 4 - 6 \beta_{1} + 6 \beta_{9} - 4 \beta_{10} ) q^{88} + ( -18 + 2 \beta_{3} + 9 \beta_{6} - 2 \beta_{9} ) q^{89} + ( 8 - 4 \beta_{9} ) q^{96} + ( 5 + 6 \beta_{1} - 6 \beta_{3} - 5 \beta_{4} - 5 \beta_{6} - 6 \beta_{7} ) q^{97} -7 \beta_{3} q^{98} + ( -7 \beta_{1} + 7 \beta_{2} + \beta_{10} - 5 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + O(q^{10})$$ $$12q + 18q^{11} - 12q^{12} - 48q^{18} + 12q^{22} - 30q^{27} + 6q^{33} - 24q^{34} + 72q^{38} - 18q^{41} + 30q^{43} - 12q^{51} + 42q^{57} + 36q^{59} + 48q^{64} - 42q^{67} - 72q^{68} - 24q^{76} + 36q^{86} + 48q^{88} - 162q^{89} + 96q^{96} + 30q^{97} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.483690 − 1.32893i −0.483690 + 1.32893i 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
−0.909039 + 1.08335i 1.42338 + 0.986906i −0.347296 1.96962i 0 −2.36307 + 0.644886i 0 2.44949 + 1.41421i 1.05203 + 2.80949i 0
11.2 0.909039 1.08335i 0.456003 1.67095i −0.347296 1.96962i 0 −1.39570 2.01297i 0 −2.44949 1.41421i −2.58412 1.52391i 0
59.1 −0.909039 1.08335i 1.42338 0.986906i −0.347296 + 1.96962i 0 −2.36307 0.644886i 0 2.44949 1.41421i 1.05203 2.80949i 0
59.2 0.909039 + 1.08335i 0.456003 + 1.67095i −0.347296 + 1.96962i 0 −1.39570 + 2.01297i 0 −2.44949 + 1.41421i −2.58412 + 1.52391i 0
83.1 −1.39273 0.245576i −1.67508 + 0.440563i 1.87939 + 0.684040i 0 2.44113 0.202225i 0 −2.44949 1.41421i 2.61181 1.47596i 0
83.2 1.39273 + 0.245576i 0.142995 1.72614i 1.87939 + 0.684040i 0 0.623050 2.36893i 0 2.44949 + 1.41421i −2.95911 0.493657i 0
131.1 −0.483690 + 1.32893i −1.56638 0.739232i −1.53209 1.28558i 0 1.74002 1.72404i 0 2.44949 1.41421i 1.90707 + 2.31583i 0
131.2 0.483690 1.32893i 1.21908 1.23038i −1.53209 1.28558i 0 −1.04543 2.21519i 0 −2.44949 + 1.41421i −0.0276864 2.99987i 0
155.1 −0.483690 1.32893i −1.56638 + 0.739232i −1.53209 + 1.28558i 0 1.74002 + 1.72404i 0 2.44949 + 1.41421i 1.90707 2.31583i 0
155.2 0.483690 + 1.32893i 1.21908 + 1.23038i −1.53209 + 1.28558i 0 −1.04543 + 2.21519i 0 −2.44949 1.41421i −0.0276864 + 2.99987i 0
203.1 −1.39273 + 0.245576i −1.67508 0.440563i 1.87939 0.684040i 0 2.44113 + 0.202225i 0 −2.44949 + 1.41421i 2.61181 + 1.47596i 0
203.2 1.39273 0.245576i 0.142995 + 1.72614i 1.87939 0.684040i 0 0.623050 + 2.36893i 0 2.44949 1.41421i −2.95911 + 0.493657i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 203.2 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.f odd 18 1 inner
216.v even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.v.a 12
3.b odd 2 1 648.2.v.a 12
4.b odd 2 1 864.2.bh.a 12
8.b even 2 1 864.2.bh.a 12
8.d odd 2 1 CM 216.2.v.a 12
24.f even 2 1 648.2.v.a 12
27.e even 9 1 648.2.v.a 12
27.f odd 18 1 inner 216.2.v.a 12
108.l even 18 1 864.2.bh.a 12
216.r odd 18 1 648.2.v.a 12
216.v even 18 1 inner 216.2.v.a 12
216.x odd 18 1 864.2.bh.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.v.a 12 1.a even 1 1 trivial
216.2.v.a 12 8.d odd 2 1 CM
216.2.v.a 12 27.f odd 18 1 inner
216.2.v.a 12 216.v even 18 1 inner
648.2.v.a 12 3.b odd 2 1
648.2.v.a 12 24.f even 2 1
648.2.v.a 12 27.e even 9 1
648.2.v.a 12 216.r odd 18 1
864.2.bh.a 12 4.b odd 2 1
864.2.bh.a 12 8.b even 2 1
864.2.bh.a 12 108.l even 18 1
864.2.bh.a 12 216.x odd 18 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 - 8 T^{6} + T^{12}$$
$3$ $$729 + 270 T^{3} + 73 T^{6} + 10 T^{9} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$T^{12}$$
$11$ $$1014049 - 3625200 T + 6653400 T^{2} - 5831244 T^{3} + 2541012 T^{4} - 637074 T^{5} + 140302 T^{6} - 29736 T^{7} + 6528 T^{8} - 1296 T^{9} + 183 T^{10} - 18 T^{11} + T^{12}$$
$13$ $$T^{12}$$
$17$ $$10156969 + 43884990 T + 71493687 T^{2} + 35815770 T^{3} + 5850975 T^{4} - 702270 T^{5} - 247376 T^{6} + 13770 T^{7} + 7803 T^{8} - 102 T^{10} + T^{12}$$
$19$ $$87254281 + 56438322 T + 66854673 T^{2} - 17650166 T^{3} + 10131579 T^{4} - 1033182 T^{5} + 362940 T^{6} - 18126 T^{7} + 9747 T^{8} - 212 T^{9} + 114 T^{10} + T^{12}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12}$$
$37$ $$T^{12}$$
$41$ $$41437894969 + 2345045760 T + 830094720 T^{2} - 623389716 T^{3} - 71266908 T^{4} - 1888506 T^{5} + 2202202 T^{6} + 351576 T^{7} + 28488 T^{8} + 1296 T^{9} + 93 T^{10} + 18 T^{11} + T^{12}$$
$43$ $$23845845241 - 27578046390 T + 12632246061 T^{2} - 3004705910 T^{3} + 503792109 T^{4} - 57712590 T^{5} + 4963332 T^{6} - 350460 T^{7} + 45882 T^{8} - 5420 T^{9} + 471 T^{10} - 30 T^{11} + T^{12}$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$260443853569 - 180441894438 T + 42050717511 T^{2} - 4631311350 T^{3} + 751181712 T^{4} - 150362694 T^{5} + 18977005 T^{6} - 1955952 T^{7} + 183153 T^{8} - 14094 T^{9} + 894 T^{10} - 36 T^{11} + T^{12}$$
$61$ $$T^{12}$$
$67$ $$805307017321 + 519259786626 T + 147790636341 T^{2} + 23838564274 T^{3} + 2498333517 T^{4} + 203605290 T^{5} + 17172516 T^{6} + 1615572 T^{7} + 190170 T^{8} + 16324 T^{9} + 975 T^{10} + 42 T^{11} + T^{12}$$
$71$ $$T^{12}$$
$73$ $$964620586801 + 92489159670 T + 55972933011 T^{2} - 3671837510 T^{3} + 1910568483 T^{4} - 61869690 T^{5} + 19227516 T^{6} - 282510 T^{7} + 143883 T^{8} - 860 T^{9} + 438 T^{10} + T^{12}$$
$79$ $$T^{12}$$
$83$ $$195930594145441 - 188210667366 T^{3} + 74262493 T^{6} - 13446 T^{9} + T^{12}$$
$89$ $$( 55225 + 12690 T + 1207 T^{2} + 54 T^{3} + T^{4} )^{3}$$
$97$ $$828247426561 - 490187828220 T + 204629473500 T^{2} - 49789788320 T^{3} + 7962301116 T^{4} - 226179270 T^{5} - 23158410 T^{6} + 1666620 T^{7} + 2700 T^{8} - 2180 T^{9} + 309 T^{10} - 30 T^{11} + T^{12}$$