# 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: $$x^{12} - 8x^{6} + 64$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{11} q^{2} + ( - \beta_{11} - \beta_{8} + \beta_{5}) q^{3} - 2 \beta_{4} q^{4} + ( - 2 \beta_{10} - \beta_1) q^{6} + (2 \beta_{9} - 2 \beta_{3}) q^{8} + ( - \beta_{10} - 2 \beta_{7} + \beta_{4}) q^{9}+O(q^{10})$$ q - b11 * q^2 + (-b11 - b8 + b5) * q^3 - 2*b4 * q^4 + (-2*b10 - b1) * q^6 + (2*b9 - 2*b3) * q^8 + (-b10 - 2*b7 + b4) * q^9 $$q - \beta_{11} q^{2} + ( - \beta_{11} - \beta_{8} + \beta_{5}) q^{3} - 2 \beta_{4} q^{4} + ( - 2 \beta_{10} - \beta_1) q^{6} + (2 \beta_{9} - 2 \beta_{3}) q^{8} + ( - \beta_{10} - 2 \beta_{7} + \beta_{4}) q^{9} + (\beta_{7} - 3 \beta_{6} + 3 \beta_{4} - \beta_{3} - \beta_1 + 3) q^{11} + (2 \beta_{6} - 2 \beta_{3} - 2) q^{12} + 4 \beta_{8} q^{16} + (3 \beta_{10} - 3 \beta_{8} - 2 \beta_{5} + 3 \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{9} - 4) q^{18} + (\beta_{10} - \beta_{8} + 3 \beta_{7} + 3 \beta_{5} - \beta_{4} + \beta_{2}) q^{19} + ( - 3 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} - 3 \beta_{5} + 3 \beta_{3} - 2 \beta_{2}) q^{22} + (4 \beta_{8} + 2 \beta_{5} - 4 \beta_{2}) q^{24} + 5 \beta_{2} q^{25} + (\beta_{9} - 5 \beta_{6} - \beta_{3}) q^{27} + 4 \beta_1 q^{32} + ( - 2 \beta_{11} - \beta_{6} + 4 \beta_{3} - 5 \beta_{2} + 1) q^{33} + (4 \beta_{10} - 3 \beta_{7} + 4 \beta_{6} - 4 \beta_{4} + 3 \beta_{3} - 4) q^{34} + (4 \beta_{11} - 2 \beta_{2}) q^{36} + ( - 6 \beta_{10} + \beta_{9} - \beta_{7} + 6 \beta_{4} + 6) q^{38} + (4 \beta_{11} - 4 \beta_{9} - 3 \beta_{8} - 3 \beta_{6} - 4 \beta_{5} + 4 \beta_{3}) q^{41} + (5 \beta_{10} + 3 \beta_{9} + 5 \beta_{6} - 3 \beta_{3} + 3 \beta_1) q^{43} + ( - 2 \beta_{11} + 6 \beta_{10} - 6 \beta_{8} + 2 \beta_{7} + 2 \beta_{5} - 6 \beta_{4}) q^{44} + ( - 4 \beta_{10} + 4 \beta_{7} + 4 \beta_{4}) q^{48} - 7 \beta_{10} q^{49} + ( - 5 \beta_{7} + 5 \beta_1) q^{50} + (5 \beta_{9} - \beta_{7} - 7 \beta_{4} + \beta_1 - 1) q^{51} + (5 \beta_{11} + 2 \beta_{8} - 5 \beta_{5}) q^{54} + ( - 2 \beta_{9} - 4 \beta_{7} + 7 \beta_{6} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_1) q^{57} + (5 \beta_{11} - 5 \beta_{9} + 3 \beta_{2} + 3) q^{59} + ( - 8 \beta_{6} + 8) q^{64} + ( - 8 \beta_{8} + 5 \beta_{7} - \beta_{5} - 4 \beta_{4} + 8 \beta_{2} - 5 \beta_1) q^{66} + (3 \beta_{11} + 7 \beta_{6} - 3 \beta_{3} - 7 \beta_{2} - 7) q^{67} + (4 \beta_{9} - 6 \beta_{8} + 4 \beta_{5} + 6 \beta_{2} - 6) q^{68} + (2 \beta_{7} + 8 \beta_{4} - 2 \beta_1) q^{72} + ( - 6 \beta_{11} + \beta_{8} + 6 \beta_{7} + 6 \beta_{5} + \beta_{4} - 6 \beta_1) q^{73} + ( - 5 \beta_{10} + 5 \beta_1) q^{75} + ( - 6 \beta_{11} - 6 \beta_{9} + 2 \beta_{2} - 2) q^{76} + (7 \beta_{8} - 4 \beta_{5} - 7 \beta_{2}) q^{81} + (3 \beta_{11} + 8 \beta_{10} - 8 \beta_{8} - 3 \beta_{5} - 3 \beta_1) q^{82} + (\beta_{11} + 18 \beta_{8} - 9 \beta_{2}) q^{83} + ( - 5 \beta_{11} + 6 \beta_{8} - 6 \beta_{6} + 5 \beta_{5} + 5 \beta_{3} + 6) q^{86} + ( - 4 \beta_{10} + 6 \beta_{9} - 6 \beta_1 + 4) q^{88} + ( - 2 \beta_{9} + 9 \beta_{6} + 2 \beta_{3} - 18) q^{89} + ( - 4 \beta_{9} + 8) q^{96} + ( - 6 \beta_{7} - 5 \beta_{6} - 5 \beta_{4} - 6 \beta_{3} + 6 \beta_1 + 5) q^{97} - 7 \beta_{3} q^{98} + ( - 5 \beta_{11} + \beta_{10} + 7 \beta_{2} - 7 \beta_1) q^{99}+O(q^{100})$$ q - b11 * q^2 + (-b11 - b8 + b5) * q^3 - 2*b4 * q^4 + (-2*b10 - b1) * q^6 + (2*b9 - 2*b3) * q^8 + (-b10 - 2*b7 + b4) * q^9 + (b7 - 3*b6 + 3*b4 - b3 - b1 + 3) * q^11 + (2*b6 - 2*b3 - 2) * q^12 + 4*b8 * q^16 + (3*b10 - 3*b8 - 2*b5 + 3*b2 - 2*b1) * q^17 + (-b9 - 4) * q^18 + (b10 - b8 + 3*b7 + 3*b5 - b4 + b2) * q^19 + (-3*b9 + 2*b8 + 2*b6 - 3*b5 + 3*b3 - 2*b2) * q^22 + (4*b8 + 2*b5 - 4*b2) * q^24 + 5*b2 * q^25 + (b9 - 5*b6 - b3) * q^27 + 4*b1 * q^32 + (-2*b11 - b6 + 4*b3 - 5*b2 + 1) * q^33 + (4*b10 - 3*b7 + 4*b6 - 4*b4 + 3*b3 - 4) * q^34 + (4*b11 - 2*b2) * q^36 + (-6*b10 + b9 - b7 + 6*b4 + 6) * q^38 + (4*b11 - 4*b9 - 3*b8 - 3*b6 - 4*b5 + 4*b3) * q^41 + (5*b10 + 3*b9 + 5*b6 - 3*b3 + 3*b1) * q^43 + (-2*b11 + 6*b10 - 6*b8 + 2*b7 + 2*b5 - 6*b4) * q^44 + (-4*b10 + 4*b7 + 4*b4) * q^48 - 7*b10 * q^49 + (-5*b7 + 5*b1) * q^50 + (5*b9 - b7 - 7*b4 + b1 - 1) * q^51 + (5*b11 + 2*b8 - 5*b5) * q^54 + (-2*b9 - 4*b7 + 7*b6 + 5*b4 + 2*b3 + 4*b1) * q^57 + (5*b11 - 5*b9 + 3*b2 + 3) * q^59 + (-8*b6 + 8) * q^64 + (-8*b8 + 5*b7 - b5 - 4*b4 + 8*b2 - 5*b1) * q^66 + (3*b11 + 7*b6 - 3*b3 - 7*b2 - 7) * q^67 + (4*b9 - 6*b8 + 4*b5 + 6*b2 - 6) * q^68 + (2*b7 + 8*b4 - 2*b1) * q^72 + (-6*b11 + b8 + 6*b7 + 6*b5 + b4 - 6*b1) * q^73 + (-5*b10 + 5*b1) * q^75 + (-6*b11 - 6*b9 + 2*b2 - 2) * q^76 + (7*b8 - 4*b5 - 7*b2) * q^81 + (3*b11 + 8*b10 - 8*b8 - 3*b5 - 3*b1) * q^82 + (b11 + 18*b8 - 9*b2) * q^83 + (-5*b11 + 6*b8 - 6*b6 + 5*b5 + 5*b3 + 6) * q^86 + (-4*b10 + 6*b9 - 6*b1 + 4) * q^88 + (-2*b9 + 9*b6 + 2*b3 - 18) * q^89 + (-4*b9 + 8) * q^96 + (-6*b7 - 5*b6 - 5*b4 - 6*b3 + 6*b1 + 5) * q^97 - 7*b3 * q^98 + (-5*b11 + b10 + 7*b2 - 7*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q + 18 q^{11} - 12 q^{12} - 48 q^{18} + 12 q^{22} - 30 q^{27} + 6 q^{33} - 24 q^{34} + 72 q^{38} - 18 q^{41} + 30 q^{43} - 12 q^{51} + 42 q^{57} + 36 q^{59} + 48 q^{64} - 42 q^{67} - 72 q^{68} - 24 q^{76} + 36 q^{86} + 48 q^{88} - 162 q^{89} + 96 q^{96} + 30 q^{97}+O(q^{100})$$ 12 * q + 18 * q^11 - 12 * q^12 - 48 * q^18 + 12 * q^22 - 30 * q^27 + 6 * q^33 - 24 * q^34 + 72 * q^38 - 18 * q^41 + 30 * q^43 - 12 * q^51 + 42 * q^57 + 36 * q^59 + 48 * q^64 - 42 * q^67 - 72 * q^68 - 24 * q^76 + 36 * q^86 + 48 * q^88 - 162 * q^89 + 96 * q^96 + 30 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} ) / 4$$ (v^4) / 4 $$\beta_{5}$$ $$=$$ $$( \nu^{5} ) / 4$$ (v^5) / 4 $$\beta_{6}$$ $$=$$ $$( \nu^{6} ) / 8$$ (v^6) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{7} ) / 8$$ (v^7) / 8 $$\beta_{8}$$ $$=$$ $$( \nu^{8} ) / 16$$ (v^8) / 16 $$\beta_{9}$$ $$=$$ $$( \nu^{9} ) / 16$$ (v^9) / 16 $$\beta_{10}$$ $$=$$ $$( \nu^{10} ) / 32$$ (v^10) / 32 $$\beta_{11}$$ $$=$$ $$( \nu^{11} ) / 32$$ (v^11) / 32
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3 $$\nu^{4}$$ $$=$$ $$4\beta_{4}$$ 4*b4 $$\nu^{5}$$ $$=$$ $$4\beta_{5}$$ 4*b5 $$\nu^{6}$$ $$=$$ $$8\beta_{6}$$ 8*b6 $$\nu^{7}$$ $$=$$ $$8\beta_{7}$$ 8*b7 $$\nu^{8}$$ $$=$$ $$16\beta_{8}$$ 16*b8 $$\nu^{9}$$ $$=$$ $$16\beta_{9}$$ 16*b9 $$\nu^{10}$$ $$=$$ $$32\beta_{10}$$ 32*b10 $$\nu^{11}$$ $$=$$ $$32\beta_{11}$$ 32*b11

## 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

$p$ $F_p(T)$
$2$ $$T^{12} - 8T^{6} + 64$$
$3$ $$T^{12} + 10 T^{9} + 73 T^{6} + \cdots + 729$$
$5$ $$T^{12}$$
$7$ $$T^{12}$$
$11$ $$T^{12} - 18 T^{11} + 183 T^{10} + \cdots + 1014049$$
$13$ $$T^{12}$$
$17$ $$T^{12} - 102 T^{10} + \cdots + 10156969$$
$19$ $$T^{12} + 114 T^{10} + \cdots + 87254281$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12}$$
$37$ $$T^{12}$$
$41$ $$T^{12} + 18 T^{11} + \cdots + 41437894969$$
$43$ $$T^{12} - 30 T^{11} + \cdots + 23845845241$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12} - 36 T^{11} + \cdots + 260443853569$$
$61$ $$T^{12}$$
$67$ $$T^{12} + 42 T^{11} + \cdots + 805307017321$$
$71$ $$T^{12}$$
$73$ $$T^{12} + 438 T^{10} + \cdots + 964620586801$$
$79$ $$T^{12}$$
$83$ $$T^{12} + \cdots + 195930594145441$$
$89$ $$(T^{4} + 54 T^{3} + 1207 T^{2} + \cdots + 55225)^{3}$$
$97$ $$T^{12} - 30 T^{11} + \cdots + 828247426561$$