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

Downloads

Learn more

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: \( x^{12} - 8x^{6} + 64 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + ( - \beta_{11} - \beta_{8} + \beta_{5}) q^{3} - 2 \beta_{4} q^{4} + ( - 2 \beta_{10} - \beta_1) q^{6} + (2 \beta_{9} - 2 \beta_{3}) q^{8} + ( - \beta_{10} - 2 \beta_{7} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 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}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 18 q^{11} - 12 q^{12} - 48 q^{18} + 12 q^{22} - 30 q^{27} + 6 q^{33} - 24 q^{34} + 72 q^{38} - 18 q^{41} + 30 q^{43} - 12 q^{51} + 42 q^{57} + 36 q^{59} + 48 q^{64} - 42 q^{67} - 72 q^{68} - 24 q^{76} + 36 q^{86} + 48 q^{88} - 162 q^{89} + 96 q^{96} + 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16\beta_{8} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{10} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 32\beta_{11} \) Copy content Toggle raw display

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:

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 \(S_{2}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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