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Properties

Label 216.2.n.b
Level $216$
Weight $2$
Character orbit 216.n
Analytic conductor $1.725$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,2,Mod(37,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( - \beta_{9} + \beta_{4}) q^{4} + ( - \beta_{8} - \beta_{6} + \beta_{5}) q^{5} + (\beta_{7} - \beta_{3} - \beta_1) q^{7} + ( - \beta_{15} - \beta_{13} + \cdots + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + ( - \beta_{9} + \beta_{4}) q^{4} + ( - \beta_{8} - \beta_{6} + \beta_{5}) q^{5} + (\beta_{7} - \beta_{3} - \beta_1) q^{7} + ( - \beta_{15} - \beta_{13} + \cdots + \beta_1) q^{8}+ \cdots + ( - \beta_{15} - \beta_{13} - 2 \beta_{9} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} - q^{4} + 6 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} - q^{4} + 6 q^{7} + 2 q^{8} - 16 q^{10} - 16 q^{14} - 9 q^{16} + 28 q^{17} + 8 q^{20} + q^{22} + 10 q^{23} + 2 q^{25} - 28 q^{26} + 4 q^{28} - 10 q^{31} - 11 q^{32} + q^{34} - 23 q^{38} + 6 q^{40} + 8 q^{41} - 18 q^{44} - 20 q^{46} - 6 q^{47} + 18 q^{49} + 23 q^{50} - 8 q^{52} - 4 q^{55} - 10 q^{56} - 14 q^{58} + 52 q^{62} + 26 q^{64} + 14 q^{65} + 39 q^{68} - 72 q^{71} - 44 q^{73} + 38 q^{74} + 5 q^{76} - 30 q^{79} + 96 q^{80} + 38 q^{82} - 7 q^{86} + 31 q^{88} - 64 q^{89} + 30 q^{92} - 12 q^{94} - 44 q^{95} - 66 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + \nu^{14} + 3 \nu^{13} + 4 \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} + 4 \nu^{8} + \cdots - 256 ) / 192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} + \nu^{14} - \nu^{13} + 2 \nu^{12} + 2 \nu^{11} - 8 \nu^{9} - 8 \nu^{8} - 12 \nu^{7} + \cdots + 64 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} - \nu^{14} + \nu^{13} + 2 \nu^{11} - 4 \nu^{10} - 8 \nu^{8} + 4 \nu^{7} - 16 \nu^{6} + \cdots - 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{14} - \nu^{13} - 2 \nu^{12} + \nu^{11} - 5 \nu^{10} - 4 \nu^{9} - 10 \nu^{8} + 8 \nu^{7} + \cdots - 32 ) / 96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} + 3 \nu^{14} + 5 \nu^{13} + 12 \nu^{12} + 18 \nu^{11} + 28 \nu^{10} + 24 \nu^{9} + 24 \nu^{8} + \cdots - 512 ) / 384 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{15} - \nu^{14} - 2 \nu^{13} - 3 \nu^{12} - 5 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 8 \nu^{8} + \cdots + 128 ) / 96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2 \nu^{15} + \nu^{14} - 5 \nu^{13} - 5 \nu^{12} - 16 \nu^{11} - 10 \nu^{10} - 28 \nu^{9} + \cdots + 704 ) / 192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2 \nu^{15} - \nu^{14} - 3 \nu^{13} - 3 \nu^{12} - 8 \nu^{11} - 6 \nu^{10} + 8 \nu^{8} + 20 \nu^{6} + \cdots + 256 ) / 96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4 \nu^{15} + \nu^{14} + 3 \nu^{13} + \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} - 8 \nu^{8} + \cdots - 448 ) / 192 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 9 \nu^{15} - 7 \nu^{14} - 9 \nu^{13} - 16 \nu^{12} - 50 \nu^{11} - 12 \nu^{10} - 32 \nu^{9} + \cdots + 1664 ) / 384 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11 \nu^{15} - 3 \nu^{14} - 13 \nu^{13} - 14 \nu^{12} - 54 \nu^{11} - 32 \nu^{10} - 40 \nu^{9} + \cdots + 2048 ) / 384 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13 \nu^{15} - 7 \nu^{14} - 17 \nu^{13} - 28 \nu^{12} - 74 \nu^{11} - 28 \nu^{10} - 56 \nu^{9} + \cdots + 2560 ) / 384 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 15 \nu^{15} + \nu^{14} + 15 \nu^{13} + 32 \nu^{12} + 62 \nu^{11} + 36 \nu^{10} + 64 \nu^{9} + \cdots - 2944 ) / 384 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{12} + \beta_{7} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{11} + \beta_{7} + \beta_{6} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + 2\beta_{9} + \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} + \beta_{13} - \beta_{11} - 2\beta_{10} - 2\beta_{9} - 5\beta_{7} - 5\beta_{5} + 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{14} + \beta_{13} - \beta_{11} - 2\beta_{10} + \beta_{7} - \beta_{6} - 2\beta_{4} - 9\beta_{3} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{15} - 2\beta_{12} + 2\beta_{9} + 8\beta_{8} - 3\beta_{6} + \beta_{5} - 2\beta_{4} + 5\beta_{3} + 2\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - \beta_{15} + 2 \beta_{14} - \beta_{13} - 2 \beta_{12} + 5 \beta_{11} + 10 \beta_{10} - 2 \beta_{9} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10 \beta_{14} - 3 \beta_{13} + 7 \beta_{11} - 2 \beta_{10} + 21 \beta_{7} + 7 \beta_{6} + 2 \beta_{4} + \cdots + 8 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3 \beta_{15} - 2 \beta_{12} - 6 \beta_{9} + 8 \beta_{8} + 17 \beta_{6} - 11 \beta_{5} + 6 \beta_{4} + \cdots + 25 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 19 \beta_{15} - 2 \beta_{14} + 19 \beta_{13} + 2 \beta_{12} - 15 \beta_{11} - 6 \beta_{10} + 6 \beta_{9} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 6 \beta_{14} + 13 \beta_{13} - 17 \beta_{11} - 42 \beta_{10} - 35 \beta_{7} - 17 \beta_{6} + \cdots - 8 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11 \beta_{15} - 42 \beta_{12} + 26 \beta_{9} - 24 \beta_{8} - 23 \beta_{6} - 11 \beta_{5} - 26 \beta_{4} + \cdots - 47 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 19 \beta_{15} + 22 \beta_{14} + 19 \beta_{13} - 22 \beta_{12} + 65 \beta_{11} + 10 \beta_{10} + \cdots + 87 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(-1 - \beta_{3}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−0.722180 + 1.21592i
−1.34532 + 0.436011i
0.587625 + 1.28635i
0.820200 + 1.15207i
−1.12494 − 0.857038i
1.41411 − 0.0174668i
−0.179748 − 1.40274i
1.05026 − 0.947078i
−0.722180 − 1.21592i
−1.34532 − 0.436011i
0.587625 − 1.28635i
0.820200 − 1.15207i
−1.12494 + 0.857038i
1.41411 + 0.0174668i
−0.179748 + 1.40274i
1.05026 + 0.947078i
−1.41411 + 0.0174668i 0 1.99939 − 0.0493999i 3.17262 + 1.83171i 0 −0.191926 − 0.332426i −2.82649 + 0.104780i 0 −4.51841 − 2.53482i
37.2 −1.05026 + 0.947078i 0 0.206086 − 1.98935i −0.602794 − 0.348023i 0 0.795065 + 1.37709i 1.66763 + 2.28452i 0 0.962695 − 0.205379i
37.3 −0.820200 − 1.15207i 0 −0.654545 + 1.88986i −1.97542 − 1.14051i 0 −0.907824 − 1.57240i 2.71411 − 0.795980i 0 0.306290 + 3.21128i
37.4 −0.587625 − 1.28635i 0 −1.30939 + 1.51178i 1.97542 + 1.14051i 0 −0.907824 − 1.57240i 2.71411 + 0.795980i 0 0.306290 − 3.21128i
37.5 0.179748 + 1.40274i 0 −1.93538 + 0.504281i 1.19115 + 0.687709i 0 1.80469 + 3.12581i −1.05526 − 2.62420i 0 −0.750573 + 1.79449i
37.6 0.722180 − 1.21592i 0 −0.956913 − 1.75622i −3.17262 − 1.83171i 0 −0.191926 − 0.332426i −2.82649 − 0.104780i 0 −4.51841 + 2.53482i
37.7 1.12494 + 0.857038i 0 0.530970 + 1.92823i −1.19115 − 0.687709i 0 1.80469 + 3.12581i −1.05526 + 2.62420i 0 −0.750573 − 1.79449i
37.8 1.34532 − 0.436011i 0 1.61979 − 1.17315i 0.602794 + 0.348023i 0 0.795065 + 1.37709i 1.66763 − 2.28452i 0 0.962695 + 0.205379i
181.1 −1.41411 − 0.0174668i 0 1.99939 + 0.0493999i 3.17262 − 1.83171i 0 −0.191926 + 0.332426i −2.82649 − 0.104780i 0 −4.51841 + 2.53482i
181.2 −1.05026 − 0.947078i 0 0.206086 + 1.98935i −0.602794 + 0.348023i 0 0.795065 − 1.37709i 1.66763 − 2.28452i 0 0.962695 + 0.205379i
181.3 −0.820200 + 1.15207i 0 −0.654545 − 1.88986i −1.97542 + 1.14051i 0 −0.907824 + 1.57240i 2.71411 + 0.795980i 0 0.306290 − 3.21128i
181.4 −0.587625 + 1.28635i 0 −1.30939 − 1.51178i 1.97542 − 1.14051i 0 −0.907824 + 1.57240i 2.71411 − 0.795980i 0 0.306290 + 3.21128i
181.5 0.179748 − 1.40274i 0 −1.93538 − 0.504281i 1.19115 − 0.687709i 0 1.80469 − 3.12581i −1.05526 + 2.62420i 0 −0.750573 − 1.79449i
181.6 0.722180 + 1.21592i 0 −0.956913 + 1.75622i −3.17262 + 1.83171i 0 −0.191926 + 0.332426i −2.82649 + 0.104780i 0 −4.51841 − 2.53482i
181.7 1.12494 − 0.857038i 0 0.530970 − 1.92823i −1.19115 + 0.687709i 0 1.80469 − 3.12581i −1.05526 − 2.62420i 0 −0.750573 + 1.79449i
181.8 1.34532 + 0.436011i 0 1.61979 + 1.17315i 0.602794 − 0.348023i 0 0.795065 − 1.37709i 1.66763 + 2.28452i 0 0.962695 − 0.205379i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.n.b 16
3.b odd 2 1 72.2.n.b ✓ 16
4.b odd 2 1 864.2.r.b 16
8.b even 2 1 inner 216.2.n.b 16
8.d odd 2 1 864.2.r.b 16
9.c even 3 1 inner 216.2.n.b 16
9.c even 3 1 648.2.d.k 8
9.d odd 6 1 72.2.n.b ✓ 16
9.d odd 6 1 648.2.d.j 8
12.b even 2 1 288.2.r.b 16
24.f even 2 1 288.2.r.b 16
24.h odd 2 1 72.2.n.b ✓ 16
36.f odd 6 1 864.2.r.b 16
36.f odd 6 1 2592.2.d.k 8
36.h even 6 1 288.2.r.b 16
36.h even 6 1 2592.2.d.j 8
72.j odd 6 1 72.2.n.b ✓ 16
72.j odd 6 1 648.2.d.j 8
72.l even 6 1 288.2.r.b 16
72.l even 6 1 2592.2.d.j 8
72.n even 6 1 inner 216.2.n.b 16
72.n even 6 1 648.2.d.k 8
72.p odd 6 1 864.2.r.b 16
72.p odd 6 1 2592.2.d.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.b ✓ 16 3.b odd 2 1
72.2.n.b ✓ 16 9.d odd 6 1
72.2.n.b ✓ 16 24.h odd 2 1
72.2.n.b ✓ 16 72.j odd 6 1
216.2.n.b 16 1.a even 1 1 trivial
216.2.n.b 16 8.b even 2 1 inner
216.2.n.b 16 9.c even 3 1 inner
216.2.n.b 16 72.n even 6 1 inner
288.2.r.b 16 12.b even 2 1
288.2.r.b 16 24.f even 2 1
288.2.r.b 16 36.h even 6 1
288.2.r.b 16 72.l even 6 1
648.2.d.j 8 9.d odd 6 1
648.2.d.j 8 72.j odd 6 1
648.2.d.k 8 9.c even 3 1
648.2.d.k 8 72.n even 6 1
864.2.r.b 16 4.b odd 2 1
864.2.r.b 16 8.d odd 2 1
864.2.r.b 16 36.f odd 6 1
864.2.r.b 16 72.p odd 6 1
2592.2.d.j 8 36.h even 6 1
2592.2.d.j 8 72.l even 6 1
2592.2.d.k 8 36.f odd 6 1
2592.2.d.k 8 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 21T_{5}^{14} + 326T_{5}^{12} - 2049T_{5}^{10} + 9318T_{5}^{8} - 18357T_{5}^{6} + 26129T_{5}^{4} - 11712T_{5}^{2} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{15} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 21 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( (T^{8} - 3 T^{7} + 14 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 40 T^{14} + \cdots + 10556001 \) Copy content Toggle raw display
$13$ \( T^{16} - 53 T^{14} + \cdots + 20736 \) Copy content Toggle raw display
$17$ \( (T^{4} - 7 T^{3} - 2 T^{2} + \cdots + 36)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 83 T^{6} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 5 T^{7} + \cdots + 90000)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 4032758016 \) Copy content Toggle raw display
$31$ \( (T^{8} + 5 T^{7} + \cdots + 92416)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 152 T^{6} + \cdots + 1327104)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 4 T^{7} + \cdots + 335241)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 20 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$47$ \( (T^{8} + 3 T^{7} + \cdots + 2178576)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 160 T^{6} + \cdots + 451584)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 131079601 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 176319369216 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 9881774573841 \) Copy content Toggle raw display
$71$ \( (T^{4} + 18 T^{3} + \cdots - 864)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 11 T^{3} + \cdots + 36)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 15 T^{7} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 31713911056 \) Copy content Toggle raw display
$89$ \( (T^{4} + 16 T^{3} + \cdots - 504)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 134 T^{6} + \cdots + 1324801)^{2} \) Copy content Toggle raw display
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