↑

Properties

Label 666.2.be.d
Level $666$
Weight $2$
Character orbit 666.be
Analytic conductor $5.318$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [666,2,Mod(125,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.125");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.be (of order \(12\), degree \(4\), 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: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.6040479020157644046336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} - \beta_{9} q^{4} + (\beta_{7} + \beta_1) q^{5} + (\beta_{13} + \beta_{6} - \beta_{5}) q^{7} - \beta_{14} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} - \beta_{9} q^{4} + (\beta_{7} + \beta_1) q^{5} + (\beta_{13} + \beta_{6} - \beta_{5}) q^{7} - \beta_{14} q^{8} + ( - \beta_{5} + \beta_{4}) q^{10} + (\beta_{15} - 2 \beta_{14} + \cdots - \beta_{3}) q^{11}+ \cdots + ( - 2 \beta_{14} - 2 \beta_{12} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{7} + 4 q^{13} + 8 q^{16} + 16 q^{19} + 20 q^{22} + 12 q^{28} - 12 q^{31} + 8 q^{34} + 12 q^{37} + 12 q^{40} - 20 q^{43} + 12 q^{46} + 20 q^{49} - 4 q^{52} - 4 q^{55} + 4 q^{61} - 132 q^{67} + 28 q^{70} - 16 q^{76} - 4 q^{79} + 12 q^{82} + 16 q^{88} + 20 q^{91} + 12 q^{94} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{12} + 32\nu^{8} + 2368\nu^{4} - 6561 ) / 12960 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{13} + 32\nu^{9} + 2368\nu^{5} - 6561\nu ) / 38880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{14} + 32\nu^{10} + 2368\nu^{6} - 6561\nu^{2} ) / 38880 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31\nu^{14} + 512\nu^{10} - 992\nu^{6} - 17577\nu^{2} ) / 116640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{12} - 32\nu^{8} + 224\nu^{4} + 3969 ) / 2592 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -31\nu^{13} - 512\nu^{9} + 992\nu^{5} + 17577\nu ) / 38880 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\nu^{12} - 64\nu^{8} - 416\nu^{4} - 7371 ) / 4320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{14} + 1079\nu^{2} ) / 1440 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{13} - 599\nu ) / 480 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 91\nu^{15} + 416\nu^{11} - 8096\nu^{7} - 85293\nu^{3} ) / 349920 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -31\nu^{15} - 512\nu^{11} + 992\nu^{7} + 17577\nu^{3} ) / 116640 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{14} - 359\nu^{2} ) / 720 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{15} - 359\nu^{3} ) / 2160 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -49\nu^{15} - 224\nu^{11} - 1024\nu^{7} + 45927\nu^{3} ) / 69984 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + 2\beta_{9} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} + 3\beta_{14} - 2\beta_{12} - 2\beta_{11} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 5\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} + \beta_{7} + 16\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{9} - 3\beta_{5} + 16\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -13\beta_{15} - 35\beta_{11} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -35\beta_{8} - 39\beta_{6} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -31\beta_{10} - 74\beta_{7} + 31\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -31\beta_{13} + 31\beta_{9} + 222\beta_{5} + 31\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -93\beta_{14} - 253\beta_{12} - 93\beta_{11} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 160\beta_{8} - 160\beta_{6} + 160\beta_{2} + 599 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 480\beta_{10} + 599\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1079\beta_{13} + 718\beta_{9} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 718\beta_{15} + 3237\beta_{14} - 718\beta_{12} - 718\beta_{11} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(-1\) \(\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
−1.73122 − 0.0537601i
1.47240 − 0.912166i
−1.47240 + 0.912166i
1.73122 + 0.0537601i
−0.912166 + 1.47240i
−0.0537601 − 1.73122i
0.0537601 + 1.73122i
0.912166 − 1.47240i
−1.73122 + 0.0537601i
1.47240 + 0.912166i
−1.47240 − 0.912166i
1.73122 − 0.0537601i
−0.912166 − 1.47240i
−0.0537601 + 1.73122i
0.0537601 − 1.73122i
0.912166 + 1.47240i
−0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.205059 + 0.765290i 0 −0.103857 − 0.179885i 0.707107 + 0.707107i 0 0.792287
125.2 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 0.653347 − 2.43832i 0 −1.76217 − 3.05217i 0.707107 + 0.707107i 0 −2.52434
125.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.653347 + 2.43832i 0 −1.76217 − 3.05217i −0.707107 − 0.707107i 0 −2.52434
125.4 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0.205059 − 0.765290i 0 −0.103857 − 0.179885i −0.707107 − 0.707107i 0 0.792287
251.1 −0.965926 − 0.258819i 0 0.866025 + 0.500000i −2.43832 + 0.653347i 0 0.762169 − 1.32012i −0.707107 − 0.707107i 0 2.52434
251.2 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 0.765290 − 0.205059i 0 −0.896143 + 1.55217i −0.707107 − 0.707107i 0 −0.792287
251.3 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.765290 + 0.205059i 0 −0.896143 + 1.55217i 0.707107 + 0.707107i 0 −0.792287
251.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 2.43832 − 0.653347i 0 0.762169 − 1.32012i 0.707107 + 0.707107i 0 2.52434
341.1 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −0.205059 − 0.765290i 0 −0.103857 + 0.179885i 0.707107 − 0.707107i 0 0.792287
341.2 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i 0.653347 + 2.43832i 0 −1.76217 + 3.05217i 0.707107 − 0.707107i 0 −2.52434
341.3 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −0.653347 − 2.43832i 0 −1.76217 + 3.05217i −0.707107 + 0.707107i 0 −2.52434
341.4 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 0.205059 + 0.765290i 0 −0.103857 + 0.179885i −0.707107 + 0.707107i 0 0.792287
467.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i −2.43832 − 0.653347i 0 0.762169 + 1.32012i −0.707107 + 0.707107i 0 2.52434
467.2 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 0.765290 + 0.205059i 0 −0.896143 − 1.55217i −0.707107 + 0.707107i 0 −0.792287
467.3 0.965926 − 0.258819i 0 0.866025 − 0.500000i −0.765290 − 0.205059i 0 −0.896143 − 1.55217i 0.707107 − 0.707107i 0 −0.792287
467.4 0.965926 − 0.258819i 0 0.866025 − 0.500000i 2.43832 + 0.653347i 0 0.762169 + 1.32012i 0.707107 − 0.707107i 0 2.52434
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.g odd 12 1 inner
111.m even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.be.d ✓ 16
3.b odd 2 1 inner 666.2.be.d ✓ 16
37.g odd 12 1 inner 666.2.be.d ✓ 16
111.m even 12 1 inner 666.2.be.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.be.d ✓ 16 1.a even 1 1 trivial
666.2.be.d ✓ 16 3.b odd 2 1 inner
666.2.be.d ✓ 16 37.g odd 12 1 inner
666.2.be.d ✓ 16 111.m even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\):

\( T_{5}^{16} - 41T_{5}^{12} + 1665T_{5}^{8} - 656T_{5}^{4} + 256 \) Copy content Toggle raw display
\( T_{7}^{8} + 4T_{7}^{7} + 17T_{7}^{6} + 16T_{7}^{5} + 43T_{7}^{4} + 26T_{7}^{3} + 98T_{7}^{2} + 20T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 41 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{8} + 4 T^{7} + 17 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 36 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 2 T^{7} - 13 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 12 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{8} - 8 T^{7} + \cdots + 21904)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 8280 T^{12} + \cdots + 26873856 \) Copy content Toggle raw display
$29$ \( T^{16} + 3050 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( (T^{8} + 6 T^{7} + \cdots + 743044)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 6 T^{7} + \cdots + 1874161)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 15352201216 \) Copy content Toggle raw display
$43$ \( (T^{8} + 10 T^{7} + \cdots + 484)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 108 T^{6} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 14 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23425600000000 \) Copy content Toggle raw display
$61$ \( (T^{8} - 2 T^{7} + \cdots + 484)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 66 T^{7} + \cdots + 45796)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 3110228525056 \) Copy content Toggle raw display
$73$ \( (T^{8} + 130 T^{6} + \cdots + 110224)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 2 T^{7} - 13 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 6044831973376 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 10875854196736 \) Copy content Toggle raw display
$97$ \( (T^{4} + 10 T^{3} + \cdots + 49)^{4} \) Copy content Toggle raw display
show more
show less