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Properties

Label 666.2.s.f
Level $666$
Weight $2$
Character orbit 666.s
Analytic conductor $5.318$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [666,2,Mod(307,666)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("666.307"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,4,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 222)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{4} + 1) q^{4} + (\beta_{7} + \beta_{5} - 2 \beta_1) q^{5} + (\beta_{7} - \beta_{5} - \beta_{4} - 1) q^{7} + (\beta_{5} + \beta_{3}) q^{8} + ( - \beta_{6} - \beta_{4} + 2 \beta_{2}) q^{10}+ \cdots + (\beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 4 q^{7} + 4 q^{10} + 4 q^{11} - 6 q^{13} - 4 q^{16} - 6 q^{17} + 6 q^{19} - 6 q^{22} + 14 q^{25} - 16 q^{26} + 4 q^{28} - 8 q^{34} + 30 q^{35} + 12 q^{37} + 4 q^{38} + 2 q^{40} + 2 q^{41}+ \cdots - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 13 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{4} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 15\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 48\beta_{5} - 13\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(1 + \beta_{4}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
0.396143 + 1.68614i
−1.26217 − 1.18614i
1.26217 + 1.18614i
−0.396143 − 1.68614i
0.396143 − 1.68614i
−1.26217 + 1.18614i
1.26217 − 1.18614i
−0.396143 + 1.68614i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i −2.92048 − 1.68614i 0 −0.896143 + 1.55217i 1.00000i 0 3.37228
307.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 2.05446 + 1.18614i 0 0.762169 − 1.32012i 1.00000i 0 −2.37228
307.3 0.866025 − 0.500000i 0 0.500000 − 0.866025i −2.05446 − 1.18614i 0 −1.76217 + 3.05217i − 1.00000i 0 −2.37228
307.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i 2.92048 + 1.68614i 0 −0.103857 + 0.179885i − 1.00000i 0 3.37228
397.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −2.92048 + 1.68614i 0 −0.896143 − 1.55217i − 1.00000i 0 3.37228
397.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.05446 − 1.18614i 0 0.762169 + 1.32012i − 1.00000i 0 −2.37228
397.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.05446 + 1.18614i 0 −1.76217 − 3.05217i 1.00000i 0 −2.37228
397.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.92048 − 1.68614i 0 −0.103857 − 0.179885i 1.00000i 0 3.37228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.s.f 8
3.b odd 2 1 222.2.j.b ✓ 8
12.b even 2 1 1776.2.bz.i 8
37.e even 6 1 inner 666.2.s.f 8
111.h odd 6 1 222.2.j.b ✓ 8
111.m even 12 1 8214.2.a.z 4
111.m even 12 1 8214.2.a.ba 4
444.p even 6 1 1776.2.bz.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.j.b ✓ 8 3.b odd 2 1
222.2.j.b ✓ 8 111.h odd 6 1
666.2.s.f 8 1.a even 1 1 trivial
666.2.s.f 8 37.e even 6 1 inner
1776.2.bz.i 8 12.b even 2 1
1776.2.bz.i 8 444.p even 6 1
8214.2.a.z 4 111.m even 12 1
8214.2.a.ba 4 111.m even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 17T_{5}^{6} + 225T_{5}^{4} - 1088T_{5}^{2} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 17 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 22 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( (T^{4} + 96 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 130 T^{6} + \cdots + 110224 \) Copy content Toggle raw display
$31$ \( T^{8} + 70 T^{6} + \cdots + 5476 \) Copy content Toggle raw display
$37$ \( T^{8} - 12 T^{7} + \cdots + 1874161 \) Copy content Toggle raw display
$41$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$43$ \( T^{8} + 302 T^{6} + \cdots + 4169764 \) Copy content Toggle raw display
$47$ \( (T^{4} - 14 T^{3} + \cdots - 3464)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 4 T^{7} + \cdots + 160000 \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + \cdots + 16384 \) Copy content Toggle raw display
$61$ \( T^{8} + 36 T^{7} + \cdots + 28217344 \) Copy content Toggle raw display
$67$ \( T^{8} + 8 T^{7} + \cdots + 749956 \) Copy content Toggle raw display
$71$ \( T^{8} + 6 T^{7} + \cdots + 20286016 \) Copy content Toggle raw display
$73$ \( (T^{4} - 10 T^{3} + \cdots - 1832)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 24 T^{7} + \cdots + 5635876 \) Copy content Toggle raw display
$83$ \( (T^{4} - 12 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 42 T^{7} + \cdots + 515290000 \) Copy content Toggle raw display
$97$ \( T^{8} + 276 T^{6} + \cdots + 927369 \) Copy content Toggle raw display
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