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Properties

Label 666.2.s.b
Level $666$
Weight $2$
Character orbit 666.s
Analytic conductor $5.318$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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 = [4,0,0,2,0,0,-2] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} - \zeta_{12}^{2} q^{7} + \zeta_{12}^{3} q^{8} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{11} + ( - \zeta_{12}^{2} + 2) q^{13} - \zeta_{12}^{3} q^{14} + (\zeta_{12}^{2} - 1) q^{16} + \cdots + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{7} + 6 q^{13} - 2 q^{16} + 12 q^{19} + 12 q^{22} - 10 q^{25} + 2 q^{28} + 12 q^{34} - 20 q^{37} - 12 q^{46} + 12 q^{49} + 6 q^{52} - 48 q^{61} - 4 q^{64} - 14 q^{67} - 20 q^{73} + 12 q^{76}+ \cdots - 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(\zeta_{12}^{2}\)

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
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i 0 0 −0.500000 + 0.866025i 1.00000i 0 0
307.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0 0 −0.500000 + 0.866025i − 1.00000i 0 0
397.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0 0 −0.500000 − 0.866025i − 1.00000i 0 0
397.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 −0.500000 − 0.866025i 1.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.e even 6 1 inner
111.h odd 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.b ✓ 4
3.b odd 2 1 inner 666.2.s.b ✓ 4
37.e even 6 1 inner 666.2.s.b ✓ 4
111.h odd 6 1 inner 666.2.s.b ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.s.b ✓ 4 1.a even 1 1 trivial
666.2.s.b ✓ 4 3.b odd 2 1 inner
666.2.s.b ✓ 4 37.e even 6 1 inner
666.2.s.b ✓ 4 111.h odd 6 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T + 37)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$43$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$61$ \( (T^{2} + 24 T + 192)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$73$ \( (T + 5)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$89$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$97$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
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