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Properties

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

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [666,2,Mod(343,666)] 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("666.343"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,0,-1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 222)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{7} - q^{8} + 2 q^{11} + 5 \zeta_{6} q^{13} + 3 q^{14} + (\zeta_{6} - 1) q^{16} + ( - 2 \zeta_{6} + 2) q^{22} + 4 q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + \cdots + 2 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 3 q^{7} - 2 q^{8} + 4 q^{11} + 5 q^{13} + 6 q^{14} - q^{16} + 2 q^{22} + 8 q^{23} + 5 q^{25} + 10 q^{26} + 3 q^{28} - 4 q^{29} - 6 q^{31} + q^{32} + 10 q^{37} + 2 q^{41} + 18 q^{43}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0 0 1.50000 + 2.59808i −1.00000 0 0
433.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 1.50000 − 2.59808i −1.00000 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
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.f.e 2
3.b odd 2 1 222.2.e.a ✓ 2
12.b even 2 1 1776.2.q.c 2
37.c even 3 1 inner 666.2.f.e 2
111.h odd 6 1 8214.2.a.a 1
111.i odd 6 1 222.2.e.a ✓ 2
111.i odd 6 1 8214.2.a.h 1
444.t even 6 1 1776.2.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.e.a ✓ 2 3.b odd 2 1
222.2.e.a ✓ 2 111.i odd 6 1
666.2.f.e 2 1.a even 1 1 trivial
666.2.f.e 2 37.c even 3 1 inner
1776.2.q.c 2 12.b even 2 1
1776.2.q.c 2 444.t even 6 1
8214.2.a.a 1 111.h odd 6 1
8214.2.a.h 1 111.i odd 6 1

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$43$ \( (T - 9)^{2} \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( (T + 3)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 7)^{2} \) Copy content Toggle raw display
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