Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [666,2,Mod(343,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
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 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

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

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + 4 \zeta_{6} q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + 4 \zeta_{6} q^{7} + q^{8} - 3 q^{10} + 6 q^{11} - 2 \zeta_{6} q^{13} - 4 q^{14} + (\zeta_{6} - 1) q^{16} + ( - 3 \zeta_{6} + 3) q^{17} - 2 \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 3) q^{20} + (6 \zeta_{6} - 6) q^{22} - 6 q^{23} + (4 \zeta_{6} - 4) q^{25} + 2 q^{26} + ( - 4 \zeta_{6} + 4) q^{28} - 3 q^{29} + 2 q^{31} - \zeta_{6} q^{32} + 3 \zeta_{6} q^{34} + (12 \zeta_{6} - 12) q^{35} + (3 \zeta_{6} + 4) q^{37} + 2 q^{38} + 3 \zeta_{6} q^{40} + 3 \zeta_{6} q^{41} - 4 q^{43} - 6 \zeta_{6} q^{44} + ( - 6 \zeta_{6} + 6) q^{46} + 6 q^{47} + (9 \zeta_{6} - 9) q^{49} - 4 \zeta_{6} q^{50} + (2 \zeta_{6} - 2) q^{52} + (6 \zeta_{6} - 6) q^{53} + 18 \zeta_{6} q^{55} + 4 \zeta_{6} q^{56} + ( - 3 \zeta_{6} + 3) q^{58} + \zeta_{6} q^{61} + (2 \zeta_{6} - 2) q^{62} + q^{64} + ( - 6 \zeta_{6} + 6) q^{65} - 2 \zeta_{6} q^{67} - 3 q^{68} - 12 \zeta_{6} q^{70} - 12 \zeta_{6} q^{71} - 10 q^{73} + (4 \zeta_{6} - 7) q^{74} + (2 \zeta_{6} - 2) q^{76} + 24 \zeta_{6} q^{77} - 14 \zeta_{6} q^{79} - 3 q^{80} - 3 q^{82} + ( - 6 \zeta_{6} + 6) q^{83} + 9 q^{85} + ( - 4 \zeta_{6} + 4) q^{86} + 6 q^{88} + ( - 3 \zeta_{6} + 3) q^{89} + ( - 8 \zeta_{6} + 8) q^{91} + 6 \zeta_{6} q^{92} + (6 \zeta_{6} - 6) q^{94} + ( - 6 \zeta_{6} + 6) q^{95} - 13 q^{97} - 9 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 3 q^{5} + 4 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + 3 q^{5} + 4 q^{7} + 2 q^{8} - 6 q^{10} + 12 q^{11} - 2 q^{13} - 8 q^{14} - q^{16} + 3 q^{17} - 2 q^{19} + 3 q^{20} - 6 q^{22} - 12 q^{23} - 4 q^{25} + 4 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} + 3 q^{34} - 12 q^{35} + 11 q^{37} + 4 q^{38} + 3 q^{40} + 3 q^{41} - 8 q^{43} - 6 q^{44} + 6 q^{46} + 12 q^{47} - 9 q^{49} - 4 q^{50} - 2 q^{52} - 6 q^{53} + 18 q^{55} + 4 q^{56} + 3 q^{58} + q^{61} - 2 q^{62} + 2 q^{64} + 6 q^{65} - 2 q^{67} - 6 q^{68} - 12 q^{70} - 12 q^{71} - 20 q^{73} - 10 q^{74} - 2 q^{76} + 24 q^{77} - 14 q^{79} - 6 q^{80} - 6 q^{82} + 6 q^{83} + 18 q^{85} + 4 q^{86} + 12 q^{88} + 3 q^{89} + 8 q^{91} + 6 q^{92} - 6 q^{94} + 6 q^{95} - 26 q^{97} - 9 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.50000 + 2.59808i 0 2.00000 + 3.46410i 1.00000 0 −3.00000
433.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.50000 2.59808i 0 2.00000 3.46410i 1.00000 0 −3.00000
\(n\): 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.d 2
3.b odd 2 1 74.2.c.b 2
12.b even 2 1 592.2.i.a 2
37.c even 3 1 inner 666.2.f.d 2
111.h odd 6 1 2738.2.a.c 1
111.i odd 6 1 74.2.c.b 2
111.i odd 6 1 2738.2.a.a 1
444.t even 6 1 592.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.c.b 2 3.b odd 2 1
74.2.c.b 2 111.i odd 6 1
592.2.i.a 2 12.b even 2 1
592.2.i.a 2 444.t even 6 1
666.2.f.d 2 1.a even 1 1 trivial
666.2.f.d 2 37.c even 3 1 inner
2738.2.a.a 1 111.i odd 6 1
2738.2.a.c 1 111.h odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 16 \) 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
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$89$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$97$ \( (T + 13)^{2} \) Copy content Toggle raw display
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