Properties

Label 666.2.a.d
Level $666$
Weight $2$
Character orbit 666.a
Self dual yes
Analytic conductor $5.318$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [666,2,Mod(1,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.a (trivial)

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: yes
Analytic conductor: \(5.31803677462\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 222)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{4} - 4 q^{5} - q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - 4 q^{5} - q^{7} + q^{8} - 4 q^{10} + q^{11} - 3 q^{13} - q^{14} + q^{16} - 3 q^{17} - 5 q^{19} - 4 q^{20} + q^{22} - 5 q^{23} + 11 q^{25} - 3 q^{26} - q^{28} - 4 q^{29} - 10 q^{31} + q^{32} - 3 q^{34} + 4 q^{35} - q^{37} - 5 q^{38} - 4 q^{40} + 6 q^{41} + 4 q^{43} + q^{44} - 5 q^{46} - 2 q^{47} - 6 q^{49} + 11 q^{50} - 3 q^{52} + 11 q^{53} - 4 q^{55} - q^{56} - 4 q^{58} + 12 q^{59} + 10 q^{61} - 10 q^{62} + q^{64} + 12 q^{65} + 14 q^{67} - 3 q^{68} + 4 q^{70} - 11 q^{73} - q^{74} - 5 q^{76} - q^{77} - 10 q^{79} - 4 q^{80} + 6 q^{82} + 9 q^{83} + 12 q^{85} + 4 q^{86} + q^{88} - 11 q^{89} + 3 q^{91} - 5 q^{92} - 2 q^{94} + 20 q^{95} + 10 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 0 1.00000 −4.00000 0 −1.00000 1.00000 0 −4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(37\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.a.d 1
3.b odd 2 1 222.2.a.c 1
4.b odd 2 1 5328.2.a.b 1
12.b even 2 1 1776.2.a.e 1
15.d odd 2 1 5550.2.a.z 1
24.f even 2 1 7104.2.a.p 1
24.h odd 2 1 7104.2.a.a 1
111.d odd 2 1 8214.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.a.c 1 3.b odd 2 1
666.2.a.d 1 1.a even 1 1 trivial
1776.2.a.e 1 12.b even 2 1
5328.2.a.b 1 4.b odd 2 1
5550.2.a.z 1 15.d odd 2 1
7104.2.a.a 1 24.h odd 2 1
7104.2.a.p 1 24.f even 2 1
8214.2.a.j 1 111.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(666))\):

\( T_{5} + 4 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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