Properties

Label 666.2.a.i
Level $666$
Weight $2$
Character orbit 666.a
Self dual yes
Analytic conductor $5.318$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( - 3 \beta + 1) q^{5} - 2 \beta q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + ( - 3 \beta + 1) q^{5} - 2 \beta q^{7} - q^{8} + (3 \beta - 1) q^{10} + ( - \beta + 3) q^{11} + ( - 3 \beta + 2) q^{13} + 2 \beta q^{14} + q^{16} + (4 \beta - 2) q^{17} + (4 \beta - 2) q^{19} + ( - 3 \beta + 1) q^{20} + (\beta - 3) q^{22} + ( - 3 \beta + 2) q^{23} + (3 \beta + 5) q^{25} + (3 \beta - 2) q^{26} - 2 \beta q^{28} + (7 \beta - 2) q^{29} + ( - \beta + 9) q^{31} - q^{32} + ( - 4 \beta + 2) q^{34} + (4 \beta + 6) q^{35} - q^{37} + ( - 4 \beta + 2) q^{38} + (3 \beta - 1) q^{40} + ( - \beta - 8) q^{41} + ( - 2 \beta - 2) q^{43} + ( - \beta + 3) q^{44} + (3 \beta - 2) q^{46} + (2 \beta - 2) q^{47} + (4 \beta - 3) q^{49} + ( - 3 \beta - 5) q^{50} + ( - 3 \beta + 2) q^{52} + ( - 4 \beta + 6) q^{53} + ( - 7 \beta + 6) q^{55} + 2 \beta q^{56} + ( - 7 \beta + 2) q^{58} + ( - 2 \beta + 8) q^{59} + (\beta + 9) q^{61} + (\beta - 9) q^{62} + q^{64} + 11 q^{65} + (5 \beta - 7) q^{67} + (4 \beta - 2) q^{68} + ( - 4 \beta - 6) q^{70} + ( - 8 \beta + 10) q^{71} + (5 \beta - 1) q^{73} + q^{74} + (4 \beta - 2) q^{76} + ( - 4 \beta + 2) q^{77} + ( - 9 \beta + 6) q^{79} + ( - 3 \beta + 1) q^{80} + (\beta + 8) q^{82} + (4 \beta + 8) q^{83} + ( - 2 \beta - 14) q^{85} + (2 \beta + 2) q^{86} + (\beta - 3) q^{88} + ( - 4 \beta + 8) q^{89} + (2 \beta + 6) q^{91} + ( - 3 \beta + 2) q^{92} + ( - 2 \beta + 2) q^{94} + ( - 2 \beta - 14) q^{95} + ( - 4 \beta + 6) q^{97} + ( - 4 \beta + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8} + q^{10} + 5 q^{11} + q^{13} + 2 q^{14} + 2 q^{16} - q^{20} - 5 q^{22} + q^{23} + 13 q^{25} - q^{26} - 2 q^{28} + 3 q^{29} + 17 q^{31} - 2 q^{32} + 16 q^{35} - 2 q^{37} + q^{40} - 17 q^{41} - 6 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} - 2 q^{49} - 13 q^{50} + q^{52} + 8 q^{53} + 5 q^{55} + 2 q^{56} - 3 q^{58} + 14 q^{59} + 19 q^{61} - 17 q^{62} + 2 q^{64} + 22 q^{65} - 9 q^{67} - 16 q^{70} + 12 q^{71} + 3 q^{73} + 2 q^{74} + 3 q^{79} - q^{80} + 17 q^{82} + 20 q^{83} - 30 q^{85} + 6 q^{86} - 5 q^{88} + 12 q^{89} + 14 q^{91} + q^{92} + 2 q^{94} - 30 q^{95} + 8 q^{97} + 2 q^{98}+O(q^{100}) \) 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
1.61803
−0.618034
−1.00000 0 1.00000 −3.85410 0 −3.23607 −1.00000 0 3.85410
1.2 −1.00000 0 1.00000 2.85410 0 1.23607 −1.00000 0 −2.85410
\(n\): 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.i 2
3.b odd 2 1 74.2.a.b 2
4.b odd 2 1 5328.2.a.bc 2
12.b even 2 1 592.2.a.g 2
15.d odd 2 1 1850.2.a.t 2
15.e even 4 2 1850.2.b.j 4
21.c even 2 1 3626.2.a.s 2
24.f even 2 1 2368.2.a.u 2
24.h odd 2 1 2368.2.a.y 2
33.d even 2 1 8954.2.a.j 2
111.d odd 2 1 2738.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.b 2 3.b odd 2 1
592.2.a.g 2 12.b even 2 1
666.2.a.i 2 1.a even 1 1 trivial
1850.2.a.t 2 15.d odd 2 1
1850.2.b.j 4 15.e even 4 2
2368.2.a.u 2 24.f even 2 1
2368.2.a.y 2 24.h odd 2 1
2738.2.a.g 2 111.d odd 2 1
3626.2.a.s 2 21.c even 2 1
5328.2.a.bc 2 4.b odd 2 1
8954.2.a.j 2 33.d even 2 1

Hecke kernels

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

\( T_{5}^{2} + T_{5} - 11 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$13$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} - 20 \) Copy content Toggle raw display
$23$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 59 \) Copy content Toggle raw display
$31$ \( T^{2} - 17T + 71 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 17T + 71 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
$61$ \( T^{2} - 19T + 89 \) Copy content Toggle raw display
$67$ \( T^{2} + 9T - 11 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T - 44 \) Copy content Toggle raw display
$73$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T - 99 \) Copy content Toggle raw display
$83$ \( T^{2} - 20T + 80 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
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