Properties

Label 666.2.a.h
Level $666$
Weight $2$
Character orbit 666.a
Self dual yes
Analytic conductor $5.318$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - 2 q^{5} + ( - \beta + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - 2 q^{5} + ( - \beta + 1) q^{7} - q^{8} + 2 q^{10} + (\beta - 1) q^{11} + (3 \beta - 1) q^{13} + (\beta - 1) q^{14} + q^{16} + ( - \beta - 1) q^{17} + ( - \beta - 1) q^{19} - 2 q^{20} + ( - \beta + 1) q^{22} + ( - 3 \beta - 1) q^{23} - q^{25} + ( - 3 \beta + 1) q^{26} + ( - \beta + 1) q^{28} + (2 \beta - 4) q^{29} + ( - 2 \beta - 4) q^{31} - q^{32} + (\beta + 1) q^{34} + (2 \beta - 2) q^{35} - q^{37} + (\beta + 1) q^{38} + 2 q^{40} + ( - 2 \beta - 6) q^{41} + (2 \beta - 4) q^{43} + (\beta - 1) q^{44} + (3 \beta + 1) q^{46} + 4 \beta q^{47} + ( - \beta - 2) q^{49} + q^{50} + (3 \beta - 1) q^{52} + (\beta - 1) q^{53} + ( - 2 \beta + 2) q^{55} + (\beta - 1) q^{56} + ( - 2 \beta + 4) q^{58} + (2 \beta - 6) q^{59} + ( - 4 \beta - 2) q^{61} + (2 \beta + 4) q^{62} + q^{64} + ( - 6 \beta + 2) q^{65} + (4 \beta - 8) q^{67} + ( - \beta - 1) q^{68} + ( - 2 \beta + 2) q^{70} + ( - 4 \beta + 8) q^{71} + (\beta + 1) q^{73} + q^{74} + ( - \beta - 1) q^{76} + (\beta - 5) q^{77} - 6 q^{79} - 2 q^{80} + (2 \beta + 6) q^{82} + ( - 7 \beta + 3) q^{83} + (2 \beta + 2) q^{85} + ( - 2 \beta + 4) q^{86} + ( - \beta + 1) q^{88} + (7 \beta - 5) q^{89} + (\beta - 13) q^{91} + ( - 3 \beta - 1) q^{92} - 4 \beta q^{94} + (2 \beta + 2) q^{95} + ( - 2 \beta + 8) q^{97} + (\beta + 2) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7} - 2 q^{8} + 4 q^{10} - q^{11} + q^{13} - q^{14} + 2 q^{16} - 3 q^{17} - 3 q^{19} - 4 q^{20} + q^{22} - 5 q^{23} - 2 q^{25} - q^{26} + q^{28} - 6 q^{29} - 10 q^{31} - 2 q^{32} + 3 q^{34} - 2 q^{35} - 2 q^{37} + 3 q^{38} + 4 q^{40} - 14 q^{41} - 6 q^{43} - q^{44} + 5 q^{46} + 4 q^{47} - 5 q^{49} + 2 q^{50} + q^{52} - q^{53} + 2 q^{55} - q^{56} + 6 q^{58} - 10 q^{59} - 8 q^{61} + 10 q^{62} + 2 q^{64} - 2 q^{65} - 12 q^{67} - 3 q^{68} + 2 q^{70} + 12 q^{71} + 3 q^{73} + 2 q^{74} - 3 q^{76} - 9 q^{77} - 12 q^{79} - 4 q^{80} + 14 q^{82} - q^{83} + 6 q^{85} + 6 q^{86} + q^{88} - 3 q^{89} - 25 q^{91} - 5 q^{92} - 4 q^{94} + 6 q^{95} + 14 q^{97} + 5 q^{98}+O(q^{100}) \) 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
2.56155
−1.56155
−1.00000 0 1.00000 −2.00000 0 −1.56155 −1.00000 0 2.00000
1.2 −1.00000 0 1.00000 −2.00000 0 2.56155 −1.00000 0 2.00000
\(n\): 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.h 2
3.b odd 2 1 666.2.a.k yes 2
4.b odd 2 1 5328.2.a.y 2
12.b even 2 1 5328.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.a.h 2 1.a even 1 1 trivial
666.2.a.k yes 2 3.b odd 2 1
5328.2.a.y 2 4.b odd 2 1
5328.2.a.bh 2 12.b even 2 1

Hecke kernels

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

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 4 \) 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
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$23$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$31$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 14T + 32 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$53$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T - 32 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T - 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$79$ \( (T + 6)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + T - 208 \) Copy content Toggle raw display
$89$ \( T^{2} + 3T - 206 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
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