Properties

Label 3864.2.a.j
Level $3864$
Weight $2$
Character orbit 3864.a
Self dual yes
Analytic conductor $30.854$
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] = [3864,2,Mod(1,3864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3864, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 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("3864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3864.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: \(30.8541953410\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - \beta q^{5} + q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta q^{5} + q^{7} + q^{9} + (2 \beta - 3) q^{11} + ( - \beta - 1) q^{13} - \beta q^{15} + (2 \beta - 3) q^{17} + ( - 2 \beta - 1) q^{19} + q^{21} + q^{23} + (\beta - 4) q^{25} + q^{27} + (2 \beta + 1) q^{29} + (2 \beta - 7) q^{31} + (2 \beta - 3) q^{33} - \beta q^{35} + (6 \beta - 5) q^{37} + ( - \beta - 1) q^{39} + ( - 6 \beta + 1) q^{41} + (5 \beta - 1) q^{43} - \beta q^{45} + ( - 2 \beta - 2) q^{47} + q^{49} + (2 \beta - 3) q^{51} + (\beta - 4) q^{53} + (\beta - 2) q^{55} + ( - 2 \beta - 1) q^{57} + ( - 9 \beta + 2) q^{59} + ( - 3 \beta - 5) q^{61} + q^{63} + (2 \beta + 1) q^{65} + ( - 3 \beta - 2) q^{67} + q^{69} + ( - \beta + 5) q^{71} + (4 \beta - 5) q^{73} + (\beta - 4) q^{75} + (2 \beta - 3) q^{77} - 7 q^{79} + q^{81} + ( - 8 \beta + 1) q^{83} + (\beta - 2) q^{85} + (2 \beta + 1) q^{87} + (13 \beta - 7) q^{89} + ( - \beta - 1) q^{91} + (2 \beta - 7) q^{93} + (3 \beta + 2) q^{95} + ( - 8 \beta + 1) q^{97} + (2 \beta - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} - 3 q^{13} - q^{15} - 4 q^{17} - 4 q^{19} + 2 q^{21} + 2 q^{23} - 7 q^{25} + 2 q^{27} + 4 q^{29} - 12 q^{31} - 4 q^{33} - q^{35} - 4 q^{37} - 3 q^{39} - 4 q^{41} + 3 q^{43} - q^{45} - 6 q^{47} + 2 q^{49} - 4 q^{51} - 7 q^{53} - 3 q^{55} - 4 q^{57} - 5 q^{59} - 13 q^{61} + 2 q^{63} + 4 q^{65} - 7 q^{67} + 2 q^{69} + 9 q^{71} - 6 q^{73} - 7 q^{75} - 4 q^{77} - 14 q^{79} + 2 q^{81} - 6 q^{83} - 3 q^{85} + 4 q^{87} - q^{89} - 3 q^{91} - 12 q^{93} + 7 q^{95} - 6 q^{97} - 4 q^{99}+O(q^{100}) \) 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
0 1.00000 0 −1.61803 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 0.618034 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 3864.2.a.j 2
4.b odd 2 1 7728.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.j 2 1.a even 1 1 trivial
7728.2.a.y 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3864))\):

\( T_{5}^{2} + T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 31 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$43$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$59$ \( T^{2} + 5T - 95 \) Copy content Toggle raw display
$61$ \( T^{2} + 13T + 31 \) Copy content Toggle raw display
$67$ \( T^{2} + 7T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} - 9T + 19 \) Copy content Toggle raw display
$73$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$79$ \( (T + 7)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 71 \) Copy content Toggle raw display
$89$ \( T^{2} + T - 211 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 71 \) Copy content Toggle raw display
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