Properties

Label 768.2.a.j
Level $768$
Weight $2$
Character orbit 768.a
Self dual yes
Analytic conductor $6.133$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,2,Mod(1,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("768.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.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: \(6.13251087523\)
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: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 192)
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta q^{5} + \beta q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta q^{5} + \beta q^{7} + q^{9} - \beta q^{15} + 6 q^{17} - 4 q^{19} - \beta q^{21} - 2 \beta q^{23} + 7 q^{25} - q^{27} - \beta q^{29} + \beta q^{31} + 12 q^{35} - 2 \beta q^{37} + 6 q^{41} - 4 q^{43} + \beta q^{45} + 2 \beta q^{47} + 5 q^{49} - 6 q^{51} - \beta q^{53} + 4 q^{57} + 12 q^{59} + 2 \beta q^{61} + \beta q^{63} + 4 q^{67} + 2 \beta q^{69} + 2 \beta q^{71} - 2 q^{73} - 7 q^{75} - 3 \beta q^{79} + q^{81} + 6 \beta q^{85} + \beta q^{87} - 6 q^{89} - \beta q^{93} - 4 \beta q^{95} - 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} + 12 q^{17} - 8 q^{19} + 14 q^{25} - 2 q^{27} + 24 q^{35} + 12 q^{41} - 8 q^{43} + 10 q^{49} - 12 q^{51} + 8 q^{57} + 24 q^{59} + 8 q^{67} - 4 q^{73} - 14 q^{75} + 2 q^{81} - 12 q^{89} - 4 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
0 −1.00000 0 −3.46410 0 −3.46410 0 1.00000 0
1.2 0 −1.00000 0 3.46410 0 3.46410 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.a.j 2
3.b odd 2 1 2304.2.a.s 2
4.b odd 2 1 768.2.a.k 2
8.b even 2 1 768.2.a.k 2
8.d odd 2 1 inner 768.2.a.j 2
12.b even 2 1 2304.2.a.u 2
16.e even 4 2 192.2.d.a 4
16.f odd 4 2 192.2.d.a 4
24.f even 2 1 2304.2.a.s 2
24.h odd 2 1 2304.2.a.u 2
48.i odd 4 2 576.2.d.b 4
48.k even 4 2 576.2.d.b 4
80.i odd 4 2 4800.2.d.o 4
80.j even 4 2 4800.2.d.o 4
80.k odd 4 2 4800.2.k.j 4
80.q even 4 2 4800.2.k.j 4
80.s even 4 2 4800.2.d.j 4
80.t odd 4 2 4800.2.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.2.d.a 4 16.e even 4 2
192.2.d.a 4 16.f odd 4 2
576.2.d.b 4 48.i odd 4 2
576.2.d.b 4 48.k even 4 2
768.2.a.j 2 1.a even 1 1 trivial
768.2.a.j 2 8.d odd 2 1 inner
768.2.a.k 2 4.b odd 2 1
768.2.a.k 2 8.b even 2 1
2304.2.a.s 2 3.b odd 2 1
2304.2.a.s 2 24.f even 2 1
2304.2.a.u 2 12.b even 2 1
2304.2.a.u 2 24.h odd 2 1
4800.2.d.j 4 80.s even 4 2
4800.2.d.j 4 80.t odd 4 2
4800.2.d.o 4 80.i odd 4 2
4800.2.d.o 4 80.j even 4 2
4800.2.k.j 4 80.k odd 4 2
4800.2.k.j 4 80.q even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(768))\):

\( T_{5}^{2} - 12 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{19} + 4 \) 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
$5$ \( T^{2} - 12 \) Copy content Toggle raw display
$7$ \( T^{2} - 12 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 48 \) Copy content Toggle raw display
$29$ \( T^{2} - 12 \) Copy content Toggle raw display
$31$ \( T^{2} - 12 \) Copy content Toggle raw display
$37$ \( T^{2} - 48 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 48 \) Copy content Toggle raw display
$53$ \( T^{2} - 12 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 48 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 48 \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 108 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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