Properties

Label 3192.2.a.q
Level $3192$
Weight $2$
Character orbit 3192.a
Self dual yes
Analytic conductor $25.488$
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] = [3192,2,Mod(1,3192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3192, 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("3192.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3192.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: \(25.4882483252\)
Analytic rank: \(1\)
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: \( 1 \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + (\beta + 1) q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + (\beta + 1) q^{5} + q^{7} + q^{9} + ( - 2 \beta - 2) q^{13} + ( - \beta - 1) q^{15} + (\beta - 3) q^{17} - q^{19} - q^{21} + (2 \beta - 1) q^{25} - q^{27} + ( - 5 \beta - 1) q^{29} + 2 q^{31} + (\beta + 1) q^{35} - 10 q^{37} + (2 \beta + 2) q^{39} + ( - 4 \beta - 2) q^{41} + (4 \beta - 2) q^{43} + (\beta + 1) q^{45} + ( - 5 \beta - 3) q^{47} + q^{49} + ( - \beta + 3) q^{51} + ( - \beta - 5) q^{53} + q^{57} + 8 q^{59} - 2 q^{61} + q^{63} + ( - 4 \beta - 8) q^{65} + ( - 2 \beta - 10) q^{67} + (\beta + 3) q^{71} + 6 \beta q^{73} + ( - 2 \beta + 1) q^{75} + 4 q^{79} + q^{81} + (3 \beta + 5) q^{83} - 2 \beta q^{85} + (5 \beta + 1) q^{87} + (4 \beta - 2) q^{89} + ( - 2 \beta - 2) q^{91} - 2 q^{93} + ( - \beta - 1) q^{95} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} + 2 q^{35} - 20 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} + 2 q^{45} - 6 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 2 q^{57} + 16 q^{59} - 4 q^{61} + 2 q^{63} - 16 q^{65} - 20 q^{67} + 6 q^{71} + 2 q^{75} + 8 q^{79} + 2 q^{81} + 10 q^{83} + 2 q^{87} - 4 q^{89} - 4 q^{91} - 4 q^{93} - 2 q^{95} + 4 q^{97}+O(q^{100}) \) 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 −0.732051 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 2.73205 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(7\) \( -1 \)
\(19\) \( +1 \)

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 3192.2.a.q 2
3.b odd 2 1 9576.2.a.bj 2
4.b odd 2 1 6384.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.q 2 1.a even 1 1 trivial
6384.2.a.bs 2 4.b odd 2 1
9576.2.a.bj 2 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3192))\):

\( T_{5}^{2} - 2T_{5} - 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} + 6 \) 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
$5$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 74 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T - 66 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 22 \) Copy content Toggle raw display
$59$ \( (T - 8)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 20T + 88 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$73$ \( T^{2} - 108 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 10T - 2 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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