Properties

Label 3192.2.a.y
Level $3192$
Weight $2$
Character orbit 3192.a
Self dual yes
Analytic conductor $25.488$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7232.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 5x^{2} + 4x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_1 + 1) q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_1 + 1) q^{5} - q^{7} + q^{9} - \beta_{2} q^{11} + (\beta_{3} + \beta_{2} + 1) q^{13} + ( - \beta_1 + 1) q^{15} + ( - \beta_1 + 1) q^{17} + q^{19} - q^{21} + \beta_{2} q^{23} + ( - \beta_{3} - 2 \beta_1 + 2) q^{25} + q^{27} + (\beta_{2} + \beta_1 + 3) q^{29} + ( - \beta_{3} - 2 \beta_{2} + 1) q^{31} - \beta_{2} q^{33} + (\beta_1 - 1) q^{35} + ( - 2 \beta_{2} + 2) q^{37} + (\beta_{3} + \beta_{2} + 1) q^{39} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + (\beta_{3} - 1) q^{43} + ( - \beta_1 + 1) q^{45} + ( - \beta_{2} + 3 \beta_1 + 3) q^{47} + q^{49} + ( - \beta_1 + 1) q^{51} + ( - 2 \beta_{3} - 3 \beta_{2} + \beta_1 + 5) q^{53} + ( - 2 \beta_{3} - 2 \beta_{2} + 2) q^{55} + q^{57} + ( - 2 \beta_{3} - 2) q^{59} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{61} - q^{63} + (2 \beta_{3} + 4 \beta_{2} + 2) q^{65} + 3 \beta_{2} q^{67} + \beta_{2} q^{69} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{71} + ( - 2 \beta_{2} - 2 \beta_1) q^{73} + ( - \beta_{3} - 2 \beta_1 + 2) q^{75} + \beta_{2} q^{77} + (2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{79} + q^{81} + ( - 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{83} + ( - \beta_{3} - 2 \beta_1 + 7) q^{85} + (\beta_{2} + \beta_1 + 3) q^{87} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 6) q^{89} + ( - \beta_{3} - \beta_{2} - 1) q^{91} + ( - \beta_{3} - 2 \beta_{2} + 1) q^{93} + ( - \beta_1 + 1) q^{95} + ( - \beta_{2} + 2 \beta_1) q^{97} - \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 4 q^{13} + 4 q^{15} + 4 q^{17} + 4 q^{19} - 4 q^{21} + 8 q^{25} + 4 q^{27} + 12 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} + 4 q^{39} + 8 q^{41} - 4 q^{43} + 4 q^{45} + 12 q^{47} + 4 q^{49} + 4 q^{51} + 20 q^{53} + 8 q^{55} + 4 q^{57} - 8 q^{59} + 16 q^{61} - 4 q^{63} + 8 q^{65} + 4 q^{71} + 8 q^{75} + 4 q^{81} + 4 q^{83} + 28 q^{85} + 12 q^{87} + 24 q^{89} - 4 q^{91} + 4 q^{93} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 5x^{2} + 4x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 4\nu^{2} - \nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 5\nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} - 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{3} + 9\beta_{2} - 4\beta _1 + 13 ) / 2 \) 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
−0.652223
1.22219
3.06644
−1.63640
0 1.00000 0 −2.33660 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.314226 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 2.92238 0 −1.00000 0 1.00000 0
1.4 0 1.00000 0 3.72844 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.y 4
3.b odd 2 1 9576.2.a.cg 4
4.b odd 2 1 6384.2.a.bz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.y 4 1.a even 1 1 trivial
6384.2.a.bz 4 4.b odd 2 1
9576.2.a.cg 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3192))\):

\( T_{5}^{4} - 4T_{5}^{3} - 6T_{5}^{2} + 24T_{5} + 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 8 \) Copy content Toggle raw display
\( T_{17}^{4} - 4T_{17}^{3} - 6T_{17}^{2} + 24T_{17} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} - 6 T^{2} + 24 T + 8 \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} - 20 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} - 6 T^{2} + 24 T + 8 \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{3} + 18 T^{2} + \cdots - 136 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} - 52 T^{2} + 160 T - 64 \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} - 112 T^{2} + \cdots - 3088 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} - 20 T^{2} - 32 T + 64 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} - 46 T^{2} + \cdots + 1312 \) Copy content Toggle raw display
$53$ \( T^{4} - 20 T^{3} - 14 T^{2} + \cdots - 8648 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} - 80 T^{2} - 512 T + 512 \) Copy content Toggle raw display
$61$ \( T^{4} - 16 T^{3} - 168 T^{2} + \cdots - 15728 \) Copy content Toggle raw display
$67$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 4 T^{3} - 110 T^{2} + \cdots + 1088 \) Copy content Toggle raw display
$73$ \( T^{4} - 144 T^{2} + 160 T + 752 \) Copy content Toggle raw display
$79$ \( T^{4} - 136 T^{2} - 768 T - 1024 \) Copy content Toggle raw display
$83$ \( T^{4} - 4 T^{3} - 302 T^{2} + \cdots + 3616 \) Copy content Toggle raw display
$89$ \( T^{4} - 24 T^{3} + 48 T^{2} + \cdots - 7312 \) Copy content Toggle raw display
$97$ \( T^{4} - 48 T^{2} + 32 T + 368 \) Copy content Toggle raw display
show more
show less