Properties

Label 3168.2.a.bd
Level $3168$
Weight $2$
Character orbit 3168.a
Self dual yes
Analytic conductor $25.297$
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] = [3168,2,Mod(1,3168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3168.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3168.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: \(25.2966073603\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 352)
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.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{5} + q^{11} + 2 q^{13} + (2 \beta - 4) q^{17} + ( - 2 \beta - 2) q^{19} + (3 \beta - 3) q^{23} + 3 \beta q^{25} + (2 \beta - 4) q^{29} + (\beta + 7) q^{31} + (3 \beta - 1) q^{37} + ( - 4 \beta + 2) q^{41} + ( - 2 \beta - 2) q^{43} - 4 q^{47} - 7 q^{49} + (4 \beta - 2) q^{53} + ( - \beta - 1) q^{55} + ( - 3 \beta - 5) q^{59} - 6 \beta q^{61} + ( - 2 \beta - 2) q^{65} + (3 \beta - 3) q^{67} + ( - 3 \beta + 11) q^{71} - 6 q^{73} + ( - 2 \beta + 10) q^{79} + (6 \beta - 2) q^{83} - 4 q^{85} + (\beta + 1) q^{89} + (6 \beta + 10) q^{95} + ( - 3 \beta + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} + 2 q^{11} + 4 q^{13} - 6 q^{17} - 6 q^{19} - 3 q^{23} + 3 q^{25} - 6 q^{29} + 15 q^{31} + q^{37} - 6 q^{43} - 8 q^{47} - 14 q^{49} - 3 q^{55} - 13 q^{59} - 6 q^{61} - 6 q^{65} - 3 q^{67} + 19 q^{71} - 12 q^{73} + 18 q^{79} + 2 q^{83} - 8 q^{85} + 3 q^{89} + 26 q^{95} - q^{97}+O(q^{100}) \) 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
2.56155
−1.56155
0 0 0 −3.56155 0 0 0 0 0
1.2 0 0 0 0.561553 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(11\) \( -1 \)

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 3168.2.a.bd 2
3.b odd 2 1 352.2.a.g 2
4.b odd 2 1 3168.2.a.bc 2
8.b even 2 1 6336.2.a.cv 2
8.d odd 2 1 6336.2.a.cw 2
12.b even 2 1 352.2.a.h yes 2
15.d odd 2 1 8800.2.a.be 2
24.f even 2 1 704.2.a.n 2
24.h odd 2 1 704.2.a.o 2
33.d even 2 1 3872.2.a.p 2
48.i odd 4 2 2816.2.c.t 4
48.k even 4 2 2816.2.c.s 4
60.h even 2 1 8800.2.a.bd 2
132.d odd 2 1 3872.2.a.ba 2
264.m even 2 1 7744.2.a.cm 2
264.p odd 2 1 7744.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.a.g 2 3.b odd 2 1
352.2.a.h yes 2 12.b even 2 1
704.2.a.n 2 24.f even 2 1
704.2.a.o 2 24.h odd 2 1
2816.2.c.s 4 48.k even 4 2
2816.2.c.t 4 48.i odd 4 2
3168.2.a.bc 2 4.b odd 2 1
3168.2.a.bd 2 1.a even 1 1 trivial
3872.2.a.p 2 33.d even 2 1
3872.2.a.ba 2 132.d odd 2 1
6336.2.a.cv 2 8.b even 2 1
6336.2.a.cw 2 8.d odd 2 1
7744.2.a.bw 2 264.p odd 2 1
7744.2.a.cm 2 264.m even 2 1
8800.2.a.bd 2 60.h even 2 1
8800.2.a.be 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3168))\):

\( T_{5}^{2} + 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} - 8 \) Copy content Toggle raw display
\( T_{19}^{2} + 6T_{19} - 8 \) Copy content Toggle raw display
\( T_{47} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$31$ \( T^{2} - 15T + 52 \) Copy content Toggle raw display
$37$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$41$ \( T^{2} - 68 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$47$ \( (T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 68 \) Copy content Toggle raw display
$59$ \( T^{2} + 13T + 4 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 144 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$71$ \( T^{2} - 19T + 52 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 18T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 2T - 152 \) Copy content Toggle raw display
$89$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$97$ \( T^{2} + T - 38 \) Copy content Toggle raw display
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