Properties

Label 3648.2.a.by
Level $3648$
Weight $2$
Character orbit 3648.a
Self dual yes
Analytic conductor $29.129$
Analytic rank $1$
Dimension $3$
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] = [3648,2,Mod(1,3648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3648, 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("3648.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.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: \(29.1294266574\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1824)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_{2} - 2) q^{5} + \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_{2} - 2) q^{5} + \beta_1 q^{7} + q^{9} - \beta_1 q^{11} + (\beta_{2} - \beta_1 - 2) q^{13} + ( - \beta_{2} - 2) q^{15} + (\beta_1 + 2) q^{17} - q^{19} + \beta_1 q^{21} + (3 \beta_{2} - \beta_1) q^{23} + (4 \beta_{2} - \beta_1 + 3) q^{25} + q^{27} + (2 \beta_1 - 2) q^{29} + ( - 3 \beta_{2} - \beta_1 - 4) q^{31} - \beta_1 q^{33} + (2 \beta_{2} - \beta_1) q^{35} + (\beta_{2} - \beta_1 - 2) q^{37} + (\beta_{2} - \beta_1 - 2) q^{39} + (2 \beta_{2} + 2) q^{41} + ( - 2 \beta_{2} + \beta_1 - 4) q^{43} + ( - \beta_{2} - 2) q^{45} + (3 \beta_{2} - 2 \beta_1) q^{47} + ( - 2 \beta_{2} + \beta_1 + 1) q^{49} + (\beta_1 + 2) q^{51} + (4 \beta_{2} - 2) q^{53} + ( - 2 \beta_{2} + \beta_1) q^{55} - q^{57} + ( - 4 \beta_{2} + 2 \beta_1) q^{59} + ( - 2 \beta_{2} - \beta_1 - 6) q^{61} + \beta_1 q^{63} + ( - 2 \beta_{2} + 2 \beta_1) q^{65} + (2 \beta_{2} - 4) q^{67} + (3 \beta_{2} - \beta_1) q^{69} + ( - 4 \beta_{2} + 2 \beta_1) q^{71} + ( - 4 \beta_{2} + \beta_1 - 2) q^{73} + (4 \beta_{2} - \beta_1 + 3) q^{75} + (2 \beta_{2} - \beta_1 - 8) q^{77} + ( - \beta_{2} - 3 \beta_1 - 4) q^{79} + q^{81} + ( - \beta_1 - 4) q^{85} + (2 \beta_1 - 2) q^{87} + ( - 4 \beta_1 - 2) q^{89} + ( - 4 \beta_1 - 8) q^{91} + ( - 3 \beta_{2} - \beta_1 - 4) q^{93} + (\beta_{2} + 2) q^{95} + ( - 2 \beta_{2} - 4 \beta_1 - 6) q^{97} - \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 5 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 5 q^{5} - q^{7} + 3 q^{9} + q^{11} - 6 q^{13} - 5 q^{15} + 5 q^{17} - 3 q^{19} - q^{21} - 2 q^{23} + 6 q^{25} + 3 q^{27} - 8 q^{29} - 8 q^{31} + q^{33} - q^{35} - 6 q^{37} - 6 q^{39} + 4 q^{41} - 11 q^{43} - 5 q^{45} - q^{47} + 4 q^{49} + 5 q^{51} - 10 q^{53} + q^{55} - 3 q^{57} + 2 q^{59} - 15 q^{61} - q^{63} - 14 q^{67} - 2 q^{69} + 2 q^{71} - 3 q^{73} + 6 q^{75} - 25 q^{77} - 8 q^{79} + 3 q^{81} - 11 q^{85} - 8 q^{87} - 2 q^{89} - 20 q^{91} - 8 q^{93} + 5 q^{95} - 12 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 6 ) / 2 \) 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
−1.86081
2.11491
−0.254102
0 1.00000 0 −4.32340 0 −1.39821 0 1.00000 0
1.2 0 1.00000 0 −1.35793 0 3.58774 0 1.00000 0
1.3 0 1.00000 0 0.681331 0 −3.18953 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3648.2.a.by 3
4.b odd 2 1 3648.2.a.bw 3
8.b even 2 1 1824.2.a.t 3
8.d odd 2 1 1824.2.a.v yes 3
24.f even 2 1 5472.2.a.bn 3
24.h odd 2 1 5472.2.a.bm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.t 3 8.b even 2 1
1824.2.a.v yes 3 8.d odd 2 1
3648.2.a.bw 3 4.b odd 2 1
3648.2.a.by 3 1.a even 1 1 trivial
5472.2.a.bm 3 24.h odd 2 1
5472.2.a.bn 3 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3648))\):

\( T_{5}^{3} + 5T_{5}^{2} + 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{3} + T_{7}^{2} - 12T_{7} - 16 \) Copy content Toggle raw display
\( T_{11}^{3} - T_{11}^{2} - 12T_{11} + 16 \) Copy content Toggle raw display
\( T_{23}^{3} + 2T_{23}^{2} - 60T_{23} - 224 \) Copy content Toggle raw display
\( T_{31}^{3} + 8T_{31}^{2} - 56T_{31} - 392 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 5 T^{2} + 2 T - 4 \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 12 T - 16 \) Copy content Toggle raw display
$11$ \( T^{3} - T^{2} - 12 T + 16 \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} - 4 T - 16 \) Copy content Toggle raw display
$17$ \( T^{3} - 5 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 2 T^{2} - 60 T - 224 \) Copy content Toggle raw display
$29$ \( T^{3} + 8 T^{2} - 28 T - 208 \) Copy content Toggle raw display
$31$ \( T^{3} + 8 T^{2} - 56 T - 392 \) Copy content Toggle raw display
$37$ \( T^{3} + 6 T^{2} - 4 T - 16 \) Copy content Toggle raw display
$41$ \( T^{3} - 4 T^{2} - 20 T + 16 \) Copy content Toggle raw display
$43$ \( T^{3} + 11 T^{2} + 8 T - 16 \) Copy content Toggle raw display
$47$ \( T^{3} + T^{2} - 90 T - 148 \) Copy content Toggle raw display
$53$ \( T^{3} + 10 T^{2} - 68 T - 424 \) Copy content Toggle raw display
$59$ \( T^{3} - 2 T^{2} - 128 T + 512 \) Copy content Toggle raw display
$61$ \( T^{3} + 15 T^{2} + 32 T - 196 \) Copy content Toggle raw display
$67$ \( T^{3} + 14 T^{2} + 40 T - 32 \) Copy content Toggle raw display
$71$ \( T^{3} - 2 T^{2} - 128 T + 512 \) Copy content Toggle raw display
$73$ \( T^{3} + 3 T^{2} - 100 T + 292 \) Copy content Toggle raw display
$79$ \( T^{3} + 8 T^{2} - 104 T - 248 \) Copy content Toggle raw display
$83$ \( T^{3} \) Copy content Toggle raw display
$89$ \( T^{3} + 2 T^{2} - 196 T + 632 \) Copy content Toggle raw display
$97$ \( T^{3} + 12 T^{2} - 196 T - 1168 \) Copy content Toggle raw display
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