Properties

Label 4620.2.a.t
Level $4620$
Weight $2$
Character orbit 4620.a
Self dual yes
Analytic conductor $36.891$
Analytic rank $0$
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] = [4620,2,Mod(1,4620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4620, 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("4620.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4620.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: \(36.8908857338\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1944.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 9x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 9x - 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 6 \) 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
−2.58423
3.28995
−0.705720
0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 4620.2.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4620.2.a.t 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4620))\):

\( T_{13}^{3} - 18T_{13} - 24 \) Copy content Toggle raw display
\( T_{17}^{3} - 27T_{17} + 42 \) Copy content Toggle raw display
\( T_{19}^{3} - 27T_{19} - 18 \) 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
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 18T - 24 \) Copy content Toggle raw display
$17$ \( T^{3} - 27T + 42 \) Copy content Toggle raw display
$19$ \( T^{3} - 27T - 18 \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} - 15 T + 88 \) Copy content Toggle raw display
$29$ \( T^{3} - 6 T^{2} - 69 T + 316 \) Copy content Toggle raw display
$31$ \( T^{3} - 18T + 24 \) Copy content Toggle raw display
$37$ \( T^{3} + 6 T^{2} - 42 T - 196 \) Copy content Toggle raw display
$41$ \( T^{3} - 12 T^{2} - 6 T + 56 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} + 3 T - 16 \) Copy content Toggle raw display
$47$ \( T^{3} - 18T + 24 \) Copy content Toggle raw display
$53$ \( T^{3} - 99T - 246 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} - 69 T - 412 \) Copy content Toggle raw display
$61$ \( T^{3} + 12 T^{2} + 21 T - 86 \) Copy content Toggle raw display
$67$ \( T^{3} - 108T + 336 \) Copy content Toggle raw display
$71$ \( T^{3} - 24 T^{2} + 138 T - 224 \) Copy content Toggle raw display
$73$ \( T^{3} - 72T + 192 \) Copy content Toggle raw display
$79$ \( T^{3} - 6 T^{2} - 150 T - 476 \) Copy content Toggle raw display
$83$ \( T^{3} - 27T + 42 \) Copy content Toggle raw display
$89$ \( T^{3} - 9T - 6 \) Copy content Toggle raw display
$97$ \( T^{3} + 12 T^{2} - 105 T - 362 \) Copy content Toggle raw display
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