Properties

Label 4840.2.a.s
Level $4840$
Weight $2$
Character orbit 4840.a
Self dual yes
Analytic conductor $38.648$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.788.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + (\beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} + (\beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9} + ( - \beta_{2} + 1) q^{13} - \beta_1 q^{15} + ( - 2 \beta_{2} - 2 \beta_1) q^{17} + (\beta_{2} + 1) q^{19} + ( - 2 \beta_{2} - 2 \beta_1 - 5) q^{21} + ( - \beta_{2} - 2 \beta_1 - 1) q^{23} + q^{25} + (\beta_{2} + 3 \beta_1 + 7) q^{27} - 2 \beta_1 q^{29} + ( - \beta_{2} - 7) q^{31} + ( - \beta_{2} + \beta_1 + 1) q^{35} + (2 \beta_1 - 2) q^{37} + (\beta_{2} + 1) q^{39} + (\beta_{2} - 2) q^{41} + (3 \beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{2} - 2 \beta_1 - 1) q^{45} + ( - 2 \beta_{2} + \beta_1 - 2) q^{47} + (\beta_{2} + 7) q^{49} + ( - 6 \beta_1 - 6) q^{51} + ( - 3 \beta_{2} - 1) q^{53} + ( - \beta_{2} + 2 \beta_1 - 1) q^{57} + ( - \beta_{2} - 2 \beta_1 - 1) q^{59} + ( - \beta_{2} - 2 \beta_1 + 8) q^{61} + ( - 3 \beta_{2} - 8 \beta_1 - 3) q^{63} + (\beta_{2} - 1) q^{65} + ( - 3 \beta_1 + 6) q^{67} + ( - \beta_{2} - 6 \beta_1 - 7) q^{69} + ( - 5 \beta_{2} - 2 \beta_1 - 1) q^{71} + ( - 2 \beta_{2} + 2) q^{73} + \beta_1 q^{75} + (2 \beta_{2} + 2 \beta_1 - 2) q^{79} + ( - \beta_{2} + 8 \beta_1 + 8) q^{81} + (\beta_{2} - 2 \beta_1 - 3) q^{83} + (2 \beta_{2} + 2 \beta_1) q^{85} + ( - 2 \beta_{2} - 4 \beta_1 - 8) q^{87} + (2 \beta_{2} + 2 \beta_1 + 5) q^{89} + (\beta_{2} + 2 \beta_1 - 9) q^{91} + (\beta_{2} - 8 \beta_1 + 1) q^{93} + ( - \beta_{2} - 1) q^{95} + (\beta_{2} - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} + 2 q^{13} - q^{15} - 4 q^{17} + 4 q^{19} - 19 q^{21} - 6 q^{23} + 3 q^{25} + 25 q^{27} - 2 q^{29} - 22 q^{31} + 3 q^{35} - 4 q^{37} + 4 q^{39} - 5 q^{41} + q^{43} - 6 q^{45} - 7 q^{47} + 22 q^{49} - 24 q^{51} - 6 q^{53} - 2 q^{57} - 6 q^{59} + 21 q^{61} - 20 q^{63} - 2 q^{65} + 15 q^{67} - 28 q^{69} - 10 q^{71} + 4 q^{73} + q^{75} - 2 q^{79} + 31 q^{81} - 10 q^{83} + 4 q^{85} - 30 q^{87} + 19 q^{89} - 24 q^{91} - 4 q^{93} - 4 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) 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
−1.87740
−0.476452
3.35386
0 −1.87740 0 −1.00000 0 4.15686 0 0.524645 0
1.2 0 −0.476452 0 −1.00000 0 −3.34364 0 −2.77299 0
1.3 0 3.35386 0 −1.00000 0 −3.81322 0 8.24835 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4840.2.a.s 3
4.b odd 2 1 9680.2.a.cd 3
11.b odd 2 1 4840.2.a.v yes 3
44.c even 2 1 9680.2.a.by 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4840.2.a.s 3 1.a even 1 1 trivial
4840.2.a.v yes 3 11.b odd 2 1
9680.2.a.by 3 44.c even 2 1
9680.2.a.cd 3 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4840))\):

\( T_{3}^{3} - T_{3}^{2} - 7T_{3} - 3 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - 17T_{7} - 53 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 8T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 7T - 3 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} - 17 T - 53 \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} - 48 T - 144 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} - 4 T + 12 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} - 20 T - 12 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 28 T + 24 \) Copy content Toggle raw display
$31$ \( T^{3} + 22 T^{2} + 152 T + 324 \) Copy content Toggle raw display
$37$ \( T^{3} + 4 T^{2} - 24 T - 80 \) Copy content Toggle raw display
$41$ \( T^{3} + 5T^{2} - T - 9 \) Copy content Toggle raw display
$43$ \( T^{3} - T^{2} - 81 T + 275 \) Copy content Toggle raw display
$47$ \( T^{3} + 7 T^{2} - 35 T + 9 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 72 T - 212 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} - 20 T - 12 \) Copy content Toggle raw display
$61$ \( T^{3} - 21 T^{2} + 115 T - 75 \) Copy content Toggle raw display
$67$ \( T^{3} - 15 T^{2} + 9 T + 351 \) Copy content Toggle raw display
$71$ \( T^{3} + 10 T^{2} - 196 T - 1996 \) Copy content Toggle raw display
$73$ \( T^{3} - 4 T^{2} - 32 T + 32 \) Copy content Toggle raw display
$79$ \( T^{3} + 2 T^{2} - 52 T + 40 \) Copy content Toggle raw display
$83$ \( T^{3} + 10 T^{2} - 12 T - 180 \) Copy content Toggle raw display
$89$ \( T^{3} - 19 T^{2} + 67 T + 159 \) Copy content Toggle raw display
$97$ \( T^{3} + 2 T^{2} - 8 T - 4 \) Copy content Toggle raw display
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