Properties

Label 4840.2.a.o
Level $4840$
Weight $2$
Character orbit 4840.a
Self dual yes
Analytic conductor $38.648$
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] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + q^{5} + ( - \beta - 1) q^{7} + 2 \beta q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + q^{5} + ( - \beta - 1) q^{7} + 2 \beta q^{9} - 2 q^{13} + (\beta + 1) q^{15} - 4 \beta q^{17} + ( - 2 \beta - 2) q^{19} + ( - 2 \beta - 3) q^{21} + (4 \beta - 2) q^{23} + q^{25} + ( - \beta + 1) q^{27} - 2 q^{29} - 4 \beta q^{31} + ( - \beta - 1) q^{35} - 4 q^{37} + ( - 2 \beta - 2) q^{39} + (6 \beta + 1) q^{41} + (\beta - 5) q^{43} + 2 \beta q^{45} + ( - \beta + 9) q^{47} + (2 \beta - 4) q^{49} + ( - 4 \beta - 8) q^{51} + ( - 4 \beta - 2) q^{53} + ( - 4 \beta - 6) q^{57} - 8 \beta q^{59} - q^{61} + ( - 2 \beta - 4) q^{63} - 2 q^{65} + ( - \beta - 5) q^{67} + (2 \beta + 6) q^{69} - 4 q^{71} - 4 q^{73} + (\beta + 1) q^{75} + (6 \beta + 6) q^{79} + ( - 6 \beta - 1) q^{81} + (8 \beta + 2) q^{83} - 4 \beta q^{85} + ( - 2 \beta - 2) q^{87} + (4 \beta - 3) q^{89} + (2 \beta + 2) q^{91} + ( - 4 \beta - 8) q^{93} + ( - 2 \beta - 2) q^{95} + ( - 8 \beta - 6) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{13} + 2 q^{15} - 4 q^{19} - 6 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 2 q^{35} - 8 q^{37} - 4 q^{39} + 2 q^{41} - 10 q^{43} + 18 q^{47} - 8 q^{49} - 16 q^{51} - 4 q^{53} - 12 q^{57} - 2 q^{61} - 8 q^{63} - 4 q^{65} - 10 q^{67} + 12 q^{69} - 8 q^{71} - 8 q^{73} + 2 q^{75} + 12 q^{79} - 2 q^{81} + 4 q^{83} - 4 q^{87} - 6 q^{89} + 4 q^{91} - 16 q^{93} - 4 q^{95} - 12 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 −0.414214 0 1.00000 0 0.414214 0 −2.82843 0
1.2 0 2.41421 0 1.00000 0 −2.41421 0 2.82843 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.o 2
4.b odd 2 1 9680.2.a.bk 2
11.b odd 2 1 4840.2.a.p yes 2
44.c even 2 1 9680.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4840.2.a.o 2 1.a even 1 1 trivial
4840.2.a.p yes 2 11.b odd 2 1
9680.2.a.bj 2 44.c even 2 1
9680.2.a.bk 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4840))\):

\( T_{3}^{2} - 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 1 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 32 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 32 \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 71 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 23 \) Copy content Toggle raw display
$47$ \( T^{2} - 18T + 79 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$59$ \( T^{2} - 128 \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 23 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 12T - 36 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 23 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T - 92 \) Copy content Toggle raw display
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