Properties

Label 8010.2.a.o
Level $8010$
Weight $2$
Character orbit 8010.a
Self dual yes
Analytic conductor $63.960$
Analytic rank $0$
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] = [8010,2,Mod(1,8010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8010 = 2 \cdot 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8010.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: \(63.9601720190\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} - 2 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} - 2 q^{7} - q^{8} + q^{10} + \beta q^{13} + 2 q^{14} + q^{16} + 2 q^{17} + \beta q^{19} - q^{20} + (2 \beta + 2) q^{23} + q^{25} - \beta q^{26} - 2 q^{28} + 4 q^{29} + ( - \beta - 2) q^{31} - q^{32} - 2 q^{34} + 2 q^{35} + ( - \beta - 4) q^{37} - \beta q^{38} + q^{40} + (4 \beta - 2) q^{41} - 2 \beta q^{43} + ( - 2 \beta - 2) q^{46} + 6 q^{47} - 3 q^{49} - q^{50} + \beta q^{52} + 2 q^{53} + 2 q^{56} - 4 q^{58} - \beta q^{59} + (2 \beta - 8) q^{61} + (\beta + 2) q^{62} + q^{64} - \beta q^{65} + (4 \beta + 4) q^{67} + 2 q^{68} - 2 q^{70} - 4 q^{71} + ( - 2 \beta + 2) q^{73} + (\beta + 4) q^{74} + \beta q^{76} + (2 \beta - 12) q^{79} - q^{80} + ( - 4 \beta + 2) q^{82} - 2 \beta q^{83} - 2 q^{85} + 2 \beta q^{86} + q^{89} - 2 \beta q^{91} + (2 \beta + 2) q^{92} - 6 q^{94} - \beta q^{95} + (2 \beta + 10) q^{97} + 3 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} - 2 q^{8} + 2 q^{10} + 4 q^{14} + 2 q^{16} + 4 q^{17} - 2 q^{20} + 4 q^{23} + 2 q^{25} - 4 q^{28} + 8 q^{29} - 4 q^{31} - 2 q^{32} - 4 q^{34} + 4 q^{35} - 8 q^{37} + 2 q^{40} - 4 q^{41} - 4 q^{46} + 12 q^{47} - 6 q^{49} - 2 q^{50} + 4 q^{53} + 4 q^{56} - 8 q^{58} - 16 q^{61} + 4 q^{62} + 2 q^{64} + 8 q^{67} + 4 q^{68} - 4 q^{70} - 8 q^{71} + 4 q^{73} + 8 q^{74} - 24 q^{79} - 2 q^{80} + 4 q^{82} - 4 q^{85} + 2 q^{89} + 4 q^{92} - 12 q^{94} + 20 q^{97} + 6 q^{98}+O(q^{100}) \) 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
−2.44949
2.44949
−1.00000 0 1.00000 −1.00000 0 −2.00000 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 −2.00000 −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(1\)
\(89\) \(-1\)

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 8010.2.a.o 2
3.b odd 2 1 8010.2.a.t yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8010.2.a.o 2 1.a even 1 1 trivial
8010.2.a.t yes 2 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8010))\):

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 6 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 6 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 20 \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 10 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 92 \) Copy content Toggle raw display
$43$ \( T^{2} - 24 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 40 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 80 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 20 \) Copy content Toggle raw display
$79$ \( T^{2} + 24T + 120 \) Copy content Toggle raw display
$83$ \( T^{2} - 24 \) Copy content Toggle raw display
$89$ \( (T - 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 20T + 76 \) Copy content Toggle raw display
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