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Properties

Label 8010.2.a.bk
Level $8010$
Weight $2$
Character orbit 8010.a
Self dual yes
Analytic conductor $63.960$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.63199488.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 5x^{4} + 14x^{3} + 4x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - \beta_1 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + q^{5} - \beta_1 q^{7} - q^{8} - q^{10} + ( - \beta_{2} - 2) q^{11} + ( - \beta_{3} + \beta_{2} + 2) q^{13} + \beta_1 q^{14} + q^{16} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{17} + (\beta_{5} - 2 \beta_{4} + \beta_{2} + \cdots + 1) q^{19}+ \cdots + ( - 2 \beta_{5} - \beta_{4} - 3 \beta_{3} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{5} + q^{7} - 6 q^{8} - 6 q^{10} - 9 q^{11} + 6 q^{13} - q^{14} + 6 q^{16} + q^{17} - 5 q^{19} + 6 q^{20} + 9 q^{22} - 8 q^{23} + 6 q^{25} - 6 q^{26} + q^{28} - 6 q^{29} - 2 q^{31} - 6 q^{32} - q^{34} + q^{35} - 4 q^{37} + 5 q^{38} - 6 q^{40} - 11 q^{41} - 4 q^{43} - 9 q^{44} + 8 q^{46} - 15 q^{47} + q^{49} - 6 q^{50} + 6 q^{52} - 24 q^{53} - 9 q^{55} - q^{56} + 6 q^{58} - 30 q^{59} + 15 q^{61} + 2 q^{62} + 6 q^{64} + 6 q^{65} + q^{67} + q^{68} - q^{70} - 16 q^{71} + 32 q^{73} + 4 q^{74} - 5 q^{76} - 10 q^{77} - 9 q^{79} + 6 q^{80} + 11 q^{82} - 13 q^{83} + q^{85} + 4 q^{86} + 9 q^{88} + 6 q^{89} - 8 q^{92} + 15 q^{94} - 5 q^{95} + 4 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 5x^{4} + 14x^{3} + 4x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 13\nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{5} - 4\nu^{4} - 2\nu^{3} + 18\nu^{2} - 8\nu - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{5} - 7\nu^{4} - 7\nu^{3} + 32\nu^{2} - 4\nu - 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{5} + 7\nu^{4} + 7\nu^{3} - 32\nu^{2} + 6\nu + 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -3\nu^{5} + 10\nu^{4} + 11\nu^{3} - 44\nu^{2} + 6\nu + 11 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 3\beta_{4} + 3\beta_{3} + \beta_{2} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 3\beta_{4} + 8\beta_{3} + \beta_{2} + 10\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{5} + 22\beta_{4} + 24\beta_{3} + 7\beta_{2} + 26\beta _1 + 34 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
3.08852
−0.520497
−2.04330
0.787517
−0.374845
2.06261
−1.00000 0 1.00000 1.00000 0 −4.03867 −1.00000 0 −1.00000
1.2 −1.00000 0 1.00000 1.00000 0 −0.868572 −1.00000 0 −1.00000
1.3 −1.00000 0 1.00000 1.00000 0 −0.574947 −1.00000 0 −1.00000
1.4 −1.00000 0 1.00000 1.00000 0 0.317253 −1.00000 0 −1.00000
1.5 −1.00000 0 1.00000 1.00000 0 1.27965 −1.00000 0 −1.00000
1.6 −1.00000 0 1.00000 1.00000 0 4.88528 −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bk ✓ 6
3.b odd 2 1 8010.2.a.bl yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8010.2.a.bk ✓ 6 1.a even 1 1 trivial
8010.2.a.bl yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8010))\):

\( T_{7}^{6} - T_{7}^{5} - 21T_{7}^{4} + 4T_{7}^{3} + 28T_{7}^{2} + 4T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{6} + 9T_{11}^{5} + 19T_{11}^{4} - 16T_{11}^{3} - 56T_{11}^{2} + 32 \) Copy content Toggle raw display
\( T_{13}^{6} - 6T_{13}^{5} - 8T_{13}^{4} + 56T_{13}^{3} + 40T_{13}^{2} - 48T_{13} - 32 \) Copy content Toggle raw display
\( T_{17}^{6} - T_{17}^{5} - 49T_{17}^{4} + 40T_{17}^{3} + 208T_{17}^{2} - 240T_{17} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - T^{5} - 21 T^{4} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( T^{6} + 9 T^{5} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{5} + \cdots - 32 \) Copy content Toggle raw display
$17$ \( T^{6} - T^{5} + \cdots + 48 \) Copy content Toggle raw display
$19$ \( T^{6} + 5 T^{5} + \cdots + 324 \) Copy content Toggle raw display
$23$ \( T^{6} + 8 T^{5} + \cdots + 208 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} + \cdots + 3152 \) Copy content Toggle raw display
$31$ \( T^{6} + 2 T^{5} + \cdots + 8992 \) Copy content Toggle raw display
$37$ \( T^{6} + 4 T^{5} + \cdots + 512 \) Copy content Toggle raw display
$41$ \( T^{6} + 11 T^{5} + \cdots - 16432 \) Copy content Toggle raw display
$43$ \( T^{6} + 4 T^{5} + \cdots - 2048 \) Copy content Toggle raw display
$47$ \( T^{6} + 15 T^{5} + \cdots + 61984 \) Copy content Toggle raw display
$53$ \( T^{6} + 24 T^{5} + \cdots - 896 \) Copy content Toggle raw display
$59$ \( T^{6} + 30 T^{5} + \cdots - 274224 \) Copy content Toggle raw display
$61$ \( T^{6} - 15 T^{5} + \cdots - 4364 \) Copy content Toggle raw display
$67$ \( T^{6} - T^{5} + \cdots + 30864 \) Copy content Toggle raw display
$71$ \( T^{6} + 16 T^{5} + \cdots - 576 \) Copy content Toggle raw display
$73$ \( T^{6} - 32 T^{5} + \cdots + 355584 \) Copy content Toggle raw display
$79$ \( T^{6} + 9 T^{5} + \cdots + 9104 \) Copy content Toggle raw display
$83$ \( T^{6} + 13 T^{5} + \cdots - 3712 \) Copy content Toggle raw display
$89$ \( (T - 1)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} - 4 T^{5} + \cdots + 7424 \) Copy content Toggle raw display
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