Properties

Label 6039.2.a.b
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, 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("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.2661761.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 6x^{4} + 3x^{3} + 9x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 671)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{5} q^{4} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{4} + \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{5} q^{4} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{4} + \beta_1 + 1) q^{8} + (\beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{10} + q^{11} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{13} + ( - \beta_{4} - \beta_{2}) q^{14} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{16} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{17} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{19} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{20} - \beta_{3} q^{22} + (2 \beta_{5} + \beta_{4} + 1) q^{23} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{25} + (\beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{26} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{28} + \beta_{4} q^{29} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{31} + ( - \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{32} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 5 \beta_1 - 3) q^{34} + (\beta_{5} - \beta_{3} + \beta_1 - 1) q^{35} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{37} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{38} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{40} + (\beta_{5} + 2 \beta_{4} - 2 \beta_{2} + \beta_1 + 1) q^{41} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{43} - \beta_{5} q^{44} + (2 \beta_{4} + 3 \beta_{3} - 3 \beta_1 - 1) q^{46} + ( - 4 \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{47} + (\beta_{2} - \beta_1 - 4) q^{49} + (4 \beta_{5} - \beta_{4} + 2 \beta_{2} + 4 \beta_1 - 2) q^{50} + (2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 1) q^{52} + (4 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{53} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{55} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2}) q^{56} + ( - \beta_1 + 1) q^{58} + (3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{59} - q^{61} + (2 \beta_{5} - \beta_{4} + \beta_{2} + 2 \beta_1) q^{62} + (\beta_{4} + 3 \beta_{3} + 3 \beta_{2} - \beta_1 - 1) q^{64} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2) q^{65} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 1) q^{67} + (\beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{68} + ( - \beta_{5} + 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{70} + ( - 4 \beta_{4} - \beta_{3} + 3) q^{71} + (5 \beta_{5} - 3 \beta_{4} + 3 \beta_{2} + 5 \beta_1 - 1) q^{73} + ( - \beta_{5} + 4 \beta_{3} - \beta_{2} + 4 \beta_1 - 2) q^{74} + (\beta_{4} - 2) q^{76} + (\beta_1 - 1) q^{77} + (5 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{79} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 4) q^{80} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{82} + (\beta_{4} + 3 \beta_{3} - 4 \beta_{2} - 6 \beta_1 - 2) q^{83} + (2 \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_1 + 1) q^{85} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{86} + ( - \beta_{4} + \beta_1 + 1) q^{88} + ( - \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + \beta_1 + 4) q^{89} + (2 \beta_{5} - 2 \beta_{4} - \beta_{2} + 2 \beta_1) q^{91} + ( - \beta_{5} + \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 6) q^{92} + (\beta_{5} - 3 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 5) q^{94} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{95} + (\beta_{4} - 4 \beta_{2} - \beta_1 - 4) q^{97} + ( - \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + \beta_{2} - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{4} + q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{4} + q^{5} - 5 q^{7} + 6 q^{8} - 7 q^{10} + 6 q^{11} - 4 q^{13} - q^{14} - 10 q^{16} + 5 q^{17} - 3 q^{19} + 6 q^{20} + 3 q^{23} - 11 q^{25} - q^{26} + 4 q^{28} + q^{29} - 10 q^{31} - 3 q^{32} - 19 q^{34} - 7 q^{35} - 19 q^{37} + 3 q^{38} + 5 q^{40} + 7 q^{41} - 2 q^{43} + 2 q^{44} - 7 q^{46} - 5 q^{47} - 25 q^{49} - 17 q^{50} + 2 q^{52} + 9 q^{53} + q^{55} + 4 q^{56} + 5 q^{58} + 5 q^{59} - 6 q^{61} - 3 q^{62} - 6 q^{64} + 11 q^{65} - 14 q^{67} - 18 q^{68} + 7 q^{70} + 14 q^{71} - 14 q^{73} - 6 q^{74} - 11 q^{76} - 5 q^{77} + 5 q^{79} - 26 q^{80} + q^{82} - 17 q^{83} + 2 q^{85} + 7 q^{86} + 6 q^{88} + 25 q^{89} - 4 q^{91} - 35 q^{92} + 30 q^{94} - 3 q^{95} - 24 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 6x^{4} + 3x^{3} + 9x^{2} - x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - \nu^{4} - 5\nu^{3} + 2\nu^{2} + 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} + 5\beta_{2} + 6\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + \beta_{4} + 7\beta_{3} + 8\beta_{2} + 19\beta _1 + 8 \) 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
−0.303283
2.36588
−1.34697
0.387870
−1.68584
1.58234
−1.78997 0 1.20400 3.29725 0 −1.30328 1.42482 0 −5.90199
1.2 −1.54773 0 0.395474 −0.422675 0 1.36588 2.48338 0 0.654188
1.3 −0.782747 0 −1.38731 0.742408 0 −2.34697 2.65140 0 −0.581118
1.4 0.255699 0 −1.93462 −2.57819 0 −0.612130 −1.00608 0 −0.659240
1.5 1.57580 0 0.483130 0.593176 0 −2.68584 −2.39028 0 0.934724
1.6 2.28896 0 3.23932 −0.631975 0 0.582341 2.83675 0 −1.44656
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)
\(61\) \(1\)

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 6039.2.a.b 6
3.b odd 2 1 671.2.a.b 6
33.d even 2 1 7381.2.a.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.b 6 3.b odd 2 1
6039.2.a.b 6 1.a even 1 1 trivial
7381.2.a.h 6 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 7T_{2}^{4} - 2T_{2}^{3} + 12T_{2}^{2} + 5T_{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 7 T^{4} - 2 T^{3} + 12 T^{2} + \cdots - 2 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} - 9 T^{4} + 3 T^{3} + 6 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 5 T^{5} + 4 T^{4} - 11 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} - 12 T^{4} - 63 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( T^{6} - 5 T^{5} - 39 T^{4} + 143 T^{3} + \cdots - 926 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} - 9 T^{4} - 40 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} - 43 T^{4} + 74 T^{3} + \cdots + 131 \) Copy content Toggle raw display
$29$ \( T^{6} - T^{5} - 9 T^{4} + 12 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$31$ \( T^{6} + 10 T^{5} + 25 T^{4} - 21 T^{3} + \cdots + 7 \) Copy content Toggle raw display
$37$ \( T^{6} + 19 T^{5} + 32 T^{4} + \cdots - 2993 \) Copy content Toggle raw display
$41$ \( T^{6} - 7 T^{5} - 89 T^{4} + 204 T^{3} + \cdots + 724 \) Copy content Toggle raw display
$43$ \( T^{6} + 2 T^{5} - 74 T^{4} + 155 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$47$ \( T^{6} + 5 T^{5} - 110 T^{4} + \cdots - 26800 \) Copy content Toggle raw display
$53$ \( T^{6} - 9 T^{5} - 82 T^{4} + \cdots - 1714 \) Copy content Toggle raw display
$59$ \( T^{6} - 5 T^{5} - 59 T^{4} + \cdots - 4793 \) Copy content Toggle raw display
$61$ \( (T + 1)^{6} \) Copy content Toggle raw display
$67$ \( T^{6} + 14 T^{5} - 57 T^{4} + \cdots - 27503 \) Copy content Toggle raw display
$71$ \( T^{6} - 14 T^{5} - 56 T^{4} + \cdots - 10877 \) Copy content Toggle raw display
$73$ \( T^{6} + 14 T^{5} - 200 T^{4} + \cdots + 467006 \) Copy content Toggle raw display
$79$ \( T^{6} - 5 T^{5} - 163 T^{4} + \cdots - 80764 \) Copy content Toggle raw display
$83$ \( T^{6} + 17 T^{5} - 171 T^{4} + \cdots - 59872 \) Copy content Toggle raw display
$89$ \( T^{6} - 25 T^{5} + 115 T^{4} + \cdots - 547 \) Copy content Toggle raw display
$97$ \( T^{6} + 24 T^{5} + 96 T^{4} + \cdots + 24269 \) Copy content Toggle raw display
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