Properties

Label 6039.2.a.a
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.24217.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 5x^{3} - x^{2} + 3x + 1 \) 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} - \beta_{2} - 2 \beta_1) q^{4} + \beta_{2} q^{5} + ( - \beta_{4} + 2 \beta_{2} + \beta_1 - 1) q^{7} + (2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{4} - \beta_{2} - 2 \beta_1) q^{4} + \beta_{2} q^{5} + ( - \beta_{4} + 2 \beta_{2} + \beta_1 - 1) q^{7} + (2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{8} + ( - \beta_{3} + 1) q^{10} - q^{11} + ( - \beta_{4} + \beta_1 - 2) q^{13} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{14} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{16} + ( - \beta_{2} + 1) q^{17} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{19} + ( - \beta_{2} - \beta_1) q^{20} + \beta_1 q^{22} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{23} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{25} + ( - \beta_{4} + \beta_{2} + 4 \beta_1 - 1) q^{26} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{28} + (\beta_{4} + 3 \beta_{3} - 4 \beta_{2} - \beta_1 + 3) q^{29} + (\beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{31} + ( - 2 \beta_{4} - \beta_{3} - 1) q^{32} + (\beta_{3} - \beta_1 - 1) q^{34} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{35} + (3 \beta_{4} + \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{37} + ( - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 7 \beta_1 - 1) q^{38} + (\beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{40} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{41} + (3 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 2) q^{43} + ( - \beta_{4} + \beta_{2} + 2 \beta_1) q^{44} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 1) q^{46} + ( - \beta_{4} + \beta_{2} + 3 \beta_1 + 1) q^{47} + (\beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1) q^{49} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} + 5 \beta_1 - 1) q^{50} + ( - 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + 7 \beta_1 - 2) q^{52} + ( - 3 \beta_{4} - 5 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{53} - \beta_{2} q^{55} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 6) q^{56} + (\beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 5 \beta_1 - 3) q^{58} + (3 \beta_{3} - 2 \beta_1 + 2) q^{59} + q^{61} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 5) q^{62} + ( - 4 \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 2) q^{64} + ( - 2 \beta_{2} - 1) q^{65} + ( - 4 \beta_{4} + 4 \beta_1 - 1) q^{67} + (\beta_{4} - \beta_1) q^{68} + ( - 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + \beta_1 - 3) q^{70} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{71} + (\beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{73} + (3 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} - 6 \beta_1) q^{74} + ( - 3 \beta_{4} - \beta_{3} + 6 \beta_{2} + 11 \beta_1 - 3) q^{76} + (\beta_{4} - 2 \beta_{2} - \beta_1 + 1) q^{77} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 - 5) q^{79} + (2 \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{80} + ( - \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{82} + (\beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 1) q^{83} + (\beta_{3} - \beta_1 - 2) q^{85} + (3 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} - 8 \beta_1 - 1) q^{86} + ( - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{88} + ( - 2 \beta_{3} - 5 \beta_{2} - \beta_1 + 4) q^{89} + (2 \beta_{4} + \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 2) q^{91} + ( - 4 \beta_{4} - 2 \beta_{3} + 3 \beta_1 - 3) q^{92} + ( - 3 \beta_{4} - \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 4) q^{94} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{95} + ( - 2 \beta_{4} - 7 \beta_{3} + 4 \beta_{2} + 6 \beta_1 - 1) q^{97} + ( - \beta_{4} + \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 2 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{2} + 2 q^{5} - q^{7} + 6 q^{8} + 5 q^{10} - 5 q^{11} - 10 q^{13} + 3 q^{14} + 2 q^{16} + 3 q^{17} - 13 q^{19} - 2 q^{22} - 15 q^{25} - 9 q^{26} - 12 q^{28} + 7 q^{29} - 13 q^{31} - q^{32} - 3 q^{34} + 13 q^{35} - 6 q^{37} - 9 q^{38} - 5 q^{40} + 9 q^{41} + 2 q^{43} - 7 q^{46} + 3 q^{47} - 6 q^{49} - 9 q^{50} - 12 q^{52} - 3 q^{53} - 2 q^{55} - 28 q^{56} - 15 q^{58} + 14 q^{59} + 5 q^{61} - 31 q^{62} + 6 q^{64} - 9 q^{65} - 5 q^{67} - 5 q^{70} - 3 q^{71} - 4 q^{73} - 2 q^{74} - 19 q^{76} + q^{77} - 27 q^{79} + 2 q^{80} + 11 q^{82} + 3 q^{83} - 8 q^{85} - 5 q^{86} - 6 q^{88} + 12 q^{89} + 4 q^{91} - 13 q^{92} - 18 q^{94} - 7 q^{95} - 5 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{4} - 5\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{4} + 5\nu^{2} + \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} + \nu^{3} + 5\nu^{2} - 3\nu - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{4} - \nu^{3} - 9\nu^{2} + 3\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{4} + 5\beta_{3} - 4\beta _1 + 8 \) 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
−0.369680
2.17442
−0.722813
0.878095
−1.96003
−1.33536 0 −0.216816 −1.70504 0 −3.82358 2.96025 0 2.27684
1.2 −0.714533 0 −1.48944 1.45989 0 1.23480 2.49332 0 −1.04314
1.3 0.339328 0 −1.88486 −0.383484 0 0.840700 −1.31824 0 −0.130127
1.4 1.26073 0 −0.410549 2.13883 0 2.81011 −3.03906 0 2.69649
1.5 2.44983 0 4.00166 0.489803 0 −2.06203 4.90374 0 1.19993
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.a 5
3.b odd 2 1 671.2.a.a 5
33.d even 2 1 7381.2.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.a 5 3.b odd 2 1
6039.2.a.a 5 1.a even 1 1 trivial
7381.2.a.g 5 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 2T_{2}^{4} - 3T_{2}^{3} + 4T_{2}^{2} + 2T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 2 T^{4} - 3 T^{3} + 4 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 2 T^{4} - 3 T^{3} + 6 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$7$ \( T^{5} + T^{4} - 14 T^{3} - T^{2} + 37 T - 23 \) Copy content Toggle raw display
$11$ \( (T + 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 10 T^{4} + 34 T^{3} + 41 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$17$ \( T^{5} - 3 T^{4} - T^{3} + 5 T^{2} - 1 \) Copy content Toggle raw display
$19$ \( T^{5} + 13 T^{4} + 49 T^{3} + 36 T^{2} + \cdots - 5 \) Copy content Toggle raw display
$23$ \( T^{5} - 35 T^{3} - 101 T^{2} + \cdots + 59 \) Copy content Toggle raw display
$29$ \( T^{5} - 7 T^{4} - 67 T^{3} + \cdots + 2885 \) Copy content Toggle raw display
$31$ \( T^{5} + 13 T^{4} + 8 T^{3} - 175 T^{2} + \cdots + 313 \) Copy content Toggle raw display
$37$ \( T^{5} + 6 T^{4} - 32 T^{3} - 139 T^{2} + \cdots + 97 \) Copy content Toggle raw display
$41$ \( T^{5} - 9 T^{4} + T^{3} + 122 T^{2} + \cdots + 43 \) Copy content Toggle raw display
$43$ \( T^{5} - 2 T^{4} - 66 T^{3} + \cdots - 1037 \) Copy content Toggle raw display
$47$ \( T^{5} - 3 T^{4} - 22 T^{3} + 107 T^{2} + \cdots + 17 \) Copy content Toggle raw display
$53$ \( T^{5} + 3 T^{4} - 130 T^{3} + \cdots - 2347 \) Copy content Toggle raw display
$59$ \( T^{5} - 14 T^{4} + 27 T^{3} + \cdots - 655 \) Copy content Toggle raw display
$61$ \( (T - 1)^{5} \) Copy content Toggle raw display
$67$ \( T^{5} + 5 T^{4} - 86 T^{3} + \cdots + 1249 \) Copy content Toggle raw display
$71$ \( T^{5} + 3 T^{4} - 73 T^{3} + 259 T^{2} + \cdots + 149 \) Copy content Toggle raw display
$73$ \( T^{5} + 4 T^{4} - 118 T^{3} + \cdots + 431 \) Copy content Toggle raw display
$79$ \( T^{5} + 27 T^{4} + 259 T^{3} + \cdots + 1285 \) Copy content Toggle raw display
$83$ \( T^{5} - 3 T^{4} - 143 T^{3} + \cdots - 3457 \) Copy content Toggle raw display
$89$ \( T^{5} - 12 T^{4} - 75 T^{3} + \cdots - 3995 \) Copy content Toggle raw display
$97$ \( T^{5} + 5 T^{4} - 227 T^{3} + \cdots + 62147 \) Copy content Toggle raw display
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