Properties

Label 6032.2.a.n
Level $6032$
Weight $2$
Character orbit 6032.a
Self dual yes
Analytic conductor $48.166$
Analytic rank $0$
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] = [6032,2,Mod(1,6032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6032, 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("6032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6032.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: \(48.1657624992\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1220776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 8x^{3} - x^{2} + 12x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 754)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{4} + \beta_{3} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{4} + \beta_{3} + \beta_1 + 2) q^{9} + ( - \beta_{4} - \beta_{2} - 2) q^{11} - q^{13} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{15} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{17} + ( - \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{19} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 1) q^{21} + 2 \beta_{3} q^{23} + (2 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 1) q^{25} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 1) q^{27} - q^{29} - 4 \beta_1 q^{31} + (2 \beta_{4} - 2 \beta_{2} - 6 \beta_1) q^{33} + (\beta_{4} + 3 \beta_{2} + 5 \beta_1 + 2) q^{35} + (\beta_{4} + \beta_{3} + \beta_1 - 1) q^{37} + ( - \beta_{2} - \beta_1 + 1) q^{39} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 1) q^{41} + ( - 3 \beta_{2} - \beta_1 - 1) q^{43} + (\beta_{4} + 3 \beta_{2} + 6 \beta_1 + 2) q^{45} + ( - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{47} + (2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 2) q^{49} + (3 \beta_{4} + \beta_{3} - 3 \beta_1 + 5) q^{51} + ( - 2 \beta_{4} - 2 \beta_{3} + 2) q^{53} + ( - \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 6 \beta_1 - 1) q^{55} + (4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{57} + (\beta_{4} + \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{59} + ( - \beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{61} + (4 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 9) q^{63} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{65} + (\beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{67} - 4 q^{69} + ( - 3 \beta_{2} - \beta_1 - 1) q^{71} + ( - \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{73} + (3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 8 \beta_1 + 10) q^{75} + ( - 4 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 7) q^{77} + ( - 3 \beta_{3} + 2 \beta_1) q^{79} + (2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 3) q^{81} + (\beta_{4} + \beta_{3} + 3 \beta_{2} + 4) q^{83} + (\beta_{4} + \beta_{3} + \beta_1 - 1) q^{85} + ( - \beta_{2} - \beta_1 + 1) q^{87} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 2) q^{89} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{91} + ( - 4 \beta_{4} - 4 \beta_{3} - 8 \beta_{2} - 4 \beta_1 - 4) q^{93} + ( - \beta_{3} + \beta_{2} + 5) q^{95} + (\beta_{4} + \beta_{3} - 2 \beta_{2} - 6 \beta_1 - 1) q^{97} + ( - 7 \beta_{4} - 6 \beta_{3} - 5 \beta_{2} - 2 \beta_1 - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} + 6 q^{5} + 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{3} + 6 q^{5} + 2 q^{7} + 10 q^{9} - 13 q^{11} - 5 q^{13} + 9 q^{15} - 5 q^{17} - 7 q^{19} + 7 q^{21} - 2 q^{23} + 11 q^{25} - 3 q^{27} - 5 q^{29} - 2 q^{33} + 17 q^{35} - 5 q^{37} + 3 q^{39} + q^{41} - 11 q^{43} + 17 q^{45} - 7 q^{47} + 17 q^{49} + 27 q^{51} + 10 q^{53} - 11 q^{55} + 16 q^{57} + 6 q^{59} + 3 q^{61} + 57 q^{63} - 6 q^{65} - 8 q^{67} - 20 q^{69} - 11 q^{71} + 3 q^{73} + 56 q^{75} - 40 q^{77} + 3 q^{79} + 21 q^{81} + 26 q^{83} - 5 q^{85} + 3 q^{87} + 11 q^{89} - 2 q^{91} - 36 q^{93} + 28 q^{95} - 9 q^{97} - 51 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 8x^{3} - x^{2} + 12x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 6\nu^{2} - \nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 8\nu^{2} - \nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} + 6\nu^{2} - 9\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{3} + 8\beta_{2} + 7\beta _1 + 14 \) 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
−1.24900
1.51741
−2.40898
−0.351552
2.49212
0 −3.08772 0 −0.439989 0 3.69528 0 6.53400 0
1.2 0 −2.49807 0 0.302546 0 −1.77512 0 3.24036 0
1.3 0 −0.775569 0 3.80318 0 −1.76508 0 −2.39849 0
1.4 0 0.461095 0 −1.87641 0 −2.97475 0 −2.78739 0
1.5 0 2.90026 0 4.21067 0 4.81967 0 5.41153 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(1\)
\(29\) \(1\)

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 6032.2.a.n 5
4.b odd 2 1 754.2.a.i 5
12.b even 2 1 6786.2.a.bo 5
52.b odd 2 1 9802.2.a.x 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
754.2.a.i 5 4.b odd 2 1
6032.2.a.n 5 1.a even 1 1 trivial
6786.2.a.bo 5 12.b even 2 1
9802.2.a.x 5 52.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):

\( T_{3}^{5} + 3T_{3}^{4} - 8T_{3}^{3} - 26T_{3}^{2} - 4T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{5} - 6T_{5}^{4} + 31T_{5}^{2} + 4T_{5} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 3 T^{4} - 8 T^{3} - 26 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{5} - 6 T^{4} + 31 T^{2} + 4 T - 4 \) Copy content Toggle raw display
$7$ \( T^{5} - 2 T^{4} - 24 T^{3} + 9 T^{2} + \cdots + 166 \) Copy content Toggle raw display
$11$ \( T^{5} + 13 T^{4} + 43 T^{3} - 34 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$13$ \( (T + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} + 5 T^{4} - 26 T^{3} - 8 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$19$ \( T^{5} + 7 T^{4} - 35 T^{3} - 142 T^{2} + \cdots - 416 \) Copy content Toggle raw display
$23$ \( T^{5} + 2 T^{4} - 52 T^{3} + 64 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$29$ \( (T + 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} - 128 T^{3} + 64 T^{2} + \cdots - 4096 \) Copy content Toggle raw display
$37$ \( T^{5} + 5 T^{4} - 28 T^{3} - 84 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$41$ \( T^{5} - T^{4} - 39 T^{3} - 72 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( T^{5} + 11 T^{4} - 34 T^{3} + \cdots + 5504 \) Copy content Toggle raw display
$47$ \( T^{5} + 7 T^{4} - 122 T^{3} + \cdots + 3328 \) Copy content Toggle raw display
$53$ \( T^{5} - 10 T^{4} - 76 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{5} - 6 T^{4} - 134 T^{3} + \cdots + 208 \) Copy content Toggle raw display
$61$ \( T^{5} - 3 T^{4} - 103 T^{3} - 320 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$67$ \( T^{5} + 8 T^{4} - 42 T^{3} + \cdots + 2864 \) Copy content Toggle raw display
$71$ \( T^{5} + 11 T^{4} - 34 T^{3} + \cdots + 5504 \) Copy content Toggle raw display
$73$ \( T^{5} - 3 T^{4} - 149 T^{3} + \cdots + 9032 \) Copy content Toggle raw display
$79$ \( T^{5} - 3 T^{4} - 185 T^{3} + \cdots - 3064 \) Copy content Toggle raw display
$83$ \( T^{5} - 26 T^{4} + 194 T^{3} + \cdots + 17536 \) Copy content Toggle raw display
$89$ \( T^{5} - 11 T^{4} - 81 T^{3} + \cdots - 5536 \) Copy content Toggle raw display
$97$ \( T^{5} + 9 T^{4} - 269 T^{3} + \cdots + 46688 \) Copy content Toggle raw display
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