Properties

Label 6032.2.a.p
Level $6032$
Weight $2$
Character orbit 6032.a
Self dual yes
Analytic conductor $48.166$
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] = [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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.149169.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 6x^{3} - 3x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1508)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} - \beta_{2} q^{5} + ( - \beta_{4} + \beta_{3} - \beta_1) q^{7} + (\beta_{4} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} - \beta_{2} q^{5} + ( - \beta_{4} + \beta_{3} - \beta_1) q^{7} + (\beta_{4} + \beta_{2} - 1) q^{9} + (\beta_{4} - \beta_{3} + 2 \beta_1) q^{11} - q^{13} + ( - 2 \beta_{4} - \beta_{2} - \beta_1 + 2) q^{15} + (\beta_{4} + \beta_{3} + 2 \beta_{2} - 3) q^{17} + (\beta_{4} - 2 \beta_{3} + \beta_1) q^{19} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{21} + ( - 3 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 3) q^{23} + ( - \beta_{3} - 2) q^{25} + ( - \beta_{4} + 2 \beta_{2} + \beta_1) q^{27} + q^{29} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{31} + (2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{33} + (\beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{35} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{37} - \beta_{4} q^{39} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{41} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{43} + ( - 2 \beta_{4} + \beta_{3} - \beta_1 - 1) q^{45} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} - 1) q^{47} + (2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{49} + (\beta_{4} + 3 \beta_{2} - 1) q^{51} + ( - 3 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 2) q^{53} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{55} + (3 \beta_{4} - \beta_{3} + \beta_{2} + 4 \beta_1 - 1) q^{57} + ( - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 4 \beta_1 - 1) q^{59} + ( - 5 \beta_{4} + \beta_{3} - 5 \beta_1 - 2) q^{61} + ( - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 1) q^{63} + \beta_{2} q^{65} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 5 \beta_1 + 6) q^{67} + ( - 4 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{69} + (5 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{71} + (2 \beta_{4} + 2 \beta_{3} - \beta_1 - 6) q^{73} + ( - \beta_{4} + 2 \beta_1 - 1) q^{75} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 5) q^{77} + (2 \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{79} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{81} + ( - 4 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 5 \beta_1 + 1) q^{83} + ( - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 3) q^{85} + \beta_{4} q^{87} + ( - 2 \beta_{3} + 4 \beta_1 - 3) q^{89} + (\beta_{4} - \beta_{3} + \beta_1) q^{91} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 1) q^{93} + ( - \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{95} + ( - 4 \beta_{4} - 3 \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{97} + (2 \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{3} - 2 q^{5} - 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{3} - 2 q^{5} - 3 q^{7} - q^{9} + 3 q^{11} - 5 q^{13} + 4 q^{15} - 10 q^{17} + 4 q^{19} - 7 q^{21} + 5 q^{23} - 9 q^{25} + 2 q^{27} + 5 q^{29} + 5 q^{31} + 3 q^{33} - 15 q^{37} - 2 q^{39} - q^{41} + 16 q^{43} - 10 q^{45} - 7 q^{47} - 12 q^{49} + 3 q^{51} - q^{53} + q^{55} + 4 q^{57} - 5 q^{59} - 21 q^{61} - 4 q^{63} + 2 q^{65} + 29 q^{67} - 20 q^{69} + 6 q^{71} - 28 q^{73} - 7 q^{75} - 26 q^{77} + 4 q^{79} - 23 q^{81} - 8 q^{83} - 17 q^{85} + 2 q^{87} - 13 q^{89} + 3 q^{91} - 11 q^{93} - 5 q^{95} - 12 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 4\nu^{2} + \nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{4} + 6\beta_{3} + \beta_{2} + 3\beta _1 + 14 \) 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
−1.25464
2.54970
0.752046
−1.81853
−0.228573
0 −1.67244 0 −2.46948 0 −0.171230 0 −0.202960 0
1.2 0 −1.26799 0 −0.875787 0 0.951283 0 −1.39220 0
1.3 0 0.818715 0 2.14842 0 −3.18647 0 −2.32971 0
1.4 0 1.59684 0 1.04695 0 2.12560 0 −0.450107 0
1.5 0 2.52487 0 −1.85010 0 −2.71918 0 3.37497 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.p 5
4.b odd 2 1 1508.2.a.b 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1508.2.a.b 5 4.b odd 2 1
6032.2.a.p 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):

\( T_{3}^{5} - 2T_{3}^{4} - 5T_{3}^{3} + 8T_{3}^{2} + 6T_{3} - 7 \) Copy content Toggle raw display
\( T_{5}^{5} + 2T_{5}^{4} - 6T_{5}^{3} - 11T_{5}^{2} + 6T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 2 T^{4} - 5 T^{3} + 8 T^{2} + \cdots - 7 \) Copy content Toggle raw display
$5$ \( T^{5} + 2 T^{4} - 6 T^{3} - 11 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{5} + 3 T^{4} - 7 T^{3} - 16 T^{2} + \cdots + 3 \) Copy content Toggle raw display
$11$ \( T^{5} - 3 T^{4} - 16 T^{3} + 51 T^{2} + \cdots - 67 \) Copy content Toggle raw display
$13$ \( (T + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} + 10 T^{4} + 8 T^{3} - 75 T^{2} + \cdots - 29 \) Copy content Toggle raw display
$19$ \( T^{5} - 4 T^{4} - 27 T^{3} + 103 T^{2} + \cdots - 633 \) Copy content Toggle raw display
$23$ \( T^{5} - 5 T^{4} - 47 T^{3} + 345 T^{2} + \cdots - 27 \) Copy content Toggle raw display
$29$ \( (T - 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} - 5 T^{4} - 55 T^{3} + 39 T^{2} + \cdots + 127 \) Copy content Toggle raw display
$37$ \( T^{5} + 15 T^{4} + 8 T^{3} + \cdots - 4493 \) Copy content Toggle raw display
$41$ \( T^{5} + T^{4} - 123 T^{3} - 44 T^{2} + \cdots - 337 \) Copy content Toggle raw display
$43$ \( T^{5} - 16 T^{4} + 45 T^{3} + \cdots + 2319 \) Copy content Toggle raw display
$47$ \( T^{5} + 7 T^{4} - 25 T^{3} - 209 T^{2} + \cdots + 51 \) Copy content Toggle raw display
$53$ \( T^{5} + T^{4} - 192 T^{3} + \cdots + 18559 \) Copy content Toggle raw display
$59$ \( T^{5} + 5 T^{4} - 150 T^{3} + \cdots + 333 \) Copy content Toggle raw display
$61$ \( T^{5} + 21 T^{4} - 7 T^{3} + \cdots + 9509 \) Copy content Toggle raw display
$67$ \( T^{5} - 29 T^{4} + 205 T^{3} + \cdots - 6421 \) Copy content Toggle raw display
$71$ \( T^{5} - 6 T^{4} - 149 T^{3} + \cdots + 479 \) Copy content Toggle raw display
$73$ \( T^{5} + 28 T^{4} + 228 T^{3} + \cdots - 3489 \) Copy content Toggle raw display
$79$ \( T^{5} - 4 T^{4} - 87 T^{3} + \cdots - 7603 \) Copy content Toggle raw display
$83$ \( T^{5} + 8 T^{4} - 329 T^{3} + \cdots - 51399 \) Copy content Toggle raw display
$89$ \( T^{5} + 13 T^{4} - 30 T^{3} + \cdots + 697 \) Copy content Toggle raw display
$97$ \( T^{5} + 12 T^{4} - 217 T^{3} + \cdots - 39099 \) Copy content Toggle raw display
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