Properties

Label 6084.2.a.y
Level $6084$
Weight $2$
Character orbit 6084.a
Self dual yes
Analytic conductor $48.581$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6084,2,Mod(1,6084)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6084, 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("6084.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6084.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: \(48.5809845897\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2028)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{5} + (\beta_{2} - 2 \beta_1 - 1) q^{7} + (3 \beta_{2} - \beta_1 + 3) q^{11} + ( - \beta_1 - 4) q^{17} + (2 \beta_{2} + 3 \beta_1) q^{19} + (5 \beta_{2} - 4 \beta_1 + 3) q^{23}+ \cdots + (6 \beta_{2} - 4 \beta_1 - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{5} - 6 q^{7} + 5 q^{11} - 13 q^{17} + q^{19} - 9 q^{25} + 4 q^{29} - 2 q^{31} + 4 q^{35} + 2 q^{37} + 11 q^{41} - 9 q^{43} + 19 q^{47} + 5 q^{49} - 5 q^{53} - 15 q^{55} + 10 q^{59} + 7 q^{61}+ \cdots - 13 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
−1.24698
0.445042
0 0 0 −2.24698 0 −3.35690 0 0 0
1.2 0 0 0 −0.554958 0 1.04892 0 0 0
1.3 0 0 0 0.801938 0 −3.69202 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(13\) \( -1 \)

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 6084.2.a.y 3
3.b odd 2 1 2028.2.a.j yes 3
12.b even 2 1 8112.2.a.co 3
13.b even 2 1 6084.2.a.bb 3
13.d odd 4 2 6084.2.b.r 6
39.d odd 2 1 2028.2.a.i 3
39.f even 4 2 2028.2.b.f 6
39.h odd 6 2 2028.2.i.l 6
39.i odd 6 2 2028.2.i.m 6
39.k even 12 4 2028.2.q.j 12
156.h even 2 1 8112.2.a.ch 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2028.2.a.i 3 39.d odd 2 1
2028.2.a.j yes 3 3.b odd 2 1
2028.2.b.f 6 39.f even 4 2
2028.2.i.l 6 39.h odd 6 2
2028.2.i.m 6 39.i odd 6 2
2028.2.q.j 12 39.k even 12 4
6084.2.a.y 3 1.a even 1 1 trivial
6084.2.a.bb 3 13.b even 2 1
6084.2.b.r 6 13.d odd 4 2
8112.2.a.ch 3 156.h even 2 1
8112.2.a.co 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6084))\):

\( T_{5}^{3} + 2T_{5}^{2} - T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{3} + 6T_{7}^{2} + 5T_{7} - 13 \) Copy content Toggle raw display
\( T_{11}^{3} - 5T_{11}^{2} - 8T_{11} + 41 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 2T^{2} - T - 1 \) Copy content Toggle raw display
$7$ \( T^{3} + 6 T^{2} + \cdots - 13 \) Copy content Toggle raw display
$11$ \( T^{3} - 5 T^{2} + \cdots + 41 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 13 T^{2} + \cdots + 71 \) Copy content Toggle raw display
$19$ \( T^{3} - T^{2} + \cdots - 83 \) Copy content Toggle raw display
$23$ \( T^{3} - 49T + 91 \) Copy content Toggle raw display
$29$ \( T^{3} - 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} + \cdots + 13 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} + \cdots - 13 \) Copy content Toggle raw display
$41$ \( T^{3} - 11 T^{2} + \cdots + 211 \) Copy content Toggle raw display
$43$ \( T^{3} + 9 T^{2} + \cdots - 757 \) Copy content Toggle raw display
$47$ \( T^{3} - 19 T^{2} + \cdots + 601 \) Copy content Toggle raw display
$53$ \( T^{3} + 5 T^{2} + \cdots - 419 \) Copy content Toggle raw display
$59$ \( T^{3} - 10 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$61$ \( T^{3} - 7 T^{2} + \cdots + 1673 \) Copy content Toggle raw display
$67$ \( T^{3} + 15 T^{2} + \cdots + 97 \) Copy content Toggle raw display
$71$ \( T^{3} - 36 T^{2} + \cdots - 1637 \) Copy content Toggle raw display
$73$ \( T^{3} + 2 T^{2} + \cdots - 281 \) Copy content Toggle raw display
$79$ \( T^{3} - T^{2} + \cdots - 503 \) Copy content Toggle raw display
$83$ \( T^{3} - 28 T^{2} + \cdots - 287 \) Copy content Toggle raw display
$89$ \( T^{3} - 19 T^{2} + \cdots + 1231 \) Copy content Toggle raw display
$97$ \( T^{3} + 13 T^{2} + \cdots - 13 \) Copy content Toggle raw display
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