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Properties

Label 3042.2.b.l
Level $3042$
Weight $2$
Character orbit 3042.b
Analytic conductor $24.290$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3042,2,Mod(1351,3042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3042.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{2} - 3 \beta_1) q^{7} - \beta_1 q^{8} + \beta_{3} q^{10} + ( - \beta_{2} + 3 \beta_1) q^{11} + ( - \beta_{3} + 3) q^{14} + q^{16} + 3 \beta_{3} q^{17}+ \cdots + (6 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 12 q^{14} + 4 q^{16} - 12 q^{22} - 12 q^{23} + 8 q^{25} + 12 q^{29} + 12 q^{35} - 12 q^{38} - 4 q^{43} - 20 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{56} + 40 q^{61} - 24 q^{62} - 4 q^{64}+ \cdots + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
−0.866025 − 0.500000i
0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
− 1.00000i 0 −1.00000 − 1.73205i 0 4.73205i 1.00000i 0 −1.73205
1351.2 − 1.00000i 0 −1.00000 1.73205i 0 1.26795i 1.00000i 0 1.73205
1351.3 1.00000i 0 −1.00000 − 1.73205i 0 − 1.26795i − 1.00000i 0 1.73205
1351.4 1.00000i 0 −1.00000 1.73205i 0 − 4.73205i − 1.00000i 0 −1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3042.2.b.l 4
3.b odd 2 1 1014.2.b.d 4
13.b even 2 1 inner 3042.2.b.l 4
13.c even 3 1 234.2.l.a 4
13.d odd 4 1 3042.2.a.s 2
13.d odd 4 1 3042.2.a.v 2
13.e even 6 1 234.2.l.a 4
39.d odd 2 1 1014.2.b.d 4
39.f even 4 1 1014.2.a.h 2
39.f even 4 1 1014.2.a.j 2
39.h odd 6 1 78.2.i.b ✓ 4
39.h odd 6 1 1014.2.i.f 4
39.i odd 6 1 78.2.i.b ✓ 4
39.i odd 6 1 1014.2.i.f 4
39.k even 12 2 1014.2.e.h 4
39.k even 12 2 1014.2.e.j 4
52.i odd 6 1 1872.2.by.k 4
52.j odd 6 1 1872.2.by.k 4
156.l odd 4 1 8112.2.a.bq 2
156.l odd 4 1 8112.2.a.bx 2
156.p even 6 1 624.2.bv.d 4
156.r even 6 1 624.2.bv.d 4
195.x odd 6 1 1950.2.bc.c 4
195.y odd 6 1 1950.2.bc.c 4
195.bf even 12 1 1950.2.y.a 4
195.bf even 12 1 1950.2.y.h 4
195.bl even 12 1 1950.2.y.a 4
195.bl even 12 1 1950.2.y.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.b ✓ 4 39.h odd 6 1
78.2.i.b ✓ 4 39.i odd 6 1
234.2.l.a 4 13.c even 3 1
234.2.l.a 4 13.e even 6 1
624.2.bv.d 4 156.p even 6 1
624.2.bv.d 4 156.r even 6 1
1014.2.a.h 2 39.f even 4 1
1014.2.a.j 2 39.f even 4 1
1014.2.b.d 4 3.b odd 2 1
1014.2.b.d 4 39.d odd 2 1
1014.2.e.h 4 39.k even 12 2
1014.2.e.j 4 39.k even 12 2
1014.2.i.f 4 39.h odd 6 1
1014.2.i.f 4 39.i odd 6 1
1872.2.by.k 4 52.i odd 6 1
1872.2.by.k 4 52.j odd 6 1
1950.2.y.a 4 195.bf even 12 1
1950.2.y.a 4 195.bl even 12 1
1950.2.y.h 4 195.bf even 12 1
1950.2.y.h 4 195.bl even 12 1
1950.2.bc.c 4 195.x odd 6 1
1950.2.bc.c 4 195.y odd 6 1
3042.2.a.s 2 13.d odd 4 1
3042.2.a.v 2 13.d odd 4 1
3042.2.b.l 4 1.a even 1 1 trivial
3042.2.b.l 4 13.b even 2 1 inner
8112.2.a.bq 2 156.l odd 4 1
8112.2.a.bx 2 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3042, [\chi])\):

\( T_{5}^{2} + 3 \) Copy content Toggle raw display
\( T_{7}^{4} + 24T_{7}^{2} + 36 \) Copy content Toggle raw display
\( T_{17}^{2} - 27 \) Copy content Toggle raw display
\( T_{23}^{2} + 6T_{23} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$37$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$53$ \( (T + 3)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 20 T + 73)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 168T^{2} + 6084 \) Copy content Toggle raw display
$71$ \( T^{4} + 72T^{2} + 324 \) Copy content Toggle raw display
$73$ \( (T^{2} + 147)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 104)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 168T^{2} + 4356 \) Copy content Toggle raw display
$89$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$97$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
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