↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3042,2,Mod(1351,3042)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,0,0,0,0,12,0,0,0,0,0,4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 234)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} + 3 \beta_1 q^{5} + \beta_1 q^{8} + 3 q^{10} + q^{16} + 3 \beta_{3} q^{17} - 4 \beta_{2} q^{19} - 3 \beta_1 q^{20} - 4 q^{25} + 3 \beta_{3} q^{29} + 4 \beta_{2} q^{31} - \beta_1 q^{32}+ \cdots - 7 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 12 q^{10} + 4 q^{16} - 16 q^{25} - 12 q^{40} - 16 q^{43} + 28 q^{49} - 20 q^{61} - 4 q^{64} + 16 q^{79} + 36 q^{82} - 48 q^{94}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 3.00000i 0 0 1.00000i 0 3.00000
1351.2 − 1.00000i 0 −1.00000 3.00000i 0 0 1.00000i 0 3.00000
1351.3 1.00000i 0 −1.00000 − 3.00000i 0 0 − 1.00000i 0 3.00000
1351.4 1.00000i 0 −1.00000 − 3.00000i 0 0 − 1.00000i 0 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.b even 2 1 inner
39.d odd 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.m 4
3.b odd 2 1 inner 3042.2.b.m 4
13.b even 2 1 inner 3042.2.b.m 4
13.c even 3 1 234.2.l.b ✓ 4
13.d odd 4 1 3042.2.a.t 2
13.d odd 4 1 3042.2.a.u 2
13.e even 6 1 234.2.l.b ✓ 4
39.d odd 2 1 inner 3042.2.b.m 4
39.f even 4 1 3042.2.a.t 2
39.f even 4 1 3042.2.a.u 2
39.h odd 6 1 234.2.l.b ✓ 4
39.i odd 6 1 234.2.l.b ✓ 4
52.i odd 6 1 1872.2.by.i 4
52.j odd 6 1 1872.2.by.i 4
156.p even 6 1 1872.2.by.i 4
156.r even 6 1 1872.2.by.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.2.l.b ✓ 4 13.c even 3 1
234.2.l.b ✓ 4 13.e even 6 1
234.2.l.b ✓ 4 39.h odd 6 1
234.2.l.b ✓ 4 39.i odd 6 1
1872.2.by.i 4 52.i odd 6 1
1872.2.by.i 4 52.j odd 6 1
1872.2.by.i 4 156.p even 6 1
1872.2.by.i 4 156.r even 6 1
3042.2.a.t 2 13.d odd 4 1
3042.2.a.t 2 39.f even 4 1
3042.2.a.u 2 13.d odd 4 1
3042.2.a.u 2 39.f even 4 1
3042.2.b.m 4 1.a even 1 1 trivial
3042.2.b.m 4 3.b odd 2 1 inner
3042.2.b.m 4 13.b even 2 1 inner
3042.2.b.m 4 39.d odd 2 1 inner

Hecke kernels

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

\( T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{17}^{2} - 27 \) Copy content Toggle raw display
\( T_{23} \) 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
$5$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T + 5)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
show more
show less