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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,2,Mod(293,342)] 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("342.293"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.764411904.5
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{4} - \beta_1) q^{3} + q^{4} + ( - \beta_{3} + \beta_{2} + 1) q^{5} + (\beta_{4} - \beta_1) q^{6} + ( - \beta_{7} - \beta_{6} - 2 \beta_{3} + \cdots - 2) q^{7} + q^{8} + (\beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 1) q^{9}+ \cdots + ( - 3 \beta_{7} - \beta_{6} + 4 \beta_{5} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 12 q^{5} - 8 q^{7} + 8 q^{8} + 12 q^{10} - 8 q^{14} + 8 q^{16} - 12 q^{17} + 8 q^{19} + 12 q^{20} - 12 q^{21} - 8 q^{28} - 12 q^{29} + 24 q^{31} + 8 q^{32} - 12 q^{33} - 12 q^{34}+ \cdots + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 9\nu^{4} - 3\nu^{2} + 18 ) / 45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 3\nu^{4} - 6\nu^{2} - 9 ) / 45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} - 6\nu^{5} + 57\nu^{3} - 27\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 4\nu^{4} + 18\nu^{2} - 33 ) / 15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 6\nu^{5} - 21\nu^{3} + 27\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 3\nu^{5} - 6\nu^{3} - 9\nu ) / 45 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 3\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{3} + 6\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{7} + 6\beta_{6} + 12\beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{5} - 30\beta_{3} + 6\beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -24\beta_{7} + 9\beta_{6} + 9\beta_{4} - 12\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(-\beta_{3}\) \(1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
1.27970 + 1.16721i
1.69185 − 0.370982i
−1.69185 + 0.370982i
−1.27970 − 1.16721i
1.27970 − 1.16721i
1.69185 + 0.370982i
−1.69185 − 0.370982i
−1.27970 + 1.16721i
1.00000 −1.65068 − 0.524648i 1.00000 0.275255 − 0.158919i −1.65068 − 0.524648i −0.629018 − 1.08949i 1.00000 2.44949 + 1.73205i 0.275255 − 0.158919i
293.2 1.00000 −0.524648 − 1.65068i 1.00000 2.72474 − 1.57313i −0.524648 − 1.65068i −2.16721 − 3.75371i 1.00000 −2.44949 + 1.73205i 2.72474 − 1.57313i
293.3 1.00000 0.524648 + 1.65068i 1.00000 2.72474 − 1.57313i 0.524648 + 1.65068i 0.167207 + 0.289611i 1.00000 −2.44949 + 1.73205i 2.72474 − 1.57313i
293.4 1.00000 1.65068 + 0.524648i 1.00000 0.275255 − 0.158919i 1.65068 + 0.524648i −1.37098 − 2.37461i 1.00000 2.44949 + 1.73205i 0.275255 − 0.158919i
335.1 1.00000 −1.65068 + 0.524648i 1.00000 0.275255 + 0.158919i −1.65068 + 0.524648i −0.629018 + 1.08949i 1.00000 2.44949 − 1.73205i 0.275255 + 0.158919i
335.2 1.00000 −0.524648 + 1.65068i 1.00000 2.72474 + 1.57313i −0.524648 + 1.65068i −2.16721 + 3.75371i 1.00000 −2.44949 − 1.73205i 2.72474 + 1.57313i
335.3 1.00000 0.524648 − 1.65068i 1.00000 2.72474 + 1.57313i 0.524648 − 1.65068i 0.167207 − 0.289611i 1.00000 −2.44949 − 1.73205i 2.72474 + 1.57313i
335.4 1.00000 1.65068 − 0.524648i 1.00000 0.275255 + 0.158919i 1.65068 − 0.524648i −1.37098 + 2.37461i 1.00000 2.44949 − 1.73205i 0.275255 + 0.158919i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 293.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.n.d yes 8
3.b odd 2 1 1026.2.n.d 8
9.c even 3 1 1026.2.j.d 8
9.d odd 6 1 342.2.j.d ✓ 8
19.d odd 6 1 342.2.j.d ✓ 8
57.f even 6 1 1026.2.j.d 8
171.i odd 6 1 1026.2.n.d 8
171.t even 6 1 inner 342.2.n.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.j.d ✓ 8 9.d odd 6 1
342.2.j.d ✓ 8 19.d odd 6 1
342.2.n.d yes 8 1.a even 1 1 trivial
342.2.n.d yes 8 171.t even 6 1 inner
1026.2.j.d 8 9.c even 3 1
1026.2.j.d 8 57.f even 6 1
1026.2.n.d 8 3.b odd 2 1
1026.2.n.d 8 171.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 6T_{5}^{3} + 13T_{5}^{2} - 6T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 6T^{4} + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 6 T^{3} + 13 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{8} - 22 T^{6} + \cdots + 361 \) Copy content Toggle raw display
$13$ \( (T^{4} + 36 T^{2} + 300)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 12 T^{7} + \cdots + 36481 \) Copy content Toggle raw display
$19$ \( T^{8} - 8 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} + 80 T^{6} + \cdots + 1600 \) Copy content Toggle raw display
$29$ \( T^{8} + 12 T^{7} + \cdots + 2954961 \) Copy content Toggle raw display
$31$ \( T^{8} - 24 T^{7} + \cdots + 13995081 \) Copy content Toggle raw display
$37$ \( T^{8} + 312 T^{6} + \cdots + 19184400 \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} + \cdots + 91809 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + 12 T^{2} + \cdots - 20)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 24 T^{7} + \cdots + 44521 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + \cdots + 5377761 \) Copy content Toggle raw display
$59$ \( T^{8} + 66 T^{6} + \cdots + 81225 \) Copy content Toggle raw display
$61$ \( T^{8} - 4 T^{7} + \cdots + 744769 \) Copy content Toggle raw display
$67$ \( T^{8} + 192 T^{6} + \cdots + 230400 \) Copy content Toggle raw display
$71$ \( T^{8} + 24 T^{7} + \cdots + 1320201 \) Copy content Toggle raw display
$73$ \( T^{8} - 4 T^{7} + \cdots + 10336225 \) Copy content Toggle raw display
$79$ \( T^{8} + 264 T^{6} + \cdots + 32400 \) Copy content Toggle raw display
$83$ \( T^{8} + 24 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$89$ \( T^{8} + 12 T^{7} + \cdots + 2331729 \) Copy content Toggle raw display
$97$ \( T^{8} + 264 T^{6} + \cdots + 2396304 \) Copy content Toggle raw display
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