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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,2,Mod(28,171)] 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("171.28"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{9}]$

$q$-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 primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{18}^{2} q^{4} + ( - 2 \zeta_{18}^{5} + \cdots + 3 \zeta_{18}) q^{7} + ( - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + \cdots + 4) q^{13} + 4 \zeta_{18}^{4} q^{16} + (5 \zeta_{18}^{3} - 3) q^{19} + \cdots + 19 \zeta_{18} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 15 q^{13} - 3 q^{19} - 30 q^{28} - 39 q^{43} - 21 q^{49} - 12 q^{52} + 42 q^{61} + 24 q^{64} + 33 q^{67} + 51 q^{73} - 12 q^{79} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(\zeta_{18}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
−0.173648 − 0.984808i
−0.173648 + 0.984808i
−0.766044 − 0.642788i
−0.766044 + 0.642788i
0.939693 − 0.342020i
0.939693 + 0.342020i
0 0 1.87939 − 0.684040i 0 0 0.418748 − 0.725293i 0 0 0
55.1 0 0 1.87939 + 0.684040i 0 0 0.418748 + 0.725293i 0 0 0
73.1 0 0 −0.347296 − 1.96962i 0 0 −2.47178 − 4.28125i 0 0 0
82.1 0 0 −0.347296 + 1.96962i 0 0 −2.47178 + 4.28125i 0 0 0
100.1 0 0 −1.53209 + 1.28558i 0 0 2.05303 + 3.55596i 0 0 0
118.1 0 0 −1.53209 − 1.28558i 0 0 2.05303 − 3.55596i 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 28.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.e even 9 1 inner
57.l odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.u.b ✓ 6
3.b odd 2 1 CM 171.2.u.b ✓ 6
19.e even 9 1 inner 171.2.u.b ✓ 6
19.e even 9 1 3249.2.a.u 3
19.f odd 18 1 3249.2.a.v 3
57.j even 18 1 3249.2.a.v 3
57.l odd 18 1 inner 171.2.u.b ✓ 6
57.l odd 18 1 3249.2.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.u.b ✓ 6 1.a even 1 1 trivial
171.2.u.b ✓ 6 3.b odd 2 1 CM
171.2.u.b ✓ 6 19.e even 9 1 inner
171.2.u.b ✓ 6 57.l odd 18 1 inner
3249.2.a.u 3 19.e even 9 1
3249.2.a.u 3 57.l odd 18 1
3249.2.a.v 3 19.f odd 18 1
3249.2.a.v 3 57.j even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 21 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 15 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 19)^{3} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + 93 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( (T^{3} - 111 T - 323)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 39 T^{5} + \cdots + 5041 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} - 42 T^{5} + \cdots + 516961 \) Copy content Toggle raw display
$67$ \( T^{6} - 33 T^{5} + \cdots + 1014049 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} - 51 T^{5} + \cdots + 73441 \) Copy content Toggle raw display
$79$ \( T^{6} + 12 T^{5} + \cdots + 1923769 \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} - 6859 T^{3} + 47045881 \) Copy content Toggle raw display
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