Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,3,Mod(5,18)] 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("18.5"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 18.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.490464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + (3 \beta_{2} - 6) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{6} + ( - 6 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{7}+ \cdots + (36 \beta_{3} - 27 \beta_{2} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 18 q^{5} - 12 q^{6} + 2 q^{7} + 12 q^{9} + 18 q^{11} + 12 q^{12} - 10 q^{13} + 36 q^{14} + 18 q^{15} - 8 q^{16} - 24 q^{18} - 40 q^{19} - 36 q^{20} - 42 q^{21} - 12 q^{22} + 18 q^{23} + 4 q^{25}+ \cdots + 126 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−1.22474 + 0.707107i
1.22474 − 0.707107i
−1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 2.44949 + 1.73205i 1.00000 − 1.73205i −4.50000 − 2.59808i −4.22474 − 0.389270i −3.17423 − 5.49794i 2.82843i 3.00000 + 8.48528i 7.34847
5.2 1.22474 − 0.707107i −2.44949 + 1.73205i 1.00000 − 1.73205i −4.50000 − 2.59808i −1.77526 + 3.85337i 4.17423 + 7.22999i − 2.82843i 3.00000 − 8.48528i −7.34847
11.1 −1.22474 − 0.707107i 2.44949 − 1.73205i 1.00000 + 1.73205i −4.50000 + 2.59808i −4.22474 + 0.389270i −3.17423 + 5.49794i − 2.82843i 3.00000 − 8.48528i 7.34847
11.2 1.22474 + 0.707107i −2.44949 − 1.73205i 1.00000 + 1.73205i −4.50000 + 2.59808i −1.77526 − 3.85337i 4.17423 − 7.22999i 2.82843i 3.00000 + 8.48528i −7.34847
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.3.d.a ✓ 4
3.b odd 2 1 54.3.d.a 4
4.b odd 2 1 144.3.q.c 4
5.b even 2 1 450.3.i.b 4
5.c odd 4 2 450.3.k.a 8
8.b even 2 1 576.3.q.f 4
8.d odd 2 1 576.3.q.e 4
9.c even 3 1 54.3.d.a 4
9.c even 3 1 162.3.b.a 4
9.d odd 6 1 inner 18.3.d.a ✓ 4
9.d odd 6 1 162.3.b.a 4
12.b even 2 1 432.3.q.d 4
15.d odd 2 1 1350.3.i.b 4
15.e even 4 2 1350.3.k.a 8
24.f even 2 1 1728.3.q.c 4
24.h odd 2 1 1728.3.q.d 4
36.f odd 6 1 432.3.q.d 4
36.f odd 6 1 1296.3.e.g 4
36.h even 6 1 144.3.q.c 4
36.h even 6 1 1296.3.e.g 4
45.h odd 6 1 450.3.i.b 4
45.j even 6 1 1350.3.i.b 4
45.k odd 12 2 1350.3.k.a 8
45.l even 12 2 450.3.k.a 8
72.j odd 6 1 576.3.q.f 4
72.l even 6 1 576.3.q.e 4
72.n even 6 1 1728.3.q.d 4
72.p odd 6 1 1728.3.q.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a ✓ 4 1.a even 1 1 trivial
18.3.d.a ✓ 4 9.d odd 6 1 inner
54.3.d.a 4 3.b odd 2 1
54.3.d.a 4 9.c even 3 1
144.3.q.c 4 4.b odd 2 1
144.3.q.c 4 36.h even 6 1
162.3.b.a 4 9.c even 3 1
162.3.b.a 4 9.d odd 6 1
432.3.q.d 4 12.b even 2 1
432.3.q.d 4 36.f odd 6 1
450.3.i.b 4 5.b even 2 1
450.3.i.b 4 45.h odd 6 1
450.3.k.a 8 5.c odd 4 2
450.3.k.a 8 45.l even 12 2
576.3.q.e 4 8.d odd 2 1
576.3.q.e 4 72.l even 6 1
576.3.q.f 4 8.b even 2 1
576.3.q.f 4 72.j odd 6 1
1296.3.e.g 4 36.f odd 6 1
1296.3.e.g 4 36.h even 6 1
1350.3.i.b 4 15.d odd 2 1
1350.3.i.b 4 45.j even 6 1
1350.3.k.a 8 15.e even 4 2
1350.3.k.a 8 45.k odd 12 2
1728.3.q.c 4 24.f even 2 1
1728.3.q.c 4 72.p odd 6 1
1728.3.q.d 4 24.h odd 2 1
1728.3.q.d 4 72.n even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(18, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} - 6T^{2} + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + \cdots + 2809 \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} + 10 T^{3} + \cdots + 36481 \) Copy content Toggle raw display
$17$ \( T^{4} + 360T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} + 20 T - 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 18 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{4} - 18 T^{3} + \cdots + 2025 \) Copy content Toggle raw display
$31$ \( T^{4} - 38 T^{3} + \cdots + 15625 \) Copy content Toggle raw display
$37$ \( (T^{2} - 64 T + 808)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 126 T^{3} + \cdots + 455625 \) Copy content Toggle raw display
$43$ \( T^{4} + 46 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( T^{4} - 54 T^{3} + \cdots + 408321 \) Copy content Toggle raw display
$53$ \( T^{4} + 9000 T^{2} + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} - 126 T^{3} + \cdots + 2954961 \) Copy content Toggle raw display
$61$ \( T^{4} - 62 T^{3} + \cdots + 966289 \) Copy content Toggle raw display
$67$ \( T^{4} + 106 T^{3} + \cdots + 5396329 \) Copy content Toggle raw display
$71$ \( T^{4} + 7704 T^{2} + 2396304 \) Copy content Toggle raw display
$73$ \( (T^{2} + 104 T + 760)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 14 T^{3} + \cdots + 1692601 \) Copy content Toggle raw display
$83$ \( T^{4} + 378 T^{3} + \cdots + 131262849 \) Copy content Toggle raw display
$89$ \( T^{4} + 22824 T^{2} + 36144144 \) Copy content Toggle raw display
$97$ \( T^{4} - 14 T^{3} + \cdots + 110986225 \) Copy content Toggle raw display
show more
show less