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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3920,1,Mod(79,3920)] 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("3920.79"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3920, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 3, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3920.bt (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,1,0,0,0,1,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{5})\)
Artin image: $C_3\times D_4$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{5} + \zeta_{6} q^{9} + \zeta_{6}^{2} q^{25} + 2 q^{29} - 2 q^{41} + \zeta_{6}^{2} q^{45} + 2 \zeta_{6} q^{61} + \zeta_{6}^{2} q^{81} - 2 \zeta_{6} q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} + q^{9} - q^{25} + 4 q^{29} - 4 q^{41} - q^{45} + 2 q^{61} - q^{81} - 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0 0.500000 − 0.866025i 0 0 0 0.500000 − 0.866025i 0
1439.1 0 0 0 0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 RM by \(\Q(\sqrt{5}) \)
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
7.c even 3 1 inner
28.g odd 6 1 inner
35.j even 6 1 inner
140.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.bt.b 2
4.b odd 2 1 CM 3920.1.bt.b 2
5.b even 2 1 RM 3920.1.bt.b 2
7.b odd 2 1 3920.1.bt.a 2
7.c even 3 1 80.1.h.a ✓ 1
7.c even 3 1 inner 3920.1.bt.b 2
7.d odd 6 1 3920.1.j.a 1
7.d odd 6 1 3920.1.bt.a 2
20.d odd 2 1 CM 3920.1.bt.b 2
21.h odd 6 1 720.1.j.a 1
28.d even 2 1 3920.1.bt.a 2
28.f even 6 1 3920.1.j.a 1
28.f even 6 1 3920.1.bt.a 2
28.g odd 6 1 80.1.h.a ✓ 1
28.g odd 6 1 inner 3920.1.bt.b 2
35.c odd 2 1 3920.1.bt.a 2
35.i odd 6 1 3920.1.j.a 1
35.i odd 6 1 3920.1.bt.a 2
35.j even 6 1 80.1.h.a ✓ 1
35.j even 6 1 inner 3920.1.bt.b 2
35.l odd 12 2 400.1.b.a 1
56.k odd 6 1 320.1.h.a 1
56.p even 6 1 320.1.h.a 1
84.n even 6 1 720.1.j.a 1
105.o odd 6 1 720.1.j.a 1
105.x even 12 2 3600.1.e.a 1
112.u odd 12 2 1280.1.e.a 2
112.w even 12 2 1280.1.e.a 2
140.c even 2 1 3920.1.bt.a 2
140.p odd 6 1 80.1.h.a ✓ 1
140.p odd 6 1 inner 3920.1.bt.b 2
140.s even 6 1 3920.1.j.a 1
140.s even 6 1 3920.1.bt.a 2
140.w even 12 2 400.1.b.a 1
168.s odd 6 1 2880.1.j.a 1
168.v even 6 1 2880.1.j.a 1
280.bf even 6 1 320.1.h.a 1
280.bi odd 6 1 320.1.h.a 1
280.br even 12 2 1600.1.b.a 1
280.bt odd 12 2 1600.1.b.a 1
420.ba even 6 1 720.1.j.a 1
420.bp odd 12 2 3600.1.e.a 1
560.cr even 12 2 1280.1.e.a 2
560.cs odd 12 2 1280.1.e.a 2
840.cg odd 6 1 2880.1.j.a 1
840.cv even 6 1 2880.1.j.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.1.h.a ✓ 1 7.c even 3 1
80.1.h.a ✓ 1 28.g odd 6 1
80.1.h.a ✓ 1 35.j even 6 1
80.1.h.a ✓ 1 140.p odd 6 1
320.1.h.a 1 56.k odd 6 1
320.1.h.a 1 56.p even 6 1
320.1.h.a 1 280.bf even 6 1
320.1.h.a 1 280.bi odd 6 1
400.1.b.a 1 35.l odd 12 2
400.1.b.a 1 140.w even 12 2
720.1.j.a 1 21.h odd 6 1
720.1.j.a 1 84.n even 6 1
720.1.j.a 1 105.o odd 6 1
720.1.j.a 1 420.ba even 6 1
1280.1.e.a 2 112.u odd 12 2
1280.1.e.a 2 112.w even 12 2
1280.1.e.a 2 560.cr even 12 2
1280.1.e.a 2 560.cs odd 12 2
1600.1.b.a 1 280.br even 12 2
1600.1.b.a 1 280.bt odd 12 2
2880.1.j.a 1 168.s odd 6 1
2880.1.j.a 1 168.v even 6 1
2880.1.j.a 1 840.cg odd 6 1
2880.1.j.a 1 840.cv even 6 1
3600.1.e.a 1 105.x even 12 2
3600.1.e.a 1 420.bp odd 12 2
3920.1.j.a 1 7.d odd 6 1
3920.1.j.a 1 28.f even 6 1
3920.1.j.a 1 35.i odd 6 1
3920.1.j.a 1 140.s even 6 1
3920.1.bt.a 2 7.b odd 2 1
3920.1.bt.a 2 7.d odd 6 1
3920.1.bt.a 2 28.d even 2 1
3920.1.bt.a 2 28.f even 6 1
3920.1.bt.a 2 35.c odd 2 1
3920.1.bt.a 2 35.i odd 6 1
3920.1.bt.a 2 140.c even 2 1
3920.1.bt.a 2 140.s even 6 1
3920.1.bt.b 2 1.a even 1 1 trivial
3920.1.bt.b 2 4.b odd 2 1 CM
3920.1.bt.b 2 5.b even 2 1 RM
3920.1.bt.b 2 7.c even 3 1 inner
3920.1.bt.b 2 20.d odd 2 1 CM
3920.1.bt.b 2 28.g odd 6 1 inner
3920.1.bt.b 2 35.j even 6 1 inner
3920.1.bt.b 2 140.p odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3920, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{41} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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