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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [720,1,Mod(79,720)] 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("720.79"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 720.bu (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: \(0.359326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.10497600.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{3} - \zeta_{12}^{4} q^{5} + (\zeta_{12}^{3} + \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9} + \zeta_{12}^{5} q^{15} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{21} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{23} + \cdots - q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 2 q^{9} - 2 q^{25} + 2 q^{29} - 2 q^{41} + 4 q^{45} - 4 q^{49} - 2 q^{61} - 6 q^{69} - 2 q^{81} - 4 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{12}^{4}\)

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
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0 −0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0.866025 − 1.50000i 0 0.500000 − 0.866025i 0
79.2 0 0.866025 − 0.500000i 0 0.500000 + 0.866025i 0 −0.866025 + 1.50000i 0 0.500000 − 0.866025i 0
319.1 0 −0.866025 − 0.500000i 0 0.500000 − 0.866025i 0 0.866025 + 1.50000i 0 0.500000 + 0.866025i 0
319.2 0 0.866025 + 0.500000i 0 0.500000 − 0.866025i 0 −0.866025 − 1.50000i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
45.j even 6 1 inner
180.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.1.bu.a ✓ 4
3.b odd 2 1 2160.1.bu.a 4
4.b odd 2 1 inner 720.1.bu.a ✓ 4
5.b even 2 1 inner 720.1.bu.a ✓ 4
5.c odd 4 1 3600.1.cc.a 2
5.c odd 4 1 3600.1.cc.b 2
8.b even 2 1 2880.1.bu.c 4
8.d odd 2 1 2880.1.bu.c 4
9.c even 3 1 inner 720.1.bu.a ✓ 4
9.d odd 6 1 2160.1.bu.a 4
12.b even 2 1 2160.1.bu.a 4
15.d odd 2 1 2160.1.bu.a 4
20.d odd 2 1 CM 720.1.bu.a ✓ 4
20.e even 4 1 3600.1.cc.a 2
20.e even 4 1 3600.1.cc.b 2
36.f odd 6 1 inner 720.1.bu.a ✓ 4
36.h even 6 1 2160.1.bu.a 4
40.e odd 2 1 2880.1.bu.c 4
40.f even 2 1 2880.1.bu.c 4
45.h odd 6 1 2160.1.bu.a 4
45.j even 6 1 inner 720.1.bu.a ✓ 4
45.k odd 12 1 3600.1.cc.a 2
45.k odd 12 1 3600.1.cc.b 2
60.h even 2 1 2160.1.bu.a 4
72.n even 6 1 2880.1.bu.c 4
72.p odd 6 1 2880.1.bu.c 4
180.n even 6 1 2160.1.bu.a 4
180.p odd 6 1 inner 720.1.bu.a ✓ 4
180.x even 12 1 3600.1.cc.a 2
180.x even 12 1 3600.1.cc.b 2
360.z odd 6 1 2880.1.bu.c 4
360.bk even 6 1 2880.1.bu.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.bu.a ✓ 4 1.a even 1 1 trivial
720.1.bu.a ✓ 4 4.b odd 2 1 inner
720.1.bu.a ✓ 4 5.b even 2 1 inner
720.1.bu.a ✓ 4 9.c even 3 1 inner
720.1.bu.a ✓ 4 20.d odd 2 1 CM
720.1.bu.a ✓ 4 36.f odd 6 1 inner
720.1.bu.a ✓ 4 45.j even 6 1 inner
720.1.bu.a ✓ 4 180.p odd 6 1 inner
2160.1.bu.a 4 3.b odd 2 1
2160.1.bu.a 4 9.d odd 6 1
2160.1.bu.a 4 12.b even 2 1
2160.1.bu.a 4 15.d odd 2 1
2160.1.bu.a 4 36.h even 6 1
2160.1.bu.a 4 45.h odd 6 1
2160.1.bu.a 4 60.h even 2 1
2160.1.bu.a 4 180.n even 6 1
2880.1.bu.c 4 8.b even 2 1
2880.1.bu.c 4 8.d odd 2 1
2880.1.bu.c 4 40.e odd 2 1
2880.1.bu.c 4 40.f even 2 1
2880.1.bu.c 4 72.n even 6 1
2880.1.bu.c 4 72.p odd 6 1
2880.1.bu.c 4 360.z odd 6 1
2880.1.bu.c 4 360.bk even 6 1
3600.1.cc.a 2 5.c odd 4 1
3600.1.cc.a 2 20.e even 4 1
3600.1.cc.a 2 45.k odd 12 1
3600.1.cc.a 2 180.x even 12 1
3600.1.cc.b 2 5.c odd 4 1
3600.1.cc.b 2 20.e even 4 1
3600.1.cc.b 2 45.k odd 12 1
3600.1.cc.b 2 180.x even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(720, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$89$ \( (T + 1)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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