Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [60,3,Mod(23,60)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(60, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("60.23");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 60 = 2^{2} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 60.l (of order \(4\), degree \(2\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(1.63488158616\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −1.99497 | − | 0.141758i | −2.06649 | + | 2.17477i | 3.95981 | + | 0.565605i | −3.07600 | − | 3.94185i | 4.43087 | − | 4.04566i | −5.18766 | − | 5.18766i | −7.81952 | − | 1.68970i | −0.459255 | − | 8.98827i | 5.57774 | + | 8.29993i |
23.2 | −1.84549 | + | 0.770813i | −2.78107 | − | 1.12501i | 2.81170 | − | 2.84506i | 3.86232 | + | 3.17529i | 5.99962 | − | 0.0674770i | 4.75159 | + | 4.75159i | −2.99596 | + | 7.41783i | 6.46869 | + | 6.25748i | −9.57545 | − | 2.88285i |
23.3 | −1.81610 | − | 0.837725i | 2.69303 | + | 1.32197i | 2.59643 | + | 3.04278i | −3.21472 | + | 3.82956i | −3.78336 | − | 4.65685i | 3.54241 | + | 3.54241i | −2.16636 | − | 7.70110i | 5.50478 | + | 7.12021i | 9.04638 | − | 4.26182i |
23.4 | −1.75394 | − | 0.961083i | 0.903948 | − | 2.86057i | 2.15264 | + | 3.37137i | 4.95584 | − | 0.663068i | −4.33472 | + | 4.14852i | −7.30016 | − | 7.30016i | −0.535443 | − | 7.98206i | −7.36576 | − | 5.17162i | −9.32953 | − | 3.59999i |
23.5 | −1.68375 | + | 1.07935i | 2.99716 | + | 0.130491i | 1.67002 | − | 3.63470i | 1.65103 | − | 4.71955i | −5.18731 | + | 3.01526i | 1.91561 | + | 1.91561i | 1.11122 | + | 7.92245i | 8.96594 | + | 0.782204i | 2.31412 | + | 9.72856i |
23.6 | −1.07935 | + | 1.68375i | 0.130491 | + | 2.99716i | −1.67002 | − | 3.63470i | −1.65103 | + | 4.71955i | −5.18731 | − | 3.01526i | −1.91561 | − | 1.91561i | 7.92245 | + | 1.11122i | −8.96594 | + | 0.782204i | −6.16449 | − | 7.87394i |
23.7 | −0.961083 | − | 1.75394i | −0.903948 | + | 2.86057i | −2.15264 | + | 3.37137i | 4.95584 | − | 0.663068i | 5.88605 | − | 1.16377i | 7.30016 | + | 7.30016i | 7.98206 | + | 0.535443i | −7.36576 | − | 5.17162i | −5.92596 | − | 8.05500i |
23.8 | −0.837725 | − | 1.81610i | −2.69303 | − | 1.32197i | −2.59643 | + | 3.04278i | −3.21472 | + | 3.82956i | −0.144815 | + | 5.99825i | −3.54241 | − | 3.54241i | 7.70110 | + | 2.16636i | 5.50478 | + | 7.12021i | 9.64792 | + | 2.63013i |
23.9 | −0.770813 | + | 1.84549i | −1.12501 | − | 2.78107i | −2.81170 | − | 2.84506i | −3.86232 | − | 3.17529i | 5.99962 | + | 0.0674770i | −4.75159 | − | 4.75159i | 7.41783 | − | 2.99596i | −6.46869 | + | 6.25748i | 8.83710 | − | 4.68034i |
23.10 | −0.141758 | − | 1.99497i | 2.06649 | − | 2.17477i | −3.95981 | + | 0.565605i | −3.07600 | − | 3.94185i | −4.63154 | − | 3.81429i | 5.18766 | + | 5.18766i | 1.68970 | + | 7.81952i | −0.459255 | − | 8.98827i | −7.42783 | + | 6.69532i |
23.11 | 0.141758 | + | 1.99497i | 2.17477 | − | 2.06649i | −3.95981 | + | 0.565605i | 3.07600 | + | 3.94185i | 4.43087 | + | 4.04566i | 5.18766 | + | 5.18766i | −1.68970 | − | 7.81952i | 0.459255 | − | 8.98827i | −7.42783 | + | 6.69532i |
23.12 | 0.770813 | − | 1.84549i | 2.78107 | + | 1.12501i | −2.81170 | − | 2.84506i | 3.86232 | + | 3.17529i | 4.21989 | − | 4.26527i | −4.75159 | − | 4.75159i | −7.41783 | + | 2.99596i | 6.46869 | + | 6.25748i | 8.83710 | − | 4.68034i |
23.13 | 0.837725 | + | 1.81610i | 1.32197 | + | 2.69303i | −2.59643 | + | 3.04278i | 3.21472 | − | 3.82956i | −3.78336 | + | 4.65685i | −3.54241 | − | 3.54241i | −7.70110 | − | 2.16636i | −5.50478 | + | 7.12021i | 9.64792 | + | 2.63013i |
23.14 | 0.961083 | + | 1.75394i | −2.86057 | + | 0.903948i | −2.15264 | + | 3.37137i | −4.95584 | + | 0.663068i | −4.33472 | − | 4.14852i | 7.30016 | + | 7.30016i | −7.98206 | − | 0.535443i | 7.36576 | − | 5.17162i | −5.92596 | − | 8.05500i |
23.15 | 1.07935 | − | 1.68375i | −2.99716 | − | 0.130491i | −1.67002 | − | 3.63470i | 1.65103 | − | 4.71955i | −3.45469 | + | 4.90562i | −1.91561 | − | 1.91561i | −7.92245 | − | 1.11122i | 8.96594 | + | 0.782204i | −6.16449 | − | 7.87394i |
23.16 | 1.68375 | − | 1.07935i | −0.130491 | − | 2.99716i | 1.67002 | − | 3.63470i | −1.65103 | + | 4.71955i | −3.45469 | − | 4.90562i | 1.91561 | + | 1.91561i | −1.11122 | − | 7.92245i | −8.96594 | + | 0.782204i | 2.31412 | + | 9.72856i |
23.17 | 1.75394 | + | 0.961083i | 2.86057 | − | 0.903948i | 2.15264 | + | 3.37137i | −4.95584 | + | 0.663068i | 5.88605 | + | 1.16377i | −7.30016 | − | 7.30016i | 0.535443 | + | 7.98206i | 7.36576 | − | 5.17162i | −9.32953 | − | 3.59999i |
23.18 | 1.81610 | + | 0.837725i | −1.32197 | − | 2.69303i | 2.59643 | + | 3.04278i | 3.21472 | − | 3.82956i | −0.144815 | − | 5.99825i | 3.54241 | + | 3.54241i | 2.16636 | + | 7.70110i | −5.50478 | + | 7.12021i | 9.04638 | − | 4.26182i |
23.19 | 1.84549 | − | 0.770813i | 1.12501 | + | 2.78107i | 2.81170 | − | 2.84506i | −3.86232 | − | 3.17529i | 4.21989 | + | 4.26527i | 4.75159 | + | 4.75159i | 2.99596 | − | 7.41783i | −6.46869 | + | 6.25748i | −9.57545 | − | 2.88285i |
23.20 | 1.99497 | + | 0.141758i | −2.17477 | + | 2.06649i | 3.95981 | + | 0.565605i | 3.07600 | + | 3.94185i | −4.63154 | + | 3.81429i | −5.18766 | − | 5.18766i | 7.81952 | + | 1.68970i | 0.459255 | − | 8.98827i | 5.57774 | + | 8.29993i |
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
12.b | even | 2 | 1 | inner |
15.e | even | 4 | 1 | inner |
20.e | even | 4 | 1 | inner |
60.l | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 60.3.l.a | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 60.3.l.a | ✓ | 40 |
4.b | odd | 2 | 1 | inner | 60.3.l.a | ✓ | 40 |
5.b | even | 2 | 1 | 300.3.l.g | 40 | ||
5.c | odd | 4 | 1 | inner | 60.3.l.a | ✓ | 40 |
5.c | odd | 4 | 1 | 300.3.l.g | 40 | ||
12.b | even | 2 | 1 | inner | 60.3.l.a | ✓ | 40 |
15.d | odd | 2 | 1 | 300.3.l.g | 40 | ||
15.e | even | 4 | 1 | inner | 60.3.l.a | ✓ | 40 |
15.e | even | 4 | 1 | 300.3.l.g | 40 | ||
20.d | odd | 2 | 1 | 300.3.l.g | 40 | ||
20.e | even | 4 | 1 | inner | 60.3.l.a | ✓ | 40 |
20.e | even | 4 | 1 | 300.3.l.g | 40 | ||
60.h | even | 2 | 1 | 300.3.l.g | 40 | ||
60.l | odd | 4 | 1 | inner | 60.3.l.a | ✓ | 40 |
60.l | odd | 4 | 1 | 300.3.l.g | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
60.3.l.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
60.3.l.a | ✓ | 40 | 3.b | odd | 2 | 1 | inner |
60.3.l.a | ✓ | 40 | 4.b | odd | 2 | 1 | inner |
60.3.l.a | ✓ | 40 | 5.c | odd | 4 | 1 | inner |
60.3.l.a | ✓ | 40 | 12.b | even | 2 | 1 | inner |
60.3.l.a | ✓ | 40 | 15.e | even | 4 | 1 | inner |
60.3.l.a | ✓ | 40 | 20.e | even | 4 | 1 | inner |
60.3.l.a | ✓ | 40 | 60.l | odd | 4 | 1 | inner |
300.3.l.g | 40 | 5.b | even | 2 | 1 | ||
300.3.l.g | 40 | 5.c | odd | 4 | 1 | ||
300.3.l.g | 40 | 15.d | odd | 2 | 1 | ||
300.3.l.g | 40 | 15.e | even | 4 | 1 | ||
300.3.l.g | 40 | 20.d | odd | 2 | 1 | ||
300.3.l.g | 40 | 20.e | even | 4 | 1 | ||
300.3.l.g | 40 | 60.h | even | 2 | 1 | ||
300.3.l.g | 40 | 60.l | odd | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(60, [\chi])\).