Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [39,4,Mod(2,39)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("39.2");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 39 = 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 39.k (of order \(12\), degree \(4\), 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: | \(2.30107449022\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −1.37323 | + | 5.12497i | 0.625377 | + | 5.15838i | −17.4513 | − | 10.0755i | 5.80773 | + | 5.80773i | −27.2953 | − | 3.87861i | 20.4169 | − | 5.47069i | 45.5877 | − | 45.5877i | −26.2178 | + | 6.45187i | −37.7398 | + | 21.7891i |
2.2 | −1.13863 | + | 4.24944i | −3.90424 | − | 3.42883i | −9.83302 | − | 5.67709i | −8.66436 | − | 8.66436i | 19.0161 | − | 12.6867i | −1.76245 | + | 0.472246i | 10.4342 | − | 10.4342i | 3.48625 | + | 26.7740i | 46.6841 | − | 26.9531i |
2.3 | −0.908381 | + | 3.39012i | 2.68627 | − | 4.44791i | −3.73959 | − | 2.15905i | 12.1413 | + | 12.1413i | 12.6388 | + | 13.1472i | −3.81863 | + | 1.02320i | −9.13753 | + | 9.13753i | −12.5679 | − | 23.8966i | −52.1894 | + | 30.1316i |
2.4 | −0.614396 | + | 2.29296i | −3.60382 | + | 3.74332i | 2.04803 | + | 1.18243i | −2.27968 | − | 2.27968i | −6.36910 | − | 10.5633i | −25.9009 | + | 6.94012i | −17.3981 | + | 17.3981i | −1.02489 | − | 26.9805i | 6.62784 | − | 3.82659i |
2.5 | −0.532104 | + | 1.98584i | 4.60622 | + | 2.40472i | 3.26777 | + | 1.88665i | −4.41602 | − | 4.41602i | −7.22639 | + | 7.86767i | 3.85828 | − | 1.03382i | −17.1153 | + | 17.1153i | 15.4346 | + | 22.1534i | 11.1193 | − | 6.41973i |
2.6 | −0.0560169 | + | 0.209058i | −5.01878 | − | 1.34604i | 6.88764 | + | 3.97658i | 10.0436 | + | 10.0436i | 0.562537 | − | 0.973816i | 17.5010 | − | 4.68938i | −2.44149 | + | 2.44149i | 23.3764 | + | 13.5109i | −2.66232 | + | 1.53709i |
2.7 | 0.0560169 | − | 0.209058i | 1.34369 | − | 5.01941i | 6.88764 | + | 3.97658i | −10.0436 | − | 10.0436i | −0.974079 | − | 0.562081i | 17.5010 | − | 4.68938i | 2.44149 | − | 2.44149i | −23.3890 | − | 13.4891i | −2.66232 | + | 1.53709i |
2.8 | 0.532104 | − | 1.98584i | −0.220561 | + | 5.19147i | 3.26777 | + | 1.88665i | 4.41602 | + | 4.41602i | 10.1921 | + | 3.20040i | 3.85828 | − | 1.03382i | 17.1153 | − | 17.1153i | −26.9027 | − | 2.29007i | 11.1193 | − | 6.41973i |
2.9 | 0.614396 | − | 2.29296i | 5.04372 | − | 1.24934i | 2.04803 | + | 1.18243i | 2.27968 | + | 2.27968i | 0.234152 | − | 12.3326i | −25.9009 | + | 6.94012i | 17.3981 | − | 17.3981i | 23.8783 | − | 12.6027i | 6.62784 | − | 3.82659i |
2.10 | 0.908381 | − | 3.39012i | −5.19514 | + | 0.102423i | −3.73959 | − | 2.15905i | −12.1413 | − | 12.1413i | −4.37194 | + | 17.7052i | −3.81863 | + | 1.02320i | 9.13753 | − | 9.13753i | 26.9790 | − | 1.06421i | −52.1894 | + | 30.1316i |
2.11 | 1.13863 | − | 4.24944i | −1.01733 | − | 5.09559i | −9.83302 | − | 5.67709i | 8.66436 | + | 8.66436i | −22.8117 | − | 1.47892i | −1.76245 | + | 0.472246i | −10.4342 | + | 10.4342i | −24.9301 | + | 10.3678i | 46.6841 | − | 26.9531i |
2.12 | 1.37323 | − | 5.12497i | 4.15460 | + | 3.12078i | −17.4513 | − | 10.0755i | −5.80773 | − | 5.80773i | 21.6992 | − | 17.0066i | 20.4169 | − | 5.47069i | −45.5877 | + | 45.5877i | 7.52142 | + | 25.9312i | −37.7398 | + | 21.7891i |
11.1 | −5.07899 | − | 1.36091i | 4.03642 | − | 3.27221i | 17.0159 | + | 9.82412i | 0.833167 | − | 0.833167i | −24.9541 | + | 11.1263i | −6.38907 | − | 23.8443i | −43.3091 | − | 43.3091i | 5.58535 | − | 26.4160i | −5.36552 | + | 3.09778i |
11.2 | −3.88819 | − | 1.04184i | −4.68804 | − | 2.24104i | 7.10439 | + | 4.10172i | 0.168873 | − | 0.168873i | 15.8932 | + | 13.5978i | 5.26279 | + | 19.6410i | −0.579067 | − | 0.579067i | 16.9555 | + | 21.0122i | −0.832548 | + | 0.480672i |
11.3 | −3.31082 | − | 0.887131i | −2.02268 | + | 4.78631i | 3.24631 | + | 1.87426i | 9.09982 | − | 9.09982i | 10.9428 | − | 14.0522i | −7.29644 | − | 27.2307i | 10.3043 | + | 10.3043i | −18.8175 | − | 19.3624i | −38.2006 | + | 22.0551i |
11.4 | −2.98884 | − | 0.800858i | 3.86434 | + | 3.47374i | 1.36361 | + | 0.787281i | −6.51442 | + | 6.51442i | −8.76795 | − | 13.4773i | 4.63030 | + | 17.2805i | 14.0588 | + | 14.0588i | 2.86629 | + | 26.8474i | 24.6877 | − | 14.2534i |
11.5 | −0.933566 | − | 0.250148i | 0.0424052 | − | 5.19598i | −6.11923 | − | 3.53294i | −11.3224 | + | 11.3224i | −1.33935 | + | 4.84018i | −3.58928 | − | 13.3954i | 10.2963 | + | 10.2963i | −26.9964 | − | 0.440673i | 13.4025 | − | 7.73791i |
11.6 | −0.385245 | − | 0.103226i | 4.91747 | − | 1.67883i | −6.79045 | − | 3.92047i | 9.27643 | − | 9.27643i | −2.06773 | + | 0.139148i | 2.08748 | + | 7.79057i | 4.46744 | + | 4.46744i | 21.3631 | − | 16.5112i | −4.53127 | + | 2.61613i |
11.7 | 0.385245 | + | 0.103226i | −3.91264 | + | 3.41924i | −6.79045 | − | 3.92047i | −9.27643 | + | 9.27643i | −1.86028 | + | 0.913360i | 2.08748 | + | 7.79057i | −4.46744 | − | 4.46744i | 3.61755 | − | 26.7566i | −4.53127 | + | 2.61613i |
11.8 | 0.933566 | + | 0.250148i | −4.52105 | − | 2.56127i | −6.11923 | − | 3.53294i | 11.3224 | − | 11.3224i | −3.58000 | − | 3.52204i | −3.58928 | − | 13.3954i | −10.2963 | − | 10.2963i | 13.8798 | + | 23.1592i | 13.4025 | − | 7.73791i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
13.f | odd | 12 | 1 | inner |
39.k | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 39.4.k.a | ✓ | 48 |
3.b | odd | 2 | 1 | inner | 39.4.k.a | ✓ | 48 |
13.f | odd | 12 | 1 | inner | 39.4.k.a | ✓ | 48 |
39.k | even | 12 | 1 | inner | 39.4.k.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
39.4.k.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
39.4.k.a | ✓ | 48 | 3.b | odd | 2 | 1 | inner |
39.4.k.a | ✓ | 48 | 13.f | odd | 12 | 1 | inner |
39.4.k.a | ✓ | 48 | 39.k | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(39, [\chi])\).