Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [108,6,Mod(35,108)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(108, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("108.35");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 108 = 2^{2} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 108.h (of order \(6\), degree \(2\), not 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: | \(17.3214525398\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 36) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
35.1 | −5.65362 | + | 0.191212i | 0 | 31.9269 | − | 2.16208i | −3.68052 | + | 2.12495i | 0 | −13.9322 | − | 8.04377i | −180.089 | + | 18.3284i | 0 | 20.4019 | − | 12.7174i | ||||||
35.2 | −5.52836 | + | 1.19885i | 0 | 29.1255 | − | 13.2554i | 20.1454 | − | 11.6310i | 0 | 156.832 | + | 90.5473i | −145.125 | + | 108.198i | 0 | −97.4272 | + | 88.4515i | ||||||
35.3 | −5.48009 | − | 1.40307i | 0 | 28.0628 | + | 15.3779i | 77.8058 | − | 44.9212i | 0 | −124.034 | − | 71.6112i | −132.210 | − | 123.647i | 0 | −489.410 | + | 137.005i | ||||||
35.4 | −5.12358 | − | 2.39768i | 0 | 20.5022 | + | 24.5695i | −61.0162 | + | 35.2277i | 0 | −181.157 | − | 104.591i | −46.1351 | − | 175.042i | 0 | 397.087 | − | 34.1946i | ||||||
35.5 | −4.87383 | + | 2.87155i | 0 | 15.5084 | − | 27.9909i | −70.1957 | + | 40.5275i | 0 | −89.9985 | − | 51.9607i | 4.79147 | + | 180.956i | 0 | 225.745 | − | 399.094i | ||||||
35.6 | −4.63825 | − | 3.23831i | 0 | 11.0267 | + | 30.0402i | −61.0162 | + | 35.2277i | 0 | 181.157 | + | 104.591i | 46.1351 | − | 175.042i | 0 | 397.087 | + | 34.1946i | ||||||
35.7 | −3.95514 | − | 4.04436i | 0 | −0.713689 | + | 31.9920i | 77.8058 | − | 44.9212i | 0 | 124.034 | + | 71.6112i | 132.210 | − | 123.647i | 0 | −489.410 | − | 137.005i | ||||||
35.8 | −3.75843 | + | 4.22779i | 0 | −3.74838 | − | 31.7797i | −12.8631 | + | 7.42651i | 0 | 15.2989 | + | 8.83282i | 148.446 | + | 103.595i | 0 | 16.9473 | − | 82.2944i | ||||||
35.9 | −3.65969 | + | 4.31354i | 0 | −5.21330 | − | 31.5725i | 51.8636 | − | 29.9435i | 0 | 33.8827 | + | 19.5622i | 155.268 | + | 93.0578i | 0 | −60.6425 | + | 333.300i | ||||||
35.10 | −2.66122 | − | 4.99179i | 0 | −17.8359 | + | 26.5684i | −3.68052 | + | 2.12495i | 0 | 13.9322 | + | 8.04377i | 180.089 | + | 18.3284i | 0 | 20.4019 | + | 12.7174i | ||||||
35.11 | −1.78768 | + | 5.36696i | 0 | −25.6084 | − | 19.1888i | 52.1793 | − | 30.1257i | 0 | −185.316 | − | 106.992i | 148.765 | − | 103.136i | 0 | 68.4038 | + | 333.899i | ||||||
35.12 | −1.72594 | − | 5.38713i | 0 | −26.0423 | + | 18.5957i | 20.1454 | − | 11.6310i | 0 | −156.832 | − | 90.5473i | 145.125 | + | 108.198i | 0 | −97.4272 | − | 88.4515i | ||||||
35.13 | −0.253656 | + | 5.65116i | 0 | −31.8713 | − | 2.86690i | −84.3255 | + | 48.6853i | 0 | −87.9661 | − | 50.7872i | 24.2857 | − | 179.383i | 0 | −253.739 | − | 488.886i | ||||||
35.14 | −0.216257 | + | 5.65272i | 0 | −31.9065 | − | 2.44488i | 0.143132 | − | 0.0826372i | 0 | 192.634 | + | 111.217i | 20.7202 | − | 179.830i | 0 | 0.436172 | + | 0.826955i | ||||||
35.15 | 0.0499164 | − | 5.65663i | 0 | −31.9950 | − | 0.564717i | −70.1957 | + | 40.5275i | 0 | 89.9985 | + | 51.9607i | −4.79147 | + | 180.956i | 0 | 225.745 | + | 399.094i | ||||||
35.16 | 1.64008 | + | 5.41389i | 0 | −26.6203 | + | 17.7584i | −38.3603 | + | 22.1474i | 0 | 102.940 | + | 59.4323i | −139.801 | − | 114.994i | 0 | −182.817 | − | 171.355i | ||||||
35.17 | 1.78216 | − | 5.36879i | 0 | −25.6478 | − | 19.1360i | −12.8631 | + | 7.42651i | 0 | −15.2989 | − | 8.83282i | −148.446 | + | 103.595i | 0 | 16.9473 | + | 82.2944i | ||||||
35.18 | 1.90579 | − | 5.32616i | 0 | −24.7359 | − | 20.3011i | 51.8636 | − | 29.9435i | 0 | −33.8827 | − | 19.5622i | −155.268 | + | 93.0578i | 0 | −60.6425 | − | 333.300i | ||||||
35.19 | 2.01287 | + | 5.28662i | 0 | −23.8967 | + | 21.2825i | 78.5398 | − | 45.3450i | 0 | −69.6792 | − | 40.2293i | −160.614 | − | 83.4942i | 0 | 397.812 | + | 323.937i | ||||||
35.20 | 3.75408 | − | 4.23165i | 0 | −3.81374 | − | 31.7719i | 52.1793 | − | 30.1257i | 0 | 185.316 | + | 106.992i | −148.765 | − | 103.136i | 0 | 68.4038 | − | 333.899i | ||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
9.d | odd | 6 | 1 | inner |
36.h | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 108.6.h.a | 56 | |
3.b | odd | 2 | 1 | 36.6.h.a | ✓ | 56 | |
4.b | odd | 2 | 1 | inner | 108.6.h.a | 56 | |
9.c | even | 3 | 1 | 36.6.h.a | ✓ | 56 | |
9.d | odd | 6 | 1 | inner | 108.6.h.a | 56 | |
12.b | even | 2 | 1 | 36.6.h.a | ✓ | 56 | |
36.f | odd | 6 | 1 | 36.6.h.a | ✓ | 56 | |
36.h | even | 6 | 1 | inner | 108.6.h.a | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
36.6.h.a | ✓ | 56 | 3.b | odd | 2 | 1 | |
36.6.h.a | ✓ | 56 | 9.c | even | 3 | 1 | |
36.6.h.a | ✓ | 56 | 12.b | even | 2 | 1 | |
36.6.h.a | ✓ | 56 | 36.f | odd | 6 | 1 | |
108.6.h.a | 56 | 1.a | even | 1 | 1 | trivial | |
108.6.h.a | 56 | 4.b | odd | 2 | 1 | inner | |
108.6.h.a | 56 | 9.d | odd | 6 | 1 | inner | |
108.6.h.a | 56 | 36.h | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(108, [\chi])\).