Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [195,2,Mod(11,195)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(195, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 0, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("195.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 195 = 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 195.bg (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: | \(1.55708283941\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(18\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −2.45997 | − | 0.659148i | −0.394058 | − | 1.68663i | 3.88494 | + | 2.24297i | −0.707107 | + | 0.707107i | −0.142367 | + | 4.40880i | −0.856074 | − | 3.19491i | −4.47674 | − | 4.47674i | −2.68944 | + | 1.32926i | 2.20555 | − | 1.27338i |
11.2 | −2.29212 | − | 0.614171i | 1.55515 | + | 0.762566i | 3.14455 | + | 1.81551i | 0.707107 | − | 0.707107i | −3.09624 | − | 2.70302i | −1.17252 | − | 4.37590i | −2.73676 | − | 2.73676i | 1.83698 | + | 2.37181i | −2.05506 | + | 1.18649i |
11.3 | −2.13499 | − | 0.572069i | −1.66401 | − | 0.480688i | 2.49887 | + | 1.44273i | 0.707107 | − | 0.707107i | 3.27767 | + | 1.97820i | 0.438491 | + | 1.63647i | −1.38389 | − | 1.38389i | 2.53788 | + | 1.59974i | −1.91418 | + | 1.10515i |
11.4 | −2.06524 | − | 0.553380i | 1.71528 | − | 0.240414i | 2.22694 | + | 1.28572i | −0.707107 | + | 0.707107i | −3.67552 | − | 0.452691i | 0.858972 | + | 3.20573i | −0.863953 | − | 0.863953i | 2.88440 | − | 0.824757i | 1.85164 | − | 1.06905i |
11.5 | −1.47996 | − | 0.396555i | −0.525689 | + | 1.65035i | 0.300984 | + | 0.173773i | 0.707107 | − | 0.707107i | 1.43245 | − | 2.23399i | 0.218736 | + | 0.816332i | 1.79028 | + | 1.79028i | −2.44730 | − | 1.73514i | −1.32690 | + | 0.766085i |
11.6 | −1.10861 | − | 0.297051i | −1.55196 | + | 0.769034i | −0.591272 | − | 0.341371i | −0.707107 | + | 0.707107i | 1.94897 | − | 0.391547i | −0.0589315 | − | 0.219935i | 2.17721 | + | 2.17721i | 1.81717 | − | 2.38702i | 0.993954 | − | 0.573859i |
11.7 | −0.584891 | − | 0.156721i | 1.60715 | + | 0.645804i | −1.41451 | − | 0.816670i | 0.707107 | − | 0.707107i | −0.838798 | − | 0.629600i | 1.15475 | + | 4.30957i | 1.55569 | + | 1.55569i | 2.16587 | + | 2.07581i | −0.524399 | + | 0.302762i |
11.8 | −0.433679 | − | 0.116204i | 1.50707 | − | 0.853659i | −1.55748 | − | 0.899210i | −0.707107 | + | 0.707107i | −0.752784 | + | 0.195086i | −0.961038 | − | 3.58664i | 1.20590 | + | 1.20590i | 1.54253 | − | 2.57305i | 0.388826 | − | 0.224489i |
11.9 | −0.424461 | − | 0.113734i | −1.34964 | − | 1.08557i | −1.56482 | − | 0.903449i | −0.707107 | + | 0.707107i | 0.449405 | + | 0.614282i | 0.145569 | + | 0.543270i | 1.18291 | + | 1.18291i | 0.643078 | + | 2.93026i | 0.380561 | − | 0.219717i |
11.10 | 0.424461 | + | 0.113734i | −0.265309 | − | 1.71161i | −1.56482 | − | 0.903449i | 0.707107 | − | 0.707107i | 0.0820551 | − | 0.756687i | 0.145569 | + | 0.543270i | −1.18291 | − | 1.18291i | −2.85922 | + | 0.908210i | 0.380561 | − | 0.219717i |
11.11 | 0.433679 | + | 0.116204i | −1.49283 | + | 0.878333i | −1.55748 | − | 0.899210i | 0.707107 | − | 0.707107i | −0.749473 | + | 0.207442i | −0.961038 | − | 3.58664i | −1.20590 | − | 1.20590i | 1.45706 | − | 2.62240i | 0.388826 | − | 0.224489i |
11.12 | 0.584891 | + | 0.156721i | −0.244293 | + | 1.71474i | −1.41451 | − | 0.816670i | −0.707107 | + | 0.707107i | −0.411620 | + | 0.964648i | 1.15475 | + | 4.30957i | −1.55569 | − | 1.55569i | −2.88064 | − | 0.837797i | −0.524399 | + | 0.302762i |
11.13 | 1.10861 | + | 0.297051i | 1.44198 | − | 0.959522i | −0.591272 | − | 0.341371i | 0.707107 | − | 0.707107i | 1.88363 | − | 0.635393i | −0.0589315 | − | 0.219935i | −2.17721 | − | 2.17721i | 1.15864 | − | 2.76723i | 0.993954 | − | 0.573859i |
11.14 | 1.47996 | + | 0.396555i | 1.69209 | + | 0.369915i | 0.300984 | + | 0.173773i | −0.707107 | + | 0.707107i | 2.35754 | + | 1.21847i | 0.218736 | + | 0.816332i | −1.79028 | − | 1.79028i | 2.72633 | + | 1.25186i | −1.32690 | + | 0.766085i |
11.15 | 2.06524 | + | 0.553380i | −1.06585 | + | 1.36527i | 2.22694 | + | 1.28572i | 0.707107 | − | 0.707107i | −2.95675 | + | 2.22980i | 0.858972 | + | 3.20573i | 0.863953 | + | 0.863953i | −0.727941 | − | 2.91034i | 1.85164 | − | 1.06905i |
11.16 | 2.13499 | + | 0.572069i | 0.415718 | − | 1.68142i | 2.49887 | + | 1.44273i | −0.707107 | + | 0.707107i | 1.84944 | − | 3.35200i | 0.438491 | + | 1.63647i | 1.38389 | + | 1.38389i | −2.65436 | − | 1.39800i | −1.91418 | + | 1.10515i |
11.17 | 2.29212 | + | 0.614171i | −0.117173 | + | 1.72808i | 3.14455 | + | 1.81551i | −0.707107 | + | 0.707107i | −1.32991 | + | 3.88901i | −1.17252 | − | 4.37590i | 2.73676 | + | 2.73676i | −2.97254 | − | 0.404970i | −2.05506 | + | 1.18649i |
11.18 | 2.45997 | + | 0.659148i | −1.26364 | − | 1.18458i | 3.88494 | + | 2.24297i | 0.707107 | − | 0.707107i | −2.32769 | − | 3.74695i | −0.856074 | − | 3.19491i | 4.47674 | + | 4.47674i | 0.193547 | + | 2.99375i | 2.20555 | − | 1.27338i |
41.1 | −0.698381 | + | 2.60640i | −1.27348 | − | 1.17399i | −4.57351 | − | 2.64052i | −0.707107 | − | 0.707107i | 3.94925 | − | 2.49931i | 2.70283 | − | 0.724221i | 6.26025 | − | 6.26025i | 0.243511 | + | 2.99010i | 2.33683 | − | 1.34917i |
41.2 | −0.649589 | + | 2.42430i | −1.00663 | + | 1.40950i | −3.72321 | − | 2.14960i | 0.707107 | + | 0.707107i | −2.76316 | − | 3.35597i | −2.77349 | + | 0.743154i | 4.08041 | − | 4.08041i | −0.973391 | − | 2.83769i | −2.17357 | + | 1.25491i |
See all 72 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 | 195.2.bg.a | ✓ | 72 |
3.b | odd | 2 | 1 | inner | 195.2.bg.a | ✓ | 72 |
5.b | even | 2 | 1 | 975.2.bo.e | 72 | ||
5.c | odd | 4 | 1 | 975.2.bp.h | 72 | ||
5.c | odd | 4 | 1 | 975.2.bp.i | 72 | ||
13.f | odd | 12 | 1 | inner | 195.2.bg.a | ✓ | 72 |
15.d | odd | 2 | 1 | 975.2.bo.e | 72 | ||
15.e | even | 4 | 1 | 975.2.bp.h | 72 | ||
15.e | even | 4 | 1 | 975.2.bp.i | 72 | ||
39.k | even | 12 | 1 | inner | 195.2.bg.a | ✓ | 72 |
65.o | even | 12 | 1 | 975.2.bp.h | 72 | ||
65.s | odd | 12 | 1 | 975.2.bo.e | 72 | ||
65.t | even | 12 | 1 | 975.2.bp.i | 72 | ||
195.bc | odd | 12 | 1 | 975.2.bp.i | 72 | ||
195.bh | even | 12 | 1 | 975.2.bo.e | 72 | ||
195.bn | odd | 12 | 1 | 975.2.bp.h | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
195.2.bg.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
195.2.bg.a | ✓ | 72 | 3.b | odd | 2 | 1 | inner |
195.2.bg.a | ✓ | 72 | 13.f | odd | 12 | 1 | inner |
195.2.bg.a | ✓ | 72 | 39.k | even | 12 | 1 | inner |
975.2.bo.e | 72 | 5.b | even | 2 | 1 | ||
975.2.bo.e | 72 | 15.d | odd | 2 | 1 | ||
975.2.bo.e | 72 | 65.s | odd | 12 | 1 | ||
975.2.bo.e | 72 | 195.bh | even | 12 | 1 | ||
975.2.bp.h | 72 | 5.c | odd | 4 | 1 | ||
975.2.bp.h | 72 | 15.e | even | 4 | 1 | ||
975.2.bp.h | 72 | 65.o | even | 12 | 1 | ||
975.2.bp.h | 72 | 195.bn | odd | 12 | 1 | ||
975.2.bp.i | 72 | 5.c | odd | 4 | 1 | ||
975.2.bp.i | 72 | 15.e | even | 4 | 1 | ||
975.2.bp.i | 72 | 65.t | even | 12 | 1 | ||
975.2.bp.i | 72 | 195.bc | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(195, [\chi])\).