Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [192,2,Mod(13,192)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(192, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([0, 15, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 192.r (of order \(16\), degree \(8\), 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: | \(1.53312771881\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{16})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{16}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −1.40448 | + | 0.165612i | 0.831470 | − | 0.555570i | 1.94515 | − | 0.465199i | 3.01271 | − | 0.599265i | −1.07578 | + | 0.917991i | −1.64361 | − | 3.96803i | −2.65488 | + | 0.975505i | 0.382683 | − | 0.923880i | −4.13205 | + | 1.34060i |
13.2 | −1.37379 | + | 0.335725i | −0.831470 | + | 0.555570i | 1.77458 | − | 0.922428i | −1.61488 | + | 0.321219i | 0.955743 | − | 1.04238i | 0.265757 | + | 0.641595i | −2.12821 | + | 1.86299i | 0.382683 | − | 0.923880i | 2.11065 | − | 0.983439i |
13.3 | −1.34188 | + | 0.446493i | 0.831470 | − | 0.555570i | 1.60129 | − | 1.19828i | −1.15816 | + | 0.230372i | −0.867675 | + | 1.11675i | 1.66636 | + | 4.02294i | −1.61371 | + | 2.32291i | 0.382683 | − | 0.923880i | 1.45125 | − | 0.826242i |
13.4 | −1.21525 | − | 0.723297i | −0.831470 | + | 0.555570i | 0.953684 | + | 1.75798i | −1.45351 | + | 0.289121i | 1.41229 | − | 0.0737597i | −0.968670 | − | 2.33858i | 0.112572 | − | 2.82619i | 0.382683 | − | 0.923880i | 1.97550 | + | 0.699962i |
13.5 | −0.968999 | + | 1.03007i | −0.831470 | + | 0.555570i | −0.122081 | − | 1.99627i | 2.48770 | − | 0.494834i | 0.233418 | − | 1.39482i | −1.03562 | − | 2.50021i | 2.17459 | + | 1.80863i | 0.382683 | − | 0.923880i | −1.90087 | + | 3.04199i |
13.6 | −0.744400 | + | 1.20244i | 0.831470 | − | 0.555570i | −0.891737 | − | 1.79020i | −4.01045 | + | 0.797728i | 0.0490954 | + | 1.41336i | −1.73589 | − | 4.19080i | 2.81642 | + | 0.260360i | 0.382683 | − | 0.923880i | 2.02616 | − | 5.41616i |
13.7 | −0.643008 | − | 1.25958i | 0.831470 | − | 0.555570i | −1.17308 | + | 1.61984i | 1.10082 | − | 0.218966i | −1.23443 | − | 0.690066i | −0.899420 | − | 2.17139i | 2.79462 | + | 0.436020i | 0.382683 | − | 0.923880i | −0.983641 | − | 1.24577i |
13.8 | −0.392875 | − | 1.35855i | −0.831470 | + | 0.555570i | −1.69130 | + | 1.06748i | −0.00368381 | 0.000732755i | 1.08143 | + | 0.911321i | 1.78909 | + | 4.31925i | 2.11469 | + | 1.87832i | 0.382683 | − | 0.923880i | 0.00244276 | + | 0.00471675i | |
13.9 | −0.187202 | + | 1.40177i | 0.831470 | − | 0.555570i | −1.92991 | − | 0.524828i | 2.51755 | − | 0.500772i | 0.623128 | + | 1.26953i | 0.484569 | + | 1.16985i | 1.09697 | − | 2.60704i | 0.382683 | − | 0.923880i | 0.230676 | + | 3.62277i |
13.10 | 0.431051 | − | 1.34692i | −0.831470 | + | 0.555570i | −1.62839 | − | 1.16118i | −2.72905 | + | 0.542841i | 0.389904 | + | 1.35940i | −1.23163 | − | 2.97341i | −2.26594 | + | 1.69279i | 0.382683 | − | 0.923880i | −0.445193 | + | 3.90980i |
13.11 | 0.688951 | − | 1.23505i | 0.831470 | − | 0.555570i | −1.05069 | − | 1.70178i | 2.41656 | − | 0.480684i | −0.113315 | − | 1.40967i | 1.33165 | + | 3.21488i | −2.82565 | + | 0.125218i | 0.382683 | − | 0.923880i | 1.07122 | − | 3.31574i |
13.12 | 0.711584 | + | 1.22215i | −0.831470 | + | 0.555570i | −0.987297 | + | 1.73932i | 3.50591 | − | 0.697369i | −1.27065 | − | 0.620845i | 0.519328 | + | 1.25377i | −2.82826 | + | 0.0310494i | 0.382683 | − | 0.923880i | 3.34704 | + | 3.78851i |
13.13 | 0.929504 | + | 1.06584i | −0.831470 | + | 0.555570i | −0.272044 | + | 1.98141i | −4.03340 | + | 0.802293i | −1.36501 | − | 0.369811i | −0.0612023 | − | 0.147755i | −2.36474 | + | 1.55177i | 0.382683 | − | 0.923880i | −4.60418 | − | 3.55324i |
13.14 | 1.11696 | + | 0.867416i | 0.831470 | − | 0.555570i | 0.495179 | + | 1.93773i | −0.668642 | + | 0.133001i | 1.41063 | + | 0.100683i | 0.440731 | + | 1.06402i | −1.12772 | + | 2.59388i | 0.382683 | − | 0.923880i | −0.862210 | − | 0.431434i |
13.15 | 1.14999 | − | 0.823123i | −0.831470 | + | 0.555570i | 0.644937 | − | 1.89316i | 1.87933 | − | 0.373823i | −0.498876 | + | 1.32330i | −0.0424280 | − | 0.102430i | −0.816635 | − | 2.70797i | 0.382683 | − | 0.923880i | 1.85351 | − | 1.97681i |
13.16 | 1.39610 | − | 0.225635i | 0.831470 | − | 0.555570i | 1.89818 | − | 0.630017i | −1.24882 | + | 0.248406i | 1.03546 | − | 0.963239i | −0.409753 | − | 0.989232i | 2.50789 | − | 1.30786i | 0.382683 | − | 0.923880i | −1.68743 | + | 0.628577i |
37.1 | −1.40493 | − | 0.161763i | −0.555570 | + | 0.831470i | 1.94767 | + | 0.454533i | 0.573343 | − | 2.88239i | 0.915039 | − | 1.07829i | −0.457164 | + | 1.10369i | −2.66281 | − | 0.953649i | −0.382683 | − | 0.923880i | −1.27177 | + | 3.95682i |
37.2 | −1.35486 | − | 0.405391i | 0.555570 | − | 0.831470i | 1.67132 | + | 1.09850i | −0.550464 | + | 2.76737i | −1.08979 | + | 0.901306i | −1.44187 | + | 3.48098i | −1.81909 | − | 2.16585i | −0.382683 | − | 0.923880i | 1.86767 | − | 3.52626i |
37.3 | −1.27221 | + | 0.617652i | 0.555570 | − | 0.831470i | 1.23701 | − | 1.57156i | −0.108368 | + | 0.544803i | −0.193240 | + | 1.40095i | 0.796770 | − | 1.92357i | −0.603056 | + | 2.76339i | −0.382683 | − | 0.923880i | −0.198632 | − | 0.760034i |
37.4 | −1.27201 | + | 0.618052i | −0.555570 | + | 0.831470i | 1.23602 | − | 1.57234i | −0.391022 | + | 1.96580i | 0.192800 | − | 1.40101i | −0.156914 | + | 0.378825i | −0.600450 | + | 2.76396i | −0.382683 | − | 0.923880i | −0.717582 | − | 2.74219i |
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
64.i | even | 16 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 192.2.r.a | ✓ | 128 |
3.b | odd | 2 | 1 | 576.2.bd.b | 128 | ||
4.b | odd | 2 | 1 | 768.2.r.a | 128 | ||
64.i | even | 16 | 1 | inner | 192.2.r.a | ✓ | 128 |
64.j | odd | 16 | 1 | 768.2.r.a | 128 | ||
192.q | odd | 16 | 1 | 576.2.bd.b | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
192.2.r.a | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
192.2.r.a | ✓ | 128 | 64.i | even | 16 | 1 | inner |
576.2.bd.b | 128 | 3.b | odd | 2 | 1 | ||
576.2.bd.b | 128 | 192.q | odd | 16 | 1 | ||
768.2.r.a | 128 | 4.b | odd | 2 | 1 | ||
768.2.r.a | 128 | 64.j | odd | 16 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(192, [\chi])\).