Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [138,4,Mod(13,138)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(138, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 14]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("138.13");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 138 = 2 \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 138.e (of order \(11\), degree \(10\), 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: | \(8.14226358079\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | 1.30972 | + | 1.51150i | 2.52376 | + | 1.62192i | −0.569259 | + | 3.95929i | −4.97931 | + | 10.9032i | 0.853889 | + | 5.93893i | 10.5666 | + | 3.10264i | −6.73003 | + | 4.32513i | 3.73874 | + | 8.18669i | −23.0016 | + | 6.75388i |
13.2 | 1.30972 | + | 1.51150i | 2.52376 | + | 1.62192i | −0.569259 | + | 3.95929i | −3.70472 | + | 8.11221i | 0.853889 | + | 5.93893i | −24.4006 | − | 7.16466i | −6.73003 | + | 4.32513i | 3.73874 | + | 8.18669i | −17.1138 | + | 5.02505i |
13.3 | 1.30972 | + | 1.51150i | 2.52376 | + | 1.62192i | −0.569259 | + | 3.95929i | 5.61304 | − | 12.2908i | 0.853889 | + | 5.93893i | 18.2993 | + | 5.37317i | −6.73003 | + | 4.32513i | 3.73874 | + | 8.18669i | 25.9291 | − | 7.61348i |
25.1 | −1.68251 | − | 1.08128i | −0.426945 | − | 2.96946i | 1.66166 | + | 3.63853i | −11.8265 | − | 3.47259i | −2.49249 | + | 5.45779i | 13.0481 | − | 15.0583i | 1.13852 | − | 7.91857i | −8.63544 | + | 2.53559i | 16.1434 | + | 18.6305i |
25.2 | −1.68251 | − | 1.08128i | −0.426945 | − | 2.96946i | 1.66166 | + | 3.63853i | 4.71256 | + | 1.38373i | −2.49249 | + | 5.45779i | 3.70690 | − | 4.27800i | 1.13852 | − | 7.91857i | −8.63544 | + | 2.53559i | −6.43272 | − | 7.42375i |
25.3 | −1.68251 | − | 1.08128i | −0.426945 | − | 2.96946i | 1.66166 | + | 3.63853i | 19.5175 | + | 5.73085i | −2.49249 | + | 5.45779i | −20.5565 | + | 23.7235i | 1.13852 | − | 7.91857i | −8.63544 | + | 2.53559i | −26.6416 | − | 30.7461i |
31.1 | 0.284630 | − | 1.97964i | 1.24625 | + | 2.72890i | −3.83797 | − | 1.12693i | −7.03109 | − | 8.11432i | 5.75696 | − | 1.69040i | −5.50559 | − | 3.53823i | −3.32332 | + | 7.27706i | −5.89375 | + | 6.80175i | −18.0647 | + | 11.6095i |
31.2 | 0.284630 | − | 1.97964i | 1.24625 | + | 2.72890i | −3.83797 | − | 1.12693i | −2.69854 | − | 3.11428i | 5.75696 | − | 1.69040i | 22.2299 | + | 14.2863i | −3.32332 | + | 7.27706i | −5.89375 | + | 6.80175i | −6.93325 | + | 4.45573i |
31.3 | 0.284630 | − | 1.97964i | 1.24625 | + | 2.72890i | −3.83797 | − | 1.12693i | 13.1550 | + | 15.1817i | 5.75696 | − | 1.69040i | −15.4591 | − | 9.93493i | −3.32332 | + | 7.27706i | −5.89375 | + | 6.80175i | 33.7986 | − | 21.7210i |
49.1 | 0.284630 | + | 1.97964i | 1.24625 | − | 2.72890i | −3.83797 | + | 1.12693i | −7.03109 | + | 8.11432i | 5.75696 | + | 1.69040i | −5.50559 | + | 3.53823i | −3.32332 | − | 7.27706i | −5.89375 | − | 6.80175i | −18.0647 | − | 11.6095i |
49.2 | 0.284630 | + | 1.97964i | 1.24625 | − | 2.72890i | −3.83797 | + | 1.12693i | −2.69854 | + | 3.11428i | 5.75696 | + | 1.69040i | 22.2299 | − | 14.2863i | −3.32332 | − | 7.27706i | −5.89375 | − | 6.80175i | −6.93325 | − | 4.45573i |
49.3 | 0.284630 | + | 1.97964i | 1.24625 | − | 2.72890i | −3.83797 | + | 1.12693i | 13.1550 | − | 15.1817i | 5.75696 | + | 1.69040i | −15.4591 | + | 9.93493i | −3.32332 | − | 7.27706i | −5.89375 | − | 6.80175i | 33.7986 | + | 21.7210i |
55.1 | 1.91899 | − | 0.563465i | −1.96458 | − | 2.26725i | 3.36501 | − | 2.16256i | −1.52360 | − | 10.5969i | −5.04752 | − | 3.24384i | 0.567261 | − | 1.24213i | 5.23889 | − | 6.04600i | −1.28083 | + | 8.90839i | −8.89475 | − | 19.4768i |
55.2 | 1.91899 | − | 0.563465i | −1.96458 | − | 2.26725i | 3.36501 | − | 2.16256i | 1.45978 | + | 10.1530i | −5.04752 | − | 3.24384i | −13.4424 | + | 29.4347i | 5.23889 | − | 6.04600i | −1.28083 | + | 8.90839i | 8.52215 | + | 18.6609i |
55.3 | 1.91899 | − | 0.563465i | −1.96458 | − | 2.26725i | 3.36501 | − | 2.16256i | 1.75662 | + | 12.2176i | −5.04752 | − | 3.24384i | 12.8585 | − | 28.1562i | 5.23889 | − | 6.04600i | −1.28083 | + | 8.90839i | 10.2551 | + | 22.4555i |
73.1 | −0.830830 | − | 1.81926i | −2.87848 | + | 0.845198i | −2.61944 | + | 3.02300i | −16.6025 | − | 10.6698i | 3.92916 | + | 4.53450i | 3.87830 | + | 26.9742i | 7.67594 | + | 2.25386i | 7.57128 | − | 4.86577i | −5.61729 | + | 39.0691i |
73.2 | −0.830830 | − | 1.81926i | −2.87848 | + | 0.845198i | −2.61944 | + | 3.02300i | −3.38842 | − | 2.17761i | 3.92916 | + | 4.53450i | −3.30636 | − | 22.9962i | 7.67594 | + | 2.25386i | 7.57128 | − | 4.86577i | −1.14644 | + | 7.97366i |
73.3 | −0.830830 | − | 1.81926i | −2.87848 | + | 0.845198i | −2.61944 | + | 3.02300i | 3.54028 | + | 2.27520i | 3.92916 | + | 4.53450i | −0.484390 | − | 3.36901i | 7.67594 | + | 2.25386i | 7.57128 | − | 4.86577i | 1.19782 | − | 8.33101i |
85.1 | 1.30972 | − | 1.51150i | 2.52376 | − | 1.62192i | −0.569259 | − | 3.95929i | −4.97931 | − | 10.9032i | 0.853889 | − | 5.93893i | 10.5666 | − | 3.10264i | −6.73003 | − | 4.32513i | 3.73874 | − | 8.18669i | −23.0016 | − | 6.75388i |
85.2 | 1.30972 | − | 1.51150i | 2.52376 | − | 1.62192i | −0.569259 | − | 3.95929i | −3.70472 | − | 8.11221i | 0.853889 | − | 5.93893i | −24.4006 | + | 7.16466i | −6.73003 | − | 4.32513i | 3.73874 | − | 8.18669i | −17.1138 | − | 5.02505i |
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 138.4.e.c | ✓ | 30 |
23.c | even | 11 | 1 | inner | 138.4.e.c | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
138.4.e.c | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
138.4.e.c | ✓ | 30 | 23.c | even | 11 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} + 4 T_{5}^{29} + 35 T_{5}^{28} + 797 T_{5}^{27} + 81670 T_{5}^{26} - 589815 T_{5}^{25} + \cdots + 51\!\cdots\!21 \) acting on \(S_{4}^{\mathrm{new}}(138, [\chi])\).