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 | −1.00284 | + | 2.19592i | −0.853889 | − | 5.93893i | 19.8790 | + | 5.83701i | −6.73003 | + | 4.32513i | 3.73874 | + | 8.18669i | −4.63257 | + | 1.36025i |
13.2 | 1.30972 | + | 1.51150i | −2.52376 | − | 1.62192i | −0.569259 | + | 3.95929i | 2.20263 | − | 4.82309i | −0.853889 | − | 5.93893i | −18.0866 | − | 5.31070i | −6.73003 | + | 4.32513i | 3.73874 | + | 8.18669i | 10.1749 | − | 2.98763i |
13.3 | 1.30972 | + | 1.51150i | −2.52376 | − | 1.62192i | −0.569259 | + | 3.95929i | 3.30855 | − | 7.24472i | −0.853889 | − | 5.93893i | −0.555799 | − | 0.163197i | −6.73003 | + | 4.32513i | 3.73874 | + | 8.18669i | 15.2837 | − | 4.48769i |
25.1 | −1.68251 | − | 1.08128i | 0.426945 | + | 2.96946i | 1.66166 | + | 3.63853i | −17.7871 | − | 5.22276i | 2.49249 | − | 5.45779i | 3.76192 | − | 4.34149i | 1.13852 | − | 7.91857i | −8.63544 | + | 2.53559i | 24.2796 | + | 28.0202i |
25.2 | −1.68251 | − | 1.08128i | 0.426945 | + | 2.96946i | 1.66166 | + | 3.63853i | 1.85372 | + | 0.544301i | 2.49249 | − | 5.45779i | −14.6603 | + | 16.9188i | 1.13852 | − | 7.91857i | −8.63544 | + | 2.53559i | −2.53035 | − | 2.92018i |
25.3 | −1.68251 | − | 1.08128i | 0.426945 | + | 2.96946i | 1.66166 | + | 3.63853i | 15.3096 | + | 4.49530i | 2.49249 | − | 5.45779i | 7.46958 | − | 8.62035i | 1.13852 | − | 7.91857i | −8.63544 | + | 2.53559i | −20.8978 | − | 24.1174i |
31.1 | 0.284630 | − | 1.97964i | −1.24625 | − | 2.72890i | −3.83797 | − | 1.12693i | −10.3477 | − | 11.9419i | −5.75696 | + | 1.69040i | −5.39451 | − | 3.46684i | −3.32332 | + | 7.27706i | −5.89375 | + | 6.80175i | −26.5860 | + | 17.0858i |
31.2 | 0.284630 | − | 1.97964i | −1.24625 | − | 2.72890i | −3.83797 | − | 1.12693i | 0.957858 | + | 1.10543i | −5.75696 | + | 1.69040i | −12.8395 | − | 8.25145i | −3.32332 | + | 7.27706i | −5.89375 | + | 6.80175i | 2.46098 | − | 1.58158i |
31.3 | 0.284630 | − | 1.97964i | −1.24625 | − | 2.72890i | −3.83797 | − | 1.12693i | 8.16051 | + | 9.41773i | −5.75696 | + | 1.69040i | 20.8971 | + | 13.4298i | −3.32332 | + | 7.27706i | −5.89375 | + | 6.80175i | 20.9665 | − | 13.4743i |
49.1 | 0.284630 | + | 1.97964i | −1.24625 | + | 2.72890i | −3.83797 | + | 1.12693i | −10.3477 | + | 11.9419i | −5.75696 | − | 1.69040i | −5.39451 | + | 3.46684i | −3.32332 | − | 7.27706i | −5.89375 | − | 6.80175i | −26.5860 | − | 17.0858i |
49.2 | 0.284630 | + | 1.97964i | −1.24625 | + | 2.72890i | −3.83797 | + | 1.12693i | 0.957858 | − | 1.10543i | −5.75696 | − | 1.69040i | −12.8395 | + | 8.25145i | −3.32332 | − | 7.27706i | −5.89375 | − | 6.80175i | 2.46098 | + | 1.58158i |
49.3 | 0.284630 | + | 1.97964i | −1.24625 | + | 2.72890i | −3.83797 | + | 1.12693i | 8.16051 | − | 9.41773i | −5.75696 | − | 1.69040i | 20.8971 | − | 13.4298i | −3.32332 | − | 7.27706i | −5.89375 | − | 6.80175i | 20.9665 | + | 13.4743i |
55.1 | 1.91899 | − | 0.563465i | 1.96458 | + | 2.26725i | 3.36501 | − | 2.16256i | −1.77045 | − | 12.3137i | 5.04752 | + | 3.24384i | 8.86624 | − | 19.4144i | 5.23889 | − | 6.04600i | −1.28083 | + | 8.90839i | −10.3358 | − | 22.6323i |
55.2 | 1.91899 | − | 0.563465i | 1.96458 | + | 2.26725i | 3.36501 | − | 2.16256i | 1.07844 | + | 7.50075i | 5.04752 | + | 3.24384i | 1.48666 | − | 3.25533i | 5.23889 | − | 6.04600i | −1.28083 | + | 8.90839i | 6.29593 | + | 13.7862i |
55.3 | 1.91899 | − | 0.563465i | 1.96458 | + | 2.26725i | 3.36501 | − | 2.16256i | 1.91350 | + | 13.3087i | 5.04752 | + | 3.24384i | −11.4577 | + | 25.0888i | 5.23889 | − | 6.04600i | −1.28083 | + | 8.90839i | 11.1710 | + | 24.4610i |
73.1 | −0.830830 | − | 1.81926i | 2.87848 | − | 0.845198i | −2.61944 | + | 3.02300i | −14.1264 | − | 9.07847i | −3.92916 | − | 4.53450i | −1.02856 | − | 7.15376i | 7.67594 | + | 2.25386i | 7.57128 | − | 4.86577i | −4.77952 | + | 33.2423i |
73.2 | −0.830830 | − | 1.81926i | 2.87848 | − | 0.845198i | −2.61944 | + | 3.02300i | 0.809240 | + | 0.520067i | −3.92916 | − | 4.53450i | −0.360364 | − | 2.50638i | 7.67594 | + | 2.25386i | 7.57128 | − | 4.86577i | 0.273798 | − | 1.90431i |
73.3 | −0.830830 | − | 1.81926i | 2.87848 | − | 0.845198i | −2.61944 | + | 3.02300i | 8.44042 | + | 5.42433i | −3.92916 | − | 4.53450i | 2.02267 | + | 14.0680i | 7.67594 | + | 2.25386i | 7.57128 | − | 4.86577i | 2.85573 | − | 19.8621i |
85.1 | 1.30972 | − | 1.51150i | −2.52376 | + | 1.62192i | −0.569259 | − | 3.95929i | −1.00284 | − | 2.19592i | −0.853889 | + | 5.93893i | 19.8790 | − | 5.83701i | −6.73003 | − | 4.32513i | 3.73874 | − | 8.18669i | −4.63257 | − | 1.36025i |
85.2 | 1.30972 | − | 1.51150i | −2.52376 | + | 1.62192i | −0.569259 | − | 3.95929i | 2.20263 | + | 4.82309i | −0.853889 | + | 5.93893i | −18.0866 | + | 5.31070i | −6.73003 | − | 4.32513i | 3.73874 | − | 8.18669i | 10.1749 | + | 2.98763i |
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.d | ✓ | 30 |
23.c | even | 11 | 1 | inner | 138.4.e.d | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
138.4.e.d | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
138.4.e.d | ✓ | 30 | 23.c | even | 11 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{30} + 2 T_{5}^{29} - 165 T_{5}^{28} - 137 T_{5}^{27} + 21974 T_{5}^{26} - 565839 T_{5}^{25} + 6329128 T_{5}^{24} + 51001204 T_{5}^{23} + 638237736 T_{5}^{22} - 25323100482 T_{5}^{21} + \cdots + 11\!\cdots\!49 \)
acting on \(S_{4}^{\mathrm{new}}(138, [\chi])\).