Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [143,3,Mod(15,143)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(143, base_ring=CyclotomicField(60))
chi = DirichletCharacter(H, H._module([12, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("143.15");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 143 = 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 143.x (of order \(60\), degree \(16\), 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: | \(3.89646778035\) |
Analytic rank: | \(0\) |
Dimension: | \(416\) |
Relative dimension: | \(26\) over \(\Q(\zeta_{60})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{60}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
15.1 | −2.32625 | − | 2.87268i | −3.36943 | + | 0.716195i | −2.00921 | + | 9.45258i | −1.04665 | + | 6.60829i | 9.89555 | + | 8.01326i | −4.33114 | − | 6.66936i | 18.6540 | − | 9.50466i | 2.61823 | − | 1.16571i | 21.4183 | − | 12.3659i |
15.2 | −2.30707 | − | 2.84900i | −2.29818 | + | 0.488494i | −1.96256 | + | 9.23311i | 0.820972 | − | 5.18342i | 6.69379 | + | 5.42053i | 6.63766 | + | 10.2211i | 17.7672 | − | 9.05286i | −3.17889 | + | 1.41533i | −16.6616 | + | 9.61957i |
15.3 | −2.23118 | − | 2.75527i | 5.15210 | − | 1.09511i | −1.78174 | + | 8.38240i | −1.24740 | + | 7.87576i | −14.5126 | − | 11.7520i | 1.40858 | + | 2.16903i | 14.4354 | − | 7.35519i | 17.1229 | − | 7.62362i | 24.4830 | − | 14.1353i |
15.4 | −2.00444 | − | 2.47527i | 1.26393 | − | 0.268657i | −1.27756 | + | 6.01046i | 0.107818 | − | 0.680737i | −3.19847 | − | 2.59007i | −2.16442 | − | 3.33292i | 6.08659 | − | 3.10128i | −6.69656 | + | 2.98150i | −1.90112 | + | 1.09761i |
15.5 | −1.72816 | − | 2.13410i | 4.29634 | − | 0.913216i | −0.736192 | + | 3.46351i | 1.15494 | − | 7.29200i | −9.37364 | − | 7.59062i | 2.16355 | + | 3.33158i | −1.12334 | + | 0.572373i | 9.40268 | − | 4.18634i | −17.5577 | + | 10.1370i |
15.6 | −1.58502 | − | 1.95733i | −4.81165 | + | 1.02275i | −0.487229 | + | 2.29223i | 1.37420 | − | 8.67637i | 9.62842 | + | 7.79694i | −5.98498 | − | 9.21606i | −3.71749 | + | 1.89416i | 13.8841 | − | 6.18159i | −19.1607 | + | 11.0624i |
15.7 | −1.54727 | − | 1.91072i | −1.40193 | + | 0.297989i | −0.425165 | + | 2.00024i | −0.194031 | + | 1.22506i | 2.73854 | + | 2.21762i | −0.619325 | − | 0.953677i | −4.28291 | + | 2.18225i | −6.34531 | + | 2.82511i | 2.64097 | − | 1.52477i |
15.8 | −1.33739 | − | 1.65154i | 1.01179 | − | 0.215062i | −0.107320 | + | 0.504903i | −0.928523 | + | 5.86247i | −1.70833 | − | 1.38338i | 5.59682 | + | 8.61835i | −6.59663 | + | 3.36115i | −7.24445 | + | 3.22544i | 10.9239 | − | 6.30690i |
15.9 | −1.03738 | − | 1.28106i | 3.02155 | − | 0.642250i | 0.266698 | − | 1.25471i | −0.0624081 | + | 0.394029i | −3.95725 | − | 3.20452i | −6.65322 | − | 10.2451i | −7.75900 | + | 3.95341i | 0.495372 | − | 0.220554i | 0.569514 | − | 0.328809i |
15.10 | −0.901785 | − | 1.11361i | −4.82609 | + | 1.02582i | 0.404732 | − | 1.90411i | −0.855758 | + | 5.40304i | 5.49446 | + | 4.44933i | 0.898384 | + | 1.38339i | −7.59250 | + | 3.86857i | 14.0170 | − | 6.24075i | 6.78860 | − | 3.91940i |
15.11 | −0.634076 | − | 0.783019i | −2.74209 | + | 0.582849i | 0.620581 | − | 2.91960i | 0.688364 | − | 4.34616i | 2.19507 | + | 1.77754i | 2.22937 | + | 3.43293i | −6.27056 | + | 3.19501i | −1.04257 | + | 0.464182i | −3.83960 | + | 2.21679i |
15.12 | −0.455608 | − | 0.562629i | 1.72143 | − | 0.365902i | 0.722674 | − | 3.39991i | 1.32469 | − | 8.36377i | −0.990165 | − | 0.801820i | 1.29304 | + | 1.99110i | −4.82238 | + | 2.45713i | −5.39246 | + | 2.40088i | −5.30923 | + | 3.06529i |
15.13 | −0.197927 | − | 0.244419i | 5.51004 | − | 1.17119i | 0.811081 | − | 3.81584i | −0.316909 | + | 2.00089i | −1.37685 | − | 1.11495i | −0.829205 | − | 1.27686i | −2.21412 | + | 1.12815i | 20.7669 | − | 9.24602i | 0.551780 | − | 0.318570i |
15.14 | −0.0330410 | − | 0.0408022i | −2.28061 | + | 0.484758i | 0.831074 | − | 3.90989i | −0.708282 | + | 4.47192i | 0.0951328 | + | 0.0770370i | −3.34268 | − | 5.14728i | −0.374113 | + | 0.190620i | −3.25572 | + | 1.44954i | 0.205867 | − | 0.118857i |
15.15 | 0.155357 | + | 0.191850i | 3.01364 | − | 0.640570i | 0.818976 | − | 3.85298i | −0.234160 | + | 1.47843i | 0.591086 | + | 0.478652i | 5.96535 | + | 9.18584i | 1.74626 | − | 0.889766i | 0.449812 | − | 0.200269i | −0.320015 | + | 0.184761i |
15.16 | 0.584280 | + | 0.721526i | −0.197077 | + | 0.0418900i | 0.652430 | − | 3.06944i | 0.391863 | − | 2.47413i | −0.145373 | − | 0.117721i | −5.04086 | − | 7.76225i | 5.90484 | − | 3.00866i | −8.18482 | + | 3.64412i | 2.01410 | − | 1.16284i |
15.17 | 0.631628 | + | 0.779996i | −4.72698 | + | 1.00475i | 0.622207 | − | 2.92726i | 0.298225 | − | 1.88292i | −3.76940 | − | 3.05240i | 6.21428 | + | 9.56915i | 6.25335 | − | 3.18624i | 13.1129 | − | 5.83824i | 1.65704 | − | 0.956691i |
15.18 | 0.794977 | + | 0.981715i | −0.472113 | + | 0.100351i | 0.499871 | − | 2.35171i | −1.45627 | + | 9.19455i | −0.473835 | − | 0.383704i | 1.25181 | + | 1.92761i | 7.20828 | − | 3.67280i | −8.00909 | + | 3.56588i | −10.1841 | + | 5.87982i |
15.19 | 1.18503 | + | 1.46339i | 3.22906 | − | 0.686357i | 0.0944359 | − | 0.444286i | 0.243108 | − | 1.53492i | 4.83092 | + | 3.91200i | −1.51394 | − | 2.33126i | 7.47323 | − | 3.80780i | 1.73380 | − | 0.771939i | 2.53427 | − | 1.46316i |
15.20 | 1.31081 | + | 1.61872i | −3.50993 | + | 0.746058i | −0.0703772 | + | 0.331099i | 1.03607 | − | 6.54148i | −5.80853 | − | 4.70365i | −1.94462 | − | 2.99445i | 6.79532 | − | 3.46239i | 3.54109 | − | 1.57659i | 11.9469 | − | 6.89756i |
See next 80 embeddings (of 416 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
13.f | odd | 12 | 1 | inner |
143.x | odd | 60 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 143.3.x.a | ✓ | 416 |
11.c | even | 5 | 1 | inner | 143.3.x.a | ✓ | 416 |
13.f | odd | 12 | 1 | inner | 143.3.x.a | ✓ | 416 |
143.x | odd | 60 | 1 | inner | 143.3.x.a | ✓ | 416 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
143.3.x.a | ✓ | 416 | 1.a | even | 1 | 1 | trivial |
143.3.x.a | ✓ | 416 | 11.c | even | 5 | 1 | inner |
143.3.x.a | ✓ | 416 | 13.f | odd | 12 | 1 | inner |
143.3.x.a | ✓ | 416 | 143.x | odd | 60 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(143, [\chi])\).