Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,2,Mod(317,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([2, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.317");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 630.bt (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: | \(5.03057532734\) |
Analytic rank: | \(0\) |
Dimension: | \(192\) |
Relative dimension: | \(48\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
317.1 | −0.965926 | − | 0.258819i | −1.73191 | − | 0.0221408i | 0.866025 | + | 0.500000i | 1.08506 | + | 1.95516i | 1.66717 | + | 0.469637i | −2.53684 | − | 0.751284i | −0.707107 | − | 0.707107i | 2.99902 | + | 0.0766917i | −0.542058 | − | 2.16937i |
317.2 | −0.965926 | − | 0.258819i | −1.73140 | − | 0.0476508i | 0.866025 | + | 0.500000i | −2.20409 | − | 0.376807i | 1.66007 | + | 0.494145i | −2.58006 | + | 0.585890i | −0.707107 | − | 0.707107i | 2.99546 | + | 0.165005i | 2.03146 | + | 0.934428i |
317.3 | −0.965926 | − | 0.258819i | −1.58790 | + | 0.691796i | 0.866025 | + | 0.500000i | 2.19402 | + | 0.431615i | 1.71284 | − | 0.257245i | 2.55054 | − | 0.703393i | −0.707107 | − | 0.707107i | 2.04284 | − | 2.19700i | −2.00755 | − | 0.984761i |
317.4 | −0.965926 | − | 0.258819i | −1.51438 | + | 0.840620i | 0.866025 | + | 0.500000i | 0.611756 | − | 2.15076i | 1.68035 | − | 0.420025i | −1.08314 | − | 2.41388i | −0.707107 | − | 0.707107i | 1.58672 | − | 2.54604i | −1.14757 | + | 1.91914i |
317.5 | −0.965926 | − | 0.258819i | −1.45309 | − | 0.942613i | 0.866025 | + | 0.500000i | 0.260263 | − | 2.22087i | 1.15961 | + | 1.28658i | 2.64011 | − | 0.172669i | −0.707107 | − | 0.707107i | 1.22296 | + | 2.73941i | −0.826198 | + | 2.07783i |
317.6 | −0.965926 | − | 0.258819i | −1.39826 | − | 1.02218i | 0.866025 | + | 0.500000i | 1.21681 | + | 1.87600i | 1.08606 | + | 1.34925i | 2.15901 | + | 1.52928i | −0.707107 | − | 0.707107i | 0.910289 | + | 2.85856i | −0.689806 | − | 2.12701i |
317.7 | −0.965926 | − | 0.258819i | −1.09766 | − | 1.33983i | 0.866025 | + | 0.500000i | −2.04805 | + | 0.897484i | 0.713485 | + | 1.57827i | −0.0959969 | + | 2.64401i | −0.707107 | − | 0.707107i | −0.590284 | + | 2.94135i | 2.21055 | − | 0.336828i |
317.8 | −0.965926 | − | 0.258819i | −1.08875 | + | 1.34708i | 0.866025 | + | 0.500000i | −1.18556 | − | 1.89590i | 1.40030 | − | 1.01939i | 0.500796 | + | 2.59792i | −0.707107 | − | 0.707107i | −0.629257 | − | 2.93326i | 0.654473 | + | 2.13815i |
317.9 | −0.965926 | − | 0.258819i | −0.989308 | + | 1.42171i | 0.866025 | + | 0.500000i | −1.72555 | + | 1.42213i | 1.32356 | − | 1.11722i | 1.90960 | − | 1.83123i | −0.707107 | − | 0.707107i | −1.04254 | − | 2.81303i | 2.03483 | − | 0.927070i |
317.10 | −0.965926 | − | 0.258819i | −0.624647 | − | 1.61549i | 0.866025 | + | 0.500000i | −0.455787 | + | 2.18912i | 0.185243 | + | 1.72212i | −0.715343 | − | 2.54721i | −0.707107 | − | 0.707107i | −2.21963 | + | 2.01823i | 1.00684 | − | 1.99656i |
317.11 | −0.965926 | − | 0.258819i | −0.196486 | + | 1.72087i | 0.866025 | + | 0.500000i | −0.145784 | + | 2.23131i | 0.635185 | − | 1.61138i | −2.10331 | + | 1.60502i | −0.707107 | − | 0.707107i | −2.92279 | − | 0.676254i | 0.718322 | − | 2.11755i |
317.12 | −0.965926 | − | 0.258819i | −0.186889 | − | 1.72194i | 0.866025 | + | 0.500000i | 2.17367 | − | 0.524538i | −0.265150 | + | 1.71164i | 0.387487 | − | 2.61722i | −0.707107 | − | 0.707107i | −2.93014 | + | 0.643623i | −2.23537 | − | 0.0559240i |
317.13 | −0.965926 | − | 0.258819i | 0.0615400 | − | 1.73096i | 0.866025 | + | 0.500000i | 2.23012 | + | 0.162985i | −0.507448 | + | 1.65605i | −1.31208 | + | 2.29749i | −0.707107 | − | 0.707107i | −2.99243 | − | 0.213046i | −2.11195 | − | 0.734629i |
317.14 | −0.965926 | − | 0.258819i | 0.241443 | + | 1.71514i | 0.866025 | + | 0.500000i | 2.06314 | − | 0.862238i | 0.210695 | − | 1.71919i | 1.66641 | + | 2.05501i | −0.707107 | − | 0.707107i | −2.88341 | + | 0.828216i | −2.21600 | + | 0.298878i |
317.15 | −0.965926 | − | 0.258819i | 0.499395 | + | 1.65849i | 0.866025 | + | 0.500000i | −2.06108 | − | 0.867151i | −0.0531288 | − | 1.73124i | −0.585433 | − | 2.58017i | −0.707107 | − | 0.707107i | −2.50121 | + | 1.65649i | 1.76641 | + | 1.37105i |
317.16 | −0.965926 | − | 0.258819i | 0.729061 | − | 1.57114i | 0.866025 | + | 0.500000i | −1.92875 | − | 1.13134i | −1.11086 | + | 1.32891i | 2.62221 | − | 0.352128i | −0.707107 | − | 0.707107i | −1.93694 | − | 2.29091i | 1.57022 | + | 1.59198i |
317.17 | −0.965926 | − | 0.258819i | 0.754597 | − | 1.55903i | 0.866025 | + | 0.500000i | −1.96329 | + | 1.07027i | −1.13239 | + | 1.31061i | −2.51410 | − | 0.824195i | −0.707107 | − | 0.707107i | −1.86117 | − | 2.35288i | 2.17340 | − | 0.525664i |
317.18 | −0.965926 | − | 0.258819i | 1.05933 | + | 1.37034i | 0.866025 | + | 0.500000i | 0.141250 | + | 2.23160i | −0.668564 | − | 1.59782i | 2.52582 | + | 0.787538i | −0.707107 | − | 0.707107i | −0.755646 | + | 2.90327i | 0.441144 | − | 2.19212i |
317.19 | −0.965926 | − | 0.258819i | 1.32691 | + | 1.11325i | 0.866025 | + | 0.500000i | −1.59332 | − | 1.56887i | −0.993564 | − | 1.41874i | −2.14973 | + | 1.54229i | −0.707107 | − | 0.707107i | 0.521364 | + | 2.95435i | 1.13297 | + | 1.92779i |
317.20 | −0.965926 | − | 0.258819i | 1.45299 | − | 0.942773i | 0.866025 | + | 0.500000i | 0.148657 | − | 2.23112i | −1.64749 | + | 0.534587i | −2.28350 | − | 1.33628i | −0.707107 | − | 0.707107i | 1.22236 | − | 2.73968i | −0.721048 | + | 2.11662i |
See next 80 embeddings (of 192 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
63.n | odd | 6 | 1 | inner |
315.bx | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 630.2.bt.a | ✓ | 192 |
5.c | odd | 4 | 1 | inner | 630.2.bt.a | ✓ | 192 |
7.c | even | 3 | 1 | 630.2.cd.a | yes | 192 | |
9.d | odd | 6 | 1 | 630.2.cd.a | yes | 192 | |
35.l | odd | 12 | 1 | 630.2.cd.a | yes | 192 | |
45.l | even | 12 | 1 | 630.2.cd.a | yes | 192 | |
63.n | odd | 6 | 1 | inner | 630.2.bt.a | ✓ | 192 |
315.bx | even | 12 | 1 | inner | 630.2.bt.a | ✓ | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
630.2.bt.a | ✓ | 192 | 1.a | even | 1 | 1 | trivial |
630.2.bt.a | ✓ | 192 | 5.c | odd | 4 | 1 | inner |
630.2.bt.a | ✓ | 192 | 63.n | odd | 6 | 1 | inner |
630.2.bt.a | ✓ | 192 | 315.bx | even | 12 | 1 | inner |
630.2.cd.a | yes | 192 | 7.c | even | 3 | 1 | |
630.2.cd.a | yes | 192 | 9.d | odd | 6 | 1 | |
630.2.cd.a | yes | 192 | 35.l | odd | 12 | 1 | |
630.2.cd.a | yes | 192 | 45.l | even | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(630, [\chi])\).