Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,2,Mod(23,630)]
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([10, 9, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.23");
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.cd (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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −0.707107 | − | 0.707107i | −1.72986 | + | 0.0870655i | 1.00000i | −1.30535 | − | 1.81551i | 1.28476 | + | 1.16163i | 0.500654 | − | 2.59795i | 0.707107 | − | 0.707107i | 2.98484 | − | 0.301222i | −0.360736 | + | 2.20678i | ||
23.2 | −0.707107 | − | 0.707107i | −1.72848 | + | 0.111237i | 1.00000i | 1.72320 | − | 1.42499i | 1.30087 | + | 1.14356i | 2.49778 | + | 0.872417i | 0.707107 | − | 0.707107i | 2.97525 | − | 0.384541i | −2.22610 | − | 0.210865i | ||
23.3 | −0.707107 | − | 0.707107i | −1.66933 | + | 0.461890i | 1.00000i | −1.22527 | + | 1.87049i | 1.50700 | + | 0.853788i | 0.471054 | + | 2.60348i | 0.707107 | − | 0.707107i | 2.57332 | − | 1.54209i | 2.18903 | − | 0.456238i | ||
23.4 | −0.707107 | − | 0.707107i | −1.66709 | − | 0.469915i | 1.00000i | 0.917891 | + | 2.03899i | 0.846529 | + | 1.51109i | −1.29507 | − | 2.30712i | 0.707107 | − | 0.707107i | 2.55836 | + | 1.56678i | 0.792736 | − | 2.09083i | ||
23.5 | −0.707107 | − | 0.707107i | −1.48287 | − | 0.895033i | 1.00000i | 1.60407 | − | 1.55787i | 0.415666 | + | 1.68143i | −0.441381 | + | 2.60867i | 0.707107 | − | 0.707107i | 1.39783 | + | 2.65444i | −2.23583 | − | 0.0326704i | ||
23.6 | −0.707107 | − | 0.707107i | −1.26995 | − | 1.17780i | 1.00000i | −2.02633 | + | 0.945500i | 0.0651609 | + | 1.73082i | −2.61782 | + | 0.383406i | 0.707107 | − | 0.707107i | 0.225564 | + | 2.99151i | 2.10140 | + | 0.764265i | ||
23.7 | −0.707107 | − | 0.707107i | −0.913950 | + | 1.47129i | 1.00000i | 0.187599 | + | 2.22818i | 1.68662 | − | 0.394099i | 1.77346 | − | 1.96337i | 0.707107 | − | 0.707107i | −1.32939 | − | 2.68937i | 1.44291 | − | 1.70822i | ||
23.8 | −0.707107 | − | 0.707107i | −0.863799 | + | 1.50128i | 1.00000i | 0.347488 | − | 2.20890i | 1.67237 | − | 0.450769i | −2.23833 | + | 1.41062i | 0.707107 | − | 0.707107i | −1.50770 | − | 2.59361i | −1.80764 | + | 1.31622i | ||
23.9 | −0.707107 | − | 0.707107i | −0.853456 | − | 1.50719i | 1.00000i | −2.22267 | + | 0.244427i | −0.462258 | + | 1.66923i | 2.34249 | − | 1.22994i | 0.707107 | − | 0.707107i | −1.54323 | + | 2.57264i | 1.74450 | + | 1.39883i | ||
23.10 | −0.707107 | − | 0.707107i | −0.717442 | + | 1.57648i | 1.00000i | 2.15148 | − | 0.609219i | 1.62204 | − | 0.607429i | −0.673710 | − | 2.55854i | 0.707107 | − | 0.707107i | −1.97055 | − | 2.26206i | −1.95211 | − | 1.09054i | ||
23.11 | −0.707107 | − | 0.707107i | −0.568379 | + | 1.63614i | 1.00000i | −2.22302 | − | 0.241222i | 1.55883 | − | 0.755019i | 2.52346 | + | 0.795090i | 0.707107 | − | 0.707107i | −2.35389 | − | 1.85989i | 1.40134 | + | 1.74248i | ||
23.12 | −0.707107 | − | 0.707107i | −0.299381 | − | 1.70598i | 1.00000i | 2.04921 | + | 0.894851i | −0.994616 | + | 1.41801i | −1.82080 | + | 1.91955i | 0.707107 | − | 0.707107i | −2.82074 | + | 1.02148i | −0.816252 | − | 2.08176i | ||
23.13 | −0.707107 | − | 0.707107i | 0.176209 | − | 1.72306i | 1.00000i | 2.05547 | − | 0.880375i | −1.34299 | + | 1.09379i | 1.34585 | − | 2.27787i | 0.707107 | − | 0.707107i | −2.93790 | − | 0.607238i | −2.07595 | − | 0.830914i | ||
23.14 | −0.707107 | − | 0.707107i | 0.386634 | + | 1.68835i | 1.00000i | −1.36579 | + | 1.77049i | 0.920450 | − | 1.46723i | −2.64536 | − | 0.0453818i | 0.707107 | − | 0.707107i | −2.70103 | + | 1.30554i | 2.21768 | − | 0.286164i | ||
23.15 | −0.707107 | − | 0.707107i | 0.623928 | + | 1.61577i | 1.00000i | 2.08758 | + | 0.801264i | 0.701339 | − | 1.58371i | 0.153256 | + | 2.64131i | 0.707107 | − | 0.707107i | −2.22143 | + | 2.01625i | −0.909560 | − | 2.04272i | ||
23.16 | −0.707107 | − | 0.707107i | 0.638220 | − | 1.61018i | 1.00000i | −1.09810 | − | 1.94787i | −1.58986 | + | 0.687278i | 0.498510 | + | 2.59836i | 0.707107 | − | 0.707107i | −2.18535 | − | 2.05530i | −0.600878 | + | 2.15382i | ||
23.17 | −0.707107 | − | 0.707107i | 0.970368 | + | 1.43471i | 1.00000i | −1.84064 | − | 1.26967i | 0.328338 | − | 1.70065i | −1.20135 | − | 2.35728i | 0.707107 | − | 0.707107i | −1.11677 | + | 2.78439i | 0.403736 | + | 2.19932i | ||
23.18 | −0.707107 | − | 0.707107i | 1.18452 | + | 1.26369i | 1.00000i | 0.379698 | − | 2.20359i | 0.0559834 | − | 1.73115i | 2.56402 | − | 0.652527i | 0.707107 | − | 0.707107i | −0.193831 | + | 2.99373i | −1.82666 | + | 1.28969i | ||
23.19 | −0.707107 | − | 0.707107i | 1.27109 | − | 1.17657i | 1.00000i | −1.89933 | + | 1.18006i | −1.73076 | − | 0.0668357i | 2.22881 | − | 1.42562i | 0.707107 | − | 0.707107i | 0.231353 | − | 2.99107i | 2.17746 | + | 0.508599i | ||
23.20 | −0.707107 | − | 0.707107i | 1.27573 | − | 1.17154i | 1.00000i | −0.562088 | + | 2.16427i | −1.73048 | − | 0.0736737i | −2.58208 | + | 0.576951i | 0.707107 | − | 0.707107i | 0.254982 | − | 2.98914i | 1.92783 | − | 1.13291i | ||
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.j | odd | 6 | 1 | inner |
315.bv | 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.cd.a | yes | 192 |
5.c | odd | 4 | 1 | inner | 630.2.cd.a | yes | 192 |
7.c | even | 3 | 1 | 630.2.bt.a | ✓ | 192 | |
9.d | odd | 6 | 1 | 630.2.bt.a | ✓ | 192 | |
35.l | odd | 12 | 1 | 630.2.bt.a | ✓ | 192 | |
45.l | even | 12 | 1 | 630.2.bt.a | ✓ | 192 | |
63.j | odd | 6 | 1 | inner | 630.2.cd.a | yes | 192 |
315.bv | even | 12 | 1 | inner | 630.2.cd.a | yes | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
630.2.bt.a | ✓ | 192 | 7.c | even | 3 | 1 | |
630.2.bt.a | ✓ | 192 | 9.d | odd | 6 | 1 | |
630.2.bt.a | ✓ | 192 | 35.l | odd | 12 | 1 | |
630.2.bt.a | ✓ | 192 | 45.l | even | 12 | 1 | |
630.2.cd.a | yes | 192 | 1.a | even | 1 | 1 | trivial |
630.2.cd.a | yes | 192 | 5.c | odd | 4 | 1 | inner |
630.2.cd.a | yes | 192 | 63.j | odd | 6 | 1 | inner |
630.2.cd.a | yes | 192 | 315.bv | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(630, [\chi])\).