Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,2,Mod(157,630)]
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([4, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.157");
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.cg (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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
157.1 | −0.965926 | + | 0.258819i | −1.71321 | + | 0.254780i | 0.866025 | − | 0.500000i | −1.99551 | − | 1.00893i | 1.58889 | − | 0.689510i | −0.221143 | + | 2.63649i | −0.707107 | + | 0.707107i | 2.87017 | − | 0.872982i | 2.18864 | + | 0.458077i |
157.2 | −0.965926 | + | 0.258819i | −1.70832 | − | 0.285724i | 0.866025 | − | 0.500000i | 1.73208 | + | 1.41418i | 1.72406 | − | 0.166158i | 2.64401 | + | 0.0960094i | −0.707107 | + | 0.707107i | 2.83672 | + | 0.976216i | −2.03908 | − | 0.917694i |
157.3 | −0.965926 | + | 0.258819i | −1.64267 | + | 0.549224i | 0.866025 | − | 0.500000i | 1.99630 | − | 1.00736i | 1.44454 | − | 0.955663i | −2.31389 | + | 1.28293i | −0.707107 | + | 0.707107i | 2.39671 | − | 1.80438i | −1.66756 | + | 1.48971i |
157.4 | −0.965926 | + | 0.258819i | −1.43992 | − | 0.962609i | 0.866025 | − | 0.500000i | −1.93341 | + | 1.12335i | 1.64000 | + | 0.557129i | 1.38573 | − | 2.25383i | −0.707107 | + | 0.707107i | 1.14677 | + | 2.77217i | 1.57679 | − | 1.58548i |
157.5 | −0.965926 | + | 0.258819i | −1.26782 | + | 1.18010i | 0.866025 | − | 0.500000i | −1.44041 | + | 1.71033i | 0.919188 | − | 1.46802i | 2.43402 | − | 1.03708i | −0.707107 | + | 0.707107i | 0.214733 | − | 2.99231i | 0.948666 | − | 2.02485i |
157.6 | −0.965926 | + | 0.258819i | −1.20057 | − | 1.24845i | 0.866025 | − | 0.500000i | −0.928450 | − | 2.03420i | 1.48278 | + | 0.895181i | 1.70605 | + | 2.02223i | −0.707107 | + | 0.707107i | −0.117263 | + | 2.99771i | 1.42330 | + | 1.72459i |
157.7 | −0.965926 | + | 0.258819i | −1.16870 | − | 1.27833i | 0.866025 | − | 0.500000i | 2.19640 | − | 0.419297i | 1.45974 | + | 0.932291i | −1.80421 | − | 1.93515i | −0.707107 | + | 0.707107i | −0.268265 | + | 2.98798i | −2.01304 | + | 0.973481i |
157.8 | −0.965926 | + | 0.258819i | −1.01214 | + | 1.40555i | 0.866025 | − | 0.500000i | 0.201115 | + | 2.22701i | 0.613868 | − | 1.61962i | −2.51031 | − | 0.835662i | −0.707107 | + | 0.707107i | −0.951147 | − | 2.84523i | −0.770654 | − | 2.09907i |
157.9 | −0.965926 | + | 0.258819i | −0.798392 | + | 1.53707i | 0.866025 | − | 0.500000i | −1.46535 | − | 1.68900i | 0.373365 | − | 1.69133i | −1.21582 | − | 2.34985i | −0.707107 | + | 0.707107i | −1.72514 | − | 2.45436i | 1.85257 | + | 1.25219i |
157.10 | −0.965926 | + | 0.258819i | −0.697697 | − | 1.58531i | 0.866025 | − | 0.500000i | −2.05976 | + | 0.870292i | 1.08423 | + | 1.35072i | −2.59232 | + | 0.529026i | −0.707107 | + | 0.707107i | −2.02644 | + | 2.21214i | 1.76432 | − | 1.37374i |
157.11 | −0.965926 | + | 0.258819i | −0.231316 | + | 1.71654i | 0.866025 | − | 0.500000i | 1.97412 | + | 1.05016i | −0.220838 | − | 1.71791i | 1.21830 | + | 2.34856i | −0.707107 | + | 0.707107i | −2.89299 | − | 0.794125i | −2.17866 | − | 0.503437i |
157.12 | −0.965926 | + | 0.258819i | −0.0309423 | − | 1.73177i | 0.866025 | − | 0.500000i | 2.05295 | − | 0.886221i | 0.478104 | + | 1.66476i | 2.61918 | − | 0.374056i | −0.707107 | + | 0.707107i | −2.99809 | + | 0.107170i | −1.75363 | + | 1.38737i |
157.13 | −0.965926 | + | 0.258819i | 0.100845 | + | 1.72911i | 0.866025 | − | 0.500000i | 1.99067 | − | 1.01845i | −0.544936 | − | 1.64409i | 1.32746 | − | 2.28864i | −0.707107 | + | 0.707107i | −2.97966 | + | 0.348745i | −1.65925 | + | 1.49897i |
157.14 | −0.965926 | + | 0.258819i | 0.419695 | − | 1.68043i | 0.866025 | − | 0.500000i | 0.445332 | + | 2.19127i | 0.0295336 | + | 1.73180i | 0.601210 | − | 2.57654i | −0.707107 | + | 0.707107i | −2.64771 | − | 1.41054i | −0.997301 | − | 2.00135i |
157.15 | −0.965926 | + | 0.258819i | 0.512590 | − | 1.65446i | 0.866025 | − | 0.500000i | −0.725830 | − | 2.11499i | −0.0669173 | + | 1.73076i | −2.64367 | − | 0.104885i | −0.707107 | + | 0.707107i | −2.47450 | − | 1.69612i | 1.24850 | + | 1.85506i |
157.16 | −0.965926 | + | 0.258819i | 0.579077 | + | 1.63238i | 0.866025 | − | 0.500000i | −0.445639 | − | 2.19121i | −0.981837 | − | 1.42688i | −2.40360 | + | 1.10575i | −0.707107 | + | 0.707107i | −2.32934 | + | 1.89055i | 0.997581 | + | 2.00121i |
157.17 | −0.965926 | + | 0.258819i | 0.810488 | + | 1.53072i | 0.866025 | − | 0.500000i | −0.924726 | + | 2.03590i | −1.17905 | − | 1.26879i | 0.515728 | + | 2.59500i | −0.707107 | + | 0.707107i | −1.68622 | + | 2.48126i | 0.366288 | − | 2.20586i |
157.18 | −0.965926 | + | 0.258819i | 0.973845 | + | 1.43235i | 0.866025 | − | 0.500000i | −2.22463 | + | 0.225873i | −1.31138 | − | 1.13149i | 2.44785 | − | 1.00400i | −0.707107 | + | 0.707107i | −1.10325 | + | 2.78977i | 2.09037 | − | 0.793954i |
157.19 | −0.965926 | + | 0.258819i | 1.29834 | + | 1.14644i | 0.866025 | − | 0.500000i | 1.62901 | + | 1.53177i | −1.55082 | − | 0.771341i | −1.13641 | − | 2.38926i | −0.707107 | + | 0.707107i | 0.371355 | + | 2.97693i | −1.96995 | − | 1.05796i |
157.20 | −0.965926 | + | 0.258819i | 1.38183 | − | 1.04429i | 0.866025 | − | 0.500000i | −2.17615 | − | 0.514181i | −1.06446 | + | 1.36635i | 2.32202 | + | 1.26816i | −0.707107 | + | 0.707107i | 0.818901 | − | 2.88607i | 2.23508 | − | 0.0665681i |
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.k | odd | 6 | 1 | inner |
315.cg | 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.cg.a | yes | 192 |
5.c | odd | 4 | 1 | inner | 630.2.cg.a | yes | 192 |
7.d | odd | 6 | 1 | 630.2.bw.a | ✓ | 192 | |
9.c | even | 3 | 1 | 630.2.bw.a | ✓ | 192 | |
35.k | even | 12 | 1 | 630.2.bw.a | ✓ | 192 | |
45.k | odd | 12 | 1 | 630.2.bw.a | ✓ | 192 | |
63.k | odd | 6 | 1 | inner | 630.2.cg.a | yes | 192 |
315.cg | even | 12 | 1 | inner | 630.2.cg.a | yes | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
630.2.bw.a | ✓ | 192 | 7.d | odd | 6 | 1 | |
630.2.bw.a | ✓ | 192 | 9.c | even | 3 | 1 | |
630.2.bw.a | ✓ | 192 | 35.k | even | 12 | 1 | |
630.2.bw.a | ✓ | 192 | 45.k | odd | 12 | 1 | |
630.2.cg.a | yes | 192 | 1.a | even | 1 | 1 | trivial |
630.2.cg.a | yes | 192 | 5.c | odd | 4 | 1 | inner |
630.2.cg.a | yes | 192 | 63.k | odd | 6 | 1 | inner |
630.2.cg.a | yes | 192 | 315.cg | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(630, [\chi])\).