Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [882,2,Mod(5,882)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([35, 29]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("882.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 882.bk (of order \(42\), degree \(12\), 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: | \(7.04280545828\) |
Analytic rank: | \(0\) |
Dimension: | \(672\) |
Relative dimension: | \(56\) over \(\Q(\zeta_{42})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{42}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −0.781831 | + | 0.623490i | −1.73071 | − | 0.0681459i | 0.222521 | − | 0.974928i | 1.83461 | + | 1.25081i | 1.39561 | − | 1.02580i | 0.211960 | − | 2.63725i | 0.433884 | + | 0.900969i | 2.99071 | + | 0.235881i | −2.21422 | + | 0.165933i |
5.2 | −0.781831 | + | 0.623490i | −1.69372 | + | 0.362376i | 0.222521 | − | 0.974928i | −1.25279 | − | 0.854138i | 1.09826 | − | 1.33933i | 2.59469 | − | 0.517268i | 0.433884 | + | 0.900969i | 2.73737 | − | 1.22753i | 1.51202 | − | 0.113310i |
5.3 | −0.781831 | + | 0.623490i | −1.65319 | + | 0.516682i | 0.222521 | − | 0.974928i | −2.27516 | − | 1.55118i | 0.970370 | − | 1.43471i | −1.77406 | − | 1.96283i | 0.433884 | + | 0.900969i | 2.46608 | − | 1.70835i | 2.74593 | − | 0.205779i |
5.4 | −0.781831 | + | 0.623490i | −1.63652 | + | 0.567289i | 0.222521 | − | 0.974928i | 3.21299 | + | 2.19058i | 0.925780 | − | 1.46388i | −2.29563 | + | 1.31532i | 0.433884 | + | 0.900969i | 2.35637 | − | 1.85675i | −3.87782 | + | 0.290602i |
5.5 | −0.781831 | + | 0.623490i | −1.54555 | − | 0.781835i | 0.222521 | − | 0.974928i | 2.77812 | + | 1.89409i | 1.69583 | − | 0.352373i | 2.33331 | + | 1.24726i | 0.433884 | + | 0.900969i | 1.77747 | + | 2.41674i | −3.35297 | + | 0.251270i |
5.6 | −0.781831 | + | 0.623490i | −1.52352 | − | 0.823950i | 0.222521 | − | 0.974928i | −3.67482 | − | 2.50545i | 1.70486 | − | 0.305708i | −0.560178 | + | 2.58577i | 0.433884 | + | 0.900969i | 1.64221 | + | 2.51060i | 4.43521 | − | 0.332373i |
5.7 | −0.781831 | + | 0.623490i | −1.34273 | + | 1.09411i | 0.222521 | − | 0.974928i | 0.806214 | + | 0.549667i | 0.367626 | − | 1.69259i | 0.824300 | + | 2.51407i | 0.433884 | + | 0.900969i | 0.605860 | − | 2.93819i | −0.973035 | + | 0.0729189i |
5.8 | −0.781831 | + | 0.623490i | −1.23334 | − | 1.21609i | 0.222521 | − | 0.974928i | −0.255877 | − | 0.174454i | 1.72248 | + | 0.181809i | −2.52813 | − | 0.780116i | 0.433884 | + | 0.900969i | 0.0422313 | + | 2.99970i | 0.308823 | − | 0.0231431i |
5.9 | −0.781831 | + | 0.623490i | −1.17019 | + | 1.27697i | 0.222521 | − | 0.974928i | −0.474915 | − | 0.323792i | 0.118716 | − | 1.72798i | −2.64285 | − | 0.123929i | 0.433884 | + | 0.900969i | −0.261299 | − | 2.98860i | 0.573185 | − | 0.0429542i |
5.10 | −0.781831 | + | 0.623490i | −0.724077 | + | 1.57344i | 0.222521 | − | 0.974928i | 2.44331 | + | 1.66582i | −0.414917 | − | 1.68162i | 0.419163 | − | 2.61234i | 0.433884 | + | 0.900969i | −1.95142 | − | 2.27858i | −2.94888 | + | 0.220988i |
5.11 | −0.781831 | + | 0.623490i | −0.569639 | − | 1.63570i | 0.222521 | − | 0.974928i | 2.43032 | + | 1.65696i | 1.46520 | + | 0.923677i | −1.41335 | − | 2.23661i | 0.433884 | + | 0.900969i | −2.35102 | + | 1.86352i | −2.93320 | + | 0.219813i |
5.12 | −0.781831 | + | 0.623490i | −0.488019 | + | 1.66188i | 0.222521 | − | 0.974928i | 0.689539 | + | 0.470120i | −0.654615 | − | 1.60358i | 2.62337 | − | 0.343387i | 0.433884 | + | 0.900969i | −2.52368 | − | 1.62205i | −0.832218 | + | 0.0623661i |
5.13 | −0.781831 | + | 0.623490i | −0.485664 | − | 1.66257i | 0.222521 | − | 0.974928i | 0.154754 | + | 0.105510i | 1.41630 | + | 0.997041i | 0.0786204 | + | 2.64458i | 0.433884 | + | 0.900969i | −2.52826 | + | 1.61490i | −0.186776 | + | 0.0139969i |
5.14 | −0.781831 | + | 0.623490i | −0.310768 | − | 1.70394i | 0.222521 | − | 0.974928i | −0.786622 | − | 0.536309i | 1.30536 | + | 1.13844i | 2.58652 | − | 0.556718i | 0.433884 | + | 0.900969i | −2.80685 | + | 1.05906i | 0.949389 | − | 0.0711469i |
5.15 | −0.781831 | + | 0.623490i | −0.240845 | + | 1.71522i | 0.222521 | − | 0.974928i | −2.31965 | − | 1.58151i | −0.881125 | − | 1.49118i | 2.04849 | + | 1.67442i | 0.433884 | + | 0.900969i | −2.88399 | − | 0.826206i | 2.79963 | − | 0.209803i |
5.16 | −0.781831 | + | 0.623490i | 0.227043 | − | 1.71711i | 0.222521 | − | 0.974928i | −3.47346 | − | 2.36816i | 0.893088 | + | 1.48405i | 0.805676 | − | 2.52010i | 0.433884 | + | 0.900969i | −2.89690 | − | 0.779714i | 4.19218 | − | 0.314161i |
5.17 | −0.781831 | + | 0.623490i | 0.307117 | + | 1.70461i | 0.222521 | − | 0.974928i | 1.25571 | + | 0.856126i | −1.30292 | − | 1.14123i | −0.752068 | + | 2.53661i | 0.433884 | + | 0.900969i | −2.81136 | + | 1.04703i | −1.51554 | + | 0.113574i |
5.18 | −0.781831 | + | 0.623490i | 0.366631 | + | 1.69280i | 0.222521 | − | 0.974928i | −2.85326 | − | 1.94532i | −1.34209 | − | 1.09490i | −2.64286 | − | 0.123709i | 0.433884 | + | 0.900969i | −2.73116 | + | 1.24127i | 3.44365 | − | 0.258066i |
5.19 | −0.781831 | + | 0.623490i | 0.760180 | − | 1.55632i | 0.222521 | − | 0.974928i | −1.95730 | − | 1.33446i | 0.376016 | + | 1.69074i | −1.48234 | + | 2.19150i | 0.433884 | + | 0.900969i | −1.84425 | − | 2.36616i | 2.36230 | − | 0.177030i |
5.20 | −0.781831 | + | 0.623490i | 0.978483 | − | 1.42919i | 0.222521 | − | 0.974928i | 2.35227 | + | 1.60375i | 0.126073 | + | 1.72746i | 2.64481 | − | 0.0707346i | 0.433884 | + | 0.900969i | −1.08514 | − | 2.79687i | −2.83900 | + | 0.212754i |
See next 80 embeddings (of 672 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
441.bn | even | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 882.2.bk.a | yes | 672 |
9.d | odd | 6 | 1 | 882.2.bc.a | ✓ | 672 | |
49.h | odd | 42 | 1 | 882.2.bc.a | ✓ | 672 | |
441.bn | even | 42 | 1 | inner | 882.2.bk.a | yes | 672 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
882.2.bc.a | ✓ | 672 | 9.d | odd | 6 | 1 | |
882.2.bc.a | ✓ | 672 | 49.h | odd | 42 | 1 | |
882.2.bk.a | yes | 672 | 1.a | even | 1 | 1 | trivial |
882.2.bk.a | yes | 672 | 441.bn | even | 42 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(882, [\chi])\).