Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,2,Mod(103,630)]
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, 9, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.103");
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.bw (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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
103.1 | −0.707107 | + | 0.707107i | −1.71556 | − | 0.238440i | − | 1.00000i | 0.973575 | + | 2.01300i | 1.38169 | − | 1.04448i | 0.404878 | + | 2.61459i | 0.707107 | + | 0.707107i | 2.88629 | + | 0.818117i | −2.11182 | − | 0.734981i | |
103.2 | −0.707107 | + | 0.707107i | −1.67295 | − | 0.448583i | − | 1.00000i | −0.239265 | + | 2.22323i | 1.50015 | − | 0.865760i | −0.403547 | − | 2.61479i | 0.707107 | + | 0.707107i | 2.59755 | + | 1.50092i | −1.40288 | − | 1.74125i | |
103.3 | −0.707107 | + | 0.707107i | −1.53699 | + | 0.798544i | − | 1.00000i | 2.11145 | − | 0.736070i | 0.522158 | − | 1.65147i | 2.64560 | − | 0.0283592i | 0.707107 | + | 0.707107i | 1.72466 | − | 2.45470i | −0.972537 | + | 2.01350i | |
103.4 | −0.707107 | + | 0.707107i | −1.38967 | + | 1.03383i | − | 1.00000i | 2.18560 | + | 0.472406i | 0.251617 | − | 1.71368i | −2.55035 | − | 0.704070i | 0.707107 | + | 0.707107i | 0.862380 | − | 2.87338i | −1.87949 | + | 1.21141i | |
103.5 | −0.707107 | + | 0.707107i | −1.37690 | + | 1.05079i | − | 1.00000i | −2.23501 | − | 0.0688901i | 0.230594 | − | 1.71663i | −2.60633 | + | 0.455003i | 0.707107 | + | 0.707107i | 0.791690 | − | 2.89365i | 1.62910 | − | 1.53168i | |
103.6 | −0.707107 | + | 0.707107i | −1.34565 | − | 1.09051i | − | 1.00000i | 1.81952 | − | 1.29974i | 1.72263 | − | 0.180414i | 1.89634 | − | 1.84497i | 0.707107 | + | 0.707107i | 0.621572 | + | 2.93490i | −0.367541 | + | 2.20565i | |
103.7 | −0.707107 | + | 0.707107i | −1.12396 | − | 1.31785i | − | 1.00000i | −1.44027 | − | 1.71045i | 1.72662 | + | 0.137102i | −2.58180 | − | 0.578175i | 0.707107 | + | 0.707107i | −0.473446 | + | 2.96241i | 2.22789 | + | 0.191049i | |
103.8 | −0.707107 | + | 0.707107i | −0.949637 | − | 1.44851i | − | 1.00000i | −2.17592 | − | 0.515133i | 1.69575 | + | 0.352758i | 2.58119 | − | 0.580908i | 0.707107 | + | 0.707107i | −1.19638 | + | 2.75112i | 1.90286 | − | 1.17435i | |
103.9 | −0.707107 | + | 0.707107i | −0.721761 | + | 1.57450i | − | 1.00000i | −0.997387 | + | 2.00130i | −0.602980 | − | 1.62370i | 0.960462 | − | 2.46526i | 0.707107 | + | 0.707107i | −1.95812 | − | 2.27283i | −0.709876 | − | 2.12040i | |
103.10 | −0.707107 | + | 0.707107i | −0.688185 | − | 1.58947i | − | 1.00000i | 1.94881 | − | 1.09641i | 1.61054 | + | 0.637302i | −0.414054 | + | 2.61315i | 0.707107 | + | 0.707107i | −2.05280 | + | 2.18769i | −0.602738 | + | 2.15330i | |
103.11 | −0.707107 | + | 0.707107i | −0.374933 | + | 1.69098i | − | 1.00000i | −0.935649 | − | 2.03090i | −0.930588 | − | 1.46082i | 2.14864 | + | 1.54381i | 0.707107 | + | 0.707107i | −2.71885 | − | 1.26801i | 2.09767 | + | 0.774460i | |
103.12 | −0.707107 | + | 0.707107i | 0.0979337 | − | 1.72928i | − | 1.00000i | −1.65263 | + | 1.50626i | 1.15354 | + | 1.29204i | 0.935970 | + | 2.47466i | 0.707107 | + | 0.707107i | −2.98082 | − | 0.338709i | 0.103501 | − | 2.23367i | |
103.13 | −0.707107 | + | 0.707107i | 0.272070 | + | 1.71055i | − | 1.00000i | −1.78688 | + | 1.34427i | −1.40192 | − | 1.01716i | −0.883234 | + | 2.49397i | 0.707107 | + | 0.707107i | −2.85196 | + | 0.930777i | 0.312975 | − | 2.21406i | |
103.14 | −0.707107 | + | 0.707107i | 0.379200 | + | 1.69003i | − | 1.00000i | 0.792977 | − | 2.09074i | −1.46317 | − | 0.926898i | −1.57559 | − | 2.12544i | 0.707107 | + | 0.707107i | −2.71241 | + | 1.28172i | 0.917656 | + | 2.03909i | |
103.15 | −0.707107 | + | 0.707107i | 0.757106 | − | 1.55782i | − | 1.00000i | 1.59411 | + | 1.56806i | 0.566187 | + | 1.63690i | 2.62505 | + | 0.330352i | 0.707107 | + | 0.707107i | −1.85358 | − | 2.35886i | −2.23599 | + | 0.0184185i | |
103.16 | −0.707107 | + | 0.707107i | 0.894028 | − | 1.48348i | − | 1.00000i | −1.22625 | − | 1.86984i | 0.416805 | + | 1.68115i | 1.29629 | − | 2.30643i | 0.707107 | + | 0.707107i | −1.40143 | − | 2.65255i | 2.18927 | + | 0.455093i | |
103.17 | −0.707107 | + | 0.707107i | 0.929118 | − | 1.46176i | − | 1.00000i | 1.34455 | − | 1.78667i | 0.376634 | + | 1.69061i | −2.51243 | − | 0.829274i | 0.707107 | + | 0.707107i | −1.27348 | − | 2.71629i | 0.312627 | + | 2.21411i | |
103.18 | −0.707107 | + | 0.707107i | 1.20754 | + | 1.24171i | − | 1.00000i | 2.21622 | − | 0.297244i | −1.73188 | − | 0.0241584i | −0.456895 | + | 2.60600i | 0.707107 | + | 0.707107i | −0.0836789 | + | 2.99883i | −1.35692 | + | 1.77729i | |
103.19 | −0.707107 | + | 0.707107i | 1.45946 | − | 0.932722i | − | 1.00000i | 0.207084 | + | 2.22646i | −0.372462 | + | 1.69153i | −2.32758 | + | 1.25792i | 0.707107 | + | 0.707107i | 1.26006 | − | 2.72254i | −1.72077 | − | 1.42791i | |
103.20 | −0.707107 | + | 0.707107i | 1.47488 | + | 0.908144i | − | 1.00000i | −2.17109 | − | 0.535116i | −1.68505 | + | 0.400744i | −1.11920 | − | 2.39737i | 0.707107 | + | 0.707107i | 1.35055 | + | 2.67881i | 1.91358 | − | 1.15681i | |
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.t | odd | 6 | 1 | inner |
315.bs | 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.bw.a | ✓ | 192 |
5.c | odd | 4 | 1 | inner | 630.2.bw.a | ✓ | 192 |
7.d | odd | 6 | 1 | 630.2.cg.a | yes | 192 | |
9.c | even | 3 | 1 | 630.2.cg.a | yes | 192 | |
35.k | even | 12 | 1 | 630.2.cg.a | yes | 192 | |
45.k | odd | 12 | 1 | 630.2.cg.a | yes | 192 | |
63.t | odd | 6 | 1 | inner | 630.2.bw.a | ✓ | 192 |
315.bs | even | 12 | 1 | inner | 630.2.bw.a | ✓ | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
630.2.bw.a | ✓ | 192 | 1.a | even | 1 | 1 | trivial |
630.2.bw.a | ✓ | 192 | 5.c | odd | 4 | 1 | inner |
630.2.bw.a | ✓ | 192 | 63.t | odd | 6 | 1 | inner |
630.2.bw.a | ✓ | 192 | 315.bs | even | 12 | 1 | inner |
630.2.cg.a | yes | 192 | 7.d | odd | 6 | 1 | |
630.2.cg.a | yes | 192 | 9.c | even | 3 | 1 | |
630.2.cg.a | yes | 192 | 35.k | even | 12 | 1 | |
630.2.cg.a | yes | 192 | 45.k | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(630, [\chi])\).