Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [633,2,Mod(2,633)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(633, base_ring=CyclotomicField(210))
chi = DirichletCharacter(H, H._module([105, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("633.2");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 633 = 3 \cdot 211 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 633.bd (of order \(210\), degree \(48\), 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: | \(5.05453044795\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{210}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | 0 | 1.38587 | + | 1.03892i | 1.99910 | + | 0.0598309i | 0 | 0 | 1.24511 | − | 2.23309i | 0 | 0.841282 | + | 2.87963i | 0 | ||||||||||
17.1 | 0 | 1.21555 | − | 1.23387i | 1.89265 | − | 0.646419i | 0 | 0 | 0.415774 | − | 0.505657i | 0 | −0.0448782 | − | 2.99966i | 0 | ||||||||||
29.1 | 0 | −0.797856 | + | 1.53734i | 1.19964 | − | 1.60027i | 0 | 0 | −0.152905 | − | 2.55218i | 0 | −1.72685 | − | 2.45316i | 0 | ||||||||||
35.1 | 0 | 0.0518151 | + | 1.73128i | −0.503174 | + | 1.93567i | 0 | 0 | −0.969587 | − | 2.36578i | 0 | −2.99463 | + | 0.179412i | 0 | ||||||||||
41.1 | 0 | 0.103584 | + | 1.72895i | 1.74682 | + | 0.973978i | 0 | 0 | 3.36271 | + | 3.41339i | 0 | −2.97854 | + | 0.358183i | 0 | ||||||||||
92.1 | 0 | −1.05954 | + | 1.37018i | 1.54483 | + | 1.27023i | 0 | 0 | −2.75838 | − | 2.33810i | 0 | −0.754760 | − | 2.90350i | 0 | ||||||||||
116.1 | 0 | −1.69940 | − | 0.334729i | 1.29320 | − | 1.52566i | 0 | 0 | −2.09071 | + | 1.47171i | 0 | 2.77591 | + | 1.13768i | 0 | ||||||||||
131.1 | 0 | −0.797856 | − | 1.53734i | 1.19964 | + | 1.60027i | 0 | 0 | −0.152905 | + | 2.55218i | 0 | −1.72685 | + | 2.45316i | 0 | ||||||||||
149.1 | 0 | 1.21555 | + | 1.23387i | 1.89265 | + | 0.646419i | 0 | 0 | 0.415774 | + | 0.505657i | 0 | −0.0448782 | + | 2.99966i | 0 | ||||||||||
152.1 | 0 | 1.54911 | + | 0.774769i | −0.0299188 | − | 1.99978i | 0 | 0 | 3.49968 | − | 0.909735i | 0 | 1.79947 | + | 2.40040i | 0 | ||||||||||
155.1 | 0 | 0.410635 | − | 1.68267i | 0.894626 | + | 1.78875i | 0 | 0 | 0.108174 | + | 3.61437i | 0 | −2.66276 | − | 1.38193i | 0 | ||||||||||
158.1 | 0 | 1.66255 | + | 0.485714i | −1.99642 | + | 0.119608i | 0 | 0 | −4.07609 | − | 2.51885i | 0 | 2.52816 | + | 1.61505i | 0 | ||||||||||
164.1 | 0 | −1.63053 | + | 0.584273i | 1.68544 | + | 1.07670i | 0 | 0 | −0.155203 | + | 0.635979i | 0 | 2.31725 | − | 1.90535i | 0 | ||||||||||
167.1 | 0 | 1.67634 | + | 0.435761i | −0.386512 | + | 1.96230i | 0 | 0 | 2.52222 | + | 0.418842i | 0 | 2.62022 | + | 1.46097i | 0 | ||||||||||
191.1 | 0 | −1.69940 | + | 0.334729i | 1.29320 | + | 1.52566i | 0 | 0 | −2.09071 | − | 1.47171i | 0 | 2.77591 | − | 1.13768i | 0 | ||||||||||
218.1 | 0 | −0.206797 | − | 1.71966i | −1.05137 | − | 1.70136i | 0 | 0 | 0.654570 | + | 0.00979307i | 0 | −2.91447 | + | 0.711241i | 0 | ||||||||||
233.1 | 0 | 0.656838 | + | 1.60267i | 0.327636 | − | 1.97298i | 0 | 0 | −4.51071 | + | 1.61634i | 0 | −2.13713 | + | 2.10539i | 0 | ||||||||||
296.1 | 0 | −1.32126 | + | 1.11994i | −1.77517 | − | 0.921285i | 0 | 0 | −2.45417 | − | 0.716984i | 0 | 0.491454 | − | 2.95947i | 0 | ||||||||||
302.1 | 0 | −1.70865 | + | 0.283741i | −1.15123 | + | 1.63544i | 0 | 0 | 2.57954 | + | 4.03795i | 0 | 2.83898 | − | 0.969629i | 0 | ||||||||||
317.1 | 0 | 1.38587 | − | 1.03892i | 1.99910 | − | 0.0598309i | 0 | 0 | 1.24511 | + | 2.23309i | 0 | 0.841282 | − | 2.87963i | 0 | ||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
211.p | odd | 210 | 1 | inner |
633.bd | even | 210 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 633.2.bd.a | ✓ | 48 |
3.b | odd | 2 | 1 | CM | 633.2.bd.a | ✓ | 48 |
211.p | odd | 210 | 1 | inner | 633.2.bd.a | ✓ | 48 |
633.bd | even | 210 | 1 | inner | 633.2.bd.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
633.2.bd.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
633.2.bd.a | ✓ | 48 | 3.b | odd | 2 | 1 | CM |
633.2.bd.a | ✓ | 48 | 211.p | odd | 210 | 1 | inner |
633.2.bd.a | ✓ | 48 | 633.bd | even | 210 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\).