Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [252,4,Mod(25,252)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(252, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("252.25");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 252.i (of order \(3\), degree \(2\), 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: | \(14.8684813214\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 | 0 | −5.19456 | − | 0.128780i | 0 | −5.68089 | + | 9.83959i | 0 | 18.4438 | + | 1.68138i | 0 | 26.9668 | + | 1.33791i | 0 | ||||||||||
25.2 | 0 | −5.19442 | − | 0.133994i | 0 | 7.81356 | − | 13.5335i | 0 | 6.71488 | − | 17.2601i | 0 | 26.9641 | + | 1.39204i | 0 | ||||||||||
25.3 | 0 | −4.72803 | + | 2.15539i | 0 | 4.93620 | − | 8.54976i | 0 | −6.20989 | + | 17.4481i | 0 | 17.7086 | − | 20.3815i | 0 | ||||||||||
25.4 | 0 | −4.58842 | − | 2.43852i | 0 | −6.20694 | + | 10.7507i | 0 | −18.4402 | + | 1.72023i | 0 | 15.1073 | + | 22.3779i | 0 | ||||||||||
25.5 | 0 | −4.33327 | + | 2.86754i | 0 | 0.863703 | − | 1.49598i | 0 | −16.5772 | − | 8.25811i | 0 | 10.5544 | − | 24.8517i | 0 | ||||||||||
25.6 | 0 | −4.01985 | − | 3.29253i | 0 | 3.23563 | − | 5.60427i | 0 | 4.58917 | + | 17.9427i | 0 | 5.31845 | + | 26.4710i | 0 | ||||||||||
25.7 | 0 | −2.96592 | + | 4.26654i | 0 | −5.16462 | + | 8.94539i | 0 | 8.74826 | − | 16.3238i | 0 | −9.40665 | − | 25.3084i | 0 | ||||||||||
25.8 | 0 | −2.55972 | − | 4.52193i | 0 | 6.02006 | − | 10.4271i | 0 | −8.82278 | − | 16.2837i | 0 | −13.8957 | + | 23.1497i | 0 | ||||||||||
25.9 | 0 | −1.93223 | + | 4.82354i | 0 | −7.79077 | + | 13.4940i | 0 | 0.496367 | + | 18.5136i | 0 | −19.5330 | − | 18.6403i | 0 | ||||||||||
25.10 | 0 | −1.41439 | − | 4.99995i | 0 | 2.84184 | − | 4.92220i | 0 | 15.2960 | + | 10.4419i | 0 | −22.9990 | + | 14.1438i | 0 | ||||||||||
25.11 | 0 | −1.19189 | + | 5.05761i | 0 | 8.50761 | − | 14.7356i | 0 | 18.0631 | + | 4.08939i | 0 | −24.1588 | − | 12.0562i | 0 | ||||||||||
25.12 | 0 | −0.977623 | − | 5.10336i | 0 | −6.36172 | + | 11.0188i | 0 | 5.55536 | − | 17.6674i | 0 | −25.0885 | + | 9.97831i | 0 | ||||||||||
25.13 | 0 | 0.340754 | + | 5.18497i | 0 | 4.29795 | − | 7.44428i | 0 | −11.6662 | − | 14.3840i | 0 | −26.7678 | + | 3.53359i | 0 | ||||||||||
25.14 | 0 | 1.24769 | + | 5.04413i | 0 | 1.32728 | − | 2.29891i | 0 | −14.0290 | + | 12.0908i | 0 | −23.8866 | + | 12.5870i | 0 | ||||||||||
25.15 | 0 | 1.30312 | − | 5.03010i | 0 | 4.53303 | − | 7.85144i | 0 | −15.6380 | + | 9.92235i | 0 | −23.6038 | − | 13.1096i | 0 | ||||||||||
25.16 | 0 | 2.05679 | − | 4.77175i | 0 | −7.73619 | + | 13.3995i | 0 | 10.1757 | + | 15.4743i | 0 | −18.5392 | − | 19.6290i | 0 | ||||||||||
25.17 | 0 | 3.26868 | + | 4.03928i | 0 | −0.275008 | + | 0.476327i | 0 | 18.2696 | − | 3.03661i | 0 | −5.63149 | + | 26.4062i | 0 | ||||||||||
25.18 | 0 | 3.59994 | + | 3.74705i | 0 | −10.9363 | + | 18.9422i | 0 | −13.2752 | − | 12.9139i | 0 | −1.08080 | + | 26.9784i | 0 | ||||||||||
25.19 | 0 | 3.62259 | − | 3.72516i | 0 | 10.2794 | − | 17.8044i | 0 | 18.5152 | − | 0.432294i | 0 | −0.753703 | − | 26.9895i | 0 | ||||||||||
25.20 | 0 | 4.15765 | − | 3.11672i | 0 | −4.98332 | + | 8.63137i | 0 | −18.3914 | + | 2.18061i | 0 | 7.57211 | − | 25.9165i | 0 | ||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.h | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 252.4.i.a | ✓ | 48 |
3.b | odd | 2 | 1 | 756.4.i.a | 48 | ||
7.c | even | 3 | 1 | 252.4.l.a | yes | 48 | |
9.c | even | 3 | 1 | 252.4.l.a | yes | 48 | |
9.d | odd | 6 | 1 | 756.4.l.a | 48 | ||
21.h | odd | 6 | 1 | 756.4.l.a | 48 | ||
63.h | even | 3 | 1 | inner | 252.4.i.a | ✓ | 48 |
63.j | odd | 6 | 1 | 756.4.i.a | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.4.i.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
252.4.i.a | ✓ | 48 | 63.h | even | 3 | 1 | inner |
252.4.l.a | yes | 48 | 7.c | even | 3 | 1 | |
252.4.l.a | yes | 48 | 9.c | even | 3 | 1 | |
756.4.i.a | 48 | 3.b | odd | 2 | 1 | ||
756.4.i.a | 48 | 63.j | odd | 6 | 1 | ||
756.4.l.a | 48 | 9.d | odd | 6 | 1 | ||
756.4.l.a | 48 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(252, [\chi])\).