Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,2,Mod(299,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.299");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.q (of order \(6\), degree \(2\), not 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: | \(4.19214610612\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
299.1 | −1.36696 | + | 2.36764i | −1.70405 | − | 0.310170i | −2.73715 | − | 4.74088i | 0 | 3.06374 | − | 3.61059i | 1.39384 | + | 2.24882i | 9.49844 | 2.80759 | + | 1.05709i | 0 | ||||||
299.2 | −1.36696 | + | 2.36764i | −0.583411 | + | 1.63084i | −2.73715 | − | 4.74088i | 0 | −3.06374 | − | 3.61059i | −1.39384 | − | 2.24882i | 9.49844 | −2.31926 | − | 1.90290i | 0 | ||||||
299.3 | −1.12521 | + | 1.94891i | 1.42544 | − | 0.983931i | −1.53217 | − | 2.65380i | 0 | 0.313682 | + | 3.88518i | −2.22667 | − | 1.42897i | 2.39522 | 1.06376 | − | 2.80507i | 0 | ||||||
299.4 | −1.12521 | + | 1.94891i | 1.56483 | − | 0.742502i | −1.53217 | − | 2.65380i | 0 | −0.313682 | + | 3.88518i | 2.22667 | + | 1.42897i | 2.39522 | 1.89738 | − | 2.32378i | 0 | ||||||
299.5 | −0.846473 | + | 1.46613i | −1.53466 | − | 0.803015i | −0.433034 | − | 0.750036i | 0 | 2.47637 | − | 1.57028i | 2.01688 | − | 1.71236i | −1.91969 | 1.71033 | + | 2.46470i | 0 | ||||||
299.6 | −0.846473 | + | 1.46613i | −0.0718963 | + | 1.73056i | −0.433034 | − | 0.750036i | 0 | −2.47637 | − | 1.57028i | −2.01688 | + | 1.71236i | −1.91969 | −2.98966 | − | 0.248842i | 0 | ||||||
299.7 | −0.450666 | + | 0.780577i | 0.248842 | − | 1.71408i | 0.593800 | + | 1.02849i | 0 | 1.22583 | + | 0.966719i | −2.64365 | − | 0.105498i | −2.87309 | −2.87616 | − | 0.853070i | 0 | ||||||
299.8 | −0.450666 | + | 0.780577i | 1.60886 | + | 0.641538i | 0.593800 | + | 1.02849i | 0 | −1.22583 | + | 0.966719i | 2.64365 | + | 0.105498i | −2.87309 | 2.17686 | + | 2.06429i | 0 | ||||||
299.9 | −0.442404 | + | 0.766266i | −1.72165 | + | 0.189492i | 0.608557 | + | 1.05405i | 0 | 0.616465 | − | 1.40308i | 0.206062 | − | 2.63771i | −2.84653 | 2.92819 | − | 0.652481i | 0 | ||||||
299.10 | −0.442404 | + | 0.766266i | −1.02493 | + | 1.39625i | 0.608557 | + | 1.05405i | 0 | −0.616465 | − | 1.40308i | −0.206062 | + | 2.63771i | −2.84653 | −0.899028 | − | 2.86212i | 0 | ||||||
299.11 | 0.442404 | − | 0.766266i | 1.02493 | − | 1.39625i | 0.608557 | + | 1.05405i | 0 | −0.616465 | − | 1.40308i | 0.206062 | − | 2.63771i | 2.84653 | −0.899028 | − | 2.86212i | 0 | ||||||
299.12 | 0.442404 | − | 0.766266i | 1.72165 | − | 0.189492i | 0.608557 | + | 1.05405i | 0 | 0.616465 | − | 1.40308i | −0.206062 | + | 2.63771i | 2.84653 | 2.92819 | − | 0.652481i | 0 | ||||||
299.13 | 0.450666 | − | 0.780577i | −1.60886 | − | 0.641538i | 0.593800 | + | 1.02849i | 0 | −1.22583 | + | 0.966719i | −2.64365 | − | 0.105498i | 2.87309 | 2.17686 | + | 2.06429i | 0 | ||||||
299.14 | 0.450666 | − | 0.780577i | −0.248842 | + | 1.71408i | 0.593800 | + | 1.02849i | 0 | 1.22583 | + | 0.966719i | 2.64365 | + | 0.105498i | 2.87309 | −2.87616 | − | 0.853070i | 0 | ||||||
299.15 | 0.846473 | − | 1.46613i | 0.0718963 | − | 1.73056i | −0.433034 | − | 0.750036i | 0 | −2.47637 | − | 1.57028i | 2.01688 | − | 1.71236i | 1.91969 | −2.98966 | − | 0.248842i | 0 | ||||||
299.16 | 0.846473 | − | 1.46613i | 1.53466 | + | 0.803015i | −0.433034 | − | 0.750036i | 0 | 2.47637 | − | 1.57028i | −2.01688 | + | 1.71236i | 1.91969 | 1.71033 | + | 2.46470i | 0 | ||||||
299.17 | 1.12521 | − | 1.94891i | −1.56483 | + | 0.742502i | −1.53217 | − | 2.65380i | 0 | −0.313682 | + | 3.88518i | −2.22667 | − | 1.42897i | −2.39522 | 1.89738 | − | 2.32378i | 0 | ||||||
299.18 | 1.12521 | − | 1.94891i | −1.42544 | + | 0.983931i | −1.53217 | − | 2.65380i | 0 | 0.313682 | + | 3.88518i | 2.22667 | + | 1.42897i | −2.39522 | 1.06376 | − | 2.80507i | 0 | ||||||
299.19 | 1.36696 | − | 2.36764i | 0.583411 | − | 1.63084i | −2.73715 | − | 4.74088i | 0 | −3.06374 | − | 3.61059i | 1.39384 | + | 2.24882i | −9.49844 | −2.31926 | − | 1.90290i | 0 | ||||||
299.20 | 1.36696 | − | 2.36764i | 1.70405 | + | 0.310170i | −2.73715 | − | 4.74088i | 0 | 3.06374 | − | 3.61059i | −1.39384 | − | 2.24882i | −9.49844 | 2.80759 | + | 1.05709i | 0 | ||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
15.d | odd | 2 | 1 | inner |
21.g | even | 6 | 1 | inner |
35.i | odd | 6 | 1 | inner |
105.p | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.2.q.g | 40 | |
3.b | odd | 2 | 1 | inner | 525.2.q.g | 40 | |
5.b | even | 2 | 1 | inner | 525.2.q.g | 40 | |
5.c | odd | 4 | 1 | 525.2.t.h | ✓ | 20 | |
5.c | odd | 4 | 1 | 525.2.t.i | yes | 20 | |
7.d | odd | 6 | 1 | inner | 525.2.q.g | 40 | |
15.d | odd | 2 | 1 | inner | 525.2.q.g | 40 | |
15.e | even | 4 | 1 | 525.2.t.h | ✓ | 20 | |
15.e | even | 4 | 1 | 525.2.t.i | yes | 20 | |
21.g | even | 6 | 1 | inner | 525.2.q.g | 40 | |
35.i | odd | 6 | 1 | inner | 525.2.q.g | 40 | |
35.k | even | 12 | 1 | 525.2.t.h | ✓ | 20 | |
35.k | even | 12 | 1 | 525.2.t.i | yes | 20 | |
105.p | even | 6 | 1 | inner | 525.2.q.g | 40 | |
105.w | odd | 12 | 1 | 525.2.t.h | ✓ | 20 | |
105.w | odd | 12 | 1 | 525.2.t.i | yes | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
525.2.q.g | 40 | 1.a | even | 1 | 1 | trivial | |
525.2.q.g | 40 | 3.b | odd | 2 | 1 | inner | |
525.2.q.g | 40 | 5.b | even | 2 | 1 | inner | |
525.2.q.g | 40 | 7.d | odd | 6 | 1 | inner | |
525.2.q.g | 40 | 15.d | odd | 2 | 1 | inner | |
525.2.q.g | 40 | 21.g | even | 6 | 1 | inner | |
525.2.q.g | 40 | 35.i | odd | 6 | 1 | inner | |
525.2.q.g | 40 | 105.p | even | 6 | 1 | inner | |
525.2.t.h | ✓ | 20 | 5.c | odd | 4 | 1 | |
525.2.t.h | ✓ | 20 | 15.e | even | 4 | 1 | |
525.2.t.h | ✓ | 20 | 35.k | even | 12 | 1 | |
525.2.t.h | ✓ | 20 | 105.w | odd | 12 | 1 | |
525.2.t.i | yes | 20 | 5.c | odd | 4 | 1 | |
525.2.t.i | yes | 20 | 15.e | even | 4 | 1 | |
525.2.t.i | yes | 20 | 35.k | even | 12 | 1 | |
525.2.t.i | yes | 20 | 105.w | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):
\( T_{2}^{20} + 17 T_{2}^{18} + 190 T_{2}^{16} + 1211 T_{2}^{14} + 5569 T_{2}^{12} + 15953 T_{2}^{10} + \cdots + 4761 \) |
\( T_{11}^{20} - 50 T_{11}^{18} + 2059 T_{11}^{16} - 19244 T_{11}^{14} + 122755 T_{11}^{12} - 461399 T_{11}^{10} + \cdots + 76176 \) |
\( T_{13}^{10} - 72T_{13}^{8} + 1950T_{13}^{6} - 24651T_{13}^{4} + 142515T_{13}^{2} - 286443 \) |