Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,3,Mod(124,525)]
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([0, 3, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.124");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.s (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: | \(14.3052138789\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 105) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
124.1 | −3.44796 | − | 1.99068i | −0.866025 | − | 1.50000i | 5.92561 | + | 10.2635i | 0 | 6.89592i | 6.61672 | + | 2.28451i | − | 31.2586i | −1.50000 | + | 2.59808i | 0 | |||||||
124.2 | −3.12287 | − | 1.80299i | 0.866025 | + | 1.50000i | 4.50153 | + | 7.79688i | 0 | − | 6.24573i | 6.55404 | − | 2.45857i | − | 18.0409i | −1.50000 | + | 2.59808i | 0 | ||||||
124.3 | −2.54146 | − | 1.46731i | 0.866025 | + | 1.50000i | 2.30602 | + | 3.99415i | 0 | − | 5.08293i | −4.43411 | + | 5.41652i | − | 1.79613i | −1.50000 | + | 2.59808i | 0 | ||||||
124.4 | −2.17524 | − | 1.25588i | −0.866025 | − | 1.50000i | 1.15446 | + | 1.99958i | 0 | 4.35049i | −1.85004 | + | 6.75110i | 4.24760i | −1.50000 | + | 2.59808i | 0 | ||||||||
124.5 | −1.19112 | − | 0.687692i | 0.866025 | + | 1.50000i | −1.05416 | − | 1.82586i | 0 | − | 2.38224i | −2.42539 | − | 6.56639i | 8.40129i | −1.50000 | + | 2.59808i | 0 | |||||||
124.6 | −0.499806 | − | 0.288563i | 0.866025 | + | 1.50000i | −1.83346 | − | 3.17565i | 0 | − | 0.999611i | 6.80420 | + | 1.64406i | 4.42478i | −1.50000 | + | 2.59808i | 0 | |||||||
124.7 | 0.499806 | + | 0.288563i | −0.866025 | − | 1.50000i | −1.83346 | − | 3.17565i | 0 | − | 0.999611i | −6.80420 | − | 1.64406i | − | 4.42478i | −1.50000 | + | 2.59808i | 0 | ||||||
124.8 | 1.19112 | + | 0.687692i | −0.866025 | − | 1.50000i | −1.05416 | − | 1.82586i | 0 | − | 2.38224i | 2.42539 | + | 6.56639i | − | 8.40129i | −1.50000 | + | 2.59808i | 0 | ||||||
124.9 | 2.17524 | + | 1.25588i | 0.866025 | + | 1.50000i | 1.15446 | + | 1.99958i | 0 | 4.35049i | 1.85004 | − | 6.75110i | − | 4.24760i | −1.50000 | + | 2.59808i | 0 | |||||||
124.10 | 2.54146 | + | 1.46731i | −0.866025 | − | 1.50000i | 2.30602 | + | 3.99415i | 0 | − | 5.08293i | 4.43411 | − | 5.41652i | 1.79613i | −1.50000 | + | 2.59808i | 0 | |||||||
124.11 | 3.12287 | + | 1.80299i | −0.866025 | − | 1.50000i | 4.50153 | + | 7.79688i | 0 | − | 6.24573i | −6.55404 | + | 2.45857i | 18.0409i | −1.50000 | + | 2.59808i | 0 | |||||||
124.12 | 3.44796 | + | 1.99068i | 0.866025 | + | 1.50000i | 5.92561 | + | 10.2635i | 0 | 6.89592i | −6.61672 | − | 2.28451i | 31.2586i | −1.50000 | + | 2.59808i | 0 | ||||||||
199.1 | −3.44796 | + | 1.99068i | −0.866025 | + | 1.50000i | 5.92561 | − | 10.2635i | 0 | − | 6.89592i | 6.61672 | − | 2.28451i | 31.2586i | −1.50000 | − | 2.59808i | 0 | |||||||
199.2 | −3.12287 | + | 1.80299i | 0.866025 | − | 1.50000i | 4.50153 | − | 7.79688i | 0 | 6.24573i | 6.55404 | + | 2.45857i | 18.0409i | −1.50000 | − | 2.59808i | 0 | ||||||||
199.3 | −2.54146 | + | 1.46731i | 0.866025 | − | 1.50000i | 2.30602 | − | 3.99415i | 0 | 5.08293i | −4.43411 | − | 5.41652i | 1.79613i | −1.50000 | − | 2.59808i | 0 | ||||||||
199.4 | −2.17524 | + | 1.25588i | −0.866025 | + | 1.50000i | 1.15446 | − | 1.99958i | 0 | − | 4.35049i | −1.85004 | − | 6.75110i | − | 4.24760i | −1.50000 | − | 2.59808i | 0 | ||||||
199.5 | −1.19112 | + | 0.687692i | 0.866025 | − | 1.50000i | −1.05416 | + | 1.82586i | 0 | 2.38224i | −2.42539 | + | 6.56639i | − | 8.40129i | −1.50000 | − | 2.59808i | 0 | |||||||
199.6 | −0.499806 | + | 0.288563i | 0.866025 | − | 1.50000i | −1.83346 | + | 3.17565i | 0 | 0.999611i | 6.80420 | − | 1.64406i | − | 4.42478i | −1.50000 | − | 2.59808i | 0 | |||||||
199.7 | 0.499806 | − | 0.288563i | −0.866025 | + | 1.50000i | −1.83346 | + | 3.17565i | 0 | 0.999611i | −6.80420 | + | 1.64406i | 4.42478i | −1.50000 | − | 2.59808i | 0 | ||||||||
199.8 | 1.19112 | − | 0.687692i | −0.866025 | + | 1.50000i | −1.05416 | + | 1.82586i | 0 | 2.38224i | 2.42539 | − | 6.56639i | 8.40129i | −1.50000 | − | 2.59808i | 0 | ||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
35.i | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.3.s.j | 24 | |
5.b | even | 2 | 1 | inner | 525.3.s.j | 24 | |
5.c | odd | 4 | 1 | 105.3.n.b | ✓ | 12 | |
5.c | odd | 4 | 1 | 525.3.o.m | 12 | ||
7.d | odd | 6 | 1 | inner | 525.3.s.j | 24 | |
15.e | even | 4 | 1 | 315.3.w.b | 12 | ||
35.i | odd | 6 | 1 | inner | 525.3.s.j | 24 | |
35.k | even | 12 | 1 | 105.3.n.b | ✓ | 12 | |
35.k | even | 12 | 1 | 525.3.o.m | 12 | ||
35.k | even | 12 | 1 | 735.3.h.b | 12 | ||
35.l | odd | 12 | 1 | 735.3.h.b | 12 | ||
105.w | odd | 12 | 1 | 315.3.w.b | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.3.n.b | ✓ | 12 | 5.c | odd | 4 | 1 | |
105.3.n.b | ✓ | 12 | 35.k | even | 12 | 1 | |
315.3.w.b | 12 | 15.e | even | 4 | 1 | ||
315.3.w.b | 12 | 105.w | odd | 12 | 1 | ||
525.3.o.m | 12 | 5.c | odd | 4 | 1 | ||
525.3.o.m | 12 | 35.k | even | 12 | 1 | ||
525.3.s.j | 24 | 1.a | even | 1 | 1 | trivial | |
525.3.s.j | 24 | 5.b | even | 2 | 1 | inner | |
525.3.s.j | 24 | 7.d | odd | 6 | 1 | inner | |
525.3.s.j | 24 | 35.i | odd | 6 | 1 | inner | |
735.3.h.b | 12 | 35.k | even | 12 | 1 | ||
735.3.h.b | 12 | 35.l | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\):
\( T_{2}^{24} - 46 T_{2}^{22} + 1327 T_{2}^{20} - 23878 T_{2}^{18} + 314989 T_{2}^{16} - 2905264 T_{2}^{14} + \cdots + 49787136 \) |
\( T_{11}^{12} - 20 T_{11}^{11} + 649 T_{11}^{10} - 1808 T_{11}^{9} + 99307 T_{11}^{8} + 399706 T_{11}^{7} + \cdots + 15854839056 \) |
\( T_{13}^{12} - 978 T_{13}^{10} + 326655 T_{13}^{8} - 49084380 T_{13}^{6} + 3468233295 T_{13}^{4} + \cdots + 732611029329 \) |