Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,3,Mod(349,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.349");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.e (of order \(2\), degree \(1\), 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: | \(24\) |
Twist minimal: | no (minimal twist has level 105) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
349.1 | − | 1.71214i | −1.73205 | 1.06857 | 0 | 2.96552i | −6.15534 | − | 3.33344i | − | 8.67811i | 3.00000 | 0 | ||||||||||||||
349.2 | 1.71214i | −1.73205 | 1.06857 | 0 | − | 2.96552i | −6.15534 | + | 3.33344i | 8.67811i | 3.00000 | 0 | |||||||||||||||
349.3 | − | 3.80460i | 1.73205 | −10.4750 | 0 | − | 6.58976i | 6.55866 | + | 2.44621i | 24.6348i | 3.00000 | 0 | ||||||||||||||
349.4 | 3.80460i | 1.73205 | −10.4750 | 0 | 6.58976i | 6.55866 | − | 2.44621i | − | 24.6348i | 3.00000 | 0 | |||||||||||||||
349.5 | − | 0.112974i | −1.73205 | 3.98724 | 0 | 0.195676i | −1.98374 | − | 6.71303i | − | 0.902349i | 3.00000 | 0 | ||||||||||||||
349.6 | 0.112974i | −1.73205 | 3.98724 | 0 | − | 0.195676i | −1.98374 | + | 6.71303i | 0.902349i | 3.00000 | 0 | |||||||||||||||
349.7 | − | 2.79155i | −1.73205 | −3.79273 | 0 | 4.83510i | 5.63139 | + | 4.15782i | − | 0.578591i | 3.00000 | 0 | ||||||||||||||
349.8 | 2.79155i | −1.73205 | −3.79273 | 0 | − | 4.83510i | 5.63139 | − | 4.15782i | 0.578591i | 3.00000 | 0 | |||||||||||||||
349.9 | − | 3.50369i | −1.73205 | −8.27584 | 0 | 6.06857i | 2.03600 | + | 6.69736i | 14.9812i | 3.00000 | 0 | |||||||||||||||
349.10 | 3.50369i | −1.73205 | −8.27584 | 0 | − | 6.06857i | 2.03600 | − | 6.69736i | − | 14.9812i | 3.00000 | 0 | ||||||||||||||
349.11 | − | 3.80460i | −1.73205 | −10.4750 | 0 | 6.58976i | −6.55866 | + | 2.44621i | 24.6348i | 3.00000 | 0 | |||||||||||||||
349.12 | 3.80460i | −1.73205 | −10.4750 | 0 | − | 6.58976i | −6.55866 | − | 2.44621i | − | 24.6348i | 3.00000 | 0 | ||||||||||||||
349.13 | − | 2.91758i | 1.73205 | −4.51225 | 0 | − | 5.05339i | 3.36195 | − | 6.13981i | 1.49451i | 3.00000 | 0 | ||||||||||||||
349.14 | 2.91758i | 1.73205 | −4.51225 | 0 | 5.05339i | 3.36195 | + | 6.13981i | − | 1.49451i | 3.00000 | 0 | |||||||||||||||
349.15 | − | 1.71214i | 1.73205 | 1.06857 | 0 | − | 2.96552i | 6.15534 | − | 3.33344i | − | 8.67811i | 3.00000 | 0 | |||||||||||||
349.16 | 1.71214i | 1.73205 | 1.06857 | 0 | 2.96552i | 6.15534 | + | 3.33344i | 8.67811i | 3.00000 | 0 | ||||||||||||||||
349.17 | − | 0.112974i | 1.73205 | 3.98724 | 0 | − | 0.195676i | 1.98374 | − | 6.71303i | − | 0.902349i | 3.00000 | 0 | |||||||||||||
349.18 | 0.112974i | 1.73205 | 3.98724 | 0 | 0.195676i | 1.98374 | + | 6.71303i | 0.902349i | 3.00000 | 0 | ||||||||||||||||
349.19 | − | 2.91758i | −1.73205 | −4.51225 | 0 | 5.05339i | −3.36195 | − | 6.13981i | 1.49451i | 3.00000 | 0 | |||||||||||||||
349.20 | 2.91758i | −1.73205 | −4.51225 | 0 | − | 5.05339i | −3.36195 | + | 6.13981i | − | 1.49451i | 3.00000 | 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.b | odd | 2 | 1 | inner |
35.c | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.3.e.c | 24 | |
5.b | even | 2 | 1 | inner | 525.3.e.c | 24 | |
5.c | odd | 4 | 1 | 105.3.h.a | ✓ | 12 | |
5.c | odd | 4 | 1 | 525.3.h.d | 12 | ||
7.b | odd | 2 | 1 | inner | 525.3.e.c | 24 | |
15.e | even | 4 | 1 | 315.3.h.d | 12 | ||
20.e | even | 4 | 1 | 1680.3.s.c | 12 | ||
35.c | odd | 2 | 1 | inner | 525.3.e.c | 24 | |
35.f | even | 4 | 1 | 105.3.h.a | ✓ | 12 | |
35.f | even | 4 | 1 | 525.3.h.d | 12 | ||
105.k | odd | 4 | 1 | 315.3.h.d | 12 | ||
140.j | odd | 4 | 1 | 1680.3.s.c | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.3.h.a | ✓ | 12 | 5.c | odd | 4 | 1 | |
105.3.h.a | ✓ | 12 | 35.f | even | 4 | 1 | |
315.3.h.d | 12 | 15.e | even | 4 | 1 | ||
315.3.h.d | 12 | 105.k | odd | 4 | 1 | ||
525.3.e.c | 24 | 1.a | even | 1 | 1 | trivial | |
525.3.e.c | 24 | 5.b | even | 2 | 1 | inner | |
525.3.e.c | 24 | 7.b | odd | 2 | 1 | inner | |
525.3.e.c | 24 | 35.c | odd | 2 | 1 | inner | |
525.3.h.d | 12 | 5.c | odd | 4 | 1 | ||
525.3.h.d | 12 | 35.f | even | 4 | 1 | ||
1680.3.s.c | 12 | 20.e | even | 4 | 1 | ||
1680.3.s.c | 12 | 140.j | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 46T_{2}^{10} + 807T_{2}^{8} + 6676T_{2}^{6} + 25567T_{2}^{4} + 34878T_{2}^{2} + 441 \)
acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\).