Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6300,2,Mod(1349,6300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 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("6300.1349");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6300.dd (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: | \(50.3057532734\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 1260) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1349.1 | 0 | 0 | 0 | 0 | 0 | −2.63285 | − | 0.260926i | 0 | 0 | 0 | ||||||||||||||||
1349.2 | 0 | 0 | 0 | 0 | 0 | −2.08318 | + | 1.63107i | 0 | 0 | 0 | ||||||||||||||||
1349.3 | 0 | 0 | 0 | 0 | 0 | −1.98249 | − | 1.75207i | 0 | 0 | 0 | ||||||||||||||||
1349.4 | 0 | 0 | 0 | 0 | 0 | −0.390758 | − | 2.61674i | 0 | 0 | 0 | ||||||||||||||||
1349.5 | 0 | 0 | 0 | 0 | 0 | −0.380350 | + | 2.61827i | 0 | 0 | 0 | ||||||||||||||||
1349.6 | 0 | 0 | 0 | 0 | 0 | −0.0290059 | − | 2.64559i | 0 | 0 | 0 | ||||||||||||||||
1349.7 | 0 | 0 | 0 | 0 | 0 | 0.0290059 | + | 2.64559i | 0 | 0 | 0 | ||||||||||||||||
1349.8 | 0 | 0 | 0 | 0 | 0 | 0.380350 | − | 2.61827i | 0 | 0 | 0 | ||||||||||||||||
1349.9 | 0 | 0 | 0 | 0 | 0 | 0.390758 | + | 2.61674i | 0 | 0 | 0 | ||||||||||||||||
1349.10 | 0 | 0 | 0 | 0 | 0 | 1.98249 | + | 1.75207i | 0 | 0 | 0 | ||||||||||||||||
1349.11 | 0 | 0 | 0 | 0 | 0 | 2.08318 | − | 1.63107i | 0 | 0 | 0 | ||||||||||||||||
1349.12 | 0 | 0 | 0 | 0 | 0 | 2.63285 | + | 0.260926i | 0 | 0 | 0 | ||||||||||||||||
4049.1 | 0 | 0 | 0 | 0 | 0 | −2.63285 | + | 0.260926i | 0 | 0 | 0 | ||||||||||||||||
4049.2 | 0 | 0 | 0 | 0 | 0 | −2.08318 | − | 1.63107i | 0 | 0 | 0 | ||||||||||||||||
4049.3 | 0 | 0 | 0 | 0 | 0 | −1.98249 | + | 1.75207i | 0 | 0 | 0 | ||||||||||||||||
4049.4 | 0 | 0 | 0 | 0 | 0 | −0.390758 | + | 2.61674i | 0 | 0 | 0 | ||||||||||||||||
4049.5 | 0 | 0 | 0 | 0 | 0 | −0.380350 | − | 2.61827i | 0 | 0 | 0 | ||||||||||||||||
4049.6 | 0 | 0 | 0 | 0 | 0 | −0.0290059 | + | 2.64559i | 0 | 0 | 0 | ||||||||||||||||
4049.7 | 0 | 0 | 0 | 0 | 0 | 0.0290059 | − | 2.64559i | 0 | 0 | 0 | ||||||||||||||||
4049.8 | 0 | 0 | 0 | 0 | 0 | 0.380350 | + | 2.61827i | 0 | 0 | 0 | ||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
21.g | even | 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 | 6300.2.dd.b | 24 | |
3.b | odd | 2 | 1 | 6300.2.dd.c | 24 | ||
5.b | even | 2 | 1 | inner | 6300.2.dd.b | 24 | |
5.c | odd | 4 | 1 | 1260.2.cg.b | yes | 12 | |
5.c | odd | 4 | 1 | 6300.2.ch.b | 12 | ||
7.d | odd | 6 | 1 | 6300.2.dd.c | 24 | ||
15.d | odd | 2 | 1 | 6300.2.dd.c | 24 | ||
15.e | even | 4 | 1 | 1260.2.cg.a | ✓ | 12 | |
15.e | even | 4 | 1 | 6300.2.ch.c | 12 | ||
21.g | even | 6 | 1 | inner | 6300.2.dd.b | 24 | |
35.i | odd | 6 | 1 | 6300.2.dd.c | 24 | ||
35.k | even | 12 | 1 | 1260.2.cg.a | ✓ | 12 | |
35.k | even | 12 | 1 | 6300.2.ch.c | 12 | ||
35.k | even | 12 | 1 | 8820.2.d.b | 12 | ||
35.l | odd | 12 | 1 | 8820.2.d.a | 12 | ||
105.p | even | 6 | 1 | inner | 6300.2.dd.b | 24 | |
105.w | odd | 12 | 1 | 1260.2.cg.b | yes | 12 | |
105.w | odd | 12 | 1 | 6300.2.ch.b | 12 | ||
105.w | odd | 12 | 1 | 8820.2.d.a | 12 | ||
105.x | even | 12 | 1 | 8820.2.d.b | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1260.2.cg.a | ✓ | 12 | 15.e | even | 4 | 1 | |
1260.2.cg.a | ✓ | 12 | 35.k | even | 12 | 1 | |
1260.2.cg.b | yes | 12 | 5.c | odd | 4 | 1 | |
1260.2.cg.b | yes | 12 | 105.w | odd | 12 | 1 | |
6300.2.ch.b | 12 | 5.c | odd | 4 | 1 | ||
6300.2.ch.b | 12 | 105.w | odd | 12 | 1 | ||
6300.2.ch.c | 12 | 15.e | even | 4 | 1 | ||
6300.2.ch.c | 12 | 35.k | even | 12 | 1 | ||
6300.2.dd.b | 24 | 1.a | even | 1 | 1 | trivial | |
6300.2.dd.b | 24 | 5.b | even | 2 | 1 | inner | |
6300.2.dd.b | 24 | 21.g | even | 6 | 1 | inner | |
6300.2.dd.b | 24 | 105.p | even | 6 | 1 | inner | |
6300.2.dd.c | 24 | 3.b | odd | 2 | 1 | ||
6300.2.dd.c | 24 | 7.d | odd | 6 | 1 | ||
6300.2.dd.c | 24 | 15.d | odd | 2 | 1 | ||
6300.2.dd.c | 24 | 35.i | odd | 6 | 1 | ||
8820.2.d.a | 12 | 35.l | odd | 12 | 1 | ||
8820.2.d.a | 12 | 105.w | odd | 12 | 1 | ||
8820.2.d.b | 12 | 35.k | even | 12 | 1 | ||
8820.2.d.b | 12 | 105.x | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{12} + 12 T_{11}^{11} + 38 T_{11}^{10} - 120 T_{11}^{9} - 636 T_{11}^{8} + 1320 T_{11}^{7} + 10748 T_{11}^{6} + 10056 T_{11}^{5} - 37340 T_{11}^{4} - 57600 T_{11}^{3} + 131040 T_{11}^{2} + 375840 T_{11} + 272484 \)
acting on \(S_{2}^{\mathrm{new}}(6300, [\chi])\).