Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [300,2,Mod(109,300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(300, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.109");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 300.o (of order \(10\), degree \(4\), 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: | \(2.39551206064\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
109.1 | 0 | −0.951057 | + | 0.309017i | 0 | −2.10592 | + | 0.751722i | 0 | 0.595901i | 0 | 0.809017 | − | 0.587785i | 0 | ||||||||||||
109.2 | 0 | −0.951057 | + | 0.309017i | 0 | 0.913250 | + | 2.04107i | 0 | − | 4.62675i | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||
109.3 | 0 | −0.951057 | + | 0.309017i | 0 | 0.971442 | − | 2.01403i | 0 | − | 1.04684i | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||
109.4 | 0 | 0.951057 | − | 0.309017i | 0 | −1.98828 | − | 1.02311i | 0 | − | 3.54704i | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||
109.5 | 0 | 0.951057 | − | 0.309017i | 0 | −1.64247 | + | 1.51733i | 0 | 3.78808i | 0 | 0.809017 | − | 0.587785i | 0 | ||||||||||||
109.6 | 0 | 0.951057 | − | 0.309017i | 0 | 2.23394 | − | 0.0974182i | 0 | − | 1.31873i | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||
169.1 | 0 | −0.587785 | − | 0.809017i | 0 | −1.74098 | + | 1.40321i | 0 | 1.57893i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||||
169.2 | 0 | −0.587785 | − | 0.809017i | 0 | −0.900274 | − | 2.04683i | 0 | 0.957526i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||||
169.3 | 0 | −0.587785 | − | 0.809017i | 0 | 1.99921 | + | 1.00158i | 0 | − | 3.80992i | 0 | −0.309017 | + | 0.951057i | 0 | |||||||||||
169.4 | 0 | 0.587785 | + | 0.809017i | 0 | −0.921600 | − | 2.03732i | 0 | 4.41540i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||||
169.5 | 0 | 0.587785 | + | 0.809017i | 0 | 0.892889 | − | 2.05006i | 0 | − | 4.13266i | 0 | −0.309017 | + | 0.951057i | 0 | |||||||||||
169.6 | 0 | 0.587785 | + | 0.809017i | 0 | 1.28878 | + | 1.82730i | 0 | 2.44380i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||||
229.1 | 0 | −0.587785 | + | 0.809017i | 0 | −1.74098 | − | 1.40321i | 0 | − | 1.57893i | 0 | −0.309017 | − | 0.951057i | 0 | |||||||||||
229.2 | 0 | −0.587785 | + | 0.809017i | 0 | −0.900274 | + | 2.04683i | 0 | − | 0.957526i | 0 | −0.309017 | − | 0.951057i | 0 | |||||||||||
229.3 | 0 | −0.587785 | + | 0.809017i | 0 | 1.99921 | − | 1.00158i | 0 | 3.80992i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||||
229.4 | 0 | 0.587785 | − | 0.809017i | 0 | −0.921600 | + | 2.03732i | 0 | − | 4.41540i | 0 | −0.309017 | − | 0.951057i | 0 | |||||||||||
229.5 | 0 | 0.587785 | − | 0.809017i | 0 | 0.892889 | + | 2.05006i | 0 | 4.13266i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||||
229.6 | 0 | 0.587785 | − | 0.809017i | 0 | 1.28878 | − | 1.82730i | 0 | − | 2.44380i | 0 | −0.309017 | − | 0.951057i | 0 | |||||||||||
289.1 | 0 | −0.951057 | − | 0.309017i | 0 | −2.10592 | − | 0.751722i | 0 | − | 0.595901i | 0 | 0.809017 | + | 0.587785i | 0 | |||||||||||
289.2 | 0 | −0.951057 | − | 0.309017i | 0 | 0.913250 | − | 2.04107i | 0 | 4.62675i | 0 | 0.809017 | + | 0.587785i | 0 | ||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.e | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 300.2.o.a | ✓ | 24 |
3.b | odd | 2 | 1 | 900.2.w.c | 24 | ||
5.b | even | 2 | 1 | 1500.2.o.c | 24 | ||
5.c | odd | 4 | 1 | 1500.2.m.c | 24 | ||
5.c | odd | 4 | 1 | 1500.2.m.d | 24 | ||
25.d | even | 5 | 1 | 1500.2.o.c | 24 | ||
25.d | even | 5 | 1 | 7500.2.d.g | 24 | ||
25.e | even | 10 | 1 | inner | 300.2.o.a | ✓ | 24 |
25.e | even | 10 | 1 | 7500.2.d.g | 24 | ||
25.f | odd | 20 | 1 | 1500.2.m.c | 24 | ||
25.f | odd | 20 | 1 | 1500.2.m.d | 24 | ||
25.f | odd | 20 | 1 | 7500.2.a.m | 12 | ||
25.f | odd | 20 | 1 | 7500.2.a.n | 12 | ||
75.h | odd | 10 | 1 | 900.2.w.c | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
300.2.o.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
300.2.o.a | ✓ | 24 | 25.e | even | 10 | 1 | inner |
900.2.w.c | 24 | 3.b | odd | 2 | 1 | ||
900.2.w.c | 24 | 75.h | odd | 10 | 1 | ||
1500.2.m.c | 24 | 5.c | odd | 4 | 1 | ||
1500.2.m.c | 24 | 25.f | odd | 20 | 1 | ||
1500.2.m.d | 24 | 5.c | odd | 4 | 1 | ||
1500.2.m.d | 24 | 25.f | odd | 20 | 1 | ||
1500.2.o.c | 24 | 5.b | even | 2 | 1 | ||
1500.2.o.c | 24 | 25.d | even | 5 | 1 | ||
7500.2.a.m | 12 | 25.f | odd | 20 | 1 | ||
7500.2.a.n | 12 | 25.f | odd | 20 | 1 | ||
7500.2.d.g | 24 | 25.d | even | 5 | 1 | ||
7500.2.d.g | 24 | 25.e | even | 10 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(300, [\chi])\).