Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [237,2,Mod(29,237)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(237, base_ring=CyclotomicField(78))
chi = DirichletCharacter(H, H._module([39, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("237.29");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 237 = 3 \cdot 79 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 237.n (of order \(78\), degree \(24\), 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: | \(1.89245452790\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{78}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 | 0 | −1.56482 | − | 0.742517i | −1.83996 | − | 0.783933i | 0 | 0 | 0.193617 | + | 2.39838i | 0 | 1.89734 | + | 2.32381i | 0 | ||||||||||
35.1 | 0 | −0.139372 | − | 1.72643i | 1.59889 | − | 1.20148i | 0 | 0 | 3.90354 | + | 1.85225i | 0 | −2.96115 | + | 0.481234i | 0 | ||||||||||
47.1 | 0 | 1.24916 | − | 1.19983i | 1.97410 | + | 0.320823i | 0 | 0 | 0.0572213 | + | 0.0165743i | 0 | 0.120798 | − | 2.99757i | 0 | ||||||||||
53.1 | 0 | −1.73065 | + | 0.0697427i | 1.89707 | − | 0.633336i | 0 | 0 | −2.34544 | − | 3.70902i | 0 | 2.99027 | − | 0.241400i | 0 | ||||||||||
59.1 | 0 | 0.548485 | + | 1.64291i | −1.69038 | − | 1.06893i | 0 | 0 | 5.15347 | + | 1.05209i | 0 | −2.39833 | + | 1.80223i | 0 | ||||||||||
68.1 | 0 | 0.678906 | + | 1.59345i | −1.99351 | + | 0.160933i | 0 | 0 | −1.50887 | + | 2.00794i | 0 | −2.07817 | + | 2.16361i | 0 | ||||||||||
74.1 | 0 | −1.04052 | − | 1.38468i | 0.857385 | + | 1.80690i | 0 | 0 | −1.94747 | + | 4.57087i | 0 | −0.834652 | + | 2.88155i | 0 | ||||||||||
77.1 | 0 | −0.277840 | + | 1.70962i | 0.556435 | + | 1.92104i | 0 | 0 | 0.936046 | + | 0.764258i | 0 | −2.84561 | − | 0.950004i | 0 | ||||||||||
86.1 | 0 | −0.925722 | + | 1.46391i | −0.400051 | − | 1.95958i | 0 | 0 | 3.55040 | + | 0.143076i | 0 | −1.28608 | − | 2.71035i | 0 | ||||||||||
107.1 | 0 | 1.34166 | − | 1.09543i | 1.38545 | + | 1.44240i | 0 | 0 | −0.790218 | − | 4.86241i | 0 | 0.600077 | − | 2.93937i | 0 | ||||||||||
113.1 | 0 | −0.925722 | − | 1.46391i | −0.400051 | + | 1.95958i | 0 | 0 | 3.55040 | − | 0.143076i | 0 | −1.28608 | + | 2.71035i | 0 | ||||||||||
116.1 | 0 | 1.24916 | + | 1.19983i | 1.97410 | − | 0.320823i | 0 | 0 | 0.0572213 | − | 0.0165743i | 0 | 0.120798 | + | 2.99757i | 0 | ||||||||||
122.1 | 0 | 0.678906 | − | 1.59345i | −1.99351 | − | 0.160933i | 0 | 0 | −1.50887 | − | 2.00794i | 0 | −2.07817 | − | 2.16361i | 0 | ||||||||||
149.1 | 0 | −0.139372 | + | 1.72643i | 1.59889 | + | 1.20148i | 0 | 0 | 3.90354 | − | 1.85225i | 0 | −2.96115 | − | 0.481234i | 0 | ||||||||||
161.1 | 0 | −1.73065 | − | 0.0697427i | 1.89707 | + | 0.633336i | 0 | 0 | −2.34544 | + | 3.70902i | 0 | 2.99027 | + | 0.241400i | 0 | ||||||||||
164.1 | 0 | 1.69705 | + | 0.346455i | −0.0805319 | + | 1.99838i | 0 | 0 | −0.419301 | − | 1.25596i | 0 | 2.75994 | + | 1.17590i | 0 | ||||||||||
188.1 | 0 | −1.56482 | + | 0.742517i | −1.83996 | + | 0.783933i | 0 | 0 | 0.193617 | − | 2.39838i | 0 | 1.89734 | − | 2.32381i | 0 | ||||||||||
197.1 | 0 | −0.277840 | − | 1.70962i | 0.556435 | − | 1.92104i | 0 | 0 | 0.936046 | − | 0.764258i | 0 | −2.84561 | + | 0.950004i | 0 | ||||||||||
206.1 | 0 | 1.34166 | + | 1.09543i | 1.38545 | − | 1.44240i | 0 | 0 | −0.790218 | + | 4.86241i | 0 | 0.600077 | + | 2.93937i | 0 | ||||||||||
212.1 | 0 | 1.66367 | + | 0.481887i | −1.26489 | + | 1.54921i | 0 | 0 | −3.78300 | + | 3.63362i | 0 | 2.53557 | + | 1.60340i | 0 | ||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
79.h | odd | 78 | 1 | inner |
237.n | even | 78 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 237.2.n.a | ✓ | 24 |
3.b | odd | 2 | 1 | CM | 237.2.n.a | ✓ | 24 |
79.h | odd | 78 | 1 | inner | 237.2.n.a | ✓ | 24 |
237.n | even | 78 | 1 | inner | 237.2.n.a | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
237.2.n.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
237.2.n.a | ✓ | 24 | 3.b | odd | 2 | 1 | CM |
237.2.n.a | ✓ | 24 | 79.h | odd | 78 | 1 | inner |
237.2.n.a | ✓ | 24 | 237.n | even | 78 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(237, [\chi])\).