Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [168,2,Mod(37,168)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(168, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("168.37");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 168 = 2^{3} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 168.bc (of order \(6\), degree \(2\), 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: | \(1.34148675396\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −1.40107 | − | 0.192386i | 0.866025 | − | 0.500000i | 1.92598 | + | 0.539091i | −2.93503 | − | 1.69454i | −1.30955 | + | 0.533922i | −1.85242 | − | 1.88906i | −2.59471 | − | 1.12583i | 0.500000 | − | 0.866025i | 3.78617 | + | 2.93882i |
37.2 | −1.33630 | + | 0.462932i | −0.866025 | + | 0.500000i | 1.57139 | − | 1.23723i | −1.56250 | − | 0.902108i | 0.925802 | − | 1.06906i | 2.63683 | − | 0.217074i | −1.52709 | + | 2.38076i | 0.500000 | − | 0.866025i | 2.50558 | + | 0.482155i |
37.3 | −1.31787 | − | 0.513056i | −0.866025 | + | 0.500000i | 1.47355 | + | 1.35228i | −0.0402223 | − | 0.0232224i | 1.39783 | − | 0.214614i | −1.97032 | + | 1.76574i | −1.24814 | − | 2.53814i | 0.500000 | − | 0.866025i | 0.0410933 | + | 0.0512403i |
37.4 | −1.00926 | − | 0.990649i | 0.866025 | − | 0.500000i | 0.0372299 | + | 1.99965i | 0.586448 | + | 0.338586i | −1.36937 | − | 0.353295i | 2.23683 | + | 1.41301i | 1.94338 | − | 2.05506i | 0.500000 | − | 0.866025i | −0.256462 | − | 0.922687i |
37.5 | −0.938973 | + | 1.05751i | 0.866025 | − | 0.500000i | −0.236659 | − | 1.98595i | 1.98722 | + | 1.14732i | −0.284419 | + | 1.38532i | −1.05630 | − | 2.42574i | 2.32238 | + | 1.61448i | 0.500000 | − | 0.866025i | −3.07926 | + | 1.02420i |
37.6 | −0.446345 | + | 1.34193i | −0.866025 | + | 0.500000i | −1.60155 | − | 1.19793i | −1.98722 | − | 1.14732i | −0.284419 | − | 1.38532i | −1.05630 | − | 2.42574i | 2.32238 | − | 1.61448i | 0.500000 | − | 0.866025i | 2.42662 | − | 2.15461i |
37.7 | −0.189716 | − | 1.40143i | −0.866025 | + | 0.500000i | −1.92802 | + | 0.531748i | 3.09843 | + | 1.78888i | 0.865014 | + | 1.11882i | 0.993295 | + | 2.45222i | 1.11098 | + | 2.60110i | 0.500000 | − | 0.866025i | 1.91917 | − | 4.68162i |
37.8 | 0.267238 | + | 1.38873i | 0.866025 | − | 0.500000i | −1.85717 | + | 0.742246i | 1.56250 | + | 0.902108i | 0.925802 | + | 1.06906i | 2.63683 | − | 0.217074i | −1.52709 | − | 2.38076i | 0.500000 | − | 0.866025i | −0.835229 | + | 2.41097i |
37.9 | 0.268038 | − | 1.38858i | 0.866025 | − | 0.500000i | −1.85631 | − | 0.744384i | −1.23074 | − | 0.710569i | −0.462163 | − | 1.33656i | 1.39545 | − | 2.24783i | −1.53120 | + | 2.37811i | 0.500000 | − | 0.866025i | −1.31657 | + | 1.51852i |
37.10 | 0.491996 | − | 1.32587i | −0.866025 | + | 0.500000i | −1.51588 | − | 1.30465i | −3.08781 | − | 1.78275i | 0.236856 | + | 1.39424i | −2.38336 | + | 1.14873i | −2.47560 | + | 1.36799i | 0.500000 | − | 0.866025i | −3.88289 | + | 3.21694i |
37.11 | 0.867144 | + | 1.11717i | −0.866025 | + | 0.500000i | −0.496121 | + | 1.93749i | 2.93503 | + | 1.69454i | −1.30955 | − | 0.533922i | −1.85242 | − | 1.88906i | −2.59471 | + | 1.12583i | 0.500000 | − | 0.866025i | 0.652012 | + | 4.74833i |
37.12 | 0.902242 | − | 1.08902i | 0.866025 | − | 0.500000i | −0.371918 | − | 1.96512i | 3.08781 | + | 1.78275i | 0.236856 | − | 1.39424i | −2.38336 | + | 1.14873i | −2.47560 | − | 1.36799i | 0.500000 | − | 0.866025i | 4.72740 | − | 1.75421i |
37.13 | 1.06853 | − | 0.926418i | −0.866025 | + | 0.500000i | 0.283500 | − | 1.97980i | 1.23074 | + | 0.710569i | −0.462163 | + | 1.33656i | 1.39545 | − | 2.24783i | −1.53120 | − | 2.37811i | 0.500000 | − | 0.866025i | 1.97336 | − | 0.380919i |
37.14 | 1.10325 | + | 0.884778i | 0.866025 | − | 0.500000i | 0.434335 | + | 1.95227i | 0.0402223 | + | 0.0232224i | 1.39783 | + | 0.214614i | −1.97032 | + | 1.76574i | −1.24814 | + | 2.53814i | 0.500000 | − | 0.866025i | 0.0238287 | + | 0.0612079i |
37.15 | 1.30853 | − | 0.536416i | 0.866025 | − | 0.500000i | 1.42451 | − | 1.40384i | −3.09843 | − | 1.78888i | 0.865014 | − | 1.11882i | 0.993295 | + | 2.45222i | 1.11098 | − | 2.60110i | 0.500000 | − | 0.866025i | −5.01398 | − | 0.678758i |
37.16 | 1.36256 | + | 0.378724i | −0.866025 | + | 0.500000i | 1.71314 | + | 1.03207i | −0.586448 | − | 0.338586i | −1.36937 | + | 0.353295i | 2.23683 | + | 1.41301i | 1.94338 | + | 2.05506i | 0.500000 | − | 0.866025i | −0.670840 | − | 0.683446i |
109.1 | −1.40107 | + | 0.192386i | 0.866025 | + | 0.500000i | 1.92598 | − | 0.539091i | −2.93503 | + | 1.69454i | −1.30955 | − | 0.533922i | −1.85242 | + | 1.88906i | −2.59471 | + | 1.12583i | 0.500000 | + | 0.866025i | 3.78617 | − | 2.93882i |
109.2 | −1.33630 | − | 0.462932i | −0.866025 | − | 0.500000i | 1.57139 | + | 1.23723i | −1.56250 | + | 0.902108i | 0.925802 | + | 1.06906i | 2.63683 | + | 0.217074i | −1.52709 | − | 2.38076i | 0.500000 | + | 0.866025i | 2.50558 | − | 0.482155i |
109.3 | −1.31787 | + | 0.513056i | −0.866025 | − | 0.500000i | 1.47355 | − | 1.35228i | −0.0402223 | + | 0.0232224i | 1.39783 | + | 0.214614i | −1.97032 | − | 1.76574i | −1.24814 | + | 2.53814i | 0.500000 | + | 0.866025i | 0.0410933 | − | 0.0512403i |
109.4 | −1.00926 | + | 0.990649i | 0.866025 | + | 0.500000i | 0.0372299 | − | 1.99965i | 0.586448 | − | 0.338586i | −1.36937 | + | 0.353295i | 2.23683 | − | 1.41301i | 1.94338 | + | 2.05506i | 0.500000 | + | 0.866025i | −0.256462 | + | 0.922687i |
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
8.b | even | 2 | 1 | inner |
56.p | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 168.2.bc.a | ✓ | 32 |
3.b | odd | 2 | 1 | 504.2.cj.e | 32 | ||
4.b | odd | 2 | 1 | 672.2.bk.a | 32 | ||
7.c | even | 3 | 1 | inner | 168.2.bc.a | ✓ | 32 |
7.c | even | 3 | 1 | 1176.2.c.e | 16 | ||
7.d | odd | 6 | 1 | 1176.2.c.f | 16 | ||
8.b | even | 2 | 1 | inner | 168.2.bc.a | ✓ | 32 |
8.d | odd | 2 | 1 | 672.2.bk.a | 32 | ||
12.b | even | 2 | 1 | 2016.2.cr.e | 32 | ||
21.h | odd | 6 | 1 | 504.2.cj.e | 32 | ||
24.f | even | 2 | 1 | 2016.2.cr.e | 32 | ||
24.h | odd | 2 | 1 | 504.2.cj.e | 32 | ||
28.f | even | 6 | 1 | 4704.2.c.f | 16 | ||
28.g | odd | 6 | 1 | 672.2.bk.a | 32 | ||
28.g | odd | 6 | 1 | 4704.2.c.e | 16 | ||
56.j | odd | 6 | 1 | 1176.2.c.f | 16 | ||
56.k | odd | 6 | 1 | 672.2.bk.a | 32 | ||
56.k | odd | 6 | 1 | 4704.2.c.e | 16 | ||
56.m | even | 6 | 1 | 4704.2.c.f | 16 | ||
56.p | even | 6 | 1 | inner | 168.2.bc.a | ✓ | 32 |
56.p | even | 6 | 1 | 1176.2.c.e | 16 | ||
84.n | even | 6 | 1 | 2016.2.cr.e | 32 | ||
168.s | odd | 6 | 1 | 504.2.cj.e | 32 | ||
168.v | even | 6 | 1 | 2016.2.cr.e | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
168.2.bc.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
168.2.bc.a | ✓ | 32 | 7.c | even | 3 | 1 | inner |
168.2.bc.a | ✓ | 32 | 8.b | even | 2 | 1 | inner |
168.2.bc.a | ✓ | 32 | 56.p | even | 6 | 1 | inner |
504.2.cj.e | 32 | 3.b | odd | 2 | 1 | ||
504.2.cj.e | 32 | 21.h | odd | 6 | 1 | ||
504.2.cj.e | 32 | 24.h | odd | 2 | 1 | ||
504.2.cj.e | 32 | 168.s | odd | 6 | 1 | ||
672.2.bk.a | 32 | 4.b | odd | 2 | 1 | ||
672.2.bk.a | 32 | 8.d | odd | 2 | 1 | ||
672.2.bk.a | 32 | 28.g | odd | 6 | 1 | ||
672.2.bk.a | 32 | 56.k | odd | 6 | 1 | ||
1176.2.c.e | 16 | 7.c | even | 3 | 1 | ||
1176.2.c.e | 16 | 56.p | even | 6 | 1 | ||
1176.2.c.f | 16 | 7.d | odd | 6 | 1 | ||
1176.2.c.f | 16 | 56.j | odd | 6 | 1 | ||
2016.2.cr.e | 32 | 12.b | even | 2 | 1 | ||
2016.2.cr.e | 32 | 24.f | even | 2 | 1 | ||
2016.2.cr.e | 32 | 84.n | even | 6 | 1 | ||
2016.2.cr.e | 32 | 168.v | even | 6 | 1 | ||
4704.2.c.e | 16 | 28.g | odd | 6 | 1 | ||
4704.2.c.e | 16 | 56.k | odd | 6 | 1 | ||
4704.2.c.f | 16 | 28.f | even | 6 | 1 | ||
4704.2.c.f | 16 | 56.m | even | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(168, [\chi])\).