Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [756,2,Mod(107,756)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(756, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("756.107");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 756.be (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: | \(6.03669039281\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(32\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
107.1 | −1.41403 | − | 0.0228601i | 0 | 1.99895 | + | 0.0646498i | 0.158956 | + | 0.0917731i | 0 | 1.36747 | + | 2.26495i | −2.82510 | − | 0.137113i | 0 | −0.222670 | − | 0.133404i | ||||||
107.2 | −1.38037 | − | 0.307526i | 0 | 1.81086 | + | 0.849002i | 3.20305 | + | 1.84928i | 0 | −2.45439 | − | 0.987919i | −2.23856 | − | 1.72882i | 0 | −3.85269 | − | 3.53771i | ||||||
107.3 | −1.37308 | + | 0.338582i | 0 | 1.77072 | − | 0.929803i | −3.53194 | − | 2.03916i | 0 | 1.92589 | − | 1.81410i | −2.11654 | + | 1.87623i | 0 | 5.54007 | + | 1.60410i | ||||||
107.4 | −1.32269 | + | 0.500480i | 0 | 1.49904 | − | 1.32396i | 2.34508 | + | 1.35394i | 0 | 1.75134 | + | 1.98313i | −1.32015 | + | 2.50144i | 0 | −3.77945 | − | 0.617175i | ||||||
107.5 | −1.28676 | − | 0.586718i | 0 | 1.31152 | + | 1.50994i | −1.46375 | − | 0.845096i | 0 | −1.83823 | + | 1.90287i | −0.801716 | − | 2.71243i | 0 | 1.38767 | + | 1.94625i | ||||||
107.6 | −1.27826 | + | 0.605023i | 0 | 1.26789 | − | 1.54675i | −1.71449 | − | 0.989859i | 0 | −2.59406 | − | 0.520431i | −0.684877 | + | 2.74426i | 0 | 2.79044 | + | 0.227993i | ||||||
107.7 | −1.15149 | − | 0.821011i | 0 | 0.651880 | + | 1.89078i | −1.46375 | − | 0.845096i | 0 | 1.83823 | − | 1.90287i | 0.801716 | − | 2.71243i | 0 | 0.991666 | + | 2.17488i | ||||||
107.8 | −1.04167 | + | 0.956516i | 0 | 0.170154 | − | 1.99275i | 0.835158 | + | 0.482179i | 0 | 2.03771 | − | 1.68753i | 1.72885 | + | 2.23854i | 0 | −1.33117 | + | 0.296571i | ||||||
107.9 | −0.956512 | − | 1.04167i | 0 | −0.170171 | + | 1.99275i | 3.20305 | + | 1.84928i | 0 | 2.45439 | + | 0.987919i | 2.23856 | − | 1.72882i | 0 | −1.13740 | − | 5.10539i | ||||||
107.10 | −0.743813 | + | 1.20281i | 0 | −0.893484 | − | 1.78933i | 1.64673 | + | 0.950741i | 0 | −0.905127 | − | 2.48611i | 2.81680 | + | 0.256236i | 0 | −2.36842 | + | 1.27353i | ||||||
107.11 | −0.726812 | − | 1.21315i | 0 | −0.943489 | + | 1.76347i | 0.158956 | + | 0.0917731i | 0 | −1.36747 | − | 2.26495i | 2.82510 | − | 0.137113i | 0 | −0.00419589 | − | 0.259540i | ||||||
107.12 | −0.669754 | + | 1.24556i | 0 | −1.10286 | − | 1.66844i | −1.64673 | − | 0.950741i | 0 | 0.905127 | + | 2.48611i | 2.81680 | − | 0.256236i | 0 | 2.28712 | − | 1.41435i | ||||||
107.13 | −0.393322 | − | 1.35842i | 0 | −1.69060 | + | 1.06859i | −3.53194 | − | 2.03916i | 0 | −1.92589 | + | 1.81410i | 2.11654 | + | 1.87623i | 0 | −1.38085 | + | 5.59989i | ||||||
107.14 | −0.307532 | + | 1.38037i | 0 | −1.81085 | − | 0.849017i | −0.835158 | − | 0.482179i | 0 | −2.03771 | + | 1.68753i | 1.72885 | − | 2.23854i | 0 | 0.922423 | − | 1.00454i | ||||||
107.15 | −0.227919 | − | 1.39573i | 0 | −1.89611 | + | 0.636225i | 2.34508 | + | 1.35394i | 0 | −1.75134 | − | 1.98313i | 1.32015 | + | 2.50144i | 0 | 1.35523 | − | 3.58168i | ||||||
107.16 | −0.115165 | − | 1.40952i | 0 | −1.97347 | + | 0.324653i | −1.71449 | − | 0.989859i | 0 | 2.59406 | + | 0.520431i | 0.684877 | + | 2.74426i | 0 | −1.19777 | + | 2.53059i | ||||||
107.17 | 0.115165 | + | 1.40952i | 0 | −1.97347 | + | 0.324653i | 1.71449 | + | 0.989859i | 0 | 2.59406 | + | 0.520431i | −0.684877 | − | 2.74426i | 0 | −1.19777 | + | 2.53059i | ||||||
107.18 | 0.227919 | + | 1.39573i | 0 | −1.89611 | + | 0.636225i | −2.34508 | − | 1.35394i | 0 | −1.75134 | − | 1.98313i | −1.32015 | − | 2.50144i | 0 | 1.35523 | − | 3.58168i | ||||||
107.19 | 0.307532 | − | 1.38037i | 0 | −1.81085 | − | 0.849017i | 0.835158 | + | 0.482179i | 0 | −2.03771 | + | 1.68753i | −1.72885 | + | 2.23854i | 0 | 0.922423 | − | 1.00454i | ||||||
107.20 | 0.393322 | + | 1.35842i | 0 | −1.69060 | + | 1.06859i | 3.53194 | + | 2.03916i | 0 | −1.92589 | + | 1.81410i | −2.11654 | − | 1.87623i | 0 | −1.38085 | + | 5.59989i | ||||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
12.b | even | 2 | 1 | inner |
21.h | odd | 6 | 1 | inner |
28.g | odd | 6 | 1 | inner |
84.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 756.2.be.e | ✓ | 64 |
3.b | odd | 2 | 1 | inner | 756.2.be.e | ✓ | 64 |
4.b | odd | 2 | 1 | inner | 756.2.be.e | ✓ | 64 |
7.c | even | 3 | 1 | inner | 756.2.be.e | ✓ | 64 |
12.b | even | 2 | 1 | inner | 756.2.be.e | ✓ | 64 |
21.h | odd | 6 | 1 | inner | 756.2.be.e | ✓ | 64 |
28.g | odd | 6 | 1 | inner | 756.2.be.e | ✓ | 64 |
84.n | even | 6 | 1 | inner | 756.2.be.e | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
756.2.be.e | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
756.2.be.e | ✓ | 64 | 3.b | odd | 2 | 1 | inner |
756.2.be.e | ✓ | 64 | 4.b | odd | 2 | 1 | inner |
756.2.be.e | ✓ | 64 | 7.c | even | 3 | 1 | inner |
756.2.be.e | ✓ | 64 | 12.b | even | 2 | 1 | inner |
756.2.be.e | ✓ | 64 | 21.h | odd | 6 | 1 | inner |
756.2.be.e | ✓ | 64 | 28.g | odd | 6 | 1 | inner |
756.2.be.e | ✓ | 64 | 84.n | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):
\( T_{5}^{32} - 49 T_{5}^{30} + 1478 T_{5}^{28} - 28081 T_{5}^{26} + 389462 T_{5}^{24} - 3870311 T_{5}^{22} + \cdots + 4477456 \) |
\( T_{19}^{32} - 134 T_{19}^{30} + 11525 T_{19}^{28} - 580090 T_{19}^{26} + 21075775 T_{19}^{24} + \cdots + 2702336256 \) |