Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [663,2,Mod(157,663)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(663, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("663.157");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 663 = 3 \cdot 13 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 663.j (of order \(4\), 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: | \(5.29408165401\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
157.1 | − | 2.28916i | 0.707107 | − | 0.707107i | −3.24026 | −1.12612 | + | 1.12612i | −1.61868 | − | 1.61868i | −2.07325 | − | 2.07325i | 2.83915i | − | 1.00000i | 2.57786 | + | 2.57786i | ||||||
157.2 | − | 2.27457i | 0.707107 | − | 0.707107i | −3.17368 | 2.96612 | − | 2.96612i | −1.60837 | − | 1.60837i | 2.96539 | + | 2.96539i | 2.66962i | − | 1.00000i | −6.74666 | − | 6.74666i | ||||||
157.3 | − | 2.18004i | −0.707107 | + | 0.707107i | −2.75257 | −2.15243 | + | 2.15243i | 1.54152 | + | 1.54152i | 2.93384 | + | 2.93384i | 1.64064i | − | 1.00000i | 4.69239 | + | 4.69239i | ||||||
157.4 | − | 1.91545i | −0.707107 | + | 0.707107i | −1.66893 | −1.35359 | + | 1.35359i | 1.35442 | + | 1.35442i | −0.811625 | − | 0.811625i | − | 0.634143i | − | 1.00000i | 2.59273 | + | 2.59273i | |||||
157.5 | − | 1.44794i | 0.707107 | − | 0.707107i | −0.0965417 | 0.705234 | − | 0.705234i | −1.02385 | − | 1.02385i | −1.70047 | − | 1.70047i | − | 2.75610i | − | 1.00000i | −1.02114 | − | 1.02114i | |||||
157.6 | − | 1.09385i | −0.707107 | + | 0.707107i | 0.803491 | 1.54655 | − | 1.54655i | 0.773469 | + | 0.773469i | −1.02590 | − | 1.02590i | − | 3.06660i | − | 1.00000i | −1.69169 | − | 1.69169i | |||||
157.7 | − | 0.518644i | 0.707107 | − | 0.707107i | 1.73101 | −0.144430 | + | 0.144430i | −0.366737 | − | 0.366737i | 2.76683 | + | 2.76683i | − | 1.93506i | − | 1.00000i | 0.0749079 | + | 0.0749079i | |||||
157.8 | 0.0854616i | −0.707107 | + | 0.707107i | 1.99270 | −0.427664 | + | 0.427664i | −0.0604305 | − | 0.0604305i | 2.11132 | + | 2.11132i | 0.341222i | − | 1.00000i | −0.0365488 | − | 0.0365488i | |||||||
157.9 | 0.303671i | 0.707107 | − | 0.707107i | 1.90778 | 1.45526 | − | 1.45526i | 0.214728 | + | 0.214728i | −1.92916 | − | 1.92916i | 1.18668i | − | 1.00000i | 0.441920 | + | 0.441920i | |||||||
157.10 | 0.685099i | −0.707107 | + | 0.707107i | 1.53064 | −2.82114 | + | 2.82114i | −0.484438 | − | 0.484438i | −3.41418 | − | 3.41418i | 2.41884i | − | 1.00000i | −1.93276 | − | 1.93276i | |||||||
157.11 | 1.08778i | 0.707107 | − | 0.707107i | 0.816737 | −0.495535 | + | 0.495535i | 0.769176 | + | 0.769176i | 1.02138 | + | 1.02138i | 3.06399i | − | 1.00000i | −0.539032 | − | 0.539032i | |||||||
157.12 | 1.15222i | 0.707107 | − | 0.707107i | 0.672395 | −2.59606 | + | 2.59606i | 0.814741 | + | 0.814741i | 2.15133 | + | 2.15133i | 3.07918i | − | 1.00000i | −2.99123 | − | 2.99123i | |||||||
157.13 | 1.49170i | −0.707107 | + | 0.707107i | −0.225164 | 0.731803 | − | 0.731803i | −1.05479 | − | 1.05479i | −0.0312384 | − | 0.0312384i | 2.64752i | − | 1.00000i | 1.09163 | + | 1.09163i | |||||||
157.14 | 1.81235i | −0.707107 | + | 0.707107i | −1.28460 | −1.46563 | + | 1.46563i | −1.28152 | − | 1.28152i | 0.830570 | + | 0.830570i | 1.29656i | − | 1.00000i | −2.65623 | − | 2.65623i | |||||||
157.15 | 2.52895i | −0.707107 | + | 0.707107i | −4.39556 | 0.699465 | − | 0.699465i | −1.78823 | − | 1.78823i | −3.42121 | − | 3.42121i | − | 6.05825i | − | 1.00000i | 1.76891 | + | 1.76891i | ||||||
157.16 | 2.57244i | 0.707107 | − | 0.707107i | −4.61745 | 2.47817 | − | 2.47817i | 1.81899 | + | 1.81899i | −0.373629 | − | 0.373629i | − | 6.73323i | − | 1.00000i | 6.37494 | + | 6.37494i | ||||||
625.1 | − | 2.57244i | 0.707107 | + | 0.707107i | −4.61745 | 2.47817 | + | 2.47817i | 1.81899 | − | 1.81899i | −0.373629 | + | 0.373629i | 6.73323i | 1.00000i | 6.37494 | − | 6.37494i | |||||||
625.2 | − | 2.52895i | −0.707107 | − | 0.707107i | −4.39556 | 0.699465 | + | 0.699465i | −1.78823 | + | 1.78823i | −3.42121 | + | 3.42121i | 6.05825i | 1.00000i | 1.76891 | − | 1.76891i | |||||||
625.3 | − | 1.81235i | −0.707107 | − | 0.707107i | −1.28460 | −1.46563 | − | 1.46563i | −1.28152 | + | 1.28152i | 0.830570 | − | 0.830570i | − | 1.29656i | 1.00000i | −2.65623 | + | 2.65623i | ||||||
625.4 | − | 1.49170i | −0.707107 | − | 0.707107i | −0.225164 | 0.731803 | + | 0.731803i | −1.05479 | + | 1.05479i | −0.0312384 | + | 0.0312384i | − | 2.64752i | 1.00000i | 1.09163 | − | 1.09163i | ||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.c | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 663.2.j.a | ✓ | 32 |
17.c | even | 4 | 1 | inner | 663.2.j.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
663.2.j.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
663.2.j.a | ✓ | 32 | 17.c | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 44 T_{2}^{30} + 868 T_{2}^{28} + 10148 T_{2}^{26} + 78320 T_{2}^{24} + 420752 T_{2}^{22} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\).