Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [675,2,Mod(136,675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("675.136");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 675.h (of order \(5\), 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: | \(5.38990213644\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
136.1 | −1.95344 | + | 1.41926i | 0 | 1.18360 | − | 3.64275i | −1.22237 | − | 1.87238i | 0 | 0.0108985 | 1.36561 | + | 4.20292i | 0 | 5.04522 | + | 1.92272i | ||||||||
136.2 | −1.92473 | + | 1.39840i | 0 | 1.13104 | − | 3.48099i | 0.154240 | + | 2.23074i | 0 | 1.19432 | 1.22050 | + | 3.75632i | 0 | −3.41634 | − | 4.07790i | ||||||||
136.3 | −1.17685 | + | 0.855035i | 0 | 0.0358679 | − | 0.110390i | 2.21985 | − | 0.268806i | 0 | −0.960718 | −0.846861 | − | 2.60637i | 0 | −2.38260 | + | 2.21440i | ||||||||
136.4 | −0.634202 | + | 0.460775i | 0 | −0.428135 | + | 1.31766i | −2.22112 | − | 0.258104i | 0 | 4.26911 | −0.820110 | − | 2.52404i | 0 | 1.52757 | − | 0.859747i | ||||||||
136.5 | −0.324639 | + | 0.235864i | 0 | −0.568275 | + | 1.74897i | −0.0534510 | + | 2.23543i | 0 | −3.13164 | −0.476037 | − | 1.46509i | 0 | −0.509905 | − | 0.738315i | ||||||||
136.6 | 0.324639 | − | 0.235864i | 0 | −0.568275 | + | 1.74897i | 0.0534510 | − | 2.23543i | 0 | −3.13164 | 0.476037 | + | 1.46509i | 0 | −0.509905 | − | 0.738315i | ||||||||
136.7 | 0.634202 | − | 0.460775i | 0 | −0.428135 | + | 1.31766i | 2.22112 | + | 0.258104i | 0 | 4.26911 | 0.820110 | + | 2.52404i | 0 | 1.52757 | − | 0.859747i | ||||||||
136.8 | 1.17685 | − | 0.855035i | 0 | 0.0358679 | − | 0.110390i | −2.21985 | + | 0.268806i | 0 | −0.960718 | 0.846861 | + | 2.60637i | 0 | −2.38260 | + | 2.21440i | ||||||||
136.9 | 1.92473 | − | 1.39840i | 0 | 1.13104 | − | 3.48099i | −0.154240 | − | 2.23074i | 0 | 1.19432 | −1.22050 | − | 3.75632i | 0 | −3.41634 | − | 4.07790i | ||||||||
136.10 | 1.95344 | − | 1.41926i | 0 | 1.18360 | − | 3.64275i | 1.22237 | + | 1.87238i | 0 | 0.0108985 | −1.36561 | − | 4.20292i | 0 | 5.04522 | + | 1.92272i | ||||||||
271.1 | −0.813148 | + | 2.50261i | 0 | −3.98383 | − | 2.89442i | −1.55748 | − | 1.60445i | 0 | 3.64251 | 6.22536 | − | 4.52299i | 0 | 5.28178 | − | 2.59311i | ||||||||
271.2 | −0.626486 | + | 1.92813i | 0 | −1.70715 | − | 1.24032i | 1.52132 | − | 1.63878i | 0 | −1.89708 | 0.180674 | − | 0.131267i | 0 | 2.20668 | + | 3.95997i | ||||||||
271.3 | −0.601697 | + | 1.85183i | 0 | −1.44921 | − | 1.05292i | 1.17636 | + | 1.90162i | 0 | 1.11838 | −0.328714 | + | 0.238825i | 0 | −4.22930 | + | 1.03423i | ||||||||
271.4 | −0.402108 | + | 1.23756i | 0 | 0.248165 | + | 0.180302i | −2.22558 | − | 0.216297i | 0 | −2.28898 | −2.42839 | + | 1.76433i | 0 | 1.16261 | − | 2.66732i | ||||||||
271.5 | −0.0972368 | + | 0.299264i | 0 | 1.53793 | + | 1.11737i | 1.59400 | − | 1.56817i | 0 | 3.04319 | −0.993071 | + | 0.721508i | 0 | 0.314302 | + | 0.629511i | ||||||||
271.6 | 0.0972368 | − | 0.299264i | 0 | 1.53793 | + | 1.11737i | −1.59400 | + | 1.56817i | 0 | 3.04319 | 0.993071 | − | 0.721508i | 0 | 0.314302 | + | 0.629511i | ||||||||
271.7 | 0.402108 | − | 1.23756i | 0 | 0.248165 | + | 0.180302i | 2.22558 | + | 0.216297i | 0 | −2.28898 | 2.42839 | − | 1.76433i | 0 | 1.16261 | − | 2.66732i | ||||||||
271.8 | 0.601697 | − | 1.85183i | 0 | −1.44921 | − | 1.05292i | −1.17636 | − | 1.90162i | 0 | 1.11838 | 0.328714 | − | 0.238825i | 0 | −4.22930 | + | 1.03423i | ||||||||
271.9 | 0.626486 | − | 1.92813i | 0 | −1.70715 | − | 1.24032i | −1.52132 | + | 1.63878i | 0 | −1.89708 | −0.180674 | + | 0.131267i | 0 | 2.20668 | + | 3.95997i | ||||||||
271.10 | 0.813148 | − | 2.50261i | 0 | −3.98383 | − | 2.89442i | 1.55748 | + | 1.60445i | 0 | 3.64251 | −6.22536 | + | 4.52299i | 0 | 5.28178 | − | 2.59311i | ||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
75.j | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 675.2.h.b | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 675.2.h.b | ✓ | 40 |
25.d | even | 5 | 1 | inner | 675.2.h.b | ✓ | 40 |
75.j | odd | 10 | 1 | inner | 675.2.h.b | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
675.2.h.b | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
675.2.h.b | ✓ | 40 | 3.b | odd | 2 | 1 | inner |
675.2.h.b | ✓ | 40 | 25.d | even | 5 | 1 | inner |
675.2.h.b | ✓ | 40 | 75.j | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{40} + 18 T_{2}^{38} + 194 T_{2}^{36} + 1663 T_{2}^{34} + 12932 T_{2}^{32} + 80442 T_{2}^{30} + 418854 T_{2}^{28} + 1915409 T_{2}^{26} + 7601379 T_{2}^{24} + 23323386 T_{2}^{22} + 54398827 T_{2}^{20} + \cdots + 15625 \)
acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\).