Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,4,Mod(68,175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([9, 10]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("175.68");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 175.o (of order \(12\), 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: | \(10.3253342510\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
68.1 | −1.24155 | − | 4.63351i | 1.84884 | + | 0.495394i | −12.9998 | + | 7.50544i | 0 | − | 9.18166i | 4.86236 | − | 17.8706i | 23.7807 | + | 23.7807i | −20.2099 | − | 11.6682i | 0 | |||||
68.2 | −0.951619 | − | 3.55149i | −7.36493 | − | 1.97343i | −4.77930 | + | 2.75933i | 0 | 28.0344i | 0.130416 | + | 18.5198i | −6.45116 | − | 6.45116i | 26.9651 | + | 15.5683i | 0 | ||||||
68.3 | −0.575247 | − | 2.14685i | 9.51748 | + | 2.55020i | 2.65014 | − | 1.53006i | 0 | − | 21.8996i | 9.19207 | + | 16.0781i | −17.3821 | − | 17.3821i | 60.6963 | + | 35.0430i | 0 | |||||
68.4 | −0.320201 | − | 1.19501i | 0.325765 | + | 0.0872885i | 5.60269 | − | 3.23471i | 0 | − | 0.417242i | −17.8942 | − | 4.77454i | −12.6579 | − | 12.6579i | −23.2842 | − | 13.4431i | 0 | |||||
68.5 | 0.320201 | + | 1.19501i | −0.325765 | − | 0.0872885i | 5.60269 | − | 3.23471i | 0 | − | 0.417242i | 17.8942 | + | 4.77454i | 12.6579 | + | 12.6579i | −23.2842 | − | 13.4431i | 0 | |||||
68.6 | 0.575247 | + | 2.14685i | −9.51748 | − | 2.55020i | 2.65014 | − | 1.53006i | 0 | − | 21.8996i | −9.19207 | − | 16.0781i | 17.3821 | + | 17.3821i | 60.6963 | + | 35.0430i | 0 | |||||
68.7 | 0.951619 | + | 3.55149i | 7.36493 | + | 1.97343i | −4.77930 | + | 2.75933i | 0 | 28.0344i | −0.130416 | − | 18.5198i | 6.45116 | + | 6.45116i | 26.9651 | + | 15.5683i | 0 | ||||||
68.8 | 1.24155 | + | 4.63351i | −1.84884 | − | 0.495394i | −12.9998 | + | 7.50544i | 0 | − | 9.18166i | −4.86236 | + | 17.8706i | −23.7807 | − | 23.7807i | −20.2099 | − | 11.6682i | 0 | |||||
82.1 | −4.63351 | + | 1.24155i | −0.495394 | + | 1.84884i | 12.9998 | − | 7.50544i | 0 | − | 9.18166i | −17.8706 | − | 4.86236i | −23.7807 | + | 23.7807i | 20.2099 | + | 11.6682i | 0 | |||||
82.2 | −3.55149 | + | 0.951619i | 1.97343 | − | 7.36493i | 4.77930 | − | 2.75933i | 0 | 28.0344i | 18.5198 | − | 0.130416i | 6.45116 | − | 6.45116i | −26.9651 | − | 15.5683i | 0 | ||||||
82.3 | −2.14685 | + | 0.575247i | −2.55020 | + | 9.51748i | −2.65014 | + | 1.53006i | 0 | − | 21.8996i | 16.0781 | − | 9.19207i | 17.3821 | − | 17.3821i | −60.6963 | − | 35.0430i | 0 | |||||
82.4 | −1.19501 | + | 0.320201i | −0.0872885 | + | 0.325765i | −5.60269 | + | 3.23471i | 0 | − | 0.417242i | −4.77454 | + | 17.8942i | 12.6579 | − | 12.6579i | 23.2842 | + | 13.4431i | 0 | |||||
82.5 | 1.19501 | − | 0.320201i | 0.0872885 | − | 0.325765i | −5.60269 | + | 3.23471i | 0 | − | 0.417242i | 4.77454 | − | 17.8942i | −12.6579 | + | 12.6579i | 23.2842 | + | 13.4431i | 0 | |||||
82.6 | 2.14685 | − | 0.575247i | 2.55020 | − | 9.51748i | −2.65014 | + | 1.53006i | 0 | − | 21.8996i | −16.0781 | + | 9.19207i | −17.3821 | + | 17.3821i | −60.6963 | − | 35.0430i | 0 | |||||
82.7 | 3.55149 | − | 0.951619i | −1.97343 | + | 7.36493i | 4.77930 | − | 2.75933i | 0 | 28.0344i | −18.5198 | + | 0.130416i | −6.45116 | + | 6.45116i | −26.9651 | − | 15.5683i | 0 | ||||||
82.8 | 4.63351 | − | 1.24155i | 0.495394 | − | 1.84884i | 12.9998 | − | 7.50544i | 0 | − | 9.18166i | 17.8706 | + | 4.86236i | 23.7807 | − | 23.7807i | 20.2099 | + | 11.6682i | 0 | |||||
143.1 | −4.63351 | − | 1.24155i | −0.495394 | − | 1.84884i | 12.9998 | + | 7.50544i | 0 | 9.18166i | −17.8706 | + | 4.86236i | −23.7807 | − | 23.7807i | 20.2099 | − | 11.6682i | 0 | ||||||
143.2 | −3.55149 | − | 0.951619i | 1.97343 | + | 7.36493i | 4.77930 | + | 2.75933i | 0 | − | 28.0344i | 18.5198 | + | 0.130416i | 6.45116 | + | 6.45116i | −26.9651 | + | 15.5683i | 0 | |||||
143.3 | −2.14685 | − | 0.575247i | −2.55020 | − | 9.51748i | −2.65014 | − | 1.53006i | 0 | 21.8996i | 16.0781 | + | 9.19207i | 17.3821 | + | 17.3821i | −60.6963 | + | 35.0430i | 0 | ||||||
143.4 | −1.19501 | − | 0.320201i | −0.0872885 | − | 0.325765i | −5.60269 | − | 3.23471i | 0 | 0.417242i | −4.77454 | − | 17.8942i | 12.6579 | + | 12.6579i | 23.2842 | − | 13.4431i | 0 | ||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
7.d | odd | 6 | 1 | inner |
35.i | odd | 6 | 1 | inner |
35.k | even | 12 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 175.4.o.a | ✓ | 32 |
5.b | even | 2 | 1 | inner | 175.4.o.a | ✓ | 32 |
5.c | odd | 4 | 2 | inner | 175.4.o.a | ✓ | 32 |
7.d | odd | 6 | 1 | inner | 175.4.o.a | ✓ | 32 |
35.i | odd | 6 | 1 | inner | 175.4.o.a | ✓ | 32 |
35.k | even | 12 | 2 | inner | 175.4.o.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
175.4.o.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
175.4.o.a | ✓ | 32 | 5.b | even | 2 | 1 | inner |
175.4.o.a | ✓ | 32 | 5.c | odd | 4 | 2 | inner |
175.4.o.a | ✓ | 32 | 7.d | odd | 6 | 1 | inner |
175.4.o.a | ✓ | 32 | 35.i | odd | 6 | 1 | inner |
175.4.o.a | ✓ | 32 | 35.k | even | 12 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{32} - 739 T_{2}^{28} + 430246 T_{2}^{24} - 80374023 T_{2}^{20} + 11478799782 T_{2}^{16} - 296436161763 T_{2}^{12} + 6269585321601 T_{2}^{8} - 14542274767104 T_{2}^{4} + \cdots + 30601961865216 \)
acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\).