Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [429,2,Mod(109,429)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(429, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("429.109");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 429 = 3 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 429.m (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: | \(3.42558224671\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
109.1 | −1.92050 | − | 1.92050i | 1.00000 | 5.37666i | −0.179544 | + | 0.179544i | −1.92050 | − | 1.92050i | −1.23995 | + | 1.23995i | 6.48489 | − | 6.48489i | 1.00000 | 0.689629 | ||||||||
109.2 | −1.63677 | − | 1.63677i | 1.00000 | 3.35804i | 0.440340 | − | 0.440340i | −1.63677 | − | 1.63677i | 2.71876 | − | 2.71876i | 2.22281 | − | 2.22281i | 1.00000 | −1.44147 | ||||||||
109.3 | −1.43677 | − | 1.43677i | 1.00000 | 2.12863i | −2.46780 | + | 2.46780i | −1.43677 | − | 1.43677i | 0.806735 | − | 0.806735i | 0.184806 | − | 0.184806i | 1.00000 | 7.09133 | ||||||||
109.4 | −1.20600 | − | 1.20600i | 1.00000 | 0.908895i | −0.200991 | + | 0.200991i | −1.20600 | − | 1.20600i | −3.33407 | + | 3.33407i | −1.31588 | + | 1.31588i | 1.00000 | 0.484791 | ||||||||
109.5 | −1.10479 | − | 1.10479i | 1.00000 | 0.441107i | 1.95513 | − | 1.95513i | −1.10479 | − | 1.10479i | 1.11517 | − | 1.11517i | −1.72224 | + | 1.72224i | 1.00000 | −4.32001 | ||||||||
109.6 | −0.555689 | − | 0.555689i | 1.00000 | − | 1.38242i | −2.09993 | + | 2.09993i | −0.555689 | − | 0.555689i | −1.20106 | + | 1.20106i | −1.87957 | + | 1.87957i | 1.00000 | 2.33382 | |||||||
109.7 | −0.290762 | − | 0.290762i | 1.00000 | − | 1.83092i | 1.55280 | − | 1.55280i | −0.290762 | − | 0.290762i | 0.786011 | − | 0.786011i | −1.11388 | + | 1.11388i | 1.00000 | −0.902988 | |||||||
109.8 | 0.290762 | + | 0.290762i | 1.00000 | − | 1.83092i | 1.55280 | − | 1.55280i | 0.290762 | + | 0.290762i | −0.786011 | + | 0.786011i | 1.11388 | − | 1.11388i | 1.00000 | 0.902988 | |||||||
109.9 | 0.555689 | + | 0.555689i | 1.00000 | − | 1.38242i | −2.09993 | + | 2.09993i | 0.555689 | + | 0.555689i | 1.20106 | − | 1.20106i | 1.87957 | − | 1.87957i | 1.00000 | −2.33382 | |||||||
109.10 | 1.10479 | + | 1.10479i | 1.00000 | 0.441107i | 1.95513 | − | 1.95513i | 1.10479 | + | 1.10479i | −1.11517 | + | 1.11517i | 1.72224 | − | 1.72224i | 1.00000 | 4.32001 | ||||||||
109.11 | 1.20600 | + | 1.20600i | 1.00000 | 0.908895i | −0.200991 | + | 0.200991i | 1.20600 | + | 1.20600i | 3.33407 | − | 3.33407i | 1.31588 | − | 1.31588i | 1.00000 | −0.484791 | ||||||||
109.12 | 1.43677 | + | 1.43677i | 1.00000 | 2.12863i | −2.46780 | + | 2.46780i | 1.43677 | + | 1.43677i | −0.806735 | + | 0.806735i | −0.184806 | + | 0.184806i | 1.00000 | −7.09133 | ||||||||
109.13 | 1.63677 | + | 1.63677i | 1.00000 | 3.35804i | 0.440340 | − | 0.440340i | 1.63677 | + | 1.63677i | −2.71876 | + | 2.71876i | −2.22281 | + | 2.22281i | 1.00000 | 1.44147 | ||||||||
109.14 | 1.92050 | + | 1.92050i | 1.00000 | 5.37666i | −0.179544 | + | 0.179544i | 1.92050 | + | 1.92050i | 1.23995 | − | 1.23995i | −6.48489 | + | 6.48489i | 1.00000 | −0.689629 | ||||||||
307.1 | −1.92050 | + | 1.92050i | 1.00000 | − | 5.37666i | −0.179544 | − | 0.179544i | −1.92050 | + | 1.92050i | −1.23995 | − | 1.23995i | 6.48489 | + | 6.48489i | 1.00000 | 0.689629 | |||||||
307.2 | −1.63677 | + | 1.63677i | 1.00000 | − | 3.35804i | 0.440340 | + | 0.440340i | −1.63677 | + | 1.63677i | 2.71876 | + | 2.71876i | 2.22281 | + | 2.22281i | 1.00000 | −1.44147 | |||||||
307.3 | −1.43677 | + | 1.43677i | 1.00000 | − | 2.12863i | −2.46780 | − | 2.46780i | −1.43677 | + | 1.43677i | 0.806735 | + | 0.806735i | 0.184806 | + | 0.184806i | 1.00000 | 7.09133 | |||||||
307.4 | −1.20600 | + | 1.20600i | 1.00000 | − | 0.908895i | −0.200991 | − | 0.200991i | −1.20600 | + | 1.20600i | −3.33407 | − | 3.33407i | −1.31588 | − | 1.31588i | 1.00000 | 0.484791 | |||||||
307.5 | −1.10479 | + | 1.10479i | 1.00000 | − | 0.441107i | 1.95513 | + | 1.95513i | −1.10479 | + | 1.10479i | 1.11517 | + | 1.11517i | −1.72224 | − | 1.72224i | 1.00000 | −4.32001 | |||||||
307.6 | −0.555689 | + | 0.555689i | 1.00000 | 1.38242i | −2.09993 | − | 2.09993i | −0.555689 | + | 0.555689i | −1.20106 | − | 1.20106i | −1.87957 | − | 1.87957i | 1.00000 | 2.33382 | ||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
13.d | odd | 4 | 1 | inner |
143.g | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 429.2.m.b | ✓ | 28 |
11.b | odd | 2 | 1 | inner | 429.2.m.b | ✓ | 28 |
13.d | odd | 4 | 1 | inner | 429.2.m.b | ✓ | 28 |
143.g | even | 4 | 1 | inner | 429.2.m.b | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
429.2.m.b | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
429.2.m.b | ✓ | 28 | 11.b | odd | 2 | 1 | inner |
429.2.m.b | ✓ | 28 | 13.d | odd | 4 | 1 | inner |
429.2.m.b | ✓ | 28 | 143.g | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{28} + 115T_{2}^{24} + 4521T_{2}^{20} + 76475T_{2}^{16} + 564863T_{2}^{12} + 1562521T_{2}^{8} + 556319T_{2}^{4} + 14641 \)
acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).