Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [210,3,Mod(37,210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("210.37");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 210.v (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: | \(5.72208555157\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −1.36603 | + | 0.366025i | −1.67303 | − | 0.448288i | 1.73205 | − | 1.00000i | −4.60436 | − | 1.94932i | 2.44949 | −4.28537 | − | 5.53495i | −2.00000 | + | 2.00000i | 2.59808 | + | 1.50000i | 7.00318 | + | 0.977504i | ||
37.2 | −1.36603 | + | 0.366025i | −1.67303 | − | 0.448288i | 1.73205 | − | 1.00000i | −1.25847 | − | 4.83903i | 2.44949 | 5.72194 | + | 4.03230i | −2.00000 | + | 2.00000i | 2.59808 | + | 1.50000i | 3.49032 | + | 6.14961i | ||
37.3 | −1.36603 | + | 0.366025i | −1.67303 | − | 0.448288i | 1.73205 | − | 1.00000i | −0.438103 | + | 4.98077i | 2.44949 | 6.08450 | − | 3.46105i | −2.00000 | + | 2.00000i | 2.59808 | + | 1.50000i | −1.22463 | − | 6.96421i | ||
37.4 | −1.36603 | + | 0.366025i | −1.67303 | − | 0.448288i | 1.73205 | − | 1.00000i | 4.98663 | − | 0.365451i | 2.44949 | −3.26382 | − | 6.19254i | −2.00000 | + | 2.00000i | 2.59808 | + | 1.50000i | −6.67809 | + | 2.32445i | ||
37.5 | −1.36603 | + | 0.366025i | 1.67303 | + | 0.448288i | 1.73205 | − | 1.00000i | −4.60928 | + | 1.93765i | −2.44949 | −1.45688 | − | 6.84672i | −2.00000 | + | 2.00000i | 2.59808 | + | 1.50000i | 5.58717 | − | 4.33400i | ||
37.6 | −1.36603 | + | 0.366025i | 1.67303 | + | 0.448288i | 1.73205 | − | 1.00000i | −2.69832 | − | 4.20940i | −2.44949 | 1.39786 | + | 6.85901i | −2.00000 | + | 2.00000i | 2.59808 | + | 1.50000i | 5.22672 | + | 4.76250i | ||
37.7 | −1.36603 | + | 0.366025i | 1.67303 | + | 0.448288i | 1.73205 | − | 1.00000i | 2.01909 | + | 4.57420i | −2.44949 | −6.19056 | + | 3.26756i | −2.00000 | + | 2.00000i | 2.59808 | + | 1.50000i | −4.43240 | − | 5.50943i | ||
37.8 | −1.36603 | + | 0.366025i | 1.67303 | + | 0.448288i | 1.73205 | − | 1.00000i | 4.87077 | − | 1.12941i | −2.44949 | 6.92053 | − | 1.05182i | −2.00000 | + | 2.00000i | 2.59808 | + | 1.50000i | −6.24021 | + | 3.32563i | ||
67.1 | 0.366025 | − | 1.36603i | −0.448288 | − | 1.67303i | −1.73205 | − | 1.00000i | −4.97092 | − | 0.538514i | −2.44949 | 3.26756 | − | 6.19056i | −2.00000 | + | 2.00000i | −2.59808 | + | 1.50000i | −2.55510 | + | 6.59329i | ||
67.2 | 0.366025 | − | 1.36603i | −0.448288 | − | 1.67303i | −1.73205 | − | 1.00000i | −1.45729 | + | 4.78292i | −2.44949 | −1.05182 | + | 6.92053i | −2.00000 | + | 2.00000i | −2.59808 | + | 1.50000i | 6.00018 | + | 3.74136i | ||
67.3 | 0.366025 | − | 1.36603i | −0.448288 | − | 1.67303i | −1.73205 | − | 1.00000i | 0.626586 | − | 4.96058i | −2.44949 | −6.84672 | − | 1.45688i | −2.00000 | + | 2.00000i | −2.59808 | + | 1.50000i | −6.54694 | − | 2.67163i | ||
67.4 | 0.366025 | − | 1.36603i | −0.448288 | − | 1.67303i | −1.73205 | − | 1.00000i | 4.99461 | − | 0.232109i | −2.44949 | 6.85901 | + | 1.39786i | −2.00000 | + | 2.00000i | −2.59808 | + | 1.50000i | 1.51109 | − | 6.90772i | ||
67.5 | 0.366025 | − | 1.36603i | 0.448288 | + | 1.67303i | −1.73205 | − | 1.00000i | −4.09442 | − | 2.86979i | 2.44949 | −3.46105 | + | 6.08450i | −2.00000 | + | 2.00000i | −2.59808 | + | 1.50000i | −5.41887 | + | 4.54267i | ||
67.6 | 0.366025 | − | 1.36603i | 0.448288 | + | 1.67303i | −1.73205 | − | 1.00000i | −2.17682 | + | 4.50127i | 2.44949 | −6.19254 | − | 3.26382i | −2.00000 | + | 2.00000i | −2.59808 | + | 1.50000i | 5.35208 | + | 4.62118i | ||
67.7 | 0.366025 | − | 1.36603i | 0.448288 | + | 1.67303i | −1.73205 | − | 1.00000i | 3.99034 | − | 3.01284i | 2.44949 | −5.53495 | − | 4.28537i | −2.00000 | + | 2.00000i | −2.59808 | + | 1.50000i | −2.65505 | − | 6.55368i | ||
67.8 | 0.366025 | − | 1.36603i | 0.448288 | + | 1.67303i | −1.73205 | − | 1.00000i | 4.81996 | + | 1.32965i | 2.44949 | 4.03230 | + | 5.72194i | −2.00000 | + | 2.00000i | −2.59808 | + | 1.50000i | 3.58056 | − | 6.09751i | ||
163.1 | 0.366025 | + | 1.36603i | −0.448288 | + | 1.67303i | −1.73205 | + | 1.00000i | −4.97092 | + | 0.538514i | −2.44949 | 3.26756 | + | 6.19056i | −2.00000 | − | 2.00000i | −2.59808 | − | 1.50000i | −2.55510 | − | 6.59329i | ||
163.2 | 0.366025 | + | 1.36603i | −0.448288 | + | 1.67303i | −1.73205 | + | 1.00000i | −1.45729 | − | 4.78292i | −2.44949 | −1.05182 | − | 6.92053i | −2.00000 | − | 2.00000i | −2.59808 | − | 1.50000i | 6.00018 | − | 3.74136i | ||
163.3 | 0.366025 | + | 1.36603i | −0.448288 | + | 1.67303i | −1.73205 | + | 1.00000i | 0.626586 | + | 4.96058i | −2.44949 | −6.84672 | + | 1.45688i | −2.00000 | − | 2.00000i | −2.59808 | − | 1.50000i | −6.54694 | + | 2.67163i | ||
163.4 | 0.366025 | + | 1.36603i | −0.448288 | + | 1.67303i | −1.73205 | + | 1.00000i | 4.99461 | + | 0.232109i | −2.44949 | 6.85901 | − | 1.39786i | −2.00000 | − | 2.00000i | −2.59808 | − | 1.50000i | 1.51109 | + | 6.90772i | ||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.c | even | 3 | 1 | inner |
35.l | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 210.3.v.a | ✓ | 32 |
5.c | odd | 4 | 1 | inner | 210.3.v.a | ✓ | 32 |
7.c | even | 3 | 1 | inner | 210.3.v.a | ✓ | 32 |
35.l | odd | 12 | 1 | inner | 210.3.v.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
210.3.v.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
210.3.v.a | ✓ | 32 | 5.c | odd | 4 | 1 | inner |
210.3.v.a | ✓ | 32 | 7.c | even | 3 | 1 | inner |
210.3.v.a | ✓ | 32 | 35.l | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{16} + 16 T_{11}^{15} + 646 T_{11}^{14} + 8632 T_{11}^{13} + 246126 T_{11}^{12} + \cdots + 34757981448100 \) acting on \(S_{3}^{\mathrm{new}}(210, [\chi])\).