Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [210,3,Mod(83,210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("210.83");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 210.k (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: | \(5.72208555157\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
83.1 | −1.00000 | − | 1.00000i | −2.99336 | + | 0.199427i | 2.00000i | 4.37611 | + | 2.41861i | 3.19279 | + | 2.79394i | −5.12625 | − | 4.76671i | 2.00000 | − | 2.00000i | 8.92046 | − | 1.19391i | −1.95750 | − | 6.79472i | ||
83.2 | −1.00000 | − | 1.00000i | −2.84642 | − | 0.947561i | 2.00000i | −4.64638 | − | 1.84693i | 1.89886 | + | 3.79398i | −6.25771 | + | 3.13705i | 2.00000 | − | 2.00000i | 7.20426 | + | 5.39432i | 2.79945 | + | 6.49331i | ||
83.3 | −1.00000 | − | 1.00000i | −2.35150 | + | 1.86291i | 2.00000i | −1.91622 | − | 4.61824i | 4.21441 | + | 0.488596i | 6.26242 | + | 3.12763i | 2.00000 | − | 2.00000i | 2.05914 | − | 8.76127i | −2.70202 | + | 6.53446i | ||
83.4 | −1.00000 | − | 1.00000i | −2.28799 | − | 1.94039i | 2.00000i | 4.58449 | − | 1.99562i | 0.347606 | + | 4.22838i | 5.57668 | + | 4.23091i | 2.00000 | − | 2.00000i | 1.46981 | + | 8.87917i | −6.58010 | − | 2.58887i | ||
83.5 | −1.00000 | − | 1.00000i | −2.14733 | − | 2.09499i | 2.00000i | −1.13661 | + | 4.86910i | 0.0523328 | + | 4.24232i | 6.60685 | − | 2.31291i | 2.00000 | − | 2.00000i | 0.222012 | + | 8.99726i | 6.00571 | − | 3.73249i | ||
83.6 | −1.00000 | − | 1.00000i | −1.21742 | + | 2.74188i | 2.00000i | 3.32079 | − | 3.73796i | 3.95930 | − | 1.52446i | −6.61294 | − | 2.29543i | 2.00000 | − | 2.00000i | −6.03579 | − | 6.67602i | −7.05875 | + | 0.417165i | ||
83.7 | −1.00000 | − | 1.00000i | −0.282815 | + | 2.98664i | 2.00000i | −3.28357 | + | 3.77070i | 3.26945 | − | 2.70382i | −3.67639 | + | 5.95686i | 2.00000 | − | 2.00000i | −8.84003 | − | 1.68933i | 7.05427 | − | 0.487124i | ||
83.8 | −1.00000 | − | 1.00000i | −0.00829838 | − | 2.99999i | 2.00000i | −3.67015 | − | 3.39558i | −2.99169 | + | 3.00829i | 1.45834 | − | 6.84640i | 2.00000 | − | 2.00000i | −8.99986 | + | 0.0497901i | 0.274569 | + | 7.06574i | ||
83.9 | −1.00000 | − | 1.00000i | 0.00829838 | + | 2.99999i | 2.00000i | 3.67015 | + | 3.39558i | 2.99169 | − | 3.00829i | 6.84640 | − | 1.45834i | 2.00000 | − | 2.00000i | −8.99986 | + | 0.0497901i | −0.274569 | − | 7.06574i | ||
83.10 | −1.00000 | − | 1.00000i | 0.282815 | − | 2.98664i | 2.00000i | 3.28357 | − | 3.77070i | −3.26945 | + | 2.70382i | −5.95686 | + | 3.67639i | 2.00000 | − | 2.00000i | −8.84003 | − | 1.68933i | −7.05427 | + | 0.487124i | ||
83.11 | −1.00000 | − | 1.00000i | 1.21742 | − | 2.74188i | 2.00000i | −3.32079 | + | 3.73796i | −3.95930 | + | 1.52446i | 2.29543 | + | 6.61294i | 2.00000 | − | 2.00000i | −6.03579 | − | 6.67602i | 7.05875 | − | 0.417165i | ||
83.12 | −1.00000 | − | 1.00000i | 2.14733 | + | 2.09499i | 2.00000i | 1.13661 | − | 4.86910i | −0.0523328 | − | 4.24232i | 2.31291 | − | 6.60685i | 2.00000 | − | 2.00000i | 0.222012 | + | 8.99726i | −6.00571 | + | 3.73249i | ||
83.13 | −1.00000 | − | 1.00000i | 2.28799 | + | 1.94039i | 2.00000i | −4.58449 | + | 1.99562i | −0.347606 | − | 4.22838i | −4.23091 | − | 5.57668i | 2.00000 | − | 2.00000i | 1.46981 | + | 8.87917i | 6.58010 | + | 2.58887i | ||
83.14 | −1.00000 | − | 1.00000i | 2.35150 | − | 1.86291i | 2.00000i | 1.91622 | + | 4.61824i | −4.21441 | − | 0.488596i | −3.12763 | − | 6.26242i | 2.00000 | − | 2.00000i | 2.05914 | − | 8.76127i | 2.70202 | − | 6.53446i | ||
83.15 | −1.00000 | − | 1.00000i | 2.84642 | + | 0.947561i | 2.00000i | 4.64638 | + | 1.84693i | −1.89886 | − | 3.79398i | −3.13705 | + | 6.25771i | 2.00000 | − | 2.00000i | 7.20426 | + | 5.39432i | −2.79945 | − | 6.49331i | ||
83.16 | −1.00000 | − | 1.00000i | 2.99336 | − | 0.199427i | 2.00000i | −4.37611 | − | 2.41861i | −3.19279 | − | 2.79394i | 4.76671 | + | 5.12625i | 2.00000 | − | 2.00000i | 8.92046 | − | 1.19391i | 1.95750 | + | 6.79472i | ||
167.1 | −1.00000 | + | 1.00000i | −2.99336 | − | 0.199427i | − | 2.00000i | 4.37611 | − | 2.41861i | 3.19279 | − | 2.79394i | −5.12625 | + | 4.76671i | 2.00000 | + | 2.00000i | 8.92046 | + | 1.19391i | −1.95750 | + | 6.79472i | |
167.2 | −1.00000 | + | 1.00000i | −2.84642 | + | 0.947561i | − | 2.00000i | −4.64638 | + | 1.84693i | 1.89886 | − | 3.79398i | −6.25771 | − | 3.13705i | 2.00000 | + | 2.00000i | 7.20426 | − | 5.39432i | 2.79945 | − | 6.49331i | |
167.3 | −1.00000 | + | 1.00000i | −2.35150 | − | 1.86291i | − | 2.00000i | −1.91622 | + | 4.61824i | 4.21441 | − | 0.488596i | 6.26242 | − | 3.12763i | 2.00000 | + | 2.00000i | 2.05914 | + | 8.76127i | −2.70202 | − | 6.53446i | |
167.4 | −1.00000 | + | 1.00000i | −2.28799 | + | 1.94039i | − | 2.00000i | 4.58449 | + | 1.99562i | 0.347606 | − | 4.22838i | 5.57668 | − | 4.23091i | 2.00000 | + | 2.00000i | 1.46981 | − | 8.87917i | −6.58010 | + | 2.58887i | |
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
15.e | even | 4 | 1 | inner |
105.k | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 210.3.k.a | ✓ | 32 |
3.b | odd | 2 | 1 | 210.3.k.b | yes | 32 | |
5.c | odd | 4 | 1 | 210.3.k.b | yes | 32 | |
7.b | odd | 2 | 1 | inner | 210.3.k.a | ✓ | 32 |
15.e | even | 4 | 1 | inner | 210.3.k.a | ✓ | 32 |
21.c | even | 2 | 1 | 210.3.k.b | yes | 32 | |
35.f | even | 4 | 1 | 210.3.k.b | yes | 32 | |
105.k | odd | 4 | 1 | inner | 210.3.k.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
210.3.k.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
210.3.k.a | ✓ | 32 | 7.b | odd | 2 | 1 | inner |
210.3.k.a | ✓ | 32 | 15.e | even | 4 | 1 | inner |
210.3.k.a | ✓ | 32 | 105.k | odd | 4 | 1 | inner |
210.3.k.b | yes | 32 | 3.b | odd | 2 | 1 | |
210.3.k.b | yes | 32 | 5.c | odd | 4 | 1 | |
210.3.k.b | yes | 32 | 21.c | even | 2 | 1 | |
210.3.k.b | yes | 32 | 35.f | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{16} - 12 T_{23}^{15} + 72 T_{23}^{14} + 11600 T_{23}^{13} + 1114532 T_{23}^{12} + \cdots + 46\!\cdots\!76 \) acting on \(S_{3}^{\mathrm{new}}(210, [\chi])\).