Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [105,4,Mod(4,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 105.q (of order \(6\), 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: | \(6.19520055060\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −4.85916 | + | 2.80544i | −2.59808 | − | 1.50000i | 11.7410 | − | 20.3360i | 2.40235 | − | 10.9192i | 16.8326 | −9.44251 | − | 15.9323i | 86.8672i | 4.50000 | + | 7.79423i | 18.9597 | + | 59.7978i | ||||
4.2 | −4.10597 | + | 2.37058i | −2.59808 | − | 1.50000i | 7.23930 | − | 12.5388i | −2.66182 | + | 10.8589i | 14.2235 | 15.6813 | + | 9.85381i | 30.7161i | 4.50000 | + | 7.79423i | −14.8124 | − | 50.8961i | ||||
4.3 | −3.89316 | + | 2.24772i | 2.59808 | + | 1.50000i | 6.10447 | − | 10.5732i | −0.533135 | − | 11.1676i | −13.4863 | −3.71892 | + | 18.1430i | 18.9210i | 4.50000 | + | 7.79423i | 27.1772 | + | 42.2790i | ||||
4.4 | −3.00489 | + | 1.73487i | −2.59808 | − | 1.50000i | 2.01958 | − | 3.49802i | −10.2956 | − | 4.35891i | 10.4092 | −12.4845 | + | 13.6798i | − | 13.7431i | 4.50000 | + | 7.79423i | 38.4994 | − | 4.76357i | |||
4.5 | −2.89323 | + | 1.67041i | −2.59808 | − | 1.50000i | 1.58052 | − | 2.73755i | 9.21296 | + | 6.33414i | 10.0224 | −6.15758 | − | 17.4667i | − | 16.1660i | 4.50000 | + | 7.79423i | −37.2358 | − | 2.93674i | |||
4.6 | −2.21373 | + | 1.27810i | 2.59808 | + | 1.50000i | −0.732945 | + | 1.26950i | 5.74885 | − | 9.58910i | −7.66857 | −2.98784 | − | 18.2777i | − | 24.1966i | 4.50000 | + | 7.79423i | −0.470588 | + | 28.5752i | |||
4.7 | −2.01833 | + | 1.16528i | 2.59808 | + | 1.50000i | −1.28423 | + | 2.22435i | 7.90497 | + | 7.90642i | −6.99170 | 18.1257 | + | 3.80267i | − | 24.6305i | 4.50000 | + | 7.79423i | −25.1681 | − | 6.74622i | |||
4.8 | −1.96118 | + | 1.13229i | 2.59808 | + | 1.50000i | −1.43586 | + | 2.48698i | −5.60388 | + | 9.67453i | −6.79371 | −18.1088 | − | 3.88231i | − | 24.6198i | 4.50000 | + | 7.79423i | 0.0358556 | − | 25.3186i | |||
4.9 | −1.50072 | + | 0.866440i | −2.59808 | − | 1.50000i | −2.49856 | + | 4.32764i | 10.6609 | − | 3.36827i | 5.19864 | −14.9159 | + | 10.9780i | − | 22.5225i | 4.50000 | + | 7.79423i | −13.0806 | + | 14.2919i | |||
4.10 | −0.761268 | + | 0.439518i | −2.59808 | − | 1.50000i | −3.61365 | + | 6.25902i | 0.571597 | − | 11.1657i | 2.63711 | 18.5184 | − | 0.260966i | − | 13.3854i | 4.50000 | + | 7.79423i | 4.47240 | + | 8.75133i | |||
4.11 | −0.755385 | + | 0.436122i | −2.59808 | − | 1.50000i | −3.61960 | + | 6.26932i | −7.84576 | + | 7.96518i | 2.61673 | 10.7711 | − | 15.0659i | − | 13.2923i | 4.50000 | + | 7.79423i | 2.45278 | − | 9.43848i | |||
4.12 | 0.755385 | − | 0.436122i | 2.59808 | + | 1.50000i | −3.61960 | + | 6.26932i | 10.8209 | − | 2.81204i | 2.61673 | −10.7711 | + | 15.0659i | 13.2923i | 4.50000 | + | 7.79423i | 6.94757 | − | 6.84341i | ||||
4.13 | 0.761268 | − | 0.439518i | 2.59808 | + | 1.50000i | −3.61365 | + | 6.25902i | −9.95559 | − | 5.08784i | 2.63711 | −18.5184 | + | 0.260966i | 13.3854i | 4.50000 | + | 7.79423i | −9.81508 | + | 0.502454i | ||||
4.14 | 1.50072 | − | 0.866440i | 2.59808 | + | 1.50000i | −2.49856 | + | 4.32764i | −8.24745 | + | 7.54848i | 5.19864 | 14.9159 | − | 10.9780i | 22.5225i | 4.50000 | + | 7.79423i | −5.83680 | + | 18.4741i | ||||
4.15 | 1.96118 | − | 1.13229i | −2.59808 | − | 1.50000i | −1.43586 | + | 2.48698i | 11.1803 | − | 0.0158333i | −6.79371 | 18.1088 | + | 3.88231i | 24.6198i | 4.50000 | + | 7.79423i | 21.9087 | − | 12.6904i | ||||
4.16 | 2.01833 | − | 1.16528i | −2.59808 | − | 1.50000i | −1.28423 | + | 2.22435i | 2.89467 | + | 10.7991i | −6.99170 | −18.1257 | − | 3.80267i | 24.6305i | 4.50000 | + | 7.79423i | 18.4264 | + | 18.4231i | ||||
4.17 | 2.21373 | − | 1.27810i | −2.59808 | − | 1.50000i | −0.732945 | + | 1.26950i | −11.1788 | + | 0.184097i | −7.66857 | 2.98784 | + | 18.2777i | 24.1966i | 4.50000 | + | 7.79423i | −24.5116 | + | 14.6951i | ||||
4.18 | 2.89323 | − | 1.67041i | 2.59808 | + | 1.50000i | 1.58052 | − | 2.73755i | 0.879049 | + | 11.1457i | 10.0224 | 6.15758 | + | 17.4667i | 16.1660i | 4.50000 | + | 7.79423i | 21.1612 | + | 30.7788i | ||||
4.19 | 3.00489 | − | 1.73487i | 2.59808 | + | 1.50000i | 2.01958 | − | 3.49802i | 1.37289 | − | 11.0957i | 10.4092 | 12.4845 | − | 13.6798i | 13.7431i | 4.50000 | + | 7.79423i | −15.1243 | − | 35.7232i | ||||
4.20 | 3.89316 | − | 2.24772i | −2.59808 | − | 1.50000i | 6.10447 | − | 10.5732i | −9.40488 | − | 6.04552i | −13.4863 | 3.71892 | − | 18.1430i | − | 18.9210i | 4.50000 | + | 7.79423i | −50.2033 | − | 2.39667i | |||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
35.j | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 105.4.q.b | ✓ | 44 |
5.b | even | 2 | 1 | inner | 105.4.q.b | ✓ | 44 |
7.c | even | 3 | 1 | inner | 105.4.q.b | ✓ | 44 |
35.j | even | 6 | 1 | inner | 105.4.q.b | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.4.q.b | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
105.4.q.b | ✓ | 44 | 5.b | even | 2 | 1 | inner |
105.4.q.b | ✓ | 44 | 7.c | even | 3 | 1 | inner |
105.4.q.b | ✓ | 44 | 35.j | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{44} - 119 T_{2}^{42} + 8222 T_{2}^{40} - 380331 T_{2}^{38} + 13122870 T_{2}^{36} + \cdots + 38\!\cdots\!16 \) acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\).