Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,3,Mod(40,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 7]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.40");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 363.g (of order \(10\), degree \(4\), not 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: | \(9.89103359628\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
40.1 | −2.19808 | + | 3.02540i | 0.535233 | − | 1.64728i | −3.08541 | − | 9.49592i | 0.450251 | − | 0.327126i | 3.80719 | + | 5.24014i | −11.4499 | + | 3.72029i | 21.2846 | + | 6.91580i | −2.42705 | − | 1.76336i | 2.08124i | ||
40.2 | −1.82934 | + | 2.51787i | −0.535233 | + | 1.64728i | −1.75712 | − | 5.40785i | −4.58660 | + | 3.33236i | −3.16851 | − | 4.36108i | −9.13303 | + | 2.96750i | 4.99088 | + | 1.62164i | −2.42705 | − | 1.76336i | − | 17.6445i | |
40.3 | −1.22082 | + | 1.68031i | −0.535233 | + | 1.64728i | −0.0969856 | − | 0.298491i | 1.35053 | − | 0.981217i | −2.11452 | − | 2.91039i | 2.87537 | − | 0.934266i | −7.28135 | − | 2.36585i | −2.42705 | − | 1.76336i | 3.46720i | ||
40.4 | −0.0729469 | + | 0.100403i | 0.535233 | − | 1.64728i | 1.23131 | + | 3.78958i | −3.68632 | + | 2.67827i | 0.126348 | + | 0.173903i | 4.82146 | − | 1.56659i | −0.942426 | − | 0.306213i | −2.42705 | − | 1.76336i | − | 0.565488i | |
40.5 | 0.0729469 | − | 0.100403i | 0.535233 | − | 1.64728i | 1.23131 | + | 3.78958i | −3.68632 | + | 2.67827i | −0.126348 | − | 0.173903i | −4.82146 | + | 1.56659i | 0.942426 | + | 0.306213i | −2.42705 | − | 1.76336i | 0.565488i | ||
40.6 | 1.22082 | − | 1.68031i | −0.535233 | + | 1.64728i | −0.0969856 | − | 0.298491i | 1.35053 | − | 0.981217i | 2.11452 | + | 2.91039i | −2.87537 | + | 0.934266i | 7.28135 | + | 2.36585i | −2.42705 | − | 1.76336i | − | 3.46720i | |
40.7 | 1.82934 | − | 2.51787i | −0.535233 | + | 1.64728i | −1.75712 | − | 5.40785i | −4.58660 | + | 3.33236i | 3.16851 | + | 4.36108i | 9.13303 | − | 2.96750i | −4.99088 | − | 1.62164i | −2.42705 | − | 1.76336i | 17.6445i | ||
40.8 | 2.19808 | − | 3.02540i | 0.535233 | − | 1.64728i | −3.08541 | − | 9.49592i | 0.450251 | − | 0.327126i | −3.80719 | − | 5.24014i | 11.4499 | − | 3.72029i | −21.2846 | − | 6.91580i | −2.42705 | − | 1.76336i | − | 2.08124i | |
94.1 | −3.55657 | + | 1.15560i | −1.40126 | − | 1.01807i | 8.07771 | − | 5.86880i | −0.171980 | + | 0.529301i | 6.16016 | + | 2.00156i | 7.07642 | + | 9.73985i | −13.1546 | + | 18.1058i | 0.927051 | + | 2.85317i | − | 2.08124i | |
94.2 | −2.95993 | + | 0.961741i | 1.40126 | + | 1.01807i | 4.60019 | − | 3.34223i | 1.75192 | − | 5.39187i | −5.12675 | − | 1.66578i | 5.64452 | + | 7.76902i | −3.08454 | + | 4.24550i | 0.927051 | + | 2.85317i | 17.6445i | ||
94.3 | −1.97533 | + | 0.641823i | 1.40126 | + | 1.01807i | 0.253912 | − | 0.184478i | −0.515856 | + | 1.58764i | −3.42137 | − | 1.11167i | −1.77708 | − | 2.44594i | 4.50012 | − | 6.19388i | 0.927051 | + | 2.85317i | − | 3.46720i | |
94.4 | −0.118031 | + | 0.0383504i | −1.40126 | − | 1.01807i | −3.22361 | + | 2.34209i | 1.40805 | − | 4.33353i | 0.204435 | + | 0.0664249i | −2.97983 | − | 4.10138i | 0.582452 | − | 0.801676i | 0.927051 | + | 2.85317i | 0.565488i | ||
94.5 | 0.118031 | − | 0.0383504i | −1.40126 | − | 1.01807i | −3.22361 | + | 2.34209i | 1.40805 | − | 4.33353i | −0.204435 | − | 0.0664249i | 2.97983 | + | 4.10138i | −0.582452 | + | 0.801676i | 0.927051 | + | 2.85317i | − | 0.565488i | |
94.6 | 1.97533 | − | 0.641823i | 1.40126 | + | 1.01807i | 0.253912 | − | 0.184478i | −0.515856 | + | 1.58764i | 3.42137 | + | 1.11167i | 1.77708 | + | 2.44594i | −4.50012 | + | 6.19388i | 0.927051 | + | 2.85317i | 3.46720i | ||
94.7 | 2.95993 | − | 0.961741i | 1.40126 | + | 1.01807i | 4.60019 | − | 3.34223i | 1.75192 | − | 5.39187i | 5.12675 | + | 1.66578i | −5.64452 | − | 7.76902i | 3.08454 | − | 4.24550i | 0.927051 | + | 2.85317i | − | 17.6445i | |
94.8 | 3.55657 | − | 1.15560i | −1.40126 | − | 1.01807i | 8.07771 | − | 5.86880i | −0.171980 | + | 0.529301i | −6.16016 | − | 2.00156i | −7.07642 | − | 9.73985i | 13.1546 | − | 18.1058i | 0.927051 | + | 2.85317i | 2.08124i | ||
112.1 | −3.55657 | − | 1.15560i | −1.40126 | + | 1.01807i | 8.07771 | + | 5.86880i | −0.171980 | − | 0.529301i | 6.16016 | − | 2.00156i | 7.07642 | − | 9.73985i | −13.1546 | − | 18.1058i | 0.927051 | − | 2.85317i | 2.08124i | ||
112.2 | −2.95993 | − | 0.961741i | 1.40126 | − | 1.01807i | 4.60019 | + | 3.34223i | 1.75192 | + | 5.39187i | −5.12675 | + | 1.66578i | 5.64452 | − | 7.76902i | −3.08454 | − | 4.24550i | 0.927051 | − | 2.85317i | − | 17.6445i | |
112.3 | −1.97533 | − | 0.641823i | 1.40126 | − | 1.01807i | 0.253912 | + | 0.184478i | −0.515856 | − | 1.58764i | −3.42137 | + | 1.11167i | −1.77708 | + | 2.44594i | 4.50012 | + | 6.19388i | 0.927051 | − | 2.85317i | 3.46720i | ||
112.4 | −0.118031 | − | 0.0383504i | −1.40126 | + | 1.01807i | −3.22361 | − | 2.34209i | 1.40805 | + | 4.33353i | 0.204435 | − | 0.0664249i | −2.97983 | + | 4.10138i | 0.582452 | + | 0.801676i | 0.927051 | − | 2.85317i | − | 0.565488i | |
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
11.c | even | 5 | 3 | inner |
11.d | odd | 10 | 3 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 363.3.g.h | 32 | |
11.b | odd | 2 | 1 | inner | 363.3.g.h | 32 | |
11.c | even | 5 | 1 | 363.3.c.d | ✓ | 8 | |
11.c | even | 5 | 3 | inner | 363.3.g.h | 32 | |
11.d | odd | 10 | 1 | 363.3.c.d | ✓ | 8 | |
11.d | odd | 10 | 3 | inner | 363.3.g.h | 32 | |
33.f | even | 10 | 1 | 1089.3.c.j | 8 | ||
33.h | odd | 10 | 1 | 1089.3.c.j | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
363.3.c.d | ✓ | 8 | 11.c | even | 5 | 1 | |
363.3.c.d | ✓ | 8 | 11.d | odd | 10 | 1 | |
363.3.g.h | 32 | 1.a | even | 1 | 1 | trivial | |
363.3.g.h | 32 | 11.b | odd | 2 | 1 | inner | |
363.3.g.h | 32 | 11.c | even | 5 | 3 | inner | |
363.3.g.h | 32 | 11.d | odd | 10 | 3 | inner | |
1089.3.c.j | 8 | 33.f | even | 10 | 1 | ||
1089.3.c.j | 8 | 33.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{32} - 28 T_{2}^{30} + 546 T_{2}^{28} - 9212 T_{2}^{26} + 144443 T_{2}^{24} - 1551116 T_{2}^{22} + 14465556 T_{2}^{20} - 120719536 T_{2}^{18} + 848100901 T_{2}^{16} - 3507362544 T_{2}^{14} + \cdots + 6561 \)
acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\).