Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [336,2,Mod(155,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.155");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 336.s (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: | \(2.68297350792\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
155.1 | −1.33839 | − | 0.456860i | −1.70974 | + | 0.277087i | 1.58256 | + | 1.22291i | −0.459458 | + | 0.459458i | 2.41489 | + | 0.410265i | 1.00000 | −1.55938 | − | 2.35973i | 2.84645 | − | 0.947494i | 0.824840 | − | 0.405024i | ||
155.2 | −1.31976 | − | 0.508179i | 1.45100 | − | 0.945839i | 1.48351 | + | 1.34134i | 1.69539 | − | 1.69539i | −2.39561 | + | 0.510910i | 1.00000 | −1.27623 | − | 2.52413i | 1.21078 | − | 2.74482i | −3.09906 | + | 1.37594i | ||
155.3 | −1.30954 | + | 0.533943i | 1.26449 | + | 1.18367i | 1.42981 | − | 1.39844i | 1.97913 | − | 1.97913i | −2.28791 | − | 0.874897i | 1.00000 | −1.12571 | + | 2.59476i | 0.197870 | + | 2.99347i | −1.53501 | + | 3.64849i | ||
155.4 | −1.29139 | + | 0.576471i | 1.71763 | − | 0.223071i | 1.33536 | − | 1.48889i | −2.27022 | + | 2.27022i | −2.08953 | + | 1.27823i | 1.00000 | −0.866165 | + | 2.69254i | 2.90048 | − | 0.766304i | 1.62302 | − | 4.24045i | ||
155.5 | −1.20363 | − | 0.742471i | −0.505560 | − | 1.65663i | 0.897474 | + | 1.78733i | −2.00763 | + | 2.00763i | −0.621486 | + | 2.36934i | 1.00000 | 0.246807 | − | 2.81764i | −2.48882 | + | 1.67505i | 3.90706 | − | 0.925846i | ||
155.6 | −1.15900 | + | 0.810376i | −1.50282 | − | 0.861124i | 0.686583 | − | 1.87846i | −1.19856 | + | 1.19856i | 2.43961 | − | 0.219802i | 1.00000 | 0.726504 | + | 2.73353i | 1.51693 | + | 2.58823i | 0.417853 | − | 2.36042i | ||
155.7 | −0.809995 | − | 1.15927i | 0.594785 | + | 1.62672i | −0.687816 | + | 1.87801i | 0.0563417 | − | 0.0563417i | 1.40404 | − | 2.00716i | 1.00000 | 2.73425 | − | 0.723812i | −2.29246 | + | 1.93510i | −0.110952 | − | 0.0196788i | ||
155.8 | −0.786529 | + | 1.17532i | −1.09085 | + | 1.34538i | −0.762743 | − | 1.84884i | 1.07379 | − | 1.07379i | −0.723260 | − | 2.34028i | 1.00000 | 2.77290 | + | 0.557705i | −0.620087 | − | 2.93522i | 0.417477 | + | 2.10661i | ||
155.9 | −0.720078 | + | 1.21716i | 0.399703 | − | 1.68530i | −0.962977 | − | 1.75291i | 2.09830 | − | 2.09830i | 1.76347 | + | 1.70005i | 1.00000 | 2.82699 | + | 0.0901270i | −2.68048 | − | 1.34724i | 1.04303 | + | 4.06491i | ||
155.10 | −0.641365 | − | 1.26042i | −1.71202 | − | 0.262630i | −1.17730 | + | 1.61678i | 2.76208 | − | 2.76208i | 0.767010 | + | 2.32631i | 1.00000 | 2.79289 | + | 0.446945i | 2.86205 | + | 0.899256i | −5.25287 | − | 1.70987i | ||
155.11 | −0.277681 | + | 1.38668i | 0.490427 | + | 1.66117i | −1.84579 | − | 0.770111i | −2.27018 | + | 2.27018i | −2.43970 | + | 0.218793i | 1.00000 | 1.58044 | − | 2.34568i | −2.51896 | + | 1.62936i | −2.51763 | − | 3.77840i | ||
155.12 | −0.103263 | − | 1.41044i | −1.15087 | + | 1.29441i | −1.97867 | + | 0.291291i | −0.186466 | + | 0.186466i | 1.94453 | + | 1.48957i | 1.00000 | 0.615171 | + | 2.76072i | −0.350987 | − | 2.97940i | 0.282253 | + | 0.243743i | ||
155.13 | 0.103263 | + | 1.41044i | 1.29441 | − | 1.15087i | −1.97867 | + | 0.291291i | 0.186466 | − | 0.186466i | 1.75690 | + | 1.70684i | 1.00000 | −0.615171 | − | 2.76072i | 0.350987 | − | 2.97940i | 0.282253 | + | 0.243743i | ||
155.14 | 0.277681 | − | 1.38668i | 1.66117 | + | 0.490427i | −1.84579 | − | 0.770111i | 2.27018 | − | 2.27018i | 1.14134 | − | 2.16733i | 1.00000 | −1.58044 | + | 2.34568i | 2.51896 | + | 1.62936i | −2.51763 | − | 3.77840i | ||
155.15 | 0.641365 | + | 1.26042i | −0.262630 | − | 1.71202i | −1.17730 | + | 1.61678i | −2.76208 | + | 2.76208i | 1.98942 | − | 1.42906i | 1.00000 | −2.79289 | − | 0.446945i | −2.86205 | + | 0.899256i | −5.25287 | − | 1.70987i | ||
155.16 | 0.720078 | − | 1.21716i | −1.68530 | + | 0.399703i | −0.962977 | − | 1.75291i | −2.09830 | + | 2.09830i | −0.727043 | + | 2.33910i | 1.00000 | −2.82699 | − | 0.0901270i | 2.68048 | − | 1.34724i | 1.04303 | + | 4.06491i | ||
155.17 | 0.786529 | − | 1.17532i | 1.34538 | − | 1.09085i | −0.762743 | − | 1.84884i | −1.07379 | + | 1.07379i | −0.223917 | − | 2.43923i | 1.00000 | −2.77290 | − | 0.557705i | 0.620087 | − | 2.93522i | 0.417477 | + | 2.10661i | ||
155.18 | 0.809995 | + | 1.15927i | 1.62672 | + | 0.594785i | −0.687816 | + | 1.87801i | −0.0563417 | + | 0.0563417i | 0.628122 | + | 2.36759i | 1.00000 | −2.73425 | + | 0.723812i | 2.29246 | + | 1.93510i | −0.110952 | − | 0.0196788i | ||
155.19 | 1.15900 | − | 0.810376i | −0.861124 | − | 1.50282i | 0.686583 | − | 1.87846i | 1.19856 | − | 1.19856i | −2.21589 | − | 1.04394i | 1.00000 | −0.726504 | − | 2.73353i | −1.51693 | + | 2.58823i | 0.417853 | − | 2.36042i | ||
155.20 | 1.20363 | + | 0.742471i | −1.65663 | − | 0.505560i | 0.897474 | + | 1.78733i | 2.00763 | − | 2.00763i | −1.61861 | − | 1.83851i | 1.00000 | −0.246807 | + | 2.81764i | 2.48882 | + | 1.67505i | 3.90706 | − | 0.925846i | ||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.f | odd | 4 | 1 | inner |
48.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 336.2.s.d | ✓ | 48 |
3.b | odd | 2 | 1 | inner | 336.2.s.d | ✓ | 48 |
4.b | odd | 2 | 1 | 1344.2.s.d | 48 | ||
12.b | even | 2 | 1 | 1344.2.s.d | 48 | ||
16.e | even | 4 | 1 | 1344.2.s.d | 48 | ||
16.f | odd | 4 | 1 | inner | 336.2.s.d | ✓ | 48 |
48.i | odd | 4 | 1 | 1344.2.s.d | 48 | ||
48.k | even | 4 | 1 | inner | 336.2.s.d | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
336.2.s.d | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
336.2.s.d | ✓ | 48 | 3.b | odd | 2 | 1 | inner |
336.2.s.d | ✓ | 48 | 16.f | odd | 4 | 1 | inner |
336.2.s.d | ✓ | 48 | 48.k | even | 4 | 1 | inner |
1344.2.s.d | 48 | 4.b | odd | 2 | 1 | ||
1344.2.s.d | 48 | 12.b | even | 2 | 1 | ||
1344.2.s.d | 48 | 16.e | even | 4 | 1 | ||
1344.2.s.d | 48 | 48.i | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{48} + 696 T_{5}^{44} + 196224 T_{5}^{40} + 29529376 T_{5}^{36} + 2599544544 T_{5}^{32} + 136749727616 T_{5}^{28} + 4177756384768 T_{5}^{24} + 67930945379840 T_{5}^{20} + \cdots + 40960000 \)
acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\).