Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,2,Mod(143,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([27, 34]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("950.143");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 950.bb (of order \(36\), degree \(12\), 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: | \(7.58578819202\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{36})\) |
Twist minimal: | no (minimal twist has level 190) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
143.1 | −0.573576 | + | 0.819152i | −0.243772 | + | 2.78632i | −0.342020 | − | 0.939693i | 0 | −2.14260 | − | 1.79785i | 0.966713 | + | 3.60782i | 0.965926 | + | 0.258819i | −4.74974 | − | 0.837508i | 0 | ||||
143.2 | −0.573576 | + | 0.819152i | −0.197676 | + | 2.25945i | −0.342020 | − | 0.939693i | 0 | −1.73745 | − | 1.45789i | −0.925273 | − | 3.45316i | 0.965926 | + | 0.258819i | −2.11161 | − | 0.372333i | 0 | ||||
143.3 | −0.573576 | + | 0.819152i | 0.0801625 | − | 0.916262i | −0.342020 | − | 0.939693i | 0 | 0.704578 | + | 0.591211i | −0.322803 | − | 1.20472i | 0.965926 | + | 0.258819i | 2.12131 | + | 0.374045i | 0 | ||||
143.4 | −0.573576 | + | 0.819152i | 0.0833140 | − | 0.952283i | −0.342020 | − | 0.939693i | 0 | 0.732278 | + | 0.614454i | 0.790865 | + | 2.95155i | 0.965926 | + | 0.258819i | 2.05452 | + | 0.362268i | 0 | ||||
143.5 | −0.573576 | + | 0.819152i | 0.277971 | − | 3.17723i | −0.342020 | − | 0.939693i | 0 | 2.44319 | + | 2.05008i | −1.05257 | − | 3.92825i | 0.965926 | + | 0.258819i | −7.06307 | − | 1.24541i | 0 | ||||
143.6 | 0.573576 | − | 0.819152i | −0.255151 | + | 2.91639i | −0.342020 | − | 0.939693i | 0 | 2.24262 | + | 1.88178i | −0.0607618 | − | 0.226766i | −0.965926 | − | 0.258819i | −5.48583 | − | 0.967300i | 0 | ||||
143.7 | 0.573576 | − | 0.819152i | −0.0418747 | + | 0.478630i | −0.342020 | − | 0.939693i | 0 | 0.368052 | + | 0.308832i | 0.971585 | + | 3.62600i | −0.965926 | − | 0.258819i | 2.72709 | + | 0.480860i | 0 | ||||
143.8 | 0.573576 | − | 0.819152i | 0.0201525 | − | 0.230344i | −0.342020 | − | 0.939693i | 0 | −0.177128 | − | 0.148628i | −1.23887 | − | 4.62353i | −0.965926 | − | 0.258819i | 2.90177 | + | 0.511661i | 0 | ||||
143.9 | 0.573576 | − | 0.819152i | 0.0709170 | − | 0.810585i | −0.342020 | − | 0.939693i | 0 | −0.623316 | − | 0.523024i | 0.211848 | + | 0.790626i | −0.965926 | − | 0.258819i | 2.30240 | + | 0.405976i | 0 | ||||
143.10 | 0.573576 | − | 0.819152i | 0.205957 | − | 2.35409i | −0.342020 | − | 0.939693i | 0 | −1.81023 | − | 1.51896i | −0.0727825 | − | 0.271628i | −0.965926 | − | 0.258819i | −2.54492 | − | 0.448738i | 0 | ||||
193.1 | −0.0871557 | − | 0.996195i | −1.35298 | + | 2.90149i | −0.984808 | + | 0.173648i | 0 | 3.00836 | + | 1.09496i | 1.43361 | + | 0.384135i | 0.258819 | + | 0.965926i | −4.65969 | − | 5.55320i | 0 | ||||
193.2 | −0.0871557 | − | 0.996195i | −0.507694 | + | 1.08875i | −0.984808 | + | 0.173648i | 0 | 1.12886 | + | 0.410871i | −2.08332 | − | 0.558224i | 0.258819 | + | 0.965926i | 1.00073 | + | 1.19263i | 0 | ||||
193.3 | −0.0871557 | − | 0.996195i | −0.186197 | + | 0.399301i | −0.984808 | + | 0.173648i | 0 | 0.414010 | + | 0.150687i | −1.97694 | − | 0.529719i | 0.258819 | + | 0.965926i | 1.80359 | + | 2.14944i | 0 | ||||
193.4 | −0.0871557 | − | 0.996195i | 0.684861 | − | 1.46869i | −0.984808 | + | 0.173648i | 0 | −1.52279 | − | 0.554250i | 3.13506 | + | 0.840038i | 0.258819 | + | 0.965926i | 0.240351 | + | 0.286439i | 0 | ||||
193.5 | −0.0871557 | − | 0.996195i | 1.36202 | − | 2.92085i | −0.984808 | + | 0.173648i | 0 | −3.02844 | − | 1.10226i | −1.04490 | − | 0.279979i | 0.258819 | + | 0.965926i | −4.74793 | − | 5.65836i | 0 | ||||
193.6 | 0.0871557 | + | 0.996195i | −1.08832 | + | 2.33390i | −0.984808 | + | 0.173648i | 0 | −2.41987 | − | 0.880761i | 3.72976 | + | 0.999386i | −0.258819 | − | 0.965926i | −2.33430 | − | 2.78191i | 0 | ||||
193.7 | 0.0871557 | + | 0.996195i | −0.602383 | + | 1.29181i | −0.984808 | + | 0.173648i | 0 | −1.33940 | − | 0.487501i | 1.39748 | + | 0.374454i | −0.258819 | − | 0.965926i | 0.622444 | + | 0.741800i | 0 | ||||
193.8 | 0.0871557 | + | 0.996195i | 0.202610 | − | 0.434499i | −0.984808 | + | 0.173648i | 0 | 0.450504 | + | 0.163970i | −3.80940 | − | 1.02072i | −0.258819 | − | 0.965926i | 1.78062 | + | 2.12207i | 0 | ||||
193.9 | 0.0871557 | + | 0.996195i | 0.715886 | − | 1.53522i | −0.984808 | + | 0.173648i | 0 | 1.59177 | + | 0.579359i | 4.29599 | + | 1.15111i | −0.258819 | − | 0.965926i | 0.0839459 | + | 0.100043i | 0 | ||||
193.10 | 0.0871557 | + | 0.996195i | 0.772202 | − | 1.65599i | −0.984808 | + | 0.173648i | 0 | 1.71699 | + | 0.624934i | −2.34531 | − | 0.628423i | −0.258819 | − | 0.965926i | −0.217652 | − | 0.259388i | 0 | ||||
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
19.f | odd | 18 | 1 | inner |
95.r | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 950.2.bb.e | 120 | |
5.b | even | 2 | 1 | 190.2.r.a | ✓ | 120 | |
5.c | odd | 4 | 1 | 190.2.r.a | ✓ | 120 | |
5.c | odd | 4 | 1 | inner | 950.2.bb.e | 120 | |
19.f | odd | 18 | 1 | inner | 950.2.bb.e | 120 | |
95.o | odd | 18 | 1 | 190.2.r.a | ✓ | 120 | |
95.r | even | 36 | 1 | 190.2.r.a | ✓ | 120 | |
95.r | even | 36 | 1 | inner | 950.2.bb.e | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
190.2.r.a | ✓ | 120 | 5.b | even | 2 | 1 | |
190.2.r.a | ✓ | 120 | 5.c | odd | 4 | 1 | |
190.2.r.a | ✓ | 120 | 95.o | odd | 18 | 1 | |
190.2.r.a | ✓ | 120 | 95.r | even | 36 | 1 | |
950.2.bb.e | 120 | 1.a | even | 1 | 1 | trivial | |
950.2.bb.e | 120 | 5.c | odd | 4 | 1 | inner | |
950.2.bb.e | 120 | 19.f | odd | 18 | 1 | inner | |
950.2.bb.e | 120 | 95.r | even | 36 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{120} - 36 T_{3}^{116} + 120 T_{3}^{115} - 192 T_{3}^{113} + 2682 T_{3}^{112} - 4320 T_{3}^{111} + 7200 T_{3}^{110} + 8988 T_{3}^{109} - 1056716 T_{3}^{108} - 337368 T_{3}^{107} - 240768 T_{3}^{106} + \cdots + 110075314176 \)
acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).