Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [660,2,Mod(151,660)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(660, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 0, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("660.151");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 660.be (of order \(10\), degree \(4\), 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: | \(5.27012653340\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(24\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
151.1 | −1.39999 | + | 0.200103i | 0.951057 | − | 0.309017i | 1.91992 | − | 0.560284i | 0.809017 | + | 0.587785i | −1.26963 | + | 0.622929i | 0.767370 | − | 2.36172i | −2.57574 | + | 1.16857i | 0.809017 | − | 0.587785i | −1.25023 | − | 0.661004i |
151.2 | −1.39898 | − | 0.207008i | 0.951057 | − | 0.309017i | 1.91430 | + | 0.579201i | 0.809017 | + | 0.587785i | −1.39448 | + | 0.235432i | −0.370335 | + | 1.13977i | −2.55816 | − | 1.20657i | 0.809017 | − | 0.587785i | −1.01012 | − | 0.989774i |
151.3 | −1.33488 | − | 0.467017i | −0.951057 | + | 0.309017i | 1.56379 | + | 1.24682i | 0.809017 | + | 0.587785i | 1.41386 | + | 0.0316600i | −0.589915 | + | 1.81557i | −1.50518 | − | 2.39467i | 0.809017 | − | 0.587785i | −0.805432 | − | 1.16245i |
151.4 | −1.32024 | + | 0.506913i | −0.951057 | + | 0.309017i | 1.48608 | − | 1.33850i | 0.809017 | + | 0.587785i | 1.09898 | − | 0.890080i | −1.07719 | + | 3.31525i | −1.28348 | + | 2.52045i | 0.809017 | − | 0.587785i | −1.36605 | − | 0.365918i |
151.5 | −0.965670 | + | 1.03319i | −0.951057 | + | 0.309017i | −0.134965 | − | 1.99544i | 0.809017 | + | 0.587785i | 0.599133 | − | 1.28103i | 0.660835 | − | 2.03384i | 2.19200 | + | 1.78749i | 0.809017 | − | 0.587785i | −1.38854 | + | 0.268262i |
151.6 | −0.909042 | + | 1.08335i | 0.951057 | − | 0.309017i | −0.347286 | − | 1.96962i | 0.809017 | + | 0.587785i | −0.529777 | + | 1.31123i | −0.0472177 | + | 0.145321i | 2.44948 | + | 1.41423i | 0.809017 | − | 0.587785i | −1.37221 | + | 0.342126i |
151.7 | −0.891899 | − | 1.09750i | −0.951057 | + | 0.309017i | −0.409031 | + | 1.95773i | 0.809017 | + | 0.587785i | 1.18739 | + | 0.768176i | 0.00165851 | − | 0.00510436i | 2.51343 | − | 1.29718i | 0.809017 | − | 0.587785i | −0.0764651 | − | 1.41214i |
151.8 | −0.768176 | − | 1.18739i | 0.951057 | − | 0.309017i | −0.819810 | + | 1.82426i | 0.809017 | + | 0.587785i | −1.09750 | − | 0.891899i | −0.00165851 | + | 0.00510436i | 2.79587 | − | 0.427913i | 0.809017 | − | 0.587785i | 0.0764651 | − | 1.41214i |
151.9 | −0.364252 | + | 1.36650i | −0.951057 | + | 0.309017i | −1.73464 | − | 0.995501i | 0.809017 | + | 0.587785i | −0.0758471 | − | 1.41218i | −0.494503 | + | 1.52193i | 1.99220 | − | 2.00777i | 0.809017 | − | 0.587785i | −1.09789 | + | 0.891419i |
151.10 | −0.198732 | + | 1.40018i | 0.951057 | − | 0.309017i | −1.92101 | − | 0.556521i | 0.809017 | + | 0.587785i | 0.243674 | + | 1.39306i | −1.45429 | + | 4.47583i | 1.16100 | − | 2.57916i | 0.809017 | − | 0.587785i | −0.983783 | + | 1.01596i |
151.11 | −0.0316600 | − | 1.41386i | 0.951057 | − | 0.309017i | −1.99800 | + | 0.0895256i | 0.809017 | + | 0.587785i | −0.467017 | − | 1.33488i | 0.589915 | − | 1.81557i | 0.189833 | + | 2.82205i | 0.809017 | − | 0.587785i | 0.805432 | − | 1.16245i |
151.12 | 0.235432 | − | 1.39448i | −0.951057 | + | 0.309017i | −1.88914 | − | 0.656611i | 0.809017 | + | 0.587785i | 0.207008 | + | 1.39898i | 0.370335 | − | 1.13977i | −1.36040 | + | 2.47978i | 0.809017 | − | 0.587785i | 1.01012 | − | 0.989774i |
151.13 | 0.269956 | + | 1.38821i | 0.951057 | − | 0.309017i | −1.85425 | + | 0.749510i | 0.809017 | + | 0.587785i | 0.685723 | + | 1.23684i | 1.07349 | − | 3.30387i | −1.54104 | − | 2.37175i | 0.809017 | − | 0.587785i | −0.597570 | + | 1.28176i |
151.14 | 0.275572 | + | 1.38710i | −0.951057 | + | 0.309017i | −1.84812 | + | 0.764494i | 0.809017 | + | 0.587785i | −0.690723 | − | 1.23406i | 0.550425 | − | 1.69403i | −1.56972 | − | 2.35286i | 0.809017 | − | 0.587785i | −0.592378 | + | 1.28417i |
151.15 | 0.622929 | − | 1.26963i | −0.951057 | + | 0.309017i | −1.22392 | − | 1.58178i | 0.809017 | + | 0.587785i | −0.200103 | + | 1.39999i | −0.767370 | + | 2.36172i | −2.77069 | + | 0.568589i | 0.809017 | − | 0.587785i | 1.25023 | − | 0.661004i |
151.16 | 0.737307 | + | 1.20681i | 0.951057 | − | 0.309017i | −0.912758 | + | 1.77957i | 0.809017 | + | 0.587785i | 1.07414 | + | 0.919900i | −0.917712 | + | 2.82443i | −2.82058 | + | 0.210569i | 0.809017 | − | 0.587785i | −0.112849 | + | 1.40970i |
151.17 | 0.890080 | − | 1.09898i | 0.951057 | − | 0.309017i | −0.415515 | − | 1.95636i | 0.809017 | + | 0.587785i | 0.506913 | − | 1.32024i | 1.07719 | − | 3.31525i | −2.51984 | − | 1.28467i | 0.809017 | − | 0.587785i | 1.36605 | − | 0.365918i |
151.18 | 0.919900 | + | 1.07414i | −0.951057 | + | 0.309017i | −0.307569 | + | 1.97621i | 0.809017 | + | 0.587785i | −1.20681 | − | 0.737307i | 0.917712 | − | 2.82443i | −2.40566 | + | 1.48754i | 0.809017 | − | 0.587785i | 0.112849 | + | 1.40970i |
151.19 | 1.23406 | + | 0.690723i | 0.951057 | − | 0.309017i | 1.04580 | + | 1.70479i | 0.809017 | + | 0.587785i | 1.38710 | + | 0.275572i | −0.550425 | + | 1.69403i | 0.113047 | + | 2.82617i | 0.809017 | − | 0.587785i | 0.592378 | + | 1.28417i |
151.20 | 1.23684 | + | 0.685723i | −0.951057 | + | 0.309017i | 1.05957 | + | 1.69627i | 0.809017 | + | 0.587785i | −1.38821 | − | 0.269956i | −1.07349 | + | 3.30387i | 0.147351 | + | 2.82459i | 0.809017 | − | 0.587785i | 0.597570 | + | 1.28176i |
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
11.d | odd | 10 | 1 | inner |
44.g | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 660.2.be.b | ✓ | 96 |
4.b | odd | 2 | 1 | inner | 660.2.be.b | ✓ | 96 |
11.d | odd | 10 | 1 | inner | 660.2.be.b | ✓ | 96 |
44.g | even | 10 | 1 | inner | 660.2.be.b | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
660.2.be.b | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
660.2.be.b | ✓ | 96 | 4.b | odd | 2 | 1 | inner |
660.2.be.b | ✓ | 96 | 11.d | odd | 10 | 1 | inner |
660.2.be.b | ✓ | 96 | 44.g | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{96} + 74 T_{7}^{94} + 3283 T_{7}^{92} + 112572 T_{7}^{90} + 3322475 T_{7}^{88} + \cdots + 60523872256 \) acting on \(S_{2}^{\mathrm{new}}(660, [\chi])\).