Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1716,2,Mod(157,1716)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1716, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 8, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1716.157");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1716.z (of order \(5\), 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: | \(13.7023289869\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(7\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
157.1 | 0 | 0.309017 | + | 0.951057i | 0 | −2.66097 | − | 1.93331i | 0 | 1.32690 | − | 4.08378i | 0 | −0.809017 | + | 0.587785i | 0 | ||||||||||
157.2 | 0 | 0.309017 | + | 0.951057i | 0 | −2.42313 | − | 1.76050i | 0 | −1.22736 | + | 3.77743i | 0 | −0.809017 | + | 0.587785i | 0 | ||||||||||
157.3 | 0 | 0.309017 | + | 0.951057i | 0 | −2.11469 | − | 1.53641i | 0 | 0.0695547 | − | 0.214067i | 0 | −0.809017 | + | 0.587785i | 0 | ||||||||||
157.4 | 0 | 0.309017 | + | 0.951057i | 0 | 0.236907 | + | 0.172123i | 0 | −0.645989 | + | 1.98815i | 0 | −0.809017 | + | 0.587785i | 0 | ||||||||||
157.5 | 0 | 0.309017 | + | 0.951057i | 0 | 0.299270 | + | 0.217433i | 0 | 1.43233 | − | 4.40826i | 0 | −0.809017 | + | 0.587785i | 0 | ||||||||||
157.6 | 0 | 0.309017 | + | 0.951057i | 0 | 1.77307 | + | 1.28821i | 0 | −1.20885 | + | 3.72047i | 0 | −0.809017 | + | 0.587785i | 0 | ||||||||||
157.7 | 0 | 0.309017 | + | 0.951057i | 0 | 3.27151 | + | 2.37689i | 0 | 0.680469 | − | 2.09427i | 0 | −0.809017 | + | 0.587785i | 0 | ||||||||||
313.1 | 0 | −0.809017 | − | 0.587785i | 0 | −1.20740 | + | 3.71600i | 0 | 0.900522 | − | 0.654268i | 0 | 0.309017 | + | 0.951057i | 0 | ||||||||||
313.2 | 0 | −0.809017 | − | 0.587785i | 0 | −0.558385 | + | 1.71853i | 0 | −2.68818 | + | 1.95307i | 0 | 0.309017 | + | 0.951057i | 0 | ||||||||||
313.3 | 0 | −0.809017 | − | 0.587785i | 0 | −0.373730 | + | 1.15022i | 0 | 0.357194 | − | 0.259516i | 0 | 0.309017 | + | 0.951057i | 0 | ||||||||||
313.4 | 0 | −0.809017 | − | 0.587785i | 0 | 0.152253 | − | 0.468586i | 0 | 2.52781 | − | 1.83656i | 0 | 0.309017 | + | 0.951057i | 0 | ||||||||||
313.5 | 0 | −0.809017 | − | 0.587785i | 0 | 0.360665 | − | 1.11001i | 0 | −3.79821 | + | 2.75956i | 0 | 0.309017 | + | 0.951057i | 0 | ||||||||||
313.6 | 0 | −0.809017 | − | 0.587785i | 0 | 1.09186 | − | 3.36039i | 0 | −1.00142 | + | 0.727576i | 0 | 0.309017 | + | 0.951057i | 0 | ||||||||||
313.7 | 0 | −0.809017 | − | 0.587785i | 0 | 1.15278 | − | 3.54788i | 0 | 0.775227 | − | 0.563235i | 0 | 0.309017 | + | 0.951057i | 0 | ||||||||||
625.1 | 0 | −0.809017 | + | 0.587785i | 0 | −1.20740 | − | 3.71600i | 0 | 0.900522 | + | 0.654268i | 0 | 0.309017 | − | 0.951057i | 0 | ||||||||||
625.2 | 0 | −0.809017 | + | 0.587785i | 0 | −0.558385 | − | 1.71853i | 0 | −2.68818 | − | 1.95307i | 0 | 0.309017 | − | 0.951057i | 0 | ||||||||||
625.3 | 0 | −0.809017 | + | 0.587785i | 0 | −0.373730 | − | 1.15022i | 0 | 0.357194 | + | 0.259516i | 0 | 0.309017 | − | 0.951057i | 0 | ||||||||||
625.4 | 0 | −0.809017 | + | 0.587785i | 0 | 0.152253 | + | 0.468586i | 0 | 2.52781 | + | 1.83656i | 0 | 0.309017 | − | 0.951057i | 0 | ||||||||||
625.5 | 0 | −0.809017 | + | 0.587785i | 0 | 0.360665 | + | 1.11001i | 0 | −3.79821 | − | 2.75956i | 0 | 0.309017 | − | 0.951057i | 0 | ||||||||||
625.6 | 0 | −0.809017 | + | 0.587785i | 0 | 1.09186 | + | 3.36039i | 0 | −1.00142 | − | 0.727576i | 0 | 0.309017 | − | 0.951057i | 0 | ||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1716.2.z.g | ✓ | 28 |
11.c | even | 5 | 1 | inner | 1716.2.z.g | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1716.2.z.g | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
1716.2.z.g | ✓ | 28 | 11.c | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{28} + 2 T_{5}^{27} + 26 T_{5}^{26} + 65 T_{5}^{25} + 492 T_{5}^{24} + 1280 T_{5}^{23} + \cdots + 2560000 \) acting on \(S_{2}^{\mathrm{new}}(1716, [\chi])\).