Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [861,2,Mod(379,861)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(861, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("861.379");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 861 = 3 \cdot 7 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 861.n (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: | \(6.87511961403\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
379.1 | −2.05092 | − | 1.49008i | −1.00000 | 1.36789 | + | 4.20994i | 0.749014 | + | 2.30523i | 2.05092 | + | 1.49008i | −0.809017 | + | 0.587785i | 1.90094 | − | 5.85050i | 1.00000 | 1.89881 | − | 5.84392i | ||||
379.2 | −1.75068 | − | 1.27194i | −1.00000 | 0.829009 | + | 2.55143i | −0.874705 | − | 2.69207i | 1.75068 | + | 1.27194i | −0.809017 | + | 0.587785i | 0.456542 | − | 1.40509i | 1.00000 | −1.89283 | + | 5.82553i | ||||
379.3 | −1.25194 | − | 0.909585i | −1.00000 | 0.121966 | + | 0.375373i | −0.134104 | − | 0.412730i | 1.25194 | + | 0.909585i | −0.809017 | + | 0.587785i | −0.767655 | + | 2.36260i | 1.00000 | −0.207523 | + | 0.638691i | ||||
379.4 | −0.789928 | − | 0.573916i | −1.00000 | −0.323428 | − | 0.995408i | 0.165515 | + | 0.509404i | 0.789928 | + | 0.573916i | −0.809017 | + | 0.587785i | −0.919248 | + | 2.82915i | 1.00000 | 0.161610 | − | 0.497385i | ||||
379.5 | −0.748328 | − | 0.543692i | −1.00000 | −0.353640 | − | 1.08839i | 1.24839 | + | 3.84215i | 0.748328 | + | 0.543692i | −0.809017 | + | 0.587785i | −0.898783 | + | 2.76617i | 1.00000 | 1.15474 | − | 3.55393i | ||||
379.6 | −0.0474178 | − | 0.0344510i | −1.00000 | −0.616972 | − | 1.89885i | 0.137025 | + | 0.421721i | 0.0474178 | + | 0.0344510i | −0.809017 | + | 0.587785i | −0.0723857 | + | 0.222780i | 1.00000 | 0.00803128 | − | 0.0247177i | ||||
379.7 | 0.578443 | + | 0.420264i | −1.00000 | −0.460059 | − | 1.41592i | −1.04672 | − | 3.22147i | −0.578443 | − | 0.420264i | −0.809017 | + | 0.587785i | 0.770831 | − | 2.37237i | 1.00000 | 0.748399 | − | 2.30333i | ||||
379.8 | 1.04389 | + | 0.758430i | −1.00000 | −0.103545 | − | 0.318680i | 0.753340 | + | 2.31854i | −1.04389 | − | 0.758430i | −0.809017 | + | 0.587785i | 0.931067 | − | 2.86553i | 1.00000 | −0.972048 | + | 2.99166i | ||||
379.9 | 1.63189 | + | 1.18564i | −1.00000 | 0.639302 | + | 1.96757i | −0.682008 | − | 2.09901i | −1.63189 | − | 1.18564i | −0.809017 | + | 0.587785i | −0.0429009 | + | 0.132035i | 1.00000 | 1.37570 | − | 4.23397i | ||||
379.10 | 2.07596 | + | 1.50828i | −1.00000 | 1.41670 | + | 4.36014i | 1.18425 | + | 3.64474i | −2.07596 | − | 1.50828i | −0.809017 | + | 0.587785i | −2.04939 | + | 6.30739i | 1.00000 | −3.03882 | + | 9.35253i | ||||
631.1 | −0.871631 | + | 2.68260i | −1.00000 | −4.81859 | − | 3.50091i | 1.18938 | + | 0.864139i | 0.871631 | − | 2.68260i | 0.309017 | + | 0.951057i | 9.02766 | − | 6.55898i | 1.00000 | −3.35485 | + | 2.43744i | ||||
631.2 | −0.644982 | + | 1.98505i | −1.00000 | −1.90639 | − | 1.38507i | 0.173347 | + | 0.125944i | 0.644982 | − | 1.98505i | 0.309017 | + | 0.951057i | 0.601856 | − | 0.437274i | 1.00000 | −0.361810 | + | 0.262870i | ||||
631.3 | −0.520464 | + | 1.60182i | −1.00000 | −0.676923 | − | 0.491813i | −0.682871 | − | 0.496135i | 0.520464 | − | 1.60182i | 0.309017 | + | 0.951057i | −1.58507 | + | 1.15162i | 1.00000 | 1.15013 | − | 0.835619i | ||||
631.4 | −0.460795 | + | 1.41818i | −1.00000 | −0.180874 | − | 0.131413i | 3.42821 | + | 2.49074i | 0.460795 | − | 1.41818i | 0.309017 | + | 0.951057i | −2.14304 | + | 1.55701i | 1.00000 | −5.11202 | + | 3.71410i | ||||
631.5 | −0.138407 | + | 0.425972i | −1.00000 | 1.45574 | + | 1.05766i | −3.36253 | − | 2.44302i | 0.138407 | − | 0.425972i | 0.309017 | + | 0.951057i | −1.37672 | + | 1.00025i | 1.00000 | 1.50606 | − | 1.09421i | ||||
631.6 | −0.0301873 | + | 0.0929069i | −1.00000 | 1.61031 | + | 1.16996i | 1.86597 | + | 1.35571i | 0.0301873 | − | 0.0929069i | 0.309017 | + | 0.951057i | −0.315371 | + | 0.229130i | 1.00000 | −0.182283 | + | 0.132436i | ||||
631.7 | 0.410384 | − | 1.26303i | −1.00000 | 0.191201 | + | 0.138916i | −1.52489 | − | 1.10790i | −0.410384 | + | 1.26303i | 0.309017 | + | 0.951057i | 2.40272 | − | 1.74568i | 1.00000 | −2.02510 | + | 1.47132i | ||||
631.8 | 0.492891 | − | 1.51696i | −1.00000 | −0.440198 | − | 0.319822i | −1.96225 | − | 1.42566i | −0.492891 | + | 1.51696i | 0.309017 | + | 0.951057i | 1.87868 | − | 1.36494i | 1.00000 | −3.12985 | + | 2.27397i | ||||
631.9 | 0.754720 | − | 2.32279i | −1.00000 | −3.20772 | − | 2.33054i | 0.0540505 | + | 0.0392700i | −0.754720 | + | 2.32279i | 0.309017 | + | 0.951057i | −3.88253 | + | 2.82082i | 1.00000 | 0.132009 | − | 0.0959102i | ||||
631.10 | 0.817488 | − | 2.51597i | −1.00000 | −4.04378 | − | 2.93798i | 2.32159 | + | 1.68673i | −0.817488 | + | 2.51597i | 0.309017 | + | 0.951057i | −6.41719 | + | 4.66236i | 1.00000 | 6.14164 | − | 4.46217i | ||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
41.d | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 861.2.n.e | ✓ | 40 |
41.d | even | 5 | 1 | inner | 861.2.n.e | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
861.2.n.e | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
861.2.n.e | ✓ | 40 | 41.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 3 T_{2}^{39} + 24 T_{2}^{38} + 62 T_{2}^{37} + 303 T_{2}^{36} + 708 T_{2}^{35} + 2759 T_{2}^{34} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\).