Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [775,2,Mod(51,775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(775, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([0, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("775.51");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 775.bl (of order \(15\), degree \(8\), 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: | \(6.18840615665\) |
Analytic rank: | \(0\) |
Dimension: | \(88\) |
Relative dimension: | \(11\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
51.1 | −0.728325 | − | 2.24155i | −1.23933 | − | 1.37642i | −2.87607 | + | 2.08958i | 0 | −2.18268 | + | 3.78052i | −0.412834 | − | 3.92786i | 2.96507 | + | 2.15425i | −0.0449980 | + | 0.428127i | 0 | ||||
51.2 | −0.710118 | − | 2.18552i | 1.44998 | + | 1.61037i | −2.65419 | + | 1.92838i | 0 | 2.48983 | − | 4.31251i | 0.00344974 | + | 0.0328221i | 2.38107 | + | 1.72995i | −0.177253 | + | 1.68645i | 0 | ||||
51.3 | −0.406696 | − | 1.25168i | −2.10309 | − | 2.33571i | 0.216726 | − | 0.157461i | 0 | −2.06825 | + | 3.58232i | 0.446441 | + | 4.24760i | −2.41472 | − | 1.75440i | −0.719001 | + | 6.84084i | 0 | ||||
51.4 | −0.390313 | − | 1.20126i | 0.0873606 | + | 0.0970238i | 0.327350 | − | 0.237834i | 0 | 0.0824529 | − | 0.142813i | 0.175285 | + | 1.66773i | −2.45718 | − | 1.78524i | 0.311804 | − | 2.96661i | 0 | ||||
51.5 | −0.111255 | − | 0.342407i | 1.77458 | + | 1.97087i | 1.51317 | − | 1.09938i | 0 | 0.477408 | − | 0.826895i | −0.492083 | − | 4.68185i | −1.12732 | − | 0.819045i | −0.421610 | + | 4.01135i | 0 | ||||
51.6 | −0.0136821 | − | 0.0421092i | 1.46288 | + | 1.62470i | 1.61645 | − | 1.17442i | 0 | 0.0483994 | − | 0.0838302i | 0.378239 | + | 3.59871i | −0.143211 | − | 0.104049i | −0.186026 | + | 1.76992i | 0 | ||||
51.7 | 0.0170192 | + | 0.0523797i | −1.21604 | − | 1.35055i | 1.61558 | − | 1.17379i | 0 | 0.0500455 | − | 0.0866813i | −0.227901 | − | 2.16834i | 0.178092 | + | 0.129392i | −0.0316464 | + | 0.301095i | 0 | ||||
51.8 | 0.315467 | + | 0.970908i | −0.470915 | − | 0.523004i | 0.774891 | − | 0.562991i | 0 | 0.359231 | − | 0.622206i | 0.224881 | + | 2.13960i | 2.44287 | + | 1.77485i | 0.261813 | − | 2.49098i | 0 | ||||
51.9 | 0.542143 | + | 1.66854i | 0.426864 | + | 0.474080i | −0.872089 | + | 0.633610i | 0 | −0.559603 | + | 0.969261i | −0.389029 | − | 3.70137i | 1.30870 | + | 0.950823i | 0.271046 | − | 2.57883i | 0 | ||||
51.10 | 0.670122 | + | 2.06242i | −2.21494 | − | 2.45994i | −2.18650 | + | 1.58858i | 0 | 3.58917 | − | 6.21662i | −0.321728 | − | 3.06104i | −1.23275 | − | 0.895646i | −0.831765 | + | 7.91371i | 0 | ||||
51.11 | 0.815637 | + | 2.51027i | 0.0352620 | + | 0.0391624i | −4.01817 | + | 2.91937i | 0 | −0.0695474 | + | 0.120460i | 0.206303 | + | 1.96284i | −6.33506 | − | 4.60269i | 0.313295 | − | 2.98080i | 0 | ||||
76.1 | −0.728325 | + | 2.24155i | −1.23933 | + | 1.37642i | −2.87607 | − | 2.08958i | 0 | −2.18268 | − | 3.78052i | −0.412834 | + | 3.92786i | 2.96507 | − | 2.15425i | −0.0449980 | − | 0.428127i | 0 | ||||
76.2 | −0.710118 | + | 2.18552i | 1.44998 | − | 1.61037i | −2.65419 | − | 1.92838i | 0 | 2.48983 | + | 4.31251i | 0.00344974 | − | 0.0328221i | 2.38107 | − | 1.72995i | −0.177253 | − | 1.68645i | 0 | ||||
76.3 | −0.406696 | + | 1.25168i | −2.10309 | + | 2.33571i | 0.216726 | + | 0.157461i | 0 | −2.06825 | − | 3.58232i | 0.446441 | − | 4.24760i | −2.41472 | + | 1.75440i | −0.719001 | − | 6.84084i | 0 | ||||
76.4 | −0.390313 | + | 1.20126i | 0.0873606 | − | 0.0970238i | 0.327350 | + | 0.237834i | 0 | 0.0824529 | + | 0.142813i | 0.175285 | − | 1.66773i | −2.45718 | + | 1.78524i | 0.311804 | + | 2.96661i | 0 | ||||
76.5 | −0.111255 | + | 0.342407i | 1.77458 | − | 1.97087i | 1.51317 | + | 1.09938i | 0 | 0.477408 | + | 0.826895i | −0.492083 | + | 4.68185i | −1.12732 | + | 0.819045i | −0.421610 | − | 4.01135i | 0 | ||||
76.6 | −0.0136821 | + | 0.0421092i | 1.46288 | − | 1.62470i | 1.61645 | + | 1.17442i | 0 | 0.0483994 | + | 0.0838302i | 0.378239 | − | 3.59871i | −0.143211 | + | 0.104049i | −0.186026 | − | 1.76992i | 0 | ||||
76.7 | 0.0170192 | − | 0.0523797i | −1.21604 | + | 1.35055i | 1.61558 | + | 1.17379i | 0 | 0.0500455 | + | 0.0866813i | −0.227901 | + | 2.16834i | 0.178092 | − | 0.129392i | −0.0316464 | − | 0.301095i | 0 | ||||
76.8 | 0.315467 | − | 0.970908i | −0.470915 | + | 0.523004i | 0.774891 | + | 0.562991i | 0 | 0.359231 | + | 0.622206i | 0.224881 | − | 2.13960i | 2.44287 | − | 1.77485i | 0.261813 | + | 2.49098i | 0 | ||||
76.9 | 0.542143 | − | 1.66854i | 0.426864 | − | 0.474080i | −0.872089 | − | 0.633610i | 0 | −0.559603 | − | 0.969261i | −0.389029 | + | 3.70137i | 1.30870 | − | 0.950823i | 0.271046 | + | 2.57883i | 0 | ||||
See all 88 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
31.g | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 775.2.bl.d | ✓ | 88 |
5.b | even | 2 | 1 | 775.2.bl.e | yes | 88 | |
5.c | odd | 4 | 2 | 775.2.ck.d | 176 | ||
31.g | even | 15 | 1 | inner | 775.2.bl.d | ✓ | 88 |
155.u | even | 30 | 1 | 775.2.bl.e | yes | 88 | |
155.w | odd | 60 | 2 | 775.2.ck.d | 176 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
775.2.bl.d | ✓ | 88 | 1.a | even | 1 | 1 | trivial |
775.2.bl.d | ✓ | 88 | 31.g | even | 15 | 1 | inner |
775.2.bl.e | yes | 88 | 5.b | even | 2 | 1 | |
775.2.bl.e | yes | 88 | 155.u | even | 30 | 1 | |
775.2.ck.d | 176 | 5.c | odd | 4 | 2 | ||
775.2.ck.d | 176 | 155.w | odd | 60 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} + 32 T_{2}^{86} + 3 T_{2}^{85} + 608 T_{2}^{84} + 92 T_{2}^{83} + 8924 T_{2}^{82} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\).