Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,2,Mod(29,350)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.29");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 350.m (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: | \(2.79476407074\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 | −0.951057 | − | 0.309017i | −0.840443 | + | 1.15677i | 0.809017 | + | 0.587785i | −1.82116 | + | 1.29744i | 1.15677 | − | 0.840443i | 1.00000i | −0.587785 | − | 0.809017i | 0.295277 | + | 0.908768i | 2.13296 | − | 0.671171i | ||
29.2 | −0.951057 | − | 0.309017i | 0.310301 | − | 0.427092i | 0.809017 | + | 0.587785i | 1.73700 | + | 1.40813i | −0.427092 | + | 0.310301i | 1.00000i | −0.587785 | − | 0.809017i | 0.840930 | + | 2.58812i | −1.21685 | − | 1.87597i | ||
29.3 | −0.951057 | − | 0.309017i | 1.06039 | − | 1.45950i | 0.809017 | + | 0.587785i | 0.717612 | − | 2.11779i | −1.45950 | + | 1.06039i | 1.00000i | −0.587785 | − | 0.809017i | −0.0786699 | − | 0.242121i | −1.33692 | + | 1.79238i | ||
29.4 | 0.951057 | + | 0.309017i | −0.974986 | + | 1.34195i | 0.809017 | + | 0.587785i | 2.09432 | + | 0.783465i | −1.34195 | + | 0.974986i | − | 1.00000i | 0.587785 | + | 0.809017i | 0.0768108 | + | 0.236399i | 1.74971 | + | 1.39230i | |
29.5 | 0.951057 | + | 0.309017i | 1.26372 | − | 1.73935i | 0.809017 | + | 0.587785i | 0.915083 | + | 2.04025i | 1.73935 | − | 1.26372i | − | 1.00000i | 0.587785 | + | 0.809017i | −0.501328 | − | 1.54293i | 0.239823 | + | 2.22317i | |
29.6 | 0.951057 | + | 0.309017i | 1.41709 | − | 1.95046i | 0.809017 | + | 0.587785i | −0.0248168 | − | 2.23593i | 1.95046 | − | 1.41709i | − | 1.00000i | 0.587785 | + | 0.809017i | −0.869087 | − | 2.67478i | 0.667338 | − | 2.13416i | |
169.1 | −0.951057 | + | 0.309017i | −0.840443 | − | 1.15677i | 0.809017 | − | 0.587785i | −1.82116 | − | 1.29744i | 1.15677 | + | 0.840443i | − | 1.00000i | −0.587785 | + | 0.809017i | 0.295277 | − | 0.908768i | 2.13296 | + | 0.671171i | |
169.2 | −0.951057 | + | 0.309017i | 0.310301 | + | 0.427092i | 0.809017 | − | 0.587785i | 1.73700 | − | 1.40813i | −0.427092 | − | 0.310301i | − | 1.00000i | −0.587785 | + | 0.809017i | 0.840930 | − | 2.58812i | −1.21685 | + | 1.87597i | |
169.3 | −0.951057 | + | 0.309017i | 1.06039 | + | 1.45950i | 0.809017 | − | 0.587785i | 0.717612 | + | 2.11779i | −1.45950 | − | 1.06039i | − | 1.00000i | −0.587785 | + | 0.809017i | −0.0786699 | + | 0.242121i | −1.33692 | − | 1.79238i | |
169.4 | 0.951057 | − | 0.309017i | −0.974986 | − | 1.34195i | 0.809017 | − | 0.587785i | 2.09432 | − | 0.783465i | −1.34195 | − | 0.974986i | 1.00000i | 0.587785 | − | 0.809017i | 0.0768108 | − | 0.236399i | 1.74971 | − | 1.39230i | ||
169.5 | 0.951057 | − | 0.309017i | 1.26372 | + | 1.73935i | 0.809017 | − | 0.587785i | 0.915083 | − | 2.04025i | 1.73935 | + | 1.26372i | 1.00000i | 0.587785 | − | 0.809017i | −0.501328 | + | 1.54293i | 0.239823 | − | 2.22317i | ||
169.6 | 0.951057 | − | 0.309017i | 1.41709 | + | 1.95046i | 0.809017 | − | 0.587785i | −0.0248168 | + | 2.23593i | 1.95046 | + | 1.41709i | 1.00000i | 0.587785 | − | 0.809017i | −0.869087 | + | 2.67478i | 0.667338 | + | 2.13416i | ||
239.1 | −0.587785 | − | 0.809017i | −2.21568 | − | 0.719918i | −0.309017 | + | 0.951057i | 0.562587 | + | 2.16414i | 0.719918 | + | 2.21568i | − | 1.00000i | 0.951057 | − | 0.309017i | 1.96390 | + | 1.42686i | 1.42014 | − | 1.72719i | |
239.2 | −0.587785 | − | 0.809017i | 0.195447 | + | 0.0635047i | −0.309017 | + | 0.951057i | 0.0419655 | − | 2.23567i | −0.0635047 | − | 0.195447i | − | 1.00000i | 0.951057 | − | 0.309017i | −2.39288 | − | 1.73853i | −1.83337 | + | 1.28015i | |
239.3 | −0.587785 | − | 0.809017i | 1.85325 | + | 0.602159i | −0.309017 | + | 0.951057i | 1.98854 | + | 1.02259i | −0.602159 | − | 1.85325i | − | 1.00000i | 0.951057 | − | 0.309017i | 0.644904 | + | 0.468550i | −0.341542 | − | 2.20983i | |
239.4 | 0.587785 | + | 0.809017i | −3.21572 | − | 1.04485i | −0.309017 | + | 0.951057i | −2.16379 | − | 0.563933i | −1.04485 | − | 3.21572i | 1.00000i | −0.951057 | + | 0.309017i | 6.82209 | + | 4.95654i | −0.815611 | − | 2.08201i | ||
239.5 | 0.587785 | + | 0.809017i | −0.331395 | − | 0.107677i | −0.309017 | + | 0.951057i | −1.25842 | + | 1.84834i | −0.107677 | − | 0.331395i | 1.00000i | −0.951057 | + | 0.309017i | −2.32882 | − | 1.69199i | −2.23502 | + | 0.0683444i | ||
239.6 | 0.587785 | + | 0.809017i | 1.47802 | + | 0.480239i | −0.309017 | + | 0.951057i | 2.21108 | − | 0.333354i | 0.480239 | + | 1.47802i | 1.00000i | −0.951057 | + | 0.309017i | −0.473123 | − | 0.343744i | 1.56933 | + | 1.59286i | ||
309.1 | −0.587785 | + | 0.809017i | −2.21568 | + | 0.719918i | −0.309017 | − | 0.951057i | 0.562587 | − | 2.16414i | 0.719918 | − | 2.21568i | 1.00000i | 0.951057 | + | 0.309017i | 1.96390 | − | 1.42686i | 1.42014 | + | 1.72719i | ||
309.2 | −0.587785 | + | 0.809017i | 0.195447 | − | 0.0635047i | −0.309017 | − | 0.951057i | 0.0419655 | + | 2.23567i | −0.0635047 | + | 0.195447i | 1.00000i | 0.951057 | + | 0.309017i | −2.39288 | + | 1.73853i | −1.83337 | − | 1.28015i | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.e | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 350.2.m.a | ✓ | 24 |
25.e | even | 10 | 1 | inner | 350.2.m.a | ✓ | 24 |
25.f | odd | 20 | 1 | 8750.2.a.z | 12 | ||
25.f | odd | 20 | 1 | 8750.2.a.bb | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
350.2.m.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
350.2.m.a | ✓ | 24 | 25.e | even | 10 | 1 | inner |
8750.2.a.z | 12 | 25.f | odd | 20 | 1 | ||
8750.2.a.bb | 12 | 25.f | odd | 20 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 13 T_{3}^{22} + 30 T_{3}^{21} + 88 T_{3}^{20} - 390 T_{3}^{19} + 119 T_{3}^{18} + \cdots + 400 \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\).