Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(47,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([10, 5, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("930.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 930.bk (of order \(20\), 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: | \(7.42608738798\) |
Analytic rank: | \(0\) |
Dimension: | \(512\) |
Relative dimension: | \(64\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | −0.156434 | + | 0.987688i | −1.73203 | − | 0.00940555i | −0.951057 | − | 0.309017i | 0.500643 | + | 2.17930i | 0.280238 | − | 1.70923i | −0.753670 | + | 0.384014i | 0.453990 | − | 0.891007i | 2.99982 | + | 0.0325813i | −2.23079 | + | 0.153561i |
47.2 | −0.156434 | + | 0.987688i | −1.70896 | − | 0.281880i | −0.951057 | − | 0.309017i | −2.22941 | − | 0.172430i | 0.545749 | − | 1.64382i | −0.413436 | + | 0.210656i | 0.453990 | − | 0.891007i | 2.84109 | + | 0.963442i | 0.519064 | − | 2.17499i |
47.3 | −0.156434 | + | 0.987688i | −1.69850 | + | 0.339248i | −0.951057 | − | 0.309017i | −1.78466 | − | 1.34721i | −0.0693673 | − | 1.73066i | −0.684621 | + | 0.348832i | 0.453990 | − | 0.891007i | 2.76982 | − | 1.15243i | 1.60981 | − | 1.55194i |
47.4 | −0.156434 | + | 0.987688i | −1.67858 | + | 0.427044i | −0.951057 | − | 0.309017i | 0.419593 | + | 2.19635i | −0.159198 | − | 1.72472i | 3.17140 | − | 1.61591i | 0.453990 | − | 0.891007i | 2.63527 | − | 1.43366i | −2.23495 | + | 0.0708430i |
47.5 | −0.156434 | + | 0.987688i | −1.52046 | + | 0.829577i | −0.951057 | − | 0.309017i | 2.03771 | − | 0.920729i | −0.581511 | − | 1.63152i | −3.90942 | + | 1.99195i | 0.453990 | − | 0.891007i | 1.62360 | − | 2.52268i | 0.590625 | + | 2.15666i |
47.6 | −0.156434 | + | 0.987688i | −1.48019 | − | 0.899467i | −0.951057 | − | 0.309017i | 0.191858 | − | 2.22782i | 1.11995 | − | 1.32126i | 3.04103 | − | 1.54948i | 0.453990 | − | 0.891007i | 1.38192 | + | 2.66276i | 2.17038 | + | 0.538004i |
47.7 | −0.156434 | + | 0.987688i | −1.22292 | − | 1.22657i | −0.951057 | − | 0.309017i | 2.20307 | − | 0.382712i | 1.40277 | − | 1.01599i | −0.157227 | + | 0.0801110i | 0.453990 | − | 0.891007i | −0.00893248 | + | 2.99999i | 0.0333634 | + | 2.23582i |
47.8 | −0.156434 | + | 0.987688i | −1.20093 | − | 1.24811i | −0.951057 | − | 0.309017i | −1.74392 | + | 1.39955i | 1.42061 | − | 0.990898i | −2.75830 | + | 1.40542i | 0.453990 | − | 0.891007i | −0.115534 | + | 2.99777i | −1.10951 | − | 1.94139i |
47.9 | −0.156434 | + | 0.987688i | −1.00125 | + | 1.41333i | −0.951057 | − | 0.309017i | 1.95036 | + | 1.09365i | −1.23929 | − | 1.21002i | 1.24759 | − | 0.635678i | 0.453990 | − | 0.891007i | −0.994977 | − | 2.83020i | −1.38529 | + | 1.75527i |
47.10 | −0.156434 | + | 0.987688i | −0.999352 | + | 1.41467i | −0.951057 | − | 0.309017i | −2.01289 | + | 0.973791i | −1.24092 | − | 1.20835i | −2.22585 | + | 1.13413i | 0.453990 | − | 0.891007i | −1.00259 | − | 2.82751i | −0.646917 | − | 2.14044i |
47.11 | −0.156434 | + | 0.987688i | −0.893454 | − | 1.48383i | −0.951057 | − | 0.309017i | −0.275456 | − | 2.21904i | 1.60532 | − | 0.650333i | −4.52946 | + | 2.30788i | 0.453990 | − | 0.891007i | −1.40348 | + | 2.65146i | 2.23481 | + | 0.0750690i |
47.12 | −0.156434 | + | 0.987688i | −0.776954 | − | 1.54801i | −0.951057 | − | 0.309017i | 0.00309216 | + | 2.23607i | 1.65050 | − | 0.525226i | 3.71530 | − | 1.89304i | 0.453990 | − | 0.891007i | −1.79268 | + | 2.40547i | −2.20902 | − | 0.346744i |
47.13 | −0.156434 | + | 0.987688i | −0.678564 | + | 1.59360i | −0.951057 | − | 0.309017i | −1.22925 | − | 1.86787i | −1.46783 | − | 0.919503i | −0.247778 | + | 0.126249i | 0.453990 | − | 0.891007i | −2.07910 | − | 2.16271i | 2.03717 | − | 0.921913i |
47.14 | −0.156434 | + | 0.987688i | −0.343275 | + | 1.69769i | −0.951057 | − | 0.309017i | 2.03327 | − | 0.930487i | −1.62309 | − | 0.604626i | 3.29626 | − | 1.67953i | 0.453990 | − | 0.891007i | −2.76433 | − | 1.16555i | 0.600958 | + | 2.15380i |
47.15 | −0.156434 | + | 0.987688i | −0.212558 | + | 1.71896i | −0.951057 | − | 0.309017i | −1.82187 | + | 1.29645i | −1.66454 | − | 0.478845i | 1.93565 | − | 0.986265i | 0.453990 | − | 0.891007i | −2.90964 | − | 0.730756i | −0.995488 | − | 2.00225i |
47.16 | −0.156434 | + | 0.987688i | −0.115485 | − | 1.72820i | −0.951057 | − | 0.309017i | −2.08414 | − | 0.810158i | 1.72499 | + | 0.156287i | 2.37194 | − | 1.20856i | 0.453990 | − | 0.891007i | −2.97333 | + | 0.399160i | 1.12621 | − | 1.93175i |
47.17 | −0.156434 | + | 0.987688i | 0.207488 | − | 1.71958i | −0.951057 | − | 0.309017i | 1.69493 | − | 1.45849i | 1.66595 | + | 0.473934i | 0.759514 | − | 0.386992i | 0.453990 | − | 0.891007i | −2.91390 | − | 0.713583i | 1.17539 | + | 1.90223i |
47.18 | −0.156434 | + | 0.987688i | 0.219461 | + | 1.71809i | −0.951057 | − | 0.309017i | 0.873512 | + | 2.05839i | −1.73127 | − | 0.0520095i | −2.14611 | + | 1.09350i | 0.453990 | − | 0.891007i | −2.90367 | + | 0.754108i | −2.16970 | + | 0.540754i |
47.19 | −0.156434 | + | 0.987688i | 0.345889 | − | 1.69716i | −0.951057 | − | 0.309017i | −0.410025 | + | 2.19815i | 1.62216 | + | 0.607126i | −0.710855 | + | 0.362199i | 0.453990 | − | 0.891007i | −2.76072 | − | 1.17406i | −2.10695 | − | 0.748844i |
47.20 | −0.156434 | + | 0.987688i | 0.690891 | − | 1.58829i | −0.951057 | − | 0.309017i | 1.41201 | + | 1.73384i | 1.46066 | + | 0.930848i | −2.17193 | + | 1.10665i | 0.453990 | − | 0.891007i | −2.04534 | − | 2.19467i | −1.93339 | + | 1.12340i |
See next 80 embeddings (of 512 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
31.d | even | 5 | 1 | inner |
93.l | odd | 10 | 1 | inner |
155.s | odd | 20 | 1 | inner |
465.bj | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 930.2.bk.a | ✓ | 512 |
3.b | odd | 2 | 1 | inner | 930.2.bk.a | ✓ | 512 |
5.c | odd | 4 | 1 | inner | 930.2.bk.a | ✓ | 512 |
15.e | even | 4 | 1 | inner | 930.2.bk.a | ✓ | 512 |
31.d | even | 5 | 1 | inner | 930.2.bk.a | ✓ | 512 |
93.l | odd | 10 | 1 | inner | 930.2.bk.a | ✓ | 512 |
155.s | odd | 20 | 1 | inner | 930.2.bk.a | ✓ | 512 |
465.bj | even | 20 | 1 | inner | 930.2.bk.a | ✓ | 512 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
930.2.bk.a | ✓ | 512 | 1.a | even | 1 | 1 | trivial |
930.2.bk.a | ✓ | 512 | 3.b | odd | 2 | 1 | inner |
930.2.bk.a | ✓ | 512 | 5.c | odd | 4 | 1 | inner |
930.2.bk.a | ✓ | 512 | 15.e | even | 4 | 1 | inner |
930.2.bk.a | ✓ | 512 | 31.d | even | 5 | 1 | inner |
930.2.bk.a | ✓ | 512 | 93.l | odd | 10 | 1 | inner |
930.2.bk.a | ✓ | 512 | 155.s | odd | 20 | 1 | inner |
930.2.bk.a | ✓ | 512 | 465.bj | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(930, [\chi])\).