Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [403,2,Mod(9,403)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([20, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("403.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 403 = 13 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 403.bk (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: | \(3.21797120146\) |
Analytic rank: | \(0\) |
Dimension: | \(280\) |
Relative dimension: | \(35\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | −1.83126 | − | 2.03382i | −0.500677 | + | 0.556058i | −0.573858 | + | 5.45990i | −0.330308 | − | 0.572109i | 2.04780 | −3.67674 | − | 2.67131i | 7.72715 | − | 5.61410i | 0.255062 | + | 2.42676i | −0.558690 | + | 1.71947i | ||
9.2 | −1.65225 | − | 1.83501i | 2.14947 | − | 2.38723i | −0.428272 | + | 4.07474i | −1.22323 | − | 2.11870i | −7.93205 | −1.37387 | − | 0.998173i | 4.18946 | − | 3.04382i | −0.765054 | − | 7.27901i | −1.86675 | + | 5.74525i | ||
9.3 | −1.59232 | − | 1.76846i | −1.58986 | + | 1.76572i | −0.382880 | + | 3.64286i | 1.32032 | + | 2.28686i | 5.65416 | 2.08439 | + | 1.51440i | 3.20149 | − | 2.32602i | −0.276520 | − | 2.63091i | 1.94183 | − | 5.97634i | ||
9.4 | −1.55327 | − | 1.72508i | 1.03676 | − | 1.15144i | −0.354201 | + | 3.36999i | 0.252831 | + | 0.437917i | −3.59669 | 1.69707 | + | 1.23299i | 2.60770 | − | 1.89460i | 0.0626468 | + | 0.596045i | 0.362727 | − | 1.11636i | ||
9.5 | −1.46781 | − | 1.63017i | −2.12808 | + | 2.36348i | −0.293925 | + | 2.79651i | −0.951364 | − | 1.64781i | 6.97649 | −1.54320 | − | 1.12120i | 1.44087 | − | 1.04685i | −0.743696 | − | 7.07579i | −1.28979 | + | 3.96956i | ||
9.6 | −1.44960 | − | 1.60995i | 0.384158 | − | 0.426651i | −0.281525 | + | 2.67853i | −1.83417 | − | 3.17687i | −1.24376 | 0.633388 | + | 0.460183i | 1.21509 | − | 0.882816i | 0.279132 | + | 2.65576i | −2.45578 | + | 7.55812i | ||
9.7 | −1.32661 | − | 1.47335i | 0.650439 | − | 0.722386i | −0.201810 | + | 1.92010i | 1.17248 | + | 2.03079i | −1.92721 | 0.184244 | + | 0.133861i | −0.111198 | + | 0.0807903i | 0.214815 | + | 2.04383i | 1.43665 | − | 4.42155i | ||
9.8 | −1.24715 | − | 1.38510i | −0.607940 | + | 0.675186i | −0.154065 | + | 1.46583i | 1.19707 | + | 2.07339i | 1.69340 | −2.18489 | − | 1.58741i | −0.793286 | + | 0.576356i | 0.227301 | + | 2.16262i | 1.37893 | − | 4.24390i | ||
9.9 | −0.918709 | − | 1.02033i | −0.334947 | + | 0.371997i | 0.0120106 | − | 0.114273i | −0.585861 | − | 1.01474i | 0.687278 | 3.13175 | + | 2.27535i | −2.34917 | + | 1.70677i | 0.287394 | + | 2.73437i | −0.497134 | + | 1.53002i | ||
9.10 | −0.769629 | − | 0.854759i | 1.59325 | − | 1.76948i | 0.0707719 | − | 0.673349i | −0.220512 | − | 0.381938i | −2.73870 | −1.17517 | − | 0.853812i | −2.49107 | + | 1.80987i | −0.279042 | − | 2.65490i | −0.156753 | + | 0.482435i | ||
9.11 | −0.736014 | − | 0.817427i | 2.06902 | − | 2.29788i | 0.0825875 | − | 0.785768i | 1.51346 | + | 2.62138i | −3.40117 | 2.69694 | + | 1.95944i | −2.48286 | + | 1.80390i | −0.685820 | − | 6.52514i | 1.02886 | − | 3.16652i | ||
9.12 | −0.718707 | − | 0.798205i | −0.717726 | + | 0.797115i | 0.0884655 | − | 0.841693i | 1.01665 | + | 1.76088i | 1.15210 | −3.07573 | − | 2.23465i | −2.47334 | + | 1.79699i | 0.193323 | + | 1.83935i | 0.674874 | − | 2.07705i | ||
9.13 | −0.669055 | − | 0.743061i | −1.14502 | + | 1.27167i | 0.104552 | − | 0.994745i | −1.77181 | − | 3.06886i | 1.71101 | −2.25690 | − | 1.63973i | −2.42696 | + | 1.76329i | 0.00750493 | + | 0.0714046i | −1.09491 | + | 3.36979i | ||
9.14 | −0.628504 | − | 0.698025i | −1.76109 | + | 1.95588i | 0.116836 | − | 1.11162i | −0.604953 | − | 1.04781i | 2.47210 | 2.24460 | + | 1.63080i | −2.36917 | + | 1.72130i | −0.410474 | − | 3.90540i | −0.351181 | + | 1.08082i | ||
9.15 | −0.165338 | − | 0.183626i | 1.65099 | − | 1.83361i | 0.202675 | − | 1.92832i | −1.68082 | − | 2.91126i | −0.609670 | 0.0202552 | + | 0.0147163i | −0.787407 | + | 0.572085i | −0.322771 | − | 3.07096i | −0.256681 | + | 0.789984i | ||
9.16 | −0.142826 | − | 0.158624i | 0.434282 | − | 0.482319i | 0.204295 | − | 1.94373i | 1.97676 | + | 3.42385i | −0.138534 | 0.326262 | + | 0.237043i | −0.682872 | + | 0.496136i | 0.269555 | + | 2.56464i | 0.260774 | − | 0.802579i | ||
9.17 | −0.133023 | − | 0.147737i | 0.0833490 | − | 0.0925684i | 0.204926 | − | 1.94974i | −0.998333 | − | 1.72916i | −0.0247631 | −1.37062 | − | 0.995817i | −0.636973 | + | 0.462788i | 0.311964 | + | 2.96813i | −0.122660 | + | 0.377509i | ||
9.18 | −0.0525666 | − | 0.0583811i | −1.70784 | + | 1.89675i | 0.208412 | − | 1.98291i | 1.96469 | + | 3.40295i | 0.200510 | 2.32100 | + | 1.68631i | −0.253832 | + | 0.184420i | −0.367355 | − | 3.49515i | 0.0953908 | − | 0.293583i | ||
9.19 | 0.117605 | + | 0.130614i | 0.508986 | − | 0.565287i | 0.205828 | − | 1.95832i | 0.00796452 | + | 0.0137949i | 0.133693 | 3.75325 | + | 2.72689i | 0.564372 | − | 0.410040i | 0.253104 | + | 2.40812i | −0.000865141 | 0.00266263i | |||
9.20 | 0.169063 | + | 0.187763i | −1.38560 | + | 1.53887i | 0.202384 | − | 1.92556i | 0.388050 | + | 0.672122i | −0.523196 | −2.21450 | − | 1.60893i | 0.804576 | − | 0.584559i | −0.134633 | − | 1.28095i | −0.0605950 | + | 0.186492i | ||
See next 80 embeddings (of 280 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
403.bk | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 403.2.bk.a | yes | 280 |
13.c | even | 3 | 1 | 403.2.bj.a | ✓ | 280 | |
31.g | even | 15 | 1 | 403.2.bj.a | ✓ | 280 | |
403.bk | even | 15 | 1 | inner | 403.2.bk.a | yes | 280 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
403.2.bj.a | ✓ | 280 | 13.c | even | 3 | 1 | |
403.2.bj.a | ✓ | 280 | 31.g | even | 15 | 1 | |
403.2.bk.a | yes | 280 | 1.a | even | 1 | 1 | trivial |
403.2.bk.a | yes | 280 | 403.bk | even | 15 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).