Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [805,2,Mod(484,805)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("805.484");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 805 = 5 \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 805.c (of order \(2\), degree \(1\), 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: | \(6.42795736271\) |
Analytic rank: | \(0\) |
Dimension: | \(42\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
484.1 | − | 2.80022i | − | 0.208644i | −5.84124 | −1.70159 | − | 1.45072i | −0.584249 | 1.00000i | 10.7563i | 2.95647 | −4.06234 | + | 4.76483i | ||||||||||||
484.2 | − | 2.72900i | − | 3.06108i | −5.44746 | −2.23566 | + | 0.0429229i | −8.35371 | − | 1.00000i | 9.40815i | −6.37023 | 0.117137 | + | 6.10111i | |||||||||||
484.3 | − | 2.70441i | 1.97894i | −5.31383 | 0.525494 | + | 2.17344i | 5.35185 | − | 1.00000i | 8.96197i | −0.916187 | 5.87788 | − | 1.42115i | ||||||||||||
484.4 | − | 2.68001i | 3.08395i | −5.18247 | 1.08626 | − | 1.95449i | 8.26504 | 1.00000i | 8.52906i | −6.51078 | −5.23807 | − | 2.91118i | |||||||||||||
484.5 | − | 2.57149i | 0.285863i | −4.61257 | 1.95531 | − | 1.08478i | 0.735095 | − | 1.00000i | 6.71821i | 2.91828 | −2.78950 | − | 5.02807i | ||||||||||||
484.6 | − | 2.38755i | − | 1.09554i | −3.70041 | 2.13213 | + | 0.673804i | −2.61567 | 1.00000i | 4.05982i | 1.79978 | 1.60874 | − | 5.09058i | ||||||||||||
484.7 | − | 2.35921i | 2.50751i | −3.56589 | −2.09438 | − | 0.783299i | 5.91575 | − | 1.00000i | 3.69426i | −3.28760 | −1.84797 | + | 4.94110i | ||||||||||||
484.8 | − | 2.20248i | 2.83822i | −2.85093 | −0.287811 | + | 2.21747i | 6.25112 | 1.00000i | 1.87416i | −5.05547 | 4.88394 | + | 0.633899i | |||||||||||||
484.9 | − | 2.11197i | − | 1.59044i | −2.46043 | −1.77982 | + | 1.35361i | −3.35897 | − | 1.00000i | 0.972410i | 0.470497 | 2.85878 | + | 3.75893i | |||||||||||
484.10 | − | 1.67541i | − | 3.24787i | −0.806990 | −0.276266 | + | 2.21894i | −5.44150 | 1.00000i | − | 1.99878i | −7.54864 | 3.71762 | + | 0.462857i | |||||||||||
484.11 | − | 1.61751i | − | 2.88096i | −0.616354 | 2.14065 | + | 0.646219i | −4.65999 | − | 1.00000i | − | 2.23807i | −5.29992 | 1.04527 | − | 3.46254i | ||||||||||
484.12 | − | 1.53505i | 0.943091i | −0.356385 | 0.00201580 | − | 2.23607i | 1.44769 | 1.00000i | − | 2.52303i | 2.11058 | −3.43248 | − | 0.00309436i | ||||||||||||
484.13 | − | 1.50084i | − | 0.816695i | −0.252529 | 2.06282 | − | 0.863005i | −1.22573 | − | 1.00000i | − | 2.62268i | 2.33301 | −1.29523 | − | 3.09597i | ||||||||||
484.14 | − | 1.30923i | 2.43985i | 0.285909 | 0.245062 | − | 2.22260i | 3.19433 | − | 1.00000i | − | 2.99279i | −2.95286 | −2.90990 | − | 0.320843i | |||||||||||
484.15 | − | 1.22234i | − | 1.28664i | 0.505878 | 1.74857 | − | 1.39374i | −1.57271 | 1.00000i | − | 3.06304i | 1.34457 | −1.70363 | − | 2.13735i | |||||||||||
484.16 | − | 0.817573i | 1.89500i | 1.33157 | −1.04173 | + | 1.97858i | 1.54930 | − | 1.00000i | − | 2.72381i | −0.591014 | 1.61764 | + | 0.851693i | |||||||||||
484.17 | − | 0.685218i | 1.12744i | 1.53048 | −2.23604 | − | 0.0108956i | 0.772540 | 1.00000i | − | 2.41915i | 1.72889 | −0.00746587 | + | 1.53218i | ||||||||||||
484.18 | − | 0.556064i | 2.94180i | 1.69079 | 1.39046 | + | 1.75118i | 1.63583 | 1.00000i | − | 2.05232i | −5.65418 | 0.973767 | − | 0.773184i | ||||||||||||
484.19 | − | 0.475047i | − | 0.568069i | 1.77433 | −0.0751241 | − | 2.23481i | −0.269859 | − | 1.00000i | − | 1.79298i | 2.67730 | −1.06164 | + | 0.0356875i | ||||||||||
484.20 | − | 0.294301i | − | 3.07644i | 1.91339 | 1.51307 | − | 1.64640i | −0.905400 | 1.00000i | − | 1.15171i | −6.46449 | −0.484536 | − | 0.445298i | |||||||||||
See all 42 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 805.2.c.c | ✓ | 42 |
5.b | even | 2 | 1 | inner | 805.2.c.c | ✓ | 42 |
5.c | odd | 4 | 1 | 4025.2.a.bd | 21 | ||
5.c | odd | 4 | 1 | 4025.2.a.be | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
805.2.c.c | ✓ | 42 | 1.a | even | 1 | 1 | trivial |
805.2.c.c | ✓ | 42 | 5.b | even | 2 | 1 | inner |
4025.2.a.bd | 21 | 5.c | odd | 4 | 1 | ||
4025.2.a.be | 21 | 5.c | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} + 72 T_{2}^{40} + 2390 T_{2}^{38} + 48524 T_{2}^{36} + 674123 T_{2}^{34} + 6793094 T_{2}^{32} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\).