Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [187,2,Mod(69,187)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(187, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([8, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("187.69");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 187 = 11 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 187.g (of order \(5\), 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: | \(1.49320251780\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
69.1 | −2.14992 | + | 1.56201i | 0.943884 | + | 2.90498i | 1.56424 | − | 4.81425i | 1.58671 | + | 1.15281i | −6.56686 | − | 4.77110i | −0.547865 | + | 1.68616i | 2.51450 | + | 7.73883i | −5.12092 | + | 3.72057i | −5.21199 | ||
69.2 | −1.73247 | + | 1.25872i | 0.331943 | + | 1.02161i | 0.799065 | − | 2.45927i | −2.02945 | − | 1.47448i | −1.86100 | − | 1.35210i | 1.12029 | − | 3.44788i | 0.387671 | + | 1.19313i | 1.49354 | − | 1.08512i | 5.37193 | ||
69.3 | −1.36392 | + | 0.990946i | −0.936920 | − | 2.88354i | 0.260271 | − | 0.801032i | 2.28514 | + | 1.66025i | 4.13532 | + | 3.00449i | 0.207430 | − | 0.638405i | −0.603152 | − | 1.85631i | −5.00995 | + | 3.63994i | −4.76196 | ||
69.4 | −1.28769 | + | 0.935561i | 0.00348887 | + | 0.0107376i | 0.164835 | − | 0.507309i | 2.53950 | + | 1.84506i | −0.0145383 | − | 0.0105627i | −0.112651 | + | 0.346704i | −0.721344 | − | 2.22007i | 2.42695 | − | 1.76328i | −4.99625 | ||
69.5 | −0.415893 | + | 0.302164i | 0.814643 | + | 2.50721i | −0.536370 | + | 1.65078i | −3.29479 | − | 2.39381i | −1.09640 | − | 0.796578i | −0.581623 | + | 1.79005i | −0.593447 | − | 1.82644i | −3.19543 | + | 2.32161i | 2.09360 | ||
69.6 | 0.905302 | − | 0.657740i | −0.440186 | − | 1.35475i | −0.231085 | + | 0.711206i | 1.95004 | + | 1.41679i | −1.28958 | − | 0.936932i | 1.26946 | − | 3.90701i | 0.950176 | + | 2.92434i | 0.785461 | − | 0.570671i | 2.69726 | ||
69.7 | 1.31285 | − | 0.953841i | −0.637980 | − | 1.96350i | 0.195728 | − | 0.602389i | −2.95777 | − | 2.14895i | −2.71044 | − | 1.96925i | −0.195368 | + | 0.601281i | 0.685306 | + | 2.10916i | −1.02126 | + | 0.741992i | −5.93286 | ||
69.8 | 1.61769 | − | 1.17532i | 1.01397 | + | 3.12067i | 0.617507 | − | 1.90049i | −1.06499 | − | 0.773760i | 5.30807 | + | 3.85654i | 1.03762 | − | 3.19345i | 0.00105288 | + | 0.00324042i | −6.28340 | + | 4.56516i | −2.63224 | ||
69.9 | 2.11405 | − | 1.53595i | 0.334210 | + | 1.02859i | 1.49204 | − | 4.59203i | −0.441442 | − | 0.320727i | 2.28641 | + | 1.66117i | −1.38827 | + | 4.27266i | −2.28388 | − | 7.02906i | 1.48074 | − | 1.07582i | −1.42585 | ||
86.1 | −0.841398 | + | 2.58956i | −1.62645 | + | 1.18168i | −4.37981 | − | 3.18212i | 0.802755 | + | 2.47063i | −1.69155 | − | 5.20605i | −2.56930 | − | 1.86671i | 7.51983 | − | 5.46348i | 0.321909 | − | 0.990733i | −7.07326 | ||
86.2 | −0.768807 | + | 2.36614i | 1.93066 | − | 1.40271i | −3.38954 | − | 2.46265i | −1.37389 | − | 4.22840i | 1.83471 | + | 5.64664i | −2.07663 | − | 1.50876i | 4.40735 | − | 3.20213i | 0.832818 | − | 2.56315i | 11.0613 | ||
86.3 | −0.709792 | + | 2.18452i | 0.676145 | − | 0.491248i | −2.65027 | − | 1.92553i | 0.327553 | + | 1.00811i | 0.593217 | + | 1.82573i | 2.54751 | + | 1.85087i | 2.37098 | − | 1.72262i | −0.711203 | + | 2.18886i | −2.43472 | ||
86.4 | −0.343162 | + | 1.05614i | −2.58082 | + | 1.87507i | 0.620355 | + | 0.450715i | 0.625693 | + | 1.92569i | −1.09471 | − | 3.36916i | 2.66865 | + | 1.93889i | −2.48572 | + | 1.80598i | 2.21766 | − | 6.82526i | −2.24851 | ||
86.5 | −0.135862 | + | 0.418139i | −0.0312072 | + | 0.0226734i | 1.46165 | + | 1.06195i | 0.919117 | + | 2.82875i | −0.00524076 | − | 0.0161294i | −2.52440 | − | 1.83408i | −1.35401 | + | 0.983744i | −0.926591 | + | 2.85175i | −1.30768 | ||
86.6 | 0.0881614 | − | 0.271333i | −1.31211 | + | 0.953301i | 1.55218 | + | 1.12773i | −1.06412 | − | 3.27502i | 0.142985 | + | 0.440062i | 3.21023 | + | 2.33237i | 0.904452 | − | 0.657123i | −0.114211 | + | 0.351506i | −0.982436 | ||
86.7 | 0.282168 | − | 0.868424i | 1.40032 | − | 1.01739i | 0.943492 | + | 0.685487i | −0.0971410 | − | 0.298969i | −0.488401 | − | 1.50314i | −2.74134 | − | 1.99170i | 2.33897 | − | 1.69936i | −0.00124648 | + | 0.00383627i | −0.287042 | ||
86.8 | 0.648313 | − | 1.99530i | −2.15638 | + | 1.56670i | −1.94289 | − | 1.41159i | 1.10893 | + | 3.41293i | 1.72804 | + | 5.31835i | 0.243832 | + | 0.177154i | −0.681533 | + | 0.495163i | 1.26837 | − | 3.90365i | 7.52877 | ||
86.9 | 0.780378 | − | 2.40176i | 1.77278 | − | 1.28800i | −3.54141 | − | 2.57299i | 0.678152 | + | 2.08714i | −1.71003 | − | 5.26293i | 0.932422 | + | 0.677444i | −4.85721 | + | 3.52897i | 0.556762 | − | 1.71354i | 5.54201 | ||
103.1 | −2.14992 | − | 1.56201i | 0.943884 | − | 2.90498i | 1.56424 | + | 4.81425i | 1.58671 | − | 1.15281i | −6.56686 | + | 4.77110i | −0.547865 | − | 1.68616i | 2.51450 | − | 7.73883i | −5.12092 | − | 3.72057i | −5.21199 | ||
103.2 | −1.73247 | − | 1.25872i | 0.331943 | − | 1.02161i | 0.799065 | + | 2.45927i | −2.02945 | + | 1.47448i | −1.86100 | + | 1.35210i | 1.12029 | + | 3.44788i | 0.387671 | − | 1.19313i | 1.49354 | + | 1.08512i | 5.37193 | ||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 187.2.g.f | ✓ | 36 |
11.c | even | 5 | 1 | inner | 187.2.g.f | ✓ | 36 |
11.c | even | 5 | 1 | 2057.2.a.bd | 18 | ||
11.d | odd | 10 | 1 | 2057.2.a.be | 18 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
187.2.g.f | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
187.2.g.f | ✓ | 36 | 11.c | even | 5 | 1 | inner |
2057.2.a.bd | 18 | 11.c | even | 5 | 1 | ||
2057.2.a.be | 18 | 11.d | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(187, [\chi])\):
\( T_{2}^{36} + 4 T_{2}^{35} + 24 T_{2}^{34} + 61 T_{2}^{33} + 247 T_{2}^{32} + 556 T_{2}^{31} + 2130 T_{2}^{30} + 4561 T_{2}^{29} + 16506 T_{2}^{28} + 33778 T_{2}^{27} + 104146 T_{2}^{26} + 181950 T_{2}^{25} + 476916 T_{2}^{24} + \cdots + 609961 \) |
\( T_{3}^{36} + T_{3}^{35} + 25 T_{3}^{34} + 42 T_{3}^{33} + 372 T_{3}^{32} + 443 T_{3}^{31} + 4054 T_{3}^{30} + 2386 T_{3}^{29} + 35694 T_{3}^{28} + 6965 T_{3}^{27} + 257219 T_{3}^{26} + 36751 T_{3}^{25} + 1669862 T_{3}^{24} + \cdots + 625 \) |