Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,2,Mod(16,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([0, 6, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.bg (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: | \(4.19214610612\) |
Analytic rank: | \(0\) |
Dimension: | \(160\) |
Relative dimension: | \(20\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −2.54817 | + | 1.13452i | −0.669131 | + | 0.743145i | 3.86776 | − | 4.29558i | 2.21934 | + | 0.272971i | 0.861946 | − | 2.65280i | −1.05082 | − | 2.42812i | −3.25839 | + | 10.0283i | −0.104528 | − | 0.994522i | −5.96495 | + | 1.82231i |
16.2 | −2.43994 | + | 1.08633i | −0.669131 | + | 0.743145i | 3.43493 | − | 3.81488i | −1.94782 | − | 1.09817i | 0.825337 | − | 2.54013i | 0.181785 | + | 2.63950i | −2.58614 | + | 7.95931i | −0.104528 | − | 0.994522i | 5.94555 | + | 0.563491i |
16.3 | −1.90213 | + | 0.846881i | −0.669131 | + | 0.743145i | 1.56262 | − | 1.73546i | −1.16209 | + | 1.91038i | 0.643416 | − | 1.98023i | 1.98170 | − | 1.75295i | −0.215734 | + | 0.663961i | −0.104528 | − | 0.994522i | 0.592568 | − | 4.61794i |
16.4 | −1.86259 | + | 0.829278i | −0.669131 | + | 0.743145i | 1.44327 | − | 1.60292i | −0.348898 | − | 2.20868i | 0.630042 | − | 1.93907i | 1.58384 | − | 2.11930i | −0.0988780 | + | 0.304315i | −0.104528 | − | 0.994522i | 2.48146 | + | 3.82453i |
16.5 | −1.73660 | + | 0.773186i | −0.669131 | + | 0.743145i | 1.07971 | − | 1.19914i | 0.215068 | + | 2.22570i | 0.587426 | − | 1.80791i | −2.62167 | + | 0.356150i | 0.226976 | − | 0.698561i | −0.104528 | − | 0.994522i | −2.09437 | − | 3.69887i |
16.6 | −1.50205 | + | 0.668755i | −0.669131 | + | 0.743145i | 0.470657 | − | 0.522717i | 2.21252 | + | 0.323643i | 0.508085 | − | 1.56372i | −1.94373 | + | 1.79497i | 0.658790 | − | 2.02755i | −0.104528 | − | 0.994522i | −3.53976 | + | 0.993509i |
16.7 | −1.30094 | + | 0.579215i | −0.669131 | + | 0.743145i | 0.0186901 | − | 0.0207574i | −2.15784 | + | 0.586291i | 0.440057 | − | 1.35436i | 0.667942 | + | 2.56005i | 0.867823 | − | 2.67088i | −0.104528 | − | 0.994522i | 2.46763 | − | 2.01258i |
16.8 | −0.766251 | + | 0.341157i | −0.669131 | + | 0.743145i | −0.867508 | + | 0.963466i | 0.938720 | − | 2.02948i | 0.259193 | − | 0.797714i | 1.21717 | + | 2.34915i | 0.854422 | − | 2.62964i | −0.104528 | − | 0.994522i | −0.0269227 | + | 1.87535i |
16.9 | −0.650196 | + | 0.289486i | −0.669131 | + | 0.743145i | −0.999309 | + | 1.10984i | 2.08872 | + | 0.798271i | 0.219936 | − | 0.676893i | 0.659935 | − | 2.56213i | 0.768334 | − | 2.36469i | −0.104528 | − | 0.994522i | −1.58917 | + | 0.0856230i |
16.10 | 0.0671031 | − | 0.0298762i | −0.669131 | + | 0.743145i | −1.33465 | + | 1.48228i | 1.65633 | − | 1.50219i | −0.0226984 | + | 0.0698585i | −2.32293 | − | 1.26649i | −0.0906711 | + | 0.279057i | −0.104528 | − | 0.994522i | 0.0662647 | − | 0.150287i |
16.11 | 0.104767 | − | 0.0466452i | −0.669131 | + | 0.743145i | −1.32946 | + | 1.47652i | 0.449213 | + | 2.19048i | −0.0354385 | + | 0.109069i | 2.46049 | + | 0.972613i | −0.141288 | + | 0.434840i | −0.104528 | − | 0.994522i | 0.149238 | + | 0.208536i |
16.12 | 0.246594 | − | 0.109791i | −0.669131 | + | 0.743145i | −1.28951 | + | 1.43214i | −1.94989 | + | 1.09450i | −0.0834131 | + | 0.256719i | −2.11046 | + | 1.59561i | −0.327575 | + | 1.00817i | −0.104528 | − | 0.994522i | −0.360664 | + | 0.483977i |
16.13 | 0.337888 | − | 0.150438i | −0.669131 | + | 0.743145i | −1.24672 | + | 1.38463i | −1.99129 | − | 1.01724i | −0.114295 | + | 0.351762i | 1.72521 | − | 2.00591i | −0.441543 | + | 1.35893i | −0.104528 | − | 0.994522i | −0.825864 | − | 0.0441485i |
16.14 | 1.26263 | − | 0.562160i | −0.669131 | + | 0.743145i | −0.0600440 | + | 0.0666857i | 1.57128 | + | 1.59093i | −0.427100 | + | 1.31448i | −2.04775 | + | 1.67532i | −0.892525 | + | 2.74691i | −0.104528 | − | 0.994522i | 2.87831 | + | 1.12545i |
16.15 | 1.27220 | − | 0.566420i | −0.669131 | + | 0.743145i | −0.0405990 | + | 0.0450897i | −0.413100 | − | 2.19758i | −0.430336 | + | 1.32444i | −0.306993 | + | 2.62788i | −0.886782 | + | 2.72924i | −0.104528 | − | 0.994522i | −1.77030 | − | 2.56177i |
16.16 | 1.43006 | − | 0.636705i | −0.669131 | + | 0.743145i | 0.301423 | − | 0.334764i | −1.95826 | + | 1.07945i | −0.483734 | + | 1.48878i | −0.280963 | − | 2.63079i | −0.749561 | + | 2.30691i | −0.104528 | − | 0.994522i | −2.11314 | + | 2.79051i |
16.17 | 1.63302 | − | 0.727066i | −0.669131 | + | 0.743145i | 0.799857 | − | 0.888331i | 0.337843 | − | 2.21040i | −0.552386 | + | 1.70007i | −2.03088 | − | 1.69574i | −0.444468 | + | 1.36793i | −0.104528 | − | 0.994522i | −1.05540 | − | 3.85525i |
16.18 | 1.77124 | − | 0.788606i | −0.669131 | + | 0.743145i | 1.17712 | − | 1.30733i | 2.19192 | − | 0.442138i | −0.599141 | + | 1.84397i | 2.64021 | + | 0.171129i | −0.144284 | + | 0.444060i | −0.104528 | − | 0.994522i | 3.53374 | − | 2.51169i |
16.19 | 2.34270 | − | 1.04304i | −0.669131 | + | 0.743145i | 3.06206 | − | 3.40076i | 1.20119 | + | 1.88604i | −0.792445 | + | 2.43890i | −0.520919 | − | 2.59396i | 2.04148 | − | 6.28303i | −0.104528 | − | 0.994522i | 4.78124 | + | 3.16554i |
16.20 | 2.41357 | − | 1.07459i | −0.669131 | + | 0.743145i | 3.33230 | − | 3.70090i | −2.13490 | − | 0.664998i | −0.816416 | + | 2.51267i | 2.61882 | + | 0.376541i | 2.43296 | − | 7.48789i | −0.104528 | − | 0.994522i | −5.86731 | + | 0.689119i |
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
25.d | even | 5 | 1 | inner |
175.q | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.2.bg.a | ✓ | 160 |
7.c | even | 3 | 1 | inner | 525.2.bg.a | ✓ | 160 |
25.d | even | 5 | 1 | inner | 525.2.bg.a | ✓ | 160 |
175.q | even | 15 | 1 | inner | 525.2.bg.a | ✓ | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
525.2.bg.a | ✓ | 160 | 1.a | even | 1 | 1 | trivial |
525.2.bg.a | ✓ | 160 | 7.c | even | 3 | 1 | inner |
525.2.bg.a | ✓ | 160 | 25.d | even | 5 | 1 | inner |
525.2.bg.a | ✓ | 160 | 175.q | even | 15 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} + 2 T_{2}^{159} - 28 T_{2}^{158} - 62 T_{2}^{157} + 307 T_{2}^{156} + 728 T_{2}^{155} + \cdots + 390625 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).