Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(169,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.169");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.bh (of order \(6\), degree \(2\), 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: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
169.1 | −2.39459 | − | 1.38252i | 1.22002 | + | 1.22945i | 2.82270 | + | 4.88906i | 2.16705 | + | 0.551264i | −1.22171 | − | 4.63073i | 0.866025 | + | 0.500000i | − | 10.0797i | −0.0231036 | + | 2.99991i | −4.42706 | − | 4.31603i | |
169.2 | −2.27796 | − | 1.31518i | 0.801507 | − | 1.53544i | 2.45940 | + | 4.25981i | −2.22338 | + | 0.237830i | −3.84519 | + | 2.44355i | 0.866025 | + | 0.500000i | − | 7.67750i | −1.71517 | − | 2.46134i | 5.37757 | + | 2.38239i | |
169.3 | −2.11868 | − | 1.22322i | 0.00483785 | + | 1.73204i | 1.99254 | + | 3.45119i | −2.11025 | − | 0.739493i | 2.10842 | − | 3.67557i | −0.866025 | − | 0.500000i | − | 4.85641i | −2.99995 | + | 0.0167587i | 3.56638 | + | 4.14805i | |
169.4 | −2.08930 | − | 1.20626i | −1.72484 | − | 0.157901i | 1.91013 | + | 3.30844i | −0.387093 | − | 2.20231i | 3.41324 | + | 2.41051i | 0.866025 | + | 0.500000i | − | 4.39141i | 2.95013 | + | 0.544707i | −1.84780 | + | 5.06823i | |
169.5 | −2.01113 | − | 1.16112i | −0.0358898 | − | 1.73168i | 1.69642 | + | 2.93828i | 1.53191 | − | 1.62888i | −1.93852 | + | 3.52430i | −0.866025 | − | 0.500000i | − | 3.23452i | −2.99742 | + | 0.124299i | −4.97219 | + | 1.49716i | |
169.6 | −1.85508 | − | 1.07103i | 1.44646 | − | 0.952757i | 1.29422 | + | 2.24165i | 1.12216 | + | 1.93410i | −3.70374 | + | 0.218235i | −0.866025 | − | 0.500000i | − | 1.26045i | 1.18451 | − | 2.75626i | −0.0102188 | − | 4.78979i | |
169.7 | −1.46667 | − | 0.846784i | −1.23864 | − | 1.21069i | 0.434088 | + | 0.751862i | 2.19600 | + | 0.421388i | 0.791480 | + | 2.82455i | 0.866025 | + | 0.500000i | 1.91682i | 0.0684429 | + | 2.99922i | −2.86400 | − | 2.47758i | ||
169.8 | −1.39315 | − | 0.804334i | 1.72917 | + | 0.0997919i | 0.293908 | + | 0.509063i | −2.18712 | + | 0.465298i | −2.32873 | − | 1.52986i | 0.866025 | + | 0.500000i | 2.27174i | 2.98008 | + | 0.345115i | 3.42124 | + | 1.11095i | ||
169.9 | −1.36108 | − | 0.785823i | −1.05946 | − | 1.37023i | 0.235034 | + | 0.407091i | −2.16120 | + | 0.573754i | 0.365260 | + | 2.69755i | −0.866025 | − | 0.500000i | 2.40451i | −0.755074 | + | 2.90342i | 3.39245 | + | 0.917396i | ||
169.10 | −1.09189 | − | 0.630403i | 1.01772 | + | 1.40151i | −0.205184 | − | 0.355388i | 0.0910240 | + | 2.23421i | −0.227721 | − | 2.17188i | −0.866025 | − | 0.500000i | 3.03901i | −0.928486 | + | 2.85270i | 1.30907 | − | 2.49690i | ||
169.11 | −1.03279 | − | 0.596282i | −0.463032 | + | 1.66901i | −0.288895 | − | 0.500381i | 1.52125 | − | 1.63884i | 1.47342 | − | 1.44764i | 0.866025 | + | 0.500000i | 3.07418i | −2.57120 | − | 1.54561i | −2.54835 | + | 0.785481i | ||
169.12 | −0.814131 | − | 0.470039i | −1.50236 | + | 0.861916i | −0.558127 | − | 0.966705i | −0.721394 | − | 2.11650i | 1.62826 | + | 0.00445709i | −0.866025 | − | 0.500000i | 2.92952i | 1.51420 | − | 2.58983i | −0.407530 | + | 2.06219i | ||
169.13 | −0.719518 | − | 0.415414i | 1.08800 | − | 1.34768i | −0.654862 | − | 1.13425i | 1.82483 | − | 1.29228i | −1.34268 | + | 0.517712i | 0.866025 | + | 0.500000i | 2.74981i | −0.632505 | − | 2.93256i | −1.84983 | + | 0.171758i | ||
169.14 | −0.293339 | − | 0.169359i | 0.530399 | + | 1.64884i | −0.942635 | − | 1.63269i | −2.22372 | − | 0.234648i | 0.123660 | − | 0.573497i | 0.866025 | + | 0.500000i | 1.31601i | −2.43735 | + | 1.74909i | 0.612564 | + | 0.445440i | ||
169.15 | −0.101400 | − | 0.0585435i | −1.22623 | − | 1.22326i | −0.993145 | − | 1.72018i | −1.51285 | − | 1.64660i | 0.0527266 | + | 0.195826i | 0.866025 | + | 0.500000i | 0.466742i | 0.00728996 | + | 2.99999i | 0.0570063 | + | 0.255533i | ||
169.16 | −0.0813489 | − | 0.0469668i | −1.71108 | + | 0.268687i | −0.995588 | − | 1.72441i | −1.28952 | + | 1.82678i | 0.151814 | + | 0.0585067i | 0.866025 | + | 0.500000i | 0.374905i | 2.85561 | − | 0.919491i | 0.190699 | − | 0.0880417i | ||
169.17 | 0.0813489 | + | 0.0469668i | 1.71108 | − | 0.268687i | −0.995588 | − | 1.72441i | 2.22680 | − | 0.203371i | 0.151814 | + | 0.0585067i | −0.866025 | − | 0.500000i | − | 0.374905i | 2.85561 | − | 0.919491i | 0.190699 | + | 0.0880417i | |
169.18 | 0.101400 | + | 0.0585435i | 1.22623 | + | 1.22326i | −0.993145 | − | 1.72018i | −0.669567 | − | 2.13347i | 0.0527266 | + | 0.195826i | −0.866025 | − | 0.500000i | − | 0.466742i | 0.00728996 | + | 2.99999i | 0.0570063 | − | 0.255533i | |
169.19 | 0.293339 | + | 0.169359i | −0.530399 | − | 1.64884i | −0.942635 | − | 1.63269i | 0.908650 | − | 2.04312i | 0.123660 | − | 0.573497i | −0.866025 | − | 0.500000i | − | 1.31601i | −2.43735 | + | 1.74909i | 0.612564 | − | 0.445440i | |
169.20 | 0.719518 | + | 0.415414i | −1.08800 | + | 1.34768i | −0.654862 | − | 1.13425i | −2.03156 | + | 0.934211i | −1.34268 | + | 0.517712i | −0.866025 | − | 0.500000i | − | 2.74981i | −0.632505 | − | 2.93256i | −1.84983 | − | 0.171758i | |
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
45.j | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.bh.c | ✓ | 64 |
3.b | odd | 2 | 1 | 945.2.bh.c | 64 | ||
5.b | even | 2 | 1 | inner | 315.2.bh.c | ✓ | 64 |
9.c | even | 3 | 1 | inner | 315.2.bh.c | ✓ | 64 |
9.d | odd | 6 | 1 | 945.2.bh.c | 64 | ||
15.d | odd | 2 | 1 | 945.2.bh.c | 64 | ||
45.h | odd | 6 | 1 | 945.2.bh.c | 64 | ||
45.j | even | 6 | 1 | inner | 315.2.bh.c | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.bh.c | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
315.2.bh.c | ✓ | 64 | 5.b | even | 2 | 1 | inner |
315.2.bh.c | ✓ | 64 | 9.c | even | 3 | 1 | inner |
315.2.bh.c | ✓ | 64 | 45.j | even | 6 | 1 | inner |
945.2.bh.c | 64 | 3.b | odd | 2 | 1 | ||
945.2.bh.c | 64 | 9.d | odd | 6 | 1 | ||
945.2.bh.c | 64 | 15.d | odd | 2 | 1 | ||
945.2.bh.c | 64 | 45.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 49 T_{2}^{62} + 1327 T_{2}^{60} - 24802 T_{2}^{58} + 353233 T_{2}^{56} - 4023391 T_{2}^{54} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).