Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1260,2,Mod(529,1260)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1260, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1260.529");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1260.by (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: | \(10.0611506547\) |
Analytic rank: | \(0\) |
Dimension: | \(92\) |
Relative dimension: | \(46\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
529.1 | 0 | −1.73178 | − | 0.0308374i | 0 | −1.82997 | − | 1.28499i | 0 | 2.58090 | − | 0.582198i | 0 | 2.99810 | + | 0.106807i | 0 | ||||||||||
529.2 | 0 | −1.72845 | + | 0.111624i | 0 | −1.03305 | + | 1.98313i | 0 | 1.19778 | + | 2.35909i | 0 | 2.97508 | − | 0.385872i | 0 | ||||||||||
529.3 | 0 | −1.70896 | + | 0.281868i | 0 | 1.75673 | + | 1.38344i | 0 | −2.30807 | − | 1.29337i | 0 | 2.84110 | − | 0.963403i | 0 | ||||||||||
529.4 | 0 | −1.69964 | + | 0.333494i | 0 | −0.812314 | − | 2.08330i | 0 | −1.14400 | + | 2.38564i | 0 | 2.77756 | − | 1.13364i | 0 | ||||||||||
529.5 | 0 | −1.69435 | − | 0.359407i | 0 | 2.14707 | − | 0.624586i | 0 | −2.64530 | + | 0.0486853i | 0 | 2.74165 | + | 1.21792i | 0 | ||||||||||
529.6 | 0 | −1.61264 | + | 0.631969i | 0 | 0.196400 | + | 2.22743i | 0 | 2.20782 | − | 1.45793i | 0 | 2.20123 | − | 2.03828i | 0 | ||||||||||
529.7 | 0 | −1.58514 | − | 0.698092i | 0 | −2.16138 | − | 0.573097i | 0 | −1.25382 | − | 2.32979i | 0 | 2.02534 | + | 2.21315i | 0 | ||||||||||
529.8 | 0 | −1.52923 | + | 0.813293i | 0 | 1.98915 | − | 1.02141i | 0 | 2.53981 | − | 0.741175i | 0 | 1.67711 | − | 2.48743i | 0 | ||||||||||
529.9 | 0 | −1.44879 | − | 0.949220i | 0 | 1.64783 | + | 1.51151i | 0 | 1.91428 | + | 1.82634i | 0 | 1.19796 | + | 2.75043i | 0 | ||||||||||
529.10 | 0 | −1.42888 | − | 0.978925i | 0 | −1.55644 | + | 1.60546i | 0 | −2.23582 | + | 1.41461i | 0 | 1.08341 | + | 2.79754i | 0 | ||||||||||
529.11 | 0 | −1.38444 | − | 1.04083i | 0 | 0.795436 | − | 2.08980i | 0 | 0.820288 | − | 2.51538i | 0 | 0.833332 | + | 2.88194i | 0 | ||||||||||
529.12 | 0 | −1.24055 | + | 1.20874i | 0 | 0.639876 | − | 2.14256i | 0 | −1.73123 | − | 2.00071i | 0 | 0.0779117 | − | 2.99899i | 0 | ||||||||||
529.13 | 0 | −1.23553 | − | 1.21386i | 0 | 0.563662 | − | 2.16386i | 0 | 2.16061 | + | 1.52701i | 0 | 0.0530816 | + | 2.99953i | 0 | ||||||||||
529.14 | 0 | −1.10932 | + | 1.33019i | 0 | 1.31494 | + | 1.80858i | 0 | −0.204856 | + | 2.63781i | 0 | −0.538805 | − | 2.95122i | 0 | ||||||||||
529.15 | 0 | −1.05137 | + | 1.37646i | 0 | −2.12213 | + | 0.704663i | 0 | 2.07068 | + | 1.64690i | 0 | −0.789262 | − | 2.89432i | 0 | ||||||||||
529.16 | 0 | −0.664746 | + | 1.59941i | 0 | −0.917391 | + | 2.03921i | 0 | −0.626206 | − | 2.57058i | 0 | −2.11622 | − | 2.12640i | 0 | ||||||||||
529.17 | 0 | −0.651915 | − | 1.60468i | 0 | −1.74822 | + | 1.39417i | 0 | 2.36449 | − | 1.18709i | 0 | −2.15001 | + | 2.09223i | 0 | ||||||||||
529.18 | 0 | −0.587940 | + | 1.62921i | 0 | 2.15987 | − | 0.578741i | 0 | −1.02962 | + | 2.43719i | 0 | −2.30865 | − | 1.91576i | 0 | ||||||||||
529.19 | 0 | −0.474433 | − | 1.66581i | 0 | −1.30394 | − | 1.81652i | 0 | −2.63906 | + | 0.188075i | 0 | −2.54983 | + | 1.58063i | 0 | ||||||||||
529.20 | 0 | −0.398582 | − | 1.68557i | 0 | 2.12867 | − | 0.684661i | 0 | −0.0952265 | − | 2.64404i | 0 | −2.68227 | + | 1.34367i | 0 | ||||||||||
See all 92 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
63.h | even | 3 | 1 | inner |
315.r | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1260.2.by.b | ✓ | 92 |
3.b | odd | 2 | 1 | 3780.2.by.b | 92 | ||
5.b | even | 2 | 1 | inner | 1260.2.by.b | ✓ | 92 |
7.c | even | 3 | 1 | 1260.2.di.b | yes | 92 | |
9.c | even | 3 | 1 | 1260.2.di.b | yes | 92 | |
9.d | odd | 6 | 1 | 3780.2.di.b | 92 | ||
15.d | odd | 2 | 1 | 3780.2.by.b | 92 | ||
21.h | odd | 6 | 1 | 3780.2.di.b | 92 | ||
35.j | even | 6 | 1 | 1260.2.di.b | yes | 92 | |
45.h | odd | 6 | 1 | 3780.2.di.b | 92 | ||
45.j | even | 6 | 1 | 1260.2.di.b | yes | 92 | |
63.h | even | 3 | 1 | inner | 1260.2.by.b | ✓ | 92 |
63.j | odd | 6 | 1 | 3780.2.by.b | 92 | ||
105.o | odd | 6 | 1 | 3780.2.di.b | 92 | ||
315.r | even | 6 | 1 | inner | 1260.2.by.b | ✓ | 92 |
315.br | odd | 6 | 1 | 3780.2.by.b | 92 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1260.2.by.b | ✓ | 92 | 1.a | even | 1 | 1 | trivial |
1260.2.by.b | ✓ | 92 | 5.b | even | 2 | 1 | inner |
1260.2.by.b | ✓ | 92 | 63.h | even | 3 | 1 | inner |
1260.2.by.b | ✓ | 92 | 315.r | even | 6 | 1 | inner |
1260.2.di.b | yes | 92 | 7.c | even | 3 | 1 | |
1260.2.di.b | yes | 92 | 9.c | even | 3 | 1 | |
1260.2.di.b | yes | 92 | 35.j | even | 6 | 1 | |
1260.2.di.b | yes | 92 | 45.j | even | 6 | 1 | |
3780.2.by.b | 92 | 3.b | odd | 2 | 1 | ||
3780.2.by.b | 92 | 15.d | odd | 2 | 1 | ||
3780.2.by.b | 92 | 63.j | odd | 6 | 1 | ||
3780.2.by.b | 92 | 315.br | odd | 6 | 1 | ||
3780.2.di.b | 92 | 9.d | odd | 6 | 1 | ||
3780.2.di.b | 92 | 21.h | odd | 6 | 1 | ||
3780.2.di.b | 92 | 45.h | odd | 6 | 1 | ||
3780.2.di.b | 92 | 105.o | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{46} - T_{11}^{45} + 144 T_{11}^{44} - 233 T_{11}^{43} + 12041 T_{11}^{42} - 23805 T_{11}^{41} + 686061 T_{11}^{40} - 1575159 T_{11}^{39} + 29596896 T_{11}^{38} - 74834889 T_{11}^{37} + \cdots + 252019943018496 \)
acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\).