Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1098,2,Mod(367,1098)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1098, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1098.367");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1098 = 2 \cdot 3^{2} \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1098.e (of order \(3\), 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: | \(8.76757414194\) |
Analytic rank: | \(0\) |
Dimension: | \(34\) |
Relative dimension: | \(17\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
367.1 | −0.500000 | + | 0.866025i | −1.70593 | + | 0.299644i | −0.500000 | − | 0.866025i | −2.02139 | − | 3.50116i | 0.593468 | − | 1.62720i | 1.31426 | − | 2.27636i | 1.00000 | 2.82043 | − | 1.02235i | 4.04279 | ||||
367.2 | −0.500000 | + | 0.866025i | −1.68083 | + | 0.418122i | −0.500000 | − | 0.866025i | −1.31261 | − | 2.27350i | 0.478309 | − | 1.66470i | −2.37545 | + | 4.11441i | 1.00000 | 2.65035 | − | 1.40558i | 2.62522 | ||||
367.3 | −0.500000 | + | 0.866025i | −1.53282 | + | 0.806520i | −0.500000 | − | 0.866025i | 1.59654 | + | 2.76529i | 0.0679412 | − | 1.73072i | −2.01441 | + | 3.48906i | 1.00000 | 1.69905 | − | 2.47249i | −3.19308 | ||||
367.4 | −0.500000 | + | 0.866025i | −1.37904 | − | 1.04798i | −0.500000 | − | 0.866025i | 1.65700 | + | 2.87001i | 1.59709 | − | 0.670294i | 0.254712 | − | 0.441174i | 1.00000 | 0.803491 | + | 2.89040i | −3.31400 | ||||
367.5 | −0.500000 | + | 0.866025i | −1.02303 | + | 1.39765i | −0.500000 | − | 0.866025i | 0.182999 | + | 0.316964i | −0.698884 | − | 1.58479i | 1.22896 | − | 2.12861i | 1.00000 | −0.906832 | − | 2.85966i | −0.365998 | ||||
367.6 | −0.500000 | + | 0.866025i | −0.887881 | − | 1.48717i | −0.500000 | − | 0.866025i | −2.04970 | − | 3.55018i | 1.73187 | − | 0.0253439i | −1.01857 | + | 1.76421i | 1.00000 | −1.42333 | + | 2.64086i | 4.09940 | ||||
367.7 | −0.500000 | + | 0.866025i | −0.177122 | − | 1.72297i | −0.500000 | − | 0.866025i | −0.951924 | − | 1.64878i | 1.58070 | + | 0.708094i | −0.486861 | + | 0.843268i | 1.00000 | −2.93726 | + | 0.610350i | 1.90385 | ||||
367.8 | −0.500000 | + | 0.866025i | −0.143239 | + | 1.72612i | −0.500000 | − | 0.866025i | −0.792589 | − | 1.37280i | −1.42324 | − | 0.987108i | −2.02040 | + | 3.49944i | 1.00000 | −2.95897 | − | 0.494496i | 1.58518 | ||||
367.9 | −0.500000 | + | 0.866025i | −0.0462544 | + | 1.73143i | −0.500000 | − | 0.866025i | 0.379433 | + | 0.657197i | −1.47634 | − | 0.905774i | 1.58694 | − | 2.74866i | 1.00000 | −2.99572 | − | 0.160173i | −0.758865 | ||||
367.10 | −0.500000 | + | 0.866025i | −0.0352391 | − | 1.73169i | −0.500000 | − | 0.866025i | 1.66276 | + | 2.87999i | 1.51731 | + | 0.835328i | −1.00835 | + | 1.74651i | 1.00000 | −2.99752 | + | 0.122047i | −3.32553 | ||||
367.11 | −0.500000 | + | 0.866025i | 0.935531 | − | 1.45766i | −0.500000 | − | 0.866025i | 0.939094 | + | 1.62656i | 0.794608 | + | 1.53903i | 1.66888 | − | 2.89058i | 1.00000 | −1.24956 | − | 2.72738i | −1.87819 | ||||
367.12 | −0.500000 | + | 0.866025i | 1.20191 | − | 1.24716i | −0.500000 | − | 0.866025i | −0.417846 | − | 0.723730i | 0.479118 | + | 1.66447i | −0.864413 | + | 1.49721i | 1.00000 | −0.110822 | − | 2.99795i | 0.835692 | ||||
367.13 | −0.500000 | + | 0.866025i | 1.24732 | + | 1.20175i | −0.500000 | − | 0.866025i | −1.50626 | − | 2.60891i | −1.66440 | + | 0.479334i | −0.0606085 | + | 0.104977i | 1.00000 | 0.111601 | + | 2.99792i | 3.01251 | ||||
367.14 | −0.500000 | + | 0.866025i | 1.40109 | + | 1.01830i | −0.500000 | − | 0.866025i | 0.830957 | + | 1.43926i | −1.58242 | + | 0.704234i | 0.867433 | − | 1.50244i | 1.00000 | 0.926133 | + | 2.85347i | −1.66191 | ||||
367.15 | −0.500000 | + | 0.866025i | 1.53940 | + | 0.793888i | −0.500000 | − | 0.866025i | 1.46258 | + | 2.53326i | −1.45723 | + | 0.936212i | −2.14457 | + | 3.71451i | 1.00000 | 1.73948 | + | 2.44422i | −2.92515 | ||||
367.16 | −0.500000 | + | 0.866025i | 1.55524 | − | 0.762377i | −0.500000 | − | 0.866025i | −1.78057 | − | 3.08403i | −0.117384 | + | 1.72807i | 1.95142 | − | 3.37997i | 1.00000 | 1.83756 | − | 2.37136i | 3.56113 | ||||
367.17 | −0.500000 | + | 0.866025i | 1.73088 | + | 0.0635888i | −0.500000 | − | 0.866025i | −0.378482 | − | 0.655549i | −0.920511 | + | 1.46719i | −2.37896 | + | 4.12049i | 1.00000 | 2.99191 | + | 0.220130i | 0.756963 | ||||
733.1 | −0.500000 | − | 0.866025i | −1.70593 | − | 0.299644i | −0.500000 | + | 0.866025i | −2.02139 | + | 3.50116i | 0.593468 | + | 1.62720i | 1.31426 | + | 2.27636i | 1.00000 | 2.82043 | + | 1.02235i | 4.04279 | ||||
733.2 | −0.500000 | − | 0.866025i | −1.68083 | − | 0.418122i | −0.500000 | + | 0.866025i | −1.31261 | + | 2.27350i | 0.478309 | + | 1.66470i | −2.37545 | − | 4.11441i | 1.00000 | 2.65035 | + | 1.40558i | 2.62522 | ||||
733.3 | −0.500000 | − | 0.866025i | −1.53282 | − | 0.806520i | −0.500000 | + | 0.866025i | 1.59654 | − | 2.76529i | 0.0679412 | + | 1.73072i | −2.01441 | − | 3.48906i | 1.00000 | 1.69905 | + | 2.47249i | −3.19308 | ||||
See all 34 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1098.2.e.f | ✓ | 34 |
9.c | even | 3 | 1 | inner | 1098.2.e.f | ✓ | 34 |
9.c | even | 3 | 1 | 9882.2.a.bm | 17 | ||
9.d | odd | 6 | 1 | 9882.2.a.bk | 17 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1098.2.e.f | ✓ | 34 | 1.a | even | 1 | 1 | trivial |
1098.2.e.f | ✓ | 34 | 9.c | even | 3 | 1 | inner |
9882.2.a.bk | 17 | 9.d | odd | 6 | 1 | ||
9882.2.a.bm | 17 | 9.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1098, [\chi])\):
\( T_{5}^{34} + 5 T_{5}^{33} + 71 T_{5}^{32} + 250 T_{5}^{31} + 2477 T_{5}^{30} + 7296 T_{5}^{29} + \cdots + 6323430400 \) |
\( T_{7}^{34} + 11 T_{7}^{33} + 141 T_{7}^{32} + 920 T_{7}^{31} + 7281 T_{7}^{30} + 36593 T_{7}^{29} + \cdots + 97575641641 \) |