Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [637,2,Mod(97,637)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("637.97");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 637.bd (of order \(12\), 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: | \(5.08647060876\) |
Analytic rank: | \(0\) |
Dimension: | \(112\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
97.1 | −0.698623 | − | 2.60730i | −1.59953 | − | 0.923490i | −4.57787 | + | 2.64304i | 0.611546 | + | 0.611546i | −1.29034 | + | 4.81563i | 0 | 6.27204 | + | 6.27204i | 0.205668 | + | 0.356227i | 1.16724 | − | 2.02172i | ||
97.2 | −0.698623 | − | 2.60730i | 1.59953 | + | 0.923490i | −4.57787 | + | 2.64304i | −0.611546 | − | 0.611546i | 1.29034 | − | 4.81563i | 0 | 6.27204 | + | 6.27204i | 0.205668 | + | 0.356227i | −1.16724 | + | 2.02172i | ||
97.3 | −0.593883 | − | 2.21640i | −0.730443 | − | 0.421721i | −2.82768 | + | 1.63256i | −0.109871 | − | 0.109871i | −0.500906 | + | 1.86941i | 0 | 2.05269 | + | 2.05269i | −1.14430 | − | 1.98199i | −0.178268 | + | 0.308770i | ||
97.4 | −0.593883 | − | 2.21640i | 0.730443 | + | 0.421721i | −2.82768 | + | 1.63256i | 0.109871 | + | 0.109871i | 0.500906 | − | 1.86941i | 0 | 2.05269 | + | 2.05269i | −1.14430 | − | 1.98199i | 0.178268 | − | 0.308770i | ||
97.5 | −0.527888 | − | 1.97010i | −2.81776 | − | 1.62683i | −1.87060 | + | 1.07999i | −2.21395 | − | 2.21395i | −1.71757 | + | 6.41007i | 0 | 0.230723 | + | 0.230723i | 3.79318 | + | 6.56999i | −3.19299 | + | 5.53043i | ||
97.6 | −0.527888 | − | 1.97010i | 2.81776 | + | 1.62683i | −1.87060 | + | 1.07999i | 2.21395 | + | 2.21395i | 1.71757 | − | 6.41007i | 0 | 0.230723 | + | 0.230723i | 3.79318 | + | 6.56999i | 3.19299 | − | 5.53043i | ||
97.7 | −0.382384 | − | 1.42708i | −2.31749 | − | 1.33800i | −0.158275 | + | 0.0913803i | 2.34842 | + | 2.34842i | −1.02326 | + | 3.81886i | 0 | −1.89845 | − | 1.89845i | 2.08049 | + | 3.60352i | 2.45338 | − | 4.24938i | ||
97.8 | −0.382384 | − | 1.42708i | 2.31749 | + | 1.33800i | −0.158275 | + | 0.0913803i | −2.34842 | − | 2.34842i | 1.02326 | − | 3.81886i | 0 | −1.89845 | − | 1.89845i | 2.08049 | + | 3.60352i | −2.45338 | + | 4.24938i | ||
97.9 | −0.304268 | − | 1.13554i | −0.530560 | − | 0.306319i | 0.535168 | − | 0.308979i | 0.927001 | + | 0.927001i | −0.186406 | + | 0.695677i | 0 | −2.17625 | − | 2.17625i | −1.31234 | − | 2.27304i | 0.770594 | − | 1.33471i | ||
97.10 | −0.304268 | − | 1.13554i | 0.530560 | + | 0.306319i | 0.535168 | − | 0.308979i | −0.927001 | − | 0.927001i | 0.186406 | − | 0.695677i | 0 | −2.17625 | − | 2.17625i | −1.31234 | − | 2.27304i | −0.770594 | + | 1.33471i | ||
97.11 | −0.201540 | − | 0.752157i | −2.47059 | − | 1.42640i | 1.20693 | − | 0.696820i | 0.942403 | + | 0.942403i | −0.574953 | + | 2.14575i | 0 | −1.86860 | − | 1.86860i | 2.56922 | + | 4.45003i | 0.518904 | − | 0.898767i | ||
97.12 | −0.201540 | − | 0.752157i | 2.47059 | + | 1.42640i | 1.20693 | − | 0.696820i | −0.942403 | − | 0.942403i | 0.574953 | − | 2.14575i | 0 | −1.86860 | − | 1.86860i | 2.56922 | + | 4.45003i | −0.518904 | + | 0.898767i | ||
97.13 | −0.0641123 | − | 0.239270i | −1.55865 | − | 0.899886i | 1.67891 | − | 0.969320i | −2.60215 | − | 2.60215i | −0.115387 | + | 0.430632i | 0 | −0.689884 | − | 0.689884i | 0.119590 | + | 0.207136i | −0.455788 | + | 0.789448i | ||
97.14 | −0.0641123 | − | 0.239270i | 1.55865 | + | 0.899886i | 1.67891 | − | 0.969320i | 2.60215 | + | 2.60215i | 0.115387 | − | 0.430632i | 0 | −0.689884 | − | 0.689884i | 0.119590 | + | 0.207136i | 0.455788 | − | 0.789448i | ||
97.15 | 0.0305678 | + | 0.114081i | −0.864337 | − | 0.499025i | 1.71997 | − | 0.993026i | 0.483986 | + | 0.483986i | 0.0305082 | − | 0.113858i | 0 | 0.332887 | + | 0.332887i | −1.00195 | − | 1.73543i | −0.0404190 | + | 0.0700078i | ||
97.16 | 0.0305678 | + | 0.114081i | 0.864337 | + | 0.499025i | 1.71997 | − | 0.993026i | −0.483986 | − | 0.483986i | −0.0305082 | + | 0.113858i | 0 | 0.332887 | + | 0.332887i | −1.00195 | − | 1.73543i | 0.0404190 | − | 0.0700078i | ||
97.17 | 0.244980 | + | 0.914278i | −0.510427 | − | 0.294695i | 0.956161 | − | 0.552040i | −1.45945 | − | 1.45945i | 0.144389 | − | 0.538867i | 0 | 2.07756 | + | 2.07756i | −1.32631 | − | 2.29724i | 0.976808 | − | 1.69188i | ||
97.18 | 0.244980 | + | 0.914278i | 0.510427 | + | 0.294695i | 0.956161 | − | 0.552040i | 1.45945 | + | 1.45945i | −0.144389 | + | 0.538867i | 0 | 2.07756 | + | 2.07756i | −1.32631 | − | 2.29724i | −0.976808 | + | 1.69188i | ||
97.19 | 0.280827 | + | 1.04806i | −2.30381 | − | 1.33011i | 0.712486 | − | 0.411354i | 2.92094 | + | 2.92094i | 0.747060 | − | 2.78806i | 0 | 2.16567 | + | 2.16567i | 2.03837 | + | 3.53056i | −2.24104 | + | 3.88160i | ||
97.20 | 0.280827 | + | 1.04806i | 2.30381 | + | 1.33011i | 0.712486 | − | 0.411354i | −2.92094 | − | 2.92094i | −0.747060 | + | 2.78806i | 0 | 2.16567 | + | 2.16567i | 2.03837 | + | 3.53056i | 2.24104 | − | 3.88160i | ||
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
13.f | odd | 12 | 1 | inner |
91.bc | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 637.2.bd.c | ✓ | 112 |
7.b | odd | 2 | 1 | inner | 637.2.bd.c | ✓ | 112 |
7.c | even | 3 | 1 | 637.2.x.c | 112 | ||
7.c | even | 3 | 1 | 637.2.bb.c | 112 | ||
7.d | odd | 6 | 1 | 637.2.x.c | 112 | ||
7.d | odd | 6 | 1 | 637.2.bb.c | 112 | ||
13.f | odd | 12 | 1 | inner | 637.2.bd.c | ✓ | 112 |
91.w | even | 12 | 1 | 637.2.bb.c | 112 | ||
91.x | odd | 12 | 1 | 637.2.x.c | 112 | ||
91.ba | even | 12 | 1 | 637.2.x.c | 112 | ||
91.bc | even | 12 | 1 | inner | 637.2.bd.c | ✓ | 112 |
91.bd | odd | 12 | 1 | 637.2.bb.c | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
637.2.x.c | 112 | 7.c | even | 3 | 1 | ||
637.2.x.c | 112 | 7.d | odd | 6 | 1 | ||
637.2.x.c | 112 | 91.x | odd | 12 | 1 | ||
637.2.x.c | 112 | 91.ba | even | 12 | 1 | ||
637.2.bb.c | 112 | 7.c | even | 3 | 1 | ||
637.2.bb.c | 112 | 7.d | odd | 6 | 1 | ||
637.2.bb.c | 112 | 91.w | even | 12 | 1 | ||
637.2.bb.c | 112 | 91.bd | odd | 12 | 1 | ||
637.2.bd.c | ✓ | 112 | 1.a | even | 1 | 1 | trivial |
637.2.bd.c | ✓ | 112 | 7.b | odd | 2 | 1 | inner |
637.2.bd.c | ✓ | 112 | 13.f | odd | 12 | 1 | inner |
637.2.bd.c | ✓ | 112 | 91.bc | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):
\( T_{2}^{56} - 105 T_{2}^{52} + 8 T_{2}^{49} + 7501 T_{2}^{48} - 216 T_{2}^{47} - 420 T_{2}^{46} - 1704 T_{2}^{45} - 279952 T_{2}^{44} + 15616 T_{2}^{43} - 3148 T_{2}^{42} + 2328 T_{2}^{41} + 7470684 T_{2}^{40} - 137736 T_{2}^{39} + \cdots + 58081 \) |
\( T_{3}^{112} - 112 T_{3}^{110} + 6652 T_{3}^{108} - 272672 T_{3}^{106} + 8572342 T_{3}^{104} - 218614832 T_{3}^{102} + 4681717696 T_{3}^{100} - 86173122064 T_{3}^{98} + 1386034819121 T_{3}^{96} + \cdots + 28\!\cdots\!96 \) |