Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [637,2,Mod(92,637)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("637.92");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 637.w (of order \(7\), degree \(6\), 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: | \(162\) |
Relative dimension: | \(27\) over \(\Q(\zeta_{7})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
92.1 | −1.64981 | + | 2.06880i | 0.0373896 | + | 0.0180059i | −1.11300 | − | 4.87639i | 1.69442 | + | 0.815989i | −0.0989361 | + | 0.0476451i | 2.20489 | + | 1.46235i | 7.15641 | + | 3.44634i | −1.86940 | − | 2.34415i | −4.48358 | + | 2.15918i |
92.2 | −1.60516 | + | 2.01281i | −2.57103 | − | 1.23814i | −1.02981 | − | 4.51189i | −1.31474 | − | 0.633146i | 6.61905 | − | 3.18757i | −0.0925892 | + | 2.64413i | 6.09552 | + | 2.93545i | 3.20673 | + | 4.02111i | 3.38477 | − | 1.63002i |
92.3 | −1.55889 | + | 1.95478i | 1.69142 | + | 0.814547i | −0.946002 | − | 4.14471i | 0.780772 | + | 0.376000i | −4.22900 | + | 2.03658i | −1.80494 | + | 1.93447i | 5.07139 | + | 2.44225i | 0.326959 | + | 0.409994i | −1.95213 | + | 0.940098i |
92.4 | −1.50697 | + | 1.88968i | 2.72555 | + | 1.31256i | −0.854886 | − | 3.74550i | −3.20148 | − | 1.54175i | −6.58762 | + | 3.17243i | −1.89391 | − | 1.84746i | 4.01081 | + | 1.93150i | 3.83535 | + | 4.80937i | 7.73794 | − | 3.72639i |
92.5 | −1.40295 | + | 1.75924i | −2.32362 | − | 1.11900i | −0.681629 | − | 2.98641i | 3.41692 | + | 1.64550i | 5.22852 | − | 2.51792i | −2.48146 | − | 0.917792i | 2.15548 | + | 1.03802i | 2.27660 | + | 2.85477i | −7.68860 | + | 3.70264i |
92.6 | −1.27694 | + | 1.60123i | 1.03070 | + | 0.496360i | −0.488322 | − | 2.13948i | −0.0782783 | − | 0.0376968i | −2.11093 | + | 1.01657i | 0.138811 | − | 2.64211i | 0.358894 | + | 0.172834i | −1.05450 | − | 1.32230i | 0.160318 | − | 0.0772049i |
92.7 | −0.928616 | + | 1.16445i | −1.16240 | − | 0.559785i | −0.0485689 | − | 0.212794i | −2.30199 | − | 1.10858i | 1.73127 | − | 0.833735i | −2.33750 | − | 1.23939i | −2.39089 | − | 1.15139i | −0.832643 | − | 1.04410i | 3.42855 | − | 1.65110i |
92.8 | −0.911679 | + | 1.14321i | −1.60612 | − | 0.773466i | −0.0307265 | − | 0.134622i | −2.27795 | − | 1.09700i | 2.34850 | − | 1.13098i | 2.64144 | + | 0.150997i | −2.45291 | − | 1.18126i | 0.110899 | + | 0.139063i | 3.33086 | − | 1.60406i |
92.9 | −0.807265 | + | 1.01228i | −0.741630 | − | 0.357150i | 0.0720117 | + | 0.315504i | 0.711905 | + | 0.342835i | 0.960227 | − | 0.462421i | −0.971640 | + | 2.46088i | −2.71057 | − | 1.30534i | −1.44801 | − | 1.81575i | −0.921740 | + | 0.443887i |
92.10 | −0.706465 | + | 0.885879i | 1.43197 | + | 0.689598i | 0.159353 | + | 0.698171i | 3.16254 | + | 1.52300i | −1.62253 | + | 0.781371i | 2.63659 | + | 0.220013i | −2.77282 | − | 1.33532i | −0.295490 | − | 0.370533i | −3.58341 | + | 1.72568i |
92.11 | −0.633585 | + | 0.794490i | 2.34732 | + | 1.13041i | 0.215257 | + | 0.943102i | 2.05977 | + | 0.991933i | −2.38533 | + | 1.14871i | −2.62531 | + | 0.328254i | −2.71678 | − | 1.30833i | 2.36163 | + | 2.96138i | −2.09312 | + | 1.00799i |
92.12 | −0.403219 | + | 0.505621i | 2.54540 | + | 1.22580i | 0.351975 | + | 1.54210i | −3.71879 | − | 1.79087i | −1.64615 | + | 0.792742i | 2.60419 | − | 0.467092i | −2.08698 | − | 1.00504i | 3.10602 | + | 3.89482i | 2.40499 | − | 1.15818i |
92.13 | −0.146099 | + | 0.183202i | 0.315054 | + | 0.151722i | 0.432824 | + | 1.89632i | −1.68254 | − | 0.810269i | −0.0738247 | + | 0.0355521i | 0.399609 | − | 2.61540i | −0.832883 | − | 0.401095i | −1.79423 | − | 2.24989i | 0.394260 | − | 0.189865i |
92.14 | 0.0552404 | − | 0.0692692i | −2.67810 | − | 1.28970i | 0.443295 | + | 1.94220i | −3.26519 | − | 1.57243i | −0.237276 | + | 0.114266i | −0.177815 | + | 2.63977i | 0.318672 | + | 0.153464i | 3.63840 | + | 4.56241i | −0.289292 | + | 0.139316i |
92.15 | 0.161166 | − | 0.202095i | −2.41515 | − | 1.16307i | 0.430174 | + | 1.88471i | 1.29813 | + | 0.625147i | −0.624290 | + | 0.300642i | −2.63945 | + | 0.182451i | 0.916003 | + | 0.441124i | 2.60973 | + | 3.27250i | 0.335553 | − | 0.161594i |
92.16 | 0.251865 | − | 0.315829i | −0.214027 | − | 0.103070i | 0.408730 | + | 1.79076i | 0.747465 | + | 0.359960i | −0.0864582 | + | 0.0416361i | 1.94492 | + | 1.79368i | 1.39643 | + | 0.672485i | −1.83529 | − | 2.30138i | 0.301946 | − | 0.145409i |
92.17 | 0.258756 | − | 0.324470i | −2.07294 | − | 0.998275i | 0.406716 | + | 1.78194i | 2.95478 | + | 1.42295i | −0.860297 | + | 0.414297i | 2.39153 | + | 1.13164i | 1.43125 | + | 0.689256i | 1.43005 | + | 1.79323i | 1.22627 | − | 0.590542i |
92.18 | 0.431982 | − | 0.541688i | 2.32009 | + | 1.11729i | 0.338224 | + | 1.48186i | 1.38659 | + | 0.667747i | 1.60746 | − | 0.774112i | −2.10133 | − | 1.60760i | 2.19727 | + | 1.05815i | 2.26399 | + | 2.83895i | 0.960693 | − | 0.462646i |
92.19 | 0.505100 | − | 0.633375i | 0.463406 | + | 0.223165i | 0.299004 | + | 1.31002i | −2.43329 | − | 1.17181i | 0.375413 | − | 0.180790i | −2.60368 | + | 0.469932i | 2.44054 | + | 1.17530i | −1.70553 | − | 2.13866i | −1.97125 | + | 0.949304i |
92.20 | 0.789572 | − | 0.990091i | 0.188017 | + | 0.0905443i | 0.0881841 | + | 0.386360i | 3.30397 | + | 1.59111i | 0.238100 | − | 0.114663i | −0.667783 | − | 2.56009i | 2.73409 | + | 1.31667i | −1.84332 | − | 2.31145i | 4.18406 | − | 2.01494i |
See next 80 embeddings (of 162 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
49.e | even | 7 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 637.2.w.a | ✓ | 162 |
49.e | even | 7 | 1 | inner | 637.2.w.a | ✓ | 162 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
637.2.w.a | ✓ | 162 | 1.a | even | 1 | 1 | trivial |
637.2.w.a | ✓ | 162 | 49.e | even | 7 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{162} - 3 T_{2}^{161} + 44 T_{2}^{160} - 128 T_{2}^{159} + 1069 T_{2}^{158} - 3072 T_{2}^{157} + \cdots + 346328719009 \) acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\).