Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [228,2,Mod(67,228)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(228, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 0, 17]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("228.67");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 228 = 2^{2} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 228.w (of order \(18\), 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: | \(1.82058916609\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 | −1.37612 | − | 0.326044i | −0.766044 | + | 0.642788i | 1.78739 | + | 0.897349i | −1.25586 | − | 0.457097i | 1.26374 | − | 0.634786i | −1.18216 | − | 0.682521i | −2.16708 | − | 1.81762i | 0.173648 | − | 0.984808i | 1.57918 | + | 1.03849i |
67.2 | −1.33191 | + | 0.475423i | −0.766044 | + | 0.642788i | 1.54795 | − | 1.26644i | 3.39140 | + | 1.23437i | 0.714703 | − | 1.22033i | −1.70277 | − | 0.983096i | −1.45963 | + | 2.42270i | 0.173648 | − | 0.984808i | −5.10387 | − | 0.0317150i |
67.3 | −0.669968 | − | 1.24545i | −0.766044 | + | 0.642788i | −1.10229 | + | 1.66882i | −2.74412 | − | 0.998777i | 1.31378 | + | 0.523422i | 3.79172 | + | 2.18915i | 2.81693 | + | 0.254784i | 0.173648 | − | 0.984808i | 0.594545 | + | 4.08681i |
67.4 | −0.662654 | + | 1.24936i | −0.766044 | + | 0.642788i | −1.12178 | − | 1.65578i | 1.27204 | + | 0.462984i | −0.295448 | − | 1.38301i | 3.79268 | + | 2.18970i | 2.81201 | − | 0.304291i | 0.173648 | − | 0.984808i | −1.42135 | + | 1.28243i |
67.5 | −0.585310 | − | 1.28741i | −0.766044 | + | 0.642788i | −1.31482 | + | 1.50706i | 0.378363 | + | 0.137713i | 1.27590 | + | 0.609980i | −3.35222 | − | 1.93540i | 2.70978 | + | 0.810613i | 0.173648 | − | 0.984808i | −0.0441675 | − | 0.567712i |
67.6 | 0.370262 | − | 1.36488i | −0.766044 | + | 0.642788i | −1.72581 | − | 1.01073i | 3.29537 | + | 1.19942i | 0.593693 | + | 1.28356i | 2.21173 | + | 1.27694i | −2.01853 | + | 1.98130i | 0.173648 | − | 0.984808i | 2.85721 | − | 4.05370i |
67.7 | 0.541986 | + | 1.30624i | −0.766044 | + | 0.642788i | −1.41250 | + | 1.41592i | −3.72560 | − | 1.35601i | −1.25482 | − | 0.652253i | 0.752955 | + | 0.434719i | −2.61509 | − | 1.07765i | 0.173648 | − | 0.984808i | −0.247958 | − | 5.60144i |
67.8 | 0.866706 | − | 1.11751i | −0.766044 | + | 0.642788i | −0.497640 | − | 1.93710i | −2.00728 | − | 0.730589i | 0.0543836 | + | 1.41317i | −3.36529 | − | 1.94295i | −2.59603 | − | 1.12278i | 0.173648 | − | 0.984808i | −2.55616 | + | 1.60994i |
67.9 | 1.15291 | + | 0.819021i | −0.766044 | + | 0.642788i | 0.658410 | + | 1.88852i | 1.35681 | + | 0.493837i | −1.40964 | + | 0.113671i | −0.113130 | − | 0.0653155i | −0.787646 | + | 2.71654i | 0.173648 | − | 0.984808i | 1.15981 | + | 1.68060i |
67.10 | 1.36774 | − | 0.359580i | −0.766044 | + | 0.642788i | 1.74140 | − | 0.983622i | 0.0388803 | + | 0.0141513i | −0.816613 | + | 1.15462i | 1.39317 | + | 0.804345i | 2.02809 | − | 1.97151i | 0.173648 | − | 0.984808i | 0.0582665 | + | 0.00537462i |
79.1 | −1.41413 | − | 0.0150734i | 0.939693 | − | 0.342020i | 1.99955 | + | 0.0426317i | −0.175493 | + | 0.995269i | −1.33401 | + | 0.469498i | 2.28954 | − | 1.32186i | −2.82698 | − | 0.0904268i | 0.766044 | − | 0.642788i | 0.263172 | − | 1.40480i |
79.2 | −1.13985 | + | 0.837107i | 0.939693 | − | 0.342020i | 0.598505 | − | 1.90835i | 0.316000 | − | 1.79212i | −0.784799 | + | 1.17647i | 0.258690 | − | 0.149354i | 0.915287 | + | 2.67624i | 0.766044 | − | 0.642788i | 1.14001 | + | 2.30727i |
79.3 | −0.663081 | + | 1.24913i | 0.939693 | − | 0.342020i | −1.12065 | − | 1.65655i | −0.379809 | + | 2.15400i | −0.195865 | + | 1.40058i | −3.95398 | + | 2.28283i | 2.81232 | − | 0.301406i | 0.766044 | − | 0.642788i | −2.43878 | − | 1.90271i |
79.4 | −0.578676 | − | 1.29040i | 0.939693 | − | 0.342020i | −1.33027 | + | 1.49345i | −0.540387 | + | 3.06469i | −0.985121 | − | 1.01466i | 1.31815 | − | 0.761035i | 2.69694 | + | 0.852357i | 0.766044 | − | 0.642788i | 4.26739 | − | 1.07615i |
79.5 | −0.488367 | − | 1.32721i | 0.939693 | − | 0.342020i | −1.52300 | + | 1.29633i | 0.683388 | − | 3.87569i | −0.912849 | − | 1.08014i | −0.181667 | + | 0.104885i | 2.46429 | + | 1.38825i | 0.766044 | − | 0.642788i | −5.47761 | + | 0.985755i |
79.6 | −0.0863378 | + | 1.41158i | 0.939693 | − | 0.342020i | −1.98509 | − | 0.243745i | 0.432248 | − | 2.45140i | 0.401656 | + | 1.35598i | 1.95860 | − | 1.13080i | 0.515453 | − | 2.78106i | 0.766044 | − | 0.642788i | 3.42301 | + | 0.821799i |
79.7 | 0.869785 | − | 1.11511i | 0.939693 | − | 0.342020i | −0.486948 | − | 1.93981i | 0.234529 | − | 1.33008i | 0.435940 | − | 1.34535i | −1.38264 | + | 0.798268i | −2.58665 | − | 1.14422i | 0.766044 | − | 0.642788i | −1.27920 | − | 1.41841i |
79.8 | 1.13356 | + | 0.845604i | 0.939693 | − | 0.342020i | 0.569909 | + | 1.91708i | 0.162908 | − | 0.923898i | 1.35441 | + | 0.406908i | 2.27351 | − | 1.31261i | −0.975067 | + | 2.65504i | 0.766044 | − | 0.642788i | 0.965918 | − | 0.909537i |
79.9 | 1.24913 | − | 0.663079i | 0.939693 | − | 0.342020i | 1.12065 | − | 1.65654i | −0.711633 | + | 4.03587i | 0.947012 | − | 1.05032i | 1.83842 | − | 1.06142i | 0.301424 | − | 2.81232i | 0.766044 | − | 0.642788i | 1.78718 | + | 5.51320i |
79.10 | 1.38401 | + | 0.290702i | 0.939693 | − | 0.342020i | 1.83099 | + | 0.804669i | −0.0217506 | + | 0.123354i | 1.39997 | − | 0.200190i | −3.23383 | + | 1.86705i | 2.30019 | + | 1.64594i | 0.766044 | − | 0.642788i | −0.0659622 | + | 0.164400i |
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
76.k | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 228.2.w.a | ✓ | 60 |
3.b | odd | 2 | 1 | 684.2.cf.c | 60 | ||
4.b | odd | 2 | 1 | 228.2.w.b | yes | 60 | |
12.b | even | 2 | 1 | 684.2.cf.b | 60 | ||
19.f | odd | 18 | 1 | 228.2.w.b | yes | 60 | |
57.j | even | 18 | 1 | 684.2.cf.b | 60 | ||
76.k | even | 18 | 1 | inner | 228.2.w.a | ✓ | 60 |
228.u | odd | 18 | 1 | 684.2.cf.c | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
228.2.w.a | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
228.2.w.a | ✓ | 60 | 76.k | even | 18 | 1 | inner |
228.2.w.b | yes | 60 | 4.b | odd | 2 | 1 | |
228.2.w.b | yes | 60 | 19.f | odd | 18 | 1 | |
684.2.cf.b | 60 | 12.b | even | 2 | 1 | ||
684.2.cf.b | 60 | 57.j | even | 18 | 1 | ||
684.2.cf.c | 60 | 3.b | odd | 2 | 1 | ||
684.2.cf.c | 60 | 228.u | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{60} - 120 T_{7}^{58} + 8037 T_{7}^{56} - 234 T_{7}^{55} - 370070 T_{7}^{54} + 29574 T_{7}^{53} + 12934554 T_{7}^{52} - 2059938 T_{7}^{51} - 359272269 T_{7}^{50} + 96327522 T_{7}^{49} + \cdots + 54\!\cdots\!89 \)
acting on \(S_{2}^{\mathrm{new}}(228, [\chi])\).