Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [360,2,Mod(11,360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(360, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("360.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 360.bm (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: | \(2.87461447277\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −1.40162 | + | 0.188277i | 1.67034 | − | 0.458221i | 1.92910 | − | 0.527787i | 0.500000 | + | 0.866025i | −2.25492 | + | 0.956740i | −3.22730 | − | 1.86328i | −2.60451 | + | 1.10297i | 2.58007 | − | 1.53077i | −0.863865 | − | 1.11970i |
11.2 | −1.33590 | − | 0.464076i | −1.29784 | + | 1.14700i | 1.56927 | + | 1.23992i | 0.500000 | + | 0.866025i | 2.26609 | − | 0.929977i | −3.14000 | − | 1.81288i | −1.52097 | − | 2.38467i | 0.368796 | − | 2.97725i | −0.266049 | − | 1.38896i |
11.3 | −1.31577 | − | 0.518408i | 0.424591 | + | 1.67920i | 1.46251 | + | 1.36421i | 0.500000 | + | 0.866025i | 0.311848 | − | 2.42956i | 3.88456 | + | 2.24275i | −1.21710 | − | 2.55317i | −2.63944 | + | 1.42595i | −0.208931 | − | 1.39870i |
11.4 | −1.21368 | + | 0.725941i | −0.452164 | + | 1.67199i | 0.946019 | − | 1.76211i | 0.500000 | + | 0.866025i | −0.664985 | − | 2.35750i | −0.550736 | − | 0.317967i | 0.131031 | + | 2.82539i | −2.59109 | − | 1.51203i | −1.23552 | − | 0.688104i |
11.5 | −1.17035 | + | 0.793896i | 1.72054 | − | 0.199393i | 0.739459 | − | 1.85828i | 0.500000 | + | 0.866025i | −1.85534 | + | 1.59929i | 3.97204 | + | 2.29326i | 0.609850 | + | 2.76190i | 2.92049 | − | 0.686124i | −1.27271 | − | 0.616609i |
11.6 | −1.14743 | − | 0.826684i | −0.952268 | − | 1.44678i | 0.633188 | + | 1.89712i | 0.500000 | + | 0.866025i | −0.103374 | + | 2.44731i | −2.56287 | − | 1.47968i | 0.841782 | − | 2.70026i | −1.18637 | + | 2.75545i | 0.142215 | − | 1.40704i |
11.7 | −0.877098 | − | 1.10937i | 0.875353 | − | 1.49458i | −0.461398 | + | 1.94605i | 0.500000 | + | 0.866025i | −2.42581 | + | 0.339801i | 1.05351 | + | 0.608247i | 2.56358 | − | 1.19502i | −1.46751 | − | 2.61656i | 0.522192 | − | 1.31427i |
11.8 | −0.760571 | + | 1.19228i | −1.71670 | + | 0.230091i | −0.843062 | − | 1.81363i | 0.500000 | + | 0.866025i | 1.03134 | − | 2.22179i | 1.06767 | + | 0.616420i | 2.80356 | + | 0.374228i | 2.89412 | − | 0.789993i | −1.41283 | − | 0.0625343i |
11.9 | −0.645247 | + | 1.25843i | 0.801968 | − | 1.53520i | −1.16731 | − | 1.62400i | 0.500000 | + | 0.866025i | 1.41448 | + | 1.99981i | −4.07138 | − | 2.35061i | 2.79690 | − | 0.421104i | −1.71369 | − | 2.46237i | −1.41246 | + | 0.0704168i |
11.10 | −0.582600 | − | 1.28863i | 1.36390 | + | 1.06761i | −1.32115 | + | 1.50152i | 0.500000 | + | 0.866025i | 0.581150 | − | 2.37955i | −1.02179 | − | 0.589933i | 2.70461 | + | 0.827699i | 0.720424 | + | 2.91221i | 0.824689 | − | 1.14886i |
11.11 | −0.148442 | − | 1.40640i | −0.857820 | + | 1.50471i | −1.95593 | + | 0.417537i | 0.500000 | + | 0.866025i | 2.24356 | + | 0.983078i | −0.661806 | − | 0.382094i | 0.877566 | + | 2.68884i | −1.52829 | − | 2.58154i | 1.14376 | − | 0.831755i |
11.12 | −0.116736 | − | 1.40939i | −1.41981 | − | 0.992033i | −1.97275 | + | 0.329051i | 0.500000 | + | 0.866025i | −1.23242 | + | 2.11687i | 4.17077 | + | 2.40800i | 0.694050 | + | 2.74195i | 1.03174 | + | 2.81700i | 1.16220 | − | 0.805790i |
11.13 | −0.0418375 | + | 1.41359i | 1.46420 | + | 0.925262i | −1.99650 | − | 0.118282i | 0.500000 | + | 0.866025i | −1.36920 | + | 2.03108i | 0.947055 | + | 0.546782i | 0.250732 | − | 2.81729i | 1.28778 | + | 2.70954i | −1.24513 | + | 0.670565i |
11.14 | 0.0747682 | + | 1.41224i | −0.478797 | − | 1.66456i | −1.98882 | + | 0.211181i | 0.500000 | + | 0.866025i | 2.31495 | − | 0.800631i | 1.88846 | + | 1.09030i | −0.446938 | − | 2.79289i | −2.54151 | + | 1.59397i | −1.18565 | + | 0.770869i |
11.15 | 0.475189 | + | 1.33199i | −1.73205 | + | 0.00397907i | −1.54839 | + | 1.26589i | 0.500000 | + | 0.866025i | −0.828349 | − | 2.30518i | −2.40441 | − | 1.38819i | −2.42193 | − | 1.46090i | 2.99997 | − | 0.0137839i | −0.915942 | + | 1.07752i |
11.16 | 0.884203 | − | 1.10371i | −1.18636 | − | 1.26196i | −0.436372 | − | 1.95181i | 0.500000 | + | 0.866025i | −2.44183 | + | 0.193576i | −2.20775 | − | 1.27465i | −2.54009 | − | 1.24417i | −0.185091 | + | 2.99428i | 1.39795 | + | 0.213885i |
11.17 | 0.913976 | − | 1.07919i | −1.45808 | + | 0.934888i | −0.329296 | − | 1.97270i | 0.500000 | + | 0.866025i | −0.323726 | + | 2.42800i | 1.98473 | + | 1.14588i | −2.42989 | − | 1.44763i | 1.25197 | − | 2.72627i | 1.39159 | + | 0.251932i |
11.18 | 0.951037 | + | 1.04667i | 1.28580 | − | 1.16048i | −0.191057 | + | 1.99085i | 0.500000 | + | 0.866025i | 2.43749 | + | 0.242161i | 1.13105 | + | 0.653010i | −2.26548 | + | 1.69340i | 0.306582 | − | 2.98429i | −0.430929 | + | 1.34696i |
11.19 | 0.959041 | + | 1.03935i | 0.0478668 | + | 1.73139i | −0.160481 | + | 1.99355i | 0.500000 | + | 0.866025i | −1.75361 | + | 1.71022i | −1.21691 | − | 0.702581i | −2.22590 | + | 1.74510i | −2.99542 | + | 0.165752i | −0.420580 | + | 1.35023i |
11.20 | 1.02877 | − | 0.970379i | 1.62132 | − | 0.609351i | 0.116731 | − | 1.99659i | 0.500000 | + | 0.866025i | 1.07667 | − | 2.20018i | −0.518944 | − | 0.299612i | −1.81736 | − | 2.16730i | 2.25738 | − | 1.97591i | 1.35476 | + | 0.405751i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
72.l | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 360.2.bm.b | yes | 48 |
3.b | odd | 2 | 1 | 1080.2.bm.a | 48 | ||
4.b | odd | 2 | 1 | 1440.2.cc.b | 48 | ||
8.b | even | 2 | 1 | 1440.2.cc.a | 48 | ||
8.d | odd | 2 | 1 | 360.2.bm.a | ✓ | 48 | |
9.c | even | 3 | 1 | 1080.2.bm.b | 48 | ||
9.d | odd | 6 | 1 | 360.2.bm.a | ✓ | 48 | |
12.b | even | 2 | 1 | 4320.2.cc.a | 48 | ||
24.f | even | 2 | 1 | 1080.2.bm.b | 48 | ||
24.h | odd | 2 | 1 | 4320.2.cc.b | 48 | ||
36.f | odd | 6 | 1 | 4320.2.cc.b | 48 | ||
36.h | even | 6 | 1 | 1440.2.cc.a | 48 | ||
72.j | odd | 6 | 1 | 1440.2.cc.b | 48 | ||
72.l | even | 6 | 1 | inner | 360.2.bm.b | yes | 48 |
72.n | even | 6 | 1 | 4320.2.cc.a | 48 | ||
72.p | odd | 6 | 1 | 1080.2.bm.a | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
360.2.bm.a | ✓ | 48 | 8.d | odd | 2 | 1 | |
360.2.bm.a | ✓ | 48 | 9.d | odd | 6 | 1 | |
360.2.bm.b | yes | 48 | 1.a | even | 1 | 1 | trivial |
360.2.bm.b | yes | 48 | 72.l | even | 6 | 1 | inner |
1080.2.bm.a | 48 | 3.b | odd | 2 | 1 | ||
1080.2.bm.a | 48 | 72.p | odd | 6 | 1 | ||
1080.2.bm.b | 48 | 9.c | even | 3 | 1 | ||
1080.2.bm.b | 48 | 24.f | even | 2 | 1 | ||
1440.2.cc.a | 48 | 8.b | even | 2 | 1 | ||
1440.2.cc.a | 48 | 36.h | even | 6 | 1 | ||
1440.2.cc.b | 48 | 4.b | odd | 2 | 1 | ||
1440.2.cc.b | 48 | 72.j | odd | 6 | 1 | ||
4320.2.cc.a | 48 | 12.b | even | 2 | 1 | ||
4320.2.cc.a | 48 | 72.n | even | 6 | 1 | ||
4320.2.cc.b | 48 | 24.h | odd | 2 | 1 | ||
4320.2.cc.b | 48 | 36.f | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{48} - 96 T_{7}^{46} + 5325 T_{7}^{44} + 1140 T_{7}^{43} - 198840 T_{7}^{42} + \cdots + 15\!\cdots\!64 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).