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 | −1.40729 | + | 0.139770i | 1.36390 | + | 1.06761i | 1.96093 | − | 0.393394i | −0.500000 | − | 0.866025i | −2.06862 | − | 1.31180i | 1.02179 | + | 0.589933i | −2.70461 | + | 0.827699i | 0.720424 | + | 2.91221i | 0.824689 | + | 1.14886i |
11.2 | −1.39929 | − | 0.204905i | 0.875353 | − | 1.49458i | 1.91603 | + | 0.573443i | −0.500000 | − | 0.866025i | −1.53112 | + | 1.91198i | −1.05351 | − | 0.608247i | −2.56358 | − | 1.19502i | −1.46751 | − | 2.61656i | 0.522192 | + | 1.31427i |
11.3 | −1.29220 | + | 0.574647i | −0.857820 | + | 1.50471i | 1.33956 | − | 1.48512i | −0.500000 | − | 0.866025i | 0.243800 | − | 2.43733i | 0.661806 | + | 0.382094i | −0.877566 | + | 2.68884i | −1.52829 | − | 2.58154i | 1.14376 | + | 0.831755i |
11.4 | −1.28964 | − | 0.580361i | −0.952268 | − | 1.44678i | 1.32636 | + | 1.49692i | −0.500000 | − | 0.866025i | 0.388429 | + | 2.41850i | 2.56287 | + | 1.47968i | −0.841782 | − | 2.70026i | −1.18637 | + | 2.75545i | 0.142215 | + | 1.40704i |
11.5 | −1.27893 | + | 0.603598i | −1.41981 | − | 0.992033i | 1.27134 | − | 1.54392i | −0.500000 | − | 0.866025i | 2.41464 | + | 0.411747i | −4.17077 | − | 2.40800i | −0.694050 | + | 2.74195i | 1.03174 | + | 2.81700i | 1.16220 | + | 0.805790i |
11.6 | −1.10684 | − | 0.880287i | 0.424591 | + | 1.67920i | 0.450191 | + | 1.94867i | −0.500000 | − | 0.866025i | 1.00823 | − | 2.23237i | −3.88456 | − | 2.24275i | 1.21710 | − | 2.55317i | −2.63944 | + | 1.42595i | −0.208931 | + | 1.39870i |
11.7 | −1.06985 | − | 0.924887i | −1.29784 | + | 1.14700i | 0.289169 | + | 1.97898i | −0.500000 | − | 0.866025i | 2.44934 | − | 0.0267591i | 3.14000 | + | 1.81288i | 1.52097 | − | 2.38467i | 0.368796 | − | 2.97725i | −0.266049 | + | 1.38896i |
11.8 | −0.537760 | − | 1.30798i | 1.67034 | − | 0.458221i | −1.42163 | + | 1.40676i | −0.500000 | − | 0.866025i | −1.49759 | − | 1.93836i | 3.22730 | + | 1.86328i | 2.60451 | + | 1.10297i | 2.58007 | − | 1.53077i | −0.863865 | + | 1.11970i |
11.9 | −0.513744 | + | 1.31760i | −1.18636 | − | 1.26196i | −1.47213 | − | 1.35382i | −0.500000 | − | 0.866025i | 2.27224 | − | 0.914825i | 2.20775 | + | 1.27465i | 2.54009 | − | 1.24417i | −0.185091 | + | 2.99428i | 1.39795 | − | 0.213885i |
11.10 | −0.477617 | + | 1.33112i | −1.45808 | + | 0.934888i | −1.54376 | − | 1.27153i | −0.500000 | − | 0.866025i | −0.548047 | − | 2.38739i | −1.98473 | − | 1.14588i | 2.42989 | − | 1.44763i | 1.25197 | − | 2.72627i | 1.39159 | − | 0.251932i |
11.11 | −0.325988 | + | 1.37613i | 1.62132 | − | 0.609351i | −1.78746 | − | 0.897203i | −0.500000 | − | 0.866025i | 0.310014 | + | 2.42979i | 0.518944 | + | 0.299612i | 1.81736 | − | 2.16730i | 2.25738 | − | 1.97591i | 1.35476 | − | 0.405751i |
11.12 | 0.0218455 | − | 1.41404i | −0.452164 | + | 1.67199i | −1.99905 | − | 0.0617811i | −0.500000 | − | 0.866025i | 2.35439 | + | 0.675906i | 0.550736 | + | 0.317967i | −0.131031 | + | 2.82539i | −2.59109 | − | 1.51203i | −1.23552 | + | 0.688104i |
11.13 | 0.102357 | − | 1.41050i | 1.72054 | − | 0.199393i | −1.97905 | − | 0.288749i | −0.500000 | − | 0.866025i | −0.105136 | − | 2.44723i | −3.97204 | − | 2.29326i | −0.609850 | + | 2.76190i | 2.92049 | − | 0.686124i | −1.27271 | + | 0.616609i |
11.14 | 0.158141 | + | 1.40534i | 0.284002 | + | 1.70861i | −1.94998 | + | 0.444484i | −0.500000 | − | 0.866025i | −2.35627 | + | 0.669321i | −2.24682 | − | 1.29720i | −0.933024 | − | 2.67011i | −2.83869 | + | 0.970497i | 1.13799 | − | 0.839626i |
11.15 | 0.524388 | + | 1.31340i | −0.0300586 | − | 1.73179i | −1.45003 | + | 1.37746i | −0.500000 | − | 0.866025i | 2.25877 | − | 0.947610i | −3.45090 | − | 1.99238i | −2.56954 | − | 1.18215i | −2.99819 | + | 0.104110i | 0.875243 | − | 1.11083i |
11.16 | 0.652259 | − | 1.25481i | −1.71670 | + | 0.230091i | −1.14912 | − | 1.63693i | −0.500000 | − | 0.866025i | −0.831012 | + | 2.30422i | −1.06767 | − | 0.616420i | −2.80356 | + | 0.374228i | 2.89412 | − | 0.789993i | −1.41283 | + | 0.0625343i |
11.17 | 0.767212 | − | 1.18802i | 0.801968 | − | 1.53520i | −0.822770 | − | 1.82292i | −0.500000 | − | 0.866025i | −1.20857 | − | 2.13058i | 4.07138 | + | 2.35061i | −2.79690 | − | 0.421104i | −1.71369 | − | 2.46237i | −1.41246 | − | 0.0704168i |
11.18 | 0.893891 | + | 1.09588i | 1.59734 | + | 0.669699i | −0.401917 | + | 1.95920i | −0.500000 | − | 0.866025i | 0.693939 | + | 2.34914i | 2.53202 | + | 1.46186i | −2.50632 | + | 1.31086i | 2.10301 | + | 2.13948i | 0.502116 | − | 1.32207i |
11.19 | 0.962641 | + | 1.03601i | −1.57528 | − | 0.720074i | −0.146644 | + | 1.99462i | −0.500000 | − | 0.866025i | −0.770419 | − | 2.32518i | 1.68164 | + | 0.970893i | −2.20761 | + | 1.76818i | 1.96299 | + | 2.26863i | 0.415893 | − | 1.35168i |
11.20 | 1.20329 | − | 0.743030i | 1.46420 | + | 0.925262i | 0.895814 | − | 1.78816i | −0.500000 | − | 0.866025i | 2.44936 | + | 0.0254133i | −0.947055 | − | 0.546782i | −0.250732 | − | 2.81729i | 1.28778 | + | 2.70954i | −1.24513 | − | 0.670565i |
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.a | ✓ | 48 |
3.b | odd | 2 | 1 | 1080.2.bm.b | 48 | ||
4.b | odd | 2 | 1 | 1440.2.cc.a | 48 | ||
8.b | even | 2 | 1 | 1440.2.cc.b | 48 | ||
8.d | odd | 2 | 1 | 360.2.bm.b | yes | 48 | |
9.c | even | 3 | 1 | 1080.2.bm.a | 48 | ||
9.d | odd | 6 | 1 | 360.2.bm.b | yes | 48 | |
12.b | even | 2 | 1 | 4320.2.cc.b | 48 | ||
24.f | even | 2 | 1 | 1080.2.bm.a | 48 | ||
24.h | odd | 2 | 1 | 4320.2.cc.a | 48 | ||
36.f | odd | 6 | 1 | 4320.2.cc.a | 48 | ||
36.h | even | 6 | 1 | 1440.2.cc.b | 48 | ||
72.j | odd | 6 | 1 | 1440.2.cc.a | 48 | ||
72.l | even | 6 | 1 | inner | 360.2.bm.a | ✓ | 48 |
72.n | even | 6 | 1 | 4320.2.cc.b | 48 | ||
72.p | odd | 6 | 1 | 1080.2.bm.b | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
360.2.bm.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
360.2.bm.a | ✓ | 48 | 72.l | even | 6 | 1 | inner |
360.2.bm.b | yes | 48 | 8.d | odd | 2 | 1 | |
360.2.bm.b | yes | 48 | 9.d | odd | 6 | 1 | |
1080.2.bm.a | 48 | 9.c | even | 3 | 1 | ||
1080.2.bm.a | 48 | 24.f | even | 2 | 1 | ||
1080.2.bm.b | 48 | 3.b | odd | 2 | 1 | ||
1080.2.bm.b | 48 | 72.p | odd | 6 | 1 | ||
1440.2.cc.a | 48 | 4.b | odd | 2 | 1 | ||
1440.2.cc.a | 48 | 72.j | odd | 6 | 1 | ||
1440.2.cc.b | 48 | 8.b | even | 2 | 1 | ||
1440.2.cc.b | 48 | 36.h | even | 6 | 1 | ||
4320.2.cc.a | 48 | 24.h | odd | 2 | 1 | ||
4320.2.cc.a | 48 | 36.f | odd | 6 | 1 | ||
4320.2.cc.b | 48 | 12.b | even | 2 | 1 | ||
4320.2.cc.b | 48 | 72.n | even | 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 \(S_{2}^{\mathrm{new}}(360, [\chi])\).