Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [273,2,Mod(11,273)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(273, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 8, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("273.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 273.bw (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: | \(2.17991597518\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(32\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −0.700434 | − | 2.61406i | 1.08921 | + | 1.34671i | −4.61063 | + | 2.66195i | −2.04884 | − | 0.548984i | 2.75745 | − | 3.79054i | −0.908491 | + | 2.48488i | 6.36068 | + | 6.36068i | −0.627240 | + | 2.93370i | 5.74030i | ||
11.2 | −0.679659 | − | 2.53652i | −1.73115 | − | 0.0557742i | −4.23995 | + | 2.44794i | 2.98598 | + | 0.800092i | 1.03512 | + | 4.42901i | 2.50266 | + | 0.858313i | 5.37724 | + | 5.37724i | 2.99378 | + | 0.193107i | − | 8.11779i | |
11.3 | −0.614189 | − | 2.29218i | −0.572173 | − | 1.63481i | −3.14483 | + | 1.81567i | 0.203686 | + | 0.0545776i | −3.39587 | + | 2.31561i | −2.22424 | − | 1.43274i | 2.73738 | + | 2.73738i | −2.34524 | + | 1.87079i | − | 0.500408i | |
11.4 | −0.590022 | − | 2.20199i | 1.60162 | − | 0.659394i | −2.76859 | + | 1.59845i | −1.90854 | − | 0.511391i | −2.39697 | − | 3.13771i | 0.185390 | − | 2.63925i | 1.92936 | + | 1.92936i | 2.13040 | − | 2.11220i | 4.50431i | ||
11.5 | −0.588963 | − | 2.19804i | 1.52334 | − | 0.824281i | −2.75245 | + | 1.58913i | 3.96614 | + | 1.06272i | −2.70899 | − | 2.86289i | −1.45144 | + | 2.21209i | 1.89590 | + | 1.89590i | 1.64112 | − | 2.51132i | − | 9.34364i | |
11.6 | −0.548543 | − | 2.04719i | −0.721270 | + | 1.57473i | −2.15804 | + | 1.24594i | −0.706275 | − | 0.189246i | 3.61942 | + | 0.612770i | 2.22477 | − | 1.43193i | 0.737165 | + | 0.737165i | −1.95954 | − | 2.27161i | 1.54969i | ||
11.7 | −0.504901 | − | 1.88432i | −1.22779 | − | 1.22170i | −1.56368 | + | 0.902789i | −3.20968 | − | 0.860032i | −1.68215 | + | 2.93037i | 0.850713 | + | 2.50525i | −0.268188 | − | 0.268188i | 0.0149148 | + | 2.99996i | 6.48229i | ||
11.8 | −0.428681 | − | 1.59986i | 1.25532 | + | 1.19339i | −0.643729 | + | 0.371657i | 0.428701 | + | 0.114870i | 1.37112 | − | 2.51992i | 2.29163 | + | 1.32228i | −1.47180 | − | 1.47180i | 0.151659 | + | 2.99616i | − | 0.735103i | |
11.9 | −0.375380 | − | 1.40094i | −1.57757 | + | 0.715025i | −0.0896663 | + | 0.0517688i | 0.283932 | + | 0.0760792i | 1.59390 | + | 1.94168i | −2.00629 | + | 1.72476i | −1.94493 | − | 1.94493i | 1.97748 | − | 2.25601i | − | 0.426329i | |
11.10 | −0.329383 | − | 1.22927i | 0.639267 | − | 1.60976i | 0.329428 | − | 0.190195i | 1.27845 | + | 0.342559i | −2.18940 | − | 0.255606i | 2.63584 | + | 0.228814i | −2.14209 | − | 2.14209i | −2.18268 | − | 2.05814i | − | 1.68440i | |
11.11 | −0.305471 | − | 1.14003i | 1.47664 | + | 0.905280i | 0.525688 | − | 0.303506i | 1.64653 | + | 0.441185i | 0.580979 | − | 1.95996i | −1.12852 | − | 2.39300i | −2.17571 | − | 2.17571i | 1.36093 | + | 2.67355i | − | 2.01186i | |
11.12 | −0.250255 | − | 0.933963i | 0.138617 | + | 1.72650i | 0.922391 | − | 0.532543i | −4.14513 | − | 1.11068i | 1.57779 | − | 0.561527i | −2.32521 | − | 1.26230i | −2.09562 | − | 2.09562i | −2.96157 | + | 0.478645i | 4.14935i | ||
11.13 | −0.248591 | − | 0.927753i | −1.67503 | − | 0.440774i | 0.933123 | − | 0.538739i | 2.67091 | + | 0.715669i | 0.00746719 | + | 1.66358i | −1.47717 | − | 2.19498i | −2.09011 | − | 2.09011i | 2.61144 | + | 1.47662i | − | 2.65586i | |
11.14 | −0.179441 | − | 0.669682i | 0.594756 | − | 1.62673i | 1.31578 | − | 0.759664i | −1.48044 | − | 0.396684i | −1.19612 | − | 0.106395i | −2.58806 | + | 0.549511i | −1.72532 | − | 1.72532i | −2.29253 | − | 1.93502i | 1.06261i | ||
11.15 | −0.0262469 | − | 0.0979549i | −0.748033 | + | 1.56219i | 1.72314 | − | 0.994858i | 2.20563 | + | 0.590996i | 0.172658 | + | 0.0322707i | 1.71000 | − | 2.01889i | −0.286094 | − | 0.286094i | −1.88089 | − | 2.33714i | − | 0.231564i | |
11.16 | −0.00573734 | − | 0.0214121i | −1.43178 | − | 0.974676i | 1.73163 | − | 0.999754i | 2.15869 | + | 0.578419i | −0.0126552 | + | 0.0362495i | 1.44048 | + | 2.21924i | −0.0626912 | − | 0.0626912i | 1.10001 | + | 2.79105i | − | 0.0495405i | |
11.17 | 0.00573734 | + | 0.0214121i | −1.43178 | + | 0.974676i | 1.73163 | − | 0.999754i | −2.15869 | − | 0.578419i | −0.0290845 | − | 0.0250654i | 1.44048 | + | 2.21924i | 0.0626912 | + | 0.0626912i | 1.10001 | − | 2.79105i | − | 0.0495405i | |
11.18 | 0.0262469 | + | 0.0979549i | −0.748033 | − | 1.56219i | 1.72314 | − | 0.994858i | −2.20563 | − | 0.590996i | 0.133391 | − | 0.114276i | 1.71000 | − | 2.01889i | 0.286094 | + | 0.286094i | −1.88089 | + | 2.33714i | − | 0.231564i | |
11.19 | 0.179441 | + | 0.669682i | 0.594756 | + | 1.62673i | 1.31578 | − | 0.759664i | 1.48044 | + | 0.396684i | −0.982671 | + | 0.690199i | −2.58806 | + | 0.549511i | 1.72532 | + | 1.72532i | −2.29253 | + | 1.93502i | 1.06261i | ||
11.20 | 0.248591 | + | 0.927753i | −1.67503 | + | 0.440774i | 0.933123 | − | 0.538739i | −2.67091 | − | 0.715669i | −0.825325 | − | 1.44444i | −1.47717 | − | 2.19498i | 2.09011 | + | 2.09011i | 2.61144 | − | 1.47662i | − | 2.65586i | |
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
91.bd | odd | 12 | 1 | inner |
273.bw | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 273.2.bw.b | yes | 128 |
3.b | odd | 2 | 1 | inner | 273.2.bw.b | yes | 128 |
7.c | even | 3 | 1 | 273.2.bv.b | ✓ | 128 | |
13.f | odd | 12 | 1 | 273.2.bv.b | ✓ | 128 | |
21.h | odd | 6 | 1 | 273.2.bv.b | ✓ | 128 | |
39.k | even | 12 | 1 | 273.2.bv.b | ✓ | 128 | |
91.bd | odd | 12 | 1 | inner | 273.2.bw.b | yes | 128 |
273.bw | even | 12 | 1 | inner | 273.2.bw.b | yes | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.bv.b | ✓ | 128 | 7.c | even | 3 | 1 | |
273.2.bv.b | ✓ | 128 | 13.f | odd | 12 | 1 | |
273.2.bv.b | ✓ | 128 | 21.h | odd | 6 | 1 | |
273.2.bv.b | ✓ | 128 | 39.k | even | 12 | 1 | |
273.2.bw.b | yes | 128 | 1.a | even | 1 | 1 | trivial |
273.2.bw.b | yes | 128 | 3.b | odd | 2 | 1 | inner |
273.2.bw.b | yes | 128 | 91.bd | odd | 12 | 1 | inner |
273.2.bw.b | yes | 128 | 273.bw | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} + 6 T_{2}^{126} - 208 T_{2}^{124} - 1320 T_{2}^{122} + 25278 T_{2}^{120} + 163512 T_{2}^{118} + \cdots + 269517889 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).