Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [663,2,Mod(110,663)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(663, base_ring=CyclotomicField(24))
chi = DirichletCharacter(H, H._module([12, 10, 15]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("663.110");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 663 = 3 \cdot 13 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 663.cf (of order \(24\), degree \(8\), 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: | \(5.29408165401\) |
Analytic rank: | \(0\) |
Dimension: | \(640\) |
Relative dimension: | \(80\) over \(\Q(\zeta_{24})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{24}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
110.1 | −1.36681 | + | 2.36739i | 0.0409062 | + | 1.73157i | −2.73634 | − | 4.73949i | 1.62588 | + | 3.92522i | −4.15520 | − | 2.26988i | 0.172306 | + | 1.30879i | 9.49301 | −2.99665 | + | 0.141664i | −11.5148 | − | 1.51595i | ||
110.2 | −1.35757 | + | 2.35138i | 0.549761 | − | 1.64249i | −2.68600 | − | 4.65229i | 0.528427 | + | 1.27574i | 3.11578 | + | 3.52249i | 0.526117 | + | 3.99626i | 9.15548 | −2.39553 | − | 1.80595i | −3.71712 | − | 0.489368i | ||
110.3 | −1.32818 | + | 2.30048i | −1.30504 | − | 1.13880i | −2.52813 | − | 4.37886i | 0.291822 | + | 0.704520i | 4.35312 | − | 1.48968i | −0.218505 | − | 1.65971i | 8.11856 | 0.406266 | + | 2.97236i | −2.00833 | − | 0.264401i | ||
110.4 | −1.31688 | + | 2.28090i | 1.56064 | + | 0.751258i | −2.46834 | − | 4.27530i | 0.0697335 | + | 0.168352i | −3.76873 | + | 2.57036i | −0.518512 | − | 3.93849i | 7.73453 | 1.87122 | + | 2.34489i | −0.475824 | − | 0.0626434i | ||
110.5 | −1.28851 | + | 2.23177i | 1.10924 | − | 1.33026i | −2.32052 | − | 4.01926i | −1.20903 | − | 2.91885i | 1.53955 | + | 4.18962i | −0.227238 | − | 1.72605i | 6.80601 | −0.539168 | − | 2.95115i | 8.07205 | + | 1.06271i | ||
110.6 | −1.27834 | + | 2.21416i | −1.40104 | + | 1.01838i | −2.26833 | − | 3.92886i | −0.553348 | − | 1.33590i | −0.463851 | − | 4.40395i | −0.307457 | − | 2.33537i | 6.48543 | 0.925799 | − | 2.85358i | 3.66526 | + | 0.482540i | ||
110.7 | −1.20385 | + | 2.08512i | 0.662912 | + | 1.60017i | −1.89850 | − | 3.28829i | −1.11848 | − | 2.70024i | −4.13460 | − | 0.544107i | 0.0587604 | + | 0.446330i | 4.32661 | −2.12110 | + | 2.12155i | 6.97681 | + | 0.918515i | ||
110.8 | −1.17604 | + | 2.03696i | −1.67248 | + | 0.450352i | −1.76613 | − | 3.05902i | 0.521848 | + | 1.25985i | 1.04955 | − | 3.93640i | 0.232315 | + | 1.76460i | 3.60397 | 2.59437 | − | 1.50641i | −3.17998 | − | 0.418652i | ||
110.9 | −1.17216 | + | 2.03024i | −0.556238 | + | 1.64030i | −1.74791 | − | 3.02746i | −0.821377 | − | 1.98298i | −2.67821 | − | 3.05199i | 0.524203 | + | 3.98172i | 3.50665 | −2.38120 | − | 1.82480i | 4.98870 | + | 0.656775i | ||
110.10 | −1.14218 | + | 1.97832i | −1.42818 | − | 0.979946i | −1.60917 | − | 2.78716i | 0.593397 | + | 1.43259i | 3.56990 | − | 1.70612i | 0.0392968 | + | 0.298488i | 2.78313 | 1.07941 | + | 2.79908i | −3.51188 | − | 0.462348i | ||
110.11 | −1.10600 | + | 1.91565i | 1.69205 | − | 0.370072i | −1.44647 | − | 2.50536i | 1.00569 | + | 2.42794i | −1.16248 | + | 3.65068i | −0.315622 | − | 2.39739i | 1.97518 | 2.72609 | − | 1.25236i | −5.76337 | − | 0.758762i | ||
110.12 | −1.09736 | + | 1.90069i | 1.73138 | + | 0.0481161i | −1.40841 | − | 2.43944i | 0.840948 | + | 2.03023i | −1.99141 | + | 3.23802i | 0.524066 | + | 3.98068i | 1.79270 | 2.99537 | + | 0.166615i | −4.78166 | − | 0.629517i | ||
110.13 | −1.03251 | + | 1.78836i | −0.447337 | − | 1.67329i | −1.13216 | − | 1.96096i | −0.690499 | − | 1.66701i | 3.45433 | + | 0.927688i | 0.340463 | + | 2.58607i | 0.545839 | −2.59978 | + | 1.49705i | 3.69417 | + | 0.486347i | ||
110.14 | −0.997743 | + | 1.72814i | 1.20097 | − | 1.24806i | −0.990983 | − | 1.71643i | −0.412947 | − | 0.996942i | 0.958568 | + | 3.32070i | −0.0921696 | − | 0.700097i | −0.0359885 | −0.115326 | − | 2.99778i | 2.13487 | + | 0.281061i | ||
110.15 | −0.977494 | + | 1.69307i | 0.390475 | − | 1.68746i | −0.910988 | − | 1.57788i | 1.61273 | + | 3.89347i | 2.47530 | + | 2.31058i | −0.259830 | − | 1.97361i | −0.348037 | −2.69506 | − | 1.31782i | −8.16835 | − | 1.07538i | ||
110.16 | −0.938225 | + | 1.62505i | −0.0494020 | + | 1.73135i | −0.760531 | − | 1.31728i | 0.367649 | + | 0.887583i | −2.76718 | − | 1.70467i | −0.497341 | − | 3.77768i | −0.898702 | −2.99512 | − | 0.171064i | −1.78731 | − | 0.235303i | ||
110.17 | −0.861486 | + | 1.49214i | 1.33159 | + | 1.10764i | −0.484315 | − | 0.838859i | 0.264202 | + | 0.637841i | −2.79990 | + | 1.03270i | 0.127074 | + | 0.965223i | −1.77702 | 0.546262 | + | 2.94985i | −1.17935 | − | 0.155265i | ||
110.18 | −0.842846 | + | 1.45985i | 1.71216 | − | 0.261710i | −0.420780 | − | 0.728812i | −1.12550 | − | 2.71721i | −1.06103 | + | 2.72009i | 0.277857 | + | 2.11053i | −1.95277 | 2.86302 | − | 0.896182i | 4.91535 | + | 0.647118i | ||
110.19 | −0.836096 | + | 1.44816i | −1.72422 | − | 0.164558i | −0.398114 | − | 0.689553i | −1.20374 | − | 2.90608i | 1.67992 | − | 2.35936i | 0.598949 | + | 4.54947i | −2.01294 | 2.94584 | + | 0.567468i | 5.21492 | + | 0.686557i | ||
110.20 | −0.789656 | + | 1.36772i | −1.07437 | + | 1.35858i | −0.247113 | − | 0.428012i | 1.33211 | + | 3.21599i | −1.00978 | − | 2.54225i | 0.167968 | + | 1.27585i | −2.37809 | −0.691459 | − | 2.91923i | −5.45050 | − | 0.717571i | ||
See next 80 embeddings (of 640 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
221.bf | odd | 24 | 1 | inner |
663.cf | even | 24 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 663.2.cf.a | ✓ | 640 |
3.b | odd | 2 | 1 | inner | 663.2.cf.a | ✓ | 640 |
13.f | odd | 12 | 1 | 663.2.ck.a | yes | 640 | |
17.d | even | 8 | 1 | 663.2.ck.a | yes | 640 | |
39.k | even | 12 | 1 | 663.2.ck.a | yes | 640 | |
51.g | odd | 8 | 1 | 663.2.ck.a | yes | 640 | |
221.bf | odd | 24 | 1 | inner | 663.2.cf.a | ✓ | 640 |
663.cf | even | 24 | 1 | inner | 663.2.cf.a | ✓ | 640 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
663.2.cf.a | ✓ | 640 | 1.a | even | 1 | 1 | trivial |
663.2.cf.a | ✓ | 640 | 3.b | odd | 2 | 1 | inner |
663.2.cf.a | ✓ | 640 | 221.bf | odd | 24 | 1 | inner |
663.2.cf.a | ✓ | 640 | 663.cf | even | 24 | 1 | inner |
663.2.ck.a | yes | 640 | 13.f | odd | 12 | 1 | |
663.2.ck.a | yes | 640 | 17.d | even | 8 | 1 | |
663.2.ck.a | yes | 640 | 39.k | even | 12 | 1 | |
663.2.ck.a | yes | 640 | 51.g | odd | 8 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(663, [\chi])\).