Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [460,2,Mod(3,460)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, base_ring=CyclotomicField(44))
chi = DirichletCharacter(H, H._module([22, 33, 32]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("460.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 460 = 2^{2} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 460.w (of order \(44\), degree \(20\), 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: | \(3.67311849298\) |
Analytic rank: | \(0\) |
Dimension: | \(1360\) |
Relative dimension: | \(68\) over \(\Q(\zeta_{44})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{44}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −1.41421 | − | 0.00122474i | 0.959477 | + | 2.57246i | 2.00000 | + | 0.00346408i | 1.75727 | − | 1.38275i | −1.35375 | − | 3.63918i | 0.698270 | + | 0.151899i | −2.82842 | − | 0.00734842i | −3.42969 | + | 2.97185i | −2.48685 | + | 1.95335i |
3.2 | −1.40653 | + | 0.147185i | −0.487655 | − | 1.30745i | 1.95667 | − | 0.414043i | 0.154780 | − | 2.23070i | 0.878342 | + | 1.76720i | −0.318907 | − | 0.0693739i | −2.69118 | + | 0.870359i | 0.795620 | − | 0.689408i | 0.110624 | + | 3.16034i |
3.3 | −1.40026 | + | 0.198151i | 0.585996 | + | 1.57112i | 1.92147 | − | 0.554927i | −2.01963 | − | 0.959740i | −1.13187 | − | 2.08386i | 4.26825 | + | 0.928502i | −2.58061 | + | 1.15779i | 0.142232 | − | 0.123245i | 3.01818 | + | 0.943697i |
3.4 | −1.39894 | − | 0.207265i | −0.826183 | − | 2.21508i | 1.91408 | + | 0.579903i | −2.13268 | + | 0.672078i | 0.696675 | + | 3.27001i | −2.06386 | − | 0.448965i | −2.55750 | − | 1.20797i | −1.95677 | + | 1.69555i | 3.12279 | − | 0.498170i |
3.5 | −1.37680 | + | 0.323160i | 0.184673 | + | 0.495127i | 1.79114 | − | 0.889851i | −1.56442 | + | 1.59768i | −0.414262 | − | 0.622010i | −2.72912 | − | 0.593685i | −2.17846 | + | 1.80397i | 2.05620 | − | 1.78171i | 1.63758 | − | 2.70524i |
3.6 | −1.37121 | + | 0.346108i | −0.338395 | − | 0.907273i | 1.76042 | − | 0.949171i | 1.81802 | + | 1.30185i | 0.778024 | + | 1.12694i | 1.65566 | + | 0.360166i | −2.08538 | + | 1.91080i | 1.55862 | − | 1.35055i | −2.94346 | − | 1.15588i |
3.7 | −1.35719 | − | 0.397537i | 0.361954 | + | 0.970436i | 1.68393 | + | 1.07907i | −0.0598253 | − | 2.23527i | −0.105456 | − | 1.46096i | −3.20911 | − | 0.698099i | −1.85644 | − | 2.13392i | 1.45651 | − | 1.26208i | −0.807408 | + | 3.05746i |
3.8 | −1.32376 | − | 0.497658i | 0.189809 | + | 0.508896i | 1.50467 | + | 1.31756i | 0.243658 | + | 2.22275i | 0.00199564 | − | 0.768116i | 3.80918 | + | 0.828636i | −1.33613 | − | 2.49294i | 2.04430 | − | 1.77140i | 0.783626 | − | 3.06365i |
3.9 | −1.29922 | − | 0.558604i | −0.594892 | − | 1.59497i | 1.37592 | + | 1.45149i | 2.23464 | − | 0.0798607i | −0.118062 | + | 2.40452i | −1.24509 | − | 0.270853i | −0.976810 | − | 2.65440i | 0.0772235 | − | 0.0669146i | −2.94789 | − | 1.14452i |
3.10 | −1.28874 | − | 0.582374i | 1.09678 | + | 2.94057i | 1.32168 | + | 1.50105i | 0.110052 | + | 2.23336i | 0.299057 | − | 4.42835i | −1.66107 | − | 0.361343i | −0.829123 | − | 2.70417i | −5.17679 | + | 4.48572i | 1.15882 | − | 2.94230i |
3.11 | −1.19585 | + | 0.754944i | 0.865316 | + | 2.32000i | 0.860120 | − | 1.80560i | −0.780937 | + | 2.09527i | −2.78626 | − | 2.12111i | −0.367846 | − | 0.0800199i | 0.334552 | + | 2.80857i | −2.36639 | + | 2.05049i | −0.647923 | − | 3.09519i |
3.12 | −1.18443 | + | 0.772733i | −1.12425 | − | 3.01422i | 0.805766 | − | 1.83050i | 1.84293 | + | 1.26634i | 3.66078 | + | 2.70140i | −3.65616 | − | 0.795348i | 0.460113 | + | 2.79075i | −5.55434 | + | 4.81286i | −3.16137 | − | 0.0757989i |
3.13 | −1.13266 | − | 0.846813i | −0.616030 | − | 1.65164i | 0.565816 | + | 1.91829i | −1.87330 | − | 1.22097i | −0.700881 | + | 2.39240i | 2.66779 | + | 0.580342i | 0.983562 | − | 2.65191i | −0.0811762 | + | 0.0703396i | 1.08787 | + | 2.96927i |
3.14 | −1.13162 | − | 0.848200i | 0.616030 | + | 1.65164i | 0.561114 | + | 1.91967i | −1.87330 | − | 1.22097i | 0.703812 | − | 2.39154i | −2.66779 | − | 0.580342i | 0.993302 | − | 2.64827i | −0.0811762 | + | 0.0703396i | 1.08423 | + | 2.97060i |
3.15 | −1.11682 | + | 0.867590i | −0.106315 | − | 0.285041i | 0.494574 | − | 1.93788i | 1.46736 | − | 1.68726i | 0.366034 | + | 0.226102i | 3.49451 | + | 0.760183i | 1.12894 | + | 2.59336i | 2.19730 | − | 1.90397i | −0.174928 | + | 3.15744i |
3.16 | −1.06351 | + | 0.932176i | 0.317081 | + | 0.850128i | 0.262098 | − | 1.98275i | 2.06623 | − | 0.854811i | −1.12969 | − | 0.608542i | −4.65012 | − | 1.01157i | 1.56953 | + | 2.35299i | 1.64507 | − | 1.42546i | −1.40062 | + | 2.83519i |
3.17 | −1.04207 | + | 0.956077i | −0.816083 | − | 2.18800i | 0.171833 | − | 1.99260i | −2.19626 | + | 0.420065i | 2.94232 | + | 1.49982i | 3.93457 | + | 0.855914i | 1.72602 | + | 2.24073i | −1.85412 | + | 1.60660i | 1.88705 | − | 2.53753i |
3.18 | −1.03221 | + | 0.966716i | 1.10382 | + | 2.95946i | 0.130921 | − | 1.99571i | −1.01263 | − | 1.99363i | −4.00033 | − | 1.98770i | −1.74183 | − | 0.378912i | 1.79415 | + | 2.18656i | −5.27271 | + | 4.56883i | 2.97253 | + | 1.07892i |
3.19 | −0.921863 | − | 1.07246i | −1.09678 | − | 2.94057i | −0.300338 | + | 1.97732i | 0.110052 | + | 2.23336i | −2.14257 | + | 3.88705i | 1.66107 | + | 0.361343i | 2.39747 | − | 1.50072i | −5.17679 | + | 4.48572i | 2.29373 | − | 2.17688i |
3.20 | −0.902008 | − | 1.08921i | 0.594892 | + | 1.59497i | −0.372763 | + | 1.96495i | 2.23464 | − | 0.0798607i | 1.20066 | − | 2.08664i | 1.24509 | + | 0.270853i | 2.47649 | − | 1.36639i | 0.0772235 | − | 0.0669146i | −2.10265 | − | 2.36196i |
See next 80 embeddings (of 1360 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
20.e | even | 4 | 1 | inner |
23.c | even | 11 | 1 | inner |
92.g | odd | 22 | 1 | inner |
115.k | odd | 44 | 1 | inner |
460.w | even | 44 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 460.2.w.a | ✓ | 1360 |
4.b | odd | 2 | 1 | inner | 460.2.w.a | ✓ | 1360 |
5.c | odd | 4 | 1 | inner | 460.2.w.a | ✓ | 1360 |
20.e | even | 4 | 1 | inner | 460.2.w.a | ✓ | 1360 |
23.c | even | 11 | 1 | inner | 460.2.w.a | ✓ | 1360 |
92.g | odd | 22 | 1 | inner | 460.2.w.a | ✓ | 1360 |
115.k | odd | 44 | 1 | inner | 460.2.w.a | ✓ | 1360 |
460.w | even | 44 | 1 | inner | 460.2.w.a | ✓ | 1360 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
460.2.w.a | ✓ | 1360 | 1.a | even | 1 | 1 | trivial |
460.2.w.a | ✓ | 1360 | 4.b | odd | 2 | 1 | inner |
460.2.w.a | ✓ | 1360 | 5.c | odd | 4 | 1 | inner |
460.2.w.a | ✓ | 1360 | 20.e | even | 4 | 1 | inner |
460.2.w.a | ✓ | 1360 | 23.c | even | 11 | 1 | inner |
460.2.w.a | ✓ | 1360 | 92.g | odd | 22 | 1 | inner |
460.2.w.a | ✓ | 1360 | 115.k | odd | 44 | 1 | inner |
460.2.w.a | ✓ | 1360 | 460.w | even | 44 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(460, [\chi])\).