Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [475,2,Mod(6,475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(90))
chi = DirichletCharacter(H, H._module([36, 70]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("475.6");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 475.bc (of order \(45\), degree \(24\), 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: | \(3.79289409601\) |
Analytic rank: | \(0\) |
Dimension: | \(1152\) |
Relative dimension: | \(48\) over \(\Q(\zeta_{45})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{45}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6.1 | −2.29973 | − | 1.43703i | 2.30476 | − | 0.660880i | 2.34697 | + | 4.81199i | 2.22377 | − | 0.234179i | −6.25005 | − | 1.79217i | 1.23280 | − | 2.13527i | 0.950677 | − | 9.04509i | 2.33103 | − | 1.45659i | −5.45060 | − | 2.65708i |
6.2 | −2.29691 | − | 1.43527i | −2.18186 | + | 0.625639i | 2.33907 | + | 4.79580i | 1.19746 | + | 1.88841i | 5.90951 | + | 1.69453i | 0.140645 | − | 0.243604i | 0.944410 | − | 8.98546i | 1.82496 | − | 1.14036i | −0.0400903 | − | 6.05619i |
6.3 | −2.19199 | − | 1.36971i | −1.37858 | + | 0.395301i | 2.05198 | + | 4.20718i | −2.03188 | − | 0.933532i | 3.56327 | + | 1.02175i | 1.36546 | − | 2.36504i | 0.724330 | − | 6.89154i | −0.799932 | + | 0.499853i | 3.17519 | + | 4.82937i |
6.4 | −2.08209 | − | 1.30103i | 0.278378 | − | 0.0798235i | 1.76566 | + | 3.62014i | 1.47314 | − | 1.68222i | −0.683459 | − | 0.195979i | −1.61228 | + | 2.79255i | 0.520393 | − | 4.95121i | −2.47302 | + | 1.54532i | −5.25582 | + | 1.58594i |
6.5 | −2.04037 | − | 1.27497i | 2.34026 | − | 0.671059i | 1.66084 | + | 3.40523i | −1.72152 | − | 1.42702i | −5.63058 | − | 1.61454i | −2.03230 | + | 3.52005i | 0.449834 | − | 4.27989i | 2.48235 | − | 1.55115i | 1.69313 | + | 5.10653i |
6.6 | −1.91392 | − | 1.19595i | −2.99079 | + | 0.857597i | 1.35606 | + | 2.78033i | 0.364394 | − | 2.20618i | 6.74979 | + | 1.93547i | −1.35748 | + | 2.35123i | 0.257940 | − | 2.45413i | 5.66524 | − | 3.54003i | −3.33590 | + | 3.78666i |
6.7 | −1.89200 | − | 1.18225i | 2.66437 | − | 0.763997i | 1.30520 | + | 2.67605i | −2.06433 | + | 0.859385i | −5.94423 | − | 1.70448i | 2.28413 | − | 3.95623i | 0.227928 | − | 2.16859i | 3.97105 | − | 2.48139i | 4.92172 | + | 0.814604i |
6.8 | −1.79524 | − | 1.12179i | −0.560140 | + | 0.160617i | 1.08772 | + | 2.23017i | −1.94606 | + | 1.10130i | 1.18576 | + | 0.340012i | −1.52823 | + | 2.64697i | 0.106498 | − | 1.01326i | −2.25619 | + | 1.40982i | 4.72906 | + | 0.205964i |
6.9 | −1.76581 | − | 1.10340i | −0.597589 | + | 0.171356i | 1.02386 | + | 2.09922i | 1.42897 | − | 1.71990i | 1.24431 | + | 0.356799i | 1.89729 | − | 3.28621i | 0.0730432 | − | 0.694959i | −2.21639 | + | 1.38496i | −4.42104 | + | 1.46028i |
6.10 | −1.68487 | − | 1.05282i | 1.58487 | − | 0.454455i | 0.853608 | + | 1.75016i | 1.47769 | + | 1.67822i | −3.14876 | − | 0.902893i | −0.289478 | + | 0.501391i | −0.0109594 | + | 0.104272i | −0.238856 | + | 0.149254i | −0.722847 | − | 4.38334i |
6.11 | −1.48496 | − | 0.927909i | −0.380214 | + | 0.109025i | 0.467363 | + | 0.958236i | 0.439600 | + | 2.19243i | 0.665769 | + | 0.190906i | 1.66674 | − | 2.88687i | −0.170928 | + | 1.62627i | −2.41147 | + | 1.50685i | 1.38158 | − | 3.66359i |
6.12 | −1.42766 | − | 0.892103i | −3.13595 | + | 0.899221i | 0.365632 | + | 0.749656i | −1.46779 | + | 1.68689i | 5.27928 | + | 1.51381i | −0.405461 | + | 0.702280i | −0.205169 | + | 1.95206i | 6.48147 | − | 4.05007i | 3.60039 | − | 1.09889i |
6.13 | −1.30322 | − | 0.814342i | 1.85231 | − | 0.531140i | 0.158485 | + | 0.324943i | −0.982153 | − | 2.00882i | −2.84649 | − | 0.816218i | 0.367774 | − | 0.637004i | −0.263190 | + | 2.50409i | 0.604785 | − | 0.377912i | −0.355909 | + | 3.41775i |
6.14 | −1.05096 | − | 0.656713i | 3.06632 | − | 0.879252i | −0.203496 | − | 0.417228i | 1.91224 | − | 1.15903i | −3.79999 | − | 1.08963i | −0.887411 | + | 1.53704i | −0.319211 | + | 3.03709i | 6.08506 | − | 3.80237i | −2.77084 | − | 0.0377008i |
6.15 | −0.978277 | − | 0.611295i | −2.30987 | + | 0.662346i | −0.293398 | − | 0.601556i | 2.15159 | − | 0.608806i | 2.66459 | + | 0.764058i | 0.629622 | − | 1.09054i | −0.321864 | + | 3.06233i | 2.35267 | − | 1.47011i | −2.47701 | − | 0.719678i |
6.16 | −0.954295 | − | 0.596310i | −0.197845 | + | 0.0567310i | −0.321648 | − | 0.659477i | −2.23191 | + | 0.136321i | 0.222632 | + | 0.0638386i | 0.295393 | − | 0.511636i | −0.321554 | + | 3.05938i | −2.50822 | + | 1.56731i | 2.21119 | + | 1.20082i |
6.17 | −0.918651 | − | 0.574037i | −1.08630 | + | 0.311492i | −0.362341 | − | 0.742910i | −0.569939 | − | 2.16221i | 1.17674 | + | 0.337424i | −1.35582 | + | 2.34835i | −0.320054 | + | 3.04511i | −1.46112 | + | 0.913011i | −0.717615 | + | 2.31348i |
6.18 | −0.764890 | − | 0.477956i | −1.97717 | + | 0.566943i | −0.520128 | − | 1.06642i | 0.973170 | + | 2.01319i | 1.78329 | + | 0.511350i | −2.15040 | + | 3.72461i | −0.300419 | + | 2.85829i | 1.04362 | − | 0.652125i | 0.217850 | − | 2.00500i |
6.19 | −0.730424 | − | 0.456420i | −0.0193247 | + | 0.00554125i | −0.551542 | − | 1.13083i | 2.18285 | − | 0.484947i | 0.0166443 | + | 0.00477268i | −1.43830 | + | 2.49120i | −0.293334 | + | 2.79088i | −2.54380 | + | 1.58954i | −1.81574 | − | 0.642078i |
6.20 | −0.695663 | − | 0.434699i | −2.62473 | + | 0.752630i | −0.581758 | − | 1.19278i | −1.90436 | − | 1.17193i | 2.15310 | + | 0.617391i | 2.44900 | − | 4.24179i | −0.285284 | + | 2.71430i | 3.77862 | − | 2.36115i | 0.815352 | + | 1.64309i |
See next 80 embeddings (of 1152 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
25.d | even | 5 | 1 | inner |
475.bc | even | 45 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 475.2.bc.a | ✓ | 1152 |
19.e | even | 9 | 1 | inner | 475.2.bc.a | ✓ | 1152 |
25.d | even | 5 | 1 | inner | 475.2.bc.a | ✓ | 1152 |
475.bc | even | 45 | 1 | inner | 475.2.bc.a | ✓ | 1152 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
475.2.bc.a | ✓ | 1152 | 1.a | even | 1 | 1 | trivial |
475.2.bc.a | ✓ | 1152 | 19.e | even | 9 | 1 | inner |
475.2.bc.a | ✓ | 1152 | 25.d | even | 5 | 1 | inner |
475.2.bc.a | ✓ | 1152 | 475.bc | even | 45 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(475, [\chi])\).