Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [475,2,Mod(2,475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(180))
chi = DirichletCharacter(H, H._module([9, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("475.2");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 475.bi (of order \(180\), degree \(48\), 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: | \(2304\) |
Relative dimension: | \(48\) over \(\Q(\zeta_{180})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{180}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −2.62434 | − | 0.903631i | −2.47135 | − | 0.570556i | 4.49457 | + | 3.51154i | 1.68444 | − | 1.47060i | 5.97008 | + | 3.73052i | 0.0634865 | − | 0.236935i | −5.59878 | − | 8.62136i | 3.08565 | + | 1.50497i | −5.74942 | + | 2.33723i |
2.2 | −2.56561 | − | 0.883409i | 0.702914 | + | 0.162281i | 4.22590 | + | 3.30163i | −1.98966 | + | 1.02042i | −1.66004 | − | 1.03731i | 0.919986 | − | 3.43344i | −4.96961 | − | 7.65253i | −2.22863 | − | 1.08697i | 6.00613 | − | 0.860317i |
2.3 | −2.37287 | − | 0.817046i | 3.28556 | + | 0.758531i | 3.38695 | + | 2.64617i | −0.200815 | − | 2.22703i | −7.17646 | − | 4.48435i | 0.308540 | − | 1.15149i | −3.14110 | − | 4.83687i | 7.52313 | + | 3.66928i | −1.34308 | + | 5.44854i |
2.4 | −2.35508 | − | 0.810920i | 0.00625138 | + | 0.00144324i | 3.31280 | + | 2.58824i | −1.87745 | − | 1.21457i | −0.0135522 | − | 0.00846832i | −1.26696 | + | 4.72836i | −2.98990 | − | 4.60405i | −2.69635 | − | 1.31510i | 3.43662 | + | 4.38288i |
2.5 | −2.28668 | − | 0.787366i | 0.766057 | + | 0.176858i | 3.03292 | + | 2.36958i | 2.20863 | + | 0.349212i | −1.61247 | − | 1.00759i | 0.287418 | − | 1.07266i | −2.43524 | − | 3.74994i | −2.14082 | − | 1.04415i | −4.77547 | − | 2.53753i |
2.6 | −2.23217 | − | 0.768598i | −2.11065 | − | 0.487282i | 2.81582 | + | 2.19996i | 0.259366 | + | 2.22097i | 4.33681 | + | 2.70994i | −0.637349 | + | 2.37862i | −2.02295 | − | 3.11507i | 1.52102 | + | 0.741853i | 1.12809 | − | 5.15694i |
2.7 | −2.15383 | − | 0.741624i | 2.53352 | + | 0.584910i | 2.51296 | + | 1.96334i | −0.432784 | + | 2.19379i | −5.02300 | − | 3.13872i | −0.361347 | + | 1.34856i | −1.47513 | − | 2.27150i | 3.38023 | + | 1.64865i | 2.55911 | − | 4.40408i |
2.8 | −2.11303 | − | 0.727576i | −2.93221 | − | 0.676953i | 2.35952 | + | 1.84346i | −1.89401 | + | 1.18859i | 5.70331 | + | 3.56382i | 0.732295 | − | 2.73296i | −1.21018 | − | 1.86352i | 5.44318 | + | 2.65482i | 4.86688 | − | 1.13350i |
2.9 | −1.79315 | − | 0.617429i | −0.497419 | − | 0.114838i | 1.25813 | + | 0.982958i | 1.00275 | + | 1.99862i | 0.821041 | + | 0.513043i | 0.500493 | − | 1.86786i | 0.416679 | + | 0.641630i | −2.46214 | − | 1.20087i | −0.564078 | − | 4.20295i |
2.10 | −1.68202 | − | 0.579164i | −1.00903 | − | 0.232952i | 0.917723 | + | 0.717004i | 2.04123 | − | 0.912897i | 1.56228 | + | 0.976222i | −0.783209 | + | 2.92298i | 0.809393 | + | 1.24636i | −1.73251 | − | 0.845003i | −3.96210 | + | 0.353300i |
2.11 | −1.65294 | − | 0.569153i | 1.76866 | + | 0.408326i | 0.832254 | + | 0.650228i | 0.672217 | − | 2.13263i | −2.69108 | − | 1.68157i | −0.575171 | + | 2.14657i | 0.898671 | + | 1.38383i | 0.265032 | + | 0.129265i | −2.32493 | + | 3.14252i |
2.12 | −1.63974 | − | 0.564606i | −0.536741 | − | 0.123916i | 0.793930 | + | 0.620286i | −0.505939 | − | 2.17808i | 0.810149 | + | 0.506237i | 1.26758 | − | 4.73067i | 0.937428 | + | 1.44351i | −2.42365 | − | 1.18209i | −0.400151 | + | 3.85713i |
2.13 | −1.53158 | − | 0.527365i | 1.56218 | + | 0.360658i | 0.491602 | + | 0.384081i | −2.11036 | − | 0.739181i | −2.20241 | − | 1.37622i | −0.0254288 | + | 0.0949017i | 1.21407 | + | 1.86950i | −0.386047 | − | 0.188288i | 2.84236 | + | 2.24504i |
2.14 | −1.31509 | − | 0.452821i | 2.79569 | + | 0.645436i | −0.0516128 | − | 0.0403243i | 1.94765 | + | 1.09847i | −3.38431 | − | 2.11475i | 1.08161 | − | 4.03663i | 1.56465 | + | 2.40935i | 4.70291 | + | 2.29376i | −2.06393 | − | 2.32653i |
2.15 | −1.18387 | − | 0.407639i | −3.17084 | − | 0.732046i | −0.340643 | − | 0.266139i | −0.821350 | − | 2.07976i | 3.45545 | + | 2.15921i | −0.594087 | + | 2.21716i | 1.65866 | + | 2.55411i | 6.82196 | + | 3.32729i | 0.124582 | + | 2.79698i |
2.16 | −1.15799 | − | 0.398726i | −2.70353 | − | 0.624158i | −0.394073 | − | 0.307884i | 2.13344 | − | 0.669642i | 2.88178 | + | 1.80073i | 0.726074 | − | 2.70974i | 1.66762 | + | 2.56791i | 4.22309 | + | 2.05974i | −2.73750 | − | 0.0752243i |
2.17 | −1.08643 | − | 0.374089i | −1.56658 | − | 0.361674i | −0.535624 | − | 0.418476i | −1.77819 | + | 1.35575i | 1.56669 | + | 0.978977i | −0.847061 | + | 3.16128i | 1.67699 | + | 2.58234i | −0.373010 | − | 0.181929i | 2.43905 | − | 0.807730i |
2.18 | −0.967384 | − | 0.333097i | −0.527689 | − | 0.121827i | −0.751143 | − | 0.586857i | −2.21092 | + | 0.334424i | 0.469898 | + | 0.293625i | 0.292070 | − | 1.09002i | 1.64563 | + | 2.53405i | −2.43277 | − | 1.18654i | 2.25020 | + | 0.412934i |
2.19 | −0.865537 | − | 0.298028i | 2.22234 | + | 0.513069i | −0.915688 | − | 0.715414i | 2.21532 | − | 0.303886i | −1.77061 | − | 1.10640i | −0.390977 | + | 1.45915i | 1.57648 | + | 2.42757i | 1.97919 | + | 0.965317i | −2.00801 | − | 0.397204i |
2.20 | −0.507373 | − | 0.174703i | −2.17799 | − | 0.502828i | −1.34912 | − | 1.05404i | 1.04478 | + | 1.97698i | 1.01721 | + | 0.635621i | 0.724035 | − | 2.70213i | 1.08488 | + | 1.67056i | 1.79441 | + | 0.875194i | −0.184708 | − | 1.18559i |
See next 80 embeddings (of 2304 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.f | odd | 18 | 1 | inner |
25.f | odd | 20 | 1 | inner |
475.bi | even | 180 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 475.2.bi.a | ✓ | 2304 |
19.f | odd | 18 | 1 | inner | 475.2.bi.a | ✓ | 2304 |
25.f | odd | 20 | 1 | inner | 475.2.bi.a | ✓ | 2304 |
475.bi | even | 180 | 1 | inner | 475.2.bi.a | ✓ | 2304 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
475.2.bi.a | ✓ | 2304 | 1.a | even | 1 | 1 | trivial |
475.2.bi.a | ✓ | 2304 | 19.f | odd | 18 | 1 | inner |
475.2.bi.a | ✓ | 2304 | 25.f | odd | 20 | 1 | inner |
475.2.bi.a | ✓ | 2304 | 475.bi | even | 180 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(475, [\chi])\).