Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,3,Mod(31,175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([12, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("175.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 175.v (of order \(30\), 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: | \(4.76840462631\) |
Analytic rank: | \(0\) |
Dimension: | \(304\) |
Relative dimension: | \(38\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −2.49483 | + | 2.77079i | −4.88519 | + | 0.513454i | −1.03498 | − | 9.84720i | 0.0370158 | − | 4.99986i | 10.7650 | − | 14.8168i | −6.93002 | + | 0.987337i | 17.8011 | + | 12.9332i | 14.7981 | − | 3.14543i | 13.7612 | + | 12.5764i |
31.2 | −2.46635 | + | 2.73916i | 3.90575 | − | 0.410511i | −1.00199 | − | 9.53333i | −4.87433 | + | 1.11397i | −8.50848 | + | 11.7109i | −5.89171 | − | 3.77992i | 16.6567 | + | 12.1018i | 6.28303 | − | 1.33550i | 8.97045 | − | 16.0990i |
31.3 | −2.40308 | + | 2.66889i | −0.888631 | + | 0.0933988i | −0.930071 | − | 8.84903i | 2.13906 | + | 4.51934i | 1.88618 | − | 2.59610i | −0.0558604 | + | 6.99978i | 14.2303 | + | 10.3389i | −8.02239 | + | 1.70521i | −17.2020 | − | 5.15141i |
31.4 | −2.23580 | + | 2.48310i | −0.901848 | + | 0.0947881i | −0.748904 | − | 7.12534i | −4.52006 | − | 2.13753i | 1.78098 | − | 2.45131i | 6.60425 | − | 2.32031i | 8.55454 | + | 6.21524i | −7.99898 | + | 1.70024i | 15.4136 | − | 6.44471i |
31.5 | −2.10006 | + | 2.33235i | 3.01791 | − | 0.317195i | −0.611504 | − | 5.81807i | 3.10424 | − | 3.91965i | −5.59798 | + | 7.70496i | −5.63331 | + | 4.15521i | 4.69763 | + | 3.41303i | 0.203842 | − | 0.0433280i | 2.62292 | + | 15.4717i |
31.6 | −2.06481 | + | 2.29321i | 4.51294 | − | 0.474329i | −0.577233 | − | 5.49201i | 3.92442 | + | 3.09821i | −8.23064 | + | 11.3285i | 5.63678 | − | 4.15051i | 3.80029 | + | 2.76107i | 11.3383 | − | 2.41003i | −15.2080 | + | 2.60227i |
31.7 | −1.91067 | + | 2.12201i | −3.47027 | + | 0.364740i | −0.434170 | − | 4.13085i | −1.55857 | + | 4.75088i | 5.85656 | − | 8.06086i | −2.25597 | − | 6.62651i | 0.354846 | + | 0.257811i | 3.10641 | − | 0.660287i | −7.10353 | − | 12.3847i |
31.8 | −1.70001 | + | 1.88805i | 0.145330 | − | 0.0152748i | −0.256596 | − | 2.44134i | 4.49189 | − | 2.19612i | −0.218223 | + | 0.300358i | −2.33140 | − | 6.60035i | −3.17604 | − | 2.30753i | −8.78244 | + | 1.86677i | −3.48986 | + | 12.2144i |
31.9 | −1.65998 | + | 1.84360i | −4.90771 | + | 0.515821i | −0.225195 | − | 2.14259i | 4.96258 | − | 0.610552i | 7.19574 | − | 9.90409i | 6.87919 | + | 1.29487i | −3.70416 | − | 2.69123i | 15.0162 | − | 3.19179i | −7.11219 | + | 10.1625i |
31.10 | −1.40049 | + | 1.55540i | −0.438533 | + | 0.0460917i | −0.0397882 | − | 0.378560i | −0.744880 | − | 4.94420i | 0.542470 | − | 0.746646i | 4.32167 | + | 5.50665i | −6.12855 | − | 4.45265i | −8.61314 | + | 1.83078i | 8.73342 | + | 5.76572i |
31.11 | −1.38496 | + | 1.53816i | 3.56615 | − | 0.374817i | −0.0296908 | − | 0.282489i | −3.08750 | + | 3.93286i | −4.36245 | + | 6.00440i | 2.60110 | + | 6.49879i | −6.22236 | − | 4.52081i | 3.77359 | − | 0.802101i | −1.77329 | − | 10.1959i |
31.12 | −1.12383 | + | 1.24813i | −4.12211 | + | 0.433251i | 0.123257 | + | 1.17272i | −4.98024 | − | 0.444119i | 4.09177 | − | 5.63185i | −5.21929 | + | 4.66465i | −7.03730 | − | 5.11290i | 8.00074 | − | 1.70061i | 6.15124 | − | 5.71689i |
31.13 | −0.967647 | + | 1.07468i | 2.37369 | − | 0.249485i | 0.199516 | + | 1.89826i | −4.47801 | − | 2.22428i | −2.02877 | + | 2.79237i | −6.34778 | − | 2.95054i | −6.91285 | − | 5.02248i | −3.23118 | + | 0.686808i | 6.72353 | − | 2.66011i |
31.14 | −0.907746 | + | 1.00815i | 5.11223 | − | 0.537317i | 0.225742 | + | 2.14779i | −1.47546 | − | 4.77734i | −4.09891 | + | 5.64167i | 4.76870 | − | 5.12440i | −6.76029 | − | 4.91164i | 17.0429 | − | 3.62257i | 6.15565 | + | 2.84912i |
31.15 | −0.805329 | + | 0.894408i | 0.182928 | − | 0.0192266i | 0.266702 | + | 2.53750i | 2.29201 | + | 4.44372i | −0.130121 | + | 0.179096i | −6.99969 | + | 0.0657773i | −6.37910 | − | 4.63469i | −8.77024 | + | 1.86417i | −5.82033 | − | 1.52867i |
31.16 | −0.492153 | + | 0.546592i | −3.83732 | + | 0.403319i | 0.361566 | + | 3.44007i | −1.97600 | + | 4.59298i | 1.66810 | − | 2.29594i | 6.41025 | + | 2.81223i | −4.43843 | − | 3.22471i | 5.75904 | − | 1.22412i | −1.53799 | − | 3.34051i |
31.17 | −0.259618 | + | 0.288335i | −3.00636 | + | 0.315981i | 0.402378 | + | 3.82837i | 1.63247 | − | 4.72600i | 0.689396 | − | 0.948872i | −0.900309 | − | 6.94186i | −2.46389 | − | 1.79012i | 0.135019 | − | 0.0286991i | 0.938850 | + | 1.69765i |
31.18 | −0.229715 | + | 0.255125i | 0.760744 | − | 0.0799574i | 0.405794 | + | 3.86088i | 4.46427 | + | 2.25174i | −0.154355 | + | 0.212452i | 6.90683 | − | 1.13829i | −2.18918 | − | 1.59053i | −8.23099 | + | 1.74955i | −1.59998 | + | 0.621685i |
31.19 | −0.202397 | + | 0.224785i | 4.72650 | − | 0.496775i | 0.408550 | + | 3.88710i | 4.80168 | − | 1.39423i | −0.844963 | + | 1.16299i | −2.30801 | + | 6.60856i | −1.93529 | − | 1.40607i | 13.2896 | − | 2.82480i | −0.658444 | + | 1.36154i |
31.20 | 0.192889 | − | 0.214225i | −2.42365 | + | 0.254736i | 0.409428 | + | 3.89544i | 1.90879 | − | 4.62131i | −0.412925 | + | 0.568343i | −2.66665 | + | 6.47217i | 1.84633 | + | 1.34144i | −2.99412 | + | 0.636419i | −0.621814 | − | 1.30031i |
See next 80 embeddings (of 304 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
25.d | even | 5 | 1 | inner |
175.v | odd | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 175.3.v.a | ✓ | 304 |
7.d | odd | 6 | 1 | inner | 175.3.v.a | ✓ | 304 |
25.d | even | 5 | 1 | inner | 175.3.v.a | ✓ | 304 |
175.v | odd | 30 | 1 | inner | 175.3.v.a | ✓ | 304 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
175.3.v.a | ✓ | 304 | 1.a | even | 1 | 1 | trivial |
175.3.v.a | ✓ | 304 | 7.d | odd | 6 | 1 | inner |
175.3.v.a | ✓ | 304 | 25.d | even | 5 | 1 | inner |
175.3.v.a | ✓ | 304 | 175.v | odd | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(175, [\chi])\).