Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [300,2,Mod(67,300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(300, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([10, 0, 13]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.67");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 300.w (of order \(20\), 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: | \(2.39551206064\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(30\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 | −1.38407 | + | 0.290430i | −0.987688 | − | 0.156434i | 1.83130 | − | 0.803952i | 0.979302 | + | 2.01022i | 1.41246 | − | 0.0703383i | −0.0808033 | + | 0.0808033i | −2.30116 | + | 1.64459i | 0.951057 | + | 0.309017i | −1.93925 | − | 2.49786i |
67.2 | −1.37859 | − | 0.315398i | 0.987688 | + | 0.156434i | 1.80105 | + | 0.869612i | 2.22681 | + | 0.203301i | −1.31228 | − | 0.527175i | 2.91192 | − | 2.91192i | −2.20864 | − | 1.76689i | 0.951057 | + | 0.309017i | −3.00574 | − | 0.982600i |
67.3 | −1.37682 | − | 0.323077i | 0.987688 | + | 0.156434i | 1.79124 | + | 0.889636i | −2.11409 | + | 0.728446i | −1.30932 | − | 0.534481i | 0.284106 | − | 0.284106i | −2.17879 | − | 1.80357i | 0.951057 | + | 0.309017i | 3.14605 | − | 0.319922i |
67.4 | −1.36970 | − | 0.352033i | −0.987688 | − | 0.156434i | 1.75215 | + | 0.964358i | 1.21468 | − | 1.87738i | 1.29776 | + | 0.561967i | −0.466793 | + | 0.466793i | −2.06042 | − | 1.93769i | 0.951057 | + | 0.309017i | −2.32465 | + | 2.14383i |
67.5 | −1.29457 | + | 0.569294i | 0.987688 | + | 0.156434i | 1.35181 | − | 1.47398i | 2.22495 | − | 0.222669i | −1.36769 | + | 0.359770i | −3.70249 | + | 3.70249i | −0.910882 | + | 2.67774i | 0.951057 | + | 0.309017i | −2.75359 | + | 1.55491i |
67.6 | −1.28400 | + | 0.592739i | 0.987688 | + | 0.156434i | 1.29732 | − | 1.52216i | −1.17964 | − | 1.89959i | −1.36092 | + | 0.384580i | 1.39751 | − | 1.39751i | −0.763519 | + | 2.72342i | 0.951057 | + | 0.309017i | 2.64062 | + | 1.73986i |
67.7 | −1.12003 | − | 0.863445i | −0.987688 | − | 0.156434i | 0.508925 | + | 1.93417i | −1.70279 | + | 1.44932i | 0.971166 | + | 1.02803i | 1.87554 | − | 1.87554i | 1.10004 | − | 2.60575i | 0.951057 | + | 0.309017i | 3.15857 | − | 0.153014i |
67.8 | −0.976661 | + | 1.02281i | −0.987688 | − | 0.156434i | −0.0922667 | − | 1.99787i | −0.698282 | − | 2.12424i | 1.12464 | − | 0.857431i | −1.04258 | + | 1.04258i | 2.13355 | + | 1.85687i | 0.951057 | + | 0.309017i | 2.85467 | + | 1.36046i |
67.9 | −0.723992 | + | 1.21484i | 0.987688 | + | 0.156434i | −0.951672 | − | 1.75907i | −1.94762 | + | 1.09854i | −0.905121 | + | 1.08663i | −2.52433 | + | 2.52433i | 2.82599 | + | 0.117421i | 0.951057 | + | 0.309017i | 0.0755097 | − | 3.16138i |
67.10 | −0.665012 | − | 1.24810i | 0.987688 | + | 0.156434i | −1.11552 | + | 1.66001i | −0.495035 | + | 2.18058i | −0.461578 | − | 1.33677i | −0.602522 | + | 0.602522i | 2.81369 | + | 0.288356i | 0.951057 | + | 0.309017i | 3.05079 | − | 0.832260i |
67.11 | −0.664619 | − | 1.24831i | −0.987688 | − | 0.156434i | −1.11656 | + | 1.65930i | −1.95630 | − | 1.08300i | 0.461157 | + | 1.33691i | −2.24842 | + | 2.24842i | 2.81342 | + | 0.291015i | 0.951057 | + | 0.309017i | −0.0517207 | + | 3.16185i |
67.12 | −0.557275 | + | 1.29979i | −0.987688 | − | 0.156434i | −1.37889 | − | 1.44868i | −1.94762 | + | 1.09854i | 0.753745 | − | 1.19661i | 2.52433 | − | 2.52433i | 2.65139 | − | 0.984953i | 0.951057 | + | 0.309017i | −0.342508 | − | 3.14367i |
67.13 | −0.344199 | − | 1.37169i | 0.987688 | + | 0.156434i | −1.76305 | + | 0.944266i | 1.92573 | − | 1.13647i | −0.125382 | − | 1.40864i | 0.176718 | − | 0.176718i | 1.90208 | + | 2.09335i | 0.951057 | + | 0.309017i | −2.22172 | − | 2.25032i |
67.14 | −0.253401 | + | 1.39133i | 0.987688 | + | 0.156434i | −1.87158 | − | 0.705127i | −0.698282 | − | 2.12424i | −0.467933 | + | 1.33456i | 1.04258 | − | 1.04258i | 1.45532 | − | 2.42529i | 0.951057 | + | 0.309017i | 3.13246 | − | 0.433253i |
67.15 | −0.194555 | − | 1.40077i | −0.987688 | − | 0.156434i | −1.92430 | + | 0.545051i | 1.56105 | + | 1.60098i | −0.0269690 | + | 1.41396i | 1.26539 | − | 1.26539i | 1.13787 | + | 2.58945i | 0.951057 | + | 0.309017i | 1.93889 | − | 2.49814i |
67.16 | 0.170201 | − | 1.40393i | −0.987688 | − | 0.156434i | −1.94206 | − | 0.477901i | 0.0997280 | − | 2.23384i | −0.387729 | + | 1.36002i | 2.03014 | − | 2.03014i | −1.00148 | + | 2.64519i | 0.951057 | + | 0.309017i | −3.11920 | − | 0.520213i |
67.17 | 0.275181 | + | 1.38718i | −0.987688 | − | 0.156434i | −1.84855 | + | 0.763453i | −1.17964 | − | 1.89959i | −0.0547899 | − | 1.41315i | −1.39751 | + | 1.39751i | −1.56773 | − | 2.35419i | 0.951057 | + | 0.309017i | 2.31047 | − | 2.15910i |
67.18 | 0.300359 | + | 1.38195i | −0.987688 | − | 0.156434i | −1.81957 | + | 0.830163i | 2.22495 | − | 0.222669i | −0.0804769 | − | 1.41192i | 3.70249 | − | 3.70249i | −1.69377 | − | 2.26520i | 0.951057 | + | 0.309017i | 0.976003 | + | 3.00789i |
67.19 | 0.578573 | + | 1.29045i | 0.987688 | + | 0.156434i | −1.33051 | + | 1.49324i | 0.979302 | + | 2.01022i | 0.369579 | + | 1.36507i | 0.0808033 | − | 0.0808033i | −2.69674 | − | 0.853002i | 0.951057 | + | 0.309017i | −2.02748 | + | 2.42679i |
67.20 | 0.611319 | − | 1.27526i | 0.987688 | + | 0.156434i | −1.25258 | − | 1.55918i | −0.726283 | + | 2.11483i | 0.803287 | − | 1.16393i | 3.39156 | − | 3.39156i | −2.75409 | + | 0.644208i | 0.951057 | + | 0.309017i | 2.25297 | + | 2.21904i |
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
25.f | odd | 20 | 1 | inner |
100.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 300.2.w.a | ✓ | 240 |
3.b | odd | 2 | 1 | 900.2.bj.f | 240 | ||
4.b | odd | 2 | 1 | inner | 300.2.w.a | ✓ | 240 |
12.b | even | 2 | 1 | 900.2.bj.f | 240 | ||
25.f | odd | 20 | 1 | inner | 300.2.w.a | ✓ | 240 |
75.l | even | 20 | 1 | 900.2.bj.f | 240 | ||
100.l | even | 20 | 1 | inner | 300.2.w.a | ✓ | 240 |
300.u | odd | 20 | 1 | 900.2.bj.f | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
300.2.w.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
300.2.w.a | ✓ | 240 | 4.b | odd | 2 | 1 | inner |
300.2.w.a | ✓ | 240 | 25.f | odd | 20 | 1 | inner |
300.2.w.a | ✓ | 240 | 100.l | even | 20 | 1 | inner |
900.2.bj.f | 240 | 3.b | odd | 2 | 1 | ||
900.2.bj.f | 240 | 12.b | even | 2 | 1 | ||
900.2.bj.f | 240 | 75.l | even | 20 | 1 | ||
900.2.bj.f | 240 | 300.u | odd | 20 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(300, [\chi])\).