Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [196,2,Mod(3,196)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(196, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([21, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("196.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 196.p (of order \(42\), degree \(12\), 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: | \(1.56506787962\) |
Analytic rank: | \(0\) |
Dimension: | \(312\) |
Relative dimension: | \(26\) over \(\Q(\zeta_{42})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{42}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | −1.39719 | − | 0.218763i | −0.168622 | + | 2.25011i | 1.90429 | + | 0.611309i | 1.45940 | − | 2.14054i | 0.727840 | − | 3.10695i | 2.45399 | − | 0.988908i | −2.52692 | − | 1.27070i | −2.06807 | − | 0.311712i | −2.50733 | + | 2.67149i |
3.2 | −1.39358 | − | 0.240682i | −0.0422877 | + | 0.564290i | 1.88414 | + | 0.670821i | −0.402891 | + | 0.590932i | 0.194746 | − | 0.776207i | −1.63732 | + | 2.07826i | −2.46426 | − | 1.38832i | 2.64986 | + | 0.399402i | 0.703688 | − | 0.726544i |
3.3 | −1.39325 | + | 0.242579i | 0.179404 | − | 2.39399i | 1.88231 | − | 0.675947i | 1.57236 | − | 2.30623i | 0.330774 | + | 3.37895i | −0.993249 | − | 2.45224i | −2.45857 | + | 1.39837i | −2.73249 | − | 0.411856i | −1.63125 | + | 3.59458i |
3.4 | −1.31819 | + | 0.512232i | 0.119444 | − | 1.59387i | 1.47524 | − | 1.35044i | −0.497555 | + | 0.729779i | 0.658979 | + | 2.16220i | 2.00509 | + | 1.72615i | −1.25290 | + | 2.53579i | 0.440352 | + | 0.0663724i | 0.282055 | − | 1.21685i |
3.5 | −1.09300 | + | 0.897413i | −0.229583 | + | 3.06358i | 0.389299 | − | 1.96175i | −0.107063 | + | 0.157032i | −2.49836 | − | 3.55452i | −1.48187 | + | 2.19182i | 1.33499 | + | 2.49355i | −6.36629 | − | 0.959564i | −0.0239032 | − | 0.267716i |
3.6 | −1.08499 | − | 0.907078i | 0.0626803 | − | 0.836411i | 0.354421 | + | 1.96835i | −1.74354 | + | 2.55731i | −0.826697 | + | 0.850644i | 2.29203 | − | 1.32159i | 1.40090 | − | 2.45713i | 2.27084 | + | 0.342274i | 4.21141 | − | 1.19313i |
3.7 | −0.819528 | + | 1.15255i | −0.0308976 | + | 0.412300i | −0.656747 | − | 1.88910i | 1.18688 | − | 1.74083i | −0.449876 | − | 0.373503i | −1.42242 | − | 2.23086i | 2.71550 | + | 0.791233i | 2.79746 | + | 0.421649i | 1.03372 | + | 2.79460i |
3.8 | −0.815561 | + | 1.15536i | 0.141975 | − | 1.89453i | −0.669721 | − | 1.88454i | −2.20782 | + | 3.23827i | 2.07308 | + | 1.70914i | −2.57933 | + | 0.589130i | 2.72352 | + | 0.763183i | −0.602593 | − | 0.0908263i | −1.94077 | − | 5.19184i |
3.9 | −0.809118 | − | 1.15988i | −0.234586 | + | 3.13033i | −0.690655 | + | 1.87696i | −1.00654 | + | 1.47633i | 3.82063 | − | 2.26072i | −1.58100 | − | 2.12142i | 2.73588 | − | 0.717607i | −6.77745 | − | 1.02154i | 2.52678 | − | 0.0270517i |
3.10 | −0.769816 | − | 1.18633i | 0.157628 | − | 2.10340i | −0.814767 | + | 1.82651i | 1.95056 | − | 2.86094i | −2.61668 | + | 1.43223i | 2.57201 | + | 0.620295i | 2.79407 | − | 0.439496i | −1.43295 | − | 0.215983i | −4.89560 | − | 0.111610i |
3.11 | −0.209076 | − | 1.39867i | −0.0654786 | + | 0.873751i | −1.91257 | + | 0.584859i | −0.948216 | + | 1.39078i | 1.23578 | − | 0.0910975i | 0.250153 | + | 2.63390i | 1.21790 | + | 2.55279i | 2.20734 | + | 0.332703i | 2.14349 | + | 1.03547i |
3.12 | −0.200389 | − | 1.39994i | 0.204591 | − | 2.73007i | −1.91969 | + | 0.561068i | −1.04779 | + | 1.53683i | −3.86295 | + | 0.260662i | −2.46706 | − | 0.955837i | 1.17015 | + | 2.57502i | −4.44495 | − | 0.669969i | 2.36144 | + | 1.15889i |
3.13 | −0.187997 | + | 1.40166i | −0.141975 | + | 1.89453i | −1.92931 | − | 0.527017i | −2.20782 | + | 3.23827i | −2.62880 | − | 0.555168i | 2.57933 | − | 0.589130i | 1.10141 | − | 2.60517i | −0.602593 | − | 0.0908263i | −4.12390 | − | 3.70340i |
3.14 | −0.183177 | + | 1.40230i | 0.0308976 | − | 0.412300i | −1.93289 | − | 0.513738i | 1.18688 | − | 1.74083i | 0.572509 | + | 0.118852i | 1.42242 | + | 2.23086i | 1.07448 | − | 2.61639i | 2.79746 | + | 0.421649i | 2.22376 | + | 1.98324i |
3.15 | 0.190830 | + | 1.40128i | 0.229583 | − | 3.06358i | −1.92717 | + | 0.534812i | −0.107063 | + | 0.157032i | 4.33674 | − | 0.262912i | 1.48187 | − | 2.19182i | −1.11718 | − | 2.59844i | −6.36629 | − | 0.959564i | −0.240477 | − | 0.120058i |
3.16 | 0.262609 | − | 1.38962i | −0.0991378 | + | 1.32290i | −1.86207 | − | 0.729853i | 1.15756 | − | 1.69784i | 1.81229 | + | 0.485170i | 1.35716 | − | 2.27115i | −1.50321 | + | 2.39590i | 1.22625 | + | 0.184828i | −2.05535 | − | 2.05444i |
3.17 | 0.617894 | + | 1.27209i | −0.119444 | + | 1.59387i | −1.23641 | + | 1.57203i | −0.497555 | + | 0.729779i | −2.10134 | + | 0.832897i | −2.00509 | − | 1.72615i | −2.76373 | − | 0.601480i | 0.440352 | + | 0.0663724i | −1.23578 | − | 0.182008i |
3.18 | 0.752674 | − | 1.19728i | 0.0991378 | − | 1.32290i | −0.866965 | − | 1.80232i | 1.15756 | − | 1.69784i | −1.50927 | − | 1.11441i | −1.35716 | + | 2.27115i | −2.81043 | − | 0.318561i | 1.22625 | + | 0.184828i | −1.16152 | − | 2.66385i |
3.19 | 0.856332 | + | 1.12548i | −0.179404 | + | 2.39399i | −0.533391 | + | 1.92756i | 1.57236 | − | 2.30623i | −2.84800 | + | 1.84813i | 0.993249 | + | 2.45224i | −2.62618 | + | 1.05031i | −2.73249 | − | 0.411856i | 3.94207 | − | 0.205244i |
3.20 | 1.09910 | − | 0.889932i | −0.204591 | + | 2.73007i | 0.416041 | − | 1.95625i | −1.04779 | + | 1.53683i | 2.20472 | + | 3.18270i | 2.46706 | + | 0.955837i | −1.28366 | − | 2.52036i | −4.44495 | − | 0.669969i | 0.216045 | + | 2.62159i |
See next 80 embeddings (of 312 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
49.h | odd | 42 | 1 | inner |
196.p | even | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 196.2.p.a | ✓ | 312 |
4.b | odd | 2 | 1 | inner | 196.2.p.a | ✓ | 312 |
49.h | odd | 42 | 1 | inner | 196.2.p.a | ✓ | 312 |
196.p | even | 42 | 1 | inner | 196.2.p.a | ✓ | 312 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
196.2.p.a | ✓ | 312 | 1.a | even | 1 | 1 | trivial |
196.2.p.a | ✓ | 312 | 4.b | odd | 2 | 1 | inner |
196.2.p.a | ✓ | 312 | 49.h | odd | 42 | 1 | inner |
196.2.p.a | ✓ | 312 | 196.p | even | 42 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(196, [\chi])\).