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Properties

Label 196.2.p.a
Level $196$
Weight $2$
Character orbit 196.p
Analytic conductor $1.565$
Analytic rank $0$
Dimension $312$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

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.

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 13 q^{2} - 13 q^{4} - 22 q^{5} - 14 q^{6} - 4 q^{8} - 4 q^{9} - 20 q^{10} + 9 q^{12} - 28 q^{13} - 51 q^{14} - 17 q^{16} - 22 q^{17} - 12 q^{18} - 14 q^{20} - 34 q^{21} - 18 q^{22} - 44 q^{24} - 48 q^{25}+ \cdots + 183 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.26
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(196, [\chi])\).