↑

Properties

Label 441.3.j.i
Level $441$
Weight $3$
Character orbit 441.j
Analytic conductor $12.016$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,3,Mod(263,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.263");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.j (of order \(6\), degree \(2\), not 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: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 96 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 96 q^{4} - 4 q^{9} - 36 q^{11} - 68 q^{15} + 192 q^{16} - 200 q^{18} - 144 q^{23} + 120 q^{25} - 348 q^{30} + 356 q^{36} + 84 q^{37} + 224 q^{39} + 288 q^{44} - 168 q^{46} - 288 q^{50} - 236 q^{51} + 72 q^{53} + 444 q^{57} + 1064 q^{60} - 888 q^{64} + 336 q^{67} + 1776 q^{72} - 252 q^{74} - 732 q^{78} + 24 q^{79} - 1084 q^{81} + 60 q^{85} - 1188 q^{86} + 864 q^{92} - 776 q^{93} - 268 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1 − 3.88398i −1.40667 + 2.64977i −11.0853 4.45755 + 2.57357i 10.2917 + 5.46347i 0 27.5193i −5.04257 − 7.45470i 9.99569 − 17.3130i
263.2 − 3.88398i 1.40667 − 2.64977i −11.0853 −4.45755 − 2.57357i −10.2917 − 5.46347i 0 27.5193i −5.04257 − 7.45470i −9.99569 + 17.3130i
263.3 − 3.05297i −2.15145 + 2.09076i −5.32060 0.236509 + 0.136549i 6.38301 + 6.56830i 0 4.03176i 0.257465 − 8.99632i 0.416879 − 0.722056i
263.4 − 3.05297i 2.15145 − 2.09076i −5.32060 −0.236509 − 0.136549i −6.38301 − 6.56830i 0 4.03176i 0.257465 − 8.99632i −0.416879 + 0.722056i
263.5 − 2.16791i −2.17308 − 2.06826i −0.699816 1.63423 + 0.943525i −4.48380 + 4.71104i 0 − 7.15449i 0.444583 + 8.98901i 2.04547 − 3.54286i
263.6 − 2.16791i 2.17308 + 2.06826i −0.699816 −1.63423 − 0.943525i 4.48380 − 4.71104i 0 − 7.15449i 0.444583 + 8.98901i −2.04547 + 3.54286i
263.7 − 1.95935i −2.85441 + 0.923238i 0.160944 −6.52428 − 3.76680i 1.80895 + 5.59278i 0 − 8.15275i 7.29526 − 5.27059i −7.38048 + 12.7834i
263.8 − 1.95935i 2.85441 − 0.923238i 0.160944 6.52428 + 3.76680i −1.80895 − 5.59278i 0 − 8.15275i 7.29526 − 5.27059i 7.38048 − 12.7834i
263.9 − 1.79796i −2.37631 − 1.83116i 0.767339 7.82681 + 4.51881i −3.29236 + 4.27251i 0 − 8.57149i 2.29368 + 8.70282i 8.12464 − 14.0723i
263.10 − 1.79796i 2.37631 + 1.83116i 0.767339 −7.82681 − 4.51881i 3.29236 − 4.27251i 0 − 8.57149i 2.29368 + 8.70282i −8.12464 + 14.0723i
263.11 − 0.0364909i −0.191407 − 2.99389i 3.99867 −3.35494 − 1.93698i −0.109250 + 0.00698464i 0 − 0.291879i −8.92673 + 1.14610i −0.0706821 + 0.122425i
263.12 − 0.0364909i 0.191407 + 2.99389i 3.99867 3.35494 + 1.93698i 0.109250 − 0.00698464i 0 − 0.291879i −8.92673 + 1.14610i 0.0706821 − 0.122425i
263.13 0.467585i −2.99270 + 0.209099i 3.78136 −4.35325 − 2.51335i −0.0977716 − 1.39934i 0 3.63845i 8.91256 − 1.25154i 1.17521 − 2.03552i
263.14 0.467585i 2.99270 − 0.209099i 3.78136 4.35325 + 2.51335i 0.0977716 + 1.39934i 0 3.63845i 8.91256 − 1.25154i −1.17521 + 2.03552i
263.15 1.04847i −2.29737 + 1.92927i 2.90071 2.27557 + 1.31380i −2.02278 − 2.40872i 0 7.23519i 1.55581 − 8.86450i −1.37748 + 2.38586i
263.16 1.04847i 2.29737 − 1.92927i 2.90071 −2.27557 − 1.31380i 2.02278 + 2.40872i 0 7.23519i 1.55581 − 8.86450i 1.37748 − 2.38586i
263.17 1.69745i −1.06438 − 2.80484i 1.11868 5.73513 + 3.31118i 4.76106 − 1.80672i 0 8.68868i −6.73420 + 5.97081i −5.62055 + 9.73508i
263.18 1.69745i 1.06438 + 2.80484i 1.11868 −5.73513 − 3.31118i −4.76106 + 1.80672i 0 8.68868i −6.73420 + 5.97081i 5.62055 − 9.73508i
263.19 2.86154i −2.61965 − 1.46200i −4.18841 0.898805 + 0.518925i 4.18358 − 7.49622i 0 − 0.539142i 4.72510 + 7.65986i −1.48492 + 2.57197i
263.20 2.86154i 2.61965 + 1.46200i −4.18841 −0.898805 − 0.518925i −4.18358 + 7.49622i 0 − 0.539142i 4.72510 + 7.65986i 1.48492 − 2.57197i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 263.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.j.i 48
7.b odd 2 1 inner 441.3.j.i 48
7.c even 3 1 441.3.n.i 48
7.c even 3 1 441.3.r.i ✓ 48
7.d odd 6 1 441.3.n.i 48
7.d odd 6 1 441.3.r.i ✓ 48
9.d odd 6 1 441.3.n.i 48
63.i even 6 1 inner 441.3.j.i 48
63.j odd 6 1 inner 441.3.j.i 48
63.n odd 6 1 441.3.r.i ✓ 48
63.o even 6 1 441.3.n.i 48
63.s even 6 1 441.3.r.i ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.3.j.i 48 1.a even 1 1 trivial
441.3.j.i 48 7.b odd 2 1 inner
441.3.j.i 48 63.i even 6 1 inner
441.3.j.i 48 63.j odd 6 1 inner
441.3.n.i 48 7.c even 3 1
441.3.n.i 48 7.d odd 6 1
441.3.n.i 48 9.d odd 6 1
441.3.n.i 48 63.o even 6 1
441.3.r.i ✓ 48 7.c even 3 1
441.3.r.i ✓ 48 7.d odd 6 1
441.3.r.i ✓ 48 63.n odd 6 1
441.3.r.i ✓ 48 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{24} + 72 T_{2}^{22} + 2232 T_{2}^{20} + 39114 T_{2}^{18} + 427824 T_{2}^{16} + 3043386 T_{2}^{14} + \cdots + 8281 \) Copy content Toggle raw display
\( T_{5}^{48} - 360 T_{5}^{46} + 74949 T_{5}^{44} - 10508352 T_{5}^{42} + 1102831704 T_{5}^{40} + \cdots + 36\!\cdots\!76 \) Copy content Toggle raw display