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Properties

Label 441.3.j.f
Level $441$
Weight $3$
Character orbit 441.j
Analytic conductor $12.016$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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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: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 19 q^{3} - 24 q^{4} - 12 q^{5} + 8 q^{6} - 37 q^{9} - 25 q^{10} + 24 q^{11} - 40 q^{12} + 18 q^{13} + 53 q^{15} - 24 q^{16} + 6 q^{17} + 40 q^{18} - 3 q^{19} + 39 q^{20} - 59 q^{22} + 81 q^{23} - 126 q^{24}+ \cdots - 103 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1 − 3.69072i 1.07647 + 2.80021i −9.62138 −5.05096 − 2.91617i 10.3348 − 3.97296i 0 20.7469i −6.68241 + 6.02872i −10.7628 + 18.6416i
263.2 − 3.22356i 2.32685 − 1.89361i −6.39137 4.79167 + 2.76647i −6.10417 − 7.50076i 0 7.70873i 1.82848 − 8.81230i 8.91790 − 15.4463i
263.3 − 2.41355i −1.94613 − 2.28311i −1.82523 −5.93347 − 3.42569i −5.51041 + 4.69707i 0 − 5.24892i −1.42519 + 8.88644i −8.26808 + 14.3207i
263.4 − 1.46555i −0.993737 + 2.83063i 1.85217 −0.998268 − 0.576350i 4.14843 + 1.45637i 0 − 8.57663i −7.02497 − 5.62581i −0.844669 + 1.46301i
263.5 − 1.29088i 2.37377 − 1.83445i 2.33363 −4.18841 − 2.41818i −2.36806 − 3.06425i 0 − 8.17595i 2.26955 − 8.70914i −3.12158 + 5.40673i
263.6 − 0.513687i 2.25960 + 1.97337i 3.73613 6.08350 + 3.51231i 1.01370 − 1.16073i 0 − 3.97395i 1.21160 + 8.91807i 1.80423 − 3.12502i
263.7 0.0767494i −0.891547 − 2.86446i 3.99411 3.53076 + 2.03848i 0.219846 − 0.0684257i 0 0.613543i −7.41029 + 5.10760i −0.156452 + 0.270983i
263.8 1.28539i 2.79035 + 1.10178i 2.34778 −6.82011 − 3.93759i −1.41622 + 3.58669i 0 8.15936i 6.57215 + 6.14872i 5.06133 − 8.76649i
263.9 2.15495i −0.867675 + 2.87178i −0.643801 −1.62693 − 0.939308i −6.18854 − 1.86979i 0 7.23243i −7.49428 − 4.98355i 2.02416 − 3.50595i
263.10 2.74500i 0.432548 − 2.96865i −3.53503 −2.32531 − 1.34252i 8.14895 + 1.18734i 0 1.27635i −8.62581 − 2.56817i 3.68521 − 6.38297i
263.11 2.87176i 2.93949 − 0.599520i −4.24700 6.53753 + 3.77444i 1.72168 + 8.44150i 0 − 0.709334i 8.28115 − 3.52456i −10.8393 + 18.7742i
275.1 − 2.87176i 2.93949 + 0.599520i −4.24700 6.53753 − 3.77444i 1.72168 − 8.44150i 0 0.709334i 8.28115 + 3.52456i −10.8393 − 18.7742i
275.2 − 2.74500i 0.432548 + 2.96865i −3.53503 −2.32531 + 1.34252i 8.14895 − 1.18734i 0 − 1.27635i −8.62581 + 2.56817i 3.68521 + 6.38297i
275.3 − 2.15495i −0.867675 − 2.87178i −0.643801 −1.62693 + 0.939308i −6.18854 + 1.86979i 0 − 7.23243i −7.49428 + 4.98355i 2.02416 + 3.50595i
275.4 − 1.28539i 2.79035 − 1.10178i 2.34778 −6.82011 + 3.93759i −1.41622 − 3.58669i 0 − 8.15936i 6.57215 − 6.14872i 5.06133 + 8.76649i
275.5 − 0.0767494i −0.891547 + 2.86446i 3.99411 3.53076 − 2.03848i 0.219846 + 0.0684257i 0 − 0.613543i −7.41029 − 5.10760i −0.156452 − 0.270983i
275.6 0.513687i 2.25960 − 1.97337i 3.73613 6.08350 − 3.51231i 1.01370 + 1.16073i 0 3.97395i 1.21160 − 8.91807i 1.80423 + 3.12502i
275.7 1.29088i 2.37377 + 1.83445i 2.33363 −4.18841 + 2.41818i −2.36806 + 3.06425i 0 8.17595i 2.26955 + 8.70914i −3.12158 − 5.40673i
275.8 1.46555i −0.993737 − 2.83063i 1.85217 −0.998268 + 0.576350i 4.14843 − 1.45637i 0 8.57663i −7.02497 + 5.62581i −0.844669 − 1.46301i
275.9 2.41355i −1.94613 + 2.28311i −1.82523 −5.93347 + 3.42569i −5.51041 − 4.69707i 0 5.24892i −1.42519 − 8.88644i −8.26808 − 14.3207i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 263.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.f 22
7.b odd 2 1 63.3.j.b ✓ 22
7.c even 3 1 441.3.n.f 22
7.c even 3 1 441.3.r.f 22
7.d odd 6 1 63.3.n.b yes 22
7.d odd 6 1 441.3.r.g 22
9.d odd 6 1 441.3.n.f 22
21.c even 2 1 189.3.j.b 22
21.g even 6 1 189.3.n.b 22
63.i even 6 1 63.3.j.b ✓ 22
63.j odd 6 1 inner 441.3.j.f 22
63.l odd 6 1 189.3.n.b 22
63.n odd 6 1 441.3.r.f 22
63.o even 6 1 63.3.n.b yes 22
63.s even 6 1 441.3.r.g 22
63.t odd 6 1 189.3.j.b 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.j.b ✓ 22 7.b odd 2 1
63.3.j.b ✓ 22 63.i even 6 1
63.3.n.b yes 22 7.d odd 6 1
63.3.n.b yes 22 63.o even 6 1
189.3.j.b 22 21.c even 2 1
189.3.j.b 22 63.t odd 6 1
189.3.n.b 22 21.g even 6 1
189.3.n.b 22 63.l odd 6 1
441.3.j.f 22 1.a even 1 1 trivial
441.3.j.f 22 63.j odd 6 1 inner
441.3.n.f 22 7.c even 3 1
441.3.n.f 22 9.d odd 6 1
441.3.r.f 22 7.c even 3 1
441.3.r.f 22 63.n odd 6 1
441.3.r.g 22 7.d odd 6 1
441.3.r.g 22 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 \(S_{3}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{22} + 56 T_{2}^{20} + 1326 T_{2}^{18} + 17369 T_{2}^{16} + 138193 T_{2}^{14} + 690216 T_{2}^{12} + \cdots + 2187 \) Copy content Toggle raw display
\( T_{5}^{22} + 12 T_{5}^{21} - 94 T_{5}^{20} - 1704 T_{5}^{19} + 6369 T_{5}^{18} + 161319 T_{5}^{17} + \cdots + 112049133289443 \) Copy content Toggle raw display