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Properties

Label 441.3.r.h
Level $441$
Weight $3$
Character orbit 441.r
Analytic conductor $12.016$
Analytic rank $0$
Dimension $24$
CM no
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(50,441)]
 
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([5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.50");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.r (of order \(6\), degree \(2\), 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: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 63)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{3} + 24 q^{4} + 18 q^{5} + 14 q^{6} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{3} + 24 q^{4} + 18 q^{5} + 14 q^{6} + 26 q^{9} - 18 q^{11} - 4 q^{12} - 10 q^{15} - 48 q^{16} - 62 q^{18} + 24 q^{19} + 18 q^{20} - 24 q^{22} + 72 q^{23} - 54 q^{24} + 54 q^{25} + 124 q^{27} + 54 q^{29} - 212 q^{30} - 30 q^{31} + 126 q^{32} + 178 q^{33} - 60 q^{34} + 124 q^{36} + 84 q^{37} + 144 q^{38} + 92 q^{39} + 60 q^{40} - 180 q^{41} - 60 q^{43} + 118 q^{45} - 168 q^{46} - 378 q^{47} - 436 q^{48} - 378 q^{50} + 168 q^{51} + 18 q^{52} - 514 q^{54} + 132 q^{55} - 232 q^{57} + 90 q^{58} + 90 q^{59} + 76 q^{60} + 324 q^{64} + 126 q^{65} - 202 q^{66} + 6 q^{67} + 738 q^{68} + 432 q^{69} - 246 q^{72} + 72 q^{73} - 792 q^{74} - 40 q^{75} - 84 q^{76} + 28 q^{78} - 6 q^{79} - 34 q^{81} + 108 q^{82} + 558 q^{83} + 126 q^{85} + 90 q^{86} - 428 q^{87} + 168 q^{88} + 488 q^{90} + 774 q^{92} - 738 q^{93} + 354 q^{94} + 648 q^{95} + 280 q^{96} + 270 q^{97} + 296 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1 −3.28587 + 1.89710i 2.99829 + 0.101368i 5.19798 − 9.00316i 1.68242 + 0.971344i −10.0443 + 5.35497i 0 24.2675i 8.97945 + 0.607858i −7.37094
50.2 −2.95047 + 1.70346i −2.25172 − 1.98236i 3.80352 − 6.58789i −6.84828 − 3.95386i 10.0205 + 2.01321i 0 12.2888i 1.14047 + 8.92745i 26.9409
50.3 −2.50746 + 1.44768i −2.16846 + 2.07310i 2.19156 − 3.79590i 6.82498 + 3.94040i 2.43614 − 8.33747i 0 1.10929i 0.404480 − 8.99091i −22.8178
50.4 −1.73625 + 1.00242i 0.111068 − 2.99794i 0.00971148 − 0.0168208i 4.27746 + 2.46959i 2.81237 + 5.31652i 0 − 7.98046i −8.97533 − 0.665951i −9.90232
50.5 −0.649615 + 0.375055i −2.92707 + 0.657482i −1.71867 + 2.97682i −2.68085 − 1.54779i 1.65487 − 1.52492i 0 − 5.57882i 8.13543 − 3.84899i 2.32203
50.6 0.296130 − 0.170971i 2.98677 + 0.281486i −1.94154 + 3.36284i 7.71344 + 4.45336i 0.932598 − 0.427294i 0 2.69555i 8.84153 + 1.68146i 3.04558
50.7 0.526549 − 0.304003i −1.22623 − 2.73795i −1.81516 + 3.14396i −0.914466 − 0.527967i −1.47801 − 1.06889i 0 4.63929i −5.99274 + 6.71469i −0.642015
50.8 0.744550 − 0.429866i 2.65028 − 1.40570i −1.63043 + 2.82399i −5.58239 − 3.22299i 1.36900 − 2.18588i 0 6.24239i 5.04801 − 7.45102i −5.54182
50.9 1.64693 − 0.950855i −1.53158 + 2.57959i −0.191750 + 0.332121i −1.42048 − 0.820116i −0.0695945 + 5.70470i 0 8.33614i −4.30852 − 7.90169i −3.11925
50.10 2.27188 − 1.31167i −2.79644 − 1.08624i 1.44095 − 2.49580i 7.02923 + 4.05833i −7.77795 + 1.20021i 0 2.93316i 6.64018 + 6.07519i 21.2927
50.11 2.65531 − 1.53305i 2.10962 + 2.13295i 2.70046 − 4.67733i −0.225868 − 0.130405i 8.87162 + 2.42952i 0 − 4.29534i −0.0989900 + 8.99946i −0.799667
50.12 2.98832 − 1.72531i 1.04547 − 2.81194i 3.95337 − 6.84744i −0.855181 − 0.493739i −1.72725 − 10.2067i 0 − 13.4807i −6.81397 − 5.87961i −3.40741
344.1 −3.28587 − 1.89710i 2.99829 − 0.101368i 5.19798 + 9.00316i 1.68242 − 0.971344i −10.0443 − 5.35497i 0 − 24.2675i 8.97945 − 0.607858i −7.37094
344.2 −2.95047 − 1.70346i −2.25172 + 1.98236i 3.80352 + 6.58789i −6.84828 + 3.95386i 10.0205 − 2.01321i 0 − 12.2888i 1.14047 − 8.92745i 26.9409
344.3 −2.50746 − 1.44768i −2.16846 − 2.07310i 2.19156 + 3.79590i 6.82498 − 3.94040i 2.43614 + 8.33747i 0 − 1.10929i 0.404480 + 8.99091i −22.8178
344.4 −1.73625 − 1.00242i 0.111068 + 2.99794i 0.00971148 + 0.0168208i 4.27746 − 2.46959i 2.81237 − 5.31652i 0 7.98046i −8.97533 + 0.665951i −9.90232
344.5 −0.649615 − 0.375055i −2.92707 − 0.657482i −1.71867 − 2.97682i −2.68085 + 1.54779i 1.65487 + 1.52492i 0 5.57882i 8.13543 + 3.84899i 2.32203
344.6 0.296130 + 0.170971i 2.98677 − 0.281486i −1.94154 − 3.36284i 7.71344 − 4.45336i 0.932598 + 0.427294i 0 − 2.69555i 8.84153 − 1.68146i 3.04558
344.7 0.526549 + 0.304003i −1.22623 + 2.73795i −1.81516 − 3.14396i −0.914466 + 0.527967i −1.47801 + 1.06889i 0 − 4.63929i −5.99274 − 6.71469i −0.642015
344.8 0.744550 + 0.429866i 2.65028 + 1.40570i −1.63043 − 2.82399i −5.58239 + 3.22299i 1.36900 + 2.18588i 0 − 6.24239i 5.04801 + 7.45102i −5.54182
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d 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.r.h 24
7.b odd 2 1 63.3.r.a ✓ 24
7.c even 3 1 441.3.j.g 24
7.c even 3 1 441.3.n.h 24
7.d odd 6 1 441.3.j.h 24
7.d odd 6 1 441.3.n.g 24
9.d odd 6 1 inner 441.3.r.h 24
21.c even 2 1 189.3.r.a 24
63.i even 6 1 441.3.n.g 24
63.j odd 6 1 441.3.n.h 24
63.l odd 6 1 189.3.r.a 24
63.l odd 6 1 567.3.b.a 24
63.n odd 6 1 441.3.j.g 24
63.o even 6 1 63.3.r.a ✓ 24
63.o even 6 1 567.3.b.a 24
63.s even 6 1 441.3.j.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.r.a ✓ 24 7.b odd 2 1
63.3.r.a ✓ 24 63.o even 6 1
189.3.r.a 24 21.c even 2 1
189.3.r.a 24 63.l odd 6 1
441.3.j.g 24 7.c even 3 1
441.3.j.g 24 63.n odd 6 1
441.3.j.h 24 7.d odd 6 1
441.3.j.h 24 63.s even 6 1
441.3.n.g 24 7.d odd 6 1
441.3.n.g 24 63.i even 6 1
441.3.n.h 24 7.c even 3 1
441.3.n.h 24 63.j odd 6 1
441.3.r.h 24 1.a even 1 1 trivial
441.3.r.h 24 9.d odd 6 1 inner
567.3.b.a 24 63.l odd 6 1
567.3.b.a 24 63.o even 6 1

Hecke kernels

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

\( T_{2}^{24} - 36 T_{2}^{22} + 828 T_{2}^{20} - 198 T_{2}^{19} - 11558 T_{2}^{18} + 4860 T_{2}^{17} + \cdots + 281961 \) Copy content Toggle raw display
\( T_{5}^{24} - 18 T_{5}^{23} - 15 T_{5}^{22} + 2214 T_{5}^{21} - 3426 T_{5}^{20} - 190926 T_{5}^{19} + \cdots + 148046413824 \) Copy content Toggle raw display