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Properties

Label 441.3.j.g
Level $441$
Weight $3$
Character orbit 441.j
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(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: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{3} - 48 q^{4} - 18 q^{5} + 14 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{3} - 48 q^{4} - 18 q^{5} + 14 q^{6} - 10 q^{9} + 18 q^{11} + 26 q^{12} - 10 q^{15} + 96 q^{16} + 108 q^{17} + 70 q^{18} - 12 q^{19} + 18 q^{20} - 24 q^{22} - 72 q^{23} + 84 q^{24} + 54 q^{25} + 144 q^{26} + 124 q^{27} + 54 q^{29} + 250 q^{30} + 60 q^{31} - 152 q^{33} - 60 q^{34} + 124 q^{36} - 42 q^{37} - 144 q^{38} - 148 q^{39} + 60 q^{40} - 180 q^{41} - 60 q^{43} - 72 q^{44} - 20 q^{45} + 84 q^{46} - 436 q^{48} - 378 q^{50} - 96 q^{51} + 18 q^{52} - 324 q^{53} + 68 q^{54} + 132 q^{55} - 232 q^{57} + 90 q^{58} + 214 q^{60} + 324 q^{64} + 230 q^{66} - 12 q^{67} - 738 q^{68} + 432 q^{69} - 504 q^{72} - 36 q^{73} + 792 q^{74} + 212 q^{75} - 84 q^{76} + 28 q^{78} + 12 q^{79} + 504 q^{80} + 146 q^{81} - 54 q^{82} + 558 q^{83} + 126 q^{85} - 90 q^{86} - 242 q^{87} + 168 q^{88} + 522 q^{89} + 488 q^{90} + 774 q^{92} + 696 q^{93} - 708 q^{94} - 464 q^{96} + 270 q^{97} + 296 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1 − 3.45061i −2.95794 − 0.500562i −7.90674 0.855181 + 0.493739i −1.72725 + 10.2067i 0 13.4807i 8.49887 + 2.96127i 1.70370 − 2.95090i
263.2 − 3.06609i 0.792382 + 2.89346i −5.40092 0.225868 + 0.130405i 8.87162 − 2.42952i 0 4.29534i −7.74426 + 4.58546i 0.399833 − 0.692532i
263.3 − 2.62334i 0.457514 − 2.96491i −2.88190 −7.02923 − 4.05833i −7.77795 − 1.20021i 0 − 2.93316i −8.58136 − 2.71297i −10.6464 + 18.4400i
263.4 − 1.90171i 2.99978 − 0.0365957i 0.383500 1.42048 + 0.820116i −0.0695945 − 5.70470i 0 − 8.33614i 8.99732 − 0.219558i 1.55962 − 2.70135i
263.5 − 0.859732i −2.54252 + 1.59236i 3.26086 5.58239 + 3.22299i 1.36900 + 2.18588i 0 − 6.24239i 3.92877 − 8.09721i 2.77091 − 4.79936i
263.6 − 0.608006i −1.75802 − 2.43092i 3.63033 0.914466 + 0.527967i −1.47801 + 1.06889i 0 − 4.63929i −2.81872 + 8.54721i 0.321007 − 0.556001i
263.7 − 0.341942i −1.24961 + 2.72736i 3.88308 −7.71344 − 4.45336i 0.932598 + 0.427294i 0 − 2.69555i −5.87696 − 6.81626i −1.52279 + 2.63755i
263.8 0.750111i 2.03293 − 2.20617i 3.43733 2.68085 + 1.54779i 1.65487 + 1.52492i 0 5.57882i −0.734396 − 8.96999i −1.16101 + 2.01093i
263.9 2.00485i −2.65183 − 1.40278i −0.0194230 −4.27746 − 2.46959i 2.81237 − 5.31652i 0 7.98046i 5.06439 + 7.43989i 4.95116 − 8.57566i
263.10 2.89536i 2.87959 − 0.841394i −4.38313 −6.82498 − 3.94040i 2.43614 + 8.33747i 0 − 1.10929i 7.58411 − 4.84574i 11.4089 − 19.7608i
263.11 3.40691i −0.590919 − 2.94123i −7.60704 6.84828 + 3.95386i 10.0205 − 2.01321i 0 − 12.2888i −8.30163 + 3.47605i −13.4704 + 23.3315i
263.12 3.79420i −1.41136 + 2.64728i −10.3960 −1.68242 − 0.971344i −10.0443 − 5.35497i 0 − 24.2675i −5.01615 − 7.47250i 3.68547 − 6.38343i
275.1 − 3.79420i −1.41136 − 2.64728i −10.3960 −1.68242 + 0.971344i −10.0443 + 5.35497i 0 24.2675i −5.01615 + 7.47250i 3.68547 + 6.38343i
275.2 − 3.40691i −0.590919 + 2.94123i −7.60704 6.84828 − 3.95386i 10.0205 + 2.01321i 0 12.2888i −8.30163 − 3.47605i −13.4704 − 23.3315i
275.3 − 2.89536i 2.87959 + 0.841394i −4.38313 −6.82498 + 3.94040i 2.43614 − 8.33747i 0 1.10929i 7.58411 + 4.84574i 11.4089 + 19.7608i
275.4 − 2.00485i −2.65183 + 1.40278i −0.0194230 −4.27746 + 2.46959i 2.81237 + 5.31652i 0 − 7.98046i 5.06439 − 7.43989i 4.95116 + 8.57566i
275.5 − 0.750111i 2.03293 + 2.20617i 3.43733 2.68085 − 1.54779i 1.65487 − 1.52492i 0 − 5.57882i −0.734396 + 8.96999i −1.16101 − 2.01093i
275.6 0.341942i −1.24961 − 2.72736i 3.88308 −7.71344 + 4.45336i 0.932598 − 0.427294i 0 2.69555i −5.87696 + 6.81626i −1.52279 − 2.63755i
275.7 0.608006i −1.75802 + 2.43092i 3.63033 0.914466 − 0.527967i −1.47801 − 1.06889i 0 4.63929i −2.81872 − 8.54721i 0.321007 + 0.556001i
275.8 0.859732i −2.54252 − 1.59236i 3.26086 5.58239 − 3.22299i 1.36900 − 2.18588i 0 6.24239i 3.92877 + 8.09721i 2.77091 + 4.79936i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 263.12
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.g 24
7.b odd 2 1 441.3.j.h 24
7.c even 3 1 441.3.n.h 24
7.c even 3 1 441.3.r.h 24
7.d odd 6 1 63.3.r.a ✓ 24
7.d odd 6 1 441.3.n.g 24
9.d odd 6 1 441.3.n.h 24
21.g even 6 1 189.3.r.a 24
63.i even 6 1 441.3.j.h 24
63.i even 6 1 567.3.b.a 24
63.j odd 6 1 inner 441.3.j.g 24
63.k odd 6 1 189.3.r.a 24
63.n odd 6 1 441.3.r.h 24
63.o even 6 1 441.3.n.g 24
63.s even 6 1 63.3.r.a ✓ 24
63.t odd 6 1 567.3.b.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.r.a ✓ 24 7.d odd 6 1
63.3.r.a ✓ 24 63.s even 6 1
189.3.r.a 24 21.g even 6 1
189.3.r.a 24 63.k odd 6 1
441.3.j.g 24 1.a even 1 1 trivial
441.3.j.g 24 63.j odd 6 1 inner
441.3.j.h 24 7.b odd 2 1
441.3.j.h 24 63.i even 6 1
441.3.n.g 24 7.d odd 6 1
441.3.n.g 24 63.o even 6 1
441.3.n.h 24 7.c even 3 1
441.3.n.h 24 9.d odd 6 1
441.3.r.h 24 7.c even 3 1
441.3.r.h 24 63.n odd 6 1
567.3.b.a 24 63.i even 6 1
567.3.b.a 24 63.t odd 6 1

Hecke kernels

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

\( T_{2}^{24} + 72 T_{2}^{22} + 2232 T_{2}^{20} + 38986 T_{2}^{18} + 421524 T_{2}^{16} + 2917782 T_{2}^{14} + \cdots + 281961 \) 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