Properties

Label 441.3.r.f
Level $441$
Weight $3$
Character orbit 441.r
Analytic conductor $12.016$
Analytic rank $0$
Dimension $22$
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: \(22\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 6 q^{2} - 11 q^{3} + 12 q^{4} + 12 q^{5} + 8 q^{6} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 6 q^{2} - 11 q^{3} + 12 q^{4} + 12 q^{5} + 8 q^{6} + 17 q^{9} + 50 q^{10} - 24 q^{11} + 20 q^{12} + 18 q^{13} + 53 q^{15} + 12 q^{16} + 16 q^{18} + 6 q^{19} + 39 q^{20} - 59 q^{22} - 81 q^{23} + 141 q^{24} + 57 q^{25} + 97 q^{27} - 63 q^{29} + 58 q^{30} + 29 q^{31} - 246 q^{32} + 73 q^{33} + 99 q^{34} + 76 q^{36} + 40 q^{37} - 48 q^{38} - 124 q^{39} + 105 q^{40} + 51 q^{41} + 65 q^{43} - 143 q^{45} - 150 q^{46} + 261 q^{47} + 113 q^{48} + 63 q^{50} - 63 q^{51} + 46 q^{52} - 52 q^{54} + 100 q^{55} + 224 q^{57} + 40 q^{58} - 102 q^{59} + 379 q^{60} - 78 q^{61} + 106 q^{64} + 225 q^{65} - 340 q^{66} - 132 q^{67} + 27 q^{68} + 297 q^{69} + 288 q^{72} + 2 q^{73} - 342 q^{74} - 541 q^{75} - 233 q^{76} - 440 q^{78} + 140 q^{79} + 65 q^{81} - 314 q^{82} - 255 q^{83} + 102 q^{85} - 504 q^{86} + 568 q^{87} + 408 q^{88} - 418 q^{90} - 1239 q^{92} - 174 q^{93} - 261 q^{94} - 642 q^{95} + 142 q^{96} - 178 q^{97} - 103 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1 −2.48702 + 1.43588i −1.98894 + 2.24591i 2.12350 3.67801i −6.53753 3.77444i 1.72168 8.44150i 0 0.709334i −1.08822 8.93397i 21.6786
50.2 −2.37724 + 1.37250i −2.78720 1.10973i 1.76751 3.06142i 2.32531 + 1.34252i 8.14895 1.18734i 0 1.27635i 6.53700 + 6.18608i −7.37042
50.3 −1.86624 + 1.07747i 2.92087 + 0.684463i 0.321900 0.557548i 1.62693 + 0.939308i −6.18854 + 1.86979i 0 7.23243i 8.06302 + 3.99846i −4.04832
50.4 −1.11318 + 0.642694i −0.441006 + 2.96741i −1.17389 + 2.03324i 6.82011 + 3.93759i −1.41622 3.58669i 0 8.15936i −8.61103 2.61729i −10.1227
50.5 −0.0664669 + 0.0383747i −2.03492 2.20433i −1.99705 + 3.45900i −3.53076 2.03848i 0.219846 + 0.0684257i 0 0.613543i −0.718171 + 8.97130i 0.312905
50.6 0.444866 0.256844i 0.579189 + 2.94356i −1.86806 + 3.23558i −6.08350 3.51231i 1.01370 + 1.16073i 0 3.97395i −8.32908 + 3.40976i −3.60846
50.7 1.11793 0.645440i −2.77557 + 1.13852i −1.16681 + 2.02098i 4.18841 + 2.41818i −2.36806 + 3.06425i 0 8.17595i 6.40756 6.32006i 6.24316
50.8 1.26920 0.732774i 2.94827 + 0.554715i −0.926086 + 1.60403i 0.998268 + 0.576350i 4.14843 1.45637i 0 8.57663i 8.38458 + 3.27090i 1.68934
50.9 2.09020 1.20678i −1.00417 2.82695i 0.912615 1.58070i 5.93347 + 3.42569i −5.51041 4.69707i 0 5.24892i −6.98329 + 5.67747i 16.5362
50.10 2.79169 1.61178i −2.80334 + 1.06831i 3.19568 5.53509i −4.79167 2.76647i −6.10417 + 7.50076i 0 7.70873i 6.71743 5.98966i −17.8358
50.11 3.19625 1.84536i 1.88682 + 2.33236i 4.81069 8.33236i 5.05096 + 2.91617i 10.3348 + 3.97296i 0 20.7469i −1.87982 + 8.80149i 21.5255
344.1 −2.48702 1.43588i −1.98894 2.24591i 2.12350 + 3.67801i −6.53753 + 3.77444i 1.72168 + 8.44150i 0 0.709334i −1.08822 + 8.93397i 21.6786
344.2 −2.37724 1.37250i −2.78720 + 1.10973i 1.76751 + 3.06142i 2.32531 1.34252i 8.14895 + 1.18734i 0 1.27635i 6.53700 6.18608i −7.37042
344.3 −1.86624 1.07747i 2.92087 0.684463i 0.321900 + 0.557548i 1.62693 0.939308i −6.18854 1.86979i 0 7.23243i 8.06302 3.99846i −4.04832
344.4 −1.11318 0.642694i −0.441006 2.96741i −1.17389 2.03324i 6.82011 3.93759i −1.41622 + 3.58669i 0 8.15936i −8.61103 + 2.61729i −10.1227
344.5 −0.0664669 0.0383747i −2.03492 + 2.20433i −1.99705 3.45900i −3.53076 + 2.03848i 0.219846 0.0684257i 0 0.613543i −0.718171 8.97130i 0.312905
344.6 0.444866 + 0.256844i 0.579189 2.94356i −1.86806 3.23558i −6.08350 + 3.51231i 1.01370 1.16073i 0 3.97395i −8.32908 3.40976i −3.60846
344.7 1.11793 + 0.645440i −2.77557 1.13852i −1.16681 2.02098i 4.18841 2.41818i −2.36806 3.06425i 0 8.17595i 6.40756 + 6.32006i 6.24316
344.8 1.26920 + 0.732774i 2.94827 0.554715i −0.926086 1.60403i 0.998268 0.576350i 4.14843 + 1.45637i 0 8.57663i 8.38458 3.27090i 1.68934
344.9 2.09020 + 1.20678i −1.00417 + 2.82695i 0.912615 + 1.58070i 5.93347 3.42569i −5.51041 + 4.69707i 0 5.24892i −6.98329 5.67747i 16.5362
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.11
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.f 22
7.b odd 2 1 441.3.r.g 22
7.c even 3 1 441.3.j.f 22
7.c even 3 1 441.3.n.f 22
7.d odd 6 1 63.3.j.b 22
7.d odd 6 1 63.3.n.b yes 22
9.d odd 6 1 inner 441.3.r.f 22
21.g even 6 1 189.3.j.b 22
21.g even 6 1 189.3.n.b 22
63.i even 6 1 63.3.n.b yes 22
63.j odd 6 1 441.3.n.f 22
63.k odd 6 1 189.3.j.b 22
63.n odd 6 1 441.3.j.f 22
63.o even 6 1 441.3.r.g 22
63.s even 6 1 63.3.j.b 22
63.t odd 6 1 189.3.n.b 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.j.b 22 7.d odd 6 1
63.3.j.b 22 63.s even 6 1
63.3.n.b yes 22 7.d odd 6 1
63.3.n.b yes 22 63.i even 6 1
189.3.j.b 22 21.g even 6 1
189.3.j.b 22 63.k odd 6 1
189.3.n.b 22 21.g even 6 1
189.3.n.b 22 63.t odd 6 1
441.3.j.f 22 7.c even 3 1
441.3.j.f 22 63.n odd 6 1
441.3.n.f 22 7.c even 3 1
441.3.n.f 22 63.j odd 6 1
441.3.r.f 22 1.a even 1 1 trivial
441.3.r.f 22 9.d odd 6 1 inner
441.3.r.g 22 7.b odd 2 1
441.3.r.g 22 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 \(S_{3}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{22} - 6 T_{2}^{21} - 10 T_{2}^{20} + 132 T_{2}^{19} + 63 T_{2}^{18} - 1884 T_{2}^{17} + 887 T_{2}^{16} + 15735 T_{2}^{15} - 12503 T_{2}^{14} - 93906 T_{2}^{13} + 112326 T_{2}^{12} + 336231 T_{2}^{11} - 485559 T_{2}^{10} + \cdots + 2187 \) Copy content Toggle raw display
\( T_{5}^{22} - 12 T_{5}^{21} - 94 T_{5}^{20} + 1704 T_{5}^{19} + 6369 T_{5}^{18} - 161319 T_{5}^{17} - 40288 T_{5}^{16} + 8740143 T_{5}^{15} - 10719116 T_{5}^{14} - 345688017 T_{5}^{13} + \cdots + 112049133289443 \) Copy content Toggle raw display