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Properties

Label 441.3.n.f
Level $441$
Weight $3$
Character orbit 441.n
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(128,441)]
 
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, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.128");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.n (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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 6 q^{2} - 8 q^{3} + 12 q^{4} + 8 q^{6} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 6 q^{2} - 8 q^{3} + 12 q^{4} + 8 q^{6} + 20 q^{9} - 25 q^{10} + 20 q^{12} + 18 q^{13} + 53 q^{15} + 12 q^{16} - 6 q^{17} - 56 q^{18} - 3 q^{19} + 39 q^{20} - 59 q^{22} - 15 q^{24} - 114 q^{25} + 3 q^{26} + 97 q^{27} - 63 q^{29} - 20 q^{30} + 29 q^{31} + 246 q^{32} - 77 q^{33} + 99 q^{34} + 76 q^{36} - 20 q^{37} + 200 q^{39} - 210 q^{40} + 51 q^{41} + 65 q^{43} + 54 q^{44} - 71 q^{45} + 75 q^{46} - 261 q^{47} + 113 q^{48} + 63 q^{50} - 78 q^{51} - 92 q^{52} - 63 q^{53} + 485 q^{54} + 100 q^{55} + 224 q^{57} - 80 q^{58} + 102 q^{59} + 103 q^{60} - 78 q^{61} + 106 q^{64} - 225 q^{65} + 401 q^{66} - 132 q^{67} + 297 q^{69} - 66 q^{72} - q^{73} + 245 q^{75} - 233 q^{76} - 440 q^{78} + 140 q^{79} - 96 q^{80} + 104 q^{81} + 157 q^{82} - 255 q^{83} + 102 q^{85} + 136 q^{87} - 816 q^{88} + 720 q^{89} - 418 q^{90} - 1239 q^{92} + 210 q^{93} - 261 q^{94} + 642 q^{95} - 539 q^{96} - 178 q^{97} - 103 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
128.1 −3.19625 + 1.84536i −2.96329 − 0.467854i 4.81069 − 8.33236i 5.83234i 10.3348 − 3.97296i 0 20.7469i 8.56223 + 2.77278i −10.7628 − 18.6416i
128.2 −2.79169 + 1.61178i 0.476488 + 2.96192i 3.19568 − 5.53509i − 5.53294i −6.10417 − 7.50076i 0 7.70873i −8.54592 + 2.82264i 8.91790 + 15.4463i
128.3 −2.09020 + 1.20678i 2.95029 − 0.543839i 0.912615 − 1.58070i 6.85138i −5.51041 + 4.69707i 0 − 5.24892i 8.40848 − 3.20897i −8.26808 − 14.3207i
128.4 −1.26920 + 0.732774i −1.95453 − 2.27592i −0.926086 + 1.60403i 1.15270i 4.14843 + 1.45637i 0 − 8.57663i −1.35961 + 8.89671i −0.844669 − 1.46301i
128.5 −1.11793 + 0.645440i 0.401800 + 2.97297i −1.16681 + 2.02098i 4.83636i −2.36806 − 3.06425i 0 − 8.17595i −8.67711 + 2.38908i −3.12158 − 5.40673i
128.6 −0.444866 + 0.256844i −2.83879 + 0.970187i −1.86806 + 3.23558i − 7.02462i 1.01370 − 1.16073i 0 − 3.97395i 7.11748 − 5.50832i 1.80423 + 3.12502i
128.7 0.0664669 − 0.0383747i 2.92647 + 0.660129i −1.99705 + 3.45900i − 4.07697i 0.219846 − 0.0684257i 0 0.613543i 8.12846 + 3.86370i −0.156452 − 0.270983i
128.8 1.11318 − 0.642694i −2.34935 + 1.86563i −1.17389 + 2.03324i 7.87519i −1.41622 + 3.58669i 0 8.15936i 2.03887 − 8.76601i 5.06133 + 8.76649i
128.9 1.86624 − 1.07747i −2.05320 − 2.18732i 0.321900 − 0.557548i 1.87862i −6.18854 − 1.86979i 0 7.23243i −0.568739 + 8.98201i 2.02416 + 3.50595i
128.10 2.37724 − 1.37250i 2.35466 + 1.85892i 1.76751 − 3.06142i 2.68504i 8.14895 + 1.18734i 0 1.27635i 2.08880 + 8.75425i 3.68521 + 6.38297i
128.11 2.48702 − 1.43588i −0.950543 + 2.84543i 2.12350 − 3.67801i − 7.54889i 1.72168 + 8.44150i 0 − 0.709334i −7.19294 − 5.40941i −10.8393 − 18.7742i
410.1 −3.19625 − 1.84536i −2.96329 + 0.467854i 4.81069 + 8.33236i − 5.83234i 10.3348 + 3.97296i 0 − 20.7469i 8.56223 − 2.77278i −10.7628 + 18.6416i
410.2 −2.79169 − 1.61178i 0.476488 − 2.96192i 3.19568 + 5.53509i 5.53294i −6.10417 + 7.50076i 0 − 7.70873i −8.54592 − 2.82264i 8.91790 − 15.4463i
410.3 −2.09020 − 1.20678i 2.95029 + 0.543839i 0.912615 + 1.58070i − 6.85138i −5.51041 − 4.69707i 0 5.24892i 8.40848 + 3.20897i −8.26808 + 14.3207i
410.4 −1.26920 − 0.732774i −1.95453 + 2.27592i −0.926086 − 1.60403i − 1.15270i 4.14843 − 1.45637i 0 8.57663i −1.35961 − 8.89671i −0.844669 + 1.46301i
410.5 −1.11793 − 0.645440i 0.401800 − 2.97297i −1.16681 − 2.02098i − 4.83636i −2.36806 + 3.06425i 0 8.17595i −8.67711 − 2.38908i −3.12158 + 5.40673i
410.6 −0.444866 − 0.256844i −2.83879 − 0.970187i −1.86806 − 3.23558i 7.02462i 1.01370 + 1.16073i 0 3.97395i 7.11748 + 5.50832i 1.80423 − 3.12502i
410.7 0.0664669 + 0.0383747i 2.92647 − 0.660129i −1.99705 − 3.45900i 4.07697i 0.219846 + 0.0684257i 0 − 0.613543i 8.12846 − 3.86370i −0.156452 + 0.270983i
410.8 1.11318 + 0.642694i −2.34935 − 1.86563i −1.17389 − 2.03324i − 7.87519i −1.41622 − 3.58669i 0 − 8.15936i 2.03887 + 8.76601i 5.06133 − 8.76649i
410.9 1.86624 + 1.07747i −2.05320 + 2.18732i 0.321900 + 0.557548i − 1.87862i −6.18854 + 1.86979i 0 − 7.23243i −0.568739 − 8.98201i 2.02416 − 3.50595i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 128.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.n 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.n.f 22
7.b odd 2 1 63.3.n.b yes 22
7.c even 3 1 441.3.j.f 22
7.c even 3 1 441.3.r.f 22
7.d odd 6 1 63.3.j.b ✓ 22
7.d odd 6 1 441.3.r.g 22
9.d odd 6 1 441.3.j.f 22
21.c even 2 1 189.3.n.b 22
21.g even 6 1 189.3.j.b 22
63.i even 6 1 441.3.r.g 22
63.j odd 6 1 441.3.r.f 22
63.k odd 6 1 189.3.n.b 22
63.l odd 6 1 189.3.j.b 22
63.n odd 6 1 inner 441.3.n.f 22
63.o even 6 1 63.3.j.b ✓ 22
63.s even 6 1 63.3.n.b yes 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.o even 6 1
63.3.n.b yes 22 7.b odd 2 1
63.3.n.b yes 22 63.s even 6 1
189.3.j.b 22 21.g even 6 1
189.3.j.b 22 63.l odd 6 1
189.3.n.b 22 21.c even 2 1
189.3.n.b 22 63.k odd 6 1
441.3.j.f 22 7.c even 3 1
441.3.j.f 22 9.d odd 6 1
441.3.n.f 22 1.a even 1 1 trivial
441.3.n.f 22 63.n odd 6 1 inner
441.3.r.f 22 7.c even 3 1
441.3.r.f 22 63.j odd 6 1
441.3.r.g 22 7.d odd 6 1
441.3.r.g 22 63.i even 6 1

Hecke kernels

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

\( T_{2}^{22} + 6 T_{2}^{21} - 10 T_{2}^{20} - 132 T_{2}^{19} + 63 T_{2}^{18} + 1884 T_{2}^{17} + \cdots + 2187 \) Copy content Toggle raw display
\( T_{13}^{22} - 18 T_{13}^{21} + 1051 T_{13}^{20} - 8096 T_{13}^{19} + 547897 T_{13}^{18} + \cdots + 75\!\cdots\!29 \) Copy content Toggle raw display