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Properties

Label 441.3.r.g
Level $441$
Weight $3$
Character orbit 441.r
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(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 algebraic \(q\)-expansion of this newform has not been computed, 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

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 −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:
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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.g 22
7.b odd 2 1 441.3.r.f 22
7.c even 3 1 63.3.j.b ✓ 22
7.c even 3 1 63.3.n.b yes 22
7.d odd 6 1 441.3.j.f 22
7.d odd 6 1 441.3.n.f 22
9.d odd 6 1 inner 441.3.r.g 22
21.h odd 6 1 189.3.j.b 22
21.h odd 6 1 189.3.n.b 22
63.g even 3 1 189.3.j.b 22
63.h even 3 1 189.3.n.b 22
63.i even 6 1 441.3.n.f 22
63.j odd 6 1 63.3.n.b yes 22
63.n odd 6 1 63.3.j.b ✓ 22
63.o even 6 1 441.3.r.f 22
63.s even 6 1 441.3.j.f 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.j.b ✓ 22 7.c even 3 1
63.3.j.b ✓ 22 63.n odd 6 1
63.3.n.b yes 22 7.c even 3 1
63.3.n.b yes 22 63.j odd 6 1
189.3.j.b 22 21.h odd 6 1
189.3.j.b 22 63.g even 3 1
189.3.n.b 22 21.h odd 6 1
189.3.n.b 22 63.h even 3 1
441.3.j.f 22 7.d odd 6 1
441.3.j.f 22 63.s even 6 1
441.3.n.f 22 7.d odd 6 1
441.3.n.f 22 63.i even 6 1
441.3.r.f 22 7.b odd 2 1
441.3.r.f 22 63.o even 6 1
441.3.r.g 22 1.a even 1 1 trivial
441.3.r.g 22 9.d odd 6 1 inner

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
\( T_{5}^{22} + 12 T_{5}^{21} - 94 T_{5}^{20} - 1704 T_{5}^{19} + 6369 T_{5}^{18} + 161319 T_{5}^{17} + \cdots + 112049133289443 \) Copy content Toggle raw display