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Properties

Label 441.3.k.b
Level $441$
Weight $3$
Character orbit 441.k
Analytic conductor $12.016$
Analytic rank $0$
Dimension $28$
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(31,441)]
 
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([2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.k (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: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{2} + 3 q^{3} - 23 q^{4} - 12 q^{6} - 16 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{2} + 3 q^{3} - 23 q^{4} - 12 q^{6} - 16 q^{8} + 9 q^{9} + 6 q^{10} - 14 q^{11} + 3 q^{12} - 15 q^{13} - 18 q^{15} - 27 q^{16} + 33 q^{17} + 33 q^{18} + 6 q^{19} - 108 q^{20} - 10 q^{22} - 68 q^{23} - 42 q^{24} - 62 q^{25} - 54 q^{26} + 81 q^{27} + 70 q^{29} - 6 q^{30} - 45 q^{31} + 153 q^{32} + 114 q^{33} - 12 q^{34} - 174 q^{36} + 9 q^{37} - 120 q^{39} + 234 q^{41} + 30 q^{43} + 51 q^{44} - 276 q^{45} - 22 q^{46} + 111 q^{47} - 147 q^{48} + 241 q^{50} - 6 q^{51} + 148 q^{53} - 378 q^{54} + 189 q^{57} - 34 q^{58} - 42 q^{59} + 456 q^{60} - 120 q^{61} - 48 q^{64} + 114 q^{65} + 447 q^{66} - 34 q^{67} - 78 q^{69} - 350 q^{71} - 339 q^{72} + 6 q^{73} - 718 q^{74} + 123 q^{75} - 72 q^{76} - 375 q^{78} - 82 q^{79} + 609 q^{80} - 3 q^{81} + 18 q^{82} - 738 q^{83} + 3 q^{85} - 34 q^{86} - 3 q^{87} - 50 q^{88} - 21 q^{89} + 543 q^{90} + 288 q^{92} + 252 q^{93} + 3 q^{94} + 507 q^{95} + 582 q^{96} + 57 q^{97} - 162 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.67789 + 2.90618i −2.39606 + 1.80524i −3.63061 − 6.28839i − 8.51666i −1.22605 − 9.99238i 0 10.9439 2.48221 − 8.65093i 24.7510 + 14.2900i
31.2 −1.67756 + 2.90562i 2.99109 + 0.230993i −3.62842 − 6.28461i − 0.888628i −5.68892 + 8.30348i 0 10.9271 8.89328 + 1.38184i 2.58202 + 1.49073i
31.3 −1.32841 + 2.30087i −0.187748 + 2.99412i −1.52933 − 2.64888i 9.20400i −6.63967 − 4.40939i 0 −2.50096 −8.92950 − 1.12428i −21.1772 − 12.2267i
31.4 −1.12025 + 1.94033i −1.92745 − 2.29890i −0.509909 − 0.883189i − 1.93444i 6.61983 − 1.16455i 0 −6.67708 −1.56985 + 8.86203i 3.75345 + 2.16706i
31.5 −0.840995 + 1.45665i 1.62217 − 2.52360i 0.585454 + 1.01404i 2.34462i 2.31176 + 4.48526i 0 −8.69742 −3.73715 − 8.18741i −3.41529 − 1.97182i
31.6 −0.227576 + 0.394173i 2.69941 + 1.30889i 1.89642 + 3.28469i − 4.37081i −1.13025 + 0.766164i 0 −3.54692 5.57364 + 7.06644i 1.72285 + 0.994690i
31.7 −0.178911 + 0.309883i −2.99942 + 0.0589959i 1.93598 + 3.35322i 4.59004i 0.518348 − 0.940026i 0 −2.81677 8.99304 − 0.353907i −1.42238 − 0.821210i
31.8 0.198068 − 0.343064i −1.78343 + 2.41234i 1.92154 + 3.32820i − 2.97240i 0.474346 + 1.08964i 0 3.10693 −2.63876 − 8.60447i −1.01972 − 0.588737i
31.9 0.662399 − 1.14731i 1.57692 − 2.55212i 1.12246 + 1.94415i − 7.23514i −1.88353 − 3.49973i 0 8.27324 −4.02667 − 8.04897i −8.30093 − 4.79254i
31.10 0.826674 − 1.43184i 2.53185 + 1.60927i 0.633221 + 1.09677i 7.86923i 4.39723 − 2.29486i 0 8.70726 3.82051 + 8.14885i 11.2675 + 6.50529i
31.11 0.902282 − 1.56280i −0.540538 − 2.95090i 0.371774 + 0.643931i 5.75495i −5.09938 − 1.81779i 0 8.56004 −8.41564 + 3.19015i 8.99383 + 5.19259i
31.12 1.41697 − 2.45427i −0.222472 + 2.99174i −2.01561 − 3.49114i − 2.39855i 7.02729 + 4.78521i 0 −0.0884848 −8.90101 − 1.33116i −5.88667 − 3.39867i
31.13 1.62718 − 2.81835i −2.71323 − 1.27999i −3.29541 − 5.70782i − 3.39483i −8.02237 + 5.56407i 0 −8.43145 5.72324 + 6.94582i −9.56782 − 5.52398i
31.14 1.91801 − 3.32210i 2.84892 − 0.940039i −5.35756 − 9.27956i 0.216546i 2.34136 − 11.2674i 0 −25.7594 7.23265 − 5.35619i 0.719386 + 0.415338i
313.1 −1.67789 − 2.90618i −2.39606 − 1.80524i −3.63061 + 6.28839i 8.51666i −1.22605 + 9.99238i 0 10.9439 2.48221 + 8.65093i 24.7510 − 14.2900i
313.2 −1.67756 − 2.90562i 2.99109 − 0.230993i −3.62842 + 6.28461i 0.888628i −5.68892 − 8.30348i 0 10.9271 8.89328 − 1.38184i 2.58202 − 1.49073i
313.3 −1.32841 − 2.30087i −0.187748 − 2.99412i −1.52933 + 2.64888i − 9.20400i −6.63967 + 4.40939i 0 −2.50096 −8.92950 + 1.12428i −21.1772 + 12.2267i
313.4 −1.12025 − 1.94033i −1.92745 + 2.29890i −0.509909 + 0.883189i 1.93444i 6.61983 + 1.16455i 0 −6.67708 −1.56985 − 8.86203i 3.75345 − 2.16706i
313.5 −0.840995 − 1.45665i 1.62217 + 2.52360i 0.585454 − 1.01404i − 2.34462i 2.31176 − 4.48526i 0 −8.69742 −3.73715 + 8.18741i −3.41529 + 1.97182i
313.6 −0.227576 − 0.394173i 2.69941 − 1.30889i 1.89642 − 3.28469i 4.37081i −1.13025 − 0.766164i 0 −3.54692 5.57364 − 7.06644i 1.72285 − 0.994690i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.k 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.k.b 28
7.b odd 2 1 63.3.k.a ✓ 28
7.c even 3 1 63.3.t.a yes 28
7.c even 3 1 441.3.l.b 28
7.d odd 6 1 441.3.l.a 28
7.d odd 6 1 441.3.t.a 28
9.c even 3 1 441.3.t.a 28
21.c even 2 1 189.3.k.a 28
21.h odd 6 1 189.3.t.a 28
63.g even 3 1 63.3.k.a ✓ 28
63.h even 3 1 441.3.l.a 28
63.k odd 6 1 inner 441.3.k.b 28
63.l odd 6 1 63.3.t.a yes 28
63.n odd 6 1 189.3.k.a 28
63.o even 6 1 189.3.t.a 28
63.t odd 6 1 441.3.l.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.k.a ✓ 28 7.b odd 2 1
63.3.k.a ✓ 28 63.g even 3 1
63.3.t.a yes 28 7.c even 3 1
63.3.t.a yes 28 63.l odd 6 1
189.3.k.a 28 21.c even 2 1
189.3.k.a 28 63.n odd 6 1
189.3.t.a 28 21.h odd 6 1
189.3.t.a 28 63.o even 6 1
441.3.k.b 28 1.a even 1 1 trivial
441.3.k.b 28 63.k odd 6 1 inner
441.3.l.a 28 7.d odd 6 1
441.3.l.a 28 63.h even 3 1
441.3.l.b 28 7.c even 3 1
441.3.l.b 28 63.t odd 6 1
441.3.t.a 28 7.d odd 6 1
441.3.t.a 28 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - T_{2}^{27} + 40 T_{2}^{26} - 29 T_{2}^{25} + 977 T_{2}^{24} - 620 T_{2}^{23} + \cdots + 1034289 \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display