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Properties

Label 441.3.k.c
Level $441$
Weight $3$
Character orbit 441.k
Analytic conductor $12.016$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $4$

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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: \(96\)
Relative dimension: \(48\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 96 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 96 q^{4} - 8 q^{9} + 48 q^{11} + 104 q^{15} - 192 q^{16} + 128 q^{18} + 192 q^{23} - 480 q^{25} - 144 q^{29} + 336 q^{30} - 120 q^{32} + 136 q^{36} + 208 q^{39} - 192 q^{44} - 912 q^{50} - 480 q^{51} - 48 q^{53} - 256 q^{57} - 1272 q^{60} + 768 q^{64} - 168 q^{65} + 1344 q^{71} + 80 q^{72} + 336 q^{74} + 920 q^{78} + 24 q^{79} + 696 q^{81} - 120 q^{85} - 816 q^{86} - 1584 q^{92} + 784 q^{93} - 672 q^{95} - 672 q^{99}+O(q^{100}) \) 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 −1.95544 + 3.38693i −2.28555 + 1.94325i −5.64751 − 9.78178i 4.44440i −2.11239 − 11.5409i 0 28.5300 1.44752 − 8.88283i −15.0528 − 8.69076i
31.2 −1.95544 + 3.38693i 2.28555 − 1.94325i −5.64751 − 9.78178i − 4.44440i 2.11239 + 11.5409i 0 28.5300 1.44752 − 8.88283i 15.0528 + 8.69076i
31.3 −1.75139 + 3.03349i −0.261939 + 2.98854i −4.13472 − 7.16155i 0.119517i −8.60697 − 6.02869i 0 14.9549 −8.86278 − 1.56563i −0.362553 − 0.209320i
31.4 −1.75139 + 3.03349i 0.261939 − 2.98854i −4.13472 − 7.16155i − 0.119517i 8.60697 + 6.02869i 0 14.9549 −8.86278 − 1.56563i 0.362553 + 0.209320i
31.5 −1.68596 + 2.92017i −1.07479 − 2.80086i −3.68492 − 6.38247i 4.36228i 9.99104 + 1.58357i 0 11.3628 −6.68964 + 6.02069i −12.7386 − 7.35464i
31.6 −1.68596 + 2.92017i 1.07479 + 2.80086i −3.68492 − 6.38247i − 4.36228i −9.99104 − 1.58357i 0 11.3628 −6.68964 + 6.02069i 12.7386 + 7.35464i
31.7 −1.51376 + 2.62191i −2.85433 − 0.923457i −2.58293 − 4.47377i 5.21800i 6.74199 − 6.08591i 0 3.52967 7.29446 + 5.27171i −13.6811 − 7.89879i
31.8 −1.51376 + 2.62191i 2.85433 + 0.923457i −2.58293 − 4.47377i − 5.21800i −6.74199 + 6.08591i 0 3.52967 7.29446 + 5.27171i 13.6811 + 7.89879i
31.9 −1.38685 + 2.40210i −0.388293 + 2.97477i −1.84672 − 3.19861i − 2.65547i −6.60717 − 5.05828i 0 −0.850313 −8.69846 − 2.31016i 6.37869 + 3.68274i
31.10 −1.38685 + 2.40210i 0.388293 − 2.97477i −1.84672 − 3.19861i 2.65547i 6.60717 + 5.05828i 0 −0.850313 −8.69846 − 2.31016i −6.37869 − 3.68274i
31.11 −1.29999 + 2.25165i −2.68466 + 1.33888i −1.37995 − 2.39015i 6.20726i 0.475339 − 7.78545i 0 −3.22423 5.41479 − 7.18888i −13.9766 − 8.06939i
31.12 −1.29999 + 2.25165i 2.68466 − 1.33888i −1.37995 − 2.39015i − 6.20726i −0.475339 + 7.78545i 0 −3.22423 5.41479 − 7.18888i 13.9766 + 8.06939i
31.13 −0.947236 + 1.64066i −2.97748 − 0.366867i 0.205489 + 0.355918i − 4.85008i 3.42228 − 4.53753i 0 −8.35647 8.73082 + 2.18468i 7.95734 + 4.59417i
31.14 −0.947236 + 1.64066i 2.97748 + 0.366867i 0.205489 + 0.355918i 4.85008i −3.42228 + 4.53753i 0 −8.35647 8.73082 + 2.18468i −7.95734 − 4.59417i
31.15 −0.793889 + 1.37506i −2.56239 − 1.56018i 0.739481 + 1.28082i − 5.74216i 4.17959 − 2.28481i 0 −8.69937 4.13166 + 7.99559i 7.89580 + 4.55864i
31.16 −0.793889 + 1.37506i 2.56239 + 1.56018i 0.739481 + 1.28082i 5.74216i −4.17959 + 2.28481i 0 −8.69937 4.13166 + 7.99559i −7.89580 − 4.55864i
31.17 −0.574972 + 0.995880i −2.69930 + 1.30911i 1.33881 + 2.31890i 5.51529i 0.248301 − 3.44088i 0 −7.67890 5.57244 − 7.06738i −5.49257 − 3.17114i
31.18 −0.574972 + 0.995880i 2.69930 − 1.30911i 1.33881 + 2.31890i − 5.51529i −0.248301 + 3.44088i 0 −7.67890 5.57244 − 7.06738i 5.49257 + 3.17114i
31.19 −0.516223 + 0.894124i −0.614760 + 2.93634i 1.46703 + 2.54097i 0.568971i −2.30809 − 2.06547i 0 −7.15903 −8.24414 − 3.61029i −0.508730 − 0.293715i
31.20 −0.516223 + 0.894124i 0.614760 − 2.93634i 1.46703 + 2.54097i − 0.568971i 2.30809 + 2.06547i 0 −7.15903 −8.24414 − 3.61029i 0.508730 + 0.293715i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.g even 3 1 inner
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.c 96
7.b odd 2 1 inner 441.3.k.c 96
7.c even 3 1 441.3.l.c ✓ 96
7.c even 3 1 441.3.t.c 96
7.d odd 6 1 441.3.l.c ✓ 96
7.d odd 6 1 441.3.t.c 96
9.c even 3 1 441.3.t.c 96
63.g even 3 1 inner 441.3.k.c 96
63.h even 3 1 441.3.l.c ✓ 96
63.k odd 6 1 inner 441.3.k.c 96
63.l odd 6 1 441.3.t.c 96
63.t odd 6 1 441.3.l.c ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.3.k.c 96 1.a even 1 1 trivial
441.3.k.c 96 7.b odd 2 1 inner
441.3.k.c 96 63.g even 3 1 inner
441.3.k.c 96 63.k odd 6 1 inner
441.3.l.c ✓ 96 7.c even 3 1
441.3.l.c ✓ 96 7.d odd 6 1
441.3.l.c ✓ 96 63.h even 3 1
441.3.l.c ✓ 96 63.t odd 6 1
441.3.t.c 96 7.c even 3 1
441.3.t.c 96 7.d odd 6 1
441.3.t.c 96 9.c even 3 1
441.3.t.c 96 63.l odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 72 T_{2}^{46} + 2952 T_{2}^{44} + 12 T_{2}^{43} + 82560 T_{2}^{42} + 1044 T_{2}^{41} + \cdots + 532130857729 \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display