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Properties

Label 144.2.x.e
Level $144$
Weight $2$
Character orbit 144.x
Analytic conductor $1.150$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,2,Mod(13,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 9, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 28 q^{6} - 8 q^{8} - 20 q^{10} - 2 q^{11} + 8 q^{12} - 16 q^{13} - 4 q^{14} - 20 q^{15} - 10 q^{16} - 16 q^{17} + 28 q^{19} + 12 q^{20} - 16 q^{21} - 8 q^{22}+ \cdots + 28 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.39409 + 0.237718i 1.28937 − 1.15652i 1.88698 − 0.662802i −0.326078 − 1.21694i −1.52257 + 1.91880i 0.707732 − 0.408609i −2.47306 + 1.37258i 0.324925 − 2.98235i 0.743871 + 1.61901i
13.2 −1.31447 + 0.521699i 0.241506 + 1.71513i 1.45566 − 1.37152i 0.531653 + 1.98415i −1.21223 − 2.12849i 1.54969 − 0.894715i −1.19790 + 2.56223i −2.88335 + 0.828427i −1.73397 − 2.33075i
13.3 −1.17822 − 0.782180i 1.34849 + 1.08700i 0.776390 + 1.84315i −0.777246 − 2.90072i −0.738585 − 2.33549i 1.04527 − 0.603486i 0.526922 − 2.77891i 0.636858 + 2.93162i −1.35312 + 4.02562i
13.4 −0.709450 − 1.22339i −0.705067 − 1.58205i −0.993361 + 1.73587i −0.679606 − 2.53632i −1.43525 + 1.98496i −0.614293 + 0.354662i 2.82838 − 0.0162452i −2.00576 + 2.23090i −2.62076 + 2.63082i
13.5 −0.671979 + 1.24436i 1.72444 − 0.162237i −1.09689 − 1.67237i 0.592492 + 2.21121i −0.956902 + 2.25485i −2.67054 + 1.54184i 2.81813 − 0.241128i 2.94736 − 0.559536i −3.14970 − 0.748611i
13.6 −0.595953 − 1.28251i 1.66966 − 0.460673i −1.28968 + 1.52863i 0.722679 + 2.69708i −1.58586 − 1.86683i 2.89314 − 1.67035i 2.72908 + 0.743037i 2.57556 − 1.53834i 3.02835 − 2.53418i
13.7 −0.174567 + 1.40340i 0.215523 − 1.71859i −1.93905 − 0.489974i −0.733432 − 2.73721i 2.37424 + 0.602473i 1.14487 − 0.660988i 1.02612 − 2.63573i −2.90710 − 0.740791i 3.96942 − 0.551472i
13.8 0.193628 + 1.40090i −1.53873 − 0.795173i −1.92502 + 0.542506i 0.646846 + 2.41406i 0.816012 − 2.30957i −2.82197 + 1.62927i −1.13273 − 2.59170i 1.73540 + 2.44712i −3.25660 + 1.37359i
13.9 0.282685 + 1.38567i 0.944122 + 1.45211i −1.84018 + 0.783419i 0.131764 + 0.491749i −1.74526 + 1.71874i 2.40518 − 1.38863i −1.60575 − 2.32842i −1.21727 + 2.74194i −0.644156 + 0.321592i
13.10 0.327151 − 1.37585i −0.844796 + 1.51206i −1.78594 − 0.900224i −0.891015 − 3.32531i 1.80399 + 1.65699i 3.95817 − 2.28525i −1.82285 + 2.16269i −1.57264 − 2.55476i −4.86664 + 0.138025i
13.11 0.562928 − 1.29735i 0.388965 − 1.68781i −1.36622 − 1.46063i 0.226831 + 0.846545i −1.97072 − 1.45474i 0.567074 − 0.327400i −2.66403 + 0.950239i −2.69741 − 1.31300i 1.22595 + 0.182265i
13.12 0.932590 − 1.06314i 1.51401 + 0.841300i −0.260550 − 1.98296i 0.0185225 + 0.0691269i 2.30637 − 0.825017i −1.28192 + 0.740118i −2.35115 − 1.57228i 1.58443 + 2.54747i 0.0907658 + 0.0447750i
13.13 1.04877 + 0.948722i −1.13227 + 1.31071i 0.199854 + 1.98999i 0.00302457 + 0.0112878i −2.43100 + 0.300433i −1.05753 + 0.610563i −1.67835 + 2.27665i −0.435936 − 2.96816i −0.00753693 + 0.0147079i
13.14 1.06791 + 0.927124i 1.71768 − 0.222657i 0.280882 + 1.98018i −0.798307 − 2.97932i 2.04077 + 1.35472i −1.78208 + 1.02889i −1.53591 + 2.37507i 2.90085 − 0.764906i 1.90968 − 3.92179i
13.15 1.24122 − 0.677764i −1.37530 + 1.05287i 1.08127 − 1.68251i 1.10094 + 4.10876i −0.993456 + 2.23898i 1.63313 − 0.942891i 0.201751 − 2.82122i 0.782911 − 2.89604i 4.15129 + 4.35372i
13.16 1.26390 − 0.634467i −1.68301 − 0.409248i 1.19490 − 1.60381i −0.884514 − 3.30105i −2.38681 + 0.550563i −2.63210 + 1.51965i 0.492680 − 2.78519i 2.66503 + 1.37753i −3.21235 − 3.61102i
13.17 1.30272 + 0.550382i −1.21980 − 1.22967i 1.39416 + 1.43399i −0.0468197 − 0.174734i −0.912266 − 2.27327i 4.04791 − 2.33706i 1.02696 + 2.63540i −0.0241875 + 2.99990i 0.0351772 − 0.253398i
13.18 1.41328 − 0.0512684i 0.543291 − 1.64464i 1.99474 − 0.144914i 0.430214 + 1.60558i 0.683506 − 2.35219i −3.62762 + 2.09441i 2.81171 − 0.307071i −2.40967 − 1.78703i 0.690330 + 2.24708i
61.1 −1.40500 + 0.161164i −1.68781 + 0.388965i 1.94805 − 0.452871i −0.846545 − 0.226831i 2.30869 − 0.818511i −0.567074 − 0.327400i −2.66403 + 0.950239i 2.69741 − 1.31300i 1.22595 + 0.182265i
61.2 −1.38700 − 0.276075i 0.841300 + 1.51401i 1.84757 + 0.765835i −0.0691269 − 0.0185225i −0.748909 − 2.33220i 1.28192 + 0.740118i −2.35115 − 1.57228i −1.58443 + 2.54747i 0.0907658 + 0.0447750i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
16.e even 4 1 inner
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.x.e ✓ 72
3.b odd 2 1 432.2.y.e 72
4.b odd 2 1 576.2.bb.e 72
9.c even 3 1 inner 144.2.x.e ✓ 72
9.d odd 6 1 432.2.y.e 72
12.b even 2 1 1728.2.bc.e 72
16.e even 4 1 inner 144.2.x.e ✓ 72
16.f odd 4 1 576.2.bb.e 72
36.f odd 6 1 576.2.bb.e 72
36.h even 6 1 1728.2.bc.e 72
48.i odd 4 1 432.2.y.e 72
48.k even 4 1 1728.2.bc.e 72
144.u even 12 1 1728.2.bc.e 72
144.v odd 12 1 576.2.bb.e 72
144.w odd 12 1 432.2.y.e 72
144.x even 12 1 inner 144.2.x.e ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.e ✓ 72 1.a even 1 1 trivial
144.2.x.e ✓ 72 9.c even 3 1 inner
144.2.x.e ✓ 72 16.e even 4 1 inner
144.2.x.e ✓ 72 144.x even 12 1 inner
432.2.y.e 72 3.b odd 2 1
432.2.y.e 72 9.d odd 6 1
432.2.y.e 72 48.i odd 4 1
432.2.y.e 72 144.w odd 12 1
576.2.bb.e 72 4.b odd 2 1
576.2.bb.e 72 16.f odd 4 1
576.2.bb.e 72 36.f odd 6 1
576.2.bb.e 72 144.v odd 12 1
1728.2.bc.e 72 12.b even 2 1
1728.2.bc.e 72 36.h even 6 1
1728.2.bc.e 72 48.k even 4 1
1728.2.bc.e 72 144.u even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 4 T_{5}^{71} + 8 T_{5}^{70} - 48 T_{5}^{69} - 362 T_{5}^{68} + 1784 T_{5}^{67} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display