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Properties

Label 444.2.k.d
Level $444$
Weight $2$
Character orbit 444.k
Analytic conductor $3.545$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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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] = [444,2,Mod(31,444)] 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("444.31"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 36 q^{3} + 2 q^{5} - 6 q^{8} + 36 q^{9} + 12 q^{11} - 16 q^{14} + 2 q^{15} + 12 q^{16} - 6 q^{17} + 12 q^{19} - 6 q^{20} - 24 q^{22} + 10 q^{23} - 6 q^{24} + 36 q^{27} + 8 q^{28} + 14 q^{29} + 20 q^{32}+ \cdots + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.41406 + 0.0205801i 1.00000 1.99915 − 0.0582032i −0.488057 + 0.488057i −1.41406 + 0.0205801i − 2.09703i −2.82573 + 0.123446i 1.00000 0.680099 − 0.700188i
31.2 −1.41121 − 0.0921066i 1.00000 1.98303 + 0.259964i −0.266641 + 0.266641i −1.41121 − 0.0921066i 2.75168i −2.77453 − 0.549514i 1.00000 0.400846 − 0.351727i
31.3 −1.22904 + 0.699616i 1.00000 1.02107 − 1.71971i 2.69841 − 2.69841i −1.22904 + 0.699616i 0.0895973i −0.0518016 + 2.82795i 1.00000 −1.42860 + 5.20430i
31.4 −1.16159 − 0.806666i 1.00000 0.698580 + 1.87403i −1.33996 + 1.33996i −1.16159 − 0.806666i 1.31810i 0.700253 − 2.74037i 1.00000 2.63738 − 0.475582i
31.5 −1.14132 − 0.835092i 1.00000 0.605243 + 1.90622i 2.31462 − 2.31462i −1.14132 − 0.835092i − 4.97464i 0.901092 − 2.68105i 1.00000 −4.57465 + 0.708811i
31.6 −0.857832 + 1.12433i 1.00000 −0.528247 − 1.92898i 0.174272 − 0.174272i −0.857832 + 1.12433i 2.80515i 2.62196 + 1.06081i 1.00000 0.0464434 + 0.345435i
31.7 −0.471495 − 1.33330i 1.00000 −1.55538 + 1.25729i −2.85013 + 2.85013i −0.471495 − 1.33330i − 4.14724i 2.40970 + 1.48099i 1.00000 5.14390 + 2.45626i
31.8 −0.352760 − 1.36951i 1.00000 −1.75112 + 0.966219i 1.71736 − 1.71736i −0.352760 − 1.36951i 2.69078i 1.94097 + 2.05733i 1.00000 −2.95775 − 1.74612i
31.9 0.0484004 + 1.41339i 1.00000 −1.99531 + 0.136817i 1.42514 − 1.42514i 0.0484004 + 1.41339i − 3.24756i −0.289949 − 2.81353i 1.00000 2.08324 + 1.94529i
31.10 0.197969 − 1.40029i 1.00000 −1.92162 − 0.554427i −1.88022 + 1.88022i 0.197969 − 1.40029i 3.25963i −1.15678 + 2.58106i 1.00000 2.26062 + 3.00507i
31.11 0.376838 + 1.36308i 1.00000 −1.71599 + 1.02732i −2.47002 + 2.47002i 0.376838 + 1.36308i 0.376967i −2.04697 − 1.95189i 1.00000 −4.29764 − 2.43605i
31.12 0.563166 − 1.29724i 1.00000 −1.36569 − 1.46113i 2.38255 − 2.38255i 0.563166 − 1.29724i − 1.05815i −2.66455 + 0.948772i 1.00000 −1.74898 − 4.43252i
31.13 0.647695 + 1.25718i 1.00000 −1.16098 + 1.62853i 1.35559 − 1.35559i 0.647695 + 1.25718i 5.05885i −2.79931 − 0.404768i 1.00000 2.58223 + 0.826207i
31.14 0.937129 − 1.05915i 1.00000 −0.243577 − 1.98511i −0.362196 + 0.362196i 0.937129 − 1.05915i − 0.539693i −2.33079 − 1.60232i 1.00000 0.0441937 + 0.723043i
31.15 1.18804 + 0.767175i 1.00000 0.822884 + 1.82287i −0.272886 + 0.272886i 1.18804 + 0.767175i − 1.36826i −0.420842 + 2.79694i 1.00000 −0.533552 + 0.114849i
31.16 1.30348 − 0.548584i 1.00000 1.39811 − 1.43013i −0.466535 + 0.466535i 1.30348 − 0.548584i − 4.08674i 1.03786 − 2.63113i 1.00000 −0.352185 + 0.864052i
31.17 1.37908 + 0.313264i 1.00000 1.80373 + 0.864033i 1.57791 − 1.57791i 1.37908 + 0.313264i − 0.0465570i 2.21682 + 1.75662i 1.00000 2.67037 − 1.68177i
31.18 1.39752 − 0.216672i 1.00000 1.90611 − 0.605606i −2.24921 + 2.24921i 1.39752 − 0.216672i 3.21511i 2.53260 − 1.25935i 1.00000 −2.65596 + 3.63064i
43.1 −1.41406 − 0.0205801i 1.00000 1.99915 + 0.0582032i −0.488057 − 0.488057i −1.41406 − 0.0205801i 2.09703i −2.82573 − 0.123446i 1.00000 0.680099 + 0.700188i
43.2 −1.41121 + 0.0921066i 1.00000 1.98303 − 0.259964i −0.266641 − 0.266641i −1.41121 + 0.0921066i − 2.75168i −2.77453 + 0.549514i 1.00000 0.400846 + 0.351727i
See all 36 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 31.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
148.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 444.2.k.d yes 36
4.b odd 2 1 444.2.k.c ✓ 36
37.d odd 4 1 444.2.k.c ✓ 36
148.g even 4 1 inner 444.2.k.d yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.k.c ✓ 36 4.b odd 2 1
444.2.k.c ✓ 36 37.d odd 4 1
444.2.k.d yes 36 1.a even 1 1 trivial
444.2.k.d yes 36 148.g even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\):

\( T_{5}^{36} - 2 T_{5}^{35} + 2 T_{5}^{34} + 560 T_{5}^{32} - 1192 T_{5}^{31} + 1264 T_{5}^{30} + \cdots + 2768896 \) Copy content Toggle raw display
\( T_{11}^{18} - 6 T_{11}^{17} - 58 T_{11}^{16} + 424 T_{11}^{15} + 865 T_{11}^{14} - 10114 T_{11}^{13} + \cdots + 65536 \) Copy content Toggle raw display