Properties

Label 544.2.bu.a
Level $544$
Weight $2$
Character orbit 544.bu
Analytic conductor $4.344$
Analytic rank $0$
Dimension $560$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [544,2,Mod(75,544)] 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("544.75"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 10, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 544 = 2^{5} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 544.bu (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.34386186996\)
Analytic rank: \(0\)
Dimension: \(560\)
Relative dimension: \(70\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 560 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 24 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9} - 8 q^{10} - 8 q^{11} + 32 q^{12} - 8 q^{13} - 8 q^{14} - 16 q^{15} - 16 q^{18} - 8 q^{20} - 8 q^{21} - 56 q^{22} - 8 q^{23}+ \cdots - 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1 −1.41416 − 0.0123877i −1.71930 − 0.341991i 1.99969 + 0.0350362i −2.32812 − 0.463091i 2.42713 + 0.504927i −1.70457 − 2.55107i −2.82745 − 0.0743184i 0.0674060 + 0.0279205i 3.28659 + 0.683725i
75.2 −1.40422 + 0.167797i −2.14568 − 0.426802i 1.94369 − 0.471249i −1.96091 − 0.390050i 3.08463 + 0.239287i 1.98014 + 2.96349i −2.65030 + 0.987883i 1.65013 + 0.683506i 2.81901 + 0.218682i
75.3 −1.39804 − 0.213285i 2.65646 + 0.528403i 1.90902 + 0.596362i 1.10375 + 0.219550i −3.60113 − 1.30531i 0.885220 + 1.32482i −2.54168 − 1.24090i 4.00594 + 1.65932i −1.49626 − 0.542353i
75.4 −1.38871 + 0.267358i 2.06649 + 0.411050i 1.85704 − 0.742567i 3.95218 + 0.786138i −2.97966 − 0.0183370i −1.51065 − 2.26085i −2.38036 + 1.52771i 1.32978 + 0.550812i −5.69862 − 0.0350696i
75.5 −1.37773 − 0.319156i 1.16659 + 0.232048i 1.79628 + 0.879421i −2.21126 − 0.439846i −1.53318 − 0.692022i 0.707459 + 1.05879i −2.19412 − 1.78490i −1.46456 − 0.606642i 2.90613 + 1.31172i
75.6 −1.35410 − 0.407949i −1.74718 − 0.347536i 1.66716 + 1.10480i 2.37286 + 0.471992i 2.22408 + 1.18336i −0.0547325 − 0.0819129i −1.80679 − 2.17613i 0.160231 + 0.0663698i −3.02054 − 1.60713i
75.7 −1.34784 + 0.428167i −2.99817 − 0.596373i 1.63335 − 1.15420i 2.00468 + 0.398755i 4.29640 − 0.479903i −1.45602 − 2.17909i −1.70730 + 2.25502i 5.86172 + 2.42801i −2.87272 + 0.320879i
75.8 −1.34498 + 0.437064i 2.01686 + 0.401179i 1.61795 − 1.17569i −4.20360 − 0.836147i −2.88798 + 0.341920i −0.371646 − 0.556207i −1.66226 + 2.28842i 1.13514 + 0.470192i 6.01921 − 0.712638i
75.9 −1.33526 + 0.465911i 0.924622 + 0.183919i 1.56585 − 1.24423i 0.477443 + 0.0949693i −1.32030 + 0.185211i −0.441549 − 0.660825i −1.51113 + 2.39092i −1.95054 − 0.807940i −0.681759 + 0.0956367i
75.10 −1.32127 − 0.504224i 0.163271 + 0.0324766i 1.49152 + 1.33243i −0.272402 − 0.0541841i −0.199349 − 0.125235i −2.26085 − 3.38360i −1.29885 − 2.51256i −2.74604 − 1.13745i 0.332596 + 0.208943i
75.11 −1.29945 + 0.558068i −0.0189162 − 0.00376266i 1.37712 − 1.45036i 1.20684 + 0.240055i 0.0266804 − 0.00566713i 2.46163 + 3.68408i −0.980095 + 2.65319i −2.77129 − 1.14791i −1.70219 + 0.361559i
75.12 −1.19311 − 0.759270i 3.24075 + 0.644625i 0.847019 + 1.81178i −1.91667 − 0.381250i −3.37712 − 3.22971i −2.01421 − 3.01448i 0.365045 − 2.80477i 7.31527 + 3.03008i 1.99733 + 1.91014i
75.13 −1.15379 − 0.817780i −2.01314 − 0.400439i 0.662471 + 1.88710i 1.76742 + 0.351562i 1.99528 + 2.10833i 2.03778 + 3.04975i 0.778877 − 2.71907i 1.12076 + 0.464234i −1.75174 − 1.85099i
75.14 −1.14656 + 0.827894i 0.289563 + 0.0575976i 0.629183 − 1.89845i −0.667910 − 0.132855i −0.379684 + 0.173688i −1.21533 − 1.81887i 0.850325 + 2.69758i −2.69111 − 1.11469i 0.875786 − 0.400632i
75.15 −1.09578 + 0.894021i 3.24031 + 0.644538i 0.401454 − 1.95929i −0.0924219 − 0.0183839i −4.12689 + 2.19064i 1.44310 + 2.15975i 1.31175 + 2.50586i 7.31255 + 3.02896i 0.117709 − 0.0624825i
75.16 −1.08777 − 0.903747i −0.525679 − 0.104564i 0.366482 + 1.96614i −3.28374 − 0.653177i 0.477318 + 0.588822i 1.26866 + 1.89869i 1.37824 − 2.46991i −2.50623 − 1.03812i 2.98165 + 3.67818i
75.17 −1.04564 − 0.952171i 1.88279 + 0.374511i 0.186743 + 1.99126i 3.30440 + 0.657286i −1.61213 − 2.18435i 2.69808 + 4.03797i 1.70076 − 2.25996i 0.633013 + 0.262203i −2.82938 − 3.83364i
75.18 −1.04515 + 0.952713i −1.68516 − 0.335199i 0.184677 − 1.99146i −2.48148 − 0.493598i 2.08059 − 1.25514i −1.64159 − 2.45682i 1.70427 + 2.25731i −0.0442405 − 0.0183250i 3.06378 − 1.84826i
75.19 −1.01324 − 0.986579i 0.553979 + 0.110193i 0.0533230 + 1.99929i 3.61599 + 0.719265i −0.452601 − 0.658196i −2.47027 − 3.69702i 1.91843 − 2.07837i −2.47689 − 1.02596i −2.95426 − 4.29625i
75.20 −0.971365 + 1.02784i −2.85622 − 0.568137i −0.112900 − 1.99681i −1.55503 − 0.309314i 3.35838 − 2.38386i 1.46439 + 2.19161i 2.16206 + 1.82359i 5.06357 + 2.09740i 1.82842 − 1.29786i
See next 80 embeddings (of 560 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 75.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
544.bu even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 544.2.bu.a ✓ 560
17.e odd 16 1 544.2.bz.a yes 560
32.h odd 8 1 544.2.bz.a yes 560
544.bu even 16 1 inner 544.2.bu.a ✓ 560
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.bu.a ✓ 560 1.a even 1 1 trivial
544.2.bu.a ✓ 560 544.bu even 16 1 inner
544.2.bz.a yes 560 17.e odd 16 1
544.2.bz.a yes 560 32.h odd 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(544, [\chi])\).