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Properties

Label 544.2.bd.b
Level $544$
Weight $2$
Character orbit 544.bd
Analytic conductor $4.344$
Analytic rank $0$
Dimension $128$
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(69,544)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("544.69"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 544 = 2^{5} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 544.bd (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.34386186996\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 20 q^{6} - 24 q^{8} - 16 q^{12} - 20 q^{16} - 20 q^{18} - 48 q^{22} - 8 q^{23} + 68 q^{24} + 20 q^{26} - 24 q^{27} + 20 q^{30} + 48 q^{31} - 24 q^{35} + 8 q^{36} + 68 q^{38} - 24 q^{39} - 36 q^{40}+ \cdots - 136 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1 −1.41395 − 0.0273756i −0.343657 − 0.829662i 1.99850 + 0.0774154i 2.82705 + 1.17100i 0.463201 + 1.18251i −0.551065 − 0.551065i −2.82366 − 0.164172i 1.55108 − 1.55108i −3.96525 − 1.73313i
69.2 −1.41078 + 0.0985307i 0.875214 + 2.11295i 1.98058 − 0.278010i −1.94173 − 0.804291i −1.44292 − 2.89467i −2.71566 − 2.71566i −2.76677 + 0.587358i −1.57725 + 1.57725i 2.81860 + 0.943355i
69.3 −1.37075 + 0.347933i −0.846083 − 2.04263i 1.75789 − 0.953855i −3.45794 − 1.43232i 1.87046 + 2.50554i 1.87474 + 1.87474i −2.07774 + 1.91912i −1.33514 + 1.33514i 5.23831 + 0.760222i
69.4 −1.33838 + 0.456876i 0.502476 + 1.21308i 1.58253 − 1.22295i 0.0822346 + 0.0340627i −1.22673 − 1.39400i 3.73184 + 3.73184i −1.55929 + 2.35979i 0.902229 − 0.902229i −0.125624 − 0.00801783i
69.5 −1.26627 − 0.629734i −0.330180 − 0.797125i 1.20687 + 1.59482i −1.39863 − 0.579332i −0.0838800 + 1.21730i −0.237221 − 0.237221i −0.523909 − 2.77948i 1.59493 − 1.59493i 1.40622 + 1.61435i
69.6 −1.24990 − 0.661625i 1.13988 + 2.75191i 1.12450 + 1.65393i −3.02227 − 1.25186i 0.395998 − 4.19379i 1.22449 + 1.22449i −0.311237 − 2.81125i −4.15239 + 4.15239i 2.94927 + 3.56431i
69.7 −1.21759 + 0.719358i −1.28383 − 3.09944i 0.965047 − 1.75177i 2.78329 + 1.15288i 3.79278 + 2.85031i −2.24686 − 2.24686i 0.0851164 + 2.82715i −5.83697 + 5.83697i −4.21824 + 0.598454i
69.8 −0.989905 + 1.00999i 1.05342 + 2.54319i −0.0401773 − 1.99960i −0.188868 − 0.0782316i −3.61139 − 1.45356i −0.313018 − 0.313018i 2.05935 + 1.93883i −3.23678 + 3.23678i 0.265975 − 0.113314i
69.9 −0.911396 + 1.08137i −0.691727 − 1.66998i −0.338716 − 1.97111i −1.22093 − 0.505726i 2.43630 + 0.773997i 0.0299653 + 0.0299653i 2.44020 + 1.43018i −0.189014 + 0.189014i 1.65963 − 0.859359i
69.10 −0.818572 − 1.15323i −0.0536177 − 0.129444i −0.659879 + 1.88800i −3.50023 − 1.44984i −0.105389 + 0.167793i −2.01400 − 2.01400i 2.71746 − 0.784477i 2.10744 − 2.10744i 1.19319 + 5.22337i
69.11 −0.780220 − 1.17952i 0.998357 + 2.41025i −0.782513 + 1.84056i 1.73773 + 0.719793i 2.06399 − 3.05810i 1.04993 + 1.04993i 2.78151 − 0.513058i −2.69125 + 2.69125i −0.506808 − 2.61128i
69.12 −0.741522 − 1.20422i 0.199073 + 0.480606i −0.900290 + 1.78591i 2.37294 + 0.982902i 0.431138 − 0.596108i 2.54907 + 2.54907i 2.81821 − 0.240146i 1.92997 − 1.92997i −0.575955 − 3.58638i
69.13 −0.294733 + 1.38316i −1.24455 − 3.00462i −1.82627 − 0.815325i −0.182136 − 0.0754430i 4.52268 − 0.835858i 3.05170 + 3.05170i 1.66599 − 2.28571i −5.35750 + 5.35750i 0.158031 − 0.229687i
69.14 −0.157873 + 1.40537i 0.282548 + 0.682131i −1.95015 − 0.443740i −1.34787 − 0.558306i −1.00326 + 0.289396i −0.110028 − 0.110028i 0.931496 − 2.67064i 1.73585 − 1.73585i 0.997420 − 1.80612i
69.15 −0.145762 − 1.40668i 0.277352 + 0.669587i −1.95751 + 0.410081i −0.147838 − 0.0612364i 0.901469 − 0.487746i −0.272034 − 0.272034i 0.862183 + 2.69382i 1.74990 − 1.74990i −0.0645910 + 0.216887i
69.16 −0.0222405 − 1.41404i −1.02696 − 2.47929i −1.99901 + 0.0628977i 4.01303 + 1.66225i −3.48298 + 1.50730i 2.41481 + 2.41481i 0.133399 + 2.82528i −2.97093 + 2.97093i 2.26123 − 5.71154i
69.17 0.167882 + 1.40421i −0.0740901 − 0.178869i −1.94363 + 0.471484i 3.36841 + 1.39524i 0.238732 − 0.134067i 0.795103 + 0.795103i −0.988364 − 2.65012i 2.09482 − 2.09482i −1.39372 + 4.96421i
69.18 0.433377 − 1.34617i −0.497273 − 1.20052i −1.62437 − 1.16680i 1.36390 + 0.564945i −1.83162 + 0.149136i −3.09184 − 3.09184i −2.27468 + 1.68102i 0.927343 − 0.927343i 1.35160 − 1.59121i
69.19 0.569875 + 1.29431i 1.11923 + 2.70207i −1.35049 + 1.47519i 3.06285 + 1.26868i −2.85950 + 2.98848i −2.11503 − 2.11503i −2.67896 − 0.907275i −3.92717 + 3.92717i 0.103382 + 4.68727i
69.20 0.599873 + 1.28068i −0.804966 − 1.94336i −1.28030 + 1.53650i −0.256881 − 0.106403i 2.00595 − 2.19668i −1.21608 − 1.21608i −2.73579 − 0.717964i −1.00735 + 1.00735i −0.0178265 − 0.392812i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 544.2.bd.b ✓ 128
32.g even 8 1 inner 544.2.bd.b ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.bd.b ✓ 128 1.a even 1 1 trivial
544.2.bd.b ✓ 128 32.g even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{128} + 8 T_{3}^{125} - 56 T_{3}^{123} + 16 T_{3}^{122} - 520 T_{3}^{121} + 33008 T_{3}^{120} + \cdots + 17\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(544, [\chi])\). Copy content Toggle raw display