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Properties

Label 580.2.be.e
Level $580$
Weight $2$
Character orbit 580.be
Analytic conductor $4.631$
Analytic rank $0$
Dimension $960$
Inner twists $8$

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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] = [580,2,Mod(19,580)] 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("580.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(580, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([14, 14, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 580.be (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [960,0] 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: \(4.63132331723\)
Analytic rank: \(0\)
Dimension: \(960\)
Relative dimension: \(80\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 960 q - 28 q^{4} - 28 q^{5} - 28 q^{6} - 56 q^{9} - 4 q^{10} + 8 q^{14} - 68 q^{16} - 66 q^{20} + 8 q^{21} + 60 q^{24} - 52 q^{25} + 28 q^{26} - 56 q^{29} - 104 q^{30} + 56 q^{34} - 284 q^{36} + 44 q^{40}+ \cdots - 28 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41420 − 0.00575538i 0.140751 − 0.402244i 1.99993 + 0.0162785i −0.643662 − 2.14142i −0.201366 + 0.568044i 3.64162 − 1.75371i −2.82822 − 0.0345315i 2.20351 + 1.75724i 0.897943 + 3.03211i
19.2 −1.41252 − 0.0692268i 0.635461 − 1.81604i 1.99042 + 0.195568i 1.85870 + 1.24307i −1.02332 + 2.52120i −1.72662 + 0.831494i −2.79796 − 0.414034i −0.548707 − 0.437579i −2.53940 − 1.88453i
19.3 −1.41105 + 0.0944669i 0.800177 − 2.28677i 1.98215 − 0.266596i −1.03389 + 1.98269i −0.913069 + 3.30235i 0.497528 − 0.239597i −2.77174 + 0.563430i −2.24356 − 1.78918i 1.27158 − 2.89536i
19.4 −1.41075 + 0.0989745i 0.441789 − 1.26256i 1.98041 − 0.279256i −2.23541 − 0.0542720i −0.498290 + 1.82488i −3.37467 + 1.62516i −2.76621 + 0.589969i 0.946616 + 0.754901i 3.15897 − 0.144685i
19.5 −1.40624 − 0.149942i −0.802152 + 2.29242i 1.95503 + 0.421709i −2.13056 + 0.678756i 1.47175 − 3.10342i 1.56074 − 0.751612i −2.68602 − 0.886166i −2.26624 − 1.80727i 3.09786 − 0.635035i
19.6 −1.38642 + 0.279019i 1.00294 − 2.86623i 1.84430 − 0.773672i 1.66528 − 1.49226i −0.590755 + 4.25362i 3.29136 − 1.58504i −2.34109 + 1.58723i −4.86388 − 3.87881i −1.89241 + 2.53354i
19.7 −1.37797 + 0.318101i −0.285112 + 0.814802i 1.79762 − 0.876670i −0.668030 − 2.13395i 0.133687 − 1.21347i −1.69810 + 0.817764i −2.19821 + 1.77985i 1.76288 + 1.40585i 1.59934 + 2.72802i
19.8 −1.37124 − 0.345973i −0.843252 + 2.40988i 1.76061 + 0.948825i 1.26283 − 1.84534i 1.99005 − 3.01278i −1.31620 + 0.633850i −2.08595 − 1.91019i −2.75094 − 2.19380i −2.37008 + 2.09350i
19.9 −1.36868 + 0.355962i −1.03092 + 2.94620i 1.74658 − 0.974397i 1.49733 + 1.66073i 0.362267 − 4.39938i 2.83691 − 1.36618i −2.04367 + 1.95536i −5.27182 − 4.20414i −2.64052 − 1.74001i
19.10 −1.31254 − 0.526525i 0.0519439 − 0.148447i 1.44554 + 1.38217i 1.43170 + 1.71762i −0.146340 + 0.167494i 3.79074 − 1.82552i −1.16959 − 2.57528i 2.32616 + 1.85505i −0.974800 − 3.00828i
19.11 −1.31025 + 0.532214i −0.315085 + 0.900461i 1.43350 − 1.39466i 2.13529 − 0.663741i −0.0663986 − 1.34752i 1.82561 − 0.879170i −1.13598 + 2.59028i 1.63394 + 1.30303i −2.44450 + 2.00609i
19.12 −1.26448 − 0.633323i −0.597714 + 1.70817i 1.19780 + 1.60164i −0.274798 + 2.21912i 1.83762 − 1.78139i −1.68449 + 0.811207i −0.500238 − 2.78384i −0.215086 − 0.171525i 1.75289 − 2.63199i
19.13 −1.25394 − 0.653934i 0.402185 − 1.14938i 1.14474 + 1.63999i 1.44596 − 1.70563i −1.25593 + 1.17825i −1.25186 + 0.602865i −0.362994 − 2.80504i 1.18617 + 0.945943i −2.92853 + 1.19320i
19.14 −1.24398 − 0.672684i 1.07758 − 3.07954i 1.09499 + 1.67362i −1.20447 − 1.88395i −3.41205 + 3.10603i −2.06536 + 0.994627i −0.236336 − 2.81854i −5.97689 − 4.76641i 0.231037 + 3.15383i
19.15 −1.23645 − 0.686437i −0.133329 + 0.381031i 1.05761 + 1.69749i −2.13732 − 0.657156i 0.426408 − 0.379604i 0.231423 − 0.111448i −0.142457 − 2.82484i 2.21809 + 1.76886i 2.19159 + 2.27968i
19.16 −1.21671 + 0.720836i −0.665447 + 1.90174i 0.960791 − 1.75410i −2.20239 + 0.386598i −0.561182 − 2.79355i −2.92479 + 1.40851i 0.0954131 + 2.82682i −0.828292 − 0.660541i 2.40101 − 2.05794i
19.17 −1.17736 + 0.783463i 0.665447 − 1.90174i 0.772371 − 1.84484i −2.20239 + 0.386598i 0.706469 + 2.76039i 2.92479 − 1.40851i 0.536005 + 2.77717i −0.828292 − 0.660541i 2.29013 − 2.18066i
19.18 −1.04800 + 0.949573i 0.315085 − 0.900461i 0.196621 − 1.99031i 2.13529 − 0.663741i 0.524844 + 1.24288i −1.82561 + 0.879170i 1.68389 + 2.27256i 1.63394 + 1.30303i −1.60752 + 2.72321i
19.19 −0.990152 − 1.00975i 0.599016 − 1.71189i −0.0391963 + 1.99962i −1.75723 + 1.38280i −2.32170 + 1.09017i 2.33669 − 1.12529i 2.05793 − 1.94035i −0.226253 − 0.180431i 3.13621 + 0.405189i
19.20 −0.971287 − 1.02791i −0.814806 + 2.32858i −0.113202 + 1.99679i 0.413510 − 2.19750i 3.18498 − 1.42417i 4.22928 − 2.03672i 2.16248 − 1.82310i −2.41289 − 1.92421i −2.66047 + 1.70935i
See next 80 embeddings (of 960 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 19.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
29.f odd 28 1 inner
116.l even 28 1 inner
145.s odd 28 1 inner
580.be even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 580.2.be.e ✓ 960
4.b odd 2 1 inner 580.2.be.e ✓ 960
5.b even 2 1 inner 580.2.be.e ✓ 960
20.d odd 2 1 inner 580.2.be.e ✓ 960
29.f odd 28 1 inner 580.2.be.e ✓ 960
116.l even 28 1 inner 580.2.be.e ✓ 960
145.s odd 28 1 inner 580.2.be.e ✓ 960
580.be even 28 1 inner 580.2.be.e ✓ 960
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.be.e ✓ 960 1.a even 1 1 trivial
580.2.be.e ✓ 960 4.b odd 2 1 inner
580.2.be.e ✓ 960 5.b even 2 1 inner
580.2.be.e ✓ 960 20.d odd 2 1 inner
580.2.be.e ✓ 960 29.f odd 28 1 inner
580.2.be.e ✓ 960 116.l even 28 1 inner
580.2.be.e ✓ 960 145.s odd 28 1 inner
580.2.be.e ✓ 960 580.be even 28 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(580, [\chi])\):

\( T_{3}^{480} + 14 T_{3}^{478} - 479 T_{3}^{476} - 7644 T_{3}^{474} + 123270 T_{3}^{472} + \cdots + 45\!\cdots\!24 \) Copy content Toggle raw display
\( T_{13}^{480} + 622 T_{13}^{478} + 201299 T_{13}^{476} + 44995802 T_{13}^{474} + 7787656963 T_{13}^{472} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display