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Properties

Label 1000.2.bd.a
Level $1000$
Weight $2$
Character orbit 1000.bd
Analytic conductor $7.985$
Analytic rank $0$
Dimension $2960$
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] = [1000,2,Mod(29,1000)] 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("1000.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1000, base_ring=CyclotomicField(50)) chi = DirichletCharacter(H, H._module([0, 25, 31])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.bd (of order \(50\), degree \(20\), 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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(2960\)
Relative dimension: \(148\) over \(\Q(\zeta_{50})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{50}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2960 q - 20 q^{2} - 20 q^{4} - 20 q^{6} - 50 q^{7} - 35 q^{8} - 40 q^{9} - 20 q^{10} - 20 q^{12} - 30 q^{14} - 40 q^{15} - 40 q^{17} - 25 q^{18} - 20 q^{20} + 5 q^{22} - 40 q^{23} - 15 q^{24} - 40 q^{25}+ \cdots - 115 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.41421 − 0.00203185i −2.92155 + 1.60614i 1.99999 + 0.00574692i 2.11287 + 0.731957i 4.13496 − 2.26548i 0.590916 − 0.813326i −2.82840 − 0.0121910i 4.34832 − 6.85185i −2.98657 − 1.03944i
29.2 −1.41420 − 0.00544908i −1.58195 + 0.869687i 1.99994 + 0.0154122i −1.42035 + 1.72703i 2.24194 − 1.22129i 1.69036 − 2.32658i −2.82824 − 0.0326938i 0.138744 − 0.218625i 2.01807 − 2.43463i
29.3 −1.41340 + 0.0480003i 1.77607 − 0.976404i 1.99539 − 0.135687i 1.90822 + 1.16564i −2.46343 + 1.46530i −2.55880 + 3.52189i −2.81377 + 0.287560i 0.593587 − 0.935343i −2.75302 − 1.55591i
29.4 −1.41190 − 0.0809313i −0.729503 + 0.401047i 1.98690 + 0.228533i 1.81725 − 1.30292i 1.06244 − 0.507198i 0.856648 − 1.17908i −2.78680 − 0.483468i −1.23615 + 1.94785i −2.67121 + 1.69252i
29.5 −1.41069 + 0.0997881i −1.23898 + 0.681133i 1.98008 − 0.281540i −1.81317 − 1.30860i 1.67984 − 1.08450i −1.35792 + 1.86901i −2.76519 + 0.594754i −0.536360 + 0.845168i 2.68840 + 1.66509i
29.6 −1.40533 + 0.158249i −0.297933 + 0.163790i 1.94991 − 0.444783i −2.16003 − 0.578152i 0.392775 − 0.277327i −1.28226 + 1.76488i −2.66989 + 0.933639i −1.54554 + 2.43539i 3.12705 + 0.470673i
29.7 −1.40205 − 0.185077i 2.09920 − 1.15404i 1.93149 + 0.518973i −1.94519 − 1.10283i −3.15677 + 1.22952i 2.03349 − 2.79886i −2.61200 − 1.08510i 1.46733 − 2.31215i 2.52314 + 1.90624i
29.8 −1.40132 + 0.190557i 1.09925 − 0.604320i 1.92738 − 0.534061i −1.37052 + 1.76683i −1.42525 + 1.05631i 1.88665 − 2.59675i −2.59910 + 1.11566i −0.764323 + 1.20438i 1.58385 − 2.73705i
29.9 −1.39728 + 0.218210i 1.07077 − 0.588662i 1.90477 − 0.609799i 0.473257 − 2.18541i −1.36771 + 1.05618i −1.63453 + 2.24974i −2.52843 + 1.26770i −0.807450 + 1.27234i −0.184394 + 3.15690i
29.10 −1.39137 + 0.253168i −2.72371 + 1.49737i 1.87181 − 0.704500i −1.47591 − 1.67979i 3.41059 − 2.77295i 2.25056 − 3.09763i −2.42602 + 1.45410i 3.56898 − 5.62382i 2.47880 + 1.96355i
29.11 −1.38623 − 0.279952i 1.58311 − 0.870323i 1.84325 + 0.776154i 1.47337 − 1.68201i −2.43820 + 0.763271i 0.136564 − 0.187964i −2.33788 − 1.59195i 0.141299 − 0.222652i −2.51331 + 1.91918i
29.12 −1.37490 − 0.331127i 0.305041 − 0.167698i 1.78071 + 0.910535i −0.479378 + 2.18408i −0.474930 + 0.129560i −2.09675 + 2.88593i −2.14680 − 1.84154i −1.54255 + 2.43067i 1.38231 − 2.84416i
29.13 −1.36369 + 0.374642i 0.0516299 − 0.0283838i 1.71929 − 1.02179i 1.07054 + 1.96315i −0.0597733 + 0.0580493i 1.50763 − 2.07507i −1.96177 + 2.03752i −1.60562 + 2.53005i −2.19536 − 2.27605i
29.14 −1.36332 − 0.375984i −1.88977 + 1.03891i 1.71727 + 1.02517i 2.23402 − 0.0957776i 2.96697 − 0.705839i −1.93108 + 2.65790i −1.95574 − 2.04330i 0.884407 − 1.39360i −3.08168 − 0.709379i
29.15 −1.36132 − 0.383154i 2.30479 − 1.26707i 1.70639 + 1.04319i 0.965606 + 2.01683i −3.62304 + 0.841798i 0.721096 − 0.992503i −1.92324 − 2.07392i 2.09912 − 3.30768i −0.541743 − 3.11553i
29.16 −1.33250 + 0.473753i 2.59153 − 1.42471i 1.55112 − 1.26255i 2.05456 − 0.882483i −2.77826 + 3.12617i 1.98344 − 2.72997i −1.46873 + 2.41720i 3.07877 − 4.85136i −2.31963 + 2.14926i
29.17 −1.32805 − 0.486079i −0.933129 + 0.512992i 1.52745 + 1.29108i −0.423403 − 2.19562i 1.48860 − 0.227707i 2.01038 − 2.76705i −1.40098 − 2.45709i −0.999912 + 1.57561i −0.504942 + 3.12170i
29.18 −1.32233 − 0.501442i −2.50837 + 1.37899i 1.49711 + 1.32614i −1.98291 + 1.03347i 4.00838 − 0.565676i −1.83821 + 2.53007i −1.31469 − 2.50431i 2.78284 − 4.38506i 3.14029 − 0.372265i
29.19 −1.30316 + 0.549346i −0.623770 + 0.342921i 1.39644 − 1.43177i 2.15211 + 0.606986i 0.624489 − 0.789545i −0.524496 + 0.721907i −1.03324 + 2.63295i −1.33599 + 2.10518i −3.13798 + 0.391253i
29.20 −1.29946 − 0.558045i 2.89362 − 1.59078i 1.37717 + 1.45031i −1.75231 + 1.38903i −4.64785 + 0.452378i −0.949757 + 1.30723i −0.980234 − 2.65314i 4.23495 − 6.67322i 3.05219 − 0.827108i
See next 80 embeddings (of 2960 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 29.148
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
125.h even 50 1 inner
1000.bd even 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.bd.a ✓ 2960
8.b even 2 1 inner 1000.2.bd.a ✓ 2960
125.h even 50 1 inner 1000.2.bd.a ✓ 2960
1000.bd even 50 1 inner 1000.2.bd.a ✓ 2960
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.bd.a ✓ 2960 1.a even 1 1 trivial
1000.2.bd.a ✓ 2960 8.b even 2 1 inner
1000.2.bd.a ✓ 2960 125.h even 50 1 inner
1000.2.bd.a ✓ 2960 1000.bd even 50 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1000, [\chi])\).