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Properties

Label 425.2.k.b
Level $425$
Weight $2$
Character orbit 425.k
Analytic conductor $3.394$
Analytic rank $0$
Dimension $76$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,2,Mod(86,425)] 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("425.86"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.k (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(19\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 6 q^{2} - 7 q^{3} - 26 q^{4} + 13 q^{5} + 12 q^{6} + 22 q^{7} + 10 q^{8} - 24 q^{9} - 6 q^{10} - 16 q^{11} - 15 q^{12} - q^{13} - 19 q^{14} - 3 q^{15} + 30 q^{16} - 19 q^{17} + 6 q^{18} + 18 q^{19}+ \cdots - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1 −2.26173 + 1.64324i 0.672306 − 2.06914i 1.79714 − 5.53104i −2.15401 + 0.600183i 1.87953 + 5.78461i 2.13787 3.29638 + 10.1452i −1.40231 − 1.01884i 3.88556 − 4.89702i
86.2 −1.92608 + 1.39938i 0.431982 − 1.32950i 1.13349 − 3.48853i 2.16658 − 0.553118i 1.02845 + 3.16524i 4.93434 1.22719 + 3.77690i 0.846082 + 0.614714i −3.39899 + 4.09722i
86.3 −1.89465 + 1.37655i −0.866934 + 2.66815i 1.07680 − 3.31404i −2.13509 + 0.664386i −2.03029 − 6.24859i −2.04332 1.07439 + 3.30663i −3.94040 − 2.86287i 3.13069 − 4.19783i
86.4 −1.59626 + 1.15975i 0.848137 − 2.61030i 0.584991 − 1.80042i −0.617518 − 2.14911i 1.67345 + 5.15034i −2.32069 −0.0651963 − 0.200653i −3.66726 − 2.66442i 3.47815 + 2.71437i
86.5 −1.38672 + 1.00751i −0.537884 + 1.65544i 0.289884 − 0.892173i 0.0918514 − 2.23418i −0.921979 − 2.83756i 3.20986 −0.562476 − 1.73112i −0.0241012 − 0.0175105i 2.12360 + 3.19073i
86.6 −1.23201 + 0.895109i −0.658347 + 2.02618i 0.0985994 − 0.303458i 1.71600 + 1.43365i −1.00256 − 3.08558i −1.63215 −0.791021 − 2.43451i −1.24495 − 0.904509i −3.39740 − 0.230264i
86.7 −1.18469 + 0.860728i −0.207649 + 0.639078i 0.0446047 − 0.137279i −0.708744 + 2.12077i −0.304073 − 0.935839i 4.82170 −0.839706 − 2.58435i 2.06175 + 1.49795i −0.985767 − 3.12250i
86.8 −1.14899 + 0.834793i 0.0501431 − 0.154325i 0.00527402 − 0.0162318i 1.95968 − 1.07687i 0.0712150 + 0.219177i −3.62244 −0.870263 − 2.67839i 2.40575 + 1.74788i −1.35269 + 2.87325i
86.9 −0.571302 + 0.415075i 0.998425 − 3.07284i −0.463935 + 1.42785i 2.18305 + 0.484058i 0.705056 + 2.16994i 1.24893 −0.764052 − 2.35151i −6.01842 − 4.37264i −1.44810 + 0.629585i
86.10 −0.382111 + 0.277620i 0.549532 − 1.69129i −0.549098 + 1.68995i −0.894807 + 2.04922i 0.259553 + 0.798821i 1.24434 −0.551255 − 1.69659i −0.131414 − 0.0954780i −0.226990 − 1.03145i
86.11 0.337793 − 0.245421i −0.235041 + 0.723382i −0.564161 + 1.73631i −0.399103 − 2.20016i 0.0981377 + 0.302037i −3.97689 0.493607 + 1.51917i 1.95901 + 1.42331i −0.674779 − 0.645251i
86.12 0.557913 − 0.405348i −0.826041 + 2.54229i −0.471074 + 1.44982i 0.0377362 + 2.23575i 0.569653 + 1.75321i −0.350888 0.751069 + 2.31155i −3.35386 − 2.43672i 0.927309 + 1.23206i
86.13 0.589613 − 0.428379i 0.387406 − 1.19231i −0.453899 + 1.39696i −2.22905 + 0.177058i −0.282342 − 0.868960i 2.48707 0.781226 + 2.40437i 1.15552 + 0.839535i −1.23843 + 1.05927i
86.14 0.742425 − 0.539403i 0.213285 − 0.656424i −0.357795 + 1.10118i 2.18854 − 0.458598i −0.195729 − 0.602392i 1.08853 0.895506 + 2.75609i 2.04165 + 1.48335i 1.37745 − 1.52098i
86.15 1.23279 − 0.895677i −0.0898893 + 0.276651i 0.0995081 − 0.306255i −1.61292 + 1.54871i 0.136975 + 0.421565i −2.08667 0.790138 + 2.43179i 2.35860 + 1.71362i −0.601248 + 3.35389i
86.16 1.31354 − 0.954345i −0.933504 + 2.87303i 0.196587 − 0.605031i 1.82226 − 1.29591i 1.51566 + 4.66473i 3.76582 0.684274 + 2.10598i −4.95583 − 3.60062i 1.15687 − 3.44129i
86.17 1.55652 − 1.13088i 0.803039 − 2.47150i 0.525836 − 1.61836i −1.36443 − 1.77154i −1.54502 − 4.75508i 3.93026 0.177386 + 0.545938i −3.03639 − 2.20607i −4.12715 − 1.21443i
86.18 1.71096 − 1.24309i 0.882251 − 2.71529i 0.764097 − 2.35165i 1.52540 + 1.63498i −1.86584 − 5.74248i −1.34454 −0.308905 − 0.950711i −4.16738 − 3.02778i 4.64233 + 0.901189i
86.19 1.80693 − 1.31281i −0.436131 + 1.34227i 0.923486 − 2.84220i 2.23360 + 0.105046i 0.974092 + 2.99795i −2.63704 −0.682224 − 2.09967i 0.815568 + 0.592544i 4.17386 − 2.74248i
171.1 −0.847347 + 2.60787i 0.638461 + 0.463869i −4.46494 − 3.24397i 2.23317 + 0.113866i −1.75071 + 1.27196i 2.23574 7.80642 − 5.67170i −0.734593 − 2.26085i −2.18922 + 5.72732i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.19
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.k.b ✓ 76
25.d even 5 1 inner 425.2.k.b ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.k.b ✓ 76 1.a even 1 1 trivial
425.2.k.b ✓ 76 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{76} + 6 T_{2}^{75} + 50 T_{2}^{74} + 226 T_{2}^{73} + 1177 T_{2}^{72} + 4356 T_{2}^{71} + \cdots + 5958450481 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display