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Properties

Label 578.2.f.b
Level $578$
Weight $2$
Character orbit 578.f
Analytic conductor $4.615$
Analytic rank $0$
Dimension $224$
Inner twists $2$

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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] = [578,2,Mod(35,578)] 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("578.35"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(34)) chi = DirichletCharacter(H, H._module([14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 578.f (of order \(17\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [224] 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: \(4.61535323683\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(14\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 14 q^{2} - 2 q^{3} - 14 q^{4} + 28 q^{5} - 2 q^{6} - 4 q^{7} - 14 q^{8} - 24 q^{9} - 6 q^{10} - 18 q^{11} - 2 q^{12} - 12 q^{13} - 4 q^{14} - 22 q^{15} - 14 q^{16} - 16 q^{17} - 24 q^{18} - 12 q^{19}+ \cdots - 162 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 0.739009 + 0.673696i −0.863421 − 3.03461i 0.0922684 + 0.995734i −2.58044 − 3.41705i 1.40633 − 2.82429i 3.72316 − 2.30528i −0.602635 + 0.798017i −5.91271 + 3.66100i 0.395087 − 4.26366i
35.2 0.739009 + 0.673696i −0.776677 − 2.72974i 0.0922684 + 0.995734i 2.08196 + 2.75695i 1.26504 − 2.54054i −2.06878 + 1.28093i −0.602635 + 0.798017i −4.29758 + 2.66095i −0.318765 + 3.44002i
35.3 0.739009 + 0.673696i −0.671242 − 2.35917i 0.0922684 + 0.995734i −0.0369385 − 0.0489145i 1.09331 − 2.19566i −1.15729 + 0.716564i −0.602635 + 0.798017i −2.56447 + 1.58785i 0.00565559 − 0.0610336i
35.4 0.739009 + 0.673696i −0.641416 − 2.25434i 0.0922684 + 0.995734i 0.981061 + 1.29913i 1.04473 − 2.09810i 1.36765 − 0.846815i −0.602635 + 0.798017i −2.11999 + 1.31264i −0.150209 + 1.62101i
35.5 0.739009 + 0.673696i −0.478415 − 1.68145i 0.0922684 + 0.995734i −2.07071 − 2.74207i 0.779235 − 1.56492i −3.86237 + 2.39148i −0.602635 + 0.798017i −0.0477546 + 0.0295684i 0.317044 − 3.42145i
35.6 0.739009 + 0.673696i −0.243655 − 0.856360i 0.0922684 + 0.995734i −0.241021 − 0.319163i 0.396862 − 0.797007i 1.19234 − 0.738266i −0.602635 + 0.798017i 1.87667 − 1.16198i 0.0369023 − 0.398239i
35.7 0.739009 + 0.673696i −0.0667563 − 0.234624i 0.0922684 + 0.995734i 2.34602 + 3.10663i 0.108732 − 0.218363i 2.86816 − 1.77589i −0.602635 + 0.798017i 2.50006 − 1.54797i −0.359195 + 3.87633i
35.8 0.739009 + 0.673696i 0.0191600 + 0.0673403i 0.0922684 + 0.995734i −1.81559 − 2.40423i −0.0312075 + 0.0626731i −0.0632371 + 0.0391547i −0.602635 + 0.798017i 2.54648 − 1.57672i 0.277982 − 2.99990i
35.9 0.739009 + 0.673696i 0.254148 + 0.893239i 0.0922684 + 0.995734i 1.02934 + 1.36307i −0.413953 + 0.831330i −4.17844 + 2.58718i −0.602635 + 0.798017i 1.81737 − 1.12527i −0.157601 + 1.70078i
35.10 0.739009 + 0.673696i 0.263973 + 0.927767i 0.0922684 + 0.995734i 0.148902 + 0.197178i −0.429955 + 0.863466i 2.02926 − 1.25646i −0.602635 + 0.798017i 1.75958 − 1.08949i −0.0227982 + 0.246031i
35.11 0.739009 + 0.673696i 0.402911 + 1.41609i 0.0922684 + 0.995734i 0.640893 + 0.848679i −0.656256 + 1.31794i −0.499754 + 0.309435i −0.602635 + 0.798017i 0.707687 − 0.438181i −0.0981260 + 1.05895i
35.12 0.739009 + 0.673696i 0.734416 + 2.58120i 0.0922684 + 0.995734i −1.63281 − 2.16219i −1.19621 + 2.40230i −3.44007 + 2.13000i −0.602635 + 0.798017i −3.57259 + 2.21206i 0.249997 − 2.69790i
35.13 0.739009 + 0.673696i 0.772126 + 2.71374i 0.0922684 + 0.995734i 1.81938 + 2.40925i −1.25763 + 2.52566i −0.404704 + 0.250582i −0.602635 + 0.798017i −4.21756 + 2.61140i −0.278562 + 3.00616i
35.14 0.739009 + 0.673696i 0.890370 + 3.12932i 0.0922684 + 0.995734i −0.422629 − 0.559652i −1.45022 + 2.91244i 4.20146 − 2.60143i −0.602635 + 0.798017i −6.44926 + 3.99321i 0.0647081 − 0.698311i
69.1 0.0922684 + 0.995734i −2.91551 + 1.80521i −0.982973 + 0.183750i −0.179895 + 0.632266i −2.06651 − 2.73651i 0.134925 − 0.270965i −0.273663 − 0.961826i 3.90420 − 7.84069i −0.646168 − 0.120790i
69.2 0.0922684 + 0.995734i −1.91770 + 1.18739i −0.982973 + 0.183750i 0.521586 − 1.83319i −1.35927 − 1.79996i 0.632668 − 1.27057i −0.273663 − 0.961826i 0.930473 − 1.86864i 1.87349 + 0.350216i
69.3 0.0922684 + 0.995734i −1.63995 + 1.01541i −0.982973 + 0.183750i 1.14782 − 4.03416i −1.16239 − 1.53926i −2.00558 + 4.02774i −0.273663 − 0.961826i 0.321146 − 0.644948i 4.12286 + 0.770696i
69.4 0.0922684 + 0.995734i −1.46777 + 0.908807i −0.982973 + 0.183750i −1.03169 + 3.62602i −1.04036 − 1.37766i 1.45243 − 2.91687i −0.273663 − 0.961826i −0.00878502 + 0.0176427i −3.70575 − 0.692724i
69.5 0.0922684 + 0.995734i −1.33634 + 0.827425i −0.982973 + 0.183750i −0.509699 + 1.79141i −0.947197 − 1.25429i −1.73964 + 3.49368i −0.273663 − 0.961826i −0.236050 + 0.474052i −1.83079 − 0.342235i
69.6 0.0922684 + 0.995734i −1.11647 + 0.691289i −0.982973 + 0.183750i 0.00113889 − 0.00400277i −0.791354 − 1.04792i 1.42111 − 2.85398i −0.273663 − 0.961826i −0.568591 + 1.14189i 0.00409078 0.000764700i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 35.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
289.f even 17 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 578.2.f.b ✓ 224
289.f even 17 1 inner 578.2.f.b ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
578.2.f.b ✓ 224 1.a even 1 1 trivial
578.2.f.b ✓ 224 289.f even 17 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{224} + 2 T_{3}^{223} + 35 T_{3}^{222} + 35 T_{3}^{221} + 674 T_{3}^{220} + 200 T_{3}^{219} + \cdots + 41\!\cdots\!84 \) acting on \(S_{2}^{\mathrm{new}}(578, [\chi])\). Copy content Toggle raw display