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Properties

Label 409.2.h.a
Level $409$
Weight $2$
Character orbit 409.h
Analytic conductor $3.266$
Analytic rank $0$
Dimension $512$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [409,2,Mod(5,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(34)) chi = DirichletCharacter(H, H._module([2])) 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("409.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 409 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 409.h (of order \(17\), degree \(16\), 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: \(3.26588144267\)
Analytic rank: \(0\)
Dimension: \(512\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 512 q - 13 q^{2} - 11 q^{3} - 73 q^{4} - 9 q^{5} - 27 q^{6} - 22 q^{7} + q^{8} - 31 q^{9} + 5 q^{10} - 60 q^{11} - 38 q^{12} + 7 q^{13} - 37 q^{14} + 3 q^{15} - 53 q^{16} + 3 q^{17} + 16 q^{18} + 17 q^{19}+ \cdots - 103 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.29352 + 1.42009i −1.84651 − 0.345172i 2.35210 − 4.72365i 0.488755 + 1.71780i 4.72517 − 1.83054i 1.48483 0.815610 + 8.80184i 0.493031 + 0.191001i −3.56039 − 3.24572i
5.2 −2.13981 + 1.32491i 2.58288 + 0.482823i 1.93191 − 3.87980i 0.0170437 + 0.0599023i −6.16656 + 2.38894i 2.84476 0.542040 + 5.84955i 3.64071 + 1.41042i −0.115836 − 0.105598i
5.3 −1.99154 + 1.23311i −0.753952 − 0.140938i 1.55419 − 3.12123i −0.455844 − 1.60212i 1.67532 − 0.649021i −2.82620 0.321333 + 3.46774i −2.24884 − 0.871204i 2.88342 + 2.62859i
5.4 −1.70588 + 1.05624i 0.418329 + 0.0781992i 0.902915 − 1.81330i 1.04145 + 3.66033i −0.796215 + 0.308455i −1.26503 0.00475048 + 0.0512659i −2.62853 − 1.01830i −5.64277 − 5.14406i
5.5 −1.69947 + 1.05227i −2.80321 − 0.524011i 0.889465 − 1.78629i −0.597121 − 2.09866i 5.31538 − 2.05919i 1.00255 −0.000832348 − 0.00898247i 4.78598 + 1.85410i 3.22315 + 2.93829i
5.6 −1.68467 + 1.04310i 0.805609 + 0.150594i 0.858567 − 1.72424i 0.381398 + 1.34047i −1.51427 + 0.586631i 3.86107 −0.0134983 − 0.145670i −2.17109 − 0.841085i −2.04078 − 1.86042i
5.7 −1.65031 + 1.02183i 2.36246 + 0.441621i 0.787920 − 1.58236i −0.853653 − 3.00028i −4.35007 + 1.68522i −2.90115 −0.0416089 − 0.449032i 2.58878 + 1.00290i 4.47458 + 4.07911i
5.8 −1.19429 + 0.739476i 2.36313 + 0.441746i −0.0119617 + 0.0240224i −0.166845 − 0.586399i −3.14893 + 1.21990i −0.333050 −0.262696 − 2.83494i 2.59183 + 1.00408i 0.632890 + 0.576955i
5.9 −1.16780 + 0.723072i −1.09147 − 0.204032i −0.0505488 + 0.101516i 0.226710 + 0.796803i 1.42216 − 0.550946i −2.47979 −0.267840 − 2.89045i −1.64773 − 0.638333i −0.840898 − 0.766580i
5.10 −1.00350 + 0.621340i −2.45047 − 0.458072i −0.270531 + 0.543299i 0.896577 + 3.15114i 2.74366 − 1.06290i 4.86474 −0.283902 − 3.06379i 2.99754 + 1.16125i −2.85764 − 2.60509i
5.11 −1.00100 + 0.619792i 0.241764 + 0.0451934i −0.273621 + 0.549506i −0.624145 − 2.19364i −0.270015 + 0.104605i 3.32282 −0.283948 − 3.06429i −2.74101 − 1.06187i 1.98437 + 1.80899i
5.12 −0.923108 + 0.571564i 3.17385 + 0.593295i −0.366034 + 0.735095i 1.08870 + 3.82638i −3.26891 + 1.26638i −1.16207 −0.282623 − 3.04999i 6.92391 + 2.68234i −3.19201 − 2.90990i
5.13 −0.524017 + 0.324458i 0.552299 + 0.103242i −0.722155 + 1.45028i 0.0445750 + 0.156665i −0.322912 + 0.125097i −2.28728 −0.205870 − 2.22169i −2.50304 − 0.969684i −0.0741892 − 0.0676324i
5.14 −0.345026 + 0.213631i 1.17349 + 0.219364i −0.818072 + 1.64291i −1.04892 − 3.68657i −0.451748 + 0.175008i 0.828287 −0.143607 − 1.54977i −1.46845 − 0.568881i 1.14947 + 1.04788i
5.15 −0.334195 + 0.206925i −1.94971 − 0.364465i −0.822608 + 1.65202i −0.677315 − 2.38052i 0.727001 − 0.281642i 1.15113 −0.139468 − 1.50510i 0.871130 + 0.337478i 0.718943 + 0.655403i
5.16 −0.174924 + 0.108308i −1.80189 − 0.336831i −0.872609 + 1.75244i 0.986873 + 3.46850i 0.351675 − 0.136239i −0.978877 −0.0751298 − 0.810779i 0.335919 + 0.130136i −0.548295 − 0.499837i
5.17 0.0870788 − 0.0539169i −2.95487 − 0.552360i −0.886801 + 1.78094i −0.0489916 − 0.172188i −0.287088 + 0.111218i −3.21583 0.0377012 + 0.406860i 5.62872 + 2.18058i −0.0135500 − 0.0123524i
5.18 0.131697 − 0.0815436i 1.39847 + 0.261420i −0.880782 + 1.76885i 0.609236 + 2.14124i 0.205492 − 0.0796081i 3.07712 0.0568260 + 0.613251i −0.910036 − 0.352550i 0.254840 + 0.232317i
5.19 0.396783 − 0.245678i 3.05158 + 0.570439i −0.794398 + 1.59537i −0.230067 − 0.808601i 1.35096 − 0.523364i 0.0434321 0.162862 + 1.75757i 6.18931 + 2.39775i −0.289942 − 0.264317i
5.20 0.640153 − 0.396366i −2.17173 − 0.405966i −0.638787 + 1.28286i −0.494266 − 1.73716i −1.55115 + 0.600918i 4.02178 0.238502 + 2.57385i 1.75417 + 0.679568i −1.00496 − 0.916140i
See next 80 embeddings (of 512 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
409.h even 17 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 409.2.h.a ✓ 512
409.h even 17 1 inner 409.2.h.a ✓ 512
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
409.2.h.a ✓ 512 1.a even 1 1 trivial
409.2.h.a ✓ 512 409.h even 17 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(409, [\chi])\).