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Properties

Label 409.2.g.b
Level $409$
Weight $2$
Character orbit 409.g
Analytic conductor $3.266$
Analytic rank $0$
Dimension $132$
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(49,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([7])) 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("409.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 409 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 409.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [132] 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: \(3.26588144267\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(33\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 132 q - 6 q^{3} + 58 q^{4} - 4 q^{6} + 56 q^{9} - 18 q^{10} - 2 q^{11} - 30 q^{12} - 2 q^{13} - 24 q^{14} - 30 q^{15} - 58 q^{16} + 2 q^{17} - 36 q^{18} + 6 q^{19} - 6 q^{20} + 6 q^{21} - 22 q^{22} - 12 q^{23}+ \cdots - 84 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −2.32512 + 1.34241i −1.94270 − 1.12162i 2.60411 − 4.51046i −0.360691 6.02269 −0.266632 − 0.995083i 8.61350i 1.01607 + 1.75988i 0.838650 − 0.484195i
49.2 −2.31307 + 1.33545i 0.819287 + 0.473016i 2.56687 − 4.44596i 3.51810 −2.52676 0.740138 + 2.76223i 8.36995i −1.05251 − 1.82300i −8.13763 + 4.69826i
49.3 −2.05848 + 1.18846i 1.12161 + 0.647561i 1.82490 − 3.16081i −0.0835627 −3.07841 −0.396630 − 1.48024i 3.92144i −0.661330 − 1.14546i 0.172012 − 0.0993112i
49.4 −2.05768 + 1.18800i 0.999971 + 0.577334i 1.82269 − 3.15700i −4.20665 −2.74349 −0.371517 − 1.38652i 3.90944i −0.833372 − 1.44344i 8.65593 − 4.99751i
49.5 −1.78125 + 1.02841i −2.72564 − 1.57365i 1.11524 − 1.93165i 3.30630 6.47341 0.505098 + 1.88505i 0.474049i 3.45276 + 5.98036i −5.88935 + 3.40022i
49.6 −1.70608 + 0.985005i 2.68402 + 1.54962i 0.940471 − 1.62894i −1.28653 −6.10554 1.16797 + 4.35893i − 0.234544i 3.30265 + 5.72035i 2.19492 − 1.26724i
49.7 −1.65160 + 0.953553i −0.938434 − 0.541805i 0.818525 − 1.41773i −0.204581 2.06656 0.771407 + 2.87893i − 0.692182i −0.912894 − 1.58118i 0.337886 − 0.195079i
49.8 −1.55568 + 0.898172i 2.15399 + 1.24361i 0.613425 − 1.06248i 1.69452 −4.46789 −0.793944 − 2.96304i − 1.38884i 1.59312 + 2.75937i −2.63613 + 1.52197i
49.9 −1.23367 + 0.712262i −0.570226 − 0.329220i 0.0146342 − 0.0253472i 3.67480 0.937964 −0.439959 − 1.64195i − 2.80735i −1.28323 − 2.22262i −4.53350 + 2.61742i
49.10 −1.20185 + 0.693886i −0.963418 − 0.556230i −0.0370437 + 0.0641616i −3.71176 1.54384 −0.326971 − 1.22027i − 2.87836i −0.881217 − 1.52631i 4.46096 − 2.57554i
49.11 −0.853701 + 0.492884i 0.894622 + 0.516510i −0.514130 + 0.890499i −0.300721 −1.01832 −0.611290 − 2.28136i − 2.98516i −0.966435 − 1.67391i 0.256726 − 0.148221i
49.12 −0.542490 + 0.313207i 1.78346 + 1.02968i −0.803803 + 1.39223i 0.805933 −1.29001 0.179784 + 0.670964i − 2.25985i 0.620481 + 1.07471i −0.437211 + 0.252424i
49.13 −0.537814 + 0.310507i 0.648232 + 0.374257i −0.807171 + 1.39806i 3.40675 −0.464837 1.23700 + 4.61655i − 2.24456i −1.21986 − 2.11287i −1.83220 + 1.05782i
49.14 −0.352909 + 0.203752i −1.29982 − 0.750452i −0.916970 + 1.58824i 1.88589 0.611624 −0.185005 − 0.690448i − 1.56235i −0.373645 − 0.647171i −0.665547 + 0.384254i
49.15 −0.337421 + 0.194810i −2.36537 − 1.36565i −0.924098 + 1.60058i −2.80788 1.06417 0.910277 + 3.39720i − 1.49934i 2.22999 + 3.86246i 0.947437 − 0.547003i
49.16 −0.282598 + 0.163158i 0.695665 + 0.401643i −0.946759 + 1.63983i −3.00846 −0.262125 0.686628 + 2.56253i − 1.27052i −1.17737 − 2.03926i 0.850187 − 0.490856i
49.17 0.0772619 − 0.0446072i 2.84786 + 1.64421i −0.996020 + 1.72516i 2.82782 0.293375 −0.197723 − 0.737912i 0.356147i 3.90688 + 6.76692i 0.218483 − 0.126141i
49.18 0.171780 − 0.0991774i 0.167202 + 0.0965340i −0.980328 + 1.69798i −1.27773 0.0382960 −1.35354 − 5.05149i 0.785615i −1.48136 − 2.56579i −0.219488 + 0.126722i
49.19 0.273325 − 0.157804i −1.63084 − 0.941565i −0.950196 + 1.64579i 0.799573 −0.594332 0.168228 + 0.627834i 1.23100i 0.273088 + 0.473002i 0.218543 − 0.126176i
49.20 0.530127 − 0.306069i 2.36967 + 1.36813i −0.812643 + 1.40754i −2.89554 1.67497 0.0978842 + 0.365309i 2.21918i 2.24355 + 3.88595i −1.53501 + 0.886236i
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.33
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
409.g even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 409.2.g.b ✓ 132
409.g even 12 1 inner 409.2.g.b ✓ 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
409.2.g.b ✓ 132 1.a even 1 1 trivial
409.2.g.b ✓ 132 409.g even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{132} - 95 T_{2}^{130} + 4746 T_{2}^{128} - 36 T_{2}^{127} - 163305 T_{2}^{126} + 3210 T_{2}^{125} + \cdots + 38775529 \) acting on \(S_{2}^{\mathrm{new}}(409, [\chi])\). Copy content Toggle raw display