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Properties

Label 209.4.e.a
Level $209$
Weight $4$
Character orbit 209.e
Analytic conductor $12.331$
Analytic rank $0$
Dimension $50$
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] = [209,4,Mod(45,209)] 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("209.45"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(209, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 209.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [50,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.3313991912\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(25\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 2 q^{2} - 6 q^{3} - 106 q^{4} - 19 q^{6} - 20 q^{7} + 126 q^{8} - 225 q^{9} - 23 q^{10} - 550 q^{11} + 176 q^{12} - 157 q^{13} - 27 q^{14} - 86 q^{15} - 362 q^{16} - 52 q^{17} + 386 q^{18} + 245 q^{19}+ \cdots + 2475 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −2.74965 − 4.76253i −4.56537 − 7.90745i −11.1211 + 19.2623i −5.54290 − 9.60058i −25.1063 + 43.4854i −11.4589 78.3220 −28.1852 + 48.8182i −30.4820 + 52.7964i
45.2 −2.59995 − 4.50324i −1.90332 − 3.29665i −9.51947 + 16.4882i 7.77173 + 13.4610i −9.89707 + 17.1422i 2.92858 57.4013 6.25474 − 10.8335i 40.4122 − 69.9960i
45.3 −2.56079 − 4.43541i 4.17337 + 7.22848i −9.11527 + 15.7881i 1.66759 + 2.88835i 21.3742 − 37.0212i 27.0630 52.3965 −21.3340 + 36.9515i 8.54068 − 14.7929i
45.4 −2.07384 − 3.59200i 3.26037 + 5.64712i −4.60162 + 7.97025i −0.180621 − 0.312844i 13.5230 − 23.4225i −11.3583 4.99068 −7.75999 + 13.4407i −0.749156 + 1.29758i
45.5 −2.02212 − 3.50241i 0.573215 + 0.992837i −4.17790 + 7.23634i −9.98094 − 17.2875i 2.31821 − 4.01526i −13.1224 1.43897 12.8428 − 22.2445i −40.3652 + 69.9146i
45.6 −1.93232 − 3.34687i −1.48337 − 2.56927i −3.46771 + 6.00625i 2.53830 + 4.39647i −5.73268 + 9.92930i −27.6157 −4.11422 9.09923 − 15.7603i 9.80962 − 16.9908i
45.7 −1.61292 − 2.79366i −4.18024 − 7.24040i −1.20301 + 2.08368i −4.61249 − 7.98907i −13.4848 + 23.3563i 26.5035 −18.0452 −21.4489 + 37.1506i −14.8792 + 25.7714i
45.8 −1.38635 − 2.40122i 1.25111 + 2.16699i 0.156092 − 0.270359i 9.06599 + 15.7028i 3.46894 − 6.00838i 19.7884 −23.0471 10.3695 − 17.9604i 25.1372 − 43.5389i
45.9 −1.17156 − 2.02921i −1.55977 − 2.70160i 1.25487 − 2.17351i 3.37040 + 5.83771i −3.65473 + 6.33019i 13.0443 −24.6257 8.63426 − 14.9550i 7.89728 − 13.6785i
45.10 −0.949028 − 1.64377i 4.33856 + 7.51461i 2.19869 − 3.80824i 4.74392 + 8.21672i 8.23484 − 14.2632i −26.4923 −23.5309 −24.1462 + 41.8225i 9.00424 − 15.5958i
45.11 −0.431708 − 0.747740i −0.536900 − 0.929938i 3.62726 − 6.28259i −8.36072 − 14.4812i −0.463568 + 0.802923i −14.6616 −13.1710 12.9235 − 22.3841i −7.21878 + 12.5033i
45.12 −0.335889 − 0.581776i 2.40763 + 4.17014i 3.77436 − 6.53738i −5.73349 − 9.93070i 1.61739 − 2.80141i 22.0005 −10.4453 1.90660 − 3.30232i −3.85163 + 6.67122i
45.13 −0.225447 − 0.390485i −4.61959 − 8.00137i 3.89835 − 6.75214i −0.910798 − 1.57755i −2.08295 + 3.60777i −19.8667 −7.12263 −29.1813 + 50.5435i −0.410673 + 0.711307i
45.14 0.553468 + 0.958635i 0.0230292 + 0.0398877i 3.38735 − 5.86706i 0.397836 + 0.689073i −0.0254918 + 0.0441532i −24.5331 16.3546 13.4989 − 23.3808i −0.440379 + 0.762759i
45.15 0.764066 + 1.32340i 1.28389 + 2.22377i 2.83241 − 4.90587i 7.04537 + 12.2029i −1.96196 + 3.39821i 13.8829 20.8816 10.2032 − 17.6725i −10.7662 + 18.6477i
45.16 0.789264 + 1.36705i −3.17263 − 5.49516i 2.75412 − 4.77028i 3.79327 + 6.57014i 5.00809 − 8.67427i 18.4171 21.3232 −6.63120 + 11.4856i −5.98779 + 10.3712i
45.17 0.903626 + 1.56513i 4.56009 + 7.89831i 2.36692 − 4.09963i −3.47289 − 6.01522i −8.24123 + 14.2742i 17.3262 23.0133 −28.0888 + 48.6513i 6.27639 − 10.8710i
45.18 1.14735 + 1.98728i −3.54242 − 6.13566i 1.36716 − 2.36798i −9.35321 − 16.2002i 8.12883 − 14.0795i 15.8464 24.6321 −11.5975 + 20.0875i 21.4629 − 37.1748i
45.19 1.52357 + 2.63889i −2.96538 − 5.13619i −0.642501 + 1.11284i 10.6596 + 18.4630i 9.03590 − 15.6506i −28.5098 20.4615 −4.08696 + 7.07882i −32.4813 + 56.2593i
45.20 1.71892 + 2.97726i 0.789088 + 1.36674i −1.90940 + 3.30717i −6.32047 − 10.9474i −2.71276 + 4.69864i 6.21877 14.3743 12.2547 − 21.2257i 21.7288 − 37.6354i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.25
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.4.e.a ✓ 50
19.c even 3 1 inner 209.4.e.a ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.4.e.a ✓ 50 1.a even 1 1 trivial
209.4.e.a ✓ 50 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{50} + 2 T_{2}^{49} + 155 T_{2}^{48} + 244 T_{2}^{47} + 13411 T_{2}^{46} + 17499 T_{2}^{45} + \cdots + 60\!\cdots\!84 \) acting on \(S_{4}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display