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Properties

Label 209.4.d.b
Level $209$
Weight $4$
Character orbit 209.d
Analytic conductor $12.331$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [209,4,Mod(208,209)] 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("209.208"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(209, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 209.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56] 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: \(12.3313991912\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 236 q^{4} + 16 q^{5} - 484 q^{9} + 74 q^{11} + 668 q^{16} - 264 q^{20} - 296 q^{23} + 1040 q^{25} - 672 q^{26} - 428 q^{36} + 700 q^{38} - 2472 q^{42} - 668 q^{44} + 248 q^{45} - 2284 q^{47} - 2880 q^{49}+ \cdots - 414 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1 −5.32196 − 6.45839i 20.3232 7.08774 34.3713i − 10.5649i −65.5837 −14.7108 −37.7207
208.2 −5.32196 6.45839i 20.3232 7.08774 − 34.3713i 10.5649i −65.5837 −14.7108 −37.7207
208.3 −5.31409 − 6.03000i 20.2396 −16.2735 32.0440i 31.5028i −65.0422 −9.36085 86.4787
208.4 −5.31409 6.03000i 20.2396 −16.2735 − 32.0440i − 31.5028i −65.0422 −9.36085 86.4787
208.5 −4.85760 − 2.62380i 15.5963 −2.07409 12.7454i − 3.96480i −36.8999 20.1156 10.0751
208.6 −4.85760 2.62380i 15.5963 −2.07409 − 12.7454i 3.96480i −36.8999 20.1156 10.0751
208.7 −4.40765 − 2.91118i 11.4274 16.1151 12.8315i − 26.0720i −15.1066 18.5250 −71.0299
208.8 −4.40765 2.91118i 11.4274 16.1151 − 12.8315i 26.0720i −15.1066 18.5250 −71.0299
208.9 −4.05308 − 9.73675i 8.42746 13.1386 39.4638i 12.0311i −1.73251 −67.8042 −53.2518
208.10 −4.05308 9.73675i 8.42746 13.1386 − 39.4638i − 12.0311i −1.73251 −67.8042 −53.2518
208.11 −3.68982 − 1.27945i 5.61474 −16.5878 4.72095i − 24.0550i 8.80118 25.3630 61.2060
208.12 −3.68982 1.27945i 5.61474 −16.5878 − 4.72095i 24.0550i 8.80118 25.3630 61.2060
208.13 −3.57974 − 8.20420i 4.81451 −11.1698 29.3689i − 10.0111i 11.4032 −40.3088 39.9850
208.14 −3.57974 8.20420i 4.81451 −11.1698 − 29.3689i 10.0111i 11.4032 −40.3088 39.9850
208.15 −3.29634 − 2.96198i 2.86587 13.0149 9.76370i 26.6922i 16.9239 18.2267 −42.9016
208.16 −3.29634 2.96198i 2.86587 13.0149 − 9.76370i − 26.6922i 16.9239 18.2267 −42.9016
208.17 −3.01648 − 6.93615i 1.09917 −3.66168 20.9228i 4.46538i 20.8162 −21.1102 11.0454
208.18 −3.01648 6.93615i 1.09917 −3.66168 − 20.9228i − 4.46538i 20.8162 −21.1102 11.0454
208.19 −1.87305 − 2.96686i −4.49167 −1.74425 5.55708i − 24.0786i 23.3976 18.1978 3.26707
208.20 −1.87305 2.96686i −4.49167 −1.74425 − 5.55708i 24.0786i 23.3976 18.1978 3.26707
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 208.56
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
19.b odd 2 1 inner
209.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.4.d.b ✓ 56
11.b odd 2 1 inner 209.4.d.b ✓ 56
19.b odd 2 1 inner 209.4.d.b ✓ 56
209.d even 2 1 inner 209.4.d.b ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.4.d.b ✓ 56 1.a even 1 1 trivial
209.4.d.b ✓ 56 11.b odd 2 1 inner
209.4.d.b ✓ 56 19.b odd 2 1 inner
209.4.d.b ✓ 56 209.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 171 T_{2}^{26} + 12933 T_{2}^{24} - 570429 T_{2}^{22} + 16287999 T_{2}^{20} + \cdots + 464885555200 \) acting on \(S_{4}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display