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Properties

Label 513.2.bp.b
Level $513$
Weight $2$
Character orbit 513.bp
Analytic conductor $4.096$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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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] = [513,2,Mod(53,513)] 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("513.53"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.bp (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,0,0,-6,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.09632562369\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{4} - 12 q^{10} + 30 q^{16} + 54 q^{19} - 6 q^{22} + 6 q^{25} + 66 q^{28} + 36 q^{31} - 12 q^{34} - 30 q^{40} - 42 q^{43} + 18 q^{46} + 6 q^{49} - 60 q^{52} + 24 q^{55} + 192 q^{58} - 12 q^{61}+ \cdots + 42 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −2.19036 − 0.797227i 0 2.63004 + 2.20686i −1.71339 − 2.04193i 0 −0.360137 + 0.623776i −1.67043 − 2.89327i 0 2.12505 + 5.83853i
53.2 −2.08756 − 0.759809i 0 2.24850 + 1.88672i 2.42252 + 2.88705i 0 2.43674 − 4.22057i −1.03880 − 1.79926i 0 −2.86355 − 7.86754i
53.3 −0.929686 − 0.338378i 0 −0.782273 − 0.656405i −0.958434 − 1.14222i 0 −0.718167 + 1.24390i 1.49451 + 2.58856i 0 0.504541 + 1.38622i
53.4 0.929686 + 0.338378i 0 −0.782273 − 0.656405i 0.958434 + 1.14222i 0 −0.718167 + 1.24390i −1.49451 − 2.58856i 0 0.504541 + 1.38622i
53.5 2.08756 + 0.759809i 0 2.24850 + 1.88672i −2.42252 − 2.88705i 0 2.43674 − 4.22057i 1.03880 + 1.79926i 0 −2.86355 − 7.86754i
53.6 2.19036 + 0.797227i 0 2.63004 + 2.20686i 1.71339 + 2.04193i 0 −0.360137 + 0.623776i 1.67043 + 2.89327i 0 2.12505 + 5.83853i
242.1 −2.19036 + 0.797227i 0 2.63004 − 2.20686i −1.71339 + 2.04193i 0 −0.360137 − 0.623776i −1.67043 + 2.89327i 0 2.12505 − 5.83853i
242.2 −2.08756 + 0.759809i 0 2.24850 − 1.88672i 2.42252 − 2.88705i 0 2.43674 + 4.22057i −1.03880 + 1.79926i 0 −2.86355 + 7.86754i
242.3 −0.929686 + 0.338378i 0 −0.782273 + 0.656405i −0.958434 + 1.14222i 0 −0.718167 − 1.24390i 1.49451 − 2.58856i 0 0.504541 − 1.38622i
242.4 0.929686 − 0.338378i 0 −0.782273 + 0.656405i 0.958434 − 1.14222i 0 −0.718167 − 1.24390i −1.49451 + 2.58856i 0 0.504541 − 1.38622i
242.5 2.08756 − 0.759809i 0 2.24850 − 1.88672i −2.42252 + 2.88705i 0 2.43674 + 4.22057i 1.03880 − 1.79926i 0 −2.86355 + 7.86754i
242.6 2.19036 − 0.797227i 0 2.63004 − 2.20686i 1.71339 − 2.04193i 0 −0.360137 − 0.623776i 1.67043 − 2.89327i 0 2.12505 − 5.83853i
269.1 −1.88303 − 1.58005i 0 0.701950 + 3.98095i −0.429081 − 0.0756585i 0 0.234271 + 0.405769i 2.51019 − 4.34778i 0 0.688428 + 0.820436i
269.2 −1.19186 − 1.00009i 0 0.0730565 + 0.414324i 3.22642 + 0.568905i 0 −0.546546 − 0.946645i −1.22858 + 2.12796i 0 −3.27649 − 3.90477i
269.3 −0.621173 − 0.521226i 0 −0.233117 − 1.32207i −3.32859 − 0.586921i 0 1.59926 + 2.77001i −1.35518 + 2.34723i 0 1.76171 + 2.09953i
269.4 0.621173 + 0.521226i 0 −0.233117 − 1.32207i 3.32859 + 0.586921i 0 1.59926 + 2.77001i 1.35518 − 2.34723i 0 1.76171 + 2.09953i
269.5 1.19186 + 1.00009i 0 0.0730565 + 0.414324i −3.22642 − 0.568905i 0 −0.546546 − 0.946645i 1.22858 − 2.12796i 0 −3.27649 − 3.90477i
269.6 1.88303 + 1.58005i 0 0.701950 + 3.98095i 0.429081 + 0.0756585i 0 0.234271 + 0.405769i −2.51019 + 4.34778i 0 0.688428 + 0.820436i
431.1 −1.88303 + 1.58005i 0 0.701950 − 3.98095i −0.429081 + 0.0756585i 0 0.234271 − 0.405769i 2.51019 + 4.34778i 0 0.688428 − 0.820436i
431.2 −1.19186 + 1.00009i 0 0.0730565 − 0.414324i 3.22642 − 0.568905i 0 −0.546546 + 0.946645i −1.22858 − 2.12796i 0 −3.27649 + 3.90477i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.f odd 18 1 inner
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 513.2.bp.b ✓ 36
3.b odd 2 1 inner 513.2.bp.b ✓ 36
19.f odd 18 1 inner 513.2.bp.b ✓ 36
57.j even 18 1 inner 513.2.bp.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
513.2.bp.b ✓ 36 1.a even 1 1 trivial
513.2.bp.b ✓ 36 3.b odd 2 1 inner
513.2.bp.b ✓ 36 19.f odd 18 1 inner
513.2.bp.b ✓ 36 57.j even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 3 T_{2}^{34} - 21 T_{2}^{32} + 271 T_{2}^{30} + 1074 T_{2}^{28} - 8514 T_{2}^{26} + \cdots + 97594641 \) acting on \(S_{2}^{\mathrm{new}}(513, [\chi])\). Copy content Toggle raw display