↑

Properties

Label 990.2.z.a
Level $990$
Weight $2$
Character orbit 990.z
Analytic conductor $7.905$
Analytic rank $0$
Dimension $32$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [990,2,Mod(161,990)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 0, 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("990.161"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.z (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,-8] 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: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{2} - 8 q^{4} - 8 q^{8} - 8 q^{16} + 4 q^{17} + 8 q^{25} - 8 q^{29} + 32 q^{31} + 32 q^{32} + 24 q^{34} + 16 q^{37} + 32 q^{41} - 20 q^{46} - 20 q^{47} + 16 q^{49} + 8 q^{50} - 40 q^{53} + 8 q^{55}+ \cdots + 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 −0.809017 − 0.587785i 0 0.309017 + 0.951057i −0.587785 − 0.809017i 0 −3.20471 + 1.04127i 0.309017 − 0.951057i 0 1.00000i
161.2 −0.809017 − 0.587785i 0 0.309017 + 0.951057i −0.587785 − 0.809017i 0 −1.65927 + 0.539131i 0.309017 − 0.951057i 0 1.00000i
161.3 −0.809017 − 0.587785i 0 0.309017 + 0.951057i −0.587785 − 0.809017i 0 1.89200 − 0.614747i 0.309017 − 0.951057i 0 1.00000i
161.4 −0.809017 − 0.587785i 0 0.309017 + 0.951057i −0.587785 − 0.809017i 0 2.97198 − 0.965656i 0.309017 − 0.951057i 0 1.00000i
161.5 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 0.587785 + 0.809017i 0 −2.25567 + 0.732912i 0.309017 − 0.951057i 0 − 1.00000i
161.6 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 0.587785 + 0.809017i 0 −1.61538 + 0.524868i 0.309017 − 0.951057i 0 − 1.00000i
161.7 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 0.587785 + 0.809017i 0 −0.756962 + 0.245952i 0.309017 − 0.951057i 0 − 1.00000i
161.8 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 0.587785 + 0.809017i 0 4.62801 − 1.50373i 0.309017 − 0.951057i 0 − 1.00000i
431.1 0.309017 − 0.951057i 0 −0.809017 − 0.587785i −0.951057 + 0.309017i 0 −1.27614 + 1.75646i −0.809017 + 0.587785i 0 1.00000i
431.2 0.309017 − 0.951057i 0 −0.809017 − 0.587785i −0.951057 + 0.309017i 0 −0.770514 + 1.06052i −0.809017 + 0.587785i 0 1.00000i
431.3 0.309017 − 0.951057i 0 −0.809017 − 0.587785i −0.951057 + 0.309017i 0 0.0728709 − 0.100298i −0.809017 + 0.587785i 0 1.00000i
431.4 0.309017 − 0.951057i 0 −0.809017 − 0.587785i −0.951057 + 0.309017i 0 1.97378 − 2.71668i −0.809017 + 0.587785i 0 1.00000i
431.5 0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0.951057 − 0.309017i 0 −1.76442 + 2.42852i −0.809017 + 0.587785i 0 − 1.00000i
431.6 0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0.951057 − 0.309017i 0 −0.616585 + 0.848657i −0.809017 + 0.587785i 0 − 1.00000i
431.7 0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0.951057 − 0.309017i 0 0.0895822 − 0.123299i −0.809017 + 0.587785i 0 − 1.00000i
431.8 0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0.951057 − 0.309017i 0 2.29143 − 3.15388i −0.809017 + 0.587785i 0 − 1.00000i
611.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −0.951057 − 0.309017i 0 −1.27614 − 1.75646i −0.809017 − 0.587785i 0 − 1.00000i
611.2 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −0.951057 − 0.309017i 0 −0.770514 − 1.06052i −0.809017 − 0.587785i 0 − 1.00000i
611.3 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −0.951057 − 0.309017i 0 0.0728709 + 0.100298i −0.809017 − 0.587785i 0 − 1.00000i
611.4 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −0.951057 − 0.309017i 0 1.97378 + 2.71668i −0.809017 − 0.587785i 0 − 1.00000i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.2.z.a ✓ 32
3.b odd 2 1 990.2.z.b yes 32
11.d odd 10 1 990.2.z.b yes 32
33.f even 10 1 inner 990.2.z.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.z.a ✓ 32 1.a even 1 1 trivial
990.2.z.a ✓ 32 33.f even 10 1 inner
990.2.z.b yes 32 3.b odd 2 1
990.2.z.b yes 32 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{32} - 4 T_{17}^{31} + 66 T_{17}^{30} - 140 T_{17}^{29} + 3945 T_{17}^{28} - 14212 T_{17}^{27} + \cdots + 15352201216 \) acting on \(S_{2}^{\mathrm{new}}(990, [\chi])\). Copy content Toggle raw display