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Properties

Label 966.2.f.b
Level $966$
Weight $2$
Character orbit 966.f
Analytic conductor $7.714$
Analytic rank $0$
Dimension $24$
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] = [966,2,Mod(461,966)] 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("966.461"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(966, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] 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: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} - 4 q^{7} + 8 q^{9} + 24 q^{16} + 16 q^{18} - 28 q^{21} + 8 q^{22} - 24 q^{25} + 4 q^{28} + 24 q^{30} - 8 q^{36} + 40 q^{37} + 72 q^{39} + 64 q^{43} + 24 q^{46} - 24 q^{51} + 16 q^{58} + 12 q^{63}+ \cdots - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
461.1 − 1.00000i −1.72384 − 0.168468i −1.00000 1.25574 −0.168468 + 1.72384i −1.98144 − 1.75325i 1.00000i 2.94324 + 0.580822i − 1.25574i
461.2 − 1.00000i −1.66919 − 0.462387i −1.00000 −1.95027 −0.462387 + 1.66919i −0.602835 + 2.57616i 1.00000i 2.57240 + 1.54363i 1.95027i
461.3 − 1.00000i −1.62776 + 0.591954i −1.00000 2.12506 0.591954 + 1.62776i 2.32692 + 1.25915i 1.00000i 2.29918 − 1.92711i − 2.12506i
461.4 − 1.00000i −0.950180 − 1.44816i −1.00000 0.575987 −1.44816 + 0.950180i 2.10044 − 1.60877i 1.00000i −1.19432 + 2.75202i − 0.575987i
461.5 − 1.00000i −0.740879 − 1.56560i −1.00000 −3.67536 −1.56560 + 0.740879i −0.229372 − 2.63579i 1.00000i −1.90220 + 2.31984i 3.67536i
461.6 − 1.00000i −0.375300 + 1.69090i −1.00000 −0.513446 1.69090 + 0.375300i −2.61371 + 0.410499i 1.00000i −2.71830 − 1.26919i 0.513446i
461.7 − 1.00000i 0.375300 − 1.69090i −1.00000 0.513446 −1.69090 − 0.375300i −2.61371 − 0.410499i 1.00000i −2.71830 − 1.26919i − 0.513446i
461.8 − 1.00000i 0.740879 + 1.56560i −1.00000 3.67536 1.56560 − 0.740879i −0.229372 + 2.63579i 1.00000i −1.90220 + 2.31984i − 3.67536i
461.9 − 1.00000i 0.950180 + 1.44816i −1.00000 −0.575987 1.44816 − 0.950180i 2.10044 + 1.60877i 1.00000i −1.19432 + 2.75202i 0.575987i
461.10 − 1.00000i 1.62776 − 0.591954i −1.00000 −2.12506 −0.591954 − 1.62776i 2.32692 − 1.25915i 1.00000i 2.29918 − 1.92711i 2.12506i
461.11 − 1.00000i 1.66919 + 0.462387i −1.00000 1.95027 0.462387 − 1.66919i −0.602835 − 2.57616i 1.00000i 2.57240 + 1.54363i − 1.95027i
461.12 − 1.00000i 1.72384 + 0.168468i −1.00000 −1.25574 0.168468 − 1.72384i −1.98144 + 1.75325i 1.00000i 2.94324 + 0.580822i 1.25574i
461.13 1.00000i −1.72384 + 0.168468i −1.00000 1.25574 −0.168468 − 1.72384i −1.98144 + 1.75325i − 1.00000i 2.94324 − 0.580822i 1.25574i
461.14 1.00000i −1.66919 + 0.462387i −1.00000 −1.95027 −0.462387 − 1.66919i −0.602835 − 2.57616i − 1.00000i 2.57240 − 1.54363i − 1.95027i
461.15 1.00000i −1.62776 − 0.591954i −1.00000 2.12506 0.591954 − 1.62776i 2.32692 − 1.25915i − 1.00000i 2.29918 + 1.92711i 2.12506i
461.16 1.00000i −0.950180 + 1.44816i −1.00000 0.575987 −1.44816 − 0.950180i 2.10044 + 1.60877i − 1.00000i −1.19432 − 2.75202i 0.575987i
461.17 1.00000i −0.740879 + 1.56560i −1.00000 −3.67536 −1.56560 − 0.740879i −0.229372 + 2.63579i − 1.00000i −1.90220 − 2.31984i − 3.67536i
461.18 1.00000i −0.375300 − 1.69090i −1.00000 −0.513446 1.69090 − 0.375300i −2.61371 − 0.410499i − 1.00000i −2.71830 + 1.26919i − 0.513446i
461.19 1.00000i 0.375300 + 1.69090i −1.00000 0.513446 −1.69090 + 0.375300i −2.61371 + 0.410499i − 1.00000i −2.71830 + 1.26919i 0.513446i
461.20 1.00000i 0.740879 − 1.56560i −1.00000 3.67536 1.56560 + 0.740879i −0.229372 − 2.63579i − 1.00000i −1.90220 − 2.31984i 3.67536i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 461.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.f.b ✓ 24
3.b odd 2 1 inner 966.2.f.b ✓ 24
7.b odd 2 1 inner 966.2.f.b ✓ 24
21.c even 2 1 inner 966.2.f.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.f.b ✓ 24 1.a even 1 1 trivial
966.2.f.b ✓ 24 3.b odd 2 1 inner
966.2.f.b ✓ 24 7.b odd 2 1 inner
966.2.f.b ✓ 24 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 24T_{5}^{10} + 178T_{5}^{8} - 536T_{5}^{6} + 640T_{5}^{4} - 256T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display