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Properties

Label 966.2.k.b
Level $966$
Weight $2$
Character orbit 966.k
Analytic conductor $7.714$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(229,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.229");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.k (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{2} - 16 q^{4} - 32 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{2} - 16 q^{4} - 32 q^{8} + 16 q^{9} - 16 q^{16} - 16 q^{18} + 4 q^{23} - 40 q^{25} - 12 q^{26} + 24 q^{29} - 24 q^{31} + 16 q^{32} + 12 q^{35} - 32 q^{36} + 8 q^{39} - 4 q^{46} + 12 q^{47} + 24 q^{49} - 80 q^{50} - 12 q^{52} + 12 q^{58} - 12 q^{59} + 32 q^{64} + 24 q^{70} + 16 q^{71} - 16 q^{72} - 84 q^{73} + 12 q^{75} - 40 q^{77} + 16 q^{78} - 16 q^{81} - 36 q^{82} - 112 q^{85} - 48 q^{87} - 8 q^{92} - 8 q^{93} + 12 q^{94} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1 0.500000 − 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i −2.02969 + 3.51552i 1.00000i 2.03303 + 1.69316i −1.00000 0.500000 − 0.866025i 2.02969 + 3.51552i
229.2 0.500000 − 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i −1.92123 + 3.32767i 1.00000i 0.661545 − 2.56171i −1.00000 0.500000 − 0.866025i 1.92123 + 3.32767i
229.3 0.500000 − 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i −0.320775 + 0.555599i 1.00000i −2.04200 + 1.68233i −1.00000 0.500000 − 0.866025i 0.320775 + 0.555599i
229.4 0.500000 − 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i −0.139134 + 0.240988i 1.00000i 2.42763 + 1.05197i −1.00000 0.500000 − 0.866025i 0.139134 + 0.240988i
229.5 0.500000 − 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i 0.139134 − 0.240988i 1.00000i −2.42763 − 1.05197i −1.00000 0.500000 − 0.866025i −0.139134 − 0.240988i
229.6 0.500000 − 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i 0.320775 − 0.555599i 1.00000i 2.04200 − 1.68233i −1.00000 0.500000 − 0.866025i −0.320775 − 0.555599i
229.7 0.500000 − 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i 1.92123 − 3.32767i 1.00000i −0.661545 + 2.56171i −1.00000 0.500000 − 0.866025i −1.92123 − 3.32767i
229.8 0.500000 − 0.866025i −0.866025 + 0.500000i −0.500000 − 0.866025i 2.02969 − 3.51552i 1.00000i −2.03303 − 1.69316i −1.00000 0.500000 − 0.866025i −2.02969 − 3.51552i
229.9 0.500000 − 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i −1.70402 + 2.95145i − 1.00000i −2.64298 − 0.121148i −1.00000 0.500000 − 0.866025i 1.70402 + 2.95145i
229.10 0.500000 − 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i −1.36115 + 2.35758i − 1.00000i 2.49823 − 0.871108i −1.00000 0.500000 − 0.866025i 1.36115 + 2.35758i
229.11 0.500000 − 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i −1.28362 + 2.22329i − 1.00000i −1.74878 − 1.98539i −1.00000 0.500000 − 0.866025i 1.28362 + 2.22329i
229.12 0.500000 − 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i −0.814185 + 1.41021i − 1.00000i −0.285113 + 2.63034i −1.00000 0.500000 − 0.866025i 0.814185 + 1.41021i
229.13 0.500000 − 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i 0.814185 − 1.41021i − 1.00000i 0.285113 − 2.63034i −1.00000 0.500000 − 0.866025i −0.814185 − 1.41021i
229.14 0.500000 − 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i 1.28362 − 2.22329i − 1.00000i 1.74878 + 1.98539i −1.00000 0.500000 − 0.866025i −1.28362 − 2.22329i
229.15 0.500000 − 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i 1.36115 − 2.35758i − 1.00000i −2.49823 + 0.871108i −1.00000 0.500000 − 0.866025i −1.36115 − 2.35758i
229.16 0.500000 − 0.866025i 0.866025 − 0.500000i −0.500000 − 0.866025i 1.70402 − 2.95145i − 1.00000i 2.64298 + 0.121148i −1.00000 0.500000 − 0.866025i −1.70402 − 2.95145i
367.1 0.500000 + 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −2.02969 − 3.51552i − 1.00000i 2.03303 − 1.69316i −1.00000 0.500000 + 0.866025i 2.02969 − 3.51552i
367.2 0.500000 + 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −1.92123 − 3.32767i − 1.00000i 0.661545 + 2.56171i −1.00000 0.500000 + 0.866025i 1.92123 − 3.32767i
367.3 0.500000 + 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −0.320775 − 0.555599i − 1.00000i −2.04200 − 1.68233i −1.00000 0.500000 + 0.866025i 0.320775 − 0.555599i
367.4 0.500000 + 0.866025i −0.866025 − 0.500000i −0.500000 + 0.866025i −0.139134 − 0.240988i − 1.00000i 2.42763 − 1.05197i −1.00000 0.500000 + 0.866025i 0.139134 − 0.240988i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.b odd 2 1 inner
161.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.k.b ✓ 32
7.d odd 6 1 inner 966.2.k.b ✓ 32
23.b odd 2 1 inner 966.2.k.b ✓ 32
161.g even 6 1 inner 966.2.k.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.k.b ✓ 32 1.a even 1 1 trivial
966.2.k.b ✓ 32 7.d odd 6 1 inner
966.2.k.b ✓ 32 23.b odd 2 1 inner
966.2.k.b ✓ 32 161.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 60 T_{5}^{30} + 2165 T_{5}^{28} + 51252 T_{5}^{26} + 900834 T_{5}^{24} + 11822508 T_{5}^{22} + \cdots + 136048896 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display