Properties

Label 966.2.j.a
Level $966$
Weight $2$
Character orbit 966.j
Analytic conductor $7.714$
Analytic rank $0$
Dimension $128$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(137,966)]
 
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([3, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.j (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: \(128\)
Relative dimension: \(64\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 64 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 64 q^{4} + 8 q^{6} - 4 q^{9} - 64 q^{16} + 4 q^{24} - 72 q^{25} + 24 q^{27} - 8 q^{36} + 4 q^{39} + 12 q^{46} + 72 q^{49} + 4 q^{54} - 96 q^{55} - 128 q^{64} - 56 q^{69} + 32 q^{70} - 40 q^{73} - 40 q^{75} - 16 q^{78} + 36 q^{81} - 32 q^{82} - 64 q^{85} - 24 q^{87} + 32 q^{93} - 8 q^{94} - 4 q^{96}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 −0.866025 + 0.500000i −1.70593 + 0.299664i 0.500000 0.866025i −0.915921 1.58642i 1.32755 1.11248i 1.74832 + 1.98579i 1.00000i 2.82040 1.02241i 1.58642 + 0.915921i
137.2 −0.866025 + 0.500000i −1.70593 + 0.299664i 0.500000 0.866025i 0.915921 + 1.58642i 1.32755 1.11248i −1.74832 1.98579i 1.00000i 2.82040 1.02241i −1.58642 0.915921i
137.3 −0.866025 + 0.500000i −1.62530 + 0.598665i 0.500000 0.866025i −1.08784 1.88420i 1.10822 1.33111i 2.11385 1.59112i 1.00000i 2.28320 1.94602i 1.88420 + 1.08784i
137.4 −0.866025 + 0.500000i −1.62530 + 0.598665i 0.500000 0.866025i 1.08784 + 1.88420i 1.10822 1.33111i −2.11385 + 1.59112i 1.00000i 2.28320 1.94602i −1.88420 1.08784i
137.5 −0.866025 + 0.500000i −1.58146 0.706395i 0.500000 0.866025i −1.53878 2.66525i 1.72278 0.178972i −0.379647 2.61837i 1.00000i 2.00201 + 2.23427i 2.66525 + 1.53878i
137.6 −0.866025 + 0.500000i −1.58146 0.706395i 0.500000 0.866025i 1.53878 + 2.66525i 1.72278 0.178972i 0.379647 + 2.61837i 1.00000i 2.00201 + 2.23427i −2.66525 1.53878i
137.7 −0.866025 + 0.500000i −1.34224 1.09471i 0.500000 0.866025i −0.485598 0.841080i 1.70977 + 0.276923i −0.839629 + 2.50899i 1.00000i 0.603233 + 2.93873i 0.841080 + 0.485598i
137.8 −0.866025 + 0.500000i −1.34224 1.09471i 0.500000 0.866025i 0.485598 + 0.841080i 1.70977 + 0.276923i 0.839629 2.50899i 1.00000i 0.603233 + 2.93873i −0.841080 0.485598i
137.9 −0.866025 + 0.500000i −1.25907 + 1.18943i 0.500000 0.866025i −1.82493 3.16086i 0.495676 1.65961i −2.63810 0.201024i 1.00000i 0.170531 2.99515i 3.16086 + 1.82493i
137.10 −0.866025 + 0.500000i −1.25907 + 1.18943i 0.500000 0.866025i 1.82493 + 3.16086i 0.495676 1.65961i 2.63810 + 0.201024i 1.00000i 0.170531 2.99515i −3.16086 1.82493i
137.11 −0.866025 + 0.500000i −0.608468 + 1.62166i 0.500000 0.866025i −0.435291 0.753947i −0.283879 1.70863i −2.60300 0.473676i 1.00000i −2.25953 1.97345i 0.753947 + 0.435291i
137.12 −0.866025 + 0.500000i −0.608468 + 1.62166i 0.500000 0.866025i 0.435291 + 0.753947i −0.283879 1.70863i 2.60300 + 0.473676i 1.00000i −2.25953 1.97345i −0.753947 0.435291i
137.13 −0.866025 + 0.500000i −0.583589 1.63077i 0.500000 0.866025i −1.70837 2.95898i 1.32079 + 1.12050i 2.46321 + 0.965707i 1.00000i −2.31885 + 1.90340i 2.95898 + 1.70837i
137.14 −0.866025 + 0.500000i −0.583589 1.63077i 0.500000 0.866025i 1.70837 + 2.95898i 1.32079 + 1.12050i −2.46321 0.965707i 1.00000i −2.31885 + 1.90340i −2.95898 1.70837i
137.15 −0.866025 + 0.500000i −0.498221 1.65885i 0.500000 0.866025i −1.35083 2.33970i 1.26090 + 1.18749i −2.62714 0.313267i 1.00000i −2.50355 + 1.65295i 2.33970 + 1.35083i
137.16 −0.866025 + 0.500000i −0.498221 1.65885i 0.500000 0.866025i 1.35083 + 2.33970i 1.26090 + 1.18749i 2.62714 + 0.313267i 1.00000i −2.50355 + 1.65295i −2.33970 1.35083i
137.17 −0.866025 + 0.500000i 0.144090 + 1.72605i 0.500000 0.866025i −2.22533 3.85438i −0.987809 1.42276i 1.67417 + 2.04869i 1.00000i −2.95848 + 0.497412i 3.85438 + 2.22533i
137.18 −0.866025 + 0.500000i 0.144090 + 1.72605i 0.500000 0.866025i 2.22533 + 3.85438i −0.987809 1.42276i −1.67417 2.04869i 1.00000i −2.95848 + 0.497412i −3.85438 2.22533i
137.19 −0.866025 + 0.500000i 0.373762 + 1.69124i 0.500000 0.866025i −0.437223 0.757293i −1.16931 1.27778i 0.914719 2.48260i 1.00000i −2.72060 + 1.26424i 0.757293 + 0.437223i
137.20 −0.866025 + 0.500000i 0.373762 + 1.69124i 0.500000 0.866025i 0.437223 + 0.757293i −1.16931 1.27778i −0.914719 + 2.48260i 1.00000i −2.72060 + 1.26424i −0.757293 0.437223i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner
161.f odd 6 1 inner
483.m 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.j.a 128
3.b odd 2 1 inner 966.2.j.a 128
7.c even 3 1 inner 966.2.j.a 128
21.h odd 6 1 inner 966.2.j.a 128
23.b odd 2 1 inner 966.2.j.a 128
69.c even 2 1 inner 966.2.j.a 128
161.f odd 6 1 inner 966.2.j.a 128
483.m even 6 1 inner 966.2.j.a 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.j.a 128 1.a even 1 1 trivial
966.2.j.a 128 3.b odd 2 1 inner
966.2.j.a 128 7.c even 3 1 inner
966.2.j.a 128 21.h odd 6 1 inner
966.2.j.a 128 23.b odd 2 1 inner
966.2.j.a 128 69.c even 2 1 inner
966.2.j.a 128 161.f odd 6 1 inner
966.2.j.a 128 483.m even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(966, [\chi])\).