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Properties

Label 966.4.g.b
Level $966$
Weight $4$
Character orbit 966.g
Analytic conductor $56.996$
Analytic rank $0$
Dimension $48$
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] = [966,4,Mod(643,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.643");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 966.g (of order \(2\), degree \(1\), 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: \(56.9958450655\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 96 q^{2} + 192 q^{4} + 384 q^{8} - 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 96 q^{2} + 192 q^{4} + 384 q^{8} - 432 q^{9} + 768 q^{16} - 864 q^{18} - 160 q^{23} + 700 q^{25} - 100 q^{29} + 1536 q^{32} - 1080 q^{35} - 1728 q^{36} + 300 q^{39} - 320 q^{46} + 704 q^{49} + 1400 q^{50} - 200 q^{58} + 3072 q^{64} - 2160 q^{70} - 1960 q^{71} - 3456 q^{72} + 180 q^{77} + 600 q^{78} + 3888 q^{81} + 168 q^{85} - 640 q^{92} - 1008 q^{93} + 1408 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
643.1 2.00000 − 3.00000i 4.00000 −7.52483 − 6.00000i −10.3481 + 15.3596i 8.00000 −9.00000 −15.0497
643.2 2.00000 − 3.00000i 4.00000 7.52483 − 6.00000i 10.3481 − 15.3596i 8.00000 −9.00000 15.0497
643.3 2.00000 3.00000i 4.00000 −7.52483 6.00000i −10.3481 − 15.3596i 8.00000 −9.00000 −15.0497
643.4 2.00000 3.00000i 4.00000 7.52483 6.00000i 10.3481 + 15.3596i 8.00000 −9.00000 15.0497
643.5 2.00000 − 3.00000i 4.00000 −1.58167 − 6.00000i 16.9416 − 7.48210i 8.00000 −9.00000 −3.16334
643.6 2.00000 − 3.00000i 4.00000 1.58167 − 6.00000i −16.9416 + 7.48210i 8.00000 −9.00000 3.16334
643.7 2.00000 3.00000i 4.00000 −1.58167 6.00000i 16.9416 + 7.48210i 8.00000 −9.00000 −3.16334
643.8 2.00000 3.00000i 4.00000 1.58167 6.00000i −16.9416 − 7.48210i 8.00000 −9.00000 3.16334
643.9 2.00000 − 3.00000i 4.00000 −11.9336 − 6.00000i −17.7009 + 5.44765i 8.00000 −9.00000 −23.8671
643.10 2.00000 − 3.00000i 4.00000 11.9336 − 6.00000i 17.7009 − 5.44765i 8.00000 −9.00000 23.8671
643.11 2.00000 3.00000i 4.00000 −11.9336 6.00000i −17.7009 − 5.44765i 8.00000 −9.00000 −23.8671
643.12 2.00000 3.00000i 4.00000 11.9336 6.00000i 17.7009 + 5.44765i 8.00000 −9.00000 23.8671
643.13 2.00000 − 3.00000i 4.00000 −1.46794 − 6.00000i 7.13994 + 17.0886i 8.00000 −9.00000 −2.93587
643.14 2.00000 − 3.00000i 4.00000 1.46794 − 6.00000i −7.13994 − 17.0886i 8.00000 −9.00000 2.93587
643.15 2.00000 3.00000i 4.00000 −1.46794 6.00000i 7.13994 − 17.0886i 8.00000 −9.00000 −2.93587
643.16 2.00000 3.00000i 4.00000 1.46794 6.00000i −7.13994 + 17.0886i 8.00000 −9.00000 2.93587
643.17 2.00000 − 3.00000i 4.00000 −6.50709 − 6.00000i 15.8106 − 9.64489i 8.00000 −9.00000 −13.0142
643.18 2.00000 − 3.00000i 4.00000 6.50709 − 6.00000i −15.8106 + 9.64489i 8.00000 −9.00000 13.0142
643.19 2.00000 3.00000i 4.00000 −6.50709 6.00000i 15.8106 + 9.64489i 8.00000 −9.00000 −13.0142
643.20 2.00000 3.00000i 4.00000 6.50709 6.00000i −15.8106 − 9.64489i 8.00000 −9.00000 13.0142
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 643.48
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.4.g.b ✓ 48
7.b odd 2 1 inner 966.4.g.b ✓ 48
23.b odd 2 1 inner 966.4.g.b ✓ 48
161.c even 2 1 inner 966.4.g.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.g.b ✓ 48 1.a even 1 1 trivial
966.4.g.b ✓ 48 7.b odd 2 1 inner
966.4.g.b ✓ 48 23.b odd 2 1 inner
966.4.g.b ✓ 48 161.c even 2 1 inner