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Properties

Label 384.9.g.b
Level $384$
Weight $9$
Character orbit 384.g
Analytic conductor $156.433$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,9,Mod(127,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 384.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: \(156.433386263\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 1344 q^{5} - 69984 q^{9} + 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} + 275520 q^{29} - 2421440 q^{37} - 4374720 q^{41} - 2939328 q^{45} - 14219104 q^{49} + 6224448 q^{53} + 3100032 q^{57} + 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} - 154517760 q^{77} + 153055008 q^{81} + 384830848 q^{85} - 182669760 q^{89} - 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1 0 − 46.7654i 0 779.371 0 4180.82i 0 −2187.00 0
127.2 0 46.7654i 0 779.371 0 − 4180.82i 0 −2187.00 0
127.3 0 − 46.7654i 0 −909.886 0 − 3388.15i 0 −2187.00 0
127.4 0 46.7654i 0 −909.886 0 3388.15i 0 −2187.00 0
127.5 0 − 46.7654i 0 −407.399 0 3273.60i 0 −2187.00 0
127.6 0 46.7654i 0 −407.399 0 − 3273.60i 0 −2187.00 0
127.7 0 − 46.7654i 0 −795.016 0 2625.43i 0 −2187.00 0
127.8 0 46.7654i 0 −795.016 0 − 2625.43i 0 −2187.00 0
127.9 0 − 46.7654i 0 −317.614 0 − 2582.66i 0 −2187.00 0
127.10 0 46.7654i 0 −317.614 0 2582.66i 0 −2187.00 0
127.11 0 − 46.7654i 0 −26.4806 0 − 1828.22i 0 −2187.00 0
127.12 0 46.7654i 0 −26.4806 0 1828.22i 0 −2187.00 0
127.13 0 − 46.7654i 0 −1083.37 0 399.103i 0 −2187.00 0
127.14 0 46.7654i 0 −1083.37 0 − 399.103i 0 −2187.00 0
127.15 0 − 46.7654i 0 425.597 0 − 584.708i 0 −2187.00 0
127.16 0 46.7654i 0 425.597 0 584.708i 0 −2187.00 0
127.17 0 − 46.7654i 0 19.7383 0 759.796i 0 −2187.00 0
127.18 0 46.7654i 0 19.7383 0 − 759.796i 0 −2187.00 0
127.19 0 − 46.7654i 0 441.863 0 − 408.575i 0 −2187.00 0
127.20 0 46.7654i 0 441.863 0 408.575i 0 −2187.00 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.9.g.b yes 32
4.b odd 2 1 inner 384.9.g.b yes 32
8.b even 2 1 384.9.g.a ✓ 32
8.d odd 2 1 384.9.g.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.9.g.a ✓ 32 8.b even 2 1
384.9.g.a ✓ 32 8.d odd 2 1
384.9.g.b yes 32 1.a even 1 1 trivial
384.9.g.b yes 32 4.b odd 2 1 inner