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Properties

Label 384.4.k.a
Level $384$
Weight $4$
Character orbit 384.k
Analytic conductor $22.657$
Analytic rank $0$
Dimension $44$
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] = [384,4,Mod(95,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.95");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.k (of order \(4\), degree \(2\), not 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: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{3} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{3} + 8 q^{7} + 4 q^{13} + 20 q^{19} + 56 q^{21} - 134 q^{27} - 4 q^{33} + 4 q^{37} - 596 q^{39} - 436 q^{43} + 252 q^{45} + 972 q^{49} - 648 q^{51} - 280 q^{55} + 916 q^{61} - 1636 q^{67} - 52 q^{69} + 1454 q^{75} - 4 q^{81} - 736 q^{85} - 1284 q^{87} + 424 q^{91} + 2084 q^{93} - 8 q^{97} + 1196 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 0 −5.19561 + 0.0749974i 0 −5.37662 − 5.37662i 0 −14.8575 0 26.9888 − 0.779314i 0
95.2 0 −5.16550 − 0.563550i 0 −13.1633 − 13.1633i 0 13.2717 0 26.3648 + 5.82204i 0
95.3 0 −4.81196 + 1.96089i 0 6.30133 + 6.30133i 0 −24.6728 0 19.3098 − 18.8714i 0
95.4 0 −4.66767 − 2.28317i 0 11.5146 + 11.5146i 0 −0.829117 0 16.5743 + 21.3142i 0
95.5 0 −4.46768 + 2.65327i 0 5.27809 + 5.27809i 0 22.9284 0 12.9203 − 23.7079i 0
95.6 0 −3.58035 − 3.76578i 0 4.71515 + 4.71515i 0 −4.67595 0 −1.36225 + 26.9656i 0
95.7 0 −2.71261 − 4.43190i 0 −3.17566 − 3.17566i 0 32.3513 0 −12.2835 + 24.0440i 0
95.8 0 −2.65327 + 4.46768i 0 −5.27809 − 5.27809i 0 22.9284 0 −12.9203 − 23.7079i 0
95.9 0 −1.96089 + 4.81196i 0 −6.30133 − 6.30133i 0 −24.6728 0 −19.3098 − 18.8714i 0
95.10 0 −1.13522 − 5.07063i 0 −11.2665 − 11.2665i 0 −30.2121 0 −24.4225 + 11.5126i 0
95.11 0 −0.151014 − 5.19396i 0 4.66675 + 4.66675i 0 −0.405799 0 −26.9544 + 1.56872i 0
95.12 0 −0.0749974 + 5.19561i 0 5.37662 + 5.37662i 0 −14.8575 0 −26.9888 − 0.779314i 0
95.13 0 0.563550 + 5.16550i 0 13.1633 + 13.1633i 0 13.2717 0 −26.3648 + 5.82204i 0
95.14 0 2.24138 − 4.68788i 0 2.69043 + 2.69043i 0 −10.6336 0 −16.9524 − 21.0147i 0
95.15 0 2.28317 + 4.66767i 0 −11.5146 − 11.5146i 0 −0.829117 0 −16.5743 + 21.3142i 0
95.16 0 3.47813 − 3.86039i 0 −13.5794 − 13.5794i 0 19.7355 0 −2.80518 − 26.8539i 0
95.17 0 3.76578 + 3.58035i 0 −4.71515 − 4.71515i 0 −4.67595 0 1.36225 + 26.9656i 0
95.18 0 3.86039 − 3.47813i 0 13.5794 + 13.5794i 0 19.7355 0 2.80518 − 26.8539i 0
95.19 0 4.43190 + 2.71261i 0 3.17566 + 3.17566i 0 32.3513 0 12.2835 + 24.0440i 0
95.20 0 4.68788 − 2.24138i 0 −2.69043 − 2.69043i 0 −10.6336 0 16.9524 − 21.0147i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.4.k.a 44
3.b odd 2 1 inner 384.4.k.a 44
4.b odd 2 1 384.4.k.b 44
8.b even 2 1 192.4.k.a 44
8.d odd 2 1 48.4.k.a ✓ 44
12.b even 2 1 384.4.k.b 44
16.e even 4 1 48.4.k.a ✓ 44
16.e even 4 1 384.4.k.b 44
16.f odd 4 1 192.4.k.a 44
16.f odd 4 1 inner 384.4.k.a 44
24.f even 2 1 48.4.k.a ✓ 44
24.h odd 2 1 192.4.k.a 44
48.i odd 4 1 48.4.k.a ✓ 44
48.i odd 4 1 384.4.k.b 44
48.k even 4 1 192.4.k.a 44
48.k even 4 1 inner 384.4.k.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.k.a ✓ 44 8.d odd 2 1
48.4.k.a ✓ 44 16.e even 4 1
48.4.k.a ✓ 44 24.f even 2 1
48.4.k.a ✓ 44 48.i odd 4 1
192.4.k.a 44 8.b even 2 1
192.4.k.a 44 16.f odd 4 1
192.4.k.a 44 24.h odd 2 1
192.4.k.a 44 48.k even 4 1
384.4.k.a 44 1.a even 1 1 trivial
384.4.k.a 44 3.b odd 2 1 inner
384.4.k.a 44 16.f odd 4 1 inner
384.4.k.a 44 48.k even 4 1 inner
384.4.k.b 44 4.b odd 2 1
384.4.k.b 44 12.b even 2 1
384.4.k.b 44 16.e even 4 1
384.4.k.b 44 48.i odd 4 1