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Properties

Label 1152.2.s.d
Level $1152$
Weight $2$
Character orbit 1152.s
Analytic conductor $9.199$
Analytic rank $0$
Dimension $24$
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] = [1152,2,Mod(383,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.383");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.s (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: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{3} + 6 q^{11} + 12 q^{15} - 8 q^{21} + 12 q^{25} - 28 q^{27} - 24 q^{29} - 36 q^{31} + 8 q^{33} + 20 q^{39} + 12 q^{41} + 42 q^{43} - 16 q^{45} + 12 q^{49} + 22 q^{51} + 4 q^{57} - 6 q^{59} - 52 q^{63} - 54 q^{67} - 24 q^{69} + 18 q^{75} - 48 q^{77} + 60 q^{79} - 4 q^{81} - 36 q^{83} + 64 q^{87} - 16 q^{93} - 100 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1 0 −1.72425 − 0.164237i 0 1.94279 + 1.12167i 0 −3.69568 + 2.13370i 0 2.94605 + 0.566370i 0
383.2 0 −1.69448 − 0.358813i 0 −0.932763 − 0.538531i 0 −1.20495 + 0.695680i 0 2.74251 + 1.21600i 0
383.3 0 −1.52581 + 0.819705i 0 −0.860638 − 0.496890i 0 3.59294 − 2.07438i 0 1.65617 − 2.50142i 0
383.4 0 −0.572688 − 1.63463i 0 0.273344 + 0.157815i 0 2.41739 − 1.39568i 0 −2.34406 + 1.87227i 0
383.5 0 −0.446598 + 1.67348i 0 −0.190721 − 0.110113i 0 2.40217 − 1.38689i 0 −2.60110 − 1.49475i 0
383.6 0 −0.194406 + 1.72111i 0 −3.35990 − 1.93984i 0 −3.77468 + 2.17932i 0 −2.92441 − 0.669187i 0
383.7 0 0.543114 − 1.64470i 0 3.09733 + 1.78824i 0 1.46519 − 0.845928i 0 −2.41005 − 1.78652i 0
383.8 0 0.991794 + 1.41998i 0 1.38781 + 0.801253i 0 1.71629 − 0.990903i 0 −1.03269 + 2.81666i 0
383.9 0 1.22009 − 1.22938i 0 1.62431 + 0.937794i 0 −0.0301976 + 0.0174346i 0 −0.0227488 − 2.99991i 0
383.10 0 1.32083 + 1.12045i 0 2.60329 + 1.50301i 0 −3.54102 + 2.04441i 0 0.489199 + 2.95985i 0
383.11 0 1.53886 + 0.794921i 0 −3.37746 − 1.94998i 0 −0.0433891 + 0.0250507i 0 1.73620 + 2.44655i 0
383.12 0 1.54352 − 0.785832i 0 −2.20740 − 1.27444i 0 0.695946 − 0.401805i 0 1.76493 − 2.42590i 0
767.1 0 −1.72425 + 0.164237i 0 1.94279 − 1.12167i 0 −3.69568 − 2.13370i 0 2.94605 − 0.566370i 0
767.2 0 −1.69448 + 0.358813i 0 −0.932763 + 0.538531i 0 −1.20495 − 0.695680i 0 2.74251 − 1.21600i 0
767.3 0 −1.52581 − 0.819705i 0 −0.860638 + 0.496890i 0 3.59294 + 2.07438i 0 1.65617 + 2.50142i 0
767.4 0 −0.572688 + 1.63463i 0 0.273344 − 0.157815i 0 2.41739 + 1.39568i 0 −2.34406 − 1.87227i 0
767.5 0 −0.446598 − 1.67348i 0 −0.190721 + 0.110113i 0 2.40217 + 1.38689i 0 −2.60110 + 1.49475i 0
767.6 0 −0.194406 − 1.72111i 0 −3.35990 + 1.93984i 0 −3.77468 − 2.17932i 0 −2.92441 + 0.669187i 0
767.7 0 0.543114 + 1.64470i 0 3.09733 − 1.78824i 0 1.46519 + 0.845928i 0 −2.41005 + 1.78652i 0
767.8 0 0.991794 − 1.41998i 0 1.38781 − 0.801253i 0 1.71629 + 0.990903i 0 −1.03269 − 2.81666i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 383.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.s.d yes 24
3.b odd 2 1 3456.2.s.b 24
4.b odd 2 1 1152.2.s.a ✓ 24
8.b even 2 1 1152.2.s.b yes 24
8.d odd 2 1 1152.2.s.c yes 24
9.c even 3 1 3456.2.s.d 24
9.d odd 6 1 1152.2.s.a ✓ 24
12.b even 2 1 3456.2.s.d 24
24.f even 2 1 3456.2.s.a 24
24.h odd 2 1 3456.2.s.c 24
36.f odd 6 1 3456.2.s.b 24
36.h even 6 1 inner 1152.2.s.d yes 24
72.j odd 6 1 1152.2.s.c yes 24
72.l even 6 1 1152.2.s.b yes 24
72.n even 6 1 3456.2.s.a 24
72.p odd 6 1 3456.2.s.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.s.a ✓ 24 4.b odd 2 1
1152.2.s.a ✓ 24 9.d odd 6 1
1152.2.s.b yes 24 8.b even 2 1
1152.2.s.b yes 24 72.l even 6 1
1152.2.s.c yes 24 8.d odd 2 1
1152.2.s.c yes 24 72.j odd 6 1
1152.2.s.d yes 24 1.a even 1 1 trivial
1152.2.s.d yes 24 36.h even 6 1 inner
3456.2.s.a 24 24.f even 2 1
3456.2.s.a 24 72.n even 6 1
3456.2.s.b 24 3.b odd 2 1
3456.2.s.b 24 36.f odd 6 1
3456.2.s.c 24 24.h odd 2 1
3456.2.s.c 24 72.p odd 6 1
3456.2.s.d 24 9.c even 3 1
3456.2.s.d 24 12.b even 2 1