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Properties

Label 1368.3.cs.a
Level $1368$
Weight $3$
Character orbit 1368.cs
Analytic conductor $37.275$
Analytic rank $0$
Dimension $40$
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] = [1368,3,Mod(809,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1368.809");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1368.cs (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: \(37.2753001645\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 16 q^{7} - 44 q^{13} - 36 q^{19} + 100 q^{25} + 56 q^{31} - 72 q^{37} - 44 q^{43} + 48 q^{49} - 116 q^{55} - 52 q^{61} - 176 q^{67} - 180 q^{73} + 76 q^{79} + 472 q^{85} + 484 q^{91} + 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
809.1 0 0 0 −7.75370 − 4.47660i 0 −8.60956 0 0 0
809.2 0 0 0 −7.55555 − 4.36220i 0 6.97290 0 0 0
809.3 0 0 0 −6.25470 − 3.61115i 0 8.10170 0 0 0
809.4 0 0 0 −4.58360 − 2.64634i 0 −0.207051 0 0 0
809.5 0 0 0 −4.41339 − 2.54807i 0 −12.5926 0 0 0
809.6 0 0 0 −2.81415 − 1.62475i 0 0.581102 0 0 0
809.7 0 0 0 −2.57694 − 1.48779i 0 −7.84769 0 0 0
809.8 0 0 0 −2.31730 − 1.33789i 0 4.66571 0 0 0
809.9 0 0 0 −2.25037 − 1.29925i 0 −2.96949 0 0 0
809.10 0 0 0 −1.78631 − 1.03132i 0 7.90500 0 0 0
809.11 0 0 0 1.78631 + 1.03132i 0 7.90500 0 0 0
809.12 0 0 0 2.25037 + 1.29925i 0 −2.96949 0 0 0
809.13 0 0 0 2.31730 + 1.33789i 0 4.66571 0 0 0
809.14 0 0 0 2.57694 + 1.48779i 0 −7.84769 0 0 0
809.15 0 0 0 2.81415 + 1.62475i 0 0.581102 0 0 0
809.16 0 0 0 4.41339 + 2.54807i 0 −12.5926 0 0 0
809.17 0 0 0 4.58360 + 2.64634i 0 −0.207051 0 0 0
809.18 0 0 0 6.25470 + 3.61115i 0 8.10170 0 0 0
809.19 0 0 0 7.55555 + 4.36220i 0 6.97290 0 0 0
809.20 0 0 0 7.75370 + 4.47660i 0 −8.60956 0 0 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 809.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.3.cs.a ✓ 40
3.b odd 2 1 inner 1368.3.cs.a ✓ 40
19.c even 3 1 inner 1368.3.cs.a ✓ 40
57.h odd 6 1 inner 1368.3.cs.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.3.cs.a ✓ 40 1.a even 1 1 trivial
1368.3.cs.a ✓ 40 3.b odd 2 1 inner
1368.3.cs.a ✓ 40 19.c even 3 1 inner
1368.3.cs.a ✓ 40 57.h odd 6 1 inner