Properties

Label 1368.2.s.a
Level $1368$
Weight $2$
Character orbit 1368.s
Analytic conductor $10.924$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1368,2,Mod(505,1368)]
 
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, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1368.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.s (of order \(3\), 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: \(10.9235349965\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (4 \zeta_{6} - 4) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + (4 \zeta_{6} - 4) q^{5} - 3 q^{11} - 2 \zeta_{6} q^{13} + ( - 2 \zeta_{6} + 2) q^{17} + (5 \zeta_{6} - 2) q^{19} + 6 \zeta_{6} q^{23} - 11 \zeta_{6} q^{25} - 4 \zeta_{6} q^{29} - 10 q^{31} + 2 q^{37} + ( - 9 \zeta_{6} + 9) q^{41} + ( - 4 \zeta_{6} + 4) q^{43} - 12 \zeta_{6} q^{47} - 7 q^{49} - 2 \zeta_{6} q^{53} + ( - 12 \zeta_{6} + 12) q^{55} + (\zeta_{6} - 1) q^{59} + 8 \zeta_{6} q^{61} + 8 q^{65} - 9 \zeta_{6} q^{67} + (6 \zeta_{6} - 6) q^{71} + ( - 9 \zeta_{6} + 9) q^{73} + ( - 4 \zeta_{6} + 4) q^{79} + 5 q^{83} + 8 \zeta_{6} q^{85} - 18 \zeta_{6} q^{89} + ( - 8 \zeta_{6} - 12) q^{95} + (\zeta_{6} - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 6 q^{11} - 2 q^{13} + 2 q^{17} + q^{19} + 6 q^{23} - 11 q^{25} - 4 q^{29} - 20 q^{31} + 4 q^{37} + 9 q^{41} + 4 q^{43} - 12 q^{47} - 14 q^{49} - 2 q^{53} + 12 q^{55} - q^{59} + 8 q^{61} + 16 q^{65} - 9 q^{67} - 6 q^{71} + 9 q^{73} + 4 q^{79} + 10 q^{83} + 8 q^{85} - 18 q^{89} - 32 q^{95} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −2.00000 + 3.46410i 0 0 0 0 0
577.1 0 0 0 −2.00000 3.46410i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.s.a 2
3.b odd 2 1 152.2.i.b 2
4.b odd 2 1 2736.2.s.a 2
12.b even 2 1 304.2.i.d 2
19.c even 3 1 inner 1368.2.s.a 2
24.f even 2 1 1216.2.i.b 2
24.h odd 2 1 1216.2.i.f 2
57.f even 6 1 2888.2.a.a 1
57.h odd 6 1 152.2.i.b 2
57.h odd 6 1 2888.2.a.d 1
76.g odd 6 1 2736.2.s.a 2
228.m even 6 1 304.2.i.d 2
228.m even 6 1 5776.2.a.e 1
228.n odd 6 1 5776.2.a.j 1
456.u even 6 1 1216.2.i.b 2
456.x odd 6 1 1216.2.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.b 2 3.b odd 2 1
152.2.i.b 2 57.h odd 6 1
304.2.i.d 2 12.b even 2 1
304.2.i.d 2 228.m even 6 1
1216.2.i.b 2 24.f even 2 1
1216.2.i.b 2 456.u even 6 1
1216.2.i.f 2 24.h odd 2 1
1216.2.i.f 2 456.x odd 6 1
1368.2.s.a 2 1.a even 1 1 trivial
1368.2.s.a 2 19.c even 3 1 inner
2736.2.s.a 2 4.b odd 2 1
2736.2.s.a 2 76.g odd 6 1
2888.2.a.a 1 57.f even 6 1
2888.2.a.d 1 57.h odd 6 1
5776.2.a.e 1 228.m even 6 1
5776.2.a.j 1 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1368, [\chi])\):

\( T_{5}^{2} + 4T_{5} + 16 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( (T - 5)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$97$ \( T^{2} + T + 1 \) Copy content Toggle raw display
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