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Properties

Label 1368.2.s.j
Level $1368$
Weight $2$
Character orbit 1368.s
Analytic conductor $10.924$
Analytic rank $0$
Dimension $6$
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] = [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: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 456)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4}) q^{5} + ( - \beta_{5} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{4}) q^{5} + ( - \beta_{5} + 1) q^{7} + (\beta_{3} - 2 \beta_1 + 2) q^{11} + (\beta_{2} - 1) q^{13} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{17}+ \cdots + ( - 3 \beta_{5} + 3 \beta_{4} + \cdots - \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{7} + 12 q^{11} - 3 q^{13} - 9 q^{25} + 12 q^{29} + 30 q^{31} - 24 q^{35} - 6 q^{37} + 12 q^{41} - 9 q^{43} - 18 q^{47} + 12 q^{49} - 6 q^{59} + 3 q^{61} + 3 q^{67} + 6 q^{71} + 9 q^{73} + 12 q^{77} - 27 q^{79} + 24 q^{83} + 24 q^{85} + 18 q^{89} - 3 q^{91} - 48 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{18}^{2} + 2\zeta_{18} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{18}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{18}^{5} + 2\zeta_{18}^{4} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{18}^{2} + 2\zeta_{18} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{18}^{5} + 2\zeta_{18}^{4} \) Copy content Toggle raw display

Display \(\zeta_{18}^j\) in terms of \(\beta_i\)

\(\zeta_{18}\)\(=\) \( ( \beta_{4} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{18}^{2}\)\(=\) \( ( -\beta_{4} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{18}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{18}^{4}\)\(=\) \( ( \beta_{5} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{18}^{5}\)\(=\) \( ( -\beta_{5} + \beta_{3} ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{18}^j\)

Character values

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

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{2}\) \(1\)

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
−0.766044 − 0.642788i
0.939693 − 0.342020i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0.939693 + 0.342020i
−0.173648 − 0.984808i
0 0 0 −1.87939 + 3.25519i 0 4.75877 0 0 0
505.2 0 0 0 0.347296 − 0.601535i 0 0.305407 0 0 0
505.3 0 0 0 1.53209 − 2.65366i 0 −2.06418 0 0 0
577.1 0 0 0 −1.87939 − 3.25519i 0 4.75877 0 0 0
577.2 0 0 0 0.347296 + 0.601535i 0 0.305407 0 0 0
577.3 0 0 0 1.53209 + 2.65366i 0 −2.06418 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 505.3
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.j 6
3.b odd 2 1 456.2.q.f ✓ 6
4.b odd 2 1 2736.2.s.x 6
12.b even 2 1 912.2.q.k 6
19.c even 3 1 inner 1368.2.s.j 6
57.f even 6 1 8664.2.a.z 3
57.h odd 6 1 456.2.q.f ✓ 6
57.h odd 6 1 8664.2.a.x 3
76.g odd 6 1 2736.2.s.x 6
228.m even 6 1 912.2.q.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.q.f ✓ 6 3.b odd 2 1
456.2.q.f ✓ 6 57.h odd 6 1
912.2.q.k 6 12.b even 2 1
912.2.q.k 6 228.m even 6 1
1368.2.s.j 6 1.a even 1 1 trivial
1368.2.s.j 6 19.c even 3 1 inner
2736.2.s.x 6 4.b odd 2 1
2736.2.s.x 6 76.g odd 6 1
8664.2.a.x 3 57.h odd 6 1
8664.2.a.z 3 57.f even 6 1

Hecke kernels

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

\( T_{5}^{6} + 12T_{5}^{4} - 16T_{5}^{3} + 144T_{5}^{2} - 96T_{5} + 64 \) Copy content Toggle raw display
\( T_{7}^{3} - 3T_{7}^{2} - 9T_{7} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 12 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{3} - 3 T^{2} - 9 T + 3)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 6 T^{2} + \cdots + 136)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 48 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( T^{6} + 9 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{6} - 12 T^{5} + \cdots + 36864 \) Copy content Toggle raw display
$31$ \( (T^{3} - 15 T^{2} + \cdots + 127)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} + \cdots - 111)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 12 T^{5} + \cdots + 36864 \) Copy content Toggle raw display
$43$ \( T^{6} + 9 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 36)^{3} \) Copy content Toggle raw display
$53$ \( T^{6} + 84 T^{4} + \cdots + 87616 \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + \cdots + 179776 \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$67$ \( T^{6} - 3 T^{5} + \cdots + 1369 \) Copy content Toggle raw display
$71$ \( T^{6} - 6 T^{5} + \cdots + 732736 \) Copy content Toggle raw display
$73$ \( T^{6} - 9 T^{5} + \cdots + 32761 \) Copy content Toggle raw display
$79$ \( T^{6} + 27 T^{5} + \cdots + 104329 \) Copy content Toggle raw display
$83$ \( (T^{3} - 12 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 18 T^{5} + \cdots + 5184 \) Copy content Toggle raw display
$97$ \( T^{6} + 6 T^{5} + \cdots + 732736 \) Copy content Toggle raw display
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