Properties

Label 1368.2.a.n
Level $1368$
Weight $2$
Character orbit 1368.a
Self dual yes
Analytic conductor $10.924$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1368,2,Mod(1,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1368.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.a (trivial)

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: yes
Analytic conductor: \(10.9235349965\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.961.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 10x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + ( - \beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + ( - \beta_{2} + 1) q^{7} + ( - \beta_1 + 2) q^{11} + ( - \beta_{2} + \beta_1 + 1) q^{13} + ( - \beta_{2} - 1) q^{17} - q^{19} + (3 \beta_{2} - \beta_1 + 3) q^{23} + (2 \beta_{2} - \beta_1 + 3) q^{25} + (\beta_{2} + \beta_1 + 3) q^{29} + (2 \beta_{2} - \beta_1 + 4) q^{35} - 2 q^{37} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{41} + ( - 2 \beta_{2} + 3 \beta_1 + 4) q^{43} + ( - 2 \beta_{2} - \beta_1) q^{47} + ( - 3 \beta_{2} + 2 \beta_1) q^{49} + (\beta_{2} + 3 \beta_1 - 1) q^{53} + (2 \beta_{2} - 3 \beta_1 + 8) q^{55} + ( - \beta_{2} + \beta_1 + 7) q^{59} + (2 \beta_{2} - \beta_1 + 2) q^{61} - 4 q^{65} + (\beta_{2} + \beta_1 + 5) q^{67} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{71} + (3 \beta_{2} - 2 \beta_1 + 3) q^{73} + ( - \beta_1 + 6) q^{77} + (2 \beta_{2} - 2 \beta_1 + 10) q^{79} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{83} + (2 \beta_{2} + \beta_1 + 4) q^{85} + (2 \beta_{2} - 4 \beta_1 - 4) q^{89} + ( - 5 \beta_{2} + 3 \beta_1 + 3) q^{91} + \beta_1 q^{95} + ( - 2 \beta_1 - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{5} + 4 q^{7} + 5 q^{11} + 5 q^{13} - 2 q^{17} - 3 q^{19} + 5 q^{23} + 6 q^{25} + 9 q^{29} + 9 q^{35} - 6 q^{37} - 8 q^{41} + 17 q^{43} + q^{47} + 5 q^{49} - q^{53} + 19 q^{55} + 23 q^{59} + 3 q^{61} - 12 q^{65} + 15 q^{67} + 12 q^{71} + 4 q^{73} + 17 q^{77} + 26 q^{79} + 6 q^{83} + 11 q^{85} - 18 q^{89} + 17 q^{91} + q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 10x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + \nu - 8 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} - \beta _1 + 8 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.29707
0.786802
−3.08387
0 0 0 −3.29707 0 −2.08387 0 0 0
1.2 0 0 0 −0.786802 0 4.29707 0 0 0
1.3 0 0 0 3.08387 0 1.78680 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.a.n 3
3.b odd 2 1 152.2.a.c 3
4.b odd 2 1 2736.2.a.bd 3
12.b even 2 1 304.2.a.g 3
15.d odd 2 1 3800.2.a.r 3
15.e even 4 2 3800.2.d.j 6
21.c even 2 1 7448.2.a.bf 3
24.f even 2 1 1216.2.a.v 3
24.h odd 2 1 1216.2.a.u 3
57.d even 2 1 2888.2.a.o 3
60.h even 2 1 7600.2.a.bv 3
228.b odd 2 1 5776.2.a.bp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.c 3 3.b odd 2 1
304.2.a.g 3 12.b even 2 1
1216.2.a.u 3 24.h odd 2 1
1216.2.a.v 3 24.f even 2 1
1368.2.a.n 3 1.a even 1 1 trivial
2736.2.a.bd 3 4.b odd 2 1
2888.2.a.o 3 57.d even 2 1
3800.2.a.r 3 15.d odd 2 1
3800.2.d.j 6 15.e even 4 2
5776.2.a.bp 3 228.b odd 2 1
7448.2.a.bf 3 21.c even 2 1
7600.2.a.bv 3 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1368))\):

\( T_{5}^{3} + T_{5}^{2} - 10T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{3} - 4T_{7}^{2} - 5T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{3} - 5T_{11}^{2} - 2T_{11} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + T^{2} - 10T - 8 \) Copy content Toggle raw display
$7$ \( T^{3} - 4 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{3} - 5 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$13$ \( T^{3} - 5 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} + \cdots - 2 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 5 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{3} - 9 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$31$ \( T^{3} \) Copy content Toggle raw display
$37$ \( (T + 2)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + 8 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$43$ \( T^{3} - 17 T^{2} + \cdots + 368 \) Copy content Toggle raw display
$47$ \( T^{3} - T^{2} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{3} + T^{2} + \cdots - 256 \) Copy content Toggle raw display
$59$ \( T^{3} - 23 T^{2} + \cdots - 376 \) Copy content Toggle raw display
$61$ \( T^{3} - 3 T^{2} + \cdots + 92 \) Copy content Toggle raw display
$67$ \( T^{3} - 15 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} + \cdots + 928 \) Copy content Toggle raw display
$73$ \( T^{3} - 4 T^{2} + \cdots + 326 \) Copy content Toggle raw display
$79$ \( T^{3} - 26 T^{2} + \cdots - 256 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} + \cdots + 736 \) Copy content Toggle raw display
$89$ \( T^{3} + 18 T^{2} + \cdots - 1024 \) Copy content Toggle raw display
$97$ \( T^{3} + 8 T^{2} + \cdots - 128 \) Copy content Toggle raw display
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