Properties

Label 1368.2.s.h
Level $1368$
Weight $2$
Character orbit 1368.s
Analytic conductor $10.924$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 - 1) q^{5} + (\beta_{3} + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 - 1) q^{5} + (\beta_{3} + 2) q^{7} + (\beta_{3} + 1) q^{11} + (2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{13} + ( - 4 \beta_1 - 4) q^{17} + (\beta_{3} + 2 \beta_{2} - 2) q^{19} + (\beta_{3} + \beta_{2} + \beta_1) q^{23} + (2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{25} + ( - 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1) q^{29} + ( - 3 \beta_{3} - 2) q^{31} + ( - \beta_{2} + 3 \beta_1 + 3) q^{35} + 7 q^{37} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + (\beta_{2} + 4 \beta_1 + 4) q^{43} + 2 \beta_1 q^{47} + (4 \beta_{3} + 2) q^{49} + (5 \beta_{3} + 5 \beta_{2} + \beta_1) q^{53} + (4 \beta_1 + 4) q^{55} + (3 \beta_{2} - 3 \beta_1 - 3) q^{59} + 11 \beta_1 q^{61} + ( - \beta_{3} + 9) q^{65} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{67} + 6 \beta_{2} q^{71} + ( - 2 \beta_{2} + \beta_1 + 1) q^{73} + (3 \beta_{3} + 7) q^{77} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{79} + 8 q^{83} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{85} + ( - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{89} + (5 \beta_{3} + 5 \beta_{2} - 12 \beta_1) q^{91} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 7) q^{95} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} + 8 q^{7} + 4 q^{11} + 2 q^{13} - 8 q^{17} - 8 q^{19} - 2 q^{23} - 2 q^{25} - 12 q^{29} - 8 q^{31} + 6 q^{35} + 28 q^{37} + 4 q^{41} + 8 q^{43} - 4 q^{47} + 8 q^{49} - 2 q^{53} + 8 q^{55} - 6 q^{59} - 22 q^{61} + 36 q^{65} + 4 q^{67} + 2 q^{73} + 28 q^{77} - 4 q^{79} + 32 q^{83} - 8 q^{85} - 6 q^{89} + 24 q^{91} + 34 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2 \) Copy content Toggle raw display

Character values

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

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

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
0.809017 1.40126i
−0.309017 + 0.535233i
0.809017 + 1.40126i
−0.309017 0.535233i
0 0 0 −1.61803 + 2.80252i 0 −0.236068 0 0 0
505.2 0 0 0 0.618034 1.07047i 0 4.23607 0 0 0
577.1 0 0 0 −1.61803 2.80252i 0 −0.236068 0 0 0
577.2 0 0 0 0.618034 + 1.07047i 0 4.23607 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.h 4
3.b odd 2 1 456.2.q.d 4
4.b odd 2 1 2736.2.s.s 4
12.b even 2 1 912.2.q.j 4
19.c even 3 1 inner 1368.2.s.h 4
57.f even 6 1 8664.2.a.q 2
57.h odd 6 1 456.2.q.d 4
57.h odd 6 1 8664.2.a.u 2
76.g odd 6 1 2736.2.s.s 4
228.m even 6 1 912.2.q.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.q.d 4 3.b odd 2 1
456.2.q.d 4 57.h odd 6 1
912.2.q.j 4 12.b even 2 1
912.2.q.j 4 228.m even 6 1
1368.2.s.h 4 1.a even 1 1 trivial
1368.2.s.h 4 19.c even 3 1 inner
2736.2.s.s 4 4.b odd 2 1
2736.2.s.s 4 76.g odd 6 1
8664.2.a.q 2 57.f even 6 1
8664.2.a.u 2 57.h odd 6 1

Hecke kernels

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

\( T_{5}^{4} + 2T_{5}^{3} + 8T_{5}^{2} - 8T_{5} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + 23 T^{2} + 38 T + 361 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$29$ \( T^{4} + 12 T^{3} + 128 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 41)^{2} \) Copy content Toggle raw display
$37$ \( (T - 7)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} + 32 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + 53 T^{2} - 88 T + 121 \) Copy content Toggle raw display
$47$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} + 128 T^{2} + \cdots + 15376 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + 72 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$61$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + 17 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$71$ \( T^{4} + 180 T^{2} + 32400 \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} + 23 T^{2} + 38 T + 361 \) Copy content Toggle raw display
$79$ \( T^{4} + 4 T^{3} + 57 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$83$ \( (T - 8)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + 72 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} + 92 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
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