Properties

Label 1386.2.k.e
Level $1386$
Weight $2$
Character orbit 1386.k
Analytic conductor $11.067$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,2,Mod(793,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.793");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.k (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: \(11.0672657201\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + (3 \zeta_{6} - 2) q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + (3 \zeta_{6} - 2) q^{7} + q^{8} + (\zeta_{6} - 1) q^{11} - q^{13} + ( - \zeta_{6} + 3) q^{14} - \zeta_{6} q^{16} + (6 \zeta_{6} - 6) q^{17} - 2 \zeta_{6} q^{19} + q^{22} - 6 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + \zeta_{6} q^{26} + ( - 2 \zeta_{6} - 1) q^{28} - 9 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + (\zeta_{6} - 1) q^{32} + 6 q^{34} - 2 \zeta_{6} q^{37} + (2 \zeta_{6} - 2) q^{38} + 6 q^{41} - 4 q^{43} - \zeta_{6} q^{44} + (6 \zeta_{6} - 6) q^{46} - 6 \zeta_{6} q^{47} + ( - 3 \zeta_{6} - 5) q^{49} - 5 q^{50} + ( - \zeta_{6} + 1) q^{52} + (3 \zeta_{6} - 2) q^{56} + 9 \zeta_{6} q^{58} + (3 \zeta_{6} - 3) q^{59} - 11 \zeta_{6} q^{61} - 4 q^{62} + q^{64} + (11 \zeta_{6} - 11) q^{67} - 6 \zeta_{6} q^{68} + (2 \zeta_{6} - 2) q^{73} + (2 \zeta_{6} - 2) q^{74} + 2 q^{76} + ( - 2 \zeta_{6} - 1) q^{77} - 5 \zeta_{6} q^{79} - 6 \zeta_{6} q^{82} + 6 q^{83} + 4 \zeta_{6} q^{86} + (\zeta_{6} - 1) q^{88} - 18 \zeta_{6} q^{89} + ( - 3 \zeta_{6} + 2) q^{91} + 6 q^{92} + (6 \zeta_{6} - 6) q^{94} - 13 q^{97} + (8 \zeta_{6} - 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8} - q^{11} - 2 q^{13} + 5 q^{14} - q^{16} - 6 q^{17} - 2 q^{19} + 2 q^{22} - 6 q^{23} + 5 q^{25} + q^{26} - 4 q^{28} - 18 q^{29} + 4 q^{31} - q^{32} + 12 q^{34} - 2 q^{37} - 2 q^{38} + 12 q^{41} - 8 q^{43} - q^{44} - 6 q^{46} - 6 q^{47} - 13 q^{49} - 10 q^{50} + q^{52} - q^{56} + 9 q^{58} - 3 q^{59} - 11 q^{61} - 8 q^{62} + 2 q^{64} - 11 q^{67} - 6 q^{68} - 2 q^{73} - 2 q^{74} + 4 q^{76} - 4 q^{77} - 5 q^{79} - 6 q^{82} + 12 q^{83} + 4 q^{86} - q^{88} - 18 q^{89} + q^{91} + 12 q^{92} - 6 q^{94} - 26 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
793.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −0.500000 2.59808i 1.00000 0 0
991.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 + 2.59808i 1.00000 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.k.e 2
3.b odd 2 1 154.2.e.c 2
7.c even 3 1 inner 1386.2.k.e 2
7.c even 3 1 9702.2.a.br 1
7.d odd 6 1 9702.2.a.bs 1
12.b even 2 1 1232.2.q.d 2
21.c even 2 1 1078.2.e.k 2
21.g even 6 1 1078.2.a.c 1
21.g even 6 1 1078.2.e.k 2
21.h odd 6 1 154.2.e.c 2
21.h odd 6 1 1078.2.a.e 1
84.j odd 6 1 8624.2.a.u 1
84.n even 6 1 1232.2.q.d 2
84.n even 6 1 8624.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.c 2 3.b odd 2 1
154.2.e.c 2 21.h odd 6 1
1078.2.a.c 1 21.g even 6 1
1078.2.a.e 1 21.h odd 6 1
1078.2.e.k 2 21.c even 2 1
1078.2.e.k 2 21.g even 6 1
1232.2.q.d 2 12.b even 2 1
1232.2.q.d 2 84.n even 6 1
1386.2.k.e 2 1.a even 1 1 trivial
1386.2.k.e 2 7.c even 3 1 inner
8624.2.a.k 1 84.n even 6 1
8624.2.a.u 1 84.j odd 6 1
9702.2.a.br 1 7.c even 3 1
9702.2.a.bs 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} + 36 \) Copy content Toggle raw display
\( T_{23}^{2} + 6T_{23} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$67$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$97$ \( (T + 13)^{2} \) Copy content Toggle raw display
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