Properties

Label 1638.2.j.e
Level $1638$
Weight $2$
Character orbit 1638.j
Analytic conductor $13.079$
Analytic rank $0$
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] = [1638,2,Mod(235,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.235");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.j (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: \(13.0794958511\)
Analytic rank: \(0\)
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 546)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 1) q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 1) q^{7} + q^{8} + ( - \zeta_{6} + 1) q^{10} + (3 \zeta_{6} - 3) q^{11} - q^{13} + (2 \zeta_{6} - 3) q^{14} - \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} - q^{20} + 3 q^{22} + 6 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + \zeta_{6} q^{26} + (\zeta_{6} + 2) q^{28} - 9 q^{29} + (5 \zeta_{6} - 5) q^{31} + (\zeta_{6} - 1) q^{32} + 2 q^{34} + ( - 2 \zeta_{6} + 3) q^{35} + 8 \zeta_{6} q^{37} + \zeta_{6} q^{40} + 4 q^{41} - 3 \zeta_{6} q^{44} + ( - 6 \zeta_{6} + 6) q^{46} + (3 \zeta_{6} - 8) q^{49} - 4 q^{50} + ( - \zeta_{6} + 1) q^{52} + ( - \zeta_{6} + 1) q^{53} - 3 q^{55} + ( - 3 \zeta_{6} + 1) q^{56} + 9 \zeta_{6} q^{58} + (7 \zeta_{6} - 7) q^{59} + 4 \zeta_{6} q^{61} + 5 q^{62} + q^{64} - \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} - 2 \zeta_{6} q^{68} + ( - \zeta_{6} - 2) q^{70} - 6 q^{71} + (6 \zeta_{6} - 6) q^{73} + ( - 8 \zeta_{6} + 8) q^{74} + (3 \zeta_{6} + 6) q^{77} + 13 \zeta_{6} q^{79} + ( - \zeta_{6} + 1) q^{80} - 4 \zeta_{6} q^{82} + 3 q^{83} - 2 q^{85} + (3 \zeta_{6} - 3) q^{88} - 8 \zeta_{6} q^{89} + (3 \zeta_{6} - 1) q^{91} - 6 q^{92} - 15 q^{97} + (5 \zeta_{6} + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + q^{5} - q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + q^{5} - q^{7} + 2 q^{8} + q^{10} - 3 q^{11} - 2 q^{13} - 4 q^{14} - q^{16} - 2 q^{17} - 2 q^{20} + 6 q^{22} + 6 q^{23} + 4 q^{25} + q^{26} + 5 q^{28} - 18 q^{29} - 5 q^{31} - q^{32} + 4 q^{34} + 4 q^{35} + 8 q^{37} + q^{40} + 8 q^{41} - 3 q^{44} + 6 q^{46} - 13 q^{49} - 8 q^{50} + q^{52} + q^{53} - 6 q^{55} - q^{56} + 9 q^{58} - 7 q^{59} + 4 q^{61} + 10 q^{62} + 2 q^{64} - q^{65} + 4 q^{67} - 2 q^{68} - 5 q^{70} - 12 q^{71} - 6 q^{73} + 8 q^{74} + 15 q^{77} + 13 q^{79} + q^{80} - 4 q^{82} + 6 q^{83} - 4 q^{85} - 3 q^{88} - 8 q^{89} + q^{91} - 12 q^{92} - 30 q^{97} + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.500000 2.59808i 1.00000 0 0.500000 0.866025i
1171.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −0.500000 + 2.59808i 1.00000 0 0.500000 + 0.866025i
\(n\): 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 1638.2.j.e 2
3.b odd 2 1 546.2.i.f 2
7.c even 3 1 inner 1638.2.j.e 2
21.g even 6 1 3822.2.a.l 1
21.h odd 6 1 546.2.i.f 2
21.h odd 6 1 3822.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.f 2 3.b odd 2 1
546.2.i.f 2 21.h odd 6 1
1638.2.j.e 2 1.a even 1 1 trivial
1638.2.j.e 2 7.c even 3 1 inner
3822.2.a.f 1 21.h odd 6 1
3822.2.a.l 1 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} + 4 \) 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
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$41$ \( (T - 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$83$ \( (T - 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$97$ \( (T + 15)^{2} \) Copy content Toggle raw display
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