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Properties

Label 1638.2.r.j
Level $1638$
Weight $2$
Character orbit 1638.r
Analytic conductor $13.079$
Analytic rank $0$
Dimension $2$
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] = [1638,2,Mod(757,1638)]
 
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, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.757");
 
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.r (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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 2 q^{5} - \zeta_{6} q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 2 q^{5} - \zeta_{6} q^{7} + q^{8} + (2 \zeta_{6} - 2) q^{10} + ( - 3 \zeta_{6} - 1) q^{13} + q^{14} + (\zeta_{6} - 1) q^{16} + \zeta_{6} q^{17} - 6 \zeta_{6} q^{19} - 2 \zeta_{6} q^{20} + ( - 3 \zeta_{6} + 3) q^{23} - q^{25} + ( - \zeta_{6} + 4) q^{26} + (\zeta_{6} - 1) q^{28} + ( - 6 \zeta_{6} + 6) q^{29} + 5 q^{31} - \zeta_{6} q^{32} - q^{34} - 2 \zeta_{6} q^{35} + ( - 2 \zeta_{6} + 2) q^{37} + 6 q^{38} + 2 q^{40} + (10 \zeta_{6} - 10) q^{41} - 3 \zeta_{6} q^{43} + 3 \zeta_{6} q^{46} - 6 q^{47} + (\zeta_{6} - 1) q^{49} + ( - \zeta_{6} + 1) q^{50} + (4 \zeta_{6} - 3) q^{52} - 7 q^{53} - \zeta_{6} q^{56} + 6 \zeta_{6} q^{58} - 7 \zeta_{6} q^{59} - 11 \zeta_{6} q^{61} + (5 \zeta_{6} - 5) q^{62} + q^{64} + ( - 6 \zeta_{6} - 2) q^{65} + ( - 13 \zeta_{6} + 13) q^{67} + ( - \zeta_{6} + 1) q^{68} + 2 q^{70} - 3 \zeta_{6} q^{71} + 12 q^{73} + 2 \zeta_{6} q^{74} + (6 \zeta_{6} - 6) q^{76} - 4 q^{79} + (2 \zeta_{6} - 2) q^{80} - 10 \zeta_{6} q^{82} + 15 q^{83} + 2 \zeta_{6} q^{85} + 3 q^{86} + (11 \zeta_{6} - 11) q^{89} + (4 \zeta_{6} - 3) q^{91} - 3 q^{92} + ( - 6 \zeta_{6} + 6) q^{94} - 12 \zeta_{6} q^{95} - 12 \zeta_{6} q^{97} - \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 4 q^{5} - q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + 4 q^{5} - q^{7} + 2 q^{8} - 2 q^{10} - 5 q^{13} + 2 q^{14} - q^{16} + q^{17} - 6 q^{19} - 2 q^{20} + 3 q^{23} - 2 q^{25} + 7 q^{26} - q^{28} + 6 q^{29} + 10 q^{31} - q^{32} - 2 q^{34} - 2 q^{35} + 2 q^{37} + 12 q^{38} + 4 q^{40} - 10 q^{41} - 3 q^{43} + 3 q^{46} - 12 q^{47} - q^{49} + q^{50} - 2 q^{52} - 14 q^{53} - q^{56} + 6 q^{58} - 7 q^{59} - 11 q^{61} - 5 q^{62} + 2 q^{64} - 10 q^{65} + 13 q^{67} + q^{68} + 4 q^{70} - 3 q^{71} + 24 q^{73} + 2 q^{74} - 6 q^{76} - 8 q^{79} - 2 q^{80} - 10 q^{82} + 30 q^{83} + 2 q^{85} + 6 q^{86} - 11 q^{89} - 2 q^{91} - 6 q^{92} + 6 q^{94} - 12 q^{95} - 12 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
0.500000 − 0.866025i
0.500000 + 0.866025i
−0.500000 − 0.866025i 0 −0.500000 + 0.866025i 2.00000 0 −0.500000 + 0.866025i 1.00000 0 −1.00000 − 1.73205i
1387.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 2.00000 0 −0.500000 − 0.866025i 1.00000 0 −1.00000 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.r.j 2
3.b odd 2 1 546.2.l.f ✓ 2
13.c even 3 1 inner 1638.2.r.j 2
39.h odd 6 1 7098.2.a.u 1
39.i odd 6 1 546.2.l.f ✓ 2
39.i odd 6 1 7098.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.f ✓ 2 3.b odd 2 1
546.2.l.f ✓ 2 39.i odd 6 1
1638.2.r.j 2 1.a even 1 1 trivial
1638.2.r.j 2 13.c even 3 1 inner
7098.2.a.c 1 39.i odd 6 1
7098.2.a.u 1 39.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 \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17}^{2} - T_{17} + 1 \) 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
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$31$ \( (T - 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$43$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( (T + 7)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$67$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$71$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$73$ \( (T - 12)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T - 15)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
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