Properties

Label 1638.2.m.d
Level $1638$
Weight $2$
Character orbit 1638.m
Analytic conductor $13.079$
Analytic rank $1$
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(289,1638)]
 
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, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.289");
 
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.m (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: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - 2 \zeta_{6} q^{5} + ( - \zeta_{6} - 2) q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - 2 \zeta_{6} q^{5} + ( - \zeta_{6} - 2) q^{7} + q^{8} - 2 \zeta_{6} q^{10} + 2 \zeta_{6} q^{11} + ( - \zeta_{6} - 3) q^{13} + ( - \zeta_{6} - 2) q^{14} + q^{16} - 4 q^{17} + (5 \zeta_{6} - 5) q^{19} - 2 \zeta_{6} q^{20} + 2 \zeta_{6} q^{22} - 2 q^{23} + ( - \zeta_{6} + 1) q^{25} + ( - \zeta_{6} - 3) q^{26} + ( - \zeta_{6} - 2) q^{28} + (8 \zeta_{6} - 8) q^{29} + q^{32} - 4 q^{34} + (6 \zeta_{6} - 2) q^{35} + 7 q^{37} + (5 \zeta_{6} - 5) q^{38} - 2 \zeta_{6} q^{40} + (10 \zeta_{6} - 10) q^{41} - 7 \zeta_{6} q^{43} + 2 \zeta_{6} q^{44} - 2 q^{46} - 12 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} + ( - \zeta_{6} + 1) q^{50} + ( - \zeta_{6} - 3) q^{52} + (6 \zeta_{6} - 6) q^{53} + ( - 4 \zeta_{6} + 4) q^{55} + ( - \zeta_{6} - 2) q^{56} + (8 \zeta_{6} - 8) q^{58} + ( - \zeta_{6} + 1) q^{61} + q^{64} + (8 \zeta_{6} - 2) q^{65} + 8 \zeta_{6} q^{67} - 4 q^{68} + (6 \zeta_{6} - 2) q^{70} - 8 \zeta_{6} q^{71} + (9 \zeta_{6} - 9) q^{73} + 7 q^{74} + (5 \zeta_{6} - 5) q^{76} + ( - 6 \zeta_{6} + 2) q^{77} - 8 \zeta_{6} q^{79} - 2 \zeta_{6} q^{80} + (10 \zeta_{6} - 10) q^{82} - 14 q^{83} + 8 \zeta_{6} q^{85} - 7 \zeta_{6} q^{86} + 2 \zeta_{6} q^{88} + 6 q^{89} + (6 \zeta_{6} + 5) q^{91} - 2 q^{92} - 12 \zeta_{6} q^{94} + 10 q^{95} - 17 \zeta_{6} q^{97} + (5 \zeta_{6} + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 5 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 5 q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{11} - 7 q^{13} - 5 q^{14} + 2 q^{16} - 8 q^{17} - 5 q^{19} - 2 q^{20} + 2 q^{22} - 4 q^{23} + q^{25} - 7 q^{26} - 5 q^{28} - 8 q^{29} + 2 q^{32} - 8 q^{34} + 2 q^{35} + 14 q^{37} - 5 q^{38} - 2 q^{40} - 10 q^{41} - 7 q^{43} + 2 q^{44} - 4 q^{46} - 12 q^{47} + 11 q^{49} + q^{50} - 7 q^{52} - 6 q^{53} + 4 q^{55} - 5 q^{56} - 8 q^{58} + q^{61} + 2 q^{64} + 4 q^{65} + 8 q^{67} - 8 q^{68} + 2 q^{70} - 8 q^{71} - 9 q^{73} + 14 q^{74} - 5 q^{76} - 2 q^{77} - 8 q^{79} - 2 q^{80} - 10 q^{82} - 28 q^{83} + 8 q^{85} - 7 q^{86} + 2 q^{88} + 12 q^{89} + 16 q^{91} - 4 q^{92} - 12 q^{94} + 20 q^{95} - 17 q^{97} + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 0 1.00000 −1.00000 + 1.73205i 0 −2.50000 + 0.866025i 1.00000 0 −1.00000 + 1.73205i
1621.1 1.00000 0 1.00000 −1.00000 1.73205i 0 −2.50000 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
91.h 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.m.d yes 2
3.b odd 2 1 1638.2.m.b 2
7.c even 3 1 1638.2.p.b yes 2
13.c even 3 1 1638.2.p.b yes 2
21.h odd 6 1 1638.2.p.e yes 2
39.i odd 6 1 1638.2.p.e yes 2
91.h even 3 1 inner 1638.2.m.d yes 2
273.s odd 6 1 1638.2.m.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.m.b 2 3.b odd 2 1
1638.2.m.b 2 273.s odd 6 1
1638.2.m.d yes 2 1.a even 1 1 trivial
1638.2.m.d yes 2 91.h even 3 1 inner
1638.2.p.b yes 2 7.c even 3 1
1638.2.p.b yes 2 13.c even 3 1
1638.2.p.e yes 2 21.h odd 6 1
1638.2.p.e yes 2 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 13 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$43$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$73$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( (T + 14)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
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