Properties

Label 1638.2.a.w
Level $1638$
Weight $2$
Character orbit 1638.a
Self dual yes
Analytic conductor $13.079$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1638,2,Mod(1,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.1");
 
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.a (trivial)

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: yes
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta q^{5} + q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + \beta q^{5} + q^{7} - q^{8} - \beta q^{10} + ( - \beta - 2) q^{11} + q^{13} - q^{14} + q^{16} - \beta q^{17} + (\beta + 2) q^{19} + \beta q^{20} + (\beta + 2) q^{22} + (\beta - 2) q^{23} + (\beta + 5) q^{25} - q^{26} + q^{28} + \beta q^{29} - q^{32} + \beta q^{34} + \beta q^{35} + ( - \beta + 8) q^{37} + ( - \beta - 2) q^{38} - \beta q^{40} + (2 \beta + 2) q^{41} + (3 \beta - 2) q^{43} + ( - \beta - 2) q^{44} + ( - \beta + 2) q^{46} + 8 q^{47} + q^{49} + ( - \beta - 5) q^{50} + q^{52} + 2 q^{53} + ( - 3 \beta - 10) q^{55} - q^{56} - \beta q^{58} + ( - 4 \beta + 4) q^{59} + (\beta + 4) q^{61} + q^{64} + \beta q^{65} - 2 \beta q^{67} - \beta q^{68} - \beta q^{70} + 8 q^{71} + ( - \beta - 4) q^{73} + (\beta - 8) q^{74} + (\beta + 2) q^{76} + ( - \beta - 2) q^{77} + ( - 2 \beta + 4) q^{79} + \beta q^{80} + ( - 2 \beta - 2) q^{82} + ( - 2 \beta + 8) q^{83} + ( - \beta - 10) q^{85} + ( - 3 \beta + 2) q^{86} + (\beta + 2) q^{88} + ( - 4 \beta - 2) q^{89} + q^{91} + (\beta - 2) q^{92} - 8 q^{94} + (3 \beta + 10) q^{95} + (4 \beta + 2) q^{97} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} - 2 q^{8} - q^{10} - 5 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - q^{17} + 5 q^{19} + q^{20} + 5 q^{22} - 3 q^{23} + 11 q^{25} - 2 q^{26} + 2 q^{28} + q^{29} - 2 q^{32} + q^{34} + q^{35} + 15 q^{37} - 5 q^{38} - q^{40} + 6 q^{41} - q^{43} - 5 q^{44} + 3 q^{46} + 16 q^{47} + 2 q^{49} - 11 q^{50} + 2 q^{52} + 4 q^{53} - 23 q^{55} - 2 q^{56} - q^{58} + 4 q^{59} + 9 q^{61} + 2 q^{64} + q^{65} - 2 q^{67} - q^{68} - q^{70} + 16 q^{71} - 9 q^{73} - 15 q^{74} + 5 q^{76} - 5 q^{77} + 6 q^{79} + q^{80} - 6 q^{82} + 14 q^{83} - 21 q^{85} + q^{86} + 5 q^{88} - 8 q^{89} + 2 q^{91} - 3 q^{92} - 16 q^{94} + 23 q^{95} + 8 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.70156
3.70156
−1.00000 0 1.00000 −2.70156 0 1.00000 −1.00000 0 2.70156
1.2 −1.00000 0 1.00000 3.70156 0 1.00000 −1.00000 0 −3.70156
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)
\(13\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.a.w 2
3.b odd 2 1 546.2.a.i 2
12.b even 2 1 4368.2.a.bg 2
21.c even 2 1 3822.2.a.bt 2
39.d odd 2 1 7098.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.i 2 3.b odd 2 1
1638.2.a.w 2 1.a even 1 1 trivial
3822.2.a.bt 2 21.c even 2 1
4368.2.a.bg 2 12.b even 2 1
7098.2.a.bh 2 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1638))\):

\( T_{5}^{2} - T_{5} - 10 \) Copy content Toggle raw display
\( T_{11}^{2} + 5T_{11} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} + T_{17} - 10 \) Copy content Toggle raw display
\( T_{19}^{2} - 5T_{19} - 4 \) 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
$5$ \( T^{2} - T - 10 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 5T - 4 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + T - 10 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 8 \) Copy content Toggle raw display
$29$ \( T^{2} - T - 10 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 15T + 46 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 32 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 92 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 160 \) Copy content Toggle raw display
$61$ \( T^{2} - 9T + 10 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 40 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 9T + 10 \) Copy content Toggle raw display
$79$ \( T^{2} - 6T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 8T - 148 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 148 \) Copy content Toggle raw display
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