Properties

Label 1638.2.a.v
Level $1638$
Weight $2$
Character orbit 1638.a
Self dual yes
Analytic conductor $13.079$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). 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 \( q - q^{2} + q^{4} - \beta q^{5} - q^{7} - q^{8} + \beta q^{10} + \beta q^{11} - q^{13} + q^{14} + q^{16} + (\beta - 2) q^{17} + (\beta + 4) q^{19} - \beta q^{20} - \beta q^{22} + ( - \beta - 6) q^{23} + (\beta + 3) q^{25} + q^{26} - q^{28} + (\beta - 4) q^{29} + 2 \beta q^{31} - q^{32} + ( - \beta + 2) q^{34} + \beta q^{35} + (\beta - 2) q^{37} + ( - \beta - 4) q^{38} + \beta q^{40} + (2 \beta - 4) q^{41} + ( - 3 \beta + 4) q^{43} + \beta q^{44} + (\beta + 6) q^{46} + ( - 2 \beta - 6) q^{47} + q^{49} + ( - \beta - 3) q^{50} - q^{52} + (2 \beta - 4) q^{53} + ( - \beta - 8) q^{55} + q^{56} + ( - \beta + 4) q^{58} - 2 q^{59} + ( - 3 \beta - 2) q^{61} - 2 \beta q^{62} + q^{64} + \beta q^{65} - 4 \beta q^{67} + (\beta - 2) q^{68} - \beta q^{70} - 2 \beta q^{71} + (\beta - 6) q^{73} + ( - \beta + 2) q^{74} + (\beta + 4) q^{76} - \beta q^{77} - 2 \beta q^{79} - \beta q^{80} + ( - 2 \beta + 4) q^{82} + 6 q^{83} + (\beta - 8) q^{85} + (3 \beta - 4) q^{86} - \beta q^{88} + (2 \beta + 8) q^{89} + q^{91} + ( - \beta - 6) q^{92} + (2 \beta + 6) q^{94} + ( - 5 \beta - 8) q^{95} + ( - 4 \beta - 2) q^{97} - q^{98} +O(q^{100}) \) 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 \( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8} + q^{10} + q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} + 9 q^{19} - q^{20} - q^{22} - 13 q^{23} + 7 q^{25} + 2 q^{26} - 2 q^{28} - 7 q^{29} + 2 q^{31} - 2 q^{32} + 3 q^{34} + q^{35} - 3 q^{37} - 9 q^{38} + q^{40} - 6 q^{41} + 5 q^{43} + q^{44} + 13 q^{46} - 14 q^{47} + 2 q^{49} - 7 q^{50} - 2 q^{52} - 6 q^{53} - 17 q^{55} + 2 q^{56} + 7 q^{58} - 4 q^{59} - 7 q^{61} - 2 q^{62} + 2 q^{64} + q^{65} - 4 q^{67} - 3 q^{68} - q^{70} - 2 q^{71} - 11 q^{73} + 3 q^{74} + 9 q^{76} - q^{77} - 2 q^{79} - q^{80} + 6 q^{82} + 12 q^{83} - 15 q^{85} - 5 q^{86} - q^{88} + 18 q^{89} + 2 q^{91} - 13 q^{92} + 14 q^{94} - 21 q^{95} - 8 q^{97} - 2 q^{98}+O(q^{100}) \) 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
3.37228
−2.37228
−1.00000 0 1.00000 −3.37228 0 −1.00000 −1.00000 0 3.37228
1.2 −1.00000 0 1.00000 2.37228 0 −1.00000 −1.00000 0 −2.37228
\(n\): 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.v 2
3.b odd 2 1 1638.2.a.x yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.a.v 2 1.a even 1 1 trivial
1638.2.a.x yes 2 3.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + T_{5} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 8 \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} - 6 \) Copy content Toggle raw display
\( T_{19}^{2} - 9T_{19} + 12 \) 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
$5$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$19$ \( T^{2} - 9T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 13T + 34 \) Copy content Toggle raw display
$29$ \( T^{2} + 7T + 4 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$37$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$43$ \( T^{2} - 5T - 68 \) Copy content Toggle raw display
$47$ \( T^{2} + 14T + 16 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$59$ \( (T + 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 7T - 62 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 128 \) Copy content Toggle raw display
$71$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$73$ \( T^{2} + 11T + 22 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 116 \) Copy content Toggle raw display
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