Properties

Label 1840.2.a.q
Level $1840$
Weight $2$
Character orbit 1840.a
Self dual yes
Analytic conductor $14.692$
Analytic rank $1$
Dimension $3$
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] = [1840,2,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(14.6924739719\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{3} + q^{5} + (\beta_{2} - 2) q^{7} + (\beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{3} + q^{5} + (\beta_{2} - 2) q^{7} + (\beta_1 + 1) q^{9} + ( - \beta_1 - 1) q^{11} + (\beta_1 + 2) q^{13} + ( - \beta_{2} - 1) q^{15} + (2 \beta_{2} - 3 \beta_1 - 1) q^{17} + (\beta_{2} + 2 \beta_1 - 2) q^{19} + (3 \beta_{2} - \beta_1 - 1) q^{21} + q^{23} + q^{25} + (2 \beta_{2} - 2 \beta_1 + 1) q^{27} + (\beta_{2} - 5 \beta_1) q^{29} + ( - 2 \beta_{2} + \beta_1 - 4) q^{31} + (\beta_{2} + 2 \beta_1 + 2) q^{33} + (\beta_{2} - 2) q^{35} + ( - 2 \beta_{2} + 6 \beta_1) q^{37} + ( - 2 \beta_{2} - 2 \beta_1 - 3) q^{39} + (\beta_{2} + 4 \beta_1 - 3) q^{41} + (4 \beta_1 - 4) q^{43} + (\beta_1 + 1) q^{45} + (\beta_{2} - \beta_1) q^{47} + ( - 6 \beta_{2} + \beta_1) q^{49} + (3 \beta_{2} + 4 \beta_1 - 2) q^{51} + ( - 4 \beta_{2} + 2) q^{53} + ( - \beta_1 - 1) q^{55} + (3 \beta_{2} - 5 \beta_1 - 3) q^{57} + (4 \beta_{2} - 6 \beta_1 - 2) q^{59} + (4 \beta_{2} - 3 \beta_1 - 3) q^{61} + (\beta_{2} - \beta_1 - 1) q^{63} + (\beta_1 + 2) q^{65} + (6 \beta_{2} - 2 \beta_1 + 4) q^{67} + ( - \beta_{2} - 1) q^{69} + ( - 5 \beta_{2} - 5) q^{71} + ( - \beta_{2} - 3 \beta_1 + 2) q^{73} + ( - \beta_{2} - 1) q^{75} + ( - \beta_{2} + \beta_1 + 1) q^{77} + (2 \beta_{2} + 2 \beta_1 - 4) q^{79} + (\beta_{2} - \beta_1 - 8) q^{81} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{83} + (2 \beta_{2} - 3 \beta_1 - 1) q^{85} + (\beta_{2} + 9 \beta_1 + 2) q^{87} + ( - 4 \beta_{2} + 4 \beta_1 - 8) q^{89} + (2 \beta_{2} - \beta_1 - 3) q^{91} + (2 \beta_{2} + 9) q^{93} + (\beta_{2} + 2 \beta_1 - 2) q^{95} + ( - 2 \beta_{2} - \beta_1 + 1) q^{97} + ( - \beta_{2} - 2 \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9} - 3 q^{11} + 6 q^{13} - 2 q^{15} - 5 q^{17} - 7 q^{19} - 6 q^{21} + 3 q^{23} + 3 q^{25} + q^{27} - q^{29} - 10 q^{31} + 5 q^{33} - 7 q^{35} + 2 q^{37} - 7 q^{39} - 10 q^{41} - 12 q^{43} + 3 q^{45} - q^{47} + 6 q^{49} - 9 q^{51} + 10 q^{53} - 3 q^{55} - 12 q^{57} - 10 q^{59} - 13 q^{61} - 4 q^{63} + 6 q^{65} + 6 q^{67} - 2 q^{69} - 10 q^{71} + 7 q^{73} - 2 q^{75} + 4 q^{77} - 14 q^{79} - 25 q^{81} - 16 q^{83} - 5 q^{85} + 5 q^{87} - 20 q^{89} - 11 q^{91} + 25 q^{93} - 7 q^{95} + 5 q^{97} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
2.11491
−1.86081
−0.254102
0 −2.47283 0 1.00000 0 −0.527166 0 3.11491 0
1.2 0 −1.46260 0 1.00000 0 −1.53740 0 −0.860806 0
1.3 0 1.93543 0 1.00000 0 −4.93543 0 0.745898 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)
\(23\) \( -1 \)

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 1840.2.a.q 3
4.b odd 2 1 920.2.a.i 3
5.b even 2 1 9200.2.a.ci 3
8.b even 2 1 7360.2.a.cf 3
8.d odd 2 1 7360.2.a.bw 3
12.b even 2 1 8280.2.a.bl 3
20.d odd 2 1 4600.2.a.v 3
20.e even 4 2 4600.2.e.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.i 3 4.b odd 2 1
1840.2.a.q 3 1.a even 1 1 trivial
4600.2.a.v 3 20.d odd 2 1
4600.2.e.q 6 20.e even 4 2
7360.2.a.bw 3 8.d odd 2 1
7360.2.a.cf 3 8.b even 2 1
8280.2.a.bl 3 12.b even 2 1
9200.2.a.ci 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 7 \) Copy content Toggle raw display
\( T_{7}^{3} + 7T_{7}^{2} + 11T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} + 3T_{11}^{2} - T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} + \cdots - 7 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 7 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{3} + 3T^{2} - T - 2 \) Copy content Toggle raw display
$13$ \( T^{3} - 6 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$17$ \( T^{3} + 5 T^{2} + \cdots - 148 \) Copy content Toggle raw display
$19$ \( T^{3} + 7 T^{2} + \cdots - 106 \) Copy content Toggle raw display
$23$ \( (T - 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + T^{2} + \cdots - 148 \) Copy content Toggle raw display
$31$ \( T^{3} + 10 T^{2} + \cdots - 53 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} + \cdots - 481 \) Copy content Toggle raw display
$43$ \( T^{3} + 12 T^{2} + \cdots - 256 \) Copy content Toggle raw display
$47$ \( T^{3} + T^{2} - 6T - 4 \) Copy content Toggle raw display
$53$ \( T^{3} - 10 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$59$ \( T^{3} + 10 T^{2} + \cdots - 1184 \) Copy content Toggle raw display
$61$ \( T^{3} + 13 T^{2} + \cdots - 214 \) Copy content Toggle raw display
$67$ \( T^{3} - 6 T^{2} + \cdots + 1184 \) Copy content Toggle raw display
$71$ \( T^{3} + 10 T^{2} + \cdots - 875 \) Copy content Toggle raw display
$73$ \( T^{3} - 7 T^{2} + \cdots + 236 \) Copy content Toggle raw display
$79$ \( T^{3} + 14 T^{2} + \cdots - 224 \) Copy content Toggle raw display
$83$ \( T^{3} + 16 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$89$ \( T^{3} + 20 T^{2} + \cdots - 256 \) Copy content Toggle raw display
$97$ \( T^{3} - 5 T^{2} + \cdots + 56 \) Copy content Toggle raw display
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