Properties

Label 2368.2.a.be
Level $2368$
Weight $2$
Character orbit 2368.a
Self dual yes
Analytic conductor $18.909$
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] = [2368,2,Mod(1,2368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2368, 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("2368.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.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: \(18.9085751986\)
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 296)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} - \beta_{2} q^{5} + (\beta_{2} - \beta_1 - 2) q^{7} + (\beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} - \beta_{2} q^{5} + (\beta_{2} - \beta_1 - 2) q^{7} + (\beta_1 + 1) q^{9} - 3 \beta_1 q^{11} + (3 \beta_1 - 1) q^{13} + (\beta_{2} - \beta_1 - 3) q^{15} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{17} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{19} + ( - 3 \beta_{2} - \beta_1) q^{21} + (\beta_1 - 3) q^{23} + ( - 2 \beta_{2} + \beta_1 - 2) q^{25} + ( - 2 \beta_{2} + 2 \beta_1 - 1) q^{27} + ( - \beta_1 + 3) q^{29} + ( - \beta_{2} - 6) q^{31} + ( - 6 \beta_1 - 3) q^{33} + (4 \beta_{2} - 2) q^{35} + q^{37} + ( - \beta_{2} + 6 \beta_1 + 2) q^{39} + (\beta_{2} - 2 \beta_1 - 5) q^{41} + (4 \beta_{2} + 2 \beta_1) q^{43} + ( - \beta_{2} - \beta_1 - 1) q^{45} + ( - \beta_{2} - 3 \beta_1 - 4) q^{47} + ( - 5 \beta_{2} + 3 \beta_1 + 1) q^{49} + (2 \beta_1 - 6) q^{51} + (3 \beta_{2} - 5 \beta_1 + 2) q^{53} + (3 \beta_1 + 3) q^{55} + (4 \beta_{2} + 2 \beta_1 - 2) q^{57} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{59} + (3 \beta_{2} - 2 \beta_1 - 4) q^{61} + ( - 2 \beta_1 - 4) q^{63} + (\beta_{2} - 3 \beta_1 - 3) q^{65} + (5 \beta_{2} - 6 \beta_1) q^{67} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{69} + (\beta_{2} - 3 \beta_1 + 2) q^{71} + ( - 3 \beta_1 - 2) q^{73} - 7 q^{75} + (3 \beta_{2} + 3 \beta_1 + 6) q^{77} + (2 \beta_{2} - \beta_1 + 1) q^{79} + (\beta_{2} - \beta_1 - 8) q^{81} + ( - 3 \beta_{2} + 5 \beta_1 - 4) q^{83} + ( - 2 \beta_{2} + 4) q^{85} + (3 \beta_{2} - 2 \beta_1 + 2) q^{87} + ( - 4 \beta_{2} + 4) q^{89} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{91} + ( - 5 \beta_{2} - \beta_1 - 9) q^{93} + ( - 6 \beta_{2} + 4) q^{95} + (6 \beta_{2} + 2 \beta_1 + 2) q^{97} + ( - 3 \beta_{2} - 3 \beta_1 - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + q^{5} - 7 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} + q^{5} - 7 q^{7} + 3 q^{9} - 3 q^{13} - 10 q^{15} - 4 q^{17} + 8 q^{19} + 3 q^{21} - 9 q^{23} - 4 q^{25} - q^{27} + 9 q^{29} - 17 q^{31} - 9 q^{33} - 10 q^{35} + 3 q^{37} + 7 q^{39} - 16 q^{41} - 4 q^{43} - 2 q^{45} - 11 q^{47} + 8 q^{49} - 18 q^{51} + 3 q^{53} + 9 q^{55} - 10 q^{57} - 2 q^{59} - 15 q^{61} - 12 q^{63} - 10 q^{65} - 5 q^{67} - 3 q^{69} + 5 q^{71} - 6 q^{73} - 21 q^{75} + 15 q^{77} + q^{79} - 25 q^{81} - 9 q^{83} + 14 q^{85} + 3 q^{87} + 16 q^{89} - 8 q^{91} - 22 q^{93} + 18 q^{95} - 24 q^{99}+O(q^{100}) \) 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
−0.254102
−1.86081
2.11491
0 −1.93543 0 2.93543 0 −4.68133 0 0.745898 0
1.2 0 1.46260 0 −0.462598 0 0.323404 0 −0.860806 0
1.3 0 2.47283 0 −1.47283 0 −2.64207 0 3.11491 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(37\) \( -1 \)

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 2368.2.a.be 3
4.b odd 2 1 2368.2.a.bb 3
8.b even 2 1 592.2.a.i 3
8.d odd 2 1 296.2.a.c 3
24.f even 2 1 2664.2.a.p 3
24.h odd 2 1 5328.2.a.bn 3
40.e odd 2 1 7400.2.a.k 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.2.a.c 3 8.d odd 2 1
592.2.a.i 3 8.b even 2 1
2368.2.a.bb 3 4.b odd 2 1
2368.2.a.be 3 1.a even 1 1 trivial
2664.2.a.p 3 24.f even 2 1
5328.2.a.bn 3 24.h odd 2 1
7400.2.a.k 3 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2368))\):

\( T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 7 \) Copy content Toggle raw display
\( T_{5}^{3} - T_{5}^{2} - 5T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 7T_{7}^{2} + 10T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{3} - 36T_{11} + 27 \) 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
$5$ \( T^{3} - T^{2} - 5T - 2 \) Copy content Toggle raw display
$7$ \( T^{3} + 7 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( T^{3} - 36T + 27 \) Copy content Toggle raw display
$13$ \( T^{3} + 3 T^{2} + \cdots - 62 \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{3} + 9 T^{2} + \cdots + 14 \) Copy content Toggle raw display
$29$ \( T^{3} - 9 T^{2} + \cdots - 14 \) Copy content Toggle raw display
$31$ \( T^{3} + 17 T^{2} + \cdots + 148 \) Copy content Toggle raw display
$37$ \( (T - 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + 16 T^{2} + \cdots + 47 \) Copy content Toggle raw display
$43$ \( T^{3} + 4 T^{2} + \cdots - 232 \) Copy content Toggle raw display
$47$ \( T^{3} + 11 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$53$ \( T^{3} - 3 T^{2} + \cdots - 292 \) Copy content Toggle raw display
$59$ \( T^{3} + 2 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$61$ \( T^{3} + 15 T^{2} + \cdots - 52 \) Copy content Toggle raw display
$67$ \( T^{3} + 5 T^{2} + \cdots - 944 \) Copy content Toggle raw display
$71$ \( T^{3} - 5 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$73$ \( T^{3} + 6 T^{2} + \cdots - 37 \) Copy content Toggle raw display
$79$ \( T^{3} - T^{2} + \cdots + 32 \) Copy content Toggle raw display
$83$ \( T^{3} + 9 T^{2} + \cdots + 112 \) Copy content Toggle raw display
$89$ \( T^{3} - 16T^{2} + 64 \) Copy content Toggle raw display
$97$ \( T^{3} - 244T + 256 \) Copy content Toggle raw display
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