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Properties

Label 2340.2.bp.e
Level $2340$
Weight $2$
Character orbit 2340.bp
Analytic conductor $18.685$
Analytic rank $0$
Dimension $4$
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] = [2340,2,Mod(1477,2340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2340, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2340.1477");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2340.bp (of order \(4\), 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: \(18.6849940730\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1 - 1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1 - 1) q^{5} + (\beta_{3} - \beta_{2} + 1) q^{11} + (2 \beta_{2} - 3) q^{13} + ( - 2 \beta_{3} - \beta_{2} + 1) q^{17} + ( - 3 \beta_{3} + \beta_{2} - 1) q^{19} + (5 \beta_{2} + \beta_1 + 5) q^{23} + 5 q^{25} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{29} + (\beta_{2} - 3 \beta_1 + 1) q^{31} + (3 \beta_{3} - 3 \beta_1) q^{37} + ( - 7 \beta_{2} - 2 \beta_1 - 7) q^{41} + ( - \beta_{2} - 3 \beta_1 - 1) q^{43} + (4 \beta_{3} - 4 \beta_1 - 4) q^{47} - 7 q^{49} + ( - 4 \beta_{3} - 5 \beta_{2} + 5) q^{53} + (3 \beta_{3} - \beta_{2} + 1) q^{55} + ( - 5 \beta_{2} - \beta_1 - 5) q^{59} + (3 \beta_{3} - 3 \beta_1 - 4) q^{61} + ( - 5 \beta_{3} - 2 \beta_{2} + \cdots + 3) q^{65}+ \cdots + ( - 6 \beta_{3} - 4 \beta_{2} - 6 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{11} - 12 q^{13} - 10 q^{19} + 18 q^{23} + 20 q^{25} + 10 q^{31} + 12 q^{37} - 24 q^{41} + 2 q^{43} - 28 q^{49} + 12 q^{53} + 10 q^{55} - 18 q^{59} - 4 q^{61} - 6 q^{71} + 24 q^{83} - 20 q^{85} - 30 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu - 1 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(1171\) \(2081\)
\(\chi(n)\) \(\beta_{2}\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1477.1
0.618034i
− 1.61803i
− 0.618034i
1.61803i
0 0 0 −2.23607 0 0 0 0 0
1477.2 0 0 0 2.23607 0 0 0 0 0
1513.1 0 0 0 −2.23607 0 0 0 0 0
1513.2 0 0 0 2.23607 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.2.bp.e 4
3.b odd 2 1 260.2.r.b yes 4
5.c odd 4 1 2340.2.u.f 4
12.b even 2 1 1040.2.cd.j 4
13.d odd 4 1 2340.2.u.f 4
15.d odd 2 1 1300.2.r.b 4
15.e even 4 1 260.2.m.b ✓ 4
15.e even 4 1 1300.2.m.b 4
39.f even 4 1 260.2.m.b ✓ 4
60.l odd 4 1 1040.2.bg.j 4
65.f even 4 1 inner 2340.2.bp.e 4
156.l odd 4 1 1040.2.bg.j 4
195.j odd 4 1 1300.2.r.b 4
195.n even 4 1 1300.2.m.b 4
195.u odd 4 1 260.2.r.b yes 4
780.u even 4 1 1040.2.cd.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.m.b ✓ 4 15.e even 4 1
260.2.m.b ✓ 4 39.f even 4 1
260.2.r.b yes 4 3.b odd 2 1
260.2.r.b yes 4 195.u odd 4 1
1040.2.bg.j 4 60.l odd 4 1
1040.2.bg.j 4 156.l odd 4 1
1040.2.cd.j 4 12.b even 2 1
1040.2.cd.j 4 780.u even 4 1
1300.2.m.b 4 15.e even 4 1
1300.2.m.b 4 195.n even 4 1
1300.2.r.b 4 15.d odd 2 1
1300.2.r.b 4 195.j odd 4 1
2340.2.u.f 4 5.c odd 4 1
2340.2.u.f 4 13.d odd 4 1
2340.2.bp.e 4 1.a even 1 1 trivial
2340.2.bp.e 4 65.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2340, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{4} - 6T_{11}^{3} + 18T_{11}^{2} - 12T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 100 \) Copy content Toggle raw display
$19$ \( T^{4} + 10 T^{3} + \cdots + 100 \) Copy content Toggle raw display
$23$ \( T^{4} - 18 T^{3} + \cdots + 1444 \) Copy content Toggle raw display
$29$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{4} - 10 T^{3} + \cdots + 100 \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T - 36)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 24 T^{3} + \cdots + 3844 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$47$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$59$ \( T^{4} + 18 T^{3} + \cdots + 1444 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 44)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 188T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$73$ \( T^{4} + 140T^{2} + 400 \) Copy content Toggle raw display
$79$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 100 \) Copy content Toggle raw display
$97$ \( T^{4} + 368 T^{2} + 30976 \) Copy content Toggle raw display
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