Properties

Label 5040.2.t.p
Level $5040$
Weight $2$
Character orbit 5040.t
Analytic conductor $40.245$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5040,2,Mod(1009,5040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5040.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.t (of order \(2\), degree \(1\), not 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: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i + 2) q^{5} + i q^{7} - 3 q^{11} - i q^{13} - 7 i q^{17} - 6 i q^{23} + (4 i + 3) q^{25} - 5 q^{29} - 2 q^{31} + (2 i - 1) q^{35} + 2 i q^{37} - 2 q^{41} - 4 i q^{43} - 3 i q^{47} - q^{49} + 6 i q^{53} + \cdots + 7 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 6 q^{11} + 6 q^{25} - 10 q^{29} - 4 q^{31} - 2 q^{35} - 4 q^{41} - 2 q^{49} - 12 q^{55} - 20 q^{59} - 16 q^{61} + 2 q^{65} - 16 q^{71} - 10 q^{79} + 14 q^{85} + 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times\).

\(n\) \(2017\) \(2801\) \(3151\) \(3601\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
1.00000i
1.00000i
0 0 0 2.00000 1.00000i 0 1.00000i 0 0 0
1009.2 0 0 0 2.00000 + 1.00000i 0 1.00000i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5040.2.t.p 2
3.b odd 2 1 560.2.g.b 2
4.b odd 2 1 315.2.d.a 2
5.b even 2 1 inner 5040.2.t.p 2
12.b even 2 1 35.2.b.a 2
15.d odd 2 1 560.2.g.b 2
15.e even 4 1 2800.2.a.l 1
15.e even 4 1 2800.2.a.w 1
20.d odd 2 1 315.2.d.a 2
20.e even 4 1 1575.2.a.a 1
20.e even 4 1 1575.2.a.k 1
24.f even 2 1 2240.2.g.h 2
24.h odd 2 1 2240.2.g.g 2
28.d even 2 1 2205.2.d.b 2
60.h even 2 1 35.2.b.a 2
60.l odd 4 1 175.2.a.a 1
60.l odd 4 1 175.2.a.c 1
84.h odd 2 1 245.2.b.a 2
84.j odd 6 2 245.2.j.d 4
84.n even 6 2 245.2.j.e 4
120.i odd 2 1 2240.2.g.g 2
120.m even 2 1 2240.2.g.h 2
140.c even 2 1 2205.2.d.b 2
420.o odd 2 1 245.2.b.a 2
420.w even 4 1 1225.2.a.a 1
420.w even 4 1 1225.2.a.i 1
420.ba even 6 2 245.2.j.e 4
420.be odd 6 2 245.2.j.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 12.b even 2 1
35.2.b.a 2 60.h even 2 1
175.2.a.a 1 60.l odd 4 1
175.2.a.c 1 60.l odd 4 1
245.2.b.a 2 84.h odd 2 1
245.2.b.a 2 420.o odd 2 1
245.2.j.d 4 84.j odd 6 2
245.2.j.d 4 420.be odd 6 2
245.2.j.e 4 84.n even 6 2
245.2.j.e 4 420.ba even 6 2
315.2.d.a 2 4.b odd 2 1
315.2.d.a 2 20.d odd 2 1
560.2.g.b 2 3.b odd 2 1
560.2.g.b 2 15.d odd 2 1
1225.2.a.a 1 420.w even 4 1
1225.2.a.i 1 420.w even 4 1
1575.2.a.a 1 20.e even 4 1
1575.2.a.k 1 20.e even 4 1
2205.2.d.b 2 28.d even 2 1
2205.2.d.b 2 140.c even 2 1
2240.2.g.g 2 24.h odd 2 1
2240.2.g.g 2 120.i odd 2 1
2240.2.g.h 2 24.f even 2 1
2240.2.g.h 2 120.m even 2 1
2800.2.a.l 1 15.e even 4 1
2800.2.a.w 1 15.e even 4 1
5040.2.t.p 2 1.a even 1 1 trivial
5040.2.t.p 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5040, [\chi])\):

\( T_{11} + 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 49 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{29} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 49 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T + 10)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T + 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 49 \) Copy content Toggle raw display
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