Properties

Label 5040.2.t.n
Level $5040$
Weight $2$
Character orbit 5040.t
Analytic conductor $40.245$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + 2 i ) q^{5} + i q^{7} +O(q^{10})\) \( q + ( 1 + 2 i ) q^{5} + i q^{7} + 2 q^{11} -2 i q^{13} + 2 q^{19} -8 i q^{23} + ( -3 + 4 i ) q^{25} + 2 q^{29} + 6 q^{31} + ( -2 + i ) q^{35} + 8 i q^{37} + 10 q^{41} -12 i q^{47} - q^{49} -2 i q^{53} + ( 2 + 4 i ) q^{55} + 2 q^{61} + ( 4 - 2 i ) q^{65} -4 i q^{67} + 14 q^{71} + 2 i q^{73} + 2 i q^{77} + 4 q^{79} -16 i q^{83} -6 q^{89} + 2 q^{91} + ( 2 + 4 i ) q^{95} + 2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{5} + 4q^{11} + 4q^{19} - 6q^{25} + 4q^{29} + 12q^{31} - 4q^{35} + 20q^{41} - 2q^{49} + 4q^{55} + 4q^{61} + 8q^{65} + 28q^{71} + 8q^{79} - 12q^{89} + 4q^{91} + 4q^{95} + O(q^{100}) \)

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.

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 1.00000 2.00000i 0 1.00000i 0 0 0
1009.2 0 0 0 1.00000 + 2.00000i 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.n 2
3.b odd 2 1 1680.2.t.c 2
4.b odd 2 1 2520.2.t.c 2
5.b even 2 1 inner 5040.2.t.n 2
12.b even 2 1 840.2.t.a 2
15.d odd 2 1 1680.2.t.c 2
15.e even 4 1 8400.2.a.u 1
15.e even 4 1 8400.2.a.br 1
20.d odd 2 1 2520.2.t.c 2
60.h even 2 1 840.2.t.a 2
60.l odd 4 1 4200.2.a.k 1
60.l odd 4 1 4200.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.a 2 12.b even 2 1
840.2.t.a 2 60.h even 2 1
1680.2.t.c 2 3.b odd 2 1
1680.2.t.c 2 15.d odd 2 1
2520.2.t.c 2 4.b odd 2 1
2520.2.t.c 2 20.d odd 2 1
4200.2.a.k 1 60.l odd 4 1
4200.2.a.x 1 60.l odd 4 1
5040.2.t.n 2 1.a even 1 1 trivial
5040.2.t.n 2 5.b even 2 1 inner
8400.2.a.u 1 15.e even 4 1
8400.2.a.br 1 15.e even 4 1

Hecke kernels

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

\( T_{11} - 2 \)
\( T_{13}^{2} + 4 \)
\( T_{17} \)
\( T_{19} - 2 \)
\( T_{29} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 - 2 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( ( -6 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( 144 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -14 + T )^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 256 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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