# Properties

 Label 5040.2.t.j Level 5040 Weight 2 Character orbit 5040.t Analytic conductor 40.245 Analytic rank 1 Dimension 2 CM no Inner twists 2

# Related objects

## 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 630) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + 2 i ) q^{5} -i q^{7} +O(q^{10})$$ $$q + ( 1 + 2 i ) q^{5} -i q^{7} -6 q^{11} + 2 i q^{13} -2 i q^{17} + 4 q^{19} -4 i q^{23} + ( -3 + 4 i ) q^{25} -2 q^{29} + 2 q^{31} + ( 2 - i ) q^{35} + 10 i q^{37} + 6 q^{41} + 2 i q^{43} -2 i q^{47} - q^{49} -6 i q^{53} + ( -6 - 12 i ) q^{55} -4 q^{59} -12 q^{61} + ( -4 + 2 i ) q^{65} + 10 i q^{67} -12 q^{71} -2 i q^{73} + 6 i q^{77} -16 q^{79} -12 i q^{83} + ( 4 - 2 i ) q^{85} -14 q^{89} + 2 q^{91} + ( 4 + 8 i ) q^{95} -18 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + O(q^{10})$$ $$2q + 2q^{5} - 12q^{11} + 8q^{19} - 6q^{25} - 4q^{29} + 4q^{31} + 4q^{35} + 12q^{41} - 2q^{49} - 12q^{55} - 8q^{59} - 24q^{61} - 8q^{65} - 24q^{71} - 32q^{79} + 8q^{85} - 28q^{89} + 4q^{91} + 8q^{95} + O(q^{100})$$

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.j 2
3.b odd 2 1 5040.2.t.i 2
4.b odd 2 1 630.2.g.e yes 2
5.b even 2 1 inner 5040.2.t.j 2
12.b even 2 1 630.2.g.b 2
15.d odd 2 1 5040.2.t.i 2
20.d odd 2 1 630.2.g.e yes 2
20.e even 4 1 3150.2.a.k 1
20.e even 4 1 3150.2.a.br 1
60.h even 2 1 630.2.g.b 2
60.l odd 4 1 3150.2.a.l 1
60.l odd 4 1 3150.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.g.b 2 12.b even 2 1
630.2.g.b 2 60.h even 2 1
630.2.g.e yes 2 4.b odd 2 1
630.2.g.e yes 2 20.d odd 2 1
3150.2.a.k 1 20.e even 4 1
3150.2.a.l 1 60.l odd 4 1
3150.2.a.v 1 60.l odd 4 1
3150.2.a.br 1 20.e even 4 1
5040.2.t.i 2 3.b odd 2 1
5040.2.t.i 2 15.d odd 2 1
5040.2.t.j 2 1.a even 1 1 trivial
5040.2.t.j 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 $$S_{2}^{\mathrm{new}}(5040, [\chi])$$:

 $$T_{11} + 6$$ $$T_{13}^{2} + 4$$ $$T_{17}^{2} + 4$$ $$T_{19} - 4$$ $$T_{29} + 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 + 6 T + 11 T^{2} )^{2}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{2}$$
$23$ $$1 - 30 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 2 T + 31 T^{2} )^{2}$$
$37$ $$1 + 26 T^{2} + 1369 T^{4}$$
$41$ $$( 1 - 6 T + 41 T^{2} )^{2}$$
$43$ $$1 - 82 T^{2} + 1849 T^{4}$$
$47$ $$1 - 90 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 12 T + 61 T^{2} )^{2}$$
$67$ $$1 - 34 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$1 - 142 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 16 T + 79 T^{2} )^{2}$$
$83$ $$1 - 22 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 14 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} )$$