# Properties

 Label 5040.2.t.a Level $5040$ Weight $2$ Character orbit 5040.t Analytic conductor $40.245$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + i ) q^{5} -i q^{7} +O(q^{10})$$ $$q + ( -2 + i ) q^{5} -i q^{7} - q^{11} -i q^{13} -3 i q^{17} -4 q^{19} + 2 i q^{23} + ( 3 - 4 i ) q^{25} - q^{29} + 6 q^{31} + ( 1 + 2 i ) q^{35} -2 i q^{37} + 10 q^{41} -9 i q^{47} - q^{49} + 14 i q^{53} + ( 2 - i ) q^{55} -6 q^{59} -4 q^{61} + ( 1 + 2 i ) q^{65} + 10 i q^{67} -16 q^{71} + 10 i q^{73} + i q^{77} -11 q^{79} + 4 i q^{83} + ( 3 + 6 i ) q^{85} + 12 q^{89} - q^{91} + ( 8 - 4 i ) q^{95} + 19 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} + O(q^{10})$$ $$2q - 4q^{5} - 2q^{11} - 8q^{19} + 6q^{25} - 2q^{29} + 12q^{31} + 2q^{35} + 20q^{41} - 2q^{49} + 4q^{55} - 12q^{59} - 8q^{61} + 2q^{65} - 32q^{71} - 22q^{79} + 6q^{85} + 24q^{89} - 2q^{91} + 16q^{95} + O(q^{100})$$

## 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 −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: 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.a 2
3.b odd 2 1 560.2.g.d 2
4.b odd 2 1 2520.2.t.a 2
5.b even 2 1 inner 5040.2.t.a 2
12.b even 2 1 280.2.g.a 2
15.d odd 2 1 560.2.g.d 2
15.e even 4 1 2800.2.a.k 1
15.e even 4 1 2800.2.a.u 1
20.d odd 2 1 2520.2.t.a 2
24.f even 2 1 2240.2.g.b 2
24.h odd 2 1 2240.2.g.a 2
60.h even 2 1 280.2.g.a 2
60.l odd 4 1 1400.2.a.d 1
60.l odd 4 1 1400.2.a.j 1
84.h odd 2 1 1960.2.g.a 2
120.i odd 2 1 2240.2.g.a 2
120.m even 2 1 2240.2.g.b 2
420.o odd 2 1 1960.2.g.a 2
420.w even 4 1 9800.2.a.p 1
420.w even 4 1 9800.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.a 2 12.b even 2 1
280.2.g.a 2 60.h even 2 1
560.2.g.d 2 3.b odd 2 1
560.2.g.d 2 15.d odd 2 1
1400.2.a.d 1 60.l odd 4 1
1400.2.a.j 1 60.l odd 4 1
1960.2.g.a 2 84.h odd 2 1
1960.2.g.a 2 420.o odd 2 1
2240.2.g.a 2 24.h odd 2 1
2240.2.g.a 2 120.i odd 2 1
2240.2.g.b 2 24.f even 2 1
2240.2.g.b 2 120.m even 2 1
2520.2.t.a 2 4.b odd 2 1
2520.2.t.a 2 20.d odd 2 1
2800.2.a.k 1 15.e even 4 1
2800.2.a.u 1 15.e even 4 1
5040.2.t.a 2 1.a even 1 1 trivial
5040.2.t.a 2 5.b even 2 1 inner
9800.2.a.p 1 420.w even 4 1
9800.2.a.bb 1 420.w even 4 1

## Hecke kernels

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

 $$T_{11} + 1$$ $$T_{13}^{2} + 1$$ $$T_{17}^{2} + 9$$ $$T_{19} + 4$$ $$T_{29} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 + 4 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$( -6 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$81 + T^{2}$$
$53$ $$196 + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$( 4 + T )^{2}$$
$67$ $$100 + T^{2}$$
$71$ $$( 16 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 11 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( -12 + T )^{2}$$
$97$ $$361 + T^{2}$$