# Properties

 Label 5040.2.t.s 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - i ) q^{5} + i q^{7} +O(q^{10})$$ $$q + ( 2 - i ) q^{5} + i q^{7} + 3 q^{11} + i q^{13} -5 i q^{17} -8 q^{19} + 2 i q^{23} + ( 3 - 4 i ) q^{25} - q^{29} + 2 q^{31} + ( 1 + 2 i ) q^{35} -10 i q^{37} + 6 q^{41} + 4 i q^{43} -11 i q^{47} - q^{49} -6 i q^{53} + ( 6 - 3 i ) q^{55} + 10 q^{59} + ( 1 + 2 i ) q^{65} -10 i q^{67} -10 i q^{73} + 3 i q^{77} -7 q^{79} + 12 i q^{83} + ( -5 - 10 i ) q^{85} + 8 q^{89} - q^{91} + ( -16 + 8 i ) q^{95} -3 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} + O(q^{10})$$ $$2 q + 4 q^{5} + 6 q^{11} - 16 q^{19} + 6 q^{25} - 2 q^{29} + 4 q^{31} + 2 q^{35} + 12 q^{41} - 2 q^{49} + 12 q^{55} + 20 q^{59} + 2 q^{65} - 14 q^{79} - 10 q^{85} + 16 q^{89} - 2 q^{91} - 32 q^{95} + O(q^{100})$$

## 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.s 2
3.b odd 2 1 560.2.g.a 2
4.b odd 2 1 1260.2.k.c 2
5.b even 2 1 inner 5040.2.t.s 2
12.b even 2 1 140.2.e.a 2
15.d odd 2 1 560.2.g.a 2
15.e even 4 1 2800.2.a.a 1
15.e even 4 1 2800.2.a.bf 1
20.d odd 2 1 1260.2.k.c 2
20.e even 4 1 6300.2.a.c 1
20.e even 4 1 6300.2.a.t 1
24.f even 2 1 2240.2.g.e 2
24.h odd 2 1 2240.2.g.f 2
60.h even 2 1 140.2.e.a 2
60.l odd 4 1 700.2.a.a 1
60.l odd 4 1 700.2.a.j 1
84.h odd 2 1 980.2.e.b 2
84.j odd 6 2 980.2.q.c 4
84.n even 6 2 980.2.q.f 4
120.i odd 2 1 2240.2.g.f 2
120.m even 2 1 2240.2.g.e 2
420.o odd 2 1 980.2.e.b 2
420.w even 4 1 4900.2.a.b 1
420.w even 4 1 4900.2.a.w 1
420.ba even 6 2 980.2.q.f 4
420.be odd 6 2 980.2.q.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 12.b even 2 1
140.2.e.a 2 60.h even 2 1
560.2.g.a 2 3.b odd 2 1
560.2.g.a 2 15.d odd 2 1
700.2.a.a 1 60.l odd 4 1
700.2.a.j 1 60.l odd 4 1
980.2.e.b 2 84.h odd 2 1
980.2.e.b 2 420.o odd 2 1
980.2.q.c 4 84.j odd 6 2
980.2.q.c 4 420.be odd 6 2
980.2.q.f 4 84.n even 6 2
980.2.q.f 4 420.ba even 6 2
1260.2.k.c 2 4.b odd 2 1
1260.2.k.c 2 20.d odd 2 1
2240.2.g.e 2 24.f even 2 1
2240.2.g.e 2 120.m even 2 1
2240.2.g.f 2 24.h odd 2 1
2240.2.g.f 2 120.i odd 2 1
2800.2.a.a 1 15.e even 4 1
2800.2.a.bf 1 15.e even 4 1
4900.2.a.b 1 420.w even 4 1
4900.2.a.w 1 420.w even 4 1
5040.2.t.s 2 1.a even 1 1 trivial
5040.2.t.s 2 5.b even 2 1 inner
6300.2.a.c 1 20.e even 4 1
6300.2.a.t 1 20.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 $$S_{2}^{\mathrm{new}}(5040, [\chi])$$:

 $$T_{11} - 3$$ $$T_{13}^{2} + 1$$ $$T_{17}^{2} + 25$$ $$T_{19} + 8$$ $$T_{29} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 - 4 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$25 + T^{2}$$
$19$ $$( 8 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$121 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( -10 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$100 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 7 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -8 + T )^{2}$$
$97$ $$9 + T^{2}$$
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