# Properties

 Label 5040.2.t.z Level $5040$ Weight $2$ Character orbit 5040.t Analytic conductor $40.245$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{5} + \beta_{1} q^{7} +O(q^{10})$$ $$q -\beta_{5} q^{5} + \beta_{1} q^{7} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( -4 \beta_{1} - \beta_{3} + \beta_{4} ) q^{13} + ( -4 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{23} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{25} -2 q^{29} + ( 4 + \beta_{2} - \beta_{5} ) q^{31} -\beta_{3} q^{35} + ( 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} ) q^{37} + ( 4 - \beta_{2} + \beta_{5} ) q^{41} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{43} - q^{49} + ( 4 \beta_{1} - \beta_{3} + \beta_{4} ) q^{53} + ( -6 - 8 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{55} + ( 4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{59} + ( -2 + 4 \beta_{2} - 4 \beta_{5} ) q^{61} + ( -2 + 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{65} + ( 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{67} + ( -4 - 3 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{71} + ( -8 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{73} + ( \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{77} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{79} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{83} + ( 2 + 6 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{85} + ( -3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{89} + ( 4 - \beta_{2} + \beta_{5} ) q^{91} + ( 10 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{95} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{5} + O(q^{10})$$ $$6 q + 2 q^{5} - 4 q^{11} - 4 q^{19} - 2 q^{25} - 12 q^{29} + 28 q^{31} + 20 q^{41} - 6 q^{49} - 36 q^{55} + 16 q^{59} + 4 q^{61} - 8 q^{65} - 36 q^{71} + 8 q^{79} + 16 q^{85} - 12 q^{89} + 20 q^{91} - 12 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-7 \nu^{5} + 10 \nu^{4} - 5 \nu^{3} - 30 \nu^{2} - 32 \nu + 13$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-9 \nu^{5} + 3 \nu^{4} + 10 \nu^{3} - 32 \nu^{2} - 74 \nu - 3$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$-10 \nu^{5} + 11 \nu^{4} - 17 \nu^{3} - 10 \nu^{2} - 72 \nu - 11$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$12 \nu^{5} - 27 \nu^{4} + 25 \nu^{3} + 12 \nu^{2} + 68 \nu - 65$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-19 \nu^{5} + 37 \nu^{4} - 30 \nu^{3} - 42 \nu^{2} - 54 \nu + 55$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{2} - 4 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_{1} - 4$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{5} - 5 \beta_{4} - 5 \beta_{3} + \beta_{2} - 14$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{5} - 11 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} + 18 \beta_{1} - 18$$$$)/2$$

## 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
 0.403032 − 0.403032i 0.403032 + 0.403032i 1.45161 + 1.45161i 1.45161 − 1.45161i −0.854638 + 0.854638i −0.854638 − 0.854638i
0 0 0 −1.48119 1.67513i 0 1.00000i 0 0 0
1009.2 0 0 0 −1.48119 + 1.67513i 0 1.00000i 0 0 0
1009.3 0 0 0 0.311108 2.21432i 0 1.00000i 0 0 0
1009.4 0 0 0 0.311108 + 2.21432i 0 1.00000i 0 0 0
1009.5 0 0 0 2.17009 0.539189i 0 1.00000i 0 0 0
1009.6 0 0 0 2.17009 + 0.539189i 0 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1009.6 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.z 6
3.b odd 2 1 1680.2.t.j 6
4.b odd 2 1 2520.2.t.k 6
5.b even 2 1 inner 5040.2.t.z 6
12.b even 2 1 840.2.t.d 6
15.d odd 2 1 1680.2.t.j 6
15.e even 4 1 8400.2.a.di 3
15.e even 4 1 8400.2.a.dl 3
20.d odd 2 1 2520.2.t.k 6
60.h even 2 1 840.2.t.d 6
60.l odd 4 1 4200.2.a.bn 3
60.l odd 4 1 4200.2.a.bp 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.d 6 12.b even 2 1
840.2.t.d 6 60.h even 2 1
1680.2.t.j 6 3.b odd 2 1
1680.2.t.j 6 15.d odd 2 1
2520.2.t.k 6 4.b odd 2 1
2520.2.t.k 6 20.d odd 2 1
4200.2.a.bn 3 60.l odd 4 1
4200.2.a.bp 3 60.l odd 4 1
5040.2.t.z 6 1.a even 1 1 trivial
5040.2.t.z 6 5.b even 2 1 inner
8400.2.a.di 3 15.e even 4 1
8400.2.a.dl 3 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 $$S_{2}^{\mathrm{new}}(5040, [\chi])$$:

 $$T_{11}^{3} + 2 T_{11}^{2} - 36 T_{11} - 104$$ $$T_{13}^{6} + 60 T_{13}^{4} + 560 T_{13}^{2} + 64$$ $$T_{17}^{6} + 128 T_{17}^{4} + 5376 T_{17}^{2} + 73984$$ $$T_{19}^{3} + 2 T_{19}^{2} - 60 T_{19} - 200$$ $$T_{29} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$125 - 50 T + 15 T^{2} - 12 T^{3} + 3 T^{4} - 2 T^{5} + T^{6}$$
$7$ $$( 1 + T^{2} )^{3}$$
$11$ $$( -104 - 36 T + 2 T^{2} + T^{3} )^{2}$$
$13$ $$64 + 560 T^{2} + 60 T^{4} + T^{6}$$
$17$ $$73984 + 5376 T^{2} + 128 T^{4} + T^{6}$$
$19$ $$( -200 - 60 T + 2 T^{2} + T^{3} )^{2}$$
$23$ $$92416 + 6464 T^{2} + 144 T^{4} + T^{6}$$
$29$ $$( 2 + T )^{6}$$
$31$ $$( -40 + 52 T - 14 T^{2} + T^{3} )^{2}$$
$37$ $$102400 + 7936 T^{2} + 176 T^{4} + T^{6}$$
$41$ $$( 8 + 20 T - 10 T^{2} + T^{3} )^{2}$$
$43$ $$4096 + 2816 T^{2} + 112 T^{4} + T^{6}$$
$47$ $$T^{6}$$
$53$ $$1600 + 1584 T^{2} + 92 T^{4} + T^{6}$$
$59$ $$( 256 - 64 T - 8 T^{2} + T^{3} )^{2}$$
$61$ $$( 104 - 212 T - 2 T^{2} + T^{3} )^{2}$$
$67$ $$65536 + 8192 T^{2} + 256 T^{4} + T^{6}$$
$71$ $$( -1352 - 52 T + 18 T^{2} + T^{3} )^{2}$$
$73$ $$10816 + 5424 T^{2} + 284 T^{4} + T^{6}$$
$79$ $$( 64 - 48 T - 4 T^{2} + T^{3} )^{2}$$
$83$ $$65536 + 8192 T^{2} + 192 T^{4} + T^{6}$$
$89$ $$( 232 - 124 T + 6 T^{2} + T^{3} )^{2}$$
$97$ $$40000 + 4400 T^{2} + 124 T^{4} + T^{6}$$