# Properties

 Label 5040.2.k.f Level $5040$ Weight $2$ Character orbit 5040.k Analytic conductor $40.245$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5040.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.2446026187$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.10070523904.11 Defining polynomial: $$x^{8} - 10 x^{4} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1260) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{2} + \beta_{6} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{2} + \beta_{6} ) q^{7} + ( \beta_{5} + \beta_{6} ) q^{11} + ( \beta_{1} - \beta_{7} ) q^{13} + ( \beta_{1} + \beta_{7} ) q^{17} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{19} + ( 5 - \beta_{4} ) q^{23} + ( 3 \beta_{2} - \beta_{4} ) q^{25} + \beta_{2} q^{29} + ( 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{31} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{35} + ( -\beta_{2} - \beta_{5} - \beta_{6} ) q^{37} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{41} + ( -5 \beta_{2} + \beta_{5} + \beta_{6} ) q^{43} + ( -4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{47} + ( -1 - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{49} + ( 7 + \beta_{4} ) q^{53} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{55} + ( -4 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{59} + ( 1 + 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{65} + ( -4 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} ) q^{67} + ( \beta_{5} + \beta_{6} ) q^{71} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{73} + ( -7 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{77} + ( -4 + 4 \beta_{4} ) q^{79} + ( -4 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{83} + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{85} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{89} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{91} + ( -3 - \beta_{2} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{95} + ( 5 \beta_{1} + 4 \beta_{2} - 4 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 40 q^{23} + 16 q^{35} - 8 q^{49} + 56 q^{53} + 8 q^{65} - 56 q^{77} - 32 q^{79} + 8 q^{85} - 16 q^{91} - 24 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 10 x^{4} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - \nu^{2}$$$$)/18$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 7 \nu$$$$)/6$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 19 \nu^{2}$$$$)/18$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + 3 \nu^{5} + 9 \nu^{4} + \nu^{3} - \nu^{2} - 3 \nu - 45$$$$)/36$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - 3 \nu^{5} + 9 \nu^{4} - \nu^{3} + \nu^{2} + 3 \nu - 45$$$$)/36$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} - 10 \nu^{3}$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{6} + 2 \beta_{5} + 5$$ $$\nu^{5}$$ $$=$$ $$6 \beta_{3} + 7 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{4} + 19 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{7} + 20 \beta_{6} - 20 \beta_{5} + 20 \beta_{3} + 20 \beta_{2} + 20 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1889.1
 0.420861 + 1.68014i 0.420861 − 1.68014i −1.68014 − 0.420861i −1.68014 + 0.420861i 1.68014 − 0.420861i 1.68014 + 0.420861i −0.420861 + 1.68014i −0.420861 − 1.68014i
0 0 0 −1.95522 1.08495i 0 −2.37608 1.16372i 0 0 0
1889.2 0 0 0 −1.95522 + 1.08495i 0 −2.37608 + 1.16372i 0 0 0
1889.3 0 0 0 −1.08495 1.95522i 0 0.595188 + 2.57794i 0 0 0
1889.4 0 0 0 −1.08495 + 1.95522i 0 0.595188 2.57794i 0 0 0
1889.5 0 0 0 1.08495 1.95522i 0 −0.595188 2.57794i 0 0 0
1889.6 0 0 0 1.08495 + 1.95522i 0 −0.595188 + 2.57794i 0 0 0
1889.7 0 0 0 1.95522 1.08495i 0 2.37608 + 1.16372i 0 0 0
1889.8 0 0 0 1.95522 + 1.08495i 0 2.37608 1.16372i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1889.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g 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.k.f 8
3.b odd 2 1 5040.2.k.e 8
4.b odd 2 1 1260.2.f.a 8
5.b even 2 1 5040.2.k.e 8
7.b odd 2 1 inner 5040.2.k.f 8
12.b even 2 1 1260.2.f.b yes 8
15.d odd 2 1 inner 5040.2.k.f 8
20.d odd 2 1 1260.2.f.b yes 8
20.e even 4 2 6300.2.d.f 16
21.c even 2 1 5040.2.k.e 8
28.d even 2 1 1260.2.f.a 8
35.c odd 2 1 5040.2.k.e 8
60.h even 2 1 1260.2.f.a 8
60.l odd 4 2 6300.2.d.f 16
84.h odd 2 1 1260.2.f.b yes 8
105.g even 2 1 inner 5040.2.k.f 8
140.c even 2 1 1260.2.f.b yes 8
140.j odd 4 2 6300.2.d.f 16
420.o odd 2 1 1260.2.f.a 8
420.w even 4 2 6300.2.d.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.f.a 8 4.b odd 2 1
1260.2.f.a 8 28.d even 2 1
1260.2.f.a 8 60.h even 2 1
1260.2.f.a 8 420.o odd 2 1
1260.2.f.b yes 8 12.b even 2 1
1260.2.f.b yes 8 20.d odd 2 1
1260.2.f.b yes 8 84.h odd 2 1
1260.2.f.b yes 8 140.c even 2 1
5040.2.k.e 8 3.b odd 2 1
5040.2.k.e 8 5.b even 2 1
5040.2.k.e 8 21.c even 2 1
5040.2.k.e 8 35.c odd 2 1
5040.2.k.f 8 1.a even 1 1 trivial
5040.2.k.f 8 7.b odd 2 1 inner
5040.2.k.f 8 15.d odd 2 1 inner
5040.2.k.f 8 105.g even 2 1 inner
6300.2.d.f 16 20.e even 4 2
6300.2.d.f 16 60.l odd 4 2
6300.2.d.f 16 140.j odd 4 2
6300.2.d.f 16 420.w even 4 2

## Hecke kernels

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

 $$T_{11}^{2} + 14$$ $$T_{13}^{4} - 12 T_{13}^{2} + 8$$ $$T_{23}^{2} - 10 T_{23} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$625 + 22 T^{4} + T^{8}$$
$7$ $$2401 + 196 T^{2} - 10 T^{4} + 4 T^{6} + T^{8}$$
$11$ $$( 14 + T^{2} )^{4}$$
$13$ $$( 8 - 12 T^{2} + T^{4} )^{2}$$
$17$ $$( 8 + 12 T^{2} + T^{4} )^{2}$$
$19$ $$( 648 + 52 T^{2} + T^{4} )^{2}$$
$23$ $$( 18 - 10 T + T^{2} )^{4}$$
$29$ $$( 2 + T^{2} )^{4}$$
$31$ $$( 72 + 76 T^{2} + T^{4} )^{2}$$
$37$ $$( 144 + 32 T^{2} + T^{4} )^{2}$$
$41$ $$( 3528 - 140 T^{2} + T^{4} )^{2}$$
$43$ $$( 1296 + 128 T^{2} + T^{4} )^{2}$$
$47$ $$( 2592 + 104 T^{2} + T^{4} )^{2}$$
$53$ $$( 42 - 14 T + T^{2} )^{4}$$
$59$ $$( 288 - 152 T^{2} + T^{4} )^{2}$$
$61$ $$T^{8}$$
$67$ $$( 576 + 176 T^{2} + T^{4} )^{2}$$
$71$ $$( 14 + T^{2} )^{4}$$
$73$ $$( 72 - 76 T^{2} + T^{4} )^{2}$$
$79$ $$( -96 + 8 T + T^{2} )^{4}$$
$83$ $$( 1568 + 168 T^{2} + T^{4} )^{2}$$
$89$ $$( 72 - 20 T^{2} + T^{4} )^{2}$$
$97$ $$( 31752 - 364 T^{2} + T^{4} )^{2}$$