# Properties

 Label 5040.2.k.b Level 5040 Weight 2 Character orbit 5040.k Analytic conductor 40.245 Analytic rank 0 Dimension 4 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: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{5} + ( \beta_{1} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q -\beta_{3} q^{5} + ( \beta_{1} - \beta_{3} ) q^{7} -4 \beta_{1} q^{11} -2 \beta_{3} q^{13} -\beta_{2} q^{17} -\beta_{2} q^{19} -4 q^{23} + 5 q^{25} + 2 \beta_{1} q^{29} -2 \beta_{2} q^{31} + ( 5 - \beta_{2} ) q^{35} + 7 \beta_{1} q^{37} -2 \beta_{3} q^{41} -\beta_{1} q^{43} -3 \beta_{2} q^{47} + ( 3 - 2 \beta_{2} ) q^{49} + 4 q^{53} + 4 \beta_{2} q^{55} + 2 \beta_{3} q^{59} + 3 \beta_{2} q^{61} + 10 q^{65} -5 \beta_{1} q^{67} -\beta_{1} q^{71} -6 \beta_{3} q^{73} + ( 8 + 4 \beta_{2} ) q^{77} -6 q^{79} + 4 \beta_{2} q^{83} + 5 \beta_{1} q^{85} + 2 \beta_{3} q^{89} + ( 10 - 2 \beta_{2} ) q^{91} + 5 \beta_{1} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 16q^{23} + 20q^{25} + 20q^{35} + 12q^{49} + 16q^{53} + 40q^{65} + 32q^{77} - 24q^{79} + 40q^{91} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 8 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 3$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{2} + 4 \beta_{1}$$

## 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
 − 0.874032i 0.874032i 2.28825i − 2.28825i
0 0 0 −2.23607 0 −2.23607 1.41421i 0 0 0
1889.2 0 0 0 −2.23607 0 −2.23607 + 1.41421i 0 0 0
1889.3 0 0 0 2.23607 0 2.23607 1.41421i 0 0 0
1889.4 0 0 0 2.23607 0 2.23607 + 1.41421i 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
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.b 4
3.b odd 2 1 5040.2.k.c 4
4.b odd 2 1 630.2.d.b 4
5.b even 2 1 5040.2.k.c 4
7.b odd 2 1 inner 5040.2.k.b 4
12.b even 2 1 630.2.d.c yes 4
15.d odd 2 1 inner 5040.2.k.b 4
20.d odd 2 1 630.2.d.c yes 4
20.e even 4 2 3150.2.b.b 8
21.c even 2 1 5040.2.k.c 4
28.d even 2 1 630.2.d.b 4
35.c odd 2 1 5040.2.k.c 4
60.h even 2 1 630.2.d.b 4
60.l odd 4 2 3150.2.b.b 8
84.h odd 2 1 630.2.d.c yes 4
105.g even 2 1 inner 5040.2.k.b 4
140.c even 2 1 630.2.d.c yes 4
140.j odd 4 2 3150.2.b.b 8
420.o odd 2 1 630.2.d.b 4
420.w even 4 2 3150.2.b.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.d.b 4 4.b odd 2 1
630.2.d.b 4 28.d even 2 1
630.2.d.b 4 60.h even 2 1
630.2.d.b 4 420.o odd 2 1
630.2.d.c yes 4 12.b even 2 1
630.2.d.c yes 4 20.d odd 2 1
630.2.d.c yes 4 84.h odd 2 1
630.2.d.c yes 4 140.c even 2 1
3150.2.b.b 8 20.e even 4 2
3150.2.b.b 8 60.l odd 4 2
3150.2.b.b 8 140.j odd 4 2
3150.2.b.b 8 420.w even 4 2
5040.2.k.b 4 1.a even 1 1 trivial
5040.2.k.b 4 7.b odd 2 1 inner
5040.2.k.b 4 15.d odd 2 1 inner
5040.2.k.b 4 105.g even 2 1 inner
5040.2.k.c 4 3.b odd 2 1
5040.2.k.c 4 5.b even 2 1
5040.2.k.c 4 21.c even 2 1
5040.2.k.c 4 35.c odd 2 1

## Hecke kernels

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

 $$T_{11}^{2} + 32$$ $$T_{13}^{2} - 20$$ $$T_{23} + 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 5 T^{2} )^{2}$$
$7$ $$1 - 6 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 10 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 6 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 24 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 28 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{4}$$
$29$ $$( 1 - 50 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 22 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 24 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 62 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 84 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 4 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 4 T + 53 T^{2} )^{4}$$
$59$ $$( 1 + 98 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 32 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 84 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 140 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 34 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 6 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 6 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 158 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 97 T^{2} )^{4}$$