# Properties

 Label 4032.2.a.bq Level $4032$ Weight $2$ Character orbit 4032.a Self dual yes Analytic conductor $32.196$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4032.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} - q^{7} +O(q^{10})$$ $$q + \beta q^{5} - q^{7} + \beta q^{11} -2 q^{13} + \beta q^{17} -4 q^{19} + \beta q^{23} + 7 q^{25} + 4 q^{31} -\beta q^{35} -2 q^{37} + 3 \beta q^{41} -4 q^{43} -2 \beta q^{47} + q^{49} + 2 \beta q^{53} + 12 q^{55} -2 \beta q^{59} + 10 q^{61} -2 \beta q^{65} -4 q^{67} + 3 \beta q^{71} + 14 q^{73} -\beta q^{77} -8 q^{79} + 12 q^{85} -\beta q^{89} + 2 q^{91} -4 \beta q^{95} + 14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{7} + O(q^{10})$$ $$2q - 2q^{7} - 4q^{13} - 8q^{19} + 14q^{25} + 8q^{31} - 4q^{37} - 8q^{43} + 2q^{49} + 24q^{55} + 20q^{61} - 8q^{67} + 28q^{73} - 16q^{79} + 24q^{85} + 4q^{91} + 28q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
0 0 0 −3.46410 0 −1.00000 0 0 0
1.2 0 0 0 3.46410 0 −1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.a.bq 2
3.b odd 2 1 inner 4032.2.a.bq 2
4.b odd 2 1 4032.2.a.bt 2
8.b even 2 1 1008.2.a.n 2
8.d odd 2 1 63.2.a.b 2
12.b even 2 1 4032.2.a.bt 2
24.f even 2 1 63.2.a.b 2
24.h odd 2 1 1008.2.a.n 2
40.e odd 2 1 1575.2.a.q 2
40.k even 4 2 1575.2.d.i 4
56.e even 2 1 441.2.a.g 2
56.h odd 2 1 7056.2.a.cm 2
56.k odd 6 2 441.2.e.j 4
56.m even 6 2 441.2.e.i 4
72.l even 6 2 567.2.f.j 4
72.p odd 6 2 567.2.f.j 4
88.g even 2 1 7623.2.a.bi 2
120.m even 2 1 1575.2.a.q 2
120.q odd 4 2 1575.2.d.i 4
168.e odd 2 1 441.2.a.g 2
168.i even 2 1 7056.2.a.cm 2
168.v even 6 2 441.2.e.j 4
168.be odd 6 2 441.2.e.i 4
264.p odd 2 1 7623.2.a.bi 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b 2 8.d odd 2 1
63.2.a.b 2 24.f even 2 1
441.2.a.g 2 56.e even 2 1
441.2.a.g 2 168.e odd 2 1
441.2.e.i 4 56.m even 6 2
441.2.e.i 4 168.be odd 6 2
441.2.e.j 4 56.k odd 6 2
441.2.e.j 4 168.v even 6 2
567.2.f.j 4 72.l even 6 2
567.2.f.j 4 72.p odd 6 2
1008.2.a.n 2 8.b even 2 1
1008.2.a.n 2 24.h odd 2 1
1575.2.a.q 2 40.e odd 2 1
1575.2.a.q 2 120.m even 2 1
1575.2.d.i 4 40.k even 4 2
1575.2.d.i 4 120.q odd 4 2
4032.2.a.bq 2 1.a even 1 1 trivial
4032.2.a.bq 2 3.b odd 2 1 inner
4032.2.a.bt 2 4.b odd 2 1
4032.2.a.bt 2 12.b even 2 1
7056.2.a.cm 2 56.h odd 2 1
7056.2.a.cm 2 168.i even 2 1
7623.2.a.bi 2 88.g even 2 1
7623.2.a.bi 2 264.p odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4032))$$:

 $$T_{5}^{2} - 12$$ $$T_{11}^{2} - 12$$ $$T_{13} + 2$$ $$T_{17}^{2} - 12$$ $$T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-12 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-12 + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$-12 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$-108 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$-48 + T^{2}$$
$59$ $$-48 + T^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$-108 + T^{2}$$
$73$ $$( -14 + T )^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$-12 + T^{2}$$
$97$ $$( -14 + T )^{2}$$