# Properties

 Label 4032.2.a.x Level 4032 Weight 2 Character orbit 4032.a Self dual yes Analytic conductor 32.196 Analytic rank 1 Dimension 1 CM no Inner twists 1

# 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{7} + O(q^{10})$$ $$q + q^{7} + 2q^{11} + 2q^{13} - 4q^{17} - 4q^{19} - 6q^{23} - 5q^{25} - 2q^{29} + 6q^{37} - 8q^{41} - 8q^{43} - 4q^{47} + q^{49} - 6q^{53} + 14q^{61} + 4q^{67} - 2q^{71} - 2q^{73} + 2q^{77} - 4q^{79} - 12q^{83} + 2q^{91} + 6q^{97} + O(q^{100})$$

## 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
 0
0 0 0 0 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

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.a.x 1
3.b odd 2 1 1344.2.a.e 1
4.b odd 2 1 4032.2.a.q 1
8.b even 2 1 2016.2.a.i 1
8.d odd 2 1 2016.2.a.d 1
12.b even 2 1 1344.2.a.p 1
21.c even 2 1 9408.2.a.ch 1
24.f even 2 1 672.2.a.c 1
24.h odd 2 1 672.2.a.g yes 1
48.i odd 4 2 5376.2.c.j 2
48.k even 4 2 5376.2.c.y 2
84.h odd 2 1 9408.2.a.w 1
168.e odd 2 1 4704.2.a.z 1
168.i even 2 1 4704.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.c 1 24.f even 2 1
672.2.a.g yes 1 24.h odd 2 1
1344.2.a.e 1 3.b odd 2 1
1344.2.a.p 1 12.b even 2 1
2016.2.a.d 1 8.d odd 2 1
2016.2.a.i 1 8.b even 2 1
4032.2.a.q 1 4.b odd 2 1
4032.2.a.x 1 1.a even 1 1 trivial
4704.2.a.k 1 168.i even 2 1
4704.2.a.z 1 168.e odd 2 1
5376.2.c.j 2 48.i odd 4 2
5376.2.c.y 2 48.k even 4 2
9408.2.a.w 1 84.h odd 2 1
9408.2.a.ch 1 21.c even 2 1

## Atkin-Lehner signs

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 5 T^{2}$$
$7$ $$1 - T$$
$11$ $$1 - 2 T + 11 T^{2}$$
$13$ $$1 - 2 T + 13 T^{2}$$
$17$ $$1 + 4 T + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 6 T + 23 T^{2}$$
$29$ $$1 + 2 T + 29 T^{2}$$
$31$ $$1 + 31 T^{2}$$
$37$ $$1 - 6 T + 37 T^{2}$$
$41$ $$1 + 8 T + 41 T^{2}$$
$43$ $$1 + 8 T + 43 T^{2}$$
$47$ $$1 + 4 T + 47 T^{2}$$
$53$ $$1 + 6 T + 53 T^{2}$$
$59$ $$1 + 59 T^{2}$$
$61$ $$1 - 14 T + 61 T^{2}$$
$67$ $$1 - 4 T + 67 T^{2}$$
$71$ $$1 + 2 T + 71 T^{2}$$
$73$ $$1 + 2 T + 73 T^{2}$$
$79$ $$1 + 4 T + 79 T^{2}$$
$83$ $$1 + 12 T + 83 T^{2}$$
$89$ $$1 + 89 T^{2}$$
$97$ $$1 - 6 T + 97 T^{2}$$