# Properties

 Label 4032.2.a.z Level $4032$ Weight $2$ Character orbit 4032.a Self dual yes Analytic conductor $32.196$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{7} + O(q^{10})$$ $$q + q^{7} + 4q^{11} + 4q^{13} + 2q^{17} - 6q^{19} + 8q^{23} - 5q^{25} + 2q^{29} + 4q^{31} - 10q^{37} + 10q^{41} + 4q^{43} + 4q^{47} + q^{49} - 2q^{53} - 10q^{59} + 8q^{61} - 8q^{67} - 6q^{73} + 4q^{77} + 16q^{79} - 2q^{83} - 18q^{89} + 4q^{91} - 2q^{97} + O(q^{100})$$

## 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

## Atkin-Lehner signs

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

## 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.z 1
3.b odd 2 1 448.2.a.b 1
4.b odd 2 1 4032.2.a.p 1
8.b even 2 1 2016.2.a.g 1
8.d odd 2 1 2016.2.a.e 1
12.b even 2 1 448.2.a.f 1
21.c even 2 1 3136.2.a.y 1
24.f even 2 1 224.2.a.a 1
24.h odd 2 1 224.2.a.b yes 1
48.i odd 4 2 1792.2.b.b 2
48.k even 4 2 1792.2.b.f 2
84.h odd 2 1 3136.2.a.f 1
120.i odd 2 1 5600.2.a.c 1
120.m even 2 1 5600.2.a.t 1
168.e odd 2 1 1568.2.a.h 1
168.i even 2 1 1568.2.a.b 1
168.s odd 6 2 1568.2.i.b 2
168.v even 6 2 1568.2.i.k 2
168.ba even 6 2 1568.2.i.j 2
168.be odd 6 2 1568.2.i.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.a 1 24.f even 2 1
224.2.a.b yes 1 24.h odd 2 1
448.2.a.b 1 3.b odd 2 1
448.2.a.f 1 12.b even 2 1
1568.2.a.b 1 168.i even 2 1
1568.2.a.h 1 168.e odd 2 1
1568.2.i.b 2 168.s odd 6 2
1568.2.i.c 2 168.be odd 6 2
1568.2.i.j 2 168.ba even 6 2
1568.2.i.k 2 168.v even 6 2
1792.2.b.b 2 48.i odd 4 2
1792.2.b.f 2 48.k even 4 2
2016.2.a.e 1 8.d odd 2 1
2016.2.a.g 1 8.b even 2 1
3136.2.a.f 1 84.h odd 2 1
3136.2.a.y 1 21.c even 2 1
4032.2.a.p 1 4.b odd 2 1
4032.2.a.z 1 1.a even 1 1 trivial
5600.2.a.c 1 120.i odd 2 1
5600.2.a.t 1 120.m even 2 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{11} - 4$$ $$T_{13} - 4$$ $$T_{17} - 2$$ $$T_{19} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$-4 + T$$
$13$ $$-4 + T$$
$17$ $$-2 + T$$
$19$ $$6 + T$$
$23$ $$-8 + T$$
$29$ $$-2 + T$$
$31$ $$-4 + T$$
$37$ $$10 + T$$
$41$ $$-10 + T$$
$43$ $$-4 + T$$
$47$ $$-4 + T$$
$53$ $$2 + T$$
$59$ $$10 + T$$
$61$ $$-8 + T$$
$67$ $$8 + T$$
$71$ $$T$$
$73$ $$6 + T$$
$79$ $$-16 + T$$
$83$ $$2 + T$$
$89$ $$18 + T$$
$97$ $$2 + T$$