# Properties

 Label 4032.2.a.m 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 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{5} + q^{7} + O(q^{10})$$ $$q - 2 q^{5} + q^{7} + 4 q^{11} - 6 q^{13} - 2 q^{17} - 4 q^{19} + 8 q^{23} - q^{25} - 2 q^{29} - 2 q^{35} + 10 q^{37} + 6 q^{41} - 4 q^{43} + q^{49} + 6 q^{53} - 8 q^{55} - 4 q^{59} - 6 q^{61} + 12 q^{65} + 4 q^{67} + 8 q^{71} + 10 q^{73} + 4 q^{77} + 4 q^{83} + 4 q^{85} + 6 q^{89} - 6 q^{91} + 8 q^{95} - 14 q^{97} + O(q^{100})$$

## 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 −2.00000 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.m 1
3.b odd 2 1 1344.2.a.i 1
4.b odd 2 1 4032.2.a.e 1
8.b even 2 1 1008.2.a.j 1
8.d odd 2 1 126.2.a.a 1
12.b even 2 1 1344.2.a.q 1
21.c even 2 1 9408.2.a.bw 1
24.f even 2 1 42.2.a.a 1
24.h odd 2 1 336.2.a.d 1
40.e odd 2 1 3150.2.a.bo 1
40.k even 4 2 3150.2.g.r 2
48.i odd 4 2 5376.2.c.e 2
48.k even 4 2 5376.2.c.bc 2
56.e even 2 1 882.2.a.b 1
56.h odd 2 1 7056.2.a.k 1
56.k odd 6 2 882.2.g.h 2
56.m even 6 2 882.2.g.j 2
72.l even 6 2 1134.2.f.g 2
72.p odd 6 2 1134.2.f.j 2
84.h odd 2 1 9408.2.a.n 1
120.i odd 2 1 8400.2.a.k 1
120.m even 2 1 1050.2.a.i 1
120.q odd 4 2 1050.2.g.a 2
168.e odd 2 1 294.2.a.g 1
168.i even 2 1 2352.2.a.l 1
168.s odd 6 2 2352.2.q.i 2
168.v even 6 2 294.2.e.c 2
168.ba even 6 2 2352.2.q.n 2
168.be odd 6 2 294.2.e.a 2
264.p odd 2 1 5082.2.a.d 1
312.h even 2 1 7098.2.a.f 1
840.b odd 2 1 7350.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 24.f even 2 1
126.2.a.a 1 8.d odd 2 1
294.2.a.g 1 168.e odd 2 1
294.2.e.a 2 168.be odd 6 2
294.2.e.c 2 168.v even 6 2
336.2.a.d 1 24.h odd 2 1
882.2.a.b 1 56.e even 2 1
882.2.g.h 2 56.k odd 6 2
882.2.g.j 2 56.m even 6 2
1008.2.a.j 1 8.b even 2 1
1050.2.a.i 1 120.m even 2 1
1050.2.g.a 2 120.q odd 4 2
1134.2.f.g 2 72.l even 6 2
1134.2.f.j 2 72.p odd 6 2
1344.2.a.i 1 3.b odd 2 1
1344.2.a.q 1 12.b even 2 1
2352.2.a.l 1 168.i even 2 1
2352.2.q.i 2 168.s odd 6 2
2352.2.q.n 2 168.ba even 6 2
3150.2.a.bo 1 40.e odd 2 1
3150.2.g.r 2 40.k even 4 2
4032.2.a.e 1 4.b odd 2 1
4032.2.a.m 1 1.a even 1 1 trivial
5082.2.a.d 1 264.p odd 2 1
5376.2.c.e 2 48.i odd 4 2
5376.2.c.bc 2 48.k even 4 2
7056.2.a.k 1 56.h odd 2 1
7098.2.a.f 1 312.h even 2 1
7350.2.a.f 1 840.b odd 2 1
8400.2.a.k 1 120.i odd 2 1
9408.2.a.n 1 84.h odd 2 1
9408.2.a.bw 1 21.c even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$2 + T$$
$7$ $$-1 + T$$
$11$ $$-4 + T$$
$13$ $$6 + T$$
$17$ $$2 + T$$
$19$ $$4 + T$$
$23$ $$-8 + T$$
$29$ $$2 + T$$
$31$ $$T$$
$37$ $$-10 + T$$
$41$ $$-6 + T$$
$43$ $$4 + T$$
$47$ $$T$$
$53$ $$-6 + T$$
$59$ $$4 + T$$
$61$ $$6 + T$$
$67$ $$-4 + T$$
$71$ $$-8 + T$$
$73$ $$-10 + T$$
$79$ $$T$$
$83$ $$-4 + T$$
$89$ $$-6 + T$$
$97$ $$14 + T$$