# Properties

 Label 4032.2.a.bw Level 4032 Weight 2 Character orbit 4032.a Self dual yes Analytic conductor 32.196 Analytic rank 1 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 224) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{5} + q^{7} +O(q^{10})$$ $$q + 2 \beta q^{5} + q^{7} -4 \beta q^{11} + ( -4 + 2 \beta ) q^{13} + ( 2 - 4 \beta ) q^{17} + ( -2 + 2 \beta ) q^{19} -4 q^{23} + ( -1 + 4 \beta ) q^{25} + ( 2 - 4 \beta ) q^{29} + ( 4 - 4 \beta ) q^{31} + 2 \beta q^{35} + ( -2 + 4 \beta ) q^{37} + ( 2 + 4 \beta ) q^{41} -4 \beta q^{43} + ( -4 - 4 \beta ) q^{47} + q^{49} -10 q^{53} + ( -8 - 8 \beta ) q^{55} + ( -6 - 2 \beta ) q^{59} + ( -8 - 2 \beta ) q^{61} + ( 4 - 4 \beta ) q^{65} -4 q^{67} + ( -8 + 8 \beta ) q^{71} + ( 10 - 8 \beta ) q^{73} -4 \beta q^{77} + 8 \beta q^{79} + ( -6 - 2 \beta ) q^{83} + ( -8 - 4 \beta ) q^{85} + 6 q^{89} + ( -4 + 2 \beta ) q^{91} + 4 q^{95} + ( 6 + 4 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + 2q^{7} + O(q^{10})$$ $$2q + 2q^{5} + 2q^{7} - 4q^{11} - 6q^{13} - 2q^{19} - 8q^{23} + 2q^{25} + 4q^{31} + 2q^{35} + 8q^{41} - 4q^{43} - 12q^{47} + 2q^{49} - 20q^{53} - 24q^{55} - 14q^{59} - 18q^{61} + 4q^{65} - 8q^{67} - 8q^{71} + 12q^{73} - 4q^{77} + 8q^{79} - 14q^{83} - 20q^{85} + 12q^{89} - 6q^{91} + 8q^{95} + 16q^{97} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 0 0 −1.23607 0 1.00000 0 0 0
1.2 0 0 0 3.23607 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.bw 2
3.b odd 2 1 448.2.a.j 2
4.b odd 2 1 4032.2.a.bv 2
8.b even 2 1 2016.2.a.r 2
8.d odd 2 1 2016.2.a.o 2
12.b even 2 1 448.2.a.i 2
21.c even 2 1 3136.2.a.bf 2
24.f even 2 1 224.2.a.d yes 2
24.h odd 2 1 224.2.a.c 2
48.i odd 4 2 1792.2.b.k 4
48.k even 4 2 1792.2.b.m 4
84.h odd 2 1 3136.2.a.by 2
120.i odd 2 1 5600.2.a.bk 2
120.m even 2 1 5600.2.a.z 2
168.e odd 2 1 1568.2.a.k 2
168.i even 2 1 1568.2.a.v 2
168.s odd 6 2 1568.2.i.v 4
168.v even 6 2 1568.2.i.m 4
168.ba even 6 2 1568.2.i.n 4
168.be odd 6 2 1568.2.i.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 24.h odd 2 1
224.2.a.d yes 2 24.f even 2 1
448.2.a.i 2 12.b even 2 1
448.2.a.j 2 3.b odd 2 1
1568.2.a.k 2 168.e odd 2 1
1568.2.a.v 2 168.i even 2 1
1568.2.i.m 4 168.v even 6 2
1568.2.i.n 4 168.ba even 6 2
1568.2.i.v 4 168.s odd 6 2
1568.2.i.w 4 168.be odd 6 2
1792.2.b.k 4 48.i odd 4 2
1792.2.b.m 4 48.k even 4 2
2016.2.a.o 2 8.d odd 2 1
2016.2.a.r 2 8.b even 2 1
3136.2.a.bf 2 21.c even 2 1
3136.2.a.by 2 84.h odd 2 1
4032.2.a.bv 2 4.b odd 2 1
4032.2.a.bw 2 1.a even 1 1 trivial
5600.2.a.z 2 120.m even 2 1
5600.2.a.bk 2 120.i odd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4032))$$:

 $$T_{5}^{2} - 2 T_{5} - 4$$ $$T_{11}^{2} + 4 T_{11} - 16$$ $$T_{13}^{2} + 6 T_{13} + 4$$ $$T_{17}^{2} - 20$$ $$T_{19}^{2} + 2 T_{19} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T )^{2}$$
$11$ $$1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$1 + 6 T + 30 T^{2} + 78 T^{3} + 169 T^{4}$$
$17$ $$1 + 14 T^{2} + 289 T^{4}$$
$19$ $$1 + 2 T + 34 T^{2} + 38 T^{3} + 361 T^{4}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$1 + 38 T^{2} + 841 T^{4}$$
$31$ $$1 - 4 T + 46 T^{2} - 124 T^{3} + 961 T^{4}$$
$37$ $$1 + 54 T^{2} + 1369 T^{4}$$
$41$ $$1 - 8 T + 78 T^{2} - 328 T^{3} + 1681 T^{4}$$
$43$ $$1 + 4 T + 70 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 + 12 T + 110 T^{2} + 564 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 10 T + 53 T^{2} )^{2}$$
$59$ $$1 + 14 T + 162 T^{2} + 826 T^{3} + 3481 T^{4}$$
$61$ $$1 + 18 T + 198 T^{2} + 1098 T^{3} + 3721 T^{4}$$
$67$ $$( 1 + 4 T + 67 T^{2} )^{2}$$
$71$ $$1 + 8 T + 78 T^{2} + 568 T^{3} + 5041 T^{4}$$
$73$ $$1 - 12 T + 102 T^{2} - 876 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$1 + 14 T + 210 T^{2} + 1162 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 6 T + 89 T^{2} )^{2}$$
$97$ $$1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4}$$