# Properties

 Label 4032.2.b.b Level 4032 Weight 2 Character orbit 4032.b 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - 4 \zeta_{6} ) q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 - 4 \zeta_{6} ) q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + ( -2 + 4 \zeta_{6} ) q^{11} + ( 2 - 4 \zeta_{6} ) q^{13} -2 q^{19} + ( -2 + 4 \zeta_{6} ) q^{23} -7 q^{25} + 6 q^{29} -8 q^{31} + ( 2 + 8 \zeta_{6} ) q^{35} + 2 q^{37} + ( -4 + 8 \zeta_{6} ) q^{41} + ( -6 + 12 \zeta_{6} ) q^{43} + ( 5 - 8 \zeta_{6} ) q^{49} + 6 q^{53} + 12 q^{55} + 6 q^{59} + ( 2 - 4 \zeta_{6} ) q^{61} -12 q^{65} + ( 2 - 4 \zeta_{6} ) q^{67} + ( 2 - 4 \zeta_{6} ) q^{71} + ( -4 + 8 \zeta_{6} ) q^{73} + ( -2 - 8 \zeta_{6} ) q^{77} + ( -2 + 4 \zeta_{6} ) q^{79} -6 q^{83} + ( -4 + 8 \zeta_{6} ) q^{89} + ( 2 + 8 \zeta_{6} ) q^{91} + ( -4 + 8 \zeta_{6} ) q^{95} + ( -8 + 16 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{7} + O(q^{10})$$ $$2q - 4q^{7} - 4q^{19} - 14q^{25} + 12q^{29} - 16q^{31} + 12q^{35} + 4q^{37} + 2q^{49} + 12q^{53} + 24q^{55} + 12q^{59} - 24q^{65} - 12q^{77} - 12q^{83} + 12q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$1793$$ $$3781$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3583.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 3.46410i 0 −2.00000 + 1.73205i 0 0 0
3583.2 0 0 0 3.46410i 0 −2.00000 1.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.b.b 2
3.b odd 2 1 448.2.f.a 2
4.b odd 2 1 4032.2.b.h 2
7.b odd 2 1 4032.2.b.h 2
8.b even 2 1 1008.2.b.b 2
8.d odd 2 1 1008.2.b.g 2
12.b even 2 1 448.2.f.c 2
21.c even 2 1 448.2.f.c 2
24.f even 2 1 112.2.f.a 2
24.h odd 2 1 112.2.f.b yes 2
28.d even 2 1 inner 4032.2.b.b 2
48.i odd 4 2 1792.2.e.c 4
48.k even 4 2 1792.2.e.a 4
56.e even 2 1 1008.2.b.b 2
56.h odd 2 1 1008.2.b.g 2
84.h odd 2 1 448.2.f.a 2
120.i odd 2 1 2800.2.k.b 2
120.m even 2 1 2800.2.k.e 2
120.q odd 4 2 2800.2.e.c 4
120.w even 4 2 2800.2.e.b 4
168.e odd 2 1 112.2.f.b yes 2
168.i even 2 1 112.2.f.a 2
168.s odd 6 1 784.2.p.a 2
168.s odd 6 1 784.2.p.b 2
168.v even 6 1 784.2.p.e 2
168.v even 6 1 784.2.p.f 2
168.ba even 6 1 784.2.p.e 2
168.ba even 6 1 784.2.p.f 2
168.be odd 6 1 784.2.p.a 2
168.be odd 6 1 784.2.p.b 2
336.v odd 4 2 1792.2.e.c 4
336.y even 4 2 1792.2.e.a 4
840.b odd 2 1 2800.2.k.b 2
840.u even 2 1 2800.2.k.e 2
840.bm even 4 2 2800.2.e.b 4
840.bp odd 4 2 2800.2.e.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.f.a 2 24.f even 2 1
112.2.f.a 2 168.i even 2 1
112.2.f.b yes 2 24.h odd 2 1
112.2.f.b yes 2 168.e odd 2 1
448.2.f.a 2 3.b odd 2 1
448.2.f.a 2 84.h odd 2 1
448.2.f.c 2 12.b even 2 1
448.2.f.c 2 21.c even 2 1
784.2.p.a 2 168.s odd 6 1
784.2.p.a 2 168.be odd 6 1
784.2.p.b 2 168.s odd 6 1
784.2.p.b 2 168.be odd 6 1
784.2.p.e 2 168.v even 6 1
784.2.p.e 2 168.ba even 6 1
784.2.p.f 2 168.v even 6 1
784.2.p.f 2 168.ba even 6 1
1008.2.b.b 2 8.b even 2 1
1008.2.b.b 2 56.e even 2 1
1008.2.b.g 2 8.d odd 2 1
1008.2.b.g 2 56.h odd 2 1
1792.2.e.a 4 48.k even 4 2
1792.2.e.a 4 336.y even 4 2
1792.2.e.c 4 48.i odd 4 2
1792.2.e.c 4 336.v odd 4 2
2800.2.e.b 4 120.w even 4 2
2800.2.e.b 4 840.bm even 4 2
2800.2.e.c 4 120.q odd 4 2
2800.2.e.c 4 840.bp odd 4 2
2800.2.k.b 2 120.i odd 2 1
2800.2.k.b 2 840.b odd 2 1
2800.2.k.e 2 120.m even 2 1
2800.2.k.e 2 840.u even 2 1
4032.2.b.b 2 1.a even 1 1 trivial
4032.2.b.b 2 28.d even 2 1 inner
4032.2.b.h 2 4.b odd 2 1
4032.2.b.h 2 7.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 12$$ $$T_{11}^{2} + 12$$ $$T_{19} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 2 T^{2} + 25 T^{4}$$
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 - 10 T^{2} + 121 T^{4}$$
$13$ $$1 - 14 T^{2} + 169 T^{4}$$
$17$ $$( 1 - 17 T^{2} )^{2}$$
$19$ $$( 1 + 2 T + 19 T^{2} )^{2}$$
$23$ $$1 - 34 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 8 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 34 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} )$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 - 6 T + 53 T^{2} )^{2}$$
$59$ $$( 1 - 6 T + 59 T^{2} )^{2}$$
$61$ $$1 - 110 T^{2} + 3721 T^{4}$$
$67$ $$( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$
$71$ $$1 - 130 T^{2} + 5041 T^{4}$$
$73$ $$1 - 98 T^{2} + 5329 T^{4}$$
$79$ $$1 - 146 T^{2} + 6241 T^{4}$$
$83$ $$( 1 + 6 T + 83 T^{2} )^{2}$$
$89$ $$1 - 130 T^{2} + 7921 T^{4}$$
$97$ $$( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$