# Properties

 Label 4032.2.p.f Level 4032 Weight 2 Character orbit 4032.p Analytic conductor 32.196 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4032.p (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 448) 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} +O(q^{10})$$ $$q + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{11} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{13} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{17} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{19} -3 q^{25} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{31} + ( \zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{35} + ( -4 + 8 \zeta_{24}^{4} ) q^{37} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{41} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{43} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{47} + ( 5 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} + ( -8 + 16 \zeta_{24}^{4} ) q^{53} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{55} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{59} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{61} -6 q^{65} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{67} -6 \zeta_{24}^{6} q^{71} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{73} + ( 2 + 6 \zeta_{24} + 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 6 \zeta_{24}^{5} ) q^{77} + 14 \zeta_{24}^{6} q^{79} + ( 7 \zeta_{24} - 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} ) q^{83} + ( 4 - 8 \zeta_{24}^{4} ) q^{85} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{89} + ( -3 \zeta_{24} - 12 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{91} + 6 \zeta_{24}^{6} q^{95} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 24q^{25} + 40q^{49} - 48q^{65} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i −0.258819 − 0.965926i 0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i −0.965926 − 0.258819i
0 0 0 −1.41421 0 −2.44949 1.00000i 0 0 0
1567.2 0 0 0 −1.41421 0 −2.44949 + 1.00000i 0 0 0
1567.3 0 0 0 −1.41421 0 2.44949 1.00000i 0 0 0
1567.4 0 0 0 −1.41421 0 2.44949 + 1.00000i 0 0 0
1567.5 0 0 0 1.41421 0 −2.44949 1.00000i 0 0 0
1567.6 0 0 0 1.41421 0 −2.44949 + 1.00000i 0 0 0
1567.7 0 0 0 1.41421 0 2.44949 1.00000i 0 0 0
1567.8 0 0 0 1.41421 0 2.44949 + 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1567.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.d even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.p.f 8
3.b odd 2 1 448.2.e.b 8
4.b odd 2 1 inner 4032.2.p.f 8
7.b odd 2 1 inner 4032.2.p.f 8
8.b even 2 1 inner 4032.2.p.f 8
8.d odd 2 1 inner 4032.2.p.f 8
12.b even 2 1 448.2.e.b 8
21.c even 2 1 448.2.e.b 8
24.f even 2 1 448.2.e.b 8
24.h odd 2 1 448.2.e.b 8
28.d even 2 1 inner 4032.2.p.f 8
48.i odd 4 2 1792.2.f.j 8
48.k even 4 2 1792.2.f.j 8
56.e even 2 1 inner 4032.2.p.f 8
56.h odd 2 1 inner 4032.2.p.f 8
84.h odd 2 1 448.2.e.b 8
168.e odd 2 1 448.2.e.b 8
168.i even 2 1 448.2.e.b 8
336.v odd 4 2 1792.2.f.j 8
336.y even 4 2 1792.2.f.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.e.b 8 3.b odd 2 1
448.2.e.b 8 12.b even 2 1
448.2.e.b 8 21.c even 2 1
448.2.e.b 8 24.f even 2 1
448.2.e.b 8 24.h odd 2 1
448.2.e.b 8 84.h odd 2 1
448.2.e.b 8 168.e odd 2 1
448.2.e.b 8 168.i even 2 1
1792.2.f.j 8 48.i odd 4 2
1792.2.f.j 8 48.k even 4 2
1792.2.f.j 8 336.v odd 4 2
1792.2.f.j 8 336.y even 4 2
4032.2.p.f 8 1.a even 1 1 trivial
4032.2.p.f 8 4.b odd 2 1 inner
4032.2.p.f 8 7.b odd 2 1 inner
4032.2.p.f 8 8.b even 2 1 inner
4032.2.p.f 8 8.d odd 2 1 inner
4032.2.p.f 8 28.d even 2 1 inner
4032.2.p.f 8 56.e even 2 1 inner
4032.2.p.f 8 56.h odd 2 1 inner

## Hecke kernels

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

 $$T_{5}^{2} - 2$$ $$T_{11}^{2} - 12$$ $$T_{13}^{2} - 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 8 T^{2} + 25 T^{4} )^{4}$$
$7$ $$( 1 - 10 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 10 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 + 8 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 - 10 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 - 20 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 23 T^{2} )^{8}$$
$29$ $$( 1 - 29 T^{2} )^{8}$$
$31$ $$( 1 + 38 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 - 10 T + 37 T^{2} )^{4}( 1 + 10 T + 37 T^{2} )^{4}$$
$41$ $$( 1 + 14 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 74 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 70 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 + 86 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 - 68 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 + 104 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 26 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 106 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 122 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 + 38 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 - 68 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 38 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 - 170 T^{2} + 9409 T^{4} )^{4}$$