# Properties

 Label 4032.2.p.h Level 4032 Weight 2 Character orbit 4032.p Analytic conductor 32.196 Analytic rank 0 Dimension 8 CM discriminant -56 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: 8.0.629407744.1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 448) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} -\beta_{4} q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{5} -\beta_{4} q^{7} + ( -\beta_{1} + \beta_{6} ) q^{13} + \beta_{5} q^{19} -3 \beta_{2} q^{23} + ( 5 - \beta_{7} ) q^{25} + ( 2 \beta_{3} + \beta_{5} ) q^{35} -7 q^{49} + ( 4 \beta_{3} - \beta_{5} ) q^{59} + ( \beta_{1} + 2 \beta_{6} ) q^{61} + ( 6 - 3 \beta_{7} ) q^{65} -6 \beta_{4} q^{71} -2 \beta_{4} q^{79} + ( -\beta_{3} - 2 \beta_{5} ) q^{83} + ( -3 \beta_{3} + 2 \beta_{5} ) q^{91} + ( 9 \beta_{2} - 6 \beta_{4} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 40q^{25} - 56q^{49} + 48q^{65} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} - 10 \nu^{3} + 28 \nu$$$$)/12$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{3} - 4 \nu$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} - \nu^{4} + 4 \nu^{2} - 7$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{5} + 2 \nu^{3} + 4 \nu$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} + 2 \nu^{3} + 10 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} + 2 \nu^{4} + 2 \nu^{2} + 4$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + 2 \beta_{4} - \beta_{2} + 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{6} + \beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{7} + 3 \beta_{2}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{6} - 3 \beta_{5} + 5 \beta_{3} + 2 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{4} + 5 \beta_{2} + 10$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-\beta_{6} + \beta_{5} + 7 \beta_{3} + 4 \beta_{1}$$$$)/2$$

## 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
 0.767178 + 1.18804i 0.767178 − 1.18804i 1.38255 + 0.297594i 1.38255 − 0.297594i −1.38255 − 0.297594i −1.38255 + 0.297594i −0.767178 − 1.18804i −0.767178 + 1.18804i
0 0 0 −3.91044 0 2.64575i 0 0 0
1567.2 0 0 0 −3.91044 0 2.64575i 0 0 0
1567.3 0 0 0 −2.16991 0 2.64575i 0 0 0
1567.4 0 0 0 −2.16991 0 2.64575i 0 0 0
1567.5 0 0 0 2.16991 0 2.64575i 0 0 0
1567.6 0 0 0 2.16991 0 2.64575i 0 0 0
1567.7 0 0 0 3.91044 0 2.64575i 0 0 0
1567.8 0 0 0 3.91044 0 2.64575i 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
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
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

## Twists

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

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

## Hecke kernels

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

 $$T_{5}^{4} - 20 T_{5}^{2} + 72$$ $$T_{11}$$ $$T_{13}^{4} - 52 T_{13}^{2} + 648$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 22 T^{4} + 625 T^{8} )^{2}$$
$7$ $$( 1 + 7 T^{2} )^{4}$$
$11$ $$( 1 + 11 T^{2} )^{8}$$
$13$ $$( 1 + 310 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 17 T^{2} )^{8}$$
$19$ $$( 1 - 650 T^{4} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 10 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 29 T^{2} )^{8}$$
$31$ $$( 1 + 31 T^{2} )^{8}$$
$37$ $$( 1 - 37 T^{2} )^{8}$$
$41$ $$( 1 - 41 T^{2} )^{8}$$
$43$ $$( 1 + 43 T^{2} )^{8}$$
$47$ $$( 1 + 47 T^{2} )^{8}$$
$53$ $$( 1 - 53 T^{2} )^{8}$$
$59$ $$( 1 - 1130 T^{4} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 7370 T^{4} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 67 T^{2} )^{8}$$
$71$ $$( 1 + 110 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 73 T^{2} )^{8}$$
$79$ $$( 1 - 130 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 - 13130 T^{4} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 89 T^{2} )^{8}$$
$97$ $$( 1 - 97 T^{2} )^{8}$$