# Properties

 Label 4032.2.p.a Level 4032 Weight 2 Character orbit 4032.p Analytic conductor 32.196 Analytic rank 0 Dimension 4 CM no Inner twists 4

# 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1344) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{7} + ( -\beta_{1} - 2 \beta_{2} ) q^{11} -4 q^{13} -\beta_{3} q^{17} + 2 \beta_{1} q^{19} + 3 \beta_{1} q^{23} -5 q^{25} -\beta_{3} q^{29} + ( \beta_{1} + 2 \beta_{2} ) q^{31} + 2 \beta_{3} q^{37} -\beta_{3} q^{41} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{43} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{47} + ( 5 - \beta_{3} ) q^{49} -\beta_{3} q^{53} -6 \beta_{1} q^{59} + 4 q^{61} -3 \beta_{1} q^{71} + ( 12 - \beta_{3} ) q^{77} + \beta_{1} q^{79} + 6 \beta_{1} q^{83} -\beta_{3} q^{89} + 4 \beta_{2} q^{91} -2 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 16q^{13} - 20q^{25} + 20q^{49} + 16q^{61} + 48q^{77} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2}$$$$/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - \nu^{2} + 3 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + 6 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{1}$$$$/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{3} - 6 \beta_{2} - 3 \beta_{1}$$$$)/4$$

## 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
 1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i −1.22474 − 1.22474i
0 0 0 0 0 −2.44949 1.00000i 0 0 0
1567.2 0 0 0 0 0 −2.44949 + 1.00000i 0 0 0
1567.3 0 0 0 0 0 2.44949 1.00000i 0 0 0
1567.4 0 0 0 0 0 2.44949 + 1.00000i 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
4.b odd 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.a 4
3.b odd 2 1 1344.2.p.a 4
4.b odd 2 1 inner 4032.2.p.a 4
7.b odd 2 1 4032.2.p.d 4
8.b even 2 1 4032.2.p.d 4
8.d odd 2 1 4032.2.p.d 4
12.b even 2 1 1344.2.p.a 4
21.c even 2 1 1344.2.p.b yes 4
24.f even 2 1 1344.2.p.b yes 4
24.h odd 2 1 1344.2.p.b yes 4
28.d even 2 1 4032.2.p.d 4
56.e even 2 1 inner 4032.2.p.a 4
56.h odd 2 1 inner 4032.2.p.a 4
84.h odd 2 1 1344.2.p.b yes 4
168.e odd 2 1 1344.2.p.a 4
168.i even 2 1 1344.2.p.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.p.a 4 3.b odd 2 1
1344.2.p.a 4 12.b even 2 1
1344.2.p.a 4 168.e odd 2 1
1344.2.p.a 4 168.i even 2 1
1344.2.p.b yes 4 21.c even 2 1
1344.2.p.b yes 4 24.f even 2 1
1344.2.p.b yes 4 24.h odd 2 1
1344.2.p.b yes 4 84.h odd 2 1
4032.2.p.a 4 1.a even 1 1 trivial
4032.2.p.a 4 4.b odd 2 1 inner
4032.2.p.a 4 56.e even 2 1 inner
4032.2.p.a 4 56.h odd 2 1 inner
4032.2.p.d 4 7.b odd 2 1
4032.2.p.d 4 8.b even 2 1
4032.2.p.d 4 8.d odd 2 1
4032.2.p.d 4 28.d even 2 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}^{2} - 24$$ $$T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 5 T^{2} )^{4}$$
$7$ $$1 - 10 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 4 T + 13 T^{2} )^{4}$$
$17$ $$( 1 - 10 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 22 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 10 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 34 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 38 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 22 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 58 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 10 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 2 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 82 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 26 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 4 T + 61 T^{2} )^{4}$$
$67$ $$( 1 + 67 T^{2} )^{4}$$
$71$ $$( 1 - 106 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 73 T^{2} )^{4}$$
$79$ $$( 1 - 154 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 22 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 154 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 98 T^{2} + 9409 T^{4} )^{2}$$