# Properties

 Label 4032.2.k.b Level 4032 Weight 2 Character orbit 4032.k Analytic conductor 32.196 Analytic rank 0 Dimension 4 CM discriminant -7 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4032.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{7} + ( -\beta_{1} + \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{3} ) q^{23} -5 q^{25} + ( -2 \beta_{1} + \beta_{3} ) q^{29} + 4 \beta_{2} q^{37} + 2 \beta_{2} q^{43} + 7 q^{49} + ( -2 \beta_{1} - \beta_{3} ) q^{53} -4 q^{67} + ( \beta_{1} - 3 \beta_{3} ) q^{71} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{77} -8 q^{79} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 20q^{25} + 28q^{49} - 16q^{67} - 32q^{79} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} - 16 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} - 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{3} - 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-16 \beta_{3} + 15 \beta_{1}$$$$)/6$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3905.1
 − 1.16372i 1.16372i − 2.57794i 2.57794i
0 0 0 0 0 −2.64575 0 0 0
3905.2 0 0 0 0 0 −2.64575 0 0 0
3905.3 0 0 0 0 0 2.64575 0 0 0
3905.4 0 0 0 0 0 2.64575 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
21.c 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.k.b 4
3.b odd 2 1 inner 4032.2.k.b 4
4.b odd 2 1 4032.2.k.c 4
7.b odd 2 1 CM 4032.2.k.b 4
8.b even 2 1 1008.2.k.a 4
8.d odd 2 1 63.2.c.a 4
12.b even 2 1 4032.2.k.c 4
21.c even 2 1 inner 4032.2.k.b 4
24.f even 2 1 63.2.c.a 4
24.h odd 2 1 1008.2.k.a 4
28.d even 2 1 4032.2.k.c 4
40.e odd 2 1 1575.2.b.a 4
40.k even 4 2 1575.2.g.d 8
56.e even 2 1 63.2.c.a 4
56.h odd 2 1 1008.2.k.a 4
56.k odd 6 2 441.2.p.b 8
56.m even 6 2 441.2.p.b 8
72.l even 6 2 567.2.o.f 8
72.p odd 6 2 567.2.o.f 8
84.h odd 2 1 4032.2.k.c 4
120.m even 2 1 1575.2.b.a 4
120.q odd 4 2 1575.2.g.d 8
168.e odd 2 1 63.2.c.a 4
168.i even 2 1 1008.2.k.a 4
168.v even 6 2 441.2.p.b 8
168.be odd 6 2 441.2.p.b 8
280.n even 2 1 1575.2.b.a 4
280.y odd 4 2 1575.2.g.d 8
504.be even 6 2 567.2.o.f 8
504.co odd 6 2 567.2.o.f 8
840.b odd 2 1 1575.2.b.a 4
840.bm even 4 2 1575.2.g.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.c.a 4 8.d odd 2 1
63.2.c.a 4 24.f even 2 1
63.2.c.a 4 56.e even 2 1
63.2.c.a 4 168.e odd 2 1
441.2.p.b 8 56.k odd 6 2
441.2.p.b 8 56.m even 6 2
441.2.p.b 8 168.v even 6 2
441.2.p.b 8 168.be odd 6 2
567.2.o.f 8 72.l even 6 2
567.2.o.f 8 72.p odd 6 2
567.2.o.f 8 504.be even 6 2
567.2.o.f 8 504.co odd 6 2
1008.2.k.a 4 8.b even 2 1
1008.2.k.a 4 24.h odd 2 1
1008.2.k.a 4 56.h odd 2 1
1008.2.k.a 4 168.i even 2 1
1575.2.b.a 4 40.e odd 2 1
1575.2.b.a 4 120.m even 2 1
1575.2.b.a 4 280.n even 2 1
1575.2.b.a 4 840.b odd 2 1
1575.2.g.d 8 40.k even 4 2
1575.2.g.d 8 120.q odd 4 2
1575.2.g.d 8 280.y odd 4 2
1575.2.g.d 8 840.bm even 4 2
4032.2.k.b 4 1.a even 1 1 trivial
4032.2.k.b 4 3.b odd 2 1 inner
4032.2.k.b 4 7.b odd 2 1 CM
4032.2.k.b 4 21.c even 2 1 inner
4032.2.k.c 4 4.b odd 2 1
4032.2.k.c 4 12.b even 2 1
4032.2.k.c 4 28.d even 2 1
4032.2.k.c 4 84.h odd 2 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{43}^{2} - 28$$ $$T_{67} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 5 T^{2} )^{4}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$1 - 206 T^{4} + 14641 T^{8}$$
$13$ $$( 1 - 13 T^{2} )^{4}$$
$17$ $$( 1 + 17 T^{2} )^{4}$$
$19$ $$( 1 - 19 T^{2} )^{4}$$
$23$ $$1 - 734 T^{4} + 279841 T^{8}$$
$29$ $$1 + 1234 T^{4} + 707281 T^{8}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 - 38 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$( 1 + 58 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$1 - 5582 T^{4} + 7890481 T^{8}$$
$59$ $$( 1 + 59 T^{2} )^{4}$$
$61$ $$( 1 - 61 T^{2} )^{4}$$
$67$ $$( 1 + 4 T + 67 T^{2} )^{4}$$
$71$ $$1 + 2914 T^{4} + 25411681 T^{8}$$
$73$ $$( 1 - 73 T^{2} )^{4}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{4}$$
$83$ $$( 1 + 83 T^{2} )^{4}$$
$89$ $$( 1 + 89 T^{2} )^{4}$$
$97$ $$( 1 - 97 T^{2} )^{4}$$