# Properties

 Label 4032.2.b.m Level $4032$ Weight $2$ Character orbit 4032.b Analytic conductor $32.196$ Analytic rank $0$ Dimension $4$ CM discriminant -7 Inner twists $8$

# 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Defining polynomial: $$x^{4} - 3x^{2} + 4$$ x^4 - 3*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 252) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{7}+O(q^{10})$$ q + b3 * q^7 $$q + \beta_{3} q^{7} - \beta_1 q^{11} + 2 \beta_1 q^{23} + 5 q^{25} - \beta_{2} q^{29} + 6 q^{37} + 2 \beta_{3} q^{43} - 7 q^{49} + \beta_{2} q^{53} + 6 \beta_{3} q^{67} - 4 \beta_1 q^{71} + \beta_{2} q^{77} + 6 \beta_{3} q^{79}+O(q^{100})$$ q + b3 * q^7 - b1 * q^11 + 2*b1 * q^23 + 5 * q^25 - b2 * q^29 + 6 * q^37 + 2*b3 * q^43 - 7 * q^49 + b2 * q^53 + 6*b3 * q^67 - 4*b1 * q^71 + b2 * q^77 + 6*b3 * q^79 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 20 q^{25} + 24 q^{37} - 28 q^{49}+O(q^{100})$$ 4 * q + 20 * q^25 + 24 * q^37 - 28 * q^49

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 3x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu^{3} - 2\nu$$ 2*v^3 - 2*v $$\beta_{2}$$ $$=$$ $$-2\nu^{3} + 10\nu$$ -2*v^3 + 10*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 3$$ 2*v^2 - 3
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 8$$ (b2 + b1) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 3 ) / 2$$ (b3 + 3) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{2} + 5\beta_1 ) / 8$$ (b2 + 5*b1) / 8

## 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
 −1.32288 + 0.500000i 1.32288 − 0.500000i 1.32288 + 0.500000i −1.32288 − 0.500000i
0 0 0 0 0 2.64575i 0 0 0
3583.2 0 0 0 0 0 2.64575i 0 0 0
3583.3 0 0 0 0 0 2.64575i 0 0 0
3583.4 0 0 0 0 0 2.64575i 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
4.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.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.b.m 4
3.b odd 2 1 inner 4032.2.b.m 4
4.b odd 2 1 inner 4032.2.b.m 4
7.b odd 2 1 CM 4032.2.b.m 4
8.b even 2 1 252.2.b.c 4
8.d odd 2 1 252.2.b.c 4
12.b even 2 1 inner 4032.2.b.m 4
21.c even 2 1 inner 4032.2.b.m 4
24.f even 2 1 252.2.b.c 4
24.h odd 2 1 252.2.b.c 4
28.d even 2 1 inner 4032.2.b.m 4
56.e even 2 1 252.2.b.c 4
56.h odd 2 1 252.2.b.c 4
84.h odd 2 1 inner 4032.2.b.m 4
168.e odd 2 1 252.2.b.c 4
168.i even 2 1 252.2.b.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.b.c 4 8.b even 2 1
252.2.b.c 4 8.d odd 2 1
252.2.b.c 4 24.f even 2 1
252.2.b.c 4 24.h odd 2 1
252.2.b.c 4 56.e even 2 1
252.2.b.c 4 56.h odd 2 1
252.2.b.c 4 168.e odd 2 1
252.2.b.c 4 168.i even 2 1
4032.2.b.m 4 1.a even 1 1 trivial
4032.2.b.m 4 3.b odd 2 1 inner
4032.2.b.m 4 4.b odd 2 1 inner
4032.2.b.m 4 7.b odd 2 1 CM
4032.2.b.m 4 12.b even 2 1 inner
4032.2.b.m 4 21.c even 2 1 inner
4032.2.b.m 4 28.d even 2 1 inner
4032.2.b.m 4 84.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 $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}$$ T5 $$T_{11}^{2} + 16$$ T11^2 + 16 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 7)^{2}$$
$11$ $$(T^{2} + 16)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 64)^{2}$$
$29$ $$(T^{2} - 112)^{2}$$
$31$ $$T^{4}$$
$37$ $$(T - 6)^{4}$$
$41$ $$T^{4}$$
$43$ $$(T^{2} + 28)^{2}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} - 112)^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} + 252)^{2}$$
$71$ $$(T^{2} + 256)^{2}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + 252)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$