# Properties

 Label 4032.2.h.a Level 4032 Weight 2 Character orbit 4032.h Analytic conductor 32.196 Analytic rank 1 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.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2016) 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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{2} q^{7} +O(q^{10})$$ $$q + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{2} q^{7} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{11} -6 q^{13} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{17} -8 \zeta_{8}^{2} q^{19} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{23} -3 q^{25} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{29} + 4 \zeta_{8}^{2} q^{31} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{35} -8 q^{37} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{41} + 12 \zeta_{8}^{2} q^{43} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{47} - q^{49} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{53} + 4 \zeta_{8}^{2} q^{55} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{59} -2 q^{61} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{65} -14 \zeta_{8}^{2} q^{67} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{71} + 2 q^{73} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{77} + 10 \zeta_{8}^{2} q^{79} -16 q^{85} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{89} -6 \zeta_{8}^{2} q^{91} + ( -16 \zeta_{8} + 16 \zeta_{8}^{3} ) q^{95} -6 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 24q^{13} - 12q^{25} - 32q^{37} - 4q^{49} - 8q^{61} + 8q^{73} - 64q^{85} - 24q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
575.1
 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i
0 0 0 2.82843i 0 1.00000i 0 0 0
575.2 0 0 0 2.82843i 0 1.00000i 0 0 0
575.3 0 0 0 2.82843i 0 1.00000i 0 0 0
575.4 0 0 0 2.82843i 0 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b 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.h.a 4
3.b odd 2 1 inner 4032.2.h.a 4
4.b odd 2 1 inner 4032.2.h.a 4
8.b even 2 1 2016.2.h.d 4
8.d odd 2 1 2016.2.h.d 4
12.b even 2 1 inner 4032.2.h.a 4
24.f even 2 1 2016.2.h.d 4
24.h odd 2 1 2016.2.h.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.h.d 4 8.b even 2 1
2016.2.h.d 4 8.d odd 2 1
2016.2.h.d 4 24.f even 2 1
2016.2.h.d 4 24.h odd 2 1
4032.2.h.a 4 1.a even 1 1 trivial
4032.2.h.a 4 3.b odd 2 1 inner
4032.2.h.a 4 4.b odd 2 1 inner
4032.2.h.a 4 12.b even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{2} + 8$$ $$T_{11}^{2} - 2$$ $$T_{13} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 2 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 + 20 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 6 T + 13 T^{2} )^{4}$$
$17$ $$( 1 - 6 T + 17 T^{2} )^{2}( 1 + 6 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 26 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 44 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 40 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 46 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 8 T + 37 T^{2} )^{4}$$
$41$ $$( 1 - 10 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 58 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 86 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 104 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 82 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 2 T + 61 T^{2} )^{4}$$
$67$ $$( 1 + 62 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 20 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 2 T + 73 T^{2} )^{4}$$
$79$ $$( 1 - 58 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{4}$$
$89$ $$( 1 + 22 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 6 T + 97 T^{2} )^{4}$$