# Properties

 Label 4032.2.v.a Level 4032 Weight 2 Character orbit 4032.v 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.v (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1008) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{8} q^{5} - q^{7} +O(q^{10})$$ $$q + 2 \zeta_{8} q^{5} - q^{7} -2 \zeta_{8}^{3} q^{11} + ( 2 + 2 \zeta_{8}^{2} ) q^{13} + ( -5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{23} -\zeta_{8}^{2} q^{25} + 2 \zeta_{8}^{3} q^{29} -8 \zeta_{8}^{2} q^{31} -2 \zeta_{8} q^{35} + ( 5 - 5 \zeta_{8}^{2} ) q^{37} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{41} + ( 3 + 3 \zeta_{8}^{2} ) q^{43} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{47} + q^{49} + 4 q^{55} + 4 \zeta_{8}^{3} q^{59} + ( -2 - 2 \zeta_{8}^{2} ) q^{61} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{65} + ( 5 - 5 \zeta_{8}^{2} ) q^{67} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{71} + 14 \zeta_{8}^{2} q^{73} + 2 \zeta_{8}^{3} q^{77} -16 \zeta_{8}^{2} q^{79} -4 \zeta_{8} q^{83} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{89} + ( -2 - 2 \zeta_{8}^{2} ) q^{91} -6 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{7} + O(q^{10})$$ $$4q - 4q^{7} + 8q^{13} + 20q^{37} + 12q^{43} + 4q^{49} + 16q^{55} - 8q^{61} + 20q^{67} - 8q^{91} - 24q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$1793$$ $$3781$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-\zeta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1583.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 0 0 −1.41421 1.41421i 0 −1.00000 0 0 0
1583.2 0 0 0 1.41421 + 1.41421i 0 −1.00000 0 0 0
3599.1 0 0 0 −1.41421 + 1.41421i 0 −1.00000 0 0 0
3599.2 0 0 0 1.41421 1.41421i 0 −1.00000 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
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.v.a 4
3.b odd 2 1 inner 4032.2.v.a 4
4.b odd 2 1 1008.2.v.b 4
12.b even 2 1 1008.2.v.b 4
16.e even 4 1 1008.2.v.b 4
16.f odd 4 1 inner 4032.2.v.a 4
48.i odd 4 1 1008.2.v.b 4
48.k even 4 1 inner 4032.2.v.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.b 4 4.b odd 2 1
1008.2.v.b 4 12.b even 2 1
1008.2.v.b 4 16.e even 4 1
1008.2.v.b 4 48.i odd 4 1
4032.2.v.a 4 1.a even 1 1 trivial
4032.2.v.a 4 3.b odd 2 1 inner
4032.2.v.a 4 16.f odd 4 1 inner
4032.2.v.a 4 48.k even 4 1 inner

## Hecke kernels

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

 $$T_{5}^{4} + 16$$ $$T_{11}^{4} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 8 T^{2} + 25 T^{4} )( 1 + 8 T^{2} + 25 T^{4} )$$
$7$ $$( 1 + T )^{4}$$
$11$ $$1 + 82 T^{4} + 14641 T^{8}$$
$13$ $$( 1 - 4 T + 8 T^{2} - 52 T^{3} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 17 T^{2} )^{4}$$
$19$ $$( 1 + 361 T^{4} )^{2}$$
$23$ $$( 1 + 4 T^{2} + 529 T^{4} )^{2}$$
$29$ $$1 + 1234 T^{4} + 707281 T^{8}$$
$31$ $$( 1 + 2 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 74 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 6 T + 18 T^{2} - 258 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 62 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 2809 T^{4} )^{2}$$
$59$ $$1 + 3442 T^{4} + 12117361 T^{8}$$
$61$ $$( 1 + 4 T + 8 T^{2} + 244 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 10 T + 50 T^{2} - 670 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 44 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 50 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 98 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 + 8722 T^{4} + 47458321 T^{8}$$
$89$ $$( 1 + 50 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 6 T + 97 T^{2} )^{4}$$