# Properties

 Label 4032.2.c.f Level $4032$ Weight $2$ Character orbit 4032.c Analytic conductor $32.196$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4032.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 448) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{5} + q^{7} +O(q^{10})$$ $$q + 4 i q^{5} + q^{7} -2 i q^{11} -4 i q^{13} -2 q^{17} + 6 i q^{19} -11 q^{25} -8 i q^{29} -8 q^{31} + 4 i q^{35} -8 i q^{37} -10 q^{41} + 2 i q^{43} -8 q^{47} + q^{49} + 8 q^{55} + 10 i q^{59} + 4 i q^{61} + 16 q^{65} + 2 i q^{67} -8 q^{71} -6 q^{73} -2 i q^{77} + 8 q^{79} -6 i q^{83} -8 i q^{85} -10 q^{89} -4 i q^{91} -24 q^{95} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{7} + O(q^{10})$$ $$2 q + 2 q^{7} - 4 q^{17} - 22 q^{25} - 16 q^{31} - 20 q^{41} - 16 q^{47} + 2 q^{49} + 16 q^{55} + 32 q^{65} - 16 q^{71} - 12 q^{73} + 16 q^{79} - 20 q^{89} - 48 q^{95} + 4 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2017.1
 − 1.00000i 1.00000i
0 0 0 4.00000i 0 1.00000 0 0 0
2017.2 0 0 0 4.00000i 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
8.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.c.f 2
3.b odd 2 1 448.2.b.b yes 2
4.b odd 2 1 4032.2.c.b 2
8.b even 2 1 inner 4032.2.c.f 2
8.d odd 2 1 4032.2.c.b 2
12.b even 2 1 448.2.b.a 2
21.c even 2 1 3136.2.b.a 2
24.f even 2 1 448.2.b.a 2
24.h odd 2 1 448.2.b.b yes 2
48.i odd 4 1 1792.2.a.d 1
48.i odd 4 1 1792.2.a.e 1
48.k even 4 1 1792.2.a.a 1
48.k even 4 1 1792.2.a.h 1
84.h odd 2 1 3136.2.b.c 2
168.e odd 2 1 3136.2.b.c 2
168.i even 2 1 3136.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.b.a 2 12.b even 2 1
448.2.b.a 2 24.f even 2 1
448.2.b.b yes 2 3.b odd 2 1
448.2.b.b yes 2 24.h odd 2 1
1792.2.a.a 1 48.k even 4 1
1792.2.a.d 1 48.i odd 4 1
1792.2.a.e 1 48.i odd 4 1
1792.2.a.h 1 48.k even 4 1
3136.2.b.a 2 21.c even 2 1
3136.2.b.a 2 168.i even 2 1
3136.2.b.c 2 84.h odd 2 1
3136.2.b.c 2 168.e odd 2 1
4032.2.c.b 2 4.b odd 2 1
4032.2.c.b 2 8.d odd 2 1
4032.2.c.f 2 1.a even 1 1 trivial
4032.2.c.f 2 8.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 $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}^{2} + 16$$ $$T_{11}^{2} + 4$$ $$T_{13}^{2} + 16$$ $$T_{17} + 2$$ $$T_{23}$$ $$T_{31} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$16 + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$64 + T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$T^{2}$$
$59$ $$100 + T^{2}$$
$61$ $$16 + T^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$( 6 + T )^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$( -2 + T )^{2}$$