# Properties

 Label 4032.2.c.j Level 4032 Weight 2 Character orbit 4032.c Analytic conductor 32.196 Analytic rank 0 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1344) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{5} + q^{7} +O(q^{10})$$ $$q + 2 i q^{5} + q^{7} + 2 i q^{11} + 6 i q^{13} + 6 q^{17} + 8 i q^{19} -8 q^{23} + q^{25} + 8 q^{31} + 2 i q^{35} -4 i q^{37} -6 q^{41} -6 i q^{43} -12 q^{47} + q^{49} -12 i q^{53} -4 q^{55} + 4 i q^{59} + 10 i q^{61} -12 q^{65} -2 i q^{67} + 6 q^{73} + 2 i q^{77} -8 i q^{83} + 12 i q^{85} + 6 q^{89} + 6 i q^{91} -16 q^{95} -10 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{7} + O(q^{10})$$ $$2q + 2q^{7} + 12q^{17} - 16q^{23} + 2q^{25} + 16q^{31} - 12q^{41} - 24q^{47} + 2q^{49} - 8q^{55} - 24q^{65} + 12q^{73} + 12q^{89} - 32q^{95} - 20q^{97} + O(q^{100})$$

## 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 2.00000i 0 1.00000 0 0 0
2017.2 0 0 0 2.00000i 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.j 2
3.b odd 2 1 1344.2.c.d yes 2
4.b odd 2 1 4032.2.c.e 2
8.b even 2 1 inner 4032.2.c.j 2
8.d odd 2 1 4032.2.c.e 2
12.b even 2 1 1344.2.c.a 2
24.f even 2 1 1344.2.c.a 2
24.h odd 2 1 1344.2.c.d yes 2
48.i odd 4 1 5376.2.a.a 1
48.i odd 4 1 5376.2.a.k 1
48.k even 4 1 5376.2.a.f 1
48.k even 4 1 5376.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.c.a 2 12.b even 2 1
1344.2.c.a 2 24.f even 2 1
1344.2.c.d yes 2 3.b odd 2 1
1344.2.c.d yes 2 24.h odd 2 1
4032.2.c.e 2 4.b odd 2 1
4032.2.c.e 2 8.d odd 2 1
4032.2.c.j 2 1.a even 1 1 trivial
4032.2.c.j 2 8.b even 2 1 inner
5376.2.a.a 1 48.i odd 4 1
5376.2.a.f 1 48.k even 4 1
5376.2.a.h 1 48.k even 4 1
5376.2.a.k 1 48.i odd 4 1

## Hecke kernels

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

 $$T_{5}^{2} + 4$$ $$T_{11}^{2} + 4$$ $$T_{13}^{2} + 36$$ $$T_{17} - 6$$ $$T_{23} + 8$$ $$T_{31} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$7$ $$( 1 - T )^{2}$$
$11$ $$1 - 18 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$17$ $$( 1 - 6 T + 17 T^{2} )^{2}$$
$19$ $$1 + 26 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 8 T + 23 T^{2} )^{2}$$
$29$ $$( 1 - 29 T^{2} )^{2}$$
$31$ $$( 1 - 8 T + 31 T^{2} )^{2}$$
$37$ $$1 - 58 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$1 - 50 T^{2} + 1849 T^{4}$$
$47$ $$( 1 + 12 T + 47 T^{2} )^{2}$$
$53$ $$1 + 38 T^{2} + 2809 T^{4}$$
$59$ $$1 - 102 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} )$$
$67$ $$1 - 130 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 6 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 79 T^{2} )^{2}$$
$83$ $$1 - 102 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 6 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{2}$$