# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{7} +O(q^{10})$$ $$q -\beta q^{7} + 2 \beta q^{11} + 2 \beta q^{23} + 5 q^{25} -2 q^{29} -6 q^{37} -2 \beta q^{43} -7 q^{49} -10 q^{53} + 6 \beta q^{67} + 2 \beta q^{71} + 14 q^{77} + 6 \beta q^{79} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 10q^{25} - 4q^{29} - 12q^{37} - 14q^{49} - 20q^{53} + 28q^{77} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3583.1
 0.5 + 1.32288i 0.5 − 1.32288i
0 0 0 0 0 2.64575i 0 0 0
3583.2 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 $$\Q(\sqrt{-7})$$
4.b odd 2 1 inner
28.d 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.b.e 2
3.b odd 2 1 448.2.f.b 2
4.b odd 2 1 inner 4032.2.b.e 2
7.b odd 2 1 CM 4032.2.b.e 2
8.b even 2 1 252.2.b.a 2
8.d odd 2 1 252.2.b.a 2
12.b even 2 1 448.2.f.b 2
21.c even 2 1 448.2.f.b 2
24.f even 2 1 28.2.d.a 2
24.h odd 2 1 28.2.d.a 2
28.d even 2 1 inner 4032.2.b.e 2
48.i odd 4 2 1792.2.e.b 4
48.k even 4 2 1792.2.e.b 4
56.e even 2 1 252.2.b.a 2
56.h odd 2 1 252.2.b.a 2
84.h odd 2 1 448.2.f.b 2
120.i odd 2 1 700.2.g.a 2
120.m even 2 1 700.2.g.a 2
120.q odd 4 2 700.2.c.d 4
120.w even 4 2 700.2.c.d 4
168.e odd 2 1 28.2.d.a 2
168.i even 2 1 28.2.d.a 2
168.s odd 6 2 196.2.f.b 4
168.v even 6 2 196.2.f.b 4
168.ba even 6 2 196.2.f.b 4
168.be odd 6 2 196.2.f.b 4
336.v odd 4 2 1792.2.e.b 4
336.y even 4 2 1792.2.e.b 4
840.b odd 2 1 700.2.g.a 2
840.u even 2 1 700.2.g.a 2
840.bm even 4 2 700.2.c.d 4
840.bp odd 4 2 700.2.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.d.a 2 24.f even 2 1
28.2.d.a 2 24.h odd 2 1
28.2.d.a 2 168.e odd 2 1
28.2.d.a 2 168.i even 2 1
196.2.f.b 4 168.s odd 6 2
196.2.f.b 4 168.v even 6 2
196.2.f.b 4 168.ba even 6 2
196.2.f.b 4 168.be odd 6 2
252.2.b.a 2 8.b even 2 1
252.2.b.a 2 8.d odd 2 1
252.2.b.a 2 56.e even 2 1
252.2.b.a 2 56.h odd 2 1
448.2.f.b 2 3.b odd 2 1
448.2.f.b 2 12.b even 2 1
448.2.f.b 2 21.c even 2 1
448.2.f.b 2 84.h odd 2 1
700.2.c.d 4 120.q odd 4 2
700.2.c.d 4 120.w even 4 2
700.2.c.d 4 840.bm even 4 2
700.2.c.d 4 840.bp odd 4 2
700.2.g.a 2 120.i odd 2 1
700.2.g.a 2 120.m even 2 1
700.2.g.a 2 840.b odd 2 1
700.2.g.a 2 840.u even 2 1
1792.2.e.b 4 48.i odd 4 2
1792.2.e.b 4 48.k even 4 2
1792.2.e.b 4 336.v odd 4 2
1792.2.e.b 4 336.y even 4 2
4032.2.b.e 2 1.a even 1 1 trivial
4032.2.b.e 2 4.b odd 2 1 inner
4032.2.b.e 2 7.b odd 2 1 CM
4032.2.b.e 2 28.d 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 $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}$$ $$T_{11}^{2} + 28$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 5 T^{2} )^{2}$$
$7$ $$1 + 7 T^{2}$$
$11$ $$( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$13$ $$( 1 - 13 T^{2} )^{2}$$
$17$ $$( 1 - 17 T^{2} )^{2}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$( 1 + 6 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 41 T^{2} )^{2}$$
$43$ $$( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} )$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 + 10 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 - 61 T^{2} )^{2}$$
$67$ $$( 1 - 4 T + 67 T^{2} )( 1 + 4 T + 67 T^{2} )$$
$71$ $$( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} )$$
$73$ $$( 1 - 73 T^{2} )^{2}$$
$79$ $$( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} )$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 - 89 T^{2} )^{2}$$
$97$ $$( 1 - 97 T^{2} )^{2}$$