# Properties

 Label 4032.2.b.p Level 4032 Weight 2 Character orbit 4032.b Analytic conductor 32.196 Analytic rank 0 Dimension 8 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: no (minimal twist has level 224) 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 $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{5} + ( \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{7} +O(q^{10})$$ $$q + ( \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{5} + ( \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{7} + 2 \zeta_{16}^{4} q^{11} + ( -3 \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{13} + ( -4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} ) q^{17} + ( -3 \zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} + 3 \zeta_{16}^{7} ) q^{19} + ( 4 \zeta_{16}^{2} - 2 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{23} + ( 1 + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{25} + ( -2 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{29} + ( -4 \zeta_{16} + 4 \zeta_{16}^{7} ) q^{31} + ( \zeta_{16} + 2 \zeta_{16}^{2} + \zeta_{16}^{3} - 4 \zeta_{16}^{4} - \zeta_{16}^{5} + 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{35} + ( 2 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{37} + ( 6 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{41} + ( -4 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 4 \zeta_{16}^{6} ) q^{43} + ( -4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} ) q^{47} + ( 1 + 2 \zeta_{16} - 4 \zeta_{16}^{2} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 4 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{49} -2 q^{53} + ( 2 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{55} + ( \zeta_{16} + 5 \zeta_{16}^{3} - 5 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{59} + ( \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{61} + ( 4 - 6 \zeta_{16}^{2} + 6 \zeta_{16}^{6} ) q^{65} + ( 4 \zeta_{16}^{2} - 10 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{67} + ( -2 \zeta_{16}^{2} + 6 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{71} + ( 6 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{73} + ( -2 - 2 \zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{77} + ( -2 \zeta_{16}^{2} + 10 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{79} + ( 3 \zeta_{16} - 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{83} -8 q^{85} + ( -2 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{89} + ( 5 \zeta_{16} - 6 \zeta_{16}^{2} + \zeta_{16}^{3} + 4 \zeta_{16}^{4} - \zeta_{16}^{5} - 6 \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{91} + ( -2 \zeta_{16}^{2} + 8 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{95} + ( 4 \zeta_{16} - 8 \zeta_{16}^{3} - 8 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{25} - 16q^{29} + 16q^{37} + 8q^{49} - 16q^{53} + 32q^{65} - 16q^{77} - 64q^{85} + O(q^{100})$$

## 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
 −0.382683 − 0.923880i 0.382683 − 0.923880i −0.923880 + 0.382683i 0.923880 + 0.382683i −0.923880 − 0.382683i 0.923880 − 0.382683i −0.382683 + 0.923880i 0.382683 + 0.923880i
0 0 0 2.61313i 0 −2.61313 + 0.414214i 0 0 0
3583.2 0 0 0 2.61313i 0 2.61313 0.414214i 0 0 0
3583.3 0 0 0 1.08239i 0 −1.08239 2.41421i 0 0 0
3583.4 0 0 0 1.08239i 0 1.08239 + 2.41421i 0 0 0
3583.5 0 0 0 1.08239i 0 −1.08239 + 2.41421i 0 0 0
3583.6 0 0 0 1.08239i 0 1.08239 2.41421i 0 0 0
3583.7 0 0 0 2.61313i 0 −2.61313 0.414214i 0 0 0
3583.8 0 0 0 2.61313i 0 2.61313 + 0.414214i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3583.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.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.p 8
3.b odd 2 1 448.2.f.d 8
4.b odd 2 1 inner 4032.2.b.p 8
7.b odd 2 1 inner 4032.2.b.p 8
8.b even 2 1 2016.2.b.b 8
8.d odd 2 1 2016.2.b.b 8
12.b even 2 1 448.2.f.d 8
21.c even 2 1 448.2.f.d 8
24.f even 2 1 224.2.f.a 8
24.h odd 2 1 224.2.f.a 8
28.d even 2 1 inner 4032.2.b.p 8
48.i odd 4 1 1792.2.e.f 8
48.i odd 4 1 1792.2.e.g 8
48.k even 4 1 1792.2.e.f 8
48.k even 4 1 1792.2.e.g 8
56.e even 2 1 2016.2.b.b 8
56.h odd 2 1 2016.2.b.b 8
84.h odd 2 1 448.2.f.d 8
168.e odd 2 1 224.2.f.a 8
168.i even 2 1 224.2.f.a 8
168.s odd 6 2 1568.2.p.a 16
168.v even 6 2 1568.2.p.a 16
168.ba even 6 2 1568.2.p.a 16
168.be odd 6 2 1568.2.p.a 16
336.v odd 4 1 1792.2.e.f 8
336.v odd 4 1 1792.2.e.g 8
336.y even 4 1 1792.2.e.f 8
336.y even 4 1 1792.2.e.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.f.a 8 24.f even 2 1
224.2.f.a 8 24.h odd 2 1
224.2.f.a 8 168.e odd 2 1
224.2.f.a 8 168.i even 2 1
448.2.f.d 8 3.b odd 2 1
448.2.f.d 8 12.b even 2 1
448.2.f.d 8 21.c even 2 1
448.2.f.d 8 84.h odd 2 1
1568.2.p.a 16 168.s odd 6 2
1568.2.p.a 16 168.v even 6 2
1568.2.p.a 16 168.ba even 6 2
1568.2.p.a 16 168.be odd 6 2
1792.2.e.f 8 48.i odd 4 1
1792.2.e.f 8 48.k even 4 1
1792.2.e.f 8 336.v odd 4 1
1792.2.e.f 8 336.y even 4 1
1792.2.e.g 8 48.i odd 4 1
1792.2.e.g 8 48.k even 4 1
1792.2.e.g 8 336.v odd 4 1
1792.2.e.g 8 336.y even 4 1
2016.2.b.b 8 8.b even 2 1
2016.2.b.b 8 8.d odd 2 1
2016.2.b.b 8 56.e even 2 1
2016.2.b.b 8 56.h odd 2 1
4032.2.b.p 8 1.a even 1 1 trivial
4032.2.b.p 8 4.b odd 2 1 inner
4032.2.b.p 8 7.b odd 2 1 inner
4032.2.b.p 8 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 $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}^{4} + 8 T_{5}^{2} + 8$$ $$T_{11}^{2} + 4$$ $$T_{19}^{4} - 40 T_{19}^{2} + 392$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 12 T^{2} + 78 T^{4} - 300 T^{6} + 625 T^{8} )^{2}$$
$7$ $$1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8}$$
$11$ $$( 1 - 18 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 - 12 T^{2} + 366 T^{4} - 2028 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} )^{2}( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 36 T^{2} + 1038 T^{4} + 12996 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 4 T + 30 T^{2} + 116 T^{3} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 60 T^{2} + 2310 T^{4} + 57660 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 4 T + 46 T^{2} - 148 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 4 T^{2} + 3238 T^{4} - 6724 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 100 T^{2} + 5686 T^{4} - 184900 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 124 T^{2} + 7750 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 2 T + 53 T^{2} )^{8}$$
$59$ $$( 1 + 132 T^{2} + 10926 T^{4} + 459492 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 236 T^{2} + 21358 T^{4} - 878156 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - 4 T^{2} - 3818 T^{4} - 17956 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 196 T^{2} + 18534 T^{4} - 988036 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 132 T^{2} + 8742 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 100 T^{2} + 11782 T^{4} - 624100 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 196 T^{2} + 19150 T^{4} + 1350244 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 324 T^{2} + 41958 T^{4} - 2566404 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 68 T^{2} + 19462 T^{4} - 639812 T^{6} + 88529281 T^{8} )^{2}$$