# Properties

 Label 4032.2.k.f Level 4032 Weight 2 Character orbit 4032.k 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.k (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 504) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{16} + 2 \zeta_{16}^{7} ) q^{5} + ( 1 + \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} +O(q^{10})$$ $$q + ( -2 \zeta_{16} + 2 \zeta_{16}^{7} ) q^{5} + ( 1 + \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} + ( \zeta_{16}^{2} + 2 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{11} + ( -2 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{13} + ( -2 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{17} + ( -2 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{19} + ( -3 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 3 \zeta_{16}^{6} ) q^{23} + ( 3 + 4 \zeta_{16}^{2} - 4 \zeta_{16}^{6} ) q^{25} + ( \zeta_{16}^{2} - 4 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{29} + ( 4 \zeta_{16} + 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{31} + ( -4 \zeta_{16} - 2 \zeta_{16}^{3} + 4 \zeta_{16}^{4} + 2 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{35} -4 q^{37} + ( 2 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{41} + ( -2 + 6 \zeta_{16}^{2} - 6 \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} + ( -\zeta_{16}^{2} + 4 \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{53} + ( -2 \zeta_{16} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{55} + ( 8 \zeta_{16}^{2} + 8 \zeta_{16}^{4} + 8 \zeta_{16}^{6} ) q^{65} + ( 4 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{67} + ( \zeta_{16}^{2} + 6 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{71} + ( -6 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 6 \zeta_{16}^{7} ) q^{73} + ( 2 \zeta_{16} + 3 \zeta_{16}^{2} + 4 \zeta_{16}^{4} + 3 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{77} + ( -8 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{79} + ( 8 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 8 \zeta_{16}^{7} ) q^{83} + ( 8 + 12 \zeta_{16}^{2} - 12 \zeta_{16}^{6} ) q^{85} + ( -2 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{89} + ( -2 \zeta_{16} - 4 \zeta_{16}^{2} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} + 4 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{91} -8 \zeta_{16}^{4} q^{95} + ( -2 \zeta_{16} + 6 \zeta_{16}^{3} + 6 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{7} + O(q^{10})$$ $$8q + 8q^{7} + 24q^{25} - 32q^{37} - 16q^{43} - 8q^{49} + 32q^{67} - 64q^{79} + 64q^{85} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3905.1
 0.923880 + 0.382683i 0.923880 − 0.382683i 0.382683 − 0.923880i 0.382683 + 0.923880i −0.382683 − 0.923880i −0.382683 + 0.923880i −0.923880 + 0.382683i −0.923880 − 0.382683i
0 0 0 −3.69552 0 2.41421 1.08239i 0 0 0
3905.2 0 0 0 −3.69552 0 2.41421 + 1.08239i 0 0 0
3905.3 0 0 0 −1.53073 0 −0.414214 2.61313i 0 0 0
3905.4 0 0 0 −1.53073 0 −0.414214 + 2.61313i 0 0 0
3905.5 0 0 0 1.53073 0 −0.414214 2.61313i 0 0 0
3905.6 0 0 0 1.53073 0 −0.414214 + 2.61313i 0 0 0
3905.7 0 0 0 3.69552 0 2.41421 1.08239i 0 0 0
3905.8 0 0 0 3.69552 0 2.41421 + 1.08239i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3905.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c 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.k.f 8
3.b odd 2 1 inner 4032.2.k.f 8
4.b odd 2 1 4032.2.k.e 8
7.b odd 2 1 inner 4032.2.k.f 8
8.b even 2 1 504.2.k.a 8
8.d odd 2 1 1008.2.k.c 8
12.b even 2 1 4032.2.k.e 8
21.c even 2 1 inner 4032.2.k.f 8
24.f even 2 1 1008.2.k.c 8
24.h odd 2 1 504.2.k.a 8
28.d even 2 1 4032.2.k.e 8
56.e even 2 1 1008.2.k.c 8
56.h odd 2 1 504.2.k.a 8
56.j odd 6 2 3528.2.bl.b 16
56.p even 6 2 3528.2.bl.b 16
84.h odd 2 1 4032.2.k.e 8
168.e odd 2 1 1008.2.k.c 8
168.i even 2 1 504.2.k.a 8
168.s odd 6 2 3528.2.bl.b 16
168.ba even 6 2 3528.2.bl.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.k.a 8 8.b even 2 1
504.2.k.a 8 24.h odd 2 1
504.2.k.a 8 56.h odd 2 1
504.2.k.a 8 168.i even 2 1
1008.2.k.c 8 8.d odd 2 1
1008.2.k.c 8 24.f even 2 1
1008.2.k.c 8 56.e even 2 1
1008.2.k.c 8 168.e odd 2 1
3528.2.bl.b 16 56.j odd 6 2
3528.2.bl.b 16 56.p even 6 2
3528.2.bl.b 16 168.s odd 6 2
3528.2.bl.b 16 168.ba even 6 2
4032.2.k.e 8 4.b odd 2 1
4032.2.k.e 8 12.b even 2 1
4032.2.k.e 8 28.d even 2 1
4032.2.k.e 8 84.h odd 2 1
4032.2.k.f 8 1.a even 1 1 trivial
4032.2.k.f 8 3.b odd 2 1 inner
4032.2.k.f 8 7.b odd 2 1 inner
4032.2.k.f 8 21.c 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 $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}^{4} - 16 T_{5}^{2} + 32$$ $$T_{43}^{2} + 4 T_{43} - 68$$ $$T_{67}^{2} - 8 T_{67} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 4 T^{2} + 22 T^{4} + 100 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 - 4 T + 10 T^{2} - 28 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 32 T^{2} + 466 T^{4} - 3872 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 20 T^{2} + 310 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 12 T^{2} + 582 T^{4} - 3468 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 44 T^{2} + 1078 T^{4} - 15884 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 48 T^{2} + 1346 T^{4} - 25392 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 80 T^{2} + 3154 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 4 T^{2} - 122 T^{4} + 3844 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 4 T + 37 T^{2} )^{8}$$
$41$ $$( 1 + 84 T^{2} + 3558 T^{4} + 141204 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 4 T + 18 T^{2} + 172 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 124 T^{2} + 7750 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 176 T^{2} + 13234 T^{4} - 494384 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{8}$$
$61$ $$( 1 - 61 T^{2} )^{8}$$
$67$ $$( 1 - 8 T + 118 T^{2} - 536 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 208 T^{2} + 20610 T^{4} - 1048528 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 132 T^{2} + 8742 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 16 T + 190 T^{2} + 1264 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 12 T^{2} + 13302 T^{4} + 82668 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 276 T^{2} + 34854 T^{4} + 2186196 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 228 T^{2} + 31686 T^{4} - 2145252 T^{6} + 88529281 T^{8} )^{2}$$