# Properties

 Label 4032.2.c.n Level 4032 Weight 2 Character orbit 4032.c Analytic conductor 32.196 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Learn more about

## 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}$$ 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 a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + q^{7} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + q^{7} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( 1 - 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{13} -2 q^{17} + ( 1 - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{19} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{23} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{25} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{29} + 4 q^{31} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{35} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{37} + 2 q^{41} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{43} + ( 4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{47} + q^{49} + 12 \zeta_{12}^{3} q^{53} + ( 8 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{55} + ( 5 - 10 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{59} + ( -7 + 14 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{61} + ( -8 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{65} + 8 \zeta_{12}^{3} q^{67} + ( 4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{71} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{73} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{77} + ( -4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{79} + ( 5 - 10 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{83} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{85} + ( 2 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{89} + ( 1 - 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{91} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{95} + ( 2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} + O(q^{10})$$ $$4q + 4q^{7} - 8q^{17} + 4q^{25} + 16q^{31} + 8q^{41} + 16q^{47} + 4q^{49} + 32q^{55} - 32q^{65} + 16q^{71} + 24q^{73} - 16q^{79} + 8q^{89} + 8q^{97} + O(q^{100})$$

## 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
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 0 0 2.73205i 0 1.00000 0 0 0
2017.2 0 0 0 0.732051i 0 1.00000 0 0 0
2017.3 0 0 0 0.732051i 0 1.00000 0 0 0
2017.4 0 0 0 2.73205i 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.n 4
3.b odd 2 1 448.2.b.d yes 4
4.b odd 2 1 4032.2.c.k 4
8.b even 2 1 inner 4032.2.c.n 4
8.d odd 2 1 4032.2.c.k 4
12.b even 2 1 448.2.b.c 4
21.c even 2 1 3136.2.b.h 4
24.f even 2 1 448.2.b.c 4
24.h odd 2 1 448.2.b.d yes 4
48.i odd 4 1 1792.2.a.i 2
48.i odd 4 1 1792.2.a.s 2
48.k even 4 1 1792.2.a.k 2
48.k even 4 1 1792.2.a.q 2
84.h odd 2 1 3136.2.b.g 4
168.e odd 2 1 3136.2.b.g 4
168.i even 2 1 3136.2.b.h 4

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

 $$T_{5}^{4} + 8 T_{5}^{2} + 4$$ $$T_{11}^{4} + 32 T_{11}^{2} + 64$$ $$T_{13}^{4} + 56 T_{13}^{2} + 484$$ $$T_{17} + 2$$ $$T_{23}^{2} - 12$$ $$T_{31} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8}$$
$7$ $$( 1 - T )^{4}$$
$11$ $$1 - 12 T^{2} + 86 T^{4} - 1452 T^{6} + 14641 T^{8}$$
$13$ $$1 + 4 T^{2} + 42 T^{4} + 676 T^{6} + 28561 T^{8}$$
$17$ $$( 1 + 2 T + 17 T^{2} )^{4}$$
$19$ $$1 - 52 T^{2} + 1290 T^{4} - 18772 T^{6} + 130321 T^{8}$$
$23$ $$( 1 + 34 T^{2} + 529 T^{4} )^{2}$$
$29$ $$1 - 84 T^{2} + 3254 T^{4} - 70644 T^{6} + 707281 T^{8}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{4}$$
$37$ $$1 - 116 T^{2} + 5910 T^{4} - 158804 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 - 2 T + 41 T^{2} )^{4}$$
$43$ $$1 - 140 T^{2} + 8406 T^{4} - 258860 T^{6} + 3418801 T^{8}$$
$47$ $$( 1 - 8 T + 62 T^{2} - 376 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 38 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$1 - 84 T^{2} + 8426 T^{4} - 292404 T^{6} + 12117361 T^{8}$$
$61$ $$1 + 52 T^{2} + 7530 T^{4} + 193492 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 - 70 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 8 T + 110 T^{2} - 568 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 - 164 T^{2} + 17802 T^{4} - 1129796 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 4 T - 10 T^{2} - 356 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{2}$$
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