# Properties

 Label 4032.2.c.q Level 4032 Weight 2 Character orbit 4032.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.897122304.10 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{5} - q^{7} +O(q^{10})$$ $$q -\beta_{5} q^{5} - q^{7} + \beta_{5} q^{11} + ( -\beta_{4} - \beta_{6} ) q^{13} + \beta_{1} q^{17} + \beta_{4} q^{19} -\beta_{2} q^{23} + ( -5 - \beta_{7} ) q^{25} -\beta_{3} q^{29} + 2 q^{31} + \beta_{5} q^{35} + ( \beta_{4} + 4 \beta_{6} ) q^{37} -\beta_{1} q^{41} + ( 2 \beta_{4} + \beta_{6} ) q^{43} + ( -\beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( 10 + \beta_{7} ) q^{55} + \beta_{3} q^{59} + ( \beta_{4} + \beta_{6} ) q^{61} + ( \beta_{1} - 3 \beta_{2} ) q^{65} -5 \beta_{6} q^{67} + ( 2 \beta_{1} - \beta_{2} ) q^{71} + ( 6 - \beta_{7} ) q^{73} -\beta_{5} q^{77} + ( 4 - \beta_{7} ) q^{79} + ( -\beta_{3} - 2 \beta_{5} ) q^{83} + ( -\beta_{4} + 4 \beta_{6} ) q^{85} + ( -\beta_{1} + 2 \beta_{2} ) q^{89} + ( \beta_{4} + \beta_{6} ) q^{91} + 2 \beta_{2} q^{95} + ( 2 - \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{7} + O(q^{10})$$ $$8q - 8q^{7} - 40q^{25} + 16q^{31} + 8q^{49} + 80q^{55} + 48q^{73} + 32q^{79} + 16q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8 x^{6} + 51 x^{4} - 104 x^{2} + 169$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$6 \nu^{7} + 17 \nu^{5} + 85 \nu^{3} - 403 \nu$$$$)/663$$ $$\beta_{2}$$ $$=$$ $$($$$$-14 \nu^{7} + 255 \nu^{5} - 1377 \nu^{3} + 4771 \nu$$$$)/663$$ $$\beta_{3}$$ $$=$$ $$($$$$32 \nu^{7} - 204 \nu^{5} + 1632 \nu^{3} - 676 \nu$$$$)/663$$ $$\beta_{4}$$ $$=$$ $$($$$$32 \nu^{6} - 204 \nu^{4} + 1632 \nu^{2} - 2002$$$$)/663$$ $$\beta_{5}$$ $$=$$ $$($$$$40 \nu^{7} - 255 \nu^{5} + 1377 \nu^{3} - 845 \nu$$$$)/663$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{6} - 16 \nu^{4} + 76 \nu^{2} - 104$$$$)/39$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{6} + 400$$$$)/51$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 3 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} - 6 \beta_{6} + 8 \beta_{4} + 16$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{5} + 5 \beta_{3}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$4 \beta_{7} - 24 \beta_{6} + 19 \beta_{4} - 38$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-32 \beta_{5} + 27 \beta_{3} + 5 \beta_{2} + 81 \beta_{1}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$51 \beta_{7} - 400$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$204 \beta_{5} - 151 \beta_{3} + 53 \beta_{2} + 453 \beta_{1}$$$$)/8$$

## 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
 1.30421 + 0.752986i −1.30421 + 0.752986i −2.07341 − 1.19709i 2.07341 − 1.19709i 2.07341 + 1.19709i −2.07341 + 1.19709i −1.30421 − 0.752986i 1.30421 − 0.752986i
0 0 0 4.11439i 0 −1.00000 0 0 0
2017.2 0 0 0 4.11439i 0 −1.00000 0 0 0
2017.3 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.4 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.5 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.6 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.7 0 0 0 4.11439i 0 −1.00000 0 0 0
2017.8 0 0 0 4.11439i 0 −1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2017.8 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
8.b even 2 1 inner
24.h odd 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.q 8
3.b odd 2 1 inner 4032.2.c.q 8
4.b odd 2 1 4032.2.c.r yes 8
8.b even 2 1 inner 4032.2.c.q 8
8.d odd 2 1 4032.2.c.r yes 8
12.b even 2 1 4032.2.c.r yes 8
24.f even 2 1 4032.2.c.r yes 8
24.h odd 2 1 inner 4032.2.c.q 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4032.2.c.q 8 1.a even 1 1 trivial
4032.2.c.q 8 3.b odd 2 1 inner
4032.2.c.q 8 8.b even 2 1 inner
4032.2.c.q 8 24.h odd 2 1 inner
4032.2.c.r yes 8 4.b odd 2 1
4032.2.c.r yes 8 8.d odd 2 1
4032.2.c.r yes 8 12.b even 2 1
4032.2.c.r yes 8 24.f even 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 20 T_{5}^{2} + 52$$ $$T_{11}^{4} + 20 T_{11}^{2} + 52$$ $$T_{13}^{4} + 32 T_{13}^{2} + 64$$ $$T_{17}^{4} - 44 T_{17}^{2} + 52$$ $$T_{23}^{4} - 60 T_{23}^{2} + 468$$ $$T_{31} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 2 T^{4} + 625 T^{8} )^{2}$$
$7$ $$( 1 + T )^{8}$$
$11$ $$( 1 - 24 T^{2} + 338 T^{4} - 2904 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 20 T^{2} + 246 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 + 24 T^{2} + 290 T^{4} + 6936 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{4}( 1 + 8 T + 19 T^{2} )^{4}$$
$23$ $$( 1 + 32 T^{2} + 882 T^{4} + 16928 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 4 T + 14 T^{2} - 116 T^{3} + 841 T^{4} )^{2}( 1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 2 T + 31 T^{2} )^{8}$$
$37$ $$( 1 + 4 T^{2} - 330 T^{4} + 5476 T^{6} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 120 T^{2} + 6530 T^{4} + 201720 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 60 T^{2} + 4550 T^{4} + 132540 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{8}$$
$59$ $$( 1 - 108 T^{2} + 9110 T^{4} - 375948 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 212 T^{2} + 18486 T^{4} - 788852 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - 34 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 + 96 T^{2} + 12338 T^{4} + 483936 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 8 T + 126 T^{2} - 632 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 - 92 T^{2} + 8982 T^{4} - 633788 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 120 T^{2} + 5570 T^{4} + 950520 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{4}$$