# Properties

 Label 4032.2.h.g Level 4032 Weight 2 Character orbit 4032.h 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.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.5473632256.1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 2016) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{5} + \beta_{6} ) q^{5} -\beta_{1} q^{7} +O(q^{10})$$ $$q + ( \beta_{5} + \beta_{6} ) q^{5} -\beta_{1} q^{7} + ( -\beta_{4} - 2 \beta_{7} ) q^{11} + ( 1 - \beta_{2} ) q^{13} + ( -\beta_{5} - 3 \beta_{6} ) q^{17} + 4 \beta_{1} q^{19} + ( \beta_{4} + 2 \beta_{7} ) q^{23} + ( -4 - \beta_{2} ) q^{25} + ( -2 \beta_{5} - 3 \beta_{6} ) q^{29} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{31} + ( \beta_{4} + \beta_{7} ) q^{35} + ( 7 - \beta_{2} ) q^{37} + ( -\beta_{5} - 5 \beta_{6} ) q^{41} + 4 \beta_{1} q^{43} + ( -2 \beta_{4} + 4 \beta_{7} ) q^{47} - q^{49} + ( -2 \beta_{5} - 3 \beta_{6} ) q^{53} + ( -10 \beta_{1} + 2 \beta_{3} ) q^{55} + ( -2 \beta_{4} - 4 \beta_{7} ) q^{59} + 2 q^{61} -8 \beta_{6} q^{65} + ( \beta_{1} + \beta_{3} ) q^{67} + ( 3 \beta_{4} + 2 \beta_{7} ) q^{71} + ( -3 - \beta_{2} ) q^{73} + ( \beta_{5} + 2 \beta_{6} ) q^{77} + ( -\beta_{1} + 3 \beta_{3} ) q^{79} + 4 \beta_{4} q^{83} + ( 11 + 3 \beta_{2} ) q^{85} + ( -\beta_{5} - \beta_{6} ) q^{89} + ( -\beta_{1} - \beta_{3} ) q^{91} + ( -4 \beta_{4} - 4 \beta_{7} ) q^{95} + ( -3 - \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{13} - 32q^{25} + 56q^{37} - 8q^{49} + 16q^{61} - 24q^{73} + 88q^{85} - 24q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 49 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 65 \nu^{2}$$$$)/144$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{4} + 49$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 33 \nu^{2}$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 65 \nu^{3} + 144 \nu$$$$)/144$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 65 \nu^{3} + 144 \nu$$$$)/144$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{7} + 16 \nu^{5} + 181 \nu^{3} + 464 \nu$$$$)/576$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 16 \nu^{5} - 181 \nu^{3} + 464 \nu$$$$)/576$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 9 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{7} - 4 \beta_{6} + 5 \beta_{5} - 5 \beta_{4}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$9 \beta_{2} - 49$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$36 \beta_{7} + 36 \beta_{6} - 29 \beta_{5} - 29 \beta_{4}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-65 \beta_{3} - 297 \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-260 \beta_{7} + 260 \beta_{6} - 181 \beta_{5} + 181 \beta_{4}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
575.1
 −1.10418 − 1.10418i 1.10418 − 1.10418i −1.81129 − 1.81129i 1.81129 − 1.81129i 1.81129 + 1.81129i −1.81129 + 1.81129i 1.10418 + 1.10418i −1.10418 + 1.10418i
0 0 0 3.62258i 0 1.00000i 0 0 0
575.2 0 0 0 3.62258i 0 1.00000i 0 0 0
575.3 0 0 0 2.20837i 0 1.00000i 0 0 0
575.4 0 0 0 2.20837i 0 1.00000i 0 0 0
575.5 0 0 0 2.20837i 0 1.00000i 0 0 0
575.6 0 0 0 2.20837i 0 1.00000i 0 0 0
575.7 0 0 0 3.62258i 0 1.00000i 0 0 0
575.8 0 0 0 3.62258i 0 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 575.8 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
4.b odd 2 1 inner
12.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.h.g 8
3.b odd 2 1 inner 4032.2.h.g 8
4.b odd 2 1 inner 4032.2.h.g 8
8.b even 2 1 2016.2.h.e 8
8.d odd 2 1 2016.2.h.e 8
12.b even 2 1 inner 4032.2.h.g 8
24.f even 2 1 2016.2.h.e 8
24.h odd 2 1 2016.2.h.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.h.e 8 8.b even 2 1
2016.2.h.e 8 8.d odd 2 1
2016.2.h.e 8 24.f even 2 1
2016.2.h.e 8 24.h odd 2 1
4032.2.h.g 8 1.a even 1 1 trivial
4032.2.h.g 8 3.b odd 2 1 inner
4032.2.h.g 8 4.b odd 2 1 inner
4032.2.h.g 8 12.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 $$S_{2}^{\mathrm{new}}(4032, [\chi])$$:

 $$T_{5}^{4} + 18 T_{5}^{2} + 64$$ $$T_{11}^{4} - 26 T_{11}^{2} + 16$$ $$T_{13}^{2} - 2 T_{13} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 2 T^{2} + 34 T^{4} - 50 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$( 1 + 18 T^{2} + 170 T^{4} + 2178 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 2 T + 10 T^{2} - 26 T^{3} + 169 T^{4} )^{4}$$
$17$ $$( 1 - 26 T^{2} + 322 T^{4} - 7514 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 22 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 + 66 T^{2} + 1994 T^{4} + 34914 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 32 T^{2} + 850 T^{4} - 26912 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 20 T^{2} + 934 T^{4} + 19220 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 14 T + 106 T^{2} - 518 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 66 T^{2} + 3074 T^{4} - 110946 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 70 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 20 T^{2} - 2282 T^{4} + 44180 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 128 T^{2} + 8626 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 132 T^{2} + 8870 T^{4} + 459492 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 2 T + 61 T^{2} )^{8}$$
$67$ $$( 1 - 232 T^{2} + 22366 T^{4} - 1041448 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 130 T^{2} + 14154 T^{4} + 655330 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 6 T + 138 T^{2} + 438 T^{3} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 8 T^{2} + 11886 T^{4} - 49928 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 44 T^{2} + 9910 T^{4} + 303116 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 338 T^{2} + 44386 T^{4} - 2677298 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 6 T + 186 T^{2} + 582 T^{3} + 9409 T^{4} )^{4}$$