# Properties

 Label 4032.2.h.f 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: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 1008) 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} + \zeta_{24}^{6} q^{7} +O(q^{10})$$ $$q + ( 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} + \zeta_{24}^{6} q^{7} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{11} -2 q^{13} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{17} + ( -2 + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{19} + ( -\zeta_{24} - \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{23} + ( -3 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{25} + ( \zeta_{24} - \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{29} + ( -2 + 4 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{31} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{35} + ( 4 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{37} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{41} -4 \zeta_{24}^{6} q^{43} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{47} - q^{49} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{53} + ( -6 + 12 \zeta_{24}^{4} + 10 \zeta_{24}^{6} ) q^{55} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{59} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{61} + ( -4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{65} + ( -2 + 4 \zeta_{24}^{4} + 12 \zeta_{24}^{6} ) q^{67} + ( \zeta_{24} + \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{71} + ( 4 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{73} + ( -\zeta_{24} + \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{77} + ( 6 - 12 \zeta_{24}^{4} ) q^{79} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{83} + 4 q^{85} + ( 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{89} -2 \zeta_{24}^{6} q^{91} + ( -12 \zeta_{24} - 12 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{95} + ( 8 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 16q^{13} - 24q^{25} + 32q^{37} - 8q^{49} + 32q^{73} + 32q^{85} + 64q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
575.1
 0.965926 − 0.258819i −0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 − 0.965926i −0.258819 + 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i
0 0 0 3.86370i 0 1.00000i 0 0 0
575.2 0 0 0 3.86370i 0 1.00000i 0 0 0
575.3 0 0 0 1.03528i 0 1.00000i 0 0 0
575.4 0 0 0 1.03528i 0 1.00000i 0 0 0
575.5 0 0 0 1.03528i 0 1.00000i 0 0 0
575.6 0 0 0 1.03528i 0 1.00000i 0 0 0
575.7 0 0 0 3.86370i 0 1.00000i 0 0 0
575.8 0 0 0 3.86370i 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.f 8
3.b odd 2 1 inner 4032.2.h.f 8
4.b odd 2 1 inner 4032.2.h.f 8
8.b even 2 1 1008.2.h.b 8
8.d odd 2 1 1008.2.h.b 8
12.b even 2 1 inner 4032.2.h.f 8
24.f even 2 1 1008.2.h.b 8
24.h odd 2 1 1008.2.h.b 8
56.e even 2 1 7056.2.h.k 8
56.h odd 2 1 7056.2.h.k 8
168.e odd 2 1 7056.2.h.k 8
168.i even 2 1 7056.2.h.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.h.b 8 8.b even 2 1
1008.2.h.b 8 8.d odd 2 1
1008.2.h.b 8 24.f even 2 1
1008.2.h.b 8 24.h odd 2 1
4032.2.h.f 8 1.a even 1 1 trivial
4032.2.h.f 8 3.b odd 2 1 inner
4032.2.h.f 8 4.b odd 2 1 inner
4032.2.h.f 8 12.b even 2 1 inner
7056.2.h.k 8 56.e even 2 1
7056.2.h.k 8 56.h odd 2 1
7056.2.h.k 8 168.e odd 2 1
7056.2.h.k 8 168.i even 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 16 T_{5}^{2} + 16$$ $$T_{11}^{4} - 28 T_{11}^{2} + 4$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 4 T^{2} + 6 T^{4} - 100 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$( 1 + 16 T^{2} + 114 T^{4} + 1936 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 2 T + 13 T^{2} )^{8}$$
$17$ $$( 1 - 52 T^{2} + 1206 T^{4} - 15028 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 44 T^{2} + 1014 T^{4} - 15884 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 64 T^{2} + 1890 T^{4} + 33856 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 64 T^{2} + 2514 T^{4} - 53824 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 28 T^{2} + 390 T^{4} - 26908 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 8 T + 42 T^{2} - 296 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 52 T^{2} + 3606 T^{4} - 87412 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 70 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 22 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 - 56 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 110 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 + 110 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 44 T^{2} + 2550 T^{4} + 197516 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 160 T^{2} + 14754 T^{4} + 806560 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 8 T + 150 T^{2} - 584 T^{3} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 50 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 - 116 T^{2} + 10230 T^{4} - 799124 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 340 T^{2} + 44694 T^{4} - 2693140 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 16 T + 246 T^{2} - 1552 T^{3} + 9409 T^{4} )^{4}$$