# Properties

 Label 6048.2.h.f Level 6048 Weight 2 Character orbit 6048.h Analytic conductor 48.294 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6048 = 2^{5} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6048.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.2935231425$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + \zeta_{24}^{6} q^{7} +O(q^{10})$$ $$q + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + \zeta_{24}^{6} q^{7} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{11} + 4 q^{13} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{17} -\zeta_{24}^{6} q^{19} + ( -\zeta_{24} - \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{23} + ( -1 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{25} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{29} + ( -1 + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{31} + ( \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{35} + ( -10 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{37} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 7 \zeta_{24}^{7} ) q^{41} + ( -2 + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{43} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{47} - q^{49} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{53} -3 \zeta_{24}^{6} q^{55} + ( -2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{59} + ( -2 + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{61} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{65} + ( -4 + 8 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{67} + ( 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{71} + ( -4 + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{73} + ( -\zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{77} + ( 2 - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{79} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{83} + ( 9 - 14 \zeta_{24}^{2} + 7 \zeta_{24}^{6} ) q^{85} + ( -9 \zeta_{24} + 9 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{89} + 4 \zeta_{24}^{6} q^{91} + ( -\zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{95} + ( -4 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 32q^{13} - 8q^{25} - 80q^{37} - 8q^{49} - 16q^{61} - 32q^{73} + 72q^{85} - 32q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$$.

 $$n$$ $$2593$$ $$3781$$ $$3809$$ $$4159$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2591.1
 0.258819 − 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i 0.965926 − 0.258819i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 + 0.965926i
0 0 0 3.34607i 0 1.00000i 0 0 0
2591.2 0 0 0 3.34607i 0 1.00000i 0 0 0
2591.3 0 0 0 0.896575i 0 1.00000i 0 0 0
2591.4 0 0 0 0.896575i 0 1.00000i 0 0 0
2591.5 0 0 0 0.896575i 0 1.00000i 0 0 0
2591.6 0 0 0 0.896575i 0 1.00000i 0 0 0
2591.7 0 0 0 3.34607i 0 1.00000i 0 0 0
2591.8 0 0 0 3.34607i 0 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2591.8 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 6048.2.h.f 8
3.b odd 2 1 inner 6048.2.h.f 8
4.b odd 2 1 inner 6048.2.h.f 8
12.b even 2 1 inner 6048.2.h.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.h.f 8 1.a even 1 1 trivial
6048.2.h.f 8 3.b odd 2 1 inner
6048.2.h.f 8 4.b odd 2 1 inner
6048.2.h.f 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 $$S_{2}^{\mathrm{new}}(6048, [\chi])$$:

 $$T_{5}^{4} + 12 T_{5}^{2} + 9$$ $$T_{11}^{4} - 12 T_{11}^{2} + 9$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 8 T^{2} + 39 T^{4} - 200 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$( 1 + 32 T^{2} + 471 T^{4} + 3872 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 4 T + 13 T^{2} )^{8}$$
$17$ $$( 1 - 16 T^{2} + 450 T^{4} - 4624 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 37 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 + 40 T^{2} + 783 T^{4} + 21160 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 52 T^{2} + 1590 T^{4} - 43732 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 110 T^{2} + 4899 T^{4} - 105710 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 20 T + 171 T^{2} + 740 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 16 T^{2} + 3279 T^{4} - 26896 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 116 T^{2} + 6294 T^{4} - 214484 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 76 T^{2} + 5430 T^{4} + 167884 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 98 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 220 T^{2} + 19014 T^{4} + 765820 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 4 T + 18 T^{2} + 244 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$( 1 - 140 T^{2} + 10806 T^{4} - 628460 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 248 T^{2} + 25215 T^{4} + 1250168 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 8 T + 54 T^{2} + 584 T^{3} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 260 T^{2} + 28614 T^{4} - 1622660 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 124 T^{2} + 14550 T^{4} + 854236 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 64 T^{2} + 15999 T^{4} - 506944 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 8 T + 162 T^{2} + 776 T^{3} + 9409 T^{4} )^{4}$$