# Properties

 Label 6048.2.h.e 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: $$2^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{5} + \zeta_{24}^{6} q^{7} +O(q^{10})$$ $$q + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{5} + \zeta_{24}^{6} q^{7} + ( 2 \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{11} + ( 3 + \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{13} + ( 1 - \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{17} -4 \zeta_{24}^{6} q^{19} + ( -\zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{23} -3 q^{25} + ( -3 + 6 \zeta_{24}^{4} ) q^{29} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{31} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{35} + ( -4 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{37} + ( -2 + 4 \zeta_{24}^{4} ) q^{41} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{43} + ( 4 \zeta_{24} + 8 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{47} - q^{49} + ( -3 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{53} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{55} + ( -2 \zeta_{24} + 10 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 5 \zeta_{24}^{6} ) q^{59} + ( 8 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{61} + ( 4 + 6 \zeta_{24} - 6 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - 6 \zeta_{24}^{5} ) q^{65} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{67} + ( -3 \zeta_{24} + 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{71} -4 q^{73} + ( -2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{77} + 10 \zeta_{24}^{6} q^{79} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{83} + ( 4 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{85} + ( -7 + \zeta_{24} - \zeta_{24}^{3} + 14 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{89} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{91} + ( -8 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} ) q^{95} + ( -6 - 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 24q^{13} - 24q^{25} - 32q^{37} - 8q^{49} + 64q^{61} - 32q^{73} + 32q^{85} - 48q^{97} + O(q^{100})$$

## 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 −0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 + 0.258819i −0.258819 + 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 − 0.258819i
0 0 0 2.82843i 0 1.00000i 0 0 0
2591.2 0 0 0 2.82843i 0 1.00000i 0 0 0
2591.3 0 0 0 2.82843i 0 1.00000i 0 0 0
2591.4 0 0 0 2.82843i 0 1.00000i 0 0 0
2591.5 0 0 0 2.82843i 0 1.00000i 0 0 0
2591.6 0 0 0 2.82843i 0 1.00000i 0 0 0
2591.7 0 0 0 2.82843i 0 1.00000i 0 0 0
2591.8 0 0 0 2.82843i 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.e 8
3.b odd 2 1 inner 6048.2.h.e 8
4.b odd 2 1 inner 6048.2.h.e 8
12.b even 2 1 inner 6048.2.h.e 8

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

 $$T_{5}^{2} + 8$$ $$T_{11}^{4} - 40 T_{11}^{2} + 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 2 T^{2} + 25 T^{4} )^{4}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$( 1 + 4 T^{2} - 138 T^{4} + 484 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 6 T + 29 T^{2} - 78 T^{3} + 169 T^{4} )^{4}$$
$17$ $$( 1 - 58 T^{2} + 1395 T^{4} - 16762 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 22 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 + 82 T^{2} + 2715 T^{4} + 43378 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 31 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 70 T^{2} + 2763 T^{4} + 67270 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 8 T + 66 T^{2} + 296 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 70 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 142 T^{2} + 8523 T^{4} - 262558 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 28 T^{2} - 1530 T^{4} + 61852 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 142 T^{2} + 9795 T^{4} - 398878 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 70 T^{2} + 5787 T^{4} + 243670 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 16 T + 162 T^{2} - 976 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$( 1 - 254 T^{2} + 25083 T^{4} - 1140206 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 242 T^{2} + 24507 T^{4} + 1219922 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 4 T + 73 T^{2} )^{8}$$
$79$ $$( 1 - 58 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 154 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 - 58 T^{2} + 15507 T^{4} - 459418 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 12 T + 134 T^{2} + 1164 T^{3} + 9409 T^{4} )^{4}$$