# Properties

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

# 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: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{5} + \zeta_{24}^{6} q^{7} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{5} + \zeta_{24}^{6} q^{7} + ( -1 - \zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{11} + ( -1 + 2 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{13} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{17} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{19} + ( -1 + \zeta_{24} - 6 \zeta_{24}^{2} + \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{23} + ( -1 - 2 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{25} + ( 2 + 3 \zeta_{24} - 3 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{29} + ( 4 \zeta_{24}^{5} + \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{31} + ( 1 - \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{35} + ( 5 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{37} + ( -1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{41} + ( -1 - 6 \zeta_{24} + 6 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{43} + ( 2 - \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{47} - q^{49} + ( -2 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{53} + ( 3 - 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{55} + ( -8 - \zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{59} + ( -2 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{61} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{65} + ( -5 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 10 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{67} + ( 9 + 2 \zeta_{24}^{2} - \zeta_{24}^{5} - \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{71} + ( 3 + 2 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{73} + ( -1 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{77} + ( 3 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{79} + ( 4 - 3 \zeta_{24} + 4 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 7 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{83} + ( 1 - 2 \zeta_{24} - 6 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{85} + ( 3 + 4 \zeta_{24} - 4 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{89} + ( -1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{91} + ( 7 + 3 \zeta_{24} + 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - \zeta_{24}^{6} + 5 \zeta_{24}^{7} ) q^{95} + ( 6 + 4 \zeta_{24} - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{11} - 8q^{13} - 8q^{23} - 8q^{25} + 8q^{35} + 40q^{37} + 16q^{47} - 8q^{49} - 64q^{59} - 16q^{61} + 72q^{71} + 24q^{73} + 32q^{83} + 8q^{85} + 56q^{95} + 48q^{97} + O(q^{100})$$

## 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 −0.965926 + 0.258819i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.258819 + 0.965926i
0 0 0 3.24969i 0 1.00000i 0 0 0
2591.2 0 0 0 2.66390i 0 1.00000i 0 0 0
2591.3 0 0 0 2.21441i 0 1.00000i 0 0 0
2591.4 0 0 0 1.19980i 0 1.00000i 0 0 0
2591.5 0 0 0 1.19980i 0 1.00000i 0 0 0
2591.6 0 0 0 2.21441i 0 1.00000i 0 0 0
2591.7 0 0 0 2.66390i 0 1.00000i 0 0 0
2591.8 0 0 0 3.24969i 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
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.a 8
3.b odd 2 1 6048.2.h.g yes 8
4.b odd 2 1 6048.2.h.g yes 8
12.b even 2 1 inner 6048.2.h.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.h.a 8 1.a even 1 1 trivial
6048.2.h.a 8 12.b even 2 1 inner
6048.2.h.g yes 8 3.b odd 2 1
6048.2.h.g yes 8 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(6048, [\chi])$$:

 $$T_{5}^{8} + 24 T_{5}^{6} + 194 T_{5}^{4} + 600 T_{5}^{2} + 529$$ $$T_{11}^{4} + 4 T_{11}^{3} - 12 T_{11}^{2} + 4 T_{11} + 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 16 T^{2} + 174 T^{4} - 1280 T^{6} + 7379 T^{8} - 32000 T^{10} + 108750 T^{12} - 250000 T^{14} + 390625 T^{16}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$( 1 + 4 T + 32 T^{2} + 136 T^{3} + 463 T^{4} + 1496 T^{5} + 3872 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 4 T + 36 T^{2} + 116 T^{3} + 602 T^{4} + 1508 T^{5} + 6084 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 56 T^{2} + 1330 T^{4} - 16184 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 26 T^{2} + 795 T^{4} - 9386 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 4 T + 32 T^{2} + 40 T^{3} + 655 T^{4} + 920 T^{5} + 16928 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$1 - 64 T^{2} + 4452 T^{4} - 163520 T^{6} + 6050150 T^{8} - 137520320 T^{10} + 3148815012 T^{12} - 38068692544 T^{14} + 500246412961 T^{16}$$
$31$ $$1 - 116 T^{2} + 7098 T^{4} - 315760 T^{6} + 11036483 T^{8} - 303445360 T^{10} + 6555152058 T^{12} - 102950426996 T^{14} + 852891037441 T^{16}$$
$37$ $$( 1 - 10 T + 91 T^{2} - 370 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$1 - 288 T^{2} + 37342 T^{4} - 2869632 T^{6} + 143812995 T^{8} - 4823851392 T^{10} + 105519567262 T^{12} - 1368030021408 T^{14} + 7984925229121 T^{16}$$
$43$ $$1 - 40 T^{2} + 5668 T^{4} - 165496 T^{6} + 14065894 T^{8} - 306002104 T^{10} + 19377764068 T^{12} - 252854521960 T^{14} + 11688200277601 T^{16}$$
$47$ $$( 1 - 4 T + 96 T^{2} - 188 T^{3} + 2209 T^{4} )^{4}$$
$53$ $$1 - 136 T^{2} + 6300 T^{4} + 118408 T^{6} - 20822362 T^{8} + 332608072 T^{10} + 49710030300 T^{12} - 3014353113544 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 + 32 T + 584 T^{2} + 7136 T^{3} + 63778 T^{4} + 421024 T^{5} + 2032904 T^{6} + 6572128 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 4 T + 78 T^{2} + 244 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$1 - 200 T^{2} + 27492 T^{4} - 2608600 T^{6} + 201429158 T^{8} - 11710005400 T^{10} + 553994618532 T^{12} - 18091676433800 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 - 36 T + 760 T^{2} - 10392 T^{3} + 103479 T^{4} - 737832 T^{5} + 3831160 T^{6} - 12884796 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 12 T + 228 T^{2} - 1644 T^{3} + 20234 T^{4} - 120012 T^{5} + 1215012 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$1 - 424 T^{2} + 86788 T^{4} - 11475640 T^{6} + 1070068678 T^{8} - 71619469240 T^{10} + 3380399629828 T^{12} - 103069081140904 T^{14} + 1517108809906561 T^{16}$$
$83$ $$( 1 - 16 T + 88 T^{2} - 560 T^{3} + 8322 T^{4} - 46480 T^{5} + 606232 T^{6} - 9148592 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$1 - 496 T^{2} + 119790 T^{4} - 18269696 T^{6} + 1929877235 T^{8} - 144714262016 T^{10} + 7515893049390 T^{12} - 246502720316656 T^{14} + 3936588805702081 T^{16}$$
$97$ $$( 1 - 24 T + 444 T^{2} - 5928 T^{3} + 63110 T^{4} - 575016 T^{5} + 4177596 T^{6} - 21904152 T^{7} + 88529281 T^{8} )^{2}$$