# Properties

 Label 6048.2.c.a Level 6048 Weight 2 Character orbit 6048.c Analytic conductor 48.294 Analytic rank 0 Dimension 2 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.2935231425$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1512) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{5} + q^{7} +O(q^{10})$$ $$q + 2 i q^{5} + q^{7} -5 i q^{13} - q^{17} + 4 i q^{19} -5 q^{23} + q^{25} -9 i q^{29} + 7 q^{31} + 2 i q^{35} -2 i q^{37} -2 q^{41} + 5 i q^{43} + q^{49} -9 i q^{53} -i q^{59} -6 i q^{61} + 10 q^{65} -9 i q^{67} -15 q^{71} + 14 q^{79} -4 i q^{83} -2 i q^{85} + 13 q^{89} -5 i q^{91} -8 q^{95} -14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{7} + O(q^{10})$$ $$2q + 2q^{7} - 2q^{17} - 10q^{23} + 2q^{25} + 14q^{31} - 4q^{41} + 2q^{49} + 20q^{65} - 30q^{71} + 28q^{79} + 26q^{89} - 16q^{95} - 28q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3025.1
 − 1.00000i 1.00000i
0 0 0 2.00000i 0 1.00000 0 0 0
3025.2 0 0 0 2.00000i 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.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.c.a 2
3.b odd 2 1 6048.2.c.b 2
4.b odd 2 1 1512.2.c.b yes 2
8.b even 2 1 inner 6048.2.c.a 2
8.d odd 2 1 1512.2.c.b yes 2
12.b even 2 1 1512.2.c.a 2
24.f even 2 1 1512.2.c.a 2
24.h odd 2 1 6048.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.a 2 12.b even 2 1
1512.2.c.a 2 24.f even 2 1
1512.2.c.b yes 2 4.b odd 2 1
1512.2.c.b yes 2 8.d odd 2 1
6048.2.c.a 2 1.a even 1 1 trivial
6048.2.c.a 2 8.b even 2 1 inner
6048.2.c.b 2 3.b odd 2 1
6048.2.c.b 2 24.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 4$$ $$T_{17} + 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$7$ $$( 1 - T )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$1 - T^{2} + 169 T^{4}$$
$17$ $$( 1 + T + 17 T^{2} )^{2}$$
$19$ $$1 - 22 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 5 T + 23 T^{2} )^{2}$$
$29$ $$1 + 23 T^{2} + 841 T^{4}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$1 - 61 T^{2} + 1849 T^{4}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$1 - 25 T^{2} + 2809 T^{4}$$
$59$ $$1 - 117 T^{2} + 3481 T^{4}$$
$61$ $$1 - 86 T^{2} + 3721 T^{4}$$
$67$ $$1 - 53 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 15 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 73 T^{2} )^{2}$$
$79$ $$( 1 - 14 T + 79 T^{2} )^{2}$$
$83$ $$1 - 150 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 13 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 14 T + 97 T^{2} )^{2}$$