# Properties

 Label 6192.2.a.ba Level 6192 Weight 2 Character orbit 6192.a Self dual yes Analytic conductor 49.443 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6192 = 2^{4} \cdot 3^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6192.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$49.4433689316$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{5} + O(q^{10})$$ $$q + 4q^{5} + 3q^{11} - 5q^{13} + 3q^{17} + 2q^{19} - q^{23} + 11q^{25} + 6q^{29} + q^{31} - 5q^{41} + q^{43} + 4q^{47} - 7q^{49} + 5q^{53} + 12q^{55} - 12q^{59} + 2q^{61} - 20q^{65} + 3q^{67} + 2q^{71} + 2q^{73} + 8q^{79} + 15q^{83} + 12q^{85} + 4q^{89} + 8q^{95} + 7q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 4.00000 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6192.2.a.ba 1
3.b odd 2 1 688.2.a.b 1
4.b odd 2 1 387.2.a.e 1
12.b even 2 1 43.2.a.a 1
20.d odd 2 1 9675.2.a.b 1
24.f even 2 1 2752.2.a.f 1
24.h odd 2 1 2752.2.a.b 1
60.h even 2 1 1075.2.a.h 1
60.l odd 4 2 1075.2.b.b 2
84.h odd 2 1 2107.2.a.a 1
132.d odd 2 1 5203.2.a.a 1
156.h even 2 1 7267.2.a.a 1
516.h odd 2 1 1849.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.a.a 1 12.b even 2 1
387.2.a.e 1 4.b odd 2 1
688.2.a.b 1 3.b odd 2 1
1075.2.a.h 1 60.h even 2 1
1075.2.b.b 2 60.l odd 4 2
1849.2.a.d 1 516.h odd 2 1
2107.2.a.a 1 84.h odd 2 1
2752.2.a.b 1 24.h odd 2 1
2752.2.a.f 1 24.f even 2 1
5203.2.a.a 1 132.d odd 2 1
6192.2.a.ba 1 1.a even 1 1 trivial
7267.2.a.a 1 156.h even 2 1
9675.2.a.b 1 20.d odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$43$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6192))$$:

 $$T_{5} - 4$$ $$T_{7}$$ $$T_{11} - 3$$ $$T_{13} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 4 T + 5 T^{2}$$
$7$ $$1 + 7 T^{2}$$
$11$ $$1 - 3 T + 11 T^{2}$$
$13$ $$1 + 5 T + 13 T^{2}$$
$17$ $$1 - 3 T + 17 T^{2}$$
$19$ $$1 - 2 T + 19 T^{2}$$
$23$ $$1 + T + 23 T^{2}$$
$29$ $$1 - 6 T + 29 T^{2}$$
$31$ $$1 - T + 31 T^{2}$$
$37$ $$1 + 37 T^{2}$$
$41$ $$1 + 5 T + 41 T^{2}$$
$43$ $$1 - T$$
$47$ $$1 - 4 T + 47 T^{2}$$
$53$ $$1 - 5 T + 53 T^{2}$$
$59$ $$1 + 12 T + 59 T^{2}$$
$61$ $$1 - 2 T + 61 T^{2}$$
$67$ $$1 - 3 T + 67 T^{2}$$
$71$ $$1 - 2 T + 71 T^{2}$$
$73$ $$1 - 2 T + 73 T^{2}$$
$79$ $$1 - 8 T + 79 T^{2}$$
$83$ $$1 - 15 T + 83 T^{2}$$
$89$ $$1 - 4 T + 89 T^{2}$$
$97$ $$1 - 7 T + 97 T^{2}$$