# Properties

 Label 6768.2.a.h Level $6768$ Weight $2$ Character orbit 6768.a Self dual yes Analytic conductor $54.043$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6768 = 2^{4} \cdot 3^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6768.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$54.0427520880$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 141) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{7} + O(q^{10})$$ $$q - 4q^{7} + 6q^{13} + 6q^{17} - 2q^{19} + 4q^{23} - 5q^{25} - 8q^{29} - 6q^{31} - 6q^{37} + 8q^{41} + 6q^{43} + q^{47} + 9q^{49} - 2q^{53} + 12q^{59} + 2q^{61} + 2q^{67} - 10q^{73} + 4q^{79} + 4q^{83} + 10q^{89} - 24q^{91} - 18q^{97} + O(q^{100})$$

## 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 0 0 −4.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$47$$ $$-1$$

## 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 6768.2.a.h 1
3.b odd 2 1 2256.2.a.l 1
4.b odd 2 1 423.2.a.e 1
12.b even 2 1 141.2.a.b 1
24.f even 2 1 9024.2.a.bk 1
24.h odd 2 1 9024.2.a.i 1
60.h even 2 1 3525.2.a.k 1
84.h odd 2 1 6909.2.a.e 1
564.f odd 2 1 6627.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.2.a.b 1 12.b even 2 1
423.2.a.e 1 4.b odd 2 1
2256.2.a.l 1 3.b odd 2 1
3525.2.a.k 1 60.h even 2 1
6627.2.a.b 1 564.f odd 2 1
6768.2.a.h 1 1.a even 1 1 trivial
6909.2.a.e 1 84.h odd 2 1
9024.2.a.i 1 24.h odd 2 1
9024.2.a.bk 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6768))$$:

 $$T_{5}$$ $$T_{7} + 4$$ $$T_{11}$$ $$T_{13} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$T$$
$13$ $$-6 + T$$
$17$ $$-6 + T$$
$19$ $$2 + T$$
$23$ $$-4 + T$$
$29$ $$8 + T$$
$31$ $$6 + T$$
$37$ $$6 + T$$
$41$ $$-8 + T$$
$43$ $$-6 + T$$
$47$ $$-1 + T$$
$53$ $$2 + T$$
$59$ $$-12 + T$$
$61$ $$-2 + T$$
$67$ $$-2 + T$$
$71$ $$T$$
$73$ $$10 + T$$
$79$ $$-4 + T$$
$83$ $$-4 + T$$
$89$ $$-10 + T$$
$97$ $$18 + T$$