# Properties

 Label 6800.2.a.r Level $6800$ Weight $2$ Character orbit 6800.a Self dual yes Analytic conductor $54.298$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6800 = 2^{4} \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$54.2982733745$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 170) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 2q^{9} + O(q^{10})$$ $$q + q^{3} - 2q^{9} + 6q^{11} - 3q^{13} - q^{17} + 7q^{19} - 8q^{23} - 5q^{27} - 5q^{29} - 5q^{31} + 6q^{33} - 8q^{37} - 3q^{39} - 4q^{43} + 3q^{47} - 7q^{49} - q^{51} - 9q^{53} + 7q^{57} - 5q^{59} - 3q^{61} - 2q^{67} - 8q^{69} + 15q^{71} + 11q^{73} - 8q^{79} + q^{81} + 4q^{83} - 5q^{87} - q^{89} - 5q^{93} + 9q^{97} - 12q^{99} + O(q^{100})$$

## 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 1.00000 0 0 0 0 0 −2.00000 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$17$$ $$1$$

## 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 6800.2.a.r 1
4.b odd 2 1 850.2.a.h 1
5.b even 2 1 6800.2.a.g 1
5.c odd 4 2 1360.2.e.b 2
12.b even 2 1 7650.2.a.s 1
20.d odd 2 1 850.2.a.d 1
20.e even 4 2 170.2.c.a 2
60.h even 2 1 7650.2.a.cb 1
60.l odd 4 2 1530.2.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.c.a 2 20.e even 4 2
850.2.a.d 1 20.d odd 2 1
850.2.a.h 1 4.b odd 2 1
1360.2.e.b 2 5.c odd 4 2
1530.2.d.b 2 60.l odd 4 2
6800.2.a.g 1 5.b even 2 1
6800.2.a.r 1 1.a even 1 1 trivial
7650.2.a.s 1 12.b even 2 1
7650.2.a.cb 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6800))$$:

 $$T_{3} - 1$$ $$T_{7}$$ $$T_{11} - 6$$ $$T_{13} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-6 + T$$
$13$ $$3 + T$$
$17$ $$1 + T$$
$19$ $$-7 + T$$
$23$ $$8 + T$$
$29$ $$5 + T$$
$31$ $$5 + T$$
$37$ $$8 + T$$
$41$ $$T$$
$43$ $$4 + T$$
$47$ $$-3 + T$$
$53$ $$9 + T$$
$59$ $$5 + T$$
$61$ $$3 + T$$
$67$ $$2 + T$$
$71$ $$-15 + T$$
$73$ $$-11 + T$$
$79$ $$8 + T$$
$83$ $$-4 + T$$
$89$ $$1 + T$$
$97$ $$-9 + T$$