# Properties

 Label 6800.2.a.b 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 34) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{3} - 4q^{7} + q^{9} + O(q^{10})$$ $$q - 2q^{3} - 4q^{7} + q^{9} - 6q^{11} - 2q^{13} + q^{17} + 4q^{19} + 8q^{21} + 4q^{27} + 4q^{31} + 12q^{33} + 4q^{37} + 4q^{39} + 6q^{41} + 8q^{43} + 9q^{49} - 2q^{51} + 6q^{53} - 8q^{57} - 4q^{61} - 4q^{63} + 8q^{67} - 2q^{73} + 24q^{77} - 8q^{79} - 11q^{81} - 6q^{89} + 8q^{91} - 8q^{93} - 14q^{97} - 6q^{99} + O(q^{100})$$

## 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 −2.00000 0 0 0 −4.00000 0 1.00000 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.b 1
4.b odd 2 1 850.2.a.e 1
5.b even 2 1 272.2.a.d 1
12.b even 2 1 7650.2.a.ci 1
15.d odd 2 1 2448.2.a.k 1
20.d odd 2 1 34.2.a.a 1
20.e even 4 2 850.2.c.b 2
40.e odd 2 1 1088.2.a.l 1
40.f even 2 1 1088.2.a.d 1
60.h even 2 1 306.2.a.a 1
85.c even 2 1 4624.2.a.a 1
120.i odd 2 1 9792.2.a.bj 1
120.m even 2 1 9792.2.a.y 1
140.c even 2 1 1666.2.a.m 1
220.g even 2 1 4114.2.a.a 1
260.g odd 2 1 5746.2.a.b 1
340.d odd 2 1 578.2.a.a 1
340.n odd 4 2 578.2.b.a 2
340.ba odd 8 4 578.2.c.e 4
340.bg even 16 8 578.2.d.e 8
1020.b even 2 1 5202.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a 1 20.d odd 2 1
272.2.a.d 1 5.b even 2 1
306.2.a.a 1 60.h even 2 1
578.2.a.a 1 340.d odd 2 1
578.2.b.a 2 340.n odd 4 2
578.2.c.e 4 340.ba odd 8 4
578.2.d.e 8 340.bg even 16 8
850.2.a.e 1 4.b odd 2 1
850.2.c.b 2 20.e even 4 2
1088.2.a.d 1 40.f even 2 1
1088.2.a.l 1 40.e odd 2 1
1666.2.a.m 1 140.c even 2 1
2448.2.a.k 1 15.d odd 2 1
4114.2.a.a 1 220.g even 2 1
4624.2.a.a 1 85.c even 2 1
5202.2.a.d 1 1020.b even 2 1
5746.2.a.b 1 260.g odd 2 1
6800.2.a.b 1 1.a even 1 1 trivial
7650.2.a.ci 1 12.b even 2 1
9792.2.a.y 1 120.m even 2 1
9792.2.a.bj 1 120.i odd 2 1

## Hecke kernels

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

 $$T_{3} + 2$$ $$T_{7} + 4$$ $$T_{11} + 6$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$2 + T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$6 + T$$
$13$ $$2 + T$$
$17$ $$-1 + T$$
$19$ $$-4 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$-4 + T$$
$37$ $$-4 + T$$
$41$ $$-6 + T$$
$43$ $$-8 + T$$
$47$ $$T$$
$53$ $$-6 + T$$
$59$ $$T$$
$61$ $$4 + T$$
$67$ $$-8 + T$$
$71$ $$T$$
$73$ $$2 + T$$
$79$ $$8 + T$$
$83$ $$T$$
$89$ $$6 + T$$
$97$ $$14 + T$$