# Properties

 Label 6800.2.a.n Level $6800$ Weight $2$ Character orbit 6800.a Self dual yes Analytic conductor $54.298$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4 q^{7} - 3 q^{9}+O(q^{10})$$ q + 4 * q^7 - 3 * q^9 $$q + 4 q^{7} - 3 q^{9} + 2 q^{13} - q^{17} + 4 q^{19} + 4 q^{23} + 6 q^{29} - 4 q^{31} + 2 q^{37} - 6 q^{41} + 4 q^{43} + 9 q^{49} - 6 q^{53} + 12 q^{59} - 10 q^{61} - 12 q^{63} + 4 q^{67} + 4 q^{71} + 6 q^{73} - 12 q^{79} + 9 q^{81} - 4 q^{83} + 10 q^{89} + 8 q^{91} - 2 q^{97}+O(q^{100})$$ q + 4 * q^7 - 3 * q^9 + 2 * q^13 - q^17 + 4 * q^19 + 4 * q^23 + 6 * q^29 - 4 * q^31 + 2 * q^37 - 6 * q^41 + 4 * q^43 + 9 * q^49 - 6 * q^53 + 12 * q^59 - 10 * q^61 - 12 * q^63 + 4 * q^67 + 4 * q^71 + 6 * q^73 - 12 * q^79 + 9 * q^81 - 4 * q^83 + 10 * q^89 + 8 * q^91 - 2 * q^97

## 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 −3.00000 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.n 1
4.b odd 2 1 425.2.a.d 1
5.b even 2 1 272.2.a.b 1
12.b even 2 1 3825.2.a.d 1
15.d odd 2 1 2448.2.a.o 1
20.d odd 2 1 17.2.a.a 1
20.e even 4 2 425.2.b.b 2
40.e odd 2 1 1088.2.a.i 1
40.f even 2 1 1088.2.a.h 1
60.h even 2 1 153.2.a.c 1
68.d odd 2 1 7225.2.a.g 1
85.c even 2 1 4624.2.a.d 1
120.i odd 2 1 9792.2.a.i 1
120.m even 2 1 9792.2.a.n 1
140.c even 2 1 833.2.a.a 1
140.p odd 6 2 833.2.e.b 2
140.s even 6 2 833.2.e.a 2
220.g even 2 1 2057.2.a.e 1
260.g odd 2 1 2873.2.a.c 1
340.d odd 2 1 289.2.a.a 1
340.n odd 4 2 289.2.b.a 2
340.ba odd 8 4 289.2.c.a 4
340.bg even 16 8 289.2.d.d 8
380.d even 2 1 6137.2.a.b 1
420.o odd 2 1 7497.2.a.l 1
460.g even 2 1 8993.2.a.a 1
1020.b even 2 1 2601.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 20.d odd 2 1
153.2.a.c 1 60.h even 2 1
272.2.a.b 1 5.b even 2 1
289.2.a.a 1 340.d odd 2 1
289.2.b.a 2 340.n odd 4 2
289.2.c.a 4 340.ba odd 8 4
289.2.d.d 8 340.bg even 16 8
425.2.a.d 1 4.b odd 2 1
425.2.b.b 2 20.e even 4 2
833.2.a.a 1 140.c even 2 1
833.2.e.a 2 140.s even 6 2
833.2.e.b 2 140.p odd 6 2
1088.2.a.h 1 40.f even 2 1
1088.2.a.i 1 40.e odd 2 1
2057.2.a.e 1 220.g even 2 1
2448.2.a.o 1 15.d odd 2 1
2601.2.a.g 1 1020.b even 2 1
2873.2.a.c 1 260.g odd 2 1
3825.2.a.d 1 12.b even 2 1
4624.2.a.d 1 85.c even 2 1
6137.2.a.b 1 380.d even 2 1
6800.2.a.n 1 1.a even 1 1 trivial
7225.2.a.g 1 68.d odd 2 1
7497.2.a.l 1 420.o odd 2 1
8993.2.a.a 1 460.g even 2 1
9792.2.a.i 1 120.i odd 2 1
9792.2.a.n 1 120.m even 2 1

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{7} - 4$$ T7 - 4 $$T_{11}$$ T11 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

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