# Properties

 Label 819.2.a.f Level $819$ Weight $2$ Character orbit 819.a Self dual yes Analytic conductor $6.540$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{2} + 2q^{4} + 3q^{5} - q^{7} + O(q^{10})$$ $$q + 2q^{2} + 2q^{4} + 3q^{5} - q^{7} + 6q^{10} + 6q^{11} - q^{13} - 2q^{14} - 4q^{16} - 4q^{17} + 5q^{19} + 6q^{20} + 12q^{22} - 3q^{23} + 4q^{25} - 2q^{26} - 2q^{28} + 5q^{29} - 3q^{31} - 8q^{32} - 8q^{34} - 3q^{35} - 4q^{37} + 10q^{38} + 6q^{41} - q^{43} + 12q^{44} - 6q^{46} - 7q^{47} + q^{49} + 8q^{50} - 2q^{52} + 9q^{53} + 18q^{55} + 10q^{58} - 8q^{59} - 10q^{61} - 6q^{62} - 8q^{64} - 3q^{65} - 6q^{67} - 8q^{68} - 6q^{70} + 8q^{71} - 13q^{73} - 8q^{74} + 10q^{76} - 6q^{77} + 3q^{79} - 12q^{80} + 12q^{82} - 15q^{83} - 12q^{85} - 2q^{86} - 3q^{89} + q^{91} - 6q^{92} - 14q^{94} + 15q^{95} + 7q^{97} + 2q^{98} + O(q^{100})$$

## 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
2.00000 0 2.00000 3.00000 0 −1.00000 0 0 6.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$13$$ $$1$$

## 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 819.2.a.f 1
3.b odd 2 1 91.2.a.a 1
7.b odd 2 1 5733.2.a.l 1
12.b even 2 1 1456.2.a.g 1
15.d odd 2 1 2275.2.a.h 1
21.c even 2 1 637.2.a.a 1
21.g even 6 2 637.2.e.d 2
21.h odd 6 2 637.2.e.e 2
24.f even 2 1 5824.2.a.t 1
24.h odd 2 1 5824.2.a.s 1
39.d odd 2 1 1183.2.a.b 1
39.f even 4 2 1183.2.c.b 2
273.g even 2 1 8281.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.a 1 3.b odd 2 1
637.2.a.a 1 21.c even 2 1
637.2.e.d 2 21.g even 6 2
637.2.e.e 2 21.h odd 6 2
819.2.a.f 1 1.a even 1 1 trivial
1183.2.a.b 1 39.d odd 2 1
1183.2.c.b 2 39.f even 4 2
1456.2.a.g 1 12.b even 2 1
2275.2.a.h 1 15.d odd 2 1
5733.2.a.l 1 7.b odd 2 1
5824.2.a.s 1 24.h odd 2 1
5824.2.a.t 1 24.f even 2 1
8281.2.a.l 1 273.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(819))$$:

 $$T_{2} - 2$$ $$T_{5} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$T$$
$5$ $$-3 + T$$
$7$ $$1 + T$$
$11$ $$-6 + T$$
$13$ $$1 + T$$
$17$ $$4 + T$$
$19$ $$-5 + T$$
$23$ $$3 + T$$
$29$ $$-5 + T$$
$31$ $$3 + T$$
$37$ $$4 + T$$
$41$ $$-6 + T$$
$43$ $$1 + T$$
$47$ $$7 + T$$
$53$ $$-9 + T$$
$59$ $$8 + T$$
$61$ $$10 + T$$
$67$ $$6 + T$$
$71$ $$-8 + T$$
$73$ $$13 + T$$
$79$ $$-3 + T$$
$83$ $$15 + T$$
$89$ $$3 + T$$
$97$ $$-7 + T$$