# Properties

 Label 804.2.a.d Level 804 Weight 2 Character orbit 804.a Self dual yes Analytic conductor 6.420 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$804 = 2^{2} \cdot 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 804.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.41997232251$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} - 3q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - q^{5} - 3q^{7} + q^{9} - 2q^{11} - 2q^{13} - q^{15} - 4q^{17} - 4q^{19} - 3q^{21} + 7q^{23} - 4q^{25} + q^{27} - 8q^{29} + 3q^{31} - 2q^{33} + 3q^{35} - 3q^{37} - 2q^{39} + q^{41} - 11q^{43} - q^{45} + 2q^{49} - 4q^{51} + 11q^{53} + 2q^{55} - 4q^{57} - 3q^{59} + 8q^{61} - 3q^{63} + 2q^{65} - q^{67} + 7q^{69} + 8q^{71} - 9q^{73} - 4q^{75} + 6q^{77} + q^{81} + 11q^{83} + 4q^{85} - 8q^{87} - 6q^{89} + 6q^{91} + 3q^{93} + 4q^{95} - 6q^{97} - 2q^{99} + O(q^{100})$$

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

## 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 804.2.a.d 1
3.b odd 2 1 2412.2.a.d 1
4.b odd 2 1 3216.2.a.c 1
12.b even 2 1 9648.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.a.d 1 1.a even 1 1 trivial
2412.2.a.d 1 3.b odd 2 1
3216.2.a.c 1 4.b odd 2 1
9648.2.a.m 1 12.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$67$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(804))$$.