# Properties

 Label 1002.2.a.a Level 1002 Weight 2 Character orbit 1002.a Self dual Yes Analytic conductor 8.001 Analytic rank 0 Dimension 1 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1002 = 2 \cdot 3 \cdot 167$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1002.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$8.00101028253$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut -\mathstrut q^{3}$$ $$\mathstrut +\mathstrut q^{4}$$ $$\mathstrut +\mathstrut q^{6}$$ $$\mathstrut -\mathstrut q^{8}$$ $$\mathstrut +\mathstrut q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut -\mathstrut q^{3}$$ $$\mathstrut +\mathstrut q^{4}$$ $$\mathstrut +\mathstrut q^{6}$$ $$\mathstrut -\mathstrut q^{8}$$ $$\mathstrut +\mathstrut q^{9}$$ $$\mathstrut -\mathstrut q^{12}$$ $$\mathstrut -\mathstrut 4q^{13}$$ $$\mathstrut +\mathstrut q^{16}$$ $$\mathstrut +\mathstrut 6q^{17}$$ $$\mathstrut -\mathstrut q^{18}$$ $$\mathstrut +\mathstrut 4q^{23}$$ $$\mathstrut +\mathstrut q^{24}$$ $$\mathstrut -\mathstrut 5q^{25}$$ $$\mathstrut +\mathstrut 4q^{26}$$ $$\mathstrut -\mathstrut q^{27}$$ $$\mathstrut +\mathstrut 6q^{29}$$ $$\mathstrut -\mathstrut q^{32}$$ $$\mathstrut -\mathstrut 6q^{34}$$ $$\mathstrut +\mathstrut q^{36}$$ $$\mathstrut -\mathstrut 4q^{37}$$ $$\mathstrut +\mathstrut 4q^{39}$$ $$\mathstrut -\mathstrut 10q^{41}$$ $$\mathstrut +\mathstrut 10q^{43}$$ $$\mathstrut -\mathstrut 4q^{46}$$ $$\mathstrut +\mathstrut 8q^{47}$$ $$\mathstrut -\mathstrut q^{48}$$ $$\mathstrut -\mathstrut 7q^{49}$$ $$\mathstrut +\mathstrut 5q^{50}$$ $$\mathstrut -\mathstrut 6q^{51}$$ $$\mathstrut -\mathstrut 4q^{52}$$ $$\mathstrut +\mathstrut 12q^{53}$$ $$\mathstrut +\mathstrut q^{54}$$ $$\mathstrut -\mathstrut 6q^{58}$$ $$\mathstrut +\mathstrut 6q^{59}$$ $$\mathstrut +\mathstrut 6q^{61}$$ $$\mathstrut +\mathstrut q^{64}$$ $$\mathstrut +\mathstrut 10q^{67}$$ $$\mathstrut +\mathstrut 6q^{68}$$ $$\mathstrut -\mathstrut 4q^{69}$$ $$\mathstrut -\mathstrut 4q^{71}$$ $$\mathstrut -\mathstrut q^{72}$$ $$\mathstrut +\mathstrut 10q^{73}$$ $$\mathstrut +\mathstrut 4q^{74}$$ $$\mathstrut +\mathstrut 5q^{75}$$ $$\mathstrut -\mathstrut 4q^{78}$$ $$\mathstrut +\mathstrut q^{81}$$ $$\mathstrut +\mathstrut 10q^{82}$$ $$\mathstrut +\mathstrut 14q^{83}$$ $$\mathstrut -\mathstrut 10q^{86}$$ $$\mathstrut -\mathstrut 6q^{87}$$ $$\mathstrut -\mathstrut 6q^{89}$$ $$\mathstrut +\mathstrut 4q^{92}$$ $$\mathstrut -\mathstrut 8q^{94}$$ $$\mathstrut +\mathstrut q^{96}$$ $$\mathstrut +\mathstrut 14q^{97}$$ $$\mathstrut +\mathstrut 7q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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
−1.00000 −1.00000 1.00000 0 1.00000 0 −1.00000 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.

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$167$$ $$-1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1002))$$.