# Properties

 Label 1050.2.a.l Level $1050$ Weight $2$ Character orbit 1050.a Self dual yes Analytic conductor $8.384$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} + q^{9} + 6q^{11} - q^{12} + q^{13} - q^{14} + q^{16} - 3q^{17} + q^{18} - 4q^{19} + q^{21} + 6q^{22} + 3q^{23} - q^{24} + q^{26} - q^{27} - q^{28} + 3q^{29} + 5q^{31} + q^{32} - 6q^{33} - 3q^{34} + q^{36} + 10q^{37} - 4q^{38} - q^{39} + 9q^{41} + q^{42} + q^{43} + 6q^{44} + 3q^{46} - q^{48} + q^{49} + 3q^{51} + q^{52} - 9q^{53} - q^{54} - q^{56} + 4q^{57} + 3q^{58} + 9q^{59} + 11q^{61} + 5q^{62} - q^{63} + q^{64} - 6q^{66} + 4q^{67} - 3q^{68} - 3q^{69} - 12q^{71} + q^{72} + 10q^{73} + 10q^{74} - 4q^{76} - 6q^{77} - q^{78} - 10q^{79} + q^{81} + 9q^{82} - 9q^{83} + q^{84} + q^{86} - 3q^{87} + 6q^{88} - 6q^{89} - q^{91} + 3q^{92} - 5q^{93} - q^{96} - 14q^{97} + q^{98} + 6q^{99} + O(q^{100})$$

## 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
1.00000 −1.00000 1.00000 0 −1.00000 −1.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$

## 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 1050.2.a.l yes 1
3.b odd 2 1 3150.2.a.a 1
4.b odd 2 1 8400.2.a.ci 1
5.b even 2 1 1050.2.a.j 1
5.c odd 4 2 1050.2.g.e 2
7.b odd 2 1 7350.2.a.cz 1
15.d odd 2 1 3150.2.a.bg 1
15.e even 4 2 3150.2.g.a 2
20.d odd 2 1 8400.2.a.a 1
35.c odd 2 1 7350.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.a.j 1 5.b even 2 1
1050.2.a.l yes 1 1.a even 1 1 trivial
1050.2.g.e 2 5.c odd 4 2
3150.2.a.a 1 3.b odd 2 1
3150.2.a.bg 1 15.d odd 2 1
3150.2.g.a 2 15.e even 4 2
7350.2.a.r 1 35.c odd 2 1
7350.2.a.cz 1 7.b odd 2 1
8400.2.a.a 1 20.d odd 2 1
8400.2.a.ci 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1050))$$:

 $$T_{11} - 6$$ $$T_{13} - 1$$ $$T_{17} + 3$$ $$T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$-6 + T$$
$13$ $$-1 + T$$
$17$ $$3 + T$$
$19$ $$4 + T$$
$23$ $$-3 + T$$
$29$ $$-3 + T$$
$31$ $$-5 + T$$
$37$ $$-10 + T$$
$41$ $$-9 + T$$
$43$ $$-1 + T$$
$47$ $$T$$
$53$ $$9 + T$$
$59$ $$-9 + T$$
$61$ $$-11 + T$$
$67$ $$-4 + T$$
$71$ $$12 + T$$
$73$ $$-10 + T$$
$79$ $$10 + T$$
$83$ $$9 + T$$
$89$ $$6 + T$$
$97$ $$14 + T$$