# Properties

 Label 1050.2.a.f 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} + q^{9} - 2q^{11} + q^{12} - 7q^{13} + q^{14} + q^{16} + 7q^{17} - q^{18} + 8q^{19} - q^{21} + 2q^{22} + 5q^{23} - q^{24} + 7q^{26} + q^{27} - q^{28} + 9q^{29} + q^{31} - q^{32} - 2q^{33} - 7q^{34} + q^{36} + 2q^{37} - 8q^{38} - 7q^{39} + 11q^{41} + q^{42} - 3q^{43} - 2q^{44} - 5q^{46} - 4q^{47} + q^{48} + q^{49} + 7q^{51} - 7q^{52} - 3q^{53} - q^{54} + q^{56} + 8q^{57} - 9q^{58} + 7q^{59} - 5q^{61} - q^{62} - q^{63} + q^{64} + 2q^{66} + 12q^{67} + 7q^{68} + 5q^{69} - 4q^{71} - q^{72} - 10q^{73} - 2q^{74} + 8q^{76} + 2q^{77} + 7q^{78} - 6q^{79} + q^{81} - 11q^{82} + 9q^{83} - q^{84} + 3q^{86} + 9q^{87} + 2q^{88} - 10q^{89} + 7q^{91} + 5q^{92} + q^{93} + 4q^{94} - q^{96} - 10q^{97} - q^{98} - 2q^{99} + O(q^{100})$$

## 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
−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.f 1
3.b odd 2 1 3150.2.a.bd 1
4.b odd 2 1 8400.2.a.bb 1
5.b even 2 1 1050.2.a.n yes 1
5.c odd 4 2 1050.2.g.b 2
7.b odd 2 1 7350.2.a.i 1
15.d odd 2 1 3150.2.a.r 1
15.e even 4 2 3150.2.g.p 2
20.d odd 2 1 8400.2.a.cb 1
35.c odd 2 1 7350.2.a.cm 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.a.f 1 1.a even 1 1 trivial
1050.2.a.n yes 1 5.b even 2 1
1050.2.g.b 2 5.c odd 4 2
3150.2.a.r 1 15.d odd 2 1
3150.2.a.bd 1 3.b odd 2 1
3150.2.g.p 2 15.e even 4 2
7350.2.a.i 1 7.b odd 2 1
7350.2.a.cm 1 35.c odd 2 1
8400.2.a.bb 1 4.b odd 2 1
8400.2.a.cb 1 20.d odd 2 1

## Hecke kernels

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

 $$T_{11} + 2$$ $$T_{13} + 7$$ $$T_{17} - 7$$ $$T_{19} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$2 + T$$
$13$ $$7 + T$$
$17$ $$-7 + T$$
$19$ $$-8 + T$$
$23$ $$-5 + T$$
$29$ $$-9 + T$$
$31$ $$-1 + T$$
$37$ $$-2 + T$$
$41$ $$-11 + T$$
$43$ $$3 + T$$
$47$ $$4 + T$$
$53$ $$3 + T$$
$59$ $$-7 + T$$
$61$ $$5 + T$$
$67$ $$-12 + T$$
$71$ $$4 + T$$
$73$ $$10 + T$$
$79$ $$6 + T$$
$83$ $$-9 + T$$
$89$ $$10 + T$$
$97$ $$10 + T$$