# Properties

 Label 1050.2.a.r 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

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

## 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.r yes 1
3.b odd 2 1 3150.2.a.n 1
4.b odd 2 1 8400.2.a.d 1
5.b even 2 1 1050.2.a.b 1
5.c odd 4 2 1050.2.g.i 2
7.b odd 2 1 7350.2.a.cb 1
15.d odd 2 1 3150.2.a.y 1
15.e even 4 2 3150.2.g.h 2
20.d odd 2 1 8400.2.a.ck 1
35.c odd 2 1 7350.2.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.a.b 1 5.b even 2 1
1050.2.a.r yes 1 1.a even 1 1 trivial
1050.2.g.i 2 5.c odd 4 2
3150.2.a.n 1 3.b odd 2 1
3150.2.a.y 1 15.d odd 2 1
3150.2.g.h 2 15.e even 4 2
7350.2.a.bi 1 35.c odd 2 1
7350.2.a.cb 1 7.b odd 2 1
8400.2.a.d 1 4.b odd 2 1
8400.2.a.ck 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1050))$$:

 $$T_{11} - 2$$ $$T_{13} + 1$$ $$T_{17} - 3$$ $$T_{19}$$

## Hecke characteristic polynomials

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