# Properties

 Label 1050.2.a.h 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: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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.h 1
3.b odd 2 1 3150.2.a.w 1
4.b odd 2 1 8400.2.a.p 1
5.b even 2 1 210.2.a.c 1
5.c odd 4 2 1050.2.g.d 2
7.b odd 2 1 7350.2.a.p 1
15.d odd 2 1 630.2.a.b 1
15.e even 4 2 3150.2.g.e 2
20.d odd 2 1 1680.2.a.q 1
35.c odd 2 1 1470.2.a.q 1
35.i odd 6 2 1470.2.i.b 2
35.j even 6 2 1470.2.i.f 2
40.e odd 2 1 6720.2.a.k 1
40.f even 2 1 6720.2.a.bp 1
60.h even 2 1 5040.2.a.i 1
105.g even 2 1 4410.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.c 1 5.b even 2 1
630.2.a.b 1 15.d odd 2 1
1050.2.a.h 1 1.a even 1 1 trivial
1050.2.g.d 2 5.c odd 4 2
1470.2.a.q 1 35.c odd 2 1
1470.2.i.b 2 35.i odd 6 2
1470.2.i.f 2 35.j even 6 2
1680.2.a.q 1 20.d odd 2 1
3150.2.a.w 1 3.b odd 2 1
3150.2.g.e 2 15.e even 4 2
4410.2.a.l 1 105.g even 2 1
5040.2.a.i 1 60.h even 2 1
6720.2.a.k 1 40.e odd 2 1
6720.2.a.bp 1 40.f even 2 1
7350.2.a.p 1 7.b odd 2 1
8400.2.a.p 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1050))$$:

 $$T_{11} - 4$$ $$T_{13} - 2$$ $$T_{17} + 2$$ $$T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$-4 + T$$
$13$ $$-2 + T$$
$17$ $$2 + T$$
$19$ $$4 + T$$
$23$ $$-8 + T$$
$29$ $$-6 + T$$
$31$ $$8 + T$$
$37$ $$-2 + T$$
$41$ $$-2 + T$$
$43$ $$-12 + T$$
$47$ $$-8 + T$$
$53$ $$6 + T$$
$59$ $$-4 + T$$
$61$ $$2 + T$$
$67$ $$12 + T$$
$71$ $$-8 + T$$
$73$ $$-14 + T$$
$79$ $$T$$
$83$ $$12 + T$$
$89$ $$-2 + T$$
$97$ $$10 + T$$