# Properties

 Label 1050.2.a.a Level $1050$ Weight $2$ Character orbit 1050.a Self dual yes Analytic conductor $8.384$ Analytic rank $1$ 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: $$1$$ 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

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

## 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.a 1
3.b odd 2 1 3150.2.a.ba 1
4.b odd 2 1 8400.2.a.cn 1
5.b even 2 1 210.2.a.d 1
5.c odd 4 2 1050.2.g.h 2
7.b odd 2 1 7350.2.a.bd 1
15.d odd 2 1 630.2.a.f 1
15.e even 4 2 3150.2.g.o 2
20.d odd 2 1 1680.2.a.b 1
35.c odd 2 1 1470.2.a.m 1
35.i odd 6 2 1470.2.i.h 2
35.j even 6 2 1470.2.i.d 2
40.e odd 2 1 6720.2.a.cc 1
40.f even 2 1 6720.2.a.bb 1
60.h even 2 1 5040.2.a.ba 1
105.g even 2 1 4410.2.a.f 1

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

 $$T_{11}$$ $$T_{13} + 2$$ $$T_{17} - 6$$ $$T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$T$$
$13$ $$2 + T$$
$17$ $$-6 + T$$
$19$ $$4 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$4 + T$$
$37$ $$2 + T$$
$41$ $$-6 + T$$
$43$ $$8 + T$$
$47$ $$-12 + T$$
$53$ $$6 + T$$
$59$ $$12 + T$$
$61$ $$-2 + T$$
$67$ $$8 + T$$
$71$ $$T$$
$73$ $$14 + T$$
$79$ $$16 + T$$
$83$ $$12 + T$$
$89$ $$-6 + T$$
$97$ $$14 + T$$