# Properties

 Label 2850.2.a.b Level $2850$ Weight $2$ Character orbit 2850.a Self dual yes Analytic conductor $22.757$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.7573645761$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 −2.00000 −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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 2850.2.a.b 1
3.b odd 2 1 8550.2.a.w 1
5.b even 2 1 570.2.a.j 1
5.c odd 4 2 2850.2.d.l 2
15.d odd 2 1 1710.2.a.k 1
20.d odd 2 1 4560.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.j 1 5.b even 2 1
1710.2.a.k 1 15.d odd 2 1
2850.2.a.b 1 1.a even 1 1 trivial
2850.2.d.l 2 5.c odd 4 2
4560.2.a.d 1 20.d odd 2 1
8550.2.a.w 1 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2850))$$:

 $$T_{7} + 2$$ $$T_{11} + 4$$ $$T_{13} + 6$$ $$T_{23}$$

## Hecke characteristic polynomials

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