# Properties

 Label 2850.2.a.bb 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

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

## 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.bb 1
3.b odd 2 1 8550.2.a.n 1
5.b even 2 1 570.2.a.a 1
5.c odd 4 2 2850.2.d.r 2
15.d odd 2 1 1710.2.a.q 1
20.d odd 2 1 4560.2.a.t 1

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

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

## Hecke characteristic polynomials

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