# Properties

 Label 2850.2.a.h Level $2850$ Weight $2$ Character orbit 2850.a Self dual yes Analytic conductor $22.757$ Analytic rank $1$ 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: $$1$$ 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

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

## 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 −4.00000 −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.h 1
3.b odd 2 1 8550.2.a.s 1
5.b even 2 1 570.2.a.i 1
5.c odd 4 2 2850.2.d.c 2
15.d odd 2 1 1710.2.a.e 1
20.d odd 2 1 4560.2.a.x 1

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

 $$T_{7} + 4$$ $$T_{11}$$ $$T_{13} - 6$$ $$T_{23} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$T$$
$13$ $$-6 + T$$
$17$ $$2 + T$$
$19$ $$-1 + T$$
$23$ $$8 + T$$
$29$ $$-2 + T$$
$31$ $$8 + T$$
$37$ $$10 + T$$
$41$ $$-6 + T$$
$43$ $$-8 + T$$
$47$ $$T$$
$53$ $$-2 + T$$
$59$ $$12 + T$$
$61$ $$2 + T$$
$67$ $$-12 + T$$
$71$ $$8 + T$$
$73$ $$2 + T$$
$79$ $$16 + T$$
$83$ $$-4 + T$$
$89$ $$-6 + T$$
$97$ $$-10 + T$$